A dual-weighted approach to order reduction in 4d-var data assimilation

4D-VarDataAssimilation

D.N.Daescu∗

DepartmentofMathematicsandStatistics

PortlandStateUniversity,P.O.Box751,Portland,OR97207I.M.Navon

DepartmentofMathematicsandSchoolofComputationalScience

FloridaStateUniversity,Tallahassee,FL32306,U.S.A.

November16,2006

Correspondingauthoraddress:Dr.DacianN.Daescu,DepartmentofMathematicsandStatistics,PortlandStateUniversity,P.O.Box751,Portland,OR97207,U.S.A.;E-mail:daescu@pdx.edu

∗

Abstract

Strategiestoachieveorderreductioninfourdimensionalvariationaldataassimilation(4D-Var)searchforanoptimallowrankstatesubspacefortheanalysisupdate.Acom-monfeatureofthereductionmethodsproposedinatmosphericandoceanographicstudiesisthattheoptimalitycriteriatocomputethebasisfunctionsreliesonthemodeldynamicsonly,withoutproperlyaccountingforthespecificdetailsofthedataassimilationsystem(DAS).Inthisstudyageneralframeworkoftheproperorthogonaldecomposition(POD)methodisconsideredandacost-effectiveapproachisproposedtoincorporateDASin-formationintotheorderreductionprocedure.Thesensitivitiesofthecostfunctionalin4D-Vardataassimilationwithrespecttothetimevaryingmodelstateareobtainedfromabackwardintegrationoftheadjointmodel.Thisinformationisfurtherusedtodefineappropriateweightsandtoimplementadual-weightedproperorthogonaldecomposition(DWPOD)methodtoorderreduction.TheuseofaweightedensembledatameanandweightedsnapshotsusingtheadjointDASisanovelelementinreducedorder4D-Vardataassimilation.Numericalresultsarepresentedwithaglobalshallow-watermodelbasedontheLin-Roodflux-formsemi-Lagrangianscheme.Asimplified4D-VarDASisconsideredinthetwin-experimentsframeworkwithinitialconditionsspecifiedfromtheECMWFERA-40datasets.AcomparativeanalysiswiththestandardPODmethodshowsthatthereducedDWPODbasisprovidesanincreasedefficiencyinrepresentingaspecifiedmodelforecastaspectandasatooltoperformreducedorderoptimalcontrol.Thisap-proachrepresentsafirststeptowardthedevelopmentofanorderreductionmethodologythatcombinesinanoptimalfashionthemodeldynamicsandthecharacteristicsofthe4D-VarDAS.

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1.Introduction

Implementationofmoderndataassimilationtechniquesasformulatedinthecontextofestimationtheory(Jazwinski1970,Lorenc1986,Daley1991,Bennett1992,Cohn1997,Kalnay2002)isoftenhamperedbythehighcomputationalcosttoobtaintheanalysisstateandtodynamicallyevolvetheerrorstatistics.Acharacteristicfeatureoftheglobaloceanandatmosphericcirculationmodelsisthelargedimensionalityofthediscretestatevector,typicallyintherange106−107.Thisdimensionislikelytoincreaseinthenearfuturewhenclimatemodelsareenvisagedtorunatahorizontalresolutionashighas1/4degreeinforecastanddataassimilationmode.Toaccommodatetheserequirements,computationallyefficienttechniquestoassimilateaneverincreasingamountofobservationaldataintomodelsmustbedeveloped.

SignificanteffortshavebeendedicatedtoeasethecomputationalburdenofKalmanfilterbasedalgorithmsthroughvarioussimplifyingassumptions.Statereductiontech-niquesandlow-rankapproximationsoftheerrorcovariancematrixaredescribedintheworkof(Dee1990),TodlingandCohn(1994),Caneetal.(1996),Phametal.(1998),HoteitandPham(2003).EnsembleKalmanfilter(EnKF)methodsbuildontheoriginalworkofEvensen(1994)toprovidetheanalysisstateanderrorcovarianceusinganen-sembleofmodelforecasts(Moltenietal.1996,Burgersetal.1998,Anderson2001).AreviewoftheEnKFandlow-rankfilterscanbefoundintheworkofEvensen(2003)andNergeretal.(2005)whoemphasizethatacommonfeatureofthesemethodsisthattheiranalysisstepoperatesinalow-dimensionalsubspaceofthetrueerrorspace.

Infourdimensionalvariationaldataassimilation(4D-Var)theanalysisstateisob-tainedbysolvingalarge-scaleoptimizationproblem(LeDimetandTalagrand1986)withtheinitialconditionsofthediscretemodelascontrolparameters.Theincrementalap-proach(Courtieretal.1994)iscurrentlyusedatnumericalweatherpredictioncenters

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implementing4D-Var(Rabieretal.2000).Computationalsavingsarefurtherachievedbyrunningacoarseresolutiontangentlinearandadjointmodelsintheinnerloopoftheminimization.Implementationissuesandastudyontheconvergenceoftheincremental4D-VarmethodareprovidedbyTr´emolet(2004,2005).

