DART Tutorial Sec’on 6: Other Updates for an Observed Variable · −4 −2 0 2 4 0 0.2 0.4 0.6 Probability Prior Ensemble Obs. Likelihood Random Draws from Obs. Proceed to associate

TheNa'onalCenterforAtmosphericResearchissponsoredbytheNa'onalScienceFounda'on.Anyopinions,findingsandconclusionsorrecommenda'onsexpressedinthispublica'onarethoseoftheauthor(s)anddonotnecessarilyreflecttheviewsoftheNa'onalScienceFounda'on.

©UCAR

DARTTutorialSec'on6:OtherUpdatesforanObservedVariable

Don’tknowmuchaboutproper'esofthissample.Maynaivelyassumeitisrandomdrawfrom‘truth’.

Ensemblefilters:Priorisavailableasfinitesample.

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ProductofTwoGaussians

p A | BC( ) = p(B | AC)p(A |C)p(B |C)

= p(B | AC)p(A |C)p(B | x)p(x |C)dx∫

DARTTutorialSec'on6:Slide2

Howcanwetakeproductofsamplewithcon'nuouslikelihood?

Fitacon'nuous(Gaussianfornow)distribu'ontosample.−4 −2 0 2 40

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Prior Ensemble


p A | BC( ) = p(B | AC)p(A |C)p(B |C)



Observa'onlikelihoodusuallycon'nuous(nearlyalwaysGaussian).

IfObs.Likelihoodisn’tGaussian,cangeneralizemethodsbelow.Forinstance,canfitsetofGaussiankernelstoobs.likelihood.

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Obs. Likelihood

Prior Ensemble


p A | BC( ) = p(B | AC)p(A |C)p(B |C)



ProductofpriorGaussianfitandObs.likelihoodisGaussian.

Compu'ngcon'nuousposteriorissimple.BUT,needtohaveaSAMPLEofthisPDF.

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Obs. Likelihood

Posterior PDF

Prior Ensemble


p A | BC( ) = p(B | AC)p(A |C)p(B |C)



SamplingPosteriorPDF:

Therearemanywaystodothis.

Exactproper'esofdifferentmethodsmaybeunclear.Trialanderrors'llbestwaytoseehowtheyperform.Willinteractwithproper'esofpredic'onmodels,etc.

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Random Sample

Justdrawarandomsample(filter_kind=5in&assim_tools_nml).

NOTE:Whentryingfilter_kindsotherthan1,sort_obs_incin&assim_tools_nmlshouldbe.true.(seesec'on10).



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Random Sample; Exact Mean


Can‘playgames’withthissampletoimprove(modify)itsproper'es.

Example: Adjustthemeanofthesampletobeexact. Canalsoadjustthevariancetobeexact.



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Random Sample; Exact Mean and Var.


Can‘playgames’withthissampletoimprove(modify)itsproper'es.

Example: Adjustthemeanofthesampletobeexact. Canalsoadjustthevariancetobeexact.



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Random Sample; Exact Mean and Var.

Mightalsowanttoeliminaterareextremeoutliers.

NOTE:Proper'esoftheseadjustedsamplescanbequitedifferent.Howtheseproper'esinteractwiththerestoftheassimila'onisanopenques'on.




Constructa‘determinis'c’samplewithcertainfeatures.

Forinstance:Samplecouldhaveexactmeanandvariance.

Thisisinsufficienttoconstrainensemble,needotherconstraints.

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Constructa‘determinis'c’samplewithcertainfeatures(filter_kind=6in&assim_tools_nml;manuallyadjustkurtosis).

Example:Exactsamplemeanandvariance.

Samplekurtosisis3(expectedvalueforGaussianinlargesamplelimit)(Constructedbystar'nguniformly-spacedandadjus'ngquadra'cally).

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Kurtosis 3



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Kurtosis 3

Kurtosis 2

Example:Exactsamplemeanandvariance.

