Sequential methods for ocean data assimilation. From theory to practical implementations (I)
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1 Sequenl ehods or ocen d sslon Fro heory o rccl leenons I P. BRASSEUR CNRS/LEGI, Grenoble, Frnce Perre.Brsseur@hg.ng.r J. Bllbrer, L. Berlne, F. Brol, J.M. Brnkr, G. Broque, V. Crlle, F. Csrucco, F. Debos, D. Rozer, Y. Ourères, T.Pendu, J. Verron E. Blyo, S. Cre, Ph D.T., C. Rober C.E. Tesu, B. Trnchn, N. Ferry, E. Rey, M. Benkrn F. Durnd, L. Gourdeu, Ch. Mes, P. De Mey A. Brh, C. Rck, M. Grégore L. Pren
2 Ocen d sslon D sslon nvolves he ol cobnon o esureens wh he underlyng dyncl rncles governng he syse under observon. D sslon cn serve severl ocenogrhc objecves: Ocen se eson n sce & e 4D ; Deecon o odel errors ; Eson o budges & odel reers ; Inlson, redcon, onorng ; Ol desgn o cole observon syses ; Theores: ol conrol VAR nd ol eson Kln
3 MERCATOR Asslon Syses Increenl leenon sregy OI Kln lers 3D/4D-VAR Reserch 993 SOFA 998 SEEK 999 OPAVAR R&D DEV ? OP ?? SAM- SAM- SAM-3
4 OUTLINE Se-o-he-r. Inroducon. Kln ler: undenls 3. Aled ocen d sslon: secc ssues 4. Slcons o he KF Ol Inerolon Advnced ssues 5. Sce reducon: se nd error sub-sces 6. Low rnk lers: SEEK nd EnKF 7. Objecve vldon nd dve schees 8. Iroved eorl sreges : FGAT nd IAU
5 «Sequenl» d sslon? Mehods/lgorhs «Vronl» vs. «Sequenl» Tlgrnd 997, Ide e l., 997 «Soohers» vs. «Flers» «Globl roble» vs. «sub-robles» Schröer 004 SEQ VAR! In non-lner world led ocen DA: he de h VAR ehods cn solve DA whou slng no surobles y be sledng
6 . Kln Fler undenls Proble denon Noons: Ide e l. 997 M y : eson o he «rue» se vecor e, denson n y : orecs o he se vecor e, usng he lner odel M : observons vlble e, denson How cn he rue se be bes esed ro hs ror noron?
7 . Kln Fler undenls Mulvre eson y ncolee d M erec odel
8 . Kln Fler undenls Uncernes & PDFs M e : error on se ese e ; unknown quny, bu ssue 0, P ~ e T N P η M : error on odel orecs beween e nd e ; ssue: 0,Q ~ e T η N η Q η
9 . Kln Fler undenls Error dgr M M η M M Se sce
10 . Kln Fler undenls Forecs error η η M M M Assung d nd, odel lnery nd uncorreled nl nd odellng errors, he orecs error s dsrbued s : P ~,P 3 e 0 T N Q M MP P M M M 4 T T T T T ηη η wh The eson error s led by: - unsble odel dyncs M ; - odellng errors Q.
11 . Kln Fler undenls Observons nd errors M y o y H : observon vlble e A robbly dsrbuon or he observon error s ssued: o 0, R ~ e o T o N R 5
12 . Kln Fler undenls Ol eson 6 P P P P y y y gven by 5 gven by 3 sclng cor Usng Byes rule e : { } H R H P H R H P ~ 7 e e e T T T T P P y y y y y The bes ese o s he vlue o whch ze 7,.e. he nu o : H R H P 8 y y T T J
13 . Kln Fler undenls Kln gn T δ J P H R y H 9 0 Equon 9 cn be solved or usng sle lgebr *, ledng o: T T P H HP H R y H Kln gn K Noe: he orecs nd nlyss equons cn be eended o wekly nonlner odels nd observon oeror. M H * Hn : use r equly [ X X X X ] X X X [ X X X X ] X X
14 . Kln Fler undenls Asslon cycle «sub-roble» P Q M MP P T M Forecs R H HP H P K T T H H P K I P K y Anlyss y
15 . Kln Fler undenls Asslon sequence Sequenl sslon reeed orecs/nlyss cycles 3 The bes ese gven e s nluenced by ll revous observons Kln «ler», nd he nlyss error covrnce relecs he coeon beween hs ccuulon o s noron nd he error growh due o odel erecons.
