dart tutorial sec’on 6: other updates for an observed variable · −4 −2 0 2 4 0 0.2 0.4 0.6...
TRANSCRIPT
TheNa'onalCenterforAtmosphericResearchissponsoredbytheNa'onalScienceFounda'on.Anyopinions,findingsandconclusionsorrecommenda'onsexpressedinthispublica'onarethoseoftheauthor(s)anddonotnecessarilyreflecttheviewsoftheNa'onalScienceFounda'on.
©UCAR
DARTTutorialSec'on6:OtherUpdatesforanObservedVariable
Don’tknowmuchaboutproper'esofthissample.Maynaivelyassumeitisrandomdrawfrom‘truth’.
Ensemblefilters:Priorisavailableasfinitesample.
−4 −2 0 2 40
0.2
0.4
0.6
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∫
DARTTutorialSec'on6:Slide2
Howcanwetakeproductofsamplewithcon'nuouslikelihood?
Fitacon'nuous(Gaussianfornow)distribu'ontosample.−4 −2 0 2 40
0.2
0.4
0.6
Prob
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∫
DARTTutorialSec'on6:Slide3
Observa'onlikelihoodusuallycon'nuous(nearlyalwaysGaussian).
IfObs.Likelihoodisn’tGaussian,cangeneralizemethodsbelow.Forinstance,canfitsetofGaussiankernelstoobs.likelihood.
−4 −2 0 2 40
0.2
0.4
0.6
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∫
DARTTutorialSec'on6:Slide4
ProductofpriorGaussianfitandObs.likelihoodisGaussian.
Compu'ngcon'nuousposteriorissimple.BUT,needtohaveaSAMPLEofthisPDF.
−4 −2 0 2 40
0.2
0.4
0.6
Prob
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∫
DARTTutorialSec'on6:Slide5
SamplingPosteriorPDF:
Therearemanywaystodothis.
Exactproper'esofdifferentmethodsmaybeunclear.Trialanderrors'llbestwaytoseehowtheyperform.Willinteractwithproper'esofpredic'onmodels,etc.
−2 −1 0 1 2 30
0.2
0.4
0.6Posterior PDF
Prob
abilit
y
DARTTutorialSec'on6:Slide6
SamplingPosteriorPDF:
−2 −1 0 1 2 30
0.2
0.4
0.6Posterior PDF
Prob
abilit
y
Random Sample
Justdrawarandomsample(filter_kind=5in&assim_tools_nml).
NOTE:Whentryingfilter_kindsotherthan1,sort_obs_incin&assim_tools_nmlshouldbe.true.(seesec'on10).
DARTTutorialSec'on6:Slide7
SamplingPosteriorPDF:
−2 −1 0 1 2 30
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.
DARTTutorialSec'on6:Slide8
SamplingPosteriorPDF:
−2 −1 0 1 2 30
0.2
0.4
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.
DARTTutorialSec'on6:Slide9
SamplingPosteriorPDF:
−2 −1 0 1 2 30
0.2
0.4
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).
DARTTutorialSec'on6:Slide10
SamplingPosteriorPDF:
Constructa‘determinis'c’samplewithcertainfeatures.
Forinstance:Samplecouldhaveexactmeanandvariance.
Thisisinsufficienttoconstrainensemble,needotherconstraints.
−3 −2 −1 0 1 2 3 40
0.2
0.4
0.6Posterior PDF
Prob
abilit
y
DARTTutorialSec'on6:Slide11
SamplingPosteriorPDF:
Constructa‘determinis'c’samplewithcertainfeatures(filter_kind=6in&assim_tools_nml;manuallyadjustkurtosis).
Example:Exactsamplemeanandvariance.
Samplekurtosisis3(expectedvalueforGaussianinlargesamplelimit)(Constructedbystar'nguniformly-spacedandadjus'ngquadra'cally).
−3 −2 −1 0 1 2 3 40
0.2
0.4
0.6Posterior PDF
Prob
abilit
y
Kurtosis 3
DARTTutorialSec'on6:Slide12
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. DARTTutorialSec'on6:Slide13
Constructa‘determinis'c’samplewithcertainfeatures(filter_kind=6in&assim_tools_nml;manuallyadjustkurtosis).