Althoughrunningacoarseresolutionmodelprovidesacertainstatereduction,theissueoffindinganoptimallow-dimensionalstatesubspaceforthe4D-Varminimizationproblemisanopenquestionwherethecurrentstateofresearchisatanincipientstage.Mathematicalfoundationsofapproximationtheoryforlarge-scaledynamicalsystemsandflowcontrolarepresentedbyAntoulas(2005)andGunzburger(2003).Asubstantialamountofworkwasdoneintheclimateresearchcommunitytobuildreducedmodelsoftheatmosphericdynamicswithaminimalnumberofdegreesoffreedom.Theproperorthogonaldecomposition(POD)method(alsoknownasthemethodofempiricalorthog-onalfunctions-EOFs,Karhunen-Lo`evedecomposition)hasbeenwidelyusedinfluiddynamics(Holmes,LumleyandBerkooz1996,Sirovich1987)andatmosphericflowmod-eling(Selten1995,1997,AchatzandOpsteegh2003)toobtainbasisfunctionsforreducedorderdynamics.ShortcomingsofthePOD/EOFsreducedmodelsarediscussedbyAubryetal.(1993)andinpracticeotherchoicesshouldbealsoconsidered.Inparticular,princi-palinteractionpatterns(Hasselmann1988)haveshownthepotentialtoachieveimprovedresultswhencomparedtoEOFs(AchatzandSchmitz1997,Kwasniok2004,CrommelinandMajda2004).Whilethesestudieswereonlyconcernedwiththeconstructionandanalysisofreducedmodelstotheatmosphericflow,thedevelopmentandimplementationofoptimalorder-reductionstrategiesinthecontextof4D-Varatmosphericdataassimi-lationisafarmoredifficulttask.

Foroceanicmodels,initialeffortsonreducedorder4D-VarwereputforwardbyBlayoetal.(1998)andDurbiano(2001).TheuseofEOFstoidentifyalow-rankcontrolspacehasshownpromisingresultsinthestudiesofRobertetal.(2005),HoteitandK¨ohl(2006),

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Caoetal.(2006).Thepotentialuseofthereducedorder4D-Varasapreconditionerto4D-VardataassimilationwasconsideredbyRobertetal.(2006).Acommonfeatureofthereductionmethodsusedinthesestudiesisthattheoptimalitycriteriatocomputethebasisfunctionsreliesonthemodeldynamicsonly,withoutproperlyaccountingforthespecificdetailsofthedataassimilationsystem(DAS).Assuch,theefficiencyofthereducedbasismaybeimpairedbythelackofinformationontheoptimizationproblemathand.

MeyerandMatthies(2003)usedadjointmodelingtoimprovetheefficiencyofthePODapproachtomodelreductionwhentargetingascalaraspectofthemodeldynamics.AmethodtoachievebalancedmodelreductionoflinearsystemsusingPODandpotentialextensionstononlineardynamicsarediscussedbyWillcoxandPeraire(2002).Agoal-oriented,model-constrainedoptimizationframeworktoreductionoflarge-scalemodelsispresentedintheworkofBui-Thanhetal.(2007).

InthisworkweconsideranovelmethodtoincorporateDASinformationintotheorderreductionprocedurebyimplementingadual-weightedproperorthogonaldecomposition(DWPOD)method.TheDWPODmethodsearchestoprovidean”enriched”setofbasisfunctionsthatcombineinformationfrombothmodeldynamicsandDAS.TheuseofaweightedensembledatameanandweightedsnapshotsusingtheadjointDASisanovelelementinreducedorder4D-Vardataassimilation.ThetraditionalPODbasisconsistsonthemodesthatcapturemostofthe”energy”ofthedynamicalsystemwhereastheDWPODbasismayincludelowerenergymodesthataremoresignificanttotherepresen-tationofthe4D-Varcostfunctional.TheDWPODprocedureisshowntobecost-effectivesinceitprovidesasubstantialqualitativeimprovementascomparedtothestandardPODapproachattheadditionalcomputationalexpenseofasingleadjointmodelintegration.Henceforth,thepaperisorganizedasfollows:inSection2the4D-Vardataassimi-lationproblemisbrieflyrevisited.AgeneralPODframeworktoreducedorder4D-Var

4

andthedualweightedPODapproacharedescribedinSection3.NumericalexperimentswithafinitevolumeglobalshallowwatermodelareprovidedinSection4.ConcludingremarksandfurtherresearchdirectionsarepresentedinSection5.

2.The4D-Vardataassimilationproblem

The4D-Vardataassimilationsearchesforanoptimalestimate(analysis)xa0tothem−dimensionalvectorofthediscretemodelinitialconditionsbysolvingalarge-scaleoptimizationproblem

x0∈R

minJ(x0);m

xa0=ArgminJ

(1)

Thecostfunctional

N

11󰀄1T−1

J=(x0−xb)B(x0−xb)+(Hkxk−yk)TR−k(Hkxk−yk)22k=1

(2)

includesthedistancetoaprior(background)estimatetoinitialconditionsxbandthedistanceofthemodelforecastxk=M(x0)toobservationsyk,k=1,2,...Ntimedis-tributedovertheanalysisinterval[t0,tN].ThemodelMisnonlinearandforsimplicityweassumealinearrepresentationoftheobservationaloperatorHkthatmapsthestatespaceintotheobservationspaceattimetk.Statisticalinformationontheerrorsinthebackgroundanddataisusedtodefineappropriateweights:BisthecovariancematrixofthebackgrounderrorsandRkisthecovariancematrixoftheobservationalerrors.AnaccurateestimationofthematrixBisdifficulttoprovideand,givenitshugedimension-ality,simplifyingapproximationsarerequiredforthepracticalimplementation(Lorencetal.2000).Informationontheerrorsstatisticsmaybeobtainedusingdifferencesbe-tweenforecastswithdifferentinitializationtimesasinthe”NMC”method(ParrishandDerber1992)orensemblemethodsbasedonaperturbedforecast-analysissystem.Re-centadvancesinmodelingflow-dependentbackgrounderrorvariancesarediscussedby

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KucukkaracaandFisher(2006).

