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DARTTutorialPart IV:OtherUpdatesforanObservedVariable

Don’tknowmuchaboutproper'esofthissample.Maynaivelyassumeitisrandomdrawfrom‘truth’.

Ensemblefilters:Priorisavailableasfinitesample.

−4 −2 0 2 40

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Prob

abilit

y

Prior Ensemble

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∫

Howcanwetakeproductofsamplewithcon'nuouslikelihood?

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

0.2

0.4

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abilit

y

Prior PDF

Prior Ensemble

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∫

Observa'onlikelihoodusuallycon'nuous(nearlyalwaysGaussian).

IfObs.likelihoodisn’tGaussian,cangeneralizemethodsbelow.

−4 −2 0 2 40

0.2

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Prob

abilit

y

Prior PDF

Obs. Likelihood

Prior Ensemble

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∫

ProductofpriorGaussianfitandObs.likelihoodisGaussian.

Compu'ngcon'nuousposteriorissimple.BUT,needtohaveaSAMPLEofthisPDF.

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abilit

y

Prior PDF

Obs. Likelihood

Posterior PDF

Prior Ensemble

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∫

SamplingPosteriorPDF:

Therearemanywaystodothis.

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

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0.6Posterior PDF

Prob

abilit

y

SamplingPosteriorPDF:

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0.2

0.4

0.6Posterior PDF

Prob

abilit

y

Random Sample

Justdrawarandomsample(filter_kind=5in&assim_tools_nml).

SamplingPosteriorPDF:

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0.2

0.4

0.6Posterior PDF

Prob

abilit

y

Random Sample; Exact Mean

Justdrawarandomsample(filter_kind=5in&assim_tools_nml).

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

Example: Adjustthemeanofthesampletobeexact. Canalsoadjustthevariancetobeexact.

SamplingPosteriorPDF:

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0.2

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0.6Posterior PDF

Prob

abilit

y

Random Sample; Exact Mean and Var.

Justdrawarandomsample(filter_kind=5in&assim_tools_nml).

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

Example: Adjustthemeanofthesampletobeexact. Canalsoadjustthevariancetobeexact.

SamplingPosteriorPDF:

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0.2

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0.6Posterior PDF

Prob

abilit

y

Random Sample; Exact Mean and Var.

Mightalsowanttoeliminaterareextremeoutliers.

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

Justdrawarandomsample(filter_kind=5in&assim_tools_nml).

SamplingPosteriorPDF:

Constructa‘determinis'c’samplewithcertainfeatures.

Forinstance:Samplecouldhaveexactmeanandvariance.

Thisisinsufficienttoconstrainensemble,needotherconstraints.

−3 −2 −1 0 1 2 3 40

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Prob

abilit

y

SamplingPosteriorPDF:

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

Example:Exactsamplemeanandvariance.

Samplekurtosis(related to the sharpness/tailedness of a distribution) is3, which is the expectedvaluefora normal distribution. Startby assuming auniformly-spacedsample andadjus'ngquadra'cally.

−3 −2 −1 0 1 2 3 40

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0.6Posterior PDF

Prob

abilit

y

Kurtosis 3

SamplingPosteriorPDF:

−3 −2 −1 0 1 2 3 40

0.2

0.4

0.6Posterior PDF

Prob

abilit

y

Kurtosis 3

Kurtosis 2

Example:Exactsamplemeanandvariance.

Samplekurtosis2:lessextremeoutliers,lessdensenearmean.Avoidingoutliersmightbeniceincertainapplica'ons.Samplingheavilynearmeanmightbenice.

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

SamplingPosteriorPDF:

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0.6

Prior Ensemble

Prob

abilit

yFirsttwomethodsdependonlyonmeanandvarianceofpriorsample.

Example:Supposepriorsampleis(significantly)bimodal?

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

SamplingPosteriorPDF:

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0.2

0.4

Prior Ensemble

Prob

abilit

y

Prior PDFObs. Likelihood

Posterior PDF

Random Posterior Ensemble

Firsttwomethodsdependonlyonmeanandvarianceofpriorsample.

Example:Supposepriorsampleis(significantly)bimodal?

