TheEnsembleKalmanfilter

PartI:Theory

AlisonFowlerandSanitaVetra-Carvalho

Data-assimilationtrainingcourse.5th-8thMarch2019,UniversityofReading1

Recapofproblemwewishtosolve

•  Givenpriorknowledgeofthestateofasystemandasetofobservations,wewishtoestimatethestateofthesystematagiventime.Thisisknownastheposteriororanalysis.

•  Bayes’theoremstates

Figure:1DexampleofBayes’theorem.

Moderaterain HeavyrainNorain

Forexamplethiscouldberainfallamountinagivengridbox.A-prioriweareunsureiftherewillbemoderateorheavyrainfall.Theobservationonlygivesprobabilitytotherainfallbeingmoderate.ApplyingBayes’theoremwecannowbecertainthattherainfallwasmoderateandtheuncertaintyisreducedcomparedtoboththeobservationsandoura-prioriestimate.

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Recapof4DVar

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xaxb

*Observationsbackgrounduncertainty,characterisedbyBobservationuncertainty,characterisedbyR

timeAssimilationwindow

•  4DVaraimstofindthemostlikelystateattimet0,givenaninitialestimate,xb,andawindowofobservations.

t0

M(xa)analysisuncertainty.

forecast

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Recapof4DVar

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• J(thecostfunction)isderivedassumingGaussianerrordistributionsandaperfectmodel.

Recapof4DVar:whydoanydifferent?

Advantages•  MetOfficeandECMWFbothusemethodsbasedon4DVarfortheiratmospheric

assimilation.•  Gaussianandnear-linearassumptionmakesthisanefficientalgorithm.•  Minimisationofthecostfunctionisawellposedproblem(theB-matrixisdesignedto

befullrank).•  Analysisisconsistentwiththemodel(balanced).•  Lotsoftheoryandtechniquestomodifythebasicalgorithmtomakeitapragmatic

methodforvariousapplications,e.g.incremental4DVar,preconditioning,controlvariabletransforms,weakconstraint4DVar...

Disadvantages•  Gaussianassumptionisnotalwaysvalid.•  ReliesonthevalidityofTLandperfectmodelassumption.Thistendstorestrictthe

lengthoftheassimilationwindow.•  DevelopmentofTLmodel,M,andadjoint,MT,isverytimeconsuminganddifficultto

updateasthenon-linearmodelisdeveloped.•  B-matrixispredominatelystatic.

Thismotivatesadifferentapproach…

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SequentialDA

•  Insteadofassimilatingallobservationsatonetime,assimilatethemsequentiallyintime.

•  Thiscanbeshowntobeequivalenttothevariationalproblem,assumingalinearmodelandallerrorcovariancesaretreatedconsistently.

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time

xb

Assimilationwindow

*ObservationsM(xb)M(xa)forecasta-prioriuncertainty,characterisedbyPfobservationuncertainty,characterisedbyRanalysisuncertainty.

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SequentialDA-TheKalmanequations

•  Ateachobservationtimektheprioruncertainty,p(xk),isupdatedtofindtheposterior,p(xk|yk).

•  RecallBayes’theorem:

•  LetusassumethedistributionsareGaussian, Notethechangeinnotation!Seefinalslideforalistofallnotationused

•  ThisimpliesthattheposteriorisalsoGaussian•  Themeanof isisgivenby

where•  Andthecovarianceisgivenby

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•  Needtobeabletoevolvetheuncertaintyinthestatefromoneobservationtimetothenext.

•  Thatistogofromp(xk-1|yk-1)top(xk),or

•  TheExtendedKalmanfilter(EKF,GrewalandAndrews(2008))doesthisusingthenon-linearmodelanditsTLandadjoint.

TheKalmanfilter

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Predictionstep•  Evolvemeanstatefromtimen-1totimeofobservation,n.

where•  Evolvecovariancetotimeofobservationallowingformodelerror,

Observationupdatestep•  Updatemeanstategivenobservation

where•  Updateerrorcovariancegivenobservation

TheKalmanfilteralgorithm

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MotivationfortheensembleKalmanfilter(EnKF)

•  TheEKFstillneedstheTLandadjointmodeltopropagatethecovariancematrix.

•  Duetothesizeofthismatrixformostenvironmentalapplications,theEKFisnotfeasibleinpractice.

•  Analternativeapproachtoexplicitlyevolvingthefullcovariancematrixistoinsteadestimateitusingasampleofevolvedstates(knownastheensemble).