3.AgeneralPODframeworktoreduced-order4D-Vardataassimilation

Thespecificationofthebasisfunctionsliesatthecoreofthereduced-order4D-Varprocedure.Theproperorthogonaldecomposition(POD)methodprovidesanoptimallow-rankrepresentationofanensembledataset{x(1),x(2),...,x(n)},x(i)∈Rmthatmaybecollectedfromobservationaldataand/orthestateevolutionatvariousinstantsintimet1,t2,...,tn(methodofsnapshots,Sirovich1987).TheuseofdataweightingasatooltoimprovetheperformanceofthePODmethodwaspreviouslyconsideredinmodelreductionfordynamicalsystems.GrahamandKevrekidis(1996)proposedanensembleaveragebasedonthearc-lengthinthephasespaceandemphasizedthatthechoiceoftheensembleaverage(weights)forthePODmethodcanhaveasignificantimpactontheselectionofthedominantmodes.AweightedPOD(w-POD)approachisdiscussedbyChristensenetal.(2000)whoconsiderincludingmultiplecopiesofan”important”snapshotintheensembledataset.KunischandVolkwein(2002)usethetimedistributionofthesnapshots∆ti=ti+1−titospecifyweightsandprovideadetailedtheoreticalframeworkanderrorestimateswithapplicationstoNavier-Stokesequations.Wedefinetheweightedensembleaverageofthedataas

n󰀄i=1

x=

ωix(i)

󰀃n

i=1

(3)

ωi=1andareusedto

wherethesnapshotweightsωiaresuchthat0<ωi<1,

assignadegreeofimportancetoeachmemberoftheensemble.Timeweightingisusuallyconsideredandinthestandardapproachωi=1/n.Amodifiedm×ndimensionalmatrixisobtainedbysubtractingthemeanfromeachsnapshot

X=x

󰀅

(1)

−x,x

(2)

−x,...x

(n)

−x󰀆

(4)

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andtheweightedcovariancematrixC∈Rm×misdefined

C=XWXT

(5)

whereW=diag{ω1,...,ωn}isthediagonalmatrixofweights.Sincethemetriconthestatespaceisoftenrelatedtothephysicalpropertiesofthesystem,weconsiderageneral

Tm×m

norm󰀒x󰀒2isasymmetricpositivedefinitematrix.A=󰀏x,x󰀐A=xAx,whereA∈R

ForthestandardEuclideannormAistheidentitymatrixandforthetotalenergymetricAisadiagonalmatrix.

ThePODbasisoforderk≤nprovidesanoptimalrepresentationoftheensembledatainak-dimensionalstatesubspacebyminimizingtheaveragedprojectionerror

n󰀄

min

{ψ1,ψ2,...,ψk}i=1

󰀇󰀇2

󰀇󰀇(i)(i)

ωj󰀇(x−x)−Pψ,k(x−x)󰀇

A

(6)

subjecttotheA-orthonormalityconstraint󰀏ψi,ψj󰀐A=δi,j,1≤i,j≤k,wherePψ,kistheprojectionoperatorontothek-dimensionalspaceSpan{ψ1,ψ2,...,ψk}

Pψ,k(x)=

k󰀄i=1

󰀏x,ψi󰀐Aψi

ThePODmodesarem-dimensionaleigenvectorstotheeigenvalueproblem

2

CAψi=σiψi

(7)

andsinceinpracticethenumberofsnapshotsismuchsmallerthanthestatedimension,n󰀈m,anefficientwaytocomputethereducedbasisistosolvethen-dimensionaleigenvalueproblem

2

W2XTAXW2µi=σiµi

11

(8)

toobtaintheorthonormal(Euclidean)eigenvectorsµi∈RnthentocomputethePODmodesas

11

ψi=XW2µi

σi

(9)

7

From(8)and(9)thecloserelationshipwiththesingularvaluedecomposition(GolubandVanLoan1996)

A2XW2=UΣVT

1

1

(10)

isestablished:σ1≥σ2≥...σn≥0arethesingularvalues,µitherightsingularvectorsandA2ψitheleftsingularvectors.Thefractionoftotalinformation(”energy”)capturedbythedominantkmodesisI(k)=(

󰀃k

2

i=1σi)/(

1

󰀃n

i=1

2σi)andinpractice,givenatolerance

0<γ≤1inthevicinityoftheunity,kisselectedsuchthatI(k)≥γ.

a.Thereducedorder4D-Var

Thek-dimensionalreducedordercontrolproblemisobtainedbyprojectingx0−xontothePODspace

k󰀄i=1

Pψ,k(x0−x)=Ψη=

ηiψi

(11)

wherethematrixΨ=[ψ1,...,ψk]∈Rm×khasthePODbasisvectorsascolumnsandη=(η1,...ηk)T∈Rkisthecoordinatesvectorinthereducedspace