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

EnsembleFilterAlgorithms:

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0.2

0.4

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abilit

y

Prior Ensemble

‘Classical’MonteCarloalgorithmfordataassimila'on

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

EnsembleFilterAlgorithms:

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abilit

y

Prior Ensemble

Again,fitaGaussiantothesample.

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

EnsembleFilterAlgorithms:

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Prob

abilit

y

Prior Ensemble

Obs. Likelihood

Again,fitaGaussiantothesample.Aretherewaystodothiswithoutcompu'ngpriorsamplestats?

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

EnsembleFilterAlgorithms:

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Prob

abilit

y

Prior Ensemble

Obs. Likelihood

Random Draws from Obs.

Generatearandomdrawfromtheobserva'onlikelihood.Associateitwiththefirstsampleofthepriorensemble.

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

EnsembleFilterAlgorithms:

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Prob

abilit

y

Prior Ensemble

Obs. Likelihood

Random Draws from Obs.

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

Adjus'ngthevariancemayfurtherimproveperformance.Outliersarea poten'alproblem,butcanberemoved.

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

EnsembleFilterAlgorithms:

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Probability

Foreachpriormean/obs.pair,findmeanofposteriorPDF.

DARTTutorialSec'on6:Slide24

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

EnsembleFilterAlgorithms:

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Probability

Priorsamplestandarddevia'ons'llmeasuresuncertaintyofpriormeanes'mate.

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

EnsembleFilterAlgorithms:

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Probability

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

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

EnsembleFilterAlgorithms:

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Prob

abilit

y

Posterior PDF

Takeproductoftheprior/obsdistribu'onsforfirstsample.ThisisthestandardGaussianproduct.

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

EnsembleFilterAlgorithms:

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Probability

Meanofproductisrandomsampleofposterior.Productofrandomsamplesisrandomsampleofproduct.

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

EnsembleFilterAlgorithms:

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Probability

Repeatthisopera'onforeachjointpriorpair.

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

EnsembleFilterAlgorithms:

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Probability

Posteriorsamplemaintainsmuchofpriorsamplestructure.(Thisismoreapparentforlargerensemblesizes.)

Posteriorsamplemeanandvarianceconvergeto‘exact’forlargesamples.

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

EnsembleFilterAlgorithms:EnsembleKernelFilter(EKF)

(filter_kind=3in&assim_tools_nml).

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Prob

abilit

y

Prior Ensemble

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

EnsembleFilterAlgorithms:

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Prob

abilit

y

Prior Ensemble

Prior PDF

Usually,kernelwidthsareafunc'onofthesamplevariance.

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

EnsembleFilterAlgorithms:

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0.8

Prob

abilit

y

Prior Ensemble

Prior PDFObs. Likelihood

Usually,kernelwidthsareafunc'onofthesamplevariance.

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

EnsembleFilterAlgorithms:

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Prob

abilit

y

Obs. Likelihood

Example Kernels: Half as Wide as Prior PDF

ApproximatepriorasasumofGaussianscenteredoneachsample.

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

EnsembleFilterAlgorithms:

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Prob

abilit

y

Obs. Likelihood

Example Kernels: Half as Wide as Prior PDF

Normalized Sum of Kernels

Thees'mateoftheprioristhenormalizedsumofallkernels.

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

EnsembleFilterAlgorithms:

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Prob

abilit

y

Obs. Likelihood

Applydistribu'velawtotakeproduct: productofthesumisthesumoftheproducts. Otherwise, the

product cannot be analytically determined.

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

EnsembleFilterAlgorithms:

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Prob

abilit

y

Obs. Likelihood

ComputeproductoffirstkernelwithObs.likelihood.

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

EnsembleFilterAlgorithms:

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Prob

abilit

y

Obs. Likelihood

But,cannolongerignoretheweighttermforproductofGaussians.Kernelswithmeanfurtherfromobserva'ongetlessweight.

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

EnsembleFilterAlgorithms:

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Prob

abilit

y

Obs. Likelihood

Con'nuetotakeproductsforeachkernelinturn.Closerkernelsdominateposterior.

