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I have completed this docu-) Tj 1 0 0 1 130.099 599.799 Tm (ment in accordance with the Department of Statistics instructions and with ) Tj 1 0 0 1 129.849 573.149 Tm 102 Tz (in the limits set by the School and the University of London. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12 Tf 102 Tz 3 Tr 1 0 0 1 146.65 493.5 Tm 101 Tz (Signature: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12 Tf 101 Tz 3 Tr 1 0 0 1 404.899 493.5 Tm 96 Tz (Date: ) Tj 1 0 0 1 429.35 493.699 Tm 839 Tz (\t) Tj 1 0 0 1 454.55 494.449 Tm 96 Tz (2-19 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12 Tf 96 Tz 3 Tr 1 0 0 1 308.399 139.25 Tm 68 Tz (1 ) Tj ET EMC endstream endobj 4 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 5 0 obj <> stream 0  ,, ֔j]jYESxQCVA]_m&$l%9M X*賕V#æs " &ڜXFkB#79#J;!ڰn//x~GGAw0YO-uv#'Fi992M! ?s%1 D f͓vU+t/]oO7)#!AjZhNI_Ｓ6㜘v?Κa+3a£!}H[ᲱUPh;ulB?Hqh,/ cIma+'rڠXMQEanʚЄmQ*D/0^5=ȗ E:(6*v@uWoJrD2hɰVl"8{p iʘoFj ё)% /@3>B {\~V}Gjnqp4P8zlTǽmX3 I %=d:R8$m j3yI{NTX'*=F|a Wҟj4 |E001ie@p5`Ț,<(OsVGQ/O}ސҝGWSGHb-B8Y\a t 8k}LkYj9rʡtsqTʣ<Ӧ9󩅶.r8<߈ΑGp;CY؝cpŕ`Ӡ vӱQ({bjy?}* (ԇFQײ"uB78WkRqjcL`H&\턅=fm2@v0O (Y#aH܁/6&4ԔJ32H۠sEpXf;CkHb K 3`Cwg8 ʸf/8ԗ]j.qVqh@5?g%n523=rq&A.\mTwM-JEFڜanz+ #qvHtoꅿp5Xo3Vrw|܃CL]WDYJLƚ׋_Ow=`FH;5[Z~Pȶ4B_Xǁ3/QcIva[0>NsJ pK󤝞Vh4)sPA-CXc@Ƨtx$bUcouYUh}|A%-*jQ ~`خs\śr FߘPک mQ̓zsTJ.nRDg:N_25q@2<,t̏چCGŧߺN o 5YA/VQ3Կ_U>lLsBox w^H2Dwc*(M4)1dM2xg:mK ų)\/'X:j}IOWQz>½vȜ .gQ'%WȭhA!wV"ٻKI]Ѳl7*PPH=.;xe1I'-D m`mxIesbHp؟I}o; C>ǖ;8*AE`KDzt\v*ԍV:2FLPm> l<@o|Hc*NZDC ]KToMHw)Jsz vS9_JfʖoY7eQwNR%K5A<,y%\Y9x )]pIG頗v2|FGlq7UVw O֭NMU2v خ%C&=Vdw會Oj oma[Q~B<#čef-O<,7[~6:I)( ޹vA^~&I*,-hXN(ˆw2CL쑉tR洊1ˢfRi'ZCkX*.Hɷ^՗B3{TtwDEcW+"W0z#7!0[Rc{2.ܶZn8=*JrpAj0NW{vѼ'BiNm.9ssqeV 5 VAw)|]j խF=S/|V[|cuURi#%$q{ !T7+M[ ՞4,練$_K#"i=zgd7Ç͊ ĺ1I\lء(Ϋۦٵ4Ae;& ln93l$^φC4kl+-JO eҁSX]cLUք=d҈H`F?ϕ! 9E)bף'w_=!E{ 3sڞ~R+Ƴ~}[]GLQl) 0RJ/FE`()^:t 2'-7:; h_K=csxk;B_MWֱP"p;:|KcR"{~"-H)O)ϟRdgK#Bp2*+*RaxuΒMDcx#{0ّS 1g٦4"|UvO:Bj`}P-Fϥ51n}'ɮ*1 #<z6^Jƽs`ԛԊZ'$e_u hw"{ K:8 b*.=;-%3IJe2 h[̴ҌOC,WAuݡD$i /[?E\Zυf+U.V‚_S@:UO;0`F|2I<}`%dnb}ؕ,9@CS2~Nk]}޶vtFy+*gN)NNW+;ٸܺj,*>8y}ame+_8N I,Yx 杻% ހƶ \<gةذ92pkd*ܬ?iC3[0,egQm£?;0:n]-bºe;z߳l 8}Wl~=~LtDޒ=>   -#;IrksObia jf:>@_6귳Ķķ*VҥI܄@A⥳ݰ<{H~xj|ugN(swߑH[]xy#Ȟ]2VrCv>FnKUJvgBtY\r6+E K>-Q n_2ﭣ:G)+)-5nF%X9p]͇Xx#>P)e["ENh[Pv$="}yήM9S*FY+o(&VmyN{]J#}@K9Hk/9j/dGD'7eI }b+^ yJ չ,},Wչ8I&2}m HgI,U ҨOF?݈WhײBg3ϑEqG%fUTay_պV--%eqpgOp]cͦ!Ԛ:;_S)(>˫({@3~f^xk: h(aMmFxݷALWwN@cYd@)d㵏|?<T.'"?Ap+*Mmms6o}&Nc0bC&"[:pMJj J{9-~\yV}C $7q4RY7lҽK3ť% kXDi'R[Nʱ8%0\(ciGK)oj[լ#v6k}- t<l ad-eۅL얾I€3a25jcl9, otЕ\D33PWnr5u-E 3!I@5|GϨrdRSv M"sr endstream endobj 6 0 obj <> endobj 7 0 obj [8 0 R] endobj 8 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 618 0 0 840 0 0 cm /ImagePart_2022 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 293.05 620.649 Tm 109 Tz 3 Tr /OPExtFont3 15.5 Tf (Abstract ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15.5 Tf 109 Tz 3 Tr 1 0 0 1 158.4 567.1 Tm 94 Tz /OPExtFont3 11 Tf (Physical processes such as the weather are usually modelled using ) Tj 1 0 0 1 158.15 549.85 Tm 96 Tz (nonlinear dynamical systems. Statistical methods are found to be ) Tj 1 0 0 1 158.15 532.549 Tm 93 Tz (difficult to draw the dynamical information from the observations of ) Tj 1 0 0 1 158.15 515.299 Tm (nonlinear dynamics. This thesis is focusing on combining statistical ) Tj 1 0 0 1 158.15 497.75 Tm 92 Tz (methods with dynamical insight to improve the nonlinear estimate of ) Tj 1 0 0 1 158.15 480.5 Tm 91 Tz (the initial states, parameters and future states. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 158.15 458.399 Tm 94 Tz (In the perfect model scenario \(PMS\), method based on the Indistin-) Tj 1 0 0 1 157.9 441.35 Tm 89 Tz (guishable States theory is introduced to produce initial conditions that ) Tj 1 0 0 1 158.15 423.85 Tm (are consistent with both observations and model dynamics. Our meth-) Tj 1 0 0 1 157.9 406.3 Tm 95 Tz (ods are demonstrated to outperform the variational method, Four-) Tj 1 0 0 1 157.9 388.8 Tm 90 Tz (dimensional Variational Assimilation, and the sequential method, En-) Tj 1 0 0 1 157.699 371.5 Tm 92 Tz (semble Kalman Filter. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 157.699 349.449 Tm 91 Tz (Problem of parameter estimation of deterministic nonlinear models is ) Tj 1 0 0 1 157.9 331.899 Tm (considered within the perfect model scenario where the mathematical ) Tj 1 0 0 1 157.449 314.649 Tm 92 Tz (structure of the model equations are correct, but the true parameter ) Tj 1 0 0 1 157.449 297.35 Tm 89 Tz (values are unknown. Traditional methods like least squares are known ) Tj 1 0 0 1 157.449 279.85 Tm 88 Tz (to be not optimal as it base on the wrong assumption that the distribu-) Tj 1 0 0 1 157.449 262.299 Tm 90 Tz (tion of forecast error is Gaussian IID. We introduce two approaches to ) Tj 1 0 0 1 157.699 244.799 Tm 92 Tz (address the shortcomings of traditional methods. The first approach ) Tj 1 0 0 1 157.449 227.299 Tm 91 Tz (forms the cost function based on probabilistic forecasting; the second ) Tj 1 0 0 1 157.449 210 Tm 92 Tz (approach focuses on the geometric properties of trajectories in short ) Tj 1 0 0 1 157.449 192.5 Tm 91 Tz (term while noting the global behaviour of the model in the long term. ) Tj 1 0 0 1 157.449 174.95 Tm 95 Tz (Both methods are tested on a variety of nonlinear models, the true ) Tj 1 0 0 1 157.199 157.7 Tm 91 Tz (parameter values are well identified. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 157.449 135.35 Tm 94 Tz (Outside perfect model scenario, to estimate the current state of the ) Tj 1 0 0 1 156.949 117.85 Tm 93 Tz (model one need to account the uncertainty from both observatiOnal ) Tj ET EMC endstream endobj 9 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 10 0 obj <> stream 0  ,,''f҂VT,oycW{.6EOyk0GD7O'o(Ӓ@@q'ߤ* _q\a1*Ë.o ^$1bP_ ۾[:-(抴18É"ŌA1-,Ί(`ETvJycgTHAc붪lpf"w ${ɟBU-RdRq L-L%fb-w֧ٓypdsz&cKT Lu d3o HTMPCc+Jv<䒹Z(QV0!is׾Vw4T[I<, _@{|pz '>-P|:^l)W*oA爘B[y( fR->Ma'wul;OpX+i"+@h(ozh$Bn  A>K_vnݟn(c1 En4hOHQØ0ߓ7DA)Ċ`Z "YR+ F']Щ]3>EA7 r')o!dM9!)@U yAދP:T _XU:c3ajMZmW;kʼsY{%$Qc?>+%Fm~6bj+ ) Fע(9mj(1X3Sj; 0/4M9$Sgg97Kc>>L<;`#(ɜ EkH`0yU[C!doȰfkEIsz`TT^cNTK/^̛>e4uaqVޔ euOVa< $ Cq8<(ZO"Fʟ4T= Bʦ1ZG0rٯ:E7yJTarꤻ]m5㛩`"_H&WeF̠m3 USӴ㲆p>],X7+Ѣ'5:VxEʁ`vs PFg]tepߺԭ!ĻaiK(tcN{񨳖;p2^CꯕN8r BIMí%Fj:_Y@f?VCgS!dߣB%qϨ&N?9W8fGN^,"p[mj,'E{ӡy8EbH ,˲XGPе՟˸ ; C7J.We#0 ]6^8wKeRgN'4yH'd43Μzs%Uf,N8dX­AW?Wiq볣W%:+Dm~y'aU'9JvqT#(ؼM60d-\G*象1BO©mH24e ?Rg׫C)i\,w9$ T{a:].AmDA*O٭Aÿ㏑eZ-D? "r G3#Q` =tD!?MsNV\ tC<&Xlp7ɏ.:Rhwků|~?_H~Cb0!WUJ|Eq5[}fTq [Dz7 WM{s8eT͏rx|*LKk̨@F1}WU+8ϭ$eMO%w$@g;eHʕ=LckoC恘%bW"u3Ih[f1Qw6 wtcه!^S uXLBԁxbbP BHYJ8ƾ5p{ﱩyy A"uSB"aen\Z 3^n*^+a:Fon"6?HͼRzφ;PACRPSyN;[8ždιHvOޞ:\@&/~*`#z)H'P7 Blڹ9cW#;,?O/4t#&Y>: bNDX/| $? 8u-lCqxWxhY׹R(J10Y䂶[:|,W.B}>QuH~*Y6Ѿ7\?$[tG+Ԁ3?R3)F~ )I?.JMG >;{ ӻa[&)o~y;9~ d.zuQ6Cǀ>ڍضճ9XЯӎI^ݡ@S7c?tƘS[r*Y@9C W&n N ǁW7-SXFS *!G؃liC, ^`{y,P:1Nv60cWiwwU ĺ5:y؋Vǵkc5[?E|KnJ^3Ӭ辭 1Ӿ< bTW*lX4qWl{= Zzp @%jXn^  Fb~Ӊ?c\-b|ũIm>+[{3a5#k:RqJ=Vne\{jtuD >f&9ء8żR fؐkW➳5l{\%~P-0D=Eژ?F%b<܅LpN?$ܠ䖤UY:5~ořei : 024N­Ef#,PC xvv=3$͐@GM%Piˉ&2HbڭW΅,Sz8hV|hȱ=5|NO+#^gX8ҟa12j'olǭm="^ʐ,:5 M#xC*6Ӯ 3`KuCʈ ,!%[ձ :>,XTp V< p|7M[='2iq8i!?.![[8]GD3u9 0sP=xQj⑾txwoZ4_b o H+PPq>7PC1]8V5ܫٌhO\.CH0NMtk+y5-ד't4xNO rHL{Wk-P(fS /</   x/31h>1z#}=R7"fwRpÖ9ױx*Ֆ3u[zl2c"L?ePh9W%!pN`#RϩK.4Zu&ozEr?,~og @ꟃFPLwGӚTb)rKVkNpIN>.׻uZNYYtӪD1J## >`> 꼕!/$0"o|q~*+C:r1;\>d﵇\&d 8Dp፽Kv7Q)d^jqe"ԠSk'*YSP[r n*SnXjAbVRPS?GsJo7ICjHLaBa[g$k\YZ;#MIZ#j_~"V ˮUU*LbYmzH+at8AT>2l&++GP۾`dzҰz.+nٱ԰䜴ܽ<K1b?885$کy?5+!}g(M}F]x؈niKTCgxvk<" rt(vJi%>:^j3& ^0 $Ѩ݁YФ^O\4:V&~T#սtч(9Cɴ:8G{4H{']ƤY3kE}6X }k YBnUhOԜƔ_/–QbkMB+j['>rM*EM#3iI&zoŴĊ*bɸsQp(-Ш Pbaj`4jZ%Sl Ul 37B)01qhHx +  6Ƚ(Y?5t͊!-ggBy[?gq״hk10a1xbSdV(G=Ogbp6yoʰ ۖf?1@3]]SA+b UxZqw 6O!@FSjsus#.M:Qw[g~~ɟRg;'gJݾ0r9(p3:{w+*a_,vP0 *4MI)M4 mC+/cK-Xr#@ 9V?XĦ KQr4B]ڴ-Mar*^ԯߛ9;e?YglG;R  ra 5h{`Ǡ(22|r{Unj<'0w Q'5d#HoټҚ0Ha6)$weKk@P\Zh2IE}[Z;;7 П q'C({~]U@|3|ַx\IP:m0˸}EAD> u/){`,(_"q\e% h?T@#0!-6Hte sd07oūهΠT @bžWGhw"Ȭ6+v kl[A(JU:(Z2g|7xZi{ka<.` nXl jo@ɃEŘYGdLuLDvpk(͞%M]?Mhl@l0Ͽ?'l c(F027A{؀')P|:0$d=bG>PwU-We@)!BE1Da2O$"(t:Cl-*A嗙J~E= /9VrtV~x՚Z*1-(5F!,QKlER*DpV,/B/3subY`H}v]$*oJyv)^}gx Sw]01a"R`LD!`ݰmC毫/qk㖅apO8|¥ӆr~(&5dDSNS395V .Uovqw۹WJ ۞}pX!0I%n2wfnafЉ{-t4m?a+Ƥ)3KcGEҕjfzԂ\9&i3 3΁0`u@Ir;8iP'c̆UH`?dTjVE\ی3tF|1Y|UV)jڻ'_"͸ e n,% `Z=ʷeJQgcbS?]Bt醉vۭ]aJa}"vH6tg"Txr4t0bk'<e&>*(B-!R᝭f[b#2dRǵ{ .*ͦe1KkB endstream endobj 11 0 obj <> endobj 12 0 obj [13 0 R] endobj 13 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 616 0 0 841 0 0 cm /ImagePart_2023 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 156.699 679.5 Tm 91 Tz 3 Tr /OPExtFont3 11 Tf (noise and model inadequacy. Methods assuming the model is perfect ) Tj 1 0 0 1 157.199 662.2 Tm 92 Tz (are either inapplicable or unable to produce the optimal results. It is ) Tj 1 0 0 1 156.699 644.899 Tm 94 Tz (almost certain that no trajectory of the model is consistent with an ) Tj 1 0 0 1 156.699 627.399 Tm 91 Tz (infinite series of observations. There are pseudo-orbits, however, that ) Tj 1 0 0 1 156.949 610.1 Tm 94 Tz (are consistent with observations and these can be used to estimate ) Tj 1 0 0 1 156.699 593.1 Tm 91 Tz (the model states. Applying the Indistinguishable States Gradient De-) Tj 1 0 0 1 156.5 575.799 Tm 90 Tz (scent algorithm with certain stopping criteria is introduced to find rel-) Tj 1 0 0 1 156.699 558.299 Tm (evant pseudo-orbits. The difference between Weakly Constraint Four-) Tj 1 0 0 1 156.699 541.25 Tm 92 Tz (dimensional Variational Assimilation \(WC4DVAR\) method and Indis-) Tj 1 0 0 1 156.699 523.7 Tm 90 Tz (tinguishable States Gradient Descent method is discussed. By testing ) Tj 1 0 0 1 156.699 506.449 Tm 95 Tz (on two system-model pairs, our method is shown to produce more ) Tj 1 0 0 1 156.699 488.899 Tm 97 Tz (consistent results than the WC4DVAR method. Ensemble formed ) Tj 1 0 0 1 156.699 471.649 Tm 90 Tz (from the pseudo-orbit generated by Indistinguishable States Gradient ) Tj 1 0 0 1 156.699 454.35 Tm 93 Tz (Descent method is shown to outperform the Inverse Noise ensemble ) Tj 1 0 0 1 156.25 437.1 Tm 91 Tz (in estimating the current states. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 156.699 415 Tm 95 Tz (Outside perfect model scenario, we demonstrate that forecast with ) Tj 1 0 0 1 156.5 397.5 Tm (relevant adjustment can produce better forecast than ignoring the ) Tj 1 0 0 1 156.5 379.699 Tm 94 Tz (existence of model error and using the model directly to make fore-) Tj 1 0 0 1 156.699 362.449 Tm 91 Tz (casts. Measurement based on probabilistic forecast skill is suggested ) Tj 1 0 0 1 156 345.149 Tm 93 Tz (to measure the predictability outside PMS. ) Tj ET EMC endstream endobj 14 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 15 0 obj <> stream 0 ,,  jnk0+{<[Oz4qas ȃ卸"(7#aⱖYJt:l!VJ{(Pc`경cDg_֖+1Dˆq&+K A 5M_")Cwa/H#1~B !I|W^E~T_GsΛ~a!6SL9w5f<,! f1>}ހ%~tdj@03M2?phw}ApBRko]} "I:$934>K7 mòߛnWI]"Is}?BtR DOdžWj αBhҀTBK+IPO@B>{3Uff׵Ҳ%% X`[۽]+<#ݷ-#I'9Ⱦ`zFL9!ZÞq -,3 ŀa&?Až:uZ#۵)ik% zֲnoN%W@uf{v; kHb\eE ijG0t@:KXs<7 m6]eHh0#a4PjTf|BXreƙR"cGwf){!&8kQ}2sCjO2#{C^J5L=0ui2j/ߌ-xIRq0W5~ v)i=YwVnà؃S N}`B MJ}3pW{0nƨ~VlKSbqh/$~/Z_IqW^hf2K+ok:7U-x Bc=##f)J=MC؞"|^:L J/V)f%ElzɏyA- D.*x$\2?"Fl[TS+灾܍"ј.7VdE~W&9~wQNS_"箷MZcy>ɳҕ'Բ~հ۳$U*KUyJk$}'73G69Fd-3ꖎd텪mNièn7syA[5Ϟ?1~M\13v2n4T%(|;s>\/V g k ڛ"w&㜀83ǟ(=CX<\k* Pe݊ ȆYnwRU?Ϸ8h<~OkYԉvZŀB4b1\WvO ٢#Ed0e,kp& ` 5S'p>Ǥ&&ܴ0*ۖ?DZ&ڿ/DMU2&Գ!K<"zZơNcY#]&yqRP>T~)*,8, KI鳡B;?-؅`~V>hMPkX= 9(3uLeT6_?EuC/D[[8K 2r&)o*5a#K.ȕuECCIp;,!$9)kFg~Fߐh7g~0V [O2ֻ2ݜkDVJh#ynn ɑR͂uOI\G*\\PSdZ|euyvg N MCxd2OY2q0|!S9 ŌOքBXGx7&!^xyi~"G81JgBP{ tkOMwP]5x&8Pޕ[y.E?Kp$&`k# Sџ n^P2!;txVƻvzyQXw3y#^*);j iyw|@&rDx29ȅa 6.a5@+Ȁ!"aZj,,fBbddP4%?" ?-?Djyn$L0r.}-CH!Hgqk``ƾnՎp{iW D6'7]D{JaQv< `ѤnF sDa1+ui QYM0뼹" !3:Pn@cfpR;oi^}1QLt륑!^Wz">߳Zbn MMxj:VPVKwb|2r: Z@v8!Ї0EN/,6=5a>Dq U؏~ PbcDņ◪y:-}G +ĴDS-2(g =H6+c;{E.rjA,!Y=—hM:]Ļj/^ٞET+0T9R4ۭUW[4aOWP+mq۫U{qwH3bh=l :H02Gr.;jKVYr}S*9^H{"Vhӈ|j@4OPֆ  ,CWQH%E?=.般兝@0$[(:40tgjTy1,QJ,= N?)ڼ_lV'^(71"lOr^/nPB>3$.)%<j؊8409i95oJJ/,H"(eE(CR踛l"yaG=x>G^0vk!#\ *}${Juv_a˜\YOwC,υ" "'PRZPR> i#8ss6d OZZ>~^̶1uM~OR@\@_]D 8DĞ\lQct3L$LK Zmfd{G\"3݉ԝH'x&w*X4?̸~d vG \'A 4Xes ˙ k 3* bX!?s30$oAfQ kdM|/LZI\IO(V!TAseKe;Gz7)dƛ.G  | ,wҊ٦K'1UPgyzFiހ#+>!K1{[E 7gƸ(IbI;70I,˦<ipKZ6cblpfݖ/٘ccK14C8T#6wI;ѝS^WωGL924Z.`+RPV{vKo5Τ 8dڋ&C 0oԘ㹛<]*b yPyx~_A1U"lb%5y05G.g0Db< NuCI0wqrڈaǾL*ڍ8BLz峳pHGE:bs%NsGx2[3~Ζ&',2fcbAjfqc):Oga}H5ȽOVhue]/*;)Co69J &,M]-(=+пwѼ,+Z$TOgYg@/5ä%]Hg鼛6n1gj|"w3LH=F>P$Wcng3Y ka|͉xRs};,VDQl !+KL'Li0(%5_?}rAPR69%[ S.x21Tv,h@|4 WU%اEצWL8ZXq,[qyN4^ʹyR CJ"= YPTz$T78:P%vp[ NdGُ, W9yq:ia]G:% m#rj^ Xp2HMf۲'Y%wH8dǫTǛD=O`&:*  < `g9G1jωF j dԠک Ɋdύ*켎d.5:~1= R(*Ws!AMx*^SMfn)X&o0LF!DA~-zmA1x Ѫl$<$#/Ql}# i/&V~0?"bud@jAtA&; tw,aj l[l"x?ΫQ-Xx3Mo'[P8,jS[ļm#8a! !MkHRQj=0tbj9}/y)4 *q~$cBZZZ=Oc*2F>eJ1b̷);| "cOy>է}4D .#{sm# 7Jr97 #<Ms{ (BQhrkQ(Q~ֵU;H8q%ͷԳr"_4x|nN棷 ni"4aZK"pKsZwPeHRסѓ:bDD$=B ^+0ؤ kJ;ZAׁ0g]<1;3b3 $Qsy)޷H0G I+eǏX%\dmz`= ل I$:l)`)t.٭-|nx*֙#/2*?);_C?%m^ 1*.ެɔJ6V1աSo `>D: swZ[*H^PUON9.^Ql 'HfԸ3b 5+< J9XڜңӠ@8}9_M+E,zQEPaL &b(_q;"/5v;I. 6/}%%J- ECEtGD~OE\* 3KF'Fc ;B9Zr Ub}j]τ]]kx;\TSbp<<IzCnF82DxP3Z3)ה8ʕ۫ޒQJ'CJF;T^vmVʒ+DMIwZD6"M s C\u*8(L/8QlB%HSud#q .1) X%0M0*M *du%E͎=JpF>Dh`#uΰ^բ endstream endobj 16 0 obj <> endobj 17 0 obj [18 0 R] endobj 18 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 616 0 0 840 0 0 cm /ImagePart_2024 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 130.3 595.899 Tm 101 Tz 3 Tr /OPExtFont3 23 Tf (Contents ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 23 Tf 101 Tz 3 Tr 1 0 0 1 129.349 529.45 Tm 74 Tz /OPExtFont3 11 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 74 Tz 3 Tr 1 0 0 1 145.9 529.2 Tm 105 Tz (Introduction ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 105 Tz 3 Tr 1 0 0 1 520.1 529.2 Tm 63 Tz (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 63 Tz 3 Tr 1 0 0 1 129.099 501.6 Tm 77 Tz (2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 77 Tz 3 Tr 1 0 0 1 145.9 501.35 Tm 102 Tz (Background ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 102 Tz 3 Tr 1 0 0 1 519.35 500.649 Tm 74 Tz (5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 74 Tz 3 Tr 1 0 0 1 145.9 484.1 Tm 90 Tz /OPExtFont5 12.5 Tf (2.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr 1 0 0 1 171.599 484.1 Tm 102 Tz (Dynamical system ) Tj 1 0 0 1 266.149 484.1 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 484.1 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 520.1 483.35 Tm 82 Tz (5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 82 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 82 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 82 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 82 Tz 3 Tr 1 0 0 1 145.9 466.55 Tm 93 Tz (2.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 93 Tz 3 Tr 1 0 0 1 171.849 466.8 Tm 101 Tz (Flow and Map ) Tj 1 0 0 1 248.4 466.8 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 466.8 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 520.549 466.3 Tm 77 Tz (6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 77 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 77 Tz 3 Tr 1 0 0 1 145.9 449.3 Tm 93 Tz (2.3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 93 Tz 3 Tr 1 0 0 1 172.099 449.3 Tm 96 Tz (Chaos ) Tj 1 0 0 1 212.65 449.3 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 449.3 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 520.549 448.8 Tm 82 Tz (7 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 82 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 82 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 82 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 82 Tz 3 Tr 1 0 0 1 145.9 432 Tm 94 Tz (2.4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 94 Tz 3 Tr 1 0 0 1 171.599 432 Tm 104 Tz (Analytical systems ) Tj 1 0 0 1 275.05 432 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 432 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 520.549 431.5 Tm 74 Tz (8 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 74 Tz 3 Tr 1 0 0 1 171.849 414.5 Tm 95 Tz (2.4.1 ) Tj 1 0 0 1 193.9 414.5 Tm 446 Tz (\t) Tj 1 0 0 1 207.849 414.5 Tm 101 Tz (Logistic map ) Tj 1 0 0 1 284.149 414.5 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 414.5 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 520.549 414 Tm 77 Tz (8 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 77 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 77 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 77 Tz 3 Tr 1 0 0 1 171.599 397.199 Tm 98 Tz (2.4.2 ) Tj 1 0 0 1 194.15 397.199 Tm 438 Tz (\t) Tj 1 0 0 1 207.849 397.199 Tm 96 Tz (Henon map ) Tj 1 0 0 1 274.8 397.199 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 397.199 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 171.599 379.899 Tm 98 Tz (2.4.3 ) Tj 1 0 0 1 194.15 379.899 Tm 438 Tz (\t) Tj 1 0 0 1 207.849 379.899 Tm 100 Tz (Ikeda map ) Tj 1 0 0 1 266.149 379.899 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 379.899 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 515.299 379.699 Tm 82 Tz (11 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 82 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 82 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 82 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 82 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 82 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 82 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 82 Tz 3 Tr 1 0 0 1 171.599 362.649 Tm 98 Tz (2.4.4 ) Tj 1 0 0 1 194.15 362.649 Tm 430 Tz (\t) Tj 1 0 0 1 207.599 362.399 Tm 100 Tz (Moore-Spiegel system ) Tj 1 0 0 1 319.449 362.399 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 362.399 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 515.299 362.399 Tm 83 Tz (12 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr 1 0 0 1 171.849 344.899 Tm 95 Tz (2.4.5 ) Tj 1 0 0 1 193.9 344.899 Tm 446 Tz (\t) Tj 1 0 0 1 207.849 344.899 Tm 99 Tz (Lorenz96 system ) Tj 1 0 0 1 301.699 345.1 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 345.1 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 515.299 345.1 Tm 83 Tz (12 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr 1 0 0 1 145.449 327.6 Tm 93 Tz (2.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 93 Tz 3 Tr 1 0 0 1 171.349 327.6 Tm 101 Tz (Nonlinear dynamics modelling ) Tj 1 0 0 1 328.55 327.35 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 327.35 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 515.5 327.6 Tm 84 Tz (14 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr 1 0 0 1 171.599 310.1 Tm 94 Tz (2.5.1 ) Tj 1 0 0 1 193.449 310.1 Tm 460 Tz (\t) Tj 1 0 0 1 207.849 310.1 Tm 100 Tz (Delay reconstruction ) Tj 1 0 0 1 319.699 310.1 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 310.1 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 515.5 310.1 Tm 68 Tz (1.4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 68 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 68 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 68 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 68 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 68 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 68 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 68 Tz 3 Tr 1 0 0 1 171.599 292.549 Tm 98 Tz (2.5.2 ) Tj 1 0 0 1 194.15 292.549 Tm 430 Tz (\t) Tj 1 0 0 1 207.599 292.799 Tm 99 Tz (Analogue models ) Tj 1 0 0 1 301.899 292.799 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 292.799 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 515.75 292.799 Tm 79 Tz (15 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 79 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 79 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 79 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 79 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 79 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 79 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 79 Tz 3 Tr 1 0 0 1 171.349 275.299 Tm 98 Tz (2.5.3 ) Tj 1 0 0 1 193.9 275.299 Tm 438 Tz (\t) Tj 1 0 0 1 207.599 275.049 Tm 103 Tz (Radial Basis Functions ) Tj 1 0 0 1 328.55 275.049 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 275.049 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 515.299 274.799 Tm 83 Tz (16 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 83 Tz 3 Tr 1 0 0 1 171.599 258 Tm 98 Tz (2.5.4 ) Tj 1 0 0 1 194.15 258 Tm 438 Tz (\t) Tj 1 0 0 1 207.849 257.75 Tm 102 Tz (Summary ) Tj 1 0 0 1 265.899 257.75 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 257.75 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 515.5 257.5 Tm 88 Tz (17 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr 1 0 0 1 128.65 229.7 Tm 74 Tz /OPExtFont3 11 Tf (3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 74 Tz 3 Tr 1 0 0 1 145.199 229.7 Tm 108 Tz (Nowcasting in PMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 108 Tz 3 Tr 1 0 0 1 513.85 228.95 Tm 82 Tz (19 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 82 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 82 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 82 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 82 Tz 3 Tr 1 0 0 1 145.199 212.399 Tm 89 Tz /OPExtFont5 12.5 Tf (3.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 89 Tz 3 Tr 1 0 0 1 171.099 212.399 Tm 101 Tz (Perfect Model Scenario ) Tj 1 0 0 1 292.55 212.399 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 212.399 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 515.299 212.149 Tm 85 Tz (20 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 85 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 85 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 85 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 85 Tz 3 Tr 1 0 0 1 145.449 194.899 Tm 93 Tz (3.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 93 Tz 3 Tr 1 0 0 1 171.099 194.899 Tm 103 Tz (Indistinguishable States ) Tj 1 0 0 1 301.899 194.649 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 194.649 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 515.049 194.899 Tm 88 Tz (22 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr 1 0 0 1 145.449 177.6 Tm 93 Tz (3.3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 93 Tz 3 Tr 1 0 0 1 171.099 177.35 Tm 102 Tz (Nowcasting using indistinguishable states ) Tj 1 0 0 1 382.3 177.35 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 177.35 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 515.049 177.1 Tm 90 Tz (25 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr 1 0 0 1 171.099 160.1 Tm 97 Tz (3.3.1 ) Tj 1 0 0 1 193.449 160.1 Tm 444 Tz (\t) Tj 1 0 0 1 207.349 160.1 Tm 103 Tz (Reference trajectory ) Tj 1 0 0 1 319.449 160.1 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 160.1 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 515.299 159.85 Tm 90 Tz (27 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr 1 0 0 1 171.099 142.549 Tm 98 Tz (3.3.2 ) Tj 1 0 0 1 193.699 142.549 Tm 436 Tz (\t) Tj 1 0 0 1 207.349 142.299 Tm 103 Tz (Finding a reference trajectory via ISGD ) Tj 1 0 0 1 417.85 142.299 Tm 1336 Tz (\t) Tj 1 0 0 1 459.6 142.299 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 515.299 142.299 Tm 88 Tz (28 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr 1 0 0 1 170.9 125.299 Tm 98 Tz (3.3.3 ) Tj 1 0 0 1 193.449 125.049 Tm 444 Tz (\t) Tj 1 0 0 1 207.349 125.299 Tm 101 Tz (Form the ensemble via ISIS ) Tj 1 0 0 1 355.199 125.299 Tm 2000 Tz (\t) Tj 1 0 0 1 459.6 125.299 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 515.299 125.049 Tm 88 Tz (29 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr 1 0 0 1 322.55 52.799 Tm 106 Tz (iii ) Tj ET EMC endstream endobj 19 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 20 0 obj <> stream 0  ,,ggb7 [ d<-c E K<ށ4L™8IQwz+1G>t?ڰ3ѵ.[P۲2< w pஃXd8]Kӳ iO?VTlv!Ч@zM9 :⁰_ZLa?FWݵl8+,ڐFk/6*f֢)G5k\Yݤ-Һ6 SENi, +|gX|'XtJc5%I)G:nQWAX@HI @%? ]`M7x-|nb8|0c1BI.oni ޹7>$'_j%EAj'MyQUߥz+XMwYf 琱PԃBGcC5)0">bЪ⪻j2d~fȅqU UtJ g7 eY-Nus1mU&DL E@ SϣfX?bDC5[ JkeCPvM/OV~v~fߪ|`@ Y=o(@'̐IA3 Nb(2걙 sapǮ& x?^kV-m'JГz0$u|jt*=Y)SEt=0&{Psop 93\$ztb~E`a\jo݃o+4mV6x69B_Zϩw8P 7]Mw^цtd t8KpUuMPƒBjorl:ʮ)ƺ6^jXD1Z6_&:I8,m qI2>riʚMyVTVsP bsa>?VRrx↺Z% Rh:՘!U$hV4./ꆓW=jPw]^=7὞V<ux"rp%؂d9BU9gIz6+ L7B Rr:\XwykXa]WK\N oPY+Tb2goѿRw%#~[npn@qftem3)?6SڬH~-mzW ~WWY/ 1]eoYy3=&y?DBjk|xD_pawY{>/AцIcvr[}%u7d࠸I6!Vz,/Zhۈ&QIFlR!D@ >՝37։"܃΋ҲڽFٱ)S=3Pkά>4;n+/eɊ.5 䃭.]IU!ׯ;D~R瑅̱$wi"Wi[.s;CRWskRژ_tQ96Q4!K d2Yןhi/@s|ĴޓV*ocj'ֳʼ_Oj_L4v7޵7h4<>dKvlS%>P=roX@z 6)Hb͡V3z{5H)*>p}OkR 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Tm 110 Tz (3.4.1 Methodology ) Tj 1 0 0 1 282.699 645.149 Tm 2000 Tz (\t) Tj 1 0 0 1 497.75 645.149 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 170.65 627.649 Tm 102 Tz (3.4.2 Differences between ISGD and 4DVAR ) Tj 1 0 0 1 407.5 627.649 Tm 2000 Tz (\t) Tj 1 0 0 1 498 627.649 Tm 171 Tz ( 33 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 171 Tz 3 Tr 1 0 0 1 144.699 610.35 Tm 107 Tz (3.5 Ensemble Kalman Filter ) Tj 1 0 0 1 300.699 610.35 Tm 2000 Tz (\t) Tj 1 0 0 1 497.75 610.1 Tm 175 Tz ( 36 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 175 Tz 3 Tr 1 0 0 1 170.65 593.299 Tm 113 Tz (3.5.1 Kalman Filter ) Tj 1 0 0 1 282.699 593.299 Tm 2000 Tz (\t) Tj 1 0 0 1 513.35 592.85 Tm 90 Tz (37 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr 1 0 0 1 170.4 575.799 Tm 109 Tz (3.5.2 Extended Kalman Filter ) Tj 1 0 0 1 336 576.049 Tm 2000 Tz (\t) Tj 1 0 0 1 497.75 575.549 Tm 159 Tz ( 4\(\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 159 Tz 3 Tr 1 0 0 1 170.4 558.299 Tm 108 Tz (3.5.3 Ensemble Kalman Filter ) Tj 1 0 0 1 336 558.299 Tm 2000 Tz (\t) Tj 1 0 0 1 498 558.299 Tm 168 Tz ( 41 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 168 Tz 3 Tr 1 0 0 1 144.5 541 Tm 107 Tz (3.6 Perfect Ensemble ) Tj 1 0 0 1 255.599 541 Tm 2000 Tz (\t) Tj 1 0 0 1 513.1 541 Tm 93 Tz /OPExtFont2 11.5 Tf (44 ) Tj 1 0 0 1 523.899 541 Tm 32 Tz /OPExtFont5 12.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 144.699 523.7 Tm 116 Tz (3.7 Results ) Tj 1 0 0 1 210.949 523.7 Tm 2000 Tz (\t) Tj 1 0 0 1 498 523.7 Tm 173 Tz ( 48 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 173 Tz 3 Tr 1 0 0 1 170.4 506.449 Tm 106 Tz (3.7.1 ISGD vs 4DVAR ) Tj 1 0 0 1 300.699 506.449 Tm 2000 Tz (\t) Tj 1 0 0 1 498 506.449 Tm 173 Tz ( 48 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 173 Tz 3 Tr 1 0 0 1 170.4 489.149 Tm 110 Tz (3.7.2 ISIS vs EnKF ) Tj 1 0 0 1 282.699 489.399 Tm 2000 Tz (\t) Tj 1 0 0 1 498.25 489.399 Tm 168 Tz ( 51 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 168 Tz 3 Tr 1 0 0 1 170.4 471.899 Tm 105 Tz (3.7.3 ISIS vs Dynamically consistent ensemble ) Tj 1 0 0 1 416.649 471.899 Tm 2000 Tz (\t) Tj 1 0 0 1 498 471.899 Tm 174 Tz ( 54 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 174 Tz 3 Tr 1 0 0 1 144.5 454.6 Tm 108 Tz (3.8 Conclusions ) Tj 1 0 0 1 238.099 454.6 Tm 2000 Tz (\t) Tj 1 0 0 1 498 454.6 Tm 171 Tz ( 61 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 171 Tz 3 Tr 1 0 0 1 126.95 426.5 Tm 125 Tz (4 Parameter estimation ) Tj 1 0 0 1 267.1 426.5 Tm 2000 Tz (\t) Tj 1 0 0 1 512.399 426.3 Tm 96 Tz (63 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 96 Tz 3 Tr 1 0 0 1 144 409 Tm 105 Tz (4.1 Technical statement of the problem ) Tj 1 0 0 1 354.25 409 Tm 2000 Tz (\t) Tj 1 0 0 1 498.25 409 Tm 176 Tz ( 64 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 176 Tz 3 Tr 1 0 0 1 144 391.5 Tm 109 Tz (4.2 Least Squares estimates ) Tj 1 0 0 1 300.5 391.5 Tm 2000 Tz (\t) Tj 1 0 0 1 498.25 391.25 Tm 173 Tz ( 65 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 173 Tz 3 Tr 1 0 0 1 143.75 373.949 Tm 107 Tz (4.3 Forecast based parameter estimation ) Tj 1 0 0 1 363.1 373.949 Tm 2000 Tz (\t) Tj 1 0 0 1 498 373.949 Tm 178 Tz ( 67 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 178 Tz 3 Tr 1 0 0 1 169.699 356.899 Tm 108 Tz (4.3.1 Ensemble forecast ) Tj 1 0 0 1 300.5 356.899 Tm 2000 Tz (\t) Tj 1 0 0 1 498.25 356.199 Tm 173 Tz ( 68 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 173 Tz 3 Tr 1 0 0 1 169.699 339.399 Tm 108 Tz (4.3.2 Ensemble interpretation ) Tj 1 0 0 1 336.25 339.399 Tm 2000 Tz (\t) Tj 1 0 0 1 498.25 339.399 Tm 174 Tz ( 70 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 174 Tz 3 Tr 1 0 0 1 169.699 321.899 Tm 107 Tz (4.3.3 Scoring probabilistic forecasts ) Tj 1 0 0 1 362.899 321.899 Tm 2000 Tz (\t) Tj 1 0 0 1 498 321.899 Tm 174 Tz ( 73 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 174 Tz 3 Tr 1 0 0 1 169.699 304.6 Tm 118 Tz (4.3.4 Results ) Tj 1 0 0 1 246.699 304.6 Tm 2000 Tz (\t) Tj 1 0 0 1 498.25 304.35 Tm 173 Tz ( 75 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 173 Tz 3 Tr 1 0 0 1 143.75 286.85 Tm 106 Tz (4.4 Parameter estimation by exploiting dynamical coherence ) Tj 1 0 0 1 461.3 286.85 Tm 1174 Tz (\t) Tj 1 0 0 1 498 286.85 Tm 174 Tz ( 78 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 174 Tz 3 Tr 1 0 0 1 169.699 269.549 Tm 110 Tz (4.4.1 Shadowing time ) Tj 1 0 0 1 291.35 269.549 Tm 2000 Tz (\t) Tj 1 0 0 1 498.25 269.299 Tm 174 Tz ( 79 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 174 Tz 3 Tr 1 0 0 1 169.199 252.299 Tm 106 Tz (4.4.2 Further insight of Pseudo-orbits ) Tj 1 0 0 1 371.75 252.299 Tm 2000 Tz (\t) Tj 1 0 0 1 498.25 252.049 Tm 174 Tz ( 82 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 174 Tz 3 Tr 1 0 0 1 169.449 235 Tm 118 Tz (4.4.3 Results ) Tj 1 0 0 1 241.699 235 Tm 2000 Tz (\t) Tj 1 0 0 1 514.1 234.75 Tm 87 Tz /OPExtFont2 11.5 Tf (83 ) Tj 1 0 0 1 524.149 234.75 Tm 32 Tz /OPExtFont5 12.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 169.449 217.5 Tm 108 Tz (4.4.4 Application in partial observational case ) Tj 1 0 0 1 416.649 217.25 Tm 2000 Tz (\t) Tj 1 0 0 1 498.5 217.25 Tm 174 Tz ( 86 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 174 Tz 3 Tr 1 0 0 1 143.75 199.7 Tm 111 Tz (4.5 Outside PMS ) Tj 1 0 0 1 246.699 199.7 Tm 2000 Tz (\t) Tj 1 0 0 1 498.699 199.5 Tm 175 Tz ( 89 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 175 Tz 3 Tr 1 0 0 1 143.5 182.45 Tm 109 Tz (4.6 Conclusions ) Tj 1 0 0 1 237.849 182.45 Tm 2000 Tz (\t) Tj 1 0 0 1 514.299 182.45 Tm 78 Tz (9\(\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 78 Tz 3 Tr 1 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3 Tr /OPExtFont3 11 Tf (CONTENTS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 114 Tz 3 Tr 1 0 0 1 128.15 677.5 Tm 74 Tz (5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 74 Tz 3 Tr 1 0 0 1 144.949 678.25 Tm 107 Tz (Nowcasting Outside PMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 107 Tz 3 Tr 1 0 0 1 511.449 678.5 Tm 82 Tz (92 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 82 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 82 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 82 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 82 Tz 3 Tr 1 0 0 1 145.199 660.95 Tm 89 Tz /OPExtFont5 12.5 Tf (5.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 89 Tz 3 Tr 1 0 0 1 170.65 660.95 Tm 100 Tz (Imperfect Model Scenario ) Tj 1 0 0 1 309.35 660.95 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 660.95 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 512.899 661.2 Tm 88 Tz (94 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr 1 0 0 1 145.199 643.7 Tm 91 Tz (5.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 91 Tz 3 Tr 1 0 0 1 170.9 643.7 Tm 101 Tz (IS methods in IPMS ) Tj 1 0 0 1 282.699 643.7 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 643.7 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 512.899 643.899 Tm 85 Tz (98 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 85 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 85 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 85 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 85 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 85 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 85 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 85 Tz 3 Tr 1 0 0 1 170.9 626.399 Tm 95 Tz (5.2.1 ) Tj 1 0 0 1 192.949 626.399 Tm 446 Tz (\t) Tj 1 0 0 1 206.9 626.399 Tm 102 Tz (Assuming the model is perfect when it is not ) Tj 1 0 0 1 434.149 626.399 Tm 1451 Tz (\t) Tj 1 0 0 1 479.5 626.399 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 513.1 626.649 Tm 86 Tz (98 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr 1 0 0 1 171.099 609.1 Tm 97 Tz (5.2.2 ) Tj 1 0 0 1 193.449 609.1 Tm 430 Tz (\t) Tj 1 0 0 1 206.9 609.1 Tm 99 Tz (Model error ) Tj 1 0 0 1 274.1 609.1 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 609.1 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 512.899 609.1 Tm 88 Tz (98 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr 1 0 0 1 170.65 591.85 Tm 99 Tz (5.2.3 ) Tj 1 0 0 1 193.449 591.85 Tm 430 Tz (\t) Tj 1 0 0 1 206.9 591.85 Tm 99 Tz (Pseudo-orbit ) Tj 1 0 0 1 273.85 591.85 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 591.85 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 512.899 591.85 Tm 88 Tz (99 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr 1 0 0 1 170.65 574.299 Tm 99 Tz (5.2.4 ) Tj 1 0 0 1 193.449 574.299 Tm 422 Tz (\t) Tj 1 0 0 1 206.65 574.299 Tm 101 Tz (Adjusted ISGD method in IPMS ) Tj 1 0 0 1 380.649 574.1 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 574.1 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 508.55 574.549 Tm 79 Tz (101 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 79 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 79 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 79 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 79 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 79 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 79 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 79 Tz 3 Tr 1 0 0 1 170.9 556.799 Tm 96 Tz (5.2.5 ) Tj 1 0 0 1 193.199 556.799 Tm 438 Tz (\t) Tj 1 0 0 1 206.9 557.049 Tm 101 Tz (ISGD with stopping criteria ) Tj 1 0 0 1 353.75 557.049 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 557.049 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 508.1 557.049 Tm 84 Tz (105 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr 1 0 0 1 145.199 539.5 Tm 89 Tz (5.3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 89 Tz 3 Tr 1 0 0 1 170.4 539.75 Tm 100 Tz (Weak constraint 4DVAR Method ) Tj 1 0 0 1 345.1 539.75 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 539.75 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 508.1 540 Tm 84 Tz (111 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr 1 0 0 1 170.9 522.25 Tm 95 Tz (5.3.1 ) Tj 1 0 0 1 192.949 522.25 Tm 430 Tz (\t) Tj 1 0 0 1 206.4 522.7 Tm 100 Tz (Methodology ) Tj 1 0 0 1 282.949 522.7 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 522.7 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 508.3 522.7 Tm 84 Tz (111 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr 1 0 0 1 170.9 504.949 Tm 98 Tz (5.3.2 ) Tj 1 0 0 1 193.449 504.949 Tm 430 Tz (\t) Tj 1 0 0 1 206.9 505.199 Tm 96 Tz (Differences between ) Tj 1 0 0 1 308.149 505.199 Tm 60 Tz /OPExtFont6 12 Tf (1) Tj 1 0 0 1 311.75 505.199 Tm 37 Tz /OPExtFont4 12 Tf (-) Tj 1 0 0 1 314.149 505.199 Tm 101 Tz /OPExtFont6 12 Tf (SGDc ) Tj 1 0 0 1 348 505.199 Tm 98 Tz /OPExtFont5 12.5 Tf (and WC4DVAR ) Tj 1 0 0 1 434.149 504.949 Tm 1451 Tz (\t) Tj 1 0 0 1 479.5 504.949 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 508.3 505.199 Tm 87 Tz (113 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr 1 0 0 1 145.199 487.449 Tm 93 Tz (5.4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 93 Tz 3 Tr 1 0 0 1 170.65 487.699 Tm 100 Tz (Methods of forming an ensemble in IPMS ) Tj 1 0 0 1 381.1 487.899 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 487.899 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 508.8 487.899 Tm 84 Tz (116 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr 1 0 0 1 170.9 470.399 Tm 95 Tz (5.4.1 ) Tj 1 0 0 1 192.949 470.399 Tm 452 Tz (\t) Tj 1 0 0 1 207.099 470.399 Tm 102 Tz (Gaussian perturbation ) Tj 1 0 0 1 327.35 470.399 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 470.399 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 508.3 470.649 Tm 87 Tz (116 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr 1 0 0 1 170.65 452.899 Tm 99 Tz (5.4.2 ) Tj 1 0 0 1 193.449 452.899 Tm 430 Tz (\t) Tj 1 0 0 1 206.9 453.1 Tm 101 Tz (Perturbing with imperfection error ) Tj 1 0 0 1 389.75 453.1 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 453.1 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 508.3 453.35 Tm 87 Tz (116 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr 1 0 0 1 170.65 435.6 Tm 99 Tz (5.4.3 ) Tj 1 0 0 1 193.449 435.6 Tm 422 Tz (\t) Tj 1 0 0 1 206.65 435.6 Tm 103 Tz (Perturbing the pseudo-orbit and applying ) Tj 1 0 0 1 417.1 435.85 Tm 92 Tz /OPExtFont6 12 Tf (ISG_Dc ) Tj 1 0 0 1 461.05 435.85 Tm 614 Tz (\t) Tj 1 0 0 1 479.5 435.85 Tm 33 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 12 Tf 33 Tz 3 Tr 1 0 0 1 508.55 435.85 Tm 88 Tz /OPExtFont5 12.5 Tf (117 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr 1 0 0 1 144.949 418.1 Tm 89 Tz (5.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 89 Tz 3 Tr 1 0 0 1 170.65 417.85 Tm 102 Tz (Results ) Tj 1 0 0 1 211.199 417.85 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 417.85 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 508.8 418.55 Tm 86 Tz (118 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr 1 0 0 1 170.9 401.05 Tm 95 Tz (5.5.1 ) Tj 1 0 0 1 192.949 401.05 Tm 446 Tz (\t) Tj 1 0 0 1 206.9 401.05 Tm 86 Tz /OPExtFont6 12 Tf (_TSGDc ) Tj 1 0 0 1 246.699 401.05 Tm 98 Tz /OPExtFont5 12.5 Tf (vs WC4DVAR ) Tj 1 0 0 1 327.85 400.8 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 400.8 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 508.8 401.3 Tm 84 Tz (118 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr 1 0 0 1 170.65 383.5 Tm 99 Tz (5.5.2 ) Tj 1 0 0 1 193.449 383.5 Tm 422 Tz (\t) Tj 1 0 0 1 206.65 383.75 Tm 100 Tz (Evaluate ensemble nowcast ) Tj 1 0 0 1 345.35 383.75 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 383.75 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 508.55 384 Tm 86 Tz (121 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr 1 0 0 1 144.949 366 Tm 91 Tz (5.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 91 Tz 3 Tr 1 0 0 1 170.65 366 Tm 97 Tz (Conclusions ) Tj 1 0 0 1 238.3 366.25 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 366.25 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 508.8 366.5 Tm 87 Tz (126 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr 1 0 0 1 127.9 337.899 Tm 77 Tz /OPExtFont3 11 Tf (6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 77 Tz 3 Tr 1 0 0 1 144.699 338.149 Tm 107 Tz (Forecast and predictability outside PMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 107 Tz 3 Tr 1 0 0 1 506.399 338.399 Tm 85 Tz (128 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 85 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 85 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 85 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 85 Tz 3 Tr 1 0 0 1 144.949 320.649 Tm 89 Tz /OPExtFont5 12.5 Tf (6.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 89 Tz 3 Tr 1 0 0 1 170.4 320.649 Tm 101 Tz (Forecasting using imperfect model ) Tj 1 0 0 1 345.6 320.649 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 320.649 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 509.05 320.899 Tm 87 Tz (129 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr 1 0 0 1 170.65 302.899 Tm 96 Tz (6.1.1 ) Tj 1 0 0 1 192.949 302.899 Tm 438 Tz (\t) Tj 1 0 0 1 206.65 303.1 Tm 102 Tz (Problem setting up ) Tj 1 0 0 1 309.85 303.1 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 303.1 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 509.05 303.35 Tm 86 Tz (129 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr 1 0 0 1 170.4 285.35 Tm 99 Tz (6.1.2 ) Tj 1 0 0 1 193.199 285.6 Tm 430 Tz (\t) Tj 1 0 0 1 206.65 285.6 Tm 103 Tz (Ignoring the fact that the model is wrong ) Tj 1 0 0 1 416.649 285.85 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 285.85 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 509.05 285.85 Tm 87 Tz (130 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr 1 0 0 1 170.4 268.1 Tm 99 Tz (6.1.3 ) Tj 1 0 0 1 193.199 268.1 Tm 430 Tz (\t) Tj 1 0 0 1 206.65 268.1 Tm 102 Tz (Forecast with model error adjustment ) Tj 1 0 0 1 399.1 268.299 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 268.299 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 509.3 268.299 Tm 84 Tz (130 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 84 Tz 3 Tr 1 0 0 1 170.4 250.799 Tm 100 Tz (6.1.4 ) Tj 1 0 0 1 193.449 250.799 Tm 422 Tz (\t) Tj 1 0 0 1 206.65 250.799 Tm 102 Tz (Forecast with imperfection error adjustment ) Tj 1 0 0 1 434.649 250.799 Tm 1435 Tz (\t) Tj 1 0 0 1 479.5 250.799 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 509.3 250.799 Tm 86 Tz (140 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr 1 0 0 1 144.949 233.5 Tm 91 Tz (6.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 91 Tz 3 Tr 1 0 0 1 170.4 233.5 Tm 105 Tz (Predictability outside PMS ) Tj 1 0 0 1 309.85 233.5 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 233.5 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 509.3 233.75 Tm 86 Tz (146 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr 1 0 0 1 170.4 216 Tm 98 Tz (6.2.1 ) Tj 1 0 0 1 192.949 216 Tm 446 Tz (\t) Tj 1 0 0 1 206.9 216 Tm 99 Tz (Lyapunov Exponents ) Tj 1 0 0 1 318.949 216 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 216 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 509.3 216.25 Tm 88 Tz (147 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr 1 0 0 1 170.65 198.25 Tm 99 Tz (6.2.2 ) Tj 1 0 0 1 193.449 198.25 Tm 436 Tz (\t) Tj 1 0 0 1 207.099 198.5 Tm 100 Tz (q-pling time ) Tj 1 0 0 1 274.3 198.25 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 198.25 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 509.3 198.7 Tm 87 Tz (148 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr 1 0 0 1 170.65 180.95 Tm 99 Tz (6.2.3 ) Tj 1 0 0 1 193.449 180.95 Tm 422 Tz (\t) Tj 1 0 0 1 206.65 180.95 Tm 104 Tz (Predictability measured by skill score ) Tj 1 0 0 1 399.6 180.95 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 180.95 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 509.5 180.95 Tm 86 Tz (150 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr 1 0 0 1 144.699 163.45 Tm 93 Tz (6.3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 93 Tz 3 Tr 1 0 0 1 170.65 163.45 Tm 97 Tz (Conclusions ) Tj 1 0 0 1 238.8 163.45 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 163.45 Tm 32 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 509.5 163.7 Tm 87 Tz (154 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 87 Tz 3 Tr 1 0 0 1 127.9 135.6 Tm 77 Tz /OPExtFont3 11 Tf (7 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 77 Tz 3 Tr 1 0 0 1 145.199 135.1 Tm 99 Tz (Conclusions ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 99 Tz 3 Tr 1 0 0 1 506.899 135.6 Tm 86 Tz (156 ) Tj ET EMC endstream endobj 29 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 30 0 obj <> stream 0  ,,x  jQr;ݒ_g1j kp6E+0dcE*PJV5؝i*'zS.Kt $QJX$ه('nǮElH|, `&_1,|Ou]jRaw@ؑNΆwh.P@-xjNuaq6r-K돐3Gm6όpRH6leG0dv +މ_HAg'ܦ28ܼjrGRG\5u@& 9ۧ?6 :#cؗxq=2~6]a&PHb4&v,3@(xӽ-rd/W> EwФ385IYۚ)VEt4]<&TqU uCX//q puޢ),.(=ł e*izcpyL:͔+ήR|r+ijTCM~$RR4Lx@MFFe> t9Qb^? 6M<`pF9Cy=B߈ɜkeuE.(Ca&bG1FhN?pxULxh#yBroivQXahJ CՖQN%?'OO)t t8ƈ@}I{mC2y1Dmc#sƎ,_t4x kis)| Չiئjx[m2'>R>) !h: #I0K(קh{WEt#DsgOX-J羢1 h#jH(i @$nS\ajT i]` <8bB#R\mRxP6FˌT3எr<;)wT'% g 8tBǤnӜr^C !&p KtzrdG]/4n`P4tw*ޛ9oY6y f&P&:Kۅ14`ԀD3!_n%>v2 %rJ1._'Tp !OFJ10~ f^`JQntB7 ׹sbX[¹`e| G8s-D>&ZE?}? 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xt:]5ま53Q#Ëxl>ŃԱ!!")<>9N^%%*E MλP'>Zz~- ,1Qϡ-Hр4`N%(_ Ya'OCt,{:-M?sMfZS8J0Ԧ¤A('YPgJS1l&ϛXP6!c'4 iK.F9E/Ә*kFFh_hXԙՆhet91@tշJkގt>J*Fט%&MEGfO^,&,01=K%C]V@&˜T ;̀ 4tH*vm ɘC҃/~lk&`!@PzҥI5sPͼ)O19Ӭ4z 9I`߇F3HUtwMրuG:s#1(GE<1v=WsG!Nĺ#xZLyi.S=]-Mqsnjjj Dp//yD3*SjuGzsx"Fa|CC 3嫏'W)h F#nѣ:jqY(a&֐XtX^+ I0$6\NαK.De`Dz9vItگQ;bHyB<3k`B>]z8P W>]?j{d͉dWk՘pl0-qmyōט|&/=7!zQleࠂTA e9lWda0&(9_l%4"dm q+kV1ZHx*M\6=aCKP+hGNH$5|N40n?? endstream endobj 31 0 obj <> endobj 32 0 obj [33 0 R] endobj 33 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 615 0 0 840 0 0 cm /ImagePart_2027 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 451.449 721.2 Tm 114 Tz 3 Tr /OPExtFont3 11 Tf (CONTENTS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 114 Tz 3 Tr 1 0 0 1 127.9 678.25 Tm 110 Tz (A Gradient Descent Algorithm ) Tj 1 0 0 1 306.5 678.25 Tm 2000 Tz (\t) Tj 1 0 0 1 504.949 678.25 Tm 86 Tz (160 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 86 Tz 3 Tr 1 0 0 1 127.9 650.399 Tm 109 Tz (B Experiments Details ) Tj 1 0 0 1 259.449 650.649 Tm 2000 Tz (\t) Tj 1 0 0 1 505.199 650.399 Tm 85 Tz (162 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 85 Tz 3 Tr 1 0 0 1 127.9 622.299 Tm 103 Tz (References ) Tj 1 0 0 1 187.699 622.549 Tm 2000 Tz (\t) Tj 1 0 0 1 505.199 622.1 Tm 86 Tz (166 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 86 Tz 3 Tr 1 0 0 1 321.1 52.1 Tm 98 Tz (vi ) Tj ET EMC endstream endobj 34 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 35 0 obj <> stream 0  ,,PPb6u*KRr=N Ӽy]jcHK1y2:p\ NcRw&gst)[e?![IJ 4N_5Жy|婹})>`lGBbYc475loh-mpeIek5}O8FIj ސzbD.A, xOCdRGW[F[c+pǻ?5}ȐϘ$}";P?/7jr(c0'zLZ!Htjd&һArV̿N&(jD3HMJu$zc^ bz | &I_b^:U(OI˿=7ޞ?0ƾJ \6Ls eMȅGov! d/^j|ȍsPj n ,wyixڪMox(C_2?Y!eub_w'%Pjw>$08 d:*Or︛(#P@#3cHE.37!qkXUa@/ ouӥ Ccw t8/8 v sg~L4JC] ̶A Da5E~cN{{qgAOy`=O`-e]-EvuI1cf{>۱ ޭզwyCww;t71c_e»)AUϿY]ÿ)SukƼZOEsWj?u8FpNy^(U;Yo;'9Bs =mF UĨX ֍k•C![Ռ*?ܗ/)7sI'.@1m2Ӧ5*ԄU?G.#svl)X=ZR9Yg^b93▲=2` 9:y9tgOd$Kg5壙W <Ӊ`N fP%58V#SɬcxMv+gawܣjqOv xȝQ)vag1`,[S-!s|]L rr<2YuF!ʨ kIJwDN*I3k˅Ѕ#k䨶J  D4>g|Z;v bp}zǐ4K ^>:|ṔM`iej/(.r]j %7</ݦܖF -  j4n<4#k7 ^=e`m9"Y9|U^3Rgm`A]fovSq|3>0hG HE$uẸigGYe)> A 9Ś|>VJȢf& "^ۣyyX҆:H19$y82+:1)BH"o*<=dsnR}Sz\6W5@EKY%?iXͫ ? endstream endobj 36 0 obj <> endobj 37 0 obj [38 0 R] endobj 38 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 610 0 0 841 0 0 cm /ImagePart_2028 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 121.9 596.2 Tm 105 Tz 3 Tr /OPExtFont3 22.5 Tf (Chapter 1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 105 Tz 3 Tr 1 0 0 1 120.95 542.45 Tm 107 Tz (Introduction ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 107 Tz 3 Tr 1 0 0 1 120.7 480.75 Tm 88 Tz /OPExtFont3 11 Tf (Nonlinear dynamical systems are frequently used to model physical processes such ) Tj 1 0 0 1 120.7 457.5 Tm 92 Tz (as the dynamics of breeding population, the electronic circuit and weather. The ) Tj 1 0 0 1 120.5 434.449 Tm 93 Tz (ultimate goal we have in mind is forecasting the future states of the system. Of ) Tj 1 0 0 1 120.5 411.649 Tm (course there are many operational details involved, but the mathematical prin-) Tj 1 0 0 1 120.25 388.6 Tm 96 Tz (ciple is simple, first estimate the state of the model of the dynamical system, ) Tj 1 0 0 1 120.25 365.1 Tm 91 Tz (then integrate this initial condition forward to obtain a forecast. When the equa-) Tj 1 0 0 1 120 341.8 Tm 93 Tz (tions of motion that describe the system are known, which is the perfect model ) Tj 1 0 0 1 120 318.75 Tm 95 Tz (scenario case, the key to the problem is the accurate estimation of state given ) Tj 1 0 0 1 120 295.7 Tm (observations. But given a perfect model of a chaotic system and a set of noisy ) Tj 1 0 0 1 120.25 272.45 Tm 96 Tz (observations of arbitrary duration, it is not possible to determine the state of ) Tj 1 0 0 1 120 249.149 Tm 92 Tz (this system precisely. Traditional approaches to statistical estimation are rarely ) Tj 1 0 0 1 120 225.899 Tm 93 Tz (optimal when applied to nonlinear models. Even in the perfect model class sce-) Tj 1 0 0 1 120 202.6 Tm 91 Tz (nario, likelihood methods have difficulty in estimating either the initial condition ) Tj 1 0 0 1 120 179.1 Tm 94 Tz (or the model parameters. The question is besides getting information from the ) Tj 1 0 0 1 120 156.049 Tm 91 Tz (observations, how much the information we can draw from the nonlinear system ) Tj 1 0 0 1 119.75 133 Tm 94 Tz (itself \(that is, information implicit in the equations\). Our aim is to enhance the ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 94 Tz 3 Tr 1 0 0 1 316.8 53.1 Tm 57 Tz /OPExtFont3 11.5 Tf (1 ) Tj ET EMC endstream endobj 39 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 40 0 obj <> stream 0 ,,55juxL/)JXȐFLD)G"aGB˺,0BcMvEВl@v);J=uj <#>KB^^4Z1b4?O# Y'3z1h![?WBnKLheط5 y=+#@E _-ְJ6|h@\ @@m m0G0,?4K[kmOZCPRIc/XGuzKM9y7^PmUÃűLL<`X^9o;m~AE<ԳjJrwKx0uŴ9FUeQ αGѩGIYiͳp ȅNExJ*LXs-:hxLrj9ђcEm7S`X*KU<. )MMkgY}sۮq[@ y'J{h=}puxh{D6,G#Of0\~+`C6?KkDhz;`:W6KĈETLX/)*Ke)ڼ0f)*$'"r)؟p+}1e¡zH0$ѫP?_8/1+O0Ym >[7Ҏn:WnEt5)llw7XŔe3ƽuɞb #zj G$X9=>01,&Y3oքבv+jȘ&鼅7-wS]83+8KI\&65:4;mtޯwS$ LfVԙ?Uk$ZuRS`r?J.h)$TG^b,d+WzqVLrEPCKBI?<ߺ4O={"|3t/['XWq!vfI0˿Hr+|3~Iq,X`qVw]EH4%" |?zB2 r)na,X {o3f^'h~458Ԣw'U&ja?K;@ Ft0pԛЭUYo<ͯ0GnIn" HA4= `Ðc҃jQxōxNaKP%W L ,Qy,>ؑanJcw"~C?^M(ocpS_WXa1sK|xw:ZUn|R+H8h2s]e UoEJ(WL75-]Z6H)#}J/7}/S̈(KELFip!B3CK ?*@9ؠ$GJ˵2X aP2H?OG4jJm}lyǒx#וp$sgk긘 GnNu5NEul`4?mȇaXXj-Z)v͓ n$)fV$ qwvIiɺsMN<_ׄh޴,9r'Zk[e-Vnf,n5\ٹ=T q.G06>qEhQ%?thz^_Ɓf,K5RC7hNDR@8cu K,J`פY{: d-pk Y"5a8Xs}iDEo0Daf8?PhHF"ֆKWuMzgRaؓV#]W$t|n!?2 k?suOOE]}A9`Rc9VYMFS7skggaM,,@bdd7" P o5]f8B5=gW.pyW{#]gb4>|<"X5rt՞~(Lr#$1k'#c=!k IFLG9Le7aVy"QV( xy#}}lȊ_3\ttǭ rc,T$澮l ;/R?4PQ͡6@iYD.d#Ct't>gQa~g[Ô]T99[fGݔcEk{w2_:mU {*)J)!'b'%'U GSxʪ%b?Ίf00 {#J|kz3*c{41A (RK@t |8lX9q7I/[)" ruSq ":AY#F6{NP.P9h.Sİml1S;K*;T|j>ٗ¾筲ϣ0³*7%0ܲ|ݦeQ4&(ڱ%մݸɻ墨(v<;켻`~ &͘UwNE,)w-č"ʙ愿jÉ(+l 1J2 OѮseį"d*;9tT?nTMJгKaoqrddWe?&iU82NmSp"G3]>&Zb"3kݚNWPbNyw齸 FM 4nmK Rq,sY (N:QVg̀؝U#x3T3KH5FO߈G6 b2V 36#!cβJ0`_݅ }>Lгύwoi4Mi-p3WzQ~OΥE }[+y)csXwlqPY8PlH^W>4[)NGr\9~Po||urvCiTDB 2W0HQѤ^c6b'-D!GŻj4TM;7pWKMp2҆굯tFl> q>ө,-pk+o U+kž&y5/$kx:/AhW幷}3:̳Yʺuh3ooZ Z,+KRX%;AO.xI#j~UJ @*>^1?n >enLU.:L}ř#MuJM ^1uRK]-`2 ,A mXC%6G2nb0:1r.7 z21hnA:323t gq=JUReaG\2`6 .8dFL2yG'wam5FF G}0 DZZB{$H93XHοR`:7-lc W->ŵ G8NuU8~ӔIYk?jF IUS2xw" gdߘcZ6kXj朔wPbUr2#&]"fn`fڬsпcɼD95M< &eOʊ]  [O6hxb@>j S$pǥq#ذ>ͰVòϰ&թ(-̾J<4n~#fv ;Y$u-CDbz6fIL;?ԙc(/t83)h`Tg>cGԍT3"(6.п>ސ0i3ϭQ(IJ%0&##5߽r|8PFǶn2:>ΘPMLj>|!>-ɿ՟D&ћհͱ9._l/ru縿^6{(eGfG.Ud3N]y Oȵ+С>/4 rPOfm(4*["uoZcCNX> RGC ɂLmnZ 53BvV2WMeuEn:{drVWx)UZp S[ۥ"U.sO$ʲ+ŸLh*ŚT`]JYY0J o#ykzR,<Z)AOV8OKSC`0S7]xaoTUX$>q;>-Cy丝>6qtKto4=o|t?$tHJWu}y \h+,軐j< nq=2]/aIw8ȼf7,NUa"k8h(:i6.G8سY(y6ID~B+5dtHwj⋍C-BS}B Z,SÆhl/oM0a F ex[{ĄD|ſt7p-}3aTy*WZ\.ba+Lipd{.@S98!Aph[wc z־zV#x|8w!RQHL@آgw%  lQڞrW"Y-cq|i]eXZav2{B Aw\ϒw bT@<m!f3zT:BЧ4/ E.aqxz9I" aΖ_w$^+V_Jb2za ɍ@ ?'ZT]d.H\Y[dQy$GpStORQ ?(0[fNDx69NwNJ*;-"r+T|TF1/Sn S>ڟy{cPfYWC$AUS鑦V H`%_zCϙ4("%LrS6ߖW@w/q ~3EL /Dm-g=,m=l4 ndvw@cNs2}˼ ZUǨw7K믯KD+DNj>Y#*]ߺ +O44Ѥg}gic% s@[e3~p)5Uy3=dmiUwx-6ϦD>?77&)ڽ"ψZx<+RPؙ l;9,agM^.d" k_>sgd%Rq]`QrmKtcW:GBLVS#1)011h~J}|BE K_N~<՞?V n2|+C2)Ƥ,2 zh.5fEeXN7*~0[K+67T1 Qz{KBMh7=#OLS4ʆFOyOM@.ݪGҋ} ;4Ri6*?-NvIPn2@$J_]ןWZu|τPm(Ƚ';-?hǧv04wo\4嫎D׈OmGyc|[ i 9_qH\4q<[vŅ~3kki@clRGToti³sAJOk#Kzđ$1ӄo`f$;r endstream endobj 41 0 obj <> endobj 42 0 obj [43 0 R] endobj 43 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 842 0 0 cm /ImagePart_2029 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 123.849 679.049 Tm 90 Tz 3 Tr /OPExtFont3 11 Tf (balance between the information contained in the dynamic equations and the in-) Tj 1 0 0 1 123.849 656 Tm (formation in the observations themselves. Outside perfect model scenario, things ) Tj 1 0 0 1 123.849 633.2 Tm 92 Tz (become more difficult. The uncertainty of the initial conditions comes from both ) Tj 1 0 0 1 123.849 610.149 Tm 93 Tz (observational noise and model inadequacy. To estimate the future states of the ) Tj 1 0 0 1 123.599 587.1 Tm (model by interacting the initial condition forward will eventually fail to shadow ) Tj 1 0 0 1 123.599 563.85 Tm 92 Tz (the observations no matter what initial condition is used. To produce more con-) Tj 1 0 0 1 123.599 540.799 Tm 91 Tz (sistent estimate of the current or future states, information from the model error ) Tj 1 0 0 1 123.599 517.5 Tm 98 Tz (need to be extracted. This chapter provides an overview of the thesis. Some ) Tj 1 0 0 1 123.599 494.5 Tm 90 Tz (terms undoubtly are new to the reader, all terms are defined in the later chapters ) Tj 1 0 0 1 123.599 471.449 Tm (when they are first used. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 141.349 448.399 Tm 102 Tz (Outline of the thesis: In Chapter 2. Some terminologies of dynamical ) Tj 1 0 0 1 123.349 425.35 Tm 90 Tz (system are introduced and general properties of nonlinear dynamical systems are ) Tj 1 0 0 1 123.599 402.1 Tm 89 Tz (illustrated. An overview of the systems and models used in the thesis is presented. ) Tj 1 0 0 1 123.599 378.8 Tm 93 Tz (Other than details on the system-model pairs, nothing new is presented in this ) Tj 1 0 0 1 123.599 355.5 Tm 86 Tz (chapter. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 86 Tz 3 Tr 1 0 0 1 140.65 332.25 Tm 93 Tz (In Chapter 3. we consider the nowcasting problem in the perfect model sce-) Tj 1 0 0 1 123.099 309.2 Tm 94 Tz (nario. We illustrate a new ensemble filter approach within the context of indis-) Tj 1 0 0 1 123.099 285.7 Tm 93 Tz (tinguishable states \(48\), using Gradient Descent to find a model trajectory from ) Tj 1 0 0 1 123.099 262.649 Tm 92 Tz (which an ensemble is formed. An introduction of traditional variational method, ) Tj 1 0 0 1 123.349 239.1 Tm 96 Tz (Four-dimensional Variational Assimilation \(4DVAR\), is presented. The differ-) Tj 1 0 0 1 123.099 215.85 Tm 91 Tz (ence between our method and 4DVAR is discussed. Results presented show that ) Tj 1 0 0 1 123.099 192.799 Tm 93 Tz (4DVAR is only applicable to short assimilation windows while our method does ) Tj 1 0 0 1 122.9 169.5 Tm (not have such shortcoming. The popular sequential method, Ensemble Kalman ) Tj 1 0 0 1 122.9 146.25 Tm 98 Tz (Filter, is also applied to solve the nowcasting problem. For the first time we ) Tj 1 0 0 1 122.9 122.95 Tm 94 Tz (demonstrate that the indistinguishable states approach systematically outper- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 94 Tz 3 Tr 1 0 0 1 319.899 53.1 Tm 70 Tz (2 ) Tj ET EMC endstream endobj 44 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 45 0 obj <> stream 0 ,,ffb7!e.qí&dWM(R JjR t῱ ^9kMǍ4g1o+c`w"FN>Wp|N]wU62Ԧ.Y3ck\od ٓRtXX;תC\i]Yu'Ztn]'VA?Yܿ}/;G _F郘F#Wr9$`J(+w>7OFΗ.}/;x)6'eڐB[յ7}mb|grMiI4aڝ,r`S5yF{rkhK %;DC-Yz HO؆v?F<2,oa;;z&jV|R? x +_ \}gޥϪE( l=qg+0n`79(D7: qxsi)%.i D [nU9x4}^P>wվ3ݳ51u$mad'OaІ2Ă<ݞxwt(7_cےW2*;@;*$Kq`%Ƞ'GQx0zj:_i֟2c ̵p;c"sC5$ 'qL:h=OU+HaZ- SxIwjzگmbS+ }vFԻ?쉞 +4SWJs6W{tA`)>bTeMdb)n.Ptc#*'4ěeϳwDt=:svҬʨ糏!"7C:Mř\w-$aj cc<߬ !, MqG{b%ڼ]^^3Ge+ÛI8r|tIm҅i soI-2I[{Hlmxc˞l ]zv}m5M?F^}iS"^b8+rw'V"c'O@U]FFxqquR* ٔt&P)snh͖ ԄS9U9.lT.K1ܒg/ ހPĆΔgPOX]n9#D-a/^TsyQRuo3HkgVs$f(p{֓53-'8b!a R$,N#EG0l%DXZΫ\`ZIΚQw^uPyvwflI# XSOM9h%-S4 L؞=YbOTVo~[J"_O{!*}ퟧwT胜\@+6-KI;.關k)7INQm2rhL51^n&WynaqŤ /+w m&27_CBI%W!.17vVzJǶ+fC!g˯ >WzKL^4oUΜmY5UޜC]Yr 5d,:ƻwh6A$케L&8~YVS˓wYUq.AX&(yLyt٧1=luqfvb٩\el)#b|}G# $ h&r4(.C了Oy&=P?B[?Q+f̭%߀sJ0B,^U-%?0xB2< Lmp u=5| 2̓G*>,јHpʳ氨5NJr>䬙َ԰9볾FѰq::[ľi3e)!;"9qH$L0h7jԤ:۪=0LG\nwq.z9 YqI1?;8tH>9tl9W/;J]X0eA?zf)\)"&ȯOw@ s@0|dA%4gX0u~619lQ0i3h#X_1E:PݗjМbi/@ZyBVjk!zC`c\/^]Fe!YqG/Q`\0KuRs=#-u^fT_΢uavn=r Fj@6 46TfKM)7ZDU?Qj&mP?ܙ^(RNmer.>ɻyq~BL nKӴ!rTk.4W<6> JycF˼8.Ԫ/XqOmO{L7ڴ}@7 WF mfPǯqB@dqb7|pQ07t fbSxɊ 2i?cݼoz3dg:YGi<Lzs3j`G@>,nc6E(j#SցZWlD- gi)p?(ߨ~| |z(mX&^-W-ϯ D3O1uH?#2ڲ)u1AF$rKC"Za-Dh暂گe,UrҤqDm'?C$c8.&kތS=o~챛~n[SHe}R|ֲ=UyF՟cAtv@JVCȫŠXOo01|AsRCQUtwɄQG'B9Y8aFV)e9e?DAY{\``~^Q2Wr5an%t–G\aGM-kENgb7^-oBb}j޾SiZ# ?Щi!/SwX+՚$lʐmx3K/ķهk\!*MJg (ța)(m@8}>9P=_J'&X[Gz~iAd2ܫ>R @0_5©%_*x|,2Zh~]SN2~o<xT,m3!' OkЯaƧowPWPaG%+!!=1խij13^v> R )ľ~0.:*^3ii<>W6D%J r[Rtb|ܒ>̃ѤlRYwAO6u]tu@ %WUqLRwi¹SOIKg։(U9TlQf$ZPO-."u&tEaYgbKŊip2q{I7!¼LD5eF>nWnsN /~N~N |׽*M_gh l&-ZV; WLZLn=\:/"r^ƒU0|; p*NZ Xg [l:-q*7gJkHtqaAjFz(m_13ӼT^:|> &TLyat8g6}@@F%~U jSrT;Շd˜O~a3Ab Hhʻ󫠒<)0Kp%Igx*a&~ + >vXŸ()MXη[|b.*3Fzq~;SrM-S=1 >ua[0zyitпGۅ +ǘJ䟶*7=\jŰ"٢@AI%pdqƇpdfQ?&~8 r9.OEAA'i08 ]p@C]tak?8b7〞 W׎Ox5 1AS#j 2N^RX58؏YՓv迵Tg˪/M=?q=姃.(6[:멁3PܴŸɤ-l<W W#&Xv.6^ 1Zm%Z#ҕ3S3rL@$^U_Zh\.yr h8X NFi֝m}_wnilM$-֏/ܕ6YWRDe{ uIUI컆k7t>f|sy%N8p6s8oAsQ"Cͩ$Nw% gTBtQM^j@zy5 T |\$!kQ mpM誳;(/Z@浟=([Vv[VfCk"B\M^xyĨ֌UvW70x Anrs(g|2a.$Lp{0jeMlI=TT S3Tu 4e&n%b]uj 66N-HOz|\q'xFZ5oPȇ@ 1S%BM=p;|E\9vb6=,m7AOFމ[if64.xE/2?$}Ir)J =ۖ(\xLw!wFF5TM26S!cO&/E~*@g\ཐʅ8t۬UaGYSC5qtŒ(/Uy4Eւz2:ST[]9fbޯ  kM2) 22o毡i<DėS"Α9igdm=.d.> ,-[ao#иM޿p 8㒕^%0[VLJn=ցFTW v;H9τh;:Tg{s^ڤqml z9%Tۻv12LkdgB( !dwAq|~Ȕv6.V# ^|$ADZ`.ug]ν &6K%{K?1"xflr8\̷+& 3s}Ͼg%Kiwp-3g~pBO'L A}u3ݣV#"|<8a* [ q֯~slߌ wƎk;L,+nsYjV}4=7⁔YU]I{#}PcLi#^p eBYf/%q= endstream endobj 46 0 obj <> endobj 47 0 obj [48 0 R] endobj 48 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 841 0 0 cm /ImagePart_2030 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 123.599 679.25 Tm 92 Tz 3 Tr /OPExtFont3 11 Tf (forms the Ensemble Kalman Filter in both low dimension Ikeda Map and higher ) Tj 1 0 0 1 123.599 656.2 Tm 89 Tz (dimension Lorenz96 system. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 140.4 633.149 Tm 99 Tz (In Chapter 4. we provide new results to solve the problem of parameter ) Tj 1 0 0 1 123.599 610.1 Tm 93 Tz (estimation of deterministic nonlinear models within the perfect model scenario ) Tj 1 0 0 1 123.349 587.1 Tm 95 Tz (where the mathematical structure of the model equations are correct, but the ) Tj 1 0 0 1 123.349 564.049 Tm 92 Tz (true parameter values are unknown. Traditional parameter estimation methods ) Tj 1 0 0 1 123.099 541 Tm 89 Tz (like least squares often base on the assumption that the forecast error is Gaussian ) Tj 1 0 0 1 123.099 517.5 Tm 91 Tz (distributed. Unlike linear models, when one put a Gaussian uncertainty through ) Tj 1 0 0 1 123.099 494.449 Tm (the nonlinear model, one will get non-Gaussian forecast error. Results show that ) Tj 1 0 0 1 122.9 471.399 Tm 88 Tz (the least squares estimates may even reject the true parameter value of the system ) Tj 1 0 0 1 123.099 448.35 Tm 94 Tz (in preference for incorrect parameter values \(64\). Two new approaches are in-) Tj 1 0 0 1 123.099 425.1 Tm 91 Tz (troduced to address the shortcomings of traditional methods. The first approach ) Tj 1 0 0 1 122.9 402.05 Tm 92 Tz (forms the cost function based on probabilistic forecasting; the second approach ) Tj 1 0 0 1 122.9 378.75 Tm 91 Tz (focuses on the geometric properties of trajectories in short term while noting the ) Tj 1 0 0 1 122.9 355.699 Tm 96 Tz (global behaviour of the model in the long term. Both methods are tested on a ) Tj 1 0 0 1 122.9 332.449 Tm 91 Tz (variety of nonlinear models, the true parameter values are well identified. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 139.699 309.399 Tm 94 Tz (In Chapter 5. we) Tj 1 0 0 1 226.55 309.149 Tm 50 Tz /OPExtFont3 3 Tf (, ) Tj 1 0 0 1 228.699 309.149 Tm 91 Tz /OPExtFont3 11 Tf (consider the nowcasting problem outside the perfect model ) Tj 1 0 0 1 122.65 285.899 Tm 96 Tz (scenario. Outside perfect model scenario, to estimate the current state of the ) Tj 1 0 0 1 122.65 262.6 Tm 95 Tz (model one need to account the uncertainty from both observational noise and ) Tj 1 0 0 1 122.9 239.299 Tm 90 Tz (model inadequacy. Methods assuming the model is perfect are shown to be either ) Tj 1 0 0 1 122.65 216.049 Tm 94 Tz (inapplicable or unable to produce the optimal results. It is almost certain that ) Tj 1 0 0 1 122.4 192.75 Tm 95 Tz (no trajectory of the model is consistent with an infinite series of observations. ) Tj 1 0 0 1 122.65 169.7 Tm 91 Tz (There are pseudo-orbits \(50\), however, that are consistent with observations and ) Tj 1 0 0 1 122.15 146.45 Tm 94 Tz (these can be used to estimate the model states. Applying the Indistinguishable ) Tj 1 0 0 1 122.65 123.149 Tm 95 Tz (States Gradient Descent algorithm with a stopping criteria is found to be able ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 95 Tz 3 Tr 1 0 0 1 318.949 53.299 Tm 74 Tz (3 ) Tj ET EMC endstream endobj 49 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 50 0 obj <> stream 0 ,,RRb7 l8_+uDW9 b`SiŒ##DX *0 PI a+f.-JLEDK՝lO!.3N_*jrN0ֹ a{oMf& vԹ {AV&dq[+5$PrӉ{@;%.tM?U\{T&O$tj(B7c_F2+i};&%٧4S(q骩z֜f $zs{0uғJ5jj1|~H- [׎1RAkXdz"þ'~+Q" sس9ҜSsZ= ӸĠwN1`R: k(֏X^l3QvyZ%0fKo@ąk e}ͧz =ޟ.`!StM@cFȧ Q;F&r6)jd+:zIm"aXȠC-'Lbn{qU2 m:/(KQpŀUIiT;P7=zPfg/u;.}ć:̽V/qLY, &LAY0G="TJIܾ\./ KOhSPd@vR@r:cxw{`Rxn:Hcû_?0y?%5Ꚇ-hx(*-Zaڹ`? LuIs49zfy; 6p{mrɓ4=< H% vQuTBp~ddXδRP4ݏAlxVbq3 '_FP)Km7Dl QMmmF&!;ű6P|4Jհ- Mc>^*f Uyx+fUK1此 Z?lIOB{T,)>z^fsO/ڐ)Ჲ֦b;VKv(טk}2qyqlϫ\cB-cJo: WdX?" "/^:2;1Ps { _b_,٠*-3xDVAہ>o}i"x=s+Hv]b4ZѼ)Ǣ4學粉z{"bƁG}v"?-Z RMo>h"Q_%>l)TNK8~~0/NHՂW? 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(.$BXv UT01@Z[n[n%%q$X5eAᔗoGW}Kbès{D͍@15PX&*4m0;L`kqk۟dT.̢x%gV R}wDjgci`q9l )g<<SFyh5 \N;7;=q, "*ZE8%;ѳtwŒa{eU*_V%8%]\1x){m,1\oW DK _y.;5 f;l믁w'@;RKw4X;P$9Z0.F \Z/iR 20kvR&(RVO.CV j:\BH!f<OXWS!4Lc8md{oF-ح9*ƌ74)º0Ox ͠s[3;`D̔ ўg=.TfX!X!F_^'zx0Y%Z]铙≯# [ 9 R[V*[`W 3*(^OΟLs.ْĦ647ƫb0'&i򮺎(ykfy#7!d(9$ v ϕXdĨDN|%w6; _8a0B;_ika=X?$nySfr2dӹӻD+čq5o_N6B=dzG=)q/\wzmU2%O,A,:F,>.XWH8 ;W|EO<@N։p5b@RQgcmEqV=Qq0I5dwA!Ys11g 21^ e+-Gp@#alVdU-KZ{E [LJ4~s# "Obh-P[Q;˰$hak jw# 52xrKnI&`>%T@J7 J9ceʯ>B;l[I௩;K6?˳ÑKWYO oDw  )1n* KQ6ĝe-XN֕KNZ|J[x`[Xj&*%OцPߕKS4K8'Jb[]Tm`F%is\[ pmZu# RFܡ}_ ݔ05ʄn)EX3Ph\d]sPdNQE8>k/Q.<0*^kuօO99LQ>\J8z nu:XҪ-ftlǣWdmnCs @Vϭ1IqVzFؕ*) Y/wyPCI1D.Em8VR?Vscyd3E( Ag"wi]SuCma$9^E7WMŢP,^)7ho-fK.IZ 3vUCF=>>,<ڌ̣|_ pۦ`1|Q}?.6eL)m?Kssk%2!~/ڰ"ߜ Kf aUZ(DMHM5Odpu[$['9z]pR\ ;T#ηMıޕ7js<9چ=Tb$i нv VK+^UC;h/<%̱ɡZ gƐJ㽘TYD/1+sUUHUc&->=!نȴڑ-4-dzڴʱDLպٰ>m!ad<+tF*>,W@Wx ެ V_ZVI#gB [[=08CfX5Sg-u #u hSvay?Fr3]ll7\k<{ ԽnŁ֠9YVzw[Y _7QN~ޔCF 8)ћh\ΙזS+a>@ۊOĭqItzIwbK(,)T=Z`hTle~PRlEvjk}Wo}mGy{x ɑǰe޲/~!x l;$Z"cʍZdYKyGkoL 5fΡ " /g1 - wep:mܕyAAicאּp@[NQP?ugwEx̅*ѷW6Gj88,܊C$p,3*߻-7zxMeH(4LqbR$⽬0!?7Vunmf w{V^۶Ҳ%NMp2a PGNB -O:b,.(5\n{ߍ\3eogYdupZo{ؿ+&@HfƂTO4,L̓{$:ihQ [{?Mb5 ; L"!!14җ\!zRݢD-|u v1bU:٢C-$S吴)s^e:[@2/Mكm!)A?ּ%Ub@h(s^WtB˥Q|<;Pkm=J֟+kˍ$4fzxŮR|iekA^!*g2U&7ڿ~r.`3s.4_SBP}<.$~'q+mi_z)w D9QZr?r}Y5g$], >*ٿʎi9ϻ S6oTRA܃\}jV ,r$C-;>} ÄOO/k+i1eUXK z?OfLl'Ʈ M"BԤ&[ot2\[0Lw,D׽Xf ;eӁ<\)PZKDk7m tv#$$86^ivn,4 ϘUFzh?_؎+cc`gKFU|MҲXL8˽4xk"FrzOB[ ͝\RN"„&F,! w'%Xu|>L NE LA9xɃ%Pe@+OԍQX8!΃ѱ[ΝksoL\G s(rO Hq a$$(![@ XNWF49rwsw>F[>";s0v|$&mbţ:|`;tMx?-|CYv[H 5*ȁ0l)MrjMN'R.Ie7OFִ>0K߅q?{/^T|xݣy2%glwQYhe2DOPIuI1*.U cF5[aF07b9]Ezj –Ekp\rr(T*A{`m4` Jn?oՋYX 0'YL$"g!``jAH=Q\f>;'ƄػAKT.7O;. )pkנ 3tt|:`Pd(pzoL%GjgN sQ9C2x--D0aQǀF6qJ{؛_ endstream endobj 51 0 obj <> endobj 52 0 obj [53 0 R] endobj 53 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 841 0 0 cm /ImagePart_2031 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 123.849 679.25 Tm 92 Tz 3 Tr /OPExtFont3 11 Tf (to produce more consistent pseudo-orbit and estimates of the model error than ) Tj 1 0 0 1 123.599 656.45 Tm (the Indistinguishable States approach introduced in the PMS and the approach ) Tj 1 0 0 1 123.849 633.399 Tm 90 Tz (introduced in Judd and Smith 2004. An introduction of Weak Constraint 4DVAR, ) Tj 1 0 0 1 123.599 610.6 Tm 91 Tz (is presented. Although the Weak Constraint 4DVAR method accounts the model ) Tj 1 0 0 1 123.599 587.299 Tm (inadequacy by introducing the model error term in the cost function, like 4DVAR ) Tj 1 0 0 1 123.599 564.299 Tm 94 Tz (method it still suffers from the increasing density of local minimums. Our new ) Tj 1 0 0 1 123.599 541.25 Tm 89 Tz (method is shown to produce more consistent results than the WC4DVAR method. ) Tj 1 0 0 1 123.599 517.95 Tm 94 Tz (Ensemble formed from the pseudo-orbit generated by Indistinguishable States ) Tj 1 0 0 1 123.849 494.899 Tm 92 Tz (Gradient Descent method is shown to outperform the Inverse Noise ensemble in ) Tj 1 0 0 1 123.599 471.899 Tm 90 Tz (estimating the current states. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 140.4 448.85 Tm 91 Tz (In Chapter 6. we consider the problem of estimating the future states outside ) Tj 1 0 0 1 123.349 425.3 Tm 93 Tz (the perfect model scenario. We demonstrate that forecast with relevant adjust-) Tj 1 0 0 1 123.349 402.3 Tm 92 Tz (ment can produce better forecast than ignoring the existence of model error and ) Tj 1 0 0 1 123.599 379 Tm 90 Tz (using the model directly to make forecasts. The adjustment can be obtained from ) Tj 1 0 0 1 123.349 355.949 Tm (the estimates of the model error using Indistinguishable States Gradient Descent ) Tj 1 0 0 1 123.349 332.699 Tm 96 Tz (with a stopping criteria. Methods of interpreting predictability are discussed. ) Tj 1 0 0 1 123.099 309.649 Tm 92 Tz (We suggest using the probability forecast skill to measure the predictability out-) Tj 1 0 0 1 123.099 286.1 Tm 96 Tz (side PMS. Traditional ways of evaluating the predictability of one model, e.g. ) Tj 1 0 0 1 123.099 262.85 Tm 93 Tz (Lyapunov exponents and doubling time, are discussed. Measurement based on ) Tj 1 0 0 1 123.099 239.549 Tm 90 Tz (probabilistic forecast skill is suggested to measure the predictability outside PMS. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 139.9 216.5 Tm 93 Tz (A bullet point list of new results is on Page 157. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 319.449 53.299 Tm 76 Tz (4 ) Tj ET EMC endstream endobj 54 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 55 0 obj <> stream 0 ,,FFb7 +`wGJ{ZKƺ;2RZQZœ8ьk+]!لhƈ.W_Y! kǚ{#KhvݚϚܤ%FJP`xDɧ!fIbm}v4(! %TxbF)Rg$dVbՖtqz|e xc}~9XL:@}DI+nI-((-}IaFC4[֕"IUwWo6:?]7U tARTl)Ζ)S]oh#L".B}ZqQ[ECx[E-ZKY&|ú+ihu簔br 09Q=c Gb2I|@,VO[,)I9'ƫ&{_ɲ,||mD34\H55qŏ|mzIJ9Ļc sGN]M=iPezeHrJ65%+:jWyO;Fo4y]=%[ýqL{mStZs%1*iZ_I}1ծqi(+ 9i(/ޚdg^Nぁ Tq^lwfJzjFm`Μ5!&>5e[m,=gN_*8.˾g: u4 }]hZBV'3`[tr@U-?ӫ "WcFfZFx?&^U8h@b7igN~UQݝWR@ Xfhy@4v/&Y9zAk'QAB:8]4 :/Y٬LM#?5q8jq=F܁ έ06nz7bLvpܿP'4QVhCʂ?#,u>-A~ܓ=n+l`WElc/qB(wrf߭5'Vg@G# ROf&oҭXhQ:n8/Am8^*8&?ͽ灵+noCh+FpwdI)lCONzyYoȖsaL&5[! 1b7n5d ivhoY<;>p FI =2%%o 3{4=gacSuCTʐ BVO?8wAB8ƴ7pc͛i8MZ|N*qiKJgByٻ?+|?taxr ԋ͓ / ikL2Z*ͤ'ҊNSыJ0T n',G,F4GVɕBkfWvck~e3|dϽ4h#<h HUVむ=ZZlhtCa~ñٛm_O =y>2VS~4п<*.APcMeW5ݢ )D> y4O0~mwy=nZ]dT[ ySح-5FÑOxI, bA5nEUn/dz'ނ 4.vw؛i*#[,s AOFzIUBD)HmbF~_@F߭5@  \+ b%h kq$ ,\inoFj~[??Cug*W0 X 4Jk "DSGz@ qy.'G8P 8o5(4h8[3+H TdmM׽ dGk,vnxQt$䭂gl\QAk4S5 Y*ZAΟXѬo`Y7s$' Gե(C)Wzmp+( WM5u ?U$P+#BPຕvt;0::j\k%U#LtvA)ДdW0A)j}=a%~W >/ϩ f皑pUio34#eut@X 1~$9[\LTy E=TpfǙjԪРfK%7M~VNȷ݋ܪS+ -qcp,ƑEۥ6=<o4(t6p}QpuVTaK~n;XS$g8bv3ԗ~HE>s=?th \seMPGFץW̥iBKq&CM^J])`I\^ui/$MGc xW-!za`r&qI\b~n}!O?yۛ(Y %` Ί=,XbuC@r3 !3-`|tD$pL"}rG$w_z r%|Ӭ!ι"NҊf,a/O50_Iرyi eHw+D;"Djt:$D{{q12A4b09LT 9i\L{kTDčmԓ2_?jtӌ c+dȤztbuFF}M67y%_·HMl\kAh0W(uv- <KlęQ\N%ކZ{: ?.ig>C- Ge= lhÎZh\tX*>v{r39Za΅`ks\Y?&QDirGe⬝E4O9>C1- 3^芺˸jt0oCK,P4 \wy~RupID~=iZu{z+d[ʼnf'U葉%A!bF*|+hEz|$x`0Yctn{ Õ6(9jɥ@oϵ}߳DΫIW?YEe}-8V$yFMq`RrbV:vkq: CkE0QsҞc (ta'_-B`f͑q?  /[n啨=Kz Vس7jBp'ےbNPeڞCQ88K_x~[CJwCpjࠐ>cI~c#s QwQZ |^n܃`wT[Qh41Beўe \lR_%Pb[L򫛽J@IE,ɀ_n"9}~b@"P~[2JN%kq˙?lߺC%z>xތ~ b98E^<z0ݕ\Tvص|W v d|}@k`F&YLCBl :>n@guONqRuҖ Φ46g6X?h>u ,*DždF4=Z/k1aW3=_d8Y'QH}H3G kVzۍ332?4X)@}| E4-YeOfN.#C^Od̹V`{#s>WuXe$DY,FWJGsXoq'Xw0>$`X\4̦mV&\Ϻ@5m+}F1JJ~86\'Rr ʱ~Y0+ӎّ;DU3syxfhX[]dHӿ-"Gg3N\XBbqn3(DRAV)lЌYsI*EASWPrv2q|BupyY'zj~p _?یH5#.[T&],3L-U*_&ϡw zÐCPjG^a١R2f%cDh yWSQuH+D LO1p<4O%4hph٫$eMb( IYJ Z] ~^~8ʍ^)'6#}9W7(?2 |&"@%qtI+ lN*v]{}M/dJslIuyYz6%䰨Ő;䰿ɛ5'.Gْ3҉*1Ƅ/O*Զ03!Y2{8ӴᲵჸ4컻hαʵeDZ2׶`'ҙ.::3!3λ¹;-ڳلňҴ++٧ᴥĵ>mx9'<ܭR H;#sKCQ#de&Y<Ϲ$b 7$"BB\@l)@lJKq1 +51Zm`_9픦geTA$ǝ;hY͚c= {UN_atN@RdAXmWV,ɑK%w}aF_U D'Z2qϑ!J&_ƛ/,/lK\ $M~':H(<^>l (fꕞGki7HګG$,õMJё4+˳':;N Q)B'R*jM.fN 4&i +BмDLֆЉ=ե*mewML'vAvҟ](<{JU7Vg,TR*Y*(㍆HIV nXOHT9NVZQΨ+I$z:cv][&ETl&k]?! ftl(&us+faBZĚS0I ;7;g.D@_u2jݗ0Ѱj_ #PQ [\3b'CKvfרSU}\h;S?_ևTJD!u4<="-&\>TߓD(CW_Zlx(K_ b$#.r8&` 9C=bG8,Pڲ6xnc9<ЎmAxդ&5}9}S;E c Ǹ&OӔ3E)mE>xJ<n pT$"Q1IwNroJ8dž̩dW]Wg|5Ez eRߎZ) ̽JV> endobj 57 0 obj [58 0 R] endobj 58 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 611 0 0 841 0 0 cm /ImagePart_2032 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 124.099 583.25 Tm 105 Tz 3 Tr /OPExtFont3 22.5 Tf (Chapter 2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 105 Tz 3 Tr 1 0 0 1 122.9 516.049 Tm 103 Tz (Background ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 103 Tz 3 Tr 1 0 0 1 122.65 454.35 Tm 91 Tz /OPExtFont3 11 Tf (In this chapter we will first introduce some terminology of dynamical system and ) Tj 1 0 0 1 122.4 431.1 Tm 89 Tz (the properties of nonlinear dynamical systems. Details of the systems used in this ) Tj 1 0 0 1 122.4 408.05 Tm 90 Tz (thesis are then provided. In the end, some relevant nonlinear dynamics modelling ) Tj 1 0 0 1 122.4 385 Tm (methods are described. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 122.9 332.449 Tm 113 Tz /OPExtFont3 15.5 Tf (2.1 Dynamical system ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15.5 Tf 113 Tz 3 Tr 1 0 0 1 121.9 297.649 Tm 106 Tz /OPExtFont3 11 Tf (A ) Tj 1 0 0 1 136.3 297.399 Tm 92 Tz /OPExtFont4 11 Tf (Dynamical system ) Tj 1 0 0 1 232.099 297.399 Tm 101 Tz /OPExtFont3 11 Tf (is a system that evolves in time. The set of rules that ) Tj 1 0 0 1 122.15 274.1 Tm 94 Tz (determine the evolution of the state of the system in time are called ) Tj 1 0 0 1 468 274.1 Tm 89 Tz /OPExtFont4 11 Tf (Dynamics. ) Tj 1 0 0 1 121.9 250.85 Tm 92 Tz /OPExtFont3 11 Tf (For example we write x) Tj 1 0 0 1 236.65 250.85 Tm 68 Tz (t ) Tj 1 0 0 1 239.5 250.85 Tm 170 Tz ( = \(x) Tj 1 0 0 1 279.1 250.85 Tm 52 Tz (0) Tj 1 0 0 1 284.149 250.85 Tm 91 Tz (\) where ) Tj 1 0 0 1 323.5 250.85 Tm 123 Tz /OPExtFont4 11 Tf (F ) Tj 1 0 0 1 335.5 250.85 Tm 89 Tz /OPExtFont3 11 Tf (represents the dynamics, x represents ) Tj 1 0 0 1 121.7 227.799 Tm 90 Tz (the ) Tj 1 0 0 1 143.5 227.799 Tm 84 Tz /OPExtFont4 11 Tf (state ) Tj 1 0 0 1 171.099 227.799 Tm 95 Tz /OPExtFont3 11 Tf (of the system, ) Tj 1 0 0 1 248.4 227.799 Tm 112 Tz /OPExtFont3 12.5 Tf (x e ) Tj 1 0 0 1 274.55 227.549 Tm 96 Tz /OPExtFont3 11 Tf (S where 5 denotes the ) Tj 1 0 0 1 395.5 227.549 Tm 87 Tz /OPExtFont4 11 Tf (state space, ) Tj 1 0 0 1 457.199 227.549 Tm 94 Tz /OPExtFont3 11 Tf (which is the ) Tj 1 0 0 1 121.9 204.5 Tm 97 Tz (collection of all possible states \(typically S = Rm\) and t is the time evolution. ) Tj 1 0 0 1 121.7 181.25 Tm 95 Tz (The starting state x) Tj 1 0 0 1 222.25 181.25 Tm 58 Tz (o ) Tj 1 0 0 1 225.849 181.25 Tm 93 Tz ( is called the ) Tj 1 0 0 1 294 181.25 Tm 95 Tz /OPExtFont4 11 Tf (initial condition. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11 Tf 95 Tz 3 Tr 1 0 0 1 138.699 157.7 Tm 94 Tz /OPExtFont3 11 Tf (Mathematically dynamical system can be categorised into two types, deter-) Tj 1 0 0 1 121.7 134.45 Tm 95 Tz (ministic and stochastic. The evolution of a ) Tj 1 0 0 1 344.149 134.45 Tm 91 Tz /OPExtFont4 11 Tf (stochastic dynamical system ) Tj 1 0 0 1 487.199 134.45 Tm 88 Tz /OPExtFont3 11 Tf (is irre- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 318.25 52.35 Tm 63 Tz (5 ) Tj ET EMC endstream endobj 59 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 60 0 obj <> stream 0 ,,77[{n!Yoy,`r=dt6%YGޯ8i3b0G.DXNB[]@؏xߏQ0g\m>;Y7tr}k)@\ Xh#ipյ 6 h$gL{݄&p6;8*1U.M8/7ePr4?8?H/DeFC9ZQyʯsD<W \-k SӨޒYUgvtFL\kDȮX/%d׬N: uEQܖ`+\1xvBm*wΎ3b{ۂYXRT,l6n F}aBlI"sWQH吤;xcw&e%MӨ2@9y0(@t!Ʌw/oGy1H{phf {$б/3i})f#\7nh8471TJ08j{Ǧ[轴,6 gO \REή!t͓XXw )2:P 0]+4g>e/$J|ʽ&p4Utrt'('gdU+Ԥk)AJxϿs' ="[QC 'o8ۨ4#HNabc+em||l>3 Dy#CGD9D ) =3Q3ⲳz6>aҼ׹?Bmr]rgGNQVUb;x3v 5ٻ^UHťa'} .mi1xP):w*3'$>g.(&*G@D I3C21-/SY`F]h)A--J̜L!SB^zr>BGNGa m/nNBWN>t2Xwʜw ~4Aƙx %!>J'M |jYU=xSp7$^%ZWAj*r诜FsiH"m=>2Vօ_2ջ\""鳧հdwŶղсA䰨Գݳ GLUL ݆8C:}ie V?խ !2 D^.iuY;'?ϾwD"Hv8o$`9Ky3@DXH*WrOHEE(IN񊉥P Jk ;AV9^$7/4Et =;x7*xVo&+[\?=Q#T4gBn:ljNK ހXثfpMэ}2n(5J<aqiv't6X\d68|5|s¬2y tPNO}רpi@o,swμwA\YT[^*V/iux/6_v>wz/̐R`cYך8c\Y'+彬V}J؀*cqDxC|-9lCi?,I snd"nc|!YmQ f^QYFFEK 'Ca.κ_ۄrd8{9 ੮&=Zpì }xs~50ANOi:v@iWߛ?,zK.2lG~ʕԕwH4ӑ^h̻ܨ\ fkʾժ m% +zsX<[ ;ie~ZT<@g‡ kG pG'>9(o)r#%ŦTΦ8vs`%i ͸˽ddǰ+>p8}0u02pJ`7LhpB- `Vy퉜>f{m^$*f%!KO>#XFZC1yB%=2ݒMO:o^(>;\)0 ʢ~.+mCl. G#\~`SCH@\%@d+7u yia9iSJ$1Kp=~|%+Rf>H@l}?wqQaREn,\ڜ>:f:ð t@:N>W- 1!?{_C">Pv&JN,7¥>27tsf$gSL·bݫR`,7m:FXJ)jjO~?LH"q́澵r"H :/o0Eh Mh4gZPXѫIТ<FΊ5EΎN&H'@vɩڴ0y_Б+rjcA &.WSl$>7[, }K^0xf aN,V.Ԣj8aWuTTZ};"[&ֹP?5Dܻ"ZMOoR#m  |Ke=貹G6vD}f0!H$?B]2bkp*Gz{n֣Ic+*ZRuzNHg]^T\][1bŪ/GKPM[jW'NO>bKpTKiRT-G7Pvm7qWI?Ǎ6T9Bì% U՟S .U%a5z?Yğ}$@Rݖm U0FjWwhibl,T8ݩ.(q7Nu0rĘ"e(_[TAfFB*jS.O~g$?L(/N@?.Նr!STSg3-UJ5<7;J ov{{ͭQ#a[F&s򔏽*KT,yL!Ud>3DX󶕋tw‚E);2x08+1Ył!ӓ=ðg,Ef|m{Lk.*RJ!*o yf?%DDe]Ջ^]o(OȪrGSfke[|sr0jhg0v:~* JTrxz󤍅x䔜3E&3/oD*9 +XSDڡwڇt`f`[` ՗=9e(z2|zgdӮTٷkӝ)An<^?d^]!_g|K֌8S+0h3 Qyom#s xj!: f!$J]Y\33DWcZ$b2;2\i^Ac #+p=G4%:ĦW+\@_7ۍ@SL,߭K+{IVrDI ѿ8C7Cd%@z'<3l$C}:4~NhyWF)X}Cŷ}M &ce7biV<ō+tR_Mэ7AÞ4hqa},]DNvgMD=ZM7r4S#rg] ,hvW2'jvgee)4&/0ԟtrt) j [ E\K~ aL5D|B| ޮw7e cr3gd%s9]Pr^?k_(s=Pٓ!::8a:`fah<:@ҫ'!Ҹyڝ*;J . -1{#+]a11U8mBung@5XLMfgF֋~}0F$8M\y (j뱝tOU $18U%]@wY/Ȓ:,Xƚ[ Z]Y7QDtY ;VĮ{6.M(%ˁ04O]xI<^42z_{#n\x)agQߍo[[ObdX!UGK@=*X&X=I궙Z9AJpfzl/i){̒J[] 5NePeO/A:7 ~4o:af+)̴C\ f_W|?h9'n!96͚s2̺PDR̃B<ͷ 33 W-L"FIbw@!fv; +׆:.˰ ˾H4PPSeLP?P6^;yf,l*'*2Lt.Sp|^W6S'"4ӂؿVi'hkóIDr/I8>-Lh\/Iuhzkn0{ʯ{bSdwkh]^#,̶ vaJa3>m5L2'WIϊܥeq)&8 yʜ(V5{-'HqaRrLv%B,yZŢl6_Q޾i$t=@Why %H#X?V \8v.7=[W%bҦDKsEQ3g]ɜoK.5!ָb6U1sZ'nU;{*G t a-y  vbU5`ȶ ;C[ʌ5j3R߮3 !V9kxTW(7+ўjF.Zoh?48p8ȾݤQrsn;61#rt;y yF+lhQB 7HTzA*ʘ~:rDp}W'+㟴F4{Tzv~y`6E[arzik%4a"yUi,sUʝzZ PNZa*u lx5cp/ >,u!%7+O[ɉBG{gPNqtI Ж.LFJ`nBYImeӴ#+\zsoJ\o1zҁOEO=7 ː+h3d6z@K Z;T^2O؃:ݳ hi'pN5i/Sdk]\|!68(°٠E+bGNH>6^+LhSs,1)tB,i?Z EsȬ ˬF^EY{=(S-Q=ON`f ߣ垜wz Kg08hF$6!6$5`O7! s2gvhCS''Du-?yPgonN5Z!rk>1|_泙#x(`y+1g?T0WKSE|/ҌHt_l[ܞ'{DJֽS?we,{%XV֡F:hB*VXZ0ynF\t: ;u7`NLʧZ+OP6,Y& R@byoB]"i!HH5^bҨ%O%7p\qd3fyQI^psҗGϞa1 qBۗ7*[͵F>3W|Zx) GCѶ`ܽg(Q3NnS#HDTŗjzk7Q$*1٬ݬ&6j7\߱-J%>-!- bqKW d'LX12_PQ}6 2{![hrHQ_" Íf? ׮<_)tczw%l1iC!qڂ{hU*k[bŷ?%Mu}iOxqXUC{T^d O~R*{5b */4[Ũ#tPUC\Hk>iϲ洸鳦##> endobj 62 0 obj [63 0 R] endobj 63 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 841 0 0 cm /ImagePart_2033 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 416.399 722.2 Tm 120 Tz 3 Tr /OPExtFont2 11.5 Tf (2.2 Flow and Map ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 120 Tz 3 Tr 1 0 0 1 123.099 679 Tm 104 Tz /OPExtFont5 13 Tf (ducibly random. A ) Tj 1 0 0 1 226.8 679 Tm 92 Tz /OPExtFont4 11 Tf (deterministic dynamical system, ) Tj 1 0 0 1 390 679.25 Tm 101 Tz /OPExtFont5 13 Tf (on the other hand, is one ) Tj 1 0 0 1 122.9 656.2 Tm 100 Tz (for which the dynamics and initial condition define the future state unambigu-) Tj 1 0 0 1 123.099 633.149 Tm 97 Tz (ously. In this thesis, we will only study the case where the system is deterministic ) Tj 1 0 0 1 123.099 610.35 Tm 99 Tz (and especially ) Tj 1 0 0 1 196.3 610.35 Tm 89 Tz /OPExtFont4 11 Tf (nonlinear. ) Tj 1 0 0 1 250.3 610.35 Tm 96 Tz /OPExtFont5 13 Tf (The evolution of a nonlinear system involves nonlinear ) Tj 1 0 0 1 122.9 587.1 Tm 98 Tz (dynamics and the observed behaviour of system can be irregular. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 123.599 534.5 Tm 113 Tz /OPExtFont3 16 Tf (2.2 Flow and Map ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 16 Tf 113 Tz 3 Tr 1 0 0 1 122.9 500.199 Tm 103 Tz /OPExtFont5 13 Tf (Dynamical systems may evolve either continuously or discretely in time. The ) Tj 1 0 0 1 122.9 476.899 Tm 99 Tz (continuous dynamical system, called ) Tj 1 0 0 1 307.199 476.899 Tm 87 Tz /OPExtFont4 11 Tf (flow, ) Tj 1 0 0 1 334.55 476.899 Tm 100 Tz /OPExtFont5 13 Tf (is usually represented as a set of first ) Tj 1 0 0 1 122.65 453.899 Tm 97 Tz (order ordinary differential equations of the form ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 279.1 419.3 Tm 102 Tz /OPExtFont4 11.5 Tf (dx\(t\) ) Tj 1 0 0 1 303.6 417.899 Tm 139 Tz /OPExtFont7 11 Tf ( = ) Tj 1 0 0 1 320.899 419.3 Tm 113 Tz /OPExtFont8 11 Tf (F \(x\) ) Tj 1 0 0 1 344.149 419.3 Tm 30 Tz /OPExtFont4 11 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11 Tf 30 Tz 3 Tr 1 0 0 1 287.05 403.5 Tm 86 Tz (dt ) Tj 1 0 0 1 296.399 416.1 Tm 2000 Tz (\t) Tj 1 0 0 1 497.3 411.399 Tm 87 Tz /OPExtFont3 11 Tf (\(2.1\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 122.4 368.899 Tm 99 Tz /OPExtFont5 13 Tf (where the state x and the dynamics ) Tj 1 0 0 1 300.949 368.899 Tm 119 Tz /OPExtFont4 11 Tf (F ) Tj 1 0 0 1 312.949 369.149 Tm 96 Tz /OPExtFont5 13 Tf (are defined for all real values of time t ) Tj 1 0 0 1 501.1 369.399 Tm 98 Tz /OPExtFont2 10 Tf (E ) Tj 1 0 0 1 511.199 369.399 Tm 97 Tz /OPExtFont5 13 Tf (R ) Tj 1 0 0 1 122.65 345.899 Tm 100 Tz (and {x) Tj 1 0 0 1 156.949 345.899 Tm 53 Tz /OPExtFont3 13 Tf (t) Tj 1 0 0 1 161.05 346.1 Tm 69 Tz /OPExtFont5 13 Tf (}) Tj 1 0 0 1 166.099 345.899 Tm 52 Tz /OPExtFont3 13 Tf (t) Tj 1 0 0 1 166.3 345.899 Tm 68 Tz (T) Tj 1 0 0 1 169.199 345.899 Tm 88 Tz /OPExtFont5 13 Tf (_) Tj 1 0 0 1 175.449 345.899 Tm 49 Tz /OPExtFont3 13 Tf (o ) Tj 1 0 0 1 179.05 345.899 Tm 100 Tz /OPExtFont5 13 Tf ( forms an unbroken trajectory in the system state space. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 3 Tr 1 0 0 1 139.449 322.85 Tm 104 Tz (The evolution of a discrete dynamical system, called ) Tj 1 0 0 1 418.8 322.85 Tm 84 Tz /OPExtFont4 11 Tf (map, ) Tj 1 0 0 1 448.3 322.85 Tm 106 Tz /OPExtFont5 13 Tf (takes place at ) Tj 1 0 0 1 122.4 299.549 Tm 99 Tz (regular time intervals. The mathematical form of a map is defined by ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 279.35 257.1 Tm 134 Tz /OPExtFont2 9 Tf (Xt+i = ) Tj 1 0 0 1 315.85 254.2 Tm 130 Tz /OPExtFont8 11 Tf (F\(xt\) ) Tj 1 0 0 1 342.5 265.95 Tm 2000 Tz (\t) Tj 1 0 0 1 497.5 257.1 Tm 93 Tz /OPExtFont5 13 Tf (\(2.2\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 122.15 214.6 Tm 97 Tz (where time ) Tj 1 0 0 1 180.949 214.6 Tm 89 Tz /OPExtFont4 11 Tf (t ) Tj 1 0 0 1 188.9 214.35 Tm 98 Tz /OPExtFont2 10 Tf (E ) Tj 1 0 0 1 198.949 214.35 Tm 84 Tz /OPExtFont5 13 Tf (Z. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 84 Tz 3 Tr 1 0 0 1 139.199 191.1 Tm 97 Tz (For continuous dynamical systems, solving the ordinary differential equations ) Tj 1 0 0 1 122.65 167.799 Tm 98 Tz (analytically may prove difficult, or even impossible. One can, however, study the ) Tj 1 0 0 1 122.15 144.5 Tm 100 Tz (flow by numerical procedures. In this thesis, continuous dynamical systems are ) Tj 1 0 0 1 122.15 121.25 Tm 98 Tz (simulated by 4th-order Runge-Kutta approximation and we define the numerical ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 318.699 52.85 Tm 75 Tz (6 ) Tj ET EMC endstream endobj 64 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 65 0 obj <> stream 0 ,,xxb7 [6bY-ְoo:>NZl7y8 nrĢ\to=7Ⱥ8G-R/] Md|9Z;M.lGI (v+k6bu'0\1s'9E{F)G NMDZk?R) I87FP]88~| ;OxPThD4s"WY]ݷ4EKwmEyb[Uꃔ쌷F2sÃIC$ lELw?r٨M2:!E2$~ 7ϰ`x."e<\*DQ=ZP>&'ɞ"q{-M5{ Cu`x((C(i+1Ↄ|lhFփ*ͧ0ڟE #t~#/ -D9zT8%^[c]>L N>Ae?nK0G+l%_[`)7:vX8T ,J{f#gV={&؞2Aϓ;moh2y - 8siDf(;kP ],djL@B)?ki~+c,;%z`sJFܒ"g9SWy_OM+«WN [dܼ9>Օi8U$u '_ޠxQ!@쇯^B aml_éooF}Z:pQ:4k]H#8ԭi=!Uqm4(<-d`2LN?A1lPjJX- ZgbĦ+~}fBLŐ ;m5ITP-7K$w` *Di ́E~-dѿUnhEZ7*m2F\z\JӒS6p1F}0/&@&\b|֣ ,6(#(|A0l+5R\lT8[+ ǁQ$iM|A6kɳNbX3Nt`wfa((X5;CэancILzEAGƶuӂ϶ r@MXG\HlxSR3XekUj:גGg@[:T^5]`꽄Hh!YT_+䲵*zgf,@p 8B#KP* +suo\ru߻u}XSOsJKc+G R Lēd J1sŮumM0Vy Q1!f՜z2t&jLT-7LTu.~f?Eֶ) :a5IL#ieX Hq㗎UBj|4 %1ɽI:89A4+!VXҳrnf81p12 k3:bŢwαa)*'=ۇW<=ć pz-(c55?}/ggΛl+κBv)|[ُtXKrz/qrPmU9}QAB$Ռq1TZ FیS襪0w2 Z?ppzN@$Kqjp /|vd~r0ro!M]Ӌ.p vdԊŃ8rԛ 8ce4C+-J6v7g&֜jOpֺU(܈r%lge8n\w@yşmKF'} Ad[oIQ>}W-s-QW7mQѶ`V'!P:u2`6lwGnq"(})#G>Q_ L:/b7y:/a؎ ȑ}=(65/Kn~ V{BN~DJi|v}$H8ʦP/[~<.YA"DLvh \21.6mWl9WU?ua@jNfv ΋N"ojl8ƒue0jI[xMlq 9SIY]5f\ ~mϬ"^CXiqؔla%\K}2WfS {并$Hð`#t0-.MZ1@@,75ڋtqaGVPזl|2^?7\/s#vdwlW33hqxq z%_A~o_@1t@C{A}E 2+0#{gh/Y[gPWӡLFs7Θlg\?ɒZW˪+~u˘FI>:mzvҔY2JޑOV"%|1fҭv_\j:]Eeu$I+s,WWT%yyj/b~C2iaj%V68vN$/1BPWUvX>tJ l?c7Is&#$b?gr]ځKKvtFXGF1e V5߁JBң }E; ]F ,w=,*ם)$} vO(^tёwKp4?"sw.iavEޮ-U pğ2xryąhk pE]\XxÙR-F+ء0@Vae[JnRyтMv ߅v&nez|+Z.HÏ|dTX!="$Rn NvC 1wo>F@2_p*d4^BYrÓ"\J~*]G*Xi~Qh/A$l*#Z8_/GIq%syfRʒ r%[L ~y!qzL~HW:dC ?:qR.,"w): J-[d\b%yŠNVRe~,NrhtjN1+ָ@^H9v7uj'h\HTT40mLPi#9,adPfnp9dY=۲jOd`Ad4.K} 6ږZlFmixތMx<3cn侞B;\ީ"hH2Mȝܷۜg`-M(${)f(0ơ#X]M)bx/g<[>GԲe&E(jk.걟FWEېamPť9l E8 Lë}3g^;t/CVPMpu4c#{B@dʧ9XǪ=s +5` :i\Q oKdC:t2&!#TOGhEQ)_*;nPYj@X잦W`Sr_^My3qg'nPoC`k! s}~T7B6>צ'#i{ ?vG,ǒ;[Gyt!.abbDeFZbuѲz'F=tx'Q{m7TGu?OEeM"4Y;e{IqC1?oIP>v]SbD=-/1f |4gjVmy֨5RTWlhGr ާoܰ,.h +VG@bEдOǑ" V\ $-5f7͡"|"me|$/ "!]݌^E!\ti!HGB^$1*]b#tnl=-tՐL@^:pB#՘ Be͈͡u%$ir@ar7$WxM 6+$B]( $fLi"0<1~ rZпXUǩ-Gl*C)$gjG&fGŸcX.dxc,+$׺ZLYfSi? B^wAp@||I8󱎘"whƾ ]1HxD^ʜO3Vf\cQel>z'ÕkT*E5:/fp &-/ui8 !m됷-#c(2~6fDd-L#9#LmrǑ"Y[&QIw.p %D0}ڇ~뎌>m.f4L UF=."J*^s1;8jr\| +=)OIYU" I>\VntBN[=}6ޙ5.9&b|[mya_`S>t:[i ziL-M|~u~@E„U:dݸϣoNU"ߨA|L׊HD8e1^U׿bD{C)ՇʂH˘e~YJ3"(Y[{v4ʉgqw[q=2Hpj|qA?0֖qrdTUR%)Bt]C46t GOv]CK1b˥.u_ ~k/_Xqح6)Ժfnd4jxd씆dvq3joV2]^HtF EΠa򽆇I@u39&}JlnKXT̼ml@XqczV1 YR.\U!oM(6:{; !m_>=!w5?d ř%Ug\DDU9DcG9p5@f8F{=gHf &KUyRυ6\owF()fR@= "wVS"m.pnoZ/K;Ph5%$a9E7 KA&񵘴ğǎ BFtF9䨡g!jklˑGIw=!ys/AǴ;>6ZG9F@͉|ZCËeq+Cx]Z۳Yc\("IQɳg-f[YgD=2@qG?H\}7~mb~67=WXdt/Ji+V|>kK n++ 6F:sn8<(t rÈ=Ha!VAZ@M%g( l\|K0_6=Ȍ\-> nSܺ8cw)w&atfEBMjW n?VsyaΔA$#Z P ,|Hgcφayd>g~JcM5ɷOy8hzx;.μHPm=/V>H\>AlK_ +lKi%̓$y }ac &0˻#6u1vfٷ5;F%70_biP ~lY+䊏@fMmwGaJԆClܛmH2Oy6sD?kǽ* @v2_Xrqi O,r 09rP(}D97BU?. dN6FpeEjQR;Nwʪ Oa~Io!{qjsKGhxzIJ)nEKJ\3l̤h~7 n.o4! Ewj?~|z.3㟖O,Q@5 ({ͽ!F]_HJtZ/6 ņ$u{'ʯ ix23P{4㿩~(w2;X2#U3YdFXh=2l0!s_ qHYe9ZX#06g<]ܑ1g !B][g>>i,o MΥc;<łAyI6:K1C:h2ɋ鯏103oN,Ui  \fXr,0KF@IQx#^)qi#݁P7$] @Kw?~OydGM㔖Eu ϑ(M _D>GaEl!'V|Is \ՎF,ީœ_a 0&@͡q ;AOn5`mcjؤl[ xUö|MN##?93ÙƙΤ`)4*~,bsfU'Bi7|e}|dY`Z |98b>h}2ܮ -C٪%72 endstream endobj 66 0 obj <> endobj 67 0 obj [68 0 R] endobj 68 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 611 0 0 840 0 0 cm /ImagePart_2034 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 463.199 721.45 Tm 98 Tz 3 Tr /OPExtFont3 11 Tf (2.3 Chaos ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 98 Tz 3 Tr 1 0 0 1 122.4 678.25 Tm 93 Tz (realization to be the ) Tj 1 0 0 1 226.8 678.25 Tm 82 Tz /OPExtFont4 11 Tf (system. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11 Tf 82 Tz 3 Tr 1 0 0 1 122.9 626.149 Tm 118 Tz /OPExtFont3 15.5 Tf (2.3 Chaos ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15.5 Tf 118 Tz 3 Tr 1 0 0 1 122.65 591.85 Tm 93 Tz /OPExtFont3 11 Tf (Given the state space S of a deterministic dynamical system, A subset A C S is ) Tj 1 0 0 1 122.4 568.549 Tm 89 Tz (an invariant set ) Tj 1 0 0 1 203.5 568.799 Tm 35 Tz (1 ) Tj 1 0 0 1 205.9 568.799 Tm 94 Tz ( for the dynamics ) Tj 1 0 0 1 298.55 568.799 Tm 119 Tz /OPExtFont4 11 Tf (F ) Tj 1 0 0 1 311.05 568.799 Tm 107 Tz /OPExtFont3 11 Tf (if F) Tj 1 0 0 1 330.5 568.799 Tm 81 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 334.55 568.799 Tm 103 Tz /OPExtFont3 11 Tf (\(x\) E A for x ) Tj 1 0 0 1 403.899 568.799 Tm 86 Tz /OPExtFont5 11 Tf (E ) Tj 1 0 0 1 414.5 568.799 Tm 96 Tz /OPExtFont3 11 Tf (A and all t. A closed ) Tj 1 0 0 1 122.15 545.75 Tm 94 Tz (invariant set A C is called an attracting set if there is some neighbourhood U ) Tj 1 0 0 1 122.4 522.7 Tm 111 Tz (of A such that Ft\(x\) E U for t > 0 when Ft\(x\) A as t oo, for all x E U ) Tj 1 0 0 1 123.099 499.449 Tm 99 Tz (\(35\). The attracting set, also called ) Tj 1 0 0 1 314.649 499.699 Tm 97 Tz /OPExtFont4 11 Tf (on attractor ) Tj 1 0 0 1 378.5 499.699 Tm 89 Tz /OPExtFont3 11 Tf (or ) Tj 1 0 0 1 394.55 499.699 Tm 92 Tz /OPExtFont4 11 Tf (invariant measure ) Tj 1 0 0 1 488.649 499.699 Tm 97 Tz /OPExtFont3 11 Tf (of the ) Tj 1 0 0 1 122.15 476.149 Tm 94 Tz (dynamical system, describe the long term behaviour of the dynamical system. ) Tj 1 0 0 1 122.15 453.1 Tm 95 Tz (The probability distribution of states in the set of invariant measure is called ) Tj 1 0 0 1 122.4 430.1 Tm 91 Tz (unconditional probability distribution, which can be treated as prior distribution ) Tj 1 0 0 1 122.15 407.05 Tm 95 Tz (of the states before any state information is available. The invariant measure ) Tj 1 0 0 1 122.15 383.75 Tm 92 Tz (is, however, rarely known analytically, but can be approximated by evolving the ) Tj 1 0 0 1 121.9 360.5 Tm 94 Tz (system forwards over a long period of time if the system dynamics are known. ) Tj 1 0 0 1 122.15 337.449 Tm 93 Tz (We define the observed invariant measure to be ) Tj 1 0 0 1 368.149 337.449 Tm 88 Tz /OPExtFont4 11 Tf (climatology. ) Tj 1 0 0 1 431.75 337.449 Tm 94 Tz /OPExtFont3 11 Tf (Without knowing ) Tj 1 0 0 1 122.15 314.149 Tm 95 Tz (the dynamics of the system, the distribution of all previously observed states, ) Tj 1 0 0 1 122.15 290.899 Tm 90 Tz (termed ) Tj 1 0 0 1 161.05 290.899 Tm 91 Tz /OPExtFont4 11 Tf (sample climatology, ) Tj 1 0 0 1 261.6 291.1 Tm 92 Tz /OPExtFont3 11 Tf (is usually treated as the estimate the unconditional ) Tj 1 0 0 1 121.9 267.85 Tm 91 Tz (probability distribution. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 139.449 244.549 Tm 90 Tz (Given a nonlinear system whose long term dynamics converges to the attract-) Tj 1 0 0 1 122.15 221.299 Tm 93 Tz (ing set A, ) Tj 1 0 0 1 175.449 221.299 Tm 84 Tz /OPExtFont4 11 Tf (chaos ) Tj 1 0 0 1 205.449 221.299 Tm 91 Tz /OPExtFont3 11 Tf (is often observed from the phenomena, ) Tj 1 0 0 1 402.949 221.5 Tm 90 Tz /OPExtFont4 11 Tf (sensitive dependence on ) Tj 1 0 0 1 122.65 197.75 Tm 95 Tz (initial conditions, ) Tj 1 0 0 1 214.3 198 Tm /OPExtFont3 11 Tf (where points that are initially close are separated on length ) Tj 1 0 0 1 121.9 174.7 Tm 94 Tz (scales commensurate with the range of the dynamics over relatively short lead ) Tj 1 0 0 1 121.9 151.7 Tm (times. Mathematically, for every initial condition x) Tj 1 0 0 1 376.8 151.7 Tm 54 Tz (o ) Tj 1 0 0 1 380.149 151.7 Tm 110 Tz /OPExtFont5 11 Tf ( E ) Tj 1 0 0 1 395.5 151.7 Tm 95 Tz /OPExtFont3 11 Tf (A, and any lei > 0, there ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 95 Tz 3 Tr 1 0 0 1 138.25 130.1 Tm 100 Tz /OPExtFont5 11 Tf ('We assume that A can not be decomposed into smaller invariant sets ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 11 Tf 3 Tr 1 0 0 1 318.25 53.049 Tm 76 Tz /OPExtFont3 11 Tf (7 ) Tj ET EMC endstream endobj 69 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 70 0 obj <> stream 0 ,,+b7!ںQ!DE"n|s׌e:.ڰlWWK'akzw$)G{2f<=BhYJ< FŝH`v9=;ȬO9UO;"207/< ⏛:WkXT$W P:+Pu|)0@.y~eCy%l n!fj,LtzÙm!'}T#͔ TlBm6Jhp)Of'$_`-D(|R9^p<_4i{HhQcbҠ~DB2VUe?U0⭂G.Ehƒ$P]x4d4(i{É{S0K5oGrs1:]̏F=#Ak>_h1L[]d; |H n?6` Fl=> '?MtpA|%p+ !t6P섳ci}6lRۊB?8 w !S0fS/5]9HguC5a{v mYt6V_yug*ο3A2z])2o"+3j2P楄у8cN>Å0p%biKk7 Vȣ Sm&c}٨+/{tIeV| {ؘdN5$90sf̚I{{܂)01ͪ3 \xV#Mᅂ"ý`2vNT]7a(ry*{;},-fWX:T䤹nS_6q".H-74\HmG|Zm|DB%vj׷FPI-6x:4&AL?*Evm960P~ ?m@GƗ '< -,dma)۽QPNLG#I\(CJZ(w9d+\ ̶8)뭵*=rk]F(G-S8N!ePP%Y6zbYE{ RfudB|Y:$!u5 c]֯- Oʆ%%Z'C&)b(ك 1(h־.&"vwm.f17'."Fayᠼso63Pօpi\$~QLZp_vmgҰm.Gt8hƮM?NFV, OKK%Dj0 8/Q{DQgE}_\~CHFͣ7YtWۏ@5ٛ33d5snt78[NI!0Ζfkm2W|U=@? ]yJ<\2С+bj:+hwdҤ{?pW` u:\t|:4MtpNWWCbd*Z@m{:ygf*"|QԹk5̢09WPQnl'ZVXf?) 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D-CQhz1}D=BPy@W7Ym!I0`Ew Q97V ^|C$qOum׃ 1K"4aj$lf+ȹr.)dďb1nVWKn&\è<]F #fC1VK `U ,"2 耚$O붚h&Q1,N: 1!Fޑ1vڟi(]C-Y>{K aݰG뙗NJZSœs)@:.!ΞdD ԅJS3@=m8|5]!W/_~5a/l'c@0̃" 2:y[b/(@Ltբ1 ކ)VP\jV}a[Lҽ,?칺`57{{Z^(rbq-]:FmvMg ശn+6>rz0l66'ܵ7Z嶡!7<`)!\Hh35#IvFSTɞ:6؍|JAvʮCjB:ckACީ=+},3_Hvއcⲁ,(J;#E:O]ܾDA fMs/.a e )u-,84WI8t?yFyT@ƒ\-#ݶ߭~> P[씸nws]9 Vtv&&6+ tPp"e7YӰK(*\=Q/ fAYwކ=NM]2gQCyD#oLF& MU iTa%pěv1AH5<. ׎qMLX1C yn-#T7X ] /=NSϴbϾs-2:#V#}DB|Μw2:= ku؋!:iMzMEWTroJqEk0G+؜K;{$3U[p'Xys}u^,5ruBR`!AO5M˞wzJ 3|ϪƣwĘ|&  A  h3+biTOea[?"xpW nfg[CU\.S9%ax`tg;Ncw$j'(&{)HxԨzx J]'~ƽ;'#9R"BdFEg0%H#)\|/dd%H]ȁKFpGAy1/}D ͏ɤj]3*< e겗r0ݯh]]08Jm6ؗa uPH**su<+:G=e^ƪ<=C±M'5eP }@0[VDM!sLK?4̐&mm A1Z.Pf`m"(lDT !JNVyƵ x-1$2%$hϻo 8uW q5C(Ha }nNj%$QgZNK%#<"ìNe^ľTUW":v/s_(9"}W~yA=ط–%e*J_n6w Bs&7RXe{<\DA  ]ف X\unj˵ro#AoqG1ɧddj,A2mV6hJgQ3PKJ9;ji={Qc)_>Q5UF`0ɋzp.!`>y/ uf{)2y& +w wTXCs["@Y,5?Τg> jvx2W?i2mb`6`-Q\ʞLJ*>r ӕ("'Dlw[SJy9WqvKwpيO>j^\e/&t T՗/XA Hx%LiMtx}m6p#V75)mkPb~s{"=ӼE&Ehp*"3{@cBʾfKɄ\+ g|`E^&MktꟑxjŋllA'*L@!W~ԟVʃّ{|.D3'"j‹]P^ 7g2͙.^/2zGW?Z ѥnhJߖ@ 4yp2ƀ4<`NgzQએ! jީ=.s'd6f= }ZG)jp]5BȚ5pJ?[EZzZ짡O))![Jb;>fݱFF̏ 47uomU8y {M`ӟYzʒ 1Mp}lv(_?Jfxա+Eh^Oaѥ2*OԵ.J 3FMUWWA?44J'C%a?[62TьvϢܺ+qlBف2ދ@ƺ? 3m\kLIxXsb~y xmubRXN`» *skC;ygvJ@%ıLkNMrNh*>lZL%N%zs޷ ΆJ 3Q^d^cdQI\N2ކClEaݚ͖=6ęo1 ԰QZ\N.lY|8S̋^ܣ`ZWRގASMׅ /0jOy_3{1fϱ\ 65Q&B3 ?%W6/V-J_6=jJx~JȰDTEG8w4GɎ rmKF!7C<3g.І P8yCF## ![U05jߗ~B)N=wX.!g"p@URx* teC ڣgFmSc$Vxp-;|bR>73VY?&M;/%z%~.mWx [^r>W!n W`]MR@6'͌kym35Q%4#p;ffk09:**\9K599T.cwu;5]yuMի8`%@;Sd&ْ10- endstream endobj 71 0 obj <> endobj 72 0 obj [73 0 R] endobj 73 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 840 0 0 cm /ImagePart_2035 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 392.649 721.7 Tm 104 Tz 3 Tr /OPExtFont3 11 Tf (2.4 Analytical systems ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 104 Tz 3 Tr 1 0 0 1 123.599 678.7 Tm 86 Tz (exists ) Tj 1 0 0 1 155.75 678.95 Tm 103 Tz /OPExtFont6 12 Tf (S > ) Tj 1 0 0 1 179.75 678.95 Tm 93 Tz /OPExtFont3 11 Tf (0 such that for some ) Tj 1 0 0 1 289.699 678.7 Tm 116 Tz /OPExtFont6 12 Tf (t > ) Tj 1 0 0 1 312.5 678.7 Tm 103 Tz /OPExtFont3 11 Tf (0, II Ft\(x) Tj 1 0 0 1 360.25 678.7 Tm 62 Tz (o ) Tj 1 0 0 1 364.1 678.7 Tm 128 Tz ( \) F) Tj 1 0 0 1 414.5 678.5 Tm 81 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 418.8 678.7 Tm 102 Tz /OPExtFont3 11 Tf (\(xo\) II> S. Another ) Tj 1 0 0 1 123.349 655.899 Tm 91 Tz (property of chaotic system is ) Tj 1 0 0 1 269.05 655.899 Tm 96 Tz /OPExtFont6 12 Tf (recurrent ) Tj 1 0 0 1 316.3 655.7 Tm 93 Tz /OPExtFont3 11 Tf (but not periodic. A system is recurrent if ) Tj 1 0 0 1 123.099 632.899 Tm (the state of the system returns to itself, i.e. for any initial condition ) Tj 1 0 0 1 465.1 632.899 Tm 94 Tz /OPExtFont3 10.5 Tf (xo E ) Tj 1 0 0 1 490.8 632.899 Tm 97 Tz /OPExtFont3 11 Tf (A, we ) Tj 1 0 0 1 123.349 609.85 Tm 105 Tz (require that x) Tj 1 0 0 1 201.349 609.85 Tm 58 Tz (o ) Tj 1 0 0 1 204.949 609.85 Tm 99 Tz ( Ft\(x) Tj 1 0 0 1 243.849 609.85 Tm 56 Tz (0) Tj 1 0 0 1 248.9 609.85 Tm 99 Tz (\) Il< ) Tj 1 0 0 1 274.1 609.85 Tm 67 Tz /OPExtFont6 12 Tf (e ) Tj 1 0 0 1 281.75 609.85 Tm 91 Tz /OPExtFont3 11 Tf (for any ) Tj 1 0 0 1 321.1 609.85 Tm 95 Tz /OPExtFont2 11 Tf (E > ) Tj 1 0 0 1 340.8 609.85 Tm 93 Tz /OPExtFont3 11 Tf (0' \(Note t could be very large\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 123.599 557.5 Tm 114 Tz /OPExtFont3 15.5 Tf (2.4 Analytical systems ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15.5 Tf 114 Tz 3 Tr 1 0 0 1 122.9 522.95 Tm 95 Tz /OPExtFont3 11 Tf (In order to demonstrate that our results is rather general than restricted in a ) Tj 1 0 0 1 122.9 499.899 Tm 92 Tz (particular system, methods will be applied to a variety of systems with different ) Tj 1 0 0 1 122.9 476.649 Tm 90 Tz (properties. In this section, we define those analytical systems that will be used to ) Tj 1 0 0 1 122.9 453.35 Tm 89 Tz (illustrate the questions to be addressed and discuss the difference among different ) Tj 1 0 0 1 122.65 430.55 Tm 87 Tz (methods. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 123.099 385.699 Tm 114 Tz /OPExtFont3 13 Tf (2.4.1 Logistic map ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 114 Tz 3 Tr 1 0 0 1 122.9 354.949 Tm 93 Tz /OPExtFont3 11 Tf (The logistic map is a one dimensional map first introduced by Hutchinson \(14\) ) Tj 1 0 0 1 122.65 331.449 Tm 94 Tz (in order to investigate the role of explicit delays in ecological models. It is then ) Tj 1 0 0 1 122.9 308.399 Tm (applied in modelling the dynamics of breeding population to capture the effect ) Tj 1 0 0 1 122.15 285.1 Tm (that the growth rate of the population varies according to the size of the popu-) Tj 1 0 0 1 122.15 262.1 Tm 93 Tz (lation \(GO\). The mathematical form of the logistic map is defined by ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 265.449 219.6 Tm (x) Tj 1 0 0 1 271.699 219.6 Tm 94 Tz (i+i ) Tj 1 0 0 1 284.149 219.6 Tm 108 Tz ( = ax) Tj 1 0 0 1 312.949 219.35 Tm 72 Tz (i ) Tj 1 0 0 1 315.35 219.35 Tm 82 Tz ( \(1 x) Tj 1 0 0 1 347.05 219.35 Tm 72 Tz (i) Tj 1 0 0 1 350.649 219.35 Tm 60 Tz (\) , ) Tj 1 0 0 1 356.899 219.35 Tm 2000 Tz (\t) Tj 1 0 0 1 497.5 219.35 Tm 90 Tz (\(2.3\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 122.15 176.899 Tm 94 Tz (where x) Tj 1 0 0 1 162.25 176.899 Tm 72 Tz (i ) Tj 1 0 0 1 164.65 176.899 Tm 97 Tz ( represents the population at year i. Logistic map is a non-invertible ) Tj 1 0 0 1 122.15 154.1 Tm 92 Tz (map as each state ) Tj 1 0 0 1 216.949 153.85 Tm 104 Tz /OPExtFont6 12 Tf (x) Tj 1 0 0 1 223.199 153.85 Tm 55 Tz /OPExtFont4 12 Tf (n ) Tj 1 0 0 1 228 153.85 Tm 94 Tz /OPExtFont3 11 Tf ( has two preimages. The invariant measure of the logistic ) Tj 1 0 0 1 121.9 130.549 Tm 95 Tz (map strongly depends on the parameter value of a. Figure 2.1 shows how the ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 95 Tz 3 Tr 1 0 0 1 318.25 53.049 Tm 70 Tz (8 ) Tj ET EMC endstream endobj 74 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 75 0 obj <> stream 0 ,,b6%Erյ 毡3[Q)z'C醟}a RUD]K"N[?3U~m;8k;EjikafeMpݘ M2x0ŖagmCSD3'u( + a9Å>_MT?Gdyy~$v@&ZU9ġQ^L`Wv\Up/@ `)F + M3Cd൯1A,IݵG\?X6 +nACEġ9f ts˞i*R'Yxz#׭_H)^r10|mJ-yn"}4-FRe⎈Zǫ1Ae $E ꗳi4l))י]0mi~ }8%_@f+4S݅ijmvnGBw@Lޗ18NOس2!rߎN5j)|=yO-;9`g_2|'IiJ /Pfe nl̶WhfTW}beȮdTAh9t3PtC~x#(٠qJ9b1t^2ռ0:w_A~Vo*N{- 0B+xyoqpZ?OjZ G4ۜjymjZx-7QߚY Q> D]NA<S׵ljNv R)}ѭk$r4ꯉkcSVk̻܇Vs?ڙ%h]vA!ͺg%^XRTL Li̩t0}HX=LT8wMj{J(ó-6aw T jkŸj_!}Fy$SS[|P]s6M :ڻ}JՋo]L}zUL5wq1;w?2ʇ)m,< C+1>^BNjRo e-bP_,ƛ0Ձ:Ps?ir҃ED(ǧb&R_ƉK1eY tYJ.YOA#*dQ%}Vf8`Cz, 6ҳ!݂sR zU8"|Ms2?c~ AbơYh鵺~b-~7eRIQwG {/cmL -G*wt96'R?׶cL-{NFkx:XpюF@$I JNg&/Q-382\lR>{|{xa)WMrڿ>>VXTaŪ5p#zc+W::!٤AcdbP86̇,7mps)_0ޑ? 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IH4N \Ϳ,[Y;d[. mg"j#WW|t1|IEU&5cu!  ..Z\z',P⌈LJ\?ʬy5E-&_/7A3g)3#ٗf]qX{pINT:!y^*5$m=N0u--X*0K?ޕ(gk| vp#A"5 ^9#g@DVʷ/TOQ̇%; Qփ374T4_lBH5o}xM,kl u[Ou Wפ=W0|`?&HM&qT\(587y,?53_1xz:G`FVkz+X2UiI*J+N7o"G \ pt/ӭn"SK'YWŨM.K3vΖ;:$YfF*!]1lӊ0(> endobj 77 0 obj [78 0 R] endobj 78 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 840 0 0 cm /ImagePart_2036 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 391.899 721.899 Tm 104 Tz 3 Tr /OPExtFont3 11 Tf (2.4 Analytical systems ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 104 Tz 3 Tr 1 0 0 1 122.4 678.95 Tm 92 Tz (system behaviour changes corresponding to the value of a. For a=4, a change of ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 204.25 552 Tm 37 Tz /OPExtFont3 17.5 Tf (....._-1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 17.5 Tf 37 Tz 3 Tr 1 0 0 1 218.65 539.75 Tm 18 Tz /OPExtFont3 10 Tf ( 111 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 18 Tz 3 Tr 1 0 0 1 177.599 513.6 Tm 79 Tz /OPExtFont3 12.5 Tf (.--C:7411 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 12.5 Tf 79 Tz 3 Tr 1 0 0 1 206.4 443.5 Tm 35 Tz /OPExtFont5 5.5 Tf (I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 5.5 Tf 35 Tz 3 Tr 1 0 0 1 227.75 617.75 Tm 117 Tz /OPExtFont9 3 Tf (,,) Tj 1 0 0 1 230.15 617.75 Tm 233 Tz /OPExtFont3 3 Tf (1 ) Tj 1 0 0 1 234.5 617.75 Tm 100 Tz /OPExtFont9 3 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr 1 0 0 1 244.55 600 Tm 85 Tz /OPExtFont10 5 Tf (o ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 5 Tf 85 Tz 3 Tr 1 0 0 1 238.099 554.399 Tm 23 Tz /OPExtFont10 13 Tf (I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 13 Tf 23 Tz 3 Tr 1 0 0 1 246.699 539.75 Tm 36 Tz /OPExtFont9 4.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 36 Tz 3 Tr 1 0 0 1 238.3 443.05 Tm 51 Tz /OPExtFont5 5.5 Tf (I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 5.5 Tf 51 Tz 3 Tr 1 0 0 1 256.8 608.899 Tm 88 Tz /OPExtFont11 4 Tf (-1 ) Tj 1 0 0 1 260.649 608.899 Tm 40 Tz /OPExtFont11 7 Tf ( ) Tj 1 0 0 1 254.4 616.1 Tm 43 Tz /OPExtFont3 35 Tf (fi ) Tj 1 0 0 1 263.75 616.1 Tm 100 Tz /OPExtFont9 3 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr 1 0 0 1 255.599 563.049 Tm 40 Tz /OPExtFont3 17.5 Tf (r ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 17.5 Tf 40 Tz 3 Tr 1 0 0 1 255.099 525.35 Tm 19 Tz /OPExtFont3 10 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 19 Tz 3 Tr 1 0 0 1 273.85 555.35 Tm 12 Tz /OPExtFont3 17.5 Tf (, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 17.5 Tf 12 Tz 3 Tr 1 0 0 1 264 525.35 Tm 11 Tz /OPExtFont3 10 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 11 Tz 3 Tr 1 0 0 1 263.05 530.399 Tm 47 Tz /OPExtFont12 12.5 Tf (,IIS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 12.5 Tf 47 Tz 3 Tr 1 0 0 1 264.949 502.1 Tm 50 Tz /OPExtFont3 10 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 50 Tz 3 Tr 1 0 0 1 270.5 443.3 Tm 35 Tz /OPExtFont5 5.5 Tf (I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 5.5 Tf 35 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 5.5 Tf 35 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 5.5 Tf 35 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 5.5 Tf 35 Tz 3 Tr 1 0 0 1 291.35 538.799 Tm 22 Tz /OPExtFont3 10 Tf (I ) Tj 1 0 0 1 292.1 538.799 Tm 247 Tz (\t) Tj 1 0 0 1 300 538.799 Tm 19 Tz (Ii. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 19 Tz 3 Tr 1 0 0 1 302.399 443.3 Tm 48 Tz /OPExtFont5 5.5 Tf (I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 5.5 Tf 48 Tz 3 Tr 1 0 0 1 302.149 531.85 Tm 29 Tz /OPExtFont3 10 Tf (q ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 29 Tz 3 Tr 1 0 0 1 302.899 494.399 Tm 23 Tz /OPExtFont3 12 Tf (I) Tj 1 0 0 1 304.3 494.399 Tm 13 Tz (i) Tj 1 0 0 1 303.85 494.399 Tm 26 Tz (i ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 12 Tf 26 Tz 3 Tr 1 0 0 1 313.899 546 Tm 22 Tz /OPExtFont3 10 Tf (I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 22 Tz 3 Tr 1 0 0 1 313.699 538.799 Tm 30 Tz (P) Tj 1 0 0 1 316.1 538.799 Tm 12 Tz /OPExtFont5 10 Tf (I ) Tj 1 0 0 1 316.55 538.799 Tm 158 Tz /OPExtFont3 10 Tf (\t) Tj 1 0 0 1 321.6 538.799 Tm 4 Tz (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 4 Tz 3 Tr 1 0 0 1 317.5 515.5 Tm 13 Tz /OPExtFont3 12 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 12 Tf 13 Tz 3 Tr 1 0 0 1 330 610.799 Tm 15 Tz /OPExtFont11 7 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 7 Tf 15 Tz 3 Tr 1 0 0 1 327.6 573.85 Tm 30 Tz (-,1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 7 Tf 30 Tz 3 Tr 1 0 0 1 334.3 443.3 Tm 38 Tz /OPExtFont5 5.5 Tf (I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 5.5 Tf 38 Tz 3 Tr 1 0 0 1 343.899 553.2 Tm 14 Tz /OPExtFont3 17.5 Tf (II ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 17.5 Tf 14 Tz 3 Tr 1 0 0 1 338.649 518.399 Tm 45 Tz /OPExtFont3 10 Tf ($ ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 45 Tz 3 Tr 1 0 0 1 338.649 515.5 Tm 42 Tz /OPExtFont3 12 Tf (or ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 12 Tf 42 Tz 3 Tr 1 0 0 1 341.75 484.55 Tm 26 Tz (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 12 Tf 26 Tz 3 Tr 1 0 0 1 347.75 546.25 Tm 17 Tz /OPExtFont3 10 Tf (r ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 17 Tz 3 Tr 1 0 0 1 354.949 484.55 Tm 23 Tz /OPExtFont3 12 Tf (I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 12 Tf 23 Tz 3 Tr 1 0 0 1 372.25 546.25 Tm 31 Tz /OPExtFont3 10 Tf (' ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 31 Tz 3 Tr 1 0 0 1 372 519.6 Tm 16 Tz (1111 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 16 Tz 3 Tr 1 0 0 1 372 515.5 Tm 26 Tz /OPExtFont3 12 Tf (i ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 12 Tf 26 Tz 3 Tr 1 0 0 1 366.25 443.3 Tm 48 Tz /OPExtFont5 5.5 Tf (I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 5.5 Tf 48 Tz 3 Tr 1 0 0 1 384.5 635.049 Tm 292 Tz /OPExtFont9 3 Tf (m..,) Tj 1 0 0 1 407.75 633.6 Tm 111 Tz /OPExtFont3 3 Tf (,1 ) Tj 1 0 0 1 410.899 633.6 Tm 100 Tz /OPExtFont9 3 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr 1 0 0 1 375.1 546.25 Tm 21 Tz /OPExtFont3 10 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 21 Tz 3 Tr 1 0 0 1 374.149 534.95 Tm 23 Tz (11 ) Tj 1 0 0 1 377.05 534.95 Tm 277 Tz (\t) Tj 1 0 0 1 385.899 542.649 Tm 139 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 139 Tz 3 Tr 1 0 0 1 402.699 487.699 Tm 75 Tz /OPExtFont5 5.5 Tf (' ) Tj 1 0 0 1 403.699 491.75 Tm 25 Tz /OPExtFont3 12 Tf (,!1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 12 Tf 25 Tz 3 Tr 1 0 0 1 402.949 486 Tm 62 Tz /OPExtFont5 5.5 Tf (; ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 5.5 Tf 62 Tz 3 Tr 1 0 0 1 398.149 443.5 Tm 38 Tz (I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 5.5 Tf 38 Tz 3 Tr 1 0 0 1 409.699 561.85 Tm 6 Tz /OPExtFont3 17.5 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 17.5 Tf 6 Tz 3 Tr 1 0 0 1 417.6 537.35 Tm 31 Tz /OPExtFont3 10 Tf (' ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 31 Tz 3 Tr 1 0 0 1 412.1 494.899 Tm 29 Tz /OPExtFont3 7.5 Tf (,) Tj 1 0 0 1 414.25 494.899 Tm 9 Tz /OPExtFont3 12 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 12 Tf 9 Tz 3 Tr 1 0 0 1 411.1 492 Tm 43 Tz /OPExtFont5 5.5 Tf (II ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 5.5 Tf 43 Tz 3 Tr 1 0 0 1 408.699 487.699 Tm 58 Tz (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 5.5 Tf 58 Tz 3 Tr 1 0 0 1 420.5 631.7 Tm 486 Tz /OPExtFont9 3 Tf (,! ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 486 Tz 3 Tr 1 0 0 1 429.1 606.25 Tm 34 Tz /OPExtFont11 7 Tf (I ) Tj 1 0 0 1 429.1 574.299 Tm 16 Tz (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 7 Tf 16 Tz 3 Tr 1 0 0 1 429.85 443.3 Tm 48 Tz /OPExtFont5 5.5 Tf (I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 5.5 Tf 48 Tz 3 Tr 1 0 0 1 438.25 574.299 Tm 35 Tz /OPExtFont11 7 Tf (ti ) Tj 1 0 0 1 439.899 574.299 Tm 128 Tz (\t) Tj 1 0 0 1 443.05 574.299 Tm 24 Tz (11 ) Tj 1 0 0 1 445.199 574.299 Tm 234 Tz (\t) Tj 1 0 0 1 450.949 574.299 Tm 23 Tz (- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 7 Tf 23 Tz 3 Tr 1 0 0 1 444.5 568.549 Tm 12 Tz /OPExtFont3 17.5 Tf (, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 17.5 Tf 12 Tz 3 Tr 1 0 0 1 443.3 558.25 Tm 18 Tz (i ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 17.5 Tf 18 Tz 3 Tr 1 0 0 1 441.35 538.299 Tm 44 Tz /OPExtFont3 10 Tf (HI ) Tj 1 0 0 1 442.55 534.95 Tm 22 Tz (' ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 22 Tz 3 Tr 1 0 0 1 441.35 629.5 Tm 163 Tz /OPExtFont9 3 Tf (,,,,,, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 163 Tz 3 Tr 1 0 0 1 438.25 492.699 Tm 59 Tz /OPExtFont5 5.5 Tf (d ) Tj 1 0 0 1 439.899 492.699 Tm 261 Tz (\t) Tj 1 0 0 1 443.5 494.899 Tm 18 Tz /OPExtFont3 12 Tf (.. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 12 Tf 18 Tz 3 Tr 1 0 0 1 438 487.699 Tm 74 Tz /OPExtFont5 5.5 Tf (t ) Tj 1 0 0 1 439.199 487.699 Tm 698 Tz (\t) Tj 1 0 0 1 448.8 487.699 Tm 36 Tz (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 5.5 Tf 36 Tz 3 Tr 1 0 0 1 456.25 628.549 Tm 69 Tz /OPExtFont9 3 Tf (, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 69 Tz 3 Tr 1 0 0 1 460.8 538.799 Tm 67 Tz /OPExtFont5 8 Tf (I ) Tj 1 0 0 1 462.699 538.799 Tm 31 Tz /OPExtFont3 10 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 31 Tz 3 Tr 1 0 0 1 456.25 525.85 Tm 18 Tz /OPExtFont5 8 Tf (1 ) Tj 1 0 0 1 456.949 525.85 Tm 31 Tz /OPExtFont3 10 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 31 Tz 3 Tr 1 0 0 1 460.8 522.5 Tm 28 Tz (III ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 28 Tz 3 Tr 1 0 0 1 462 443.3 Tm 25 Tz /OPExtFont5 5.5 Tf (I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 5.5 Tf 25 Tz 3 Tr 1 0 0 1 469.199 610.799 Tm 28 Tz /OPExtFont11 7 Tf (II ) Tj 1 0 0 1 469.199 605.049 Tm 19 Tz (11 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 7 Tf 19 Tz 3 Tr 1 0 0 1 473.75 574.299 Tm 59 Tz (r' ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 7 Tf 59 Tz 3 Tr 1 0 0 1 473.75 568.549 Tm 13 Tz /OPExtFont3 17.5 Tf (, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 17.5 Tf 13 Tz 3 Tr 1 0 0 1 475.899 525.85 Tm 25 Tz /OPExtFont3 10 Tf (i) Tj 1 0 0 1 477.6 525.85 Tm 41 Tz /OPExtFont5 10 Tf (t ) Tj 1 0 0 1 478.8 525.85 Tm 31 Tz /OPExtFont3 10 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 31 Tz 3 Tr 1 0 0 1 460.8 510.949 Tm 97 Tz /OPExtFont3 12 Tf (H ) Tj 1 0 0 1 470.399 626.899 Tm 341 Tz /OPExtFont9 3 Tf (, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 341 Tz 3 Tr 1 0 0 1 477.6 610.799 Tm 43 Tz /OPExtFont11 7 Tf (t ) Tj 1 0 0 1 477.85 605.049 Tm 27 Tz /OPExtFont11 4 Tf (1 ) Tj 1 0 0 1 478.55 605.049 Tm 40 Tz /OPExtFont11 7 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 7 Tf 40 Tz 3 Tr 1 0 0 1 477.85 539.299 Tm 25 Tz /OPExtFont5 8 Tf (1) Tj 1 0 0 1 479.05 539.299 Tm 11 Tz /OPExtFont3 10 Tf (1 ) Tj 1 0 0 1 479.75 539.299 Tm 180 Tz (\t) Tj 1 0 0 1 485.5 539.5 Tm 18 Tz /OPExtFont5 10 Tf (11 ) Tj 1 0 0 1 491.75 539.5 Tm 31 Tz /OPExtFont3 10 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 31 Tz 3 Tr 1 0 0 1 482.149 534.95 Tm 79 Tz (NN ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 79 Tz 3 Tr 1 0 0 1 483.1 525.85 Tm 35 Tz (I ) Tj 1 0 0 1 484.3 525.85 Tm 166 Tz (\t) Tj 1 0 0 1 489.6 525.85 Tm 33 Tz (11 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 33 Tz 3 Tr 1 0 0 1 489.6 522.5 Tm 119 Tz /OPExtFont7 4 Tf (i ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont7 4 Tf 119 Tz 3 Tr 1 0 0 1 489.6 510.699 Tm 20 Tz /OPExtFont3 7.5 Tf (1) Tj 1 0 0 1 490.8 510.699 Tm /OPExtFont3 12 Tf (11 ) Tj 1 0 0 1 493.199 506.899 Tm 18 Tz (. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 12 Tf 18 Tz 3 Tr 1 0 0 1 492.949 482.649 Tm 48 Tz /OPExtFont5 5.5 Tf (I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 5.5 Tf 48 Tz 3 Tr 1 0 0 1 199.199 434.149 Tm 87 Tz /OPExtFont11 7.5 Tf (3.55 ) Tj 1 0 0 1 214.099 439.899 Tm 729 Tz (\t) Tj 1 0 0 1 233.3 433.899 Tm 86 Tz (3.6 ) Tj 1 0 0 1 243.849 439.699 Tm 738 Tz (\t) Tj 1 0 0 1 263.3 433.899 Tm 85 Tz (3.65 ) Tj 1 0 0 1 277.899 439.8 Tm 748 Tz (\t) Tj 1 0 0 1 297.6 433.899 Tm 82 Tz (3.7 ) Tj 1 0 0 1 307.699 439.8 Tm 736 Tz (\t) Tj 1 0 0 1 327.1 433.899 Tm 87 Tz (3.75 ) Tj 1 0 0 1 342 439.699 Tm 729 Tz (\t) Tj 1 0 0 1 361.199 433.899 Tm 84 Tz (3.8 ) Tj 1 0 0 1 371.5 433.899 Tm 738 Tz (\t) Tj 1 0 0 1 390.949 433.899 Tm 86 Tz (3.85 ) Tj 1 0 0 1 405.6 439.699 Tm 738 Tz (\t) Tj 1 0 0 1 425.05 433.899 Tm 84 Tz (3.9 ) Tj 1 0 0 1 435.35 439.8 Tm 738 Tz (\t) Tj 1 0 0 1 454.8 433.699 Tm 84 Tz (3.95 ) Tj 1 0 0 1 469.199 439.699 Tm 875 Tz (\t) Tj 1 0 0 1 492.25 433.899 Tm 75 Tz (4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 7.5 Tf 75 Tz 3 Tr 1 0 0 1 332.149 423.1 Tm 62 Tz /OPExtFont3 12 Tf (a ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 12 Tf 62 Tz 3 Tr 1 0 0 1 192.699 392.399 Tm 93 Tz /OPExtFont3 11 Tf (Figure 2.1: The bifurcation diagram of logistic map ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 121.45 343.199 Tm 92 Tz (variables \(substitute ) Tj 1 0 0 1 227.5 343.199 Tm 108 Tz /OPExtFont6 11.5 Tf (x ) Tj 1 0 0 1 237.849 343.199 Tm 97 Tz /OPExtFont3 11 Tf (with sin) Tj 1 0 0 1 280.55 342.949 Tm 65 Tz /OPExtFont5 11 Tf (2) Tj 1 0 0 1 285.6 342.949 Tm 97 Tz /OPExtFont3 11 Tf (\(7\)\) transforms the logistic map into the tent ) Tj 1 0 0 1 121.7 319.7 Tm 92 Tz (map, which is proven to be chaotic \(69\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 138.699 296.649 Tm (The logistic map was also used as a computer random number generator by ) Tj 1 0 0 1 121.7 273.6 Tm (Ulam and Neumann \(1947\) who studied the logistic map in its equivalent form ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 162.699 621.1 Tm 86 Tz /OPExtFont11 7.5 Tf (0.9 ) Tj 1 0 0 1 162.699 601.2 Tm 84 Tz (0.8 ) Tj 1 0 0 1 162.699 580.799 Tm 82 Tz (0.7 ) Tj 1 0 0 1 162.5 560.899 Tm 84 Tz (0.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 7.5 Tf 84 Tz 3 Tr 1 0 0 1 152.4 541.2 Tm 105 Tz (x 0.5 ) Tj 1 0 0 1 162.5 520.549 Tm 82 Tz (0.4 ) Tj 1 0 0 1 162.5 500.899 Tm (0.3 ) Tj 1 0 0 1 162.5 480.5 Tm (0.2 ) Tj 1 0 0 1 162.5 460.3 Tm 74 Tz (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 7.5 Tf 74 Tz 3 Tr 1 0 0 1 168.699 440.149 Tm 80 Tz (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 7.5 Tf 80 Tz 3 Tr 1 0 0 1 169.699 433.899 Tm 86 Tz (3.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 7.5 Tf 86 Tz 3 Tr 1 0 0 1 297.6 230.899 Tm 91 Tz /OPExtFont3 11 Tf (= 1 ax) Tj 1 0 0 1 341.3 231.1 Tm 68 Tz (n) Tj 1 0 0 1 341.75 231.1 Tm 65 Tz /OPExtFont5 11 Tf (2) Tj 1 0 0 1 347.75 231.1 Tm 34 Tz /OPExtFont3 11 Tf (. ) Tj 1 0 0 1 348.949 231.1 Tm 2000 Tz (\t) Tj 1 0 0 1 497.05 231.1 Tm 88 Tz (\(2.4\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 317.75 53.299 Tm 70 Tz (9 ) Tj ET EMC endstream endobj 79 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 80 0 obj <> stream 0 ,,yfҁa[6di%= xu`}"ᆅ|z`ΠkcgH/11c[Z5H\Ъx2{f.pٜowCD8KUgl")t=i:~xNjV2)}Lrlȯ[YB#$%{ ;ϡb=D"+/|aN; xۣw #4_ 4~鐌ۘoMj)W}Qhޣ%;ۻQ,>H&Ħ/ڲե1쳢=9#'H,ASȥ07o=7H4<F \p;6Yɠ&@*Cϡ noi߈&YoL45s`Q˂F[P(Jk`=< Nbb!yur?x~ǵr#ol2Vک 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The two dimensional Henon map is defined by ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 0 840 Tm 32 Tz (\t) Tj 1 0 0 1 255.599 581.299 Tm 95 Tz (Xn+i = 1 aX,) Tj 1 0 0 1 333.6 582 Tm 28 Tz /OPExtFont3 12.5 Tf (L) Tj 1 0 0 1 331.899 582 Tm 58 Tz /OPExtFont5 12.5 Tf (2 ) Tj 1 0 0 1 335.3 582.25 Tm 103 Tz ( + Y) Tj 1 0 0 1 356.649 582.5 Tm 48 Tz /OPExtFont3 12.5 Tf (r) Tj 1 0 0 1 359.3 582.5 Tm 78 Tz /OPExtFont5 12.5 Tf (, ) Tj 1 0 0 1 361.449 587.649 Tm 2000 Tz (\t) Tj 1 0 0 1 494.149 582.25 Tm 96 Tz (\(2.5\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 96 Tz 3 Tr 1 0 0 1 0 840 Tm 33 Tz /OPExtFont2 12 Tf (\t) Tj 1 0 0 1 302.149 552.95 Tm 94 Tz (Y) Tj 1 0 0 1 308.399 553.899 Tm 116 Tz /OPExtFont5 8 Tf (rt+ 1 = ) Tj 1 0 0 1 340.1 555.85 Tm 94 Tz /OPExtFont6 13 Tf (bX) Tj 1 0 0 1 353.75 556.799 Tm 65 Tz /OPExtFont8 13 Tf (ri) Tj 1 0 0 1 360 557.299 Tm 36 Tz /OPExtFont6 13 Tf (. ) Tj 1 0 0 1 361.199 557.299 Tm 2000 Tz (\t) Tj 1 0 0 1 494.149 556.299 Tm 96 Tz /OPExtFont5 12.5 Tf (\(2.6\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 96 Tz 3 Tr 1 0 0 1 120.5 514.1 Tm 101 Tz (The parameter values used in Henon \(1976\) were a = 1.4 and ) Tj 1 0 0 1 426.5 514.1 Tm 85 Tz /OPExtFont6 13 Tf (b = ) Tj 1 0 0 1 446.649 514.1 Tm 99 Tz /OPExtFont5 12.5 Tf (0.3 in order to ) Tj 1 0 0 1 120.25 490.8 Tm 102 Tz (produce chaotic behaviour. Figure 2.2 shows the attractor of Henon Map in the ) Tj 1 0 0 1 120.25 467.5 Tm (state space. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 218.65 208.299 Tm 103 Tz (Figure 2.2: The attractor of Henon Map ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 103 Tz 3 Tr 1 0 0 1 313.699 52.1 Tm 85 Tz (10 ) Tj ET EMC endstream endobj 84 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 85 0 obj <> stream 0 ,,]((b72u={_1kɕFLq23\~hl2NJ=yh\S DHLiE$5˄u>E7 @|./@hr,1XD76v0Yg`dm_%_m7A^0{7#"$ yԡfQ"mRR ߊjɤ@ /m=#(D/5*~kDhWlxlz}zΎA!`(67 a"Ŝ|+Z֢UF ? 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With real variables it has the form ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 101 Tz 3 Tr 1 0 0 1 0 839 Tm 33 Tz /OPExtFont5 12 Tf (\t) Tj 1 0 0 1 228 581.7 Tm 111 Tz (X) Tj 1 0 0 1 237.099 581.5 Tm 74 Tz /OPExtFont3 12 Tf (ri+i ) Tj 1 0 0 1 251.75 580.75 Tm 248 Tz /OPExtFont5 12 Tf ( ) Tj 1 0 0 1 269.5 580.75 Tm 40 Tz /OPExtFont3 12 Tf (-) Tj 1 0 0 1 271.199 581 Tm 103 Tz /OPExtFont5 12 Tf (y + u\(X) Tj 1 0 0 1 308.649 581.7 Tm 51 Tz /OPExtFont3 12 Tf (n) Tj 1 0 0 1 312.5 581.7 Tm 94 Tz /OPExtFont9 3 Tf (, ) Tj 1 0 0 1 316.3 581.7 Tm 85 Tz /OPExtFont5 13 Tf (cos 0 Y) Tj 1 0 0 1 360.25 581.7 Tm 52 Tz /OPExtFont3 13 Tf (ri ) Tj 1 0 0 1 365.3 581.7 Tm 89 Tz /OPExtFont5 13 Tf ( sin ) Tj 1 0 0 1 383.75 581.7 Tm 86 Tz /OPExtFont8 15 Tf (0\) ) Tj 1 0 0 1 393.35 581.95 Tm 2000 Tz (\t) Tj 1 0 0 1 496.3 581.7 Tm 92 Tz /OPExtFont5 13 Tf (\(2.7\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 0 839 Tm 30 Tz (\t) Tj 1 0 0 1 248.4 554.1 Tm 90 Tz (Yn+1 = u\(X) Tj 1 0 0 1 305.75 555.299 Tm 53 Tz /OPExtFont3 13 Tf (n ) Tj 1 0 0 1 310.3 555.799 Tm 90 Tz /OPExtFont5 13 Tf ( sin 0 + Y cos 0\), ) Tj 1 0 0 1 393.6 556.049 Tm 2000 Tz (\t) Tj 1 0 0 1 496.55 555.799 Tm 91 Tz (\(2.8\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 122.65 513.549 Tm 93 Tz (where ) Tj 1 0 0 1 155.5 513.549 Tm 82 Tz /OPExtFont8 15 Tf (0 ) Tj 1 0 0 1 165.349 513.549 Tm 90 Tz /OPExtFont5 13 Tf (=13 ) Tj 1 0 0 1 187.449 513.549 Tm 70 Tz /OPExtFont8 15 Tf ( a/ ) Tj 1 0 0 1 211.699 513.549 Tm 93 Tz /OPExtFont5 13 Tf (\(1 + ) Tj 1 0 0 1 232.099 513.549 Tm 981 Tz (\t) Tj 1 0 0 1 264 513.799 Tm 101 Tz (Y) Tj 1 0 0 1 270.5 513.799 Tm 55 Tz /OPExtFont3 13 Tf (n) Tj 1 0 0 1 273.35 513.799 Tm 41 Tz (2) Tj 1 0 0 1 278.149 513.799 Tm 86 Tz /OPExtFont5 13 Tf (\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 86 Tz 3 Tr 1 0 0 1 122.65 490.05 Tm 107 Tz (With the parameter a = 6, /3 = 0.4,7 = ) Tj 1 0 0 1 341.5 490.3 Tm 102 Tz /OPExtFont5 12.5 Tf (1, u = ) Tj 1 0 0 1 378.25 490.05 Tm 103 Tz /OPExtFont5 13 Tf (0.83, the system is believed ) Tj 1 0 0 1 122.4 467.25 Tm 102 Tz (to be chaotic. Figure 2.3 shows the attractor of Ikeda Map in the state space. ) Tj 1 0 0 1 122.15 444.199 Tm (An imperfect model of Ikeda Map is obtained by replacing the trigonometric ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 102 Tz 3 Tr 1 0 0 1 209.3 224.85 Tm 83 Tz /OPExtFont9 7 Tf (O ) Tj 1 0 0 1 213.849 230 Tm 1286 Tz (\t) Tj 1 0 0 1 238.8 224.85 Tm 125 Tz (0.2 ) Tj 1 0 0 1 251.05 229.899 Tm 1088 Tz (\t) Tj 1 0 0 1 272.149 224.6 Tm 123 Tz (0.4 ) Tj 1 0 0 1 284.149 229.899 Tm 1064 Tz (\t) Tj 1 0 0 1 304.8 224.85 Tm 125 Tz (0.6 ) Tj 1 0 0 1 317.05 224.85 Tm 1088 Tz (\t) Tj 1 0 0 1 338.149 224.85 Tm 123 Tz (0.8 ) Tj 1 0 0 1 350.149 229.899 Tm 1312 Tz (\t) Tj 1 0 0 1 375.6 224.6 Tm 59 Tz /OPExtFont11 7 Tf (1 ) Tj 1 0 0 1 378.25 229.899 Tm 1084 Tz (\t) Tj 1 0 0 1 404.899 224.6 Tm 118 Tz /OPExtFont9 7 Tf (1.2 ) Tj 1 0 0 1 416.399 229.899 Tm 1113 Tz (\t) Tj 1 0 0 1 438 224.6 Tm 98 Tz (1 .4 ) Tj 1 0 0 1 449.5 230 Tm 1113 Tz (\t) Tj 1 0 0 1 471.1 224.6 Tm 113 Tz (1 6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 7 Tf 113 Tz 3 Tr 1 0 0 1 324.699 215.5 Tm (X ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 7 Tf 113 Tz 3 Tr 1 0 0 1 223.449 184.5 Tm 100 Tz /OPExtFont5 13 Tf (Figure 2.3: The attractor of Ikeda Map ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 3 Tr 1 0 0 1 172.099 409.649 Tm 62 Tz /OPExtFont11 6.5 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6.5 Tf 62 Tz 3 Tr 1 0 0 1 163.9 373.649 Tm 123 Tz /OPExtFont9 7 Tf (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 7 Tf 123 Tz 3 Tr 1 0 0 1 171.349 337.899 Tm 89 Tz /OPExtFont13 8.5 Tf (O ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 8.5 Tf 89 Tz 3 Tr 1 0 0 1 158.65 302.1 Tm 98 Tz /OPExtFont9 7 Tf (-) Tj 1 0 0 1 163.9 302.1 Tm 121 Tz (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 7 Tf 121 Tz 3 Tr 1 0 0 1 158.4 230.6 Tm 98 Tz (-) Tj 1 0 0 1 164.15 230.6 Tm 120 Tz (1.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 7 Tf 120 Tz 3 Tr 1 0 0 1 167.05 224.6 Tm 147 Tz (-0 2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 7 Tf 147 Tz 3 Tr 1 0 0 1 121.7 135.549 Tm 97 Tz /OPExtFont5 13 Tf (functions in Equation 2.7 with truncated power series \(50\). The truncations used ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 315.6 51.799 Tm 84 Tz /OPExtFont5 12.5 Tf (11 ) Tj ET EMC endstream endobj 89 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 90 0 obj <> stream 0 ,,0jQqJ srI{hmNl|P$P({3W`J8o(zEKDw\Csoљ930.!c֙-Tp y=Kq%9VXPzdJSֻL DwYssPg{tOw%ctm%Q=H#o'z(%8-0{iܔDǑ3mZxMGb$ 7M[=dQI:u\P=R3&Lzb 5#T[3@ݣL!1RQC,]7#ucǔ0z1_0L`rc?!~9-rOLIOAJL~n ƣ/GUȣb[u27 |Hc~Q|6zߠX7~w i3Sk]*%h B{zMOóhopydZ(KAVm3 ͵krVsE4;ŷ=?y?g2. s4r .Ry۞8!%N[yTpCb$GeyF@Rk:ERq1S[ C|q#CVV!i '5Jsau#.McU*WC Colclk`Op[.u?c5J$Sҥ'BA_24g5 ƫ:Q;OZpg!oS/,wi6=:rq?0y '6ӊ1c![zR. `ȠOz6QFʨ\JefxeD'o͂h0PL~סe+v5 } 90kc4-c+Dgfz }>&?>8k7S!!N3u!J*l/KY i][ZïDEi2;Y 5zA`P+&6mBUٙDSL~hD =R B2[boL(AL@AS'NF;5ydgPXɂo&Ʒ_.o@FBJ&vH.sHz*]X8Վex0Yn9J- ֵrIz . 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We call this model truncated Ikeda model. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 123.349 475.649 Tm 129 Tz /OPExtFont2 13.5 Tf (2.4.4 Moore-Spiegel system ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 13.5 Tf 129 Tz 3 Tr 1 0 0 1 122.9 445.149 Tm 99 Tz /OPExtFont5 13 Tf (The Moore-Spiegel Flow was introduced by Moore and Spiegel \(66\) as a model ) Tj 1 0 0 1 122.9 421.899 Tm 98 Tz (of the nonlinear oscillator dynamics. The flow is defined by: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 0 839 Tm 33 Tz /OPExtFont6 12 Tf (\t) Tj 1 0 0 1 352.1 379.149 Tm 96 Tz (dx I dy = y ) Tj 1 0 0 1 402.5 379.149 Tm 2000 Tz (\t) Tj 1 0 0 1 491.75 379.149 Tm 86 Tz /OPExtFont3 11 Tf (\(2.11\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 86 Tz 3 Tr 1 0 0 1 0 839 Tm 33 Tz /OPExtFont6 12 Tf (\t) Tj 1 0 0 1 354.949 353.25 Tm 96 Tz (dy I dt = z ) Tj 1 0 0 1 402.5 353.25 Tm 2000 Tz (\t) Tj 1 0 0 1 491.5 353.25 Tm 93 Tz /OPExtFont5 13 Tf (\(2.12\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 0 839 Tm 30 Tz /OPExtFont8 13 Tf (\t) Tj 1 0 0 1 220.3 327.3 Tm 104 Tz (dz/dt ) Tj 1 0 0 1 251.05 327.3 Tm 91 Tz /OPExtFont5 13 Tf (= ) Tj 1 0 0 1 263.5 327.3 Tm 98 Tz /OPExtFont6 12 Tf (z \(T R + Rx) Tj 1 0 0 1 356.399 327.3 Tm 55 Tz (2) Tj 1 0 0 1 361.199 327.3 Tm 99 Tz (\)y Tx. ) Tj 1 0 0 1 402 327.3 Tm 2000 Tz (\t) Tj 1 0 0 1 491.75 327.3 Tm 93 Tz /OPExtFont5 13 Tf (\(2.13\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 122.15 284.85 Tm 95 Tz (We use the forth order Runge-Kutta scheme to simulate the differential equations. ) Tj 1 0 0 1 122.65 261.799 Tm 106 Tz (The simulation time step is 0.01 time unit. Figure 2.4 shows an attractor of ) Tj 1 0 0 1 122.65 238.5 Tm 96 Tz (Moore-Spiegel system for ) Tj 1 0 0 1 250.8 238.299 Tm 110 Tz /OPExtFont6 12 Tf (T = ) Tj 1 0 0 1 274.8 238.299 Tm 96 Tz /OPExtFont5 13 Tf (36 and ) Tj 1 0 0 1 311.75 238.299 Tm 108 Tz /OPExtFont6 12 Tf (R = ) Tj 1 0 0 1 336.25 238.299 Tm 76 Tz /OPExtFont3 11 Tf (100 ) Tj 1 0 0 1 356.399 238.299 Tm 100 Tz /OPExtFont5 13 Tf (in the state space. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 3 Tr 1 0 0 1 122.65 193.649 Tm 131 Tz /OPExtFont2 13.5 Tf (2.4.5 Lorenz96 system ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 13.5 Tf 131 Tz 3 Tr 1 0 0 1 122.4 162.899 Tm 94 Tz /OPExtFont5 13 Tf (A system of nonlinear ODEs was introduced by Lorenz \(63\) in 1995. The variables ) Tj 1 0 0 1 122.15 139.649 Tm 101 Tz (involved in the system are analogous to some atmospheric variables regionally ) Tj 1 0 0 1 122.15 116.35 Tm 103 Tz (distributed around the earth. For the system containing m variables x) Tj 1 0 0 1 482.399 116.1 Tm 80 Tz /OPExtFont2 13 Tf (l) Tj 1 0 0 1 487.449 116.1 Tm 203 Tz /OPExtFont5 13 Tf (, x) Tj 1 0 0 1 512.399 116.1 Tm 66 Tz /OPExtFont2 13 Tf (n) Tj 1 0 0 1 516.7 116.1 Tm 84 Tz /OPExtFont5 13 Tf (, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 84 Tz 3 Tr 1 0 0 1 316.3 51.299 Tm 80 Tz (12 ) Tj ET EMC endstream endobj 94 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 95 0 obj <> stream 0 ,,j !"h$SFL|Ÿs 5[8rK#~ Z6ñR^R* We|e3% 9oAB;^K@H&INl%Þݹ#EE1 hqg:/W|GgQ|~j\1 YT E?E B% < kH>;oWO8;A^ 5!0;.3zN4曍[9|џ8V=mhKcr (3-c(HcQu{`g)%&\9K-!j{#^_~ʬA Eb``'|T$}"-D H0#A~xMÄdYNI{už4b0WaLW~6W}a˟st) 1'[\JY͖Gq`i :%E +nxy&M|)SYoOm4ِ9nI )UxXsc43Dܐ4?;ka`1= ]~;n edOk| ^)S&v;x^s}KW̯?iI4 ~2A8ϯC4[NHVxtWͼ+½w-*$uވOBпMnH8c;6j~ң_ ,L&h 5>l)8_a{n\hTw%AR6Xg}*_$L97NZ>;* / 2ZpiIOIG Q|#[ *qV[7Exvז W-<~s[X )Zhw7&~V,*FOzYt.Vi&tfR6s~NVee@hў9&IqXzw%9j-74@<]SlQGHnF; ƭ`r@S,I1__5=jQnMHh^aMSo"YF@|0}~L .W^6XI 1D&Z)(῞3-]m.=YWi~t7|j*fU^q!_C6MN0%=QȊKTvL.a&wz`p_wn f ?ReQz{ V3|vrJ^i&gy߃iU_ b3<ߊUoKPGԵB `?^Ծ0s9j'm=?F_Ŭ" p* MӘ5!<Ǐ-gyC<:qNl v:մ6:;u:+(Ձ޳?ѿ肴軶ij۰OڳF5@۽JFm+ӂ`%ʅ4pvֿ-~[@]1gEReڝCȂA h%?IB$mJl]t 3 w{6.%~ákxt'3Fy|֕0MFr4? ;|hC'\- \RˁMBKqA]}Q9R]F t/NWyCL22K *SO M hfgE Vּ[,tYϽZl!//Hq΄y<64A:r;l9b'A*$Vw]WJ>QjX&L5l\' yep&llMٵfya*+%䁽xzd;z"l0I%qc ^0sM 7e!08ujCE˽ m徣rHy ybc67Eb;@[h>AG>w?6X#`,r\ HpR{ P+$eȮmϞŻs;! m^-XJr.iULtF5eEҢLMVS  } ~MS%}:ҫvnT!O a9%XEl.2c}>P&|¯om{xV _ ) \X0kZ+s)6;.P;lDxF!} F;u2ц潿 Xv[HFq0_ 9]\0W&j,j>aLYف}z )6+DFإ}>rZ-d Ex |#u}y6>&0ާhdz?ܛd(壛E";ڳ@#$#ѳϼg<4l2^! pSX6vc4P0j e*)Gc3B2@%ϰp*@}G S}er3Y뙩7j%ȖeY\2 Ĝ20ca\K1d t { }heo#Fy2ӭŚ&UKA:q>dܓ!^SǜvkT~oKQ:ͳ9tУR$#U(*}h۟ QycZo DrD)YVB8RQ&ɑ63'KȘ Sp͢ DO.(1m97 sϞ"":lFo'Ʉ8+jłyȆQ ;*fAdC \LG[mڿ_`|c!5*£0ؖIajIJM -S~wF1`Xt2cd;Q2_PZŒ׬UԍN%b;7:07⛯Ïy Y+O olrcg4͡ lX&swܻ_%>7g2v4Q&VZ-2OJL_SaHhG9JrAy~xY1:Y&n! 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We call the ODEs of equation 2.14 as Lorenz96 Model I. As a simulation ) Tj 1 0 0 1 122.9 270.2 Tm 94 Tz (to the weather model, Lorenz \(63\) assume the time unit of the Lorenz96 Model ) Tj 1 0 0 1 122.9 247.149 Tm 93 Tz (I equal to 5 days as the doubling time of the Lorenz96 Model I is roughly equal ) Tj 1 0 0 1 122.9 223.899 Tm (that of the current state of the art weather model. In the thesis, we will use the ) Tj 1 0 0 1 122.9 200.6 Tm 91 Tz (same scaling in all the experiments related to Lorenz96 model. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 139.699 177.299 Tm (In addition to equation 2.14 Lorenz also introduced another set of ODEs. The ) Tj 1 0 0 1 122.4 154.049 Tm 94 Tz (second set of ODEs consists of m x n "fast" small scale variables in addition to ) Tj 1 0 0 1 122.4 130.75 Tm 96 Tz (the m "slow" variables. The time scale of those variables are shorter than the ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 96 Tz 3 Tr 1 0 0 1 316.55 52.299 Tm 72 Tz (13 ) Tj ET EMC endstream endobj 99 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 100 0 obj <> stream 0 ,, jRInMZtůQ2`MUY)i ܛe/"8@7kWF tph\S).(J EN uQE 2mR^-o\/mwy̓2DcF 6ʗ$"@^! 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MPBGpC7YBg44rG&vP Y9_m$nX+2؉KtG zJtQP)v#E=lBYoZ9 endstream endobj 101 0 obj <> endobj 102 0 obj [103 0 R] endobj 103 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 839 0 0 cm /ImagePart_2041 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 327.6 720.2 Tm 104 Tz 3 Tr /OPExtFont3 11 Tf (2.5 Nonlinear dynamics modelling ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 104 Tz 3 Tr 1 0 0 1 124.099 677.25 Tm 100 Tz /OPExtFont5 13 Tf (variable x) Tj 1 0 0 1 172.8 677.25 Tm 68 Tz /OPExtFont3 13 Tf (i ) Tj 1 0 0 1 175.449 677.25 Tm 97 Tz /OPExtFont5 13 Tf ( in Model I. The equations of the two sets of ODEs are ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 188.9 642.899 Tm 138 Tz /OPExtFont6 8 Tf (dpi ) Tj 1 0 0 1 203.05 642.899 Tm 2000 Tz (\t) Tj 1 0 0 1 385.699 642.899 Tm 75 Tz /OPExtFont6 10.5 Tf (ii,f) Tj 1 0 0 1 393.35 642.899 Tm 48 Tz /OPExtFont4 10.5 Tf (z) Tj 1 0 0 1 397.199 642.899 Tm 86 Tz /OPExtFont6 10.5 Tf (6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 10.5 Tf 86 Tz 3 Tr 1 0 0 1 208.55 633.549 Tm 64 Tz /OPExtFont11 6.5 Tf (1-=) Tj 1 0 0 1 361.449 635.5 Tm 127 Tz /OPExtFont6 10.5 Tf (F ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 10.5 Tf 127 Tz 3 Tr 1 0 0 1 191.5 627.549 Tm 94 Tz /OPExtFont14 10 Tf (dt ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont14 10 Tf 94 Tz 3 Tr 1 0 0 1 405.85 622.299 Tm 93 Tz /OPExtFont15 8 Tf (j=1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont15 8 Tf 93 Tz 3 Tr 1 0 0 1 209.3 604.299 Tm 101 Tz /OPExtFont12 10.5 Tf (dyj) Tj 1 0 0 1 224.4 604.5 Tm 110 Tz /OPExtFont7 7 Tf (.i ) Tj 1 0 0 1 228.25 604.5 Tm 2000 Tz /OPExtFont7 11 Tf (\t) Tj 1 0 0 1 411.1 607.399 Tm 89 Tz (hoc ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont7 11 Tf 89 Tz 3 Tr 1 0 0 1 214.3 591.799 Tm 135 Tz /OPExtFont14 10 Tf (dt ) Tj 1 0 0 1 241.699 599.5 Tm 2000 Tz (\t) Tj 1 0 0 1 313.199 598.299 Tm 71 Tz ( ) Tj 1 0 0 1 320.399 598.299 Tm 1039 Tz (\t) Tj 1 0 0 1 356.899 598.299 Tm 72 Tz ( ) Tj 1 0 0 1 364.1 598.299 Tm 1072 Tz (\t) Tj 1 0 0 1 401.75 599.25 Tm 108 Tz (-- ) Tj 1 0 0 1 416.899 599.5 Tm 38 Tz /OPExtFont7 10 Tf (1) Tj 1 0 0 1 411.6 599.5 Tm 128 Tz /OPExtFont14 10 Tf (7:xi ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont14 10 Tf 128 Tz 3 Tr 1 0 0 1 123.849 557.5 Tm 100 Tz /OPExtFont5 13 Tf (Let us call the ODEs of equation 2.15 and 2.16 to be Lorenz96 Model II. The ) Tj 1 0 0 1 123.849 534.45 Tm 103 Tz (small-scale variables y) Tj 1 0 0 1 235.9 534.45 Tm 92 Tz /OPExtFont3 13 Tf (i) Tj 1 0 0 1 240.25 534.45 Tm 42 Tz /OPExtFont5 13 Tf (,) Tj 1 0 0 1 242.15 534.45 Tm 61 Tz /OPExtFont3 13 Tf (i ) Tj 1 0 0 1 244.55 534.45 Tm 106 Tz /OPExtFont5 13 Tf ( have the cyclic boundary conditions as well \(that is ) Tj 1 0 0 1 123.849 511.649 Tm 101 Tz (y) Tj 1 0 0 1 129.099 511.649 Tm 72 Tz /OPExtFont3 13 Tf (n+i) Tj 1 0 0 1 144.5 511.649 Tm 33 Tz /OPExtFont5 13 Tf (,) Tj 1 0 0 1 146.15 511.649 Tm 68 Tz /OPExtFont3 13 Tf (i ) Tj 1 0 0 1 148.8 511.649 Tm 117 Tz /OPExtFont5 13 Tf ( A set of n small-scale variables are coupled to every large scale ) Tj 1 0 0 1 123.599 488.6 Tm 104 Tz (variable. The constants ) Tj 1 0 0 1 250.8 488.6 Tm 81 Tz /OPExtFont6 10.5 Tf (b ) Tj 1 0 0 1 261.1 488.6 Tm 93 Tz /OPExtFont5 13 Tf (and ) Tj 1 0 0 1 284.149 488.6 Tm 97 Tz /OPExtFont6 10.5 Tf (c ) Tj 1 0 0 1 294.25 488.6 Tm 106 Tz /OPExtFont5 13 Tf (are set to be 10, in that case the dynamics, ) Tj 1 0 0 1 123.599 465.3 Tm 100 Tz (represented by the small scale variables, is 10 times as fast and 1/10 as large as ) Tj 1 0 0 1 123.599 442.05 Tm 96 Tz (that represented by the large scale variables. In the thesis the coupling coefficients ) Tj 1 0 0 1 123.849 419 Tm 95 Tz /OPExtFont6 11.5 Tf (h) Tj 1 0 0 1 129.849 419 Tm 65 Tz /OPExtFont4 11.5 Tf (x ) Tj 1 0 0 1 133.9 419 Tm 105 Tz /OPExtFont5 13 Tf ( and h) Tj 1 0 0 1 167.5 419 Tm 58 Tz /OPExtFont3 13 Tf (g ) Tj 1 0 0 1 171.599 418.75 Tm 99 Tz /OPExtFont5 13 Tf ( are set to be 1. The design of Lorenz96 Model ) Tj 1 0 0 1 411.6 418.75 Tm 89 Tz /OPExtFont3 11 Tf (I ) Tj 1 0 0 1 419.75 418.75 Tm 93 Tz /OPExtFont5 13 Tf (and ) Tj 1 0 0 1 441.35 418.75 Tm 109 Tz /OPExtFont3 11 Tf (II ) Tj 1 0 0 1 453.85 418.75 Tm 99 Tz /OPExtFont5 13 Tf (is to simulate ) Tj 1 0 0 1 123.599 395.949 Tm 101 Tz (the reality that the model is built on the m dimensional \(slow dynamics\) space ) Tj 1 0 0 1 123.599 372.699 Tm 99 Tz (while the underlying system is also contain m x n fast dynamics variables which ) Tj 1 0 0 1 123.349 349.649 Tm (one can not observe. In this thesis, both Lorenz96 Model I and II are simulated ) Tj 1 0 0 1 123.599 326.35 Tm 102 Tz (by the forth order Runge-Kutta scheme with simulation time step 0.001 time ) Tj 1 0 0 1 123.349 303.1 Tm 96 Tz (unit. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 96 Tz 3 Tr 1 0 0 1 123.849 250.75 Tm 112 Tz /OPExtFont3 15.5 Tf (2.5 Nonlinear dynamics modelling ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15.5 Tf 112 Tz 3 Tr 1 0 0 1 123.599 210.45 Tm 111 Tz /OPExtFont3 13 Tf (2.5.1 Delay reconstruction ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 111 Tz 3 Tr 1 0 0 1 123.099 179.5 Tm 100 Tz /OPExtFont5 13 Tf (In reality, the state of the unknown dynamical system is observed in the obser- ) Tj 1 0 0 1 122.9 156.2 Tm 99 Tz (vation space 0. It is often the case that the observation space is not sufficient to ) Tj 1 0 0 1 122.9 132.899 Tm 98 Tz (express the dynamics of the system unambiguously, for example, only one com- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 434.399 632.1 Tm 154 Tz /OPExtFont9 7 Tf (i ) Tj 1 0 0 1 436.8 632.1 Tm 2000 Tz (\t) Tj 1 0 0 1 492 635.5 Tm 93 Tz /OPExtFont5 13 Tf (\(2.15\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 492 599.5 Tm (\(2.16\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 316.8 52.299 Tm 83 Tz (14 ) Tj ET EMC endstream endobj 104 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 105 0 obj <> /FirstChar 0 /FontDescriptor 106 0 R /LastChar 130 /Subtype/TrueType /ToUnicode 107 0 R /Type/Font /Widths[0 0 0 0 0 0 0 0 0 600 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 600 600 600 0 0 0 0 600 600 600 0 600 600 600 600 0 600 600 600 600 600 600 600 600 600 600 600 600 0 600 0 600 0 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 0 600 600 0 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 0 600 0 0 600 600 600]>> endobj 106 0 obj <> endobj 107 0 obj <> stream /CIDInit /ProcSet findresource begin 12 dict begin begincmap /CIDSystemInfo << /Registry (CourierNewPS-ItalicMTOPExtFont15) /Ordering (UCS) /Supplement 0 >> def /CMapName /CourierNewPS-ItalicMTOPExtFont15 def /CMapType 2 def 1 begincodespacerange <00> endcodespacerange 16 beginbfrange <09> <09> <0009> <0A> <0A> <000D> <0D> <0D> <000D> <20> <22> <0020> <27> <29> <0027> <2B> <2E> <002B> <30> <3B> <0030> <3D> <3D> <003D> <3F> <3F> <003F> <41> <5B> <0041> <5D> <5E> <005D> <60> <7B> <0060> <7D> <7D> <007D> <80> <80> <2014> <81> <81> <00B0> <82> <82> <2022> endbfrange endcmap CMapName currentdict /CMap defineresource pop end end endstream endobj 108 0 obj <> stream 0 ,,:b6%/e\ Jzv4OlK%%nl:g+ْu> > %rTP/mn_ ~UmŹ=C_44V"GΝx"q(!Ŷ2ʀNd$(͈E>M`R,2g_7\-Ouv} y?痄b_r0S~Ț^ 6FHHĺOvjOBmDlCq(ZNm}15ZCnLCDqU)c;7& Ob)c3<-ZIiEynK0ʑ) ZGfJ'"۠<Ƴ=wOkspi`0ryYU,d&5;R$脖aHk])ؘ({kB7v2OXw5l|/ۺAGY;L /7zGcqCd ?8D; |WV%ʍeYr/ד.˕%]ۇNȶEƤџse `a@7P5a05 9䲸S1BKuǯvL{'%x0Pj\8 &i嗕 lH3%06=(yԠ2 z@^d675)ӹU-En#%s1]"Âڳ#}(ތZ1!$`='܁Z9Dܵ#bK78f'I#(20wU6ͻw&ZT$ΌuC"ٺsU7w#_B-@7|s!!Uaӡ URD!KH28# UK¨TW9iC UL vĞ5n"fs)Cfu/&Ʉ;sVzNteQ1R A+/B4:v\yc Ƚ ʭR \aP16sSM]j\wxB!:lEHGhmmո3O`Aw兏dOCY8O (JEʰO"bĥ5龠}R=Bk19B U_lТ?{絘!ҿd5IuUߌC]n\By-(gu'U]MKX&)($]\:̻ z|*[ȁb[ȑ$9?HΕ;V˪Jw;I/5 &ӌB扮4}_@z_io\̟ y"Vx+,X-3٦v²&ߣ8,)_&?')ǶG\IvuQZ\|ʟ+^'ӵ‰6zC!ϺETh4\#LSL1]_6!@jBnYyM9MҩO 1* Aqkx" o_t)3Ul:ߢwr6\Y8 MxacR.|2΁+ t dj(mn@Lv`Λicm Xq 0Z%ngWJ gxJ'c_a8$x0%fF Qujd\nsyzj%"E]")D_KC2`!>?k\XLr(}ajPs.I${@*Xs& A?];#Q0_ht?i!3t \Mp&b q!FvèNU|fNP\V1y+&mos "ާ"7ޞ`؁rtmn#GFag;Y}p#A_A#AGLe!9/t2l>f&`1)]ǟ6BW) wJLv),Bcί#҇{'^{\._4mje.W|) >{r.m_ 5v|ٕt:;iЩ?w.T?}~76tr/ς:z+57Wwʺl)*坆q3E*X,ŕb&bPʔ4gd[ [5dV8/#VZ Sft l˚1Y)*S"* lտJn CO˻ꦐK̥-!8n-쪊>ؘO *i>NiT2=CM*H@ v}Qw6}VqLq+ 5q?2c(OWjA;Z");P5*Y=ϜmL~1;Irz~NVC( Rsm.(xqA m$JxgqFc38wNKrb$aqPO&J5 @vZTBo@􎓇Ȇ_"\szSg4I}@}M)Ah0!kӱ#zBX;^/$K@]L[z~ :%q? 1EΚ0 {A-H=՞Adk1Z<<[+ҬԿ%[4{Kcmù}H/U f{iMt/x߆9 qӍXǦ}{ cRZI;5V /v ,;s-?7TLAjF #smg:w'3.و‘lϸ0D.8Ӟs`_ ݠrlBK}{_.E+ *kG]C6,4`P`QtMq#dٲ5qU??M/\`i1cq/Bhcy#KɇUlLmM1B 9ܫ?nBʓs~Is&[M5cNUI՞RӟktʬSxK(O5{\z(}U«c'@|.ɽ#f@%~pffDg H)Üq!XI|CQCv`_.iwoZxfqԯnK9j͏( :BCa.[O0}zѻa*ve$Forj{y(loPR C0YtتN fB2I0O umP|kV#I G1Ǫb#XѴ&I!kv%re`^Jd]lbҔ-r6 |[P~8/onnh{Ңa't;1Y{ۤo1s7& k?~-Oz^4١tUxy7!e>*|ZÜDdunF]rxriqcCg˨+(fq ;פgFS7#"R3qSF +?F1@a!?Û]ZZjRj't? Kuj}$BsWd) (.{_0jlC#!^A6@* ɧW@q8|zTЦ78sW4TL^>874(7=#FkQcEE,z|43\bK?y^(0c%I^dbj f^ع _"ޞ甇%6/ tM&tb8@Y2JxD>usO{1E@ee !Oөr閯3q*_F"T 9g}]3ePݡx,QF$ 7W"N0Xr0};⺩B5WqFbUf2}o'e[ÁWc[Ҧo4i3PMz))Öe/p3]b6?S ĩ`ֆ- u|h :g3D{hl\jnYOƫBr1$(Km2 C2"wrkfС+jC?Oͨ.[^ Ѐl}bLN)G@h^ :-=G B\L" '_耩wˁMt @[FJ\@Y_Nm8f,Ea~^^0+gy|Mxr3u&}d/Do[~?WnbG!ӋD%x:CG*[MN*;oC3{xgH仕O'ʚ׿}jvu-q3ĎOWPXBs=Lxf~6h:u0iJrj\} #G?McOs|dwn[TzR #6[[%2L.˿"*ĝA;WtȿP\ $œmp;icQ1su:Mˆ)_ZF\ k6*J2V'uŸgLDhU ⦴~5h3f7x"&)̄|8>u3#idkXvnSwiO&$95Ag}욇sW8 ֛`sC>yu*62Z*{HY,52sϢ(#z%rr';6rOR8jĢO$8Mİ}2cƉȼ9V)*(ޛ {uE6׏+wc֯㾷8:PB  ]/Qoݩ ;X6K"i=4 ~Id }vSdfJ|{VXPd=YZ+9()ΟC1N{ׇf IThjSzBrtjM@hk rZ/S˿sERM~Z/Z4`LlY\Ԍdgk.[n o0%I &pZgR5 V/c|j\@ariabۣ+#r S Qȱ71`]4t|cC})cDV0ջiDO-WqʼL.:@t=nI /8f½OcsWW \-$X| gv-O4"Ɂ;[uف7{iS9=ADt$dصaˇ8V9'3N}JNK?(묈6;#JKQѢ?]fd=X&;[[f + &ZO0 *s2F/T׍ xAsMeSLxHfcujR%< ƫ2_Av?w1Oyqf(-i2L2,|bd &"atoV}J9Dׄ(0EnEKPwzPFיHNbig"F K ܚ>)S3~ʵUҡ͈.ѱt->#ǴζM,1ꉏ<tUfq\D$2 UN&̶-ܣnFK8ʻC%My)g_fˋT7y˵j4 vTEkdˌIŷd"J3qeVF3QMu uZn!ell ƄsR[loV+f=ܿ*QҖ]N4K9՞A. xؤYrw\+{ov)˸ߢZ2p=Cl]G;}JFP(.61Zbe.EG5~,jMnYVT鲽f Ͽl7H=T=1/=1[ O*.-o1 6dn߁Ӧ+e+1 M\#BI$TV΋c]uV>KH*?{5AO=`9C~xK-!@z(C9={RI AſZFX~]+<$"ڃE^dKZk•fЛW%9wC D'n\uĮ& ll "3C888ت]&8(WC\ZCOm'-+ZJx:U SHd`!*toYT-B^D+Zvp $333RfxIp&$_{ /YS A իn7R*OM4R| Jm?8ۚR UiBe+EXTUS,6I p#4Gr@cyMA岮 8O0.49Wߥj9٧YRmd V k?6Ǜ4CfW3Դ5&{:bGI9aGXWyoNu6g{Ju]s³ 6| F:`lC?M>@(Ks) yԳ/n&V,ͼAGO{ o)[ǫF4.0dn endstream endobj 109 0 obj <> endobj 110 0 obj [111 0 R] endobj 111 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 840 0 0 cm /ImagePart_2042 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 326.899 721.45 Tm 111 Tz 3 Tr /OPExtFont5 13 Tf (2.5 Nonlinear dynamics modelling ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 111 Tz 3 Tr 1 0 0 1 123.849 678.5 Tm 99 Tz (ponent of the system state may be measured. Rather than model in observation ) Tj 1 0 0 1 123.599 655.45 Tm (space 0, it is therefore usual to reconstruct the dynamics of the system in a fur-) Tj 1 0 0 1 123.849 632.649 Tm 97 Tz (ther space: the model state space M. How can we construct a higher dimensional ) Tj 1 0 0 1 123.849 609.6 Tm 102 Tz (model state space given the observation is scalar? Takens' Theorem \(89\) tells ) Tj 1 0 0 1 123.849 586.299 Tm (us that we do not have to measure all the state space variables of the system. ) Tj 1 0 0 1 123.599 563.299 Tm 98 Tz (We can reconstruct an equivalent dynamical system using delays of the observed ) Tj 1 0 0 1 123.849 540.25 Tm 101 Tz (component, such method is called delay reconstruction \(79; 81\). Given a time ) Tj 1 0 0 1 123.599 517.2 Tm 96 Tz (series of scalar observations, s) Tj 1 0 0 1 269.5 517.2 Tm 53 Tz /OPExtFont3 13 Tf (t) Tj 1 0 0 1 274.1 517.45 Tm 95 Tz /OPExtFont5 13 Tf (, t = 1, ..., n, recorded with uniform sampling time, ) Tj 1 0 0 1 123.849 494.149 Tm 104 Tz (a trajectory of model state x) Tj 1 0 0 1 270.699 494.149 Tm 53 Tz /OPExtFont3 13 Tf (t ) Tj 1 0 0 1 273.35 494.399 Tm 102 Tz /OPExtFont5 13 Tf ( can be reconstructed in ) Tj 1 0 0 1 402 494.149 Tm 111 Tz /OPExtFont4 12 Tf (M ) Tj 1 0 0 1 418.55 494.399 Tm 96 Tz /OPExtFont5 13 Tf (dimensions from the ) Tj 1 0 0 1 123.599 471.1 Tm (single observable s) Tj 1 0 0 1 215.05 471.1 Tm 53 Tz /OPExtFont3 13 Tf (t) Tj 1 0 0 1 219.599 471.1 Tm 99 Tz /OPExtFont5 13 Tf (, by delay reconstructions. This yields a series of vectors ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 246 427.699 Tm 124 Tz /OPExtFont3 8.5 Tf (xt = \(St St ) Tj 1 0 0 1 298.3 427.199 Tm 62 Tz () Tj 1 0 0 1 304.1 426.5 Tm 110 Tz /OPExtFont9 5 Tf (Td 7 7 ) Tj 1 0 0 1 323.05 426.5 Tm 92 Tz /OPExtFont8 8.5 Tf (St \(Al ) Tj 1 0 0 1 349.449 426.25 Tm 94 Tz /OPExtFont14 6.5 Tf (1\)Td ) Tj 1 0 0 1 371.5 426 Tm 149 Tz /OPExtFont7 4.5 Tf (\)1 ) Tj 1 0 0 1 377.5 434.05 Tm 2000 Tz (\t) Tj 1 0 0 1 492 428.899 Tm 93 Tz /OPExtFont5 13 Tf (\(2.17\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 123.349 386.399 Tm (where ) Tj 1 0 0 1 157.199 386.399 Tm 105 Tz /OPExtFont8 8.5 Tf (Td ) Tj 1 0 0 1 171.599 386.399 Tm /OPExtFont5 13 Tf (is called the delay time. To predict a fixed period in the future, we ) Tj 1 0 0 1 123.599 363.35 Tm 95 Tz (consider a third time scale, ) Tj 1 0 0 1 255.599 363.35 Tm 109 Tz /OPExtFont3 7 Tf (Tp, ) Tj 1 0 0 1 271.449 363.35 Tm 98 Tz /OPExtFont5 13 Tf (the prediction time. Each state x) Tj 1 0 0 1 432.25 363.35 Tm 48 Tz /OPExtFont3 13 Tf (t ) Tj 1 0 0 1 434.649 363.35 Tm 98 Tz /OPExtFont5 13 Tf ( on the trajectory ) Tj 1 0 0 1 123.099 340.3 Tm 103 Tz (has a scalar image s) Tj 1 0 0 1 225.099 340.3 Tm 66 Tz /OPExtFont3 13 Tf (t+) Tj 1 0 0 1 234.25 340.3 Tm 101 Tz /OPExtFont5 13 Tf (,, and we wish to construct a predictor to determine this ) Tj 1 0 0 1 123.349 316.799 Tm 99 Tz (image for any x. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 123.599 271.899 Tm 112 Tz /OPExtFont3 13 Tf (2.5.2 Analogue models ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 112 Tz 3 Tr 1 0 0 1 123.099 241.45 Tm 99 Tz /OPExtFont5 13 Tf (Analogue modelling is a popular and straightforward method which is effective ) Tj 1 0 0 1 122.9 218.149 Tm 100 Tz (to systems whose trajectories are recurrent in state space. Extracting the spatial ) Tj 1 0 0 1 122.9 194.899 Tm 99 Tz (information of the system dynamics requires sufficient historical data to form a ) Tj 1 0 0 1 122.9 171.35 Tm (learning set from which neighbours of the preimage of the state to be predicted ) Tj 1 0 0 1 123.099 148.299 Tm 102 Tz (are defined. In this thesis the nearest neighbour is determined by the distance ) Tj 1 0 0 1 122.9 125.049 Tm 99 Tz (between the current state and its neighbour. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 316.8 53.049 Tm 80 Tz (15 ) Tj ET EMC endstream endobj 112 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 113 0 obj <> stream 0 ,, b6MhurYS=yK0VJ:,<+٨' ћSAtDHVb}q h`4%j-X;(ɒT<䶈DY1-G΋V:kWؽ.yg;&bc^9ړ pCSGgNid4.kCdO^1U5å#Ba!OjӮJ.cBDX5n"7O=96lbƙJy:A SB i:,/t@քh w\7*U/7K㞨e3)m$L^! ċnOvz4ahyoɬPEk: 7/=dϒN`{`t=9f|HƇ7)Υ5ڰρU<"juz8aeQB&߅H{;|dGр;?!U1K q4ɾD1lϣ*Q~q$s 5c0r=ןAv4&uҰ]P\iJ o4&Y[nu82@dT&0A}|̈t ȥa|^2{>h@41YQ2+5 `tC!"qf@:0{P\cǎs _UF[Z GnƘ g+&ܧ2*e3"BC9ڡWgO{X+7rbָM:ӻ/O}Ü ZO6\1-L5TpR|1*<rr~p&Cj?.3b6jQi>(~T{KfM?mQ tcFj4Y RM?={fH>CSZoBE#ia~ lqFa)co2wH4!hGھHoG~1NIՓW;FZ+rs4eܭsRP BϺDT.ס_U20Ʒx#ȦiaunD_: Eo/>Ɠ_Y!d#Mh/YhL9 1}ݬ>x1yͳᯛLU$-Rw !&% Z瞞x'zHaU-= HfNa;X˪?Y.VW5J25"Xt<_\vYǬo(w%fHzְy;qӊo)B(56p/2?}ŸEL/j{>jʭ@$'>T~z,+08ݟa^eV2B,}q"UpQ.T{P"T81DiI%x@6ďtةV;fXvj^r A0W.~JZP!\QkGeL "٬AeɕZWH ̽kFb<:ɬ"* W-}݋\i+bF"7ш.ͿgʷE~4̀ 1R< s[;G_9^*zc wb#E^ "nY^,Mߚxb&h+JVՍ|2EL~<-&۪&lcVn^E6-;nl/o;GL p^TVl  Ӫ%J~k6>3M~w@tp%n KDGZ ~15x ʹ@M~!U'>!›n˜nhu_³}*m'~v'3-̺JL!/*s#ұ\DچFir5up8SdrxVD'+)x.gU : (c v>+7obh&wK;Ȼqך\Z$v}7eT纊mx7JL{):O IGrG1n5_ڢQr|166Et zOo4T 9`Wh"<[sAFShoIHz)*eˑгޝf"i s2xŽ:UdZ;j)1䕥ʳ-4OQ)h4Sfx,cfgbjqP 7f܋b#+󾦷3ZȈ ޔZ";,n\$Oޢ, @,Aх`^(f/eR^: H,@ otܹekX> 7 {/D2<1ΎèHa@:lzV>8kBUDL ߧ._ʰpwZ0R2ьB t ܹRhȃ6Z^]!)oW 3.eR|.b,T9D,pu! 6(Ȟ,#LZw}Fn y4.pX&]1Ӳ'Fp(J>1P8_|:1 %WRyzՄ;z-L4F5pjf[Ea1|-->̉NݿYȐ8 ?ayNf8Fȥݰ*Vb1Xs"ӈKwv> mrDmyIf(5bMh}Y XU_?n|Q)@SH]s-W QqoCue~|ϟ?/l:0^~ ?VZx8΂&1n?rp*DtJeS>cZB8|Ef5*2$!E Fg΢KjAI,Zg!A-_F?2+ћ5޸whl+j#m]mex&{X6_N-'H vYQӟ5(ׇ%7RNեa*d#SOg6ǀ"@[ EH U 9;?;Be@'59Y0W(l`MTa:dBt2SpcS:  MSev&Bm8#:3=SiQl>GZ4cUa P{SwfrI7(P^'>TFyO[17ﳝ5 U*%_A1wֺМ䧳g6nrK&&)3 `$хNrv„?2s#Q?[o:?[Ȭ2 zɔL2U3e1F@9o'DM='܂R!=<^`֤䇒J}X uÅ[>EVAA #C s5]Lj͉7UG8D  \! Q11;ys;q19Tm?6ݣv*kƴ*M$O ?|:]XDXU3(E"I?Mל&YFh[Ҩ:V'/d4-RzԍxD>۳ҵ/&36<{q-UC :QNOhr$CL'8E!J 5>l#iM-}nX4Pi2i )R(D3:波='|?-9T*[Z W(Qgc 6sl'ʶسxJ'#% `K=ą)+5*m[5ղZb"Ͼ.85=:>3&EZ4:3*O!I=K4U_@ }J(Exw5Z#*YEmޒ^xʟN4V}<mmL Ib-f>3@Flͭ娕Y+iW!,>!tL9 h$1Fvk䵖$Mk,/ V"y2-k4xCŒ8h[l)\ErHR%Kړ߲}-`3QsMY:9QՐ5M/#QYV~rRΊϛCRbMT~#MJ]Qmd)e5m5bZPljK$>.")j&˳47,(!<6[JAl;$=V.o$=@BWǜB8p3*{gӞ{ O~f5+x&7A-%a?D\ z LgӖSK } P0D1 ,` H"lk~nD#q0Usf*yɊq=l lm襏y;ZDZY倲jwfHp IJkXfW- lޣ kʇ4l_H RFI#LY$F- WsC6}tfZ뗃nfuʂM0`NˢH6/|4d0_cjxρrvUC*L1c4]}&gȡ0HL: WtbearN*țQQt!pM>|VF3 CaS'5!/Q:O[ ,/=YvЯ\K+N#>Hm|C2kr.$?;­^@WipU fO]k( d#/_U!cG|ʠmk*kS<ʙ *),AˬO~Bqicᓄ4=M ɭ%آ-Q\z(ssR|SIDdG+9UANԃ$hV7u,fhJ`0좉'Y[Ȅx0dI?260ꏛ&ć@8g %xפ)RdlT* z2l>~wusPMJ sԒ^ÃaT?-Y`h099dME ':?I2Q06W[kQ[yo#WbmZ-s1Amː o9- {nj{W.# ojԙ0ˇO(/=yן,~^OPRgٳw3Gĉ|2' ;w <5"op[f/$;>Zlw뙼q"'W~Fzh|G/u\1a-8DG 0 8$; G9ע]+NI?VSdX95~rKRuP-i91B[Q 4[S'qQgP %e%B8yEK_3hsmY C`]?EsaBU}q׊ڏX\ʫ Nm=QE3q $=*XkcҪTd:˂ϽU%u"^_S'\Ҵ?wwki< B<!uneLP-aXLR]Eȴ{6IQOxs> endobj 115 0 obj [116 0 R] endobj 116 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 839 0 0 cm /ImagePart_2043 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 327.1 720.45 Tm 97 Tz 3 Tr /OPExtFont3 10 Tf (2.5 ) Tj 1 0 0 1 347.75 719.95 Tm 105 Tz /OPExtFont3 11 Tf (Nonlinear dynamics modelling ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 105 Tz 3 Tr 1 0 0 1 141.349 677.25 Tm 99 Tz () Tj 1 0 0 1 152.4 677 Tm 91 Tz (Local analogue ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 152.4 649.399 Tm 89 Tz (For local analogue, we firstly find the nearest neighbour in the model space. ) Tj 1 0 0 1 152.15 626.1 Tm 91 Tz (We then report the nearest neighbour's image as the prediction. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 141.349 594.2 Tm 99 Tz () Tj 1 0 0 1 152.15 593.7 Tm 93 Tz (Local Random Analogue ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 152.4 565.899 Tm 88 Tz (We are not always lucky enough to determine whether or not the data from a ) Tj 1 0 0 1 152.4 542.6 Tm 90 Tz (stochastic process or deterministic process. Paparella et al. \(70\) introduced ) Tj 1 0 0 1 152.4 519.799 Tm 92 Tz (a hybrid approach, Random Analogue Prediction\(RAP\), which exploits the ) Tj 1 0 0 1 152.15 496.75 Tm 95 Tz (deterministic nature of the process while incorporating variations in the ) Tj 1 0 0 1 152.15 473.5 Tm 93 Tz (local probability distribution function, thereby adhering to the stochastic ) Tj 1 0 0 1 151.9 450.449 Tm 90 Tz (nature of each observed trajectory. To produce the Local Random Analogue ) Tj 1 0 0 1 152.15 427.399 Tm 91 Tz (prediction we firstly define a local neighbourhood in the model state space, ) Tj 1 0 0 1 152.15 404.35 Tm 94 Tz (usually with a fixed radius or fixed number of k nearest neighbours. We ) Tj 1 0 0 1 151.9 381.1 Tm 92 Tz (then select a near neighbour randomly from the ) Tj 1 0 0 1 395.3 381.3 Tm 107 Tz /OPExtFont6 11.5 Tf (k ) Tj 1 0 0 1 405.35 381.3 Tm 89 Tz /OPExtFont3 11 Tf (nearest neighbours and ) Tj 1 0 0 1 151.9 357.8 Tm 92 Tz (report its image as the prediction. The probability of selecting a particular ) Tj 1 0 0 1 151.9 334.75 Tm (neighbour can be based on the distance between the preimage of the state ) Tj 1 0 0 1 151.699 311.5 Tm 93 Tz (to be predicted and that neighbour or treat the ) Tj 1 0 0 1 390.949 311.5 Tm 104 Tz /OPExtFont6 11.5 Tf (k ) Tj 1 0 0 1 400.8 311.5 Tm 87 Tz /OPExtFont3 11 Tf (neighbours equally. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 123.849 266.6 Tm 110 Tz /OPExtFont3 13 Tf (2.5.3 Radial Basis Functions ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 110 Tz 3 Tr 1 0 0 1 123.599 235.899 Tm 89 Tz /OPExtFont3 11 Tf (Analogue models, when considered as a kind of local models, require constructing ) Tj 1 0 0 1 123.599 212.85 Tm 93 Tz (a new local predictor for each initial condition by searching the learning set. As ) Tj 1 0 0 1 123.599 189.299 Tm (a result, a large amount of computational resources are needed. Global models ) Tj 1 0 0 1 123.349 166.049 Tm 95 Tz (can cover the entire domain once the model is constructed. In this section we ) Tj 1 0 0 1 123.099 143 Tm 92 Tz (illustrate the Radial Basis Functions as an example of global model. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 140.15 119.95 Tm 94 Tz (The Radial Basis Functions are a global interpolation technique. They con- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 94 Tz 3 Tr 1 0 0 1 317.05 52.049 Tm 75 Tz (16 ) Tj ET EMC endstream endobj 117 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 118 0 obj <> stream 0 ,,nnb6[!S ?&Ő E&L2Q|XCJ^&2 mi{!m8GDkBQڭe;j SHW*#o)NknH m˥S_>>-y-ޠO|XmHl?yvN&)d~mE,E7:Pgedȑ/XiPڑZ= )|[ 1ܳ6I[B7KJ}d?͖D_B?uJ]$3: fwH>)>e! !by F{ ETѹȟ^H!QYTcEjH)@b[Z`}bѧLLL t^*۞Z(QUH$sH#fp2=2 3#HՐP1R "@ Xع&xҋA_lI( a~J\eaOX]ŁZ77a@]õo)1M*2]b^6-nvE&a3udzqJb'Q(̀߆IUc)_^<#~+5AlŒE6gk`h&,{! ,!$~̾2v\)Ciu)SΨmZ4X[^;-T5*Ǭ3\6t{d:=춐;xx/``/~U^D"k>`|d>JjYl|f[iܽ9pp|yz_֖U`XnLTn#nl8?Q"eucfgUg.h֛e,X_ۀ@B^BO))꺗AiE7ekiVpU4Cʹ ydTl g&/j sbp5}UwCgnt^C˱3v"5;L@nVMx8 us$ gOT< ~ gB<"Yt6ٴ:iތ+gVV%4H~leGG [OϨ/G*|ڶPb0N3j\mlp|Sj*vx>f+{exZ pe$nOM?p[(RUkYZ^ Q8Q t9j.$iCfD_ustWur#Uv KIp/'kpfr WFh5i#X:SDeyD|=p ZI +X]\}k'O\L KRH>#e1UrE Efk.2-7G,෹ /b=?m\ĞGa=emW*HN͋FM]'$ ]0WP6b'f!e(>lـ[~M1;9bFo.dĪ#C!d") zP,7,iydKгGڨdY, voU$)u Q U:J> o9V#5~v[ftW>&HL4 &V1E SMAI?!wլm$I{ D7 M8i0Ȼǿ$[ ܍luyq31k0;c) }B+TF>lb3!l=^l>u@dSB2?0/mohH2rg~vN{d_ 1bsud@63\|5& [F$M%] OR[ttC{fM&% e?GeMy%EhQG'}chM/)a)SrNt6)hn2I K. 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Pˎ)] H6uL|˪{&2dGlB,d[}cfȪ}"$(s1s4TP`qW9_f@EV)˺o(q=}ƪGD"/Ql^ՕV|R8iV=p73ʆj?Ara<ڽ1]RvF1 < /;q'bD=ZS+'OoT&fnuh%B8^YRc'$®B=᯼BU@ 39T#Gìl>jf8*M(^@8*[ vU¶쀼k>Ț+PG kYtq!ˇ0 HѼ*0N#)X<0/BxcptHKaEBœqm c/rnLS:S4V [ G ;bl6(zl*R:S Z׉Z $/-aB'>L$ΧEWNZWۗ+n:9[)@W ~wsIAm4ɨK=1wkYjyhЮ/BHzaof.CЭ+UV3|U#:f[ypC-4WRK@į _ /'XE?e6r 2 pj ja&L#n4>ճ?a-<X  :ţrl+fWKnZ}tr%tyz7j>{K09rv B/)P/e,p񩫍ץ[*1#+`Ϙn) gcc\BM0 )P* _/ ZaC(8Jh2(LUes}tHsXƗ5dY-Os'{J0X$L'5 BO2/0 yywB."kD#-&q+~֍*7#F: endstream endobj 119 0 obj <> endobj 120 0 obj [121 0 R] endobj 121 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 613 0 0 839 0 0 cm /ImagePart_2044 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 328.1 720.7 Tm 104 Tz 3 Tr /OPExtFont3 11 Tf (2.5 Nonlinear dynamics modelling ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 104 Tz 3 Tr 1 0 0 1 124.799 677.95 Tm 94 Tz (struct a predictor \(map\), F\(x\) : Ern ) Tj 1 0 0 1 299.05 677.7 Tm 525 Tz (\t) Tj 1 0 0 1 317.5 677.7 Tm 90 Tz (R' which estimates the scalar observation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 125.049 654.899 Tm 74 Tz /OPExtFont4 10 Tf (s ) Tj 1 0 0 1 133.699 654.899 Tm 93 Tz /OPExtFont3 11 Tf (for any x based on ) Tj 1 0 0 1 230.9 654.899 Tm 64 Tz /OPExtFont4 10 Tf (rb) Tj 1 0 0 1 237.849 654.899 Tm 115 Tz /OPExtFont8 10 Tf (e ) Tj 1 0 0 1 241.199 654.899 Tm 91 Tz /OPExtFont3 11 Tf ( centres, denoted as c) Tj 1 0 0 1 348.699 654.899 Tm 117 Tz (j) Tj 1 0 0 1 354.25 654.899 Tm 41 Tz (, ) Tj 1 0 0 1 358.1 654.899 Tm 172 Tz /OPExtFont4 10 Tf (j ) Tj 1 0 0 1 367.449 654.899 Tm 88 Tz /OPExtFont3 11 Tf (= 1, ..., rb, where c) Tj 1 0 0 1 456.949 654.899 Tm 109 Tz (j ) Tj 1 0 0 1 460.55 654.899 Tm 125 Tz /OPExtFont2 10.5 Tf ( E ) Tj 1 0 0 1 476.399 654.899 Tm 96 Tz /OPExtFont3 11 Tf (Rm. The ) Tj 1 0 0 1 124.549 631.899 Tm (predictor F\(x\) is defined by ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 96 Tz 3 Tr 1 0 0 1 288.5 602.85 Tm 95 Tz /OPExtFont3 7 Tf (nc ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 7 Tf 95 Tz 3 Tr 1 0 0 1 245.05 589.149 Tm 100 Tz /OPExtFont3 13 Tf (F\(x\) = ) Tj 1 0 0 1 280.55 588.899 Tm 746 Tz (\t) Tj 1 0 0 1 311.5 587.7 Tm 73 Tz (-0\(11 x - ) Tj 1 0 0 1 351.1 587.5 Tm 429 Tz (\t) Tj 1 0 0 1 368.899 586.75 Tm 55 Tz /OPExtFont9 16 Tf (ID, ) Tj 1 0 0 1 380.149 594.6 Tm 2000 Tz (\t) Tj 1 0 0 1 492.25 589.399 Tm 87 Tz /OPExtFont3 11 Tf (\(2.18\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 285.35 576.45 Tm 106 Tz /OPExtFont3 8.5 Tf (j=1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 8.5 Tf 106 Tz 3 Tr 1 0 0 1 124.299 545 Tm 94 Tz /OPExtFont3 11 Tf (where 0\(\) are radial basis functions \(14; 15; 79\), II II is the Euclidean norm. ) Tj 1 0 0 1 124.299 522.2 Tm 95 Tz (Typical choices of radial bases functions include 0\(r\) = r, r) Tj 1 0 0 1 423.35 521.95 Tm 65 Tz /OPExtFont5 11 Tf (3) Tj 1 0 0 1 428.649 521.95 Tm 93 Tz /OPExtFont3 11 Tf (, and e) Tj 1 0 0 1 463.899 521.95 Tm 147 Tz /OPExtFont5 11 Tf (-) Tj 1 0 0 1 469.699 521.95 Tm 69 Tz /OPExtFont3 11 Tf (r) Tj 1 0 0 1 473.75 521.95 Tm 56 Tz /OPExtFont5 11 Tf (2) Tj 1 0 0 1 477.85 521.95 Tm 86 Tz /OPExtFont3 11 Tf (/a where ) Tj 1 0 0 1 124.299 499.149 Tm 89 Tz (the constant ) Tj 1 0 0 1 188.9 499.149 Tm 96 Tz /OPExtFont4 10 Tf (a ) Tj 1 0 0 1 199.199 498.899 Tm 90 Tz /OPExtFont3 11 Tf (reflects the average spacing of the centres c) Tj 1 0 0 1 411.35 498.899 Tm 95 Tz (j) Tj 1 0 0 1 416.649 498.899 Tm 90 Tz (. In the simplest case ) Tj 1 0 0 1 124.299 475.899 Tm 91 Tz (the centres are chosen to cover the region of state space. To determine the value ) Tj 1 0 0 1 124.299 452.85 Tm 90 Tz (of A) Tj 1 0 0 1 143.05 452.85 Tm 109 Tz (j) Tj 1 0 0 1 148.8 452.85 Tm 86 Tz (, we assume ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 86 Tz 3 Tr 1 0 0 1 287.05 410.6 Tm 116 Tz (F\(x\) ) Tj 1 0 0 1 310.1 410.6 Tm 484 Tz (\t) Tj 1 0 0 1 327.1 410.85 Tm 76 Tz (s) Tj 1 0 0 1 332.149 410.85 Tm 72 Tz (i) Tj 1 0 0 1 336.25 410.85 Tm 34 Tz (. ) Tj 1 0 0 1 337.449 410.85 Tm 2000 Tz (\t) Tj 1 0 0 1 492.699 410.35 Tm 85 Tz (\(2.19\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 85 Tz 3 Tr 1 0 0 1 124.099 368.35 Tm 94 Tz (The A) Tj 1 0 0 1 153.349 368.35 Tm 101 Tz (j ) Tj 1 0 0 1 156.699 368.1 Tm 92 Tz ( are then determined by solving a linear minimisation problem, i.e. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 291.85 325.899 Tm 104 Tz (b = AA. ) Tj 1 0 0 1 332.399 325.899 Tm 2000 Tz (\t) Tj 1 0 0 1 492.5 325.899 Tm 86 Tz (\(2.20\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 86 Tz 3 Tr 1 0 0 1 485.05 283.649 Tm 34 Tz (, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 34 Tz 3 Tr 1 0 0 1 123.599 283.899 Tm 96 Tz (where A = [A) Tj 1 0 0 1 189.599 283.899 Tm 101 Tz (i) Tj 1 0 0 1 195.099 283.899 Tm 76 Tz (, ..., A) Tj 1 0 0 1 220.55 283.399 Tm 62 Tz (n) Tj 1 0 0 1 225.599 283.649 Tm 103 Tz (j, A is defined by A) Tj 1 0 0 1 329.5 283.899 Tm 91 Tz (ij ) Tj 1 0 0 1 335.5 283.899 Tm 1464 Tz (\t) Tj 1 0 0 1 386.899 283.899 Tm 80 Tz ( c) Tj 1 0 0 1 403.449 283.899 Tm 101 Tz (j ) Tj 1 0 0 1 406.8 283.399 Tm 97 Tz ( II\) and b = [Si ., ) Tj 1 0 0 1 504 281.5 Tm 76 Tz /OPExtFont4 10 Tf (sn) Tj 1 0 0 1 512.899 281.5 Tm 113 Tz /OPExtFont8 10 Tf (i) Tj 1 0 0 1 517.2 280.75 Tm 55 Tz /OPExtFont4 10 Tf (] ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 10 Tf 55 Tz 3 Tr 1 0 0 1 123.349 260.6 Tm 90 Tz /OPExtFont3 11 Tf (where ) Tj 1 0 0 1 155.3 260.6 Tm 96 Tz /OPExtFont4 10 Tf (n) Tj 1 0 0 1 161.75 260.6 Tm 41 Tz /OPExtFont8 10 Tf (1 ) Tj 1 0 0 1 163.699 260.6 Tm 94 Tz /OPExtFont4 10 Tf ( is ) Tj 1 0 0 1 177.599 260.35 Tm 87 Tz /OPExtFont3 11 Tf (the size of the learning set based on which the model is constructed \(14; ) Tj 1 0 0 1 124.299 237.1 Tm 85 Tz (15; 79\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 85 Tz 3 Tr 1 0 0 1 123.599 192.45 Tm 96 Tz /OPExtFont3 13 Tf (2.5.4 ) Tj 1 0 0 1 171.599 192.45 Tm 123 Tz /OPExtFont2 13.5 Tf (Summary ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 13.5 Tf 123 Tz 3 Tr 1 0 0 1 123.349 161.5 Tm 94 Tz /OPExtFont3 11 Tf (In this chapter, some terminologies of dynamical system and its properties are ) Tj 1 0 0 1 123.349 138.45 Tm (defined; details of the systems used in this thesis are then provided and some ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 94 Tz 3 Tr 1 0 0 1 317.05 53.25 Tm 73 Tz (17 ) Tj ET EMC endstream endobj 122 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 123 0 obj <> stream 0 ,, b7 >_$` Sx>Qey02y;l: (ḱ vzOa^TXჾ]Lй5)+c4o;jeĪ63 x%$8H@zGz7oȋh.fFL䭗 BAȕ`Xh .q3|ӽXmh'fd  WqZML|Pdkn PU]űiI6MR!U0"˨蠨~HϩO.w)X`uxꮸ_fN7>\L3ULR 7sD&dcwV9 \-ԊiQ}x ;Q;F|sc H:?$\L?b>\p,GPmqKwJ 8ЯX"##tabR'y dtD>©r?M/2& l6"y%AM@bd՝Р|C͹v$5*1%3!L f ;,!BӰM_X4!{-.|/!̲w\rc6=kO2sLj_Pbх]?.ClIg t.e/p[:D‘-Z_Q/ ✿)|lׂ2¦%=?@r[UKV_bYA<_HPB_,jib;uXRB?S_sbdͷAgkO ~0C'8,#Z`1͹+,IS>v]'ECX"P44CC0 /= aݎ?0l\Vt 7Ϸn|.><76/Om^ڧBY# ˘Qj=q/ BM3FPTQC$c>ُC"ݱzҰOڐ#@2<fm9wx y(xcQ0U*(@עOKF;<!?Kâk`krђg4&GA$[X`^ %VuhgɼNXvE.xa0B7;@S(hi͈p&z}Mtr]Vh{|g Üśf<>( ~";Fv[JݯXȃd5•8M/t!`.Vpom8RKW|ECEfn/ CiDkƥñA#ˎh$'>BI6e']M$u3y,OnmS5"죮#,svjF2SOƳDĴG3J^ ݥPO'2DB :^WcTZσ..ݕ^1eiܧ5I|Aw¯H͚J'?b0Êڄq|צ7IFP^'4K5`2q'L=sZkoTG]#|GsvV#EPiؙ^R~AcK8;:v.?m{i 7>"fX , kd%3 ^)* %B}Dެ p01;2*V5zydA+gtm˜4'67^\Ь]aj= ?3ɰk l[n,V}_U5 )`ϊ3JMY(J5FUށZIif2c۠\XO&l)^p%bo_$M˙Iiyz"TcT\aS0d*sfq9QU0D(bQ}3/fsɁt1[I䧵94vU|ӡf^73J+1м(G}6T+! 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#_hYW-]* vrLG^}F(J2#Y&/0`?qTusF9* i`oZ4%^RJ<<+T;-9Q&p{ɭ sYTcHL3TN`ȏ{⣊7B$vhk^n$ںݶ?^g׍ J=0͹M VIG-q,:bwV~PApS}%Nxګ!қsLř@yLVQ,a.g{)}Mk3]wR)+5¥ c˗> endobj 125 0 obj [126 0 R] endobj 126 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 613 0 0 839 0 0 cm /ImagePart_2045 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 328.8 719 Tm 109 Tz 3 Tr /OPExtFont3 10.5 Tf (2.5 Nonlinear dynamics modelling ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 109 Tz 3 Tr 1 0 0 1 125.75 675.799 Tm 90 Tz /OPExtFont3 11 Tf (relevant nonlinear dynamics modelling methods are described. Although nothing ) Tj 1 0 0 1 125.75 652.75 Tm 94 Tz (new is presented in this chapter, the content of this chapter provide the back-) Tj 1 0 0 1 125.75 629.7 Tm 90 Tz (ground knowledge of the thesis. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 318.5 51.299 Tm 75 Tz (18 ) Tj ET EMC endstream endobj 127 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 128 0 obj <> stream 0 ,, ~b6|Y׽b: | %4HNsCo&V]pl*Ciȴ:gWoiXOO¾z^We0^ٟSegySkK~PJD):(6:50 ZhoLKKBx[({ƘĆ'v4ҬA>9dbc=Z;dn#-`| R`U2NJ U y.[&=Iű݊3ޫ>i43 Ι\g2CwW/Kn_*>'DU4"FPMTzIC`9\MӨN]qO%MRe,Uӊ{*; D"'~gjC+d!uryϗ")e,37gl`YAW5nUcy};RR!L/b`ߦY.)Dg8;W$ʠ?vVW+uݻ_W.גN̗f$XNV̌猡!Ķ![< gkz ćOuA65e$:qq"lML ôlF(7d;&7o|ݶzDW@X%W#%][D&ԜbL(a+51_~b%IƠuj}DDyln|/2 nRx3GM!Zd (S 8A>Ttb#c0kh(ӹ0 jYco 䙈?*gY@"O} [!YI66X.L*K;}?[ +wokzb"χ>5-zubiy+ Nsh^6c)Q}. ØL oz&y[Y4GJVAGJ1u"D@,>A(#$yfaMhD799x#]\ |{se>{ͦݴ6],3$)>+k / 2&xz7*Gy PaĖ}I8#*Ir}r51)G%~LAWb*‡.dG2=*wPН%Nnp|454~ - 30j 5\ť5@[ 혈k@!>7@=J:er&@#¼ɯ~Rjgʋ=ňp"u+0Ᏻe]~տ`ሀ}}=uʶf\rhsٿ9-n>uI|;Ue 0lycj I^7L"t |֯jWmm÷'͌}|ݯZqrWv#)c(Aq G3.niI)NJZۨ,*.]͜Drcg( G%ZB#66eެ>y78^sX2AIMG¨2{I9 @؉Ggc14a$WKKKr̿FDs9i Xf<7#|i?D0>!š5 HJQt3=nq[ bL"uUW;5%DJs!ʅvfi{Շ}iG<p쇲L,1A{2v&^>LK첳T)=boQʴcΰ™.P=< u Ftd=;(\|%my/ĺm0V`{&8_y P:f1pbXpFA95JX'*e*熈ж:T~I}^U!{)r9z޴B+m&leQnE~5NbOC9fԓe8[ = #'Z.N5JR]dVr/wˢnrXRO[{woV !uraM)JѻWBI-U/X\)oQJT ZS#]Q| jhTwfpVZ ^,kz|J=6b-&J'24):PireL}ĀQ-rY򧳨J}7a56';np*Z/BWDMW*G]!iY㷖bP# X+O>mGQ9dr* gx~ApUlЏOrsub&M33O!̍2ӿ!s/໿H?sBB h:`si0PKݕiӤYMA6r(T.QpuЁ۞8;F#x`pxx],Gozg]W`wAJ!񱘢Ȃo!~n|Y.NbUeD3r endstream endobj 129 0 obj <> endobj 130 0 obj [131 0 R] endobj 131 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 614 0 0 839 0 0 cm /ImagePart_2046 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 126.95 580.75 Tm 105 Tz 3 Tr /OPExtFont3 22.5 Tf (Chapter 3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 105 Tz 3 Tr 1 0 0 1 126.25 513.549 Tm 109 Tz (Nowcasting in PMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 109 Tz 3 Tr 1 0 0 1 125.299 452.1 Tm 93 Tz /OPExtFont3 11 Tf (The quality of forecasts from dynamical nonlinear models depends both on the ) Tj 1 0 0 1 125.299 429.1 Tm 97 Tz (model and on the quality of the initial conditions. This chapter is concerned ) Tj 1 0 0 1 125.299 406.05 Tm 94 Tz (with the identification of the current state of a nonlinear chaotic system given ) Tj 1 0 0 1 125.299 382.75 Tm 96 Tz (both previous and current observations in the Perfect Model Scenario \(PMS\). ) Tj 1 0 0 1 125.299 359.5 Tm 97 Tz (It has been shown that even under the ideal conditions of a perfect model of ) Tj 1 0 0 1 125.299 336.449 Tm 93 Tz (a deterministic nonlinear system and infinite past observations, uncertainty in ) Tj 1 0 0 1 125.049 313.399 Tm 95 Tz (the observations makes identification of the exact state impossible \(48\). Such ) Tj 1 0 0 1 125.049 290.1 Tm 92 Tz (limitations mean that.a single "best guess" prediction is not an ideal solution to ) Tj 1 0 0 1 125.049 267.1 Tm 95 Tz (the problem of accurate estimation of the initial state. Instead an ensemble of ) Tj 1 0 0 1 124.799 243.799 Tm 94 Tz (initial conditions better accounts for uncertainty in the observations. Here we ) Tj 1 0 0 1 125.049 220.5 Tm 95 Tz (define the problem of state estimation of the current state conditioned on the ) Tj 1 0 0 1 124.549 197.7 Tm 92 Tz (past as a ) Tj 1 0 0 1 174.5 197.5 Tm 96 Tz /OPExtFont6 12 Tf (nowcasting ) Tj 1 0 0 1 232.3 197.5 Tm 94 Tz /OPExtFont3 11 Tf (problem. In the PMS, there are states that are consistent ) Tj 1 0 0 1 124.549 174.45 Tm 92 Tz (with model's dynamics and those states that are not. Those consistent states lie ) Tj 1 0 0 1 124.799 151.399 Tm 93 Tz (on the model's attractor. States off the model's attractor are pulled towards the ) Tj 1 0 0 1 124.799 128.1 Tm 94 Tz (attractor. For nonlinear chaotic systems, this collapse onto the attractor dom- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 94 Tz 3 Tr 1 0 0 1 318.25 51.799 Tm 72 Tz (19 ) Tj ET EMC endstream endobj 132 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 133 0 obj <> stream 0 ,,!""b1ŬjOCeDZVNo S>B8G}gP_û)d1y|рZ(^;\"wHE:C UWo/3̯D!!  ?>ʢ=8a|pW[PBT1x?N岮bxu]@آᱬ:w l.Iò81l챨Xҽ:%W/ЭV "QqcE p`tx܋n JcñNw 33驻$A9`zFV?N'W˹`bEH,XwMXLn}Bzp:&ͻo.ƮQA3A|])ݧDR6p[z]tN']a ӫD,=RK-m?ĪZz-SmY֋45f٩Iי݃/"DjzPaFG/(!B%+jR<7lly+5eJ҃#P`>b 2tN)H"( I 8(.I`{YR Y|)Gc?6Vä(+\n3=c!edgrWФǾn)(nj=*_$e/K)1Tv.M9UϞf3pZXMgr;p)28S߫ !bMaM0bUӁ** c coS*aS^ ?HmbXX(XO@[ȻZi$+1 %I& E&$ĹP &S;߉wcAv󍡯homTdyTjg+1] FƫRCEul1b&ڊ#"_EE\eFpz&婛<ZX9C'u䬽 gx.B),o8#WX 9n\cDZ]+߭v@;_E5yDTU\?G׌߀& %-}$rHhZoIup{ιADB#&#_4ӁԞMG.oVt5MUm7}',;vNrLhg,!voe_Fdz;3=꺾^֑M l.҆~m/З'©jAap "{z>/ʉGh_qTȮ1Xn+=|6}C5,; IV)6fVkfĝZ9h()q~̫d+rjXӨF2"ᕗ,!RVM٧3g墮#9 "WlȏRe.<#IuO [X2WK(_'z?%&N,W[K{/ M  5fis<5ij&\YnYr(K-~>kXS%Plo' ݠA^4K~i]vצmR-6[9/U?egB W?5a~}UT5~괊_jm}hݱ'Εey <zNA@*]|"x9V%ۂh 2}3jLkglPʠJH@r.2&ꙃ~{KEͧ!._جCM#{_FU| Lq#kw1B{q'rmēXw J&eunAW=]]l-ףVz~piq0dId(,"l0p,!8 ,(`V}PWh``AkmXs ͵2\iSgP -WR3_OEoZdq6Wپ33VB,W% 6T=Nq[%Wn|bM(̎ utqP t)BjFx,,% qҵ+FJf?Ĥݗde|R|n"Mc/`%iˀi*MS%8NR ttO$qZn7_yUݞI:697 5,> ݨ%IP ܅W&|0 ɒSoO:j7Ck3z$6`1*ܔj>Wc\ RG[t2k{&+Yn耖F^ ɥDt+&jǥ'%'A`m/@ DpsϲҠh\u9N92 B*BFJu`-gǬT? \wuo!}E6ЇPۜ"EJtqP%i\IQMx4Br{V=a6~IaCJA}fZ`OP/q]o .*:_oOÎ?2fOC6&GrM@7/!W$̓ iQ#">J#0zrݣe\nWˠ8A.^Q!Oܮ۾t8ްixW@ ѣv [e40uQ=Rs?V_ '%wwNUH\jK97(aФ8~Rr{!8k8jc?.0Uy(@[,(D|呬Rϣ͒ٹ(%Z0U!Yފ#4|EFkS]9RrK,ALSV/|H48 MbF~}[;@5z{!K=rWGb Q$uB9s+%/D580nrw,5E&G=y cwl7i Mc]d a q!\b:F[k0E##m7gϙ#6O+Y)C;4|ʱPםQwoP瀛#j_N.kjqC³G| hY4*8ɸ`^ sݥJagޘ(a]}8 <EevO1{%-uPSQœ^uU"'8weMmGH_r;8. B6uɠemAIԧGЋõ@1JӌMM }q *" r^TYnbI'6BabyrW#LB3@OhJ^ &"oא RܴvRAߣָ-yYط͓z) m>p u4F*Tfl(,Hq˻vlW 1$Р5hKt¿t5 ,QHeNlכD Ua"kByKͤFD{<2k;EX@zhYp u^^ϵ2 sp-fLRKKg8l^/pP5#>y$׭7ρ+36mnCmfB'ƂDEX/ǔ|n,  y /9M-{@<ݻSx17;};T Co0! ܙA͌u.xC14Gymb˘zE!p3pY6Z~~N!5~$ Ǔ, 's6> m^њM^wǓr_T #s=aw.|X|^Ssp'./>:hhiQʇI5v]dCQr8m0&_A}㡈 -mLGZ?^9I* )^;KZ\pL@>oʁNZV5Cl_CL7XTn;ƏVia^!Puy?mvRM|nn="]6 ޅ^.6m YɮXq JQdžs'sn1tЎ2"EYe|zR; <󄒠=kCn=!ƩPjVhR|Vln2D1 X܎Қ&/]򫾓i3.vf{ r|YN+wX ?yf$/qýx6(Qɢir!#k̻`ƚaޚ[fif  KʍPN6sCI.hd]H^RU(eݍ11'0&jSV,ZRa} E-݁h4NY!pbd?J\j 4rX>.'ƲπԟudYŞ9a8'ܸչ6?Śڴ<:? qx}q8ޏp΍*MWweI՛?eqWc _ n<MG^vs8L\CM~{7PĮzl=EdMM];c Hp'~A`Y #$S%6"f og.wO  BA_FJ3Dz#dŔC YRG4첔̽QBEKB`I-_wq @)TZ>~pI{gKUڷ掏PSoǓ4DGR T )xDέ]Z&*єx[ރ<6I-_ 9T] `3ip'lꪢ£W!X_7V5;ڨ8UGx}ISUa P،گ}gjmX1 ubSdy<>9^]ϐф訊> Œc.g5>Tr,ɝQFE9ßRTq}Fӏ&Dpyi~њ^n(:vD[ SvrTzf}> endobj 135 0 obj [136 0 R] endobj 136 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 613 0 0 839 0 0 cm /ImagePart_2047 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 368.899 719 Tm 106 Tz 3 Tr /OPExtFont3 11 Tf (3.1 Perfect Model Scenario ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 106 Tz 3 Tr 1 0 0 1 126.5 676.049 Tm 95 Tz (Mates the model's dynamics. Intuitively, it make sense in state estimation to ) Tj 1 0 0 1 126.25 653 Tm 91 Tz (identify those states that are not only consistent with observations but also con-) Tj 1 0 0 1 126.25 630.2 Tm 93 Tz (sistent with the model's dynamics. The perfect model scenario is firstly defined ) Tj 1 0 0 1 126.25 607.149 Tm 94 Tz (in Section 3.1. The theory of Indistinguishable States \(IS\) is then described in ) Tj 1 0 0 1 126.5 584.35 Tm 89 Tz (Section 3.2. In Section 3.3, we introduce our methodology to address the problem ) Tj 1 0 0 1 126 561.1 Tm 95 Tz (of nowcasting in PMS by first producing a reference trajectory by the method ) Tj 1 0 0 1 126.25 538.049 Tm 90 Tz (called Indistinguishable States Gradient Descent \(ISGD\) and then an ensemble of ) Tj 1 0 0 1 126 514.75 Tm 92 Tz (initial conditions being formed by Indistinguishable States Importance Sampler ) Tj 1 0 0 1 126.7 491.949 Tm 93 Tz (\(ISIS\). Other state estimation methods including Four-dimensional Variational ) Tj 1 0 0 1 125.75 468.899 Tm 96 Tz (Assimilation \(4DVAR\), Ensemble Kalman Filter \(EnKF\) and Perfect ensemble ) Tj 1 0 0 1 126 445.899 Tm 93 Tz (are described in Section 3.4, 3.5 and 3.6 respectively. Comparison are made in ) Tj 1 0 0 1 126 422.6 Tm 91 Tz (Section 3.7 i\) between ISGD method and 4DVAR method relative to the reference ) Tj 1 0 0 1 125.75 399.55 Tm 93 Tz (trajectory \(defined in Section 3.3\) they produce; ii\) between the initial condition ) Tj 1 0 0 1 125.75 376.5 Tm (ensemble generated by ISIS and that produced by EnKF; iii\) between the initial ) Tj 1 0 0 1 125.5 353.5 Tm 89 Tz (condition ensemble generated by ISIS and that of a perfect ensemble. It is the first ) Tj 1 0 0 1 125.5 330.449 Tm 91 Tz (time that IS theory is applied to produce analysis and initial condition ensemble ) Tj 1 0 0 1 125.75 307.149 Tm 93 Tz (and contrast with 4DVAR method and Ensemble Kalman Filter method. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 126 254.6 Tm 132 Tz /OPExtFont2 16 Tf (3.1 Perfect Model Scenario ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 16 Tf 132 Tz 3 Tr 1 0 0 1 125.299 220.049 Tm 97 Tz /OPExtFont3 11 Tf (Let R) Tj 1 0 0 1 152.65 220.049 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 155.3 219.799 Tm 158 Tz /OPExtFont3 11 Tf ( EI) Tj 1 0 0 1 179.5 219.549 Tm 73 Tz (n) Tj 1 0 0 1 184.099 219.799 Tm 94 Tz (' to be the state of a deterministic dynamical system at time ) Tj 1 0 0 1 492.25 219.799 Tm 85 Tz /OPExtFont4 11.5 Tf (t ) Tj 1 0 0 1 500.399 219.799 Tm 98 Tz /OPExtFont2 10 Tf (E ) Tj 1 0 0 1 511.199 219.799 Tm 88 Tz /OPExtFont3 11 Tf (Z. ) Tj 1 0 0 1 125.299 196.75 Tm 92 Tz (The evolution of the system is given by ) Tj 1 0 0 1 320.149 196.299 Tm 78 Tz /OPExtFont2 14 Tf (.P\(R) Tj 1 0 0 1 340.1 196.5 Tm 63 Tz /OPExtFont5 14 Tf (t) Tj 1 0 0 1 344.399 197.25 Tm 137 Tz /OPExtFont3 11 Tf (, : Rth IR) Tj 1 0 0 1 411.35 197.25 Tm 30 Tz (71) Tj 1 0 0 1 415.449 194.35 Tm 96 Tz (/ and iit+i = F\(xt, a\), ) Tj 1 0 0 1 125.049 173.5 Tm 93 Tz (where F donates the system dynamics that evolves the state forward in time in ) Tj 1 0 0 1 125.049 150.45 Tm 96 Tz (the system space R' and the system's parameters are contained in the vector ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 96 Tz 3 Tr 1 0 0 1 125.299 127.399 Tm 53 Tz /OPExtFont13 18 Tf (a) Tj 1 0 0 1 135.349 126.899 Tm 94 Tz /OPExtFont2 10 Tf (E) Tj 1 0 0 1 145.199 127.399 Tm 76 Tz /OPExtFont3 14 Tf (W. ) Tj 1 0 0 1 141.099 127.149 Tm 22 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 14 Tf 22 Tz 3 Tr 1 0 0 1 318.25 51.799 Tm 75 Tz /OPExtFont3 11 Tf (20 ) Tj ET EMC endstream endobj 137 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 138 0 obj <> stream 0 ,,=b6Qj14۬}a|{쮂zri `zJ1 /\ڏn2'6K|HNj,]8rnnBֵUo0IZTi?e/;NP!+7{He-5PrrE54A^=rE0QdIa+s% iV䬮c+Zɀ)!\QlkI2*ǂ{]&ij4Eӫ}JqO[Jna$孋LX&˖!ɣ0F5A;P?F8.{f| 4?(Ql/1V=2н_ՑuGAF?4 uU!=*=7\KAr*/ M5P|LנE@2mʌ~xr"Xey EcE# 8\וwVx %zXv ,"Dܿ$BgX7R s<5_iS(+KŹwXܧ5)B%/~2)rAq!ׂ{N[`}{"rJDӔj²2É7n4~ER<\ c)r /.7Z%Upbe*KٍЦ:`ٜK#9H "(w1GB;y 0Rt8bZWdTʩ)`2ى2uM5(vto5,E:Ct**KIZM Iх!sR);`5*0O4U7VC׆+>C8 8>Bfi-׍ YŻ<z{+ bg F̛T2ʟU@tBdRHN{~m`t(nu{UnV \{LvzeCp$#<G̴~Di {'$MW#vKoJd.V9<}>["NyVK͞=v=ƙ Z:vH 'A Z/qϞH/3B&K /w0|.wy^ ˳t rKA`xw8tH%BϜ-gڌ҈eɌrP7VU.A=TJy iۦ[fTB} si|yP@aDNT~VL.iC0^ؽmMXx19`yPQUG| |$ 3P %#241ʛx_vrFĭ,z0_B`H/@ac"ra5.\A\jXv9>oOIgq9n7PS^_).OWJV{9=qv }kmD]f.ެokkIx>bP4UAwb uBOSyQ)՝KI?u&'/@,KœYהR Ҧ!~7x c`_|{ӝ8?a[M,5`rKGɘOsb-U\Xd~Bʺ&$,2ح;r%GzFC$̿$s *wNG徒~x: $V=6a5.x;5ZtB)"05O;M@b~k<ƈqS#1Ӏ?$;+I?rXs}8F>t`Us7O-A;ɬM; =?Mj^u{7͛7=eB!$>Ցʳ(4%;N3Lϵ;٭<:d_RshX"Wue0. B_TL&ZڕLЛ S.Daw :=fMK8CpL JN>kCjy ?y0Gx${о;O rM#1# }ay(~}8-2W8b o b%ڲn@[8K^l ڶMdǣ,uGCsoL 9"6/_B"N&vg7`s9Ǜ SWbz+9 N+OPmYwܯ~5%|MwczzXNEE)9s/#Bv.h5[ӑjM3e1 ]Sy9Ӗ@l#Nj GsVzd^oaeOU汼-֬5Xe;$ǐnBN*bu" _>xPɍ BEX)nTV7+:S8mr7gL[G5 S.XswP3pɾcN* L°4`ab\^yxo *:T| Hϰ <=A57(mb˥H'G<.!m4(eLjР[{~LAVHks#"Ա7tm}1 `u}o6h/+$E{Ȑԋ<&C.!"Ǧ^&jů"ZS SՈo\0t%6B|4q ~-=xVبی5iJ:iM~qŏh~#Elu3tihRB )D^gA}'*]A2FQxDߌ3@pz+Շ8UҨ¥Qs ɎTvRޠT7дh}ta#Tp4Powc hnIU:1$I1x4Qqx(wz;7G \|˛$/Ptk,%(j*7=Dˢn`f JJ+8< V_u1;V7-]TMC|tLx&WkZx]ߢJH@Μ},ZYlx#Z6%wَ`\AIC ೖYn~HR!"Q'=W2Ln0҈jn_iŬ6*yӏ+1 *ܑ 5yBO "7~_~\'zʶ=y,2[N=\-{V֯]xY׾̾23M͐x['QU"")$j:aÃ.cWj ٨b A5.|9uHz[.! ~~&Tb0a#Jt ճ7gGqKXgW=&"XI`k1jee44"ȄcOsg?@;Db[ӽjI mE6p,h9?pgFo("ĭ╬âuL F?*y*tIU_ c(%:gK]0Me?FviW3\H`*ݻA4Uq{61mu }^Z:|<hT=iy"m`1G\ZW:T726Dh <y<誧Swǿ߇Q~:\)7ߜ3mV%O{',&Fb6_%J\!<sH 69=עDoR 4rV5"4o6zX]>Kh1+4Ul5H綠Ө:OcU̼, o+Ƞ }osm_s?11RO+0->5#1ռFӁ4+Hiy;>:hc3Zvv.y.Ӟj|RF;hC'R V̒u($\s]7HXjs;fwv"9%O.$ϖ.$ ٠7E\&CMVQ`3`a?H|YfLɑ2EH5A ' N ?mf`8R E$aƌJn"9!L8<^.]v`#Q[}Oj,2]EQLX_e{8r Hok[~:jH@.YiZU\|2 u8zOyZ*t/e[_{PZ"A}7"3Kg$>Ըm+GՀݵ9$&)ڳ场ht7elj('^PW9}DLw os9]2Z=cPZz*"\02,G%6&k˼ @QIh4x9-OfAiD&4UF@L\s( A/?0D=;pIFfOyVb|,F~Mg` O@8 Dc{Jj *!]N`13_NލWЯ졢!M:p ps2\ ]dz p ׂ\I&T:qfjWlhĨgi끄9 0D9kg ?g3 M2,ϠD MC+:&8RZ>hͿ2Ա"e3ǝyjc15<5T0Fq'LH#Y@I7>?<@\ quNX%c.7?lJIJz｢PՎ\Kc=@<>R6 2u_zqҿ^(MS T\P[<ͧIEc!|~ iE"aR/Y'K..*d&!;-  P8C P|w UyYDg͵O!]CgO+q˜fk? q&v3 ŒQ@>&4ai3n|'hY“6Hތ; /;/o"hҐo I!mRgzᴦ5 ͵]󖵷^+~TuRwM藈~N+J>ZFQxㅰ :$qW)ekmHNe_omѭ OPWm?5X2WKq]6Esʅ@,gPB& $FÅfuÿV!@@$٬(?to؅; )~Gۙg76z@ G0`m[cea$6&U˼o.OG調Ί3'n{gi2U:,MU(vK+F[Cod r-"vw;-0[ YV?A+n7I/Ɉ!x͌Avfyq_&QVdsX颜OX&w@V-W-&ϋI\TLhKsK3iU@¿aP+"j"rOQ^roN諊3O沴X`>7k_,Bb Q;"KpJ\PCpiVg{_jaHQѼyēgi d쓪Y#иF'Ce\{CeY.736̹?ȱ'X1CJ# M<= 5vB ü6Jl4tܐX.VZЎ(o?dʯreܥRIlo$G.e%vd 2yNrE))T;^ ٶXopW(mMЮ0Qs57fVVA֡F+厖i;!c,.Ydh%s=&^M&W(]Yz74էqXK8T`3JЍ{2Q}G/LS <۠4gR:45x@}A_a~s@'YO~OEOuBSHD,WRq: +uZ Ü»ڰ%]LMmICJp_6xTPJy OR "(>|O8!LLaқv{mY$-Xp}~GEfLb|i_IC w_2Ä;{IThrݙ*G}$wx^tLhLlP_|tC$OFAI&)^C&<}W q=*jޝy}roȵ}9qb<#O-c_+w%j..0mμGݡ$FcVV$p-bq(>p4EmCݓ#UXaZQ'۴ .S[0G";fO'`:{{*m%[P[vj*A'Z` % endstream endobj 139 0 obj <> endobj 140 0 obj [141 0 R] endobj 141 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 613 0 0 839 0 0 cm /ImagePart_2048 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 368.649 719 Tm 106 Tz 3 Tr /OPExtFont3 11 Tf (3.1 Perfect Model Scenario ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 106 Tz 3 Tr 1 0 0 1 143.05 676.299 Tm 93 Tz (Define x) Tj 1 0 0 1 184.55 675.799 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 187.199 675.799 Tm 93 Tz /OPExtFont3 10.5 Tf ( E IR"' ) Tj 1 0 0 1 222 676.049 Tm /OPExtFont3 11 Tf (to be the state of the deterministic dynamical model at time ) Tj 1 0 0 1 125.75 653 Tm 108 Tz /OPExtFont6 12 Tf (t ) Tj 1 0 0 1 133.9 653 Tm 98 Tz /OPExtFont3 11 Tf (E Z. The model is defined by F\(x) Tj 1 0 0 1 304.1 653 Tm 81 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 308.649 653 Tm 111 Tz /OPExtFont3 11 Tf (, a\) : > JR and x) Tj 1 0 0 1 416.649 653 Tm 81 Tz /OPExtFont5 11 Tf (t+1 ) Tj 1 0 0 1 429.35 653 Tm 121 Tz /OPExtFont3 11 Tf ( = F\(x) Tj 1 0 0 1 466.1 652.75 Tm 89 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 470.399 652.75 Tm /OPExtFont3 11 Tf (, a\), where ) Tj 1 0 0 1 125.75 630.2 Tm 111 Tz /OPExtFont6 12 Tf (F ) Tj 1 0 0 1 138.699 629.95 Tm 93 Tz /OPExtFont3 11 Tf (donates the model dynamics that evolves state forward in time in the model ) Tj 1 0 0 1 125.5 607.149 Tm 89 Tz (space Rrn and a\) ) Tj 1 0 0 1 211.9 606.899 Tm 93 Tz /OPExtFont2 10.5 Tf (E ) Tj 1 0 0 1 222 607.149 Tm 91 Tz /OPExtFont3 11 Tf (W donates the model parameters. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 142.8 584.1 Tm 98 Tz (We define the observation at time t to be s) Tj 1 0 0 1 366.5 584.1 Tm 67 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 368.649 584.1 Tm 116 Tz /OPExtFont3 11 Tf ( = h\(R) Tj 1 0 0 1 406.1 583.899 Tm 89 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 410.399 583.899 Tm 86 Tz /OPExtFont3 11 Tf (\) + r/) Tj 1 0 0 1 435.35 583.899 Tm 98 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 440.149 583.899 Tm 111 Tz /OPExtFont3 11 Tf (, where is the ) Tj 1 0 0 1 125.5 561.1 Tm 96 Tz (true state of the system. ) Tj 1 0 0 1 262.1 561.1 Tm 98 Tz /OPExtFont6 12 Tf (h\(.\) ) Tj 1 0 0 1 285.6 560.85 Tm 95 Tz /OPExtFont3 11 Tf (is the observation operator which projects the ) Tj 1 0 0 1 125.5 538.049 Tm 94 Tz (state in the model space into observational space. For simplicity, we take ) Tj 1 0 0 1 503.3 537.799 Tm 97 Tz /OPExtFont6 12 Tf (h\(.\) ) Tj 1 0 0 1 125.5 514.75 Tm 95 Tz /OPExtFont3 11 Tf (to be the identity. Unless otherwise stated, it is assumed that all components ) Tj 1 0 0 1 125.75 491.699 Tm 91 Tz (of si) Tj 1 0 0 1 146.4 491.699 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 149.05 491.949 Tm 100 Tz /OPExtFont3 11 Tf ( are observed, i.e. s) Tj 1 0 0 1 253.9 491.5 Tm 90 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 256.8 491.5 Tm 123 Tz /OPExtFont3 11 Tf ( E Ie. The n) Tj 1 0 0 1 333.35 491.25 Tm 98 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 336.5 491.25 Tm 143 Tz /OPExtFont2 10.5 Tf ( E ) Tj 1 0 0 1 355.449 491.5 Tm 102 Tz /OPExtFont3 11 Tf (R) Tj 1 0 0 1 364.1 491.5 Tm 81 Tz /OPExtFont5 11 Tf (th ) Tj 1 0 0 1 371.3 491.5 Tm 92 Tz /OPExtFont3 11 Tf ( represent observational noise ) Tj 1 0 0 1 126 469.149 Tm (\(or ) Tj 1 0 0 1 143.75 468.899 Tm 98 Tz /OPExtFont6 12 Tf (measurement error\); ) Tj 1 0 0 1 246.699 468.699 Tm 90 Tz /OPExtFont3 11 Tf (otherwise stated the rh are taken to be independent and ) Tj 1 0 0 1 125.049 445.649 Tm 91 Tz (identically distributed. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 142.3 422.6 Tm 88 Tz (In the ) Tj 1 0 0 1 174 422.6 Tm 99 Tz /OPExtFont6 12 Tf (Perfect Model Scenario\(PMS\), ) Tj 1 0 0 1 324.949 422.35 Tm 88 Tz /OPExtFont3 11 Tf (we assume i\) the system state and model ) Tj 1 0 0 1 125.049 399.3 Tm 93 Tz (states evolve according to the same structure of the dynamics, i.e. F = ) Tj 1 0 0 1 480.949 399.3 Tm 102 Tz /OPExtFont6 12 Tf (F. ) Tj 1 0 0 1 498 399.3 Tm 94 Tz /OPExtFont3 11 Tf (Note ) Tj 1 0 0 1 125.049 376.3 Tm 95 Tz (that it does not require the system parameters a and the model parameters a ) Tj 1 0 0 1 124.799 353.25 Tm 92 Tz (having the same values. In this chapter, however, we focus on the case that not ) Tj 1 0 0 1 125.049 330.199 Tm 90 Tz (only the model class ) Tj 1 0 0 1 228.699 330.199 Tm 114 Tz /OPExtFont6 12 Tf (F ) Tj 1 0 0 1 241.199 330.199 Tm 92 Tz /OPExtFont3 11 Tf (but also the model parameters a are identical to those of ) Tj 1 0 0 1 124.799 307.149 Tm 95 Tz (the system. ii\) the system state 5) Tj 1 0 0 1 293.75 306.899 Tm 56 Tz /OPExtFont5 11 Tf (-) Tj 1 0 0 1 293.75 306.899 Tm 94 Tz /OPExtFont3 11 Tf (c and the model state x share the same state ) Tj 1 0 0 1 124.549 283.649 Tm 98 Tz (space, i.e. fit = m. iii\) model state and system state correspond exactly and ) Tj 1 0 0 1 124.549 260.35 Tm 92 Tz (iv\) the noise model is independent and identically-distributed and the statistical ) Tj 1 0 0 1 124.549 237.299 Tm 90 Tz (characteristics of the observational noise are known exactly. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 141.599 214.299 Tm 96 Tz (The problem of nowcasting in the PMS will be interpreted as how to form ) Tj 1 0 0 1 124.799 191.25 Tm 94 Tz (an ensemble to estimate the current state R) Tj 1 0 0 1 347.5 191 Tm 68 Tz /OPExtFont5 11 Tf (o ) Tj 1 0 0 1 351.35 191 Tm 95 Tz /OPExtFont3 11 Tf ( given the history of observations ) Tj 1 0 0 1 124.549 168.2 Tm 76 Tz (s) Tj 1 0 0 1 129.349 168.2 Tm 82 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 133.699 168.2 Tm 92 Tz /OPExtFont3 11 Tf (, t = ) Tj 1 0 0 1 158.15 167.7 Tm 105 Tz /OPExtFont6 12 Tf (N +1,..., ) Tj 1 0 0 1 215.05 167.7 Tm 91 Tz /OPExtFont3 11 Tf (0, a perfect model class with perfect parameter values and the ) Tj 1 0 0 1 124.299 144.899 Tm (parameters of the observational noise model. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 317.5 51.799 Tm 74 Tz (21 ) Tj ET EMC endstream endobj 142 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 143 0 obj <> stream 0 ,,b6%xҌu. 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KoÄlJQ*GA/ )ǎ `5$'2\s z[nU) }?jK@4~Z%K67Xؓk S Q2DeGo YH)k){ 7 U yWuApRڂ5)Pz[?EThnږnO_ߊpTG =?x[ (wHrn|oϡ9`ZZ_3=WQۄfvr0aQ?2 S:a Ksa[#v웄9xH|VONHu#\@pgl(s YdiLIǵvbB良V@,r{ W\r/Q8PG;ϛ U IܻUz kM}d]MRM-An:=~|m+M*y!4 <ʂd/9Dr Xh I#XͯEw5u 08͵?KIFl>1Քc [i.e͊3د\o baFf͵eB,|eFzh)%(^vHFK{xF=tݨNtk~pѠxNsgi 0;NI:VsЧBI^?}r_COõ3}+\D[\F\r1Z;S#O:"C!X72+ډO+-6٢es#?[1ͧAF 7L b/Al JA'X4補5?E<뎄-ʫjwtyA -"J.!G3"OOzS[DD1oADbu*e5-.ѡ )%;^|ǚx>;YM{L?AӕPٮ%03HTe-PmAN-B$@{ڸuƲO(n0:@S:QzHPߩAAg2)B6o%o2we@g!r ghrK*-nA 0922Di5!Gvs(ci tO>9|?6,%+'ܣ<0QXJ&z^A lEb`:?ms^ /;FԨqz%%ϯ&c!Rq))q0:r,_!1✍<}1RtlÝ3\h'b 6ٸ vQ/aR 2mJb lʢnn,+[DxT^F: GݝqY^ŭ1EJ+%-Kv";o>!u:4¤ |)Z?Jn"FωG̶X+%pəh'3$5.[=#oHҮªFm>AW6!DM"0b_2HCc]7MtpfG'8ق@N'^ֈ."@s$fH/зA>w6xxWVX0 8 jB]HffgBt~g!Yʧ["姒hI,,+$g2}Oͷ\"y 3O@붉d/ِ)Ew"wczynqdҟP@QWuAUL\}_/>aE/ &zn, [؞&V[iQDڰ +WE-1Q!g+ kf|I 1?JJ|0k[N&MVXN |k1U+13[X4%.e[4m(XD ;HHѤ%់e! 7 zn5+X5@{q^ٓo"DiCUOGm>U L/ $cͦ.^5$GJ (e$%j#&@3%{7 'L=p 8=nZ9U]9_  YEm=*OK}~!$?^[/v^s4(;U7Lܴ/L EEQIg ^ UNױ:5.: n!zuY'Bg{Ѹ&ij*^~bY"Ԫ'atun 8/\ُђIӮ^C!