Samplekurtosis2:lessextremeoutliers,lessdensenearmean.Avoidingoutliersmightbeniceincertainapplica'ons.Samplingheavilynearmeanmightbenice. DARTTutorialSec'on6:Slide13

Constructa‘determinis'c’samplewithcertainfeatures(filter_kind=6in&assim_tools_nml;manuallyadjustkurtosis).


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Prob

abilit

yFirsttwomethodsdependonlyonmeanandvarianceofpriorsample.

Example:Supposepriorsampleis(significantly)bimodal?

Mightwanttoretainaddi'onalinforma'onfromprior.RecallthatEnsembleAdjustmentFiltertriedtodothis(Sec'on1).



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abilit

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Prior PDFObs. Likelihood

Posterior PDF

Random Posterior Ensemble

Firsttwomethodsdependonlyonmeanandvarianceofpriorsample.

Example:Supposepriorsampleis(significantly)bimodal?

Mightwanttoretainaddi'onalinforma'onfromprior.RecallthatEnsembleAdjustmentFiltertriedtodothis(Sec'on1).


EnsembleFilterAlgorithms:

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‘Classical’MonteCarloAlgorithmforDataAssimila'on.Warning:earliestreferences(Evensen1994)have

subop'malalgorithm(moreinaminute).

EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).



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Again,fitaGaussiantothesample.Aretherewaystodothiswithoutcompu'ngpriorsamplestats?




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Obs. Likelihood

Again,fitaGaussiantothesample.Aretherewaystodothiswithoutcompu'ngpriorsamplestats?




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Prior Ensemble

Obs. Likelihood

Random Draws from Obs.

Generatearandomdrawfromtheobserva'onlikelihood.Associateitwiththefirstsampleofthepriorensemble.




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Obs. Likelihood


Proceedtoassociatearandomdrawfromobs.witheachpriorsample.Thishasbeencalled‘perturbed’observa'ons.

Algorithmsome'mescalled‘perturbedobs.’ensembleKalmanfilter.




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Obs. Likelihood


Proceedtoassociatearandomdrawfromobs.witheachpriorsample.Thishasbeencalled‘perturbed’observa'ons.

Algorithmsome'mescalled‘perturbedobs.’ensembleKalmanfilter.




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Obs. Likelihood


Proceedtoassociatearandomdrawfromobs.witheachpriorsample.Earliestpublica'onsassociatedmeanofobs.likelihoodwitheachprior.

Thisresultedininsufficientvarianceinposterior.




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Prior Ensemble

Obs. Likelihood


Havesampleofjointpriordistribu'onforobserva'onandpriorMEAN.Adjus'ngthemeanofobs.sampletobeexactimprovesperformance.

Adjus'ngthevariancemayfurtherimproveperformance.Outliersarepoten'alproblem,butcanberemoved.




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Foreachpriormean/obs.pair,findmeanofposteriorPDF.




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Priorsamplestandarddevia'ons'llmeasuresuncertaintyofpriormeanes'mate.




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Priorsamplestandarddevia'ons'llmeasuresuncertaintyofpriormeanes'mate.Obs.likelihoodstandarddevia'onmeasuresuncertaintyofobs.es'mate.




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Posterior PDF

Takeproductoftheprior/obsdistribu'onsforfirstsample.ThisisthestandardGaussianproduct.




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Meanofproductisrandomsampleofposterior.Productofrandomsamplesisrandomsampleofproduct.




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Repeatthisopera'onforeachjointpriorpair.




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Posteriorsamplemaintainsmuchofpriorsamplestructure.(Thisismoreapparentforlargerensemblesizes.)

Posteriorsamplemeanandvarianceconvergeto‘exact’forlargesamples.



EnsembleFilterAlgorithms:EnsembleKernelFilter(EKF)

(filter_kind=3in&assim_tools_nml).

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Canretainmorecorrectinforma'onaboutnon-Gaussianpriors.Canalsobeusedforobs.likelihoodterminproduct(notshownhere).