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

EnsembleFilterAlgorithms:

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Prob

abilit

y

Obs. LikelihoodNormalized Sum of Posteriors

Finalposteriorisweight-normalizedsumofkernelproducts.

PosteriorissomewhatdifferentthanforensembleadjustmentorensembleKalmanfilter(muchlessdensityinleolobe.)

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

EnsembleFilterAlgorithms:

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Prob

abilit

y

Obs. LikelihoodNormalized Sum of Posteriors

Posterior Ensemble

Formingsampleoftheposteriorcanbeproblema'c.Randomsampleissimple.

Determinis'csamplingismuchmoretrickyhere(fewresultsavailable).

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

EnsembleFilterAlgorithms:

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0

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PriorProb

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.

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PriorProb

abilit

y D

ensi

ty

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

EnsembleFilterAlgorithms:

DARTTutorialSec'on6:Slide60

RankHistogramFilter(filter_kind=8in&assim_tools_nml).

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0

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PriorProb

abilit

y D

ensi

ty

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

EnsembleFilterAlgorithms:RankHistogramFilter(filter_kind=8in&assim_tools_nml).

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PriorProb

abilit

y D

ensi

ty

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

EnsembleFilterAlgorithms:RankHistogramFilter(filter_kind=8in&assim_tools_nml).

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0

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PriorProb

abilit

y D

ensi

ty

Step2:Uselikelihoodtocomputeweightforeachensemblemember.• Canapproximateinteriorlikelihoodwithlinearfit;forefficiency.

EnsembleFilterAlgorithms:RankHistogramFilter(filter_kind=8in&assim_tools_nml).

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0

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PriorProb

abilit

y D

ensi

ty

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

(DisplayedproductnormalizedtomakeposterioraPDF).

EnsembleFilterAlgorithms:RankHistogramFilter(filter_kind=8in&assim_tools_nml).

−3 −2 −1 0 1 2 3

0

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PriorProb

abilit

y D

ensi

ty

Step3:Computecon'nuousposteriordistribu'on.• ProductofpriorGaussiankernelwithlikelihoodfortails.

EnsembleFilterAlgorithms:RankHistogramFilter(filter_kind=8in&assim_tools_nml).

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PriorProb

abilit

y D

ensi

ty

RHF Posterior

Step4:Computeupdatedensemblemembers:• (ens_size+1)-1ofposteriormassbetweeneachensemblepair.• (ens_size+1)-1ineachtail.• Uninforma'veobserva'on(not shown) would havenoimpact.

EnsembleFilterAlgorithms:RankHistogramFilter(filter_kind=8in&assim_tools_nml).

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0

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PriorProb

abilit

y D

ensi

ty

RHF Posterior

EAKF Posterior

ComparetostandardensembleadjustmentKalman filter(EAKF).NearlyGaussiancase,differencesaresmall.

EnsembleFilterAlgorithms:RankHistogramFilter(filter_kind=8in&assim_tools_nml).

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1

0

1

2

3

PriorProb

abilit

y D

ensi

ty

RHF Posterior

EAKF Posterior

RankHistogramgetsridofprior outlierthatisinconsistentwithobs. EAKFcan’tgetridofthis prior outlier.

LargepriorvariancefromoutliercausesEAKFtoshioallmemberstoomuchtowardsobserva'on (with mean off the page).

EnsembleFilterAlgorithms:RankHistogramFilter(filter_kind=8in&assim_tools_nml).

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5

Prob

abilit

y D

ensi

ty

0

0.5

1

1.5

PriorRHF Posterior

EAKF Posterior

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

EnsembleFilterAlgorithms:RankHistogramFilter(filter_kind=8in&assim_tools_nml).

DealingwithEnsembleFilterErrorsFix1,2,3independently,HARDbutongoing.O\en,ensemblefilters...1-4:Varianceinfla'on,Increaseprioruncertaintytogiveobsmoreimpact.5.‘Localiza'on’:onlyletobs.impactasetof‘nearby’statevariables.O\ensmoothlydecreaseimpactto0asfunc'onofdistance.