ηi=ψTiA(x0−x),

η=ΨTA(x0−x)

(12)

Thelarge-scale4D-Varoptimization(1)isthusreplacedbythereducedorder4D-Varproblemoffindingtheoptimalcoefficientsη

ˆ(η):=J(x+Ψη);J

ˆ(η)minkJη∈R

(13)

Ifηadenotesthesolutionto(13),anapproximationtotheanalysis(1)isobtainedas

a

xa0≈x+Ψη

(14)

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Itshouldbenoticedthatinthereduced-order4D-Varasformulatedin(13)onlytheinitialconditionsareprojectedintothePODstatesubspaceandthecostfunctionaliscomputedusingthefullmodeldynamics.Thegradientofthecost(13)isexpressedas

ˆ(η)=ΨT(∇x0J)|∇ηJx0=x+Ψη

(15)

anditsevaluationrequiresintegrationofthefulladjointmodel.Secondorderderivativesinthereducedspacemaybecomputedifafullsecondorderadjointmodelisavailable(DaescuandNavon2006).Consequently,computationalsavingsmaybeachievedonlybyadrasticreductioninthenumberofiterationsduetothelowdimensionoftheoptimizationproblem(13).

OncethePODbasisisselected,areducedmodelapproachtoorderreductionmaybealsoconsideredbyprojectingthefullmodeldynamicsintothePODspace.Theprojectedˆ(t)=x+Ψη(t)evolvesintimeaccordingtothedifferentialequationssystemstatex

ˆ󰀄(t)=ΨΨTAM(ˆxx,t)ˆ(0)=ΨΨTA(x(0)−x)+xx

(16)(17)

andthecoefficientsη(t)maybeobtainedbyintegratingthereducedmodelequations

η󰀄(t)=ΨTAM(x+Ψη(t),t)η(0)=ΨTA(x(0)−x)

(18)(19)

SuchapproachmayresultinsignificantcomputationalsavingswhenGalerkintypenumericalschemesareimplemented(Ravindran2002,KunischandVolkwein1999)oranimplicittimeintegrationschemetofinitedifferences/finitevolumesemi-discretizationisconsidered(vanDorenetal.2006).However,forfinitedifferenceandfinitevolumenu-mericalmethodswithexplicittimeschemes,integrationofthereducedmodelequations(18-19)willrequireingeneralanincreasedCPUtimeduetothecostofrepeatedpro-jectionoperations(unlessanalyticsimplificationscanbemade).Anadditionalissuein

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thereducedmodelapproachisthattheprojection(16-17)introducesamodelerrorthatisdifficulttoquantify(RathinamandPetzold2003)andthustoaccountforinthereduced4D-Vardataassimilation.

b.Thedual-weightedPODbasis

ThespecificationoftheweightsωitothesnapshotsmayhaveasignificantimpactonwhichmodesareselectedasdominantandthusinsertedintothePODbasis.Thedual-weightedapproachweproposemakesuseofthetimevaryingsensitivitiesofthe4D-Varcostfunctionalwithrespecttoperturbationsinthestateatthetimeinstantsti,i=1,...,nwhenthesnapshotsaretaken.

Theuseoftheadjointmodelingtoidentify”target”regionswhereobservationaldataisofmostbenefittoaforecastaspectJ(x)iswellestablishedinthecontextoftargetedobservationsforhighimpactweatherevents(Langlandetal.1999).Byanalogy,thedual-weightedapproachmaybetaughtasatargetingintimeprocedure(ratherthantargetingthestatespaceatagiventime)thatassignsweightstotimedistributedsnapshotsdata.Forsimplicityofthepresentation,weassumeacostfunctionalJ(x(t))definedintermsofthestateattimet.Theimpactofsmallerrors/perturbationsδxiinthestatevectoratasnapshottimeti≤tonJmaybeestimatedusingthetangentlinearmodelM(ti,t)anditsadjointmodelM∗(t,ti)

δJ≈󰀏J󰀄(x(t)),δx(t)󰀐=󰀏J󰀄(x(t)),M(ti,t)δx(ti)󰀐=󰀏M∗(t,ti)J󰀄(x(t)),δx(ti)󰀐

=󰀏A−1M∗(t,ti)J󰀄(x(t)),δx(ti)󰀐A

Thedual-weightsωitothesnapshotsareobtainedasnormalizedvalues

αi=󰀒A−1M∗(t,ti)J󰀄(x(t))󰀒A,

αi

ωi=󰀃n,

i=1,2,...n

(21)(20)

j=1αj

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andprovideameasureoftherelativeimpactofthestateerrors󰀒δx(ti)󰀒Aonthecostfunctional.Alargevalueofωiindicatethatstateerrorsattiplayanimportantroleintherepresentationofthecostfunctionalandanincreasedweightisassignedtothefittosnapshotdatax(i)inthereduced-basisoptimizationproblem(6).Theweights(21)aredeterminedbythe4D-Vardataassimilationcostfunctional(2)suchthatinformationfromtheDASisincorporateddirectlyintotheoptimalitycriteriathatidentifiesthereduced-spacebasisfunctions.TheDWPODbasisisthusadjustedtothe4D-Varoptimizationproblemathand.