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ExtendedKalmanfilterapproachExplicitlyevolvethemeanandcovariancesforwardintimeusingM,MandMT.

EnsembleKalmanfilterapproachSamplefromtheinitialtimeuncertainty,evolveeachstateforwardintimeusingM,thenestimatethemeanandcovariancefromtheevolvedsample.

Time1 Time2

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EnKFalgorithms•  TheEnKF(Envensen1994)mergesKFtheorywithMonteCarloestimation

methods.•  TherearemanymanydifferentflavoursofEnKF.•  EnKFalgorithmscanbegeneralisedintotwomaincategories:

–  Stochasticalgorithms(e.g.theperturbedobservationKalmanfilter)–  Deterministicalgorithms(e.g.theensembletransformKalmanfilter)

•  AllEnKFmethodscanberepresentedbythesamebasicschematic:

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xb

Assimilationwindow12

Predictionstep–  Evolveeachensemblememberforwardusingthenon-linearmodelwith

addednoise.

–  Reconstructtheensemblemean

–  Anditscovariance

–  Notethereisnoneedtoeverexplicitlycompute ,justtheperturbationmatrix ,,whichisgenerallyofasmallerdimension.

TheperturbedobservationKalmanFilter

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Filteringstep–  Updatetheensembleusingperturbedobservations

Where and

TheperturbedobservationKalmanFilter

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Derivedfromtheensemble

TheperturbedobservationKalmanFilter

•  TheadvantagesoftheperturbedobservationKFisthatitisverysimpletoimplementandunderstandfortoymodels.However…–  Itisnecessarytoperturbtheobservationsinorderforthevarianceof

theensembleaftertheupdatesteptocorrectlyrepresenttheuncertaintyintheanalysis.

–  Thisintroducesadditionalsamplingnoise.

•  Thismotivatesthedevelopmentofsquare-rootordeterministicformsoftheEnKFthatdonotneedtoperturbtheobservations.

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•  TheideaofESRFistocreateanupdatedensemblewithcovarianceconsistentwith

•  Recallthattheensemblecovariancematrixisgivenby

•  Wecanwriteasimilarexpressionfortheanalysiserrorcovariancematrixintermsoftheanalysisperturbations

•  Insteadofupdatingeachensemblememberseparately,asintheperturbedobservationKF,theESRFgeneratesthenewensemblesimultaneouslybyupdatingXfinsteadofx(i),f

EnsembleSquareRootFilter

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Predictionstep–  ThisisthesameasfortheperturbedobservationensembleKF.–  Computetheevolvedensemble,,itsmean,,andits

perturbationmatrix,.–  Computetheforecast-observationensemble

•  Transformtheforecastensembletoobservationspace

•  fromthiscancomputethemean

•  andperturbationmatrix

EnsembleSquareRootFilter

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Filteringstep•  Updateensemblemean

–  where

•  Updateperturbationmatrix

–  NeedtodefinethematrixT.

EnsembleSquareRootFilter

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EnsembleSquareRootFilterTheTmatrix•  ThematrixTischosensuchthat

•  ThisdoesnotuniquelydefineTwhichiswhytherearesomanydifferentvariantsoftheESRF,e.g.theEnsembleAdjustmentKalmanFilter(Anderson(2001),andtheEnsembleTransformKalmanFilterBishopetal.(2001))

•  Tippetetal.(2003)reviewseveralsquarerootfiltersandcomparetheirnumericalefficiency.Showthatalthoughtheyleadtodifferentensemblestheyallspanthesamesubspace.

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EnsembleSquareRootFilterAnexpressionforTcanbefoundbyrearranging using

,and= .

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TheEnsembleTransformKalmanFilter

•  FirstintroducedbyBishopetal.(2001),laterrevisedbyWangetal.(2004).

•  TiscomputedusingtheMorrison-WoodburyidentitytorewritethepreviousexpressionforTTT.

•  TherevisionbyWangetal.highlightedthatanyTwhichsatisfiestheestimateoftheanalysiserrorcovariancedoesnotnecessarilyleadtoanunbiasedanalysisensemble,seeLivingsetal.(2008)forconditionsthatTmustsatisfyfortheanalysisensembletobecentredonthemean.

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Modelerror•  TheensembleKalmanfilterallowsforanimperfectmodelbyaddingnoise

ateachtimestepofthemodelevolution.

