Didier Auroux Jacques Blum

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Didier Auroux, Jacques Blum Laboratoire MIP - Institut de Mathematiques de Toulouse Universite Paul Sabatier Toulouse 3 France The Back and Forth Nudging (BFN) algorithm Laboratoire J.A. Dieudonne, Universite de Nice 19 Octobre 2006

  • dx˜ dt

  • bfn algorithm

  • backward model

  • observation operator

  • xobs ?

  • ups-tlse

  • universite de nice

  • forward nudging


Publié le : dimanche 1 octobre 2006
Lecture(s) : 58
Source : math.unice.fr
Nombre de pages : 42
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DidierAurouxUniversityofNiceSophiaAntipolisauroux@unice.frWorkincollaborationwith:J.Blum(U.Nice)+workofE.Cosme(U.Grenoble)INSU-CNRSLEFE-ASSIMBFN/BFN2,Villefranche-sur-Mer,France.BackandForthNudgingorSEEK(BFN,BF-SEEK)September29,2009
td=ForwardnudgingF(X)+K(XbosXdX(0)=0whereKisthenudging(orgain)matrix.X,H(X)),0<t<T,Inthelinearcase(whereFisamatrix),theforwardnudgingiscalledLuenbergerorasymptoticobserver.–Meteorology:Hoke-Anthes(1976)–Oceanography(QGmodel):Verron-Holland(1989)–Atmosphere(meso-scale):Stauffer-Seaman(1990)–Optimaldeterminationofthenudgingcoeffcients:Zou-Navon-LeDimet(1992),Stauffer-Bao(1993),Vidard-LeDimet-Piacentini(2003)LEFE/ASSIM2009,Villefranche-sur-Mer,September29200931/1
Forwardnudging:linearcaseLuenbergerobserver,orasymptoticobserver(Luenberger,1966)ddXt=FX+K(XobsHX),dXˆ=FXˆ,Xobs=HXˆ.tdddt(XXˆ)=(FKH)(XXˆ)IfFKHisaHurwitzmatrix,i.e.itsspectrumisstrictlyincludedinthehalf-plane{λC;Re(λ)<0},thenXXˆwhent+.LEFE/ASSIM2009,Villefranche-sur-Mer,September29200931/2
BackwardnudgingHowtorecovertheinitialstatefromthefinalsolution?Backwardmodel:dX˜=F(X˜),T>t>0,tdX˜(T)=X˜T.Ifweapplynudgingtothisbackwardmodel:td=F(˜X)dX˜X˜(T)=X˜T.K(XbosLEFE/ASSIM2009,Villefranche-sur-Mer,September292009H˜X),T>t>0,/331
BFN:BackandForthNudgingalgorithmIterativealgorithm(forwardandbackwardresolutions):X˜0(0)=Xb(firstguess)[Auroux-Blum(05)]Xdtdk=F(Xk)+K(XobsH(Xk))Xk(0)=X˜k1(0)˜Xdtdk=F(X˜k)K(XobsH(X˜k))X˜k(T)=Xk(T)IfXkandX˜kconvergetowardsthesamelimitX,andifK=K,thenXsatisfiesthestateequationandfitstotheobservations.ELEFA/SSIM0290,Vlielrfnahces-urM-er,Septebmer920290/431
31/5900292rebmetpeS,reM-rus-ehcnarfelliV,9002MISSA/EFELXHsboX,)XHsboX(1Rh2tΔ+iX,XFh2tΔinXX,nXXh2[Auroux-Blum(08)]n1+nXX=FXn+1+K(XobsHXn+1).tΔImplicitdiscretizationofthedirectmodelequationwithnudging:K=kHTR1whereRisthecovariancematrixoftheerrorsofobservation.bychoosingVariationalinterpretation:directnudgingisacompromisebetweenthemini-mizationoftheenergyofthesystemandthequadraticdistancetotheobser-vations:1nimXChoiceofthedirectnudgingmatrixK,i
ChoiceofthebackwardnudgingmatrixKThefeedbacktermhasadoublerole:stabilizationofthebackwardresolutionofthemodel(irreversiblesystem)feedbacktotheobservationsIfthesystemisobservable,i.e.rank[H,HF,...,HFN1]=N,thenthereexistsamatrixKsuchthatFKHisaHurwitzmatrix(poleassignmentmethod).Simplersolution:onecandefineK=kHTR1,wherekise.g.thesmallestvaluemakingthebackwardnumericalintegrationstable.LEFE/ASSIM2009,Villefranche-sur-Mer,September29200931/6
Shallowwatermodelτxtu(f+ζ)v+xB=ru+νΔuhρ0τtv+(f+ζ)u+yB=yrv+νΔvhρ0th+x(hu)+y(hv)=0ζ=xvyuistherelativevorticity;1B=gh+(u2+v2)istheBernoullipotential;2g=0.02m.s2isthereducedgravity;f=f0+βyistheCoriolisparameter(intheβ-planeapproximation),withf0=7.105s1andβ=2.1011m1.s1;τ=(τxy)istheforcingtermofthemodel(e.g.thewindstress),withamaximumamplitudeofτ0=0.05s2;ρ0=103kg.m3isthewaterdensity;r=9.108s1isthefrictioncoefficient.ν=5m2.s1istheviscosity(ordissipation)coefficient.LEFE/ASSIM2009,Villefranche-sur-Mer,September29200931/7
Shallowwatermodel2DshallowwatermodelState=heighthandhorizontalvelocity(u,v)uNemiraclpramateres:[Auroux(09)](runexample)Domain:L=2000km×2000km;Rigidboundaryandno-slipBC;Timestep=1800s;Assimilationperiod:15days;Forecastperiod:15+45daysObservations:ofhonly(satelliteobs),every5gridpointsineachspacedirection,every24hours.Background:truestateonemonthbeforethebeginningoftheassimilationperiod+whitegaussiannoise(10%)ComparisonBFN-4DVAR:seaheighth;velocity:uandv.LEFE/ASSIM2009,Villefranche-sur-Mer,September29200931/8
miT007 006 005 004 003 002 001 0 3.0 52.0 2.0 51.0 1.0 50.0 0 uspets emiT007 006 005 004 003 002 001 0 52.0 2.0 0 hspets emiT007 006 005 004 003 002 001 0 4.0 53.0 3.0 52.0 2.0 51.0 1.0 50.0 0 .sbotcefrep-ecnegrevnoCv51.0 1.0 50.0 RelativedifferencebetweentheBFNiterates(5firstiterations)andthetruesolutionversusthetimesteps,forh,uandv.31/9900292rebmetpeS,reM-rus-ehcnarfelliV,9002MISSA/EFELspets e
Comparaison-perfectobs.RelativeerrorBackgroundstateBFN(5iterations,converged)4D-VAR(5iterations)revnoc,snoitareti81(RAV-D4)degrevnoc,snoitareti81(RAV-D4huv37.6%21.5%30.3%0.44%1.78%2.41%13%460%744%4...13%460%744%40.61%2.43%3.46%Relativeerrorofthebackgroundstateandvariousidentifiedinitialconditionsforthethreevariables.LEFE/ASSIM2009,Villefranche-sur-Mer,September292009131/0
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