Fromtheimplementationpointofview,theevaluationofalldual-weightsrequiresonlyoneadjointmodelintegrationtoobtainthebackwardtrajectoriesoftheadjointvariables(influencefunctions)λ(τ)=M∗(t,τ)J󰀄(x(t)),t0≤τ≤t.Sinceinthe4D-Vardataassimilationcontexttheadjointmodelisalreadyavailable,littleadditionalsoftwaredevelopmentisrequiredandtheincreasedcomputationalcostofimplementingDWPODoverthestandardPODmethodismodest.Inthenumericalexperimentssectionwecomparetheperformanceofthesetwomethodsfirstastoolstoprovideareducedorderrepresentationofaforecastoutput,thenastoolstoperformreducedorder4D-Vardataassimilation.

4.NumericalExperiments

Thenumericalexperimentsareperformedwithatwo-dimensionalglobalshallow-water(SW)modelusingtheexplicitflux-formsemi-Lagrangian(FFSL)schemeofLinandRood(1997).ThefinitevolumeFFSLschemeisofparticularimportancesinceitprovidesthehorizontaldiscretizationtothefinite-volumedynamicalcoreofNCARCommunityAtmosphereModel(CAM)andNASAGEOS-5dataassimilationandforecastingsystem(Lin2004).TheadjointmodeltotheSW-FFSLschemeusedinthisstudywasdeveloped

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intheworkofAkellaandNavon(2006)withtheaidofTAMCsoftware(GieringandKaminski1998).

InputdataobtainedfromtheECMWFERA-40atmosphericdatasetsisusedtospecifytheSWmodelstatevariablesattheinitialtime:geopotentialheighthandthezonalandmeridionalwindvelocities(u,v).Weconsidera2.5◦×2.5◦resolution(144×72gridcells)suchthatthedimensionofthediscretestatevectorx=(h,u,v)is∼3×104.Thetimeintegrationisperformedwithaconstanttimestep∆t=450susingastaggered’CD-grid’systemwiththeprognosticvariablesupdatedontheD-grid(LinandRood1997).PointvaluesofthemodeloutputareobtainedbyconvertingthewindsfromtheD-gridtoanunstaggeredA-grid.

Asareferenceinitialstatexrefweconsiderthe500mbECMWFERA-40datavalid0for06hUTC15March2002.Theconfigurationofthegeopotentialheightattheinitialtimeanda24hSWmodelforecastisdisplayedinFig.1.Onthediscretestatespaceweconsideratotalenergynorm

1g22

󰀒x󰀒2=󰀒u󰀒+󰀒v󰀒+󰀒h󰀒2A

2h󰀁

󰀂

(22)

where󰀒·󰀒denotestheEuclideannorm,gisthegravitationalconstantandhisthemeanheightofthereferencedataattheinitialtime,suchthatAisadiagonalmatrixwithblockconstantentriesg/2h,1/2,1/2.

Togeneratethesetofsnapshotsweintroducedsmallrandomperturbationsδx0inthereferenceinitialconditionsandperformedafullmodelintegrationstartingwithxref0+δx0.Thestateevolutionx(ti)=Mt0,ti(xref+δx0)wasstoredateachtimestepandusedto0definetheensembledatasetx(i)=x(ti),i=1,2,...n.ThisdatasetisthenusedbythePODandDWPODmethodstoidentifyanappropriatereducedorderstatesubspace.InthestandardPODapproach,alltheweightsaresetωi=1/nandthePODbasisoforderk12

theproblemathand.

a.Reduced-orderrepresentationofaforecastaspect

InthefirstsetofexperimentsweconsiderthePODandDWPODmethodsastoolstoprovideareduced-orderrepresentationofascalaraspectofthemodelforecast.Thetargetfunctionalistakenasameasureofthetimeintegratedenergyofthesystemfora24hforecastinitiatedfromxref0,J(x)=

󰀃n

i=1

󰀒xi󰀒2A.Forthe24hperiod,theensemble

datasetincludes193snapshots.Thevariance(”energy”)I(k)capturedbytheleadingPODandDWPODmodesfromtheensembledataasafunctionofthedimensionkofthereducedspaceisdisplayedinFig.2,andselectednumericalvaluesareprovidedinTable1.ItisnoticedthatforthesamedimensionkofthereducedspaceasimilaramountofvarianceiscapturedbythePODandDWPODfromthedatasetandweighteddataset,respectively.Ineachcasethedominantmodeprovides∼78%oftheinformation,firsttenmodes∼99%,anduptoasmallfraction,mostoftheinformationiscontainedintheleading25modes.However,thek−dimensionalbasesΨpodandΨdwpodidentifiedbythePODandrespectivelyDWPODaredistinctandinparticular,highermodesofsamerankmaydiffersignificantlyfromthePODbasistotheDWPODbasis.InFig.3isoplethsofthePODandDWPODmodesaredisplayedusingtheenergynormtoprovidepointvalues.AcloseresemblanceisnoticedbetweenthedominantPODandDWPODmodes,whereashigherPODandDWPODmodesofsamerankhaveacompletelydifferentstructure.