SamplingPosteriorPDF:
−4 −2 0 2 40
0.2
0.4
0.6
Prior Ensemble
Prob
abilit
yFirsttwomethodsdependonlyonmeanandvarianceofpriorsample.
Example:Supposepriorsampleis(significantly)bimodal?
Mightwanttoretainaddi'onalinforma'onfromprior.RecallthatEnsembleAdjustmentFiltertriedtodothis(Sec'on1).
DARTTutorialSec'on6:Slide14
SamplingPosteriorPDF:
−4 −2 0 2 40
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).
DARTTutorialSec'on6:Slide15
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
Prob
abilit
y
Prior Ensemble
‘Classical’MonteCarloAlgorithmforDataAssimila'on.Warning:earliestreferences(Evensen1994)have
subop'malalgorithm(moreinaminute).
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
DARTTutorialSec'on6:Slide16
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
Prob
abilit
y
Prior Ensemble
Again,fitaGaussiantothesample.Aretherewaystodothiswithoutcompu'ngpriorsamplestats?
DARTTutorialSec'on6:Slide17
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
Prob
abilit
y
Prior Ensemble
Obs. Likelihood
Again,fitaGaussiantothesample.Aretherewaystodothiswithoutcompu'ngpriorsamplestats?
DARTTutorialSec'on6:Slide18
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
Prob
abilit
y
Prior Ensemble
Obs. Likelihood
Random Draws from Obs.
Generatearandomdrawfromtheobserva'onlikelihood.Associateitwiththefirstsampleofthepriorensemble.
DARTTutorialSec'on6:Slide19
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
Prob
abilit
y
Prior Ensemble
Obs. Likelihood
Random Draws from Obs.
Proceedtoassociatearandomdrawfromobs.witheachpriorsample.Thishasbeencalled‘perturbed’observa'ons.
Algorithmsome'mescalled‘perturbedobs.’ensembleKalmanfilter.
DARTTutorialSec'on6:Slide20
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
Prob
abilit
y
Prior Ensemble
Obs. Likelihood
Random Draws from Obs.
Proceedtoassociatearandomdrawfromobs.witheachpriorsample.Thishasbeencalled‘perturbed’observa'ons.
Algorithmsome'mescalled‘perturbedobs.’ensembleKalmanfilter.
DARTTutorialSec'on6:Slide21
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
Prob
abilit
y
Prior Ensemble
Obs. Likelihood
Random Draws from Obs.
Proceedtoassociatearandomdrawfromobs.witheachpriorsample.Earliestpublica'onsassociatedmeanofobs.likelihoodwitheachprior.
Thisresultedininsufficientvarianceinposterior.
DARTTutorialSec'on6:Slide22
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
Prob
abilit
y
Prior Ensemble
Obs. Likelihood
Random Draws from Obs.
Havesampleofjointpriordistribu'onforobserva'onandpriorMEAN.Adjus'ngthemeanofobs.sampletobeexactimprovesperformance.
Adjus'ngthevariancemayfurtherimproveperformance.Outliersarepoten'alproblem,butcanberemoved.
DARTTutorialSec'on6:Slide23
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
Probability
Foreachpriormean/obs.pair,findmeanofposteriorPDF.
DARTTutorialSec'on6:Slide24
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
Probability
Priorsamplestandarddevia'ons'llmeasuresuncertaintyofpriormeanes'mate.
DARTTutorialSec'on6:Slide25
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
Probability
Priorsamplestandarddevia'ons'llmeasuresuncertaintyofpriormeanes'mate.Obs.likelihoodstandarddevia'onmeasuresuncertaintyofobs.es'mate.
DARTTutorialSec'on6:Slide26
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
Prob
abilit
y
Posterior PDF
Takeproductoftheprior/obsdistribu'onsforfirstsample.ThisisthestandardGaussianproduct.
DARTTutorialSec'on6:Slide27
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
Probability
Meanofproductisrandomsampleofposterior.Productofrandomsamplesisrandomsampleofproduct.
DARTTutorialSec'on6:Slide28
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
Probability
Repeatthisopera'onforeachjointpriorpair.
DARTTutorialSec'on6:Slide29
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
Probability
Repeatthisopera'onforeachjointpriorpair.
DARTTutorialSec'on6:Slide30
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
Probability
Repeatthisopera'onforeachjointpriorpair.