ɰT> endobj 145 0 obj [146 0 R] endobj 146 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 839 0 0 cm /ImagePart_2049 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 366.25 718.75 Tm 102 Tz 3 Tr /OPExtFont3 11 Tf (3.2 Indistinguishable States ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 102 Tz 3 Tr 1 0 0 1 126.7 675.799 Tm 110 Tz /OPExtFont3 15.5 Tf (3.2 Indistinguishable States ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15.5 Tf 110 Tz 3 Tr 1 0 0 1 126.5 641.5 Tm 98 Tz /OPExtFont3 11 Tf (Given a perfect model, an ideal point forecast is possible if we initialise the ) Tj 1 0 0 1 125.75 618.7 Tm 96 Tz (model with the true state of the system. For periodical system, the state can ) Tj 1 0 0 1 125.75 595.399 Tm 97 Tz (be identified uniquely when t co. For chaotic systems, noisy observations ) Tj 1 0 0 1 126 572.35 Tm 94 Tz (prevent us from identifying the true state of the system precisely, nonetheless ) Tj 1 0 0 1 125.75 549.299 Tm (one can find a set of states that are ) Tj 1 0 0 1 310.8 549.299 Tm 97 Tz /OPExtFont6 12 Tf (indistinguishable ) Tj 1 0 0 1 396.25 549.299 Tm 94 Tz /OPExtFont3 11 Tf (from the true state given ) Tj 1 0 0 1 125.75 526.299 Tm 99 Tz (the perfect model and the noise model \(48\). In this section we describe the ) Tj 1 0 0 1 125.75 503.25 Tm 92 Tz (background knowledge of Indistinguishable States Theory following the work of ) Tj 1 0 0 1 125.75 480.199 Tm 96 Tz (Judd and Smith in \(48\). Figure 3.1 \(reproduced from Figure 1 in \(48\)\) shows ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 96 Tz 3 Tr 1 0 0 1 125.299 233.25 Tm (Figure 3.1: Following Judd and Smith \(2001\), Suppose x) Tj 1 0 0 1 417.35 233 Tm 90 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 420.25 233 Tm 100 Tz /OPExtFont3 11 Tf ( is the true state of ) Tj 1 0 0 1 125.049 219.299 Tm 95 Tz (the system and y) Tj 1 0 0 1 212.65 219.299 Tm 89 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 215.5 219.299 Tm 95 Tz /OPExtFont3 11 Tf ( some other state where x) Tj 1 0 0 1 347.75 219.299 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 350.399 219.299 Tm 102 Tz /OPExtFont3 11 Tf ( and y) Tj 1 0 0 1 384.949 219.299 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 387.6 219.299 Tm 136 Tz /OPExtFont2 10 Tf ( E ) Tj 1 0 0 1 404.649 219.299 Tm 106 Tz /OPExtFont3 11 Tf (R) Tj 1 0 0 1 413.3 219.299 Tm 49 Tz (2) Tj 1 0 0 1 418.3 219.299 Tm 95 Tz (. The circles centred ) Tj 1 0 0 1 125.299 205.149 Tm 99 Tz (on x) Tj 1 0 0 1 148.3 205.149 Tm 90 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 151.199 205.149 Tm 103 Tz /OPExtFont3 11 Tf ( and y) Tj 1 0 0 1 186.25 205.149 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 188.9 204.899 Tm 94 Tz /OPExtFont3 11 Tf ( represent the bounded measurement error. When an observation ) Tj 1 0 0 1 125.049 191 Tm 97 Tz (falls in the overlap of the two circles \(e.g., at a\), then the states x) Tj 1 0 0 1 462 191 Tm 90 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 464.899 191 Tm 101 Tz /OPExtFont3 11 Tf ( and y) Tj 1 0 0 1 498.949 191.25 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 501.6 191.25 Tm 97 Tz /OPExtFont3 11 Tf ( are ) Tj 1 0 0 1 125.049 176.85 Tm 96 Tz (indistinguishable given this single observation. If the observation falls in the ) Tj 1 0 0 1 124.799 162.899 Tm 97 Tz (region about x) Tj 1 0 0 1 199.9 162.899 Tm 82 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 204.25 162.899 Tm 96 Tz /OPExtFont3 11 Tf (, but outside the overlap region \(e.g., at ) Tj 1 0 0 1 413.5 162.899 Tm 25 Tz /OPExtFont4 11 Tf (/) Tj 1 0 0 1 414.949 162.899 Tm 95 Tz /OPExtFont4 11.5 Tf (3\), ) Tj 1 0 0 1 433.899 162.899 Tm 93 Tz /OPExtFont3 11 Tf (then on the basis ) Tj 1 0 0 1 125.049 149 Tm 97 Tz (of this observation one can reject y) Tj 1 0 0 1 308.649 149 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 311.3 149 Tm 101 Tz /OPExtFont3 11 Tf ( being the true state, i.e., x) Tj 1 0 0 1 459.1 149 Tm 74 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 461.5 149 Tm 108 Tz /OPExtFont3 11 Tf ( and y) Tj 1 0 0 1 498 149 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 500.649 149 Tm 102 Tz /OPExtFont3 11 Tf ( are ) Tj 1 0 0 1 124.799 135.1 Tm 90 Tz (distinguishable given the observation. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 318.25 51.549 Tm 73 Tz (22 ) Tj ET EMC endstream endobj 147 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 148 0 obj <> stream 0 ,, b7 [xPa x̜!6T渚^e;~Lc5y,jT̷UIx܏ʼn # (@IgmuOl &0LP^}F(oE>˟X[hL"^- L5Ɉ<3,Thc1׫y]=F'쬮j7_1?)BrXLJqV:uq['vuD~<0L ٺ.ʵvfcbEM3Z;cZRD{J-ȺF9Bߖa1($vV25ujjbv+E!}rs p,ZCCmۗ9v*V#oO3j_/\NɎ5ȱg&vB*Ill6XTr ="av Z.G -p9dh"^D¾]kR>7%<LטE.$WH=DaHrO#xE9KFgP똈\).pg` 2dZ'{Ҋ3}!@?72m/q :-A \ӽ4(D RA@>z" cg@ _BE iE 9g2ltء{~"hfu+&Ѵ2%Z,nKE)YnDnJ\AƛZ!Ŝh}w}c֠ACʿ ,˃]Yɩ'HTzJ?|$#e1Y^֝P7a D_{3ujDUt<1M*{yㅛlN߯H>Ɖ lhhfs܌.\gc}j))U@oBUE#Nk]4BPQc| B2j K[C6mEn@ꡒBQ-# [m9϶0yܵ<&t2PO۟blbȻ tRKA ?yuQWaqvRu=գ4UX>2RX; )_:/ rI%\?HJ|_7X(<ȐeGhnNEWp "nAih1aD6^|9czD]aCng% } yS FX"wzmG7Ďg79uLA_SWw8ͷe|>¶/4BϲMŴރݸڹЈ,)k578N& <@|/' &| +jgD y~?y@/2 |_x*YΆ2H`o}a~UB}3zPD(NF̰ 8ԷFD d+jOg-ॐb8N6viv- éMV{+5tVexDD|+) J跨zrdYUGYظBTl_VJL(5Kc]t½fni!f BFBBc6J6: \9֒vY;ep4V{3)`F `tAEJZHRYkƂD˜{1  oj/[r6(zi8 "g6m:{Ox{^%47_QuJpBJFw۫K%`nh2%/Lz:] i9^$WR;J *;R_ݚtx-)R; W7Eun\?xm=4MQ>։j K7e!QQf Խ%w>}js ٤e]DQ u-HEpxz鍧i> in(qO%>0ף8={cvMęڮeo)"}\CKt=MɣUTw[$/4QԒ#h8>E~H s#ޅcjfH0,֨vrs# > ,p94o˚wiR\ -Rw[+@u^thm$1'Q:% 1t-\-fA1evfUBEOEXUMإL3ړB^`AUc= ͣ7bg-~v5:\5㛄+-\'##9   r>"ĩĸ-i9ǖڲ4#1:`,Ͳѐ<{C(1L* >A}Xiqr v9<-L95Î0cRr0LK{Q922]yk@Ԣ{ G׸Q9RX8 >]͖nJ{7A p'jCni$4@t\lB)?zr؈{'Ӗq뱝1Ds[GH-KM1ӗV.\TE̪VQS NIE/CXs8cqE Q0ʜoWqSbF Ι)>($^,Ä :' YdIN eLy|C&B_P^eV^3 @a 잚F@PnDEۍ+G(Ctwkْuqː<쩅6lu̧_l'nNQoqRmaR҇I`!06>kŒ|Xz rмOᦝ-2eypuFRPΛzRib]~eI|p 浿Γ.p%ZD(ݡ u1)+G_41 zk0ǩӸ #'2nMXLZº)Q wbl*oճΊ{[G|y֤|^R9^|dId Ǡøk.8gASow3~?U+ԯϤ.(S=CO.]4̫/annWQk RHd↮w7r9w)A,v0g.ݲG4 ؿT'Cxq֧,Oz%>%IŸʱ᳧:/=tt۴t<áч틶!麋?Ji'Q![ ][Ig~I BpPq="-OX%r2%\ #r #u:f3) Lo8kLWJ@Brb `ySo؜zHK5 /dR6v6FIRpk@f (24oM7lGDj!=ѵ5*f3qsD/"/vs>-0a)yQc~p`rD咞*֞;57֞޴yisaFnmDr.g0KW$Z$ҧ1)YI) 2(tzuh$NSa.PZ&>zL.s0s})? 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Vvƀ9 V A>J}6>fH$·iĐ- BEivzUHlڬ"( 'ﻳ,w Dܭr?vdLvnW` oY^]hvVr B'΄7:n u~BFwD }5=>TU?JD̘fAVRT+ ),<0vioՑO-h<}[t>uaeL.\7# *9FC:I@Gm?S-XVm oz/.j̎6k.$F=<ZLK>gC_&y"y4|]FVa3!NAr]!PIR:jאk 0jpܐH[ǵT 4+"/=OL’0!YstVrxo׍PC>ܸSRc /*|! |ske{g#J$[8g0RQL"ݔݻ!IWC^j^c)W{ؒ!-U:N50"zؒ1Y)b`ǼsqH/qB\ۢ>Oe t4bd`'#G 2BD/im9F9+2|b(7ާvV,s}8/6m|vjv ma!\,@4i# @4^B:`0m`n+hNoBq=:TR˓ܺ [.O+DԽ^!< j#?pe UWwp`"[S X-iC뀁xT\As{lPp)5@~_R|F '61caP}}bUQRPCq٠|Pμ"ը W/|_Bc86D#KxޝgEhZ_nL  -y|110@Hn(DsUU"*x-;W8.Xb(uuY9(%VT]ctUR 1PʫNZfS)(BrFs||J:WGr2ف%H;i:[|\^AANC*sG+ZI)XO'1c@ ,{&t4nJxZr"3xMK8XECԦT[ UR*vJ2]③-இ?z!J1C T?VX':8^>_,Vi|HgIk ܉tZlPz(,u`wwt஭cᲰD"fu54IYS4/bƐ>65aXcEIKoIj!Hpۢ GSI*v}elDn 7 q$=-cJxkbtJg ~ĵ!L\L],r["B\C o8\*(,Q'-!v - lJPykkк5Q=ĺ;xJSR\dY(kEjˈ_Ga$_2yʋ&xfg-UeJLp?@tcD簨ZZu⇬Gvs1Һm}xJ 51%#Xq6I&g ~*'_~L8~M\dQ oD+*竐5F2Ka c~BrGkH4^ef_^]=͆ܤbAr'$<*~T@32{щ|k5cI!iSgC˒}"v7I>tۭ'/+g16YZϳ8璼zFfzm8y(gpH>q y͚X9ᐽLT.0# "] />KR.kmr7nV{"NEtg~Q*ZCQYluh[mzuCB/+tMM 4./0AH_vq)>ŏp`%қUq+h=; O6y+fQ,9*%h#0Y#v!cl,L|IsnngLq賾1bU >~XQѸX[O͸Oz\6!YwnV;/.\.H1u4]} ~9+I"554HKTtžNX%ykW)7T ɑ'yCPlT^iuJw{z\5s*5 G}q2RWXaN٨N ظntםl *2,1绮Q럐Iû(W༬[Pſ\eRmR*l:J'l;4 *z4X),ucqLͲ,Pmk*+i%=CrhrS90;.(s%xs~Ųsj_,}e½gc罗׼1Bǟ^6EM.MRN ?fUj!w_BG$p|S&yt\ ˠlMt+&oZ1?XW g9;a<7|:r#yЮQ4=VB=BMk;d#aDsAd$gaJ0Hٻ\ͫd!,YE MWQ/•XBEoB`ź ;Sj/{G+~g{ܵyB=eSh F cYuMWz(wޒ0*8z3Q˦f,VRl?ʨݫx;QzZou!q(ENb`lձ?cN7Un[tY`_ĩܼ{52u˄]=G}By`X%dAwj)1cԩ9PR?~'hTyV/ns 8I{vz_0T 8JI1?x{6 3/Oy+@.ƈH9>X:$!B}$?pѐrH BQ"6/M'-C3^zw [\]Sl[j.VJ'oK=Nr%Pos P"ӿėf54SeGIͷ7+ެO6j$7ơSҿ_8}aiyY1'/x2SnL^%4FHIƐB\tn~ZD坔~J$Z'P:"ĺq9-]uGq›0JJBJ%JA@r9 ^VT5EJ7>6,h0⬌ F.F7C6a*mT: ,c'?D!@. Z6ک& |+؁ QPNlybxsU1r$P*Y3>%,Tx}Fͧy^"z]-~ ѿ$N? R p~O>cU:A \ں Cڸ(Ԍ"=@΁=Z_<7ceU'<`ɫg¢"b^-[!;uvd}Tq?@哖Q5|"9Ԓn;6`{mZ7qn_^V!铥^5kY<۩h흫xhR1@09Ju[,z_"YOdR/0ʯ1>T̹ThIҁ;{' ;P\֬jP# :6'11yxIvMRem>5 w 9ff%\x6xΦyi ڬfaZ2 v*6FpB6G9.`}Kš@Y-86R2.Bi(L!B` :oG$AEwl,H :}3}X,IfAŐ!*<;.7 endstream endobj 149 0 obj <> endobj 150 0 obj [151 0 R] endobj 151 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 839 0 0 cm /ImagePart_2050 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 364.8 720.7 Tm 102 Tz 3 Tr /OPExtFont3 11 Tf (3.2 Indistinguishable States ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 102 Tz 3 Tr 1 0 0 1 124.799 677.5 Tm 88 Tz (that based on one single observation s) Tj 1 0 0 1 308.649 677.7 Tm 57 Tz (t ) Tj 1 0 0 1 311.05 677.7 Tm 94 Tz ( of state x) Tj 1 0 0 1 360.699 677.7 Tm 74 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 365.3 677.7 Tm 89 Tz /OPExtFont3 11 Tf (, there exist many states y) Tj 1 0 0 1 492.25 677.95 Tm 74 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 494.649 677.95 Tm 87 Tz /OPExtFont3 11 Tf ( each ) Tj 1 0 0 1 125.049 654.899 Tm 92 Tz (of which is indistinguishable from x) Tj 1 0 0 1 302.399 654.899 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 305.05 654.899 Tm 92 Tz /OPExtFont3 11 Tf ( because of the observational uncertainty if ) Tj 1 0 0 1 125.049 631.899 Tm 91 Tz (the overlap region in Figure 3.1 covers the observation s) Tj 1 0 0 1 403.449 632.1 Tm 74 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 407.3 632.1 Tm 95 Tz /OPExtFont3 11 Tf (. Notice that given the ) Tj 1 0 0 1 124.799 608.85 Tm 93 Tz (bounded noise model, x) Tj 1 0 0 1 244.099 608.85 Tm 74 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 246.5 608.85 Tm 101 Tz /OPExtFont3 11 Tf ( and y) Tj 1 0 0 1 280.8 608.85 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 283.449 608.85 Tm 92 Tz /OPExtFont3 11 Tf ( are indistinguishable as there exist the overlap ) Tj 1 0 0 1 124.799 585.799 Tm 93 Tz (region in Figure 3.1. However, a particular realization of observation, e.g. /3 in ) Tj 1 0 0 1 125.049 562.5 Tm 91 Tz (Figure 3.1, could distinguish x) Tj 1 0 0 1 275.75 562.5 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 278.399 562.5 Tm 99 Tz /OPExtFont3 11 Tf ( from y) Tj 1 0 0 1 316.1 562.5 Tm 89 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 320.399 562.5 Tm 34 Tz /OPExtFont3 11 Tf (. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 34 Tz 3 Tr 1 0 0 1 141.849 539.7 Tm 93 Tz (We describe the statistical background of Indistinguishable State Theory in ) Tj 1 0 0 1 124.799 516.7 Tm (the following. For the convenience of explanation, x) Tj 1 0 0 1 384 516.7 Tm 90 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 388.55 516.7 Tm 92 Tz /OPExtFont3 11 Tf (, y) Tj 1 0 0 1 400.3 516.7 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 402.949 516.7 Tm 96 Tz /OPExtFont3 11 Tf ( and s) Tj 1 0 0 1 435.35 516.7 Tm 90 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 438.25 516.7 Tm 95 Tz /OPExtFont3 11 Tf ( are scalars. Let ) Tj 1 0 0 1 124.799 493.649 Tm 91 Tz (the probability density function of the observational noise be ) Tj 1 0 0 1 425.3 493.649 Tm 99 Tz /OPExtFont6 11.5 Tf (p\(\), ) Tj 1 0 0 1 449.75 493.899 Tm 90 Tz /OPExtFont3 11 Tf (the joint prob-) Tj 1 0 0 1 125.049 470.35 Tm 93 Tz (ability density of ) Tj 1 0 0 1 211.699 470.35 Tm 112 Tz /OPExtFont6 11.5 Tf (x) Tj 1 0 0 1 217.9 470.35 Tm 92 Tz /OPExtFont8 11.5 Tf (t ) Tj 1 0 0 1 220.55 470.35 Tm 95 Tz /OPExtFont3 11 Tf ( and y) Tj 1 0 0 1 252.5 470.35 Tm 89 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 255.349 470.6 Tm 91 Tz /OPExtFont3 11 Tf ( being indistinguishable is then defined by ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 252 395.699 Tm 130 Tz /OPExtFont16 34.5 Tf (f) Tj 1 0 0 1 265.449 402.899 Tm 99 Tz /OPExtFont6 11.5 Tf (P\(st xt\)P\(st y) Tj 1 0 0 1 351.35 405.1 Tm 74 Tz /OPExtFont8 11.5 Tf (t) Tj 1 0 0 1 354.949 405.3 Tm 102 Tz /OPExtFont6 11.5 Tf (\)ds) Tj 1 0 0 1 369.85 405.3 Tm 83 Tz /OPExtFont8 11.5 Tf (t) Tj 1 0 0 1 374.149 405.55 Tm 50 Tz /OPExtFont6 11.5 Tf (. ) Tj 1 0 0 1 375.6 405.55 Tm 2000 Tz (\t) Tj 1 0 0 1 498.699 405.55 Tm 87 Tz /OPExtFont3 11 Tf (\(3.1\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 124.299 363.1 Tm 91 Tz (This joint density function depends only on the difference between xt and y) Tj 1 0 0 1 495.6 363.1 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 498.25 363.3 Tm 91 Tz /OPExtFont3 11 Tf ( and ) Tj 1 0 0 1 124.549 339.8 Tm (the distribution of the measurement error s) Tj 1 0 0 1 339.6 339.8 Tm 74 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 342 339.8 Tm 79 Tz /OPExtFont3 11 Tf ( ) Tj 1 0 0 1 357.1 340.05 Tm 108 Tz /OPExtFont6 11.5 Tf (x) Tj 1 0 0 1 363.35 340.05 Tm 100 Tz /OPExtFont8 11.5 Tf (t) Tj 1 0 0 1 367.899 340.05 Tm 41 Tz /OPExtFont6 11.5 Tf (, ) Tj 1 0 0 1 373.899 340.05 Tm 84 Tz /OPExtFont3 11 Tf (since ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 84 Tz 3 Tr 1 0 0 1 143.05 264.45 Tm 134 Tz /OPExtFont16 34.5 Tf (f) Tj 1 0 0 1 156.25 272.35 Tm 97 Tz /OPExtFont3 11 Tf (gst ) Tj 1 0 0 1 189.599 272.1 Tm 129 Tz /OPExtFont8 10.5 Tf (xt\)P\(st ) Tj 1 0 0 1 225.849 272.6 Tm 90 Tz /OPExtFont3 11 Tf ( Yt\)dst = ) Tj 1 0 0 1 276.25 273.799 Tm 122 Tz (\t) Tj 1 0 0 1 294 273.1 Tm 94 Tz /OPExtFont6 11.5 Tf (P\(st ) Tj 1 0 0 1 315.85 273.1 Tm 65 Tz /OPExtFont3 11 Tf ( ) Tj 1 0 0 1 326.649 271.899 Tm 105 Tz /OPExtFont6 11.5 Tf (xt\)P\(st ) Tj 1 0 0 1 373.899 272.1 Tm 90 Tz /OPExtFont3 11 Tf (xt + xt ) Tj 1 0 0 1 421.699 272.85 Tm 102 Tz /OPExtFont6 11.5 Tf (Yt\)d\(st ) Tj 1 0 0 1 468.25 274.5 Tm 115 Tz /OPExtFont3 11 Tf (xt\), \(3.2\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 115 Tz 3 Tr 1 0 0 1 124.549 231.799 Tm 96 Tz (The ) Tj 1 0 0 1 147.099 232.049 Tm 103 Tz /OPExtFont6 11.5 Tf (indistinguishability ) Tj 1 0 0 1 241.699 232.049 Tm 88 Tz /OPExtFont3 11 Tf (of two states ) Tj 1 0 0 1 304.8 232.049 Tm 107 Tz /OPExtFont6 11.5 Tf (x) Tj 1 0 0 1 311.05 232.049 Tm 83 Tz /OPExtFont8 11.5 Tf (t ) Tj 1 0 0 1 313.449 232.049 Tm 89 Tz /OPExtFont3 11 Tf ( and y) Tj 1 0 0 1 343.199 232.049 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 345.85 232.299 Tm 88 Tz /OPExtFont3 11 Tf ( can be quantified by the normalised ) Tj 1 0 0 1 124.299 208.5 Tm 90 Tz (density function ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 244.3 151.149 Tm 249 Tz /OPExtFont8 11.5 Tf (f ) Tj 1 0 0 1 250.55 151.149 Tm 104 Tz /OPExtFont6 11.5 Tf ( p\(s) Tj 1 0 0 1 268.8 151.149 Tm 92 Tz /OPExtFont8 11.5 Tf (t ) Tj 1 0 0 1 271.449 151.149 Tm 98 Tz /OPExtFont6 11.5 Tf ( x) Tj 1 0 0 1 292.8 150.899 Tm 92 Tz /OPExtFont8 11.5 Tf (t) Tj 1 0 0 1 297.1 150.899 Tm 102 Tz /OPExtFont6 11.5 Tf (\)p\(s) Tj 1 0 0 1 316.1 150.899 Tm 90 Tz /OPExtFont8 11.5 Tf (t ) Tj 1 0 0 1 318.699 150.899 Tm 99 Tz /OPExtFont6 11.5 Tf ( x) Tj 1 0 0 1 340.1 151.399 Tm 83 Tz /OPExtFont8 11.5 Tf (t ) Tj 1 0 0 1 342.5 151.399 Tm 108 Tz /OPExtFont6 11.5 Tf ( +) Tj 1 0 0 1 381.35 150.2 Tm 133 Tz /OPExtFont8 11.5 Tf (yt\)d\(st ) Tj 1 0 0 1 428.149 149.7 Tm 106 Tz /OPExtFont6 11.5 Tf (xt\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 11.5 Tf 106 Tz 3 Tr 1 0 0 1 180.25 143 Tm 104 Tz (q\(x) Tj 1 0 0 1 196.3 143 Tm 92 Tz /OPExtFont8 11.5 Tf (t ) Tj 1 0 0 1 198.949 143 Tm 96 Tz /OPExtFont6 11.5 Tf ( y) Tj 1 0 0 1 219.349 143.25 Tm 92 Tz /OPExtFont8 11.5 Tf (t) Tj 1 0 0 1 223.199 143 Tm 90 Tz /OPExtFont6 11.5 Tf (\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 11.5 Tf 90 Tz 3 Tr 1 0 0 1 267.6 134.1 Tm 96 Tz /OPExtFont3 10.5 Tf (f p\(st ) Tj 1 0 0 1 310.1 134.35 Tm 114 Tz /OPExtFont6 11 Tf (xt\)p\(st ) Tj 1 0 0 1 346.1 134.35 Tm 68 Tz /OPExtFont3 10.5 Tf ( ) Tj 1 0 0 1 357.6 134.35 Tm 129 Tz /OPExtFont8 11 Tf (xt\)d\(st ) Tj 1 0 0 1 393.85 134.35 Tm 66 Tz /OPExtFont8 10.5 Tf ( ) Tj 1 0 0 1 404.899 134.35 Tm 110 Tz /OPExtFont8 11 Tf (xt\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont8 11 Tf 110 Tz 3 Tr 1 0 0 1 498.949 143.25 Tm 88 Tz /OPExtFont3 11 Tf (\(3.3\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 317.75 53.7 Tm 75 Tz (23 ) Tj ET EMC endstream endobj 152 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 153 0 obj <> stream 0 ,,R}}b7 \Rp-!?TO3K3Y Or:N;u9 a1y˶=^R=to`,͈,?.%K\=qBtsZdA?\+"0d|dR9EfQyx((NC,EjuVc=5F18W *e^NII|+6 c=dTK_>_ c?U͒r0j36܊NKv#uT\W<4!1bVOFa*W؍__^krC.yiqZՈaV1],Ӷ/8 ]*x@4To{d{j2ܸO]ԓfcd `( 12tS_qb KL[iQ2h [x ]q DlmoU8|wFVvaǔOf|$|><7Kw2a/Km74/AQQ`DZ _&vwIC@ nr$En0EW^<鴂w4f CѦ"+J^غ;T*-(Ou</uFGc`c;% RFIGbLzT7fIb0 vaGfN;Mpʴ7p[nU0J6{|M5ߪJ0-p-qfyvA"4Zu*5"?H/‡&Xt@$‹rd0eLl*5ac<:NPvtm:ujE>bVz/np5[*2.m`Hr3iC }6[2GB{quTIEj6]/MbaM?7 l%6iE) tR*ʭ68֫Mc&F cu,gϓiD /А> mB 3N[Q\U/B,dP`Ip/'ЀE }@%%-ΐ4e7ƗZ:'ߒF#v8סkq} I46$8 R95TYq$-"#0V2 0vA`jeJ%=Ԟ{~m& H 5STqƌ7vĩkU%Սog;[ B\ouDpW_ EG+@:-/(͝Jo[ud6 W; v*td:opW9 p/-jB(@# p1F({ZJBVt&-IH; %E)T z q:TrֱL`.tvi6974,w s?h4}o4˕BAcf@Qo y|>*E Jh7=7<G9%?|CEtâr;$ Վ6b OY.$|g_4&Ńtyc@;))I ļ৬ 9y-_7ӯ>wlC|=&p*+WF>#Ԁ~QfezjaUۅN\s~QcߩVv?"ݥt?]=7r ta;6z#)Vnm+:cTqp?Xψ@[ ^낹/C > 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U.ΨUnUT&J\6*wQG3Fݵ{H0ħ/-br! 7@?1/.wjDSLh(wս?'AX\WSn4)CHrHi+yػvUMSi"~NUj =~>2}N2pm WF o0@;QO O~l '4Bdx.ıWS!k0NL W%\ T6DʉõiyN#bZÒuI2= i3Y˚b\ {N*,:zw8e~ANu%*2B (.$씒*gWp&LRt$*Wh޲U !/½ 805;2k8*zMn<|5| lCUqIMOF8?[ܠC<8M^בwQbuƼ{p>W5$ޫ )=Y s0A٩9Dlanp2EUena.ye&&dk%T)Kkb#\'>m>1~;"/eƭ8xXeQtҭgbd"6yAe]'ى"x I7+ƑѫXX\[LXv$#^܍QCIN8o6Ɠխ̟zFؼD^%z6 mwbp=oBrlptē8lu_ly#ZV(*o2c~C:ϰۛpr2 ĄlVV sޟo|iwj#ϤŒ8Y_v[;w0'/PP">50"&4tʴӗB85U$,֐n<o/? 7wC,}cbC2Zv=c cb@NA]x}ݚ@&;s2θa%8[?Qgr߹Q ;1Ogm=k x6=nG`bls%˧ڋEs %fL1hn>f6 ]ޠ6r ')<@3ؕـGHXU1"`m $7bgVbp}m!~aqFyuGG2SRN+X;'{uEi ̯T;}x1qILc76G\k.UH(ti19Nr 0~<}5:NSoS'}1h#1(U"#-A$0MgG[Do2CD7?z/ V`=B57~-p2y4L)$^!y_Rtbz /q3wx2jd,HX7Q+6QuE@@\Gyr]A' GqgnYTSW@} Re|sZla=_e$8D!w8m% g|=Q=Eɸ1n߰K]{3b1\ũ'WUOfQ.(`^?|>宮.pA#֙hȱ4fTN jK= wch^V⅒8Nx"荎R*:vКI w0H8>`3|22-\*{ujAMZ#paf1^GuYr #[闛f{cKj7fX/aoRJp"7HG{vHku־(^ӹ}蝩Y]w3!gϯd`6mK/وPI;[g>XtT0qCL/K y9VId~R|_\hت ^T XaY=((FhBzՄ nHh|q8ܡ 5[Okn=p;iLZE߇2Dk(7YBg=XܧK$K ' ֧[s{Vchd$ ^ZD85IO;bfD&73k*w?  /PuS|ݛ){EnjlԽ`mz7sbXaaQjtD3cUW2|!8Uj+եkr6*ug}4|z~xSncx&5.ϯk%m RݿJx xfJeMUVnlw0I. l+GJ6M$#9}?lKԘ\pf14F]d]qNeЎ _${9.I)Z%~")8^gmt}5z7K]0v9h.tӾ%s8t恋mv8=΁49PB(Ys+ڣ-G:e3Mw&:TܜV8kISvЭE8^>ejuͦ%efa>仳&U4:Ұ??0/ܳܝҐ9ݖ°EǶo+Єp0& ;h&:Բ"4?tu=rjk$^N& ~ NY=X1{-KuG3rC)[=~w""~~ j4BU_ uK)gvpU_-J_(<"*``ddғe`A*qo`(.,Ji"9lR;~#W>6\$ -1`wG f9t"  uܠ\ډt\ݰMuQknmk!͝cBzE NҬDjyq!G$jTNͧY Ռu[a A)BǴ^G |Y<=W[6ģ XǯNS8LeZ%%/7Ԧ.ֈPJuB;aEU5_僻njux@xD&fA DZχ;3#N1O;n$${w[KqZ^0 W\*izĽ㽍f*X?ſc-[de#@VJ[+4p1PDff.ոUis dlȳ,xmyuăS|5gswu} b :E3ز?3Vp=OôZxAbWx?l +KOx$OQٶhإMo&Ǽ2]*8v~߇|*f-m Q~,FHBj}.c~[Q e z,G4HR[ =$%ۜ}' w,cTX&k a4HLe X56ӕHidq}!ѾC1JC[=SX_V;Б> endobj 155 0 obj [156 0 R] endobj 156 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 610 0 0 840 0 0 cm /ImagePart_2051 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 360.699 720.25 Tm 102 Tz 3 Tr /OPExtFont3 11 Tf (3.2 Indistinguishable States ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 102 Tz 3 Tr 1 0 0 1 120.7 677.299 Tm 92 Tz (This function is called ) Tj 1 0 0 1 235.699 677.299 Tm 80 Tz /OPExtFont6 12 Tf (q ) Tj 1 0 0 1 246.25 677.299 Tm 83 Tz /OPExtFont4 11 Tf (density. ) Tj 1 0 0 1 289.199 677.299 Tm 92 Tz /OPExtFont3 11 Tf (The normalisation implies the constraint that ) Tj 1 0 0 1 120.5 654.5 Tm 93 Tz (when x) Tj 1 0 0 1 156.949 654.5 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 159.599 654.5 Tm 121 Tz /OPExtFont3 11 Tf ( = y) Tj 1 0 0 1 183.349 654.5 Tm 82 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 187.699 654.5 Tm 93 Tz /OPExtFont3 11 Tf (, the density function reaches its maximum value of 1: in no case ) Tj 1 0 0 1 120.7 631.45 Tm 95 Tz (that x) Tj 1 0 0 1 151.449 631.45 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 154.099 631.45 Tm 94 Tz /OPExtFont3 11 Tf ( is distinguishable from itself. If ) Tj 1 0 0 1 318.5 631.2 Tm 101 Tz /OPExtFont6 12 Tf (q\(x) Tj 1 0 0 1 334.3 631.2 Tm 88 Tz /OPExtFont8 12 Tf (t ) Tj 1 0 0 1 336.949 631.2 Tm 296 Tz /OPExtFont4 3 Tf ( ) Tj 1 0 0 1 352.3 631.2 Tm 99 Tz /OPExtFont6 12 Tf (y) Tj 1 0 0 1 357.6 631.2 Tm 79 Tz /OPExtFont8 12 Tf (t) Tj 1 0 0 1 361.699 631.2 Tm 101 Tz /OPExtFont6 12 Tf (\) = ) Tj 1 0 0 1 381.6 631.45 Tm 90 Tz /OPExtFont3 11 Tf (0, then the states ) Tj 1 0 0 1 471.6 631.2 Tm 96 Tz /OPExtFont4 11 Tf (x) Tj 1 0 0 1 477.85 630.95 Tm /OPExtFont8 11 Tf (t ) Tj 1 0 0 1 480.5 630.5 Tm /OPExtFont3 11 Tf ( and y) Tj 1 0 0 1 512.899 631.45 Tm 74 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 515.299 608.399 Tm 1 Tz /OPExtFont3 11 Tf ( are distinguishable with probability one, any particular realization of observation ) Tj 1 0 0 1 120.5 585.35 Tm 93 Tz (will only be consistent with either ) Tj 1 0 0 1 293.3 585.35 Tm 92 Tz /OPExtFont4 11 Tf (x) Tj 1 0 0 1 299.3 585.35 Tm 103 Tz /OPExtFont8 11 Tf (t ) Tj 1 0 0 1 302.149 585.35 Tm 101 Tz /OPExtFont3 11 Tf ( or y) Tj 1 0 0 1 326.149 585.35 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 328.8 585.35 Tm 97 Tz /OPExtFont3 11 Tf ( but not both. A value ) Tj 1 0 0 1 448.1 585.35 Tm 98 Tz /OPExtFont6 12 Tf (q\(x) Tj 1 0 0 1 463.899 585.35 Tm 88 Tz /OPExtFont8 12 Tf (t ) Tj 1 0 0 1 466.55 585.35 Tm 289 Tz /OPExtFont4 3 Tf ( ) Tj 1 0 0 1 481.899 585.35 Tm 99 Tz /OPExtFont6 12 Tf (y) Tj 1 0 0 1 486.949 585.35 Tm 88 Tz /OPExtFont8 12 Tf (t) Tj 1 0 0 1 491.05 585.35 Tm 100 Tz /OPExtFont6 12 Tf (\) > ) Tj 1 0 0 1 510.949 585.35 Tm 66 Tz /OPExtFont3 11 Tf (0 ) Tj 1 0 0 1 120.5 562.299 Tm 92 Tz (indicates that x) Tj 1 0 0 1 198.25 562.299 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 200.9 562.299 Tm 97 Tz /OPExtFont3 11 Tf ( and y) Tj 1 0 0 1 233.5 562.299 Tm 74 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 235.9 562.299 Tm 91 Tz /OPExtFont3 11 Tf ( are indistinguishable given the noise model. One should ) Tj 1 0 0 1 120.5 539.299 Tm 93 Tz (notice that there might be some particular observations that can distinguish x) Tj 1 0 0 1 512.649 539.299 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 515.299 516.25 Tm 2 Tz /OPExtFont3 11 Tf ( from y) Tj 1 0 0 1 153.349 516.25 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 156 516.25 Tm 95 Tz /OPExtFont3 11 Tf ( for example, in the bounded noise case, if \)3 in Figure 3.1 is observed ) Tj 1 0 0 1 120.25 493.199 Tm 104 Tz (x) Tj 1 0 0 1 126.7 493.199 Tm 90 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 129.599 493.199 Tm 106 Tz /OPExtFont3 11 Tf ( and y) Tj 1 0 0 1 165.099 493.199 Tm 74 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 167.5 493.199 Tm 96 Tz /OPExtFont3 11 Tf ( are distinguishable. Therefore particular realizations will give extra ) Tj 1 0 0 1 120 470.149 Tm 92 Tz (information to distinguish x) Tj 1 0 0 1 258.949 470.149 Tm 74 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 261.35 470.149 Tm 96 Tz /OPExtFont3 11 Tf ( and y) Tj 1 0 0 1 293.5 470.149 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 296.149 470.149 Tm 91 Tz /OPExtFont3 11 Tf ( besides the ) Tj 1 0 0 1 359.05 470.149 Tm 80 Tz /OPExtFont6 12 Tf (q ) Tj 1 0 0 1 368.399 470.149 Tm 84 Tz /OPExtFont3 11 Tf (density. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 84 Tz 3 Tr 1 0 0 1 137.5 446.649 Tm 82 Tz (Such ) Tj 1 0 0 1 164.9 446.899 Tm 75 Tz /OPExtFont6 12 Tf (q ) Tj 1 0 0 1 174 446.899 Tm 91 Tz /OPExtFont3 11 Tf (density can be generalised to a sequence of observations. Any system ) Tj 1 0 0 1 120.25 423.6 Tm 93 Tz (state x) Tj 1 0 0 1 154.3 423.85 Tm 60 Tz /OPExtFont5 11 Tf (o ) Tj 1 0 0 1 157.699 423.85 Tm 93 Tz /OPExtFont3 11 Tf ( defines a trajectory \(we will often drop the subscript for x) Tj 1 0 0 1 449.75 423.85 Tm 65 Tz /OPExtFont5 11 Tf (0 ) Tj 1 0 0 1 453.1 424.1 Tm 92 Tz /OPExtFont3 11 Tf ( afterwards\), ) Tj 1 0 0 1 120 401.05 Tm 93 Tz (that goes infinite past and terminates at ) Tj 1 0 0 1 327.1 401.05 Tm 88 Tz /OPExtFont4 11 Tf (x. ) Tj 1 0 0 1 343.449 401.05 Tm 92 Tz /OPExtFont3 11 Tf (Given a time series of observations ) Tj 1 0 0 1 120.25 378 Tm 75 Tz (s) Tj 1 0 0 1 125.299 378 Tm 81 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 129.349 378 Tm 41 Tz /OPExtFont3 11 Tf (, ) Tj 1 0 0 1 134.15 378 Tm 110 Tz /OPExtFont4 12 Tf (t = ) Tj 1 0 0 1 155.75 378 Tm 90 Tz /OPExtFont3 11 Tf (0, 1, 2, ..., it follows from the independence of the measurement error ) Tj 1 0 0 1 120 354.699 Tm (that by considering all the states on the trajectory, the indistinguishability of two ) Tj 1 0 0 1 120 331.449 Tm (state ) Tj 1 0 0 1 148.099 331.449 Tm 92 Tz /OPExtFont4 11 Tf (x ) Tj 1 0 0 1 158.15 331.449 Tm 93 Tz /OPExtFont3 11 Tf (and y is then given by the product ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 250.55 289.2 Tm 103 Tz /OPExtFont4 11 Tf (Q\(x, ) Tj 1 0 0 1 275.3 288.5 Tm 131 Tz /OPExtFont3 11 Tf (y\) = fl) Tj 1 0 0 1 317.5 286.299 Tm 121 Tz /OPExtFont8 10.5 Tf (q\(xt ) Tj 1 0 0 1 340.1 287.049 Tm 237 Tz /OPExtFont5 10 Tf (- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 10 Tf 237 Tz 3 Tr 1 0 0 1 301.199 276 Tm 90 Tz (t<0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 10 Tf 90 Tz 3 Tr 1 0 0 1 494.399 289.2 Tm 88 Tz /OPExtFont3 11 Tf (\(3.4\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 120 244.549 Tm 92 Tz (Similar to the single observation case, If ) Tj 1 0 0 1 323.05 244.549 Tm 103 Tz /OPExtFont4 11 Tf (Q\(x, ) Tj 1 0 0 1 347.5 244.549 Tm 95 Tz /OPExtFont3 11 Tf (y\) > 0, then the trajectory ending ) Tj 1 0 0 1 120 221.299 Tm 88 Tz (at ) Tj 1 0 0 1 132.699 221.299 Tm 97 Tz /OPExtFont4 11 Tf (x ) Tj 1 0 0 1 142.55 221.299 Tm 90 Tz /OPExtFont3 11 Tf (and the trajectory ending at y are not distinguishable, given the noise model. ) Tj 1 0 0 1 120 198 Tm 92 Tz (Therefore the set of indistinguishable states of x is defined as ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 234.699 155.75 Tm 91 Tz (111\(x\) = {y ) Tj 1 0 0 1 289.899 155.75 Tm 95 Tz /OPExtFont2 10 Tf (E ) Tj 1 0 0 1 295.699 155.75 Tm 662 Tz (\t) Tj 1 0 0 1 312.25 155.5 Tm 34 Tz /OPExtFont3 11 Tf (: ) Tj 1 0 0 1 318 155.5 Tm 107 Tz /OPExtFont6 12 Tf (Q\(x,y\) > 01. ) Tj 1 0 0 1 381.6 155.5 Tm 2000 Tz (\t) Tj 1 0 0 1 494.399 155.75 Tm 89 Tz /OPExtFont3 11 Tf (\(3.5\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 312.949 53.5 Tm 79 Tz (24 ) Tj ET EMC endstream endobj 157 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 158 0 obj <> stream 0 ,,#b7 >``d=t -n?cR%+k~)Ux[H`003ڰԚ-<b&|,:>cBIh?x %)Ê^`v?xT?lnm8 QSŪV.]YM^hBYJ44ѣ-;M]W/զjj.XYR k1Úr4f+0H8@oyV^!>l11䳻Zpwkt~~|}lip;A%#L|"לnXµ JY`2@UƟKaL\.ީQgZ2X0arPz+ 6,@{VLn HE!}ܫO\C~(>yB <|_Gic b^bH U|{CgQ$286uw :5wµO=sxoS,sl~oC0_?IC8@z&W h35rZ/H=uϿ9])aRQ+x%Y<&s[ $NpKbrZpY; A8X(gE D0K/U8-Aׁp 99TYUH[ˆ+3/?Zx}s۵(fc 〯Z DϲʒndIU9ʶꍳk@ 9wDgY!%<~^ &IO $ D#;>@t*v%6:Sb`FZ B9KCcyM?+;Ù?*6Ƥ-NO|RXJ T_G(HEBYƍ ԣ.Br LR(n.@{;,:awuI1 +mGtJ#a{-=.[ѤBҾ* n!E4!*O=MM/ZŖU䇠TBԲ_nJ;G/t~Ԭ  mLRwK;O.J?ܧEwr-Ƴ)k 9*zHRVM 1f2>`l[=ZdP>/M D{;8nQӛM=B: {;A:ҙ(~ǔ%! ҇ǎ5<핵N'[I;5%v%Gu"58&!U.p]~Wo*Lxkp_>-ڔD'g=i2tƝ Y;b;Z4^ 5V?$.o;k~#@m`^$,sưgbGZE]‘6ag>6W?NUhXd݀Ned 0<{ŁmʽVEɺ)}w%+Mԝ;I0AYڧ=3J1SS7];]Fm^ZSjtvf\0qRCȬGqW:aP?xDyi|IAXN[66h_]dzi;c0]wi!zRa6OF3#l+i3d'*m&1h5|NdÿY ?U2M7Vbv>|?l1$Xp:(Fz9U [K)? ufӎ~̘zq cI`Q=8>"*wp;7?k|"?sţV~K$ 0 jeL6E)~,S^ȡ1ŪWZ6s5x_T V˿<T ]9=lh&g)R"A42K$co{ ?[1ˍ(xȞe(qxWWNllECŃlBg`NX5iER w-c2 ]c:+0䩠90NFYA" TWj.J½0Y!~'c s}^SWss ԡEX,[*^x4x.т)۝nAcCgʐ_v=w8 d{r-29e{`C铖޵~`t*Jj[JԤ%?ہxF\'Y?^vID)_գdMKpB 8 <\NO>FRhk48bZ:X4ʒuZ,VT MG?lDž1Se~DP:`]v{3f{JX -  >Ϫ*5|!TYC{&^ Z>D_#Ļ8 ye_EQm?Uiךsmܩ@::Iү'JŠrIJ AՄC qMz2DROʀ7v!F.SKc\x!˪eq==_QgYm>Ѱ_$A;My ^n84{c@dɯ& `ooUɖYb, j.H=IBR(~7t h,Vj8t)TQ$ubuj"jV_IVƁC9ʄ3 -cU>afL9ս%T$Wa%!1[;X(1'@kZve%ϾD'K&NSD&v(\s=!4-M49o)Q˳g: sr8 ϋy @n̪P (v~t+ē|u Q/=D@ 7sm:7"h*r?@Øa] sNЩ̳en:E]}3v^"'f&:! c=P}z`q??;@:,#0%JXZFcӋM?45(=40W.$y=/ˢVZK.ˮ\չl7#J%ɀdCWcC`,)RCad Xyz3\ V@Krݘ9t /c$ߙ2Fڥ:Au/\69֍^Mg<5=G/&&$[ Y6f teҥ4)f[죃0'.'č+(Nje'3'Yq;,HF/9 $#}O)p:oo%[ )K{ T$tS}u}Qt?5lQ'f|h]Pd^EG ./}cL!h̩]@o!z24yj]ploŪ!9%>^xY)s18J+췰d0#w} z x&h Qv3Fcv+գZ#UWL%6OeHj5Ӹ&€bȨ'#IU܎("xs\ɜr3(pc\~Pk֢<)WgHmw{/W&nfTIRfUև;Z!^"aNJ ,sUKtT[_z aoWFw Jj<[~Gx-p A&Ԙ}}R_lkAh&#GjpR :}c&xIy=G'd9Z6Xo+Tx^ߨ|^!'$̇8NaT~%\nAƾUڃ{L#GZ S8{UoOwżWa*}fՅ 0d:?ʮ:yb\n0:{|p 8i/TWf4|@̌8` mwKnv7X؍f{?#gE,K(:70w. OΞ.g DPzkxJ8,,ŹAwH#R+Rmfs7ʭ%)hzi2h-j);}ͩա/Bd?$,j"XPVblAj[sWsh%3ZQ|m8b7CϿYl|HYănbP^%ʏ XB/<Ղ\i'5Eg.:ܱu!;Wom ҩJ,J*y3gݤ=`~K} XldQ;c m4zOTQ޲H怶:vtˡuW?v7'@N0U0ELUfYWMnX M ]B澍r:,+~L: r̩R0{47:xtC錖l6k O9)H[Bnӧ۟g?m!`蠪sE%0ZmG)_x" %z`pn0 uSI.%qhAi5sդg2# z`2:vϲJ! Пky֙7Td":Q[310NV ThI屯B.5{\f|2j"R⸻<& E'9qsO&`PEt%bԨDDJ~pX>Y#73$tǓ>TḿܳH4)-w1qf;'/7y)ίY{T0# <+W41Fi9坊PIҊȸ|HjG^bT1=o38_eE~WhAq7]Jn030L `r  -yei/͵]]R@ 0885G?xT+tg?VP7B>P2L͠C ;MpP/*,q `4v„w*MokO߸@M;p0ْAz1ӃfĨ>Vor^i vyg*7EVٓ%چ r^YlUִb+RdＨȵwO^'.]{.zhg#ɻ,VlŰe9l]\?&9iG_bF 6Ά3 ?L79*ᚁe&7<(9 I_hj^ YJv݀X޶]`O/iG%noG2fns`4G7{|c/ebRU\irkʧ ڂS<ا\e߳sW[h<'u}oLy . .Ey>NzqUYZѬc lŸ'l;'ӛ=$!4g AgLC~%mnstyb@\CM)d0!欓'ծ.3Mm^FKI{4ٍthO1 .@Hh2`؞JV]ĘKRf9rڬJ{vvGle3\bG5c5UNDч7s#)^ ZhF;cE[k:tjh^ɼbJLFF 4; lI<Ȝۈ~Bojɺi>r"0N:xv z0qR{I ͸49oPsF:i|X0ջmꥱ="cT@gU,w$/9}|PlX xK_B\/̏6~]Mf#A֨4BLvuj"پ Gވ_XI{21hMun۾kO6S+`K挍o7ްK㣨#nW^5Q2aHp52[F5ad8xܛ YyY CrmҜk1ǃ ?3/D}r<Og1ܓZVW{ϬNmpDʇ @V84)2%;L[uڢ:r9:IC~1Q<ViZ:~@>v hBakS (Ǘ 8Q˯́DSH "wJobIql,n\-mr[Z r¯bK:6rV ܗĺ(ʨAzqɬf;'L6*0(m,7I -:.㫿c/ƈp$ _9ӹto50>ïCۇIj}fɛ,bɏC,ɋɲjd1NvAj>$[D]sDG7&j uV">e8+P_b endstream endobj 159 0 obj <> endobj 160 0 obj [161 0 R] endobj 161 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 609 0 0 839 0 0 cm /ImagePart_2052 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 263.3 719 Tm 102 Tz 3 Tr /OPExtFont3 11 Tf (3.3 Nowcasting using indistinguishable states ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 102 Tz 3 Tr 1 0 0 1 122.65 676.299 Tm 92 Tz (As showed in \(48\), for three typical measurement error densities ) Tj 1 0 0 1 447.1 676.299 Tm 145 Tz /OPExtFont8 9.5 Tf (p\(.\) ) Tj 1 0 0 1 469.449 676.299 Tm 87 Tz /OPExtFont3 11 Tf (\(Gaussian ) Tj 1 0 0 1 122.9 653.5 Tm 90 Tz (error density, Uniform error and non-uniform bounded error\), IHI\(x\) is non-trivial ) Tj 1 0 0 1 122.9 630.45 Tm 92 Tz (and is a subset of the unstable set of ) Tj 1 0 0 1 309.35 630.45 Tm 95 Tz /OPExtFont4 10.5 Tf (x. ) Tj 1 0 0 1 324.699 630.45 Tm 91 Tz /OPExtFont3 11 Tf (In practice, only finite observations are ) Tj 1 0 0 1 122.9 607.399 Tm 93 Tz (available. The ) Tj 1 0 0 1 196.099 607.399 Tm /OPExtFont4 10.5 Tf (Q ) Tj 1 0 0 1 207.849 607.399 Tm 90 Tz /OPExtFont3 11 Tf (density used in the later application is calculated within a finite ) Tj 1 0 0 1 122.4 584.35 Tm 91 Tz (time interval. This requires a reference trajectory as discussed in Section 3.3.1. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 122.9 532.049 Tm 108 Tz /OPExtFont3 15.5 Tf (3.3 Nowcasting using indistinguishable states ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15.5 Tf 108 Tz 3 Tr 1 0 0 1 122.4 497.699 Tm 92 Tz /OPExtFont3 11 Tf (In this section we introduce a new methodology to address the problem of now-) Tj 1 0 0 1 122.4 474.449 Tm 89 Tz (casting in the perfect model scenario by applying the Indistinguishable States \(IS\) ) Tj 1 0 0 1 122.15 451.399 Tm 90 Tz (theory. An illustration of this methodology is depicted in the schematic flowchart ) Tj 1 0 0 1 122.15 428.35 Tm (of Figure 3.2. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 139.449 405.1 Tm (Given a sequence of observations, we firstly identify a trajectory of the model, ) Tj 1 0 0 1 122.15 382.3 Tm 94 Tz (here termed a ) Tj 1 0 0 1 197.75 382.05 Tm 98 Tz /OPExtFont4 10.5 Tf (reference trajectory' ) Tj 1 0 0 1 301.449 382.3 Tm 97 Tz /OPExtFont3 11 Tf (in order to apply the IS theory to form an ) Tj 1 0 0 1 122.15 359.25 Tm 93 Tz (ensemble of initial conditions: The reference trajectory is discussed in detail in ) Tj 1 0 0 1 122.15 335.949 Tm 90 Tz (Section 3.3.1. The Indistinguishable States Gradient Descent \(ISGD\) \(48\) method ) Tj 1 0 0 1 121.9 312.7 Tm 94 Tz (is suggested to find the reference trajectory. Based on the reference trajectory, ) Tj 1 0 0 1 121.9 289.649 Tm 90 Tz (we introduce a method called Indistinguishable States Importance Sampler \(ISIS\) ) Tj 1 0 0 1 121.9 266.6 Tm 94 Tz (to form an ) Tj 1 0 0 1 179.5 266.6 Tm 89 Tz /OPExtFont4 10.5 Tf (Nees ) Tj 1 0 0 1 206.65 266.6 Tm 91 Tz /OPExtFont3 11 Tf (member ensemble of initial conditions \(details are discussed in ) Tj 1 0 0 1 122.15 243.549 Tm (Section 3.3.3\). The ISIS method includes two procedures, i\) draw ) Tj 1 0 0 1 444.25 243.549 Tm 89 Tz /OPExtFont4 10.5 Tf (Nees ) Tj 1 0 0 1 470.649 243.549 Tm /OPExtFont3 11 Tf (candidate ) Tj 1 0 0 1 121.7 220.299 Tm 95 Tz (trajectories from the set of indistinguishable states of the reference trajectory ) Tj 1 0 0 1 121.9 197 Tm 90 Tz (according to ) Tj 1 0 0 1 185.75 197 Tm 99 Tz /OPExtFont4 10.5 Tf (Q ) Tj 1 0 0 1 198 197 Tm 91 Tz /OPExtFont3 11 Tf (density; ii\) use the end point of each candidate trajectories as the ) Tj 1 0 0 1 121.9 173.7 Tm 92 Tz (ensemble member of the estimation of current state and weight them according ) Tj 1 0 0 1 121.45 150.7 Tm (to the likelihood of the observations. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 138 131.95 Tm 93 Tz /OPExtFont3 9.5 Tf ('In practice particular model trajectory chosen to be the reference trajectory will depend ) Tj 1 0 0 1 121.45 120.7 Tm 90 Tz (on the details of algorithm. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 9.5 Tf 90 Tz 3 Tr 1 0 0 1 314.399 52.299 Tm 75 Tz /OPExtFont3 11 Tf (25 ) Tj ET EMC endstream endobj 162 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 163 0 obj <> stream 0 ,,8b7 3 e0YٍzQ>Ce?o蒒FMqx1GP|9PP [D*0? #waq&7! #ˆȏ[jׁt8eRݷ#[3zX'ѱU7ZI%6?Ǜ,`ՑVvNY`IN#-БN;$OEt!J cמNR׭XJQUӓ[k=НnN⼛9[܍ 0jO]C|-r[^hKӵ;4չ1!̔k^/6fXt<|gv#h/d@PG/>afolACwiW׼8s & 0xmUAy|wԢ7js\:[UuGnFhu$gHv%`( ^aRJS3 PRdD(gBwES3$dы*l;ܰEys33ϡ ALZyDHHr:t~h: pIBaȾPܥ+t]uee eqANJ$@Cl]|.ylPFX<\C}4w:8NwȦkN:ayNhwTa{&e\xfm{v`^!J|*]=)eG%5+z& [`I-KÒk@{ިn/ m9/[& 3 '1V* OYA}"tf Ubw5 ,N >_!iv`\J @BLƇRaZn:!^xkhudih&)_:}gEd5;:b9|pERe!Ifܹvh$;roQ[#xܟpD$tҐ +PQngo(SO^ 'ui4Gb;m%_J@fC[P ;MpU%~zB$pCX _zډrA?ɵ_:8I˵Oir\EģW}DbF];/f!&WH Dc%BHLia={e>6ġ%ś=8;ڹe,ܱԯ:ȉ=<mR$z @M1?+2?%5=+>֗={>^ }+ XW3wjJkU~ { .ٲt0҇u3rPI0h'cμs.mq#Ysq0zzp&Iۖ ,uO9_eGpR٣,~2;|u?yG NZZ7UçaMS֤ZAURwc41ԛD1_ j4LB*K,3E?ri'9G뤰ض,R"LN7Iԇ,QeW>} \궹 eS !/5o-I"l/ 6">gU*QsbnJO{mLnm 4`-ǝ fZ[JC4"9)z-01;j8Vx<{s"؁;gO(x듚*Jl?2e4Q,E$dG}z(gj=,=()G#ǩI/wChq[s߈yL) GIϔM~e JlӰIk0IP |Nֿ8*RJ.I|~6zSZUalkFa;27`OIFW:a ^|ᒺAYj%ԕ0J~_Q}SǙCqo ԁϡË Aud-C?^aO{\S+ꪎ]ɕ$.71!a̕9V(d|RldY`u{]Gq]eLx:ʓk)ШٲT4Koܘi% bUD($|M[BMF4_힦[:*)K,2?)42G.Q`jvbkP/FI{xe!.+¹C=5c|$%Ni*}4SJIpsz! 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Zy6 t,pgqD69? 2Xԉyh<+T]V*˩kPau5=^h$aIH/ƥ7jkYns!A]Gb?g@-bbAAY»Q 87{wl=l5Vŋ~9@k:ѿ6ؾ7(]\ rEg.yXfE|;ashp2-S;P'%&hPf֙lPlp .J:nzZֳ~ZOz1LPRZN Zuatʓh7&bnC[ToUXgxy.2#ky沖B:N,pk:[)~0~/8-=$eӥGbl!3Nԛv*ɴ78#qgwOUm~X$ա*ആBf5xoz {J'_h DL0Q Vomb4գI)̕"KvEuoG|B4aextVۊpm/Eps~Z\Y3)nɃ/ 5;i̺$Q ~fS$ICMWE+,&l/n䵐9vN;O1gjM%FjZmƛ) ̓džnu`FI,W]tj$^ӉүFdڇ+s~0ʅm޻9}RA︶H*e1!M%M)JKԕh®ErpHX pZJ wn= ?؝pZoeܧhC5mx4{{t P8 <`lmHx%.%9Z"H0}VB +b;c^ M8cc2l٥5:ߗ1U#ui(lY#n3=^ʱ(gvy_<C3Ihil?7K53dRoR:O `bEtٔeG]2O 2@OpMl& ]f1+yP-8BxZM%&z#:+b mBts: e=Yٞwsń$Ѡq U<Z=[B[15M@ f0\ԹߖÿdQ،ێ(ytC&j Ѽ2I y݂rBX׭{6FB݆[p(Bx- ሠG M"#=)6̪OD9ΞMEy 9Zjs7%2̈́ݬ;\f[pU]u?D㵺J qUCS}AqXH0*>as6"_+YUl3S%,H#[82Hy "3yw[{]k{ )cW'N~=eϜ{)ZnN:06+? _|V aPܛ#h{06LuyE3؈qf)zsdP>oYpn!q* 7W8SAY:,JDV=q0Pݏy/iٟYJ(t`c[ǘxG,p@^|Ѳ?#?S>7 4+#ҌN5ll՘./fLo4fG<BV =Y40KZtBK h$ܞ(|?4)zLϖ+8SnmEMpЗy3D*ݩ~xhsNͧ?dQ)طUM}:b":5Ob|"z^-ɤ6r8٣>V?mD S.npjI eL^ ;&^8sWqBG˭/nAr'a`תiۃ65k|܏,hd/KqX9l7)f1)֒??+4ܥe2k#:{F q6Hawyzv]K e`Z/=b_/҂$Ƴ4) ?X>&q &P$D=U|\M8k\rt]) ҝPg*m;f._zQ]>O`b7&`9F| e[)lDvC $£^z|#m-gmQ2ϋIz#홅n7A7yɆMo47eE\+Q3aa{u> ߑmG3P-Xr m៑p@%GЅ\p 1du:*vf" {&&%(nԀpo_^*K)u edGrx8<3w hD\j.Bf=&)D1#reRN"5ۈ?.<ޜWdpxS%UZiCm]?^kvњ!IʜFMpE=z2f'\d笔o̍%8_ٱOQGY|zTQ9A_EO^{EW%ruDERc]BsLͫ+lDZYdCIIa  s-/< E8>#g\yI>0^4Zw KAS}gn8'Cgq&'A" ycoP76FxM`~[볃BSl-zl_ӨQW9`%> endobj 165 0 obj [166 0 R] endobj 166 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 609 0 0 839 0 0 cm /ImagePart_2053 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 216.25 502.5 Tm 97 Tz 3 Tr /OPExtFont2 21 Tf (A reference trajectory ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 21 Tf 97 Tz 3 Tr 1 0 0 1 237.099 478.3 Tm (where ) Tj 1 0 0 1 294.699 478.3 Tm 62 Tz /OPExtFont8 21.5 Tf (I ) Tj 1 0 0 1 304.1 478.3 Tm 84 Tz /OPExtFont2 21 Tf (= NI 2,...,0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 21 Tf 84 Tz 3 Tr 1 0 0 1 302.649 426.199 Tm 98 Tz /OPExtFont2 22 Tf (Q\(z\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 22 Tf 98 Tz 3 Tr 1 0 0 1 216.25 378.699 Tm 138 Tz /OPExtFont2 21 Tf (N') Tj 1 0 0 1 243.349 378.699 Tm 50 Tz /OPExtFont5 21 Tf (s ) Tj 1 0 0 1 247.199 378.699 Tm 96 Tz /OPExtFont2 21 Tf ( candidate trajectories ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 21 Tf 96 Tz 3 Tr 1 0 0 1 237.849 345.55 Tm 60 Tz /OPExtFont2 4.5 Tf ( ) Tj 1 0 0 1 259.449 345.3 Tm 104 Tz /OPExtFont2 21 Tf (where j = ) Tj 1 0 0 1 342.699 345.3 Tm 745 Tz (\t) Tj 1 0 0 1 381.85 345.1 Tm 97 Tz (Ar" ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 21 Tf 97 Tz 3 Tr 1 0 0 1 263.05 719.95 Tm 102 Tz /OPExtFont3 11 Tf (3.3 Nowcasting using indistinguishable states ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 102 Tz 3 Tr 1 0 0 1 207.349 639.549 Tm 96 Tz /OPExtFont2 21 Tf (A sequence of observations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 21 Tf 96 Tz 3 Tr 1 0 0 1 202.8 611.95 Tm 98 Tz (between ) Tj 1 0 0 1 279.1 612.2 Tm /OPExtFont8 21.5 Tf (t ) Tj 1 0 0 1 288.699 611.95 Tm 90 Tz /OPExtFont2 21 Tf (= N+1 and ) Tj 1 0 0 1 397.199 613.399 Tm 120 Tz /OPExtFont8 21.5 Tf (t ) Tj 1 0 0 1 407.75 613.899 Tm 94 Tz /OPExtFont2 21 Tf (= 0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 21 Tf 94 Tz 3 Tr 1 0 0 1 299.75 560.6 Tm 99 Tz (ISGD ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 21 Tf 99 Tz 3 Tr 1 0 0 1 219.849 248.1 Tm 95 Tz /OPExtFont11 19 Tf (A) Tj 1 0 0 1 231.599 248.1 Tm 71 Tz /OPExtFont9 19 Tf (re) Tj 1 0 0 1 244.099 248.1 Tm 50 Tz /OPExtFont2 21 Tf ('''') Tj 1 0 0 1 251.75 248.1 Tm 31 Tz /OPExtFont5 21 Tf (s ) Tj 1 0 0 1 254.15 248.35 Tm 97 Tz /OPExtFont2 21 Tf ( member ensemble of ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 21 Tf 97 Tz 3 Tr 1 0 0 1 233.5 214.75 Tm 96 Tz (initial conditions -) Tj 1 0 0 1 381.85 215 Tm 46 Tz /OPExtFont5 21 Tf (1) Tj 1 0 0 1 390.5 215 Tm 16 Tz /OPExtFont2 21 Tf (1= ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 21 Tf 16 Tz 3 Tr 1 0 0 1 164.15 154.049 Tm 106 Tz /OPExtFont2 11.5 Tf (Figure 3.2: Schematic flowchart of the IS nowcasting algorithm ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 106 Tz 3 Tr 1 0 0 1 315.1 52.75 Tm 77 Tz /OPExtFont3 11 Tf (26 ) Tj ET EMC endstream endobj 167 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 168 0 obj <> stream 0 ,,(j]J<Šs$P\Ui-R$H8J%m|b<1`qy HRv(3FWH Y1B j. &R=ò8Wn(cw|0|ƱfHy(pzoJqS CH%Ńhx. . *G6‚:̐k>w𼮰Y<`OKOV(qZl@٢E"|gq`fbꖑFvH]A:fL}&7ú$-TG\RMff!Jd%߽!‘dQ+^lZR Un{R2ƌNVԖb"*Ց%?FbTu+K߲@c.Yk^?`>йya<4;ԆؗJ6i@0=._+pjmi;nD4o,Ɂ{@H1b "yܥeSv)3=² ~3}pWgB4_`O\BgҍFsaFHCV*4Z-y*[pg9?'llAرy/vhRcotvZM<^șH\bETQ=iG8‡C5/v2 dIe?&zfHe.!\jyR6a5#4Շ{B&nJh\諺p\n::@) (#`*FB*nTvȹW)N8rӅ*%ĪV ;fg$2c#AM[4bx8wG^.W-4w["A;);|p%sU%Uѕj8K&MoSltՋT-J_n\_Z詏=VJّ£i] Vaò\fJw0cSR4JEJsjF- PewsEU˰;|uB?b:KugB?S_mĨWv,c]y)B}1p8'6O/ QpQїWjOsAIni+Ky:GL!_9jQ.YGʟEASbmU0@bHSo1/6|@2EЌѳR))>.>ԭ&af[ ?db@+_+>+^xcs ⏳D b]Ӱ0C6!VT8XM 4CHv5$k!,("~ϼ=`"$ׇ +N0 tQc>T"Q,,1xc|40Cqr%%]H?Oz7]*Ueɲp.sYmFDZ914FJ4Q:G[m[zONq.oıAYTF'uB5E;cs G{*}wȑ{XI*Ov:[Am~bP: u(I徑%cseLd^vQ-ڪ'hAa,é}&ziJQw&玫_yPKL܅UOSփ͊x>ˊQgыKlq"5زPad-+P\\(cQ h(sT"3e32=#z@?Qnq=*FL/[?t'B DNH$2s 49+Q }i]mUݪD|~7⭳ 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h|}BBbe^+q>G?jhS&.%j7CX,TCEu&W6-QNBjMRRJ6 x'ܼeaר14+?Zs'x2yxqW+kL\~Fd5\}j>wi)uİu <ʤ$:ڲ;ӕQ7gOcbn2Ϭy&wAymL2w K=r̭ҞB_dh7lw4 !0y )<; i[]ݕ:m΋ÿ(. ia\8'b2inw%"A|7bO.nL0UxtMs ]TBғӒH91+)t*ڥB?Rga7i$k"VqiмmØ19Rv'+:о/KʗKM],7@[WQ )u D׾AA)1pi{36X;oFmb-K*ZgKԥq"E:?AH'Pz-mY(+K$d _P iָ$+ZE]#nk}^",T81hޫ03:7B)"2}aiE-,'8"B$ӌ.Fڿ oNS)).<m+0q?}Y 7{tB !~d9_Hr V|<Щ N}2;&̎6y<Vol˃ʋnh3UopygSm]g% OZѦXO0$iAmvx(1}6~SJhԂjAs732X8&R8_t8o6#r` JWI=BS+ L+J 'Lh:yªŇe2g{nNeLȑ(;ya kqs1?14|Lp"%vPtry;߂biH |! . \p?IwкpY| ~: KPˎp'BÍtǟ38%+ˎ6CHK(K `*J0ːi5ʂKHS(S *`XUv߸`7*([D( xyc-LS* s2$ڛbi&ۥK|Xzm6fZ"6{Վoi+UH9S*^'32urOz va`^.}F2/μ$\WY*Sn(\@^[-c(etT=;q6T`ʮŒg%b] kK ,$S04 MI"rYi^@] 0Fˋ*-i1 YUX^)pSbКlJC3&@o=/YAAne~HR1RV0$zWsgw H &d <@ѓDNu.,Y]]D&4܄πD2 /8unrCEn`G'yPJ>o7ƽC^4牐eG/w/fyqww.7꼦Q~FE;-tS M MI4.Í:NzP2+~evr!x(PPYy,B|-s?,?m훮&OEKD jIGm`?.ee$uqIajTv6?ӄBcm82tHQ?T? 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FnIP'LADPĉ:P؄䊇َFI<SxEڪ&Mz 2niTz~"ڈ*0!6j=bwTAl~5at(]#úaG:`/~<,V|Qx #Q5w.K ]E\c@ ,xq갋T-C(;0dMXZvyY5QֹNOx1p}+Sze /.s#Dvh;stxTɗIy% hmR9I (@='[U]5{n814mQ=U>ݼ<lhW_f%D%NG@v(>"9"\4pVrSݴhO유C Hi#U9d-Ҁj2,.]t-C>Ӊ؋O2UE=ž 3`Ib nh:T;qT]u۷HS#7Tv mNm"vX1&|?I1lRTW"<.;j7.JWDpFAغ{4mky|h懤sZJ6%q"\o[H;.D}rZ9&Π]c~W*8L;+KqrҐ F[iU !n_M2 ZX$`?]]L6 *GrD!$̴sOۇA;"nJb@-O-ź1.m+az^l[o~2Lj \yPY7+Z;k*?]`dɜ5Z}*ƝwTIBЇצveqee@tǩd'  r ًh13YYm"-D[;?cF*|?㔫=Tn}Qs /]˂g/uxDҌ" B:>s;c{Å##51&)ZiR=R&|sSk$?5DGRp1I@I/7: ֞lu[@Vyj9^x *,9#W{SspI}ϔd E]kӣ)~NZYWl#c˙ťd`F6/4 3&\Ϟ6M@L 2P73ZKwr2>eCĂ/ endstream endobj 169 0 obj <> endobj 170 0 obj [171 0 R] endobj 171 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 608 0 0 838 0 0 cm /ImagePart_2054 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 263.05 718.25 Tm 102 Tz 3 Tr /OPExtFont3 11 Tf (3.3 Nowcasting using indistinguishable states ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 102 Tz 3 Tr 1 0 0 1 123.099 675.5 Tm 114 Tz /OPExtFont3 13 Tf (3.3.1 Reference trajectory ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 114 Tz 3 Tr 1 0 0 1 122.4 645.049 Tm 94 Tz /OPExtFont3 11 Tf (In our nowcasting methods, we define a ) Tj 1 0 0 1 329.05 644.549 Tm 97 Tz /OPExtFont4 10.5 Tf (reference trajectory ) Tj 1 0 0 1 427.899 644.799 Tm 98 Tz /OPExtFont3 11 Tf (to be the ) Tj 1 0 0 1 479.75 644.799 Tm 88 Tz /OPExtFont4 10.5 Tf (analysis ) Tj 1 0 0 1 122.65 622 Tm 92 Tz /OPExtFont3 11 Tf (about which an ensemble can be formed. Generally, any model trajectory might ) Tj 1 0 0 1 122.4 598.95 Tm 95 Tz (be a reference trajectory. The quality of the ensemble depends largely on how ) Tj 1 0 0 1 123.599 576.149 Tm 94 Tz ("good" the reference trajectory is ) Tj 1 0 0 1 292.8 576.149 Tm 35 Tz (1) Tj 1 0 0 1 297.1 576.149 Tm 34 Tz (. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 34 Tz 3 Tr 1 0 0 1 139.449 552.899 Tm 90 Tz (In the PMS, as we discussed in Section 3.2, there is a set of indistinguishable ) Tj 1 0 0 1 122.15 530.1 Tm 99 Tz (states of the true state, i.e. H\(ii\). Let the reference trajectory end at x. One ) Tj 1 0 0 1 122.15 506.8 Tm 94 Tz (can form an ensemble of initial condition by drawing members from the set of ) Tj 1 0 0 1 122.15 483.75 Tm 89 Tz (indistinguishable states of the model state x, i.e. II-1[\(x\). It is desired that such set ) Tj 1 0 0 1 122.15 460.5 Tm (of indistinguishable states IHI\(x\) contains the true state x) Tj 1 0 0 1 398.899 460.5 Tm 120 Tz /OPExtFont3 3 Tf (, ) Tj 1 0 0 1 404.399 460.25 Tm 88 Tz /OPExtFont3 11 Tf (which means Q\(51, x\) > ) Tj 1 0 0 1 122.4 437.699 Tm 93 Tz (0. And symmetrically the model state x is in the set of indistinguishable states ) Tj 1 0 0 1 122.15 414.899 Tm 92 Tz (of true state Elf\(*\) ) Tj 1 0 0 1 215.3 414.899 Tm 49 Tz (2) Tj 1 0 0 1 220.3 414.649 Tm 93 Tz (. Therefore the desirable reference trajectory we are looking ) Tj 1 0 0 1 121.9 391.35 Tm 92 Tz (for acts as a proxy of the true state. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 138.699 368.3 Tm 90 Tz (We suggest using the Indistinguishable States Gradient Descent\(ISGD\) method \(48\) ) Tj 1 0 0 1 121.7 345.3 Tm 96 Tz (to find a reference trajectory which use the information both from model dy-) Tj 1 0 0 1 121.45 322.25 Tm 93 Tz (namics and the observations \(details are discussed in the following section\). In ) Tj 1 0 0 1 121.45 299.2 Tm 92 Tz (practice, the set of indistinguishable states of the reference trajectory we obtain ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 121.7 275.899 Tm 93 Tz (by ISGD method, almost surely, does not contain the true state, nor would any ) Tj 1 0 0 1 121.45 252.649 Tm (other methods due to the fact that only finite sample is available. We are, how-) Tj 1 0 0 1 121.45 229.35 Tm 94 Tz (ever, interested in whether the reference trajectory we obtain provides a better ) Tj 1 0 0 1 121.45 206.299 Tm 90 Tz (ensemble of estimates of current states. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 137.3 187.6 Tm 93 Tz /OPExtFont3 9.5 Tf ('or "are". We might take more than one reference trajectory in future work ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 9.5 Tf 93 Tz 3 Tr 1 0 0 1 137.3 176.1 Tm 88 Tz /OPExtFont5 7.5 Tf (2) Tj 1 0 0 1 141.599 176.1 Tm 91 Tz /OPExtFont3 9.5 Tf (1t does not mean that the set of indistinguishable states of the true state and that of the ) Tj 1 0 0 1 121.45 164.549 Tm (reference trajectory are identical but overlapped ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 9.5 Tf 91 Tz 3 Tr 1 0 0 1 313.899 51.5 Tm (27 ) Tj ET EMC endstream endobj 172 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 173 0 obj <> stream 0 ,,ڔjQr#x$듮~TWoNec6GáS*LP"ln%H A4)E',='c?6*x;H0R3wr\"]ƫ.S=UFiefl"Z[/WQy$k_8PTT'}|jY>S8@ r# TP DO6ϫ? xLUA#<|Y2K?|@pꦔL@/ە.WmD#nS>޵[/) g]l?PԎR+mܘ}<*[yoTm(ibt7x'C~B~UlX#?9h辱QIS]in``cZ1!3@ӿ } ^H'SO}: \p,•[iunOvX]ø,9:H7~s4; SF`Q@nؑsġquDlAE !JNO3z&_]}*iHPFl]{4#vPi 2P. 崔lv f`RW ~hK|ABO`a ᇊXl`_~긠:Wȇȣ-;)W :ԯnpsYkC3XC@=g?KLX'RCKH t ( H_-VX4i[pFR߾ʁtxղԷ]e~~|8[_pwvh&H8:T])%(#+IpMoh1s4e uIBF; E#D;N1rʶ<~OW7C)N&]Q|zwݺrn=+Ow,KIAc0KGvaAƑ#\)~5ÃT4FA~8CyF3md7ܧ{<ΧGm[G7;#fIEEF3>*oO >`acP35;4sc= bh_C_bBȘ%M>q:ԕϚ4kǚˍHN/=~Z}`=۵ћҾH o>+M7ӷ+9ˁD6lcm=YXy4$\e^=d 8e~{ zo''}ُ SRl3a`I{r1]7ձ_ %1sRl4(cZI`XcɁ. 2gno_ꡣ(.QL@|P4dY_e>-&AyCCH$$6_Wj>T =l͎U.vu‚oT?U0xpb\g=j?/l8Z}2ASrDKRhU-BcqTO5ɜvTޣ9v pIԻ@ )3:MOȊ}ӕbmo 4%Rd2@16EJ;5u pV{}[NbXoij2d;,\wk&ﬗ`f&"b6^īd |eD~p R}.: 9ۤlF36!I%$rN%vndݰm@ݞlf N0G[/.+c<@s :,T !'O|a5p:8[M HAYji3(Ϩ,M%ahLFds&īhrMjaAT \9&6};1~) 2SploUma*vGLhg]fbA@@īzy;Qt &LR0; 6[ FXx9.HM?R{ W]9Ztp jz䲆 N# ×:O.ӶK]¬x`Y&{ ?w_~۹D ӔȦ"ewSi;ρ(qod1oRh':ϕ9nHjP}a2N!|5 Ui~B_<,= ѸN 7RPAz bO^gi{3ucS8/H' 7vص6܌R?9X Z,#]D5q%g@FźAEŢ\=br7?f5bhGG@̃1N 0{YR DMۨ:P$u QSKW^Y !O 5EnmI}Q +nAwhN7K1,;5Dzb!GyӇh6H6B,!}a>+́ H Q&BN,-:G7\`bR]+)K@s_uY+w1 P`L:r@$a?0m{4!3⛣(e*KtbhQDDwDY6xYEŬQ;x_JPC^T^*ۦO&=8.\{C%VewÏ 49 32"dzsbY_Qw! @'=ࡨCݑ kx!3il7@:7,'[y2 zxW);)I "x M% _4ڟJn/pY~$N;BX;VqMN+&!uj9#q"#Ĵi f'Ij{ozOIAq,s'4(WV!yN,gBw Мƴ+V"KeadLBάkAX<ށ$ܱ\Eq0LIɷ X١2uZ)ǝ%2k#p`9M.冀+$] -ȭp> a;%(Y,Yw{i. n%(؛)=^ ݍyI \AJkDoHRMNd ]ѥ1m %\\zv 42Ҳ문[%E +Jp@9URU;eKf5R}; YtF_@8휸(sѿClfte+V)rR lQ*cy$1{ѿ|zq{'XZAC90 B)ˊK3y|mt~ E=)l{8ý޺ZIZ9Ķ1rMYmpV);Eùs]ǡ’+X?*clď=ٗ`#ey 0hm'/C'肾hdġ`'E ܣ1V01%V]eOoq`G7hru-xe&#;AŋtEu&I64G XTPj8ڏ>v6b.x `/h@+sZ,qRV-'oUbBTn  pCaf8yĞ}zk%hrJJDv44PqS?/0Q'! Y ąos-yt^h.U[WK>Կқ0Hǎ5!{ޔT=j϶(ďe܆>F$R<?ng癟W*B`DaB*je O,mi<M߉;=ۈo,hwӄ[ HpIbU,CHƞ&ʁtIaKW ܰo0ZmgZ/;@GM u}2FM[x:> ձ˵2|'=9峾_Ԟ':ғ̶Ĵҡ3ǎb;eᾣ4<ةH\n/\ɾ#}̂hi^{_y㕙x-UӮr$kZDe"0|qxiWUE)=eDORV$Z :4ʦ˗"W7)\4Lhf_HMғWSq{IhVoP>K* Jq g;N_ xOu=L!T )k"LM@m0|O-)ԦX!y  -Yhmf'{aN Gv9;XV7u,ꝮuUiiQVbS+ i;4標5긊aUߪJm<ʸ.0}T%t\!ڲ\1Y9L-Glh!ƢV،E մcх*fvFacC9`*vOō |L~HHcX4 hfu3d[2k“ xd^3``:!=WrE~V_m靃IޱZ܋t:o4ZSu~h$6E9sU DzN6" o8;[STgmqa_: ]Ȇt[7O]_b/R?u*U7YbgNnV`ifжFg$^NmBodX\4fg7#.oWMySj䮇Ђ7j /xpuAO4+v4rs;x sEߞrz)f_-Je7}_1ŽNdv_V+hmG8uAq T1/]F(YS|V]Qʗ c82 \+Ih9J ^l>2vҸCksl_טyΓ;ؙXǜ,/}}*F6]oԵ-t}P0hPL:5&ps@H6>eIJ%k1#YB=(}js`4aH 3$`#`,aۈ: U,X韂pR*>ԥtB8ZȋeLvmh:ujf4Rx;n12X&3W؇ԉ{[%Cwg `!<ϦJ &y=фC v>BGydW6H1 xbP1w'HǓ$lwo}l,X&ܶ1#tT%Н $$MաT ޙ MҘ; ];&\U"@4-ČA8srܑ[k<7K0}i/1ߩvWgAC}. B! Sg#~\WjW@735t~%fGe

2i5H~KB"ƮvQSPcѡOLuB9ƓM+MgZ\x Pa"H I`P"Hb:r $(bŏBFS^RP!{$tiԾzxń2{L›0&xJ@+R_*owj8kHF[t?4bɽTMf3@%m!nj}A$(+ =L ҠgGV?RqX7?}'˾ʛ7R\7?Ά R}dD,%N7+9}SNUǕ޽@jXf:p,,%\K(@q\?Ϯ.;z/{U?v0.YM}tji Pk(/Hւw2BR,1t|U>z 8[|Oj.3B3S{5cȃBrĴiZ"ܳLW<(+pzf~$g[8y">BGZw E/+UrLls2toʶW dq7b~?{yN}Zz1j8aLS吽3kzgɘ5KTv[z/#NQym8\OW? # ^$=@ޡ<ˡ Gm؂ٮ:4-Y ?3)9J&ǝK Q^ĭS=z>"Dw}V^}I,ŒL@`O_ｯՠퟷ,nΜFWrK]^ʉ͘y&ǽVsZ^jTEK6Cyt*KxeOe "x `jF4IemA+^4tMx $X)5lߨU pSlx msЯԱZ mU:WXxO'R_XJop,K0-Tb_F#rSmZ] ;3%%r9jl@wT?G%9-`u3 .}'dPKHLv"5qS K <؇r#\( AJHO(`KNq9C fe ؀>QCȞwCiL7X&>g$J2(1No3 eOp| 2NuޓhxQ̲>ne Tgֹ%uB TF7KBSnHr_P{w}A'?eD.p,ڤMK\6AT|Bl#5ھ'! G+ϫ=RE|o*>O*ظ:o2GF*zCcÉ:HgR|6A Sh(%NQ) o/ (-|99Ƹ`*'ħ27p4W 6D>P 3BZ-cgY.v ZnlsȂ1T]G<~&r|uUACHn&1OjdRֽ?(pZ)cR$uL>[&/lާ{a44NL c;\? Ig;[90j'̀hA;DrEQ.U;,}\q@ uh|LG,dAZc`2#TA[*-b'[DJKd$d@X`R/ᰅ1y$,~+gGǿo&OX1;R{YQM_]ꕷU!A 33&3 ux7K/y'-eVF/ 8, =Az+k82co)Mj9}msRǎYVY8ا> i)m3l endstream endobj 174 0 obj <> endobj 175 0 obj [176 0 R] endobj 176 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 609 0 0 839 0 0 cm /ImagePart_2055 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 263.3 719.5 Tm 126 Tz 3 Tr /OPExtFont2 11 Tf (3.3 Nowcasting using indistinguishable states ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11 Tf 126 Tz 3 Tr 1 0 0 1 123.099 676.75 Tm 108 Tz /OPExtFont3 13.5 Tf (3.3.2 Finding a reference trajectory via ISGD ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13.5 Tf 108 Tz 3 Tr 1 0 0 1 122.9 646.299 Tm 89 Tz /OPExtFont3 11 Tf (Given a sequence of observations and a perfect model, we apply Indistinguishable ) Tj 1 0 0 1 122.9 623 Tm 93 Tz (States Gradient Descent algorithm \(48\) to find a reference trajectory. Judd and ) Tj 1 0 0 1 122.65 600.2 Tm 92 Tz (Smith \(2001\) demonstrate that the states produced by the ISGD method reflect ) Tj 1 0 0 1 122.15 577.399 Tm 90 Tz (the set of indistinguishable states of the true state. Here we give a brief introduc-) Tj 1 0 0 1 122.4 554.1 Tm 93 Tz (tion of how to apply such method \(see \(18\) for more details\). Let the dimension ) Tj 1 0 0 1 122.15 531.1 Tm 89 Tz (of our model state space be m and the number of observations be n; the sequence ) Tj 1 0 0 1 122.15 508.05 Tm 88 Tz (space is an m x n dimensional space in which a single point can be thought of as a ) Tj 1 0 0 1 121.9 485 Tm 89 Tz (particular series of n states u) Tj 1 0 0 1 263.3 485 Tm 95 Tz /OPExtFont5 11 Tf (i) Tj 1 0 0 1 267.6 485 Tm 52 Tz /OPExtFont2 11 Tf (, ) Tj 1 0 0 1 271.449 485 Tm 96 Tz /OPExtFont3 11 Tf (i = n+ 1, ) Tj 1 0 0 1 328.8 485 Tm 94 Tz /OPExtFont2 11 Tf (.., 0. ) Tj 1 0 0 1 353.05 484.75 Tm 87 Tz /OPExtFont3 11 Tf (Some points in sequence space are ) Tj 1 0 0 1 121.7 461.949 Tm 91 Tz (trajectories of the model, some are not. We define a ) Tj 1 0 0 1 375.6 461.699 Tm 103 Tz /OPExtFont6 11.5 Tf (pseudo-orbit ) Tj 1 0 0 1 439.199 461.699 Tm 87 Tz /OPExtFont3 11 Tf (to be a sequence ) Tj 1 0 0 1 121.9 438.899 Tm 95 Tz (of model states that at each step differ from trajectories of the model, that is, ) Tj 1 0 0 1 121.7 415.899 Tm 86 Tz (u) Tj 1 0 0 1 128.65 415.899 Tm 83 Tz /OPExtFont5 11 Tf (i+1 ) Tj 1 0 0 1 141.099 415.899 Tm 171 Tz /OPExtFont3 11 Tf ( F\(u) Tj 1 0 0 1 178.3 415.899 Tm 87 Tz /OPExtFont5 11 Tf (i) Tj 1 0 0 1 181.9 415.649 Tm 92 Tz /OPExtFont3 11 Tf (\)\). Particularly the observations being points of interest which, with ) Tj 1 0 0 1 121.7 392.6 Tm 91 Tz (probability one, are not a trajectory but a pseudo-orbit. We define the mismatch ) Tj 1 0 0 1 121.45 369.55 Tm 92 Tz (to be: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 0 839 Tm 28 Tz (\t) Tj 1 0 0 1 289.199 301.399 Tm 109 Tz (F\(ui\) ) Tj 1 0 0 1 315.85 312.899 Tm 2000 Tz (\t) Tj 1 0 0 1 495.6 303.799 Tm 88 Tz (\(3.6\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 138.25 261.799 Tm 98 Tz (Model trajectories with probability 1 have e) Tj 1 0 0 1 368.149 261.799 Tm 95 Tz /OPExtFont5 11 Tf (i ) Tj 1 0 0 1 370.55 261.549 Tm 109 Tz /OPExtFont3 11 Tf ( = 0. We apply a gradient ) Tj 1 0 0 1 121.2 238.5 Tm 94 Tz (descent \(GD\) algorithm \(details of GD can be found in appendix\), initialised at ) Tj 1 0 0 1 120.95 215.25 Tm 93 Tz (the observations, i.e. u) Tj 1 0 0 1 236.65 215.25 Tm 85 Tz /OPExtFont5 11 Tf (i ) Tj 1 0 0 1 238.8 215.25 Tm 128 Tz /OPExtFont3 11 Tf ( = ) Tj 1 0 0 1 256.1 215.25 Tm 100 Tz /OPExtFont2 11 Tf (s) Tj 1 0 0 1 261.1 215.25 Tm 52 Tz /OPExtFont3 11 Tf (t) Tj 1 0 0 1 265.199 215.25 Tm 61 Tz /OPExtFont2 11 Tf (, ) Tj 1 0 0 1 271.899 215.25 Tm 93 Tz /OPExtFont3 11 Tf (and evolving the GD algorithm so as to minimise ) Tj 1 0 0 1 120.7 191.95 Tm 94 Tz (the sum of the squared mismatch errors. It has been proven \(18\) that the cost ) Tj 1 0 0 1 120.7 169.149 Tm 89 Tz (function ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 0 839 Tm 28 Tz (\t) Tj 1 0 0 1 275.75 126.45 Tm 110 Tz (C\(u\) = ) Tj 1 0 0 1 311.5 126.45 Tm 2000 Tz (\t) Tj 1 0 0 1 495.6 126.2 Tm 88 Tz (\(3.7\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 313.899 52.299 Tm 75 Tz (28 ) Tj ET EMC endstream endobj 177 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 178 0 obj <> stream 0 ,,!.j;~>1.D84z.[ Ay+|V&^Nce!%.Xe[3ZbL_Y\R|5%O9:h|Sr5 G>3k A<3U xb _dewiVFa`1[³.&ƉfuXy#`"J/QJ:XG= P`te7soTq;G$حf*:O|l}WcͅEBWW,Hg9P?Pl܈IYMP{!4J/x\gqM8*]kWrZs{磧o\S܅&OOÖ#8u5B;bwnA쒀3o)V9"̓xw@]T+A2Rp?+3v9y;\zRSKc@y'FV477$HJ4Q͸ @Uj)vw7%t|+U'D.3zǡ…p`%a:-/ʺCaFo vshVuHXGϾ 7_2{3iڏmqqAɀQ2z UB-P)??Ol\ve\+U%,95-1N1kH3WK.8C_6yrS Qi"زoMxgj'|vm (w%>g$[] *$,k%l|F3?8\] C(^vzWXcħ-b|¬3B,t>W%+]qٓ}- lr@wiy;c~AlwbaJ"/,\63#^V y'(3Sx"7ڽQ9B FdZ%!e,Qv} heru:eݝmw}=EX\H״ꏓ}|ߍ=d&X"~~@J yA_Hz_`RL؆n l`rnJ ~]k>8R5hr9e+Ar{s7V2pfܭȋ#ct\4U\9B'7[BffMN <_z1 JM$+dSo%dǜ)&Q=yз ;)suURU.ӎ@&lw4ۢkE|Oڳ"fHFn̽M=y!¡+F_g#T 6"<]tKzBm"L jRٙ *˚~8chrִ+`V3梄'uCzVOOfy FOziADu\h!w.^ ? ͬ3 eґu!4 ءqfo**!K$Qο\}<ü:?=Z9@@3DL$ߚTnRg- C_¾p轌rRNVT9=S7z:_2͙fYc]6yh2%+O# >h? u?eב?C{"p_O~B>?k[cXő ׹o39em#n0Au T*'h_ͅmu!lR%A;>[xvz+yZEsx禍cYcrA^"j*QAá0T4} Y}ѲMQ%N;X@{ĵIDK%ڎKԯބ쓃U\0y}grjH\Tത1zqc[>p!'=~]|MP'5MGmicG΁}`'TD! |I,Qr<‚M߯ u'e7_AR>MIE}ܾҼZpy"?sB0{ |HP~&TFb(Rflk9ls ^t>Jt&1ó$RLr{-ԱƵܪ=1®ϡ2ڃʫ004;<¨lѠ$'t9Z%6K%ZhWLE=:nO=~[Ywo6,'pcK d)uE/&pxP4C8i(d*Z kR(}[daː;E T$Ek3}-&XjbӀJ|D!7{@kX@-N-8T\}V9&N=El=;XKޔ $!kY:!q PP"(7 =.S<~wӝkhI*ٰ('7sX' 1a2m'Cp3)GjR :N}`ղr}7sM/Cp(>'R>ī"1 12_D}Ec$&cIl&m%㢍(|޺xP)[5Qܩ8&5@#ub/طň̀g7.WzjgÙȯ+֑~;a h_kSVUdZQSmȚQw Y=Ք.'vS%u_+ut&ߡze4D(G2=ትƓimh;Dghľ u1E7ǹ)WMH|Īv"܌@To~Hhͼv{DB$܈}w2Pכ;|D)&߿LJ3LM} ڣ-ո`p5ZL!"HYiY(󠔈w5H <'e7bDW 46JA5ӵ'WkI"M`*>eXю-]N?vGZɫWCiƣ @mfg.P*W܇~1JY%tk'w}B /#J>ayAH3ĔvT <2-{{҈CUs%cJEHFg珠P]LN;)Юtk5l8b#gۃ[D~a{ŤK ~8bғ`"O1ʄ8#uO b ~uxt.Fy(͉R*~ F[osuqg?}qXA_!f] BCyY~lqwMU9Cs#(u*l3Mu: k'J[sCԜjx:7~{d?:?^tː@yKL+wgg110]HD6̭\>1\){Ӏ8e뒛IS:ȡ /l~[}dIV,wF.]@cƮ-!N ygzSƙݔ&_Z M@#<HEM#YKaߺ w6Z҈pMSߍݶmާm h7\Dԋ5dbm[HM78'Ypr##4E9+:+20uk=.s^Y8[ M⪏OnaAmvXi|#@z)^=7/΄S{jtrdh<_Ҹ:pAIgQVX7őcCB4Wu3 pbcp5S@D`$ |7h6&2.L|dźryae_V)糁/q⌹A֡4Qx#}M#9w򩿨Ymy-Cd^>nB_)B7: r3s`|z1 a?@n!m'%e2_(yn>f&Ԛ-=y 79*t3ElFKT,pNp+|s O+Ti0ZyGDO|eeE3缱u"/%Z՗}8ox?@;WV~c8Ѩr5cNϱnεcXkz3c3-*?HX1+ Ą%oշ?tjy CζO"4g8vOT%3${7 gŬQ>= F^גz>֎V/ZYj/]q^Xu d?ܥ | h]n%R`У*B}nE漫mj"V HBĔՉ|\V+»`3$Xn_ߵfu.53s>?6ؤ$ѳ+=Җ;%6P强(Ӷȳՙy+<[I?0H4SErt%;HQ2~'9/q݆ElTao}P\ rݯZiOщd籞DYHЁ&@+UܙUkBj 4X2 |dvH" dy}HŻJ(4b -zyb{~Bba>N77QZ~A ky5?ăg⢴\|28u%ꆽdWT̙6_ÙTCܗU 8F~$UV;a ㍝ oT1 00f<lj>9.YA P{F \P S??sXGŢ׻x2(V"Cp\Dvrt-- U,Yg)BIi޾Nxy9ӧĴ. ϋ` 0dµM cgp08d=Xӧ3AWN%_ w 8_e8(+8;Ͻ4d6Փa }ښQݪbX (*%FĴ V6-,ԋp|,^u%-]چ+$TAaEVr.b26вĶ3kn͟:넉13oji;I1?ͱxh۠Zޕږ lgIr%1-%  9 K(٥.ZUļSkԍ12YLs)D.} \=FA"&p{>88C#,E2G*C ; ,c|;COH4&>YÑ lCxY%)?  ; q7IARj]ar XC ѣan[ě <1t^ ZURRcwUE@rUGs(<]4tI3#u!܊I:,0Kɑ5XF=ЋOM]% x'Q鯡^"0DAe6Q8E"A"ȒT}o]8!#NsF+%'~F0nsӚQĠS xHYhGKQetuqӧB͵ΧX[:]-N |h£yTwUKq5s(gpdi2yiuwq#`igPPD%-Y./% B "sC[~lbBtigH@ 9E'uӸ5|O!4#7>8l造tKE ? KC)k0-1P[Z77'aO`F.$`7C_R%ZezԄS  Fk<3G"[2wjnY9 _YjPS #mCdSPKÏعbk2[0Q%)Z`9,s$0 TцnOEN0q9d8GhϤeg85ҋRwBˉ-d]W`CFwz :2uh^{<`V ^EI7UD < >J DA]֡Q GcJV;e* w Y$^M{SXNDY/Mn6( DnLi\ThVGBݿV ۀU*}I*A\2\WЃy*4ʖ֛/qbl$܇qQzzUq614<ݸLUX|hCZ ѫz,k|i 9NOkPոUj=&8nd^HhÕGN\-:skG[>SKh,1.s%}fnȇKTUZaP"cUCBWb,~2SG 2VE>eV힅V ,P4Y8o3d?`Be:H)Ik]uS)W yK+ kq}녡 [kS~́jtDژYe{v1VqHrl,y%_;¥2W!*w׌ojRs{*7C+ 襈 E߶j]r]3 Y +Ï͹uȃ.1d0r!tcjdT* [1JdUr;5,bLLL^{"Ga&ҏ_&ҥGN9m)ÄÐxI3'5BFRaJa=䌦V )#7q (U}$2j8 e';Gd`je?nRKZAg!K59+&߳@\D̕bɓ /wW=H iz_/>i&c%ʡ@<,؄+X1ͳ1U7 GrVPJK6{fLwo7 z^Gt5a`_z2<oU9][z݄'% l ӕR{l _Opa3KQscxgrYЇHxlXo.sm?ˠVOJ˩@Y[r-e sȨQ$R)#DIKvTBg LGb]%.ڿ)H{;r/Z:rOmPu8wEdyx!q||vNu5&)"R=bSt endstream endobj 179 0 obj <> endobj 180 0 obj [181 0 R] endobj 181 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 837 0 0 cm /ImagePart_2056 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 262.3 717.5 Tm 102 Tz 3 Tr /OPExtFont3 11 Tf (3.3 Nowcasting using indistinguishable states ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 102 Tz 3 Tr 1 0 0 1 121.9 674.75 Tm 93 Tz (has no local minima, while at points along every segment of trajectory the cost ) Tj 1 0 0 1 121.9 651.5 Tm 96 Tz (function has the value of zero. As the minimisation runs deeper and deeper, ) Tj 1 0 0 1 121.7 629.149 Tm 92 Tz (the pseudo-orbit u_) Tj 1 0 0 1 220.099 627.95 Tm 66 Tz (n) Tj 1 0 0 1 225.349 628.45 Tm 91 Tz (+1, -, uo is closer to be a trajectory of the model. In other ) Tj 1 0 0 1 121.7 605.649 Tm 94 Tz (words, the GD algorithm takes us from the observations towards a model tra-) Tj 1 0 0 1 120.95 582.85 Tm 97 Tz (jectory. In practice, the GD algorithm is run for a finite time and thus not a ) Tj 1 0 0 1 121.45 560.049 Tm 93 Tz (trajectory but a pseudo-orbit is obtained. We denote the pseudo-orbit obtained ) Tj 1 0 0 1 121.45 537 Tm (from finite GD runs as y) Tj 1 0 0 1 243.099 537 Tm 72 Tz (i) Tj 1 0 0 1 246.949 536.75 Tm 91 Tz (, i = n + 1, ..., 0. In order to find our reference trajec-) Tj 1 0 0 1 121.2 513.7 Tm (tory close to the pseudo-orbit obtained from GD algorithm, we iterate the middle ) Tj 1 0 0 1 121.2 490.899 Tm 100 Tz (point y-) Tj 1 0 0 1 163.449 490.899 Tm 38 Tz (7) Tj 1 0 0 1 166.099 490.899 Tm 39 Tz (,) Tj 1 0 0 1 167.05 490.899 Tm 66 Tz (12 ) Tj 1 0 0 1 176.15 488.05 Tm 83 Tz ( 1) Tj 1 0 0 1 189.849 490.699 Tm 94 Tz (forward to create a segment of model trajectory z) Tj 1 0 0 1 436.8 490.449 Tm 72 Tz (i) Tj 1 0 0 1 441.1 490.199 Tm 87 Tz (, i = n/2, ..., ) Tj 1 0 0 1 121.9 467.899 Tm 116 Tz (\(yn/2 z_) Tj 1 0 0 1 186.699 467.899 Tm 63 Tz (n) Tj 1 0 0 1 192 467.899 Tm 46 Tz (/) Tj 1 0 0 1 195.849 467.899 Tm 49 Tz (2) Tj 1 0 0 1 200.65 467.649 Tm 93 Tz (\). We treat such model trajectory to be the reference trajectory, ) Tj 1 0 0 1 121.2 444.6 Tm 94 Tz (in Meteorology this trajectory might be called "the analysis". It is important to ) Tj 1 0 0 1 121.2 421.55 Tm 95 Tz (notice that although the GD algorithm can be applied to any length of obser-) Tj 1 0 0 1 120.7 398.5 Tm 92 Tz (vation window, the reference trajectory will likely diverge from the pseudo-orbit ) Tj 1 0 0 1 120.7 375.25 Tm 91 Tz (when n is large due to the consequence of sensitivity to initial conditions. In the ) Tj 1 0 0 1 120.7 352.449 Tm 90 Tz (results shown in section 3.7, n is adjusted to provide the reference trajectory that ) Tj 1 0 0 1 120.7 329.399 Tm 92 Tz (is close to the pseudo-orbit y) Tj 1 0 0 1 264 329.399 Tm 80 Tz (i) Tj 1 0 0 1 268.3 329.399 Tm 21 Tz (. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 21 Tz 3 Tr 1 0 0 1 120.7 284.75 Tm 111 Tz /OPExtFont3 13 Tf (3.3.3 Form the ensemble via ISIS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 111 Tz 3 Tr 1 0 0 1 120.5 254.049 Tm 92 Tz /OPExtFont3 11 Tf (In a fully Bayesian treatment one could use the natural measure as a prior and ) Tj 1 0 0 1 120.25 230.75 Tm 94 Tz (then update given the observations and the inverse noise model. Inasmuch as ) Tj 1 0 0 1 120 207.7 Tm 89 Tz (natural measure cannot be phrased analytically, in general, this approach is com-) Tj 1 0 0 1 120 184.7 Tm 93 Tz (putationally intractable due to the cost of estimating the prior. The idea of ISIS ) Tj 1 0 0 1 120 161.649 Tm 90 Tz (is to select the ensemble members using the set of Indistinguishable States of the ) Tj 1 0 0 1 120 138.35 Tm 91 Tz (reference trajectory as an importance sampler \(53\). In order to do this, we firstly ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 136.3 117 Tm 87 Tz /OPExtFont3 10 Tf (lif n is odd, we take 3) Tj 1 0 0 1 227.5 117 Tm 68 Tz (,) Tj 1 0 0 1 230.65 117 Tm 55 Tz (\() Tj 1 0 0 1 233.5 117 Tm 115 Tz (-) Tj 1 0 0 1 239.05 117 Tm 34 Tz (7) Tj 1 0 0 1 241.199 117 Tm 29 Tz (.) Tj 1 0 0 1 241.699 117 Tm 84 Tz (0) Tj 1 0 0 1 246.949 117 Tm 43 Tz (_) Tj 1 0 0 1 250.099 117 Tm 62 Tz (1\)/2 ) Tj 1 0 0 1 263.5 117 Tm 31 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 31 Tz 3 Tr 1 0 0 1 312.5 51.95 Tm 77 Tz /OPExtFont3 11 Tf (29 ) Tj ET EMC endstream endobj 182 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 183 0 obj <> stream 0 ,,(b7!{hqMsOCnmYܐ :~Tf;p׵vm~S״N'M=N 4 6'QMt|9$I,+Idfm.wcr)::4LrT 8FNklZ+R]k< 9c`> Bq"[ЊmxNwϳ :'`#q TymYe 4yEÓvI[Ec_`\QY>ʷy?^Y wEgWmOd;Chkab16[g%C(!vuyY\{Axc8g"yNEC;dc,1a[uj5%K5X$ Jz gѨ}Y|_D`Tm=Ae{mc*vrUv0]m #Kzٓ`Hf]cvop xK y||k3,Zxq0XnKnL4CGϟdh"R6]ܰC6=Xyt?1a]aѺU>5Z-gz #q?7R1^>wzD+oBΦ!̠zu9&G~5?'J% EzHp+_KXDr3[gNh\:棹.\$F!~-q@_)_F]َ[29*aqTu)1euugs#tkd0:|~8%VJjNx4496БqS7\3B0qwj~ҋT{J1[ϩ5q019ȳvg|K-c[Qo\p#^OYw +HwtƎ^ 8r=OqIŤȝHmW?b r*&@Zk&@>TRr{Yp-KW~!X;ӆCS~ŪۂvŒQυnvbg£aHhk 6Ps"=K~7-2V3=n vxL5$ma6s6[糂G}< Nx%bZޕ1Uahvr5BȜ8t3!vJY:+vB~M~澅*=V[T-|;<~;G lL~M w \}"` R}Kֲ_JBwTZghژ%/bӝ1stRW\)z, ]h}j}O;9>whb[ B [g$wyF:fj0 HQj#>R6d;#3!xjW7,?ۥ#'dQ=CTT;R1)u -(x6cn?.2m(4}1QaE4CFbS& A֙rWCAs C̕gwSOf-޸E;xY@훖!"כGR0$8V~/O8(C/*JhCRӼ< fSÝPGMZ Xzɥm*3Ve՞oPKBD!ʎĽZr&;aU %0!ӱa"K ˫6PgOƟ̃1J>49p17yI *}%h.0LM K^vQ wf:)+YS#>̪ `:H"IdF:dx1T{iAt~~)Z|Ri"!)^3UvMT, D콦3pjC%@nyL/vn*E[Ƈf@kZ8™N`ǛMmʕv#yiwG>7 +du4j}_@̬l(>ڙ:&l>1'##ZsXˈKafP;0_C[q`q7$I8w/(H̳8 1_NG$Oofs}%zpv4L N gݾ,V)$:uKr+ۋ_ )+RluTiy遑~r/6||V 'oGeSdc2p2RiA۾&3~9m,= 9Cl}S=MWlk<&fY}8eM{IEq_S?Н.e4>M+0۰б԰ݾj/T?<K͋-)O&r0J)5P$˘Tl(,kY~M^'|+V%,4v JA@;,OQ(\#z;`h a:E:4)_BϼvaK,kvaUPAߦ|H7 'eWLxIL!C]tdN`/VuoD/a؝WV>Wd]6nS}&AiVjf3//7OO@`wI(ȉ2%gsih^P=r;W/пm#cN-~QgKuHz#7{'ATbdVejJ~$ ls M`}CtˠY!HPq BG Z4!p,Xrm̳|/' "ӉIo.R` oNVo3-5LH=*#xpOO}4[JъWXIqBł,?yi;x1vb'@Mw %,%b6u #jyV!AhTsϹW*x`g|/e2Ov8c{ !: U 7͵'Y Ji|"`8~넀s]T# I0QGsX@+eJEu $t7շW?;#ґ[IQMH¯lg 6~Y܏&E'|lq:+wR$EB^YeF]mIxnsF%f6#o'؛-s|9^%> F ڒ  Nze !_-D hòxUvQ\䲪j/jb]7\b0![)ϟpW{wsY"`b'q'k+>ozFIujɰՍ8{q~(-ȓ. Sа:?k{. ą6AWsM湣ۙ;NQ{i2bmI&WW3J'C9 ?3h!]Cz\]5P t8EQrЅpkŏ% Z" U~y^-1/AWЋ_}'25Hsš2%-c[yN?jLY>s(b*H)ig Tn)t KxoVpnp"2Y7Ġ٨ q L/$6{˻o`_AﴨZ%eV8Q":)BA9=|R"T>'e=&޸nC&( jIt/}s7r̠2kA7hAy6dV yIAF*Q[-h1bRgݏdV*oYaʆ,lt| KB+h1EgDiq#{KInƽ띳E1bJKAT3`u =f}QMl~غoLrT-2btP#{R8bHP¶p7ùp?'A4਌0!4TLs |&i_007!'$rNL_؀hӔ\ 1y࣯gF 1V=d@Wml~ZDe:)+K{Rx"m4ݪ=Y$}gyRC0DAK戶 vhNۣwRNT`8<(5_BÓ[Ι4vW CHBKw8NdV?ǺTPM|Y> ;:bcGyx?Z0lkǭeFd1l"kzF,.sǥxq~Z`|Qq]JS1g<8h؞)W~nU.hS =uH?-mnuhnW'X^ɎU25UMl!/<5Xm^xAKyqxLH~r*3sBXшA>_ 9r);Erd6~b|GI7a qFgX.)}ʶ)҇Q s{Ŧ89 Y~'qIOJ1_^+bI%sl`v NG!*ZJf!@0Mcj$UP( @cq#A4y;$ av;h S:!ʬ*bBŜ)p?SyD{nˣ⨜Y6:N8 ZMs YNs]q,_ ZiAySHtv8&iGdf^x{S ,^er$:OR0risKTlnB_Ryw0sT+v̖-'ꖆ =. waL]ŲU¹NM@<E"y Y7:-"RlTSY&y)(Iwl bi7%. 4efP4Xʭ`{ [1!`@TsCT9"j잹>wiRL]_ ?vF_Fl5wyTZ'{&{`ޟ @Pኔ'B:P#,oxiNz_{?$wIw|Gb*iEwPj@7LzρvDtmhr߭> X)zuoBިy -h/ K3*. * xY΀TgA#) j +z-_u5r' z WZJTkd-C,| 6Vd%_ømIB$pxNG)29k׸ռH$V#VV@M4 L|铭w%zf {\1MCUIi5QEA'6vq_(lq /JbTQT_Jl]1r+_& xހ,*>"#6 D3KX81E_ME o Vq##Ivhó?z׫=r Ss&T>IÖ:ق.ԏ*~\G8NmB.įV̉G(wvټ"07Ć^?r5 ս&` YY{b:z+D:&N )"J?1]۟Ԕ|gJTƄ$R@g4E=ʱJkHk$ز//v^7pV3Sz;vOv NsBmjqq|8|ظCb$?UͿyA(d؁h-I74$bVT*̗ۂ:1:WVmuThCNj~*xAy|2LK&+leo I`{E+&y>JS{i~=p8 GSme^u:{|lv]Fw[X%r z튓OO7/9~g49L/o[eJsҨJK6i dr(Lj%ANråC(DRDxKa4e"3u:#l ^R/[NjϫHa05 ȩP"_sPkJJ4E#,^Y/9(k{i]h^(bWFz~ˍWc ['Ir/KwB"x8m` XZ66rS7mҐrh3u*[$[3,zX Wb#p=:({riR$RiLl?j+cBeSn;!Oҏ-8B״bq]ca+@h3?y r;)J4R_=NUZᰢEIh@噺%NUgr=MN4u(_M, h#.۳uUhLP+&D{ē5\auF wjN;? y"o{Y|Dwzp1[k޶GBc˫o VgP&{1ANV6oPg4m!n҇)G@Y|@E_h{QDa\>+*V8OJ Tn;QWЭI1h`jJEP5^PV BXACh䅃,lQ ƿ& ԓψZgEfq/8)t'*]ndϥ.j~Y+}!s{L[HA*]6#%y[eB LtdXGa9d*_Re1A8GC29|\4{qJ[a6UDW4tÅ'ވ :joɓ;! ~h|Ow;?S X>{?(jk:ЬYbRO"'MM΢- s:̊]QRr*UcOmj!x]+ M'BTps@HW}'妈\J,?rJz #\IL+Z5S&[38+g`ȓy7ӓ1lެpbQDRt@Yf SsbG[Т\E4PSJ܇{/OnAƇ8Y@4\'&9 <A9,7zr\ x$ olF`ÆBy 8T6y䨤I=F`vq5wS{b? 2SU1(585|M$eP>"Z/$<9y>p8)ǡʄ\Ų?$«,b4ǹYG͖DYw߲Ѱ(㒲.油춝³ᰫ;ճ2g8S+Qy!9ȽrňFĔunB mMA#KLw+7ynMey/J٣d.C :ޱW_7E[f~TQp~$&h<,e6Qd`0⏁, 8v;?֥=ߑJn3GNhSvMNW2W#@դ#榣FJU'Mw@'5>{ʻ`txݝ윾 1Ėu>7'l.@aW 5q0TU)rvG.@\} IB'f&]yi&y-i$`_,iZrRAy;Hth⌫~W@Q-8H4n~;.n1n6^uCll r;*H*q Ϊ*CNfg%mRU$Yl8wX+ $Adn (N}V5S(.k3T\[1,›R:" +n.YUP7I8GOy-/W3)1t;>3r3=8(,!2䒃D:ʻ66ұ뱿~PC8?~D,$4݅9;:´րu<'syJa2XYܔ,Zz6s_:<`hsӇoR0Ov m䋫1ܥM:\9srpl'"jU \+,{'Rm4I"ZPȂyxb5ŵXJͱ,]1ɂg;XCrM;ڊ .5=vJvd=|K!w]?J endstream endobj 184 0 obj <> endobj 185 0 obj [186 0 R] endobj 186 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 613 0 0 838 0 0 cm /ImagePart_2057 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 262.8 718.95 Tm 102 Tz 3 Tr /OPExtFont3 11 Tf (3.3 Nowcasting using indistinguishable states ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 102 Tz 3 Tr 1 0 0 1 122.4 676 Tm 97 Tz /OPExtFont5 13 Tf (generate a large number of model trajectories, called candidate trajectories, from ) Tj 1 0 0 1 122.15 652.95 Tm 98 Tz (Which ensemble members can be selected. Ensemble members are drawn from ) Tj 1 0 0 1 122.4 629.899 Tm 100 Tz (the candidate trajectories according to their Q density relative to the reference ) Tj 1 0 0 1 122.15 607.1 Tm 105 Tz (trajectory. There are many ways to produce candidate trajectories. Here we ) Tj 1 0 0 1 122.4 584.1 Tm 103 Tz (suggest two methods of producing candidate trajectories. i\) Sample the local ) Tj 1 0 0 1 122.15 561.299 Tm 99 Tz (space around the reference trajectory. One can perturb the starting point of the ) Tj 1 0 0 1 122.15 538.25 Tm (reference trajectory and iterate the perturbed point forward to create candidate ) Tj 1 0 0 1 122.15 514.95 Tm 96 Tz (trajectories. ii\) Perturb the whole segment of observations s) Tj 1 0 0 1 412.55 514.95 Tm 55 Tz /OPExtFont3 13 Tf (i) Tj 1 0 0 1 416.649 514.95 Tm 90 Tz /OPExtFont5 13 Tf (, i = n+ 1, ..., 0 and ) Tj 1 0 0 1 122.15 491.899 Tm 99 Tz (apply the ISGD onto the perturbed orbit to produce the candidate trajectories, ) Tj 1 0 0 1 121.9 468.899 Tm (i.e. the same way that we produce the reference trajectory. Although method ii\) ) Tj 1 0 0 1 122.15 445.85 Tm 98 Tz (may produce more informative candidates, it is obviously much more expensive ) Tj 1 0 0 1 121.9 422.8 Tm 99 Tz (than method i\) since the ISGD involves a large number of model runs. The re-) Tj 1 0 0 1 121.9 399.75 Tm 97 Tz (sults shown in section 3.7 are produced by using method i\) to generate candidate ) Tj 1 0 0 1 121.7 376.699 Tm 99 Tz (trajectories. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 138.949 353.449 Tm 91 Tz (Given ) Tj 1 0 0 1 171.849 353.699 Tm 118 Tz /OPExtFont4 11.5 Tf (N) Tj 1 0 0 1 181.9 353.699 Tm 75 Tz /OPExtFont6 11.5 Tf (cand ) Tj 1 0 0 1 198.699 353.449 Tm 100 Tz /OPExtFont5 13 Tf ( number of candidate trajectories, the Q density is then used to ) Tj 1 0 0 1 121.7 330.649 Tm 98 Tz (measure the indistinguishability between the candidate trajectories and reference ) Tj 1 0 0 1 121.7 307.6 Tm 97 Tz (trajectory. Since only a segment of reference trajectory is obtained, the Q density ) Tj 1 0 0 1 121.7 284.549 Tm 102 Tz (is calculated over the time interval \(-) Tj 1 0 0 1 310.8 284.549 Tm 26 Tz /OPExtFont3 13 Tf (7) Tj 1 0 0 1 311.3 284.549 Tm 41 Tz (2) Tj 1 0 0 1 310.55 284.549 Tm 53 Tz (2) Tj 1 0 0 1 314.649 284.549 Tm 75 Tz /OPExtFont5 13 Tf (:, ) Tj 1 0 0 1 322.1 284.549 Tm 86 Tz /OPExtFont3 11 Tf (0\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 86 Tz 3 Tr 1 0 0 1 138.699 261.299 Tm 97 Tz /OPExtFont5 13 Tf (To form an ) Tj 1 0 0 1 200.65 261.299 Tm 121 Tz /OPExtFont4 7.5 Tf (Nens ) Tj 1 0 0 1 228.5 261.299 Tm 100 Tz /OPExtFont5 13 Tf (member ensemble estimate of current state, we randomly ) Tj 1 0 0 1 121.45 238 Tm 96 Tz (draw ) Tj 1 0 0 1 150.949 238 Tm 98 Tz /OPExtFont4 7.5 Tf (IV ens ) Tj 1 0 0 1 179.05 238 Tm 99 Tz /OPExtFont5 13 Tf (trajectories from ) Tj 1 0 0 1 267.85 242.1 Tm 60 Tz /OPExtFont3 13 Tf (Neared ) Tj 1 0 0 1 294.949 238 Tm 103 Tz /OPExtFont5 13 Tf ( candidate trajectories according to their Q ) Tj 1 0 0 1 121.45 214.95 Tm 102 Tz (density, i.e. the larger its Q density is, the more likely the candidate trajectory ) Tj 1 0 0 1 121.45 191.899 Tm 100 Tz (is chosen. And the end point of each selected candidate trajectory is treated as ) Tj 1 0 0 1 121.45 168.899 Tm 101 Tz (the ensemble member. As the Q density depends not on the observations but ) Tj 1 0 0 1 121.45 145.6 Tm 97 Tz (on the noise model, in order to take account of the information ) Tj 1 0 0 1 437.05 145.85 Tm 83 Tz /OPExtFont3 11 Tf (in ) Tj 1 0 0 1 450 145.85 Tm 100 Tz /OPExtFont5 13 Tf (the particular ) Tj 1 0 0 1 121.2 122.549 Tm 99 Tz (observations we have, we weight the ensemble members using the likelihood of ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 313.899 52.7 Tm 87 Tz (30 ) Tj ET EMC endstream endobj 187 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 188 0 obj <> stream 0 ,,b6cJtt] 3 z Y/ |E buZEa҃S\# <a _Uq,a4\roQ#2pg]rs^`U_x7ox_G#6T2na2ϫ>n5ؕL]`&(mA`kAX 9ULnuכr$FK];&<y42kXͥŜ)W=rh<{%ߣ:!рZJUрۅmvӉe@ PMsНNZ &yL!pOvg9oC7NU@9Rkb{$ëk*PrKs Z}~VQ|B=F*rLbh0wlۡѨZVp49J7~1J(ȅ:X^lK2:h5_?+:-˼Tx>j2k-08J+= J!| -|s̲,8A- taJEi9 %kL a]Dh7fwǮ2tmm%ǬBax.޺ N4Mo~<ܦqxhBسz¸=ΓZ`v/F) 'ˍKÝuUvF5DRi&HoC7k06SYқ5򾙗G|cpPX:>jPĎ%=3r$ݱ! -#Lvs\B Yg.>< h9(;PZ&j~fhU&|8I^{V2 hQayhzE*s;C$2~%F9! ?uA9t|0h(3!{ P >ѱ:)s(V$"۱`NR.ĝH]ao ^'Ҽnxx{˫Z|rK磌'O#V&jJz}$Vl]&P4׫I6rņh qY箿;5M!h!/(+"RSZ:|[ q\rQ㋲-dr(~Jѱ/e y 7j a2L@ƚ-:[ ߨYvˀ)~5p,$9/N}g"uzY*K]lT~eyCSanP`Ճ0>~ \FtOFʾo%dQbfE9x9o #:ф-Tλ<Qǖ:k5ÒC:K|nƼ†0-D0Hi;ns?d:ֹ҃"D FaT M_^*:?:Tlh7X$AAzQyO\ʼnOI2+5oty3)k80޿Зktx H=qkCHG/_$S2En3ExTR@1jI ) ~71VH݁5/1(ɪ Cj`ƥ_cN'z+ӭHOF?Dh$*B' }?gܫ(kb$7)- x߹E1 /! ! }7{T:sȿT#."8^xI` !-ܐƢ| y8xQ& mM!b*\[z\vRl19}X{3 ~b ~4ԧ)Bi\4l6nSb FaK Enw+TSI״y \%C[ ,TO#f,>h#=ʳհP2:%9cľ5L7$)<)w ۟ \[ n>+>G8xH\~+E̔,cEXy>1Bmh=YLwzN EQ(]qkj \-ι4}ؘ5Z%:/!.PϑQZ.3:?,?{#p뤍JG2P;@zl#8O?OƼkj}Imb<%z$E@>7-iMk* #CڭyODO9ʹEw ?{pmesOPP 3&IYzn/gY*D*)@6+j#8GF)GWE˷H5&.cKU: %:)mT*t]TA!ϊ?OU]:n6w%3yoMJeHەTmP,3A^Wj x<$⾹h&Wo1ȵd#좫>gCYl] d~۰5&s᎝ڲʷͰѣ$<=B&"%H \zE좻I5u k\[1Ux8}=(gd1kzmQ4f2Kſ{9{+ @B5|Chrz8~|ui5*GyF"sn5 :Lg%:Sh@Q:`2f >ltqE%l^qĎGG%]XM`_"co5L}(į P7Ո1m{yL?C{h :WiM#?CelBRDMgFwBXhР8ϪXI%}\ Z)vgY%E1.ttpmB'(RqlM34HCY 75m4AB4gm<͑݁bJϸ%e#>8.{#+Os&A Qj[#OVa~K "C-&+/a i0Tbv9eH.<.*k#M5t3S;EDL]8BN56{~ 0uC@ (o6k<"_R4Ck:(LBVٮpz^ǾH )҇@+FtkbQq׮ޟ!A{n/ptYAIkל70:U K@Z`Z}x2/f TC*v^K:[Fp xUXpƭu/kB=(Y:8oxs3xԆbr*yjr\Ѭ )F#EFz߀v@u#vb/$Gëvv_0}f*.//'Sdm ڟX'wxC>e^jͦasPfGus׮ 8HŌY8W0-gG0>wx;FW|\W >RZ 1 b@F% Ov5Ð}JNJ TCSET8~s|H<6w37dn=ng?vvN">kJ3p р9 ?^y]͟]"@10Y{51S.U-*s>`Uwس-@e p* |-bC)49ǹCLM| ڨq2A#0,'hƵ%^ #To?w*+hA1;KF=CTh`6DOCJYĻjI(dmzGm`"RL 1E><;DZm5f=?6۴-^QD^& ߟ> ̪7UԈP]3gQJɒf'yJ"VɮqbSA zdNV@||*yrx[iXl.V,(541I(}o8*{ d~+92|Q ${])R̻&(C'$6[b$~N7}v$؍!6Dn; ְadKGNEoOԻLo0<{x;c+|@'X%Aj(UQO bWV_ء6;>赓qq沅Ñs4'<5='nJi >'4q%#alZl®9il'n?$ӽ{j+mz13AaacA2֊Z(xi_CkOqrHDp#UQ~#r\:+ODcڣ6.μ8Ԕ\yiA,,a@ $цaf t+6HKO.*Knk׳[ oZYV 7{*8YcR$MDD""\ Fnm,*>mv8]4\p Efj!>ݍ1ʡϳƳ<ϸoVu&P Ҧjj#RVv^} bHj>?_᱾3?\T[&^M֎0Nf>f``pBo-0ү6eM7T̷_sF5:3 I&\;py0> rG'y_Xa裣τ* v. 8ε] 4@qF 6F6_=w_{)o0h|]ص(O i.Z sJow#-BQy}wab`O VAS!FW5(T૏:]4\ǯ{kVU*9RޟqUܧl- " 1Yk3'ӧ{+dFtkI\&pB32YzPe^j/ ~W~G\(]Vɜ,#p,"~ msݝSzb/fE~I')^kW x}%zMcqHbw)v?z<'.*̽%f#fWV%3.Au jsr9@7 &KaPǃ +Sy=%˄f\@N$䭏$ -.VSdcK*iV Sʽt)T@WfaʅʎkA@'WYIJ*`PGNRυ*$&:W%6 bh=>EOhaSь9ѐn@y9ei;њO R!@ڭ)>^<\6HOSU$,¹3_$gp լ?e_lZXuidV{a:TV.:Vz.o5Iv]~HLEaL EԃdL1IqNh,xj%E?ˋG1\c' bI?CQ`YxrR&`܅m=5Ɇo'Tq=h6pz7vB>bϪA X%B h{ %=wco#)ϢJuFv8;U8VT,7-z-Vi7 ]#4p94n`cm6tM0׮9"4M tn?6/8h* ,Gm G%Qt ~NE S/YY3-y'##* vCƅ-6v `TFݣU#wf' tz?UllŘ/etic~w]BHI ћeONt06 xPvm&I =rKsN_ؠ+]UEG\7I&KՅ8D]t4MkC@yYO4!4e'52Nv<-19H xUl/5nIj lLG?Hƍom?V?54ٜ֮FX7C>_ UήF΍JrpTdkaJ\p!'X؞qCzUu4c5V7m]3bly+ <,WaJ0sK2m-%@kL{)[beńp$,ɝMM4@N< sP@lqꏡM:V1B"խ@ \gZ۫s|kS8+UL>$%lox|fR ր̶KeRʐz.%baV%Q]~usekxe-SGL`*2۪AͣV?.ko^+ gGDCx^?Q/ K٩DKQ9P'b*;|aR7JԆ&dU a>98F0^B;Q?& :aݘ+{V}޻n|3c3fDA_K;‚v4Jf$].lWBȡ*CF@KeF=u!bPl=x]!Ċ},WZW~RKh&3dJ7 BghבB)Mn5̔X8gk69+M[r'K#Un!|ISb|qxwS#>ǢS-=ޛ幍)񷧳?77x'A,;Դ'88掶E3ᱫ/(#<>ղ3ذ`/@Ė<5@'~ endstream endobj 189 0 obj <> endobj 190 0 obj [191 0 R] endobj 191 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 614 0 0 838 0 0 cm /ImagePart_2058 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 262.3 718.25 Tm 102 Tz 3 Tr /OPExtFont3 11 Tf (3.3 Nowcasting using indistinguishable states ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 102 Tz 3 Tr 1 0 0 1 121.9 675.5 Tm 94 Tz (the observations over the time interval \(-) Tj 1 0 0 1 330.5 675.299 Tm 38 Tz (1) Tj 1 0 0 1 331.199 675.299 Tm 60 Tz /OPExtFont2 11 Tf (2) Tj 1 0 0 1 332.899 675.299 Tm 64 Tz /OPExtFont3 11 Tf (-) Tj 1 0 0 1 333.1 675.299 Tm 32 Tz (1) Tj 1 0 0 1 337.899 675.299 Tm 92 Tz (, 0\). The likelihood function is ) Tj 1 0 0 1 491.05 675.5 Tm 102 Tz /OPExtFont2 11 Tf (given ) Tj 1 0 0 1 122.15 652.25 Tm 95 Tz (by: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11 Tf 95 Tz 3 Tr 1 0 0 1 261.1 618.399 Tm 60 Tz /OPExtFont3 11 Tf (1 ) Tj 1 0 0 1 278.399 624.649 Tm 75 Tz /OPExtFont11 7 Tf (0 ) Tj 1 0 0 1 285.1 624.649 Tm 100 Tz /OPExtFont9 3 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr 1 0 0 1 265.199 618.399 Tm ( ) Tj 1 0 0 1 217.699 610.5 Tm 112 Tz /OPExtFont3 11 Tf (L\(z) Tj 1 0 0 1 234.949 610.5 Tm 51 Tz /OPExtFont9 11 Tf (3) Tj 1 0 0 1 240 610.5 Tm 112 Tz /OPExtFont3 12 Tf (\) = -) Tj 1 0 0 1 260.649 604.25 Tm 61 Tz (9 ) Tj 1 0 0 1 265.199 604.25 Tm 702 Tz (\t) Tj 1 0 0 1 292.1 610.5 Tm 94 Tz (\(z) Tj 1 0 0 1 301.449 610.5 Tm 28 Tz /OPExtFont9 12 Tf (.7 ) Tj 1 0 0 1 304.3 610.5 Tm 136 Tz /OPExtFont3 12 Tf ( - ) Tj 1 0 0 1 319.899 610.5 Tm 71 Tz /OPExtFont3 11 Tf (s ) Tj 1 0 0 1 328.55 610.5 Tm 170 Tz /OPExtFont3 10.5 Tf (yr) Tj 1 0 0 1 347.05 610.5 Tm 92 Tz /OPExtFont12 10.5 Tf (-1) Tj 1 0 0 1 355.899 610.5 Tm 98 Tz /OPExtFont3 10.5 Tf (.\(zi) Tj 1 0 0 1 367.899 610.5 Tm 53 Tz /OPExtFont11 10.5 Tf (t ) Tj 1 0 0 1 370.1 610.5 Tm 196 Tz /OPExtFont3 10.5 Tf ( S) Tj 1 0 0 1 390.949 610.5 Tm 58 Tz /OPExtFont11 10.5 Tf (t) Tj 1 0 0 1 394.8 610.5 Tm 96 Tz /OPExtFont3 10.5 Tf (\), ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 96 Tz 3 Tr 1 0 0 1 301.699 608.299 Tm 60 Tz (t ) Tj 1 0 0 1 304.1 608.299 Tm 609 Tz (\t) Tj 1 0 0 1 324.5 608.1 Tm 65 Tz (t ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 65 Tz 3 Tr 1 0 0 1 494.149 610.25 Tm 88 Tz /OPExtFont3 11 Tf (\(3.8\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 272.149 596.549 Tm 63 Tz /OPExtFont2 10.5 Tf (= ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 10.5 Tf 63 Tz 3 Tr 1 0 0 1 286.3 593.7 Tm 85 Tz /OPExtFont3 5.5 Tf (2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 5.5 Tf 85 Tz 3 Tr 1 0 0 1 121.7 562.7 Tm 88 Tz /OPExtFont3 11 Tf (where ) Tj 1 0 0 1 153.849 562.7 Tm 172 Tz /OPExtFont4 10 Tf (j ) Tj 1 0 0 1 163.699 562.5 Tm 90 Tz /OPExtFont3 11 Tf (E {1, ..., ) Tj 1 0 0 1 205.449 566.799 Tm 152 Tz /OPExtFont6 7.5 Tf (Nens} ) Tj 1 0 0 1 240.949 566.799 Tm 57 Tz (1) Tj 1 0 0 1 243.099 566.799 Tm 94 Tz /OPExtFont6 6.5 Tf (r) Tj 1 0 0 1 245.5 566.799 Tm 73 Tz (1) Tj 1 0 0 1 249.349 566.799 Tm 83 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 6.5 Tf 83 Tz 3 Tr 1 0 0 1 240.949 562.7 Tm 77 Tz /OPExtFont6 7.5 Tf (1. ) Tj 1 0 0 1 255.349 562.7 Tm 67 Tz /OPExtFont4 7.5 Tf (1 ) Tj 1 0 0 1 258.5 562.7 Tm 92 Tz /OPExtFont3 11 Tf ( is the inverse of the covariance matrix of the obser- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 121.7 539.7 Tm 93 Tz (vational noise, zi denotes the chosen candidate trajectory and ) Tj 1 0 0 1 437.5 539.7 Tm 114 Tz /OPExtFont5 11.5 Tf (zi) Tj 1 0 0 1 443.05 539.7 Tm 55 Tz /OPExtFont3 11.5 Tf (o ) Tj 1 0 0 1 446.649 539.7 Tm 94 Tz /OPExtFont3 11 Tf ( is then taken ) Tj 1 0 0 1 121.45 516.649 Tm (to be the ) Tj 1 0 0 1 169.699 516.899 Tm 155 Tz /OPExtFont4 10.5 Tf (j) Tj 1 0 0 1 175.199 516.899 Tm 110 Tz /OPExtFont8 10.5 Tf (ih ) Tj 1 0 0 1 182.65 516.649 Tm 91 Tz /OPExtFont3 11 Tf ( member of the ensemble estimates of the current state. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 122.15 472 Tm 135 Tz /OPExtFont2 13.5 Tf (3.3.4 Summary ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 13.5 Tf 135 Tz 3 Tr 1 0 0 1 121.7 441.5 Tm 91 Tz /OPExtFont3 11 Tf (In this section, a new state estimation method based on applying IS theory is in-) Tj 1 0 0 1 121.7 418.5 Tm (troduced in the perfect model scenario. A reference trajectory, which is expected ) Tj 1 0 0 1 121.45 395.449 Tm 89 Tz (to reflect the set of indistinguishable states of the true state, is identified by ISGD ) Tj 1 0 0 1 121.7 372.149 Tm 93 Tz (algorithm. Based on the reference trajectory \(analysis\), the ISIS method is then ) Tj 1 0 0 1 121.45 349.1 Tm 91 Tz (introduced to form ensemble members from model trajectories, therefore the en-) Tj 1 0 0 1 121.45 326.1 Tm 94 Tz (semble members reflect the nonlinearity of the dynamics. Our methodology is ) Tj 1 0 0 1 121.45 303.049 Tm 90 Tz (aiming to enhance balance between the extracting information from the dynamic ) Tj 1 0 0 1 121.45 280 Tm 94 Tz (equations and information in the observations. Two state-of-the-art methods, ) Tj 1 0 0 1 121.45 256.95 Tm 91 Tz (Four-dimensional Variational Assimilation and Ensemble Kalman Filter, are dis-) Tj 1 0 0 1 121.2 233.45 Tm (cussed in the following sections. Results shown in Section :3.7 demonstrate that ) Tj 1 0 0 1 121.45 210.399 Tm 93 Tz (our method outperforms those two methods. The Perfect Ensemble as the opti-) Tj 1 0 0 1 121.2 187.35 Tm 88 Tz (mal ensemble states is defined and discussed in Section ) Tj 1 0 0 1 392.899 187.35 Tm 98 Tz /OPExtFont2 11 Tf (3.6. ) Tj 1 0 0 1 415.699 187.35 Tm 89 Tz /OPExtFont3 11 Tf (Comparison between ) Tj 1 0 0 1 121.2 164.299 Tm 91 Tz (the Perfect Ensemble and our method is provide in Section 3.7. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 313.699 52.25 Tm 73 Tz (31 ) Tj ET EMC endstream endobj 192 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 193 0 obj <> stream 0 ,,b6"Bl>IrFfXso !h`$.lUeגUb2 o^D?nsO!Aa(tJ)t Bؽg&9?lj|D z%CifҟwqT*TR$/\֦;Ў66- ߜUsG'Jڪ /LZ!Q.yBẇXBf޿+ b,8] ($R&7*E`.;-q)6|P|`[v[G{QO[&,MM%D v]];\?|؃8SQۙ&͗NsD[Bc̤E-(Ʋ(K#p*ijⰭʢ+W4i{P{ 3Z~D61+tߨӠ#yjH&[P]:U'gM=1 ]=r&{e 4W'`MlVKk x969K*^`xj(\f+ ?ԧzDKO Ǧ{%C:e],N܇&+#SW4#( `ktُ5hUjPn?CQhqO*ӫm(0@ n%G,l1b <1OjrFYXBH_%}`u3E  w> TA;!ԬKFc&KO^LwS$$†bA$P[:'pCncz y+}Zo}s`9;>K!# wdY g),%Mc.:=3"z߶{iAɩrtv {LdeN'0G@^[c \߮KWP㹫%nd+EcA 6OsABT8to:TiƢ(Q&YxR"̘0]GQFht?G>^9mJ6 BʧHL2a visbll8-:] ?Qh`ZpUwee}j3dB‹:I+YAtڱ,=#!n_-h1GVFZ؇(8蘍kY!-a3S^1x尦,7C]PX;f,U[ʈ =zRߣ(E<`}B& 2^\f%"lLkۢ?9{<[- EjQ-wr3WQѢ{P<GE%S( vP+ ߚGI !p/ 48 Zoy zf.]ʰN/.=$,n)%ł(;,=n#?8Ul}:;58U}g 'z44P_:~c0ȇiJ>5ֳݶı՜3ձVF*$Sɿx#-4E!)?'O46%.#R)Yʱ9eʱԮ9,'>cIڳaY&a}ֳ,Ӏ1'*`<&moY~\w&ʇo3jVMm[xIyUef6uo@o~܋*tҭ@4< 3g[Ҭ-:G%=虋NZ:B|V)2kQΩ<ݞ<'E%Q(,xmU@D_$!-u*8.ULKV ڇ8;b7ܼq} F{|w cp*/Mrj콦C 9Ѩa[#LvFF7Mx\ι7GtYԶsE"sEN>@-UMql7u9+Qn",ȟ:sUoΤe׻HfMH?I5f`Bǔ8 ;Km6H%ZZ94o%#&%Tf`m'צ֜L{JwAօ<*59L_LKF5N's'_;D$xPak=u A|I9v]Kc o.-~. 6@Pי-$yAf||Ou?jto#σ: c7B5eG&qe>mO$+EuA5 cc\o(Dh}h]j!4;g oJjU-mB7ĒȂ7Х .4T"@qY/"/݈6?2#W=N7m^_J_n^"j>+Ahés[?Q0mTNofqBJuqN3YS_ D!O93W_z^AOsձi4{F4"|ɓ b(C&=b6NjPd%Հhbq KZ># x:>a7PPF8_*NiL5 EIknT)[So,z1﨩҇3)tnl%#¦>۬ӊ_vb(v Q(#FKQ{C~U-D(.h J7u}P&j{|NG6} \Sntu[k\2 L'=2yAg7?ڑC4l$q3?^OFm\~. g;^Kٮ$*۲oh[i" =u6G] =M9r.\A$.*5]=QhYߵ d{:-BM_Gϝqo*eǔTFPzH+se#ҳ&샓g/0n@AUވY mr } oE8N c8E9`S݄ UN-I5MtgS|SwyDU(_q٨de摻mQQœ)B3/}"7OZ$`#'!ua*'s83{f)>Cw\܍)ECYJl^ٽ:m\c6?dgk8Sb~)kXKooBy4Q]?cAR LԠ(O,vT~ k{4ZqDz;8:WiFJH9% w˕o?Ӱ8v6\Q^WNR-Ylc'RYx f&);E9xbȇs|2+t{)*A$=?Zō|Z wS'S5|zM  ? <|?`09Xp 0Q=XH׹z)_'#ʰ*ЉVd c n4(#@{^{89Z>ewߦ1oA5 XB B6XpSw J JT滏v''s%I7?;BCa.>ԝ4Wե؊CcoziB3v>+X }ϥv~@'9wۛpIm$5|;^h ~g#N̦5YpE_Pل5bRGЦ KĦހ:^Rj ֪ oݶ~#(^% H(x+.~0;Y?]D%Q3wR3 ]#yә _”)K<vz iZ6fɽHz2'%LAi`pTmtzV-p2w`w,HAOx7M?DelK((S /:黳噃2􋰭5oܴS;!v0'±n#ꊑ !isɲSge)p.@2 '=`,Qp4z-B0g EIU Ďr8i lKPw;@BCC]gcRRH AYt~S䨓Vi򛼲Fɖw \H7Q1R@[ۉt0#Wd%m -q7"ݺ-'?{YB 2*0h!J7/}\Jc&}R *єy-Z͖ߢEcy_T~heB Vw>: | &dC$-/Vpp~((9Z_OuZuԱͳMk(.Z) a-j? MUL66u=F2 r풬 ZA̴d. 3d. їU2O 峾tFDړwĎ尿˰Ӹ?ްz\EJ@47.Ŷ7C}ܟžˎEǦK+[L2z{,'7 {GOk m@wAz"c(\j[9Tz،zgʇ[$%P™53MbjDz_g8N i!v&}EЏiZ-2KkԆӽ"wt@+[߉l"SXK+̅V=mRbu57AR+}XFbG5W\pd)MY+ ;<Q=iSoc}=J3iRe =uknѾ!)̉("ECpjՔˉő[RkF,$ٗCڐfH s4}Bz$^N/F}|BDxڲ't0y0kkebN!P+ 0$7+xP\eɆ^pvmb_sϽ07Leg[ Zߴ@ahOc(Gڔ@QFIt%Q}"z,T!{<ٗji !e,#bop endstream endobj 194 0 obj <> endobj 195 0 obj [196 0 R] endobj 196 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 613 0 0 837 0 0 cm /ImagePart_2059 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 453.85 717.95 Tm 125 Tz 3 Tr /OPExtFont2 10.5 Tf (3.4 4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 10.5 Tf 125 Tz 3 Tr 1 0 0 1 123.849 675.25 Tm 121 Tz /OPExtFont3 16 Tf (3.4 4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 16 Tf 121 Tz 3 Tr 1 0 0 1 123.349 640.899 Tm 93 Tz /OPExtFont3 11 Tf (Four-dimensional Variational Assimilation \(4DVAR\) is a widely used method of ) Tj 1 0 0 1 123.099 618.1 Tm 95 Tz (noise reduction in data assimilation \(18; 19; 90\). The method provides an es-) Tj 1 0 0 1 123.099 594.85 Tm 93 Tz (timate of a system state by using the information in both model dynamics and ) Tj 1 0 0 1 123.349 571.799 Tm 90 Tz (observations. 4DVAR looks for initial conditions that are consistent with the sys-) Tj 1 0 0 1 123.099 549 Tm 93 Tz (tem trajectory by taking account the observational uncertainty of the sequence ) Tj 1 0 0 1 122.9 525.95 Tm 94 Tz (of system observations. It aims to select the initial condition which minimises ) Tj 1 0 0 1 123.099 502.899 Tm 92 Tz (a cost function which measures the misfit between the model states and obser-) Tj 1 0 0 1 122.9 480.1 Tm 93 Tz (vations. During the application of 4DVAR, the minimisation is carried out over ) Tj 1 0 0 1 122.9 456.85 Tm 92 Tz (short assimilation windows rather than across all available data \(Increasing the ) Tj 1 0 0 1 122.9 433.55 Tm 93 Tz (window length will not only increase the CPU cost but also introduce problems ) Tj 1 0 0 1 122.9 410.5 Tm 92 Tz (due to local minima \(65; 71\)\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 123.099 366.1 Tm 116 Tz /OPExtFont3 13 Tf (3.4.1 Methodology ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 116 Tz 3 Tr 1 0 0 1 122.65 335.399 Tm 90 Tz /OPExtFont3 11 Tf (Assume the observations recorded within a time interval ) Tj 1 0 0 1 400.8 335.399 Tm 121 Tz /OPExtFont8 13.5 Tf (t ) Tj 1 0 0 1 408.699 335.399 Tm /OPExtFont5 12 Tf (E \(-n, ) Tj 1 0 0 1 444.5 335.399 Tm 89 Tz /OPExtFont3 11 Tf (0\) will be used. ) Tj 1 0 0 1 122.9 312.1 Tm 101 Tz (Let x) Tj 1 0 0 1 149.5 312.1 Tm 63 Tz (t ) Tj 1 0 0 1 152.15 312.1 Tm 120 Tz ( = F\(x) Tj 1 0 0 1 188.4 312.1 Tm 63 Tz (t) Tj 1 0 0 1 192 312.1 Tm 91 Tz (_) Tj 1 0 0 1 198.25 312.1 Tm 80 Tz (i) Tj 1 0 0 1 202.8 312.1 Tm 93 Tz (\), the 4DVAR cost function is: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 211.449 245.149 Tm 118 Tz /OPExtFont6 8.5 Tf (Cldvar ) Tj 1 0 0 1 256.1 245.649 Tm 70 Tz /OPExtFont4 8.5 Tf () Tj 1 0 0 1 257.05 238.899 Tm 112 Tz /OPExtFont6 8.5 Tf (2) Tj 1 0 0 1 264.25 245.899 Tm 244 Tz (\(x, ) Tj 1 0 0 1 289.899 246.1 Tm 65 Tz /OPExtFont3 11 Tf ( ) Tj 1 0 0 1 300.699 246.35 Tm 93 Tz /OPExtFont2 10.5 Tf (xb ) Tj 1 0 0 1 313.699 244.2 Tm 65 Tz /OPExtFont3 10.5 Tf (n ) Tj 1 0 0 1 319.449 246.35 Tm 136 Tz /OPExtFont2 10.5 Tf (\)TB1\( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 10.5 Tf 136 Tz 3 Tr 1 0 0 1 339.6 244.45 Tm 88 Tz /OPExtFont3 7 Tf (n) Tj 1 0 0 1 355.199 244.2 Tm 123 Tz /OPExtFont2 10.5 Tf (x ) Tj 1 0 0 1 368.399 244.2 Tm 65 Tz /OPExtFont3 10.5 Tf (n ) Tj 1 0 0 1 377.05 246.85 Tm 95 Tz /OPExtFont2 10.5 Tf ( x) Tj 1 0 0 1 395.05 246.85 Tm 49 Tz /OPExtFont12 10.5 Tf (b) Tj 1 0 0 1 395.3 244.2 Tm 166 Tz /OPExtFont9 3 Tf () Tj 1 0 0 1 401.05 244.2 Tm 242 Tz /OPExtFont3 3 Tf (n ) Tj 1 0 0 1 405.85 247.1 Tm 210 Tz /OPExtFont9 3 Tf ( \) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 210 Tz 3 Tr 1 0 0 1 495.85 246.85 Tm 87 Tz /OPExtFont3 11 Tf (\(3.9\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 233.05 222.35 Tm 56 Tz (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 56 Tz 3 Tr 1 0 0 1 232.3 206.75 Tm 95 Tz /OPExtFont3 8.5 Tf (2 ) Tj 1 0 0 1 237.349 206.75 Tm 920 Tz (\t) Tj 1 0 0 1 262.3 213.95 Tm 111 Tz /OPExtFont2 10.5 Tf (\(H\(xt\) st\)) Tj 1 0 0 1 321.85 213.7 Tm 92 Tz /OPExtFont3 10.5 Tf (T) Tj 1 0 0 1 328.3 213.7 Tm 200 Tz /OPExtFont2 10.5 Tf (r) Tj 1 0 0 1 336.699 213.7 Tm 85 Tz /OPExtFont3 10.5 Tf (-1) Tj 1 0 0 1 347.75 213.5 Tm 108 Tz /OPExtFont2 10.5 Tf (\(1/\(xt\) St\), ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 10.5 Tf 108 Tz 3 Tr 1 0 0 1 240.949 201.25 Tm 101 Tz /OPExtFont3 7.5 Tf (t=n ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 7.5 Tf 101 Tz 3 Tr 1 0 0 1 122.65 169.549 Tm 89 Tz /OPExtFont3 11 Tf (where x_ is the model initial condition, xb , is the first guess, or background state ) Tj 1 0 0 1 122.9 146.299 Tm 88 Tz (of the model and 13.7) Tj 1 0 0 1 226.3 146.5 Tm 42 Tz (1 ) Tj 1 0 0 1 229.199 146.5 Tm 92 Tz ( is a weighting matrix that is the inverse of the covariance ) Tj 1 0 0 1 122.4 123.25 Tm 91 Tz (matrix of xb. The first term in the cost function is usually called the background ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 315.1 52.2 Tm 77 Tz (32 ) Tj ET EMC endstream endobj 197 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 198 0 obj <> stream 0 ,,Qb6 ;dnl' ƶ&USA=o^d฼=z1>oE3ؠg7׺Ť6BM,*<mD 5g[ݚLG&3tr _F%_-2BJ zW92n^5Vg~%A=rGzԺL2):f?;TvlΉCq9"|NI}\x S_?g 7ґY#A<}SesK W"5X 댢<+xijvz#HO h6xŒ˨LW"qFz[`niWjƺ)TTy3^. !aBQ&XLWu_ yn"Q}bbD~ceoilˬg"RHڴZ&u#˟vuBds/ 2UPqbu ʅdMXnm̐3#Tsũj!渢fI DK1tD$v%.ܢd1g$ݽB| 4zr3?^w~:ɻ0M ז 6OdBy[xqh?E{?ȺzkEo^g:Ӏki%hQ#s% !:=mID/ېtR>-w k/ s3v5Z JЮN5L PqQ`_,$hX:^p*6H&6HF>Ӵi!-?㘬3#B; ӹytGPf܋{49c0mfʔ/bL-q;%'+5f;,-Ch{8.9iQ $\YAZp*&:͈~^Q>X}># oM-Z-bn|&`cf/ F IN_D̊Ri+}c;w#}͉uQ3vR//}ZTHߧ\ icb8&;gKH$8n6C!g<7_؇ػT=V99Mlm4Aνۤ!|}eq_T4߲}ff]b")$cՖcd i "׆t/?O vD/Dy Spg}޻qxHh\xS;vXJ{d)ڞc$pku|u09Wp?|xʷLgev K0S`o{  $86fRϫ5tD_qpYOi 8 @;.n] $S)wa#s zN܇G'3ex,3ߠ*P˙6QYŶ ӧÎt;baIQF^4$UمwQ8kr=+tr'i+Ho][|]Mu 9'WHz$$P.݌T ^<!f%f$^8= 9!'Gtm`3k=kaDž[wl(s7`7*oUZmX]2,s'-Ly;X`fJ@"THDV'fS'4("tIZeKܕ,hzNy?i^26p6PŸJ-GkqN:ۣ͙Ǯ$b℟Lę[\tYaiqr9ч$KӺ+Ķxee(ZS5z38 .`Fޢ{:^ m<IS8p6Ճףv9yٵ4/%FGAL={ Q!QBm3U^1eYK +%ėz(T -1? %dbtO.Z2. NOh~ߍvwFTbǚO$Ll0M1^دe T|d.#̴)wLvt7mث쨖=MN=5ÃRx!+l=,YHe ZL|)Vx娸aez_S+!k9/a 8~6.aZfn4T{`#ڑD&mcx3J4 y>'zH(;T/QAs01V0'*˷1{=xt>oڞ0mhxf]ϐ~EhR9iX-4ZbA@!i@I~&[AR9YB!zAdDnQTL'XTLO3d>b  {X2]l1 W'r}9Z|<ׄ;aQqd~i:^Vp?q %%dlKo=kKEo䜣 t.Sw/*G,1`?qЇXxO:Mh!Dh@yc}݈b2[̌{߾GxRie}-9VH Vhή΃PacnM+ uk Aцo\ ҳ4K;NW{*KI}W6Ψ>җv5cGԠewaNm3wWKr&Vv59[ Ӆە/|ǡ#A c::.TB QՖyEd3RFm0-Dv Jgiw>URň핤ġۯS}'Y&kGzGciɱ;"]K򫶌g|d|XxߏH5\ &k, y N|Grֈ7wK^~TNB рwaBOyh0+~'9 @3O¢xZi?u*P ,['`>rlet>l-LڿB|v҆еuߔ'T П9/;Qx K UzAdjB#ѭB= ɽrmQ|.@b$wyTe-0 S-l@͊.9eN^-b:ȑ6,ޚOoάexl} H7UdԷ+ыh]M3uU-=Y>\HD١sTc(OϷ֏-IbSLD(AyPS`%a$ɐDdx63-}?hǻ( /Z!7giyTQJlaP.!gK(;Ώ}BȞ^7Ne4z~qYBN] uŋ2?tpvt@"<zTNAH? vF+喋fPJlK݉S}F'7P9n7Z(=u?EvSt.cc&{k^ij,30^zMʞt .XEmű6(%<Ys)6~n~ B)j Œ'CvZ怬 35DVDCBg=.c e,Pj[{9HmKXK9wHyZr\˯C ߠ fR4lSEHfբ&F1MzOrT2lR-H+8Bi=Ш>ٛ]k#q/IJy$q.v\O5 }=[&#'5{Ѥ>.d;кoxbf_ =AS}e'%FGL׼Y PPa-J4l{u@ө72¨^ qkuR{ïHZ*wF2Mh3UR`"0J?4} V ^ wt|&Afy%X1dk`y\$H vtNʒ'"YnorŪ.ݫNYp)+ʬˀy(.r߇N7*CiL<ڍ^(RR+zqJMPZٛl"I} 2YP%R]uV"<[\2XKWA#'7E,ޖ%~/fXH 'Edh3Ŵ-qe$ mtIGcϗW-@ĺc­K085p2]~/Z_xEK[KuJ?u"B88R_^ԩ~asT慨ّӓ$Lk6C< Iut9crX]#"/eQ>+5{̍tUOpAҐdh_P)8!rT4|-cG~)w 8^A\j5ʁ*_rsTUuO*R?ɵj OYKQfKDs%n9+~ϔTq G]c_B{;1E@"h4  3>O?[4'󡨍/,ʵa *pWU6ٝ[ "dm]>(}4bc:a祀s aw/`8viiw(8e?`9\>LFΊTMYUV> A2>x0 B^Q*t Ӆ5:_%ە2 J$+3(*Rg! q٭0ݠ!;;d<tz3`=6F0b]0=B:UGEX87>s454W G!4oI_17?Qi,uT  #3\]?QToIKݚUwL0K ވ9P)} I0a?ן);3f_1 ?R1l8 }jci1ǁLs&*4 3WP&&~}4~9g3u5L!=>ka ܲ8v0G ل2#p0T!T恋:@E.<>6(i གྷjS/υ'r~ +(*5xD( p^foH$H;DVbFa) {ğogc_Zt1 k ' x1Tw<sH>;79b-2$&[ I z#¹㠬ILov3͋zZ٪ 9j̅ht>v=0u,9_㗑 ;n= J)02|lLx7pdf7V %XvEGa."p7h%.|faek,ep5{f>^> I&fb`3lp`Egn.Y% iklik.9GSBXݥ ݫ(&yF\^/NRcI܉)]ϽQ_;lUlV"փwC6ރ&I{qc^jB9jMѐ ΡX ORI]=:0}߈3QfokjgnzKf-t{, 1_d>i:Ê| |Ԙ] t# gxV$Ulwxs2!QTn6]]Q@>Qh̡GЧN\fˋ `9̒!K'ShSb䆃7@Z>"}Dفl3.\sКfrS䅗h-lYƒNL `7zH7KTѢd`=A՝ȯ%뚡6e\R+ _W>*"W9O*X³FF6!d}7~f ŠzE,#6sr*>(vK5I.8Hpm4tXO{[4Ӧi3XkZc;sBӈjm.6TroW;*= p?`} j 8/r Ȱ:H"Oi k`-o]“m{?t ȃNqfݤwA*Ї9`0a9+P]*h}Zy>GlሩQ޵yQ)7i~Xn܃?·p=K/*Wus\PP̼a | ݍVrp%>3 /Uދ,()WI9~fY U7AOpcxRn+c撰R8^pl$X[pl 'LM^g֝|ʝϱ+KBDpJԓm͘t/a;PUfV w>rF`>NWs]`qsxlrbtbIL?v=t#gJ.f}>d{ uBٞ` a]Z+ iޜx't ^BtLsdgB"'Sda}lrc GLit\/ަO.'YGj !>]݈|QLx֮l!5b: Ѧf1Mz;[xKsW?rɠl LmF/ڢ!tvشfy[wp̲G:c endstream endobj 199 0 obj <> endobj 200 0 obj [201 0 R] endobj 201 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 613 0 0 837 0 0 cm /ImagePart_2060 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 453.1 718.45 Tm 115 Tz 3 Tr /OPExtFont5 12 Tf (3.4 4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12 Tf 115 Tz 3 Tr 1 0 0 1 122.15 675 Tm 125 Tz /OPExtFont2 11.5 Tf (term. s) Tj 1 0 0 1 162.25 675 Tm 55 Tz /OPExtFont3 11.5 Tf (t ) Tj 1 0 0 1 164.65 675 Tm 117 Tz /OPExtFont2 11.5 Tf ( is the observation at time t and P) Tj 1 0 0 1 345.85 675.25 Tm 75 Tz /OPExtFont3 11.5 Tf (-1 ) Tj 1 0 0 1 354.699 675.25 Tm 111 Tz /OPExtFont2 11.5 Tf ( is the inverse of the covariance ) Tj 1 0 0 1 122.4 652.2 Tm (matrix of the observational noise. Hence the second term in the cost function ) Tj 1 0 0 1 122.4 629.399 Tm 107 Tz (minimises the distance between the model trajectory and the observations. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 107 Tz 3 Tr 1 0 0 1 139.449 606.6 Tm 104 Tz (By locating a minimum of the cost function, one finds initial conditions which ) Tj 1 0 0 1 122.9 583.299 Tm 106 Tz (defines a model trajectory that has the minimum distance from the observations. ) Tj 1 0 0 1 122.9 560.299 Tm 113 Tz (Such model trajectory is expected to be found in the perfect model case and ) Tj 1 0 0 1 122.4 537.25 Tm (the longer window is looked at, the better the global minima is expected to. ) Tj 1 0 0 1 122.65 514.2 Tm 108 Tz (In practice, increasing the window length will also increase the density of local ) Tj 1 0 0 1 122.65 491.399 Tm 106 Tz (minima which makes it much harder to locate the global minima \(65; 71\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 106 Tz 3 Tr 1 0 0 1 122.9 446.5 Tm 107 Tz /OPExtFont3 13.5 Tf (3.4.2 Differences between ISGD and 4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13.5 Tf 107 Tz 3 Tr 1 0 0 1 122.4 415.8 Tm 108 Tz /OPExtFont2 11.5 Tf (The 4DVAR method aims to produce a model trajectory consistent with obser-) Tj 1 0 0 1 122.15 392.75 Tm (vations. The 4DVAR analysis, whatever it may be in practice, can also be used ) Tj 1 0 0 1 122.4 369.699 Tm 106 Tz (as a reference trajectory to form an initial condition ensemble by ISIS. Although ) Tj 1 0 0 1 122.4 346.899 Tm 108 Tz (both ISGD method and 4DVAR method use the information of both model dy-) Tj 1 0 0 1 122.4 323.649 Tm 105 Tz (namics and observations to produce the model trajectories, there are fundamental ) Tj 1 0 0 1 122.65 300.1 Tm 103 Tz (differences between them. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 103 Tz 3 Tr 1 0 0 1 139.449 264.85 Tm 111 Tz ( Both methods produce the model trajectories "close" to the observations ) Tj 1 0 0 1 150.699 242.049 Tm 110 Tz (but in a different way. The 4DVAR method tends to find a model trajec-) Tj 1 0 0 1 150.5 218.75 Tm 109 Tz (tory close to the observations as the cost function minimises the distance ) Tj 1 0 0 1 150.699 195.5 Tm 111 Tz (between the model trajectory and the observations. If one initialises the ) Tj 1 0 0 1 150.699 172.45 Tm 108 Tz (cost function with the true state of the system, the minimisation algorithm ) Tj 1 0 0 1 150.5 149.399 Tm 106 Tz (will with probability 1 move away from the trajectory in order to minimise ) Tj 1 0 0 1 150.5 126.1 Tm 112 Tz (the distance between observations and the model trajectory. Only if the ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 112 Tz 3 Tr 1 0 0 1 314.649 52.2 Tm 91 Tz /OPExtFont5 12 Tf (33 ) Tj ET EMC endstream endobj 202 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 203 0 obj <> stream 0 ,,wwb6)RprMeKOAsJY],KNRfk+nf1h<.vSWR^X u\vCNv G&Fŗ$v9j<"6 )XSYt[?Ovr*\Y˫I<@SFqaڳ t5k(vQ-2>%HWf&3~Ǣ!”Òqwgn|Z0Q)u|GaʪO^u |D~O]RPԿ8v (ʼ?pa0b-<&. Eg\+ڙC4!9;}Y@#$~|G$?ރpMB]:} Լ6bI]/Hp(^6iJ2ҍCu!pSqpH9b'gÉ]2@c4CUq'rޙ^ۀ޼":{j]V)hK @xS^ޒҔN'ͩ|NͿ2;5 Gڶ[唔H-v3 u`ZN1=B8NHY6]kR#0vg^E ;>0=f\¤UT=>6l4x8+*0YzSV|j N5{`¹mڢKJ"4G8Xtߝ$)7siC:r|y+l}+FYt,/y+TLM#}\s8n"ܮȯRa^`gTڽ&`8_ Z"}}f9CZ+:*vC /KVuO4YE3i^xjf/1v#Ժư 6 I'A,PBѱiry J9!8"Hu;i(}b͕p:" a E$C[)-F߻+TԇbGYv-m ?>)sNFA\$kE).䫬\;ÿ%k4efng]4$5ۅP6ʡ}.ͺMdb2ٴ[(u|]pfí{݌AvÕ'$AG,bʂcqa33{\v5ICrOɺ:TC"IjA(D m>}eϬf\̔-DuoeUR}}DISO% }(vZD̫фWƭHm|4h$=UR-!U2O;t{I)v"G4.E~2R:F똯3PyU$T. =]p.@2*JSmË DwmW]Xs?Z^W7y*ז%, @4Gʱ\||БcW WAl@Hz{V\h[\+Z_<Ēwa qyrW`dStpJ4AheBRқ ʾ?d;MR]Y&" t eKr3$VMcuylq1#!cJ4"O"#,d)lX0AT^%4eޡH7UiIC />~3(%՟l]:'t׃^&7-<0r̤guу/{I?@Hb,?퟽HՋux>+6ܱظ鹆#*ZD䳢2cرאΨ".q?kȱ&'<3k4r]B#{Ux}*:Ĺ;;,d> U; IBꨑX~lT̆LHYjs&͸݊]4z?l-(L[j7 KŃǑ9T8wY+z!iqzSԛu:g wB0hI f X[Eo맗406Ok.SykH^eSB 9Ck;XEKaX7f{fm+|NX(_=fla7i:=Oqr[ίTTo֏9ʔm P;% h~#(ipin l+O ISh"c߃ۺ y0@}?!11`a¢?NX%3|*0+"/wW3U9jȷrMxK {ڡ$`IJzyr)EtƭӃ~Mjɫ!+U^m7H2KouitJCXg(6BЊmH㭇 Ac[d D̂۞[+0cq86A磺0<95KGl䄟(Ր;Q{kpuRt դ2m#D~ZLlYtg'g7'-m}]ŧtѼ5=VRCmpyZrd[ε>Y(0a@scܓKB-$PŬf~Uy$yyTcxylG/$(HyAaӢ`(N.ݥ]2 4tX?{Iaw{lXCCLdŐmRcYT`n± (&#F4LxH@2@r:[eu^E2J4C5kSBWxdor|s6I]l#?q  *F9fH,; $遅*lfqV׽LzVq#|a)hY컍t /f2}e159|g$ 19`>Аᾱ^[,{<8ouPX2oA\ЇgO1`nFW٠qwI͖یE6Ҷ帣bS1opnj=VK^UA M)]6`:ZILz Jz~2ϼfq Fsq>kIa, A:N=L$8zrC.2+ry`|`7q4n3ٍ%tP&]>Kl NP)\F3"ډ7CF])KcTo.VG^W ggQ74 uO4  ^2Ds:' |3[GPۊ, F HU,4VG㏊K&N8N#N5EC&S9Qb5W[^Յ  %~${cAJ`ZoRLЍgw@r@风e@9#yW]iJ%Qn:5Ny.l+)J&u{dtil&:6.7NnGpYyڴ "Р1TtR$vܦ.+:rr_qٷ^KV5;S eE9ңEeʆ-A4x 8 sn7ֳ*׏%(whV9azw6:AQ[..SR3P/҂kuaUc! $M L/GvB5yn{K+l~&ԚPHgF}6ͿU+gqtcZnMyUs[ l!fv>eD~ԅ !" hJT7īp`*09 WA0> zsb8H}ݵ租;sJ ,1IUQdLoKH<& {6{U}D]aA}c]`bx0C)|8s OJE~=@ cB>`D)|.vk;'9oª&6w^EQ%|nǶN5=>C)tX^ ÊfFf_; %n% \M=h7-eM Z5(_EWȰz?K5i5i-eὁel_|3ʃM|ڰ`Isu+B=nOAMRy-?سW12.ܕ\rYєYXz*,̡B(_tK#q%[vt%ݜ NG /*׵:WOo^^(ciF -mM h1F,E0YЄΚBxqiߢ3S?WKՠŕT'bUFS9&\f:Cn<$t xb{P;.ȲIX-g2ύW xFc} 5E#B"@srgI3Alp* f2v9sPRȐwQi颕@vD%a۬MyǥW4gY|R`՞9$%J$41j*5x*7vmA=WL[z{33hOqxk KMٿfUGtOK0>8`/`$ʫ͉Aa e_MFv?HU^ZNJ,o꺘R!Z9D#A  i( Dn(ǼED.suxŗ{ ÷`Ӻ> ;ψxP8>=?z›;0|0{ h7T0сs`B(Ac3C;EҁPi,P@f8pv5BcI4wU܎Qc%ҿكH_=<[%,N8q[z8v%L5 >;VZ'&4h }3!1)G&Qj16O91lst/.hi aGZ+/ @yb,_ͅ>{!L3d߲LS5bh=o%q[eu6OSRRQKG\ 6OخB%.Q {*\=VyK=YL~6Ɠ3B$iQJqeՄdOiZm8ˍN~n -z'"HΜ\ vA_0}ix"lS*yg 6?J췠!2ηcMT\זDkz1RI?]EWrA<;// Vr\F ij;2{K~/lwVy7rͣ0FD. TK+|{M羽s a fcX@>bQҲwK옽 AQS|.dOd9ٗ_aXSY`Pe<((rK{]/Xq;aČ?xkۀ% ϠZ%%58 D[ bOř{.C{p`p7b~ܢp:Q@p ٜ0MHqؽrh4=J4w0o #nҕ3k!a*),kgH`<Z $όrNS gF矺2 O [AA(}6]ߏXAgQs{] <պHhو.yXޛ>Ll|7D#$> z[P-CE>@"u[.}P"wk#f.,v6 1ER^ ٰЃ׎ms A-5D 2e< >Za 1\WjR~JwK~.,ڇ8.y2bV=.9+!1bRPv\B~/@S!/Lۺ1V0;STΤc1s}PL؜D[Jk Q' o][ 4?ܚm#9VvѯY(:.yWJ;9s8,=<=S1rA1pcM$mRP@0}_u`/5[2܆z䙋6fqJqt=; (QOzMeuO Ǫϗ!2G1KAE$Z:]PАJ7rK#5Ax Nh2|-YGvCu N\~o0` Kd'rX;3 xeOzInF>S|Lm҂/rDT.$}9 8$r̩iwg]m|d+͎JgU\0hA.:/j/j,%G@/'4=WKFO(x8 cr^ G淏 ;=i[BZ.XY;QH,7xTcDmtݺEV.iIM?x>~Ac;̘f]X:bY-;qB?S9_ ,ʳg WX [62qƻBzسkQ)B3\ƹ\ׅYwsS9!(p\XNJ[7UC৵Z ĮqE7 9/"s¯g]AxaS<*mF\Amd_x2DK.C*+UN ;1¯^ذiTǶ)}DZ}cT :V4oU-ᰶ稊A~敊Sqݚecęd]D IIdx?uDe `Kms5v}*6 -T_4s8(5lqdu7N֜=´\6ǣ֠s̙$RLgGȍcw)P$CE߅8]qt!1prc饮fүuQUziQ>ҝuY9jpr(R&~}뉔 endstream endobj 204 0 obj <> endobj 205 0 obj [206 0 R] endobj 206 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 613 0 0 838 0 0 cm /ImagePart_2061 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 452.899 718.7 Tm 102 Tz 3 Tr /OPExtFont3 11.5 Tf (3.4 4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11.5 Tf 102 Tz 3 Tr 1 0 0 1 150.5 675.5 Tm 94 Tz /OPExtFont3 11 Tf (window is infinite then this does not have to happen. In practice 4DVAR ) Tj 1 0 0 1 150.25 652.7 Tm (is applied to an assimilation window with finite length, the cost function ) Tj 1 0 0 1 150.25 629.7 Tm 91 Tz (forces the resulting model trajectory to be close to the observations, which ) Tj 1 0 0 1 150.5 606.649 Tm (may cause the estimate stay further away from the true state. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 150.25 578.799 Tm 94 Tz (In the ISGD algorithm, the cost function itself does not contain any con-) Tj 1 0 0 1 150.25 555.75 Tm 92 Tz (straints to force the result staying close to the observations. The GD min-) Tj 1 0 0 1 150.25 532.95 Tm 93 Tz (imisation is, however, initialised with the observations in practice ) Tj 1 0 0 1 484.1 532.95 Tm 41 Tz /OPExtFont5 11 Tf (1) Tj 1 0 0 1 488.149 532.95 Tm 105 Tz /OPExtFont3 11 Tf (. The ) Tj 1 0 0 1 150.25 509.899 Tm 88 Tz (states one achieves is on the attracting manifold that is close to the observa-) Tj 1 0 0 1 150.25 486.899 Tm 93 Tz (tions \(48; 52\). Unlike 4DVAR method, ISGD method does not require the ) Tj 1 0 0 1 150.25 463.6 Tm (pseudo-orbit to stay close to the observations and actually ISGD method ) Tj 1 0 0 1 150.25 440.55 Tm 90 Tz (forces the pseudo-orbit, on average, to move away from the observations as ) Tj 1 0 0 1 150 417.5 Tm (the minimisation goes further and further. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 150.25 389.699 Tm 93 Tz (The results shown in section 3.7.1, indicate that 4DVAR method tends to ) Tj 1 0 0 1 150.25 366.649 Tm 90 Tz (produce the model trajectory closer to the observations than ) Tj 1 0 0 1 446.649 366.899 Tm 88 Tz /OPExtFont3 11.5 Tf (ISGD ) Tj 1 0 0 1 477.1 366.899 Tm /OPExtFont3 11 Tf (method. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 139.199 334 Tm 97 Tz ( The behaviour of the 4DVAR cost function strongly depends on the as-) Tj 1 0 0 1 150 310.7 Tm 95 Tz (similation window while ISGD does not. In practice, the number of local ) Tj 1 0 0 1 150 287.7 Tm 93 Tz (minima in the 4DVAR cost function increases with the length of the data ) Tj 1 0 0 1 150.25 264.649 Tm 94 Tz (assimilation window \(71\). The model trajectory defined by the local min-) Tj 1 0 0 1 150.25 241.85 Tm (ima stays father away from the observations than the one defined by the ) Tj 1 0 0 1 150 218.299 Tm 91 Tz (global minima of the cost function. The results trapped in the local minima ) Tj 1 0 0 1 150.25 195.299 Tm 89 Tz (are very likely inconsistent with the observations. Gauthier\(1992\), Stensrud ) Tj 1 0 0 1 150.25 172 Tm 97 Tz (and Bao \(1992\) and Miller et al. \(1994\) have performed the 4DVAR ex-) Tj 1 0 0 1 149.75 148.95 Tm 94 Tz (periments with Lorenz63 system \(61\). They all found that performance of ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 94 Tz 3 Tr 1 0 0 1 137.75 127.6 Tm 136 Tz /OPExtFont3 6.5 Tf (l) Tj 1 0 0 1 142.3 127.6 Tm 91 Tz /OPExtFont3 9.5 Tf (One may initialise the GD minimisation with better analysis if it is available ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 9.5 Tf 91 Tz 3 Tr 1 0 0 1 313.899 52.25 Tm 81 Tz /OPExtFont3 11 Tf (34 ) Tj ET EMC endstream endobj 207 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 208 0 obj <> stream 0 ,, jh]oҹrcLsDg4f ɵ9eq<d*/LѦك/]M'@_5ZUBTd.[lMGM \L.5t˛ }r_NHn$hjU y:@Q!lbZR).ZtN)L tI^ y(%M*h긾a"1|.nɕJem4(%f,\ ;]et[g!cÎ5qVsp~6ScjC"|N[,41E~D@~$@RZxYZ5Sw9޼ٺPQ;Y\ aB FitSVD b"Yx,*6G8Xv~޺Pf(PBҁŌPV]\TT]6oy Tɫ6}΢89`I!NԌ$ߍϭJ3 H?UrҐwV!1*JlW/=:M[oDw$ rmp1wi݅L+{+'wlhӉ7AY;Xݝ׈I95;GzJ1MgT?e)zl7:+Um$sd;7A/xᣎ>3]!Ck fM *8z!H& #lJpb-ߤ>mMG2y;@#]dGO32{g7Chl$q ĪAi89'7dֺ|ѷ 2 (&ꩼdz K q-V5[y9=b (;MvX@\~ny[[ES2OFI|,$AOD1 V"!1'G{ReDT㌿7*a#V){2M.BIOC!W vbo~,k3Fy9Q@;-XČW&,pE~$^6q7u5=K+s< WôUop?B>L# Z-?Dzo1ȿю-(gT_\TC&}L֔"[X\hyLw1A(H'h5*93+Rç:-8ʨψLr.p[#liP@e7 :a*/W )A歺3}Dxƥf|>о̶>[Eks< ntQuNE`Rz%:Pď:@qC*+yƹI:B<;mda%{|}(&5rg} YZ] ԀdXW`䆣mB gvoL_-&TB}y\%xSTC=A@ӕzQJ4zPmc-ٹyti; E&QS}|*M^ۿ= 6{RNb@+L:gV#mO4!8g 5RC <{Do1 bň 8Yx/o+m~qW>lWx;}d)%ιȓAuy磆w$B}Ζ/ A kS8V=H=5A["@HTބQpw`k) 'w*mijGDN *bk=+`d*" 86]‰ ?IV<}zݧF/'p;0e %?!6ߩw֥#<(ϼdͅv5٥%=FauWAhҤ0يӤ A8}/@)];_8rxeN ? I͎7>h & TaбTJضg4Ap MO*NV5,(eYkc4-).O;3h,>gfrxvjC4/mY (;fݶ<d]jljʛT^y[;eAٵr~Zd-)Ĺx1n${020ߟM%U9G/?"'^p'#E[3A];~ԻDnWc+%yeLNԭ7n`KR\hh .>a "Vw9+&C Ep\y>uhh7IMat=uh9a'l[}!eZgߠ!3.ޗP`jD9}e+gN:/հZ ʺDp=[' .}yP*]b",OPz[cbL&rcϥ+}iD|gbO g#C3HQ@FWknH$re N Z6xD/D$ Ux׿)ދl[yp7vgw,T\lцS3OE{t ,#( GpgSrjt }`vIPQ|◭|5MeMo=!W[FLv> l}`LaҩrsGe>o؄eXNa viRL t_aaE:D27}on2v,izB{*=aS? dPؗT0jsr8:ݿ@#UFz219BWlP^cdŗrE/ZiD*~~Rpqqu D6Y7O!R5npl(ȯMr{w5P2H(d*qU;Q^v`m}B-(w5D~}´%ߍsIGx\;QodGr 22@Y)༴PmG}nvIO`B.:_pȥt.bD*MQ=ؖȝ[ si:1qP%ޅDo@,))[b!baoIy.!=O!̓NOnXd )'v"Sf6#jd.D@f6N xI 3T":vZ_iZ(ecqO/)Sq9wE.sg 3%gÛ-8 }-(aMÌxm, fL߆c41k&ׅ `()R5&Vg1j[dʝ@#*GT-ѡ5XPAdn+o nTBm?nāk;WeޗC1Jp=_?>Cm@aM6_i 9b@B7IZo6@\EjZ) 4zr/=f楨[cw~r D[ b .yXbZ'^tJs"Y<'t8 0Fk<1 =zL-UE[Byeb2/S,bb |d Ý&I0Gz`u7_8y&L Z+X0{Yk'&x]0λ"Rg:S#ѫQp3cTRm@'7=L'0kg08 zmB-gh^<6,Œaa>'0E:޲Y)Ϗ*ރ-ӽ={4ɱђͫsxPȳ&ݳٱշU)r83Xy_w6 øGe_SY$ND肗H։W>WiIY6p@"1-΀+醬* ;]28^ѹW%*CPhnI툗m]&咸p[7}MrRܖpL3>CEWY,RI>MA_EMz]1z'_pˆ"Uxq}NEo@t+2J ZlNJt?i ҙ{3:Su~澍NX6>!@껜P)HBۈ㤿AEFiI#ۋPḄFdRFlf3b PX9>[%$2;(a/ڈɏʌٳyѴ/緡Dщ G($f®^! 9B*xŽ'qPe}iCO+ . ~ ,1Ϩ׼!Au.ԢK1kT b}% N ɼ^av[sL>`3[Qbf]UL\1}xvͦǜKպ`1Y~ P0OcK~MapWjͳQΖw[G[ssemm|`eW~u*#ϭ[θ"Nnb;T}dKޙkAߦr"8-hW5),J5ʍGFNIYՎ\MB˼;N嫚)G";0@0~xtwɎIm/\j 8Ffݗ˛7<mz8`-FM\΍ g^:|9$Z>'#ԚnmȇEDju\o~EqaNGv0:p0A͏BSwGV Ck%󥼅#:?:FA0 <0DHWS4-+uK;;he|5L+3MH]G* Yd! HzrUvSTa*`n_vG)E(GwlP"lN&r%t6wbUJ֓ZROIܦ|dB5ǀ b]U6SXr{7 Ex= RH'W hB$ B)8eWD*D1 ~ì2о7Xm: ԣEJ^AIF `|lvkEcNSl):rFKoLq3uT"`78gH[ʐNʾX-:aK{[bRί2?MHwqo0[ab 6al= |,AZ^͔6lf c7ncatR7~J%3XtkhM,W]b Tt S_8/H%1[ N+0ʀ0 }o\i 1W'^Ly-XNzE8uιyX S-,Ut!\K7U|A>M6V2^s2MT" 8qy2ї E]O!~,Z>4$-)m3]ޕ+* Y,RzYW1> endobj 210 0 obj [211 0 R] endobj 211 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 838 0 0 cm /ImagePart_2062 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 452.899 718.7 Tm 102 Tz 3 Tr /OPExtFont3 11.5 Tf (3.4 4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11.5 Tf 102 Tz 3 Tr 1 0 0 1 150.5 675.75 Tm 90 Tz /OPExtFont3 11 Tf (assimilation varies significantly depending on the length of the assimilation ) Tj 1 0 0 1 150.25 652.95 Tm 88 Tz (window and difficulties arises with the extension of assimilation window due ) Tj 1 0 0 1 150.25 630.149 Tm 95 Tz (to the occurrence of multiple minima in the cost function. Applying the ) Tj 1 0 0 1 150.25 607.1 Tm 93 Tz (4DVAR algorithm, one faces the dilemma of either from the difficulties of ) Tj 1 0 0 1 150.25 583.85 Tm 94 Tz (locating the global minima with long assimilation window or from losing ) Tj 1 0 0 1 150.25 561.049 Tm 90 Tz (information of model dynamics and observations by using short window. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 150.25 533.2 Tm 89 Tz (The mismatch cost function in ISGD does not introduce such shortcomings. ) Tj 1 0 0 1 150.25 510.399 Tm 92 Tz (Although the cost function itself has more than one minima, each minima ) Tj 1 0 0 1 150.25 487.1 Tm 89 Tz (represents model trajectories where the mismatch cost function equals zero. ) Tj 1 0 0 1 150.25 463.85 Tm 91 Tz (Longer assimilation windows do not bring any trouble to the minimisation ) Tj 1 0 0 1 150.25 441.05 Tm 89 Tz (algorithm using GD. On the other hand, as longer assimilation window con-) Tj 1 0 0 1 150 418 Tm (tains more information of the model dynamics and observations, the results ) Tj 1 0 0 1 150 394.949 Tm 90 Tz (in Section 3.7 show that the states obtained by ISGD method stay closer to ) Tj 1 0 0 1 150.25 371.699 Tm 93 Tz (the true state when the window length increases. The minima of the cost ) Tj 1 0 0 1 150.25 348.649 Tm (function are only model trajectories. And by initialising the minimisation ) Tj 1 0 0 1 150.25 325.6 Tm 91 Tz (algorithm with the observations, a pseudo-orbit on the attracting manifold ) Tj 1 0 0 1 150 302.299 Tm 90 Tz (which close to the observations can be found. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 138.949 269.7 Tm 97 Tz ( The aim of 4DVAR is to locate a model trajectory through the available ) Tj 1 0 0 1 150 246.899 Tm 91 Tz (observations which minimises the distances between model states and ob-) Tj 1 0 0 1 149.75 223.85 Tm 92 Tz (servations, which is the second term of the cost function. The first term of ) Tj 1 0 0 1 150 200.549 Tm (the cost function, i.e. the background term, contains xb, the estimation of ) Tj 1 0 0 1 150 177.049 Tm 94 Tz (the state at the initial time of the assimilation window. In practice x) Tj 1 0 0 1 492.699 177.299 Tm 42 Tz (b ) Tj 1 0 0 1 495.6 177.299 Tm 93 Tz ( can ) Tj 1 0 0 1 149.75 154 Tm 94 Tz (be obtained from the previous assimilation window. By having the back-) Tj 1 0 0 1 150 131.2 Tm 90 Tz (ground term in the cost function, it not only makes the minimisation faster ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 314.149 52.5 Tm 75 Tz (35 ) Tj ET EMC endstream endobj 212 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 213 0 obj <> stream 0 ,,AAb6TqG6 +apb)yoY}H :\A%R&$ȋHTp R^vϕ[YzzZ#m?/VȄ8v(4.sOrDJD2P1(A;f&w2Jsn5>ӳХ'>ž&#h,5.1z1?Kǁ⫧8dͭPj"kܱ'/96Y$ɷ*oٌN3+/&=vⲳ]b6J*.F]&+OrNEd {j3}O8iHİ֓VBywE C0?Իt c~ݤ+L Ԙ #bE\Ԃ`+L<4k7`{Cjyʰ`@װ_zfr=no'thrj͏*Vr>x1wBR%jK@ ֒P$'mJj~,73209r#9ELQK? CXNp9N!>5`Se ':I٪"í*]6E-J1'AM5sֿvXO_j0j\n9] $L`&3!w,O4*S5J41- Naa 0ѷ)o{k>.k78oޒn@F$^GN;G~;5 ϙ9pQ1W*s0Grc<冡=/ 4$ =UWoN]yėjE) \0yWo8'|1 ;o{xC5&qRgݜkvu7afD?[pHi]]zt:˂ l*OyI 9E0TQ$n-NerC/ XVߝ.^Os!1ߝ,Z^"Ǒ)b[򯒪Tq-VE.jQ `zd䌩U oݓ5(_r#O90 bI6h/k8  t&3=-8BwA0JQC*T,`; f,"0Np#':JyBDW\~Ǥ9B`g^;H?FZ,|۰) _E"y KdAO:" XZ:{^xdZ0t*DU;(E -v-LmNJV>ņ)T޼<H\)#jAi^ؠ+޹HG.е3|[Q=A&z=ȧ:SӽpLEʑЄBN{߀Rh p;+ Ab]L@W"N:v3R,RDx]<#Jy(@HGk:dسJS)+6IzX;wq%#m ._O{@T^U|3r4[죻bxKݳt5L1}=̰5 @ye6خen1 P)Zoh9U+ waW᭾0&S <KKu@ \kLt\`V.\-;F]Ng3M{#g^"Zc_{u!*0T#{QKDYB%3jFK,+!lӾ`'BbI<=}LHIE{:?T(HRsX{JS׶)ҌbtQğG`9p[ S9N>"t n9Yk8\G꯳f<  A+d0DE@}[n@0|tfOWxck|5 qS vZxtBgin-˓fw-"DkЋXƸvJ~wRMخ@t 1pk s=#0!GVD)**z.+*͢]Bj'DN~j W)4`0xG!j_.,4T/WK0VdLuY }tzP!~ Ҥd@nHQ2eê;9?v9G-v}:2hz<U UQWb!5m[Q+_B:* o0Hf##r!BLDU|5 >kV Dh7Үm4Pn4,f%6d!VE \3`WZ#kT$tPWĭ}Š mD i=E"'mk'Y-J!*ŵjKIf힛PM3^ v @v͓4)*C"4䒚8osA9ǸQp]Qe v{<=,.>^TSn XZUda{>Q+sx5'1zcl17uHY fs(EZ##8e‹DX[S|2/@bR2m2h|y!:lں 80RrH@ GfzlXKi|̄f\2)u3pG*=FuHtFwI^nd#Exv)SC ?UCu4ry':NT*oEAqŻ، Jy!ypO:Z9 Ƶ׻]yDjM7SGIŸЌt+/q+Xfs0ySi_ts.Kkd Jי:9L})5~kwPYJqa!|žG-4Wr#3^.}3eP"y8LaГYpCYګ꣐/IzI4G`F^36%l{|3[f 5 ePA=tdkn-S d[Og>jpH(]}?  'ÃfM}bnm?ҋW| "# MD賮#g]oMmL,6\;uL;$Y #Bvu.L h `Rβ"vnvݿ]@!_2bzKi5>GJbm1pL}voa#wsQNM`"Onq tBm n=LWOv˦iϾf!HO.'L|G-RI|I!uʕm^=|_}C%OÂxs:c3!7vH_SV#).fSOXmu#<񀷣{+5+y9wba\ B.\2/G|:PK%VӹZ$%D3wK' ^$ѹLOm>fG.X _Ȅ}q8X:cq ]#PT^,CCR$!]T\yBGA{!Ū  /bf\.g nK o~[ !RzW yN 5!3#0̊<1w?]R@#d (M(+̈Sg3N,Eq!R7.-iȎdXd/;&V:DWnp)S1Nw|m~m.};e|oU) m2Q%mԳa7[$ 3nȑ-EМ ¼ѤBJ.x8𣆪r9GBdhH==B`:*fuJ0km\> 8-՝6tH̆;8u3<3W wyp,6& u">k o?ГZсZԦÅ,$ʊ[靵rS˂-tG𢅝bn9_ VnzB|;l$r6k}@6Hy d쥑|8~d>|w2GNEW\ky8 ,M?\3&$YcU&PD|(+yi/ָcavMbj縮}Q*&D:;ú\[['@\h(vuP͔a蓠,S.n#ʥ5e&`m @ލ|P.Rm Ȓm*Ûۮ++i ;»yV<ʑn.80Y>a0qMf&Yio%B 2}K.l^HO{DK1' -q&OEz(1HE[8?_ =WfnhaR`Pfv&\h䀨4 |>jscyVӺŨ*c]BPEj(J: 36. ԍglChLx;_4:)Bzd‰ࡪƽ^+^}H?iOd=".TZm=Ԥ",/~VtV 柑xn0-I05,Y׭]zӚmOeo3KBlRY؝ brfG=!K*[U᫃\o+joKӲV6-w3fX JGLްc"CkU_Mz8q9ꝺWm+ yP^Sex^ eM+IJ7VXJ%J}'y K\ȟ` J2uf@zn[7Q#@Y e:ETH,(]HT)D.6ik%Ҭ -[ ; NVm&v %S*ug yVqߌ/FЬL/|rrn] $aVJ!O*cA۳ e,~'u WpEC"M|&Z0)'hYW1:s|Qհ 0dvt4[AO1U}cP^MWJzi}W!Tu!rfպJ7نw%q8@Y.͗DZi3$XVabڞ=Yemva͞gbn-L4EGJ Vhyc` ?/Kv8%@ŋFKq26>w)s H\ʕt;gv)+R% 9B:h\#ai`Ȭ%\s%*_{V!ڿ?! ET*O;Z-"Gw\Zp)?Wa=),ǻiUn]y[4kNS:+4*Q49;s7E|1;.ܒh1Ȣ;+AN=KGRL˦ XFsOǂϟ>w텨\0UztݶhqV5Ƙ9 =;{-Gaʹ)" lt"mK瑾g?*PQS,H?2Ъlsb+`8C ؑwwwҷW袁ρPB RO؅T)'8u_gW5w_;d%%%c/,HH;ltaʃGb[4>N ˛48 endstream endobj 214 0 obj <> endobj 215 0 obj [216 0 R] endobj 216 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 838 0 0 cm /ImagePart_2063 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 359.05 718.5 Tm 104 Tz 3 Tr /OPExtFont3 11 Tf (3.5 Ensemble Kalman Filter ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 104 Tz 3 Tr 1 0 0 1 150.5 675.299 Tm 92 Tz (but most importantly tries to help the minimisation algorithm avoid being ) Tj 1 0 0 1 150.699 652.5 Tm 95 Tz (trapped from the local minima. In the presence of multiple minima, the ) Tj 1 0 0 1 150.5 629.45 Tm 92 Tz (result of the minimisation will depend on the starting point of the minimi-) Tj 1 0 0 1 150.5 606.399 Tm 94 Tz (sation \(71\). When the window length is very long, the second term of the ) Tj 1 0 0 1 150.699 583.35 Tm 91 Tz (cost function dominates the cost function. But when- the window length is ) Tj 1 0 0 1 150.699 560.299 Tm 90 Tz (short, the background term forces the final estimate to stay close to the ini-) Tj 1 0 0 1 150.5 537.5 Tm (tial estimate, which means the quality of the assimilation depends critically ) Tj 1 0 0 1 150.699 514.25 Tm 94 Tz (on the initial estimate. While the ISGD method does not have to use any ) Tj 1 0 0 1 150.699 491.449 Tm 92 Tz (other initial estimates except the observation itself as the minimisation is ) Tj 1 0 0 1 150.699 468.399 Tm 91 Tz (initialised with the entire window of observations. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 139.449 434.8 Tm 92 Tz (In the sense of forming the ensemble, we can also treat the model trajectory ) Tj 1 0 0 1 122.15 411.75 Tm 90 Tz (produced by 4DVAR as a reference trajectory and form the ensemble in the same ) Tj 1 0 0 1 122.15 388.699 Tm 92 Tz (way as ISIS method. Obviously the quality of the ensemble depends strongly on ) Tj 1 0 0 1 122.4 365.449 Tm 93 Tz (the quality of the reference trajectory. In section 3.7.1, we compare the quality ) Tj 1 0 0 1 122.65 342.399 Tm 94 Tz (of the model trajectory produced by 4DVAR and the one generated by ISGD in ) Tj 1 0 0 1 122.15 319.35 Tm (both low dimensional and higher dimensional case. The results show that the ) Tj 1 0 0 1 122.4 296.299 Tm 92 Tz (reference trajectory produced by ISGD is more consistent with the observations ) Tj 1 0 0 1 122.65 273.299 Tm (and closer to the true system trajectory than the 4DVAR results. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 122.9 220.95 Tm 129 Tz /OPExtFont2 16.5 Tf (3.5 Ensemble Kalman Filter ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 16.5 Tf 129 Tz 3 Tr 1 0 0 1 122.4 186.399 Tm 90 Tz /OPExtFont3 11 Tf (Second well established class of algorithms has been defined for state estimation ) Tj 1 0 0 1 122.15 163.35 Tm 95 Tz (are sequential algorithms. In sequential algorithms, one integrates the model ) Tj 1 0 0 1 122.15 140.299 Tm 97 Tz (forward until the time that observations are available, the state at that time ) Tj 1 0 0 1 122.4 117.049 Tm 92 Tz (estimated by the model is usually called the first-guess, which is then corrected ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 314.899 51.75 Tm 77 Tz (36 ) Tj ET EMC endstream endobj 217 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 218 0 obj <> stream 0 ,,y}}eب[cͦ&xF.˻Λ[~ ^=y5r5;Uxg=u]J2]lTUpAQ_U_9[64!phLr9_ESb1Oy+=WZꌤ&EHĵĠڜ#i^7˷Z# ͠ki(F `3R'N:͇sliEQ3!}'-%1=-GEʿ!;yUqkܲk췡ǻ+谨ǹ!^/"Ϝ#-P͆,쳾3۷7"O)73U/>(D(fѾ`GK\8FNêՇ*|B&RmqyBF'Z CŹi)K=CIY4XUH ޜQ q{6~R;ȍQ# pT֬ӝKw wOq^ mD P˓jKGifvPq8t*bZC+Y,+gg(`?;,;6g<=bHIĉdZ9i"MykS腇>R ?n;Bc`T`BM[y8YV&išjaB\wiU^ ϖqʶ$~1d ˴mlt#Ss7?H<0)wcG-zLX$52$"0S1Ew)¬rST|*+\#~=FqpjZ"kXorStO0?" `H>AgPfT-GBPkE5 {0orP(Yi,9z&E:t/W} r'\NJ [A*%lYW&!zhrkwWXpʧA QZ0jSD:Nߠy4~TJ$KJhgvU[gma}Z'iw.`'/Ҟ%G4cCDn [HԐ \x/Dߺ㸇zC}xiI0BN:+Mw&:' "+;#I ZzUոGaf0L3/l>ľ( JlsIL6Ʊvp2u9eK21{?ezZ B_&OP":6GlFL66v0V1n'Knj @S vDv◢E%Z$`hZp (AٷiXf*%u^bҬhj4{_P[kbj*{?aOgfȉ]QL+)VF&DՅk KP2ʇR#~5Php\lDE TMئr6ٟ~Y-!lHV5^Ֆ k*:DZl||BW5FZ;`u wLMZs]/Qr> 辳<[H5"/=<1 &<ٴ}h{#otM.krF Z'W ssSf?],.;U|؝LUvMuWPyej8l aJ4a3"E&cSʭC:ebŜ-[{7 4/^5^ѵɯz*- ?{V=^JES<.3#4Gw5c"?s~%._4{H2 X\C3 ڻFYqC+/-i`W1?+yX&OHWmqHCg`=4o'E. H20Zzt*>בa\ XɡR)}=䘎(ϒ~V-戄*LŲ¢>^ц3܌^J8-/ JST&iIx@?gzNӾArv?PyBhi<'fpHglǽ85"x쒫ۍ(%pF,bBKP9Ռ@Qʏ [kfbvp;.B[iiPlNM tmiN _X7$E~h\K;#\^\*tx,pp7V* pfPYbj4 ksd93kqG 鲓qCƾMv^B≿ɔ +*s!Geu-iVzR̸}ȲIik n]f[]۠_r۹GL@ h$alNg_i^9 M1ibnd,{u]HYȘM&T4!rF}h~c=VkcÈ" '߄ϓ0#ÿcn^SCpp λzKҟ`*dp@ʜ?fBę^qC*/ y_M Zwܛx;qDoWBHjp6B؊mYoYe~ ̏3XC4tLykV aa(X}ٺ2a&`+ºur.|!bJ$#un9!Jڴr?r;W0||"4bfaH=iL( 3aJ}Md"fv"HQ+|D|\,nbamg7_n5{M)q]Jzm-ұGhuLAخiE+KłY}DĆ'?SLxC3r>Ҵeߴ6hЖ. d}Ǵ3aƥ?g!k!_NJ|J5/QV8`,'fG6I Ob1;vjBR &iI%==ms Кx5Wj ԛO8Y:\g!zdU@,>PS|*WU)::82YHc ̪#2z1@ IOΖ>ܔT}R؜aX;gxfkCVL2)It%وkr21տPK4mxnl L59(iӇ.ڬD_f |;qt9l pު=0a0T@,I(c9Z>.8Znd+vfZmU8j_mT9? dݻD*JmwpWJq~~ fx! מb`8PgpS3K]SĒ %ɺƯ6ީ[^{ Wd>JϽy5bt0sݹCҰ;45w{O?~+lT%?N͆ua+^ş@1 X9^@cr;IR8"q>x;6NZ ?I #ȳ=D?mV X/biĊ9Pů <:/,aYu&DbՉq#_1 8SWRg{Nܑ>x7]9S禷BZV pHF&t7W k}p&$LJSݥu+w=|z8zGU؛P B9;ZsSٟ$t".*2t,'n_5)e"F"4sx 2ǹ\z{ `fxF%@4Jי]U+ %]ܘf!Ov6s@l5IV(}#Svr6P&bUo3q FkRm͋P6(e|D>_5K<;SREF(8[ʠÀXHZ%؟f"TTAWt3$Q{yssZ (—|FS aS bJMw4z^䮕_Jʈ#3>q٘?2ٱW%l.'<񀆖ՠM'QE y5wuGCPWOL[ Q\c%Sor sۉ 0iݳ.&)EE*{x|m|S-7B\8p?SA b 9I`ݤey[Lu ih0#&j^Uԯ}r[N`EF}G`Vrn!}fk'iNc"0qŃUEo3ݕ4!xPobXz+ *#"f᪎<~r4ķ j82Sz s8ҧY\TSK$6ωoDShfE{Ths=9Z-A|ãLp+!SN.em6 nf7 pY!PYsF5-<:b j)YQ2 v/!(]D.xs]< h6X*SOXnEo{}9t#Wrm &I9[^0,S!ST=1 cAmvyu%{c6_f;Ǯ|0yZPn{,[R!0oa+heE=+|n aaRvgR-aeels\AJnZP]U1*]1}KjhH(>2d)K;y,;jT V.vf!a`tq1fY!Qk/!fJ$w;wn^XȷYl(XK' ~c[/ii9atCHcw@3u=SDpwڍ +2qmnӰ4z)a5`(բ +Gnn'nNe+(*~?"2LREӒ]<@av1CZ0O9䋯M{+}}S?bdI2|lO ׃Ю>1ŽϓǴ8]K$Kǁ+MIɯjyY"Ǵ012ٷ+p ˧0Qn,5'1r1pیꎹ&%P,AP}2Q=>)M0P, kKD+R]gIu{ȼNY?QֹېrT豴oʥbwF72bڝmM V oBa:_8!4xͼt!^+˪(CXEF^iK*M;r=([JER2,/EH#ĸ_<}A[֋xң|?v.^+W_ʁ'('ePB%Y٨7ÈL 5!I^]I 3ӨG튚t ڒ:1D,wݚ#ߛ*"l'nOi*:9>ubd LvAG?Pl)UiLX3"tiEՂ,dd-J^qmJ!)xpD$\DRO]*tpSAeR{ڋ7`g6J$lRڀ 8 HNMWM=/jflwuFũB3m;Ō-LDR gW1U$K@vzꕄ,, ʠG##\a4':{=ܸ+n`u} Yn/ɗzp&Llq`dJs:8BԏnZPyuI mp:0$dgL*@/hTpYKUWac4&? endstream endobj 219 0 obj <> endobj 220 0 obj [221 0 R] endobj 221 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 837 0 0 cm /ImagePart_2064 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 360.949 718.45 Tm 120 Tz 3 Tr /OPExtFont5 12 Tf (3.5 Ensemble Kalman Filter ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12 Tf 120 Tz 3 Tr 1 0 0 1 124.299 675.25 Tm 113 Tz /OPExtFont2 11.5 Tf (with the new observations. The sequential algorithm, used in this chapter to ) Tj 1 0 0 1 124.299 652.2 Tm 105 Tz (compare with our methods, is based on the forms of Kalman filter \(54\). Although ) Tj 1 0 0 1 124.099 629.149 Tm 114 Tz (we only provide the comparison between ISIS method and a state of the art ) Tj 1 0 0 1 124.299 606.35 Tm 106 Tz (ensemble Kalman filter scheme \(1; 2\) later in the chapter, we first provide a brief ) Tj 1 0 0 1 124.299 583.299 Tm 110 Tz (overview of other versions of Kalman Filter methods including Kalman filter, ) Tj 1 0 0 1 124.299 560.299 Tm 104 Tz (Extended Kalman filter as background information on the ensemble Kalman filter ) Tj 1 0 0 1 124.299 537.25 Tm 105 Tz (being discussed later. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 105 Tz 3 Tr 1 0 0 1 124.549 492.6 Tm 135 Tz /OPExtFont2 13.5 Tf (3.5.1 Kalman Filter ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 13.5 Tf 135 Tz 3 Tr 1 0 0 1 124.099 462.1 Tm 108 Tz /OPExtFont2 11.5 Tf (The Kalman filter ) Tj 1 0 0 1 218.65 462.1 Tm 98 Tz /OPExtFont5 12 Tf (\(54\) ) Tj 1 0 0 1 242.4 462.1 Tm 110 Tz /OPExtFont2 11.5 Tf (is a commonly used method of state estimation \(86\). It ) Tj 1 0 0 1 124.099 439.1 Tm (provides a sequential method to estimate the state of a system, with the aim of ) Tj 1 0 0 1 123.849 415.8 Tm 104 Tz (minimising the mean of the squared error of one step forecast. It gives the optimal ) Tj 1 0 0 1 124.099 392.75 Tm 107 Tz (estimate when the system dynamics are linear and the model is perfect \(86\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 107 Tz 3 Tr 1 0 0 1 140.9 369.699 Tm 112 Tz (The Kalman filter addresses the general problem of trying to estimate the ) Tj 1 0 0 1 123.599 346.449 Tm 109 Tz (state of the system ) Tj 1 0 0 1 221.3 346.449 Tm 121 Tz /OPExtFont2 10 Tf (xt E ) Tj 1 0 0 1 245.3 346.699 Tm 106 Tz /OPExtFont2 11.5 Tf (Rin, where the dynamics of the system is ) Tj 1 0 0 1 451.449 346.699 Tm 104 Tz /OPExtFont8 12.5 Tf (F: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont8 12.5 Tf 104 Tz 3 Tr 1 0 0 1 280.3 281.149 Tm 117 Tz /OPExtFont3 10 Tf (xt = ) Tj 1 0 0 1 306 280.45 Tm 75 Tz /OPExtFont5 16 Tf (fr) Tj 1 0 0 1 315.35 280.2 Tm 96 Tz /OPExtFont3 10 Tf (\() Tj 1 0 0 1 318.949 279.95 Tm 60 Tz /OPExtFont5 16 Tf (X) Tj 1 0 0 1 325.899 279.7 Tm 100 Tz /OPExtFont3 10 Tf (t-1\) ) Tj 1 0 0 1 342.949 290.399 Tm 2000 Tz (\t) Tj 1 0 0 1 491.75 281.399 Tm 95 Tz /OPExtFont2 11.5 Tf (\(3.10\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 95 Tz 3 Tr 1 0 0 1 140.9 238.899 Tm 109 Tz ( Given a linear model ) Tj 1 0 0 1 259.199 238.7 Tm 111 Tz /OPExtFont8 12 Tf (F: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont8 12 Tf 111 Tz 3 Tr 1 0 0 1 294.25 168.6 Tm 120 Tz (F\(x) Tj 1 0 0 1 313.899 168.35 Tm 88 Tz (t) Tj 1 0 0 1 318 167.149 Tm 104 Tz (\) = Axt, ) Tj 1 0 0 1 357.85 166.7 Tm 2000 Tz (\t) Tj 1 0 0 1 491.75 168.85 Tm 100 Tz /OPExtFont5 12 Tf (\(3.11\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12 Tf 3 Tr 1 0 0 1 151.699 126.35 Tm 109 Tz /OPExtFont2 11.5 Tf (where the model ) Tj 1 0 0 1 240.949 126.35 Tm 118 Tz /OPExtFont8 12 Tf (F ) Tj 1 0 0 1 254.65 126.35 Tm 112 Tz /OPExtFont2 11.5 Tf (need not be a perfect representation of the system's ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 112 Tz 3 Tr 1 0 0 1 316.1 51.5 Tm 95 Tz (37 ) Tj ET EMC endstream endobj 222 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 223 0 obj <> stream 0 ,, b7 \Hmr o™Ӹ ;D`̠d}9Uק $*QVIs)j<0oMUkMi(.5EMmLZ ȕPfuT KT%SG˫!QQ!؄M9h FW]Yc-_O }DY 9 wxp@N[4?{x9(xX}'ubV%U&au3tQEkԯq[YAZ8|?` | naLU4p?+rՅö3

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LRe~v 9!8bw+%ȬTBɧ[$ޏڴsщ h\uf@SDۃmR#l+ z5O 7v;Ҹ^Gѭf-xV:=l[yWUKC(>F`ֶ=yMt-Ȕ%_\SxBfiGXH5±X0aPL ӆ9/L_ѡ3Z!06#L.; TްX )ĕ<*wN `Y'AĊAQ;=-q%[cvcYGiT1L;ݯqEY!D9j\Āk|E܃.-p@Z/}ܮ8Lɉ3x+2|m|Š0I43C_S'cH_ ZF I@aAhE-1G  & ~'Ã|wedhw5Z&xDoG὎ \gXa4h4'O9p#zh#q/ߟ“Yo,|u~/aƂ b\A~AoFTܖ&TWB-8TH~]QW(J #FPE|3`K ^(~l\^ZJʼ߂$R@hOt9WdrD&O!JGb6Яi]d$3o Skl56' g4] Qse+P{>r:Q8ơ3ڏm RY̼F]{<J +OEI|{G.˕&uR a?BtQ#a2_ZJk!6{3pu rpPj@Y 1bt6 bsbg%ܬ 7^QypL2)P=Zv8 FNV2&k߾kV) ¹tpO`[0~ P; I˃ty9Ty/e_mi4rXFpCEe>G$aؖvn._^?PRm"ɺaI+[U5-89md%2GU?Ǡ~ mC3txaG^1)ZKu'h^k zڳ@}t) F;#K4>eb=Vg2Ĩ `pB񮢈J;HA׃%S5d,Sv RT|ʾܟ^axODgiݫQF) 'A} iWYף׭@j{Mc[9>}Rif7ګ-o ihp>QȰ7 mg4T uR )tܯ7M|D vLpic|czƨ*% rwZv)%cIa]坓Lr:"lJd7ώD-RW?A2(U_p}ع![ =0Ɉ-<@>O;x;_)_pz6WX6ǩcҋes0F6G=̰qzASbwuF2^@*cZp*k}k՜%jE7]&\ q@^[]VwVU-=C FɐQpY.8B!9xuq( dg/95]ՒJFola^eÏjHv[O fO/7pLhƥճ=FU$_-j 1~)\0ZzNܷWQBXL!U -+b,P..%0%~Þf3B+ |}ڡ##˨y }U/)gdݡPhGwRɦp!uֲK&mӴH3rcq>ޟMEF-m| vq}VmCawo endstream endobj 224 0 obj <> endobj 225 0 obj [226 0 R] endobj 226 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 838 0 0 cm /ImagePart_2065 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 359.05 718.7 Tm 104 Tz 3 Tr /OPExtFont3 11 Tf (3.5 Ensemble Kalman Filter ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 104 Tz 3 Tr 1 0 0 1 150.949 675.5 Tm 106 Tz /OPExtFont2 11.5 Tf (dynamics F. It is assumed that model space and system space are identical. ) Tj 1 0 0 1 150.699 652.7 Tm 107 Tz (Any discrepancy between the model and the system can be written as: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 107 Tz 3 Tr 1 0 0 1 273.35 582.649 Tm 99 Tz /OPExtFont3 13.5 Tf (P\(x) Tj 1 0 0 1 293.3 582.649 Tm 66 Tz /OPExtFont5 13.5 Tf (t) Tj 1 0 0 1 297.1 582.149 Tm 92 Tz /OPExtFont3 13.5 Tf (\) = ) Tj 1 0 0 1 316.55 582.899 Tm 111 Tz /OPExtFont3 11.5 Tf (F\() Tj 1 0 0 1 329.3 583.1 Tm 127 Tz /OPExtFont5 11.5 Tf (x) Tj 1 0 0 1 336.25 583.1 Tm 55 Tz /OPExtFont3 11.5 Tf (t) Tj 1 0 0 1 340.1 581.45 Tm 104 Tz (\) + ) Tj 1 0 0 1 358.1 581.45 Tm 91 Tz /OPExtFont5 11.5 Tf (wr) Tj 1 0 0 1 369.35 581.2 Tm 60 Tz /OPExtFont3 11.5 Tf (t ) Tj 1 0 0 1 372 581.2 Tm 2000 Tz (\t) Tj 1 0 0 1 489.85 582.899 Tm 95 Tz /OPExtFont2 11.5 Tf (\(3.12\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 95 Tz 3 Tr 1 0 0 1 150.5 540.899 Tm 101 Tz (where ) Tj 1 0 0 1 185.3 540.899 Tm 99 Tz /OPExtFont4 6.5 Tf (7XI ) Tj 1 0 0 1 202.55 540.899 Tm 116 Tz /OPExtFont2 11.5 Tf (is understood to reflect the model error. When defining the ) Tj 1 0 0 1 150.699 517.85 Tm 108 Tz (Kalman Filter it is also assumed that Tx; is ) Tj 1 0 0 1 369.35 517.85 Tm 99 Tz /OPExtFont3 11 Tf (IID ) Tj 1 0 0 1 390.5 517.85 Tm 108 Tz /OPExtFont2 11.5 Tf (normally distributed with ) Tj 1 0 0 1 150.5 494.8 Tm 106 Tz (zero mean and variance ) Tj 1 0 0 1 270.949 498.899 Tm 93 Tz /OPExtFont8 12 Tf (Q ) Tj 1 0 0 1 279.6 498.899 Tm 129 Tz /OPExtFont2 7.5 Tf (err ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 7.5 Tf 129 Tz 3 Tr 1 0 0 1 139.449 461.899 Tm 107 Tz /OPExtFont2 11.5 Tf ( given observations ) Tj 1 0 0 1 244.3 462.149 Tm 89 Tz /OPExtFont3 10.5 Tf (s E ) Tj 1 0 0 1 263.3 462.149 Tm 100 Tz /OPExtFont8 12 Tf (Rm) Tj 1 0 0 1 279.1 468.899 Tm 46 Tz /OPExtFont4 12 Tf (obs ) Tj 1 0 0 1 293.75 462.149 Tm 101 Tz /OPExtFont2 11.5 Tf (we have ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 101 Tz 3 Tr 1 0 0 1 288.5 392.1 Tm 116 Tz /OPExtFont3 9 Tf (St = h\(xt\) + Et ) Tj 1 0 0 1 359.75 390.399 Tm 2000 Tz (\t) Tj 1 0 0 1 489.85 392.1 Tm 106 Tz (\(3.13\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 9 Tf 106 Tz 3 Tr 1 0 0 1 150.25 349.85 Tm 103 Tz /OPExtFont2 11.5 Tf (where c) Tj 1 0 0 1 187.699 349.85 Tm 77 Tz /OPExtFont5 11.5 Tf (t ) Tj 1 0 0 1 190.3 349.85 Tm 106 Tz /OPExtFont2 11.5 Tf ( is the observational noise, assumed to be ) Tj 1 0 0 1 397.199 350.1 Tm 99 Tz /OPExtFont3 11 Tf (IID ) Tj 1 0 0 1 417.35 350.1 Tm 106 Tz /OPExtFont2 11.5 Tf (normally distributed ) Tj 1 0 0 1 150.25 326.8 Tm 105 Tz (with zero mean and variance ) Tj 1 0 0 1 293.5 326.8 Tm 83 Tz /OPExtFont3 12 Tf (F. ) Tj 1 0 0 1 309.1 326.8 Tm 106 Tz /OPExtFont2 11.5 Tf (The function ) Tj 1 0 0 1 375.85 326.8 Tm 91 Tz /OPExtFont6 12 Tf (h ) Tj 1 0 0 1 385.199 327.05 Tm 105 Tz /OPExtFont2 11.5 Tf (is the observation function, ) Tj 1 0 0 1 150.25 303.75 Tm 118 Tz (here assumed to be linear: h\(x) Tj 1 0 0 1 313.449 303.75 Tm 79 Tz /OPExtFont5 11.5 Tf (t) Tj 1 0 0 1 317.75 303.75 Tm 139 Tz /OPExtFont2 11.5 Tf (\) = Hx) Tj 1 0 0 1 360 303.75 Tm 79 Tz /OPExtFont5 11.5 Tf (t) Tj 1 0 0 1 364.3 303.75 Tm 135 Tz /OPExtFont2 11.5 Tf (. The ) Tj 1 0 0 1 402.5 303.75 Tm 87 Tz /OPExtFont8 12 Tf (in,) Tj 1 0 0 1 416.399 303.75 Tm 44 Tz /OPExtFont4 12 Tf (b5 ) Tj 1 0 0 1 422.899 303.75 Tm 122 Tz /OPExtFont2 11.5 Tf ( x m matrix ) Tj 1 0 0 1 494.899 303.75 Tm 110 Tz /OPExtFont6 12 Tf (H ) Tj 1 0 0 1 510.5 303.75 Tm 87 Tz /OPExtFont2 11.5 Tf (is ) Tj 1 0 0 1 150.25 280.5 Tm 112 Tz (a projection operator that gives the transformation from model space to ) Tj 1 0 0 1 150.25 257.45 Tm 103 Tz (observation space. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 103 Tz 3 Tr 1 0 0 1 138.5 224.299 Tm 98 Tz (We define xb ) Tj 1 0 0 1 204.5 224.549 Tm 79 Tz /OPExtFont3 10.5 Tf (E ) Tj 1 0 0 1 210.5 224.549 Tm 694 Tz (\t) Tj 1 0 0 1 233.75 224.549 Tm 107 Tz /OPExtFont2 11.5 Tf (to be the background or prior estimate of the system state ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 107 Tz 3 Tr 1 0 0 1 122.15 201.299 Tm 113 Tz (at time t, and ) Tj 1 0 0 1 194.4 201.5 Tm 114 Tz /OPExtFont3 15.5 Tf (4 ) Tj 1 0 0 1 210.25 201.5 Tm 99 Tz /OPExtFont3 10.5 Tf (E R) Tj 1 0 0 1 228.949 201.5 Tm 70 Tz (m ) Tj 1 0 0 1 235.9 201.5 Tm 112 Tz /OPExtFont2 11.5 Tf ( to be the analysis or a ) Tj 1 0 0 1 355.199 201.5 Tm 120 Tz /OPExtFont8 13 Tf (posteriori ) Tj 1 0 0 1 407.05 201.5 Tm 108 Tz /OPExtFont2 11.5 Tf (estimate of the system ) Tj 1 0 0 1 121.7 178.25 Tm 115 Tz (state x) Tj 1 0 0 1 156 178.25 Tm 86 Tz /OPExtFont5 11.5 Tf (t) Tj 1 0 0 1 160.3 178.25 Tm 109 Tz /OPExtFont2 11.5 Tf (. The estimation error is then defined by ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 109 Tz 3 Tr 1 0 0 1 314.399 52 Tm 91 Tz (38 ) Tj ET EMC endstream endobj 227 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 228 0 obj <> stream 0 ,,;;#yrl!zDћ/ ,b*‘Υk(](XĤYT5;tlo>S1SDQ`NiߡGFH$#$0elN6{L&B=+ej|]QCpzs(ItSjn-,T-ֺɱ &zÂUY!(Ro3}R+h^ k4^lζRLZa67Y%Si9޴zw[= ut$aM\%zh.Bʪ{ރ{5v(4}/uh%C::^G%o ~wT}~(fwD uu\hy`vR&!@Ve뺳pJ4PNNh>2X%ۘw;Y/1;e* ~N/t3nlq:MvqK!zCCqnxtXlOfdߥ N𾍠 /.>9AlT.CAzX.ُlaT# "Ѽ׾Η9( [ cU)&Hedpʗr| MBt6Q -toK %*|t?WG {I+wZ>W$u*@X(w?a܇u֬g9-\0EJm'a>{{eȇ:8!lׇ2rr4\Awq`ާcDdB6\2MK/v/2D(͛Do )uFpern7,#$/{N< Ql"'XيFa)R;ڰ]a#Ed2M^K)͹HWȜӿp$ol VuDu%MJz}:\T1lI[\c^=dZT-Y';ҳWGyc,!QbYrK~VB9@Lj  տZ-s3(sz"F9At\xQ*R]jj`@)D]9~L؝ ڈ 8aGy+BfVҾLy .g}Akn[0x3ZQm?6t$!'5 \V҈hLlLz$txj3r@mj?2.$ۍC|< ÎK0i'902: p`ĸŽ]3=0CqxןmS,cZ?dա@|cq #$(gCXZGG~#b3RW@xzNst55!l$vN8QO &ǺVirg#^v(P :ajI5 qH65BzkE2U+<^fڻ:9 S;w%[S 0˻,49e^!VZD3n@kʲ/3b,fgAjz3#IlaД%xqS-|[m =4TRe.Bs|*$#:dQj *KݖUQ0bV< LyJtG{v;DI< ѾmK}:\8WVBqWݭ2-dzmO򻎜M!9aw[x ncUfqg<8]:KS S߽ۈ3pH 52vwʜn a6vڧ`_:ɘt C_}em¼y~-0\2etؒ#S7tFR#3V$=Tti#\X6\O1~-8\AY:019PR^%aϧSA՗[[̾ k ߋDžhKB=nU 3pJڢ=O]aD-H @ kl)' CF񶁀[`: ~9pr7~~%hyRmӚ,$8ם[/QЌw 3H-S +jfAe8&$DٛeTJkKɠӟC9YZ]ҬobFuDE*=[LSe7oٷmoFսZ6ctL[Gfo^ZƽPt q`hf[ Z$*fU7~zeV @|?kP9:ɍL'r}ҋX:\FUT\iU|66[ Yvס|̌ɦsn,LC|ͅ'YR\m_d @ԲN}414; _.> K^9:zD0->*QM$^'`%jiP әAopAgr\U+RNWϳy=7Xˡ9R09~0g)@taj+M0.YcF㒥6D>{mȇoFLCnpIE^ɞY+ȿ \jC_FDFp+*vso/|0n#b^eӝS{0SLN+_b[ Dubiաh'Dv3(23]7Z+ , GZmCBȑ 6yH+kH?ip= zC"OC(Wߔ'lfFKV]ڹ?g4o]+ Y>mZ·bZQsq H=JB 0<9Z^+*!r9=Fnd|M;͉(>u@U ΜPWKRtķn,o'UF k2VuUFpè UI(w[))v{d[`uElQ-֕3,(]=R;Н_j1ǻy+R65 D\AӜ1We R3K:JYMV7EiԂ}rB@'o > .>gk:=,ȡC!-մ6wot8 [+~")w$DAd SVɁkD&ɉБTYz]:p~{dow˂b¿|6Kiĸa* kQ[.o?a@PZ=b?OI)2( !q#=Mg6?OOJ!]mΪEQ9 ^Y.T-W0 Xn3# Ś[C[}Dnav%W-qZǨEXU_\[ =9ۦ*g ެsmPA?$ɓyӾzHVE@ihLq;U-F~>7lA}\rbxqݓx}H[揳&CptǏ] fp#̅u VǢNjW‘+3O_>P|ƿH,mJ" S|*lK@ 6uzM]itu]Zd.6e΀6 Pb6% T >Y"pw($?>22) pe7ۢ ru@V=) WL(mEI{zodQ&_`aBk6gL LE,N7{<h<@^.:ӣGb5AGqUɧH|$9=6@_r'15'o^` ,T϶$wObBٯrme*^YR9<;Ռ$O`PN)Ws?ή73Tqz?LLma5#DyWyx!{|^xs:" Ĭ4WA9⌖DXD&S|3TT#!=fOcx}jL*IG O^ot3 b4g0;xl"t9M1O dn)<6 p@LzcĩZŦ_s> E3>(^'r{٪=xsԵd֓;BuNlE-H*L~HLDK=ͤ[P/Qjqku*̳a2[(0?v@w0Xv;2pu9%B~i m)VeIqrúOurp辀KNMJށzd1e_|$#8F(Eچt׺2N]hHv,,#C qKkU-hN+r̷y_}\oP[5~w^-k-jθ%2f)9qi0߅i2:oÄd-Oi^E=9NPgp( ܰ8Q xwQ_y*߱6/R_Pȓ"aI]KҔ̏4T2*K29wެE%5)YE0ƥϖif _C02]jE,VdKC~~7{MpHORcerX`| Vq4/$ŅU m8s8SFW6a[@wә2}cvq(jSk [?t"ȆV1!j|EDZ ,~8:逽S˵}T{#ĥ?8 Rlox6+sxབྷgp}+հ _٘be/Edcmڟ9} < Q@6}sJ-Sm^oU%kbmۖ?= n'8_ԃ><.<?-${G@Bw`mOvf`G\b3gF QxA}&u~UyNbL:*B#8q@^ޛ>O!t >Ff%ԧ"ڙ>.]o _Y`i@ze^W^y}'P؄?R7*'AmHTê&;ٓ1j1f7vl&gc2:>#j*v2ڱ.*`<ݍs~=sxÇQ]Bdc-8օ4T-16y6+L-zkW'?c9,+l}[|kXaXئp@r#]>.? E[I endstream endobj 229 0 obj <> endobj 230 0 obj [231 0 R] endobj 231 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 838 0 0 cm /ImagePart_2066 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 359.75 718.95 Tm 104 Tz 3 Tr /OPExtFont3 11 Tf (3.5 Ensemble Kalman Filter ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 104 Tz 3 Tr 1 0 0 1 278.899 633.75 Tm 87 Tz (e) Tj 1 0 0 1 284.399 633.75 Tm 63 Tz (t ) Tj 1 0 0 1 287.05 633.75 Tm 115 Tz ( = x) Tj 1 0 0 1 310.1 633.75 Tm 57 Tz (t ) Tj 1 0 0 1 312.5 633.75 Tm 93 Tz ( x) Tj 1 0 0 1 334.8 633.75 Tm 51 Tz (t) Tj 1 0 0 1 334.55 633.75 Tm 46 Tz (b) Tj 1 0 0 1 339.35 633.75 Tm 34 Tz (, ) Tj 1 0 0 1 279.1 608.1 Tm 71 Tz (ec) Tj 1 0 0 1 284.899 608.1 Tm 57 Tz (t) Tj 1 0 0 1 287.3 608.1 Tm 106 Tz (' = x) Tj 1 0 0 1 311.3 608.299 Tm 57 Tz (t ) Tj 1 0 0 1 313.699 608.299 Tm 79 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 79 Tz 3 Tr 1 0 0 1 123.099 566.1 Tm 91 Tz (and the error covariances are given by ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 490.1 634 Tm 86 Tz (\(3.14\) ) Tj 1 0 0 1 490.1 608.1 Tm 97 Tz /OPExtFont5 12.5 Tf (\(3.15\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 97 Tz 3 Tr 1 0 0 1 273.35 500.8 Tm 120 Tz /OPExtFont4 8.5 Tf (Pt ) Tj 1 0 0 1 282 505.6 Tm 65 Tz (b ) Tj 1 0 0 1 285.35 505.6 Tm 633 Tz (\t) Tj 1 0 0 1 301.449 505.6 Tm 141 Tz (E) Tj 1 0 0 1 310.8 505.6 Tm 119 Tz /OPExtFont8 9 Tf (\() Tj 1 0 0 1 314.899 505.6 Tm 99 Tz /OPExtFont4 9 Tf (et) Tj 1 0 0 1 320.649 505.6 Tm 84 Tz /OPExtFont8 9 Tf (b) Tj 1 0 0 1 324.5 505.6 Tm 103 Tz /OPExtFont4 9 Tf (et) Tj 1 0 0 1 330.25 505.6 Tm 116 Tz /OPExtFont8 9 Tf (bT\)) Tj 1 0 0 1 345.6 505.6 Tm 44 Tz /OPExtFont4 9 Tf (, ) Tj 1 0 0 1 346.8 505.6 Tm /OPExtFont8 9 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont8 9 Tf 44 Tz 3 Tr 1 0 0 1 490.1 501.05 Tm 87 Tz /OPExtFont3 11 Tf (\(3.16\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 272.899 474.899 Tm 122 Tz /OPExtFont4 8.5 Tf (Pa ) Tj 1 0 0 1 285.6 479.699 Tm 633 Tz (\t) Tj 1 0 0 1 301.699 479.699 Tm 114 Tz (E\(eaetaT\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 8.5 Tf 114 Tz 3 Tr 1 0 0 1 490.3 475.1 Tm 86 Tz /OPExtFont3 11 Tf (\(3.17\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 86 Tz 3 Tr 1 0 0 1 122.9 432.899 Tm 88 Tz (Pb and ) Tj 1 0 0 1 160.099 432.899 Tm 150 Tz /OPExtFont8 9 Tf (Pa ) Tj 1 0 0 1 177.099 433.1 Tm 88 Tz /OPExtFont3 11 Tf (are often called background-error covariance and analysis-error covari-) Tj 1 0 0 1 122.9 409.85 Tm 92 Tz (ance. The Kalman filter provides an estimate of the updated state ) Tj 1 0 0 1 452.899 410.1 Tm 114 Tz /OPExtFont3 15.5 Tf (4 ) Tj 1 0 0 1 468.5 410.1 Tm 87 Tz /OPExtFont3 11 Tf (as a linear ) Tj 1 0 0 1 122.9 386.55 Tm 91 Tz (combination of the first guess estimate xt and a weighted difference between the ) Tj 1 0 0 1 122.9 363.3 Tm 94 Tz (actual observation and the prediction H4, i.e. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 94 Tz 3 Tr 1 0 0 1 251.05 298.5 Tm 98 Tz /OPExtFont5 13 Tf (xa ) Tj 1 0 0 1 262.1 298.5 Tm 915 Tz (\t) Tj 1 0 0 1 291.85 298.5 Tm 96 Tz (+ K) Tj 1 0 0 1 312.25 298.5 Tm 63 Tz (t) Tj 1 0 0 1 316.55 298.5 Tm 94 Tz (\(s, rixt) Tj 1 0 0 1 359.75 298.5 Tm 70 Tz (t) Tj 1 0 0 1 361.699 298.5 Tm 99 Tz ('\), ) Tj 1 0 0 1 370.55 303.649 Tm 2000 Tz (\t) Tj 1 0 0 1 490.8 298.7 Tm 87 Tz /OPExtFont3 11 Tf (\(3.18\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 122.4 255.5 Tm 92 Tz (where the m x mb) Tj 1 0 0 1 216.5 255.75 Tm 41 Tz (5 ) Tj 1 0 0 1 219.349 255.75 Tm 102 Tz ( matrix K) Tj 1 0 0 1 270.5 255.75 Tm 62 Tz (t) Tj 1 0 0 1 275.05 256 Tm 92 Tz (, often called the Kalman gain, can be derived by ) Tj 1 0 0 1 122.65 232.7 Tm 94 Tz (minimising the posterior error covariance P. The Kalman gain is given by: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 94 Tz 3 Tr 1 0 0 1 240.5 167.45 Tm 100 Tz /OPExtFont3 13.5 Tf (K) Tj 1 0 0 1 249.849 167.45 Tm 67 Tz /OPExtFont5 13.5 Tf (t ) Tj 1 0 0 1 252.5 167.45 Tm 98 Tz /OPExtFont3 13.5 Tf ( = ) Tj 1 0 0 1 269.05 167.45 Tm 120 Tz /OPExtFont4 10.5 Tf (ppliT\(Hp:HT ) Tj 1 0 0 1 345.1 167.45 Tm 75 Tz /OPExtFont3 13.5 Tf (+11\)) Tj 1 0 0 1 368.649 167.45 Tm 66 Tz (-1) Tj 1 0 0 1 379.899 167.45 Tm 27 Tz (. ) Tj 1 0 0 1 381.1 172.6 Tm 2000 Tz (\t) Tj 1 0 0 1 491.05 167.7 Tm 87 Tz /OPExtFont3 11 Tf (\(3.19\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 122.4 124.95 Tm 93 Tz (The application of the Kalman filter is as follows ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 315.35 52 Tm 77 Tz (39 ) Tj ET EMC endstream endobj 232 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 233 0 obj <> stream 0 ,,BBb6|Y|5D%D4$Hi:abMIOe |&aoF\; =FJol/C3jCUr&8d?Sh[]1} _ *V0_4&_jBt+=6-/ϝ+cb *Õ|沊VuEG4YޯIBQo6&@Q*KvӻΩד5Z74DC\}5թvIИG W+o `$OE11i$Z91Gk~z?\P<0W( H%h9CP`G%!p3bct4ڌ .R^8-ϖ ɇ'ۅTΰ7#GjfNj?%؂ yBOx%sly:YIޫBDYcuT_йmoo@yCbЫyxO9ف,tFˡTHNB7^&9onﾞ;Rb@Y)Dz5dRK,1Ɉgbm@TjO{V#وd)C\ =GMUٻ4]dGH}1lKoknpqͽfz:[+x7?'8 \D-KS$ۉ}*v$A둔T!*~*-_yG=9:dTWEʪ9Em6G "aD7E.}qNzn ƹ"Z} NmoT`3v*4g|LHQa;0%` [Ex(垅oO=$q2Wq-x/5||vNb{X O+/͓93`ΕfN onRg?A/ Q_Xݻ$%-X>nk฻1T@NU^Kdzfzԋrfrp_Vl#UX DtH qP]\dٌIi%Z7{ t3"^b͗ ~@Rې-8>Brk!ͣC_GiG/LOr;$muY&}yVI̒=2يU# h 0;FsE&\ + 9(+H$''Du>Uwx k39Bis}r޴g?F~ڴir.U JA6;KGKwiV#^6cwdB/:Ay-Z-Kԡs놆.*O n+kC\yмG'5m565 vk`'a ?_ҹa:SZ2UQv+ g\DMwwvФݟf9GbDHhSFxD4M;^u/v^䚃7J3_g\+=tJvp /1m8Bh)N|CEu7;2aJtcJ򧗜6y)]Z>HU W A9z4NNW^ B9_Cザ89H:(>wm+Gdd332y SJWWG?[M]UbaHrR['i=c q~Ŷ [$:p=|{* 5=>\_ H=?)*KpA p3,31 `UH!#FcAlB ?m2H^.0uɞG{BI_ hzrIU~yRTH& 4̨?^-f:-8rj݈F zܐ"f8* sle4G} i(_Ȯ1C@g$̏Sonb>UӮ|m hy (;.;ۻ 9ZlBd1KsjÑ[_X5QIW9nI˔ٹutzjaE{]XD `ΈC'k2Rbn>{0 '!v^+ Mh 489=ͫ΍fgqЖ %U`Nݠ6L t޸"4e礕,uVrV$D1C10ܱB 5@)&o[c]r lpⷁۚD{k <*M̆ݫ/^j7΄ʴzUYU5|kkrax`kiFհ{GZ nv,/=']AEskF;]P9Aٯ̕#E)UPNJY.y ^ؕ#Frį=FrykpKAv~m! Guphm(>e͏?~t*%/!(PV'8*r FyŰH&/w'߀pzE1P4_:JGO9 !B"p>vZ})cg} .juYy{F#sJhޕ=(}(˺z}"9ٗByjbyk|M ?b$5;D./WίRWu@lĺ\.8g\b IahÍ(ȭhn8׾nز#]M6tWDԎ PD:aKh"bUg7<``2Bү#N|h/AL|ȇusW/wy}$q> -T%o`Of%dFoƗC[oA^{?.TP\['D3xI )]crESfMb\($jU0J5N8"&G*ŏXhZLtC"OsaĮ]ǭzk_8f(eVL'4m>fG|r^ \RK" ?5)nogLGqT'zd'|; l5nzݝG`P1+<8fF }j/ p.f+{9t0l 66-xXrg wc#!^ݧdݝw[x ʐP5zSȍ,;FUw+ 7ۡčuq[R:ktZ_l!d%!JM4WCSv%VL3l,Z,g- UepD_-޵h4_tau]MK,;Pί3-E 31/=\4ӻӇ[ڵY*&Ǹ3OyƆH/;JELm+)GB{A;Dm ){rb<6M~,xotS]& P1u9~&&5]dOޓ sW=o0*;7v9.HhR:3cΤO6~zƷ@{iZna—ce:@Gh?F? ܊|Ϩk0c3t/r+5G4rqĈ=F~>*y)% ZTFh# Q}.h<`6?Ɓ'a9ݦ-/B=(Iֻ呰.xyF b?r}k}.Q% QYݿL+_nC wAv+B/LEmQ:ky]n呸2aNz׽/ľ0=]E߀F`{P)Ѣh3Vyny㟗W> endobj 235 0 obj [236 0 R] endobj 236 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 611 0 0 838 0 0 cm /ImagePart_2067 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 359.5 718.7 Tm 104 Tz 3 Tr /OPExtFont3 11 Tf (3.5 Ensemble Kalman Filter ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 104 Tz 3 Tr 1 0 0 1 315.85 630.649 Tm 85 Tz /OPExtFont14 9 Tf (xe) Tj 1 0 0 1 322.8 630.649 Tm 67 Tz /OPExtFont12 9 Tf (b) Tj 1 0 0 1 330.25 630.649 Tm 94 Tz /OPExtFont6 12 Tf (= ) Tj 1 0 0 1 342 630.649 Tm 105 Tz /OPExtFont4 11.5 Tf (F \(4-1\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 105 Tz 3 Tr 1 0 0 1 489.85 633.75 Tm 95 Tz /OPExtFont2 11.5 Tf (\(3.20\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 95 Tz 3 Tr 1 0 0 1 273.35 607.85 Tm 90 Tz /OPExtFont6 12 Tf (Pb ) Tj 1 0 0 1 285.35 612.149 Tm 536 Tz (\t) Tj 1 0 0 1 301.449 612.399 Tm 104 Tz (A) Tj 1 0 0 1 309.6 612.399 Tm 131 Tz (p) Tj 1 0 0 1 316.55 612.399 Tm 88 Tz /OPExtFont8 12 Tf (t) Tj 1 0 0 1 318.5 612.399 Tm 42 Tz /OPExtFont6 12 Tf (a, ) Tj 1 0 0 1 326.399 612.399 Tm 34 Tz /OPExtFont14 12 Tf (1 ) Tj 1 0 0 1 329.05 612.399 Tm 117 Tz /OPExtFont6 9.5 Tf ( AT ) Tj 1 0 0 1 355.899 607.1 Tm 515 Tz (\t) Tj 1 0 0 1 368.149 612.649 Tm 73 Tz /OPExtFont6 12 Tf (err ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 12 Tf 73 Tz 3 Tr 1 0 0 1 489.6 608.1 Tm 95 Tz /OPExtFont2 11.5 Tf (\(3.21\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 95 Tz 3 Tr 1 0 0 1 242.15 555.049 Tm 96 Tz /OPExtFont4 11.5 Tf (Kt ) Tj 1 0 0 1 258.5 555.049 Tm 94 Tz /OPExtFont6 12 Tf (= ) Tj 1 0 0 1 270.25 555.75 Tm 139 Tz /OPExtFont4 11.5 Tf (P) Tj 1 0 0 1 279.1 555.75 Tm 62 Tz /OPExtFont8 11.5 Tf (b ) Tj 1 0 0 1 282 555.75 Tm 85 Tz /OPExtFont4 11.5 Tf ( H) Tj 1 0 0 1 293.05 555.75 Tm /OPExtFont8 11.5 Tf (T ) Tj 1 0 0 1 298.8 555.75 Tm 89 Tz /OPExtFont4 11.5 Tf ( \(H P) Tj 1 0 0 1 321.1 555.75 Tm 92 Tz /OPExtFont8 11.5 Tf (t) Tj 1 0 0 1 322.8 555.75 Tm 71 Tz (b ) Tj 1 0 0 1 326.149 555.75 Tm 83 Tz /OPExtFont4 11.5 Tf ( H) Tj 1 0 0 1 336.949 555.75 Tm 85 Tz /OPExtFont8 11.5 Tf (T ) Tj 1 0 0 1 342.699 555.75 Tm 101 Tz /OPExtFont6 12 Tf ( + ) Tj 1 0 0 1 357.1 555.75 Tm 138 Tz /OPExtFont9 12 Tf (r\)) Tj 1 0 0 1 369.85 555.75 Tm 67 Tz /OPExtFont11 12 Tf (-1 ) Tj 1 0 0 1 378.699 555.75 Tm 30 Tz /OPExtFont9 12 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 12 Tf 30 Tz 3 Tr 1 0 0 1 490.1 556.25 Tm 94 Tz /OPExtFont2 11.5 Tf (\(3.22\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 94 Tz 3 Tr 1 0 0 1 273.6 529.35 Tm 110 Tz /OPExtFont9 12 Tf (= ) Tj 1 0 0 1 285.35 529.6 Tm 138 Tz /OPExtFont2 12 Tf (x:+K) Tj 1 0 0 1 319.199 529.85 Tm 75 Tz /OPExtFont5 12 Tf (t) Tj 1 0 0 1 323.5 529.85 Tm 94 Tz /OPExtFont2 12 Tf (\(s) Tj 1 0 0 1 332.399 529.85 Tm 68 Tz /OPExtFont5 12 Tf (t ) Tj 1 0 0 1 334.8 530.1 Tm 75 Tz /OPExtFont9 12 Tf ( ) Tj 1 0 0 1 350.149 530.1 Tm 119 Tz /OPExtFont6 12 Tf (h\(4\)\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 12 Tf 119 Tz 3 Tr 1 0 0 1 490.1 530.299 Tm 95 Tz /OPExtFont2 11.5 Tf (\(3.23\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 95 Tz 3 Tr 1 0 0 1 287.05 504.149 Tm 119 Tz /OPExtFont4 11.5 Tf (Pt ) Tj 1 0 0 1 303.85 504.149 Tm 94 Tz /OPExtFont6 12 Tf (= ) Tj 1 0 0 1 316.55 504.149 Tm 86 Tz /OPExtFont4 11.5 Tf (\(1 ) Tj 1 0 0 1 325.449 504.149 Tm 426 Tz (\t) Tj 1 0 0 1 340.1 504.149 Tm 115 Tz (K) Tj 1 0 0 1 349.699 504.149 Tm 52 Tz /OPExtFont14 11.5 Tf (t) Tj 1 0 0 1 353.3 504.149 Tm 116 Tz /OPExtFont4 11.5 Tf (H\)p) Tj 1 0 0 1 374.899 504.149 Tm 52 Tz /OPExtFont14 11.5 Tf (t) Tj 1 0 0 1 376.3 504.149 Tm 81 Tz /OPExtFont12 7 Tf (b ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 7 Tf 81 Tz 3 Tr 1 0 0 1 490.1 504.399 Tm 95 Tz /OPExtFont2 11.5 Tf (\(3.24\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 95 Tz 3 Tr 1 0 0 1 139.699 461.899 Tm 107 Tz (The equations above describe two phases, the first two equations are respon-) Tj 1 0 0 1 122.9 438.899 Tm 110 Tz (sible for projecting the current state and error covariance estimates forward in ) Tj 1 0 0 1 122.9 415.85 Tm 109 Tz (time to obtain the first guess estimates for the next time step. Equations \(3.22-) Tj 1 0 0 1 122.9 393.05 Tm 107 Tz (3.24\) are responsible for updating the estimates using the new observation. This ) Tj 1 0 0 1 122.9 369.5 Tm 112 Tz (results in the recursive nature of the Kalman filter. By doing so, the Kalman ) Tj 1 0 0 1 122.65 346.699 Tm 111 Tz (filter estimates the current state using the information of all past observations ) Tj 1 0 0 1 122.9 323.45 Tm 108 Tz (although not the same time. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 108 Tz 3 Tr 1 0 0 1 123.099 278.549 Tm 109 Tz /OPExtFont3 13.5 Tf (3.5.2 Extended Kalman Filter ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13.5 Tf 109 Tz 3 Tr 1 0 0 1 122.9 248.1 Tm 112 Tz /OPExtFont2 11.5 Tf (The Kalman filter addresses the state estimation problem of a process that is ) Tj 1 0 0 1 122.9 224.799 Tm 113 Tz (governed by a linear dynamics. But it is often the case that the process to be ) Tj 1 0 0 1 122.9 202 Tm 108 Tz (assimilated and \(or\) the observation operator is non-linear. A Kalman filter that ) Tj 1 0 0 1 122.9 178.7 Tm 110 Tz (linearises about the current mean and covariance is referred to as an extended ) Tj 1 0 0 1 122.9 155.45 Tm 106 Tz (Kalman filter \(EKF\) \(29; 30; 47\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 106 Tz 3 Tr 1 0 0 1 139.199 132.399 Tm 110 Tz (In the extended Kalman filter, the state transition model ) Tj 1 0 0 1 427.899 132.649 Tm 111 Tz /OPExtFont4 11.5 Tf (F ) Tj 1 0 0 1 440.899 132.649 Tm 105 Tz /OPExtFont2 11.5 Tf (and observation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 105 Tz 3 Tr 1 0 0 1 314.649 52.5 Tm 93 Tz (40 ) Tj ET EMC endstream endobj 237 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 238 0 obj <> stream 0 ,,#b7 #xˊj> !Pwf˯iEW<'`8- z~c/A[R  Erņ%v9 71SįxLkE4;Fh jNf/|Ti ,* n}s $: &YG35  Gƛe EbJ'6hJ!+VeTNʘ/Qpu+mmdyv+Ł?ZKߏttzcfkkB|y ){;~W#:cM`C06o7F't #w ,Iĸ,uu`= OZZe"`Ofi$\-r8V1h%g)^dle^R_ ɦu>֪ _T%*L+gS:9"⻰o¢=?b]Pc[OK90#NIr퀚LM_)Cyt%ID%"0ŧ,PV)w.Y6 0)F]]o!EQrgW>RY[(Odn6S}R{ wQWj} 2CmD)j*1ߣpvlv2}bͫ5@Y5.ʯ 민|E [ gqY͌D^wf>g EV~x6 .aнpm~yR;0giA\]:fd4!CS}ÄGR/5sŘI +|2Nab>l?óPDKt-t.4\{P`UBgܸMbfx^m~ѥRw)]xA.K mҬ3<IL:iY%3 e_m U[2{,hC ~oйˮS2!^]%q{#!yZ.4fɗ Ne ?'w3,H< zmx/ʫm&yę;ch<'7r!5q1hOK~YŸۘ:G%Fk IXdI2̎ <|qKD]v˦aSZi1|G<̰d)=8#ǀ_â! 1@6/JXS&2IM $0VzƯ(AN4*v}[s-FdbӘR&<&5UIKbN=ԙRg w*F*niӮ"ҚN~5Tr\hoaÍe↻Tx'?`r2˖҃ &° @;ʂ9|>K])O q5otΞ!@5WW]Oi5lȶ;} >FbUhe6+|Q4qНhD.)] h2;U(wɪ?qhnIո+RMĻpc9𾃲t NG %Я 2'7{Ci ˀ)e-폏>MOo[+>ς Z93 5!V?KMtg?qX;t ECO [~bJ $lCG$ l]]  ʫz g9 )ZTq*<@i. !b9[zi@ |ǞpB'!)%+D4kS풃ċp|wrJ".Tˍ*T5+;[yy! w0|[V괖MT?VI yͷ(#d[tX7-oyP̳0yNT#|N~)Px (zY|=+3I޽c$8PzNrfjD-g}=cQˏYvhhxX!8FH!WQ&R *5`/Ei 4=nCLl<͜O1fheFx AF$j} ,z7o/MI{ʟ6lڵ(k+%[{QhN_o#;l}ǂ𫽀Џt(x =TGϋβA7b-6d";;BQl *tRcE7RkCx\"/R[Y.@v yP=}qV۟Wd{C0ICxCquR=kXDeCFj6&H:k֭LOΠzm^]Y⡁ȣ W3564[C%v<ͩ!c5q@\DDþe^2O޺:Z)XTV#<;2˵E6z\]5/"pztyp' ~PpdVQo<ϯH"1oErtmвEkeI5k5 0^k1䦆ux5t={Muf kΟp'u@mN GYr Q=RA3,zӦ^`&gȯ紊3x&K )Km7$k4< ]öc)7yN%^\l@ׄIˠF$($+J[}9]wRtC`<sȕ%DN3ܔ J./q@m_p(:}b{StPIcl d^Zm<3dTz(9]3`\Z, .0^1+:'E@RJG?'86 Ʌot2EP_a,_#Ke2̰r5%~ <N D,S>KFog( ztrǜ 6a}.2Ɗ~"D\ :G卟1nH"wqP{*m4Ue~:xUR=ntJ?Z)LJ WzO yu"Lў] 8Cv؆cW eS ?Ě>Ѻ.ú<˓IkF0H=V޼Ǒ ih(.rj\8f>mG+I?fd?+ Y\Nu'0 cیO5&M[f6ϔ$+ X1 DWHI犟x)kTz ۘ4u+r<= 6$*ϤΈiy,=j4&Q}o(*B VćÈa%f,6JQ0 ITh6̗-=CK+ _ #{-ڦ)@ WF٬AERr`~&f?dJ6š^ ̓:f^V:ȋ#v|B쨑uq޴HeZjsd][w= 5`*|lOFRVu;vS }9D2$Rd'd"Δ|߿0i6p V1Hڄ1tKŠ`,kG* lU*L7kXHpӁ?+ň*U3%%-juSK6UzX(0%"U.$ca 8؍pݑ~#yC_{6Xu\t|ٶ\d7B׳Uq!uB /MYt?-fgiȨ#VGU'_{ 풢cyOeQFgr2*˞&U Z:&yDO\Zl˃S3JŢ`ؿdrt_d4:FъdU&`SL/cš@ ?tCWAKr?J @_Ub2fgZexV]DUJ7 }T~#QJY /fzSFmE+[j Pw\`9fj^&]#fH J6 UA͊Nt%O%WG9̣΀o3B@yI* |Vt9Ċ@to%D&K a5f)/J[u#'WB'M 1hv>-c{r@>uKrm op 8+RrXƣ>&F)ua b*ڿcmqz]K2Ee `5'8`n%FM ;4K5ٝ:A(tLE0 -ro6֙s="1wƨJ*эQײk-31|AF~T0pS^Eѿ*/2HYkצ쪐;tw  7 p'g[HYqY{9~ŷA- R4qUŁOpWQQ]`;]kSެeA.Y*!X}Y.?(Q 2vg!yVr'ï4ι\#⿰"k9yµdJwkÿ \ЍQ.M׍?a)#b;z).(,*TĔSjN^|,cca6cT>19-4Pn3YuȒ45p&[u_-I$g vL,b2T5YnӹWѩpUBp6?%,x\d>NQ/5g0#MA1訴ʛ@#vxwV8pםpqaT@jV'/ ِ GáydŁ@Ml?{FN)lq^ܐ. /zkn :#j:=U(/Fil\GTOhĊ?sL"}rRa[6:cŕjMn_"Ȋ3 )SS(݇f@qOhoR7S֖h.!yx?ﹶwʖ@DO]8\C[d-MLEk< :(Ѡ瑔X#JM8yqr)}ρ> '- ^h7!6MݠQQ@df!s*<Dy'|կUVϬy4v] hI2=I@C#S#|"$YĤ@ 2{+)BOM:kSZvfEOv O*}#Zag篎H,lsp7;޻_9G-vuލx1!רx a]PyJn毓@;.`%g()b/ai߂P+!P@f fx_y. HTG Rl3ac۬1Sx&9NhЖ)s /Ic)tBe>cy@5 J'/T PoWt`M9A{ClQx_t5L m&b٧o}}dKA|]" bݑWmer(f^ՌG>68x,+&#)@.~3Čz?׷^pKB`yXwNڵsL ws>d\ckxPjU_`|I&= ퟆS׻|B]/>hz7-,ߐ jxۻWc8hk y>=͔;FR{Jwq?2U# i.>4i>^ 9Ŏ w#3ak̓V]v %TCvgy"=d@_&"cY:Q9C1=u'11:,x=X)gQۭB7OJ,;9w/},515>KDn5#Foy HWD1%\-Qtp)iAkZVa.p8n18[&8e/\jp$&lϿ֐O@$M Y\qL4 ]^jSF^T^bD endstream endobj 239 0 obj <> endobj 240 0 obj [241 0 R] endobj 241 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 611 0 0 839 0 0 cm /ImagePart_2068 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 359.3 719.95 Tm 105 Tz 3 Tr /OPExtFont3 11 Tf (3.5 Ensemble Kalman Filter ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 105 Tz 3 Tr 1 0 0 1 122.65 676.75 Tm 90 Tz (model ) Tj 1 0 0 1 156.949 676.75 Tm 88 Tz /OPExtFont4 10.5 Tf (h ) Tj 1 0 0 1 167.3 676.75 Tm 93 Tz /OPExtFont3 11 Tf (need not be linear functions of the state. It is, however, assumed that ) Tj 1 0 0 1 122.9 653.95 Tm 89 Tz (these functions are differentiable. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 139.699 631.149 Tm 98 Tz (The equations of EKF differs from KF in that ) Tj 1 0 0 1 381.85 631.149 Tm 114 Tz /OPExtFont4 10.5 Tf (H ) Tj 1 0 0 1 396.5 631.149 Tm 89 Tz /OPExtFont3 11 Tf (represents the ) Tj 1 0 0 1 470.899 631.149 Tm 76 Tz /OPExtFont4 10.5 Tf (mobs ) Tj 1 0 0 1 497.05 631.149 Tm 102 Tz /OPExtFont3 11 Tf (x m ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 102 Tz 3 Tr 1 0 0 1 122.9 608.1 Tm 94 Tz (Jacobian matrix of ) Tj 1 0 0 1 221.75 608.1 Tm 118 Tz /OPExtFont4 11 Tf (h: H = ---) Tj 1 0 0 1 266.649 608.1 Tm 81 Tz (h ) Tj 1 0 0 1 275.75 608.1 Tm 96 Tz /OPExtFont3 11 Tf ( instead of the linear projection operator and A ) Tj 1 0 0 1 266.899 604.049 Tm 75 Tz /OPExtFont4 10.5 Tf (ox ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 10.5 Tf 75 Tz 3 Tr 1 0 0 1 122.65 585.1 Tm 98 Tz /OPExtFont3 11 Tf (is the m x m Jacobian matrix of model ) Tj 1 0 0 1 331.449 585.1 Tm 137 Tz /OPExtFont4 10.5 Tf (F: A =) Tj 1 0 0 1 380.399 581 Tm 69 Tz (a ) Tj 1 0 0 1 384.949 585.1 Tm 105 Tz /OPExtFont3 11 Tf ( often referred to as the ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 105 Tz 3 Tr 1 0 0 1 384.949 581 Tm 102 Tz /OPExtFont8 8.5 Tf (x ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont8 8.5 Tf 102 Tz 3 Tr 1 0 0 1 122.4 562.049 Tm 91 Tz /OPExtFont3 11 Tf (transition matrix. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 139.699 539 Tm 96 Tz (Similar to the Kalman filter, model errors are required to be uncorrelated ) Tj 1 0 0 1 122.4 515.7 Tm 94 Tz (with the growth of analysis errors through the model dynamics. This becomes ) Tj 1 0 0 1 122.9 492.899 Tm 93 Tz (a fundamental flaw of EKF as the distributions of the initial uncertainty are no ) Tj 1 0 0 1 122.4 469.899 Tm (longer normal after going through the nonlinear model. The linear assumption ) Tj 1 0 0 1 122.4 446.85 Tm 92 Tz (of error growth in EKF results in an overestimate of background error variance. ) Tj 1 0 0 1 122.4 423.55 Tm 94 Tz (Furthermore, estimating the model error covariance ) Tj 1 0 0 1 391.899 427.899 Tm 97 Tz /OPExtFont4 10.5 Tf (Q ) Tj 1 0 0 1 400.8 427.899 Tm 110 Tz /OPExtFont5 9 Tf (err ) Tj 1 0 0 1 418.1 423.55 Tm 95 Tz /OPExtFont3 11 Tf (may be particularly ) Tj 1 0 0 1 122.15 400.5 Tm 91 Tz (difficult while the accuracy of the assimilation strongly depends on ) Tj 1 0 0 1 456.949 400.5 Tm 119 Tz /OPExtFont4 11.5 Tf (Q") Tj 1 0 0 1 473.05 400.5 Tm 53 Tz /OPExtFont8 11.5 Tf (T ) Tj 1 0 0 1 476.649 400.5 Tm 97 Tz /OPExtFont3 11 Tf ( \(37\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 97 Tz 3 Tr 1 0 0 1 122.65 355.899 Tm 111 Tz /OPExtFont3 13 Tf (3.5.3 Ensemble Kalman Filter ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 111 Tz 3 Tr 1 0 0 1 122.15 325.399 Tm 95 Tz /OPExtFont3 11 Tf (The ensemble Kalman filter \(EnKF\) was first introduced by Evensen \(23\) as a ) Tj 1 0 0 1 122.15 302.1 Tm (method for avoiding the expensive calculation of the forecast error covariance ) Tj 1 0 0 1 122.15 278.85 Tm 96 Tz (matrix necessary for both KF and EKF in Numerical Weather Prediction. The ) Tj 1 0 0 1 121.9 255.549 Tm 92 Tz (mechanism of the EnKF's production of an analysis follows from the methods of ) Tj 1 0 0 1 121.9 232.5 Tm 94 Tz (the KF and EKF. It differs only in its method of using an ensemble to estimate ) Tj 1 0 0 1 121.9 209.5 Tm 96 Tz (the forecast error covariance matrix. No assumptions about linearity of error ) Tj 1 0 0 1 122.15 186.2 Tm 90 Tz (growth are made. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 138.949 163.149 Tm 93 Tz (There are two general classes of ensemble Kalman filter, stochastic \(37; 41; ) Tj 1 0 0 1 121.9 140.1 Tm 91 Tz /OPExtFont3 10.5 Tf (42; 43\) ) Tj 1 0 0 1 159.599 139.649 Tm 92 Tz /OPExtFont3 11 Tf (and deterministic \(1; 7\). Both filters propagate the ensemble of analyses ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 314.649 53.25 Tm 77 Tz (41 ) Tj ET EMC endstream endobj 242 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 243 0 obj <> stream 0 ,,b7 ¢:$8D4?dhV/9wPIF[۬B~y9-ERk".|m!ڍ6^:s ؈ta{iUj&9b"]Gv٩ T a\ 8&95O Z d7P&]Vdh%VE AmK fU`H#agXҊoYRQT'ܝ'q?.kHF;Z{n<}r֖  3^] BE7ʠ Y1N 9 # aSc3 )[meuV|uz+U!ڰ껒F?λ1Mn.΂RDu8P)Y]CJ ar+#}(:j\h3vh^hyT[gW֐l@֞iI[}rZs64-+I2l-,p驑G:m4a9F|.8eMXtphq!2e>N"f K`9F;]Hفv{!l:U=]?1 ݏODZTE2ns4_^~$a,v,S,]3?*W%G̙sN:Dy38F]Ŀ [j{9PXЂ3'΢`[~^{XmT:#RfMslg{ic@e' *g?>X+ q"@tY7M]Nan"^hV>0ʇ88ҺY!v$vʳoD`ˠ6u$5t.#[jA9S^%pt\ӡgsva9u}#[`j4f4/֖# 愍46BEkfPI|} G/;LLk.cC{c|[_ xYm*{6,>Wnl-|hh,'N`5 lbZ'KDB+@G9kWybtL+,O}:ԩI^AOdO< ptj?Zt鉣05gws$BۭTSLH:_}#4ΰd B廩yTKց,&5@3ND(JJatEVaؕuPjZ"4L }ǵUz| Gj(1E?O)SYu;䀛x?2b36Ѻg*dc/!3ȭ} .6dfܾR\PT-79@ ۋ"SОigI;rڕ6'ˏB&M2OWEelR_<} 3DJ؁b[_x7:2^#NBZ0BLs@LP3Ye粡3x=쀜۶;DQ!ߋaE)K3gwL!S %/3 $wwɩg:HaD)Ov P$]p"]K.*a(չ`S]Syoe> "ǐdJ¹*ޜ˦P%9ꗷrfiPgs6m3s R)fƀ_\&&;Uj}BCl[qf"[aL|1%Ϧp p֖.*+Wibg0/(on" =m`$ƦG%QHy>VLe2@WD˥i 4<N}`@yr8tS/U*d<9ѹ*' |nZ-]444UrcZ?hQw,b(xSZy]hq"Ty+Q( @Ҋqտm@ipDX};޺jr,+ W X `?~29ܢ֕yl }n!}q2gL5hjMAKqd6+7m+YvGRƐ6<jS1W)Ī *߇!I"ZP^gybIDw'aÙO\z q@Yw?l~ 挛[J;{r0D<ʱD4,gM4Jm (f`yGs0Qw 1-alF[ `=f:&fЮ$_^PR؟nm%m[h]G.fx4,!Ǹ25DW`f*s}=ۊ+y61:F-oר.ߘ%Ɯ+%5z<7kl88Ko(8VX H 41,~0&:eNg.?+9Խ?)4,%QFٕ53/4hnكP(o^T*ʀ }KH{ + [wEM-+g'4&VT(­?}u\ W'v#Em*M'lCy=O!ߦ]t2PKh吻wӷgBK0遒m@"a\p$n҄y{K Y1ΥQ-_{=ŠkX y/c v_[u_XpO!=US%vsnj_::mcM~mCB#Jz(TUR|tg!ܑFX_֘-`VѰ< (bluD3U.B|p@<*bj?/FsVm^Ju1L7lfRv=Ceb̆):m/!>xA[=T{*M: U2j>⿿Ѱ斃k(U>5bbtrxue!Ú!LǃÙt_m7z"_ԭ2?!% 9FirzUbŭ }lD1}m[ R E -'?djn& MJ# >X b)m 7nZԕci砍)GgBՊ+_?/F_,EIKYǣ%/{'Ô v3؞ڛֲ:C`Гq|29+ܧSm,æ(CCź@RΖƋMg>԰ȯ䇰%U<.:?i>w)*oy&tt+ovkFh3gqxn}.g~5LsֿѿL$dL@!3 k65Y RU*-Zzgd3W{!"sa6zRʣK6PneM!T90ί:U8\UQál۴ctog.EG_Sg Km^;kMNm쀲DG)$4nܽ31V?@Ô?IBx望%ԙ% ;^yMù \ϳd޵KA ~JIGuc$RM!/M'/Q#ƂmjSClz}S|\/n;Kd8\E.sĽ owOB9/?#g#u阆IAk n{Xߔ'>SmqQ?ftLڟ 08mP3hiDr bnFl.}H;aRx=Xb9{0jhXBmqBzZ7oTΊeA~E"6QACPק>s]N $3,!'\puaJf KmcnRQpuHЈvJ4dr T*0-Z\8`KJ:uF`7H8S%, h'5J}A0D|Lg8)I#q*!0:ٍ"'d'wR6s/"ՓQGTckT>8ZmFì!FSwKߙ?yƴ$Fb7V`$y29t˒ڲ!iiǘP6pBKE"7J0z#3^|h1 DWWW}`0Jff]**Vm-  b />=b)e?s2h>ymV%ȴ&0)q6U{y}Ka^KhF#4k˞,b[1]gaPb'r, d\+zvm""]:?.CS:wAŃH_yICQIL. Vf[LIؿ2#M̵J=M,O\و줊ߥ UHe:+FxμМ9j,kkUoN Jy/ߣi-W =){{ܛ̛}:!gd <4^'$ۺYt (:b _ބ9jYRiiGV,^VM{Mi#ĒT%RIn`E7V%:C@-5)ӌ'D @8l' SamM1 kx;P IW$+9Zo֨K ʶkTQGQ=ʆ-yZIGBu`p zx9ݑ5WûPi\'.&db9\v9{ ,$Qp}WX CJgXMuУI wU{U̪\e/p̈́X&a}C}D.bbx /ate]l`sE% 89V^*"؈?]&s sWnlj^v;igh4{ nqY&=b.;WSx^hݦvFjBTyé^ƈYP/T 'VRњygWm!o5`.u(dQkDw-Hkܣ =)Y/Nn؝ӳUߙ ;B5}^bH J[_|: N -Ecdl_)IO+"RAkm_V2K:ƥ5jƵ*Ȃ;0jRf.9_P.7r)yiŇ6;W:刑ps522EFyhIX.8_쾱>㾔4_[ (ʁL">[lKbEٸCe7y_bGpN 낁qAG"bѮ{VM ȳ9.lg+L{VW~Yȅs#R߈]pXgP Ӧ^]IMSP[_AG /.};JV&zZs RЗG ?:oNHQOaN<iypa( ZdxڈpEJN_KùӍn J&Yls9\ oQ{Xo>> Vmpp>F]':{:ujn+=IUrj%Y('ȉJ|\ۂۅOpW)^^;YM6 |V&uc!w+~Yc91Jz,BB>!Lsgag'i_BƔQnE͟sFq#D@"y")`UT9Ta´,o3=!5,. 86p;s}E36؆sT}v{"N6kVI]HCLFMiJ*30ҋT*I'I_ r-Lx̣^iHUh>&Z^ɦs&]R`9M3U xqk#I8_g:آtTA\QI75 eRO]"]űf hqa@Pxb@E I?È-''ƋGnp%6m&,̭ή̈́Yxj ;^cl='d}̒L4w/?W'0 4"cJ]4&R(pӫvoˏEH&y'M`vA\HY#F)4QàgH11)A4 WXu^JOŜҘ.W4@,|*ʼlwaێ4-RuC">H{$j[0ᦲ/A4G-+vY)=nW%vjcבߝ{xg Y:Rl%g>7* ~K}:p{ }$,x!.5& V_Z y3]gjez=R#xk\?-4 8],ȺV6GNy[ϡlTֈ-4jzkىKwp a' ~! A;`P;(D~'| I0L^si$o7/kCSoR{"jі=]9xC-XG1MU7w;W&= \}t&OLypH?ErXiOrMu ˵q7HXښP, #{܀ޢgcݾ: ;ɨ#EDrF6D5e݁?Ĩ|]y䇨4km3\zN|mcS:BM//c4cʜb`xe mv/@g(T;tbMr/G[YuGhYBUT5$кwӋraez!SʼnzPPDIQ 0Nix endstream endobj 244 0 obj <> endobj 245 0 obj [246 0 R] endobj 246 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 610 0 0 838 0 0 cm /ImagePart_2069 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 264.699 439.35 Tm 66 Tz 3 Tr /OPExtFont2 11.5 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 66 Tz 3 Tr 1 0 0 1 220.3 431.199 Tm 95 Tz /OPExtFont12 13 Tf (P = ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 13 Tf 95 Tz 3 Tr 1 0 0 1 245.5 423.3 Tm 113 Tz /OPExtFont4 11.5 Tf (N) Tj 1 0 0 1 255.349 423.3 Tm 69 Tz /OPExtFont12 11.5 Tf (ens ) Tj 1 0 0 1 267.6 423.3 Tm 100 Tz /OPExtFont7 3 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont7 3 Tf 3 Tr 1 0 0 1 489.6 431.449 Tm 87 Tz /OPExtFont3 11 Tf (\(3.26\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 359.05 718.95 Tm 121 Tz /OPExtFont2 11.5 Tf (3.5 Ensemble Kalman Filter ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 121 Tz 3 Tr 1 0 0 1 122.4 675.75 Tm 90 Tz /OPExtFont3 11 Tf (with non-linear models, the primary difference is whether or not random noise is ) Tj 1 0 0 1 122.4 652.95 Tm 91 Tz (applied during the update step to simulate observational uncertainty \(37\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 139.199 630.149 Tm 94 Tz (Let ) Tj 1 0 0 1 158.9 630.149 Tm 145 Tz /OPExtFont6 10.5 Tf (X ) Tj 1 0 0 1 172.55 630.149 Tm 120 Tz /OPExtFont3 11 Tf (= ) Tj 1 0 0 1 180.5 630.149 Tm 718 Tz (\t) Tj 1 0 0 1 205.699 630.149 Tm 96 Tz (..., xr) Tj 1 0 0 1 234 630.149 Tm 46 Tz (e) Tj 1 0 0 1 237.349 630.149 Tm 109 Tz ('\) be an ) Tj 1 0 0 1 284.149 630.149 Tm 96 Tz /OPExtFont6 10.5 Tf (/Yens ) Tj 1 0 0 1 310.3 630.149 Tm 90 Tz /OPExtFont3 11 Tf (member ensemble state estimation at time ) Tj 1 0 0 1 122.4 607.1 Tm 103 Tz /OPExtFont6 10.5 Tf (t. ) Tj 1 0 0 1 134.4 607.1 Tm 90 Tz /OPExtFont3 11 Tf (The ensemble mean ) Tj 1 0 0 1 235.699 607.1 Tm 88 Tz /OPExtFont4 15.5 Tf (X ) Tj 1 0 0 1 249.599 607.1 Tm /OPExtFont3 11 Tf (is defined as ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 282.5 541.85 Tm 82 Tz /OPExtFont3 15.5 Tf (= ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15.5 Tf 82 Tz 3 Tr 1 0 0 1 321.35 556.5 Tm 131 Tz /OPExtFont4 6.5 Tf (N'ns ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 6.5 Tf 131 Tz 3 Tr 1 0 0 1 304.55 549.75 Tm 53 Tz /OPExtFont13 12 Tf (1 ) Tj 1 0 0 1 323.05 549.5 Tm 193 Tz /OPExtFont13 6.5 Tf (r1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 6.5 Tf 193 Tz 3 Tr 1 0 0 1 489.6 541.85 Tm 87 Tz /OPExtFont3 11 Tf (\(3.25\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 295.449 534.149 Tm 113 Tz /OPExtFont3 7 Tf (IVens ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 7 Tf 113 Tz 3 Tr /OPExtFont13 10 Tf 1 0 0 1 6364 5856 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 113 Tz 3 Tr 1 0 0 1 2409 6082 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 113 Tz 3 Tr 1 0 0 1 5899 6082 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 113 Tz 3 Tr 1 0 0 1 324 528.399 Tm 112 Tz /OPExtFont5 8 Tf (i=1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 8 Tf 112 Tz 3 Tr /OPExtFont13 10 Tf 1 0 0 1 8289 6082 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 112 Tz 3 Tr 1 0 0 1 122.15 496.25 Tm 90 Tz /OPExtFont3 11 Tf (and the variance ) Tj 1 0 0 1 207.599 496.5 Tm 87 Tz /OPExtFont4 15.5 Tf (P ) Tj 1 0 0 1 220.3 496.5 Tm 90 Tz /OPExtFont3 11 Tf (of a finite ensemble is given: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 122.15 388.949 Tm 94 Tz (The EnKF uses the variance of nonlinear ensemble forecast ) Tj 1 0 0 1 427.899 389.199 Tm 82 Tz /OPExtFont4 15.5 Tf (P ) Tj 1 0 0 1 441.1 389.199 Tm 94 Tz /OPExtFont3 11 Tf (to estimate the ) Tj 1 0 0 1 122.15 365.899 Tm 88 Tz (background-error covariance ) Tj 1 0 0 1 265.199 365.899 Tm 105 Tz /OPExtFont6 12 Tf (P) Tj 1 0 0 1 273.85 365.899 Tm 46 Tz /OPExtFont12 12 Tf (b ) Tj 1 0 0 1 276.949 365.899 Tm 100 Tz /OPExtFont7 3 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont7 3 Tf 3 Tr 1 0 0 1 139.199 330.399 Tm 93 Tz /OPExtFont3 11 Tf ( Stochastic update methodology ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 150.5 302.299 Tm 90 Tz (The traditional ensemble Kalman filter \(37; 41; ) Tj 1 0 0 1 382.8 302.549 Tm 96 Tz /OPExtFont2 11.5 Tf (42; 43\) ) Tj 1 0 0 1 419.3 302.549 Tm 88 Tz /OPExtFont3 11 Tf (involves a stochastic ) Tj 1 0 0 1 150.25 279.299 Tm 93 Tz (update method. This algorithm updates each member according to differ-) Tj 1 0 0 1 150.25 256.25 Tm 92 Tz (ent perturbed observations. As the perturbation involves randomness, the ) Tj 1 0 0 1 150.25 233.2 Tm 90 Tz (update is considered stochastic method. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 150 205.1 Tm 89 Tz (We define the perturbed observations "S) Tj 1 0 0 1 343.699 205.1 Tm 65 Tz (i ) Tj 1 0 0 1 345.85 205.1 Tm 121 Tz ( = ) Tj 1 0 0 1 362.399 205.1 Tm 111 Tz /OPExtFont2 11.5 Tf (s + m ) Tj 1 0 0 1 395.5 205.1 Tm 85 Tz /OPExtFont3 11 Tf (where 7j) Tj 1 0 0 1 435.1 205.1 Tm 80 Tz (i ) Tj 1 0 0 1 437.75 205.1 Tm 105 Tz ( N ) Tj 1 0 0 1 454.3 205.35 Tm 107 Tz /OPExtFont2 11.5 Tf (N\(0, ) Tj 1 0 0 1 479.5 205.35 Tm 77 Tz /OPExtFont3 17 Tf (r\). ) Tj 1 0 0 1 150.25 177.049 Tm 90 Tz /OPExtFont3 11 Tf (For each ensemble member x) Tj 1 0 0 1 293.75 177.049 Tm 80 Tz (i) Tj 1 0 0 1 293.75 177.049 Tm 63 Tz (t ) Tj 1 0 0 1 296.399 177.049 Tm 91 Tz ( the update equations are: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 314.399 52.7 Tm /OPExtFont2 11.5 Tf (42 ) Tj ET EMC endstream endobj 247 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 248 0 obj <> stream 0 ,,b7(/R}oìQO z_CQ^y+DzBf ;79L灟$ة2 QؼB#(1ӄ+|;)+׃x9׻u\l3<9۲䊯;{~יQDEX ۜƖwsgsf vѐcǎs6-)U bƽWCH:K-WdZ -Gp0:df~&&н:HJzCcuFe1m)4p6V~KPb-p   ĉp,<%p+ \`ՍZ%gMGx4l&Lqn[Y+:(! %Mf[]/ ߳BN'swDPm0"#( ͌;4 O͇Ff[Κgc*vR8G5xgaЮ G F_8Ipv-G솷+]@VGVEQb #!4Yw(4m(c qLuJqpeøqt$l1'*-PSy3Ɩ [ :֍D{vݤFP0sp{B+_m 9fTa.p-GSAVp᧼hqĭZl֟p9ѹg 1e2AJ* HƄSKz\ rK̩/"MFPi*AELakPؓ*q;p'y:n\X r޽@@3H[޽ -;6?BI֧q5Œmme{-'"?*9:JQ]vuE߉0g Ԣ!w_g)?EV >,ʜ\kjy-T5!S#SQg D-3 eyP\H 1@mAUdm"e`7>Yk;8^ppO?^' +R9 V"1QWągaSo˥~fqhP Lѳx'i9IyxƺZd zr|8&[%t[Ny&eK:}#,$% 7 [Fk{~I:K &w>j0^WNX{-̜yW ߵ g v&S^NNw]%dHԤ3NS1ϚbX;sDJmX3*`)} $uqɪ^qWN2uR:7, F,E]od^ /Gj ><]&.sC=D̖=-&s\{ .X bvE i3F  !j$TyaeUPI#|pQѴU̘0Y/ц a?o!Xxȴ)}YlnC s u!~*Е޵ax>[!˔&ʪ}6h9+6?lj񲹱ʰڃXCOʼǚ8*59&/-d9³ힵ<#I'O#y_Fh0.ig4FUEU\iW-߳Tn 1X`ƸF`ik~=vtn|=TdKE,Ya2N{"hIKPյ0Z3+O ֏K'GOϋ}tPBohr4ZQɅy|h.#U%aK+N߶G&u֋Q48G 뎫ijDwDGob${V8PL+ %FFCqz@M)z(*=)aP)C[zCB)#48ؑ0 hA[Q(d sQPн+D8Ao?k32nn#mVMJx'bɐËf%q#j| u14;FI8 VIRϜ5Menqй#YOBxAg\א4 щ5|Gsqv,+M@L 7'䱙\^^bQɠW$m"ÎC?[֫HUޟk%ɛ+A04q0)Ma~>/Cf-RH_.iYRrMlV7llΌ+:kdI|~=f# 8]~ƺYE|a/PfzV[<j.j&yngG$.~))aY46ˍxJ;!NꆑБT΄XTecPXp(-zzP+GZ!>ԤP/# $>{T\]^ irNwlE._Q' c1c7?Yä2s3#wQC+(\,;xm/6Nåd_̂3ѩ.):1&,=;kS.Xjg\3_o 485BU d_-(?T~[}m|AVqE~b3[p.`rlw ;JT$Kڵ倲 T7.^Y:c PqoxVʏڕ^9͐~n؂z-PqL@r2ʆ0IWaEAYhP^D+X٭DyK h`[dTԎ5p@n6kz{Q eŠ0SR(s}D?Ie xKY| Bbh5ݔ.Cfvsq?r줻V`_7.6&JBӛcG.2aM.'!bӽ€´|Ğw8Gss$p ^I00n+ 9, 32:E koq:1O8Ow /^!jxQ6jo沠]5.7<ܔ|_@v-%?.kQ />900G>pdB,qQ^w b~VC Vu=@MbuEi&Qf&gWV((ScoT8C~y;JQsCy*h h|QOqX%/'yT}HT}0\eO*{ܓ-4/õmxYȾ'VIqLV_]? tC5d/X&Q&_%SǣǚC J6vLX #5^CQ>I`%<=ɉvq*~mqw;EeLBX\CqժfJEa]Vo եd`d,Fj; ߊꑢv֒)Pկn|kX,9z!6,5E>0TQ[k\ak+C|*Ԓ12a2 _2S5S;`x_*7"œȆM˨@>ҏ4-2vD?LPmEʿ90g.-aoLQ(ʫ.d)8 D EAUs! O^|ZN@V.B6M|K`f(@zE]ֽ x)s(¥-Y͘a%+yN٧q,%5XGJK\&;/ag`ĔRT@)<˹EnQ(4H c/X "#I^'ߒ @Ј` MB%fWLF{uVQ>l3_'E" BZ?٩~t)*t^Gt4Vf)l^CQsa*,#Z~* aͶݤsU ~ݤTo [9PK3. a_KU(qrli-?xΘͲU^U\ԣMlĭJ BWB:FPjNy endstream endobj 249 0 obj <> endobj 250 0 obj [251 0 R] endobj 251 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 610 0 0 838 0 0 cm /ImagePart_2070 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 358.8 718.7 Tm 104 Tz 3 Tr /OPExtFont3 11 Tf (3.5 Ensemble Kalman Filter ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 104 Tz 3 Tr 1 0 0 1 0 838 Tm 28 Tz (\t) Tj 1 0 0 1 313.449 633.75 Tm 117 Tz (+ k\(s) Tj 1 0 0 1 344.399 634 Tm 72 Tz (i ) Tj 1 0 0 1 346.8 634 Tm 105 Tz ( h\(4\)\) ) Tj 1 0 0 1 390.949 642.899 Tm 2000 Tz (\t) Tj 1 0 0 1 489.35 634 Tm 85 Tz (\(3.27\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 85 Tz 3 Tr 1 0 0 1 0 838 Tm 25 Tz /OPExtFont4 13 Tf (\t) Tj 1 0 0 1 256.8 607.85 Tm 96 Tz (k = Pb ) Tj 1 0 0 1 295.199 608.1 Tm 93 Tz /OPExtFont6 12.5 Tf (HT \(H ) Tj 1 0 0 1 326.149 608.1 Tm 75 Tz /OPExtFont4 13 Tf (Pb ) Tj 1 0 0 1 339.1 608.1 Tm 97 Tz /OPExtFont6 12.5 Tf (HT + ) Tj 1 0 0 1 369.35 608.1 Tm 104 Tz /OPExtFont3 11 Tf (F\)) Tj 1 0 0 1 381.85 608.299 Tm 81 Tz (-1 ) Tj 1 0 0 1 390.949 617.7 Tm 2000 Tz (\t) Tj 1 0 0 1 489.6 608.1 Tm 85 Tz (\(3.28\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 85 Tz 3 Tr 1 0 0 1 150.25 561.049 Tm 94 Tz (As we can see from the equation, the perturbed observations are used to ) Tj 1 0 0 1 150.5 538 Tm 89 Tz (update the ensemble states, similar to the Kalman gain ) Tj 1 0 0 1 421.899 538.25 Tm 131 Tz /OPExtFont6 11 Tf (K ) Tj 1 0 0 1 435.35 538.5 Tm 94 Tz /OPExtFont3 11 Tf (in EKF \(10\), but ) Tj 1 0 0 1 150.5 515.2 Tm 90 Tz (using the ensemble to estimate the background-error covariance matrix. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 150.699 487.1 Tm 93 Tz (If unperturbed observations are used in \(15\) without other modifications ) Tj 1 0 0 1 150.5 463.85 Tm 95 Tz (to the algorithm, the analysis error variance ) Tj 1 0 0 1 381.35 464.1 Tm 102 Tz /OPExtFont6 11 Tf (Pa ) Tj 1 0 0 1 399.6 464.3 Tm 92 Tz /OPExtFont3 11 Tf (will be underestimated, ) Tj 1 0 0 1 150.699 441.05 Tm 94 Tz (and observations will not be adequately weighted by the Kalman gain in ) Tj 1 0 0 1 150.25 418 Tm (subsequent assimilation cycles \(37\). Adding noise to the observations in ) Tj 1 0 0 1 150.5 394.949 Tm 92 Tz (the EnKF can, however, introduce spurious observation background error ) Tj 1 0 0 1 150.699 371.899 Tm (correlations that can bias the analysis-error covariances, especially when ) Tj 1 0 0 1 150.5 348.899 Tm 90 Tz (the ensemble size is small \(92\). Such shortage is overcome by deterministic ) Tj 1 0 0 1 150.5 325.35 Tm (update methodology. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 139.199 292.5 Tm 94 Tz ( Deterministic update methodology ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 94 Tz 3 Tr 1 0 0 1 150.699 265.1 Tm 91 Tz (Deterministic algorithm like \(1; 7\) update in a way that generates the same ) Tj 1 0 0 1 150.949 242.1 Tm 89 Tz (analysis error covariance without adding stochastic noise. There are a num-) Tj 1 0 0 1 150.5 218.799 Tm 90 Tz (ber of different approaches, here in the case we are only going to talk about ) Tj 1 0 0 1 150.699 195.5 Tm 88 Tz (one. Here we briefly describe one of the methods called the ensemble square-) Tj 1 0 0 1 150.699 172.25 Tm 91 Tz (root filter\(EnSRF\) \(92\) which is mathematically equivalent to the Ensemble ) Tj 1 0 0 1 150.5 149.2 Tm 96 Tz (Adjustment Kalman filter \(1\). We use this method to produce ensemble ) Tj 1 0 0 1 150.5 126.149 Tm 90 Tz (results comparing with the results obtained from IS method in Section 3.7. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 314.649 52.25 Tm 79 Tz (43 ) Tj ET EMC endstream endobj 252 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 253 0 obj <> stream 0 ,,b6F 0`QA%!97KHj b *o0?쥍esm*AШ!DV/G-B⟨8뜧ҀCpтw:O6M҃tC|݋7w.}̄;%Sk!.; BPB,x_A}aR_ 70z?pNPN.4*}| ;nV V_q3mFں8K4ʔӣvo ^iҲMgy[H\f()!ˤ!AMT3)ۿ |S{ʴxTDT:YQ H:2ʨ[J 9Czg;wV/9rW/[ RgI &~0HP*m;]mfgdlTSvftnuʼn3FW9UhOC'x/ѥ12i#x6ġLHa>E wOK27dpWqWtn.d\ Ä)&|E(]aSZ7̒5S*\r#C^FQB~?9 ,Pٗe W@y*K Wq/=8u"ě+R˘@c}8T%"# Lw>)%Bg1BKqPJmvk~ƴwVjڞ| Ez(,V Hрʔ}?ŭXa0ScmFeBxkA#uЦPtpwG^YM8WHj>RY64V.&]:Ef _o畑YMW.qzz$m2'7Tӑ )S8N%=\eZ]P :a/w X|Xxte>٭A`ݚ LBD8e;j-;Z͋o= gP]ǥb4bbX p|Z~y0*.ԛz7;$F#+#0vYñjZU#=?[%v=wL08o$ F' BCuσWL'=[gQ8iVh"V-| WC kٴSAiP.f==n:!meR-._K!ɶJ͖Mjo3`frL mQט>uc{ʨqEm-)*`!zPՔB%HKtn1oj4`8&yLH|x::w7[CNVF*G*K+'OY#j,—LT?N&4%U h%-xک\+7:ӥ0VBfI /MF"e2Յ݌a!=:W HD{OB)>4@"KԶ&0l$00< 7u1n›{ng˸ b)=cs\+@R|bt}SsKAc50 x@m|75[u;GS3~@}t"ibGGcn? #ĩ֚X6iߵ^F()y.<j"Z5\iL֤Ώ9H&iYkKuT 7##={4ieM;#h[("PojXt#G:*B>VQJ%m~{ =_߼el$C"2Jΐq̈́1#0U0&&T?z{b5)V`䦻WBblr!OQoeHT(~bJȬ4 In[y莼B~$H^">H2S1e@ؤ_cBf h1 9Tj>9ޜ+َ۪KaH+Bq*7VOq.Q-񚩦mUf9q7]`d%i -~ӁC/1,,sr&8gPK B{CvO }z1G7kii|mn>xڲ< d0X#>*;0ܚ,شcǴ՝Œ॰5=Ӥ3ɪ#3<3ᙃ_QLE뻓Uc !2aiK,؎ӏiP-%ÿW6ޅ'řHP(+)br+rs.ndz/mPt©cY썽jάEj4lD'쭓{ZS=iVRweOP7Xw&t *r\(9Mwp=axg?tK1UX{iLKDVA9y ɨdĽrg74lq-!!kFpúr_Ux4֭gG &N#I䶹lxvv>|88]_.d\r9\UݯZ>~ۜ93F5V][P& (;ZUcJ bd;4 -M!{ezfYx0 s}ܪOG4GqiƉ&{h&p^)̀8qr*LmGJE;D~%Adgh-$/xIv#f Ffq.!2ImbL//mHYc6 qi<&eǾ)TTѸ \^Vچ s-(\^yS꼝csrj[yo/ Zuahlz:!FAD?g0 ǭ@D`!)\|o1g=$0z_{ 9Q ^ފ|UZLdDw5'./N؎bx+!')d,:$"Fo8i2D'l\j ѹte^#z  ~ʝ|f1VS=ʗs|2($;2V`Q5-#mi 6t|ߦ6&A*$Z Ŭk9;z/.(pcvKsIzM2,DVK٪1r_cvGlvR-` ' mLht 3` &0m8Wbx~j LuzW9ci?5*H4 Hk,!]+;lE 7ܛs!dÂL nEtG|1yڬ־fDi}ޭ~s\ٶ[}hm8B̊z/f"ש+5^wហSi͞NR'tQr32<5 kXKdm/SӘ3dmIybƈף1ޫlmKP\)3b=s"_E:bY͕`vNGOZ0Ӭ6VF>3?aXb: `[Ys})F@2wrq!x\[QզC}2 M*lŬ,*:φlPv5ƽ6 ϏFZ]$Fc{_|m铛VLB^}>q6iЗ8_mNEWq SmČ3Fhlf,QH>3XdWLApz! Px7OwZuM#SRxH7F2 ,Օ5k FOFnl2 M )= vIY!^ԣ5QՏbGG 2wfڊ';vS^±AwC#Ip]9Me |Y}@ojG|=L_ <+b3aL:)q~e=׬œi$ȧ(W"tcvehY:'؈n7pp(Rl&<%QU@asGT x7OކJ6~pSit7ROj&Bi]=;KOV2!s%PY@Do_)R:mqno bӆʪuoq/ASs.?'WYIa.jj_yZH!@r ǵXIO>ݽ!/\u" N7hYK*K9k9QS1g_   '-}*iw6#{Ƞ(~]GtaLI;z)!Rǔ^3οKzW_OTixb"bNNo{čZ/!&q?0f+gOBon<1nWUԪ?9 jNV΋%<b֦rӎֵ :z$jXJ zKz- A A7ڣ+;/IE13""u4E*H!ǥ"I ¨sIkS_526"sc"21{B>~ʱճڲ'/ű(ʢ7ɸ i5ğdTer?+gjϓ_WxK*sÞ&v2MQs2:$r'虐QlfW$xY'yCV`q{!la^)DVdzdDa#ոT ?EdCRaAu LJY9h1& KG9^֣ j'!;~m'fz1ӦOw7R~}p#'FK_dGnz' %!)ȼHVC>gĢM֜IC3;uǾ\gBmbVtp-%U7E# :TIW0 t$MeS ،:|+~qid%:kz0Vqq+n?n֑oҿL_m!"ԛV}[7:/ʶ6Mx8=] aNoAS!fĒvm%O\qWwJkt<_vP@?]F8zqʓ,b͆"MoV9mZ#L횻sl]]!wKڥ/_9l bk7$CF*:AZwŪ}?%9Hte8j~S:Xf 6.{$%IzxXq#QNЍ T"%d ǚ0tM,' j4#ΚSs_|WK_/I $)Wu%{3Am6BR<5wD m!#kk0ho G\_i&HJLʫ1uh`&}>OY;dV+t"jAJν>+eo=u`XiqZkBYMrjDIn]~"l,7"s̨dIIW/4Aֿx aÄO*>G/;W E;{̼ؔ\5NtPUh/M^&R,ԛFSx^6°q~JZ=7 k]lо`0Y¸21{_"lD͐,qS(ɰ%f41gز\?겝nZ$Ap7'݄*hPK:xO qv: 6Xu;8L6v/"gn7I)XeH7/Te p=1XeaO{Mx$5<04%[|7|7#=dV~p~щɍAb; ~T&qQwf&UTS|!x>{MCXQ1CHPNSiEcFFB[4/%1Ƶs!4B ġv++NKyO0 _/0Ѣ<=v4`??Rl/ NL ¢mS = + @V3{ov4{${P )m! \}тjX.6pzG'I^m34sKoJ] R9= &Rb<CE MB%Hl^4堟 h 2DlZ9¶cE]-HJg1;Eve ,LhPI]t 9ÊD'Щ+Z11Wd . zlR}}[Px%H/(<,xgPx0G=J\0sVWG6s83J-ۭt@e訷 ;>, aƅ<~tכInHSy]K{idޢh&&҅$DŽINF#_G2ې祳B^*!/G'a@ѷNÉ%G(%ETԜVYj2g#*_goAcrz*~o pċv`#Hl9"/h._{!̓x'hٔ0DcշE0?6ǯ1IoX:c9nsZ3O vD&/S-2DzBDƙ?* ٽ6:r-A 5G_%t&D3Z7<Y.1EMU,Ha2VgX_h7$M>ޑG8@0jh֪E0|hMAX*+/& :]"ݜn -dDJyg e-Svx._Q@{HS endstream endobj 254 0 obj <> endobj 255 0 obj [256 0 R] endobj 256 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 614 0 0 833 0 0 cm /ImagePart_2071 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 400.1 714.899 Tm 103 Tz 3 Tr /OPExtFont3 11 Tf (3.6 Perfect Ensemble ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 103 Tz 3 Tr 1 0 0 1 152.4 671.95 Tm 89 Tz (Generally the EnSRF updates the ensemble mean and the deviation of each ) Tj 1 0 0 1 152.4 649.149 Tm 90 Tz (ensemble member from the the mean separately: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 290.149 579.299 Tm 77 Tz (Ra ) Tj 1 0 0 1 305.3 579.299 Tm 161 Tz /OPExtFont9 9 Tf (=X) Tj 1 0 0 1 324 579.299 Tm 62 Tz /OPExtFont12 9 Tf (b) Tj 1 0 0 1 341.75 579.299 Tm 112 Tz /OPExtFont4 11 Tf (k\(S ) Tj 1 0 0 1 364.55 579.299 Tm 78 Tz /OPExtFont2 11 Tf ( h\(5C) Tj 1 0 0 1 393.1 579.299 Tm 51 Tz /OPExtFont12 11 Tf (b) Tj 1 0 0 1 397.449 579.299 Tm 120 Tz /OPExtFont9 9 Tf (\)\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 9 Tf 120 Tz 3 Tr 1 0 0 1 282.5 553.399 Tm 172 Tz /OPExtFont6 12.5 Tf (- ) Tj 1 0 0 1 289.699 553.399 Tm 590 Tz (\t) Tj 1 0 0 1 308.149 553.399 Tm 91 Tz (= ) Tj 1 0 0 1 319.699 553.399 Tm 79 Tz /OPExtFont8 13.5 Tf (\)4, ) Tj 1 0 0 1 333.85 553.399 Tm 166 Tz /OPExtFont6 12.5 Tf (- ) Tj 1 0 0 1 344.399 553.149 Tm 79 Tz /OPExtFont8 13.5 Tf (\)cb ) Tj 1 0 0 1 358.55 553.149 Tm 139 Tz /OPExtFont6 12.5 Tf (- kh\(x\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 12.5 Tf 139 Tz 3 Tr 1 0 0 1 489.85 579.1 Tm 87 Tz /OPExtFont3 11 Tf (\(3.29\) ) Tj 1 0 0 1 489.6 553.149 Tm (\(3.30\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 489.6 523.399 Tm (\(3.31\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 244.8 523.649 Tm 110 Tz /OPExtFont5 17 Tf (k= ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 17 Tf 110 Tz 3 Tr 1 0 0 1 311.5 514.299 Tm 104 Tz /OPExtFont8 13.5 Tf (HPb.riT +r ) Tj 1 0 0 1 373.449 541.649 Tm 144 Tz /OPExtFont6 12.5 Tf (\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 12.5 Tf 144 Tz 3 Tr 1 0 0 1 272.649 523.649 Tm 94 Tz /OPExtFont5 17 Tf (\(1+ ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 17 Tf 94 Tz 3 Tr 1 0 0 1 151.699 473.949 Tm 86 Tz /OPExtFont3 11 Tf (Here ) Tj 1 0 0 1 178.3 473.949 Tm 136 Tz /OPExtFont8 13.5 Tf (k ) Tj 1 0 0 1 192.5 473.699 Tm 94 Tz /OPExtFont3 11 Tf (is the Kalman gain as in Eq.\(16\) and k is called the ) Tj 1 0 0 1 457.899 473.25 Tm /OPExtFont6 11.5 Tf (reduced ) Tj 1 0 0 1 497.05 473.5 Tm 89 Tz /OPExtFont3 11 Tf (gain ) Tj 1 0 0 1 151.699 450.699 Tm 90 Tz (and is used to update deviations from the ensemble mean. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 151.449 422.85 Tm 95 Tz (We can see that in order to obtain the correct analysis-error covariance ) Tj 1 0 0 1 150.949 400.05 Tm 91 Tz (with unperturbed observations, a modified Kalman gain, which is reduced ) Tj 1 0 0 1 151.199 377.25 Tm 92 Tz (relative to the traditional Kalman gain, has to be used to update the error ) Tj 1 0 0 1 151.199 354.199 Tm 94 Tz (covariance. Consequently, deviations from the mean are reduced less in ) Tj 1 0 0 1 150.949 331.399 Tm 91 Tz (the analysis using ) Tj 1 0 0 1 245.5 331.149 Tm 123 Tz /OPExtFont8 13.5 Tf (K ) Tj 1 0 0 1 260.899 331.149 Tm 97 Tz /OPExtFont3 11 Tf (than using K. In the stochastic EnKF, the excess ) Tj 1 0 0 1 150.699 308.1 Tm 89 Tz (variance reduction caused by using ) Tj 1 0 0 1 326.649 307.899 Tm 116 Tz /OPExtFont8 13.5 Tf (K ) Tj 1 0 0 1 340.55 307.899 Tm 91 Tz /OPExtFont3 11 Tf (to update deviations from the mean ) Tj 1 0 0 1 150.949 284.85 Tm (is compensated for by the introduction of noise to the observations \(37\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 123.099 233 Tm 129 Tz /OPExtFont5 17 Tf (3.6 Perfect Ensemble ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 17 Tf 129 Tz 3 Tr 1 0 0 1 122.9 198.7 Tm 92 Tz /OPExtFont3 11 Tf (Given a model ) Tj 1 0 0 1 197.5 198.7 Tm 109 Tz /OPExtFont6 11.5 Tf (F, ) Tj 1 0 0 1 212.9 198.45 Tm 91 Tz /OPExtFont3 11 Tf (there is a set of states consistent with the long term dynamics ) Tj 1 0 0 1 122.65 175.149 Tm 94 Tz (of the model, in the system with an attractor, this set will reflect the invariant ) Tj 1 0 0 1 122.4 152.35 Tm 97 Tz (measure on the attractor. The probability distribution of states in the set of ) Tj 1 0 0 1 122.4 129.549 Tm 93 Tz (invariant measure is called unconditional probability distribution. Generally a ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 313.699 51.299 Tm 80 Tz (44 ) Tj ET EMC endstream endobj 257 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 258 0 obj <> stream 0  ,,b6 {Mf c"BJi!%:Ɋ0VHW-ac3\`moX|u@nqk:b9)IZv=bmzR1l YQ.T[n["ub̔m- ȳP"C+^PcBKdǐnew˪8g$[c%vܩuܰiAp4+x1W8`+|JGLU{ۄo_ ^wI&H[SY26%k}v.sҴ9gHBv'  $C)쾀^r}F vkvea5BZy evkUa(ջQ8vhZFIf_@'f"1BN,2رTW|5zݯu/#xFK-cX8VLwIr*84 Ys13 ĝjJ믯=鯅%0cp.dш!HxYP*`MjWs]IQ\'Cd!Q?Mb ]dK,$'l$I[|OH ,3qW>JZ7@C=v[yTFZ:#Ça(HUwC/3:oDZ8rn2Ӫ;Dw ǘcce?N ~_Q .F +Ր>8Z/w̜E9 nˈu|eZ@@YOi[#@apuN 쭡 C r5Hh/u+K qu;@9?3kwzL|e!F*"X2o.Ș\Set[z37pA3NF)2(ʪr447Ų kg)s"hPN#flU1٧hʽĹ/w^Ҳ>cW6L4|#6kG%&7Zѷ>K}!ϗ!4"ܻ0曽2\Ⱥe6dz(<[ dWl60z][r@t%Էja+coD_j`X:B-ͦ'w/GI6Yi|A0%1MAom+mߋ`KyQeakNF A/Naݠ]#ķ \5o Z m#t/doh~+Mx 8Y-uWhBpٕ=qSc כ0ک-|kj5(X*)ÖpB/n0[gh~yr4%Dr&T֪-)&yD髥}L{oh X_LQG*a 3P{veVڅLnSXo! ͹-u(ĄmfB'"Lasm& U!n8zhN`'n?֨7[Z~>Bo]U@4N-b|toMy\k+Ӌ;['S 46bP#,isOν[a'J78߿Ա{-:wgw|3)Se2f۔] ?#&T%qf5Up&C)9Oy]3"rt$.Pc㊇ &m5(" C (E1 aV2-k錃ꪴ,:8e.,gN!O ˅&ȗ~:^LȂ7X$:JC2/@|A(.U] tH{}6rlJ}ϼuY:Љ*ٽϾ:G^=*. ֙n EAJRq <8GC/7mE4\J,!~ Kv8jau;KDmQrP͛|*gg d.+J۱ C%mS U1A^PkN7]DH\j 苙aVw%cV@+uQRK߮pD T꿰 ίwtGƎY)%M8_3RXTomǟ-KR>5x i -TRv|3r`֧kR{: E&Ej$ ].U$TX?E!2j*^UĥsAuz$iуB ia6i7ڣD6F֙+ע_ ٻoqI;:LꉀVjO@ 13b$&h;-뮺*!T|ZdG)Xt<eF CtPsrP >OB3sB8r X Y*~g𼸄@7^we{ZnX]P3Ay+@sĻzWb cN<Mzj'CFľ? _39 DAbkS0fyS-ND^{@ܭ:BSͰwv2wEn=P._pQڪ{oz&xtjwUbʠ@>G !~zl<_"Pz.$xLSˣ-X li/\ I4Npk,'!ibrKd?WlB¹CA)oeIoӲ>λ:ML:Z/d_>zg8~-*`q;} .\q*N e/~b*<8zGkXo Љf7&X{Oywrm0Jٵnz<{_~pX1o7V!\p҄ <%#f _׮ Rxб Njy>u=+`#2j 5a/(G6'aASJb 31WYΩ" ~K%uEa5Ή2#?+0*i>dxMv"ssś1jӾ?L\'9SH,u4fa]I}i@ 7&qXUSa./(`9+IS0%qlVId1gw~kp(:{s N#ZGP+Y2zh`vmgahzx:[#(ysj<4Sf"8:~c4 @El~ &F$h(G]n;(=7%C^B3ZL9m2ܙ"걎g G}^J/m:==\rFhar3:Ŏ.EVP!FhMhzMcQ,"\zjVWnX@a~_@1%枱_-&̸w@B?+2dxxrC@9]*w c+>N&֌X"yܔb+2!-1KnCˊ[ =2?HZ{䖘\X "NOxZ rS倲4Kb+ݠ˜FEΐ9Yq8PP |'|O|!M(-V x$N}V)s7`/ 5%:vnk|wWt(Jcfy;`m(seaUSEf5<_hv[ xWG.s&~0>QN`@P6.dڭNm/xe(ɿ)be+SSwqg AŃA=0#c) zֺK 'z6@N /D뙳h'ϐ0_ި|eMA\W1} 0ohUz*>zs̥. P -8~}&-{߶?HچO{uP)s[RU’=4htb:8}+)7}B+޲d7NvÿJ1wT[oBSqݎ_b^3L.AbC~1VBP):U]vȂHc/݈D(hP[L ZvO9OL.K*_%[y Mk,  cnlbAC8壂>)F fɅgvG1{OS9ҡmCxU#Is!WX:A5-dܖ0|pZ+Hdt`wiY:/>"yK)oXksEյ>" Y%%E(r?pj+5W/q~IlE!8š!dݤ&x3Ր1+&)Kf;CU7 ' 7 -L,y5:7X&\H?3(*m…F 6izB7q7,BV,FK #f!yZ.󟸼n[d\ۥp%>P. Q[J^ބA.03NhC/'$Ƈ k57RS47v(! dpLJ\[0{OS,`r6O郉 \?/#?gIl&DŽ瀜29)w1m _TSpim)1)7ͷ՜i RW>}\| }&gm]P V޿}NdDQnoQfؑ m !pZ*2,5/K2uyGŅ' WC Cܔ9? y7Pae=,TQ#Qzu}8MNL勴@&.qZyġ*~1YxC{ 3s!I e8M $H؀$ K $r=fP!%~\Fdk1dt;lAD7n>p^ ቭeNW-C!g~$Oh ]W?ՊGEO5 t~ށ 15\q} *i)h$I)v8+bG#C8>^d* -L'Z8s7޽, !%{ik "c6u(!&kԥ1kM8C^?mV5?t]O mP}0nY$Y7௱Jk' Ќ=zh qc\l@ŗS>nXnG΍?lW nn鋭tm]t+[|ۏe.$)I^6A0BƖ`-ç/Z*(%)W^rnWa%B͋e-Ri>PNhdʂ۪ 77!Z*Gd!+`3H7eK@&B"<~s8j2ŀői҂dzLvd*MA@⃿oHTaS_ mdpKc>=etY$a֬)7֩(oOr|D}5ٌDh XXO¯$rid԰ό[_0oӓ/*w}a P_ao-3mV;Kh^N v<@OxOaLڵlWԠpS8^e9%c"\N m{8)!v47v`DgQ䬟XiK/` Zk?s~FM_" PiFVބƅfulɁMLzVd( ~ȠeSLU&QGbta*=6 B&8VpWMXe@ZWfOdD7XEl΃mXAob+ȩhGϘZ>4U,<*{poNg7ſL/=vh;ǽ'5Mypy~& ZXrڳg[yUQB03 ݍ1 ewC "qLMHkM:`|M:=RЧoF+x q1j{l|=25ʊ=a.\d-% }wb޳%@W㳼epiCMQFY_Ajfȶ1M^ra?.AO 5%cd5X`F!d\$!1K\-]% ,asj|'!%@t{m\&E]AO~3{E-I"5z]N5( 1U,yx> endobj 260 0 obj [261 0 R] endobj 261 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 614 0 0 835 0 0 cm /ImagePart_2072 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 396.5 716.45 Tm 103 Tz 3 Tr /OPExtFont3 11 Tf (3.6 Perfect Ensemble ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 103 Tz 3 Tr 1 0 0 1 120.5 673.25 Tm 104 Tz /OPExtFont6 11.5 Tf (perfect ensemble ) Tj 1 0 0 1 207.849 673.5 Tm 95 Tz /OPExtFont3 11 Tf (\(SO\), is an ensemble of initial conditions which are not only ) Tj 1 0 0 1 120.7 650.45 Tm 93 Tz (consistent with the observational noise, but also consistent with the long term ) Tj 1 0 0 1 120.7 627.649 Tm 97 Tz (dynamics \(as in "on the attractor"\). The ensemble members are drawn from ) Tj 1 0 0 1 120.7 604.85 Tm 91 Tz (the posterior probability distribution of the model states given the observations. ) Tj 1 0 0 1 120.7 581.799 Tm 92 Tz (If only one observation so is considered the posterior distribution of the current ) Tj 1 0 0 1 120.5 558.75 Tm 95 Tz (state x) Tj 1 0 0 1 154.8 558.75 Tm 68 Tz /OPExtFont5 11 Tf (o ) Tj 1 0 0 1 158.65 558.75 Tm 92 Tz /OPExtFont3 11 Tf ( given the observation can be derived from ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 239.5 515.1 Tm 101 Tz (p\(xo I so\) a p\(so xo\)) Tj 1 0 0 1 347.05 515.1 Tm 56 Tz (1) Tj 1 0 0 1 354.25 515.1 Tm 94 Tz (\(xo\), ) Tj 1 0 0 1 375.35 514.85 Tm 2000 Tz (\t) Tj 1 0 0 1 486.949 517 Tm 86 Tz (\(3.32\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 86 Tz 3 Tr 1 0 0 1 120.5 474.5 Tm 88 Tz (where ) Tj 1 0 0 1 153.099 474.5 Tm 104 Tz /OPExtFont6 11.5 Tf (p\(s ) Tj 1 0 0 1 179.05 474.5 Tm 93 Tz /OPExtFont3 11 Tf (x\) is the probability density function of the observational noise and ) Tj 1 0 0 1 120.7 451.5 Tm 94 Tz (41\)\(x\) is the unconditional probability density function of x. Figure 3.3 shows ) Tj 1 0 0 1 120.7 428.699 Tm 99 Tz (an example using Ikeda Map. In Figure 3.3, states \(black\) on the attractor ) Tj 1 0 0 1 120.7 405.899 Tm 94 Tz (are consistent with the long term dynamics of the Ikeda Map and those black ) Tj 1 0 0 1 120.25 382.85 Tm 89 Tz (states that inside the bounded noise region are members of the perfect ensemble. ) Tj 1 0 0 1 120.25 359.55 Tm 97 Tz (If a segment of n observations s_) Tj 1 0 0 1 292.55 359.55 Tm 93 Tz /OPExtFont5 11 Tf (n+i) Tj 1 0 0 1 309.1 359.8 Tm 162 Tz /OPExtFont3 11 Tf (, s_) Tj 1 0 0 1 339.6 359.8 Tm 51 Tz /OPExtFont5 11 Tf (1) Tj 1 0 0 1 344.649 359.8 Tm 67 Tz /OPExtFont3 11 Tf (, s) Tj 1 0 0 1 354 359.8 Tm 59 Tz /OPExtFont5 11 Tf (o ) Tj 1 0 0 1 357.35 359.8 Tm 96 Tz /OPExtFont3 11 Tf ( is given, the perfect ensemble ) Tj 1 0 0 1 120.25 336.5 Tm 95 Tz (of current states are those states at ) Tj 1 0 0 1 309.85 336.75 Tm 106 Tz /OPExtFont8 13.5 Tf (t ) Tj 1 0 0 1 319.899 336.75 Tm 99 Tz /OPExtFont3 11 Tf (= 0 that are consistent with the long ) Tj 1 0 0 1 120.25 313.7 Tm 92 Tz (term dynamics and their trajectories backwards in time are consistent with the ) Tj 1 0 0 1 120 290.7 Tm 91 Tz (sequence of the observations. That is, the posterior distribution is then given by ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 225.099 248.45 Tm 150 Tz /OPExtFont8 9.5 Tf (p ) Tj 1 0 0 1 232.3 248.45 Tm 95 Tz /OPExtFont3 12 Tf (\(x) Tj 1 0 0 1 242.65 248.45 Tm 49 Tz (o ) Tj 1 0 0 1 246 248.45 Tm 114 Tz /OPExtFont3 11 Tf ( S\) oc ) Tj 1 0 0 1 279.6 248.7 Tm 356 Tz (\t) Tj 1 0 0 1 316.3 248.7 Tm 143 Tz /OPExtFont8 9.5 Tf (p ) Tj 1 0 0 1 322.8 248.7 Tm 80 Tz /OPExtFont3 12 Tf (\(s) Tj 1 0 0 1 331.449 248.7 Tm 59 Tz (i ) Tj 1 0 0 1 333.6 248.7 Tm 93 Tz ( I x) Tj 1 0 0 1 351.6 248.7 Tm 52 Tz (i ) Tj 1 0 0 1 353.5 248.7 Tm 60 Tz ( \) \(I\) \(x0 \) , ) Tj 1 0 0 1 389.05 248.899 Tm 2000 Tz (\t) Tj 1 0 0 1 486.699 248.7 Tm 101 Tz /OPExtFont5 12 Tf (\(3.33\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12 Tf 101 Tz 3 Tr 1 0 0 1 120 205 Tm 90 Tz /OPExtFont3 11 Tf (Figure 3.4 shows examples of a segment observations are considered in the Ikeda ) Tj 1 0 0 1 120 182.2 Tm 97 Tz (Map case. As more observations are considered, the set of perfect ensemble ) Tj 1 0 0 1 120 159.149 Tm 94 Tz (becomes more concentrated to the true state of the system and stay the same ) Tj 1 0 0 1 120 136.35 Tm 93 Tz (attracting manifold as the true sate. When n approaches to infinity, the perfect ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 311.5 52.1 Tm 77 Tz (45 ) Tj ET EMC endstream endobj 262 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 263 0 obj <> stream 0 ,,b7(A3k,2Qw{gvoLT>[iQGD@*rЋrnH:K4WA5Ci%)`?