InthePODapproachthereducedorderrepresentationoftheinitialstateis

ref

ˆ0=x+ΨpodΨTxpodA(x0−x)13

withx=

1n󰀃n

i=1

x(i)andintheDWPODapproachtheinitialstateisrepresentedas

refωˆ0=xω+ΨdwpodΨTxdwpodA(x0−x)

withtheweightedmeanxωcomputedaccordingto(3),(21).Sinceinpracticethedimen-sionkofthereducedspaceisdeterminedbyspecifyingathresholdvalue0<γ<1suchthatI(k)≥γ,itisofinteresttoanalyzetheerrorinthereduced-orderrepresentationofthetargetfunctional|J(x)−J(ˆx)|asthedimensionofthereducedspacevaries.ThenumericalresultsusingPODandDWPODbasesofdimensionk=5,10,15,20,25aredisplayedinFig.4anditisnoticedthattheDWPODbasisprovidedasignificantlyimprovedaccuracyascomparedtothePODbasis.Forexample,projectionoftheinitialconditionsinthe10-dimensionalDWPODspaceprovidedqualitativeresultssimilartothe15-dimensionalPODspace,whereastherepresentationinthe15-dimensionalDWPODspaceprovidedoneorderofmagnitudegaininaccuracyoverthe15-dimensionalPODspace.

ThereducedDWPODspaceprovidednotonlyanimprovedrepresentationofJ(ˆx)butalsoamoreaccuratestateforecasttrajectory,asmeasuredinthetotalenergynorm.ˆi󰀒AisdisplayedinFig.4foreachtimestepduringthemodelTheforecasterror󰀒xref−xiintegration.ItisnoticedtheincreasedefficiencyoftheDWPODbasisthatprovidedqual-itativeresultssimilartothePODbasiswhilerequiringfewerbasisvectors.Inparticular,theerrorsinthe5-dimensionalDWPODspaceareclosetotheerrorsinthe10-dimensionalPODspace,the10-dimensionalDWPODspaceprovidedforecasterrorsnearlyidenticaltotheerrorsinthe15-dimensionalPODspace,andthe15-dimensionalDWPODprovidedoneorderofmagnitudegaininforecastaccuracyoverthe15-dimensionalPODspace.Asthedimensionofthereducedspaceincreases,eachbasiscapturespracticallyalloftheinformationfromtheensembledata.LittleimprovementinforecastaccuracymaybeachievedbyincreasingtheDWPODdimensionfrom20to25andthestateforecasterrorusinga25-dimensionalDWPODanda25-dimensionalPODbasisprovidedoverlapping

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graphs(visuallyindistinguishable)inFig.4.

WhileforbothPODandDWPODmethodsthestatereductionfrom∼3×104to∼20isremarkable,inpracticalapplicationsitisimportanttoobtainaccuratereducedorderrepresentationsusingasmall(thesmallest)numberofbasisvectors.TheenhancedefficiencyoftheDWPODoverPODinprovidingaccurateresultsforsmalldimensionalbasesisthusadesirablepropertythatmaybecomeincreasinglysignificantasthedimen-sionofthefullmodelstateincreases.Thedimensionofthereducedspaceisalsocrucialintheefficiencyofthereducedorder4D-Vardataassimilationthataimstoperformaminimalnumberofiterationstoachieveacertainaccuracygainintheanalysis.

b.Dataassimilationexperiments

Toanalyzethepotentialcomputationalsavingsofthereducedorderprocedure,4D-Vardataassimilationexperimentsaresetupinatwinexperimentsframework.Asabackgroundestimatexbtotheinitialconditionsweconsider500mbECMWFERA-40datavalidfor00hUTC15March2002,sixhourspriortothereferencestatexref0.Theerrorsinthebackgroundterm,averagedoverthelongitudinalcoordinate,aredisplayedinFig.5.Adataassimilationtimeinterval[t0,t0+24h]isconsideredwithfourdatasetsat6h,12h,18h,and24hprovidedbyamodelintegrationinitiatedfromxref0.Twodataassimilationexperimentsaresetup:thefirstexperiment,hereafterreferredtoasDAS-I,isamodelinversionproblemwheredataisprovidedforalldiscretestatecomponentsandnobackgroundtermisincludedinthecostfunctional(2);inthesecondexperiment,hereafterreferredtoasDAS-II,thebackgroundtermisincludedinthecostanddataisprovidedatevery4thgridpointonthelongitudinalandlatitudinaldirections(∼6%ofthestateis”observed”everysixhours).ThedistancetothebackgroundandobservationsismeasuredintheA-normthatcorrespondstodiagonalmatricesBandR.Toemphasize

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thefittodata,aweightfactorof0.01isassignedtothedistancetobackgroundinDAS-II.Dataassimilationexperimentsperformedinthefullmodelspaceresultedinaslowconvergenceforthelargescaleoptimizationproblem(1).Theminimizationprocessusingahigh-performancelimitedmemoryquasi-NewtonL-BFGSalgorithm(LiuandNocedal19)isdisplayedinFig.6anditisnoticedthatalargenumberofiterationsisrequiredtoapproachtheoptimalpoint.AslowerconvergencerateisobservedforDAS-IversusDAS-IIduetotheincreasednumberofdataconstraintsandtheabsenceoftheregular-izationprovidedbythebackgroundterm.

c.Reduced-order4D-Vardataassimilation

Twinreduced-order4D-VardataassimilationexperimentswereimplementedusingthePODandDWPODbasesrespectively.ItshouldbenoticedthatwhilethePODbasisvectorsremainunchangedforbothDAS-IandDAS-IIexperiments,inthedual-weightedapproachthereducedbasisisadjustedtotheoptimizationproblemathand.AsshowninFig.7,thedualweightstothesnapshotdataaredistinctfromDAS-ItoDAS-IIand,asanillustrativeexample,inFig.8isoplethsoftheDWPODmodeofrank10revealadifferentconfigurationinDAS-IthaninDAS-II.

