DART Tutorial Part IV - Mesoscale Research...
Transcript of DART Tutorial Part IV - Mesoscale Research...
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DARTTutorialPart IV: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∫
Howcanwetakeproductofsamplewithcon'nuouslikelihood?
Fitacon'nuous(Gaussianfornow)distribu'ontosample.−4 −2 0 2 40
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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.
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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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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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SamplingPosteriorPDF:
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Random Sample
Justdrawarandomsample(filter_kind=5in&assim_tools_nml).
SamplingPosteriorPDF:
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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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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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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.
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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.
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Kurtosis 3
SamplingPosteriorPDF:
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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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Prior Ensemble
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Example:Supposepriorsampleis(significantly)bimodal?
Mightwanttoretainaddi'onalinforma'onfromprior.RecallthatEnsembleAdjustmentFiltertriedtodothis(Sec'on1).
SamplingPosteriorPDF:
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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
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
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Again,fitaGaussiantothesample.
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
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Obs. Likelihood
Again,fitaGaussiantothesample.Aretherewaystodothiswithoutcompu'ngpriorsamplestats?
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
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Prior Ensemble
Obs. Likelihood
Random Draws from Obs.
Generatearandomdrawfromtheobserva'onlikelihood.Associateitwiththefirstsampleofthepriorensemble.
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
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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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Foreachpriormean/obs.pair,findmeanofposteriorPDF.
DARTTutorialSec'on6:Slide24
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
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Priorsamplestandarddevia'ons'llmeasuresuncertaintyofpriormeanes'mate.
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
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Priorsamplestandarddevia'ons'llmeasuresuncertaintyofpriormeanes'mate.Obs.likelihoodstandarddevia'onmeasuresuncertaintyofobs.es'mate.
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
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Posterior PDF
Takeproductoftheprior/obsdistribu'onsforfirstsample.ThisisthestandardGaussianproduct.
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
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Meanofproductisrandomsampleofposterior.Productofrandomsamplesisrandomsampleofproduct.
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
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Repeatthisopera'onforeachjointpriorpair.
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
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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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Prior Ensemble
Canretainmorecorrectinforma'onaboutnon-Gaussianpriors.Canalsobeusedforobs.likelihoodterminproduct(notshownhere).
EnsembleFilterAlgorithms:
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Prior PDF
Usually,kernelwidthsareafunc'onofthesamplevariance.
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
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Prior Ensemble
Prior PDFObs. Likelihood
Usually,kernelwidthsareafunc'onofthesamplevariance.
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
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Obs. Likelihood
Example Kernels: Half as Wide as Prior PDF
ApproximatepriorasasumofGaussianscenteredoneachsample.
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
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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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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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Obs. Likelihood
ComputeproductoffirstkernelwithObs.likelihood.
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
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Obs. Likelihood
But,cannolongerignoretheweighttermforproductofGaussians.Kernelswithmeanfurtherfromobserva'ongetlessweight.
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
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Obs. Likelihood
Con'nuetotakeproductsforeachkernelinturn.Closerkernelsdominateposterior.
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
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Obs. LikelihoodNormalized Sum of Posteriors
Finalposteriorisweight-normalizedsumofkernelproducts.
PosteriorissomewhatdifferentthanforensembleadjustmentorensembleKalmanfilter(muchlessdensityinleolobe.)
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
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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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RankHistogramFilter(filter_kind=8in&assim_tools_nml).
Goal:Wanttohandlenon-Gaussianpriorsorobserva'onlikelihoods.Lowinforma'oncontentobs.mustyieldsmallincrements.
MustperformwellforGaussianpriors.Mustbecomputa'onallyefficient.
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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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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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Step2:Uselikelihoodtocomputeweightforeachensemblemember.• Analogoustoclassicalpar'clefilter.• Canbeextendedtonon-Gaussianobs.likelihoods.
EnsembleFilterAlgorithms:RankHistogramFilter(filter_kind=8in&assim_tools_nml).
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Step2:Uselikelihoodtocomputeweightforeachensemblemember.• Canapproximateinteriorlikelihoodwithlinearfit;forefficiency.
EnsembleFilterAlgorithms:RankHistogramFilter(filter_kind=8in&assim_tools_nml).
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Step3:Computecon'nuousposteriordistribu'on.• Approximatelikelihoodwithtrapezoidalquadrature,takeproduct.
(DisplayedproductnormalizedtomakeposterioraPDF).
EnsembleFilterAlgorithms:RankHistogramFilter(filter_kind=8in&assim_tools_nml).
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Step3:Computecon'nuousposteriordistribu'on.• ProductofpriorGaussiankernelwithlikelihoodfortails.
EnsembleFilterAlgorithms:RankHistogramFilter(filter_kind=8in&assim_tools_nml).
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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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RHF Posterior
EAKF Posterior
ComparetostandardensembleadjustmentKalman filter(EAKF).NearlyGaussiancase,differencesaresmall.
EnsembleFilterAlgorithms:RankHistogramFilter(filter_kind=8in&assim_tools_nml).
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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).
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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.
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1. Model Error
2. h errors; Representativeness
3. ‘Gross’ Obs. Error 4. Sampling Error; Gaussian Assumption
5. Sampling Error; Assuming Linear Statistical Relation
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DARTTutorialSec'on9:Slide3
Model/FilterError:FilterDivergenceandVarianceInfla'on
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1.Historyofobserva'onsandphysicalsystem=>‘true’distribu'on.
Model/FilterError:FilterDivergenceandVarianceInfla'on
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Variance Deficient PDF
1.Historyofobserva'onsandphysicalsystem=>‘true’distribu'on.2.Samplingerror,somemodelerrorsleadtoinsufficientpriorvariance.3.Canleadto‘filterdivergence’:prioristooconfident,obs.Ignored.
Model/FilterError:FilterDivergenceandVarianceInfla'on
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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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"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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"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
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Observa'onoutsideprior:dangeroffilterdivergence.
Observa'oninred.Priorensembleingreen.
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A\erinfla'ng,observa'onisinpriorcloud:filterdivergenceavoided.
Observa'oninred.Priorensembleingreen.
Inflatedensembleinmagenta.
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Priordistribu'onissignificantly‘curved’.
Observa'oninred.Priorensembleingreen.
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Inflatedprioroutsidealractor.Posteriorwillalsobeoffalractor.
Observa'oninred.Priorensembleingreen.
Inflatedensembleinmagenta.
Canleadtotransientoff-alractorbehavioror…model‘blow-up’.