_d9U.{EcRjǔۅ̂7 ko5_! 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W՞oikDUZ"%C*}qQ^nϞtL<ԛV)>Xh n/{\1dMiho} 䫪dB^472x$R~{0lqN,Fؔ3 C^^9:p_CX@>44ܕ%ы848?)xi3^]^:z @dvɣ#pP F)Ylj*49EQ"Uh<ɴE2Z@c(Kzv϶92xVЧ}?3(/ q85C 0oe.:3ZgNF!blrs@ZNs_5$(pֲ[P% I!0G0# S7}}@qtۥ)x1'Aݏk1r=c?$uف)MiHc4OM"gW zhd{[d@|3'wS?:E>'4ٱ+㲫aɷ*ƻ6t7ΩЊD۱2HذQD"l[Y%VP뉞PXf0=Y"h\BΰHҵ~bfI%|Xs߅_ fZ?r1 Avմ,BA'4X.CuV[5rѬ?YJm2Gփxȃ6mZSyR2rfx{xA8z-1Tl@it{KCQA;j*) ΝI~^'\pra,r"JMI/jص3ѵMry]8[!(?J+lsg1~e@Ƶʏni f<slϧvq HKEDWP^gk1 !OlAhx[ ΌA0ac] ޜ_ϓc ~M1̽,{ph]lYi;-eP"δPs.G7ҔН^c 1f+ R$s#.32 >Ba߽|R?rSzMhfbmszrcdN@]&:#Loε0pÝڶ {7J ͡Vx,msX"`ɓzR%9^QGmM?jd,t6+Voңn(:&&r㥱62|l=JA̵Idmc඘UQz8ɸ' hR7>vUmHߒ({OG! Ak?Vwf%9kOĨs+) oɑ;[.\+"]AKN,XP:9%v<FGܫ;vL=χ? 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(b4?C v]͂gրxa&$sCPYaFy a-_%K3{62Ēho͒?ꉴA᯸(BJ^.u 1yҋ{?Ϫ|T{)| T\ D>!<\ᔐo)h-w7Y&ՉQVǩ*X0}0Pʲag;QT[l6x>UD6W!L{t]{50 G:SZ8|R H{fnq L{9o{zES Ћ{}uΣ+z`5 eb0UC$+ 2#@@J.6l3Me,)(ZmsJ=-,gsswPm^K=6PBPV=2V7!qeWacXu[2|}"z:⫗`tIxeSx(;F/+i ^*Ka,/~ˁ Jծl^9cd1"Ң <Ţ}qt񵧲rԼ/MrBɝ^eI0Y09!gU`y3Qy}ia'^3拪Q"53[1rx8g+~nbΣx#C`_[𞄃,IÂҴBN`5$xv1tWG^BHȳvپl:^ J`gg`@I3X@ {6u<ԈF:_澉rcݥ0^xA( ~HC<ܖfLտpp5HNMy2z>HwA a?37~*?vAQ mZ3vuE4׭*l@G?$0>n ØB*|.I[ÂL%<:/"x fB;.ĹQ-N'}8+rx!LkD\p1THw$sA]e1 G1вo2mM̨%q<:btzɦ9 1rs"L\LI DѿTZpZ*Jr/50m엎YXbv!?+<'~>H) 咻0$1HFp.dӴzUBSAHEjUw R٬mh4>y,Jr@&D,ر3.\ rӳ%kzYWʋb;0QJmPAoxdkjF: Et*d퀹,R~s",.n8:3 i&/! /1rhf Õ?[]S0߾8#T4PQ( &Ѧ6᧊ÂOx L!㇜nD+֕TzQ\W, \3gS[{ dݖe!EAcsK]XSplOy#[0NĎo.e0Ƴ(9G\%hu Y磛8Qy1%DbD*UF J{J#C@Di2 3ߦd}G>n࡫KtM#.+b֝? qe;,L\-`P:.2܅I%d(g,sl.?~νx {1b8 %>1JÌ5?+"j"[@Qd%t8'1pI俍cz,W~~WCn[oAn~xnj%p+ \#1c i=9Wumf S'K.dZbꢤէE8;vr=/;&ͣX_F)m0ƾh9K%#HtC; {@ckѓj%H=n:W@P&x9hժ>;#1]Tն)h}…CEcM,\P*,(z~ْ_ 9LN)%#x47NC>қo5d\8I$V]\[x5h̆=8[cҽicȡUss/|)jBJ*VGoO4Rտ` &EW,ZyYzpweog~@xNbe"RB~/ΧJcx<,ccX\zgT*pi jѭq/(.]-}i,:O1vbH5GK -@'"IVv==NbO>2+9vj>V ~6#&J]2;#X} endstream endobj 264 0 obj <> endobj 265 0 obj [266 0 R] endobj 266 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 835 0 0 cm /ImagePart_2073 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 192.5 495.399 Tm 86 Tz 3 Tr /OPExtFont17 5.5 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont17 5.5 Tf 86 Tz 3 Tr 1 0 0 1 227.3 494.899 Tm 85 Tz /OPExtFont11 6 Tf (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 85 Tz 3 Tr 1 0 0 1 302.899 495.399 Tm 80 Tz (1.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 80 Tz 3 Tr 1 0 0 1 154.8 605.1 Tm 81 Tz (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 81 Tz 3 Tr 1 0 0 1 146.15 590.2 Tm 95 Tz (-0.2 ) Tj 1 0 0 1 145.9 575.1 Tm 97 Tz (-0.4 ) Tj 1 0 0 1 145.9 560.2 Tm 94 Tz (-0.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 94 Tz 3 Tr 1 0 0 1 145.9 545.1 Tm 95 Tz (-0.8 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 95 Tz 3 Tr 1 0 0 1 145.9 515.549 Tm 94 Tz (-1.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 94 Tz 3 Tr 1 0 0 1 360 660.299 Tm 99 Tz (-) Tj 1 0 0 1 364.1 660.299 Tm 84 Tz (1.12 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 84 Tz 3 Tr 1 0 0 1 360.25 627.899 Tm 93 Tz (-1.14 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 93 Tz 3 Tr 1 0 0 1 360 595.5 Tm 95 Tz (-1.16 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 95 Tz 3 Tr 1 0 0 1 360.25 563.1 Tm 99 Tz (-) Tj 1 0 0 1 364.3 563.1 Tm 84 Tz (1.18 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 84 Tz 3 Tr 1 0 0 1 363.85 530.7 Tm 93 Tz (-1.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 93 Tz 3 Tr 1 0 0 1 399.85 712.1 Tm 123 Tz /OPExtFont2 11 Tf (3.6 Perfect Ensemble ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11 Tf 123 Tz 3 Tr 1 0 0 1 128.65 457.699 Tm 111 Tz (Figure 3.3: Example of perfect ensemble for the Ikeda Map when only one ob-) Tj 1 0 0 1 128.65 444.3 Tm 110 Tz (servation is considered. The observational noise is uniformly bounded. In panel ) Tj 1 0 0 1 128.65 430.6 Tm 112 Tz (a, the black dots indicate samples from the Ikeda Map attractor, the blue circle ) Tj 1 0 0 1 128.65 416.899 Tm (denotes the bounded noise region where the single observation is the centre of ) Tj 1 0 0 1 128.65 403.5 Tm 109 Tz (the circle. Panel b is the zoom-in plot of the bounded) Tj 1 0 0 1 385.199 403.5 Tm 11 Tz /OPExtFont3 11 Tf (-) Tj 1 0 0 1 387.6 403.5 Tm 107 Tz /OPExtFont2 11 Tf (noise region. The red cross ) Tj 1 0 0 1 128.65 389.8 Tm 112 Tz (denotes the true state of the system ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11 Tf 112 Tz 3 Tr 1 0 0 1 128.4 357.399 Tm 106 Tz (ensemble becomes actually "perfect" , that is the ensemble members are consistent ) Tj 1 0 0 1 128.15 334.85 Tm 112 Tz (with infinite past observations which is the best ensemble one can obtain from ) Tj 1 0 0 1 128.15 312.049 Tm 110 Tz (the past observations. One might conjecture that this perfect ensemble is the set ) Tj 1 0 0 1 128.4 289.7 Tm 113 Tz (of indistinguishable states of the true state. In order to avoid confusion, in our ) Tj 1 0 0 1 128.4 267.149 Tm 110 Tz (thesis we call the perfect ensemble that based on finite number of observations, ) Tj 1 0 0 1 129.099 244.6 Tm 92 Tz /OPExtFont4 10.5 Tf (dynamically consistent ensemble. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 10.5 Tf 92 Tz 3 Tr 1 0 0 1 145.199 222.049 Tm 104 Tz (\(I\)\(x\) ) Tj 1 0 0 1 172.099 222.299 Tm 111 Tz /OPExtFont2 11 Tf (can be known to be very complicated fractal without being known ex-) Tj 1 0 0 1 128.15 199.7 Tm 115 Tz (plicitly. In practice, to form the dynamically consistent ensemble, we simply ) Tj 1 0 0 1 128.15 176.899 Tm 110 Tz (integrate the system of interest and collect the states that are consistent with the ) Tj 1 0 0 1 127.9 154.35 Tm 109 Tz (observations considered \(80\). For bounded noise model, consistent means within ) Tj 1 0 0 1 128.15 131.549 Tm 113 Tz (the bounded region about the observations. For unbounded noise, one can, for ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11 Tf 113 Tz 3 Tr 1 0 0 1 316.8 61.5 Tm 93 Tz (46 ) Tj ET EMC endstream endobj 267 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 268 0 obj <> stream 0 ,,2jJ7->($7K+2^ m^7Ԃ-`@5YFVMw4njXNn_8]9Ё-+7zF^wYahϓ-efߞGegVx85kPkL$0JB , x-ja[ʔ-vPX<6;v!0WR19;ʲ& Nڢb&?ڹgVڍ<[1mlh(_%^Og3v:U,@6`>ҹpaKU+pa_rEuTCO;fq9wOlGYE'XQ嚬;'˵rm23 : ((pUvK5EGA'޼b v௖yO㓮PvA5ӺrĢ. 5h_4zI*1V:Izf)} ȅg7x ?T8h@1y (3r[֕\ +Zl`45Mzhݼ'ڨ~H9 # `OAʢ+e?G¥WOSD?Ȕ[w (8QI>8-pĈx:佀m"ܱRg+< 726Da?^"AJw+bwf!A/?Z(R0`l^iZ e @+ 6e5 _ | Bpmyid=]j#~  yCه}0y]mAѧ=U(|TYCgUN= H&"LTϜP" 0LxW_hcȣ%8P7q1:jlhBc!Q/8lb7)]=o3lIXD Y5CՁ(T/"0kу:HQw"qJ^-Cx,itCV].8R N1xcUAɽYKiY?ڣLL;U*D7nk{jX尶x\uX}C.o]FJǺP gt QM5sr=L`f职7d7T $.&H1j~">; DK(<O 6?srDy˜=i&m[? 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X9:nP=)҄cƱ2{ݲNHrwK8:@cẙ(Ȑ{ 'z;6 O־@lϼL4|p%ϏC61w$&wRxCQR@pf[*jI*!c ?˴!IUY‡پf J-ӳ}r= x'7GY k*T"{zWx#71)eLBo;\5XSk]{_8o wVzU.Tvȍ~}*v=&|)Y5c9?5LC!/ӈ* x ai.y- NS)rz|zYClw5=o3Tv2GTҍE[wH;yN{b!{nbksg׋m]lߧR}\+n׭;c+(4= ^D#9) { q9 ?gyavrsb/G.6(HVJANH4S]F0} K#WrAA g🣅κ De nVIrD^'O!7'B!–5`Kд?`,;ڕZʱʅ"<4CEǾ- kmOHMzT7aib:]6F~ yQP"ȼz:'a$HvzΡjBƻVnI՛g>oH~ 5%>@dPk"p,A~ު ڼ[ X3 EjN^WүJq=qu Dپ9N<֫@z[hnIMR[2ypQ3YV+gjaA(<!Yhi }:;lN n%sS&b a1a 5 zqp3QjM14([oV gТC|ۮKn?-uxFCC*T*r`KSԡ3skbg;z<xj[~t l:Rhq>Ǚp >5.^zkVexmIxÛr:"HM˾杙X{q)ehIw(^Ű=w^tǛ 7GL,|hm nvvݒ,HWN5F=i-s6"£~Pk)? pImޏ9(KzA)YiбD4Ɩ_({JX̎P)b#ίY@]5$Izɽ5w2RSe[erq[Nv67tu3,KmU7;ڢKl2#[#b"V.cX[v_(ȏ7,t5AQ̖"=3@XQs;@\ɨXC~}:s!.yvX^ J`!"PYsV1=IfN%C'<'@\M Q6\mA,G[$6&KA endstream endobj 269 0 obj <> endobj 270 0 obj [271 0 R] endobj 271 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 835 0 0 cm /ImagePart_2074 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 140.9 644.2 Tm 114 Tz 3 Tr /OPExtFont9 6 Tf (-1.12 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 114 Tz 3 Tr 1 0 0 1 141.099 611.549 Tm 97 Tz (-) Tj 1 0 0 1 145.199 611.549 Tm 98 Tz (1.14 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 98 Tz 3 Tr 1 0 0 1 141.099 578.899 Tm 112 Tz (-1.16 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 112 Tz 3 Tr 1 0 0 1 141.099 546.5 Tm 114 Tz (-1.18 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 114 Tz 3 Tr 1 0 0 1 144.699 513.399 Tm 97 Tz (-) Tj 1 0 0 1 148.8 513.399 Tm 94 Tz (1.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 94 Tz 3 Tr 1 0 0 1 205.699 479.1 Tm 100 Tz (1.02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 3 Tr 1 0 0 1 239.3 478.85 Tm 98 Tz (1.04 ) Tj 1 0 0 1 240.25 471.399 Tm 71 Tz (X ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 71 Tz 3 Tr 1 0 0 1 272.649 478.6 Tm 94 Tz (1,06 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 94 Tz 3 Tr 1 0 0 1 305.5 478.85 Tm 98 Tz (1.08 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 98 Tz 3 Tr 1 0 0 1 141.099 441.649 Tm 99 Tz /OPExtFont9 6.5 Tf (-) Tj 1 0 0 1 145.199 441.399 Tm 90 Tz (1.12 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 90 Tz 3 Tr 1 0 0 1 141.349 409 Tm 97 Tz /OPExtFont9 6 Tf (-) Tj 1 0 0 1 145.449 409 Tm 98 Tz (1.14 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 98 Tz 3 Tr 1 0 0 1 141.099 344.199 Tm 97 Tz (-) Tj 1 0 0 1 145.199 344.199 Tm 98 Tz (1.18 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 98 Tz 3 Tr 1 0 0 1 144.949 311.799 Tm 97 Tz (-) Tj 1 0 0 1 149.05 311.799 Tm 94 Tz (1.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 94 Tz 3 Tr 1 0 0 1 424.55 276.75 Tm 96 Tz (1.02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 96 Tz 3 Tr 1 0 0 1 457.699 277.25 Tm (1.04 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 96 Tz 3 Tr 1 0 0 1 490.3 277 Tm 98 Tz (1.06 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 98 Tz 3 Tr 1 0 0 1 522.95 277 Tm 96 Tz (1.08 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 96 Tz 3 Tr 1 0 0 1 358.8 643.7 Tm 97 Tz (-) Tj 1 0 0 1 362.649 643.7 Tm 98 Tz (1.12 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 98 Tz 3 Tr 1 0 0 1 359.05 611.299 Tm 114 Tz (-1.14 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 114 Tz 3 Tr 1 0 0 1 359.05 547 Tm 97 Tz (-) Tj 1 0 0 1 363.1 547 Tm 98 Tz (1.18 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 98 Tz 3 Tr 1 0 0 1 362.649 514.6 Tm 116 Tz (-1.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 116 Tz 3 Tr 1 0 0 1 359.3 441.899 Tm 112 Tz (-1.12 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 112 Tz 3 Tr 1 0 0 1 359.5 409.5 Tm 97 Tz (-) Tj 1 0 0 1 363.6 409.5 Tm 98 Tz (1.14 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 98 Tz 3 Tr 1 0 0 1 359.5 344.699 Tm 97 Tz (-) Tj 1 0 0 1 363.35 344.699 Tm 96 Tz (1.18 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 96 Tz 3 Tr 1 0 0 1 363.1 312.5 Tm 114 Tz (-1.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 114 Tz 3 Tr 1 0 0 1 398.399 696.049 Tm 111 Tz /OPExtFont5 12.5 Tf (3.6 Perfect Ensemble ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 111 Tz 3 Tr 1 0 0 1 127.7 238.85 Tm 98 Tz (Figure 3.4: Following Figure 3.3, examples of perfect ensemble are shown for the ) Tj 1 0 0 1 127.45 225.149 Tm 101 Tz (Ikeda Map when more than one observation is considered. The perfect ensem-) Tj 1 0 0 1 127.7 211.5 Tm 100 Tz (ble of different number observations are considered are plotted separately. Two ) Tj 1 0 0 1 127.7 197.799 Tm 104 Tz (observations are considered in panel \(a\), 4 in panel \(b\), 6 in panel \(c\) and 8 in ) Tj 1 0 0 1 127.45 184.1 Tm 102 Tz (panel \(d\). 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In practice we use ) Tj 1 0 0 1 120 650.2 Tm 106 Tz (`within four standard deviations' \(39\) i.e. the states, treated to be consistent with ) Tj 1 0 0 1 119.299 627.149 Tm 107 Tz (observations, never farther than 4a from the observations. Although the dynam-) Tj 1 0 0 1 119.049 604.35 Tm 105 Tz (ically consistent ensemble produce a desirable ensemble state of the current state ) Tj 1 0 0 1 119.049 581.299 Tm 108 Tz (where the ensemble members are consistent with both model dynamics and the ) Tj 1 0 0 1 119.299 558.75 Tm 107 Tz (observations, it is extremely costly to construct such ensemble, when the model ) Tj 1 0 0 1 119.049 535.7 Tm 106 Tz (states are in the high dimensional state space. Even in low dimensional systems, ) Tj 1 0 0 1 118.799 512.45 Tm 105 Tz (it is prohibitively costly when a relative long observation window is considered. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 105 Tz 3 Tr 1 0 0 1 119.299 460.35 Tm 117 Tz /OPExtFont3 15.5 Tf (3.7 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15.5 Tf 117 Tz 3 Tr 1 0 0 1 118.799 426.05 Tm 102 Tz /OPExtFont2 11.5 Tf (In this section we first compare the ISGD method with 4DVAR method by looking ) Tj 1 0 0 1 119.049 403 Tm 109 Tz (at the model trajectory each produces. We then compare the ISIS method with ) Tj 1 0 0 1 118.549 379.949 Tm 106 Tz (Ensemble Kalman Filter by comparing ensemble members in the state space and ) Tj 1 0 0 1 118.549 356.699 Tm 105 Tz (evaluating them using the new e-ball method defined in Section 3.7.2. Finally we ) Tj 1 0 0 1 118.549 333.899 Tm (compare our met hod with the perfect ensemble. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 105 Tz 3 Tr 1 0 0 1 118.799 289.5 Tm 125 Tz /OPExtFont2 13.5 Tf (3.7.1 IS GD vs 4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 13.5 Tf 125 Tz 3 Tr 1 0 0 1 118.549 258.75 Tm 108 Tz /OPExtFont2 11.5 Tf (Since the 4DVAR method produces a model trajectory, we can use such model ) Tj 1 0 0 1 118.099 235.7 Tm 106 Tz (trajectory as a reference trajectory to form the ensemble in the same way as ISIS ) Tj 1 0 0 1 117.849 212.7 Tm 107 Tz (method. Here instead of comparing the ensemble nowcasting results, we simply ) Tj 1 0 0 1 117.849 189.649 Tm 106 Tz (compare the trajectory produced by 4DVAR with the reference trajectory gener-) Tj 1 0 0 1 118.099 166.6 Tm 111 Tz (ated by ISGD. We apply both methods to Ikeda Map \(Experiment A\) and the ) Tj 1 0 0 1 118.549 143.549 Tm 106 Tz (18 dimensional Lorenz96 Model I \(Experiment B\). For each case three different ) Tj 1 0 0 1 117.599 120.5 Tm 111 Tz (length assimilation windows are tested. For Ikeda Map, the assimilation win- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 111 Tz 3 Tr 1 0 0 1 309.35 51.899 Tm 91 Tz (48 ) Tj ET EMC endstream endobj 277 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 278 0 obj <> stream 0 ,,~b6z,[w9hxG QGEմ&R9,!c9G7tfESF1 Yʦ ~_W#ps߶fi7ՔE!,AX(ƕG=(dz&wBX GJ棧Yn-JE <Fϔ7[uX^YV|qPq?}K|,I~f8P)=<ߠnY ~uQ*v$ou[|!-{%ri,+3==[=7;!)4ĥ1/`\ũ.j4Jڃ>REuMFHeXdXޮ1YY/ۘtK bTz6WxP*ltad4tbppgɡj!Ψ0Im6@$o.VԍTlqh'bHxq>Ӛ0Ǘ䲵۹6<|2˖BFP76.1yGbCW7D/F?ܠ@6Ţ>E17 )ryt-TM֨xY!Lmoܻq2SL}6fuQH팖9$~"QKemWV_܄`eOpL d{7mƍsAͷ3 N a94. dEæ x ss Ƞ `=ވdQ?PJ%Ef* L'.(m ~:~vX1 7W\}eSZ? c ShJ n{3[o`Ѫ./ G<uͣn_HVzW➠pBh9[?U71딢oKojO05(Ob RQy"(o*VMaNs `T8_8~[Jok7H#,#I9J34!V׌ u&n e4[:gHBfHoVj XJ *I9ݧc藑}2Ź~-5E\P^5πŹ}|pu|yj%#z(_-*WY6R7`J=&%م g9 `돴=KMo9GoZ෻@j[r!`"}+Afq>?S"EY+=rǷy^hҭHE_ 9Ĝ Psg:J `SKrcL!KÅi{~ͺ;}D#&ew!<ȮbDﬥO`RU2eܤ5eb5&TCd/[qD^.LU؏v v(Ϫ{@(6ѱr(u,$.V ޒSRAi[j$k rV<|qbJ8/٤t*E@6}(mJ _4NtTr'+j;'diekh YN&۴v;B0azr[hMw][bu)i $m %)d fk{')/VaO!}%4jXM6[$˽!g$s `6콴.uz&}JguGױMij=clhCEChQ03⨇ogS"ATA!=2tZ'y$R2~L;U6$,y)l4,gHa T+9nTGMCѼ¬ 1-KPENAW*AC YOܢh2>V tF(iXbJպ<sS.W],oƻn&~>PU5W,Ϳ;t8_d#/ۢ[kJ}+sx,-Wx5G[xfrm Q&߄cߊfK .HEA?;BȎkIWM^.FRn`yQwa >3p5`ʻ3,~VydR!JB]qU4R=/"xcy⹞L*v[ IUQ~աc':DS0˘%,b;PEKlxٱbjM*Ģ&- Rc [:(ƫK[z뤼*K6C+Ls.~^Ɲ3-` zD rڣisr*¦QHbOW0W(S>!l)L kKu[Y>FwfQx$ /6Rӱ9\=>]Ij@NqeN@Rfg>w4IY1\o IɹCU1? tѰT5XNOBMl$tDB2B/>QY8d+.0kG|V^t4bU$ Eeu@v Q &.?#D[Ęa2#ڛx1F~iR={a9(혗b] J,ҫzպ "9P~ov,P 5+Mw ifn@)oE'OpgQب3"l%K6j>~t"PbMA/w<{Ns| C )A9%#T~5_bǙt3p̓,ek$X'ω f &F eP`)_j[>Ae"$7WhNQj(isxG/qyLI5xTaܶT#@/>DxrfeyBr  )F{U$uT=W{RERo5h)U>w0sJHڟE[5#nj샊T1lWʬ=Bݢ-߸Ou28(;ҷmIċSƝ | o,?[Y}c|$r7<۠ӴJ)M4Q*/`=Xζr?i'/ @^j)Kz%^:\uTM1 &ʫN kB(o1}5_CѮ__`Br 7 z*=+q HdaL婜ƫ6,nUJ+SpMBtn0Ry̿z.;-x~&nfQ ثk x$:K_EVnM+>ly*ZYdvg1uR#u ~_ +H =wͲ >S܅ =x[Wvd[%7Z}}ќ9e"_z D14;!)EVYaNZ^|ZuM텰Al2&e1:#Uֆ?` ,S]za^$6p%S~Re]b"Βσn>$ՓVd}/NyBך.v9't;.VPB4fhaص,/m#v]I )SHF3 Rdon[O\YC hm>"SeeҐs+]P l2d)0. Rbyq\~C&|:}(h"ېF v$8iy]6v>9krGX8JYC{xW/rm3XUAN4"28shߒM#cPF=|fN+O׵ϨLuR&?5]IZubѻ1& O4عWIbW I,RTLԲ0/ULZ\ĺZEpо#^Nu ;ALm`xP// YzRz1zZ0.ٗT ZHR}Vk>gCHDbnmRy9$s/\kFFMķ 9ofBA4\tCvZYHm$<`2*|Go4̊x%q*W= c4MN&߶;a,Nɇfl%a_tZs!Ec1עw< w>Kusd]U&^rmTLTDb}za%Q5qlD꾉~/UmIȝ [@p XxU!h;8*)E(3w-3Ycل#zJ}C舺 eUז7l1A[d|} a T,! l$c%В,CkME,D ;:XMB`LXMvaY'Tl-ti4\wj qW:ub',trfGEz-xV,I&yRd_VC .]&{@רnhjQ_K%AT#@E+±(zHe*D}/Ђ#vKA'?rmݮ v 6z++pZH1B- w1ҖJ-yqŖH.!E̾BAO$/k_hVK4€ݷip Pp'2TزΦR5[b'Y}l!Bz =כ J"TEwc֘Jz>-Vb5ʜ Opv=ڶqDHi60+g21l!40}]o\llwÍq c_]4nD,T@-aH%W8D8ùVy_R4sCs>T!͞g5ٲ ubgpK"K+#3ߗT+Dq\9kʡL>W})lj㬚Y 5mW o/s0HhѴaaΙ.Vi)P]C qV 9.?uْHٴw_NcblHp]p薰h=Xc8;B_#*/Fޕ*90@I7ι0g2#0:5grH6CJ-!WeIB U() )Px8<4h׭KD B ܔ913S cmMn*klᤕ^D&S-D;kŁHnGDEOS)E*sxXXDgd0#v:+t]s9OpV1k0(iFb֥v9ڄNsV0ߝDA+ V=5i=B=@f}Nń[BHPhZON7rK^m  8@v{ Ƙ g]&1s9-|T)`lMN"gTsC8!!_74V&҈͇ dнznVAzGd/17qk7uxȉYS.ǝ$dZ0XcoC5Do/J OYMAZ=+󓫧-jvb%&s{"f j ɛǺ̪Z0V ^cqcSfGaj4]}F#jj0G$A<-Ed>ٔ#tm%5^OW% ,u<=fFj@+<#'KOr~ʵ4)D U;+a?o_sIݘ4%aw0I[j*-pӀą'ǮDyq Rպ27Z\2g_|!^(Kk,KwPs\VΉ)x Ὄ5Q R?+|!V԰a"w@L5izLCė^r &x"Kn!:CAE:ő#;<]#.SZNje1f#lwYP z\bP8 \r\Le%샍lߩ[A7W35( TkoVZoԶĪIXYtaJ𸳜3O4QJRR: mn_m9Ē3~Z`,^^CRv_O2Mww7*oZ!?7 rJq|jCh"n52rm΀|!:73b}SP~![XNd KbՐpS:}dCg'4I *1oDŽn^yѱ<5w5E(8?ش<y/pȴiv=%+"B"} rshXGWjh z({qCf`m']ͱekfg'ϰa(3XG 0W2?vYUBEcR_ [XR3dTAkzτx _ endstream endobj 279 0 obj <> endobj 280 0 obj [281 0 R] endobj 281 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 835 0 0 cm /ImagePart_2076 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 450.5 716.899 Tm 3 Tr /OPExtFont3 11 Tf (3.7 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 3 Tr 1 0 0 1 118.099 673.95 Tm 93 Tz (dows are 4 steps, 6 steps and 8 steps. For Lorenz96, the assimilation windows ) Tj 1 0 0 1 118.099 650.899 Tm 92 Tz (are 8 hours \(short window\), 16 hours \(median window\) and 32 hours \(long win-) Tj 1 0 0 1 117.849 627.899 Tm 91 Tz (dow\). An hour indicates 0.01 Lorenz96 time unit \(see Section 2.4\). Details of the ) Tj 1 0 0 1 117.849 605.1 Tm 92 Tz (experiments are listed in Appendix B Table B.1 ) Tj 1 0 0 1 357.1 605.1 Tm 87 Tz /OPExtFont6 12 Tf (& ) Tj 1 0 0 1 369.6 605.1 Tm 84 Tz /OPExtFont3 11 Tf (B.2. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 84 Tz 3 Tr 1 0 0 1 134.65 582.299 Tm 90 Tz (We use the second term of the 4DVAR cost function, i.e. the distance between ) Tj 1 0 0 1 117.849 559.25 Tm (observations and model trajectory \(equation 3.34\), and the distance between true ) Tj 1 0 0 1 117.599 536.45 Tm 96 Tz (states and model trajectory \(equation 3.35\) as diagnostic tools to look at the ) Tj 1 0 0 1 117.849 513.149 Tm 91 Tz (quality the model trajectories generated by each method. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 254.65 445.5 Tm 139 Tz /OPExtFont2 9 Tf (\(h\(xti\) sti\)Tril\(h\(xti\) sti\), ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 9 Tf 139 Tz 3 Tr 1 0 0 1 484.1 448.35 Tm 87 Tz /OPExtFont3 11 Tf (\(3.34\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 289.699 374.199 Tm 63 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 63 Tz 3 Tr 1 0 0 1 484.1 373.699 Tm 87 Tz (\(3.35\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 134.15 331.949 Tm 95 Tz (From Table 3.1 and 3.2, we can see that when the assimilation window is ) Tj 1 0 0 1 116.9 308.899 Tm 92 Tz (short for both Ikeda and Lorenz96 experiments, both 4DVAR and ISGD tend to ) Tj 1 0 0 1 117.099 285.899 Tm 93 Tz (generate model trajectories that are closer to the true states than to the obser-) Tj 1 0 0 1 116.9 262.6 Tm 89 Tz (vations ) Tj 1 0 0 1 156.699 262.6 Tm 38 Tz (1) Tj 1 0 0 1 161.5 262.85 Tm 94 Tz (. This is expected as both methods can be treated as noise reduction ) Tj 1 0 0 1 117.099 239.799 Tm 93 Tz (method. For ISGD method, the larger window length is considered, the better ) Tj 1 0 0 1 117.099 216.75 Tm 94 Tz (model trajectories are produced. We expect the ensemble formed based on the ) Tj 1 0 0 1 117.099 193.7 Tm 92 Tz (reference trajectory to produce better ensemble forecast when the reference tra-) Tj 1 0 0 1 116.15 170.7 Tm 93 Tz (jectory is closer to the true states of the system. For 4DVAR method, when the ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 132.949 150.049 Tm 91 Tz /OPExtFont3 9.5 Tf ('Although the trajectories is slightly father away from the observations, they are still con-) Tj 1 0 0 1 116.9 138.299 Tm 89 Tz (sistent with the observational noise. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 9.5 Tf 89 Tz 3 Tr 1 0 0 1 132.5 126.75 Tm 83 Tz /OPExtFont3 6.5 Tf (2) Tj 1 0 0 1 137.05 126.75 Tm 90 Tz /OPExtFont3 9.5 Tf (closer to the true states of the system ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 9.5 Tf 90 Tz 3 Tr 1 0 0 1 308.399 52.85 Tm 79 Tz /OPExtFont3 11 Tf (49 ) Tj ET EMC endstream endobj 282 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 283 0 obj <> stream 0 ,,b7!V;ΰmTraeU{.yG Rf-y/T.͹}?;2~9C=0q¤0Qu3PSЫgpAQ >#bۉimt[)KcߋBKd} 0dV4,sh.n=`ꤍ'X9p}hn[B4v~%1h=\?9gѩ  ^8AMmT"MuNMX`xWhr]|Ζ_h|g#+}v0Rb B,lN 8pLVej`v Mt .}5A ]cm$ Wu9;'Vhfm˧f]/Z"\"Z(J*N:|r_[#qEKvAM\*5RLAur-Vif^lSM>a9w#+xJM:8oMa L$_4{+H?oADM WŎ}F n]M|BZ>FF0x|X **ly6H`%rooNVKL|uQI <=L}'ŅnINz!`erO tZOr }d>7w^Ӿ \sR/`(Hʳ$<d A+@VSz >:D;ض˻1ᑱfYD%>^g=::w ^T3-eZGGX~"B(|>e :|i<У:Q03ur=W??pc 3 uY{~b_=+g:maxUV{'O h-ۀS_3٣y#ws  l|e T.!ɊX\Se dq\QGk _ 'M:gCPg tճ泾F8ѿ<[~Vmy{懬KωMj&vpG[ 4jk pKSbCƑ2BЛ|~A*]W= tFƞDwxD[)ġFw@SjSU,z  o<1b&,)܉zOsS鍟Ic'&5++I?ΧZ49Dt3CsQ50kyS!?/ش;%'"jBх GV߭ubm8`,9ȮX~%$Ӧ`ww4OUSuuM[%gQ`x+@= f!D2~?*Mqʷ;$=7M+(B=o!*Y_XDYdHY$w!8#Y9gvmnk0]_w 㠺ewA2]?!Z'NjOg!4QG&S2678 jO)}(הp|5j5gqY-֪@!a!uC1x웋~μ/,dIynho`^iǽ3zHQ tlJÜo)"L Y1$3+ܦqAΈ~ SDՓ$ebPUÁ{(qWV{Xj} ǯ"%"%0҉O.M;w.&] yh \㮅Vx 4(ʹ4f~Pk] -~:wN"%J Ǿ\ >I1*٦2%YZ"$4Ϯ<˟á00I6Jm2vD鑴 +Шu;~;X%_Q`=V3lbaOKUY?P'NjJrM=rphM(pq,D,wUe_>**Wr-ɮ8ry. 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Pg2/tx{u3HT6 ;v|n,/ т8 oAFl TKv֮J}! ߼Zb}=WmDtVePvɓ:ѻxbԕFb=ߋӤ: Nqގ5OI=XDoj^!l ԐwS#P=Ż;;hł$7$ `DK*G"ϋ}x tP~LBIyeʨgspSx53N#᫥˖xᘰB'֏d|HsLΎȆL7/:^C=!!kZn6kK%cQ'w99h-,t&bD[`߶wċ"ϥ; ~'gRF-@0:>"+F_6s:&\m)c$ϵq$u&^RG(OaY('Qӑr+PS-UPW;Ye,@e{Kɢ_JtULGfC1F`(\\l;nh]#R-7>S>%$]+]Euw˭p=vD 5 L\hXay 7aS$bU `u"#X2ѝad3bDF93lS,"HquF*.Os1Z|dQ瑉"aerl7E9|*@LwTcO=ܯB"k}R'tflQPcU_`ڑ N'H`ҧ8@ ͌!u}7."³Y5-"jG7?},*2J~E2mwqj^ʾ_|'nRbBuӬxn4UZ|wec~dt.>2ϝt |KEr[cߧZI4&x-/)2#n:,Q2ԧZzg s8p-qXƺ8I>}g`I63}e.P$2@(ͨng{"PgSbiS8zϋoi{CBaѩ?~nOCa2}#pH,ɋMGڼK|/^p|'tOTLo='.]|C۵ș x)Ul|5^ksxNknmzeuS8pR|ǥE$;[ud=d淓r$d9R\=(u[09]5d?O!JTIcJW8NmV-cBX}dTCu?AI\tdE'sYalZW z\o°Ywx@:ݿxIoE&{7{Dz#@3-{5ʛ;QEXJTkn/MK}`q ?4vcKb 6lNv(hGAu>"S O޷*\g.sWVR@d!nc\Uyvnadg[^sZi3k-vyu]R@xqj{T-*ʞkdJ+2cPE@#a ,g \/10S8U v!4La K]&qQ{<ŷ5cS P?a+ÁA.{FqKQO~t:/ʫ?N+8"a* sU9`E[#Hq+\6Io1cql-|ϙVxbrxyO\1Û\*Xr>=$_~K)*ISA(m.DQK)X)t1 9Й6.Jejr$gbR*>ZqS=)(?\/Mpt*AS*-4O Lg ?ҾfT3Y.D9[J_ݬpmJ:pNr-9OYr&Bk?i;ͽXhp ]OUq9c-h|L=S%$-{ôRzNO">㥖#6^߲ LS_ k7枞Bt GDY:9R5 $-fdh%Ka&/ ;7sg31쓇%mόwW@Mdmbư3wV'JǫhLʱQ_i䩩)E:ubn|t5خm飾vsү)OKLwItjӎVB /uGc˥뺼 [e}Cj&XGd$YQW,uZS'FʹZĎ(7TB*Xi*A9~p*1ʺb ov2)VRO(YFD3|x@v`]tt'cyR(yF$,vcFK: :?jEO&t Ї_ѳgoQWgWYGU:lQ%5%9$}@FDHӍXYYexә*c+Vm!|Frϵ\H2dMZ elo&}I-{M2 ]ʷA$2|`vOyavϱ6$ Rr5lb߂^õ\C*2 }_5Va?zW:T ďY:_1!CV"gީi:,@̆+o)N\ YNMM 9yk a_eH b PxvֹM3!uwft&VGjaWe~~(^z"Eӧ? CaPƵ*-zI9͂nI=WAu' EuqPm't,!*.? ) 0_ɡ0gI;~jSpPdM47@fj2@ >ZTSkȤ=b]THVqpKI?R!RCQ@挷 ^CT(tF BL٭Z~!U=S2d Yi([Np%MST9QU$YPB>-o`ܳ8ȳ۽9&6pѮn53П {Iők߿`M5$-@H<6;Cb.%>AZgz9s֙$Ϛ^֋S!xF98e(~Y$M"*~;[{`®dTahƜfxh΅!&wr畢lh'jq2}3~tT=ClWnP1C 3I^N5QDZVadbR]w5vO~Q\F4ߡB%FOvS(Piqkd_I$$Fɚ؏/v}& w_*w$wŷ/ussCDax؋r[$7F٬*"c)օjAϏu+NA=) ҅YfEƋ|%RH8@vAWlxʄ9{x\JŃ#oMj#GfNp/ՆXTcnȽclG%acN$x=&z֍vА\ 琀H2Кr B3Slf$T $E_tu!Y37_b]{pnS 0DN hhr .|/Ej(THPğI 1ؖ N2"8n&.WY=j,|m(@y0ZmLd'V Ʒ 0jK^P2߱KrvzƽJKr>O~V )hN%BB򵐴\{!q /_ pC?Hl8k#!c9C14 0dJO9,_T 6h0SIϷlqȨUx"Pizx auA-3H%(Em;#= bSIs5xKJܹ7\E5flDS`NHa3 UC%MdǴcW; _zC3EdcJn4i~p\=UQC\vu=rrWxzYclq!XDܚI׵߲R: B*:>Rz ? endstream endobj 284 0 obj <> endobj 285 0 obj [286 0 R] endobj 286 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 835 0 0 cm /ImagePart_2077 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 451.699 716.45 Tm 3 Tr /OPExtFont3 11 Tf (3.7 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 3 Tr 1 0 0 1 164.9 675.149 Tm 104 Tz /OPExtFont2 11.5 Tf (Window length ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 104 Tz 3 Tr 1 0 0 1 274.55 674.899 Tm 107 Tz (a\) ) Tj 1 0 0 1 292.1 674.899 Tm 108 Tz /OPExtFont3 10.5 Tf (Distance from observations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 108 Tz 3 Tr 1 0 0 1 262.8 661.25 Tm 102 Tz (Average ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 102 Tz 3 Tr 1 0 0 1 344.649 661.25 Tm 96 Tz (Lower ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 96 Tz 3 Tr 1 0 0 1 420.699 661 Tm 92 Tz (Upper ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 92 Tz 3 Tr 1 0 0 1 251.5 647.299 Tm 79 Tz (4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 79 Tz 3 Tr 1 0 0 1 293.05 647.299 Tm 80 Tz (ISGD ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 80 Tz 3 Tr 1 0 0 1 327.35 647.299 Tm 79 Tz (4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 79 Tz 3 Tr 1 0 0 1 368.899 647.299 Tm 81 Tz (ISGD ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 81 Tz 3 Tr 1 0 0 1 403.449 647.299 Tm 78 Tz (4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 78 Tz 3 Tr 1 0 0 1 444.949 647.299 Tm 81 Tz (ISGD ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 81 Tz 3 Tr 1 0 0 1 185.5 633.149 Tm 104 Tz /OPExtFont2 11.5 Tf (4 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 104 Tz 3 Tr 1 0 0 1 252.25 632.899 Tm 91 Tz (1.58 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 91 Tz 3 Tr 1 0 0 1 293.3 632.899 Tm 90 Tz (1.66 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 90 Tz 3 Tr 1 0 0 1 328.1 632.899 Tm 89 Tz (1.51 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 89 Tz 3 Tr 1 0 0 1 369.1 632.899 Tm 91 Tz (1.59 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 91 Tz 3 Tr 1 0 0 1 404.149 632.899 Tm (1.63 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 91 Tz 3 Tr 1 0 0 1 444.949 632.7 Tm (1.73 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 91 Tz 3 Tr 1 0 0 1 185.75 619 Tm 104 Tz (6 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 104 Tz 3 Tr 1 0 0 1 252 618.75 Tm 94 Tz (11.06 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 94 Tz 3 Tr 1 0 0 1 293.3 618.75 Tm 91 Tz (1.77 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 91 Tz 3 Tr 1 0 0 1 327.6 618.75 Tm 95 Tz (8.17 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 95 Tz 3 Tr 1 0 0 1 369.1 618.75 Tm 88 Tz (1.71 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 88 Tz 3 Tr 1 0 0 1 404.149 618.5 Tm 91 Tz (14.28 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 91 Tz 3 Tr 1 0 0 1 444.699 618.5 Tm 93 Tz (1.83 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 93 Tz 3 Tr 1 0 0 1 185.5 604.85 Tm 104 Tz (8 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 104 Tz 3 Tr 1 0 0 1 251.5 604.85 Tm 96 Tz (51.84 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 96 Tz 3 Tr 1 0 0 1 293.05 604.6 Tm 90 Tz (1.85 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 90 Tz 3 Tr 1 0 0 1 327.1 604.6 Tm 97 Tz (46.16 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 97 Tz 3 Tr 1 0 0 1 368.899 604.35 Tm 92 Tz (1.80 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 92 Tz 3 Tr 1 0 0 1 403.449 604.35 Tm 96 Tz (58.54 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 96 Tz 3 Tr 1 0 0 1 444.699 604.1 Tm 93 Tz (1.90 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 93 Tz 3 Tr 1 0 0 1 164.65 589 Tm 105 Tz (Window length ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 105 Tz 3 Tr 1 0 0 1 294.949 589 Tm 100 Tz (b\) ) Tj 1 0 0 1 312.25 589 Tm 109 Tz /OPExtFont3 10.5 Tf (Distance from truth ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 109 Tz 3 Tr 1 0 0 1 262.55 575.1 Tm 102 Tz (Average ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 102 Tz 3 Tr 1 0 0 1 344.649 575.299 Tm 95 Tz (Lower ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 95 Tz 3 Tr 1 0 0 1 420.5 574.85 Tm 92 Tz (Upper ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 92 Tz 3 Tr 1 0 0 1 251.3 561.399 Tm 79 Tz (4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 79 Tz 3 Tr 1 0 0 1 292.8 561.149 Tm 81 Tz (ISGD ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 81 Tz 3 Tr 1 0 0 1 327.1 561.149 Tm 79 Tz (4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 79 Tz 3 Tr 1 0 0 1 368.899 561.149 Tm 80 Tz (ISGD ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 80 Tz 3 Tr 1 0 0 1 403.449 560.899 Tm 77 Tz (4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 77 Tz 3 Tr 1 0 0 1 444.699 560.899 Tm 81 Tz (ISGD ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 81 Tz 3 Tr 1 0 0 1 185.3 546.75 Tm 105 Tz /OPExtFont2 11.5 Tf (4 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 105 Tz 3 Tr 1 0 0 1 251.3 546.75 Tm 95 Tz (0.52 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 95 Tz 3 Tr 1 0 0 1 292.55 546.75 Tm 91 Tz (0.61 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 91 Tz 3 Tr 1 0 0 1 327.1 546.75 Tm 95 Tz (0.48 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 95 Tz 3 Tr 1 0 0 1 368.399 546.5 Tm 92 Tz (0.55 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 92 Tz 3 Tr 1 0 0 1 403.449 546.5 Tm (0.55 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 92 Tz 3 Tr 1 0 0 1 444.25 546.5 Tm 95 Tz (0.67 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 95 Tz 3 Tr 1 0 0 1 185.75 532.35 Tm 104 Tz (6 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 104 Tz 3 Tr 1 0 0 1 251.5 532.6 Tm 91 Tz (9.51 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 91 Tz 3 Tr 1 0 0 1 292.3 532.6 Tm 94 Tz (0.39 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 94 Tz 3 Tr 1 0 0 1 327.35 532.6 Tm 95 Tz (6.70 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 95 Tz 3 Tr 1 0 0 1 368.649 532.35 Tm 92 Tz (0.36 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 92 Tz 3 Tr 1 0 0 1 404.149 532.35 Tm (12.59 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 92 Tz 3 Tr 1 0 0 1 444.25 532.35 Tm 94 Tz (0.42 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 94 Tz 3 Tr 1 0 0 1 185.5 518.45 Tm 104 Tz (8 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 104 Tz 3 Tr 1 0 0 1 251.3 518.45 Tm 96 Tz (50.04 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 96 Tz 3 Tr 1 0 0 1 292.3 518.45 Tm 94 Tz (0.28 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 94 Tz 3 Tr 1 0 0 1 327.1 518.2 Tm 96 Tz (43.59 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 96 Tz 3 Tr 1 0 0 1 368.649 518.2 Tm 91 Tz (0.25 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 91 Tz 3 Tr 1 0 0 1 403.449 518.2 Tm 95 Tz (55.77 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 95 Tz 3 Tr 1 0 0 1 444.25 518.2 Tm 94 Tz (0.31 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 94 Tz 3 Tr 1 0 0 1 118.799 494.699 Tm 109 Tz (Table 3.1: a\) Distance between the observations and the model trajectory gen-) Tj 1 0 0 1 118.799 481 Tm 106 Tz (erated by 4DVAR and ISGD for Ikeda experiment, b\) Distance between the true ) Tj 1 0 0 1 118.549 467.1 Tm (states and the model trajectory generated by 4DVAR and ISGD for Ikeda exper-) Tj 1 0 0 1 118.549 453.149 Tm (iment, Average: average distance, Lower and Upper are the 90 percent bootstrap ) Tj 1 0 0 1 118.799 439 Tm 104 Tz (re-sampling bounds, the noise model is N\(0, 0.05\) and the statistics are calculated ) Tj 1 0 0 1 118.549 425.1 Tm 107 Tz (based on 1024 assimilations and 512 bootstrap samples are used to calculate the ) Tj 1 0 0 1 118.549 411.149 Tm (error bars \(Details of the experiment are listed in Appendix B Table B.1\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 107 Tz 3 Tr 1 0 0 1 118.299 377.8 Tm 109 Tz (window length is relatively long, it suffers from the multiple local minima and ) Tj 1 0 0 1 118.299 354.75 Tm 108 Tz (produces the model trajectory which is both inconsistent with observations and ) Tj 1 0 0 1 118.099 331.699 Tm (far away from the truth although we expect to obtain more information of both ) Tj 1 0 0 1 118.099 308.45 Tm 105 Tz (observation and model dynamics from the longer window of observations. As we ) Tj 1 0 0 1 118.099 285.649 Tm 104 Tz (discussed in Section ) Tj 1 0 0 1 220.099 285.399 Tm 80 Tz /OPExtFont3 11 Tf (3.4, ) Tj 1 0 0 1 241.9 285.399 Tm 106 Tz /OPExtFont2 11.5 Tf (applying the 4DVAR algorithm, one faces the dilemma ) Tj 1 0 0 1 117.849 262.35 Tm 108 Tz (of either from the difficulties of locating the global minima with long assimila-) Tj 1 0 0 1 117.849 239.549 Tm 106 Tz (tion window or from losing information of model dynamics and observations by ) Tj 1 0 0 1 117.849 216.5 Tm 107 Tz (using short window. Without introducing such shortcomings, our ISGD method ) Tj 1 0 0 1 117.599 193.5 Tm 108 Tz (produces better model trajectories. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 108 Tz 3 Tr 1 0 0 1 309.35 52.1 Tm 91 Tz (50 ) Tj ET EMC endstream endobj 287 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 288 0 obj <> stream 0 ,,zzV.6[Wޥl䧉DrpCE4J 7Jzy##>}k*HsHgl?kf GC#*.B b&,~3V g`sazwB=`ԝ*IFm/4jr>X5t H0Ȑ[>A}+;6 #w*??'ez*lhHX;^6Me3CDddXnĿ4eS xDkJi.m]@7%^z=DC/yBd6*F^ܯ>2ڧ=S ssn|Jʄ0Xhz*{G*6̗B#7Q|/ff "n*Tj(mhzI4EamXkq\@ E@?h]e36JCcpRC"e&qs66U5E~#o)|fv,(+jTկ:8徺a_i8"0Q'p[9ۿCiiCV/ {VqeC^C)`˃mi1A̙o}CU3J_o:O@</54ͫa1@mJ@FD<0ٶ!b%Xœ@Ū}~۸."DL|F(d}0OAIq# 'U(-T'؉ `Ϲ';Y'@e^NGS:p o Y~ eW4ʓ&$ j)]C$@UH{h$ xA3^nL V(!,[ J{us*oCҺ9H&INZk3 e{G3 3bF9~X]\7z 9_t\>/'( lU/\ γiU%W Q=-u'l|뻾g00;L蒸c'֨% Zeg: ߫C۹6&C`-;2=Xaɧ<, 6dh=L ) {qN‰l|44G= bZ-f:HtUmU#T~ySC߭vBq Vp:rNu; ZX;}͋<pv8W vLb湩#Jic/ekb5~л߀pDTeHf9.W:3ݰ''tbӛ ##pzuZC}JQ W`u 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q"Rn;u .{rTZU 8g?S+q$˅s !-&J< .-Vʷ31H|[y, d- WxhK4W88C:5w|y'Z0V)h^8=ؗbX8]zM(LF'J{\Z_兗L᫨X#CE(pTu!s^)L/' ǑMq[g=l9U[dG]sҠUߑThDf@iwoo7?m/Ԝ,NH̖2BP+7xD5sEJtvJNG2a=ӷ{$,PG ~GyS钷)=#k { ߄ ǭ9/LxM6mY 4Dw%zx.b>ʵ8mQN۠D^AxXf iLDrQNiU ?J05SIJoF1|5-l1VĊ\A1w]*h?Q;;q<]y(8FsU˛Tk[JCbgiط(hL!0 :P{AIN}8۫0]^,OT+:;Mp EKD ]K`d< _OI`sߟHٵ=R_ qIAKkU6 <ꩦJ=>?1H0[ă&JagUZ5 ϦXxd|=a$'>|\bq$і(,\lh,k9%tyR-vrU h>0k6Xr|FD"y9jKb,{$K5@5ƌh,:Ucm|a/ݴD5`)b7k`$! @jV(|WYdToD\Umx^1lH6?Bb[GLYRNLBBGgåW ɎG:sw WUYD-f !"l.THR.URt\;>*mA #"Sa2sف FpֲcX̜5V񬻦>B:^5\»{˨p'nK`Sį91H緁܁w xu3NkDd$c{+u<4`CYuQMb1xx6:z ćd'YT{$=gBL]D^XN*>E -Tz6Ňw֎#Zj:c|xNI(5[^4K#i`v@jأweOڥ?\3^]6xN4Mcl]-94]+<.ԻEwaĀg'b~Tӆ].$У7$d՛<_wҜ/6ѕ"%Kslt޽V\wX t챛tJkɻv@[^OiE '63od!G-# HQ R*bR,:#l*]xúb8SZ/ns6B_I06\f|Yڽ1VqC4 BG3LS"A##N1jQqiuꕻ E7q+ ?߬Fa@,_VwMqe%@|>T|2n0zשSv(nR1QeٱS•7hWC@_w,G"_o7caNA,̓pFڄ&c[N{M}|'Loax6^w.mu"s?8O @e;E(% w\Z~EH]ƚ\RQRU8vM_,pOd~chk"T^sa# Nıkt3.w{3m^44H[ꋾXh 次X (Yv9$e.Μ )v|G7][kss݃ ;`|uX׊}ͨ簚2xϏPunm0u]C0*3Ma{ٲ7]ƴ$Fvltg5GAbl,Vu0.h]2+G㬽R> endobj 290 0 obj [291 0 R] 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/OPExtFont5 13 Tf 86 Tz 3 Tr 1 0 0 1 369.1 633.149 Tm 88 Tz (16.4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 404.149 633.149 Tm 85 Tz (16.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 85 Tz 3 Tr 1 0 0 1 445.199 633.149 Tm 87 Tz (16.8 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 87 Tz 3 Tr 1 0 0 1 182.15 619 Tm 92 Tz (16 hours ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 252.25 619 Tm 88 Tz (16.8 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 293.3 619 Tm 87 Tz (17.0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 87 Tz 3 Tr 1 0 0 1 328.55 619 Tm 85 Tz (16.7 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 85 Tz 3 Tr 1 0 0 1 369.1 619 Tm 87 Tz (16.9 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 87 Tz 3 Tr 1 0 0 1 404.149 619 Tm 88 Tz (16.9 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 444.949 619.25 Tm (17.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 181.699 604.85 Tm 93 Tz (32 hours ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 251.75 604.85 Tm 89 Tz (28.3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 293.5 604.85 Tm 85 Tz (17.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 85 Tz 3 Tr 1 0 0 1 327.85 604.85 Tm 88 Tz (27.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 369.1 604.85 Tm 86 Tz (17.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 86 Tz 3 Tr 1 0 0 1 403.699 604.85 Tm 91 Tz (28.9 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 444.949 604.85 Tm 88 Tz (17.3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 165.099 589 Tm 96 Tz (Window length ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 96 Tz 3 Tr 1 0 0 1 294.949 589.25 Tm 85 Tz /OPExtFont13 11.5 Tf (b\) Distance from truth ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 85 Tz 3 Tr 1 0 0 1 263.3 575.299 Tm 83 Tz (Average ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 83 Tz 3 Tr 1 0 0 1 345.1 575.299 Tm (Lower ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 83 Tz 3 Tr 1 0 0 1 420.699 575.299 Tm 84 Tz (Upper ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 84 Tz 3 Tr 1 0 0 1 251.75 561.399 Tm 85 Tz (4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 85 Tz 3 Tr 1 0 0 1 293.5 561.399 Tm 80 Tz (ISGD ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 80 Tz 3 Tr 1 0 0 1 327.85 561.399 Tm 83 Tz (4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 83 Tz 3 Tr 1 0 0 1 369.1 561.649 Tm 80 Tz (ISGD ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 80 Tz 3 Tr 1 0 0 1 403.699 561.649 Tm 84 Tz (4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 84 Tz 3 Tr 1 0 0 1 445.199 561.649 Tm 80 Tz (ISGD ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 80 Tz 3 Tr 1 0 0 1 184.8 547 Tm 92 Tz /OPExtFont5 13 Tf (8 hours ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 251.75 547.25 Tm 88 Tz (2.73 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 293.05 547 Tm 87 Tz (0.93 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 87 Tz 3 Tr 1 0 0 1 327.85 547 Tm 88 Tz (2.68 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 368.899 547 Tm (0.89 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 403.699 547 Tm 89 Tz (2.78 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 444.5 547.25 Tm (0.96 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 182.15 532.6 Tm 92 Tz (16 hours ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 252.5 532.85 Tm 86 Tz (1.35 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 86 Tz 3 Tr 1 0 0 1 293.05 532.85 Tm (0.41 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 86 Tz 3 Tr 1 0 0 1 328.3 532.85 Tm 87 Tz (1.33 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 87 Tz 3 Tr 1 0 0 1 368.649 532.85 Tm 91 Tz (0.40 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 404.149 532.85 Tm 88 Tz (1.37 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 444.5 532.85 Tm 91 Tz (0.42 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 181.699 518.45 Tm 93 Tz (32 hours ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 252.25 518.45 Tm 89 Tz (11.76 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 293.3 518.45 Tm 86 Tz (0.19 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 86 Tz 3 Tr 1 0 0 1 328.55 518.45 Tm 88 Tz (11.17 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 368.899 518.45 Tm 89 Tz (0.18 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 404.399 518.7 Tm (12.46 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 444.699 518.7 Tm 91 Tz (0.20 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 119.299 494.899 Tm 100 Tz (Table 3.2: a\) Distance between the observations and the model trajectory gen-) Tj 1 0 0 1 119.5 481 Tm 97 Tz (erated by 4DVAR and ISGD for Lorenz96 experiment, b\) Distance between the ) Tj 1 0 0 1 119.299 467.3 Tm 94 Tz (true states and the model trajectory generated by 4DVAR and ISGD for Lorenz96 ) Tj 1 0 0 1 119.5 453.399 Tm 103 Tz (experiment, Average: average distance, Lower and Upper are the 90 percent ) Tj 1 0 0 1 119.5 439.5 Tm 102 Tz (bootstrap re-sampling bounds, the noise model is N\(0, 0.4\) and the statistics ) Tj 1 0 0 1 119.5 425.55 Tm 100 Tz (are calculated based on 1024 assimilations and 512 bootstrap samples are used ) Tj 1 0 0 1 119.299 411.399 Tm 101 Tz (to calculate the error bars \(Details of the experiment are listed in Appendix B ) Tj 1 0 0 1 119.299 397.5 Tm 97 Tz (Table B.2\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 119.5 366.5 Tm 124 Tz /OPExtFont5 15 Tf (3.7.2 ISIS vs EnKF ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 15 Tf 124 Tz 3 Tr 1 0 0 1 119.5 336.3 Tm 96 Tz /OPExtFont5 13 Tf (In this section we first explore the low dimensional case in order to provide easily ) Tj 1 0 0 1 119.049 313 Tm 98 Tz (visualised evidence. Then we evaluate the nowcasts using c-ball method defined ) Tj 1 0 0 1 119.299 289.7 Tm 91 Tz (on following. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 135.849 259 Tm 101 Tz ( Compare the results in the state space ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 101 Tz 3 Tr 1 0 0 1 146.9 232.1 Tm 98 Tz (We applied both ISIS and EnKF in the 2 dimensional Ikeda Map \(Experi-) Tj 1 0 0 1 146.9 209.1 Tm 97 Tz (ment C\) and plot the ensemble results in the state space \(The details of the ) Tj 1 0 0 1 147.099 186.049 Tm 99 Tz (experiments are given in Appendix B Table B.3\). Four nowcast examples ) Tj 1 0 0 1 147.349 163 Tm 96 Tz (are plotted in Figure 3.5. In all panels of Figure 3.5, the ensemble, produced ) Tj 1 0 0 1 146.9 139.7 Tm 98 Tz (by ISIS method, not only stays closer to the true state but also reflects the ) Tj 1 0 0 1 146.9 116.899 Tm 100 Tz (structure of the model's attractor as the ensemble members lies along the ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 3 Tr 1 0 0 1 311.05 52.35 Tm 80 Tz (51 ) Tj ET EMC endstream endobj 292 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 293 0 obj <> stream 0 ,,'єb7!_$I#Ds0j>8vhp|*~#͌;ͩ&V㮇T# h}[\ʙjPUԧ%Yn!'Ɨ@t֐Z=p2@c uBsV;t bH'g O#X@ǨDavulwlhrgG‡cMU>72.}F]XdE[M=1#@!>"C15ddTm]ߍδwV/>I1'cPqwW,ՙ_ݘa}z4%sj:ľH#!`r PI7 S7 4;'2y-y2x|Mvtm[^A`kN$L6Jۉ`')kq?R uڄuH-l#Am K۵?Kh ΉKYpfVB _bC!f|~p7I¾F}3Y=Eŵ ŀU;I˴LV7M#gt; ( Y#2X G^K|n e׶rJd99:qrd#!#4ag?20,0ClF'B;OBnb/6Q><#*eC El fNv5Cmx\J<`oR8U/ؙ,֛DK`S3olq=l|#%A-ж G:.˓% %7qxȠ$X-XJ& 8)JE5$%zVҭ2ȍj \euP„QŊg65 3`k1{(ncX|* UZ <98&&ςܳ٢Ϸ*%6v%zcώpA46˼:o @|y h'~>Xֈ%>T-4(K4K%uvq?v Ozu2hD1$czoq4DF&%UߩF=M/^7ً&>8 0h9񄳨1|H*rZ;BR ^xM/oYk2qjAˇ.I_JKTDS<~48yxt1T>%TDJf}zϮDY"ƂAq8*( Nr)S `^h260{:tF_QQZtx]l;cp*Ut:,97?<7]: y 3`crUOlƃ~}KщsbQ2ȠCe%#,{k}8ff{ڡu?é%/бW,B*-DnӉ~u8Z/= X,݉e&9'Ft ojJƕ@& I9q]7ipo=֡!.մ~ qcmW40[\_;WhO +nX8e*M@*ῃfb*Բ~>KQ;vk]OzF.hsbMH4;?kSW)R ~4P# "cYc kJAEJAޞ%SctmdnU'v$+,Lg[wh7w$FTkֶЊâdu#fήᩘ\9SuX7|&y>!opCz)E7s?iwy+ ;UtHL@ӿ}"ljXk> ؘg?C]' _F`D]y~t5%Y;%8u㸦n/!Jee3>w`K'"p?κ>Pe#g1Xl&ù֧_hwO&lhLQ6Vs ^/ʶ҈ NlxƊ-^&!r)Bv#չļ yIeƫ{Z XlxiUp _I}݁kqx1S_QSjP(\ƥMAP;[珞Cg_6TLI,*vM`@)X}]va4 /\]pI#r;z"Aj #-8 =f˞Lǁd7>fJ ~6 p,"C!eMt0kmm \T8 JM*-eoww>X,ݖ[^DRz؊X "Je 9A&a ҏ'_@̢/3-$!lk16s1B. )%MR)]ܢ2x:7Aǐ11v 2ɱ`D7ZV**M}wcw%Ҷ] tI#(MM I!mԾ91P05? 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The EnKF ensemble, however, has its own structure as ) Tj 1 0 0 1 148.3 651.7 Tm 91 Tz (the ensemble members do not lie along the model attractor. In the top two ) Tj 1 0 0 1 148.099 628.899 Tm 89 Tz (panels of Figure 3.5, the EnKF ensemble manage to cover the true state and ) Tj 1 0 0 1 148.099 605.85 Tm 92 Tz (tends to stay close to the model's attractor. While in bottom two panels of ) Tj 1 0 0 1 148.3 582.549 Tm 91 Tz (Figure 3.5, the ensemble members are systematically off the attractor and ) Tj 1 0 0 1 148.099 559.75 Tm (tend to stay close to the observations and not covering the true state. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 137.3 527.1 Tm 92 Tz ( Evaluate both methods via 6-ball ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 148.099 499.5 Tm 95 Tz (Here we introduce a simple new probabilistic evaluation method, which ) Tj 1 0 0 1 148.099 476.5 Tm 94 Tz (evaluate the ensemble forecasts without transforming it into probability ) Tj 1 0 0 1 148.099 453.449 Tm 88 Tz (distribution. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 148.099 425.6 Tm 93 Tz (Given the verification corresponding to the forecast at time t, in this case ) Tj 1 0 0 1 147.849 402.8 Tm 95 Tz (the verification is the true state at ) Tj 1 0 0 1 327.6 402.8 Tm 111 Tz /OPExtFont6 12.5 Tf (t = ) Tj 1 0 0 1 350.149 402.8 Tm 94 Tz /OPExtFont3 11 Tf (0. One can draw a hyper-sphere ) Tj 1 0 0 1 147.599 379.75 Tm 91 Tz (with radius E \(hereafter ) Tj 1 0 0 1 270 379.75 Tm 83 Tz /OPExtFont6 12.5 Tf (6-ball\) ) Tj 1 0 0 1 304.1 379.75 Tm 93 Tz /OPExtFont3 11 Tf (around the verification. For any methods, ) Tj 1 0 0 1 147.599 356.699 Tm (one can record the probability mass that is inside different size of 6-ball. ) Tj 1 0 0 1 147.849 333.699 Tm 92 Tz (One can compare the result between two methods by simply counting the ) Tj 1 0 0 1 147.599 310.399 Tm 94 Tz (proportion of times one method beats the other. If the methods tie, both ) Tj 1 0 0 1 147.349 287.35 Tm 90 Tz (methods win. When the size of the 6-ball is very small, we expect neither of ) Tj 1 0 0 1 147.349 264.299 Tm 91 Tz (the methods to be able to have ensemble members inside the 6-ball. When ) Tj 1 0 0 1 147.349 241.5 Tm 94 Tz (the size of the 6-ball is big enough, we expect all the ensemble members ) Tj 1 0 0 1 147.099 218.25 Tm 92 Tz (will fall inside the c-ball. In both cases, both methods wins. When the size ) Tj 1 0 0 1 147.349 195.45 Tm 95 Tz (of the 6-ball is neither too large nor too small, we can investigate which ) Tj 1 0 0 1 147.099 172.149 Tm 89 Tz (method produces ensemble forecasts assigns more probability mass around ) Tj 1 0 0 1 147.349 149.1 Tm 97 Tz (the verification. The advantage of the 6-ball method is that it is simple ) Tj 1 0 0 1 147.349 126.1 Tm 93 Tz (and easy to implement. Note the weakness of this method is that it is not ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 311.05 52.649 Tm 75 Tz (52 ) Tj ET EMC endstream endobj 297 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 298 0 obj <> stream 0 ,,ggj?ǰL,X̴{XGv mBUw׀gAe6Jw*qB^l܀k ar:%Pb2*lbx100uvh٭ g(X-JiÉKʡ;I4Do1v{#UŻO&//Հ/"X0۪,eķXxk1i [-1Ϗ<9DfbwSF6Va T6ՇoBl0]e ~4qB"pO0AK{6/%RKB1ˁ;#ZRo4i#f}ѥy{-@=MD lT<lX?:ڗiS1z(<+,O^ưj}ƕPǩ/ eC _Cڬl! r E#X#@0ٿ(6*ג"MAKkр⁴BPxBVb'Ԇ|5Ԓ9& OSɇh<̼6a*hjզ urE] @IK} 4&ϧ*'9~Fcj>8c A1>H\M)N"jzo}% 4[Pr 6>ݰ̚1!IcJ\+썘鰿f)Lc渿qzO_WڗɳșIZ"жٱ촥߯ȲڳұB/ݳϟMu5ݳٝb9c$OtY/8+>ͽ<g`x9&fjQRJ(!@GpUx7aPRH/ iC!@ؐ6AYDRT.plmj_'Yj{鶾`'gHmy]=mօՇkn֨oL#9!JYR u,1иʽ\&Hڷ QQ s;L9+T?a}UꛨZ;dQ*7pXomsBą z\MD«WFV\A&42O>z2~WOMi/ w-W!u Mp5*?!crDzZy |b6 +w(G²/f^t8(\c&๩ W[-X"O4n”0Y6˅Ll@H0iĀJdV/Ax>9>˰Xub-ڏQa~~Ej7,Od!l7wfL؉y/IDgf]FU͓= "zWp)::\LdRWSfp'ґ4]5f{>E>cZkVbY*p05-Ym38'&"8c l=Oƈ42f32yzWµ;j(1U|~zϖ{8|9 2n~B),!OnMU[ ")A-9/#PJ0` ;s΁-mFmw+={4kٱM:d-ƽN؊E]SLHt4QXMz$ P) st>tݐXEg74$i? >7$xifϏsJ_~˲_2T%5Zo44.` rW`zDR,l M|} (I>0Lj.N. _gΦ 5'9$:qJx;mr1/VP&ns+18ev8(nLz3,;{Ylà@S)-=F:o90gυ_S7jy/X0$m(L|[g(zL$Oe u}Jz$AZu㡰Ff2 :Kx̹6Mp-`_8Bj$zFk\p)MZvu_Iڱbi }ᐴrp!y$3IUcrD,Re{{Y ,NnOlo` ʛn *ޞm'uXͪfU2MEoLq`2"n'Ve,RFIEo{jv8<ӎ$r6IUc]ROE>nꌠ^6ׁ\*Mo]uya8Fa BwcdoOLHsr42Xq`1RYk 㔉;sND VT??51 2~+*^ȳNǎ H@;R2u@OJc Xhk3+=Cާ#JxBHpejEBn*LP-O}zw {pi+VvddUyMC{4=bygϤϼ(d}cK+W6]㵆Lp%o<5+gRjmOvh?v RnwX /UP=` _4{xƱ쇿 ^IC&rԗHTw(.=m'? 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o^yV|Ҁp{wh3KOy~倂/Ƥigzf{{="hw%t( lV& .(3R;[얜RT\F`hΞC; ׾ )ltl* D{w3{ՙZ0ݼ߂ nkvM W9["իǁQ g!-I|;ߘa [C9jgnY"_uŠzF{_TԧD]Hx=8 h9wŮ+JMsS6$*h1aBP MRg 0U0H՝h)lB•}R$3˯E<`!υY35Bf}І\sd)hv~݆;b8l8uBz B#+C40L*"/O)$ջѿKʄゴ3v۱.uq&2qG&6Դ}bޱ(!#糦0bղ)Pf'V0]mZXhփS*.Ql$iCjvL]aj}ҚG+J#^ Xy9A)!4XJ*)NW$ry X2 糥lY -n[Ssb#mBXt vaےMM IXI˷爫6!WyU͌g ̹.HnyWgR,f9=;~Q^LX?4bÂ]cO3a-kYU fXy.^%׌cv}2>e'υko# 8. ҕ{sOn𤶂&ɬ 7,[:i"C8:K CF3tB+QMOtvp@"tC>cKbVd]@?T{nZO?o @GReH1 lw +6y~]K }sօ!OFo4qgm1 hSL /jH Bڸ0BOSLy! zҞ_3ˍOJu1F*p(|~ OQ0ioK g~4ukC|]0ݴZƏΧ ux5I~8QՖSxrZ#t7m ۨ_XI;  ډ֦[CL7603kȡʽ?jΰlIvx%v}I$.nGߗgcj\n{7Ee6إêF*:?BG=9^r;ʳ!@X[^uS`}C)K%{ 1yQ5G;nvG:d"\%CM8@iLF$vtܯ]&!K@l Es|?-T $M_۔J$_ل6U_b"FԠ `!HDЬK!:Z endstream endobj 299 0 obj <> endobj 300 0 obj [301 0 R] endobj 301 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 500 0 0 796 0 0 cm /ImagePart_2080 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 250.8 640.95 Tm 82 Tz 3 Tr /OPExtFont18 5.5 Tf (3.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 5.5 Tf 82 Tz 3 Tr ET EMC /Span 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Tj 1 0 0 1 341.05 505.35 Tm 37 Tz /OPExtFont9 4.5 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 37 Tz 3 Tr 1 0 0 1 345.6 335.199 Tm 108 Tz /OPExtFont9 5 Tf (-0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 108 Tz 3 Tr 1 0 0 1 363.85 504.899 Tm 95 Tz (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 95 Tz 3 Tr 1 0 0 1 376.55 335.199 Tm 74 Tz /OPExtFont19 5.5 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 5.5 Tf 74 Tz 3 Tr 1 0 0 1 387.1 504.899 Tm 66 Tz (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 5.5 Tf 66 Tz 3 Tr 1 0 0 1 400.1 335.199 Tm 63 Tz (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 5.5 Tf 63 Tz 3 Tr 1 0 0 1 412.8 504.649 Tm 78 Tz (0.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 5.5 Tf 78 Tz 3 Tr 1 0 0 1 426 335.199 Tm 73 Tz (0.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 5.5 Tf 73 Tz 3 Tr 1 0 0 1 438.699 504.649 Tm (0.3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 5.5 Tf 73 Tz 3 Tr 1 0 0 1 451.699 335.199 Tm (0.3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 5.5 Tf 73 Tz 3 Tr 1 0 0 1 476.649 650.799 Tm 87 Tz (x10) Tj 1 0 0 1 486.949 650.549 Tm 73 Tz /OPExtFont1 5.5 Tf (-3 ) Tj 1 0 0 1 490.55 650.799 Tm 53 Tz /OPExtFont19 5.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 5.5 Tf 53 Tz 3 Tr 1 0 0 1 478.1 633.75 Tm 75 Tz (3.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 5.5 Tf 75 Tz 3 Tr 1 0 0 1 478.1 615.75 Tm 79 Tz (3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 5.5 Tf 79 Tz 3 Tr 1 0 0 1 478.3 597.75 Tm 73 Tz (2.5 ) Tj 1 0 0 1 478.3 580.25 Tm 74 Tz (2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 5.5 Tf 74 Tz 3 Tr 1 0 0 1 478.8 562.25 Tm 68 Tz (1.5 ) Tj 1 0 0 1 478.8 544.7 Tm 37 Tz (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 5.5 Tf 37 Tz 3 Tr 1 0 0 1 478.1 526.7 Tm 75 Tz (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 5.5 Tf 75 Tz 3 Tr 1 0 0 1 476.899 481.6 Tm 96 Tz /OPExtFont12 4.5 Tf (x ) Tj 1 0 0 1 481.699 481.35 Tm 151 Tz /OPExtFont9 4.5 Tf (10' ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 151 Tz 3 Tr 1 0 0 1 84.7 335.699 Tm 86 Tz /OPExtFont9 5 Tf (1.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 86 Tz 3 Tr 1 0 0 1 110.9 335.449 Tm 100 Tz (1.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 3 Tr 1 0 0 1 137.05 335.449 Tm 96 Tz (1.3 ) Tj 1 0 0 1 143.75 335.699 Tm 1386 Tz (\t) Tj 1 0 0 1 162.949 335.449 Tm 93 Tz (1.4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 93 Tz 3 Tr 1 0 0 1 188.65 335.449 Tm 100 Tz (1.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 3 Tr 1 0 0 1 214.55 335.199 Tm 70 Tz /OPExtFont19 5.5 Tf (1.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 5.5 Tf 70 Tz 3 Tr 1 0 0 1 99.099 505.1 Tm 103 Tz /OPExtFont9 5 Tf (0.9 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 103 Tz 3 Tr 1 0 0 1 69.849 417.5 Tm 100 Tz (0.4 ) Tj 1 0 0 1 69.849 400.25 Tm 96 Tz (0.3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 96 Tz 3 Tr 1 0 0 1 69.849 382.949 Tm 100 Tz (0.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 3 Tr 1 0 0 1 69.849 365.899 Tm 86 Tz (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 86 Tz 3 Tr 1 0 0 1 74.15 348.649 Tm 95 Tz (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 95 Tz 3 Tr 1 0 0 1 372.25 691.35 Tm 93 Tz /OPExtFont0 11 Tf (3.7 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont0 11 Tf 93 Tz 3 Tr 1 0 0 1 383.05 328.949 Tm 57 Tz /OPExtFont13 7 Tf (x ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 7 Tf 57 Tz 3 Tr 1 0 0 1 46.1 299.899 Tm 92 Tz /OPExtFont3 11 Tf (Figure 3.5: Ensemble results from both EnKF and ISIS for the Ikeda Map \(Ex-) Tj 1 0 0 1 45.6 286.25 Tm 89 Tz (periment C\). The true state of the system is centred in the picture located by the ) Tj 1 0 0 1 45.6 272.799 Tm 88 Tz (cross; the square is the corresponding observation; the background dots indicate ) Tj 1 0 0 1 45.6 259.1 Tm 92 Tz (samples from the Ikeda Map attractor. The EnKF ensemble is depicted by 512 ) Tj 1 0 0 1 45.6 245.7 Tm 90 Tz (purple dots. Since the EnKF ensemble members are equally weighted, the same ) Tj 1 0 0 1 45.6 232 Tm (colour is given. The ISIS ensemble is depicted by 512 coloured dots. The colour-) Tj 1 0 0 1 45.6 218.1 Tm 92 Tz (ing indicates their relative likelihood weights. Each panel is an example of one ) Tj 1 0 0 1 45.6 204.649 Tm 83 Tz (nowcast. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 83 Tz 3 Tr 1 0 0 1 73.2 174.399 Tm 88 Tz (proper \(12\). We will discuss the weakness of the ) Tj 1 0 0 1 304.1 174.399 Tm 91 Tz /OPExtFont2 11.5 Tf (E-ball ) Tj 1 0 0 1 333.1 174.399 Tm 87 Tz /OPExtFont3 11 Tf (method and compare ) Tj 1 0 0 1 73.2 151.85 Tm 90 Tz (it with the proper Ignorance Score in Section ) Tj 1 0 0 1 295.199 152.1 Tm 75 Tz /OPExtFont0 11 Tf (6.1.3. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont0 11 Tf 75 Tz 3 Tr 1 0 0 1 72.95 124.5 Tm 92 Tz /OPExtFont3 11 Tf (We compare our ISIS method with the EnKF method in both low dimen- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 234.5 41.45 Tm 75 Tz (53 ) Tj ET EMC endstream endobj 302 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 303 0 obj <> stream 0' ,,##kcާt|[I#h-~X8[)ǑfMCSgG ni@ɚj˃!ZgoO᥆^خj{""SK0EL"/X/>_θ>&RkKpp]urFG>ypbpa!&hkh䗸cC**V)_(u؟KYò p8?+8]YDb_uZhzA V?L 5C- ǒ9:),c%TMH)GBuM#$1 Gkb<@0FZP(3K`ƘF*2F|Mnzn)h.EOϨ xR[~ )J&)ڴf{xx14;a0~ 1**GnWP>*j{r̀$qHMKZA26,! ӂy\ hlBF|>[f=6ɴn4&ыwkBnV~sW'%Q";QN^`_x4Leەԃ2*4'o#qt,O=^lh Ogtʝ;d#d:ת0 {Btf+>wF*,bΐ1܏1kQc-qL6XX6@ =~x6-`GtrC);EԍnIc \߼_X?w 2>ʞMsο G\!;H6$˝QBt\FuP.rGzv+ІRXRBQ /1 pɹbxRЈX?θ LUHG\aGP`QxN,k | ;KB#:;7ytҕcYj%xZr|kec@jxtv ^@L=Pa`qDŽQBm>BH ,}|Ͳ%86( 7h"0&\+ܔJe2B*8gkBDf{\4$=7AC |FĀ22^J:[!D4' 쀡tL~vc߄6_vQ(*oN}UH! y۾{}$7;68uo4|FI% o7^{_oׅ?3ufc=dLO7]1VJ٪kaH1u&ҕL y,-}V(Tx\>b(CQt0Jݖ~_Fa34 D'X= (_cբVw II Ilˣwy_FH/VOcrB11¾OۇCڰZ2cPJ$E&O5p9b~=xZ؝)[%nw3R!!b J}3ۉ˛ϓLG4(ͧNR{2E;w))~>wܸF/h[5ZvJᜠ/U qp$HW"Z0H1'umP.y6ޥt6D`߫ IrG]ѣUqnQUBd|%5RTT\#]14 9y-zNL&^܈~"%|4Kt 0zFDHRT-NYUSAWLw؂2=W(|{)*ichF9x)ͬzM= H%, 3^2t&Ԑ#4uq=[%JMp^-;cF40 0yjl:ƧeABT:A}m?+;qS0l&d|7q1MI;,&q<(҅ @Kj5q3vGBg1Xy熧žKi^d{ک@'?9Uuxw>ޱ8.ub ڭRKUC΍FqnKԀIr/HB-g].k Ɔv{Ę1w~j4!7MHCa(٤;hT>YC+%U+1̵k̒=,n9uȰ1D*ЍvcP44H$`U.? !jܫ( 8֚/8-Dx@tf ,4$WdUȋ>¬A񮛺*TЎ=۳$.ײRӏХ˲:⸿+n<]KUT-%1\ 26xrX T@VU/!Ȉ^> ,|ZtmGi<3tmsWrY87@j%]eZ(Qϙ4QbY,sN<m0fIyVMYIا߸Mn+`f9 {}UԢ]p(x8S~yi]`2%B;(:SwH¯&5'B6$ހ}e[ù vQ*&(A?<9` $rKI~Dj[gzA3(M+d*]/'=))<[PJWJ-yGDžZq!o('[#uF~諵8`+dpI\SQMӪ g ,{ Y=bgofpd !/RV\ʿ펄-@Ix=VSd'q ~ Y=՗e/C.4_b~^Ejd^B33 #A r}!e9X7ky|h>|b>,88-&,^ז; sVofdݹi`W 3*CalFi/+|*8v W/h1.gsAD:ǏI _ .xK3~.S3)0 e2V3n.ze:W6vI.az]brfrGH.â9S-t60 E).\A?Ȼu PۀQJO+(~bw_ҫm eO UʰC1 R`y*,&ecڷ(KSjk3X;~ɻnB"djU+aJ+aE,BLA]ݙ;ϰ~\3/Kd)|ʠh"պξYR[;[SI-_pl6jw0m Dֹ= ~KL޼d~[z51 N*zXr5?܈xeuНuvzG0D+]$+lثjwd+68_Ԗ5YAg_* q46Ν!#oą)ss,(.3N";e?D"D2U",ċ# 47EQNOFe(({:m/Ew0'o=ץP1i^Œbuo(N8} M|493n4?P867ڟU~l=5' Rdd%+[&9Ho{d{bT^u[G뙲K" d{+$9!c^$搦P+y~ O^d D辐 go #[@ 'Om w+0r,]>1D;}sֱ:v!Eڄ 7sKm`b*3 E &w7.v:C|çY [F[RkND^=^3!;#_?cԀ釪b՞ݰQ9b /+f=t2=m"X`HkwtJa QL"rg%lM3>$,C:9܃6';;\"ۇ=oL='մɟ)1[8>Mñۤ+<ه= 2dDmҌȓBv+π3+ME^k"b m 3@_c7 d9f gXqGǹg DS6%2'"ٻ5<ƅgN}EnƷ h#^BAԝXT4}%tWNW2BJ1x8Z|4݂D_ɲXβE_dϋzRYS+B E<GZr>9;&D\JjݺK0ﳧ^"Ҵ:ڰ6ɬg9"C﷡k,#詹UR[nܬU%pnu[Ʉ~a̳$>݌88qd$<]BQeO h[ͻ){0 >AWM'A}%K g ~ȣHaz}ݾ\f,no`&`]Eٵ')ω9[6T5Eׅm:1yH.mxGQ=RW.}'d(Y'&,LҸx9'⬁o&ٹ]s,5B)>V'z4m6-2C-FKYnAo #SSs4e|5CcRf>~Vk*OV3U +O!/'q!Q>f4b]het@;AYYzm]p7_^ҕ%qj ђn]!_VH˰Ƅdэ02ì±FZm+`ugAƉ?o6Sdj-?o3{`5mL\(<_ThcYmq`Ms=KxEg w)0ye 40AT 4ƴAd3["f&iBJUS|A^94mq`TkB9ޛ*HOi :.eCPixyUNdKC2(k02 AGAWotb;8p@e7%7 I6JN[Uo"AP@:FYu}WF[PFGp}xg"4dR[ ^D#${ڟ3ddT'z0[oy<S1)>ewp=u{Hs#Ր*-9< <#|wNU}:KrCسCS,mp>IuޏOH!gHI48(@y(nUG,.P-xSz/Xjҙ/B-3`uZMK Js\幠8;? >/Y'pYT(ѱY;*8H3 Nig>K,am{2 ٭sLx\e*e٠,7YԫC mSbO~o GˁI2O{iQZeKȺ6"Jx i]řľ$JFozN`Zͨ-$!p%]^0~-;hr{E(^y_g: &/|qF^9o8vGsh_L$]S7VG Hdv u쳸f*l"k%x՛Ђ)71J8$#[Jokoc,Lr+wB)Z6 owQ t;";%$\ `ߑ,= [Vyqؗ_3ԯMq~]4.}IK^꩛kj3SL2ћP6Qy75j fq^O& %POJkVP(ʍ^.qpw:.x{W"X?U,J8JB" $+M{ݗ\!/v@ l6*Z&|EVd#n9*ֻ-!3C匭&!iY;$o,`ilD(d1.Ϩ((J#sOKy?kO_odZ-_XOeo7OpZ)vHb8e"Ǩ*.^R7zKXGo0SXzҫGňO ?23mwK3XWD0wE=$w CyL2.&@np.-s P9zێ ՗%*"ni$0b"Ǟ[5nh>l}'PAs~ m5EPyD>QMm ^vXwzyx 'n~hqk9vO+.m`c0s^jG~>nL$z`?¦ ! 6ӔS:%A!9' f=↋UӟL\lNUXgT6cp;;iX#M.4GVZcAsdPUrs5ez;BM܂6^xGbi=3~ Wq ٽM ܉7tSӢоBu$$nDLb3p@Հϛůwȍ:q5s[@H&M39oB0kO8+!G$i VOW-e :5)~ߍM} /iT}O,o9%2v\zM*2ňt-/ʦ41r IFDKzw8jIȎ;xhy/W0y^?/Yba(eP4ݛȊWH' >qUxaAeJ(.g6~"78gVk͂%h!vBmF:7cD  0~HuԹxQ~ Ib$ѱ/A=v><3 D+U%3%e9c1E`ҾAG4[QNd ,.[[Y̓h,vlԐW>?QG).œ7n%[wg|~JM H47zDɫ_eQ08'V0Ѽ:z_^uX{:é,/jkzY/3kDhyBQdi(}}1p6+du9$IiY0o@FQg[4!_/N(3gk4*7wlx8 _C)"]@zQRTj D6n:| BtE=*cmzkU>JӰ+±<:Qlo񰻃fPEc 5Qn\8#1@&2vRN8 s?ZA`6N!:cĞ'R V#м)ltKו,mZ;_V8$e[iWU@4[B`eb,ngM˄DMUCCȉd?Lgw%\HD5WG" & !i$z?j}f ҙMdgS"`dWxyB(U=uui?1u?Sp:,:EerJo,\VtL|AFKO}AnPs9}#j[Av^Sy,;9/c"ɶD? kxBxA"fhj6wg|^if{0u^v 5h*tj,v^%C\B7RZ?S|G¢m"g^`:LST:+y=! {}ƞcoc5u}#٬S 2Q%~Q0si e'сYG '*?x\ r9Gvv) .w8K6WTb=냱Sv*j4e|$Da qIfoaf͠n/^i endstream endobj 304 0 obj <> endobj 305 0 obj [306 0 R] endobj 306 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 835 0 0 cm /ImagePart_2081 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 452.149 715.95 Tm 3 Tr /OPExtFont3 11 Tf (3.7 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 3 Tr 1 0 0 1 147.599 673 Tm 96 Tz /OPExtFont5 13 Tf (sional Ikeda Map \(Experiment C\) and higher dimensional Lorenz 96 model ) Tj 1 0 0 1 148.099 649.95 Tm 98 Tz (I \(Experiment D\). The details of the experiments are given in Appendix B ) Tj 1 0 0 1 147.849 627.149 Tm 101 Tz (Table l3.3 & B.I. In both cases we evaluate the nowcasting performance ) Tj 1 0 0 1 147.599 604.1 Tm 100 Tz (using E-ball. Figure 3.6 shows the comparison between EnKF and ISIS. ) Tj 1 0 0 1 147.599 581.1 Tm (From the figures, it appears that the ensemble generated by ISIS outper-) Tj 1 0 0 1 147.599 558.049 Tm 94 Tz (forms the one generated by EnKF for almost all different sizes of the epsilon ) Tj 1 0 0 1 147.349 535 Tm 96 Tz (balls in both higher dimensional Lorenz96 and low dimensional Ikeda Map ) Tj 1 0 0 1 147.349 512.2 Tm 95 Tz (experiments. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 119.5 467.55 Tm 120 Tz /OPExtFont5 15 Tf (3.7.3 ISIS vs Dynamically consistent ensemble ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 15 Tf 120 Tz 3 Tr 1 0 0 1 119.299 437.1 Tm 98 Tz /OPExtFont5 13 Tf (In this section, we compare the nowcasting performance of ISIS ensemble with ) Tj 1 0 0 1 119.049 414.3 Tm 96 Tz (that of the dynamically consistent ensemble \(DCEn\). For the purpose of simplic-) Tj 1 0 0 1 119.049 391.25 Tm 94 Tz (ity and efficiency, in the following experiments only uniform bounded noise model ) Tj 1 0 0 1 118.799 367.949 Tm 96 Tz (is used to create the observations. Since finding the perfect ensemble members in ) Tj 1 0 0 1 118.549 345.149 Tm 99 Tz (the high dimensional case is extremely cost, we will only compare the results in ) Tj 1 0 0 1 118.549 322.1 Tm 97 Tz (the low dimensional Ikeda Map \(Experiment E\). Similar to the previous section, ) Tj 1 0 0 1 118.549 299.1 Tm 101 Tz (we first compare both methods by looking at the ensemble results in the state ) Tj 1 0 0 1 118.299 276.049 Tm 103 Tz (space and then we compare them by the c ball method. As we mentioned in ) Tj 1 0 0 1 118.299 253 Tm 97 Tz (section 3.6, the more observations are considered, the better DCEn member can ) Tj 1 0 0 1 118.549 229.95 Tm 101 Tz (be found. In Figure 3.7, 3.8, 3.9, 3.10, we compare the ISIS results \(with fixed ) Tj 1 0 0 1 118.299 206.899 Tm 98 Tz (window length, i.e. each window contains 12 observations\) with the results pro-) Tj 1 0 0 1 118.549 183.899 Tm 97 Tz (duced by DCEn where different number of observations are considered. Details ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 134.65 163.25 Tm 98 Tz /OPExtFont5 11 Tf ('Note we expect both methods wins when the size of the 6-ball is very small or very large. ) Tj 1 0 0 1 117.849 151.7 Tm (In the Ikeda experiment, it happens when the size of the 6-ball less than 0.001 or larger than 1 ) Tj 1 0 0 1 118.099 140.2 Tm 101 Tz (although it is not seen in panel a\) of Figure 3.6. And in the Lorenz96 experiment it happens ) Tj 1 0 0 1 118.099 128.7 Tm 99 Tz (when the size of the 6-ball larger than 6 although it is not seen in panel b\) of Figure 3.6. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 11 Tf 99 Tz 3 Tr 1 0 0 1 310.1 51.649 Tm 86 Tz /OPExtFont5 13 Tf (54 ) Tj ET EMC endstream endobj 307 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 308 0 obj <> stream 0 ,,kRr,&X'FnRy;&tKV@j ]W8Ӹ4%u b a-4m$raؘ}(5<"Z/Pm5e*zmD=ȖlJ!3u mC帇B,*v.~|g>,^Iv*Ԧ#+ܹIb^+-ʰ!3Ő"$c3βr }آe@mBF5 (\KT~N}~m xfڵ%d0G`^諯ި2aFoJzy=dqdXFtm"U"O J؟5i&uBx7Af2:kVOʹTa"2.B\4zb_W"DI)Tӏ`VB({Rquau2@jC镂}EdX C}R (f @RdĿdbB&qn$Â*4ٴ)3!{M@Jgf*~ C?Mb:ֱ4 #* }'CryJِ mE- =n95̠~?GsT7\q$v=)׾ʇ(G,f(ą寽I-fJY ԩz Q8s؄ܫCT2EWI>5$cƤ.z{V4]"EE2\>ڂjHb9R[vYtGd]R0k!QGZi07P1y{R4_d"Y\;d,Q{W,SbB2'~T#iU*Nnлy_dƱ NJwjK$D?9lC3m/Rh6B+lTI;zϨr7'2Dꀪ . /.Tem"IHmFBC5z17׮|=e.KKAhea5#r/gCgc ;eLbfIZt (!]Qu/*,q)| mWQv+ @Y`Xi㶡ˇĔG *?\~)(xL}ǟ\uCeϺKX0;P#{ϓ+xYGG҇[xh'(홯3{\wg1mlu y[;S*r:Mc}B㨸R&d_`_YۑIJlmZY]5 3mj^ 0hDiމ['K ^:ycx*]ڪ0x7ntzh9m5ڒ6vhHBJ$tKG*nV6&}Y@ExfmA $+U`ՃsJpbxϵp#4bv39V ǘjsOxA'']ob/ׁfUtEuNE[ӿB䈴LT͛w"JjR EbOdbVl=~3 h4S %;TN¦79+Uj,RJo([ߛ&̤N*\g( qT'uAmb>+״F_3`'Vm)%3~B/iG7ҷWݗG\%E@7|eXC^Kd-*z l%"T&¿(ac9hudWKpCUcq$>/*sҚ ė~5o[* Bmf{ ' dĿ˝{ n*ڷHLUUُڊr¨4V:ξ#cS0t#r80vGO/AfƁ3D۠wm\ `?;IҴ.e ?Xdx17c@>d5aYE'erنu눓On/on|$6G ͬ-93$rB̧e@Hk U04z:+4_=69IP|DʑG?r-,6#/hD&hUqHƙ>F6ffG_+C_ʵ#{:Fwyĸ˪H>COWy;}:뚴bHyu`AAH;\SQijgs:[r12/'ff?`R4M^۸ܡg;xʐDb-7A,ͦ/Wid 1 j裏;[yR aXtɪQCFb㒢UT҉hsQZ;'nG˯^=cOѸNNs}9CyM@/)€DE+}șD@0/ph(&SCܪ>⑇M*' nd"xcvkt[{`j'IѦ)$Z6ŀ0)yڳ*㨾t8 ?Mg>x}\y ¤ bd& c@KVGArpt@OeUӘ3c?嫪o(N5],~귍 V9{5DfO؂YѼM㇈R6WDP L?&& 6iQ 0>Z+o 1fb},[VVE9.jx4 >68-#(ٷɿ%1ӫ)=r<yo𤺶 ]7 ycZ.s]J佪;} 1Pdzbjviff9)GJ d U^3Hu0BTJ?U /Sy)Iٹn<@ FC^Uma0l@ /#^H=E]dih:?FO[`4,Kcn}i&Oby5.n<`~dscK61H?rgh 7Dv_&P=4~DSx¯D@>&,"Yvp t`RtVq @T>S0Kۘ;qگ'nFJ*Qhzؘ/IS"2}OAc~hce.B}?Ktc؈SUTL.-t*m,4!G6gΨ~v2|#%D?:ꢲu6eW=sCζK 3Iek=*%QG>T9,/O^y.ܐ`Q/K ~[4KsqCN$CA)+o5\6h02>iU"y;B/m&Swòs1MjR9$q+3KLNY>Y>hvy(Se'X;hMPXQ!玷\4e{kٗ 1A?-%V]8UYhHon?۱Β 7Grܸ?zէ^FsaH8P "_۴:-2یB\u }Yg"Eg>ݵVW&tCڴd!ɟ}%G Mv +A(|sr[`ỉ*(*߮R zи, $ͧ<-JH"RfѺ,YʞͪLrK j^bkVIf:]/0C2fpiAK5FH/&]] UDmI|N`EJ9_׍fvbOgAzȰ&z畑{xhUP3If1ODCbQ*Z&ɘ% X=RR2,-;ns(?U)lє\_1V3,9Z[ 0 PR11w&kTb#a\ݏ~hm>Ի+6w|-V~ktQ@uS* t)-J߉ .K D7'MY/gWH#}% 7=\9 Cȁm ??VWBࡤ ˝nc!""C@~lO{A3+K7Ծ*)D,l@}Y/uBS(~UJb.`΀^У`5"S4 U?NQx&dFB!%mixʋ>5}]٧r^J1W7}.tk"qZvӦLhwvS4H Tr jw t#RUa*Utzo5EӍ*I J{'HuizAuhb85dJ3! 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{,ߜRhcR*ՠ0g,X rl[ɺ(C#bs)Cm#TTwRBwPgӇnX%u08kwƇk>fT_eRbw/ endstream endobj 309 0 obj <> endobj 310 0 obj [311 0 R] endobj 311 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 510 0 0 792 0 0 cm /ImagePart_2082 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 1 0 0 1 5755 7190 Tm 3 Tr ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 3 Tr 1 0 0 1 5851 7190 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 3 Tr 1 0 0 1 6335 7190 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 3 Tr 1 0 0 1 5755 8203 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 3 Tr 1 0 0 1 5851 8203 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 3 Tr 1 0 0 1 320.149 372.25 Tm 143 Tz /OPExtFont9 6.5 Tf (EnKF>=ISIS ) Tj 1 0 0 1 320.149 361.199 Tm (ISIS>=EnKF ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 143 Tz 3 Tr /OPExtFont13 10 Tf 1 0 0 1 5755 8338 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 143 Tz 3 Tr 1 0 0 1 5851 8338 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 143 Tz 3 Tr 1 0 0 1 5755 8554 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 143 Tz 3 Tr 1 0 0 1 5851 8554 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 143 Tz 3 Tr 1 0 0 1 5755 8702 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 143 Tz 3 Tr 1 0 0 1 5851 8702 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 143 Tz 3 Tr 1 0 0 1 6335 8702 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 143 Tz 3 Tr 1 0 0 1 97.2 468.5 Tm 86 Tz /OPExtFont9 8.5 Tf (o) Tj 1 0 0 1 102.5 462.949 Tm 78 Tz /OPExtFont11 8.5 Tf (o ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 8.5 Tf 78 Tz 3 Tr 1 0 0 1 330.699 462.699 Tm 125 Tz /OPExtFont9 6.5 Tf (0.4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 125 Tz 3 Tr 1 0 0 1 388.8 462.5 Tm (0 5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 125 Tz 3 Tr 1 0 0 1 91.2 410.649 Tm 126 Tz /OPExtFont9 6 Tf (0.9 ) Tj 1 0 0 1 91.2 393.1 Tm (0.8 ) Tj 1 0 0 1 91.2 375.85 Tm 123 Tz (0.7 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 123 Tz 3 Tr 1 0 0 1 79.9 368.649 Tm 89 Tz /OPExtFont11 4 Tf (CO ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 89 Tz 3 Tr 1 0 0 1 79.9 358.55 Tm 162 Tz /OPExtFont9 6 Tf ( 0.6 ) Tj 1 0 0 1 78.25 349.449 Tm 192 Tz /OPExtFont9 6.5 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 192 Tz 3 Tr 1 0 0 1 79.9 341.05 Tm 156 Tz /OPExtFont9 6 Tf (S 0.5 ) Tj 1 0 0 1 78.25 333.6 Tm 77 Tz /OPExtFont9 6.5 Tf (*-E ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 77 Tz 3 Tr 1 0 0 1 84.95 323.75 Tm 90 Tz (-) Tj 1 0 0 1 83.5 323.75 Tm 138 Tz (\) o.4 ) Tj 1 0 0 1 79.9 314.649 Tm 96 Tz (0_ ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 96 Tz 3 Tr 1 0 0 1 90.95 306.25 Tm 116 Tz (0.3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 116 Tz 3 Tr 1 0 0 1 90.95 288.949 Tm 126 Tz /OPExtFont9 6 Tf (0.2 ) Tj 1 0 0 1 90.95 271.7 Tm 109 Tz (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 109 Tz 3 Tr 1 0 0 1 90 624.7 Tm 130 Tz /OPExtFont9 6.5 Tf (0.9 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 130 Tz 3 Tr 1 0 0 1 90 607.2 Tm (0.8 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 130 Tz 3 Tr 1 0 0 1 90 589.899 Tm (0.7 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 130 Tz 3 Tr 1 0 0 1 77.299 586.299 Tm 121 Tz /OPExtFont12 4 Tf (CI, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 4 Tf 121 Tz 3 Tr 1 0 0 1 90 572.649 Tm 130 Tz /OPExtFont9 6.5 Tf (0.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 130 Tz 3 Tr 1 0 0 1 312 541.899 Tm 133 Tz (EnKF>=ISIS ) Tj 1 0 0 1 312.25 532.549 Tm (ISIS>=EnKF ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 133 Tz 3 Tr 1 0 0 1 97.7 254.149 Tm 80 Tz /OPExtFont9 8.5 Tf (o) Tj 1 0 0 1 102.25 248.899 Tm 74 Tz /OPExtFont11 8.5 Tf (o ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 8.5 Tf 74 Tz 3 Tr 1 0 0 1 147.099 248.899 Tm 119 Tz /OPExtFont9 6.5 Tf (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 119 Tz 3 Tr 1 0 0 1 295.899 248.899 Tm 106 Tz (2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 106 Tz 3 Tr 1 0 0 1 340.55 248.649 Tm 116 Tz (2.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 116 Tz 3 Tr 1 0 0 1 392.149 248.649 Tm 120 Tz (3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 120 Tz 3 Tr 1 0 0 1 77.299 564.7 Tm 172 Tz /OPExtFont13 6 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 6 Tf 172 Tz 3 Tr 1 0 0 1 77.299 555.35 Tm 196 Tz /OPExtFont9 5.5 Tf (0 ) Tj 1 0 0 1 83.299 555.35 Tm 170 Tz /OPExtFont9 6.5 Tf ( 0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 170 Tz 3 Tr 1 0 0 1 77.299 537.85 Tm 134 Tz (o_ 0.4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 134 Tz 3 Tr 1 0 0 1 77.299 533.299 Tm 236 Tz /OPExtFont11 4 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 236 Tz 3 Tr 1 0 0 1 77.299 526.299 Tm 83 Tz /OPExtFont11 4.5 Tf (CD_ ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 83 Tz 3 Tr 1 0 0 1 89.75 520.549 Tm 130 Tz /OPExtFont9 6.5 Tf (0.3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 130 Tz 3 Tr 1 0 0 1 89.75 503.3 Tm (0.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 130 Tz 3 Tr 1 0 0 1 89.75 486 Tm 111 Tz (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 111 Tz 3 Tr 1 0 0 1 375.85 690.5 Tm 98 Tz /OPExtFont3 11 Tf (3.7 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 98 Tz 3 Tr 1 0 0 1 48.7 209.049 Tm 91 Tz (Figure 3.6: Compare the EnKF and ISIS results via &ball, the blue line denotes ) Tj 1 0 0 1 48.5 195.6 Tm (the proportion of EnKF method wins and the red line denotes the proportion of ) Tj 1 0 0 1 48.5 181.899 Tm 88 Tz (ISIS method wins a\) Ikeda experiment, Noise level 0.05 \(Details of the experiment ) Tj 1 0 0 1 48.5 168.5 Tm 96 Tz (are listed in Appendix B Table ) Tj 1 0 0 1 209.75 168.5 Tm 89 Tz /OPExtFont20 12.5 Tf (B.3\); ) Tj 1 0 0 1 239.05 168.25 Tm 93 Tz /OPExtFont3 11 Tf (b\) Lorenz96 experiment, Noise level 0.5 ) Tj 1 0 0 1 49.2 154.799 Tm 92 Tz (\(Details of the experiment are listed in Appendix B Table ) Tj 1 0 0 1 332.649 154.799 Tm 88 Tz /OPExtFont20 12.5 Tf (B.4\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont20 12.5 Tf 88 Tz 3 Tr 1 0 0 1 237.099 40.299 Tm 75 Tz /OPExtFont3 11 Tf (55 ) Tj ET EMC endstream endobj 312 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 313 0 obj <> stream 0O ,,j`÷ڑue`>~\ƙ22Ӽ Xd-qCiFލ6]k4pPinIwN\Р^k+t5ԇdž;fu?ɴB<"r\rv#.瑯?#x(0 G;p@Ilʲ{H1g_Nq . 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It appears that when the ) Tj 1 0 0 1 122.15 650.7 Tm 105 Tz (DCEn is constructed by considering a small number of observations \(e.g. 1 or 2\), ) Tj 1 0 0 1 121.9 627.649 Tm 108 Tz (the ISIS ensemble built on 12 observations outperforms the DCEn as shown in ) Tj 1 0 0 1 121.9 604.85 Tm 109 Tz (Figure 3.7, 3.8. From the first four panels of Figure 3.7 and 3.8, we found that ) Tj 1 0 0 1 121.45 581.549 Tm 110 Tz (some of the DCEn members lie on the same model's attractor as the true state ) Tj 1 0 0 1 121.7 558.75 Tm 104 Tz (does, some are not while the ISIS ensemble seems to be lying on the right model's ) Tj 1 0 0 1 121.7 535.7 Tm (attractor. And by evaluating the nowcast ensemble using e-ball method, we found ) Tj 1 0 0 1 121.45 512.899 Tm 108 Tz (the ISIS ensemble assigns more probability mass around the true state than the ) Tj 1 0 0 1 121.45 489.899 Tm 112 Tz (DCEn for almost all different sizes of E ball. This is due to the fact that lim-) Tj 1 0 0 1 121.45 466.85 Tm 107 Tz (ited dynamical information are contained in such short window of observations. ) Tj 1 0 0 1 121.2 443.8 Tm (When more observations are considered the DCEn outperforms the ) Tj 1 0 0 1 457.699 443.3 Tm 90 Tz /OPExtFont3 11 Tf (ISIS ) Tj 1 0 0 1 481.899 443.3 Tm 99 Tz /OPExtFont2 11.5 Tf (ensem-) Tj 1 0 0 1 121.2 420.75 Tm 108 Tz (ble. Figure 3.9 shows that even using half window-size of the observations, the ) Tj 1 0 0 1 121.2 397.949 Tm 105 Tz (DCEn outperforms the ISIS ensemble. The DCEn seems to be more concentrated ) Tj 1 0 0 1 121.2 375.149 Tm 109 Tz (and closer to the true state than the ISIS ensemble and assign more probability ) Tj 1 0 0 1 120.7 351.899 Tm (mass around the true state. Using the same length of the observations, with no ) Tj 1 0 0 1 120.5 328.85 Tm 106 Tz (surprise the DCEn again wins. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 106 Tz 3 Tr 1 0 0 1 137.5 305.549 Tm 107 Tz (As we discussed in Section 3.6, the DCEn is the optimal ensemble estimates ) Tj 1 0 0 1 120.5 282.5 Tm 109 Tz (one may achieve. It is expected to outperform any other state estimation meth-) Tj 1 0 0 1 120.5 259.7 Tm 105 Tz (ods. Although our ISIS ensemble, with no doubt, underperforms the dynamically ) Tj 1 0 0 1 120 236.7 Tm 108 Tz (consistent ensemble, it seems to have similar structure as the dynamically con-) Tj 1 0 0 1 120 213.649 Tm 106 Tz (sistent ensemble does. Note the ISGD algorithm is run for finite time, with more ) Tj 1 0 0 1 120.25 190.35 Tm 108 Tz (ISGD iterations, we conjecture the ISIS ensemble will converges to the DCEn. ) Tj 1 0 0 1 120 167.299 Tm (In practice, DCEn is computationally inapplicable while our IS methods can be ) Tj 1 0 0 1 120 144.5 Tm 106 Tz (applied in both low dimensional and high dimensional systems \(53\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 106 Tz 3 Tr 1 0 0 1 311.75 51.899 Tm 90 Tz (56 ) Tj ET EMC endstream endobj 317 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 318 0 obj <> stream 0 ,,[[b7!2n[]("`[d[q&'m11"ðLxuvZX[-̄*ճ.ӡ9Q?2 >7(]{iXpEBd4f8d$~ G Uvuۻtqn#(S- (&߶@mgSԬRزrI:-E+4>3}YY=Dg@ )'U!͏֬qSH&ˡ i6v?vD{4ʞjM q(Q@ *s7(9=a= _1@CWM+s0/MOf/96^=*C2TWϡy{~&FNIU"Y<422:.FI/Y)4rpVo/"o(Dm#`ٯ%ưOG~H`ƯZIKFGZdn !M>eԊAs2HsP.v9odz/a2 nOd){p哢BrT DER2D}pnܹxL>̻tjq]ڬ,l]8X)U`_mE=tE⑜=57>~ǭUh͍UXdS(YԩQ%\oF=XbPF,D}+`Hq ;Wd\w ?ci &*cjd*fݖ85,r~$:HL Cp~G&;> '3ަv`Y\@w H)$&);N\G,+=ZwkPe`Wo sPGH"~Y?s٪FMV#eZbY6{?h"dh.G:UꍜTzPH8gaj+3^95zf FτIW.=ʗ}%ҫ/<+xXDa9:'*QQjM~AwχDk(bx~֦q?~k,L9BP궷K6<# )j%-qjW'ykaAv=Blqsch侌mxUZ'\hQӰ*3:c 82J83rIZ|u4Ual-ib7߂=:Z{q0Bx|)'Wf۶i +e$1hDD@Xg98GT:J D8R&z{ j |UIS@jɔT$_#JջJ]kb{œsޭN5a0LldI!xSiD@ijdƥ{#mmxtBKaѡby,+ R\9!-TR<qSSWKT'*5ENno AL‰;;~G %y6==\BBt3/t >*cR2Ff O(ԓ\LDHXtwi ~;(wUwOן5x~ 55Ԫ١\Òd{ 04dnk/O0+MםYD`VRe[ukVCyiUh,Ul 47! k1Δ8A2M*e='v5 DD/1Ǒ%HF(=N|W4 .J.@ -+{)ݬȴFj~IuN)Lf8̪W9I+Hd0PU)J+XpF}H~{aEǿLDZ^ C3m"k[< SZ.4?)#ݖˋ["W#c*JntcVD:yݠTf ME48RsKK!hzvEo}ȍk?hYG@{&'Hk6x 9^nb%xl?aCXCǡ 8*}zzrtTR`;Nx6[c`h= #Z2ЁjqJ<m@04'z QhcGumB޼~W˵ƻ4ѷ8'UIW|)Rƥh|G'1״OmCYvS4*ԍ5JW6qBW%#e'ǩ?05V:y,C `PlxUtChԬb~/{s7N|eG!|:`\:Ì(C:A-<-@Nv9 %THF?.~Upv'4d##n8ipE]Lr!QLjF46W²X]`C\b^#(%H1JRᆮ@ovtT K c2^U Nk{ȉH~^ _f4DfL2z&M>gSS/2d)sڒ\wVe] !usrr[Џ_B E8 (4zub}bNfK1l\* ֥7q* ZtO5Nv~5sPö͉bԹ5"wSCUaߋBqZކǠ;nnɡvc*J(*v, Q |+C1LQWGj(H1F:?xg}<6vuG q̙gvt;D4_Y]tM4Fb"9@_ ]hU /J`y2X(1tܟ~]͢rB8R~fAK6!~< IASF+7&=kx&uY_=̏YfcUV}M-"-s MbC,Xx U)tJӊx,r5w9[@iqkEBɴ܎1>R29oWj!Gi4(7o[s}ҩ/ީ:bwcy9{eGy+T-?>6\iQZv eH%"Jfi-3-a;{bʏ.kZ޼&TF|¢*Q  0h^rXHa^^^‚4;}ӖnO4dN*j$/  ^D@? $6Ԑϧtêueľ{ba Y1l #JсCKQX,?Ŏk(IjGIϿ:ԟ:s#m/]^] Ԝޝ4b` ,JlZJknmxݬ:1$%#Bpβ""rHĸsmNu8@ IIwS6BL2=PQ^ wpf z9{yS&;Ow&L$Իj頷|(xa9tRmBm50QRvfqɀyèEUy{⟶Y?ƻZdW;ZSwȫO ])PI\9M(+J7(xpe}D.}:9J~K%zVt$B1џ:|B00I m8?RC[su']=WmNop9!Q919Cr|5cAT 6w`0B&Nrka`x`Qq;N[9̻7qcy"5D}畬6Ѐ>XLr 9D7k6s. 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ct͌"#ҎK>r+>{Q0?XR_x~e,?UwuE]$Z[̀f*%RFЃt_˒;bN8$ 2zN`12o{Կ˫,q|YR0~3J )̉ !-\>Ƀr=iAqNlo!jw9ۘi&Aۍgt3_Fu T%}z{=£D*!&ϤH%6ܑ+͕hEX"za6|4خ endstream endobj 319 0 obj <> endobj 320 0 obj [321 0 R] endobj 321 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 483 0 0 786 0 0 cm /ImagePart_2084 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 98.15 566.649 Tm 303 Tz 3 Tr /OPExtFont9 3 Tf (. ) Tj 1 0 0 1 105.599 566.899 Tm 824 Tz /OPExtFont9 6 Tf (\t) Tj 1 0 0 1 119.299 566.649 Tm 136 Tz (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 136 Tz 3 Tr 1 0 0 1 92.15 556.1 Tm 74 Tz /OPExtFont9 7.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 7.5 Tf 74 Tz 3 Tr 1 0 0 1 365.05 574.299 Tm 75 Tz /OPExtFont18 4.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4.5 Tf 75 Tz 3 Tr 1 0 0 1 368.399 565.7 Tm 95 Tz /OPExtFont17 5.5 Tf (4tom T: ) Tj 1 0 0 1 378.699 556.799 Tm 163 Tz /OPExtFont1 4.5 Tf (s ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 163 Tz 3 Tr 1 0 0 1 57.6 631.2 Tm 96 Tz (-) Tj 1 0 0 1 60.7 631.45 Tm 98 Tz (1.02 ) Tj 1 0 0 1 57.6 619.45 Tm 114 Tz (-1.04 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 114 Tz 3 Tr 1 0 0 1 57.6 607.7 Tm 96 Tz (-) Tj 1 0 0 1 60.7 607.899 Tm 98 Tz (1.08 ) Tj 1 0 0 1 57.6 595.899 Tm 114 Tz (-1.08 ) Tj 1 0 0 1 60.25 584.399 Tm 108 Tz (-1.1 ) Tj 1 0 0 1 57.6 572.899 Tm 114 Tz (-1.12 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 114 Tz 3 Tr 1 0 0 1 57.6 561.1 Tm 96 Tz (-) Tj 1 0 0 1 60.7 561.1 Tm 98 Tz (1,14 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 98 Tz 3 Tr 1 0 0 1 57.6 549.35 Tm 96 Tz (-) Tj 1 0 0 1 60.7 549.35 Tm 98 Tz (1.18 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 98 Tz 3 Tr 1 0 0 1 57.6 537.85 Tm 96 Tz (-) Tj 1 0 0 1 60.7 537.85 Tm 98 Tz (1.18 ) Tj 1 0 0 1 60.25 525.85 Tm 114 Tz (-1.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 114 Tz 3 Tr 1 0 0 1 260.399 633.85 Tm 96 Tz (-) Tj 1 0 0 1 262.8 634.1 Tm 109 Tz (0.66 ) Tj 1 0 0 1 260.399 622.299 Tm 117 Tz (-0.86 ) Tj 1 0 0 1 263.05 610.549 Tm 120 Tz (-0.7 ) Tj 1 0 0 1 260.399 598.799 Tm 114 Tz (-0.72 ) Tj 1 0 0 1 260.399 587.049 Tm (-0.74 ) Tj 1 0 0 1 260.399 575.299 Tm (-0.76 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 114 Tz 3 Tr 1 0 0 1 260.399 563.75 Tm 96 Tz (-) Tj 1 0 0 1 262.8 563.75 Tm 106 Tz (0.78 ) Tj 1 0 0 1 263.05 552 Tm 117 Tz (-0.8 ) Tj 1 0 0 1 260.399 540.25 Tm 114 Tz (-0.82 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 114 Tz 3 Tr 1 0 0 1 260.399 528.7 Tm 96 Tz (-) Tj 1 0 0 1 262.8 528.7 Tm 106 Tz (0.84 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 106 Tz 3 Tr 1 0 0 1 77.75 518.899 Tm 157 Tz (0.31 0.32 0.33 034 0.35 0.38 037 0.38 0.39 0.4 ) Tj 1 0 0 1 225.349 518.899 Tm 2000 Tz (\t) Tj 1 0 0 1 275.5 518.649 Tm 148 Tz (0.06 0.07 0.08 0.09 0.1 ) Tj 1 0 0 1 344.399 518.649 Tm 693 Tz (\t) Tj 1 0 0 1 353.05 518.649 Tm 145 Tz (0.11 0.12 0.13 0.14 0.15 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 145 Tz 3 Tr 1 0 0 1 96.5 454.1 Tm 132 Tz /OPExtFont12 4 Tf (" ) Tj 1 0 0 1 102.7 454.1 Tm 1395 Tz /OPExtFont2 4 Tf (\t) Tj 1 0 0 1 116.65 454.1 Tm 66 Tz /OPExtFont12 4 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 4 Tf 66 Tz 3 Tr 1 0 0 1 73.7 374.149 Tm 147 Tz /OPExtFont1 4.5 Tf (1.14 1.15 1.18 1.17 1.18 1.19 ) Tj 1 0 0 1 160.3 374.149 Tm 637 Tz (\t) Tj 1 0 0 1 168.25 374.149 Tm 99 Tz (1.2 ) Tj 1 0 0 1 174.5 374.149 Tm 653 Tz (\t) Tj 1 0 0 1 182.65 374.149 Tm 90 Tz (1.21 ) Tj 1 0 0 1 190.55 374.149 Tm 617 Tz (\t) Tj 1 0 0 1 198.25 374.149 Tm 149 Tz (122 123 ) Tj 1 0 0 1 222.5 376.55 Tm 2000 Tz (\t) Tj 1 0 0 1 273.85 373.899 Tm 132 Tz (-003 -0.02 -0.01 ) Tj 1 0 0 1 316.3 373.899 Tm 810 Tz (\t) Tj 1 0 0 1 326.399 373.899 Tm 95 Tz (0 ) Tj 1 0 0 1 328.8 373.899 Tm 790 Tz (\t) Tj 1 0 0 1 338.649 373.899 Tm 147 Tz (0.01 0.02 0.03 0.04 0.05 0.06 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 147 Tz 3 Tr 1 0 0 1 60.25 490.8 Tm 101 Tz (0.32 ) Tj 1 0 0 1 62.899 479.05 Tm 99 Tz (0.3 ) Tj 1 0 0 1 60 467.3 Tm 94 Tz /OPExtFont21 4.5 Tf (028 ) Tj 1 0 0 1 60.25 455.75 Tm 101 Tz /OPExtFont1 4.5 Tf (0.26 ) Tj 1 0 0 1 60.25 444.25 Tm 117 Tz (024 ) Tj 1 0 0 1 60 432.5 Tm 103 Tz (0.22 ) Tj 1 0 0 1 62.899 420.949 Tm 123 Tz (02 ) Tj 1 0 0 1 60 409.199 Tm 103 Tz (0.18 ) Tj 1 0 0 1 60 397.449 Tm (0.18 ) Tj 1 0 0 1 60 385.699 Tm (0.14 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 103 Tz 3 Tr 1 0 0 1 189.849 304.8 Tm 158 Tz (1.88r1 ens wins ) Tj 1 0 0 1 190.099 298.55 Tm 169 Tz (IS ) Tj 1 0 0 1 200.65 298.55 Tm 154 Tz /OPExtFont21 4.5 Tf (ens ) Tj 1 0 0 1 217.199 298.55 Tm 170 Tz /OPExtFont1 4.5 Tf (wins ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 170 Tz 3 Tr 1 0 0 1 84.95 338.899 Tm 146 Tz /OPExtFont19 4.5 Tf (0.9 ) Tj 1 0 0 1 84.95 327.6 Tm (0.8 ) Tj 1 0 0 1 84.95 316.55 Tm (0.7 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 146 Tz 3 Tr 1 0 0 1 73.9 305.3 Tm 99 Tz /OPExtFont1 14.5 Tf (W) Tj ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 14.5 Tf 99 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 14.5 Tf 99 Tz 3 Tr 1 0 0 1 73.7 778.299 Tm 103 Tz ( ) Tj 1 0 0 1 84.95 305.3 Tm 146 Tz /OPExtFont19 4.5 Tf (0.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 146 Tz 3 Tr 1 0 0 1 84.95 293.75 Tm (0.5 ) Tj 1 0 0 1 73.7 282.699 Tm 166 Tz (g) Tj 1 0 0 1 78.7 282.699 Tm 113 Tz /OPExtFont1 4.5 Tf (- ) Tj 1 0 0 1 80.4 282.699 Tm 168 Tz /OPExtFont19 4.5 Tf ( 0.4 ) Tj 1 0 0 1 84.95 271.2 Tm 144 Tz (0.3 ) Tj 1 0 0 1 84.95 259.899 Tm (0.2 ) Tj 1 0 0 1 84.7 248.649 Tm 129 Tz (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 129 Tz 3 Tr 1 0 0 1 381.6 234 Tm 147 Tz (0.05 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 147 Tz 3 Tr 1 0 0 1 260.399 493.699 Tm 96 Tz /OPExtFont1 4.5 Tf (-) Tj 1 0 0 1 262.8 493.699 Tm 124 Tz (096 ) Tj 1 0 0 1 260.149 481.899 Tm 117 Tz (-0.08 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 117 Tz 3 Tr 1 0 0 1 260.399 458.899 Tm 96 Tz (-) Tj 1 0 0 1 262.8 458.899 Tm 106 Tz (0.12 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 106 Tz 3 Tr 1 0 0 1 260.399 447.35 Tm 96 Tz (-) Tj 1 0 0 1 262.8 447.35 Tm 106 Tz (0.14 ) Tj 1 0 0 1 260.399 435.6 Tm 114 Tz (-0.18 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 114 Tz 3 Tr 1 0 0 1 267.6 424.1 Tm 55 Tz /OPExtFont12 4 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 4 Tf 55 Tz 3 Tr 1 0 0 1 263.05 412.3 Tm 136 Tz /OPExtFont1 4.5 Tf (-02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 136 Tz 3 Tr 1 0 0 1 260.149 400.3 Tm 96 Tz (-) Tj 1 0 0 1 262.8 400.3 Tm 121 Tz (022 ) Tj 1 0 0 1 260.149 388.8 Tm 130 Tz (-024 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 130 Tz 3 Tr 1 0 0 1 316.1 448.1 Tm 99 Tz /OPExtFont7 7.5 Tf () Tj 1 0 0 1 319.449 448.1 Tm 74 Tz (1 ) Tj 1 0 0 1 322.55 444.5 Tm 48 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont7 7.5 Tf 48 Tz 3 Tr 1 0 0 1 371.05 682.549 Tm 99 Tz /OPExtFont3 11 Tf (3.7 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 99 Tz 3 Tr 1 0 0 1 44.399 200.649 Tm 92 Tz (Figure 3.7: Dynamically consistent ensemble built on 1 observation compared ) Tj 1 0 0 1 44.149 186.95 Tm 89 Tz (with ISIS ensemble built on 12 observations, the noise model is U\(-0.025,0.025\), ) Tj 1 0 0 1 44.399 173.299 Tm 93 Tz (each ensemble contains 64 ensemble members. The top four panels following ) Tj 1 0 0 1 44.399 159.6 Tm 91 Tz (Figure 3.5, plot the ensemble in the state space. The ISIS ensemble is depicted ) Tj 1 0 0 1 44.399 145.899 Tm 96 Tz (by green dots. The DCEn is depicted by purple dots. The bottom panel fol-) Tj 1 0 0 1 44.149 132.5 Tm 91 Tz (lowing Figure 3.6 compare the DCEn and ISIS results via &ball. \(Details of the ) Tj 1 0 0 1 44.149 118.799 Tm (experiment are listed in Appendix B Table B.5\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 232.8 32.899 Tm 77 Tz (57 ) Tj ET EMC endstream endobj 322 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 323 0 obj <> /FirstChar 0 /FontDescriptor 324 0 R /LastChar 128 /Subtype/TrueType /ToUnicode 325 0 R /Type/Font /Widths[0 0 0 0 0 0 0 0 0 341 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 341 402 587 0 0 0 0 332 543 543 0 0 361 479 361 0 710 710 710 710 710 710 710 710 710 710 402 402 0 0 0 616 0 776 761 723 830 683 650 811 837 545 555 770 637 947 846 850 732 850 782 710 681 812 763 1128 763 736 691 543 0 543 0 0 710 667 699 588 699 664 422 699 712 341 402 670 341 1058 712 685 699 699 497 593 455 712 648 979 668 650 596 710 0 710 0 0 1000]>> endobj 324 0 obj <> endobj 325 0 obj <> stream /CIDInit /ProcSet findresource begin 12 dict begin begincmap /CIDSystemInfo << /Registry (Verdana-BoldItalicOPExtFont21) /Ordering (UCS) /Supplement 0 >> def /CMapName /Verdana-BoldItalicOPExtFont21 def /CMapType 2 def 1 begincodespacerange <00> endcodespacerange 13 beginbfrange <09> <09> <0009> <0A> <0A> <000D> <0D> <0D> <000D> <20> <22> <0020> <27> <29> <0027> <2C> <2E> <002C> <30> <3B> <0030> <3F> <3F> <003F> <41> <5B> <0041> <5D> <5D> <005D> <60> <7B> <0060> <7D> <7D> <007D> <80> <80> <2014> endbfrange endcmap CMapName currentdict /CMap defineresource pop end end endstream endobj 326 0 obj <> stream 0 ,,2}}kc f+osX˺95ydk%ϻ¡/S[3r]yRan*R'\!j_<9euR]|]C(QDS`AOճ滾5h0?&à1&ڼ۳͌հհ=@3حKż<0jr@Ŏ6KOV So;t*` ^8X_n \'x#4ٲZ_W7v|0/#E7\ķ&. ڋ52MĞEN+4_ q0 pZ?= \/xk18 dBm xe ILD.՗eLj0r ƻ cwp˜ ?s lU}ȈmĐ}U>S6F.lsAё=oW;o܈&7]{x r6[=<W'S5!1"+5EGƵtw\6ITz# :|2W9ISj*ŵD,t$Xsxf6X}9[-hkM!?)CY4 wĮ^U wGUt X\<.al8CY(w qu8'T̓ < W"' {ޛ^aEɋAnj,4*3Kn8&j/ƮezQN}QfNW#|&T^K"6 R,%Q[AH<0)"W)ɇ@W6&_g9~*À" C~MO.ea._@a7b |:'R1[ISq/jz{Ż 3y>;<i{5>pkY'=쒍Fr{+u;t̏?->^I(ZaENct>pm&/YSu/ l<'Jvp1&#K?FhUMV޳78{,X [UcIJ/߆Ҫ(o_(s#cy_g[nCK(7Skli†V-r| Mqt0߉zҶ( RR<*4$Fovtz /N%"ѯ;l0 /WGz:2Kp̪ V^rSI]eU0IP{;"Å_'ݶ[e7Id]="ORRf?+ɘ|[zyCXό"M'^7tCqSOۗoZG2&S !aRbÈj~WZ6CЈ \ .e#nzV̱^.\ *5 ],nbE/}&&э!ڦ 29P3U 6Q݄^N+ $Z iYn?v@~h͂ETiD0V,>ڎX0sK5U <M+wn}vQGƇoI`"@̸| լ/J{Xm3s@mא8SH_L*z6VhA>J3+#Yr uR$ AS{[VYdP}N!b)߽|e#r"rf=M炆L 18M>,cW kvL,^IgKL^6^umoBo  dVXYVԑ  OpuSr7"' Ě8ٙZӬ^KQ6<̋UTDhX( k#A6)K@0Ciŭ9&A}gώj *޻D 7=KTň_;N!yEI=5M5RP=p:ԫfl4? aQyE0_!>NQ`ĩծUMu{=y+s0X*4 )M &G3bbGC/]SeB[ sXS{ nm;mTFޡӀy'T0F+I7SD6ۗcG:bWD{yOf$dOʒ+%ċ<oV8{;2ͦqu"bBӺ(ۛZAc8tzW~3S?cS$Zwm 99ڰ*`%"QqOR|.|in\28pHetl;g>嵺rB4ޫ:D|݆! 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%w-3TvO*K{>s*]n '\.Gw6G>fMȵJէXX>)$8#9 !aرrz JÚ󈇫Sxsk(g D eO-#"wê;DCy>yhIRP:NE(r};F8r_ DԳt-1FZ:ZSFlׅbj#2E}A=s5ؓR:l&[df46FBQED B'ɭ5y U"ͺ*Վ%if(ϟ+Z|s.嫇烉X2b$,PgcuxFR#4sԎ\JY4ydԐ/sV UҴ14H߷^y:AI_pq{'ϫB]S}o= n'%Ir<-zhr޻ׅkU={Ve7F࿝s^h&{Tr F̷)]YaIw]6ED,X⟛kPſ`r:ƶ5t??оgO:=T 1u4 GϽ}5&rCp9Up=~q,j|ruWK2\3GTJU[1Nnz drx^h(zיld_x_$0/ 碗)[,Į=|wC7W^<"àPl@oVK&×j>BO(BPE8MxL#$o~r&G/g{=C\JB%gS_dqPP5b!YY-#Ǐ&Wk5R Jz'W1RLH7ScSbfmbTI z`+Tf3F!E{:'PUfH(KhU<8R27FvTVZ0Ke"fX`6Uv2&Gzf]Wf(Xf{(zvYQ˱O|:oHvdAxa&x ]k)-UTGE"|VhbbqbY=4QϖG d}#T|9Rqوb,6A=ʋ\󋊾B a_S FQ2%!K4 B-ΟG6bu)|WM (H qXiKzN٘e1]k2;Oޥ4v[] 1[UIlclE~hC#쨼o@^~,1NuT  _t3b]B:8Ll=001u6<_n\dxSNJ♹繓:hإ '4(߃][ђ|g~.5 XbJ-VҧaM5eSe t&^JĄ]?{7:8, Q\^@" 6>Aw'V.No&Jyj.C?xEb.xҺWFM/H.hz Y hWl&bR!On^` -U&q F@`=Gi|ױ9jv@H ̹ʌ>}8"_k jUy2R#HmR]d~JoTo}hzkNv1]r8VJH6 D'4a&{.S\둷sS`aHկ7Ů@qz>n-^"Pg0IcFVNMe^QMBB*GX8{j{jPElŝ|#axΫ ˓LTW Z[JyTCD?<7@ $361K@p kV`49=kMv0Np{mR;iq-##uی-3翴<$XZߔ0;*p097VeC6>.a'Z8K3c0 ܗ**4O}uN^ąE+ X(B#FmT#9ݢP|7;k_}DdI.[Oîlq`^ET>ඏ*1LaQP4T*hJ=-:R^ȘM7}l˗fgJBZ9PgsM*-MIF LRw-ZßW.>N4-PEbg0b|ΕR^;86E $C׼aasꚳQXHbO9Պ* 7 Eȴ w9}XXt[jI=5B皉 fk# BcVwvӯye~%n~?]FTp]4y:kXbﺎ>f< hQBd)7X4!NsE'!>V&kRM4Dĵl-}wwTԘvT,w~jJ;pK_kBڢ꩛P./ix MF_qn6dan3jخ֩P؍E#* IVQl\!u9Fm[sUgQ7h{gnVW?)VPMۥ`EkICuucl u +Št|iu&mpdw?.s\r]'|A/eBܢ7nJ^ &MaNjPă:b Z3h endstream endobj 327 0 obj <> endobj 328 0 obj [329 0 R] endobj 329 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 481 0 0 791 0 0 cm /ImagePart_2085 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 305.75 301.149 Tm 194 Tz 3 Tr /OPExtFont9 4.5 Tf (Per! ens wins ) Tj 1 0 0 1 305.75 294.899 Tm (IS ens wins ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 194 Tz 3 Tr 1 0 0 1 70.549 352.05 Tm 146 Tz /OPExtFont19 4.5 Tf (0.9 ) Tj 1 0 0 1 70.549 341 Tm 149 Tz (0.8 ) Tj 1 0 0 1 70.549 329.949 Tm (0.7 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 149 Tz 3 Tr 1 0 0 1 310.55 246.899 Tm 145 Tz (0.04 ) Tj 1 0 0 1 326.899 246.899 Tm 2000 Tz (\t) Tj 1 0 0 1 368.649 247.149 Tm 149 Tz (0.05 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 149 Tz 3 Tr 1 0 0 1 135.849 246.7 Tm 134 Tz (0.01 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 134 Tz 3 Tr 1 0 0 1 70.549 318.199 Tm 149 Tz (0.6 ) Tj 1 0 0 1 70.549 307.149 Tm (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 149 Tz 3 Tr 1 0 0 1 59.5 295.899 Tm 160 Tz (g) Tj 1 0 0 1 64.549 295.899 Tm 113 Tz /OPExtFont1 4.5 Tf (- ) Tj 1 0 0 1 66.25 295.899 Tm 192 Tz /OPExtFont19 4 Tf ( 0.4 ) Tj 1 0 0 1 70.549 284.35 Tm 146 Tz /OPExtFont19 4.5 Tf (0.3 ) Tj 1 0 0 1 70.549 273.1 Tm 149 Tz (0.2 ) Tj 1 0 0 1 70.549 261.799 Tm 131 Tz (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 131 Tz 3 Tr 1 0 0 1 78.25 250.299 Tm 134 Tz (0 ) Tj 1 0 0 1 87.599 246.899 Tm 80 Tz /OPExtFont1 4.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 80 Tz 3 Tr 1 0 0 1 193.9 246.7 Tm 149 Tz /OPExtFont19 4.5 Tf (0.02 ) Tj 1 0 0 1 210.699 246.7 Tm 2000 Tz (\t) Tj 1 0 0 1 252.25 246.899 Tm 149 Tz (0.03 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 149 Tz 3 Tr 1 0 0 1 201.099 241.649 Tm 150 Tz (size ) Tj 1 0 0 1 216 241.649 Tm 177 Tz /OPExtFont19 4 Tf ( of ) Tj 1 0 0 1 226.3 241.649 Tm 163 Tz /OPExtFont19 4.5 Tf ( eps-ball ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 163 Tz 3 Tr 1 0 0 1 356.899 696.899 Tm 103 Tz /OPExtFont3 10.5 Tf (3.7 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 103 Tz 3 Tr 1 0 0 1 104.4 580.5 Tm 156 Tz /OPExtFont9 5.5 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 156 Tz 3 Tr 1 0 0 1 62.899 532.75 Tm 154 Tz /OPExtFont9 4.5 Tf (0.31 0.32 0.33 0.34 0.35 0.38 0.37 0.38 0.39 0.4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 154 Tz 3 Tr 1 0 0 1 45.6 508.3 Tm 103 Tz (0.02 ) Tj 1 0 0 1 52.299 496.5 Tm 95 Tz (0 ) Tj 1 0 0 1 42.95 484.5 Tm 112 Tz (-0.02 ) Tj 1 0 0 1 42.95 473 Tm 114 Tz (-0.04 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 114 Tz 3 Tr 1 0 0 1 42.95 461 Tm 96 Tz (-) Tj 1 0 0 1 45.6 461 Tm 103 Tz (0.06 ) Tj 1 0 0 1 36.7 449.5 Tm 125 Tz (. -0.08 ) Tj 1 0 0 1 45.85 437.699 Tm 105 Tz (-0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 105 Tz 3 Tr 1 0 0 1 42.95 425.949 Tm 96 Tz (-) Tj 1 0 0 1 45.6 425.949 Tm 103 Tz (0.12 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 103 Tz 3 Tr 1 0 0 1 42.95 414.449 Tm 96 Tz (-) Tj 1 0 0 1 45.35 414.449 Tm 106 Tz (0.14 ) Tj 1 0 0 1 42.95 402.699 Tm 114 Tz (-0.16 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 114 Tz 3 Tr 1 0 0 1 58.799 387.55 Tm 98 Tz (1.06 ) Tj 1 0 0 1 67.45 387.55 Tm 557 Tz (\t) Tj 1 0 0 1 74.4 387.55 Tm 98 Tz (1.07 ) Tj 1 0 0 1 83.049 387.55 Tm 557 Tz (\t) Tj 1 0 0 1 90 387.55 Tm 98 Tz (1.06 ) Tj 1 0 0 1 98.65 387.55 Tm 557 Tz (\t) Tj 1 0 0 1 105.599 387.55 Tm 98 Tz (1.09 ) Tj 1 0 0 1 114.25 387.55 Tm 653 Tz (\t) Tj 1 0 0 1 122.4 387.55 Tm 87 Tz (1.1 ) Tj 1 0 0 1 127.9 387.55 Tm 734 Tz (\t) Tj 1 0 0 1 137.05 387.55 Tm 93 Tz (1.11 ) Tj 1 0 0 1 145.199 387.8 Tm 577 Tz (\t) Tj 1 0 0 1 152.4 387.8 Tm 101 Tz (1.12 ) Tj 1 0 0 1 161.3 387.55 Tm 557 Tz (\t) Tj 1 0 0 1 168.25 387.8 Tm 98 Tz (1.13 ) Tj 1 0 0 1 176.9 387.8 Tm 557 Tz (\t) Tj 1 0 0 1 183.849 387.8 Tm 98 Tz (1.14 ) Tj 1 0 0 1 192.5 387.8 Tm 557 Tz (\t) Tj 1 0 0 1 199.449 387.8 Tm 98 Tz (1.15 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 98 Tz 3 Tr 1 0 0 1 346.3 581.7 Tm 48 Tz /OPExtFont3 8.5 Tf (:,.S) Tj 1 0 0 1 355.449 581.7 Tm 141 Tz (: ) Tj 1 0 0 1 359.3 582.45 Tm 1219 Tz /OPExtFont9 3 Tf ( ) Tj 1 0 0 1 371.5 581.7 Tm 255 Tz /OPSUFont0 3 Tf ( ) Tj 1 0 0 1 376.1 581.7 Tm 100 Tz /OPExtFont9 3 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr 1 0 0 1 261.35 533 Tm 144 Tz /OPExtFont9 4.5 Tf (0.06 0.07 0.08 0.09 ) Tj 1 0 0 1 317.3 533 Tm 613 Tz (\t) Tj 1 0 0 1 324.949 533 Tm 91 Tz (0.1 ) Tj 1 0 0 1 330.699 533 Tm 673 Tz (\t) Tj 1 0 0 1 339.1 533 Tm 95 Tz (0.11 ) Tj 1 0 0 1 347.5 533 Tm 577 Tz (\t) Tj 1 0 0 1 354.699 533 Tm 143 Tz (0.12 0.13 0,14 0.15 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 143 Tz 3 Tr 1 0 0 1 318.699 440.85 Tm 255 Tz /OPSUFont0 3 Tf () Tj 1 0 0 1 323.5 434.1 Tm 744 Tz /OPExtFont5 3 Tf (t ) Tj 1 0 0 1 330.949 434.1 Tm 100 Tz /OPExtFont9 3 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr 1 0 0 1 261.35 387.8 Tm 159 Tz /OPExtFont9 4.5 Tf (0.27 0.28 0.29 0.3 0.310.32 0.33 034 0.35 0.36 ) Tj 1 0 0 1 336.699 382.3 Tm 56 Tz (X ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 56 Tz 3 Tr 1 0 0 1 246 647.95 Tm 87 Tz /OPExtFont19 4.5 Tf (-0.66 ) Tj 1 0 0 1 246 636.2 Tm 89 Tz (-0.68 ) Tj 1 0 0 1 248.65 624.899 Tm 120 Tz /OPExtFont9 4.5 Tf (-0.7 ) Tj 1 0 0 1 246 612.899 Tm 117 Tz (-0.72 ) Tj 1 0 0 1 246.25 601.399 Tm 114 Tz (-0.74 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 114 Tz 3 Tr 1 0 0 1 246.25 589.649 Tm 96 Tz (-) Tj 1 0 0 1 248.65 589.649 Tm 106 Tz (0.76 ) Tj 1 0 0 1 246.25 577.899 Tm 114 Tz (-0.78 ) Tj 1 0 0 1 248.9 566.1 Tm 117 Tz (-0.8 ) Tj 1 0 0 1 246.25 554.35 Tm 114 Tz (-0.82 ) Tj 1 0 0 1 246.25 542.6 Tm (-0.84 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 114 Tz 3 Tr 1 0 0 1 246.25 507.3 Tm (-0.06 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 114 Tz 3 Tr 1 0 0 1 246.5 495.55 Tm 96 Tz (-) Tj 1 0 0 1 248.9 495.55 Tm 124 Tz (008 ) Tj 1 0 0 1 249.099 483.8 Tm 108 Tz (-0.1 ) Tj 1 0 0 1 246.25 472.05 Tm 117 Tz (-0.12 ) Tj 1 0 0 1 246.25 460.3 Tm (-0.14 ) Tj 1 0 0 1 246.5 448.5 Tm 114 Tz (-0.16 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 114 Tz 3 Tr 1 0 0 1 246.5 437 Tm 96 Tz (-) Tj 1 0 0 1 248.9 437 Tm 106 Tz (0.18 ) Tj 1 0 0 1 249.099 425 Tm 118 Tz (-0.2 ) Tj 1 0 0 1 246.5 413.5 Tm 130 Tz (-022 ) Tj 1 0 0 1 246.5 401.699 Tm 114 Tz (-0.24 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 114 Tz 3 Tr 1 0 0 1 42.95 645.1 Tm 96 Tz (-) Tj 1 0 0 1 46.1 645.1 Tm 95 Tz (1.02 ) Tj 1 0 0 1 42.95 633.299 Tm 114 Tz (-1.04 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 114 Tz 3 Tr 1 0 0 1 42.95 621.549 Tm 96 Tz (-) Tj 1 0 0 1 46.1 621.549 Tm 95 Tz (1.06 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 95 Tz 3 Tr 1 0 0 1 42.95 609.799 Tm 96 Tz (-) Tj 1 0 0 1 46.1 609.799 Tm 95 Tz (1.08 ) Tj 1 0 0 1 45.6 598.299 Tm 105 Tz (-1.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 105 Tz 3 Tr 1 0 0 1 42.95 586.5 Tm 96 Tz (-) Tj 1 0 0 1 46.1 586.5 Tm 98 Tz (1.12 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 98 Tz 3 Tr 1 0 0 1 42.7 575 Tm 96 Tz (-) Tj 1 0 0 1 46.1 575 Tm 98 Tz (1.14 ) Tj 1 0 0 1 42.95 563.25 Tm 114 Tz (-1.18 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 114 Tz 3 Tr 1 0 0 1 42.95 551.5 Tm 96 Tz (-) Tj 1 0 0 1 46.1 551.5 Tm 98 Tz (1.18 ) Tj 1 0 0 1 45.6 539.7 Tm 114 Tz (-1.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 114 Tz 3 Tr 1 0 0 1 30.25 213.299 Tm 91 Tz /OPExtFont3 11 Tf (Figure 3.8: Dynamically consistent ensemble built on 2 observations compared ) Tj 1 0 0 1 30 199.649 Tm 89 Tz (with ISIS ensemble built on 12 observations, the noise model is U\(-0.025,0.025\), ) Tj 1 0 0 1 30.25 185.95 Tm 93 Tz (each ensemble contains 64 ensemble members. The top four panels following ) Tj 1 0 0 1 30.25 172.299 Tm 91 Tz (Figure 3.5, plot the ensemble in the state space. The ISIS ensemble is depicted ) Tj 1 0 0 1 30.25 158.6 Tm 97 Tz (by green dots. The DCEn is depicted by purple dots. The bottom panel fol-) Tj 1 0 0 1 30 144.899 Tm 92 Tz (lowing Figure 3.6 compare the DCEn and ISIS results via c-ball. \(Details of the ) Tj 1 0 0 1 30.25 131 Tm (experiment are listed in Appendix B Table B.5\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 219.849 44.85 Tm 75 Tz (58 ) Tj ET EMC endstream endobj 330 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 331 0 obj <> /FontDescriptor 332 0 R /Subtype/Type1 /ToUnicode 333 0 R /Type/Font>> endobj 332 0 obj <> endobj 333 0 obj <> stream /CIDInit /ProcSet findresource begin 12 dict begin begincmap /CIDSystemInfo << /Registry (CourierOPSUFont0) /Ordering (UCS) /Supplement 0 >> def /CMapName /CourierOPSUFont0 def /CMapType 2 def 1 begincodespacerange <00> endcodespacerange 2 beginbfrange <20> <20> <0020> <80> <80> <2751> endbfrange endcmap CMapName currentdict /CMap defineresource pop end end endstream endobj 334 0 obj <> stream 0 ,,&j޲S55X8 TF["-\[{jO[CK>2B;4I4SGWQoŗZӴI!ڐN0gߔ5{@$薽K#@Nc,D)4iiԹ_-80HtHI !yV=Wi8uP)"΀u>rOх+#z285p=r75~.?9us]὏a(kowSzJWg;? c`fu^Yp2a|)ufen^#B +qю\lJGіDP C`zflfh“-+ V R8I6 ~J[CtUD#-B?p|3i;oLdWK[2 { x1nJ~Wbi @1b[ܒg((kabRux4UwU%4>Fv9#3qh_}Ag(&4MmJUnC|d~8hh^}5e˝-~XGbH 3۳m BZg`.iWRp]0TI<#Hihr8ONM7݊ #;*y@mc;2:R`)ěr\M EۤZ>(ZGY=|FdD$ėB^ 62t@(IܧcR+wps5ʭ_*ƍ'4 BoTV!g_#j=U'I109hojR>h )"IUfb]^a=2.mȬ总 8֖H޵҂.M9lY{BfzI>}Ix.v7^"0Vϗ\O[] HhB:RM*>w~ &AZQm@4=|z } 2/tG=ZgWpkoGro&h*C}G-|V]oGD]&){ϲv3M2%9я$5MH n26l~$$j^ఱsɺF@msI.w(:fO@'£pbG| v?ٲ[9&" ^R%O3?zCi9\|i^k>6EfwyU Za֎]Yb |@wĜ/yk$a&[ x&}Q%N:6Wg!~,y~++E M-٤*Oa"~?r{0U$G,BؙkK(x"T1}h}3@7w$c/qyͣР-i ??ǓMmY|͘yӏ0n徇^#OHD|xJXe}piOZV\LK7STA6)gg=vol&"([.n\k oASv~gaq|v@J*2xu[ )G.6duA:jQf9/&qi-@H@YRNX$-A寜Q܀[%}oY}A6C={guUs:Kq8IBbƤ~V":<1Z˧%{dnDiD+pV.onAmLrƬ*ЗE63e| 'A3nrG/eHkͧg(JP<ٳ 3ł^ǥBQ{g/>t^t6Õsffs.Ξl:ՃK9HY;hp]E1YJ˺ͯJz,s"\1!%ԁBO&}e\x5AFnX#h-fȌHg.1N&stsTyR8%t5sQ~O?7(צ)e$8>{qSǮ^mfs9bRQdP݅ w t;a+<(_`5_[7o۬W߆Ds*aӃ2r߁cQ#tj nl+7|?`i̕,RgNI'Qέq%1*ҳŏ|;Ӧ l(YٝG[`ds_ϜBh#c_M|?e$DfGøRjow8MM^s$~^Jc 4fМ:a-`_iOU03]ZՊU&C^ne-y7=h#k2, e䯄LDe nD54bbzvG?+w,25A - 8=v_6筸LVk]3(=PDaN¿r ^}l;|Bo=Eatk [&amՋ}(wBp^b菻|HE! >7Tձ􄯽Ұ³-!E0ܰcXw"G JQ?G0ay wB7@!`ްzZ`4{)M{ߔb^ON/^//hrם_?ѡ~%z>3{Yei {tUE4#l4)ԿCL "`Hހ/MA2ӏשrx?.Z9ҏ״Y?' mɳ3-Mɶ p96p@-pE+~DūN%)W*GX&P<~)|x3+x̜Ok`.`= mwN$(́F[dv7VXf I 2]kzᑜ$W}\MI0++=i,rږpHcvY7tf !G/+;Isu`4LGI.3.5ߘO<7TY&g'u ORĎ(j8=ף4G| 2,g7l%+s- <[͛`17},2},оq[A0\x$R^w~̭nDp /36sO؃)B{~@^v%IbpW4鹣ڒpѳKݴͰ!ռǨW̺jڲ= po]CEANۦ*l :=!?p@Dm7`&r8|r\4Nx;MLLxAҁ՞`tKj D]WƇN_ 0h0sn3#8{P63]a–R3(ݭ 1*O,9x:.Sÿ$DĪLquhCdzH> ؝ 5L$P!PGHpHeÞ_i}G_oap7Ü ] ڇOPgg5nɳL9eRY.%Ub*x9@n hIPFcٌ,/OGcWڶt>S&iل7k7] nǾ]RZoҟhAO83YΓVNG0lt (ز-1,?ZqhfCot0,vP ¿B,V)%Bn|fr tI<%疿Ca $Qjޝa$wYS|Ӝ(ismmZz_g?*9l*>vQ|$D?9CJ˧p:f )sY*B󏁡+Ԗ+6& GUA} !sk7SFC3|lIƱ9H!FG%k>4UXD ML:ᗒuZd@RvCeGB&gyآ(Oy!m_hl\ o@PDj~TOSg;lk`L[Oߧ|ЂvWdˑSO{9O u#<[knFب<>AUxŁĿ;]Szh*G!sse}z`r1yWfnn (a.FӘ7Nka'aơZJ;ϖ,S{fo2FPjB@L|{%N]duGu{Bg֘gqx8@A׆K/ا% hb5Gl(K5I*{>I<Iq2 q3}&ETWo!jK2"[7x~OTcj`P~58[',Wbx ;] M'Œ[nNK+.ϕXQA<. ȅ# g->ҶѶa>*k*[[-kW#Vcoz7z=`޿j*"H/?x}t'w6rZS7O0٘D)O z~Oּc9YϜz}xYrd>̙8RcWœP{FP= E4*Wplszl}3\i4~c.8Jw?K jf[% Pu6 v+=vBYo!_ƽñٞLT0k@yC1⪂33٫6dH ba seSGR]N2pQpV$#lÚ o[H`VIpcփg]|Ae}*0UӠ ZWj|JʽnA!CG#dmHKRz bsj~-IcruŸiztڙ\̲*oc_LAYa @ _}  ZRL̂ѐgc[L2_-_9lUf+Lz8cvj5kglX1k-f߿YU#;zBW -A+ [YRTP?ɡcMB|鼡+޷_m4\ĺρn]Uz!:Ĝ0{EQ@UU9c+mTur;+Xד. 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q 1 0 0 1 0 0 cm 500 0 0 789 0 0 cm /ImagePart_2086 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 116.9 566.75 Tm 136 Tz 3 Tr /OPExtFont9 6 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 136 Tz 3 Tr 1 0 0 1 54.95 631.299 Tm 96 Tz /OPExtFont9 4.5 Tf (-) Tj 1 0 0 1 58.1 631.299 Tm 101 Tz (1.02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 101 Tz 3 Tr 1 0 0 1 54.95 619.549 Tm 96 Tz (-) Tj 1 0 0 1 58.1 619.549 Tm 101 Tz (1.04 ) Tj 1 0 0 1 54.95 607.799 Tm 119 Tz (-1.08 ) Tj 1 0 0 1 54.95 596.049 Tm 117 Tz (-1.08 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 117 Tz 3 Tr 1 0 0 1 54.95 561.25 Tm 96 Tz (-) Tj 1 0 0 1 58.1 561 Tm 101 Tz (1.14 ) Tj 1 0 0 1 54.95 549.25 Tm 117 Tz (-1.16 ) Tj 1 0 0 1 54.95 537.5 Tm (-1.18 ) Tj 1 0 0 1 57.6 525.7 Tm 120 Tz (-1.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 120 Tz 3 Tr 1 0 0 1 258.25 634.2 Tm 117 Tz (-0.66 ) Tj 1 0 0 1 258.25 622.7 Tm (-0.68 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 117 Tz 3 Tr 1 0 0 1 261.1 610.899 Tm 96 Tz (-) Tj 1 0 0 1 263.5 610.899 Tm 107 Tz (0.7 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 107 Tz 3 Tr 1 0 0 1 258.25 598.899 Tm 96 Tz (-) Tj 1 0 0 1 260.649 598.899 Tm 109 Tz (0.72 ) Tj 1 0 0 1 258.25 587.399 Tm 117 Tz (-0.74 ) Tj 1 0 0 1 258.25 575.899 Tm 154 Tz (-0/6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 154 Tz 3 Tr 1 0 0 1 258.25 564.1 Tm 96 Tz (-) Tj 1 0 0 1 260.899 564.1 Tm 106 Tz (0.78 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 106 Tz 3 Tr 1 0 0 1 261.1 552.1 Tm 96 Tz (-) Tj 1 0 0 1 263.75 552.1 Tm 103 Tz (0.8 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 103 Tz 3 Tr 1 0 0 1 258.25 540.35 Tm 96 Tz (-) Tj 1 0 0 1 260.899 540.35 Tm 106 Tz (0.82 ) Tj 1 0 0 1 258.25 528.85 Tm 117 Tz (-0.84 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 117 Tz 3 Tr 1 0 0 1 75.349 519 Tm 157 Tz (0.31 0.32 0.33 034 0.35 0.36 0.37 0.38 0.39 04 ) Tj 1 0 0 1 223.449 519 Tm 2000 Tz (\t) Tj 1 0 0 1 273.6 519 Tm 143 Tz (0.06 0.07 0.04 0.09 ) Tj 1 0 0 1 329.3 519 Tm 633 Tz (\t) Tj 1 0 0 1 337.199 519 Tm 91 Tz (0.1 ) Tj 1 0 0 1 342.949 519 Tm 653 Tz (\t) Tj 1 0 0 1 351.1 519.25 Tm 146 Tz (0.11 0.12 0.13 0.14 0.15 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 146 Tz 3 Tr 1 0 0 1 68.9 373.8 Tm 126 Tz (-0.03 -0.02 -0.01 ) Tj 1 0 0 1 110.9 373.8 Tm 826 Tz (\t) Tj 1 0 0 1 121.2 373.8 Tm 105 Tz (0 ) Tj 1 0 0 1 123.849 373.8 Tm 790 Tz (\t) Tj 1 0 0 1 133.699 373.8 Tm 154 Tz (0.01 0.02 0.03 004 0.05 006 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 154 Tz 3 Tr 1 0 0 1 281.75 373.8 Tm 131 Tz (0.68 0.69 ) Tj 1 0 0 1 306.5 373.8 Tm 613 Tz (\t) Tj 1 0 0 1 314.149 373.8 Tm 103 Tz (0.7 ) Tj 1 0 0 1 320.649 373.8 Tm 633 Tz (\t) Tj 1 0 0 1 328.55 374.05 Tm 148 Tz (0.71 0.72 0.73 0.74 0.75 0.76 0.77 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 148 Tz 3 Tr 1 0 0 1 54.95 493.8 Tm 117 Tz (-0.06 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 117 Tz 3 Tr 1 0 0 1 54.95 482.05 Tm 96 Tz (-) Tj 1 0 0 1 57.6 482.05 Tm 106 Tz (0.08 ) Tj 1 0 0 1 57.6 470.5 Tm 111 Tz (-0.1 ) Tj 1 0 0 1 54.95 458.75 Tm 117 Tz (-0.12 ) Tj 1 0 0 1 54.95 447.25 Tm (-0.14 ) Tj 1 0 0 1 54.95 435.25 Tm (-0.18 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 117 Tz 3 Tr 1 0 0 1 54.95 423.5 Tm 96 Tz (-) Tj 1 0 0 1 57.6 423.5 Tm 106 Tz (0.18 ) Tj 1 0 0 1 57.6 411.949 Tm 120 Tz (-0.2 ) Tj 1 0 0 1 54.95 400.199 Tm 117 Tz (-0.22 ) Tj 1 0 0 1 54.95 388.449 Tm 114 Tz (-0.24 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 114 Tz 3 Tr 1 0 0 1 258.25 484.199 Tm 117 Tz (-0.88 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 117 Tz 3 Tr 1 0 0 1 261.1 472.699 Tm 118 Tz (-0.9 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 118 Tz 3 Tr 1 0 0 1 258.25 460.699 Tm 96 Tz (-) Tj 1 0 0 1 261.1 460.699 Tm 121 Tz (022 ) Tj 1 0 0 1 258.25 449.149 Tm 117 Tz (-0.94 ) Tj 1 0 0 1 252.25 437.149 Tm 118 Tz (a. -0.96 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 118 Tz 3 Tr 1 0 0 1 258.25 425.649 Tm 96 Tz (-) Tj 1 0 0 1 260.899 425.649 Tm 106 Tz (0.98 ) Tj 1 0 0 1 264.949 414.1 Tm 113 Tz (-1 ) Tj 1 0 0 1 258.25 402.1 Tm 117 Tz (-1.02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 117 Tz 3 Tr 1 0 0 1 258.25 390.85 Tm 96 Tz (-) Tj 1 0 0 1 261.35 390.85 Tm 118 Tz (104 ) Tj 1 0 0 1 261.35 379.1 Tm 101 Tz (1 06 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 101 Tz 3 Tr 1 0 0 1 319.449 295.3 Tm 181 Tz /OPExtFont1 4.5 Tf (Pert ens wins ) Tj 1 0 0 1 319.699 289.1 Tm 172 Tz (IS ens wins. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 172 Tz 3 Tr 1 0 0 1 82.799 339 Tm 187 Tz /OPExtFont9 4.5 Tf (0.0 ) Tj 1 0 0 1 82.799 327.5 Tm (0.0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 187 Tz 3 Tr 1 0 0 1 82.799 316.199 Tm 159 Tz /OPExtFont17 4 Tf (0.7 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont17 4 Tf 159 Tz 3 Tr 1 0 0 1 70.099 304.899 Tm 70 Tz /OPExtFont9 4.5 Tf (' ) Tj 1 0 0 1 70.799 304.899 Tm 962 Tz (\t) Tj 1 0 0 1 82.799 304.899 Tm 187 Tz (0.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 187 Tz 3 Tr 1 0 0 1 71.5 293.649 Tm 100 Tz /OPExtFont9 4 Tf () Tj 1 0 0 1 82.799 293.649 Tm 140 Tz (0 . ) Tj 1 0 0 1 90.5 293.649 Tm 161 Tz /OPExtFont9 4.5 Tf (5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 161 Tz 3 Tr 1 0 0 1 71.5 282.1 Tm 198 Tz (- 0.4 ) Tj 1 0 0 1 82.799 270.6 Tm 183 Tz (0.3 ) Tj 1 0 0 1 82.549 259.549 Tm 191 Tz (0.2 ) Tj 1 0 0 1 82.799 248.049 Tm 161 Tz (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 161 Tz 3 Tr 1 0 0 1 380.399 233.149 Tm 191 Tz /OPExtFont1 4.5 Tf (0.05 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 191 Tz 3 Tr 1 0 0 1 369.1 682.899 Tm 93 Tz /OPExtFont0 11 Tf (3.7 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont0 11 Tf 93 Tz 3 Tr 1 0 0 1 42.25 199.549 Tm 91 Tz /OPExtFont3 11 Tf (Figure 3.9: Dynamically consistent ensemble built on 6 observations compared ) Tj 1 0 0 1 41.75 185.899 Tm 89 Tz (with ISIS ensemble built on 12 observations, the noise model is U\(-0.025,0.025\), ) Tj 1 0 0 1 41.75 172.2 Tm 93 Tz (each ensemble contains 64 ensemble members. The top four panels following ) Tj 1 0 0 1 42 158.5 Tm 91 Tz (Figure 3.5, plot the ensemble in the state space. The ISIS ensemble is depicted ) Tj 1 0 0 1 42 145.1 Tm 97 Tz (by green dots. The DCEn is depicted by purple dots. The bottom panel fol-) Tj 1 0 0 1 42 131.399 Tm 91 Tz (lowing Figure 3.6 compare the DCEn and ISIS results via e-ball. \(Details of the ) Tj 1 0 0 1 41.75 117.7 Tm 92 Tz (experiment are listed in Appendix B Table B.5\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 231.099 31.1 Tm 77 Tz (59 ) Tj ET EMC endstream endobj 338 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 339 0 obj <> stream 0' ,,UUkek 0^J+P. 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J-2z;,RD](tҊ*+4?M9/a ڱ4~[)Z Z}Mo@=H/Of[ qfhAR(:GKZŸüT(O2;Ɨ Iv-upb}sGvͶNI2*Ù1J("MR G`Dd9@,4WT/#G\L8sbk.4jdIy>WHp jsYM-r e2{j 2 ~M2/Fq`5UX8x#uXNBx?hY#z04Q]ZvCZkFuA!Rg M쳤/Z?Nòg/56B`4T).$ @d:r9AIB^z3BSPZCQ D) #ܜN+(1x:6~a?ڄuQCmnrcM7CHhlm2^lM_e:C[UF͠QCr z0)@_I(#߿/IL_KzCKEb&lGVXR }4v$GbN:$=X0Kpm|fL5FCR]G\`r:g4>;CDoGb|6,F6a8m2[(\Vj%5ϗd d\z"a.(eg90&݇qP~t֭'V-f=ΒNK;PT` dHM9\@1 p`/MO"Mon);k=6PyΞm7eXiUkK5.T-sJyOYSl$Y2a2br#$"'&[ANT]%Ik )X>j,bbA31hfG$3r4P6$ eJ$t<] ,mCfq3r~)S[)) 6sJkD|?G:k0k{BżR2a  䏤*S'Q2OL74ٝ,ܱ5ٲԮ`ѓᎲ\5-卸PFNbT5-<.x:T)<ݼP,}zR%HbIᤇ /χL 㬀hȘs`lQʢ%)z4䡳9s endstream endobj 340 0 obj <> endobj 341 0 obj [342 0 R] endobj 342 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 484 0 0 783 0 0 cm /ImagePart_2087 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 380.399 580.899 Tm 143 Tz 3 Tr /OPExtFont9 6 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 143 Tz 3 Tr 1 0 0 1 270.25 529.799 Tm 113 Tz /OPExtFont19 4.5 Tf (0.06 0.07 0.08 0.09 ) Tj 1 0 0 1 326.149 529.799 Tm 518 Tz (\t) Tj 1 0 0 1 334.1 529.799 Tm 68 Tz (0.1 ) Tj 1 0 0 1 339.6 529.799 Tm 547 Tz (\t) Tj 1 0 0 1 348 529.799 Tm 118 Tz (0.11 0.12 0.13 014 0.15 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 118 Tz 3 Tr 1 0 0 1 255.349 412.899 Tm 99 Tz (-) Tj 1 0 0 1 258.25 412.899 Tm 77 Tz (1.02 ) Tj 1 0 0 1 255.349 401.149 Tm 86 Tz (-1.04 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 86 Tz 3 Tr 1 0 0 1 258.25 389.649 Tm 90 Tz (106) Tj 1 0 0 1 268.1 389.649 Tm 98 Tz /OPExtFont9 5.5 Tf ( ) Tj 1 0 0 1 270 389.649 Tm 2000 Tz (\t) Tj 1 0 0 1 422.899 389.649 Tm 65 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 65 Tz 3 Tr 1 0 0 1 278.399 384.6 Tm 125 Tz /OPExtFont19 4.5 Tf (0.68 0.69 0/ 0/1 0.72 0.73 0.74 0.75 0.79 0.77 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 125 Tz 3 Tr 1 0 0 1 377.05 243.95 Tm 152 Tz (0.05 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 152 Tz 3 Tr 1 0 0 1 255.349 644.75 Tm 86 Tz (-0.66 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 86 Tz 3 Tr 1 0 0 1 255.099 633.25 Tm 99 Tz (-) Tj 1 0 0 1 257.5 633.25 Tm 89 Tz (OAS ) Tj 1 0 0 1 257.75 621.5 Tm (-0.7 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 89 Tz 3 Tr 1 0 0 1 255.099 609.7 Tm 88 Tz (-0.72 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 88 Tz 3 Tr 1 0 0 1 255.099 597.95 Tm 89 Tz (-0.74 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 89 Tz 3 Tr 1 0 0 1 255.099 586.45 Tm 99 Tz (-) Tj 1 0 0 1 257.75 586.45 Tm 81 Tz (0.76 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 81 Tz 3 Tr 1 0 0 1 255.349 574.7 Tm 86 Tz (-0.78 ) Tj 1 0 0 1 257.75 562.7 Tm 89 Tz (-0.8 ) Tj 1 0 0 1 255.349 551.149 Tm 98 Tz (-092 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 98 Tz 3 Tr 1 0 0 1 255.099 539.649 Tm 88 Tz (-0.64 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 88 Tz 3 Tr 1 0 0 1 255.349 495.25 Tm 80 Tz (43.88 ) Tj 1 0 0 1 257.75 483.5 Tm 89 Tz (-0.9 ) Tj 1 0 0 1 255.349 471.699 Tm 86 Tz (-0.92 ) Tj 1 0 0 1 255.349 459.949 Tm (-0.94 ) Tj 1 0 0 1 255.349 448.199 Tm 98 Tz (-096 ) Tj 1 0 0 1 255.349 436.449 Tm (-016 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 98 Tz 3 Tr 1 0 0 1 374.899 443.399 Tm 136 Tz /OPExtFont9 6 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 136 Tz 3 Tr 1 0 0 1 366 693.7 Tm 94 Tz /OPExtFont0 11 Tf (3.7 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont0 11 Tf 94 Tz 3 Tr 1 0 0 1 38.899 210.35 Tm 88 Tz /OPExtFont3 11 Tf (Figure 3.10: Dynamically consistent ensemble built on 12 observations compared ) Tj 1 0 0 1 38.649 196.7 Tm 89 Tz (with ISIS ensemble built on 12 observations, the noise model is U\(-0.025,0.025\), ) Tj 1 0 0 1 38.649 183 Tm 93 Tz (each ensemble contains 64 ensemble members. The top four panels following ) Tj 1 0 0 1 38.899 169.299 Tm 91 Tz (Figure 3.5, plot the ensemble in the state space. The ISIS ensemble is depicted ) Tj 1 0 0 1 38.649 155.649 Tm 97 Tz (by green dots. The DCEn is depicted by purple dots. The bottom panel fol-) Tj 1 0 0 1 38.649 141.95 Tm 90 Tz (lowing Figure ) Tj 1 0 0 1 108.25 141.95 Tm 74 Tz /OPExtFont0 11 Tf (3.6 ) Tj 1 0 0 1 126.25 141.95 Tm 92 Tz /OPExtFont3 11 Tf (compare the DCEn and ISIS results via &ball. \(Details of the ) Tj 1 0 0 1 38.399 128.299 Tm (experiment are listed in Appendix B Table B.5\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 227.75 41.899 Tm 75 Tz (60 ) Tj ET EMC endstream endobj 343 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 344 0 obj <> stream 0 ,,kekcR ߔrH4`k QoYmO%\!%WVgc4s{["Vb )]=*6Zۖݰ<˼04i]5AUw3f~[̕M3nHISCT%8d< w XRi}mL{f|p4~@ͻ< 61|K[UxpdloZȘpwSQu]j)ϸ42< ŽfcL#_35i7+pa.?ti̕Lf5zҡ߱x&Zv2:Y:]J[C}yoXL!@eFgY} gsie=$ fe^I܍ˡ?;]$S.DI xykH$i L&Q Q_B@ʎ\Q@ƉRsԠ ռXiQoȳ ǵ ~"Gw5_,&ZߺW/z6bh.߽gHQ e>Q*sFf_} H0Dl*mІ|{y@HE? /D{$ Nġ"SJatKvLԇwP24F5t-/x|Ƥ@}x ||@X;ۼwY\*O^r+=oSPUvQyO8V^Yƌە3?fS6ƨdo٭pV:`7R0:`ay6~;E]:6 `Q/;LN So[_D,׷i֢_a6pLJ'NrLjA[)r\{)ZY=Co|XTaduņլgCe+Hfl6[򅻼bNА\T1*G"^2}A$f0:IQ Kʻ-B{3:_.BG4\@o&=yUVEC]UG`HPwDC]lwTr V&hl%" `뇓=aXsV3lC :AhNڽ}jNUG W\^~15D%a&|"V!dzkW1}g-t7p VV`ީ*ȳgYJY60t@ЖiI9ׯ]ҒNLvx}x"`gn(n*KO6L类N ]q/+&~yy%xēG-p>x c7Wo' O -Y{J(AĮK B2{upһFDΡd^[!A\F,pBTLD%f4Mh N55/&歰d56zil <੪A uxC=烪f xqb}?½P޿ӣ3Hλ03V[M_8 ae9x`J؎/e6S|gٓfwf;{eX[?pN74kK:/}שgaxC7c!Bz'zw:Z/]Eŋ .e$Hr't][Iǫx@PЉfɖe+zsfvSHMaO %"0͌n@H>u\'0ߑ#Vt:ᕚY8@+dp-س4"cu&6!Rnhv):b$5IE 2 o[D,v %CCC&Vm5`([_^~.߄m8x,![x Xǁ&He*z &atoX% D*hT{gf{.Wt&bG(a_'|/6T{ bq[YF$=݆uޝWpr*%KSA8Y~&;vғ3{G416\bw"KJݙDIsͷнU|a^,AyN 'J.K> X/>5װ戰_1\sS֪(c[DzϖfME?:RA5W. y.W-;3Z#]"^\k5n> Xm4n^dέQ ߌ(kLi}mIԫr.$SYTͶqFň37buC?_f]ˏ dPh ')nNG)0 ߱i_gHUsX)p;`kMt:yDriF` +D7*XgZRek}gj?PMm\Y<TYPҬF~?쯶j=F2HMTJ uiDy23HJq`޿/x _5E6;+=RcΕ hw%<}{ nGA.k9Ϙ͛T/}9DX%|#^Lgdׯ@w\AydZp'ߥmaPB4F+(Ҡ[V tk@uI[ qi|O@M`1GLY)ա+0l _U RbTG:& ' g ZS1OgU`GW#:R<`o8㆙ X!<4R; IW:Jc=yqTpk۩jdOUd̃,H* ?z#K&Bd JmK@([Zy9v ?\Dخƒy[+Q/JZ%9 uy2)O?h9bF9;#U8ώGMw>Ӧ<۶=A4LI>:h2\+g.ӻa(-hD\Y|wo uo`V?u|[MW-Fp^:}X)Af!^x]J^Һé Nj'7aXf9aPzݏ-f2U}4Gt$yk e^_̔kV_a>~:T Fh[ULIl U|a컗\ȫ$:sѮYJ.T 7K _FDfJ=@j Bk!1-'fч^9f7@q3 >Zڸ)PT3#Oɉc R p-B΃ci~/tv?W!::[cU$?x >N³Je;>}+ƵoDϑ.uaҎi}#vi* s> ^(/U$.%+KWȜe!*[6MO\9  6C6{pʼ^]6$q5QIB8U4O3b/rGLpyx@PL6 ń %C/s2!2=V 9nhHKBMޏN _,zDCmza4>$:tk!颺#KW#5YtJلټ_åb0:02n鞫·sX}H1c@o sdl^*ǫ=؄p-)7JKT1"Han^p#.o(}APڕ.@{"A Dz jsqk a؃'UEp_ Z-!( rH`e*YT~BD}k>Kr҈kU߀ 3$^'!vAcr8flɓYY]=7xi#̗Zn|y52ra:$?JDƇpWJF dYYٗ>ef)3f9=M i?yJG4ڒ#8®Z3.24T,`tM|]v$󴫩FO|Ko{8b ߎO+(, dה-s0 G4I/=9 Rj[Ay)g l@&t&Ƞ4XXҮs')T :Y~-r )˩5_D_{0,[ԍr9iYQ-" T)g1B[T8/rcP ? ;xz!рwtj O-jWJXH'ÏGےq(k0[9b)+j{BLN?4嘞:@Y.$1ZV6|74ZI9y*Y bX>1-{Ŵ8p J(: PS7ZE}%6T  AݩGK[W`.EctʉK7v?ml p}Ǧ꽰,/Q>w ̔pN2 @]q2C$bQGf-/ߗ>UUD䱺qr a^EǏrp ^vxIve-HF RDL|q$&AOJRەֶ:ȥB! 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Vjc.H2ŨFkvWJa َ .DXCr$徠QiȤNY̫+?l0PG\ CQ|BuA `]_?_2P5IvƊLwϏ la Rg|IMnvoU((1RoaƟ&% r2= z:dfڪrUaho[4waEoSu`:e0~W {V*1 w%u>KMLVk}8?vrEP$~Xj61yU%: yvlH* s{ÃlP nTP5@+, NP$2&ٙ}}4-|@8 6,H4lj%2IVЗdkFk#_XVU58РYmqS/ J'L]O:d1/mf 7fR;4dn>j9T7Hg>^Ўɯ~N(%n7YRS>auw ZVYMP}(0ǹ Z$)V#$lXA1Pӡ-3$c~2gվ:?rKwY*o:y86cʶ}3Ƌ~UW aGL_ &~M@* ?BbooEU:҇aza)JJer] xbJۗO0"۰2IA&r#0WVYц^=;,HQڵVGL$IMީb52N :*|gJ34T0EfQt[8_#')-5d?cxCy`m"6}Ih~?1UujtJCF.( *ǏjāasbRTQ$7%)QU/9 }j?s1v)=jJM+Uc҈ ,kОU!(HQ47R"@xKr0RQyUez Z-Fu|yh͖7~?6ALÇ 2k|W zh r *2TҢ$q)HlƜ%eP=Q\r\sLncП/XgpWGlU8壅{ef!+J3g3B9)Ϧ_SLf;m}qEAi:kc42cW]̰ +Ln"+S-.v t]D%hΨVt3s)ŮrJ؊CcGo"-RPO U!wG $vcQd\[ ffsJO1|r .㳶/P[tf[X;}I M.h}_8΀UW`۰" A܋ʷ+ *>r'=GYFyZEQ˹-罂9 Ys5 ֫Vl`@_f%,ƻ]!շ(mK;Deb@z BL,JpWYRP z`W| BXT\E LK߸\wukCi5 fyHiFp3cr_#%QNjRӻ[W#`RsFrY3 a1Tf{VP~NL X?O%=RxcEQ%0Yg{_;&XJY8O"fpc 6TޝtƂQ+FTM~V (Z]<OXx~+JmadO' 믿GAWR֬w^ |Q?\!v~c;x+2GLQG8 Tn֊J?d#o_x(3Gpm󫛰#DDN:Bz? ~qƳ$KA+9'BR9]Tq\Y⎶(we}P)HА !*Ѡ瑸{VͥB|0* N;I1]_y H͟\ )^ ^R,Bc#绖E(\Փg 2_sw(ot/Z]4!g Kcu>#]b5_WX3E 2D ۻ{ qCyiޔ^(۸|2mm}ǘT#$!o,)dΩTPX~Zyb)FĴUrx:F/&~P͠_%#4'\\"V:R[ iJʈBPK-]ތJOq{ǹ͞S#FP%{[Uy*-_০λ*YKֹ0<#yu@z[ UɈP-jn$˅IJ8fZ~JY Lۣ4)4k;,4=dX N2f.alV kB&G^3xbU NI‚άSRj^kM2$Y;-:7z hHTd7@P;TS>йú03y 9U=F-Ҡk+dVOn endstream endobj 345 0 obj <> endobj 346 0 obj [347 0 R] endobj 347 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 607 0 0 838 0 0 cm /ImagePart_2088 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 423.6 718.95 Tm 99 Tz 3 Tr /OPExtFont3 11 Tf (3.8 Conclusions ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 99 Tz 3 Tr 1 0 0 1 116.65 676 Tm 112 Tz /OPExtFont3 15.5 Tf (3.8 Conclusions ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15.5 Tf 112 Tz 3 Tr 1 0 0 1 116.15 641.7 Tm /OPExtFont2 11.5 Tf (In this chapter, we considered the problem of estimating the current states of ) Tj 1 0 0 1 115.7 618.899 Tm 109 Tz (the model in the perfect model scenario. Based on the Indistinguishable States ) Tj 1 0 0 1 115.9 595.6 Tm 106 Tz (Theory, reviewed in Section 3.2, a new methodology is introduced to address the ) Tj 1 0 0 1 115.7 572.799 Tm 103 Tz (nowcasting problem. Our methodology involves first applying the ISGD algorithm ) Tj 1 0 0 1 115.7 549.75 Tm 106 Tz (to identify a reference trajectory which reflects the set of indistinguishable states ) Tj 1 0 0 1 115.7 526.7 Tm 111 Tz (of the true state. The ISIS method is then introduced to form the ensemble by ) Tj 1 0 0 1 115.7 503.699 Tm 112 Tz (selecting the model trajectories from the set of indistinguishable states of the ) Tj 1 0 0 1 115.7 480.399 Tm 105 Tz (reference trajectory. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 105 Tz 3 Tr 1 0 0 1 132.5 457.6 Tm 106 Tz (The well established 4DVAR method is reviewed and the difference between ) Tj 1 0 0 1 115.45 434.55 Tm (4DVAR method and ISGD method is discussed. Applying both method to Ikeda ) Tj 1 0 0 1 115.7 411.5 Tm 111 Tz (Map and Lorenz96 Model I, we demonstrate that the ISGD method produces ) Tj 1 0 0 1 115.7 388.5 Tm 106 Tz (more consistent results than 4DVAR method. This result comes with no surprise ) Tj 1 0 0 1 115.7 365.199 Tm 110 Tz (due to the fundamental shortcoming of the 4DVAR method, i.e. one faces the ) Tj 1 0 0 1 115.45 342.399 Tm 109 Tz (dilemma of either from the difficulties of locating the global minima with long ) Tj 1 0 0 1 115.7 319.35 Tm 108 Tz (assimilation window or from losing information of model dynamics and obser-) Tj 1 0 0 1 115.45 296.299 Tm 112 Tz (vations by using short window. The widely used sequential method EnKF is ) Tj 1 0 0 1 115.2 273.049 Tm 108 Tz (reviewed and discussed. Comparisons between ISIS method and EnKF method ) Tj 1 0 0 1 114.95 250 Tm 105 Tz (have been made in low dimensional Ikeda map and higher dimensional Lorenz96 ) Tj 1 0 0 1 115.2 226.7 Tm 112 Tz (model. By looking at the ensemble results in the state space, we find that the ) Tj 1 0 0 1 115.2 203.7 Tm 110 Tz (structure of the ensemble obtained by ISIS method is more consistent with the ) Tj 1 0 0 1 115.2 180.399 Tm (model dynamics than that of the ensemble produced by EnKF method. A new ) Tj 1 0 0 1 114.95 157.6 Tm 105 Tz (simple evaluation method, &ball, is introduced to evaluate the nowcasting results ) Tj 1 0 0 1 115.2 134.299 Tm 110 Tz (of both methods. We find that in both Ikeda Map and Lorenz96 model experi- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 110 Tz 3 Tr 1 0 0 1 307.699 52.95 Tm 75 Tz /OPExtFont3 11 Tf (61 ) Tj ET EMC endstream endobj 348 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 349 0 obj <> stream 0 ,,AEEb7 oF$%?'ܧ3ufoylԶUb׃\67@~̌\ H '20Um(@,.PuG2@yk8ZV_IC/?Ū-LfXY蘵o?^(i? 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Although the DCEn outperforms the ISIS ensemble, we found the ) Tj 1 0 0 1 115.2 584.549 Tm (ensembles they produce have similar structure as both methods produce the en-) Tj 1 0 0 1 114.95 561.75 Tm (sembles that reflect the dynamical information of the model. In practice, DCEn ) Tj 1 0 0 1 114.95 538.7 Tm (is computationally inapplicable while our IS methods can be applied in both low ) Tj 1 0 0 1 114.95 515.7 Tm 101 Tz (dimensional and high dimensional systems \(53\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 307.699 52.7 Tm 75 Tz /OPExtFont3 11 Tf (62 ) Tj ET EMC endstream endobj 353 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 354 0 obj <> stream 0 ,, jeܰ*VSӄ>]1YI>;ZO+'=Alp0H\2YQ8ҞL7(аMi+Ic* kb_d!6ӲAݤ9hE)n|?nldD(뺿W282]̗_~XH1Lp%ˊl:c"\J'm6G{ ;,͵kD2X H޽@;Qv7l7 (IoPG,DZ.M*`oj[χ0 xQidpMdT:x<m;[_~^ ~gk"kϪKaYHLS*q46EY3MoA+ ƒ(Vs (LlirO P07߿l՝꨽2Y,6mJ1i#`-,EHHr *W)Yl&X+tæTl`[͞x]{Uؚq~gb䖴/o:.ꥁ#6S(<_I b^-Ȼqov)4W,ƴysr5$)HH>EfM]e8|bh$ (@2t$Aߛ.Dһp[-a;/ӮO8'{wBҍh E,xjyATპKZ\-SKu,+KYO_Fv;<"ҮGV:yK *"KZc8ϕ.a8߫46H |e=d+ӱh[rr[AJm8yU[N4-g\wmSw-: p"h'q}MׅAUfvvX|I>io>  fD,zȾq2iO'Kgtπ074MR@f3ڇC'=`x1Φa8P~Gb|f-S2D)x-(>r -ԙǁI.H _(gX/Ǯ,+sQW)-r3DenMJnNm6feH yke(2: Ar7ian5y-_ /O`I4IF+eD"Q>A@)0vfO;s'0.Ifկ#i;eP^vI왛zKq>eXۀeikYQU~ӣ@xF [."<+z:[{:v ''H4|0 #" *7Kᴖ~I; RT)#. fT>GUə8>ni0/H;wЗy2wq{=D ;(MpT dDS&ÍFlfhTIU%:-`lz oFa3YF!G??rS@` c3P:alH;B~@X͓XoXBuSd胷`vUI~Æ ӷہPYs!Ɣ+r\*8rR$a{ӲBO:2ǐryM(8^sEHǚ_:_εjZ),KsF˅E]I'\WU71ܡke'OEwa'6"i >å:󵍓s%8s 33o,.ԩ8P?].+Ԋ\#m7ukV$pjَS.`1J,$R%󵘽Gr ̶hK00K x~ Ylkn)TO0%"?T`+!Os4<!{Z7Y+̀(^d-=fr.jm%$͊W_ V?%\‡D;ČSF:vFKFiX<V1y+!:DU vmk1'#{T&? gEGGScMַń@8>XµK|+(Κ?ɄSY/^*6s 8ZB Eܟ@vuhW5`i>|-vg iM὇3~j+0*BtW"̈P+/7n{bW)r3ԡXݜ3GdTB67 ƛa/L?%#70]eeR lt89n08C%yQ1Evט?QNFɧv7넉1ЕOfц/K5ƹ`. ii8=JDE*"uߝz# ,kuXMGnX? g7@OOЏҵqEܹY ^ᇊᏢT` XO%RgRہ`ص+,TŲq#e_Kg`˃ч̞vYY(yd3Wŕc᱉SV ;~P*lzR wyqY8JS* s*kO8t6zBٕavҖ+3G\%m4PCaC {!z>qJo Ź" ooNס,|4y%{4x u!7ae]y x2\'  :'Ãx#{5 oX -ʑ$lb,W]*F?D<6j`.MKTyB%]m['Ὤ|y9!Ygy`|\ O 6̊۸m-qz,2 tVx +6&Pjl˓X*'<=2Tm՝I3LCT4#(@W *VD} Iw2/k&|FuBY^;1wҴ7/ AjzRP}4bNM+QΡsAġ5:`R{o1"l%sN77ӼVJقJP0snjKB\xC/*N"s`U+P@ SˌA`}} Jha[[rP/o5kc0YhcŇ eI~6abE]4J :9˄Q:{&'hEv0Dc73.f;Ґ`~c5BRhT8½y1N.doh9E@^iCkFq׾9ڬ<rU; h H\n&Zr<;>Dޫe!*> u;5M!Nq?-`EMR_ e"y DT24{7@GrdxހhyVv 0>:#ZgF/߀ [X0]vv˭x$ _Q#?)6[kh|[wPKh3GM((ċ.ʿ%ĝTQk(9hUكŸHɿGӱImG ZG xϻx%?D4x0aK endstream endobj 355 0 obj <> endobj 356 0 obj [357 0 R] endobj 357 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 605 0 0 839 0 0 cm /ImagePart_2090 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 115.2 581.7 Tm 106 Tz 3 Tr /OPExtFont3 22.5 Tf (Chapter 4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 106 Tz 3 Tr 1 0 0 1 114.25 515 Tm 107 Tz (Parameter estimation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 107 Tz 3 Tr 1 0 0 1 113.75 453.3 Tm 91 Tz /OPExtFont3 11 Tf (In this chapter we consider the problem of parameter estimation of deterministic ) Tj 1 0 0 1 113.75 430.3 Tm 94 Tz (nonlinear models. The future evolution of the nonlinear dynamical models de-) Tj 1 0 0 1 113.75 407.25 Tm 92 Tz (pend strongly on the initial conditions and parameter specifications. As forecast ) Tj 1 0 0 1 113.75 384.199 Tm 89 Tz (errors of nonlinear models will not be Gaussian distributed even if the observation ) Tj 1 0 0 1 113.75 360.899 Tm (errors are drawn from Gaussian distribution, tradition methods like least squares ) Tj 1 0 0 1 114 337.899 Tm 94 Tz (are not optimal. Methods have been developed to address the shortcomings of ) Tj 1 0 0 1 113.5 314.6 Tm 92 Tz (traditional methods, for example estimating model parameters by incorporating ) Tj 1 0 0 1 113.5 291.549 Tm (the global behaviour of the model into the selection criteria \(64\). Two new alter-) Tj 1 0 0 1 113.5 268.5 Tm 88 Tz (native approaches are introduced in this chapter within the perfect model scenario ) Tj 1 0 0 1 114 245.5 Tm 92 Tz (\(PMS\) where the mathematical structure of the model equations are correct and ) Tj 1 0 0 1 113.299 222.2 Tm 91 Tz (the noise model is known, but the true parameter values are unknown. The first ) Tj 1 0 0 1 113.5 198.899 Tm 94 Tz (approach forms the cost function based on probabilistic forecasting, we call it ) Tj 1 0 0 1 113.5 175.899 Tm 92 Tz (Forecast Based Estimates. The second approach focuses on the geometric prop-) Tj 1 0 0 1 113.299 152.85 Tm 91 Tz (erties of trajectories in short term while noting the global behaviour of the model ) Tj 1 0 0 1 113.049 129.549 Tm (in the long term, we call this method Dynamical Coherent Estimates. Dynamical ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 306 53.25 Tm 75 Tz (63 ) Tj ET EMC endstream endobj 358 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 359 0 obj <> stream 0 ,,K77b7 Bwѭ_-=M2&Y ϥl%-z~{[B"GiCKa+}]3;޽7h)# bֿq dyY/guG]07g\$LN%0\CIَ9%II]5+h2[s:[1H_ ̉:VUǼ>-9a4ր=DU|vHCf%50 QMbCT~!؇Ҁ?Os'b$0b' 3"9:%7wLJe)VgwF:.ܚCwن>N.hps*1WCp!b2}LM&m(/g"O;QD˹$dbx~gyk ;HX@fHx++m)2D^TOӬ|,b a-΁d{kGxw(̓Qۦ4wv#)U)(DHuNt6oϲ4:nd3fT7!Mh=kSQ\gxLiwy.ӌO`/MV9C1˩4&Jdw/GųS13'"gSɠpbu)MOh4Ͷ9fӵʁIA̅V7Be.VKkf#:V%; ;w΅lm$,ޡ8‚&j+%U wm>w`CcT.:2>a 3y( aEx {eWĝǝl fbKؕ|p>[( %m?Ez~Xq.r"&7.$paOX7KߢbhWU=Ԧ'C/W|"nF01أm4INKᙀw <6״BnY>f==sERA:_0 =_ RHdED J5/vq^Nwyа,>&د]I]ii>g{oq1.hs mС3邽;Y^XoU)bLQ{o"?hLbRY-RnRCZz8*y, G3wy+.1N@q$?9}S UxS"W(ح) ^7#5I|A9g`(I^x`iD|'ql`9kG"ڌŨ$BӷfZҚ:]tȻS8E6S*lIW?٤{2-% !z.[옻0;nO`˭RXeb4Pބa3H{EHÉ^bT1gЫoVo7w',|`k (JUdAARg{8ߎ 6ZDl?\rr?bG&h%0 G9]ͩڷUC"Z;|i}??֝'|r$LZ@NAsuxeγ{g;mFaXo' O&_5w^}vtg8@'N ;4u05/AVڍkErM g`:Ќf,'C+Y@c:T$]{<"(5E-84uőϔvY9u`1sr!9d6Hg#Lݹ B˖=cלujҊΖTҀ, 2=KyZ&7,6f&&*R.;tuU]ԚΕ&y6Zi hm.sRL^FZpӱH+5&+u9Н>G6.Z@6\cw/r\>.!j XA/?h]/RV(~?fP\vmX]ܳy$쬨; ~YIy#o1; ^CZAxaY^`QCP[iyG| %xۍWD?hoQ*ܖܰW%=Gʑ",b@%)/fazWvo_@jC ]l, JiS NJ @^)[7J=^[ůݯ0}̸FJ&F #jj\|.b|SQF50nJYNjᙏh;Za&)@Z lqf`B r\$m$eP2Sb3n᫣hWʼnz$ZT )a]We*{ԑzW';"FU a'ȃ.40GF!B1__̈L쁙Oj4uUq=epaSjZ`ÄfcA+b2I;'/|ߛ\Cm}E*jpdk7z2VFY@#()D1FS6}uG|K lj> ,U$نʴ&vj&XuϦ̘-<ҐWy+w;jmC{hܸbW\)Z %O7E1)̛[tHc.m9%*9ᦳٻ>һΤS‚i!͟6?3<+mA MլGz*l e<`7Ē5_=-|e= M(/ŞN~nfU\q,;D?>WL։9DJԔ+IV ㆷ毣":gyo귣P4[&0>E߬6̙6*Ҍus_Q_& g"N7wz 66P^NLK^Bhj6!G&ZC!UysLp_׏gLAr`^E3uZ=5~P?cc(DM]Cb;f ɓ֊'*<0-3w9cFX-ŕOs.-| ($nl0t5M+e?>ư'$}1*"6:>7Ђ-ܵi BRmQ6e7$;VW_Q|5f[s屆3~B`;jt7U%G#Wo[zk>tyj-1L|:to$t8MwEkVJ`#q0kI6WuHae ;С]A3 :%&3SL~ރF4_50K9>ն'|"=Q)"in uAhSl@^T[zr EA/&&=.oOiLI5wSPsx!~^t&3`#pi-Ym5I4d`hv6n'1&`oHӉKKMP`/`  5 Ɉ'à YmU" WA}8σk/'B {^G`,Q0h&g>J]^wbka‰?m&I`5~`iVJdPrTIS?V?Ej6ԜZAPGÀ"eb>2nͺ𨈱m\WF=1qT_wRddYEkvb VG 9vpIzot:6w4 8՟9A g޹8՚xNMy0"ǯfBb- $&łVڽw( 7SBˊM;=8eCf{ gp8)q+:m{-8("8D5Pdِݷ _ x 2T  GkBif^VpA*i\lpBq[}-4r.d 垛1osp4bOK+uaVQz[RG7o?$ˋqs<ZسY+zS[D-M혪|_Mm2?<܇pGEl@jM$U0[nsаsU,8LHX7:9)z{s&Ԡ7/ς'w-r1HbڳDXuCRjLLyfPJ@` V Pz0^(&a,N?;moxĚ[|^2wvm/ :0JzHr,,|nHtNS~3yӅ'|.ˢ ѿ,8_hi8NuHBSRN'֪O,a v~5aVm!1j$;% Q緸i|.k^BA@a,6H}bYq]"|SN DD#cv b 4Z=V:hn ~J±Icx(*XǗF?x[]` 1|&?O`f[o'z`vճq,^aE&&K`K#G_7zHT(gc ;FBՋN{{/#p?Y_mSB4WJ〓\*ƎLTb~f9d>pꌉ!)%Ii/Finűm#=^ ,uNfxF%GL`)Ep 9AEë.8.i_jc>ٰ'c9n%ժ"9{Nҝy+?&'ߋ~̰ҙ;B^柴7#jLg'HBZb#޺Fdz,yCDX=01~(jYA5q~ _h/ɬZNUM3]qݏWI{@WMNg_}dLѓ𶈴w KQfH">!'<E]pDȖ.uy ;^:+ NZJ΍/ɡhnəZ fm~$--MX5hM*ֱi"޵6&[r/[xռy%XdY,NX:u!^|;?iN2m23,!N. j?YlE(zng1'AL{;H"$%cCN!emvۿhIc| S^AصQS?(N ablC M1`ʹxz'(c*ȬmsUp-'X!ٍਲ,54L4ܯqp':z3g98te@=ji ]NC6dY5,h2 Dl;m\RZh銨@Ve$ 塟||Ʊmac@ @DZ0>B=73O$֣֞N,X67Ѵ,ԶtRŠTw%/9<"0FI#SQTT,/cbҞU$o-j ߍApv/pdL~=U}nMSUbYKBw1jc|A+% ɘ?rr;gmz^I;qa> endobj 361 0 obj [362 0 R] endobj 362 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 605 0 0 838 0 0 cm /ImagePart_2091 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 289.199 718.95 Tm 104 Tz 3 Tr /OPExtFont3 11 Tf (4.1 Technical statement of the problem ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 104 Tz 3 Tr 1 0 0 1 115.7 675.75 Tm 92 Tz (Coherent Estimates is also applicable to the case that only partial observations ) Tj 1 0 0 1 115.2 652.95 Tm 95 Tz (are available. We will first define the problem of parameter estimation in Sec-) Tj 1 0 0 1 115.2 629.899 Tm 92 Tz (tion 4.1. The traditional Least Squares estimates method is then described and ) Tj 1 0 0 1 115.2 606.899 Tm 90 Tz (discussed in section 4.2. Forecast Based Estimates and Dynamical Coherent Es-) Tj 1 0 0 1 115.2 584.1 Tm (timates are presented in section 4.3 and section 4.4 respectively. Our approaches ) Tj 1 0 0 1 114.95 561.049 Tm (are compared with Least Squares Estimates and the numerical results are shown ) Tj 1 0 0 1 114.95 538 Tm 89 Tz (on several nonlinear models. Fundamental challenges remain in estimating model ) Tj 1 0 0 1 114.95 514.7 Tm 91 Tz (parameters when the system is not a member of the model class. Discussions of ) Tj 1 0 0 1 114.95 491.699 Tm 96 Tz (applying both methods to the case that the model structure is imperfect and ) Tj 1 0 0 1 114.7 468.649 Tm 90 Tz (defining optimal parameter values are presented in section 4.5. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 114.7 416.3 Tm 108 Tz /OPExtFont3 16 Tf (4.1 Technical statement of the problem) Tj 1 0 0 1 449.3 416.3 Tm 44 Tz /OPExtFont3 3 Tf (. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 3 Tf 44 Tz 3 Tr 1 0 0 1 114.7 381.75 Tm 90 Tz /OPExtFont3 11 Tf (Suppose the evolution of a system state x) Tj 1 0 0 1 315.85 381.75 Tm 72 Tz (i ) Tj 1 0 0 1 318.25 381.75 Tm 98 Tz /OPExtFont3 10.5 Tf ( E ) Tj 1 0 0 1 333.1 381.75 Tm 93 Tz /OPExtFont3 11 Tf (le is governed by finite dimensional ) Tj 1 0 0 1 114.25 358.699 Tm 90 Tz (discrete deterministic nonlinear dynamics: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 288 314.799 Tm 78 Tz /OPExtFont6 14.5 Tf (= ) Tj 1 0 0 1 295.699 314.799 Tm 1211 Tz (\t) Tj 1 0 0 1 339.6 314.799 Tm 33 Tz (, ) Tj 1 0 0 1 340.8 321.649 Tm 2000 Tz (\t) Tj 1 0 0 1 488.149 316.5 Tm 88 Tz /OPExtFont3 11 Tf (\(4.1\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 114 274.25 Tm 93 Tz (where the system's parameters are contained in the vector a ) Tj 1 0 0 1 422.649 274.25 Tm 105 Tz /OPExtFont3 10.5 Tf (E R) Tj 1 0 0 1 442.55 274.25 Tm 37 Tz /OPExtFont2 10.5 Tf (1) Tj 1 0 0 1 446.149 274.25 Tm 43 Tz /OPExtFont3 10.5 Tf (. ) Tj 1 0 0 1 455.75 274.25 Tm 94 Tz /OPExtFont3 11 Tf (In the Per-) Tj 1 0 0 1 114 250.95 Tm 95 Tz (fect Model Scenario \(see section 3.1\), the model state space and system state ) Tj 1 0 0 1 114 227.899 Tm 98 Tz (space are identical. F\(x, a\) of a model is known to match that of the system ) Tj 1 0 0 1 113.75 204.899 Tm 90 Tz (exactly, i.e. ) Tj 1 0 0 1 176.4 204.899 Tm 84 Tz /OPExtFont8 16.5 Tf (F = ) Tj 1 0 0 1 202.099 204.899 Tm 98 Tz /OPExtFont8 13.5 Tf (F. ) Tj 1 0 0 1 220.3 204.649 Tm 93 Tz /OPExtFont3 11 Tf (Suppose the value for the vector of model parameters a is ) Tj 1 0 0 1 114 181.85 Tm 92 Tz (unknown and must be estimated from observations s) Tj 1 0 0 1 379.899 181.6 Tm 72 Tz (i ) Tj 1 0 0 1 382.3 181.6 Tm 95 Tz ( of the state variables ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 95 Tz 3 Tr 1 0 0 1 113.5 158.799 Tm 91 Tz (Assuming additive measurement error m yields observations s) Tj 1 0 0 1 421.449 158.549 Tm 72 Tz (i ) Tj 1 0 0 1 423.85 158.549 Tm 116 Tz ( = x) Tj 1 0 0 1 447.1 158.799 Tm 66 Tz (i ) Tj 1 0 0 1 449.3 158.549 Tm 96 Tz ( + m, where ) Tj 1 0 0 1 113.5 135.299 Tm 99 Tz (m is IID distributed. Without measurement error, l ) Tj 1 0 0 1 389.5 135.049 Tm 85 Tz /OPExtFont8 13.5 Tf (+ ) Tj 1 0 0 1 402.5 135.299 Tm 89 Tz /OPExtFont3 11 Tf (1 sequential measure- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 306.5 52 Tm 79 Tz (64 ) Tj ET EMC endstream endobj 363 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 364 0 obj <> stream 0 ,,jj"^%#)qQ :gA*Hf/rG`'n !9Q/烉/=[Ne>4~uP1e|nh ݃]]! 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N=UHh^R F iX ໧@ ta&qNI4s|'1 @iS=Ls&tgY;: "w]u]eqRQҒ"m<!D4lW8J] ]@k'q1rǦc5ؔk|fy4Fa*1^V,5JYeG.2]\ޥD>Gs?F\dTmp(0 Za`Ϭ0_?~*̨I3irsnS-9eza04]) d|dkP5%Dgzp=Q38 @LJ 8]#x&_<Ȓ*@r0'zj<1[ endstream endobj 365 0 obj <> endobj 366 0 obj [367 0 R] endobj 367 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 604 0 0 839 0 0 cm /ImagePart_2092 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 354.25 720.2 Tm 102 Tz 3 Tr /OPExtFont3 11 Tf (4.2 Least Squares estimates ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 102 Tz 3 Tr 1 0 0 1 114.95 677 Tm 88 Tz (ments s) Tj 1 0 0 1 153.099 677 Tm 80 Tz (i) Tj 1 0 0 1 157.449 677 Tm 41 Tz (, ) Tj 1 0 0 1 158.9 677 Tm 1182 Tz (\t) Tj 1 0 0 1 200.4 677 Tm 79 Tz (s) Tj 1 0 0 1 205.199 677 Tm 89 Tz (i+) Tj 1 0 0 1 214.8 677 Tm 91 Tz (1 would, in general, be sufficient to identify the true param- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 114.95 653.95 Tm 94 Tz (eter a \(64\). In the presence of observational noise, the true state of the system ) Tj 1 0 0 1 114.7 631.149 Tm 52 Tz (3) Tj 1 0 0 1 117.849 631.149 Tm 48 Tz (-) Tj 1 0 0 1 118.299 631.149 Tm 93 Tz (4 can not be determined precisely even infinite observations are provided and ) Tj 1 0 0 1 114.95 608.1 Tm 91 Tz (the parameter values are known exactly \(48\). As we will see, this also makes the ) Tj 1 0 0 1 114.7 585.1 Tm 90 Tz (problem of parameter estimation much harder. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 131.75 562.049 Tm 92 Tz (In this chapter we focus on addressing the problem of parameter estimation ) Tj 1 0 0 1 114.95 539 Tm 91 Tz (in the perfect model scenario. Our aim is to extract the information from a finite ) Tj 1 0 0 1 114.7 515.95 Tm (series of observations given the exact noise model and the functional form of the ) Tj 1 0 0 1 114.5 492.899 Tm (dynamic model to determine the model parameter values. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 131.5 469.899 Tm 95 Tz (We never identify the true model parameter precisely of course; rather we ) Tj 1 0 0 1 114.25 446.85 Tm 89 Tz (introduce two methods for extracting significant information on parameter values, ) Tj 1 0 0 1 114.5 423.8 Tm 92 Tz (one via evaluating the probabilistic forecast performance that they produce; the ) Tj 1 0 0 1 114.7 400.5 Tm 91 Tz (other via the trajectories they admit. And how to report the parameter estimates ) Tj 1 0 0 1 114.25 377.5 Tm 89 Tz (based on our methods are discussed. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 114.5 324.899 Tm 107 Tz /OPExtFont3 16 Tf (4.2 Least Squares estimates ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 16 Tf 107 Tz 3 Tr 1 0 0 1 114.25 290.6 Tm 92 Tz /OPExtFont3 11 Tf (The famous least squares method \(8; 27; 56\) estimates the parameter by testing ) Tj 1 0 0 1 114 267.299 Tm 89 Tz (the error in the forecast initialised at observations. The one-step least squares\(LS\) ) Tj 1 0 0 1 114.25 244.299 Tm 95 Tz (estimate gives the value of parameter which minimises the least squares cost ) Tj 1 0 0 1 114 221.25 Tm 86 Tz (function. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 86 Tz 3 Tr 1 0 0 1 305.5 193.399 Tm 105 Tz /OPExtFont4 8 Tf (N -1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 8 Tf 105 Tz 3 Tr 1 0 0 1 256.1 177.299 Tm 120 Tz (CLS) Tj 1 0 0 1 276.25 177.1 Tm 99 Tz /OPExtFont3 11 Tf (\(a\) ) Tj 1 0 0 1 289.199 186 Tm 1176 Tz (\t) Tj 1 0 0 1 330.5 178.75 Tm 101 Tz (rr) Tj 1 0 0 1 340.55 178.75 Tm 72 Tz (i ) Tj 1 0 0 1 342.949 178.75 Tm 71 Tz ( , ) Tj 1 0 0 1 348 184.049 Tm 2000 Tz (\t) Tj 1 0 0 1 487.699 178.75 Tm 88 Tz (\(4.2\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 307.699 165.299 Tm 103 Tz /OPExtFont4 8 Tf (i=1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 8 Tf 103 Tz 3 Tr 1 0 0 1 306.25 53.5 Tm 77 Tz /OPExtFont3 11 Tf (65 ) Tj ET EMC endstream endobj 368 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 369 0 obj <> stream 0 ,,kbbb7! fg(q>ڱ擳-IѤ;虰MQX$YG'6ul"nf+/Pǭ.FH'.4= o}й)ymLrcV*ڽ,{đt0Ȍ0E&f^ nыm ip:݈&p0ZQ"k #ҪwBܯr_sXKb{ WEpdoi$EӃ_*-eFHBg('IZФэ,)WR(ź![ K^s8,=u)pi-|CZklb(.rMpP[GlC K"@ k_s& pτ%!вi{W:籗sm؁~D?iqYđΕ(X-VUoT"vaFcRWa//Ec;7v[yzyYMaO9N(/1CË-75:#CB5d=>+SQd 2̱]B8S)˓U0 o F%RK`Y َ0$#+c"mxҀ8j1NW7fwamܘGs霒ץ?ãQxfV*SNEMtE9`FLj漄X؟3)O ȂLyXT"L,ZTH7Rg; ]?RA -SP4v_ȤteOXřT-qk(ow_OSa Z_g5^ Y0wfHhJ 9:jW*"ҽB^# 7'%Rp;(1\*񽖿7WD2*G$MR%̞ |%Kf-E[('{U".Ϸw?\g1%SO6 B,Dw@CeKW}$WK'*챖;ր/mBӧphQĠ6}=hӯ?68/:5^-Ջ IP8I[L⪠Wae6OHx{Tz)c-yna!PMԄD?C1״Fiȕ3 :g*(P3 Mjwȟ珏YEU$qlBH[">դڰڹ<@NuYaKf>䁅كM%6:Ӭ+.<끶>˖Pm~}sG:t.p3Rz4J\H k톥/yC{/mdgV%YכK8b\y8Cb2^qkrHhfࠣ &qzAf3m]`&P[-=ȘB,lOv)v׫ $$񋙯PW .kzA!6\@;A!-Tw4+k6\wLE XR<"]-&|+;#N ĝx6l=y9QMQD}ORrlmݎQ©i[12FhQm^af1J,Xi5xfٟ\t<}lO$$_ALf M/ tnaNq=I>3ˎ՚;n)ﲰ'Í."a  QaЉ]HVt(fq(^nzt)lOȞ-CF8>;qOH *=V䊳TI UzI:m֖ǩ&2`|07'?-siOwjqBձKDm}AVͻHkEof.l[3 jIu6ZKmsH"UB˴DZ<#p?AVRcn3xu[8֋j]q# h;2R~GNmp,\<2r3,^ y^g 6'#}nY nQ1)H ѥ#߹)XZEިt$uQ.hp5o&ԏd8k˞q.Iq )jĉe/c*rdg$hCjzlDpZ]"uyȭxzfѦpD#7%b9R |^:j^Eg\ t__svrCmXVu>_"~-iނ 2䜩9.Q^K,a|z+> D"ʹaw.[(#B >G=|b| ;򹖜>4pQ̻Q 2)wh q6%{U7/qN ]FD5_j-O* ǓOKƷz x\wŇ$>8ؕlzWbJ7.Τ8܂IFc9:sֻ>RU {s9ywDQs)mⒽݏKv:+]ɖw[DeB᧰IZ(nL#pc& V\MKYAdi}̈ތ[Ƽ`3vh[I(!#mm[9^S Tb~NvEJcdBFi!rٹ/XE͍[e41]fDA7 ZOV'%JȚcf6kIe8D bԴMzѡ!O5e磎[iDzZk|U^ά `?!Fډ;Hԍ"=.ص`ӱ􊙻彦Pv>H9癭6̪5<4VXmO.$R%^.ӆou< 2!<&)<`{Ƿ|>#:+ icM 6`,`&?7)E=F焕@fݕEL%9q?PsJ A"v_\=~F((>/ 7]1“X^mֵ]Nʶ)im|ͩ{ãʞF,GQ!h' "!,FZ:e4 1d'ľSb2؆:Oԉ~\/Uy>9 d)ډ<6QVzD&crF,[ vPX DpUHG)jv4Gޝ#"A&HA9a6< a!ڑqi$}gpBRV7KlNmw74E!f,WoCk  O'ÃQiw/)vvQq, sXB:]P|m%iH0ܠ!5XsVW ry{Ax~O>]ʘĿ$9bZJꙑDpqXqmr,"H hЁiPoe0 ˉEJi9Y L8j~n| yoTMsYʎ|3ڕ_O}>3,-b0rۯ#EcW$0DIV_?ZP@[p2hIaB4 k".rRZDPEݝYdIW-ҵD.C*`8%efdؙf,eSjZZ0)bDڻ}ϿD'x'aƭXey)k鰔&<@WS8;'E:^#Ex}-zYaK"ܫbki8_r{$#KQjY^e̝~Ʃм)ih:,y5o#དྷՁW21(4 楟b-/oiVfWn>k[~"B2eS>Uؒlrtf`#q@ӜNh{aodϣCצ0:㯬Y^Ra,V U=X E3YL&Xoz/?87 '܊0 >ˡ=.է!,V^=XrЎzR-9/x̑[mQ838I~`-~GCu]S%;'U_q/)ᰩ1:?RBI2cI=iHL$Z9bC(y')~|bNt MK  H3VUD"]BcW|%(Q&,^1Sk^  ȓjF77>k:!j-֢&޴)z7<{\m9S}=|lμfMȮՌʝ`K G MioޙƂ%yap^%ǧWYL 9)V숣*?[]0]?OPZ #1O.gjUh:*XA=R%V7^΃Pj+uO<\IF< c9I0 mwmt!CjS+,tnTFBT XnΞ]u0OAi\⮑S_ʜG3Nd|,j+[翖<2JPϒ{2'!7dPZ?Ӓ=Ś ^{̄yBP+5@|R)GY~+5d{-]<_X{UG@^8D Mp0uۄdxLDiR2p(DUBtU] 7,:3N73 Cx'E8P YHtd5s6"ϘUB t3b͈Ŵ:{^7@{2r_P$(9%2P2ssEږx%)%\6Կ;}/*r FdIK fV3P6i(:9/p̉N 4qGJGsM uF3Hw/[ƹ}$ F+o,f*a`P$J*Hj`Wnm^J\F}TZf0^+H~R[ L\4531q=*ƻ핼V|+)FؿJJ`>{ @is"ґ&ۿgGBFmO-T,%:' kW^<=-c]X@.xBV=Ԗk_6Qy|}8coKe†- i<r$=m9% *~d_ JWxWmKgƓ}5<"i$l޸(!KX Jœ,3abaVKQ\}-~+oiN v#܊a9Mx=YEq"jp9pa1Tơ̮TpuM>m&@u]&CR4?V\ endstream endobj 370 0 obj <> endobj 371 0 obj [372 0 R] endobj 372 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 605 0 0 839 0 0 cm /ImagePart_2093 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 354.699 719.5 Tm 102 Tz 3 Tr /OPExtFont3 11 Tf (4.2 Least Squares estimates ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 102 Tz 3 Tr 1 0 0 1 115.45 676.299 Tm 88 Tz (where ) Tj 1 0 0 1 148.099 676.299 Tm 105 Tz /OPExtFont4 11 Tf (ern = s) Tj 1 0 0 1 186.949 676.299 Tm 99 Tz /OPExtFont8 11 Tf (i+i ) Tj 1 0 0 1 199.199 676.299 Tm 98 Tz /OPExtFont4 11 Tf ( F\(s) Tj 1 0 0 1 233.05 676.299 Tm 85 Tz /OPExtFont8 11 Tf (i) Tj 1 0 0 1 236.65 676.299 Tm 50 Tz /OPExtFont4 11 Tf (, ) Tj 1 0 0 1 241.449 676.5 Tm 93 Tz /OPExtFont3 11 Tf (a\), the one-step prediction error. The LS cost function ) Tj 1 0 0 1 115.7 653.5 Tm 98 Tz (can be derived from Maximum Likelihood Estimate\(MLE\) \(17\). Assume the ) Tj 1 0 0 1 115.7 630.45 Tm 92 Tz (observational noise and the forecast error, i.e. s) Tj 1 0 0 1 353.3 630.7 Tm 83 Tz /OPExtFont5 11 Tf (i+1 ) Tj 1 0 0 1 365.75 630.7 Tm 97 Tz /OPExtFont3 11 Tf ( F\(s) Tj 1 0 0 1 400.1 630.7 Tm 85 Tz /OPExtFont5 11 Tf (i) Tj 1 0 0 1 404.149 630.7 Tm 92 Tz /OPExtFont3 11 Tf (, a\), are IID Gaussian ) Tj 1 0 0 1 115.7 607.649 Tm 90 Tz (distributed with mean 0 and standard deviation ) Tj 1 0 0 1 351.85 607.649 Tm 87 Tz /OPExtFont4 11 Tf (a. ) Tj 1 0 0 1 366.949 607.649 Tm 89 Tz /OPExtFont3 11 Tf (Given the observations s) Tj 1 0 0 1 486 607.649 Tm 82 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 490.55 607.649 Tm 71 Tz /OPExtFont3 11 Tf (, t ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 71 Tz 3 Tr 1 0 0 1 115.9 584.35 Tm 67 Tz (1, ) Tj 1 0 0 1 122.9 584.35 Tm 889 Tz (\t) Tj 1 0 0 1 154.099 584.35 Tm 92 Tz (the likelihood function of parameter a is then given by ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 238.8 527.5 Tm 52 Tz (1 ) Tj 1 0 0 1 242.4 527.5 Tm 1216 Tz (\t) Tj 1 0 0 1 285.1 525.799 Tm 75 Tz (, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 75 Tz 3 Tr 1 0 0 1 177.849 517.899 Tm 113 Tz (L\(a\) = ) Tj 1 0 0 1 217.9 517.149 Tm 91 Tz /OPExtFont5 11 Tf (\(27r0) Tj 1 0 0 1 238.099 516.7 Tm 34 Tz /OPExtFont3 11 Tf (_) Tj 1 0 0 1 240.5 516.7 Tm 92 Tz /OPExtFont5 11 Tf (2\)N) Tj 1 0 0 1 256.3 516.2 Tm 51 Tz /OPExtFont3 11 Tf (/) Tj 1 0 0 1 259.899 516.2 Tm 69 Tz /OPExtFont5 11 Tf (2 ) Tj 1 0 0 1 263.5 515.95 Tm 95 Tz /OPExtFont3 11 Tf ( expl) Tj 1 0 0 1 289.449 515.25 Tm 65 Tz ( ) Tj 1 0 0 1 296.649 515.25 Tm 28 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 28 Tz 3 Tr 1 0 0 1 341.05 519.549 Tm 75 Tz (s) Tj 1 0 0 1 346.1 519.549 Tm 95 Tz /OPExtFont5 11 Tf (t+i ) Tj 1 0 0 1 358.55 519.1 Tm 82 Tz /OPExtFont3 11 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 82 Tz 3 Tr 1 0 0 1 488.649 519.299 Tm 87 Tz (\(4.3\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 115.2 476.35 Tm 92 Tz (By minimising the log likelihood function, i.e. ) Tj 1 0 0 1 346.8 476.35 Tm 110 Tz /OPExtFont4 11 Tf (log\(L\(a\)\), ) Tj 1 0 0 1 400.1 476.35 Tm 90 Tz /OPExtFont3 11 Tf (the Least Squares cost ) Tj 1 0 0 1 114.95 453.3 Tm 96 Tz (function is then derived. IVIcsharry and Smith \(64\) proved that even with an ) Tj 1 0 0 1 114.95 430.3 Tm 88 Tz (infinite amount of data the optimal least squares solution is biased when it applied ) Tj 1 0 0 1 114.95 407.25 Tm 97 Tz (to the 1-D Logistic Map. Figure 4.1 plots the least square estimates against ) Tj 1 0 0 1 114.95 383.949 Tm 93 Tz (different noise level ) Tj 1 0 0 1 217.699 383.949 Tm 38 Tz (1 ) Tj 1 0 0 1 220.3 383.949 Tm 98 Tz ( for both Logistic map and Ikeda map. Figure 4.2 plots ) Tj 1 0 0 1 114.95 360.899 Tm 93 Tz (the Least Squares cost function in the parameter space for both Moore-Spiegel ) Tj 1 0 0 1 114.95 337.899 Tm 96 Tz (System and Henon Map experiments, given the noise level fixed. We can see ) Tj 1 0 0 1 114.95 314.6 Tm 91 Tz (from both figures that Least Squares Estimates systematically rejects the correct ) Tj 1 0 0 1 114.95 291.549 Tm (parameter value and from Figure 4.1, the higher the noise level is, the more bias ) Tj 1 0 0 1 114.7 268.5 Tm 96 Tz (in the estimate \(20\). The 911Թmethod fails simply because the assumption of ) Tj 1 0 0 1 114.5 245.25 Tm 93 Tz (Independent Normal Distributed \(IND\) forecast errors does not hold even if the ) Tj 1 0 0 1 114.5 222.45 Tm 91 Tz (noise is IND. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 130.8 203.7 Tm 71 Tz /OPExtFont3 6.5 Tf (1) Tj 1 0 0 1 134.9 203.7 Tm 91 Tz /OPExtFont3 9.5 Tf (The different noise levels in the plots are defined by the ratio between the standard devi-) Tj 1 0 0 1 114.7 192.2 Tm (ation of the observation noise and the standard deviation of the signal ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 9.5 Tf 91 Tz 3 Tr 1 0 0 1 307.199 52.5 Tm 79 Tz /OPExtFont3 11 Tf (66 ) Tj ET EMC endstream endobj 373 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 374 0 obj <> stream 0 ,,הb72usj>p p=TH]cϜ mJt{[m`:oRDZ8Kd_@Cso55 .,Gzl3Fukl7~v׺ 4Դi_:r:x׾ }CD kvc](&>(1QȵWoiX -ZI:a2h.῱=A*Lز01ts^:sLtcQ28Փ(3h >=9=FXҰ+ȱ"-[ ½;fQ<ާJULeX"#ie9 $0VBn˼ͅEp~nq)z`-x`p!G\:2O)ʙF=x kl[X)׶޵ K)\ثW.! ~Y#rHb`Gb!f"A\w\yH Fx'\ГI";]RvG7xk+V_5':Y/x7XF1gl^Q޿f/L &W;? 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(09 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 3.5 Tf 130 Tz 3 Tr /OPExtFont13 10 Tf 1 0 0 1 5107 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 5242 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 6418 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 6504 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 6566 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 6619 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 6682 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 6739 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 6826 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 6883 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 6941 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 6998 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 7056 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 7118 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 7162 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 7205 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 7248 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 7291 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 7334 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 7378 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 7435 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 7493 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 7550 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 7618 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 7685 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 7781 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 7848 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 7906 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 7978 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 8078 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 8141 3014 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 130 Tz 3 Tr 1 0 0 1 393.1 625.049 Tm 104 Tz /OPExtFont1 4.5 Tf (\(b\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4.5 Tf 104 Tz 3 Tr 1 0 0 1 255.349 615.899 Tm 160 Tz /OPExtFont12 3.5 Tf (07- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 3.5 Tf 160 Tz 3 Tr 1 0 0 1 273.85 536 Tm 132 Tz (01 ) Tj 1 0 0 1 278.899 536 Tm 960 Tz /OPExtFont2 3.5 Tf (\t) Tj 1 0 0 1 288.5 536 Tm 144 Tz /OPExtFont12 3.5 Tf (02 ) Tj 1 0 0 1 294 536 Tm 889 Tz /OPExtFont2 3.5 Tf (\t) Tj 1 0 0 1 302.899 536 Tm 150 Tz /OPExtFont12 3.5 Tf (03 ) Tj 1 0 0 1 308.649 536 Tm 910 Tz /OPExtFont2 3.5 Tf (\t) Tj 1 0 0 1 317.75 536 Tm 132 Tz /OPExtFont12 3.5 Tf (04 ) Tj 1 0 0 1 322.8 536 Tm 934 Tz /OPExtFont2 3.5 Tf (\t) Tj 1 0 0 1 332.149 536 Tm 150 Tz /OPExtFont12 3.5 Tf (05 ) Tj 1 0 0 1 337.899 536 Tm 890 Tz /OPExtFont2 3.5 Tf (\t) Tj 1 0 0 1 346.8 536 Tm 123 Tz /OPExtFont18 3.5 Tf (06 ) Tj 1 0 0 1 352.3 536 Tm 889 Tz /OPExtFont22 3.5 Tf (\t) Tj 1 0 0 1 361.199 536 Tm 129 Tz /OPExtFont18 3.5 Tf (02 ) Tj 1 0 0 1 366.949 536 Tm 890 Tz /OPExtFont22 3.5 Tf (\t) Tj 1 0 0 1 375.85 536 Tm 123 Tz /OPExtFont18 3.5 Tf (08 ) Tj 1 0 0 1 381.35 536 Tm 889 Tz /OPExtFont22 3.5 Tf (\t) Tj 1 0 0 1 390.25 536 Tm 107 Tz /OPExtFont18 3.5 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 3.5 Tf 107 Tz 3 Tr 1 0 0 1 321.6 529.5 Tm 110 Tz /OPExtFont18 4 Tf (a) Tj 1 0 0 1 324.699 529.5 Tm 108 Tz /OPExtFont18 3.5 Tf (nous) Tj 1 0 0 1 334.1 529.75 Tm 128 Tz (ia) Tj 1 0 0 1 338.149 529.75 Tm 82 Tz (9ignffil ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 3.5 Tf 82 Tz 3 Tr 1 0 0 1 205.9 696.549 Tm 102 Tz /OPExtFont3 11 Tf (4.3 Forecast based parameter estimation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 102 Tz 3 Tr 1 0 0 1 42.5 501.899 Tm 89 Tz (Figure 4.1: Parameter estimation using LS cost functions for different noise level, ) Tj 1 0 0 1 42.5 488 Tm (the black shading reflects the 95% limits and the red solid line is the mean, they ) Tj 1 0 0 1 42.25 474.55 Tm 86 Tz (are calculated from 1000 realizations and each cost function is calculated based on ) Tj 1 0 0 1 42.25 460.649 Tm 90 Tz (the observations with length 100, the blue flat line indicates the true parameter ) Tj 1 0 0 1 42 446.949 Tm 94 Tz (value \(a\) Logistic Map for a = 1.85 \(b\) Ikeda Map for u = 0.83 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 94 Tz 3 Tr 1 0 0 1 42.5 414.3 Tm 112 Tz /OPExtFont3 15 Tf (4.3 Forecast based parameter estimation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15 Tf 112 Tz 3 Tr 1 0 0 1 42.25 380.25 Tm 92 Tz /OPExtFont3 11 Tf (In this section we address the parameter estimation problem by looking at the ) Tj 1 0 0 1 42.25 357.699 Tm (forecast performance of different parameter values. Given the same initial con-) Tj 1 0 0 1 42 334.899 Tm 91 Tz (ditions, the forecast performance varies as different parameter values are used. ) Tj 1 0 0 1 42.25 312.1 Tm 89 Tz (An illustration of the procedure used to obtain the forecast skill score is depicted ) Tj 1 0 0 1 42.25 289.299 Tm 92 Tz (in the schematic flow chart of Figure ) Tj 1 0 0 1 228.25 289.299 Tm /OPExtFont20 12.5 Tf (4.3 ) Tj 1 0 0 1 247.449 289.5 Tm /OPExtFont3 11 Tf (\(Details of each step of the procedure ) Tj 1 0 0 1 42.25 266.7 Tm 90 Tz (are described in the following sections\): An ensemble of initial conditions is first ) Tj 1 0 0 1 42.25 243.899 Tm 91 Tz (formed to account the initial uncertainty. The forecast ensemble at lead time ) Tj 1 0 0 1 422.649 244.149 Tm 113 Tz /OPExtFont6 12 Tf (N ) Tj 1 0 0 1 42 221.1 Tm 91 Tz /OPExtFont3 11 Tf (is obtained by iterating the initial condition ensemble ) Tj 1 0 0 1 309.6 221.35 Tm 113 Tz /OPExtFont6 12 Tf (N ) Tj 1 0 0 1 323.5 221.35 Tm 90 Tz /OPExtFont3 11 Tf (times forward through ) Tj 1 0 0 1 42 198.549 Tm 91 Tz (the model for given parameter values. The ensemble forecast is then interpreted ) Tj 1 0 0 1 42.25 175.75 Tm 93 Tz (as a continuous forecast distribution by standard kernel dressing. In order to ) Tj 1 0 0 1 42 153.2 Tm 92 Tz (evaluate the probabilistic forecast in a more robust way, we blend the forecast ) Tj 1 0 0 1 42.25 130.399 Tm 94 Tz (distribution with the sample climatology, i.e. the historical distribution of the ) Tj 1 0 0 1 42 107.6 Tm 90 Tz (data. In the end we evaluate the forecast distribution via a probabilistic forecast ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 232.3 41.85 Tm 79 Tz (67 ) Tj ET EMC endstream endobj 378 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 379 0 obj <> stream 0 !,, 66kRr,[UES8If23{[Y^ivf1^j{L\ FyB믶$$ kJȔ'Z N2mNe/zb'ܵFv99 2f qCo|Wr7$q?p.nI][W倅+s|~d;]hauţ7CD!q #}:-X)4k8.*MAg,!BD❚s5=Nnu_fY ?Q{r(67J5Ůքs痖ݝoŽ$ZtQB%~9 o3dHRDɺ b,N0s{HuV{ܜ+ɆY^{P=V&z#ŮιNߙԻ頪)C5*pV"l +IF@wiI *z@T^sM lA^CC¦Դ@=4ԯ/ט _ M۞w?>G~1S*@xvQr"H6GZXԤ?&tqHhZVYWD ѺgFdimL\Ṍʋ x~@1g0p lN\ J^YئtsU0-KJH05DLQ5y9[+f nL9IJ6bN}frl/V<`I0'7yb;#إoy%w24^d+O%5C#LZA/M:l>,awX:+g4'IwRuP@ޑR^쌊_O[ dHT8gHgE}g@rwyŸ]E,V&]0&@{r&{I,Zat;7x vOm GhS&@i5(j]߹Z.zJQJnw!Ϛ"ڕ NnU@l/ħ|T^7 Ρ!] N9J>V,wHHly9K82|v#WڮWcނ3 G!4/_b9n&u"Ws"wbjS'ekÛB^8?8oNRMtڅT<N]w!@7,,(/a>;܏$WV!z=[:s.m0rhUy>/L 7,/<׬k! 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|>)/\^̶)} ׭>n f?؍cJN_Ҿ7>|ޮ8H>FPz?'xN6P-v+gRDRk{] %~q||EP)ɒ@#͓wS  f'a6Cf|$kCo,3m-7.͡&?Ry_s%js&#O )ZWN,q(GqJi\;}+%ʼg[CԪPl"W "ˌ^A LŌyN]hfB2R.]s45n$i 5ۼ _H@ lfy͟OX+HR]Rn#;M[]cz8/u@‡OR^R*[6LRD]I9*GlolvO' M^+6q'U;&k#"p VYMf\p𴕗\K3xLP>ێ KOHDԸ1茓}Jހޤ&$bZn'\[tSZÀ endstream endobj 380 0 obj <> endobj 381 0 obj [382 0 R] endobj 382 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 605 0 0 838 0 0 cm /ImagePart_2095 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 273.6 509.199 Tm 77 Tz 3 Tr /OPExtFont12 7 Tf (104 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 7 Tf 77 Tz 3 Tr 1 0 0 1 337.899 510.899 Tm 86 Tz /OPExtFont5 13 Tf (1.3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 86 Tz 3 Tr 1 0 0 1 396.25 511.1 Tm 83 Tz /OPExtFont2 13 Tf (1.4 ) Tj 1 0 0 1 400.3 499.85 Tm 92 Tz (a ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 13 Tf 92 Tz 3 Tr 1 0 0 1 455.05 511.1 Tm 76 Tz /OPExtFont3 11 Tf (1.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 76 Tz 3 Tr 1 0 0 1 129.349 670.5 Tm 82 Tz /OPExtFont12 7 Tf (0.085 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 7 Tf 82 Tz 3 Tr 1 0 0 1 129.099 631.6 Tm 84 Tz (0.084 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 7 Tf 84 Tz 3 Tr 1 0 0 1 122.15 603.75 Tm 77 Tz /OPExtFont12 8.5 Tf (2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 8.5 Tf 77 Tz 3 Tr 1 0 0 1 121.2 593.2 Tm 85 Tz /OPExtFont12 6 Tf (tn) Tj 1 0 0 1 122.15 592.95 Tm 62 Tz /OPExtFont9 6 Tf (3 ) Tj 1 0 0 1 125.75 592.7 Tm 92 Tz /OPExtFont12 7 Tf ( 0.083 ) Tj 1 0 0 1 121.2 578.799 Tm 70 Tz (cc ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 7 Tf 70 Tz 3 Tr 1 0 0 1 128.9 553.85 Tm 84 Tz (0.082 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 7 Tf 84 Tz 3 Tr 1 0 0 1 316.8 600.899 Tm 83 Tz /OPExtFont10 12 Tf (b ) Tj 1 0 0 1 327.35 594.149 Tm 89 Tz /OPExtFont5 13 Tf (0.3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 321.6 521.2 Tm 91 Tz (0.25 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 490.8 650.299 Tm 80 Tz /OPExtFont3 11 Tf (0.18 ) Tj 1 0 0 1 490.8 623.899 Tm (0.17 ) Tj 1 0 0 1 490.55 597.299 Tm (0.16 ) Tj 1 0 0 1 490.55 570.649 Tm 81 Tz (0.15 ) Tj 1 0 0 1 490.8 543.75 Tm 80 Tz (0.14 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 80 Tz 3 Tr 1 0 0 1 321.85 666.399 Tm (0.35 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 80 Tz 3 Tr 1 0 0 1 282.699 718.7 Tm 103 Tz (4.3 Forecast based parameter estimation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 103 Tz 3 Tr 1 0 0 1 115.9 472.949 Tm /OPExtFont5 13 Tf (Figure 4.2: LS cost function in the parameter space, \(a\) Moore-Spiegel Flow ) Tj 1 0 0 1 115.7 458.8 Tm 102 Tz (with true parameter value R=100 \(vertical line\), Noise level=0.05; \(b\) Henon ) Tj 1 0 0 1 115.7 444.899 Tm 97 Tz (Map with true parameter values a=1.4 and b=0.3 \(white plus\), Noise level=0.05. ) Tj 1 0 0 1 115.9 430.949 Tm (In each case, LS cost function is calculated based on 2048 observations. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 115.45 397.85 Tm 99 Tz (skill score, Ignorance. Such forecast score is treated as a cost function to obtain ) Tj 1 0 0 1 115.7 374.8 Tm 98 Tz (the estimate of the unknown parameter. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 115.7 330.149 Tm 106 Tz /OPExtFont3 13.5 Tf (4.3.1 Ensemble forecast ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13.5 Tf 106 Tz 3 Tr 1 0 0 1 115.2 299.45 Tm 98 Tz /OPExtFont5 13 Tf (Even with perfect knowledge of the model class of the system and the observa-) Tj 1 0 0 1 115.2 276.149 Tm 100 Tz (tional noise model, it is not possible to disentangle uncertainty in the dynamics ) Tj 1 0 0 1 114.95 253.1 Tm 97 Tz (from uncertainty in a given set of observations. Any parameter values, except the ) Tj 1 0 0 1 115.2 230.1 Tm 98 Tz (true parameter values, being used will introduce extra uncertainty in the dynam-) Tj 1 0 0 1 114.95 207.049 Tm 102 Tz (ics. In order to partially account for those uncertainty in the initial condition, ) Tj 1 0 0 1 114.95 184 Tm 97 Tz (we suggest using ensemble forecast. An ) Tj 1 0 0 1 314.149 183.75 Tm 88 Tz /OPExtFont4 11 Tf (ensemble forecast ) Tj 1 0 0 1 401.75 183.5 Tm 99 Tz /OPExtFont5 13 Tf (is a forecast initialised ) Tj 1 0 0 1 114.7 160.5 Tm 102 Tz (with an ensemble of initial states. Methods, like ISIS, EnKF and Dynamically ) Tj 1 0 0 1 115.2 137.45 Tm 100 Tz (Consistent ensemble \(introduced in Chapter 3\) can be used to form an ensem- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 3 Tr 1 0 0 1 307.699 52 Tm 75 Tz /OPExtFont3 11 Tf (68 ) Tj ET EMC endstream endobj 383 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 384 0 obj <> stream 0 ,,PIIj*S/GN?֨%`e)Y'(%j@Vf^w a%օwmcǚ5d̫zX HD!c WeeȘ y1ڗPDd~Bi`u3xrY"\0mP pyjs0_@6R+D2Nܲ\dvJ@xL*K$w^V2Bf @Q4CYTI3 3xd>\B.KKPo|e?ieMH]$/0'0-oP `oLnQfzZt 4[ {Dž*8aFܟKz`b@ER9#tD|fi$yMhS6sN}i: VSq&k 9e*O [[ ԷھunN t+Di*=K@3ZfxEBs$/GAX kc@o\X!tQ+K2nCcj\(毜;TJHM <{'0MO17=˥Iե5 v| 1@|Ha|̖lȕ<]yV=A}Lq6n_T%aΫp6IB$ ({0fE v*&em\l?JXxT%m7_=6ZxEj\$s1ތ3=zK7r>Yuܲ6'djVuFݨ)~ʨfc*YC3 ;4~V ߬w2^W&+{X[%78KY| Ҝ;vaMyN~9$`1srߠGw'iVAH,x8QSK[C4@z3ݨҚ#79/u2G凋 Xһ9X%w&ן<#x\hi6jeH.c]F 0wFGrf? )d1ЋN `YN=QY~`o&FUMs=k|S F[c }&}Z) oZ"×Wz{>4DJkX9VG]?rL}flt,̟^7ۡ-|k(HaJj@Vr4?oj z >#o9":NLM!jssngY,b?6>Ւ &DUϜY`=O^);rT\7I,jɒ<"}qIgi\z.s/` _rAܯzKL]jQӛ d p2ƭ!~:I_:EȂah+ {bl,շnK0P~u7KEWY.ME,{BE|K:v@N_68u* QQRT֙tVZ7Vei4 /s(*sŅ<3 x$Cɩ~V:@*.ζ `YȔFݵ%5]2,0WYg9AU{»N@TP/]fS~i0|nqP\fD.وMN=k箘{>s?}'h.٧,6@GM\Zf٫Ϸl_~qPF|*nimBSQq,N:@J:|꫄gw&lq)+ܩ-ehgAUpNгY{\{ v"^[2G PqǟJSUS|4[թkpOHM u5SSˮj2U P$X5 Uk\8k @MǷɖ$Rw sAc^6R%SV_j092E*K%ܼXW+K:']3?_̦#L9, *  >{)]^NWջf arѴjք̴-RjT\|3pPY )Cp4[j:7v!O?P6QC2C_[PI ԛ^mY?"ӍFlX܄~z(K()}SM^3-wZ~gAč$\B{WќVGE3 Q6Oy;[ sƮTC!Q&eR[ FړV0qT׹ks~HXvOwNίNJwbs3MZy 09w+(⩅>N!ew:C"T c=hRΝ4fČ*sڶq+G|.]pߔ7kyˆUbqp>we8hqs͸3Ylje/nIC{4p g+ gClx%= =m#8ͷ[)t%SJN563!m 9r.48Uѭ|x6#dI< L؎ x4̞{>8[/O'ZN.©]N44z߈x\5٨:;}Dw.\|lMGYO= ZkR.D~AD~ (NM-N/:f[ode1,^N꾯&[ L"te8\UC+<~tEJ='-#G.kK elɥFyn'^ԑI+9tkh|E<@TEGUHmAʀwA1G赑^CUcƀi1 ,^T$,BcjljTJ~ Ư DxcJhF2`&0C3OzDضoN֐7ȶUeqV}C˘c6,n9[C\>an?ώA)0+̵h`[(FE `2HIHF1. 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Zpm8N_!4_qȝo*j."B7*1 lrhL3}[s 4獬,?ܑ9G5.&@7<dūc_QyZ}~ ܈|HfZ# C~yj-zZw;ij&hf%]ہ^3db)pqc+ݕ2CmTO1?3XC=mvHR0}s$yQeNd r!V, QMht&Z, ;`gS!lⳄ>>,l7h稪e)HB>"*HZ̧+Xm[ yuTu>"xW9i)_u+cOդMπOAbOC, B+j#ml?4A@x]S!K[_` KFoێʝv2*>38v04[B:ZC:)(Q,{-zIo_˂K}XV\rgܞ-b'px妚%J/ |i;cqGdK%v\)TrD-骉;'k ^?/w!jfsYdP86Ȟ͖9 HXAxJ잟T7#b(ü67Jչג6Ɔꭓ7˲]쒀,Eكj4hqEz.JdY05W<`]HBS U of f +y~u)+ϐC_Qye]*k(RH}Om_WBdϹn(uѭxuFY6i%Z%+DZ! i0@;u,ggT nNٺvz-9mEY`2 ^4^@Vf:M2I̽y̲"1sFKrOTs ֯`)@FzӅM2s80Y8t3M+&xTQ<9 vDZs5q_?Hi(mkiʢ㏤elT'gJ!oK Cz*UKWI+Qu6fO+b^e@;֍T-r&/69HԀ/x_f{um=j䃳jPPJP>9]ct*kmˑSQCO,YĐC)$ \XLS>`:jƀC@j~~zQs{p$ +'@84Nt7/nKT:QGD0Oe{E1]HAC])fؼ;m. !Aࣕ2;%C$ygR[2"SA*-z0M ʭI?xs)# cj}i`Iwpzzpn0Z B82`cY{(mс9UNV]}|au؉(V.xLTKXhٺYsb >XTkXWJvV_/v`gM 줯 NкnR6ٳigk:t$dѧL[RhJټkӆ~@~?K]F”]u60ӣ1~Q[Ņ.3C)MSz,j)Q'sM=Dz/}!Y@+GPG )ҬK/ }ǰV1beѲWpr' (> jW4f_j!9t׎3Oc]ofߚfCܷ)SO䦳x]R k }Ncys1.6Vr.vŦ4,`5T 5Ƭ<F/$U#֙X7 q`? U TM> endobj 386 0 obj [387 0 R] endobj 387 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 605 0 0 838 0 0 cm /ImagePart_2096 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 282.5 718.7 Tm 103 Tz 3 Tr /OPExtFont3 11 Tf (4.3 Forecast based parameter estimation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 103 Tz 3 Tr 1 0 0 1 249.599 642.649 Tm 96 Tz /OPExtFont2 17 Tf (An initial condition ) Tj 1 0 0 1 282.699 623.899 Tm 95 Tz (ensemble ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 17 Tf 95 Tz 3 Tr 1 0 0 1 296.399 592.95 Tm 86 Tz /OPExtFont20 14 Tf (Mo del ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont20 14 Tf 86 Tz 3 Tr 1 0 0 1 297.1 578.549 Tm 101 Tz (Ffr, ) Tj 1 0 0 1 319.899 578.549 Tm 100 Tz /OPExtFont23 11 Tf (a\) ) Tj 1 0 0 1 346.8 578.549 Tm 740 Tz (\t) Tj 1 0 0 1 374.399 578.549 Tm 26 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont23 11 Tf 26 Tz 3 Tr 1 0 0 1 252.5 545.899 Tm 95 Tz /OPExtFont2 17 Tf (Forecast ensemble ) Tj 1 0 0 1 268.3 527.2 Tm 96 Tz (at lead time N ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 17 Tf 96 Tz 3 Tr 1 0 0 1 294.699 496.949 Tm 99 Tz /OPExtFont20 14 Tf (Kernel ) Tj 1 0 0 1 251.3 482.8 Tm 822 Tz (\t) Tj 1 0 0 1 280.1 482.55 Tm 107 Tz ( dressing ) Tj 1 0 0 1 346.55 482.55 Tm 891 Tz (\t) Tj 1 0 0 1 377.75 482.55 Tm 28 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont20 14 Tf 28 Tz 3 Tr 1 0 0 1 223.199 450.149 Tm 97 Tz /OPExtFont2 17 Tf (Model forecast distribution ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 17 Tf 97 Tz 3 Tr 1 0 0 1 292.1 424.949 Tm 91 Tz /OPExtFont3 11 Tf (Blending ) Tj 1 0 0 1 302.649 412 Tm 92 Tz (with ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 228 369.5 Tm 97 Tz /OPExtFont2 17 Tf (Final forecast distribution ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 17 Tf 97 Tz 3 Tr 1 0 0 1 292.1 343.6 Tm 90 Tz /OPExtFont3 11 Tf (Evaluate ) Tj 1 0 0 1 305.75 330.649 Tm 83 Tz (the ) Tj 1 0 0 1 250.8 317.45 Tm 827 Tz (\t) Tj 1 0 0 1 279.85 317.45 Tm 117 Tz ( forecast ) Tj 1 0 0 1 347.05 317.45 Tm 827 Tz (\t) Tj 1 0 0 1 376.1 317.45 Tm 28 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 28 Tz 3 Tr 1 0 0 1 257.5 287.7 Tm 96 Tz /OPExtFont2 17 Tf (Cost function for ) Tj 1 0 0 1 272.899 268.95 Tm 98 Tz (parameter ) Tj 1 0 0 1 343.899 268.95 Tm 90 Tz /OPExtFont6 17.5 Tf (a ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 17.5 Tf 90 Tz 3 Tr 1 0 0 1 114.95 209.2 Tm 96 Tz /OPExtFont3 11 Tf (Figure 4.3: Schematic flowchart of obtaining forecast based cost function for ) Tj 1 0 0 1 114.95 195.299 Tm 91 Tz (parameter estimation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 114.5 162.149 Tm 93 Tz (ble of initial states. Here we adopt another simple method, called Inverse Noise ) Tj 1 0 0 1 115.2 139.1 Tm 92 Tz (\(Defined in the following paragraph\), to form the initial condition ensemble. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 307.199 52.5 Tm 77 Tz (69 ) Tj ET EMC endstream endobj 388 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 389 0 obj <> endobj 390 0 obj <> endobj 391 0 obj <> stream /CIDInit /ProcSet findresource begin 12 dict begin begincmap /CIDSystemInfo << /Registry (BookmanOldStyle-BoldItalicOPExtFont23) /Ordering (UCS) /Supplement 0 >> def /CMapName /BookmanOldStyle-BoldItalicOPExtFont23 def /CMapType 2 def 1 begincodespacerange <00> endcodespacerange 12 beginbfrange <09> <09> <0009> <0A> <0A> <000D> <0D> <0D> <000D> <20> <22> <0020> <27> <29> <0027> <2C> <2E> <002C> <30> <3B> <0030> <3F> <3F> <003F> <41> <5B> <0041> <5D> <5D> <005D> <60> <7B> <0060> <7D> <7D> <007D> endbfrange endcmap CMapName currentdict /CMap defineresource pop end end endstream endobj 392 0 obj <> stream 0 ,,#jb6󸄿6L34 澦c1n.S|aD vG](' *7Zevl60,s6xVԶ:`aS wX=j+HGVl.5k2H8Nܪs=>*MM+Q'kHO,|*4~!$ rT8Sfq*L 3gSb*݅Bro6VFAq ?h%Ρglnځ@X*Mx$aUr361+ 46"C%-0݁F;L;tL͂)nbk՗`Qc]$!Pq$#yuw\I+W:JfWL=?Ѳ-IBIH5rC#9+hˑ>1>00řĭ6╂Ie債zl$X꺘<ý~9‡ ΆVej6(@'?Ҝ,ioti:H(>,z#߸Ԯ %t9!Khju`WI=?:+0[ [N`HgU= ?)2o4 8R1`JK ڡqv]eJmŗaX R]Q|9b!zT^Q9w +a+IHw%^>X( h?fctM)qS41WolcΡ~dJDy_C]YP[3()ܬ-92<e N7bUSw6<"h!Iq}#P 씉t )3Gb@s L^*22iO(eA;O%Ax_kҥXdv5η)oB䭷-WfoAS7SwL,1Ga˪,ө4կᨲ˞<`߰dvMZxp3pOGvMFҰi-r>: ToԨ1c}TWޮ8~j pV"lQ=fB#Gmn Uc9sAQ:&`~wHMNB/a:62HIJUY!e.xpt(U*WCLrnLݲH;FVR'ܰ!EG{HDyvBZ{.P ?VB9!ɱ teq2`-w^YNyt%rYOosBh3YTf8i,'4$mEWE큋qKIJe^)cp.]GF@fw?E74HxM_wmKUB:Q|YJ 9aL92ͰW+ݢFSYF hJw%@;s1._E*1PFEJ .scˡjω2t!pҟ>s1O {g\:d΅Fp9 x\ ~WK{>@->&uׁ(d$X _zWPR6o[k3W!x o`x@#Zm ݝ;{ j9ǘT4f0VlYdVHvϴv4 JS7F]fW*:s)fP;2GοBR( Wx,}m`@DVJ3&n.F!Vsv%0hBjUP՟:Gj2h7Ŷ俷oSZmKㄾptmZft-ْ(|ssLTSaD WƬϓmZb-NKPϠZ.df#~P~iQs5XІ(Fg yF4~si:@ A)Y2u. {4/Q0$ި633%cRt.F+ %6?R=`F#Ջ+4a5 Jr0I1"@1šȬĐgc~w!5~(d]m}Z?E0VXriSTzu.#ScF4Y**Dc1 ȚFdLoHV!00om3 nb㵧A){q7ubZ׾.CLz(_Ί)TKxYϛO '~?]Դ$E*o|gJ)IKV%B<$v'ne#*oo]W|W%^m1'W˕< ]?L0yvl_J}D0 imn(=Lp.!zgLMSn%!:U3\[!5JP% JÀeBpf.R\ؕAsBFz;Ϣ|"c5%phF|Vy#F ߫w!|?k$=O@OICfLXiǣg-z Ew>u%~)w|fvѧ!- u5ڗ)—˵k;g\25Lvyw_t?b,kk/=] TT<{pX`i( ޺mCMO^|BU{J~n8T @\7Yp5ܨoK})$+zS΁z[Lz:>HNzpẮe1g jp5nnO5kp 5Km\䷛g<=*{l*Y|$bC#87NRc*{ gr<[b(&.hqfj [%1UO#dt0O Ov~ٗҰ딫;x 1sg܂* ",vCf+ N۹YYn\{^ɪ!J X&f#ٍ0m^?0wdqn)"F@˕Y!1\EīIG>+dGh?ؽף8spuQ`-& wM:r}ɦ#]dkՋ'SOIx2ۃ_ gzJLU`p/-}lS_L,@G_1xlTu:%2$`PUjgd%QRrs|Vk/U(ܕ+I2()NAJdsQUb&PM`㓥.ә }}Ne 9`("maFX1%3 k=yc2<| ROj %]駺mڠVJΕPP?j!RYf }cm?O,ĂZIPӆ8)L1)Z%G B,BA~*YU4,8Ҋ︡l{--ߘHWTBB:/ӂuOb-DݷkI{JgGXgPvPeNXgMQ5 fo-_aCq#LbjD"u.ƃޥ#p GǙ?&Պf'%'/WJ]Pfva.Mdwhi rhUAtwuQX.VTXRl7 S:pU*b/=^#u'f|֨vPoG!͔Zc*m"$@IzW9Qȿpu&*N>tlco1T E7/w{;?6B.]WcrQsuǠc퐠)%# TNhM)*{g<-gBŘw"Eh3DyG*rpAMR t7!z[i•&൘.h煫Ͻ{v^_jk5^sk`k_rwk'5I}í k<9Bj/=Z\v*lA8[#j[Z)i 6cށ7id)[l XgӒ{ۯ%d8))hWع.eR7!Qit?#[ono= wLZyAUzn^Ɓ #l0?a"(ZCO U7PrGT$olmtDD1 :jY*>g*o rԶUVHUuתBaXL} 3lIe7p%m'1:FkMݧg}u,44\ZAƾX{9$5ǽO;~ 0\Rϼ,Rќ3+ Ow jlK ~H9eKONmR/! g5,!>ܔւѶ>ESUfhILvdX"7Ph@")DCP*IT&gP,qG,/g!S;uK"B ! cY `l!4z-jWRr"Vm^Q#_ &V0R9$0agNHmܼ4)mLH`j<d҄P WSגecϪr^gC]wDL (k/zvEI0 wӝ^M" t6hjOZ7{,^F2ڨ1ܠ!XkWSӺ T7n/R>W}/xa66SVĠ؇$2:>2$6.ﾗN]"FGΊq|`$?`bGxu>Z˿qÜ*VP|#_?V1VJ_s?pD "b_},%y_~fyܝq2$+$Q:Q:nnv֛!x\ǾSA0IiTgz1%XAQ }?2 zW,HPr5>V}6pd,( "Qlm fZ{M˯ؖ4Y F.fi>">4^T&92 L2W3f MzULΛzU<9b$Mpͭ#UNAڳ](pt 1)qmy|?a'4M@G~3~%ðdv@m3ki!6s-eS D2ޖl?B"qc̝OOGwyrO[@4YՄ_F5P^֚lRw;M樧c Zy9\mY0:RR_ al~D4>$[0%f=ExhRMf_@aHWt:=-3+׊iߑD981$Ҋ rڛ|2M 4H2uZ_tHB&܉fy ޔ^󜹱kN 6$8*( ijl<-&'?4MZ/p`2hxrڂk$0Y֯kcfhN 1EfpP1^H!vΚ [ā4qe¥=1!m"ʟ U4%szJʞ1Yڤ}D >Ev0 \ {lJQE낣q ]e}Ȁ?Ǘa'Bg5.uA|eY]ԓU pcbT|kEa ׁK)#=L69 aLx\Q=Hz4߂Żus}sZxmA Moh,/ZT0^k|ahsљyبБ鴈DoHš}I10}H?$}~R)+EYv A\slc7)řR~d=:}΁\gCe$f/)=syqW<1^'Ej5 0͒J"U8S}=[ R77һ($[ 4O;u&#y:ͧN1@5d #>O_[CN߅f3k{W!MUViǾW67q1\ZXk "IyOOPL\YJSp@#|RY"5+:uTXԕ%f s\?m8dW5JIA ‹\^mMuy!M׮q|7%4m `f4,ҞclR9 &72iҞzp @}}r4|)44kQfEܚJP-42"/GE6Tur?3!|%c{%9pڋCM;V;J;9\oOvCf}af\p_T9J1!GՑX@$Z`Dz{h,7lg5hɤA{b  (Ņ'\m9PIoI=\=.Cv\+"9u֋FU# Q<6TJ|J;z,2 _k$YO+.y*oKuKcz}=nK^][\LL^"hxbKpTϫJاLڏA%n›`}(RbѤ4a qʌjg^] ͞縇?*"ҙ0Zlf.tí'AnAcU_/bzX:JMKZ-fK^W1d|uEъf.4KK$Qʹ} -9x{oOpki[;ݣT^R&ptXVWm}lf6ZԤ.!NvfZ/k m.5fsM$Сb3 w?00vAxK*A<+Q4(G-@9Y " 3?!Xnu^f~dP%nʦ_ te)q5{j b%KE|21l&%SɮٸAVY?H"\6-W1+׫e+(-i?ΚA-ĒB(nsʰ@/:<PXTm4tPKs!Soh],W~Fo8Ƴ`LU6e^mBxS6H ltN@|kOvNE8J=` x[ jQ Ln,1Ʒ/] f L<~YCS5ey`E#TN PۜIy6KK`FOphtifRwIiaY[֚j3Cs1ߌ*)TiLC]Dћv&û-,9#ۜ͠] fU(q]pZޗԓB8XR}8[ZR1֋)8٠؍ MH)zԦ5Ì$1orOO%@w1N>ᯐ;ڥ9.ߴ2<*+`O!>-0J[z ф\x1p]\k&El_FR 3d'RB=3Aq|wu{3_%j)L+S,9v@+BO1swrqxI#~(t;초2pns P'd%"δzC0Xc|q?罵>}vhދU~\?RF_XRRHyevWD&蜇`od!sA oM-L_muq '%:Nc~^SGS_$pyA?e{ܾ,ye+G,, v􂹻:3̷ZK}-9 [Py萕9p ?& endstream endobj 393 0 obj <> endobj 394 0 obj [395 0 R] endobj 395 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 607 0 0 836 0 0 cm /ImagePart_2097 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 285.1 717.899 Tm 103 Tz 3 Tr /OPExtFont3 11 Tf (4.3 Forecast based parameter estimation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 103 Tz 3 Tr 1 0 0 1 136.3 674.95 Tm 94 Tz (Given a model of the observational noise, one can add random draws from ) Tj 1 0 0 1 118.549 652.149 Tm 89 Tz (the inverse of the observational noise model to the observation to define ensemble ) Tj 1 0 0 1 118.549 629.1 Tm 93 Tz (members. As each ensemble member is an independent draw from the inverse ) Tj 1 0 0 1 118.549 606.1 Tm 89 Tz (observational noise distribution, each ensemble member is equally weighted. This ) Tj 1 0 0 1 118.549 582.799 Tm 93 Tz (Inverse Noise method is an easy way to form the ensemble although the initial ) Tj 1 0 0 1 118.299 560 Tm 91 Tz (states are not guaranteed to ) Tj 1 0 0 1 262.1 560 Tm 38 Tz (1 ) Tj 1 0 0 1 264.699 560 Tm 92 Tz ( be consistent with the long term model dynamics ) Tj 1 0 0 1 118.099 537.2 Tm 96 Tz (i.e. the ensemble members are not on the attracting manifold of the model \(if ) Tj 1 0 0 1 118.099 513.899 Tm 92 Tz (there is one\). For purposes of illustration and simplicity, most results shown in ) Tj 1 0 0 1 118.099 490.899 Tm (section 4.3.4 are obtained by using Inverse Noise instead of other sophisticated ) Tj 1 0 0 1 118.099 467.85 Tm 91 Tz (state estimation methods \(Discussed in chapter 3\) to form the ensemble. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 117.849 423.199 Tm 110 Tz /OPExtFont3 13 Tf (4.3.2 Ensemble interpretation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 110 Tz 3 Tr 1 0 0 1 117.849 392.699 Tm 93 Tz /OPExtFont3 11 Tf (Ensemble members are often transformed into a distribution function which is ) Tj 1 0 0 1 117.599 369.449 Tm 92 Tz (easier to express the information contains in the ensemble members and it can ) Tj 1 0 0 1 117.599 346.149 Tm 94 Tz (be evaluated by forecast skill scores. Continuous forecast distributions can be ) Tj 1 0 0 1 117.599 323.1 Tm 95 Tz (produced from an ensemble by kernel dressing the ensemble forecast. In this ) Tj 1 0 0 1 117.349 300.1 Tm (section we give a brief introduction to standard kernel dressing which will be ) Tj 1 0 0 1 117.599 277.049 Tm 96 Tz (used to explain the problem in this section \(see \(13; 75\) for more details\). We ) Tj 1 0 0 1 117.349 254 Tm 88 Tz (define an ) Tj 1 0 0 1 165.599 253.75 Tm 131 Tz /OPExtFont6 8.5 Tf (Nens ) Tj 1 0 0 1 192.25 254 Tm 93 Tz /OPExtFont3 11 Tf (member ensemble at time t to be X) Tj 1 0 0 1 367.899 254 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 370.55 253.75 Tm 265 Tz /OPExtFont3 11 Tf ( ..., ) Tj 1 0 0 1 420.25 253.75 Tm 110 Tz /OPExtFont6 20.5 Tf (xr] ) Tj 1 0 0 1 453.1 254 Tm 92 Tz /OPExtFont3 11 Tf (and treat all ) Tj 1 0 0 1 117.349 230.7 Tm 90 Tz (ensemble members as exchangeable. In other words, the ensemble interpretation ) Tj 1 0 0 1 117.099 207.45 Tm 92 Tz (methods do not depend on the ordering of the ensemble members \(13\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 134.15 184.649 Tm (A standard kernel dressing approach is to transform the ensemble members ) Tj 1 0 0 1 117.099 161.6 Tm (into a probability density function: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 133.449 140.7 Tm 90 Tz /OPExtFont3 9.5 Tf ('guaranteed not to, in the case of dispersive dynamics. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 9.5 Tf 90 Tz 3 Tr 1 0 0 1 309.6 51.2 Tm 79 Tz /OPExtFont3 11 Tf (70 ) Tj ET EMC endstream endobj 396 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 397 0 obj <> stream 0 ,,b2~%$NVp|٤ ̮x>yr&aD@SBwܙtZrلND/Vت]*JoqT@!I]>Mp-RBf:irA ͷ==p/H*!$]ԅ ?5ς8ϐk:I''/ 1uY>R;~(֧`aO Y,05T^)=,XeAqD٨kn`{lRADBdqstV(+v%`?SA)Έ۔Ux00-CԐ_=eGK=x Bdmb={E8cIX@~ARvQ.`ЮPB1oHZS݄5KRԊu'd_ ah!Ekl~$U} Z y;,sdq9I3aWԌj, g%/j޵BZ1i96!K-Xg%KȻ?vt|z\':DhK;UÈj8鏘!&XH[I4G-٣@}gb"&A' h7D\W?&M{$ΤMPIM >z쌼UZ5r %f8q oKCEzZm%鐥H'42 KSYD#ej\P%=cK#K7S OY]+caz^`j9@,Oxr a͎>a3E:}ECkLeo֏k-tٞ}Lcd q-oV#u7FhYz~e.ʳ6d,iz4Y$؅THk0WZ-7Sg"qkrV11j7y6^[5\D&2cԚnI8at W_rlxُMUqXjW ӄ&I^;]W!ݴL_Vեؿ|R0Z;/AUX񍄎 ֣ATC!Ĥy2}p]*KF7in,#_2? nn!FQ WwU(Er(kbWo+j裡X|Ow2Sh>V|4-[ MQ@(|%5lxD`f5d1~B"]qȼ jŧgapZ`Lq R rɏKӘKS<(( Hv.#s46]t4Om]z b_J|^+Ciݹ>!dŘUFɉVd-À+臘;Hu|@"J2Ì\vvٌj'?{)S.25ނG݋gcQE(#Wztd utqz L`d&Ե}.ղ6EJ0xT7gV9*#M!n"Hue W(mw8i28&Anmtt^BG%NMh wyDx5NA_:Lh ׹`W, 7^/=vqD̾G:(HwsOrBY:"̕n4 ђ^k~7SEnJw[)3hG(C)Pk:d\C,,xB]}T8F>zeh_bVvfg$Ix.'[~- F-w2]FtϵE]n#ulF!}bE4{IқU%"Vpyy9\=ӈ}Nm*6ܶ$Ɔ0<k~W^Y^>pyjVmvkԇxCjKqM\,ZB WRVgIK>\A[w-n|@N.9֭2dSj|`7G+XXR5 |V n],}e_fvdz])лq[{)Kl hl){j%UV3酚?"`fGnw\z8['4E`3g#]QahQ2W-HPҺ4+ML]#~Yu$Qwy׈KbKDba#,v'k]yv9 ̷G}9rvZW;aSyP|5'|_+*;NVdmBZm:E6̆0X\ nˎ&bݕ]FCԙZUMmjN~^tp!O ܔI) Wa]su*poJl}ܔ- 3 jopIkol5[˪e$\eyN{u@mESKk3BJ"'%~h>QQMA.lBddeo Tt8l^祊 a~s?0F77z*4X`9<)2*݋Q NzP5!BG<{&̈zի("‘~b%(fK&M8367`ggjǸN)w/x\u]pCgy1*TWˆ=^%*( (,ݳfx)$,vN_,3HinQv$]㙊~_p=0OX 3.{yn]9 7i=B]0hfj8eD9<۲7c%]i@jA̅XY&CԦ1b1+l X`+Dh-q^5pBN7 /8Icay`<uӶ20畛+JBp)iٔjR&a c`fQnT!"c.*Ј߷)?EC*+?Zk)Et#AoSE1d)BoT( H+` 8m `uc^^ 2eYhVFs؍oOitpo>/݁NmXx~&V}^`g%B6ّ8pgDW%"뵄-F'VQmދvUѓ6~I+iea3N1Z~t+! 7je$[S1{7~AŴ`(/MR_UQ'f\{>ayے1jdAAGK,S!;JJͭe,d 8PcX%x?E:w0s5-K]l|2'Cfwg5* 7u 2c'-o;3={j/50IM\J;J]Cf9g^Wo_*oua✌GCbv5E%}Q4 ճOI.G\Qhp.J,TAҾo;]@Zb0Q#>#ÚYqzS8p=;9#>~koAy Яfl?z1'ʍ p mxK`^{Sp䉅CAjFRAÏxjZRRA?'Q$a>΍+B(wΛuo2qse–2iM3!+_sshkB@u9LA)1 Q&GN, ah;iWV\ia#UW)0:"yQnvś[:=z'z k]7J^p sDZ38VTxA2fd%(C2w;ghs[o7l[N<-f֤4sɍ;4d'j]L+S"KŗPHׄ.L#}vOIpEh @ [)E3Ϝ^E4Qn`PꝗU1O86Wf)YzaP5E!+t}}fۃ2,`g^WtQ|ѣ.tt]-H| oe/3KB~#;z>)|is~ apW˃vĥ?WtPS/%jf2ͽl*k;Zc,{HMLY"C"7C;˂FY- ORo>ԓY)JAp Oѐ&WV6/^Xͩ* l; qSCDbb ͆ EX&dd$ # vTV$ۂKsdid]`4zXK!Y eM.A(^t3pBꔟ`Bi.љ@MM Dq [ ~ݛa/E6 1FU&^x}O=E D[723+~ή ,0|r؂9rW* qe<]{ )(,7J~c0wRk76h.5Yh4DskRDcj'~ς ' 7&:%SQ(00TZ3]pe;b@d;,k 4wbG OGi eKH%勏IbۻN64zX4f;F=@-EM7>-5(EאD zod!(Q3 ,9Xfiu`aM0FvCVnE<8yI*K//'ءJ9eFƻTyb]<.)Y"N@etԙH.`DjYeB 7A>83hƸ=z2C'g(Z[$NA7 @dT36R$>caެ%p8R*o# :8$rCVGOWH_.AG SUtIUQ W67.{R7W>מr\f~>ځ2vA`p*r]#{)\>1gk]1;OE [|:WwL\ww?pT=4c "tJWJ %n4AI6Q0EY*A}_09WK-?>Ŋ{yww/fh: < yL1`62;8bS w@GVA D Г*aNKBe 8_[~U1պiR jׄվ[?`DOiGmBFa+oR H\y4HY"ؐ*O9mAڮ>2;TJ`Z)aQ7ÕPݹD_^\q'e, sB r-oL5jȯ\=çm=-sek4-Ia&ݣﬔKmHa7#ʧ>dpq,9"cɼ7nư#6fX4vyS'[3/X8 jm5"CE~Dw!U ÒVZ:"_:N;/a>02znO|L'q[nre ?;(K+}~n!ΠZ' ܊2Tx拻'n Ӧ9ĂHk)IemF0yټ8hfYmOs(:I&oՎsĜ'P2K"{ν"~Jؑd#1z.7N.IgXYBC @Ë1KA}6}".. tZu~ ri*GEIaPx ^GO4W8^=A'^m]yΤ(:y@WgW/gI],/ver`u)aO,*XPaj^EM5E7PŶ;۔oNr+3$_06ѧuXᡳYZ!KUE2xDedH#}Y, P&JUډ\=xiwR]] {OD(vبLA\j.)(vuĀZURٟmj j=?ӳqm X\40='X%"x4|qK;\g)C4ƔF5XHY϶]PxQR1?t< $%[eDGUⳘqi%vl(/y1w tI, _L/u 8~b =I髶5V}gQ4`e]bZlv!rMܷL=@չbChJ9zH$P ``un+glkV $#wߴ bhͣL61-)zye (m}9$asxkx u2 ~o41|k(q=Cݖ8"$gǮQہS_4h[VeTS"Kl dv?-ڳ#|ދ8zDm q^ ]ib}RJz5Wt;(ع`f$R2솧u{vMYn gתTґA]fzկ2wڅԔ`;4aunx?4I9le9kՖZ>I 4qHmDK#q# [5w;084 tƿN(ь{=Q3_Au'^Is]s#6/9ѩF4 1qN.l5~h ֧S[wRͻ`؎`Og)nwY6iTכ>ײӭʰڎu°lj60"ĉ<.;|{ %DMC$*k?:S肥&kS=񂙼NeC-BqL{(· tfn+;w<8%UӭrU̬A>OpD(?uO -"S n{Z*m>‹p~Eض\ *ԛ.TG 7v%zK ~?uKW%0 `*[P4$Ch1h*J+՛W}{M'NLG 0_f|PUo/vzS)m^?&NS>7j\94%ٳdZ)aS\[hms;OUj[3rq3\1i$:/,i endstream endobj 398 0 obj <> endobj 399 0 obj [400 0 R] endobj 400 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 606 0 0 836 0 0 cm /ImagePart_2098 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 284.149 718.149 Tm 103 Tz 3 Tr /OPExtFont3 11 Tf (4.3 Forecast based parameter estimation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 103 Tz 3 Tr 1 0 0 1 490.55 633.2 Tm 89 Tz (\(4.4\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 117.599 591.2 Tm 88 Tz (where ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 295.699 557.35 Tm 69 Tz /OPExtFont6 11 Tf (1 ) Tj 1 0 0 1 299.5 557.35 Tm 1368 Tz /OPExtFont6 12.5 Tf (\t) Tj 1 0 0 1 342.25 556.649 Tm 64 Tz (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 12.5 Tf 64 Tz 3 Tr 1 0 0 1 245.75 546.299 Tm 111 Tz (K\(\(\) = ) Tj 1 0 0 1 296.649 541.049 Tm 66 Tz (TIT) Tj 1 0 0 1 310.1 546.549 Tm 90 Tz (exP\() Tj 1 0 0 1 332.399 546.549 Tm 174 Tz /OPExtFont4 12.5 Tf (-) Tj 1 0 0 1 341.3 546.549 Tm 96 Tz /OPExtFont6 12.5 Tf (2) Tj 1 0 0 1 348.5 546.799 Tm 55 Tz /OPExtFont3 11 Tf (5) Tj 1 0 0 1 354.25 546.549 Tm 52 Tz (2) Tj 1 0 0 1 359.05 546.549 Tm 91 Tz (\), ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 490.8 548.95 Tm 88 Tz (\(4.5\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 117.349 506.949 Tm 90 Tz (where y is a random variable corresponding to the density function ) Tj 1 0 0 1 448.8 506.949 Tm 92 Tz /OPExtFont6 12.5 Tf (p ) Tj 1 0 0 1 458.399 506.949 Tm 84 Tz /OPExtFont3 11 Tf (and ) Tj 1 0 0 1 480 506.949 Tm 119 Tz /OPExtFont6 12.5 Tf (KO ) Tj 1 0 0 1 505.449 506.699 Tm 79 Tz /OPExtFont3 11 Tf (is ) Tj 1 0 0 1 117.349 483.899 Tm 87 Tz (the kernel density function, for standard kernel dressing we use standard Gaussian ) Tj 1 0 0 1 117.099 460.899 Tm 91 Tz (density to be the kernel density function. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 134.4 437.85 Tm 89 Tz (In this case a standard kernel dressed ensemble is a sum of Gaussian kernels. ) Tj 1 0 0 1 117.349 415.05 Tm 95 Tz (Each ensemble member is replaced by a Gaussian kernel centred at x) Tj 1 0 0 1 477.35 414.8 Tm 65 Tz (i) Tj 1 0 0 1 481.199 414.8 Tm 117 Tz (. The ) Tj 1 0 0 1 117.099 392 Tm 92 Tz (width of each kernel, called the ) Tj 1 0 0 1 277.199 391.75 Tm 94 Tz /OPExtFont6 12.5 Tf (kernel width, ) Tj 1 0 0 1 343.699 391.75 Tm 93 Tz /OPExtFont3 11 Tf (is given by the standard deviation ) Tj 1 0 0 1 117.099 368.5 Tm (of the Gaussian kernel. The kernel width as one of the parameters of ensemble ) Tj 1 0 0 1 116.9 345.449 Tm 94 Tz (interpretation can be determined by optimising the expected performance, for ) Tj 1 0 0 1 117.099 322.399 Tm 92 Tz (example the ignorance score introduced in the next section, based on a training ) Tj 1 0 0 1 116.65 299.35 Tm 90 Tz (set of ensemble and its verification pairs. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 133.449 276.299 Tm 94 Tz (We are aware that the variance of the standard kernel dressed ensemble is ) Tj 1 0 0 1 117.099 253.299 Tm 93 Tz (always larger than the variance of the raw ensemble, no matter how the kernel ) Tj 1 0 0 1 116.65 229.75 Tm 94 Tz (width is actually determined \(93\). When the ensemble is over dispersive, or in ) Tj 1 0 0 1 116.9 206.5 Tm 91 Tz (other words, the ensemble members are further away from each other than from ) Tj 1 0 0 1 116.4 183.7 Tm 89 Tz (the verification, the standard kernel dressing may even under-performs the Gaus-) Tj 1 0 0 1 116.4 160.649 Tm 92 Tz (sian fit \(93\). In practice, ensembles tend to be under dispersive. Many advanced ) Tj 1 0 0 1 116.65 137.6 Tm 91 Tz (and complicated dressing methods exist, for example Brocker \(13\) introduced an ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 309.1 51.45 Tm 74 Tz (71 ) Tj ET EMC endstream endobj 401 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 402 0 obj <> stream 0 ,,ffb7!Yne/orvvM#bض(3NƬ…lq–ʖ8:ՠ{ct  i$҈~(EZt ˼c` xZQޑ4/ N9w{c6-:Yp+%_I}Xy`|tde7X̎(r5Az*/՛nC]2# Lqx3gbvIJ^$ބ Irg"9OZe ܐ4zL" %̓?f&{-+PIcJ-}/70?纗'eߵQ,v r;:?Sڑ(Gޡ]4"8G"iv<޻4A,3dbﵼc{ G8qd1o@1닺nPņ}f`ځJ=띹%jn *AQ.8bنX#WPO^54=/|Fs+j=5YY֖M(e3m 4tkO^աzvͻ?|BHv=#>g3 ʞ}(JxIKTˌ90s0}(+H4vUWvٖt4 昱Fb߰kI t ^m蟽.biVQ2A;Q  u\$F-'/-}QfW|1e ",qFB>`:3"'6f|eːZB~nBqv'L#JpTI);EF\j!a_WEUUX7t Kp50"v% I7xZ#bwo^(S>)K0$[AjƏi,]F*ȷ.g3ügR{CAkH`$>|?dв ,@l_.]IgdKPkW;uw;[[2%vBk,_2"bM> 0#PD$""#dS(/FQ~Mo:Hg "褯EY/+TfcUӱC$f*dG쪩&^tRKT鸓~φ?LyX)| ,'}/2ǥv/HVOl+؎{owZA$s"Åwd3>]uS@OZ \Z *د*i-+:HMGU.^iHuf`X+ph\`;>G|,'){hڍ5uPKG}1ҝijB3§ 7}&fc)VO{\wɴxiX崮wU-9{#+kk >* ˄2~:[^Pvr'“ C_%Jl%q8Dme^ЧyҀaTd:A_~j|!=7њY=*dhlHsG$Mo‘ " v ǓA1xՕZU{6N}O=FVF/ ! ŗxovxCl֎@1zC,7 RD葦҇X(>i)(, TJ/֔OWg. xֹу54OIf7^0۸zYXEj?9N뉵 ) PyYЉ5dnǎڤ[1):p^y,!lƻW6\Nz$,$/컠-5F'$Hc.a4~ 17oPqJԭe~f/>˂l~CA+ޢ[(߂lݗhAX -`5ukLP|,a Qk_ϠYczz$ϤC fr^cb^-Cׂ <ۚ?dP#2;-]X _q!(&?%@D%DNŽ{]P`6|%~mqN8*˳;+/jNUajhrftί|t6.RUl\J~@p cC-G4n$ѝ0Ͷ"~T{?OGs( :ǃuX(W/3U-Q{_),8)gI5u n<*m2l'iHWpb3Ȝw1VyuO0_Ctɞ9x~~$8$=] < fgZoQCN۰ptWU>ަ]*~{M]b,(>Q{V?U/\q:۳ YRjyFzQv1+yV҄~#ᯩr$7i&C67N: b@m;.lSk%>`_9۽ل6{f_2k/c'RF?:WRSմ*(3OMϛ/ \CAG4&/4f$^0(,2ymaS&9*ޢ^V 9휳mIY N` 0i*;f }Q>XCsT-\^.4~jOM}ԦE$fO5gad0ՆójE{޹(5b7 YC5Z5wqs+{硴>2}BF/~޿eͣ?CRZ-H7@% cD0V(~\4|i_G$`JkP_%Vi+O^r^D"Y #1Kvhq2'K,UxR|Fov=P! i_*fv:9ms/m?ځ=Y CJQ)=rɀ&gw-ƁC8|e*l25)5h(V\-]2i #`dI̭>Z6&K۽tl{ݚ{bly4)g| gETV\{§_ Z=]^guoa3yP] "ۓPvڙTa-PfmWj&j6<[q~&Rf#{ [X kv.ˢ;:vL`qU ˉٹII-d`L?{$Zw>;/8;o<-x짒P{N]pG36+r-h:ɿu1f㧼S/ɥr/\'BFW-d|oˆhC#M/1P&{3[b'AjMe ]*TlKNsCg֠$LXņ$ o.'m6Vp2T `RͲx{rw-:CӴEl+tT 7Sh֒E:߮`ia44F?A$bҿmR-RB8Qz/]4&bڂػ46:4(  qYJkV}ٺT2gȁrDiwmGM3VSI*lь)=q6XoG4ք{.+G(ʴWL)Zֲ7ʼn*8J-/ D$MWdyanwv|p:JqWcp:a3s׭W `P_;>D 5c-] N! %DWIzN4`fD4t[IN{g~4G-=oTnp(P'|w( @[:hIĸcm)2:7vjC}B )ldA}/pӕww[ARLUĘ]E.,1G1wQk;."kփe1saeHs{D2ر^aj`ۈ 2kxgK .wsc;_.`^%<4Z~[y7&ɿy1z>kʆ*Uc˵0.00 V}*"ϲ)J )[o5V q 1l]OX֦'- 3Y]dA,JF, ӯMNs*64t$ H<1s.Ac?JhӖc3魳Ʊmg0ڜ"ra n'qNH.,Ah t~@T#|Pq<Nb<;)dQk؛rO1aո;HzvDt92t!:r7O^ zj-c%pkq0 яw9]ڷPA ܥl0\о>"_T݋ -^(˙$=nxyE%>U,Ak {ij،'de QYCG̲гWX;y=R ]"U[G`k~+ #b8q ׊t2`"V}OOXxfpr;dI*l'#rŌP@uw{:x ^Zۭb$VAQnt+[sm0O=K:j;}}rEG`e0KyXَ_[ʪ~#K rgu]K9NxS]dL^ u|C@&G4u5Ctt%ݭPU1I+/BSyl kwFR=E0.n곔E|$OD^dI$У[%:&\UqNi,q:J78wCxQEyLםh >4\'V`vh=4"a H)8 Mҟ{_Ln} uZ$w*Rk`4axw>7vqE[ ZlrX#?uOĘV%Հ?N). tɩ|lM5!Em?VjXBFj` ZTC:G?*lE!q>֊2? &E<(#MuKnAV 8>;]U"XWITiii4P`y ,A'IU%>/!0ܾ,*G<:d4UTj}35y;!&\ѯ5A.ByiGߺ7bS) Wmʠ1 q;/^w7$u3zy%W2 YUљv2 <(yrV Ŝ يNd*$m:U$:zZMw~Nb+Wd=k4n; VtnHڋYYS8Y4o:|mV*~9 IjvBM P)J 뙞P-zjZKwl[u{vp8 0CPԪdB Oj6v~+g x\}0&vR#` ʶ(1?\ؗhUo4Kbs^'oFc%0uFJyS@rf nx8iU6¶oIcRG4P =Xf2[i,#܌6[{,)K{ݳD:h{2@gM>!Ё 3K#-*Y9j DZ! G윐2Otmg?