Thelowdimensionalityofthereducedspacedallowedtheimplementationofafullquasi-NewtonBFGSalgorithmtosolvetheoptimizationproblem(13).TheminimizationprocessisdisplayedinFig.9anditisnoticedthatonlyfewiterationswererequiredtoreachtheoptimalpointforeachoftheDAS-IandDAS-IIexperiments.Forexample,inDAS-IIexperiments3to4iterationsarepracticallysufficienttoreachaclosevicinityoftheoptimalpointandthecomputationalsavingsofthereduced-order4D-Vararethussignificant.Tofacilitatethequalitativeanalysis,thetotalenergyerrorsintheretrievedinitialconditions,averagedoverthelongitudinaldirection,aredisplayedinFig.10and

16

Fig.11forDAS-Iandrespectively,DAS-II.BycomparisontoFig.5,thereduced4D-Vardataassimilationisabletoprovideanalysiserrorsthatareloweredbyanorderofmagni-tudeascomparedtotheerrorsinthebackgroundestimate.Forthe5-and10-dimensionalspaces,theanalysiserrorscorrespondingtotheDWPODspacehavemuchlowervaluesascomparedtotheanalysiserrorsforthePODspaceshowingthatthedualweightedapproachtoorderreductionisofsignificantbenefit.Inparticular,forthe10-dimensionalspaces,intheDAS-IexperimentsonenoticesanerrorreductionbyasmuchasafactorofthreeintheDWPODspaceascomparedtothePODspace,whereasintheDAS-IIexperiments,theanalysiserrorisreducedbyasmuchasafactoroftwointheDWPODspaceascomparedtothePODspace.Increasingthedimensionofthereducedspacefrom10to15provestobeoflittlebenefittotheanalysisthusindicatingthatfurtherimprovementsareconstrainedbythelimitedinformationprovidedbythesnapshotdata.

5.Conclusionsandfurtherresearch

Thecomputationalburdenofthelarge-scale4D-Varoptimizationproblemmaybesignificantlyreducedbyperformingtheoptimizationinalowordercontrolspace.Anoptimalorderreductionapproachto4D-Vardataassimilationmustcaptureaccuratelythepropertiesofthefulldynamicalmodelthataremostrelevanttoaspecificdataassim-ilationsystem.Todate,studiesonreducedorder4D-Varhaveconsideredloworderstatesubspacesbasedonthepropertiesoftheflowonly,withoutproperlytakingintoaccountthecharacteristicsoftheDAS.Inthisworkanadjoint-modelapproachisproposedtodi-rectlyincorporateinformationfromtheDASintotheoptimalitycriteriathatdefinesthereducedspacebasis.ThedualweightedPODmethodisnovelinreducedorder4D-VardataassimilationandreliesonaweightedensembledatameanandweightedsnapshotswithweightsdeterminedbytheadjointDAS.Thenumericalexperimentspresentedwith

17

afinitevolumeglobalshallowwatermodelindicatethattheDWPODapproachmaysig-nificantlyimprovetheefficiencyofthereducedbasisascomparedtothestandardPODmethod.TheDWPODspacewasfoundtoincreasetheaccuracyintherepresentationofaforecastaspectbyasmuchasanorderofmagnitudeversusthePODspacerepresenta-tion.In4D-Vardataassimilationtwinexperiments,optimizationintheDWPODspaceprovidedareductionintheanalysiserrorsbyasmuchasafactoroftwowhencomparedtothePOD-basedoptimization.Thedual-weightedapproachisthuscost-effectivesincetheadditionalcomputationalrequirementstoidentifytheDWPODbasisconsistofasingleadjointmodelintegrationtoevaluatethedualweightstothesnapshotdata.

Thisworkrepresentsafirststeptowardthedevelopmentofanorderreductionmethod-ologythatcombinesinanoptimalfashionthemodeldynamicsandthecharacteristicsofthe4D-VarDAS.Themathematicalformulationofthedual-weightedPODapproachtomodelreductionissoundhowever,takingintoaccountthesimplicityoftheshallowwatermodelusedinthisstudy,theenhancedefficiencyoftheDWPODmethodremainstobevalidatedfornumericalweatherpredictionandgeneralcirculationmodelsinanoperationaldataassimilationenvironment.