****

1. Model Error

2. h errors; Representativeness

3. ‘Gross’ Obs. Error 4. Sampling Error; Gaussian Assumption

5. Sampling Error; Assuming Linear Statistical Relation

tk

tk+1

tk+2

h h h

DARTTutorialSec'on9:Slide3

Model/FilterError:FilterDivergenceandVarianceInfla'on

−4 −3 −2 −1 00

0.5

1

Prob

abilit

y

"TRUE" Prior PDF

1.Historyofobserva'onsandphysicalsystem=>‘true’distribu'on.

Model/FilterError:FilterDivergenceandVarianceInfla'on

−4 −3 −2 −1 00

0.5

1

Prob

abilit

y

"TRUE" Prior PDF

Variance Deficient PDF

1.Historyofobserva'onsandphysicalsystem=>‘true’distribu'on.2.Samplingerror,somemodelerrorsleadtoinsufficientpriorvariance.3.Canleadto‘filterdivergence’:prioristooconfident,obs.Ignored.

Model/FilterError:FilterDivergenceandVarianceInfla'on

−4 −3 −2 −1 00

0.5

1

Prob

abilit

y

"TRUE" Prior PDF

Variance Deficient PDF

1.Historyofobserva'onsandphysicalsystem=>‘true’distribu'on.2.Samplingerror,somemodelerrorsleadtoinsufficientpriorvariance.3.Canleadto‘filterdivergence’:prioristooconfident,obs.Ignored.

Naïvesolu'onisvarianceinfla'on:justincreasespreadofprior.Forensemblememberi, inflate xi( ) = λ xi − x( )+ x

Model/FilterError:FilterDivergenceandVarianceInfla'on

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Prob

abilit

y

"TRUE" Prior PDF Error in Mean (from model)

1.Historyofobserva'onsandphysicalsystem=>‘true’distribu'on.2.Mostmodelerrorsalsoleadtoerroneousshi\inen'redistribu'on.3.Again,priorcanbeviewedasbeingTOOCERTAIN.

Model/FilterError:FilterDivergenceandVarianceInfla'on

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Prob

abilit

y

"TRUE" Prior PDF Error in Mean (from model)

Variance Inflated

Infla'ngcanamelioratethis.Obviously,ifweknewE(error),we’dcorrectforitdirectly.

1.Historyofobserva'onsandphysicalsystem=>‘true’distribu'on.2.Mostmodelerrorsalsoleadtoerroneousshi\inen'redistribu'on.3.Again,priorcanbeviewedasbeingTOOCERTAIN.

PhysicalSpaceVarianceInfla'on

Inflateallstatevariablesbysameamountbeforeassimila'on.Capabili'es:1. Canbeeffec'veforavarietyofmodels.2. Canmaintainlinearbalances.3. Prior continues to resemble that from the first guess.

4. Simpleandcomputationally cheap.

Liabili'es:

1. Statevariablesnotconstrainedbyobserva'onscan‘blowup’.ForinstanceunobservedregionsnearthetopofAGCMs.

2. Magnitudeofλnormallyselectedbytrialanderror.

PhysicalSpaceVarianceInfla'oninLorenz63

−20

0

20 −20

0

2010

20

30

40

Observa'onoutsideprior:dangeroffilterdivergence.

Observa'oninred.Priorensembleingreen.

PhysicalSpaceVarianceInfla'oninLorenz63

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0

20 −20

0

2010

20

30

40

A\erinfla'ng,observa'onisinpriorcloud:filterdivergenceavoided.

Observa'oninred.Priorensembleingreen.

Inflatedensembleinmagenta.

PhysicalSpaceVarianceInfla'oninLorenz63

−20

0

20 −20

0

2010

20

30

40

Priordistribu'onissignificantly‘curved’.

Observa'oninred.Priorensembleingreen.

PhysicalSpaceVarianceInfla'oninLorenz63

−20

0

20 −20

0

2010

20

30

40

Inflatedprioroutsidealractor.Posteriorwillalsobeoffalractor.

Observa'oninred.Priorensembleingreen.

Inflatedensembleinmagenta.

Canleadtotransientoff-alractorbehavioror…model‘blow-up’.