$N 󶟑4W5`yo=`UW$֣Ԅ*aÒژJQ^٭[G-Iԡ4F)51a5: XџoTo^bB|lƶsםѹ;36ǒ}$]8Rj-oZ]#%~{HT%:8Ъ#ĥ'ٮ;9HD!k_u %,[2 6 iXQ@>l)ʸ=;J5001<_t0` 㭫Xۅ齦ƾѾ$@6,A-_6E%K6|INTzp88 `.[p=ݜhmԓ&_ Q#E7< _Dovd$̀ .JM!w2pH#$&+}ﳷBYah 0J8bFͧ\= lBAZs}ZQXt ?}au?9'k%\XcOu C)"x͈8CGCZ4Z<'|*_4)[mo{\ tVL$`畆LM!S+y{I(|~v endstream endobj 403 0 obj <> endobj 404 0 obj [405 0 R] endobj 405 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 603 0 0 837 0 0 cm /ImagePart_2099 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 283.449 717.7 Tm 103 Tz 3 Tr /OPExtFont3 11 Tf (4.3 Forecast based parameter estimation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 103 Tz 3 Tr 1 0 0 1 117.099 675.25 Tm 90 Tz (improved kernel dressing, called "affine kernel dressing" that is more flexible and ) Tj 1 0 0 1 117.099 652.2 Tm 91 Tz (robust kernel dressing method. In this chapter we use standard kernel dressing ) Tj 1 0 0 1 117.099 629.149 Tm (to produce the results as it is straightforward to understand and implement. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 133.9 606.1 Tm 88 Tz (For any finite ensemble, there remains the chance that the verification lies out-) Tj 1 0 0 1 116.9 583.299 Tm 91 Tz (side the range of the ensemble. Even if the verification is selected from the same ) Tj 1 0 0 1 116.9 560.299 Tm 93 Tz (distribution as the ensemble itself, the probability of this happening is ) Tj 1 0 0 1 496.3 564.35 Tm 84 Tz /OPExtFont2 8 Tf (2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 8 Tf 84 Tz 3 Tr 1 0 0 1 488.649 558.1 Tm 126 Tz /OPExtFont4 5.5 Tf (Nens ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 5.5 Tf 126 Tz 3 Tr 1 0 0 1 117.099 537.25 Tm 94 Tz /OPExtFont3 11 Tf (Given the nonlinearity of the model, these points may be very far from the en-) Tj 1 0 0 1 116.4 514.2 Tm 96 Tz (semble, and appear as "outliers" or "bad busts". Those outliers will affect the ) Tj 1 0 0 1 116.4 491.149 Tm 94 Tz (kernel width significantly by making them wider in order to make the forecast ) Tj 1 0 0 1 116.4 468.1 Tm 92 Tz (distributions cover them which therefore degrades the performance of probabil-) Tj 1 0 0 1 116.15 445.1 Tm 95 Tz (ity forecast where the outliers do not appear. To overcome such problems, we ) Tj 1 0 0 1 116.15 422.05 Tm 91 Tz (combine the forecast distribution with the sample climatology. As we mentioned ) Tj 1 0 0 1 115.9 399.25 Tm (in Section 2.3, the sample climatology ) Tj 1 0 0 1 308.399 403.3 Tm 38 Tz (1 ) Tj 1 0 0 1 311.05 398.75 Tm 93 Tz ( is the distribution of the historical data ) Tj 1 0 0 1 115.9 375.699 Tm 88 Tz (which can also be treated as an estimate of observed invariant measure of the sys-) Tj 1 0 0 1 115.7 352.449 Tm 93 Tz (tem. The probability density function of climatology can be approximated from ) Tj 1 0 0 1 115.7 329.649 Tm 90 Tz (the historical data simply by kernel dressing the historical data. In this thesis we ) Tj 1 0 0 1 115.7 306.6 Tm 91 Tz (use standard kernel dressing to approximate the density function of climatology. ) Tj 1 0 0 1 115.7 283.299 Tm 90 Tz (The probabilistic forecast can be improved on average by blending model forecast ) Tj 1 0 0 1 115.45 260.299 Tm 91 Tz (distribution, which is obtained from the dressed ensemble, with the climatology. ) Tj 1 0 0 1 115.45 237.25 Tm 92 Tz (By blending with the climatology, defines the forecast distribution to be: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 234.699 192.6 Tm 111 Tz (PO = aPm + \() Tj 1 0 0 1 319.899 191.899 Tm 60 Tz (1 ) Tj 1 0 0 1 324 191.899 Tm 89 Tz ( cf\)Pc\('\) ) Tj 1 0 0 1 370.8 203.549 Tm 2000 Tz (\t) Tj 1 0 0 1 488.899 194.5 Tm 88 Tz (\(4.6\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 114.7 153.25 Tm 89 Tz (where ) Tj 1 0 0 1 146.4 153.25 Tm 104 Tz /OPExtFont6 11.5 Tf (p) Tj 1 0 0 1 152.4 153.25 Tm 66 Tz /OPExtFont4 11.5 Tf (m ) Tj 1 0 0 1 159.099 152.75 Tm 89 Tz /OPExtFont3 11 Tf ( is the density function generated by dressing the ensemble and p) Tj 1 0 0 1 477.85 152.5 Tm 54 Tz (c ) Tj 1 0 0 1 480.949 152.5 Tm 91 Tz ( is the ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 131.05 132.35 Tm 65 Tz /OPExtFont3 6.5 Tf (1) Tj 1 0 0 1 134.9 132.1 Tm 98 Tz /OPExtFont5 11 Tf (We will often drop the word "sample" afterwards ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 11 Tf 98 Tz 3 Tr 1 0 0 1 307.699 51.7 Tm 77 Tz /OPExtFont3 11 Tf (72 ) Tj ET EMC endstream endobj 406 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 407 0 obj <> stream 0 ,,a``j0Ile~]Qk*H 8TgTet*u\~EJU*%-3:1}mKelMݰcM^@£6iƭXq`]춼PzM"!*W^2#L_CֱuVlQk/`XwÍh4QGSC0PU'6}i7U7eb;:Wln֞cz YXv0.[p$k֖TiArMe=U]dOH '7ZAr̅S֍_gF.f_Ar ja*x[Pgᯂ|[U-L̊bbHۊ<,wg]h#HP#|f%Ɵo|a8^!pǏ4!_р/S(徜/D虴r,˗+K^a\m7gHG-!W>ʶļ`T_?]u_^mC. Z[Rر/mE;S^c=mTÉKKc!1-"JTkST8xTL9Joo6RϹ^Y.vӾ tKD(D!<L%mfף`WqV<*#!rKo?6/ msZsLck!QC;}νkVǗWī-"|~Er\]NPQ/B+q5gI>g+²HC~˷zUUHI%\5"]ɇ Z][u|-TgjPZB{ν_7U@v`WzŨ)}04 J(ϘEZ_^sHx4;79}O0:5p-Kbꅶģ'vwߵSZKV|v_max^D:(2v~ ^9Qm)m S(u8 ?4D =~S4qů ZkdtDIr9RUΰﶈ3u;T0=%ɱ4׶Z1< #œ{%bW*@g\EYE zPpF-SRSbC!Zwr? rԧ8QCE,:9ՠ̊eE"I k8fyjint'D(jc ۛ$_ŽzyG,2o<*BʵЗ6nU  RG5Pgw_nmG ifKQ%QwZ+jt܍T@ )]sBN C0Ext~gY;_;T(9w4MKZ isF=>Ʉޥױ0X(gͳ)je+c.J>q.&(@'2@-lyP10tyE1maz'ISQx3|)*8tzM&Ȃk(w&{>Ԟ3G?8E&˳V(&o̒ -T7IzOz s(sx.h$}`@ʔX*VƢS4(&Bu^*QLQ<ù#97+C: T`Đ" eԎj{]j UNֺ5m8qh7ݺNz`r(e~ŝH~=E f!qMa#^mm!H<57yQ8Zi6KuG4/hBy`ވ޼w}C-^*hAdCp3q(T KfOӡt) i^bh/V˴En+$52uqs8U1|Dby}Pz#`VӳV 9JT+fҞGp&N9nmfH U\%{@lfK߽s .L$\_sQJ.T{?7WU>;'t\Y.,{wcm f_b퍯V\ܖa_qײMBM퇍-#K&.uq&p2t++YY`l?&,ku'{!r燰}K|N~ 6+Oi1gX~L*7ZUvF7*|u!aOf70%z۔K]`ƳxsF uU %{8w(1nrl&ǭdL7tYO4LܻHCې;MNi(aFp& /v _ǻk`3RPHUNG2#U d'16;\g^鱛aD7+WIz&]VDNOCgʠygEVFstdjs-PG߭]0ӍBL*Yl31:pm\<˵G0/@R}1Ƞ_߱D5?t5x>ݲѳKζ48]9&4ʹ{S~Eb}buj"2,L8,6t/m9D Q͸fm^M`LV<0SG*>5D?o u~Υu嵎ڧB,MhC@a^;04Rbx9bcɒʽFCslסx dPntxHCd-?7=qY7~Otl0u=}Q1v`QڞYa6) _ud\w'-lQPBqz|cJV¿fʼ8)ݵ::#̴.ݰֹmQYn<3ò2=>k@NTuW(rZ}'-= SDR.>Kph#4%!R܆&'O +^zaWˇ7M] P0/#>pyFZ^aQch=roVy)e&_uavͩm"@v! ^ aaPx2WN6Ku  (ŅznZ/V{mUbP5aL6}t̜ Rj84qKH^ sP|,'g4)}ysșXAvꦹ r]!j mx?Mj(؜b #0z%oDh Vz}<+]zVV<}$_ RH=^LƝtҮE޴=l iS½ZZ$C;z[Fcgi R;SkN=y>Cnuxwri\i 0*_nrHX)ʯkYª8.nZ40!+'%-f,=UʂyM^ ^%X*F"֟*OF$rO n@b?$f&OyDŽ t&'za]0[-$V 8]}JBS-MA<) PeE>aelAr(E"ht~.HDzj0$liJy=II[T#Uj#ʱ 9ۊIm1e5\hnXYtFcY1Qޕ A/=>u y61NsJ9 }zh;'!V:jz*VSw[zNz\p@Kj*R` eMJVV0f`EFҪS+*N7Q~O6|J ǝ)JlC"/li+ZSoV4 pc.Qu `Ҁ G'Jc1D,SWT Vw.D[L@Gk[dcid 03b>QVw6UB`7Z[I}Y.8g-v} lR&8\Չ]DK%^sm#fL?ҲNx{o2 Ҕ"#5=DIbr- .5 JoRBunXZ9x> |Z;yԦ5BQ;h+—4'}$0A9aQZ6gp1MQ}0ECEMrDW2|mc8ziF$$D{{Ŕ= _ް˪)Bp"Fn/P-UYf4hr{ ռ%ooXVS i*J̰4,tT=lG?x}b*qjE ` :aBBZh46IVDм~02'7s.( T^kMs1gĸ l#.e_ K. 9 Gu ҹ$ ˗J]'G$ѩʷ k]p)_%_̀$o&)6m;j6s&t+ npZޜo " 9X%B?S}A+WJ1ZoͷX^Z F5vp.M9O[%\H)'n$d gbÑ.[H,)TXY3b_yDĽl v_!;|I1o9݈!3H]oT*̓Az#5 h[oU\e0>K~/.x||lTyqonmEٴZx5w- 330bq7zM ;M^u9+$\HLa;mOjwB iIQޱҡQIww]g)ɣg!0R$`7̎M2E^=t024071^ux UnIf}E.ɬYP74/% ޹o5($ۼ\FH"fK x^DO3Vg$ȢPxCD[>^ v. N'r513 l|(·96|?;AoOrvk(ǜ%lN>s j|OMgvo1bT[ͷf3M\)nſJnoݒc-nlq_Ý:ҭmHMטLʝA3+c\\%a$s^p#;U2G|aisD]5R :Ғ|k9@藬hdL/VK70 kͣHdiKO\vԨ5̩P*0*$(YCHA2fuj:+89F&qj&w98cśFtcTk1HeG!f<qUMuST| \@_KgU tVOAА[.;kvY={BY|R?'p' XRԌcq,x7_Ԝ%~|p_zDh _Rqae)E>3mC]†/ߴM< I͸zіE`H +5$O}g+Y1g Iο0`9C(lRYՎc@=XOնHea%.7AFoˎ$y9׈j}/J?oNM#4Z9׮5{ҧ'ATdDYqZMq3q7sdnsW68K;^$+5diƙKDX^0&ƨK$P[B$dBad3숕5)Jٹ/K4%==ni=IƪMІ3 4ݙvS|^P C%QR[TcT0!ؒ@+C0v$aJnҝ ŃN8c6Qą3<,b]0?N{? <=YPQY;cL֪lRȂ6 R|Xnw:x"lw8|Ր<ݒa*@="[4L0:9+g<.F(Z Y,cdopc\ɵR&(3qm=s"bU-[#26ܩq.3\D2ZatV= N0X:EF5 E9V,sF9;y/qrmh?Ԫ＠-ZG puW LYK/B2yh8woK8c0g0NF؋B|$юTk2:(40Qd7Z|bRZR (?YS9MeJ endstream endobj 408 0 obj <> endobj 409 0 obj [410 0 R] endobj 410 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 602 0 0 838 0 0 cm /ImagePart_2100 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 283.199 718.5 Tm 103 Tz 3 Tr /OPExtFont3 11 Tf (4.3 Forecast based parameter estimation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 103 Tz 3 Tr 1 0 0 1 117.099 675.5 Tm 92 Tz (estimate of climatological density, the subscript m denotes the model and ) Tj 1 0 0 1 487.449 675.5 Tm 85 Tz /OPExtFont6 12 Tf (c ) Tj 1 0 0 1 496.55 675.5 Tm 88 Tz /OPExtFont3 11 Tf (the ) Tj 1 0 0 1 117.099 652.5 Tm 96 Tz (climatology. a ) Tj 1 0 0 1 194.15 652.5 Tm 102 Tz /OPExtFont5 11.5 Tf (E [0, ) Tj 1 0 0 1 219.849 652.5 Tm 92 Tz /OPExtFont3 11 Tf (1], called blending parameter, denotes the weight assign to ) Tj 1 0 0 1 116.9 629.45 Tm 91 Tz (the model forecast distribution. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 133.9 606.399 Tm 94 Tz (Note that comparing forecast performance of different models may provide ) Tj 1 0 0 1 117.099 583.6 Tm 93 Tz (a misleading comparison without blending climatology. As it might be the case ) Tj 1 0 0 1 116.9 560.549 Tm 97 Tz (that, without blending climatology Model A outperforms Model B while with ) Tj 1 0 0 1 116.9 537.5 Tm 91 Tz (blending climatology this is not the case. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 116.65 492.649 Tm 109 Tz /OPExtFont3 13 Tf (4.3.3 Scoring probabilistic forecasts ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 109 Tz 3 Tr 1 0 0 1 116.4 462.399 Tm 95 Tz /OPExtFont3 11 Tf (A probability forecast describes our expectation of how likely an event is on a ) Tj 1 0 0 1 116.65 439.35 Tm 92 Tz (particular occasion. One may wish to ask whether a probability forecast is right ) Tj 1 0 0 1 116.65 416.3 Tm 94 Tz (or wrong. Unlike point forecasts, however, single probability forecasts have no ) Tj 1 0 0 1 116.4 393.3 Tm 96 Tz (such clear sense of "right" and "wrong". One can only measure how good the ) Tj 1 0 0 1 116.4 370 Tm 95 Tz (probabilistic forecasts are by looking at a large set of forecasts. Conventional ) Tj 1 0 0 1 116.4 346.699 Tm 90 Tz (diagnostics for evaluating deterministic forecasts, measures such as "root-mean-) Tj 1 0 0 1 116.4 323.7 Tm 92 Tz (square error", are not useful with probabilistic forecasts \(37\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 133.199 300.649 Tm (A probabilistic forecast skill score is a function S\(p\(y\), Y\), where Y is the ver-) Tj 1 0 0 1 116.4 277.35 Tm 93 Tz (ification and p\(y\) is a probability density. Following Good \(1950\), Roulston. and ) Tj 1 0 0 1 116.4 254.299 Tm 89 Tz (Smith \(2001\) introduced a measure of the quality of the forecasting scheme, which ) Tj 1 0 0 1 116.15 231.299 Tm 90 Tz (is called Ignorance. Ignorance is a logarithmic scoring rule that can be calculated ) Tj 1 0 0 1 116.4 208 Tm 94 Tz (for real forecasts and realizations. It is equivalent to the expected returns that ) Tj 1 0 0 1 115.9 184.95 Tm 92 Tz (would be obtained by placing bets proportional to the forecast probabilities \(75\). ) Tj 1 0 0 1 115.9 161.899 Tm 94 Tz (And Ignorance is the only proper local score for continuous variables \(12; 75\). ) Tj 1 0 0 1 116.15 138.649 Tm 92 Tz (The Ignorance Score is given by: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 308.899 51.75 Tm 77 Tz (73 ) Tj ET EMC endstream endobj 411 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 412 0 obj <> stream 0 ,,~~b6 @]߇ E*\uo`u1~|Piڋp4VK{ ZÌj'b'M c"U\5`N7;kcJZXHJ{2j2K?dJ|AFCB%E؞" 4,Oѥ(ܵ6즯`8 !bi^;|7"Cwo4~&>nGZ^8^fnP Ow gnҺaKr4)!E58ٓ?Nv~/i"0b(dmp3Q |-Mhk>6jiYA]v6Us:+mXvaFzr{E@qی"$=".&VTxMs]V0>;+mɐe Of̄U X]p@έtӖ7* G@3b(fyU[v7Dj&ݘŝ^ޠIkl9QB(1WZF=ZAB (bf9n4M=@k *9^P/sO92c2O0y4Z}5\NŐR!Nr)hTV sa㕒fx%[)u [U? C`"ߑak dS tgpfDUy_w+f4[eh&e~S|#fRk~ tB c+T%Ce]9xMSOȸy 8}˲t~2λC}FN&ⱼ5卥X ~BViOMVA\5!/qʱwa?HFȬ,hv 3έ\&PU liO{7E<1NHح=Zp@P9Nud~,L>B屫۲J$H,?{M˾WrPtg$柟.,Χ셳ܧ$2vⅺ,ij՚öı+ոj!ıg7Ҭvް왜ԨJt6╉-5<En= 2%p5֒z#PSaM" 2RU=~.P]-΢Ԁ9$qs:7bn ߄lᧂW$L-]F)lygڗdMv]ft/RXZȻcr]fGq1rtkMo\ńC q>ҰcjͿkڅO4'1<={"fA|aTdi#$3@UG8ϙ'uڐpgw%*# Sl69$쵾J3z&0hٛΩyၩP{gOq90^39.ri@;`Cn9`wh$MeDڢLm#7 HRHm$琋*$ #*J:OK2sC^kIҢ?ZIUJ;kwo,r )_qL&v'=5OƒfGU1=zT8l 8hVgex%| i[at~f2xJ-=O5;J UZ͔ZT lV\J67}G~0X?ˊxI`3[~"&}A̒mMˬIl`GAWG9LJlgo+yKuqFVqiSvXm:^_ *r!M![nd# uy5Givepc)BQ)Xl`%oyu~zQo|me M1hXQ +HV>)—o bo: j@+{†YC.]Ceb"+ w.,Z,.tLsߡư)Ы%n#U`t*Qik(-q,.W⸙t4:EqV>߅TuK$t{fA8"Cl u\2Rks]<w Ҳ ڄW1o2v D5gzCYfaKB Nc :ʁ青]C4"*6R5k4ۀ}\~L^"5d>Ẁ(+Ļ+#Ε^lͿ&|7(h_!:Qŭ|Ug82@!~7^ 檯TgnVP`,Aga]:wU ? 󺐞ܯoONE`MqocN7mƅJ+V-vw}^BO hr4ˁ' 3/8?Ylńo;Fg}ٳ(QO"Å||,ōh[EBةDv=Ri-mWLg, t8VbStQŸi'dQ<6)|7.7VtVO`oMYJ_HpK\aΛT; ^*L5cC^HfgL&ı{rL\-~7{Jy?Y"zW\h!I2Vw~w#i=8`O|땧3N܀69HXm uCs0(z)Q#ךv17vV(&#ၺvy+p-vʶ%\He{o3 8W` @Gm˗{%);9u͋D֣,'O `#6{K/%r4~m_Mѯ&LR;U: ާ+E>mLtG懮 m\zAbREf$hP:a% &n 3vt#p*Y5 $gfrM2Cy i:0Xu~'ap 5i *ҩdF_4%b/bﲫbN*:2Ep#%u fP eY% SZڕ;ZBj?\V94,۫36!_X&:(!(¦Ա:Ⲱ6ّ?J!*$ڰLM=fﾱ< ⪪ۙX?m-Azr0!˘\zbus&n.KQzjpJ㲎璎WapӢ[\XJ\tCȑ@kOBFeui[p 9tf=,34#征g8 NB)HMzIٴ¢~.e3JLS+o jJe@܃g}"[*|>_>)Db;^Kǚ.8 lw wz M '7AͧuJ*CуZ'Uod6 sR)%kQ\IeY`'Hqml4>r?内@Є$FosT!4089ʍD[^~FReCG$oq VӃJIzn|}I!QKq=$C!{k3qk#&wR CUrG)b2g4侌h@܃lpi!֫(TPڝz֐0UA 5Da)H SG:{&t&[z,q}ixj9u!(2[n߿Ҋ!I3|04xQn|M)UƩEnKqo!Ⱦ1go+W'v0RP3OFgF̣f4{I6.C[^FY ѿ`/2Ȍ6sٮ+'lz`Л3ǿ 0B>('DE iɟE]o!9x%/RQtI(Rʠk= 1[bX#Ҷ汫UVM]hjU1v0 ?Cs>iGwE@ZQUREք{N }: qa4(F9\/EY+^Wӳ 5hj=z:tPPq  u'Ã;/L:&6Ch6˸*d|O m{=j/P"5Z.݉O fݞW<AL5~LWR†Ȗ"2otWl]X F׆ka=f6*c@1ZI [C\ϵM4J$he7!*|?Q`*YC;daC'2#-$;Jљ=]ѡǩP&ս鴡\Ⱙy<).oH4MW[yGuNy'/W^Csצ$!1p%m.Geg[ DUYڗTJJ,$4(0E  P.+T~nq٘?% k}PjwO9BCx#%,o,h~13j앱4 YPr>?ͳK/. 3Ȯl2>9Ԧr,p $juTwk'OFh _y!l z\sH<SU+ ep2Gg#ѵ(ß.o  {T]^ZpNĢX}| $IRrRIruTc> ~9lⰭ%?7jνٵ5;ܔmZ099P.#9'w.Gkb"+)#1c ҭ5ĥWmF xrj/hW&/n{MP~?1Qfv)Q7?Г{8v4 y$`;f<1Y1f P~fl+!LT}5Gg[2OTȈ>p#'JG)\ƤT>'p8R Nq?+X;}\~@n䁰XVZVY+AYցYW*jE#$5^cx`n៤*;F Kj;̪)劕)A Ke ԪM& oE|`!F>VA[nI`܁J51N&/yL>k2n&0{"\ Paw; 97-])mLH!sMk /Xm? ):bi<ciK9t }z7&?^ {cdO|3܏ِhj6mF(TGݙG+uUK3䞠5`eqE[ s;t_)xNsS?q]PYi onou+{aYvx;_A=P [z%Q(0).\E=.ew΍Lܰn+A3>(8+3z岊vܛhy=i'ǯֽ2*< STI/h( a$ݵ!voaBr?G(oNơww\7sQ%[FiFR+.L?{86z7[DOta~gŽV޲\>]3!&ެ<ʐtc:3t79B|M_f6EC//~5lvgx@tWa$V57/.nX [Hh ݙ-\ux{Ganv`S*^MLw,^E|ѱҹYQV*0)4do&՘L$6?髃6(%LҲNwҢAVٯ'𯯰ʱhLfɨ幱Q*7°$~< ?> endobj 414 0 obj [415 0 R] endobj 415 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 604 0 0 839 0 0 cm /ImagePart_2101 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 284.149 719.7 Tm 103 Tz 3 Tr /OPExtFont3 11 Tf (4.3 Forecast based parameter estimation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 103 Tz 3 Tr 1 0 0 1 245.5 631.899 Tm 98 Tz (S\(P\(Y\), ) Tj 1 0 0 1 279.6 631.899 Tm 567 Tz (\t) Tj 1 0 0 1 299.5 634.5 Tm 104 Tz (= log\(p\(Y\)\) ) Tj 1 0 0 1 365.75 643.399 Tm 2000 Tz (\t) Tj 1 0 0 1 490.8 634.5 Tm 88 Tz (\(4.7\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 117.849 592.75 Tm 90 Tz (The difference between the Ignorance scores of two forecast schemes, reflects the ) Tj 1 0 0 1 117.849 569.7 Tm 91 Tz (expected wealth doubling time under a Kelly Betting. ) Tj 1 0 0 1 381.85 569.7 Tm 35 Tz (1 ) Tj 1 0 0 1 384.25 569.7 Tm 92 Tz ( We employ the ignorance ) Tj 1 0 0 1 117.599 546.45 Tm (score to evaluate the probabilistic forecast in this thesis. In practice, we have to ) Tj 1 0 0 1 117.849 523.649 Tm 95 Tz (go to empirical since we have limited data. Given N forecast-verification pairs ) Tj 1 0 0 1 118.099 498.699 Tm 87 Tz /OPExtFont6 11.5 Tf (\(Pt, ) Tj 1 0 0 1 136.099 498.699 Tm 98 Tz /OPExtFont3 11 Tf (Yt, t = ) Tj 1 0 0 1 175.199 500.6 Tm 109 Tz /OPExtFont6 11.5 Tf (1,...,N\) \(forecast-verification pair ) Tj 1 0 0 1 348 500.35 Tm 95 Tz /OPExtFont3 11 Tf (are a forecast and what actually ) Tj 1 0 0 1 117.349 477.3 Tm 93 Tz (happened, for example a forecast probability distribution of the temperature in ) Tj 1 0 0 1 117.349 454.5 Tm 91 Tz (London Heathrow and the temperature actually observed\), the empirical average ) Tj 1 0 0 1 117.349 431.5 Tm 90 Tz (Ignorance skill score is given by: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 219.349 364.3 Tm 94 Tz /OPExtFont6 11.5 Tf (SEmp\(P\(Y\), ) Tj 1 0 0 1 274.1 362.1 Tm 118 Tz /OPExtFont3 11 Tf (Y\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 118 Tz 3 Tr 1 0 0 1 336.25 363.3 Tm 239 Tz /OPExtFont4 3 Tf () Tj 1 0 0 1 344.899 363.3 Tm 104 Tz /OPExtFont6 11.5 Tf (log\(P\(Y\)\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 11.5 Tf 104 Tz 3 Tr 1 0 0 1 490.3 365.949 Tm 89 Tz /OPExtFont3 11 Tf (\(4.8\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 116.65 323.95 Tm 90 Tz (This empirical average Ignorance skill score is used as a cost function to estimate ) Tj 1 0 0 1 116.4 300.899 Tm 88 Tz (the parameter values of the model in the results shown in next section. In practice, ) Tj 1 0 0 1 116.4 277.649 Tm 95 Tz (we can get an idea how accurate of uncertainty in our empirical ignorance by ) Tj 1 0 0 1 116.4 254.6 Tm 91 Tz (bootstrapping. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 132.699 233.7 Tm 97 Tz /OPExtFont5 11 Tf ('In a Kelly betting contest \(57\), one bets all of one's wealth on every outcome in proportion ) Tj 1 0 0 1 116.15 222.2 Tm 98 Tz (to the forecast probability of that outcome. More precisely, a fraction wi of ones wealth, where ) Tj 1 0 0 1 116.4 210.899 Tm 97 Tz (co, is the forecast probability of event ) Tj 1 0 0 1 276.25 210.7 Tm 76 Tz /OPExtFont6 10.5 Tf (.E) Tj 1 0 0 1 282.949 210.7 Tm 37 Tz (2) Tj 1 0 0 1 284.649 210.7 Tm 26 Tz (, ) Tj 1 0 0 1 289.449 210.7 Tm 97 Tz /OPExtFont5 11 Tf (occurring, should be wagered on the ith outcome. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 11 Tf 97 Tz 3 Tr 1 0 0 1 308.649 53.25 Tm 77 Tz /OPExtFont3 11 Tf (74 ) Tj ET EMC endstream endobj 416 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 417 0 obj <> stream 0 ,,V4LW0hWY[ ?iRun_L 5$>2a#F/)sÙTAE% J$zFiIhNCgA ؂T\=tGd#V]#sGy\78Rs*u/}^}BJ!AQc΢{=1W҉vf f]2g7~H8g+p7PKb}jc7>bH)'>$t~geZtGZ.[f Dr F"X]W;ޝB?#oxrH']Y֙;!XpE%%JBn.N\6U[V%DbqvvFlف&Z`p޷q+Cf((:P|NXOOZꪺtL^ĎKB3p|"Īg#~KaXL_F6ÐQ qIoٍ}r`Y<6|1gZQCV^ vxҕIbOq[j[0PՕjBHWdwus'ZJQE)JaaM6]T{!jcth7Kw|%}D0 iȥD[s 9-5%yZrm*KIA4JSw9_Ie~R>b  i?xK/82 7,XolDI~|Nzs5ެ0 dqti}M00s /mQT{35ʡԍα]Z|nP9v*q@Ux['Im AIv-$5gd=JD2zpvP_2z^BQ%58*7`NhˎR#݃suZ;CoAî*L[m΃QQh2;*)P 2,KGMG }P vA]&8OKʼRAn3" 2 E ;W=aӇFjOn )P̢%zg i/?yyIԳX(iܪY,8.!\вW*cZЊ͐}Ie͝G"(g;K[| 6LPnbMe/$Jq8&M }·RDQ"u[wе=@!\1EWr&gMubl+ޫFuT/8>.huKQp#Q$oHTҰ ~šE9]&_NlЊ/,Q,pٴ@w'DڧQ6\oqT=Ńh,rKse앉X)T>hlS<>tC 6u^}|؞] /$̘?QM~ibQe>YvL}(&|`swDЦvrU ^osJYzWmam gXyz%k=@.jۭhN2Jln/E?ĆO 78TL*xN% Q^|mBcV̷j\N) ~<Ґ#P#'Đ 4L722ӝV Mj;qȐEMUL4twqѠvP򦄭2횹k I #Ń=+f.:LCQH!1(Uh@γw8+r!$M|#}!B' )jkK1E&[s\`\ #ÌL,D:Z&'NP0Kȃ^$83(h?e*t*~^AQzr" L>}B [ze~0"؊A" IKe$&Á&*hVEi7lt>; rɜ ?Zb9~Q397A > Ի廥Mo4$밪ܔ'Ѷ16Kp≕IKBCRzpiIg넉whJeS䎣7e,!D_-QPI8^ ?dVJ`:GDgb#]uqmKDq$6WXgM24c4Rm~_@-$يݷ9q۝6POJ?D`~$\)N4Ѱ8N/(."kL_;R*?gD-ߊ`NM8 *z3suܪMhQ{I+^W)(؝کu;r!-NR< D';AH蹌Vqk IJEΰjSmD,|ʜ|^FXϹv+jܿ~VKȱNR5[EL.Ekx%`=H\esFz)NΥ5dݾh &Bq;:X![QJ-J/>/ٰю<&W"cYUSr|/2+Q`|ǨB`,@N?I YY.eڣs1PTW uTtQH9;޴oC]論B{)`a3AzȜ}h= zOe[mW?QA>3̃0P&/<ɧ[ 0tC 1nDs,r(ۇ+NMѶu6#0=eUKå(9- >^9ɴP#LxE'KCv! ~D_!"KՈl|[y%Ky9냶$L <Mɫ:['&oZl`O a i "JvnH!HٱZQn#Ұp J G]./: oNeUc/`n_ِ7^*3}xyê%5IM`\a2`\xnbGM0hpb2ғA]C9! hd٫TDz zetp/3pA4#:lےک`ޚuB1QUҖQ%|6E'm@~@2_f1ZLYkǐo4 <6^R7cAy o&6= 3K@,\,3D-aoiE\n5zNH4HZئu)Xu6Hu/<Ҧ8a'k}?CPw$}2w5CQrw"IWHXV/֣pQqi,\o@!Zq %Bd^:bhS=<=>VoBDʽOq Hv?gtwYFP|Ѕ AmT""p/M&*_Ui FJVl)8 A*>98J1ĒõĿ4,/1"8^]Ć1ٞXճݺf%.`njqb`9콶a(%i(ձʾ#Ih:4'Pc^ԕ?,m8?5@qUAm(i@IC (x6>xАKr18 ۵0NX\F&4P67!'4m $ӦEaJ B2q'ID8\\'4iDvphA,X 1=-JIu,7ξ7 Qҟvu^ ^f8, c8. Mf V;UkÆx s tp㷝uyl|PlJC$FD.WElocsEU#7(6E-. Esj:*EZX%.C h\SQX7xzZONEuҤ7!7ŕrR/%pɭ &ݿ (*uJ\≁^SJnKB:7>.P3|3 m4\a{`ЂLc@Ad9+"Zo(o&4,)X&a}z NqF?]%r ܛtao^\ OZ  o Wf9׮]ƀ #4PF% PIMXQ^S.Ge@fCPcտPwG8C危Y؉  ˈ(ņ&T-?dZ/Ϧ#K LbI q"$K`tO떅?!w [w#" C@USaE&Uվ8hZEYsw'sjzN^.2Tqӫ(Y R Y"pP2e36>Вp cHC3n/7^>h$sԵ8}·pz +e{+b$PI3r5]<lS"t:@i?+A>ȷ8Xܰڳ5ܲԤ9)<3ԍ-_8/'I?D;MpQ]g{\ysxslNI9. x  5#j#G ݰnW^jEH,^sbTGai3>4@C-wl)7w! bY6>4lS׬Lr5)Aq|6! Yb *FoX ErNף-Ba߫[tk\U=} :jT ϋl,!VKR?p&;&,Z0 šxRdwb-tZ. NYQW+-} kv r".*qGGygy3oEE};^\)nK7mDK& LW"`sID},'zkf& zDk3E[mxG]174rL<&{N[|qP! ~@~JSpi)NKX^5k<:ɞJBGff$&RRki, hmAC"qBg}.: D{ÓIU.`@6`'߾t=t]p[;k=HFx!v[4(UɡmWwRXpmgC~2$/S8}6Ur1 Od&T :Y^\PiNO Byl`dP|Vk² aO5g/? #Ae0pm[&_#U 0\ h/0P liK 4z zM41LZ8IUfCby#`h?Q:__'&xҿ%Q+%ep.B0$ˀu1 Ju2X'xVh39淼C0 +?)BnunWY4OI_Wv\24WGW;NBf`U{v{*=Yvê%-SBcʹ,Qp Vxc<*cXU1- U6ʁsbA*jċԯs;Njk!,NB{fӚȇsY`RVLq,3y3mUQ}w8ewf<+zϛ~틘'Wӄpix_~s}5v#U"fÔV+\ם8|p<{|_ /5d[wf$hZ fHp V~I= dS_Q>N@UO0nh`%2 'aV&"=E&~eK?5dF/rPK":.g9'\CKi9^爙q!&li Z %,eŪ3+\b?,!`^F djz=y=)]ưQ 8;²ςd ):aKƾWh&òc&VhYG,LW*u.޻sxB̘eS,$d+瑒qU5̊2WHCJ=ٯ(o"2 da*jX$ڸ=\p/k+;xO.d͋4νPAHѮ do(1 cNˠ\ʽac?( \^}, NraYa&ˏ3J ]~ &Bi@  @( {_ o)NnjqrvZ_Jr2OSiUmb 3X¢ItVuNw}5;orCk-.uW SqU͡gnQc»TYr@=uB^[%~ktqB>YRhJ+wJ/S+rc䒔^ 'ET&e q.St~57*1AFdy>ŬL6%^CV=ۉpx{-!Y),S]BUCgd܄r D@~{D2>6=8> endobj 419 0 obj [420 0 R] endobj 420 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 622 0 0 839 0 0 cm /ImagePart_2102 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 287.3 720.45 Tm 103 Tz 3 Tr /OPExtFont3 11 Tf (4.3 Forecast based parameter estimation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 103 Tz 3 Tr 1 0 0 1 121.45 677.25 Tm 132 Tz /OPExtFont5 15 Tf (4.3.4 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 15 Tf 132 Tz 3 Tr 1 0 0 1 121.45 647.25 Tm 93 Tz /OPExtFont3 11 Tf (In order to demonstrate the effectiveness of our method, parameter estimation ) Tj 1 0 0 1 121.45 624.2 Tm 90 Tz (by forecast performance is applied to both one and two dimensional systems and ) Tj 1 0 0 1 121.2 601.149 Tm 92 Tz (results are compared with Least Squares estimates. Figure 4.4a shows the cost ) Tj 1 0 0 1 121.2 578.1 Tm 93 Tz (function, i.e. Ignorance score, based on probabilistic forecast at lead time 4 for ) Tj 1 0 0 1 121.2 555.1 Tm 95 Tz (the logistic map, where initial condition ensemble is formed by Inverse Noise. ) Tj 1 0 0 1 121.2 532.299 Tm 94 Tz (Results of different noise levels are plotted separately. When the noise level is ) Tj 1 0 0 1 121.2 509.5 Tm 90 Tz (relative large for example 1/8, the information contained in the forecast is unable ) Tj 1 0 0 1 121.2 486.199 Tm 91 Tz (to tell the difference between the parameter values. When the noise level is small ) Tj 1 0 0 1 121.45 462.899 Tm (enough, estimates obtained by looking at the Ignorance score of the probabilistic ) Tj 1 0 0 1 121.2 439.899 Tm 93 Tz (forecast well identifies the parameter values as the minimum ignorance occurs ) Tj 1 0 0 1 121.45 417.1 Tm 94 Tz (at the vertical line that marks the true parameter value. Figure 4.4b plots the ) Tj 1 0 0 1 121.2 393.8 Tm 96 Tz (Ignorance cost function of forecast at lead time 4 in the parameter space for ) Tj 1 0 0 1 121.2 370.75 Tm 93 Tz (the Henon map, same observations are used as Figure 4.2b. The low ignorance ) Tj 1 0 0 1 121.2 347.5 Tm 91 Tz (region \(black\) captures the true parameter values. Comparing with LS estimates ) Tj 1 0 0 1 121.45 324.45 Tm 88 Tz (\(Figure 4.2b\), using Ignorance as a cost function produces more consistent results. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 138 301.399 Tm 91 Tz (The forecast based parameter estimate results shown in Figure 4.4a and Fig-) Tj 1 0 0 1 120.7 278.1 Tm 95 Tz (ure 4.4b are based on the probabilistic forecast at lead time 4. The particular ) Tj 1 0 0 1 120.7 255.299 Tm 90 Tz (lead time was chosen because the cost functions at such lead time produce more ) Tj 1 0 0 1 121.2 232.049 Tm 92 Tz (consistent results. Figure 4.5 shows the forecast based parameter estimates for ) Tj 1 0 0 1 120.95 209 Tm 100 Tz (different lead times. Note there is a bias at short lead time. Also note that ) Tj 1 0 0 1 121.2 185.5 Tm 91 Tz (although estimates at longer lead time provides more consistent results, the cost ) Tj 1 0 0 1 120.7 162.45 Tm 89 Tz (function becomes less sharp as lead time gets larger. Examining graphs of several ) Tj 1 0 0 1 120.7 139.399 Tm 91 Tz (lead times \(Figure 4.5\), it was found that those of lead time 4 were consistent for ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 313.699 53.95 Tm 75 Tz (75 ) Tj ET EMC endstream endobj 421 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 422 0 obj <> stream 0 ,,M||b7!Ҩ5W/yeU$UoŒdkTs΄(0~$' Wr79l¨2vDձ?Tzpo},5$Ås8S`OӮkJ怼7 _~jۈcaSL+Px>5ԙI~$Ĭ_aU׀./q^b]E LaV<`zc] R|5y7GwHHo{':k!Ȅ0"VU@I*Mi;$u$u֮"MH?!Ntդ=LtWPF\>) B9k|PY<7I d硲J3'_Rgz0ڤY|R= hn>u|>iE$M{O/맠RȞ6tǯwG::] MHw O^j[Ȅ.33s퇮,&DkA-4]\d)i砎BIx}9z-a1ҳeV), U]+R8?Bc2H'2C+%ݔ Dsޜ.;v_^ӏX7^vQ> A)Wk?Ơ>xa`p7U,6G Šǚ+-c;=IrV̢zcU~]*O_G0iqhӲU7unY4m]eצ,B'xЄ!&#vۑXa%}iWbI"pCɐS/!h@‡R!im&rIS"[Ł3”ߍJS"mլX7/dއd[m_y%e`Vy^LϪaÂJk6/-`"Y!KT0r]*ܦ ‹ږ{q>},!=r2-~1$wmB4E& ʑk ;u>L벹ǢM$(N< S_X.Bc]n p3;{^~oCI?xO]!;(zw1񝮸hp0E}<(+J)T=FNqN$aSs}m!41svdބ#jnJb~^iȽ [}1+ž7X9Qe%/x"ՌbO}]O}D%}x9ˤw2i݊uV`e\86N'NC@ oY|R?][@=) Taw0ş iI PoQV=u_-iϓ;ir'?ǪG\S6QsG%GGܼTk>*`6fcZ?,!)LvQ(hϡ׊κ_Dgr;@$˾ ԾK;Qei/}|v2[#Mm ֆ&L2郈H_˅Z|ܔ@$( )|qЎaYnC8$ . 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VsA3_&hSQSO_Lɬ'.V~׌cEi Ok1rFͮcZp:??8C ̛ĢX(A>TNjJ`;@پ_[{7m =8o{t%f%Ӄ@c;Erۑ.F⬃Ї!)=tv@6ҪF%_=c\3i+T /AY껶RCy]I9O {o2Hb-%Q'E.?\L#P;mkݹiBuɴS5/ZW´O˳YXJ|ޒ n/GUV|b|TRbQ ҮeZ[ ܌as^r0lǡ戶研{Lw`L³7Spe3t]7A#_mghXG( {<Z50w$badC'gɃ#a\HiClX:s>0°6 cIg[4.R%e"`&]SPX1F޷0bZ:- pZƌ΅n ^{WUCqTYW ?;ݻLdg ޺7{N|^|S0"A4R[J PZ .:հ(t<Ҫ$hBC8vKsНK0rah$;_@x pwBdOGU5Da44= %ݼ I!3A^.iKCH}DPK~z*VI@1K ?(RDj>`q-Tol!O[S3'ԡ@lO=GbЧP{Xeѕ=2ŗ63o6}fWo'te[xU3Ǟ 2,+D Ƀ:Sѣ+IW)n8zHOy d%@l ^%k. conY)o9% NSq'!Oʉ08Zj#ߎ" c;s>21ْӀY`#F_L#_ubyff [#}w#ռyoB22 EPt0)ly{ n"nYTN⃩ d Y@qmcɆq@gfʥ{~i0ODTcˌG<:A5Ƃ,xhXZgCŞ^#=Nn<ƶFK01[Gn2X5#ҠhFu^v\w㗯$lw ڦQ̎mgذE(t,ujԚ5gDO:A@xxy7vL"V5^ʶI1룞!]@h4ڎ9x"NsQKa3r0~ :8/C2^ x.9k\U -M#$5ELS2i? endstream endobj 423 0 obj <> endobj 424 0 obj [425 0 R] endobj 425 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 621 0 0 839 0 0 cm /ImagePart_2103 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 357.6 668.1 Tm 79 Tz 3 Tr /OPExtFont3 11 Tf (0.35 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 79 Tz 3 Tr 1 0 0 1 357.35 606.7 Tm 81 Tz /OPExtFont11 9.5 Tf (b) Tj 1 0 0 1 363.35 606.7 Tm 78 Tz /OPExtFont3 11 Tf (0.3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 78 Tz 3 Tr 1 0 0 1 357.85 545.25 Tm 79 Tz (0.25 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 79 Tz 3 Tr 1 0 0 1 373.699 536.85 Tm 75 Tz (1.3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 75 Tz 3 Tr 1 0 0 1 431.5 537.1 Tm 74 Tz (1.4 ) Tj 1 0 0 1 435.1 527.5 Tm 79 Tz (a ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 79 Tz 3 Tr 1 0 0 1 488.899 537.1 Tm 74 Tz (1.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 74 Tz 3 Tr 1 0 0 1 303.35 638.35 Tm 87 Tz /OPExtFont9 6.5 Tf (\(a\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 87 Tz 3 Tr 1 0 0 1 155.75 668.1 Tm 76 Tz /OPExtFont11 6 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 76 Tz 3 Tr 1 0 0 1 138.5 655.149 Tm 99 Tz /OPExtFont9 6.5 Tf () Tj 1 0 0 1 147.099 655.149 Tm 75 Tz (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 75 Tz 3 Tr 1 0 0 1 139.699 647.7 Tm 113 Tz /OPExtFont11 4 Tf (C, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 113 Tz 3 Tr 1 0 0 1 138.5 642.45 Tm 175 Tz /OPExtFont9 6.5 Tf ({) Tj 1 0 0 1 142.099 642.45 Tm 66 Tz (,) Tj 1 0 0 1 140.9 642.45 Tm 108 Tz (2 1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 108 Tz 3 Tr 1 0 0 1 138.699 633.1 Tm 87 Tz /OPExtFont9 9.5 Tf (a ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 9.5 Tf 87 Tz 3 Tr 1 0 0 1 139.699 628.049 Tm 99 Tz /OPExtFont9 6.5 Tf (E) Tj 1 0 0 1 146.9 629.95 Tm 77 Tz (1.5 ) Tj 1 0 0 1 138.699 628.049 Tm 41 Tz (. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 41 Tz 3 Tr 1 0 0 1 186.949 605.5 Tm 93 Tz (noise level=1/8 ) Tj 1 0 0 1 169.449 597.549 Tm 110 Tz ( noise level=1/16 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 110 Tz 3 Tr 1 0 0 1 169.449 589.399 Tm 100 Tz () Tj 1 0 0 1 186.949 589.649 Tm 93 Tz (noise level=1/32 ) Tj 1 0 0 1 186.949 581.7 Tm (noise level=1/64 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 93 Tz 3 Tr 1 0 0 1 169.449 573.549 Tm 100 Tz () Tj 1 0 0 1 186.949 573.549 Tm 94 Tz (noise level."' /128 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 94 Tz 3 Tr 1 0 0 1 139.449 564.899 Tm 56 Tz /OPExtFont3 11 Tf (2 ) Tj 1 0 0 1 151.699 564.899 Tm 68 Tz /OPExtFont9 6.5 Tf (4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 68 Tz 3 Tr 1 0 0 1 146.9 552.7 Tm 77 Tz (4.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 77 Tz 3 Tr 1 0 0 1 139.9 615.799 Tm 99 Tz (o) Tj 1 0 0 1 152.15 615.799 Tm 66 Tz (2 ) Tj 1 0 0 1 158.9 615.799 Tm 55 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 55 Tz 3 Tr 1 0 0 1 139.9 604.299 Tm 99 Tz /OPExtFont9 12 Tf (>) Tj ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 12 Tf 99 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 12 Tf 99 Tz 3 Tr 1 0 0 1 139.449 831.549 Tm 91 Tz ( ) Tj 1 0 0 1 147.099 604.299 Tm 74 Tz /OPExtFont9 6.5 Tf (2.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 74 Tz 3 Tr 1 0 0 1 138.5 590.6 Tm 332 Tz (i) Tj 1 0 0 1 140.9 590.6 Tm 66 Tz (a) Tj 1 0 0 1 141.849 590.6 Tm 122 Tz (! 3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 122 Tz 3 Tr 1 0 0 1 139.449 578.1 Tm 80 Tz (`t'' 3.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 80 Tz 3 Tr 1 0 0 1 292.3 716.1 Tm 101 Tz /OPExtFont3 11 Tf (4.3 Forecast based parameter estimation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 101 Tz 3 Tr 1 0 0 1 129.099 501.55 Tm 103 Tz /OPExtFont5 12.5 Tf (Figure 4.4: Parameter estimation based on ignorance score, 64-member initial ) Tj 1 0 0 1 129.349 487.899 Tm 98 Tz (condition ensemble is formed by inverse noise, the kernel parameter and blending ) Tj 1 0 0 1 129.099 474.199 Tm 99 Tz (parameter is trained based on 2048 forecasts and the empirical ignorance score is ) Tj 1 0 0 1 129.349 460.5 Tm 100 Tz (calculated base on another 2048 forecasts, the ignorance relative to climatology, ) Tj 1 0 0 1 129.099 446.85 Tm 102 Tz (i.e. 0 represents climatology, is plotted in the parameter space \(a\) Logistic Map ) Tj 1 0 0 1 129.099 433.399 Tm 105 Tz (with true parameter value a=1.85, results of different noise levels are plotted ) Tj 1 0 0 1 129.349 419.5 Tm 102 Tz (separately; \(b\) Henon Map with true parameter values a=1.4 and b=0.3, Noise ) Tj 1 0 0 1 129.349 405.3 Tm 96 Tz (level=0.05 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 96 Tz 3 Tr 1 0 0 1 129.349 375.55 Tm 101 Tz (this particular example and so these are presented. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 145.9 353 Tm 107 Tz (The short lead time bias is due to the fact our initial condition ensemble ) Tj 1 0 0 1 129.349 330.449 Tm 106 Tz (does not contain the information of the model dynamics as explained in the ) Tj 1 0 0 1 129.349 307.649 Tm 98 Tz (following. The Logistic Map is a nonlinear chaotic map when a=1.85. A randomly ) Tj 1 0 0 1 129.349 285.1 Tm 99 Tz (observed state is expected to be on the attractor of the Logistic Map. It is almost ) Tj 1 0 0 1 129.349 262.5 Tm (always true that neither the observation itself \(in the case that observational noise ) Tj 1 0 0 1 129.349 239.95 Tm 98 Tz (exists\) nor the initial ensemble members formed by inverse noise lie on the model ) Tj 1 0 0 1 129.599 217.399 Tm 101 Tz (attractor. Using ensemble members not consistent with the long time dynamics ) Tj 1 0 0 1 129.349 194.35 Tm 107 Tz (cause the estimates to be biased. Figure 4.6 shows the dynamical consistent ) Tj 1 0 0 1 129.349 171.549 Tm 100 Tz (ensemble produces unbiased results at both short and long lead time. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 146.15 149.25 Tm 99 Tz (Producing dynamical consistent ensembles, however, can be extremely costly. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 99 Tz 3 Tr 1 0 0 1 319.449 63.1 Tm 73 Tz /OPExtFont3 11 Tf (76 ) Tj ET EMC endstream endobj 426 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 427 0 obj <> stream 0  ,,-YZZj`l/Pd>%7:*-w- = ΪESgW/yak&(<_O26/fu6dXPv@XV^uΡ=*,=wlRfV{J"b'9^\:_'SE2FV!/J' ߌ$y/ؕsWhX9h3U4|5kۛz=x9lI))8p<%s >!ӂk|$ef}ΟC>}\v#Mgl\ =|A D @+,̹yڌNN9>[:kS\a(-_Qۂ95Lr̯K P50TSuAh,Z盐tۡ:p0DYZ ;jYRXDl"`r)&;KUNU`<[b: Yޢn` r|hXry씙үyq*Z1a"BsM@\,YFԼ2l5HS_F(@oa]) liQ?6$;zS~[wn*97Zzf2+Y-\U i/ -UewBFTf3ķ*ȿcW9lс%˲R}ԮڠDsƖ9^S.&,3cZ&5-8*d7a|ϒGʈĖ;2x/#|f Hg,\vtl7ڄ37l%pJƨ8BynȈx!&42{'n ăڙ䟑-xPʐ{8t+ km?$OP[ &b̭,@œ#'&:&D, wfX(oe@7fX|q.(BL2 ЄϑhMX4]Gڐ:ѝ72%(?UWe}۲(HJ t7Jcϩ{~w"K~Xq]&yo=k7M$\րB qv}"*(EJIRǼ%7F"bE L*]4ffK.lM`r5  j{>⼷*?PȱE@.±Ĕ08)<;mf+ _u߾Du)'kKp[ WtY'~\z=C=޽'ބBjJ{^{DaK8kPtT̜.$lṽ #Hq,8+Q5dE+,c~/0_s bĀݦzyvQv^tWu0a%*`p2%8w[۷MOjX4ֽҢs8抖Ul=L.X7%Bdi<\CQ`*po,fNy1K"Q@.= )-0C blm_,K(vppS l|QV9Pq5IԒ-A|>-jVͼ&׹>=%9iɳ+͞Gt}<7|,I=O|Vb>(/iipW8jR#1+m=|wr`GHGGr2N umPh!tQ8u4d^\3ބwG6 xmКC9Ԩ4<5@z¶#$zAUِN/V"b,f4 7O(лfq\,KF{g.F[}!iSJ+qbBsɭ'_F颅fvnr_gT\}ʅ1VftɥJ4;繬'\w , 2g>Hye$DEQ6o2qS|f7@1vfeGqWkwON:@DC2,w~G;94dz2AAQ9y+?ؕ5N09ddY`3 KSۆCՌ, 㙍$! р: M^7$SC `ƀĉbsP X)2 D,SR0AEV7Q,, AK}ݵ\“08x9~}Wp,dXWVALXM6aNV 2mf[O>&:S;4?P3uI eit DdN#XmM>1OF"HqrV;(#߲AY9+q@_uW}SL^c%x{~j _G u{@a Sg庹Fjf.\s1яȍ ǶWDwAlF q1(w[j@ؤխ#](QB*@ps3sV%l$"AgqR%Jt2w܂XKư'fnؗt8&+xĖ5>ٻw=4Q"g!'P9U6ȮXSLiцnR3 Ը0'Yk]R\D!hZeE$v7}k2HW޵&}@(NqN9 @<5IaɐA)\w..{~xIS,i䐁JԾ6[ 9x:['!kJCm',OL gXP=u<ИgfҠOS4!,V)bz|ġ~eZ$ϋm "*W7us\wOcn 6Teص[}JWʔH< 7q2h{n@ALʋ><~unvSLNel싹6 I.tREUKi9 =.QoP#R>>\$1{ @D6>UcZo8 ;+z.\ Fb GFx5M~>˸߁^ކ?1z J.y>'/57ѱCfʰշɲ4)<.e;R -Mg$" H)mdGHU>aczI:+ igc0(깂avž1OceyVahP 괂QsB 0!sdD:WnT_|=E#ྪ>O8K-4.nvIEgW5Jć(CHP(Avlѯ,]J4!W>o "LQ<8K?\פ"ם! 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D8ZOB)zap\c 3#$n dLO5@*G?Zt#)Ȑl˸w% 6E">\Hy' q*33%.w\DH3M])MhBTQִatǟ"VR5e0]&U AZL`Eyªӵ8t&~Sopb"'vAxR] *`ùEa"Ų'*hfÄsoք/o6-2ߒ۰!Ns!h&\MeR*Jͧ)TC9IN6Ki,SXH=ܧ+QrH0My E @*ah8E1l寅:.#FpL0ߘjQH ^+KmkHVY8B )<>;):5@q~[a [zfސ$ͣđha'iܢSA9O+λٝ@dSpSÜ?ƶ7eƵ7_CL7`Y> $Ih/A: 46>շ[o@> endobj 429 0 obj [430 0 R] endobj 430 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 621 0 0 839 0 0 cm /ImagePart_2104 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 265.699 535.399 Tm 79 Tz 3 Tr /OPExtFont9 6.5 Tf (1.9 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 79 Tz 3 Tr 1 0 0 1 291.6 535.149 Tm 90 Tz (1.95 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 90 Tz 3 Tr 1 0 0 1 137.75 598.5 Tm 47 Tz /OPExtFont11 8 Tf (rs ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 8 Tf 47 Tz 3 Tr 1 0 0 1 150.25 593.5 Tm 89 Tz /OPExtFont9 6.5 Tf (_3 - ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 89 Tz 3 Tr 1 0 0 1 137.75 585.549 Tm 66 Tz (a, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 66 Tz 3 Tr 1 0 0 1 161.75 576.7 Tm 99 Tz (-) Tj 1 0 0 1 179.3 576.2 Tm 89 Tz (noise level=1 /8 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 89 Tz 3 Tr 1 0 0 1 162 568.75 Tm 99 Tz (-) Tj 1 0 0 1 179.3 568.5 Tm 90 Tz (noise level=1 /16 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 90 Tz 3 Tr 1 0 0 1 162 560.6 Tm 99 Tz (-) Tj 1 0 0 1 179.3 560.35 Tm 90 Tz (noise level=1 /32 ) Tj 1 0 0 1 179.3 552.7 Tm 85 Tz (noise level=.1 ) Tj 1 0 0 1 214.8 552.899 Tm 81 Tz /OPExtFont11 6 Tf (/64 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 81 Tz 3 Tr 1 0 0 1 162 544.75 Tm 99 Tz /OPExtFont9 6.5 Tf (-) Tj 1 0 0 1 179.3 544.5 Tm 89 Tz (noise level=1 /128 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 89 Tz 3 Tr 1 0 0 1 150 540.45 Tm 115 Tz (-5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 115 Tz 3 Tr 1 0 0 1 154.8 535.899 Tm 85 Tz (1.7 ) Tj 1 0 0 1 162.5 535.899 Tm 999 Tz (\t) Tj 1 0 0 1 180.5 535.399 Tm 88 Tz (1.75 ) Tj 1 0 0 1 191.75 535.399 Tm 1013 Tz (\t) Tj 1 0 0 1 210 535.399 Tm 85 Tz (1.8 ) Tj 1 0 0 1 217.699 535.399 Tm 1038 Tz (\t) Tj 1 0 0 1 236.4 535.399 Tm 87 Tz (1.85 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 87 Tz 3 Tr 1 0 0 1 224.65 528.45 Tm 96 Tz (parameter a ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 96 Tz 3 Tr 1 0 0 1 300 654.7 Tm 87 Tz (\(a\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 87 Tz 3 Tr 1 0 0 1 370.55 667.899 Tm 82 Tz (0 - ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 82 Tz 3 Tr 1 0 0 1 362.149 655.149 Tm 104 Tz (-0.5 ) Tj 1 0 0 1 375.6 654.899 Tm 80 Tz (\t) Tj 1 0 0 1 377.05 654.899 Tm 55 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 55 Tz 3 Tr 1 0 0 1 354.699 647.95 Tm 56 Tz /OPExtFont11 8 Tf (rn ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 8 Tf 56 Tz 3 Tr 1 0 0 1 381.1 580.75 Tm 99 Tz /OPExtFont9 6.5 Tf (-) Tj 1 0 0 1 398.399 580.75 Tm 88 Tz (noise level=1 /8 ) Tj 1 0 0 1 398.399 572.85 Tm 89 Tz (noise level=1 /16 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 89 Tz 3 Tr 1 0 0 1 381.1 564.899 Tm 99 Tz (-) Tj 1 0 0 1 398.399 564.899 Tm 88 Tz (noise lever=1 /32 ) Tj 1 0 0 1 398.149 557 Tm 92 Tz (noise ) Tj 1 0 0 1 414.699 557 Tm 81 Tz /OPExtFont12 6.5 Tf (lever=1 ) Tj 1 0 0 1 433.899 557 Tm 90 Tz /OPExtFont9 6.5 Tf (/64 ) Tj 1 0 0 1 398.149 549.1 Tm 93 Tz (noise level=1/128 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 93 Tz 3 Tr 1 0 0 1 366.949 540.2 Tm 126 Tz /OPExtFont1 6 Tf (-5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 6 Tf 126 Tz 3 Tr 1 0 0 1 371.5 535.149 Tm 109 Tz /OPExtFont9 6.5 Tf (17 ) Tj 1 0 0 1 379.449 535.149 Tm 999 Tz (\t) Tj 1 0 0 1 397.449 535.149 Tm 90 Tz (1.75 ) Tj 1 0 0 1 408.949 535.149 Tm 999 Tz (\t) Tj 1 0 0 1 426.949 534.899 Tm 88 Tz (1.8 ) Tj 1 0 0 1 434.899 534.899 Tm 999 Tz (\t) Tj 1 0 0 1 452.899 534.899 Tm 90 Tz (1.85 ) Tj 1 0 0 1 464.399 534.899 Tm 985 Tz (\t) Tj 1 0 0 1 482.149 534.7 Tm 85 Tz (1.9 ) Tj 1 0 0 1 489.85 534.7 Tm 1024 Tz (\t) Tj 1 0 0 1 508.3 534.899 Tm 103 Tz /OPExtFont9 5.5 Tf (1.95 ) Tj 1 0 0 1 519.35 534.899 Tm 1355 Tz (\t) Tj 1 0 0 1 540 534.7 Tm 73 Tz /OPExtFont9 6.5 Tf (2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 73 Tz 3 Tr 1 0 0 1 441.35 527.95 Tm 97 Tz (parameter a ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 97 Tz 3 Tr 1 0 0 1 512.649 646.049 Tm 87 Tz (\(b\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 87 Tz 3 Tr 1 0 0 1 354.699 590.85 Tm 110 Tz /OPExtFont12 6 Tf (2 ) Tj 1 0 0 1 366.949 590.85 Tm 116 Tz /OPExtFont9 6.5 Tf (-3 ) Tj 1 0 0 1 354.699 578.6 Tm 73 Tz /OPExtFont12 8.5 Tf (2 ) Tj 1 0 0 1 361.899 578.6 Tm 105 Tz /OPExtFont9 6.5 Tf (-3.5 ) Tj 1 0 0 1 354.699 565.899 Tm 81 Tz /OPExtFont9 7.5 Tf (g ) Tj 1 0 0 1 367.199 565.899 Tm 99 Tz /OPExtFont11 6 Tf (-4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 99 Tz 3 Tr 1 0 0 1 361.699 552.899 Tm 116 Tz /OPExtFont1 6 Tf (-4.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 6 Tf 116 Tz 3 Tr 1 0 0 1 153.849 668.6 Tm 120 Tz (0- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 6 Tf 120 Tz 3 Tr 1 0 0 1 145.199 604.75 Tm 102 Tz /OPExtFont9 6.5 Tf (-2.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 102 Tz 3 Tr 1 0 0 1 137.75 579.1 Tm 110 Tz (c) Tj 1 0 0 1 137.75 579.1 Tm 138 Tz /OPExtFont10 6.5 Tf ( ) Tj 1 0 0 1 141.349 579.1 Tm 143 Tz ( -3.5 ) Tj 1 0 0 1 137.75 565.399 Tm 267 Tz (c -) Tj 1 0 0 1 153.599 565.399 Tm 104 Tz (4 ) Tj 1 0 0 1 156.699 565.399 Tm 67 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 6.5 Tf 67 Tz 3 Tr 1 0 0 1 137.75 553.649 Tm 176 Tz /OPExtFont10 7.5 Tf (-) Tj 1 0 0 1 144.949 553.649 Tm 128 Tz /OPExtFont10 6.5 Tf (-4.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 6.5 Tf 128 Tz 3 Tr 1 0 0 1 294.5 715.399 Tm 101 Tz /OPExtFont3 11 Tf (4.3 Forecast based parameter estimation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 101 Tz 3 Tr 1 0 0 1 153.349 506.6 Tm 80 Tz /OPExtFont9 6.5 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 80 Tz 3 Tr 1 0 0 1 144.699 493.899 Tm 107 Tz (-0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 107 Tz 3 Tr 1 0 0 1 136.3 490.05 Tm 106 Tz (S) Tj 1 0 0 1 140.65 490.05 Tm 80 Tz /OPExtFont10 6.5 Tf (.) Tj 1 0 0 1 140.65 490.05 Tm 128 Tz (; ) Tj 1 0 0 1 137.5 486.199 Tm 65 Tz /OPExtFont1 6 Tf (er\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 6 Tf 65 Tz 3 Tr 1 0 0 1 137.5 481.149 Tm 28 Tz /OPExtFont0 12.5 Tf (:1 ) Tj 1 0 0 1 141.099 481.149 Tm 204 Tz (\t) Tj 1 0 0 1 149.75 481.149 Tm 107 Tz /OPExtFont1 6 Tf (-1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 6 Tf 107 Tz 3 Tr 1 0 0 1 137.5 466.3 Tm 68 Tz /OPExtFont9 7.5 Tf (E ) Tj 1 0 0 1 144.699 468.199 Tm 105 Tz /OPExtFont9 6.5 Tf (-1.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 105 Tz 3 Tr 1 0 0 1 137.5 454.05 Tm 46 Tz /OPExtFont3 13 Tf (o ) Tj 1 0 0 1 140.9 455.5 Tm 213 Tz (\t) Tj 1 0 0 1 149.75 455.5 Tm 116 Tz /OPExtFont9 6.5 Tf (-2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 116 Tz 3 Tr 1 0 0 1 137.5 446.85 Tm 58 Tz (a\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 58 Tz 3 Tr 1 0 0 1 144.5 442.5 Tm 107 Tz (-2.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 107 Tz 3 Tr 1 0 0 1 136.099 434.6 Tm 63 Tz /OPExtFont3 13 Tf (a ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 63 Tz 3 Tr 1 0 0 1 149.75 428.85 Tm 108 Tz /OPExtFont9 6.5 Tf (-3 ) Tj 1 0 0 1 137.3 416.6 Tm 77 Tz /OPExtFont12 8.5 Tf (2 ) Tj 1 0 0 1 144.5 416.6 Tm 107 Tz /OPExtFont9 6.5 Tf (-3.5 ) Tj 1 0 0 1 137.3 404.1 Tm 159 Tz (S -4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 159 Tz 3 Tr 1 0 0 1 144.699 391.149 Tm 103 Tz (-4.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 103 Tz 3 Tr 1 0 0 1 149.75 378.199 Tm 112 Tz (-5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 112 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 112 Tz 3 Tr 1 0 0 1 371.05 506.35 Tm 95 Tz /OPExtFont9 5 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 95 Tz 3 Tr 1 0 0 1 364.1 493.649 Tm 87 Tz /OPExtFont9 6.5 Tf (-0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 87 Tz 3 Tr 1 0 0 1 356.649 489.55 Tm 111 Tz /OPExtFont12 5.5 Tf (a. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 5.5 Tf 111 Tz 3 Tr 1 0 0 1 355.449 480.899 Tm 132 Tz /OPExtFont9 6.5 Tf (2 ) Tj 1 0 0 1 360.25 480.899 Tm 438 Tz (\t) Tj 1 0 0 1 368.149 480.699 Tm 78 Tz (-1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 78 Tz 3 Tr 1 0 0 1 301.449 475.149 Tm 96 Tz /OPExtFont9 5 Tf (\(C\) ) Tj 1 0 0 1 308.149 474.899 Tm 2000 Tz (\t) Tj 1 0 0 1 355.699 474.699 Tm 146 Tz /OPExtFont12 6 Tf (2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 146 Tz 3 Tr 1 0 0 1 363.85 468.199 Tm 108 Tz (-t.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 108 Tz 3 Tr 1 0 0 1 356.399 453.55 Tm 118 Tz (o ) Tj 1 0 0 1 360.25 453.55 Tm 526 Tz /OPExtFont2 6 Tf (\t) Tj 1 0 0 1 368.149 453.55 Tm 107 Tz /OPExtFont9 6 Tf (-2 ) Tj 1 0 0 1 373.899 453.55 Tm 65 Tz /OPExtFont9 5.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 65 Tz 3 Tr 1 0 0 1 356.149 446.35 Tm 89 Tz (to ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 89 Tz 3 Tr 1 0 0 1 356.649 442.3 Tm 107 Tz (o -2.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 107 Tz 3 Tr 1 0 0 1 368.149 428.6 Tm 131 Tz (-) Tj 1 0 0 1 371.05 428.6 Tm 78 Tz (3 ) Tj 1 0 0 1 373.449 428.6 Tm 65 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 65 Tz 3 Tr 1 0 0 1 356.399 416.85 Tm 83 Tz /OPExtFont12 8.5 Tf (2 ) Tj 1 0 0 1 363.85 416.85 Tm 101 Tz /OPExtFont9 5.5 Tf (-3.5 ) Tj 1 0 0 1 356.399 410.35 Tm 110 Tz /OPExtFont12 6 Tf (2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 110 Tz 3 Tr 1 0 0 1 356.399 403.899 Tm 121 Tz /OPExtFont9 5.5 Tf (c ) Tj 1 0 0 1 359.75 403.899 Tm 534 Tz (\t) Tj 1 0 0 1 367.899 403.899 Tm 108 Tz (-4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 108 Tz 3 Tr 1 0 0 1 356.399 398.85 Tm 75 Tz /OPExtFont11 6 Tf (rn ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 75 Tz 3 Tr 1 0 0 1 363.85 391.149 Tm 104 Tz /OPExtFont9 5.5 Tf (-4.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 104 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 104 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 104 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 104 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 104 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 104 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 104 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 104 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 104 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 104 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 104 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 104 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 104 Tz 3 Tr 1 0 0 1 520.299 477.8 Tm 90 Tz /OPExtFont9 6.5 Tf (\(d\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 90 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 90 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 90 Tz 3 Tr 1 0 0 1 185.5 435.8 Tm 93 Tz (noise level=1/8 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 93 Tz 3 Tr 1 0 0 1 405.85 435.3 Tm 90 Tz /OPExtFont9 5.5 Tf (nolso leve1.1/B ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 90 Tz 3 Tr 1 0 0 1 167.5 427.899 Tm 121 Tz /OPExtFont9 6.5 Tf (- noise level=1/16 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 121 Tz 3 Tr 1 0 0 1 388.3 428.35 Tm 116 Tz /OPExtFont9 5.5 Tf (--- noise revel-1/1B ) Tj 1 0 0 1 388.3 420.899 Tm 129 Tz (- noise level-1/32 ) Tj 1 0 0 1 405.85 413.699 Tm 91 Tz (noise leve1.1/64 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 91 Tz 3 Tr 1 0 0 1 167.5 419.949 Tm 112 Tz /OPExtFont9 6.5 Tf (- noise leve1=1 /32 ) Tj 1 0 0 1 185.5 412.05 Tm 89 Tz (noise level=1 /64 ) Tj 1 0 0 1 185.5 404.1 Tm (noise leve1.1 /128 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 89 Tz 3 Tr 1 0 0 1 388.3 406.75 Tm 127 Tz /OPExtFont9 5.5 Tf (- noise level-1/128 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 127 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 127 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 127 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 127 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 127 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 127 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 127 Tz 3 Tr 1 0 0 1 154.099 373.399 Tm 105 Tz /OPExtFont9 6.5 Tf (17 ) Tj 1 0 0 1 161.75 377.949 Tm 1013 Tz (\t) Tj 1 0 0 1 180 373.399 Tm 87 Tz (1.75 ) Tj 1 0 0 1 191.05 373.399 Tm 1024 Tz (\t) Tj 1 0 0 1 209.5 373.399 Tm 82 Tz (1.8 ) Tj 1 0 0 1 216.949 373.149 Tm 1052 Tz (\t) Tj 1 0 0 1 235.9 373.399 Tm 87 Tz (1.85 ) Tj 1 0 0 1 246.949 373.399 Tm 1027 Tz (\t) Tj 1 0 0 1 265.449 373.399 Tm 82 Tz (1.9 ) Tj 1 0 0 1 272.899 373.399 Tm 1024 Tz (\t) Tj 1 0 0 1 291.35 373.149 Tm 87 Tz (1.95 ) Tj 1 0 0 1 302.399 377.949 Tm 2000 Tz (\t) Tj 1 0 0 1 375.85 373.899 Tm 86 Tz /OPExtFont9 5.5 Tf (7 ) Tj 1 0 0 1 378.5 373.899 Tm 1289 Tz (\t) Tj 1 0 0 1 398.149 373.899 Tm 89 Tz (1.75 ) Tj 1 0 0 1 407.75 373.899 Tm 1293 Tz (\t) Tj 1 0 0 1 427.449 373.899 Tm 81 Tz (1.8 ) Tj 1 0 0 1 433.699 373.899 Tm 1322 Tz (\t) Tj 1 0 0 1 453.85 373.899 Tm 93 Tz /OPExtFont9 5 Tf (1.85 ) Tj 1 0 0 1 462.949 373.899 Tm 1440 Tz (\t) Tj 1 0 0 1 482.899 373.399 Tm 86 Tz (1.9 ) Tj 1 0 0 1 488.899 373.399 Tm 1454 Tz (\t) Tj 1 0 0 1 509.05 373.399 Tm 71 Tz /OPExtFont9 6.5 Tf (1.95 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 71 Tz 3 Tr 1 0 0 1 223.9 366.449 Tm 92 Tz /OPExtFont11 6 Tf (parameter a ) Tj 1 0 0 1 258.25 370.149 Tm 2000 Tz (\t) Tj 1 0 0 1 441.1 366.899 Tm 97 Tz /OPExtFont9 6.5 Tf (parameter a ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 97 Tz 3 Tr 1 0 0 1 130.3 339.3 Tm 90 Tz /OPExtFont3 11 Tf (Figure 4.5: Following Figure 4.4a, Parameter estimation using ignorance for Lo-) Tj 1 0 0 1 129.849 325.399 Tm 93 Tz (gistic Map with alpha=1.85 ) Tj 1 0 0 1 271.699 325.149 Tm /OPExtFont5 13 Tf (\(a\) ) Tj 1 0 0 1 289.199 324.899 Tm /OPExtFont3 11 Tf (Lead time 1 forecast Ignorance\(b\) Lead time 2 ) Tj 1 0 0 1 129.849 311.5 Tm 94 Tz (forecast Ignorance \(c\) Lead time 4 forecast Ignorance \(d\) Lead time 6 forecast ) Tj 1 0 0 1 129.599 298.049 Tm 87 Tz (Ignorance. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 129.599 265.149 Tm 94 Tz (There are other data assimilation methods which can form informative initial ) Tj 1 0 0 1 129.849 242.6 Tm 88 Tz (ensemble, for example Indistinguishable States methods introduced in Chapter 3. ) Tj 1 0 0 1 129.599 220.049 Tm 92 Tz (Using such methods may produce more skillful forecasts which may also help ) Tj 1 0 0 1 129.849 197.25 Tm 90 Tz (distinguish different parameter values. Nevertheless, when it is costly to run the ) Tj 1 0 0 1 129.599 174.7 Tm (model, as with weather or climate models, Inverse Noise provides a much faster ) Tj 1 0 0 1 129.599 151.899 Tm 96 Tz (and cheaper way to form the ensemble. It is presented here to illustrate the ) Tj 1 0 0 1 129.349 129.299 Tm 91 Tz (methodology for estimating ) Tj 1 0 0 1 266.899 129.299 Tm 96 Tz /OPExtFont5 13 Tf (parameter ) Tj 1 0 0 1 320.149 129.1 Tm 89 Tz /OPExtFont3 11 Tf (in a nonlinear deterministic model. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 318.949 62.6 Tm 79 Tz (77 ) Tj ET EMC endstream endobj 431 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 432 0 obj <> stream 0  ,,0  j]w>1[@| AROrN)%!mN0FI{1tلyLn(B΅aY|MPE:=v@uN|qwT{Mτ1`+RQNhn4 s+)QKSDrt@=qCy*P[fzM60tpXO_Rt}OYԒz=||أŀy7pUlrB# `9O' :&tyV]3s%;$3o*TsBq?X+mù;B55:6KpӍGOw۟E)<;ѝĢ~v7S%g eELnO7Pʭz%{ s';UKobR޲t _mgrsG[ڬ9!EC_}ƺ#nhm}Pei7[w/NᾪptL텎 Uk#TDG9Y(ZP&4e 5Uhu?@AIv Yz?\4|7L/ t{f7nI?'ކwulBBW>x7m*d bUZ 6D(Rd?Ki2D ZFt esLk]NiK9fhGښ(51לx]/D`5]zĆ@ KjcGNbΨG<37h;oK;8̍*z?q ro"Bf״`R֝!PM`VWߗ9lQ9w(z.Hw+ `m'"c;iQ 6(8UQbbz*库k;&;icR.C#$zwDlګ'Z M>\®*_SY#Z̴8coXw!|RUf};6ǽ\ỊOki`nRQepF6_3uKτqU}qFSr1\h=d)dےmf;kh4ĺ`>,2`u~ zC/)[כ4eeHtkL?C*dǞ4taK|RZPz@3*WcwV/v89Uxw|Df7}~-A =yv{ \ʡ[7gﾋaGcEyBUQQVnX# 5? u >R()\}W+$!Ahr"DK\SLr̽3O~"~nܘzͨC#͏ ϶}CG6a{-zb/lJD-ҏnfo@2a'faFV^LEcdm0,:%=60š0‘ZO:%'L`JrO;|;n`YwxKC7]](5#cg '؍}RĤS⮓*N+"@܌n|YHyzמX'(5֥ vǂ`36E- ۣ?Ċ]ӄW&#]'X\ѩ\2Ptu5oA> ܷ=)X=$3n7).Zո銜=<d;$]>k4ʳݖ?10ѵY94]mѻ.ݷ!5ܱάׅ,-><aS[6KgH|wܲIy =r L{]"_dAu5Isn|CPߏR^ejI 6\~xe 1B֤Id]F^Ktx  \|FrwF <'1*V 0b7eQd!rL'+/BKN=Os; hGѸIRT퓑+uW+AXA}umN3T3bG*,ySQ4x^Ql,@[)ۨ?2n-nbsgLi9Ymig8-wf/8WΟRrĘDB %LMdli D>niO'Dswrz,g鲏(aOs~g7͹x :0Yih@ p3CYönf Re82U!!wvjr hcH(ůNLm !LFO;Tܧ̟7hEx̍0g ]$zX\M)b;#:M #lJعֶ2ͰB\C_&z:|{@fLpG8|||E^*ڝ'"NBz Rě E-rniV4z89cIH#Yj m4,sZBBXD%ƀi|y*"g3O8Rjۿi>ʴܐʮ0Dz-f0##ө. 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m!)~q$rwR GtYW<U~^u&[$ @Öqz, )a̚y ْƞD/ëXdc;= ^-Y!܌hy_nBtn)#9ˮ AU{ҽY4khKOX4ɉgd/<ݾrPoGd <@jf.F*0X&+f~Lz|kMngulL׻>(Z`ؖmC@x4ZtF٭ߓHdj]h R NbYV`"/J83Ō:楖6 C,:SZ5fapra!G7ݴhGQjNs2痤-K:+a:i}}yv;Ny0Sec2Fo'R^auo2% Oovb ijB endstream endobj 433 0 obj <> endobj 434 0 obj [435 0 R] endobj 435 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 621 0 0 839 0 0 cm /ImagePart_2105 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 249.099 665.25 Tm 117 Tz 3 Tr /OPExtFont9 6.5 Tf (-- noise level=1/8 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 117 Tz 3 Tr 1 0 0 1 249.099 657.299 Tm 99 Tz (-) Tj 1 0 0 1 266.899 657.299 Tm 89 Tz (noise level=1116 ) Tj 1 0 0 1 249.349 649.399 Tm 120 Tz (- noise level=1/32 ) Tj 1 0 0 1 266.899 641.5 Tm 93 Tz (noise level=1/64 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 93 Tz 3 Tr 1 0 0 1 249.349 633.549 Tm 99 Tz (-) Tj 1 0 0 1 266.899 633.549 Tm 93 Tz (noise level=1/128 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 93 Tz 3 Tr 1 0 0 1 465.6 593.7 Tm 99 Tz (-) Tj 1 0 0 1 483.1 593.7 Tm 91 Tz (noise level=1/8 ) Tj 1 0 0 1 465.6 585.799 Tm 118 Tz (- noise level=1/16 ) Tj 1 0 0 1 465.6 577.899 Tm 119 Tz (- noise level=1/32 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 119 Tz 3 Tr 1 0 0 1 483.1 569.95 Tm 91 Tz (noise level=1/64 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 91 Tz 3 Tr 1 0 0 1 465.6 562.049 Tm 99 Tz (-) Tj 1 0 0 1 483.1 562.049 Tm 92 Tz (noise level=1/128 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 92 Tz 3 Tr 1 0 0 1 176.4 715.399 Tm 108 Tz /OPExtFont3 10.5 Tf (4.4 Parameter estimation by exploiting dynamical coherence ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 108 Tz 3 Tr 1 0 0 1 125.299 500.85 Tm 98 Tz (Figure 4.6: Follow Figure 4.5, Parameter estimation using forecast Ignorance ) Tj 1 0 0 1 125.5 487.399 Tm 94 Tz (Score for logistic map with a=1.85, initial condition ensemble formed by dynam-) Tj 1 0 0 1 125.299 473.699 Tm 95 Tz (ical consistent ensemble, a\) based on lead time 1 forecast b\) based on lead time ) Tj 1 0 0 1 125.5 460.05 Tm 94 Tz (4 forecast. Note scale change on y axis from Figure 4.4a and 4.5. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 94 Tz 3 Tr 1 0 0 1 125.75 427.649 Tm 109 Tz /OPExtFont3 15.5 Tf (4.4 Parameter estimation by exploiting dynam-) Tj 1 0 0 1 167.3 393.1 Tm 103 Tz (ical coherence ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15.5 Tf 103 Tz 3 Tr 1 0 0 1 125.299 359.5 Tm 96 Tz /OPExtFont3 10.5 Tf (In this section we introduce a second new parameter estimation method which ) Tj 1 0 0 1 125.75 336.899 Tm 98 Tz (aims to balance the information provided by the dynamic equations and that ) Tj 1 0 0 1 125.5 314.35 Tm 97 Tz (from the observations. We consider this method as a "geometric" approach as ) Tj 1 0 0 1 125.5 292.049 Tm 95 Tz (emphasis is placed on model trajectories and their distributions rather than on ) Tj 1 0 0 1 125.5 269.25 Tm 93 Tz (traditional summary test statistics using observations and forecasts at particular ) Tj 1 0 0 1 125.5 246.45 Tm (lead times. This study is made in cooperation with Milena C. Cuellar, Leonard A. ) Tj 1 0 0 1 125.75 223.649 Tm 92 Tz (Smith and Kevin Judd and some of the principal results are presented in \(20; 85\). ) Tj 1 0 0 1 142.3 201.1 Tm 98 Tz (For each parameter value, model trajectories and pseudo-orbits are firstly ) Tj 1 0 0 1 125.75 178.299 Tm 97 Tz (obtained by applying ISGD method upon the observations \(see chapter 3\), the ) Tj 1 0 0 1 125.75 155.7 Tm 92 Tz (parameter values are then evaluated upon how well the corresponding trajectories ) Tj 1 0 0 1 125.75 132.899 Tm 93 Tz (and pseudo-orbits mimic the observations. Instead of looking at only one statistic ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 93 Tz 3 Tr 1 0 0 1 315.6 62.85 Tm 77 Tz (78 ) Tj ET EMC endstream endobj 436 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 437 0 obj <> stream 0  ,,'ttj[ATj\X[V.z TCo۬OpXH|'7p^%OmK᷂gt!+w%<섀8 WyhG(J2A{L%l3K~'ZGQLCbvc= { ~Qٌn!3z2`>aA{v"d˝Pe\v|p@ `9&ԁKj$(ew&7YX#ʹ띿9JZU4?]E{"@u6\>lU{7k›9cg4,eN[t(LVׄfw|T WWoªC#5yb}>bb2hr7yr`^X Җξ=hhQ=OQYcXZ8lMv(ބJ3ʼ8+]:Z2Hk[E;F^]_{Kͧ;@8ؗtqyb+6BSBΟe@1*RQ(4wN9MogAE $1r]amM7)O/j[JR{wRP^QuDߙzvA u9o;6$+DdMLZ@pS*dVy@ PVץ@UcxIpW2Fq93+l`_5AR9Q5\ ȒS O0c $A$.B#Yi2\&"];>d\W3LNgZ%|QcUBFrN dWf~J†I恥؏#ztP5S:f(M淀68Lus/ԩ5@&̢=Qd*fZ"~h#RzlI #3Kbyɔ2A!u~v7JǝܾZΘ,p̉}i,RGaÒw)!=^us>=Kǰ}ŷ䰪cyܴӨb0BJq|'KD>EU= mmzk8nLqvsW~mtܫȦRe\a68ָܐl`đ;Sd9Ī4v=mq3#m?LvLɫ|;ǾKS7|\Z7,A U=[-Ćb1F &-7BOx!&@^{NWwRkṕ ٥IgkaP 8xpzp"74Qc6NHP$GQMbtޠ+40 &hɾ8݅H0 S"YKB g ;ƦÆ `9w$LRokIw#,ֵzQ+l\ $ۄ͏YI% JՐVLo,vp^6BS?"Ҏ ydg\jt_̒.ANvo{A2KAꅧ`'c2`:rTcNJ0=R%Η:lsMH &G u{PxyӼ , l\Gڋz~&~-|G!u̢៷mRB= G jt F Jwx,nx$Kh3HΠ:?95fb_pos' {:\JY2[e\ :|*!(ǫjҙз:,vbAn]Wg;*NijSR9qpC3Ԙ+S5IdMfinQHZdcy>Oڏ xpäҴ1uiI $9.cRkNjAOm \~ K͔fq˿RIpiƎE1}H%ký7?cQA=5u=duZvy5bMC*i,ŀNi-"A 9$֊nnC%Qȁyd싇7f RN'ޯN^;Pwm>a+\91~2.`f"$9!&Fˊyɻtgjհ!ԭz eKu;.n$.qx[5&Av#nYym㡲]QDX+XT =?> L($}A C3)0{9qy͒M3X;~ ﴻFTecdy>Յޤ@z2n#!=wǢB/Aw>썁KkRkʤ V}ɧɧI(6GBU$(3'4<KNf8ϰDѰF{c~.B91rnbp_%֭7oHk _ehLPf*j4,x/"zJx^i$&|xY&"V-:.En瘜ǞĶP Y,a`_Qmp5/ MF6ޠy`ˁ/"N.JuGoQ7P߈W6B!5M̯Ƞ &,=iV؁J$q)[CX)5N<KTсdW9}}0|{חe%V˙QtˮE~ȦW Bma.$)跻^xSrUGYTP!Iwr~ލhHPBtQ'#-ԩ-zR.T w-4iH2M =`YNr;@ \TO^PϢ@@;(|1NDǸ{O⮇m:22Mfү#^^|ُ"Mw 'uq=xhQ ;.!9O&~.d~u [0*d VqIs\1ub c$u6#8Yc_ /Zt} ZV.{iǻ~keJ-LDmF#Z\<ņY;_e`m|a3*n ,b":E1 p噓>V4MX֤؂ET S^0$O! ~^Wb/DhKsH$%Vm+b*Et=VCڢW<)~XeN_jWry vl@NZtRPX̹ư2HwU$oDjԕ\)X|*jX+ը0*5D^Ғdف_ T:gJ'lڪi)ޅVKogy8#u7p]Dґd. zGrT]" .PI C4ZztG/=Gs|q(&t7^/Fm>W'ռhZ (cRYT ԯtؐB>N䑹ԆD[~cbhWDCbX qϑ&O~x~R>rһ&L)Qob0|uFQ3 0al2y|lk?zpkxB,A zA)d<()*Xi)xW+&Ә1d?Ts O/OI0V?wt'p $U1LU; !{Z@g(G{9t 6>9& 8|/H-@X q|p\=#$f#G¾ 9{e+HJ_n0-?qCқ</u7FƆ],T RrF7D" [P , Ԩѭ$BC؀Z|[Ӯ!N.Nγ{Xm+w%Y2Xy'f[LyXV"DM͒Ru|{olIyDcO~pѹ—ˬ?x\7zA} ,z%KsL1ԥ8_ /e_2@?.%^7E{@zbuԓ؅>ZM:2Z]/Q-w11T\ 㛛6}#q. 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[1Ó(œ@;Hi2$T㥣(.~ vgleZKZþ.S/[_:AhsTːv0"ag>`|)Q%DHn~2:`$EL`9y-8OC:Cq(+O$_FZ`V>yG]eNOx܂ZG^_'?j}h1x~mTz;D„DkYfxV$J$;̒&:wE ,%7ZuSo dYϒ?"4}oRr Ů,GvuzqH,| iŢ3VvG oFQ|%m* ZӼf|"\s`drqztk~L&nWqL)Y4+4w^{ 8t*@K֌2r:&6H$L&&._i(}RUw0|Xׁ8F=6zvsp[.5ԈJ(dcGPIVZUJ5ub}jPA~]fcAuH\ub/VxO2c9[Bmq16J|@nIxy’p/LF|.L]60u!k: %e.8}Mwun,36hV }K{QI(qik~{ஈ5{B*wڗp[ 4wtBL!_7ڋFP[D/ 1G!>BgJ*$}y/˷ㅳ,7\'rfB}\[l޼\T]wg!8)U_XH rN߅uh?^;ԺȝݾP5?Ebo$UmcCmL}qxh󘥡vPTqJ7 ވbved;(0=^bZ\5 !1 ;`BZ$*QiT)!@eP 9uj]?EbU(tɕnؘ]K\̽u\ \0ƎBn'v^xU6 ɧ'QF GlLxW<އH+a /L%5ùlxqe#Λ'#xhB=0 \Uu=ۛ(BnTw [jL,%%Ч[Zʔ4uZsk''R\gzYEJX"֯‹ė9kn.7KEQZF̃HI4w Qt~,*mBYh6nW/댦lI70 ohӥZa&OWN&#fvm*T:K[dg"E_vlBB?uĺ !OoH,6:R%Zx-eII:fڼMX7 KG=͑P:6#c|"_w(YK>%. ˆ0 dp)/܏u{\TKDHhn=y5_yK裙Xݐw}y)|7Y3d1F2D +)clk_:z>q/JK[P;K{]z,,JaU& m"|Z=}iP"RPh oIش#j)*mY9,~:'#9fAM׵m?SA/v3Gv6`$9ž=HcBMZɉK>Gk3ӷ_sn{i_m$B#?KZ'e޽: mE0|[kp~$$T;lx;Y^ME+WLEGVoW[+Ǹ =ZFK3'n^a dtšH@#gr8SeZAVbFQbn/&ڿGn3P&H|krTfki!>'OX̾P)NozarQCRK(ET &$)"W2B z|m;RեbJrPxX@݈/j8m9c\$SI%Mӭ$qTPA9 8N –g ^ D1!lU>YKH*y}D+ ^lgτr^ ]]20`~^D=q> b+)[Ô_+q8u1*CasG˿D{g h_5MLݳ^m endstream endobj 438 0 obj <> endobj 439 0 obj [440 0 R] endobj 440 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 622 0 0 839 0 0 cm /ImagePart_2106 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 175.9 719 Tm 105 Tz 3 Tr /OPExtFont3 11 Tf (4.4 Parameter estimation by exploiting dynamical coherence ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 105 Tz 3 Tr 1 0 0 1 123.599 676.049 Tm 92 Tz (or measurement, we measure i\) the consistency between model trajectories and ) Tj 1 0 0 1 123.599 653.25 Tm 93 Tz (observations by shadowing time; ii\) how well model pseudo-orbits approximate ) Tj 1 0 0 1 123.599 630.2 Tm 92 Tz (relevant trajectories by the mismatch error and iii\) the consistency between the ) Tj 1 0 0 1 123.349 607.399 Tm 94 Tz (implied noise distribution \(corresponding to the model pseudo-orbits\) and the ) Tj 1 0 0 1 123.349 584.35 Tm 88 Tz (noise model. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 140.4 561.299 Tm 94 Tz (Within the perfect model scenario, there exists a parameter set \(for the dy-) Tj 1 0 0 1 123.349 538.299 Tm 93 Tz (namic model and the noise model\) which admits the true trajectory which did, ) Tj 1 0 0 1 123.099 515.25 Tm 91 Tz (in fact, generate the observed data. Our method is aiming to identify such set by ) Tj 1 0 0 1 123.099 492.199 Tm 90 Tz (exploiting dynamical coherence. Outside PMS the preferred cost function will un-) Tj 1 0 0 1 123.099 469.149 Tm (doubtedly depend upon the application; parameters which admit long shadowing ) Tj 1 0 0 1 123.099 445.899 Tm 91 Tz (times seem a good choice for forecast models. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 123.099 401.25 Tm 112 Tz /OPExtFont3 13 Tf (4.4.1 Shadowing time ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 112 Tz 3 Tr 1 0 0 1 122.9 370.5 Tm 94 Tz /OPExtFont3 11 Tf (Although superficially similar, the question of whether a model shadows a set ) Tj 1 0 0 1 122.9 347.5 Tm 95 Tz (of observations is a fundamentally different notion from the traditional ques-) Tj 1 0 0 1 122.9 324.45 Tm 93 Tz (tion of whether or not one mathematical system can shadow the trajectories of ) Tj 1 0 0 1 122.9 301.399 Tm 94 Tz (another \(26; 31; 33; 58; 77; 83\). Traditional shadowing \(77\) involves two well-) Tj 1 0 0 1 122.9 278.1 Tm 95 Tz (defined mathematical systems. Our ultimate interest here is between a set of ) Tj 1 0 0 1 122.9 255.1 Tm 93 Tz (observations and a proposed model. Given a segment of observations s) Tj 1 0 0 1 478.8 255.299 Tm 65 Tz /OPExtFont2 11 Tf (o) Tj 1 0 0 1 484.3 255.299 Tm 74 Tz /OPExtFont3 11 Tf (, ..., ) Tj 1 0 0 1 503.05 255.299 Tm 92 Tz /OPExtFont6 12 Tf (s) Tj 1 0 0 1 508.1 255.299 Tm 71 Tz /OPExtFont8 12 Tf (N) Tj 1 0 0 1 516.5 255.299 Tm 40 Tz /OPExtFont6 12 Tf (, ) Tj 1 0 0 1 122.65 232.049 Tm 92 Tz /OPExtFont3 11 Tf (we are interested whether there exists a model trajectory \(for a given parameter ) Tj 1 0 0 1 122.4 208.75 Tm 99 Tz (value\) x) Tj 1 0 0 1 163.9 208.75 Tm 65 Tz /OPExtFont2 11 Tf (0) Tj 1 0 0 1 169.699 208.75 Tm 82 Tz /OPExtFont3 11 Tf (, ..., x) Tj 1 0 0 1 195.349 208.75 Tm 84 Tz /OPExtFont2 11 Tf (N ) Tj 1 0 0 1 202.099 208.75 Tm 96 Tz /OPExtFont3 11 Tf ( that the residuals defined by the trajectory and the observa-) Tj 1 0 0 1 122.4 185.5 Tm 92 Tz (tions, i.e. si x) Tj 1 0 0 1 202.55 185.7 Tm 70 Tz /OPExtFont2 11 Tf (i) Tj 1 0 0 1 206.4 185.7 Tm 88 Tz /OPExtFont3 11 Tf (, i = 0, ..., ) Tj 1 0 0 1 255.349 185.5 Tm 111 Tz /OPExtFont6 12 Tf (N, ) Tj 1 0 0 1 272.649 185.5 Tm 90 Tz /OPExtFont3 11 Tf (are consistent with the observational noise model. ) Tj 1 0 0 1 122.4 162.45 Tm 93 Tz (For an observation s) Tj 1 0 0 1 226.55 162.45 Tm 58 Tz (o ) Tj 1 0 0 1 230.15 162.7 Tm 97 Tz ( at initial time t = 0, the corresponding shadowing time ) Tj 1 0 0 1 122.4 139.399 Tm 225 Tz (;) Tj 1 0 0 1 131.05 139.399 Tm 51 Tz /OPExtFont2 11 Tf (0 ) Tj 1 0 0 1 133.9 139.399 Tm 100 Tz /OPExtFont3 11 Tf ( is the largest ) Tj 1 0 0 1 212.9 139.399 Tm 128 Tz /OPExtFont6 12 Tf (K ) Tj 1 0 0 1 228.699 139.399 Tm 97 Tz /OPExtFont3 11 Tf (such that, there is some model state x) Tj 1 0 0 1 430.3 139.399 Tm 69 Tz /OPExtFont2 11 Tf (0) Tj 1 0 0 1 436.1 139.399 Tm 95 Tz /OPExtFont3 11 Tf (, the time series ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 95 Tz 3 Tr 1 0 0 1 315.1 52.75 Tm 77 Tz (79 ) Tj ET EMC endstream endobj 441 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 442 0 obj <> stream 0 ! ,,Ob7 BT}@Ѣ%z*\"T}YF-SEdIe!Bm?Hj*<%B}9 x!)V$$7ި$ ?9rq3ͫ$O\\Bm}#/` ?L]?NòA@_//BXE= T,6'8x7'_3`zVՁ(_\ 뎲T)HsL*P|)3v2,{ G{'K仅̲ 6y!OZ'ÿLy6jj ""x-xy0 ]nT﫩:znC $r;kA%j-1mIYb]IAːƟ!bXAYS\qؗG7_L5(Vz)32jz-7~kM[|̓h͚ن&BOGs [ݑ1J#xF]6On5r(6}g}f-B:sa)w4w=苑,(^| 1r&m9ɔ=:2$V@M?%Ŭ!#.=ܕ}؎לj7v,nR8W @h- *Gv_x MC*o,e-L6bJJ&NQMZ߬q{ hoٯf' :,QkMqRNBPfb_ތ5 ,gq Kvʲ$Dx?^K a!#m'(*Zo(Ml/JODb5,RHV; "rsk3;:E#0X[Шv:j?&'8| `4C<AN:-XR*^ v[Zeu\d,kW\L,UV.XДT`Δx<; obl w$_e {L}LMReSFp"8v6JaqNl>}(`Gﭮ Ȕb'Jq[rF$v*G. $Tr3c.D͊~|Ry bm>XGVAo^U{.lk1.7:s+!e4VGn%޴2"|[ ]P7go7e rx 6nt>C8'\ #ٔB+p¯ܟ cg謯h\(CH;l@VDcT9mg;v+s+'hBq UDF`W -̬ӓ;WkfZ&F *Mk_>![3 wGC\B1Yg/;xd5-N\vpJx2׈ 7=t0>Qա+/2r!f(9i; n~z$IbGI>x7Ց`:q⾭%c/L]e8v"$)N J_d8dPq]Yw &rSG篿<\mߓ4էZ!+n*<0S8wږ7gZLOɊvN}^LMLzGvk.u"M%lAHDvP>9݁( 0Z|qYbk:kɍԁeATda:K'_I!_OS BqkX/ގ٦y|vQC(ʠn_^xt #R2_ {%0Wt! 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'Ã&f60+#ʯ<X\˾Y|""zohYa4)x:;NK3,L ~F hG:6 \T̽ 4.tXnͲBD2KjWlS _Jc"Z~T,G@E1fy4FDh 6!=k7 bqvr dH"+JG-Rs(?75N2//5V&H=7pr# (Bג)xq`6nNE۬y>?l`p"PjJѪ>FʞC*=lZJK_͍@mx6]/2ŀ }.=9S Ǫđ0bS0n.h*BM*r>@,ufZ=>8f? 0R_,5H -t8f~ DSS)o1(m2>G#ikcXY^M I{'ݻsȒGODSw@?(! /2u~SVL6gre [oݦlpyS[O ~deuNe WxY (sGMnY-z3rVo=_2?D3Lnݪ:t4fU6mǪ fK\wR9[XLh%.r`H-:nA8r?*b"6Ćqx\Y$sn{ºJ]L zus%l&\beggs.ci' z3Up4=Y>_cjxg26y^Aړ1@dpxCxДj A,ŲaM6{㎡o5\8 8 #iYYz-Ǖ:݃n[tv` IܱRicq $h'іQLTaSNj;Zk!297 9G9[)SVJmhO!!ۼV% j'u1-Ѕ4{O(8->y]i{k ͵ 4ݫ5Ơn`E.A ɋ tVLkjq[upKIVe @ Vv1 \{H=ʿ_uHllSʔ^cʂΛ R>8KIxHfʹ8C`xoo+j[g zgț`o_nU*(Onݡo}QeWmCxPk@-cy櫷N?ne`prٙQlf$a<r61ohH^Js_ q탵 ^< &6RG'>i$ VV1U]LT{wFm'r|̢RX}dvJ`7BS?Sƪ(q/H7ۙ x3Lf6"9X[Enhl*?NCcDpj@!nG紐bgʡ;DWݧhk)bJ1@DiPbr[Xn4HOi 0z$wtێ{^lSJ3Mydr7DIϺQaZm#  ,M󴝥 t2M ȸ%vp3q$@kcǬ P`3FOQƸO0@س[fPBviLn% ك[`XY] LvLڴc ӧm cأ`0QϷG&'kUkӊ ējCzb/'t[9·*ݎwĹ3 @14eC"j{tErg*콲w+m(>IlJn> *Ej3T6lX&Os AX` ɠ2|qZ^Vr=nn*T,48kj0~~`( /,OygAC kF#/kޜ.y rx=zHyHL[I͝9/4[ ϔ&KqN 5{ $$F `DMcW{rn rݠ8PYb:q5g̬߫t?H4M<N)l%8qmxT+_G,E{ܼ+*yE~0GF7=ƜvSuT\]EzbfiZ419 &.#$0Ћ0~ߊ9,h$`.yrͤ%Pezk~3dj1cE-w&xmNG%;Yu"ʹ zL`Hw&3]$O[-nwӳ1+$ Ub=Wp52L~)kH R8*:Ἤv&m|?UN$l*'Jm`B}Hj4:7L IpV&xx7 6N]_/!)A[WXdkǜĎ{$_/Bw"Û`ķ&n~DKojR9c~0WAk endstream endobj 443 0 obj <> endobj 444 0 obj [445 0 R] endobj 445 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 622 0 0 839 0 0 cm /ImagePart_2107 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 176.15 719.7 Tm 105 Tz 3 Tr /OPExtFont3 11 Tf (4.4 Parameter estimation by exploiting dynamical coherence ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 105 Tz 3 Tr 1 0 0 1 123.599 676.75 Tm 95 Tz (ri = si F\() Tj 1 0 0 1 183.849 676.75 Tm 65 Tz (\)) Tj 1 0 0 1 188.15 676.75 Tm 109 Tz (\(x) Tj 1 0 0 1 198.949 676.75 Tm 68 Tz /OPExtFont5 11 Tf (o) Tj 1 0 0 1 204.699 676.75 Tm 91 Tz /OPExtFont3 11 Tf (, a\), i = 0, ) Tj 1 0 0 1 255.099 676.75 Tm 116 Tz /OPExtFont6 12 Tf (...,K ) Tj 1 0 0 1 282.949 676.75 Tm 90 Tz /OPExtFont3 11 Tf (is consistent with the noise model. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 140.9 653.7 Tm 92 Tz (In order to calculate the shadowing time, one must evaluate the consistency ) Tj 1 0 0 1 123.849 630.7 Tm 96 Tz (between a series of residuals and the noise model in some way. For uniform ) Tj 1 0 0 1 123.849 607.899 Tm 99 Tz (bounded noise this is straightforward: A series of residuals r) Tj 1 0 0 1 448.3 608.1 Tm 86 Tz /OPExtFont5 11 Tf (K ) Tj 1 0 0 1 455.3 608.1 Tm 88 Tz /OPExtFont3 11 Tf ( is consistent ) Tj 1 0 0 1 123.599 584.85 Tm 94 Tz (with the noise model when every residual is inside the bound. For unbounded ) Tj 1 0 0 1 123.599 562.049 Tm 90 Tz (noise model, for example Gaussian distribution, there are a variety of approaches ) Tj 1 0 0 1 123.849 539 Tm (to test whether a series points are drawn from the given distribution, for example ) Tj 1 0 0 1 124.299 515.95 Tm 96 Tz (Chi-Square test and Kolmogorov-Smirnov test. In our methodology we adopt ) Tj 1 0 0 1 124.299 492.699 Tm 94 Tz (a simple method based on threshold exceedance to do the test. Given that the ) Tj 1 0 0 1 123.599 469.649 Tm 89 Tz (noise model is unbounded, any observation is conceivable; we look for relevant \(9\) ) Tj 1 0 0 1 123.599 446.6 Tm (shadows within a certain probability bound. For purposes of illustration and sim-) Tj 1 0 0 1 123.599 423.55 Tm 91 Tz (plicity, we use the scalar to illustrate the procedure. We test the null hypothesis ) Tj 1 0 0 1 123.599 400.5 Tm 92 Tz (that the set of residuals \(r) Tj 1 0 0 1 252.5 400.5 Tm 85 Tz /OPExtFont5 11 Tf (i) Tj 1 0 0 1 256.3 400.5 Tm 84 Tz /OPExtFont3 11 Tf (, i = 0, 1, 2, ) Tj 1 0 0 1 325.899 400.5 Tm 114 Tz /OPExtFont6 12 Tf (K\) ) Tj 1 0 0 1 344.399 400.75 Tm 89 Tz /OPExtFont3 11 Tf (is consistent in distribution with in-) Tj 1 0 0 1 123.849 377.5 Tm 91 Tz (dependent draw from the noise distribution. The shadowing time is then defined ) Tj 1 0 0 1 123.849 353.699 Tm 93 Tz (to be the largest ) Tj 1 0 0 1 209.3 353.949 Tm 125 Tz /OPExtFont6 12 Tf (K ) Tj 1 0 0 1 223.9 354.199 Tm 92 Tz /OPExtFont3 11 Tf (that the null hypothesis is not rejected at the 99.9% signifi-) Tj 1 0 0 1 123.849 330.899 Tm (cant level. To accept the null hypothesis, we require both that the 90% isopleth ) Tj 1 0 0 1 123.599 307.899 Tm (of the residual distribution falls below the 99) Tj 1 0 0 1 347.75 308.1 Tm 57 Tz (t) Tj 1 0 0 1 350.899 308.35 Tm 92 Tz (h percentile of the distributions of ) Tj 1 0 0 1 123.599 284.6 Tm 90 Tz (90% isopleths given ) Tj 1 0 0 1 224.15 284.85 Tm 120 Tz /OPExtFont6 12 Tf (K ) Tj 1 0 0 1 238.3 285.1 Tm 90 Tz /OPExtFont3 11 Tf (draws from a Gaussian distribution, ) Tj 1 0 0 1 420 285.1 Tm 93 Tz /OPExtFont6 12 Tf (and ) Tj 1 0 0 1 440.399 285.299 Tm 91 Tz /OPExtFont3 11 Tf (that the median ) Tj 1 0 0 1 123.599 261.799 Tm 92 Tz (of the residual distribution falls below the corresponding 90) Tj 1 0 0 1 421.899 262.049 Tm 67 Tz (th ) Tj 1 0 0 1 429.6 262.299 Tm 94 Tz ( percentile for the ) Tj 1 0 0 1 123.599 238.5 Tm 95 Tz (median of the noise model \(Note: The thresholds will vary with the size of the ) Tj 1 0 0 1 123.599 215.5 Tm (data set and the noise model\). Together this implies that the chance rejection ) Tj 1 0 0 1 123.599 192.2 Tm 90 Tz (rate is 0.001, which will yield good results as long as the shadowing times we test ) Tj 1 0 0 1 123.599 168.899 Tm 91 Tz (are below 100 \(as they are in the results presented in section 4.4.3\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 140.4 146.1 Tm 93 Tz (For a given observation time, we are most interested in the trajectory which ) Tj 1 0 0 1 123.599 122.6 Tm 94 Tz (shadows the longest and is consistent with the observation made at that time. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 94 Tz 3 Tr 1 0 0 1 316.3 52.75 Tm 77 Tz (80 ) Tj ET EMC endstream endobj 446 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 447 0 obj <> stream 0 ! ,,b6gSOXÆkQTx{xpwIFH "Eɝ6.-gVSq' (G\=tZ3qEMUP夓zZ4mD J&+4D&yˡ7jF$e rgNΡ'&V5EЦ[LATC wk" CH~J"[;XD.T2O ES.;)i9.j2_vxb٩AvBr@>R! uov.R*[~?|2 kN;Zw*`exW-fKO\|BZry}4w(L0E(̈ v~{[1+pͮqSlUNN& N[ʝ(=%2y{{U!d)j,^b$.J۔l?=U`y˧hOg&,}Կ`*9B3'6-Ux"ij\LhD"οp,B-`j'ř.[NACPCe<6^t@z/x %/+ ^!K j0O:^% 30t Y%E dA!k*c~_wJB8O% ]H ǩ}%D>o$=ۧS:[f#U'Ϳ,_۩?'Ǝl֨|‚QӔ>!&6~<({{뵆?/H5~owNIdסϫ]?w6LqEy#fײ:dkHeq[?[] X44HKYts\7L;YΑ1a,F֪&ƐyVby4Nbn3}Ař&RKsyVDʬEGI a>$@=/`8[y?el]VKQOŇU'MY=`&LukbE|ٖ?vK{"M nqޔFq/J3 _56S\f@a) |m 2#4sFvarT$X2]b{Ot#W^65̧2L 9y3ܕrP60Ig5͇^ 3#h7eݷ 2(OK\UZԴ 58Ðd:T JlLΞB.$4ilSG/6Ft\QzpIϬ>SiRLBQywLPϣvA3SRi+BUAM2Nb x Zw6]*l)hj3$M*u'fȢ *Y8D;!f%"TBssT{WXШ&hS Bbˆ8 ,?դX9o&Zrk Rsv<;xm g\i~C45~ah{ jߦ0Bf d.:Qa ny ^ Rrc+ hy,MbJ_YPJMqQVSIXS`;$ bsc,%YEXf)綸oEd:,Y9S*{>2nh5U &֯HAE$"@6%0oqh8}},Z!wHh ((N?+E\KJ/$ylf7OI9Qӓ>;nM QemبZg}qۋ/i~@Z.68j|6*W,[8 =Ѡd)32IyJ<UPsfL&d8Ņ~ TG.9cjbUܻPO`ogGO#9JK@kLSMnh˙:s$=4]NpŲ]Ƞj*ۤq:AWqX:D+X?K sq>>o!^kѕ|!j"~Fa+:F,WC4h\gxÙEqeG}S Ie|:ݥ!ʔ($z,@Og8[)ɷ!s6YpL廀OS}1Z+Oْ"dFI >iTX7Sm w;zو帞$@liK7S¢HQ)E#<ֻ RkeB1) <:N`att9U5S X<{FkcHor7tƛmzיuذkn ĸ(N"QI4'V*`fTtד,QJnBS0J7_36.ThM߯zͮJ^e>Dž0{J/ڹδ.3+ݴʹY?\T۳eѷ!&Ǹ(9Uղ8ĸlópr=t$gʰ޳-ǾE`3ڛ:-1,N"Ñϻ<7ZJVm`e낏fZmZIm)PVH1qy; @qw~BP(…`3i̥nFwIle*_(IDN{QAN,~vY z}~)Vx<(ķ*cPЮot ,8ԇ}Rp"Dž$/;Ѿp#~m" U&); WC`]~!bc9*~ʞcEt=B,)qHBT i08<-uBL}* .2)oE“4SR$grܟ) wfr:1pd+$e\rğq@$ ؠ*֔xDSO%8 WČ J|]dg>].\ݨ$11sv4,@|,ð0e||>gb{ 9?̓uUɿ=vbЌ<Q}֙݀m:Kob PcpJ>:#`g 𡊥f !AږV-i0~|*r[& e m[(@+Y$jp25ȷVFAO/P1lD^OvZ* ry?(fJ3Q5ҸUY}/ng.֗A%USȊK†=>7zTP] ;!A;Il8d|=i~?XI+ܲ2lPNNo#;dcBۢZ o)NSZdi">7=Ӎ;Zm ckюu`8?P[Iqs`͈oON[_LMVa&_l߭0(W[pygž I/%iLfSxp$Ffdҥ86]A &m&%}ᶮ׈wpk6n0cȄ0iKNEsrNjePק*w=KXW٧wȶ>o) k^uӁzV7j? f! >s7B **sz6:6 J>m1'09I={GgHO]jۋFgQ\&ç޴#"w~S!qEۨ'VةQ?[V_k>[h-[Nvn~2,Y#{\Db b'`Dg}oc 1"at"" {˕>cp8psxW]@F UEy߇޹bW1/de4,crO%ݺ·ɹܤɸ(< ZQo7;RDbAo"郩)=odܛ=NAc%;kg 1mMF)CBDASFFr ul;`G- :fF`gͫJS pqXcWx\O56[NtxVXsElO-FSr 5`!laT$ K lqa2 pk@x"&f;Iڨ!TWX\ەz; k~!vj/',uNrO-W]!#-FC̈WBT`+KD(vaNP |tcdjqgY~#gAuDtHG<;cbO'fj m*rބnf̎J&_8;_+YzEe4Tl}S?biu_a/xcO߅򋤟2p4}B;}dɅvm2 uL0St,l+ 0[0IT;8.H,?TcQXYǧ}qo}њqO#&'Jii~Qj>ao</W))b*ES '︝掎S&^q~h^F +r c+挿DZ~8hNeᑓwzw-iu\'XρJA-n(#Jyhy?:P:*Z"B,}h>OzEF~v5cUʤ ᠾ,14>rRo5yBّ 9Rxz)O݃rtTgBb}H*c~KYNu ~5QJgo"(곶vB# xJv&axMV,@(zO&]Sn6?y=r̈·j~/4?o9,1Mskp <{;(]n>O%4ٱʲaڵ^sg+Ł±\02 N0&4>/ZE,δ]|btd\ V!l4ȟlfAдe{d' Ty F#M0еgA(㆏w̶ܦȿsS ūEײ[&ǵ/<<<ѝ8/ZǧnwVSgzc*pbS=w!M\St+hV J-LK cb9Vi-c37 LHr.QJQ׊!cONj{4̀"vc~A̒+ޛ:#I3?~SJmE Af 8Wk٫-FHDjɹkly̌=! ۘD*W HyB=5=Zu`n Z4Ir6VTP~dUsw|Bݓ-CY^QIz1٫X-bL!9wNtY*X&b8"4hn|!}ָg"5ogm&TV%sEUJCX|4+rc3Jfe0?Qa$NC(XO%skԜ!,"ZBuœt^LR>yB=lA+qBFY $x٦ԑOMǔ,U7ThIP~4d7 ؔ)zZD_ wZ`0V)en"" {cMS耾qvSHu^ۂHvI?>%ڬ'z,4ڀ((eߥ\>#f챫9엷ꂨ?+<O)[u A"%!B^4bkzggH@U!P2?'F#T Ҥl;"h0ێ<\O|䋦u,u<;Qaks8C>YdBH1]g0\7Su s_|L#oPQUhLEtѳ* 2o|F\PotVw FͲ-ư9NҰb̍KƉZDg _Vkljѡc)"Kys|Ax=ߞXA|sK}%4G^_}-jYgbĄM[ d: '큧6o]z4XNAJ`<B#-PCC(M K(\[vdF0sֈ]o̓󪕕|tG&o1V4-߂ߴ%r)Uߧ;1~N.zÑ.(l fh"bs_!npE] UjoQ )gJ\KaRu4./+kER`6 "Ms!-mz84&ED9_=o*RJK2 p*0f|nv^|oYuxCY@G}AOo]ǘoLBܲ/v߅15" h0&/n9؈ʳtqLMg$.&Srz^<}8&fLAqWANFr; gٸC/t&,gK,Fs7VеC_"\\!g]ANfkkG>ɨdm't*2 . @Ϋ+;3g|U?(ZOV 0?mBo5di@E7ZxXj3w6Ԋɮ *`[k8ϑjYGwyQ-T9-KNn|[( ﱟpQpTPDÚ2qMCRڤayaO,TU[PUSb^z jyYD' "Q;14Wdw9~<&b.r eAʛY9b"Eٓ1.LcuZ Bgc''qes ƣ 8.U,cfAGR`FR]NO23tHPDZlR;s"->5A& "<8{hҭ endstream endobj 448 0 obj <> endobj 449 0 obj [450 0 R] endobj 450 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 626 0 0 836 0 0 cm /ImagePart_2108 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 179.75 717.45 Tm 110 Tz 3 Tr /OPExtFont3 10.5 Tf (4.4 Parameter estimation by exploiting dynamical coherence ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 110 Tz 3 Tr 1 0 0 1 127.45 674.5 Tm 94 Tz /OPExtFont3 11 Tf (In practice we consider only a finite set of candidate trajectory segments. Call ) Tj 1 0 0 1 127.45 651.45 Tm 90 Tz (these candidates ) Tj 1 0 0 1 215.05 651.45 Tm 97 Tz /OPExtFont3 12.5 Tf (4, j = 1, ..., ) Tj 1 0 0 1 282.699 651.45 Tm 105 Tz /OPExtFont8 15 Tf (iv, ) Tj 1 0 0 1 301.199 651.45 Tm 93 Tz /OPExtFont3 11 Tf (where .AT, is the number of candidates \(the ) Tj 1 0 0 1 127.2 628.649 Tm 86 Tz (subscript ) Tj 1 0 0 1 176.15 628.649 Tm 81 Tz /OPExtFont6 12 Tf (c ) Tj 1 0 0 1 185.05 628.649 Tm 92 Tz /OPExtFont3 11 Tf (denotes candidate\). For each observation define the shadowing time ) Tj 1 0 0 1 127.2 605.85 Tm 136 Tz /OPExtFont4 7 Tf (T) Tj 1 0 0 1 132.25 605.85 Tm 104 Tz /OPExtFont6 7 Tf (s ) Tj 1 0 0 1 135.099 605.85 Tm 100 Tz /OPExtFont2 15 Tf ( = ) Tj 1 0 0 1 151.699 605.85 Tm 89 Tz /OPExtFont3 11 Tf (max) Tj 1 0 0 1 172.3 605.85 Tm 66 Tz (x ) Tj 1 0 0 1 176.4 605.85 Tm 141 Tz /OPExtFont4 7 Tf ( T) Tj 1 0 0 1 184.55 605.85 Tm 115 Tz /OPExtFont6 7 Tf (s ) Tj 1 0 0 1 187.699 605.85 Tm 92 Tz /OPExtFont3 11 Tf ( \(xi) Tj 1 0 0 1 199.9 605.85 Tm 59 Tz (c) Tj 1 0 0 1 204.25 605.6 Tm 88 Tz (\) where the maximum is taken over all candidates x) Tj 1 0 0 1 450.5 605.85 Tm 62 Tz (e ) Tj 1 0 0 1 454.1 605.85 Tm 86 Tz ( values tested. ) Tj 1 0 0 1 127.2 582.799 Tm 90 Tz (Instead of random sampling around the observations, we derive more useful can-) Tj 1 0 0 1 127.2 559.75 Tm 94 Tz (didates from relative pseudo-orbits. Following section 3.3, given a sequence of ) Tj 1 0 0 1 126.95 536.7 Tm 95 Tz (observations, a pseudo-orbit of the model can be derived by ISGD method. Of ) Tj 1 0 0 1 126.95 513.45 Tm 90 Tz (course the quality of the pseudo-orbit strongly depends on the parameter values, ) Tj 1 0 0 1 126.7 490.399 Tm 95 Tz (which also links the quality of the parameter value to the candidates used to ) Tj 1 0 0 1 126.95 467.6 Tm 91 Tz (calculate shadowing time. Points along a pseudo-orbit can be used as candidate ) Tj 1 0 0 1 126.7 444.55 Tm 93 Tz (initial conditions of trajectory segments. In the results presented below in sec-) Tj 1 0 0 1 126.7 421.5 Tm 91 Tz (tion 4.4.3, only three candidates per observation were tested: the corresponding ) Tj 1 0 0 1 126.7 398.5 Tm 93 Tz (point on the pseudo-orbit, the image of the previous point on the pseudo-orbit, ) Tj 1 0 0 1 126.7 375.199 Tm 91 Tz (and the point midway between these two. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 143.5 352.149 Tm 95 Tz (As for each observation s) Tj 1 0 0 1 273.6 352.149 Tm 63 Tz (t) Tj 1 0 0 1 277.699 351.899 Tm 96 Tz (, it has its corresponding shadowing time, for a ) Tj 1 0 0 1 126.5 328.899 Tm 97 Tz (segment of observations we have a distribution of shadowing time. Our idea ) Tj 1 0 0 1 126.25 306.1 Tm 95 Tz (is using the shadowing time distribution to estimate the parameter values by ) Tj 1 0 0 1 126.5 283.049 Tm 93 Tz (identifying the interest area in the parameter space. The parameter estimation ) Tj 1 0 0 1 126 259.75 Tm 91 Tz (method introduced in section 4.3 quantifies how well the dynamics of the model ) Tj 1 0 0 1 126.5 236.7 Tm 92 Tz (mimic the observations at a fixed lead time. The shadowing time distribution is ) Tj 1 0 0 1 126.5 213.7 Tm (a different flavour of quality statistic, quantifying the time scales over which the ) Tj 1 0 0 1 126.25 190.399 Tm (dynamics of the system reflect those of the data. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 318.5 51.2 Tm 73 Tz (81 ) Tj ET EMC endstream endobj 451 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 452 0 obj <> stream 0 3 ,,Дb6\U}(o[IB/dˍP !_)[r(q[T>rF{#qKfUTJ #LKtjg!vXi).mn5pH'jpؗ$)i,JΖ ay!O2)JcBq.}A,xiK&Ѹݱ|ANak-T]B-MP 9iY-xg<09r%3&Y^p5sj<Ŋ +ab•/Ktu _d֛IZ)o#4 ؁vv+w&MuF\)chu( EYNB+ q7KQANhf}z̃7N<;߲e*r6D. ]#"J)! ږ'B+_mJzz[~?e8`z\axͼ0t_[ػ9ҕj?,O Ґy&Fod _L]r=9iE_hW;ɻƃI1~ eLind;`y%x".m|wOZjH)H{2lڗ51wG8A/+!פ,05v|̳Ɠ^e3Fl 늵`s.\|H#m5v|'$ML½H43 <C ̖3K߮vTz0ՔS7ɮ$?>oH&ڊՁ&;'1mnzI M ,Uwc*HWB}b \0 # X'U4Q 7GUJ;ᄾc#ab+~*s[:$3X^LBK^jbu^ p+}z)$,injΰ27bR>\eg5V>{FtA| Bh3{*&lu*mX5 K7$ɴe)25``eHn?.^(:gQb4Yo2UR^yUkB D[+6/d9:QWT1&S v~h[us7~zJ 'EG`;XF)ہ .j5ŃVӡd dr; FP}RQ06 @>ˢfCS3C"¬ n#=Yp{܈cofņ $Zǃ*X^= +VԹjgAzxf4;dߦg|g'A;lwWs98J}Z$4 sX|:75$B'^NӑL_h/d3ꢲb9`4 j1-Z=xKo<#ȢꮟiƫL`AE̗sdY |٢N<- Ϋ&-(:sJU7GA3 /F5爖}^4ZXw%P&`JrJ*sJIR@vHݱ&L a' $5U39nyW¼ߧ>9-Te2غRwHw"^ ÑSME*俨>h7`>,UCbHg5D^/ׂ/M  M:  0Wr/oSQBU.Kg* P'0U*4: hE᢫IgbWW>{di'L# c]~NȊ7_,n' 4SN[Yv ye6mHJ]̆eN["G0ouU ÷ LQG8 dDբn{fͯ ^Ϳ4`{p_!"zӐ;#Jk!&MX cX=i '*qXƥ68:fn}( Ue"Ih;@yXG=:s-ˌ}쳫+PdRr;ɫ ٭F##\V1#kۢE=?d}szU̼e27*|Fv:%*Dzz8V2÷٩aծfy@w2߫e"tvlKGi`EXSLiRȾ6;Y5Y#H>3(m+G+!"wMP\ZnNv%dO5 5Rٵ43l$1fjqƵ(e|ۣi)Ų8JtMpyc!vq]"3ޡϊ 2x;_`/lzv7_oZ<ȝ6j9~H؅S8bbeŹ3im{g#6Ju|3Z71\ιehx_a)ΤdzKkrԨo䷱u|t`UvN*ai f$x4.ǃ#8I3 Nz|ϝ%!)}w d+L]Pgg.a00o`\hPw7z|GH߸>cVcEP(ZS@.Rb6E뉃tBQdTw ^\ͧG~`<(֛7v G 6|c;] $BOCvL{wBY1ZZ[.+8aXo/U޷㲧x84 ~`E31.53L39wr<8'j30V^߃oif(M>UdV n̿,dj((/'gMwC. kXj)m#,=$pKV!b$@fkgHQWrl\%̿`q:sϓ@~ڣMLçLf][.ԍx V&YAp7n"mq&z"` Ρ&YIOqm53KoN$^rZhwNTM7eߕahTi(Z]bn+L41}MS.G=hLr%qz230+i="w\q&8-6ڝR9=hW3LYsjf\?L9n)1=H1g>&(xH.'Ku?ԉXNahX*Tiʡp _^/׶ Kf&%Ҳ8B!^̤Q NNs Ol~B5n ~(d5R?c5Wf Wvr;{/Rʏ?^f9B?1.3S ގVY`ApҽfȚyePb8(qj/aNN-<{Q'Z\t$O>>kAdM5Qě ҅jm=$ `2T`F;OUY`ر]!U28Q.sz*B_OZ&#`p\V!YA7 B?&ɯyw6f~ w :xU-rB~ ܦ˩k{AR.uVXG499~;*Γs̙* 3ԈbW']2%öBSxF$1;2A ;Ugf@_:'SRȺ+єm3]..rNW.Z'y $M{4NZU'"2bN{|d]rEH8axI^F=jG~gƁ /.B;hl Ō];K=_ {9(߿,zzcADOw}&4{s)of7(g֓^dn 4_Lgi-(tOtȱ\Q]ˎ94qj-fG_~x03@d&}ޯ$ޣQP6swW[e*JɗTy4E{8-GemɠSy3tLf֫iNU)[\ xBE<ð]n9ECeG't>qZV˰`<wTݥgleQbéoDof.7LCF(iyZJn[+! eT"+z/VR .OW솷}|Ӑv;РRZc;J> p<%rhf&mA`mG<#$Μg~؇mEPq=0MdY| /^.Хs۵9čdFj)[@m8o=vj򷯰a7w^ell׮ߡ j]}F]b:?ڙ.(T {GF) (/J$D!V;Jrk1 |>1ٻ ډ;j\sPſ_K.cW?KpX׹dY0~wr&oh nLSkx'clLH-ASd+Yq Zf8/xya8?>r{Wo@|./#v%٥'¤C5oRM t|;7_tƇFŢ۳9-CW‹`5ವ/:K+f±1ʱʳ+2Ϫ`<|2p+_ϨoRh*B '߉Z7'-}lUJb܇kln/ bB**`g|Y+@ܨ=&𛡢$U (vBvYJoD Dɻ})IkD:y??fX.tzȻ{*SPU\bC 95ASAרʈ/Ōݰno_2pauG!MY+Sz*>* G7l5tyLrltOvp~ݣ }jGoVaUӾ5Lڷɵڷ=J0-ʙ5Fl찿$˶_ɓ:\0Hg'ƳHڊ0Գףδ7"ܛ'7dS?%1l0P°pE L*.CWMf&ڮ"h?2 pHn7:ZPS ÎPgB Gh<+5|Fem>\x3NvܼcM"C# ‰``&[hV0 &.NP  c 0=SHBo2+Y6f/r<s\:"AeE[A.8-\eiW.xvdrC d/20w<;C,hUQ_2)t?8[k94;-Rծq2[GBWf0>J| ,#k ^oe#(g%N'(K+J/"-/^$`4nM;O/DҦD+t&81a@󋸎~5 4Hi3yBhɓM0sLy bݛO=U⧹4tXu0 g$Wg],.V%sU?CY8,KU0r(3Ia+hn!D#d|hjFh-3 tҲͯ4D?D.c'-w=gTosVS6R "Ӧcfct-|M}=G]ȋGC@!7R9$lMc.t?_[.JK"n+YM,7 `QUd֋H2{w{a w/Nn : %7خ=^Ebm-}7Qt}dwk541l_JGgxKBk##66(y_g.d m9QC`h:@T ! S-#hיXI/ !#}axO}|33lbcnTjQ nyyWGE4hwU i."cuv7bng(n/+QCM6 "Z^͹ՂU ;hucϐ!';&<8w)* Ecgrϴ:+(KL/!?{[(5/JK/I$](}D:xcA*X'DDx*d#C@._О>NW[w9?r:8ٌCvp/=l#3 ЪhQ?LjߓwZeHw(StGh_pq#l]1rm_;ObzU2˓7|>Ͳ3o} DUS˔[Nese3 Eƌe86͍* eAp̧SʚMi-hMG=Yrɮ)馡;jYJ[Nh]fflj0}"uWՎ*5_L]< 'V-;ÞJ eUyw`|S "0K=WƫA񕺄9_6;gf[V戭dzg_s~B-͟w~o %9M3NrYM>K $7C)Ii9Hdgh+D=`S@|\R@'۰ts([xg~v'~Vz&P"n~f¯} A.ˣC>;l  FFƉJ٪tẘTJ:qbӤXeq@)!H%|1sD[4jh;-ibXQآ6(c}c{*V zbp~#c#t_ zQ.(Ye?~<Ezqk 󘣇͡ !4c9 G%@,j㟮xEjUdP/#FZͧ#暔dinFOJ( XN|7 hG__|3{*jRzDSG`qImR+d62zw9}(2* _?7:К1B*YJ̮j<_ON?.Z5QG:Z5}6ѿ␈"+^AD*2s*Q(-*'^=|?2Z7ICFt0r= Im B7HZݜ>>: > endobj 454 0 obj [455 0 R] endobj 455 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 625 0 0 836 0 0 cm /ImagePart_2109 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 180.5 717.45 Tm 110 Tz 3 Tr /OPExtFont3 10.5 Tf (4.4 Parameter estimation by exploiting dynamical coherence ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 110 Tz 3 Tr 1 0 0 1 127.9 674.5 Tm 109 Tz /OPExtFont3 13 Tf (4.4.2 Further insight of Pseudo-orbits ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 109 Tz 3 Tr 1 0 0 1 127.9 644 Tm 94 Tz /OPExtFont3 11 Tf (As the model pseudo-orbits obtained by the ISGD method strongly depend on ) Tj 1 0 0 1 127.7 620.95 Tm 91 Tz (the parameter values, the dynamical information contained in the pseudo-orbits ) Tj 1 0 0 1 127.9 597.899 Tm 90 Tz (can help highlight areas where the estimates can be considered as candidates for ) Tj 1 0 0 1 129.099 575.35 Tm 93 Tz ("good" estimations in the parameter space \(20\). In this section we extract such ) Tj 1 0 0 1 127.7 552.1 Tm 91 Tz (information by looking at the remaining mismatch error and the implied noise of ) Tj 1 0 0 1 127.45 529.049 Tm 90 Tz (the model pseudo-orbits. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 144.5 505.75 Tm (As the ISGD algorithm, introduced in Section 3.3, is iterated for a finite num-) Tj 1 0 0 1 127.45 482.949 Tm 94 Tz (ber of steps, the minimum of the mismatch cost function, i.e 0, is not reached ) Tj 1 0 0 1 127.7 459.899 Tm 92 Tz (and therefore a model pseudo-orbit is obtained instead of model trajectory. The ) Tj 1 0 0 1 127.2 436.899 Tm 97 Tz (remaining mismatch error after a fix number of iterations of the ISGD algo-) Tj 1 0 0 1 127.2 413.85 Tm 90 Tz (rithm indicates how well the model pseudo-orbit converges to a model trajectory. ) Tj 1 0 0 1 127.2 390.8 Tm 93 Tz (For each parameter value, the speed of convergence also indicates how easily a ) Tj 1 0 0 1 126.95 367.75 Tm 91 Tz (corresponding model trajectory can be found. Therefore the magnitude of the re-) Tj 1 0 0 1 127.2 344.699 Tm (maining mismatch as a quality of the model pseudo-orbit can be used to identify ) Tj 1 0 0 1 126.95 321.7 Tm 90 Tz (the interesting areas in the parameter space. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 144 298.649 Tm 93 Tz (As the model pseudo-orbit can be treated as the estimate of the true states ) Tj 1 0 0 1 126.5 275.35 Tm 92 Tz (in the model space, the quality of the pseudo-orbit can also be evaluated by the ) Tj 1 0 0 1 126.95 252.299 Tm (consistency of the corresponding implied noise \(defined in section 3.3\) distribu-) Tj 1 0 0 1 126.5 229.049 Tm (tion with the noise model. With finite ISGD iterations, it generally appears to be ) Tj 1 0 0 1 126.5 206 Tm (the case that the final pseudo-orbit obtained corresponding to the true parame-) Tj 1 0 0 1 126.5 182.95 Tm 90 Tz (ter values has an implied noise level no more than the true noise level, inasmuch ) Tj 1 0 0 1 126.5 159.7 Tm 94 Tz (as we initialise the ISGD algorithm with the observations and aim to explicitly ) Tj 1 0 0 1 126 136.649 Tm 95 Tz (minimise the mismatch cost function. The pseudo-orbit corresponding to the ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 95 Tz 3 Tr 1 0 0 1 319.199 51.2 Tm 75 Tz (82 ) Tj ET EMC endstream endobj 456 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 457 0 obj <> stream 0 0 ,,cj8BtX{]' V :kr{, ^a<0b6ˉ46H,EDm{ufZwoOb )찼ʥ|hS%$/AO%f!.b5_(_ ACf8H7EWOpW% # yWrNCG4lZA}7-q)C[xW>|7fG WH8c,BRt wq}xfGxZ5RCX e%|'?F뚇V)D7Es32u3 $v[*;1|h!jpmLwܬ5Xel]t( +u{Bþ3ހd9 .&Q@Pk-TknrdsOOa9M#s(*շ5<7K{MPFa M i]]_zT¯ x #ѮrIr=+:XRyQa1Wcnȉ?QJ?3:$R0!) _e>0 mI~/Lݻ]07,.%y&HڼD;<0nAL‹G,w/j0x _*ş1"hJު6&J&7KM@GGU] "j0LKCwܣ>+mٿ?"*܏YA+~ I )pCo1ůAL-ݳI6ZXaY5i:r@֋"_p9n ф\f{׈F⍐1*"ApO"#wB 7`Hn29(;[uwrI@+ %X(F򽢕  LAbj_@%%2j2,wj +AHSח{XFB'oގ`/T%}^m T4AD,2zLn. !C}S3yKC2k>K\`stdeZap: W}kMSrj7GF9#:0O t$PѻBe8w؟#Cgy7\jHͤ(3l  @>T>n#S\C'8'd#䩋G͊U5Qɚ b9K*ꗶerEe4_"@. |yא%+vj_`E[l[v-I6H|C+ED'' XxZ7n6tQ/bڇNΥ34]ԟ >ͥj}ʩt5Q/೒Bd]9y6HSg@2JG GSU ௘bÆa:PX dPM(8sOg-QHle5(bW385upl\zo0<#=D_! e+6n^Fxx7M$ J_z41媐>34<^*xf1{./f+,ՏFv잝J]ikt }LRRM&?C5Ti⋿ةiU;*ܸ&R3%VȗTf 9X[-20' _f{l%X[*iEX:(@޸)XϗI\ 2! 3%) Lt,G$Uʍ n} f Mh~b&{M ,ݽ}t!;b3tYKPU@wɿUNl!#'EWAP'B̩N1XuGpf0!ۡ-+AMWCBl% ِ2" B}Hxf?62ZFas4CC;]0*ej ʞ H;>nǼW~44 `j{;,Ԣy R]y65T\`]B 3HƗB3P#;m灾qӒ5.0<?;:FDv 'Qlf{YfwnRk$Q%l.t=-\I.Sp~Թ4Ι~םnж{/dN()lagl8I$!bW  $lQXoxgQ#%!hʓhLo@\t.#P{1&<^ > ?:!c+932PȤt5!gBKVq]ۅ_mXZfe-qC'њ$W΃{jF B1}ZCr+֕nmtKlY: b`:10{/*QM:ک8>uuLlgeRkɌ5!RTJaYuHO*DO;4da$c]]̢ Wاr&;GXZ~] ۂ+h91]25>`v"XwkL 6QVoZt7"@#ȭǦ4dv6X6Osړ'#,:Jr8D 1 2߱,|; e> K6=*8ϧ]DxUs^DG~^z< ̭T(ax?>MJ(NU0jӖ le \' cv!ORzť]YxcD PpKvhd[gkVCz6̑NE%m݈ [glJ.~$+mAV.  E&hEE~6W8ݲ{fŇ6GMFt}Mf^(Xfk8CCp٦?d1Jw=.,'H,7FhѾVnۃec\_C *%|/P1ѕ1.Xyj=X$kr^G݊;`D^Œ?Ir8r2A/L͚<V3BA;/Jx }M}}YYqYva =7 Q|bv?,kג/zz7cZέ9"PaDWa]4?K/%4zPaBB;0WTR#=̥<Q9\:nj.ֲbN(]*OAAH-_Tamz PkNƊ/6@Pl H7S/s?}k K(%3͋N 6\?"|0ԫDT'ΣJjoZHh#-CY ߖF%4fk  0HGVlͺ=,co67KBR[+/_Nyt[rk#yr..I ?.q^]{-i]#d\bgnS^o{/sT7,50ZӋX)6l!&(g6MS0~ 4dB;!ܟ=' P5~$O ryu:"1ό $%ܰsI0e"AZQ?9REt ^܊&c=5_ K0ch}7nfYe d*F!^w.ى{]~0 }xʐ_ńgwR#׽9lMH~Leu#?e듦US]G@P0,R:\r5V 3{#وsf):.EDp.:'B}2$4WHwHČ~QCϬy/iUJTQ;@`Uh;=Ir_]@L=Vv{!S4oAH}N؍@ux>卖bdy3ƞD %J "xOj0/K ;%X@ЄO- JW}S8d˄]:;DZD)Y s13{~ OY^݂f#񻪍!q hSJѽUMh }9'"C_NkX43%)ˈ{>B/T\Mr}_N'ME 1INiWE!QBb'"6 q&Gi%ExbE,WTb\&;`,+2`jV޿&i kX hxWuk&Љ=H}{T7)NGPO12_8NSf L_nh!B08wbsz%>ٔ,D!@ܿ3W^%97(xٜ5#tvY c[g[Jq؁}]0{1NUV'TUK9n;pGp ~jft: VgR5%x<]EqC#լ!XafBSyp9~:o/FUh@ǰ^)%]b7}/!'4exZfTc4΍hĥBSTLpEçZ49ްʙ9,T[Vc`@TiX2E$,b $qOkMn7(v Acj+Z{sĂjajA"ZLNbwE|jYn:lH@l}X;zC~,{[25G.B)aZv]5NUqٔ5){ ,U(ˢ|J<Ɓq俫~aSv]֙?mXbm ӕnDN2#01f-*z}?4 o &mԲtӃWIA_?IB6~ .D.΍_4<X_?xX0[hc0Vv3Z/EfvL`ͧ6Sþga^\3kPƼ~-uM?ft k?6$/T&&!,y;~J"Wդ<6/DX^esg 4cR14هeXp!-(Ȣ1fڬC`:sÞMe+ssJڰ~n*ؠwڻPi*I ![m+{773 _I|G}G0WUM5B5|MX~`^VdZf3fԊj_2Y\<3и]*;Z"<[Vqr4o&7 :L.)m:=sHrudHH3-3؛(Z9:ONCx1Ҽ&>Oʱ$ϛ[$)!̱x(!/Eآﰭ2@s+䧕Ʊx' }3=5#I2s5}yQQycb5B|IrLH'KUҾ 1Ğ/D(~S?Au:ԂƛO&E@ZJe7Jl|!W^E5!?ãם joIx{ꁄܧٿ1K[瑃 PZj=,Io30|ՀL-rk/5Qb1"Tf_+^hycȣa!kkeo7 "c -[hSpP%t6$`_uVA$N 3Ensh>qڡOaksj.T1 " `:ahҰrtn}Ч΁򫣆"iRSM (J1x?}usgSFWY\ @7LY'ArT]6 pٍ̄Qc. tmh @3\M?kr+xKcEȀ='-"`Z;b?22.4gR&<:j* F4 HG""(@ꓥ[XCfǂ#j*f\ @l"ہltb;>{E[Gߡ?n7!yS"ZD`5V舡RR ZWo(_*0P1w gZ_ҕ0dXE,ֶ!!Fy/$/VYA&S3t ,Q0aIx>B=uIm(CL{r{^m͜v ,H'R] W"3+弿*"R($L zۚnص0:EM'd͔ʬJ Fs<:58Us2e*sO4c?{ E.݈$/УaGűY~_Z#}}ٓ49josR`0Ǭɳ0(ޟ+;"cGv#eHW7WJ:W4lfB4%H-,ibYx )ZAFaKcQ3iBs?Xb>i{y%J?x6m =v*k$A^''z[dvx7pdv֙)S'C:7 B!+?)e23Xݟ.Qeh-r=n$csz%UJ_c_J-j;ە_z3{- W6JCpn]ĥO=}.Ů 7Dnd93*240a17czZkZ(Ǭb= GD11י3%jE> endobj 459 0 obj [460 0 R] endobj 460 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 622 0 0 839 0 0 cm /ImagePart_2110 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 185.05 716.6 Tm 108 Tz 3 Tr /OPExtFont3 10.5 Tf (4.4 Parameter estimation by exploiting dynamical coherence ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 108 Tz 3 Tr 1 0 0 1 133.449 674.6 Tm 104 Tz /OPExtFont5 12.5 Tf (incorrect parameter values usually have an implied noise level larger than the ) Tj 1 0 0 1 133.699 652.049 Tm 98 Tz (true noise level as the implied noise has contributions from not only the observa-) Tj 1 0 0 1 133.699 629.5 Tm 101 Tz (tional uncertainty but also the inadequacy of the model dynamics caused by the ) Tj 1 0 0 1 133.449 606.899 Tm 102 Tz (incorrect parameter values. Figure 4.7 shows the standard deviation of implied ) Tj 1 0 0 1 133.449 584.35 Tm 103 Tz (noise changes as a function of number of ISGD iterations for Ikeda Map \(the ) Tj 1 0 0 1 133.699 562.049 Tm (ISGD algorithm is applied 1024 observations\). For the true parameter values, ) Tj 1 0 0 1 133.699 539.25 Tm (the implied noise level converges to the real noise level very fast. The implied ) Tj 1 0 0 1 133.449 516.899 Tm 99 Tz (noise level corresponding to the incorrect parameter values slowly converges to a ) Tj 1 0 0 1 133.449 494.35 Tm 100 Tz (relative larger noise level. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 259.449 462.199 Tm 170 Tz /OPExtFont9 4 Tf (u 0.83 ) Tj 1 0 0 1 259.449 456.199 Tm 154 Tz (u=0.825 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 154 Tz 3 Tr 1 0 0 1 210.699 456.199 Tm 155 Tz (0.026 ) Tj 1 0 0 1 210.699 443.949 Tm 158 Tz (0.025 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 158 Tz 3 Tr 1 0 0 1 202.8 431.699 Tm 140 Tz (3.1 0.024 ) Tj 1 0 0 1 201.349 419.949 Tm 170 Tz /OPExtFont9 3 Tf (-) Tj 1 0 0 1 202.55 419.949 Tm 178 Tz /OPExtFont9 4 Tf (o 0.023 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 178 Tz 3 Tr 1 0 0 1 201.349 407.949 Tm 90 Tz /OPExtFont5 12.5 Tf (g ) Tj 1 0 0 1 210.699 407.5 Tm 158 Tz /OPExtFont9 4 Tf (0.022 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 158 Tz 3 Tr 1 0 0 1 201.599 400.75 Tm 107 Tz (73 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 107 Tz 3 Tr 1 0 0 1 210.699 395.5 Tm 146 Tz (0.021 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 146 Tz 3 Tr 1 0 0 1 210.699 371.5 Tm 155 Tz (0.019 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 155 Tz 3 Tr 1 0 0 1 210.699 359.5 Tm 158 Tz (0.018) Tj 1 0 0 1 226.8 359.5 Tm 139 Tz (0 ) Tj 1 0 0 1 230.15 360.8 Tm 2000 Tz (\t) Tj 1 0 0 1 274.3 355.649 Tm 151 Tz (500 ) Tj 1 0 0 1 284.399 359.149 Tm 2000 Tz (\t) Tj 1 0 0 1 323.75 355.899 Tm 148 Tz (1000 ) Tj 1 0 0 1 336.949 355.899 Tm 2000 Tz (\t) Tj 1 0 0 1 374.399 355.899 Tm 151 Tz (1500 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 151 Tz 3 Tr 1 0 0 1 296.649 350.85 Tm 158 Tz (number of ISGD itoralion6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 158 Tz 3 Tr 1 0 0 1 133.699 323 Tm 105 Tz /OPExtFont5 12.5 Tf (Figure 4.7: The standard deviation of implied noise as a function of number ) Tj 1 0 0 1 133.449 309.1 Tm 102 Tz (of ISGD iterations for Ikeda Map with true parameter value u=0.83, the black ) Tj 1 0 0 1 133.199 295.399 Tm 105 Tz (horizontal line denotes the noise level. The statistics for tests using different ) Tj 1 0 0 1 132.949 281.5 Tm 101 Tz (parameter values are plotted separately. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 133.449 225.299 Tm 108 Tz /OPExtFont3 13.5 Tf (4.4.3 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13.5 Tf 108 Tz 3 Tr 1 0 0 1 133.449 195.799 Tm 100 Tz /OPExtFont5 12.5 Tf (Panels in Figure 4.8 show the standard deviation of mismatch and implied noise ) Tj 1 0 0 1 133.449 172.75 Tm (and the isopleths of shadowing time in the parameter space for both Ikeda Map ) Tj 1 0 0 1 133.199 150.2 Tm 106 Tz (and Moore-Spiegel System, the true parameter value is denoted by a vertical ) Tj 1 0 0 1 133.199 127.649 Tm 101 Tz (line. These figures establish that our approach can be effective in 2-dimensional ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 424.8 355.899 Tm 154 Tz /OPExtFont9 4 Tf (2000) Tj 1 0 0 1 431.3 355.649 Tm 63 Tz (. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 63 Tz 3 Tr 1 0 0 1 323.05 64.049 Tm 88 Tz /OPExtFont5 12.5 Tf (83 ) Tj ET EMC endstream endobj 461 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 462 0 obj <> stream 0 ! ,,"єjQqlv]-t.o:He:z*HeAIXa($z|nϬjɷ,WG M(ǐ[* xXp&~ͩoS@}VFp>jRe!Cbu1IJ" SJ[9;\8 ?oىIKWzFJMJgϘDʤ1y$勠4.jYxJAϡW\uv#V7Z xHo= ;pxk U)!7uvLh$[D, Pr5ccN N=Txe4l}5=P/-,ߖXX|T. *ʑZLZ8tTЋ`TN|Lg_DwE-zKC@^} #p,p#U沵7Yg-?+0d/RY^(B$\[T_z˱|} kNk;p֔ 0"T4Fy64nY*HQxBۻ@3S_-N'ۻК䐡>6xq˪pt8)R%{ʂP24.-?g+O2k9slv[-k?l7ﺽnu_@$}`i5D V "^/mHO")0>ڸTrʱ#/)Qp K w7uD齅j'0qi'(a:YOܝRgi'OOLw(rȣU'pjA=u!g)b`'r7İbP~\-%MxJVOj*,4&(Zk.+M 8ng>Ѩ&5VeY(݂T ڲVS @&4`PR^ ۝͓gFXZZV,筼u_]gnsiF@ >N[2(aOsz»Mu4I,e#jPo!֞9Xq灩-s ߏeL:Uj}Q1=Kgcj]R%6 QHZ$ÏPC*zXeKӖ_}}+.kd֮iAv%T.Y. |RiT{sj[E`U^ ! m8A2\\Rf^qMIaY´ٳuE M̊>نYQ8#e3,? )zX=4L*:*Pm[㝡 <aFU-rxsM~k센F(oY>NԨ+Ȉ݇ w&Z`'ህ(MA@JȪD 2Qť!LmƁ6H{kD [lTꅯ+i۰5!}q ee/"2@R C>-۰ɜh>GhU‰֢0A;ɲ00A+~f'4dyH #v5&I 0/n4w//YYRpt2ye[U=WlP{OV왦$B,CT,:+m7( iH0aEԑߍTKi'YYjφy!D (-%9JYbS.h3YWd|ψjJJH΄y8Q {҈i%Zx0Y7lf0lZJ^M]f~ɕN^?yU j\ N w`#rA-[p, 7(;UhhԄd\`ga%gSӦ̝uQS-m {2ƠE.HmIm"0Fv xG;`#[]0Pe[s^2Ƿ+̜lY<,k0m2^8tO*1HxMh -ȇչ#U*1ّzbH[ )]uZ)@,H}ߦ=NsK/6B [MiN.xDG=~$JmsAX3Dž7/S0J tkҡNjVB$.ݘK^M)qoZﱞ1잌dXծ+:Ev6(͎^W!"QktCU:, >hpڙ9[2[jbBiq7W9UE>/CjV0\a-ǹ*qyg_[*-_H;vHH[hp0&"GLdSSmgwz^NtsWĉoREl5: *5/CCQt6mYj٤n bfᔁ" E5p=8WXSIMH&KpDW+>L)@lpr'@t.0Uss: a$TF<[(e8vr}*JkI`]8p޷]s`BHMBG7LY"Q89fz՗*B{l=#1 NX@,q-3=&k}Tlu@Y|Єė/wI#?wQegC^|17~: j+*@0R##djcMPp>xoQQ˖j&G+l3 |&F 3J Ǒ-[_# \a6INt?x#nx鑙Sױ^~I^E^_}4 !7ڃE["*~lYҖZ`筀DVYh89K+0-U'*[ɤ]1r1_؄qo[tVHurnD#Hș*^b7˧9'κ#ꤢʃ\4} ֕8DpI[bx#xX*&|EVx%-!s8Si˚j5*ͪ 7xp,.nf~.MnW%1#"4gC,H$|4%Q/"t `z2G3l0vAvDy$o:221r-lgx͉``pV6wkI=C Iy~<`VuZvZ\/V.j^@'Bdi\0ln]8]~p =l8*Q޳(n1ݗ"eKFylA]BeV8|5; J &m'2Nj%ԇ !á.* _[DPfՐ.tL7=UDs0)o%8Q^z̃ Y- KsȽ3 J9ǽzBJL$%`39?{i}Pe͑w8pd4I@>ǮB.帿Sڰv?ʓ"ѶٕuK'5y?ذOV:WճҰաzZ:%t&־$mEx ^J\d+3nPwEv0Ũ|xmR?EG1 cyv咉{ӳC TB:G#eGi-yKb&mY5U:[~bпPhQj />Zm5i'-m DPΎ=P;0V̥J5? Ҥ{M90DO*6D=y@W)wA`Oz(J k p jIRZ9_iܶӹ0O=eE{edJun w{YBt] er7`MUj%ʇ;LbQ]rعXRP{s8vLayges7ql]+W!M)CzRyBUUn#ꇨ04y3f˪0{f7+vږn$.Q@Ta~s."t 5غ@lMT6 ^dq`)%)V)IUs./`K+l3$LJ4DN %-+&oe0;?N^ai0L!eN|-kgZc?ŰK[{T[hE3KoQ/9El) #;eD~7ȔI37v%շȀwYA "lBb #*qa2_vŧ>XӃX)¿܊gmB32uL]؞ZC5/ `Cj׸#>ͼ*( {3p}={Gx_dR Rz@6 zޝX._ vTL KuR8)b؅RiB0^ݫ& T:B2h.7G=ٝ5}4zv:;IIKyT7ئn=qwd8]fk5T \ZuD/(V\_D?dO&]uvπmm@2U `63ߑi]r^mڐTqj~S(HuUT3S{Po)#/{WU"<&4\RkfZ OD.S'dj)B {6BM#OSml%:nΡm|kӗ"]S{J5 {9٭Z-%sَKeYgƽcgBҲ`92%2 tuz|g7U p:~[v蠷)үoRw$B.zhe|%D}`1!4@8]PqHFLN4Vo 4 -W6*;`:cn8MR;V)/-__x_7?_zB"Qn4pȥvC`ڑvWׅ0gbB UHf{b+\w۪RkһցL͝dN$JdZ 4yu͛^fA*- :WO/MlI 5">\w)_-e6|]2wr{xhyRYvyv#O{Aw2*_TMU!6oFC_֏K|6kPVf\3F`%0'z?l_]|bC$yNAI¡mofJ#D]jf z;ey 2|˜~ޝh-~ n Ę1 a@m9|n,; eZr |z ԏ (?<&(TnRe"gG6J^a 3ܝ̈́h&$]rv"`9蛤A$kΙu_`ulvw 5v8\ɰ_3ZX##RLr"=EHK@Ƶ 0sBP!R1Y}86M"킓<$eE;fԪZ3x5~6jL~ J1\p:Zӥ~ igC[.)7: &>#U W-=& c@42]81bdn'˦U35/}ԮÔ7K>ܷ1|ݟ32.Cɡ'ٰ&b5X7(>')ܲ⊗ܱąv<,2~*7qdC1(|qȼ 0\ğCgp C' Jo;qRCv(1hVד :/c|W R9A3UȧХQohzV4Ѭ/V6.:<E;N?jy D"(Ϫè}H"]V# X@IY©x5J O'/X]&Y .dHkow!]zG-qo"<։8 3gcnYo,;ϩ0mbsx{M -5sJ"m5* QsP*YfG/mqC>ݧn_f@q#{esݼ"4;=7T qSPѽ֤0?6^ԫH/qARE#@#* ng| +)_ӕinLcB>1 (4٧D0H}fu4zO4gY x]AƔ=?R6YVbx4@!4[AA"g7N,Ԏ/vH B=wԪ?p}aG `DdtKBYe+i5#[†RCNcw 5z : $I̠F  ! (&k-MbX@_:4;6i%% .cD+4!@=U,ԫ@0Kf"yƔn&B~432I6ۉdPlQ,3r['sȔ[o%7]%ԯY|`5:lmV\BOHK~%c'z@fK S .TJ؀zkg ΪoIS33LaMO&OԼp:,P@E<ˢZ5cu/ şfЗM ]樤`ᶉg[4=zfhYŧQ$0kR%nJ`U-Gi%!,B_vχb{x|WVg }j-L:yQ/_S%)M06l+A7slt!6 eV[ 1l}శ^TX԰;?sR 恵G6Mz+۰,rx&ܼk?ch4`f\t(]p Mh9~B I?6ܺ8URtC-[tۢ NRګ:U Șh<[)4i7{gR&1"&|z2^4h܊8ǜ]Er4 L s}= ?]VIbBId#9r ?k0g;9^6pm%0zրV EMST-We&}eyą\bPe5@`')(xhg"wiҦ jCb4h<ъU'ח<:}.!ڍf쪿.}uX>]oW¿ ! [>ch+QJP,9@U]s1sֱV F12>'XHߨ^4vfs8òZC" <,PxkI,=٭{p>$rcJN#,r%)BLJ^BDOD |,g\ ෶C9&ɥƣM uh] 7"Č0u|L*{;=2mshhM(A^lĎ^#/3@ @P%sX#?}JXhUۻ(^.(BȠɑaVdi՜ 銴i-\pq۸9gcJ},z=28|+&\2rꫩ G#p!V[搘"V2^FDN[[>T "؉'ǗߝOqK4sMKAz:'dU{oV+EzдLD5pKdAzo980SK]LꄟA\wFA6$#/ .]$ KHO5 )ǿ<8JcR$¬2u}⽬p!­I+ rl䴗H1tZF|گ;_"P@I zbrp2ߒE+ц0S!SbkҤ55rPb洯h YTQ|䛶li@I`s^ GRY&t0][vv} Qٮd71ImFW6Q3 Sx (I<}S3h ۜe`91CQ4T>&BM% _&7|g<i/ 72z9 acz{WX.oǬS&ԙ`LdpYo{J"e[S3p. csī6Hx3 AU{gGapAj΍ݪ?)څ]<PȭIMDquQ~k/2CU5=kuc8J yihV3SfBI Zp@C5P Y|?}!f1BKPMvCJoRTɡvş*e%ƨ endstream endobj 463 0 obj <> endobj 464 0 obj [465 0 R] endobj 465 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 622 0 0 839 0 0 cm /ImagePart_2111 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 177.599 718.75 Tm 105 Tz 3 Tr /OPExtFont3 11 Tf (4.4 Parameter estimation by exploiting dynamical ) Tj 1 0 0 1 463.449 718.75 Tm 27 Tz (-) Tj 1 0 0 1 465.1 719 Tm 99 Tz (coherence ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 99 Tz 3 Tr 1 0 0 1 125.299 675.549 Tm 91 Tz (chaotic maps, 3-dimensional chaotic flows. Before discussing these individually, ) Tj 1 0 0 1 125.049 652.5 Tm 89 Tz (note that in each case the vicinity of the true parameter value is clearly indicated. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 141.849 629.7 Tm 93 Tz (The distribution of shadowing time for several isopleths are shown for both ) Tj 1 0 0 1 125.299 606.7 Tm 97 Tz (the Ikeda system \(panel a\) and the Moore-Spiegel third order ODE \(panel d\) ) Tj 1 0 0 1 125.049 583.899 Tm 93 Tz (in Figure 4.8. The median and 90% contour provide good parameter estimates, ) Tj 1 0 0 1 125.049 560.85 Tm 95 Tz (while the 99% contour suffers from sampling effects. The choice of isopleth is ) Tj 1 0 0 1 125.049 537.799 Tm 90 Tz (not critical, although sampling noise will, of course, become an issue for extreme ) Tj 1 0 0 1 124.799 514.75 Tm 98 Tz (values of the distribution. Thresholds will vary with the size of the data set ) Tj 1 0 0 1 125.049 491.699 Tm 92 Tz (and the noise model; a simple bootstrap re-sampling approach can identify how ) Tj 1 0 0 1 125.049 468.899 Tm 93 Tz (high an isopleth can be robustly estimated. In addition to shadowing time, the ) Tj 1 0 0 1 125.049 445.649 Tm 95 Tz (vicinity of the true parameter value also provide small mismatch error \(panel ) Tj 1 0 0 1 125.75 422.6 Tm 93 Tz (\(b\) and \(e\)\) and their implied noise level is consistent with the true noise model ) Tj 1 0 0 1 125.75 399.55 Tm (\(panel \(c\) and \(f\)\). Note in Figure 4.8e, the true parameter does not provide the ) Tj 1 0 0 1 125.049 376.5 Tm 92 Tz (smallest mismatch error. As we mentioned in Section 4.4.2, the mismatch error ) Tj 1 0 0 1 125.049 353.25 Tm 93 Tz (is obtained by fixed number of iterations for each parameter value. It indicates ) Tj 1 0 0 1 125.049 330.199 Tm 96 Tz (how easily a corresponding model trajectory can be found. It is possible that ) Tj 1 0 0 1 125.049 307.399 Tm 92 Tz (for some parameter values other than the truth, it is easier for the pseudo-orbit ) Tj 1 0 0 1 125.049 284.1 Tm 94 Tz (to converge to a model trajectory under Gradient Descent. However the model ) Tj 1 0 0 1 124.799 260.85 Tm 95 Tz (trajectory may not consistent with the observations which can be testified by ) Tj 1 0 0 1 125.049 237.799 Tm 94 Tz (looking at the shadowing time distribution and implied noise distribution. We ) Tj 1 0 0 1 124.799 214.75 Tm (suggest looking at the distribution of shadowing time, the mismatch error and ) Tj 1 0 0 1 125.049 191.7 Tm 92 Tz (the distribution of implied noise together instead of looking at only one of them. ) Tj 1 0 0 1 125.5 168.7 Tm 95 Tz (Comparing with results shown in Figure 4.1\(b\) and Figure 4.2\(a\), our method ) Tj 1 0 0 1 125.049 145.149 Tm 89 Tz (outperforms least squares estimate approach significantly. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 142.099 122.1 Tm (Figure 4.9 shows the results for simultaneous estimation of the two parameter ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 318.25 52.75 Tm 75 Tz (84 ) Tj ET EMC endstream endobj 466 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 467 0 obj <> stream 0 " ,,b6} Ft3u IIP@ۖG?[5~ڊoA%:dr96&'4Tl† ;6SdkEk$q Y fM ֣$YF܋5R) )؃Mtpbs7Yʄt! )Pӑ]%ȽBf;,\GiJ\軎Bo5+ޣ]V l"Gf) "ବEXx5yr~^zgOxybi))f'0=VBѦs(S-HLvP;/Ӷ&IH*$IisxH+zI7+LvSknKJ?tHR]Z<+qZIq9f [nF[!u5Ȥa"2eOu Tjb^Z~?k6[ KH\4> +"ʶİӬ&Ƃڐ SfLHSi=s2?il}{`{j^LV8*xZ8Kٖ d]EXIG9zuܙ%;N "P |MNF{|- \qTU;6nRy-wVzPu }i !UL!?rBvVAoan[yk]_OI8v ȁ_ZlWwN6)7 jMDAL}o B} ma i -Nlg+iX򂯪BM4@ZSi]rs̾dR>Bχ߽عّڸOMİ#jD4<.%' Ę ]5˔-t 9$FŇh_'4cLdx]Ô谙(ICPKVG ]nI1e}f!|?$wy,bx\5\\(V%vQvרS gpedvk%{ 㯦H$ɦiˢ!1M- RP8'&MMsuAzJ0S>3\+_ I`)F D700cYx/odx'Y7N(+n3!a9zE Zdm1f|YHP{?ۦ(yֳPtV5a(hR xIe4cN-%}OK?? ^?ӥ$9Aca!0|Z2GT-;zڽc!'I?(~$~ |Ȝ8SkGUްC2UʈӑxinR7c,_Bo_B>ll_xE`=$,P h]+SJiv $x*)>)7:ñQ']' eZSn[E2(Y!DDfd;ö:p 4Efw!.kTqհ؜^5QcCN,t>4 4F&B#LsKIu9"Ho HoFB2)SY3 :<.o](]z &$8*O@WS^a:yW ; $ \y]x|U'mXNKf⩈zc5Nm| ہcK$Bp{:#ߘ?z8 Tma`4jU J|b-@InB6۸bg+?Aiފy@֛. nI1C>pghzEٚ`x[7%UuYpU"ks |: :By:^aV#JR,F-w٪LU> Ҩ6ϹcIt7}oT* eu-~#+(PiSfS!1f;K^3C"tAdE7hO% w[yV/jw쳞VQ=pk(r[[ bfhE0C5qX hl)!2 Ydq D Jz5lk?i Ҋ>֔30ݿS?,6Й*+7%<)|[&Ljp4O P*NW`=[1z 2rIe-qʚ~UIA14qmw\pgUۺU=ΔOx7XjBa|}ZPWPRr%vk]ӝS0*"j@WJlrX'̤Jю+PB>v:ϣOJ$@=򇺢qyGM^Tz37*ʖǛʷEydrl  YzG:X͆{R}ڛ (&*CN@*2vB77iV!v&w@ա{!5Bm\ }8 vlz'(@6䎼ǝZ,2_ g~0 ɰ}Dڏ"D}ۤ=8HCm)$U􂧫bBoԬԷ؝<vB< Y^B˓Qr+uZ~%5՝ZdV|>r.|@w p@yN>!0{IZZ{L;@(F9{QM 9#WHd^Mڅ ?l;ƕuAk/:WXUE>ZIu\鍍^ uM3;|aQ,׀~il?Bv9 :A} 1(.K&y1[׎+e\Fn H-mr٨`$cdp}xNVkE)mԥJ}ɶCqZUyrcmwh}atmUS函CN̰-AUYmXkzmI֭%g$(ݢg=cȿI]ʘ=]mh xv-Â, 9iiX21?\DXe=kөoJture"j &K@~m~թuzM7ŋ~AuO>Nۖ\B wCY +XC^f is3@4Zة-Ý$J >a&K^sx( W,0e!D%F*œJY< 6/:wgvnz}"u+YKv-OhX£ <|.4CLa~ =xgjY$qN OQַ8W8t3#tp::ۀ48a*~{jǯL6HxWq9BfEJNߴ T߱d2+lcזQv^> ˔.=F  y̒24&3:o:I7Ѡ2m t=2"o 9ݞ<`8c{Ɵ'q֢\t-|pQ9EE5J=5ݰh˥h.;~)`-\"3nqdP5SgPHG²mR´5U|@>&z/'pA)5lLY1>s`My0!Q~Vӆ5#JbO~ kk{6PA~vƶikMxyV3َB2dj m|fL*;ĺKNc7cEpjri4<\<Ӌk|#f OCC1S-?i !rȎ휛Ի Ej#6]|特{{3z%\ґ欬! 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Ge^^{XPwFی a3 DZ5'P!c/Ғ2L1X6AKݽ,I/k7"s0DLdz0׿ ;APvz}Īzּ {OwRp+ja([On9 Prc4>ƒ8(F+2J8tn,2}hUgj * Üɩ7YE?qFf5"7q9>T^QFV[w07"h-x5>D~j4|7 =au`'G^i8aG`8DJu9ѩ5G9f qϻ(Ó'7ab,z\?%W)d]:V޲׌.0chyG#}rIJ=..- ܪ8.@ ruJM[ QO9lOĊ+[!RhJ ؤ823^U9O~c qqEMs2Ye U,0VŠ W9so$AS*z` V.GqX7)~ٚ`9JLEqїE{&ۄF!yYHQf.zBr)n~#n`?čw;SMuLH:̽zAu+ 1掣x(7Z}ot.CZNc2K amAnUBWS g~C]!B+I$ZOlT~2']=!ˬ}P39DуIjq.5`lc}(vdCzJ}^xdI|JZ22Diz`EàҖԝQ=Z1l'3Fh:IcY;AqJ_>}B'\$l, R8n# kpYwQ<$4{] 0i/eh&qePZ3v {{luHqZRt%&)&'?+6K62 \@ >qƀղ3uegĺ (+WjÊU` Bz?V9z6 @ 툦]gIPDg7ZP_YnjIS,yDCf쟈cW2ke˽TQ-)wX- bN"7uCk;edAUNq)z0o+천'+#~zD泻m8d 4@G,FqCr,5(LX˚nQLdM!>EB!(I*9ou=axYx 1TN?fy.<&nALƻJW\IK!7!sv̭%u_,n_:t^a۲9rE R'c^O Mi+dN'~B,R4] FK9i7sG]\%0F"A$M3gz[εnD~X#oT^蒅!Qsu"[hF 9 }j^ (SŠ'!l& E"=!OZ_T0N)gld}I!]o;?njL endstream endobj 468 0 obj <> endobj 469 0 obj [470 0 R] endobj 470 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 622 0 0 839 0 0 cm /ImagePart_2112 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 152.15 487.649 Tm 99 Tz 3 Tr /OPExtFont9 4.5 Tf (3.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 99 Tz 3 Tr 1 0 0 1 156 473.5 Tm 105 Tz (3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 105 Tz 3 Tr 1 0 0 1 146.4 458.6 Tm 24 Tz /OPExtFont3 13.5 Tf (E ) Tj 1 0 0 1 152.15 458.6 Tm 103 Tz /OPExtFont9 4.5 Tf (2.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 103 Tz 3 Tr 1 0 0 1 156 442.3 Tm 105 Tz (2 ) Tj 1 0 0 1 158.65 442.3 Tm 80 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 80 Tz 3 Tr 1 0 0 1 146.15 441.3 Tm 40 Tz /OPExtFont13 11 Tf (S ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11 Tf 40 Tz 3 Tr 1 0 0 1 152.9 429.8 Tm 91 Tz /OPExtFont9 4.5 Tf (1.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 91 Tz 3 Tr 1 0 0 1 152.4 401 Tm 99 Tz (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 99 Tz 3 Tr 1 0 0 1 185.5 699.299 Tm 115 Tz /OPExtFont5 12.5 Tf (4.4 Parameter estimation by exploiting dynamical coherence ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 115 Tz 3 Tr 1 0 0 1 353.75 345.1 Tm 88 Tz /OPExtFont9 4 Tf (1.053 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 88 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 88 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 88 Tz 3 Tr 1 0 0 1 465.85 343.399 Tm 139 Tz (MOM-Spiegel ) Tj 1 0 0 1 501.6 343.149 Tm 129 Tz /OPExtFont9 4.5 Tf (Sysied ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 129 Tz 3 Tr 1 0 0 1 353.75 330.449 Tm 100 Tz /OPExtFont9 4 Tf (0525 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 3 Tr 1 0 0 1 353.75 316.049 Tm 91 Tz (1.052 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 91 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 91 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 91 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 91 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 91 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 91 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 91 Tz 3 Tr 1 0 0 1 353.75 301.899 Tm 102 Tz (0515 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 102 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 102 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 102 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 102 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 102 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 102 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 102 Tz 3 Tr 1 0 0 1 353.75 287.5 Tm 83 Tz (1.051 ) Tj 1 0 0 1 360.25 287.5 Tm 90 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 90 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 90 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 90 Tz 3 Tr 1 0 0 1 353.75 273.1 Tm 102 Tz (0505 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 102 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 102 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 102 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 102 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 102 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 102 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 102 Tz 3 Tr 1 0 0 1 354.949 258.2 Tm 68 Tz (O. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 68 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 68 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 68 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 68 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 68 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 68 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 68 Tz 3 Tr 1 0 0 1 350.149 243.799 Tm (O. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 68 Tz 3 Tr 1 0 0 1 361.899 240.45 Tm 103 Tz (95 ) Tj 1 0 0 1 366.5 240.45 Tm 992 Tz (\t) Tj 1 0 0 1 377.5 240.45 Tm 103 Tz (96 ) Tj 1 0 0 1 382.1 240.45 Tm 1015 Tz (\t) Tj 1 0 0 1 393.35 240.45 Tm 107 Tz (97 ) Tj 1 0 0 1 398.149 240.45 Tm 974 Tz (\t) Tj 1 0 0 1 408.949 240.45 Tm 97 Tz (98 ) Tj 1 0 0 1 413.3 240.45 Tm 992 Tz (\t) Tj 1 0 0 1 424.3 240.45 Tm 107 Tz (99 ) Tj 1 0 0 1 429.1 240.45 Tm 934 Tz (\t) Tj 1 0 0 1 439.449 240.7 Tm 96 Tz (100 ) Tj 1 0 0 1 445.899 240.7 Tm 825 Tz (\t) Tj 1 0 0 1 455.05 240.7 Tm 86 Tz (101 ) Tj 1 0 0 1 460.8 240.7 Tm 911 Tz (\t) Tj 1 0 0 1 470.899 240.7 Tm 92 Tz (102 ) Tj 1 0 0 1 477.1 240.7 Tm 848 Tz (\t) Tj 1 0 0 1 486.5 240.7 Tm 92 Tz (103 ) Tj 1 0 0 1 492.699 240.7 Tm 825 Tz (\t) Tj 1 0 0 1 501.85 240.7 Tm 96 Tz (104 ) Tj 1 0 0 1 508.3 240.7 Tm 825 Tz (\t) Tj 1 0 0 1 517.45 240.7 Tm 96 Tz (105 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 96 Tz 3 Tr 1 0 0 1 134.4 206.1 Tm 112 Tz /OPExtFont5 12.5 Tf (Figure 4.8: Parameter estimations for Ikeda Map with u=0.83 and noise ) Tj 1 0 0 1 134.15 192.2 Tm 101 Tz (level=0.02; Moore-Spiegel System with R=100 and noise level=0.05, the results ) Tj 1 0 0 1 134.65 178.5 Tm 103 Tz (are calculated base on 1024 observations, \(a\) and \(d\) The median \(solid\), 90% ) Tj 1 0 0 1 135.099 165.1 Tm 99 Tz (\(dashed\) and 99% \(dash-dot\) shadowing isopleths; \(b\) and \(e\) standard deviation ) Tj 1 0 0 1 134.4 151.149 Tm (of the mismatch; \(c\) and \(1\) standard deviation of the implied noise, the horizon-) Tj 1 0 0 1 134.4 137.5 Tm 101 Tz (tal line denotes the real noise model. The vertical line represents the location of ) Tj 1 0 0 1 134.4 123.549 Tm 100 Tz (the unknown true parameter. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 324.25 46.5 Tm 85 Tz (85 ) Tj ET EMC endstream endobj 471 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 472 0 obj <> stream 0 " ,,2dV¼|"=tdKT^PLm8e)8k[QN;F"X!T㘷~Naslp@1װ7.^(d _+6:Bw3?.!WmК=f3h~QR/=Y dG8]{5KgL7sa5\]\n $zC [U2GӬ F9Un*'o|HufgkhBuEjZ әY ޞg}`\¢YH*dt5YmqxDv^cF'-Hmڭp蘭j7MۿgFe\TdKsξ/@Qh$:ak荡6+1*ꠙ<6njYMj Ƃ_萡#&h#33fuMvF]ȽC`^Dxwաnm٩Tg%MH#OY BnT?zy2GҭS'h;Q6O;9C,gǻHȸkx9,Eī8ɴ)%kƨBA~<?$c[dZCPWk}(OR-WSp\A*xNI_!FwЬ:Q(1?2 xߝ2; E܃F*a 앓cY@:TU1(f''Cԋܸ Rq8V?|J'.{D[8`mvvY 5X[rP Yg#aK"B@xes@ӂׂ߳n=.RxŴhk"Bf]$kh:m7e|%4'O!GvۯCSܧUAP,=9j? E.? 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The ) Tj 1 0 0 1 126.5 608.399 Tm 96 Tz (fine structure \("tongues"\) in panel \(c\) is due to sensitivity to the parameters, ) Tj 1 0 0 1 126.5 585.35 Tm 93 Tz (nevertheless its minima are in the relevant regions. Contrasting panels \(c\) and ) Tj 1 0 0 1 127.2 562.299 Tm 90 Tz (\(d\) of figure 4.9 reveals that shadowing times provide information complimentary ) Tj 1 0 0 1 126.25 539.5 Tm 95 Tz (to that obtained by estimating the invariant measure \(the ) Tj 1 0 0 1 426 539.5 Tm 120 Tz /OPExtFont8 8.5 Tf (CML ) Tj 1 0 0 1 453.1 539.5 Tm 107 Tz /OPExtFont3 11 Tf (of \(64\)\). The ) Tj 1 0 0 1 126.25 516.5 Tm 92 Tz (shadowing time distribution provide complimentary information quantifying the ) Tj 1 0 0 1 126.25 493.449 Tm 91 Tz (time scales on which the model dynamics reflects the observed behaviour. Com-) Tj 1 0 0 1 126 470.149 Tm 89 Tz (paring the results with Figure 4.2b, our method provides more consistent results. ) Tj 1 0 0 1 126.25 447.1 Tm (Statistics of the shadowing time distribution provide and unambiguous indication ) Tj 1 0 0 1 126.25 424.1 Tm 91 Tz (of the range of relevant parameter values. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 126 379.449 Tm 110 Tz /OPExtFont3 13 Tf (4.4.4 Application in partial observational case ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 110 Tz 3 Tr 1 0 0 1 125.75 348.699 Tm 90 Tz /OPExtFont3 11 Tf (Here we consider the case of parameter estimation in higher dimensional systems ) Tj 1 0 0 1 125.5 325.7 Tm 94 Tz (where the state vector is not completely observed, i.e. some components of the ) Tj 1 0 0 1 125.5 302.399 Tm 92 Tz (system are unobserved. In such case, i\) we firstly estimate the unobserved com-) Tj 1 0 0 1 125.5 279.6 Tm 93 Tz (ponents by simply random draw from the climatology of observed components. ) Tj 1 0 0 1 125.5 256.549 Tm 97 Tz (ii\) We then initialise the ISGD algorithm with the observed components and ) Tj 1 0 0 1 125.299 233.299 Tm 94 Tz (the estimates of the unobserved components. A pseudo-orbit is obtained after ) Tj 1 0 0 1 125.5 209.75 Tm 98 Tz (a small number of ISGD iterations. iii\) We then update the estimates of the ) Tj 1 0 0 1 125.299 186.7 Tm 94 Tz (unobserved components with the relative components of the pseudo-orbit. Re-) Tj 1 0 0 1 125.049 163.7 Tm 97 Tz (peating ii\) and iii\) several times in order to obtain an "good" estimates of the ) Tj 1 0 0 1 125.299 140.399 Tm 89 Tz (unobserved components. In the end we run a large number ISGD iterations to ob- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 318 53.299 Tm 77 Tz (86 ) Tj ET EMC endstream endobj 476 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 477 0 obj <> stream 0 $ ,,Ub6v~V)%KЙUhyE>jݒN.İ4"C8y6nTd=6@yjMF,;'Q-&>l>}4RURnTF=Y$Rτqt,Ij|v Ls;d%h k wPL١ 's\v5kb6n`⣌`>ʳ]"۰<×PFS*d!nEGSlN!BoTq>/MB :0 ?9e.o[qm;uƒBיt.-1>aԳ䲵.^i$ǝ_sЎдᄛ^2ԻTݍm<{ρ.o_ZPQ¶U>2]TeH3,.e31h)ıA9 -pWFEdbi%Ȯ3C੐@jABk@ӚϜQ9 ]E6<&:%~ Ԥ!hC^=KFskֺU눤3N,Z\9̄9¾s"5+,jqF#^8VCJH {ӾGwzgOY0#,Ts0^=m=HK.iv$ !Ɉ7#'Y)PNO(V LQf7VY;)#Mw&) |4 .{M'ٍهwKMMW&f&?O?fw·E 8q1~4u2h֏ ttEUɴAфZzt7W[Pw9-ȍƈx=rWCS<0V+BkMJ/RD܃ڭc@F=q3xWޙ L~6/>O&s/sOn) 2` cʳQCielgVd$ꁄw4M _ ē`nZCE{5J}<۫~@|b:Fס"c\Ƞt Q [%.W67<}=wI̊J7W~6qr lޖJQ=J eN܅KN޷Or˝;=ǖI1ŷ; D"T&̶E9~x;]QyyGxQۙǮaB4Q9A=eUt "<K4Ojl}JH6ToFkauu=%sOPZJٶ_aq`Q/*A%gAbVg˨vKKtP͂x+wl N%ͽ|O$B<#xǛ+ + _|Pv #L 8t%k\`lޣTInh-\`TXiXjUI|+^·Kwe*oKYSphuEN+ʧ2[pKU;Joi%.a #ԂnW?_?$MzB\V,RJ!݁2D0c GY%{~f\MLcu|4 jУ!!7 ҾL1edo[ H-HG`^5W_uRa pXz}dž8> OdדmC+W)x:7?8jޫԎ40;1tuF+A8ͨg6nNTƞlegQoh1d^n"h(AF&_^`6M2.ǡefw"V&:f尳{)E0U|fmޙ1GRy cOyUBn>Z'VH|9TX2|p Ui|5cAaI5?-}"!㾒Kh<{|+%Bl6%L/떕""#)e71WQ?Ȥ#!ݏxX\IY_k.aX}6T wy]4 44I0<ᳲ6G>ZdXkXe? 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ΞwH U͵Hr^+0^j8Cfw [zV5xɻ@,]v+O2Uī?I$5i'bVS+1J){/)r$vGi߼[j8^ 6SsU3з쳝)c^z0|E-YK5WkwݒJ|!05ymatI$㧐 ~]Bbt[@C$̿-ƍρXI䳩|ӾѶ8ױV<$a_F,;,5'>|k[Y%)mC Uӈ<_>'챑a]>vUD|A]#"7ĉSfpaг8udنEx.`X`MIrq8-'",l.MMwFaTx>6^ )"3>1@#~/H RNCO}òe+˝ŨQA>'#zɻ;jڼӱ+[9*N/ճ[6>jl.HVbA/ M|ĸ|F #ZPVW@hV$%3nzfbW4gP*"ӑ,qt.aU$5MB[F~"}@rf$`Hnz\07*صN=tIWEg?>& xՂ#w_S^՞J&`}yxyp~Cb⼹d?4iki"xk{`6KP m} z>ViuSB@IPdc`dwwz ~G W>-5W=&·9st=r#n'ndtj_?۲&mm%4JM ]}Tk/`lZRXZ oZ>L ( 2ڃ ex_+'ЈLDD oJ"Ck`l#ais_k tdi0">!p* 썢P/ 7í ^:]nN+8S`xي1O Ze-qc6Mc_Z)(Tǰ&,㨫Ї>,NٔM4>kr;5`C!4ur*xa 0|zЕ哵"!%7O1)|X*;x9t$ĕG"lB>zLQtgF>Gp@  vvv;.T`z{/QaY;[/)'3V1*6oS4D$H=9Rx/`ֆ1HO]<'ZK"<נc¸"S'o@ ÖgoH-/S4 nV{??nlBXnF ^⡆{_Ԁ ˝6 $i48UBg؁K yz7&"45pw?W8~m{f/Ü}$K4Nl2 a=Fib A2F?kӝrAHW-sZX(hv(mdKO2~I! tI#3We36կǓAyG5( f=7dB[FhZBrln֣ٜZb4":EoOh÷DS{dcJe .Ϝew6,"j=W M6#P4MCa8'$.P%[ڒQX7N@ыΪB:Vˑ0$4-66oA&QK!`m+3LdS˧M튰++Rgv}&'Y0d<~1CԖ'>Nhj+)J8zěAo-=Qݍ/m9y endstream endobj 478 0 obj <> endobj 479 0 obj [480 0 R] endobj 480 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 500 0 0 803 0 0 cm /ImagePart_2114 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 252.25 450.899 Tm 97 Tz 3 Tr /OPExtFont9 8 Tf (0.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 8 Tf 97 Tz 3 Tr 1 0 0 1 409.899 444.899 Tm 90 Tz (1.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 8 Tf 90 Tz 3 Tr 1 0 0 1 252.25 583.649 Tm 100 Tz (0.35 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 8 Tf 3 Tr 1 0 0 1 248.15 517.399 Tm 71 Tz /OPExtFont13 9.5 Tf (b ) Tj 1 0 0 1 252.25 517.399 Tm 118 Tz /OPExtFont9 8 Tf ( 0.3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 8 Tf 118 Tz 3 Tr 1 0 0 1 231.349 352.75 Tm 90 Tz (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 8 Tf 90 Tz 3 Tr 1 0 0 1 230.65 281.95 Tm 91 Tz (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 8 Tf 91 Tz 3 Tr 1 0 0 1 87.849 706.5 Tm 110 Tz /OPExtFont3 10.5 Tf (4.4 Parameter estimation by exploiting dynamical coherence ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 110 Tz 3 Tr 1 0 0 1 34.549 242.85 Tm 96 Tz (Figure ) Tj 1 0 0 1 71.5 242.85 Tm 91 Tz /OPExtFont3 11.5 Tf (4.9: Information from a pseudo-orbit determined via gradient descent ) Tj 1 0 0 1 34.549 228.899 Tm 89 Tz (applied to a 1024 observations of the flexion map with a noise level of 0.05. \(a\) ) Tj 1 0 0 1 34.299 214.75 Tm 86 Tz (standard deviation of the mismatch, \(b\) the implied noise level, \(c\) a cost function ) Tj 1 0 0 1 34.549 200.85 Tm 90 Tz (based on the model's invariant measure \(after Fig.4\(b\) of ref \(1) Tj 1 0 0 1 352.1 200.85 Tm 12 Tz (, ) Tj 1 0 0 1 352.55 200.85 Tm 122 Tz ( \(d\) median ) Tj 1 0 0 1 34.1 186.7 Tm 87 Tz (of shadowing time distribution. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11.5 Tf 87 Tz 3 Tr 1 0 0 1 226.8 40.5 Tm 84 Tz /OPExtFont3 10.5 Tf (87 ) Tj ET EMC endstream endobj 481 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 482 0 obj <> stream 0& ,,#..jɲN'].]VcrZLoG%A髛Mh}O(90].`ȏ|/P ]DГ.֦q*, C:q&D#j Ңm`X(?K"lYW9h0D̺'{..ϝt1d}P0T `%Q}?S7 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='߭]qY4^()6b Q=k-:]]D{Tʺ1@53@a{l}y޿M6 4ʠb1@;EJinkwZV Q>.7ROCM=7j,k>dH=)ٝDtFENae\)MQfjɧO06ϥ;~G6Yƅ4kWјQq. |.^9Gվж Z$9d MWI}fTQ\ˊUúuQx0aTs`Xҡ&ߩ1=՛ \DcH=@YWvz0%Msj+48uX eh\MU҈] Y^oNj2u`zw bvCꎯs 79[O0hP % l%_}կMF(Hu_SV=mE)f܋=gݟ~ܺ!cH+5q uLRp(_]`_`<<d&lz TH4/V?xifE~i4+oRfs }S^& ZuGdcj3Z7=gYBiܵa(1.2˗/b8 e[ޝ 9R~|1ݹ*2< 8 fP0Q,̔s,j2<};F pNBϺV,ZWEWWƊ}Dߙx鏝r>Q*`t2iU+ê.(pYa ^GMһ/Ĵ|/Ǥhx,hi,C^Ѓ؅Ur@IuEǝꨭZ+9dsN/ n "BOaOת [lP5 l[/6Eҭ N/xa"R;R +T-*fsÎ yH޼Cf/KcuV3T]p tp/ endstream endobj 483 0 obj <> endobj 484 0 obj [485 0 R] endobj 485 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 622 0 0 839 0 0 cm /ImagePart_2115 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 219.099 298.049 Tm 91 Tz 3 Tr /OPExtFont9 5 Tf (Parameter F ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 91 Tz 3 Tr 1 0 0 1 149.3 476.35 Tm 85 Tz /OPExtFont9 4 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 85 Tz 3 Tr 1 0 0 1 151.449 472.75 Tm (9 ) Tj 1 0 0 1 153.349 472.75 Tm 1083 Tz (\t) Tj 1 0 0 1 165.349 472.75 Tm 103 Tz (9.2 ) Tj 1 0 0 1 171.099 472.75 Tm 911 Tz (\t) Tj 1 0 0 1 181.199 472.75 Tm 103 Tz (9.4 ) Tj 1 0 0 1 186.949 472.75 Tm 934 Tz (\t) Tj 1 0 0 1 197.3 472.75 Tm 94 Tz (9.6 ) Tj 1 0 0 1 202.55 472.75 Tm 952 Tz (\t) Tj 1 0 0 1 213.099 472.75 Tm 99 Tz (9.8 ) Tj 1 0 0 1 218.65 472.75 Tm 1037 Tz (\t) Tj 1 0 0 1 230.15 472.75 Tm 86 Tz (10 ) Tj 1 0 0 1 234 472.75 Tm 929 Tz (\t) Tj 1 0 0 1 244.3 472.75 Tm 95 Tz (10.2 ) Tj 1 0 0 1 251.75 472.75 Tm 758 Tz (\t) Tj 1 0 0 1 260.149 472.75 Tm 95 Tz (10.4 ) Tj 1 0 0 1 267.6 472.75 Tm 780 Tz (\t) Tj 1 0 0 1 276.25 472.75 Tm 92 Tz (10.6 ) Tj 1 0 0 1 283.449 472.75 Tm 780 Tz (\t) Tj 1 0 0 1 292.1 472.75 Tm 92 Tz (10.8 ) Tj 1 0 0 1 299.3 472.75 Tm 929 Tz (\t) Tj 1 0 0 1 309.6 472.75 Tm 69 Tz (11 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 69 Tz 3 Tr 1 0 0 1 221.05 467.949 Tm 121 Tz (Pamela, F ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 121 Tz 3 Tr 1 0 0 1 168.5 584.6 Tm 156 Tz /OPExtFont9 3 Tf (\(9\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 156 Tz 3 Tr 1 0 0 1 149.3 601.399 Tm 96 Tz /OPExtFont9 4 Tf (4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 96 Tz 3 Tr 1 0 0 1 146.15 586.049 Tm 95 Tz (3.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 95 Tz 3 Tr 1 0 0 1 140.9 554.6 Tm 43 Tz /OPExtFont3 10.5 Tf (e ) Tj 1 0 0 1 143.3 554.6 Tm 142 Tz /OPExtFont9 4 Tf ( 25 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 142 Tz 3 Tr 1 0 0 1 149.05 533.7 Tm 123 Tz /OPExtFont9 3.5 Tf (2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 123 Tz 3 Tr 1 0 0 1 146.4 523.149 Tm 86 Tz /OPExtFont9 4 Tf (1.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 86 Tz 3 Tr 1 0 0 1 145.9 491.699 Tm 99 Tz (0.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 99 Tz 3 Tr 1 0 0 1 138.949 388.3 Tm 72 Tz /OPExtFont13 5.5 Tf (E ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 5.5 Tf 72 Tz 3 Tr 1 0 0 1 138 367.649 Tm 163 Tz /OPExtFont9 5 Tf (1 ) Tj 1 0 0 1 148.55 367.649 Tm 75 Tz /OPExtFont11 5 Tf (2 ) Tj 1 0 0 1 150.949 367.649 Tm 72 Tz /OPExtFont9 5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 72 Tz 3 Tr 1 0 0 1 144.5 324.45 Tm 86 Tz (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 86 Tz 3 Tr 1 0 0 1 144.699 428.1 Tm 98 Tz /OPExtFont9 6 Tf (as ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 98 Tz 3 Tr 1 0 0 1 144.699 393.55 Tm 89 Tz /OPExtFont9 5 Tf (2.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 89 Tz 3 Tr 1 0 0 1 145.199 359 Tm 79 Tz (1.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 79 Tz 3 Tr 1 0 0 1 159.349 315.1 Tm 20 Tz /OPExtFont9 4 Tf () Tj 1 0 0 1 160.55 315.1 Tm 68 Tz /OPExtFont9 5 Tf ( ) Tj 1 0 0 1 161.75 315.1 Tm 2000 Tz (\t) Tj 1 0 0 1 296.399 318.7 Tm 155 Tz (..... ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 155 Tz 3 Tr 1 0 0 1 148.3 307.399 Tm 79 Tz /OPExtFont11 5 Tf (o ) Tj 1 0 0 1 151.199 303.549 Tm 77 Tz /OPExtFont9 5 Tf (9 ) Tj 1 0 0 1 153.349 303.549 Tm 833 Tz (\t) Tj 1 0 0 1 164.9 303.549 Tm 89 Tz (9.2 ) Tj 1 0 0 1 171.099 303.549 Tm 711 Tz (\t) Tj 1 0 0 1 180.949 303.549 Tm 89 Tz /OPExtFont9 6 Tf (a4 ) Tj 1 0 0 1 186.949 303.549 Tm 592 Tz (\t) Tj 1 0 0 1 196.8 303.549 Tm 86 Tz /OPExtFont9 5 Tf (9.6 ) Tj 1 0 0 1 202.8 303.549 Tm 711 Tz (\t) Tj 1 0 0 1 212.65 303.549 Tm 86 Tz (9.8 ) Tj 1 0 0 1 218.65 303.549 Tm 797 Tz (\t) Tj 1 0 0 1 229.699 303.549 Tm 86 Tz (10 ) Tj 1 0 0 1 234.5 303.549 Tm 675 Tz (\t) Tj 1 0 0 1 243.849 303.549 Tm 86 Tz (10.2 ) Tj 1 0 0 1 252.25 303.549 Tm 537 Tz (\t) Tj 1 0 0 1 259.699 303.549 Tm 86 Tz (10.4 ) Tj 1 0 0 1 268.1 303.549 Tm 534 Tz (\t) Tj 1 0 0 1 275.5 303.549 Tm 86 Tz (10.6 ) Tj 1 0 0 1 283.899 303.549 Tm 537 Tz (\t) Tj 1 0 0 1 291.35 303.549 Tm 86 Tz (10.8 ) Tj 1 0 0 1 299.75 303.549 Tm 693 Tz (\t) Tj 1 0 0 1 309.35 303.549 Tm 64 Tz (11 ) Tj 1 0 0 1 312.949 303.549 Tm 2000 Tz (\t) Tj 1 0 0 1 355.899 307.399 Tm 72 Tz /OPExtFont11 5 Tf (o ) Tj 1 0 0 1 358.55 303.549 Tm 86 Tz /OPExtFont9 5 Tf (9 ) Tj 1 0 0 1 360.949 303.549 Tm 833 Tz (\t) Tj 1 0 0 1 372.5 303.549 Tm 86 Tz (9.2 ) Tj 1 0 0 1 378.5 303.549 Tm 707 Tz (\t) Tj 1 0 0 1 388.3 303.549 Tm 89 Tz (9.4 ) Tj 1 0 0 1 394.55 303.549 Tm 711 Tz (\t) Tj 1 0 0 1 404.399 303.549 Tm 82 Tz (9.6 ) Tj 1 0 0 1 410.149 303.549 Tm 729 Tz (\t) Tj 1 0 0 1 420.25 303.549 Tm 89 Tz (9.8 ) Tj 1 0 0 1 426.5 303.549 Tm 779 Tz (\t) Tj 1 0 0 1 437.3 303.549 Tm 81 Tz (10 ) Tj 1 0 0 1 441.85 303.549 Tm 675 Tz (\t) Tj 1 0 0 1 451.199 303.549 Tm 81 Tz (10.2 ) Tj 1 0 0 1 459.1 303.549 Tm 574 Tz (\t) Tj 1 0 0 1 467.05 303.549 Tm 83 Tz (10.4 ) Tj 1 0 0 1 475.199 303.549 Tm 555 Tz (\t) Tj 1 0 0 1 482.899 303.549 Tm 83 Tz (10.6 ) Tj 1 0 0 1 491.05 303.549 Tm 552 Tz (\t) Tj 1 0 0 1 498.699 303.549 Tm 84 Tz (10.8 ) Tj 1 0 0 1 506.899 303.549 Tm 693 Tz (\t) Tj 1 0 0 1 516.5 303.549 Tm 64 Tz (11 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 64 Tz 3 Tr 1 0 0 1 148.55 410.85 Tm 86 Tz (3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 86 Tz 3 Tr 1 0 0 1 148.8 445.149 Tm 77 Tz (4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 77 Tz 3 Tr 1 0 0 1 167.3 429.1 Tm 79 Tz /OPExtFont19 4.5 Tf (\(C\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 79 Tz 3 Tr 1 0 0 1 351.85 324.7 Tm 89 Tz /OPExtFont9 5 Tf (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 89 Tz 3 Tr 1 0 0 1 355.699 445.399 Tm 93 Tz (4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 93 Tz 3 Tr 1 0 0 1 352.1 428.1 Tm 97 Tz /OPExtFont9 6 Tf (as ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 97 Tz 3 Tr 1 0 0 1 355.899 410.85 Tm 86 Tz /OPExtFont9 5 Tf (3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 86 Tz 3 Tr 1 0 0 1 352.1 393.55 Tm 89 Tz (2.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 89 Tz 3 Tr 1 0 0 1 355.899 376.3 Tm 79 Tz (2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 79 Tz 3 Tr 1 0 0 1 373.899 426.699 Tm 84 Tz /OPExtFont19 4.5 Tf (\(d\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 84 Tz 3 Tr 1 0 0 1 178.8 719 Tm 110 Tz /OPExtFont3 10.5 Tf (4.4 Parameter estimation by exploiting dynamical coherence ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 110 Tz 3 Tr 1 0 0 1 126.5 675.799 Tm 97 Tz /OPExtFont5 13 Tf (taro the pseudo-orbit which is used to calculate the shadowing time distribution. ) Tj 1 0 0 1 126.5 653 Tm 98 Tz (In such cases the shadowing-time is determined without placing any constraints ) Tj 1 0 0 1 126 629.95 Tm 97 Tz (whatsoever on the value taken by the unobserved component\(s\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 125.75 266.6 Tm 101 Tz (Figure 4.10: Shadowing time isopleths as in Figure 4.8 for 8-D Lorenz96 with ) Tj 1 0 0 1 125.5 252.7 Tm 102 Tz (parameter F=10 given only partial observations, a\) the 8th component of the ) Tj 1 0 0 1 125.5 238.5 Tm 96 Tz (state vector is not observed; b\) none of the 2nd, 5th or 8th variables are observed ) Tj 1 0 0 1 125.75 224.6 Tm (only the other five components; c\) only 2nd, 5th or 8th variables are observed; d\) ) Tj 1 0 0 1 125.75 210.7 Tm 97 Tz (all the components of the state vector are observed. In this experiment the noise ) Tj 1 0 0 1 125.299 196.75 Tm (level is 0.2. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 142.3 163.899 Tm 96 Tz /OPExtFont3 10.5 Tf (Figure ) Tj 1 0 0 1 178.3 163.649 Tm 99 Tz /OPExtFont5 13 Tf (4.10 shows the result of the application in the 8-D Lorenz96 system. ) Tj 1 0 0 1 125.299 140.6 Tm 100 Tz (Panel \(d\) shows the isopleths for the 8 dimension Lorenz96 system with states ) Tj 1 0 0 1 125.049 117.299 Tm 99 Tz (fulled observed. It appears that our method provides good parameter estimates ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 318.25 52.299 Tm 86 Tz (88 ) Tj ET EMC endstream endobj 486 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 487 0 obj <> stream 0 ! ,,%o++j^7Ї;C/hg ZPqƿ(0hY[@M۩^;a˵Pl7hE뻲q73iݞqYG*4mԔaSЛs|;T1C <0L e+3KW$[{4}=_੯݇w]U݌$yؾkAt S孉9,V8ԥgI t 8~py}1琎ٛz:$"obg38y^fowg6APҶ MRW~ )=:1(hBr-TLf N`3}m7+1)RspJ/4;J?EF&Ld !_ts(@x[c=G&}=ժd6 ưb!2)nM} \KScgl s'R3kxF _<̛ytke(gY._U-:9`Wi ߋPpd *̖uM*Ƚ;#φ;d],Qz6Dy01|t/o\wa/(bg :ݠS>Yp-:а )Q>Y~yP ezpJߎE ̗N >Ƿ1юޓLjvX#ړʲ|0,S}lowZtµ3tyLp\4_6\}Ex ʠ9[zԩ.oĝnoŊDHkǷh^J[`m8H'"6Q[x+! 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M`/Vثr 7A:"}D\Lf F8$_dI~0\Ӿ0°0;S*E"Xk@IH;%s~?N m޹a=7lIU;"-po쏀urfDeA> Ve0f/!hx/`èa#:Pr.׆c5#ju)/ \;> ›v?cH|~? pSq9ufE:{ׂ͈jNxJ,NG"*~3E(O8L*Â?:ҡUFL:.SSWO3 Ublo[yip̾ۜCW0.0RU`ٓӁ!k p4!7#*!VCHm9)V"SJC AGⷍ'.כI\;quv7nAh$[׏%c:B&|V^"y;pYYU7dbL1c&8KqC]'=Cm Aۥa|4\0QIC!+fK@3rҐ/k&]^;uf ]h: 7JsN@3]M/[a6#p Hl;yN1#f$MM>}eEs`Up1qiDd1HuȮiqYBNU.NOR'K aO@ЫgJxQk\3!aZKKq`d9|I*!bLXleu4I۱؄/f1V}{^ P50('t~T_W%: [zκqFT[l֌#lV f, 5Y0`uLh%]]@c۳ɠ7Ɍ`ڤXNǍh, =1TUUp?L1)`&K,/4R ecuqV~\]{ sázbVBEfR=剎 燺-}I!cvu& Ao+Bk4[jz0e"U>w a0+h`ͮ +֕Q6ZtaA)L$c,N ˞fc(7H1u;C "rngLA\XS6sتz_8eb)']t[֊`%ME L2ԅY+yށ"fP=?ph&hC+E)cajGm#&:z-n}k)%d`{ yrMǴzN6Ëm=.ZQ@ Jot9$ )DY/l̂ xm38ס6ތ T]rsp5^R'S瀎gE=(yGfM|M׳$Qc pPx7yٍ/J-s$xu/LˍrM#bЂs?ngͻʜߏ:rsJ|ckv"xO%⸝ 1u*gfP#tJ6sK)wyFMq[k'm>}gR2˪`*D_pa)` ZYi-#a>ʱʮ4V@)p7#ս407ɪ?<ƒ5ozUvff}ׯp3o4rE7|870՘'EC@f*QğD..eC weg R\#0Ӵi+UFˏ[FvIB+YgܼH!df1n1r]NmQϺ,->^8FJ ިy 3SӀh-HC{ƒ"k,?)M+OKd:xD@DF*͔V\F\6H8ed$FŢ@tJ4WH% (ۓY ҰB~ >$U G-|y^ OѴ% G98%h"? 1 ?۫1fʲMLc)~I˒n#Fv}Ϫ3321zsaC̓@L.쐜 0!><ѝo8"kh:kW!5$ตzzJX p޸k*DƋ>_Hl?*rӝ`ÔYWG;'J;8)mw͎ZOZW>?m׼{G􄠔k0#.l}6td@p?5+!__*VDGn RZp$yoNΧc y'27V}8x$v` O(bC<<%p@!o}$gvGXGS\r1夔:i4piQ7P4jꩶHWȲtE%^'8NIvlѥTUXrNǰ 0 |sR꣌aJ $*dBpQa >x-;m_%Ǒ 4\;aWhxNVK앻~gaf׍#h$̽ᖅ3|N׍xsXso܉t047)q$jKw ctgLӱ KCSjӁSS\4 4PIl LԈU3 ʹ١H4b.4L~5 :]W?tt1l,N7ȥ!V4t~@yiWJmŽLߔ?Q<h;Qv&->ba?htw{5)2T+?'L.6vɠq xr<r@'!-իYAɪ^JU?MZ}Q+gѴ2{vUBB- *g Jދ(7\\D vӨ/: >R=G IZ5U*J87#H(X}+n endstream endobj 488 0 obj <> endobj 489 0 obj [490 0 R] endobj 490 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 622 0 0 840 0 0 cm /ImagePart_2116 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 424.55 719.299 Tm 105 Tz 3 Tr /OPExtFont3 11 Tf (4.5 Outside PMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 105 Tz 3 Tr 1 0 0 1 126.5 676.1 Tm 90 Tz (in the higher dimensional model case. In panel \(a\) seven of the eight components ) Tj 1 0 0 1 126.5 653.049 Tm 99 Tz (of the state vector are observed, in panel \(b\) 5 of the eight components are ) Tj 1 0 0 1 126.5 630.25 Tm 98 Tz (observed and in panel \(c\) only 3 components are observed. In all cases, the ) Tj 1 0 0 1 126.5 607.2 Tm 95 Tz (correct parameter values are well indicated although the length of shadowing ) Tj 1 0 0 1 126.5 584.149 Tm 89 Tz (time decreases as less components are observed. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 126.5 532.1 Tm 127 Tz /OPExtFont5 18 Tf (4.5 Outside PMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 18 Tf 127 Tz 3 Tr 1 0 0 1 126.25 497.5 Tm 93 Tz /OPExtFont3 11 Tf (Large forecast-verification archives and lower observational noise level contain ) Tj 1 0 0 1 126.25 474.5 Tm 96 Tz (more information and thus yield better parameter estimates when the model ) Tj 1 0 0 1 125.75 451.449 Tm 89 Tz (structure is perfect. When the model class does not admit on empirically adequate ) Tj 1 0 0 1 126 428.649 Tm 95 Tz (model, the notation of a "true" parameter value is lost. It is important to note ) Tj 1 0 0 1 126.25 405.35 Tm 91 Tz (that even if the true parameter values are unknown, they are well defined within ) Tj 1 0 0 1 126 382.1 Tm 89 Tz (PMS; the question of defining optimal parameter values when the model structure ) Tj 1 0 0 1 125.75 358.8 Tm 91 Tz (is imperfect is more complex. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 143.05 336.25 Tm 90 Tz (The experiment of forming probabilistic forecast to estimate parameter values ) Tj 1 0 0 1 125.75 312.95 Tm 92 Tz (is also useful at identifying "best" parameter in an imperfect model if a notation ) Tj 1 0 0 1 126 289.899 Tm 91 Tz (of best is defined as best forecast performance at certain lead time. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 143.05 266.649 Tm 95 Tz (The geometric approach using shadowing time and additional statistics of ) Tj 1 0 0 1 125.75 243.35 Tm 96 Tz (the pseudo-orbit is also useful to identify parameter values which can mimic ) Tj 1 0 0 1 125.75 220.299 Tm 94 Tz (the dynamics, quantify the time scales on which they can shadow and extract ) Tj 1 0 0 1 125.75 197.5 Tm 93 Tz (information for improving the model class itself. Even in systems as unwieldily ) Tj 1 0 0 1 125.75 174.25 Tm 89 Tz (as multi-million-dimensional operational climate models, variations in parameters ) Tj 1 0 0 1 125.75 150.95 Tm 91 Tz (over the relevant range of uncertainties yield demonstrably nonlinear effects \(87\) ) Tj 1 0 0 1 125.75 127.7 Tm (in the most basic summary statistics \(i.e. climate sensitivity\). The ISGD methods ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 318.699 52.799 Tm 77 Tz (89 ) Tj ET EMC endstream endobj 491 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 492 0 obj <> stream 0 ! ,,IIb7!nj4΅7U$s|4Y1t;":a 3PZ kM[OziMUo;9~V #RyqYTؤ%@$Oe]Cx~Ö .SeCIg_cD1lCT<\BIe%;/ݩ5 7! pmcu3Bj{|φ'>F;=Kk^,cw7f=[͙r`0.*| ;PK(>J5rYiƱQ-5Á؏m( PެTM@ZY WO#0%׿ֿ72.=6 !y[`"xKUt֢a8^0#$]gzËJٌdۋFcc-Q({ R Hs/jHzrfM2Fsӫ2`NGaE_MaA&7u*wg,)+Dj{;r-/͋W#1`CzUލְe fb5XCټ|*K=lE7=ɋY$\v`6ڏ:HuDb{zL8 qU5rZ~>7%d>j:˪Ỹơ{kO\;Ա4c6 yܭ3=p$ʛ03ki}a!I>S'׿Fni5΀/N [F|Dl8߀V}u7ákkܪ@:)Bb*z+W%v=a Vx#ND㦒i}C&{O>o5 d׿I.kFOJWjf7j7>D@[3؍$P9Y{X3( .)ٻA` ?/^+Y7n)Ys:i]4X"LI_EChcNz5SRAqܰ,: ysrIecI%[]Lc=K";!#@twqO\EۉA~xPLW(;+ICY%vO?t RHO{҅hр :>>7. \O.]ﾴ"Kb;Tpy 9dGe |k\|ءI$[\QZKrt$0r_XD ឥv-ӰI5Qq,D4x7. C|C$eV%,x7ؾVj3؎&P` ذuIǿo37e+3hnǸ'Xz?U.:*C1˩h,n/m6k/rOdP-|E5!$D+zb]ڐ૔x}4Mn.B+Œħѩ Hr]+P1݂ϳI[9xOc Qm;}Updv=Sv%jA6M@pkɎZkm(UddՉ?8S:#3+zcɭP"=ICIoX\%*P3ɕgO˅o3(Aé0.E (v؂`ܲ #p$EFu,Z(1cZ"Nbxvb2Vlkаd~0D6S2Ɗ"z.T J6Ԕ&jNd&Q#7(5呒Vy} 0,J,S):@eD/8ΛpKfݒkXishSx Fsh $R]-8N-*mpb9708VGQy1}G5k(L& c3ED^b VT#'V1Ȓ;+#^͍G}φ_Dpi\ @H$s.oYʵ}MYb1{2 XGp0kr+ DzŠ`hY.KD9qYZ+Ymgwz#IΨ :wN7KtN>]oso/)$3XņnuޤE(HGU8_%ܲbJ6V88@jW=t*GWhZox{'AuTV7Jm!Lb`-EL@|\g_o0b@GSbvp)9_Evy_B.ֵm(BrjXLpKaMG*PdBSpǥҳV.W=')@m?& ]\m4 r a[+6䧟_(ּ*LGAЗMx oE9:ib(!-{ztԪ(5`}oLuLϢWle1&3ֳҷT2o(RM}VRz՝;>{\ޜ Y\yj+d.*S]!T(E#W>ҭ7XF|>\Y"?2{%ஔV2PV+ ;FzheUV h;82_Yb"Y4d,! [$'-җR]jG}DKz'[(]rVO$ ?|6e35M'Arz`/:4mkȯ478 {ZR{ J/^Nܥ5 3Ma0+G8h¤3z%j)}tRp_mj! ,(ֽ,SbA?s "Uyj'McQW7(Ū6 q9؂졅j|>ssIR_o8i삛gWSO 6"R᫋=>[|.?$lh//J;XE[NL <@\~}?`|B5 `)pE+1Jn28>sb$ 1QFvR{Rзۉ Ϭ5/ 8Vę6өhj(5YnK䚎ˑc×V0Oǟ0H,gQ ;jп@|:(嘦"lCAi!榆lC'wes*a(+Uru$+ϥs߹Sy[}vL#SڊR!p<ϝ@~%I;'`)fqƘ`1c5\.$*t܁œt?dr5z0+1\'oJL?ɟ)Ucr|LUuw餞3'PQ*߮dq`<Gp>[&:OVˤkm"Kǀ@~VM) Ǟ6$|և9)ĞD Nntm_㫹s͡hDb 3ۗ$??WC͍&Q]hXXfǗ  < ! ]%} qm.˓) I"s"TG>:#:H?Li6q*MWO{ꈉq,^ߧsK|^~ylG`xeA :'^@ wc8Dv1MSM&'{f,rqʡSpRx |tS4aV'ʹ7^D+D-;~g=V-#iÒDMĔNv㎔/}&ӊ3p1k@02i(m(r/߶x@"D̴ nuҘ*V38Y 2@o148Vc0F x^d&2HsglXP'i0k{-j0oF ]# 翐i8X1r2DP sObtJFa߫8Ԁ??.@*&]dx,nȎȞäCL"#"5/+#[Blb{YToI+\L.X2#G87|Tp\pc9q$H'lT_v#DvY@dHz Wg8QL9QȆG )H 4-Is2Q/I+aQ έ4D=r[Ҩ%ʺg9Aɽ5%Ę6ͥs(-kv,|tQ81Ճ4 3,{ fr4#YVTWa٬x3/d\?=؈vsT#/x/aYJOw~T 8s\\$`n3Q 0]V~6k๼ډW̏zZ A_̩X7d h }o;"x𥻏"ۈ^TY0xS+k։6<[,KsIG|%t|#aP :qݳj0ױj<6+b8Y5-,EEѝW/Ieꋌ!X7P8‰a^78ߗ&/aBtMwbuN I(0 jJuY;FVR:,TS21H1t ӲB,F }~Q$K!οGowC&>Vz !&x%mFh'4 wb嘦:ss15=kKIw,Vg"iTLzڳ  Rr&?"(0Dy@Ξtq'{(cz!aw.1G-fe+dԕ1»?˽ە_6Dp=R(lS{2*( 0h-6ԍ8KG\ Xı]z_q{| (qq1O%LXΒOn s*$p 43i y!(&H%lX_16} B t*U3ie&j)K6/ q}(?*Q%ߗme @{lC2aU u+׃*(2pLX+O4CY{m /%wO1[ S0H"uNvY/mxq3>ӈNՅ.\cٷsI/`Iap-62> }*IN W[ |FFwk^!l2X_mbBAiJs,K l4#yi:e#,"f5.IHniyT]Tk-؎3yN03x֭$q ?Ax= S6 vV%2.%/,i6ÄJo"è5ll Qk)G_k^{ΑkxMI4WI\jNtG-ϘTsFb/C,|3obapn5;Vl:[K?BZ<d(x!Q~솢M>Pƒvо/ECGv`H;f,I`~E uhBwpL>B3 _CLfypΙ~SEOYqU E7D<23P32] ETe3=)ԁJpF=̱FyjW q1Q^GԽxxd0\z*2H5/ƂP=% AvF^ T8Di>>qш ^ ^C vm4^kao endstream endobj 493 0 obj <> endobj 494 0 obj [495 0 R] endobj 495 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 623 0 0 836 0 0 cm /ImagePart_2117 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 433.199 717.45 Tm 3 Tr /OPExtFont3 11 Tf (4.6 Conclusions ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 3 Tr 1 0 0 1 125.75 674.25 Tm 93 Tz (have been used on models of this level of complication \(51\). Outside PMS there ) Tj 1 0 0 1 125.75 651.2 Tm 90 Tz (may be no single optimal parameters, of course, but even in this case shadowing ) Tj 1 0 0 1 125.75 628.399 Tm 92 Tz (times have the advantage of providing information on likely lead times at which ) Tj 1 0 0 1 126 605.6 Tm 93 Tz (a forecast will have utility. Timescales on which the dynamics of the model are ) Tj 1 0 0 1 126 582.549 Tm 90 Tz (consistent with the noise model and the observations can be of use in setting the ) Tj 1 0 0 1 125.5 559.5 Tm 88 Tz (window of observations to be used, and the effectiveness of, variational approaches ) Tj 1 0 0 1 125.75 536.25 Tm 92 Tz (to data assimilation \(51\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 125.75 484.149 Tm 109 Tz /OPExtFont3 16 Tf (4.6 Conclusions ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 16 Tf 109 Tz 3 Tr 1 0 0 1 125.75 449.85 Tm 93 Tz /OPExtFont3 11 Tf (In this chapter, we considered the problem of estimating the parameter values ) Tj 1 0 0 1 125.75 426.8 Tm 97 Tz (of the model in the perfect model scenario. Traditional linear method, Least ) Tj 1 0 0 1 125.75 403.5 Tm 89 Tz (Squares estimates, is unable to produce consistent results due to the fact that the ) Tj 1 0 0 1 125.5 380.5 Tm 90 Tz (assumption of Independent Normal Distributed\(IND\) forecast does not hold when ) Tj 1 0 0 1 125.5 357.449 Tm 91 Tz (the model is nonlinear. To address the shortcomings of traditional methods, two ) Tj 1 0 0 1 125.5 334.399 Tm 92 Tz (new alternative approaches, Forecast Based estimates and Dynamical Coherent ) Tj 1 0 0 1 125.299 311.1 Tm 90 Tz (estimates, are introduced in this chapter. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 142.55 288.1 Tm 91 Tz (For Forecast Based estimates, we estimate the parameter values based upon ) Tj 1 0 0 1 125.299 264.799 Tm 99 Tz (the probabilistic skill of the model as a function of parameter values. This ) Tj 1 0 0 1 125.049 241.5 Tm 94 Tz (straightforward procedure has been shown to yield good parameter estimation ) Tj 1 0 0 1 125.049 218.5 Tm 89 Tz (in several chaotic maps. Forecast based estimation using Inverse Noise ensembles ) Tj 1 0 0 1 125.049 195.7 Tm 93 Tz (is straightforward to implement and relatively computationally inexpensive. We ) Tj 1 0 0 1 125.049 172.399 Tm 90 Tz (have shown that it can suffer biases when the ensemble is not distributed consis-) Tj 1 0 0 1 125.049 149.35 Tm 95 Tz (tently with respect to the models long term dynamics \(invariant measure\). We ) Tj 1 0 0 1 125.049 126.1 Tm 91 Tz (have also shown that, for addition computational investment to sample a perfect ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 318 50.95 Tm 77 Tz (90 ) Tj ET EMC endstream endobj 496 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 497 0 obj <> stream 0 $ ,,DDV8=X.92CnվsjqO"*Vr2pcʪU8 Ml̃ϰq,5iLK`QT#?ɡ(<_3^T!ؿ/Nb-l4XwPj Oy3Ubб,7Q軎_ XN=.@y(oJ3;Ӿfkj*Y*, dgFex5Ŕ)+ 3mhNlȋfA3_4 ^MxڧF.J, =Y 7@RBЯq[::FB)xOo°>RD_6:uCJFNNjǃf+6GM o^}/1p:%!]VWYR &y*j%['_=ukD4CRD=ǥ"Q"zX|ꆕIY{)aq4*ru/yH>:}oL|nZ; M  vn+&N.803xkF䒻j;fիbH0=5 %gJ+zDʊG ydqC a0CU ]5e&e{=᜵S ^OG}S$)P(yR9I 3VEg] ] ;%.(CTA.8y3ZCe:''Ӏ|Atf(&(Jim!jJ"ќ8r6sbJ{OQSԣq;a4rt$%tLfzi9;@IJ";U8g8juʣzXaXoN䌋´Cs:'R[Օp#At 5bu)9DÝ9zPP A. H E'jk E y~tvGKQ{CBpZ_,4't RxP1)ޔ{J@:fakeyiҘsrj\ Iػ/Kj>Պ,'eK25ij#"k׫b67~ձ3>"s³ܴ5\ߌ(EAo bFFnTm7ȗ&HjtO .+_Uvt@._^и͟p?@#xɁONEsuIG?SH .V+V!7N=떗Ѵ9(T4!0)Md =G[zE$uz֛t0| h g-}4xy]DlwtAj5(K]+`(uEy,؀t˭Mk)o&tu WtIK2Mlnq$?Ki+w#<ɔfCtxUeitu-VIo r+6ۀ7kC%S97qܰ8ñǵ%͇OR]()j}̈́ʸ.^)vEAʱǴl񛴡v=¹j>Ͽiʰ"<6jlb 8 ٲkwlz˪f|@Ɩ+(wf.x+F.YИ#7Tr%b<%5t>edxja ×±MtQFʃDt₼|4u d) 1j](j/U`\ui"xv'J.*G=(ʅS\*%N]3!qxV1G-ty_s^OPgywV֟mxb{DpSz g[[Hy`7՗I1_hVlmggC_  z $ n r{_)/l9f3J:CtwAd9okHT[;(d-("ʰBo+ɕ"0.pXwx׊:D je6:SMZEA;3^> 8%X(:Ro UL`a Qx7Wz39Z!yYxo@P9Iof?nm+=*ZIx.^#|ro&T]}g.)y#!5J ,4eS; ~mGر'0v^|qz)ߘ7RBjsg7utOP潧TV W>R쑹\8k5(CD  P`0t<_rJs"MUh?9 yc5qݣ ]4ɕ:2ׁPRMW;:w2y]w7f4r΂՟+˿FЁFӜ@YAt;]e:R й_~YEiO'2fkgrGZTx֦~J.t{y7dÉs˜uvH_,8:"kD gDhεCybJ }9ych*=eCuljEQ9aw^ @zN)ds&'ߘ?5sVnR*Y6A '(rp!=)Ԡ'iM0Ul:$5ħ]ޖ9Q'D3 $o1\.oImU5K.(j󲲵Ԇ{u4`B,=,Z6}awT|xv9Y4q#jvԗ _n:~YJt4MKM!QN=d57yv׌!\b@-Sy.}{!$sr}ˎ B,s:`6uv>ZxTÒ`@paj.(r(n+M_AeqsY!U rugcI3o7 Iw[Uhs=L]oOFGMlc"iB;Y)ws~m1!l 0"zI8iu 7t^q\2j酦A(ܹNp{* |48q őy[%8]ӉƤQ=-Y|DQzkvb k }Ric;~P=%Hw^**S݀+H>-fщzL67J`p_Z Hز·Hr8d}^Sz'#>$g'u0@ErM RyE[p11Q|5OZZ.%36IjrvҤlpg ;tv:* ew< ~iHگFZrRz"7td /X#BTPiJbw%@zJ|VZ%QQ$);2&zP*UlZƒdQ m۝d2E@}dtaxdc֌6+B"z8Pcħg~x& &C@ ӈRxGv*6C~cIt8'ʮabg12"͓t lreN{pyڊ(Dg?*[T))JHOTD8SIJ'5@z5@[L,Ta[>>B?ߠQp$eX5\w sہ9F)yi'(s{jJJȁmjMZ8]@F0F_KC,^ÈC}'D2_r^k5lv1C իB+]#Di@AP0"Xa&jƆ`y̹~`΅NlVQۅ<[I5Q@ҏY]tށ,%@qT&R]s~{|;"+CS/T4^ppC66>S-F/mc͐wg5RkF'-f~:C?AHR4Loc}+ ]C5$0# _,|()HV$4Z,hT3W6~k%NO#՝t!JC?em+ vay4%4KZGM-ލf1ŮB?jQDj]qhF)QٵNNgH h],dgUMH(uc%HLmp ?&yi]E /y\"E53Ww5-^3htLi[,鷙e dX.G 2铓& l:qvz3Av/nZܪvGTDߖB/|E$.;)^KDLs1o~ܗ}LρB;dQ dC̤WU+8̈́P 8:0гBo_3i~SiyS>;_<Ǫ6 JbUgfnV^"|]+j7:)yA3o?AP 1MpQK5./G [Yћ!vfb8>N| 7kwdzN :?}o" a!Q/Qa@5n^F endstream endobj 498 0 obj <> endobj 499 0 obj [500 0 R] endobj 500 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 600 0 0 836 0 0 cm /ImagePart_2118 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 411.85 717.45 Tm 99 Tz 3 Tr /OPExtFont3 11 Tf (4.6 Conclusions ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 99 Tz 3 Tr 1 0 0 1 104.15 674.7 Tm 89 Tz (ensemble this bias can be removed. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 121.2 651.45 Tm 93 Tz (Dynamical Coherent estimates is presented which focuses on the geometry ) Tj 1 0 0 1 104.15 628.899 Tm 92 Tz (of trajectories of the model rather than the forecast performance at a given lead ) Tj 1 0 0 1 104.15 606.1 Tm 90 Tz (time. We estimate the parameter values based upon i\) the ability of model trajec-) Tj 1 0 0 1 103.9 582.799 Tm 92 Tz (tories to shadow by looking the shadowing time distribution; ii\) how well model ) Tj 1 0 0 1 103.9 559.75 Tm 89 Tz (pseudo-orbits approximate relevant trajectories by measuring the mismatch error ) Tj 1 0 0 1 103.9 536.7 Tm 91 Tz (of the pseudo-orbits; iii\) the consistency of the distribution of implied-noise with ) Tj 1 0 0 1 103.7 513.7 Tm (the noise model. ISGD method is applied to obtain candidates with longer shad-) Tj 1 0 0 1 103.7 490.649 Tm 96 Tz (owing time and the model pseudo-orbit. The technique is illustrated for both ) Tj 1 0 0 1 103.45 467.85 Tm 92 Tz (flows and maps, applied in 1, 2, 3 and 18 dimensional dynamical systems, and ) Tj 1 0 0 1 103.45 444.55 Tm 89 Tz (shown to be effective in a case of incomplete observation where some components ) Tj 1 0 0 1 103.45 421.5 Tm 92 Tz (of the state are not observed at all. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 120.7 398.5 Tm 95 Tz (Outside PMS, although the optimal estimates of the parameter is not well ) Tj 1 0 0 1 103.2 375.449 Tm 90 Tz (defined, we suggest our approaches may still be able to produce robust results. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 295.449 51.45 Tm 72 Tz (91 ) Tj ET EMC endstream endobj 501 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 502 0 obj <> stream 0 ,,b6|-\LS.z3nh??N1i&{Rq,s@R`oG+ǣڝqޏ  t"Lד|:hyQk>E<|yxNm r#6)/,M$fr]dtGaN>pL()}"a 'ז#;fPeQfvyP >&Vƞ@=82Rr+(-`nYӬDcoZA tv{ p#S5ƲomyikQ뷒>ߓ b$کDI5Zym%ÌYMcrA)h&!Ϳ!h3]TF֙"Io,q= PN]!4dFKUc/`Zc9Br*y4|wڊ~E ،HkH@ "sn{Eέ9LD_"ZRxoȢx߭|!hr]9]h^gIq ( }H`~dn|m㱴\! O.*))>E3uێ7H5.>}e=pOW$dm~LOG'v^@`Paw@&({]q*K7vHŗo, /@Yg ^ɇowL-yBVZ5[eDZO)LR[ ã8)>@_MK;kE_ޏAf?Շ>)}٪C>~z Dŀ[qA༱ LJ2Ν+?%Q+4¼dz } +Leuq6G0jGN-zC@WDπT$'I]XpY-t FBϖz9rBuz|en (Zȇ7FD|fNOFnݴ ҅GHIoChqުdǩ3G|ﭩ9׏g͵ } "fJok{(O~TI?wl1XܿTN:LvS#/!a&UTP8mاM+s_)tAN#|joƮ-EwO[Nu Avr~/ހ=ŹJE'N9{ 8ffd?l%b[=2PZ>S/AIx/}֐R$&*wSG?׾Y +xKVᖹ|7ͿSO G2HykpɍBtrd)RH7gO$FQAyəp_G{h.t) N͚Щ\8ZFS}Ut#?Id^sE=l\iki y%35 Zx+('3Umo& rnH{b_彍0&yK֝Y/883Z+"eD}\Q_iq[$vd_ X~{M1k>97 zPBg#^}Cް%5[p$x+@Dj[ADvyc,rήBlJ$|=T7K+f<&8s/Z&_<_ZkQ{'8KphkPk⌍m!E'3b݄3Ga8cAbArvpV"Zy"0q{'/]y&Jn <|LMHWakIf/HΣA C`dSꉦ e0Q &M<%ac(|UàvqW܏3ecqcr2`סGRy@*2mLV2`ʆIl# dN`!`4] I#Yjb)Υl,:o./.QT%l$`4i&B?7IQf̻s"5a H%/?ȪNv 湍ߊ궒R(;.5p¼:Xڪz-#;?ľ&ڟWQ61xJZZU$,ƘQVd؟Zy|yDlvʳ+G{D0Z @Vi'2tm}-]q#6KD*v.KFG+ruCc7A&\b_Xd![U%L^f \u> +ȁ&ܳ$P_"KCn4>QqȔg6Zl]Oh.hU82= ?Ff,mG<LD&X$z_1\5!צ ~qJ 8ێgȉ1ܑᒎ>Wfy'KSs%6_Q? endstream endobj 503 0 obj <> endobj 504 0 obj [505 0 R] endobj 505 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 605 0 0 837 0 0 cm /ImagePart_2119 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 108.95 579.95 Tm 105 Tz 3 Tr /OPExtFont3 22.5 Tf (Chapter 5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 105 Tz 3 Tr 1 0 0 1 108.25 513.25 Tm 108 Tz (Nowcasting Outside PMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 108 Tz 3 Tr 1 0 0 1 107.299 451.55 Tm 90 Tz /OPExtFont3 11 Tf (When forecasting real systems, for example the Earth's atmosphere as in weather ) Tj 1 0 0 1 107.299 428.5 Tm 94 Tz (forecasting, there is no reason to believe that a perfect model exists. Generally ) Tj 1 0 0 1 107.299 405.5 Tm 93 Tz (the model class from which the particular model equations are drawn does not ) Tj 1 0 0 1 107.299 382.449 Tm 94 Tz (contain a process that is able to generated the data. In this case we are in the ) Tj 1 0 0 1 107.299 359.399 Tm 95 Tz (Imperfect Model Scenario \(IPMS\), and it is crucial to distinguish the model\(s\) ) Tj 1 0 0 1 107.049 336.1 Tm 91 Tz (from the system which generated the data. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 124.299 313.1 Tm 92 Tz (In the Perfect Model Scenario, given the infinite CPU power, one may be able ) Tj 1 0 0 1 107.049 289.799 Tm 91 Tz (to form a perfect ensemble \(80\), whose members are drawn from the same distri-) Tj 1 0 0 1 107.049 266.75 Tm 92 Tz (bution as the system state. In the IPMS, however, such a perfect ensemble does ) Tj 1 0 0 1 107.049 243.7 Tm 93 Tz (not exist. Any ensemble data assimilation scheme is expected to result with an ) Tj 1 0 0 1 107.049 220.45 Tm 91 Tz (probabilistically unreliable state estimation. This chapter is concerned with how ) Tj 1 0 0 1 106.799 197.399 Tm (to forecast the current state using ensemble methods given the observations and ) Tj 1 0 0 1 106.799 174.1 Tm 90 Tz (imperfect model. In the IPMS, model state space and the system state are usually ) Tj 1 0 0 1 106.799 151.299 Tm 91 Tz (different. In this chapter we are aiming to estimate the initial states of the model ) Tj 1 0 0 1 106.799 128.049 Tm (for the purpose of forecasting. In this case not only the observational uncertainty ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 300 51.7 Tm 75 Tz (92 ) Tj ET EMC endstream endobj 506 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 507 0 obj <> stream 0 ,,[& ̽ѠƛQE{QLzn n*XBoRnvl'!fO$[S,xKi^uKdCE޺+8ۿ[4D!5"(iY[`|X2(J yz57sv,N }_̛1rg:>:glv`|sf+3h8dT+r>>RE+k{]qn֎W M ^5uE"Pu sԂ 2roAx@^OSkPOLe$uAHƥ'=qª(11sbhiQc5d^>pnG3ibT=y]Ep'@cSGe8um2doUa["uȎ=ou2OJ8Ja =fX]?\TfFxnjZR|,=e&i6 O%1o `s:囆KCg4w0 =? Ҏ&YN9]B6RY%2.'̖z?f|6)|wU1)L'2P"~|j!'(ҴkEek5Sw(t-s_**4Y4H1?6S8YL!t*U/x$q5o2#߲l)]e~uQk^ 'wڭ*b".CIV²A< Vo:;CYE]Ƨ_ް0Bi&95|3SN扛m+Z2sIU*Y5T6}LSCz$WCA$􊐎~&y6;0%B! HK#SQo)> &7iQ2[kћ` Nrk{t1ެ {%ha GshngI`KzPt84̅) k8Qhka 0u 1i>.!w8YҋUyž8,GÉ~H4ζ/Q  $&j7/GƬ_$g3/ 1Lǔ%yJ?6$aFF:ʬW_e>6|5HH{8Z)d@7d+ Θ 8/ܪ>23[*0ؘ<4yxgL(C~ mJĞl#gM)X|"SgN?t!Fvr)R3dL@0kD-VT:) MB0yҲ @AQ\{V8X-^.W[HeX-qPGiވ{M};!k\. fNF&,{1*ϩctPn(\/u1hLSQ:`׍S JSBlZtݥ%1IM9q;A ¨ήxD+*qo㿤1ßv;|RgwNkPkˉ$$n,+m+[6^Ex_?PIKĉv M@r2gb6@$1mQZ:H "xzYYQ%-gg낓'i\T&}^o:X'+|icjs(ǐ۟vՓV ]Cl*<0 zB4M:W=(wn&@wG@mV]+,L|y .7^پ{]#U>"%$q?$U>*f/ټ"ǰ°ճ[$O$3՟3-4ۓfU (E2_>g]C4v#JzK>amHK-&F" xu{/Q!훍aC|"%heG)z痛2Ŕ//uuUڦ5ܘb 3T֑pB}fOV3|Ycr6U8t&b\>90d/۰hlG!άہ&1<Ň[#Mf 2?4ך\*tP5 w!O]"C)Mmw&A9Pݫ>ҳr<6]7Gǫ>>uctȚK= aLL4>G#LRR +tZ%Yz,\%Rn@JdNԌHiL~D۾UEM `ᄞquʸbGNf\2G.gbay_~C;^|Qhb%)gm0:wZ.&Gԕ_fqWbQrؓx./OQLMEY2{eeۅs?3W` QbdBظ>\+, H@y\a$[f+1XSɽ!~jĚ[ǏI h$(!L^ftvx ^|}܈='* },{d>/ <[ڑ@N!5G_H)0D"/c4:macFS]y'SH44tMAn޲R1P {# X؜R#>/qjLo[KKtw.KOWêt{4X\6oǮtzh%ay(&NA:>TNiVD!7}Ф'',12@1|W㚭޿t(_/eB 61)ks+bWyE1Jr̢4B&„̹ܧ2$' >`5&_ͬcJQ#"/ȑs&p%$Uwa*xjVMªms.Qݽo ơl(o qFsGgN̉? cXX4/lJeN%.6Pa7 Y*w gREW@7Re_k0I! +ʸNe-K85Cu[ &ձ=1FLY8=ʫY U#,s`9-yLz~f6 \MJqr`7]3CtB׃| :,J^ص?;5ֱ :ۏ o/s}?IH* U"~Rr7&W=X^u,ޙ"aA;@jC1$@o'Waڭ,O55Il ]mLpqFU\3+?50\l7e oEsJPOd}> J;:lnqblp8n?e{r "\?gGÉO㤋O:;Mn#%sUOMvP?pU$"wo%d9 ˒[l8/vpDF&ؚ;~I 9 PO^|+(-rt26(P!`JqI&]F+deye#*.Pd·t`l€m҃0N!P_0['tM׮?wyBh{W]_7UK%QyIC݇f'M$>ܚC&,ڈ~ZY:1Jt&YLM)K:QѐIJ)ԲtqK͂%[TŸHVl Y2ڼ{3e jWCGE οf#ϒp[N=+ƟgYɋRImw!C)#Z/h:r-n;I OI؋`r3fLR|RUM7 "o؊ kG`F^H_BVwO "u75 =!7!,.8VWf/:&bc]*$lg>x5ܪ\'ĴνꝴKÒo֐aԓ6 Im9:rJ;y1jU*C}~6atE0,'g'Q^ 5WzX-Yc,A׀?)X} <" |fJC]_u#>^_;LAyM!^Rض2un rieyL,?ة5o# 衺DX[{@aWؖ hiO@^Ȓ-]zg{ 3A3HG^*#@3)U6 R*_$yl4ޗs?%Tޭ5P@4Zh 9_+Ώ- \fˠѫ!; f굁ӞHf,?Fɭ^I> endobj 509 0 obj [510 0 R] endobj 510 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 607 0 0 837 0 0 cm /ImagePart_2120 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 108.95 675.25 Tm 90 Tz 3 Tr /OPExtFont3 11 Tf (but also the model inadequacy need to be considered when an ensemble of initial ) Tj 1 0 0 1 109.2 652.2 Tm 89 Tz (conditions is constructed. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 126 629.399 Tm 97 Tz (In the Imperfect Model Scenario, methods assuming the model is perfect ) Tj 1 0 0 1 108.95 606.35 Tm 93 Tz (may be inapplicable and in any event they would seem unlikely to produce the ) Tj 1 0 0 1 108.95 583.299 Tm 92 Tz (optimal results. It is almost certain that no trajectory of the model is consistent ) Tj 1 0 0 1 108.7 560.049 Tm 95 Tz (with an infinite series of observations \(50\), thus there is no consistent way to ) Tj 1 0 0 1 108.95 537.25 Tm 93 Tz (estimate the model states using trajectories. There are pseudo-orbits, however, ) Tj 1 0 0 1 108.7 513.95 Tm 89 Tz (that are consistent with observations and these can be used to estimate the model ) Tj 1 0 0 1 108.95 491.149 Tm 92 Tz (state \(50\). In this chapter we applying the same ISGD algorithm as discussed in ) Tj 1 0 0 1 108.95 468.1 Tm 91 Tz (previous chapter, but with a new stopping criteria to find relevant pseudo-orbits ) Tj 1 0 0 1 108.95 444.85 Tm 92 Tz (outside PMS. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 125.5 421.8 Tm (The Imperfect Model Scenario is defined and two system-model pairs are set ) Tj 1 0 0 1 108.7 399 Tm 89 Tz (up in Section 5.1. Section 5.2 discusses various Indistinguishable States methods ) Tj 1 0 0 1 108.7 375.949 Tm 90 Tz (of finding a pseudo-orbit and demonstrates that our new methodology, i.e. apply-) Tj 1 0 0 1 108.7 352.899 Tm (ing the ISGD method with certain stopping criteria, can find better pseudo-orbits. ) Tj 1 0 0 1 108.95 329.649 Tm 93 Tz (Other method, such as Weakly Constraint 4DVAR, is discussed and compared ) Tj 1 0 0 1 108.5 306.6 Tm 91 Tz (with our method in Section 5.3. Results of comparing the pseudo-orbit produced ) Tj 1 0 0 1 108.7 283.1 Tm 93 Tz (by ISGD method and WC4DVAR method are presented in Section 5.5.1. This is ) Tj 1 0 0 1 108.5 260.299 Tm 92 Tz (the first time IS methods and the WC4DVAR method are compared in the IMPS. ) Tj 1 0 0 1 108.7 237.25 Tm 91 Tz (Methods of forming the ensemble based on the pseudo-orbits are introduced and ) Tj 1 0 0 1 108.7 213.95 Tm 87 Tz (discussed in Section 5.4 and the results of nowcasting is presented in Section 5.5.2. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 301.899 51.5 Tm 74 Tz (93 ) Tj ET EMC endstream endobj 511 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 512 0 obj <> stream 0 ,,R11b73&͇mX^lږ.RE!:)1p@Q3$udy F8rG^ҝ\FtpF]7y zVsA6Cm Bsc27ÿ@U2Q+˘ԏ&]=vk;@c\=wTJ{"Ju.S872 G@7kDAEĴg޽ WZͩ|W C"B0B<_nf< eO$je/<$GUO^%GHڅ.Q =~̷!0 -AN?JfO9A[+ĵ/1Zdʃ9yg$ $XOD,Yg7˲Y;* "4k9= (^PLzwPw* g)PǑGgϺ4z&|iJ Lrѹ'ַՓ#I!:]c Ӊ 9{gx…?hWfe7'>>x%Y|oژNRmɫ* ʯ )γs<60D!94=&?f 2@pPWd PԎD#Eѷo@Ӳf¸N5 QCc&r_ks H*g)`>EE2p1CjP?#|NY/p{el@͟~GwDWD=bhzlkt>s6#r)d+ 1s?Pt K*phwJa u׃At5һ(c~L o_ "+@QAA:m;5FlBG 4 \ .NYu? Ѵg}jլ*Dl[kl 8gBo_ =տ߷(1!ςak9yn\`R)h4tNSOn{H 9fޗ`7̩#s9vt'H 2O7](MkV`SۭX{e:ȳvyؾp)Mdi!A;|y Dq{JgBRRs.~2G.T3R[\N2E6HْJxNT^<2Cii,!#˧4 .āhet!!aBi.]1sS5́d>srOמ=ޞ丵rA _DỎU;C > cxa>__G־ |֏xMsYXETEbUrU{[8aͷ\çfY/{]e.×p?ťm[HζCl?"i02b{Q67nEnaVɍSձ?{/V&ǃcJ+edO2 fLl(-:Rl ^~Y;)&4 ?2hj^39>~/_7#8Dzщ!- qTulMQRf֨yQ59#@0?̴^: fODv6퓣ܼ(nV8) Aas)*v<ԺXS.g硱rWW" p2[Ϙ)eH1&ry(~?nͲ{^`"&^DD(J HZ{(s_9COοٚc&{ k RVmXUk2- YiCSw Y;@_4 ruǹ#Ν)Ma"soLB&kjU [> ˩!ܱ#"׳۔F0𰨄,^868)ܸ?2<j[/`wÎ[ezhxJ^D2WA߀e{ο>((㰭Ԇ>DZ>2z&*%2蝊#j<ɳvj(2l:2sX8eQצAg=7> w`Xf;I/^w-K*aCC^R̤tM2O糐ɷ{BfG~;Ne$}69o.I&m Lf8p jkNab C`jAn͕Pr:0u؂`=dь6`h ^i]&;bqMax'Y?aÑU)Լ@2ѤͰy=/7CmI| MKxAz|i I̗DpYwu˚RAWK0icqJ#ZO3.hv Q>h\HJjэI |$ Njk^+&w6|C3ƁܰڸIuʔ>SŲմ׿ݜ³16[ۨ<8\JrEq!S Cbm@Df~C@ǝOnE͋Eu$ƒh8ozb`~+Vv Ku,GȚBVK؁}Fi}v<s|aL DuhTt-~:;P)%*} ktT Gm3F]^d:B*W2Eg=Ԇ!A|m|z{ҊXj_VI>/R6,)Ezj eSkiϷBe2;(:8 V9ZyXkHr$޷"@d5@5-M=re!*''#ܢVo9Pw$?2-Ǐk.H]l-z^al ﴮq l=ev&C= BQc b8zjc۩z32?p`b14-6ryWBԹW5$82ch8ISr9"X'дj+lPPf8/)a[ ,} ߜX*s>!R¾c s[r`qUCynJ0:DlfhcgZ٢6IW]kc˄kA]7Bkb  'R~v):GS LK7I/2j߾ʏeiTU5Zԡ"F]CS*E_n$uuevb^4F, kE%ǞVGX$ r{P|%$OTu؆Of0q;+v&p6̴MRV!Vd1Id |1a])3yW`l4?.0ϥ`jP8O[Bes Yv L,oseq_4aLdC6_@SkL$væp/b<`Id $'y^HL)\.K?)ܟ!cqSݖ* {mys?e=D"%~9nFx$ "7le<$Sl4|Db=t8ڸܒ~CE;rnع fCxnmG.$ϓ!"Hm0l>3>C-"/^$ &,JG ~1WiscN`ʏ*$~hcКZpU{&)yѵ_7KEFt,oErEP]\O݋،PַK5<$z &%j:%_A!S;CwU0\P5? Td{\3˜B6(]^E=KUOiPǹ8&Nrl "e&R0 cG{/_(GB.ԆxW}mԃhY ]wӼkxj2p0~-l+afD\KR2)++\2*Q.Ƭ^-,;9 0- liUxm?čRą܊C+K@]7u݂vj'X#%c^V~8a [zxO-PMjCُ߶# FT twD帝*2u SFS$7"P'˹8r1TXOx GiBTъㅀ`CsoIziB00=$8) Fk12׹2A/ e 3C&o5>:^Tz>H NbL`n2gTqK$EN]B.!y^^eJ-Z#'gl8%4Uۚgfas-C8*?ƈ < 4P|r;N0T;S6XiĶc©Ӵp]Md!Exoy~/~gj#m#󜭬he DWGpw<]yV ,oOOVb}˳|$uK2r8; ٚwg{[ omk,!sTK\~ "xϗs54!Cs:AHȐVO$:<\  }UT#r q]1 6uHj5lްcYCz@ƥbEJ\N 'Q~L(c]Q{%ݓ]8)J,@A=uK4ýKM+}5%/@WCԥ~sIcl ՠ$HN1ʠ,<Չ朚$wWX#SP#|{+P^k:0YoW?cC $ķ _lB[R8-.VS ʀ<{N_^CZJ7Y>g˜M }Y;Nfwö};/qĜj[6. 2~ ASh!=V/ݪ^'وG.c D[$\!zlٴ-dwx[L9/T^d06k|_;xA[bDK1ayn1-F%߃ [SaSڧ!=waK.?"^[a=Le R"+gWk`g 5ރ򦚒M&Y@0e ݑlJdm'-F44![Xd gO|@ swJTdڨ\[蒼 ^k 0U6 Փ9ytBpߢzbߦ(==9AV> Fz~P;:xMB; xQNOK5"KJcڋ^l%㕫[-X vugf8U8Uv endstream endobj 513 0 obj <> endobj 514 0 obj [515 0 R] endobj 515 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 610 0 0 837 0 0 cm /ImagePart_2121 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 341.75 718.7 Tm 106 Tz 3 Tr /OPExtFont3 11 Tf (5.1 Imperfect Model Scenario ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 106 Tz 3 Tr 1 0 0 1 114.5 675.95 Tm 127 Tz /OPExtFont2 16.5 Tf (5.1 Imperfect Model Scenario ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 16.5 Tf 127 Tz 3 Tr 1 0 0 1 113.75 641.649 Tm 96 Tz /OPExtFont3 11 Tf (Outside pure mathematics, the perfect model scenario is a fiction. Arguably, ) Tj 1 0 0 1 113.5 618.85 Tm 92 Tz (there is no perfect model for any physical dynamical system \(50\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 130.3 595.299 Tm 94 Tz (In the Imperfect Model Scenario \(IPMS\), we define a nonlinear system with ) Tj 1 0 0 1 113.299 572.5 Tm 95 Tz (state space Wh, the evolution operator of the system is ) Tj 1 0 0 1 401.3 572.5 Tm 104 Tz /OPExtFont4 11.5 Tf (F, ) Tj 1 0 0 1 418.3 572.5 Tm 92 Tz /OPExtFont3 11 Tf (i.e. 5c) Tj 1 0 0 1 448.1 572.299 Tm 95 Tz /OPExtFont5 11 Tf (t+i ) Tj 1 0 0 1 460.55 571.1 Tm 117 Tz /OPExtFont3 11 Tf ( = E\(Rt\) ) Tj 1 0 0 1 113.049 549.7 Tm 92 Tz (where ic) Tj 1 0 0 1 154.099 549.5 Tm 81 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 156.699 549.5 Tm 101 Tz /OPExtFont3 11 Tf ( E Rth is the state of the system. An observation s) Tj 1 0 0 1 431.05 549.5 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 433.699 549.5 Tm 100 Tz /OPExtFont3 11 Tf ( of the system ) Tj 1 0 0 1 113.299 526.7 Tm 85 Tz (state "X) Tj 1 0 0 1 149.05 526.2 Tm 74 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 151.449 526.45 Tm 104 Tz /OPExtFont3 11 Tf ( at time ) Tj 1 0 0 1 199.9 526.7 Tm 129 Tz /OPExtFont6 10 Tf (t ) Tj 1 0 0 1 209.05 526.45 Tm 96 Tz /OPExtFont3 11 Tf (is defined by s) Tj 1 0 0 1 284.149 525.95 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 286.8 525.95 Tm 119 Tz /OPExtFont3 11 Tf ( = h\(R) Tj 1 0 0 1 325.199 525.95 Tm 82 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 329.3 526.2 Tm 108 Tz /OPExtFont3 11 Tf (\) m where st E 0, m represents ) Tj 1 0 0 1 113.049 503.649 Tm 91 Tz (the observational noise, in this thesis we assume ) Tj 1 0 0 1 360.699 503.399 Tm 84 Tz /OPExtFont2 7.5 Tf (T) Tj 1 0 0 1 364.55 503.399 Tm 153 Tz /OPExtFont5 7.5 Tf (it ) Tj 1 0 0 1 370.55 503.399 Tm 98 Tz /OPExtFont3 11 Tf ( are IID distributed; h\(.\) is ) Tj 1 0 0 1 112.799 480.35 Tm 94 Tz (the observation operator, which projects the system state into the observation ) Tj 1 0 0 1 112.799 457.3 Tm 92 Tz (space 0. For simplicity, we take ) Tj 1 0 0 1 274.1 457.1 Tm 116 Tz /OPExtFont6 10 Tf (h\(.\) ) Tj 1 0 0 1 295.699 457.1 Tm 92 Tz /OPExtFont3 11 Tf (to be the identity. Consider a model, which ) Tj 1 0 0 1 112.799 434.05 Tm 91 Tz (represents the system approximately, with the form x) Tj 1 0 0 1 374.899 433.8 Tm 84 Tz /OPExtFont5 11 Tf (t+) Tj 1 0 0 1 384.699 433.8 Tm 106 Tz /OPExtFont3 11 Tf (1 = F\(x) Tj 1 0 0 1 423.85 433.8 Tm 89 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 427.899 433.8 Tm 94 Tz /OPExtFont3 11 Tf (\), where xt E M, ) Tj 1 0 0 1 112.799 411.25 Tm 91 Tz (M is the model state space. Assume the system state x can also be projected into ) Tj 1 0 0 1 112.549 388.199 Tm 94 Tz (the model state space by a projection operator ) Tj 1 0 0 1 353.3 388.199 Tm 112 Tz /OPExtFont6 10 Tf (g\(\), ) Tj 1 0 0 1 378.5 388.199 Tm 114 Tz /OPExtFont3 11 Tf (i.e. x = ) Tj 1 0 0 1 424.55 388.199 Tm 97 Tz /OPExtFont6 10 Tf (g\(51\). ) Tj 1 0 0 1 456.25 388.199 Tm 90 Tz /OPExtFont3 11 Tf (In general, ) Tj 1 0 0 1 112.549 365.149 Tm 96 Tz (we don't know the property of this projection operator, we don't know even if ) Tj 1 0 0 1 112.549 342.1 Tm (5"c exists. We are just going to assume that it maps the states of the system ) Tj 1 0 0 1 112.549 318.85 Tm 93 Tz (into somehow relevant states in the model. For the purposes of illustration and ) Tj 1 0 0 1 112.299 295.549 Tm 88 Tz (simplicity, unless otherwise stated, we assume ) Tj 1 0 0 1 337.449 295.549 Tm 116 Tz /OPExtFont6 10 Tf (g\(.\) ) Tj 1 0 0 1 358.3 295.549 Tm 91 Tz /OPExtFont3 11 Tf (is one-to-one identity. A better ) Tj 1 0 0 1 112.299 272.5 Tm (understanding of ) Tj 1 0 0 1 201.099 272.5 Tm 110 Tz /OPExtFont6 10 Tf (g\(\) ) Tj 1 0 0 1 223.699 272.5 Tm 95 Tz /OPExtFont3 11 Tf (is beyond the scope of this thesis but it is an important ) Tj 1 0 0 1 112.099 249.5 Tm 93 Tz (point for additional work. Our aim is to estimate the current state of the model ) Tj 1 0 0 1 112.099 226.45 Tm 104 Tz (x) Tj 1 0 0 1 118.799 226.45 Tm 64 Tz /OPExtFont5 11 Tf (o ) Tj 1 0 0 1 122.4 226.45 Tm 91 Tz /OPExtFont3 11 Tf ( given the previous and current observations s) Tj 1 0 0 1 352.1 225.95 Tm 81 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 356.399 225.95 Tm 90 Tz /OPExtFont3 11 Tf (, t = ) Tj 1 0 0 1 380.399 225.95 Tm 81 Tz /OPExtFont5 14 Tf (n + 1, ..., ) Tj 1 0 0 1 434.899 226.2 Tm 71 Tz /OPExtFont3 11 Tf (0. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 71 Tz 3 Tr 1 0 0 1 129.099 203.149 Tm 89 Tz (In the imperfect model scenario, the model is inadequate. Following Smith and ) Tj 1 0 0 1 112.299 179.899 Tm 91 Tz (Judd \(2004\), two types of model inadequacy are investigated. One is structurally ) Tj 1 0 0 1 112.099 156.85 Tm 90 Tz (incorrect model inadequacy, the other is ignored subspace model inadequacy. For ) Tj 1 0 0 1 111.849 133.549 Tm 91 Tz (each type of model inadequacy, an example is given, where both the true system ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 305.05 51.95 Tm 75 Tz (94 ) Tj ET EMC endstream endobj 516 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 517 0 obj <> stream 0 ,,rrb6PchI#s|>IxO# VP}ZhF뾄!܆H9Q*IðP`!?dVByAdQ.ۉI0\m+{T{jM[4N/q"Q!};7S\SdwNT1K9|Iw#PER$yL8 ySPތ84*m7D#5Uq$cV@Z5q66e=Ere( Zɫ:蹬U(Q>L P3Vy)Խ=Ozw8_; &^s_;vsۂ>B%$ ^t ,)E_FNv;93f攃ЦC@V.*#5[k=~\.[薾Ttbb/ah[| { 0wSk>dNNFVrRp(;˩TЗ|8ʫ>" jōX ECqO4WNڔjεlHchi;-Dtx{cǭ`Q)$O'Y1QoO]{7IR:bAf.v)lus_@P#{7YGB~k0VhiWA#X- @t8inj[ĕJ!X$q7[N"Se^->,K೧1ۇ,ԕ37ړ8q&zRJ{™?6o:\D!f owPu ïM7[:K@j ӌ *n ^xa=p$YBpSFr2? n z9P K5I[(k/Xpz" o1Ǻ5M/8/rdT:!̸v6*8{K35מw#"|?UtH>t(̵Lȉ)7L"|i!t"ܼȉEPΒZFo>Nwy7R"{04Z.jk 4GdQiS1[!rSp`Bv>=>ӖEo.>&7;}sx@ӌ^ caZ36KHL|Vh?e'Xh h3`>Ѷ1{C77(ij*{ɫՎ'uU Yz9EQT_]̻}[N.!/c~/nab|X!E] ƕ!NusZ3ug\{^ o8W4F"NM%^XVᗶO-?2c0b =킶w~(b÷Ѕu~o/z ?OWm=] 2S-pZ1R$Ed7,@1kt 0Q w<4 iDO#@.S_}XأY}twDZX><)4[)ݾ{1*N^([JaVdMJywu\)TW|{z[B }Л?]ѫ](Wb16/yR4K p[.EP0Y½{CJ4 M[ EH:j@K@NJ0pTCX孅@py"ͰJرTJ("23߻Q;4 p$ۙa[ΜS.OPW`(ELbTJg'wŸAg=,_YJO_+<%[w樟iG֞FQAg*b5Q{o&e i5nh qs*dDOq՚DFЎz916 #<_E zg C,zl!݂ڴ%F4TE[r-a~e{a-+s$~}oa|_@(l(5!CzBU:h9'E>tN`>S ŒAt@,%!t?assO߆oe9w)y}68"h}w=,K?ϊ3հ)>nհ&<ׁ͍MrA7ÜL5M!G|gt,+n$8huat'kj]c).w}Y棼%8g{n-WF[x{zWűN݅]ƘD&%\cܼwmh"S-Jڜ0'*]bOYf`ڳd'lev^( uu!ؕe>~-}FҒk6§C)BȏK';%bzU) 9@EHXD1(iY`E8tS_XihhϤMm .)Gq=}rHjQ B+FuR tH &݋{kj3o,(RI..lXG\zӀ0H?*z k{;R3I?ynf}L^CrD" @v*AMckI ;EwgTiSL/p{ioVA%͝c=MK'j7B*@IgCn= tm?'|_d6zhKQP$3$j=qw)ZԎTxIxFV&hM >6L{`X"ynjb_שJ8ZPYp0\$#E6HVXw-wϖ;80ln{^,3'ܮ=\P&)}]c:Y,~Վ{ }"d!k-aFv*)&17IQ;2l9,zXłWZOym]|-& IPB$}PC Tt,< ع̆ԈI XQ~O"Ȁq!:;u J[z`>ٶı=eҫ upM+tPWl IsEHjw׫؍V0R<P3' *p 6lfR.`Ư,Eɢ+Kݑ24+īJj<2fs_"Ԗb]w9ZB>=߱TS2ĺr7~e١.[ݨLc*C{= .)>;*HAs$kBbeWە bl_()ġn#zɿ֥|P͞o d Pԟ#ooq`/X4T4s\2$ ?"?BcVYEvxfC?wa"CHށ 英)/-_es3JZ 3qR#8ZEkvٛ'1][ |jd6, լtWzE"k'+*'`B0)U[Ztd;_qI"] ނڥ}]fQ<&LP9guj{`eVk׆ȿj 6hk`AE6#'(6Xv.5?gAAp榌WHU#2O0"`$ZcrMAi ')xYj& eR/%Yt޿;avr2N!Lf U~]7_40gBI s|,b#yN[\|!%R,>ec\uL]̈}9)$18'rҜB3맰LZFqĄ@5a{J/"}}%0ϖ"$DMIԘ)"H! .R^ŐgO#Wy=C5zP(H"`:1[ȕB.C(b=w HiVS[ gqf8R œnCW.ܴnӍJ3vd %,ܾYM%{iGܳ~dinqTNQ͢-8k驟!Uo%.y]^L<¼A|3Fpoex0= l ~uϛ?~;(U$Ao.lX.{uTBKe%$un!ǯL.3&}N2N{&J]-.^]#Q,kĆvNlqT, 5Ui[{:66e8_ഴw^La U] Uaq AKtQ#ed ~D|BoC`c>§j/|] myEXE:n:{*A!oDIZJs+EiQzKP"),\̷e.sFO">v^OWUuJM.l6m&MhpqTg@ YMloژ(^/*@beJ.3icwٶʹO(2"qg@|$}a 3r s.䠶#:kW9z:(gę!xуoMin) eQ^cĢC,}>4Ϛz |}hoNm2zP<֜<0-MR1<0$7K`p$$LuhQDWpYlu;/4b_nPڔeDty5NEl !6@hN$ؠp?1cE0fyְ \,?TEkÐ n&ג#@Xb‡+S.˨lB5}Iz[X[1*SliNUNHCD(HTN]* %ƅm ]wE$Sml(Kxخyѐ<Ƴ:c10tƶltBYݚ'z(l='3[fl/M[6!ށEB:PBqBYІ+_>xx&bUyj`>Ȯ;0m}Ǧ &΁NKs6*?͖yИAX:Yع@) ᩼Ke5u1^"X aJF 6[#0Q[}:O{EV?SΠ: jlyl\J9\w"5Dm732@4d[g~dJlS8= &;eW DoMFݧBܾ4^ p]]zb#.B'(U)/I"[Ѭvt>eى7=+=!dѹ4Ѵ洱5ڳ7|E¬nx$4ҶݘQ"(%Ȣ%u9ȹo-<`y7?4LW2qiWM F*J} eɤ3h'/h̴VvBYpl"YݟSJ!QȪ:]f_1=(4\SW),2j xة:El@]o> endobj 519 0 obj [520 0 R] endobj 520 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 611 0 0 837 0 0 cm /ImagePart_2122 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 342.949 718.2 Tm 125 Tz 3 Tr /OPExtFont5 10.5 Tf (5.1 ) Tj 1 0 0 1 363.35 717.95 Tm 107 Tz /OPExtFont3 11 Tf (Imperfect Model Scenario ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 107 Tz 3 Tr 1 0 0 1 115.7 675.25 Tm 98 Tz /OPExtFont5 13 Tf (and a class of models are listed. These model-system pairs are used to construct ) Tj 1 0 0 1 115.7 651.95 Tm 97 Tz (and compare the state estimation methods in the following sections. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 132.5 616.45 Tm 103 Tz ( Structurally inadequacy ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 103 Tz 3 Tr 1 0 0 1 143.5 588.85 Tm 97 Tz (This type of model inadequacy appears where the system dynamics are not ) Tj 1 0 0 1 143.75 566.049 Tm 103 Tz (known in detail and its mathematical structure is different from that of ) Tj 1 0 0 1 143.5 543 Tm 101 Tz (the model. Here we use the Ikeda Map and truncated Ikeda model as an ) Tj 1 0 0 1 143.75 519.7 Tm 98 Tz (example of this case \(50\). The Ikeda system is a two dimensional map \(see ) Tj 1 0 0 1 143.5 496.699 Tm 96 Tz (section 2.4\), ) Tj 1 0 0 1 206.9 496.449 Tm 77 Tz /OPExtFont6 15 Tf (P : ) Tj 1 0 0 1 255.599 496.699 Tm 100 Tz /OPExtFont5 13 Tf (W. The mathematical functions of the system are: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 3 Tr 1 0 0 1 0 837 Tm 30 Tz (\t) Tj 1 0 0 1 239.05 454.449 Tm 92 Tz (x) Tj 1 0 0 1 245.3 454.449 Tm 51 Tz /OPExtFont2 13 Tf (Th+1 ) Tj 1 0 0 1 259.899 454.449 Tm 140 Tz /OPExtFont5 13 Tf ( = + ) Tj 1 0 0 1 296.399 454.699 Tm 92 Tz /OPExtFont8 13 Tf (u\(x, ) Tj 1 0 0 1 321.35 454.899 Tm 97 Tz /OPExtFont5 13 Tf (cos y, sin 0\) ) Tj 1 0 0 1 394.8 454.899 Tm 2000 Tz (\t) Tj 1 0 0 1 488.399 454.899 Tm 92 Tz (\(5.1\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 257.05 427.1 Tm 104 Tz /OPExtFont5 10.5 Tf (Yn+i = ) Tj 1 0 0 1 293.75 428.05 Tm 108 Tz /OPExtFont4 10.5 Tf (u\(x) Tj 1 0 0 1 310.55 428.5 Tm 63 Tz (r) Tj 1 0 0 1 313.199 428.5 Tm 68 Tz (, ) Tj 1 0 0 1 318.25 428.75 Tm 93 Tz /OPExtFont5 13 Tf (sin ) Tj 1 0 0 1 334.1 428.5 Tm 74 Tz /OPExtFont8 13 Tf (0 ) Tj 1 0 0 1 342 428.3 Tm 88 Tz /OPExtFont5 13 Tf (+ ) Tj 1 0 0 1 349.699 428.3 Tm 509 Tz (\t) Tj 1 0 0 1 366.25 428.5 Tm 84 Tz (cos ) Tj 1 0 0 1 383.3 427.3 Tm 96 Tz /OPExtFont8 13 Tf (0\), ) Tj 1 0 0 1 395.3 427.1 Tm 2000 Tz (\t) Tj 1 0 0 1 488.399 429.25 Tm 92 Tz /OPExtFont5 13 Tf (\(5.2\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 143.3 386.3 Tm 93 Tz (where ) Tj 1 0 0 1 177.099 386.3 Tm 85 Tz /OPExtFont8 13 Tf (0 = 13 ) Tj 1 0 0 1 224.4 386.5 Tm 102 Tz /OPExtFont5 13 Tf (a/\(1 + x) Tj 1 0 0 1 268.55 386.5 Tm 73 Tz /OPExtFont2 13 Tf (n) Tj 1 0 0 1 269.05 386.5 Tm 55 Tz /OPExtFont5 13 Tf (2 ) Tj 1 0 0 1 272.399 386.5 Tm 110 Tz ( + y) Tj 1 0 0 1 295.199 386.5 Tm 33 Tz /OPExtFont2 13 Tf (Th) Tj 1 0 0 1 295.899 386.5 Tm 55 Tz /OPExtFont5 13 Tf (2) Tj 1 0 0 1 301.199 386.75 Tm 105 Tz (\) and the parameter values used are a = ) Tj 1 0 0 1 143.3 363 Tm 93 Tz (6, 3 ) Tj 1 0 0 1 165.349 363 Tm 78 Tz /OPExtFont6 15 Tf (= ) Tj 1 0 0 1 178.8 363.25 Tm 99 Tz /OPExtFont5 13 Tf (0.4, -y = 1, u = 0.83. The imperfect model ) Tj 1 0 0 1 396.5 363.5 Tm 109 Tz /OPExtFont8 13 Tf (F ) Tj 1 0 0 1 409.899 363.699 Tm 100 Tz /OPExtFont5 13 Tf (is obtained by using ) Tj 1 0 0 1 143.05 340.199 Tm 98 Tz (the truncated polynomial to replace the trigonometric function in ) Tj 1 0 0 1 470.899 340.699 Tm 87 Tz /OPExtFont6 15 Tf (F, ) Tj 1 0 0 1 486.949 340.449 Tm 92 Tz /OPExtFont5 13 Tf (i.e. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 213.599 270.1 Tm 85 Tz (cos ) Tj 1 0 0 1 230.9 270.1 Tm 78 Tz /OPExtFont8 13 Tf (0 ) Tj 1 0 0 1 239.5 270.1 Tm 92 Tz /OPExtFont5 13 Tf (= cos\(w ) Tj 1 0 0 1 281.3 270.1 Tm 111 Tz /OPExtFont9 9 Tf (+ 7r\) ) Tj 1 0 0 1 302.149 270.1 Tm 1540 Tz (\t) Tj 1 0 0 1 340.55 270.1 Tm 93 Tz /OPExtFont8 13 Tf (+ w) Tj 1 0 0 1 358.8 270.1 Tm 49 Tz /OPExtFont7 13 Tf (3) Tj 1 0 0 1 363.85 270.35 Tm 73 Tz /OPExtFont8 13 Tf (/6 ) Tj 1 0 0 1 388.55 270.35 Tm 111 Tz /OPExtFont5 13 Tf (2/120 ) Tj 1 0 0 1 422.899 270.35 Tm 2000 Tz (\t) Tj 1 0 0 1 488.649 270.1 Tm 93 Tz (\(5.3\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 223.699 244.2 Tm 90 Tz (sin 0 = sin\(w + 7r\) ) Tj 1 0 0 1 309.6 244.2 Tm 1447 Tz (\t) Tj 1 0 0 1 356.649 249 Tm 76 Tz /OPExtFont2 13 Tf (w) Tj 1 0 0 1 364.55 249 Tm 74 Tz /OPExtFont5 13 Tf (2/) Tj 1 0 0 1 375.35 249 Tm 135 Tz /OPExtFont2 13 Tf (2 w) Tj 1 0 0 1 402 249.25 Tm 76 Tz /OPExtFont5 13 Tf (4/) Tj 1 0 0 1 412.8 249.25 Tm 79 Tz /OPExtFont2 13 Tf (24 ) Tj 1 0 0 1 423.1 252.95 Tm 2000 Tz /OPExtFont5 13 Tf (\t) Tj 1 0 0 1 488.649 244.45 Tm 93 Tz (\(5.4\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 142.8 196.899 Tm 97 Tz (where the change of variable to w was suggested by Judd and Smith \(2004\) ) Tj 1 0 0 1 143.05 173.649 Tm 92 Tz (since ) Tj 1 0 0 1 170.4 173.649 Tm 74 Tz /OPExtFont8 13 Tf (0 ) Tj 1 0 0 1 179.05 173.899 Tm 94 Tz /OPExtFont5 13 Tf (has the approximate range 1 to 5.5, and ) Tj 1 0 0 1 396.25 174.1 Tm 113 Tz /OPExtFont3 7.5 Tf (-7V ) Tj 1 0 0 1 414.699 174.1 Tm 96 Tz /OPExtFont5 13 Tf (is conveniently near ) Tj 1 0 0 1 143.05 150.85 Tm 99 Tz (the middle of this range. In this case, the model state and the system state ) Tj 1 0 0 1 142.8 127.549 Tm (share the same state space. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 307.699 51.25 Tm 84 Tz (95 ) Tj ET EMC endstream endobj 521 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 522 0 obj <> stream 0 ,,NNb6jY$JwE>0;:ϰѻ<otR' /" dw~ qk]tחe^iZ\'hՉ:OFq>Yr0>7$+ōP|qӉ=1!~ߠpyZ7W%/T,~)҆F M`nA'ܐ >qJ~g\lJB@ #.R f~1`LiscnK͎5zh%0g ogdQIv3 /E{ P=ukgnrr $ip;`0}|ڪJO7T-% fO+ xjUŲӗu2Igy?rqqݱPJRnSuF2{Be=o*I2tn#md3g4JW t^`aFU/U|#HT:gu_d<J[OgqCZvv(ef^Ha~cpd'U*#᝻8gƚFCS lY0~h#2[ljMcoB\ .w~ eN],XVS700"&cA23bƨ{YvWqё|4JA dNU7wF 6jw_=٫N^ &,]'D0Y.B1+ p o:oV Rv3oG%#iа(ֈDdfB݁GGGS nH`^^JH9?_ Qu8{bz5J Bsj/΂O5Ѐ=^!t߮#(2DOj2qGZ;9fqvgfgT&Fj/Xmw 0H1 ֦5_S?n+j@E\v 9ĕ4 <ٰ!oS6{_?Aǐ: ?Z}ه!P[p̫6<]i1w?D䎀u'"$Ln$o=4lT‷AO} $7e[rN){uC#48€w<~9n|҂K,^ ]N75JRcCã:Pd_xuG~C"ewKDk]B;FvCmCſ%ZUW&$VƭÏbz7YN[gd,0ib4+?ӇUv=FqD3v!9Q!fsnKh 5\-I.|礰ϯP:I]m9$Ud\~)_ł$Qm-L9rfҔ= -Hr޷^g%oyǨ1gvx̦rĞ>4u,,c/Y@i(zh0r|rHd//F eל %$寃ޔ5֑FY|$δSy#ϕ5RH(Mr95hOҝ[]UīG02z$]40ZĨђݠ}%v/ ٧c6f>g"&ܷ_#c0g:dѨ7Qw"}JBz"4N=o;z= xctalz ?tQXu)_`ݐgbE`Okv4vGqQ`h~#T1.bUo*62N9$V-i-kr1sБ;e&U Iy:p'{)O Ii1uƁxh}j!όz19qaKrP ]ሩ,L:OJM7O *]!d4 bH"|ҡ.Vc7Ƥ ^nEZlhuo"~48U|+xnǕ杬=:k58XT @7ɝZ:c3Ws /&P ~q)-n4X-sar82&]v69;SPJij}']KyS:29.g19B4]Gk27PTAZcnwRש7$e,.t&9 'j*53pVȝ8, h3޶TQf>#]<,!cBadHα0[xRQ1>8#峾-3ٰP8f@P?Z.A oOkX$Q/k2| Y ԓc*ߌpѤ,5j &olrK9Kfv* {.'?S3g&iqx(x{ SҹW8 Л§$xmtQl#i-rj~Mxb/.g?Hw:2XbU>&3y]azn*Rrц +iUM ]&ğH RW#uZY0gC)'õ@闧v\אPQDEW|ssM`1sq~DI; {S$ۤ"왈If?v07P2k4g\dJFzCHRX.ОJ*KTaA0G8Wzy ^$bP 3p4 ƛ9%@fg>=E0Nf룞VU7&Dn-U!GNQ&(fG`:7>Ž ](=-NeQpǃVĀp'JYwJ %[ϩ?)Ol$Q vmX95w} f?=t㼄ԺI LnE،)=Ů{/7h-kdZUcUZ ȓOʏ呐.sYmZ$K*3D`ф \kc0}`~mrڰm rQqu(Rs?+Qn_$Ӷu'2$AOڏZ}UӷG&+Myh>1% Ź}W$EQ|w ꎃhGvDξ"h<І ߫2=h')OK>LX\b:Β# BjF\tJF㪜1]Ge~0zzAXl^ ^}\Lk*vrGnGy6朗$W ҩ'mXO|"Pj#+JjM@ȊX^|+RmOm4BE  d,͑:_ZQ#um`y);L kjMo/%NZ y~`{Yk&egT9iE][!'H .hw١@v y=Vì~l}4 FG1 E YR74w!!æ`i%e|.tni$6($ SA jnx^R"Xt8Z'}o%>A$kWs9Vwhݺkk q)JŰ/2r@S&K)TVתAOLJ;HCmZhvroxյ]^L>93<['1SءjK- M)-7>R.f7~r\Yhfə.KF(99n`&qù{o>> pe wMT˸,OҾa鉤FAw Y1>c=e!ȸԒ5~fAc : @*^􌛃jOsd w}^ x5Z=ŰkEQXtȟ/y F \C>r\r?;7EcG`i򢵵&-oq5he"^q\x3_ӐОY공oΚ&|šR? ItQq%֋G`(Ai:g!FbSU ǧLHυha,| aT^yLEF`+]kGt*ZKߞUՉ&"^i\<˫z& [~b&1n5_#ѕ'pmPxh$mH:e j jY_N2|iD}*ɗOGl}Gw\}Ak澑]r+ow:^8n (8`LOWᑧοmBĬ0+q1Ɂ8{CۢؔDr\VSgoqqW2!6mi5 uo]$c$g'de97Th,\;bĆ [?em|߲?^Z{7ǕppO@+[|qZd5|<ݣr8 MߔuY QQL÷mB 2.,'VpA\!v ܆s̕7+X,}B<{xC"R)jJdb:MR@x'y@ى~\'B,MB+7Y!34Bu?eSD:E$zN:fļ gH]V`e筐 bFNk69EM#a Zу}0xkk0լUZm^3ke l],Z`9T?5Cd5M&,ͮjÒ¶wArhʡ`ǧ6UP¨ncL~Ѕ!M^Cf5'Q!|юAId={ ,e c?MnŔf_94z^A] mL0 5Б58Yc;2nON8ı1<<Z:^pA=)(}dN2AtmΖ9\# endstream endobj 523 0 obj <> endobj 524 0 obj [525 0 R] endobj 525 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 612 0 0 837 0 0 cm /ImagePart_2123 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 343.899 718.2 Tm 126 Tz 3 Tr /OPExtFont2 11 Tf (5.1 Imperfect Model Scenario ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11 Tf 126 Tz 3 Tr 1 0 0 1 145.199 675.25 Tm 97 Tz /OPExtFont5 13 Tf (Generally, the truncated Ikeda model is a good approximation to the Ikeda ) Tj 1 0 0 1 144.699 652.2 Tm (system. The model error is relevantly small but space correlated. Figure 5.1 ) Tj 1 0 0 1 145.449 629.149 Tm (\(following Figure 1 of \(51\)\)\) shows the one-step forecast error between the ) Tj 1 0 0 1 144.5 606.35 Tm 98 Tz (Ikeda system and the truncated Ikeda model. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 161.75 568.7 Tm 52 Tz /OPExtFont2 11 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11 Tf 52 Tz 3 Tr 1 0 0 1 153.099 528.6 Tm 111 Tz /OPExtFont3 7.5 Tf (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 7.5 Tf 111 Tz 3 Tr 1 0 0 1 161.3 488.75 Tm 94 Tz /OPExtFont13 8.5 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 8.5 Tf 94 Tz 3 Tr 1 0 0 1 147.599 448.449 Tm 125 Tz /OPExtFont3 7.5 Tf (-0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 7.5 Tf 125 Tz 3 Tr 1 0 0 1 155.5 408.85 Tm 97 Tz /OPExtFont2 11 Tf (-1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11 Tf 97 Tz 3 Tr 1 0 0 1 147.349 368.75 Tm 125 Tz /OPExtFont3 7.5 Tf (-1.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 7.5 Tf 125 Tz 3 Tr 1 0 0 1 156.699 362.3 Tm (-0 2 ) Tj 1 0 0 1 175.199 362.05 Tm 1124 Tz (\t) Tj 1 0 0 1 202.099 362.05 Tm 97 Tz (0 ) Tj 1 0 0 1 206.65 362.05 Tm 1103 Tz (\t) Tj 1 0 0 1 233.05 362.05 Tm 110 Tz (0.2 ) Tj 1 0 0 1 246 362.05 Tm 932 Tz (\t) Tj 1 0 0 1 268.3 362.05 Tm 112 Tz (0.4 ) Tj 1 0 0 1 281.5 362.05 Tm 934 Tz (\t) Tj 1 0 0 1 303.85 362.05 Tm 108 Tz (0.6 ) Tj 1 0 0 1 316.55 362.05 Tm 942 Tz (\t) Tj 1 0 0 1 339.1 362.05 Tm 109 Tz (0.8 ) Tj 1 0 0 1 351.85 362.05 Tm 1132 Tz (\t) Tj 1 0 0 1 378.949 362.05 Tm 56 Tz (1 ) Tj 1 0 0 1 381.6 367.8 Tm 1203 Tz (\t) Tj 1 0 0 1 410.399 362.05 Tm 104 Tz (1.2 ) Tj 1 0 0 1 422.649 362.05 Tm 952 Tz (\t) Tj 1 0 0 1 445.449 362.05 Tm 106 Tz (1.4 ) Tj 1 0 0 1 457.899 367.8 Tm 963 Tz (\t) Tj 1 0 0 1 480.949 362.05 Tm 102 Tz (1.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 7.5 Tf 102 Tz 3 Tr 1 0 0 1 115.9 320.049 Tm 97 Tz /OPExtFont5 13 Tf (Figure 5.1: The one-step prediction errors for the truncated Ikeda map. The lines ) Tj 1 0 0 1 115.45 306.1 Tm 98 Tz (show the prediction error for 512 points by linking the prediction to the target. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 132.5 252.35 Tm ( Ignored-subspace model inadequacy ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 143.75 224.049 Tm 100 Tz (This type of model inadequacy appears where some component\(s\) of the ) Tj 1 0 0 1 143.5 201 Tm 102 Tz (system dynamics is\(are\) unknown, unobservable, or not included in the ) Tj 1 0 0 1 143.75 177.7 Tm 107 Tz (model. In this case, the system state space and model state space are ) Tj 1 0 0 1 143.75 154.899 Tm 91 Tz (different. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 143.75 126.6 Tm 99 Tz (Here we use the Lorenz96 flows \(63\) as an example of this case. We treat ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 307.899 51.5 Tm 85 Tz (96 ) Tj ET EMC endstream endobj 526 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 527 0 obj <> stream 0 ,,5b7 ,#9ـ\˸{ނ~hu.A&!pYBNa i< nIjJ5qЗٛf|[T4Uj :@3=Ya%FF wU*EtH:J|ϷrZ b吐E{^WB/{:%'/>]azb{㸔^tW9~ZTXLH1n c{7e8'cFæ L,>qf}܃_ܾeC輸TĽTx"4\-(R`ɚ'TH_Vsax'B鉷ב>ʆ"yW s(s%\Gm0n^Wςe jI1fۂҚaaCsmhoݍ fc `9sSGs9e5DR'Yڬ@Ј1y:Qɏom\Y \T2֐CL4ZH[fbW3ZLXP_(t5VF/?g]tT&NџRcmo`ݏ׮n_98L%9517҂~(Ud}nƳ⋿3Z;ǖ Pë3WCV1XG1@y /nHrBk5pljԠ=VH/@ݸ"7 K|qrB|jzB"(n2Һ]ɋ|lȪ+ڱDJgt%lP}9Y9q1%e[) ;.:\ Z]oP?t>EXɘw zPX)\"@cq?I89׎sât:.te0>R ^2ƃ9 x70Z>7ac98f>n43{tNd.I6i{b|DG>^&KqsvBi]mLR=Tc|zT70m. f3o|?V .ő j,O18aѰD͍iqMhtݮQۇr6Ddc:7!f:E[h740xGN5G OPa% %aԯ֫Cn'finNͬ'ZNd> T5/:@|bHya%ÕRK 1l*"ڎsiO'i2o5Nyuv~1]\,h;1EtezW3Ԯpfw~(':SEBuՊ$*:}7I\y!ͯ =+N߯\l~gXd{ )xZ)~r 1k2pe; N悺aF^lZ@#;k_aGA(:PEYC:PΒf<שӢ h^ H/oO25)/sT B ,~/s}C!giֈ) SWP D~%x{*5=ZNO_ܕkd9ti"Jb #blsU!HvfZkl؟z ?(% H.6}UUŇ%iZ-#܏{lTR=ޠ(llR-@j]n[Kp~lv¾e9cdr-?$#m{v *BHZ[:?;7,`M*bB!ƿ3!Mѭ4*yʜF3 +[6uBeÚ*8 &IbS"I# ΢2 mdX!Zl] :km?>y;d. kij*pPʯĿeD4V>C #Ī˰3.ڱGm6K3r#W2>+>ޠ.ٰ> gbYRg'diQ_$qX@`%vB݈& )Vҥg=<}&*<Ebs+t2>pw-_~$QVMXUJ戠F/fO#2\z-jզސVi8>?yzDYFT2 1$U`tW]p,lP (iS&[$i9E^cs^Vߛ4xfz@LުX=RyO<@#^F"1ps(M WX4Г$A"Z|LoxM!I7Q Cb{D^$,eZO۾=]g"4[7 jI.ϛ˪delc$wRP34e$B?`c1)I0YT'R8TB9Qͼu}aM2s?iO sl~шUj<%A( ]6TUuG g]#E͊ʝ2TSce .hZg?ݩo'}EXcAgP>$В<ّ;Nb^k$E}8V/.U .W0wM* #Kۇ([OY6F2mN⤊*ԬÚZ}{Nzմˉ {޷@a{,Jl޶ԟddo5z4QaaCYQfOgRX\n/FzFRdtdxLO7i pc8ѕsI_-cP䮒A`xŸIh gCo|z >ٌ(C$ť'cX7JnyzzuEM1jU.EЬׯFv̄'t n=ҔhE'(HM HL>_mn7;(x\kei;aFk";]A#1͌q_ ; M2#ЈxER3,\X6duZ:0HX*ߙm취}F72.Upl^pBDU>by b=]L.pe{NA^)2;3t2%|*(v_SulSlUM/ Ij!Wk6 rCʋquz'Ir^BiX9ĝɰmo%ρ%6\yvFn,`;YI;sWj%t@%r%aB{mӯ[̅9wd Ȩ}tԑg7ٖХdU1eK(&3(C]\vAv*!򆀠JrӺur vnH bq lyѤ\3N]X>C#F-rgBTDr#S6fu!(e-vDW5]X884l$G9qp9FykIWFǴhX^PFu%^0U8> xᦳ2e@A'h p]ws WE.ݩy욈p"}^pb;j3G펽Uv[}}w=@-&8»$57s4~8$(҇1VXmkٗ75 3_&ss"(ȷa@O ؀nЮ-Qle>CE.a#_;-CвUwuJ\ V 0F8tibF#jtɋ+:Vqn0rG)gye!L/> `2@ yDMڶ1?v*R,q0.(LS}_=3q'E⠐p5 젻޵3]fZo!>Jz-hZR6#Ve2E2lPN. t0Фcp`^onmxDλJM^0ɲph\^８ad~C mIӶf(382");@$p0x#* |-M>66ӝ0Ј\Ljw"A2󳢰崥9dWBx1jqgBXZRݜV;3mL(LԉQi r 0S O+mnNxV.zq-eoRd w\yr]Bt4/meQR釿v0v\a`\hrV)CZS/3-/;U_frkT1CXTka5^Rm[3"fÔķ_ma$rH5A@$aE1gf?~`"[)HotM4NiM]B gĺWg"8}aO1äƂ+e&܁9X$`bmR7c'9WZe(OrgU}dnZ{=igssBχ@KVW4N"vIOe!OUW&\@83*Y?vE y]JG7ৗ ؍OsylG=fv aQ 0xPk $F~9CF K~h e2T'I&#b_Ou ~I-vh!ζݫwMr>m<2KЃ7yizpؿhfo:Ά#۴eHeXyXBA=.7 LfxEWG9٭| }@x$~=ԛdTjZ*"H\1=@[:}_#mk> endobj 529 0 obj [530 0 R] endobj 530 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 611 0 0 837 0 0 cm /ImagePart_2124 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 0 837 Tm 36 Tz 3 Tr /OPExtFont6 11 Tf (\t) Tj 1 0 0 1 390.949 566.049 Tm 104 Tz (h ) Tj 1 0 0 1 412.55 566.049 Tm 392 Tz (\t) Tj 1 0 0 1 423.35 566.049 Tm 36 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 11 Tf 36 Tz 3 Tr 1 0 0 1 194.15 566.5 Tm 111 Tz (dx) Tj 1 0 0 1 206.4 566.5 Tm 70 Tz (i ) Tj 1 0 0 1 208.55 566.5 Tm 2000 Tz (\t) Tj 1 0 0 1 397.449 566.299 Tm 79 Tz (x) Tj 1 0 0 1 402.5 566.299 Tm 88 Tz (c ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 11 Tf 88 Tz 3 Tr 1 0 0 1 197.05 550.7 Tm 133 Tz (dt = ) Tj 1 0 0 1 222 557.399 Tm 2000 Tz (\t) Tj 1 0 0 1 288.949 558.1 Tm 127 Tz (xi_ixi+i xi F ) Tj 1 0 0 1 396.699 551.149 Tm 97 Tz /OPExtFont8 11 Tf (b ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont8 11 Tf 97 Tz 3 Tr 1 0 0 1 411.1 545.649 Tm 95 Tz /OPExtFont6 11 Tf (J-1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 11 Tf 95 Tz 3 Tr 1 0 0 1 0 837 Tm 36 Tz (\t) Tj 1 0 0 1 215.05 531 Tm 104 Tz (dy) Tj 1 0 0 1 225.849 531 Tm 146 Tz /OPExtFont8 11 Tf (j) Tj 1 0 0 1 229.699 531.25 Tm 43 Tz /OPExtFont6 11 Tf (,) Tj 1 0 0 1 231.599 531.25 Tm 95 Tz /OPExtFont8 11 Tf (i ) Tj 1 0 0 1 234 531.25 Tm 2000 Tz /OPExtFont6 11 Tf (\t) Tj 1 0 0 1 416.899 531 Tm 100 Tz (h) Tj 1 0 0 1 422.649 531 Tm 86 Tz /OPExtFont8 11 Tf (u) Tj 1 0 0 1 427.699 531 Tm 93 Tz /OPExtFont6 11 Tf (e ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 11 Tf 93 Tz 3 Tr 1 0 0 1 239.5 520.45 Tm 103 Tz (= ) Tj 1 0 0 1 247.199 520.45 Tm 2000 Tz (\t) Tj 1 0 0 1 329.75 520.2 Tm 107 Tz (Yi+2,i\) cYjo: ) Tj 1 0 0 1 391.699 520.2 Tm 889 Tz (\t) Tj 1 0 0 1 416.149 520.2 Tm 189 Tz ('x ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 11 Tf 189 Tz 3 Tr 1 0 0 1 0 837 Tm 36 Tz (\t) Tj 1 0 0 1 220.099 515.399 Tm 109 Tz (dt ) Tj 1 0 0 1 229.449 515.399 Tm 2000 Tz (\t) Tj 1 0 0 1 422.149 515.399 Tm 79 Tz (b ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 11 Tf 79 Tz 3 Tr 1 0 0 1 489.1 559.1 Tm 92 Tz /OPExtFont5 13 Tf (\(5.5\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 489.1 523.299 Tm (\(5.6\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 343.699 717.7 Tm 102 Tz (5.1 ) Tj 1 0 0 1 364.3 717.7 Tm 115 Tz /OPExtFont5 12.5 Tf (Imperfect Model Scenario ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 115 Tz 3 Tr 1 0 0 1 144 674.5 Tm 104 Tz /OPExtFont5 13 Tf (the Lorenz96 model II as the system that generates the data \(details of ) Tj 1 0 0 1 144 651.5 Tm 97 Tz (Lorenz96 models can be found in section 2.4\). The mathematical functions ) Tj 1 0 0 1 144 628.45 Tm 98 Tz (of the system are ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 143.75 476.5 Tm 109 Tz (for i = 1, n. The system used in our experiments containing n = 18 ) Tj 1 0 0 1 143.75 453.25 Tm 102 Tz (variables x) Tj 1 0 0 1 199.199 453.25 Tm 47 Tz (1) Tj 1 0 0 1 204.25 453.25 Tm 89 Tz (, ..., ) Tj 1 0 0 1 229.449 451.55 Tm 96 Tz (is ) Tj 1 0 0 1 236.9 453.25 Tm 104 Tz ( with cyclic boundary conditions \(where x) Tj 1 0 0 1 456.25 453.5 Tm 80 Tz (ni ) Tj 1 0 0 1 470.899 453.5 Tm 131 Tz ( = x) Tj 1 0 0 1 499.899 453.5 Tm 43 Tz (1) Tj 1 0 0 1 504.25 453.5 Tm 94 Tz (\). ) Tj 1 0 0 1 143.75 430.199 Tm 97 Tz (Like the large scale variables x) Tj 1 0 0 1 292.55 430.199 Tm 80 Tz (i) Tj 1 0 0 1 296.649 430.199 Tm 105 Tz (, the small-scale variables have the cyclic ) Tj 1 0 0 1 143.5 407.149 Tm 104 Tz (boundary conditions as well\(that is y) Tj 1 0 0 1 334.55 407.149 Tm 77 Tz (m+i) Tj 1 0 0 1 351.85 407.149 Tm 51 Tz (,) Tj 1 0 0 1 353.75 407.149 Tm 80 Tz (i ) Tj 1 0 0 1 356.149 407.149 Tm 130 Tz ( = y) Tj 1 0 0 1 382.55 407.149 Tm 112 Tz (i) Tj 1 0 0 1 386.649 407.149 Tm 42 Tz (,) Tj 1 0 0 1 388.8 407.149 Tm 83 Tz (i+i) Tj 1 0 0 1 402.5 407.149 Tm 102 Tz (\) \(in our experiments ) Tj 1 0 0 1 143.5 383.899 Tm 97 Tz (m = 5\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 143.5 356.05 Tm 95 Tz (The Lorenz96 model I is treated as the imperfect model \(details of Lorenz96 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 143.5 333 Tm 98 Tz (models can be found in section 2.4\). From the mathematical function ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 228 270.6 Tm 108 Tz /OPExtFont6 11 Tf (dx) Tj 1 0 0 1 239.75 270.85 Tm 95 Tz /OPExtFont8 11 Tf (i ) Tj 1 0 0 1 242.15 270.85 Tm 36 Tz /OPExtFont6 11 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 11 Tf 36 Tz 3 Tr 1 0 0 1 230.65 262.899 Tm 112 Tz (dt = xi_2xi_1 + x) Tj 1 0 0 1 328.8 262.899 Tm 95 Tz /OPExtFont8 11 Tf (i) Tj 1 0 0 1 332.149 262.899 Tm 91 Tz /OPExtFont6 11 Tf (_) Tj 1 0 0 1 338.649 262.899 Tm 105 Tz /OPExtFont8 11 Tf (i) Tj 1 0 0 1 342.699 262.899 Tm 119 Tz /OPExtFont6 11 Tf (x) Tj 1 0 0 1 348.949 262.899 Tm 101 Tz /OPExtFont8 11 Tf (i+i ) Tj 1 0 0 1 361.449 262.899 Tm 107 Tz /OPExtFont6 11 Tf ( x) Tj 1 0 0 1 383.75 262.899 Tm 85 Tz /OPExtFont8 11 Tf (i ) Tj 1 0 0 1 385.899 262.899 Tm 117 Tz /OPExtFont6 11 Tf ( + F ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 11 Tf 117 Tz 3 Tr 1 0 0 1 489.35 262.899 Tm 93 Tz /OPExtFont5 13 Tf (\(5.7\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 143.3 215.649 Tm 97 Tz (one can see that the small dynamical variables ) Tj 1 0 0 1 369.35 215.649 Tm 117 Tz /OPExtFont6 11 Tf (y ) Tj 1 0 0 1 378.25 215.649 Tm 96 Tz /OPExtFont5 13 Tf (in the system equation \( 5.5 ) Tj 1 0 0 1 143.5 192.6 Tm 100 Tz (& 5.6\) are not included in the Lorenz96 model I. The magnitude of error ) Tj 1 0 0 1 143.3 169.549 Tm 96 Tz (made by the imperfect model depends on the coupling parameter ) Tj 1 0 0 1 464.899 169.549 Tm 94 Tz /OPExtFont6 11 Tf (/i) Tj 1 0 0 1 470.899 169.549 Tm 78 Tz /OPExtFont8 11 Tf (x ) Tj 1 0 0 1 475.199 169.549 Tm 91 Tz /OPExtFont6 11 Tf ( ,h) Tj 1 0 0 1 485.75 169.549 Tm 105 Tz /OPExtFont8 11 Tf (y ) Tj 1 0 0 1 489.6 169.549 Tm 100 Tz /OPExtFont5 13 Tf ( and ) Tj 1 0 0 1 143.3 146.5 Tm 95 Tz (in our experiments we set both fi) Tj 1 0 0 1 303.6 146.5 Tm 68 Tz (x ) Tj 1 0 0 1 307.699 146.5 Tm 99 Tz ( and b) Tj 1 0 0 1 339.35 146.5 Tm 75 Tz (y ) Tj 1 0 0 1 343.449 146.5 Tm 98 Tz ( to be ) Tj 1 0 0 1 376.55 146.5 Tm 75 Tz /OPExtFont5 12.5 Tf (1. ) Tj 1 0 0 1 389.5 146.5 Tm 96 Tz /OPExtFont5 13 Tf (In this system and model ) Tj 1 0 0 1 143.05 123.25 Tm 98 Tz (pair setting, the model state space and the system state space are different. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 307.899 51.25 Tm 87 Tz (97 ) Tj ET EMC endstream endobj 531 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 532 0 obj <> stream 0 ,,||b7 \H!a//sJ>xxN&d.ٌVs kʑ!"0(SG;,ےsmˊ p {߀ljKGT",|.rk[Saѕ؊ͼl:0D*i㥡y%^H|s5]qDcعlXz[aeWBOΙ6R ]^^NC]F}O$bOPv#c:?\bd*AT (CX1h~e类H-XJ3z=wǣ-࠵6 "Z;3&B٠["%mcc.FsD."oJZ},@c|ľ ^Jp&3 kfB .=`[h9pFL,ܦe8 ?W妯7M]Xl~$'ɭҪpPcNJtkS|PN^ꮭ6_8#1f&"~+>:6?\eScGɰ `$2n[5p*]'-1,aIȰg_6^/}vLC<1EZyCH'ש.v|4Pj Hm.rIV o6n`aCF>6Bǯk\h)$JȦ# ^ؕ! >6#'"F >z]Kmz@EodLG3EC?\wBmu-ڊR]W\A{z,$.DSDԖz·6WOd%_3JZko~lyޥ ^rZ4(i78Mӝ*Lӊ~Vokz[ƣ箴}7"Yi6Jj{ObnthYte%x8 VGWXJLY!X5,x O_=]t5b*]ly#ٸpx\ۤșnvGכBH^?3(0sH! dxUw(rvq!hX0:sU`Kz7G۵۠zڌ+!n=59je":KHdqUV/(NDK`8!K b>z>{.Priʐp bE@'8">]GuCOYJBm(sW D)Z\\ DM:ޓ Wns此RLTm=&&lgr?˛jEp8^+ o9$Z ' [aKZ*8h=혐9s0̴{ik;?UU_3@<̿B3hwQ [; eqX DdGH!)]_F‘-1! #mgw UCvC(dp*F 80ݜ6*ZfxLm?!ŔA vC3ce*(Q IN"y)̉.0lU,H mS !>x?t@GA,? 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S3lCGAޏ`(bҐlnl :q* q%]TWj2jX|6ìp(N4NNGEx00C #$@78x;b;{cF H@iΥ(Ye`uLx>~( 7l̫o8(5-+ =ŘrY4'N/bdwg%R{ TlUzJT S1TP2_gF7S Eu!oi  F Sk݇lGi ` l)[pRߦѵ4i 󑄊tQtaƔ aquha9*ͱRϑ"d-Q,  ,͑Xvku(a:SЁVz CmND*7PƑ0"4 .{L5ؽP8 vDA"Dйvy%]'Ilh<"A$ .QԳn_d@nCiGM'[,?j릏 E_>> 3/L<ZIwDVl,Y0MDq !cBu]2E/4ba7Xr?+ڙawyx\3rP'GfU˫C),:*$1l*BWL nl1P7ޫ0H)e ΄$rE%  BoMStĥuҝɥXɆo2o{0ՁuE1ʐ)č"ѽO oE9[%|]ңIʦ2Z2X̥>./v㴘Yq!T %fx.Ʌ~6z p ()bX4_}b@mRΈI0 Km wͻrf4=L PI鈡7Qyba(O>x#iD n4/F&(V倹jg椇& c768מ((ҁ5-'ODq6=]PID|XT*k1e<`jVm3UWV:,OC$!d dCxu֩B5ZϋD T%o[a(VR! hlmX/Cf tX6D&9 V |e@SU #g$}a"jaƧ}>.1DS#K6p+II-3&gBn ṡrh k>%`}k(ltr(_-DT^ P9pI׉'t[ "_\?E)lh4>q74A='05$dfFj-㼫MXYԓyfYw]pYo)Z_x_%)r: }Ikb@',/یV(ZPLnнѢw,`B#ڎG `FT#uYowWF--BRt=fgMbJظ7X@^By5ޯGpnP{/NcU1S~kGVLE!ԘVE63`mL^)f1@㭑p?lg8 ̆ibͤp)i|flؼ֩6݅6¨vyqe`1"C,miTK N {ဖ ψdQU(~+Ē\XnW;" 2 "tؔOTnRFq/#!NLڜ9t59Nv4>!?.Aqӝ&2*ԙʫNߺڳ#(԰;T0ud@Cnh 5a_Ao~Qr CP7d}z =yR%:Ӭ/`,h{s~U¥WKyR4֩șpW/v$1TC#Ut'H^o.C |BS5KcK򧙶/byj: Nsއũ-vQ@r3FsX})M:gAz `VaRf~zv030(Dq {lČgs}W7FdVɇ"QGfJK[-(:T 3[>HgR 5Tj=8ߤZp?)Hf\J#-z~.ӡ j+H8l"fn1)?uKgӕ>G"LUa3wh0EgC5$h\2W"p ^,|~?w !IBQf1:vxO"CQd{εé(pyO=Ig"C* A[ 42>c=/a,EH )oTm<2b+B1,Tp8{PHSVbtSH(vs]Rw+ٳܙ$ȳԵ҅VՃɳn)O:+%.ٸ)D(۰> endobj 534 0 obj [535 0 R] endobj 535 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 609 0 0 837 0 0 cm /ImagePart_2125 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 371.75 719.149 Tm 98 Tz 3 Tr /OPExtFont3 10 Tf (5.2 ) Tj 1 0 0 1 392.399 718.899 Tm 112 Tz /OPExtFont5 13 Tf (IS methods in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr 1 0 0 1 114.95 675.5 Tm 131 Tz /OPExtFont2 16.5 Tf (5.2 IS methods in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 16.5 Tf 131 Tz 3 Tr 1 0 0 1 114.5 635.649 Tm 110 Tz /OPExtFont3 13 Tf (5.2.1 Assuming the model is perfect when it is not ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 110 Tz 3 Tr 1 0 0 1 113.75 605.399 Tm 103 Tz /OPExtFont5 13 Tf (What will happen if one ignores the model inadequacy and assumes that the ) Tj 1 0 0 1 114 582.6 Tm 96 Tz (model is perfect. Here we investigate whether this would degrade state estimation ) Tj 1 0 0 1 114.25 559.299 Tm 100 Tz (of the nonlinear system. And if so, how do the results from the perfect model ) Tj 1 0 0 1 114 536.299 Tm 97 Tz (scenario, as shown in chapter 3, change when applied to imperfect models? ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 130.8 513.25 Tm 98 Tz (In the Perfect Model Scenario, there are a set of indistinguishable states H\(i) Tj 1 0 0 1 503.05 513.5 Tm 22 Tz /OPExtFont3 13 Tf (.) Tj 1 0 0 1 505.699 513.7 Tm 63 Tz /OPExtFont5 13 Tf (\) ) Tj 1 0 0 1 114 490.199 Tm 98 Tz (that can not be distinguished from the system state ) Tj 1 0 0 1 370.1 490.449 Tm 52 Tz /OPExtFont6 10.5 Tf (53 ) Tj 1 0 0 1 380.399 490.449 Tm 97 Tz /OPExtFont5 12.5 Tf (\(48\). ) Tj 1 0 0 1 407.75 490.449 Tm 95 Tz /OPExtFont5 13 Tf (In IPMS, however, it ) Tj 1 0 0 1 114 467.149 Tm 98 Tz (is not necessary that IHI\(i.\) contains states other than itself. Even for the state ) Tj 1 0 0 1 503.5 467.399 Tm 39 Tz /OPExtFont6 10.5 Tf (FC ) Tj 1 0 0 1 114 444.1 Tm 97 Tz /OPExtFont5 13 Tf (itself, the projection ) Tj 1 0 0 1 216.699 448.449 Tm 68 Tz /OPExtFont3 13 Tf (i) Tj 1 0 0 1 224.4 444.35 Tm 97 Tz /OPExtFont5 13 Tf (of the system trajectory defined by x into the model space ) Tj 1 0 0 1 114 421.1 Tm 103 Tz (is not a trajectory of the model, which means no state of model is consistent ) Tj 1 0 0 1 113.75 397.8 Tm 101 Tz (with the observations. This situation can arise even when the model trajectory ) Tj 1 0 0 1 114 374.75 Tm 100 Tz (remains in proximity to \(the observed part of\) the system trajectory \(50\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 3 Tr 1 0 0 1 130.8 351.949 Tm (If we ignore the model inadequacy and apply the ISGD algorithm to find ) Tj 1 0 0 1 504.5 352.199 Tm 98 Tz /OPExtFont5 12.5 Tf (a ) Tj 1 0 0 1 113.75 328.899 Tm 103 Tz /OPExtFont5 13 Tf (model trajectory, we will find that the minimisation converges very slowly to ) Tj 1 0 0 1 113.75 305.649 Tm 99 Tz (zero when the window length is very long, which implies no model trajectory is ) Tj 1 0 0 1 115.2 282.35 Tm 103 Tz ("close" to the observations. In the results shown in 5.2.5, the results of state ) Tj 1 0 0 1 114 259.549 Tm 98 Tz (estimation by applying ISGD algorithm to minimise the mismatch degrade after ) Tj 1 0 0 1 114 235.799 Tm (certain iterations of gradient descent. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 114.25 191.399 Tm 117 Tz /OPExtFont3 13 Tf (5.2.2 Model error ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 117 Tz 3 Tr 1 0 0 1 114 160.899 Tm 95 Tz /OPExtFont5 13 Tf (In this chapter we will consider the point-wise model error to be Sc) Tj 1 0 0 1 433.199 160.7 Tm 32 Tz /OPExtFont3 13 Tf (-) Tj 1 0 0 1 436.1 160.7 Tm 68 Tz /OPExtFont5 13 Tf (ri+) Tj 1 0 0 1 448.3 160.7 Tm 97 Tz (i F\(i) Tj 1 0 0 1 484.1 160.7 Tm 68 Tz (n) Tj 1 0 0 1 490.1 160.899 Tm 106 Tz (\). It ) Tj 1 0 0 1 113.75 137.649 Tm 99 Tz (might be reasonable to assume the observational noise is IID distributed. But it ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 129.849 117 Tm 95 Tz /OPExtFont3 9 Tf (lassume the projection is one-to-one identity ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 9 Tf 95 Tz 3 Tr 1 0 0 1 306.699 52.2 Tm 87 Tz /OPExtFont5 13 Tf (98 ) Tj ET EMC endstream endobj 536 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 537 0 obj <> stream 0 ,,WEb +o{-0}=jIɪtƪ K.=9bPב} - d斿D+s;vE1f$) ĒG7ܤ7;ȍ~yGvL!LsmXA;W]9{X8ϓ۾ֻ]gB~> n뚶%PB芗ܼS#,Zwcg2Ї%^\^R/Lj`M^™Elrho=CvpTPwT^XNX[)0Kt۲f<( TH)v:1y ^( X@6'?H^E=֕bOS:Vʔ=Ulƈ‘ *{HA Rľ3L['· }2gDrF/}'Qoæpkc-,!5)&mԧΐmTtA)mtvƼQG[Yo8iSߤY^䎵?Q} T0au؅pXx@Džp zs}l/Q{B='QS+l6H LS\!g6{45kq!z) 08)TA*_d)'Ԣ?ei̺S}~iVt x7E/:s;ا\5t\G e,/O]uEʗJ{ڕKKwsYUt|`+ǯA I`,x}I0["W~{`Ҿ%2AG^hEz|GīmE8MZpM]oH޷dFxi' $\v#њBVkAfpr6zccÓz J*h(& ZS5f2 ]l7BS #yї7 yg㊣q ;t{\4 !S</n_:85 S~$o+͜*m״po${-aN 7ώ7듂B~"~l>Oȓ:QJ` kE,G-[ʐ,\{:T7$ERm)c f{_G$05ӽT:Tgm$߰T qG#F>Le^:r3&(,LIv!*~ şU޷ L;'jCCPr*a-r5ע)|_GIoyY:A$!"G#c&; IqV,N Cl/I˱lzt5@1rޟN{9)S&GJY8ed^nWv&sQي:< a! O寰sJ 6CGeD}yTK9/L!`Qw;VƟ|"~=hysi:||OW!d]ld{hIg  );&J7),/C7Z 1༟5Hg9)$Q\ ۋP20+L@ 3…O+E${Z>C0yJz.x{BVU]7wY@'. v L ǨGJN&Ic,>$X Z}wm 䲷`N;8?x֓Xeqay],%QusBn:E#-s17C 9Nħ9Se.-f!JDqcr[yb舢 ߽?cmNg6nVf}9 ڲ[/=C{#8yFt\fT!_znrd"[y1%_`qv߰ b/zd47+gsyOom+?؆^8\'3㛖ٍJl%ׁfR ز B>ĺ$GQA7N3!G t:':]W]k &SҨ+ >01;.kJ74g2Z:] AJV6]%K+xEO×PQ0^5&·o+`i6tAdWgɏN&qR- L)2qp*T$ry}_2w=bRѠQqEmX  IXwhf` pV-ǎ Z'JY[g1!gQإA *[n'䦱`.cP%FPjm帐.A16؋AjJqLx?k|ǤpLjeJcE$ɐ5>$ƚSYr&d hatjO.' BT0&`Fj&F:mRV{7 =|` ȕxLmz;3aRGHJI|Co#+T(66cm_T J !:ݲOfJ+w(,biH8`:W:*mӺ U+lI1Fv^ $.u,@!`꟱K͓ TU/5}XEW!nÍ/NVl;IA╪sdf3v%2H #gfa&(ֽl}[?UQIUYۋ-jQa8?S(8#wۗBz.\]JRh֮WF[eNzi,% "w#֝!` ?1iX)9KZBlA8(f~K-YG,8<⮏qeKUP}T|z%SӔJC񴽣Jlҹ9wI;}^u"Yai~cczA9gJB Yfylk`9p4\F9nc8㥦nei8p8.Ֆ>\anE_k$uXlC^ƐMNu n"&=w}z^#BEAYl8;YndZ ܂p;W(.% ,Y7XO; 0v?shm3aQI ?&Mxߠ[ɹNbˍ9[-!k 4R߱ ~$ OC0Lwnڐųhn=JXi'+Ѵ b}jq+}$gn $8TKSRfBr] (jfI@/0*['0О1wJJ&\z! j8^ EeaDܕL( Ƀ8ckKxd~ǟL7uCrٯFFn|,c؟+.ս 7x?vg=&X9kMRrog]&F2Q%nJ biDXEw!X e=Nw͑-ZVѥ.FyVX)$9&{8^#-24G/{٬8[vԲg}Jik֥€tKĕ pGu5;w77Hm/݈&xX 1AyU= <54sBNL vnUFpFQB\L|ē T'hVw~(PZh!ÝM=jOJ& {"yVcf*`jra)6W[9`=U^:F#.X$+~=̯ 'jr<W) xt ˱ci9L"*rG#uQtSFE`/[9U!˱#=dʯmFdlV$e5 WlI 3 6笣5[Cgs =b=uȁ!aG C, QezV3$f@Tɦ%Zi.*Q_T huG fڳ;bdP5h.x:Gqn{XV1J0@p5^R ƒN{䖸#kfmµ[ŇdR$1\W_(E 7: hя| "ZHTAH)8#Ǟda$rhj o1h~!=%I@&@dQMq>dOmg=Zf3f~-2%Vf8&׶ ɒ=>c%vS،7)b[I1(hOk4/)BKDI񐅸z0tB1EMŧ cbIWYnM5+&7v 9pnW8&R'!ZG"~ .㥖p@@U^!kc~!3P)E)  Lq6͞02:2:R+d%,\݅y?KUc8`8ӣo5t%ڽ?7]w *>pE \780=.3)y{cG3kg 5.e'Oдh6A^J2{x}xa!wS rZv.dN,ҀՏ:j\|,ژ|\l45Vf D#ܐGDB4v aGŀ߲…ٕn"<6G}Hgw(P9 gQ4Ġ_+Y9n ̸9|ж5B=k{7Oh?lƧ̓OJ_bs;sD%21u-̰1l+ xݕ'Tl_4J~0 vQ J(j1E QM(np^liq4HN+d ɲ')gSCj;<X#4]$PaH?dnȼ_ޯ>zv-o$tQYJ}C#uRyoJQ>fYS%[Z Sp@GWχ@ 5^,`n4pW6&0TE_Y4̎?OѿH lr%\#qZbթށ|q7ck)i8eF3V*5dвq}۾G\jC"]vDP+EfIﻻ},øD\-d`$3-Sqv5Ł Ǵ2:bUrEj JjS 'YL<MaI`-'Lp$,g{[?c`MI)Zx|@}Wk}diR0=Pl:/*"`FV2|CuS/ uNSXpQ]Syx&C;{}zwm u~#,HBtPf~ԌrSQ$wquTnh=RW/8lZ69A %9w^sҐ.UPPmD7n\(ETat5w ȕr0}׀HWtr Ŀ"ʈa h~"} r{^*p~ 0k N)-_\~8MV 9ןkC625@ )8=xpl.E~ e\=D:ɭaկ`lCs4e endstream endobj 538 0 obj <> endobj 539 0 obj [540 0 R] endobj 540 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 608 0 0 836 0 0 cm /ImagePart_2126 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 372.699 717.899 Tm 112 Tz 3 Tr /OPExtFont5 13 Tf (5.2 IS methods in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr 1 0 0 1 114.7 674.95 Tm 96 Tz (is almost certain that the model error of a nonlinear model is not IID distributed. ) Tj 1 0 0 1 114.95 651.899 Tm 99 Tz (For example in Figure 5.1, the model error between the truncated Ikeda model ) Tj 1 0 0 1 114.95 628.899 Tm 98 Tz (and Ikeda system is spatially correlated; there are regions where the model error ) Tj 1 0 0 1 114.7 606.1 Tm (is small and regions where it is not. Understanding the distribution of the model ) Tj 1 0 0 1 114.95 583.049 Tm 99 Tz (error aids in model development. If systematic model errors are identified, one ) Tj 1 0 0 1 114.7 560 Tm 98 Tz (can improve the model by correcting some of the errors. In this chapter, we are ) Tj 1 0 0 1 114.7 536.95 Tm 97 Tz (less interested in improving the model than in how to obtain states of the model ) Tj 1 0 0 1 114.5 513.899 Tm 96 Tz (for initial conditions which, for insistence, serve the purpose of forecast given the ) Tj 1 0 0 1 114.25 490.899 Tm 100 Tz (imperfect model. Therefore, we assume that the model we use to approximate ) Tj 1 0 0 1 114.25 467.85 Tm 102 Tz (the system is the best model one can achieve and the model errors have been ) Tj 1 0 0 1 114.25 444.8 Tm 97 Tz (reduced to the minimum given the available information. In the later section, we ) Tj 1 0 0 1 114.25 421.75 Tm (will discuss how the information about the model error can also help to improve ) Tj 1 0 0 1 114 398.949 Tm 99 Tz (the quality of estimates of future states. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 130.8 375.699 Tm 98 Tz (In the IPMS, to estimate the current state of the model, one need to account ) Tj 1 0 0 1 114 352.649 Tm 100 Tz (the uncertainty from both observational noise and model inadequacy. Without ) Tj 1 0 0 1 114 329.6 Tm (the observational noise, the model error can be derived from the observations ) Tj 1 0 0 1 113.75 306.799 Tm 101 Tz (directly. In the presence of observational noise, compounding of model error ) Tj 1 0 0 1 113.75 283.5 Tm 103 Tz (and observational noise prevent us identifying either of them precisely. Such ) Tj 1 0 0 1 113.5 260.25 Tm 101 Tz (unsolvable problem also causes the state estimation more or less biased in the ) Tj 1 0 0 1 113.5 236.95 Tm 95 Tz (IPMS. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 113.5 192.299 Tm 113 Tz /OPExtFont3 13 Tf (5.2.3 Pseudo-orbit ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 113 Tz 3 Tr 1 0 0 1 113.5 161.85 Tm 99 Tz /OPExtFont5 13 Tf (Since no state of the model has a trajectory consistent with an infinite sequence ) Tj 1 0 0 1 113.049 138.549 Tm 101 Tz (of observations of the system in the IPMS, any model trajectory must eventu- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 101 Tz 3 Tr 1 0 0 1 305.75 51.899 Tm 86 Tz (99 ) Tj ET EMC endstream endobj 541 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 542 0 obj <> stream 0 ,,UU.12 "jh̀+;MR@כG("ێapR/o͢Lp㘘TrqLU<6ZPp%#{3Id^SkʒXXV2nE,J .AJ)U@V϶=5%˘Sv^bf} )@ SPen ›@}JwfrM^T{1 "'ڇ '\nԣq[#>J<;/Qky*NPxC%P|? Jqc BO3 ^{uv< nvKȚ:1+3dMxz!; AyDPO8fAwxǢyr`; .jo'eV>9;<c4b05XH?T{DjesƸՑ)Ne4l$ȝz}sW֜"qo(ɇ $h3?\&S=t ۺv/~hݙR)&uj|BYgmtt %Ӽ =i םɌotestJ4{Qpj>c6z}B=y,-ϴC2 CrQ gF^.60rK$IU?r&0ez@AJdMM  ë@y@)*l5A_8wY 1<^>dR;yDSU:[+SigSOJvIM۠X6GL٧^լ]X.DN 2Jf7{+_>4a P@h/N ҆xœmQ>' + ަB XC]N)@ܓU?}xhFzs HuS[rv:^ͻD1!YH7Hݐz>~2`T+-X躽pXz/'sJ(QSwd0$tb-63_?MnZ(ReCY!^XqL&iԛGTII^{"W>kDjA1C)Whl$#:nV5z q):ެph"a+ȯ; `6-鞦v劜8k 讕N,N#^IqZ9u:r?1OYVgzz^s7a ]}^N2d1'ov Mȩϫv DzC;௃L ˖Ut}ޝ[~|Ѐ'L V9R I).ubędi[1h`6'E΃`.h(E>3ޏlˠiPv0B@"nt0\[f΅{᨝VCS4\Aޣ␧P.bǝBH31ѐ$ RߡKB77qʒ ZGv]EI)"#Xvk wj mIG `:88ׄ獖5|M7@m1:Ԣl-,薛Y7\s9ݸY1/e $(=Fhfu5xC|:P*#[Aچ uR%dW$B+gnՋi*]339: #سK['Su5B=|=B_yH)W)?Yb˫X[F"5RP3¿ 4%k2%TX&X P}GjMYreSŊD[K|t/2 E|bߪPu_R8T( d8` @er)5c@zŀ3Y+mmYO.TF',Hj; z4cƗkjsqRx"\ze乓!oo%<}lLJ5-;U:>dާUh8IxMh}ܸO:\LB`^p|W^7{}7,bƿҡ2RG-Fc5dhY+'J+D4 n|M 1y/yuN4G?~M Zo>P_B4Jxg}͎.YAdtG** &B-5݌֮(mhK`49-?/pjb6_{"b2{\RO/ _%wl>n)!)!<Cÿ\Eip+-qpnfԤ" yuP.@Q?G"t6uLзܴр}Z4 J>,fM#+[BݵP;[6$pWWhdĿQ:%1AǤ B -|DF}H";[ya v3H#U#③]WL + Mѡ;%bZ#E*1)ppOHSuA^i c Y@%Qjg̩ e@"AxFn)nhSҵq{!sJ ,zeG΁3.dD+^>qk|`N./\ߧ,7Oh(++֜"z䐐]z[G֫7 ҄ztM7l8O~n/[ ʋ)=k`GߤvOF.AI"LBR rl4[THztbGXԦEhPg1:pnD|][[ЋzrNEjZzbs8dYK #9J{fo2}@Lp:B V3{ZeM—a6ƼGo)cl暂ȱ՘GN qFJIiPK YTD+u'5/?)d/,q18  [B3gf((YGR.`HF%KĜR? nͼZ8L<`TËQw nbmPQ ZJɛF˞w[(^ɳʆ0Zy,@Yln }YllG>ɖP<:RhZ=IJb>JC۟0+q,3cיѱldpx$ /UR3Ϻ #Ari-]w"/GDn3l 76/8*~69,NNVRjeF.u&΋}Y LTo樍Flj+Q4} Im 6֎&C!p w*Yu;ij?(J7=ޏb4z]@Svd;E &SC7x.)# 9ORQȢlۨ[| ~։c}Pqf:Dp@ZcBL֗x+(LT:uD@ 5F.Vhՙ_W)X w|nb!LWwKSdɉF\AK|z5ETFj̡N ~swep7`7fGu.2Мu]9RW_ֆf9+Zw^UTrF[h0PQaE s%o^fC&kхުYL;d73@(5p1f4mVebņ&k9M "J;Xp ĸԍLUik\.ꑉb>9;"?gHjBA t ,zH*K;[ .\%I]ڥ4yoV"ɘ8Z8"׳5PC;o=:ڊ_Nzl±+ =jzau{bIcجѢԠ15;6!gIAE7o`R=?x-ņs8gغ[MpIfOA$e }EP$dષ3LNJ8Ud s T =ڻewޝFPZ<&> ,t^GW,j>_S,wMU8iٷ+ rK$Aq׿oߧc1ņu[WdZj(݂joVrڑFTmf>z 3,h#SzQh9k\FwןK`+8^mTI!,k9ȽjODU6hA% ir2z^>gn2)DXI"yhٜ\9ly U|\E$O6Pګj'Oh 4 '.7eM-/XvKXuv hb;R}A5*N3``xA+1 6D51AOoXf7RᔽVeCٜ;2h]t3Yww<@Ty+ٿ*nVRWw1GT+(kRR&^P!߀:*eŠ RTk3ʾ[Y{p2g9Z5z5?Чsi5z&Diy 'm!١ŲUf;o_gs2o(dk&lyDr#3^C1mCu3#VÜx~orH`&B "`RLJ4 K[*z6?GBeSu0DkȆ1 tmùLĞ=nvYU4F47J#Ba$H>10M2 Ǖ@ZFYGlTCLa!.l XXG9p,F҉Dy{Kr>uN6Rp;!vA@@%z\󮛩U7LQ qYhN)F&5 3 Dž>qUfJ;L(F_|LZ(;ǶayYfn\x2@W$Q:+^ -F˼zVq~M|K6)eJ^A <^ Hrʵx@殼~zRoJ.輱"HI>jtNilA-F=мyGzBw餌bgVEjrVeFA:#)bJ/+$HP=ugL^߹"D[! 9BV'nW}d\?Ck,BkM,ȎdF:}hJUoNn$lGHOlsBc[D4G͛s8u34v^XqDSl(_ҕI4D޳OڎoXp)qV*t lqS]ÿ%x^Ӯ} a7e@6!CL;&!R⚣z[S1|FB 3lN[y-;k%?ihrqGCd@@,I[7+Sz̥ԙp"TeK}B Ǔl9B%sEvdZi cr. |f 9]}LB$lj<, j#G1A)\ %7݌ rBZ_\6BtM搅2r{H-E8(#'԰D7n NSAob-āV[7{iSiac=Ns;^E ,&:&P؀?u9|0=]w|UP#_5j[I߭~vqnmeqDO7꤫*CQ<>Es4z8mLtEݱ8?Qq_)1?ΎG.vXk)K{Bi(l]y_Q}nCdێIpuÁ }'k8 ɉHN\}4F`:W rm3lVHQ> endobj 544 0 obj [545 0 R] endobj 545 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 609 0 0 837 0 0 cm /ImagePart_2127 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 372.25 718.45 Tm 107 Tz 3 Tr /OPExtFont3 11 Tf (5.2 IS methods in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 107 Tz 3 Tr 1 0 0 1 114.95 675.5 Tm 92 Tz (ally be unable to maintain consistency between the observations and the model ) Tj 1 0 0 1 114.95 652.7 Tm 93 Tz (dynamics. There are pseudo-orbits, however, that are consistent with observa-) Tj 1 0 0 1 114.7 629.899 Tm (tions and these can be used to provide better estimates of the projection of the ) Tj 1 0 0 1 114.25 606.85 Tm 90 Tz (system state. ) Tj 1 0 0 1 188.9 606.85 Tm 98 Tz /OPExtFont6 12 Tf (Pseudo-orbits ) Tj 1 0 0 1 261.6 606.85 Tm 95 Tz /OPExtFont3 11 Tf (\(50\) are sequences of states of the model x) Tj 1 0 0 1 480.5 606.85 Tm 74 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 482.899 606.85 Tm 100 Tz /OPExtFont3 11 Tf ( that ) Tj 1 0 0 1 114.7 583.799 Tm 97 Tz (at each step differ from trajectories of the model, that is, x) Tj 1 0 0 1 420.949 583.799 Tm 78 Tz /OPExtFont5 11 Tf (t+1 ) Tj 1 0 0 1 433.199 583.799 Tm 204 Tz /OPExtFont3 11 Tf ( F\(x) Tj 1 0 0 1 474 583.799 Tm 82 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 478.1 583.799 Tm 110 Tz /OPExtFont3 11 Tf (\) We ) Tj 1 0 0 1 114.5 560.75 Tm 95 Tz (define the imperfection error of the pseudo-orbit x) Tj 1 0 0 1 371.3 560.75 Tm 67 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 373.449 560.5 Tm 99 Tz /OPExtFont3 11 Tf ( to be co) Tj 1 0 0 1 420 560.299 Tm 67 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 422.149 559.799 Tm 106 Tz /OPExtFont3 11 Tf ( = xt+i F\(xt\) ) Tj 1 0 0 1 114.5 537.7 Tm 92 Tz (Note the imperfection error does not necessarily correspond to the model error, ) Tj 1 0 0 1 114.25 514.7 Tm 94 Tz (however the projection of a system trajectory ) Tj 1 0 0 1 347.5 514.7 Tm 35 Tz (1 ) Tj 1 0 0 1 349.899 514.7 Tm 94 Tz ( in the model state space forms ) Tj 1 0 0 1 114.5 491.649 Tm 95 Tz (a pseudo-orbit of the model where the imperfection error is exactly the model ) Tj 1 0 0 1 114.25 468.6 Tm 97 Tz (error in the model state space. Recall that in the PMS, there are a set of in-) Tj 1 0 0 1 114.25 445.55 Tm 92 Tz (distinguishable states Eff\(x\) of the system state ) Tj 1 0 0 1 353.3 445.55 Tm 101 Tz /OPExtFont6 12 Tf (x. ) Tj 1 0 0 1 370.1 445.55 Tm 90 Tz /OPExtFont3 11 Tf (Each indistinguishable state ) Tj 1 0 0 1 114.25 422.3 Tm 94 Tz (defines a system trajectory that consistent with both the observations and the ) Tj 1 0 0 1 113.75 399.25 Tm 93 Tz (system dynamics. In the IPMS, the system trajectories are pseudo-orbits of the ) Tj 1 0 0 1 113.75 376.449 Tm 91 Tz (model in the model space and these "true pseudo-orbits" are consistent with the ) Tj 1 0 0 1 114 353.149 Tm 93 Tz (observations and the model dynamics and most important the imperfection er-) Tj 1 0 0 1 114 330.1 Tm 91 Tz (ror reflects the model error exactly. Unfortunately, such desirable pseudo-orbits ) Tj 1 0 0 1 113.75 307.1 Tm 94 Tz (cannot be found in the Imperfect Model Scenario, because of the confounding ) Tj 1 0 0 1 113.5 283.799 Tm (between observational noise and model error. One can, however, find relevant ) Tj 1 0 0 1 114.25 260.75 Tm 92 Tz (\(useful\) pseudo-orbits of the model that are consistent with observational noise ) Tj 1 0 0 1 113.75 237.7 Tm 95 Tz (and the imperfection error of those pseudo-orbits can be treated as estimates ) Tj 1 0 0 1 113.5 214.7 Tm 94 Tz (of the model error or at least provide some information about the model error. ) Tj 1 0 0 1 113.5 191.399 Tm 92 Tz (Methods, adopted based on ISGD method, of finding relevant pseudo-orbits are ) Tj 1 0 0 1 113.299 168.1 Tm 90 Tz (introduced and discussed in the next two sections. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 129.349 149.899 Tm 95 Tz /OPExtFont3 9 Tf ('assume the system states are one-to-one identically projected onto the model space ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 9 Tf 95 Tz 3 Tr 1 0 0 1 303.6 52.45 Tm 76 Tz /OPExtFont3 11 Tf (100 ) Tj ET EMC endstream endobj 546 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 547 0 obj <> stream 0 ,,gb7 ^᰼t;c>CDIPd=YiCmn8Mgi|$ KL,e{7:I^w%@rV7r =yEd\b횷_!]]5^"b(f{vM]u{BVS7<^YoȱzϨ !H|-Stx2M`F3tFCCAM*V {l*HTΤy8N=}y+ͿMj߀6(SxR0 &#(}}B ~lA&ILlp'C0|16zNW$HNw by܎ESZ9?SB+.eMP(B>Q8ڵ灾;,*<6SY%,A뱽۵1VO}ŝ]e6_"V-pƱXD-3sdl 9migD\pHd8ω0U=Hƌ>a2uTi,RuR,z {EEn( tzsDn(ԮCWMkQD̓g#ٴ磩"n)'D! KhˢS9_Xqp*,b1܄e{\=Ox/0x8x@N#`!-h)Da&P}2w@>1~ I KXLAzKe"#O[Eu G;fZ¦r+vUrv%zO"ۍk`Vl,- ճQ*ș&z劎grJQ$Uu3\c82KI J{b`dh`{?Whoن)tCTEqTΆԂ]t8_dp w{}հ%/I |Z׌vߚjtw&Gfoسo9?3`M3WrǟI.Z1~Ɩ™u"5"rZwHgOXuP\"J#F-qJ7nbz,_}'q9R<3U*M(sW);n>.o)3iq>"k2v}Ab%ܙYDұ`MM;J ozXl?b(ȝA|)$eVkge A |`[Pb> ㇿZZ"Nwcq ".=ƪrBM&֑~RI,KG$R34:21f$=xeq(יּrv^c﫶4+$5%* tU ގ2=DfCZ}tjM0:B-*0ȀfvhZ`:qG.tp|>$-pD{]VQaS bѾFEL*c}SRi;Yc ڀ# TwTm=~*`GL^ɨ ao\{A (*ۏwxg7h*,9{jU7Q^ѮݵO1(Brf`S{ a1y#Ulɭ3X>NrbguX+>x;N'(Jt2I c uRpm zU[XO9]KQXiX?LMZ9=w["=7_vM(:=~Jb~r0h z`G' S5xշhwjɣ {t儶ÒzeCs$P'K8Ǭ| 7PW6ݖmOrp`&G9vC]t__a5zhcȷYK L:H, {}|Ͻ>??zW}Jn>&jB'\oh8ϜF?3H]U8ZڜG۞ FꡫF]iĭHGހ.*]rɕ˙BdbRnQ>gvws6P7HSIxIeWkWO;x1~w"6k CP'4#AdWˍn~ZM*a3hB2N&-Y&Af@~Vrޮ^uOf[516TP_]G&Z51Aa S`Zg)kN:Na1]I3S5lX7ʂN;MYp| Q]VTgߌ­UAOl~yN-`hM6Q(RJ$I).WQJlD;vXd@\KqUpK/ݵk.uc.4/}|qGy^a˙^ oM)eF1Bԟ@CѥX=>':cm}N>EQQXy^EU:/|ُgNU6JN-O6 4BUW65#A8oS), s(SNYQ :?p:>Q*Q:=rWiE-U)gD4qMHkW%l_G.; l'm+2/)'W ֑Eh ZC {dxufZ+_# OYh?pe|[v-2Jz%7ˁ [ p!$֍"SPdM[{ Pp}YP3 .QM.Z]Ca=&EiCLa&MhtPNq_V2=NPS> bܰWc2wހ%f8Ұ앚#` BM|0bjxq+k D&| /VO_nHhR[U ҙ'Z1~]!ߠ4{rT3_Όȩ=y6&SDLػ5EᱍG%1y$Z2t&ecWcUSNo->C5#d%\8oHluӱP Zu_eLdX2CYQ% 8=!a3[;enh.߫#n!#9r6h"^p;3[ՒEj ar]d~VV 3х!h(p^Q魑+8@b`^}+>лDUsد`_؎6E2 0 > Kp3ΤzMdBGMTgʺgǂ!{xպB}3sӏ4.XZߕcGxN w_BI} 8~Εb!M#_h?0Hh֧}``Y$sZPa4:.ҽjQ,k9M4H7oSx^\KK>*{eٜ}h'C 2׿ %I.сp3=͢rq -U0G9B+cXR/LaTE>Bɯ kvA 2Sm}Y1+[/[YMDru/+t&LbMt"hs Nn/q/;*&#<s;@'=$ Eq=%t~v@,q"=Լ"/8nXBI7; ދ/n rЙo^)5 Vo!W8ޭfMsK,l7SOq;4@CVqV)>1hZT >^B#ʴU T\nY¡s7^le}Ѻ',ZEVfwڥE|A"mg;9Oe5X# $H ,!SZñi˟6q<#É>Ӝ.XЊ Ѣ $XtLDy> i G.9?hď%picͫ4l9qad+蔃:(h| 6]Bۻd>ebDI(pd}%:|QOK-NdoI| *׽X [ 8R;_đ/f*DD&8o=Ds5>{,+l i:`fgV1r|ˮ[D ʕ,ڮGu"_]Xs"y WRky`" c/aU05Wjb%qIƀADq , U.{oOuQ1`y OցUCkz @էL.*m#v҂rVS`Cö+dP$x]۩Re]Kt8.jq7BF^Q/<ۚaq@ MZ6 |Tv>i6GA,6=1XrƐky?ᕢv NEO|/6i]2 kA25/9a7\0ե{SL-|EDPn"Pϲ m)[𒲥p3oN yLPWYLj9~h/CSy".N:6lv\;kBB|)$SYV/Q[&|jlQ^ "¹98i/Nm\,[MԎ}^FN2Nz"C=w M\m%۾\] ܉ؐ6-r"IN%MʂhEH_aSJe(SdmeLK4[D2=XۚY\=uJY< |pg8tn;̶d3_3f2ZngfX-msi 96xKFޢX›]ː&uhk*YLVZu3]1f[.NqW"t43zݘ$tkQ5Sj'j)eZ`0 ^_d*te4QB["iݜ!2EX4z;N#_ ۝4Y>sr(w$7YtT;˭`et {{L \N%&A4d!ԠB1YJf~"4-@ÛS`;<[#ؽ&}ǸP%E%pE B kJfRpQ'5eY|1q%YE  U Z|;Sڜg  _l%[P1ӁP] j/ߓE߼pǯ'F1) ujA}-Y= v#&7ǭ,+h$Ɣ";7Y=u5R6ߘT`W^s_)6|PkkB2bl ' Hܻ!`(SOSg1 *~GFQ&&S$GYoE:fn,$>cׅ'oV(Jr=.n o];ގdo`}.˾Gݱ.P]fg;)En}kmNbG˩wCYn]Kk.{ \F% "F|7kCC{^-o5e>d͋n~m*t&߈0v ?*uS(yyPcP[7m8bpa'ԦzQdÐ́f'vynwb8 `+ZeWMiLJbQ}c0Kqy O%_Is-=.>AƗ8YL Iw9|3qޣsrC=<6x]XA{7F^>zs&\5-5`8}VI`]uer> ش'|uRRNMC/e]xRÃp3'AV󖣹L}JuSi@'V#KG3εìdG 4v5d!HC/|m*I]j(6 dLV,ÌK82gŀvl? 3!3(Hу|;ٴ 5퓯9If~$ְICjM'g¨&?`';æ>_ֱ5^|U *se'Y{A"'{"uE}3gdqc@b&V셋U6ڝ(@ mdwAapu~Big"{kr&nm#?}m 5WC|Zfp#(g42)ژ: v ˌEEkUd;tj[ j7J\e%~>KK^&ܡrLׄ6L SyߣgA_hLʑz@]-JۺRBaq-Շ/50.O5פ6`IOi+yM,3sݡBD'qai#BzdXʊJv%ap)pr6ɶ5@beR])bF `L94^߄')9Sc:I#kG+x߿q5oálM9"M~΄!-b6ύ (qpS\2bzZࡳ8'p7  S?d$79!Q(v& c 4μ?YVG#r:c1@ +v(U\X0`W^pKP|nfZUP. ҭt F0VM]v nkCmE6mDBbj.Wi%{ۡU+T/: K~0I d NȜB #0eQNt3El3xIЏWlK~}U<?,ڔAЯ$|#TAg;]OdZ8 TM$Q/`RQjEiy);Ư%ի+~[[M9/=tKDHG:&z1t"eT>#^Ize|z$;]p[:|W=7A;oO|/R rVgZ[8> endobj 549 0 obj [550 0 R] endobj 550 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 611 0 0 839 0 0 cm /ImagePart_2128 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 373.199 720.7 Tm 112 Tz 3 Tr /OPExtFont5 13 Tf (5.2 IS methods in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr 1 0 0 1 115.45 677.5 Tm 126 Tz /OPExtFont2 14 Tf (5.2.4 Adjusted ISGD method in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 14 Tf 126 Tz 3 Tr 1 0 0 1 115.2 647.25 Tm 97 Tz /OPExtFont5 13 Tf (Judd and Smith \(2004\) introduced a method of finding relevant pseudo-orbits by ) Tj 1 0 0 1 114.95 624.2 Tm 95 Tz (adjusting the ISGD method to include the model imperfection. A brief description ) Tj 1 0 0 1 114.95 601.149 Tm 94 Tz (of this method is given here in order to introduce and compare with a new method ) Tj 1 0 0 1 114.7 578.1 Tm 98 Tz (introduced in the next section. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 131.5 554.85 Tm 100 Tz (Similar to the ISGD method introduced in section 3.3, Following Judd and ) Tj 1 0 0 1 114.95 532.049 Tm 101 Tz (Smith \(2004\), Gradient Descent algorithm is applied to minimise the adjusted ) Tj 1 0 0 1 114.5 509.25 Tm 104 Tz (mismatch error by including the model imperfection error term. For a finite ) Tj 1 0 0 1 114.5 485.949 Tm 94 Tz (sequence of observations, ) Tj 1 0 0 1 240.949 485.949 Tm 68 Tz /OPExtFont3 10 Tf (S) Tj 1 0 0 1 246 485.949 Tm 91 Tz /OPExtFont5 10 Tf (t) Tj 1 0 0 1 250.3 485.949 Tm 53 Tz /OPExtFont3 10 Tf (, ) Tj 1 0 0 1 254.65 485.949 Tm 99 Tz /OPExtFont6 12 Tf (t = N +1, ..., 1, ) Tj 1 0 0 1 350.899 485.949 Tm 95 Tz /OPExtFont5 13 Tf (0, we define the ) Tj 1 0 0 1 431.75 486.199 Tm 101 Tz /OPExtFont6 12 Tf (adjust mismatch ) Tj 1 0 0 1 115.45 462.699 Tm 96 Tz (error ) Tj 1 0 0 1 144.25 462.699 Tm /OPExtFont5 13 Tf (for a sequence of pseudo-orbit z) Tj 1 0 0 1 301.699 462.699 Tm 68 Tz (t ) Tj 1 0 0 1 304.3 462.699 Tm 102 Tz ( to be ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 102 Tz 3 Tr 1 0 0 1 235.9 397.649 Tm 106 Tz /OPExtFont3 10 Tf (e) Tj 1 0 0 1 241.9 397.649 Tm 91 Tz /OPExtFont5 10 Tf (t ) Tj 1 0 0 1 244.55 396.699 Tm 125 Tz /OPExtFont3 10 Tf ( = zt+i ) Tj 1 0 0 1 286.8 396.199 Tm 239 Tz /OPExtFont3 3 Tf ( ) Tj 1 0 0 1 297.6 397.399 Tm 97 Tz /OPExtFont3 10 Tf (wt) Tj 1 0 0 1 309.35 395.949 Tm 200 Tz /OPExtFont3 3 Tf (-) Tj 1 0 0 1 311.75 395.949 Tm 55 Tz /OPExtFont3 10 Tf (F1 ) Tj 1 0 0 1 323.3 395.949 Tm 246 Tz /OPExtFont3 3 Tf ( ) Tj 1 0 0 1 334.3 395.699 Tm 126 Tz /OPExtFont3 10 Tf (F\(zi\) ) Tj 1 0 0 1 359.75 395.5 Tm 2000 Tz (\t) Tj 1 0 0 1 488.649 397.399 Tm 92 Tz /OPExtFont5 13 Tf (\(5.8\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 114.25 355.149 Tm 88 Tz (where co) Tj 1 0 0 1 155.3 355.149 Tm 75 Tz (t ) Tj 1 0 0 1 158.15 355.149 Tm 98 Tz ( is the imperfection error. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 131.3 331.899 Tm 96 Tz (Define the implied noise, 5 to be the difference between the pseudo-orbit and ) Tj 1 0 0 1 114 308.6 Tm 97 Tz (the observations, i.e. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 273.35 243.299 Tm 98 Tz (St = s) Tj 1 0 0 1 303.35 243.299 Tm 70 Tz (t ) Tj 1 0 0 1 306 243.299 Tm 79 Tz ( z) Tj 1 0 0 1 326.649 243.299 Tm 70 Tz (t) Tj 1 0 0 1 330.699 243.299 Tm 51 Tz (. ) Tj 1 0 0 1 332.149 248.35 Tm 2000 Tz (\t) Tj 1 0 0 1 489.1 243.1 Tm 93 Tz (\(5.9\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 114 200.6 Tm 98 Tz (Hence fore, the mismatch equation 5.8 can be written as ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 205.449 133.399 Tm 106 Tz /OPExtFont5 11 Tf (et =1 st+i ) Tj 1 0 0 1 251.05 133.399 Tm 1805 Tz (\t) Tj 1 0 0 1 300.699 133.649 Tm 82 Tz (cot+1 ) Tj 1 0 0 1 322.1 133.649 Tm 574 Tz (\t) Tj 1 0 0 1 337.899 133.649 Tm 125 Tz /OPExtFont8 12 Tf (F\(st ) Tj 1 0 0 1 362.899 133.649 Tm 109 Tz /OPExtFont5 11 Tf ( \(50 ) Tj 1 0 0 1 399.85 133.649 Tm 2000 Tz (\t) Tj 1 0 0 1 483.6 135.1 Tm 109 Tz (\(5.10\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 11 Tf 109 Tz 3 Tr 1 0 0 1 305.05 52.5 Tm 82 Tz /OPExtFont5 13 Tf (101 ) Tj ET EMC endstream endobj 551 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 552 0 obj <> stream 0 ,,||b6`P?# _$7Fߔ_kIlXғ\R,x&KRA>c@w0s2Bj0#WL Wv5YሁBt'4{6@8Pj_EwMT Mlb_BuamyDRG;ҭT& 'bm}LiB`Qy ȫD##OUVDy4t-m'ߞxƹ4 BG;ɯ@gH4;*{}qZ\/R(pT*=Y>Gl?T5LLQStl)ҖsUɃ]` %nvӼtÎn]b/tNY$*Lڌ.!tge>4J͚OMyMjփ32A>/A+yG};D"QM roOnDti5C)&h!*ӌQx'69 ˡ:f C¸*SfS!6x9PF!~ ^ a`{_+xCهn3Qj[ Ѡ*|K8Z&#ut {WRˌWr5Q^e2iD#^U@TEzi-@ydX`&|\FGG~4Jw\:3hv8T%7=`,(gb=' ſ5yQ; bbYU.e/B_ >צ#2)鳧٣ͼӲձceʌe :>pل%?'{̆c/x5|$96xxJj1&)$!=Б|>;;ǫ/>Ȱ,٫z_v)R7z=iu9}W 꼭/ vAo0@Omggmٗ:=(<0v<+vt ߤz,a®S<򥙀bn2kk0 nqA&0 ȨTaf/OhR.^$~&Eb6Ǎ_9[dskrG$TV&~̫TAFJ;7bYSz9Rƈ ,"yM?DJ`}r(>81A[pUJ5grkY!wTfzLco ]&8"(L I]WG?x3i5ZNxa1qJX17j!0\P5\cLC hWUPp7@S sLe^glTTц.7>& AVnP!},kd*qI$Xg/x|}ͽwK ? ][EC]Kab8+Y?6˼kCpGx`įer&qXMlڎAp8cʬ 0V]1".X<,M3Ͱ^NAfi4G }2xGƖ$poTX>HPͧU!@~sħQa)ݗSCJxb'skzecϪgLK|BO`!FAy|vٳ-7s8s}|޶f"ϣ `2SVhAOb,W.GDG/ + ~Uc(;(^}NuJeKPX AA1Tm45Ӄ%g8jʊL߮旋HƓ v֏&}sY$SEh?:_NzB"q9 x(nccuE܍f]ėź`+B;\aMz /+>+m]',!FYWBuvJ qh<;ȃTںjח~*N}[t7܌Ҿw^#Mj1NY]CO3~}T4ԊDAtb%߉.zbSwbHT>\ִّj5̀ʨ*$Ɇ/X[rٚOlh>w<'&J+D4<J$vdu 3_䚶ym "a1}[3N}D)H02uTnW-XquJ h6)pss`"X~8֤ $a&`-I60ߏOYn]P)D[LwX6dC=e+v ݷ? 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Fp2gvDLGP2_}*gibCeP^N a{@-heޔnz D:7] ٿ1nDMߌS9K% :s-'تnx1=X'W*bStv؁? 6_ySA^s! >o)rnr3EtAi6[jgLwp_PWil~C NUW[@lfTעa' rдN*} Ֆ|iq|jЊS]GXi E@!A 9 =M)m9ʑgC^g-@#(6SےӗaRiWǡR"/,%[j'KFѻ*vnbAS?wX pWr4Cƽ,kjtM~{Bq#:ȶ1"ui_!3<`6y쑿[5ڛmjeX;̗3Q/<©gsVɆ E1wI ȁH.|/)%h.NecN2iYL"+ܟ3浛p[:ܗ4tJ؝]O\gfm?oC` o8bBTl|S}ZIpn0&ODv|*Y >k*o$F hpW.٧QrDgb#2:<"HdaIƝ5SMӋ;DΘrzFL y~\iN#^P\HqYd" ߩcGYkp|0DApobHB7:g*<2PaO2ޯQ<x>Ĵj3tϣٷdޤuo-C`zɲsL[]{移v;wY?5WJ^@T aVgR|tG^_$B'\m]ZɄΆ`0 : %On|p5ޭ:qڠL; ]gdimf1aX z[Qn<)$ fYr U'|m÷.ţ%}I84n,`]Gyƶr5u׃KĔ#458iRƶ:/a\WX^1!,Lˮr~v. ylL_v{SlU`2-!H݇r1b)F]@-MZѷhEO]\s!XBhvF؃Pqv\@Bqn)|}UU;.Y^SJkP䅎yHB 8#uRF{bro`2Z6PBBd%VF\ttA31u}5LYA^=t54e.}pٟ<&b&R4[4b\s- 4_qX-/lxfSX oE|;)•'!g9D cBsYZ 8/2| _3t D Yn)(S+Պ $G0VzhUƊ5VG`U7RժZ:)8&Il Am\,j>CJ9g;x ?sgESWqRuVT"f rW\\,-mJTb<\?L|^6n7DHB:ȕg&V~ uueI-@ent{tlJ1N,!tR3B{>V圉$TVl[t xn뒦!Ehf,St)}'_цV1q(ю |_PT'g;f ,7s2毲$B[Mҁ$ϔˡ|޻ڱ66α¼!6IUoߎսDbI]߭R pQlkQ^~ -bP`D 'txs+sq\n2Iz3 þ|Ǻ}v:[@WwrSo-'9"Ig$I>i'ʌ;<"\xsB0/[`'ޘ 7h-7)Oh`'Zk{CKk8ʖ439f6ޓ0 0 Ss+kvU87]6O;2īVX J~VM3-WolpM`LMt4jRCA[tXjl, J(ΰ i4j(zN9;(T5C&]+&k`*=\әu#4\I)kwYB&|&5U 3_ »{E_VԣCp,{! ̓XxPP\)Sʷ5G/df^pM?:!a@0l5ڒ3˓L˖f [}Ka'3a< U,OJ^@yk.4k uЊl^i endstream endobj 553 0 obj <> endobj 554 0 obj [555 0 R] endobj 555 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 604 0 0 839 0 0 cm /ImagePart_2129 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 370.3 719.7 Tm 117 Tz 3 Tr /OPExtFont5 12.5 Tf (5.2 IS methods in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 117 Tz 3 Tr 1 0 0 1 112.549 676.75 Tm 92 Tz /OPExtFont3 11 Tf (Consider cost function CM\(\(5, w\), where ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 317.5 626.1 Tm 87 Tz /OPExtFont11 6.5 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6.5 Tf 87 Tz 3 Tr 1 0 0 1 243.349 611.7 Tm 104 Tz /OPExtFont4 12 Tf (CM\(6,w\) = ) Tj 1 0 0 1 313.199 609.299 Tm 320 Tz (\t) Tj 1 0 0 1 324.699 609.299 Tm 234 Tz ( ) Tj 1 0 0 1 333.1 609.299 Tm 109 Tz /OPExtFont4 8.5 Tf (et ) Tj 1 0 0 1 345.35 609.549 Tm 86 Tz /OPExtFont3 11 Tf (et, ) Tj 1 0 0 1 356.899 616.899 Tm 2000 Tz (\t) Tj 1 0 0 1 479.5 611.7 Tm 97 Tz /OPExtFont5 12.5 Tf (\(5.11\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 97 Tz 3 Tr 1 0 0 1 307.899 598.299 Tm 135 Tz /OPExtFont4 8.5 Tf (t=-N ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 8.5 Tf 135 Tz 3 Tr 1 0 0 1 111.849 568.299 Tm 92 Tz /OPExtFont3 11 Tf (is defined in order to find relevant pseudo-orbit by GD algorithm. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 128.65 545.5 Tm 93 Tz (Following Judd and Smith \(2004\), one can find a pseudo-orbit from the se-) Tj 1 0 0 1 111.849 522.45 Tm 91 Tz (quence of observations by applying Gradient Descent to minimise the cost func-) Tj 1 0 0 1 111.599 498.899 Tm 89 Tz (tion ) Tj 1 0 0 1 135.349 498.899 Tm 117 Tz /OPExtFont8 12.5 Tf (CM\(o, ) Tj 1 0 0 1 171.849 498.899 Tm 102 Tz /OPExtFont5 12.5 Tf (w\). ) Tj 1 0 0 1 194.65 498.899 Tm 92 Tz /OPExtFont3 11 Tf (It is necessarily that ) Tj 1 0 0 1 300.699 498.899 Tm 111 Tz /OPExtFont8 12.5 Tf (CM\(6, ) Tj 1 0 0 1 337.199 498.899 Tm 94 Tz /OPExtFont5 12.5 Tf (w\) ) Tj 1 0 0 1 354.5 498.899 Tm /OPExtFont3 11 Tf (attains a minimum of zero. To ) Tj 1 0 0 1 111.599 475.649 Tm 96 Tz (solve the minimisation by gradient descent, one need to solve the differential ) Tj 1 0 0 1 111.599 452.85 Tm 87 Tz (equations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 334.55 395.25 Tm 78 Tz /OPExtFont4 14.5 Tf (aL ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 14.5 Tf 78 Tz 3 Tr 1 0 0 1 271.699 387.55 Tm 239 Tz /OPExtFont7 3 Tf ( ) Tj 1 0 0 1 278.899 387.55 Tm 2000 Tz (\t) Tj 1 0 0 1 324.949 387.55 Tm 256 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont7 3 Tf 256 Tz 3 Tr 1 0 0 1 282.5 379.399 Tm 62 Tz /OPExtFont4 14.5 Tf (ab) Tj 1 0 0 1 293.05 379.399 Tm 25 Tz (- ) Tj 1 0 0 1 294.25 379.399 Tm 935 Tz (\t) Tj 1 0 0 1 334.8 379.399 Tm 59 Tz (aw ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 14.5 Tf 59 Tz 3 Tr 1 0 0 1 480 387.8 Tm 97 Tz /OPExtFont5 12.5 Tf (\(5.12\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 97 Tz 3 Tr 1 0 0 1 111.099 345.1 Tm 91 Tz /OPExtFont3 11 Tf (to compute the asymptotic values of \(\(5, ) Tj 1 0 0 1 311.05 345.3 Tm 98 Tz /OPExtFont5 12.5 Tf (w\) ) Tj 1 0 0 1 328.3 345.3 Tm 92 Tz /OPExtFont3 11 Tf (by initialising the cost function with ) Tj 1 0 0 1 111.349 322.299 Tm 100 Tz (both 8 and w equal to 0. The resulting values of S and ) Tj 1 0 0 1 411.1 322.299 Tm 95 Tz /OPExtFont5 12.5 Tf (w ) Tj 1 0 0 1 424.8 322.299 Tm 93 Tz /OPExtFont3 11 Tf (defines a certain ) Tj 1 0 0 1 111.099 298.75 Tm 88 Tz (pseudo-orbit. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 127.9 275.7 Tm 89 Tz (Ignoring the model inadequacy, one may attempt to minimise ) Tj 1 0 0 1 427.449 275.7 Tm 117 Tz /OPExtFont8 12.5 Tf (CM\(o, ) Tj 1 0 0 1 464.149 275.7 Tm 88 Tz /OPExtFont3 11 Tf (0\), which ) Tj 1 0 0 1 111.349 252.7 Tm 92 Tz (equals to applying the ISGD method \(see Section 3.3.2\) to look for model trajec-) Tj 1 0 0 1 110.9 229.149 Tm 95 Tz (tory assuming that the model is perfect. Although the cost function ) Tj 1 0 0 1 461.5 229.149 Tm 112 Tz /OPExtFont8 12.5 Tf (CM\(5, ) Tj 1 0 0 1 498.699 229.149 Tm 83 Tz /OPExtFont3 11 Tf (0\) ) Tj 1 0 0 1 111.349 205.899 Tm 90 Tz (always has the minimum of zero regardless the model is perfect or not, the model ) Tj 1 0 0 1 110.9 182.85 Tm 94 Tz (trajectory obtained when ) Tj 1 0 0 1 242.65 182.85 Tm 111 Tz /OPExtFont8 12.5 Tf (CM\(6, ) Tj 1 0 0 1 279.6 182.85 Tm 95 Tz /OPExtFont3 11 Tf (0\) reaches 0, is expected to be far away from ) Tj 1 0 0 1 110.9 159.799 Tm 93 Tz (the true states and be inconsistent with the observations as long as ) Tj 1 0 0 1 457.899 159.549 Tm 111 Tz /OPExtFont4 12 Tf (N ) Tj 1 0 0 1 472.3 159.549 Tm 92 Tz /OPExtFont3 11 Tf (is large ) Tj 1 0 0 1 110.9 136.5 Tm 85 Tz (enough. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 85 Tz 3 Tr 1 0 0 1 302.149 52.049 Tm 87 Tz /OPExtFont5 12.5 Tf (102 ) Tj ET EMC endstream endobj 556 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 557 0 obj <> stream 0 ,,--eبbo+GhsO[]loɵuF}n4By[:qV&φ:*hd&_;}Ů 7Ł(11D5ɔo{6cTJPﻰ7bnj|$1S0"H5dǫ,=ŹwR*zʌw2xRžaL?$adk29G z긖&(B.7#cm e+ce}Q TU{6& w~8 8e2o~蠲X ;ԠU@2=TփF%\E_~֮کVnqỉX¡CۿqB:q {2nK MP> Ջ2|ThDQ1CxuP #b$0J[TZRIMkG\9XJ[~\Q+pBg8ReVed:ZK1E&F~n qeAmܜIy8-إkO#dcr@6UwLN se,Yy\4!WL*7dIU&QeM(^j!oJAEa|WBa[l(K?'M,Kx}wv60v])dJOM.QrwwutE3@' *wA,,P) &su,4U+-q( >! 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HuK¶ ĦFt; +?"8,7ݧosL y֓SNEX/8SxGdO/ [")Z+@{0iR^&u&6 cBr}3Qna::'idYMs*q*rP/Yxs& Xij/e(z~b t WMnȝ?d.K%ƒ}ٟ\]~ Τ$^!  F A4n*4^&)b rB R%N 8Dv4oz)z@pA| ϊHO{Ѐ[!^Y=EA4ֺR)*'D@}zx(2_xTX>ճ+#DžMݵ4j<6¡euۣQUGFCM YHL)"xt,6ngB,z0 >W/+=yxj s1{3uM/Qix'dȵk}èMW+v^LOxY-3I1Ҩk xc"*Ζ~bsxɉ> $7۱AＹ$Oϰ[:?%ͷX;?c"[2,5ѕQ%ný0!#'i=9α$^3ҳ([QT!(U(n9LUr;ٽ?zAvVPNƪ>ܵX?0>0(;}(/Ӯ,Ķ(9&εطʶyb.<D7ۆӈ)$Д5 z~HBB8dg 꿃D #T*R0L/6۬>C}.ܚ&w͋O8ə)=SldcB֮J4_Fj%t b P/gm5JZ} I\Tq@$!dT͹ Tgfk~MhNOͧ\7 K-gv G?fcPNwK>:a qcQ~e5W|MȒ|W;kBM伔ϲ;>?W9yDF{k\r&KQ +<'yO]5uN"< 4iJ,Ng*vEә%RMG.+.' ٹ7{'>!$sqQCٶ_9ZN%87"J%䂛gK# ՜ěˬ\ك*w2$]3 hd:`Fyt.y1*Ff--qKXl&|؝ivne0ѵo%R巶ZEAH}-ky@d^ա|/Q&#YŐ6(Pay'NA3԰VC A䅾)JIa͇5@/Ҍ.y \~:MtHnho*Ey`@b*("OqMҘ- $dŦ =Jb URPǤ_CYqn@ :>\lBg23,MyNg0h!mz^T5"qxY̹/Da_Kfl#렴wKj]Z@& !w,0ۖ)cAy =_wB *cso? r(ՕG_ &OYhn;JN=BĴ=H_wu@aQ,+P0(>%fzA])ۢ 0wBt#d(/oM2b8z'D"^jauU!,t!~ }>fHF>!CKV aCduUD3e4Lһ\*0nLn0V;wPӛFbU=~OuQeϛa@@yca?Vh2Z Mw> endobj 559 0 obj [560 0 R] endobj 560 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 604 0 0 838 0 0 cm /ImagePart_2130 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 371.5 720.149 Tm 112 Tz 3 Tr /OPExtFont5 13 Tf (5.2 IS methods in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr 1 0 0 1 130.3 676.95 Tm 102 Tz (Recall that the relevant pseudo-orbits we are looking for are those consis-) Tj 1 0 0 1 113.299 653.899 Tm 98 Tz (tent with the observations and their corresponding imperfection errors contains ) Tj 1 0 0 1 113.5 630.899 Tm 101 Tz (information about the model error or somehow reflects the model error. The ) Tj 1 0 0 1 113.5 607.85 Tm 96 Tz (implied noise ) Tj 1 0 0 1 183.599 607.85 Tm 88 Tz /OPExtFont6 12 Tf (6 ) Tj 1 0 0 1 194.15 608.1 Tm 93 Tz /OPExtFont3 11 Tf (provides an estimate of the observational noise and the imper-) Tj 1 0 0 1 113.299 585.049 Tm 92 Tz (fection error w provides an estimate of the model ) Tj 1 0 0 1 361.699 585.049 Tm 97 Tz /OPExtFont5 13 Tf (error. In order to improve the ) Tj 1 0 0 1 113.049 561.75 Tm 95 Tz (method and find ) Tj 1 0 0 1 198.25 561.75 Tm 93 Tz /OPExtFont6 12.5 Tf (better ) Tj 1 0 0 1 228 562 Tm 96 Tz /OPExtFont5 13 Tf (pseudo-orbits, we measure the quality of the pseudo-orbit ) Tj 1 0 0 1 113.049 538.7 Tm 99 Tz (by looking at the RMS distance between pseudo-orbit and the projection of the ) Tj 1 0 0 1 113.049 515.7 Tm 98 Tz (true trajectory of the system and testing the statistical consistency both between ) Tj 1 0 0 1 113.049 492.649 Tm (the implied noise and the observation noise and between the imperfection error ) Tj 1 0 0 1 113.5 469.6 Tm 104 Tz (and the model error. We are aware that when the ) Tj 1 0 0 1 377.05 469.6 Tm 92 Tz /OPExtFont3 11 Tf (system is nonlinear, linear ) Tj 1 0 0 1 112.799 446.3 Tm 87 Tz (measurement ) Tj 1 0 0 1 182.4 446.55 Tm 103 Tz /OPExtFont5 13 Tf (like RMS has systematic bias \(64\) \(see Chapter 4\). The distance ) Tj 1 0 0 1 112.799 423.3 Tm 98 Tz (between the pseudo-orbit and the true states may not reflect forecast skill in the ) Tj 1 0 0 1 113.049 400.25 Tm (Imperfect Model Scenario. We only use this measurement as a diagnostic tool to ) Tj 1 0 0 1 112.799 376.949 Tm (help explain how to construct a better method to locate relevant pseudo-orbit. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 129.599 353.699 Tm 100 Tz (We investigate the quality of the pseudo-orbits generated by minimising the ) Tj 1 0 0 1 112.799 330.649 Tm 96 Tz (cost function ) Tj 1 0 0 1 181.9 330.649 Tm 108 Tz /OPExtFont6 12 Tf (CM\(5, ) Tj 1 0 0 1 218.9 330.899 Tm 101 Tz /OPExtFont5 13 Tf (0\) and CM\(S, w\) in both Ikeda and Lorenz96 system and ) Tj 1 0 0 1 112.549 307.6 Tm 97 Tz (model pairs experiments. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 119.049 281.45 Tm 91 Tz /OPExtFont3 11 Tf (Cost function ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 196.8 281.899 Tm 87 Tz /OPExtFont5 13 Tf (No. of GD runs ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 87 Tz 3 Tr 1 0 0 1 281.3 280.95 Tm 94 Tz /OPExtFont6 12 Tf (CM\(O, ) Tj 1 0 0 1 316.3 280.95 Tm 60 Tz /OPExtFont5 13 Tf (\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 60 Tz 3 Tr 1 0 0 1 334.1 281.45 Tm 97 Tz (std of ) Tj 1 0 0 1 366.25 281.45 Tm 45 Tz /OPExtFont3 11 Tf (\(5) Tj 1 0 0 1 370.8 281.45 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 373.449 281.45 Tm 28 Tz /OPExtFont3 11 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 28 Tz 3 Tr 1 0 0 1 385.899 281.45 Tm 85 Tz (std of co) Tj 1 0 0 1 424.8 281.45 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 427.449 281.45 Tm 28 Tz /OPExtFont3 11 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 28 Tz 3 Tr 1 0 0 1 439.899 281.45 Tm 89 Tz (RMS ) Tj 1 0 0 1 466.55 281.7 Tm 90 Tz /OPExtFont5 13 Tf (distance ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 128.65 267.049 Tm 64 Tz (C./14) Tj 1 0 0 1 147.849 267.049 Tm 23 Tz /OPExtFont3 13 Tf (-) Tj 1 0 0 1 150.5 266.799 Tm 91 Tz /OPExtFont5 13 Tf (\(6, 0\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 222.25 267.049 Tm 88 Tz (4096 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 289.199 267.049 Tm 90 Tz (0.025 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 341.75 267.049 Tm (0.051 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 404.649 267.049 Tm 74 Tz (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 74 Tz 3 Tr 1 0 0 1 456.5 267.049 Tm 92 Tz (0.0154 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 127.2 252.649 Tm 95 Tz (CM\(S, co\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 225.849 252.649 Tm 84 Tz (128 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 84 Tz 3 Tr 1 0 0 1 286.3 252.649 Tm 91 Tz (0.0002 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 341.5 252.649 Tm 93 Tz (0.037 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 397.449 252.649 Tm 91 Tz (0.02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 459.35 252.649 Tm (0.012 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 112.549 229.35 Tm 100 Tz (Table 5.1: Statistics of the pseudo-orbits obtained by minimising the ) Tj 1 0 0 1 461.3 229.35 Tm 87 Tz /OPExtFont3 11 Tf (cost ) Tj 1 0 0 1 485.05 229.35 Tm 90 Tz /OPExtFont5 13 Tf (func-) Tj 1 0 0 1 112.299 215.2 Tm 105 Tz (tion CM\(6, 0\) and ) Tj 1 0 0 1 213.599 215.2 Tm 72 Tz /OPExtFont6 12 Tf (C114) Tj 1 0 0 1 232.3 215.2 Tm 37 Tz /OPExtFont4 12 Tf (-) Tj 1 0 0 1 235.199 215.2 Tm 92 Tz /OPExtFont6 12 Tf (\(6, ) Tj 1 0 0 1 250.3 215.2 Tm 102 Tz /OPExtFont5 13 Tf (w\) for the experiment of Ikeda system-model pair. ) Tj 1 0 0 1 112.549 201.5 Tm 97 Tz (Minimisations are applied upon 4096 observations, the noise level is 0.05 and the ) Tj 1 0 0 1 112.299 187.35 Tm 98 Tz (sample standard deviation of the model error is 0.018. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 129.349 150.649 Tm 101 Tz (From Table 5.1 and ) Tj 1 0 0 1 236.15 150.649 Tm 97 Tz /OPExtFont3 11 Tf (5.2 one can see that firstly in both experiments, it is ) Tj 1 0 0 1 112.099 127.35 Tm 90 Tz (much more difficult ) Tj 1 0 0 1 212.15 127.35 Tm 96 Tz /OPExtFont5 13 Tf (to minimise ) Tj 1 0 0 1 274.3 127.35 Tm 109 Tz /OPExtFont6 12 Tf (CM\(S, ) Tj 1 0 0 1 311.3 127.35 Tm 100 Tz /OPExtFont5 13 Tf (0\) than ) Tj 1 0 0 1 351.6 127.35 Tm 108 Tz /OPExtFont6 12 Tf (CM\(S, ) Tj 1 0 0 1 388.1 127.35 Tm 99 Tz /OPExtFont5 13 Tf (w\) and the cost function ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 303.6 52 Tm 84 Tz (103 ) Tj ET EMC endstream endobj 561 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 562 0 obj <> stream 0 ,,b7!͚eO#}6yesIQ~;^@{z$ eRرoG{} *:t49Ksjj0O# ׍\kdRPF#΍ܒvm^umESȝ<իqP'sp'I: ,-@F1Q3y 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Kk u   7u( T# +/6 ,DJ[ԇV5ki@p"ˁSHZgn WJ+OFX8lur=@L",H"CWG͜qfsX?ɭt[Us`JNً3y٨9  6 TYSNVzJ[[ے3ʘUTg/o ;٤BdKRK Զz@P|}"Dflօ1M+ӛ\#ttc@_ܫ`|bvW56A|$F&GKar#I=L盼Xo y-b4a4u)h%CG` ^Bi-  7z} bGsÐSlvzn}jkqCG}[@Z%3CcsX9VO|ЕSPُzNOl\ۨkRfakgDԭ IZj~K035 @8E&эP>3屴՗$|f,*j5ttK7W;e |]1/P Qw; h.BJ=Z.|W3p1;((iц t~I!WHbZc;X#rid svA9 NX]8Yik9\wL =]MO@+o*uvب ` CXRҽqX$Çկ}B^&xzbG_,2Cu(lAdN]7䴄G1 ̓Ӻ^=3r\j)[Π=WSz z;\>SM"+IWs@EQ\~Q7x;ܞ>fpԺ;)> ˷8~}.};Gn'駍d{&Lw tnw7؛Eu}$ֆ չkCX3c_m`;=r1YC+^hDr !s$l#۶ <-bh|uM:ӥl A.)/ Hy,.zWfbۍrhLFl^`9ML o†* jG%6M3Um({qjKC5N3'PJ+ GX'8@5x!= '_.rBF:ʓ`0cJ61ʠ Plarί`|b{CuS-%@d ʹ%X,h@CŁ [gzwj Sۂz(m,]v3p$K?XYHb27.cۢ=Ih#ť?U( $1Ed'?f? #:ٿO Kzebx/""#f9'9 ӼqSa;PPH+M$\ 0ubp>6|\j>SƂ+ r*qwN^M~`;*Xpt;B?/7Ci4n*tAZwi:]XލgC~2Ձf}1(@` Pr, m"-Hx3eY|jniDΛ=v Y)D\c. і>> endobj 564 0 obj [565 0 R] endobj 565 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 605 0 0 839 0 0 cm /ImagePart_2131 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 371.75 720.45 Tm 117 Tz 3 Tr /OPExtFont5 12.5 Tf (5.2 IS methods in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 117 Tz 3 Tr 1 0 0 1 119.5 678.7 Tm 95 Tz /OPExtFont5 13 Tf (Cost function ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 197.3 678.7 Tm 87 Tz (No. of GD runs ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 87 Tz 3 Tr 1 0 0 1 281.75 678.7 Tm 113 Tz /OPExtFont8 12 Tf (CM\(5,-\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont8 12 Tf 113 Tz 3 Tr 1 0 0 1 334.1 678.7 Tm 96 Tz /OPExtFont5 13 Tf (std of 5) Tj 1 0 0 1 370.8 678.7 Tm 48 Tz /OPExtFont3 13 Tf (t ) Tj 1 0 0 1 373.199 678.7 Tm 30 Tz /OPExtFont5 13 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 30 Tz 3 Tr 1 0 0 1 385.699 678.7 Tm 94 Tz (std of w) Tj 1 0 0 1 424.55 678.7 Tm 48 Tz /OPExtFont3 13 Tf (t ) Tj 1 0 0 1 426.949 678.7 Tm 30 Tz /OPExtFont5 13 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 30 Tz 3 Tr 1 0 0 1 439.449 678.7 Tm 91 Tz (RMS distance ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 129.349 664.5 Tm 114 Tz /OPExtFont8 12 Tf (CM\(S, ) Tj 1 0 0 1 166.099 664.5 Tm 85 Tz /OPExtFont5 13 Tf (0\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 85 Tz 3 Tr 1 0 0 1 222 664.299 Tm 90 Tz (4096 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 292.1 664.299 Tm 89 Tz (0.11 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 344.899 664.299 Tm 88 Tz (1.69 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 404.399 664.299 Tm 74 Tz (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 74 Tz 3 Tr 1 0 0 1 462.25 664.299 Tm 88 Tz (1.38 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 127.7 650.1 Tm 102 Tz (CM\(6, w\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 102 Tz 3 Tr 1 0 0 1 225.849 650.1 Tm 85 Tz (128 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 85 Tz 3 Tr 1 0 0 1 286.55 649.899 Tm 90 Tz (0.0006 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 344.649 649.899 Tm 89 Tz (0.63 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 397.199 650.1 Tm (0.46 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 461.75 650.1 Tm (0.52 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 113.299 627.1 Tm 95 Tz (Table 5.2: Statistics of the pseudo-orbits obtained by minimising the cost function ) Tj 1 0 0 1 113.5 613.149 Tm 104 Tz /OPExtFont8 12 Tf (CM\(c5, ) Tj 1 0 0 1 150.25 613.149 Tm 98 Tz /OPExtFont5 13 Tf (0\) and ) Tj 1 0 0 1 185.5 613.149 Tm 122 Tz /OPExtFont8 12 Tf (CM\(S,co\) ) Tj 1 0 0 1 238.8 613.149 Tm 96 Tz /OPExtFont5 13 Tf (for the experiment of Lorenz96 system and model pair. ) Tj 1 0 0 1 113.049 599.5 Tm 99 Tz (The length of the sequence of observations is 102.4 time unit and the sampling ) Tj 1 0 0 1 113.049 585.549 Tm (rate is 0.025 time unit. The noise level is 1 and the sample standard deviation of ) Tj 1 0 0 1 112.799 571.649 Tm 97 Tz (the model error is 0.25. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 113.299 540.45 Tm 100 Tz (CM\(o, 0\) does not appear to converge to zero. Secondly for both methods the ) Tj 1 0 0 1 112.549 517.399 Tm 98 Tz (standard deviations of the implied noise and the imperfection error are very dif-) Tj 1 0 0 1 112.549 494.35 Tm 101 Tz (ferent from that of the observational noise and the model error \(We are aware ) Tj 1 0 0 1 112.549 471.3 Tm (that neither the model error nor the imperfection error is IID, there are infor-) Tj 1 0 0 1 112.799 448.3 Tm 99 Tz (mation of them beyond the second moment of their distribution. For simplicity ) Tj 1 0 0 1 112.549 425 Tm 100 Tz (we use the standard deviation, as a diagnostic tool, to test consistency between ) Tj 1 0 0 1 112.299 401.699 Tm 98 Tz (imperfection error and model error\). The pseudo-orbit generated by minimising ) Tj 1 0 0 1 112.549 378.699 Tm 115 Tz /OPExtFont8 12 Tf (CM\(S, ) Tj 1 0 0 1 149.5 378.699 Tm 100 Tz /OPExtFont5 13 Tf (0\) stays too far away from the observations as the standard deviation of ) Tj 1 0 0 1 112.099 355.649 Tm 97 Tz (implied noise is much larger than that of the observational noise, which indicates ) Tj 1 0 0 1 112.099 332.6 Tm 101 Tz (that the pseudo-orbit obtained by minimising ) Tj 1 0 0 1 348.699 332.6 Tm 115 Tz /OPExtFont8 12 Tf (CM\(S, ) Tj 1 0 0 1 385.449 332.35 Tm 102 Tz /OPExtFont5 13 Tf (0\) is not consistent with ) Tj 1 0 0 1 112.099 309.299 Tm 99 Tz (observations. While the pseudo-orbit generated by ) Tj 1 0 0 1 369.35 309.299 Tm 116 Tz /OPExtFont8 12 Tf (CM\(5, w\) ) Tj 1 0 0 1 422.649 309.299 Tm 99 Tz /OPExtFont5 13 Tf (seems to stay too ) Tj 1 0 0 1 112.299 286.049 Tm 100 Tz (close to the observations according to the standard deviation of implied noise. ) Tj 1 0 0 1 112.099 262.75 Tm 98 Tz (The standard deviation of the imperfection error is larger than that of the model ) Tj 1 0 0 1 112.099 239.5 Tm 102 Tz (error between the system and the model which indicates that the model error ) Tj 1 0 0 1 111.849 216.2 Tm (is over-estimated by the imperfection error. From the RMS distance between ) Tj 1 0 0 1 111.599 192.899 Tm 101 Tz (pseudo-orbit and true states in table 5.1 & 5.2, minimising ) Tj 1 0 0 1 411.85 192.899 Tm 123 Tz /OPExtFont8 12 Tf (CM\(o, ) Tj 1 0 0 1 448.55 192.899 Tm 98 Tz /OPExtFont5 13 Tf (w\) produces ) Tj 1 0 0 1 111.599 169.649 Tm 101 Tz (pseudo-orbit closer to the truth. But apparently this method doesn't tackle the ) Tj 1 0 0 1 111.349 146.35 Tm 99 Tz (problem of confounding between observational error and model error very well ) Tj 1 0 0 1 111.349 123.1 Tm 103 Tz (as neither the implied noise is a good estimate of observational noise nor the ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 103 Tz 3 Tr 1 0 0 1 302.649 51.549 Tm 86 Tz (104 ) Tj ET EMC endstream endobj 566 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 567 0 obj <> stream 0 ,,yyjhjl> q ¬~CaB@3 g~dG{s ١S@yot69nK+>pb՞sZrc[ת0V^9 ©'mQKBwYufD lG:ȊXvA`s^Jp<8zF?1&8ljo|Оo>c{Uv'e,FlFMcΎr~ߚ[N/C NA" [燀Vt傜a^H l7 Y3'P<Yު.])Y ɨ^!xCmֲyx !m{>#{xj4Wi>z9;.'RVUo~ [H57.%U23AP` M֘X\/MG}%>cg1.khJx. 5F.qy#$CA~̺ٗ}hej1bsVҏ&쯶⤤Vh;nl,}䟥>􇨺XQʛ;hEiU%a XҍP˸X-n[Agش/ok^Ш?WHۣ6V{5oe`Q; 7J;ak=(^PCKWl>ϫ*\]9d^͙Ҥm-U)*ތ.ǷiH}$FcQ%v+~V9qz*L#lGf?%f(NN}Hh}FHݣb3$S `2+bJfzS6ob\8  4@'#)CLd:jslw^-Vm {||wr{ 7}i4e UkcM6ghPGK~i0'g7jVLZ-JM{?sS Et B'kL !5i#Ze4LqCTR笚9yd Ctx6Jt5 =J۷O8GvΧשQcCxvнb<"MZ(^_8E HGd f/ nEJӔjoir8E_uf7.R^ΟqF "d}aMX#wH(V [;*R1~D.NVO}Y?Q֬g=k/m̰>g/p64õ1G<RԪNvIic *rML;[_ R{Pl?rG6d&?Ły ))vOF5Pz&)Y}~2%c緻Z J mjRG԰^|BiAP!̄hvU%Gɔ7p%mMƯB/'hDuOh2|bYU5SJ&ALX~P]$`_lTOɶLSXLۏɳ"o,2 $L?P?5>4f 7%җF9oHMa>̯$v 4nTsPC;zGnbQ_jhCH>k-lY:` XiHhU ɫ(DBqϞJ^ gLE1Z,(7Le':xBitafͶ%If HR~=ڥ‡vɬEbzD0JmQe;TeVNooY]mi4M J&r?b\qshGe"\8id"3\!Y8X]S7N@*JIJ{A,<6'B5Iwdr<[pa2ѲBN S{pxfbV/(r&B 6Ï\&ɀ. 8\]c[AC= LervF#QYyY b#U3&g}c)8^EGt!r-6˜'?q@O}!b'88|p/TdG**~ qK/E_-mt@0WlR.j$\h0d6lbޒgwLa.O}0#6ewz6-gݻKW4QAk)R;Ɉ,D<'ƆoS%;h(6`&웓@n/uC9qcZya Sɫy>*xF)Ǯ8&Qn~_0\8%{)SkpwK!:U)=foCKkWѼ "@.nGQ{*'0L#[Tp~Z l忛M θ:4 xـŅ3J}-$F ?IrcG*J:D#EwW e*4Jû?_Ts'1[ݽoiXgCbCžI۩֏BݗohL2kpVLv Rs=Ċ3mtnjlh˵ݱӯfJQg(_RE,鬐fI1'.y% g|fwǠ%k<+0Nu,xWLo[k~pA!DL7퇚= !^[&'=2 mL=wnXuuql:*\ˑ)TT)ܡ^j- %d*u+0sBY8ˌpAm˄\c@~Vyef-^w`L_V=C;k`O'1e#n5ܼ͆a"Ͷ},ckmZA񳳦ѩK<sJ cy+Pk2`5pVSk:@YJ3&NIosoV|pm.;ܓqg7쎢- sw{Q;w\gg,̭2 L:=?; w`t`;V駅iF$u r\IZ]^r/vʡp?Ibï @➣eWC?*۫5)|8d؊~dX!uT..qLr>ӺyFN.9'+[ ɞUd<3?h?$V!12%*]$6^<-%lzł^@绕v.2?`'8T2W\W?s'%:PI4h lGxrG5* ]WaIFՖɮb\ώ`!"?/}i}(lhu(ۖڥn"YUOo[39H[(qD vrR򔈾7֚Ce's[Kp4OФ #VIj{Xf]ҺW5LW' ^Dx?V*2$w|03wJFX~!(֘;e+{0=5Ζ\qpP](.upkiJԓⳝb5mmw֨ǗӰOg@vI5'6%N^CF69N inl5̗w_o۩ЅY U԰4᭍_8ugWzfݐm2b2)pȇ-1a!w݈>5 ?N%ےLW!r bByon0+꺞%9Jk5ti不dbu=f YcQ}WV@? W `DE^1ډ4'gw/G,<1iR9 g!0h;.D";R)< g7.X7v{ mu0{*BYj>QL/(] ]1;JXwHt1pJsy"";s\ɪb7Ymte=nxl/1R N`x$x+ʶ>ͲϰQ森3{/r;&z2":{#\.ǻu8ܧ3R0-ۗ>y,Q-vķɮfVP?@vV`L Ry 5oØ\OٟYdZʽٖnUjS,y"'D P *\V _ìȦV+RMQ'\l+/G>5Vzf* !Y>X43_w8꩕@ 3= T{!(n&S߭T|o!: HT|<+5Yb{R= ^ c_ngT@T$rVȆH f!83w.VΜmqGVw`>)xt)(FNmTjB(ה-?(2H "n?Acģ,qOf<B)|0 FbB&\P4JAna w YI|3 Dy|TZWc`hy1.,9ri]=dR"YA\&d ־.4:Fso6.vP|W'׉zZ2F @јٍh'!y9mN8  鳑c/ۥmB0#}e(H$u--MDM 62q5E7":wG.̗ ݑ– gbMadpgPa`4O Ju~d|'Qx̆\ֵ# ÆZ`Ģ}ޘyhZYyy"il?Wis`;(X$K #i`$:Tmf(FDQ$k*zEi*27fh0wTr7Rޛ \9569,yak<&rMo\`;(̋Sйd>pv] Lis(BnqSXfe&|0(%ytW(<#ܻF TB,:HIy ^3c1j/$[@7Іjsl70d5?;Cj[BnJ!ԷL 鸓ka##am7f:.+{+PLAV)n.BL3e\/Ug[-[VM*4#QAG4 N]I ׁYCQdy ŏw4 ~N xyD &sm:1F&ztn pO(1>2ٳ3䤼7f׀HV/ل'!єK;^ύŜuTZU96)&KG*ֲ*վ,cժ!O@qlOd * m'fNmr'<" N\1m:3MGЖ(~j9),7+YG؉r* Lw-[E{'> W/AzcM[ &;Qb/ &ȈӞ2>`QF֭S/n3( #o;cYֺXjS_.;(quol-k餫\{o-$j;ԝvc{,/*B !'% @3ϩ5_ !63GtBmS奢j1 畵{UZ{%OKwh3EU|}y;IB\9 'jck3a3{OWLGЕ&UJB 1ͷi '0W|r;r8bRdž}KU;J솢wee^ޘ)/Yڀ#H 9.JlTh8-h1tПI t+0Cv D-j)$6A>GR Px|0gӞ4GG-mCBc'-eEZxpw +ƈ/Jyko!_y$]P9P;5rFJ6hJO0+_֬U5t{(jY5LCE$z=ņ~wB/vWƨ^sQTʊ vN&4#EQGii b}  jC*a;`$5<{#[DŽ! v2*@:cL&!-w,Di76ujx|ūw?PDgR:UX+) ڂTVLxEPtE ַo62֏ǜ tKX'6VK7Pv 7J>-< -5:[ rf_;%LBDe9RWUv8 :$JWiRT>R${Lm /d[U^ģ ,xC\dqke g8eSH,Yf耝f/[1" 466/Oz\mvن,:YB ygY q螦ʹYɩБgULgOE8kRٵLFRͼ'VJf,DW%}TfHݛmB +T CR(>XNWJ(U:&ݍsjo kHH.VrN vn !z&Ą4ҢȄ86bs'f!eOD i].65eRZwgC!߾TR!q(y?-nc6bXU tVƛ$f#v7ui-wyܺWEl)@ )Ëu( ̉MV'0F1jxcRNnJ)Q.C ~u rY(Lf,Ea#x8=g) Vz^g^94P̮J?t3'OoW} ;|o_d߈.[c6LLu8M7BJ6RBs+ *ie 'nǧ0#V>峦2"m|(,''$=2"4xl6zffYnݿ endstream endobj 568 0 obj <> endobj 569 0 obj [570 0 R] endobj 570 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 607 0 0 840 0 0 cm /ImagePart_2132 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 373.449 720.7 Tm 107 Tz 3 Tr /OPExtFont3 11 Tf (5.2 IS methods in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 107 Tz 3 Tr 1 0 0 1 114.95 677.75 Tm 91 Tz (imperfection error is a good estimate of model error which may indicate that the ) Tj 1 0 0 1 115.2 654.7 Tm (estimates of the model states are highly biased. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 115.2 610.299 Tm 113 Tz /OPExtFont3 13 Tf (5.2.5 ISGD with stopping criteria ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 113 Tz 3 Tr 1 0 0 1 114.95 579.6 Tm 95 Tz /OPExtFont3 11 Tf (The GD method introduced by Judd and Smith \(2004\) is unable to produce a ) Tj 1 0 0 1 114.95 556.549 Tm (desirable estimation of the projection of the system state as the implied noise ) Tj 1 0 0 1 114.95 533.5 Tm 94 Tz (and imperfection error of the relevant pseudo-orbit are not consistent with the ) Tj 1 0 0 1 114.5 510.5 Tm 93 Tz (observational noise and model error. Confounding between observational noise ) Tj 1 0 0 1 114.7 487.449 Tm 95 Tz (and model error makes it impossible to produce pseudo-orbits whose implied ) Tj 1 0 0 1 114.5 464.399 Tm 93 Tz (noise and imperfection error are consistent with observational noise and model ) Tj 1 0 0 1 114.25 441.1 Tm 94 Tz (error respectively. We found that applying the ISGD method with proper stop-) Tj 1 0 0 1 114.25 417.85 Tm 92 Tz (ping criteria can, however, reduce such inconsistency and obtain less bias state ) Tj 1 0 0 1 114.5 394.8 Tm 88 Tz (estimation results. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 131.3 371.75 Tm 91 Tz (As we mentioned in the previous section, applying the ISGD method is equiv-) Tj 1 0 0 1 114.25 348.699 Tm 89 Tz (alent to minimise ) Tj 1 0 0 1 201.849 348.699 Tm 109 Tz /OPExtFont6 12 Tf (CM\(6, ) Tj 1 0 0 1 238.8 348.699 Tm 90 Tz /OPExtFont3 11 Tf (0\) cost function, i.e. the mismatch cost function defined ) Tj 1 0 0 1 113.75 325.7 Tm 94 Tz (in Section 3.12 and Equation 5.11 are the same. Examples shown in previous ) Tj 1 0 0 1 113.75 302.149 Tm 93 Tz (section demonstrate that the minimisation does not converge to zero easily and ) Tj 1 0 0 1 113.75 278.899 Tm 90 Tz (the pseudo-orbit produced eventually is not consistent with the observations and ) Tj 1 0 0 1 113.5 255.85 Tm 92 Tz (stays farther away from the true pseudo-orbit than even the observations. When ) Tj 1 0 0 1 113.75 232.799 Tm 88 Tz (the ) Tj 1 0 0 1 132.699 232.799 Tm 89 Tz /OPExtFont6 12 Tf (C./V/\(6, ) Tj 1 0 0 1 169.699 232.799 Tm 90 Tz /OPExtFont3 11 Tf (0\) is greater than zero after finite iterations of GD, the mismatch error ) Tj 1 0 0 1 114 209.5 Tm 92 Tz (e) Tj 1 0 0 1 120 209.5 Tm 70 Tz /OPExtFont2 11 Tf (t ) Tj 1 0 0 1 122.15 209.5 Tm 92 Tz /OPExtFont3 11 Tf ( is actually the imperfection error. In other words, minimising the ) Tj 1 0 0 1 453.35 209.299 Tm 109 Tz /OPExtFont6 12 Tf (CM\(S, ) Tj 1 0 0 1 490.3 209.299 Tm 90 Tz /OPExtFont3 11 Tf (0\) is ) Tj 1 0 0 1 113.75 186 Tm 91 Tz (actually minimising the imperfection error. If the imperfection error goes to zero, ) Tj 1 0 0 1 113.5 162.95 Tm 96 Tz (the pseudo-orbit becomes a model trajectory. Since we treat the imperfection ) Tj 1 0 0 1 113.5 139.45 Tm 92 Tz (error as the estimate of the model error which is known to exist when the model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 304.55 52.299 Tm 76 Tz (105 ) Tj ET EMC endstream endobj 571 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 572 0 obj <> stream 0 ,,}}b7!͔K^+k3.v1Zk+3j5t '8#yQ}Kq)ߪ՛Jۚ2הDc9z O!vЌ?c 1.m> m~٨!fJno O54ҊKEهk Dٱcؑ[$IRy" . Xk.:(1{rP|R rY4hm P Gi2uJNy}4M9|EeΛsGsy"΀Xv;K!_7dF4'Ã)B- :]mCO;7 (l (D~+9UhU~@ V tx+iR}2 RN l27$K-g1]ˤK||A&B& 7x`܋fc+q/Ga)|w{nScŖIVĨ[DH!,5%;W>#|LU @>X1RŸNxĽ{?ԏVkس E*uh;[Nׇ/V9#H:0+nl"- GfQhu~%5]FÊ/C]$7-jݜWH;G8F5%;uV>h96Tsc+]td4 Jtvd,0vu|cu'1VxG"@EԹ;/#dMu](J)?=1#+ܕ8^d5m)Z$-H*pTh-Ak"6晲XaxVC![$ŧ&2441@ЗԲo޾}Pjm #tܞB+g$n"鸴ͨӇ>t0žۊŽ~ p4[7ЅԁRuqYM2 x&=eRXp6*2F}9]2Su98qço߉߽1D;i_SSHb|KFtx CLRmeGa!rYن MW>:4z33 Ks|[C3 +mP%XMILQ1İx M$Xq~3hu\*YH) QdGa,xw2j;86 CkfBϟP+Sm'YIvy̜qrv.S=zҩ׉.J#iJFoV_3/NWJ1FK$eS(p]@)_59]Thj(BRe}+{>9X!.poY!)7y0 LKjX, ?aH(hB: Q\u{}> ,M&:Lxᰨ5~=DŽ@Գ.wܿĶmBDWFTә0Mغ_=8Vu qU#r{ME?vOTF~"DD#!#&aØޢqFO60Α5Y6JĿKAo/fɬ?8fk;b?n'gq dtWPia=6bA %^IW ^4ŀ7IFQ5BR<(/) _+Ǟӱ$@"Iv/ ;;"snr| M|Vh;[X_X_Vº]d6 +TyH0cT;;俼RG)?𑠸ҟa'ǵAdzpF#>//BO~a+t_fsW(g#I&c!f҃0Y>h2mbFIT_ȭ)0e.XS ' NM1AgӰ+!Q8NkZpYd@nj4 i2LY&lwV ~∖:m~zu uzje^UۅRӣs@6=8~NACY`ݜJGHZ({m}7 : >'&ll."G 0ٶv- jhK'aXQgv5ӶwTMPG&4ky,GGEl_ "cW>r, tߡCIL'vR4“I@ ~ed:rM̎, ~c7CXlB$يPowWCˠLa_$e(LhGTؖ ϝsiN~C9cWT)D%BlIexےmI @՘:J\Zqh6fcqiw%gN|33ե#h]"C=Y 9+k-o+C|wԀ?KA8 3 ,'.l+? xs{=]M=5>K/q=܅YSJP?&Q;cq"eL!{X: ( .9ѻ͢h/VR"Y 2?ldF7G3~2_tI` +[&@sVkoaV%T]L터/OD7R53?kP~ Cp^?@Kڎw0uw]UZC3s}{).q4$%[g;.ajxϷ ӽ)hAvZ5 ԀлxRZ|/$6ː1"oa{]gTG| ~ <_1e[J|{(14\k&E1^T{okf '!(:1MN * X9<3 [fU+íNJKgch RJU)LTÞm@1y0} HS%#nfc$o77GC@[V>:%|ѱہߖ=4˳*<5hK`Wޟj92S N v{B{e@E #߭ݭQȉ]p}ҕvt%{Aɢ@g 5ɏV[@I/3_\zoul" (}tP/dt@sXu M[رqo.h~7Ɠ ̢W XHZ+DΉ)*Y",HʟHԭA:Q|m]ްegMz?fBQ8!]ȾK`AAع06rO0L-oX !n}vI qn [V;b'1[9~ј%uC]Կ&zZ.5P<{wTBm)羧k'_S?kՈ,gc=Be? f153RyY,FR'6yL'=?j:/q' KZT]ڶA˗Q-D\G8 .wCf6>mצO銠gȠ.Ne*zS@đR;sRှ5Ĥe D m7yk%,*O!DruDdH8 3񝛺ۻ%QfgPȴ 7VcsB[VH{=sO6+  ()a3Qox.cHl|GLA_C-j䫝f$x! ! ֗eVww&´סenփ+QG: ; B+h{m˽+nQ&T0IPIZք = PU>  ,w:\b Tz&vAs!ŝ8,UdA XlӒV{\ dXCF $ME}l^Z쑬)V-,sgXs2 !\;ރᇇyOԟɟyǂPjSDKa8JSrCqxz40DDy#r9ʫIu>"%q ulmn\Sϑas74%#m.vZ*+  J] y?zS@Z)A,ilbVjDp:U4G# =\"-ZKiۥ6I$HUFzWw큷 `P2+j-X;0"%7;:ij玞&\f׈^1ؚx1On%@-B끴x, )9O}mȌCS,2X1S!]W[.D7+(:\Vݎe]D;Av̧8hg^YG93EE 9c)Gmt:iC6Q{ϞT0+!ԑ{e ɱ~YsAkQQ?uQT9b0g.~n!t aUr,OIÁR/)8پ(3(}0$@ KH8{Ët4QyJڽo`Be +-I]OiJO0ތY.J|,me+FÒy.7/Ė @#/~׍i0Zz0{$3W#ٔ`'U 5%)Pp_܀}sad)~cqo9؟/bR\T%| 8*!B=b\1jjcݞM(znr Qv\dt[;_]_ٵ| hdW*})Qv1Y8UzbH k:]_v%u˔#*(9P&RNp5AA!zXknj⢐h]E!y>u5+0?F |'kM*opoCTZK**`X#Q>( ]s9{!tg1BFDn@-*ӱ1X] &-2Zp\(7Rǘdkoi$r,RDQ ؀#}.+mj2:^П|ϡ_JR`x !B?Cy58O-Un"m\%J^8a3%n +C ,ҿܿKaLW}lvi{`ԟTd[i^Z8Xo8Cje0֮C֯ BA]( (ZFf*Uf xL^9.-yuU$P(SXA/J2m錦BUn),ƳOhQވ2wwuLxI>:BgTc}%ˈC,>2`dibKS]nAj0V bꞔS>NZAF!Jyx~$B}6FlRd tY2ebݦq3WcڭMg@d>6B#+Ädx0d_H>Yv8c0oTNu)^t]]hUI/Cx3MhmFr΁T5XRCqˀv KcA @ݧݗǰSB /Jfe.MJaT?1{)d {3P 24 }[κK/ޔRW$ԯt7,EVTK,8cӘ8#/5B72 j-.@Fҳ/֛NbUc:=}MnȑIK<38Hz20w4)2DK֐I5B;sЀyi~3A_'rMN$q$6TJd]QAD\$=,jJ z_xlZ͸q&nvo& Oĥ ,~\JMd!{\J ծ@颳ʍtauv-P{Wz]˝Od3(<G2Zƌ<]-%Eft7~NS>CA p.RWx9cUG645fh}^SKfSWTs*oې !Cy-k"+„%uNDQ#yHXx[fiY|1=?s39ӻ5a #ZTdV{AeMCGqKl5\~ - d;sr; P$&s5˼OeH6w6~J;;O\9f)<(<]'ט)'7T7ﺟ ̓GS"2? &dMt' m:pѪy졔b[l㪴&lXN1k@O endstream endobj 573 0 obj <> endobj 574 0 obj [575 0 R] endobj 575 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 603 0 0 839 0 0 cm /ImagePart_2133 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 371.75 720.45 Tm 117 Tz 3 Tr /OPExtFont5 12.5 Tf (5.2 IS methods in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 117 Tz 3 Tr 1 0 0 1 113.299 677.25 Tm 93 Tz /OPExtFont3 11 Tf (is imperfect. Our purpose is not minimising the imperfection error but produc-) Tj 1 0 0 1 113.049 654.45 Tm 94 Tz (ing better or more consistent estimate of the model error. Figure ) Tj 1 0 0 1 442.8 654.45 Tm /OPExtFont5 12.5 Tf (5.2 ) Tj 1 0 0 1 460.8 654.45 Tm 88 Tz /OPExtFont3 11 Tf (shows the ) Tj 1 0 0 1 112.799 631.649 Tm 95 Tz (statistics of pseudo-orbit changes as a function of the number of iterations of ) Tj 1 0 0 1 113.5 608.6 Tm 92 Tz (Gradient Descent minimising mismatch cost function ) Tj 1 0 0 1 385.449 608.6 Tm 108 Tz /OPExtFont6 12 Tf (CM\(S, ) Tj 1 0 0 1 422.149 608.6 Tm 96 Tz /OPExtFont3 11 Tf (0\) in both higher ) Tj 1 0 0 1 113.299 585.299 Tm 91 Tz (dimensional Lorenz96 system-model pair experiment and low dimensional Ikeda ) Tj 1 0 0 1 112.799 562.299 Tm 92 Tz (system-model pair experiment \(Details of the experiments are list in Appendix B ) Tj 1 0 0 1 112.799 539.5 Tm 90 Tz (Table B.6. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 129.599 516.2 Tm 92 Tz (Figure 5.2 shows that as the Gradient Descent minimisation iterates further ) Tj 1 0 0 1 113.049 493.149 Tm 93 Tz (and further, the standard deviation of implied noise is getting larger and larger ) Tj 1 0 0 1 112.549 469.899 Tm 92 Tz (which indicates that the pseudo-orbit is moving farther away from the observa-) Tj 1 0 0 1 112.799 446.85 Tm 91 Tz (tions. By comparing the standard deviation of implied noise with that of the real ) Tj 1 0 0 1 112.549 423.55 Tm 94 Tz (noise model, we found that at the beginning of the minimisation, the observa-) Tj 1 0 0 1 112.299 400.5 Tm 92 Tz (tional noise is underestimated by the implied noise since the pseudo-orbit stays ) Tj 1 0 0 1 112.299 377.5 Tm (too close to the observations. This makes sense because the minimisation algo-) Tj 1 0 0 1 112.299 354.199 Tm 90 Tz (rithm is initialised at the observations. As the minimisation proceeds, the implied ) Tj 1 0 0 1 112.099 331.149 Tm (noise becomes more consistent with the observational noise and the pseudo-orbit ) Tj 1 0 0 1 112.099 308.1 Tm 94 Tz (gets closer to the true pseudo-orbit. After a certain number of iterations, how-) Tj 1 0 0 1 112.099 284.6 Tm 91 Tz (ever, the implied noise tends to overestimated of the observational noise and the ) Tj 1 0 0 1 112.099 261.299 Tm 94 Tz (distance between the pseudo-orbit and the projection of true system trajectory ) Tj 1 0 0 1 111.849 238.299 Tm 95 Tz (gets larger. This is due to the model inadequacy. The minimisation makes the ) Tj 1 0 0 1 111.599 215.25 Tm 93 Tz (imperfection error smaller. When the imperfection error of the pseudo-orbit be-) Tj 1 0 0 1 112.099 191.7 Tm 92 Tz (comes smaller than the actual model error, the implied noise has to compensate ) Tj 1 0 0 1 111.849 168.45 Tm 91 Tz (for the imperfection error to account for the uncertainty caused by the model in-) Tj 1 0 0 1 111.849 145.149 Tm (adequacy which makes implied noise too large and the pseudo-orbit inconsistent ) Tj 1 0 0 1 111.349 122.1 Tm (with the observations. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 302.899 51.799 Tm 87 Tz /OPExtFont5 12.5 Tf (106 ) Tj ET EMC endstream endobj 576 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 577 0 obj <> stream 0 ,,6mmb7!>[JZ 0:㓼LH(O{u"9ф|Ýu1E: -Pw@FANڅ3;֖L)_簃'`8c{H)5y=3UfZӷ.杅 }4x퇇el4 q^I'bj5Bdv:A4TT$yˇ6H{uXIIu11up8Far0HN8#cmf0(Z+(*h2ܸnO\Z)3 b7iDMt `4CQ$[5ݒ(C jp^j'Iwm7I?5b'օ`%4Cwy"w=Ygķ~h{(Y'knj&%U}>nEѲohX3UpcdW y0[[%!U5mix:L?3t/4<VlG]R&y@~Y}Q[2:]P7̮Z/hxWD%XrtF8890UsQzOߏYqPL8ea^4mm2}(X]y,[tWÍ!i,ԥ*ʟ ]{0U6+Z\9WL68ۗWۍqspC.񧵮\m 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LiZocqzYm==gxiViZ%\B݌i.]wbn0Hd;Q;>חk>7+^ߙ崡ǙՋ+ľq9f>&d<`Œ1Po{.%EYD΂6c9YɄoR u's7̀,㕨}'_Su؅[ZVΩ 'wh\DL,+_\q1omvPPr\⹬0~K@-.Ы5tZű֐="j{ر kdE$`::d3/@@%dhS׸3FCO1/w/p;yOsC\iGIMN]'yQ/9`c_3k2>XߐZ)eT:ԫf ڇ[$N9}EnD)3\!j-j]!lts#}\GSFgn&Gjr?`MV؞Gpprsbs\A +,Fz{)>S⨈e{{m)50͟-$ɕJ o$כ﹔mP@- V7Nq 8AwP}VjSZ5+M: z뾆nl^Փ d䘷|eUj׶U%l: xBAt=l/wa[J{ Rq>͜`!-#չY#>Gg,zhcO[%)Z``1" ɟaTƈF4![/5 0&*,Bֵ̜F၂M1E]+F."(Wn|֠~;[Wg?'VoeN:,3{OB\`t=*64KEIl|.n(w$ veZ/3!tHI+K=l/gvZ cîU<ZȠD/w_w?[*g( j/agx:9%$t!m}eyΉmDxt }G=olXhIw 10S7:}@11F#a.x*&iuy $q>JÏTY17X:X^ﹿ\ّJCw u߯0`&FإwNdi6%]A"fs10 7.G@úE{ GaFЖ3bU?ƆvP)C,&ޤm< ;X>Ž,:n_-.:۞j7~IѼGEێod1ѐ%J2[WAXdY0AHl¼ Y'fu-:<^!UgU|:hʋԓ4hzRM dUk8N0|3е5 иsj#Ew,a;wrv7C)W7·o\|u^jޜuKI̟ƘLZCqX36EP -TXH'K:К`#Z;+P]|*C7H]HC XW09odmY d"ȩ9p@vw4cQ$Pqh-"RNK:-wϼxrN1;$xO~>~,dȊGQ=aġ1iz~bt" SG *E,6QaV]2acQ=ꍶbG{1}D) /I_;= x A0nDՓꄷե%c'FL؜v{L̜g )VS KG=.k񈖀oSW"Ҡ(OPfo9^{ aVk(vq$3+j:0'{L' endstream endobj 578 0 obj <> endobj 579 0 obj [580 0 R] endobj 580 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 604 0 0 839 0 0 cm /ImagePart_2134 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 319.699 659 Tm 88 Tz 3 Tr /OPExtFont11 4 Tf (0.015 ) Tj 1 0 0 1 314.399 621.299 Tm 115 Tz (o 0.01 ) Tj 1 0 0 1 313.699 583.649 Tm 70 Tz /OPExtFont11 7.5 Tf (3 ) Tj 1 0 0 1 317.05 583.649 Tm 99 Tz /OPExtFont11 4 Tf ( 0.005 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 99 Tz 3 Tr /OPExtFont13 10 Tf 1 0 0 1 6605 3538 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 99 Tz 3 Tr 1 0 0 1 6624 3538 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 99 Tz 3 Tr 1 0 0 1 6605 3560 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 99 Tz 3 Tr 1 0 0 1 342.5 640.75 Tm 86 Tz /OPExtFont5 7.5 Tf (la\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 7.5 Tf 86 Tz 3 Tr /OPExtFont13 10 Tf 1 0 0 1 6605 4315 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 86 Tz 3 Tr 1 0 0 1 6624 4315 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 86 Tz 3 Tr 1 0 0 1 146.9 519.1 Tm 112 Tz /OPExtFont12 3.5 Tf (4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 3.5 Tf 112 Tz 3 Tr 1 0 0 1 351.6 542.1 Tm 85 Tz /OPExtFont11 4 Tf (100 ) Tj 1 0 0 1 358.1 542.1 Tm 1125 Tz (\t) Tj 1 0 0 1 373.899 542.1 Tm 88 Tz (202 ) Tj 1 0 0 1 380.649 542.1 Tm 1160 Tz (\t) Tj 1 0 0 1 396.949 542.35 Tm 91 Tz (300 ) Tj 1 0 0 1 403.899 542.35 Tm 1128 Tz (\t) Tj 1 0 0 1 419.75 542.1 Tm 88 Tz (400 ) Tj 1 0 0 1 426.5 545.5 Tm 1160 Tz (\t) Tj 1 0 0 1 442.8 542.1 Tm 87 Tz (500 ) Tj 1 0 0 1 449.5 545.5 Tm 1146 Tz (\t) Tj 1 0 0 1 465.6 542.1 Tm 91 Tz (600 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 91 Tz 3 Tr 1 0 0 1 381.6 536.85 Tm 88 Tz (number 01 iterations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 88 Tz 3 Tr 1 0 0 1 162.25 506.35 Tm 97 Tz (\(b\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 97 Tz 3 Tr 1 0 0 1 135.849 469.399 Tm 51 Tz /OPExtFont9 7.5 Tf (e ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 7.5 Tf 51 Tz 3 Tr 1 0 0 1 140.9 455.699 Tm 94 Tz /OPExtFont11 4 Tf (-6.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 94 Tz 3 Tr 1 0 0 1 144.25 441.1 Tm 146 Tz /OPExtFont9 3.5 Tf (-7 ) Tj 1 0 0 1 148.8 441.1 Tm 91 Tz /OPExtFont12 3.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 3.5 Tf 91 Tz 3 Tr 1 0 0 1 140.9 430.3 Tm 215 Tz /OPExtFont9 3 Tf (-) Tj 1 0 0 1 143.5 430.3 Tm 108 Tz /OPExtFont12 3.5 Tf (7.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 3.5 Tf 108 Tz 3 Tr 1 0 0 1 144.25 417.8 Tm 127 Tz /OPExtFont9 4 Tf (-8 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 127 Tz 3 Tr 1 0 0 1 181.9 401.5 Tm 82 Tz /OPExtFont11 4 Tf (100 ) Tj 1 0 0 1 188.15 401.5 Tm 751 Tz (\t) Tj 1 0 0 1 198.699 401.5 Tm 85 Tz (150 ) Tj 1 0 0 1 205.199 401.5 Tm 751 Tz (\t) Tj 1 0 0 1 215.75 401.5 Tm 91 Tz (200 ) Tj 1 0 0 1 222.699 401.5 Tm 737 Tz (\t) Tj 1 0 0 1 233.05 401.5 Tm 87 Tz (250 ) Tj 1 0 0 1 239.75 401.5 Tm 737 Tz (\t) Tj 1 0 0 1 250.099 401.5 Tm 91 Tz (300 ) Tj 1 0 0 1 257.05 401.5 Tm 751 Tz (\t) Tj 1 0 0 1 267.6 401.25 Tm 87 Tz (350 ) Tj 1 0 0 1 274.3 401.25 Tm 737 Tz (\t) Tj 1 0 0 1 284.649 401.25 Tm 87 Tz (400 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 87 Tz 3 Tr 1 0 0 1 200.65 396.449 Tm 88 Tz (number ol Iterations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 88 Tz 3 Tr 1 0 0 1 143.05 378.199 Tm 84 Tz (0.2 ) Tj 1 0 0 1 140.9 366.699 Tm 103 Tz (018 ) Tj 1 0 0 1 140.9 355.399 Tm 87 Tz (0.16 ) Tj 1 0 0 1 140.9 344.1 Tm 84 Tz (0.14 ) Tj 1 0 0 1 140.9 332.85 Tm 87 Tz (0.12 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 87 Tz 3 Tr 1 0 0 1 135.099 321.299 Tm 46 Tz /OPExtFont3 10 Tf (2 ) Tj 1 0 0 1 143.05 321.299 Tm 103 Tz /OPExtFont12 3.5 Tf (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 3.5 Tf 103 Tz 3 Tr 1 0 0 1 140.9 309.799 Tm 84 Tz /OPExtFont11 4 Tf (0.08 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 84 Tz 3 Tr 1 0 0 1 135.099 298.049 Tm 60 Tz /OPExtFont11 7.5 Tf (1 ) Tj 1 0 0 1 140.9 298.299 Tm 103 Tz /OPExtFont11 4 Tf (006 ) Tj 1 0 0 1 134.9 292.5 Tm 113 Tz (S ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 113 Tz 3 Tr 1 0 0 1 140.9 286.75 Tm 84 Tz (0.04 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 84 Tz 3 Tr 1 0 0 1 140.9 275.5 Tm (0.02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 84 Tz 3 Tr 1 0 0 1 140.9 405.1 Tm 94 Tz (-8.5) Tj 1 0 0 1 149.05 401.5 Tm 84 Tz (0 ) Tj 1 0 0 1 151.199 401.5 Tm 71 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 71 Tz 3 Tr 1 0 0 1 322.1 518.6 Tm 89 Tz (0.03 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 89 Tz 3 Tr 1 0 0 1 465.6 401.5 Tm 87 Tz (600 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4 Tf 87 Tz 3 Tr 1 0 0 1 165.349 542.6 Tm 85 Tz (50 ) Tj 1 0 0 1 169.699 542.6 Tm 868 Tz (\t) Tj 1 0 0 1 181.9 542.6 Tm 82 Tz (100 ) Tj 1 0 0 1 188.15 542.6 Tm 769 Tz (\t) Tj 1 0 0 1 198.949 542.6 Tm 78 Tz (150 ) Tj 1 0 0 1 204.949 542.6 Tm 787 Tz (\t) Tj 1 0 0 1 216 542.6 Tm 113 Tz /OPExtFont12 3.5 Tf (200 ) Tj 1 0 0 1 222.5 542.6 Tm 1080 Tz /OPExtFont2 3.5 Tf (\t) Tj 1 0 0 1 233.3 542.85 Tm 108 Tz /OPExtFont12 3.5 Tf (250 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 3.5 Tf 108 Tz 3 Tr 1 0 0 1 201.349 538.299 Tm 129 Tz (nurnboi Mile.lions ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 3.5 Tf 129 Tz 3 Tr 1 0 0 1 141.599 583.899 Tm 102 Tz (D.35 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 3.5 Tf 102 Tz 3 Tr 1 0 0 1 143.75 545.95 Tm 108 Tz (0.3) Tj 1 0 0 1 149.5 545.95 Tm 219 Tz /OPExtFont9 3.5 Tf (. ) Tj 1 0 0 1 151.449 545.95 Tm 45 Tz (\t) Tj 1 0 0 1 151.9 545.95 Tm 91 Tz /OPExtFont12 3.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 3.5 Tf 91 Tz 3 Tr 1 0 0 1 143.75 621.799 Tm 108 Tz (0.4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 3.5 Tf 108 Tz 3 Tr 1 0 0 1 250.3 542.6 Tm 85 Tz /OPExtFont11 4 Tf (300 ) Tj 1 0 0 1 256.8 545.5 Tm 769 Tz (\t) Tj 1 0 0 1 267.6 542.85 Tm 113 Tz /OPExtFont12 3.5 Tf (350 ) Tj 1 0 0 1 274.1 542.85 Tm 1079 Tz /OPExtFont2 3.5 Tf (\t) Tj 1 0 0 1 284.899 542.6 Tm 112 Tz /OPExtFont12 3.5 Tf (400 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 3.5 Tf 112 Tz 3 Tr 1 0 0 1 378 701.7 Tm 120 Tz /OPExtFont2 11.5 Tf (5.2 IS methods in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 120 Tz 3 Tr 1 0 0 1 123.349 226.75 Tm 111 Tz (Figure 5.2: Statistics of the pseudo-orbit as a function of the number of Gra-) Tj 1 0 0 1 123.099 213.1 Tm 106 Tz (dient Descent iterations for both higher dimension Lorenz96 system-model pair ) Tj 1 0 0 1 123.599 199.149 Tm 104 Tz (experiment \(left\) and low dimension Ikeda system-model pair experiment \(right\). ) Tj 1 0 0 1 124.099 185.25 Tm 111 Tz (\(a\) is the standard deviation of the implied noise \(the flat line is the standard ) Tj 1 0 0 1 123.349 171.549 Tm 104 Tz (deviation of the noise model\); \(b\) is standard deviation of the model imperfection ) Tj 1 0 0 1 123.349 157.899 Tm 106 Tz (error \(the flat line is the sample standard deviation of the model error\); \(c\) is the ) Tj 1 0 0 1 123.099 144.2 Tm (RMS distance between pseudo-orbit and the true pseudo-orbit. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 106 Tz 3 Tr 1 0 0 1 311.5 46.299 Tm 89 Tz (107 ) Tj ET EMC endstream endobj 581 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 582 0 obj <> stream 0 ,,#R??b7 ?S>9~6s= Dx4%#Q#1aiOPL nw϶X ?" 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PO48N)Ix@vutH*f9ttUc*RJ1,]~u+ҘZC }Ц%\Ic .ާ zG>aA3IpiI R{!y4<I[@Y饉3ĆA^t^9./ ּ֫cs8"Ț98ы P w,ZcmI+B0iZ4@R`٤ZӀ*C. )ռ+[ylAc %X̞;+FrImI:j˵DI9,!`^mk;l:γG6ea]*]5Ki3q}PeɎ|Ǥv\NPw\w5*+#G< T3nu<*1faCns$RbSF|y|%ǁē*,?u!c._Ibbq> ֊fE3,5X \9Khz $yߊEZ"Bl yۀ<`<mTۿtd7kHc,ШKf[6~2S:[$]?Sa-^nėZ!A!"cgbݜ2I+p/pvnpd)i5DN4"p\K0 `^&17>$96;4;Wk cpyF4?ZGNU(sȐIϢS=JI#6{ 2#릉*IL< r1 @p&UT(tvfu?UTwyȴ1ysYU@C;N+i`[{vW9U?`1"eʭ YɚgC`w[/I6dzDR[ E GPċTh̝|s櫣p؄ȽL0()^[ʨjc,Σm=׀.,oJ]>Ro13tO hä67 ÏQF6E|,cl[2jHLȳ}_~̭(uHwQ֔ɒCEKmwgC)^J`+rKc!'\i8TFΕF+PXhcSzR#uadXŪ Pa,DY!~ߩ@ũ䩉ca$y Ob+>o6+b{ɬHu'hfyT.C2A'\ Swk+"a~B" < .lO[9]Nx]ZZ$@B ΄+ 0ƒmZb'# bV%uY`1Q1n n мۊHz6V0'RFF=6S0L^P b>MP4j%5N9f̉Kv$g) J\mUƘ9QIRNx"d*dp.7u*XSy&EFOLyxL6rIOmo ށX(K,'zڱJߣ6ntqs!ئo̜:^C3VCXduJQwm6i(F,?2TAql7 Kt/ ;ViY+T+{&8 FZIvX4`1Cg=Ỉ)[7G}O~0#TގkL/Xz{Ub<DYakZe*L[-o^fGc)-=*˛5irUmg a"?h2sn~G7뢛aQ^a߇AQ^y24ױ∬~7yMI7!d! ;6fW`m| 6Kی miq4MGa+jpt457 g$/O_x3_vT$E8bɩZ/!rtieQ$w8' X1Uev%4S3Sgbe| 2,(`y"BwKpjD}R`AG(,/7@](]@uL!i#[cSJ׶jРn kt^ڬtOzH9+'WwCn39A8'FHJ|ݢY, 91ӅPc9xc6+)62z`; ^1ˏl;ݤ3>௘xHᬰR"<@獐\/*K&T.ն픵@t^.hk~VUQ %g /{wTL*BD`Q1>w ]*gbanlxiwsBH2Ktum4`I'$#OwB  V R(`AO0 raS%>Էuz3V{U^_z-~쭸 mu@Ïg;q3v 8Ϲ..e._ _nsӼB4{ ͜Ek/AQ!Z&&$=FUB6B&>Ŵo c> -ץ `y\̲bʿ0hUIPVX~H&+KTx<[Agrp?J/DE d:*QzTה b63a7U=c1xbAw`ߛ83KNA*'INS*.c,acFx.>Rp(dXvM29Q?G gz4׹ZU<9sY]^3XgRLAOf0;io6ıqa1ې1d^TV5yqXJՋK IDhV^ި&= ~>C4fݍ!#@ӷҸ*1B =;CUAOePׁs/XAΦdF6alS;stl' ba*"K E]Z7c2=9ڒqw @& &? d~LZΠVGŋJ6?^ySC=T21(>1Gͭc%NYk'?.1={HG0iwԸPNCj}'N 9l#eTi"Ԍ %ggnD#-́k=3[LnD+LqA}Pܰ NdaJLhDx~_!/(t]]V $|H7u|FEd!ܒ2wHArʫKtIܐQa|# !R;Zi" u[k*p`Glڜ^o9uKP'N] ؎nF+U,x[yRoy>2{?˸MH 6*|RЛ:U}e=;oİ(칰e*ئ$߾0Ĝ",=ް'<&oLKW{"]3#h 65?[YCC|`*G߽66ǸꈀuJVhB`E/ Fe9P"r> R?&[ +ڿ:g5^$H-&n@7"{Zb9V Q=x)kI3N&g$VOjAK\cRUvzA!@M 9c$R' 'ѿ g ߧEX1Wh6/_Hkhc/T1ٟTN|H: $̓C 4!J 3yI> endobj 584 0 obj [585 0 R] endobj 585 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 603 0 0 839 0 0 cm /ImagePart_2135 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 372.949 720.45 Tm 112 Tz 3 Tr /OPExtFont5 13 Tf (5.2 IS methods in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr 1 0 0 1 131.5 677.5 Tm 100 Tz (Since the model error is neither IID nor Gaussian distributed, how well the ) Tj 1 0 0 1 114.25 654.7 Tm 103 Tz (imperfection error mimics the model error should not be judged only by the ) Tj 1 0 0 1 114.25 631.649 Tm 96 Tz (statistics of the second moment. Figure 5.1 shows that the model error is spatially ) Tj 1 0 0 1 114.5 608.6 Tm 95 Tz (correlated. As an estimation the model error, we expect the imperfection error has ) Tj 1 0 0 1 114.25 585.299 Tm 99 Tz (similar spatial correlations as the model error. Figure 5.3 plots the imperfection ) Tj 1 0 0 1 114.25 562.5 Tm 105 Tz (error in the state space with different GD iterations for the Ikeda Map. To ) Tj 1 0 0 1 114 539.7 Tm 98 Tz (make comparison easier, Figure 5.1 is re-ploted in the fourth panel. The pictures ) Tj 1 0 0 1 114 516.2 Tm 100 Tz (show that at the beginning of the minimisation, the imperfection error is larger ) Tj 1 0 0 1 114 493.149 Tm 99 Tz (than the model error in most places. The pattern of spatial correlation can only ) Tj 1 0 0 1 113.75 469.899 Tm 98 Tz (be seen around \(0.5, 1.3\), which suggests the imperfection error is not a good ) Tj 1 0 0 1 113.75 446.85 Tm 96 Tz (estimate of the model error. This is because at the beginning of the minimisation, ) Tj 1 0 0 1 113.5 423.8 Tm 101 Tz (the imperfection error contains both the observational error and model error. ) Tj 1 0 0 1 114 400.75 Tm 100 Tz (Similarly after too many iterations, the imperfection error is forced to be small ) Tj 1 0 0 1 113.75 377.5 Tm 104 Tz (and lose the spatial correlation it should have. One can see very little spatial. ) Tj 1 0 0 1 113.5 354.199 Tm 98 Tz (correlation of imperfection error in the third panel. With a intermediate number ) Tj 1 0 0 1 113.5 331.149 Tm 97 Tz (of iterations ) Tj 1 0 0 1 177.599 331.149 Tm 32 Tz /OPExtFont3 13 Tf (1) Tj 1 0 0 1 182.4 330.899 Tm 98 Tz /OPExtFont5 13 Tf (, however, the imperfection error seems better estimate the model ) Tj 1 0 0 1 113.299 308.1 Tm 99 Tz (error, the pattern in the second panel and fourth panel are very similar. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 130.099 284.85 Tm 100 Tz (It might be asked whether the imperfection error estimates the model error ) Tj 1 0 0 1 113.299 261.549 Tm 103 Tz (precisely? Unfortunately, it does not. Confounding between model error and ) Tj 1 0 0 1 113.049 238.299 Tm 104 Tz (observational noise prevents us identifying either of them precisely \(30\). We ) Tj 1 0 0 1 113.299 215 Tm 99 Tz (also found that how well the model error can be estimated strongly depends on ) Tj 1 0 0 1 112.799 191.7 Tm 101 Tz (the signal magnitude between observational noise and model error. Figure 5.4 ) Tj 1 0 0 1 112.799 168.7 Tm 98 Tz (plots the imperfection error in the state space with intermediate GD iterations at ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 128.9 147.799 Tm 65 Tz /OPExtFont3 7 Tf (1) Tj 1 0 0 1 132.949 147.799 Tm 90 Tz /OPExtFont3 9.5 Tf (The number of iterations set up based on the statistics of imperfection error, generally we ) Tj 1 0 0 1 112.799 136.299 Tm 92 Tz (match the standard deviation of the imperfection error with that of the model error ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 9.5 Tf 92 Tz 3 Tr 1 0 0 1 303.85 52.049 Tm 85 Tz /OPExtFont5 13 Tf (108 ) Tj ET EMC endstream endobj 586 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 587 0 obj <> stream 0 ,,9b6Z*aKXl3xs ln lv|NNtX* վi0WtLo(8(_cR,^OSb׉>/q1*qVssa!_Eލ=Up|/6\NoˬSg056o=,`K[P.MʾK'Thu!j'L!p?4+RZ]srB|?s߄iȚx n2,,_hpo:Z8MxKB`#ː}EՐ m|Lc|9 TK p`c_C9+G.wx׮.{0KV04JXӴT+sy-_s"ٵ}r&xm`( C DjQ4FjeGu$YX֊줦*e3TD%$^(2NJ*'J &tayb[#5=?z 8nP`{3bF2 2F1}wiqUm%Bӄ|(8XQVVЅг;ƖKKn"NK"q#3kz湩ۖ$rA/bx5m4))smctb4ˆɓ1ZPzT*-;1J$nvT*&m{9ģ|c!OSdQhBFV*:;u4xxP׸ pg4Bڌj(rL+?wlBñEHXTg(zWISh0Nd hqϻR@r#',"EFbC P;#Y4Ǧ\% -r~0]XX%RTJt@_Z!t1i?g hk݌ (fq+āǙDs B<>'As{|^~?.E\([ *_ * 9fCsGNXF輲;YBZ'i~5Dr>*;) bY87-w@!ѱR@-V_\Ө: ?Ƽc` p0SnA>#Z?ַʒjQlp6ڭ c@bb^2D>ԁ#昳l#+h8ڜڳ++nܜ<9׺DK j-6 7̄#W+v)d<<](xC;t]qLF31^fh(yZDm~Ū>( =v` ULЕKV ) #2+RPZZN̄l4as҅c?h]Ғ! 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X62R=:; ެHJzcٝ/pB}A|,VlƂ7#T.. u$ i~cgI#t /7hKq"Z:@0& rg3?i2zUݶ~b `͡ޠ~[;JU4j#K;RRä ̥k"M?%1%hX!UmX{}@1F68eNݱp ŤR&a-7>}.It$TI@6pBEX/UM8/v],ꂌzBbM(y$\Xx?k݌-O7 v-UuC%$T4YтD+iqu0 EO 5/u]7ԟ-@aD1.E3T[HY|`&R؅VUGˬ]Dߦ#1Sm@AR8l]pc4nGV߾|* eXΘ|śNRn3f䯂G=M̿ee;1OI] $UN? nީEv̆L~F68fl5AnϮ.z GL ~~@*_fN3ƍe( Zho]8w4bDs1,8n[?]mHd!١`M9y{W34udB3Y:wo 1ģ:}OB"q%l1 eoϫC?QZgCbBtکShSݛȱlТÎmW endstream endobj 588 0 obj <> endobj 589 0 obj [590 0 R] endobj 590 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 621 0 0 839 0 0 cm /ImagePart_2136 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 441.6 511.649 Tm 43 Tz 3 Tr /OPExtFont9 6 Tf (1 ) Tj 1 0 0 1 443.05 511.649 Tm 809 Tz (\t) Tj 1 0 0 1 456.5 511.899 Tm 68 Tz (1.2 ) Tj 1 0 0 1 462.25 511.899 Tm 634 Tz (\t) Tj 1 0 0 1 472.8 511.899 Tm 65 Tz (1.4 ) Tj 1 0 0 1 478.3 511.899 Tm 649 Tz (\t) Tj 1 0 0 1 489.1 511.899 Tm 69 Tz (1.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 69 Tz 3 Tr 1 0 0 1 140.15 449 Tm 77 Tz (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr 1 0 0 1 143.5 417.55 Tm 79 Tz (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 79 Tz 3 Tr 1 0 0 1 137.3 386.6 Tm 85 Tz (-0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 85 Tz 3 Tr 1 0 0 1 140.9 355.149 Tm 83 Tz /OPExtFont12 6 Tf (-1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr 1 0 0 1 378.5 719.7 Tm 112 Tz /OPExtFont5 13 Tf (5.2 IS methods in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr 1 0 0 1 339.85 671.95 Tm 96 Tz /OPExtFont12 6 Tf (1- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 96 Tz 3 Tr 1 0 0 1 140.65 641.25 Tm 74 Tz /OPExtFont9 6 Tf (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr 1 0 0 1 335.75 641.25 Tm (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr 1 0 0 1 274.3 637.149 Tm 51 Tz /OPExtFont12 6 Tf (11) Tj 1 0 0 1 277.199 637.149 Tm 102 Tz /OPExtFont7 6 Tf (, ) Tj 1 0 0 1 278.899 637.149 Tm 53 Tz /OPExtFont12 6 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 53 Tz 3 Tr 1 0 0 1 278.149 622.299 Tm 27 Tz (.) Tj 1 0 0 1 278.149 622.299 Tm 52 Tz /OPExtFont7 3 Tf () Tj 1 0 0 1 279.1 622.299 Tm 87 Tz /OPExtFont16 3 Tf (1) Tj 1 0 0 1 279.85 622.299 Tm 58 Tz /OPExtFont12 6 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 58 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 58 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 58 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 58 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 58 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 58 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 58 Tz 3 Tr 1 0 0 1 144.5 610.299 Tm 77 Tz /OPExtFont9 6 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr 1 0 0 1 276.699 615.799 Tm 97 Tz /OPExtFont11 7 Tf (f ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 7 Tf 97 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 7 Tf 97 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 7 Tf 97 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 7 Tf 97 Tz 3 Tr 1 0 0 1 339.85 610.299 Tm 65 Tz /OPExtFont12 6 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 65 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 65 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 65 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 65 Tz 3 Tr 1 0 0 1 165.849 599.7 Tm 148 Tz /OPExtFont3 3 Tf (':) Tj 1 0 0 1 167.05 599.7 Tm 41 Tz /OPExtFont3 9.5 Tf (?f) Tj 1 0 0 1 170.15 599.7 Tm 64 Tz /OPExtFont9 9.5 Tf (t ) Tj 1 0 0 1 171.849 599.7 Tm 100 Tz /OPExtFont9 3 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 3 Tr 1 0 0 1 185.05 594.7 Tm 44 Tz /OPExtFont3 3 Tf (.) Tj 1 0 0 1 185.5 594.7 Tm 52 Tz /OPExtFont3 9 Tf (2 ) Tj 1 0 0 1 190.3 594.7 Tm 444 Tz (\t) Tj 1 0 0 1 203.05 594.7 Tm 144 Tz /OPExtFont9 3 Tf (. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 144 Tz 3 Tr 1 0 0 1 138 579.1 Tm 83 Tz /OPExtFont9 6 Tf (-0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr 1 0 0 1 333.35 579.1 Tm (-0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr 1 0 0 1 141.349 548.1 Tm /OPExtFont12 6 Tf (-1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 6 Tf 83 Tz 3 Tr 1 0 0 1 137.75 516.899 Tm 86 Tz /OPExtFont9 6 Tf (-1.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr 1 0 0 1 164.15 516.899 Tm 75 Tz /OPExtFont9 3 Tf (' ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 75 Tz 3 Tr 1 0 0 1 163.199 511.899 Tm 71 Tz /OPExtFont9 6 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 71 Tz 3 Tr 1 0 0 1 180.5 516.899 Tm 1624 Tz /OPExtFont9 3 Tf (" ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 1624 Tz 3 Tr 1 0 0 1 177.599 511.899 Tm 77 Tz /OPExtFont9 6 Tf (0.2 ) Tj 1 0 0 1 184.099 511.899 Tm 619 Tz (\t) Tj 1 0 0 1 194.4 511.649 Tm 71 Tz (0.4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 71 Tz 3 Tr 1 0 0 1 213.349 516.899 Tm 120 Tz /OPExtFont9 3 Tf (' ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 120 Tz 3 Tr 1 0 0 1 210.699 511.649 Tm 77 Tz /OPExtFont9 6 Tf (0.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 77 Tz 3 Tr 1 0 0 1 230.15 516.899 Tm 75 Tz /OPExtFont9 3 Tf (' ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 75 Tz 3 Tr 1 0 0 1 227.3 511.649 Tm 74 Tz /OPExtFont9 6 Tf (0.8 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 74 Tz 3 Tr 1 0 0 1 246.25 511.899 Tm 43 Tz (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 43 Tz 3 Tr 1 0 0 1 261.1 511.899 Tm 69 Tz (1.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 69 Tz 3 Tr 1 0 0 1 277.449 511.649 Tm 68 Tz (1.4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 68 Tz 3 Tr 1 0 0 1 294.25 511.649 Tm 65 Tz (1.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 65 Tz 3 Tr 1 0 0 1 333.1 517.149 Tm 86 Tz (-1.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 86 Tz 3 Tr 1 0 0 1 337.699 511.649 Tm 83 Tz (-0 2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 83 Tz 3 Tr 1 0 0 1 142.099 511.899 Tm 85 Tz (-0.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6 Tf 85 Tz 3 Tr 1 0 0 1 118.799 280.75 Tm 99 Tz /OPExtFont5 13 Tf (Figure 5.3: Imperfection error during the Gradient Descent runs for Ikeda Map ) Tj 1 0 0 1 118.799 267.1 Tm 103 Tz (case is plotted in the state space. \(a\) after 10 GD iterations, \(b\) after 100 GD ) Tj 1 0 0 1 118.549 252.899 Tm 100 Tz (iterations, \(c\) after 400 GD iterations \(d\) the real model error in the state space ) Tj 1 0 0 1 118.549 239 Tm 93 Tz (for comparison. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 118.799 205.399 Tm 97 Tz (another two different noise levels. When the observational noise is much smaller ) Tj 1 0 0 1 118.549 182.1 Tm 98 Tz (than the model error, the model error can be well estimated by the imperfection ) Tj 1 0 0 1 118.549 159.1 Tm 102 Tz (error. When the observational noise is much bigger than the model error, the ) Tj 1 0 0 1 118.549 135.799 Tm 96 Tz (imperfection error looks very close to random. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 96 Tz 3 Tr 1 0 0 1 309.6 51.799 Tm 86 Tz (109 ) Tj ET EMC endstream endobj 591 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 592 0 obj <> stream 0  ,,#@,,j`j`q+.9e1 vjL T&k\-t&K݈VHBS9!]5xE-ÌE&U8+yy|$.nGl_!{Oa;,pb؜$ Rqo KԾEuM#k9Xν5\LVLA(D19X CJ7rx;Dv^ { Eiu=8*cZ[QBE{'XC 8r0]αqNd^L^akڼ_G) Icm}|wp/qa&YRu&eQ~tދG[ck-; X=Ɓyd#k#pw4,heFКߑtKڒ/e_䄇Ddeύ叮䲢4?vvK~4BZlX|̯ko|1F&ZiWHy G|~&! @ξ,tMgB_!O :<^ B߻>F*BfTANl[_zbFKg \:.nk5r!+%_] 9ﻹ8CIdE}eh,G[Թa?dUzg}a &δ´(*JT/E=e٧{\jNL/Z[&i L/sGPwYD8;L7AIҭ߾삽*LeE6ЛX(ajLLGraؙ!`<, 5M\&c,V.B=P da +DfL;|ꓘjJ7!t_1eH?䕒Xi`,KHxaa9US5KOB%\ֱz紋mo|-YFEMTh􁌱ɝ嚶Uòp}瓳.' ^Y?ZssK7ꋧLcI3t3< {&άcr;źdA\!/h9ݍ ٞd:)gU<XJ)K$8{<45ni$HFuh.՜cgVWy ȨVpf!Ʉ;ƥ7+ ko}$y1Eq04.{}F݄P>N:Yh 4߭~tme"sh}Ź"7rimוvTXZc~dثH?D.]|,ydk@ _J'K4|kh}Qՠ2|qoJ6"at6 fenUE,5c#SM*')`NHOV;O~6};Zy@;X;?f.7uJ1To|ҙ K{:*< Ƿ BTդ!WyV4v<> ; mOhqo/d/.ߡ\8\rNcB 29fya`Yx E4# ]0<*s'#fTz ^V OVS@L*yZ+^Dfrh]RSV!SɯNWj{\9w@ݤa6sW1(BCgd˩CZAWL@5WLjlO&,:װŋB:V%_۔TE(.Mϙ! ?1njTxe %q[&p.;^HKf "sùƙB\K)RNjg qUO+j| D \ o "aJYhbo dTdvf;o=<j32X*n܉.gVɠpJ$|I:~}f[}86\q&juNId6Wi~ZG Rca{|O>^ThWႰm3맠b"^F (;ݾ6|M d#KPKpqU. ju@xeaըQ~4*oJ$A,AB]@TL_MZPPCB71QWsfii &=9$Yxd \иc\V'#K5|# 4a7g]Y7Y(eyn`}rq`Xر} :24EpF/1!,2~|j52upCv=YnE*EsR.tyZ%fR1dm﷕ /,yޅ⾙˕|Vm@3l5t Rf!-7xiPʕ jGu({ܫp$2@+f"Zބ, kKUrJ!,1VZC |^q}r!2KLSSP֫=IfޢKeGԮKi9h{N 3z@:y#wd+r+ LwV[Ե!+-P+L{'FrLapG~ ]tRJq00W v[365ќK黹D~U=jUd9|b _@?; ӐΠ "CD;Y1e~7M@>iTLb"?J4vUFy&t`o/ăuѴHj&;J (2QX09+w.])M6 `p}X`G؈uuiucVE"pGn snI$ R]ȏ^-5cn -s+ ڊecTY1wwJ)&&8tqKxxR՚3R ]V>V( ݣJC;Tێ̴sOsล&UA߰˯.8lWz80Ӷ bؤ/#{fUt9t/P"~a(N"*{_Ou)4q(GFX/3ff(rlQ_n"_9k dt1© EK4]Jl]`iq{Y42Jǩ /"%b~y7I4[ʎG We]D_ج^5,AT)z谦CHZCp kL {7I O,HjA_Q\ÿYhۍs K+q w00>Oٿk><=S5F [qZ& ŅXȤ:ʂT=rV*n,υ%_H-%/P)CLϡlFJ5a@_;*T;{f>cQW@z12 Z>Ѳǻ<_*!{碮60Þ&5˩@|t 'pa+AF\:Xl1=]Qmc7y%O(Wbʙ In"cmWg!Y7RpNC_}3_w61%5Ad f(M6:]s;„_䪋3Sas#JfgdY@, ZNGz{. 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B;*oֳ] ?#>6mf6?ᏰQ-o4ܟ_ C6V,,;usBP\x{.{Πl9!] J\! 80/BIyQbw}l_2=q_gX3H|Tb7Z_BXbze@++w&s$87jdb)Ċ`@5O)Oơ1?h e;Ef%bBz*;Фj=K~DKV ro@ ް4%|k&İ|-s4SŊ?{m~3SThx֢ TE40U73`,|4=߅;'HLUޡK@%HϳueiN +x;^?ν~7+Y"%ͦXJS_^IR >Voʄɡ ]sSJ eu4973WAfJ]KE^‚PfwCÅX:aJl/ }?av1Cو:ic1tᴠB*-jo3?wڄNusK-'ˡnqm`4!JF5͈IPs7|q-o endstream endobj 593 0 obj <> endobj 594 0 obj [595 0 R] endobj 595 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 619 0 0 840 0 0 cm /ImagePart_2137 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 140.4 642.25 Tm 91 Tz 3 Tr /OPExtFont10 6 Tf (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 6 Tf 91 Tz 3 Tr 1 0 0 1 144 611.299 Tm 96 Tz (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 6 Tf 96 Tz 3 Tr 1 0 0 1 343.449 512.649 Tm 87 Tz (2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 6 Tf 87 Tz 3 Tr 1 0 0 1 332.649 517.899 Tm 102 Tz (-1.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 6 Tf 102 Tz 3 Tr 1 0 0 1 337.199 512.649 Tm 109 Tz (-0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 6 Tf 109 Tz 3 Tr 1 0 0 1 358.1 512.649 Tm 87 Tz (0 ) Tj 1 0 0 1 360.5 512.649 Tm 877 Tz (\t) Tj 1 0 0 1 372.5 512.649 Tm 117 Tz (02 ) Tj 1 0 0 1 378.949 512.649 Tm 738 Tz (\t) Tj 1 0 0 1 389.05 512.649 Tm 94 Tz (0.4 ) Tj 1 0 0 1 395.5 512.649 Tm 738 Tz (\t) Tj 1 0 0 1 405.6 512.899 Tm 95 Tz (0.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 6 Tf 95 Tz 3 Tr 1 0 0 1 422.149 512.649 Tm 91 Tz (0.8 ) Tj 1 0 0 1 428.399 512.649 Tm 1980 Tz (\t) Tj 1 0 0 1 455.5 512.649 Tm 84 Tz (1.2 ) Tj 1 0 0 1 461.3 512.649 Tm 771 Tz (\t) Tj 1 0 0 1 471.85 512.649 Tm 87 Tz (1.4 ) Tj 1 0 0 1 477.85 512.649 Tm 771 Tz (\t) Tj 1 0 0 1 488.399 512.899 Tm 84 Tz (1.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 6 Tf 84 Tz 3 Tr 1 0 0 1 335.05 642 Tm 94 Tz (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 6 Tf 94 Tz 3 Tr 1 0 0 1 339.1 611.299 Tm 80 Tz /OPExtFont12 5.5 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 5.5 Tf 80 Tz 3 Tr 1 0 0 1 137.75 580.1 Tm 105 Tz /OPExtFont10 6 Tf (-0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 6 Tf 105 Tz 3 Tr 1 0 0 1 377.5 720.7 Tm 112 Tz /OPExtFont5 13 Tf (5.2 IS methods in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr 1 0 0 1 119.299 475.199 Tm 95 Tz (Figure 5.4: Imperfection errors after intermediate Gradient Descent runs for Ikeda ) Tj 1 0 0 1 119.299 461.05 Tm 97 Tz (system-model pair are plotted in the state space. \(a\) Noise level=0.002, \(b\) Noise ) Tj 1 0 0 1 119.299 447.1 Tm 94 Tz (level=0.05. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 94 Tz 3 Tr 1 0 0 1 136.3 415.899 Tm 97 Tz (Generally we conclude from the above experiments that the ISGD minimisa-) Tj 1 0 0 1 118.799 392.899 Tm 99 Tz (tion with intermediate runs produces "better" pseudo-orbits than the minimisa-) Tj 1 0 0 1 118.799 369.85 Tm 98 Tz (tion with both short runs and long runs and the IS Adjusted method \(50\). When ) Tj 1 0 0 1 118.799 346.55 Tm 95 Tz (shall we stop the ISGD minimisation in order to obtain the relevant pseudo-orbit? ) Tj 1 0 0 1 119.299 323.299 Tm 99 Tz (Certain criteria need to be defined in advance to decide when to stop. Such cri-) Tj 1 0 0 1 118.549 300 Tm 102 Tz (teria have to be defined based on the meaning of "better" \(pseudo-orbit\). For ) Tj 1 0 0 1 118.549 276.95 Tm 101 Tz (example if "better" means the pseudo-orbit is more consistent with the obser-) Tj 1 0 0 1 118.549 253.899 Tm 103 Tz (vations, the stopping criteria can be built by testing the consistency between ) Tj 1 0 0 1 118.549 230.649 Tm 97 Tz (implied noise and the noise model; if "better" means the initial condition ensem-) Tj 1 0 0 1 118.549 207.35 Tm 98 Tz (ble, formed based on the pseudo-orbit, produces "better" forecast at certain lead ) Tj 1 0 0 1 118.299 184.1 Tm 101 Tz (time, the stopping criteria can be built by fitting the number of iterations with ) Tj 1 0 0 1 118.099 161.049 Tm 98 Tz (the forecast performance. We call the ISGD method with certain stopping crite-) Tj 1 0 0 1 118.099 137.5 Tm (ria to be ) Tj 1 0 0 1 162.699 137.5 Tm /OPExtFont4 12 Tf (ISCDc. ) Tj 1 0 0 1 206.9 137.5 Tm 95 Tz /OPExtFont5 13 Tf (In the experiments whose results shown in section 5.5, we stops ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 309.35 52.549 Tm 85 Tz (110 ) Tj ET EMC endstream endobj 596 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 597 0 obj <> stream 0  ,,Дb7 \Jńo@s"\$<ߌ ,:ERGo~t {%cS/-vU3@}1}xOO^"3h#pNF|1:XQ1Ue#5Ё(?p5WGbljAdw+E#)~,60 1"!sJ~b)61Rh0 Fs{~1 [Ұ}ndDu8t.r͓ȿlK/]~KSk4R+ Ю;PUnYN@ 1\!rl~Ŵi39^z!+6μm֜sön?:m5: :pLtkJ4(d ADmM |FLu6QD Xl}_dzҎTWlČ0om(jMiP|;[~kL|BŚ*ߔ|с%Kaq`naN,*Ԯ!kT~C8z"3 8+;FVOjm$M~-M;(O;N?ׂ_+H^j{ZHU6:KBo}L $.Sk[u5AL8g YBbv֖l:nymt.Dupr%Q2M˳8ѕ}軞63򼮃^7%8@ӏqVSTJzMm/:XTAJp{N<u[qR3AP%h!ƴsi~n" L\i5%{I5űa冬w8? ;}}YvƙT-םtvEt?[&Jmh؏dZYTt A^)]1dUK KeLJߚ\ Dd` H!Lv&Zʖ@#6Yzq֒_)t{ŏw cq>1֝>㘯9=`qteXn!Z3V}'nXD;7rK=*oykWAt>Õ/fd13rDJ{`\۽>5ʁ5-'}ϨbU1qvp(qXٓی9RD1OCb x%(eS MBr([q6{5vRga%xrƭ~fhD ӰE"!>E T?aV%s6Fg_Rp9⧑Z90йyw6含[:0.җ9Mh-흾Q2x+v`ƣ6qYPXq}Xi o'','&j !uԻ3`v)IE02@܅:c7I0QRynpA2ɹ6Ŋ#Q'dp4\gs EÇnQKb]2l{8鄨D1G+XF;)K}#ekR@bg+t&yjכ0=xmy[<~T~ d)3*ъ/{xyAx+f!4y;hF&ҤCV^J>Kc4;B'{$86yaY9Y`x>FJ$az/=(#g Uqr8鬉f(\0)֙8!лg%r 3NPo7vL>h.׼?\LtrP8m YwB0^-''gex㠜N =5;M-\ X &Lc&y3@.d"x6ٮӲ( Cߜ*ty6FsƇID3YE:h+:Нpc4]?~I2o|+wh|} QU&sm%chxu?ęT Gi;eQd!l_|Om: +wh.*mEҝ00m^ZGQD*M'6xJ>W"-@/J]-!Q &-w-kraRZ:n'@0>vE2:#ry=۰;є"Ojr!>7 0Œ.UlWۗAgp4ӢI%S9n(WJqgh!7}"#F '+`t(:#\CQ~uHpNɄ z )Dw{$^c{+86N0>9:c5 ɜ=f.>{A aF{KU PPs*I,=f'@+ڳo%2MT`JTr-eFI_v޼J"R(? | 7ۏ5 |wB72H't`L螚_p[wC n$"thB^S'zf- ?S=u$ ,5%΁;k"~ѮۑzN/`P6ٷ7M \pm9/RwR ,vQ;{h0Cъ#b+v0e6mI֬e / NFU EwWYZ]?PtmB!,z 9#=RCCT=g|MP; |SU='(J!ކ *t]p0)#h:\(iͶybJl+^W$0 ^3Xkثw{fd ( ? 楷*_$dS_5l ;0MU7;RW-ىV}<Ӥ&ɫ\a"N&oR>*wu:um׌C`[eI^{j䈂SVe$ +@9~w 6$lMX 9 Ю S& a yD-^-No tD?iCl|5}.fM)?G]oE[rL_snTTмtm\tRr"AJr^eֳ`8^Gs+󛌌^!%$y~SIUX-+c4-4X@{VX\q=KF~_gk_bDDCX|EΕx6zw>B]0`8)Չڰڰ3?N&/z9N< A#Du !Co& s֚&6StmwE'blUGPwuל$@^dB$?w-VmeL`YifψшeRf6Le&%Ab뮫a҉|f;@r*]S?N~@nu/C[aQ5M[ U .-E5t rdQ;)dw IHKk{41j9eCCTRMIJi T8 Ӄ"f}.).i8F /i0&p4 o;fyf M.{$ԟZlR^|Dj68|uH/l > hbW))1`u|lT娷2#9YX:=G92dzxrߩ_p)z 1L ` 3)7U{jbM`dMuFfRj j*缞GQH0Ox 4͐D}Aqږ_ {]r0, u+d"Ckv4&a<-G]c՛f NRnkΨgyz"\\:އdkcz$Z=iJVY7fnjî#y$YΚ] ׌x&inmIy^'dL֨?sB6S2z7M[m(A2YBޏ3G(E'33ç!An7Mg9N#eۻ}z#AG큤L WpcQmΰOtU DGLX,ʘ/o9atŽDy%mBhr!|SϹdPw:">L!m.%# PY_-qxlRalR&BF!dLz'xvhWzA~þ8Vp};*$+x7[ hS-&Z5;-ϊkL>i5} h9`)gŜ9])6BdMuqdYW۹d?_(PYJ4 ˬUÑDtW}<ޔO yOl!6賫)} Ze>&m0$HC˰33I;ٗ(@)4O{yQI;8 lxT# ySTGD/:7rRDnK_' ΔП#u^6"9RezG=qFtY~ 5ٞsbW]7Xȇw˞gy Ք)xyi"/ nA<^X̚%J9SPUHo?%DҼg=SKv Kc_\ s:FSCVr&rWM@hF2rfn/j@E9rD@VEcJ9 NP0b'E?{Xz滓%e(:p=8IRb^GbI'KM1R'(x:!HSE)I,wu$7/CN.;DTDW`:q #"_Ճq<Я P& MUuW'фN*Ԫ θ'F̛O"^>C@#]f!}[j ΃I.3a;K" ntˁ{K 4 sqA]߆%'ؔ ovk5~hxMp+iW%/^]aa8RZL:r39ZCgl_v^aGEiSYܭAΕ&_Z@ԘfLRV#A;n U^bHRb-k?| ^xl P`jNOqٜ)\4^]3@ W:l& QAFcއ'E2\5^)/WVqP@Jj 㣗DhM r7 VUO2{ e>M/֍n\{w 7lg4%uvGl!͒d*[_zJJQ ƣ:4MEh4ex}dC8foxhSVOӏ]؃Y;"6ҳXw d@ DRPR8]Q%XT6!X-nxgN I}~pR󧌡)'>䋗§˫:.Sf)֫]ܲD+~ȍMH΂X>exw6M n@E3 =ƈ$^QӘL A _'O8E'QVuK%X&+˺r 9BQN`4d SRR ?[&ť9ʕݎBY V09YBvoy9-$znZ#TtH9 qrllk ~# Qa;UN\2Q9wc}S"h6wfgg*kBeH0)5捷X*WBGm2(cd٪}5>Xށؖ2‘C[zg–߉9~E;JLQZsMEvɗ>4dX~}8d{q`ETeIuѣWUbƚ3]C).e. h6XY# R@Wt̰`8Bu3 :{E $^*iIFX{O@ʕ+#dD]=qCOX+9.j42YAD{N)} sz!%ڑpZJjIÄQwqB K'^&( rzݚbyV@hYxY/nҢ#=qd]_ō#lb>IS4?Ƅ?`N2;$C%YlHXo;6 oe.Yק \h pW aO$nsS/>#|LPLlK%~X-^Pφli`MxY|jO@:YdLX@LQ51o,c~?*՝[Y~_I\N /[ؗdTI節llU\RENp00*l)wcR\J.&GN+:d9/Rsc4q؄WVaΌdLФW<Vj>Mw'0DK lIkS~0[M$p2P&`ٌ_URfA–*/8,p ;²'TLoИ+H7h^bߍ ><|e#,7< LEbmsF;ƛ x_hORĵ֤LŠXMR["YHO;{\d XW%ks/³<#u7M°'_Do>dB*DE}-F[Qe˗cj,=tUSEѳgXRjnQ[4[В]sF@,":旡ccjy lj+3ʊ!l \6҇I=yHDc{P:LL4eJQH,jI;3:drQhQX g~ ^q1#:CmhVsXK55AꓳӖ a}p, f:*m s :va-B S' t]h_$G@]+qaӍv]\ku;Bv)pT%I0fݾ,gzPpψ9t̷P٦W/i޹ϼbXѻr"T1>=0 =Bt|1cn $wTŦ/5Hmn`f`39)_r>: `Kg2E71v.;>Ϲ&x62ĖVʄ'x=gڈ<6y2`PgJ.1%!9W\=,oYӭcU1sbr^t#,htAa":"V]۵YkC¿9@# D!+4kX܀bl_]ݒ[JtŨp%Dq )bSBWfsif wVzgIJ~dL@gv1þb(ʪP@6L& 9VA*7r)0ptx몦ʚ(mUe7@g\N? 3z&B)-& 3e X'jq8}ʊBVLX& 3)Bƙy^DΦG*k ur>fcه0SkhzSڢSIs997d( {&.dd鐈`$͙p =|(2qp CxYDJi2zЦMTjrqYLܣ=hYwq&"NT(s |;::Q lEqQV V?DZ-$Slty'rݖIDx5b4;!H_. ǣ-Se⃞?Xv 4fj?5Ԏ+Cᡲ{-*G:E@ SOܛ{7 6!|Ќnd yHWXbp](PKL,T$)X=dqC8I CI9r~@Hi`E !<u$@jh&ih(8K!93}*gMN畢ry<h>,@II+>M(7znZdXJ_]:նw hu=)/J1oZ@iۘMJ3}(1 ?h0`dˎ -(ą? iL5OR4rgJ}H:@I&B^4AH3peK_Tlǎp6F<::)7Nzp@ $pq6)XVe9#,m`!3饃E0*K4g=9iik-mwpYvyWÖޮk@.i!IxؐVTmM؁$Ρ"&U hꂂ6xTtM ꄘ9*(=}~3!Tw`!׺W8YB#|R4O4;;Xalr[vpBة()8ě/) ֕%6x ۄ57(庪ŷZc^j)Tv3hth$5mjvhvO {b͜o*|ԿGqXiꓥl.hpr}&=5-"Wj6z]?y<> endobj 599 0 obj [600 0 R] endobj 600 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 619 0 0 839 0 0 cm /ImagePart_2138 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 305.75 720.45 Tm 115 Tz 3 Tr /OPExtFont5 12.5 Tf (5.3 Weak constraint 4DVAR Method ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 115 Tz 3 Tr 1 0 0 1 119.299 677.5 Tm 94 Tz /OPExtFont3 11 Tf (the ISGD iterations when the implied noise becomes larger than the standard ) Tj 1 0 0 1 119.75 654.7 Tm 92 Tz (deviation of the noise model. There are many potential criteria that can be used ) Tj 1 0 0 1 119.5 631.899 Tm 91 Tz (for stopping, here we use a simple one which no doubt could be improved upon. ) Tj 1 0 0 1 119.5 608.85 Tm 95 Tz (Most importantly, in this chapter we demonstrate that using certain stopping ) Tj 1 0 0 1 119.299 585.799 Tm 90 Tz (criteria can provide more consistent state estimation results. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 120 533.25 Tm 110 Tz /OPExtFont3 16 Tf (5.3 Weak constraint 4DVAR Method ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 16 Tf 110 Tz 3 Tr 1 0 0 1 119.299 498.899 Tm 94 Tz /OPExtFont3 11 Tf (In the traditional 4DVAR method \(see section 3.4\), the model is assumed to be ) Tj 1 0 0 1 119.299 475.649 Tm (perfect and the model dynamics is treated as a strong constraint \(90\), i.e. only ) Tj 1 0 0 1 119.299 452.6 Tm (model trajectories are considered. In the Imperfect model scenario, in order to ) Tj 1 0 0 1 119.299 429.55 Tm 93 Tz (account for the model error, one should apply the model as a weak constraint, ) Tj 1 0 0 1 119.049 406.3 Tm 90 Tz (rather than as a strong constraint in the 4DVAR method \(76\). Recent research \(4; ) Tj 1 0 0 1 119.049 383.25 Tm 95 Tz (5\) shows that applying the model 'dynamics as a weak constraint in a 4DVAR ) Tj 1 0 0 1 118.799 360.199 Tm 91 Tz (data assimilation method outperforms the one with strong constraint. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 136.099 336.899 Tm 96 Tz (Here we give a brief introduction of Weak Constraint 4DVAR \(WC4DVAR\) ) Tj 1 0 0 1 118.549 313.399 Tm (method. Differences between WC4DVAR method and ) Tj 1 0 0 1 396.699 313.399 Tm 98 Tz /OPExtFont4 12 Tf (ISCDe ) Tj 1 0 0 1 437.05 313.399 Tm 92 Tz /OPExtFont3 11 Tf (method are dis-) Tj 1 0 0 1 118.799 290.35 Tm 95 Tz (cussed and comparisons are made in both low dimensional model and higher ) Tj 1 0 0 1 118.549 267.299 Tm 90 Tz (dimensional model experiments. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 119.049 222.2 Tm 125 Tz /OPExtFont5 15 Tf (5.3.1 Methodology ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 15 Tf 125 Tz 3 Tr 1 0 0 1 118.549 191.5 Tm 92 Tz /OPExtFont3 11 Tf (The weak constraint 4DVAR method looks for pseudo-orbits instead of trajecto-) Tj 1 0 0 1 118.299 167.95 Tm (ries that are consistent with sequence of system observations. Following \(Lorene ) Tj 1 0 0 1 119.049 144.899 Tm (1986\), the weak constraint 4DVAR method can be derived as follow. Given a se-) Tj 1 0 0 1 118.099 121.899 Tm (quence of observations within a time interval \(0, ) Tj 1 0 0 1 360.25 121.649 Tm 113 Tz /OPExtFont6 11.5 Tf (N\), ) Tj 1 0 0 1 381.35 121.649 Tm 76 Tz /OPExtFont3 11 Tf (s) Tj 1 0 0 1 386.399 121.649 Tm 62 Tz (o) Tj 1 0 0 1 392.149 121.649 Tm 75 Tz (, ..., ) Tj 1 0 0 1 411.1 121.649 Tm 90 Tz /OPExtFont5 12.5 Tf (s) Tj 1 0 0 1 416.149 121.649 Tm 75 Tz /OPExtFont3 12.5 Tf (N ) Tj 1 0 0 1 423.1 121.649 Tm 91 Tz /OPExtFont3 11 Tf ( and a background ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 309.6 52.049 Tm 86 Tz /OPExtFont5 12.5 Tf (111 ) Tj ET EMC endstream endobj 601 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 602 0 obj <> stream 0  ,,zzb7 p JA+Ҏh<=` ) 2GxӨQp|xq(l..MViJF Y %ϟ:⺈x\ZT W#`>\(#5]JVK[a"^$(O C#U?[}x6a0m8Di7̧aL~5#uDlv*E)'b(h"kՇ[vtaF.w\"~jbSp';;K*5)^]ѣ`G1r75GlI:LODp9v^PhΑ[2_s1yWJzg\Qp 0 k-677cI:3.ha%[N.@dqVi u;jkD:ů=tte$UUӳ}A]AQz mr`&'Y*a޻ SR]Hac$G3AsrsjfA\;z lBJJKiry/-W}qf+_?4 U̩f| ͎[UѡFDi=1o HuruS陾VJY `ފVIA%\A0XBz@穧j%+'on'ButB57|5 rs)EmJ3uϗR*t}:~ABy ac/o#ןB+ I ZhD6 G=6DBY=1"2]Iʅ@j1ga{6y $!o)=lseQqf]pʇbzͻg?cy{aL#?B3շkrkg1Y;c-w{mO  `3.-P]gqXD]e=EI rc֎ )-\nܰnt.ʉXojcs Balk%j4\HuN,6؃ҽ9DvKteF^b[ɞc$ŴYo\̍)"q̅L[fjRlCv,߀d|#?ۤ@٪0<B08fds; $w卟VZ%W{"Q7ױivQRH3;V^po49ӭyPv._yg6Li.9}I1[3|=%h6?5- \̓nI-Z{hJ9hAvH*{K=(N^oxxBa)zq\.W-UD|[7۶kzzMNIIJD@y?Ed?gM pO^b`aXEO[2ZOX4z@)k5ܗ?3 1 {)>67{Tª w)"A@!rBʾ|XwZec lITS?i),2kIqi"㿗TotpO&[Z7/XqfѩA h7 kںK ݂r$⥑Pb/mLCU?>Ӓ=~Aܥo٧*f\ kD0CHUJ-sB'<Y tx\rEYGA9Yښa= 8,Dg joBMU|dHd҈Lf %C[-j$YOV/!{>Ů'R1E3s9@6+l*Xݍ&Ҡ\+$WM.lU vf_tYE`ގX;\gy8~ Kl%Q߁;`%5) R+:gfs ѼQ;8xK@>䝢I|l4.ASUzDV p-eCQVWE:JvVj4ƥ 6czB Vv@!l NٟMA[\+>o p_y\mY̘ͪ}Oz+\! KpZGLf'SH15JN Н"E24azZ}puIQ"/ >.u<М^w=qSܾ ?:QWlv+|Q @}lՏ=*Z'tP6_paD;XP*0~<ұ)cz K^Cx|1Џ>,.$櫼gjְ܁6h֝,Y7(ΆǶ尪ɰǒ45չwW,+°"]l([L@vB/Z@}q*_ l7vS %7x!i fedhŎG'˵Z@pWŔ &;ɲfjC;'?tY|Bz^$sjMC/D&EĦ}SSR=;PaZ>_7czN"A#yRAW$.Gd"Ɛ^gΔIi1^!瞏" iQ$gp8map y']Ops0kFY@ UNyKs6UàQѡl eS/-_v,+JBo4+ũ2~h?B18ƋuF3R^um o C!bЙo pg!:),|Ki%nemσnޞ''yhN%&2;4N"659ܳ'%lMVϗ>'Ҷ+35V7/k,ɸ uJs7]-xRP9%QdwH Ep g{' ]{hN7K:Z'_0QM(3~E-pY+~heb9Qx#Go^1pVjM:s.) u8,V9h;KPPnyY7Ӥ!hn8FȪ:"?5~.$ڸ[|(2]Xo xo<6ٛܳWϧ=ֱWq%4NڲU$jU,ǴQ3%:%+[ԫ>0y鴳⨶IJäϼt*طl<ɈH* ,=i@=Y""  {/(e‘ QإTdN=/{ހ œ:P4[鍉R,ݯBY9`Gl˨&[#AH@{C9|ެ49;f * vmeF2qq^$O?*-y)ds&ȦpSh1EeWG>nJ&pW\MY}S_ݱvug!y0 Ƅ4ݲ'R=.HfvOYg_^+ĺgK ġ~ O#\211Y360||uYȁ *t0A4<&pPEah*G*,3=A#/[!$xh{w*Wr SrB;SQH柀x>{@kGq^$8~c,"샷: -_8҄i_iwp!^R2&HEH_R  T  /ٽk& Ck`(XE>P ӕpKly%MA4EåV94SXuC[Ų_0[(tw% W-фgn^dxQ()OF~RGtn̨xѠD !Ti$\H=]_(NlČo%I]wʃ%=bS:]7?E 9%Z)cވ?j+vA%ɂ۞r,UMR/m4tZdAxGxx4u<<+ԟ]1iC'& Jv3SDPռN>#!DRʠb3"M6+r)`g+ !xZV;ΔB}ٿ>je~մsWi UJJܕ::1V<* me8iPVO{@g=! YL+|"5]. { O<KKtyPzXafc+F1W|6nySjXdS[ ZC0x=&D^C]Bz 0 W6 ,I7& cED 8Zhf04S2QI>kkKu$RY0r -d>=/òٔǂW)'oUfXnDL:jwhoCPRa )zlBl߿Q38ƍه6)P usmRNXSΰR묻ecΠQSh|3}b .eIlQUϖ 4SMKm+7]^5Gէ$y=qiۧ._u,;lNG*6Jz晶`Ьy+/^Pk0^^[)WadϢ_=h5sP+bR`>h t(*z -ˢ@c`=R 3b^YLnK[+Ֆ@RlD+cۭf0P5'W)8urI{*>m+z0ͬ3sIū䐫=\Q &cyzkj 㑺V9pD p)y0Z-6B8R -eZ.^9[%mFM 6"٣/,Q_w{=3k{G`!ْ uBdX3XY@`j&ٕ" k.Bof@]O^\ly"Ž%4ߥ;hle>@uMAccI("kZ E.P$gV;4$@HN3z(b n,>ǯl*KVo!U-wbMoOEEuJ/qƮBEg{``jWr 5)MAG)l s&*s} o "WE>I誛ZXvht.,,5zmar $fLD7n H3 l)HS)nPD-k1߯@s$g #a NĉQwߩmBhCY K@XtV~`3 UYEBc +> :DʛW&[xXѝCMu%}N gShi{It_q2ћwۡYPmq %]*s* p ZCBHtx\$dF$Nevh$Y橳Rp1KT"z vۋJ͍_%E{Т0cШeGE##?xEkl $q 1l,kP<3h6F@jxgUlhc9)1rȞ“%gdͬWsGz-HFWg eŸ&@qh9 /W_APСI4 D"&iV#OȾO/m@'$pl 9=BwFp䴓@s]d MWɐ 'n52AB#+g. !\ WSDtC,5RGAjv~?UZc<(~kp펷~f hRk=o:&Dw#og%R endstream endobj 603 0 obj <> endobj 604 0 obj [605 0 R] endobj 605 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 619 0 0 839 0 0 cm /ImagePart_2139 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 315.85 300.899 Tm 128 Tz 3 Tr /OPExtFont3 7.5 Tf (\(Xi ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 7.5 Tf 128 Tz 3 Tr 1 0 0 1 290.149 309.1 Tm 60 Tz /OPExtFont3 11 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 60 Tz 3 Tr 1 0 0 1 279.85 300.2 Tm 141 Tz (+-) Tj 1 0 0 1 289.899 293.25 Tm 70 Tz (2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 70 Tz 3 Tr 1 0 0 1 306.5 720.45 Tm 102 Tz /OPExtFont3 11.5 Tf (5.3 Weak constraint 4DVAR Method ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11.5 Tf 102 Tz 3 Tr 1 0 0 1 120.5 677.5 Tm 89 Tz /OPExtFont3 11 Tf (state ) Tj 1 0 0 1 148.8 677.5 Tm 61 Tz /OPExtFont3 12 Tf (x1) Tj 1 0 0 1 155.75 677.5 Tm 45 Tz (0) Tj 1 0 0 1 157.449 677.5 Tm 37 Tz (, ) Tj 1 0 0 1 158.9 677.5 Tm 102 Tz /OPExtFont3 11 Tf ( at time ) Tj 1 0 0 1 205.699 677.5 Tm /OPExtFont4 13 Tf (t = ) Tj 1 0 0 1 227.75 677.5 Tm 95 Tz /OPExtFont3 11 Tf (0, we want to produce the optimal estimate of the model ) Tj 1 0 0 1 120.5 654.45 Tm 91 Tz (states x) Tj 1 0 0 1 159.099 654.2 Tm 52 Tz (0) Tj 1 0 0 1 164.65 654.45 Tm 91 Tz (, ..., xN. Assuming the observational noise and the model error are both ) Tj 1 0 0 1 120.5 631.649 Tm 94 Tz /OPExtFont3 11.5 Tf (IID ) Tj 1 0 0 1 142.55 631.649 Tm /OPExtFont3 11 Tf (Gaussian distributed. Follow the maximum likelihood principle, the prob-) Tj 1 0 0 1 120.7 608.6 Tm 102 Tz (ability of x) Tj 1 0 0 1 178.55 608.6 Tm 54 Tz (o) Tj 1 0 0 1 184.099 608.6 Tm 191 Tz (, x) Tj 1 0 0 1 209.5 608.6 Tm 86 Tz (N ) Tj 1 0 0 1 216.5 608.6 Tm 102 Tz ( given xo and s) Tj 1 0 0 1 299.75 608.6 Tm 54 Tz (o) Tj 1 0 0 1 304.8 608.6 Tm 163 Tz (, S) Tj 1 0 0 1 328.8 608.6 Tm 82 Tz (N) Tj 1 0 0 1 337.449 608.6 Tm 114 Tz (, i.e. p\(x) Tj 1 0 0 1 386.649 608.6 Tm 62 Tz (o) Tj 1 0 0 1 392.149 608.6 Tm 191 Tz (, x) Tj 1 0 0 1 417.6 608.6 Tm 82 Tz (N ) Tj 1 0 0 1 424.3 608.6 Tm 122 Tz ( I ) Tj 1 0 0 1 440.399 608.6 Tm 145 Tz /OPExtFont3 12 Tf (4) Tj 1 0 0 1 452.649 608.6 Tm 43 Tz (; ) Tj 1 0 0 1 454.3 608.6 Tm 71 Tz ( s) Tj 1 0 0 1 462.5 608.6 Tm 49 Tz (o) Tj 1 0 0 1 467.75 608.6 Tm 67 Tz (, ..., ) Tj 1 0 0 1 486.5 608.85 Tm 86 Tz /OPExtFont9 7 Tf (S) Tj 1 0 0 1 491.75 608.85 Tm 120 Tz /OPExtFont5 7 Tf (N) Tj 1 0 0 1 499.449 608.6 Tm 122 Tz /OPExtFont9 7 Tf (\) ) Tj 1 0 0 1 508.8 608.6 Tm 77 Tz /OPExtFont3 11 Tf (is ) Tj 1 0 0 1 120.25 585.299 Tm 93 Tz (proportional to ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 173.05 523.399 Tm 85 Tz /OPExtFont2 12 Tf (e) Tj 1 0 0 1 178.3 523.399 Tm 110 Tz /OPExtFont3 12 Tf (-) Tj 1 0 0 1 185.3 523.399 Tm 104 Tz /OPExtFont3 7.5 Tf (2\(xo) Tj 1 0 0 1 202.8 523.399 Tm 168 Tz (-) Tj 1 0 0 1 208.55 523.149 Tm 104 Tz (x8\)) Tj 1 0 0 1 221.05 523.149 Tm 109 Tz /OPExtFont12 7.5 Tf (T) Tj 1 0 0 1 226.55 523.149 Tm 77 Tz /OPExtFont3 7.5 Tf (B\(3) Tj 1 0 0 1 239.05 523.149 Tm 64 Tz /OPExtFont12 7.5 Tf (1) Tj 1 0 0 1 243.349 523.149 Tm 101 Tz /OPExtFont3 7 Tf (\(x0) Tj 1 0 0 1 255.349 523.149 Tm 189 Tz (-) Tj 1 0 0 1 261.1 523.149 Tm 165 Tz /OPExtFont3 7.5 Tf (4\) x ) Tj 1 0 0 1 283.449 523.149 Tm 1866 Tz (\t) Tj 1 0 0 1 328.1 523.149 Tm 129 Tz (\(H\(xj\)-si\)TrTi \(H\(xj\)-si\) ) Tj 1 0 0 1 434.899 525.299 Tm 153 Tz /OPExtFont2 7.5 Tf (x ) Tj 1 0 0 1 440.649 528.1 Tm 1995 Tz /OPExtFont3 7.5 Tf (\t) Tj 1 0 0 1 488.399 520.299 Tm 88 Tz /OPExtFont3 11 Tf (\(5.13\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 291.35 498.449 Tm 42 Tz /OPExtFont3 15 Tf (Z) Tj 1 0 0 1 297.85 497.25 Tm 156 Tz /OPExtFont3 7.5 Tf (EN\(xi) Tj 1 0 0 1 330 497.949 Tm 168 Tz (-) Tj 1 0 0 1 336 497.699 Tm 137 Tz (F\(xi-i\)\)) Tj 1 0 0 1 369.1 497.699 Tm 98 Tz (7) Tj 1 0 0 1 374.899 498.199 Tm 146 Tz (C2nxiF\(xi-i\)\)) Tj 1 0 0 1 442.8 498.449 Tm 63 Tz /OPExtFont2 7.5 Tf (. ) Tj 1 0 0 1 444 498.449 Tm 41 Tz /OPExtFont3 7.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 7.5 Tf 41 Tz 3 Tr 1 0 0 1 120.5 452.1 Tm 89 Tz /OPExtFont3 11 Tf (Matrices ) Tj 1 0 0 1 165.849 452.35 Tm 97 Tz /OPExtFont3 11.5 Tf (F, B ) Tj 1 0 0 1 192.949 452.35 Tm 84 Tz /OPExtFont3 11 Tf (and ) Tj 1 0 0 1 214.55 452.35 Tm 91 Tz /OPExtFont3 11.5 Tf (Q ) Tj 1 0 0 1 227.3 452.1 Tm 89 Tz /OPExtFont3 11 Tf (are observational, background and model error covariances. ) Tj 1 0 0 1 120.25 429.1 Tm 90 Tz (The weak constraint 4DVAR cost function is then derived by taking the logarithm ) Tj 1 0 0 1 120.25 406.05 Tm 91 Tz (of the above equation, i.e. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 130.55 338.85 Tm 123 Tz /OPExtFont6 7.5 Tf (C4dvar ) Tj 1 0 0 1 162.949 338.35 Tm 76 Tz /OPExtFont6 15.5 Tf (= ) Tj 1 0 0 1 176.9 348.199 Tm 88 Tz /OPExtFont3 11.5 Tf (1, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11.5 Tf 88 Tz 3 Tr 1 0 0 1 175.449 345.1 Tm 108 Tz (_) Tj 1 0 0 1 183.599 345.1 Tm 135 Tz (\(xo x) Tj 1 0 0 1 219.849 345.1 Tm 81 Tz /OPExtFont5 7.5 Tf (b ) Tj 1 0 0 1 222.949 345.1 Tm 1165 Tz (\t) Tj 1 0 0 1 244.8 345.1 Tm 83 Tz /OPExtFont24 7 Tf (1 ) Tj 1 0 0 1 253.9 345.1 Tm 1980 Tz (\t) Tj 1 0 0 1 292.3 345.1 Tm 90 Tz /OPExtFont6 7.5 Tf (b ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 7.5 Tf 90 Tz 3 Tr 1 0 0 1 219.599 339.3 Tm 143 Tz /OPExtFont5 7.5 Tf (0/ -"0 1,X0 X) Tj 1 0 0 1 292.1 340.5 Tm 77 Tz /OPExtFont3 7.5 Tf (0) Tj 1 0 0 1 297.1 341.25 Tm 132 Tz /OPExtFont5 7.5 Tf (\) ) Tj 1 0 0 1 315.85 341.949 Tm 113 Tz /OPExtFont3 11 Tf (_ ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 113 Tz 3 Tr 1 0 0 1 176.15 332.35 Tm 70 Tz (2 ) Tj 1 0 0 1 180.949 332.35 Tm 2000 Tz (\t) Tj 1 0 0 1 316.55 332.35 Tm 66 Tz (2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 66 Tz 3 Tr 1 0 0 1 342.5 340.05 Tm 105 Tz /OPExtFont3 12 Tf (\(H\(x) Tj 1 0 0 1 367.449 340.05 Tm 66 Tz (i) Tj 1 0 0 1 371.3 340.05 Tm 75 Tz (\) s) Tj 1 0 0 1 394.3 340.3 Tm 66 Tz (i) Tj 1 0 0 1 398.149 340.3 Tm 80 Tz (\)) Tj 1 0 0 1 402.25 340.3 Tm 73 Tz (T) Tj 1 0 0 1 408.5 340.3 Tm 190 Tz (r) Tj 1 0 0 1 416.149 340.3 Tm 66 Tz (i) Tj 1 0 0 1 418.55 340.3 Tm 88 Tz (-l) Tj 1 0 0 1 428.149 340.3 Tm 104 Tz (\(H\(x) Tj 1 0 0 1 452.899 340.3 Tm 72 Tz (i) Tj 1 0 0 1 456.699 340.3 Tm 76 Tz (\) s) Tj 1 0 0 1 479.75 340.3 Tm 66 Tz (i) Tj 1 0 0 1 483.6 340.3 Tm 86 Tz (\)\(5.14\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 12 Tf 86 Tz 3 Tr 1 0 0 1 326.649 326.6 Tm 113 Tz /OPExtFont3 8.5 Tf (i=) Tj 1 0 0 1 336 326.85 Tm 89 Tz /OPExtFont5 8 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 8 Tf 89 Tz 3 Tr 1 0 0 1 343.699 299.95 Tm 117 Tz /OPExtFont3 10 Tf (F\(xi_1\)\)) Tj 1 0 0 1 386.149 299.95 Tm 92 Tz (T) Tj 1 0 0 1 392.899 299.95 Tm 106 Tz (QT) Tj 1 0 0 1 409.199 299.95 Tm 46 Tz (1) Tj 1 0 0 1 414.25 299.95 Tm 127 Tz (\(xi - ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 127 Tz 3 Tr 1 0 0 1 300 287.5 Tm 112 Tz /OPExtFont5 8 Tf (i=1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 8 Tf 112 Tz 3 Tr 1 0 0 1 120 258.7 Tm 88 Tz /OPExtFont3 11 Tf (Note that although the expression of the first and the second term in the cost func-) Tj 1 0 0 1 119.75 235.399 Tm 90 Tz (tion is same as the original 4DVAR cost function \(Equation 3.9\), they are different ) Tj 1 0 0 1 119.75 211.899 Tm 91 Tz (in the sense that the estimate of the system states x) Tj 1 0 0 1 376.3 212.1 Tm 52 Tz (0) Tj 1 0 0 1 381.85 212.1 Tm 89 Tz (, ..., xN are components of a ) Tj 1 0 0 1 119.5 188.85 Tm 95 Tz (single trajectory of the model in the original 4DVAR case i.e. x) Tj 1 0 0 1 437.3 188.85 Tm 65 Tz (i ) Tj 1 0 0 1 439.449 188.85 Tm 101 Tz ( F\(xi_) Tj 1 0 0 1 484.55 188.85 Tm 80 Tz (i) Tj 1 0 0 1 489.1 189.1 Tm 98 Tz (\) = 0. ) Tj 1 0 0 1 119.5 165.799 Tm (While in the WC4DVAR case those estimates form a pseudo-orbit. And it is ) Tj 1 0 0 1 120 142.5 Tm 90 Tz (assumed that difference between x) Tj 1 0 0 1 289.199 142.5 Tm 65 Tz (i ) Tj 1 0 0 1 291.35 142.5 Tm 103 Tz ( and F\(x) Tj 1 0 0 1 336.949 142.5 Tm 72 Tz (i) Tj 1 0 0 1 340.3 142.5 Tm 91 Tz (_) Tj 1 0 0 1 346.8 142.5 Tm 80 Tz (i) Tj 1 0 0 1 351.1 142.5 Tm 91 Tz (\) is ) Tj 1 0 0 1 369.6 142.5 Tm 95 Tz /OPExtFont3 11.5 Tf (IID ) Tj 1 0 0 1 390.25 142.75 Tm 90 Tz /OPExtFont3 11 Tf (Gaussian distributed with ) Tj 1 0 0 1 119.5 119 Tm (covariance matrix ) Tj 1 0 0 1 210.5 119 Tm 87 Tz /OPExtFont3 11.5 Tf (Q. ) Tj 1 0 0 1 228 119.25 Tm 90 Tz /OPExtFont3 11 Tf (In order to make difference from the real model error which ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 310.8 52.299 Tm 74 Tz /OPExtFont3 11.5 Tf (112 ) Tj ET EMC endstream endobj 606 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 607 0 obj <> /FirstChar 0 /FontDescriptor 608 0 R /LastChar 128 /Subtype/TrueType /ToUnicode 609 0 R /Type/Font /Widths[0 0 0 0 0 0 0 0 0 277 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 277 333 474 0 0 0 0 237 333 333 0 0 277 333 277 0 556 556 556 556 556 556 556 556 556 556 333 333 0 0 0 610 0 722 722 722 722 666 610 777 722 277 556 722 610 833 722 777 666 777 722 666 610 722 666 943 666 666 610 333 0 333 583 0 333 556 610 556 610 556 333 610 610 277 277 556 277 889 610 610 610 610 389 556 333 610 556 777 556 556 500 389 0 389 0 0 1000]>> endobj 608 0 obj <> endobj 609 0 obj <> stream /CIDInit /ProcSet findresource begin 12 dict begin begincmap /CIDSystemInfo << /Registry (Arial-BoldItalicMTOPExtFont24) /Ordering (UCS) /Supplement 0 >> def /CMapName /Arial-BoldItalicMTOPExtFont24 def /CMapType 2 def 1 begincodespacerange <00> endcodespacerange 13 beginbfrange <09> <09> <0009> <0A> <0A> <000D> <0D> <0D> <000D> <20> <22> <0020> <27> <29> <0027> <2C> <2E> <002C> <30> <3B> <0030> <3F> <3F> <003F> <41> <5B> <0041> <5D> <5E> <005D> <60> <7B> <0060> <7D> <7D> <007D> <80> <80> <2014> endbfrange endcmap CMapName currentdict /CMap defineresource pop end end endstream endobj 610 0 obj <> stream 0  ,,j]YiuAKe_<$zFk5/?A^v9 H8Xhlz[W %sPA/N9lr5p:x`[V 5˓b|:^'0\еuir4^. iDz3`S>NπlfoQ2Gfn y ; 0}7.2TMs^*$UC$ٚ^H#EX7[lj#& >C1b+JwHZ nL Ho7)%;s&fRpQ-dIe ՠׇb] $u@*B.wg-v7J~jQQNxB=Zqf>nL ĄVJ?ݷ*$S\2n1LgnI6x%0C\/:<oI0"NrbߐUl_2I%T3j-r(փ;Z5G2 j]n7$4w1Ѡ[x1 9FMYssľkL늶|jQKN?w r(@N";*'b_\G??)kǐ㈴C,D#4NŢ~bB6;3KZf 2Rv῭[̝/J5EL(TXu1oExϏ9/2P])'$(8ݾl<_ xx#kcy{5p׭{[QybpUPZ,1" }2`b'.!XT gT}L볉[%of^Wy;%cbiT/[j.#2,<`3%jEyKA\3PLN.[*Y7cxb4@[{Գ^E0m6]'kQf|ߦD4vb8J8 ʧA"~/ ,tnL-Uv F ;ks;dģ Q\!TÏY\!6D kWm6Ց#AͿ?X"~XmhF@cHXYu! d(Qo0`Pm:-:A<8^E}(7E'3-pSZBZ` @J{DWxusZxbfmٷ=ݯf_HT2R?୿n  ,z'Tl䬖qۨX'qz,ݣLt$ym!+IFЎ lO=}NgQ?8(7pv|DzV4tB 8tJxyuSȀ|^ NhxOv)x~8A /Tx^31bmTC')92ZoP |hzFlM~BM'5GBE.,0Q[klLZS+[f\1#g ,aU ^Gd27((Ai #`sn5VxzB~pj:6L@䭣`K3U]p&. @-݆)|o]5XM_L_GElps subߖc[?SNdS 3 ˌ`٧~ A^ۯl}?ˡ@@7v_zna٠l}U:}شR-f/-G];yx\ FޅnǠ P˽77Y-uexˀ.C5wq.,kN-qX$1xe<ۊW d ؈ /z.-ҙj0˪<`GU5):A6{u/#y%/<xdzJJǍ=d*J&OP^}/$!YķMLE=J1\yz~`'w$}_Q䞗C4)J"KcTTB{( }Cn$,!jrOrja]W 0UU/0b_8]Ÿ6wO]U%s2Y\*[.'4ojx5x,@4<Q@qPCpRUN5IWecS&96("uLb yX':ۿ(e굃O_ҏkR78!*Ke ,lX7iMVٺsMOrWFlB>Cgy8(yn%L0Hh!&c?[f/Q ׋zzz_⫀<2^x G+׺I;wW38/ }hwݹ -Jw4Jh|z~JװD8yUbGp׹:(%Ǯ2ԓ7cF?k 3> rBs!zq=KUcօHwiOVE5j`PȫiԽG>V1}!7E |UW3oMfk}G }$yzc{8{0ĆH,kxDfo@:4<1Z1КOp59Vyi!,㵤B9,"v #x/KLJHKi}IJRlڛe\n]D F0Yd97떑nϧul3`ڰ 22u2Lp`_TkBjC l6n(^ۚK{%YwHY.h,fRi61:'bI8Tnu`/Kwb.O:ɴ/\xpojJDU?%4vqCeOEB^)˗?ŝ2<Ҩf!:>;}"J5 oj~!+h]0<nmޜ:=JFfwCAMZbs(G \ѣm6VLB: T&+9) P+oBoG3x3{`E0&BݫoK V?WNH[ۏģk5bJ^@{|YH1ڔ2Has m+~G 2{0?逡/ؿ+muH<\ gg)Ey pu#w "T4%O{Ba ]ý;ӽdZ ;O𬎯yoh=tK(%U68=*f6aK-=9KӼ)\# m/ WQjvL;sc$u66O:VK+&/E /N[ȫ偻i- dۃ}'_˸aY,]hk}Rw=_ ]&[Zh‹늸As~<+t%;bUƱEq_6'Y!õp ^F olpn2n$;#52-I`\1Fא}hVeu 9-ӐC|ل |.^ANm't~dCkHZ@i0Hmv-htUGq:: #DJ AHr΂ރ{#s}:qBcƾ0Tc` ~r^Q}TID@\ -E"m]m GQ1 L-ƝoeRu: < jlj]h_,#cw\N)9#ѱL 7k(&=}3Of;}p|!l̕rz1>ˈzXE ZqX]5YFm0 U3vXs/m7u#qiiSN[6nwvB"?&-k{W@Ql`SeBbDsŚsݘ<’"kM D.@1ፁq:qcecjӜG (xio}R(p.'cOX7PZH}R1AI5{FCag ڻFⵞӮ뙂H0Bm(#9aZv8;3>4; qx_It3 tEn1_uODsgK^E" ݵ$` O%]%`kdDsqhO1L軝<܈Jt]Wckksm ZJW.":w? ՙ"Z8fWSC@ۘK|C@r8+O$BR2MPp뢨Ag0A~y\|B-0= 8Vb4*?q>+˶<<]L\Jx2ًP|uu/@U xH'&ݷ uj4 ?^8:r\RO_D:@~j@hOBpSm~$P(UxKeH&Ip^R}`\QGP~)1.!ZB ):S9N^U+"h/TZRC'|$1At ٪l3Kz$Ftpn-VYr0%2\ ߭aNM)0dY9i_ٌQH3nNb{Z(wC .E?~:D.p7zRMFn3J!_@{FQn`9f2hꡌ5 jmԔr4bF0+lTO9z*H}B\d+PZf:y!H=黊<HΓ%eL _a |;r5DBYT9$nݎ4t`f}3Lth@zҢ+k_O Õtd @[@wKpD~~ e}p5}Ы-b{d]hi$Dž#KLp%UlM"aŃյh =?܂CI]<VU9@8r?ȌU2X &!1KUCNqcvjL.0Doà1a|pٌX0KOyr)wf"q9Ԟ^9mO2PΣB1qI+bl%͊ߦ_    -?ƛ!;kx7:9)xkiVIF8^vHu6-48M[꘎TFAǀ`J *ZB}U׋l9HAt2<9\lU'Q1Jߐ'wa#E#tU&ͰIzˊ=,KzQ2Y${jGC<w?ċc&3vx'0gMU2.=z3HoRS^WݖZմcoUXA7]U:K>0hK30^D=dQX4_x?'%Zlw|cX=cZM7da8}#= 7K#zG%MȢ; dAyKu5QT)j:9Gތjh!b{Ռ"t*?dƶF!jfQ~㯖k7| еT]y?6,lN+2 FNJNnO?Xy3bI s>lѾt bzeNffU38"Lu#ͶmT6!JO<ݽ-K+Bnѭܔe aj lv\0ѭ;R Ko~I];ۏAZըY) }3&B/PPL>@`2`"xq:o *{;TGlz#V]~iTɐszq[2wcl1:CcX^ot QШ>9$U%Rٜ>J.`;*/<{Q"t5"dA(a~%3ZJ7坰~j!۽ٱL58:qb 8\@\&]:Sm$*ّ9rL$[(Cۺ*(9zwLUBRQgDxz%Jk?'- XQkcW ҏDeBf?Fpbu+Ml}*gu/q3Z]Jf<>>ܺX|J0utd҅Ԛŷ̝h*\ige2t[88fqγ&Ed|BU"&P1{6Uy֮#]K潋R% j#d}~:w )wn$ RqwFJT4BZQX% *QxBM3O+apIU{L03hFȀ؊48b4?1%sY|=WW(z.֑BEf )TMjs=凹g? vzH P J噠ՙ godҶ:Yw d,bEНY N^^Y~<[J^i$M|P?Tlq:x{Nd#x#1˭ZCǶ31> D2w mjb/nUg71cx뮓 YQtrc_\9L@s;ӏ':PE^22Y}n! # UBN'wtJAVVL^DD5PR݄Z,10TM7eT@JWA`ؤS%Soۧ7} - 7\e1S$zY#H*+IQg3J!RŵS H\S݀&GNf*?g~~&V1Hj)+,,<&T !|0)*_%I΋Z/*$NvVKw=Q9荒|S!@&ȳ:z3!)<_3w1 fZpL.pZG[ssk {aOFJq`:2hc]$!6f(_qoGP(p lr<*(u Dpc$- P+hm->qaeK.-1ʈzܿwD1aqE c l<6UO G } d[ C4^i :8B1a:3Y }sKf~z'bسg%nKDdy+@c-9 1Og5W3O(L=A׎G _WmUgZ 2[gp8%&ći׸ =)EnϲB3Tse )$stJZ#Z+m~"Ԇ>\Zc{QXc}4G 3ayB8oմ+xFgfSeخ.")qV_Vm4L G0\pqm=Br` U_-N=J5PSR endstream endobj 611 0 obj <> endobj 612 0 obj [613 0 R] endobj 613 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 621 0 0 839 0 0 cm /ImagePart_2140 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 307.199 720.45 Tm 107 Tz 3 Tr /OPExtFont3 11 Tf (5.3 Weak constraint 4DVAR Method ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 107 Tz 3 Tr 1 0 0 1 121.2 677.95 Tm 112 Tz (is ii F\(i) Tj 1 0 0 1 178.55 677.95 Tm 80 Tz (i) Tj 1 0 0 1 181.9 677.95 Tm 96 Tz (_) Tj 1 0 0 1 188.15 677.95 Tm 95 Tz (i) Tj 1 0 0 1 192.949 677.95 Tm (\) and to be consistent with the terminology in the previous sec-) Tj 1 0 0 1 121.2 654.899 Tm 92 Tz (tion, we call the difference between x) Tj 1 0 0 1 303.85 654.899 Tm 65 Tz (i ) Tj 1 0 0 1 306 655.149 Tm 96 Tz ( and F\(xi_i\), the imperfection error of the ) Tj 1 0 0 1 120.95 631.899 Tm 92 Tz (pseudo-orbit x) Tj 1 0 0 1 193.699 632.1 Tm 55 Tz (0) Tj 1 0 0 1 199.199 632.1 Tm 193 Tz (, x) Tj 1 0 0 1 224.9 632.1 Tm 82 Tz (N ) Tj 1 0 0 1 231.599 632.1 Tm 96 Tz ( which is expected to be minimised by the third term of ) Tj 1 0 0 1 120.7 609.1 Tm 94 Tz (the cost function. Generally WC4DVAR looks for pseudo-orbit of the model by ) Tj 1 0 0 1 120.7 585.799 Tm 89 Tz (maintaining the balance that such pseudo-orbit stays close to the observation but ) Tj 1 0 0 1 120.5 563 Tm 93 Tz (with small imperfection error. Similar to the original 4DVAR, the application of ) Tj 1 0 0 1 120.7 539.95 Tm 91 Tz (WC4DVAR is carried out over short assimilation windows as increasing the win-) Tj 1 0 0 1 120.7 516.7 Tm (dow length will not only increase the CPU cost exponentially but also suffer from ) Tj 1 0 0 1 120.7 493.649 Tm 90 Tz (the increasing density of local minimums. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 120.7 449 Tm 107 Tz /OPExtFont3 13.5 Tf (5.3.2 Differences between ) Tj 1 0 0 1 309.1 448.75 Tm 115 Tz /OPExtFont8 15 Tf (ISCDc ) Tj 1 0 0 1 357.35 449 Tm 109 Tz /OPExtFont3 13.5 Tf (and WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13.5 Tf 109 Tz 3 Tr 1 0 0 1 120.7 418.3 Tm 90 Tz /OPExtFont3 11 Tf (There is some similarity between the ) Tj 1 0 0 1 303.35 418.3 Tm 99 Tz /OPExtFont4 12 Tf (ISGDc ) Tj 1 0 0 1 342.25 418.3 Tm 95 Tz /OPExtFont3 11 Tf (method and WC4DVAR method. i\) ) Tj 1 0 0 1 120.5 395.25 Tm 90 Tz (Both methods can be applied to an assimilation window to produce an estimate of ) Tj 1 0 0 1 120.5 371.949 Tm 92 Tz (model states \(analysis\); ii\) The analysis produced by both methods is a pseudo-) Tj 1 0 0 1 120.5 348.899 Tm 93 Tz (orbit of the model with its corresponding sequence of imperfection error. There ) Tj 1 0 0 1 120.5 325.649 Tm 90 Tz (are, however, fundamental differences between them. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 137.5 289.899 Tm 97 Tz ( The WC4DVAR method forces the pseudo-orbit to stay close to the ob-) Tj 1 0 0 1 148.55 266.85 Tm (servations by the second term of its cost function. As the imperfection ) Tj 1 0 0 1 148.8 243.549 Tm 92 Tz (error brings extra freedom to the pseudo-orbit, the pseudo-orbit produced ) Tj 1 0 0 1 148.8 220.049 Tm 98 Tz (by WC4DVAR might be stay too close to the observations and the dis-) Tj 1 0 0 1 148.55 197 Tm 92 Tz (tribution of the difference between pseudo-orbit and the observations, the ) Tj 1 0 0 1 148.55 173.7 Tm 96 Tz (distribution of implied noise, might not be consistent with the observa-) Tj 1 0 0 1 148.55 150.7 Tm 95 Tz (tional noise model. In the ) Tj 1 0 0 1 284.149 150.7 Tm 100 Tz /OPExtFont4 12 Tf (ISGDc ) Tj 1 0 0 1 324.699 150.7 Tm 93 Tz /OPExtFont3 11 Tf (algorithm, the cost function itself does ) Tj 1 0 0 1 148.3 127.399 Tm 92 Tz (not contains any constraints to force the pseudo-orbit staying close to the ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 311.3 52.5 Tm 75 Tz (113 ) Tj ET EMC endstream endobj 614 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 615 0 obj <> stream 0  ,,mmj%;/W-Y*q+֘\Sʣzq*bvǼL+ٖ.^$ׁͽ_TcV_KHii"n :H,b kgmh:ŋ?΅`$>dEL|*E*( u`1AA%ٲ< uӇ% B׎#k$^A4 ]CmFDg]W#yQ``ONY}##_?~`xp1e{@_\&`;Uϙ- Ch:XvmHۑ}bE gAמm?P;E:`ބksVCy}f 2,]:<7elY&vg2RNq𗓑Z<_͐VS+V[K;Ǖ8/5U]~yJ~쑣0\j*4n+ {U0E9%XJ@6I*~5yɿә^7ij: jw|WK3>* zTn7yQ\ta#U5 p/\&R \)i|;}8.1XD[sW4m B2F0wf6 x=+*60&-U Yq/|J"eu]ڱiD^\@b[=ePNYy:.0{ f {t &+h㴾U֯g'cJkfd);.)@3Wk7zFvwdRk@<ʔ!Vή;zws%%, iTmX W3/x ~~@i/V(|4)t>Ed#&n繁= w;w8o"ċ/zdq(f^^\@٬I{B _]CEtJkA%yoq4;ֱU#6ˊNVᨬ^t<ȉPdxP$Ƿ!A[uз]TI,sEm&44{Ba^PED8<4[1~E٨"<7))!͓xkȗ1μ{]q&p_-,@؁}M0'n-y LL~8c1]b4 Vf'.r;pvC~_(rYt!wemGm%}v-K-?}ܷ%Oz-:6A_X22APˌ]y C.řyl:ײrɗN<]k) LQRJ'tas`S :SntOֿHV" /1+5R KN*=ŭ}}I7MP bռ&-G_knc^gؼkJm_1"QIҤ/'7;DWiF0߂}RfFM6땡dcMuf Yq@gi")=·XZ^ocJ08֝cW@AAfQ י;.g. 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WaÓ= BxFg[YLk~%TD-%s>\B+j4f3I]ԭb[Q_TLIMt)X=d .V/(j򌑿|q~{?(—5]ڿRZ99cNy|[ܒiR><|U|ʥ -7ԜDj\G=t#"4e}ێZ .%r?s@vi4ghXڒk>s~9;%1e[i{(sJ F?jc| t#ųy>OVp8p^Ew"&W%Ɣ }ioЗ sY{\2 t4ꄶjmC];X6g0K B2.Uk٘`#~uaxX0b%/?X J6,p+n5 ;Y~D5\t*="_~O^sT?8) :΍z D-Fߪ$pQY'Z[ٝ Ne"~w3φO[KN,=E [ Bk3 Ūm9S6G üwΩ}AK:s2‰u< @CG6g/P4߮?&`W(@ǎٰ\EYW|$فZs`6kŠ/6٠ Y%JJ]=8)ʄyZCQIGy&'z-R70{3)-y3%ժV,d 6Eu>;m QцAԱ a7FI!U..pZ/έ\p+_/̥FWϋi7!iiۢcc5e%xD RHELs* GI.⟜jwɄiX+cѝ8s7Gy_ '[:r4Q)-8r o71<+Ed^]ա}aQw &(KͳftfhKћ(45rSpG}G Bo1?xKŸhΟz-Ωt O ^?2,(~ 6U ukTɢ@e=f?e%>踤/,M,d},ΖNe9kl2^;nua)U2FNhm&CgY[Y kW˼C;€"V]Y>Ius0tr3+m$-2u*z]䠝&n\84aTO!U1ҀDj9 $A-S&8_UNlc> rҕYM#*`"uanY -;zR-6EAh]4qj _3{P3Ӣ܅Antܻ5-b(Pr ~t8)QOEkJz?Ƿ9oHN%k_^6Mull@ ?S쏏d` d65oypHX7)eT_Gog-^zf|m L髊qu{s-un&GաVKf\+!=2S+ KjvLS)/\e}w(m,~^iG X{9y/?kK+C+h "&RT3Ɛs3ُ9Sdמ2e䶷§&֪9\臸"۪v|ZO O"Yu6?w:M ?H0zm7ǧ=e[͏U֡J ø` O!0{SV+a A+q {4Ugp2m+SmQ>Pg D~-R`I{&3NؠdQ'7hyKTdOǾF(fB`s'.Mbf8#Ov2z4"[j)n Òhѯ猃W )Ս7+Ũ`c@ Y\;? }qtʤbF9Hq"5gJI1@˜'PRgȾe92nCmex '+BV/>ߘJ܋|݋3?lmEw.-D+쭍{n o~{w(>PVu:!0Lc_R+޵4ph6^IIzPkGҍBWk Yo&3V0- p)3KUX+})/z |>s0B aSRDoogYXh2eucJ F\֮z4  "Jn _yx/HO-{Ck*;aZ19fH'6Yu>;$r b (V@WGD[F/ѯܸDk`Cn|0JKWgKtM;.b(kLg\9Og }|63]#iexaU61)>6GǫÑ 8Ʒ?EpjG©#RVBО2l7;1B<*}h7yuuL RuAǴa2e tپ6Ͷ2[S*kZe K׶U?kl^aE/2a4;gu}&v|sI98\1ί,*žDf oicX.4 e-[/27n:1{Nl(! ?j)wѵ֜Xg\P>ʳ|@"{ZNQeQytco,i/dѵO( xcgWyProV#!Ji HxP=ggg:'st94-t! yoj3R^sqۋ!bu:;,i8"9UUfd/KhY >$)ć)1e_V͂C3ЩFy8II^/+G'\= 3O5#qz[|3XgJ$g OA' &ÎUεe췠xz-!(FOzx@%r$;(m0m#:&nȯ$5I{AbV:E .Esd.AeI`|nזiC 5@W-,Wyw)ij94#$EOCSbc2 (¨_7-OIFH>w"6։&~:,W Hx+-`ÂO!)jWݫ{(Oe LyWk$vK3>1ibl]WM*w&. V_-mMl5s0N JA"cՓO;c6h0x @_uP-7^jƐ-4Y gt4N\MZ0?_fh9/kwK12.hXАu23p2)_TlU endstream endobj 616 0 obj <> endobj 617 0 obj [618 0 R] endobj 618 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 620 0 0 839 0 0 cm /ImagePart_2141 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 308.149 719.5 Tm 107 Tz 3 Tr /OPExtFont3 11 Tf (5.3 Weak constraint 4DVAR Method ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 107 Tz 3 Tr 1 0 0 1 150.5 676.75 Tm 93 Tz (observations. The observations are only used to initialise the GD minimi-) Tj 1 0 0 1 150 653.95 Tm 96 Tz (sation. By setting up the relevant stopping criteria, the implied noise is ) Tj 1 0 0 1 150.25 631.149 Tm 91 Tz (found to be more consistent with the observational noise. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 139.449 598.049 Tm 99 Tz () Tj 1 0 0 1 150.25 598.299 Tm 89 Tz (Both methods produce a sequence of imperfection error besides the pseudo-) Tj 1 0 0 1 150.25 575.25 Tm 91 Tz (orbit, such imperfection error could be treated as the estimation of the real ) Tj 1 0 0 1 150 552.2 Tm 94 Tz (model error. In this case, WC4DVAR can be shown not to be self consis-) Tj 1 0 0 1 149.75 529.149 Tm 95 Tz (tent \(52\). Within the process of deriving the WC4DVAR, the model error ) Tj 1 0 0 1 150 505.899 Tm 92 Tz (is assumed to be IID Gaussian distributed. Such assumption appears un-) Tj 1 0 0 1 149.75 482.85 Tm 91 Tz (likely to hold if the model is nonlinear \(52\) and can be tested after the fact. ) Tj 1 0 0 1 150 459.8 Tm 95 Tz (As we discussed in section 5.2.2, we expect this model error to be space ) Tj 1 0 0 1 150 436.75 Tm 93 Tz (correlated and not necessarily to be Gaussian distributed. Even were this ) Tj 1 0 0 1 150.25 413.699 Tm 94 Tz (assumption to hold, the covariance matrix Q has to be predetermined in ) Tj 1 0 0 1 150 390.699 Tm 93 Tz (order to initialise the WC4DVAR cost function. Without knowing the true ) Tj 1 0 0 1 149.5 367.399 Tm (states of the system, it is impossible to obtain the model error covariance ) Tj 1 0 0 1 149.5 344.1 Tm 96 Tz (matrix. Therefore an estimation has to be used. As the imperfection er-) Tj 1 0 0 1 149.5 321.1 Tm 94 Tz (ror is the estimation of the model error, we expect the imperfection error ) Tj 1 0 0 1 149.75 298.049 Tm 93 Tz (produced by WC4DVAR is IID Gaussian distributed with covariance Q. In ) Tj 1 0 0 1 149.5 274.75 Tm 88 Tz (the ) Tj 1 0 0 1 169.9 274.75 Tm 109 Tz /OPExtFont6 12 Tf (ISGD ) Tj 1 0 0 1 210.25 275 Tm 93 Tz /OPExtFont3 11 Tf (algorithm, no assumption of the model error is made and the ) Tj 1 0 0 1 149.5 251.7 Tm 92 Tz (covariance matrix of model error is never needed, the imperfection error is ) Tj 1 0 0 1 149.5 228.2 Tm 91 Tz (the remaining mismatch after certain number of GD minimisation runs. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 138.5 195.299 Tm 99 Tz () Tj 1 0 0 1 149.5 195.299 Tm 91 Tz (It is shown in section 5.5.1 that the performance of the WC4DVAR method ) Tj 1 0 0 1 149.75 172.299 Tm 92 Tz (degrades as the length of the assimilation window increases while ) Tj 1 0 0 1 482.899 172.299 Tm 109 Tz /OPExtFont6 12 Tf (ISCDc ) Tj 1 0 0 1 149.5 149 Tm 97 Tz /OPExtFont3 11 Tf (does not. In section 3.4, we discussed that the 4DVAR. method suffers ) Tj 1 0 0 1 149.05 125.95 Tm 99 Tz (from the problem of local minimums when it is applied to a long data ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 99 Tz 3 Tr 1 0 0 1 312.25 52.049 Tm 77 Tz (114 ) Tj ET EMC endstream endobj 619 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 620 0 obj <> stream 0  ,,Huuj%m*]d7L9m\z;O Phxasg`Z('> [p5:':!?B J3wd+a`#>f?s؎[qI9?SP'}w}ǏGEKQM jehwH*j90WU-AQ|p mV<),4u12<8 =bxb@wN wk %2]K1>llegˈ} -AB1] J +zӇG2uAc<}C4@ӤAÖX2sIb**ЏULWdI}f'jNt0T)q1#0E$`C?ɆV'jiS/A UVߔpwL7!(r>b ͊ۙd+a$N5,7H%3`9V[*ZEi}70)rI&\a kCdf`s[pf򦕕[8!< 5ղG/|UO+dSpyEiJP9[,I+Khr}D$uXd6V?&R72/GҮtz+8MxNؓ\4XK:J>M!E9bz_ԳSʵ@8 ]nsU1k $Zɛ* }<l9rLY8eS! [ XԴ^SA8T_SAdWǒA>a`NjVl53!]7lMF)il8T3 3K9-_[(&[7&oM @[6єa0Dia2>b"Om,(k1yD;MH|57;Ņo5cE>=D|>\#( A-z+ӡJ̨tF%0`b3̴27S[hw )"!?YZJQAIU ;t@Cq?px ܴ)ŻU A< ,3>DsqTJtMdmzo씥EQukаċϮ75.~笗à_oA `)[ltK}J 9,f7~WE[X󋉻8bB9a Ogb5dWmQ࿒8i1NrՊu:8O#Nc8= jd>?+,ݚ꧿o/ۀ eqoK-Dݳ-6CwB6z*XtQ* ;n6ͬXq軵4L=K$Qyg9FRoJVI wElq'oєd8T %ffW2QE )qb/-~ hݖpYK!Ar?!Kk8;Xڏ*1;7笁Sߨ69!i%H%Q/mMQ#*_Yzg~Ȭv׈Oz3%sLOb$iENY᳾Xji >^Fid*r;eoNqI^ $MpoԧJJ4kB1R(?%Z`$ogou5+  _h–X?¨gcfh˔\ :˾d_(nV*,N: Oqv!rzhj=SH ]L;A0RԪFtBbΦI|ހkQ}kXQ@J\{F%3qkyN],TIHFd8`庋m0:Klxu$7@bJBI;hT0 |l@jɔ0_&*9ga@2ܘ'3Jq7TM1l'#Ͳ2fA'#y>$X)2Z_=j]cZ5|Y Gk56DO@Ӌx8\'1 wZ!A>8>mxFO?krJF@="hY&g]g1ga._.nߝ䴓}/S_\c2c}lTd5ӓc3]SM" k呴4}DggR!oaBtǶ2_'T6 >xtƼ>] ͉u'EfHt9Cqb즢 N?E|/*-/z=Ī n L^ Co[ i<d{hGj9<3727gu]Kq=u .S4KnW.DiV1!4Gea/s'82 {~jfU|)Ћ8;nyԝq+@nLݥs{;$y0" 8M#7tġt! N!v6֍nZѰ'ᐏtf1h>S#/[xC-NS2 *w>59bV.UmiީVO~&€gEib z rxz ͎౸L4+hP=w8)n@Q,ChHZGLiˬ[ъ3ӡvV3prR ^`J 0xTxvx'z-_ȃ{I M#6W6pr &.7x:eJ]REO*7EL y4`I5 !~RG(:U +\͌hngʽL9|(GF=mY7* 9mRT1[םͮIx?i#'/>! <[4ڿx.Ľ-@a#>uB*GpD#3|9RCR+=420~ FNqTnĿBvE#5\b4@Ib69~Pga,^19pΘp!dbh۬%S˲}ڦph^F?TNҁ祩7 3q$wu|_}˩Pſtmq-z(rdg>ctKL"d* X7T]w=c&tk fAY!QP/{uOӏm@PͶ'ulG,ޭ`STa5 n`^0pIԐ [d\ ؤ3ΥehQInwE|rp&mZ?W b`_>Un7]o>T+:VB'C) ( ɱ2i'f Rcq}!c|vvP/=\hIN~oȄ|qezЇCMb}~4{L@kNӿSנMW}X.";s) H[8&ɁBbS՗3l0t9en] ;3i =C=m:#4n\F=1>tIi3 ,  /u'eѩa$8qB 1c%+00<]e\j!!5 CB)V v9C mpbsl}/:mtW3;Q!~Ƙe Jt4F)7|iЄ%ȣ4f,ɨB_םApŜTf8m5l;9@ R"1 /he9n}:Էe:KAQa3*+J X'R E6AD94.f&йu`JAc|1 rPȮMBbaKO90:r:hPn1 ."Uω؀E\"Z& a! zAfv{I>_y_ p]B,$pY<`k'-Y%I"F=M4>-+ۼxF].h$/WrF%:G6ܣ.&)>4 (_땽1xXHy{ՍU??@ȼR4k [@/qw Zϸe?:VN'+F^p~u!CoA:|7a-%R. DBOyoD׭bЬjҬDNhPaE?WTun#)0> !l]wP*]W̻]EN_#5#MMį lx*5y@LGmHYLLN8,%TZZZ-Vx;IS5 ZO3 [dă:ZmX4= Ӵ P`Z⸖S`ӛ]EA̍b'nRm0KhfIz(" 'B *Y& {0-RX3P(L8<+easL^P " sUyW9;ge︻MH]+CI&mSi:bw)U9N xtCI!s#/è_)fAPՓjZiBCډKY; r0}C|X]3U{(/]q߭}-f{< 1Q+٦bIIhU?[usZӂFp7 Ō\/sc8}'r6 9RC\ S4~̧eF40mx[Yɂ> (2V O[b4@=gDZJ[Pn/Ѥ9GiT6q_L|ve#_VC|k qvPD) ʞ&vQlaw vT* Sb,uBP+N$ߢe1O章>ԒITyA^DIǎ^\`"7}s*A%T+GgL@̊WMPWˁY`JaA2dIa.o #>|}SM2ZCkn{.v̰Zw^(J]|mw aƼ/jR_DUqb4ި-QTq%Y`4'_nrO䰎o3kDKk=[N]#E'| #S]*#Q Oz -zO~^9]ipԇ еn,7pmv%<1OV1u"x4^D%ES֫}ͽ}4cJvyj'@{p/G]fm2)5mȤ,zw f'{-`Y1*Lr1/ye%U蹘t:/M$G\nE߲4`A4F/#Kd XZlޡ:N_07T$š{,B ocNt/-?֌e`&i⻱NX|[ ) nkR'Q>!U@T& x^G0iQizS .`Qԙ*FF$%%T1oS|=Aa7חC4  k1~h+SrqsxHO|me@/"ҾHnj|s˜1m?: DV3vwց`FW6 }=.2ДűdiKzV3!– =\(2cs3=tmaUvί2`d0(Qf7gFtu׀. z"$*cЯՙEFUb1E07\L"B,K@ ) G>oƀ"[ZM@_ |bvmE q\b_f h?գH|̦߬-g>?deŚۖ8?wH2xvG}JJb6Kk1"qhn,+G_[ֳ)X\ IhF@>OrZ{EAZ^2ΧDȵ nTiNıʏEUy?|IGKHs)nVzXis1CVίM:p,}l3|+%@mHU71462bCoJDbW)'8G6$V1M*K,WxlqZxIINaZǛ#r^%`U gµTPb+xpV2"' P'!%)_zQݭvf,6A)D}Vu"U(Z=6}얾XlʗLf%/i* +u_LJ "R+WXnޮR4s `3th8)pX8N=CFNB|"]"] _xG/-u;+ooSؾ8[Sme}5lM3D!78ҏ2FܗG9]nz,twt'|/z(.h2ԭa#i|;J-I4훣/h&^}W^䖃[z9x=ӄL; bv5 GS|}p5F%"5,T N YXp{3j8V"=KÏ_RcghpDG;!HUخuW2ü8k n܍XCBY$"5'mR׎Lc> endobj 622 0 obj [623 0 R] endobj 623 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 620 0 0 839 0 0 cm /ImagePart_2142 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 307.899 719.95 Tm 107 Tz 3 Tr /OPExtFont3 11 Tf (5.3 Weak constraint 4DVAR Method ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 107 Tz 3 Tr 1 0 0 1 150.699 677 Tm 94 Tz (assimilation window of observations. Miller et al. \(1994\) also anticipated ) Tj 1 0 0 1 150.5 654.2 Tm 91 Tz (difficulties in finding global minima of the WC4DVAR cost function similar ) Tj 1 0 0 1 150.25 631.149 Tm 98 Tz (to those encountered in the 4DVAR case. For the WC4DVAR method, ) Tj 1 0 0 1 150.25 608.35 Tm 93 Tz (it appears to be difficult to demonstrate analytically whether the number ) Tj 1 0 0 1 150.25 585.299 Tm 97 Tz (of local minima of the cost function increases as the length of the data ) Tj 1 0 0 1 150.5 562.299 Tm 88 Tz (assimilation window increases. Results, shown in section 5.5.1, indicate that ) Tj 1 0 0 1 150.25 539.25 Tm 95 Tz (WC4DVAR method suffers from the local minima when the assimilation ) Tj 1 0 0 1 150 515.95 Tm 92 Tz (window increases. As the cost function tries to minimise the ) Tj 1 0 0 1 453.1 516.2 Tm 87 Tz /OPExtFont4 11.5 Tf (linear ) Tj 1 0 0 1 485.05 516.2 Tm 90 Tz /OPExtFont3 11 Tf (sum of ) Tj 1 0 0 1 150.25 492.899 Tm 88 Tz (the distance between the pseudo-orbit and the observations and the squared ) Tj 1 0 0 1 150.25 469.899 Tm 94 Tz (imperfection error, it might be the case that the local minima of the cost ) Tj 1 0 0 1 150.25 446.85 Tm 90 Tz (function defines the pseudo-orbit that is too far away from the observations ) Tj 1 0 0 1 150.25 423.8 Tm 91 Tz (in order to have small imperfection error. In other words, in such cases the ) Tj 1 0 0 1 150 400.5 Tm 97 Tz (WC4DVAR is trying to find a model trajectory \(i e imperfection error is ) Tj 1 0 0 1 150.25 377.5 Tm 95 Tz (0\) close to the observations while for the imperfect model of a nonlinear ) Tj 1 0 0 1 150.25 354.199 Tm 98 Tz (chaotic system, it is often the case that no model trajectory is close to ) Tj 1 0 0 1 150 331.149 Tm 90 Tz (the observations if large assimilation window is considered. Results, shown ) Tj 1 0 0 1 150 308.1 Tm 97 Tz (in section 5.5.1, suggest this might be the reason WC4DVAR performs ) Tj 1 0 0 1 150 284.6 Tm (badly when the assimilation window is large. The ) Tj 1 0 0 1 413.3 284.85 Tm 105 Tz /OPExtFont4 11.5 Tf (ISGDc ) Tj 1 0 0 1 454.55 284.85 Tm 92 Tz /OPExtFont3 11 Tf (method does ) Tj 1 0 0 1 150 261.549 Tm 91 Tz (not have this deficiency. Results, shown in section 5.5.1, demonstrate that ) Tj 1 0 0 1 150.25 238.299 Tm 96 Tz (a longer assimilation window does not cause problems; on the contrary ) Tj 1 0 0 1 150 215 Tm 92 Tz (better estimates are produced by ) Tj 1 0 0 1 317.5 215.25 Tm 103 Tz /OPExtFont4 11.5 Tf (ISGDe ) Tj 1 0 0 1 357.35 215.25 Tm 89 Tz /OPExtFont3 11 Tf (method. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 138.699 186.899 Tm 91 Tz (The analysis produced by ) Tj 1 0 0 1 267.6 187.149 Tm 106 Tz /OPExtFont4 11.5 Tf (ISCDc ) Tj 1 0 0 1 307.199 187.149 Tm 91 Tz /OPExtFont3 11 Tf (and WC4DVAR can be used to form an en-) Tj 1 0 0 1 121.45 163.899 Tm 93 Tz (semble of initial conditions. The quality of the ensemble depends on the quality ) Tj 1 0 0 1 121.7 140.85 Tm 95 Tz (of the analysis. In section 5.5.1, we compare the quality of pseudo-orbits pro-) Tj 1 0 0 1 121.45 117.299 Tm 92 Tz (duced by ) Tj 1 0 0 1 171.849 117.299 Tm 94 Tz /OPExtFont4 11.5 Tf (ISG.Dc ) Tj 1 0 0 1 212.4 117.549 Tm /OPExtFont3 11 Tf (and those produced by WC4DVAR. Our results demonstrate ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 94 Tz 3 Tr 1 0 0 1 312.5 52.5 Tm 76 Tz (115 ) Tj ET EMC endstream endobj 624 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 625 0 obj <> stream 0  ,,ssb6!ipB]i c |ESL/wOYE >籷6$'杬31ڳ4z<epĐ:?dnT,(0wpRG$"WI%Y3h؃6)=RQd}-+ k7$dQPm_j4,bxp8=sd ʎE4pbƒ):w\ڻxQ~QԤ IH$DLkVн}N0Ofo$}S76xO#y"8+m;vI+PkE_Kg⎯L4gVpj l \W6,ؐ~OOwbIwP ]!#}al23a:%#jMd<&͔$EHF<~!Fj4!{K13nl,HAwqP{%EkE[,Ml.c7[R/kĽJ ?VUh.5E1FBNbl:(?dR+p1kf_uPLu\"˂\$rEA `x ;XfZ{-V꾪A6cd{۠Dd,qz8zX_x$.u%DK3Xpl6#mA4_$Ɔʵ¯ >mG$CMkFBnsGˊX;Bɻ!53]9PRpi"ۆ=ns сl<(0Hf6za-fzQE cM3rs0ՀF \ ]ԦlLrj\p[9g#dzq`H}M d/MTK<q^6ozFV gE}8Qs<)< X"`p`qg$kg2KSN*EjӸ'D=VCЄo%{bNxvXVlsFƦ\ߖ[Yv\Zy⒮z:. b!P~cOvo27\?cB3,ٝ M9'M8M?gޛK#lT5 ?VYU_.<=i hUOXa 3yiO AF}>xD,G[߀9@d[_jG;>\:*e 8b)=zY}k}ky'S o2&cN1QS&neLT$`D%};Ȗ2{)^p1fkl#@,1ɞZV@ٞ }<;O_Qqb~h8ro5kG>WB0x&0,50ʬTӯ2Ռ65,ٰ"*.0!B}a9!q:4%ճٲ'$Ʌ˰­l,³'ٰᏰc/ճ F!_o,Fv_BZw2> S 3 c-P! {>b nnNx01xݬnN+"(?\M, 2T2,?zqqaLSy;n6ج]A:JfN d8L &fԹ@[s(]Z˪~>eh#xu o/IJurR8J6&/A&%Z`OT䒠zda+*U-tOtZKJrxĻьbP ]'!OÌԟ8p .N^_m !oG@2/NXU] bZo(ҁawΗM(ŚYLxU"uöQrúe-K-S{n(OdryrJL,G\bK#KI\李IU嵊J}5 ^rXҀg}to5 {**"^JZΝP`97oPr;LWe¯Hܖ" b/Y7[>;+L6 ?-ڊ72#:Upt0EFeR7'']c[.ڿ}}b40!>{g$09N{+tT*kGjdi+eI}l^p|"Kcb2gB'VdCQ3> e%ř.$oޡy .K&|?.~Q+#v5`n84 P~C1-]k^ 6ֈ/qq\2=ОAULDOOy"~D3@Y &SӺ=b1 "*Fuvc&OvZJMܿgzB%Hh|t6x&(Bu?m4֮Uyյ(*7`w{1 KTaS6"4d/i6##JD[%؟M,x/x~M SA3&8#*ӊw ynG7qpA'|sa ȸx_|)/;Ch!>zsLcN[}} WgRio ebV`=5rg]>ӷ+ 1u30gksLE):'c/99ΐKHSruӓ M1O]?N!+/gEEmO02Є1p<孑4>{;KO"j3KkP Nƴ"&xA >uhZX n =%. 過.:p4h!FldV[/-'<#-{lma@F j4+p o]^.% t[j@B9Pı8 .!NN w5G&NR9#P[3*2zP$(h#}tr Ȗ :Rt*BM[ٿfGacao85+<`d6(ՂʦU{ 3DM:f[ W"q3?>:x2`"09~>9R]y||zFHuB`DV{WєqyTJjS ~W7hey:>Qx:!Suͯsm5!]K xN哩XI8XG!t.lXS }ʑjW)"w^ €Y_jj,*'wrG<;T\Vx%vc$D-(WN\䈬2;v;sȾG):"uORͭs@s?0Zх8]=BQ3Or?d~j ws_ZxM\q e)v~pSs>Vpc91Ύ"4*nP{8Ȯh 8suBSUyT,]M"m|Wro-H;ڇmv=z `T)gB͔v]N=%4-IxeMÌ| |b` 0NAZ өD8zޕ$Q"0d>,u6ٯ=n ɲ1Tz}6w*v{tl1w犬1eý$ GINhcX h엿L'[}ԠegV8TR,?.001.AK /!YIIӆfUt>nM8@ͯB` ýڜq*XF-M@ ly(=I^,Uƾ9g#yTjw|kC=jK5S/S "ݱ {;j|ҷ$@5-hH4SjHnM}' z魩v\̸W÷;su0v!pwS&{E3 F#!KM(\MT%qM}(3+{&RU eAb*B5lB)<x!-m; PaX]Y| ԒP5ZS0 l|>*" ^Ku@[g/G>Y:aϒ4ԋ\H'mtt_V[k.dէli65C^TZlJwVK1GDs:EMtOCJR h!T.]gmw?Fqx .G,Ai"=cKJzBڎfjf!H3 hkZ0>m";&ķ^<ƄS̴ VGEӐൢZ3 ܩgJ x^ qu]&6Dɖ_e!g䐪CzX)cB_^5 Y4rcd2;5!9sgUp[EۺH9qI;T#8uDdS'@cَ PQC  !($)WGE!fLރbpQC~E=[!оOAWXSťmm "+rwz !S6vTz*"f< xۅk-)jq"ʍeRc۪rA KIDA 9A@Ml2**D$S  2@@IwLjv'\d=5&'*45@iEmT^/AQ2Riaecp653s x6AG;}tj:Sp\S\Oz;tNfS!+ķrZtc>[xcWQAozږDB7 rn;yIAUs ZvOט%& .kdNUd6 'v8ݖ&WvjX', >*%U*&%Úʭ.ϸB42*݃u)xsȲ6ܤZ2ڟ`r?҅|#&1.55՗SI˯,Vq29$̔>}:1_fі!RLd,U\PR4/v'VSEW̥K#V%K*8C72N{$,^3_y1-])'ע +ͧ]>1azxMc$-.˿v4эpb\_Q5Oa(ң3-x`d3{HsO)V$nO' .S'6VK eUM%2potP"Kׯ1lӛs%%@2xZbuS[DZ2PYP==t|N]ĮW!՛(+2ڴ5:߰8:hl9qd3icߎ" 6{@VI6,Z4*?KƷ*;?kL)GI.Oz)MOPLgǙybh߆ۥΝ}i+ #zߎvcW1! sQ8R}OTCJl4\e&7(霯D5%x:͏ jY< Ym0a?b盯pgGӤu ,vO !f_.o@ c)9;(n*d/?ࡱv XXi͠M`cK3 .I:mJ@yw֩&Ӛ:]?ԬA̮#ývK {ⷴV,ELk0tHr,bZe?70T3l!CvieX2#-13HmT-ԦPnֻ Úa꓄4EZ4?ύBqc< w"H6TCҰEu+jMY͍3 ? 6WOukE mj?OI@+%kZ1]c*^FTSW;[7ZL)NϹ8q2g:eYٚ9Q56{qh!㞓,n?GZ_2 So#F~0< \;H|pTPfцX{G^̍Sy:qIޏQÁjrI/ԭ[x8PR8fww[ޝ`H%o?^hīƯ]z-5iG~P=mi &Kk~hCqs 1u۔tj\T?LjpӍJ®ڡJiJmxf&ɒmP!¡@@/*w|*=zF?c,t4KARޜ&6B"cV%M  bJWXyK޺Zb*Ӟ~Y1 Wˤh*>اNO9Js:iHñ[1t|SKcxJn^j~ԅ- $ 'ݴVMs[h$r?j-'r,gc'&h'6Fb> e$#~bg9L; endstream endobj 626 0 obj <> endobj 627 0 obj [628 0 R] endobj 628 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 620 0 0 839 0 0 cm /ImagePart_2143 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 260.149 719.7 Tm 106 Tz 3 Tr /OPExtFont3 11 Tf (5.4 Methods of forming an ensemble in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 106 Tz 3 Tr 1 0 0 1 123.099 677 Tm 95 Tz (that WC4DVAR still suffers from the local minimum when applying to longer ) Tj 1 0 0 1 123.599 654.2 Tm 93 Tz (assimilation window while our ) Tj 1 0 0 1 281.05 653.95 Tm 106 Tz /OPExtFont4 10.5 Tf (ISGDC ) Tj 1 0 0 1 322.1 654.2 Tm 93 Tz /OPExtFont3 11 Tf (method doesn't have such shortcoming ) Tj 1 0 0 1 123.349 631.149 Tm 90 Tz (and produces pseudo-orbits closer to the true pseudo-orbit. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 123.599 579.799 Tm 112 Tz /OPExtFont3 15.5 Tf (5.4 Methods of forming an ensemble in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15.5 Tf 112 Tz 3 Tr 1 0 0 1 122.9 545.25 Tm 91 Tz /OPExtFont3 11 Tf (In this section, we introduce and discuss the methods of forming an ensemble of ) Tj 1 0 0 1 122.9 521.95 Tm 94 Tz (model states at t = 0 based on the pseudo-orbit, z) Tj 1 0 0 1 374.399 521.95 Tm 105 Tz /OPExtFont5 11 Tf (i) Tj 1 0 0 1 378.699 521.95 Tm 119 Tz /OPExtFont3 3 Tf (, ) Tj 1 0 0 1 382.8 521.95 Tm 105 Tz /OPExtFont3 11 Tf (i = N, 0, which can be ) Tj 1 0 0 1 122.65 499.149 Tm 92 Tz (produced by methods like ) Tj 1 0 0 1 255.349 498.699 Tm 114 Tz /OPExtFont4 10.5 Tf (ISGDc ) Tj 1 0 0 1 295.699 498.899 Tm 93 Tz /OPExtFont3 11 Tf (and WC4DVAR. Such ensemble is treated to ) Tj 1 0 0 1 122.65 476.1 Tm 90 Tz (be the solution of nowcast. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 122.9 431.699 Tm 110 Tz /OPExtFont3 13 Tf (5.4.1 Gaussian perturbation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 110 Tz 3 Tr 1 0 0 1 122.65 401 Tm 96 Tz /OPExtFont3 11 Tf (An easy way to form the ensemble is perturbing the current estimate z) Tj 1 0 0 1 487.699 401 Tm 69 Tz /OPExtFont5 11 Tf (0 ) Tj 1 0 0 1 491.3 401.25 Tm 101 Tz /OPExtFont3 11 Tf ( with ) Tj 1 0 0 1 122.9 377.949 Tm 95 Tz (Gaussian distribution. To form an Ne" member ensemble, one can draw ) Tj 1 0 0 1 496.3 377.949 Tm 85 Tz /OPExtFont4 10.5 Tf (Nens ) Tj 1 0 0 1 122.4 354.699 Tm 97 Tz /OPExtFont3 11 Tf (samples from N\(0, ) Tj 1 0 0 1 220.099 354.449 Tm 62 Tz /OPExtFont4 10.5 Tf (cr) Tj 1 0 0 1 225.849 354.449 Tm 14 Tz (.) Tj 1 0 0 1 227.3 354.699 Tm 147 Tz ('\) ) Tj 1 0 0 1 241.699 354.699 Tm 98 Tz /OPExtFont3 11 Tf (and add onto z) Tj 1 0 0 1 321.6 354.699 Tm 69 Tz /OPExtFont5 11 Tf (0) Tj 1 0 0 1 326.899 354.699 Tm 101 Tz /OPExtFont3 11 Tf (. The parameter ) Tj 1 0 0 1 420 354.699 Tm 80 Tz /OPExtFont4 10.5 Tf (o) Tj 1 0 0 1 424.55 354.699 Tm 43 Tz (- ) Tj 1 0 0 1 426 354.699 Tm 98 Tz /OPExtFont3 11 Tf ( can be chosen to ) Tj 1 0 0 1 122.15 331.649 Tm 94 Tz (obtain the best nowcast skill or simply use the standard deviation of the noise ) Tj 1 0 0 1 122.15 308.35 Tm 91 Tz (model. The problem of this method is that it assumes the error of the analysis is ) Tj 1 0 0 1 122.4 285.299 Tm (Gaussian distributed, which is often not the case even in the perfect model case. ) Tj 1 0 0 1 122.15 262.049 Tm 95 Tz (It is, however, a simple straightforward method to form the ensemble to cover ) Tj 1 0 0 1 121.9 238.75 Tm 91 Tz (the error of the analysis. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 122.15 194.6 Tm 112 Tz /OPExtFont3 13 Tf (5.4.2 Perturbing with imperfection error ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 112 Tz 3 Tr 1 0 0 1 121.9 163.649 Tm 95 Tz /OPExtFont3 11 Tf (In this section we introduce a method to form the ensemble by perturbing the ) Tj 1 0 0 1 121.9 140.6 Tm 92 Tz (image of second last component of the pseudo-orbit, i.e. F\(z_) Tj 1 0 0 1 424.55 140.6 Tm 51 Tz /OPExtFont5 11 Tf (1) Tj 1 0 0 1 428.899 140.6 Tm 89 Tz /OPExtFont3 11 Tf (\), using the histor- ) Tj 1 0 0 1 121.7 117.299 Tm 94 Tz (ical imperfection error. This method needs a large amount of historical data in ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 94 Tz 3 Tr 1 0 0 1 312.949 52.299 Tm 76 Tz (116 ) Tj ET EMC endstream endobj 629 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 630 0 obj <> stream 0  ,,b6וMfg ܮU=Fda]с3׏!xW$8Ԡx5Z7zn ~G;V)闚C .\"Z)yB^ X/n CX/-Y[#8vF[=ϴ "/42C`G{fw=dzi#CwY변o ͠ C$w},$z|y7-6a_/2NefKwbl[:u 9 ݑ,8Qh"<DN֢ygd AoPVqeW3SAD#>sPC=(Ui `& 4^DB%{O$,[N*.tdCىTYv BaWݩ2h?מB 2ި$BJW֬ TJ](ح yA FD>3&PaYU^U{cjzV)j=ljreY mq]LoO\qF @>Il-!Y*hqZ&* ]]{lK %-3Q1.>7{M8;V*iMi)+tEخex\0JLBfeeb e_ Qұ2e8MGV3;FJ(*ִQ]LRB3}굹'{)Wإ̸̜l2ۗR^`wdܭK~9OxgkH&ՅUð@$}&{?TF on?@0J~~1%?5ⴱ qSKqIW++ 2 aJW_*HGS'gHէ~cv~wPp//t ;E1(ƿ~qH: gwUvf^37´.A{F=C?3^ dCDwy3R ?^B G5eV{AD 'F#*GjmylE-*C,e"ҢyyΠGlר8X:n |pB04fW&yߤ$gIx.ӨF)Tg;U_=d% We"Hf5:k ìXnT$壃yr) #F|k|. xv)/һ"^>UQ GܬҾ,-[ TCyM EhdnjCs&Kl6ƺFQZpM(`=R b-«#Ro MX]/=aϱA}RSTluԪ}׫bkE|gG%Hh T,a$stJeW)%-= lzpOap7KosugImϋZ>).=%Y;ɚh?{RBԺ '紬+] 4wyO_w' ’lyCS_Hc-/GkE]NuG\,TzI Kō1<,†hq/?g=~]l:=h0A@k|Iͭ#J_/3gҀƘ[ .<('̐לpvϊ Hn~nT(,h/5XѲOc;\4h_5\ەBm@\1GM:bH2 $3X5 K҇9^#Z{C7g,NܱaCJDƊTos!/Ѩ굻qwEeOOCJJ@@x*3GaR!UNڎC?>/0Ѥ 0i A:P }|#%Hٮ84Zfۅ%Ej`/S#q{~P k=#][?n`}qHV̞!$>Rogc(`;O=1 iؒG7srq*9V`)(Ll;>]NPqkj(QCE∼uHo}hD%py^j :i=)j1e }zc$~a*N1/CLe`D2gW'lg`uH R歟?$F\wq5DEđrCN}HMtخ@k53KyMa65UčdžI5]uD!pw~hC.G$$U!Rd%N'_Xtchw^Hu|@Y/V;l.ldVJgsQJ<1˹50^cU!,Ҙ:ui'r7Dؖ:ϙZOvA=XUgbfk(bKGn@2I?pZ*N\~  8'}ɦͫb̒#0[YOL&JPna_o5ˠH= lRvI8/ q/fJh2R|hg7R0ueN$|R!Mk-3ݷ!N4cDaR;5 4gd8dbzlPZP-ƇteHM{LF/R:8^_~TB%*_'C kڈ != ǞM-{%<6 ]%.Jԧ $]b2:`Z"[*.!O{LiYw^ Kl!V6¯ Gy.tFF>$1E:?tyyo4QplQB_S7+' H<5W >a%ʥ< ^J@;%d!cX*Յ-\~ч}B>"1夰ܲѲK#➷ɸp><]Aog$0rR 5 cN8$3I@L&B%\aw*]na'RTLbAerΎJɐVȏl,J SMt2Y^]]@ bI|ԭ,_(lp_N=W(&X7DZ̓fnJtPbva!GG̀_Z]de9Ih^B;8&`c4pcso?VFX/qqi>Sr4e1!tAƭIxԈ)CIJRz{%ŠP "Y"C±hE"X4GJ ๰0Gw.Z A=yP\7@Mؾ]43*[m.pB9-K܃ Hq8{4h_z=x.H;RP]o]އ'F9׊NKфa3|:I럍J4ڹsC՞o*p s3D:?[(d |af, T8,<'OGqdb̤I׷HOACᘽS 2er$_iCd\* R ou;ؐ;#zD&-%5z2 :եgTݣ<˰|DFڿ.GH dEⶖ\{wApg&c ®ҫ4_r de)kqT|H؍N>o*6_Ö;+ӥ8E@ kTxV;${2߂\[s\sfdh~*2s߈b -&ܰOU?3b)%$_ζ~HWP`ݩ`㣵gBwLY>2ȴ+N >ޒ1c/FȴlzNo,.QCLm}}0l'1Qv̬ajv31h#g"WS=M_}e>lN6#W`YЇ[0^p0g~bO{fzHوS._ XiMyV*p%j!s#oc s]FH75k͉wI|j*/|&k'A8>_+7JA&ra 5[F;Rnt)9_h 9%9 مjם8ѯi3p##IE˙wS<Tauڌ?g ]@پY`#ƨEXDs ) 1q{W.]ފ6Z+ee (xTkܡO:4Sʽ7áf]ƛF;^-Ew˺:hyFtRWrl{GjCI{L(C~ʂ(fŤ@J*Nrn7&|">iDZ.tG\e~4P+Kӆ%k6 I SUv;BE;̩;Sϋv\ek~Ϯ&./Ά5<-_T%Tχfhz$dfWƵU|sk:"]ARoD{_`yQMO+sD :Uѝۜr)dDl~py`0.HZ9!RV٭͞tɈ\$v6x҄F|IF_c[5=Fsh\w/D8h=Y>xJx5 -|5t'X0Џwcqr{ i{o(<( $eԛ~WR6 1J?Eem3.vkQ;a#M L1m9ZhRp|hhN?*|H/K 7gL_(%]O8Ex+,,7WzbÄT³Fqz`ςpkIG\=Eb /Ѱ;LkJR!: x#KNW"# ʦSIͦUaYUOJ\激[ۖl?i0`[A \Ⱥ_BցDikyQuo|y;tэ^7?Vu7TQ^^,2\i,JrcpMhU tJKf>iuX AY4 ˅m-*2g[mBeqo3@ft$g (F* u|^${n M&(=CtcproRt'd&,VlhT7Sܽ3k)'ZjTynfɝ kxGE%n* 19Ihd8߃:SQ]aC* )l:hO@2{&/9LLh*՜%e7;ĩP"V豹)<)«y26-\SF &B>*Xg ✬)7?9QqŘO5~j ? E,ŀFO"L+=\6G,n|OCmNݏ*D lm}^ˠ u_馷YZ]Jǘ֟#gRՈ,Z.v~9^mqpsɧނO F*xpӖ+eLAV#!u&1U %5+F0<XeknsO625wYiĄ{9= Y9@&8z!PM#Cf>@-)275o`5_Twwb"i,77^䪆6@"3$UQrݎ9,J Ytʋ-ɟiB;c~u̓m3-HyM vdÎhVzWjL5lMQ{.1(@x}Ggһ)T&hO8QZetrۥ(x3zv.4e9-Tq|&uP.զ֙qz20Bj1L@S)1D/zׅ&\ Ͼ.%P31e iMZf'/cV "cg۳ExlۗfT3}7`$hYb?Ќ`03 Պw?< XH*p .{uڭJ7zߙ4 #iXSz?sI"g{;G" Jn`<1#`=+դwcr 8&|M_e@LyBDQXb-!7H3>^z!H`lDpP,!2s(xoF GS">6zs-?[ԹW(LϩTo#&(?~W:e|Zim+ ?Q_IBu&DZS*%uo&kcmcb$ yVD̆ߺ QjFCSh-mJOzw6/|5ztC^OR@֟=1lE 4Jgzn}:;0\ο"1ie3PȭcO8zFUP{" vt3{VѮ #Qvx=8'Z7bl}3LNõy?_LJG7^u\JDhkF3TaͰ3>-s1nM!sQfRiP2svb+ԙc;ؓBc 0-NB6扻!BUyIivW4QS\PB7\BY1ڱ5 jf:׍+^@4$d0@}g#<>aRfaP_q6RPL.%ӫV**}zA_C4c]Vr#™Kyrc6a`*cM4 Ew>ݷm-惘,*f5mW `edcùe|ѧ@@L-ްmU-k32k>>)}41w,FBS~$ND10]!LI( Lֶyw)l ,hчCYEЎ.w|TUﴎ70#fp[A=Kmk'yWcwM}73 }"05sMMGge'H;7-RI T> endobj 632 0 obj [633 0 R] endobj 633 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 620 0 0 838 0 0 cm /ImagePart_2144 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 261.85 719.45 Tm 106 Tz 3 Tr /OPExtFont3 11 Tf (5.4 Methods of forming an ensemble in IPMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 106 Tz 3 Tr 1 0 0 1 125.049 676.5 Tm 92 Tz (order to record a large set of imperfection error for future sampling. As we men-) Tj 1 0 0 1 124.799 653.7 Tm 90 Tz (tioned before, our state estimation method ) Tj 1 0 0 1 335.75 653.45 Tm 115 Tz /OPExtFont4 10.5 Tf (ISGDc ) Tj 1 0 0 1 374.899 653.7 Tm 89 Tz /OPExtFont3 11 Tf (produces a set of imperfection ) Tj 1 0 0 1 124.799 630.899 Tm 92 Tz (errors along with the pseudo-orbit. We apply the state estimation method to the ) Tj 1 0 0 1 124.549 607.85 Tm 93 Tz (historical data and record all the imperfection errors. To form an ) Tj 1 0 0 1 455.5 605.899 Tm 81 Tz /OPExtFont4 10.5 Tf (Nens ) Tj 1 0 0 1 481.199 607.85 Tm 87 Tz /OPExtFont3 11 Tf (member ) Tj 1 0 0 1 124.799 584.799 Tm 90 Tz (ensemble, we randomly draw ) Tj 1 0 0 1 269.3 582.649 Tm 76 Tz (Nens ) Tj 1 0 0 1 289.899 584.799 Tm 90 Tz ( samples from the historical set of imperfection ) Tj 1 0 0 1 124.549 561.5 Tm 95 Tz (errors and add them onto F\(z_) Tj 1 0 0 1 280.55 561.75 Tm 42 Tz (1) Tj 1 0 0 1 285.1 561.5 Tm 94 Tz (\). The advantage of this method is that the en-) Tj 1 0 0 1 124.299 538.5 Tm 92 Tz (semble members tend to cover the uncertainty of model error. The disadvantage ) Tj 1 0 0 1 124.299 515.45 Tm 96 Tz (are i\) the imperfection error is usually not IID distributed, they usually have ) Tj 1 0 0 1 124.299 492.649 Tm 95 Tz (strong spatial correlations as shown in Figure 5.3. As simple random sample ) Tj 1 0 0 1 124.299 469.35 Tm (of imperfection errors may lose this useful information; ii\) the results are also ) Tj 1 0 0 1 124.099 446.1 Tm 92 Tz (strongly depending on how good the second last component of the pseudo-orbit ) Tj 1 0 0 1 124.299 422.8 Tm 93 Tz (estimates the true state. 'We believe better methods can be found by extracting ) Tj 1 0 0 1 124.299 400 Tm (more information in the imperfection error. In this chapter we give an example ) Tj 1 0 0 1 124.099 376.949 Tm 91 Tz (to suggest that imperfection error might be useful to produce nowcast ensemble. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 124.299 332.1 Tm 110 Tz /OPExtFont3 13 Tf (5.4.3 Perturbing the pseudo-orbit and applying ) Tj 1 0 0 1 462.699 332.3 Tm 105 Tz /OPExtFont6 15 Tf (iSGDc ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 15 Tf 105 Tz 3 Tr 1 0 0 1 123.849 301.35 Tm 95 Tz /OPExtFont3 11 Tf (Another way to form the initial condition ensemble is perturbing the pseudo- ) Tj 1 0 0 1 123.849 278.1 Tm 94 Tz (orbit and applying ) Tj 1 0 0 1 222 278.1 Tm 111 Tz /OPExtFont4 10.5 Tf (ISGDc. ) Tj 1 0 0 1 268.1 278.1 Tm 92 Tz /OPExtFont3 11 Tf (As we discussed in Section 5.2.5, given a sequence ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 123.599 255.049 Tm 94 Tz (of observation, s_n\) ) Tj 1 0 0 1 225.599 252.899 Tm 149 Tz /OPExtFont2 6.5 Tf (s-n+1\) SO\) ) Tj 1 0 0 1 290.899 255.049 Tm 97 Tz /OPExtFont3 11 Tf (we can find a pseudo-orbit, z-n\) ) Tj 1 0 0 1 459.85 253.1 Tm 158 Tz /OPExtFont2 6.5 Tf (Z-n-I-1\) ZOI ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 6.5 Tf 158 Tz 3 Tr 1 0 0 1 123.599 231.75 Tm 96 Tz /OPExtFont3 11 Tf (by the ) Tj 1 0 0 1 161.05 231.75 Tm 115 Tz /OPExtFont4 10.5 Tf (ISGDc ) Tj 1 0 0 1 202.3 231.75 Tm 95 Tz /OPExtFont3 11 Tf (method. One may consider the last component of the pseudo-) Tj 1 0 0 1 123.849 208.7 Tm 97 Tz (orbit,zo, as a point estimation of the current state. To form an /V"' member ) Tj 1 0 0 1 123.599 185.45 Tm 91 Tz (ensemble, we perturb the pseudo-orbit with the distribution of the observational ) Tj 1 0 0 1 123.599 162.149 Tm 86 Tz (noise ) Tj 1 0 0 1 152.15 162.149 Tm 88 Tz /OPExtFont4 10.5 Tf (Nens ) Tj 1 0 0 1 179.3 162.149 Tm 93 Tz /OPExtFont3 11 Tf (times, apply the ) Tj 1 0 0 1 264 162.149 Tm 115 Tz /OPExtFont4 10.5 Tf (ISGDc ) Tj 1 0 0 1 303.6 162.149 Tm 91 Tz /OPExtFont3 11 Tf (method on the perturbed pseudo-orbits and ) Tj 1 0 0 1 123.349 138.899 Tm 93 Tz (finally record the last component of each pseudo-orbit produced by the ) Tj 1 0 0 1 485.05 138.899 Tm 111 Tz /OPExtFont4 10.5 Tf (ISGDe ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 10.5 Tf 111 Tz 3 Tr 1 0 0 1 314.649 52 Tm 76 Tz /OPExtFont3 11 Tf (117 ) Tj ET EMC endstream endobj 634 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 635 0 obj <> stream 0  ,,[jQq9Sl@ȧ2*4esP%ČWJV r6xֽjsUkidm0։*~-ZrbvJrlfybQtWdk 'Em$b /_ǥ")D aQ{T=1X_[.frœ ;u}報,3v)|A"v2iKb9!.1!r@v .}h}g @~%DJIJ.0>Hd!;PVߌ_9!kZ8q%Mʕ:N*ĥaYcʵI"4$c cwNe( D 򰬕Shh@Q~;8]sنQ,kA܇z4*Z ˰M3o;|tăހr%߇9S8BTF*_ ,Oݽ -mQzvYgrn/79R*o/$SwA %eOzx"XвG6݉pύ* ʔX7Od갭ƅBjjPñ&ٍ˱q\* -7 Yaeqt(@/U;.~J*t9u-@kES9%H%jRDdIl:.O1<wz}Q{(oJ1Lcʑ)GvX\,Th&[Esr DB#L#(kjcCEs;.tZ\z"]}B_۔81:[[nm](Ɛ s8>,2X"^XcB?z83!4̹r+U7\vzdd26ȓ޶)SڃXGӉՕU^~  Aax:W/2sʉf2Bki,ֽt\zB珞uڜ -BN*?%_e8bzjC"jgb®܂}&S:/լ0+$מ [xS̗DZRK^R7>(v ɒꍶ_4-.1Cy?aA8lJ:5{H]@oqi7U{p;5?XY3 &;P} *0,m {=M"H4P8p-,>%3P>v[mIĤ,ԇJjAʺ6&Oy:Zk -ri'cAMi( (&g:(nY5:aˏ2Rz(hEdB_7.r։wK^%p7!q2PբL[1klB+ =etLE J :hOSFF@(,>N6U$ݚSsHk)LF97 giA0ҞjtM3=z a3_+G*ng W[f[t-g*"{*Fx>A^,Dp߭YP;q)a'soI'+=DLEXB޾P3JKa.*~y_g^1@O|&0(_F0bg~b #ZӵrVLYbsepRƇAKhvq]2IC|ȸK JXvj8-f 4BmLlOK̥hE70K5z3JDn' wijB}A?i:+,oJJ$W:~]jDG8MhCZ$(lvj$t`gK 3l4il"[{`^8 spuCw6([+`ƞc((USz!lsi`~ߙ)NZeL( j36 qQo8U| <2Jݘi_OUo[7W \ W-K&ҟ-Kգ^\=JVX ƒG3JKm;GCӷJ :i'WzMqԷbG%(qk j03D7Y7k)0x*u9vy&Ӛ^wV]H5=aR"Ȏ` 3 ??Me;dej`wFo&b'(;~ Ʌ!ŜFOV?IYhZSiWhLM+RU5l}Iǽ%*Tvgf%y&gۉtHTf`CS!#ة=sFN4>\m,Ŀ! 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K: mӴ?`3W:t{_DC]/h[!v|̔@MVZL`Q뫆+-G/Io%GMｒw苤D(9$'Ay#)w*$O n x  `QVnm~Q 䲑"AhÚV O >H ^82*o&5S2\V M/Z [.G&w;J~x@e@4z%-~gܘ-py)f1uEc5bZ$%W!a:/۳3(_6hqRm5hV hOʒXᾖU ĜQe׏B\dAȼ$0M7,PNc GGTuW 81-ʀ?* Ko endstream endobj 636 0 obj <> endobj 637 0 obj [638 0 R] endobj 638 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 620 0 0 839 0 0 cm /ImagePart_2145 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 457.199 719.7 Tm 112 Tz 3 Tr /OPExtFont5 13 Tf (5.5 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr 1 0 0 1 124.099 676.5 Tm 97 Tz (method as one of the ensemble member. Each ensemble member can be treated ) Tj 1 0 0 1 124.099 653.7 Tm 96 Tz (equally or weighted according to the likelihood of its corresponding pseudo-orbit ) Tj 1 0 0 1 123.599 630.899 Tm 103 Tz (given the observations. The results presented in section 5.5.2, show that this ) Tj 1 0 0 1 123.599 608.1 Tm 100 Tz (method produces better nowcasting ensembles than the other two methods. It ) Tj 1 0 0 1 123.599 584.85 Tm 103 Tz (is, however, very costly to run the ) Tj 1 0 0 1 304.55 584.85 Tm 98 Tz /OPExtFont4 12 Tf (ISGDe ) Tj 1 0 0 1 345.35 584.85 Tm 99 Tz /OPExtFont5 13 Tf (method to generate each ensemble ) Tj 1 0 0 1 123.599 561.799 Tm 92 Tz (member. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 124.099 509.5 Tm 118 Tz /OPExtFont3 15.5 Tf (5.5 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15.5 Tf 118 Tz 3 Tr 1 0 0 1 123.349 474.899 Tm 93 Tz /OPExtFont5 13 Tf (In this section we first compare the ) Tj 1 0 0 1 294 474.899 Tm 90 Tz /OPExtFont4 12 Tf (ISG.Dc ) Tj 1 0 0 1 332.399 474.899 Tm 94 Tz /OPExtFont5 13 Tf (method with WC4DVAR by looking at ) Tj 1 0 0 1 123.099 451.899 Tm 99 Tz (the pseudo-orbits they provide. Results are then shown the comparison among ) Tj 1 0 0 1 123.349 428.85 Tm 101 Tz (the ensemble formation methods. Finally we compare the ensemble nowcasts ) Tj 1 0 0 1 123.349 405.3 Tm 94 Tz (based on ) Tj 1 0 0 1 170.65 405.3 Tm 90 Tz /OPExtFont4 12 Tf (ISG.Dc ) Tj 1 0 0 1 210 405.3 Tm 95 Tz /OPExtFont5 13 Tf (with an Inverse Noise ensemble. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 123.599 360.699 Tm 108 Tz /OPExtFont2 14 Tf (5.5.1 ) Tj 1 0 0 1 171.099 360.699 Tm 91 Tz /OPExtFont4 14 Tf (ISG.De ) Tj 1 0 0 1 219.099 360.899 Tm 116 Tz /OPExtFont2 14 Tf (vs WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 14 Tf 116 Tz 3 Tr 1 0 0 1 123.349 330.199 Tm 95 Tz /OPExtFont5 13 Tf (Both the ) Tj 1 0 0 1 168.949 330.199 Tm 90 Tz /OPExtFont4 12 Tf (ISG.Dc ) Tj 1 0 0 1 207.849 330.199 Tm 93 Tz /OPExtFont5 13 Tf (and WC4DVAR produce a pseudo-orbit from which an ensemble ) Tj 1 0 0 1 122.9 306.899 Tm 106 Tz (of the current state estimates can be constructed. In this section instead of ) Tj 1 0 0 1 122.9 283.649 Tm 99 Tz (comparing ensemble nowcasting results, we compare the quality of the pseudo-) Tj 1 0 0 1 122.9 260.35 Tm 96 Tz (orbit each produces. We apply both methods in the higher dimensional Lorenz 96 ) Tj 1 0 0 1 122.65 237.299 Tm 97 Tz (system-model pair experiment and the low dimensional Ikeda system-model pair ) Tj 1 0 0 1 122.65 214.299 Tm 98 Tz (experiment. And in each case, different lengths assimilation windows are tested. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 139.9 191 Tm 96 Tz (Firstly we measure the distance between observations and pseudo-orbit \(equa-) Tj 1 0 0 1 122.4 167.7 Tm 98 Tz (tion 5.15\), and the distance between true states and pseudo-orbit \(equation 5.16\) ) Tj 1 0 0 1 122.9 144.45 Tm (as diagnostic tools to look at the quality the model trajectories generated by each ) Tj 1 0 0 1 122.4 121.399 Tm 94 Tz (method. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 94 Tz 3 Tr 1 0 0 1 313.449 52.299 Tm 84 Tz (118 ) Tj ET EMC endstream endobj 639 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 640 0 obj <> stream 0  ,,rRRV4fhT9 h^KerL~x-+.ftlAw{hp~CpiЕorMP:dJuZ-rV?Lxj nC5P "<7p~=,_Z Hlɤ(9^$.#lQ3 ,%YM;\MnSG^Ͼ9dE*@+A,%綝WB Y~mXEozS.:j@x3@Pe;ѕ GX߬@x30ZXA.Q _y, ݻWr5)99Qs o5tp# )+^A0 WbqU2NH jAáIkT<#uLR-ѳ]9rM)6}o Gi0'9"n&հ 0$DKcYP Ym!?Y,t\t(p;M޼g ߡ)qJ $ [ TPq,>8#{ ikg1}fBHE@_*SU 8K]",4f(}@Rᑞ$6CKIO+{hUhNd[ /\fnvX "'K)FX^ |5ÈХ/u2*hɡywBHUtuM?T\=;V-'?:Jz"a5b[3C[D&b4Ge(V3iuFhϠi51Ȅ@25 j,FpE[AolD9 ˟NRR=f˕Ȥ M*nSBB`"X6C'1OM<(7= XFp' -ц7g*̥zj_ 5 LWdx SAj}{I}j2gZ~cHޣ&k}oXpN̸An%d>Gf kn8hh "%bB\CWCsqDuh;`|̦bҍ0I |R}w\h?c[][ĺ5ViaD u]Gn^a 6)L5ՐyhZ/ :BrDO}l[?P&/Ֆ!56%&wA QVjHPZeBVwV~aSV@%yCM$k330#Iw5&ͪc,ȟZ D(|,ÑU1/Q3nDYy|a@#ᒻ-ƚ ,2-Enَ|W8›BB;m Dʁ~pڡ.j^e +y|5w's/oA[Ÿ10w ϋ|2Я~#tE}׾⟇NSU|?>qI *_ і'9؜]?Ё![`?@{vwQ3`#-rALC$9,6J5!jq<ѮXuSZ%mʣ4يʷ'xOaԇ; .s7)3\2qe3*BUqPbXbN&vfjS9Gs]>*,)0&lmLhEav|uD`BX?:X0Rn&Hg\Mz՟o'se}-(W ˕jVZC .ߚRו [ziG;m|ِZc#'iZ^{I"b2ah$x^ՎK'M`@@6?&8oKolGIfE/>1:HiE`Dg)F[RS(=U_ֶ]VDXFT@aٺ ,> J˶qD/41G}ʶr߆~ -'NmhO+2)v*Y)1j2FT֕g=ٖ=\2!'V_VhZքho_y@ ͕OZe[7 ?x\.O+k~d&(1p"Nb,TUBL(*(1U㎣67;Yu1Ü_2RO@@݀9N7uH_A1 g`ŏaXjGx`X <Wă?5HW[ Wѧ-h# wd÷ k(5$+CJ>ޚm7:hqbΐegx!ɪT߁ :,i7/ÔK][$"zo`ڞBnް/h?JCm FY!B[%61'b `1 ;*0TCPԩuCu.|eO@Tc/{6X/<]0FX"J;XRwE?2DȽ{L2Itƛ҄iMw֝n D7B4dWhLГQ`H.eq:/˪gJ$ _aWܹbߊ}˫䄞Gs>YV%/@z5 i X#Yߙecv1%r &0hC pzZJ牘Zzs0!BnD(p`Md52 MΥvЊ+ qWp{_<߳'2vպA lfJ!CUj|x/:CAJR]_X4^A^0!gd7+tOge@C_ v0rG!aڈi=圸q|:9ËOosFUֱy0ZD泝di΢ M(a"IB} _$ X T@8騧Q b$ר| A_tbHQ\cG궳h!14h~%-4:ٰf=$_.6CV:(Z g#ڲg1!<cߎrw# X=? YtIU[0.w(m]Iヶs6 Awf% W ߣMCӎ.M)oð07ӘGBFXNF/NS| m1 Jn<L^}.$BKr>69L/ ʨЁ{GRA#@#C!V@#mɜMXn]HdEvR'@2`@$%HN ae*b1$s& W$S JÔsU,_9<6Cؗt}DfF@&djf##2 q20Zq?ՂNu%}9s;VLC`|=-O .6 gGV/j5_8},bϷKr {(W^nw%C2_\8gmWEVť5^ĻnF& ĬGN׼O-)O\-dLd((4Ҥ70M2[vg|6:mMbⴄc1^41i=?q'ad3F2nh%LɵnV\yT+Mg+Ft|suC6ى3VfL)TҠZ>Hh0 ;],>Kig/]ÒNlJwU=N!@~2rWtqVNqkvaEfUH@hCagJR]#nSCaDh1Џ }HNߪVՇ?6zo.çyG10͓x cֺ5 !,s\ $笣֐;ٱfX:ܵ=r9GZ&>^{CaL]rx9ݳ0X "UfDv`֑鈫iOlԯ jQ8YRrH~Q!>O,=9}%?rfW~]S:|W۟tl;P:4HJs4zP~@1k N^msYmx(!5cT5vl oMiON{Z>H!F'+#ie!uASJл{*| ͆"NV .Q/YJR}ǀm}ILe/XJO;t`Я15w@F#iH3(CΩKn{KN CQX!.x"j~-_nޛ7_$x 5(GQ@~2;b Ji+<#G1 UTO^*VۅANNZ 7.yl~6!J(\a|N:e/ӫҧ8CzP?\T>z;F(*W6zʝy}U:xwtbwձ>NG2O-(g6"Tl\)V] ^{\Z/PRe+ }! (>>c8A5`މv=f}95A6Ɗ6.SyȳPV ys0OP"oQM)&sdZ09:zO¼BppRg0eGr=ԀъޏR0['T^w칚pe>UR?F'ť`!Tt>Q1Ρ;' i>mi?RDPF>!I̟kYa2Pn.؏d"w`={YdZzE endstream endobj 641 0 obj <> endobj 642 0 obj [643 0 R] endobj 643 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 620 0 0 839 0 0 cm /ImagePart_2146 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 458.399 720.45 Tm 112 Tz 3 Tr /OPExtFont5 13 Tf (5.5 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr 1 0 0 1 151.699 678.899 Tm 94 Tz (Window ) Tj 1 0 0 1 156.699 665.25 Tm 96 Tz (length ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 96 Tz 3 Tr 1 0 0 1 271.899 678.899 Tm 85 Tz /OPExtFont13 11.5 Tf (Distance from observations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 85 Tz 3 Tr 1 0 0 1 227.75 665.25 Tm 98 Tz /OPExtFont5 13 Tf (Average ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 334.3 665.25 Tm 89 Tz (Lower ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 434.399 665.25 Tm 90 Tz (Upper ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 203.75 651.549 Tm 67 Tz (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 67 Tz 3 Tr 1 0 0 1 257.05 651.299 Tm 98 Tz /OPExtFont4 12 Tf (ISGDe ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 12 Tf 98 Tz 3 Tr 1 0 0 1 304.55 651.299 Tm 85 Tz /OPExtFont13 11.5 Tf (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 85 Tz 3 Tr 1 0 0 1 357.6 651.299 Tm 98 Tz /OPExtFont4 12 Tf (ISGDe ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 12 Tf 98 Tz 3 Tr 1 0 0 1 405.35 651.299 Tm 85 Tz /OPExtFont13 11.5 Tf (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 85 Tz 3 Tr 1 0 0 1 458.399 651.299 Tm 98 Tz /OPExtFont4 12 Tf (ISGDe ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 12 Tf 98 Tz 3 Tr 1 0 0 1 155.05 637.149 Tm 96 Tz /OPExtFont5 13 Tf (4 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 96 Tz 3 Tr 1 0 0 1 204.5 637.149 Tm 89 Tz (1.52 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 257.75 636.899 Tm 88 Tz (1.19 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 305.5 636.899 Tm 86 Tz (1.45 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 86 Tz 3 Tr 1 0 0 1 358.3 637.149 Tm 88 Tz (1.13 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 406.1 637.149 Tm 87 Tz (1.60 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 87 Tz 3 Tr 1 0 0 1 459.1 637.149 Tm 88 Tz (1.24 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 155.05 623 Tm 96 Tz (6 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 96 Tz 3 Tr 1 0 0 1 204.25 623 Tm 89 Tz (2.89 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 257.5 623 Tm 88 Tz (1.29 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 304.8 622.75 Tm 92 Tz (2.27 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 358.1 623 Tm 89 Tz (1.24 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 405.35 622.75 Tm 91 Tz (3.60 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 459.1 623 Tm 90 Tz (1.34 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 155.05 608.6 Tm 96 Tz (8 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 96 Tz 3 Tr 1 0 0 1 203.5 608.6 Tm 92 Tz (4.61 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 257.5 608.35 Tm 90 Tz (1.34 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 304.8 608.35 Tm 91 Tz (3.80 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 358.1 608.35 Tm 89 Tz (1.30 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 405.85 608.6 Tm 88 Tz (5.52 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 458.899 608.6 Tm 89 Tz (1.37 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 151.199 593 Tm 95 Tz (Window ) Tj 1 0 0 1 156.25 578.85 Tm 98 Tz (length ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 274.8 593 Tm 85 Tz /OPExtFont13 11.5 Tf (Distance from true states ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 85 Tz 3 Tr 1 0 0 1 227.75 578.85 Tm 98 Tz /OPExtFont5 13 Tf (Average ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 334.3 578.85 Tm 89 Tz (Lower ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 434.649 578.85 Tm 90 Tz (Upper ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 203.5 564.899 Tm 67 Tz (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 67 Tz 3 Tr 1 0 0 1 256.8 564.899 Tm 98 Tz /OPExtFont4 12 Tf (ISGDe ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 12 Tf 98 Tz 3 Tr 1 0 0 1 304.3 564.899 Tm 86 Tz /OPExtFont13 11.5 Tf (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 86 Tz 3 Tr 1 0 0 1 357.6 564.899 Tm 98 Tz /OPExtFont4 12 Tf (ISGDe ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 12 Tf 98 Tz 3 Tr 1 0 0 1 405.1 564.899 Tm 67 Tz /OPExtFont5 13 Tf (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 67 Tz 3 Tr 1 0 0 1 458.149 564.899 Tm 98 Tz /OPExtFont4 12 Tf (ISGDe ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 12 Tf 98 Tz 3 Tr 1 0 0 1 154.55 550.5 Tm 97 Tz /OPExtFont5 13 Tf (4 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 204 550.5 Tm 91 Tz (0.70 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 257.05 550.5 Tm (0.67 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 305.05 550.5 Tm 88 Tz (0,65 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 357.6 550.299 Tm 92 Tz (0.63 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 405.6 550.5 Tm 89 Tz (0.76 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 458.399 550.5 Tm 80 Tz (0.71. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 80 Tz 3 Tr 1 0 0 1 155.05 536.1 Tm 96 Tz (6 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 96 Tz 3 Tr 1 0 0 1 203.75 536.1 Tm 93 Tz (2.07 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 256.8 536.1 Tm 91 Tz (0.55 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 305.05 536.1 Tm 89 Tz (1.43 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 357.6 536.1 Tm 92 Tz (0.52 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 405.6 536.1 Tm 88 Tz (2.81 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 458.399 536.1 Tm 92 Tz (0.58 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 154.8 521.95 Tm 96 Tz (8 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 96 Tz 3 Tr 1 0 0 1 203.5 522.2 Tm 92 Tz (4.01 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 257.05 521.95 Tm 91 Tz (0.50 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 304.8 521.95 Tm (3.19 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 357.6 521.95 Tm 93 Tz (0.47 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 405.35 521.95 Tm 91 Tz (4.88 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 458.399 521.95 Tm 92 Tz (0.52 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 124.549 498.199 Tm 103 Tz (Table 5.3: Ikeda system-model pair experiment \(Experiment G\): a\) Distance ) Tj 1 0 0 1 124.299 484.3 Tm 100 Tz (between the observations and the pseudo-orbits generated by WC4DVAR and ) Tj 1 0 0 1 124.299 470.6 Tm 96 Tz /OPExtFont4 12 Tf (ISGDe, ) Tj 1 0 0 1 167.5 470.35 Tm 98 Tz /OPExtFont5 13 Tf (b\) Distance between the true states and the pseudo-orbits generated by ) Tj 1 0 0 1 124.099 456.199 Tm 95 Tz (WC4DVAR and ) Tj 1 0 0 1 207.599 456.199 Tm 98 Tz /OPExtFont4 12 Tf (ISGDe ) Tj 1 0 0 1 247.699 456.199 Tm 99 Tz /OPExtFont5 13 Tf (in Ikeda system-model pair experiment. Average: aver-) Tj 1 0 0 1 124.299 442.3 Tm 97 Tz (age distance, Lower and Upper are the 90 percent bootstrap re-sampling bounds, ) Tj 1 0 0 1 124.099 428.35 Tm 99 Tz (the statistics are calculated based on 1024 assimilations and 512 bootstrap sam-) Tj 1 0 0 1 124.099 414.199 Tm 100 Tz (ples are used to calculate the error bars. \(Details of the experiment are listed in ) Tj 1 0 0 1 124.099 400.5 Tm 98 Tz (Appendix B Table B.7\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 258.949 324.45 Tm 215 Tz /OPExtFont4 3 Tf (\() Tj 1 0 0 1 262.55 324.2 Tm 99 Tz /OPExtFont8 12.5 Tf (H ) Tj 1 0 0 1 272.149 324.2 Tm 798 Tz (\t) Tj 1 0 0 1 297.1 324.2 Tm 584 Tz /OPExtFont3 3 Tf (- ) Tj 1 0 0 1 307.899 324.2 Tm 74 Tz /OPExtFont3 11.5 Tf (st) Tj 1 0 0 1 315.85 324.2 Tm 62 Tz (i) Tj 1 0 0 1 319.899 324.2 Tm 77 Tz (\)) Tj 1 0 0 1 323.75 323.95 Tm 80 Tz (T) Tj 1 0 0 1 330.699 323.7 Tm 90 Tz (R7) Tj 1 0 0 1 345.6 323.7 Tm 40 Tz (1) Tj 1 0 0 1 350.649 323.25 Tm 101 Tz (\(H\(zt) Tj 1 0 0 1 377.75 322.75 Tm 56 Tz (i) Tj 1 0 0 1 381.6 322.75 Tm 77 Tz (\) ) Tj 1 0 0 1 388.8 322.75 Tm 601 Tz /OPExtFont3 3 Tf (- ) Tj 1 0 0 1 399.85 322.5 Tm 89 Tz /OPExtFont3 11.5 Tf (SO, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11.5 Tf 89 Tz 3 Tr 1 0 0 1 492.949 324.2 Tm 93 Tz /OPExtFont5 13 Tf (\(5.15\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 296.649 237.799 Tm 106 Tz (- RO) Tj 1 0 0 1 325.199 237.799 Tm 71 Tz /OPExtFont3 13 Tf (T ) Tj 1 0 0 1 330.949 237.799 Tm 108 Tz /OPExtFont5 13 Tf ( Ri \(zt, - Rt,\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 108 Tz 3 Tr 1 0 0 1 492.949 239.5 Tm 93 Tz (\(5.16\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 140.65 197 Tm 104 Tz (From Table 5.3 and 5.4 we can see that when the assimilation window is ) Tj 1 0 0 1 123.349 173.7 Tm 103 Tz (short, e.g. 4 steps, both WC4DVAR and ) Tj 1 0 0 1 339.6 173.7 Tm 99 Tz /OPExtFont4 12 Tf (ISGDe ) Tj 1 0 0 1 380.899 173.7 Tm 102 Tz /OPExtFont5 13 Tf (produce similar results that ) Tj 1 0 0 1 123.099 150.45 Tm 101 Tz (the pseudo-orbits are closer to the true states than the observations except the ) Tj 1 0 0 1 123.099 127.149 Tm 98 Tz (pseudo-orbit produced by ) Tj 1 0 0 1 257.75 127.149 Tm 99 Tz /OPExtFont4 12 Tf (ISGDe ) Tj 1 0 0 1 299.05 127.149 Tm 103 Tz /OPExtFont5 13 Tf (is slightly closer to the observation and the ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 103 Tz 3 Tr 1 0 0 1 314.399 52.299 Tm 86 Tz (119 ) Tj ET EMC endstream endobj 644 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 645 0 obj <> stream 0  ,,'Aϔb6 @ ٌt 2{Kq]`T~OG[ =&ۍ|ǹ(*n&&m_`\ƃ||WFhsK@џO^DCjFwQR  ~41jzUgVzhc0 |wmƫ ^+e[ 11R@T[C[=Jk]FMDp_YXnP6%mŸ!b-RpH*'-$Xo,盥!e9U'n`&fE}d%؄mċKΝe| l,ҶZ$3 i洷rc?(3h0fcec8D2 1^VЛ|`Rhc Y~ola)mG(1)\lC ~4o-kw͌?lyuk!KWYEXf܈ݺ @zVxfU`/)ulA?,%R㿭>͐pV8A1pA( >=TV]'!r# #qw 6m,-9&NGJQoךPtJ)xbsfD]sm)K>&b"|.QVX3דbk")atx&ɻ|?#ΆWd}'d h5:ܜ&^ h]dE:6ncɢ3kT[]:@f@Y0_OUʵ A IG(iA0ھZUXp4l__Jd4i.Z1^nqJqmE /ӥ 3AO7yόQHU%HON/ |֡yGk<}#${ԶHYØ ;X)SrtpܴϹ%/"ݸ1ӥW>2dE`uExk\`韉ƞe5޿?jAbs5ܦ"Z3гi5x-[wi~c/s~PI4''h~ ,cE \_mUhQ*K/KbzNi[v-J^dx.̓spJ8Ć2p*c5խS#rx, }UsJWfxR_yz5݇6&7N|zI/8T{ wI|V"Zo8,QhˆЏn9o^ܟf+kp I(Dq˖t'u/"zGG.FT`dei c#'7<qv=O+ΰ!! dUmƒ ${garA#5d=&|/Awֱ˾+j7*(֠_7?/){f@ݕ)npb}] )_3 s!R3]U3ElG zR"$O<q+QiƱ"փӓ XxHʺ BI/z?d`gmO4}뼷=>Xf[#^ံ {s<)b;F"kPg`jȧ:P˻q_V䡶ɨ#Y`?UX=SD52G{徱o 8ටQT\{G@Jz@0"tMN1Pdv6+c9[qUx־{WJWe$+MՎ@W!^B6^[ıȜMmS5Wo7H&%c-] ibh&W<4yb\ib0exSg(9c$?U 9uxK P"pr[pܚ y.*)Goӻextvo{S{{ A HYb! 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dE$bsHV02"> !L&l3=v$[Ubs7>gz%178{~mG'rD[xE (7ɗ\M1gi$P*rTZ3˙ Bk:t?c5 oh|L!V,V/>ߺQʻY_s s9SC\03 endstream endobj 646 0 obj <> endobj 647 0 obj [648 0 R] endobj 648 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 621 0 0 840 0 0 cm /ImagePart_2147 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 458.649 721.45 Tm 3 Tr /OPExtFont3 11 Tf (5.5 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 3 Tr 1 0 0 1 151.9 679.45 Tm 99 Tz /OPExtFont3 10.5 Tf (Window ) Tj 1 0 0 1 156.699 666 Tm 94 Tz (length ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 94 Tz 3 Tr 1 0 0 1 272.649 679.7 Tm 109 Tz (Distance from observations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 109 Tz 3 Tr 1 0 0 1 228.699 666 Tm 102 Tz (Average ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 102 Tz 3 Tr 1 0 0 1 335.05 666 Tm 96 Tz (Lower ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 96 Tz 3 Tr 1 0 0 1 435.35 666 Tm 92 Tz (Upper ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 92 Tz 3 Tr 1 0 0 1 204.699 652.1 Tm 75 Tz (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 75 Tz 3 Tr 1 0 0 1 258 651.85 Tm 106 Tz /OPExtFont6 12.5 Tf (ISCDc ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 12.5 Tf 106 Tz 3 Tr 1 0 0 1 305.3 651.85 Tm 74 Tz /OPExtFont3 10.5 Tf (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 74 Tz 3 Tr 1 0 0 1 358.55 651.85 Tm 105 Tz /OPExtFont4 11.5 Tf (ISCDc ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 105 Tz 3 Tr 1 0 0 1 405.85 652.1 Tm 75 Tz /OPExtFont3 10.5 Tf (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 75 Tz 3 Tr 1 0 0 1 459.1 652.1 Tm 94 Tz /OPExtFont4 11.5 Tf (ISG.Dc ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 94 Tz 3 Tr 1 0 0 1 154.55 637.7 Tm 93 Tz /OPExtFont5 13 Tf (6 hours ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 205.449 637.7 Tm 89 Tz (16.42 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 258.5 637.45 Tm (14.00 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 306 637.45 Tm 90 Tz (16.24 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 359.05 637.7 Tm 89 Tz (13.85 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 406.8 637.899 Tm 87 Tz (16.59 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 87 Tz 3 Tr 1 0 0 1 459.35 637.899 Tm 90 Tz (14.14 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 151.699 623.5 Tm 92 Tz (12 hours ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 205.199 623.5 Tm 90 Tz (20.60 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 258.5 623.5 Tm 89 Tz (14.40 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 305.75 623.5 Tm (20.41 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 359.05 623.5 Tm (14.30 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 406.3 623.5 Tm (20.78 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 459.35 623.5 Tm 90 Tz (14.50 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 151.449 609.1 Tm 93 Tz (24 hours ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 204.699 609.1 Tm 91 Tz (81.11 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 258.25 609.1 Tm 89 Tz (14.52 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 305.75 609.1 Tm 91 Tz (78.17 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 359.05 609.1 Tm 88 Tz (14.45 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 406.1 609.1 Tm 91 Tz (84.17 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 459.35 609.1 Tm 90 Tz (14.59 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 151.9 593.5 Tm 94 Tz (Window ) Tj 1 0 0 1 156.699 579.6 Tm 97 Tz (length ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 275.5 593.5 Tm 113 Tz /OPExtFont3 10.5 Tf (Distance from true states ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 113 Tz 3 Tr 1 0 0 1 228.5 579.6 Tm 101 Tz (Average ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 101 Tz 3 Tr 1 0 0 1 334.8 579.6 Tm 96 Tz (Lower ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 96 Tz 3 Tr 1 0 0 1 435.6 579.6 Tm 92 Tz (Upper ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 92 Tz 3 Tr 1 0 0 1 204.699 565.7 Tm 75 Tz (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 75 Tz 3 Tr 1 0 0 1 257.75 565.45 Tm 94 Tz /OPExtFont4 11.5 Tf (ISG.De ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 94 Tz 3 Tr 1 0 0 1 305.3 565.45 Tm 75 Tz /OPExtFont3 10.5 Tf (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 75 Tz 3 Tr 1 0 0 1 358.3 565.45 Tm 95 Tz /OPExtFont4 11.5 Tf (ISG.Dc ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 95 Tz 3 Tr 1 0 0 1 405.85 565.7 Tm 70 Tz (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 70 Tz 3 Tr 1 0 0 1 458.899 565.45 Tm 97 Tz (ISGDC ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 97 Tz 3 Tr 1 0 0 1 153.849 551.299 Tm 94 Tz /OPExtFont5 13 Tf (6 hours ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 94 Tz 3 Tr 1 0 0 1 204.949 551.049 Tm 92 Tz (5.87 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 257.5 551.299 Tm (4.15 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 305.75 551.049 Tm 89 Tz (5.76 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 358.3 551.299 Tm 91 Tz (4.08 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 406.3 551.049 Tm 88 Tz (5.98 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 458.649 551.299 Tm 92 Tz (4.23 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 151.699 537.1 Tm (12 hours ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 204.699 536.899 Tm (7.92 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 257.75 536.899 Tm (3.06 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 305.5 536.899 Tm 91 Tz (7.77 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 358.55 536.649 Tm 88 Tz (3.01 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 406.1 536.899 Tm 91 Tz (8.10 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 458.899 536.899 Tm 92 Tz (3.10 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 151.199 522.7 Tm 93 Tz (24 hours ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 204.949 522.7 Tm 91 Tz (74.29 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 258 522.7 Tm 89 Tz (2.45 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 305.5 522.7 Tm 92 Tz (71.04 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 358.55 522.7 Tm 91 Tz (2.42 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 406.1 522.7 Tm 89 Tz (77.61 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 459.1 522.7 Tm 92 Tz (2.47 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 124.549 498.949 Tm 98 Tz (Table 5.4: Lorenz96 system-model pair experiment \(Experiment H\): a\) Distance ) Tj 1 0 0 1 124.549 485.05 Tm 100 Tz (between the observations and the pseudo-orbits generated by WC4DVAR and ) Tj 1 0 0 1 124.799 470.899 Tm 98 Tz /OPExtFont4 11.5 Tf (ISGDa, ) Tj 1 0 0 1 169.449 470.899 Tm 102 Tz /OPExtFont5 13 Tf (b\) Distance between the true states and the pseudo-orbits generated ) Tj 1 0 0 1 124.549 456.949 Tm 98 Tz (by WC4DVAR and ) Tj 1 0 0 1 226.55 456.949 Tm 102 Tz /OPExtFont4 11.5 Tf (ISGDc. ) Tj 1 0 0 1 273.85 456.949 Tm 101 Tz /OPExtFont5 13 Tf (Average: average distance, Lower and Upper are ) Tj 1 0 0 1 124.549 443.05 Tm 99 Tz (the 90 percent bootstrap re-sampling bounds, the statistics are calculated based ) Tj 1 0 0 1 124.549 429.1 Tm (on 1024 assimilations and 512 bootstrap samples are used to calculate the error ) Tj 1 0 0 1 124.549 414.949 Tm (bars.