DARTTutorialSec'on6:Slide31
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
Probability
Repeatthisopera'onforeachjointpriorpair.
DARTTutorialSec'on6:Slide32
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
Probability
Posteriorsamplemaintainsmuchofpriorsamplestructure.(Thisismoreapparentforlargerensemblesizes.)
Posteriorsamplemeanandvarianceconvergeto‘exact’forlargesamples.
DARTTutorialSec'on6:Slide33
EnsembleKalmanFilter(EnKF)(filter_kind=2in&assim_tools_nml).
EnsembleFilterAlgorithms:EnsembleKernelFilter(EKF)
(filter_kind=3in&assim_tools_nml).
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Prior Ensemble
Canretainmorecorrectinforma'onaboutnon-Gaussianpriors.Canalsobeusedforobs.likelihoodterminproduct(notshownhere).
DARTTutorialSec'on6:Slide34
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Prior Ensemble
Prior PDF
Usually,kernelwidthsareafunc'onofthesamplevariance.Almostavoidsusingpriorsamplevariance.
DARTTutorialSec'on6:Slide35
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Prior Ensemble
Prior PDFObs. Likelihood
Usually,kernelwidthsareafunc'onofthesamplevariance.Almostavoidsusingpriorsamplevariance.
DARTTutorialSec'on6:Slide36
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. Likelihood
Example Kernels: Half as Wide as Prior PDF
ApproximatepriorasasumofGaussianscenteredoneachsample.
DARTTutorialSec'on6:Slide37
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. Likelihood
Example Kernels: Half as Wide as Prior PDF
ApproximatepriorasasumofGaussianscenteredoneachsample.
DARTTutorialSec'on6:Slide38
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. Likelihood
Example Kernels: Half as Wide as Prior PDF
ApproximatepriorasasumofGaussianscenteredoneachsample.
DARTTutorialSec'on6:Slide39
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. Likelihood
Example Kernels: Half as Wide as Prior PDF
ApproximatepriorasasumofGaussianscenteredoneachsample.
DARTTutorialSec'on6:Slide40
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. Likelihood
Example Kernels: Half as Wide as Prior PDF
ApproximatepriorasasumofGaussianscenteredoneachsample.
DARTTutorialSec'on6:Slide41
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. Likelihood
Example Kernels: Half as Wide as Prior PDF
Normalized Sum of Kernels
Thees'mateoftheprioristhenormalizedsumofallkernels.
DARTTutorialSec'on6:Slide42
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. Likelihood
Applydistribu'velawtotakeproduct.Productofthesumisthesumoftheproducts.
DARTTutorialSec'on6:Slide43
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. Likelihood
ComputeproductoffirstkernelwithObs.Likelihood.
DARTTutorialSec'on6:Slide44
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. Likelihood
But,cannolongerignoretheweighttermforproductofGaussians.Kernelswithmeanfurtherfromobserva'ongetlessweight.
DARTTutorialSec'on6:Slide45
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. Likelihood
Con'nuetotakeproductsforeachkernelinturn.Moredistantkernelshavesmallerimpactonposterior.
DARTTutorialSec'on6:Slide46
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. Likelihood
Con'nuetotakeproductsforeachkernelinturn.Moredistantkernelshavesmallerimpactonposterior.
DARTTutorialSec'on6:Slide47
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. Likelihood
Con'nuetotakeproductsforeachkernelinturn.Moredistantkernelshavesmallerimpactonposterior.
DARTTutorialSec'on6:Slide48
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. Likelihood
DARTTutorialSec'on6:Slide49
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. Likelihood
Con'nuetotakeproductsforeachkernelinturn.Closerkernelsdominateposterior.
DARTTutorialSec'on6:Slide50
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. Likelihood
Con'nuetotakeproductsforeachkernelinturn.Closerkernelsdominateposterior.
DARTTutorialSec'on6:Slide51
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. Likelihood
Con'nuetotakeproductsforeachkernelinturn.Closerkernelsdominateposterior.
DARTTutorialSec'on6:Slide52
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. Likelihood
Con'nuetotakeproductsforeachkernelinturn.Closerkernelsdominateposterior.
DARTTutorialSec'on6:Slide53
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. LikelihoodNormalized Sum of Posteriors
Finalposteriorisweight-normalizedsumofkernelproducts.