16 3. Aled ocen d sslon : he r o kng successul slcons «The scenc dculy o d sslon s o nd lgorhs whch sly he BLUE Bes Lner Unbsed Eson o n ordble oun o couer resources, whle reservng soe o he essenl chrcerscs.» Courer J. Meeor. Soc. Jn, 997
17 3. Aled ocen d sslon Model coley Model vrbles n HYCOM* eerure, slny, velocy, lyer hcknesses, se-surce hegh SSH * ocen crculon odel develoed Unv. M RSMAS, E. Chssgne
18 3. Aled ocen d sslon Se vecor denson HYCOM se vecor : 3D grd o he 5 sclr odel vrbles D grd or SSH R&D rooye: /3 horzonl resoluon, 9 hybrd lyers n ~ ~. 0 7 Oeronl rooye: / horzonl resoluon, 6 hybrd lyers n ~ ~ M oeror : d n n ~ rel.e. ~ 6000 Erh Sulors! The se vecor denson cn be huge The odel rnson r «M» cnno be reresened elcely. Insed, couer code s used o rnson ro e o
19 3. Aled ocen d sslon Sce observons Observed vrbles: - ro sce: se-surce hegh SSH, se-surce eerure SST TP/ERS long rck d Augus 9-6,993 AVHRR SST Augus 9-6,993
20 One week o leer d 3. Aled ocen d sslon Sce observons
21 3. Aled ocen d sslon In su observons Observed vrbles: - n su: T/S roles drng los, eld cgns, ARGO, July 004
22 3. Aled ocen d sslon Observons Observon vecor y : d ro vrous sources, deren e nd sce resoluon Rdr Alery : long-rck esureens o SSH noles JASON: obs. / 7 k, ~ 300 k equorl rcks seron, reeed every 0 dys ; ERS/ENVISAT: ~ 80 k equorl rcks seron, reeed every 35 dys AVHRR SST : weekly coose ges 4 k resoluon no clouds ARGO los : 3 3 horzonl resoluon rgeed, roles beween 000 deh o surce every 0 dys wh obs / long vercl d y s uch sller hn n d : oo ew observons! The ocen surce relvely well observed by selles: vercl erolon o d ssled he surce no he ocen s neror hs o be conssen wh vercl d roles The observed vrbles re closely reled o odel vrbles: H s nly n nerolon oeror ~ sle
23 Seccon o error covrnce r P 0? 3. Aled ocen d sslon Error covrnce r Assue bckground se nd ssoced error covrnce Consder he nlyss se wh only one d η odel grd on nd he ssoced observon error., y s sclr nd H s vecor o he or 0 The Kln gn s hen n vecor: H T T K P0H HP0 H R wh The oseror ese s correcon o he bckground usng he colun o η 0 ηη ηη P 0 [ 0,..., 0,, 0,..., 0] { P0 } ηη { P } η 0 ηη P ˆ 0 wh { P0 } η η η0 η0 { 0} η
24 3. Aled ocen d sslon Horzonl covrnce srucures Ele: Horzonl covrnce relve o SSH η on 3 o N,70 o W MERCATOR Asslon Syse - Tesu e l.004 Inluence uncon o sngle leer esureen n he sub-rocl gyre
25 3. Aled ocen d sslon Vercl covrnce srucures Mulvre rereseners n 3D sce, showng covrnce srucures conssen wh he odel dyncs Reresener uncons o SSH n ree-surce cosl odel Echevn e l., JPO, 000
26 3. Aled ocen d sslon Mulvre covrnce srucures Ele: covrnce relve o SSH η on 0 o,44 o W - Wever e l.003 The rows/colus o P should be «blnced» dynclly. Ths requres ulvre covrnces A ull-rnk reresenon o P d n n s sll ossble!