\(Details of the experiment are listed in Appendix B Table B.8\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 124.299 381.35 Tm 101 Tz (true states than that produced by WC4DVAR. As longer assimilation window ) Tj 1 0 0 1 124.299 358.3 Tm 98 Tz (being used, the pseudo-orbit generated by ) Tj 1 0 0 1 336.949 358.3 Tm 107 Tz /OPExtFont4 11.5 Tf (ISGD ) Tj 1 0 0 1 376.55 358.3 Tm 97 Tz /OPExtFont5 13 Tf (become father away from the ) Tj 1 0 0 1 124.299 335.05 Tm 101 Tz (observations and closer to the true states while the pseudo-orbit generated by ) Tj 1 0 0 1 124.099 311.75 Tm 98 Tz (WC4DVAR become father away from the observations and the true states. This ) Tj 1 0 0 1 124.099 288.7 Tm 95 Tz (is important because we expect to obtain more information from the observations ) Tj 1 0 0 1 124.299 265.45 Tm 100 Tz (and model dynamics by using longer assimilation window. In the ) Tj 1 0 0 1 457.899 265.45 Tm 113 Tz /OPExtFont4 11.5 Tf (ISGIY ) Tj 1 0 0 1 498 265.45 Tm 93 Tz /OPExtFont5 13 Tf (case, ) Tj 1 0 0 1 123.849 242.399 Tm 97 Tz (the pseudo-orbit moves closer to the true states as we expected. The WC4DVAR ) Tj 1 0 0 1 123.849 219.1 Tm 98 Tz (method, however, fails to produce better pseudo-orbit when applying on longer ) Tj 1 0 0 1 124.099 195.85 Tm 101 Tz (assimilation windows. We suggest without proof that such failure is due to in-) Tj 1 0 0 1 123.849 172.549 Tm (crease of the density of local minima of the cost function, especially when the ) Tj 1 0 0 1 123.599 149.5 Tm 100 Tz (minimisation tends to obtain small imperfection error; as the WC4DVAR cost ) Tj 1 0 0 1 123.599 126.25 Tm 97 Tz (function depends on not only the initial state but also the imperfection errors, we ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 314.649 53.5 Tm 86 Tz (120 ) Tj ET EMC endstream endobj 649 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 650 0 obj <> stream 0  ,,'Ôj~ o>6=M;ǹ&įI=oJ,)a':B$iۛ$o3v3&`xx*^t-]Pr f@|V3Njx s,̇rPE<4Pj;% gʉs4e*J<2h"¨H zʒUҵXt~@6wv?%:A% MrgUQ^#Gn2F ErzesP\x/+9C1aGnfwy4OJ|:e؝ d~-dW3^Z^%;4 /Q6l޷u&f.MbV #s^DAٕDȾ -c~qÅSC)hKx#ޛZ,QC&I|&)9,^6ࢭ>85n͍X Ce?δ?<'#"<̬3s~;@'8)=o0Wx2uWV n VĨ. /v}.}r5> -uVX.,WTz :w‡UVE>ӏǮDD12p9%!E7;2Z&ޮګmZ+jGX!od#j)/u-L1TSu8 6kԩ.|L a`kљ~<,U2<#tDA}r~<8'0L?YhQJ/!xrZg{]L)4_ڡX$־sh,4Kg׏nmp!>bCQ#QI>K}DٕORo>shbV>u0~MoBJ HR6Te+rx("vhEHƠ )ȷH65v{7ȋoy &2buXj.pt]Ƶ`mk":g@a1䚀ڜ%ަ0GAy qA;t%_{T;^ןGkBNpOC:EI$86Xlqo(7l-m%{0}[h-&Wp"ѥor1).̸q0_PK@BIOSY_} b_L;R| %2*b&u!Tq"ĥ <] r\yXj۪V ׸״w:kFË` 'm|=/yj pLm,nJ\S or9FlMr}`BN'Rox#bh䳄"Q٧2V/'ݖQwsXC}-sߙ㙠z ܜـVg1 ݎ,KE>4zjAqZ=v,ݪsFGT2?=$wnFmBHtN\1Cm)7 ->U?`R)YxpʹgV &Api/נ?Hㄔ]$R#wYƗq(ЂcNO{}lݳJTp|TF!=ڛSrUdȲyTuj{?#Yz-=W緳ڤR0⍬2@k0MgfUt!ugݕWb'!j!-mT}:1lC#qyĭ e!U4{M2tj?0 Ο`)YY7OYxDZɤ\!IkqɗWX,Xub`8˗>ef\SSWn`Zg2:߻l06yǹt:wg>{Y,뽊'5y 0I !$$3Dl'Qd9`:UW_{Mz:*IHGD4ξ1?!5AӢ }Ŵ .$*(x>6.;@? 1h ]Pȭ 4iN@vw3"yt }a{Y$}uV*xqK;Kȧj"yѳ./цi/=&:ݦiM"dŒ%ἁqUH= ڀSʅv \ 2,]UH[VKJ4NFM/mh0TfwQ^e8 sFH޶ |G#Z}zx]E"j>kp  V)+r/~O'Ձ n?&aڵq$#SZa.ؘ`u vsJxZ틕# &o1k>uќR –9%0"֒X"Cij$ ^אւ_CMKԸn6)+z; a!ʉ% ^j;_]k &T6 0ZOd"-PJtGR!mjkt~C,B@ RwS,r{xXV3h:$ cM W,9PeXh#7Ѕ?Rzi~f CR9Zyn9-@ ;m#ч ܂3-[]5?|{-.0)4cאS8+4ݷP5:7 =9GsQrפ,g[쁻]Nb]\.R!|A=9 wX[Ui9UY㹩Gz٧OcPgْiˋ9գ5p 1 7'nT55 >j \]rg<#pÄ &K-_-,)wMIs3D?a%Qzn`*v2GP~m&t[^_C@~|}t34Uz4.\/|5ɟ\4@CfPAfD냓[[ h%<%1 qjW@6J9K}Ēdj>]/:/&<+i욟- @GjL:޽L0W_I5fҼR|qv+ eEsM]؅g: eՏ+Px&Zg|\Hn M<) 'ڬNf@[d>`fzkSH*t{k^q :|Znh k4nYjĞkA&R4)|P1nd`_6l>U&Pk(rHvCp/FBlbK^$mրA*"_'LT{#r5m5p~cB[6IQ m*q8 - a=!ʲXdĚ,d1֓D#1ago7#yxt-ZgnAoj4W~(11K r1٩JqK( j+0z/XQ0<佔BI@,Uxbqݜk.?o6ScTƭ-YM݂&ՙŶ.6b\lȨ(]~񹑠`}IF`#h-fmSB$PKY_E}#ZԁX \'KaGtvTߟc𡥟)z ԖT$&܁QPiEaE&iPo4rzM-X[L=řΐ 31uH+pRc&kxR]Yԉ].hp-Ʀ^J,9P$X^CcM%;R'HgkbN2ƻ;"ڹ5};D TIڍl^'zHuzή[P5lZAAv?"n lWȚZ|jVyHOVܣקEE{`ȏq}gzҽk3,1j-|mІhvڹ \Y̜WVi3* ~BMK"h[9ż/F)K?fGht)-5Hfc3-T1 WXT&Ԓ y*۠7Ug,[~cنO/H`\kJ2*c1ğ8|A-y8GYyO2EZ94%Wo>zYQ_{)N U5JH=ÅXdRjprǛOèCm^yRW0d3'3Իʽ;S0ћu*G <\ 7r)a1V9fhdRU-vwzPw2)1FB rp'6@2l'(_nB}OR9:, ]43K/sd9IքWp#۽-: ?YŀP|vQgHtH M\$x!5xP ?chr6DEpȵPW_(TLbH脀>ſg웖Uޖs8,XȂ83FEJ*EI& 3B㦶\)NEsjVixuTIʄ2we*:r\-Oܡt&}G܄H G9vP0sR#$"Ʉ V".a Q*l@Ġs 1tpHkP%Ia{6ܾMпʼn-"73GLdvQ^9\-(vS~tX\a0\h{KS Qn4+rӥ0d估Ej,͍B^JOTo-ʔPj07N{t *O}PP 'U>׼G 6ݖ /X$E"-H aIfXK2߅LlS2L3ix6.J`_ C!JHዟO[\X|rOpC3%wM0N3$Rm42U[a88,5]伟r0\I&Q tB|s>aɊ%MJW6EڢHgPUU M7D2"*kFƺ;i)XբEN]!Zu*Rݲ.QHUPS\ b]Dlos>vb@%,2 En`2KB{QJPۋ w/V㷝ߜ J\ "gPlki|u/-,gBdz蓞FcFs6-+ u 6aytW?p C1A5`xcqPFX6wPK-5rCm`wE5~7ޒ O L6\jkBo.H ǠgX[cʼn[IS teTL5=w`4yP 3oqvg z+y6-FdKޞb,C;wbcciUA/?hmnJe]:~|z1_&Vjq$ޑ}|Vdn)jd*d`=YYbqD jKT-+ " kQUϺwtZSNl_.41 H1ީ /yư W̰8" Q@ϥ9FAp@,Cvglƺ@lV5Oc-`L 0ⅆ* 0Ȓ_8`׳*1 |pJzÓ"QdF^p>`:lH>\kd1~ZejAZ)T?9v>ȵ4LXXr*5 APd&4PZX@UZh556$*|8'}v=R= mHB.!{'0yΎjVO?njFt+d E:αI+NPh" C*+MPB+yu;M>SⒿ{b @Z4(^0]5d1**؃}9 )[PpZyRy/waGK[%Bmyaf]8ECS3LJrDJUICY%؛ݒ֖Vw9OyAS*S/7Rp{(_ `.H2u(ò w{}OUUU|l.J=!J'o 4y$;^ap]ȪU= 2[{sC"O&pY^Tܹq3M K1TZ-Y?LZLL,3חX-@iPPVC; _7@^1uA%iTgK\_Ρ\XMz}LSYکKZnC:Lz@mO4iI: j寧I;\,|[r=L$T . r(Yl0hO=/ϫ1n m9+kٝtp$ GHqؤܛNHY(_M2q_n8!&;?]Yod0Q$6(2^ ʞ[_Pik39ԉPq"vNR\EGtO~С.5ckI!d./k}!IPq#' T ?/ȑ 4.ol*'枰Nm/FL91S2!%RI9 bs.&3uOE D@\7<heoilLDUV_PI2폀 /C[ zGϞ^6H` ?~H;lPPqz W L8Cwce}ڢ}OFU)|fF/[nld4x_ҞTv$9?JB8k*f0M9W_^HS)v+SRU;eU^ŏoHX&KzMSjW8z&t;mL1T,S:AF̭{Vҧy~4(냼5IA zYk '"XaJKqa9&mB'_}hK.+/-Y;\;qZh2!_ʻ86&y#"Wx#ь릝s9:x[EQ<᫞K&y,@,]6yI~Œ ~&NJ{.qѴ{ΙFDk!1*l ֏ &TP#&8w^OIT$.7 TS2i|u;|# *JB6|Ypt,NmV;Aˬ},ZF=BPYĈwgV{Ė[U(6*鳵7# 2[lA"e)Gd.3BM o]Xϑ&IdvU^(㚃^{seR31G4ڑ!*ղf|K'#๳`)U7+[.J"We=gD09B8D=;WPGׯ٬րBKNJeWd{|m3=`mHk7^&Ikia)~ 2bz=hH(>k 6EȫpچAPR`y֮C#X Le^4}Nvp^:e—K JmM-EM˰f}+՝4F,R;f> z^QJ31Qqyw(iW'_~D2ˢM6Nwac \{;< @QwolLV3HE]*f1Quz5'js"MBsXhJ;3 l*|f3fvkG0vT` qYjз5FGUN^jڦ 0WпIQ>tf2"86;Te *eT?DԦI+BMf5=Tx4(+A":Z|SlfkB*ԓ@jFa΋B Tw~0 =?Zj'1s~Bo$؍U# r^s c18MSū$@q8jF bHrl   ,͏yZEky`pFޜ)o(& az)~$ V48ajCiu?w9^xteUb,%{H2[]BsJv&dfB$vLY HZpt5BUCƃdFhH!,=&*Qw4 =DlP@K)s[`033FGA"_?'_7]+3@t3ND/ s%k";;Gq GtDӶrPÊV2>H'88(mlڡdC‡D'gV'~v%Eh¶T' [0ם~>}|eURlq'1T P3{y՜fzYL VI jҐ1$BB3tJA z9Vw(.;p@2U! >!E c3H.L  mȓw$"-p (L0+`!oWoFԈ`_.f:e-Ȑ\hGmbLO;HLuj?_#5IJg j-!͡@θ; z܅L=5؀ټbt5 .x0k z%8kdcKFB[J3gć]M%Qj#_U`Ja~\ ځ9wͮ\P_wJ'aj47GRs`g ^p ڢ+(,yۓ6Ww L? Q؎Q).܀l0oz(`g wtK^ϲևn-IF9͘^WK_aēEW_b{4HOQjYtC@"STf65Fw{ۿJ(K>L"Dp٩+*9]uGb>6v;KƜ h-xl!.yIi*1ԩg}[8GӉ%>&ɱݳb'<\A4#ɽM'G>,*Վñ\d6ltm u'O!2gI S/2:l1O{Vy ;T@wDd8#htTaEH{Ki}P Altk'-% 4?Z^B k$(Bv;}gS5yDj#zc74Yv|TSZ_bH[VO+ѲuQ5z+V?WKNhrߜfs6="? ʦ-,_rûц,6;V\#`E/eBxf(QMK~U#l)}Bg:G*"WXyncAM^栨œx/cT]<" `Y`x5 >pc-jk󻐧hpFI`+'[pR,e5_k[C8"Q QE@AÅ<Ѕ8:v ~2N[)MGoϚ, e+wq ̰ĥL@&gDO.X7;R)'X2l#G#SLXg,/c6|#רY(!SSa=Ie H7x,.1h][I- ‡Ʌ4C >'$I\3[D?T&+om6XY""gҸ.)=>,>^ͦ Mio7P ht@t)T??UVnьF36Q^αf\ÖR%u] T#)/yn $O?VX oQRgVؕ(711!j\%߂[Jx:Y_op#oŵ rV1;60Œ Yxjo >.{#sCTk #| `FL7b7+d D=X0e;OChRTn;)c\WMa+ES-d\fAړcp7P+tKjW<0H_P)f,CP5>?& VX`*''뮅y?ObV:BW#s1u q*98l6*,%lh)TP2B I?lc /j' hGz'`=*Dbѩܼ \(T/'gD(y;*|08O밡~ڪXW=($sk T3IP'sJ]08t[?뗖V{+R=2j *UD// vC%l1A8.Fkl~S׮l܎(iN}ܛJ|6d D*|9}^ڱ;RWQ؂pI?jcc8?{ILZd-xAkݠгǨ`kQKL>(( *vrAūPS &crw u2 .bFwQBnpss7f$O2+ oWgj#]ǭ\8dxWT]ux)?;*{!g6ϘuRvU/.׾eLūXߙ@^Zlcs5%\,5)6 :fΒrD#_,%jqAV cYy}ړm!`9${Zȧ0lzf ¿fVQj ϛQO;޼ W0薯Wo%tۥʢsES$pA)7̚LΏ uz7\́aSN% ^`tBYjΖ}Hi%o`ɻ] iKySJR60@ 5fq{d ًAi endstream endobj 651 0 obj <> endobj 652 0 obj [653 0 R] endobj 653 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 621 0 0 840 0 0 cm /ImagePart_2148 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 458.899 720.7 Tm 3 Tr /OPExtFont3 11 Tf (5.5 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 3 Tr 1 0 0 1 125.75 677.5 Tm 101 Tz /OPExtFont5 13 Tf (are unable to plot the cost function against the initial state to demonstrate the ) Tj 1 0 0 1 125.75 654.7 Tm 95 Tz (appearance of local minimums. To support our suggestion, however, we apply the ) Tj 1 0 0 1 125.299 631.7 Tm 99 Tz (WC4DVAR method on different realizations of observations of the true states. ) Tj 1 0 0 1 125.299 608.649 Tm 98 Tz (If the WC4DVAR cost function does not have multiple local minimums, we ex-) Tj 1 0 0 1 125.299 585.6 Tm (pect that the pseudo-orbit produced by WC4DVAR should not varies much for ) Tj 1 0 0 1 125.299 562.549 Tm 96 Tz (different realizations of observations. Table ) Tj 1 0 0 1 340.3 562.549 Tm 76 Tz /OPExtFont3 11 Tf (5.5 ) Tj 1 0 0 1 358.1 562.549 Tm 96 Tz /OPExtFont5 13 Tf (and 5.6, shows the standard devi-) Tj 1 0 0 1 125.5 539.75 Tm 97 Tz (ation of both middle point and end point of the pseudo-orbits , The WC4DVAR ) Tj 1 0 0 1 125.049 516.5 Tm 95 Tz (method is compared with ) Tj 1 0 0 1 251.05 516.5 Tm 91 Tz /OPExtFont4 12 Tf (ISG.Dc ) Tj 1 0 0 1 289.899 516.5 Tm 98 Tz /OPExtFont5 13 Tf (method. It appears that for ) Tj 1 0 0 1 426.949 516.5 Tm 91 Tz /OPExtFont4 12 Tf (ISG.Dc ) Tj 1 0 0 1 465.6 516.5 Tm 95 Tz /OPExtFont5 13 Tf (method the ) Tj 1 0 0 1 125.049 493.449 Tm 99 Tz (standard deviation does not vary much for different length of assimilation win-) Tj 1 0 0 1 125.049 470.149 Tm 95 Tz (dows while for WC4DVAR method different realization of observations effect the ) Tj 1 0 0 1 125.049 447.1 Tm 98 Tz (results more when the assimilation window becomes larger, which also indicates ) Tj 1 0 0 1 125.049 424.1 Tm (that more local minimums appears. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 125.299 379.449 Tm 108 Tz /OPExtFont3 13 Tf (5.5.2 Evaluate ensemble nowcast ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 108 Tz 3 Tr 1 0 0 1 125.049 348.699 Tm 91 Tz /OPExtFont3 11 Tf (In this section we compare nowcast performance of the three ensemble methods ) Tj 1 0 0 1 124.549 325.45 Tm 87 Tz (based on ) Tj 1 0 0 1 170.4 325.45 Tm 102 Tz /OPExtFont4 12 Tf (ISCDc ) Tj 1 0 0 1 209.3 325.45 Tm 95 Tz /OPExtFont5 13 Tf (with the Inverse Noise ensemble. For the purpose of illustration, ) Tj 1 0 0 1 124.549 302.149 Tm 94 Tz (we call the Inverse Noise ensemble Method ) Tj 1 0 0 1 335.05 302.149 Tm 86 Tz /OPExtFont3 11 Tf (I; ) Tj 1 0 0 1 345.35 302.149 Tm 94 Tz /OPExtFont5 13 Tf (the ensemble formed by dressing the ) Tj 1 0 0 1 124.799 279.1 Tm 95 Tz (end point of the pseudo-orbit with Gaussian distribution Method II; the ensemble ) Tj 1 0 0 1 124.549 256.1 Tm 97 Tz (formed by perturbing the image of the second last component with imperfection ) Tj 1 0 0 1 124.799 232.799 Tm 99 Tz (error Method III and the ensemble formed by perturbing the pseudo-orbit and ) Tj 1 0 0 1 124.549 209.5 Tm (applying ) Tj 1 0 0 1 171.099 209.5 Tm 93 Tz /OPExtFont4 12 Tf (IS'GDc ) Tj 1 0 0 1 210.699 209.5 Tm 97 Tz /OPExtFont5 13 Tf (Method IV. The three ensemble methods based on ) Tj 1 0 0 1 466.8 209.299 Tm 100 Tz /OPExtFont4 12 Tf (ISGDc ) Tj 1 0 0 1 506.899 209.299 Tm 95 Tz /OPExtFont5 13 Tf (are ) Tj 1 0 0 1 124.299 186.25 Tm 96 Tz (introduced and discussed in section ) Tj 1 0 0 1 300.699 186 Tm 79 Tz /OPExtFont3 11 Tf (5.4. ) Tj 1 0 0 1 323.3 186 Tm 94 Tz /OPExtFont5 13 Tf (The Inverse Noise ensemble is formed by ) Tj 1 0 0 1 124.299 162.95 Tm 98 Tz (sampling the inverse noise distribution and adding onto the observations \(details ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 140.9 141.85 Tm 65 Tz /OPExtFont3 7 Tf (1) Tj 1 0 0 1 144.699 141.6 Tm 90 Tz /OPExtFont3 9.5 Tf (We expect the middle point provides better estimate of the model state than the end point ) Tj 1 0 0 1 124.299 130.1 Tm 91 Tz (as the end point only has information from the past, the middle point has information of both ) Tj 1 0 0 1 124.299 118.299 Tm 88 Tz (past and future. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 9.5 Tf 88 Tz 3 Tr 1 0 0 1 315.35 52.799 Tm 83 Tz /OPExtFont5 13 Tf (121 ) Tj ET EMC endstream endobj 654 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 655 0 obj <> stream 0  ,,Дb7 LAez>Ir-B>gdbǂMz1^Q퓘v- [pD/?L{.Mx[gMFzjZug*imNInaqʼ)J"~ޮBƍh79yS.؈NmH %ï|gX/'c1+VQ9MgلGFNCSOP}e\wxm@lw*n0$AaE]lHፁ}]~9P/3"q}^d)l4raL `Z8$#~/cW 35Iѻ(_bEH AAoX8lq֧7-v$CWGP?Y*_Qr#laXSܳRP\=E[=Z!WYh:d$x\t|٪I8W{(y8S^o`B$З9YCǟ3rH^pꝸ Fkm Ug#WB5ja[31A؞' G-le6 K_g;g?Q>Ԏd͍(4Su@Wڅ85SG{?8g)ni}Wc Z*Poa~_rW>oF{#L7/l2YEbkF81?Zkqӌ3msC14nLJΒ63put yjqFcy9 _G lpߨ<\usBʕx2io̶bh6{4 >hż;?ѱ챷=۳!7%<*mYFz)Ls'[˳|O+JhЃF[70J-k5݆P[뜙f_JG'=Ȓ:*}Ӛ )Fː}3:/ Cgѧ\[xF= A[X6Qnj}bI1F{#쇚 ! m~͘>s{a[@o_E}Ƕʟ$> E # Ez>l#0BH (ѷ[E  2/AڭL o?ka ={}xsvQ{ZsD;F(I,615EVqjsa@?gu%-3^7)57fL\_?%ISR-"XSTqO%Wݤxç,sHzrh ;9;JʘLp^0dd;4!bB+4u ϲ{xhCť>I]0n\+?` C oV,xܶخm$*aBNJlt TcV? #Vy]8(CtIB:Nyp}Qh{(Byq)^p7u|@uPmoGY^!^s%!#ν@˓(G?ӏ_ P.uk,HaQ:P@A,d#%]~\CRlwPׅtjvD_)sҨB<7-axb*rZHX63ݪn`ɈsIz@)XEVe yǁ#"T!)( hvRGZ^ھ#7b;*UjD;Qm0~ӣ=wJ,¿sU9uH]5Vs 22gF\2BZ)G:؟kv*=z! 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E3^{B)~B{@g)0%,Oׅ)`S'hy}$>F  nc拾/_^ak"qS{Ka0г:_|O=Mݷ`9^ ّ:N?bGI좤־oμ\:SQ~\jG3XOߪD>*.9`r q"ՆNçzɑ2Z$\}fvT¦QJG'EaVq냭u{7&tl- 38"pHĨZjWR᠌J^RI[PKT)F75Mī0Nz<,d/ʆYoxڛmu ĥOBL`L,Wb)I*<_W^{Ȣ]omv=HDA;hZᔪBA_˄"X>p1L=H=}$Ʈ셲'т8<0Y#i$֜wLЌM c~#vP(p*7YlWw1~̒ZYQ4j $^!ه 97:] >Y8'C/gGQ>E; Y v=.%""Tu Aq%C Mo=_I\"s?V)옹-8m952pcF% )a}^*AhܒzYf5k Yf ?.I\avF^51X}Ώq!nSY lp+#NY'$F5 G 0 ^݃8O=LvGfKk;/k?B_g]BpĂ$ACVc*v >5C7w?Î,F9D<b_o#XRkIHԃiLfaa !jP Qii<ҭQ,p}Z롕AvH}q9G]NT|Px5.w׃jx|J|̈́(O  悏0ޅj?o ˩S b!uJT\]S%͑:vCYi|.!}!243H4~nS.XZmG+nx9&6CV,ſsdu y-E LC{Iڼ,yŶkКYe P( ]ox?4G$.gO+ v5{$e;ίp27*6{wʞ?qyVSK}"4ZrVNNM3x"ыY\CqՓ`Ovn~Th2ips $$rG,Uv G8 ڋ N^/Jr*SfȄiy endstream endobj 656 0 obj <> endobj 657 0 obj [658 0 R] endobj 658 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 620 0 0 839 0 0 cm /ImagePart_2149 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 459.1 720.2 Tm 112 Tz 3 Tr /OPExtFont5 13 Tf (5.5 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 112 Tz 3 Tr 1 0 0 1 152.4 678.45 Tm 94 Tz (Window ) Tj 1 0 0 1 157.199 664.75 Tm 97 Tz (length ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 222.5 678.7 Tm 119 Tz /OPExtFont3 10 Tf (STD of the middle point of the pseudo-orbit ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 119 Tz 3 Tr 1 0 0 1 231.599 664.75 Tm 98 Tz (Median ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 98 Tz 3 Tr 1 0 0 1 306.25 664.75 Tm 118 Tz (10th percentile ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 118 Tz 3 Tr 1 0 0 1 406.3 665 Tm (90th percentile ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 118 Tz 3 Tr 1 0 0 1 204.5 650.6 Tm 80 Tz (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 80 Tz 3 Tr 1 0 0 1 258 650.6 Tm 94 Tz /OPExtFont4 11.5 Tf (ISG.Dc ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 94 Tz 3 Tr 1 0 0 1 305.3 650.6 Tm 79 Tz /OPExtFont3 10 Tf (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 79 Tz 3 Tr 1 0 0 1 358.3 650.85 Tm 105 Tz /OPExtFont4 11.5 Tf (ISCDc ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 105 Tz 3 Tr 1 0 0 1 405.85 650.85 Tm 79 Tz /OPExtFont3 10 Tf (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 79 Tz 3 Tr 1 0 0 1 458.899 650.85 Tm 104 Tz /OPExtFont4 11.5 Tf (ISGDc ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 104 Tz 3 Tr 1 0 0 1 156 636.2 Tm 96 Tz /OPExtFont5 13 Tf (4 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 96 Tz 3 Tr 1 0 0 1 204.949 636.2 Tm 91 Tz (0.0153 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 258 636.2 Tm (0.0155 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 305.5 636.2 Tm 90 Tz (0.0113 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 358.3 636.45 Tm 91 Tz (0.0105 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 406.3 636.45 Tm 90 Tz (0.0259 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 458.899 636.7 Tm 92 Tz (0.0274 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 156.25 621.799 Tm 95 Tz (6 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 204.949 621.799 Tm 90 Tz (0.0271 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 258 621.799 Tm 91 Tz (0.0126 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 305.5 622.049 Tm 90 Tz (0.0121 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 358.3 622.049 Tm 92 Tz (0.0087 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 406.3 622.049 Tm 90 Tz (0.0595 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 459.1 622.299 Tm (0.0264 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 156 607.399 Tm 95 Tz (8 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 204.699 607.649 Tm 91 Tz (0.0431 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 257.75 607.649 Tm 93 Tz (0.0126 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 305.5 607.649 Tm 91 Tz (0.0242 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 358.55 607.899 Tm 90 Tz (0.0086 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 406.1 608.1 Tm 91 Tz (0.0905 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 458.899 608.1 Tm 90 Tz (0.0262 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 152.4 592.049 Tm 93 Tz (Window ) Tj 1 0 0 1 157.199 578.35 Tm 97 Tz (length ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 231.599 592.299 Tm 119 Tz /OPExtFont3 10 Tf (STD of the end point of the pseudo-orbit ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 119 Tz 3 Tr 1 0 0 1 231.349 578.35 Tm 98 Tz (Median ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 98 Tz 3 Tr 1 0 0 1 306.25 578.35 Tm 118 Tz (10th percentile ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 118 Tz 3 Tr 1 0 0 1 406.3 578.6 Tm (90th percentile ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 118 Tz 3 Tr 1 0 0 1 204.25 564.45 Tm 80 Tz (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 80 Tz 3 Tr 1 0 0 1 257.75 564.45 Tm 116 Tz /OPExtFont4 11.5 Tf (ISCIY ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 116 Tz 3 Tr 1 0 0 1 305.05 564.45 Tm 80 Tz /OPExtFont3 10 Tf (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 80 Tz 3 Tr 1 0 0 1 358.3 564.45 Tm 105 Tz /OPExtFont4 11.5 Tf (ISCDc ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 105 Tz 3 Tr 1 0 0 1 405.6 564.7 Tm 80 Tz /OPExtFont3 10 Tf (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 80 Tz 3 Tr 1 0 0 1 459.1 564.7 Tm 101 Tz /OPExtFont4 11.5 Tf (ISGDe ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 101 Tz 3 Tr 1 0 0 1 155.5 550.049 Tm 97 Tz /OPExtFont5 13 Tf (4 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 204.949 550.049 Tm 90 Tz (0.0260 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 258 550.049 Tm (0.0301 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 305.75 550.049 Tm 91 Tz (0.0124 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 358.3 550.299 Tm 93 Tz (0.0147 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 406.3 550.299 Tm 91 Tz (0.0417 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 459.35 550.049 Tm (0.0407 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 156 535.649 Tm 95 Tz (6 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 204.699 535.899 Tm 92 Tz (0.0369 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 258.25 535.899 Tm 90 Tz (0.0296 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 305.5 535.899 Tm (0.0228 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 358.3 535.899 Tm 93 Tz (0.0147 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 406.3 535.899 Tm 90 Tz (0.0841 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 459.1 536.1 Tm 92 Tz (0.0407 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 156 521.25 Tm 95 Tz (8 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 204.949 521.5 Tm 92 Tz (0.0590 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 258 521.5 Tm (0.0299 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 305.5 521.5 Tm 90 Tz (0.0363 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 358.55 521.7 Tm 92 Tz (0.0147 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 406.3 521.7 Tm 91 Tz (0.1294 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 459.1 521.7 Tm (0.0406 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 125.5 498.199 Tm 98 Tz (Table 5.5: Ikeda system-model pair experiment, following Table 5.3: Statistics of ) Tj 1 0 0 1 125.5 484.3 Tm (the standard deviation of the pseudo-orbits' components for different lengths of ) Tj 1 0 0 1 125.75 470.35 Tm 95 Tz (assimilation window, for each assimilation window, pseudo-orbits are produced by ) Tj 1 0 0 1 125.299 456.199 Tm 94 Tz (WC4DVAR and ) Tj 1 0 0 1 208.55 456.199 Tm 105 Tz /OPExtFont4 11.5 Tf (ISGDc ) Tj 1 0 0 1 248.15 456.449 Tm 97 Tz /OPExtFont5 13 Tf (based on 512 realizations of observations. Median, 10th ) Tj 1 0 0 1 125.5 442.5 Tm 98 Tz (percentile and 90th percentile are calculated based on 512 assimilation windows. ) Tj 1 0 0 1 125.5 428.6 Tm 103 Tz (a\) Standard deviation of the middle point of the pseudo-orbit, as the chosen ) Tj 1 0 0 1 125.299 414.449 Tm 96 Tz (window length contain even numbers of components we treat ) Tj 1 0 0 1 432.949 414.699 Tm 101 Tz /OPExtFont4 11.5 Tf (\(Length/2\) - ) Tj 1 0 0 1 503.05 414.699 Tm 94 Tz /OPExtFont5 13 Tf (1 as ) Tj 1 0 0 1 125.299 400.75 Tm 98 Tz (the middle point; b\) Standard deviation of the end point of the pseudo-orbit. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 125.75 366.899 Tm 97 Tz (can be found in section 4.3.1\). We apply each method in the Ikeda system-model ) Tj 1 0 0 1 125.299 343.649 Tm 100 Tz (pair experiment with two different noise level. For each method, the ensemble ) Tj 1 0 0 1 125.299 320.35 Tm 105 Tz (estimate of the current states, i.e. nowcasting, contains 64 equally weighted ) Tj 1 0 0 1 125.299 297.549 Tm 99 Tz (ensemble members. We use both &ball method \(Figure 5.5 and Figure 5.6\) and ) Tj 1 0 0 1 125.049 274.299 Tm (ignorance skill score \(Table 5.7\) to evaluate the results. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 142.3 251 Tm 101 Tz (Figure 5.5, 5.6 and Table 5.7 shows the comparison among four ensemble ) Tj 1 0 0 1 125.299 227.95 Tm 98 Tz (nowcasting methods in both Ikeda system-model pair experiment and Lorenz96 ) Tj 1 0 0 1 125.049 204.45 Tm 100 Tz (system-model pair experiment. In both cases, the c-ball method and ignorance ) Tj 1 0 0 1 125.049 181.399 Tm 99 Tz (score indicates Method IV and III performs better than Method II and Method ) Tj 1 0 0 1 125.049 158.35 Tm 97 Tz (II performances better than Method I. As Method I and Method II use the same ) Tj 1 0 0 1 125.299 135.1 Tm 99 Tz (Gaussian distribution to form the ensemble, the difference is that the ensemble ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 316.3 52.299 Tm 84 Tz (122 ) Tj ET EMC endstream endobj 659 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 660 0 obj <> stream 0  ,,,b7 \T.L"ͩ!=]ÙܘjIļ?iۜyKӨS꺏 ~(\ސ[Ft;K\ 6( O9?ݪi8Ԡ'v*ڻ">cβu\n'xոVȭ{-7re|bu NF>?7NutbVp9"p ?P?lAɉI~t4Fue,0|Xdp<3RdjMzF@~DZM$Iv 3:Ќ2ceU5(-nZ_B U'`$S$ʆVV1g34üObd167#9RDbxqZrWV4աؗܮA۳ bڻ"!a )"ne.9 !ť=&dk9\< P!nȂWʙ ѼsO~':NRݝ_ K3(HѓpT-ȤvhSinSXl]O+ jΐ1'e3BY\TcRnCQ}?r$ĩ%կkK鰨ʝ?5鋴~Ԩ2ϰԅճ+P/=?![~.v̯cXnK㋳YǛ m}#3X/v܎x .\gKGn\CAT{r4%)x {M q'B;)M:_װ?<{M=暬A)A+kuQά7v.g%#ptnK49ҽJp"@wyѩf B!ƺ4WD Ux`Kȍ²38x\b L I<`sN՗[Av2=^[L3Q嘑!QRð:#3ò.1윧gj<bJto{m Gڰ<t7Er:(mXp $-_2N:fz㷹.ǷrihV YÀ\w8VDq@f; 7{fܥnWeZ`c\\e~ǢhsJ#%Uw$#Y?#s uK&B}I%.j6y\PCkVbѿNRx{_hZ3j~75AG]//-86t6^j#g] psB沈6*wal5`$ 4)IB m2F4s~6ׄ+ڧYLaʲWK nMcVy!ZafZ ]ˡRO{4/+)/CpV;(3{eG=|-3 )sȻxy I1(Xn%Y,?s60Ͽ~l|#\±_ |{w $IiZp L-^) 3:H:G~j8nQ0,Y"ժRy-O ғ -r^t)o95zJ˫ ȣ^^Vwp}MQKlϽ48&ԕyR+6ԧީF5Qg I-p/ihlLV;O8.}[+jR, d UYzg?%uwX&#dTmcsǷ SUa?QOj?z"S¶I!?\)m|~m|^2>d?hc "Ph,%$j"OURm5s|uC" GOy48sHiwO]R~R*9'G*piًa126?CXXAழI70|2q HRsLj lay4^9L,r$I>mDZBM`弛AuqGhvЬ.pPm>O !we|RsӖ:gmX㦝 RpM Owy9J|y6wz0xk2Hؽk6pAm)12ې =a~u%d}-FKn=mU9o&S0D"| v.9,zmq!v'mQ/?LK{Z˧@_z{cmcGʤ'[cc%jGD L[0Q_؞rjgùXqnjÍݱTO \qNˆukp VHaU AD7S@ Vϴ`7~O{K$dDoϦ pa Ƕaqq^OW:f DfM@҆9/M=..sGx`oŏw1n" Ba^^ñ֪!6j`ك3 j|E/TE刞 T @DT?X!H+)_ Q+v H4bZ aRTN>SٲH3.h/ɀ.7w4>Ë *]˃o,,`ٓlUJNdlW@ڇLu>` 9=k^΋TMNolS 4q nY:xaN|my:OT*`ǻ_SեgR=P-YLL!*nMgm f`mXzKIhEi9]sB[a}&G0]M@9w}:Y0p#CQonpw_?E3pu.5kT9?i* smsEz({ijtl1U |= o Z14JY _" ,NdRRly F*Q&EX``(}kc~ r?AH8}yf~@=V1cø h'qP5^ƪp!nIEVX}v/q:gELD706EsOHmI+bIXH_EgXG>u@)Lȃ |:j KׄrHqo@'s-W^4R*1`K':@u2 XU}} =C'N©5GW$4H4þо4yOW{H h]Lec(,;_KrnR40[Bx ]6pmǭB2)c霥k H4lv%4m 0Eu9'.n))u!4"?cV@C&7 xf rϖx EJpz8fkB1&Oqى DP"EW4n8vbįOQ}đ'.?xMum(.'ܟMhD7B >ˣWeAO" Ol0C{ ꑼfe?ɓ Ab16{"!˾׻( Hyy}[[Gov$oj.g2ejUwbr,h#@Rug摬7ECz{4/W3;`+C?ep_=6B02p_E@y0zWFL:w"کH^))1A1ÿkI) Bl(sp>Ɗz Mg8ч*W̧k5/)2ZO}6n&@NcтŤӒp n޾O/ϊ֫T,uQ}p B`.. \9ɭ"(5fRn<:A;l_ $}1&ae.u/qzHbOKK% ^˞ Hi,QA/*UN?~+O3FOlF ԝE=PqE埱aNy:G ;Ѭ>f y5Grޘx#jS]#K`aeZCxS0ZD PI&ha՝\Jn+g3eg#^KpJZcIm4,)P2e9O85TaۺmZ ginkΫ̮br7=X_ƎEj|2>& 0.CO}wv;+75WӲԏՊ}Sb{ՀG(]oZo04:zP4}n~W`qqnGLJk&n)~J+d͙Dҋ]}ܨkNQcȧu ^(35Sؑ=^Nk!%\:ı Lhdgg!';FH R-%HSJȆ[9䚌Vɱ߰,POd鈣2 "Q1lb񈓹xPo˹_{{K;A6`o3lkzi?7܅z$ :y -yePsXB샌πr_4(x #y568}^ND&:[?nb>8|lF~9KҍM;I>RWa}L&;KNx3:C:fF{Lj)R${fŅ`i6uz[Fq~5+6OS'g[ړYvd{3-p BċL#q~Wd~їAVh^RTX(XQ$vS>Hovb[D~, ZhƦ,3'VAԲi\ڦ&hֱ3ԋ<$^{Ü;-<|,~XZ<hAVAUzK+l}^s`GTPrJDoh]ZTQc} :!=+%O#z?-?iKpڗUwXj/"7=R3 @$2Hkbx~ޫuҦ|rdֺv@֧ޭ\mߙ])5d\Ťq ӂυqF^jQ YB)7"ѫPqs)* k<)Va{-_ M0UBGضILl+Oqͭ a$K>~±ʲІD妫>!6}շ|\#6K*@ʴʿƳ;k7]%< ',%ɮV9:', Rֱ2=r4v`_ wye0 P{^P`agfKT.oi zsAE#heYĭjЅҌ @zw~1.O$Gx%vyp/tԷ9ɥk:WwZ6Aqv>y@L}kC'҄k7h}={Pt_RNx?y!MY aݻn勵<?Np OT(S4edU(7S!sOGh2urn[3 Q,yN† hjg{DR %.pК)qfhM>z8]|r8vg|[m u٪$:s5aUu0w$Ř2>iq2;V<5N! 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/OPExtFont13 11.5 Tf 84 Tz 3 Tr 1 0 0 1 410.649 578.35 Tm (90th percentile ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 84 Tz 3 Tr 1 0 0 1 208.8 564.45 Tm 85 Tz (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 85 Tz 3 Tr 1 0 0 1 262.1 564.2 Tm 90 Tz /OPExtFont4 11.5 Tf (ISG_Dc ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 90 Tz 3 Tr 1 0 0 1 309.35 564.2 Tm 67 Tz /OPExtFont5 13 Tf (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 67 Tz 3 Tr 1 0 0 1 362.899 564.2 Tm 102 Tz /OPExtFont4 11.5 Tf (ISGDe ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 102 Tz 3 Tr 1 0 0 1 410.149 564.45 Tm 67 Tz /OPExtFont5 13 Tf (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 67 Tz 3 Tr 1 0 0 1 463.449 564.45 Tm 105 Tz /OPExtFont4 11.5 Tf (ISCDc ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 105 Tz 3 Tr 1 0 0 1 158.4 549.799 Tm 93 Tz /OPExtFont5 13 Tf (6 hours ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 209.05 549.799 Tm 91 Tz (0.0563 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 262.55 549.799 Tm 90 Tz (0.0480 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 310.1 549.799 Tm (0.0429 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 362.899 549.799 Tm (0.0243 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 410.399 549.799 Tm 92 Tz (0.0934 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 463.449 550.049 Tm 91 Tz (0.0744 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 156 535.649 Tm 93 Tz (12 hours ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 209.05 535.399 Tm 91 Tz (0.0743 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 262.55 535.399 Tm (0.0477 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 310.1 535.399 Tm 90 Tz (0.0573 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 362.899 535.399 Tm 92 Tz (0.0238 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 410.399 535.649 Tm 91 Tz (0.1332 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 463.449 535.899 Tm 90 Tz (0.0741 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 155.5 521.5 Tm 94 Tz (24 hours ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 94 Tz 3 Tr 1 0 0 1 209.05 521.25 Tm 93 Tz (0.2444 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 262.55 521 Tm 91 Tz (0.0477 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 310.1 521 Tm 90 Tz (0.1859 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 362.899 521 Tm 92 Tz (0.0236 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 410.399 521.5 Tm (0.3949 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 463.449 521.25 Tm 91 Tz (0.0740 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 129.099 497.949 Tm 96 Tz (Table 5.6: Lorenz96 system-model pair experiment, following Table 5.4: Statistics ) Tj 1 0 0 1 129.099 484.05 Tm 99 Tz (of the standard deviation of pseudo-orbits' components for different lengths of ) Tj 1 0 0 1 129.349 470.1 Tm 95 Tz (assimilation window, for each assimilation window, pseudo-orbits are produced by ) Tj 1 0 0 1 128.9 456.199 Tm 94 Tz (WC4DVAR and ) Tj 1 0 0 1 212.15 456.199 Tm /OPExtFont4 11.5 Tf (ISG.Dc ) Tj 1 0 0 1 251.75 456.199 Tm 97 Tz /OPExtFont5 13 Tf (based on 512 realizations of observations. Median, 10th ) Tj 1 0 0 1 129.099 442.3 Tm 98 Tz (percentile and 90th percentile are calculated based on 512 assimilation windows. ) Tj 1 0 0 1 129.349 428.35 Tm 103 Tz (a\) Standard deviation of the middle point of the pseudo-orbit, as the chosen ) Tj 1 0 0 1 128.9 414.449 Tm 96 Tz (window length contain even numbers of components we treat ) Tj 1 0 0 1 436.55 414.449 Tm 106 Tz /OPExtFont4 11.5 Tf (\(Length12\)- ) Tj 1 0 0 1 507.1 414.449 Tm 93 Tz /OPExtFont5 13 Tf (1 as ) Tj 1 0 0 1 129.099 400.3 Tm 98 Tz (the middle point; b\) Standard deviation of the end point of the pseudo-orbit. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 129.099 366.449 Tm 96 Tz (formed by Method ) Tj 1 0 0 1 225.349 366.199 Tm 85 Tz /OPExtFont3 11.5 Tf (I ) Tj 1 0 0 1 233.05 366.449 Tm 98 Tz /OPExtFont5 13 Tf (is centred at the observation while the ensemble formed by ) Tj 1 0 0 1 128.9 343.149 Tm 96 Tz (Method I is centred at the end point of the pseudo) Tj 1 0 0 1 376.3 343.149 Tm 74 Tz /OPExtFont3 11.5 Tf (-) Tj 1 0 0 1 380.149 343.649 Tm 98 Tz /OPExtFont5 13 Tf (orbit. Whichever wins merely ) Tj 1 0 0 1 128.9 320.35 Tm 103 Tz (indicates which centre tend to be closer to the true state. Here the results in-) Tj 1 0 0 1 128.9 297.1 Tm 101 Tz (dicate that the end point of the pseudo-orbit obtained by ) Tj 1 0 0 1 423.85 297.1 Tm 106 Tz /OPExtFont4 11.5 Tf (ISCDc ) Tj 1 0 0 1 464.149 297.1 Tm 97 Tz /OPExtFont5 13 Tf (method falls ) Tj 1 0 0 1 128.9 273.799 Tm 100 Tz (closer to the true state than the observation. Therefore the ) Tj 1 0 0 1 428.149 274.049 Tm 94 Tz /OPExtFont4 11.5 Tf (ISG.Dc ) Tj 1 0 0 1 467.75 274.049 Tm 96 Tz /OPExtFont5 13 Tf (method can ) Tj 1 0 0 1 128.9 250.75 Tm 98 Tz (also be treated as a useful noise reduction method. Although Method IV did the ) Tj 1 0 0 1 128.65 227.25 Tm 103 Tz (best, it is much more costly. Method ) Tj 1 0 0 1 323.75 227.5 Tm 104 Tz /OPExtFont3 11.5 Tf (III ) Tj 1 0 0 1 341.05 227.5 Tm 100 Tz /OPExtFont5 13 Tf (seems to work better than Method ) Tj 1 0 0 1 522 227.5 Tm 85 Tz /OPExtFont3 11.5 Tf (I ) Tj 1 0 0 1 128.9 203.95 Tm 81 Tz /OPExtFont6 12.5 Tf (& ) Tj 1 0 0 1 142.3 204.2 Tm 101 Tz /OPExtFont5 13 Tf (II which indicates using the imperfection error to form the initial condition ) Tj 1 0 0 1 128.65 181.149 Tm 100 Tz (ensemble is useful. And we expect such ensemble works better in the case that ) Tj 1 0 0 1 128.4 157.899 Tm 99 Tz (the observational noise is relatively large. As we discussed in section ) Tj 1 0 0 1 470.649 158.1 Tm 97 Tz /OPExtFont2 11.5 Tf (5.2.5, ) Tj 1 0 0 1 500.399 158.1 Tm 93 Tz /OPExtFont5 13 Tf (when ) Tj 1 0 0 1 128.4 134.35 Tm 100 Tz (the observational noise is relatively larger than the model error, the geometrical ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 3 Tr 1 0 0 1 319.699 51.799 Tm 85 Tz (123 ) Tj ET EMC endstream endobj 664 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 665 0 obj <> stream 0 ! ,,b7!j8HQ@]% ڸSU\-z )ɫ&@tHߖЏɔ{C?d 4ijoo n_ŎGe:#Ne0KgοIx }?߶,eE[v,+[ Q׺xŷt<;Ohr?  <O{"T 0$,jTU'1ȑ\Q0LZ΂njUL*5ku.Nྰ˘M(O*Pͼ;lĆ~$1xߡCe-M,Lg>}މ,_'yRK $jPQOcIlGR0Πf BL&%Ns'$[Kϳlp B`sSRw - n_b -}̞DF4/}1m`+du מaK=T?4+&(z~&|a='#ͳ GS?F$IFds4 ϵv-f04J~ ?/JgD7(vcJl`3CK#(d wnvG@TL'92jvo&dDf)rFuqep~eB&.S- 5 ~=uߥ^ Kj<[JO~W3p0M#2}sJ.r3v k|PNDU)(~@L+3׹@o|Ftc{#`9Rs Kf}u;aDz#=AlGNe.l ,e\`@F㰑t vޠNVzyby #>c&!?׾FƮ:]D+y) ^gнUH薭!X)-4Qg #xvMWL@&ƖNdt&5>4.`- "kBM mh(UM~>z-ۤ)i#%R<Vl,x+ is,3v޽4 _]SQ7f!G!Q܍Wp4}c=z-"SZ0x!#eICP#3(OEy~M)uU4I ~kM&0y=>*&<#<4v)!G;̼F) 0V[e$F4:W#d?*}do:T̨H  |w[0=a)@U ugZ-#vrl. 3L`9fy0ƈO'QhQ\b{rq{xuOЍXn(TȗXPK~qn xz74$5u$.wo8-]ۉTkS! 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hDCۈ<ɚ%6\O7kjN@{?Fv MuY6lqR^ \ (|L鴔]|" g ! }pC#x,u]bB)'ddVY$)m'vGD.)Yv'c)De]D9 +=xƴי3*j;q^kU9"=J0 cvd kߞ]RsiFe;s5@PӀhVchԺ\lYeQcb+ST5*\ ؙ#CHT υGP;=:򴸰/t]Y2^1Bp>.knQ5t5BpcZLw0*բ+@{$` ͳfTAPUw}J`D/ؠ5vZ CUmO,{O E67psAUeMgaUx~JY7'NRNf!([ASL:"U_g%gbWmPϻMocCŽ6 +ٌҿW6Rjx0nl,&5jv&܇c| mX$ѠT0L:Y>1s- N/c,I{%kDra| J\d i*al[`FC,H iѸ[˘ G4 1S?>B"-=[NB@aV;[(,IE >o`*H/ dOʐ|2yks !P_ގ+$mwܕ?[^5)B֋ddH$om`Z)dE *ER(SFs +^)<* к[IRUXEk`iE( Ư~Rs^hdK5,o5C~\G 7ŵ7´HμFn{ӱ}c wTgyPj{2|'Epѓ~Zv0TLC f Ïg)Q;XKlJ-x~+fz` `Y֦{*W7(Z,_MS,zQ) MzibVYڠPEp ?bX2w٩!lG5*!}_S'm_ݓN3q${ñ3 /T%쯿)4NqgKT1bA]XBUlC*A}NdO:[ b]?|ACD*8A)K@Bax*BR/ beN]c VUKxδ0%S)xHjm5CKybirSREQݴÄeA J-1 $Exx~WyXyW=Q2iy= !'yXAeRDWX~UاF1oF-ϳe-gEʨ2A1@hץhg5X z08,'V ]w^ y-!8a2$?6"2M4XƂ4P.M$ebR @ gl@U/ȄډjO2r^[=%roPlM@dz[PF\Fvp Ð j W d4Y<]IU~kQ+u'C-kpځP64HdW>ne=Wr.g3S_1u+t/˰EDK_.w.H>QJ+;%]ykw Q/?;/!Rf:kKsC+L{H~⭕.e\lrrMG7̨d\2sAz/4~@ Ε#U t֕O8%UTd^/ٶ"J2+˝&?]|%:-šnN7Aal[f se5792?411؀k[pH9пc.rl\*0N[Xbo.nIWWP[<[}\MHw$6O<[ &_QU]lWz&v%c&aI+3{Iή+\Ͼ?%bOM!GA?L;IS{".0[48E. i?@ͥT```I'Krt[~b_@Uj3Y$^8X|v=Z/PZ>o ) |5~M0whQ/wl'7zk&QrlwJI^:N{]`#B%C;y\k.L|Rƺ\QX m? -Џ*mXڞ]rK@bb.&w7X\2r64W$D8:B3$VΣVp4DBs<ϝ[XiO&zPad 5>Ԗ~ᱹ%*> endobj 667 0 obj [668 0 R] endobj 668 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 509 0 0 804 0 0 cm /ImagePart_2151 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 478.8 557.049 Tm 116 Tz 3 Tr /OPExtFont1 4 Tf (0 1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 116 Tz 3 Tr 1 0 0 1 337.449 557.299 Tm 144 Tz /OPExtFont9 3.5 Tf (0.02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 144 Tz 3 Tr 1 0 0 1 442.1 557.049 Tm 132 Tz /OPExtFont1 4 Tf (0.08 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 132 Tz 3 Tr 1 0 0 1 442.55 427.199 Tm 133 Tz (0.08 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 133 Tz 3 Tr 1 0 0 1 479.3 427.449 Tm 116 Tz (0) Tj 1 0 0 1 484.3 427.199 Tm 61 Tz /OPExtFont9 3.5 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 61 Tz 3 Tr 1 0 0 1 302.899 560.649 Tm 116 Tz /OPExtFont1 4 Tf (0) Tj 1 0 0 1 306 557.299 Tm 119 Tz (0 ) Tj 1 0 0 1 308.649 557.299 Tm 90 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 90 Tz 3 Tr 1 0 0 1 337.899 297.1 Tm 133 Tz (0.02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 133 Tz 3 Tr 1 0 0 1 443.05 297.1 Tm 132 Tz (0.08 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 132 Tz 3 Tr 1 0 0 1 479.75 297.1 Tm 146 Tz (01 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 146 Tz 3 Tr 1 0 0 1 303.35 665.049 Tm 61 Tz /OPExtFont9 3.5 Tf (1 ) Tj 1 0 0 1 298.3 654.5 Tm 148 Tz (0.9 ) Tj 1 0 0 1 298.3 644.149 Tm (0.8 ) Tj 1 0 0 1 298.3 633.85 Tm 143 Tz (0.7 ) Tj 1 0 0 1 298.3 623.5 Tm 127 Tz (O.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 127 Tz 3 Tr 1 0 0 1 298.3 602.149 Tm 125 Tz /OPExtFont1 4 Tf (0.4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 125 Tz 3 Tr 1 0 0 1 298.3 592.1 Tm 148 Tz /OPExtFont9 3.5 Tf (0.3 ) Tj 1 0 0 1 298.3 581.5 Tm (0.2 ) Tj 1 0 0 1 298.3 571.2 Tm 133 Tz (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 133 Tz 3 Tr 1 0 0 1 303.85 405.6 Tm 48 Tz (1 ) Tj 1 0 0 1 298.8 395.3 Tm 129 Tz /OPExtFont1 4 Tf (0.9 ) Tj 1 0 0 1 298.8 384.5 Tm (0.8 ) Tj 1 0 0 1 298.8 374.149 Tm (0.7 ) Tj 1 0 0 1 292.1 363.35 Tm 178 Tz (I 0.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 178 Tz 3 Tr 1 0 0 1 291.1 353.05 Tm 50 Tz /OPExtFont9 3 Tf (.) Tj 1 0 0 1 292.1 353.05 Tm 178 Tz /OPExtFont9 3.5 Tf (5 0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 178 Tz 3 Tr 1 0 0 1 298.8 342.699 Tm 125 Tz /OPExtFont1 4 Tf (0.4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 125 Tz 3 Tr 1 0 0 1 433.449 612.95 Tm 124 Tz (Method III ) Tj 1 0 0 1 433.449 606.95 Tm (Method II ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 124 Tz 3 Tr 1 0 0 1 372.5 427.199 Tm 132 Tz (0.04 ) Tj 1 0 0 1 382.8 427.199 Tm 2000 Tz (\t) Tj 1 0 0 1 407.75 427.199 Tm 133 Tz (0.06 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 133 Tz 3 Tr 1 0 0 1 376.8 422.149 Tm 129 Tz (size of epe-bell ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 129 Tz 3 Tr 1 0 0 1 436.55 465.6 Tm 94 Tz /OPExtFont19 4.5 Tf (Method ) Tj 1 0 0 1 456.25 465.6 Tm 127 Tz /OPExtFont1 4 Tf (IV ) Tj 1 0 0 1 436.8 459.85 Tm 124 Tz (Method II ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 124 Tz 3 Tr 1 0 0 1 337.699 427.199 Tm 147 Tz /OPExtFont9 3.5 Tf (0.02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 147 Tz 3 Tr 1 0 0 1 298.55 525.35 Tm 148 Tz (0.9 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 148 Tz 3 Tr 1 0 0 1 298.55 514.799 Tm (0.8 ) Tj 1 0 0 1 298.55 504.25 Tm 142 Tz (0.7 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 142 Tz 3 Tr 1 0 0 1 291.85 496.8 Tm 101 Tz /OPExtFont9 5.5 Tf (8 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 101 Tz 3 Tr 1 0 0 1 298.55 493.699 Tm 148 Tz /OPExtFont9 3.5 Tf (0.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 148 Tz 3 Tr 1 0 0 1 291.85 483.35 Tm 170 Tz (S 0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 170 Tz 3 Tr 1 0 0 1 298.55 472.8 Tm 142 Tz (0.4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 142 Tz 3 Tr 1 0 0 1 291.85 468.949 Tm 40 Tz /OPExtFont13 5.5 Tf (LLL ) Tj 1 0 0 1 298.55 462.25 Tm 148 Tz /OPExtFont9 3.5 Tf (0.3 ) Tj 1 0 0 1 298.55 451.699 Tm (0.2 ) Tj 1 0 0 1 298.55 441.1 Tm 133 Tz (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 133 Tz 3 Tr 1 0 0 1 303.1 430.8 Tm 149 Tz (0) Tj 1 0 0 1 306.25 427.199 Tm 136 Tz (0 ) Tj 1 0 0 1 308.899 427.199 Tm 100 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 3 Tr 1 0 0 1 441.35 365.75 Tm 127 Tz /OPExtFont1 4 Tf (Method IV ) Tj 1 0 0 1 441.35 360 Tm 126 Tz (Method III ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 126 Tz 3 Tr 1 0 0 1 68.4 384.5 Tm 148 Tz /OPExtFont9 3.5 Tf (O8 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 148 Tz 3 Tr 1 0 0 1 68.4 374.149 Tm 172 Tz (07 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 172 Tz 3 Tr 1 0 0 1 61.899 363.1 Tm (5 0.6 ) Tj 1 0 0 1 61.7 352.8 Tm 93 Tz /OPExtFont9 5.5 Tf (8 ) Tj 1 0 0 1 68.4 352.8 Tm 142 Tz /OPExtFont9 3.5 Tf (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 142 Tz 3 Tr 1 0 0 1 61.7 342.5 Tm 49 Tz /OPExtFont9 15.5 Tf (k ) Tj 1 0 0 1 68.15 342.5 Tm 142 Tz /OPExtFont9 3.5 Tf (0 4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 142 Tz 3 Tr 1 0 0 1 372.949 706.549 Tm 94 Tz /OPExtFont0 11 Tf (5.5 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont0 11 Tf 94 Tz 3 Tr 1 0 0 1 73.45 665.049 Tm 61 Tz /OPExtFont9 3.5 Tf (1 ) Tj 1 0 0 1 68.65 654.95 Tm 142 Tz (0.9 ) Tj 1 0 0 1 68.65 644.399 Tm (0.8 ) Tj 1 0 0 1 68.4 633.85 Tm 148 Tz (0 7 ) Tj 1 0 0 1 61.899 623.5 Tm 167 Tz (S 0.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 167 Tz 3 Tr /OPExtFont13 10 Tf 1 0 0 1 1531 2726 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 167 Tz 3 Tr 1 0 0 1 1860 2726 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 167 Tz 3 Tr 1 0 0 1 2604 2726 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 167 Tz 3 Tr 1 0 0 1 3816 2726 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 167 Tz 3 Tr 1 0 0 1 4717 2726 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 167 Tz 3 Tr 1 0 0 1 5054 2726 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 167 Tz 3 Tr 1 0 0 1 1190 3619 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 167 Tz 3 Tr 1 0 0 1 1531 3619 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 167 Tz 3 Tr 1 0 0 1 1860 3619 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 167 Tz 3 Tr 1 0 0 1 2604 3619 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 167 Tz 3 Tr 1 0 0 1 212.15 617.75 Tm 124 Tz /OPExtFont1 4 Tf (Method II ) Tj 1 0 0 1 193.9 612 Tm 1659 Tz /OPExtFont1 3 Tf (- ) Tj 1 0 0 1 210.5 612 Tm 125 Tz /OPExtFont1 4 Tf ( Method I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 125 Tz 3 Tr /OPExtFont13 10 Tf 1 0 0 1 4717 3619 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 125 Tz 3 Tr 1 0 0 1 5054 3619 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 125 Tz 3 Tr 1 0 0 1 68.4 591.85 Tm 142 Tz /OPExtFont9 3.5 Tf (0.3 ) Tj 1 0 0 1 68.4 581.5 Tm 148 Tz (0.2 ) Tj 1 0 0 1 68.4 571.2 Tm 112 Tz /OPExtFont1 4 Tf (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 112 Tz 3 Tr /OPExtFont13 10 Tf 1 0 0 1 1531 3900 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 112 Tz 3 Tr 1 0 0 1 1860 3900 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 112 Tz 3 Tr 1 0 0 1 2604 3900 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 112 Tz 3 Tr 1 0 0 1 3816 3900 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 112 Tz 3 Tr 1 0 0 1 4717 3900 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 112 Tz 3 Tr 1 0 0 1 5054 3900 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 112 Tz 3 Tr 1 0 0 1 72.7 560.649 Tm 130 Tz /OPExtFont1 4 Tf (0) Tj 1 0 0 1 75.849 557.299 Tm 119 Tz (0 ) Tj 1 0 0 1 78.5 557.299 Tm 90 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 90 Tz 3 Tr /OPExtFont13 10 Tf 1 0 0 1 1860 4733 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 90 Tz 3 Tr 1 0 0 1 2604 4733 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 90 Tz 3 Tr 1 0 0 1 3816 4733 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 90 Tz 3 Tr 1 0 0 1 4717 4733 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 90 Tz 3 Tr 1 0 0 1 5054 4733 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 90 Tz 3 Tr 1 0 0 1 107.299 557.299 Tm 132 Tz /OPExtFont1 4 Tf (0.02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 132 Tz 3 Tr 1 0 0 1 142.55 557.299 Tm 129 Tz (0.04 ) Tj 1 0 0 1 152.65 557.5 Tm 2000 Tz (\t) Tj 1 0 0 1 177.599 557.5 Tm 151 Tz /OPExtFont9 3.5 Tf (0.06 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 151 Tz 3 Tr 1 0 0 1 146.9 552.5 Tm (size of ) Tj 1 0 0 1 163.9 552.7 Tm 119 Tz /OPExtFont1 4 Tf (eps) Tj 1 0 0 1 172.55 552.7 Tm 239 Tz /OPExtFont1 3 Tf (-) Tj 1 0 0 1 175.699 552.7 Tm 114 Tz /OPExtFont1 4 Tf (ball ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 114 Tz 3 Tr 1 0 0 1 212.9 557.299 Tm 147 Tz /OPExtFont9 3.5 Tf (0.08 ) Tj 1 0 0 1 222.949 560.399 Tm 2000 Tz (\t) Tj 1 0 0 1 249.349 557.299 Tm 140 Tz /OPExtFont1 4 Tf (01 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 140 Tz 3 Tr 1 0 0 1 182.65 349.899 Tm 148 Tz /OPExtFont9 3.5 Tf (eeesee ) Tj 1 0 0 1 199.699 349.199 Tm 127 Tz /OPExtFont1 4 Tf ( Method IV ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 127 Tz 3 Tr 1 0 0 1 201.099 343.699 Tm 124 Tz (Method I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 124 Tz 3 Tr 1 0 0 1 68.15 332.149 Tm 142 Tz /OPExtFont9 3.5 Tf (0.3 ) Tj 1 0 0 1 68.15 321.35 Tm 148 Tz (0.2 ) Tj 1 0 0 1 68.15 311.05 Tm 128 Tz (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 128 Tz 3 Tr 1 0 0 1 72.7 300.5 Tm 136 Tz (0) Tj 1 0 0 1 75.599 296.899 Tm 149 Tz (0 ) Tj 1 0 0 1 78.5 300.149 Tm 2000 Tz (\t) Tj 1 0 0 1 107.049 297.1 Tm 135 Tz /OPExtFont1 4 Tf (0.02 ) Tj 1 0 0 1 117.599 297.1 Tm 2000 Tz (\t) Tj 1 0 0 1 142.55 297.1 Tm 133 Tz (0.04 ) Tj 1 0 0 1 152.9 297.1 Tm 2000 Tz (\t) Tj 1 0 0 1 177.599 297.1 Tm 135 Tz (0.06 ) Tj 1 0 0 1 188.15 297.1 Tm 2000 Tz (\t) Tj 1 0 0 1 212.9 297.1 Tm 132 Tz (0.08 ) Tj 1 0 0 1 223.199 297.1 Tm 2000 Tz (\t) Tj 1 0 0 1 249.349 297.1 Tm 151 Tz (01 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 151 Tz 3 Tr 1 0 0 1 146.65 292.1 Tm 130 Tz (size of eps-bell ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 4 Tf 130 Tz 3 Tr 1 0 0 1 44.399 264 Tm 90 Tz /OPExtFont3 11 Tf (Figure 5.5: Comparing nowcasting ensemble using &ball. Observations are gen-) Tj 1 0 0 1 44.399 250.1 Tm 94 Tz (erated by Ikeda Map with observational noise N\(0, 0.05\). The truncated Ikeda ) Tj 1 0 0 1 44.399 236.149 Tm 88 Tz (model is used to estimate the current state. We compare the nowcasting ensemble ) Tj 1 0 0 1 44.399 222.5 Tm 95 Tz (formed by Method I, Method ) Tj 1 0 0 1 194.65 222.5 Tm 82 Tz /OPExtFont0 11 Tf (II, ) Tj 1 0 0 1 210.25 222.5 Tm 92 Tz /OPExtFont3 11 Tf (Method ) Tj 1 0 0 1 252.25 222.5 Tm 93 Tz /OPExtFont0 11 Tf (III ) Tj 1 0 0 1 269.3 222.5 Tm 94 Tz /OPExtFont3 11 Tf (and Method IV. All the ensemble ) Tj 1 0 0 1 44.399 208.799 Tm 87 Tz (contains 64 ensemble members. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 44.399 178.299 Tm 95 Tz (information of the model error is hard to extract. In this case the estimation ) Tj 1 0 0 1 44.399 155.5 Tm 92 Tz (of model error, i.e. imperfection error, will more or less look like random noise. ) Tj 1 0 0 1 44.399 132.7 Tm 93 Tz (As we mentioned in section ) Tj 1 0 0 1 187.9 132.7 Tm 74 Tz /OPExtFont0 11 Tf (5.4, ) Tj 1 0 0 1 210.25 132.7 Tm 94 Tz /OPExtFont3 11 Tf (the disadvantage of Method ) Tj 1 0 0 1 354.949 132.95 Tm 91 Tz /OPExtFont0 11 Tf (III ) Tj 1 0 0 1 372.25 132.95 Tm 96 Tz /OPExtFont3 11 Tf (is that it as- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 96 Tz 3 Tr 1 0 0 1 232.55 51.1 Tm 76 Tz (124 ) Tj ET EMC endstream endobj 669 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 670 0 obj <> stream 0J ,,(j%Nx** +:WZeWkH2|&cB7;tʌYi L`-EB(6j?8?F$soEǤx[ecPYZA|ArFF6fK®=,ů>,9i^3?2ʔԁőN`h-y?f6VG ݲȺڪ07 ɫ].S]V&c;[&TSFG}Jc w󌝟U`wv?WFo5 8ޱxZF6 }/RI/"ϗgɱ3a zq^ƙpH (562 Zƍg]0 /IdЇ@P 휔 >;7Qm"6zM"HpYx[IFɉxE\vQc$$ mT'Q7'˙!K&OLAo.a_)y Pt}vJ;\|;ق>ty"j ւΛ'\˒ݙ6 K PURn-6hR՟i=S3rJ%(o:Ԉ$OwI% dZVk& 'h4:J|a6Tx!Gn]FY8K<-65W|3NnF[1 ;G"]\D}<_OwTz_xj9I6j.М] tIvwR?ӻj2&;'ťqaRT8ՊfNHDN% ˆCOl*ϋp:'BMĥ~z.btG gj-yMyƼW}Tn.%>谎I$<@xZ}c';I ^U)ֈr\LI lĢ*vɪz8i+]Yc+lc MI9Gx5gVrLn wQ|\ U:90S},ee=wv[vѵf0Ft'Oi]8eC VNtM3M%yDj1UDRgKmqE/YsBM`&ՎRjp'f=O[1jb 13+ŷCBwP96 r< dX;4*Su҉ R%O9Tc .WZeyv"g {,ezsU'p֥=xgl )U]r;ۏ3ʊ|VH( 0K}ކǑBqaFcfjw2v$WNܨ<GDRr+QBןeCևш5"ȑ29~w] Anu=tԶ!8ėڏ)nn)BUwV$׈Oy|l>ڛ*h21~_{8%iZ 6q!Qa`:J/X=*j9>5l颋1TXo(,3<|CT iYhsPƕDUnOM­3[g) FWi_ >9ұ5=t: 49㲂 CO~g|2xtW8`*F|oV>ί(>o yE&tqQŌM.8蘗8bq| [͞B'@ov QfGQ}H+gr*Acl]~HlJY b4-Qˑ]5 ?x`38Ws 7,Z-T2zz58YX~]/TfYZu5I:J ?#?23@>co ;L}uy7Ws9Wձvԟ)aLHMg"`8' e{4>ҵ!%Ҝʱ:aDM#Ġk;\9+*_qqaj rޥb͵';!S&X P(#-ϣV0?q[f&<HcR|.rC&28 7 <~ x&G!RUny~',/7*{&R 38 9Gl:3jKT,0ЪPr(2aTa*v8$9aH ZXAlǑ7 ++ DKSD%/?ec$\ț4 WlEN1?n- *"E{ 1L_p\auT1t[ Abeѥ!GtaSEt+3LzDE7uM҂뱷*O(ܡ-ud;c8Q/e)^حz/AH>xȴDZPD DYoQrs Q`#>f(Q1Dv^%Ybds9sx-tW081f7)J?,|v/nfJf N4^(?(/24(QXOy!'KXH` XRp'h$nUJnXDl37VEqDExfHxy8rh0J2齵4 Ϊ/K@))9n*VǸV=XY<zrvjJ~i8{ô& %Ng/9+u,ds*#1*Jxz +"q߀UE-P&q{d#%aH_@4с*}7Y9Ic=ߡy72%oݫln{͞L:bd>nBAQ oY=ehM/p7?Rk؈%dAPYbD [tqۦ':nMI ^_>7Ұ[i<EWOgB:/38%citz`4rΕ?𮌕]&y1Zпb4H5cVY8R.Kzd{`A.}Ƈ"3*w޿ Y*`aSRByCs Wb4m<[cOwYSM8{$Ϻ`(ΙQ"Əq'#@9[zi%Yt !"V h˚ERAz`K.*$2ܘ-Vixa\c@^gbt0\A4Cm߭QVEU*&@qOYA0DOю7kkC@$*ξJb،$.GްeU_Ҷ` ꏬXә*Y_PI/ŚtpMk-vC7"Vzu"lgXDCK+%jѰ ?Z[EV(B0, |7tN7 6y?kI|_и#.9 (gQ9BN V]ԙ Gy+E!pi|&/f ّc=XW%S>oT7؁#mUY[g[z֨>=VpcnJC m/؆7!a-%_BPŸ *P .?k54l-WGJ ހZѠB՗@nbXI҆IoZXC8`a?ᵮ2=Ѭflyݻ#BSlyK(tσedrDc⛁fHwE+;"`vRdf40mP>Ǵʳʴϸ#CKՓZƴDҝ.%̥lII(CһPݳܳjd?F‚4V>#:۳ܱD03gZL s8$|KNQ9\4΅zЇF7CJ{v>N_AtSX=qWڮ0 R+zekAH#jmB6C$h k}`+7 C֑4O+]R4(#X8 HS7IxXF}PMEvPczx]}+U8V` Ķ- F|k:э.pջLXl4CH i׷/U򺩎xVl|ޕdz8pxR~kYK2X;]C.rǏpVyRh3V;Y^{W1$ՖQe:=SF@G[t`GFY8!E:yRvtiF5-BoL'h} ϝ%>ɪZQBoS!waq)8ٓI}rB "x5@~Zjӑx aGJB `^)  ᡔ-xo/IzG<U}:akoֈb3=zpo_уwe6]{ XEm'heNRW Ge B}5]ѹ\tx=\3DX'I<BE7WAGUJ37&Ty*$':'ffeש-{cӟ[DA7PIesF*L,IbQe6=(۪JjR ֽ~Pf;8;g(8guga}9 $TJ{Q2le4h?R3+;MrQ*Qh~'VP /]9eyx*f1-^[2xcZ^)sJ3XU|WhBǀl#Vx׎&vY=\oYƠkƕz )E?JrtoNazk!m@#NSRfAQ8+ju) zJhOR\6P 4A_?Th-YW(,ΦY~+H#N-\Se%:Y$ng ߁(#y=,PYg#Z噡4>ќAWp8 :7b+E㭛3}:du[yNM8D}9A0ÑnA"hLRhI脨e܂6 /jViv^8s YsINoQ#eth4XO&g'ľ?2Y' 햂*.'VyCvut5-t`S%!B8QqK] ᾮb!t|"z 9QLiwĀ#(˓%YʔI;ItqNU0xk.DnU-FUT)M|Ow,xE+~6>XN |WZUo$Ϥzo4?E9 Jl䅓HU_ۥ`:5SHSN&pJ)'|_;x#P*hE: ixp!/bnF4g=1 =ue<[{`a %Opqf'[4̸/ 쪊=%e|9jr6ȿdP5/XCAz݉J1k}tgtEEvV?ˎ?/ʈ[J>CPsy= gs d@>ؓsy.|P:> "쌓/uK &zQk wkBZo}^Fg+!|TrgHT0%)3?} M"i#!HnVy~`r3 ӃoBFl>hs2GYUgvJPOwڅq.Rxm 7%bi|v_yDV a~RUV<%@CYzFXIqPw=Q-Ο "F/_E#7ME OO]2 l=#rimY;!v^y1p ҌšU챝ěј$7FofYj%=}Y4Ʃ,D1xD>TVjɜNߙ]#A ʈh5ua\^)WW}$7~i zz-)E7Pؐmj'\:8NTu= -В. ?h7x"g>egg"|bfMHܼ!-e#@^ԽOh,jI&@^ERI'1[8{ rurb&G*3r^ҁ zv,^on>MFtJN@4V|r̐vBeLMqGaE&#~am"7eFՊ_>0'=t d9V ΃Z=ܻ'|{1aKR/LP΢45h{jV,cۃX8ƙ2^]&>`D OO}@ђJryY.C`B!B|+6;vaJէDKeY$ d}7ƔzG*c fҘ7#TMbv1 9^6oKePKHZyݴ s! 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<^C8d|V z)[J-g*ӡ"X`>f(S8,%WCdp^+ f5T'xׯqk%rF;,썙3 Ț j#'S׌)|)Ҡ̓ǀT~ DRcQoXj67iuw]^ekPڃ/-sޏA ;qќ|ti+2AT{' 0NJĆe oaܥzV@.ڪ}zR..hM;jF70ȑ񸩵P.u}j7qB-lL0b.Dړ\ky.$",;~N֩ [4ĤcsWi t%S@ئ+ ;坄y^O> Yc_4jopϺOau:I㿫Q^Yph7C*srڷQxGW|+6 llPznSx*E,;-I5syc` hZt摫Qd$n|CCHĉKa`HycсBERnKgʾ!!hLŜQБWT۠y(ߔf ^2E7Sz>³/ڲ+;дDZAgƀ<͈u!DКӨi7vhz͆Jri٫&4=Qr-/+.96?Y 4:9eZw9TU ˿QS=%5|]K/qB)0lKSDG;,G_a^'mԫKs͹ś+2[hOcKKSjؼDz[O'ĨBӜuaD.%RC 2ӐS+XdWbI(1&m>3"v8 d\ʄԪQwB) PGQ6þ `=L*+t N#,FZk" خSpc7 \^M +a"p` Ǽ$%h+boB,&ʑJ˃#v\'Cf[ҏ>f{nϫkڻM6xnF@:+0Z\@|.sٵe쪼rAX \4zW8cԛ YӰ2x\ql+ޒQ_ }k[2kFۂ[^Zmb.Q7 T20JSQNL%}~5 Aٱ0$ K'-(`kkKÆdrܝﵫ>(H) t`YN gAnٌo0qiicje=}KcHیwM},|M@be&M.~dMi|it0A!?&?vĒBI䅀FuOp'}7@m,֓Ef2@ qS`\\GXs&`cG2uߋ gΏdPi@'Y HPjO_ĿSvދ3&s "TVc]I%\f2ʱ&VlBy_MsmP;-ZHA9G/Sr#-}npN ExPߠ˒>};MXW`DTRK߽}n;N_AWWrԗ!AN']Gsk/B n}0B BM,%K# +е'/F)R[ͰOބʖ*J}2TK؋ :I@4sA#_@<"@d+YcdYtR呗xLx[n낅iCqP'~C<&65X^1{xr O0&c:40X SiMܦ, s e' 8 ГQޞa@1zA(EGY6lCxP=Rv Mӟ:B QXy-^\1tAanX߿ 9FW>I^jKh J PRi~ap ģ&#> endobj 672 0 obj [673 0 R] endobj 673 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 512 0 0 796 0 0 cm /ImagePart_2152 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 333.6 540.899 Tm 105 Tz 3 Tr /OPExtFont18 4 Tf (0.01 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 105 Tz 3 Tr 1 0 0 1 368.399 540.649 Tm 115 Tz (0.02 ) Tj 1 0 0 1 378.699 540.649 Tm 2000 Tz /OPExtFont22 4 Tf (\t) Tj 1 0 0 1 403.449 540.649 Tm 115 Tz /OPExtFont18 4 Tf (0.03 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 115 Tz 3 Tr 1 0 0 1 372.699 535.6 Tm 123 Tz (size of eps-ball ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 123 Tz 3 Tr 1 0 0 1 438.5 540.649 Tm 110 Tz (0.04 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 110 Tz 3 Tr 1 0 0 1 473.5 540.399 Tm 119 Tz /OPExtFont9 4 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 119 Tz 3 Tr 1 0 0 1 477.85 540.399 Tm 129 Tz /OPExtFont11 3.5 Tf (05 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 3.5 Tf 129 Tz 3 Tr 1 0 0 1 299.5 519.299 Tm 27 Tz /OPExtFont19 7.5 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 7.5 Tf 27 Tz 3 Tr 1 0 0 1 294.5 508.5 Tm 116 Tz /OPExtFont18 4 Tf (0.9 ) Tj 1 0 0 1 294.5 498.149 Tm (0.8 ) Tj 1 0 0 1 294.5 487.85 Tm 113 Tz (0.7 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 113 Tz 3 Tr 1 0 0 1 287.75 477.3 Tm 73 Tz /OPExtFont0 11.5 Tf (I ) Tj 1 0 0 1 291.1 477.3 Tm 143 Tz /OPExtFont18 4 Tf ( 0.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 143 Tz 3 Tr 1 0 0 1 64.549 509.199 Tm 117 Tz (0.9 ) Tj 1 0 0 1 64.549 498.649 Tm (0.8 ) Tj 1 0 0 1 64.549 488.3 Tm 113 Tz (0.7 ) Tj 1 0 0 1 64.549 477.75 Tm (0.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 113 Tz 3 Tr 1 0 0 1 64.549 467.199 Tm (0.5 ) Tj 1 0 0 1 64.549 456.649 Tm 109 Tz (0.4 ) Tj 1 0 0 1 64.549 446.3 Tm (0.3 ) Tj 1 0 0 1 64.299 435.75 Tm 117 Tz (0.2 ) Tj 1 0 0 1 64.299 425.449 Tm 102 Tz (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 102 Tz 3 Tr 1 0 0 1 294.5 466.699 Tm 116 Tz (0.5 ) Tj 1 0 0 1 294.5 456.149 Tm 113 Tz (0.4 ) Tj 1 0 0 1 294.5 445.85 Tm (0.3 ) Tj 1 0 0 1 294.5 435.3 Tm 116 Tz (0.2 ) Tj 1 0 0 1 294.5 424.949 Tm 101 Tz (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 101 Tz 3 Tr 1 0 0 1 299.05 414.399 Tm 128 Tz /OPExtFont11 3.5 Tf (0) Tj 1 0 0 1 302.149 411.05 Tm 149 Tz /OPExtFont9 3.5 Tf (0 ) Tj 1 0 0 1 305.05 411.05 Tm 81 Tz /OPExtFont11 3.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 3.5 Tf 81 Tz 3 Tr 1 0 0 1 68.9 414.899 Tm 103 Tz (O ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 3.5 Tf 103 Tz 3 Tr 1 0 0 1 103.2 411.5 Tm 108 Tz /OPExtFont18 4 Tf (0.01 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 108 Tz 3 Tr 1 0 0 1 333.85 411.3 Tm 111 Tz /OPExtFont11 3.5 Tf (0.01 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 3.5 Tf 111 Tz 3 Tr 1 0 0 1 244.099 411.3 Tm 115 Tz /OPExtFont18 4 Tf (0.05 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 115 Tz 3 Tr 1 0 0 1 208.8 411.3 Tm (0.04 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 115 Tz 3 Tr 1 0 0 1 438.5 410.8 Tm 113 Tz (0.04 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 113 Tz 3 Tr 1 0 0 1 473.75 410.8 Tm 127 Tz /OPExtFont11 3.5 Tf (0.05 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 3.5 Tf 127 Tz 3 Tr 1 0 0 1 138.699 411.5 Tm 116 Tz /OPExtFont18 4 Tf (0.02 ) Tj 1 0 0 1 149.05 411.5 Tm 2000 Tz /OPExtFont22 4 Tf (\t) Tj 1 0 0 1 173.75 411.5 Tm 116 Tz /OPExtFont18 4 Tf (0.03 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 116 Tz 3 Tr 1 0 0 1 143.05 406.5 Tm 122 Tz (size of eps-bell ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 122 Tz 3 Tr 1 0 0 1 368.649 411.05 Tm 126 Tz /OPExtFont11 3.5 Tf (0.02 ) Tj 1 0 0 1 378.699 411.05 Tm 2000 Tz (\t) Tj 1 0 0 1 403.899 411.05 Tm 147 Tz (003 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 3.5 Tf 147 Tz 3 Tr 1 0 0 1 372.949 406.25 Tm 139 Tz (size of es Gall ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 3.5 Tf 139 Tz 3 Tr 1 0 0 1 299.75 389.199 Tm 22 Tz /OPExtFont19 7.5 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 7.5 Tf 22 Tz 3 Tr 1 0 0 1 294.699 378.649 Tm 113 Tz /OPExtFont18 4 Tf (0.9 ) Tj 1 0 0 1 294.5 368.1 Tm 116 Tz (0.8 ) Tj 1 0 0 1 294.5 357.5 Tm 113 Tz (0.7 ) Tj 1 0 0 1 288 347.199 Tm 138 Tz ( 0.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 138 Tz 3 Tr 1 0 0 1 64.299 379.1 Tm 113 Tz (0.9 ) Tj 1 0 0 1 64.299 368.55 Tm (0.8 ) Tj 1 0 0 1 64.099 358.25 Tm (0.7 ) Tj 1 0 0 1 64.099 347.699 Tm 116 Tz (0.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 116 Tz 3 Tr 1 0 0 1 64.099 337.1 Tm (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 116 Tz 3 Tr 1 0 0 1 64.099 326.55 Tm 113 Tz (0.4 ) Tj 1 0 0 1 64.099 316 Tm (0.3 ) Tj 1 0 0 1 64.099 305.449 Tm 125 Tz /OPExtFont11 3.5 Tf (0.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 3.5 Tf 125 Tz 3 Tr 1 0 0 1 64.099 295.1 Tm 101 Tz /OPExtFont18 4 Tf (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 101 Tz 3 Tr 1 0 0 1 294.5 336.649 Tm 116 Tz (0.5 ) Tj 1 0 0 1 294.5 326.1 Tm 113 Tz (0.4 ) Tj 1 0 0 1 294.5 315.5 Tm (0.3 ) Tj 1 0 0 1 294.5 304.949 Tm (0.2 ) Tj 1 0 0 1 294.5 294.649 Tm 97 Tz (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 97 Tz 3 Tr 1 0 0 1 68.65 284.55 Tm 119 Tz /OPExtFont11 3.5 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 3.5 Tf 119 Tz 3 Tr 1 0 0 1 243.599 280.7 Tm 118 Tz /OPExtFont18 4 Tf (0.05 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 118 Tz 3 Tr 1 0 0 1 208.55 280.7 Tm 130 Tz /OPExtFont11 3.5 Tf (0.04 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 3.5 Tf 130 Tz 3 Tr 1 0 0 1 103.2 281.2 Tm 184 Tz /OPExtFont9 3.5 Tf (00 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3.5 Tf 184 Tz 3 Tr 1 0 0 1 298.8 284.1 Tm 130 Tz /OPExtFont11 3.5 Tf (0) Tj 1 0 0 1 301.899 280.7 Tm 149 Tz /OPExtFont9 3.5 Tf (0 ) Tj 1 0 0 1 304.8 280.7 Tm 81 Tz /OPExtFont11 3.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 3.5 Tf 81 Tz 3 Tr 1 0 0 1 333.35 280.5 Tm 105 Tz /OPExtFont18 4 Tf (0.01 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 105 Tz 3 Tr 1 0 0 1 438.5 280.25 Tm 117 Tz /OPExtFont9 4.5 Tf (0.04 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 117 Tz 3 Tr 1 0 0 1 473.75 280.25 Tm 130 Tz /OPExtFont11 3.5 Tf (0.05 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 3.5 Tf 130 Tz 3 Tr 1 0 0 1 138.5 280.95 Tm 113 Tz /OPExtFont18 4 Tf (0.02 ) Tj 1 0 0 1 148.55 280.95 Tm 2000 Tz /OPExtFont22 4 Tf (\t) Tj 1 0 0 1 173.3 280.95 Tm 118 Tz /OPExtFont18 4 Tf (0.03 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 118 Tz 3 Tr 1 0 0 1 142.55 275.899 Tm 126 Tz (sae of eps-bell ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 126 Tz 3 Tr 1 0 0 1 368.399 280.5 Tm 115 Tz (0.02 ) Tj 1 0 0 1 378.699 280.5 Tm 2000 Tz /OPExtFont22 4 Tf (\t) Tj 1 0 0 1 403.699 280.5 Tm 113 Tz /OPExtFont18 4 Tf (0.03 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 113 Tz 3 Tr 1 0 0 1 372.699 275.7 Tm 57 Tz /OPExtFont19 7.5 Tf (size ) Tj 1 0 0 1 382.1 275.45 Tm 124 Tz /OPExtFont18 4 Tf ( of eps-ball ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 124 Tz 3 Tr 1 0 0 1 369.35 690.149 Tm 90 Tz /OPExtFont0 11.5 Tf (5.5 Results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont0 11.5 Tf 90 Tz 3 Tr 1 0 0 1 294.5 638.299 Tm 116 Tz /OPExtFont18 4 Tf (0.9 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 116 Tz 3 Tr 1 0 0 1 294.5 628 Tm (0.8 ) Tj 1 0 0 1 294.5 617.7 Tm 113 Tz (0.7 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 113 Tz 3 Tr 1 0 0 1 287.75 607.1 Tm 81 Tz /OPExtFont19 7.5 Tf (I ) Tj 1 0 0 1 291.1 607.1 Tm 143 Tz /OPExtFont18 4 Tf ( 0.6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 143 Tz 3 Tr 1 0 0 1 57.35 602.1 Tm 38 Tz /OPExtFont19 7.5 Tf (15 ) Tj 1 0 0 1 61.45 602.1 Tm 2000 Tz /OPExtFont22 4 Tf (\t) Tj 1 0 0 1 286.8 601.6 Tm 84 Tz /OPExtFont18 4 Tf (15 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 84 Tz 3 Tr 1 0 0 1 57.6 597.299 Tm 155 Tz (I 0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 155 Tz 3 Tr 1 0 0 1 64.799 586.7 Tm 123 Tz (OA ) Tj 1 0 0 1 72 586.7 Tm 2000 Tz /OPExtFont19 7.5 Tf (\t) Tj 1 0 0 1 287.3 596.549 Tm 90 Tz (1 ) Tj 1 0 0 1 292.1 596.549 Tm 130 Tz /OPExtFont18 4 Tf ( 0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 130 Tz 3 Tr 1 0 0 1 294.5 586.25 Tm 113 Tz (0.4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 113 Tz 3 Tr 1 0 0 1 64.799 576.399 Tm (0.3 ) Tj 1 0 0 1 72 576.399 Tm 2000 Tz /OPExtFont22 4 Tf (\t) Tj 1 0 0 1 294.5 575.7 Tm 113 Tz /OPExtFont18 4 Tf (0.3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 113 Tz 3 Tr 1 0 0 1 210.25 571.1 Tm 131 Tz /OPExtFont9 4 Tf (Method I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 131 Tz 3 Tr 1 0 0 1 64.799 565.85 Tm 113 Tz /OPExtFont18 4 Tf (0.2 ) Tj 1 0 0 1 72 565.85 Tm 2000 Tz /OPExtFont22 4 Tf (\t) Tj 1 0 0 1 294.5 565.1 Tm 116 Tz /OPExtFont18 4 Tf (0.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 116 Tz 3 Tr 1 0 0 1 64.799 555.5 Tm 98 Tz (0.1 ) Tj 1 0 0 1 71.049 555.5 Tm 2000 Tz /OPExtFont22 4 Tf (\t) Tj 1 0 0 1 294.5 554.799 Tm 101 Tz /OPExtFont18 4 Tf (0.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 101 Tz 3 Tr 1 0 0 1 103.45 541.35 Tm 108 Tz (0.01 ) Tj 1 0 0 1 113.049 544.5 Tm 2000 Tz /OPExtFont22 4 Tf (\t) Tj 1 0 0 1 138.949 541.35 Tm 116 Tz /OPExtFont18 4 Tf (0.02 ) Tj 1 0 0 1 149.3 541.35 Tm 2000 Tz /OPExtFont22 4 Tf (\t) Tj 1 0 0 1 173.75 541.1 Tm 118 Tz /OPExtFont18 4 Tf (0.03 ) Tj 1 0 0 1 184.3 544.25 Tm 2000 Tz /OPExtFont22 4 Tf (\t) Tj 1 0 0 1 209.05 541.1 Tm 113 Tz /OPExtFont18 4 Tf (0.04 ) Tj 1 0 0 1 219.099 544.149 Tm 2000 Tz /OPExtFont22 4 Tf (\t) Tj 1 0 0 1 244.099 540.899 Tm 118 Tz /OPExtFont18 4 Tf (0.05 ) Tj 1 0 0 1 254.65 545.7 Tm 2000 Tz /OPExtFont22 4 Tf (\t) Tj 1 0 0 1 299.05 544.25 Tm 128 Tz /OPExtFont11 3.5 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 3.5 Tf 128 Tz 3 Tr 1 0 0 1 143.3 536.299 Tm 125 Tz /OPExtFont18 4 Tf (size of fps-ball ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont18 4 Tf 125 Tz 3 Tr 1 0 0 1 40.299 247.6 Tm 91 Tz /OPExtFont3 11 Tf (Figure 5.6: Comparing nowcasting ensemble using f-ball. Observations are gen-) Tj 1 0 0 1 40.1 234.149 Tm 93 Tz (erated by Lorenz96 Model ) Tj 1 0 0 1 174 233.899 Tm 88 Tz /OPExtFont0 11.5 Tf (II ) Tj 1 0 0 1 186.949 233.899 Tm 94 Tz /OPExtFont3 11 Tf (with observational noise N\(0, 0.1\). The Lorenz96 ) Tj 1 0 0 1 40.299 220.5 Tm 92 Tz (Model ) Tj 1 0 0 1 73.2 220.5 Tm 78 Tz /OPExtFont0 11.5 Tf (I ) Tj 1 0 0 1 80.4 220.25 Tm 88 Tz /OPExtFont3 11 Tf (is used to estimate the current state. We compare the nowcasting ensem-) Tj 1 0 0 1 40.1 206.549 Tm 90 Tz (ble formed by Method ) Tj 1 0 0 1 149.3 206.549 Tm 73 Tz /OPExtFont0 11.5 Tf (I, ) Tj 1 0 0 1 159.849 206.549 Tm 92 Tz /OPExtFont3 11 Tf (Method ) Tj 1 0 0 1 200.65 206.549 Tm 78 Tz /OPExtFont0 11.5 Tf (II, ) Tj 1 0 0 1 215.05 206.549 Tm 92 Tz /OPExtFont3 11 Tf (Method ) Tj 1 0 0 1 255.849 206.299 Tm 87 Tz /OPExtFont0 11.5 Tf (III ) Tj 1 0 0 1 271.699 206.299 Tm 91 Tz /OPExtFont3 11 Tf (and Method IV. All the ensemble ) Tj 1 0 0 1 39.85 192.899 Tm 87 Tz (contains 64 ensemble members. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 39.85 162.649 Tm 92 Tz (sume the imperfection error is ) Tj 1 0 0 1 196.3 162.399 Tm 88 Tz /OPExtFont0 11.5 Tf (IID ) Tj 1 0 0 1 217.699 162.149 Tm 92 Tz /OPExtFont3 11 Tf (distributed, the assumption become less of ) Tj 1 0 0 1 40.1 139.35 Tm (a disadvantage when the observational noise is relatively larger than the model ) Tj 1 0 0 1 39.85 116.799 Tm 85 Tz (error. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 85 Tz 3 Tr 1 0 0 1 228 34.7 Tm 67 Tz /OPExtFont0 11.5 Tf (125 ) Tj ET EMC endstream endobj 674 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 675 0 obj <> stream 0V ,,&jV)xzĕ~,FKzt?3w3n| 4\37R!^zOr(@z)(1w҇?\u˜x+J>07]&c_vkyZEnz7㩢Iv/_EՍ c>g=BtB6{+]X2&TM]c$4#x)_@FwhHŀR.b}`Bot:ïDBIbz]Y3;׭M C5bVnђe7xef~:b@frښ90M h$?CkobJo(U2%MߥL46 ۞6/}orOu^ zd15R9ѹI!ue>iǹIq,ǹ ґGdYB`[UkN1ßW?OsvI.%&ERK)`"Wn"_|nɄzj3G履;VU9?&1\69-µߤ; :9zZre68>?Dyb4fpij[ ^g U0ˏnE_JA'Aa|WL/1/ BoBst*;8 W??:PZ *\C\T&̴햵 뿆Τ`#dܖ!nMK3F1c*-3ve{UuZ* ̐!L#;ƄyXjFIA tS ]?9ʤ(txDKe*zFS?/#y!ަmv97C'ޚW (d}Sׇ`$|fm?3j٥ԜQ /hc, 05e! n[fQ ҅KECMC7 = f3R?+$ h{-`&>'6zͫL5?ٞe&sÃu7u aXD>I*\Gp[&tb/ J1~,j+JEDY#Gq_s^C1nź_laioM r,ر-XsH>(8%Vɋ p)KW P|-ܒ`B]DԶAM\nyMǪTIdoE^]tL;tHKP|[_LcWPZSff+gSϏ֍6U=ZcѕjCadX8L[xjW:p{Bé`O} (p5_W8,6=t,p܀DAc y 3{ji̊%ה^*^ /f}<̿&{q9̊#uO4`)`sめ8m^ /#5|q;Ubo}Ku3r(r1 4e ꋣF/6I oOȟ)"uG*o: \f+SKhLH/<yo\ 1w:DvlZŮ7@vyMdS󡣮#I:c=g' w08 ʡG^vd>y`x _l<'? 61Hػ4}֣RфOh(IԏKCIp0n4`Fӱ9LSzHh~m{ie;h D"w 9yWlâՔV H ,8w'|N,)xwLk؛ZHmRF5$D\;pR_p%t^3m'ȷJksmKxvߊ0ò"T~*VqAbn"{]`b%f::v/|9\|٭qPLISȰ.ǖlAta1gRT$ly"Լ? 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,y(cVи3l;Kc`ggࡁVX쯬KiJ+{&*OI6sȅu_8Auu:Hl7^]k&%OҒu1#CvVeښC`KJn"S)[Rps'P۬O urHJ $dƨ^OՕԢs 'O~Le<"n *]h9p5jN2tE#P|s E{or_t0֓e$,-Y 683'd'xeJK%AdC9J GHy.\5| O&n¼ ,~NLx(M(A~Pl4mWh[,4SB.ɁTT1 x5Df>|.ّwS;8y gL`4ُуFJ)6nlN1nR\6Pip 6IMj5v1~Z Csچ%5E-!lL سJ )͹|5sjtf0n%QV.ŒhIRK`&abaYh?8!)%&9WH݂_l>E ޒ;5 %鱻(#Hef|cQdo܂pa,cs{ b:avLO7M#7al.b| -Nmm4}8b 1gj3̋nR]Rn΍$g!zDn2KUH*nv\ Cb !^!ڡg &> endobj 677 0 obj [678 0 R] endobj 678 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 618 0 0 839 0 0 cm /ImagePart_2153 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 434.899 720.2 Tm 104 Tz 3 Tr /OPExtFont3 10.5 Tf (5.6 Conclusions ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 104 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 104 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 104 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 104 Tz 3 Tr 1 0 0 1 296.399 678.45 Tm 116 Tz (Ignorance skill score ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 116 Tz 3 Tr 1 0 0 1 207.349 664.5 Tm 107 Tz (Ikeda system-model pair ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 107 Tz 3 Tr 1 0 0 1 354.949 664.75 Tm 105 Tz (Lorenz96 system-model pair ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 105 Tz 3 Tr 1 0 0 1 206.65 650.6 Tm 102 Tz (Average ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 102 Tz 3 Tr 1 0 0 1 262.55 650.6 Tm 95 Tz (Lower ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 95 Tz 3 Tr 1 0 0 1 310.1 650.6 Tm 92 Tz (Upper ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 92 Tz 3 Tr 1 0 0 1 354.949 650.6 Tm 101 Tz (Average ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 101 Tz 3 Tr 1 0 0 1 410.399 650.6 Tm 95 Tz (Lower ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 95 Tz 3 Tr 1 0 0 1 466.3 650.6 Tm 90 Tz /OPExtFont5 13 Tf (Upper ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 145.199 636.2 Tm 95 Tz (Method I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 206.4 636.2 Tm 93 Tz (-2.1863 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 259.899 636.2 Tm 92 Tz (-2.248 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 306.25 636.2 Tm 93 Tz (-2.1233 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 354.699 636.2 Tm 91 Tz (-4.4901 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 407.75 636.2 Tm 92 Tz (-4.5519 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 454.1 636.2 Tm 93 Tz (-4.4252 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 142.8 622.049 Tm 95 Tz (Method II ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 206.4 622.049 Tm 92 Tz (-2.5351 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 259.699 622.049 Tm 93 Tz (-2.5857 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 306.5 621.799 Tm (-2.4854 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 354.699 621.799 Tm 92 Tz (-4.6042 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 407.75 622.049 Tm (-4.6682 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 454.1 621.799 Tm (-4.5349 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 140.65 608.1 Tm 94 Tz (Method ) Tj 1 0 0 1 182.4 607.899 Tm 117 Tz /OPExtFont3 10.5 Tf (III ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 117 Tz 3 Tr 1 0 0 1 206.4 607.899 Tm 93 Tz /OPExtFont5 13 Tf (-2.9782 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 259.699 607.649 Tm 92 Tz (-3.0665 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 306.5 607.649 Tm 91 Tz (-2.891 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 354.5 607.899 Tm (-4.6345 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 407.5 607.899 Tm 93 Tz (-4.6886 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 453.85 607.899 Tm (-4.5966 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 140.9 593.5 Tm 95 Tz (Method IV ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 206.4 593.25 Tm 94 Tz (-3.0267 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 94 Tz 3 Tr 1 0 0 1 259.699 593.25 Tm 92 Tz (-3.0981 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 306.25 593.5 Tm 93 Tz (-2.9249 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 354.5 593.5 Tm (-4.9227 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 407.5 593.5 Tm (-4.9964 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 454.1 593.5 Tm 91 Tz (-4.8181 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 126.5 570.45 Tm 97 Tz (Table 5.7: Following Figure 5.5 and 5.6 experiments setting, Ignorance skill score ) Tj 1 0 0 1 126.5 556.5 Tm 98 Tz (of the nowcasting results of each methods for both Ikeda system-model pair ex-) Tj 1 0 0 1 126.5 542.6 Tm (periment and Lorenz96 system-model pair experiment. Average: is the empirical ) Tj 1 0 0 1 126.25 528.45 Tm 97 Tz (ignorance score over 1024 nowcasts , Lower and Upper are the 90 percent boot-) Tj 1 0 0 1 126.5 514.5 Tm 99 Tz (strap re-sampling bounds, 512 bootstrap samples are used to calculate the error ) Tj 1 0 0 1 126.5 500.6 Tm 95 Tz (bars. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 126 469.899 Tm 113 Tz /OPExtFont3 15.5 Tf (5.6 Conclusions ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15.5 Tf 113 Tz 3 Tr 1 0 0 1 126.5 435.3 Tm 97 Tz /OPExtFont5 13 Tf (In this chapter, we considered the problem of estimating the current states of the ) Tj 1 0 0 1 126.25 412.3 Tm (model outside PMS. Methods assuming the model is perfect are shown to be un-) Tj 1 0 0 1 126.25 389 Tm 95 Tz (able to produce the optimal results outside PMS. The adjusted ISGD method \(50\) ) Tj 1 0 0 1 126 365.699 Tm 99 Tz (is also found unable to produce consistent results. Using the ISGD method but ) Tj 1 0 0 1 125.75 342.699 Tm 98 Tz (with certain stopping criteria is then introduced to address the problem of now-) Tj 1 0 0 1 126 319.649 Tm 101 Tz (casting. The ) Tj 1 0 0 1 192.5 319.649 Tm /OPExtFont4 12 Tf (ISCDc ) Tj 1 0 0 1 232.099 319.649 Tm 98 Tz /OPExtFont5 13 Tf (method produces pseudo-orbit that are consistent with the ) Tj 1 0 0 1 126 296.35 Tm 97 Tz (observations and imperfect error which well estimate the model error. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 143.05 273.299 Tm 98 Tz (The well established WC4DVAR method is reviewed and the differences be-) Tj 1 0 0 1 125.75 250.049 Tm 95 Tz (tween WC4DVAR method and ISGD method are discussed. Applying both meth-) Tj 1 0 0 1 125.75 226.75 Tm 97 Tz (ods to the Ikeda system-model pair and Lorenz96 system-model pair, we demon-) Tj 1 0 0 1 125.75 203.5 Tm 101 Tz (strate that the ) Tj 1 0 0 1 199.449 203.5 Tm 102 Tz /OPExtFont4 12 Tf (ISCDc ) Tj 1 0 0 1 238.8 203.5 Tm 95 Tz /OPExtFont5 13 Tf (method produces more consistent results than WC4DVAR ) Tj 1 0 0 1 125.5 180.45 Tm 96 Tz (method. By measuring the variation of the WC4DVAR estimates based on differ-) Tj 1 0 0 1 125.75 157.149 Tm (ent sizes of assimilation window, we demonstrate that similar to 4DVAR method, ) Tj 1 0 0 1 125.5 133.899 Tm 101 Tz (the WC4DVAR method also encounters the problem that the density of local ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 101 Tz 3 Tr 1 0 0 1 316.8 52.049 Tm 86 Tz (126 ) Tj ET EMC endstream endobj 679 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 680 0 obj <> stream 0  ,,&ߔb7 )3(/ӤW>0Ɗ eBymK+eyt8 J:e\YUٸK@s'yAb6#$CK! ]&$,zTF€^o.Ef-"g@a[VOk/)N{d8F`ϥؕjtJo'* ED®23{(ʍ!51{Ja;Npmoa^;xjr*_sQ"0rk%sފkHR;|T1SOݨ @4cYPG}ݵQ[0!(uW+"* p)TY?Pw8)M#ok :i>3^Z֩aV]7fO92OP(N K}77}4! c 1η%+Mw*l?O6DBI8~Z,bi- %wE?rY `W*QA+9ǎ ^Z O(0#vS*qZioN /tz 28 :"uPI3]m*YoY!z}ozne|T̿^`6"{'sٜ\V }o 6u&Π gg-Yduѿ$"s4bdNۉm@9z"qg%kp'ZzݧC.:yxӼr흂)97:;l\8\)q-uw2? ȝHc=K1'E/x+8!ZXz*ޠd ?.Ћ X&!li׻%`3]OwFzW5owގYs~]M]=i:L'^]2wh'R$OKf>4yrЌ&E8|t&sO]ڮtۃaGϸVڰ"!W !E{O0/~dpIJ"Z'`КӢ ?E6p ( ntw.u;!*i\ '3JjaSCrs2"a 6f#8AKF1LR7">%-⛘Eh(brW*&5_9L"i.6d0wgO( &ׂJ8m!=ayWZ jcLzDyU8D (g]TYTt4Y`HGBB6ws`_IYQR7mqtn;nDMoll c^N(I 톙b-ϾG) +1~'PbN 9 IW0i(>ؘ,O1/+zkQZnT-TW~a;>uȣ?[-:\0%-O{Oz\n`޶3jB7Pݎz؄0{borݣD`Dğѿ h<+ACm)Oh\㫚߆V P| sP??ߊR/YM΍Q)лY#di~cC1ӥje΃' !`E(bt@PpwqǨi_2{ZQ:9%fIKZ=HY/[og#H)Jsgj.,N/=$5;WgǪj/ZGt~ʏ'9d"nH*n A$JtkVD]:sbr۔Jr1TP%jH*XEO-XdUXT;tSl:XLV"SAbdӠ?taգ1uX"`CD)f/~DDqƃ'eKR(b N{#t$X9 >ҩ,3 `4<8VA+<,0cVU$دb9k޸Xx 0Kxc;k#\>1 i]:]LVL<(F?K؝v'HF!+h$LR4%9P+r8\baKCwzZ[YAz`ЀڀIJlRZ f ҙ۲aɱ%EȒ9ǣ}<8c\ɼvv$ڲ[SM!7U9@&Cs,42ZFw:cH֖8y>-UclݔmBz9cHѪ3)={i wk4 qgg۝4ᯚh1l-\L?.>o6)! & g -wOr=zuy3lao B#IM) -_>M*rв;0kճѱ仴z*沨s(pΎoF@f:ԛ:O-1F2I įk^Z3B3;t[KR8&y7%oAW, Xq^ Iugfx -̘> KjlE)g8OU~y s&$"AAe=@hsOE4HYdADľe~z=XC9;}ktMH"6t|-O )$-ڀNM:X"J]Yc+V;ĹyC08i=~p1pXbXLzI^hRJ2Rp%cVV"p _WO&n>R)s4n 'X:UbjəF$*]3mHhol1'g\$5a^U58KßAc! =<{5;SϖCD:7ﲖЗ-uKy4͂}/#;nNݷQNQi9SA[=]ҷ?ֿZ(,Әĉ \BFjG+(p>G3.*8J>h`fuپK>*'r&  ǁy;1"օN])H%* xI) %#$RH,2AtKAS2`[ l.pk*]>(ȋѯd/ռ0挲}F=|C9ZP4d-Al10j_%Qbr@D@~^QJ3;OE,&`"=% Qj͌"79+YnN;W?~QtC%n}\=rv0f]-I MyA^{j(GAw:ҋ^G.D<^Oo R}  ㆥVSqbzwcWƣvn. 9]zy}f{t}6} y N(O$yMUK4RW71'u\PYl݃FșQ>ђ<6U%1VPFm5cDdޭ§ah8kGU}%"oCeO2}DVdX3rA= i AWw,IrS  p[ \?"@CPsr`jLccW$hcR}=mׯr7aF0TB>]6RUHд@]꣛y!% NKml*x YWʏ~,rΚ Ys)'A|x;ܛ}ļ(*ӊ\طQp.m<3~&i}ؙxa~ _ _jF}7KА$4B>5,JM|2:!?b%wuǝq91Ƭn/eA9\Mz9*1z/ȶѫh.Ԣ$ ՗+~I$Uȕ "1B+䰘Cgٷ_NIcfZDWCIʠCʤ[*G7~M=.hMu<)&HܝTՐLUb{ww컢cjh:n)7hd[%R7oOyP!w6lD&ʂ_,[KafW2oYRrĀ{ҰM5f'ԘܴwxAL|T( 'I 8 O`]Q{94k}07fz58Dhmfn>&Y⚴Tl}ᾱt fD?FUzX~ǿ&;|JTS"q%X-sK 30Hy{1D Y#r2vOKr8&y{ժB7>4T#8*pk;O>r@ "R6AO-VO")8ޢjD\MkWGiW!Z* M DZj(TV1T8 割G5h:Eq%;CM 8e"IPUdP3 N0\b uܘ4q%JIv \ykMD=`P'a\,IGvWG^*edZ6B %$7ip3R!Mt pGk@71kR/;9k/ఃBâ *=CK>ґCeιײQ3.bn$V'N[oX :5^usDTFu0K|qI#O(V*tHg;,%jRZHBAy => WKQ.49Bq֦=ɪM1E! =)U,{7:Ņ_ Ƽoi-JH.W?RRV`,16WJ &?7;@|l* 'oٮO`JK?T knv/p̰}naqp/a]UԿԵ+\N&"d/ԥs- ^iUq%eb vV7UO@B5L4*SXj_ދUǢf%NK=$@bQ3#iLC߸"q {&kHF~/!ΐY?h |7sb&&%{WȭO-Cy 凖5v nͱjut`ELK4#׆bJE{,=yir/8' gR3fΩ<Ôg C{w?˵RO^pz*t(¢wz<*MKkfCqhf& iO_"&?8vdᅅ߻muB6_V}ToNťj^z5:*2|F'zV3PaJT&]rxv6[XȷL{BMR$0oU;~~=5ư\T!ԁ"Y^f*osR6KVI .J;* L(ڋE.pqn/{u#bļ:.ƑJu9IMBuͥy'tҋ`A#8|۹j'<X }F XjAWgjصȇ4) wMkP}Ԟ>\MaiF eӋ -E.njYdiuo0knW7%}|9V^GҎcśssT[D%Yc ցB 'F0s`ӊ꿐@uV5A>!6۾ϱ(ѳ~"5(r,3AB%k<0z 'UdžyA-=9Waj]^q=E-{AArm%, e{wP4Qյӣ9D!,PNFݼ}lvpXkbFӢ[9;GdC ,ьlfieip$㟰.ي,{F#GXœza#pyRDxÂZPo +Oa2&کM"Vl|Hβ"5؆Qw|JDuH(/#B&F.|^F1 + "ܾ}G1ϝAeؕ8ɲ=3'Vb#pHsP=&5 _鉚*ʇbn{w tHs# zkEKrN>E?T× )\ʼ%E$[k{;>^! cVtHfnla9BZdk7@lmTJLCN}9 #!O;]&ƛ("~Gr-l:oe>oUfaӠݠ8|J]3|ҟy}$+äSsj`j֋AxuA`CڃvJ-ZF6iם'r ,'AV78E6'ZF }s1 &Sc: "#RNb<;$N?']v-i i>v _%4%ٕUyxi?]~ۍXI?$TdZXd Bm].#tIʚ$fti+{9D^Xl4{ƺU6<뵧8j<$kȍ5=1d "X1Ƃ]H# J=5؆:2BRT Qyy鞺4"}=q)ɿ1hG5|"gmNȊUA',Fh hrV Okc4٧Գ6vy 05 J;H VDRpF%0ۘqgҝJ1%׼ (p]>`o7U-hulEf .j7U\lL2 zzsE25#$^2>Ŧnc9l!Dik+n UwcJ;~4y@B61'dꄓaJS)$R-Jy[{s@WAW+=;g<;<՟fw߉H%6hY362h'܄TÓFQLP\Oˈ8 p *Aw+1:OB#=^Ƴ|5qb$&/ `엯Z(R91L Z^ƴ݋ƶdSqњ;(3@0?2lw1jKN:u#nxKeך^,QPUEbsa'v{Zo%S [[nDDz]LzmrT } :zZۺ? vnȈ)~itINa9qR8;2M<^nvm9ࢰwq)x/q*mn4h6g{ \2|y@hi 1m.ԷyO:_Gƍ3-Z,F'Wa-ﯬZ"ּ*|eM|>`sYkzl5v OH9wu8Za q‡7LUr~i!`CI6CQopMiɻ=q.zuWM&(1Ŧxe & rsEYgYgBq+6TsyX0X6R 4enk5F9=r|>\@R [&\]iP+t|;LmR%Me6ݮżӝJξ/2,k0ӌ^F\;i{-aZHiѫK'Z   ˈQ*wL  kݗm-b-l.U77jʖ eҽ^AT]SA&@ &49?"b PrBHp7G(4!;y[Xn K9\Į8{0o/Fg9H&ul3 zu<>nu)ջΦ<">@;*q ԕ>6˕ xNoz)g3es,-0dfl+Ƨ6E+3G~UfG$pCGP;|҄CJX憥vaR|^FB427 \5< EQ`< Վ3=vSx״Vc`S;O!A9x*eX1@tAյ~`0YaCOQYJM_Mw0;*/a8g_s͙҃T`oS9v iHW:4J`V{ &1PĮfuP7<ٶ3a/w<^LsTxA燓axJJ2o : %Sؙ/^'K4Z*625{vc>`-bvJL cX!^[W}$?iQ7?tJ͘_%\T.";'1<}#&};ue+H'$ؑX%!λ]HKDG2Ic42őtxG@Gyqd۱ȺK]ҁI„yla$8\pg87YZH%#I* -URMPXOI=#mև޺X[J8c6[|*j+t dopyjp3@,4fΚ6J4ᒰohZ,JmVr ŴY_)NZ\(e:БZAu6?[hR~wKO4R}H5jk\>i]]{23LS %MiPjE=U W5̂3g,8,̗*4rՀ P{®ERӑI`։(Js-ƉM8v# 9ZaJ"$GWXmk3i Fpzg/I F}-/hܗ%Y XeGŚL()x?-[&՘ OQ爗^@7G^){m8y(by[vB J/J|fhpYh?f{c`̘J8*/qL:3)&@#DO&zK"_b&'J5s4j;RJkbFvV nMU$dg!kEy_0Qm;gJ;XPbGc_}Zym7V0IN2J8vN >VOPi|i UwÌWf6.eZ߸9R0=;@fi;[H$qˆa7_Bٰ$ႋa>bd{u)0 ^8a.,YJ D4#H]ď6k'4?{Ӆ>G><3 ŷ>'Ǿvb6L@eAߘgB+"O+g_ER l5D 8JQ>]Rn^ ;,'5^͔Q9]R0nͱKc 0>byB{!\9#˩tX8p7,dEg74VEI >Q8<8g͟E)3~> jKd~u:XP0!YQǵ@>utTDe {i@|%63Rd#9yXW&X{cKىX+/H5ֶOoe'fbb"ځt8ԟǀFW,y K) v(EeD6,[/ؔ.٠6p00gj>2)#ڰõ#2԰շ/=澱ձܕ&ê4Ho?#;TF!ɴ)> endobj 682 0 obj [683 0 R] endobj 683 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 619 0 0 840 0 0 cm /ImagePart_2154 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 436.3 720.7 Tm 99 Tz 3 Tr /OPExtFont3 11 Tf (5.6 Conclusions ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 99 Tz 3 Tr 1 0 0 1 128.4 678 Tm 89 Tz (minima increases as the length of assimilation window increases. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 145.199 654.95 Tm 90 Tz (Three methods are introduced to form the initial condition ensemble based on ) Tj 1 0 0 1 128.4 632.149 Tm 93 Tz (the pseudo-orbit provided by ) Tj 1 0 0 1 277.199 632.149 Tm 116 Tz /OPExtFont8 12 Tf (ISGDc ) Tj 1 0 0 1 317.3 631.899 Tm 94 Tz /OPExtFont3 11 Tf (method. Using the information of imper-) Tj 1 0 0 1 127.9 609.1 Tm 91 Tz (fection error are found to be useful to produce better initial condition ensemble. ) Tj 1 0 0 1 128.15 586.1 Tm 92 Tz (Forming the ensemble by applying ) Tj 1 0 0 1 303.35 586.1 Tm 105 Tz /OPExtFont8 12 Tf (ISG.Dc ) Tj 1 0 0 1 343.199 585.85 Tm 91 Tz /OPExtFont3 11 Tf (on perturbed pseudo-orbit are found ) Tj 1 0 0 1 127.9 563.049 Tm (to produce the best initial condition ensemble among these three methods. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 317.05 53.5 Tm 78 Tz (127 ) Tj ET EMC endstream endobj 684 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 685 0 obj <> stream 0  ,, nb7(x2p} |w|ŔCs5Ր^"4Fbl4U %7^ QD0RP2`[y#O pdS;T\#%R)Y{},qԓf1ǸMĩ淺8C\V`sT#_ 4c~' 6[i.J',}]u3%Tt 3^& 1KO1cz5n+F);f>r/2k6 .H&ǝxbKy[Y][t2Г缸#~_XD DOe'>,=ݒG2<Յr ?RgKh~5s+;9cN\/gyMçա8Jnzu3졭ߘ ?5ai,aWG꒽>ʼ/-`3` | !*IܡT]Rwzmw3FOXR޿}EvT>)j%J?W+PSXmh #Lu/g0GgNwWAIuVvk+WRJz X5{O42fh%r>z55vwçÛS=0A`ڣtL-ٝ¾vq: ^參s=i <6PrT1}OHy8cy 6$I*FIr1g6*S ZAZ<1 9W+$)Tqw s`ZHN  SAC:\oZ>3ճܧ6ڛIH8_ϳ"dzŒª#찿+z:8~<3R_s8׳E#XŸo"u]Nmt~H,]kԠn_YNP,nh#ѧ!h? '[ۡTa D?ߴ vᎠw^:_Ҩk9 nv{V_'u Z\ۍE~P#jwvsCS%lVd'|S=y8_mmJ5&6[p1 !w sR[N7rdJn:GG8'QxtP }&`n3Xڎq-2W&"gv'u]͆`(_YpFQ}&P༕DWc_ڦ4v[0{1=DG$q yq}HVj:=C !|a/i~qE,1@?ù7IC:4D=}˥M:@Mڑe Qq -D'\؜#C5}bdA|?]l0[!.Fi]j) slGt/[Jfڀ(,ǰ/0iq=ptMtN9As]шyi ڰu9D2Դh""ذɳ^ձD*(yΗ)梵:=l(854O>4;닱C<& kv$iS,$Ӽ0$IcH*鳮)t}lfS7/gPf>"?gVp`rk⟩ug4#. C X־;DlK KtZω(.s>F&]|_YM  Ce9 xbG3No6 *Vǹ)s%C2W=nXy:{Ѷ2^m*|ZM΀7z,83.][Ly6IQ)U`ƴ~ 'L5tDb]1[Sw_lؿ6K~A;QVڙhmFo;ɍ薽)w#\ k6CzBfUda{&k; \:z|%)⃼<1rsi\( )[v, 2w'Zn0n9Ɋ5x+~JYոOD RoW@tu5OwxY|jj9!'f%Htэ S[DH厞^"=b$a_rt/Ӊb"S 0  p"0sE~KfW׮A&UĤQ O)գ.d-@Lot{Kùu, 8Z7p5m}hray7n}7f#[nvA룍/O v`CBģI, %;S4}o(=Ϲ/ZU]; ރSB)ڀPy/X$" fLjΠ5U3eƓ Adžj:f\t˭+H!KI0ٻ7%yvUD+&JB^,PJ mt$驡%>)_S%Vj{&5yLkpYFel3{|\ FϦW}vbv#G$BJ_h5T)|Ỻ^ma:4~}U7oN#?1A?\j߸Opՙ6'"' ѬsPJ!fd:YQVR1ЋAlo|-q*Xz+\_M@7 endstream endobj 686 0 obj <> endobj 687 0 obj [688 0 R] endobj 688 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 619 0 0 840 0 0 cm /ImagePart_2155 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 128.4 582.5 Tm 105 Tz 3 Tr /OPExtFont3 22.5 Tf (Chapter 6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 105 Tz 3 Tr 1 0 0 1 127.7 515.5 Tm 106 Tz (Forecast and predictability ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 106 Tz 3 Tr 1 0 0 1 127.7 468 Tm 107 Tz (outside P ) Tj 1 0 0 1 265.449 467.75 Tm 78 Tz /OPExtFont11 22.5 Tf (S ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 22.5 Tf 78 Tz 3 Tr 1 0 0 1 127.2 406.3 Tm 99 Tz /OPExtFont5 13 Tf (In this penultimate chapter we discuss how to produce better forecast based on ) Tj 1 0 0 1 126.95 383.05 Tm 104 Tz (the initial condition ensemble given the fact that our model is imperfect. By ) Tj 1 0 0 1 126.7 359.75 Tm 98 Tz (showing the results in the Ikeda system-model pair experiment, we demonstrate ) Tj 1 0 0 1 126.7 336.699 Tm 101 Tz (first that forecast with relevant adjustment, which could be obtained from the ) Tj 1 0 0 1 126.7 313.7 Tm 99 Tz (imperfection error \(see section 5.2.3\), can produce better forecast than ignoring ) Tj 1 0 0 1 126.7 290.399 Tm 98 Tz (the existence of model error. Secondly we discuss how to interpret predictability ) Tj 1 0 0 1 126.95 267.35 Tm 103 Tz (outside PMS. Traditional ways of evaluating the predictability of one model, ) Tj 1 0 0 1 126.95 244.1 Tm 98 Tz (Lyapunov exponents and doubling time for example, provide the information of ) Tj 1 0 0 1 126.7 220.799 Tm (error growth but they implicitly assume the model is perfect. Outside PMS these ) Tj 1 0 0 1 126.7 197.75 Tm 100 Tz (measurements would systematically overestimate the predictability. We suggest ) Tj 1 0 0 1 126.7 174.7 Tm 101 Tz (using the probability forecast skill to interpret the predictability. Such forecast ) Tj 1 0 0 1 126.5 151.45 Tm (skill not only depends on the system, and model, and observation method but ) Tj 1 0 0 1 126.7 128.149 Tm 104 Tz (also depends on the way that initial conditions are formed and forecasts are ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 104 Tz 3 Tr 1 0 0 1 318 53.5 Tm 82 Tz (128 ) Tj ET EMC endstream endobj 689 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 690 0 obj <> stream 0  ,,?++jSES T9{d^ndiN:1kLa `&؛;q9'u|,9]]c EOuv8TǵzO]J6:PCV16Fp0+֥ 8dopǘzz 'ubJ:R5, n#nr ߲$Q|,ɮ iuL\Z'nj넑UޔuHi(CxD4O_&apo?a mh-9|JƖ/`ܑccι) aJi] 萴EP3LR/kYdZJ>+l R6y S0?D`U$Êˠyχ iO΅gP:xgkv̔=+2+}{@lcs{1) |*F>.c!o*|s{lܧy׽j:=-$:*%Ӄ/ZԶ2$CYeꥳ<qx;]A=lu3K 84ALYJ,PX ]E7T9&MKW BU";5TQ c4Ii7Fp ]JTQ6r3\2Xn^޶ƒF~t'B40q۲-kJ|>gRdB>Jl5UXY)]2473y뱩 }ˀV [\?KLuR)QPVIE@EEV],Mphlr)b>_&!+mt Ka*XXx(B*Q{]dJHjkXlj?\;  WKٲ,eu(S\XxgIUUa 6U픲vVkQ0T@Vۆ2#cn[r!f-n:SaY'tb O*@Qq%3mXtQ$\X>r6႙bpɽe٬凁0b4W6!D v^PU>L͊ \u*>ۘy~uYlR 2et*h*:%W /Ac;e?_ 8=8cÆk_-r-F,y5IT#Ԍ߯צHTޙeXV.Ohl`ĵQ8}6D6F%@ԀG fb|loE3٨BNjلR GMM/9pf|5g¢FV$,(ƳDOPB7~-f=@GF!v+-qh`17 IHn*%cڳ%?sJS%ˆ'XsK.?i ,)X@' qb]|اTtݠ#Y~wEW9??ռnǫ]Ѳ=ֶķֵ$7=?\F)#һ<ݶILCXv Խ _RUbla!>-2ٱЁj=Õ<V3c,={nĔ qiG)C%g"6s6֮yۘ2ȍ@rq}6|\Fhd7H'^U % ,1=(G HQ!.zP i@#y7 ǻٝ` =sH>[![ oYXJ!߾Xv1 z5cZw`7̉j `V'w̞U޸- V5Ṕg~EGF#nrTOH{̣Q0W įs} :|(ICf ̜7t&k$U'HBTZcUȨQEetOs16a;r*+UN1J>ogvto&Pݎq;jMAWaVV^)qI-zh1F:dze9[AВ];}eK`qFBӳn9̈́xh!,N9Vv܁ ܦ^7χrtZ=|r.2'$R'u3 /p[#) 3 INGR^z@$hL餭вx#/aM/+[rwVq~59&އ|(аJQ} L,t\ ea%m~]y 22kMb6hdgQɞlnA˵Y%_9Z\RpHcߌ7c&]y e @OY۹ZdƉy56N^#_qbKH|?VbreKRRß߱vu$bQƻQ%v4Sv]4 =<ȥnycP5 (GC]ɦ"JV QMѠABzs{l3;*`J:4>lyt&$ZT* $ۯ~ 9eHg6JCT)o`qnv]U"7ay3B GI ڄg[X_2;Ȱ.<*lk~>|ǖ#"Fe93S+ *jWQ88IW!LC*m`E9}b1"ٜD,Ov0, XLtpC,~7.NF~xۡ&ҝQD0hlGs-\br a]7u$H 9CYq|W˜j]brYĢ,X+{^g'r ϊ&fa^t=>Y.A4B lQ<6f@}'PRv,֔~5_}GKOqhik^|$ :e|80li0r]٘3p+F X /~r8&jӳڦL r|BN(; mw^eɋL'/mI."+4wf2;;Zs'UL;jQ{_|qc_7G1V܃F*{Ng!42hnPHgjO=ي6l3F)Aewi%G^!1.,9?/95z@"KlP!=-n&gJLܵ܏p   'Y=5EXm)RVYg ߝi pRm'q㠂gțq;4C[$$ۭDKdKkPl3/Dc<6M¨$_]=)o R~CISX3.ki:ć *[73Ic[ӝ{^F|L6G;y) n,d؇BW~wvJ_k{~sM|z 9Hެ 5?ˌ@QN`*>ZE%c' cRl[zO +qI7\AW ,@YJfkuR(s,cِ9w(Tr:U <0|"5s1Џ^׸%Lyg\B &kn)"?rJ%8u)\3S`k⚯=WkWbjS#볡{eJnmech 9"Brb._'tPCD=2r]UmXH0t ht|vJ yEq]{- Lj55_KK:v~a=G%WKc^*9gҝ;ܱ_daHØC{HB)G"5|2W:\v¼dި`rK=_;hVHа{ ,#e ̹kQ܏ȩG59zfˋ\ r<Q7wR1DA>+ D}Ѥ & ֎uY2 qad yuw' AB@ ֘!W2+i)ʕmXtRmE^i7)2xV m'pX4gvp)`1˫ / =lA6 ]a{'SÕ>z= ]+ѐc,|Ҝc0"C,٦mr8 ZA.3gSz~t)5zj)cl;&n34Hn=*z(/;53uAHSJ:Ij$X{Imςu[Tթ`mt$]D'X.KWa^fFމA@kɏq LڟE$6-D|5Q<әxE,o@V1"wzg:l^v2JeTWKӳȧ\):zSyx<[lb`g{3d,/m471jl\+Z^Wh40~ٰ(ėb:r\׫>myѫC}N5֬i2ǑI$M8\)l[(e%bo)ٛfL.V1p~7qXvn܇8_u*FǢ$%2!(4 g-Gҏ ;9&>hCBpOF&lγXyc_ ~Ef컚3ElG!66(@@隕: k.2Oyԙ~]YMBWOt4+[LؿTNB6?$m=y~{ZsxaUv 9'[f"u5> endobj 692 0 obj [693 0 R] endobj 693 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 618 0 0 838 0 0 cm /ImagePart_2156 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 308.649 719.45 Tm 104 Tz 3 Tr /OPExtFont3 11 Tf (6.1 Forecasting using imperfect model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 104 Tz 3 Tr 1 0 0 1 128.4 676.95 Tm 90 Tz (determined from the ensemble. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 128.4 624.649 Tm 111 Tz /OPExtFont3 15.5 Tf (6.1 Forecasting using imperfect model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15.5 Tf 111 Tz 3 Tr 1 0 0 1 128.15 584.799 Tm 113 Tz /OPExtFont3 13 Tf (6.1.1 Problem setting up ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 113 Tz 3 Tr 1 0 0 1 127.7 554.1 Tm 94 Tz /OPExtFont3 11 Tf (We set up the forecast problem in the imperfect model scenario following \(50\). ) Tj 1 0 0 1 127.7 531.049 Tm 95 Tz (As in section 5.1, the trajectory of system states, R) Tj 1 0 0 1 387.85 530.799 Tm 89 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 392.149 530.799 Tm 107 Tz /OPExtFont3 11 Tf (, t = n, 0, ..., ) Tj 1 0 0 1 478.8 530.799 Tm 111 Tz /OPExtFont8 11.5 Tf (ml ) Tj 1 0 0 1 494.899 530.799 Tm 88 Tz /OPExtFont3 11 Tf (where ) Tj 1 0 0 1 127.2 506.3 Tm 99 Tz (Rt e R) Tj 1 0 0 1 160.3 507.05 Tm 76 Tz /OPExtFont5 11 Tf (th) Tj 1 0 0 1 168.699 507.75 Tm 91 Tz /OPExtFont3 11 Tf (, where Rh is the state space of the system, is governed by the nonlinear ) Tj 1 0 0 1 127.7 485.199 Tm 93 Tz (evolution operator F) Tj 1 0 0 1 230.9 484.949 Tm 140 Tz /OPExtFont4 3 Tf (, ) Tj 1 0 0 1 236.9 484.699 Tm 86 Tz /OPExtFont3 11 Tf (i.e. 54) Tj 1 0 0 1 266.149 484.5 Tm 74 Tz /OPExtFont5 11 Tf (4) Tj 1 0 0 1 270 484.25 Tm 105 Tz /OPExtFont3 11 Tf (4 = P\(5) Tj 1 0 0 1 309.35 484.25 Tm 50 Tz /OPExtFont5 11 Tf (.) Tj 1 0 0 1 308.649 484.699 Tm 92 Tz /OPExtFont3 11 Tf (ct\). An observation s) Tj 1 0 0 1 411.35 484.5 Tm 67 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 413.5 484.699 Tm 90 Tz /OPExtFont3 11 Tf ( of the system state Rt ) Tj 1 0 0 1 127.9 462.149 Tm 87 Tz (at time t is defined by s) Tj 1 0 0 1 239.3 461.699 Tm 74 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 241.699 461.449 Tm 120 Tz /OPExtFont3 11 Tf ( = ) Tj 1 0 0 1 258.25 461.449 Tm 90 Tz /OPExtFont6 11.5 Tf (h\(51t\)4-ri) Tj 1 0 0 1 300.25 461.699 Tm 92 Tz /OPExtFont8 11.5 Tf (t ) Tj 1 0 0 1 302.899 461.699 Tm 87 Tz /OPExtFont3 11 Tf ( where s) Tj 1 0 0 1 342.699 461.699 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 345.35 461.699 Tm 127 Tz /OPExtFont8 11.5 Tf ( e 0, Th ) Tj 1 0 0 1 388.8 461.699 Tm 88 Tz /OPExtFont3 11 Tf (represents the observational ) Tj 1 0 0 1 127.2 438.899 Tm 89 Tz (noise, which we assume is ) Tj 1 0 0 1 260.399 438.899 Tm 91 Tz /OPExtFont2 12.5 Tf (IID ) Tj 1 0 0 1 281.3 438.899 Tm /OPExtFont3 11 Tf (distributed, and ) Tj 1 0 0 1 365.3 438.899 Tm 124 Tz /OPExtFont8 11.5 Tf (h\(.\) ) Tj 1 0 0 1 387.35 438.649 Tm 91 Tz /OPExtFont3 11 Tf (is the observation operator, ) Tj 1 0 0 1 127.2 415.6 Tm 92 Tz (which projects the system state into the observation space 0. For simplicity, we ) Tj 1 0 0 1 127.45 392.8 Tm 87 Tz (take ) Tj 1 0 0 1 152.65 392.8 Tm 126 Tz /OPExtFont8 11.5 Tf (h\(.\) ) Tj 1 0 0 1 174.949 392.55 Tm 98 Tz /OPExtFont3 11 Tf (to be the identity. Let the model be x) Tj 1 0 0 1 369.35 392.55 Tm 95 Tz /OPExtFont5 11 Tf (t+i ) Tj 1 0 0 1 381.85 392.55 Tm 126 Tz /OPExtFont3 11 Tf ( = F\(x) Tj 1 0 0 1 420.25 392.55 Tm 82 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 424.1 392.55 Tm 98 Tz /OPExtFont3 11 Tf (\), where x) Tj 1 0 0 1 475.899 392.3 Tm 60 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 477.85 392.3 Tm 108 Tz /OPExtFont3 11 Tf ( E M, M ) Tj 1 0 0 1 127.2 369.5 Tm 94 Tz (is the model state space. Assume the system state k can also be projected into ) Tj 1 0 0 1 126.95 346.5 Tm 95 Tz (the model state space by a projection operator ) Tj 1 0 0 1 367.899 346.25 Tm 122 Tz /OPExtFont8 11.5 Tf (g\(\), ) Tj 1 0 0 1 393.6 346.25 Tm 104 Tz /OPExtFont3 11 Tf (i.e. x = g\(X\). In general, ) Tj 1 0 0 1 126.95 323.2 Tm 96 Tz (we don't know the property of this projection operator, we don't know even if ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 96 Tz 3 Tr 1 0 0 1 138.949 299.899 Tm 98 Tz (exists. We are just going to assume that it maps the states of the system ) Tj 1 0 0 1 126.95 276.899 Tm 93 Tz (into somehow relevant states in the model. For the purposes of illustration and ) Tj 1 0 0 1 126.7 253.85 Tm 92 Tz (simplicity, unless otherwise stated, we assume ) Tj 1 0 0 1 364.55 253.6 Tm 133 Tz /OPExtFont8 11.5 Tf (g\(.\) ) Tj 1 0 0 1 387.35 253.6 Tm 95 Tz /OPExtFont3 11 Tf (is one-to-one identity. Our ) Tj 1 0 0 1 126.95 230.549 Tm (aim is to forecast the future model states x) Tj 1 0 0 1 348.699 230.299 Tm 68 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 352.55 230.299 Tm 91 Tz /OPExtFont3 11 Tf (, t = 1, ..., n) Tj 1 0 0 1 411.1 230.299 Tm 74 Tz /OPExtFont5 11 Tf (1 ) Tj 1 0 0 1 414.949 230.299 Tm 96 Tz /OPExtFont3 11 Tf ( given the model and ) Tj 1 0 0 1 126.7 207.5 Tm 92 Tz (the previous and current observations s) Tj 1 0 0 1 325.449 207.299 Tm 89 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 330 207.299 Tm 102 Tz /OPExtFont3 11 Tf (, t = n, 0. Chapter 3 and Chapter ) Tj 1 0 0 1 126.7 184 Tm 93 Tz (5 have discussed methods to estimate the current state using ensemble. In the ) Tj 1 0 0 1 126.5 160.95 Tm 92 Tz (following sections, we will treat the ensemble for the current states as the initial ) Tj 1 0 0 1 126.5 137.7 Tm 91 Tz (condition ensemble and use them to forecast the future states x) Tj 1 0 0 1 442.8 137.7 Tm 90 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 447.35 137.7 Tm 34 Tz /OPExtFont3 11 Tf (, ) Tj 1 0 0 1 451.449 137.7 Tm 114 Tz /OPExtFont8 11.5 Tf (t = ) Tj 1 0 0 1 471.35 137.7 Tm 84 Tz /OPExtFont3 11 Tf (1, ..., nf. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 84 Tz 3 Tr 1 0 0 1 317.3 52.25 Tm 76 Tz (129 ) Tj ET EMC endstream endobj 694 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 695 0 obj <> stream 0  ,,:b7 M*( S 9!OJ`&~ VTEq2?_;A9,<3 qTuEL 0RKctپj?ٌ;at5N#?[Xlh v!7|LąJ`Ir'CP/ZfQMPk.AULhKΎ7άm;)0M]/\3 !oibF􄫶kmHԺPˍR:$ Nܖ ]w݃k|߉Uem1>54 jјé ok3ѽȳג%rk'FIw? l".^o!Ze"뤿;G/m;68ѧ8'^'7 -<Νsy>9`K%*zC'QsDZocL$n)t ^1Ӈ,:X@PgXT""oøqP,pj͑E̱Ǧ3KWs`HT#P@&fUKMDqfɪ vCdµ ١5Zk mi\3 Ag,ꦘhyO<^t? ر_No IĕBԪUBFφ8`6ua;Pq婘Y#hHlDY[Qi<l0Ppm3BJ [>Ͼ3*.͍ޤs2jKm'|TT>>-G3|*W?|\$fvۣNhbBɴ8&-1կLF n$,E6N~;I}MƱ{v#{{xa8HLCh&Ll?Y|T3i9|Uұ\'sw^G8 fAv0 W-ͬFX4ZTR4ZJ}il{Gւ.S3r XZ|BeCՉy͈o 8_<'O>B @S5Na@cFh3- ` ޴oz?7qHKKv+#K4p Jΰ()Uy;ImZ\H,4xN|JYhihc2+?? X>xXyVNMAӴا{t'!w/yۭOJ?+" cP},ge:2{EɬS>'RQnD=8;2RV WR i:1zUs+Cg0QI$%}7h/t DGm0 Her0y:P#lh݌zG5]!*_`QzJMhD'%kZ2k lL||0mŶ:ݪ]'~d//|m+_j<7~'skTZ%2l=4pg@Doz{YMӊzr msa.4,}9hҗ)FNadRXhK)Qx߬f޽>DZ4K9{0oi⫣=x#V'fa7z9x$C);y$3$heW-)otӨEBӺ߱Q醄{(Ĕ*Zs(]Ӣ -P`g-/-7q\@űsh^gP>]wnA0?Z9]hɀ#/ʛĴ?۝Vbg/*;؅nVd`ӷhU"ސL+ILobei81W%ZoСW8J66.-xt@`GwPdƃ 񙃞D7#1Š')Nn%HϷpbt]oF1Hc.V @a-v|kBnm,ovIB7/U~d: Vft=YP`S)ji!gnI\y/@E&y$Wax >'_E°ո鷫7Aճ?撥R}S#o>?,y{[nq A65m/DC$]# 1U &42?^$?NSbQz[/|TyfF?$ NRﷁU~8I6>R7{ω~ nI;ds$B'B&gs%lÀ{?I}p p2>,ʘ󱧦rL >zƾUiI OA1ovϚ uzX'92z*(; F@,=֊-)-͉/۰Y^ Α8!e[0QLC4# m'<.:5M3֭qRxK5( Q/z/ g2c4q/CGJ|"`DBN^BgpKBqHrFV/e=v^2aܨu{y^'^_/ũ0KCAoPQd|"(}ܹ6{I];h/'U<j+8XcoN[_1siAe ;f}3&@|W۟,jrƧI<|0rA3~L^n*AGc 7c/+tď:0:P[?nyz} ܷU-*HX{rC?N&hKD(}_NO3"X뎹8u}>W;-P==d+1$J-2ʱ*;ԦO6V?ir\LmfRf.L1@6Fmp]D6&ʎ!:v.5Wk1ZcKz&d"myk({!=,.hAcYa;^"./\  Wr&ql7(mIGCe?C  2  \ݽpv*t̒0;ύ Yyw&$: iچ_xcA0;}o-(GbF[l(0yE7q"Dt@e=-DT^l&T"M߲ed5ֵ}.}J4F׹˩ml`ۮ$5EJpLWV]ދprKLox.E9?ʼnAw9r3tsRr-G(z$qTlSblI}%t_ziyYttz oj:hCyn^Mf Ψ;dsM2Ħ_l'\:OLjg2K:Ly+c>>a0tk2Z\~=ncB\za'$_ng5_Ź~(-iϙ$+A:$_7P2L7ZgD = _^RusB1Ԙej̞6ݪqX30♦dĦ&!Y3*?$/F$@XҧTfمm; ܏~2?Gw!τMe,ԵJaNpa&-y泋cpt:hIe U"FNZU{3uc@Q2L2k-=nVHKW|q+Òg|utp(a*&x{lA s\'6uJM?~f/K#WѣfYr̴@åq2-'^I=gDQN:?V6@[ui3r@b.kWf&8kXX'IkXȃy[0'?`"LubB񰅟e ]jPP*NcsZ](}ﹶlx\/6sJrBspTP"`K]%2cƼaM(ЌA5`<CtSi+fܙE'IYy0.l2Fd/GiPʤLJnH;$9 } FY#[||+5 Yr45@2I$JFeJ7{h.I+ˋ谨8?-XnY&{3ԬNX^AED61bseƪ2Z] j`]^ߊ M%fCIu4#1@㭂x.{YPDsUQ;}6xdNռvT&H&y/'jE1!t?ٽA6F/^z9>].qf_U} ,'t¶voS pL!K j Iɑ 5 '0G(O9b]fZh.pgP\U]a,\h2~n`IL#nTZ!}o1zҎ%F\> endobj 697 0 obj [698 0 R] endobj 698 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 616 0 0 838 0 0 cm /ImagePart_2157 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 306.5 719.899 Tm 104 Tz 3 Tr /OPExtFont3 11 Tf (6.1 Forecasting using imperfect model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 104 Tz 3 Tr 1 0 0 1 142.8 676.95 Tm 95 Tz (The experimental results discussed in this section will based on the Ikeda ) Tj 1 0 0 1 125.75 653.899 Tm 90 Tz (system-model pair, i.e treat the Ikeda Map as the system and the truncated Ikeda ) Tj 1 0 0 1 126 631.1 Tm 91 Tz (Map as the model. Details of this system-model pair can be found in Section 5.1 ) Tj 1 0 0 1 126 608.1 Tm 84 Tz (and 2.4. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 84 Tz 3 Tr 1 0 0 1 126 563.2 Tm 111 Tz /OPExtFont3 13 Tf (6.1.2 Ignoring the fact that the model is wrong) Tj 1 0 0 1 459.35 563.2 Tm 75 Tz /OPExtFont3 3 Tf (. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 3 Tf 75 Tz 3 Tr 1 0 0 1 125.75 532.7 Tm 95 Tz /OPExtFont3 11 Tf (Given an initial condition ensemble, the simplest way to produce the forecast ) Tj 1 0 0 1 125.5 509.699 Tm 94 Tz (ensemble is to iterate the initial condition ensemble forward bythe model, we ) Tj 1 0 0 1 125.5 486.899 Tm 93 Tz (call this the ) Tj 1 0 0 1 190.55 486.899 Tm 89 Tz /OPExtFont4 11 Tf (direct forecast. ) Tj 1 0 0 1 268.1 486.899 Tm 94 Tz /OPExtFont3 11 Tf (Unfortunately no matter how good the initial con-) Tj 1 0 0 1 125.299 463.6 Tm (dition ensembles are, by simply iterating them forward the forecast ensembles ) Tj 1 0 0 1 125.5 440.55 Tm 96 Tz (are expected to move far away from the observations eventually. This failure ) Tj 1 0 0 1 125.5 417.3 Tm 93 Tz (of producing a relevant forecast results from ignoring the fact that the model is ) Tj 1 0 0 1 125.299 394.25 Tm (imperfect. Usually the invariant measure of the system in the model space and ) Tj 1 0 0 1 125.049 371.199 Tm (that of the model are rather different, and iterations of the initial condition un-) Tj 1 0 0 1 125.049 348.149 Tm 94 Tz (der the model will, however, only approach the model attractor \(if there is one\) ) Tj 1 0 0 1 125.299 324.899 Tm 90 Tz (eventually, which essentially cause the irrelevance of the forecast. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 125.299 279.75 Tm 110 Tz /OPExtFont3 13 Tf (6.1.3 Forecast with model error adjustment ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 110 Tz 3 Tr 1 0 0 1 125.049 249.299 Tm 94 Tz /OPExtFont3 11 Tf (As discussed in section 5.2, a system trajectory provides a pseudo-orbit of the ) Tj 1 0 0 1 125.049 226 Tm 95 Tz (model instead of a model trajectory in the model space. The mismatch R) Tj 1 0 0 1 496.3 226 Tm /OPExtFont5 11 Tf (t+i ) Tj 1 0 0 1 508.8 226 Tm 80 Tz /OPExtFont3 11 Tf ( ) Tj 1 0 0 1 125.049 202.95 Tm 109 Tz (F\(R) Tj 1 0 0 1 145.199 202.95 Tm 82 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 148.8 202.7 Tm 96 Tz /OPExtFont3 11 Tf (\), i.e. the model error dh, distinguishes a system trajectory from being a ) Tj 1 0 0 1 124.799 179.45 Tm 92 Tz (model trajectory. Forecasting by iterating the initial condition ensemble forward ) Tj 1 0 0 1 124.799 156.149 Tm 97 Tz (by the model ignores the existence of model error. Given an initial condition ) Tj 1 0 0 1 124.549 133.1 Tm 93 Tz (ensemble at time 0, the ideal forecast at time 1 would be obtained by adjusting ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 315.85 52 Tm 75 Tz (130 ) Tj ET EMC endstream endobj 699 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 700 0 obj <> stream 0  ,,}b7!I6Ўd Zh' nB:^ԛ/ |2CFA@Fy4 ,bdO>fal e#H%N -U,*iO82rբMQz?cGHkp+>{ VӇW2 \f ]h˺uRbq wo-]R ^4R81:HX!Y ~alS !9T'Mu 6#LGƶ 5 r&1cDڶ(v8/NZ\K[/]y/EGyhG^W`\GLK-܏!A^rscF/m@L [ b+,cj`DM1_+)%. ޥ5ݎ_N/\L _o\&'=0]C׮(h;b,O/N^@6Ь7Z]BvPce2Lk/>u{X8|Xd?Ky;"7(rԼWrD}Iɰm͢XD[/: +h%֧W*`Ori բվ_ҳ-F(QŖcw4R+C7s)cn3 LdbFӇ$|ݛ7{5jsNb6w'.7w@c2#Zߙ٤ϡr<"/@9w)L$4$#/%B%}rVʃL#`3hLF+O_aJxe%|l-mȊF -wA!MޒyCp=$;p]>=ԴvR[ySUABNT&wp&A%5@U*s$ dg哮iWʕWД4*7]|nqyD ͒S93¾(Yy%F! ըr^3,,#e?bK{itRȟlt8$)(d.o"UiBB, ۂY th֭lr<NSJQҵSaJKpXu @ όdq*tZÞr *Bƞ8X|ۓA4*èUK"4;iA;$Q$!72TׇVw- 6x,.o,}mOO警0| g)oyq'"jD{U^͗(yǠ;t#ZS9@IM'I5ΉÝNL 6:T٨agG_/|;/Û 9RnK[|Ә̿%AB׷t+d j@K5c h+̎3/8gycmtM!Iٝ'B-(=]ׁorṲ̆Ei+j0Džs5~]cנKM͋][6b;@چ>t-{2y}|!~uCJEgL.Ohnr»S,ݩ]FD{*N; @fk^WsH_mvKCblT(Q1VUy ){JS'% BfL66iPVzQ1^5|` c:K5~V%]}੻l%$cZ +|;}[vZhDZ1EȽ}ii}{q*5zM?OTt 0 +"4i AJV Sm[+gK'4$X\=KdZ9ϫ1n[8wp9Bh~6f- |w]w㳙=x4JnT٩5>ne 9;mGdO=8^eOYETQ3ߙYde\ƶCUU?턧H Kb_)9$[ECYF/]8w韤 p95'x ˣ?|@O&*o@=%̲ϒLտ n'";  ֥KX0.]/F:ߺ|aΔ0/p[&4\ (]-]'2'zc&͓bG8IQS[\"` .LDck qqlyqD:Gk&| )Q!.p[!<1h1Yi{eV2xs I 5h^_JAMAG V=oʩg$, q5" [6)߼-]ysw-_*q8eRz,8{R=@G` =z ~b`b>y'dtvą1X AӍ5px5N3WizLg3h˅@Dh(H^ ꜭPF&ΗNJˁ!>ȍ` !(Q @nh+z,7ܱRAWF6e$wǴ8f!whD!M~A⸵?zŧ wAJ|e|_H7etpR@-Mӱ"B*Eݜ$I"J7|a(#p㫙Hc^t("!넿B:zR od\k((R3ad~ B&O =n`%~4vuafY,~z QUΕм^9=]PϑEpBJ_AMw"u!y&LbBo^+غU+0;/ kGmQhk%lq$H,7CIhpLzf+`(ķЏƝk-yd`~#޴F~ɕIg&}sU[u2LRpޯi:nS^,`ʯ:MFz0ל᠈54vl%UZ39vx@I=]!VQn;uyLo%v`T 1[Wg,0&-YVl&/6Z_d"A7SY\`Ѫ'K)*;Yi)_ dT8+m=FRH=I]։6s5S9(pu3S=w4f.CX!,r], ?&Yؕ)gȸ^PA~Sa_y ᱌(D&\8U}TlaV݃[ڑH|95hk+mgFs\@ojTC0￷YSaRePV{8>>4581l3mғ&&n9wi2vD헮\̹^z7`ʨ^T21sU G'5j:PZ@e{CDlXŁI?55Aau>۳4x'<|DzLg;-Cu Ԟ NHpC#B-u"N8B]^sTh0ROQ}LoYoHnBGNSrz`R2hmw"w&nUNPܸ庒$+g4V7JgZ%f$& Cc(~.g[6c1꧎$p ي3Hc}*<?#}oivQ"qTb1l! nmkQu tYwfvXgt',_,H0&V|):11m[sJ])$75t&b/qK, HR=w'ԽXQy:^15bѼ'$V0*JƬjMN'aVOo1Wy } I+՝(\bfM}Br^{<,Kϥ+ Y2u.u/qJ: D X{}2F'Fkcfz0V("n䓯x)Yܱ-;Dx #ض3aX^-֗= |ĤKזk"Vh ǫx_FKk;+Lg9%FtvE 6.[ Ϟ!Ntؗe,tXvr ,wQN{ Hͻ8}W"86,Xf9>CXZ%}IO&6mVM{iU-KU"9L"q- &ZgӠR.O߬c whޓzGTVK󇅚Ron~W3T5lwFnSQnv]e-) WLeGBl Bͩӆ0aX0ʜB/7TxOzb!A]J*XkM,+PkNo[8܃KgmҧJ m~XkE`8tһ~{&6lYrڡBx!!%wpS6/7dz?aGSrQDɈigx頿Lk/lVą6dGb7Ĥ[i켗xܩ,{~rC_au ޠئt=']8iKyA JcP'TNIv4@`Rm끋i|}k<߳G&Pסtf5/+%ܻ~T$պդ+`CzЙtwa~Dɥ/cq4'HR(Lp{X{`GrDO~Tvy ׷-$w +1z6F%) pzp3j~Ҷ߮Lf(դl_Gr1_LvJoJ8w>s0ybal(Ay> 0; rUWݢkl4H+d/‚t&\yN;^ٽF N&CGjt4P{*o@ej@1S"jJրE6@d ZJf!F Ee@+7.Tv71 J쳪T#/FبhQ$AFx7fzjCg [|_RI[w"8x̱ @L`x&}PA@XEutW0g>}MaUG   3,!cPꦑ)c1ﶧ8߀q?ԭxэ`dM`uL݇V.nnETwtpRv nwZ& RxyDG&C <'?]BZ)s.>(|nB_~Z.^\DP#}Mq -AL&2!EzXI4IʼtQɸeEy1D,T>c&Re@ 5q9>a {Af<*q>$+7< H>:זּU_B1臀p!rah_M`Xڝ"9^:py*V: Ӳ]o7YvUAiD{B0.Acf.]}(|q}'J FwlXZZ"0X:MpES ZF(4/Ȅ,8bN u|>\뢢ڵ,C{3颧!L <@WEp ' ;Ȥ[<M.$d(u>lBI]*Ǽa+Uau^|a@T$ O{>WY̻iGϗѭ0pGVC{h5䂇ׁ1]7YH9(ޢCǼ:H;l?jR  ;,zoMqZ5=uՏȜd 9.r|.6P 1`kr b"7>59ʾ<ӭjH0Ƈ@yMGH3h4>r͂'(AR8|;IbD`ÞrnO} Ҫe?3ͦDpϠYg GR`ύ~3 =e!ftaC~D@5,9S&:P_ЎpDx=7@wԠd'k,Q96SO wCi(Oњy؝pB'8)ԀB:뙢Q X/)<RA;:MKweH-)637vKk~Z吜Xh>v.)E16(SKvYgeg5H006Í cmaBg/9z7@eF*/v3zx+VeOWݽ'bb"HH- ,;a.1ʷn=y@n:>fl&.#Kϫ3K#gFQR"`vCfUo_p^_"t/mZb0gLvTXʻn4SF|/fF)8Q0lݟ_^(zQ O_JUKzRj 9lP<|SyVfDTsKtmٮlW?8: vܻ̑.'n.I806FJgV4PQezR[i54 xz~&h`@فߛki_4I:nLm;fqsY0gߩ "SDg%qTQq~t5( 6QjU\42Id8CôfdL6b8^-hq9rTxYg}yt7A{k,PI bQ;fi:{`KH x>L+Ⱦ;ZJ&`^97Ix%qq_n$ Sfw`E(ɣDf{L?l*ڄ);h^9焀F| i޵xgxb1gw m RMqzzdskgVhۈ=FZƕk(x xLUN7;OqݲJA yI#LHL㥀.lݺkp1 ofJKXY7C嗹}["V +ҫHxzzs)BvCAe7Jn'wqR=ϖ$dHVsb@)EjD%G jMǐ$(PB[k[]֡}QjD}BfՁdv[kM5$j ׇʴ-y 3⎮0hT\)HBzSҀiaƔT;+i'jnVxLoE$\ʊ6 |"f-ED."ҿ4Q`k0fOP_`MHkqcQ>=[˘$4gE?d Fwqzk+VW&̬ ެ$z$3\[Xǖ:YҼw\wMy߃]Ak0 89bSJCð\LNz>O6* +kWP7hl]Kw %hg9$bպk<%;p~0 IчS|#"ʯuyU;M M8Ҁƽɮ0"}!Rd9cN_zU:74Y8jde"g }zگ/f% 9 &v,aG%"&/o{K>> endobj 702 0 obj [703 0 R] endobj 703 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 615 0 0 838 0 0 cm /ImagePart_2158 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 308.149 719.45 Tm 111 Tz 3 Tr /OPExtFont5 13 Tf (6.1 Forecasting using imperfect model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 111 Tz 3 Tr 1 0 0 1 127.7 676.25 Tm 95 Tz (the iteration of the initial condition ensemble with the corresponding model error. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 144.949 653.45 Tm 98 Tz (If one has a large sample of historical model errors -P) Tj 1 0 0 1 410.649 653.45 Tm 92 Tz /OPExtFont3 13 Tf (j) Tj 1 0 0 1 415.899 653.2 Tm 99 Tz /OPExtFont5 13 Tf (l, these could be used ) Tj 1 0 0 1 127.45 630.399 Tm 96 Tz (to improve the forecasting. One could, for example, adjust the iterations of initial ) Tj 1 0 0 1 127.9 607.35 Tm 97 Tz (condition ensemble with random draws from the set of relevant historical model ) Tj 1 0 0 1 127.7 584.1 Tm 98 Tz (errors \(79\). Let the initial condition ensemble be ) Tj 1 0 0 1 371.3 584.299 Tm 118 Tz /OPExtFont5 11.5 Tf (xi) Tj 1 0 0 1 378.25 584.299 Tm 52 Tz /OPExtFont3 11.5 Tf (o) Tj 1 0 0 1 383.5 584.299 Tm 114 Tz /OPExtFont5 11.5 Tf (,i = 1, ... ) Tj 1 0 0 1 427.199 584.299 Tm 85 Tz /OPExtFont4 10.5 Tf (Nens ) Tj 1 0 0 1 456.949 584.299 Tm 93 Tz /OPExtFont5 13 Tf (where ) Tj 1 0 0 1 489.35 584.299 Tm 85 Tz /OPExtFont4 10.5 Tf (Nens ) Tj 1 0 0 1 516 584.1 Tm 86 Tz /OPExtFont5 13 Tf (is ) Tj 1 0 0 1 127.2 561.049 Tm 99 Tz (the number of ensemble members. The forecast ensemble member at lead time ) Tj 1 0 0 1 127.2 538.25 Tm 104 Tz (t is then given by x) Tj 1 0 0 1 226.55 538 Tm 68 Tz /OPExtFont3 13 Tf (i) Tj 1 0 0 1 226.55 538 Tm 53 Tz (t ) Tj 1 0 0 1 229.199 538 Tm 103 Tz /OPExtFont5 13 Tf ( = F\(x1_) Tj 1 0 0 1 276.699 538 Tm 35 Tz /OPExtFont3 13 Tf (1) Tj 1 0 0 1 281.3 538 Tm 159 Tz /OPExtFont5 13 Tf (\) C) Tj 1 0 0 1 306.25 538 Tm 61 Tz /OPExtFont3 13 Tf (i) Tj 1 0 0 1 306.25 538 Tm 53 Tz (t ) Tj 1 0 0 1 308.899 538 Tm 97 Tz /OPExtFont5 13 Tf ( where C) Tj 1 0 0 1 353.3 538 Tm 57 Tz /OPExtFont3 13 Tf (t) Tj 1 0 0 1 353.5 538 Tm 61 Tz (i ) Tj 1 0 0 1 355.899 538.25 Tm 100 Tz /OPExtFont5 13 Tf ( is random drawn from the set of ) Tj 1 0 0 1 127.2 514.95 Tm 102 Tz (historical model error. we call this the forecast with ) Tj 1 0 0 1 396.5 514.95 Tm 94 Tz /OPExtFont4 10.5 Tf (random adjustment. ) Tj 1 0 0 1 501.6 514.95 Tm 95 Tz /OPExtFont5 13 Tf (This ) Tj 1 0 0 1 127.2 492.149 Tm 100 Tz (method is equivalent to transferring the deterministic model ) Tj 1 0 0 1 436.8 491.899 Tm 112 Tz /OPExtFont4 12 Tf (F ) Tj 1 0 0 1 450.5 491.899 Tm 101 Tz /OPExtFont5 13 Tf (to a stochastic ) Tj 1 0 0 1 127.2 468.899 Tm (model by adding the dynamical noise term C) Tj 1 0 0 1 354.699 468.899 Tm 53 Tz /OPExtFont3 13 Tf (t) Tj 1 0 0 1 359.05 468.899 Tm 103 Tz /OPExtFont5 13 Tf (. And this method assumes that ) Tj 1 0 0 1 127.2 445.85 Tm 99 Tz (the model error is IID distributed when usually it is not the case. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 144.5 422.55 Tm 105 Tz (Model error is usually correlated, for example see Figure 5.1. Randomly ) Tj 1 0 0 1 127.2 399.5 Tm 101 Tz (drawing from the set of global historical model error discards the geometrical ) Tj 1 0 0 1 127.2 376.5 Tm 103 Tz (information about model error. Another approach by using historical model ) Tj 1 0 0 1 126 353.449 Tm 99 Tz (error that doesn't discard geometrical information is to employ a local analogue ) Tj 1 0 0 1 126.95 330.149 Tm 98 Tz (model to determine the adjustments fit. For each model error ) Tj 1 0 0 1 434.649 330.149 Tm 67 Tz /OPExtFont4 11.5 Tf (\(D) Tj 1 0 0 1 443.3 330.149 Tm 104 Tz (i) Tj 1 0 0 1 448.8 330.149 Tm 34 Tz (, ) Tj 1 0 0 1 454.8 330.149 Tm 95 Tz /OPExtFont5 13 Tf (it corresponds ) Tj 1 0 0 1 126.7 307.1 Tm 101 Tz (to two sequential states x) Tj 1 0 0 1 254.4 306.899 Tm 44 Tz /OPExtFont3 13 Tf (3 ) Tj 1 0 0 1 258 306.899 Tm 94 Tz /OPExtFont5 13 Tf ( and 5cj) Tj 1 0 0 1 296.149 306.649 Tm 58 Tz /OPExtFont3 13 Tf (+1 ) Tj 1 0 0 1 305.5 307.1 Tm 74 Tz /OPExtFont5 13 Tf ( as c.:.7) Tj 1 0 0 1 330 307.1 Tm 41 Tz (') Tj 1 0 0 1 333.1 307.1 Tm 86 Tz /OPExtFont3 13 Tf (; ) Tj 1 0 0 1 336.699 307.1 Tm 174 Tz /OPExtFont5 13 Tf ( = ) Tj 1 0 0 1 389.75 306.899 Tm 107 Tz /OPExtFont4 11.5 Tf (F\(R) Tj 1 0 0 1 409.449 306.899 Tm 112 Tz (i) Tj 1 0 0 1 414.5 307.1 Tm 86 Tz (\). ) Tj 1 0 0 1 427.449 306.899 Tm 98 Tz /OPExtFont5 13 Tf (To construct C) Tj 1 0 0 1 500.899 306.649 Tm 43 Tz /OPExtFont3 13 Tf (t) Tj 1 0 0 1 504.699 306.899 Tm 91 Tz /OPExtFont5 13 Tf (, we ) Tj 1 0 0 1 126.95 283.6 Tm 97 Tz (can first find ) Tj 1 0 0 1 194.15 283.6 Tm 123 Tz /OPExtFont4 10.5 Tf (K ) Tj 1 0 0 1 207.849 283.6 Tm 99 Tz /OPExtFont5 13 Tf (nearest neighbours of xi from the historical set ) Tj 1 0 0 1 445.449 283.6 Tm 97 Tz /OPExtFont3 11 Tf ({ic) Tj 1 0 0 1 456.949 283.6 Tm 95 Tz (;) Tj 1 0 0 1 462 283.6 Tm 140 Tz (} ) Tj 1 0 0 1 471.35 283.6 Tm 95 Tz /OPExtFont5 13 Tf (and record ) Tj 1 0 0 1 126.7 260.549 Tm 88 Tz (their corresponding \(.7.7) Tj 1 0 0 1 236.65 260.549 Tm 86 Tz /OPExtFont3 13 Tf (j) Tj 1 0 0 1 242.15 260.549 Tm 99 Tz /OPExtFont5 13 Tf (, we then randomly choose one model error from the ) Tj 1 0 0 1 513.6 260.549 Tm 126 Tz /OPExtFont4 10.5 Tf (K ) Tj 1 0 0 1 126.7 237.5 Tm 46 Tz (CZ\) ) Tj 1 0 0 1 144 237.5 Tm 96 Tz /OPExtFont5 13 Tf (to be C) Tj 1 0 0 1 180 237.5 Tm 58 Tz /OPExtFont3 13 Tf (t) Tj 1 0 0 1 180.25 237.5 Tm 61 Tz (i) Tj 1 0 0 1 184.8 237.5 Tm 101 Tz /OPExtFont5 13 Tf (. We call this method forecast with ) Tj 1 0 0 1 367.199 237.5 Tm 92 Tz /OPExtFont4 10.5 Tf (analogue adjustment. ) Tj 1 0 0 1 476.149 237.5 Tm 98 Tz /OPExtFont5 13 Tf (There are ) Tj 1 0 0 1 126.7 214.5 Tm 95 Tz (many other analogue models one can use \(details of analogue models can be found ) Tj 1 0 0 1 126.7 190.95 Tm 102 Tz (in Section 2.5\), our interest here is not finding a better analogue model but to ) Tj 1 0 0 1 126.95 167.7 Tm 100 Tz (demonstrate that by extracting information from the model errors the forecast ) Tj 1 0 0 1 126.7 144.649 Tm 96 Tz (performance can be improved. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 96 Tz 3 Tr 1 0 0 1 128.4 121.6 Tm 103 Tz ( For computational reasons, we illustrate both methods taking only one ad- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 103 Tz 3 Tr 1 0 0 1 317.5 51.299 Tm 84 Tz (131 ) Tj ET EMC endstream endobj 704 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 705 0 obj <> stream 0  ,,b6X{7:ǻȷ k~^YAi$IԜLp*Rcõ*дg$Km%yִy=YEޥA\ps4G)q+Tnd!!,tǐגq%;UY^XBIE9Tћ|M#xWLOvx_/zJc,a'cME2fn0AW1T "$}Ec#BTT3˔So7M>U"[N=n8ްN$ \$>?irG#$KDqMLiBL&}}7.Fi'ŲǃA67W$:G捣 >X #8kn`?n!9mAʈZ!xq\amnJOV|sC~"%G_9aCV )ӆ4xffo[B {QWok<0hf᛿4ha[$%d>>^06ɃMKamB,BO9z(R0Pb"T]EA}jfe;;5u5Jǃ TflgO)~e 1ldY!ʂ2o9Jx*ن=ͽ8w=+6y% u^Fã5y͋3Q?)C&#N|󤂠McK3n2DgmAv%5kNǹܛ /]e?||+mܧӟbjqÿT'$/WH;mLPs(R Uw"T~. 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NNg7e~,EAغZ #RKx{EJ:DwL ;fDr_ymz1xË$,#r`34v7kt!=`Y A*NafVvp6PX]q`H- _;ufY*fu8tѼuNLʡM~~k N=IZF%#Qԩ_$" .BDmKBP"XFbfm"T0)e%w4LwDr]^ňFQ=8ϔ=R *]0j"rNܫaŅaASK"ԞmD=mF V˞b]Hߗ&Q+RsT:'(CE8D3GWL7~aB LlJ8iT\>Dlv,2P_!k1ˋ⹦^0E\m_11] ykᬖ"柌M p|[%oI6dMU- At\Wl yݙX <_N$+^WieoGQ}@Bb>p6]jyih7UkaeTԚU/V@"D(yQ I ~=H2,bgUK3c^ U]-dQc(K `*j n 0盍LlL(F(¹+l 314.@b婺D,R'7Elzv*vG!^Y((ApA`u:kyeO}NxIs'ӯѰ ;L*";x4q ȨUuX٧6b#2 8er4 ؔUH#glDc2!eQ`Q&÷y7?^\ve% *+O%ɓHw SxL&"4=DN[,fr0C w!66OңI"IJK31C.щs!4ͪW1KdqHo8_.~P /]@fSTt.o, {T)* 0:M_<ƬlqpGI)+X6:n5P4>ul+ױL?,Ǚ;*)i+=< |݂s8^"ɶ>}n+Fcx̥` fwԄf*C}*DԀVDHcaF*`2Kvy/1SIWEёa 2hcV:dzYi(FSK0(quy "#6|lN0򾠷ĺdFTE4aS1E=f_"]ӗۖ>U,a23V͑"zRp{b Oc#dFrJ`\yhz]^Ma9lF}kS3?i0(ܵƬϿ CX#}(a]^Gi2`סBUQ53$irU뛫 ? endstream endobj 706 0 obj <> endobj 707 0 obj [708 0 R] endobj 708 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 625 0 0 838 0 0 cm /ImagePart_2159 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 309.35 719.45 Tm 104 Tz 3 Tr /OPExtFont3 11 Tf (6.1 Forecasting using imperfect model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 104 Tz 3 Tr 1 0 0 1 127.9 676.25 Tm 89 Tz (justment ) Tj 1 0 0 1 190.3 676.25 Tm 99 Tz /OPExtFont5 13 Tf (for each ensemble member xt. Note that in future work, one could ) Tj 1 0 0 1 128.65 653.45 Tm 97 Tz (sample more than one from the set of historical model error or from the ) Tj 1 0 0 1 487.899 653.7 Tm 113 Tz /OPExtFont4 11.5 Tf (K ) Tj 1 0 0 1 501.85 653.899 Tm 94 Tz /OPExtFont5 13 Tf (local ) Tj 1 0 0 1 128.65 630.649 Tm 100 Tz (model error \(70\). Taking large samples will be more useful, but will lead to ex-) Tj 1 0 0 1 128.65 607.85 Tm 98 Tz (ponentially growing ensemble size, which is interesting but beyond the scope of ) Tj 1 0 0 1 128.65 584.549 Tm 97 Tz (current thesis. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 145.449 561.299 Tm 96 Tz (The experiment results discussed below are based on two different initial con-) Tj 1 0 0 1 128.65 538.25 Tm 95 Tz (dition ensembles. One is inverse noise \(see section 4.3.1\), which is a computation-) Tj 1 0 0 1 128.65 515.2 Tm 97 Tz (ally cheap and easy way to form the initial condition ensemble but which ignores ) Tj 1 0 0 1 128.65 492.399 Tm (the information of the dynamics. The other is the dynamical consistent ensemble ) Tj 1 0 0 1 128.9 469.1 Tm 100 Tz (\(see section 3.6\), in our experiment the ensemble members are consistent with ) Tj 1 0 0 1 127.9 445.85 Tm 99 Tz (the system dynamics and a segment of observations, si, ) Tj 1 0 0 1 406.3 445.85 Tm 116 Tz /OPExtFont3 11 Tf (i = ) Tj 1 0 0 1 426.949 445.85 Tm 87 Tz /OPExtFont5 13 Tf (5, 4, ..., O. In the ) Tj 1 0 0 1 128.4 422.8 Tm 96 Tz (imperfect model scenario, such initial condition ensemble is not achievable as the ) Tj 1 0 0 1 128.15 399.75 Tm 99 Tz (system dynamics is unknown. We use such "perfect" initial condition ensemble ) Tj 1 0 0 1 128.15 376.699 Tm 98 Tz (as an example of the best initial condition ensemble one might hope to achieve. ) Tj 1 0 0 1 128.15 353.699 Tm 97 Tz (Results shown below demonstrate that forecasting with the adjustment of model ) Tj 1 0 0 1 127.9 330.399 Tm (error improves the forecast performance in both cases. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 144.949 307.35 Tm 95 Tz (Figure 6.1 shows four examples of the one step forecast ensemble in the model ) Tj 1 0 0 1 127.9 283.85 Tm 98 Tz (state space when the initial condition ensemble is formed by inverse noise. In all ) Tj 1 0 0 1 127.7 261.049 Tm (the examples, forecasting with random adjustment produces ensemble members ) Tj 1 0 0 1 127.7 237.75 Tm 105 Tz (with too much spread. In panel \(a\), forecasting were made in a place where ) Tj 1 0 0 1 127.7 214.5 Tm 97 Tz (the model error is very small, which makes the difference between direct forecast ) Tj 1 0 0 1 127.9 191.45 Tm 101 Tz (and forecast with analogue adjustment very small. Panel \(b\) shows an example ) Tj 1 0 0 1 127.7 168.149 Tm 100 Tz (where the model error is small but not negligible, direct forecast ensemble fails ) Tj 1 0 0 1 127.7 144.899 Tm 108 Tz (to capture the true state to a slight extent. Panel \(c\) and \(d\) are cases that ) Tj 1 0 0 1 127.7 121.6 Tm 103 Tz (the model error is moderate and relatively large, in both cases direct forecast ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 103 Tz 3 Tr 1 0 0 1 318.699 52 Tm 86 Tz (132 ) Tj ET EMC endstream endobj 709 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 710 0 obj <> stream 0 - ,,b6G $ZS=2~g[WRubfNO9<$ ]ۑjÑimAM"pX.s>抶ͫ2=":*ʠy[ܱ8I<\T;q "<15SUgN~`&\p{:JKy0]D-&-Dw\LK23L+sknS>nF;`+Hg6TBCgwEg' +([M`Pz{%|]LpوϮ߷x+o Jq pއR>qM)Q]3l5cĂRB/گV-*J*v˟Y3`j ,}vX_jt>ޔhM钊4[GВjnUeΟX#Ys$|_P揰kɘٔA$m'J:$Fj [Ÿ wݩ~'_MRkܡjpPޯon,6}"$ԏЏz/P[.Jg=.a -|ā=P7|15ViƋA樻 PѾ o/YMHB_E֕4Mˁ$~?Ԅr+spQ#zzn@im=iuES/MLX0F#Wg|~c|7~jYlRĊeՌGf 2áv̻+ۄ0;Ȑ'E-Zivc3 JdXW5s-;6zeNfdvҼ)'#oZ`{b-hz0LI_I/ }K]3+I^f#';^s>OA1XOqK I'5 qjX<7J +:7,(FIE)x? 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H1SJQ _c--;_z6Qz_(+ptf*htT:/ W=辰ݤ=*3{\4)H=TB̽KX(iB2"V|$Hޟ;Z3K>SϤI^􉓛Arз/x+k[#HKI3*>,9 AVO!U!M!vȰ?v I3~,x\j i8?PZaz|jlrr[-Y.ٓ[6a=Nhfb`/lB6 I0)"I0đ>rё`A׋$Z  9}75Qgv5|8ۥ(dNX"NǝzBczŢ'Cq2 j,8Dqed5=]W]c#ϿǪm[¢|uRH>X"$pi'%S0֞BCxDy#&H"ȝr fWp4S^hvy -"ϼQ%XR V!P/Fv@"dQԉ5'4] 5tYOyqv/ LQҘlq5t3Cݥ$=`>:c'-ng&"NxǙc]| 2 V58=-KGR's3tkّOOl+ɓ :ozPS;`5~o.d U= (^ QUpmzE3 ɻ;`(h)UB빋 \@h^xf=Au9arjuI7?5޳Et#}챂6H?DreoEV\v}WhwQ4CDH18Oc\$'Trg3|M=}mNh#ٽR@#Ñ32+s8xNa>sjޕ5游%%wV"c-i'aI )08/P*e8 Cn=,y3BO+:&`8l$UU{f\8BU ,wӛjvKG͓u38w=Q9=u>xT.@kMX7Buf*xov! %ym|$Oe΃SXirH눶DJ@M٬Uވ}A08m3<ߓLXO汯Mebʇ<; 3"#؇)O)4v(s x4ԬEFǒkn^~*Xct &m'Wgpe;c A\/郍8Q s6+g;k~sY]8+5DIXy|sq?D\[$ &p_OslxXy{f.Z_,+!+ kjbAW_> endobj 712 0 obj [713 0 R] endobj 713 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 499 0 0 802 0 0 cm /ImagePart_2160 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 296.649 376 Tm 82 Tz 3 Tr /OPExtFont1 5.5 Tf (0.81 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5.5 Tf 82 Tz 3 Tr 1 0 0 1 322.55 376 Tm 92 Tz (0.82 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5.5 Tf 92 Tz 3 Tr 1 0 0 1 348.949 376 Tm 89 Tz (0.83 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5.5 Tf 89 Tz 3 Tr 1 0 0 1 374.899 376 Tm (0.84 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5.5 Tf 89 Tz 3 Tr 1 0 0 1 401.05 375.75 Tm (0.85 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5.5 Tf 89 Tz 3 Tr 1 0 0 1 427.199 375.75 Tm (0.88 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5.5 Tf 89 Tz 3 Tr 1 0 0 1 300.699 529.6 Tm 104 Tz /OPExtFont11 4.5 Tf (07 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 104 Tz 3 Tr 1 0 0 1 325.449 529.6 Tm 85 Tz (0 7) Tj 1 0 0 1 332.399 529.6 Tm 116 Tz /OPExtFont9 4.5 Tf (, ) Tj 1 0 0 1 333.85 529.6 Tm 63 Tz /OPExtFont11 4.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 63 Tz 3 Tr 1 0 0 1 377.5 529.6 Tm 87 Tz (0 73 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 87 Tz 3 Tr 1 0 0 1 403.449 529.85 Tm 89 Tz (0 74 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 89 Tz 3 Tr 1 0 0 1 268.3 496 Tm /OPExtFont1 5.5 Tf (0.18 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5.5 Tf 89 Tz 3 Tr 1 0 0 1 268.1 475.35 Tm (0.17 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5.5 Tf 89 Tz 3 Tr 1 0 0 1 268.3 454.949 Tm 92 Tz /OPExtFont11 4.5 Tf (0.16 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 92 Tz 3 Tr 1 0 0 1 268.1 434.1 Tm 91 Tz /OPExtFont1 5.5 Tf (0.15 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5.5 Tf 91 Tz 3 Tr 1 0 0 1 268.1 413.699 Tm (0.14 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5.5 Tf 91 Tz 3 Tr 1 0 0 1 268.3 393.3 Tm 89 Tz (0.13 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5.5 Tf 89 Tz 3 Tr 1 0 0 1 91.2 529.85 Tm 82 Tz /OPExtFont11 4.5 Tf (1 06 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 82 Tz 3 Tr 1 0 0 1 117.099 530.1 Tm 85 Tz (1 07 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 85 Tz 3 Tr 1 0 0 1 143.3 529.85 Tm 84 Tz (1.08 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 84 Tz 3 Tr 1 0 0 1 169.199 529.85 Tm (1.09 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 84 Tz 3 Tr 1 0 0 1 196.55 529.6 Tm 78 Tz (1.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 78 Tz 3 Tr 1 0 0 1 63.35 489.05 Tm 92 Tz (-0 4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 92 Tz 3 Tr 1 0 0 1 60.5 468.399 Tm 101 Tz (-041 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 101 Tz 3 Tr 1 0 0 1 60.5 447.75 Tm 108 Tz (-042 ) Tj 1 0 0 1 60.5 427.1 Tm 94 Tz (-0 43 ) Tj 1 0 0 1 60.25 406.949 Tm (-0 44 ) Tj 1 0 0 1 60.25 386.1 Tm (-0 45 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 94 Tz 3 Tr 1 0 0 1 87.099 376.5 Tm /OPExtFont1 5.5 Tf (-0.03 ) Tj 1 0 0 1 98.9 376.699 Tm 974 Tz (\t) Tj 1 0 0 1 113.75 376.5 Tm 93 Tz (-0.02 ) Tj 1 0 0 1 125.5 376.5 Tm 945 Tz (\t) Tj 1 0 0 1 139.9 376.5 Tm 96 Tz /OPExtFont9 6.5 Tf (-am ) Tj 1 0 0 1 150.699 376.5 Tm 2000 Tz (\t) Tj 1 0 0 1 220.8 376.699 Tm 92 Tz /OPExtFont11 4.5 Tf (0 02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 92 Tz 3 Tr 1 0 0 1 150.949 376.5 Tm 119 Tz /OPExtFont9 6.5 Tf (. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 119 Tz 3 Tr 1 0 0 1 202.099 455.699 Tm 45 Tz /OPExtFont9 3 Tf (' ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 45 Tz 3 Tr 1 0 0 1 82.799 406.949 Tm 67 Tz /OPExtFont11 6.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6.5 Tf 67 Tz 3 Tr 1 0 0 1 87.599 488.1 Tm 77 Tz /OPExtFont9 6.5 Tf (\(C\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 77 Tz 3 Tr 1 0 0 1 221.5 700.5 Tm 95 Tz /OPExtFont0 11 Tf (6.1 Forecasting using imperfect model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont0 11 Tf 95 Tz 3 Tr 1 0 0 1 43.899 341.899 Tm 90 Tz /OPExtFont3 11 Tf (Figure 6.1: One step forecast ensemble in the state space. Observations are gen-) Tj 1 0 0 1 43.7 328.5 Tm 92 Tz (erated by Ikeda Map with ) Tj 1 0 0 1 173.5 328.5 Tm 95 Tz /OPExtFont0 11 Tf (IID ) Tj 1 0 0 1 194.4 328.25 Tm 90 Tz /OPExtFont3 11 Tf (uniform bounded noise U\(0, 0.01\). The truncated ) Tj 1 0 0 1 43.7 314.3 Tm 92 Tz (Ikeda model is used to make forecast. The initial condition ensemble is formed ) Tj 1 0 0 1 43.899 300.649 Tm 91 Tz (by inverse noise with 64 ensemble members. Four 1-step forecast examples are ) Tj 1 0 0 1 43.7 286.95 Tm 90 Tz (shown in four panels. In each panel, the background dots indicate samples from ) Tj 1 0 0 1 43.7 273.049 Tm 91 Tz (the Ikeda Map attractor, the red cross denotes the true state of the system, the ) Tj 1 0 0 1 43.7 259.35 Tm 92 Tz (blue square indicates the observation, the direct forecast ensemble is depicted ) Tj 1 0 0 1 43.7 245.7 Tm 91 Tz (by purple circles, the forecast with random adjustment ensemble is depicted by ) Tj 1 0 0 1 43.7 232 Tm (orange dots and the forecast with analogue adjustment ensemble is depicted by ) Tj 1 0 0 1 43.7 218.299 Tm 85 Tz (cyan stars. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 85 Tz 3 Tr 1 0 0 1 43.45 185.7 Tm 95 Tz (ensemble fails significantly to capture the true state, while the forecast with ) Tj 1 0 0 1 43.7 162.899 Tm 93 Tz (analogue adjustment ensemble is still able to capture the true state very well. ) Tj 1 0 0 1 43.45 140.1 Tm 92 Tz (The forecast with random adjustment sometimes produces ensemble members ) Tj 1 0 0 1 43.45 117.299 Tm 94 Tz (that stay closer to the true state than all the ensemble members produced by ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 94 Tz 3 Tr 1 0 0 1 231.599 45.049 Tm 73 Tz (133 ) Tj ET EMC endstream endobj 714 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 715 0 obj <> stream 0! ,,"jٲO>{sFaoFь8u/VGeb w,Wd\ahuESxQum:,- ,pwL|bG-hu2|OfBUb,E#猑ׅ4!@4oHnٕ%)Еdm)d3 Jp 0P= }}8*;GUt ;:L١6u)ѢᏝ?Oh@+X;N0V˅)Mn0V[Si]EJ3PXPe$mt77ӲJ1KcIfuRh!"<傒NAweS!a1 YyN}"P3_J0D3?Xox68g>ӎ oj:v6s߻2eȕ‡0O e-WG_ix EH/2t6x)\lnWZ;C6O\V2=;śoUmg^6lNXz˜ٜtj543K$?x)NݎX%5 +^mUV·v0f(!!2<-U37ʽRv| q29=DlE+WQpVQuȳS^~e `;Q9\'qm?(`gn$ -$ڷ 0<hBem΀tCjhFDт˔[Ci/c7)lEi8L5B%﫱xH1>%\xϥR( mϫZn[i}ے<1FL>/-m{ `|fCWַ/Ñs{AMz4Gpaf_UX75]\gm/Fͧ0RP%Z((--N}l^s|b3b|]yD6$.(3V3͂9w$mNnT6}l`D}p/+dj4p+K cxז{ Wujq}pxG.>s\Lޒ3kxEϽݶj{pfGbd\xlzP>1|! g}ek|^}5@ne\A"v*HBD)m8PI:/җ9-tCD-unQTڛfyY7ʮaVEwJҼ$D\JENb^&$g*qDTe>]SdIX>f +_x=oO01c 4ַen 1@}}V \%iDlu5dRZt7RetaMU뵵6vNd+3BbOg1Rװ;*0iBj\P&.hyFqgA/9ik`W֠oaD8dֲjHs8nӮ^!D*$YÐ 77LbӢ4Ԩ_ȉ-Ty|1%ayHٹw0kؽ=|fA֤}[\ >腠 SnӒS߹wɯLnL J>10"~1^3恹sK6m8Gcp!WW{$Pe%$%^ pq+D ]>9,Mn~ü, 7>s"j7-,ھ'j#E=417$VfSړ1W[iMN#g`E2Dyv]+OmX`쇰.XŤ L߸F\Yf8K0vm1/rIC|̎2ņ׍TpQ3}w0 Iz TRm׼dZHD8ޒa:,Q"8&@jBg"y&FD}}$$iuS0zǨ&g7N94bd4|HT"X#ڪ%H[;N,a{d6 aI:C58 '1kTa6[o`E1# sF(Xc0pxHԕj'~a qU{_Defv i[3!G] Cߥޚn]QG+Gϋlc>gVw\a\ ne23u-]pYH[~Jk@ޢs( cݮ J7eƱ@aV0q)-_̵P2OjP^{ˈ1T\NchExyvk`h}ܕS5էeNc==}xn8eUk^@cݐ0Imvp"XNsًpxĉ0@~@'u\ieZj3mWӆ/w;"Ú3lޓ0B ɲ$~n(7)zCqm}_j%׫ XV~VGvlğE3j:cݧ}kl3$*]qX4ӶSac*hyNֳT&] I@ڗ{;xmj/Ҩ+AiU[I܆Rd ױ",=n8m%≵{qʌWTۢ>_5Mj=|Syq m8.Nˍ^6j~#9}ɹdaBn8SҰd N* b2DrWl#1LzDH,ԧAXM%*ZqVZOGW$лbo&EIZ:ƃuXƠd/ ɻǾAd5=-7!YlY$l_6?騇q& )fbaNjL+Ox6Ҁ[+7D%M9ߗf/"I ^yʴM$cN>b Gw zK0P=M3?nvAY֫+(7>pVaG%X8T{rjT"(W&Gkk2c5u_c(@z5ЩggZmeTɄr?,]Bx9 ov6.5U(_9rW፭H>)~ibu9jS~٤h Y,="A;wRU*`ZI c7ҷ> `6]௸H.ۿ{hxDR3B݄͵ <̾wu =hjU} ͓ʞG/^ncω5صщ@jD_K`Wa[@gifR¯ B%_yakE o.W1>ڥZCY@O59y}P(P?DVi/ODCXTE?ؑ\;V RRR]dXkMcO Y>W/ʴ<>ܝ_N>xz8zTX e8֘(c潳;97k]9ۯg68? 3%bu[,obQR YCjݏڙM_ >X)\\e-5c%I{7doD08 Yw?.SS C+>Zp#'"nc))Κ.>ܻ™Waaډ,BY2Nb IJLX#r %f֥߉Ѡ. 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It appears that forecasts with analogue adjustment almost always ) Tj 1 0 0 1 128.65 585.1 Tm 94 Tz (outperform the direct forecasts for different lead times and different sizes of ) Tj 1 0 0 1 516 584.85 Tm 83 Tz /OPExtFont6 11.5 Tf (e-) Tj 1 0 0 1 128.4 562.049 Tm 92 Tz (b ) Tj 1 0 0 1 134.9 562.049 Tm 90 Tz /OPExtFont3 11 Tf (all . At lead time 1 and 2, direct forecasts outperform the forecasts with random ) Tj 1 0 0 1 128.4 539 Tm 94 Tz (adjustment no matter what the size of c-ball is. At lead time 4, the proportion ) Tj 1 0 0 1 128.15 515.7 Tm 92 Tz (of wins for these two methods are close when the diameter of c-ball is less than ) Tj 1 0 0 1 128.4 492.699 Tm 90 Tz (0.03, beyond 0.03 direct forecast wins. At lead time 8 and lead time 16, however, ) Tj 1 0 0 1 128.15 469.649 Tm 92 Tz (forecast with random adjustment outperforms the direct forecast. The reason of ) Tj 1 0 0 1 128.15 446.6 Tm 93 Tz (direct forecast winning at short lead time and losing at longer lead time is that ) Tj 1 0 0 1 127.9 423.55 Tm 94 Tz (at short lead time although the direct forecast may not capture the true state, ) Tj 1 0 0 1 127.7 400.3 Tm 90 Tz (the forecast ensemble members are still stay relatively close to the true state. For ) Tj 1 0 0 1 127.7 377.25 Tm (longer lead time, a direct forecast ensemble is not only unable to capture the true ) Tj 1 0 0 1 127.45 353.949 Tm 92 Tz (state but also further way from the true state. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 144.699 330.899 Tm 89 Tz (Following Figure 6.1, Figure 6.3 shows the same four examples of the one step ) Tj 1 0 0 1 127.7 307.899 Tm 90 Tz (forecast ensemble but based on the initial condition ensemble that is a dynamical ) Tj 1 0 0 1 127.7 284.6 Tm 92 Tz (consistent ensemble in the state space. Forecasts with random adjustment still ) Tj 1 0 0 1 127.2 261.549 Tm 89 Tz (produce ensemble members with too much spread. Although the initial conditions ) Tj 1 0 0 1 127.45 238.299 Tm 93 Tz (are consistent with both observations and system dynamics, the direct forecast ) Tj 1 0 0 1 127.2 215 Tm 92 Tz (fails to capture the true state with the appearance of model error. Forecast with ) Tj 1 0 0 1 127.2 191.7 Tm 90 Tz (analogue adjustment ensemble members not only cover the true state but also lie ) Tj 1 0 0 1 127.2 168.7 Tm 92 Tz (closer to the relevant system attractor. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 144.25 145.649 Tm 90 Tz (Following Figure 6.2, Figure 6.4 shows the c-ball test for the three forecasting ) Tj 1 0 0 1 126.95 122.35 Tm 92 Tz (methods at different lead time where the initial condition ensemble is formed by ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 317.75 52.5 Tm 78 Tz (134 ) Tj ET EMC endstream endobj 719 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 720 0 obj <> stream 0 ' ,,b6,rvN%kxp#&:ϮB#.o2B{*SjZsv(P6 BbJJ\zήf@Ü1j[#.Kْksˑ,I%gHrM+HLmTF [SjHE4}s1Y&Ga*;-ޒo_qR ,yH0nw(~k]Zť[7ix?o(,3؆lQZS&U"Z,I6C[rrG-E*:BB->8 $,L5wN 17LmoxT6S>,MNlap;~w$K) nw>ٶ,1`ٴʓOIJشE?⿼/GW>3=Iׄ"bύA7]#E<\--1clhJ;bQ)z>tϮ6RJf.겸 gpxЗ)/C#6S*RkqbB9gH1im+#cV~$}+h+S) p)z!U|̿v\zn2ad ۤ#ȱ$$|hO7(\D916[[.Dؤ#sJz|OF>7bn=? *# HՀ.j}j{&D+gT:=A;z)NR"}uon9HI K,"3/Zsuk#|HwY pXZ]irgwZlYjT kao2%O+m#o::Smz޳/'b2<s*PQgT-,zA? ݺ,n=;F=!X|vY8<_"K۪rTt &3憩ǥŚ rsk꨿'茚yo_6x7GUCiJKbk. vH+sGaț?2bȽpOA qNP:,RN|LgxM⚖8@~,7dċҫYኁ`=E@cpIO]x`R" `v7xҭqšQ_)YSgf~5K~ܠ*ת3!H׼]C`7 ,;o]Liy8$Q  Z&:C5P[hw~"zx{z IoQhtRRORVxKp9ꛆ }p&L-6ݞ}| ʬ)6#ab69C:X}CʦL2x7o< gXI  n2_fr#4 bwm/R,n)Knܽkʻ-/g,9WJ:#"LD׋;]\Pu.n Lz^?Hސ/\e@t\v6҂\n@6:O` ?0KiլOo*M߬p-c]u캞]s0R[JeVmvq f'"ĵ TyE L)+1<"!y|&CkQ!K(@x3RAPa&d9u|.n} usVhesb5*?M},G)e(%@L㳹\=o삾sQ#fXK/uО:w7-^F\=df$&\CvNzxW)$XD"|8S1|S4o+,o:/iG)!ө&`h;-7P JfPTqtqDkx]f$?$-?Z6FpR[z> w3{8 ьv+/I]v )!*a=fnNr)!cZG\\c>p>D &_*K )T 'qj]uZx.%,Ͳ+]TfLj;`d|&CcoDh0jRT6Vܭ i3?4åП'JM;Ah!ĴVń%~<,߽6;ѬiqCotx`ʙbt ^£0"zgC4@ BWۈmˇoЊ2&`WNtϣ V S>j5g"rS]+ٜ vru=x. z-9_wk—#Hb]ykm:lJ SD >9y)Zt: 'tdu/8h}R3ԉuЀu(Qol'0KR:4P="?[:)JhjlhΝA$.[w+ O#u !$~@>%<3@ Z+ec!Bc_pߘ-WpwkYɄ¾/QFS.Gӟ\+Q8/tYiú8̝M:p+>tWnke @Ԛ4G=EMm_H76IqM*e_?=}0׍O=5[U79(B)q3LA݂E`32j.~|[r0k,cϻKKVwpiy kۏݚH.Kճ%p"<8z 3b0(@G>온 UJAk]TMZWή(:i`.+F @Y*4Z卤Y074+_% gᏘ~#c|8@^b.zqgO{p#@qZB6vmf3o`{޼b'5ï"ɝL gB0YFv8Ws3}˕Np+p(++<,+R@Hxys%U>5&>dHY*44@2o"zQpZT bd g7BF5:Zܨо]{k&^tUZF̑+0Ru Jy?J/W,{7]_3FЙK(Kge;Ҩ7QOk-Qa~&`.Y/ {/vkc׏PxC\2ԛs<~8>.0Ƥ#ڷ%"e󴡸<:;\܊dW7Ol:hDK???1q.بQ5H n~k#1,xg3etLIŚ, u 1(IU8V[uمл1M=yߕqʺf䍵.[@A}&6NtCxΟ~Ԟum[Gj`ӇO=.q ztv>^s0}B%~[ :yާkٙs1pǭ₉KTZ3aP0 }Agk S R >x2G|.o̒U1|"U'.+eP$sA:K7h1vF8nM|Q] ]T|Ž=ᛱ,3")t>a-c"CH";v蜰~QݍUk T Fߥq J'1-Վ>m{4`dLn! uvQ{+K%I\R9'\%Ɓr0p +bӼa/ 8pK#u172Y629Mbs_a|\yJw_/{Og,~ ,A6`R\桞Wf?I-B홛hNlS_ʾ6? X&.wܙIApAy(4k yXȵ ;d, 't O N^0ZuU(7<gw^իagDΘ 0hw/:~dz[_2[NJe /Ȅnn? /P d?߷q/Ȯ<4USol1:'ѵx+5:D=񏀇7~p Ħc;n GԏP.Ivv+ R0az>de7/UvbګpPaylt%,KԚ ß607Խwf'#l:FUZ[kC9X~vs*fM Кjeb’G'>'_'PkUd{汛P_.,9r')@z;y'=W  ' +ˍ8 ^i;&K,eKT떏vG*1̌[~qfv_ʫC)5]Mw0̧ MBEpBcZ0q(e[9է&(;ِj9zTq P^TK}^M;X lߡ9'l4]&$4oi ^w 47?pR'! e <7Ē{ZC?J1cM-iAѻy1jzv)Pi'ZGNU1m p Fh 9B ft)77;Bg)v:,EɭLNغet!㪗8FMĆd{ZY7QCȹg=vwRCC_OCJx DX@oth5:ϸB^jd鼨6 $F8]= ?:wXwq$ߚ r$ iG(U(D|ׄ"#8/݌M{H,'?KG:ft !0Q9&v7oMܤ~Vſ_1n|㻠NI#hO͸1pl_l[g`@{ (ڢvado EA;o=:9ۂ*nNZJ6|%,HWSw/_ç{'4.1 &T7㑆%sOUVr5珳4+ M!Ha{Vi+ؑ . .KF{UzDr.BZ*H;RJog]dFMN)F!B<'N"Gaaӯ n=i~{Ht3pBZC #*((3GISO(˓mlnwH}BVj83?.θ"[sHGׂ#{I Jȷ)gTAqP}ؐak(;e8S:H&RgE"FxJ sBod0X@e^,dY 38r*3N2Lg(w%FXPb6 AV^>.yhlɁ OYIB]@?$k79TV:y ikhEcolq$$9#RDFջ}w4c+~w>n?եbAFBW8ؐ /3YcǩXi@SHnsR uWɪ~Y`Us{nNTW\c.`+\KdRVsj/ '1fƋlv+qq9D !60K&MgMܢ ۃeNz%'AK˝B5ElFna`qvК]fy+x#LYx*[y!+B5y;rg|<ć@m}kNdveJ@jle6S!^X.\6 _y␛#p -F9 wI&F{ٸ T(,B f@#3j87%d[{Ej09bF;OK9g/+ƥ /o<l_} O;qVu-mwaDz3/VF*zw+ޥs({1 in2oCꙁ _d?Z( KWV聱)1U8 M?1faVFFԩю8_}qtQnO@l!n[E1CU/g G̑ BT%FY i>T;'ó˿ّc/]Yo. h )%d8v: E#[PX;j1חKzûxELAYhw2NS(>1n%*\6)v3i4+"KET5QzmȖG-iR}K5P-RPoS.i^/J ji бU2 U;#r _"t4`X=zd x9-9! Ow7TIKP L_}xx[k9$PX4nY|60g2uNe̒r EwW,q4j tϒ';8~ ;qA|䟹(ޥɆ|I^AN^r} )ql"ZpkE%noy)qe-Aߎ_° f4CWYTd[2od[&}ԭ5}Tµܯxv'H }!\I#nzdp-Wf18$vcDN䞸aȍP^)O* &~äK(vnDv@Q?S-zojT rK՜ϳq@3h4V _LOGrBHJ_g\o%UY*dmS 19#^"7JM|aΌ7?BS 2:BLq{-e'CL` XSʎ?&b+xa LGNT<ܵwB+?=7і&d-7,v-gx%;vݱ[2oM%3I~S(\nv)ך $ "\OEwsLsw3k5X6ױ\/,2ޥXp:u_TwG6er"/1Z:0>.}.>OhbԶ*qvCpZ1};>l,XB^br=q,w#^膏m3qwQ=&N7BCGG-.z!K~BsEqE#R.R{kw擊d =Ŕ`V,_^igESl"^q+#>v09G@_}m2;;4E(VAu2:f&SˑK ]$/8?,vAH_JV>𴳶Ę3<@A;*d$l{uO wNDVxEE)`.K˕#Ӗ%7oeP$YVjh*XL/4l \|gWfP ;> endobj 722 0 obj [723 0 R] endobj 723 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 625 0 0 838 0 0 cm /ImagePart_2162 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 240.949 258.649 Tm 178 Tz 3 Tr /OPExtFont2 3 Tf (0.00 ) Tj 1 0 0 1 250.3 258.649 Tm 1705 Tz (\t) Tj 1 0 0 1 267.35 259.1 Tm 175 Tz (0.0e ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 3 Tf 175 Tz 3 Tr 1 0 0 1 357.85 560.299 Tm 85 Tz /OPExtFont3 7.5 Tf (4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 7.5 Tf 85 Tz 3 Tr 1 0 0 1 364.8 547.35 Tm 75 Tz /OPExtFont9 8 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 8 Tf 75 Tz 3 Tr 1 0 0 1 420.5 519.75 Tm 127 Tz /OPExtFont3 3 Tf (0 Oa ) Tj 1 0 0 1 429.35 520 Tm 1704 Tz (\t) Tj 1 0 0 1 446.399 520 Tm 139 Tz (0 06 ) Tj 1 0 0 1 455.5 520 Tm 1680 Tz (\t) Tj 1 0 0 1 472.3 519.75 Tm 139 Tz (0 00 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 3 Tf 139 Tz 3 Tr 1 0 0 1 434.649 516.399 Tm 109 Tz (oo so 01 0 00bealt ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 3 Tf 109 Tz 3 Tr 1 0 0 1 365.05 392.8 Tm 107 Tz /OPExtFont9 6.5 Tf (-' ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 107 Tz 3 Tr 1 0 0 1 365.05 382.949 Tm 35 Tz /OPExtFont9 16 Tf (oe ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 16 Tf 35 Tz 3 Tr 1 0 0 1 364.8 374.1 Tm 102 Tz /OPExtFont9 6.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 6.5 Tf 102 Tz 3 Tr 1 0 0 1 365.05 355.1 Tm 176 Tz /OPExtFont2 3 Tf (0 ) Tj 1 0 0 1 369.35 355.35 Tm 64 Tz /OPExtFont3 3 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 3 Tf 64 Tz 3 Tr 1 0 0 1 315.1 716.1 Tm 102 Tz /OPExtFont3 11 Tf (6.1 Forecasting using imperfect model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 102 Tz 3 Tr 1 0 0 1 137.5 229.35 Tm 105 Tz /OPExtFont2 11.5 Tf (Figure 6.2: Following Figure 6.1 experiment setting, compare forecast ensemble ) Tj 1 0 0 1 137.5 215.45 Tm 108 Tz (using e-ball. Direct forecasts are compared with forecasts with random adjust-) Tj 1 0 0 1 137.5 201.75 Tm 107 Tz (ment \(left\) and forecasts with analogue adjustment \(right\). The initial condition ) Tj 1 0 0 1 137.75 188.1 Tm 102 Tz (ensemble is formed by inverse noise with 64 ensemble members. For each forecast ) Tj 1 0 0 1 137.5 174.149 Tm 106 Tz (method, 2048 forecasts are made. Each row shows the comparison for different ) Tj 1 0 0 1 137.5 160.5 Tm 109 Tz (lead time. First row denotes lead time 1, second lead time 2, third lead time ) Tj 1 0 0 1 519.6 160.5 Tm 74 Tz /OPExtFont3 11 Tf (4, ) Tj 1 0 0 1 137.5 146.799 Tm 106 Tz /OPExtFont2 11.5 Tf (forth lead time 8 and fifth lead time 16. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 106 Tz 3 Tr 1 0 0 1 325.699 61.6 Tm 87 Tz (135 ) Tj ET EMC endstream endobj 724 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 725 0 obj <> stream 0 - ,,8IMMjCAo  /(o;H⦫ksrHV 遪soMρqÜR5P`g*E]mߣȢX ##l'8(ͤ9:풂W@tؖ6ѧ>E|܅>iDnwh24t_5+6McqrH)yKI`D;tDU$ T3XX& 0WQYS=3F>Hj'ƀ9%dBV4D {5ixx#G;fYr/ĠXe582g諁f^X~?Gtň5SuD S ^F"t N#:00H:o|Vn{0Z^Ƨd LGD_? ^_gB:SVV~Ìй I $xX-k07i ʻ|$Fɇ(c֕l7ceQ1v$8/MŰgǹc7lH]](ݨZm=r=b!ֳW 7x&48P'l=&L&Tm1$@YyfqN(0H,>v2Tgj0^&. dŶwuwڣEEFIT'x9Dci_z˄;:zP^+nsv_I~nؤgC4V=43@Vm=dZ8Bx5{gVX*zŝTL))q"ZSTVQ,j nDs禝;twΙP OX p/?#CqɏU$ zj 9(ڳ~"Bá\bw<8Zw3ӊo,,!6.9a5JZ}aҕ$16W;JYcчQ %(ϻ@%b$˻jEŭ3[묮W%m=Z_F[~A#UdkM ]WzLR8&6Q3 wx)߅Րնm`vdz|[ 哩@*7]$fxu_^3뗠8xN(XkP]hA9a1N>pӍ|xlO6z7G^[=CH8!-NF4huE~LD%*/7.Sz$ G$A6O1x"Wᶨ xu̥' {ޤDJ5:-ղp&yQӢ*Ƚ{uMߺS/c̑Fmf1}*smI[$QзdQBOU=GbC(ycw;tݲw 5K:kM~q@}w2Wq^\i]cs4/E?s$$=NOf֕7ޱ$3ɑIJHuYQAG[tc>ԸIA(u(r;h|cMȡD䴀Oo^5t ‹)`xװ->6K]:ʗ]vh̩7OgzzW Y9e6vt}\@h@T\Jē$ gvnDBa7CkgQb3{σ38j?sZAP'rˀ BU@9r$ko|jhZZnnZVȋr]肳8;9X;_PqΙkG] #'z>ͪӕt$::>$&"8:bi囙þյtʳϿNK$ګѱ岵,V7]̆iYFTBTrXJ*M|ầARb':e|c9;#8uq3[IEmrNTפJb|F%6ZdAD =8q<6Э&7'»>Y[4ˇ5+OWtAFNب$3L/sDh ,|.1qb,Ekv6ûP,M &Hl8:,LK:}-4.bZ`I_fr4M5\c($#=}YTKZPA'C [I=1|fˢOb4kFu&߾i3)Q8m L^ߠw!$M o)/DV $nl2t.` W9^⑤y0轢t0 -ު(9?>R6Ci u:y" j5AM(u 2VMA(6M`4Wtf:dl6\q|aE.}>ŕO|;g[涮 "%q+M7чnXA3/WlI_Q8WjDoC6/K+g1D;~&U՛=h>3]l3%{7WZ> ǎ@!qj'@ƿCD؏oā:E4k~kZa'@Ҏ-H`2e#p9n ԐH 8]jHwȄLgBء@3z<Fj9"3)h9%F#j އA[S >М9eaaXȲQD(l" 6Se 1c"xZ/&#Ա?:>ٍ\*mA1f`Ykޡy7YeSL[Σ$r+.Ho0R#i믾?̈y,%Եb֓0Њ7+*;ՑiDX 9=ܢe6O_0zXkB˘0Nk\G݅[KFm[(2pؒ #dxuӧCE.4+f)=k 0'aWdMo"$[O}ZfSB?B*bM9eMUv:䇚..O j_sRg/pofwB+IS&Qo3K2ff5ٖp8e-=d2 镊`{(X9/?i"83ײF5i72>+4j帘3,*观Y9\մwٓ9vզp?(ƪ)<`ePp/86UǢ4"i6/:NXL8ϓ0- 8:ޅެ*|xtr*8-%$0θ7ё<0_C9X8MUe/AzU:yDn*{KJZȗ:'|D_xSKCyʍ iUef:oI )?( =+0lzaiKq)lP6V7XڋHσld@F?#MR_>ܰQ/7;<"f"ԲQ"PN$oK&AV}zH\0h:4[ ! 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_3tt"2 1BVӃ]}7AuzҨDAZwm_JhTKi^\L==~`4*,#;S;uV_=ʦBFgꠕݢ 5eQR#FEO3zPyOp3M/xq)& ǜsLSxʮhB$.d"BX?C⊹ Y3@(xLAu6tŽi)ΗdhK^TB ZjX-@1/倚`!Gerκ%Ij鳜SܢrRꕉlu9>C0lc LɻB'.zX,ǟ-ke798uj U=[.mg"ż}:R8K6xE̸,,2 yXtY2JϨg)F7C-9Xn/cdzߎU-Q#z3K5EqQ`+/fLltHbPi^ea#FΏJh_uOuU,.:~ʄQd%whC7#Y(tr%gg#Mcڃ~ pԝX1Jg3bQU;xk7BSL#БtF3g,M/RbrߨYi:7h+a3ξ`,t'*AJmQ_ʋ` hjJn:)i92ˁf#T؃:/#P fe u/Uf *9 endstream endobj 726 0 obj <> endobj 727 0 obj [728 0 R] endobj 728 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 624 0 0 838 0 0 cm /ImagePart_2163 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 390.25 392.1 Tm 3 Tr /OPExtFont11 4.5 Tf (081 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 3 Tr 1 0 0 1 416.149 392.1 Tm 92 Tz (0 82 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 92 Tz 3 Tr 1 0 0 1 442.1 392.3 Tm 91 Tz (0.83 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 91 Tz 3 Tr 1 0 0 1 468.5 392.3 Tm 106 Tz (084 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 106 Tz 3 Tr 1 0 0 1 493.899 392.1 Tm 94 Tz (0.85 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 94 Tz 3 Tr 1 0 0 1 520.299 392.1 Tm 102 Tz /OPExtFont9 5.5 Tf (086 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 102 Tz 3 Tr 1 0 0 1 359.3 655.35 Tm 97 Tz /OPExtFont11 4.5 Tf (-) Tj 1 0 0 1 361.699 655.35 Tm 86 Tz (0.94 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 86 Tz 3 Tr 1 0 0 1 359.05 635.2 Tm 97 Tz (-) Tj 1 0 0 1 361.899 635.2 Tm 89 Tz (0.95 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 89 Tz 3 Tr 1 0 0 1 359.05 614.299 Tm 95 Tz (-0.96 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 95 Tz 3 Tr 1 0 0 1 359.05 594.149 Tm 97 Tz (-) Tj 1 0 0 1 361.899 594.149 Tm 87 Tz (0.97 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 87 Tz 3 Tr 1 0 0 1 359.3 573.5 Tm 93 Tz (-0.98 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 93 Tz 3 Tr 1 0 0 1 359.3 553.35 Tm (-0.99 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 93 Tz 3 Tr 1 0 0 1 393.85 545.2 Tm 81 Tz /OPExtFont9 5.5 Tf (0.7 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 81 Tz 3 Tr 1 0 0 1 418.3 545.45 Tm 84 Tz /OPExtFont11 4.5 Tf (0.71 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 84 Tz 3 Tr 1 0 0 1 470.649 545.2 Tm 82 Tz /OPExtFont9 5.5 Tf (0.73 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 82 Tz 3 Tr 1 0 0 1 496.1 545.45 Tm 91 Tz /OPExtFont11 4.5 Tf (0.74 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 91 Tz 3 Tr 1 0 0 1 522.5 545.2 Tm 80 Tz /OPExtFont9 5.5 Tf (0.75 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 80 Tz 3 Tr 1 0 0 1 361.699 511.35 Tm 83 Tz /OPExtFont3 8 Tf (ale ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 8 Tf 83 Tz 3 Tr 1 0 0 1 361.449 491.199 Tm 94 Tz /OPExtFont11 4.5 Tf (0.17 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 94 Tz 3 Tr 1 0 0 1 361.899 470.3 Tm 85 Tz /OPExtFont9 5.5 Tf (0.16 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 85 Tz 3 Tr 1 0 0 1 361.699 449.699 Tm 87 Tz (0.15 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 87 Tz 3 Tr 1 0 0 1 361.899 429.3 Tm 104 Tz /OPExtFont12 4.5 Tf (0.14 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 4.5 Tf 104 Tz 3 Tr 1 0 0 1 361.899 408.899 Tm 89 Tz /OPExtFont9 5.5 Tf (0.13 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 89 Tz 3 Tr 1 0 0 1 184.099 544.95 Tm 91 Tz /OPExtFont11 4.5 Tf (1.06 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 91 Tz 3 Tr 1 0 0 1 210.5 545.2 Tm 86 Tz (1.07 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 86 Tz 3 Tr 1 0 0 1 262.8 545.2 Tm 87 Tz (1.09 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 87 Tz 3 Tr 1 0 0 1 289.899 545.45 Tm 78 Tz (1.1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 78 Tz 3 Tr 1 0 0 1 314.649 545.2 Tm 100 Tz (111 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 3 Tr 1 0 0 1 180.949 392.1 Tm 95 Tz (-0.03 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 95 Tz 3 Tr 1 0 0 1 207.349 392.1 Tm 98 Tz (-0 02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 98 Tz 3 Tr 1 0 0 1 233.75 392.1 Tm 86 Tz /OPExtFont9 5.5 Tf (-0.01 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5.5 Tf 86 Tz 3 Tr 1 0 0 1 288.5 392.55 Tm 97 Tz /OPExtFont11 4.5 Tf (001 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 97 Tz 3 Tr 1 0 0 1 314.399 392.3 Tm 94 Tz (0.02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 94 Tz 3 Tr 1 0 0 1 314.399 715.85 Tm 107 Tz /OPExtFont3 10.5 Tf (6.1 Forecasting using imperfect model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 107 Tz 3 Tr 1 0 0 1 138 358 Tm 112 Tz /OPExtFont2 11 Tf (Figure 6.3: Following Figure 6.1 one step forecast ensemble in the state space. ) Tj 1 0 0 1 138 344.3 Tm 109 Tz (The initial condition ensemble is formed by dynamical consistent ensemble with ) Tj 1 0 0 1 138 330.149 Tm 105 Tz (64 ensemble members. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11 Tf 105 Tz 3 Tr 1 0 0 1 138 300.399 Tm 116 Tz (a dynamical consistent ensemble. The results are almost the same as seen in ) Tj 1 0 0 1 138 277.6 Tm 113 Tz (Figure 6.2 did. Comparing with Figure 6.2, the advantage of using adjustment ) Tj 1 0 0 1 138 254.799 Tm 115 Tz (at longer lead time becomes more obvious as the dynamical consistent initial ) Tj 1 0 0 1 138 232 Tm 111 Tz (conditions are more concentrated than inverse noise ensemble which makes the ) Tj 1 0 0 1 138 209.2 Tm (forecast ensemble less likely to capture the true state using a direct forecast. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11 Tf 111 Tz 3 Tr 1 0 0 1 154.55 186.649 Tm 107 Tz (Figure 6.1 and Figure 6.3 have compared the results of three forecasting meth-) Tj 1 0 0 1 138 163.85 Tm 109 Tz (ods in the state space at lead time 1. 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Observations are generated ) Tj 1 0 0 1 138.699 216.899 Tm 99 Tz (by Ikeda Map with IID uniform bounded noise ) Tj 1 0 0 1 372.699 216.7 Tm 101 Tz /OPExtFont8 12 Tf (U\(0, ) Tj 1 0 0 1 396.25 216.7 Tm 103 Tz /OPExtFont5 12.5 Tf (0.01\). The truncated Ikeda ) Tj 1 0 0 1 138.699 203 Tm 106 Tz (model is used to make forecast. The initial condition ensemble is formed by ) Tj 1 0 0 1 138.699 189.299 Tm 99 Tz (dynamical consistent ensemble with 64 ensemble members. Each row of pictures ) Tj 1 0 0 1 138.699 175.649 Tm 105 Tz (shows the comparison for different lead time. 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We now look at their forecast performance at different lead ) Tj 1 0 0 1 131.05 654.45 Tm 92 Tz (times using the ignorance score. Following section 4.3.2, we first transform the ) Tj 1 0 0 1 130.8 631.649 Tm 89 Tz (forecast ensemble into a probability distribution by standard kernel dressing, and ) Tj 1 0 0 1 130.8 608.6 Tm (we use the historical observations for a climatology which is blended with forecast ) Tj 1 0 0 1 131.05 585.299 Tm 90 Tz (distribution generated by forecast ensemble. we evaluate the final forecast proba-) Tj 1 0 0 1 130.8 562.299 Tm 91 Tz (bility distribution by the ignorance score. Figure 6.5 plots the ignorance score of ) Tj 1 0 0 1 130.55 539.5 Tm 93 Tz (three forecasting methods for different lead times. In panel \(a\) the forecasts are ) Tj 1 0 0 1 130.3 516.45 Tm (based on an inverse noise initial condition ensemble. In panel \(b\) the forecasts ) Tj 1 0 0 1 130.8 493.399 Tm (are based on a dynamical consistent initial condition ensemble. In both cases, ) Tj 1 0 0 1 130.3 470.35 Tm 94 Tz (the forecasts with random adjustment appears slightly better than direct fore-) Tj 1 0 0 1 130.3 447.1 Tm 90 Tz (cast, and forecasts with analogue adjustment outperforms the other two methods ) Tj 1 0 0 1 130.099 424.05 Tm 93 Tz (significantly. Panel \(c\) combines the panel \(a\) and panel \(b\) in order to compare ) Tj 1 0 0 1 130.099 401 Tm 95 Tz (the difference between different initial condition ensembles. From panel \(c\), it ) Tj 1 0 0 1 130.099 377.949 Tm 93 Tz (appears that using a dynamical consistent ensemble for the initial condition is ) Tj 1 0 0 1 130.3 354.899 Tm 90 Tz (only slightly better than using inverse noise ensemble for both direct forecast and ) Tj 1 0 0 1 130.099 331.649 Tm (forecasting with random adjustment, while for the forecast with analogue adjust-) Tj 1 0 0 1 130.099 308.6 Tm 91 Tz (ment a dynamical consistent ensemble initial condition can improve the forecast ) Tj 1 0 0 1 129.599 285.1 Tm (perform significantly which indicates that the information of the initial condition ) Tj 1 0 0 1 129.849 262.299 Tm 90 Tz (is well maintained. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 146.9 239 Tm 94 Tz (From Figure 6.2, 6.4 and 6.5, it seems that c-ball test and ignorance score ) Tj 1 0 0 1 129.599 215.95 Tm (do not indicate a single best approach in the case of comparing direct forecast ) Tj 1 0 0 1 129.849 192.7 Tm 91 Tz (and forecast with random adjustment, especially at short lead times. Comparing ) Tj 1 0 0 1 129.349 169.399 Tm 90 Tz (these two methods by 6-ball test, the proportion of wins of direct forecast is never ) Tj 1 0 0 1 129.349 146.35 Tm 89 Tz (smaller than that of forecast with random adjustment for any size of 6-ball at lead ) Tj 1 0 0 1 129.099 123.1 Tm 91 Tz (time 1 and 2. By comparing the ignorance score, however, forecasts with random ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 320.149 53 Tm 76 Tz (138 ) Tj ET EMC endstream endobj 739 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 740 0 obj <> stream 0 , ,,b7 (#Ȉlg䔾AЇN~|ۤM,Km՛VqZmޒQ_Sox=?sOF;ٔcަT!_d\|:ũ@*,1)2= kDGldlU䧟oD,*鳄r֞"nh:E:JX^IdHś%b)g6wߵ@% ']ǎrɴg.[8CgG9liis](g+j]\.CN\F\3DX5MXe$_Q1~ Ѷ-؞RFԂN̅svuKC]fF;䶛K%YPW禤y) O51pz$fuؤjKxͩiWw|?Eﲓ\b1]ϸ en% 2Qb$Oֳ6NӼpİYZrx~k5oˌ/M@ wss?ZyQQ\1B-ed,;0+X3q :J > #~.Z_jiB\HiA'hM4)G3B)S_bצiF.Bn*HUMZ]Nak Hx9 Jb2D=hS~@j?IoOAgoN_/0 A `Dڈ:?s}<%߿gtq6y!T z"g۴+7j%k:'Nb(R Q:/54Y3m(X_zUiLԴtSA812h0+GJ&ޝ2-h'{e1&#l>7aάs:z/*3W-$l%ĸݘ5J@ 4+##DB;$>$³袢ř?|8(2\,w2vT|! Ks(s/qKIsfA:LJ0PHCQd)5ݳh7i`QәL$tnXvքj`pxua^a[@]2-@KI#MJ 0m LӃct]wA8_j|\> " ,:5! 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6iFʒ:dm4_Zo4*X"b'/Y3m7y%jۈu%8+Bnd/eeCWU@nҷ[L5M#6Z+c3 ¥/<KY!U8 n&? 9B+4r4R C=x'i 6~'svHƭ0a!|1G,s/m\0 MHwS}1g5dV2a8f vFݕ[2r[(&)_~,Y%t2'"geR 6ȷ`eE7Md=3ǿ/+l; jGik-8x4`9cH@wWsf:ArM/j|LF%Zͤfx"i~x) Dbl lȵGLP mw7Cf8Yg"J$XR-dT ¿trP%zM& <M__^ӊB>Ƞ dF4=H Oo)cm_] Nţp~#44o89ܟ(><%ע1l~jdcnWxxFg֪K1s-YjPk5K#8A xrPʂ^e/1+<4V͊nmD xq]j0T]`O-jrxDv>ԛM w>O˳߽{NТ{{uI? a*c: P(มES釮q.7H^zS~L2QXOT;?r vQ=x(cΗ\TCQ+%#Qӑ]i4;]7&X.(6jOOK]q`: t D;B'w 97-ݟ\N^H|ETF3 :( ĉmƥ1ucyފ(; pb u#9cZPm2@ož9iR%* sg(ӯ_j&h/^sVcf~~) 'Ѫ@M.[sT8ź%XGקaTxI<_1~#pTq:6NzhӠEBP.T35ׄxUN4f* n/5 }B'ɮ,Gm4dċDF?lK(P:kC*܃dKr!qSn'fU',t#Z4A%,r5e4هYG`ͷ);[i$"7v::gTjј:opTOw3GͲ1odw|/ ([>O°Ϩd 91kUUƬ >fr.QRS9 jv㞫f~O1USH-ylT^(3@[*A-wDK*JQ@Kʑ%(?5C:U6ī#['hG=uz C7IpKd7h=ef3 %lzaeG^GV[s}rrb{.r6;ނi!֚U&5?Дb\7-wDY?ＫLv[dF$[S3hh=T'I\ﵢ В3Z WRA. fqOïx4a)Di>NDqRr55BO?hk7`+ikSvqnݎ߃ϣE>‚P/:4xwd3ĕPn ר Ԋ?$o?qY9,GAP'0epEo(J댒g9 NQt,kf%+#nIAn4Q@K梐~f`P'?G;ZjWkk\kgrc0*;3FIJ䷤B nB[՞kGq3H `; *{ɖIhSޛ MXrj endstream endobj 741 0 obj <> endobj 742 0 obj [743 0 R] endobj 743 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 487 0 0 802 0 0 cm /ImagePart_2166 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 60.5 635.7 Tm 90 Tz 3 Tr /OPExtFont11 4.5 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 90 Tz 3 Tr 1 0 0 1 119.5 550.7 Tm 96 Tz /OPExtFont9 4.5 Tf (-) Tj 1 0 0 1 135.849 550.7 Tm 103 Tz (Direct forecast ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 103 Tz 3 Tr 1 0 0 1 119.5 544 Tm 97 Tz /OPExtFont11 4.5 Tf (-) Tj 1 0 0 1 135.849 544 Tm 92 Tz (Forecast with random adjustment ) Tj 1 0 0 1 120.25 537.049 Tm 851 Tz (\t) Tj 1 0 0 1 133.699 536.799 Tm 93 Tz ( Forecast with analogue adjustment ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 93 Tz 3 Tr 1 0 0 1 57.6 527.7 Tm 112 Tz (-5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 112 Tz 3 Tr 1 0 0 1 60.5 514.5 Tm 90 Tz (9 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 90 Tz 3 Tr 1 0 0 1 327.1 551.45 Tm 1195 Tz /OPExtFont9 4.5 Tf (\t) Tj 1 0 0 1 342 551.45 Tm 152 Tz ( ) Tj 1 0 0 1 343.899 551.45 Tm 90 Tz /OPExtFont11 4.5 Tf (Direct forecast ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 90 Tz 3 Tr 1 0 0 1 327.1 544.7 Tm 1489 Tz /OPExtFont9 3 Tf (- ) Tj 1 0 0 1 343.899 544.7 Tm 90 Tz /OPExtFont11 4.5 Tf (Forecast with ) Tj 1 0 0 1 373.449 544.7 Tm 108 Tz /OPExtFont9 4.5 Tf (randan adwalment ) Tj 1 0 0 1 327.35 537.75 Tm 927 Tz /OPExtFont11 4.5 Tf (\t) Tj 1 0 0 1 343.899 537.75 Tm 90 Tz (Forecast with analogue achustrnent ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 90 Tz 3 Tr 1 0 0 1 263.5 527.7 Tm 79 Tz (-Et ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 79 Tz 3 Tr 1 0 0 1 333.35 510.149 Tm 95 Tz /OPExtFont9 4.5 Tf (10 ) Tj 1 0 0 1 338.149 510.149 Tm 2000 Tz (\t) Tj 1 0 0 1 365.75 510.399 Tm 95 Tz (15 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 95 Tz 3 Tr 1 0 0 1 338.899 504.899 Tm 105 Tz (lead lane ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 105 Tz 3 Tr 1 0 0 1 266.399 514.25 Tm 81 Tz /OPExtFont9 6.5 Tf (s) Tj 1 0 0 1 268.8 514.25 Tm 80 Tz /OPExtFont10 6.5 Tf (o ) Tj 1 0 0 1 270.5 514.25 Tm 64 Tz (\t) Tj 1 0 0 1 271.449 514.25 Tm 67 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 6.5 Tf 67 Tz 3 Tr 1 0 0 1 301.699 510.399 Tm 95 Tz /OPExtFont9 4.5 Tf (5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 95 Tz 3 Tr 1 0 0 1 397.899 510.149 Tm 88 Tz /OPExtFont11 4.5 Tf (20 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 88 Tz 3 Tr 1 0 0 1 244.55 362.55 Tm 99 Tz /OPExtFont10 8.5 Tf (-) Tj 1 0 0 1 265.699 362.55 Tm 80 Tz (Direct forecast \(Inverse Noise IC\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 8.5 Tf 80 Tz 3 Tr 1 0 0 1 245.05 350.3 Tm 99 Tz () Tj 1 0 0 1 265.699 350.3 Tm 80 Tz (Forecast with random adjustment ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 8.5 Tf 80 Tz 3 Tr 1 0 0 1 245.05 338.8 Tm 99 Tz () Tj 1 0 0 1 265.699 338.8 Tm 80 Tz (Forecast with analogue adjustment ) Tj 1 0 0 1 245.3 327.3 Tm 928 Tz (\t) Tj 1 0 0 1 263.3 327.3 Tm 81 Tz ( Direct forecast \(Dynamical consistent IC\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 8.5 Tf 81 Tz 3 Tr 1 0 0 1 244.8 314.55 Tm 99 Tz () Tj 1 0 0 1 265.699 314.8 Tm 81 Tz (Forecast with random adjustment ) Tj 1 0 0 1 265.699 303.05 Tm 80 Tz (Forecast with analogue adjustment ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 8.5 Tf 80 Tz 3 Tr 1 0 0 1 215.3 249.299 Tm 70 Tz (1n ) Tj 1 0 0 1 220.8 250 Tm 2000 Tz (\t) Tj 1 0 0 1 280.3 248.799 Tm 77 Tz (15 ) Tj 1 0 0 1 286.3 248.799 Tm 2000 Tz (\t) Tj 1 0 0 1 345.1 248.799 Tm 83 Tz (20 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 8.5 Tf 83 Tz 3 Tr 1 0 0 1 227.75 236.299 Tm 62 Tz /OPExtFont3 11.5 Tf (lead time ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11.5 Tf 62 Tz 3 Tr 1 0 0 1 151.199 248.799 Tm 74 Tz /OPExtFont10 8.5 Tf (5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 8.5 Tf 74 Tz 3 Tr 1 0 0 1 78.7 300.149 Tm 119 Tz /OPExtFont10 7.5 Tf (-7 ) Tj 1 0 0 1 85.2 300.149 Tm 51 Tz /OPExtFont10 8.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 8.5 Tf 51 Tz 3 Tr 1 0 0 1 70.299 293.699 Tm 91 Tz /OPExtFont12 7 Tf (C ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont12 7 Tf 91 Tz 3 Tr 1 0 0 1 70.299 286.95 Tm 59 Tz /OPExtFont11 9.5 Tf (rn ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 9.5 Tf 59 Tz 3 Tr 1 0 0 1 78.95 256.25 Tm 114 Tz /OPExtFont10 7.5 Tf (-9) Tj 1 0 0 1 85.7 248.799 Tm 83 Tz (0 ) Tj 1 0 0 1 88.549 248.799 Tm 51 Tz /OPExtFont10 8.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 8.5 Tf 51 Tz 3 Tr 1 0 0 1 87.099 325.85 Tm 315 Tz /OPExtFont9 3 Tf (- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 315 Tz 3 Tr 1 0 0 1 212.15 680.799 Tm 95 Tz /OPExtFont0 11 Tf (6.1 Forecasting using imperfect model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont0 11 Tf 95 Tz 3 Tr 1 0 0 1 63.1 510.399 Tm /OPExtFont9 4.5 Tf (0 ) Tj 1 0 0 1 65.5 510.399 Tm 2000 Tz (\t) Tj 1 0 0 1 95.75 510.399 Tm 95 Tz (5 ) Tj 1 0 0 1 98.15 510.399 Tm 2000 Tz (\t) Tj 1 0 0 1 127.9 510.149 Tm 91 Tz (10 ) Tj 1 0 0 1 132.5 510.399 Tm 2000 Tz (\t) Tj 1 0 0 1 160.3 510.399 Tm 91 Tz (15 ) Tj 1 0 0 1 164.9 513.75 Tm 2000 Tz (\t) Tj 1 0 0 1 192.699 510.149 Tm 88 Tz /OPExtFont11 4.5 Tf (20 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 4.5 Tf 88 Tz 3 Tr 1 0 0 1 133.449 504.899 Tm 93 Tz /OPExtFont9 4.5 Tf (Lead t!rne ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 93 Tz 3 Tr 1 0 0 1 34.549 205.35 Tm 95 Tz /OPExtFont3 11 Tf (Figure 6.5: Following Figure ) Tj 1 0 0 1 184.8 205.35 Tm 72 Tz /OPExtFont0 11 Tf (6.1 ) Tj 1 0 0 1 204.25 205.6 Tm 93 Tz /OPExtFont3 11 Tf (experiment setting, Ignorance score of three ) Tj 1 0 0 1 34.549 191.7 Tm 96 Tz (forecasting methods relative to climatology is plotted vs lead time. The error ) Tj 1 0 0 1 34.549 177.75 Tm 97 Tz (bars are 90% bootstrap re-sampling bars. In panel \(a\), the initial condition ) Tj 1 0 0 1 34.549 164.1 Tm 89 Tz (ensemble is formed by inverse noise with 64 ensemble members. In panel \(b\), the ) Tj 1 0 0 1 34.549 150.399 Tm 92 Tz (initial condition ensemble is formed by dynamical consistent ensemble with 64 ) Tj 1 0 0 1 34.549 136.7 Tm (ensemble members. Panel \(c\) is the combination of panel \(a\) and \(b\). Ignorance ) Tj 1 0 0 1 34.549 123.049 Tm 88 Tz (is calculated based upon 2048 forecasts. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 223.199 25.1 Tm 75 Tz (139 ) Tj ET EMC endstream endobj 744 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 745 0 obj <> stream 0 ,,$j۫9<@t\ѱ:ёr`0G knU-"%9 [4q 2E.˂7pTya M]&$5g 7eêu:CZMӱ?nplRH9u_{Gnƒ՚$YU;>rE$><#mQ#)]( ^D6W\b~5yvc>%>[_?BSv"a^o׻Ja"@Ȝ9ifN9).(/UՈjnbڞdDnDbxJIt-ʍF(~IZWe@!l/":{U6KR,.oBq}ĜDt6Ev `Wjh:*`& EE]b_XS9LRV<;ã5B2f =u.tKh9qzgςDP{5a*N"=f![QmsycD]r6>rJpE(QExZӎ@cƺ| ְ<:a\_:Ҭ8f8xުoS &2-m &g/[")r3 ͉BaQ]3߮5WyK4ֆtAW' /&;x_c@{V}iTұ~` fRNn[MjO23%ڰ-:Ǖ x͹ Z>XR}rޟk"7!-QgMȵ ,M)`l[NO Sľ݁ ,C׊4UqaeD0uGs+(Oa@+B\||!#xو1;V9wI W 3}- aC]tU6u;\SP_Ci5CG̍ŵ֐V@h.p'~*b=AG@- Eu!nԯvn&{Ѫ0}ٳղ!\عW,նIJϽ޸ߋ>I<mMBH[f.f`7iޫ|P?F#5>x NaJsYv@&ò^֒*w!mK_ 0Q]|R(.qKAݐE]?rTsFQQL0؆qt#)+lqyؚ (ɬZbf-QF: pm]m4nW'#>Cj9"?-zyq0}jX?k;IIT^@^3v oB҇+ I"SN]wwC%@A-!NnEq-%lp'uok3gz9vyyCl |x*/ACvb`!5%0©'e>C` ׮_RϻHYPw%ӼW|6u/>ƧRa9_dK! 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C ]:m6Y;H)oC}xgI6TIw+J̀#4#!t~I;-f.Fx;,Djƻb]5y4+K7a^ g ֞n{^[0ѨڭSuM_u[C٣X(q0%H> -+cx;ػ}umҵ8XϺԣB{c)SAɁ' fpAhGi&sI@$}̳&f J4^R©ԙEiVGC!ѹI9,sEׁP+$x?mNЫVvQA3J}X""-5WLAL:R|,єط9{( 㛜`B'ZOwޫfv`)Ҷ, 5δMkZ 62jU,uxcGRi͆򼈡i3f; KH\EEԂ2 ߀[fK K5C3~Zq0y4麲&pՋ5{XuvڅKHe}E)V(!@BI!U"QҕNW~g"(4HP̈́Q 6WfJț3ܓ> endobj 747 0 obj [748 0 R] endobj 748 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 624 0 0 837 0 0 cm /ImagePart_2167 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 314.399 718.899 Tm 103 Tz 3 Tr /OPExtFont3 11 Tf (G.1 Forecasting using imperfect model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 103 Tz 3 Tr 1 0 0 1 134.4 675.95 Tm 94 Tz (adjustment have slightly lower ignorance than direct forecasts. The reason for ) Tj 1 0 0 1 134.4 652.899 Tm 90 Tz (such inconsistent results is that there are fundamental differences between these ) Tj 1 0 0 1 134.15 629.899 Tm 89 Tz (two evaluation methods, as now explained. The e-ball method measures the prob-) Tj 1 0 0 1 134.4 606.85 Tm 91 Tz (ability mass that stays inside different sizes of e-balls and counts the proportion ) Tj 1 0 0 1 134.15 583.799 Tm 94 Tz (of times one method beats the other \(on a tie, both win\). For each forecast, an ) Tj 1 0 0 1 134.4 561 Tm 90 Tz (&ball test only counts which method wins regardless of how significantly the win ) Tj 1 0 0 1 134.15 537.95 Tm 91 Tz (is, which means e-ball treats an overwhelming win and slight win the same. The ) Tj 1 0 0 1 133.9 514.7 Tm 90 Tz (empirical ignorance score discussed in section 4.3.3, on the other hand, averages ) Tj 1 0 0 1 133.9 491.649 Tm 94 Tz (the ignorance of each forecast, i.e how much one method wins in one forecast ) Tj 1 0 0 1 133.9 468.6 Tm (matters. In our experiments, although direct forecasts have a lager proportion ) Tj 1 0 0 1 133.699 445.55 Tm 96 Tz (of wins, it loses a lot when it loses to forecast with random adjustment. This ) Tj 1 0 0 1 133.9 422.3 Tm 92 Tz (can also be seen from Figure 6.1 and Figure 6.3, where the model error is large, ) Tj 1 0 0 1 134.15 399.25 Tm 91 Tz (direct forecast miss the target \(true state\) completely while forecast with random ) Tj 1 0 0 1 133.9 376.199 Tm 90 Tz (adjustment may produce some ensemble members are close to the true state; for ) Tj 1 0 0 1 133.699 352.699 Tm (this kind of forecast, ignorance score punishes direct forecast heavily. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 133.9 308.049 Tm 110 Tz /OPExtFont3 13 Tf (6.1.4 Forecast with imperfection error adjustment ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 110 Tz 3 Tr 1 0 0 1 133.9 277.549 Tm 92 Tz /OPExtFont3 11 Tf (It is clear that forecasts with adjustment using the model error can improve the ) Tj 1 0 0 1 133.699 254.299 Tm (forecast performance compared to direct forecast. Unfortunately, identifying the ) Tj 1 0 0 1 133.699 231.25 Tm 89 Tz (actual model error is not achievable except the noise free case. With observational ) Tj 1 0 0 1 133.699 207.95 Tm 92 Tz (noise, model error cannot be precisely determined \(see section :5.2.2\). One can, ) Tj 1 0 0 1 133.699 184.7 Tm 90 Tz (however, estimate the model error and use the estimates to improve the forecast. ) Tj 1 0 0 1 133.699 161.649 Tm 95 Tz (The ) Tj 1 0 0 1 156.699 161.649 Tm 116 Tz /OPExtFont8 12.5 Tf (ISCDc ) Tj 1 0 0 1 196.099 161.899 Tm 91 Tz /OPExtFont3 11 Tf (method we introduced in section 5.2.5 is not only a state estimation ) Tj 1 0 0 1 133.449 138.6 Tm 92 Tz (method, it also provides estimates of model error, i.e. the imperfection error \(see ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 324.5 51.5 Tm 75 Tz (140 ) Tj ET EMC endstream endobj 749 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 750 0 obj <> stream 0 + ,,>7ٕ<:ՁK9, c# H0 obmGT,:"I݈(mWv+n VdH1`Bz|%G+;WSuE[>&r$s߮\|%; vO(@xC37a4O8ZoR옴Dz[]P͸3߶Sf-h2RHJTS+޽SA~j 6H[? jC %`șśLe nA#LtB]V+E18 8eSd 2]!X8L`|H&-ڲv7S2Pn !<8}?0C.G#)϶żSmbEw,$9_Lbfr">_26֦FZt&߳ǬhrIYb7. 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In this section we consider using the imperfection error to adjust ) Tj 1 0 0 1 132.25 654.2 Tm 90 Tz (the forecast and compare with direct forecasts. The experiment results discussed ) Tj 1 0 0 1 132.25 631.149 Tm 91 Tz (below are based on initial condition ensemble obtained by inverse noise ) Tj 1 0 0 1 488.899 631.149 Tm 18 Tz (1.) Tj 1 0 0 1 492.699 631.149 Tm 34 Tz (. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 34 Tz 3 Tr 1 0 0 1 149.3 608.1 Tm 92 Tz (Figure 6.6 shows six examples of the one step forecast ensemble in the state ) Tj 1 0 0 1 132.25 585.1 Tm 91 Tz (space, the initial condition ensemble is formed by inverse noise. Instead of using ) Tj 1 0 0 1 132.25 562.299 Tm 95 Tz (the actual model error we use the model imperfection error obtained from the ) Tj 1 0 0 1 132.25 539 Tm 94 Tz /OPExtFont8 13 Tf (ISGDC ) Tj 1 0 0 1 172.8 539.25 Tm 95 Tz /OPExtFont3 11 Tf (method to adjust the forecast. In all examples, forecasts with random ) Tj 1 0 0 1 132.5 515.95 Tm 94 Tz (adjustment still produce ensemble members with too much spread. Similar to ) Tj 1 0 0 1 132.25 492.899 Tm 90 Tz (Figure 6.1, the first 4 panels present the four cases where the model error is very ) Tj 1 0 0 1 132 469.899 Tm 89 Tz (small, small, moderate and large. Forecasts with analogue adjustment outperform ) Tj 1 0 0 1 132.25 446.85 Tm 90 Tz (the direct forecast as long as the model error is not negligible. Panel \(e\) shows an ) Tj 1 0 0 1 132 423.8 Tm 89 Tz (example where the model error is small but the forecast with analogue adjustment ) Tj 1 0 0 1 132 400.5 Tm 95 Tz (ensemble is unable to capture the true state. This failure occurs because the ) Tj 1 0 0 1 132 377.5 Tm 97 Tz (model error in this case is overestimated by the imperfection error. Panel \(f\) ) Tj 1 0 0 1 131.75 354.449 Tm 94 Tz (shows an opposite example where forecasts with analogue adjustment did not ) Tj 1 0 0 1 132 331.149 Tm 91 Tz (capture the true state because the model error in this case is underestimated by ) Tj 1 0 0 1 131.75 308.1 Tm (the imperfection error. Figure 6.7 shows the comparison between three methods ) Tj 1 0 0 1 131.75 285.1 Tm (via &ball method \(see section 3.7\) at different lead time, the forecast adjustment ) Tj 1 0 0 1 131.75 261.799 Tm 96 Tz (is obtained from the imperfection error instead of model error. Similar to the ) Tj 1 0 0 1 132 238.5 Tm 94 Tz (adjustment using model error, the forecasts with analogue adjustment almost ) Tj 1 0 0 1 132 215.5 Tm 91 Tz (always outperform the direct forecast for different lead time and different sizes of ) Tj 1 0 0 1 132 192.2 Tm 93 Tz (c-ball. Direct forecasting outperforms the forecasts with random adjustment at ) Tj 1 0 0 1 131.5 169.399 Tm 91 Tz (short lead times while underperforms at longer lead times. As the result of using ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 148.099 148.5 Tm 99 Tz /OPExtFont5 11 Tf ('dynamical consistent ensemble is not employed this time since if method can improve the ) Tj 1 0 0 1 131.75 137 Tm (forecast based on inverse noise it is expected to do so for other initial condition ensemble and ) Tj 1 0 0 1 131.5 125.25 Tm (it might be too costly to form an dynamical consistent ensemble in practice ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 11 Tf 99 Tz 3 Tr 1 0 0 1 322.3 53 Tm 98 Tz (141 ) Tj ET EMC endstream endobj 754 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 755 0 obj <> stream 0 ) ,,ٔjut8n^$k0ڈAVB_9ޏˉK"9ԕHWO7GUn!S ơ6x&sN\jq L8 ޤG$Hs,1cMQa +a(eYQ9)AW,ab@~hG $Q%释w c'__6Mw(ݥPkoMt|nAݿx㰕.wF#LDӢMc,V'k^F^}j.] ;/[ 2&r/L5.=ѦwyoP|IxsL9 }rgÚ<ޱpt8c)hp}X4u7XU m { -~(Q+6VX@r4?)X]P|}.L5nrJ_9~w儡P@۶SQ x% uf75! $AU#"3R}!_OV:,/PpWP#5R8p g8*K8̝Xo:swI=ƍ@?ߥcۅ^ߛAK^2ii]EoDr2 qItKD+S5kSw̤f5voVOM32_qa~P-!K6h&mݰX ͠ S'f> W#'࿾? !M#0W|%I' ;_ڸJ馲1(GO}Zv6Lsvα*ogT:6..6[ƖXtu[ȭkD㸑w7̗Y(0_ዤs~5P IWk蘽;yۢI̜Mϲ/b@%*iǻMxɡWyhŞ+!ʼl8ReORhi=oU0KbRܩA5eUIx@Q˘#OAT Zxo| !S!APxEIJ7s&ᄼcD1W?rOư]U.pg;;FHI6f C53G?漙*/\Ӛ7n W7^q\x M@0SS8.apA.O`C>; }{v@-GIl!(z,fIp|Tk+d9ɚ=]jNg24ma#A6bRB!ze^fj)xcdy|  vc5f꺖JC͵ .Q [Pwjbc}|LWn^"`Uu'CGoиye.|g:ɷ>{r!3.kɩy>ya/G{|L`H{ف 4pj` &!tﲍ3LX͆SD^LMRqOo}Ip^ d E~qc6`ߪNmpt g2 !9!ݪtR}rOV-*ۼ$qvtؚP@o0yW܇0>eܑѝ2hꨜo*Kr.ԑX UcU? .ag×bgE9eUB)~btt>ꢉU&K]ԧrخpO&1L =o=? GC2; ֏G{,ƸUK)چW6 VԂTǺu]dꠖO(suWnuLelv#?X=ܸ,T-d~MZkNNƩ:%) 0X:GqOTZhVT9(d=Jmig6.s\Hl0 oaZ_>l~ ;!!uiɋR n{\M T[bzH4Ęi HTU ӥ<͞ li93<Ͼw",U'2> ˪b:oǩjD,k mN:RU(Kxn' \{:zA H~F&.+Ry秌edJK==G ₇~ڪNc8HóEm _2b_A.4׷νo*ד9;!!=FDcz:2LH&f:н$O,G|fԤޠAajrÞT%5xz[_bE#`.R/H=KFjBJ&"7%Jl}8 h̳j-'ZI>迂q g9*!tJQM[0i`]tjJrvX,* xtńl!=9_Qnkh'3yLg_f,Y-eey*<;prΗMsPwv箪%M]v$pڦ*6nynK>(0ki-:6 \LRGrfcO0~WԶ{GM}HSfC*U1dU'S>M b4\my<*0 OMN]"\u>:dwM*0Sذ{|8GULЬǠc +7_3cB!~?CjXk9·7}ZR=&9?[g8FLc:,,+'2)0u Jqe2?x g#ʽNk>D4[e-Icqr(ڈ(p:$YG )lq@aKbGM -wjDn =(?3i]St tT$~v(2S̊J@|! t3^єq=,Q"rOw\h;HZ@u@n+_g`Xf J0BT<O FiĎRagDb}wd|Xsxh6F6cJ|a39\=[*Sc~p}=7}% h=("Wկ`8/V zn(#^:GhΈQ=D}5EꀓsYԈ tv}Pb[ ` F3^TP ^[66?c9I4$3Í~>ݴ.xů)>0V^+̘-P9 Ma1! @2Uz\zIuͪjv?YsF#cJ ??S_{ƺWD`_"ۖ6 >A*~O`HTIuGG_ zIY }`ئT@Rh{*ǐ~^(⿋c;~F [d^Zy2y0MD$BH8\'}rwCLlbXDTɍmaN %vm} y]EWB[L|8(|\`\?DbZخ@E_ [2|XF&_Fn]tq89N_Swv猫pnRt8TG}CΓ; o`tMK6Ќſ? -=fkDs7Q,;4rJᗩ /.9qÿ)u*T ǿq^72θOwrN rЉW5(1TI7AofAa dYߏitX: )sȜaǕ]h^w;~}k]1o+)sIyr] Y[cn^7LBk8M:Iz]$;K*x 6ͱP|ZZUՠޚ&VƖB#BDX' Hs36se`cm~+.?- `^]9q5([9\bݹa?`(+;.ob)Q~|s$oa0 |ۊ)FTp;hGJqdt: kA*FahwKyLk$~\Z5'TYFjqjqmV|j&LAOP2e" iz3L'3xQ~a6&Ɔ2 t2QjHR=$+r&uڈ q|ХIVa E#-U =-՛>hɀkI@"H2;HϙrXћ\!}V_Yk !U m_+_m]8hjy  ) *Ɋ80ό6 ~-ѐ:4e nv'5?ϻQ_Rk "@hAQMЌk+?5ZpS6|߉: &w([2 G[#S7؜kNP'2mS7kbEÓ8Ysm,fOJ1"d#qF0g-ƴn:D,8+ fp̟vFF(=M\<ԝ\?9'}ZܷCS[= wg4",|Ϸ~{k*W<(WI[ Zdb˘K3Jh'$GeYmL_Mb\r/ᏐPTh@BG T,sFPT֮e$I+Iˏ>at!k5ylnƘ!UNg%Cg?i6#F؎lѥ$[K)v(&QZ5P+FQ 9F鬗}FB{P/ѠJՃiȐzA䮥HZdNN-!:~b:QF&zoOc霐T٨I ty \ܕ˖;HU~w6 9:JGGFUKY%#)?y'Wy!mO\5^kZe%%iA"=t)d a[ >}QqnVF|٘BiGO{|Xz+iWXZr8䮰v䪆WL7]J2143#"~NK3Vi lbӊ/1v}]SZ$*wi,޻wPU[<>Irr@ s`\n$%v5jv_5}=:z/<Bf(i'b,]$f Bm'jPoYSz͉>d(l6k s ^ܐ^gJfJ sseQn Oc\Fo?* kg/Mffw)g+7bHPW`3L.9Da0FrY"7v6 m37N ,L^D4=R@E`Ҡh /ɏvɥ2+SVQe,Wh!Sb \G]0NʟlԨ@Uq j=2ře^_ns?RzxeV jE'Lҭ" +6mbV&–mB UI #"0U3;9Ld+w]hJB/X%%hCrDڔή1eo[9C|LPp] Ë:KiSF<`"1MQٚn?r8F$G*akPp4EƮ/Wc "r)6Tƭ? Ѡb՗־Q9fp_A>/4)Q8m)8mbOd'' V&\/1k+MpC%޺Y?GXluM\@/E!G!5-jKɣD5!۾c1 GٓT4*\w{ьWkn(xtN 'ӫ/ :Iˏ.%U҅^b.}dEC]G($KJ;:j"kaj ̖Z{ QSvc+ X3DVL-۰FsA཈#1:'j_`œ)Ι"6j͉N 7'o%T|f%pSThc)7:0IW!!BWMk-3F^vhEPΗ@1Jǿ>"VmqB_xᵄh@}A䲼`HwΝmuAr# PeuH 4f91u6׿fץ^t7$b<(Q?y!B2z~aN)8<5$^ p/A}FNbz-{{+xi֔Lonr;4f>@ A\֮#R֒kE_mHDOP0H9XHuHl7Qo״f(k|GM#}> Rl1'تz%aソ#-tu?[iwB^ 3wB*gHMQ%,eKr$j4Kw.xr1k[ba}}+K}󎎆3 ł~HI$7C,gBh֕[$|j7FxA"wdthVr{ +O8q7b1]Yb6 n.1Y4ui]hNj?TƐ[>.y9CA5^ԓϣLFG7fq1qgƪPu5)6s1JuINR%QGnU'6V0ϕ Ļr_W1ɋg"v15$g37(^B DE2N[Z&՚tńoxTN!{5,=wh> endobj 757 0 obj [758 0 R] endobj 758 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 497 0 0 796 0 0 cm /ImagePart_2169 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 65.299 644.299 Tm 136 Tz 3 Tr /OPExtFont10 4.5 Tf (0 29 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 136 Tz 3 Tr 1 0 0 1 65.049 622.5 Tm 129 Tz (C 26 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 129 Tz 3 Tr 1 0 0 1 65.049 600.899 Tm 139 Tz (0 27 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 139 Tz 3 Tr 1 0 0 1 65.049 579.299 Tm 126 Tz (C 26 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 126 Tz 3 Tr 1 0 0 1 65.049 556.95 Tm 137 Tz (0 25 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 137 Tz 3 Tr 1 0 0 1 64.799 535.6 Tm 140 Tz (0 24 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 140 Tz 3 Tr 1 0 0 1 283.449 634.95 Tm 133 Tz (0 02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 133 Tz 3 Tr 1 0 0 1 283.199 613.35 Tm 144 Tz (001 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 144 Tz 3 Tr 1 0 0 1 280.3 569.7 Tm 155 Tz (-CCI ) Tj 1 0 0 1 279.85 548.1 Tm 157 Tz (-0 02 ) Tj 1 0 0 1 280.3 526 Tm 172 Tz (-003 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 172 Tz 3 Tr 1 0 0 1 64.299 309.75 Tm 121 Tz /OPExtFont1 5 Tf (037 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 121 Tz 3 Tr 1 0 0 1 282.949 314.55 Tm 124 Tz (025 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 124 Tz 3 Tr 1 0 0 1 282.949 292.5 Tm 103 Tz (0 24 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 103 Tz 3 Tr 1 0 0 1 64.299 287.699 Tm (0 36 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 103 Tz 3 Tr 1 0 0 1 282.949 270.649 Tm 121 Tz (023 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 121 Tz 3 Tr 1 0 0 1 58.1 265.85 Tm 106 Tz (,,, 0 35 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 106 Tz 3 Tr 1 0 0 1 282.949 248.799 Tm 103 Tz (0 22 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 103 Tz 3 Tr 1 0 0 1 64.099 244.25 Tm 120 Tz (034 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 120 Tz 3 Tr 1 0 0 1 282.949 226.95 Tm 112 Tz (021 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 112 Tz 3 Tr 1 0 0 1 64.099 222.399 Tm 103 Tz (0 33 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 103 Tz 3 Tr 1 0 0 1 286.1 205.35 Tm 129 Tz (02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 129 Tz 3 Tr 1 0 0 1 64.099 200.549 Tm 103 Tz (0 32 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 103 Tz 3 Tr 1 0 0 1 222.25 698.299 Tm 95 Tz /OPExtFont0 11 Tf (6.1 Forecasting using imperfect model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont0 11 Tf 95 Tz 3 Tr 1 0 0 1 87.099 519.5 Tm 118 Tz /OPExtFont1 5 Tf (005 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 118 Tz 3 Tr 1 0 0 1 114.95 519.299 Tm 87 Tz /OPExtFont11 6.5 Tf (aos ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6.5 Tf 87 Tz 3 Tr 1 0 0 1 142.8 519.299 Tm 101 Tz /OPExtFont9 5 Tf (0.07 ) Tj 1 0 0 1 152.65 519.299 Tm 1299 Tz (\t) Tj 1 0 0 1 170.65 519.299 Tm 115 Tz /OPExtFont1 5 Tf (008 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 115 Tz 3 Tr 1 0 0 1 198.25 519.299 Tm 118 Tz (009 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 118 Tz 3 Tr 1 0 0 1 227.05 519.299 Tm 89 Tz (0 1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 89 Tz 3 Tr 1 0 0 1 315.85 519.049 Tm 146 Tz /OPExtFont10 4.5 Tf (13 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 146 Tz 3 Tr 1 0 0 1 342 519.049 Tm 86 Tz /OPExtFont1 5 Tf (1.31 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 86 Tz 3 Tr 1 0 0 1 369.6 518.799 Tm 96 Tz (1 32 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 96 Tz 3 Tr 1 0 0 1 396.949 518.799 Tm 115 Tz (133 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 115 Tz 3 Tr 1 0 0 1 424.3 518.799 Tm (134 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 115 Tz 3 Tr 1 0 0 1 451.899 518.549 Tm 118 Tz (135 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 118 Tz 3 Tr 1 0 0 1 280.1 490.699 Tm 98 Tz /OPExtFont10 4.5 Tf (-) Tj 1 0 0 1 283.199 490.699 Tm 137 Tz (0 05 ) Tj 1 0 0 1 279.85 468.899 Tm 118 Tz /OPExtFont1 5 Tf (-0 06 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 118 Tz 3 Tr 1 0 0 1 279.85 447.05 Tm 99 Tz (-) Tj 1 0 0 1 283.199 447.05 Tm 101 Tz (0 07 ) Tj 1 0 0 1 273.85 425.449 Tm 117 Tz (." -0 08 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 117 Tz 3 Tr 1 0 0 1 280.1 403.35 Tm 99 Tz (-) Tj 1 0 0 1 283.199 403.35 Tm 103 Tz (0 09 ) Tj 1 0 0 1 282.949 382 Tm 108 Tz (-0 1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 108 Tz 3 Tr 1 0 0 1 64.549 476.55 Tm 175 Tz /OPExtFont10 4.5 Tf (-0 ) Tj 1 0 0 1 72 476.3 Tm 95 Tz /OPExtFont1 5 Tf (4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 95 Tz 3 Tr 1 0 0 1 61.45 454.5 Tm 98 Tz /OPExtFont10 4.5 Tf (-) Tj 1 0 0 1 64.799 454.5 Tm 123 Tz (0 41 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 123 Tz 3 Tr 1 0 0 1 61.45 432.899 Tm 98 Tz (-) Tj 1 0 0 1 64.549 432.899 Tm 140 Tz (0 42 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 140 Tz 3 Tr 1 0 0 1 61.45 411.05 Tm 98 Tz (-) Tj 1 0 0 1 64.549 411.05 Tm 140 Tz (0 43 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 140 Tz 3 Tr 1 0 0 1 61.45 389.199 Tm 98 Tz (-) Tj 1 0 0 1 64.549 389.449 Tm 160 Tz (044 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 160 Tz 3 Tr 1 0 0 1 61.45 367.6 Tm 98 Tz (-) Tj 1 0 0 1 64.549 367.6 Tm 137 Tz (0 45 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 137 Tz 3 Tr 1 0 0 1 90.7 357.3 Tm 127 Tz /OPExtFont1 5 Tf (-003 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 127 Tz 3 Tr 1 0 0 1 118.549 357.05 Tm 111 Tz (-0.02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 111 Tz 3 Tr 1 0 0 1 145.699 357.3 Tm 107 Tz (-0 01) Tj 1 0 0 1 157.9 357.3 Tm 96 Tz /OPExtFont25 5 Tf (x ) Tj 1 0 0 1 160.099 357.3 Tm 87 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont25 5 Tf 87 Tz 3 Tr 1 0 0 1 180.25 357.05 Tm 102 Tz /OPExtFont9 5 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 5 Tf 102 Tz 3 Tr 1 0 0 1 204.5 357.05 Tm 147 Tz /OPExtFont10 4.5 Tf (061 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 147 Tz 3 Tr 1 0 0 1 232.099 357.05 Tm 100 Tz /OPExtFont1 5 Tf (0.02 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 3 Tr 1 0 0 1 302.649 357.05 Tm 126 Tz /OPExtFont10 4.5 Tf (1 36 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 126 Tz 3 Tr 1 0 0 1 330.25 357.05 Tm 130 Tz (1 09 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 130 Tz 3 Tr 1 0 0 1 413.05 356.8 Tm 98 Tz /OPExtFont1 5 Tf (1 12 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 98 Tz 3 Tr 1 0 0 1 440.649 356.8 Tm 93 Tz (1 13 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 93 Tz 3 Tr 1 0 0 1 90.95 194.549 Tm 96 Tz (1.34 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 96 Tz 3 Tr 1 0 0 1 118.799 194.799 Tm 133 Tz /OPExtFont10 4.5 Tf (1 35 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 133 Tz 3 Tr 1 0 0 1 146.9 194.799 Tm 130 Tz (1 36 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 130 Tz 3 Tr 1 0 0 1 174.699 194.549 Tm (1 37 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 130 Tz 3 Tr 1 0 0 1 202.55 194.549 Tm 127 Tz (1 30 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 127 Tz 3 Tr 1 0 0 1 230.15 194.299 Tm 98 Tz /OPExtFont1 5 Tf (1 39 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 98 Tz 3 Tr 1 0 0 1 303.35 194.299 Tm 151 Tz /OPExtFont10 4.5 Tf (116 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 151 Tz 3 Tr 1 0 0 1 330.949 194.549 Tm 130 Tz (1 17 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 4.5 Tf 130 Tz 3 Tr 1 0 0 1 358.55 194.299 Tm 70 Tz (1 ) Tj 1 0 0 1 362.649 194.1 Tm 216 Tz /OPExtFont1 5 Tf (181 19 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 216 Tz 3 Tr 1 0 0 1 376.55 194.299 Tm 85 Tz /OPExtFont25 5.5 Tf (x ) Tj 1 0 0 1 378.699 194.1 Tm 747 Tz /OPExtFont25 5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont25 5 Tf 747 Tz 3 Tr 1 0 0 1 415.449 194.1 Tm 96 Tz /OPExtFont1 5 Tf (1.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 96 Tz 3 Tr 1 0 0 1 441.85 194.1 Tm 83 Tz (1.21 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 5 Tf 83 Tz 3 Tr 1 0 0 1 44.399 159.299 Tm 91 Tz /OPExtFont3 11 Tf (Figure 6.6: Following Figure ) Tj 1 0 0 1 187.449 159.049 Tm 74 Tz /OPExtFont0 11 Tf (6.1, ) Tj 1 0 0 1 208.8 158.799 Tm 90 Tz /OPExtFont3 11 Tf (six 1-step forecast examples are plotted in the ) Tj 1 0 0 1 44.149 145.35 Tm 91 Tz (state space. Here the adjustment is obtained from imperfection error instead of ) Tj 1 0 0 1 44.149 131.7 Tm 88 Tz (model error. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 231.599 45.75 Tm 70 Tz /OPExtFont0 11 Tf (142 ) Tj ET EMC endstream endobj 759 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 760 0 obj <> /FirstChar 0 /FontDescriptor 761 0 R /LastChar 130 /Subtype/TrueType /ToUnicode 762 0 R /Type/Font /Widths[0 0 0 0 0 0 0 0 0 228 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 228 272 389 0 0 0 0 194 272 272 318 0 228 272 228 228 456 456 456 456 456 456 456 456 456 456 272 272 0 0 0 500 0 591 591 591 591 546 500 638 591 228 456 591 500 683 591 638 546 638 591 546 500 591 546 773 546 546 500 272 0 272 0 0 272 456 500 456 500 456 272 500 500 228 228 456 228 729 500 500 500 500 318 456 272 500 456 638 456 456 410 318 0 318 0 0 272 287 456]>> endobj 761 0 obj <> endobj 762 0 obj <> stream /CIDInit /ProcSet findresource begin 12 dict begin begincmap /CIDSystemInfo << /Registry (ArialNarrow-BoldOPExtFont25) /Ordering (UCS) /Supplement 0 >> def /CMapName /ArialNarrow-BoldOPExtFont25 def /CMapType 2 def 1 begincodespacerange <00> endcodespacerange 14 beginbfrange <09> <09> <0009> <0A> <0A> <000D> <0D> <0D> <000D> <20> <22> <0020> <27> <2A> <0027> <2C> <3B> <002C> <3F> <3F> <003F> <41> <5B> <0041> <5D> <5D> <005D> <60> <7B> <0060> <7D> <7D> <007D> <80> <80> <203A> <81> <81> <2022> <82> <82> <00A7> endbfrange endcmap CMapName currentdict /CMap defineresource pop end end endstream endobj 763 0 obj <> stream 0 ,,j٭ ncwl7uI r\rΗj"/(]*MϳwB"=|f>m+e) 0$E2F22Ԑ!G={*~t$0B{on(2]@"CO%TFoHLp^Oe aj±햃 ІV"fɼ8D}/.b+Y 5DIc4;ďpBsؙ {sx]9sv"#킢csFקhoҡKҬ*vg}w Hm':%ٸw@4B)rFHX'C Z5ƷKP{;4 yz,i&FjgP MHtudM2Ll,6WtfQ[zA.9[PzL]wэW&~`asQvNN5rSƊ- /rEn=MXKBDƝ[{ siBmL}c32^. t!#5l^$ښ@]}L_]"to \)Kٟۓ"m&OyݨDBhDHV=/,VR;g$႖Im:㷭T:6 E0ET&> * njnס4,i.L0GȎZ/HtV: } W:C ˹||ԛʖdh~ڎ R/EX.&@eХ Xw a|I9K~&B P$dENc#4ܠa5D~rO_//z*(K=oH ~ "Zra11$ewACcRҌ?cQzY'Jƿ⬜.qp7C=:+WN"h9@kEBȡb bLЈyfJf7rx}ͯI(hVpjQeU+ߗgQ ^:oZRuU! Oƽ줙U@[K)l|(tʨh +GAN?H~Odޘp=$M0H݊1krx5BB}_ZQo81'4hwN8+`&W]z0?[M=TG2(׭=p97N!wWŚ_9 DܺsբEq0'!؅ΤL6}K}H #Rd-0Md.H>YDZoπqr-wQ_a9@uMOxG6TNbI^چu\YeZcLJSB@ab1i`뗈/!OW.AQCީhw_ GZ|d)[ ^VYHk&m$!hBWN9Y%S &X) ԫ<14)ÍJl-FQpAmJZ Ŵ1"j`2.~8\g7eapf¢)]vD87:aZ K\jr.cKu6<*E±'h6ti6@d]ZkSXz pX2(P&<ԍm_N5h XWLڧO)&QB̶}3j]QTqrFe`xe\q(u')u8PLPr׷_Ԩ;]Wb +ѰJlxq҄cgm 7f s)oH[%ID[-= \V&^vX?(-3\7,Ծ1ѫ {!|s4&b'./]?~y=QW|uE'ndpGrrUMG 4ξd@ +`ɜ$ջ'=Ȃ8HX̆@F NR [1<(oEmrSlXԳߩEO#oŲy\(,v޿ۡN} ϛpo?n5JBoUtitеPLj/+A!S}d~,jԥ{NycַJ`&} LA\zG9Ύ&`W!A;>$sd;4ڳ02;CkzŧtgT݅!:C LHH_ tPsa8`5" Ff@N4߸.`q]ne2;a! 5ͥ3= N |䬚H' Z@ٸiGJRm?AQ,g'ֶ= :a~i#ZGTxt@Iyܤ],i`ZPk AUNwC+؏^/`>"nfb ~3̼jU{]Jڷ~T]:mzO ςFw-}.j@v!W\Ty̭ g%B妜,(N{IU+o(o:ӝNŒ]7+^4.ssE eЃd GXBq>3r6ꁆt?67>Ӥݲƭ#pܦ0=c64-DE 6 ڎ9ճȥGװr ^ԽތiLc]} #A ݞ(ÇDt_n1PC!%U6'鞥MFpNEq `ߦKS&׉:J S[k v{ J}} }Sa~h2 !Z"h1!}~{mfǜ'OsPT]b`,?Ny @ܘWFLkqS\%A[]] N&5nY ?c2yYT7Nk9Lpm (O=@|M'hթ7d OdT3mȌ\aAswKl`uç) `3_   ()[kK43 z6bbu4 Yİ_dzq8pIPbOVհC3,ŹTlK),qQFVrKI1zL*kXLeѦ #T6?ëZ^#1Qgl~,ÑGYO") |c?Z*2*vȓCX xwe:ԯlg`1κGe$9:qT=!Vw-tNѫlSYKF2'bQ욅 -&i `ׄcvo 3"F[╟% U h>M a#'^z֢v,d?M:ImKW,i x"Oa0wPJs9rmiٛSM6PR0 uF't7P=Kg.p=Rl&@ޫ`<Lݴ7 4'N0CYŸ< UZ+(d6PՕFqyE7] v xZYմXxHv}i?|\Pa52Ix,&a7ƱjϘר)uBdz5 ߍO:౛i᫫8Ozd|zWCB4K~H՟ߐ0v18 ;FӷB[u١e9닗/&XXa6r=88dl 3pUl*@ٹU_)!j@=b&hR~^/TzMkunoⷲZbωDe|&62O{v Jw "s-2Gd#q=35#[?~}y#t!c; k`Wn[ .*2&&"XF%jUNP)i9\Hy1w /z^-[% ~6rZ5Ng?UH>`Ugiduݷ %VF-?3V!U^=@-`CN*2,gq쫱b̠E e95L.{w,\41ChHn"<9~S1-*q>5B*GҧHqFAN"ib|nqdnB3LB`l w%u"F\nnY=LvsZ;3^ztf2)d0U3]kzTGw tk05xo ~N-nj`j_ƌr|HyYR=ZVod!“Qetp<+a[ <M Hϔ96ـ*`x]OSZ91鲟\D ?.͵'Hc%p#qGTrqIpC3] #U7yr@,?yRʱ_Irh}׫F ouB/! jס`'DbV<:ӟ/ZDrV(#.|o+ Oܕ皽e(`<VA@S9V/zwNwM h]4%3MRF9 &m{]ZJsfnj9 {{@%aqѪ`9F2FnM!/*kIY; %*A4f .|y$ JWo<o_> ҙJeo'Җq*ҳ~( q@Չ}jczפP[{QH, lKal|WVNBAV"L :U3uwO?UXI  $h21Ž Tڮ-ܧbI!v]ҍZ ,q;cydҰ~x_S , اՀg`KFÞ4憜wuwS^9RՃam;,Egb>&?I|$)&bᱷsŵ/$Dz':hd#=%8ը&b,*:đH,d'9󘢰Ԃ5+`^(G7i+>51؊ִڿaȓɀ'<)Eɧ\Љc}_jr1ݘCȯ[=+b\Y?HM5.yi@{' bT F bu4Q3{n4YM-dth+ڑtq.Gpot@G8 RM;AlE`@#PSQv^1H7WY0' 6iU|vT|Aʔ.  % <O!hO?Gm2p>QaOQJqC S~+I/}P U{r8Z '« Q=~#+Pd @QJHMp rW·3!zdX.N+z?M@K_ΫEޱ`{JA3F9$ `cS?ܻdJ=њn.5g7ǫmDDO`)Q&Z{ӧkulI˧ix׬Y%4AD/Y͢ )"d{ρN"G}/L׃?vxf) ? эȷ'!e בË 9 =P83g0̴X3ρa [1*Mdfwt~#Kd9mfP:7g'# B7ʲroTr  2MT 2Ļӽ4Ak"+,\Yqw/Y1.D5"4\b姾 C8qIJ]/WfciYBMNzڟciu&YȒ90e6ytϹ,àC0:i{耗aH7?1hh9rԩxu=bRXN[ 3ЈyIP{]E:dT86l/Yo0GgB]\c.P}3r_ɳF&T\\2Lj!BWA?tVSWkHґz Bj#{ܓI&ۏ?f>(aCE'5{=#QD 2`<Lh2jȆ|WJ^gan<|X6195Ь\mhr rlg  rF?^[w/Z3 ͝n1Ŷ r+Ti9++.plYl@SY@_UZޙILG~酉"W3Y*̴GR|:I/s>r! kV:8 [lwYɄl NOE]hJ flf0.\lRshbA9)Jk$MSr0umh5WQC}X닁 l#prAM:N}r)ǵ>: {|e;55Dl'ζFxdlA'k@q{Jor+"Ibh& 9<1Dyk%Y)CJqU"N׺qA0TiwԫGʫC!eMSjNn{4x` )X0}L-( MBH@^4|?fZ0K/ @\V0WX0{dP^:I$*SOKBl-}S5$ [$+ >@M5:WX/mD%XIF&>Nt:O65Kwq-He4HP&ҁ/ 3Ȍ7+FCSh[N"4wΊ ی_;K{/ÕK7r+]  %R!t4hG),nU'r#D.– ~7M[d_MQujLYS'遪btuReF3IPMt\c@1,X*ZrqYNX:\i3MDDuuKV`h Ng~MP1KJHt|u*HB0{$011%6@*֦x~ٛ@9IXvf2P {+DQ' )lluʇ3 7Y$w&VX_|Sl'.uζhUm[*f*dO< Afc9Q$S妦ϥ`A_cֽcUKxe1]Bx@rE $'싫0MƏa6ܤO*N*B1waQ,>Ͷ9(;v衽mXa9Rlc0"vx^]^uk?Vpȋ6[DLbߴQЁ09{Y,Zq=4! Y+mn_(9pD:]R3WS7U8m1s\iuʿ endstream endobj 764 0 obj <> endobj 765 0 obj [766 0 R] endobj 766 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 624 0 0 839 0 0 cm /ImagePart_2170 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 163.9 362.35 Tm 133 Tz 3 Tr /OPExtFont3 3 Tf (0 2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 3 Tf 133 Tz 3 Tr 1 0 0 1 369.35 490.5 Tm 128 Tz (0.8 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 3 Tf 128 Tz 3 Tr 1 0 0 1 424.8 433.149 Tm 136 Tz (0.09 ) Tj 1 0 0 1 433.699 433.149 Tm 1700 Tz (\t) Tj 1 0 0 1 450.699 433.149 Tm 136 Tz (0 00 ) Tj 1 0 0 1 459.6 433.149 Tm 1729 Tz (\t) Tj 1 0 0 1 476.899 433.149 Tm 131 Tz (0.00 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 3 Tf 131 Tz 3 Tr 1 0 0 1 438.699 429.8 Tm 134 Tz (017. 01 000-) Tj 1 0 0 1 464.399 430.05 Tm 158 Tz /OPExtFont9 3 Tf (8) Tj 1 0 0 1 467.3 430.05 Tm 77 Tz /OPExtFont3 3 Tf (011 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 3 Tf 77 Tz 3 Tr 1 0 0 1 369.35 293.95 Tm 244 Tz /OPExtFont9 3 Tf (V.- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 3 Tf 244 Tz 3 Tr 1 0 0 1 405.35 275.5 Tm 625 Tz (---) Tj 1 0 0 1 421.899 275.5 Tm 35 Tz /OPExtFont3 13 Tf (0) Tj 1 0 0 1 424.8 275.5 Tm 48 Tz /OPExtFont9 13 Tf () Tj 1 0 0 1 430.3 275.5 Tm 97 Tz /OPExtFont3 13 Tf (:,=7) Tj 1 0 0 1 454.3 275.25 Tm 44 Tz /OPExtFont9 13 Tf () Tj 1 0 0 1 460.8 275.25 Tm 95 Tz /OPExtFont3 13 Tf (""") Tj 1 0 0 1 475.199 275 Tm 960 Tz /OPExtFont9 3 Tf (-) Tj 1 0 0 1 485.3 275 Tm 515 Tz (--) Tj 1 0 0 1 495.85 275 Tm 15 Tz /OPExtFont3 12.5 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 12.5 Tf 15 Tz 3 Tr 1 0 0 1 192.949 277.649 Tm 17 Tz (I ) Tj 1 0 0 1 210.25 277.649 Tm 1847 Tz (\t) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 12.5 Tf 1847 Tz 3 Tr 1 0 0 1 319.449 716.1 Tm 102 Tz /OPExtFont3 11 Tf (6.1 Forecasting using imperfect model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 102 Tz 3 Tr 1 0 0 1 142.3 229.649 Tm 103 Tz /OPExtFont2 11.5 Tf (Figure 6.7: Following Figure 6.2, comparing three forecast ensemble results using ) Tj 1 0 0 1 142.3 215.95 Tm 105 Tz (&ball. Here the adjustment is obtained from imperfection error instead of model ) Tj 1 0 0 1 142.099 202.049 Tm 100 Tz (error. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 3 Tr 1 0 0 1 142.099 171.799 Tm 109 Tz (imperfection error, forecasts using analogue adjustment outperform the direct ) Tj 1 0 0 1 141.849 149.25 Tm 106 Tz (forecast less significantly than using the actual model error. Figure 6.8 plots the ) Tj 1 0 0 1 142.099 126.2 Tm (ignorance score of three forecasting methods for different lead time. ) Tj 1 0 0 1 483.85 126.45 Tm 87 Tz /OPExtFont3 11 Tf (Similar ) Tj 1 0 0 1 522.25 126.45 Tm 107 Tz /OPExtFont2 11.5 Tf (to ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 107 Tz 3 Tr 1 0 0 1 329.75 62.35 Tm 74 Tz /OPExtFont3 11 Tf (143 ) Tj ET EMC endstream endobj 767 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 768 0 obj <> stream 0 + ,,8j~^lHu,4~P7}‰b% ahk`&ia}OvRڤCy\O|x7/|UB! !]!Bk ifp y&(˥Go52E@j9_!7¥!=eT 00g% p175L|D<#Xu)-}K\3{d޻|!pYeHSVj˥ 35^el/]z+H;GںCy"+]6'z_Y >Ot"E7S0n@G^f'8Qv"; {56;I)G^X0cTԌ7 W-VĘT(e< |6HY_*;wZx 7^a*E>/{PUlG"# K`BÜSO<TV?KSbW<"rE [7}-#{"֧cIVE1 0n#T (N9Of[]q_,v%yj%@n!rFrZ hq0gGԍ|^SV6ea@+&=}>o5J~U %n Ӓ)'qr+<8Y C z{2mQO^ Ԛ-<ځS );h(&d`=WU;")ɭdd Z΄D~gRe;anfHljÑ(s&lZzi^]/_úz]v٬1s7\9*UX |fC4US~Wbmf;7?9k[fy?!ȸǖ;#"طLJG6L6)|4)$y 셍w&eЛnMiW @r* OvnBA&iv_t<& 9Q2D[%6y0)/ m0V.I 'KxSty1)qn dgFh:bw Ip]Lsx;!㉇թU۞X*WoY,`VOV:pvWEwEgڌCg0n*^6$  *-kuP^0r8$ož@ 6~ĊP2G1i-N߶U,qA4s|h?FG D#B<I[Fޡrm ?o'&_/v /tF%ڴqS_5XW* `2l= f;[^ܖ-X0!dg9鬭K\'`8%iݐƾ%|21Y|p`s, _[iu0_,E28$]?o8;)}%x$[A$=8 kdMAZ,vЩմӪ<ɀn;AX<ΆNY>0+V +HmdslrHm_:pԮmYmp;Կ;T1 ኿5yНr81=7\. t +YٜC}j`iY_ Ko"/nEk,]ٍ.hl΃6Q22c<.Uc~n;~cOE]t.O`!fTuKI`m7rJ.x fDz{֭+SZ°[B#ʗj|R"y /55Yas#t('BI;CMTX@Vo$}aXL1k\nb)8I4 :z`F%|H LrwdkA:|1mqفSI{Oߥl:8Y7ιٯ/2iM{g>us{gcTZ =ƞ+'izh0HCVb|K74ǣO_։pVl8@VlX~'y۱YaU{d&V.