PosteriorissomewhatdifferentthanforensembleadjustmentorensembleKalmanfilter(muchlessdensityinleolobe.)
DARTTutorialSec'on6:Slide54
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Obs. LikelihoodNormalized Sum of Posteriors
Posterior Ensemble
Formingsampleoftheposteriorcanbeproblema'c.Randomsampleissimple.
Determinis'csamplingismuchmoretrickyhere(fewresultsavailable).DARTTutorialSec'on6:Slide55
EnsembleKernelFilter(EKF)(filter_kind=3in&assim_tools_nml).
EnsembleFilterAlgorithms:
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
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.
Anderson,J.L.,2010:ANon-GaussianEnsembleFilterUpdateforDataAssimila'on.Mon.Wea.Rev.,139,4186-4198.doi:10.1175/2010MWR3253.1 DARTTutorialSec'on6:Slide56
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
PriorProb
abilit
y D
ensi
ty
Step1:Getcon'nuouspriordistribu'ondensity.• Place(ens_size+1)-1massbetweenadjacentensemblemembers.• Reminiscentofrankhistogramevalua'onmethod.
EnsembleFilterAlgorithms:
DARTTutorialSec'on6:Slide57
RankHistogramFilter(filter_kind=8in&assim_tools_nml).
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
PriorProb
abilit
y D
ensi
ty
Step1:Getcon'nuouspriordistribu'ondensity.• Place(ens_size+1)-1massbetweenadjacentensemblemembers.• Reminiscentofrankhistogramevalua'onmethod.
EnsembleFilterAlgorithms:
DARTTutorialSec'on6:Slide58
RankHistogramFilter(filter_kind=8in&assim_tools_nml).
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
PriorProb
abilit
y D
ensi
ty
Step1:Getcon'nuouspriordistribu'ondensity.• Place(ens_size+1)-1massbetweenadjacentensemblemembers.• Reminiscentofrankhistogramevalua'onmethod.
EnsembleFilterAlgorithms:
DARTTutorialSec'on6:Slide59
RankHistogramFilter(filter_kind=8in&assim_tools_nml).
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
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).
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
PriorProb
abilit
y D
ensi
ty
Step1:Getcon'nuouspriordistribu'ondensity.• Par'alGaussiankernelsontails,N(tail_mean,ens_sd).• tail_meanselectedsothat(ens_size+1)-1massisintail.
EnsembleFilterAlgorithms:
DARTTutorialSec'on6:Slide61
RankHistogramFilter(filter_kind=8in&assim_tools_nml).
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
PriorProb
abilit
y D
ensi
ty
Step2:Uselikelihoodtocomputeweightforeachensemblemember.• Analogoustoclassicalpar'clefilter.• Canbeextendedtonon-Gaussianobs.likelihoods.
EnsembleFilterAlgorithms:
DARTTutorialSec'on6:Slide62
RankHistogramFilter(filter_kind=8in&assim_tools_nml).
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
PriorProb
abilit
y D
ensi
ty
Step2:Uselikelihoodtocomputeweightforeachensemblemember.• Canapproximateinteriorlikelihoodwithlinearfit;forefficiency.
EnsembleFilterAlgorithms:
DARTTutorialSec'on6:Slide63
RankHistogramFilter(filter_kind=8in&assim_tools_nml).
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
PriorProb
abilit
y D
ensi
ty
Step3:Computecon'nuousposteriordistribu'on.• Approximatelikelihoodwithtrapezoidalquadrature,takeproduct.
(DisplayedproductnormalizedtomakeposterioraPDF).
EnsembleFilterAlgorithms:
DARTTutorialSec'on6:Slide64
RankHistogramFilter(filter_kind=8in&assim_tools_nml).
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
PriorProb
abilit
y D
ensi
ty
Step3:Computecon'nuousposteriordistribu'on.• Approximatelikelihoodwithtrapezoidalquadrature,takeproduct.
(DisplayedproductnormalizedtomakeposterioraPDF).
EnsembleFilterAlgorithms:
DARTTutorialSec'on6:Slide65
RankHistogramFilter(filter_kind=8in&assim_tools_nml).
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
PriorProb
abilit
y D
ensi
ty
Step3:Computecon'nuousposteriordistribu'on.• Approximatelikelihoodwithtrapezoidalquadrature,takeproduct.