Strategiestoimplementanadaptiveupdateofthereducedbasisfunctionsasthemin-imizationalgorithmadvancestowardtheoptimalpointareatanincipientstageandthisisanareawherefutureresearchismuchneeded.EvaluationoftheHessianmatrixofthe4D-Varcostfunctionalinthereducedspaceisfeasibleusingasecondorderadjointmodel(DaescuandNavon2006)andmaybeusedtoprovidestatisticalinformationontheanalysiserrors.Thereducedorder4D-Varapproachishighlydependentonthequal-ityofthesnapshotdataandtheissueofgeneratinga”good”setofsnapshotsiscrucialforthereducedorderproceduretobeeffective.Thetwinexperimentssetupusedinthisstudyfacilitatedtheselectionofthesnapshotdatainaclosevicinityofthereferencestatetrajectory.Forpracticalapplications,anensembleofmodelforecastsmaybeusedtogen-

18

eratesnapshotstakenfrommultiplestatecalculationswithperturbationsintheinitialconditionsthatcapturethemaindirectionsofvariabilityofthemodelsuchasthebredvectorsandsingularvectorsofthetangentlinearmodel(Kalnay2002).Inthiscontextthereducedorderprocedurewillresultinahybridapproachthatcombinesinanoptimalfashionfeaturesoftheensembleandvariationalmethodsindataassimilation.

Acknowledgments.ThisresearchwassupportedbyNASAModeling,AnalysisandPredictionProgramunderawardNNG06GC67G.

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Figure1:Isoplethsofthegeopotentialheight(m)forthereferencerun:topfigure-configurationattheinitialtimespecifiedfromECMWFERA-40datasets;bottomfigure-the24hforecastoftheshallowwatermodel.

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1 FRACTION OF THE VARIANCE CAPTURED0.95DWPODPOD0.90.850.80.75 051015REDUCED BASIS DIMENSION2025Figure2:ThefractionofthevariancecapturedbythePODandDWPODmodesfromthesnapshotdataasafunctionofthedimensionofthereducedspace.

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FIRST POD MODE500−50−10005th POD MODE500−50−100010th POD MODE500−50−1000LONGITUDE100500−50100500−50100500−50FIRST DWPOD MODELATITUDE−10005th DWPOD MODE100LATITUDE−100010010th DWPOD MODELATITUDE−1000LONGITUDE100Figure3:IsoplethsofthePODandDWPODmodesofrank1,5,and10.Atotalenergynormisusedtoprovidepointvalues.

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9DWPOD8.5POD error in functional J representation ( log 10 )87.576.565.55 51015reduced space dimension20252.5DWPOD2POD error in state forecast: log10 (energy norm)1.510.50−0.5−1 020406080100time step120140160180200Figure4:Comparativeresultsforthereduced-orderPODandDWPODforecastsasthedimensionofthereducedspacevaries,k=5,10,15,20,25.Topfigure:error(log10)inthereduced-orderrepresentationofthetimeintegratedtotalenergyofthesystem.Bottom

ˆi󰀒Aofthereduced-orderrepresentationofthefigure:totalenergyerror(log10)󰀒xref−xi

forecastateachtimestepintheinterval0-24h.Theerrorsdecreaseasthedimensionofthereducedspaceincreases.

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Figure5:Zonalaveragederrorsinthebackgroundestimatetotheinitialconditions.

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FULL SPACE OPTIMIZATION: DAS−I4.55.5FULL SPACE OPTIMIZATION: DAS−IILOG 10 (COST)54.5LOG 10 (COST)010203040ITERATION NUMBER5043.543.5305101520ITERATION NUMBER25Figure6:TheiterativeminimizationprocessinthefullstatespaceforDAS-I(left)andDAS-II(right).

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111098765432x 10−3 DAS−IDAS−IIsnapshot weight1 020406080100time step120140160180200Figure7:ThedualweightstothesnapshotdatadeterminedbytheadjointmodelinDAS-IandinDAS-II

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DAS−I: DWPOD BASIS VECTOR OF RANK 1070656055latitude5045403530−60−55−50−45−40longitude−35−30−25−20DAS−II: DWPOD BASIS VECTOR OF RANK 1070656055latitude5045403530−60−55−50−45−40longitude−35−30−25−20Figure8:Isoplethsofthe10thmodeintheDWPODbasisforDAS-I(topfigure)andDAS-II(bottomfigure).AdistinctconfigurationitisnoticedsincetheDWPODbasisisadjustedtotheoptimizationproblemathand.3354.543.532.521.510.50 0DWPOD k = 5DWPOD k = 10DWPOD k = 15POD k = 5POD k = 10POD k = 15 log 10 ( cost )510iteration number1520253.8DWPOD k = 5DWPOD k = 10DWPOD k = 15POD k = 5POD k = 10POD k = 15 3.73.63.5log 10 ( cost )3.43.33.23.13 1234iteration number567Figure9:TheiterativeminimizationprocessinthereducedspaceforthePODandDWPODspacesofdimension5,10,and15.Topfigure-optimizationwithoutbackgroundtermanddenseobservations,correspondingtoDAS-I;bottomfigure-optimizationwithbackgroundtermandsparseobservations,correspondingtoDAS-II

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Figure10:Zonalaveragederrorsintheanalysisprovidedbythereducedorder4D-Vardataassimilation.ResultsfortheDAS-IexperimentswithPODandDWPODspacesofdimension5,10,and15.

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Figure11:Zonalaveragederrorsintheanalysisprovidedbythereducedorder4D-Vardataassimilation.ResultsfortheDAS-IIexperimentswithPODandDWPODspacesofdimension5,10,and15.

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Table1:FractionofthevariancecapturedbytheleadingPODandDWPODvectors

BasisDimensionPODDWPOD

10.78270.77

50.97360.9612

100.99240.9918

150.99870.9990

200.99980.9999

250.99990.9999

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