•  ThematrixQisnotexplicitlyneededinthealgorithm,onlytheeffectofthemodelerrorintheevolutionofthestate.

•  Therehavebeenmanydifferentstrategiestoincludingmodelerrorintheensemble,basedonwhereyouthinkthesourceoftheerrorlies.Afewexamplesare–  Multiphysics-differentphysicalmodelsareusedineachensemblemember–  Stochastickineticenergybackscatter-replacesupscalekineticenergyloss

duetounresolvedprocessesandnumericalintegration.–  Stochasticallyperturbedphysicaltendencies–  Perturbedparameters–  Orcombinationsoftheabove

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SummaryoftheEnsembleKalmanFilter

Advantages–  Thea-prioriuncertaintyisflow-dependent.–  Thecodecanbedevelopedseparatelyfromthedynamicalmodele.g.PDAF

systemwhichallowsforanymodeltoassimilateobservationsusingensembletechniques(seehttp://www.met.reading.ac.uk/~darc/empire).

–  Noneedtolinearisethemodel,onlylinearassumptionisthatstatisticsremainclosetoGaussian.

–  Easytoaccountformodelerror.–  Easytoparrallelise.

Disadvantages–  Sensitivetoensemblesize.Undersamplingcanleadtofilterdivergence.Ideas

tomitigatethisincludelocalisationandinflation(seenextEnKFlecture).–  AssumesGaussianstatistics,forhighlynon-linearmodelsthismaynotbea

validassumption(seeFriday’slecturesonparticlefilters)–  Theupdatedensemblemeanmaynotbeconsistentwiththemodel

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SummaryoftheEnsembleKalmanFilter

ThedifferentEnKFalgorithms–  Manydifferentalgorithmsexist.–  Stochasticmethodsupdateeachensemblememberseparatelyand

thenestimatethefirsttwosamplemomentstogivetheensemblemeanandcovariance.

–  Deterministicmethodsupdatetheensemblesimultaneouslybasedonlinear/Gaussiantheory.Mayallowforasmallerensemblethanthestochasticmethodsasavoidssomeofthesamplingerror.

EnKFvs4DVar

–  Eachmethodhasitsownadvantagesanddisadvantages-thereisnoclearwinner.

–  Hybridmethodsaimtocombinethebestbitsofboth(seetomorrow'slecturesonhybridmethods).

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FurtherreadingKalmanFilter:�GrewalandAndrews(2008)KalmanFiltering:TheoryandPracticeusingMATLAB.Wiley,NewJersey.�Kalman(1960)Anewapproachtolinearfilteringandpredictionproblems.J.BasicEngineering,82,32-45.

StochasticEnsembleKalmanFilter:�Evensen(1994)Sequentialdataassimilationwithanonlinearquasi-geostrophicmodelusingMonteCarlomethodstoforecasterrorstatistics.J.Geophys.Res.,99(C5),10143-10162.

DetermanisticEnsembleKalmanfilter:�Anderson(2001)Anensembleadjustmentfilterfordataassimilation.Mon.WeatherRev.,129,2884-2903.�Bishopetal.(2001)AdaptivesamplingwiththeensembletransformKalmanfilter.Mon.Wea.Rev.,126,1719-1724.�Tippetetal.(2003)EnsembleSquareRootFilters.Mon.Wea.Rev.,131,1485-1490.�Livingsetal.(2008)Unbiasedensemblesquarerootfilters.PhysicaD.237,1021-1028.�Wangetal.(2004)WhichIsBetter,anEnsembleofPositive–NegativePairsoraCenteredSphericalSimplexEnsemble?Mon.Wea.Rev.,132,1590-1605

Modelerror:�Berneretal.(2011)Modeluncertaintyinamesoscaleensemblepredictionsystem:Stochasticversusmultiphysicsrepresentations,Mon.WeatherRev.,139,1972–1995.

Reviews:�Bannister(2017)Areviewofoperationalmethodsofvariationalandensemble-variationaldataassimilation.Q.J.R.Meteorol.Soc.,143:607–633.�HoutekamerandZhang(2016)ReviewoftheEnsembleKalmanFilterforAtmosphericDataAssimilation.Mon.Wea.Rev.,144,4489–4532.�Vetra-Carvalhoetal.(2018)State-of-the-artstochasticdataassimilationmethodsforhigh-dimensionalnon-Gaussianproblems.TellusA,https://doi.org/10.1080/16000870.2018.1445364

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