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Prior PDF

Usually,kernelwidthsareafunc'onofthesamplevariance.Almostavoidsusingpriorsamplevariance.


EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).


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Prior PDFObs. Likelihood

Usually,kernelwidthsareafunc'onofthesamplevariance.Almostavoidsusingpriorsamplevariance.




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Obs. Likelihood

Example Kernels: Half as Wide as Prior PDF

ApproximatepriorasasumofGaussianscenteredoneachsample.




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Obs. Likelihood


Normalized Sum of Kernels

Thees'mateoftheprioristhenormalizedsumofallkernels.




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Obs. Likelihood

Applydistribu'velawtotakeproduct.Productofthesumisthesumoftheproducts.




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Obs. Likelihood

ComputeproductoffirstkernelwithObs.Likelihood.




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abilit

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Obs. Likelihood

But,cannolongerignoretheweighttermforproductofGaussians.Kernelswithmeanfurtherfromobserva'ongetlessweight.




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Obs. Likelihood

Con'nuetotakeproductsforeachkernelinturn.Moredistantkernelshavesmallerimpactonposterior.




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Obs. Likelihood




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Obs. Likelihood

Con'nuetotakeproductsforeachkernelinturn.Closerkernelsdominateposterior.




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Obs. Likelihood





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Obs. LikelihoodNormalized Sum of Posteriors

Finalposteriorisweight-normalizedsumofkernelproducts.

PosteriorissomewhatdifferentthanforensembleadjustmentorensembleKalmanfilter(muchlessdensityinleolobe.)




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Obs. LikelihoodNormalized Sum of Posteriors

Posterior Ensemble

Formingsampleoftheposteriorcanbeproblema'c.Randomsampleissimple.

Determinis'csamplingismuchmoretrickyhere(fewresultsavailable).DARTTutorialSec'on6:Slide55



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abilit

y D

ensi

ty Applyforwardoperatortoeachensemblemember.Getpriorensembleinobserva'onspace.

RankHistogramFilter(filter_kind=8in&assim_tools_nml).

Goal:Wanttohandlenon-Gaussianpriorsorobserva'onlikelihoods.Lowinforma'oncontentobs.mustyieldsmallincrements.

MustperformwellforGaussianpriors.Mustbecomputa'onallyefficient.

Anderson,J.L.,2010:ANon-GaussianEnsembleFilterUpdateforDataAssimila'on.Mon.Wea.Rev.,139,4186-4198.doi:10.1175/2010MWR3253.1 DARTTutorialSec'on6:Slide56

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Step1:Getcon'nuouspriordistribu'ondensity.•  Place(ens_size+1)-1massbetweenadjacentensemblemembers.•  Reminiscentofrankhistogramevalua'onmethod.




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Step1:Getcon'nuouspriordistribu'ondensity.•  Par'alGaussiankernelsontails,N(tail_mean,ens_sd).•  tail_meanselectedsothat(ens_size+1)-1massisintail.




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Step2:Uselikelihoodtocomputeweightforeachensemblemember.•  Analogoustoclassicalpar'clefilter.•  Canbeextendedtonon-Gaussianobs.likelihoods.




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Step2:Uselikelihoodtocomputeweightforeachensemblemember.•  Canapproximateinteriorlikelihoodwithlinearfit;forefficiency.




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Step3:Computecon'nuousposteriordistribu'on.•  Approximatelikelihoodwithtrapezoidalquadrature,takeproduct.

(DisplayedproductnormalizedtomakeposterioraPDF).




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Step3:Computecon'nuousposteriordistribu'on.•  ProductofpriorGaussiankernelwithlikelihoodfortails.•  EasyforGaussianlikelihood.•  Morequadratureifnon-Gaussianlikelihood.




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RHF Posterior

Step4:Computeupdatedensemblemembers:•  (ens_size+1)-1ofposteriormassbetweeneachensemblepair.•  (ens_size+1)-1ineachtail.•  Uninforma'veobserva'onhasnoimpact.