$|W}ϫlUF%Z#+V:,qx J|b;si\5i*^ Ǎٲ|^Arې(UΜF(&>^9*J B*l}1־Z[ITzfa!pߍjmܴ{!R rCسE2I*;b2Vr)l)MVm_nh}lA%QJx)b1]w qT֗Kt@.c""zIqb^I1mE9OGuUڷ`O*Xq3a'Xauގ}J8Kz^Vf R/$!s6l TNL{;m4[>588˴-ݺש*ɭ1շ%U/)j.˶فɰ(%Dz%c*۶41Ż6Jհ"ٶ&M)>m!Xϥxd(V|qv}Kk=DPl Z[w9I)80HDZUJNl;8g8 tME6 (EApb jsmsWn.VpK3q^lZ/;kŦg2 iĶd7\= SA9h@{l{w۔K]`^_ pct9Fխh8Mo30rinJ)9Y`'q(n jB[H, ԡiȌ"56Ԇ^'dq ;yo [/ 9B*$Ŕ펠ᚄZ/\$("詣d4}J+Pn i&θ?d5FU,͋=?K/!A m˥쉱童qGհH<@&"I}0[Lߣa `Ή6l2hb:7m5zƠJI%'cl[ h FTH:6ɹduwrĖZ^jo4~gao F꛵ݖpkE4EÀbkadD7$wE g:g6:aY+EbH[Lר;*e@z/1Tڏ mLZُlDv pQlu+:`z)# LՂ@~2vB2e 1xE;L-DH4kT NdPkVYĈ)Da{6nyWrE-;0$U`ѣZO\m#N0-0 LuGWtPC$E=EԮ@"]ycH3,eQUϴkF8|Ww*zNfVdс (ڦ 1p BmTnymJ1ZIT)SF-vuS(" ;qגtPgQùqÍ"zW!dHqaGmwx%F>#}Glbpua >?9ClՔ *:+m"y; (tj8؟(Vmj4rlhA0IJ}xq\< fIH=1Aud7BZ94>oK%)Ģ 7$@+C&ؽW*6bRՕZxwr8yXՆ[ 4@ƑW}nyzOELUhs^/N'8ţk[Y:Ϯ>Z%H_ug0c-9;p2*q$X핮 &q|914Ű]Tda4C(mlRs4~WՙIu'z[Ὓ]T"_!c,.{mRuYA|L'Q^~j\"\VH(-{^-Ag=ejפ{[,VHЌ` J~Ʉpm;ޭescQ޲| \YqnNB .;U 9vm _~_S?Bx}$Gt# |t")/@$ WRfrD4/p_8ՔՃ=赲R7^|0&!}YPHbpB"m5=V^:G3Fna j$0ߛ p'"X'_ xcvLd~'uY?wD +7M'<(DuԝX{)Za8#Wty5¡Q<8)(B0AE,QJoo4nC#@f3QXaW3*r:JL>`bZDB54۝IaJݴ7m┎>=!k홫}1w BiU_^Q!֙f|zx;E(LLc*[hg]/X3˗hy24i6TbCYvb{N+edLE.5ay֡Brz+dWN%F~ԢHj^URRMLr"h`RU_U2EpP%bj?Eld%H3Zt uI]ZNj}9'LS<~O1k YX|nz)n݆ 5 I\/?~k ;1Pگс{km,ſy/K.aGwgmK<$+Y:2T9$Ol>5qN=D,a_0 +0Q*G ͒@#}Y4O6k_[{ c S0+x63 I,J u4ж\ӏ`gJt2}sK0D/䍹AR_t'3(2Ϗי g`A(6> Ƃ1g4f&۲bZb:/b HaaGW˵NX:{#rFX˜E;;'f/[} $Aup,*>c)5kWeuJocA=疽++plmZ"PRwG3uټ^Ggnj̲e$ZDVV}j~[8+7[ \zsY"/cQKmWǔF+wv;t3֢)1~ @.2b=' bKZmҿ\3 fpuP @AaoŷOP,u\3pg_Lnv`C քן)M~VJ%ԟ"3#{[iUƹ:Eyhݣ8tͼg'ոȼp)sFSW#˷aq]1Ib9ftYf7jBj~,9(Ot;mK )t[PkҠ?~W5G[ J"f|[ hH,GkHFoH8h "i F3y4NRZD K"$CF߃r s7ltQarJP+㙪ZC5A¬d7Ot}rzN)\z & #Yc7Oд2Vh/S-}4uE}f׌+۴vD}mgcqߟJɗH;a>C P.1<ԃ/ D+i-/`%X#֓h# ϘSChF>[ڑ'"nn5ʼn"Nf~ [(Ë;QTiSEcsS/)܎(I@NH-#S?,ܠ[-yoX=0)W×ΊpCa[agمп5C-HPpd 8㪩Y?.Dz.1^fF JR3{4g*X6ngoպDW\Ӗ!dm՜HJj,Ԯ5'CQQNTHaՓ "p:90}\-q:S#ϨBn{zr Wj."YVGŽXdi=j{ڛs5d|_+Z6Q܂/cZ-OWÍJ l a 썺8NfhO'LIA\ܥYGu%xr.MzO5/0e]+CбySb\NIaUSfkΟ%jzI8P;ىqy&W(3 +7*EJm0|}Kx ^WRDH!u1Dӹ}J c҈3_0c6q_@}aXmp0-_4}3 VW+s>ׁgذ?ԬY?caU6=&&F9W%Icx>U3U\@f/nTgi/P{A:4)PPF"`1%sKz~.&MVދCko?-xp( s<;dRq?^{JnX0bD`3W-r3)o:n{eqfK&"e"2(|!Զ4iJg61aw)uw%7~]PyZ*Z:! ȿX5o߀"eJuhsH)cCbKPQg!ۅ}8ciLִuC锻ReRyf;12Rt|5S d76uk-Ohb#ev@h]Q`SԂ7cVq1B#3D᷼@1WwM^n@@Z KbU[I r$ 9d_ u+3C[T]*.e@,ĺ/7{eShFR)O0R1]6|N[P҆A Hw7EU]#|`F倞ڎL^yLN>Orf(ӭВRfwY$i.d[] d|sҗʶĉȳނ@+ŧ֥5ϊצ()5*Ә,Øw#wV<zM( vg f icr&<FT)?j%z,բܵ`q*8/zװuڠd+D&9仱N10<EwLP߅XUULbqeug,N7J 0(?47arZGiY8@[L9|9$;_O^VZfwŘGƜAymt6 ?k6@ ;Yz(ҫ";h_&I)(J_CO -ЌV%T"US+m=\kJ9Ns1C؎7 J+%P3!aO 1X}kd k ȴWVÖ8ŇH VgmKmu`: ;\T'{M &=Xo32(CIruW:&mPi#g=sT%m؅3x,]v)Iw{t" ԕ线t?/2Lj:-ǽOnfk|__ zvzu'a.j2(B$5Q# CY'ŭ·" W>!Ycd4/mR[o;[;`kG_"-z[3-9[4Ԍdo9Lh'D#fJ'ɂ47 x!+p $t:ɿ!m8g>w9$֔$^ ,GsL&|v&UP<` ׳g+,MA"az.s*U FRv쵥_@[2YTR {NdF! ocA)ցMmG7qyY"S$ګ \`ck9UEd`nc/+jJ򉻰?EUwP+Y&RipgOϞ̌q_r WvMqFrAʙwxkLOKB 9} V]zu$dޅFո; KwF: 8A4?)'2\'(%Bu"y.x=,&VИ]=@,*?} _T./vDprЦUbf"uϞ-gbfO ]q~: (١GymTݑWbSd^1f'9;^KM^7~7)B'%TM*dp }%6̃pIV+jܿpz+G/ }{g ( b4ztE/bL(hXP!@AK/[Ys/WqxgW <5lZE cukӹ&#N±.Lnfa\ߥ$G% 6>du{\9xHةEMoEZ|$@CLV]ߤ|*s-Ed0ޔpGOZ]_sp{# MU_QS #X_LN%tgٺL & N%Qj#)qȒE\a #3T[:U|¶GUDl+TI|BJL-hSu€adϷ@ɔc P:+ T*} x[~jjTϯGb gT]dꎞa΅ 1Zf_O`  JE2=X"o׭-]!wmp2S01vL^i3MR9YQ[ǔv_{+9'PvD8cq5gdZM$?5Yhq'a#)5@YPJKgكN :;*|-{cF'1'oDw 7ŒqT x7pgJLLՂ9= =qKوܩDq$ճ-vdzE*B*Cg<}G@ڞn&bνƎtQ|!#Ԑ9웞R wM;~!T  yiЏBŅdo<0T[>pU=?p:o?vP?;ǹG%tTJNX3SIXA?0Y2~!P7VM 3dLʵ]I2(iZ(%;c6|ҡVTRTͮɰ ^<ޑ~gr'UCv;E*x`#y[hxYH1۬q+# )tsf$La:w)xZ/_W3vۖfp3EK4& (U%h_ZNNpZV᭰(? v/v"f^mfSPςu2[ǷZfi Qd=حJ!%~rvksJkJiQE< `M+{ :d x%YSr))@<4Ng2`oWo[Qy?gt{0xR{86F}>dܹB:d}aβoH> Hu$w$Deu#rNt3tZ s\{%:Mʡ\`ɴ>!rJOrY尌Ka$B(0K"{Tv v-{/)F7d:e}-/y]Ž]o|-rԓnZ&  ?lwYr/dg*NN)؇ c@qӤށ:\V<tͧr6`X-[Xh%L2E*21wPΕ6[=m,n(ⳅuMF͉d(9$KU5άJǛk J.eZsfVh::CS@6R>>u[n3fus]΂zh]1RGh;WjZ荗LLzhud Q0+适_vš)9XVGY &˪M"&s􃆮ӎB #$x^ ϲSӻc+0 XS`.V C endstream endobj 769 0 obj <> endobj 770 0 obj [771 0 R] endobj 771 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 478 0 0 788 0 0 cm /ImagePart_2171 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 222 519.45 Tm 91 Tz 3 Tr /OPExtFont11 6 Tf (Direct forecast ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 91 Tz 3 Tr 1 0 0 1 200.4 510.8 Tm 878 Tz (\t) Tj 1 0 0 1 218.9 510.55 Tm 92 Tz ( Forecast with random adjustment ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 92 Tz 3 Tr 1 0 0 1 201.099 501.899 Tm 845 Tz (\t) Tj 1 0 0 1 218.9 501.699 Tm 93 Tz ( Forecast with analogue adjustment ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 93 Tz 3 Tr 1 0 0 1 210.699 682.399 Tm 95 Tz /OPExtFont0 11 Tf (6.1 Forecasting using imperfect model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont0 11 Tf 95 Tz 3 Tr 1 0 0 1 170.15 465.699 Tm 92 Tz /OPExtFont9 7 Tf (5 ) Tj 1 0 0 1 173.75 470.5 Tm 2000 Tz (\t) Tj 1 0 0 1 212.65 465.899 Tm 78 Tz /OPExtFont11 6 Tf (10 ) Tj 1 0 0 1 218.65 465.899 Tm 1766 Tz (\t) Tj 1 0 0 1 255.849 465.699 Tm 82 Tz (15 ) Tj 1 0 0 1 262.1 470.5 Tm 1742 Tz (\t) Tj 1 0 0 1 298.8 465.699 Tm 85 Tz (20 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 85 Tz 3 Tr 1 0 0 1 220.099 458 Tm 90 Tz /OPExtFont9 7 Tf (lead time ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 7 Tf 90 Tz 3 Tr 1 0 0 1 33.6 427.5 Tm 91 Tz /OPExtFont3 11 Tf (Figure 6.8: Following Figure 6.5a, Ignorance score of three forecasting methods ) Tj 1 0 0 1 33.6 413.85 Tm 92 Tz (relative to climatology is plotted vs lead time. Forecast adjustment is obtained ) Tj 1 0 0 1 33.6 400.399 Tm 90 Tz (from imperfection error. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 33.6 368 Tm 92 Tz (Figure 6.5 forecasts with random adjustment appear to be slightly better than ) Tj 1 0 0 1 33.35 345.449 Tm (direct forecasts, and forecasts with analogue adjustment outperform the other ) Tj 1 0 0 1 33.35 322.899 Tm 95 Tz (two methods significantly. Compared with Figure ) Tj 1 0 0 1 291.35 322.899 Tm 75 Tz /OPExtFont0 11 Tf (6.5, ) Tj 1 0 0 1 314.399 322.899 Tm 92 Tz /OPExtFont3 11 Tf (by using imperfection ) Tj 1 0 0 1 33.35 300.3 Tm (error the forecast with analogue adjustment gives higher ignorance score than ) Tj 1 0 0 1 33.35 277.75 Tm 89 Tz (using the actual model error. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 49.899 254.7 Tm 92 Tz (As we mentioned in section 5.2.5, the quality of the estimates of the model ) Tj 1 0 0 1 33.1 232.149 Tm 91 Tz (error using imperfection error is strongly dependent on the observational noise ) Tj 1 0 0 1 33.1 209.35 Tm 89 Tz (level. When the model error is relatively larger than the observational noise, then ) Tj 1 0 0 1 33.1 186.799 Tm 93 Tz (the model error can be well estimated by the imperfection error. On the other ) Tj 1 0 0 1 33.1 164.25 Tm 89 Tz (hand, when the model error is relatively small corresponding to the observational ) Tj 1 0 0 1 33.1 141.45 Tm 93 Tz (noise, then the model error will be poorly estimated by the imperfection error. ) Tj 1 0 0 1 33.1 118.649 Tm 91 Tz (In general the smaller the observational noise is, the better the model error can ) Tj 1 0 0 1 33.1 95.85 Tm 94 Tz (be estimated. Figure ) Tj 1 0 0 1 141.599 95.85 Tm 76 Tz /OPExtFont0 11 Tf (6.9 ) Tj 1 0 0 1 160.3 96.1 Tm 91 Tz /OPExtFont3 11 Tf (plots the ignorance score of forecast with adjustment ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 123.599 632.7 Tm 76 Tz /OPExtFont11 6 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 76 Tz 3 Tr 1 0 0 1 111.599 576.1 Tm 94 Tz (E ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 94 Tz 3 Tr 1 0 0 1 111.099 561.45 Tm 130 Tz /OPExtFont9 6 Tf (9 ) Tj 1 0 0 1 110.65 775.75 Tm 91 Tz /OPExtFont9 15 Tf ( ) Tj 1 0 0 1 120 561.45 Tm 121 Tz /OPExtFont9 6 Tf (-4 ) Tj 1 0 0 1 110.65 542.95 Tm 62 Tz /OPExtFont11 6 Tf (11 ) Tj 1 0 0 1 110.9 778.649 Tm 77 Tz /OPExtFont12 13 Tf ( ) Tj 1 0 0 1 120 542.95 Tm 102 Tz /OPExtFont11 6 Tf (-5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 102 Tz 3 Tr 1 0 0 1 111.599 532.149 Tm 142 Tz /OPExtFont13 4.5 Tf (O ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 4.5 Tf 142 Tz 3 Tr 1 0 0 1 111.599 525.45 Tm 85 Tz /OPExtFont11 6 Tf (c7, -6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 85 Tz 3 Tr 1 0 0 1 111.599 519.2 Tm 128 Tz /OPExtFont13 5 Tf (O ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 5 Tf 128 Tz 3 Tr 1 0 0 1 111.599 512.25 Tm 87 Tz /OPExtFont11 4.5 Tf (c2, ) Tj 1 0 0 1 123.599 507.699 Tm 81 Tz /OPExtFont11 6 Tf (7 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 81 Tz 3 Tr 1 0 0 1 120 489.449 Tm 99 Tz (-) Tj 1 0 0 1 123.349 489.449 Tm 82 Tz (8 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 82 Tz 3 Tr 1 0 0 1 120 471.199 Tm 99 Tz (-) Tj 1 0 0 1 123.349 471.199 Tm 82 Tz (9 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont11 6 Tf 82 Tz 3 Tr 1 0 0 1 220.55 29.35 Tm 75 Tz /OPExtFont3 11 Tf (144 ) Tj ET EMC endstream endobj 772 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 773 0 obj <> stream 0 ,,>j{G}w&ɽ61{b4q*9\ܮU;1jr׽ɱǤ>9cpC W`䊇К ^h^~yI0+0+idue,d?@4+TCw^}[ `9g'rt>Jܠ<+2Zf g0x xY<c %Ifs V28xN{Τ. pAׁr6M 0q֏ y=;U38mHh(hm+"L4Fe'iv+6vYkiE=5J' Ҏ9WJ|.= LiX6d|QEirA"ޅö +#SlaW.]/)0#J?ȟ2_M<1RM !mvՃ8@ |3r&X'7y-8vPcZ(yh4H(]3su2]3@Z$xU LCtH6ug 3)P]ae6]SV|-J,6!FS=->+~bvF`=E3|DW~n*+-NwRy}|we 8m0D"Wy>w˟2< ˓;LƎM5j}Z[xi{ s0ZFTc\ ?Wԟ`SLX]?ĪC0_$csW;"rswdȪ;O* f91ޠ`ۮ2u{ }wqfuS7pؽXi&;pgP!]@MIA뫚SUfwm;!hD 8%#I*Cػ+3E<<}=-"+2"P-LQG}uv:[ _SD JIW>Ba]> ŝ/`xq˛f&F@`#o"#y 1ܲ^]il^(^C ?QGs諲{fQb g.>kwn(}:dx];Ce-sb(D zSxf?$!j[bcx(v=lR!7Xm~+$(:dX(u |LP)!s4|&BoE g.TDlpcHq-F]Nړg LvI ЪC%9?')t$yiHRx]O'=•B~'(wP1xHD`T ;`bР,yG{V{Ր}tl؛_ucq A*zZW y('N[]w*tBleܸvudOhN;" KYV?D3۩(H"KDAꂑ#ؕI3iش?KԯK&:0=XZ$PXN)nJNdm;_<]e_X4bll ({ok7YMw y ՁIJZGn!;۱:"iB#@Nx?y!jb"}Aәچ ¬ R[kR2(~&Kk@ftl3SV-m؈pהIY0(eBKɅz'm!:4p}A$(e!TV޹EdR1@$B0־~Ve 61.yBEEjbYgo)DZh;.':wG9H 2ezou OQy.Yab[orO Ѝ"0[;B x BW;x^%E8PX:)2F)f6,- YWQy%cCu%u^@HVvU8O^^<+F鱌 + e+mPH/,UVjiÜf٫|ֳzWRYIՑɔL|1-;yXW@]ǮpDFTDM |xA@>y(&ټ8$t٣:ڞ*6.ϴٴǴk;w靶ĶxUVuD7S1*P-jcvmC/Q&ɬ"e~%sCrE+%lX[@ǿOg.own]Zb6b\Z1 iWbX&l8 QzM{EnK#0Z =ÙҾ*Xܡni|H;=G-uΚ('mHUf`c蜅O?/VcPW/TLct^  ,yɉ2-BayuL[vk[ƌi;hao=Me+=qI[!8E]"}r\?}'&X"ЩUO|NɅl3tJuEoN1%ɠa@m ^?cF\Srm+˥L y.gѲNc:Xx|Oo?*}<f!z2O_qϸ_AȑȐ`UMˍ'4>Q_\c@( 7@]NW5>[kB%!x@.)g]XWt'㷮A$ѰJb,Ǐ^H{U].Pc>bn };;P_ fC"vA~a񂷳vdu+u- hI{Q,~cm¤%{egVd@~R^4/Dd%+{pa. 4xlNSz7<:*ȹ;}=LZy0xH&$i}(;kC)׺;ud*O]!Ze퀽%6KStaE?^ -h 4S$x[KG$1ŷ~ҀANpHq yKn'-s9_ CO}Q?~2YXu%Q|4idciK^C1,qBC7⽃hܡ~ #++ylI\]lQ-HOe4"S%/.{R=[+0Zb9R<8ZP ^ }oLטX.`ޗVCc&U߷1[HpzUNr2%goKL.}_N rQ.Cޝs;>i|kZ/J+=-`ɡ)oyg!pqVXpko4#2yvf_Z@o22/aPb56LuA2:>)- VvBv1W\*u-Sh,3鳓>Eu|ӝ-m;ٲ\4teil*;LIaUd[/h2u+!JxKU\MJڪZ/ `ӘYyP_fe7faUz9o #Um3NƮ(-I]YAB!P){ͨB[]l~˛*֟ZmK۲7F&1} ,6dvEKkUo5 -r Ld]K?rkނ;cc4 32mN4?*7#>1,X ' 4rLSnAO`UϠ``QcIٟc0J[*8gUbZpeҷ_ Y@>jlMp#{ģdLuڍ#l`dYI:9`qv?r lqiqWyۖf*'r.aW8Ȟd##h|\c ct׍/.2 ֎Qaռ PmQ4G"2}M4+ad?s#-Sc1|>"[*ƒ;B 3#cc-7Hw\ ty!nA,%ag&"ٹW݉[ZTt2Pvrč[myknލ|w{')K1l.^T5"hG'fA|B@et_~gvX KRxn`ВZNoQkH"/;NnNM*,HX)VDkD [;;wĝߘD4~|1c!ܽνrTc3uDNF9")h"\4JZяM7`qy9.r_aDov.h +]~*;_[ ȿ>-O{#,ED7[nVD0Mt`~tj/xeQjЏ9k;Byiqh}()E!KlQn׆b*"qF N&͙\Tӗ1eakfÏYV{Ky6s9PܥYy %/{BTR/3%Bc}\čOW ژ0 E^BBĴ'Ћ\\kT+~uݠeٸ~c?Wyخ[ffM`~nwu^@o1W A8^F`PQkObf]?9. JpgQ$L}J\3 77wr2s5ou\=#bELpy 8? h| DeNPIi%'W[Oګ)$†s|qijixK{.-R5Pȼ%TRu/,.kZ&1HW٘1ʁo*#N@ v퇿r3dLӔ ogPM D?q[PB3 cSY9Tm"WYW038+VUtv >`L|3g ?F0$u;~`X#9Z_ j  Ї9#C?lb྽7D<ʸqX7X;ɹǛoTJ!SMFRM<9ǹv3IopV~rXA`*0ZBq SM?[{qp'xBίհmC=`3 ҍ$SlDARo ,V`vжgWn},S4t"!IɩI˲EIx&p-Wė%÷k}"6cFh]B ڛk#m6t,(z-OMK79so䇋uM $*׈v_iU]Pjg$܎mb>h=fjWIˑ$^0SJf;` 4> )<?J;z`ǧ>ā qt8 + mդexڬr6FV\ ue&-CR;9X]5J~bJˆ՟gUDכRHkSfd5~;]'& ǐGRGV4 C{:g=ocDdnnl4M~Z]6pd To&P_NY_"$_ 8]b%|K HӪ%x_.\)%|jx/6vqAluv,xjRIRiv?k)f=ǓȭM~A :$y7fUKzThQx˃dxƌ biٖ4*]MFNh DQC"`4:}_!= 杅߼Dn; 0}u ٞ=`w?L)=9+6U?LE5 ǫ]|%ݫ3<@eNS%7LRlՖ.?ab@~"=EH 8U R2jxQLTҾsmoa^6j1gqzB8;]'*#`F7g0yڣ>rو"Ѡml[C ynI;:5$ZZf&&^1-7Sgf#C8@I )4S}Cu? ;P!={UhrfR{wEzYtHml\4()f^eK}հߧg{s X\xyje+Bn +~Ҫe T@D"`(-l盟hz=d,qEVbw2vq =ͺ<Țc{>nXYxR@=pD}ş+XUJqNM@6dVFtf- MxqDƂ@ӯw+~l\7}SsnGC6lY0x^)럾޼ϊ4}yө Hڥ$ȷUN:9䰾y+ _huvNDghTWITWSRFeOD~XV&``O9ix]!#/-j>>*^H@|o{H(Մdt '6~-(8Vwp>2ԨPHӏYw!ib, |.PB|Y !U;_zsx )jQ&w]{ Do8~vjZ%f 8Yg Įd /Xw<0 }pOh?ĩ36+,L20GY;=:RgA)a-o['He8vm56hŁ->4hޕp0rIJz/&\fw7sPks;\6- 0ļP}n~KrfTAد3GJZ.QJ~B{Ǔ[(uޥ6i BytYL'ߔ+N/c~#Ս<2DFc6ә0$ݨs[x"Ei, Ps@aowkSRotv@tGC5W#<<ZH*vA&ALd@n;\w q[xdaEx9ȟQH )p+t_e*OԂN{\ twK";>VΣOJP*\weʔ(b ,&~iӯA5 n,B@EAXyM=GLomP" ͷj짓i#H)Kŏy(iQ#`ďMgL%JOў"95>{_8ctNEW }L)yzuP=.N|HH`ɏ>䖋TwW *g,87? ,Ģ؉_J.ʥlZ7;-N/.r2:q hsNy=ovЎI\\ JYt#_Nr5 LblFfi т){Tq5 6&Au۴ v]t@Z814}q W~#1M h4yu+e`C\2|7s3o kjҢ}$ǭo{-eV¶Zo*k5?z2H/;UfcseK Aٶ4 IME` em@O`Z嵇AxT )0umY:_TjyD 7/Wן)A0?6-z _J#SAdY?&G\ hH}pE+\;EmSG[rKt54ku<"c+ըuY' ZzF%"3lL+A]7{glo&ɯ8g{yc Ywh?ouwXRdkE`8y.ɋo!.26Jmg~Ovc׼z/M]V >^ov?>HQ9!')WhG6-[R:Y^1W (S8]GLs1Ѷ%qHz=jc"B@V##yĦӻhSl"G.Rj2Kz=^@Rh)`F4ieW"6=6qEH ?T]>9ϴ4m"1Jt>l`0;+RZ$ii\#F5Ptm6@һFX&r?~wO}[^Ũ)օtyN" sЗE^aP+c?wåEo=N^-9 SU/ f 4?sˉk5-kbt$:_!u36>+Ğ8B/ŋ4k:Ű=-Ѥ50(YA> endobj 775 0 obj [776 0 R] endobj 776 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 508 0 0 804 0 0 cm /ImagePart_2172 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 385.449 560.399 Tm 99 Tz 3 Tr /OPExtFont19 4.5 Tf (-) Tj 1 0 0 1 401.75 560.399 Tm 78 Tz (U\(0,0.01\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 78 Tz 3 Tr 1 0 0 1 385.449 553.7 Tm 99 Tz (-) Tj 1 0 0 1 401.75 553.7 Tm 78 Tz (U\(0,0.02\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 78 Tz 3 Tr 1 0 0 1 385.899 547.2 Tm 99 Tz (-) Tj 1 0 0 1 401.75 546.95 Tm 78 Tz (U\(0.0.04\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 78 Tz 3 Tr 1 0 0 1 222 681.6 Tm 95 Tz /OPExtFont0 11 Tf (6.1 Forecasting using imperfect model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont0 11 Tf 95 Tz 3 Tr 1 0 0 1 176.65 556.299 Tm 127 Tz /OPExtFont19 4.5 Tf (- U\(0,0.01\) ) Tj 1 0 0 1 176.65 549.6 Tm (- U\(0.0.02\) ) Tj 1 0 0 1 176.65 542.899 Tm 954 Tz (\t) Tj 1 0 0 1 192.949 542.899 Tm 80 Tz (U\(0,0.04\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 80 Tz 3 Tr 1 0 0 1 0 804 Tm 65 Tz (\t) Tj 1 0 0 1 67.9 511.699 Tm 99 Tz (-8 ) Tj 1 0 0 1 73.2 511.699 Tm 2000 Tz (\t) Tj 1 0 0 1 275.75 515.75 Tm 75 Tz (6 ) Tj 1 0 0 1 279.85 515.75 Tm 78 Tz (\t) Tj 1 0 0 1 281.05 515.75 Tm 65 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 65 Tz 3 Tr 1 0 0 1 0 804 Tm (\t) Tj 1 0 0 1 73.45 511.699 Tm 75 Tz (0 ) Tj 1 0 0 1 75.849 511.699 Tm 1955 Tz (\t) Tj 1 0 0 1 105.849 511.699 Tm 82 Tz (5 ) Tj 1 0 0 1 108.5 511.699 Tm 1906 Tz (\t) Tj 1 0 0 1 137.75 511.899 Tm 75 Tz (10 ) Tj 1 0 0 1 142.55 511.899 Tm 1798 Tz (\t) Tj 1 0 0 1 170.15 511.699 Tm 79 Tz (15 ) Tj 1 0 0 1 175.199 511.699 Tm 2000 Tz (\t) Tj 1 0 0 1 278.399 511.699 Tm 75 Tz (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 75 Tz 3 Tr 1 0 0 1 143.3 505.899 Tm 92 Tz (load Om ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 92 Tz 3 Tr /OPExtFont13 10 Tf 1 0 0 1 5621 5688 Tm ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 10 Tf 92 Tz 3 Tr 1 0 0 1 342.5 511.449 Tm 75 Tz /OPExtFont19 4.5 Tf (10 ) Tj 1 0 0 1 347.3 511.449 Tm 1782 Tz (\t) Tj 1 0 0 1 374.649 511.449 Tm 82 Tz (15 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 82 Tz 3 Tr 1 0 0 1 348 505.699 Tm 102 Tz (Mad ti* ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont19 4.5 Tf 102 Tz 3 Tr 1 0 0 1 45.35 477.35 Tm 90 Tz /OPExtFont3 11 Tf (Figure 6.9: Ignorance score of forecast with adjustment where the adjustment is ) Tj 1 0 0 1 45.35 463.699 Tm 91 Tz (generated from the observations with different noise level. The initial condition ) Tj 1 0 0 1 45.35 450 Tm (ensemble is formed by inverse noise with fixed noise level U\(0, 0.01\) so that the ) Tj 1 0 0 1 45.1 436.3 Tm 95 Tz (observations with different noise level only affect the imperfection error. The ) Tj 1 0 0 1 45.1 422.649 Tm 91 Tz (error bars are 90% bootstrapped error bars. In panel a, the forecast is made by ) Tj 1 0 0 1 45.35 408.949 Tm 89 Tz (random adjustment; in panel b, the forecast is made by analogue adjustment ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 45.1 376.8 Tm 95 Tz (where the imperfection error is produced of different noise levels. It appears ) Tj 1 0 0 1 45.1 354 Tm 91 Tz (that forecasts with random adjustment are not affected much by having higher ) Tj 1 0 0 1 45.1 331.449 Tm 92 Tz (observational noise. Doubling the observational noise, however, decreases the ) Tj 1 0 0 1 45.1 308.899 Tm 88 Tz (forecast performance of using analogue adjustment. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 61.7 286.299 Tm 89 Tz (Overall, we conclude that forecasting with adjustments can improve the fore-) Tj 1 0 0 1 45.1 263.5 Tm 92 Tz (cast performance from direct forecast as the adjustment is able to account the ) Tj 1 0 0 1 45.1 240.95 Tm 93 Tz (model inadequacy partially. The adjustments can be obtained from estimates ) Tj 1 0 0 1 45.1 218.399 Tm (of the model error. Such estimates can be obtained for example using ) Tj 1 0 0 1 399.35 218.399 Tm 110 Tz /OPExtFont8 12.5 Tf (ISGIY ) Tj 1 0 0 1 45.1 195.6 Tm 92 Tz /OPExtFont3 11 Tf (method. Forecasts with random adjustment ignores the geometric information ) Tj 1 0 0 1 44.899 172.799 Tm 90 Tz (of model error by assuming it is ) Tj 1 0 0 1 203.75 173.049 Tm 94 Tz /OPExtFont0 11 Tf (IID ) Tj 1 0 0 1 224.15 173.049 Tm 90 Tz /OPExtFont3 11 Tf (distributed. Forecast with analogue adjust-) Tj 1 0 0 1 45.1 150.25 Tm 87 Tz (ment extracts such information and as a result, outperforms forecast with random ) Tj 1 0 0 1 45.1 127.45 Tm 89 Tz (adjustment and direct forecast significantly. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 232.3 28.799 Tm 75 Tz (145 ) Tj ET EMC endstream endobj 777 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 778 0 obj <> stream 0H ,,]]k\gmN<ŷ.8Au[LSY\2_5UZQ\<>I -)fIͺ:j mL=-CO$"4B~*+҆Uݪ> 撈\( KtMS?e_ EkE7IJKv~ve&1:%*M:9F+i&C*SET#X Oo`!x0'Z~яg]G(VBVg3z;#nņHZm2x' on!kc~pzV\|iփ!0U\ųl22Y^iay1Hk֤ܮlםqoV۸mD}|DL7}..Wc ~!s1 Y!wIDz⧛m$2t_8vkpbv# qrur;6*Et&~W}2IA?peku،vMy߼sZ.Z&fr$z';3pN͚N4 \5f]Kb f8G-u$|sM7vQqt=FU3unP͎t%ɾY0ݿJ}F5sU8\IQwA?%0=[|1׵ףɻME^KUAh~I\ Wa\6KX KZ2$v4(쨲GGdy0RQVtyu%YRe:NVIn6.\v3j,)YE }9*! w㭓w"nErif|%-E:mK $Sf&Tm>c6f;"עgu?=mQzui@3~ ]O"j%Jd[poRH!yiB$^:5HDrTk1Mp0:]>ooL݇XJmTIp& x,8oLDbڬ;ݫ5{3l}n?3Bu8BF@FbN*Iouho JOlJ &9B'_$Ri mo"8p1*gȌtjC.^Wxr7fP=G;=h-zO=+S8p2FBV8e6"(2Th,R,}/mЮwP$ A6,ҁ{okk ȆRjl:}r߬MA8S/8bqaW*D]42̳Ća#{i` >u7y^ƽyk10+v}?wRgjEg̑C5V>>if&d#ͺjm}7BŤBԇ&Z9imXA}/2cM#E= =+ Ȍ[N96=쁾#ZMGEwUℿpߓb7N5/r΂5t"Y}:`)x-e5KofUwsLͨJw//?ϔ{嶸BjܖGw+8;T٩:=vVJesKNǧ %33Wr]%8+)b]ɱ]HS=OM O@Sy"fyܾ^:?6@4 kbHVH ٿ6=MϠ"9S5U˧Дח"ΩRڶtڲų<~R@A({==z@ܟ`WEa3;>\L0vCJB̏ hX҃8 .p.[S;3q@Fv3z2EQ{Զ`i ?_π!Q5^41<[j~cb3ǻ" 1U>: ͼtnfrt02?x-B:cD79=Ḕϝ+_,wh 0-mo9ᖷ4Oܘ3sVq%h]Hݬz^.+>1(oe sYm:kF4~l1hPVK?πSs̀+0vX6PI SQNnv@%+D7le:&uo쐄R|͔{ʆ0fS[$OxĊtsDdBBh =RH8, 5bM~iW-y5˚əl&Glzv].Vn+_~ g7%,@fs>xovAuHY1JZ;y]c>tأUNRrF1`DKȗѡs V$SR+Gt$e"I ]5=D1@h1D"f[PG_< Z_IXn,ܴE읙0Ң0/_G. jJfKT)X&%DF U%Z/x?9 JUW,⢾D{j05.WvWFKY| h* ,؊%Ԥݢbs7mt?9.Y ,3V_I %{?#:PԄvj)KU+{t aKkcs|cTdߧNlDžԋZRt=W$Dɕp.`U(Pٗ0t%|QW3 ?DA T@idŐ30p•k=ޘ[`L|bqB硫JdE31vuv Rp͖H7EKqϵ'FzsSV5M*RJVk Rq3_umbqP@ν͙YgMWl͠U:fTȃIr*د5XȄgBCđ?*XaX:H>lRIN-|vh37rCvK͌9is&kp}]-Vc,MNryHj®RGO݇ãoyZX=b9BWkni֛7u*Cy/׉[`r#bw&òYkؒ;:XoOخeSOX8"sp̏CpR86&`yq/:d{t1 M}'W؟/Y{| #J 'I|zvOAFBZd^s%,>$zA׼dĂr )^IaJ>k]~mH*uRG _rShչlN4R)-?z}#5zuL(4ײ2¿?/Mt}Xڊq/oiǮg6p3$1a!&m8SSk` ^utzU#ڭsꊅBs3$ So 2+?U>}¯fs%M>"ΗN?WHHWIeR&k9?oE+&ڻdNsE'$*q==9aLrth~~ͬB_8%4 Q*b NRrС \ k Gzs@r ~/)*COLۍB6\G ;bM=TnX> % I 1_>ݳp<*ӷO߀& EV,zCɩ;F2V_˷Zp$W\+@ Y )ViZ#.K`\Sdvs>?q)TaY!Yol&mn*Kڧvz:2/` di.K\^\P2Lt|tjA:2+̧"#kAa8&E2 n"+G#Gkhx"*,~Tvr*HefuRਹ|I@ 9׿-V DXOyI@Sz|̖gtĭyE U"3V\z*Rd겸t>#&L ۹E3>c־ʯ}4铆ـYT%?{fH[>"U r=HyJOj2n*Ě)+(2wb~QM7Q昆hC_ /'.O' PqXQ=*dWBV8~()+Zit%W:ծ#6cNyUS+–nJc֔J' },YxX / %D Kw/`\ޠnA])Up)L& #<ýͅ󸷪-112hn7N8L ׊#4u)u`G"x?Jm*Yt*}<.ږW^ Y;PU&"kQuhd=MWЗ# 6Ͽ Xi{N/ޛ=GcPi]4v#hS!ʝ84AʮɅ-'NHuAn@&tOeѫS#}]wP`2JqԋVNt*%+lWG}>{wɊr*BAgD'pm7XT PWƻs/Y8}spdj,e$"4fcBR=3rԓpd}Ւė:qXHKyM\Vn0[`?_I9U Ce7Zu`ST8(d}bo 4Wvy;bRL%9TY,LfA'8O^u*MxlDߧi2WT n4. @S 5QkK壨* H  Ffv <6CdljjYԒ ` =#Քt4 C}NQ0&4(IZ(w-֛&Dl\sǨX (q^O>⁗"3}I$8|2c4u"g @>b9o7a8E/YSXwʼn/ƛ+3NAZ6KyN> p^DNS3FCGRl>^5Ӭ)9/] paoԮ52P/I=L>k;Ж9;)޶z>FQ.(2AҰƤ05⬰rxWSܗ0Tt7"̣9<0.|̯!C߈MT_UoơP y"rĖ'%EN˓ ZtP'evݲИg#aCc+8|r!<ܜo:FZxBP= d€`|΀`ضGU.Yjٰ<d4D}x1ta υ.jzid{N༈#ttcJQ#MlR~Eٕ^ۤ[XP,Eg.- K(+(62:(- 2=U9$] vD(7_EX#E/P㲱|L׆к}"5kֽO)tS[WS1tgQo kG=sqL9uO1 C1J܁~Qc،&Ƹ/'&%;( NnRB,ELj8 Pw_Q}l$}Q'5)e; Dp_<ԿDzmSN?t,#$YeIxpq" c#!25"(Zc򡽓 %x&;R uNHR PWNH;G`!B$+V$Հw{eEɘDVV;`GBEf%ILo$~Se z\pRqbPȺ#Ue>d#Z4RI94=[15Yݘ+V:I7MЎ");d{;w~`+N¹SPyx跂_:}e-D LV& @2Y۾\m59RB8_:52,pqKD+l&SN6Ƒ_4gE9u  2g<@2?!M9lK=tb#5xo#0 d`(}{lj7ǎYBZQW]v$)-a< uAs)1mJTBt"u(HPiAk^oGs$ !<-Jhp,t}.ߥ;*fJǩ2_d#ӟ Ku*޻" Oe5K m :1萜JTŽGgU *yͻq< r{XEA6&Z(c}.6gNs.(ӞjW= g2\,[tuv虳{Jp;XH՜A#4B8M8}8j7M~Bo64IT^y5Qd)<6yS{()ʖ1׈yw;zsNB:4uNqonoZLdfm, [ ȫwMbBUG5k˹F(¨k+N̖JQ=Pgyѕͻ}KҸ"Ր'-֪s͋ۖBb?9t/bqA2\İ3_RXJWsΊa.Ϋ!A79~`㦪ZȎn"KNl'Mېx7b0J'-/{eCAGg\*sPvFzdz8aG7] '2rrC\&|߉D|׌5Ր"Uod)_(XI-?) [45qZi/e!꼝ys= XG:#ߑ@`CHIk- )xV $v>款(ϻ A=dU3,p XG53 {'])㚬}xY$ [44"RUٝ4KQ1Rz+tge98Pn`4Pdd MJ9mbLj;I1ҍH'6K>{5UD1>i, w&cYIՎ=j9 H|TotNcг ˠ 4d edԢbC n"AәP0#tvQŏPo*@Bߴ#DEL~8A5&~;_~g2bXH&9$LNtʨAr^4u,-cgih E<'v. { |Cmf4pRxr\ O䤪ϕCNjM -f`y0M)7VM7KI~iyiZ4>MDnAB 0ŔPy\))썞C`o~3c##f0,VE[ȔGSc~cK!F'VB|C /TRg2֏tҌJp4?hf2.EN9 .jlDҧx{nq}#4_r؍A뱫L]^h:4ʳ,?VVd"uk3D_g6U3D|qQd=95ا) H`~mkD< op(E9i]@I|@υm Mp` o6ޙ#_pOwq*\0G߅h)YHdvl5eKsO ʚes>pEس*Ԫ _BC@Zgm܈HrZ76KnK)4'7v6 5w|Y.i5-bNG+9Yib2PTT۪ Z,Ѡj>(s+7MK £ƾwZHLI*\yۂ)E"!'ƚ^z@! |_wa7jͫV\3sy{ʓ &Xr'{-Zy$Y1n"xnR1"1zBUlH{àW uD"K*UA|ZT+W ش'fa|(eү#ײQzr9_GTOG^wxۂMC$_/b&q@ QJ>+׳Gڨ@>ɘ<a|.i=LxC:'a)-B"#곳h} !B('̥:zWl] aK~Og  л_\wQʚ1,2`5V9~I;+JU֛lLtgj<]=GQAAMJT8kχ iFz@!iYdbzL8|6+Z/F2NaCY=Zh)XB0Jj |eYNa}ipOdu1'Ί`VV܃^Zte݂8h“:A9r+($Ӡ9qzq;R?(78Xn^!ڭekdψNµ:۫Iyү4RF=Sl%ƃ0pg<]Aqx\/t$H!L3k 3+f Y s# e:Ѕ:ߛr蜺lBO#qy.L0l?0TFL-1C4 fS*/+Nd[3, q!cF vRFUЎD>|J͕|w f숮Jna ,ZO1Jtћg ʑ*S~.ڣM}sgszbxl0я$m/$<|J|BP$ [fFw{@)SJ`%Z\Ss@@r-=B^LJN0C}q&BJځko+''W59:@N"A'^752p@ ݰMQ$YTO&zChzWwpWzz&{,6rP1P E)mb}PB*p Y١kO?mN 0t#XE^wJ铪;EI{O*^mG /T*qN:[m0yFlJ6o%|08 t[alQ-s9&΂̀Y;WtPaUD/1?qF]!X;1qw?Ws\[[ ֧7גZiS-d:7sOxlmGϛYCᄱD=p|edf8Ԅf A *vߡO2\7>mxNh=Z;ꨦ{9aCPeiqQu]Nρ}ދM;.9pR|0$UOYM.\OaKu{m+l*mrV^ 4:9;kO?b`zu96&2:`q`3}?[z& 3$z::hOap2YcB#E{qY.e6~8M\v B(oj3@Vu`!dS t޲< ֹ#DH5|-Ae !MUG'"~B :J{.9VSA*]u"[w">elj˃|l=4(;oX' ׄaWHZ52:&^~gp=n3WNW{\UD&lާOλjz/=i((16e7 Z3tv^Mlj"i5pXOVO>z V#foEMwÈr3~Zn"U[K조ʰk}lhrRQ=Y Ν' f+d{+_!B-i%{쏿X]BglމQQ)L1VGae)&3T@T(W`b"9?9bUI3Yp9urJ sdX0ᄾJ{7tSϳ E%%3/jVf9 g1Ynv˅wrZӅm^K4mZ]"nãs>ZI`{BJ8_K Ox@ࠕұH1 ~xe5CsIt&wuo@):-zߊxQ̲`p0XaBL$./+06 5@I^ΔmػӭW~O A5aK:ʤf: UO+_'4υ}.*B=#oMӺXkm Y0uf0iUȳ oHhyH SfNZ@~(ѱR 9-L݋|?ZFz"ݍ^i JGXx99L9F:6M,gt!v_S_6P}DTO$ ݂&梫6"$zv_mnSMe6ƟXFaml@2uNOqxH}|nꯀ~2P5̍RI$AQssZvt=_)wMh/pp~;VFGgmC;$<%Q +el=*3JRvG!1ukmn㓚͞:IN+" P$<58cг 5Da\~";W endstream endobj 779 0 obj <> endobj 780 0 obj [781 0 R] endobj 781 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 636 0 0 839 0 0 cm /ImagePart_2173 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 355.899 719.7 Tm 107 Tz 3 Tr /OPExtFont3 11 Tf (6.2 Predictability outside PMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 107 Tz 3 Tr 1 0 0 1 136.099 677 Tm 115 Tz /OPExtFont3 15.5 Tf (6.2 Predictability outside PMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15.5 Tf 115 Tz 3 Tr 1 0 0 1 135.349 642.7 Tm 91 Tz /OPExtFont3 11 Tf (If the dynamics of a deterministic system are completely understood and the ex-) Tj 1 0 0 1 135.849 619.649 Tm (act initial state is observed, then there is no limit to predictability and the future ) Tj 1 0 0 1 135.349 596.6 Tm 93 Tz (holds no surprises. When there is uncertainty in the initial condition, sensitive ) Tj 1 0 0 1 135.099 573.799 Tm 94 Tz (dependence on initial conditions restricts our ability to predict the future. The ) Tj 1 0 0 1 135.099 550.75 Tm 93 Tz (well known Lyapunov exponents \(3; 22; 68\) measures the predictability by cal-) Tj 1 0 0 1 135.349 527.7 Tm 91 Tz (culating the average exponential uncertainty growth rates. Lorenz \(62\) discussed ) Tj 1 0 0 1 135.099 504.699 Tm 94 Tz (using finite time Lyapunov exponents to measure the predictability of high di-) Tj 1 0 0 1 135.099 481.649 Tm (mensional atmospheric model. The weakness of using Lyapunov exponents is ) Tj 1 0 0 1 134.65 458.6 Tm 93 Tz (revealed by Smith et al \(82\) by comparing q-pling times which reflects the time ) Tj 1 0 0 1 134.65 435.3 Tm 91 Tz (of error growth directly. The q-pling times are used to measure the predictability ) Tj 1 0 0 1 134.9 412.5 Tm (by directly quantifying the time at which initial uncertainty increases by a factor ) Tj 1 0 0 1 134.65 389.5 Tm 101 Tz (of ) Tj 1 0 0 1 148.099 389.5 Tm 95 Tz /OPExtFont8 10.5 Tf (q. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont8 10.5 Tf 95 Tz 3 Tr 1 0 0 1 152.15 366.199 Tm 94 Tz /OPExtFont3 11 Tf (Outside PMS, there is not only uncertainty in the initial condition but also ) Tj 1 0 0 1 134.65 343.149 Tm 93 Tz (uncertainty in the dynamics. Measuring the predictability with the assumption ) Tj 1 0 0 1 134.4 319.899 Tm 96 Tz (that the model is perfect will simply overestimate the predictability. Without ) Tj 1 0 0 1 134.4 296.6 Tm 93 Tz (knowing the true state of the system, q-pling times can be used to estimate the ) Tj 1 0 0 1 134.4 273.799 Tm 92 Tz (uncertainty doubling \(quadrupling, etc\) time based on the sequence of observa-) Tj 1 0 0 1 133.9 250.75 Tm 94 Tz (tions. Knowing a particular q-pling time, however, from which the uncertainty ) Tj 1 0 0 1 134.4 227.5 Tm (growth rate can not be simply inferred, as discussed in Smith\(1996\), the rela-) Tj 1 0 0 1 133.9 204.45 Tm 128 Tz (tion re 27) Tj 1 0 0 1 200.4 204.45 Tm 54 Tz (-) Tj 1 0 0 1 202.099 204.45 Tm 60 Tz /OPExtFont2 11 Tf (g ) Tj 1 0 0 1 205.449 204.2 Tm 98 Tz /OPExtFont3 11 Tf ( may not hold. We suggest that outside PMS one could define ) Tj 1 0 0 1 133.9 180.899 Tm 95 Tz (predictability being lost when the forecast adds no new information to the cli-) Tj 1 0 0 1 133.699 157.899 Tm 94 Tz (matology \(82\). In practice, this is to say that the predictability is lost when the ) Tj 1 0 0 1 133.699 134.85 Tm 92 Tz (forecast skill score relative to climatology is arguably zero. Lyapunov Exponents ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 324.5 53.25 Tm 75 Tz (146 ) Tj ET EMC endstream endobj 782 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 783 0 obj <> stream 0 Z ,,#jQq)4KKk}9A;Fi㒜 ]J9L}`ƨ t3/#!\CU.m$%> p>JR3t4fh˩ BlO_O"cU_U DnxFZ^54 ¡?~]CUl;->A ESya CnXXO 7;I"$t!ρGԒC([,(VC#W}宰[4<w%?d9;kZZtY88R#ui^#7*f/%(e.JC#sc@:}/z3_= ZHc<͍㓊e>\;%?"Gn 6t^_CՏ_ի` bMgb rIYX'K%}N&XqG联aDLM@j跩#`?ZFXOU֍w(u\oW~K=o~֓EuyPhdG6aH@J:j-2!D۠zoYX)uT4@w:NG8Jn\~qG;2|C|߀M 6#~Xn#n8y/MLxRXxgRks33Vi&Uz ֮ fg^)A0BD\G/ t,hoN =3N"w匠\RBs UsMA=;IRhDrI+aFqq~Aώ-*B3IEl9# *!}7*8抾o[җ kDbƑ S}h 20wnfugY~Z[HaضND rlzx*oo &.2#^.&|I9%Ѹ2罴Y5j|k1q7Arr?4b;BI2R75#%]֘>{۳֊rgl kS$lEQ"%ɠpU] hc'- 2Ӫ{!o +goh'(M!>q<۴nw0Y* "E)%vjݦ,(?@;L{5p𧯂Z\V?s| /lZr9$_ Gq}L.jU߲=b՗t&6>gô3~ 6ACfݔDcקZdW-=@9 7y6B opB ] ~ţRԠSf=Ӄ"3LmPn[ X,*!͌bH *5.~.-:Qk#'4T2i~օӹ%PCr+A|CpF oVo DQd:cl\E~»G?)={ #|U:iM<@Vb32*NxcPivhF> u~H:z:$tl_clp;fN%tbǚhg^|\` $F^FbNW\Z_o5L仺b%Q S8?WH,R>1}J x tAz4zx)JU:7}5`@}B-0JGݹ;O^38]?kq E5Z A z)5T+d7ʬ,jnr# .'zJ3ɛ4f'2w3'G dExX虍юџbZހ4B6^4žm\6?W1rA]%v:wYxakva%Kg/)OC̅(uPpVpQݰK^ݮT.59Ru0e9@ "6WPNf(ͮwj'2uxNpۇhlH=d*Ks/FO!+L*+_G),X:Ýg ?Xw'Ȗ) bps7 *+5!zҿҴ^+$ӆy~*3IT]@(SznY,M{9Zhe H'Wdϲz,@i28(*0{cj4vnzB.| @[ ܹ=_XˀAie؟=r* TZ8#[9W[N\#G3@\q#?ҥxe8a0rJ"Nry h)7|B/xtYݲW򀀥|Tƺ ã;Wp4Kiuͩ"aޡGK)jiИHKbjs9d4kԛ$,v}u/O=f!5/ ?sp Dz|^6GO>"Wҟ+().!%#ș^41=D?ϏVS6<y[zc:j\0iQ"ؕXj2("pISe:D|CzA8:e0!V@tI#TR% Noז_uH"&& Ļ"ZS_Kt'9%O%|fQYH-YNfk=t+fIٸn,g"(ɯ[>L%dh(8+k7%T6^ á$pQ,{XR6IӔЯՇNKY%CGRԴ~ Hƕs;~2#LCVnx{[Zr?+}j6W]l$ | fIMA/nJ?yTv8?2c_B|258"xR=&|za3v]3hд">ٱ y+ FMBf(3ȍ*x~KhdDa_cU]~024ܘC.D'5,ҞٱZ+]: ,5 Ck@B]پvۘx*[4}/(jZ>rr[*%q'³٥x=sݰf#l"ЩG>-(ꁲϸ:/j?|9Q[GRԴM ƵLbN@cszgͣD͚#^>gϜsp|J/8$ 9=ZfҔ~[o`\Kpj5dR'^kмz4dn4#GX)i}"$[ߨT-ѐJqp^5;耢,&S<kSsKR߱ҰJڸ<>Zz#?|XGlb` }2H>_P 9'.rJGo7, !?.pѻ%#cGY݋iG?x# ʛi.h/A;lT絻nykûy3Q!u܏[pZBB^Nm^Ѕmŭ]J%aF?\b+i]hgƂ;/TjrS#j Qa}Q(mKK.% i(ҏ|E>"2<BwT8: z-;Ao1?+lņfELƫEoд)0!6'2^8U Q< cLӾy+Qf?-$S2*p"e${JvMjJa Xo=.\9^,Lob;/>ԙR0!0 S}Ovmt+Oq$ ^mxۥ,͖q+3VbA0uYٴv  9ssSG,\|`WxjH8Mvpt8*82*NQ*6cE[%RL?Q>ڄ,}97?:yɶٗ*=.!(#Ң<=m)G&qfHQlD:n0v_ڣJf[9k8:Sل:m)2I(Q"PcA~G%uLo#]If:1foAv+Z )(cXWK\(CsaX+v=˔`͛'x[WSyt)UYQ#0]G8x?&1- /=7*TeL/\U+#>iuGGF%csB%b{V&_]x_u(Uqa K[?:ٽew"IhS;^Z9Q7+`"_ [B;u0zcއ~ɚ=ܵIq=72A5%wV0 :*(ӓ0,sUAAU{/>?{a("s"{TUD^er(b%nn | {ƻogTMk봑%XU|\&FYf{B ;omG;NEi|T5fJtd_U%EDo?Qy|pkw0 2SH]]-CO"!}32rŹu"86뷼u@Dy>?bf|1a8a}C~J3%Pʻ(iE?>T!\Y~j2`w0 U/CүxM#gDɹEx]o䏆jFss [AqKФ3fTXOUucT[4NMfEL5z/w&?tc^/DPpq)W7u)պJ <=KNGԤЛd'gQsVS9r h3!ʿr,{G`NS 8D$pVɍ>V71YUVD s4醫v?@^eB&-fmkTK?p8R闥a"Y}(^ǥ?$.KDlĩ OD/a$Θ`Cxt^ux@QGNu<6yhjgh9Jrln&06(dZ-F%xZy(a|)9|m6O);g.&sSst>Fn _wXp/GPSܵl9ݦv6?SGdD-`+` 1Ж )om+!g]~UY߅#t ֬m-K ~oS$eϜX"NzR]+Y_%j1eYRIDV!j5% H`Bj{7Kqncj8`tZ} _n>7ʳ)r8>qٻӷb%`ٰؾ峦ʪư7ڑ<ŝãTj0Oq`h$}S/bϽi!,d?RWvhnGs)ýeLZlDj sid3~)|f Ή_#^[=J3膎R卨%hK<h %\-h:Oe]?q_INM!K0'1n' &7[ 熿r7,.juߑ$QiƿWVhϑ Ew6YvK3 *>涊ޣ4ʢ@uI-b㰺R?^o>P=_CO!K r=G@J(nL"єe鶂Df̴td~2~D"l?x7/X7nG&V|{=Gr4UGk !f* &ޡ&QWv@.Y}˳*puPp_[^0;ieGۘ=mlgߠ#8Wrs2I?00=}q 4 pyqZy ٳG1S+Ҟ9@y86:1I ٺ#D k-q9>Hm2bȎgIY g&F76\0T?a 0b=MCyNaM_$q_-l?l&+ tv\mA z lAFJn3ejxd'{WW Ӥ!DwUη+V0o& WJ~޼֤=`<\ƎmYcPy/o%b{D_+ٍ|I$_6 E~` x}gF2Dkg,nO3XPY Ex{:Ui TzP:d>̘ endstream endobj 784 0 obj <> endobj 785 0 obj [786 0 R] endobj 786 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 636 0 0 839 0 0 cm /ImagePart_2174 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 356.149 719 Tm 119 Tz 3 Tr /OPExtFont5 12.5 Tf (6.2 Predictability outside PMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 119 Tz 3 Tr 1 0 0 1 135.849 676.299 Tm 102 Tz (and q-pling time are discussed in Section 6.2.1 and 6.2.2. Applying forecast skill ) Tj 1 0 0 1 135.599 653.5 Tm (to measure the predictability is introduced and discussed in Section 6.2.3. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 135.849 608.85 Tm 129 Tz /OPExtFont2 13.5 Tf (6.2.1 Lyapunov Exponents ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 13.5 Tf 129 Tz 3 Tr 1 0 0 1 135.599 578.35 Tm 108 Tz /OPExtFont5 12.5 Tf (Given an initial state on the attractor of the system xo, The evolution of an ) Tj 1 0 0 1 135.099 555.549 Tm 105 Tz (infinitesimal uncertainty around x) Tj 1 0 0 1 306.949 555.549 Tm 51 Tz /OPExtFont3 12.5 Tf (o ) Tj 1 0 0 1 310.55 555.299 Tm 108 Tz /OPExtFont5 12.5 Tf ( over a finite time At is determined by the ) Tj 1 0 0 1 135.099 532.5 Tm 105 Tz (linear propagator M\(x) Tj 1 0 0 1 247.449 532.5 Tm 55 Tz /OPExtFont3 12.5 Tf (o) Tj 1 0 0 1 252.699 532.5 Tm 105 Tz /OPExtFont5 12.5 Tf (, At\) \(81\), i.e. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 105 Tz 3 Tr 1 0 0 1 254.65 490.3 Tm 101 Tz /OPExtFont6 11.5 Tf (c\(t) Tj 1 0 0 1 267.35 490.3 Tm 82 Tz /OPExtFont8 11.5 Tf (o ) Tj 1 0 0 1 270.699 490.3 Tm 111 Tz /OPExtFont5 12.5 Tf ( + At\) = M\(x) Tj 1 0 0 1 342.699 490.3 Tm 51 Tz /OPExtFont3 12.5 Tf (o) Tj 1 0 0 1 348 490.05 Tm 104 Tz /OPExtFont5 12.5 Tf (, At\)e\(t) Tj 1 0 0 1 384.25 490.05 Tm 47 Tz /OPExtFont3 12.5 Tf (o) Tj 1 0 0 1 388.8 490.05 Tm 85 Tz /OPExtFont5 12.5 Tf (\) ) Tj 1 0 0 1 391.899 490.05 Tm 2000 Tz (\t) Tj 1 0 0 1 508.55 490.05 Tm 97 Tz (\(6.1\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 97 Tz 3 Tr 1 0 0 1 134.9 448.3 Tm (For a flow, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 97 Tz 3 Tr 1 0 0 1 331.699 417.8 Tm 93 Tz (to-Ft ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 93 Tz 3 Tr 1 0 0 1 231.849 405.8 Tm 117 Tz (M\(x) Tj 1 0 0 1 255.099 405.8 Tm 51 Tz /OPExtFont3 12.5 Tf (o) Tj 1 0 0 1 260.399 405.8 Tm 103 Tz /OPExtFont5 12.5 Tf (, At\) = ) Tj 1 0 0 1 298.55 405.8 Tm 170 Tz /OPExtFont6 11.5 Tf (exp\(J\(x\(t\)\)dt\)\), ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 11.5 Tf 170 Tz 3 Tr 1 0 0 1 326.399 395 Tm 67 Tz (to ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 11.5 Tf 67 Tz 3 Tr 1 0 0 1 508.55 405.55 Tm 96 Tz /OPExtFont5 12.5 Tf (\(6.2\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 96 Tz 3 Tr 1 0 0 1 134.4 363.55 Tm 107 Tz (where J\(x\(t\)\) is the Jacobian along the trajectory. For discrete time maps, the ) Tj 1 0 0 1 134.65 340.5 Tm 104 Tz (linear propagator is simply the product of the Jacobians along the trajectory ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 104 Tz 3 Tr 1 0 0 1 223.449 298.299 Tm 114 Tz (M\(x) Tj 1 0 0 1 246.699 298.299 Tm 51 Tz /OPExtFont3 12.5 Tf (o) Tj 1 0 0 1 251.75 298.049 Tm 53 Tz /OPExtFont5 12.5 Tf (, ) Tj 1 0 0 1 256.8 298.049 Tm 107 Tz /OPExtFont6 11.5 Tf (k\) = ) Tj 1 0 0 1 282.699 297.1 Tm 118 Tz /OPExtFont5 12.5 Tf (J\(xk_1\)J\(xk-2\)-3\(xi\)J\(xo\) ) Tj 1 0 0 1 421.699 296.35 Tm 2000 Tz (\t) Tj 1 0 0 1 508.1 297.799 Tm 98 Tz (\(6.3\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 98 Tz 3 Tr 1 0 0 1 134.4 255.799 Tm 103 Tz (For a given x and At, the finite-time Lyapunov exponents \(62\) are defined by ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 103 Tz 3 Tr 1 0 0 1 279.6 213.549 Tm 106 Tz (\(x, At\) = ) Tj 1 0 0 1 324.699 213.549 Tm 676 Tz (\t) Tj 1 0 0 1 345.85 213.549 Tm 94 Tz /OPExtFont6 11.5 Tf (log) Tj 1 0 0 1 360 213.549 Tm 66 Tz /OPExtFont8 11.5 Tf (2) Tj 1 0 0 1 364.8 213.549 Tm 74 Tz /OPExtFont6 11.5 Tf (o) Tj 1 0 0 1 369.1 213.549 Tm 39 Tz /OPExtFont4 11.5 Tf (-) Tj 1 0 0 1 370.55 213.549 Tm 100 Tz /OPExtFont8 11.5 Tf (i) Tj 1 0 0 1 374.899 213.549 Tm 48 Tz /OPExtFont6 11.5 Tf (, ) Tj 1 0 0 1 376.3 213.549 Tm 2000 Tz (\t) Tj 1 0 0 1 508.55 213.299 Tm 97 Tz /OPExtFont5 12.5 Tf (\(6.4\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 97 Tz 3 Tr 1 0 0 1 133.9 171.299 Tm (where ) Tj 1 0 0 1 166.55 171.299 Tm 79 Tz /OPExtFont6 11.5 Tf (o ) Tj 1 0 0 1 180 171.1 Tm 103 Tz /OPExtFont5 12.5 Tf (are the singular values \(in rank order, i.e. with cr) Tj 1 0 0 1 420.5 170.85 Tm 69 Tz /OPExtFont3 12.5 Tf (i ) Tj 1 0 0 1 423.1 170.85 Tm 103 Tz /OPExtFont5 12.5 Tf ( > ) Tj 1 0 0 1 439.699 170.85 Tm 100 Tz /OPExtFont6 11.5 Tf (u) Tj 1 0 0 1 445.449 170.85 Tm 140 Tz /OPExtFont8 11.5 Tf (j ) Tj 1 0 0 1 448.8 170.85 Tm 103 Tz /OPExtFont5 12.5 Tf ( for i < ) Tj 1 0 0 1 489.1 170.85 Tm 123 Tz /OPExtFont6 11.5 Tf (j\) ) Tj 1 0 0 1 503.05 170.85 Tm 95 Tz /OPExtFont5 12.5 Tf (of the ) Tj 1 0 0 1 133.699 148.049 Tm 106 Tz (linear propagator M\(x, At\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 106 Tz 3 Tr 1 0 0 1 150.949 125 Tm 102 Tz (Since the singular values ) Tj 1 0 0 1 276.25 125 Tm 74 Tz /OPExtFont6 11.5 Tf (o) Tj 1 0 0 1 280.1 125 Tm 58 Tz /OPExtFont4 11.5 Tf (-) Tj 1 0 0 1 282.25 125 Tm 83 Tz /OPExtFont6 11.5 Tf (, ) Tj 1 0 0 1 289.699 124.75 Tm 102 Tz /OPExtFont5 12.5 Tf (are positive, the Lyapunov exponents tells us, on ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 324.699 52.049 Tm 88 Tz (147 ) Tj ET EMC endstream endobj 787 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 788 0 obj <> stream 0 Z ,,}}jQq$'fr~FqSCb2d3ԜRGߧKCm W4d xzfi\W4i*WTE[;iion:3 Ld6o9Zc>8-غ,!Dq.=V=qnǠ!F e{@+-ȠND˄q4Ds'kVptE/V*d*O RiK?U쫇 !YOsH7 p rBgx,]HY*Ft¥ h9sjjVB)o 5ͥ0Y̓o?Ma }U`a\kwEcabX#ȥy+ OT?+Q^ ~p}sJȿWޭ` U0yB"J =z<>=JaTzBPg1Gṗ$~Lj8"|zcڔ0'1ԅ"+2qTi:J~9{^5%GIDf:H )@`ŒGy;f;ǿ[fcD SvO=/i`)o]%2H]JY/<.>!⧯o)ohC;׷S"M2ŁtU%K\B΍lSk?'| 4SS}yv#͍CtmIo{Tl^獭D BrON7|jQ{Styz-ʴ}!9aLir#A٥5JlbTEßzOȷxGQ DZa'j?YHgiat=C:_yYMe-WuQd8nﱢD>z(Soxˮ v{/UZ(Wӟ|b70s:Kj֟KJ] ~q]M#t*UZ 0eLE8JN<;*/^ ü{Uho+#tgp ;fK5Bڳ+FXE@#[WCx*p"E_ @9Ig'J? 5ͳt{gdHSAigw*Zg:m+v}͋0&Zu#bH@Omp7Dj!Fa}V`2Q[ m j ofIo?<03tebWgr$H*І>.:NU3 b#{Cp5- w=Fy"jkk""pl6N*Ծ\ ߑ h,]RH4OPj5ֹd T Q@4reevMGэ ga_8vV.c_(鱖jVY.ˋ"fwz!^u0z5 3Iƞr40&$)Vq?RРIW}*M4oѧ C *2|Z(/wyr}CaDO|h5=Ѹ}HɡrA1O5)X1Nmn+$ yRyg yG(C"S3&`x[G佅г +{d(Nah_6PSO!ӻ yɽBmJ Q>?xĮɑU=qr7Ոg >|:&{OM4uyf@Ǒ/DᙏG;4#+hS;W6m57-]}^[wD &/.r7}%^m/+sxTk@np|Qۡ=/@<Qbb K:ƵLYHĭlS ke=b ޘenc;s"0#]{ z{x E}vmUY2 ,=g- zݻE %H"*˨oSls:Pݲҝ v|Ԍyա# mp?Q^HP$@Tf!NZyyFؚ*OeK ^% C(a5 Pėu,ĸէDX!><62u꞊'??_>Oe"i:JRv12$:8ɡDžCj>(^\sE/39Sl®o&qfE>@pQQV[vȓ:q_zlSJffe5 ː8ȗmTn*"ڭw4FJ^9[:@;2<@"( 0T5y0icִRw6+Et]:-Yy,@5,[g䉿pǩwJ1O^. /8:z`F<`a ;O 0>#.Eשּ~eP+-T$ɒ~9LH, U1>?h/rb֒xF@]-488fNJ1W?񵦢uvaԅ)( 4f-c,&$F yx:Q-@w> ,R0Ң7{)+PtְDqfAg;$s #6ɩ-<'^Գ,L{(ZDU~ EUg(NL?ʛb߽\ݖeiYrya<5EM0J Ҹ{XIu73%QڨF΄mtrGǙ#/#LbiaS=R.z0,Μ@7]U60۷d-Y>~m=^0UVo gxo<>vMJz_جhoeK:[( oΐI:L?x4XAϊ{m[szD4 o#}=­X"U Qn Kn J5>?E.0kHŒ)KW|BIg cG>7Q'_k;ZZ ͤ`y-o)A|JSB{d[lxJV#H K{DlU@ԵODN=vDry]ّNf "p-O*+46gsK2R qkfK "MT~EwG}bf?A"N粐2gc"+#kPepPRNۮ-`)6@dGimwyEb 3 Q"Hb6rU/ϱ^`7,cʺ\L,Es( VK2.'_I7' CDM;vz po8QaiԷp"eA#!c)p-t_K=V]q7UkhĞ]ujEͲO!qet+V7rp.BʁG.i (N3 լN2{^pC} `*] d3Rn<POe9)jAK;FPt|=B.bOdUoNQ.,cZV nhcwS?$`ъI_NCX8iR=#7f?ܸK[4HXQX[`Ա(% Cۚ+:hK8Pa[% 'IF[ČЁT̳JY7Dkppf<. LJH;YMiJiJMG{BU)zRFrex_.9dXMG .)._do75Fv"GyO"ٓ6BoItpBoP?2ޒ+M-cἒ{S[7G-Z&ѣO,(8\]/ Oo$Cq<4@; 9|qiGVpLQH|] [9Vn1iZ%d{+$?8ə!/+%}8XWvrPrkxBSaO_ v=*y'u5*IWyWN!pϑ;Ārj(Ie&K o5WKH!O%?X|95=[0`=xz ~>Zy* %V~l%R"{b`isbG&,VIqoqC2}[6򵔁}o+,2++oޜ"hSs Ei~mr싷r7-5m*e>%n|yyo M̱"mJ6j~W+mqjunYʽ8rUH_)EL"laJW s`f!ٕG~Am{ߚqmߘzPȫ`.`톋nȈُ?nwXLwϺ#h+mc;2p | 9m'>G!7EJE_}95&r5 ?=ňq#\[Wlq,8|2P}809K\I<V} KG@FK$O][m(2?F60Kެ`ƚ}u;E=\-̾s/3PTaivGQ'7oi:#)w@9[)+X "lOϴ. 7Sw>G(BAC m"4WW fu/_ۨ.mP:`xOZ(";8QmWC97åK҉>%dc[b#"#iW_{4,4C*f]XNF*:%gbٜ 6M.?-bQ[a ;pdl$wEjEYQz'a6gskuɬvot&,c, n[R"rl6=ʾi7 |nee$UF‶=5/4XtN 뵿7qaS]kU';Kxnh}*mq,mnx#w)OIa'm*߭Ch@@C2'e s9=n"D-ȽOlj0Īb,ץH(\3ܖ5[HuwZK4n`.]B(W|=fU:vu(3h1E JZk5K?s[m{/nm+lmЈىK>@ c5o_Na s@1ۥg-E?/JnbOǾ: sKde92VE5j\3%Wm&oQRd 5dK^I:P~dodӁo(Ih`rmXR^rp1X3H xAOơz~zŽ#sŒQB76r TQN k|R6 7E I1,]ԗ;ֿCQct&Q3FcI7,Q&o{0 (gN~ S` _1i߿k`~n;45XTvߣ"$V@,9  B Z Έ 28iNԽuehYÙ|v˓b$bFHX<$Ci.t,exϬו#?Q൥M&GC5B3vo=Rl4M20s:VIv5zֽ@!膆`xTE1L)ꤨ\+ %>ڹ!< XioBI Yzz#A)V%vpS/dwvb$p 65ѝZSVdH`zQ=Ȑ*om*܂hXoE S>ŏUr{+z`aPs;~SCph'|EM2qqޡ %'1=FEl%@;s%KUDm{)}Umݮի x[k}P}_ bK.B_do %'_OOhpPQK?Ui_hdf>>4}3Kg ̀51l..*`r=`K9=NQ=Kngհd"]' o^뎨t$"қr(ypӧ_nƕ"(& B9-Mr畀`372BE4g0Y+LzO>B$J=|ɩO, Y+βQ옰Q*KBq="qs-F vrw7UK~a.hf;X,ďK硗T{H k; *I?mm9 w>x w7Gu &Vuv&"j4rf̟Ol({.|iyX8gȠ Wi,Lv~mX.i  RE5(~)z_I3Ag._\Ʋ>c7mi*OxUSHł: +(HIt4\)9ZT:L+S̗Dz]c+]Rp(j6G{?q>ڃ͛l>ܰ31f.ܘХʶkhy/By zFpQ!qE MYF?ޚ2?TKh5sp5fJl~JvU'r87nEf\6UL@i{ ;-o.GPwLͯcpp'4%uͲ>3b]Τ<Ҿ 2 &tX-^݀K1URlA_!&ݼل]-_)`,R?sO*kc.TMx˃6q$"5hQ#=j.䮰q*S̵c1[|ܖ!Q .m=YyW{T' } &O* yETJםNH[> jGfaTQ>] uÈ1S9+ endstream endobj 789 0 obj <> endobj 790 0 obj [791 0 R] endobj 791 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 582 0 0 838 0 0 cm /ImagePart_2175 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 304.8 719.45 Tm 123 Tz 3 Tr /OPExtFont2 11.5 Tf (6.2 Predictability outside PMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 123 Tz 3 Tr 1 0 0 1 84.7 676.25 Tm 98 Tz /OPExtFont5 13 Tf (average, how fast the initial uncertainty grows exponentially over ) Tj 1 0 0 1 409.199 676.5 Tm 96 Tz /OPExtFont8 13 Tf (At. ) Tj 1 0 0 1 431.05 676.5 Tm 65 Tz /OPExtFont5 13 Tf (A) Tj 1 0 0 1 437.75 676.5 Tm 35 Tz /OPExtFont3 13 Tf (1) Tj 1 0 0 1 442.8 676.5 Tm 103 Tz /OPExtFont5 13 Tf (\(x, At\), ) Tj 1 0 0 1 84.5 653.45 Tm 100 Tz (called the maximum average exponential growth rate, reflects the error growth ) Tj 1 0 0 1 84 630.649 Tm (in the fastest growing direction. In the limit ) Tj 1 0 0 1 306.949 630.649 Tm 106 Tz /OPExtFont8 13 Tf (At ) Tj 1 0 0 1 339.85 630.399 Tm 88 Tz /OPExtFont5 13 Tf (co, A) Tj 1 0 0 1 364.3 630.399 Tm 56 Tz /OPExtFont3 13 Tf (i ) Tj 1 0 0 1 366.5 630.399 Tm 98 Tz /OPExtFont5 13 Tf ( \(x, At\) approaches the ) Tj 1 0 0 1 84.25 607.6 Tm (global Lyapunov exponents, which are the same for almost all ) Tj 1 0 0 1 394.55 607.35 Tm 120 Tz /OPExtFont2 11.5 Tf (x ) Tj 1 0 0 1 405.35 607.35 Tm 99 Tz /OPExtFont5 13 Tf (with respect to ) Tj 1 0 0 1 84.25 584.299 Tm 98 Tz (an ergodic measure \(25\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 101.299 561.299 Tm 107 Tz (It is proved \(94\) to be true that the largest finite time Lyapunov expo-) Tj 1 0 0 1 84 538.5 Tm 101 Tz (nent\(average over the invariant measure\) is large or equal to the largest global ) Tj 1 0 0 1 84 515.2 Tm 102 Tz (Lyapunov exponent. A positive global Lyapunov exponent is therefore often ) Tj 1 0 0 1 84 492.149 Tm 99 Tz (said to destroy any hope of "long-term" predictability. Actually both finite time ) Tj 1 0 0 1 84 469.1 Tm 100 Tz (Lyapunov exponents and global Lyapunov exponents reflect average rates, not ) Tj 1 0 0 1 84.25 445.85 Tm 101 Tz (average times \(81\). Smith \(94\) gives several examples of common chaotic sys-) Tj 1 0 0 1 84 423.05 Tm 97 Tz (tems to show that even the system has positive global Lyapunov exponent, there ) Tj 1 0 0 1 84 400 Tm 101 Tz (are some states on the system attractor about which every infinitesimal uncer-) Tj 1 0 0 1 83.75 376.949 Tm 100 Tz (tainty will shrink for certain finite time regardless of its orientation, which also ) Tj 1 0 0 1 83.5 353.699 Tm 99 Tz (indicates that the local dynamics of uncertainties about that initial condition are ) Tj 1 0 0 1 83.5 330.649 Tm 95 Tz (more relevant' \(62; 82\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 84 285.75 Tm 113 Tz /OPExtFont3 13 Tf (6.2.2 q-pling time ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13 Tf 113 Tz 3 Tr 1 0 0 1 83.299 255.299 Tm 103 Tz /OPExtFont5 13 Tf (In stead of averaging the error growth rate, the uncertainty q-pling time \(82\) ) Tj 1 0 0 1 83.299 232 Tm 101 Tz (measures the average of minimum time required for an uncertainty reaching a ) Tj 1 0 0 1 83.5 208.5 Tm 102 Tz (certain threshold. Given an uncertainty co at ) Tj 1 0 0 1 317.05 208.5 Tm 96 Tz /OPExtFont2 11.5 Tf (x0, ) Tj 1 0 0 1 336.949 208.5 Tm 104 Tz /OPExtFont5 13 Tf (a q-pling time \(82\) r) Tj 1 0 0 1 440.649 208.5 Tm 47 Tz /OPExtFont3 13 Tf (q) Tj 1 0 0 1 445.699 208.5 Tm 103 Tz /OPExtFont5 13 Tf (\(x) Tj 1 0 0 1 456.25 208.5 Tm 46 Tz /OPExtFont3 13 Tf (o) Tj 1 0 0 1 461.3 208.5 Tm 81 Tz /OPExtFont5 13 Tf (, c) Tj 1 0 0 1 471.1 208.5 Tm 49 Tz /OPExtFont3 13 Tf (o) Tj 1 0 0 1 476.399 208.5 Tm 63 Tz /OPExtFont5 13 Tf (\) ) Tj 1 0 0 1 83.299 185.45 Tm 100 Tz (is defined by the smallest time for which the initial uncertainty c) Tj 1 0 0 1 407.75 185.45 Tm 46 Tz /OPExtFont3 13 Tf (o ) Tj 1 0 0 1 411.1 185.45 Tm 103 Tz /OPExtFont5 13 Tf ( about ) Tj 1 0 0 1 448.55 185.45 Tm 113 Tz /OPExtFont2 11.5 Tf (x) Tj 1 0 0 1 455.3 185.45 Tm 53 Tz /OPExtFont3 11.5 Tf (0 ) Tj 1 0 0 1 459.1 185.45 Tm 102 Tz /OPExtFont5 13 Tf ( has ) Tj 1 0 0 1 83.299 162.399 Tm 98 Tz (increased by a factor ) Tj 1 0 0 1 189.849 162.399 Tm 85 Tz /OPExtFont4 10.5 Tf (q ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 10.5 Tf 85 Tz 3 Tr 1 0 0 1 130.3 119.899 Tm 69 Tz /OPExtFont5 13 Tf (T) Tj 1 0 0 1 135.099 119.899 Tm 51 Tz /OPExtFont3 13 Tf (q) Tj 1 0 0 1 140.15 119.899 Tm 90 Tz /OPExtFont5 13 Tf (\(si) Tj 1 0 0 1 150.949 120.149 Tm 49 Tz /OPExtFont3 13 Tf (o) Tj 1 0 0 1 156 120.149 Tm 81 Tz /OPExtFont5 13 Tf (, c) Tj 1 0 0 1 166.099 120.399 Tm 49 Tz /OPExtFont3 13 Tf (o) Tj 1 0 0 1 171.099 118.7 Tm 89 Tz /OPExtFont5 13 Tf (\) = mint>oft 111 A\(56 + eo\). frt\(Ro\) ) Tj 1 0 0 1 348.949 118.25 Tm 701 Tz (\t) Tj 1 0 0 1 371.75 117.299 Tm 81 Tz /OPExtFont8 13 Tf (q ) Tj 1 0 0 1 389.5 117.299 Tm 73 Tz /OPExtFont5 13 Tf (eo ) Tj 1 0 0 1 398.399 126.549 Tm 1830 Tz (\t) Tj 1 0 0 1 457.899 119.899 Tm 92 Tz (\(6.5\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 273.85 52.25 Tm 85 Tz (148 ) Tj ET EMC endstream endobj 792 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 793 0 obj <> stream 0 } ,,ڔb7 E;A8^"J]M Q!n٤멱1߽ʵ"a<e:>/4nCʈɧ ~&>@fZ1*-gH nsݿabjD2ti:D*#ojM·277x=Lsv8$bK鳐avg{tf)ׅj%VЮLxÇ2"AY?w]bI]1|&&G;6ᾶI>~ұ}ghr$۽Gjwo@0$ f7q T<mI;^y.Ѥ-IȏveqrCdr*d̗lC' _0&I8<~"W~%|d(=āyK%hZBdbm%T]p ʷ 2 9bmiަj3Ȩ_Q)X >?} t: _fv:LUg$J`AtVި 5!/V ">EƢ} 8BFSJ33_{4ΰ.{؃jչz ֑sk /µV*hv)@G'K P҉L|C gUY #@-8.5p~rI'!(nA0npa{Y̠<\smYij߄ȨԀLX]"cg:Aa(1GlVkn/N]1eO_D[K:xr/omu帪!hŒmW*qR%T#Uyՠ@,=4y—mZJM|҄-Ug}zc+0 -'SH126#n̉`lVdxn -tQdj%w$hL~|C$W lGU!~}}NVx`\T)1zɝ lܐ]^YWIxC͹ PSzk[UgG$+'JͦQ{a+gy/f(d{'ɲ{&H^<5?Ƽ?|[GDbǓǏ&5\i2~[Teh7:*C/F^BiO(2`ϭi'<}T]PH@'HAoPS·mEZ fsjPѬc)|K-ZfWR:$c*aii9;+:j.ﾤzd\Z.dW>Pf܎0kkq&'QR b4|;IN} ˶%Y#eU*!@5qL ̈XSlZ<`t8[ WgRhd$R~و vcg}P/^AXXI*caս/)@KOcM,`6P]X91v{jG.dJ٩n %n|{ #+GۀD'Lf]pXiʊazwQ,&ݛb"Igq>›n]E1]I#֚``%s֟UWu^㭱y4} )_O~ԄdFtwڐx`MV}Uzz_ι8LQRqdk\Nn:*";=4Qke)Y>cl'H$+y[ū]E/.ō`}`2ik*һRe΀8簧e35dض]2bsompz߹h. v;J;K.Qwullܧ;ȆZV5J({^"%7j;áO"O3S80 AJ1zb;&=I>16.O6,YB_DG}˛pkw89z ]X=>bxHs s~Եin$;LCو3\ybkHa ^cWYfCY -|z}J(!@@l)veڸu#+p}ҋS]т |3ǰeFWV|Lv/6Sw )~葏Yz6E8G!rznk΃;ܻp٤E[GDzqomՊF-  1 S5FW ѢݘamϿfSj1j(LWo)Ih$K?PC*kGyC$GGbv{9l;_ڛ!mrYy[<R,JG;TȂQ^5=D$7_}ZʴP r*Q@!?`׆ʎnv¨,%8N&Z$ A/'_4H҃ަ"î9(ASg{՞m5 WKH3vgB'BAq {`br3͔1dIUOI Ǯ*}Ea)#1M-(v(#|gLXE^ Ӣu[! xA8~;y i vΣJl_f1-f72}g> {pdWvlK}CۖdpOo65Lq-F Z$gpS8r>,.Z3:O:$ ʕ%-48 9 KgVFMnjGK'{q_3|9C[|mgBb2Pϓʡ:o 66ÓfS#:>7Yы'b(p5^QWx(Pq;~.[&m=};jtAIы_RͰ\%OTT@J0|FG[(ʺ2@Yf0mx6(@^nT6ɱcQ&_aT;f 9,w3h*& ozy|;<E"0ʦ R)aM r7st5D$3ܨqOPY ia>R=1^RB]22}´sKW ZAm3N\(ՖiRπu5NBc΋muaĜ9ț`dވoqω&}V4}OC`䱖~Qϲ_ +ȿ-hUJ"*%X&_PlљMl*= rKW OYFL}qo ˨#*:̓/,>|JGq" JCrЏ?YkJ مtW_cI;7ĉebu9/>E*riWYos9(?+ QxeУ`SHT%\cSWo^_ i{]drDeO`%Ff$]{WٹM/.8ezHhj.pǮ(+V&szK%@@` lb{9X>>ϳ^z5ʴ~#,e,!.wT辱3˞8(>!^نy?+-.B迢z/簭:h< f!Vӥjx>kQx="WQO˷B2Hj#iһ`0IZbٹC . f&^Pu^[Č $YCl6C52nB`WՉ ?qй cip)FuBH|_EuGBL d \}SZO NDyd]W0>rvUa at7\^N`580Ef|< 7JUKUЪrK',EюR9!χBK<U_bYW`u\ndu$ ZT7;qxHwZ2fuC*!ݕW(;-GYp#ꄯ=YBH0kH rg_n;2arDfoF|EVr:5>*_tu!\xIV.$ 'SI@k{8cU1<_<^@TH=Rm85˘`gU漵 LS*՞KȒ4pI҈c zl5 EIdGٶͦ}s T/u^b[1}r#<*_Qhq=O'F>[|0p 8R ` } N3`_t%e2I=vİKK#3@=veom[j'5Tb Nkq .WCEߏ:OHk`N' <,.DU*4Fw|d> ӭ,F2\i|mHbH<{ޓ=W!wAZ8_Dr^,Cy梠4&c 5aG @M6J \/\ E+P )/[>|.d1J~d3ǐ/9#הvYaۢ-Q,iv4T/]baO $/~sW)YUy[OjWGk;AOPw6ty s=[s:A]dl iHLhZ/1QA0#beXyb9M$B3m,0d<__Hul ra=(odp(=ۢ.r9F Noe ^ yj g-:hi[D<&},2P9Sj 1aPG>/ x-鋈|m La~l7Uauuf$kWj0ך4r&jńPʨ鶯 @5jP9:5birEC%)*nVOP$EV 1XpA{I@RT4 8r^@"]اPBgB'YU5̀L_{f3ZRw@s=,̇D==avf7Wtz9]!!=kOVW(TZչR^I-\› INZu0[frT[?5`EwU9; sdp$1:.V-}w /CO5d|10-r#0SY ]!pt(D{ 3OK@+{HŶuKU1E#OLYÌjY30̧}edOz-ߐ~*5x0:aPkv[()=Y.)yyD%+"jҬewZ&,bsf"vߟU8+( s:mp$ԋ"RVDZ+emD[/)< I];?|Ҹ?BBu{ljw{T`'Uަ yWeӏX0sNRhGZ0Q>EMKzܶ]zݢrNp?1:&QD` v$r9V1sv6aqK5`8q9Zd2*^7W63[u2VhnaBw.I粊؄E:T$lT Rl^?aAw>-0d؜|(RVr"^| T\>#&u+¬W+%ca启ᶹ=ǵnհնPuJTmӄSݒium %WgvM3O"M+6bDlnO}38Z9pA}TW @(یŽ?RO꺏a;Nk;'\3p#^e0"Bչ`)嘕>7j0a߲d k ZGY0d^#·U~`voJ_lt١lKߡ+SAg.+UkEvt3/󓐴NѮ]ʌOg?q#'ٙ|հo}fJݖ5Etb0w2=KMGxJ#([DYmLP&P):Jm,QׁK:\4n&{_-fn[ $RO/>nhT8|9-Rr[Aj& 4kU G,$ #O$KsZv*Iy)̼{dEN;(FYуpVL?=4.0>AQ bŭT7rДop x,,9ŻYGJq=eQN]3zRFVYeRe,;B,դ)\u ^<£N9[X!\)p1q~$G| TsaTi*:vZTl0G*&E Eu"Ov {rن)VLtPUͪtI]piYGa_W9#@=BPwsz3Kɖ"Z-[p= s04铞 SKeh讥,츾}"A:'EHYWii n dzs-BSڝT.~ 69 6 XmߪE*b< iaŃtyR%4%w"9"gZEjIس\_B8)4x'Rψ,c /CԌ,q.)4;JphV!U9m+/ј*>Va2o`K ;}M6 א2fs׹k@>q|[ Z;n+:t&7zq 3]prq^Ӑsʸ1<fyfbgnW(7\%YⳆT0L9ֲ4~dUYրqyӛd{0|y|TܦQ3.uNf^<滈vp?H#_(.& _);]QTl@IXcLRp4o]»Wz0T!Ӽ*qrL7Oޜd+f<NJ(mFd)ON%#fv̾1h`|AlDu UJOM$P$1ArG$xFL2*nzHm2J endstream endobj 794 0 obj <> endobj 795 0 obj [796 0 R] endobj 796 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 596 0 0 840 0 0 cm /ImagePart_2176 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 318 720.7 Tm 107 Tz 3 Tr /OPExtFont3 11 Tf (6.2 Predictability outside PMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 107 Tz 3 Tr 1 0 0 1 97.2 677.75 Tm 92 Tz (if single value is required, the average q-pling time, rq\(ii c II\), is then defined by ) Tj 1 0 0 1 97.7 654.7 Tm (averaging the q-pling time over all points x on the attractor. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 114 631.899 Tm (Although the q-pling time is often \(81\) defined based on ) Tj 1 0 0 1 403.899 629.049 Tm 67 Tz /OPExtFont8 13.5 Tf (E 11-4 ) Tj 1 0 0 1 431.75 631.7 Tm 91 Tz /OPExtFont3 11 Tf (0 in order to ) Tj 1 0 0 1 97.2 608.899 Tm 92 Tz (compare with Lyapunov exponents, the q-pling time can be calculated based on ) Tj 1 0 0 1 97.45 585.85 Tm 94 Tz (any initial uncertainty, which does not have to be infinitesimal. For Lyapunov ) Tj 1 0 0 1 96.95 563.049 Tm 93 Tz (exponents, the linearised dynamics, Equation 6.2, is based on the assumption ) Tj 1 0 0 1 96.95 540 Tm 92 Tz (that uncertainties remain effectively infinitesimal for the time scales of interest. ) Tj 1 0 0 1 97.2 516.7 Tm 94 Tz (Clearly, as long as an uncertainty is infinitesimal it can place no limit on pre-) Tj 1 0 0 1 96.95 493.449 Tm 96 Tz (dictability. Once the uncertainty becomes finite, the linearization, and hence ) Tj 1 0 0 1 96.7 470.399 Tm 92 Tz (Lyapunov exponents are, in general, irrelevant to error growth \(82\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 113.75 447.35 Tm 93 Tz (It is usually impossible to derive the Lyapunov exponents and q-pling times ) Tj 1 0 0 1 96.7 424.3 Tm 94 Tz (analytically for a nonlinear system. In practice, to estimate the global measure ) Tj 1 0 0 1 96.5 401.3 Tm 96 Tz (of them one sample initial conditions uniformly with respect to the invariant ) Tj 1 0 0 1 96.25 378 Tm 91 Tz (measure of the system. For each initial condition x, the Lyapunov exponents, i.e. ) Tj 1 0 0 1 96.25 354.699 Tm 90 Tz (the uncertainty growth rates, can be estimated by iterating the initial uncertainty ) Tj 1 0 0 1 96.25 331.699 Tm 100 Tz (about x for a fixed lead time; the q-pling times is obtained by iterating the ) Tj 1 0 0 1 96 308.649 Tm 94 Tz (initial uncertainty about x until the uncertainty has increased by a factor of ) Tj 1 0 0 1 484.8 308.399 Tm 87 Tz /OPExtFont6 11 Tf (q. ) Tj 1 0 0 1 96.25 285.35 Tm 95 Tz /OPExtFont3 11 Tf (Outside PMS, calculating the Lyapunov exponents which have to assume the ) Tj 1 0 0 1 96 262.1 Tm 94 Tz (model is perfect tells us nothing about the real predictability. Arguably, model ) Tj 1 0 0 1 95.75 239.049 Tm 90 Tz (error may be more responsible for poor predictions of real nonlinear systems than ) Tj 1 0 0 1 97.2 215.75 Tm 97 Tz ("chaos" \(82\). If the observational noise is free, one observes the projection of ) Tj 1 0 0 1 95.5 192.5 Tm 93 Tz (the system states in the model state space precisely. In that case the q-pling is ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 112.099 171.85 Tm 96 Tz /OPExtFont3 9 Tf ('assume the projection operator is one-to-one identity ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 9 Tf 96 Tz 3 Tr 1 0 0 1 286.3 53.049 Tm 77 Tz /OPExtFont3 11 Tf (149 ) Tj ET EMC endstream endobj 797 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 798 0 obj <> stream 0 ,,$b7 La~)^W<;J !l򾭲%j}tc;j)&jḘxizhұ;jq`t'E_*B VQ\r&RهF7Ja{ fca]$HYH΋+t'.yiATs˿ƴ,\(cnc}?H+a)r@<'-u 41|۱Aޛg=1=P|RaUo԰Y[$?%Xȓ~82Iog{l /D@JӼҜw[O:2#c2Dl TqWǸ[ Kf7 :L.Wmu4s%<+3igQsߵ1[UOi\(\& ^+˿We!$jzRfٜŢQgpԟYjoFB!J%Pxi;lE:2U"tGA_G!uN7׉X|}|rim ]t1.Ir^tGi(;uҜ(NIaJ~iuؚI WG`ߏ1?挐a ;N+Qe)?nr/# ZR<1M/["x-s9և~w32Q"J=HAвzug_6el!T wEIC/'h#ω;V+6PO,`<{3At֫;^Fxt?՟PeKlm^״DZ _q'9[{#2]xƸ:42/%%Zs]7m톊avxA؋'ӧŬcJlDܒq$,ۋq沒oVx=ձ+y* HUTqf:~|)x%Loq%[;)B=,0L< ">ɰ<-nZ[_Y~dvQ,dȏ M uͭw!x:k4,-rX`K|EZYmw!.0Xx1v)~u; %\;5[&?o+h,0ck~;1->$} Ǩ$H6C5_A!`1H|mL'? ,ŔGl#CR}?0dP@s𥱖}ZD@ʔ[V ĸpH1=d'fdٽ<2j76P C;XO˦WAy#BPG%ݠ5tw tѣ":]3 9tac UWSy[glOiçg0+|/"?׀l!|b|t7.Pv< ]T!-p,q%wdaj;BB躁aK;7WA.s`ɲ.a~{V&M-ވGalf8F; ]x{@U }s͉0hxB[JMٔLMʱڱ+MTG׌<5$ΙΧlw'ʝ՟JH 8_{?D,mFOO7XYXtǿ$apX=tp*4t:mvW+YF?Sixm¯G]#rf*y”Et P_;%TRbrTP9x0qդ4tkap[kvmazF:5ҫ^!+j9|_G1ΦsP~~Wc)&9_+Pa+ 9 =6,OeLYgEA uʐ!AlO~+.2%Xx C9b–لFl A6<e`2tJjNğ-Lc11TcyEMY?%(RXT^+NmL}Q-K6:>.ʨڨY׻Dndy>,TFȃ=3Qw?FpnorJ )Aɂor%.,JD|l' D5vM&T֛C4}$+IhA B]:iɾ/l=a͓Zw,TԦewC9'?ye'ݮ¨dаp`@ozji\}==Tc@gq\G24q/#!Ɋ8ϫ g0tGݏfVRy?A6(r0]L4_ Bw#:_eSRHwZ Jh}?j 49m;mgKi0oCXfxđN䚯ݕO@S,#8pJјO5SL2uenC Xm՝4!!C!Zʱ&7#\9CKuw %,LIq3LN`7*vG 'UmD)[]OGAdDnJvGxa߰rT0\Ec; v@f$& +>⊜s%J,H=v5zݚ?2JŔ1Ք@{Өъജ:8E~㑀Poa["Ѳ';L+UC}jٯß)۟Ef.ZX5 1^R Y[{ vz&f/hRwm._W/Ξ)j3FUnQ(/-&]U \ Ŕpf 1-Vnf8|F$lA R0@dY m̔ JU8 "t ߑ"m2L\8x  G> sca;vKF(֚_nʊ8I2^q0Uz,fO&CxsΣ)4/B $CmZF14{Ws zʔA0=סsѡ0W_6uWʿ‹S[z7_#BcmUZ9@a^© bM86@;6u qg؟])s_淰 5οQ {}B6QyW煮G=Ack#u4aEEs/eG 8"ˏ(Ŕ}6}+-GN9%'G1#"/bLD7p -u: Yj&^:,aj6 !~1\5-sxuQ-`CQ@BsBސTZ#+./A6/8:<[m!3XyB)MyŜb6`E '^mfb|}-9Ad+XT*MԦ ;9s[(įnL C}n|FJ+bď+g'|^wdƤF C!-3{"( X9eW9Ξt|h@zg[^PǰBEml*tt6a2k|UrY{rLu 3Ђjfu(Joa/{ RLo~>TձX@^Gd{:Š`gO=v3RI8^BEm+ArO*iG)#7Fz8Bߐx#Qfu(Y w_^a틬軀ZWG{ Ӄ} ;me&"qH6[Kv,yP=7A҂ {Q[{%<}M <#Ve?0DL 2ͨ?@ Bq@kMm'L?0li*lDV}MXJ#}`s#&{6qw8fzԗ'652C63MvdZ!sON[am9<ѶCA7(GmX5= jQL͓A90 !ȖIGT'6`XDϠnߙQLGݘQy޴9+&soב'1Q1L:ŝ['f_7D ƌͳ=a RQڄx Ý/Kl| J@z*݅&"6"50B >1?"ʲځ}&ǰx'/OOBjVld`HӆzЃ[TK anONp[8^tR|5UWgS o̬uvN }#X@:GeU _I:>]zzRhcK ' kԘrSKi qT swj2g w>E -"6JZ"m qk hu BjU.`}D!k;AQ!PWuǙ V\Ê] 6J&x'].)I?MVaT6U6 ԎKo'wA>[=v*zvwy{T7ˆcnұb,sۑ6wNOϙA n]Oq(G[?L_|#K2ݘ.F cA` ǁ_[ä-jLJ1_SnAݖ3 yH ⭉C mӽǒZH92AX5ԛUp>2E_PpWR` (L;IkKdj@bG5)@_z~ 2\5[MRBΈ,SwCgT5\m 9mŀ/~"*zf rPOOe}tItKog;R UY/bC($rj4,1αA", ~(r: 8|GdߍO6TG׵mVV)E\B1 m;T0T7> endobj 800 0 obj [801 0 R] endobj 801 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 598 0 0 838 0 0 cm /ImagePart_2177 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 323.3 718.5 Tm 123 Tz 3 Tr /OPExtFont2 11.5 Tf (6.2 Predictability outside PMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 123 Tz 3 Tr 1 0 0 1 102.7 675.299 Tm 90 Tz /OPExtFont3 11 Tf (defined by ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 161.3 631.6 Tm 84 Tz (T) Tj 1 0 0 1 166.55 631.6 Tm 52 Tz (q) Tj 1 0 0 1 171.599 631.6 Tm 80 Tz (\(Xo, c) Tj 1 0 0 1 196.3 633.299 Tm 58 Tz (o) Tj 1 0 0 1 201.099 633.049 Tm 94 Tz (\) = m,in) Tj 1 0 0 1 240.699 632.799 Tm 78 Tz (t>) Tj 1 0 0 1 250.099 632.799 Tm 100 Tz (oft ) Tj 1 0 0 1 268.55 632.549 Tm 74 Tz /OPExtFont3 13.5 Tf (HI Fi\(x) Tj 1 0 0 1 301.899 632.299 Tm 43 Tz (0) Tj 1 0 0 1 309.1 632.1 Tm 98 Tz (+ ) Tj 1 0 0 1 320.649 631.85 Tm 107 Tz /OPExtFont8 12.5 Tf (co\) ) Tj 1 0 0 1 336.949 630.899 Tm 53 Tz /OPExtFont3 13.5 Tf ( Xt 11 ) Tj 1 0 0 1 379.449 630.399 Tm 89 Tz /OPExtFont8 12.5 Tf (q ) Tj 1 0 0 1 389.05 630.399 Tm 52 Tz /OPExtFont3 13.5 Tf (II Eo ) Tj 1 0 0 1 405.1 639.75 Tm 1655 Tz (\t) Tj 1 0 0 1 476.399 633.299 Tm 87 Tz /OPExtFont3 11 Tf (\(6.6\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 102.5 591.5 Tm 98 Tz (where x is the projection of system state in the model state space. One can ) Tj 1 0 0 1 102.5 568.5 Tm 92 Tz (calculate the q-pling times of uncertainty about such model states based on the ) Tj 1 0 0 1 102.5 545.2 Tm (imperfect model ) Tj 1 0 0 1 185.5 545.2 Tm 148 Tz /OPExtFont8 12 Tf (F . ) Tj 1 0 0 1 210 545.2 Tm 90 Tz /OPExtFont3 11 Tf (When the observational noise is not free, it is reasonable to ) Tj 1 0 0 1 102.5 522.149 Tm 92 Tz (assume that the observational noise is relatively smaller than the growth of un-) Tj 1 0 0 1 102.5 499.1 Tm 90 Tz (certainty. In that case the q-pling time can be defined based on the observations: ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 162.5 456.899 Tm 73 Tz /OPExtFont3 12 Tf (7) Tj 1 0 0 1 167.05 456.899 Tm 76 Tz /OPExtFont3 15.5 Tf (,\(s) Tj 1 0 0 1 180.949 456.649 Tm 44 Tz (o) Tj 1 0 0 1 186.5 456.649 Tm 51 Tz /OPExtFont3 12 Tf (, E) Tj 1 0 0 1 195.349 456.649 Tm 37 Tz (D) Tj 1 0 0 1 200.15 456.649 Tm 109 Tz /OPExtFont3 10.5 Tf (\) = min) Tj 1 0 0 1 239.75 455.199 Tm 87 Tz /OPExtFont11 9.5 Tf (t>o{t ) Tj 1 0 0 1 263.3 456.649 Tm 481 Tz (\t) Tj 1 0 0 1 279.35 456.399 Tm 124 Tz /OPExtFont3 10.5 Tf (F) Tj 1 0 0 1 286.3 456.399 Tm 64 Tz /OPExtFont11 10.5 Tf (t) Tj 1 0 0 1 290.649 456.399 Tm 88 Tz /OPExtFont11 9.5 Tf (\(s) Tj 1 0 0 1 299.3 456.149 Tm 66 Tz (o ) Tj 1 0 0 1 303.1 456.149 Tm 93 Tz /OPExtFont3 10.5 Tf ( + co) Tj 1 0 0 1 327.35 455.899 Tm 67 Tz /OPExtFont11 9.5 Tf (\) ) Tj 1 0 0 1 334.3 455.699 Tm 70 Tz /OPExtFont3 10.5 Tf ( ) Tj 1 0 0 1 345.6 455.449 Tm 82 Tz /OPExtFont11 9.5 Tf (s) Tj 1 0 0 1 350.149 455.199 Tm 72 Tz /OPExtFont3 10.5 Tf (t ) Tj 1 0 0 1 358.3 454.949 Tm 10 Tz /OPExtFont3 15.5 Tf (1) Tj 1 0 0 1 360.949 454.699 Tm /OPExtFont3 12 Tf (1 ) Tj 1 0 0 1 375.35 454.5 Tm 87 Tz /OPExtFont6 11 Tf (q ) Tj 1 0 0 1 384.949 454.5 Tm 16 Tz /OPExtFont3 15.5 Tf (11 ) Tj 1 0 0 1 393.35 454.25 Tm 53 Tz /OPExtFont3 12 Tf (Eo 111) Tj 1 0 0 1 417.35 454.25 Tm 24 Tz /OPExtFont3 15.5 Tf (. ) Tj 1 0 0 1 418.55 462.149 Tm 1180 Tz (\t) Tj 1 0 0 1 476.899 456.899 Tm 87 Tz /OPExtFont3 11 Tf (\(6.7\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 102.25 414.899 Tm 95 Tz (Equation 6.7 uses random perturbation around the observation as the initial ) Tj 1 0 0 1 102.25 391.85 Tm 89 Tz (condition. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 119.049 368.3 Tm 91 Tz (Figure 6.10 shows the doubling time in both noise free and low observational ) Tj 1 0 0 1 102 345.3 Tm 93 Tz (noise case. It appears that the doubling time estimated by assuming the model ) Tj 1 0 0 1 102 322.25 Tm (is perfect is much longer than the doubling time estimated based on the states ) Tj 1 0 0 1 102 299.2 Tm 94 Tz (generated system and the imperfect model, which indicates treating the model ) Tj 1 0 0 1 102 275.899 Tm 93 Tz (to be perfect will essentially over-interpret the predictability of the model. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 102.25 230.799 Tm 106 Tz /OPExtFont3 13.5 Tf (6.2.3 Predictability measured by skill score ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 13.5 Tf 106 Tz 3 Tr 1 0 0 1 101.75 200.299 Tm 91 Tz /OPExtFont3 11 Tf (As we mentioned above, Lyapunov exponents measure the predictability through ) Tj 1 0 0 1 101.75 177.299 Tm 98 Tz (globally average error growth rates in the limits of large time and small un- ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 98 Tz 3 Tr 1 0 0 1 118.099 156.399 Tm 59 Tz /OPExtFont3 6.5 Tf (1 ) Tj 1 0 0 1 120.5 156.399 Tm 87 Tz /OPExtFont3 9.5 Tf ( As we discussed in the previous section, forecast with adjustment could improve the forecast ) Tj 1 0 0 1 101.75 144.649 Tm 93 Tz (performance, which indicates that it can also increase the q-pling time. Since adjusting the ) Tj 1 0 0 1 101.75 133.1 Tm 92 Tz (forecast is essentially turn the original deterministic model into a stochastic model, here the ) Tj 1 0 0 1 101.5 121.35 Tm 91 Tz (imperfect model ) Tj 1 0 0 1 173.3 121.35 Tm 110 Tz /OPExtFont4 10.5 Tf (F ) Tj 1 0 0 1 184.099 121.35 Tm 89 Tz /OPExtFont3 9.5 Tf (can represent any model including deterministic and stochastic models ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 9.5 Tf 89 Tz 3 Tr 1 0 0 1 292.8 51.049 Tm 76 Tz /OPExtFont3 11 Tf (150 ) Tj ET EMC endstream endobj 802 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 803 0 obj <> stream 0 ,,zєb7'F|m5̎Ek>S{xBwz'(#CB$!sw/ƪqThxC*h)@;&mAtTqՓ-%0rMG!xn; W(f۟8LPJd9>~P01ZSWrsI}Ɵ6 Cߢ)󑥘dDɓd3~ewCY%AX`˪m8h}>(5h]&?F/LC"uSEcེgna=ujhgqs~ ec`%q_",NK{}kᏱD_\Tg +m2x-o{+&郞ϸ*f2u7vsoc֑s͂ČOEn4+:OVjGb }iܐسS;S<'k[WhPq~Ctl/Z{+&t?|Ljezj<z`Z#7oE/g!g8:taֶvЊ]wAp mM^3ăX(_|X,9{R0{)}Kz@*"&~׈.CA3M|B@,XhcL27NCMeG 9~X/N2[M(V墌k oL&#e]^1U8{ c:PjA?ϙh̓, Bi6ߎB1ɾtGV/T6ihk`fM2R'E&x\!I]ww'CUyx+ZIbK.hM), XtB;WkHX=z!l7.ڒcԃ |?aJiC ,HʀVz)!(KfVS#Ynݟ-y7zU5#@q"V|\7۳.[C1c; ˂xCsx`wg?'n<(Qaӏ(u6aƼmcqk"uZ!簭ζٷI")?P*O=k{VKS!ȇwHx9C>l`5311;7(!)B0S' Mis OGfAl$򢣥#"ő!)ȁ[D \q)Lk8Ȟ"i#ahhX]E#.B=!h_meU(}Bqcq\\ Ooy.rZX_"sSB-(vK}kt 8Y?k{C\gG=L8 K96[x}Fi\Z[PmK<U%Ԣ4 h/~Q٢nRDz߹Y&9j5Bֈ b%u0E^;zw4B8_T:Cm~ۀzP`\4;ݯA7r2?::.~$i,40tǒLN1ђk]  v> _`xFF#zĤ(b*p$g> ՗ULǗsf&L#0v&f4θ܁.&!+/?ӵϴڶ&i݈?X((9r)d \/!Ls2ʚY^ao\# Y#Vx& |naQk} PIFO]&n9$$QQ#r:C}TO$D1&r1|.[;N+9g|`_L5#Sk) HaN5 UrTiZ-\ aɟv)q +fGphn wyﱿ4l]oI1X.C6kTa C&\6~b1"{kNT]d} `/ Ql6_-66fޗn5_lQW>MV eN;}R?(K_\R@߿a*hؔxlxÐ,Fx՛#H-<jZ?~ZelʟҌ=g.b Cr.t)]).qVɞ򐈢F-1{6Q[c3"#v)lkS'}V1A-xqOo7US7vFmh{w:OKt:f̠YdۅOBP5,phdlr'oT|}8obEE9>u}IGt9# 1.X$ZAn_ק*{Cɽ 1;wAD)t0bŸ́ ! 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Tj 1 0 0 1 367.449 518.899 Tm 105 Tz (2 ) Tj 1 0 0 1 370.1 518.899 Tm 689 Tz (\t) Tj 1 0 0 1 378.699 518.649 Tm 47 Tz (1) Tj 1 0 0 1 380.899 518.649 Tm 120 Tz (') Tj 1 0 0 1 381.6 518.649 Tm 133 Tz (4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 133 Tz 3 Tr 1 0 0 1 252.949 468.5 Tm 134 Tz (05 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4.5 Tf 134 Tz 3 Tr 1 0 0 1 249.849 419.3 Tm 170 Tz /OPExtFont9 4 Tf (-05 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 4 Tf 170 Tz 3 Tr 1 0 0 1 241.9 689.049 Tm 99 Tz /OPExtFont0 11 Tf (6.2 Predictability outside PMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont0 11 Tf 99 Tz 3 Tr 1 0 0 1 25.449 331.449 Tm 89 Tz /OPExtFont3 11 Tf (Figure 6.10: Doubling time of Ikeda system-model pair. Panels a\) and c\) estimate ) Tj 1 0 0 1 25.449 317.5 Tm (the doubling time by assuming the model is perfect. Panels b\) and d\) estimate the ) Tj 1 0 0 1 25.199 303.6 Tm (doubling time based on the states generated Ikeda Map and using the Truncated ) Tj 1 0 0 1 25.449 289.899 Tm 97 Tz (Ikeda Map as the model. a\) and b\) are noise free cases while c\) and d\) have ) Tj 1 0 0 1 25.449 276 Tm 88 Tz (observational noise ) Tj 1 0 0 1 121.45 276 Tm 115 Tz /OPExtFont6 11.5 Tf (N ) Tj 1 0 0 1 131.75 276.25 Tm 92 Tz /OPExtFont3 11 Tf (\(0,0.0001\). Note that the scale of the color bar is different ) Tj 1 0 0 1 25.449 262.299 Tm 86 Tz (in each panel. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 86 Tz 3 Tr 1 0 0 1 25.199 229.7 Tm 89 Tz (certainty ) Tj 1 0 0 1 67.9 229.7 Tm 125 Tz /OPExtFont22 11.5 Tf ( \(94\), ) Tj 1 0 0 1 99.099 229.899 Tm 95 Tz /OPExtFont3 11 Tf ( they are of limited use in PMS and inapplicable outside PMS. ) Tj 1 0 0 1 25.449 207.35 Tm (q-pling time \(82\) measures the average of minimum time required for an un-) Tj 1 0 0 1 25.199 184.799 Tm 93 Tz (certainty reaching a certain threshold. Such measurement is well defined and ) Tj 1 0 0 1 25.449 161.75 Tm (applicable for both perfect model and imperfect model scenarios. As measure-) Tj 1 0 0 1 25.199 138.95 Tm 87 Tz (ment of the average minimum time that an uncertainty doubles can not be used to ) Tj 1 0 0 1 25.199 116.149 Tm 89 Tz (infer the average minimum time that an uncertainty reaches any other threshold, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 213.849 33.6 Tm 71 Tz (151 ) Tj ET EMC endstream endobj 807 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 808 0 obj <> stream 0 ,,4L  jٶ/ I)^&JTgVz;};3AJϬ\S> |֦.n˱_{<[="gxġUY=Z<^)J y7y 6jO{ 3Ttd{e. p1}7+4q@ryӜ\ J2A3-}-d*ptҕk3C ؊a0 wLLPEdGȷIX[4}nt,؋ w|`#F^RH~,F7#1=)cx ۉ FniIW߫R˶hvV;!ұqד~zҲn_r]ӦR}6[ 19Ҟv?rP xA9ݝ}gZ2޸A;f<W'KÚN۬8w BQ3ܫ+T'ww^Wqe94H2>T^mp #">5{~[cneQ#@(yכ[B*-UjY#aomF~rRȺiC cӪQo]Sa#~[lebu Pg*0c4 e(a"bMTWn$ϧqW?S=ߔ3K^cOk޶e&RSky2ܝkRxy-S*aT&T\߀ >=2q(jnәpFˬIN{z0qK){0G|#y%~2=mky}#Ak?|zEZVX |8 A(GJ2F@Z2ߛA%y]aH7F~vT ƌxm_:f3.h@bE~;$#CG 9[`ypf*'NҢr 2:OĞz8f8ٰonC_.Pu{}vS2(BDw8`JJel?J{s' %]qkZ!y2/4/y}#^ZM'LDh}IǤJU]!>g2 B=bs| x DBE@-ܫnfGOK _ʀud<(;tDk'j?+DM<Շt9)n I+yhݴk{f$vWey[T_6D6p)hQ!λ"TIeAJ GҢE$# _VqFzC{$]&E~pF3nQ)o'OqI|c)$ŏ]bGml3tM,NUҀo͙Q`8C3Phқ%XqYc|#Ekb>Ƣy6G^Jj`fs1']^ޝCrPK;uKgN6#`=^PnOe.%4iVgP_ev]ڢhTuZWͼ,z髚\.m";Ǻkgt,4 r: E9;e3$;ǜ GAeɨ&y͓ZRÚ1tuz ªfgrKY'jɵ3c#g")qڢ!lzB|G(pOox*JM >'[\)?L5=G|*-UVy_^񒗘8CSE³k+ٍYa;Ġ%آt~\whi 6nWb71M_1g5Բ9R.כqowlmK)։Ԣl3/rX́>ڶ;5[ ^|?1#FH$[W*2.P F(x8-J.w]u6B%7e%t_PV !׸=kxYCE\!K-h>o`ԃ5Raob:N{4pV7ΨOI|EICAU`Φo0a{BOomm ־5VOzF"SFծ 6r9DV f^g wiӷ8t'%U,c yO)kj"؍ s yC>˝ۨBAdi^5T_ҊvQaWʟ?dLDV#ԉN75Ӽ ɀL1fV1mB;0mۑ0d3P4bh\뷱+i(Ůއy rGj-!@xĺq~Y_PnHF>)I[~aO^᳼ptnTIMafwHUS 0&q¾8j[[/';H3ϔ_6+}s-y{>@o ;4qJInW8m$e=kR?`XcN|Zq!5>Q`_ɫG ()Xu~j/4 aL R5PAR!ug0gxy{kL[>XJQe"9SѹԵ%"j$)B RӺ BrUAE`.1;˕nd^}p> q)}1}2QaNWK0naP %U ZS'Kw o>u" AIJh>yˊ` 6ᗨw&(W@冴!&= sd9Fd:jH Mfҽf$%Ԋ:<Ò}yL :%2ps!=ɿD mCn7$0F=6VZ+}Tћ9Eh- P4DA70PuQp |[ԥn DsKtȦ[5Ll9m} 0/wc\^E8QE qNImp~j"ǧn(;|{L̖>D-szKƃ0,픧ǰzB^B=m1V!ወhJz o?6I^'IR4w{d(s3օR2'*g6f(ZSw0B2AQ) Q>{V:5:X9-ҒU^[!A.N۩604?u_ʻa]ʖH xYm+D/Omx0Trb_ϯdθ"G,6Cf\DLi7%;V.r S$aH=4ێ*%:.쨣W V6?M`U! 򾤥]o̎0cdm|1 wzC#TwlijrD1t, ){i`#!;Eh }^C͝ LOtm9\ 1IHh=bq0]pQ*UNUѨ svr`cF-7EF2ˁl&GABBW?yf )xgCrĢxq O3 :4Y"<+',G%Aqa߁nR$hn>(U]B\X'xǚf=rx:ɨJ5Gob+3" t-)4sEg(s,5Zȼ3_OyU@Hy9?˶jqg׈IVULL~m]ӢLd!Qhem8)9#~"T&v5T ic)7|斔A㐬ePT `PUwd2CBRKdSH $xn:7Akd(I0ӽc?T`'"ёeZR(g&pFA 5yEz%P9Y8}vGCAYbD]5V4ݞ@ >i# 7n`qK/HIBm#=c?b BT:hC_#;yIF&aIeO;Wq[}{w]l5޺% \7\}L: ek\if'bi1p:.gz#pjeon%V; ýg8.Go(ϩzЋj;Ɓ  R7nae7Fl<Ŭ4O3,s:yptޯp_1I(6lTAh:lbei5!Ftp hV:CI᷂ Qs!`GaT۾r=;)'ۈ3HM5PrJD_PA"a-&xEpЄWFɗ髂oU烫FKo]%Qr6&p[:m ҃r;KH#Z p.&->ȇr,?ֲ[%\`}%IAe KAF&h  Bt}TW 4WH;7qnf˘&ϼI, 7vPGuADb!3޷bcԾaX_V\Hlk[%<~>ɴ}ʧߕȀ惥ʷdzҴ'+ƒ~=@<^O1p7sL@!y__UI8F FH}ZD |ɛpU HK"D%%,Ȝ+I!}Jǟƀ>1n=BίX~Z٬\/qfy#4JbF|ߵܿD!2/&^;r-(k׮a8g @+q݄B56*EEa;uc]>^܋-'srv4#VZI*^C4Z߾`o#S%ALWM@bVXjZb/aldExEK1G0TOC}l,}A}ᤞѱҺIEd+#Zb*#6Qq| B&aƕ@&*J$kQ,#eqqp m'V pwzZ&)_9o FHqnп"So+`c@.m:Xk u^4AH8 R8bYŊ%4 'ialE}⬣4ON{8"jE IĤ32KPsc1`ۯϘ ">'~>נc)#j446)@&F_,pQ~#/M \:?ot+9p>@QVtMmtmE-@0m"}Ɇ\нKtu\q K>*yVei劳0{媱$^Z"fN;d yE̝%̊TϢ̃6 ˭tPrbԥacxzj50}zGQ7H.H3pYQ^O}+/:7"*&{%*o<Ŵk_]c%OƖ8X\N+n_8Xq 2OdKr-Z|3KݸG~IJpovr 4G~3?9+Td{Z~Ә&Õ-c}`.&RH)cyLwn-'Jrf-)O\DTo|M0)jv(; {0ƍ_~Rd-IWh=svp=J7}fm95>@dbg]ܳNU7V P22F$p@ޟC=&w'5y .`s 7.54 WV}]XUp{ )0-X 9Zt&)ȸQJ"7:$'l޹U X)@o b׀Z@ ר }j7#6UEOpL'2:?!8pM9߆: |VI7?+x(2'p?X\xCЬz!QܪR*JVr4Э|UB{"aGIiA' GKݏ?BOBȁgzyތ}Ƀ6K0V!D'0i|qQbwۘ3Ҡs(P רȩ[n9J1q vBO~%mX=aNdRKDwE/a*Z,1)ҫx`L?V_"i1 D3I' u4u02͆-P6|UqH{#ӧsk?nwr:{\;k/E 'jZCDgvdUFZC#X B k!^ !d5> ((PR1spg_Չj"0,Jɘvk `@qGi`T%ymWvġw < :J반;c:oF!eFh'KӃ8+NZ%kX嘄1 3Z\FƇ ڱa5LblH:^Ep/՚lxHh)J%bih'#|atۚŧ04-OKm $ cY kZ!]A |nsVSR#r|M+jijG`af]v4h×쫗6xƿ6ͶI*]lI%F? 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Gi6|/zd񿃽3y!߲?LƷ"P5mgL"d ygNh_#k5^91yZ 2{C];{XjBmߕVJq]՘0mWq6yԫC˜\ސ~_PI߈^J%8Q1Wv7Ό7Wh{m ^r ķI;j3oMD 4c> -[٠y.18+x+ơVs]}(Jɰ`.UdYzgd\BK[`4y@*,gǯ=d-? Q ?D$u{USnDÆ2B?[ã5k0%(SC`ހ43*N#H%cJ¶h6C3n+L7fYzW / FDÑtNLt$F x[@94'^a˚圝82 q~̔E[J2cj6J<}/=?"WlF#dW{~.1Ϋ2)n>~c◊x]>|1m7i"R2rBq*21{)>n*6aƃsr\3 K6 6H^4.އs]G,KmvmҧQQx厨 uHora]RHSr {8+Zv8hф毻{?UV.N'r/nhllq`"E]4KX Aȧ6.6C@ Rp dY*%?.5xP\uQ$OjF{9tF9HA;j)/D=V+E ixQ^Zi9J`ay#PN-J2@c <RbA{ sжc0T]z0),cO:>^Ѯ^ʧWEfDHPÓ I΃9SRO%oX7BAf(g Ifszgϲ1C> ϪȰ4\3ı㵸XFs;i&6[k3+Ź?Nǜ1-uԘ.<v`uF;z]>b(9 k>%Yn=|U M_؀T˲X\?G(Gr8"D}}{Bq/^(Ae&G"ҁspjA|AP0#qGZB:6t @F!*=#Ʀ㹜(I^iC]UyCk$O(qN''_V)UR=VW$%n9`\3/xw@bؕIm4t;}x{ڃEcY|cV9!4~lϿVhLqdOpA5., F^ů'yl&T־%Ҁ " W4'1róUpô3ǞIr[C'nv"2m+D)2Hp \% q[: /Nx2G)0ӷ!xԘ;ӫ:#jrKa`5xR}||9~k;E؍P2RFLC=(l0$\g sIrDJXVJNRK6zQ\]8 ҪuY: `!7Fզٲ*@/  =jo3̞3V&H oD#^t4m0-p5VԌqF7c86U:7rP_~vעҍv=oȈ6n]PԜ!i"þFh;Boi>Oq[DD6s! 6(w#K=S1c<y қB[:pt"(U}/)̓؃pׁgBF4ˢ fCE,獒Ҋp&fjk]e|Sc&,le1~]`yt yPx9\baDYDa q%+f~CI$v|lL I'mC"avV[cw=eU=$g9UOk'ڑ̌q^c> \F>'% BJVEHzO&̎TU}4dv'…*sKHJQ\^ z-ԟik endstream endobj 809 0 obj <> endobj 810 0 obj [811 0 R] endobj 811 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 598 0 0 838 0 0 cm /ImagePart_2179 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 323.3 719.45 Tm 107 Tz 3 Tr /OPExtFont3 11 Tf (6.2 Predictability outside PMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 107 Tz 3 Tr 1 0 0 1 103.2 676.5 Tm 91 Tz (predictability being lost has to be defined, in advance, upon a certain threshold. ) Tj 1 0 0 1 103.2 653.7 Tm (In this section we suggest another way to measure the predictability by compar-) Tj 1 0 0 1 103.2 630.899 Tm 94 Tz (ing the model forecast performance with climatology, we call it forecast based ) Tj 1 0 0 1 102.95 607.85 Tm 86 Tz (measurement. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 86 Tz 3 Tr 1 0 0 1 120 584.799 Tm 94 Tz (As the average forecast performance degenerates with forecast lead time, it ) Tj 1 0 0 1 102.7 561.75 Tm 93 Tz (will, for sure, happens at certain lead time that the model forecast does not do ) Tj 1 0 0 1 102.7 538.7 Tm 94 Tz (better than the climatology. We define predictability being lost when this hap-) Tj 1 0 0 1 102.7 515.45 Tm 96 Tz (pens, it indicates that the model forecast does no better than random drawn ) Tj 1 0 0 1 102.5 492.399 Tm 91 Tz (from the historical observations. Such measurement can be applied to both per-) Tj 1 0 0 1 102.5 469.6 Tm (fect model and imperfect model scenarios and places no restriction on the initial ) Tj 1 0 0 1 102.7 446.3 Tm 92 Tz (condition uncertainty. Comparing with the q-pling time, the predictability being ) Tj 1 0 0 1 102.5 423.3 Tm 93 Tz (lost is better defined. Although the measurement itself places no restriction on ) Tj 1 0 0 1 102.25 400 Tm 91 Tz (the initial uncertainties and the model, the model forecast performance depends ) Tj 1 0 0 1 102.5 376.949 Tm 95 Tz (on how good the initial conditions and the model are. Similar to q-pling time, ) Tj 1 0 0 1 102.25 353.899 Tm 90 Tz (the forecast based measurement measures the predictability given the initial con-) Tj 1 0 0 1 102.25 330.899 Tm 96 Tz (ditions and the model, of course, better initial conditions or better model will ) Tj 1 0 0 1 102 307.6 Tm 91 Tz (have more predictability. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 119.049 284.299 Tm 94 Tz (To evaluate the forecast performance, we use the Ignorance score \(see sec-) Tj 1 0 0 1 102 261.299 Tm 95 Tz (tion 4.3.3\). Given an initial condition ensemble and the model \(does not have ) Tj 1 0 0 1 102 238.25 Tm 94 Tz (to be perfect\), the forecast ensemble at each lead can be produced by iterating ) Tj 1 0 0 1 102 215.2 Tm (the initial condition through the model ) Tj 1 0 0 1 303.6 215.2 Tm 21 Tz (1) Tj 1 0 0 1 307.899 214.95 Tm 94 Tz (. To calculate the imperical ignorance ) Tj 1 0 0 1 101.75 191.7 Tm 93 Tz (score, a forecast ensemble is transformed into a forecast distribution by kernel ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 118.299 171.049 Tm 136 Tz /OPExtFont3 6.5 Tf (l) Tj 1 0 0 1 122.15 170.799 Tm 90 Tz /OPExtFont3 9.5 Tf (we are aware that one may use different forecast scheme, e.g. direct forecast and forecast ) Tj 1 0 0 1 101.5 159.299 Tm (with adjustment defined in section 6.1. In this section we treat the model and forecast scheme ) Tj 1 0 0 1 102 147.75 Tm 91 Tz (as the forecast model, i.e. forecasting using a deterministic model with random adjustment is ) Tj 1 0 0 1 101.299 136.25 Tm (treated to be a stochastic model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 9.5 Tf 91 Tz 3 Tr 1 0 0 1 292.8 52.5 Tm 75 Tz /OPExtFont3 11 Tf (152 ) Tj ET EMC endstream endobj 812 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 813 0 obj <> stream 0 ,,j]F0:TKZX]t$P`m_U4%4¾mUزn54 fWICkteqz/9Gg#m!|Bl8iYNcL >w>$8W1)ٰ&2Zڴ&zʧHAQ{ZHxqrNyJp;\yS \6`:3u4ݩu]y{o4o>ڨ {qP|9!u\k_GưMo i}L?F }w9iķ/0Þb{;}W wiSFƨ1]RܚF[k9_+ch"dQ"82~a+Oõg0׶*]bwk,1(v}+pw+%ŷ)ʜFFwff G*(N-^җ#u)c8G`[_Z8!NÑq7: U͟|:cہ ae 8%v&M^?~2Rnބ/]deD8FE3 : A=_ :$Ow琦,5'\8%G7̡ö)jv|@CA[xjG :Kds(,x1 rMϊ`DԯOó_ig~\5={.?b·Ģ*Xm%\c~lm_zTy$U.oj"4}Y!`xf/ ֲQ߶ڨq}WO?q9³cΘ H<*#4M'2.*mC>a~ucɤ5ؿmYLRP`p'Nr"u/G7[EfN̷zêEӆ3\ R Zh8߁/ 0P (Y) H')2O8#õ'UOeC !pat1<5^uj7tRNQ50+gs,G4hZz- yfY\!dzhXw% Y˨@}QѶsW6s|gaZqϚ@8e>-i  秋.*̖\q$ kȶck)2q SyvfX[HN‡ipz2NKS=1FQ(7amsB8e _3`}uJZVgG4 V(9!D4|8YpϣhLfǂy Ee5xbpd `3#vM!\7wsB ض朸FL>P`[e3H3+;Zf'Wwz^}$':0iOWbɢǝح~Xuy_i Ӧ[bE\LDDO=~dO'58v0Ni꼁$>&0P`SP+G} ](""])鳩Ia]\-G˖KTʢrp'i oS@a݄ No2HrͰanҽ|B:L4I~Ε++N*HddlfägTkEx/ɊotB@t8UUdbVڶB ĺ++NV~Ё0d>\Y6Dr?2M\,#񼢑!diGW+up8Ui Yݽb`T Y+6kɶ! a j/Ǔ ~|b }jx<b!r0:Z$lva;WZ]k5*'ei<ˋ*3IRvz(mu+T:([2磟J@—_6#죅VEš̩*1^ad?<,??^t[K1F慒Wþ{bW]y`!O֖؞7ŽZ@+fXL}!]~-H?EDM:OJ<9r2>Dřåק%SO)žz\\º_9 Rf:w?R֘ާa&çA$#J!/ɕ3P46K\I6I32-7f]w;Z$o9 Y:Io3 v ':u$C4_zaݪaRrA0wMM*V)P&?v׮w`}J>γGMQϲ5;|j({wBAJW0QMm2 !,&PA.IדMjn.,'/!5 ӫ{"*>%_}ܹ97@4fjIk裻o VJWzZbruE:yT'cH*kzܒ b:psȺL=v KzWs  |R[uEHE Chx:8"r᯸eVỰ@qc(ڵ3Ul\%HS'g4e@6e<&rd (i'ƪHK1ڲ2F|is_ < 0dYF%ɣUHF_W˨eO砣A'\s~A) UuQ'N;mL;-1$o,zN-͇hS ab6p^?.lŖuҋ4egR`NI~J ޤX'")޶Fu'Y$ʛ|s+!bN@nuٟur'9[d'[`M|u4?6O9>j 5d+pIQ|24JYHiF:{zL~h_}S\۟MiL {ӊjP1Y%/N %y1&_.9E%"'lN*pwx3Q471-Mԣ0 œ8ź;p ѻFԠ($i ub)Q賭-gqIIśFŇp޽lI? ƅVX] )'OEfZu-Fuτa UV$ӛlf݈AQmB»(y2KKmt"K'- 8SՍ?`q3rjZ7{ihSkPԢMς:lp0G_"5y1H+?$ޱj ,r.[c6Avr+u7cg̬;k_W;'}=*DŎ*5y/ mG4vkXES_eQaJQ΅+emǴ] _i]F^(wHk_ +- w Dδ1 CN芆[fXAS(lgOjKob8u*K SZ1ˉTBfPFK|#{o{= )ذEUrnzfN̼NX{z.3LŹae(mʑwl!Ҡ3}w(xvt&aY-RZQwٔ)Y|LyN ;^ƥQy<՟Z;jGw pcMu`;jK?j߲)ugas }а^X# .+aW_SV@&^;WRZ3cȥW8f3uvETC{0KW~^7@opHZ~t ]( T|PzޜrRYS:mzH*N{h:Kn+CQ/RVŗ4=!~86.u':YE6p%8 7Kw1l'%*B}1ӳ|$(.nP(7׵Zni{3&[\~c[XVM[5ȅi> DtMP2%Pz-)hBE4}k{^eNƍP5ckKiw7bJ]e)p 2V>dn:W6#!\? c@$3jqtv )wb^}WLW#U`%/5s@&^ՖK^L̐`9X ._:$3(͝ #8C[QH+L1J^ʒl83kVQo}tAHDV8d | @^ K[(=ws,L!bϧ8 :I?V6{-čDPBTߥ@DߌLbF , W s*1M5*sxk Fz ٯvBQ.Xql~`[4(6aS}_"OpoHDAiYn,?*Tcl_-裆XddQ֩eUaZ1⋳ŬhK-ZdkkF*j¢r#@D)p ԕ޸}6TpW-# 8zq h)p=hrZVmbB b(X{|ͽ%xMw 5vnk.eBZjQT&*V=Dc(zcs ƻLOm6lm([ӄ)#P,%MĬE W+pA d/{:^8y_F%;L3;bqJz k+ݣnGrZ&G ev9#=ѻuEov:BVa=aVfak$=-´IDt [Dyn ڰ*_@\fo14ve y< qe7 .܅}Á8jŪ~73Rv"Y޷ 1T,-EvhꨃR4䰞{Ds[UN{Y\. k:>@ IsPў}P#y$ >)^iV QeЛn<`oL~b"ѓw 9 > !1VI9@& Is7}XzU\n\?ٽˌX5 O!]DpW^-&msA1 G͜4K<za90f8_x=KnГFIHy@1|a;A}̀fR5Y[lV9g`Vۭ[0o 4u%C/phxxk+eFD>$~;pٖ>R@i3iL#.=[)tKfuk\^*RzȰ1mÜ u y{7 ӋZ25nJ?=yB"ho^H #(˺f\~@FOD|&ڮ /:jbuI>ϰ{"u,ԇfGo֢-Nm#aT7I>KcBobRefd8\IW Vljʍ4KK*}9Jb$v{~ӯp'b}\tƍ'ie;":9 A @/ ÷&cY ʒ36XѼZV7-Eefm!qεYc0#*ՠ9nSpXKyUMZ7cثʢ%(!(p"(ݧ 9zmL>-WմʇԴǰ>)"k<ց}c:4_CATLԙ, &;/yxFa:y8p+"GD#S bߠEA0r1^ *R+ݣwY֞I]fG~8GKC0Ntr5Zs ;o2r^:LV]Fu e7$WӲCV$MUݚ%Ql'8=gd7*͋2IiPDt=\4%bM_Lڂe͊ oK-Y\kXJ urt!+зNV|^cwB8o8. 5ý h{LdulTW~ſfp@cf`ئ8xģw 7!+3+]oLuBo WzKδ5ҤMk--'~fT+) ?p7+w8> endobj 815 0 obj [816 0 R] endobj 816 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 598 0 0 837 0 0 cm /ImagePart_2180 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 324 717.7 Tm 107 Tz 3 Tr /OPExtFont3 11 Tf (6.2 Predictability outside PMS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 107 Tz 3 Tr 1 0 0 1 103.45 674.5 Tm 90 Tz (dressing \(details of kernel dressing can be found in section 4.3.2\). To simplify the ) Tj 1 0 0 1 103.45 651.7 Tm 91 Tz (comparison between model forecast and climatology, the forecast distribution is ) Tj 1 0 0 1 103.2 628.899 Tm 93 Tz (blended with climatology \(details can be found in section 4.3.2\). After blending ) Tj 1 0 0 1 103.2 606.1 Tm (with climatology, we expect the forecast will do no worse than the climatology. ) Tj 1 0 0 1 102.95 583.1 Tm 91 Tz (By looking at the Ignorance score relative to climatology, one can measure when ) Tj 1 0 0 1 102.95 559.799 Tm 90 Tz (the predictability is lost. As the lead time goes larger, one may expect the relative ) Tj 1 0 0 1 103.2 537 Tm 91 Tz (Ignorance go to zero asymptotically. When the model is perfect, given the sample ) Tj 1 0 0 1 102.95 513.7 Tm (climatology based finite number of historical observations, we expect the relative ) Tj 1 0 0 1 102.7 490.899 Tm 93 Tz (ignorance goes to the values that relevant to proportion between the size of the ) Tj 1 0 0 1 102.7 467.899 Tm 89 Tz (ensemble and the size of the historical observations as the ensemble members will ) Tj 1 0 0 1 102.95 444.6 Tm 90 Tz (eventually become random draws from the invariant measure of the model. When ) Tj 1 0 0 1 102.7 421.3 Tm 93 Tz (the model is imperfect, we expect the relative ignorance goes to 0 eventually as ) Tj 1 0 0 1 102.5 398.3 Tm 92 Tz (the invariant measure of the model is different from that of the system. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 119.5 375 Tm 89 Tz (We suggest using the forecast based measurement to measure the predictabil-) Tj 1 0 0 1 102.5 351.949 Tm 92 Tz (ity as outside PMS the predictability should depend on not only the system and ) Tj 1 0 0 1 102.5 328.899 Tm (model but also the way initial condition ensemble is constructed and the size of ) Tj 1 0 0 1 102.7 305.899 Tm 94 Tz (ensemble. Figure 6.11 shows the ignorance score \(relative to climatology\) as a ) Tj 1 0 0 1 102.25 282.6 Tm 93 Tz (function of forecast lead time in the Ikeda system-model pair experiment, fore-) Tj 1 0 0 1 102.5 259.549 Tm 90 Tz (cast based on two different initial condition ensembles with two different sizes are ) Tj 1 0 0 1 102 236.5 Tm 91 Tz (plotted separately. In all cases, the relative ignorance converges to 0 after certain ) Tj 1 0 0 1 102 213.5 Tm 90 Tz (lead time which indicates after that lead time the information in the initial condi-) Tj 1 0 0 1 102 189.95 Tm 91 Tz (tions is lost. Using the same initial condition ensemble but with larger ensemble ) Tj 1 0 0 1 102 166.7 Tm 90 Tz (size provides more predictability and delay the convergence. As the results shown ) Tj 1 0 0 1 102 143.649 Tm (in section 5.5.2, the initial condition ensemble formed by ) Tj 1 0 0 1 383.05 143.649 Tm 99 Tz /OPExtFont8 12.5 Tf (LSGDc ) Tj 1 0 0 1 422.149 143.649 Tm 91 Tz /OPExtFont3 11 Tf (produces better ) Tj 1 0 0 1 101.75 120.35 Tm 93 Tz (estimate of the current states than the Inverse Noise ensemble, the information ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 292.8 50.75 Tm 75 Tz (153 ) Tj ET EMC endstream endobj 817 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 818 0 obj <> stream 0 ,,zzb6|W'JOi{XRWF'I<;^q&Y% ҵ|N؛7) ѧz[Dҕ* h9ɡ Ps&>i0)W`_}ʒVҫx'8VI T9]7^9x,Z.xRW9VK F*Z{=Ҁj.sDf3B4Sm `ܒ_:F /ն<]E.m/μ(ORWs9CJFk N |o R]Gjߡ&H!YuxF9OD:uK.ǵ8Z2?9Y\Ɋ+BbuxcYSG-y,PCsA.낉Occ̦ݷAڲ0dBg[#ߢ"Tvh .)- +Z\X>pRpV#ds3/{\?% ng'~] 6inIt}8ӟUoH=]۹z$K = 3 vE™gB!pIz A2,Vcttp~=b&| /Gie!gY9@S== <EPI:7.ju.?d=GGͻh&yCXT?59~ƢLN^"yfKt}UMi\vT,Pz!A &lrֶDb _ A 舠%:H 'Hi 4/1_7N$@Ff¦p 1aHcC` VuYс@|!4c\^,<p sI)^Gś~^ r89_Stl q[5L54زu\$]8.ǘAnoFD֙jn!ӧϾ&[TB8W8ilVXc~aWY<0"W#n>:вY2`1.g8ˢͰ҃⳧"ѷ>g?(=;ؖ38ϰ`<RɳLp(0]kuO O=A0Hə-l| ;{Ä ^{{zp^*G4ACș3]?JUJ@c\ĥ03K3g @gn,pZ%xjwDK4uNw623nV“yhԆF2Uu @}nM` Y=MzfKZ z7gUt_ܺ.-qFB9?S!L6x OEМW)z(+mW+C/C"M6"ch\<;6}TlZHUBRs9cPur6.3T< ^6j-Xρ\iMu8\".3p# XYgāz}ffT[z]bd"gHRn.#> ym0_c#̱ fPRZs'W qҀͧCZ{b/D<8"x]s287\f `;kp (Ь1bZJGԜy-mLcqs,:#*~߇>pHvBmJRM?Ez2hN6\YGIyc/y8^q:/*Po]I`AwͿd4uҖ, ^'sɶRgKq#sÚaJ/L;;ҫ 1͏ɵDn\uWd9$~ PC[v[ܑR gIgK;WP͕r|hKΜ 4u_R=[1AZa3{#XdHrȜgpQ3oqm6_|Ң9i۬n;ylo *isFV te; ;Htob` D2 "~]{p#.žxFI\@31TıSj;W05.%mC73'-qg1j4 ӮHqaľYQiZ?APk!M3=8N𔭴,5*@]HA4vKϗmv`]ԎƬ/)8DnPIw!6 0IJГv~3Xl6s)ÝqMuړdo-ay3?JoKX/Lc$hqX]t 2,QlᢉۭCWMiB9AT~9 lvixeK?==\~0@Դ߽eĘ#̌[R:ġՓa&XK5xǻ[ @[CC'}po3{3%f`R[`W5&Ay5 /"S01G$ߙ?{ːm6R3!ʹ[X)4gt.V<΀X{qνF :B(y*Bx-\䉮 gw-euyK߰k=?*F?TWJ\% n8;zVS>(|/%ۑ+O&2k,MCe#; NX4Q6 ؑqh=R3|{1Fr G-pX/Zbɓ]v77bSH|)6Յ]8|6R]> '7md֘K PF(ɡ/=5Yp0EJ; >)VFY.HJ5QTcS)ݛFsqq($'v @ˠ|:(۷nO7qiBTvq J3͵m5t:dպ}^hv7zg&U0Oa=M+N@Cu|.Jxa]ʟQHEF^0,[fغH+.K(+Oܭ뺚CAT]f~GڻmC7P LkTi:z|t5P=*a*]`lF+?^ /K`Z(̷Sþho^HZ/OR7,5za^Vb=E엑:Rvcy -ta S'Mș?=;z k+wZ%5,mT'43RN ݀Y^'pfk<.\V$hK"f-㋷)cp$LuJLFObR_ ^rF{A(PkQwz_…*j$ӿq808`R}fO?1٩D!,$D6lt# X~_Зu_3hl@h+q@:i~@Q-F&hC(LN= ls,E}뭾}l?ܭZR]Wnm1>Zf23ɤ+j$f1 շ>]3RTJ,l%e\Л,k'S\+pv͇e 91Tw N{4bDHFr5噹YZE("jBu&ZJjR:.!s[D:]և OP Aك{͏5+ 'Fjw[L$\. #_D%+qw!W;- g.ҏUB(,;bi"* IoוY ľGr;G/-vn> D4c/-g-Vt:W$696~?f5ApZ};ԐD4 Es/}1w4ؙ8ԾDn"Tan zI\0;{N1 &O" TWs6cWH./8e0}?wta'=%k`5udo ;B1Pcj[\^\2]*%T vm"}QeآH]:I)Jl8LQy 5ѨEH@@oAlk .;(G_@Jr}DV "VR}8 B ;h<&̎;:?? # +_,Doفf}B5M-ǭ+rUg0Gf-+o{14 H-.eI&oaCv%b(s^ g+4ۻ .`HHkSL䙅wߏk(E07CJyP7ȵi.^|t`GBS鞭=u(ȑIѺ(,  }ۓXW Tʯlѽ۲9sUzlups%=xd!Wrb~ jqb:)I L \0YE-i4UhLJh՝O | SI%]uwm aw@}B)[OLeP8t,#DShyWvLNg D1W/'3P*"u\b0 ΗKue?̂`ogܙ99a.1;vF5@²WFhX"S#`r{eH87G~Ѡ'3jH0Z5Q.k*xA:&Tz K,8MfL15H{|?Lds:jbjY$ϵoC qX Nc\8ϫd23c t̛ sH8U4[R?O}U(A輥9X$Gr mvrK'8DMLm8:*Z|:^ gZ'2⬾@'Af!lQфQpD)Ce}Tٽ?[ kb! 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JZX7}ҖOY.h`QEwyP?eWg7?x'b ɝlu<0e%Qt!kfxU@ِ)s]\YFI4lK 5|Fxdϼcː`Bݷ6p(>V"$R I \^tL6n>{ ֗*8z[yt{hCJ~ blp?sO }!7,zF~ WƑ~~߶ԿUEZiܳ MaG]|'b6^<`!tM1QgT۝9ѕ~ 4c\}iK4t]yN]"rSTCؾ` KRoQ8~Cq{`'JԖ7t2%㻌OEt&i]mđ \ uhfYbδ,se /K2D&ڀ x+Jgھ/ÔB٨2`r0NLAw'l3-ٷ~*6FL/iGnߪ\fn̂]ؠ R,ܨё2(([#ZUMUΏT0lzlDRȱsb|!dN1*|pd 2p휺OyfN a.5P]}[I8g})n7evSc85Ltx"jʈ`-V@:e/V6SWO/;B%W+[-"(画Xp.Jlho}IY&JNGmlI #$L!컨&?0!G >7\ !0+.[.=F&Yhކau]Fij&\ 'UjʼnM1L )Jҏh(X"M?"j$tßIGFT0~}>֞dpAҺ|;(ڡIւQ,9LL='%Ty||TˑPF(`FOx>Ljexe œsԤ]+;{ՠF#j1j!ah LO}װޜ ASiwT˜ҟŠF2W%>m1,Zuդ3f>Ar.8(‹ڷ1|:갨ȃǁ~=/<+%ZݽZ2@^cyjB`w;TN"^G:!wcvN`2k ۆ>1p9c[lUy5y%N endstream endobj 819 0 obj <> endobj 820 0 obj [821 0 R] endobj 821 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 498 0 0 789 0 0 cm /ImagePart_2181 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 323.5 428.75 Tm 115 Tz 3 Tr /OPExtFont1 7.5 Tf (12 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 7.5 Tf 115 Tz 3 Tr 1 0 0 1 362.399 428.75 Tm 118 Tz (14 ) Tj 1 0 0 1 372.25 428.75 Tm 1398 Tz (\t) Tj 1 0 0 1 401.3 428.75 Tm 117 Tz (16 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 7.5 Tf 117 Tz 3 Tr 1 0 0 1 284.899 429 Tm 106 Tz /OPExtFont9 7.5 Tf (10 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 7.5 Tf 106 Tz 3 Tr 1 0 0 1 129.849 428.5 Tm 121 Tz /OPExtFont1 7.5 Tf (2 ) Tj 1 0 0 1 134.9 428.5 Tm 1638 Tz (\t) Tj 1 0 0 1 168.949 428.75 Tm 127 Tz (4 ) Tj 1 0 0 1 174.25 428.75 Tm 1638 Tz (\t) Tj 1 0 0 1 208.3 428.5 Tm 115 Tz (6 ) Tj 1 0 0 1 213.099 428.5 Tm 1665 Tz (\t) Tj 1 0 0 1 247.699 428.75 Tm 109 Tz /OPExtFont9 7.5 Tf (8 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 7.5 Tf 109 Tz 3 Tr 1 0 0 1 231.099 419.649 Tm 124 Tz (lead time ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 7.5 Tf 124 Tz 3 Tr 1 0 0 1 229.699 505.55 Tm 127 Tz (4 member ISGD ensemble ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 7.5 Tf 127 Tz 3 Tr 1 0 0 1 230.65 494.75 Tm 125 Tz (16 member ISGD ensemble ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 7.5 Tf 125 Tz 3 Tr 1 0 0 1 229.699 483.949 Tm 127 Tz (4 member Inverse Noise ensemble ) Tj 1 0 0 1 230.65 473.399 Tm 126 Tz (16 member Inverse Noise ensemble ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont9 7.5 Tf 126 Tz 3 Tr 0 1 -1 0 65.299 488.3 Tm 88 Tz /OPExtFont10 9.5 Tf (no) Tj 0 1 -1 0 65.299 496.699 Tm 89 Tz (ra) Tj 0 1 -1 0 65.299 503.649 Tm (nce ) Tj 0 1 -1 0 65.5 514.899 Tm 100 Tz ( re) Tj 0 1 -1 0 65.5 524.75 Tm 83 Tz (la) Tj 0 1 -1 0 65.75 530.299 Tm 76 Tz (t) Tj 0 1 -1 0 65.75 532.7 Tm 88 Tz (ive ) Tj 0 1 -1 0 65.75 541.549 Tm 96 Tz ( to ) Tj 0 1 -1 0 65.75 549.95 Tm 98 Tz ( c) Tj 0 1 -1 0 65.75 556.7 Tm 93 Tz (lima) Tj 0 1 -1 0 65.75 570.35 Tm 88 Tz (to) Tj 0 1 -1 0 65.75 576.85 Tm 92 Tz (logy ) Tj 0 1 -1 0 65.75 590.049 Tm 1 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont10 9.5 Tf 1 Tz 3 Tr 1 0 0 1 85.9 625.549 Tm 93 Tz /OPExtFont3 7.5 Tf (0 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 7.5 Tf 93 Tz 3 Tr 1 0 0 1 72.7 601.799 Tm 98 Tz (-) Tj 1 0 0 1 78 601.799 Tm 104 Tz (0.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 7.5 Tf 104 Tz 3 Tr 1 0 0 1 80.65 578.049 Tm 98 Tz (-) Tj 1 0 0 1 86.4 578.049 Tm 51 Tz (1 ) Tj 1 0 0 1 72.7 554.299 Tm 119 Tz (-1.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 7.5 Tf 119 Tz 3 Tr 1 0 0 1 80.65 530.5 Tm 98 Tz (-) Tj 1 0 0 1 85.9 530.5 Tm 93 Tz (2 ) Tj 1 0 0 1 72.7 506.75 Tm 119 Tz (-2.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 7.5 Tf 119 Tz 3 Tr 1 0 0 1 80.4 482.75 Tm 98 Tz (-) Tj 1 0 0 1 85.9 482.75 Tm 103 Tz (3 - ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 7.5 Tf 103 Tz 3 Tr 1 0 0 1 72.7 459.25 Tm 98 Tz (-) Tj 1 0 0 1 78 459.25 Tm 104 Tz (3.5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 7.5 Tf 104 Tz 3 Tr 1 0 0 1 80.4 435 Tm 99 Tz /OPExtFont19 7 Tf (-) Tj 1 0 0 1 85.7 435 Tm 91 Tz (4) Tj 1 0 0 1 90.95 428.5 Tm 96 Tz (0 ) Tj 1 0 0 1 95.75 428.5 Tm 48 Tz /OPExtFont1 7.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont1 7.5 Tf 48 Tz 3 Tr 1 0 0 1 343.899 682.7 Tm 98 Tz /OPExtFont20 13 Tf (6.3 Conclusions ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont20 13 Tf 98 Tz 3 Tr 1 0 0 1 41.5 387.5 Tm 92 Tz /OPExtFont3 11 Tf (Figure 6.11: Ignorance as a function of forecast lead time in the Ikeda system-) Tj 1 0 0 1 41.5 373.8 Tm (model pair experiment. The observations are generated by Ikeda Map with IID ) Tj 1 0 0 1 41.75 360.1 Tm 88 Tz (N\(0, 0.05\) observational noise, initial condition ensemble is built by using Inverse ) Tj 1 0 0 1 41.5 346.199 Tm (Noise and ) Tj 1 0 0 1 92.9 346.449 Tm 98 Tz /OPExtFont8 12.5 Tf (ISGLY ) Tj 1 0 0 1 131.75 346.449 Tm 84 Tz /OPExtFont3 11 Tf (ensemble. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 84 Tz 3 Tr 1 0 0 1 41.299 304.199 Tm 92 Tz (of the initial condition ensemble formed by ) Tj 1 0 0 1 258.949 304.199 Tm 109 Tz /OPExtFont8 12.5 Tf (ISGDc ) Tj 1 0 0 1 298.3 304.199 Tm 92 Tz /OPExtFont3 11 Tf (sustain longer than that of ) Tj 1 0 0 1 41.299 281.399 Tm 88 Tz (the Inverse Noise ensemble. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 42 230.299 Tm 114 Tz /OPExtFont3 15 Tf (6.3 Conclusions ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 15 Tf 114 Tz 3 Tr 1 0 0 1 41.049 196.2 Tm 91 Tz /OPExtFont3 11 Tf (In this chapter, we firstly address the problem of estimating the future states of ) Tj 1 0 0 1 41.299 173.399 Tm 90 Tz (the model outside PMS. Directly iterating the initial condition ensemble forward ) Tj 1 0 0 1 41.299 150.85 Tm 93 Tz (is unable to provide good forecast ensemble as such method ignores the exis-) Tj 1 0 0 1 41.299 128.049 Tm 94 Tz (tence of model error. Two new methods, based on adjusting the forecast with ) Tj 1 0 0 1 41.299 105.5 Tm 90 Tz (imperfection error provided by ) Tj 1 0 0 1 191.3 105.5 Tm 96 Tz /OPExtFont8 12.5 Tf (ISGDC ) Tj 1 0 0 1 229.9 105.5 Tm 90 Tz /OPExtFont3 11 Tf (method, are introduced. The first method ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 228.949 30.1 Tm 75 Tz (154 ) Tj ET EMC endstream endobj 822 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 823 0 obj <> stream 0 ,,99j*6)If*{ z[N [,pL."IaObe.l?t3e!(BAm[fZ0M4"Y3H\JRiAFhz0h(L|%S 4 y;kG_gƍ}eYm1iYW!wDAj9p cAߨ O"Ճ̗XhF_cnϖRwT][UOpY5ϞtvnDP xQa+J;qgjzz8Վhgo ?Q}:Bx\-A| h?Zw%'St>ٲ"ѵ6ؒ\/#ə5:"a71ڶ.ҸQ\⯪z+hώъѰ< ? vC,hƶ=4aMkOY#Y&vVɬ[BfOͧ+=6vy[׶Ph>Hh]' bnc0C MvпbvdxA:e:we,7 YYxPFD=!o6yThk:yhnɊG<|@ A}=6hXoS|#D?bE눚߬4&BP?OtLIy᠟?w9Eom"*36:7Dx2wj"i ;0[+oU-pT4W(úpI5кEDd,_(\& bp0O>L(0Xiz&ϙa=4t3R~w%Ajs]Γy:@W$l6;m61iUʔ9EM*7It7e;QχJ+c] eW. kbwnQdFJ``' 2*Rm;fh#P\@߅~).>Q;;a_H}޳exV}1Ǐfy^@ƽRo^\p4e_ %~ e;tV[P!MZNqYᴯf4 y֢ϳXw5i{P-Mfl,Ij*3P)y{Gxpl4UY/\VR> ӘY#6 !H'("VxA2T-Rt}UXok_)iuq5GT,*d,REh=vQr*kfIF֒HqNv} ϲ,c%&F8ͺWn` &jkd}$44R1T],OC~_t]( l0s"[xs)= 6 yqZ800OǓwfsﺽ8qRZ@*LSӓK4U(w:ޠ3<)exO X%vX-0DIG @PAFDט~EV~u?Ed/E)L+W"sgIxtp q<hywDbMY,*݌C3pK2We~%2CXv^t=d Ry.|2w3xJ:t34ݞ1Nxm QY ;7"u?U<"4߅+mPčm):D2#I:Z&D(p@FF' &2WG'+\=mFšcPkЕFq_u|^ \] GV!_ /L .Bo3FK`%$nb!fzifO|, KUKu9۟:1) bb?ac2EaL3Bs貑R2qe5;YK ̱٥6\kn\{O>xN&fZp$CI~CEYS Q.A#Hv^ y-s j?7HcbmÓz(n-:Ú;^F w-0Uř>NaS6ݸܰ쑙1mՍyg3פװ<%q Ѻa;z3c“a둍iȬI7hN!Ͻ|>@2F3S #~z0e1Vs ~GʦgIUmA3R`ݖ ơSuai}x9DK-po뻄(z3z5_(j2v~y=@ ]z6Q7*ŜUcNbX]6igG|=,m@1E-9@1q'~Gm?G /qn- 9 6ӋN.z_a^-7C|B'rN&W!_x^ב1`"E[еG$7}K(r+v_N\DJTA~r"`֯z6R/s[NB%y'̈`]- lXέ G5D֡@2[FRU^<~Y i^4(i ֢]>M ~QnR}YN 5%D))#^C4;C)Q걥Ǥs3@f"1~Z.ís=ۉ[zs>(!Z Q٦D :_ T?VxZSC׶Z^AQm B8ǂodjm:7u(Z :\\&=y?ϴˆ,sv#Cbx2҆:O*q9.L4_\jr\B@cYx<ɰKk:SCM|hz[CmL* `H{ v~UP۝?l1,,raBhR}&s8yMFNwJ=?TBKC{ hFܓq3oM(Aź lz`k$G޵9Lr̎L hՎ#OVA %+ f(;B_$sdi>\Gk>TT* HlċhA{I] o B'oE`q(.S 6Aqg%D03dqwr5ڕ}Zed]i0fƶk}$4]J 0B L$ w`SQBE*'})TK0惬%)ID : ziQ$eWphMyT_MIkDgmC%X*MMNyL߁hSk10gi>8Dc+a`+׀KT ZSKx蓆/|1{o", Vh bYQ2W@v z ;Nh@#l:9"dp/P ]$e5t`ᱤ>@ 1E 'p Oҕ:Y]lyCm%rNl7m ZH欑_o7wL%23*ć@z8MmSTynݱԑ,0u NhMzYF ۡg?+V80-tnx>$jk^>YqaewKjYǦ#PqN ;NHOsfҬMEcZ;V& Zz*Zcs'C~&:`Gj!{Zߠ팙hOBjgG\pC vy^Cw0C /QjJTbRbEN\*UJk "ͩc#8#yٴf͖Hu >N~4z!م5t)JSQ&IU Lc,k \Y *W1vTV¹wB"U*[Tz/s?1RU2YHݶMUD'H#rP)e!%uqN#CVt"y/|)p& j<t:Awq?#n~?~aGnyϨͶ" qvRY=ΔaEH:1݃lP33H)X4/rkZPۏLnᙛ*#bpT6^Xz7OH7T <ϫ ܗcyΔ.sQY.|X rєWùoղRSޏ²/]Ŵavލf*T1p,s?Ǡ \WT@'P((ݓ+`Μ}F H;ŲX/ݫI̘&{1aGtGwO&``J(,![Ұ<XJpTxy=k\G8ЄXoS=A 3YƮ)Up<§/#^|%dbחA+kY./Yٺzf'"2V;`U;A@pxW*Hdu}h{›)iГj/FOu *}ESDБfBFL3$v򩧺z5k/]ko8aAmHKbʑ3@r1b˶%mIҍcZX^"Vw`@wa.|#FB/fŸIyt}k*ziVmKG{_h] D%1cEgQl _8s#ٟ`#'ˀ I2) ٨GN*ۺ rj zݻ@1?p9VLxȬmIK[$2iHfBVd@@,JٙEH.OҊ+Iu0 z*a "~:1pYY}F18CFZz\ʀxVu˓~zj`TfϒțQ]fi6u.iztbH1WZ6~SpduC/A#Ayׄ)qɮ:tD߹fEqAe\V%C&E29 8" wue(}\XWPgίX 5:@HP?]*( Y[z+O 8g~<E RBv#g D>ƶ՟AKbhHB݉bdu7Y }=@z6 _գw  wÝ.8"TOfRXS/1t83TMcMrrrBOBfE@oo W> R-xݔ`hڄ8W垄)"\4Ke?kRenpTҔeը[*(*!-s~߹lFtS4ȔeYNYAЩ:a|** ]1$L8|DĨC+NK$-1pXr88fĕMmV=;}Di䬷@ }H,DӞsOQܻNfcG><:}qID~)20) ċxP HDRW?Wtav0'b)C+P=]d5m &Rx,?ᓻˉbjuࣃyYLb%{7(9W 0=njw`:'>{y5A00;ۦOAmk}JrccOQۜB!JO{p-_NLL?\} EIl2`'id4Yqo[A4NGtmrqv|:U_ʼc;N0 ݠ;};?9 zA8ye\fe,*&Ľl4Y Ls}Xb:m<^B@gؾp+-1`,Q7z9U#=Y͚gkBbw܊5-h-7\V>k6 w&Vr.qSUc 诺@%\³kn!Xn8֖%C` D>(DNER3';JiiJ9,%b=](3(Ͳf3t%.ͅ=(\WϷ(]yU['- 4J`D/%~?\iC4>KSmw%'폖d?j=; uvO/&he$e!6  FŨ@?]޳`˶X [;g 9f ]',$1<] e1&S-[ozk d|n^QZ`C 77uFQ[yR_k ܨT ̿E;¤==i=Ěϑ nT, osWm̉_q2Y&gi%l F hFbrܶDV'?~G~EtV(yo)mn5 K|guEw)D.P@>VZnFzչd \/-|'pv14a"JA9q)H k`4<'T@GK9Dl#$UC[Sі"qa(J^Iap5o +I0?qۼnrZsR+9lx< pcDɏ iU]IT4lQBg{ ɬPǩN=Ab]|o{"R}F2뀈ZZ>r7UҢ/rF _ݬѬ!H!qxوŕwsU1/j |XWуU- >Pmwj/PѪ!`m7ܐ ;);{U?NBW̻tQYz =}`o^Y'8 H>?SEgD̾y4Oq }Ulg،nF3Y]e&.@ +UՌvZ@ e>J(T۱ړ1[I0[ܤջy/,<~闙u a0$#ƌ{X~;şzz&l1+ Fd\Kk PHkp2o"Mjx+*9Щn`NKR5~d@k =6F@fǟ7aEfͦ5>j8 ~kbJPt6+Y5'4Y{*B*UMm.BmO>Rv~%e]48'Wsϒj|;ڑZt gzLxPڒb }r&p G%j#R$Q`We죆 jr.8zD)^=:kwgg /jabl:q>-DsyRgmͯy{EDgyyʃ֕, OT - X2`g=ݙQ` UfA([q{= r;CA->d5P() GrNmk$Y{-I aZ) qV0A..Lb9+z~vs}Fɠ_sdA+!J`_;5M^}|wm_Mk)/ CNY#Gopns0 ^}AT`8_7<fK/8JJ$NrxaC\Gh!̒HE̵W@}¤ڗ0:i@)("by q(&$SB3G1sΏ39k7pOf! h=GAo,1Rlµ5ÛY|F Ʋ%]u8p6C{r/q@F[ruvIwo|[$Z.87P_+@*'KnFV.. _>K ֝C髰J֐V ܹK"XS=ҺD nu`=E+Ղf+S S(nrٌ 4;ݭgаX-!ќPl61S3$]*d}}p."!h/#=<=Y~)ԊaCAp-.2,wn_V=Imy~YQgbB|7ZH2<^? Q`=|Y,UV]JU5u٫Y)ڠoKv@KQPm:qIP<--⻥j-ts7}4I9?զFŐU;6nGϰ7׸Lp,ûKlp;iVW ~x@ 1ZT!ѳ~<*ϐtr`#Բ"~j`k.#` œyH3qūhCimʦ!E]l < nzCSJՄ?D26 wK^ 9r'S-+e3q箩il*jo`*:WfޠHKpSׅyi"nz7NOdWuuf(3JuoWxĿ?<߷M@[Bt:`03rE~mIlP3~$9ou +k.虿 ;G(M)h/ZMoGP%J)c2QTHQ'(K &[gGr{3egFB]E^|/EnjÅ,Qz\5 endstream endobj 824 0 obj <> endobj 825 0 obj [826 0 R] endobj 826 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 599 0 0 838 0 0 cm /ImagePart_2182 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 413.75 718.7 Tm 95 Tz 3 Tr /OPExtFont3 11.5 Tf (6.3 Conclusions ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11.5 Tf 95 Tz 3 Tr 1 0 0 1 105.599 676 Tm 94 Tz /OPExtFont3 11 Tf (adjust the forecast by adding random draws of the imperfection error onto the ) Tj 1 0 0 1 105.349 653.2 Tm 90 Tz (forecast. The other method selects the imperfection error using analogue models. ) Tj 1 0 0 1 105.599 630.149 Tm 94 Tz (Applying these methods to the Ikeda system-model pair, we demonstrate that ) Tj 1 0 0 1 105.349 607.1 Tm 92 Tz (forecast with random adjustment does not provide significantly better estimates ) Tj 1 0 0 1 105.099 584.1 Tm 90 Tz (than direct forecast as it discards the geometrical information of the imperfection ) Tj 1 0 0 1 105.099 561.049 Tm 94 Tz (error. Forecast with analogue adjustment is shown to outperforms both direct ) Tj 1 0 0 1 104.9 538.25 Tm 91 Tz (forecast and forecast with random adjustment. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 122.15 515.2 Tm 94 Tz (Secondly we address the question of how to interpret predictability outside ) Tj 1 0 0 1 104.9 492.149 Tm 92 Tz (PMS. Traditional ways of evaluating the predictability, Lyapunov exponents and ) Tj 1 0 0 1 104.9 468.899 Tm 95 Tz (doubling time are discussed. Lyapunov exponents measure the predictability ) Tj 1 0 0 1 104.4 445.6 Tm 92 Tz (through globally average error growth rates in the limits of large time and small ) Tj 1 0 0 1 104.65 422.8 Tm 95 Tz (uncertainty, they are of limited use in PMS and inapplicable outside PMS. ) Tj 1 0 0 1 492 422.55 Tm 107 Tz /OPExtFont8 11 Tf (q-) Tj 1 0 0 1 104.65 399.75 Tm 95 Tz /OPExtFont3 11 Tf (pling time \(82\), which measures the average of minimum time required for an ) Tj 1 0 0 1 104.15 376.699 Tm 90 Tz (uncertainty reaching a certain threshold, is applicable for both perfect model and ) Tj 1 0 0 1 104.15 353.449 Tm 89 Tz (imperfect model scenarios. A certain threshold is, however, required in advance in ) Tj 1 0 0 1 104.15 330.149 Tm 91 Tz (order to define when the predictability is lost. We suggest using the probabilistic ) Tj 1 0 0 1 104.15 307.1 Tm 92 Tz (forecast skill to interpret the predictability. In that case, the predictability being ) Tj 1 0 0 1 104.15 284.1 Tm 97 Tz (lost is well defined. In the IPMS, such forecast skill not only depends on the ) Tj 1 0 0 1 103.9 261.049 Tm 94 Tz (system, and model, and observation method but also depends on the way that ) Tj 1 0 0 1 103.7 238 Tm 91 Tz (initial conditions are formed and forecasts are determined from the ensemble. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 294.25 52 Tm 71 Tz /OPExtFont3 11.5 Tf (155 ) Tj ET EMC endstream endobj 827 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 828 0 obj <> stream 0 ,,FTTb7 \js,x"bk♸r\9ŇeB%[:`ABQdx ;lu(lm𦹨((u6>\"AY@}`vV8{Y?ɷuCYRe$凍p ˼Z ":}~QfoC`U}NCo!q|a%4WK-.J=^)UL>κѱʰ7Y埱Ұ'!`X޼dJܱ(;o3C< ǎMk M`{2dw)GۀU@8@vTnΙ.` @Ej F ظV\W1MX+;Il`:c8\p+[R9@傞X|4VPQ ''?r#8m]io|o%Q[/uMΑ_^p\'L>?nxc`Z/nTC cA\v zejx\Xc1~ ppTM䮬JOzA+NEaP4)@܍yӕ'qr1sh{- !:%!qXI@Z2wOʞ]7&0]`?A9&L("F'Ē,-V'GAlߗbM,SyoKR]_1qb*s+ {ٱ& ~6>$ljN٦\W&[y)eNS4|R3i50 #Nk7t$ @ѓ|<#Bq- 4@tϠ%X w7gm?C*IlD"ߍ*$N&vAE3e3KkpøNyWH(ǚy*NJ g '0; RH3O0Dpgϊ€Y}":&;u}i%eЩeI3U$ d.sSb4;bh [ԣt=7"(_O-A1b=x›{[( =f8G#IO/G H}w<0rqYOmE-I6]a',(mYT?cC<&07>pݘFܝE{0E+򠏖7ps YR_ϠQKxf3 j$!wg9b~"|Å뿄jGg|uh߮quJs'M@|jnnvd1?(.>ZtA(Oh_ot .֜Wl=riھJg7}d% AT G빳CCB%8Rt;@m6nv9%\S.h*|>jp?o]HJ8'r`p>wÞ:sF)pDoWh`L};)7N=(AN-boaa62%\bl)כn0 Թ沽zn.[EU>sU(mWmK;)9FbHmkߴ*k6@r丘]Pap!߭z3Z1O{N_(Ga '9T f\<$r ҄Mԅ=MEyE޶ .0ZP,,xb*]UɚaM񾓥ygYST|N)Ton 2 W>dԎ Ļjч܃vKAiYu.,LӶ$xN7:@?PWK^IOO@jn+aH}M6WWs6&]/l.]mJX f)<}C/#tw'P‘IJ"0p[?bC5OqCpl HZ<"ߋ,D }J+;߅x% m M ᓇa+7\w*¿ ֥fؚK* JL0$=p->^?*$K|ClZC·XtuKY{0_|U,@yfv xp>挤 LA^ C'XB`HpV[$ƫ>Φ# ̖B[~{ ϫ9[*ԉRZNOÝ8+䪲8<ܜٝs5]+" ZOVvUvj-MoBPi>c2Y%:D\PQȫdkrr>(lsD@dXqRiL\mG0kOpH1rVѕ3+`e>Zh,.μcb1Vw0U'@.pg-wZ37$.)dAmnB)[y'V 6$au0(NǹgF0w;#M䬏Z\Y>('XPdw\EGwl&O=mfqE;-ӭN^ X.f+@ (j S[x֧X%U|Q~pЎ [ϖ F޺<Ǽ% u}fэ yuqXYh1z̨+gwUJ7g^lgr !5I7JGLH1O{FH]UA+"{Oԟd4tt. 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(45qY|J{K٨WZ+*Q&V<߇w3l%Eco \8|q[Gft SwXV M%8b{c6uJ zDqz-Rs"Rh& ?x\Lhs`rY zv&)0]XR$֤M Tt/>d;=VyrjmT9XxOUX'jߣ%6d'%j9|= Zzu$6Xӿ*ªx]ӵjDBg3T[#Z%ZywӨ8~4(LgJRJq곦}G`|RU3#\ W]062wR7m7~RE3QaQG!jZM;t/4Ji¨+k@æ#bc\Yۃ![JB^"ca~ U2U/6Tsȶd⮔`ɶ-nTdUu,U聥<5d-g[ i<;%)0@L$ɥۘ2.֑B*(Cnl?8"{W6? }i[s Dhe E哋)T/ 9^aoD)OIw%R {%ʁӤo|0cjHT=3б@RHd:U%Vf+r^~~fc9n0n %gG-:/qfktD_yğS p|pFRaL&m!'K4,#T{~L2t9߷Lx&mĜA8-(еN(wG&a۶"^O@|YǦԉd4^f5wΝt]\kV)NeG8-3b cvq1+p 13j냡K(Wy0UxTdٰq xK\a9zCT*;bjAwޜO?M2l#[jIb CQa㈦F{CEؚiN],u]QҴ(Rh)3giº4c9%"zJeS[0B,DήQ+v_KTQ1J;.F5^V.\US&9CōW z1&,_iW5֒zMWmQR I]OU>|cXU A3 6K 8",]/뻺k2nP2X{SfϒER>{<0`t(]vL\gRIvdy8  )LJ fѬZig7ob1 m!2ؿѧmy(Ԫ̊Vn 'iW^nx@(kJ"*s,G&9ZU8t9Fpc)6k2G@+ Ieڝ$f]ݴhxYsZS[۶*!;"kHR |kM"ݤp;$WKQm?Ȩ}N)hSl,L>A$ؿ?CULW>.LJUn|Õ4'.e f7\mBr^ \i˗paWmM͈tZ;Z(J~4bhptCI[C::cslD,A\b>pӛuDw9[4xlna}?nOY70XTkQ5e} ώڋ'Ŵ6;|^@r4Vf(f$I>4b)ƤUp#\gRzV?ဒYZ)7-@ʈTob#߾VԹ[5^LS넱rlzCXAt.Dbtd)=7D{ Oa4s][ tesWe|0eޫyh_e@doraI{$((cR\$yeYꮾ!,{?*SI@dEmsyћ fH_qC-bS8qXK LxTy MbR `e;>x3q9"*J2aE9zSvQ3at/S" 60¤?C5"O"Y)B^+jՅڶO.2bb-sL$d"my9!^\\6~=Wo:qnw z?y$>`aթHJ#GX2wP!̭Yr{5_On6d`t!kvXOIyt7zh؂vj@.n Fc3& i*,Uģ䉵yC 4=>"@E.Zp/R?GcZo/BZ*]LhvPHIF[ %bcHS_g" G )EL 7 h %cXc|[u<kd{T$m `,[tKk/T=S$ac M\dj[E׏VJ7K"*?jW+c>;}{ գZXˑF}pO.y@dĤ m);[0Qe0#ChKmdK =T*WwQ^W9;8 DJc{3p>x;n[{Wl&^^I"@s$dU4{imzuWC-ꉐQ5jPx#!F7쪁t̾sך4}i4A)ŒOa*Q8vO|c塺T2R1=+0)iaQ+TFMtRUX&#.& ~ތ# ;9ښB[%]g?^/MDo ?F3 _ U+<(g*5MI3O c{. 6#b&*DR{gsg+>ij`4N.1OoBELE;T4[+;[Z3k+5]s[V_lmA:Z24{S"#ìtIHXœ ǹ<#[Y`7AJcQd FE [oFꛈCj4ΕC9kW.;ih5•cd(Ջ=*e;Z<v[|~aUϤ=jgTz~U8OȊh_ M+q m8n01<ʮiɴ,[s,]-ʴUtO=BQ7}9ˠnaA&!\LvD*BAҦ4h r+e\A7*@ \v|īVF * ^I#ӮS_fDt_x5{- endstream endobj 829 0 obj <> endobj 830 0 obj [831 0 R] endobj 831 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 599 0 0 838 0 0 cm /ImagePart_2183 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 106.299 599.45 Tm 106 Tz 3 Tr /OPExtFont3 22.5 Tf (Chapter 7 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 106 Tz 3 Tr 1 0 0 1 106.299 551.7 Tm 101 Tz (Conclusions ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 101 Tz 3 Tr 1 0 0 1 105.099 490.25 Tm 91 Tz /OPExtFont3 11 Tf (In this thesis, we addressed several nonlinear estimation problems by combining ) Tj 1 0 0 1 104.9 467.199 Tm (statistical methods with dynamical insight. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 122.15 443.899 Tm (Methods based on Indistinguishable States theory are introduced to estimate ) Tj 1 0 0 1 104.9 420.899 Tm 95 Tz (the current state of the model in the PMS. By enhancing the balance between ) Tj 1 0 0 1 104.65 397.85 Tm 94 Tz (the information contained in the dynamic equation and the information in the ) Tj 1 0 0 1 104.9 374.8 Tm 92 Tz (observations, the IS method produces a good ensemble estimates of the current ) Tj 1 0 0 1 104.65 351.5 Tm 93 Tz (state. Our methods are applied in Ikeda Map and Lorenz96 flow, and shown to ) Tj 1 0 0 1 104.65 328.5 Tm (outperform the variational method, Four-dimensional Variational Assimilation, ) Tj 1 0 0 1 104.9 305.45 Tm 91 Tz (and the sequential method, Ensemble Kalman Filter. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 121.7 282.149 Tm 92 Tz (To estimate the model parameter, we introduced two new approaches, Fore-) Tj 1 0 0 1 104.65 259.1 Tm 93 Tz (cast Based estimates and Dynamical Coherent estimates. Forecast Based esti-) Tj 1 0 0 1 104.4 236.1 Tm 90 Tz (mates method estimate the parameter values based on the probabilistic forecast-) Tj 1 0 0 1 104.4 212.799 Tm 93 Tz (ing at a given lead time. Dynamical Coherent estimates method focuses on the ) Tj 1 0 0 1 104.4 189.5 Tm 95 Tz (geometric properties of trajectories and the property of the pseudo-orbits pro-) Tj 1 0 0 1 104.15 166.5 Tm (vided by the ISGD method. Both methods are tested on a variety of nonlinear ) Tj 1 0 0 1 104.15 143.45 Tm 91 Tz (models, the true parameter values are well identified. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 121.45 120.399 Tm 92 Tz (Outside PMS, no model trajectories are consistent with infinite observations, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 295.199 52 Tm 75 Tz (156 ) Tj ET EMC endstream endobj 832 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 833 0 obj <> stream 0 ,, b7!z7,߂Kޢψ!}$yfirF0Rkrkξ ^5XвXG:~LR\K*A".h͏,; +xgITLa\&`y5 E=xTL4LI-,tXW5[uq1Cej6QazO|x6l_z(x̦=QPW@NPgAlNBđ^Fٿ&/oG1͢$tDjdOv|j8YyJ VGjUzu c#?m%36MG TNCrU.FWP#~#NTQ~t/*U4%^z,\pnKba P-LVz 29%rh{U@J%YřÊ`+{XY)k??S:e:5(_kP5.AWʷk"k*%L>ʳ"ٱ#<]W:&^{L fmx:I]#0Ăk>9P_G'Pn)()+3 L 샮[Pe.3^oxe0 RRKpܽ#|T:=7}BvrN64Ӄj (фKΕ 7C 3IAއp<$(tp>>󥌱5ثԟ<_} ,3gJO<__BhBcUcǬaT~a8Dew%Ұ] ل\l*mV1֪L!PM%"Bt%ɼy uaעgKMuOb"^I7ȊMJ0%Cw'Lpi>馱;޴&2/|87pn*w[X3y`uj|ϮQ{0v N5u[:4j} BQP{5/gM'n.ʅhб].Ѵ-,(^a4'=uC9>S 耗쓷jh3VOϨ,yF )15qq `'ie.A^7;_]C#_*,÷`9L㙖:QLAkȋAUڏ;yZ%Cp1whnڧ9H e_;˧qFiO6ea5'Bb9ʩJC zݣ~L(^i(p4&(4( <glh|2X0S pE qTW>v .t~VQ[mk܅eÃdټt't[F} @-2<ox{۵'ixq׆EXs* Gbѿ)qh1MJOX# 'Zκ菮a*69{q.lt]; 㤤7?S#pSe؋K_qj$S?bTE[n];/E&w6FQB)1Ŵ9Qg>KT&9t8#v!YP8#BX\""Jk.—nUZ(q[EPUT;K~jl@=6'/t~P^OmGV g"U]5kǟ?o Uh³]^z#*[]_*8V"ĸv%@TTx;TvQ0 '&ARSQk9$7Vs8htC$q# CLYQx/nxxCzK\%vݘ ?EU`t=-.,$CR>q\Y^/)q>W70h.kg+S pIK.B5Ql<տFE a340Eܱu vI>ҽCQ2(9zq}zLm:ܑt~~#WBꭝ:m&Tw{DB **kW1|4b{G-kV#ݚ@Kb9 e}ydžT"  O_ݜ.'~ 3p92B>~:j<꺒c nCC?c9=0;7$5u ( ?+N|!YNc4yMW*ꝿ/u)4﹊X]JW+.fj!FUi0\"5'C3uihzۢ7Rj~]F` %ʼPDh(RԜ_ŕw&շ!y*j=wx_!h1~Q.g)- 6Gw` vOƑ&VvB)%ar"ՍK*'--:{||n}`Zd82=iلд&pۘ My!0J2eƉV2]&Ly ܤ- =o WV#f1sB~4o>9lzQڊn88(r..-9Ƞ??~+;%(ɷɴ?76\۱վҲ)י'Პy#!)ʠD8DZ#66\3u0<ܛC87/\3ka|/,{eZ%&g+O_ˊ   ⳨-j "MF2-[0Ze l r(!m]n5E.+ ^ Z% Nxg@(f]L]bm> d+bb[V:3l6SBUi9on-)'h`5 `Ci[ j7 2BR<:.]\&͛c3cy"'8V*qBadHPn{sfS2 ba?g7^Pկ,L\~А$oVJ?V;ds~𺼬pbLqGahAtրq0&.>otEL 'd,ACuĉX_k2 H*AL Z'͈ɒ@wO완c/3bn$;Q-6i-?K :^*{/=2WBw6˴bD Ě{o>G@EnkB:QV{,DIL~^ZC0&{|*Ҋf?pu8$R7z\풳2/zVzU K6k#@?j=Nhk{FF`Y b/ PKӊCտk+]5­$XtP. ~I~O#U 6L&y[I{Tp 3~j r9Ѯ`eY.ה=u%)QHkwa9lX&\_g*hEX,tmondH4QAO& 5g[gYI"-&c[I_~ j}ctT9bՠW̐:t.I+;Oa_I H42A<.$W3MzSryu2[Լ2XO+YsO̟ 0k 3?2VCݩx~Z8X?:*0`{[jlvrvq4͊9Pܗb#)aqup3mLV;e@&Yb_T}dysVjoT-&UCф|iv,UfzɽZ(t݂YQL2"vGM[MymCf?JD y[7_8sE4w&p+qAa +|rT\,3vb260@G 4Ov?WOzȌ-*L$ fH6:iSa2-d *o | c6tp]Ϗ]Epugun|0N~ݙQt}R% 4Z 滫~4M?_̓`w Qznf'۰^%-TmXRc劉$ӥSPMG+zEY4 =GU ,2A fbgIC6g>xȺ"zdN|,1Ɖn ;f[`o颛 hSګAE8QVY0/gs{ ' >Mh58^;le=ˎ&øK7.kvs{#DZ$r#D m3\cJ5Kxi Fk6= M; L%;(=AcCzJɃ)v^ x?D0f>E8A;1$\T2< Ͷ4?Paր+ob]xjCS=K E\Z4yIEh.~D0xG{}9%);\p nE݌Q @wFHV<\G%.~ɅJ[[=Ztyf$Rx7#W),\}j14hLm>5z?*󳧰웪ȣ.=%5+kѯN]:@̠u谴_鷡泑č_smʳF:1*> endobj 835 0 obj [836 0 R] endobj 836 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 599 0 0 838 0 0 cm /ImagePart_2184 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 105.099 675.75 Tm 90 Tz 3 Tr /OPExtFont3 11 Tf (there are model pseudo-orbits that are consistent with the observations and their ) Tj 1 0 0 1 105.099 652.7 Tm 93 Tz (corresponding imperfection error reflects the model error. we find applying the ) Tj 1 0 0 1 105.099 629.7 Tm 90 Tz (ISGD method with a certain stopping criteria can produce such relevant pseudo-) Tj 1 0 0 1 104.9 606.649 Tm 95 Tz (orbits. Our methods are applied in Ikeda Map and Lorenz96 flow, and shown ) Tj 1 0 0 1 104.9 583.85 Tm (to outperform the Weak Constrain Four-dimensional Variational Assimilation ) Tj 1 0 0 1 104.9 560.799 Tm 89 Tz (method. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 122.15 538 Tm 98 Tz (Given the fact that the model is imperfect, to estimate the future states ) Tj 1 0 0 1 104.9 514.7 Tm 93 Tz (requires accounting the model inadequacy. We demonstrate that using the im-) Tj 1 0 0 1 104.9 491.699 Tm 92 Tz (perfection error produced by ) Tj 1 0 0 1 250.8 491.449 Tm 110 Tz /OPExtFont6 12 Tf (ISCDc ) Tj 1 0 0 1 290.399 491.699 Tm 92 Tz /OPExtFont3 11 Tf (method to adjust the forecast can improve ) Tj 1 0 0 1 104.65 468.649 Tm 91 Tz (the forecast performance. Forecast based measurement is suggested to measure ) Tj 1 0 0 1 104.65 445.35 Tm 93 Tz (the predictability outside PMS. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 121.7 422.55 Tm 104 Tz (Main new results ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 104 Tz 3 Tr 1 0 0 1 121.7 389.899 Tm 96 Tz ( Chapter 3 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 96 Tz 3 Tr 1 0 0 1 145.199 356.8 Tm 100 Tz () Tj 1 0 0 1 157.699 357.05 Tm 90 Tz (A new ensemble filter approach within the context of indistinguishable ) Tj 1 0 0 1 157.449 334 Tm 96 Tz (states \(48\) is introduced to address the nowcasting problem in the ) Tj 1 0 0 1 157.449 310.7 Tm 90 Tz (perfect model scenario. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 145.199 282.899 Tm 100 Tz () Tj 1 0 0 1 157.699 282.899 Tm 93 Tz (For the first time, IS method is compared with 4DVAR method when ) Tj 1 0 0 1 157.449 259.85 Tm 90 Tz (both applying to Ikeda Map and Lorenz96 system which demonstrates ) Tj 1 0 0 1 157.449 236.549 Tm 93 Tz (our method outperforms the 4DVAR method in state estimation. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 144.949 208.5 Tm 100 Tz () Tj 1 0 0 1 157.449 208.5 Tm 98 Tz (For the first time, IS method is compared with Ensemble Kalman ) Tj 1 0 0 1 157.449 185.45 Tm 91 Tz (Filter method when both applying to Ikeda Map and Lorenz96 system ) Tj 1 0 0 1 157.199 162.399 Tm 96 Tz (which demonstrates our method outperforms the EnKF method in ) Tj 1 0 0 1 157.449 139.1 Tm 90 Tz (state estimation. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 294.949 51.75 Tm 77 Tz (157 ) Tj ET EMC endstream endobj 837 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 838 0 obj <> stream 0 ,,FFj'{9k{oٳi2&-27x8UxÕ3`qtFjpiU+&Q'*'V'><·氱4v5,|B(7s h-|/'A1AF-Q`kC(C_bbc4)/qSRV;h|'4\bPtDXfxZeaPiZ2WTd>iˍ@[8Qv- 0x:x7KŚ^Q#|S&96Q۲:{6K :(/s)~P\[TLa6ܸO^a:F((#E߭w:O"9xS_ :oJ4JI Lf=42 Դ"Zzn@1VvU\RQ:/ܐ*<;DꑆQ?jz81\Uiԙ}8[ΑeL=q&v!=XJhKxA8Xoqi? ~CKiخ^>k]MӦ0xL+M0Ȉp^_̻y30oMF"h-Gw!bJ|9|qw!K/8l{\{ N0LȸVc^fOBS~Bѷ$.&WP>Pe1M1A(8֐}*$%$SHHrqq !oH4ߊ'4RP4-6k;rv]pN-(9hL5[ObNxX(60 t2H yNc6t5 }6T@ Ir6&q*g8ָ`>WGQ[l íZ''s1DB򚤬n! 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v+J,9ḥz}2ӭE_&d3͑ä's[źboXqF-qȒD_Q;RиIedIJ2:RiC^-Z~jT1ׂ' 3kCd fl{y`tsvKO$$ŧDg͇rTV ۝B9֊df6ɑˆ?/pi̷< ~bZ]4-e RĶ6B@߳8lD^f0>k8>RљKY㇛H(TԸkր endstream endobj 839 0 obj <> endobj 840 0 obj [841 0 R] endobj 841 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 600 0 0 839 0 0 cm /ImagePart_2185 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 147.349 677.25 Tm 3 Tr /OPExtFont3 11 Tf () Tj 1 0 0 1 159.599 677.25 Tm 89 Tz (A new probabilistic evaluation method, &ball, is introduced to evaluate ) Tj 1 0 0 1 159.599 654.2 Tm 88 Tz (the ensemble forecasts. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 123.849 621.549 Tm 99 Tz () Tj 1 0 0 1 135.099 621.799 Tm 90 Tz (Chapter 4 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 146.9 589.149 Tm 100 Tz () Tj 1 0 0 1 159.349 589.149 Tm 91 Tz (A new parameter estimation approach based on probabilistic forecast ) Tj 1 0 0 1 159.599 566.35 Tm 88 Tz (is introduced. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 147.099 538.299 Tm 100 Tz () Tj 1 0 0 1 159.599 538.5 Tm 94 Tz (Another new parameter estimation approach, which focuses on the ) Tj 1 0 0 1 159.599 515.25 Tm 91 Tz (geometric properties of trajectories, is introduced. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 146.9 487.399 Tm 100 Tz () Tj 1 0 0 1 159.599 487.399 Tm 91 Tz (For the first time, IS method, as part of the second parameter estima-) Tj 1 0 0 1 159.099 464.35 Tm 90 Tz (tion approach, is successfully applied to partial observations. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 123.599 431.699 Tm 99 Tz () Tj 1 0 0 1 134.65 431.699 Tm 90 Tz (Chapter 5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 146.65 399.1 Tm 100 Tz () Tj 1 0 0 1 159.099 399.1 Tm 90 Tz (A new methodology, i.e. applying the IS method with stopping criteria, ) Tj 1 0 0 1 158.9 376.05 Tm 88 Tz (is introduced to address the nowcasting problem in the imperfect model ) Tj 1 0 0 1 158.9 352.5 Tm 85 Tz (scenario. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 85 Tz 3 Tr 1 0 0 1 146.65 324.899 Tm 100 Tz () Tj 1 0 0 1 158.9 324.899 Tm 99 Tz (For the first time, our methodology is compared with WC4DVAR ) Tj 1 0 0 1 158.9 301.899 Tm 90 Tz (method when both applying to Ikeda Map and Lorenz96 system which ) Tj 1 0 0 1 158.9 278.6 Tm (demonstrates our method outperforms the WC4DVAR method in terms ) Tj 1 0 0 1 158.9 255.549 Tm 89 Tz (of nowcasting. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 146.65 227.5 Tm 100 Tz () Tj 1 0 0 1 158.9 227.7 Tm 94 Tz (For the first time, we demonstrate that applying WC4DVAR method ) Tj 1 0 0 1 158.9 204.45 Tm 91 Tz (will face the problem of increasing density of local minimums. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 146.4 176.35 Tm 100 Tz () Tj 1 0 0 1 158.65 176.6 Tm 97 Tz (For the first time, IS method is applied to form ensemble of initial ) Tj 1 0 0 1 158.9 153.299 Tm 92 Tz (condition in the Imperfect Model Scenario. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 122.9 120.2 Tm 99 Tz () Tj 1 0 0 1 134.15 120.2 Tm 90 Tz (Chapter 6 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 296.399 53.5 Tm 75 Tz (158 ) Tj ET EMC endstream endobj 842 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 843 0 obj <> stream 0 ,,&&j՛ K#Qi ŸܮrD|{M%SAn4Q`:ߩ|V~3'g1ʆz[6:]"4޼]|d朤Yvv{s.%g#LX@X#&kMd@]j-ք tz2-J}Ar 5$h8joNX_`M-] zkt'o grp~ q oyu .FX c{ı@!}X2AN֔EI&pIݾ @u]8J#( l b^>;u^n[c`~Kŗ Bw_ZGDIe( 0O=nuӪx"o~%><*^9ʻ1{VQj$~%ԝ@{6V'5Ft>rqA[bHȨ L#̷W/E8<]`!M(-ľG8w\M)mv0`>-[]kClseR^2'zBp VV-vyggvMҰF# mx ı"k(>9 Db?#v\@Ġ/&CYc[{n?]2_=j_Ky-;Y|&]ʧU4Zj\xP(Fs!MMC&ثhƤ?\d^j&M* Ji@^ǹ]#4:$0Ok!N7HT%xՖ-]q/Jۤv'\bt+mP-[LA(IjAg.tdIi5Z$K@SfC؏l)_H2R7Dwp<8}@WA{ٙ Ra1A<{PaB.0Il*_蓺+uzؙ37k;c ˸=ODC-#>'q$c JVLM'eJ ?7K$!˲&FMikKZ|S`9 XO&J*#LpC+Γ 칠,8xEK|$+i0J3Ori`)L/*cky$II=g7 =t+Q?a`h %a;Z-~=bmD9~O^>9LBM/} ȫY/~ƶFB7|=E6Oա>K-Y5 ,(0_BZ}.G IDMGm*4%ۑ$t0CN6O1x1pAdE(Ji{`Cw1 ֗5=>cjnrO M8?԰Av6ˆF}3eT+m¾du䩍{tMa'(#-C-,-֓WUZt2X!>ס~XzRQ#lR}*1y7%_Y J *&{F ʼzo}7 [5r$0O Wt=Sǯij%2.5ĵ&)±-44٠Dv\\ׂK yd4羻?&gа'0cRe,ި1VឆM"(g9Y2y{uyoXYͨTöQ_"rFiY`#战8 ;tk_H0Ţ<=(a2G}ALBzcݯRl&!6Np,λU_ czqHɫh屦ߔo )$0ax  i7}I)0nwD&Vq՘{ Ag`+Zr ΆILϑ}hNsy29+{1h%%PÓP,^έ]AFJqo,i~3\׹Wjѭ}g>˂au,%x5g9Bj 0]"\BOAI ױU $H5ܤ;E"!,}V /8 NX6I4;0i| ob8`jZqZfõ^@ 9M>J%zPI^7 [,j訣_Ii QƀM'ۢ<T@0Dl:bi^[F zi$~Lmт*PP+zE֌"};raߩa]}4"2G{XLry>':řEzSiB7GNp%'iݖV)@Lju;r @}Wnq)$4 z*i2ԵLiVRdR]]ewʹ4o@g IBV¾n5%BZC;]2n)GV~\5.ZV+H:ZŔ1^]Q#B2g5YOR'Ȯ3ׇNVLc8` r֘B dw yFp^Mz[6¤247$j4x2y} * #¸,~OMb܄鹤 ^q ܐ8#ı`S="GQMehjGHȓicF]^ܵ-E"N# a\k`X##J̉=ͬ([U@B4 AP޵#ڪ; %<]chF٫qqm&І XH7A$Grhb*Nȹdh!F7-mZ)8{ { !ʈ?m֍V %RY#f T'Ȅ}j͠X5PJ DX+?*U䆩F.xaI>XѼ!.ոiL n0"0cu6ۻja[%2Б-y\/d޾~ORx;*ײS{]lwP~$z^7\VOrh&CnaP}J6MFquatW4 =|rt+l7v*]N{ E.dcB|rC sg3@eD+Ӵ5bLumM1y4 +|tx]e?alR)Ӳ LK*dDK|[íK~a~_{liik5kº딤ObW! #:wm$8^Ro\kT7,"k8%annen3jG+j) QuL!Q-Q!u ]fJwb/˷Ԯԧ9zGx07#mXW2,|Ҵ:/S{LħQ3hRjA( @Р%<>1 _x Qf}/+pIz.(_桰ǡzt_&0 _'M?> tUW5BrSO,NZϛlK`܆f=7Ax4&5Y!IU`(Jֳ;h(R|;bf'*o:[; Łx\(U.H&.-W3gl͎uBV(ŐaՏJy!+nAf. Ajk-%iЍ#=ыl*yOOHy~ ~Mʖyaw-0mm%$qOn<6 MQ2\˯Hn31ߝeL:~Fje@Mܧ1 Y.Rbh×8]apdqdIw|yRƧ|E}ILtE.> endobj 845 0 obj [846 0 R] endobj 846 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 598 0 0 837 0 0 cm /ImagePart_2186 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 148.8 675 Tm 94 Tz 3 Tr /OPExtFont3 11 Tf ( For the first time, we use the imperfection error obtained by apply-) Tj 1 0 0 1 161.05 651.95 Tm 93 Tz (ing IS method to adjust the forecast outside perfect model scenario. ) Tj 1 0 0 1 160.8 629.149 Tm 91 Tz (And by applying in the Ikeda system-model pair, we demonstrate that ) Tj 1 0 0 1 161.05 606.1 Tm 94 Tz (our method improves the forecast performance from direct forecast ) Tj 1 0 0 1 161.05 583.1 Tm 87 Tz (significantly. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 148.8 555.5 Tm 88 Tz ( Forecast based measurement is suggested to measure the predictability ) Tj 1 0 0 1 161.05 532.2 Tm 92 Tz (outside PMS. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 297.85 51.5 Tm 75 Tz (159 ) Tj ET EMC endstream endobj 847 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 848 0 obj <> stream 0 ,, njeˉ󃶳X܊^fJ fuF!*8KscvZzlD]d83(Q2u =O̎B1W I*%H-y;h чTGSm,H|Gg{٩M f!H1ʲY\5"ip"mE f]#1,L/ha6GFEdMLD[Ԣ)"h{)5 V֯|֘Juu9j`9goҾk``b=%(!`XKKSg*FbxUJg5$*Wo=¨ ;2֩)#i6 oYTy4m]K [h^P =GӪ:!@S`r 4fd$ƊOve rVt+| 1&aucN 6lw/ûDtѨ|ԫ"( @[t P5̰3l A.ɃzJdNמ>ٹLJ'i}=:nRY93qu?SM.E^֒*GLZ jT`sDr ?/LmڷY -cAf)ڊnb)FA`,${k\SlhȲ]eeO] [10oc@})2S&*}t/ !q7-o w֡W $ta umJ5XE/~ [ܜ&{(^eq*ܼv?;^ @a/$VTRSM Ϝ }k~  4`X`6űB&wg֫qSj.ۻlRSTVVƜҽ%[BA$k$xr90@.!*) />=.g yHtqJ4 ﰭ_lL>q\q{ohW_hgIYb5:ә&8l78-'QC7dΰRAĐ׹ 3gl?",ž +qAو m }IޅpZXR (o. $4-< y2u0\{e/n*ob=ca >>}}9 $,#wP,D{Wqan<#z 8޸=1J'>6\f^2ʦ\>pkRhBa:)cK~ u q-t9Ln{P,1&|vN*^)qĊ8gt)q=p}~a~:ۃ?7O7UP7!m_M;HAaP5 *%Q OY?Q)&o\nfaZKXK&N){7)K"͝]zUW*>}6=".̧FS Iޙ 1=q2otB09{N785aAe@X'1s OώU hhU1fI~Y7p2Vs^W4SiMa`Ru k%gRqpa}1L*H,4kkU {lY\eПf 'Oʒ&+TEkuT=1N|J|~f5 y."vd3<B[т}Q'[P)"gsB#Vc8w?a뚺&zL^eaΟK:SXL5$[i?,8O({pE >!j:me20:CMEF&O-X#8oTVkj&HooW""–#w#}ޖ(-, ,+]P_‘־)fAxL,؎Ksʹ?(>^:p<]W<2P,Q CwlxE]{hTC-bLdJ*ѭ*HOoDǃʹ VK ؚq㡑aZgH˚2u9Gσ8EgPa06wf L_9Y;{d O~jSZ %%Kzwu/Ջf(,]];Kf:x>V7mƳjӦʠزy[(:ٻǮ> endobj 850 0 obj [851 0 R] endobj 851 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 597 0 0 837 0 0 cm /ImagePart_2187 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 108 579.95 Tm 113 Tz 3 Tr /OPExtFont3 22.5 Tf (Appendix A ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 113 Tz 3 Tr 1 0 0 1 108.5 512.75 Tm 108 Tz (Gradient Descent Algorithm ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 108 Tz 3 Tr 1 0 0 1 107.5 451.55 Tm 95 Tz /OPExtFont3 11 Tf (In this appendix, we review the details of applying Gradient Descent \(GD\) Al-) Tj 1 0 0 1 107.5 428.5 Tm 93 Tz (gorithm \(48; 50\) to find the minimum of the mismatch cost function 3/ given a ) Tj 1 0 0 1 107.299 404.75 Tm 87 Tz (sequence of observations s_N+1, .., so. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 124.099 381.949 Tm 93 Tz (The minimum of the mismatch cost function can be obtained by solving the ) Tj 1 0 0 1 107.299 358.899 Tm 91 Tz (ordinary differential equation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 260.149 324.35 Tm 87 Tz (du ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 260.649 308.75 Tm 91 Tz /OPExtFont6 12 Tf (d ) Tj 1 0 0 1 277.449 316.45 Tm 92 Tz /OPExtFont4 11.5 Tf (= V C \(u\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 92 Tz 3 Tr 1 0 0 1 266.149 308.75 Tm 123 Tz /OPExtFont6 12 Tf (r ) Tj 1 0 0 1 331.699 316.45 Tm 29 Tz /OPExtFont4 11.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11.5 Tf 29 Tz 3 Tr 1 0 0 1 478.3 317.149 Tm 96 Tz /OPExtFont3 11 Tf (\(A.1\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 96 Tz 3 Tr 1 0 0 1 107.049 274.2 Tm 97 Tz (where C\(u\) is the mismatch cost function. In practice, we initialise the min-) Tj 1 0 0 1 107.049 251.149 Tm 100 Tz (imisation with the observations, i.e. u = s. After every iteration of the GD ) Tj 1 0 0 1 107.299 227.899 Tm 91 Tz (algorithm, the pseudo-orbit u will be updated \(for instance, one obtain ul after ) Tj 1 0 0 1 498 228.1 Tm 146 Tz /OPExtFont4 10.5 Tf (j ) Tj 1 0 0 1 107.049 204.85 Tm 92 Tz /OPExtFont3 11 Tf (iterations\). To iterate the algorithm, one need to differentiate the mismatch cost ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 297.6 51.7 Tm 75 Tz (160 ) Tj ET EMC endstream endobj 852 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 853 0 obj <> stream 0 ,,b7 ["L/1waheap "n Ŀ=l|Őh[SvCg0o+@3Q[Gl :/*ƿ3H a`6ƹ93;N>پrU7,/cBh>/;w4"G*[֬gq%7fY0d;NYlEU+0wrY\Ħ;bZv 8DR*RW: K@|ɡCIqm|?S;d7YIu[*.AC֛m208J :)C CX]X:m"]]? `J8pFFIȕdR+&Sp1jQb8,ێ[Lʢ] $VSWd cF1rXHsBQ4|i X *d2f$n\1LUC%ɺ >  \oq̮'yYAT;Et*}H`f) {5sjQ!)ET_l )սu;3pS[Yx i˵+yȐj XN22i*ՁJ^X1o<]dzSfpv@%%D;GR{ 銀ox d@A7w*|v} ̂ &_k`6ދ3wBv 4w_?]T9e8DF[&jw" }gHZg;f~38r1zB51=B:&L2ˆ,kcnUlpK6/s,̤z^C~:I+V*CXYmao5R? <2J25YжoPhXגTmlӿ3s>۪t̜Y=q(tV0_W?g.mwٲ0}MZ;iT%JLf \˃{wBr~ ChPTח$ׅR mִSz€~ź;k ^q+掻c,|=SZhj2=<3 3~@ލ4Im0?οΛblKUX'-eom~7P1y>q-ӵ&j}Yҝw~߆9ȥ?GuSܮ8_0UuY$1BdTN3Og(-9)Ҏaxws/H`?A.ڨ,~{FK"weO zuz>E5+(gbPl曒|:?+.!Hu%rsW;9)bށ?'M' dj,ef<=mM7Ў/pel"'r?TcXd`ԔNپ-G Д޲o W-SO EXGŔj r5=:J+&٪ _~$s*@)-@ѵɉ@eG/CFP"Vy|ع fnn[CڸFguDMB8="e3+fNnQI0\fjnŶ72&e]UX QolKY+ha3ǀ Y錋L#r$[G5K$nܓ{-.ԫL(Rh0QGDld;&+|?([+R1+);̊OlRRfx#lU /Ң誮}H6 C5\K-cEs;_!*T$-ȉ[ +'p70`p~8 TBK/'y8cɚj2/H HpI#4=ͭW{[viL0?7L6}EWL2C\vg])Yqгyr^ 0tĂ:5pq`|.+/{KH$|}+L_`'\t*Jք>.-< |3ʦgV;<H\la+}9wm<% q*:Ko3XwV6xQR|J6Maɻe hJ. 2V9'.5K6e7<,xRGTr,Z%`lNQ{3X֓<؊lY_+kN'PzQ^>*:7ız,ϯ'zя">ìb4*޳ <5Ifצ.+jܲ…97`Q+J\l)l ?T9 0uB(K01T  TC`>sx"}NYf{@|pTCc4 ]3-^Ǿ1!'cٲnAeM׽FUWk> = CmwRݼG8F"a?f)&p#nmpb譲V=1z͝픇⻕ P s̚4؉(&a[?Be vH$kP.⟍? Pr>K&7G383}GϾL+T)TNi'bVi 6n !ưߪh?%*ZqSwY ᐂ-JW7ehkcp%TBQ6^t=Uϗڸ^Ai7\H.mN9r`e=@lE'wX~ld|fgQ?ƒ+Y5;Re$6NOwIX=lכJ A^ݹpSͮK]sk]yKkA />?N90O884Sq7W]0d ML{ 0{& ;OJS~Qng}\G_Ś~&>K?,͵1#S,iW쟗`\a5e-?:+GӏS{sXW45'& Q|vWRBR/׿@W=qlk)4=Nګ'TildSHbK`>f1huS3). CDwJ`ZgF.vS|Vz0pů/%ꏦ\! Xk, F SwӛP\`j%!J8P0GmNH,^ ]B >T%{8}@ ?Q\V4 8 |.:{ %A7.K6ĺZܙN=% $d3TPmۮ^;_ 2 Q׽Cq)@uhy >)i C-&m8;Q6 :Q|s9jg'޷!PpʲCxb{VCBx(9">=_j]d?d4{Q2=.KtnK 폊~XGOC"QLl hiN7ڈ>`Z&%s:f>GeiRF .7J;NP:RJвT 07ھsV(Me}zs7tE^KZe#!V=hKzTlOc81<;15Yul5C9 @Y׊8P- Y/F{p lw> rh6c.WJP )&}wm&Kvr!f~3zJmMTMz_Ʉyo xZUEBaH:8ZoꦬƗBgj5<~:{—@(cɈՑ=$<.Rr3~w*)g ƞoꦷ"%JB"vIrP3G2%Tm2'2\5K;f#UܳguAv\Ov=[RYw&=x pyhD3A:~{5sap+=DSM14g},Ȝ4"D? s];4Ge[ȳ ?^"fŜoșx;{$o -:]52nbC>;B ůH3)ka^&m;sg?n;qH>-#1?܎عY}3AUI,AEmr4ȕq"=W:J'8\ sV8mȏNlsm!mnvJ*]yfteꢭs֭_ɔAKwj jr: yvk컬6m R»A~{e EȹO j^/·&-;I8 < cnܜF< ~--E wnZe#@ ;+ XK;tE4V$X٫IQaG ?`xzD]ɚ endstream endobj 854 0 obj <> endobj 855 0 obj [856 0 R] endobj 856 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 597 0 0 837 0 0 cm /ImagePart_2188 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 106.299 674.75 Tm 96 Tz 3 Tr /OPExtFont5 13 Tf (function given by ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 96 Tz 3 Tr 1 0 0 1 107.5 622.899 Tm 101 Tz /OPExtFont14 13.5 Tf (ac ) Tj 1 0 0 1 122.9 622.899 Tm 627 Tz (\t) Tj 1 0 0 1 152.65 623.149 Tm 93 Tz /OPExtFont5 11.5 Tf (2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 11.5 Tf 93 Tz 3 Tr 1 0 0 1 108.95 607.299 Tm 77 Tz /OPExtFont14 13.5 Tf (au ) Tj 1 0 0 1 141.099 607.549 Tm 114 Tz /OPExtFont4 11 Tf (N +1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11 Tf 114 Tz 3 Tr 1 0 0 1 201.349 634.7 Tm 231 Tz /OPExtFont3 3 Tf () Tj 1 0 0 1 210.5 634.7 Tm 94 Tz /OPExtFont5 11.5 Tf (\(ut+i ) Tj 1 0 0 1 249.349 635.649 Tm 132 Tz /OPExtFont8 11.5 Tf (F\(ut\)\)dtF\(ut\) ) Tj 1 0 0 1 318 634.7 Tm 2000 Tz (\t) Tj 1 0 0 1 414.5 637.799 Tm 101 Tz /OPExtFont5 11.5 Tf (t = N + 1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 11.5 Tf 101 Tz 3 Tr 1 0 0 1 201.599 614.299 Tm 93 Tz (\(u) Tj 1 0 0 1 221.3 614.299 Tm 77 Tz (t ) Tj 1 0 0 1 223.9 612.85 Tm 99 Tz ( F\(ut--1\)\) + \(ut+1 ) Tj 1 0 0 1 335.3 613.799 Tm 118 Tz /OPExtFont4 11 Tf (F\(ut\)\)d-tF\(ut\) N +1 > BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 11 Tf 118 Tz 3 Tr 1 0 0 1 106.549 554.75 Tm 93 Tz /OPExtFont5 13 Tf (where ) Tj 1 0 0 1 141.099 554.75 Tm 78 Tz /OPExtFont4 11.5 Tf (d) Tj 1 0 0 1 146.9 554.75 Tm 55 Tz (t) Tj 1 0 0 1 150.25 554.75 Tm 113 Tz (F\(u) Tj 1 0 0 1 170.4 554.75 Tm 67 Tz (t) Tj 1 0 0 1 174.25 554.75 Tm 82 Tz (\) ) Tj 1 0 0 1 183.599 555 Tm 90 Tz /OPExtFont5 13 Tf (is ) Tj 1 0 0 1 196.3 555.25 Tm 111 Tz /OPExtFont5 11.5 Tf (the ) Tj 1 0 0 1 217.9 555.5 Tm 101 Tz /OPExtFont5 13 Tf (Jacobian of the model ) Tj 1 0 0 1 336.699 555.7 Tm 116 Tz /OPExtFont4 11 Tf (F ) Tj 1 0 0 1 350.899 555.7 Tm /OPExtFont5 13 Tf (at u) Tj 1 0 0 1 373.199 555.7 Tm 53 Tz /OPExtFont3 13 Tf (t) Tj 1 0 0 1 377.3 556.45 Tm 106 Tz /OPExtFont5 13 Tf (. We solve the ordinary ) Tj 1 0 0 1 106.799 532.45 Tm 97 Tz (differential equation A.1 using the Euler approximation. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 298.1 51.7 Tm 84 Tz (161 ) Tj ET EMC endstream endobj 857 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 858 0 obj <> stream 0 ,, yb6ߊA|d(/KˈE밂l{Tqկ ;%sqsFHi=3=R/ֵ^$u b xa._GY,7A$@Aҝ:> qBW܆[JVm+t: ep$~]V lv7eئVB;@&Lr%$ G@1WO.vBk62O%9V>wc͇}v6Lyfb hj2/5qY3h` #g`>#)#J5+?|iD2l 4YsgAjS1aMWT P%)b/saVkG,u& wgV4r-æ'JK4==73ƐU2rd5o"1*uMNkAƖ$mj?zO}EGx'h%f\'hwUlyL*gbagzM~C_'4FWS^IrR>W<{P~,!P dutt=X~ H8YH㡾]"tg 4Jl 86yl·/aZė'{EǕ# V{ùPL;Xzhm'G;#U +.ĜsnUi;ys(>bFSYF˫=;\ ;,OmJ*%8;n"Eo jU,۰^,=:6Ѥν\20ێqhFPezZVAs7x5qS'nz7Ѵvv K[c { >Qc!5RT^. %4G2I0V? niG~vh-.; u'\aFR{!g ==hsl|5!KP?zԙ34RPp^8[Bwx5@a{b)94e, QQᢦPv=>{ OZ Ĉ=Ôq\y/}9+}I8y0(D2;S]ǂB1~@8+i'7(YӠH}&_1R?pˠL, C^Zg[8RUXG9{ z`y[^@pŕ/N@je .0u*K%N.U\ Dј i9Ȗ+!$YV@u 3% RE #w='6` ʦ3z I 7M^{6ԋ*wo },]Sn7rֺ/%>K5>W yVĂm$.]agv9O}qv)8$] 56 |fL$[י0d8hQ|-YpE;^q6VQoozN0l;]K1'ʲ CPhj-~lض1 - u.SHK=a),@Ef]abmh%6̡S Ƹ=MnG@xׁ R J.7H(]%Na@-|`5]W`xYR EPdA6F9_sR1s%00 g؁ p ˑ2zasFE{Mzj+bh4{}SY;C<4`Y8daZDd 6s褍\-~LPpݻ+~M A߰y 4nG8ٞ9-kDIĎ,@v>S6sT$D$:jD7=bAwҙk0%8_vLz/"qnxV kf4wR2 L0AVXK /lI~p5 qT@`gرgRlgaq_q+lD '.[~\qF r!G ߄Wjî RP DMS pӎoq5 U#BD,<ʩ,`9Ci;hSY*PB듔b )KO =mLcR+C+9w~QLcEqMpx 2:: >#}11b_Q)k xM@V}>)-s}`":B|@hLm%h兠ABAƗGH9;Gy==KXʟ j?Š~ěi:zÇMzOpokO k4 *D0qo `wiC'z܍Mrigi2Icf»^:4i{cA_>_ cN!9i]H+Uv0M^g6mD"6[;geï\Rؽ.r:B~瀩NZB忳L|'jV(ֿ*e"eQ ''jKϚt8 !S@r|T2fSƤ|r.>DB\\+7 .idc5\3 ~jdh=P;-Kmqs<]k"u{à";e 4(sVDÖz ia5v`+S(u8>   `-W&+椻}ߏD0OT/ NIiDFW@-`nnDL0EDP׊-ŴBC@XS4 >hj%K)OX85#Vi{L8"jQ" c^%oxLϡesij/f+)nsuoߛҁF- ,b$P[yr"< \6Ƚ_aE+Tt֟ΈxܵkZX@ECQhJEY֬%ʳ߀[)(?<9Z!1 v/.xFǒd & Cꍩ@\֋Ol'\dwmm<;j6keV hU,kJ'{ G|C̦GLm옜,S.E݁"D號^ӫk61@7lo&vtGAC)eU99zzTF181J 2sAx/pI-tBe\f%{>>^pDZ1W vMb*3'^u'cK endstream endobj 859 0 obj <> endobj 860 0 obj [861 0 R] endobj 861 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 597 0 0 837 0 0 cm /ImagePart_2189 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 108.25 579.95 Tm 110 Tz 3 Tr /OPExtFont3 22.5 Tf (Appendix B ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 110 Tz 3 Tr 1 0 0 1 108 512.75 Tm 106 Tz (Experiments Details ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 106 Tz 3 Tr 1 0 0 1 107.75 451.3 Tm 90 Tz /OPExtFont3 11 Tf (The tables below define the standard experiments which are used throughout the ) Tj 1 0 0 1 107.5 428.05 Tm 84 Tz (thesis. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 84 Tz 3 Tr 1 0 0 1 256.1 404.75 Tm 93 Tz /OPExtFont3 10.5 Tf (System ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 93 Tz 3 Tr 1 0 0 1 389.05 404.75 Tm 92 Tz /OPExtFont3 11 Tf (Ikeda Map ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 240.949 390.35 Tm 103 Tz /OPExtFont3 10.5 Tf (Noise model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 103 Tz 3 Tr 1 0 0 1 389.5 390.35 Tm 91 Tz /OPExtFont3 11 Tf (N\(0, 0.05\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 208.099 375.949 Tm 97 Tz /OPExtFont3 10.5 Tf (Number of ) Tj 1 0 0 1 267.85 375.949 Tm 108 Tz /OPExtFont3 11 Tf (assimilation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 108 Tz 3 Tr 1 0 0 1 389.5 375.949 Tm 78 Tz (1024 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 78 Tz 3 Tr 1 0 0 1 193.449 361.8 Tm 105 Tz /OPExtFont3 10.5 Tf (Number of bootstrap samples ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10.5 Tf 105 Tz 3 Tr 1 0 0 1 389.3 361.55 Tm 77 Tz /OPExtFont3 11 Tf (512 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 77 Tz 3 Tr 1 0 0 1 169.699 347.149 Tm 93 Tz (ISGD ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 219.099 347.149 Tm 94 Tz (no. of GD iterations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 94 Tz 3 Tr 1 0 0 1 388.8 347.149 Tm 81 Tz (4096 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 81 Tz 3 Tr 1 0 0 1 219.349 333.5 Tm 93 Tz (GD iteration step ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 389.3 333.5 Tm 76 Tz (0.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 76 Tz 3 Tr 1 0 0 1 169.449 319.1 Tm 97 Tz (4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 97 Tz 3 Tr 1 0 0 1 219.099 319.1 Tm 94 Tz (Initial GD iteration step ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 94 Tz 3 Tr 1 0 0 1 389.3 319.1 Tm 76 Tz (0.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 76 Tz 3 Tr 1 0 0 1 219.349 305.149 Tm 93 Tz (GD stops when iteration step < ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 389.3 305.149 Tm 89 Tz (5 x 10) Tj 1 0 0 1 420.25 305.149 Tm 85 Tz (-6 ) Tj 1 0 0 1 429.85 305.149 Tm 28 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 28 Tz 3 Tr 1 0 0 1 107.049 281.649 Tm 98 Tz (Table B.1: Details of Experiment ) Tj 1 0 0 1 285.35 281.899 Tm 84 Tz /OPExtFont3 12.5 Tf (A, ) Tj 1 0 0 1 302.399 282.1 Tm 98 Tz /OPExtFont3 11 Tf (note for 4DVAR method we shrink the ) Tj 1 0 0 1 107.049 267.7 Tm 91 Tz (iteration step by 2 when cost function not decrease. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 297.85 51.95 Tm 76 Tz (162 ) Tj ET EMC endstream endobj 862 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 863 0 obj <> stream 0 ,,֔b2F!7&V+D V^l#IWk)L8o/tXeOŸV2gElx W8M;JMebߠNi =mCS ;5y4+/ST;M}fjkN\S)"RZ~o82zB S>ammRV[3U NH\cQ;#6l@Uӡ͔&ʔT /~JHcRMb,Xc(c?_n颴&ſ(3q8ܺg;]!47LpmSELIc/aVRޱ]E=܋QoϥoqjQKla 2ܚ}_ZtrA pw`B62z `ީQ6wu]2t. EVSp~ᒼ:s}_- d24-=b d}[ OӫF)QP:Iw+$e{/=ELXFâܝk'l՟_*hB %5fdh|g|_fs%54@V~L?9;VɘP 2saeVRNL`-]Su/R&j>O9,%2V :]Ԩ[+2ѭKf͔{Yh~ :nm%b)^pvj֖Ox{ʃ II5M (~]Qzdef{Ogi7:1Z{)TXҟ`tyv>TDM,Âok$?;kBI(ސMli4wmn:$a0 idVenD2&>kLKa8|cQYlM9zZr44{hԬ5b.Qd&>[ࣜ0W1 SjW=tADdޕyVڏd1TGu{^ R@kG䲜Q6Y H#N9.Ϡ'zQM! \q0vaxr/o#πVm>DUaC)25(Ms>Ey_37x۽Sᄒl 3_*kRMt sx`|eX񭾣 p먥T8%V $(fn@&*ࢃ{D6U}stzqǖ XMقzW:I-`1򴻍jP;/DZ˒h A9Q^LOGV!ɴZ(ngҷ"A b]+a4m;*6:0k?haZ5wktML>˲AD)pS^ZDC&~ɨރ6Cu0&jd KT?׈)0|$0a.^+ >B@RH* Z,7W}PKHmwpE*]* /hƽ})Oл2VL)%`Zt8n 5 uc7K0B E5i)}$sFͪi1۷d.!5W/.4,vgbΠ h#OYHx C9bK2L "9=cA*5V#S1V؄ R N)3)ɼx &=wAډDLeWP'me:snӎOޞ_*_F9C^OvV:%ƨ,v{4ޠI/ee+~s{TXD1hp jwe/;F>td9!6Qvm}" sZͱdž1k6]Q+,1".ĿGU92OjFNU3aVqiâzM"k)o, 5 F}L/ځD!MJi-a4_Q m9^9 6g/Eg-mz"G@W =&cz~U~]p{%c0^*PJdNy+yK&J9woW>͗2C;d!Z>**b_=X$BL ½*Po+ v3q_4_*C <wYdb:>ǖj:۩[ 5#1UeM҆h@ĩ#UcB;t:XIVouKhgF|bVf#A"f[wt=t\^hғL<o ck|ZD?uZ35߀Y$;caT_M9P4}Ѱl4]xsms;$A T6q~7)dהLj%Dht7._i~Cq^Ů%Ʀh'I Gph: |Tn٤@1yY]2nzHSHtm"avL "qPVB[2UoCt%*K!9|pI3}'ILSq">L<aW$ÙCdˤSZ>>Ƕ7Tx]CI9s&`7)ӜIozMez1R*~k KcrwygNLYZWeA/#.J/k͓8\:)fδ6g:QX5_KRzn\Ǯ*udsI280 fjitk+끪'zLUo:!ochrG-U^|W +A~R8B < `h2o75 :O&wZZl#g94psrȝdV{Pa/]QoGd9TPC+d{"ݤ2@t{ڈcS3!r-K bvН9 絿0)AX)ST̝ ^)dRRՃ!)qNإ Fø:.m+>iތ,EqHbC)TK`7o)oaB f?7G:28 }hL>4°<4tNc$AU0z`t9vޟ + ˆQLaSM>Z͍ kږ`!o["ALC,t#ebݐRr)yN %-)sR.l+}zt풨M+`tIu v{s cs 8;UjiGXG@D{[Ub˺^FƊ0VGMGю3]_C(yf䅇ޗpqpH.(`~*<J5OB#hZpw&v|kK$$,,!PoR['9ɰ/,ߝ1؊ψ/m)&S )2LNK=0)ħkn0òN4xp WoVVzL ?8Κ lѳKW{ 0ZsvP8EkN׏mPá:tQ$V0N]jC~.yE/bvҠD$OLN0]X](/}ဥZYvY aym}SaЄ%-,PꉴqϟB/g~Ǫ;6 nU%>^}.vK} wf˂5LB54O`Xd5 1#+c)17 endstream endobj 864 0 obj <> endobj 865 0 obj [866 0 R] endobj 866 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 597 0 0 837 0 0 cm /ImagePart_2190 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 239.05 663.95 Tm 98 Tz 3 Tr /OPExtFont5 13 Tf (System ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 371.75 663.7 Tm 93 Tz (Lorenz96 Model I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 188.4 649.799 Tm 112 Tz /OPExtFont3 10 Tf (Dimension of the system ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 112 Tz 3 Tr 1 0 0 1 372.25 649.799 Tm 79 Tz (18 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 79 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 79 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 79 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 79 Tz 3 Tr 1 0 0 1 223.9 635.649 Tm 109 Tz (Noise model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 109 Tz 3 Tr 1 0 0 1 371.75 635.399 Tm 98 Tz /OPExtFont5 13 Tf (N\(0, ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 397.199 635.149 Tm 92 Tz (0.5\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 191.3 621.25 Tm 114 Tz /OPExtFont3 10 Tf (Number of assimilation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 114 Tz 3 Tr 1 0 0 1 372.25 621.25 Tm 86 Tz (1024 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 86 Tz 3 Tr 1 0 0 1 176.4 606.85 Tm 110 Tz (Number of bootstrap samples ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 110 Tz 3 Tr 1 0 0 1 372 606.85 Tm 86 Tz /OPExtFont5 13 Tf (512 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 86 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 86 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 86 Tz 3 Tr 1 0 0 1 153.099 592.45 Tm 87 Tz (ISGD ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 87 Tz 3 Tr 1 0 0 1 202.099 592.45 Tm 96 Tz (no. of GD iterations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 96 Tz 3 Tr 1 0 0 1 371.5 592.45 Tm 90 Tz (4096 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 202.8 578.5 Tm 96 Tz (GD iteration step ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 96 Tz 3 Tr 1 0 0 1 372.25 578.75 Tm 66 Tz (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 66 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 66 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 66 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 66 Tz 3 Tr 1 0 0 1 152.9 564.35 Tm 90 Tz (4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 202.3 564.35 Tm 98 Tz (Initial GD iteration step ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 372.5 564.35 Tm 62 Tz (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 62 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 62 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 62 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 62 Tz 3 Tr 1 0 0 1 202.55 550.45 Tm 97 Tz (GD stops when iteration step < ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 372.25 550.45 Tm 80 Tz (10) Tj 1 0 0 1 383.3 550.45 Tm 72 Tz /OPExtFont3 13 Tf (-6 ) Tj 1 0 0 1 392.899 550.45 Tm 30 Tz /OPExtFont5 13 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 30 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 30 Tz 3 Tr ET EMC /Span <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 30 Tz 3 Tr ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 30 Tz 3 Tr 1 0 0 1 107.75 527.649 Tm 103 Tz (Table B.2: Details of Experiment B, note for 4DVAR method we shrink the ) Tj 1 0 0 1 107.75 513.5 Tm 98 Tz (iteration step by 2 when cost function not decrease. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 254.65 465.949 Tm 99 Tz (System ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 99 Tz 3 Tr 1 0 0 1 395.3 466.199 Tm 95 Tz (Ikeda Map ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 239.5 451.8 Tm 96 Tz (Noise ) Tj 1 0 0 1 275.05 451.8 Tm 99 Tz /OPExtFont3 10 Tf (model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 99 Tz 3 Tr 1 0 0 1 395.5 451.8 Tm 100 Tz /OPExtFont5 13 Tf (N\(0,0.05\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 3 Tr 1 0 0 1 206.65 437.649 Tm 105 Tz /OPExtFont3 10 Tf (number of nowcast made ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 105 Tz 3 Tr 1 0 0 1 395.5 437.649 Tm 86 Tz /OPExtFont5 13 Tf (2048 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 86 Tz 3 Tr 1 0 0 1 160.55 395.399 Tm 92 Tz (ISIS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 202.8 423.25 Tm 97 Tz (assimilation window length ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 396 423.5 Tm 93 Tz (12 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 93 Tz 3 Tr 1 0 0 1 202.55 409.3 Tm 96 Tz (no. of GD iterations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 96 Tz 3 Tr 1 0 0 1 395.3 409.3 Tm 87 Tz (4096 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 87 Tz 3 Tr 1 0 0 1 202.3 395.399 Tm 98 Tz (perturbation of the middle points by ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 395.75 395.399 Tm 94 Tz (N\(0, 0.025\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 94 Tz 3 Tr 1 0 0 1 202.3 381.5 Tm 97 Tz (number of the perturbations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 395.3 381.5 Tm 89 Tz (4096 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 202.099 367.3 Tm 94 Tz (number of ensemble members ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 94 Tz 3 Tr 1 0 0 1 395.5 367.55 Tm 88 Tz (512 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 160.8 352.899 Tm 91 Tz (EnKF ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 202.099 352.899 Tm 94 Tz (number of ensemble members ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 94 Tz 3 Tr 1 0 0 1 395.5 353.149 Tm 88 Tz (512 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 88 Tz 3 Tr 1 0 0 1 217.449 329.899 Tm 98 Tz (Table B.3: Details of Experiment C ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 238.8 282.35 Tm (System ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 379.199 282.35 Tm 94 Tz (Lorenz96 Model I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 94 Tz 3 Tr 1 0 0 1 187.9 268.2 Tm 113 Tz /OPExtFont3 10 Tf (Dimension of the system ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 113 Tz 3 Tr 1 0 0 1 379.899 268.2 Tm 83 Tz /OPExtFont5 13 Tf (18 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 83 Tz 3 Tr 1 0 0 1 223.699 254.049 Tm 109 Tz /OPExtFont3 10 Tf (Noise model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 109 Tz 3 Tr 1 0 0 1 379.199 253.799 Tm 95 Tz /OPExtFont5 13 Tf (N\(0, 0.5\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 191.05 239.649 Tm 105 Tz /OPExtFont3 10 Tf (number of nowcast made ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 10 Tf 105 Tz 3 Tr 1 0 0 1 379.449 239.399 Tm 89 Tz /OPExtFont5 13 Tf (2048 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 89 Tz 3 Tr 1 0 0 1 144.699 197.149 Tm 92 Tz (ISIS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 186.699 225 Tm 97 Tz (assimilation window length ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 379.899 225.25 Tm 98 Tz (1.2 time units ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 186.5 211.1 Tm 96 Tz (no. of GD iterations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 96 Tz 3 Tr 1 0 0 1 379.199 211.1 Tm 90 Tz (4096 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 90 Tz 3 Tr 1 0 0 1 186.5 196.899 Tm 98 Tz (perturbation of the middle points by ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 98 Tz 3 Tr 1 0 0 1 379.449 196.899 Tm 95 Tz (N\(0, 0.25\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 95 Tz 3 Tr 1 0 0 1 186.5 183 Tm 97 Tz (number of the perturbations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 379.199 183 Tm 91 Tz (4096 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 91 Tz 3 Tr 1 0 0 1 186.5 169.1 Tm 94 Tz (number of ensemble members ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 94 Tz 3 Tr 1 0 0 1 379.699 169.1 Tm 87 Tz (512 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 87 Tz 3 Tr 1 0 0 1 144.699 154.7 Tm 92 Tz (EnKF ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 92 Tz 3 Tr 1 0 0 1 186.5 154.7 Tm 94 Tz (number of ensemble members ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 94 Tz 3 Tr 1 0 0 1 379.699 154.7 Tm 87 Tz (512 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 87 Tz 3 Tr 1 0 0 1 217.199 131.649 Tm 97 Tz (Table B.4: Details of Experiment D ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 97 Tz 3 Tr 1 0 0 1 297.85 51.25 Tm 85 Tz (163 ) Tj ET EMC endstream endobj 867 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 868 0 obj <> stream 0 ,,#JJb7 \GmT0`Ys%PtIþ3],߁龰ECx kZ? f& JoԳ"ݟoDCH뷉7V[ƚDRfDZ!뮾E&׍H5[4]~ dLHPLG |KumQhؕiIy̛۪´IkYP=| `wiEӲ%`K7-9 AH5_U6Z{ӖctϠ6a y(p4I\@lһb@E sкh`*´ʄ#4b6yHBBS% Yc>$49Q%=1# lY&!h0 '8! -4ڌ/zINgk6SS{ ն}&k߈*&heECfKƝrk szO#\ Թ|kq'/w"ΦN_ exON%<?E~XVI( Fn޲k2RjU/+<ѥ N3vokGi~KVs>ٲ:?!沪4Ᏸs*68|<C6fYLOς$&j LtI_u-S"7|C@Mij/W9yP&1@r}XeFz?PhY},r !u& GYjo'M QԞ ˃ 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["]4;0w^OBT: Wheu joN1߮x-Glٌp5/Ex4a*2F^jknیB)V9ڻz!5W 1II:|:I2Lc}`m֜7`^1p]x G_UcVQ&+T75_wWv5 Rrw-)}<F+|s_}%wW7#)ʺ1a5+;Yܪ䱷ɱ<"Pg'H4 X@(4H@ VƺB?B" s%v9΄)ް8HvoɉI&Ϋ{ %hr ]ck^ܷ" M.):i/t'=LlZ~7e>2"ej1uC .9|UYx_9fȶ%W[W6ݎ :X$ ụnۅ>9m;Sd9uXB랢^k\Kc :&79zpPl ~}=.8q{"ӂ'Pk݇1bG?^=~)X:B2ݓ@4R5㝙ҝv`csW?5E/ OΠR1e]&iQ;KY wwFw}opj5^Cq|H8a/^ѱJ?:k>@xLNjȓ[$fpȺ*N*{ X*wp-@Oן޻0qM{Eip*#RFw{RAh I*`Dj\_5Tǫi;eB7;Of7 Ũ#o-ӺieGBcz TZz1PtyIY4# EX8ŠÑ:=`HO֢S~ԡ8_[ 5-!Za#U;R3.h>\%~yxMe3$_I{%r:9kYtwp(_{sB㒒]Br҉*Mibޯ` uE)-r}]U/4Ÿ8Hko~]IL K \ =3"1aϒߣI&I4(Fޘ ]WyzufL 6#5p̚M'Gte/ endstream endobj 869 0 obj <> endobj 870 0 obj [871 0 R] endobj 871 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 598 0 0 837 0 0 cm /ImagePart_2191 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 243.349 650.75 Tm 103 Tz 3 Tr /OPExtFont5 12.5 Tf (System ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 103 Tz 3 Tr 1 0 0 1 384.5 650.299 Tm 88 Tz (Ikeda 1Vlap ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 88 Tz 3 Tr 1 0 0 1 228.699 636.35 Tm 84 Tz /OPExtFont13 11.5 Tf (Noise model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 84 Tz 3 Tr 1 0 0 1 384.699 636.1 Tm 107 Tz /OPExtFont5 12.5 Tf (U\(-0.025,0.025\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 107 Tz 3 Tr 1 0 0 1 149.5 594.1 Tm 94 Tz (ISIS ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 94 Tz 3 Tr 1 0 0 1 191.3 622.2 Tm 100 Tz (window length ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 384.949 621.95 Tm 97 Tz (12 steps ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 97 Tz 3 Tr 1 0 0 1 191.5 608.299 Tm 100 Tz (no. of GD iterations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 384.25 607.549 Tm 93 Tz (4096 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 93 Tz 3 Tr 1 0 0 1 191.5 594.1 Tm 102 Tz (perturbation of the middle points by ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 384.699 593.899 Tm 104 Tz (U\(-0.01, 0.01\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 104 Tz 3 Tr 1 0 0 1 191.5 580.2 Tm 101 Tz (number of the perturbations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 384.699 579.95 Tm 91 Tz (1024 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 91 Tz 3 Tr 1 0 0 1 191.3 566.299 Tm 98 Tz (number of ensemble members ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 98 Tz 3 Tr 1 0 0 1 384.5 566.299 Tm 90 Tz (64 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr 1 0 0 1 149.3 552.1 Tm 94 Tz (DCEn ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 94 Tz 3 Tr 1 0 0 1 191.5 552.1 Tm 98 Tz (number of ensemble members ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 98 Tz 3 Tr 1 0 0 1 384.5 552.1 Tm 90 Tz (64 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 90 Tz 3 Tr 1 0 0 1 220.55 529.1 Tm 102 Tz (Table B.5: Details of Experiment E ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 129.849 453.5 Tm 101 Tz (System ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 268.3 453.5 Tm 100 Tz (Ikeda Map ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 395.05 453.5 Tm 97 Tz (Lorenz96 Model II ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 97 Tz 3 Tr 1 0 0 1 129.599 439.3 Tm (Model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 97 Tz 3 Tr 1 0 0 1 268.1 439.1 Tm 100 Tz (Truncated Ikeda Model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 395.05 439.1 Tm 97 Tz (Lorenz96 Model I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 97 Tz 3 Tr 1 0 0 1 129.599 424.899 Tm 96 Tz (Noise model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 96 Tz 3 Tr 1 0 0 1 268.3 424.699 Tm 97 Tz (N\(0,0.01\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 97 Tz 3 Tr 1 0 0 1 395.05 424.699 Tm (N\(0,0.4\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 97 Tz 3 Tr 1 0 0 1 129.599 410.75 Tm 98 Tz (number of observations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 98 Tz 3 Tr 1 0 0 1 268.3 410.75 Tm 92 Tz (2048 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 92 Tz 3 Tr 1 0 0 1 395.5 410.5 Tm 101 Tz (102.4 time unit ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 129.599 396.6 Tm 100 Tz (sample std of model error ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 268.3 396.35 Tm 94 Tz (0.018 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 94 Tz 3 Tr 1 0 0 1 395.3 396.35 Tm 96 Tz (0.0057 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 96 Tz 3 Tr 1 0 0 1 220.099 373.3 Tm 102 Tz (Table B.6: Details of Experiment F ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 226.55 297.5 Tm (System ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 369.35 297.5 Tm 100 Tz (Ikeda Map ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 229.199 283.1 Tm 93 Tz (Model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 93 Tz 3 Tr 1 0 0 1 369.35 283.1 Tm 100 Tz (Truncated Ikeda Model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 211.449 269.149 Tm 85 Tz /OPExtFont13 11.5 Tf (Noise model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 85 Tz 3 Tr 1 0 0 1 369.85 268.7 Tm 98 Tz /OPExtFont5 12.5 Tf (N\(0, 0.05\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 98 Tz 3 Tr 1 0 0 1 178.8 254.75 Tm 86 Tz /OPExtFont13 11.5 Tf (Number of assimilation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 86 Tz 3 Tr 1 0 0 1 370.1 254.5 Tm 92 Tz /OPExtFont5 12.5 Tf (1024 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 92 Tz 3 Tr 1 0 0 1 164.15 240.35 Tm 85 Tz /OPExtFont13 11.5 Tf (Number of bootstrap samples ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 85 Tz 3 Tr 1 0 0 1 369.6 240.35 Tm 91 Tz /OPExtFont5 12.5 Tf (512 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 91 Tz 3 Tr 1 0 0 1 130.55 225.7 Tm 108 Tz /OPExtFont6 12 Tf (ISGDc ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont6 12 Tf 108 Tz 3 Tr 1 0 0 1 199.449 225.95 Tm 101 Tz /OPExtFont5 12.5 Tf (no. of GD iterations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 370.1 226.2 Tm 85 Tz (75 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 85 Tz 3 Tr 1 0 0 1 199.9 211.799 Tm 101 Tz (GD iteration step ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 369.85 211.799 Tm 93 Tz (0.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 93 Tz 3 Tr 1 0 0 1 130.099 197.649 Tm 96 Tz (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 96 Tz 3 Tr 1 0 0 1 199.449 197.399 Tm 103 Tz (Initial GD iteration step ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 103 Tz 3 Tr 1 0 0 1 369.85 197.399 Tm 91 Tz (0.2 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 91 Tz 3 Tr 1 0 0 1 199.699 183.5 Tm 101 Tz (GD stops when iteration step < ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 369.85 183.25 Tm (5 x 10) Tj 1 0 0 1 401.05 183.25 Tm 89 Tz /OPExtFont2 12.5 Tf (-6 ) Tj 1 0 0 1 410.399 183.25 Tm 32 Tz /OPExtFont5 12.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 109.2 160.2 Tm 101 Tz (Table B.7: Details of Experiment G, note for WC4DVAR method we shrink the ) Tj 1 0 0 1 108.95 146.299 Tm 102 Tz (iteration step by 2 when cost function not decrease. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 299.5 52.2 Tm 90 Tz (164 ) Tj ET EMC endstream endobj 872 0 obj <> /ProcSet 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-h֯II߂BC\'iQWKĂvM`vU jʋӀI&[fbbKG8w8ӖLU೴Z= [jc =K+}iG 5Di89$֓jНп|7 5=,i.i@k?z.A@Dfqt%T\vdc;܋zpEÆ>YgRg60!w4`FX'.2S"Hfen`kLl~?.C .j^'1V(IG%hC+ X>bWD5 (h}(^BF|43j~PdL}\ J_cUvN~0Х!AdM'zyLSi8sR%`p?T\ R]q΋*^$|=8-KcD2nz*,GJ)Ū iEu` Ъ} ~ =})Bm %Iln%(|DZYXpN}L@',V/8%z $1{& UU!L3Qf:24[ 42)bWܠJt-$nS^4m]\<ҟӎ#y TNWZ#A/Ra%2ͮ٠x5"wGtۮC, ofS"FE'} 핓!cx۠VoLUcrga g4+sR7zQMIHLh-CifR1Q"CH!`˖V0Zg zYm}Zoj&"VFҋk``qMj,bPq g9Z*p޴-LcЗRJeh)u%&%/ss;XV,ȾǭZG1UOca#ì%£'89smތ6iX2ۇK9|N>8#E4Ѩg9)g5"A ܏B;ڟ yQ؞k_MKxq:ƍL4"m!cSra^8{Q0KOɩ/`ы(]? n4U!\~2ₚ =#{!XgK: 6Yk|RZތ6޺LeL|?fL{H]?* !'U+R_A$m> ⲒzQ38l?$5!ܿ<tqg llXZ[_Y?Zb+r +' 2ܮ}ɏұ<)ő#O9{n`\ ;$/67L Cн|jęK&q7᳖r1r~zdGwOPw(?aÄȖP1 9>TT%N+Di۳,(rnL/p:ӟEKCCqX` [_{!ْp#M endstream endobj 874 0 obj <> endobj 875 0 obj [876 0 R] endobj 876 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 596 0 0 836 0 0 cm /ImagePart_2192 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 238.3 479.85 Tm 102 Tz 3 Tr /OPExtFont5 12.5 Tf (System ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 380.649 479.85 Tm 97 Tz (Lorenz96 Model II ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 97 Tz 3 Tr 1 0 0 1 240.949 465.449 Tm 92 Tz (Model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 92 Tz 3 Tr 1 0 0 1 380.649 465.449 Tm 98 Tz (Lorenz96 Model I ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 98 Tz 3 Tr 1 0 0 1 187.699 451.3 Tm 112 Tz (Dimension of the system ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 112 Tz 3 Tr 1 0 0 1 381.35 451.05 Tm 100 Tz (18 x 5 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 223.199 437.1 Tm 99 Tz (Noise ) Tj 1 0 0 1 258.25 437.1 Tm 84 Tz /OPExtFont13 11.5 Tf (model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 84 Tz 3 Tr 1 0 0 1 380.899 436.899 Tm 105 Tz /OPExtFont5 12.5 Tf (N\(0,0.1\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 105 Tz 3 Tr 1 0 0 1 190.3 422.949 Tm 86 Tz /OPExtFont13 11.5 Tf (Number of assimilation ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 86 Tz 3 Tr 1 0 0 1 381.35 422.699 Tm 92 Tz /OPExtFont5 12.5 Tf (1024 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 92 Tz 3 Tr 1 0 0 1 175.449 408.55 Tm 85 Tz /OPExtFont13 11.5 Tf (Number of bootstrap samples ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont13 11.5 Tf 85 Tz 3 Tr 1 0 0 1 380.649 408.55 Tm 92 Tz /OPExtFont5 12.5 Tf (512 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 92 Tz 3 Tr 1 0 0 1 142.3 394.149 Tm 101 Tz /OPExtFont4 12 Tf (ISCDc ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont4 12 Tf 101 Tz 3 Tr 1 0 0 1 210.699 394.149 Tm 100 Tz /OPExtFont5 12.5 Tf (no. of GD iterations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 380.649 394.149 Tm 95 Tz (4096 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 95 Tz 3 Tr 1 0 0 1 211.199 380.25 Tm 100 Tz (GD iteration step ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 381.35 380.5 Tm 65 Tz (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 65 Tz 3 Tr 1 0 0 1 141.599 365.6 Tm 96 Tz (WC4DVAR ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 96 Tz 3 Tr 1 0 0 1 210.699 365.6 Tm 103 Tz (Initial GD iteration step ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 103 Tz 3 Tr 1 0 0 1 381.35 366.1 Tm 61 Tz (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 61 Tz 3 Tr 1 0 0 1 210.949 351.699 Tm 101 Tz (GD stops when iteration step < ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 381.35 351.699 Tm 86 Tz (10) Tj 1 0 0 1 392.149 351.699 Tm 77 Tz /OPExtFont3 12.5 Tf (-6 ) Tj 1 0 0 1 402 351.699 Tm 32 Tz /OPExtFont5 12.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 108.95 328.649 Tm 101 Tz (Table B.8: Details of Experiment H, note for WC4DVAR method we shrink the ) Tj 1 0 0 1 108.7 314.95 Tm 102 Tz (iteration step by 2 when cost function not decrease. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 298.55 51.2 Tm 90 Tz (165 ) Tj ET EMC endstream endobj 877 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 878 0 obj <> stream 0 ,,˔./t몉XmpXR|qk=h:L_p,X R1uspQ7f/tV=43ɮ =(pw>c0ƂH}NĀUKY]Y /ʚtZ[NZ{NXϷM4бG L ߟ3CeկegIkoiNb+ ȓK)*Tbf XK#Qf忣4@'kKS@7\xw9Q`<[|!( ,)yRB4BZb8O?kpm?e3nWZ|B=E]uSmZ .0*w:' 0,Y$2ϸJt h>]?/(;h OWBޒ*ϗK8 O8USfo,mwx|\eF< |vA}zԪ9Zﰈt CB8_pa\K5uv嵚k"X;`Dj 2T%T8?fDwl+h}7f 28Q߸~'R6 <MLcsL9A-_Snd5mvhQhȧM~-/T) \:o^X|0;FA@j/e(9se[T7^WwGFˤ.Y)Eg1ApsY* ׎/<5/ _(9}+?΍)M Q TNh'sM2p2QFdS)^7mM&;Տe0{?p߉~M"ViNjue FSEaЬ@*DR50~|+sghJQ7F&l;G#zy(zK, Sݷ˜D]6i7g_^W\7kr9QjIPc ~xNmPnS2IYå\(ZS- #xs gW>GD4[WU:/%{g%?dDF)7(W3J)ꀈb ز,d/\Ep64\]HЄ=.?mK1{$fO39NlT -~̄<aNXEϽfy.(_s޹`.y~P0>!K Qٞ 3n#,IC`F_m7lQL`}gzcCR qH&i/R^ ݸ!bFe(4$a9TNjxE{(>4h"+oh&X|XAq;f t`F 0 -6}!5nYㄒ;@9[$R(ak%ۯtUB98:CVہ4I4`uRCt5y}$ߞsu C?9.w/q#K7ҕ{]&']b۾}r,J4[ؘ 5_8Ʋ^ܙn O.x)[=:!Inƈ Μ{ )ѻVqw8Dݯ>7e+2վ-_Զp7+<@.G5tBEx0r}OPN:ɥy׏e58-.~E.{ nM\g8Ώ#D_bm 1Y/]W}MXqH=tנA͚,̻G׋f 6iQCP\LQ« gI+_,j'Fu "'"Yёz}o0@(2Bڜ֤^8 ^HLu0M!ԅ]XgAL'r簇}]2ZBG!~qfZ*`o˒ĽO,8;2Q 8% ?KoMû\V$3:*2>E`(S}wHOٝHK0#Z9r ۸<~3w~&9o0 UGta?@ٖ-#f}?9k?#aY#}ś&ød>;YЛ!>.mcfi[C@4Ɨ4z⋗ ZHC۪YA\RQ䗔)PKrڲꡕl&O5Xmo0}Z5-K#[۵ b'd5d5rIQM*;)QCȃԼ}'K<ψVة+Cb"c* vZyOsO)bmW6y7iSr Q5ǵÖt9Z|;TJh?%HY倮*D_%W B+!|Yws#A6/(ZZmn˹`硵 uI?H%_M`Pk%9<}oCKI ˂EJS[aUm+o3m_f M00/m^Vg4/+_LN"ŎsF-Ču.O&~ɧȼYy|[iK%`[g}uŸ;7 "z "3wl2+Jj.lvҮ!]5E[ ^h\HRUnޡ}ATZDf=P⢁3$z)~^b,?VYL-; " 7!G>d2;F:rGg$iDbn"$jJe,X!΂X(p,EԆ1l5L^!vJI-:nTz[T|nS/]73I6a>Y01z'/$s*.+ȇIOh6\o:J=y_H:uyXU?_ܞRkKʜ8#MEsA+c&KxJ5VEnoͪh-S3@.zjLٱk.(B.2TϨN o)of4|F i&'͆ #$Q4  ?-R֧dNEehj5fq-jlRZq&i+NjɐJ;S MwVý\=g/SPIѝ4I r}떢1+dG 櫲4eܐ 0:4gk_@pr$ :FC/`3AP:h1m}|xxKCl)0N_ Φ7_t’5q oL826ЪV JL$`w4+Q+|i 1ᵋf灁Cv"VL4y`N1ꂬ&J an8T[#v$Dx$jс[13Bv&C|"SDp"bYǧ /jܤ])EOy}ԌP[m*bVUH3IV~o@їB <.HYPїV"WP#!8Klw @"H8J[bO' QX{_F?!z͡M0uwB;nQ^в9H>ʑs.ܭYay(sF~y'$++&R_D4 5LL~VmL.䎭[r$U7xA+D !\yBy ڐsѸ22%0K/j  㠴>yw|o$}0@.Qɯ{{ֿR*P7Yb`@ 0^tn$Lk36E9:}+P?<kdsx!wq k endstream endobj 879 0 obj <> endobj 880 0 obj [881 0 R] endobj 881 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 594 0 0 837 0 0 cm /ImagePart_2193 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 107.75 579.95 Tm 106 Tz 3 Tr /OPExtFont3 22.5 Tf (Bibliography ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 22.5 Tf 106 Tz 3 Tr 1 0 0 1 119.299 518.299 Tm 99 Tz /OPExtFont5 12.5 Tf ([1]) Tj 1 0 0 1 136.55 518.299 Tm 100 Tz (J.L. Anderson, An ensemble adjustment Kalman filter for data assimilation, ) Tj 1 0 0 1 136.55 495.25 Tm 95 Tz /OPExtFont4 11 Tf (Mon. Weather Rev., ) Tj 1 0 0 1 239.75 495.25 Tm 101 Tz /OPExtFont5 12.5 Tf (129, 2884-2903, \(2001\). 37, 41, 43 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 119.299 463.1 Tm 99 Tz /OPExtFont5 11.5 Tf ([2]) Tj 1 0 0 1 136.099 463.1 Tm 115 Tz (J.L. ) Tj 1 0 0 1 158.15 463.1 Tm 99 Tz /OPExtFont5 12.5 Tf (Anderson, A local least squares framework for ensemble filtering, ) Tj 1 0 0 1 477.85 463.1 Tm 92 Tz /OPExtFont4 11 Tf (Mon. ) Tj 1 0 0 1 138 440.05 Tm 89 Tz (Weather Rev., ) Tj 1 0 0 1 210 440.05 Tm 101 Tz /OPExtFont5 12.5 Tf (131, 634-642, \(2003\). 37 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 119.299 407.899 Tm 99 Tz ([3]) Tj 1 0 0 1 135.849 407.899 Tm 103 Tz (L. Arnold, Random Dynamical Systems, Springer, Berlin, \(1998\). 146 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 103 Tz 3 Tr 1 0 0 1 118.799 375.5 Tm 99 Tz ([4]) Tj 1 0 0 1 135.849 375.5 Tm 104 Tz (A.F. Bennett, L.M. Leslie, C.R. Hagelberg and P.E. Powers, Tropical cy-) Tj 1 0 0 1 135.849 352.449 Tm 99 Tz (clone prediction using a barotropic model initialized by a generalized inverse ) Tj 1 0 0 1 135.599 329.399 Tm 97 Tz (method. ) Tj 1 0 0 1 180.25 329.399 Tm 96 Tz /OPExtFont4 11 Tf (Mon. Wea. Rev., ) Tj 1 0 0 1 267.6 329.399 Tm 112 Tz /OPExtFont5 11.5 Tf (121, ) Tj 1 0 0 1 294 329.399 Tm 99 Tz /OPExtFont5 12.5 Tf (1714-1729, \(1993\). 111 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 99 Tz 3 Tr 1 0 0 1 118.549 297.5 Tm ([5]) Tj 1 0 0 1 135.599 297.25 Tm 101 Tz (A.F. Bennett, B.S. Chua and L.M. Leslie, Generalized inversion of a global ) Tj 1 0 0 1 135.599 274.2 Tm 102 Tz (numerical weather prediction model. ) Tj 1 0 0 1 324 274.2 Tm 97 Tz /OPExtFont4 11 Tf (Meteor. Atmos. Phys., ) Tj 1 0 0 1 441.35 273.95 Tm 110 Tz /OPExtFont5 11.5 Tf (60, ) Tj 1 0 0 1 462.699 273.95 Tm 93 Tz /OPExtFont5 12.5 Tf (165-178, ) Tj 1 0 0 1 136.099 250.899 Tm 100 Tz (\(1996\). 111 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 118.549 218.75 Tm 99 Tz ([6]) Tj 1 0 0 1 135.099 218.5 Tm 103 Tz (M.L. Berliner, Likelihood and Bayesian Prediction for Chaotic Systems, J. ) Tj 1 0 0 1 135.099 195.5 Tm (Am. Stat. Assoc. ) Tj 1 0 0 1 222.949 195.5 Tm 112 Tz /OPExtFont5 11.5 Tf (86, ) Tj 1 0 0 1 242.9 195.5 Tm 98 Tz /OPExtFont5 12.5 Tf (938-952 \(1991\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 98 Tz 3 Tr 1 0 0 1 118.299 163.299 Tm 99 Tz ([7]) Tj 1 0 0 1 135.349 163.299 Tm 103 Tz (C.H. Bishop, B.J.Etherton and S.J.Majumdar. Adaptive sampling with the ) Tj 1 0 0 1 135.099 140.299 Tm 99 Tz (Ensemble Transform Kalman Filter. Part I: Theoretical Aspects. ) Tj 1 0 0 1 449.5 140.049 Tm 93 Tz /OPExtFont4 11 Tf (Mon. Wea. ) Tj 1 0 0 1 135.599 117.25 Tm 83 Tz (Rev. ) Tj 1 0 0 1 160.8 117.25 Tm 113 Tz /OPExtFont5 11.5 Tf (129, ) Tj 1 0 0 1 186.25 117.25 Tm 102 Tz /OPExtFont5 12.5 Tf (420C436, \(2001\). 41, 43 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 296.399 52.2 Tm 88 Tz (166 ) Tj ET EMC endstream endobj 882 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 883 0 obj <> stream 0 ,, b7 %^~f՛lr>u GQFi @PLmG>K=~<U7~IM[Htc@F@UA*#ʀ+%6R3f^GgX,\Z*ĀS>U$޳~rG?=-'i[15`6֟ge\SpIPZ!B## 5$NfJ^GR[טr3uKPYzZ B2!h4b%-Dq:fa ~)5+e'Ooe$[Qah{D :k!0ÑDPY Hj#-$+ՠ}-'dma#Gھ|>3>͇"ư;C.̙R)nԲ<{mHCNGX72Ԃ/q} j򍔛f8qb&I}T1+QSO3F/g߼6߬2 hG`Qf|!t%?k|B&pwu\l;0${`*]mͿxm H't &6(Q4 xqH$ŀ Ȋ`w)CR&lcgy1Nԝ3Bt-ˏ|8NbN['-jP<{iU,zj1]W>…=W_S|e@?\9w Qn>Ĵ¬гwD<w#5M'zBP%SǶ?UYwsVm"e\18f}Jڝ>mBaՓ5]~s1Fk 6aqYcVe_r4lw] Mh0]R?zbCvUÓ"FC#e, YلRN_Au`Z].WI 'ɱr6)Ԯ'v[ƘZ[ ? bӭe~z""X:Klt&@?>ǘ+iiڲֱт.V8ڿ<449eD`G~H[t֜cӯjПgԒ|jarV|S_3 *zi 1P NsAfi!N2[[ANE6}VC}B-n^YbDۼ}]} N 0-2CV_X?SJ &s,@CJDd,LdN=u7+ Ng;_rinsx˗'1^k sO|)$:޽[ciK$Zp2s$2/ IMO+GZڈ;E_GSؙ-G82Oa'z8M^l09DNpDϙs$ι(g5 tg`Alj-w ~ico]Mwɻk}w55&p |.B̫xx*ׯz=u߲tk'Wo屛GE ,ؽC6:9֝ CQ۫WF6.ܣ,P3B(_:~=#C04cn/ b.\5M'0@-U@XyvD; P2~Ům4:9zVA}@e}HN e|XE`ƸdOI9ԺR}g]F0yVQIĮ`zK :2mƨHi5ZQXFo_uCBXE z*uX723(DHIn֝;;ݼ?AYU&Ǻ˙=FAFư4ESmt.c쓙t^cH6BL2XTLq5F?]{4C2̙dޢHwR mNeN%6Pp.rq;ā6Xo*]V VaL͠$(^ T9{ 'Ip )t$GwV{4TRSac|M<{gU,"Qq(s`Ωs3{Oo%Wg5=g~noӕQi,塬s}RǻuePQl5j\Ěfw\: ڔ4b]ǁOoҢ n$xB"DۘݽA.e _1SdQ3Ճ )㾦۩Blt2HP%y vbŃz d[;H5#ŗ| t*{ş7m5LI~{MpZ*y| ӈfu3д+9'ĤlAlłE4jt~,R#; VG.-WK %msn _vʡ`?R[N$߀#WT/= ^;Ow?X8PM22KYYӫ&LG`HXb&[b v߽7I.΂'j p[ସ7~;k&ԘYPMqCx7RS{]Va(;&h"%bVK>C˹ϊyi2UJ]5\,IhpBO>xxc3q&m؀OkɬPg@Zk[8O%ϡ3fB;~ZO 82#Ésy&ଞ zFO"WNwn7 !ߐ“U>J8N">={kU+m@VՉm_{'n0),nAηOT'EcbHyuGgaͦ=q*.ҬJC錑L  LsEj1W{∡Gɥ._ۂ.5ݴQJ3&4-4m<܅0"2Tp5{mDrcuɢL-dAdSVHβo./ZklI.)%vT)c ٧FH|=*nU5g֣t/QD6z:@]95YOv#QpLZwChCppNVN%Q^PjzV墝r)7W"!yq=p msb):{uʧp\#7N.[ϼw| 悗"^<><jpƞjJ[P h ]?2`]w;ai%ӻ-[^2_XR:$.MT; )g EUu! JpdAk%mRY,ED-0ckq2 kz@Ʉa ޸' *Z#^dT(~H |#v,S @9EOOVDHOG6-]6xdB4bsн4 F/C~bwC[CRNq E"FcĘz[κT2{Jӥg#kw{.yt! k%!L WXﭬΘfR `ms`IY}(֢Yw X7η5s q(3+U|Ѣm~r%O"gc'Z!lO Rx U %NZ:(CQ+@̄# #]vbF<+/+R-S66sȧΞh]3`GA/NQ\7D&??׻B.-xRƆPu&ֺ%\/QNuבjJTi&,xUoWiڼtʲz=Xe]Xk.c*ha&/C&$ &;i oY?|е\'p]vi?$ZW?ʯcRm6/={jr9G rm!eboC99;۲N.K8m ~ J0="ޥaEjl׶/]C~e?m]yL > ;{=)JyP=?@)/?ޏ?fVUۓV#̊$Q Ѝ/%Vׯ3BrGL!L@%ڧyj7-j"l9-V#_4]f4N YԻX !{^ll.ʔ) 쯒)p* *2WiQϞݮmp3@!6^ Z|hY* ;sr}E3+sް+F Uc7XQ*,C5lL$5nҀVxw$hLڧfZxH\[<I#"XJU>.XH?Rf.q"tr[Bk9Ya7+2D2C{Ǚz9 7w= JXfizа=IS[3  1 Ilo 읥b@qr}8V.l;WW=KRƷޭjb.af.)Ф?OK\b!H_Yۮo7Q6"?oh9ՐpA6JKSkŲm nd4Y{/Y\ж΁k咿J| VLAj?Lf:U2POa< Aӿ(*dEՌ Wü&G c G&l}?WoУaWnX Uq1T`0UoPeITNvod?Hn㯹]_Vc=%r.rq a]Eo؜2WAv|I 5~@S+6%/`(cYl$6}S[NWzD"М4~_!dfn,hؗry4oO5OA JԱiqXϻN) aLGxQ )̋ eZs3ʤQ޾v!1ŬrHkP(cQy\,@CmK;t <nHRbV1s]1hL2巶T BpgHlh`RHOGfԼ|T>pٗ)0 d4=N4nGmNI#t=pޯj~hdCFi^:V'Rڧ=,$8#IvQ !3XYU{VD+T>>6֭RQ r_! }k;"Rˎ2뱝Y^4/E6 f rS*Jk_gŶUmL3b!ݹ3/%Zz\)M~[`ڙ !_Q⾉n J[F^nHA3F(\r=$9po8 ,I|$TLˈqb0) hH98vsԛxˮkIEL ؓ@ s$=|hdRi/LFr Ϗuٮ8ML]gTʗo_tUkX`0DBUh\Q-FO#-R R_Mv1%4k sOF7iЂ, \(Y,b3(k&6 X1vA{=1"0q*<^MjPNh'Ẫ~[42k,"vI= ùBw^;lLjckWY)Z\ (IPPrmk-\LdMV?6OHj|U> endobj 885 0 obj [886 0 R] endobj 886 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 594 0 0 837 0 0 cm /ImagePart_2194 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 397.899 718.7 Tm 112 Tz 3 Tr /OPExtFont5 12.5 Tf (BIBLIOGRAPHY ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 112 Tz 3 Tr 1 0 0 1 118.799 676.2 Tm 99 Tz /OPExtFont3 11 Tf ([8]) Tj 1 0 0 1 135.099 675.95 Tm 93 Tz (A. Bjorck, Numerical Methods for Least Squares Problems, SIAM, \(1996\) ) Tj 1 0 0 1 135.349 652.899 Tm 77 Tz (65 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 77 Tz 3 Tr 1 0 0 1 118.549 621 Tm 99 Tz ([9]) Tj 1 0 0 1 135.099 620.75 Tm 91 Tz (E. Borel, ) Tj 1 0 0 1 182.9 620.75 Tm 93 Tz /OPExtFont4 11 Tf (Probability and Certainty ) Tj 1 0 0 1 310.3 620.5 Tm /OPExtFont3 11 Tf (Walker, New York, \(1963\). 80 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 93 Tz 3 Tr 1 0 0 1 112.799 588.35 Tm 99 Tz ([10]) Tj 1 0 0 1 135.349 588.1 Tm 91 Tz (C.L.Bremer and D.T.Kaplan, Markov chain Monte Carlo estimation of non-) Tj 1 0 0 1 135.099 565.299 Tm (linear dynamics from time series, Physica ) Tj 1 0 0 1 342.949 565.1 Tm 101 Tz /OPExtFont5 12.5 Tf (D 160 ) Tj 1 0 0 1 378.949 564.85 Tm 87 Tz /OPExtFont3 11 Tf (116-126 \(2001\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 112.799 532.7 Tm 99 Tz ([11]) Tj 1 0 0 1 135.349 532.2 Tm 95 Tz (G.W. Brier, Verification of forecasts expressed in terms of probabilities. ) Tj 1 0 0 1 135.099 509.399 Tm 96 Tz /OPExtFont4 11 Tf (Mon. Wea. Rev., ) Tj 1 0 0 1 222.25 509.399 Tm 101 Tz /OPExtFont5 12.5 Tf (78, ) Tj 1 0 0 1 242.15 509.149 Tm 89 Tz /OPExtFont3 11 Tf (1C3, \(1950\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 112.549 477 Tm 99 Tz ([12]) Tj 1 0 0 1 134.9 476.75 Tm 92 Tz (J. Brocker and L.A. Smith, Scoring Probabilistic Forecasts: On the Impor-) Tj 1 0 0 1 134.4 453.949 Tm (tance of Being Proper, ) Tj 1 0 0 1 251.5 453.699 Tm 89 Tz /OPExtFont4 11 Tf (Weather and Forecasting ) Tj 1 0 0 1 374.149 453.5 Tm /OPExtFont3 11 Tf (22\(2\), 382-388, \(2007\) 53, ) Tj 1 0 0 1 134.9 430.899 Tm 73 Tz (73 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 73 Tz 3 Tr 1 0 0 1 111.849 398.05 Tm 99 Tz ([13]) Tj 1 0 0 1 134.65 397.8 Tm 91 Tz (J. Brocker and L.A. Smith, From ensemble forecasts to predictive distribu-) Tj 1 0 0 1 134.15 375 Tm (tion functions, submitted to ) Tj 1 0 0 1 277.899 374.75 Tm 85 Tz /OPExtFont4 11 Tf (Tellus, ) Tj 1 0 0 1 313.199 374.75 Tm 88 Tz /OPExtFont3 11 Tf (\(2007\) 70, 71 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 111.849 342.35 Tm 99 Tz ([14]) Tj 1 0 0 1 134.15 342.35 Tm 92 Tz (D. S. Broomhead and D. Lowe. Multivariable functional interpolation and ) Tj 1 0 0 1 134.4 319.299 Tm 89 Tz (adaptive networks. ) Tj 1 0 0 1 231.099 319.299 Tm 91 Tz /OPExtFont4 11 Tf (J. Complex Systems, ) Tj 1 0 0 1 336.5 319.1 Tm 87 Tz /OPExtFont3 11 Tf (1:417-452, \(1987\). 17 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 87 Tz 3 Tr 1 0 0 1 111.349 286.45 Tm 99 Tz ([15]) Tj 1 0 0 1 133.9 286.45 Tm 91 Tz (M. Casdagli. Nonlinear prediction of chaotic time ) Tj 1 0 0 1 377.3 286.2 Tm 97 Tz /OPExtFont2 11.5 Tf (series. ) Tj 1 0 0 1 410.899 286.2 Tm 90 Tz /OPExtFont4 11 Tf (Physica D, ) Tj 1 0 0 1 466.1 285.95 Tm 80 Tz /OPExtFont3 11 Tf (35:335-) Tj 1 0 0 1 133.699 263.399 Tm 90 Tz (356, \(1989\). 17 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 111.349 230.75 Tm 99 Tz ([16]) Tj 1 0 0 1 133.699 230.299 Tm 93 Tz (M. Casdagli, S. Eubank, J. D. Farmer, and J. Gibson, State space recon-) Tj 1 0 0 1 133.199 207.5 Tm 91 Tz (struction in the presence of noise, Physica ) Tj 1 0 0 1 344.649 207.25 Tm 102 Tz /OPExtFont5 12.5 Tf (D, 51, ) Tj 1 0 0 1 380.399 207.25 Tm 88 Tz /OPExtFont3 11 Tf (52 \(1991\); ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 111.349 174.85 Tm 99 Tz ([17]) Tj 1 0 0 1 133.699 174.6 Tm 96 Tz (G. Casella and R.L. Berger, Statistical Inference, Wadsworth, Belmont, ) Tj 1 0 0 1 133.699 151.799 Tm 92 Tz (California, \(1990\). 66 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 294.5 52.899 Tm 77 Tz (167 ) Tj ET EMC endstream endobj 887 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 888 0 obj <> stream 0 ,,ɔb7*G(Z8w:fC(o ̯BFW1ˊZX$7=~ʣ*/8ɁPϿB/ mʗ ~^5Fg\.A{QTz{C)`ļ)a]өlYW簤RLV솧% U9ko0F-m.+ʹ[Gj Dŝ*Ր+\ZY(XDM`{`WQe$2N4A"+~.)22fFe]ctmXQ,.q^?7hH=1ۍf3cJ'l_K#pl'vf`[+(* o3WͭE!򘟜'-0, ) cCe_o""/UDa.Fhz6 :bi GV*.M߂a@)(-eX=F# OxN.ngт#~l@).7i)":B Eh ݯ&yvݙl X<ȺYX#:@za# N2 l* jjgk4`eKQ-inH .\iw8bQĊNmTBhC/ּ½bo<s:b߭dU9[SO@4<(_|rZTOEd~g)9jb1A72/>9G.>BpCrzsu?y|oHz3ilS;8E \ynpK(+8ڞN ℿav04ATؐä,]l>,72ƥ$)ߦ@͐q@yJ exJ[NAEd,3G*/F..,B-[jVf**&^]g^i\P~ ˨Noz  4|oq!nܞZ`AndT$p堀PޤyD|ɯ[#PĻ$;Ec?h ~7@+bMEH1q#=ԯ2|𩅬,QvKPَ,9;kWY޾!6 ?KAr-7lJ:Ť :R*f%@YKuȳOCk sg>M.=M%ub16ۻk}A; Uj OOJ?xi# u2{ wt\|*aE۠heӯ(@a2赑eaEtQ&&x!Y`? ]v1?ؚƌ9T&@.x*jA͜D"-k݀5‚QFJTJE`LtK1źXʧ/ly#K/24@o=Na^ QkE:`^&RI}%hCEz5dG&L5sI \tj`"崜~>+a_x~Ve/R!#Z-×^cyȱu[m# 25C1і@ጂSl *vfHK½0،^ VfCy6Bs5Z"{㶡(=ruIEDg|p6bTɄ^Q&z$Oh8f /p ǔ1y9kw+빤24RQ!PpGQp NA!J݄t25Ç_$NS1jA ocv&/"$_`8 n #v9S`ܕE%cv(ܖPn?Ts#6QD aKe ?s 4tmw&F4ɯiڙ=tHL,.<ނx?+7vBI00TcOKR=r 7VZgHeOä;ڝl&|Y 48:XF렍/E*!~qBHH(ꇫ;A H^~ mАU#Xk*M:r= i?c u Jt2wĮG"Y*`k=J;M=iȐ#\)+wfWeA:=llўU#2tMVmChyojȶpZ-Ckl:6}dhFD (]#6Yh8Ź)39@Oprezi5qفXO̸x}ҽy$G n!\V^IMpDEH9 *)!ז ڑXC>ӌLZfB-@nakh?S"X ŀȘ4i:Sqx*_w~7Y=zkuL!(jl 4-^*=Mԕ4*بTMjrǝQj-*EoXt#}bԸmeԱN>8q\7y|dKۢw`q鉱:ښV/rd& {P[kA@9)xD*[o92<ޤi 9y挒Ŧ+3exI)gj9Z7]E̾IwN阩6ϡQu]d՘;˹91&{ȏUlJ#;kJ|RL ;Oa12пCaqf9$KV6|6Z)W>:Աސڵ#R繏\?/I{zIc2k-'x?2Nu--#^nݓm97J"֕7ڋ\0/9y6` nH%u1֝AO`} czᯰ q?7+FMLѱ(ū\opy\c0/75\Z\HAڻ ry+LUrԺ*09, jR|JB?|n{EX=X ğgALH(/(2uXL„)I:A4޷-8q1*J ?=/=c%=w)⬄ClJ.s' /eeX]aNW'c֜T&763 zYHyka)v=ۅecVZw^Zap0 U"8 FgHܠSxua_AC,6NdNjS}/}WQy^_N0>Z&{y?w{qQ_5qe%sc;J" +~%\ԎQ*Q`юkCMЛFL(H誑rX~B:p'Df#O}sIriOٚ2fK,Uj cs`ߦZښ>! vsg_-l.Wyqkhc`G5sl0COP\9FugFLt7v-nu9pđO PD#Iϥ@ :R&1f,"skCFm+hCF%)9/#3fP+"YWgyɡRJ~#+t3 nY1j lxb!hy)ᰉU z){tԔ /3W.9׽b3T}O`:d|MBF:8TYQaF*5\rdC NP2E%ظ0疐YmY66Zq"7w`Sт6=-'?Bs" h%4OIwxxHY= g ^ IUr8Ӷ^b,/UKB'ܗ~%xXl'=5;]h"upQħhWCan*j_ +n>A,tVDV#K + <,{2XSǦ:]Mn `?AV{VĺTE>-Ъbp+&vsq#?.LĊ]YAB> ZKbbRëJlT@1YN^c3%3hj_O&3sX:ZxFXM"7Tm 2oyG:N^Z-I3<1u<#VՇzEg>0v0e?/ӥy*p`phpfu~drr3Z;Fil3_-*e?v:@ Y%J\`A?,Fgjw y7%12e/o*Y=X*. ߦv{ S^_N*l5l܊~(Z!B" ,XwijuޚDi@*o4(_uB&!.oQ~W?,`;\^a:lb%v)Bܞb.^sʡ[d gq|]*T6p_1L^gŎ)vG{t-KpAHfAL0>4^LLJ ;Ⱦ\kƐCrĈBQEr>+ -Vt^l_|-X\ kJ഼VKǁ cKypd*t/>[LtSl{FI.RwykиhPylp;~'^}.HzIz,[Y݆MN|~4)ft@j ϬNnFD{pطs6"vy~W}v!Ͳmk~ͮ_Q/-tj7 vS|A~z;Pd-Oxhw93 L=oGG6 3 䭳~8slɃVR|=c$tCYhҲLе>09ژ( ^8 ~5cfP.H3bʧ u,O.:΃ 0sxMc@u gzy>vLTBK(^  ds#!&qJ0љI=uo0sH,ӏj ܱDܒJy1nͮp蛋"' !^3[=yNq:dyLV:.ԉL(znsMDJhŊMXQxaUb<ה@ҤJg  \y5Q5k>B/%Apn+W',YN>ZQu{|X\v^;3 }LZ"qWw8X>c׻9':~*`Z4Vq/[p2B7v﨓3 h_>e'#*$6:8w[ģ]jǔ!a"#d`8ʱ`NU8 Q>]^D~_f)L‚ծ{NFn2,µ92vڎy.y@Cyz9z)K#s2+l\%KF[rMulX }Ɯɒ v|D=fqiMvQ/5o n뚨G[Ԙ @e-laJTjc>J ’UTO:0el؛}`ǫkD7ȴ_6uJtp -}AW6[:i1C2 KMYj`OI%>ٿ"³٩ɲԶĉ0,sjʶDM_0=y?OE^li'Ƨ̊$C\۹1oO3OG1L 1iS!ۛ0*~ˎR$gr!p;_۟0,͉w/bRFѪn>sk)[K!^NZU`s2e&¡jfkFǐԷ R`ZI 2ȩjRIn>͙<xtx %  Q4ޮY|WT=] }z<_Gn=LɬW$o[ekd s-Ï{2$Ij\%J@%}a]{4$*?u٢zXh ҈XVIsS2ۣUh զ~n]@E[[&pG$Tۤ$qCg j]ɡ7|r B5ѡuJ`'Bo8}hWUMH=ymIqb\ҖZ厰ߩQU1t'ڥ\yWձIķcמ~84UHP=?YBfS"?˹6ͱVVb8 9YS8j'F%9ٮf:qpV'W䃓QQQY3W زmuKN-lTy]FoY20(pGF-< \/A}Rg5O~*)?I+:z&6,(~Wi,ȭ-LeOgJ3`׸p콂w' myw0ovJxt Vbګ:j#^B"SR3 .$h_t[wO'`X=Ǘi޴2 V~P>:M.!?5!m2R)ڙPօ˱sB^$ޭǹEFʡ*/jذ20n/H kԳ x43X"ʥ$xdz OXT͟e d{a&\)"9Y~|FU5I< 2'`^`={*Qh-иJtT鴎Dܛ}CY^Ц݂&]eX*ur˝I{GQ'2SoUjÑ nָ)ڲ`'xnT$^,?:u -cTqG"ڙrѻT{5w;)T l\#9EU  d3"b@ *X8@༴4 W |;x].Ł8I6R ӕ{p@ef8׻ h΢|`exĆW.֐ YzoG`ijelpNx2} 9AD5fBQ`ٽ}ƻLB3%ZL]B2  ֣jb7%`y!țМuUOLqBi͠WRYfk`Z1jAyj^ѲOĥ8/j9 Zǽ^]@x}gK0T|kVdqfͨezR;rL[X o +7M[+=8ȥƓBHn,a)x>^m^ THTHϯ6i/.lz8#elz>[v~t@H/.])L򕨕\cPìMxGx$tUt Ҏl?gIќMMi(m B%i=ִx"pO訁S/PbS5+ endstream endobj 889 0 obj <> endobj 890 0 obj [891 0 R] endobj 891 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 593 0 0 837 0 0 cm /ImagePart_2195 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 398.399 718.45 Tm 112 Tz 3 Tr /OPExtFont5 12.5 Tf (BIBLIOGRAPHY ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 112 Tz 3 Tr 1 0 0 1 113.5 675.5 Tm 99 Tz ([18]) Tj 1 0 0 1 135.849 675.25 Tm 100 Tz (P. Courtier and 0. Talagrand, Variational assimilation of meteorological ob-) Tj 1 0 0 1 135.599 652.45 Tm 102 Tz (servations with the adjoint vorticity equation. II: Numerical results, Q.J.R. ) Tj 1 0 0 1 136.3 629.649 Tm 93 Tz /OPExtFont4 11 Tf (Meteorol Soc., ) Tj 1 0 0 1 209.5 629.899 Tm 100 Tz /OPExtFont5 12.5 Tf (113, 1311-1328, \(1987\). 32 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 113.299 597.5 Tm 99 Tz ([19]) Tj 1 0 0 1 135.599 597.5 Tm 101 Tz (P. Courtier, J.N. Thepaut and A. Hollingsworth, A strategy for operational ) Tj 1 0 0 1 135.349 574.45 Tm 99 Tz (implementation of 4DVAR, using an incremental approach, Q.J.R. ) Tj 1 0 0 1 460.3 574.7 Tm 94 Tz /OPExtFont4 11 Tf (Meteorol ) Tj 1 0 0 1 136.099 551.149 Tm 87 Tz (Soc., ) Tj 1 0 0 1 163.199 551.399 Tm 100 Tz /OPExtFont5 12.5 Tf (120, 1367-1387, \(1994\). 32 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 113.049 519.25 Tm 99 Tz ([20]) Tj 1 0 0 1 135.349 519 Tm 101 Tz (M.C. Cuellar, Perspectives and Advances in Parameter Estimation of Non-) Tj 1 0 0 1 135.099 495.949 Tm 102 Tz (linear Models. PhD Thesis, London School of Economics, \(2007\). 66, 78, ) Tj 1 0 0 1 135.349 472.899 Tm 92 Tz (82 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 92 Tz 3 Tr 1 0 0 1 113.049 441 Tm 99 Tz ([21]) Tj 1 0 0 1 135.099 440.75 Tm 101 Tz (M.E.Davies, Noise reduction by gradient descent, Int. J. Bifurcation Chaos ) Tj 1 0 0 1 135.099 417.699 Tm 100 Tz (3, 113-118, \(1992\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 113.049 385.3 Tm 99 Tz ([22]) Tj 1 0 0 1 135.099 385.3 Tm 103 Tz (J.-P. Eckmann, D. Ruelle, Rev. Mod. Phys. 57 617, \(1985\). 146 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 103 Tz 3 Tr 1 0 0 1 112.549 353.149 Tm 99 Tz ([23]) Tj 1 0 0 1 134.9 353.149 Tm 100 Tz (G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic ) Tj 1 0 0 1 134.9 330.1 Tm 102 Tz (model using Monte Carlo methods to forecast error statistics, ) Tj 1 0 0 1 444.949 330.1 Tm 89 Tz /OPExtFont4 11 Tf (J. Geophys. ) Tj 1 0 0 1 135.349 307.1 Tm 86 Tz (Res., ) Tj 1 0 0 1 162.949 307.1 Tm 101 Tz /OPExtFont5 12.5 Tf (99, 10,143-10,162, \(1994\). 41 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 112.299 274.45 Tm 99 Tz ([24]) Tj 1 0 0 1 134.65 274.45 Tm 100 Tz (G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic ) Tj 1 0 0 1 134.4 251.399 Tm 102 Tz (model using Monte Carlo methods to forecast error statistics, ) Tj 1 0 0 1 444.949 251.399 Tm 89 Tz /OPExtFont4 11 Tf (J. Geophys. ) Tj 1 0 0 1 134.9 228.35 Tm 87 Tz (Res., ) Tj 1 0 0 1 162.949 228.35 Tm 101 Tz /OPExtFont5 12.5 Tf (127, 2128-2142, \(1994\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 112.299 196.2 Tm 99 Tz ([25]) Tj 1 0 0 1 134.4 196.2 Tm 105 Tz (J.P. Eckmann and D. Ruelle. Ergodic theory of chaos. ) Tj 1 0 0 1 416.149 195.95 Tm 96 Tz /OPExtFont4 11 Tf (Rev. Mod. Phys., ) Tj 1 0 0 1 134.15 172.899 Tm 100 Tz /OPExtFont5 12.5 Tf (45:617-656, 1985. 148 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 112.099 140.5 Tm 99 Tz ([26]) Tj 1 0 0 1 134.15 140.299 Tm 102 Tz (J.D. Farmer and J.J. Sidorowich. Optimal shadowing and noise reduction. ) Tj 1 0 0 1 134.15 117.5 Tm 93 Tz /OPExtFont4 11 Tf (Physica D, ) Tj 1 0 0 1 190.8 117.5 Tm 101 Tz /OPExtFont5 12.5 Tf (47:373-392, \(1991\). 79 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 295.699 52.899 Tm 88 Tz (168 ) Tj ET EMC endstream endobj 892 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 893 0 obj <> stream 0 ,, b6Z .s3P|u{BJ!l WAc7V奰!E(#+N2bi\8z}A'pF#90@z2qiڷCB`\'S?^4%}˲&](^˷>᳦=İP'ģ'8?UQڱՇ.;iڠx<dfhqw}qWs4V:֍);QSizEmc9IuS&Y2j oIT||w}two 4L;)Рwv].$]  N~!ҐG4k3WE '. r:(gI3TEAO8oGa?e"ξ] 0Mw6]K˫B"iz3V A~кvk}a޼IFR*Ƙ1761[:gnZ%ܴ\So%UDRXXiS⍵O,L.N: bs}%9 {@(.<{7Gg[vQ#( * :RYp|r@Avu$=߻}Cp/ڿI3'dwMDwZ2֍n?Rt İ_i6ZD+>oQ?_\$0p(G40°1yu |<L%S*\zsw)t[fEgdU E/t +'tѠ~#hfoV?|PK?޳0C'3!NE1uat_ c쀦*pn6^ZNݷm`P`xKeT}'/jW+< 9`ؒ0,]KAq>9ۄ2-]{>[ұ˳f/bA1I((1TInKIKqԖFk^s> HS'y1L2WNha'UCnk]}b\ u0ll$dH÷$є>I }K1si+$ik.x#Ĩ$4TD!B@ߐ:lRbժPK⏄{ulx@VsCu.EБ@R| ^el<<:(w 6&4T׳-FOWHgعRX(Wg$·~&Ivc%kSyZꯜ/mfwPxި~QAcwϚÖ.B[{=KQխ#ry]i+#ta~y/Dgbqa.BEKP=-0~"%>!_;n J1J*b0 ҅O%Ș+0I+B&OP+c^fWMl)qN\@#ܣ-%0^F;x׍}gW[}*) wؚE/xx|Z=|cWv \P8|'v@D9Gҡʿ˪LR;wI_As υvU>A R%1ᤘٰoϔҵћԤ۰Bay/ɱ첹ʧOܰ%4ɟo9!b¸2< PyS6?F` KA߸@ݦbПc;m_*`lmIXHm]I(lEIsP[UY)Ա}*1>~l*Y~Etf%*T'8m/hgrfQޗ2̟mV 4% z R$AouMc]rB6w>0Uk VW&@U꣭IaǸ.5&Ua$w1JV?Gbj_dbV&XntAnOcJcE=α .WbI}E!Ch3[BD :20p봣ZS)A\oLb{}r.67}Ѵ e :Du1k]"+fhsBÙ3@?K!ipÙxG" F`_{,z4RK6Tc n7·! 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'bA-DRSyޗkR amd7+|!O`15[G.C3c2m1jf (ѿПHlViy{A$2e0m4n$n!GDN,[vd =ulH)#6ofTP2NR?‚OfүVd"&AWz$NQوSAj 0CQ"gM%bڅtO $+ѥulFFn1:V9-Q@U~F{$ԓPeyzhwOgg3PLQkؙ<o^fܞn*0GF0xw%\-+5GLjqc@%1\OTD*=.Z `3?;d9s·AxS5ۍf6@9=вFMs'~1޶I2˝v#HZ ƷKطjLLөSKgm97/L5bTR&4obq&6"P8O[U@N o̧k WwVZO"U?ctTjȰlU4: bxcnSLs8jt p9z9' g`ẔO}+hjk7aiuX>U4J1NO/T5ʍ .9ˎ4 7&]o6-=/Uc3zevV+zY !%yk9 6MMn-C VAE>Ҏ'%[*lW) H<,+sMeXtD)O>76shm\,]pw Β n,ȸK0MW9\F1)i;;jnA>q媊y$g?s)= eĦm8sF!E*;̫ɤ6׵?ȫh-$%<׀Ǎܝ[L=!#@N9Jq8*fy\1b&՚zcճPoPB3ݞ|HN9HYY" Z1_6HI਍V}|NtB׊2{N0ߤ%-N!Ysefz!P}PN`gfpCAQ8)$JI]m{6vF[,=KʴV:a?MJF XwnDW[F 1  w%EPFwp40b‡a=zj!o]StJŔ ,K FP"I4|_>!>HŚZS"vk >`/8&4Y8#M%Dr"?^./; K))r&E'i$9#\c+/0ϲѧ4!oY`AMdmi od' T6K,V%g~+Ї P(9Kp-)v16ef^ T)HCMj*ɭeL*X/ *P )xIbLYa Ufg6Kmw8A) $TNTr_TEfqKvȌ* ^b\8HTG(o|^SoxN$?5 o0#lϮ8ܪG Lcw'e|S\VPQ#o G;*@1-wZe 4Z f/v<-K᥇9W 6O ի-hlt 3(]- uo;߯!ܫUhZlH"W rNR˅)DOi]WtH9bU돮:s5^C3*` &WQ+R9X2X3MJ] endstream endobj 894 0 obj <> endobj 895 0 obj [896 0 R] endobj 896 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 593 0 0 837 0 0 cm /ImagePart_2196 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 399.6 718.2 Tm 112 Tz 3 Tr /OPExtFont5 12.5 Tf (BIBLIOGRAPHY ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 112 Tz 3 Tr 1 0 0 1 114.95 675.25 Tm 99 Tz ([27]) Tj 1 0 0 1 137.3 675.25 Tm 103 Tz (C. F. Gauss, Translated by G. W. Stewart. Theory of the Combination of ) Tj 1 0 0 1 137.3 652.2 Tm 104 Tz (Observations Least Subject to Errors: Part One, Part Two, Supplement. ) Tj 1 0 0 1 137.05 629.149 Tm 103 Tz (Philadelphia: Society for Industrial and Applied Mathematics, \(1995\). 65 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 103 Tz 3 Tr 1 0 0 1 114.95 597 Tm 99 Tz ([28]) Tj 1 0 0 1 136.8 596.75 Tm 109 Tz (P. Gauthier, Chaos and quadri-dimensional data assimilation, A study ) Tj 1 0 0 1 136.8 573.95 Tm 99 Tz (based on the Lorenz model. ) Tj 1 0 0 1 278.399 573.7 Tm 90 Tz /OPExtFont4 11 Tf (Telles ) Tj 1 0 0 1 309.85 573.7 Tm 103 Tz /OPExtFont5 12.5 Tf (44A, 2-17, \(1992\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 103 Tz 3 Tr 1 0 0 1 114.5 541.299 Tm 99 Tz ([29]) Tj 1 0 0 1 136.55 541.1 Tm 104 Tz (P. Gauthier, P. Courtier and P. Moll, Assimilation of simulated lidar data ) Tj 1 0 0 1 136.3 518.299 Tm 103 Tz (with a Kalman filter, ) Tj 1 0 0 1 244.8 518.049 Tm 96 Tz /OPExtFont4 11 Tf (Mon. Weather ) Tj 1 0 0 1 319.199 518.049 Tm 103 Tz /OPExtFont5 12.5 Tf (Rev., 121, 1803-20, \(1993\). 40 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 103 Tz 3 Tr 1 0 0 1 114.25 485.649 Tm 99 Tz ([30]) Tj 1 0 0 1 136.55 485.399 Tm 103 Tz (M. Ghil and P. IVIalanotte-Rizzoli, Data assimilation in meteorology and ) Tj 1 0 0 1 136.55 462.6 Tm 97 Tz (oceanography, ) Tj 1 0 0 1 210 462.35 Tm 89 Tz /OPExtFont4 11 Tf (Adv. Geophys. ) Tj 1 0 0 1 284.149 462.35 Tm 102 Tz /OPExtFont5 12.5 Tf (33, 141-266, \(1991\). 40 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 114 429.949 Tm 99 Tz ([31]) Tj 1 0 0 1 136.55 429.949 Tm 101 Tz (I. Gilmour, Nonlinear model evaluation; t-shadowing, probabilistic predic-) Tj 1 0 0 1 136.099 406.899 Tm 104 Tz (tion and weather forecasting, Ph.D. thesis, University of Oxford, \(1998\). ) Tj 1 0 0 1 136.55 383.899 Tm 86 Tz (79 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 86 Tz 3 Tr 1 0 0 1 114 351.25 Tm 99 Tz ([32]) Tj 1 0 0 1 136.099 351 Tm 100 Tz (P. Grassberger, T. Schreiber, and C. Schaffrath, Nonlinear time sequence ) Tj 1 0 0 1 136.3 328.199 Tm 104 Tz (analysis, Int. J. Bif. and Chaos 1, 521 \(1991\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 104 Tz 3 Tr 1 0 0 1 113.5 295.549 Tm 99 Tz ([33]) Tj 1 0 0 1 136.3 295.549 Tm 100 Tz (C. Grebogi, S.M. Hammel, J.A. Yorke, and T. Sauer. Shadowing of physical ) Tj 1 0 0 1 136.099 272.299 Tm 103 Tz (trajectories in chaotic dynamics: containment and refinement. ) Tj 1 0 0 1 450.699 272.299 Tm 91 Tz /OPExtFont4 11 Tf (Phys. Rev. ) Tj 1 0 0 1 136.099 249.25 Tm 89 Tz (Letts., ) Tj 1 0 0 1 170.15 249.25 Tm 101 Tz /OPExtFont5 12.5 Tf (65\(13\):1527-1530, \(1990\). 79 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 113.299 216.35 Tm 99 Tz ([34]) Tj 1 0 0 1 135.849 216.35 Tm 104 Tz (J. Guckenheimer, J. Moser and S. Newhouse, Dynamical Systems, LIME ) Tj 1 0 0 1 135.599 193.549 Tm 102 Tz (Lectures, Birkhauser, 289pp., \(1980\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 113.299 160.7 Tm 99 Tz /OPExtFont4 11 Tf ([35]) Tj 1 0 0 1 135.849 160.7 Tm 76 Tz (J. ) Tj 1 0 0 1 149.5 160.7 Tm 100 Tz /OPExtFont5 12.5 Tf (Guckenheimer and P. Holmes, ) Tj 1 0 0 1 306.25 160.7 Tm 95 Tz /OPExtFont4 11 Tf (Nonlinear Oscillations, Dynamical Sys-) Tj 1 0 0 1 136.3 137.649 Tm 94 Tz (tems, and Bifurcations of Vector Fields, ) Tj 1 0 0 1 336.25 137.399 Tm 101 Tz /OPExtFont5 12.5 Tf (Springer, New York, \(1983\). 7 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 296.899 52.7 Tm 88 Tz (169 ) Tj ET EMC endstream endobj 897 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 898 0 obj <> stream 0 ,,"Vb7 q-UrDܯ4HPpC[n%'rF@O"}A}9rh9HʀFi(1"bӠ Z7WzT_),N,6YS3*a/1?SRqTo`TW5kLjFi@(pab,&{$ 7ʼQ^`Tߥg?9ʄLkڕcPUwY!`ݏq,d< B גݢ;?od9 D==nsop!Rٴ&QO&m΋56,Z:U* 0B: i[܉ yVB{6 %k7+]'?qˏ๫ŝf`AQmhLb45rsodMsM7-wWFO1\#ŀȮ[SA^{X`a/ -඼ eUzX҈37ve9دp\>6ܤB2ߗz5!ϸ鷪rfK4‹-Wk%l$7&"ңē,-?S䇳"$6$d淆շHʰ1Շ<6BXEQP"ev~r9]'}=lD\N6LwTzԺEf.ax1bg,'rx@z?j!R`_Zoh_%5ȧ,HfD[cɪ|h~|o@XHn%%P cukRb{14 =P5=ۧZ;pb/8YĀb砽Cme#`ʽm0=FĚQ3*N^pmK b\N9("G{UЀ47Yv6GEgYQ̻x`\3},)T;+c׾ Ǡ(.V>"_e'S&.G6++m`cW6"i뻜\k׊܍;[eKO NAXs( ʞá7gZ9lېz "!p#稐>­qU|8ӕDi*?Us} -@lq.j:%+sˠ^ +!5he{"H&10b0{?B:HVO c,gv>̤j_8z`ݳ cPOP Lȭ: zU1#7ja^ρ!˱}ljӗe9fɲoWXr)F7/w+;ڂx'$_Ҧ>ʰړ{Ӹʜ̶ı&K쩩1_)yoǸ8&ނ><VZ"e8"eF,iҍ9`2K8@G`)g ^9%߁U T\i A"ؐih<uK:7W_StQ}Jb09Ǟ/'nCAAD㷁xJW٦ yl3#%?$>gR&U.$@h^ Kf)׺a5n1eb8*YdVST'W_К_<12 BA'CJ3e<)Ab& rd-j[Fl;KC눀6Lhd^SxBȞ!@L޸c{ݭhy!GQ9FOTTSb'"6&fM{ ;hnaO}}ijDKnap4cbs84[Lh$Q 3lK: S:U)-a3/Hkf;YUYZs :p,Ci8*P~ +c8~$6H3r(n33| V7˽9*I{ߥnx2EE+o];4 c_ܥ}}$2N /VQ^.Iʉt9#'wOs@0 \i=YzD.Pk4M,ڨxWrvEz~+y%2T)5f+NȓN s ZòtD Ż(p]Q0UwD9 (Yʵf$u`|Hz@c[UyV3\-(/yTZX_ԆqM bIKQO&a|1y%Dļoϛ}QZXOQzqȨɟAݭ1+WHVuxr|n7WwbFv H^aFS;A۱Etu4K9=xDZMPvzm2띠W_]>{JaЏu=*Gx-i_@XB&ؐ9Э n&:ѥvNpG82  {.2Vk OnO0߳ 6B@i' WIɞ6 yL1Vd Tw;4ND\G㻅lqeeF_ NqK`J|SS)ɫ%T3Z8+!ss%'C0G8Ev׍ڍ{& YA`f~m/`w IXΘU'dɋ@MfաE|L0zz͕EmRzU8b5À_(ƾ_&$rKAO}r 1ʗpA':U J9go k$7{**pqmh-Q;bnlX!_'cSXI[Ժ{Q)_cV)LS&s/ <5ZOB=zm)RGȣ(EcfDIQ76jUV-A]l$ خ "$1>ngXT0c'!>p>|)k ְ" |GQq ȈM*CeLN5Gm~%<8x2k$eݞt bbɁp>??Bb/d{d_JI0bv/ZYlr^Uk۞y1 W .g4&)9j:*|vv?DQ`qb5ވTIHLg›`-kU{L^Ȗtp5ib2E1h͸2B $:L- z`AYj"ŃZ|;sSFr rm $ᥐy ^%[(] b>ȺK86FߺXAIS`[[5 {ϣDQR;.CH:Y{-/wZs(xG==wEoA!?&WFs%0wU@KiN,j wM#'aZ? 5ݳ?%B W A.ۈwqC>uBUkR?&zꄋ@ș ״ntZD=q >@ jfDJ9*MrVRfcE"j@*KOgWHȨkI?#&ˢiiVt17ȠB<('ӂK򰠐[8R(ZZ}3v:4'}^`oL*Q&@QkhfΰúUNEɲ y-g? |e|D ׉,,Qor`f~Rm&Ed ,-$W`"%d3("5R&ZweBfeQ :.n5Q [\Z֪'$Pk)(5ϥO08s^$`R\=Y;F[w?:3xgg%ϽrlG-XjE YKA1_H\S y(wNmi4W:X`0$ ?BT|w /,g M!|5kzw6,!TB]7ei`3*+Y蛗]8Xc?NnјM; Pb#y\e0T "Tf)C@O0x%}]4W+~ Q#>0,fu$Ξ0r8Aֲ,ҢaJ7 djg]zi m")y -R(la5 *`Q0'\[f "Zx(qmNJ3%+>K59Fh ė́R,\̀uk/ p%}ivNe/#tir[VE''3V_ky)Vk #;Rc{eQ53~z?z}iG)p3us@T j@kdi"6g{u1*C(PnI|*pզS<S.Ln6XHGu:kI+"teUP#65kzC4|C'mobJnj‚!r~iF3Sg5ZpW3 ;<$u"7.XG>$oX,ʈ!P9w߅Zl2 j^q\Qo%I6'tԿ\^I/P]O!fA*d@(,bxГYB; Qq#=} j#[l}?q]h^DxаnC4\^h6hzGz|LtQntƸ{cfnkq,g ]4xNB)ĘX3H7{[ir_Oײ{;2Xr,_g 7u]5svjRgFl-TL g}9#& =k똎wTNVR;g)Y:a%y>Qt\A)᧸:hN { xN LgXs M?M6l bve/語jqn|_ Ɗ@uP?v1{][΅|u7Xj瓜TG,-h͝gXX;Q_ޡ4ꊟJFT喥Lrj9JRxbMSy˝Tx]X?* x{aZ0k ~A2[dIϖTj5Zt6$IHPsdm/Cfl`u)N-/1aLA[y=G{!i x6h>)i6&q#pN{W#c,BmWʏSxfj1fFr@qǿ^/o؋ (X_gH?q* ,<-AKq#` .͗ŴVd h%)?B(GA+ }f=҅d>ua4aewp/-#7>24aڃ G}Sb-x+ D,n,u؊+8Ek7ʩ]XOQ ('YLkW'-x'ײ'GYA+q)Sw~𮓥AZH`]ܬR~Wg翆8z(%*Sv*@tw::&>dC5-n@M*ÞF HgIn99Xj*6rKQ >ȦJf?xWy4VjӛrQݤR`J?h=Qn". e^\7$~5 X]ȯq66 N+X\rk֮BtPa[&++3LP2;ZoJM7]ر( !ڦO^Ag?/B  a ?G>߼/4ӥڡŰӳ0Sʣ:˜Jo~چ~U4NK 51%rә آgXyAZ"pieh5)Z {Ñ_XbO(RL9!GC04S5F*TT|UQsۢ1 rׇ_M?_2Z.`E^~b8$_‹"Ntimbv.s"(Z=*67O\=Q ;$5 2OBx)IrkNv1ΚsCXyOmk)DǪQyR_ddFfTϛ{͓3rU8 iPj iduqz/ѩ @E^HS~m`iqT3c-Zʭ3DUTVI9bXe'A}\Gbijtmnhf: 4\2 >K,Z^;:)ehaTBX~."rk;FMVI/&C ԨhQ;G5u/k :Mj3: <<SGt/Hiv^:DK[6I&-;epip_tVazkq+4p5;dUD'L5M@ ғx%lVP!Ӧde'GTG)kn^6sk g#_Vn8=1 '?6:k96mӍxcfo1MJ嬄 RK,o1v+VR|V4/xzǏ!<&T-Q@8x:7J(A脴@ڣ7f~:ʃ)cOdϿ%ucWYq^ΖretJǀV74݀kONFg6{V'QI4ip s2 yfj 9~rӥ@.e*Jlv֍w 9vsғfzr]V0-p߸% cG"/Hs"֊ދo4O~&2W&jilWxdB5BZP<$DC2wEߒ3+OcY b<)6ocyGu0כ>ocЩW1 ulmxT ?%7T50Zh(;aC=xN )ˈXϡ=ej0WEʻЉ|e iU;v]V ˥G1)]YN,%4 ɒr~FMM %ѝ ϑt=%*T`QE`ʺB^6!v&O=`j}RN+K*c`N2_uِ`/@&unnj&@ uS=K~eCj+hE`4?.Qǘp|gÝ;BN;:L W?v l{lxr5|4#2 :qit[VߏgTX܏q7n.-Q[妕Р|/dSRǜOK+JW} 5MWrK22n^Zb]XһWa@LIPpQ$79T6Yբ`B?53],u5LGGng+Zܹj{&+F.|{ؤ3mW ڟ-SgBf.o~a¬y;>F2N(b^gFS[+&6+CxO%a.Fn w0 G/-Ň^"IZZiU`A [XKed5v Z@s@TG\O!'WF#h[GV> endobj 900 0 obj [901 0 R] endobj 901 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 598 0 0 837 0 0 cm /ImagePart_2197 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 399.1 717.7 Tm 112 Tz 3 Tr /OPExtFont5 12.5 Tf (BIBLIOGRAPHY ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 112 Tz 3 Tr 1 0 0 1 114.5 674.049 Tm 99 Tz ([36]) Tj 1 0 0 1 136.55 674.5 Tm 106 Tz (T.M. Hamill, Iterpretation of Rank Histograms for Verifying Ensemble ) Tj 1 0 0 1 136.8 651.5 Tm 97 Tz (Forecasts. ) Tj 1 0 0 1 189.349 651.5 Tm 93 Tz /OPExtFont4 11 Tf (Monthly Weather Review: ) Tj 1 0 0 1 320.399 651.5 Tm 99 Tz /OPExtFont5 12.5 Tf (Vol. 129, No. 3, 9-30, \(2001\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 99 Tz 3 Tr 1 0 0 1 114.5 619.1 Tm ([37]) Tj 1 0 0 1 136.55 619.299 Tm 102 Tz (T.M. Hamill, Ensemble-based atmospheric data assimilation, Chapter 6 of ) Tj 1 0 0 1 137.05 596.049 Tm 95 Tz /OPExtFont4 11 Tf (Predictability of Weather and Climate, ) Tj 1 0 0 1 333.6 596.299 Tm 101 Tz /OPExtFont5 12.5 Tf (Cambridge Press, 124-156, \(2006\). ) Tj 1 0 0 1 136.8 573 Tm (41, 42, 43, 44, 73 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 114.7 540.1 Tm 99 Tz ([38]) Tj 1 0 0 1 137.05 540.6 Tm 102 Tz (J.A.Hansen, Adaptive observations in spatialy-extended nonlinear dynam-) Tj 1 0 0 1 136.55 517.549 Tm (ical systems, Ph.D. thesis, University of Oxford, \(1998\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 114.25 484.699 Tm 99 Tz ([39]) Tj 1 0 0 1 137.05 484.899 Tm 102 Tz (J.A. Hansen, L.A. Smith, Probabilistic noise reduction. ) Tj 1 0 0 1 413.05 485.399 Tm 91 Tz /OPExtFont4 11 Tf (Tellus series A, ) Tj 1 0 0 1 491.05 485.399 Tm 98 Tz /OPExtFont5 12.5 Tf (53 ) Tj 1 0 0 1 137.75 461.649 Tm 104 Tz (\(5\), 585-598 \(2001\). 