(DisplayedproductnormalizedtomakeposterioraPDF).
EnsembleFilterAlgorithms:
DARTTutorialSec'on6:Slide66
RankHistogramFilter(filter_kind=8in&assim_tools_nml).
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
PriorProb
abilit
y D
ensi
ty
Step3:Computecon'nuousposteriordistribu'on.• Approximatelikelihoodwithtrapezoidalquadrature,takeproduct.
(DisplayedproductnormalizedtomakeposterioraPDF).
EnsembleFilterAlgorithms:
DARTTutorialSec'on6:Slide67
RankHistogramFilter(filter_kind=8in&assim_tools_nml).
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
PriorProb
abilit
y D
ensi
ty
Step3:Computecon'nuousposteriordistribu'on.• ProductofpriorGaussiankernelwithlikelihoodfortails.• EasyforGaussianlikelihood.• Morequadratureifnon-Gaussianlikelihood.
EnsembleFilterAlgorithms:
DARTTutorialSec'on6:Slide68
RankHistogramFilter(filter_kind=8in&assim_tools_nml).
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
PriorProb
abilit
y D
ensi
ty
RHF Posterior
Step4:Computeupdatedensemblemembers:• (ens_size+1)-1ofposteriormassbetweeneachensemblepair.• (ens_size+1)-1ineachtail.• Uninforma'veobserva'onhasnoimpact.
EnsembleFilterAlgorithms:
DARTTutorialSec'on6:Slide69
RankHistogramFilter(filter_kind=8in&assim_tools_nml).
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
PriorProb
abilit
y D
ensi
ty
RHF Posterior
EAKF Posterior
ComparetostandardEnsembleAdjustmentFilter(EAKF).NearlyGaussiancase,differencesaresmall.
EnsembleFilterAlgorithms:
DARTTutorialSec'on6:Slide70
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
RankHistogramgetsridofoutlierthatisclearlyinconsistentwithobs.EAKFcan’tgetridofoutlier.LargepriorvariancefromoutliercausesEAKFtoshioallmemberstoo
muchtowardsobserva'on.
EnsembleFilterAlgorithms:
DARTTutorialSec'on6:Slide71
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
RankHistogramhandlesmul'modalpriorandcompellingobserva'on.
EAKFs'llbimodal;leomodeisinconsistentwitheverything.Lorenz63canhavepriorslikethis.
EnsembleFilterAlgorithms:
DARTTutorialSec'on6:Slide72
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:
DARTTutorialSec'on6:Slide73
RankHistogramFilter(filter_kind=8in&assim_tools_nml).
EnsembleFilterAlgorithms:
Par'clefiltermethods:Theseare‘classical’ensemblemethodsfromsta's'calliterature.Sizeofensemblesrequiredscaleshyper-exponen'allywithmodelsize.Ensembles>1000requiredformodelswith>4degreesoffreedom.Thisrulesoutnaiveapplica'ontoanymeaningfulatmosphericmodel.
DARTTutorialSec'on6:Slide74
EnsembleFilterAlgorithms:
Canusepar'clefiltersinafewdimensions.DARTprovidesaone-dimensionalpar'clefilter.Independentpar'clefilterisusedforupda'ngeachobserva'on.PROBLEM:Inconsistencybetweenupdatesfordifferentobserva'ons.Thiscanprobablybemadetoworkinsomecleverway!
Par'cleFilter(filter_kind=4in&assim_tools_nml).
DARTTutorialSec'on6:Slide75
EnsembleFilterAlgorithms:
Othernovelfilteralgorithmsavailableonrequest:
1. Quadra'cfilterbyDanHodyss,NRL2. Localizedpar'clefilterbyJonPoterjoy,NOAA
ContacttheDARTdevelopmentteamformoreinforma'on,[email protected].
DARTTutorialSec'on6:Slide76
UsingDARTDiagnos'cs
WhathappenswhenthesedifferentmethodsareusedinLorenz63?Aretheysignificantlydifferent?Dosomeworkbeuerfordifferentobserva'onsets?Cankernelfilterdealbeuerwithdis'nctbimodalityofLorenz63?
Withproperresampling,thisshouldbethecase.Somebodyclevercouldprobablymakethiswork.
DARTTutorialSec'on6:Slide77
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
DARTTutorialSec'on6:Slide78