27 3. Aled ocen d sslon Model errors Q Model vrbly ders ro observed vrbly EKE ele : ¾ Model D Mny deren odel error sources ne dscrezons, reresenon o boo oogrhy, osherc orcngs, ec whch cnno be esly quned n ers o Q r
28 3. Aled ocen d sslon Sysec odel errors Model en SSH ders ro observed en SSH Men se-level derence beween Pcc nd Alnc syseclly oo sll n he odel OPA odel Mdec e l. 003 D Nler e l., 003 Oly roeres o he KF only n he bsence o bses!
29 4. Slcons o he Kln Fler Sub-ol lers «Ol nlyss s obned when he sscs o error elds ssoced wh orecss nd observons re known nd ccurely seced. Snce hese sscs re no generlly vlble, cul leenons o sslon lgorhs re lwys sub-ol» Dee nd D Slv Q. J. R. Meeorol. Soc., 998 Error dyncs : The orecs error requres ~ n odel negrons!!! T Q T P MP M Q M MP The ~ n odel negrons re useless Q s oorly known
30 4. Slcons o he Kln Fler Ol Inerolon Slcon o he Kln ler: «Ol Inerolon» To sve cos nd eory requreens, he KF cn be sled drsclly by usng e-ndeenden «bckground» covrnce r C P B D / C D / correlons vrnces : eressed s roduc o horzonl nd vercl correlons h v { } c l. c d C, j Ele: c h l l 3 l e l
31 4. Slcons o he Kln Fler Ol Inerolon SAM Merdonl nd zonl correlon scles rescrbed n SAM
32 4. Slcons o he Kln Fler Asyoc lers Sedy-se lers Fukuor e l., 993 The KF cn be sled usng e-ndeenden covrnce r coued s he syoc l o he Rcc equon: { [ ] } T T T P H HP H R HP M Q P M P P
33 Uncy o ol eses? The se odel wh he se se o observons cn rovde deren sequences o «ol eses», deendng on he «rge» eld! M Le : «erec eddy-resolvng» ocen odel y : «erec len-resolvng» ocen observons SLA long-rck
34 Asslon o sub-grd scle d K Trge eddes ~ 50 k T P MP M 0 T T ϕ P H HP H R K Trge lens ~ 5 k T P MP M ϕ Q T T 0 P H HP H R ϕ : reresenveness error due o he observon o lens Q ϕ : odel error ssoced o he s-reresenon o lens
35 Ele : sclr, lner odel Trge odel resoluon KF equons ϕ ϕ ϕ ϕ ϕ r r r k k r r k < < < < E: O.D.E. or? Asyoc soluon / 0 > r r or r r r r ϕ ϕ ϕ ϕ ϕ ϕ E: Wh <?
36 Ele : sclr, lner odel Trge observon resoluon KF equons 0 0 k y y k k q q! ϕ ϕ Asyoc soluon ϕ q 0
37 Ele : sclr, lner odel Nuercl ele: ϕ ϕ q r Trge odel resoluon Trge d resoluon
38 Ele : sclr, lner odel Msledng leenon : 0 ϕ q ϕ r Inlzon : 0 0, Frs sslon cycle : k y y k k! / Second sslon cycle : 0! y y
39 OI vs. EKF A ull Kln ler cnno be leened no relsc ocen odels error orecs nd nlyss equons oo eensve n CPU nd eory requreens «Ol Inerolon» over-sles he rogon o errors by neglecng dyncl rncles nd sscl noron Idelzed double-gyre odel Bllbrer e l., 00 Error % OI EKF Te dys
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