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RHF Posterior

EAKF Posterior

ComparetostandardEnsembleAdjustmentFilter(EAKF).NearlyGaussiancase,differencesaresmall.




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RHF Posterior

EAKF Posterior

RankHistogramgetsridofoutlierthatisclearlyinconsistentwithobs.EAKFcan’tgetridofoutlier.LargepriorvariancefromoutliercausesEAKFtoshioallmemberstoo

muchtowardsobserva'on.




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ty

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EAKF Posterior

RankHistogramhandlesmul'modalpriorandcompellingobserva'on.

EAKFs'llbimodal;leomodeisinconsistentwitheverything.Lorenz63canhavepriorslikethis.




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EAKF Posterior

Convec've-scalemodels(andlandmodels)haveanalogousbehavior.Convec'onmayfireat‘random’loca'ons.Subsetofensembleswillbeinrightplace,restinwrongplace.Wanttoaggressivelyeliminateconvec'oninwrongplace.





Par'clefiltermethods:Theseare‘classical’ensemblemethodsfromsta's'calliterature.Sizeofensemblesrequiredscaleshyper-exponen'allywithmodelsize.Ensembles>1000requiredformodelswith>4degreesoffreedom.Thisrulesoutnaiveapplica'ontoanymeaningfulatmosphericmodel.



Canusepar'clefiltersinafewdimensions.DARTprovidesaone-dimensionalpar'clefilter.Independentpar'clefilterisusedforupda'ngeachobserva'on.PROBLEM:Inconsistencybetweenupdatesfordifferentobserva'ons.Thiscanprobablybemadetoworkinsomecleverway!

Par'cleFilter(filter_kind=4in&assim_tools_nml).



Othernovelfilteralgorithmsavailableonrequest:

1.  Quadra'cfilterbyDanHodyss,NRL2.  Localizedpar'clefilterbyJonPoterjoy,NOAA

ContacttheDARTdevelopmentteamformoreinforma'on,[email protected].


UsingDARTDiagnos'cs

WhathappenswhenthesedifferentmethodsareusedinLorenz63?Aretheysignificantlydifferent?Dosomeworkbeuerfordifferentobserva'onsets?Cankernelfilterdealbeuerwithdis'nctbimodalityofLorenz63?

Withproperresampling,thisshouldbethecase.Somebodyclevercouldprobablymakethiswork.


1.   FilteringForaOneVariableSystem2.   TheDARTDirectoryTree3.   DARTRunAmeControlandDocumentaAon4.   HowshouldobservaAonsofastatevariableimpactanunobservedstatevariable?

MulAvariateassimilaAon.5.   ComprehensiveFilteringTheory:Non-IdenAtyObservaAonsandtheJointPhaseSpace6.   OtherUpdatesforAnObservedVariable7.   SomeAddiAonalLow-OrderModels8.   DealingwithSamplingError9.   MoreonDealingwithError;InflaAon10.   RegressionandNonlinearEffects11.   CreaAngDARTExecutables12.   AdapAveInflaAon13.   HierarchicalGroupFiltersandLocalizaAon14.   QualityControl15.   DARTExperiments:ControlandDesign16.   DiagnosAcOutput17.   CreaAngObservaAonSequences18.   LostinPhaseSpace:TheChallengeofNotKnowingtheTruth19.   DART-CompliantModelsandMakingModelsCompliant20.   ModelParameterEsAmaAon21.   ObservaAonTypesandObservingSystemDesign22.   ParallelAlgorithmImplementaAon23.  Loca'onmoduledesign(notavailable)24.  Fixedlagsmoother(notavailable)25.   Asimple1DadvecAonmodel:TracerDataAssimilaAon

DARTTutorialIndextoSec'ons


DART Tutorial Sec’on 6: Other Updates for an Observed Variable · −4 −2 0 2 4 0 0.2 0.4 0.6 Probability Prior Ensemble Obs. Likelihood Random Draws from Obs. Proceed to associate

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