48 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 104 Tz 3 Tr 1 0 0 1 114.5 429 Tm 99 Tz ([40]) Tj 1 0 0 1 136.8 429.5 Tm 100 Tz (M. Henon, A two-dimensional mapping with a strange attractor, Commun. ) Tj 1 0 0 1 136.8 405.949 Tm 101 Tz (Math. Phy, s. 50 69-77 \(1976\). 10 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 114.25 373.3 Tm 99 Tz ([41]) Tj 1 0 0 1 136.8 373.8 Tm 103 Tz (P.L. Houtekamer and H.L. Mitchell. Data assimilation using an ensemble ) Tj 1 0 0 1 136.8 350.5 Tm 101 Tz (Kalman filter technique. ) Tj 1 0 0 1 261.1 350.75 Tm 95 Tz /OPExtFont4 11 Tf (Mon. Weather Rev., ) Tj 1 0 0 1 364.1 351 Tm 103 Tz /OPExtFont5 12.5 Tf (126, 796-811, \(1998\). 41, 42 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 103 Tz 3 Tr 1 0 0 1 114.5 317.649 Tm 99 Tz ([42]) Tj 1 0 0 1 136.55 317.899 Tm 101 Tz (P.L. Houtekamer and H.L. Mitchell. Reply to comment on "Data assimila-) Tj 1 0 0 1 136.55 294.85 Tm 100 Tz (tion using an ensemble Kalman filter technique." ) Tj 1 0 0 1 379.699 295.1 Tm 97 Tz /OPExtFont4 11 Tf (Mon. Weather ) Tj 1 0 0 1 453.85 295.299 Tm 103 Tz /OPExtFont5 12.5 Tf (Rev., 127, ) Tj 1 0 0 1 137.3 271.299 Tm 101 Tz (1378-9, \(1999\). 41, 42 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 114.25 238.7 Tm 99 Tz ([43]) Tj 1 0 0 1 136.8 239.149 Tm 103 Tz (P.L. Houtekamer and H.L. Mitchell. A sequential ensemble Kalman filter ) Tj 1 0 0 1 136.55 215.899 Tm 100 Tz (for atmospheric data assimilation. ) Tj 1 0 0 1 305.5 216.1 Tm 96 Tz /OPExtFont4 11 Tf (Mon. Weather ) Tj 1 0 0 1 378.699 216.35 Tm 100 Tz /OPExtFont5 12.5 Tf (Rev., 129, 123-37, \(2001\). ) Tj 1 0 0 1 136.55 192.35 Tm 98 Tz (41, 42 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 98 Tz 3 Tr 1 0 0 1 114.25 159.5 Tm 99 Tz ([44]) Tj 1 0 0 1 137.05 159.95 Tm 104 Tz (G.E. 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Ikeda, Multiple valued stationarity state and its instability of the trans-) Tj 1 0 0 1 138 651.5 Tm 102 Tz (mitted light by a ring cavity system, Optical Communications, 30 257-261 ) Tj 1 0 0 1 138.699 628.45 Tm 101 Tz (\(1979\). 11 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 115.9 595.799 Tm 99 Tz ([46]) Tj 1 0 0 1 138 596.049 Tm 101 Tz (L. Jaeger and H. Kantz, Unbiased reconstruction of the dynamics underly-) Tj 1 0 0 1 138 573 Tm 102 Tz (ing a noisy chaotic time series, Chaos ) Tj 1 0 0 1 326.899 573 Tm 99 Tz /OPExtFont5 12 Tf (6, ) Tj 1 0 0 1 339.85 573 Tm /OPExtFont5 12.5 Tf (440 \(1996\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 99 Tz 3 Tr 1 0 0 1 115.7 539.899 Tm ([47]) Tj 1 0 0 1 138 540.1 Tm 100 Tz (A.H. Jazwinski, ) Tj 1 0 0 1 217.9 540.35 Tm 98 Tz /OPExtFont6 12 Tf (Stochastic Processes and Filtering Theory. ) Tj 1 0 0 1 424.8 540.6 Tm 99 Tz /OPExtFont5 12.5 Tf (Academic Press, ) Tj 1 0 0 1 138.699 517.1 Tm 102 Tz (\(1970\). 40 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 115.7 484.449 Tm 99 Tz ([48]) Tj 1 0 0 1 138 484.899 Tm 108 Tz (K. Judd and L.A. Smith, Indistinguishable States I: The Perfect Model ) Tj 1 0 0 1 138.25 461.399 Tm 98 Tz (Scenario, ) Tj 1 0 0 1 186.699 461.399 Tm 99 Tz /OPExtFont6 12 Tf (Physica D ) Tj 1 0 0 1 240.25 461.649 Tm 107 Tz /OPExtFont5 12 Tf (151: ) Tj 1 0 0 1 267.85 462.1 Tm 102 Tz /OPExtFont5 12.5 Tf (125-141, \(2001\). 2, 19, 22, 25, 27, 28, 34, 65, 98, ) Tj 1 0 0 1 138.699 438.35 Tm 96 Tz (157, 160 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 96 Tz 3 Tr 1 0 0 1 115.7 405.5 Tm 99 Tz ([49]) Tj 1 0 0 1 138 405.949 Tm 107 Tz (K. Judd, Chaotic-time-series reconstruction by the Bayesian paradigm: ) Tj 1 0 0 1 138 383.149 Tm 122 Tz (Right results by wrong methods, Physics Review Letters, ) Tj 1 0 0 1 488.899 383.399 Tm 105 Tz /OPExtFont5 12 Tf (67, ) Tj 1 0 0 1 138 359.899 Tm 98 Tz /OPExtFont5 12.5 Tf (026212,\(2003\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 98 Tz 3 Tr 1 0 0 1 115.7 327 Tm 99 Tz ([50]) Tj 1 0 0 1 138 327.5 Tm 102 Tz (K. Judd and L.A. Smith, Indistinguishable States II: The Imperfect Model ) Tj 1 0 0 1 138 303.95 Tm 98 Tz (Scenario, ) Tj 1 0 0 1 187.449 304.2 Tm 100 Tz /OPExtFont6 12 Tf (Physica D ) Tj 1 0 0 1 242.65 304.2 Tm 106 Tz /OPExtFont5 12 Tf (196: ) Tj 1 0 0 1 271.199 304.45 Tm /OPExtFont5 12.5 Tf (224-242, \(2004\). 3, 11, 93, 94, 95, 96, 98, 100, ) Tj 1 0 0 1 138.699 280.7 Tm 99 Tz (108, 110, 126, 129, 160 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 99 Tz 3 Tr 1 0 0 1 115.45 248.299 Tm ([51]) Tj 1 0 0 1 138 248.5 Tm 104 Tz (K. Judd and L.A. 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[bɒ9*m4 #I<Ǡو,Ծ<^ش.,6V\ :_Y{p>u!!?ոcV#\Hƍ~foZkPLHғlS Y`\e"Ll6A +8b.3KuM;ToWC]`CەCw+^- kôe 8立7ƴ7~jhoO]N1HO Ɠlzbw`'zpPJ^˻xX"(eZ-oE|3_tt -7CgkE M@*Wޝ[ d@`l֪,ihaKDڜ{($Ъ/1^?rΏ.Υ =s3$r1=2d1e`G[kb.bZawvO)7 /ҴO؅ c ݻn'QiݶqݘpË$5~ہ/p*D^nE? -pN+U(qav0Ƒ=?c f_ĉiJ3SNf>qSA>VmϺEXi_>5kMd*;Uus(mo|}`]]݅N n߄}` 'vgۥxA'Ԫ#eXl0lO:FhNw"!)}R~K-܅T~ڲD瑒=a~и p@ɸJ4_32$ &kMy nlWy`u3gQ[d'V H*9mO"Թj=u-*Ǒ#W ''Bukwam`AI,q*#Q Z\{etĒ WR% qu7t9zAA”4zu݈z6BVJlo0P% fZ&C ¥pyD"x]8L8 `%Dz6 Ubnh#-1$v 8$Y?U}&3Yfjz0DskoO `|R3rH&ZSRe?pmry2#/FY = endstream endobj 909 0 obj <> endobj 910 0 obj [911 0 R] endobj 911 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 597 0 0 837 0 0 cm /ImagePart_2199 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 400.55 717.7 Tm 120 Tz 3 Tr /OPExtFont2 11.5 Tf (BIBLIOGRAPHY ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 120 Tz 3 Tr 1 0 0 1 115.9 674.75 Tm 99 Tz /OPExtFont5 12.5 Tf ([53]) Tj 1 0 0 1 138.5 674.5 Tm 102 Tz (K. Judd, C.A. Reynolds, T.E. Rosmond and L.A. 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Sci., ) Tj 1 0 0 1 449.3 229.549 Tm 96 Tz /OPExtFont5 12.5 Tf (20:130-141, ) Tj 1 0 0 1 138 206.299 Tm 101 Tz (\(1963\). 10, ) Tj 1 0 0 1 194.9 206.299 Tm 91 Tz /OPExtFont2 11.5 Tf (34 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 91 Tz 3 Tr 1 0 0 1 114.7 173.149 Tm 99 Tz /OPExtFont5 12.5 Tf ([62]) Tj 1 0 0 1 137.05 173.399 Tm 107 Tz (E.N. Lorenz. 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V۱q TCUf0%엶['ڱ[乹 -_v*-SvTb Ȱ Jc WVմh!}ek|2 l,H35=eMg&aEn=fV;/ 7QM N|-oļOkC` hF4gyl@1!DVsAŤΪGMWc_zw3)+%E(-QK|V [W*Q󦓩pݢ誔 21Y؆jV_Uq/>Q_J?n(ʖ_)'X (n9wd@c# <ϜUvMB*ЁDzk ߪqEnGëg5 1Տ(h1f/ײ2q:Z2.kC塦=8YJ%gcQh 1 =g,Ҫ~Kջ {l:̓m=vDTj|T'n&<)zG ?w0!C)V'dKQ2m1[jWE/ F$  f&u[/Z凥~3$XSHhlֵCucm5L'XDURh'!fbe{Οy4s A6 WK%9"|sr6J.Wד~'&s^!5 {j]iGvo@c1y)"WPpGY*+i߈m O;0ڪI!a?LQ5Z>ǵTG+>g!\J-D0EA2?ex-p0\1PPDZvVCL]6F4bb(G򾞘o 7+uÔHCg5< B="cJ,uQ2d3:j-7[O@SPqm݃ߕ ߸)ï.fÃ-_4!ɒmd2#ѱ%<*{3Ғeq-`!fSn>d<(9tp)ܤAw(ۿ×n/F{%-'f*gş&+v(Ԓ._5(,ǭꝳN? )Ԭ8g `;o3#q7Pڇ.yL < ʰ$dʒ ٕ0զ3?#M#q"U=j7]&%JOUUQf/1H׫RVBF:8W02Gޙ(d&EE #0ĘtRj @&lVuz)(ޭ>xswyi?ہy 3yiLak!7ѫ1\%㟻n[참-IY߰-4 KbS/)m5ީְKͼך5׎fC6r_6`*;thOÊe0ȵE _86֔}-"t }v"0;.6)8 qR8?b8(˾ kתLf0V{fkR_oq2Mj g#=e`}#:^4lN̄Tw;$sk1E:[Wb<$6 7akrއ8}s{:wHEqOEWPNMUIo`7Hԟ+mph] Lȑ"'wKS=7ZQT FLTT*)K([T63ezmѳNa--wC&uz9hq輎Ctq-*)lv?t}e X}}eJ*5xEajo*20M."~',\: x}87DNe+.4F ?|WY+T()D+m ^WdTawS5YKԄQx#ӛt-'%f>UQx2dT[v['O_Dfa&אu35u2f(U.%^@FxWV'_ endstream endobj 914 0 obj <> endobj 915 0 obj [916 0 R] endobj 916 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 594 0 0 834 0 0 cm /ImagePart_2200 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 400.55 715.2 Tm 112 Tz 3 Tr /OPExtFont5 12.5 Tf (BIBLIOGRAPHY ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 112 Tz 3 Tr 1 0 0 1 115.9 672.95 Tm 99 Tz ([63]) Tj 1 0 0 1 138 672.25 Tm 101 Tz (E.N. Lorenz, Predictability A problem partly solved, in: ECMWF Seminar ) Tj 1 0 0 1 138 649.45 Tm 100 Tz (Proceedings on Predictability, Reading, United Kingdom, ECMWF, \(1995\). ) Tj 1 0 0 1 138.699 626.899 Tm 97 Tz (12, 13, 96 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 97 Tz 3 Tr 1 0 0 1 115.45 594.5 Tm 99 Tz ([64]) Tj 1 0 0 1 137.75 594 Tm 103 Tz (P.E. McSharry and L.A. Smith, Better nonlinear models from noisy data: ) Tj 1 0 0 1 137.5 571.2 Tm (Attractors with maximum likelihood, Phys. Rev. Lett 83, \(21\): 4285-4288, ) Tj 1 0 0 1 138 548.399 Tm 102 Tz (\(1999\). 3, 63, 65, 66, 86, 87, 103, 185 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 114.95 516.25 Tm 99 Tz ([65]) Tj 1 0 0 1 137.5 515.75 Tm 111 Tz (R.N. Miller, M. Ghil and F. Gauthiez, Advanced data assimilation in ) Tj 1 0 0 1 137.3 492.699 Tm 102 Tz (strongly nonlinear dynamical systems. ) Tj 1 0 0 1 329.05 492.5 Tm 103 Tz /OPExtFont6 12 Tf (J.Atmos.Sci. ) Tj 1 0 0 1 394.55 492.25 Tm 100 Tz /OPExtFont5 12.5 Tf (51, 1037-1056, \(1994\). ) Tj 1 0 0 1 137.3 469.899 Tm 96 Tz (32, 33 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 96 Tz 3 Tr 1 0 0 1 114.95 437.3 Tm 99 Tz ([66]) Tj 1 0 0 1 137.05 436.8 Tm 106 Tz (W. D.Moore and E. A.Spiegel, A thermally excited nonlinear oscillator. ) Tj 1 0 0 1 137.05 414 Tm 102 Tz (The Astrophysical Journal. Volume 143, pp. 871-887, \(1966\). 12 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 114.5 381.85 Tm 99 Tz ([67]) Tj 1 0 0 1 136.8 381.35 Tm 100 Tz (D. Orrell, A shadow of a Doubt: Model Error, Uncertainty, and shadowing ) Tj 1 0 0 1 136.55 358.3 Tm 99 Tz (in Nonlinear Dynamical Systems. PhD Thesis, University of Oxford, \(1999\). ) Tj 1 0 0 1 137.75 335.5 Tm 79 Tz (13 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 79 Tz 3 Tr 1 0 0 1 114.25 303.1 Tm 99 Tz ([68]) Tj 1 0 0 1 136.099 302.649 Tm 103 Tz (V.I. Oseledec, Transactions of the Moscow Mathematical Society 19 197, ) Tj 1 0 0 1 136.55 279.85 Tm 102 Tz (\(1968\). 146 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 113.5 247.45 Tm 99 Tz ([69]) Tj 1 0 0 1 136.099 247.2 Tm (E. Ott, ) Tj 1 0 0 1 176.15 246.95 Tm 100 Tz /OPExtFont6 12 Tf (Chaos in Dynamical Systems, ) Tj 1 0 0 1 324.5 246.7 Tm 102 Tz /OPExtFont5 12.5 Tf (Cambridge University Press, \(1993\). ) Tj 1 0 0 1 136.099 223.899 Tm 82 Tz (9 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 82 Tz 3 Tr 1 0 0 1 113.5 191.5 Tm 99 Tz ([70]) Tj 1 0 0 1 135.849 191.049 Tm 101 Tz (F. Paparella, A. Provenzale, L.Smith, C.Taricco, R.Vio, Local Random ana-) Tj 1 0 0 1 135.599 168.25 Tm 99 Tz (logue prediction of nonlinear processes, ) Tj 1 0 0 1 333.1 168 Tm 102 Tz /OPExtFont6 12 Tf (Phys. Lett. ) Tj 1 0 0 1 391.199 167.75 Tm 99 Tz /OPExtFont5 12.5 Tf (A 235 233-240, \(1997\). ) Tj 1 0 0 1 136.3 145.45 Tm 95 Tz (16, 132 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 95 Tz 3 Tr 1 0 0 1 296.399 50.399 Tm 88 Tz (173 ) Tj ET EMC endstream endobj 917 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 918 0 obj <> stream 0 ,,Jb6)'_4ՓXuL 5'}_A?DY)"-Qr!B{F AP@&KS_'+ ?kpfɓnTnOf +hg \C&CExmjXcr&˿;'OJr8ΦZJT JwT"yHlN#K*R#q Jg]Ȭ?)1-BHDf^'D~F|XqsmE]w?قF/Kd#2P"<(mgLtKbKA0n mpq23zZkw눁hO@B V@Ih]n!n"h;Xq@ċۮ0GѩX]?҈ Q bjٖKw!O!lפ8Mw$޽ou*UL27Z8-xNRd1u݄mJMnkVv&y@|GwZgRb nd˛dՓr1709U )8rgX 8s'Lf?okrd:d>6t[/փE\ͱÛFb!djq6=ŵk8ߌzCshʶtv''-w[+/뇝sJH8E`٘6(q^ 2کzԒ҃ f}Ͽ0\.nETMZJѤ6Zi1 *OX[u|_-YnjyT/.H~һ,H39N!'hC+n?x{Vx?lY<qM)VA`[1z7RQkUUپs ` ߠ^߸|FFZ(Jyt*ؒN΢3E@Hk5X0f;χrcMƒD})G//J}Ýf˷Y|!~NCDirdP)d!CHp]&GG|U2 ·G⶘0iPS;6`*N(+q%dXE5\ء bq2)cE(WA48cqv'WxD HB-ĥ OV OIte3? 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Pires, R. Vautard, and 0. Talagrand, On extending the limits of vari-) Tj 1 0 0 1 135.099 648.95 Tm (ational assimilation in nonlinear chaotic systems. ) Tj 1 0 0 1 396.5 648.5 Tm 85 Tz /OPExtFont4 11 Tf (Tellus, ) Tj 1 0 0 1 434.149 648.5 Tm 107 Tz /OPExtFont2 11.5 Tf (48A, 96-121, ) Tj 1 0 0 1 135.599 626.149 Tm 103 Tz (\(1996\). 32, 33, 34, 36 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 103 Tz 3 Tr 1 0 0 1 113.049 595.2 Tm 99 Tz ([72]) Tj 1 0 0 1 134.9 594.95 Tm 109 Tz (V.F. Pisarenko and D. Sornette, Statistical methods of parameter estima-) Tj 1 0 0 1 134.65 571.899 Tm 107 Tz (tion for deterministically chaotic time series, Phys Rev E, 69 \(2004\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 107 Tz 3 Tr 1 0 0 1 112.549 541.2 Tm 99 Tz ([73]) Tj 1 0 0 1 134.65 540.95 Tm 113 Tz (W. H. Press, B. Flannery, S. Teukolsky, and W. Vetterling, ) Tj 1 0 0 1 451.449 540.5 Tm 93 Tz /OPExtFont4 11 Tf (Numerical ) Tj 1 0 0 1 135.099 518.149 Tm 96 Tz (Recipes in C ) Tj 1 0 0 1 201.349 518.149 Tm 103 Tz /OPExtFont2 11.5 Tf (\(CUP, Cambridge, 1992\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 103 Tz 3 Tr 1 0 0 1 112.099 486.949 Tm 99 Tz ([74]) Tj 1 0 0 1 134.65 486.5 Tm 108 Tz (D. Ridout and K. Judd, Convergence properties of gradient descent noise ) Tj 1 0 0 1 134.4 463.699 Tm 104 Tz (reduction. ) Tj 1 0 0 1 187.9 463.449 Tm 90 Tz /OPExtFont4 11 Tf (Physica ) Tj 1 0 0 1 229.199 463.449 Tm 91 Tz /OPExtFont4 12 Tf (D ) Tj 1 0 0 1 241.449 463.449 Tm 103 Tz /OPExtFont2 11.5 Tf (165 27-48, \(2001\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 103 Tz 3 Tr 1 0 0 1 112.099 432.5 Tm 99 Tz ([75]) Tj 1 0 0 1 134.4 432.25 Tm 111 Tz (M.S. Roulston and L.A. Smith, Evaluating probabilistic forecasts using ) Tj 1 0 0 1 134.15 409.699 Tm 105 Tz (information theory, ) Tj 1 0 0 1 234.5 409.199 Tm 93 Tz /OPExtFont4 11 Tf (Monthly Weather Review, ) Tj 1 0 0 1 365.75 409.199 Tm 105 Tz /OPExtFont2 11.5 Tf (130, 1653-1660, \(2002\) 70, ) Tj 1 0 0 1 134.4 386.649 Tm 89 Tz /OPExtFont5 12 Tf (73 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12 Tf 89 Tz 3 Tr 1 0 0 1 135.849 386.899 Tm 47 Tz /OPExtFont8 6.5 Tf (1 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont8 6.5 Tf 47 Tz 3 Tr 1 0 0 1 111.849 355.199 Tm 99 Tz /OPExtFont2 11.5 Tf ([76]) Tj 1 0 0 1 133.699 354.949 Tm 106 Tz (Y. Sasaki, Some basic formalisms on numerical variational analysis. ) Tj 1 0 0 1 475.699 354.699 Tm 92 Tz /OPExtFont4 11 Tf (Mon. ) Tj 1 0 0 1 136.099 332.149 Tm 89 Tz (Wea. Rev., ) Tj 1 0 0 1 192 331.899 Tm 103 Tz /OPExtFont2 11.5 Tf (98, 875-883, \(1970\). 111 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 103 Tz 3 Tr 1 0 0 1 111.599 300.95 Tm 99 Tz ([77]) Tj 1 0 0 1 133.699 300.7 Tm 112 Tz (T. Sauer, J.A. Yorke, M. Casdagli, Embedology, J. Stat. Phys. 65, 579, ) Tj 1 0 0 1 134.4 277.899 Tm 102 Tz (\(1991\). 79 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 102 Tz 3 Tr 1 0 0 1 111.349 246.7 Tm 99 Tz ([78]) Tj 1 0 0 1 133.699 246.5 Tm 111 Tz (D.S. Sivia, Data analysis - a Bayesian tutorial, Clarendon Press, Oxford ) Tj 1 0 0 1 133.449 223.45 Tm 104 Tz (University Press, \(1997\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 104 Tz 3 Tr 1 0 0 1 111.349 192.25 Tm 99 Tz ([79]) Tj 1 0 0 1 133.449 192 Tm 109 Tz (L. A. Smith. Identification and prediction of low-dimensional dynamics. ) Tj 1 0 0 1 133.699 169.45 Tm 92 Tz /OPExtFont4 11 Tf (Physica D, ) Tj 1 0 0 1 190.099 168.95 Tm 102 Tz /OPExtFont2 11.5 Tf (58:50-76, \(1992\). 15, 17, 131 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 102 Tz 3 Tr 1 0 0 1 111.099 138 Tm 99 Tz ([80]) Tj 1 0 0 1 133.199 137.75 Tm 111 Tz (L. A. Smith. Accountability and error in forecasts. In ) Tj 1 0 0 1 410.399 137.75 Tm 93 Tz /OPExtFont4 11 Tf (Proceedings of the ) Tj 1 0 0 1 133.9 114.7 Tm 95 Tz (1995 ECMWF Predictability Seminar, ) Tj 1 0 0 1 325.449 114.7 Tm 104 Tz /OPExtFont2 11.5 Tf (\(1996\). 45, 46, 92 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 104 Tz 3 Tr 1 0 0 1 294.25 49.899 Tm 91 Tz (174 ) Tj ET EMC endstream endobj 922 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 923 0 obj <> stream 0 ,,b3_-*->8iYkT':txqK:߄v r؍t_&uw./&1u,`z ^+C;pQX{B:ƦOU2 ک)\2U)h&KO(8jq=x~vgb=o+ZW4)!F Zxc 2P{y,a!eVt&I ?F?K[3+:?BfWS( rQ6jšgtXv);kC7ٯ|T,-J48I+>ɐ`*^Qr1W -6;,Н.>̺l:Oq;t20ʼ@OJU4Gp=Tǖ*B{?5TFVV'hͰiCMZ>Cy2N+SlXtir/S|$v ?vy/0)>ISFHu AfFSl43܊F?`)9uV9T˩^_LJ|Ӆi*pI-Bw"*SDPg? m  3@&e޾Y%81\1,t?ΝZ"H` ɕC`_z_ 4Q(eܠSJ`b1#7Z4j?y>Ӈîh#WQmHJT%~2R}/%??nzU+F4aN_f w"l,+Šv(I)~m5̿]I:QoPQo'jJr.^hYY*o%^*~Fm+,Tcwszd61Upv\&Ymzkr^sߜ+KO }AYe1'$Xop?hQMf{[ EXdFں=uSY.NR "â; ,N}%'@61^TOįjLn&GI% >ϯiC/ ,d#?DtVrs.( {Gj+uoS;S#Oiڱ18,R.QTmD1 ?p1 >vVԠ[>0%t}똮8vv T bLiV 7\8@Z `;AK^7 Cmcu'(- 'Jb4=7T]4A= dZ5\hֶ3uQj H)N=7,,aΙN_n|gu6&'KFBlWR=x39F"``LF^x)^&t.sυ-W1A!LpDL$bwvFz%!fvaqx ]*Ԅ0ڤj`izxwh0$Me4kUU eE //'?1v?1!.2_}_nT]ݏ@l|S9X⃏b@y\J<axIxLg?Kbj4&GK¸[X{ZsIjȌ}n`1ֱq2O[N/[סrqS4!:rCUΔe#L)Vx8 2L^S0 h;?-/,E% ^Xq)Ya}M@*@ _#`H&rZؘTS Y/UuFszwK2Igtx ~ԣn#:Vm^ǣV<`Ay]N:%5BLS:K >An^4.kc` ?v+ÃqqACҲK-mA^A@ 'F&0A,zPVx_(r\'"BXU/?8QD_tI9̜IpHգ-E<ߢ84u %B {O5mqH$a_6r)v2Uܧ>ŮO]iKO.NR c* &)BZDɿ4XaVõU?4BV49*>"ܳܭ۪Y"p.G:>+>$%+<{ӻI9́}Łh5] |lwEx KGh3q.m ?+wԮQgV=F8Ա8GSּXa 9DiGge@Pa'ę~ӕ%ܙS%Tϝ~ݴS O%!*\0lݽ<n.z}޾VSSܾ7a?I-+ "kQa_*laѺ`.9~XCh(NBPL!(@a2Y<]f]"&R@؛0x %%mW1 \ }ꍝf Ĉcu-s\T1Gx"~;V[׏ :[K4keğ{23ڴ9D 9E!q6"o@{W 8k&F 2qpBer#)ʛ\%"JO9l[O'든 ٗ5(JsνneiQb٣5Q% *YaBI+Q8XWb}{a1(C+Å'Irso|2ʗ}#&IzO +pZUɺJQ[6w{ߊº)r]v(/Qok%)IhM Tg'隿1%!MX.yU$ oF&E6FZT F3üvl>-2t@JAZ E4c@`s8Y_H DQ® &B A4 &]pIŵэPҏ?iA[菤靪HO0\B?;8R1<9)ۈU)2j ŘuX&d'y4-:Mc#y$~ /c*#D 28!Yw“x>z!ͼ _2[T=gH-lQ8"7ZoTpuYR3OiS Ȝ咢PgF fh$/}:z׮WDNv8|Is\ԫ\HRe*4 (ۖ o{4;vyA*Ky䈆ջgBeaSA;%hh?HPnvژ&g˦/๷^F4AlPM-T.[T,,d)DRbR8'ë5* {V6Ci;t.#T>V]YYT+|j:Qj7xD5SE6-Dž;U%RVA hp|sƓۻ% Ag1٘$Qz̮a޸^b;כ*=_Ra;*lz5S3E!̽;|_jBTu]2\k$9s1Co ] }|$I2D7nPs6F+-~x nx`/$KXoŐ9Cpڥ+ ̸=ڵHkC3'Bhss˪3` %b,}e cIi=:죔ͼ+>->Y;7<ʑ+cjK a窡bVG(jLd8W+\s/*WqP4t2VBc\CS)p/_b.#")< on2 V^5.iݥ6QXl~d$loQ"rt CI$V\htl'rm[6;f(6a&gw\4=aO?R :Yt@*ҥ jtI_gp9lQ'-8ܳkDn5>(N,k!@\77Xq4wc̐4uBс*=RVla; qAb/C |,L.Q˻Fo خn12LD!a21"IDZ\RgP>6aTk\%쐜N'nw8ɚL!FZxEzWDsM7d]3t?p,>bxFlpr/(v};UfVA*poFFxE࣡:Uk*ƶºy)Vu*^qLj4Y~iK$(̔B/',y^c5o_*|%`ۃ Z@4Ec@n2PtDZ:LaKGƌK6VghM-{'6YM? IrKŮxtMuC` i\f;Rҥ^:(V 23fk`Ѳ!_e{XT t>Eɘ&Ҧh,X/ _wxBj, A"@/ƉvdlqӠ1Rc#^`uVTx\vE\0+Sa*S L:d*,Q8mB&p ` ˖T~bd a"-_="̥*6R8sLWls \k%LbxTFt2N;l p.9Dp>h . 8E?-5~hp0 \f%}=1HӒՑAupt}u{!1#:h$@Df4uv^Ӆ?̩&7R|Rz !"mqnh}eہ#-JʋN>ue8 ֭Yn9xVjO39yЫ]k4c]a."DRjFbjmJMZ R*kLD"=&kb Iu[ >0ƍ-=5I%B];ij,2&šK/K,F3*<&үB3J8̨VT&}[)DMi°|ҁ=2ߪ3e+|n}51sDw?R $^Y>YrVq6څI2Tɩ uL[*jƧYEf׶ S?G@;0;xBsxnC | l_Q2{-Hm+ ^{B iO޿jjMYJor|A(=q{~!V}PE+1IUԲ&{!;t߼f .NtS7"' o%& ʪ1Z^ŜUg(4ز˲-ٴp30ώKEi쨳ʗ6>5!%ٹ,12":Ã+ܝJ(gb˳ZV2՝,*<r^|,Cr9c\d+,T}($NZD'B3-?/T+}xצ(ZpvHQc3VJD`"/?;&KF\qŦCi/PfK7W/}6cF`c>*m1%6rX"M{BE=wvo[X[oɬ2,6[e24bYmC_:͑Cj82v,$_ݬs7#Hv(hYmtHm(ƕ:@f4@) w!JK|0ђKG1DA75hr,Ojd],e#ҝ3 VvAxq0cU|9BL];磾,T Nw%}@q=CcP)A~<;{6ɄIכVM(zմ׊9z:H%> 18ڴ<@9- ɿZ>0ؙbx[-Y|YP)US$(#orҲrWYA:l!xs-%hnG@!~Tw݁JHoPb hE= oҢHLTg}({ raSq\d^}1;Ѓ'+ewcQCL v]TQqSqьHKE*Wt\PJiz{rxS=qyMn.J[6ek{Fo+c*.?5cGځ.D7“t/7\k]ىĘ/5V*72$Zcr!]w,DU' tZK&;\KHlGvgօCJTiP[Rle ^:t*] s7 "s.^M$r)~JF"ΣaYQ@gu-ja|N̬ބ[XVׇ}Ѕ[OD]zY3ׄ:[<-EÚ5N$Ҫe=_ȑ$t8?S3xIӻڤ2d Βzզ fD*N2p1Q7c5f]\?o7_K= %Fi32Di;t7nRsrqb /HݷO;x.E g{{E&&i:_b `~~k0j600: crS\I)fr:O?/)FO܍άrJJ?̃7CuH_N[UR6`C+͙}yUs5gGiLB|1> endobj 925 0 obj [926 0 R] endobj 926 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 590 0 0 834 0 0 cm /ImagePart_2202 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 396.949 714.5 Tm 112 Tz 3 Tr /OPExtFont5 12.5 Tf (BIBLIOGRAPHY ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 112 Tz 3 Tr 1 0 0 1 112.549 670.799 Tm 99 Tz ([81]) Tj 1 0 0 1 134.65 670.799 Tm 101 Tz (L.A. Smith. The Maintenance of Uncertainty, Proc International School of ) Tj 1 0 0 1 134.65 648 Tm 102 Tz (Physics "Enrico Fermi", Course CXXXIII, pg 177-246, Societa Italiana di ) Tj 1 0 0 1 134.65 624.95 Tm 103 Tz (Fisica, Bologna, Italy \(1997\). 15, 147, 148, 149 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 103 Tz 3 Tr 1 0 0 1 112.549 592.549 Tm 99 Tz ([82]) Tj 1 0 0 1 134.65 592.799 Tm 100 Tz (L. A. Smith, C. Ziehman, and K. Raedrich. Uncertainty in dymanamics and ) Tj 1 0 0 1 134.4 570 Tm 103 Tz (predictability in chaotic systems. ) Tj 1 0 0 1 303.1 570.25 Tm 98 Tz /OPExtFont4 11 Tf (Q. J. R. Meteorol. ) Tj 1 0 0 1 397.199 570.25 Tm /OPExtFont5 12.5 Tf (Soc., 125:2855C2886, ) Tj 1 0 0 1 135.349 546.95 Tm 101 Tz (\(1999\). 146, 148, 149, 151, 155 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 112.549 514.299 Tm 99 Tz ([83]) Tj 1 0 0 1 134.65 514.549 Tm 106 Tz (L.A. Smith, Disentangling Uncertainty and Error: On the Predictability ) Tj 1 0 0 1 134.4 491.5 Tm 103 Tz (of Nonlinear Systenis. Nonlinear Dynamics and Statistics, A.I.Mees, Ed., ) Tj 1 0 0 1 134.4 468.25 Tm 102 Tz (Birkhauser, 31-64 \(2000\). 13, 79 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 112.099 435.6 Tm 99 Tz ([84]) Tj 1 0 0 1 134.65 435.85 Tm 105 Tz (L.A. Smith and M.S. Roulston, Evaluating Probabilistic Forecasts Using ) Tj 1 0 0 1 134.65 412.55 Tm 98 Tz (Information Theory. ) Tj 1 0 0 1 239.75 412.8 Tm 94 Tz /OPExtFont4 11 Tf (Monthly Weather Review ) Tj 1 0 0 1 369.1 413.05 Tm 104 Tz /OPExtFont5 12.5 Tf (130, no. 6, pp. 1653-1660, ) Tj 1 0 0 1 135.349 389.3 Tm 95 Tz (\(2001\). ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 95 Tz 3 Tr 1 0 0 1 112.299 356.899 Tm 99 Tz ([85]) Tj 1 0 0 1 134.4 356.899 Tm 107 Tz (L.A. Smith, M.C. Cuellar, H. Du, and K. Judd, Identifying dynamically ) Tj 1 0 0 1 134.4 334.1 Tm 104 Tz (coherent behaviours: A geometrical approach to parameter estimation in ) Tj 1 0 0 1 134.4 310.799 Tm 99 Tz (nonlinear models, ) Tj 1 0 0 1 225.349 311.049 Tm 95 Tz /OPExtFont4 11 Tf (Physics Letter A, ) Tj 1 0 0 1 313.699 311.049 Tm 102 Tz /OPExtFont5 12.5 Tf (under review \(2009\). 78 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 112.099 278.149 Tm 99 Tz ([86]) Tj 1 0 0 1 134.4 278.399 Tm 108 Tz (H. Sorenson, Kalman Filtering: Theory and Application. IEEE Press, ) Tj 1 0 0 1 135.099 255.1 Tm 102 Tz (\(1985\). 37 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 102 Tz 3 Tr 1 0 0 1 112.099 222.5 Tm 99 Tz ([87]) Tj 1 0 0 1 134.4 222.7 Tm 102 Tz (D.A.Stainforth, et al. 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Weather Rev., ) Tj 1 0 0 1 341.75 436.55 Tm 88 Tz /OPExtFont3 11 Tf (130, ) Tj 1 0 0 1 368.399 436.55 Tm 100 Tz /OPExtFont5 12.5 Tf (1913-24, \(2002\). 43 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 112.299 403.899 Tm 99 Tz ([93]) Tj 1 0 0 1 134.65 403.899 Tm 102 Tz (D.S. Wilks, Comparison of ensemble-MOS methods in the Lorenz'96 set-) Tj 1 0 0 1 134.15 381.1 Tm 103 Tz (ting. ) Tj 1 0 0 1 161.05 381.1 Tm 92 Tz /OPExtFont4 11 Tf (Meteorological Applications, ) Tj 1 0 0 1 301.899 380.899 Tm 101 Tz /OPExtFont5 12.5 Tf (13, \(2006\). 71 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 101 Tz 3 Tr 1 0 0 1 112.299 348.5 Tm 99 Tz ([94]) Tj 1 0 0 1 134.9 348.25 Tm 104 Tz (C. Ziehmann, L.A. Smith and J. Kurths, Localized Lyapunov Exponents ) Tj 1 0 0 1 134.4 325.449 Tm 101 Tz (and the Prediction of Predictability Phys. Lett. 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9/=>B*M( C]I XWnR4n9)fyX?~> ^O]ƍXo8 *4P` PfOflWP4G`Y%sOL$ lM8w`}N=|8LM I*e e)e91TD" N'Qb/KҔ&ln `?TZ{-iʧ k=YF'먩^lǞ^ ѼijYgXm6 T,@xBq:w܊Xn1u@^h ]-짣: ]QgO, ؔA/K/FE A` t m$DY2Ɋ:TB/`!LYqU_^˜xePr(7R_/7Pknd )*W wl6> endobj 940 0 obj [941 0 R] endobj 941 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 587 0 0 834 0 0 cm /ImagePart_2205 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 395.5 715.2 Tm 120 Tz 3 Tr /OPExtFont3 11 Tf (BIBLIOGRAPHY ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 120 Tz 3 Tr 1 0 0 1 105.349 671.75 Tm 94 Tz (a ) Tj 1 0 0 1 111.349 671.75 Tm 759 Tz (\t) Tj 1 0 0 1 138 671.75 Tm 90 Tz (model parameter ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 105.349 639.6 Tm 100 Tz (B ) Tj 1 0 0 1 113.5 639.6 Tm 698 Tz (\t) Tj 1 0 0 1 138 639.35 Tm 90 Tz (covariance matrix of xb ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 105.099 607.2 Tm 92 Tz (e ) Tj 1 0 0 1 110.4 607.2 Tm 786 Tz (\t) Tj 1 0 0 1 138 607.2 Tm 89 Tz (mismatch error ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 105.099 574.299 Tm 93 Tz (u) Tj 1 0 0 1 112.099 574.299 Tm 72 Tz (i ) Tj 1 0 0 1 114.5 574.299 Tm 669 Tz (\t) Tj 1 0 0 1 138 574.549 Tm 90 Tz (component of a pseudo-orbit ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 104.9 541.899 Tm 112 Tz (x ) Tj 1 0 0 1 111.849 541.899 Tm 745 Tz (\t) Tj 1 0 0 1 138 541.899 Tm 91 Tz (model state ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 104.9 509.3 Tm 86 Tz (xa ) Tj 1 0 0 1 115.7 509.3 Tm 628 Tz (\t) Tj 1 0 0 1 137.75 509.5 Tm 91 Tz (the posteriori estimate of system state ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 104.9 477.1 Tm 77 Tz (xb ) Tj 1 0 0 1 114.95 477.1 Tm 649 Tz (\t) Tj 1 0 0 1 137.75 477.1 Tm 91 Tz (first guess or background state of the model ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 104.9 444 Tm 108 Tz (y ) Tj 1 0 0 1 111.349 444 Tm 752 Tz (\t) Tj 1 0 0 1 137.75 444.25 Tm 92 Tz (pseudo-orbit obtained by ISGD method ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 105.099 411.6 Tm 91 Tz (z ) Tj 1 0 0 1 109.9 411.6 Tm 786 Tz (\t) Tj 1 0 0 1 137.5 411.6 Tm 92 Tz (reference trajectory ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 92 Tz 3 Tr 1 0 0 1 105.099 378.699 Tm 94 Tz (a ) Tj 1 0 0 1 111.099 378.699 Tm 752 Tz (\t) Tj 1 0 0 1 137.5 378.949 Tm 90 Tz (system parameter ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 137.5 346.3 Tm (system state ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 105.099 313.7 Tm 112 Tz (F ) Tj 1 0 0 1 113.049 313.7 Tm 691 Tz (\t) Tj 1 0 0 1 137.3 313.899 Tm 89 Tz (system dynamics ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 137.5 281.049 Tm 90 Tz (dimension of the system parameter space ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 104.9 248.399 Tm 78 Tz /OPExtFont4 9 Tf (rrz ) Tj 1 0 0 1 114.25 248.649 Tm 882 Tz (\t) Tj 1 0 0 1 138 248.649 Tm 90 Tz /OPExtFont3 11 Tf (dimension of the system space ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 104.9 216 Tm 119 Tz (F ) Tj 1 0 0 1 113.299 216 Tm 689 Tz (\t) Tj 1 0 0 1 137.5 216 Tm 90 Tz (model dynamics ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 104.9 183.35 Tm 114 Tz /OPExtFont4 9 Tf (h\(\) ) Tj 1 0 0 1 137.75 183.35 Tm 90 Tz /OPExtFont3 11 Tf (observation operator ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 104.9 150.5 Tm 121 Tz (K) Tj 1 0 0 1 114.25 150.5 Tm 57 Tz (t ) Tj 1 0 0 1 116.65 150.7 Tm 116 Tz ( Kalman gain ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 116 Tz 3 Tr 1 0 0 1 137.75 118.1 Tm 91 Tz (dimension of the model parameter space ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 294.699 50.649 Tm 75 Tz (178 ) Tj ET EMC endstream endobj 942 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 943 0 obj <> stream 0 ,, }WnK`֬G Ac4@߾'徬6pCDu@۴mV:IξY٨"跀'ᎃ20oq6i Œ&f;Rh2 -0˖7v Ԛ`$/UAˍ Ch9jWR(.MpA ΑH>ß'Jn15ԙԿ(RcrUKצmۃAy0jyB,J'b蹝CPa{+470zH%q>H[IoQ3{7 ˑ;_hO4'4@%ZdMA@sOb)ɠDZ4 B&sZ4Ҳ@?\.q#bLbe1i$k^1Aݾguwr16}WXw17? O~qրI-Fi#PKtW ŎPb[$ !'84ޖ۫G|-Sq1GъRlh__OQF>iImѮW 5@sAr5"0%rH;k#nQ;'85A-4 g7 ƙX)"a'`ΥO htuڍC~~{\(Nc1@A(<hJp9wy+FؖYn;(o Z8fRH5@1b>21}}_f! ًEןv;dO^\GU =p ̆`;(˵'z6@7FO8GQtx{7XLnD3%.,M˥#3p{rPQ] N% K A8aOI ZM("pZPnb1"Kƈ"Ց_`WB5~j5 X'IS/ٍXЫ%Yh^ FY*`OD-*]$`T׉նlDj(*QeUWbх9cAްiӮ~x0S8|*yv/g~lKq21)\FyZO\A l۱ 4vȚV">K,w9V ;PG#NbOΛ%e٦ʄT8]3RE-S"ZKu["]|' u.swG(9Sb}h%_.lOKige_R OȏWwQl'l4Ƥn4"`aL5Ҝsl)  T%O˫dľ,:L1Ǵc1 wpP& +o"\34g|1B$fzhkKxz9uJdE(3۸l?R􄘓uݟA5on?Ӷ3P$yxldg$|:ayŗIĥw=Pbo=xB-2~XPYz$1&o(M_,@S`#iv[׺scxOEQhnXJP, JuZ+_vV%멽1=cX[P21h 8d$kYBq\83 e/2eϤC,64DZұ>-,搻ù1׶9D#55U|6Sͳ@82ƾ̱HѰ=x9EyyGIeq@اP2_fo7YRYZe(n83-2S!JoA5B:N̡BoL6y6";ŞYAqӋ&"1D{l"\qYk琜Wba^ < ? 'Vi}@94ojVi^8R%G'v(a_bYTw*X;Bn~1ң h7Q_F,&SfKP%I=4 (:jmŭv?MjJ`HXNjH-@Q^B$pDS_RphT{+ !ͧrKLЬv}^A{|t5 V V_}s^F$B p"SaN4u1FbC-C8&_  ݈-؇\sc4綟/̴q%`{mJ{"'O*TՏm!,u/uΣ  E ;)Ymh!f tXShy&*V+ Pr("-(,s(S|`𼯖 LKG=A dQו;/sPl#ac{S%~2oEsadv"Ih+-fB BY Yg`NcOđ F=xb)'5046n= my:{O endstream endobj 944 0 obj <> endobj 945 0 obj [946 0 R] endobj 946 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 593 0 0 834 0 0 cm /ImagePart_2206 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 396.5 715.2 Tm 112 Tz 3 Tr /OPExtFont5 12.5 Tf (BIBLIOGRAPHY ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 112 Tz 3 Tr 1 0 0 1 105.099 672 Tm 99 Tz (m ) Tj 1 0 0 1 114.7 672 Tm 745 Tz (\t) Tj 1 0 0 1 138 671.75 Tm 90 Tz /OPExtFont3 11 Tf (dimension of the model space ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 104.9 640.1 Tm 101 Tz /OPExtFont5 12.5 Tf (n ) Tj 1 0 0 1 111.349 640.1 Tm 852 Tz (\t) Tj 1 0 0 1 138 639.35 Tm 89 Tz /OPExtFont3 11 Tf (number of observations ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 89 Tz 3 Tr 1 0 0 1 105.349 606.95 Tm 93 Tz (N"nd number of candidate trajectories ) Tj 1 0 0 1 105.349 574.799 Tm 115 Tz /OPExtFont4 8 Tf (Neils ) Tj 1 0 0 1 138 574.299 Tm 88 Tz /OPExtFont3 11 Tf (number of ensemble members ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 105.099 541.45 Tm 125 Tz /OPExtFont4 8 Tf (pa ) Tj 1 0 0 1 117.349 541.45 Tm 863 Tz (\t) Tj 1 0 0 1 138 541.7 Tm 88 Tz /OPExtFont3 11 Tf (analysis-error covariance ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 104.9 509.75 Tm 87 Tz (Pb ) Tj 1 0 0 1 116.9 509.3 Tm 594 Tz (\t) Tj 1 0 0 1 137.75 509.3 Tm 88 Tz (background-error covariance ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 88 Tz 3 Tr 1 0 0 1 104.9 476.899 Tm 82 Tz /OPExtFont5 12.5 Tf (Q ) Tj 1 0 0 1 112.799 476.899 Tm 790 Tz (\t) Tj 1 0 0 1 137.5 477.1 Tm 90 Tz /OPExtFont3 11 Tf (the density function measures the indistinguishability ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 90 Tz 3 Tr 1 0 0 1 104.4 444.5 Tm 105 Tz /OPExtFont5 12.5 Tf (Y ) Tj 1 0 0 1 113.049 444.5 Tm 776 Tz (\t) Tj 1 0 0 1 137.3 444.25 Tm 91 Tz /OPExtFont3 11 Tf (verification ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 91 Tz 3 Tr 1 0 0 1 293.5 50.649 Tm 75 Tz (179 ) Tj ET EMC endstream endobj 947 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 948 0 obj <> stream 0 ,, [@_[ņ͹oN5TFߺr,xx9Ih[.p_L;,RP:NIb0d-)nz<#@5FNi*}@8@^*tmh$i?==Gb>EZ_YrGܡ>rHjL:CPԴC v0GXXS<J\c0( 2N* &{*٥0w+(F?h~+Ghսuw;z߫۸LҳڰFkitH*&!'QRK ()kWs. R+k?e.UZy$ުhʢŻjP*y-3*b\ItbcQ\ Cu_<n P;/)Oƭ[qL0|XTFklvU`ǶkaCYK=DHU%_C[e K{Dwkd"zs".ϦGo(psmv*"C';u0?5hj^S*JMD>EK{%lo-pven(~`/ +h.wA|gqF[.5 F|8`ͫvT]><;&8[V_QWiZ:v޿&~`؛Pu)Y8[u1&NQx&9I$F'@z*$%BȋԠƹѨMKЬ4i'>dM'>[#u8\FÜT pz_?]Q.0fYyCh/e(w#'hő0hqLjS4f9>͞3;)C$A BRBtq]` DZV4KAf/GzSq%j|6NOgV|` ݛ{h"'qmL:|ϚOWUCxOY)+9ʜUi6N2F>L?k ?/;N;s=ԟýMcj6!cV_6{R^r|oE'~ `k7Uj& cS{׺"/e;()gض/䏵 ɹHC1Xl܋BWS{%4UտZ`7/x;uPRӨ d<*!hm@Ic9Gfa}U绎. o F:m3Gk0P8.}uGYI^>6<gA]Y)~-T0-XQh˗'<8i&IT KBzbj66+ ^rhL:CPvj.eH4W?j~zʐ9 #ޙԣ*tf:\hFASeYM\+ѾMQY7myΰ` 4䝞e`Zl0vW쏁~Ldb9~$gSH/6VLCeq$on{u2l8q, \”c')n.3XL3]KZyLw֏_o;wDy_Eu^G_Q[0Fj=fKQ Y)j[ i(Ԝ%݅(#&VZؘ0>zc&9.+j ,w׿qp쓌CSP)B{{>v8Z彝1n%0$/6gMmhF -l@51}qT=;+-/`#"S(x5zn)#w1ݳf,Me߮0b`ǎOPD ,}R0Ixm'׻U*\MTŶC-Zf|wr]Q^a]ni8 endstream endobj 949 0 obj <> endobj 950 0 obj [951 0 R] endobj 951 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 593 0 0 834 0 0 cm /ImagePart_2207 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 105.849 576 Tm 121 Tz 3 Tr /OPExtFont2 23.5 Tf (List of Figures ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 23.5 Tf 121 Tz 3 Tr 1 0 0 1 122.4 505.199 Tm 95 Tz /OPExtFont3 11 Tf (2.1 The bifurcation diagram of logistic map ) Tj 1 0 0 1 473.5 505.449 Tm 602 Tz (\t) Tj 1 0 0 1 494.649 505.899 Tm 66 Tz (9 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 66 Tz 3 Tr 1 0 0 1 122.4 482.399 Tm 97 Tz (2.2 The attractor of Henon Map ) Tj 1 0 0 1 295.199 482.399 Tm 2000 Tz (\t) Tj 1 0 0 1 473.5 482.899 Tm 151 Tz ( 10 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 151 Tz 3 Tr 1 0 0 1 122.4 459.35 Tm 98 Tz (2.3 The attractor of Ikeda Map ) Tj 1 0 0 1 295.199 459.6 Tm 2000 Tz (\t) Tj 1 0 0 1 473.5 459.6 Tm 148 Tz ( 11 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 148 Tz 3 Tr 1 0 0 1 122.4 436.3 Tm 96 Tz (2.4 The attractor of Moore-Spiegel system for ) Tj 1 0 0 1 359.05 436.55 Tm 99 Tz /OPExtFont6 13 Tf (T = ) Tj 1 0 0 1 382.55 436.55 Tm 86 Tz /OPExtFont3 11 Tf (36 and ) Tj 1 0 0 1 419.3 436.55 Tm 97 Tz /OPExtFont6 13 Tf (R = ) Tj 1 0 0 1 443.75 436.55 Tm 74 Tz /OPExtFont3 11 Tf (100. ) Tj 1 0 0 1 472.3 436.55 Tm 34 Tz (\t) Tj 1 0 0 1 473.5 436.8 Tm 149 Tz ( 13 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 149 Tz 3 Tr 1 0 0 1 122.15 404.149 Tm 95 Tz (3.1 Following Judd and Smith \(2001\), Suppose x) Tj 1 0 0 1 371.75 404.149 Tm 90 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 374.649 404.149 Tm 95 Tz /OPExtFont3 11 Tf ( is the true state of ) Tj 1 0 0 1 147.849 380.899 Tm (the system and y) Tj 1 0 0 1 235.699 380.399 Tm 74 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 238.099 380.899 Tm 95 Tz /OPExtFont3 11 Tf ( some other state where x) Tj 1 0 0 1 370.8 380.899 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 373.449 380.899 Tm 104 Tz /OPExtFont3 11 Tf ( and y) Tj 1 0 0 1 408.5 380.149 Tm 81 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 411.1 380.149 Tm 111 Tz /OPExtFont3 11 Tf ( E R) Tj 1 0 0 1 437.3 380.649 Tm 49 Tz (2) Tj 1 0 0 1 442.3 380.899 Tm 107 Tz (. The ) Tj 1 0 0 1 147.849 357.6 Tm 93 Tz (circles centred on x) Tj 1 0 0 1 245.5 357.85 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 248.15 358.1 Tm 102 Tz /OPExtFont3 11 Tf ( and y) Tj 1 0 0 1 282.699 358.1 Tm 74 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 285.1 358.1 Tm 91 Tz /OPExtFont3 11 Tf ( represent the bounded measurement ) Tj 1 0 0 1 147.849 335.05 Tm 93 Tz (error. When an observation falls in the overlap of the two circles ) Tj 1 0 0 1 148.8 311.75 Tm 98 Tz (\(e.g., at a\), then the states x) Tj 1 0 0 1 295.699 311.75 Tm 81 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 298.3 312 Tm 99 Tz /OPExtFont3 11 Tf ( and y) Tj 1 0 0 1 332.149 312 Tm 74 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 334.55 312 Tm 91 Tz /OPExtFont3 11 Tf ( are indistinguishable given ) Tj 1 0 0 1 147.849 288.7 Tm (this single observation. If the observation falls in the region about ) Tj 1 0 0 1 147.599 265.45 Tm 101 Tz (x) Tj 1 0 0 1 154.3 265.45 Tm 74 Tz /OPExtFont5 11 Tf (t) Tj 1 0 0 1 158.65 265.7 Tm 90 Tz /OPExtFont3 11 Tf (, but outside the overlap region \(e.g., at 13\), then on the basis of ) Tj 1 0 0 1 147.599 242.899 Tm 91 Tz (this observation one can reject y) Tj 1 0 0 1 307.699 242.899 Tm 74 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 310.1 242.899 Tm 93 Tz /OPExtFont3 11 Tf ( being the true state, i.e., x) Tj 1 0 0 1 446.149 242.899 Tm 82 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 448.8 242.899 Tm 92 Tz /OPExtFont3 11 Tf ( and ) Tj 1 0 0 1 147.349 219.6 Tm 109 Tz (y) Tj 1 0 0 1 154.099 219.6 Tm 81 Tz /OPExtFont5 11 Tf (t ) Tj 1 0 0 1 156.699 219.85 Tm 135 Tz /OPExtFont3 11 Tf ( are distinguishable given the observation. 22 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 135 Tz 3 Tr 1 0 0 1 121.9 196.799 Tm 93 Tz (3.2 Schematic flowchart of the IS nowcasting algorithm ) Tj 1 0 0 1 410.399 196.799 Tm 1798 Tz (\t) Tj 1 0 0 1 473.5 196.799 Tm 28 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 28 Tz 3 Tr 1 0 0 1 294.699 50.399 Tm 75 Tz (180 ) Tj ET EMC endstream endobj 952 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 953 0 obj <> stream 0 ,,[b7!ЍE:CQk6IKByBZfo{g@j\~D?(OCtvinja3;\nVqY-5lLJtk4YFiկ2 S&;yd7/Lj-LkBv cP} c-[{ Pbl﹖/M,Pe͕T۴<9LZϬ挆ʢ8l ieo͙BMq d3 85Ѥ.3sɵ*O/e$Ѽcڲ0$DeT3nc2)&"d ̼3D4t,˜h!@&*v &b-Edm\^#:09\.<דj3j?ϏW˨?%rmR!*C+z J+ƽO%/,!U ҧ ,U!wS f".ܰ.a!˽>W=gKb̞ _ݮ̀Y?,:\i_/Ei Oh2P6ͦJNk鹪^t6/s<%%@44YTīܠ xٚcz72XT}lACuZt%hyM6*|5-Bf85Z©=MRwap^dGجT45.9ن+LuوY%;ta@:sa MU !{6Ijx hMQؿs;iӵ7׊MՍ{E+QUDYM\G -cin$ 6X]ѣ62PieEq8t#{91uT7>0d3ڸ>2fB}77>ɓHX &uԫM1=b۞((Z~\#oIGq9%+OZ<[d1oxQ:Upo(Jϲ^h$8?m@8N[w>D|5[},ع@b$ 7@ և@1y [y-B̘.)bff`|oHasl8:}zk-&T!ЦX>m0+2"~PŹLpINٝ@$|{x`VxJI׭(ޖH4A(&S܁-"ͧS ,"JdA/P{C%AyկĒs4n/F35U@ FW3:O"/,z&='_1Wl:xAuy 2TOXrv˸^!=X804'<ŷGy&פq6rMʍծ]pqWmTEbK]Sfݳ1.NlCH8\cT'UhY9SsaC ??*N_rTH W\G"3*;uTl r"!)|Vucp>m#4?E'Q8rEž#uoEQ=+M%F SevǕN4'j$OA\wHnnXtyӍys@ SiyCeB/,&P}6B4q:' [c%=Ee `mm -Y}ƭ./g<>XT`k +߶CX'ŷ@Bx2+C= T,5{v`'x I+"t G݉z%83i5S NuVwTwpVμnШ*ٛuBB[Im47ٌkCEP :H$V)8]05l10F,DŠ.o K{ /L&:0Dj.~vHt^NTѽo:@߻"~aƠ" M䮸C*qy*7znf}(@ u0CeMQ/\ oeў8tX]2=P:s2U7%b{L~wڳCY/౏4Ms(CzsRlEz G2U*<}֝/9_Im؍R[A^y)ŪOihC MԯtI}GQ%΢`\$%yp <gj'"6;"ic߀u'js) `R`b65]- NΜŚ%x"Pwql ̦WWx!6Rt8.BCz٨-ٙz=fT f8D&c0F'u=]yRC֟K*i\a8~DE'0qCCn@]zCڅE [HtMc=>PG/%n#A|쩚r]}Ⱥ @e:J|!G:,SL K$Pg|d!}= 3?݅f@SLƢhʀȕ$D5g  zF./oT%,E_1g-E`$W8(gR_ZzS8iuW_zf}j>¢?T$1߸ |%t+yiRQ{߇̱wO'nf$%"= 7S J9E^*.y rCYR٪D_ws `N{܈`֖IU4J1+dMJhr {v:iUJaSԻ~s#RLM`D% ̱7D}V lA;Q"mwZ~vzsi#[a&mUqy~3PrNp3G}v]? y*5 ;m+I/=fd o(qR𚩪?  -#z|+ghP-$}s*Dj:rIm:x6+ŧDS3EcxegJGk5__C{6o5STNjfv 1zIFTn'ioc.|ݼjbQ˨z;2kO5Zw[_jH>Ke+>fNR  -u`~esԍXNj" \Ԯ#{St*5qbFQmaƽg-@dԺrZ(`Nݨ_8Mat/v0wb,"ǎI>FR5Ut)19Pz,#ݭ\EwRd`M:VqALMEf/AU}z5 rxb(9IKKo495( s2I)`gH u6洁n}v'lV$:pjX/aEHmbwCJo-t5Xmݨil&zAM(GرZŤ{) 5DhHc`cR&pɣUHfI5'ߦ[wv{"dȏ'j8nsjžGp:eP9T񶍖:6$+Jր!HT?{̺+2%Bsx&(fJ]#锕=}+۾5zI\̱#*8^.׵.Q@3|O0#٪r+clR#'YBD'S=>>~}?k4l>-Q7,5u,RROywbFZX|]@lEb*@KЮY`5 po?, `GґZS:)4J|׻ +W$ D2D*ƅp,8N( KG7gPzj;TZȫ rvuc-c?\5[ -9;ߤ_9ZH"q*L*,*!Pey$ 澯t> ;@"`Fҹ-E7L}(P݂0 ;6afQpg$2dN%:xdڙ'ITEvpk] $ƀ\3l ݗ[$zn̸Uhnq Wa-V})<@ww7h4L4e$e1Oa=oj3c9ZH13UК\;JHpn.yG=^|q>nefv[KRG>QTP"6>=!5{-[!$^=%Zj$ l}R-D.W7L-!9*^d`FLKzBFނ՘{ݚ+%+*l?]`q##FD{qxAu/轫Iɓ 1l#PW 2(G\h/ 5_7a(T(9F!{e [A:d5*&JHu̪4uVqL 1W}$N0Ä$jX}pr12pRo \|@,kg:-[0h;!<1Dķ9ն VhNjN"rE.qvqt i@uy@ O<7V^Q~Yg'bd:#!?y1-Xl+#HHAG꼘̍W%9>(_c>Z|XfެPIorH5F>8 cW59#P{Q2 _Z endstream endobj 954 0 obj <> endobj 955 0 obj [956 0 R] endobj 956 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 592 0 0 833 0 0 cm /ImagePart_2208 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 386.149 714.45 Tm 112 Tz 3 Tr /OPExtFont5 12.5 Tf (LIST OF FIGURES ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 112 Tz 3 Tr 1 0 0 1 122.15 671.7 Tm 107 Tz (3.3 Example of perfect ensemble for the Ikeda Map when only one ) Tj 1 0 0 1 148.099 648.7 Tm 105 Tz (observation is considered. The observational noise is uniformly ) Tj 1 0 0 1 148.099 625.649 Tm 108 Tz (bounded. In panel a, the black dots indicate samples from the ) Tj 1 0 0 1 148.099 602.6 Tm 106 Tz (Ikeda Map attractor, the blue circle denotes the bounded noise ) Tj 1 0 0 1 147.599 579.799 Tm 98 Tz (region where the single observation is the centre of the circle. Panel ) Tj 1 0 0 1 147.849 556.75 Tm 102 Tz (b is the zoom-in plot of the bounded noise region. The red cross ) Tj 1 0 0 1 147.849 533.7 Tm (denotes the true state of the system ) Tj 1 0 0 1 488.149 533.95 Tm 95 Tz /OPExtFont2 11 Tf (46 ) Tj 1 0 0 1 498.699 534.2 Tm 32 Tz /OPExtFont5 12.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 121.9 510.699 Tm 103 Tz (3.4 Following Figure 3.3, examples of perfect ensemble are shown for ) Tj 1 0 0 1 147.599 487.649 Tm 99 Tz (the Ikeda Map when more than one observation is considered. The ) Tj 1 0 0 1 147.599 464.85 Tm 101 Tz (perfect ensemble of different number observations are considered ) Tj 1 0 0 1 147.849 441.8 Tm 104 Tz (are plotted separately. Two observations are considered in panel ) Tj 1 0 0 1 148.3 419 Tm 111 Tz (\(a\), 4 in panel \(b\), 6 in panel \(c\) and 8 in panel \(d\). In all the ) Tj 1 0 0 1 147.349 396.199 Tm 100 Tz (panels, the green dots are indicates the members of perfect ensemble. 47 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 3 Tr 1 0 0 1 121.9 373.149 Tm (3.5 Ensemble results from both EnKF and ISIS for the Ikeda Map \(Ex-) Tj 1 0 0 1 147.099 349.899 Tm 102 Tz (periment C\). The true state of the system is centred in the picture ) Tj 1 0 0 1 147.099 326.85 Tm (located by the cross; the square is the corresponding observation; ) Tj 1 0 0 1 146.9 303.799 Tm (the background dots indicate samples from the Ikeda Map attrac-) Tj 1 0 0 1 146.9 281 Tm 101 Tz (tor. The EnKF ensemble is depicted by 512 purple dots. Since the ) Tj 1 0 0 1 147.099 257.95 Tm 100 Tz (EnKF ensemble members are equally weighted, the same colour is ) Tj 1 0 0 1 147.099 234.899 Tm 104 Tz (given. The ISIS ensemble is depicted by 512 coloured dots. The ) Tj 1 0 0 1 147.099 211.649 Tm 102 Tz (colouring indicates their relative likelihood weights. Each panel is ) Tj 1 0 0 1 147.099 188.6 Tm 99 Tz (an example of one nowcast. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 99 Tz 3 Tr 1 0 0 1 293.75 50.1 Tm 86 Tz (181 ) Tj ET EMC endstream endobj 957 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 958 0 obj <> stream 0 ,,   j0l؄lv7x1(LުjNB(25)n83> Q2D! O~{o XZiv:.\4py< [rEMQLµ'K8tJ&Qz*zwp1K>x2_Yxb3v=j;"L%J: a0#dHkh$[Lpt8s)0ƌpױ9ߧTig"% EEg;!D6}fǟf.Evepe@:1M+w%%2}v)wSCg)Ѐ~e"$WUzmCG1eBG*"a[fc) ܬǐ~6.v|ds+!D0_.Mr};Q%M I(ٟEt k`)+.ݙ*\E7`8wM'=mA$q涄q:N˸B]"ߝOҹq;gd:z{SX zN5zqu=*/,/*cWi슋b.Gr~ 1F[~N|n)e] YC'*b][Pi;nl쵰UQw' gy_OFJDӚFCcCj7 9Gz|vEk_ѝ(<;ʁJt,EsfK%Flv (1Ÿ^KكwՑ#/QX,BJy@ûxyô?h؎#ϰ qQ54`Riac LJb5|sVln,NۡsD,=vwU &=PO*89L ^idzHpkPx'AKȧgI:P FSNtuQSPi\`E}ѕa<8m( zŘ.=u.wy,  ˼Ι% '7ʭ(L 7ˡ}tu<# }~i]ƅjGNV" IhUq˭k-d皚DXHKRPR*N9H۬Yv^+Tv:tYTsQT#}RǨDjNf3]B$2LڍV >Irk񯄱87wQj+Ꜳ;luI?V Ū W- ^ƿLd&A@ˀs`$;=#Gyn?Ujcल;H%̕'WeBs~nXCRX`ẸûEq? t/tb yǫ*!Ů[b6sH{Sgӿy$:"OLi KMhf𫅤6$ yK |j:j=&b5˴Eʹ 5}M,j^4<>糧珴ڸ5:鉡l"c.s拻խ]Vz82,"LkYq.َiX@OD X}=".īEY9<+ss_ Q 眽2  q_1AuB2i:K~3z+D[#ڞch05Zw+*nɖ(,:= N6 |0|ڠpεk 3Ns:(H"RSj::,)bck/s] gs\A@iP|ELɚd Hac m/VԻΰ*ltD/zo č܌ѹ1!p`g.)7))nߟ -9Wev6c^Wpf_2p`x󠻹OKmROVWo[jD#z|A1ɓx[M@8K*# He-"볃U_t^ehJZ褐@r :[+51` xܩxg!W[ 9~ڠ,](b$g岱s2@ JiJdSfǑ1=E]}kGMj(JP6A?P*:j'xz] nG8Byy}uS\$owtܵ Qތ"*>D]bnld.oA-8{+ f>A+1" nD񧄖k Ԁn X1"~$!0` d@&#IJ3Ir$YeO[$Y UI?N5D)SW;۫kؑ[;tѭ"%\h_HhVO-W^=_6vJ_G!i` ƞBO\l!w\$hPQVK1$nJE.LЃKi@2h"Qˠႀo'I iW?Z%̹l9"dq=S 5%5]S6 An 2aLrp҅r)zYNu[zZݎ=БԌ }eoÙ*XYM|ϳ5BC+U蛆?@Μ'c &}i*oU4TMi{ n-c|v`8[y1{!^bQ77&Gr+bLc<<}< rOe8 y概wRhs64|R!mu F)(]13G6i@idrHuq(f a;'IhfK0aG遃R=4qM9fjB,˒’'#[_ssJB:HX^R,IJϹ(w$%fm>n&ظ߅B},}h]*& A(-t+T&Ҥݳ}e kDU1a!Oo:ji?ezb6{sYqK_n lLG^O8$Si  l k-yOY%QJ:rEiϺĸl)T9Wo.} woAR fQ/K1AuZI2EMn-e@f/ֲvS58 &!}?X'\_rtͲVdcލ nqa @|WXϩϲQ\*I .V-L[F:5W!CQ4;q#hrK xf?TGpJhmpyt~sRaHn%KQG9y֏8tgIl`rY|}(K5n4VCE |EڻO!bV9̭AQb-BO>BUΡK hXU0tUd_JC=i1%qbݹ^= 泇0% 'h1%).)&85U_8xX= ɭV("'AU﭂:8ǭ<ڊX<'2HE6UM, Arc*G&3i3źT2/\ֳ""yMp0w >cJ,jܸbs%^2!踃Xi 揰zhaD[` fP:r1=>ޱhphؑtm| {Ȗy*&ǺB&j'7BқRRGWH „6,|y50D.N]XA5Ɏ."d'p8If֏&#X+Ῡ6D*aȝw|Jݳ{@=SM(pl_5 ֏깐=$QU4}v32OdW `g$ycR=,Bnz ]Y̞LX;}̆P=EsxϻR3Q bvb;qkY%5B/F3C0]`g1MTA~ut,<}Evצap :(r6Y )E-{*a0RC δpŹķŐxeuP/#-i$k?3; Ѹ#y6>vXWEFtm;*i̛rN Q.M~]^mȤpO M#j\Dw:MF͝H_na-ʁm]a$S`i_9_/O?%ik$bıY_~Ҭ7ˊb j7,EdTgSYE׼[5`bwF LRPxgO "[!ɔVAo;~-3@} ׄh^f/=Oر)wo ՕxZB!JR]=Rx;(hS|s{2:bT1{.L0H…< S7gryYtgmAR5i"5D[ueczW7IB xA{7i7!r)qZBu[BEp̯ӔY'w~>?՛A寔<2a!${@:ٮWjłfyΗ[#`Fܙ?5/ewm.VpZ#Pn Fod^_Y^ݵey1Nɓ3#h;)[{fĜTF@+{QL\#a_~.9?-$ _Qz+y\&tL endstream endobj 959 0 obj <> endobj 960 0 obj [961 0 R] endobj 961 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 592 0 0 833 0 0 cm /ImagePart_2209 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 385.699 713.95 Tm 112 Tz 3 Tr /OPExtFont5 12.5 Tf (LIST OF FIGURES ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 112 Tz 3 Tr 1 0 0 1 121.7 671 Tm 100 Tz (3.6 Compare the EnKF and ISIS results via &ball, the blue line denotes ) Tj 1 0 0 1 147.349 648.2 Tm (the proportion of EnKF method wins and the red line denotes the ) Tj 1 0 0 1 147.349 625.399 Tm 102 Tz (proportion of ISIS method wins a\) Ikeda experiment, Noise level ) Tj 1 0 0 1 147.349 602.35 Tm 98 Tz (0.05 \(Details of the experiment are listed in Appendix B Table ) Tj 1 0 0 1 446.149 602.1 Tm 106 Tz /OPExtFont5 12 Tf (B.3\); ) Tj 1 0 0 1 147.099 579.299 Tm 99 Tz /OPExtFont5 12.5 Tf (b\) Lorenz96 experiment, Noise level 0.5 \(Details of the experiment ) Tj 1 0 0 1 147.349 556.299 Tm 103 Tz (are listed in Appendix B Table B.4\) ) Tj 1 0 0 1 487.699 556.5 Tm 98 Tz /OPExtFont5 12 Tf (55. ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12 Tf 98 Tz 3 Tr 1 0 0 1 121.2 533.5 Tm 102 Tz /OPExtFont5 12.5 Tf (3.7 Dynamically consistent ensemble built on 1 observation compared ) Tj 1 0 0 1 146.9 510.449 Tm 98 Tz (with ISIS ensemble built on 12 observations, the noise model is U\(-) Tj 1 0 0 1 147.099 487.649 Tm 102 Tz (0.025,0.025\), each ensemble contains 64 ensemble members. The ) Tj 1 0 0 1 146.9 464.6 Tm 100 Tz (top four panels following Figure 3.5, plot the ensemble in the state ) Tj 1 0 0 1 146.65 441.3 Tm 105 Tz (space. The ISIS ensemble is depicted by green dots. The DCEn ) Tj 1 0 0 1 146.65 418.5 Tm 101 Tz (is depicted by purple dots. The bottom panel following Figure 3.6 ) Tj 1 0 0 1 146.65 395.699 Tm 109 Tz (compare the DCEn and ISIS results via e-ball. \(Details of the ) Tj 1 0 0 1 146.4 372.899 Tm 141 Tz (experiment are listed in Appendix B Table B.5\) 57 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 141 Tz 3 Tr 1 0 0 1 120.7 349.649 Tm 101 Tz (3.8 Dynamically consistent ensemble built on 2 observations compared ) Tj 1 0 0 1 146.4 326.6 Tm 98 Tz (with ISIS ensemble built on 12 observations, the noise model is U\(-) Tj 1 0 0 1 146.65 303.549 Tm 102 Tz (0.025,0.025\), each ensemble contains 64 ensemble members. The ) Tj 1 0 0 1 146.15 280.5 Tm 100 Tz (top four panels following Figure 3.5, plot the ensemble in the state ) Tj 1 0 0 1 146.15 257.5 Tm 105 Tz (space. The ISIS ensemble is depicted by green dots. The DCEn ) Tj 1 0 0 1 146.15 234.7 Tm 101 Tz (is depicted by purple dots. The bottom panel following Figure 3.6 ) Tj 1 0 0 1 146.15 211.649 Tm 109 Tz (compare the DCEn and ISIS results via c-ball. \(Details of the ) Tj 1 0 0 1 146.15 188.35 Tm 103 Tz (experiment are listed in Appendix B Table B.5\) ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 103 Tz 3 Tr 1 0 0 1 292.8 50.1 Tm 88 Tz (182 ) Tj ET EMC endstream endobj 962 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 963 0 obj <> stream 0 ,,Nb7! |My/% 2 2~rI/ +HAT_ѡ\ 807'8%ncTL\W#fu)ί- Ԅh6`i 6ҵdWuR4"%~|RI(LRRo?r $&fR$Z~>DK%L[';#k׷B1yH3O #϶<kl\0+bKؖמ2TwM{r7NbIǘE{oz,쭶>硽z-}ݞmgo#db@K`/{u~#+e^ݾxkDP4 e[d瘮9ݰP}Ӟo:Nޥ;b(7UrF>搓Gq;NS[8eͲӛN L'<.M/ɱ!ijC|-O>dbWȕϱ>|MG06+ N0|G$Fen}i,S6^BLofO}<ٹ Q`Ur1wQuQAjq {C wւx6KO3QctF4E}feB'mmH]#J] _ }yL6/eŃ1e;wH÷ 9W[B(䁲#Jik،!r1_#/rӲS8Lr1(ƅ>Zp.ղ$+γA_^j$nJrG7d`vvk9rƘ W3h>?8ᔊٞ)Աgɳw۳фe>+H< E޳^ɸd~^ ѢuWqHd )9IpzӐ/)u2kJK"`a0ik#s 4cfp7}gm$JT%`ښ?SXgQF0qhySl-|]qacd,6oGsue"]6/ޙZެ\?9M.5%oϡC:F1F#-tx2rA9(%~\lA4'5sBTS WC^s5-*ѝ!Joq( k[%,>[ryʏYzLH`Y>1).WG@' sQi.)E|#(W`ћ4*>4Y!HHhFN:M4?dwǀ79PY>rZA*}{nJ/ Q-L+NL*NrIؑș$qߝ+ؾ%YeR<iW4~*KE/n>㆐h$ɮ),Ǐ*1.rՔԵ#G4 < mn!fs5Q ؋͐wxg|л=di$?{Lò`WHz xS;5*ֺP^#n_'d{1=u|W#\uU*&so?²N8`"~-6n`\0+112^Q?]jxi3OLVϪH[9%$7XX>O"h:8z1a-_[-ČHN,)A")$/4j+fKV9W0=/f5;4{ :8jK(u`C5͉ ?&#ԩ~K zc;n ̇Ց9qr7m hWycN.R1=$,>eCge$.s3kҨ>Е.D\ۏfUH'́=F2,z '78v"\7ƫҤJu sx$ZA%c NHA-/ށIo[6ˊGb؟ln@&cNܾ@/Rkdn8>p%j`.3؜0ۊdVܡ~Gg7~ngWjre?vsY "SDo&-iV>]BPG)y7Ϲπ&W _8*|:w=0++!|8!V>!൩#ı<;V @gQ= y f}Z n^q#ɍǭ+?f%b}a}T{ό'ddbUJ/-r̶F.){ml]3cjeM8M͝EcD̀u'cHfCNԇ1l,<&27 a (:rt}%kUvw1vc) f0l~NO%W dr`/%֛[U/PX<< eZ I:b,1Q;.W8 " ,O/"X\!nG NAe!4z-&%ƺN7aY'w{+4wQ:MSṆ)|0`Ph66ucJwsnTwckyŶ[)нD専 lU/iԲ X15??#RJܳϤߜ*weS/󁐲`Q醽}Z3i,xt/ygӉ){Et ?Gk&%eRAjˤÐ/2ptX;yP{&c,:FR`ʮOu2ˆ SlbtgoF V/ 6\4~ R(Ӱn*At/d?s+A>tbyɛIinٍj]ÉFBW^FBSصHn7Y on8ɑg BRI/*7KC8v5=E'ݮiDT䐙񷓠\8EA] ړDO73r((OFE/dZ:" McVC&".w@ o ̺< W~~*e@>cZ588<) & G$G/%A.!pkY7/­`զrWyC6ELCD{F*MC"*KVCsKnH8[pm*8vF@~ E8,|b_Uo.eY4[hCX_ǃ?C4!b[*0lE,"'Sc V6b7lZ0c.i"@:q rp 13=\8?1 V`K<`xA.zYr¨{}Ŀo9+J#L8pFz*j226JVOG Q wzwP{ZMڦ_YN'>dsQX#"&_]Xd;) w|E]A>{7ELoP)ߋBՈ.i 5Xf "> o?lϏ$Txlˣ?.Vzk;^ R W>m#`r"_wC"yhB Y8Ð%N9-jt5VgMB~}s-Y"=?\f"0ҤJ[ӃjY-i%'UUY'6`>RgLk+)>l3Z@Y\ђÝp Q+ ]:}y"1bQMl,|gQsȇJn`;nlu/ڎpR;0e;(B 6RF^!SM]->&ȿ'U)767𴷱೦٧ߎϺ#̃ѿu}SwpW'O'lv%ʾvxnk]p/lw4Ra2Xgq?|zZ3cI4iWЦO-eQe!O ,f`5   1>.6󫰿騳գq:<6|(n3hb:dymöa8u،$O.IUp33&{uDWayR3@JHݱ~ZmSm8k;}EjT`(u|8BXogy/!<`3K[Eֈ|"]E^.g`R5Lj8-2ǫÃ*A&b|~pKS`Om鈋EO(=J{0r\Ww lUutgd1ċJSǦ[ M&6:Xv 2"n&(Ӫ>#BmaS6m|-N ,ҕ$@ނ'8&B>Z: oL 0ؚ!> !ͧCّUo'/Ӹw_{|žeX~Z:UHNdPeT'$å 0Rt&@obf0 E)Q`t1\w~y5>Dy7 [~ |{R?@%Rg;y%:<9TQ!.؀pHj1ʍ*:3 T'8mI@6pRbyV2Ttf`~ \1zSd`^l:{ENhԮ Օ>>yuPiYĖG%u0p=q J)m A`Vq&}oܲҀxXMVjKak3A$T\r+g+iTSM ϐsЅL爬07{0M1N2{>@(dx 8~ ?A.{qP ^8!xsٶ6[|G:).{))Dc4]tF)Zi[|B4xrl؁jVG"6x7)t80}аȰ \_C2@*D%yw|%)2/DUOjnmؗ=ZtM:=֊8lٰTF0|sDU9Mߋ@AA 0&$&K,Wܝ/N+&.d N1edg * 4 yVɕtRCOA]& kϯ!n<)Tʖ|ܺZ$BnnSzVk.r`+)Bfc;8:p&8D&W!tZfʯzF229N [,fpvѡ@X&CtWI6ֲK/Rך,1rvʈ q7sCE)0R ~|b4[9Ijg-L 06CKsxplAf_.(&n޲(h=~&+6z 1cv?T`QGH^sW1M~LKf7uH֎| %G+ȮFF dT|1 t:"D9,FlшJ'xܾ@Qpy!ӟ.<$AH3i<+ȂO]Xh/'0xԤLg| #\* ͕*IwGuևbbWhE qDssZ˨!4IK} |xȨ' jo<2&!2tl1[$E&.^=K:p_R( O,D77=;-Fھ'{|=`)E /+]=/X aƨMSؗT|($bE}m.ّu3#H\s#z">`k2J1RsAlC dxL :w=Oi/HͪgI$. l}qY..tVk?a>F@zXYm1\qk] +# ޼d»`w絳Z빱n.dϗQoQ F˰6=Y6=0KTFdrHcz"ɔO>MC铍T(0 .ӓ_S8ŋybU m+CV8%'E2?y=NI? 0,,p,%4AfySgۺlf{~p - 7gΡTQZmr iS5. kІ&DpHva-N<`HpJ:Y8 ;uEK˒ Y;,ʐnʲ0Lj `4@1 k}>s%ڸ0{(DR;.)Ƒ4߳L&競*ʳ:681;7p_%g< endstream endobj 964 0 obj <> endobj 965 0 obj [966 0 R] endobj 966 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 590 0 0 834 0 0 cm /ImagePart_2210 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 385.199 714.7 Tm 107 Tz 3 Tr /OPExtFont5 13 Tf (LIST OF FIGURES ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 107 Tz 3 Tr 1 0 0 1 121.45 671.75 Tm 97 Tz (3.9 Dynamically consistent ensemble built on 6 observations compared ) Tj 1 0 0 1 146.65 648.7 Tm 94 Tz (with ISIS ensemble built on 12 observations, the noise model is 15\(-) Tj 1 0 0 1 147.099 625.7 Tm 98 Tz (0.025,0.025\), each ensemble contains 64 ensemble members. The ) Tj 1 0 0 1 146.65 602.899 Tm 96 Tz (top four panels following Figure 3.5, plot the ensemble in the state ) Tj 1 0 0 1 146.65 579.85 Tm 100 Tz (space. The ISIS ensemble is depicted by green dots. The DCEn ) Tj 1 0 0 1 146.65 557.049 Tm 97 Tz (is depicted by purple dots. The bottom panel following Figure 3.6 ) Tj 1 0 0 1 146.4 534 Tm 104 Tz (compare the DCEn and ISIS results via e-ball. \(Details of the ) Tj 1 0 0 1 146.65 510.699 Tm 135 Tz (experiment are listed in Appendix B Table B.5\) 59 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 135 Tz 3 Tr 1 0 0 1 120.95 487.899 Tm 100 Tz (3.10 Dynamically consistent ensemble built on 12 observations com-) Tj 1 0 0 1 146.4 464.899 Tm 94 Tz (pared with ISIS ensemble built on 12 observations, the noise model ) Tj 1 0 0 1 146.15 442.1 Tm 96 Tz (is U\(-0.025,0.025\), each ensemble contains 64 ensemble members. ) Tj 1 0 0 1 146.15 419.3 Tm 97 Tz (The top four panels following Figure 3.5, plot the ensemble in the ) Tj 1 0 0 1 146.15 396 Tm 104 Tz (state space. The ISIS ensemble is depicted by green dots. The ) Tj 1 0 0 1 145.9 372.949 Tm 103 Tz (DCEn is depicted by purple dots. The bottom panel following ) Tj 1 0 0 1 145.9 349.899 Tm 98 Tz (Figure 3.6 compare the DCEn and ISIS results via 6-ball. \(Details ) Tj 1 0 0 1 145.9 327.1 Tm 120 Tz (of the experiment are listed in Appendix B Table B.5\) 60 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 120 Tz 3 Tr 1 0 0 1 119.75 294.5 Tm 102 Tz (4.1 Parameter estimation using LS cost functions for different noise ) Tj 1 0 0 1 145.9 271.45 Tm 101 Tz (level, the black shading reflects the 95% limits and the red solid ) Tj 1 0 0 1 145.449 248.399 Tm 102 Tz (line is the mean, they are calculated from 1000 realizations and ) Tj 1 0 0 1 145.449 225.35 Tm (each cost function is calculated based on the observations with ) Tj 1 0 0 1 145.449 202.299 Tm 103 Tz (length 100, the blue flat line indicates the true parameter value ) Tj 1 0 0 1 146.15 179.299 Tm 114 Tz (\(a\) Logistic Map for a = 1.85 \(b\) Ikeda Map for u = 0.83 67 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 114 Tz 3 Tr 1 0 0 1 292.1 50.649 Tm 85 Tz (183 ) Tj ET EMC endstream endobj 967 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 968 0 obj <> stream 0 ,,)b7 %S_W|<%JHL (:.oH02BRq!} τn:!JakQʱd"p/IlӤlO `s(ʷs ,H˓ٷCms ~&NS^P9f{L"4Rr[0{%{0**w|^_v ?SMV3[ Ȇ٤_Wqzq|w.!Q|~44'U%>U B :D-͞iVϊx oh]I=]@6E!'8kW>gd խbpicݢ|uqF5Xb7x$yNK U `B7⃰ 2@eDx> :6|^ çrVL#꾚r5W/Bt乚s! *}O&@`[~`Nsp[M׬p-8s"gaCl>5uO~ʞߗ,Cqg3j/qKfƁ|XtN(f$ bySy:98wP%ͤ tLTcidpPmw"8h{1EgK‘9bmb6Bk6#/c̨Dۥy˝nj5(ֱxHȥ(6-,`F #fY}8!)[9'U*3FGb#qq~*#Dz:)9 =-='d7"PU{nM$4vj;Z6^btB5/Q(P9@H~ݛ@dOR.j1,I} ϳjL|ayоI IOrl\&a?\ !|!Ov>Jx'gIvXJ;ry@ZS2vSZQrvj ĵmfqC%v73P"=ܰ^ )p8&&ڑ 2\n6)~7E8T90^+JUJͼ &e+ɻ]3nmp6' Qh.?]IRiN?uyd$tIg í^khxWܮ;J#}e U#51DVZ/}ZR{N~p+495Q0#Ww buqu{|}2&':/@5U'Y-ʩ|#!SzZ}Lz~3 C|^1b])7d{B8+d`H YOg>eӆZaNI?u=kS=+zc]S> Nya0UX}q<YUx`l޺)kQ8k ,JئZKV3в;ghbyU{,Ƹ reRP:?rB֐϶晱L̈DÏ(h4Nݟ2eމ5[u^bAk:G.eB.AܥM"s#p;Pa.;W_sګJ^"e*kf ɼ\9@3ǩ!4# UU^!gդE]sђ0(iF Sˍ`"D{3{$!zr /B% ƨkO v= V=0eޕA]iV0%,9ẛ @thgRNa椦Yc?!aK޷?ɼ:I YʳKzcz9ϛ̀1Hj8iqD_K+(b̧A`G#889\XEZhGbܖ ;O%.5ޗJǜ_bE eD..|u;׿" V<κ3iޚhbP,N`ZO?%}O}ب̀;۲4,a= j:1RAOvaS1J?:wM|1|IWtB1]F LQ6_ {ֵGu&6!΅KyY)ܟs<zbFqHU$6O<2Pޜk*֔R*&$NZ?PrSЄ[Vz s=3ohAMXVTHG𝟝؆&:WqC ߃')jW3OC' prOhi6Nx@Es \![*-EZEO2=Eիipj3ҙYv*1BɺE>P&a"}y%"Z+O zNp(k㸵l\«c< ޸ JCx̗kbL'Z3xt!Ea?b2Ne@ jet p98 8`ױԋ gFcL3-=2%JPR 'oLܿ< #m`ZZΉt0@A"p˸`I'oO6f64c(^&y\h_ 4xVd`D7jP{G ]<]jqK>y 6(t:% $(+޼, d%<6eLkSᶺDz:E+s nv;T"tj5F:3w&:hKTy8RM>'Hm=QnٿMy]/̞ro2 D#MUfo'reX{V"r-sF$A S2cdn ^XZ5١m Gv?Q xA)j,}TFJ̥[8)}LXAg t%븠1No_w(p;{ʙʱbTiU?*llo4Ғ2n2_zWF?Hx& ] non#d:hy:_ 2quC\DVv'F편Rv)^dƛ A\ SRiAMQ7~8o 3%j ueR;:"z rA 1qP! ٟXmzdv 6 ոRElD0GYI5,#CzHJPb( aZke8p}>G)B6YE1ɶZs4ȸB c. *@,~el^܍3}\a?d1##qw6)@(/i't4W"k7\&qtz_v=[ ^9-O {sd ~pgݝ x y|U9" )d7obYN#b= \5LIoH6'hz,tܪ?WiTm9& "T@ꃪ2Jӎ2_}צ< N6tyj4ժD:"?0)cP@J/0([%,dQS}%|ANSlJ"H ަHI,v ށF +k5''3ֶDWDo$?>2H[B@yY U[$ȣ;[k(:12Ex`2N{7<ny=_R(yqqfdנ%T{[}1Ӻ\l^*+uB@ܮ+#mDcW',cU؉-,ˇ fAv ;lMb6ӜM-ͧS,Qu#PAd :)!y~q_imمS[JٔZ %K9zUFe2i[Bduo=GVJʛqfdH_2l o puX/ ϙVB8iMƏ 9G|y<ÓJ9v7t@WP:_`ГEЍ`"'/e֯zBWW!nD7H`lw,O7+ϐ"qNI1| 3m2!Vl .Ya;+0idA; Ϯ2H_%S`Zhn@sovr_9ȫ)< -2`5ŻE ;MG+cLRft8ӱFQs 'gQ0ܮp(iǢ oZ_ Q Oes ܣby+ֲO2j!1>湨'3aq]:s"3/S4x%3)we25|)P" }`*Ak(?D l~KxEjo`|{ ؽzoƉM@3Ck M\m9MmSO)(mΉ:43'xsxkv`/S]XJEvq] |] t3-_).bPcP8a丶|w )D~{pI‚6]|y Q\$vjpR N`.I0ӂCAIP\i "?h]!JhЄALwu!V*3.=,Zv/fZ?> )y!ɦ~ol9sB}ON[#*†FxElw)@3Wpq*h,w]cNXqutYm G{+WciSoSţ9aK{u vBEՅ&)(!`Ct=-ioh 0-.U;ղ߀ϘufN@˜Hoh6'9,׷?1iUxZ+eh59L46 oKolwduyeʳڪ!\!f@I2Wkcz1'_\[_+_?4e,,rX 6<TZ86m]!N*I3RFKՕ|Wk;y.Llt!٠6Ui}\}0"˲xL4L/pUb$ +K'vhxEB翿 endstream endobj 969 0 obj <> endobj 970 0 obj [971 0 R] endobj 971 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 590 0 0 833 0 0 cm /ImagePart_2211 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 385.899 714.7 Tm 116 Tz 3 Tr /OPExtFont5 12 Tf (LIST OF FIGURES ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12 Tf 116 Tz 3 Tr 1 0 0 1 121.9 671.7 Tm 101 Tz /OPExtFont5 13 Tf (4.2 LS cost function in the parameter space, \(a\) Moore-Spiegel Flow ) Tj 1 0 0 1 147.599 648.7 Tm 97 Tz (with true parameter value R=100 \(vertical line\), Noise level=0.05; ) Tj 1 0 0 1 148.3 625.649 Tm 95 Tz (\(b\) Henon Map with true parameter values a=1.4 and b=0.3 \(white ) Tj 1 0 0 1 147.349 602.6 Tm (plus\), Noise level=0.05. In each case, LS cost function is calculated ) Tj 1 0 0 1 147.349 579.799 Tm 213 Tz (based on 2048 observations. 68 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 213 Tz 3 Tr 1 0 0 1 121.45 557 Tm 100 Tz (4.3 Schematic flowchart of obtaining forecast based cost function for ) Tj 1 0 0 1 147.099 533.95 Tm 274 Tz (parameter estimation 69 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 274 Tz 3 Tr 1 0 0 1 121.2 510.899 Tm 99 Tz (4.4 Parameter estimation based on ignorance score, 64-member initial ) Tj 1 0 0 1 147.349 487.899 Tm 94 Tz (condition ensemble is formed by inverse noise, the kernel parameter ) Tj 1 0 0 1 147.349 464.85 Tm 103 Tz (and blending parameter is trained based on 2048 forecasts and ) Tj 1 0 0 1 147.099 441.8 Tm 100 Tz (the empirical ignorance score is calculated base on another 2048 ) Tj 1 0 0 1 146.9 419 Tm 101 Tz (forecasts, the ignorance relative to climatology, i.e. 0 represents ) Tj 1 0 0 1 147.099 396.199 Tm 103 Tz (climatology, is plotted in the parameter space \(a\) Logistic Map ) Tj 1 0 0 1 146.9 373.149 Tm 99 Tz (with true parameter value a=1.85, results of different noise levels ) Tj 1 0 0 1 147.099 350.1 Tm 98 Tz (are plotted separately; \(b\) Henon Map with true parameter values ) Tj 1 0 0 1 146.9 327.1 Tm 177 Tz (a=1.4 and b=0.3, Noise level=0.05 76 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 177 Tz 3 Tr 1 0 0 1 120.95 304.299 Tm 102 Tz (4.5 Following Figure 4.4a, Parameter estimation using ignorance for ) Tj 1 0 0 1 146.65 281.25 Tm 96 Tz (Logistic Map with alpha=1.85 \(a\) Lead time ) Tj 1 0 0 1 363.1 281 Tm 73 Tz /OPExtFont5 12 Tf (1 ) Tj 1 0 0 1 370.3 281.25 Tm 94 Tz /OPExtFont5 13 Tf (forecast Ignorance\(b\) ) Tj 1 0 0 1 146.9 257.95 Tm 97 Tz (Lead time 2 forecast Ignorance \(c\) Lead time 4 forecast Ignorance ) Tj 1 0 0 1 147.349 234.899 Tm 176 Tz (\(d\) Lead time 6 forecast Ignorance. 77 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 176 Tz 3 Tr 1 0 0 1 120.5 211.899 Tm 100 Tz (4.6 Follow Figure 4.5, Parameter estimation using forecast Ignorance ) Tj 1 0 0 1 146.65 188.85 Tm 94 Tz (Score for logistic map with a=1.85, initial condition ensemble formed ) Tj 1 0 0 1 146.4 166.049 Tm 96 Tz (by dynamical consistent ensemble, a\) based on lead time 1 forecast ) Tj 1 0 0 1 146.4 143 Tm 97 Tz (b\) based on lead time 4 forecast. Note scale change on y axis from ) Tj 1 0 0 1 146.15 119.95 Tm 292 Tz (Figure 4.4a and 4.5. 78 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 292 Tz 3 Tr 1 0 0 1 293.3 50.6 Tm 85 Tz (184 ) Tj ET EMC endstream endobj 972 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 973 0 obj <> stream 0 ,,b2.[n&Ky왝)." 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The statistics for ) Tj 1 0 0 1 148.099 603.299 Tm 109 Tz (tests using different parameter values are plotted separately. . . . 83 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 109 Tz 3 Tr 1 0 0 1 121.9 580.299 Tm 101 Tz (4.8 Parameter estimations for Ikeda Map with u=0.83 and noise level) Tj 1 0 0 1 460.1 580.299 Tm 41 Tz /OPExtFont3 12.5 Tf (,) Tj 1 0 0 1 461.05 580.299 Tm 69 Tz /OPExtFont5 12.5 Tf (-----0.02; ) Tj 1 0 0 1 147.849 557.25 Tm 104 Tz (Moore-Spiegel System with R=100 and noise level=0.05, the re-) Tj 1 0 0 1 147.599 534.45 Tm 107 Tz (sults are calculated base on 1024 observations, \(a\) and \(d\) The ) Tj 1 0 0 1 147.599 511.149 Tm 102 Tz (median \(solid\), 90% \(dashed\) and 99% \(dash-dot\) shadowing iso-) Tj 1 0 0 1 147.599 488.35 Tm 103 Tz (pleths; \(b\) and \(e\) standard deviation of the mismatch; \(c\) and \(f\) ) Tj 1 0 0 1 147.599 465.3 Tm 99 Tz (standard deviation of the implied noise, the horizontal line denotes ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 99 Tz 3 Tr 1 0 0 1 147.599 442.3 Tm 105 Tz (the real noise model. The vertical line represents the location of ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 105 Tz 3 Tr 1 0 0 1 147.599 419.25 Tm 101 Tz (the unknown true parameter. ) Tj 1 0 0 1 290.399 419.25 Tm 2000 Tz (\t) Tj 1 0 0 1 488.399 419 Tm 91 Tz /OPExtFont5 12 Tf (85 ) Tj 1 0 0 1 498.699 419 Tm 32 Tz /OPExtFont5 12.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 121.45 396.449 Tm 104 Tz (4.9 Information from a pseudo-orbit determined via gradient descent ) Tj 1 0 0 1 147.849 373.649 Tm 98 Tz (applied to a 1024 observations of the Henon map with a noise level ) Tj 1 0 0 1 147.349 350.35 Tm 106 Tz (of 0.05. \(a\) standard deviation of the mismatch, \(b\) the implied ) Tj 1 0 0 1 147.099 327.3 Tm 100 Tz (noise level, \(c\) a cost function based on the model's invariant mea-) Tj 1 0 0 1 147.099 304.5 Tm 109 Tz (sure \(after Fig.4\(b\) of ref \(64\)\), \(d\) median of shadowing time ) Tj 1 0 0 1 147.099 281.25 Tm 475 Tz (distribution 87 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 475 Tz 3 Tr 1 0 0 1 121.2 258.45 Tm 103 Tz (4.10 Shadowing time isopleths as in Figure 4.8 for 8-D Lorenz96 with ) Tj 1 0 0 1 147.099 235.399 Tm 104 Tz (parameter F=10 given only partial observations, a\) the 8th com-) Tj 1 0 0 1 147.099 212.35 Tm 100 Tz (ponent of the state vector is not observed; b\) none of the 2nd, 5th ) Tj 1 0 0 1 147.349 189.299 Tm 104 Tz (or 8th variables are observed only the other five components; c\) ) Tj 1 0 0 1 147.099 166.299 Tm 101 Tz (only 2nd, 5th or 8th variables are observed; d\) all the components ) Tj 1 0 0 1 147.099 143.5 Tm (of the state vector are observed. In this experiment the noise level ) Tj 1 0 0 1 146.65 120.45 Tm 709 Tz (is 0 2 88 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 709 Tz 3 Tr 1 0 0 1 294 50.85 Tm 86 Tz (185 ) Tj ET EMC endstream endobj 977 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 978 0 obj <> stream 0 ,,XX[ᰋy8nހ˟eMg֢$tin{jMH^b\(``uu0Ȇ0\( Wپ 9`@! WOz4(TLe.Kp(wŠF1q3ɱ4̧WƇ͍~pBh4UPI|D$[„ܪAd"Vm?R73L6"&vtjT tlkO<鉱 2Dh)Y3{E@(:4k2"[ Q}UH}si8B KgUa'EUm,]T@NBD"^^]Pݜ:1tfg5ֈI3.,*ǥ= T5n:I^J>} 2Ϻ,TYq\lXJW{Dt 6&Xs λx;b\%{U%sV] yQV~qc9]5KDH"B}?va5űnNQzF`m7;9bO]\$p#cvy.Xt5mD_>lj|8B.M_W/ޚg9yk1Pq<#5Jҽ;2M[ 䊦OBcʗ*&RK8nU]!^|Oސn7"??E4+80EGلvTrR]M2d5̎$(s[)N&ah7GV(f9g d ;DU6<d"ES3ƕs!$@wx4j֩kEAL* vžirqIYh]-4][7?fƂ 231>þB Nyx1ΖgM76WT` IΙC PC|+TglWLI/-tx'0 A;ф2|DX]3oؙ~1V1a4qjZ$hwz*Uz܇QwbX8 dÃViKԃtLoKd9;1,@JLJ=15^uI8i߾[2)fu ]{UęO}Oqx]d/LfZB۶ #1oS*{q}􁱱!lUtTzOd;v$Jt"d5n*F];P>\ؓ(R^8 83|X'Dł(ľ2;P,>J,|O;P'"b J &?ۮ>JAE 9V@8𴎌//CJ<ʞD;6&|c^8Ѳ>L=bBzKE%RشQ|/)|_IͶ M7a$O I @&s0rp p@IAW ᫝_3wBnTMLpw_Hs>W1UJG _)qKG/tʤY7=*{jB=mVNۧ|4~#ٺ_TnuߚPJQ/s3iRϪfѧsu[{W-n[!lu77eJQgAf}P?}ay P?Y1KwN3@j{Gq:KMYqJ͗ D<Wdħ^/: vj)l`c@[i ߊ< HGU~[YEuLkiSq;Ib e7ě]+^2u.>jQ$%B0,<欛bc=e[xGPH.rK#"NRnLaI|(G#7cIO(cÚSBXG߿xv 9S 'Α tBHeEv`ac bvRŅ$g"VeYxWz/WL.ǩ|mcbر1ד"<"\͂80um'Z-5N@(xG~ U*$ero s &4K m yfn Knk GȽX1/Px2[L쫟, zCJBT0`{a<ס(Ȱ%F ؙlV#Էą%},vn>fG4iـ>11K(^K+ Rf;18'*}=,;H 4V<8 qTcy9uqCM;zHQ@N9߯8m3^c/hxnh&յ@/9;ExLlk6kf DQw+5+BAqCM?>ۜ]H|aFu!1 Y"_s$?K+V1"~WS-V_\\!;n",1߳O>A7)cG"41-*"`QlmhP) UXJqt$:fݲzvEl1O%vXHA6A]u)6hTLs1+ ߱cF>CxJET OM`d|Lin&4*@:L :I-DX.HfyE+sv.L)5}Ԟ"-DN›t!4Į7pA7tZUg 8AhIgK=S>Je !(ԶᘾZ &x>KQ7^Wm3te Y6w/ezRNP_[ZO D+efܪ^ǞSӐv,'~*q)O)- p]lYA9ZX$ vHsgDlb6D_`d 2@ mUS1p5"7(/> ";J؂YFCAcĠ38$; rY9z^Gvj.[+q*G=00TZ-wֿFwO)!(*:N@Mu)xCL_hWR(\e.RA~./\Đ}u y׈zK/Byuԍ{#.Εw 6 =.F1XI>v&\o˲JUv[[U,Fʼ3C⋾6 kMI[˹s}N}Y6j=fYυO&"ɣWeߍ&H3_)/,HCq˸\@ěʱGNqCpQYfOKPi>0بEiwFb&VUF[ +j>3=~$ %Ҟ ׶LXխ xvMBzRMx0D駽9?1P}2?8=/: ӌ~ Rq V7;cZiG۾[9S[k‡^jI_ROܢL񄚿_!ؘnۗ^eDyl9xPǶ tyjB(w^,Y7r6ޏFiGvMj V!V{ ;)i4(]E8^z\ |sEb9^ݠ6Й.&_2s.IԼ>ү? Y7qe1WLƳueϔ#B=[7KA,y< z3nWg,yUu-A6*x-LĞUOO!{4 cF\|sO0jc݁e}<'w^x5h߁K.48EDӱsyᴣjy̓3 4E\nM6MkxGfLF'U RП/AZ{'AH(79M#Q~Η1=S~R=-CUJMwUP$1 xehg'ϫ|B.CcFig(^ <\t'8cV E@>ȳA6Da#Ƅ[t0P\Ow =5\N~yL&_K05ȊTfՇ(T!(t^9>s #IϪ^oJHK-qm]zfW0o­WO^gKinX؞-~R=sh=bUGKg-tc rj7Zt21:b5>N8g+zGCU̧vG$ڸk%`o$-Sb,ܐ)ljfo/K@Z!G$e@ B5.U1$'a a~n9en ^Dmg5f"GͣVK4}OCI{VҎ{=t;ھIђ-9glh4 *N &)[>.QWI{vjF]cWRVM­ T\Bj۳H0N`9YrnD^0k\kԼͱQt؟aNhFa Q>_  @ 7//vvoEmthYcJcBi9c<jpGI8\KhL,>%?ۚYg$հY3"3-0t#&Z>}O$=p…K{W|O MD@{^6QUl;_J h5 y[9e9&Rtǽ~`/LC9&y]'Ds'YTj O L7`艙o<͏CN1='˓BS<0tVo7Ԫ1Gn!݈QH$'aKKVa+GKEa7úz$o8b+`4'E$$%i ~SwȘfM4o G$j'̨Qk6v9G|Dtׁ3wF @EcժAC.>e1%f⸿^ѱ⯡ʳ.( 3V:& 8aR7`8p!XZc8Ԉ+A̯~Suj' kn7Р5&ؑDˍ |;͵BOP 0z3^p7!X!7e漓+բ G]C6nIǵasx-7#LA4 וֹL֤B[f_k}@/_ڕk䳉<^FFM{S44*H;aX%ΒO4ZHీŝ621s|T&}=ii :1" Kc[?̙F6C%K.dhNE"P]N%5݈7}hW l`s؋<|i oG66N]hhgOsO{a#s1@U<7;_gVSixʄWBVC$@{{#&O*"sؤ0fǼeg>|ԮrʪrUh"CWuRbIkMqs3VUwU#϶52^ m-<ܶF#>b$L]馢 'v<* rm U`#I٫wkXj-߬1r¡`!ctJÀVrhAﻴ=@?kP g~u=-0z_nE\#`VQ8'ЃkSVs`ӿK52b!2@`2޺~eޢ?88rQ[*kT}3䴳ged<4`ZDan}@7u;{hav}旷Zҡu;8`ft4SP1م9[I+w|&1RX@U4&6GŽ(jaK(o`m$ ~aP7R3**9Qw$>nXYŶ*rq']4[ggo`_0Ft=c7NN'ٷyEC:v݇Ӄn{xoIZ̡A@OWŴ W&%^=cV)2g=wϢ-u~b) ՛+pL jG}"9DP+ۚF0ER+X.>GH_AT́`t.`D1D D7I7Gue''.)|ۗ:fjY345[*O0#>گ1#}Xy0YX8̏ EI˒HEFGfcLUĭA?rs1 J%nc<,w:y;P#W+7`Cu?İtyK4 ǃ[ͽd{.&I0Cs8yщKov Ft;/kǹܢM{Qb, У[d6~8b`& p#knD4V xL ӜW&?=C¹U ۞pB ,,[+~OT7suݷ x  a]8A.~ o |zZG2}[e_oxJ'es^noeLu  )nt`D=-L: =`)'2>5SaubLI0P9 1U6v|ni!,s%ٷp*= 6hVT^Z}԰+ڿ.ZЗ$d./`ADU5;veHRR8Pd`71Z㼐arΧ $Td aNDJ6pDz?&]F9ky4k endstream endobj 979 0 obj <> endobj 980 0 obj [981 0 R] endobj 981 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 589 0 0 833 0 0 cm /ImagePart_2213 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 386.399 714.45 Tm 107 Tz 3 Tr /OPExtFont5 13 Tf (LIST OF FIGURES ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 107 Tz 3 Tr 1 0 0 1 122.4 671.5 Tm 101 Tz (5.1 The one-step prediction errors for the truncated Ikeda map. The ) Tj 1 0 0 1 148.099 648.7 Tm 99 Tz (lines show the prediction error for 512 points by linking the pre-) Tj 1 0 0 1 148.099 625.649 Tm 282 Tz (diction to the target. 96 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 282 Tz 3 Tr 1 0 0 1 122.15 602.85 Tm 97 Tz (5.2 Statistics of the pseudo-orbit as a function of the number of Gradi-) Tj 1 0 0 1 148.099 579.799 Tm 95 Tz (ent Descent iterations for both higher dimension Lorenz96 system-) Tj 1 0 0 1 147.849 557 Tm 103 Tz (model pair experiment \(left\) and low dimension Ikeda system-) Tj 1 0 0 1 147.849 533.95 Tm 105 Tz (model pair experiment \(right\). \(a\) is the standard deviation of ) Tj 1 0 0 1 147.599 510.699 Tm 104 Tz (the implied noise \(the flat line is the standard deviation of the ) Tj 1 0 0 1 147.349 487.899 Tm 98 Tz (noise model\); \(b\) is standard deviation of the model imperfection ) Tj 1 0 0 1 147.849 465.1 Tm 100 Tz (error \(the flat line is the sample standard deviation of the model ) Tj 1 0 0 1 147.599 442.05 Tm 98 Tz (error\); \(c\) is the RMS distance between pseudo-orbit and the true ) Tj 1 0 0 1 147.349 419 Tm 368 Tz (pseudo-orbit. 107 ) Tj 1 0 0 1 210.5 419.25 Tm 30 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 30 Tz 3 Tr 1 0 0 1 121.45 396.199 Tm 96 Tz (5.3 Imperfection error during the Gradient Descent runs for Ikeda Map ) Tj 1 0 0 1 147.349 373.149 Tm 102 Tz (case is plotted in the state space. \(a\) after 10 GD iterations, \(b\) ) Tj 1 0 0 1 147.349 350.1 Tm (after 100 GD iterations, \(c\) after 400 GD iterations \(d\) the real ) Tj 1 0 0 1 147.099 327.3 Tm 134 Tz (model error in the state space for comparison. 109 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 134 Tz 3 Tr 1 0 0 1 121.45 304.299 Tm 100 Tz (5.4 Imperfection errors after intermediate Gradient Descent runs for ) Tj 1 0 0 1 147.099 281 Tm (Ikeda system-model pair are plotted in the state space. \(a\) Noise ) Tj 1 0 0 1 146.9 257.95 Tm 182 Tz (level=0.002, \(b\) Noise level=0.05 110 ) Tj 1 0 0 1 308.899 257.95 Tm 30 Tz ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 30 Tz 3 Tr 1 0 0 1 120.95 235.149 Tm 103 Tz (5.5 Comparing nowcasting ensemble using &ball. Observations are ) Tj 1 0 0 1 146.65 212.1 Tm 98 Tz (generated by Ikeda Map with observational noise N\(0, 0.05\). The ) Tj 1 0 0 1 146.65 189.1 Tm 102 Tz (truncated Ikeda model is used to estimate the current state. We ) Tj 1 0 0 1 146.65 166.049 Tm 94 Tz (compare the nowcasting ensemble formed by Method I, Method II, ) Tj 1 0 0 1 146.65 143.25 Tm 95 Tz (Method III and Method IV. All the ensemble contains 64 ensemble ) Tj 1 0 0 1 146.65 120.45 Tm 454 Tz (members. 124 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 13 Tf 454 Tz 3 Tr 1 0 0 1 293.3 50.85 Tm 82 Tz (186 ) Tj ET EMC endstream endobj 982 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 983 0 obj <> stream 0 ,,Db6#ט@/iSqfgӏwFqt5Q srZ9:LyLg;>MU' lp/Gv쥃3xQZw+W\2wmY.݉ rmxn  8>%^f?Dbjz^l*Aڜjzb_4zXc/p&.C"TǽSV!i\0[EvU>v'ђWboOo8%| =yu+ǭnhڮ($|^FsQP7  !i_!cw9U ㌿l}$V}]y B<5`\~GdA?`[!4яRMxdY)iE<]kpU_RID^&fyEHal3q:"'Z+H T<] YGhgLNk.M|vϭ|xLi%-z:zgimpz%m[wW˲;Vu9k'(=Kok ®[]PLܹp۫f莙߮(XQVvn-QIwb ^DZa Zܟ)pxw%Ґ8\_z;2C&-,a,qtKEGoZ#T捤{b  #FY*:" ,*N]댖h8 I]oF%.t'L, BR@L mj6:?xQ(:r4zݸybr.>%~70)k q>z@gfϚg΅)u:3 tW}7M6M~HM}e 7)Y#"N6LʊiȽYK x{xl]O_>U${A2{Ƨ~=+>]$oZ'#_^ĤnI=ȌK&I 7b?UfJ:\#Y<#u ! 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"$L 'A@3f1"XH/%Qp;b~SVf]^ҟSF_ZUz0k0}3MMF((7} E4;</ڌ6)<c[V=A "?Fx~<#u5)&ߎ >䳢ǂY7-(8:{>-w|]sC dH#؟ ` pO+@!c#eqyuFԚx_" 6)eŠ[<{X_1֎iB5eTE,2d@ (<1=MЄԻÑ_\utG|?eC* {?ɳXrY~j{kok!wF4mU[()F>K3_"rNCb~s7L;Cv.6 EyFud Ƶ:<:ha%}ʨ ?]Z}̎%$ ; E^.`,ˎ-xܑ : @^}wΙ^՝WrZ-Dw߱hM\DK#gd1 ypA寮E@~Fɽe:C("p475ZÐNssĆ'h;\ (O:feZ],t%2 7Pxpr%oO6c@1נ)˚ i֔6]`p{ժ\%ʠ%&>Jcz`N Ru>̽3?3VdY(`2/ήELWMb2A%I8 G\^i'}rm6CYmzȻrnZPN&jLFY+G;\о0'Ca݌< d@|-naJ@#BG7ui$̎\{B_*EtHfJ@J:AbRMpz{"Ii̸ = d$Qj^'BR\I'AD/zv&Ơļl$5EaCcknf21Άo!pu(q-qM=oUu p91c%[{VsU0I~'7v\??]5Wv&Ar36~T OՆTp(X-{hꛖK]-prYn[/r޻!GDize0U|4l(0fyXo)lY_;ˈu]GfÜ"̄cN2 N w8>Т3*L#6q,U?^!+Mآ51W m ZwGdX(>;9-2<+̭B4#pVuWH9Ypu::,Gws2zC)dߤ|۞Jݓ.Hs=#gBD|rvQB, endstream endobj 984 0 obj <> endobj 985 0 obj [986 0 R] endobj 986 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 588 0 0 832 0 0 cm /ImagePart_2214 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 386.149 713.7 Tm 121 Tz 3 Tr /OPExtFont2 11.5 Tf (LIST OF FIGURES ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 121 Tz 3 Tr 1 0 0 1 122.15 670.7 Tm 111 Tz (5.6 Comparing nowcasting ensemble using 6-ball. Observations are ) Tj 1 0 0 1 147.599 647.7 Tm 103 Tz (generated by Lorenz96 Model II with observational noise N\(0, 0.1\). ) Tj 1 0 0 1 147.599 624.899 Tm 111 Tz (The Lorenz96 Model I is used to estimate the current state. We ) Tj 1 0 0 1 147.349 602.1 Tm 104 Tz (compare the nowcasting ensemble formed by Method I, Method II, ) Tj 1 0 0 1 147.349 578.799 Tm (Method III and Method IV. All the ensemble contains 64 ensemble ) Tj 1 0 0 1 147.349 556 Tm 494 Tz (members. 125 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 494 Tz 3 Tr 1 0 0 1 121.45 523.35 Tm 113 Tz (6.1 One step forecast ensemble in the state space. Observations are ) Tj 1 0 0 1 147.099 500.3 Tm 102 Tz (generated by Ikeda Map with IID uniform bounded noise U\(0, 0.01\). ) Tj 1 0 0 1 146.9 477.5 Tm 111 Tz (The truncated Ikeda model is used to make forecast. The initial ) Tj 1 0 0 1 147.099 454.5 Tm 108 Tz (condition ensemble is formed by inverse noise with 64 ensemble ) Tj 1 0 0 1 146.9 431.449 Tm 110 Tz (members. Four 1-step forecast examples are shown in four pan-) Tj 1 0 0 1 147.099 408.399 Tm 107 Tz (els. In each panel, the background dots indicate samples from the ) Tj 1 0 0 1 146.9 385.6 Tm 105 Tz (Ikeda Map attractor, the red cross denotes the true state of the sys-) Tj 1 0 0 1 146.65 362.55 Tm 108 Tz (tem, the blue square indicates the observation, the direct forecast ) Tj 1 0 0 1 146.65 339.75 Tm 109 Tz (ensemble is depicted by purple circles, the forecast with random ) Tj 1 0 0 1 146.65 316.7 Tm (adjustment ensemble is depicted by orange dots and the forecast ) Tj 1 0 0 1 146.4 293.45 Tm 111 Tz (with analogue adjustment ensemble is depicted by cyan stars. . . 133 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont2 11.5 Tf 111 Tz 3 Tr 1 0 0 1 292.8 50.1 Tm 90 Tz (187 ) Tj ET EMC endstream endobj 987 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 988 0 obj <> stream 0 ,,{BBj\ o-NF:oQEs, =q4cZ)̕iZ 6r&gHު_貰оdŧ=X8a+ܖ#\d,y:e[FGg@ {.)*5Ȟ{dU}'D?ţ872⊻{-&%ʷ2`و𱹲*CՇϴŧIJ<׃Jɏp3{Ui9 Z&p 9hD%̏,$H4vgjy֗lo骉O1 O %<,-L _XM˺@np}'$ r] ̹6Y1we{"]I b>AU[/ E219"j懝~&KRv` \QTDŽAy'cwSn']K8mDa}270IVbgvXJ-6 wJxXt~yԚ˙oC^; 2Gpo˥wfU=B 2$>mR:5!E?]?6Ӯx;)"V 9~L fQ=aF0}4H0ڠ&j5<heMo񊿫X\`>z`d;~jhz]NmQ}S5~$)^AmZ-ri _Yn۸pvZ2H@ɗ G7J)R|E, j3.j2yֵѺЩ]3=rcː(IzI⫘@1]\l{Nҟae شdӁ|vcScNEߩl`TXŷleZ;{C#AV5K{ZsBFִ4d́%{Rj 94eիv .IA5M{;NK!C*NC.h}uj'hj45&\Gms!m ֎)[{3@P0(7z_ر#]p0v5,;UaȌ2t$ hp.O3.u]n5O^4e;6I%N+Vl? qL'֕fp)}I`F vc?n>"%3/ك&"~ڰ$Vء BhӨ֕y l8F#Dt!Wy̐wz1m2*/nU4R=Ҽ9}Vg?.~Ses>a# 7TaW8QΝB'M_~g&`,~:w񮝨|ѓ] Q}TIPJ^UP|Ǿl\'6|yYզcG qmL=13rKu{O5A5~ Ok= ~k`rX.0٦0=֌{iO Y2siC@\|{\f׃ohAtJKcz}edgP)6h.]<р6k+ ^u3?pjZIsΧDE w_uBtC/7nX8KAvcSF"H(ҽI1za ?L+Аj@:Oh*r;DAmj"y$Cl>Fp PKC%8 ewӀɁ;8.Lɹ2FRRjf(@|2XOx6IbT"aJ;#1J%E^{ٓTɒ|mLi}4SN9͂DF=KŹɇV\@ Kڡ|:ۓEp N<;tۀ#cgмo:[:u.tGzgO$lf%Ԡ]mBu)Zs"+=0!35b'S&usC>@t_tA4^X0ΊHjh N}* C7P5cu&Fg,INBGR|ǺPά6XzCSV`@NX~$%NrI-|T8ѾIO4Dj~~.PQWG'ciX9_et(L\sI#z|LI3 @bW1V?τy4)pyNa~V /0>00؆>ֶ7+O>5?DyMpx1 nP'1}:TռeaQ$AC. m,(hΣʚ"-[?(x-6F>ΫAU) 5a*U?-Ywqכ@ޱ4H/Njxc]y"Fw?%+_JB#nOP>_`+pA^J M ֨6)r Ue0nLߘXs⵼AJo)[`991A1s"2 '0Gd1KXd̨Ԇ@Kۘ?FL9C!@Ӟ4Sa̲뀮"&iN&,5Ƒ5Ql3L-;9φM( yl;:G>ox2OW] >%wTYD*4{f#G̗mЏJHQZnc9=ӄG۹ ]F^JuVoF/čI' !hM#a`E)({=K6 _[V.54'G_, fȵYe:QCL64 BOs7 IgGӧk.,4Z]rS?uO z> q#$k!}cٝ,jW26)_ p:)z)zPJr}2;ً6:?X,f<C'MWG9`A.JI,xF;=FE9q}i/MS[[yV{m@gKY>tSdji%pH?s ;} y!pj"S]n VTu]x<(+O*Bzjl@3?;j <@e2x)I_z89jWM&ti r_x%j',N:9r 96u:==*c %~dQ (!t4\dzq7%ýu`4MUwSeH1eKDh=n&5,ZkӀ)Mհ$zhMijGI1)ʤs*RenKGr.v4&IheY}WD^L >![fo-h.,g f2'&C JcsӫITi%5%Aĩ`,XalŠp%L,p*OV[C hI糕BG܈ 3ozX;wKW݌\%[IaܰOYЍjV> ^ R]JhչQik'ZMB E+v!CA;(,wKn3BcoK1v.0z2[bJTmYr{3񛀕 45.7xc޳=; Ul6t?-KQޏ-;QsD[adFxKe 0w=0W^TKs'tld Z(J˼S^*uIg U|!_7qd|5.>U"d~JJP=qC'\!&p6CxQXNF3qMCZv> cC^&:Xi[=GMud'ؕ Qrc;!xm\7C~9On,Qi ٮ; 82`HǺ4TI =*~`}F((C*~3 :G_U<()r2[hjRGgڦLݟƽz#QJwL2r|}zۍxWMaXF+ax]6ԺXB9]qcOdxclkl7,f0(k\wh2I8dF o KF*M]* )6U /jr9Q mq8oReD v\4H8dB#͔:w&4K!Id Yf%f \"e`|.GMnE) HD'/w\%%J{1ݸrp'R#Gˢ !QkI\ѻv", ɋFN,'$W㰸Af^>9<%|BZZU#Y|{;awY`5~NOe{TDcIbYOlـ{zOT0?tW>TRF*gZZ,ґq,F53Ee;[og& SOs&>dq ,𴶝65dzɶ[bSoW8dcY-l.{KQ"»! Al<> endobj 990 0 obj [991 0 R] endobj 991 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 583 0 0 832 0 0 cm /ImagePart_2215 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 379.899 713.7 Tm 114 Tz 3 Tr /OPExtFont3 11 Tf (LIST OF FIGURES ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 114 Tz 3 Tr 1 0 0 1 116.4 670.25 Tm 92 Tz (6.2 Following Figure 6.1 experiment setting, compare forecast ensem-) Tj 1 0 0 1 142.3 647.45 Tm 90 Tz (ble using 6-ball. Direct forecasts are compared with forecasts with ) Tj 1 0 0 1 142.3 624.649 Tm 91 Tz (random adjustment \(left\) and forecasts with analogue adjustment ) Tj 1 0 0 1 142.8 601.85 Tm 93 Tz (\(right\). The initial condition ensemble is formed by inverse noise ) Tj 1 0 0 1 141.599 578.549 Tm 91 Tz (with 64 ensemble members. For each forecast method, 2048 fore-) Tj 1 0 0 1 142.099 556 Tm 90 Tz (casts are made. Each row shows the comparison for different lead ) Tj 1 0 0 1 142.099 532.95 Tm 91 Tz (time. First row denotes lead time 1, second lead time 2, third lead ) Tj 1 0 0 1 141.849 509.699 Tm 126 Tz (time 4, forth lead time 8 and fifth lead time 16. 135 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 126 Tz 3 Tr 1 0 0 1 116.4 486.649 Tm 91 Tz (6.3 Following Figure 6.1 one step forecast ensemble in the state space. ) Tj 1 0 0 1 141.849 463.6 Tm 92 Tz (The initial condition ensemble is formed by dynamical consistent ) Tj 1 0 0 1 142.099 440.55 Tm 146 Tz (ensemble with 64 ensemble members. :136 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 146 Tz 3 Tr 1 0 0 1 116.4 417.75 Tm 93 Tz (6.4 Comparing forecast ensemble using 6-ball. Observations are gen-) Tj 1 0 0 1 141.849 394.949 Tm (erated by Ikeda Map with IID uniform bounded noise U\(0, 0.01\). ) Tj 1 0 0 1 141.849 371.899 Tm 94 Tz (The truncated Ikeda model is used to make forecast. The initial ) Tj 1 0 0 1 141.849 348.899 Tm 93 Tz (condition ensemble is formed by dynamical consistent ensemble ) Tj 1 0 0 1 141.849 325.85 Tm 91 Tz (with 64 ensemble members. Each row of pictures shows the com-) Tj 1 0 0 1 141.599 302.799 Tm 98 Tz (parison for different lead time. First row denotes lead time 1, ) Tj 1 0 0 1 141.599 280 Tm 94 Tz (second lead time 2, third lead time 4, forth lead time 8 and fifth ) Tj 1 0 0 1 141.599 256.7 Tm 374 Tz (lead time 16. 137 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont3 11 Tf 374 Tz 3 Tr 1 0 0 1 288.5 49.6 Tm 76 Tz (188 ) Tj ET EMC endstream endobj 992 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 993 0 obj <> stream 0  ,,77b7 \!'Cix('4d?FU GGYf6%ݰWZY l I;.wľ9M#hH8 /c!'eVNHߒ DLn)Թuj4Wj05^8H8<"N`twf1NaSO )V{/l)hT0cʇv[JɺmV 0^So5|܈&0K313Ӫ%);a|\1@v F4($X@FH.zl t<}?K&;4# /75>Xf@ GNiO x\cK/l2^°?>bN`w4.ߔSX7.-AOL5A @` Xy]]f TUG)+(?<kWњ g=ؗFW2Z9:)}^/\I4<ǟN{WKw91"H&E$_81,qNKi|fWUZ"WJV5sBw26^y//@'f@ߓ:l?k<>LLU[LuܥZ;KwE a%WPi[6B8XpO@+?p}QzE7O-hȓ$ [`gӞW7)2c*:;szP6b弳L'2n [+\ ;cwkSj"`وRlɬe3R@1.OzsŗhXo`64ʽ%oW6<`PnAL6- rdH/H7Gow@t{Cбp9FռQy?u|nL` =;H d5AT l b@ع]<ػ>VbJc(% ܰNvqF[ z~6Fr͔ulZޗ&a*77,kH9\B]1.ֵK}jg^7WwO?suH GsY %+aL3|$_\hA)'&[4@gӁm?d 0K?Մ3GLDQv}=xb͓͆rRf[P~^S%3bT'iM'lM?uD|qi/s}@)0>s1VW7ތ+ky/l+by9Zܷ\g&ΐU Md90q'j1d wOb}b.2=X#U=98օaXm‚4yTV w!H4Ǽ`L0ÂCLp?tR#0X 6V8Od[2_5<m,{aX&f?AΟJ7Sqއ@dʷ8機&ke磫AԮˊtdwj\ &#6WTW`cjb<:]T=gXL<=YC9HLfQ$Vba7B>vTZ4Ҿ]H'&:m0KiMY5($5+)?5$q;DxAٓa}MWy֚kCܭ8 %Ȅ_.Ao l6[X .~ZPSiaun9gO ;AT[žjo Buraq_݈a R|[L3A V%v9,D@?mO Jӝ|Z18d))Jȟju/P_ЮT\T(FW.^!#q5#̲(wWi&Wlk[(ӎ!  d!OJ4>Ox6&QmT2a;v͞^,թ>aOt,]^ME| ՚ևP`mcu:|1&zUW f0^oƩS3cR]SbdB.t(k65G盛@cg斠VXE_H(^Ņj=S#+o)w ^N-^U!gTiڂ᤽l?K&Cozw0D C)d:F1q1w]iQ۶@a_swr݄`} 2Y%~K?lM RX (&KM 9Vf]`K~I.C D@P[9h JӺ끣iae.KM,5`# ,yxTSSԥNLՄf[-hUBv㼕QR{eІ)E I} ufz\H: !3)~R k[n9!O qintFսXPST^47迅=A&8;AKLJm;f: jVN謪I ?L}7zQ>.ԠĿaԾY8!ٰ7ڥU]Ta(*@[[᯾kFT"8U\)BA5$T5>zeV{ַ=eV%/lMN +pCJ\U-gs#Phϖ,6Nv Aަ ,83LK O09KıTf5 ?Rm4A/)/`!@±BViC8{Y[6; ͰX`'+x[{,nl 򥿍ލ Z etphuKgWe{io=L? q# 5pub糌Zj^#g:s^ue`0-!9dc|E}$9 =$6%׏ƿ^WOA ieMځ̷2[S=R4_*+|メ#hfCq$m']VZDs c^nR:W !p~BL6nsTU*_ œE}TDSyXef֟ ˭ax}!|03M\0 & :H 8A,ɇ~c*")@DysƸqTz.w'5q:٥&5Wl:QR1 IwMfE'ΧpϾIQ_Mv[Rd^H@.vF~aʃ>86e.'1>"  : ؞Ta|~v_W磀cL6#+pjjmse%g)+Q(-]#h|w)؂$7ltQynLFNac%ZM I]+X'L2@ieºu$I%M4!sr  ̼䍪)o6<}c/&|fJ502.>x`n{ _8+s uQz1q͓4]aoW!)Pi B(+y:jDzU_捚 z،6ote|K.vҟx zKdyi&rL} If{AuB~-kJhS9G.FuqBEQ4bt??fWî [l-jիji-l =+?|1,4t3@^NK !*\YMUUSȟ )21Ki=m{yqmv}켐4Lƒz+HNV0 4ݞ]ڻCk3*N'ڨ6[7یU.'`~+esaK3%ԇty&jAl_O0eG`BJ 9'BF̏aeHBO+O0@4?-m 2f*f4e:zWޗM־X|(k5׾ aԥ r&>ٷɶ7"+4ʋȂҴʶչi'᳢ְ<#PRM#xZ[v _ۑDԵ7mx%00k =|Nڟn V&B¯e j6+d _2Ң y"r|?->{ƽTA\8uP_z(Oc8Ny%C!&fU{TM9dz}ō FrSˀ}Y,YO<$H6oGCJh1ଚE&|Tc`?N&ɽa9^62dӟ?yGԃkʨL$- 5^]{'vv޾ʑқ LNψsBibtgch 7TY9㾄Gg<6);Q!j@B S3#-k~UK9 0B?m1@G dYǼg,a 379*̉i.P_9l#?"k,V5*1ka||enL2sM7Tpo endstream endobj 994 0 obj <> endobj 995 0 obj [996 0 R] endobj 996 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 588 0 0 833 0 0 cm /ImagePart_2216 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 379.899 713.7 Tm 112 Tz 3 Tr /OPExtFont5 12.5 Tf (LIST OF FIGURES ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 112 Tz 3 Tr 1 0 0 1 116.4 670.5 Tm 103 Tz (6.5 Following Figure 6.1 experiment setting, Ignorance score of three ) Tj 1 0 0 1 142.099 647.7 Tm 100 Tz (forecasting methods relative to climatology is plotted vs lead time. ) Tj 1 0 0 1 142.3 625.149 Tm 104 Tz (The error bars are 90% bootstrap re-sampling bars. In panel \(a\), ) Tj 1 0 0 1 141.849 602.35 Tm 103 Tz (the initial condition ensemble is formed by inverse noise with 64 ) Tj 1 0 0 1 142.099 579.299 Tm 102 Tz (ensemble members. In panel \(b\), the initial condition ensemble is ) Tj 1 0 0 1 141.849 556.5 Tm 100 Tz (formed by dynamical consistent ensemble with 64 ensemble mem-) Tj 1 0 0 1 141.849 533.5 Tm 103 Tz (bers. Panel \(c\) is the combination of panel \(a\) and \(b\). Ignorance ) Tj 1 0 0 1 141.599 510.449 Tm 160 Tz (is calculated based upon 2048 forecasts. 139 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 160 Tz 3 Tr 1 0 0 1 116.15 487.649 Tm 106 Tz (6.6 Following Figure 6.1, six 1-step forecast examples are plotted in ) Tj 1 0 0 1 141.599 464.6 Tm 99 Tz (the state space. Here the adjustment is obtained from imperfection ) Tj 1 0 0 1 141.849 441.55 Tm 224 Tz (error instead of model error 142 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 224 Tz 3 Tr 1 0 0 1 116.15 418.5 Tm 106 Tz (6.7 Following Figure 6.2, comparing three forecast ensemble results ) Tj 1 0 0 1 141.599 395.699 Tm 104 Tz (using c-ball. Here the adjustment is obtained from imperfection ) Tj 1 0 0 1 141.599 372.699 Tm 224 Tz (error instead of model error 143 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 224 Tz 3 Tr 1 0 0 1 115.9 349.649 Tm 99 Tz (6.8 Following Figure 6.5a, Ignorance score of three forecasting methods ) Tj 1 0 0 1 141.349 326.85 Tm 101 Tz (relative to climatology is plotted vs lead time. Forecast adjustment ) Tj 1 0 0 1 141.099 303.799 Tm 179 Tz (is obtained from imperfection error 144 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 179 Tz 3 Tr 1 0 0 1 115.7 280.75 Tm 103 Tz (6.9 Ignorance score of forecast with adjustment where the adjustment ) Tj 1 0 0 1 140.9 257.7 Tm 102 Tz (is generated from the observations with different noise level. The ) Tj 1 0 0 1 141.099 234.899 Tm 105 Tz (initial condition ensemble is formed by inverse noise with fixed ) Tj 1 0 0 1 140.9 211.899 Tm 101 Tz (noise level U\(0, 0.01\) so that the observations with different noise ) Tj 1 0 0 1 140.9 188.85 Tm 105 Tz (level only affect the imperfection error. The error bars are 90% ) Tj 1 0 0 1 140.9 165.799 Tm 104 Tz (bootstrapped error bars. In panel a, the forecast is made by ran-) Tj 1 0 0 1 140.9 143 Tm 109 Tz (dom adjustment; in panel b, the forecast is made by analogue ) Tj 1 0 0 1 141.099 119.95 Tm 422 Tz (adjustment :145 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 422 Tz 3 Tr 1 0 0 1 287.75 50.6 Tm 87 Tz (189 ) Tj ET EMC endstream endobj 997 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 998 0 obj <> stream 0 ,,~~b7 \H^s4yfȠ4i7`).(d7G52,ONYfZ \KSNN *F: PZet[?Z6졤ԷH=7Bۈdl"M;n(W QC.+|ÀZ)١ xXyzRpi.'C3h+Zh^Zbi^ډFRpKD9jLS*.e?/Q$7 &aԙ=V9O {a8ٮ_5 D4%0kM(e_ oBj%]y%_%/bcg7b=/4It֕4xٝ6Jkhsf [htj7zI7.@,!O^3Vf-_GWj6J\ΰ>$*~Ev;@ c 3_]VTz "W†L-EE iM'BcBξn$b iE֗%R+O 59/!Q|eflX t#D`-oo Lk7 ń4a~Ta.PLѨ8~b~u^N}ZEcp.i~/ޘo_d'I4B,'s}#>B3$^o!ą~%vD(sbJR,$ނ:[Fy>NRF+#(),:7JM sx6o-%^Z :LFEB> 49E2ZD!+k-|^M$ pX;_Q}脫L vg8)tz(:fx!ZA~tVstBguJilsuv~W!8;`b=u䔠;/;Z^V [<5q]) ZLkF\ۅ=a2Z1,1unlu"01!˜;o\a6)I"Ѿ1ҝX ^t.2zh(Z'PeQ{ Obϩ5$dT[u5mzl{E?#U*t DaԹ>߿,Tw1*.^85= Tw[bzq?T cX;2K9ed3ZmG%mi2=NWUݣMZ@ttgsZ!ڄF*Eh9|^<qpȫF@ oTw&%/ zmA!?НFgGJD@+sKyI0i͕yŚKVnQU*1a^uBZt?Lˢ[F ۰TPWJ|=ؐ<*~Y[u*SA+沼6Svpa" OTRY]eF)l3Lsa>wʈ8 `{..2`?.jF%Tg|9򯓺KF2%K|pKG$JBMSap):s>a(;z&(%o9x}fӇb,̶N"DЬ`B/_DĖޙ|SDt;Z|\nAV(YGMZ}{GA tjjRtOI.liqBU@ mxa,bzY߸b”M61Bu/Ԭv"vNpJ,Abq\`R? C*cGML'OWE r$C뺁 cI_D<ƭ|Epq:j0Ax=RAVMDZL~ukb!I#OzHMfDRKAg]s,(Zpg%y4ԷpL_:u6W5DGd]Sٱћsyzͪ0J4=~ >S"oV)AtDs(qhR M[p_ msw2=ⱁxBVp4߯. }QI@YJŁĪ&Mԇ5evƘB*ص "L-hҩK7)jKl,)1ǜk7mzݕyE(4 & ׭ⴹ߁eA ^[a~c@c8kU 2FۆlBݵ`a4,h:t糸Y׷u t+4OOʽy!&G5 #}nt*gR;! 9W/Xoͫ476kD4^Rg~}g`EG"b[E[ N鋧_m 7S Ğ 1>e(Gy̒:W{z_aҴ'R)t:8smKf 5IBCN '\9}$.y-DWG1kT ,VkYʗYDzI5 {r$]<SڅaH)+g- S^~(goUQ#k5Č5:VHwV.Ep\u;0I 4+>4_.C"z.FHA]0F*>Q6k-^DDj=PSp^H9ekߞҡ,\7W~4)SnK{JF+[c\7qC1t+.p N bmYdgu?x/6RhI1'Vuޮ6NU]$j$6˜h59X'Me Lrw56| ӀUڒJYhx埍M;- #?nJpX""cմr2zF7I>j~n+?O \cn.)u wQ;n8<0f]91ld-Tz4lUR5QN6ZĬw+^zgp L->A%;`z)` HU^kq lY?o%[W0о}:S ,hBUQR+(H.N u _#D,ޤc埣 w)Ӭ۩vjiOFw:o&qEqj#QX7aVTJn4-+:93IDT*2^Dez  ֵ9uu2! e/L^I`wD+gyNx4;bp:qv+.(!^?[U:7/]Vǰ7jcG2Jmuc<<cD-P0gYq3C[OVbDm!$ݻਥ7IGY>` hsmH2F6$=#)kSV b9e$…ץM 4)ML-+cq8 i<*D'+bH\eNUWgJ?d 1rXm6?!v'<ҸZ&:fi:򼑋$g׏[2oQ$~6d,eCml Q^`+Va:*.g> b"?-tߖi_]ݖ,2 e*BI{$&+V(ڰ7JC a*lUK$N~^=G͹EUhBт(Yhc*i* Pi#9Q&*C%;"(arCw*Ʌ52 QaVД=/CpCfQxrӥ!B?rQs0c ׅ'Sf8֯;> sQi9Gv%%Y]@@=k IM M9\ ;Dw4_G[ibGd&hoTI]Uf?I,S:vbDZeWpqr{v %3k `ThWm+vñJĴ']bG/aʝF)oJ_!26P8mE-7)2sNҤ<>t5sTa'H[Ąm @eMPGT?mu5\oH'gd8cQL+a+b]4D5!טYK9bDk_KlɥKT[0l{~\\YfL{Dq,\+2}`]>:9ֽ<V/nqm@h{APfQ=[^Ź(g?|nKK.Bt-Rz ʠ׳C̨J=tµMH) (LHjT@Q۴DڅKsP=jEYkE-Hu0&U#SA8 2Whb &aQTj@f=fDncݢlrlc-_7Dyb ,W>t5:kTѻ*'^kK'oߙQ5V E'(iR=9 ^0RvKBwQi/C:mYؑn ߢUgNWxzṘ EAc͓Ox0N<rT@oBYFkZ8E#eNUɑFh|Dʋ4.O s'5\ۋ['3JZ;11}v"t:'9cQUN쑁"Cp0~ڕ|갳+yY+\C&$8ˁuhP`cmx"0fC9p je·l҆O _jﾸe[鰖j_AT.hKiQͣD;ʪOM+|Cr*= [^b9oQJ ŷ){^G5o=*8Ǜ&W4 q =1(s$ IusTyYp 2@@ӎe1V.@)iF9zu|Cipvf+W#nv4"/IrgLua;J{%U{<_SbZoePG%} &lu8S `A7i $T,71IثW;e&~{?Y%/pb!kGUW_3uyB2rg*oeq_+ 8y?P}hcotiO#~mU^Hiw ex϶eR@jD dU'^N۷!s7糕CuNP2a &7U[+ iqsyWB \}gD1z~ȚK &NEukFXFjBte{aR$J}%Zt~&Lo;C)rTЩ&~Ն3͋ #uB{ u$}Ҧʠ}m8NڎP@5:*&} CWَ9%4[,SΚf$`,XfkR"OHdK(]PўFhx 2tC\F!NƩfSte0BjË7N1N#LF8^62J6aƣYǎ0a9>]N7!p?t: 'NP\g۪Ɂ͞e߫u9 c!Cs0n&:A?,π(x= s&KI!mX.Z:}'Z3hLNdM?p -KFstpA3TwɄ&S)&J3InB$tBn]ϟ~t~Я T譲ygḙbO8v_:W?%-n/&2|E5@6f48J"Xνg`?m3cەl &qohfU]hA4HOڑ|~D5bF7KAbol*ݟHIJ[U(` や6BE\oƉإ^M endstream endobj 999 0 obj <> endobj 1000 0 obj [1001 0 R] endobj 1001 0 obj <> stream 0 0 0 rg 0 0 0 RG q 1 0 0 1 0 0 cm 591 0 0 832 0 0 cm /ImagePart_2217 Do Q /P <> BDC BT 0 0 0 rg 0 0 0 RG 1 0 0 1 382.3 712.7 Tm 112 Tz 3 Tr /OPExtFont5 12.5 Tf (LIST OF FIGURES ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 112 Tz 3 Tr 1 0 0 1 118.799 669.75 Tm 103 Tz (6.10 Doubling time of Ikeda system-model pair. Panels a\) and c\) esti-) Tj 1 0 0 1 144.5 646.95 Tm (mate the doubling time by assuming the model is perfect. Panels ) Tj 1 0 0 1 144.25 624.149 Tm 100 Tz (b\) and d\) estimate the doubling time based on the states generated ) Tj 1 0 0 1 144.25 601.35 Tm 104 Tz (Ikeda Map and using the Truncated Ikeda Map as the model. a\) ) Tj 1 0 0 1 144 578.549 Tm 99 Tz (and b\) are noise free cases while c\) and d\) have observational noise ) Tj 1 0 0 1 144 555.75 Tm 106 Tz (N\(0, 0.0001\). Note that the scale of the color bar is different in ) Tj 1 0 0 1 143.75 533.2 Tm 99 Tz (each panel. ) Tj 1 0 0 1 479.3 532.5 Tm 91 Tz /OPExtFont5 11.5 Tf (151 ) Tj 1 0 0 1 494.149 532.5 Tm 32 Tz /OPExtFont5 12.5 Tf ( ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 32 Tz 3 Tr 1 0 0 1 117.849 509.699 Tm 102 Tz (6.11 Ignorance as a function of forecast lead time in the Ikeda system-) Tj 1 0 0 1 143.3 486.649 Tm 103 Tz (model pair experiment. The observations are generated by Ikeda ) Tj 1 0 0 1 143.05 463.6 Tm 101 Tz (Map with IID N\(0, 0.05\) observational noise, initial condition en-) Tj 1 0 0 1 143.05 440.8 Tm 100 Tz (semble is built by using Inverse Noise and ) Tj 1 0 0 1 353.3 440.55 Tm 90 Tz /OPExtFont4 12 Tf (ISG.Dc ) Tj 1 0 0 1 392.899 440.55 Tm 117 Tz /OPExtFont5 12.5 Tf (ensemble. . . . 154 ) Tj ET EMC /P <> BDC BT 0 0 0 rg 0 0 0 RG /OPExtFont5 12.5 Tf 117 Tz 3 Tr 1 0 0 1 287.75 50.1 Tm 87 Tz (190 ) Tj ET EMC endstream endobj 1002 0 obj <> /ProcSet 7362 0 R /XObject<>>> endobj 1003 0 obj <> stream 0 ,,{99b7 / r"SĠBy;Ozͪ|^ZzjŪFu)`TߏAM{A E4C얽K~[Iy"kU lQ"&AA hQp`=xJ%Ik( g|i&1ZlG53u؜~ X#@>jTIU SQC_4M|i:WAZ$.TQt皰S~P ?ڇYپ̾}%e? F-~ #@mL2[{ޕ2G;C1꠵/:<|[TqXF $t@6x\#(%C3pJ ~O]@/맋ro$-{d›uNN?0DZ1F`A͔HmAV~:i#PN `>A?[%4ԕK&#u˜׸高ɮd3~ϟp2<bE=̴6N4"; 9i W;)ri,]\@f7i9i-8E}#lu5 xHu%*=M<'WЫJ)/1D52~x'w exP b1M{?d)+x#r$\FZNUoSuf+憚FOC`zB!S[ޛ LQ-wW5BT9]7:M?o7wHz';Abץ-bZy^ 'sfWeeHu83?nuh@?w䁙SфqU-eHZ6VЇ+K}=\j,a=(!q7B#B9l+: sLdgs(';Q|BI6RT;sk@~wsDr#/TyA C~pٵV+ܞ&# EcSz3dU0۝xGd[Pc-"4`K[{ ެ:ۮG|d@I=~xM{Mh*~ VxXB]T닻v^M_:d+b0 5Z]8ӊ/sjW 袽@}(Z3K[!rVe!p\E|@Z&&D<'9" H*M'HsKFW+;.|} yIs9G e}Ⱦ,Tc5cpE*!r, `U T݅{54(B#-s?p̐MhN}5:Z90N,{qr/9gd#)vuOedY}+fpcݏAi5BO|9b̎ `!e狷Z;c@2e47`n(\T5 PAlN(m@30;o_j{<]ئUxk4ݙ9x ro䮯~fn|ѷu92+9VP4o7A0Ӑݐ|ʓc}M}yvpS7HK ~2hVMr3 8JcD? 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/P 1322 0 R /S/TD /Type/StructElem>> endobj 1332 0 obj <> /K 24 /P 1331 0 R /Pg 16 0 R /S/P /Type/StructElem>> endobj 1333 0 obj <> endobj 1334 0 obj <> /K[1335 0 R] /P 1333 0 R /S/TD /Type/StructElem>> endobj 1335 0 obj <> /K[25 1336 0 R 1337 0 R] /P 1334 0 R /Pg 16 0 R /S/P /Type/StructElem>> endobj 1336 0 obj <> endobj 1337 0 obj <> endobj 1338 0 obj <> /K[1339 0 R] /P 1333 0 R /S/TD /Type/StructElem>> endobj 1339 0 obj <> /K 28 /P 1338 0 R /Pg 16 0 R /S/P /Type/StructElem>> endobj 1340 0 obj <> /K[1341 0 R] /P 1333 0 R /S/TD /Type/StructElem>> endobj 1341 0 obj <> /K 29 /P 1340 0 R /Pg 16 0 R /S/P /Type/StructElem>> endobj 1342 0 obj <> /K[1343 0 R] /P 1333 0 R /S/TD /Type/StructElem>> endobj 1343 0 obj <> /K 30 /P 1342 0 R /Pg 16 0 R /S/P /Type/StructElem>> endobj 1344 0 obj <> endobj 1345 0 obj <> /K[1346 0 R] /P 1344 0 R /S/TD /Type/StructElem>> endobj 1346 0 obj <> /K[31 1347 0 R 1348 0 R] /P 1345 0 R /Pg 16 0 R /S/P /Type/StructElem>> endobj 1347 0 obj <> endobj 1348 0 obj <> endobj 1349 0 obj <> /K[1350 0 R] /P 1344 0 R /S/TD /Type/StructElem>> endobj 1350 0 obj <> /K[34 1351 0 R 1352 0 R] /P 1349 0 R /Pg 16 0 R /S/P /Type/StructElem>> endobj 1351 0 obj <> endobj 1352 0 obj <> endobj 1353 0 obj <> /K[1354 0 R] /P 1344 0 R /S/TD /Type/StructElem>> endobj 1354 0 obj <> /K 37 /P 1353 0 R /Pg 16 0 R /S/P /Type/StructElem>> endobj 1355 0 obj <> /K[1356 0 R] /P 1344 0 R /S/TD /Type/StructElem>> endobj 1356 0 obj <> /K 38 /P 1355 0 R /Pg 16 0 R /S/P /Type/StructElem>> endobj 1357 0 obj <> endobj 1358 0 obj <> /K[1359 0 R] /P 1357 0 R /S/TD /Type/StructElem>> endobj 1359 0 obj <> /K[39 1360 0 R 1361 0 R] /P 1358 0 R /Pg 16 0 R /S/P /Type/StructElem>> endobj 1360 0 obj <> endobj 1361 0 obj <> endobj 1362 0 obj <> /K[1363 0 R] /P 1357 0 R /S/TD /Type/StructElem>> endobj 1363 0 obj <> /K[42 1364 0 R 1365 0 R] /P 1362 0 R /Pg 16 0 R /S/P /Type/StructElem>> endobj 1364 0 obj <> endobj 1365 0 obj <> endobj 1366 0 obj <> /K[1367 0 R] /P 1357 0 R /S/TD /Type/StructElem>> 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/Type/StructElem>> endobj 2417 0 obj <> /K 3 /P 2413 0 R /Pg 184 0 R /S/P /Type/StructElem>> endobj 2418 0 obj <> /K[2419 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2419 0 obj <> /K 4 /P 2418 0 R /Pg 184 0 R /S/P /Type/StructElem>> endobj 2420 0 obj <> endobj 2421 0 obj <> /K 0 /P 2420 0 R /Pg 189 0 R /S/P /Type/StructElem>> endobj 2422 0 obj <> /K 1 /P 2420 0 R /Pg 189 0 R /S/P /Type/StructElem>> endobj 2423 0 obj <> /K[2424 0 R] /P 2420 0 R /S/Table /Type/StructElem>> endobj 2424 0 obj <> endobj 2425 0 obj <> /K[2426 0 R 2427 0 R 2428 0 R] /P 2424 0 R /S/TD /Type/StructElem>> endobj 2426 0 obj <> /K 2 /P 2425 0 R /Pg 189 0 R /S/P /Type/StructElem>> endobj 2427 0 obj <> /K 3 /P 2425 0 R /Pg 189 0 R /S/P /Type/StructElem>> endobj 2428 0 obj <> /K 4 /P 2425 0 R /Pg 189 0 R /S/P /Type/StructElem>> endobj 2429 0 obj <> /K[2430 0 R] /P 2424 0 R /S/TD /Type/StructElem>> endobj 2430 0 obj <> /K 5 /P 2429 0 R /Pg 189 0 R /S/P /Type/StructElem>> endobj 2431 0 obj <> /K 6 /P 2420 0 R /Pg 189 0 R /S/P /Type/StructElem>> endobj 2432 0 obj <> /K 7 /P 2420 0 R /Pg 189 0 R /S/P /Type/StructElem>> endobj 2433 0 obj <> /K 8 /P 2420 0 R /Pg 189 0 R /S/P /Type/StructElem>> endobj 2434 0 obj <> /K 9 /P 2420 0 R /Pg 189 0 R /S/P /Type/StructElem>> endobj 2435 0 obj <> /K 10 /P 2420 0 R /Pg 189 0 R /S/P /Type/StructElem>> endobj 2436 0 obj <> /K 11 /P 2420 0 R /Pg 189 0 R /S/P /Type/StructElem>> endobj 2437 0 obj <> /K 12 /P 2420 0 R /Pg 189 0 R /S/P /Type/StructElem>> endobj 2438 0 obj <> /K[2439 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2439 0 obj <> /K 13 /P 2438 0 R /Pg 189 0 R /S/P /Type/StructElem>> endobj 2440 0 obj <> endobj 2441 0 obj <> /K 0 /P 2440 0 R /Pg 194 0 R /S/P /Type/StructElem>> endobj 2442 0 obj <> /K 1 /P 2440 0 R /Pg 194 0 R /S/P /Type/StructElem>> endobj 2443 0 obj <> /K 2 /P 2440 0 R /Pg 194 0 R /S/P /Type/StructElem>> endobj 2444 0 obj <> /K 3 /P 2440 0 R /Pg 194 0 R /S/P /Type/StructElem>> endobj 2445 0 obj <> /K 4 /P 2440 0 R /Pg 194 0 R /S/P /Type/StructElem>> endobj 2446 0 obj <> /K[2447 0 R] /P 2440 0 R /S/Table /Type/StructElem>> endobj 2447 0 obj <> endobj 2448 0 obj <> /K[2449 0 R 2450 0 R] /P 2447 0 R /S/TD /Type/StructElem>> endobj 2449 0 obj <> /K 5 /P 2448 0 R /Pg 194 0 R /S/P /Type/StructElem>> endobj 2450 0 obj <> /K 6 /P 2448 0 R /Pg 194 0 R /S/P /Type/StructElem>> endobj 2451 0 obj <> /K[2452 0 R] /P 2447 0 R /S/TD /Type/StructElem>> endobj 2452 0 obj <> /K 7 /P 2451 0 R /Pg 194 0 R /S/P /Type/StructElem>> endobj 2453 0 obj <> /K 8 /P 2440 0 R /Pg 194 0 R /S/P /Type/StructElem>> endobj 2454 0 obj <> /K 9 /P 2440 0 R /Pg 194 0 R /S/P /Type/StructElem>> endobj 2455 0 obj <> /K 10 /P 2440 0 R /Pg 194 0 R /S/P /Type/StructElem>> endobj 2456 0 obj <> /K 11 /P 2440 0 R /Pg 194 0 R /S/P /Type/StructElem>> endobj 2457 0 obj <> /K[2458 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2458 0 obj <> /K 12 /P 2457 0 R /Pg 194 0 R /S/P /Type/StructElem>> endobj 2459 0 obj <> endobj 2460 0 obj <> /K 0 /P 2459 0 R /Pg 199 0 R /S/P /Type/StructElem>> endobj 2461 0 obj <> /K 1 /P 2459 0 R /Pg 199 0 R /S/P /Type/StructElem>> endobj 2462 0 obj <> /K 2 /P 2459 0 R /Pg 199 0 R /S/P /Type/StructElem>> endobj 2463 0 obj <> /K 3 /P 2459 0 R /Pg 199 0 R /S/P /Type/StructElem>> endobj 2464 0 obj <> /K 4 /P 2459 0 R /Pg 199 0 R /S/P /Type/StructElem>> endobj 2465 0 obj <> /K 5 /P 2459 0 R /Pg 199 0 R /S/P /Type/StructElem>> endobj 2466 0 obj <> /K[2467 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2467 0 obj <> /K 6 /P 2466 0 R /Pg 199 0 R /S/P /Type/StructElem>> endobj 2468 0 obj <> endobj 2469 0 obj <> /K 0 /P 2468 0 R /Pg 204 0 R /S/P /Type/StructElem>> endobj 2470 0 obj <> /K 1 /P 2468 0 R /Pg 204 0 R /S/P /Type/StructElem>> endobj 2471 0 obj <> /K 2 /P 2468 0 R /Pg 204 0 R /S/P /Type/StructElem>> endobj 2472 0 obj <> /K 3 /P 2468 0 R /Pg 204 0 R /S/P /Type/StructElem>> endobj 2473 0 obj <> /K 4 /P 2468 0 R /Pg 204 0 R /S/P /Type/StructElem>> endobj 2474 0 obj <> /K 5 /P 2468 0 R /Pg 204 0 R /S/P /Type/StructElem>> endobj 2475 0 obj <> /K[2476 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2476 0 obj <> /K 6 /P 2475 0 R /Pg 204 0 R /S/P /Type/StructElem>> endobj 2477 0 obj <> endobj 2478 0 obj <> /K 0 /P 2477 0 R /Pg 209 0 R /S/P /Type/StructElem>> endobj 2479 0 obj <> /K 1 /P 2477 0 R /Pg 209 0 R /S/P /Type/StructElem>> endobj 2480 0 obj <> /K 2 /P 2477 0 R /Pg 209 0 R /S/P /Type/StructElem>> endobj 2481 0 obj <> /K 3 /P 2477 0 R /Pg 209 0 R /S/P /Type/StructElem>> endobj 2482 0 obj <> /K[2483 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2483 0 obj <> /K 4 /P 2482 0 R /Pg 209 0 R /S/P /Type/StructElem>> endobj 2484 0 obj <> endobj 2485 0 obj <> /K 0 /P 2484 0 R /Pg 214 0 R /S/P /Type/StructElem>> endobj 2486 0 obj <> /K 1 /P 2484 0 R /Pg 214 0 R /S/P /Type/StructElem>> endobj 2487 0 obj <> /K 2 /P 2484 0 R /Pg 214 0 R /S/P /Type/StructElem>> endobj 2488 0 obj <> /K 3 /P 2484 0 R /Pg 214 0 R /S/P /Type/StructElem>> endobj 2489 0 obj <> /K 4 /P 2484 0 R /Pg 214 0 R /S/P /Type/StructElem>> endobj 2490 0 obj <> /K[2491 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2491 0 obj <> /K 5 /P 2490 0 R /Pg 214 0 R /S/P /Type/StructElem>> endobj 2492 0 obj <> endobj 2493 0 obj <> /K 0 /P 2492 0 R /Pg 219 0 R /S/P /Type/StructElem>> endobj 2494 0 obj <> /K 1 /P 2492 0 R /Pg 219 0 R /S/P /Type/StructElem>> endobj 2495 0 obj <> /K 2 /P 2492 0 R /Pg 219 0 R /S/P /Type/StructElem>> endobj 2496 0 obj <> /K 3 /P 2492 0 R /Pg 219 0 R /S/P /Type/StructElem>> endobj 2497 0 obj <> /K 4 /P 2492 0 R /Pg 219 0 R /S/P /Type/StructElem>> endobj 2498 0 obj <> /K 5 /P 2492 0 R /Pg 219 0 R /S/P /Type/StructElem>> endobj 2499 0 obj <> /K 6 /P 2492 0 R /Pg 219 0 R /S/P /Type/StructElem>> endobj 2500 0 obj <> /K 7 /P 2492 0 R /Pg 219 0 R /S/P /Type/StructElem>> endobj 2501 0 obj <> /K 8 /P 2492 0 R /Pg 219 0 R /S/P /Type/StructElem>> endobj 2502 0 obj <> /K[2503 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2503 0 obj <> /K 9 /P 2502 0 R /Pg 219 0 R /S/P /Type/StructElem>> endobj 2504 0 obj <> endobj 2505 0 obj <> /K 0 /P 2504 0 R /Pg 224 0 R /S/P /Type/StructElem>> endobj 2506 0 obj <> /K 1 /P 2504 0 R /Pg 224 0 R /S/P /Type/StructElem>> endobj 2507 0 obj <> /K 2 /P 2504 0 R /Pg 224 0 R /S/P /Type/StructElem>> endobj 2508 0 obj <> /K 3 /P 2504 0 R /Pg 224 0 R /S/P /Type/StructElem>> endobj 2509 0 obj <> /K 4 /P 2504 0 R /Pg 224 0 R /S/P /Type/StructElem>> endobj 2510 0 obj <> /K 5 /P 2504 0 R /Pg 224 0 R /S/P /Type/StructElem>> endobj 2511 0 obj <> /K 6 /P 2504 0 R /Pg 224 0 R /S/P /Type/StructElem>> endobj 2512 0 obj <> /K 7 /P 2504 0 R /Pg 224 0 R /S/P /Type/StructElem>> endobj 2513 0 obj <> /K 8 /P 2504 0 R /Pg 224 0 R /S/P /Type/StructElem>> endobj 2514 0 obj <> /K[2515 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2515 0 obj <> /K 9 /P 2514 0 R /Pg 224 0 R /S/P /Type/StructElem>> endobj 2516 0 obj <> endobj 2517 0 obj <> /K[2518 0 R 2525 0 R 2531 0 R 2536 0 R] /P 2516 0 R /S/Table /Type/StructElem>> endobj 2518 0 obj <> endobj 2519 0 obj <> /K[2520 0 R] /P 2518 0 R /S/TD /Type/StructElem>> endobj 2520 0 obj <> /K[0 2521 0 R 2522 0 R] /P 2519 0 R /Pg 229 0 R /S/P /Type/StructElem>> endobj 2521 0 obj <> endobj 2522 0 obj <> endobj 2523 0 obj <> /K[2524 0 R] /P 2518 0 R /S/TD /Type/StructElem>> endobj 2524 0 obj <> /K 3 /P 2523 0 R /Pg 229 0 R /S/P /Type/StructElem>> endobj 2525 0 obj <> endobj 2526 0 obj <> /K[2527 0 R 2528 0 R] /P 2525 0 R /S/TD /Type/StructElem>> endobj 2527 0 obj <> /K 4 /P 2526 0 R /Pg 229 0 R /S/P /Type/StructElem>> endobj 2528 0 obj <> /K 5 /P 2526 0 R /Pg 229 0 R /S/P /Type/StructElem>> endobj 2529 0 obj <> /K[2530 0 R] /P 2525 0 R /S/TD /Type/StructElem>> endobj 2530 0 obj <> /K 6 /P 2529 0 R /Pg 229 0 R /S/P /Type/StructElem>> endobj 2531 0 obj <> endobj 2532 0 obj <> /K[2533 0 R] /P 2531 0 R /S/TD /Type/StructElem>> endobj 2533 0 obj <> /K 7 /P 2532 0 R /Pg 229 0 R /S/P /Type/StructElem>> endobj 2534 0 obj <> /K[2535 0 R] /P 2531 0 R /S/TD /Type/StructElem>> endobj 2535 0 obj <> /K 8 /P 2534 0 R /Pg 229 0 R /S/P /Type/StructElem>> endobj 2536 0 obj <> endobj 2537 0 obj <> /K[2538 0 R] /P 2536 0 R /S/TD /Type/StructElem>> endobj 2538 0 obj <> /K 9 /P 2537 0 R /Pg 229 0 R /S/P /Type/StructElem>> endobj 2539 0 obj <> /K[2540 0 R] /P 2536 0 R /S/TD /Type/StructElem>> endobj 2540 0 obj <> /K 10 /P 2539 0 R /Pg 229 0 R /S/P /Type/StructElem>> endobj 2541 0 obj <> /K 11 /P 2516 0 R /Pg 229 0 R /S/P /Type/StructElem>> endobj 2542 0 obj <> /K 12 /P 2516 0 R /Pg 229 0 R /S/P /Type/StructElem>> endobj 2543 0 obj <> /K 13 /P 2516 0 R /Pg 229 0 R /S/P /Type/StructElem>> endobj 2544 0 obj <> /K 14 /P 2516 0 R /Pg 229 0 R /S/P /Type/StructElem>> endobj 2545 0 obj <> /K 15 /P 2516 0 R /Pg 229 0 R /S/P /Type/StructElem>> endobj 2546 0 obj <> /K[2547 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2547 0 obj <> /K 16 /P 2546 0 R /Pg 229 0 R /S/P /Type/StructElem>> endobj 2548 0 obj <> endobj 2549 0 obj <> /K[2550 0 R 2553 0 R 2558 0 R 2563 0 R 2568 0 R 2573 0 R] /P 2548 0 R /S/Table /Type/StructElem>> endobj 2550 0 obj <> endobj 2551 0 obj <> /K[2552 0 R] /P 2550 0 R /S/TD /Type/StructElem>> endobj 2552 0 obj <> /K 0 /P 2551 0 R /Pg 234 0 R /S/P /Type/StructElem>> endobj 2553 0 obj <> endobj 2554 0 obj <> /K[2555 0 R] /P 2553 0 R /S/TD /Type/StructElem>> endobj 2555 0 obj <> /K 1 /P 2554 0 R /Pg 234 0 R /S/P /Type/StructElem>> endobj 2556 0 obj <> /K[2557 0 R] /P 2553 0 R /S/TD /Type/StructElem>> endobj 2557 0 obj <> /K 2 /P 2556 0 R /Pg 234 0 R /S/P /Type/StructElem>> endobj 2558 0 obj <> endobj 2559 0 obj <> /K[2560 0 R] /P 2558 0 R /S/TD /Type/StructElem>> endobj 2560 0 obj <> /K 3 /P 2559 0 R /Pg 234 0 R /S/P /Type/StructElem>> endobj 2561 0 obj <> /K[2562 0 R] /P 2558 0 R /S/TD /Type/StructElem>> endobj 2562 0 obj <> /K 4 /P 2561 0 R /Pg 234 0 R /S/P /Type/StructElem>> endobj 2563 0 obj <> endobj 2564 0 obj <> /K[2565 0 R] /P 2563 0 R /S/TD /Type/StructElem>> endobj 2565 0 obj <> /K 5 /P 2564 0 R /Pg 234 0 R /S/P /Type/StructElem>> endobj 2566 0 obj <> /K[2567 0 R] /P 2563 0 R /S/TD /Type/StructElem>> endobj 2567 0 obj <> /K 6 /P 2566 0 R /Pg 234 0 R /S/P /Type/StructElem>> endobj 2568 0 obj <> endobj 2569 0 obj <> /K[2570 0 R] /P 2568 0 R /S/TD /Type/StructElem>> endobj 2570 0 obj <> /K 7 /P 2569 0 R /Pg 234 0 R /S/P /Type/StructElem>> endobj 2571 0 obj <> /K[2572 0 R] /P 2568 0 R /S/TD /Type/StructElem>> endobj 2572 0 obj <> /K 8 /P 2571 0 R /Pg 234 0 R /S/P /Type/StructElem>> endobj 2573 0 obj <> endobj 2574 0 obj <> /K[2575 0 R] /P 2573 0 R /S/TD /Type/StructElem>> endobj 2575 0 obj <> /K 9 /P 2574 0 R /Pg 234 0 R /S/P /Type/StructElem>> endobj 2576 0 obj <> /K[2577 0 R] /P 2573 0 R /S/TD /Type/StructElem>> endobj 2577 0 obj <> /K 10 /P 2576 0 R /Pg 234 0 R /S/P /Type/StructElem>> endobj 2578 0 obj <> /K 11 /P 2548 0 R /Pg 234 0 R /S/P /Type/StructElem>> endobj 2579 0 obj <> /K 12 /P 2548 0 R /Pg 234 0 R /S/P /Type/StructElem>> endobj 2580 0 obj <> /K 13 /P 2548 0 R /Pg 234 0 R /S/P /Type/StructElem>> endobj 2581 0 obj <> /K 14 /P 2548 0 R /Pg 234 0 R /S/P /Type/StructElem>> endobj 2582 0 obj <> /K[2583 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2583 0 obj <> /K 15 /P 2582 0 R /Pg 234 0 R /S/P /Type/StructElem>> endobj 2584 0 obj <> endobj 2585 0 obj <> /K 0 /P 2584 0 R /Pg 239 0 R /S/P /Type/StructElem>> endobj 2586 0 obj <> /K 1 /P 2584 0 R /Pg 239 0 R /S/P /Type/StructElem>> endobj 2587 0 obj <> /K 2 /P 2584 0 R /Pg 239 0 R /S/P /Type/StructElem>> endobj 2588 0 obj <> /K 3 /P 2584 0 R /Pg 239 0 R /S/P /Type/StructElem>> endobj 2589 0 obj <> /K 4 /P 2584 0 R /Pg 239 0 R /S/P /Type/StructElem>> endobj 2590 0 obj <> /K 5 /P 2584 0 R /Pg 239 0 R /S/P /Type/StructElem>> endobj 2591 0 obj <> /K 6 /P 2584 0 R /Pg 239 0 R /S/P /Type/StructElem>> endobj 2592 0 obj <> /K 7 /P 2584 0 R /Pg 239 0 R /S/P /Type/StructElem>> endobj 2593 0 obj <> /K 8 /P 2584 0 R /Pg 239 0 R /S/P /Type/StructElem>> endobj 2594 0 obj <> /K 9 /P 2584 0 R /Pg 239 0 R /S/P /Type/StructElem>> endobj 2595 0 obj <> /K 10 /P 2584 0 R /Pg 239 0 R /S/P /Type/StructElem>> endobj 2596 0 obj <> /K[2597 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2597 0 obj <> /K 11 /P 2596 0 R /Pg 239 0 R /S/P /Type/StructElem>> endobj 2598 0 obj <> /K[2599 0 R 2600 0 R 2601 0 R 2602 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2599 0 obj <> /K 0 /P 2598 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2600 0 obj <> /K 1 /P 2598 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2601 0 obj <> /K 2 /P 2598 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2602 0 obj <> /K 3 /P 2598 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2603 0 obj <> endobj 2604 0 obj <> /K 4 /P 2603 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2605 0 obj <> /K 5 /P 2603 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2606 0 obj <> /K 6 /P 2603 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2607 0 obj <> /K[2608 0 R 2616 0 R 2621 0 R] /P 2603 0 R /S/Table /Type/StructElem>> endobj 2608 0 obj <> endobj 2609 0 obj <> /K[2610 0 R] /P 2608 0 R /S/TD /Type/StructElem>> endobj 2610 0 obj <> /K 7 /P 2609 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2611 0 obj <> /K[2612 0 R 2613 0 R] /P 2608 0 R /S/TD /Type/StructElem>> endobj 2612 0 obj <> /K 8 /P 2611 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2613 0 obj <> /K 9 /P 2611 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2614 0 obj <> /K[2615 0 R] /P 2608 0 R /S/TD /Type/StructElem>> endobj 2615 0 obj <> /K 10 /P 2614 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2616 0 obj <> endobj 2617 0 obj <> /K[2618 0 R] /P 2616 0 R /S/TD /Type/StructElem>> endobj 2618 0 obj <> /K 11 /P 2617 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2619 0 obj <> /K[2620 0 R] /P 2616 0 R /S/TD /Type/StructElem>> endobj 2620 0 obj <> /K 12 /P 2619 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2621 0 obj <> endobj 2622 0 obj <> /K[2623 0 R] /P 2621 0 R /S/TD /Type/StructElem>> endobj 2623 0 obj <> /K 13 /P 2622 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2624 0 obj <> /K[2625 0 R] /P 2621 0 R /S/TD /Type/StructElem>> endobj 2625 0 obj <> /K 14 /P 2624 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2626 0 obj <> /K[2627 0 R] /P 2621 0 R /S/TD /Type/StructElem>> endobj 2627 0 obj <> /K 15 /P 2626 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2628 0 obj <> /K[2629 0 R] /P 2621 0 R /S/TD /Type/StructElem>> endobj 2629 0 obj <> /K 16 /P 2628 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2630 0 obj <> /K 17 /P 2603 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2631 0 obj <> endobj 2632 0 obj <> /K 18 /P 2631 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2633 0 obj <> /K 19 /P 2631 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2634 0 obj <> /K 20 /P 2631 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2635 0 obj <> /K 21 /P 2631 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2636 0 obj <> /K[2637 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2637 0 obj <> /K 22 /P 2636 0 R /Pg 244 0 R /S/P /Type/StructElem>> endobj 2638 0 obj <> endobj 2639 0 obj <> /K 0 /P 2638 0 R /Pg 249 0 R /S/P /Type/StructElem>> endobj 2640 0 obj <> /K 1 /P 2638 0 R /Pg 249 0 R /S/P /Type/StructElem>> endobj 2641 0 obj <> /K 2 /P 2638 0 R /Pg 249 0 R /S/P /Type/StructElem>> endobj 2642 0 obj <> /K 3 /P 2638 0 R /Pg 249 0 R /S/P /Type/StructElem>> endobj 2643 0 obj <> /K 4 /P 2638 0 R /Pg 249 0 R /S/P /Type/StructElem>> endobj 2644 0 obj <> /K 5 /P 2638 0 R /Pg 249 0 R /S/P /Type/StructElem>> endobj 2645 0 obj <> /K 6 /P 2638 0 R /Pg 249 0 R /S/P /Type/StructElem>> endobj 2646 0 obj <> /K[2647 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2647 0 obj <> /K 7 /P 2646 0 R /Pg 249 0 R /S/P /Type/StructElem>> endobj 2648 0 obj <> endobj 2649 0 obj <> /K 0 /P 2648 0 R /Pg 254 0 R /S/P /Type/StructElem>> endobj 2650 0 obj <> /K 1 /P 2648 0 R /Pg 254 0 R /S/P /Type/StructElem>> endobj 2651 0 obj <> /K[2652 0 R 2659 0 R 2663 0 R] /P 2648 0 R /S/Table /Type/StructElem>> endobj 2652 0 obj <> endobj 2653 0 obj <> /K[2654 0 R 2655 0 R] /P 2652 0 R /S/TD /Type/StructElem>> endobj 2654 0 obj <> /K 2 /P 2653 0 R /Pg 254 0 R /S/P /Type/StructElem>> endobj 2655 0 obj <> /K 3 /P 2653 0 R /Pg 254 0 R /S/P /Type/StructElem>> endobj 2656 0 obj <> /K[2657 0 R 2658 0 R] /P 2652 0 R /S/TD /Type/StructElem>> endobj 2657 0 obj <> /K 4 /P 2656 0 R /Pg 254 0 R /S/P /Type/StructElem>> endobj 2658 0 obj <> /K 5 /P 2656 0 R /Pg 254 0 R /S/P /Type/StructElem>> endobj 2659 0 obj <> endobj 2660 0 obj <> /K[2661 0 R 2662 0 R] /P 2659 0 R /S/TD /Type/StructElem>> endobj 2661 0 obj <> /K 6 /P 2660 0 R /Pg 254 0 R /S/P /Type/StructElem>> endobj 2662 0 obj <> /K 7 /P 2660 0 R /Pg 254 0 R /S/P /Type/StructElem>> endobj 2663 0 obj <> endobj 2664 0 obj <> /K[2665 0 R] /P 2663 0 R /S/TD /Type/StructElem>> endobj 2665 0 obj <> /K 8 /P 2664 0 R /Pg 254 0 R /S/P /Type/StructElem>> endobj 2666 0 obj <> /K 9 /P 2648 0 R /Pg 254 0 R /S/P /Type/StructElem>> endobj 2667 0 obj <> /K 10 /P 2648 0 R /Pg 254 0 R /S/P /Type/StructElem>> endobj 2668 0 obj <> /K 11 /P 2648 0 R /Pg 254 0 R /S/P /Type/StructElem>> endobj 2669 0 obj <> /K 12 /P 2648 0 R /Pg 254 0 R /S/P /Type/StructElem>> endobj 2670 0 obj <> /K[2671 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2671 0 obj <> /K 13 /P 2670 0 R /Pg 254 0 R /S/P /Type/StructElem>> endobj 2672 0 obj <> endobj 2673 0 obj <> /K 0 /P 2672 0 R /Pg 259 0 R /S/P /Type/StructElem>> endobj 2674 0 obj <> /K 1 /P 2672 0 R /Pg 259 0 R /S/P /Type/StructElem>> endobj 2675 0 obj <> /K 2 /P 2672 0 R /Pg 259 0 R /S/P /Type/StructElem>> endobj 2676 0 obj <> /K 3 /P 2672 0 R /Pg 259 0 R /S/P /Type/StructElem>> endobj 2677 0 obj <> /K 4 /P 2672 0 R /Pg 259 0 R /S/P /Type/StructElem>> endobj 2678 0 obj <> /K 5 /P 2672 0 R /Pg 259 0 R /S/P /Type/StructElem>> endobj 2679 0 obj <> /K[2680 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2680 0 obj <> /K 6 /P 2679 0 R /Pg 259 0 R /S/P /Type/StructElem>> endobj 2681 0 obj <> /K[2682 0 R 2683 0 R 2684 0 R 2685 0 R 2686 0 R 2687 0 R 2688 0 R 2689 0 R 2690 0 R 2691 0 R 2692 0 R 2693 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2682 0 obj <> /K 0 /P 2681 0 R /Pg 264 0 R /S/P /Type/StructElem>> endobj 2683 0 obj <> /K 1 /P 2681 0 R /Pg 264 0 R /S/P /Type/StructElem>> endobj 2684 0 obj <> /K 2 /P 2681 0 R /Pg 264 0 R /S/P /Type/StructElem>> endobj 2685 0 obj <> /K 3 /P 2681 0 R /Pg 264 0 R /S/P /Type/StructElem>> endobj 2686 0 obj <> /K 4 /P 2681 0 R /Pg 264 0 R /S/P /Type/StructElem>> endobj 2687 0 obj <> /K 5 /P 2681 0 R /Pg 264 0 R /S/P /Type/StructElem>> endobj 2688 0 obj <> /K 6 /P 2681 0 R /Pg 264 0 R /S/P /Type/StructElem>> endobj 2689 0 obj <> /K 7 /P 2681 0 R /Pg 264 0 R /S/P /Type/StructElem>> endobj 2690 0 obj <> /K 8 /P 2681 0 R /Pg 264 0 R /S/P /Type/StructElem>> endobj 2691 0 obj <> /K 9 /P 2681 0 R /Pg 264 0 R /S/P /Type/StructElem>> endobj 2692 0 obj <> /K 10 /P 2681 0 R /Pg 264 0 R /S/P /Type/StructElem>> endobj 2693 0 obj <> /K 11 /P 2681 0 R /Pg 264 0 R /S/P /Type/StructElem>> endobj 2694 0 obj <> endobj 2695 0 obj <> /K 12 /P 2694 0 R /Pg 264 0 R /S/P /Type/StructElem>> endobj 2696 0 obj <> endobj 2697 0 obj <> /K 13 /P 2696 0 R /Pg 264 0 R /S/P /Type/StructElem>> endobj 2698 0 obj <> /K 14 /P 2696 0 R /Pg 264 0 R /S/P /Type/StructElem>> endobj 2699 0 obj <> /K 15 /P 2696 0 R /Pg 264 0 R /S/P /Type/StructElem>> endobj 2700 0 obj <> /K[2701 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2701 0 obj <> /K 16 /P 2700 0 R /Pg 264 0 R /S/P /Type/StructElem>> endobj 2702 0 obj <> /K[2703 0 R 2704 0 R 2705 0 R 2706 0 R 2707 0 R 2708 0 R 2709 0 R 2710 0 R 2711 0 R 2712 0 R 2713 0 R 2714 0 R 2715 0 R 2716 0 R 2717 0 R 2718 0 R 2719 0 R 2720 0 R 2721 0 R 2722 0 R 2723 0 R 2724 0 R 2725 0 R 2726 0 R 2727 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2703 0 obj <> /K 0 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2704 0 obj <> /K 1 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2705 0 obj <> /K 2 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2706 0 obj <> /K 3 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2707 0 obj <> /K 4 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2708 0 obj <> /K 5 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2709 0 obj <> /K 6 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2710 0 obj <> /K 7 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2711 0 obj <> /K 8 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2712 0 obj <> /K 9 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2713 0 obj <> /K 10 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2714 0 obj <> /K 11 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2715 0 obj <> /K 12 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2716 0 obj <> /K 13 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2717 0 obj <> /K 14 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2718 0 obj <> /K 15 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2719 0 obj <> /K 16 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2720 0 obj <> /K 17 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2721 0 obj <> /K 18 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2722 0 obj <> /K 19 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2723 0 obj <> /K 20 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2724 0 obj <> /K 21 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2725 0 obj <> /K 22 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2726 0 obj <> /K 23 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2727 0 obj <> /K 24 /P 2702 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2728 0 obj <> endobj 2729 0 obj <> /K 25 /P 2728 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2730 0 obj <> endobj 2731 0 obj <> /K 26 /P 2730 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2732 0 obj <> /K 27 /P 2730 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2733 0 obj <> /K[2734 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2734 0 obj <> /K 28 /P 2733 0 R /Pg 269 0 R /S/P /Type/StructElem>> endobj 2735 0 obj <> endobj 2736 0 obj <> /K 0 /P 2735 0 R /Pg 274 0 R /S/P /Type/StructElem>> endobj 2737 0 obj <> /K 1 /P 2735 0 R /Pg 274 0 R /S/P /Type/StructElem>> endobj 2738 0 obj <> /K 2 /P 2735 0 R /Pg 274 0 R /S/P /Type/StructElem>> endobj 2739 0 obj <> /K 3 /P 2735 0 R /Pg 274 0 R /S/P /Type/StructElem>> endobj 2740 0 obj <> /K 4 /P 2735 0 R /Pg 274 0 R /S/P /Type/StructElem>> endobj 2741 0 obj <> /K 5 /P 2735 0 R /Pg 274 0 R /S/P /Type/StructElem>> endobj 2742 0 obj <> /K[2743 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2743 0 obj <> /K 6 /P 2742 0 R /Pg 274 0 R /S/P /Type/StructElem>> endobj 2744 0 obj <> endobj 2745 0 obj <> /K 0 /P 2744 0 R /Pg 279 0 R /S/P /Type/StructElem>> endobj 2746 0 obj <> /K 1 /P 2744 0 R /Pg 279 0 R /S/P /Type/StructElem>> endobj 2747 0 obj <> /K 2 /P 2744 0 R /Pg 279 0 R /S/P /Type/StructElem>> endobj 2748 0 obj <> /K[2749 0 R] /P 2744 0 R /S/Table /Type/StructElem>> endobj 2749 0 obj <> endobj 2750 0 obj <> /K[2751 0 R] /P 2749 0 R /S/TD /Type/StructElem>> endobj 2751 0 obj <> /K 3 /P 2750 0 R /Pg 279 0 R /S/P /Type/StructElem>> endobj 2752 0 obj <> /K[2753 0 R] /P 2749 0 R /S/TD /Type/StructElem>> endobj 2753 0 obj <> /K 4 /P 2752 0 R /Pg 279 0 R /S/P /Type/StructElem>> endobj 2754 0 obj <> /K[2755 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2755 0 obj <> /K[2756 0 R] /P 2754 0 R /S/Table /Type/StructElem>> endobj 2756 0 obj <> endobj 2757 0 obj <> /K[2758 0 R] /P 2756 0 R /S/TD /Type/StructElem>> endobj 2758 0 obj <> /K 5 /P 2757 0 R /Pg 279 0 R /S/P /Type/StructElem>> endobj 2759 0 obj <> /K[2760 0 R] /P 2756 0 R /S/TD /Type/StructElem>> endobj 2760 0 obj <> /K 6 /P 2759 0 R /Pg 279 0 R /S/P /Type/StructElem>> endobj 2761 0 obj <> endobj 2762 0 obj <> /K 7 /P 2761 0 R /Pg 279 0 R /S/P /Type/StructElem>> endobj 2763 0 obj <> /K 8 /P 2761 0 R /Pg 279 0 R /S/P /Type/StructElem>> endobj 2764 0 obj <> /K 9 /P 2761 0 R /Pg 279 0 R /S/P /Type/StructElem>> endobj 2765 0 obj <> /K[2766 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2766 0 obj <> /K 10 /P 2765 0 R /Pg 279 0 R /S/P /Type/StructElem>> endobj 2767 0 obj <> endobj 2768 0 obj <> /K 0 /P 2767 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2769 0 obj <> /K[2770 0 R 2775 0 R 2782 0 R 2795 0 R 2810 0 R 2825 0 R 2840 0 R 2845 0 R 2852 0 R 2865 0 R 2880 0 R 2895 0 R] /P 2767 0 R /S/Table /Type/StructElem>> endobj 2770 0 obj <> endobj 2771 0 obj <> /K[2772 0 R] /P 2770 0 R /S/TD /Type/StructElem>> endobj 2772 0 obj <> /K 1 /P 2771 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2773 0 obj <> /K[2774 0 R] /P 2770 0 R /S/TD /Type/StructElem>> endobj 2774 0 obj <> /K 2 /P 2773 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2775 0 obj <> endobj 2776 0 obj <> /K[2777 0 R] /P 2775 0 R /S/TD /Type/StructElem>> endobj 2777 0 obj <> /K 3 /P 2776 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2778 0 obj <> /K[2779 0 R] /P 2775 0 R /S/TD /Type/StructElem>> endobj 2779 0 obj <> /K 4 /P 2778 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2780 0 obj <> /K[2781 0 R] /P 2775 0 R /S/TD /Type/StructElem>> endobj 2781 0 obj <> /K 5 /P 2780 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2782 0 obj <> endobj 2783 0 obj <> /K[2784 0 R] /P 2782 0 R /S/TD /Type/StructElem>> endobj 2784 0 obj <> /K 6 /P 2783 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2785 0 obj <> /K[2786 0 R] /P 2782 0 R /S/TD /Type/StructElem>> endobj 2786 0 obj <> /K 7 /P 2785 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2787 0 obj <> /K[2788 0 R] /P 2782 0 R /S/TD /Type/StructElem>> endobj 2788 0 obj <> /K 8 /P 2787 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2789 0 obj <> /K[2790 0 R] /P 2782 0 R /S/TD /Type/StructElem>> endobj 2790 0 obj <> /K 9 /P 2789 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2791 0 obj <> /K[2792 0 R] /P 2782 0 R /S/TD /Type/StructElem>> endobj 2792 0 obj <> /K 10 /P 2791 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2793 0 obj <> /K[2794 0 R] /P 2782 0 R /S/TD /Type/StructElem>> endobj 2794 0 obj <> /K 11 /P 2793 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2795 0 obj <> endobj 2796 0 obj <> /K[2797 0 R] /P 2795 0 R /S/TD /Type/StructElem>> endobj 2797 0 obj <> /K 12 /P 2796 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2798 0 obj <> /K[2799 0 R] /P 2795 0 R /S/TD /Type/StructElem>> endobj 2799 0 obj <> /K 13 /P 2798 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2800 0 obj <> /K[2801 0 R] /P 2795 0 R /S/TD /Type/StructElem>> endobj 2801 0 obj <> /K 14 /P 2800 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2802 0 obj <> /K[2803 0 R] /P 2795 0 R /S/TD /Type/StructElem>> endobj 2803 0 obj <> /K 15 /P 2802 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2804 0 obj <> /K[2805 0 R] /P 2795 0 R /S/TD /Type/StructElem>> endobj 2805 0 obj <> /K 16 /P 2804 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2806 0 obj <> /K[2807 0 R] /P 2795 0 R /S/TD /Type/StructElem>> endobj 2807 0 obj <> /K 17 /P 2806 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2808 0 obj <> /K[2809 0 R] /P 2795 0 R /S/TD /Type/StructElem>> endobj 2809 0 obj <> /K 18 /P 2808 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2810 0 obj <> endobj 2811 0 obj <> /K[2812 0 R] /P 2810 0 R /S/TD /Type/StructElem>> endobj 2812 0 obj <> /K 19 /P 2811 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2813 0 obj <> /K[2814 0 R] /P 2810 0 R /S/TD /Type/StructElem>> endobj 2814 0 obj <> /K 20 /P 2813 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2815 0 obj <> /K[2816 0 R] /P 2810 0 R /S/TD /Type/StructElem>> endobj 2816 0 obj <> /K 21 /P 2815 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2817 0 obj <> /K[2818 0 R] /P 2810 0 R /S/TD /Type/StructElem>> endobj 2818 0 obj <> /K 22 /P 2817 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2819 0 obj <> /K[2820 0 R] /P 2810 0 R /S/TD /Type/StructElem>> endobj 2820 0 obj <> /K 23 /P 2819 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2821 0 obj <> /K[2822 0 R] /P 2810 0 R /S/TD /Type/StructElem>> endobj 2822 0 obj <> /K 24 /P 2821 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2823 0 obj <> /K[2824 0 R] /P 2810 0 R /S/TD /Type/StructElem>> endobj 2824 0 obj <> /K 25 /P 2823 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2825 0 obj <> endobj 2826 0 obj <> /K[2827 0 R] /P 2825 0 R /S/TD /Type/StructElem>> endobj 2827 0 obj <> /K 26 /P 2826 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2828 0 obj <> /K[2829 0 R] /P 2825 0 R /S/TD /Type/StructElem>> endobj 2829 0 obj <> /K 27 /P 2828 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2830 0 obj <> /K[2831 0 R] /P 2825 0 R /S/TD /Type/StructElem>> endobj 2831 0 obj <> /K 28 /P 2830 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2832 0 obj <> /K[2833 0 R] /P 2825 0 R /S/TD /Type/StructElem>> endobj 2833 0 obj <> /K 29 /P 2832 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2834 0 obj <> /K[2835 0 R] /P 2825 0 R /S/TD /Type/StructElem>> endobj 2835 0 obj <> /K 30 /P 2834 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2836 0 obj <> /K[2837 0 R] /P 2825 0 R /S/TD /Type/StructElem>> endobj 2837 0 obj <> /K 31 /P 2836 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2838 0 obj <> /K[2839 0 R] /P 2825 0 R /S/TD /Type/StructElem>> endobj 2839 0 obj <> /K 32 /P 2838 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2840 0 obj <> endobj 2841 0 obj <> /K[2842 0 R] /P 2840 0 R /S/TD /Type/StructElem>> endobj 2842 0 obj <> /K 33 /P 2841 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2843 0 obj <> /K[2844 0 R] /P 2840 0 R /S/TD /Type/StructElem>> endobj 2844 0 obj <> /K 34 /P 2843 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2845 0 obj <> endobj 2846 0 obj <> /K[2847 0 R] /P 2845 0 R /S/TD /Type/StructElem>> endobj 2847 0 obj <> /K 35 /P 2846 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2848 0 obj <> /K[2849 0 R] /P 2845 0 R /S/TD /Type/StructElem>> endobj 2849 0 obj <> /K 36 /P 2848 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2850 0 obj <> /K[2851 0 R] /P 2845 0 R /S/TD /Type/StructElem>> endobj 2851 0 obj <> /K 37 /P 2850 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2852 0 obj <> endobj 2853 0 obj <> /K[2854 0 R] /P 2852 0 R /S/TD /Type/StructElem>> endobj 2854 0 obj <> /K 38 /P 2853 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2855 0 obj <> /K[2856 0 R] /P 2852 0 R /S/TD /Type/StructElem>> endobj 2856 0 obj <> /K 39 /P 2855 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2857 0 obj <> /K[2858 0 R] /P 2852 0 R /S/TD /Type/StructElem>> endobj 2858 0 obj <> /K 40 /P 2857 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2859 0 obj <> /K[2860 0 R] /P 2852 0 R /S/TD /Type/StructElem>> endobj 2860 0 obj <> /K 41 /P 2859 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2861 0 obj <> /K[2862 0 R] /P 2852 0 R /S/TD /Type/StructElem>> endobj 2862 0 obj <> /K 42 /P 2861 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2863 0 obj <> /K[2864 0 R] /P 2852 0 R /S/TD /Type/StructElem>> endobj 2864 0 obj <> /K 43 /P 2863 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2865 0 obj <> endobj 2866 0 obj <> /K[2867 0 R] /P 2865 0 R /S/TD /Type/StructElem>> endobj 2867 0 obj <> /K 44 /P 2866 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2868 0 obj <> /K[2869 0 R] /P 2865 0 R /S/TD /Type/StructElem>> endobj 2869 0 obj <> /K 45 /P 2868 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2870 0 obj <> /K[2871 0 R] /P 2865 0 R /S/TD /Type/StructElem>> endobj 2871 0 obj <> /K 46 /P 2870 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2872 0 obj <> /K[2873 0 R] /P 2865 0 R /S/TD /Type/StructElem>> endobj 2873 0 obj <> /K 47 /P 2872 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2874 0 obj <> /K[2875 0 R] /P 2865 0 R /S/TD /Type/StructElem>> endobj 2875 0 obj <> /K 48 /P 2874 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2876 0 obj <> /K[2877 0 R] /P 2865 0 R /S/TD /Type/StructElem>> endobj 2877 0 obj <> /K 49 /P 2876 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2878 0 obj <> /K[2879 0 R] /P 2865 0 R /S/TD /Type/StructElem>> endobj 2879 0 obj <> /K 50 /P 2878 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2880 0 obj <> endobj 2881 0 obj <> /K[2882 0 R] /P 2880 0 R /S/TD /Type/StructElem>> endobj 2882 0 obj <> /K 51 /P 2881 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2883 0 obj <> /K[2884 0 R] /P 2880 0 R /S/TD /Type/StructElem>> endobj 2884 0 obj <> /K 52 /P 2883 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2885 0 obj <> /K[2886 0 R] /P 2880 0 R /S/TD /Type/StructElem>> endobj 2886 0 obj <> /K 53 /P 2885 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2887 0 obj <> /K[2888 0 R] /P 2880 0 R /S/TD /Type/StructElem>> endobj 2888 0 obj <> /K 54 /P 2887 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2889 0 obj <> /K[2890 0 R] /P 2880 0 R /S/TD /Type/StructElem>> endobj 2890 0 obj <> /K 55 /P 2889 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2891 0 obj <> /K[2892 0 R] /P 2880 0 R /S/TD /Type/StructElem>> endobj 2892 0 obj <> /K 56 /P 2891 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2893 0 obj <> /K[2894 0 R] /P 2880 0 R /S/TD /Type/StructElem>> endobj 2894 0 obj <> /K 57 /P 2893 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2895 0 obj <> endobj 2896 0 obj <> /K[2897 0 R] /P 2895 0 R /S/TD /Type/StructElem>> endobj 2897 0 obj <> /K 58 /P 2896 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2898 0 obj <> /K[2899 0 R] /P 2895 0 R /S/TD /Type/StructElem>> endobj 2899 0 obj <> /K 59 /P 2898 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2900 0 obj <> /K[2901 0 R] /P 2895 0 R /S/TD /Type/StructElem>> endobj 2901 0 obj <> /K 60 /P 2900 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2902 0 obj <> /K[2903 0 R] /P 2895 0 R /S/TD /Type/StructElem>> endobj 2903 0 obj <> /K 61 /P 2902 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2904 0 obj <> /K[2905 0 R] /P 2895 0 R /S/TD /Type/StructElem>> endobj 2905 0 obj <> /K 62 /P 2904 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2906 0 obj <> /K[2907 0 R] /P 2895 0 R /S/TD /Type/StructElem>> endobj 2907 0 obj <> /K 63 /P 2906 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2908 0 obj <> /K[2909 0 R] /P 2895 0 R /S/TD /Type/StructElem>> endobj 2909 0 obj <> /K 64 /P 2908 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2910 0 obj <> /K 65 /P 2767 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2911 0 obj <> /K 66 /P 2767 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2912 0 obj <> /K[2913 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 2913 0 obj <> /K 67 /P 2912 0 R /Pg 284 0 R /S/P /Type/StructElem>> endobj 2914 0 obj <> endobj 2915 0 obj <> /K 0 /P 2914 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2916 0 obj <> /K[2917 0 R 2922 0 R 2929 0 R 2942 0 R 2957 0 R 2972 0 R 2987 0 R 2992 0 R 2999 0 R 3012 0 R 3027 0 R 3042 0 R] /P 2914 0 R /S/Table /Type/StructElem>> endobj 2917 0 obj <> endobj 2918 0 obj <> /K[2919 0 R] /P 2917 0 R /S/TD /Type/StructElem>> endobj 2919 0 obj <> /K 1 /P 2918 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2920 0 obj <> /K[2921 0 R] /P 2917 0 R /S/TD /Type/StructElem>> endobj 2921 0 obj <> /K 2 /P 2920 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2922 0 obj <> endobj 2923 0 obj <> /K[2924 0 R] /P 2922 0 R /S/TD /Type/StructElem>> endobj 2924 0 obj <> /K 3 /P 2923 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2925 0 obj <> /K[2926 0 R] /P 2922 0 R /S/TD /Type/StructElem>> endobj 2926 0 obj <> /K 4 /P 2925 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2927 0 obj <> /K[2928 0 R] /P 2922 0 R /S/TD /Type/StructElem>> endobj 2928 0 obj <> /K 5 /P 2927 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2929 0 obj <> endobj 2930 0 obj <> /K[2931 0 R] /P 2929 0 R /S/TD /Type/StructElem>> endobj 2931 0 obj <> /K 6 /P 2930 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2932 0 obj <> /K[2933 0 R] /P 2929 0 R /S/TD /Type/StructElem>> endobj 2933 0 obj <> /K 7 /P 2932 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2934 0 obj <> /K[2935 0 R] /P 2929 0 R /S/TD /Type/StructElem>> endobj 2935 0 obj <> /K 8 /P 2934 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2936 0 obj <> /K[2937 0 R] /P 2929 0 R /S/TD /Type/StructElem>> endobj 2937 0 obj <> /K 9 /P 2936 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2938 0 obj <> /K[2939 0 R] /P 2929 0 R /S/TD /Type/StructElem>> endobj 2939 0 obj <> /K 10 /P 2938 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2940 0 obj <> /K[2941 0 R] /P 2929 0 R /S/TD /Type/StructElem>> endobj 2941 0 obj <> /K 11 /P 2940 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2942 0 obj <> endobj 2943 0 obj <> /K[2944 0 R] /P 2942 0 R /S/TD /Type/StructElem>> endobj 2944 0 obj <> /K 12 /P 2943 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2945 0 obj <> /K[2946 0 R] /P 2942 0 R /S/TD /Type/StructElem>> endobj 2946 0 obj <> /K 13 /P 2945 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2947 0 obj <> /K[2948 0 R] /P 2942 0 R /S/TD /Type/StructElem>> endobj 2948 0 obj <> /K 14 /P 2947 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2949 0 obj <> /K[2950 0 R] /P 2942 0 R /S/TD /Type/StructElem>> endobj 2950 0 obj <> /K 15 /P 2949 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2951 0 obj <> /K[2952 0 R] /P 2942 0 R /S/TD /Type/StructElem>> endobj 2952 0 obj <> /K 16 /P 2951 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2953 0 obj <> /K[2954 0 R] /P 2942 0 R /S/TD /Type/StructElem>> endobj 2954 0 obj <> /K 17 /P 2953 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2955 0 obj <> /K[2956 0 R] /P 2942 0 R /S/TD /Type/StructElem>> endobj 2956 0 obj <> /K 18 /P 2955 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2957 0 obj <> endobj 2958 0 obj <> /K[2959 0 R] /P 2957 0 R /S/TD /Type/StructElem>> endobj 2959 0 obj <> /K 19 /P 2958 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2960 0 obj <> /K[2961 0 R] /P 2957 0 R /S/TD /Type/StructElem>> endobj 2961 0 obj <> /K 20 /P 2960 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2962 0 obj <> /K[2963 0 R] /P 2957 0 R /S/TD /Type/StructElem>> endobj 2963 0 obj <> /K 21 /P 2962 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2964 0 obj <> /K[2965 0 R] /P 2957 0 R /S/TD /Type/StructElem>> endobj 2965 0 obj <> /K 22 /P 2964 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2966 0 obj <> /K[2967 0 R] /P 2957 0 R /S/TD /Type/StructElem>> endobj 2967 0 obj <> /K 23 /P 2966 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2968 0 obj <> /K[2969 0 R] /P 2957 0 R /S/TD /Type/StructElem>> endobj 2969 0 obj <> /K 24 /P 2968 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2970 0 obj <> /K[2971 0 R] /P 2957 0 R /S/TD /Type/StructElem>> endobj 2971 0 obj <> /K 25 /P 2970 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2972 0 obj <> endobj 2973 0 obj <> /K[2974 0 R] /P 2972 0 R /S/TD /Type/StructElem>> endobj 2974 0 obj <> /K 26 /P 2973 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2975 0 obj <> /K[2976 0 R] /P 2972 0 R /S/TD /Type/StructElem>> endobj 2976 0 obj <> /K 27 /P 2975 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2977 0 obj <> /K[2978 0 R] /P 2972 0 R /S/TD /Type/StructElem>> endobj 2978 0 obj <> /K 28 /P 2977 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2979 0 obj <> /K[2980 0 R] /P 2972 0 R /S/TD /Type/StructElem>> endobj 2980 0 obj <> /K 29 /P 2979 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2981 0 obj <> /K[2982 0 R] /P 2972 0 R /S/TD /Type/StructElem>> endobj 2982 0 obj <> /K 30 /P 2981 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2983 0 obj <> /K[2984 0 R] /P 2972 0 R /S/TD /Type/StructElem>> endobj 2984 0 obj <> /K 31 /P 2983 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2985 0 obj <> /K[2986 0 R] /P 2972 0 R /S/TD /Type/StructElem>> endobj 2986 0 obj <> /K 32 /P 2985 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2987 0 obj <> endobj 2988 0 obj <> /K[2989 0 R] /P 2987 0 R /S/TD /Type/StructElem>> endobj 2989 0 obj <> /K 33 /P 2988 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2990 0 obj <> /K[2991 0 R] /P 2987 0 R /S/TD /Type/StructElem>> endobj 2991 0 obj <> /K 34 /P 2990 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2992 0 obj <> endobj 2993 0 obj <> /K[2994 0 R] /P 2992 0 R /S/TD /Type/StructElem>> endobj 2994 0 obj <> /K 35 /P 2993 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2995 0 obj <> /K[2996 0 R] /P 2992 0 R /S/TD /Type/StructElem>> endobj 2996 0 obj <> /K 36 /P 2995 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2997 0 obj <> /K[2998 0 R] /P 2992 0 R /S/TD /Type/StructElem>> endobj 2998 0 obj <> /K 37 /P 2997 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 2999 0 obj <> endobj 3000 0 obj <> /K[3001 0 R] /P 2999 0 R /S/TD /Type/StructElem>> endobj 3001 0 obj <> /K 38 /P 3000 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3002 0 obj <> /K[3003 0 R] /P 2999 0 R /S/TD /Type/StructElem>> endobj 3003 0 obj <> /K 39 /P 3002 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3004 0 obj <> /K[3005 0 R] /P 2999 0 R /S/TD /Type/StructElem>> endobj 3005 0 obj <> /K 40 /P 3004 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3006 0 obj <> /K[3007 0 R] /P 2999 0 R /S/TD /Type/StructElem>> endobj 3007 0 obj <> /K 41 /P 3006 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3008 0 obj <> /K[3009 0 R] /P 2999 0 R /S/TD /Type/StructElem>> endobj 3009 0 obj <> /K 42 /P 3008 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3010 0 obj <> /K[3011 0 R] /P 2999 0 R /S/TD /Type/StructElem>> endobj 3011 0 obj <> /K 43 /P 3010 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3012 0 obj <> endobj 3013 0 obj <> /K[3014 0 R] /P 3012 0 R /S/TD /Type/StructElem>> endobj 3014 0 obj <> /K 44 /P 3013 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3015 0 obj <> /K[3016 0 R] /P 3012 0 R /S/TD /Type/StructElem>> endobj 3016 0 obj <> /K 45 /P 3015 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3017 0 obj <> /K[3018 0 R] /P 3012 0 R /S/TD /Type/StructElem>> endobj 3018 0 obj <> /K 46 /P 3017 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3019 0 obj <> /K[3020 0 R] /P 3012 0 R /S/TD /Type/StructElem>> endobj 3020 0 obj <> /K 47 /P 3019 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3021 0 obj <> /K[3022 0 R] /P 3012 0 R /S/TD /Type/StructElem>> endobj 3022 0 obj <> /K 48 /P 3021 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3023 0 obj <> /K[3024 0 R] /P 3012 0 R /S/TD /Type/StructElem>> endobj 3024 0 obj <> /K 49 /P 3023 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3025 0 obj <> /K[3026 0 R] /P 3012 0 R /S/TD /Type/StructElem>> endobj 3026 0 obj <> /K 50 /P 3025 0 R /Pg 289 0 R /S/P 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obj <> /K 57 /P 3040 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3042 0 obj <> endobj 3043 0 obj <> /K[3044 0 R] /P 3042 0 R /S/TD /Type/StructElem>> endobj 3044 0 obj <> /K 58 /P 3043 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3045 0 obj <> /K[3046 0 R] /P 3042 0 R /S/TD /Type/StructElem>> endobj 3046 0 obj <> /K 59 /P 3045 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3047 0 obj <> /K[3048 0 R] /P 3042 0 R /S/TD /Type/StructElem>> endobj 3048 0 obj <> /K 60 /P 3047 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3049 0 obj <> /K[3050 0 R] /P 3042 0 R /S/TD /Type/StructElem>> endobj 3050 0 obj <> /K 61 /P 3049 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3051 0 obj <> /K[3052 0 R] /P 3042 0 R /S/TD /Type/StructElem>> endobj 3052 0 obj <> /K 62 /P 3051 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3053 0 obj <> /K[3054 0 R] /P 3042 0 R /S/TD /Type/StructElem>> endobj 3054 0 obj <> /K 63 /P 3053 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3055 0 obj <> /K[3056 0 R] /P 3042 0 R /S/TD /Type/StructElem>> endobj 3056 0 obj <> /K 64 /P 3055 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3057 0 obj <> /K 65 /P 2914 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3058 0 obj <> /K 66 /P 2914 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3059 0 obj <> /K 67 /P 2914 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3060 0 obj <> /K 68 /P 2914 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3061 0 obj <> /K 69 /P 2914 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3062 0 obj <> /K[3063 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 3063 0 obj <> /K 70 /P 3062 0 R /Pg 289 0 R /S/P /Type/StructElem>> endobj 3064 0 obj <> endobj 3065 0 obj <> /K 0 /P 3064 0 R /Pg 294 0 R /S/P /Type/StructElem>> endobj 3066 0 obj <> /K 1 /P 3064 0 R /Pg 294 0 R /S/P /Type/StructElem>> endobj 3067 0 obj <> /K 2 /P 3064 0 R /Pg 294 0 R /S/P /Type/StructElem>> endobj 3068 0 obj <> /K 3 /P 3064 0 R /Pg 294 0 R /S/P /Type/StructElem>> endobj 3069 0 obj <> /K 4 /P 3064 0 R /Pg 294 0 R /S/P /Type/StructElem>> endobj 3070 0 obj <> /K[3071 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 3071 0 obj <> /K 5 /P 3070 0 R /Pg 294 0 R /S/P /Type/StructElem>> endobj 3072 0 obj <> /K[3073 0 R 3153 0 R 3154 0 R 3155 0 R 3156 0 R 3157 0 R 3158 0 R 3159 0 R 3160 0 R 3161 0 R 3162 0 R 3163 0 R 3164 0 R 3165 0 R 3166 0 R 3167 0 R 3168 0 R 3169 0 R 3170 0 R 3171 0 R 3172 0 R 3173 0 R 3174 0 R 3175 0 R 3176 0 R 3177 0 R 3178 0 R 3179 0 R 3180 0 R 3181 0 R 3182 0 R 3183 0 R 3184 0 R 3185 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 3073 0 obj <> /K[3074 0 R 3081 0 R 3088 0 R 3095 0 R 3102 0 R 3107 0 R 3113 0 R 3118 0 R 3123 0 R 3130 0 R 3139 0 R 3146 0 R] /P 3072 0 R /S/Table /Type/StructElem>> endobj 3074 0 obj <> endobj 3075 0 obj <> /K[3076 0 R] /P 3074 0 R /S/TD /Type/StructElem>> endobj 3076 0 obj <> /K 0 /P 3075 0 R /Pg 299 0 R /S/P /Type/StructElem>> endobj 3077 0 obj <> /K[3078 0 R] /P 3074 0 R /S/TD /Type/StructElem>> endobj 3078 0 obj <> /K[1 3079 0 R 3080 0 R] /P 3077 0 R /Pg 299 0 R /S/P /Type/StructElem>> endobj 3079 0 obj <> endobj 3080 0 obj <> endobj 3081 0 obj <> endobj 3082 0 obj <> /K[3083 0 R] /P 3081 0 R /S/TD /Type/StructElem>> endobj 3083 0 obj <> /K[4 3084 0 R 3085 0 R] /P 3082 0 R /Pg 299 0 R /S/P /Type/StructElem>> endobj 3084 0 obj <> endobj 3085 0 obj <> endobj 3086 0 obj <> /K[3087 0 R] /P 3081 0 R /S/TD /Type/StructElem>> endobj 3087 0 obj <> /K 7 /P 3086 0 R /Pg 299 0 R /S/P /Type/StructElem>> endobj 3088 0 obj <> endobj 3089 0 obj <> /K[3090 0 R] /P 3088 0 R /S/TD /Type/StructElem>> endobj 3090 0 obj <> /K 8 /P 3089 0 R /Pg 299 0 R /S/P /Type/StructElem>> endobj 3091 0 obj <> /K[3092 0 R] /P 3088 0 R /S/TD /Type/StructElem>> endobj 3092 0 obj <> /K[9 3093 0 R 3094 0 R] /P 3091 0 R /Pg 299 0 R /S/P /Type/StructElem>> endobj 3093 0 obj <> endobj 3094 0 obj <> endobj 3095 0 obj <> endobj 3096 0 obj <> /K[3097 0 R] /P 3095 0 R /S/TD /Type/StructElem>> endobj 3097 0 obj <> /K[12 3098 0 R 3099 0 R] /P 3096 0 R /Pg 299 0 R /S/P /Type/StructElem>> endobj 3098 0 obj <> endobj 3099 0 obj <> endobj 3100 0 obj <> /K[3101 0 R] /P 3095 0 R /S/TD /Type/StructElem>> endobj 3101 0 obj <> /K 15 /P 3100 0 R /Pg 299 0 R /S/P /Type/StructElem>> endobj 3102 0 obj <> endobj 3103 0 obj <> /K[3104 0 R] /P 3102 0 R /S/TD /Type/StructElem>> endobj 3104 0 obj <> /K 16 /P 3103 0 R /Pg 299 0 R /S/P /Type/StructElem>> endobj 3105 0 obj <> /K[3106 0 R] /P 3102 0 R /S/TD /Type/StructElem>> endobj 3106 0 obj <> /K 17 /P 3105 0 R /Pg 299 0 R /S/P /Type/StructElem>> endobj 3107 0 obj <> endobj 3108 0 obj <> /K[3109 0 R 3110 0 R] /P 3107 0 R /S/TD /Type/StructElem>> endobj 3109 0 obj <> /K 18 /P 3108 0 R /Pg 299 0 R /S/P /Type/StructElem>> endobj 3110 0 obj <> /K 19 /P 3108 0 R /Pg 299 0 R /S/P /Type/StructElem>> endobj 3111 0 obj <> /K[3112 0 R] /P 3107 0 R /S/TD /Type/StructElem>> endobj 3112 0 obj <> /K 20 /P 3111 0 R /Pg 299 0 R /S/P /Type/StructElem>> endobj 3113 0 obj <> endobj 3114 0 obj <> /K[3115 0 R] /P 3113 0 R /S/TD 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endobj 3804 0 obj <> /K[3805 0 R] /P 3799 0 R /S/TD /Type/StructElem>> endobj 3805 0 obj <> /K[74 3806 0 R 3807 0 R] /P 3804 0 R /Pg 428 0 R /S/P /Type/StructElem>> endobj 3806 0 obj <> endobj 3807 0 obj <> endobj 3808 0 obj <> endobj 3809 0 obj <> /K[3810 0 R] /P 3808 0 R /S/TD /Type/StructElem>> endobj 3810 0 obj <> /K 77 /P 3809 0 R /Pg 428 0 R /S/P /Type/StructElem>> endobj 3811 0 obj <> /K[3812 0 R] /P 3808 0 R /S/TD /Type/StructElem>> endobj 3812 0 obj <> /K 78 /P 3811 0 R /Pg 428 0 R /S/P /Type/StructElem>> endobj 3813 0 obj <> endobj 3814 0 obj <> /K[3815 0 R] /P 3813 0 R /S/TD /Type/StructElem>> endobj 3815 0 obj <> /K 79 /P 3814 0 R /Pg 428 0 R /S/P /Type/StructElem>> endobj 3816 0 obj <> /K[3817 0 R] /P 3813 0 R /S/TD /Type/StructElem>> endobj 3817 0 obj <> /K 80 /P 3816 0 R /Pg 428 0 R /S/P /Type/StructElem>> endobj 3818 0 obj <> endobj 3819 0 obj <> /K[3820 0 R] /P 3818 0 R /S/TD /Type/StructElem>> endobj 3820 0 obj <> /K 81 /P 3819 0 R /Pg 428 0 R /S/P /Type/StructElem>> endobj 3821 0 obj <> endobj 3822 0 obj <> /K[3823 0 R] /P 3821 0 R /S/TD /Type/StructElem>> endobj 3823 0 obj <> /K 82 /P 3822 0 R /Pg 428 0 R /S/P /Type/StructElem>> endobj 3824 0 obj <> endobj 3825 0 obj <> /K[3826 0 R] /P 3824 0 R /S/TD /Type/StructElem>> endobj 3826 0 obj <> /K[83 3827 0 R 3828 0 R] /P 3825 0 R /Pg 428 0 R /S/P /Type/StructElem>> endobj 3827 0 obj <> endobj 3828 0 obj <> endobj 3829 0 obj <> /K[3830 0 R] /P 3824 0 R /S/TD /Type/StructElem>> endobj 3830 0 obj <> /K[86 3831 0 R 3832 0 R] /P 3829 0 R /Pg 428 0 R /S/P /Type/StructElem>> endobj 3831 0 obj <> endobj 3832 0 obj <> endobj 3833 0 obj <> /K 89 /P 3744 0 R /Pg 428 0 R /S/P /Type/StructElem>> endobj 3834 0 obj <> /K 90 /P 3744 0 R /Pg 428 0 R /S/P /Type/StructElem>> endobj 3835 0 obj <> /K 91 /P 3744 0 R /Pg 428 0 R /S/P /Type/StructElem>> endobj 3836 0 obj <> /K 92 /P 3744 0 R /Pg 428 0 R /S/P /Type/StructElem>> endobj 3837 0 obj <> /K[3838 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 3838 0 obj <> /K 93 /P 3837 0 R /Pg 428 0 R /S/P /Type/StructElem>> endobj 3839 0 obj <> /K[3840 0 R 3841 0 R 3842 0 R 3843 0 R 3844 0 R 3845 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 3840 0 obj <> /K 0 /P 3839 0 R /Pg 433 0 R /S/P /Type/StructElem>> endobj 3841 0 obj <> /K 1 /P 3839 0 R /Pg 433 0 R /S/P /Type/StructElem>> endobj 3842 0 obj <> /K 2 /P 3839 0 R /Pg 433 0 R /S/P /Type/StructElem>> endobj 3843 0 obj <> /K 3 /P 3839 0 R /Pg 433 0 R /S/P /Type/StructElem>> endobj 3844 0 obj <> /K 4 /P 3839 0 R /Pg 433 0 R /S/P /Type/StructElem>> endobj 3845 0 obj <> /K 5 /P 3839 0 R /Pg 433 0 R /S/P /Type/StructElem>> endobj 3846 0 obj <> endobj 3847 0 obj <> /K 6 /P 3846 0 R /Pg 433 0 R /S/P /Type/StructElem>> endobj 3848 0 obj <> /K 7 /P 3846 0 R /Pg 433 0 R /S/P /Type/StructElem>> endobj 3849 0 obj <> /K 8 /P 3846 0 R /Pg 433 0 R /S/P /Type/StructElem>> endobj 3850 0 obj <> /K 9 /P 3846 0 R /Pg 433 0 R /S/P /Type/StructElem>> endobj 3851 0 obj <> /K[3852 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 3852 0 obj <> /K 10 /P 3851 0 R /Pg 433 0 R /S/P /Type/StructElem>> endobj 3853 0 obj <> endobj 3854 0 obj <> /K 0 /P 3853 0 R /Pg 438 0 R /S/P /Type/StructElem>> endobj 3855 0 obj <> /K 1 /P 3853 0 R /Pg 438 0 R /S/P /Type/StructElem>> endobj 3856 0 obj <> /K 2 /P 3853 0 R /Pg 438 0 R /S/P /Type/StructElem>> endobj 3857 0 obj <> /K 3 /P 3853 0 R /Pg 438 0 R /S/P /Type/StructElem>> endobj 3858 0 obj <> /K 4 /P 3853 0 R /Pg 438 0 R /S/P /Type/StructElem>> endobj 3859 0 obj <> /K[3860 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 3860 0 obj <> /K 5 /P 3859 0 R /Pg 438 0 R /S/P /Type/StructElem>> endobj 3861 0 obj <> endobj 3862 0 obj <> /K 0 /P 3861 0 R /Pg 443 0 R /S/P /Type/StructElem>> endobj 3863 0 obj <> /K 1 /P 3861 0 R /Pg 443 0 R /S/P /Type/StructElem>> endobj 3864 0 obj <> /K 2 /P 3861 0 R /Pg 443 0 R /S/P /Type/StructElem>> endobj 3865 0 obj <> /K 3 /P 3861 0 R /Pg 443 0 R /S/P /Type/StructElem>> endobj 3866 0 obj <> /K[3867 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 3867 0 obj <> /K 4 /P 3866 0 R /Pg 443 0 R /S/P /Type/StructElem>> endobj 3868 0 obj <> endobj 3869 0 obj <> /K 0 /P 3868 0 R /Pg 448 0 R /S/P /Type/StructElem>> endobj 3870 0 obj <> /K 1 /P 3868 0 R /Pg 448 0 R /S/P /Type/StructElem>> endobj 3871 0 obj <> /K 2 /P 3868 0 R /Pg 448 0 R /S/P /Type/StructElem>> endobj 3872 0 obj <> /K[3873 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 3873 0 obj <> /K 3 /P 3872 0 R /Pg 448 0 R /S/P /Type/StructElem>> endobj 3874 0 obj <> endobj 3875 0 obj <> /K 0 /P 3874 0 R /Pg 453 0 R /S/P /Type/StructElem>> endobj 3876 0 obj <> /K 1 /P 3874 0 R /Pg 453 0 R /S/P /Type/StructElem>> endobj 3877 0 obj <> /K 2 /P 3874 0 R /Pg 453 0 R /S/P /Type/StructElem>> endobj 3878 0 obj <> /K 3 /P 3874 0 R /Pg 453 0 R /S/P /Type/StructElem>> endobj 3879 0 obj <> /K 4 /P 3874 0 R /Pg 453 0 R /S/P /Type/StructElem>> endobj 3880 0 obj <> /K[3881 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 3881 0 obj <> /K 5 /P 3880 0 R /Pg 453 0 R /S/P /Type/StructElem>> endobj 3882 0 obj <> endobj 3883 0 obj <> /K 0 /P 3882 0 R /Pg 458 0 R /S/P /Type/StructElem>> endobj 3884 0 obj <> /K 1 /P 3882 0 R /Pg 458 0 R /S/P /Type/StructElem>> endobj 3885 0 obj <> endobj 3886 0 obj <> /K 2 /P 3885 0 R /Pg 458 0 R /S/P /Type/StructElem>> endobj 3887 0 obj <> /K[3888 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 3888 0 obj <> /K 3 /P 3887 0 R /Pg 458 0 R /S/P /Type/StructElem>> endobj 3889 0 obj <> endobj 3890 0 obj <> /K 4 /P 3889 0 R /Pg 458 0 R /S/P /Type/StructElem>> endobj 3891 0 obj <> /K 5 /P 3889 0 R /Pg 458 0 R /S/P /Type/StructElem>> endobj 3892 0 obj <> /K 6 /P 3889 0 R /Pg 458 0 R /S/P /Type/StructElem>> endobj 3893 0 obj <> /K 7 /P 3889 0 R /Pg 458 0 R /S/P /Type/StructElem>> endobj 3894 0 obj <> endobj 3895 0 obj <> /K 8 /P 3894 0 R /Pg 458 0 R /S/P /Type/StructElem>> endobj 3896 0 obj <> /K 9 /P 3894 0 R /Pg 458 0 R /S/P /Type/StructElem>> endobj 3897 0 obj <> /K 10 /P 3894 0 R /Pg 458 0 R /S/P /Type/StructElem>> endobj 3898 0 obj <> /K 11 /P 3894 0 R /Pg 458 0 R /S/P /Type/StructElem>> endobj 3899 0 obj <> /K 12 /P 3894 0 R /Pg 458 0 R /S/P /Type/StructElem>> endobj 3900 0 obj <> /K 13 /P 3894 0 R /Pg 458 0 R /S/P /Type/StructElem>> endobj 3901 0 obj <> /K[3902 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 3902 0 obj <> /K 14 /P 3901 0 R /Pg 458 0 R /S/P /Type/StructElem>> endobj 3903 0 obj <> /K[3904 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 3904 0 obj <> /K 15 /P 3903 0 R /Pg 458 0 R /S/P /Type/StructElem>> endobj 3905 0 obj <> endobj 3906 0 obj <> /K 0 /P 3905 0 R /Pg 463 0 R /S/P /Type/StructElem>> endobj 3907 0 obj <> /K 1 /P 3905 0 R /Pg 463 0 R /S/P /Type/StructElem>> endobj 3908 0 obj <> /K 2 /P 3905 0 R /Pg 463 0 R /S/P /Type/StructElem>> endobj 3909 0 obj <> /K 3 /P 3905 0 R /Pg 463 0 R /S/P /Type/StructElem>> endobj 3910 0 obj <> /K[3911 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 3911 0 obj <> /K 4 /P 3910 0 R /Pg 463 0 R /S/P /Type/StructElem>> endobj 3912 0 obj <> /K[3913 0 R 3914 0 R 3915 0 R 3916 0 R 3917 0 R 3918 0 R 3919 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 3913 0 obj <> /K 0 /P 3912 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3914 0 obj <> /K 1 /P 3912 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3915 0 obj <> /K 2 /P 3912 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3916 0 obj <> /K 3 /P 3912 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3917 0 obj <> /K 4 /P 3912 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3918 0 obj <> /K 5 /P 3912 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3919 0 obj <> /K 6 /P 3912 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3920 0 obj <> endobj 3921 0 obj <> /K 7 /P 3920 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3922 0 obj <> endobj 3923 0 obj <> /K[3924 0 R 3933 0 R 3944 0 R 3955 0 R 3966 0 R 3977 0 R 3988 0 R] /P 3922 0 R /S/Table /Type/StructElem>> endobj 3924 0 obj <> endobj 3925 0 obj <> /K[3926 0 R] /P 3924 0 R /S/TD /Type/StructElem>> endobj 3926 0 obj <> /K 8 /P 3925 0 R /Pg 468 0 R /S/P 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obj <> /K[31 3964 0 R 3965 0 R] /P 3962 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3964 0 obj <> endobj 3965 0 obj <> endobj 3966 0 obj <> endobj 3967 0 obj <> /K[3968 0 R] /P 3966 0 R /S/TD /Type/StructElem>> endobj 3968 0 obj <> /K 34 /P 3967 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3969 0 obj <> /K[3970 0 R] /P 3966 0 R /S/TD /Type/StructElem>> endobj 3970 0 obj <> /K[35 3971 0 R 3972 0 R] /P 3969 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3971 0 obj <> endobj 3972 0 obj <> endobj 3973 0 obj <> /K[3974 0 R] /P 3966 0 R /S/TD /Type/StructElem>> endobj 3974 0 obj <> /K[38 3975 0 R 3976 0 R] /P 3973 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3975 0 obj <> endobj 3976 0 obj <> endobj 3977 0 obj <> endobj 3978 0 obj <> /K[3979 0 R] /P 3977 0 R /S/TD /Type/StructElem>> endobj 3979 0 obj <> /K 41 /P 3978 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3980 0 obj <> /K[3981 0 R] /P 3977 0 R /S/TD /Type/StructElem>> endobj 3981 0 obj <> /K[42 3982 0 R 3983 0 R] /P 3980 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3982 0 obj <> endobj 3983 0 obj <> endobj 3984 0 obj <> /K[3985 0 R] /P 3977 0 R /S/TD /Type/StructElem>> endobj 3985 0 obj <> /K[45 3986 0 R 3987 0 R] /P 3984 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3986 0 obj <> endobj 3987 0 obj <> endobj 3988 0 obj <> endobj 3989 0 obj <> /K[3990 0 R] /P 3988 0 R /S/TD /Type/StructElem>> endobj 3990 0 obj <> /K 48 /P 3989 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3991 0 obj <> /K[3992 0 R] /P 3988 0 R /S/TD /Type/StructElem>> endobj 3992 0 obj <> /K[49 3993 0 R 3994 0 R] /P 3991 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3993 0 obj <> endobj 3994 0 obj <> endobj 3995 0 obj <> /K[3996 0 R] /P 3988 0 R /S/TD /Type/StructElem>> endobj 3996 0 obj <> /K[52 3997 0 R 3998 0 R] /P 3995 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 3997 0 obj <> endobj 3998 0 obj <> endobj 3999 0 obj <> /K[4000 0 R 4001 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 4000 0 obj <> /K 55 /P 3999 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 4001 0 obj <> /K 56 /P 3999 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 4002 0 obj <> endobj 4003 0 obj <> /K 57 /P 4002 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 4004 0 obj <> /K[4005 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 4005 0 obj <> /K 58 /P 4004 0 R /Pg 468 0 R /S/P /Type/StructElem>> endobj 4006 0 obj <> endobj 4007 0 obj <> /K 0 /P 4006 0 R /Pg 473 0 R /S/P /Type/StructElem>> endobj 4008 0 obj <> /K 1 /P 4006 0 R /Pg 473 0 R /S/P /Type/StructElem>> endobj 4009 0 obj <> /K 2 /P 4006 0 R /Pg 473 0 R /S/P /Type/StructElem>> endobj 4010 0 obj <> /K 3 /P 4006 0 R /Pg 473 0 R /S/P /Type/StructElem>> endobj 4011 0 obj <> /K[4012 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 4012 0 obj <> /K 4 /P 4011 0 R /Pg 473 0 R /S/P /Type/StructElem>> endobj 4013 0 obj <> /K[4014 0 R 4015 0 R 4016 0 R 4017 0 R 4018 0 R 4019 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 4014 0 obj <> /K 0 /P 4013 0 R /Pg 478 0 R /S/P /Type/StructElem>> endobj 4015 0 obj <> /K 1 /P 4013 0 R /Pg 478 0 R /S/P /Type/StructElem>> endobj 4016 0 obj <> /K 2 /P 4013 0 R /Pg 478 0 R /S/P /Type/StructElem>> endobj 4017 0 obj <> /K 3 /P 4013 0 R /Pg 478 0 R /S/P /Type/StructElem>> endobj 4018 0 obj <> /K 4 /P 4013 0 R /Pg 478 0 R /S/P /Type/StructElem>> endobj 4019 0 obj <> /K 5 /P 4013 0 R /Pg 478 0 R /S/P /Type/StructElem>> endobj 4020 0 obj <> endobj 4021 0 obj <> /K 6 /P 4020 0 R /Pg 478 0 R /S/P /Type/StructElem>> endobj 4022 0 obj <> /K 7 /P 4020 0 R /Pg 478 0 R /S/P /Type/StructElem>> endobj 4023 0 obj <> /K[4024 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 4024 0 obj <> /K 8 /P 4023 0 R /Pg 478 0 R /S/P /Type/StructElem>> endobj 4025 0 obj <> /K[4026 0 R 4027 0 R 4028 0 R 4029 0 R 4030 0 R 4031 0 R 4032 0 R 4033 0 R 4034 0 R 4035 0 R 4036 0 R 4037 0 R 4038 0 R 4039 0 R 4040 0 R 4041 0 R 4042 0 R 4043 0 R 4044 0 R 4045 0 R 4046 0 R 4047 0 R 4048 0 R 4049 0 R 4050 0 R 4051 0 R 4052 0 R 4053 0 R 4054 0 R] /P 1252 0 R /S/Div 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0 R] /P 4638 0 R /S/TD /Type/StructElem>> endobj 4658 0 obj <> /K[181 4659 0 R 4660 0 R] /P 4657 0 R /Pg 588 0 R /S/P /Type/StructElem>> endobj 4659 0 obj <> endobj 4660 0 obj <> endobj 4661 0 obj <> /K[4662 0 R] /P 4638 0 R /S/TD /Type/StructElem>> endobj 4662 0 obj <> /K[184 4663 0 R 4664 0 R] /P 4661 0 R /Pg 588 0 R /S/P /Type/StructElem>> endobj 4663 0 obj <> endobj 4664 0 obj <> endobj 4665 0 obj <> /K[4666 0 R] /P 4638 0 R /S/TD /Type/StructElem>> endobj 4666 0 obj <> /K[187 4667 0 R 4668 0 R] /P 4665 0 R /Pg 588 0 R /S/P /Type/StructElem>> endobj 4667 0 obj <> endobj 4668 0 obj <> endobj 4669 0 obj <> /K[4670 0 R] /P 4638 0 R /S/TD /Type/StructElem>> endobj 4670 0 obj <> /K[190 4671 0 R 4672 0 R] /P 4669 0 R /Pg 588 0 R /S/P /Type/StructElem>> endobj 4671 0 obj <> endobj 4672 0 obj <> endobj 4673 0 obj <> /K[4674 0 R] /P 4638 0 R /S/TD /Type/StructElem>> endobj 4674 0 obj <> /K[193 4675 0 R 4676 0 R] /P 4673 0 R /Pg 588 0 R /S/P /Type/StructElem>> endobj 4675 0 obj <> endobj 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/S/P /Type/StructElem>> endobj 7035 0 obj <> /K 5 /P 7029 0 R /Pg 949 0 R /S/P /Type/StructElem>> endobj 7036 0 obj <> /K 6 /P 7029 0 R /Pg 949 0 R /S/P /Type/StructElem>> endobj 7037 0 obj <> /K[7038 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 7038 0 obj <> /K 7 /P 7037 0 R /Pg 949 0 R /S/P /Type/StructElem>> endobj 7039 0 obj <> endobj 7040 0 obj <> /K 0 /P 7039 0 R /Pg 954 0 R /S/P /Type/StructElem>> endobj 7041 0 obj <> /K 1 /P 7039 0 R /Pg 954 0 R /S/P /Type/StructElem>> endobj 7042 0 obj <> /K 2 /P 7039 0 R /Pg 954 0 R /S/P /Type/StructElem>> endobj 7043 0 obj <> /K 3 /P 7039 0 R /Pg 954 0 R /S/P /Type/StructElem>> endobj 7044 0 obj <> /K[7045 0 R] /P 1252 0 R /S/Div /Type/StructElem>> endobj 7045 0 obj <> /K 4 /P 7044 0 R /Pg 954 0 R /S/P /Type/StructElem>> endobj 7046 0 obj <> endobj 7047 0 obj <> /K 0 /P 7046 0 R /Pg 959 0 R /S/P /Type/StructElem>> endobj 7048 0 obj <> /K 1 /P 7046 0 R /Pg 959 0 R /S/P /Type/StructElem>> endobj 7049 0 obj <> /K 2 /P 7046 0 R /Pg 959 0 R 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1335 0 R 1336 0 R 1337 0 R 1339 0 R 1341 0 R 1343 0 R 1346 0 R 1347 0 R 1348 0 R 1350 0 R 1351 0 R 1352 0 R 1354 0 R 1356 0 R 1359 0 R 1360 0 R 1361 0 R 1363 0 R 1364 0 R 1365 0 R 1367 0 R 1369 0 R 1370 0 R 1371 0 R 1374 0 R 1375 0 R 1376 0 R 1378 0 R 1379 0 R 1380 0 R 1382 0 R 1384 0 R 1387 0 R 1388 0 R 1389 0 R 1391 0 R 1392 0 R 1393 0 R 1395 0 R 1397 0 R 1400 0 R 1401 0 R 1402 0 R 1404 0 R 1405 0 R 1406 0 R 1408 0 R 1410 0 R 1413 0 R 1414 0 R 1415 0 R 1417 0 R 1419 0 R 1421 0 R 1424 0 R 1425 0 R 1426 0 R 1428 0 R 1429 0 R 1430 0 R 1432 0 R 1434 0 R 1437 0 R 1438 0 R 1439 0 R 1441 0 R 1442 0 R 1443 0 R 1445 0 R 1447 0 R 1450 0 R 1451 0 R 1452 0 R 1454 0 R 1455 0 R 1456 0 R 1458 0 R 1460 0 R 1463 0 R 1464 0 R 1465 0 R 1467 0 R 1468 0 R 1469 0 R 1471 0 R 1473 0 R 1476 0 R 1478 0 R 1480 0 R 1483 0 R 1484 0 R 1485 0 R 1487 0 R 1489 0 R 1491 0 R 1494 0 R 1495 0 R 1496 0 R 1498 0 R 1500 0 R 1502 0 R 1505 0 R 1506 0 R 1507 0 R 1509 0 R 1511 0 R 1513 0 R 1516 0 R 1517 0 R 1518 0 R 1520 0 R 1521 0 R 1522 0 R 1524 0 R 1526 0 R 1529 0 R 1530 0 R 1531 0 R 1533 0 R 1534 0 R 1535 0 R 1537 0 R 1539 0 R 1542 0 R 1543 0 R 1544 0 R 1546 0 R 1547 0 R 1548 0 R 1550 0 R 1552 0 R 1554 0 R] endobj 7122 0 obj [1556 0 R 1557 0 R 1558 0 R 1559 0 R 1560 0 R 1561 0 R 1562 0 R 1563 0 R 1564 0 R 1565 0 R 1566 0 R 1567 0 R 1568 0 R 1569 0 R 1570 0 R 1571 0 R 1572 0 R 1573 0 R 1574 0 R 1575 0 R 1576 0 R 1577 0 R 1578 0 R 1579 0 R 1580 0 R 1581 0 R 1582 0 R 1583 0 R 1584 0 R 1585 0 R 1587 0 R] endobj 7123 0 obj [1592 0 R 1593 0 R 1595 0 R 1596 0 R 1597 0 R 1599 0 R 1602 0 R 1604 0 R 1606 0 R 1609 0 R 1610 0 R 1611 0 R 1613 0 R 1615 0 R 1617 0 R 1620 0 R 1621 0 R 1622 0 R 1624 0 R 1626 0 R 1628 0 R 1631 0 R 1632 0 R 1633 0 R 1635 0 R 1636 0 R 1637 0 R 1639 0 R 1641 0 R 1644 0 R 1645 0 R 1646 0 R 1648 0 R 1649 0 R 1650 0 R 1652 0 R 1654 0 R 1657 0 R 1658 0 R 1659 0 R 1661 0 R 1662 0 R 1663 0 R 1665 0 R 1667 0 R 1670 0 R 1671 0 R 1672 0 R 1674 0 R 1675 0 R 1676 0 R 1678 0 R 1680 0 R 1683 0 R 1684 0 R 1685 0 R 1687 0 R 1688 0 R 1689 0 R 1691 0 R 1693 0 R 1696 0 R 1697 0 R 1698 0 R 1700 0 R 1702 0 R 1704 0 R 1707 0 R 1708 0 R 1709 0 R 1711 0 R 1712 0 R 1713 0 R 1715 0 R 1717 0 R 1720 0 R 1721 0 R 1722 0 R 1724 0 R 1725 0 R 1726 0 R 1728 0 R 1730 0 R 1733 0 R 1734 0 R 1735 0 R 1737 0 R 1739 0 R 1741 0 R 1744 0 R 1745 0 R 1746 0 R 1748 0 R 1749 0 R 1750 0 R 1752 0 R 1754 0 R 1757 0 R 1758 0 R 1759 0 R 1761 0 R 1762 0 R 1763 0 R 1765 0 R 1767 0 R 1770 0 R 1771 0 R 1772 0 R 1774 0 R 1775 0 R 1776 0 R 1778 0 R 1780 0 R 1783 0 R 1784 0 R 1785 0 R 1787 0 R 1789 0 R 1791 0 R 1794 0 R 1795 0 R 1796 0 R 1798 0 R 1799 0 R 1800 0 R 1802 0 R 1804 0 R 1807 0 R 1808 0 R 1809 0 R 1811 0 R 1812 0 R 1813 0 R 1815 0 R 1817 0 R 1820 0 R 1821 0 R 1822 0 R 1824 0 R 1826 0 R 1828 0 R 1831 0 R 1833 0 R 1835 0 R 1838 0 R 1839 0 R 1840 0 R 1842 0 R 1844 0 R 1846 0 R 1849 0 R 1850 0 R 1851 0 R 1853 0 R 1854 0 R 1855 0 R 1857 0 R 1859 0 R 1862 0 R 1863 0 R 1864 0 R 1866 0 R 1867 0 R 1868 0 R 1870 0 R 1872 0 R 1875 0 R 1876 0 R 1877 0 R 1879 0 R 1880 0 R 1881 0 R 1883 0 R 1885 0 R 1888 0 R 1889 0 R 1890 0 R 1892 0 R 1893 0 R 1894 0 R 1896 0 R 1898 0 R 1901 0 R 1902 0 R 1903 0 R 1905 0 R 1907 0 R 1909 0 R 1912 0 R 1913 0 R 1914 0 R 1916 0 R 1917 0 R 1918 0 R 1920 0 R 1922 0 R 1925 0 R 1926 0 R 1927 0 R 1929 0 R 1930 0 R 1931 0 R 1933 0 R 1935 0 R 1938 0 R 1939 0 R 1940 0 R 1942 0 R 1943 0 R 1944 0 R 1946 0 R 1948 0 R 1951 0 R 1952 0 R 1953 0 R 1955 0 R 1957 0 R 1959 0 R 1962 0 R 1964 0 R 1966 0 R] endobj 7124 0 obj [1968 0 R 1969 0 R 1970 0 R 1971 0 R 1973 0 R] endobj 7125 0 obj [1975 0 R 1976 0 R 1977 0 R 1979 0 R] endobj 7126 0 obj [1981 0 R 1982 0 R 1983 0 R 1985 0 R] endobj 7127 0 obj [1987 0 R 1988 0 R 1989 0 R 1991 0 R] endobj 7128 0 obj [1993 0 R 1994 0 R 1995 0 R 1997 0 R] endobj 7129 0 obj [1999 0 R 2000 0 R 2001 0 R 2002 0 R 2003 0 R 2004 0 R 2006 0 R] endobj 7130 0 obj [2008 0 R 2009 0 R 2010 0 R 2011 0 R 2012 0 R 2013 0 R 2014 0 R 2015 0 R 2016 0 R 2017 0 R 2018 0 R 2020 0 R] endobj 7131 0 obj [2022 0 R 2023 0 R 2024 0 R 2025 0 R 2026 0 R 2027 0 R 2029 0 R] endobj 7132 0 obj [2031 0 R 2032 0 R 2033 0 R 2034 0 R 2035 0 R 2036 0 R 2037 0 R 2038 0 R 2040 0 R] endobj 7133 0 obj [2042 0 R 2043 0 R 2047 0 R 2048 0 R 2049 0 R 2050 0 R 2052 0 R 2054 0 R 2055 0 R 2056 0 R 2057 0 R 2059 0 R 2060 0 R 2061 0 R 2063 0 R 2064 0 R 2065 0 R 2066 0 R 2067 0 R 2069 0 R 2070 0 R 2071 0 R 2073 0 R 2074 0 R 2076 0 R 2077 0 R 2079 0 R 2080 0 R 2081 0 R 2083 0 R 2084 0 R 2085 0 R 2087 0 R 2088 0 R 2089 0 R 2090 0 R 2092 0 R 2093 0 R 2095 0 R 2096 0 R 2097 0 R 2098 0 R 2100 0 R 2101 0 R 2102 0 R 2103 0 R 2104 0 R 2105 0 R 2107 0 R 2108 0 R 2109 0 R 2110 0 R 2111 0 R 2113 0 R 2114 0 R 2115 0 R 2117 0 R 2118 0 R 2119 0 R 2120 0 R 2121 0 R 2122 0 R 2123 0 R 2125 0 R 2126 0 R 2127 0 R 2128 0 R 2129 0 R 2131 0 R 2132 0 R 2133 0 R 2134 0 R 2135 0 R 2137 0 R 2138 0 R 2139 0 R 2140 0 R 2141 0 R 2142 0 R 2143 0 R 2144 0 R 2145 0 R 2146 0 R 2147 0 R 2148 0 R 2150 0 R 2151 0 R 2153 0 R 2154 0 R 2156 0 R 2158 0 R] endobj 7134 0 obj [2160 0 R 2161 0 R 2162 0 R 2163 0 R 2164 0 R 2165 0 R 2166 0 R 2167 0 R 2169 0 R 2170 0 R 2171 0 R 2172 0 R 2173 0 R 2174 0 R 2176 0 R 2178 0 R] endobj 7135 0 obj [2180 0 R 2181 0 R 2182 0 R 2183 0 R 2184 0 R 2185 0 R 2186 0 R 2187 0 R 2188 0 R 2189 0 R 2191 0 R 2192 0 R 2193 0 R 2194 0 R 2195 0 R 2197 0 R 2199 0 R 2201 0 R] endobj 7136 0 obj [2203 0 R 2204 0 R 2205 0 R 2206 0 R 2207 0 R 2208 0 R 2209 0 R 2210 0 R 2211 0 R 2212 0 R 2213 0 R 2214 0 R 2215 0 R 2217 0 R] endobj 7137 0 obj [2219 0 R 2220 0 R 2221 0 R 2222 0 R 2224 0 R 2225 0 R 2226 0 R 2227 0 R 2228 0 R 2230 0 R 2231 0 R 2233 0 R] endobj 7138 0 obj [2235 0 R 2236 0 R 2237 0 R 2238 0 R 2239 0 R 2240 0 R 2241 0 R 2242 0 R 2243 0 R 2244 0 R 2245 0 R 2246 0 R 2248 0 R 2250 0 R 2252 0 R] endobj 7139 0 obj [2254 0 R 2255 0 R 2256 0 R 2257 0 R 2258 0 R 2259 0 R 2261 0 R] endobj 7140 0 obj [2263 0 R 2264 0 R 2265 0 R 2266 0 R 2267 0 R 2268 0 R 2269 0 R 2270 0 R 2272 0 R] endobj 7141 0 obj [2274 0 R 2275 0 R 2276 0 R 2277 0 R 2278 0 R 2279 0 R 2280 0 R 2281 0 R 2282 0 R 2283 0 R 2284 0 R 2285 0 R 2286 0 R 2287 0 R 2288 0 R 2290 0 R] endobj 7142 0 obj [2292 0 R 2293 0 R 2295 0 R] endobj 7143 0 obj [2297 0 R 2298 0 R 2299 0 R 2301 0 R] endobj 7144 0 obj [2303 0 R 2304 0 R 2305 0 R 2306 0 R 2307 0 R 2309 0 R] endobj 7145 0 obj [2311 0 R 2312 0 R 2313 0 R 2314 0 R 2315 0 R 2317 0 R] endobj 7146 0 obj [2319 0 R 2320 0 R 2321 0 R 2322 0 R 2324 0 R] endobj 7147 0 obj [2326 0 R 2327 0 R 2328 0 R 2329 0 R 2330 0 R 2331 0 R 2332 0 R 2336 0 R 2337 0 R 2338 0 R 2340 0 R 2342 0 R] endobj 7148 0 obj [2344 0 R 2345 0 R 2346 0 R 2350 0 R 2351 0 R 2353 0 R 2354 0 R 2356 0 R 2358 0 R] endobj 7149 0 obj [2360 0 R 2361 0 R 2362 0 R 2363 0 R 2364 0 R 2365 0 R 2367 0 R] endobj 7150 0 obj [2370 0 R 2371 0 R 2372 0 R 2373 0 R 2374 0 R 2376 0 R 2377 0 R 2378 0 R 2379 0 R 2380 0 R 2381 0 R 2382 0 R 2384 0 R] endobj 7151 0 obj [2386 0 R 2387 0 R 2388 0 R 2389 0 R 2390 0 R 2391 0 R 2392 0 R 2393 0 R 2395 0 R] endobj 7152 0 obj [2397 0 R 2398 0 R 2399 0 R 2400 0 R 2401 0 R 2402 0 R 2404 0 R] endobj 7153 0 obj [2406 0 R 2407 0 R 2408 0 R 2409 0 R 2410 0 R 2412 0 R] endobj 7154 0 obj [2414 0 R 2415 0 R 2416 0 R 2417 0 R 2419 0 R] endobj 7155 0 obj [2421 0 R 2422 0 R 2426 0 R 2427 0 R 2428 0 R 2430 0 R 2431 0 R 2432 0 R 2433 0 R 2434 0 R 2435 0 R 2436 0 R 2437 0 R 2439 0 R] endobj 7156 0 obj [2441 0 R 2442 0 R 2443 0 R 2444 0 R 2445 0 R 2449 0 R 2450 0 R 2452 0 R 2453 0 R 2454 0 R 2455 0 R 2456 0 R 2458 0 R] endobj 7157 0 obj [2460 0 R 2461 0 R 2462 0 R 2463 0 R 2464 0 R 2465 0 R 2467 0 R] endobj 7158 0 obj [2469 0 R 2470 0 R 2471 0 R 2472 0 R 2473 0 R 2474 0 R 2476 0 R] endobj 7159 0 obj [2478 0 R 2479 0 R 2480 0 R 2481 0 R 2483 0 R] endobj 7160 0 obj [2485 0 R 2486 0 R 2487 0 R 2488 0 R 2489 0 R 2491 0 R] endobj 7161 0 obj [2493 0 R 2494 0 R 2495 0 R 2496 0 R 2497 0 R 2498 0 R 2499 0 R 2500 0 R 2501 0 R 2503 0 R] endobj 7162 0 obj [2505 0 R 2506 0 R 2507 0 R 2508 0 R 2509 0 R 2510 0 R 2511 0 R 2512 0 R 2513 0 R 2515 0 R] endobj 7163 0 obj [2520 0 R 2521 0 R 2522 0 R 2524 0 R 2527 0 R 2528 0 R 2530 0 R 2533 0 R 2535 0 R 2538 0 R 2540 0 R 2541 0 R 2542 0 R 2543 0 R 2544 0 R 2545 0 R 2547 0 R] endobj 7164 0 obj [2552 0 R 2555 0 R 2557 0 R 2560 0 R 2562 0 R 2565 0 R 2567 0 R 2570 0 R 2572 0 R 2575 0 R 2577 0 R 2578 0 R 2579 0 R 2580 0 R 2581 0 R 2583 0 R] endobj 7165 0 obj [2585 0 R 2586 0 R 2587 0 R 2588 0 R 2589 0 R 2590 0 R 2591 0 R 2592 0 R 2593 0 R 2594 0 R 2595 0 R 2597 0 R] endobj 7166 0 obj [2599 0 R 2600 0 R 2601 0 R 2602 0 R 2604 0 R 2605 0 R 2606 0 R 2610 0 R 2612 0 R 2613 0 R 2615 0 R 2618 0 R 2620 0 R 2623 0 R 2625 0 R 2627 0 R 2629 0 R 2630 0 R 2632 0 R 2633 0 R 2634 0 R 2635 0 R 2637 0 R] endobj 7167 0 obj [2639 0 R 2640 0 R 2641 0 R 2642 0 R 2643 0 R 2644 0 R 2645 0 R 2647 0 R] endobj 7168 0 obj [2649 0 R 2650 0 R 2654 0 R 2655 0 R 2657 0 R 2658 0 R 2661 0 R 2662 0 R 2665 0 R 2666 0 R 2667 0 R 2668 0 R 2669 0 R 2671 0 R] endobj 7169 0 obj [2673 0 R 2674 0 R 2675 0 R 2676 0 R 2677 0 R 2678 0 R 2680 0 R] endobj 7170 0 obj [2682 0 R 2683 0 R 2684 0 R 2685 0 R 2686 0 R 2687 0 R 2688 0 R 2689 0 R 2690 0 R 2691 0 R 2692 0 R 2693 0 R 2695 0 R 2697 0 R 2698 0 R 2699 0 R 2701 0 R] endobj 7171 0 obj [2703 0 R 2704 0 R 2705 0 R 2706 0 R 2707 0 R 2708 0 R 2709 0 R 2710 0 R 2711 0 R 2712 0 R 2713 0 R 2714 0 R 2715 0 R 2716 0 R 2717 0 R 2718 0 R 2719 0 R 2720 0 R 2721 0 R 2722 0 R 2723 0 R 2724 0 R 2725 0 R 2726 0 R 2727 0 R 2729 0 R 2731 0 R 2732 0 R 2734 0 R] endobj 7172 0 obj [2736 0 R 2737 0 R 2738 0 R 2739 0 R 2740 0 R 2741 0 R 2743 0 R] endobj 7173 0 obj [2745 0 R 2746 0 R 2747 0 R 2751 0 R 2753 0 R 2758 0 R 2760 0 R 2762 0 R 2763 0 R 2764 0 R 2766 0 R] endobj 7174 0 obj [2768 0 R 2772 0 R 2774 0 R 2777 0 R 2779 0 R 2781 0 R 2784 0 R 2786 0 R 2788 0 R 2790 0 R 2792 0 R 2794 0 R 2797 0 R 2799 0 R 2801 0 R 2803 0 R 2805 0 R 2807 0 R 2809 0 R 2812 0 R 2814 0 R 2816 0 R 2818 0 R 2820 0 R 2822 0 R 2824 0 R 2827 0 R 2829 0 R 2831 0 R 2833 0 R 2835 0 R 2837 0 R 2839 0 R 2842 0 R 2844 0 R 2847 0 R 2849 0 R 2851 0 R 2854 0 R 2856 0 R 2858 0 R 2860 0 R 2862 0 R 2864 0 R 2867 0 R 2869 0 R 2871 0 R 2873 0 R 2875 0 R 2877 0 R 2879 0 R 2882 0 R 2884 0 R 2886 0 R 2888 0 R 2890 0 R 2892 0 R 2894 0 R 2897 0 R 2899 0 R 2901 0 R 2903 0 R 2905 0 R 2907 0 R 2909 0 R 2910 0 R 2911 0 R 2913 0 R] endobj 7175 0 obj [2915 0 R 2919 0 R 2921 0 R 2924 0 R 2926 0 R 2928 0 R 2931 0 R 2933 0 R 2935 0 R 2937 0 R 2939 0 R 2941 0 R 2944 0 R 2946 0 R 2948 0 R 2950 0 R 2952 0 R 2954 0 R 2956 0 R 2959 0 R 2961 0 R 2963 0 R 2965 0 R 2967 0 R 2969 0 R 2971 0 R 2974 0 R 2976 0 R 2978 0 R 2980 0 R 2982 0 R 2984 0 R 2986 0 R 2989 0 R 2991 0 R 2994 0 R 2996 0 R 2998 0 R 3001 0 R 3003 0 R 3005 0 R 3007 0 R 3009 0 R 3011 0 R 3014 0 R 3016 0 R 3018 0 R 3020 0 R 3022 0 R 3024 0 R 3026 0 R 3029 0 R 3031 0 R 3033 0 R 3035 0 R 3037 0 R 3039 0 R 3041 0 R 3044 0 R 3046 0 R 3048 0 R 3050 0 R 3052 0 R 3054 0 R 3056 0 R 3057 0 R 3058 0 R 3059 0 R 3060 0 R 3061 0 R 3063 0 R] endobj 7176 0 obj [3065 0 R 3066 0 R 3067 0 R 3068 0 R 3069 0 R 3071 0 R] endobj 7177 0 obj [3076 0 R 3078 0 R 3079 0 R 3080 0 R 3083 0 R 3084 0 R 3085 0 R 3087 0 R 3090 0 R 3092 0 R 3093 0 R 3094 0 R 3097 0 R 3098 0 R 3099 0 R 3101 0 R 3104 0 R 3106 0 R 3109 0 R 3110 0 R 3112 0 R 3115 0 R 3117 0 R 3120 0 R 3122 0 R 3125 0 R 3126 0 R 3127 0 R 3129 0 R 3132 0 R 3133 0 R 3134 0 R 3136 0 R 3137 0 R 3138 0 R 3141 0 R 3142 0 R 3143 0 R 3145 0 R 3148 0 R 3150 0 R 3151 0 R 3152 0 R 3153 0 R 3154 0 R 3155 0 R 3156 0 R 3157 0 R 3158 0 R 3159 0 R 3160 0 R 3161 0 R 3162 0 R 3163 0 R 3164 0 R 3165 0 R 3166 0 R 3167 0 R 3168 0 R 3169 0 R 3170 0 R 3171 0 R 3172 0 R 3173 0 R 3174 0 R 3175 0 R 3176 0 R 3177 0 R 3178 0 R 3179 0 R 3180 0 R 3181 0 R 3182 0 R 3183 0 R 3184 0 R 3185 0 R 3187 0 R 3189 0 R 3190 0 R 3191 0 R 3192 0 R 3194 0 R] endobj 7178 0 obj [3196 0 R 3197 0 R 3198 0 R 3199 0 R 3200 0 R 3202 0 R] endobj 7179 0 obj [3208 0 R 3210 0 R 3212 0 R 3215 0 R 3217 0 R 3219 0 R 3222 0 R 3224 0 R 3227 0 R 3229 0 R 3232 0 R 3234 0 R 3236 0 R 3237 0 R 3241 0 R 3243 0 R 3244 0 R 3245 0 R 3246 0 R 3247 0 R 3248 0 R 3249 0 R 3250 0 R 3251 0 R 3252 0 R 3253 0 R 3254 0 R 3255 0 R 3256 0 R 3260 0 R 3262 0 R 3264 0 R 3266 0 R 3268 0 R 3269 0 R 3270 0 R 3271 0 R 3272 0 R 3273 0 R 3274 0 R 3275 0 R 3276 0 R 3278 0 R 3279 0 R 3281 0 R] endobj 7180 0 obj [3283 0 R 3284 0 R 3285 0 R 3287 0 R] endobj 7181 0 obj [3289 0 R 3290 0 R 3291 0 R 3292 0 R 3293 0 R 3294 0 R 3295 0 R 3296 0 R 3297 0 R 3298 0 R 3299 0 R 3300 0 R 3301 0 R 3302 0 R 3303 0 R 3304 0 R 3305 0 R 3306 0 R 3307 0 R 3308 0 R 3309 0 R 3310 0 R 3311 0 R 3312 0 R 3313 0 R 3314 0 R 3315 0 R 3316 0 R 3317 0 R 3318 0 R 3320 0 R 3321 0 R 3323 0 R] endobj 7182 0 obj [3326 0 R 3327 0 R 3328 0 R 3329 0 R 3330 0 R 3331 0 R 3332 0 R 3333 0 R 3334 0 R 3336 0 R 3338 0 R 3339 0 R 3343 0 R 3344 0 R 3345 0 R 3346 0 R 3347 0 R 3348 0 R 3349 0 R 3350 0 R 3351 0 R 3353 0 R 3354 0 R 3356 0 R 3357 0 R 3358 0 R 3360 0 R 3361 0 R 3362 0 R 3363 0 R 3364 0 R 3365 0 R 3367 0 R 3369 0 R] endobj 7183 0 obj [3371 0 R 3372 0 R 3373 0 R 3374 0 R 3375 0 R 3376 0 R 3377 0 R 3378 0 R 3379 0 R 3380 0 R 3381 0 R 3382 0 R 3383 0 R 3384 0 R 3385 0 R 3386 0 R 3387 0 R 3388 0 R 3389 0 R 3390 0 R 3391 0 R 3392 0 R 3393 0 R 3394 0 R 3395 0 R 3396 0 R 3397 0 R 3398 0 R 3400 0 R 3401 0 R 3403 0 R] endobj 7184 0 obj [3405 0 R 3406 0 R 3407 0 R 3408 0 R 3409 0 R 3410 0 R 3411 0 R 3412 0 R 3413 0 R 3414 0 R 3415 0 R 3416 0 R 3417 0 R 3418 0 R 3419 0 R 3421 0 R 3422 0 R 3424 0 R] endobj 7185 0 obj [3426 0 R 3427 0 R 3428 0 R 3429 0 R 3431 0 R] endobj 7186 0 obj [3433 0 R 3434 0 R 3435 0 R 3437 0 R] endobj 7187 0 obj [3439 0 R 3440 0 R 3441 0 R 3443 0 R] endobj 7188 0 obj [3445 0 R 3446 0 R 3447 0 R 3448 0 R 3449 0 R 3450 0 R 3451 0 R 3453 0 R] endobj 7189 0 obj [3455 0 R 3456 0 R 3457 0 R 3458 0 R 3459 0 R 3460 0 R 3461 0 R 3462 0 R 3463 0 R 3464 0 R 3466 0 R] endobj 7190 0 obj [3468 0 R 3469 0 R 3470 0 R 3475 0 R 3476 0 R 3478 0 R 3480 0 R 3482 0 R 3483 0 R 3485 0 R] endobj 7191 0 obj [3487 0 R 3488 0 R 3489 0 R 3494 0 R 3496 0 R 3498 0 R 3500 0 R 3502 0 R 3504 0 R 3506 0 R 3508 0 R 3510 0 R 3512 0 R 3514 0 R 3516 0 R 3518 0 R 3520 0 R 3522 0 R 3524 0 R 3526 0 R 3528 0 R 3530 0 R 3532 0 R 3534 0 R 3536 0 R 3538 0 R 3540 0 R 3542 0 R 3544 0 R 3546 0 R 3548 0 R 3550 0 R 3552 0 R 3554 0 R 3557 0 R 3558 0 R 3559 0 R 3560 0 R 3562 0 R 3563 0 R 3564 0 R 3565 0 R 3567 0 R] endobj 7192 0 obj [3569 0 R 3570 0 R 3571 0 R 3572 0 R 3573 0 R 3574 0 R 3575 0 R 3576 0 R 3577 0 R 3578 0 R 3579 0 R 3580 0 R 3581 0 R 3583 0 R 3584 0 R 3585 0 R 3586 0 R 3587 0 R 3589 0 R] endobj 7193 0 obj [3591 0 R 3592 0 R 3593 0 R 3594 0 R 3595 0 R 3596 0 R 3597 0 R 3598 0 R 3599 0 R 3600 0 R 3601 0 R 3602 0 R 3603 0 R 3605 0 R] endobj 7194 0 obj [3607 0 R 3608 0 R 3609 0 R 3610 0 R 3611 0 R 3612 0 R 3614 0 R] endobj 7195 0 obj [3616 0 R 3621 0 R 3623 0 R 3627 0 R 3628 0 R 3630 0 R 3631 0 R 3632 0 R 3633 0 R 3635 0 R] endobj 7196 0 obj [3637 0 R 3638 0 R 3639 0 R 3640 0 R 3641 0 R 3642 0 R 3643 0 R 3644 0 R 3646 0 R] endobj 7197 0 obj [3648 0 R 3649 0 R 3650 0 R 3651 0 R 3652 0 R 3653 0 R 3655 0 R] endobj 7198 0 obj [3657 0 R 3658 0 R 3659 0 R 3664 0 R 3666 0 R 3668 0 R 3670 0 R 3671 0 R 3673 0 R] endobj 7199 0 obj [3675 0 R 3676 0 R 3677 0 R 3678 0 R 3680 0 R] endobj 7200 0 obj [3682 0 R 3683 0 R 3684 0 R 3685 0 R 3686 0 R 3687 0 R 3688 0 R 3689 0 R 3690 0 R 3691 0 R 3692 0 R 3693 0 R 3694 0 R 3695 0 R 3696 0 R 3697 0 R 3698 0 R 3699 0 R 3700 0 R 3701 0 R 3702 0 R 3703 0 R 3704 0 R 3705 0 R 3707 0 R 3709 0 R 3710 0 R 3711 0 R 3712 0 R 3714 0 R] endobj 7201 0 obj [3716 0 R 3717 0 R 3718 0 R 3719 0 R 3720 0 R 3721 0 R 3722 0 R 3723 0 R 3724 0 R 3725 0 R 3726 0 R 3727 0 R 3728 0 R 3729 0 R 3730 0 R 3731 0 R 3732 0 R 3733 0 R 3734 0 R 3735 0 R 3736 0 R 3737 0 R 3738 0 R 3739 0 R 3740 0 R 3741 0 R 3742 0 R 3743 0 R 3745 0 R 3749 0 R 3750 0 R 3751 0 R 3752 0 R 3753 0 R 3754 0 R 3755 0 R 3756 0 R 3757 0 R 3758 0 R 3759 0 R 3760 0 R 3762 0 R 3763 0 R 3764 0 R 3766 0 R 3767 0 R 3768 0 R 3769 0 R 3770 0 R 3771 0 R 3772 0 R 3773 0 R 3774 0 R 3775 0 R 3776 0 R 3777 0 R 3778 0 R 3779 0 R 3781 0 R 3782 0 R 3783 0 R 3786 0 R 3787 0 R 3788 0 R 3790 0 R 3791 0 R 3792 0 R 3794 0 R 3795 0 R 3796 0 R 3798 0 R 3801 0 R 3802 0 R 3803 0 R 3805 0 R 3806 0 R 3807 0 R 3810 0 R 3812 0 R 3815 0 R 3817 0 R 3820 0 R 3823 0 R 3826 0 R 3827 0 R 3828 0 R 3830 0 R 3831 0 R 3832 0 R 3833 0 R 3834 0 R 3835 0 R 3836 0 R 3838 0 R] endobj 7202 0 obj [3840 0 R 3841 0 R 3842 0 R 3843 0 R 3844 0 R 3845 0 R 3847 0 R 3848 0 R 3849 0 R 3850 0 R 3852 0 R] endobj 7203 0 obj [3854 0 R 3855 0 R 3856 0 R 3857 0 R 3858 0 R 3860 0 R] endobj 7204 0 obj [3862 0 R 3863 0 R 3864 0 R 3865 0 R 3867 0 R] endobj 7205 0 obj [3869 0 R 3870 0 R 3871 0 R 3873 0 R] endobj 7206 0 obj [3875 0 R 3876 0 R 3877 0 R 3878 0 R 3879 0 R 3881 0 R] endobj 7207 0 obj [3883 0 R 3884 0 R 3886 0 R 3888 0 R 3890 0 R 3891 0 R 3892 0 R 3893 0 R 3895 0 R 3896 0 R 3897 0 R 3898 0 R 3899 0 R 3900 0 R 3902 0 R 3904 0 R] endobj 7208 0 obj [3906 0 R 3907 0 R 3908 0 R 3909 0 R 3911 0 R] endobj 7209 0 obj [3913 0 R 3914 0 R 3915 0 R 3916 0 R 3917 0 R 3918 0 R 3919 0 R 3921 0 R 3926 0 R 3928 0 R 3929 0 R 3930 0 R 3932 0 R 3935 0 R 3937 0 R 3938 0 R 3939 0 R 3941 0 R 3942 0 R 3943 0 R 3946 0 R 3948 0 R 3949 0 R 3950 0 R 3952 0 R 3953 0 R 3954 0 R 3957 0 R 3959 0 R 3960 0 R 3961 0 R 3963 0 R 3964 0 R 3965 0 R 3968 0 R 3970 0 R 3971 0 R 3972 0 R 3974 0 R 3975 0 R 3976 0 R 3979 0 R 3981 0 R 3982 0 R 3983 0 R 3985 0 R 3986 0 R 3987 0 R 3990 0 R 3992 0 R 3993 0 R 3994 0 R 3996 0 R 3997 0 R 3998 0 R 4000 0 R 4001 0 R 4003 0 R 4005 0 R] endobj 7210 0 obj [4007 0 R 4008 0 R 4009 0 R 4010 0 R 4012 0 R] endobj 7211 0 obj [4014 0 R 4015 0 R 4016 0 R 4017 0 R 4018 0 R 4019 0 R 4021 0 R 4022 0 R 4024 0 R] endobj 7212 0 obj [4026 0 R 4027 0 R 4028 0 R 4029 0 R 4030 0 R 4031 0 R 4032 0 R 4033 0 R 4034 0 R 4035 0 R 4036 0 R 4037 0 R 4038 0 R 4039 0 R 4040 0 R 4041 0 R 4042 0 R 4043 0 R 4044 0 R 4045 0 R 4046 0 R 4047 0 R 4048 0 R 4049 0 R 4050 0 R 4051 0 R 4052 0 R 4053 0 R 4054 0 R 4056 0 R 4057 0 R 4058 0 R 4059 0 R 4061 0 R] endobj 7213 0 obj [4063 0 R 4064 0 R 4065 0 R 4066 0 R 4067 0 R 4068 0 R 4070 0 R] endobj 7214 0 obj [4072 0 R 4073 0 R 4074 0 R 4075 0 R 4076 0 R 4078 0 R] endobj 7215 0 obj [4080 0 R 4081 0 R 4082 0 R 4083 0 R 4085 0 R] endobj 7216 0 obj [4087 0 R 4088 0 R 4089 0 R 4090 0 R 4092 0 R] endobj 7217 0 obj [4094 0 R 4095 0 R 4096 0 R 4098 0 R] endobj 7218 0 obj [4100 0 R 4101 0 R 4102 0 R 4103 0 R 4104 0 R 4106 0 R] endobj 7219 0 obj [4108 0 R 4109 0 R 4110 0 R 4111 0 R 4112 0 R 4113 0 R 4114 0 R 4115 0 R 4116 0 R 4117 0 R 4119 0 R] endobj 7220 0 obj [4121 0 R 4122 0 R 4124 0 R 4126 0 R 4128 0 R 4130 0 R 4132 0 R 4134 0 R 4136 0 R 4137 0 R 4138 0 R 4139 0 R 4140 0 R 4142 0 R] endobj 7221 0 obj [4144 0 R 4145 0 R 4146 0 R 4147 0 R 4148 0 R 4149 0 R 4150 0 R 4151 0 R 4152 0 R 4154 0 R 4155 0 R 4157 0 R 4158 0 R 4159 0 R 4163 0 R 4164 0 R 4166 0 R 4167 0 R 4169 0 R] endobj 7222 0 obj [4171 0 R 4172 0 R 4173 0 R 4174 0 R 4175 0 R 4176 0 R 4177 0 R 4178 0 R 4179 0 R 4181 0 R] endobj 7223 0 obj [4183 0 R 4184 0 R 4185 0 R 4186 0 R 4187 0 R 4189 0 R] endobj 7224 0 obj [4191 0 R 4192 0 R 4193 0 R 4195 0 R] endobj 7225 0 obj [4197 0 R 4198 0 R 4199 0 R 4200 0 R 4201 0 R 4202 0 R 4203 0 R 4204 0 R 4205 0 R 4207 0 R 4209 0 R] endobj 7226 0 obj [4211 0 R 4212 0 R 4213 0 R 4214 0 R 4215 0 R 4216 0 R 4217 0 R 4221 0 R 4222 0 R 4223 0 R 4225 0 R 4226 0 R 4227 0 R 4229 0 R] endobj 7227 0 obj [4231 0 R 4232 0 R 4233 0 R 4237 0 R 4239 0 R 4241 0 R 4243 0 R 4245 0 R 4247 0 R 4250 0 R 4252 0 R 4254 0 R 4256 0 R 4258 0 R 4260 0 R 4263 0 R 4265 0 R 4267 0 R 4269 0 R 4271 0 R 4273 0 R 4274 0 R 4275 0 R 4277 0 R] endobj 7228 0 obj [4279 0 R 4283 0 R 4285 0 R 4287 0 R 4289 0 R 4291 0 R 4293 0 R 4296 0 R 4298 0 R 4300 0 R 4302 0 R 4304 0 R 4306 0 R 4309 0 R 4311 0 R 4313 0 R 4315 0 R 4317 0 R 4319 0 R 4320 0 R 4321 0 R 4323 0 R] endobj 7229 0 obj [4325 0 R 4326 0 R 4327 0 R 4328 0 R 4329 0 R 4331 0 R] endobj 7230 0 obj [4333 0 R 4334 0 R 4335 0 R 4337 0 R] endobj 7231 0 obj [4343 0 R 4345 0 R 4347 0 R 4350 0 R 4352 0 R 4355 0 R 4357 0 R 4358 0 R 4359 0 R 4360 0 R 4361 0 R 4362 0 R 4363 0 R 4364 0 R 4365 0 R 4366 0 R 4367 0 R 4368 0 R 4372 0 R 4373 0 R 4374 0 R 4375 0 R 4376 0 R 4377 0 R 4378 0 R 4379 0 R 4380 0 R 4381 0 R 4382 0 R 4383 0 R 4384 0 R 4385 0 R 4386 0 R 4388 0 R 4389 0 R 4391 0 R] endobj 7232 0 obj [4393 0 R 4394 0 R 4395 0 R 4396 0 R 4398 0 R] endobj 7233 0 obj [4400 0 R 4402 0 R 4403 0 R 4404 0 R 4405 0 R 4407 0 R 4412 0 R 4413 0 R 4414 0 R 4416 0 R 4417 0 R 4418 0 R 4420 0 R 4421 0 R 4422 0 R 4424 0 R 4425 0 R 4426 0 R 4428 0 R 4429 0 R 4430 0 R 4432 0 R 4433 0 R 4434 0 R 4436 0 R 4437 0 R 4438 0 R 4440 0 R 4441 0 R 4442 0 R 4444 0 R 4445 0 R 4446 0 R 4448 0 R 4451 0 R 4453 0 R 4454 0 R 4455 0 R 4457 0 R 4458 0 R 4459 0 R 4461 0 R 4462 0 R 4463 0 R 4465 0 R 4466 0 R 4467 0 R 4469 0 R 4470 0 R 4471 0 R 4473 0 R 4474 0 R 4475 0 R 4477 0 R 4478 0 R 4479 0 R 4481 0 R 4482 0 R 4483 0 R 4485 0 R 4488 0 R 4489 0 R 4490 0 R 4492 0 R 4493 0 R 4494 0 R 4496 0 R 4497 0 R 4498 0 R 4500 0 R 4501 0 R 4502 0 R 4504 0 R 4505 0 R 4506 0 R 4508 0 R 4509 0 R 4510 0 R 4512 0 R 4513 0 R 4514 0 R 4516 0 R 4518 0 R 4519 0 R 4520 0 R 4522 0 R 4523 0 R 4524 0 R 4527 0 R 4528 0 R 4529 0 R 4531 0 R 4532 0 R 4533 0 R 4535 0 R 4536 0 R 4537 0 R 4539 0 R 4540 0 R 4541 0 R 4543 0 R 4544 0 R 4545 0 R 4547 0 R 4548 0 R 4549 0 R 4551 0 R 4552 0 R 4553 0 R 4555 0 R 4557 0 R 4558 0 R 4559 0 R 4561 0 R 4562 0 R 4563 0 R 4566 0 R 4568 0 R 4569 0 R 4570 0 R 4572 0 R 4573 0 R 4574 0 R 4576 0 R 4577 0 R 4578 0 R 4580 0 R 4581 0 R 4582 0 R 4584 0 R 4585 0 R 4586 0 R 4588 0 R 4589 0 R 4590 0 R 4592 0 R 4594 0 R 4595 0 R 4596 0 R 4598 0 R 4601 0 R 4602 0 R 4603 0 R 4605 0 R 4607 0 R 4608 0 R 4609 0 R 4611 0 R 4612 0 R 4613 0 R 4615 0 R 4616 0 R 4617 0 R 4619 0 R 4620 0 R 4621 0 R 4623 0 R 4624 0 R 4625 0 R 4627 0 R 4628 0 R 4629 0 R 4631 0 R 4632 0 R 4633 0 R 4635 0 R 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