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TheNa'onalCenterforAtmosphericResearchissponsoredbytheNa'onalScienceFounda'on.Anyopinions,findingsandconclusionsorrecommenda'onsexpressedinthispublica'onarethoseoftheauthor(s)anddonotnecessarilyreflecttheviewsoftheNa'onalScienceFounda'on.
©UCAR
DARTTutorialSec'on9:MoreonDealingwithError:Infla'on
SomeErrorSourcesinEnsembleFilters
****
1.ModelError
2.herrors;Representa'veness
3.‘Gross’Obs.Error 4.SamplingError;GaussianAssump'on
5.SamplingError;AssumingLinearSta's'calRela'on
tk
tk+1
tk+2h h h
DARTTutorialSec'on9:Slide2
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.
DARTTutorialSec'on9:Slide4
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.
DARTTutorialSec'on9:Slide5
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
DARTTutorialSec'on9:Slide6
Model/FilterError:FilterDivergenceandVarianceInfla'on
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
"TRUE" Prior PDF
1.Historyofobserva'onsandphysicalsystem=>‘true’distribu'on.
DARTTutorialSec'on9:Slide7
Model/FilterError:FilterDivergenceandVarianceInfla'on
−4 −2 0 2 40
0.2
0.4
0.6
0.8
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.
DARTTutorialSec'on9:Slide8
Model/FilterError:FilterDivergenceandVarianceInfla'on
−4 −2 0 2 40
0.2
0.4
0.6
0.8
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.
DARTTutorialSec'on9:Slide9
PhysicalSpaceVarianceInfla'on
Inflateallstatevariablesbysameamountbeforeassimila'on.Capabili'es:1. Canbeeffec'veforavarietyofmodels.2. Canmaintainlinearbalances.3. Staysonlocalflatmanifolds.
4. Simpleandcheap.
Liabili'es:1. Statevariablesnotconstrainedbyobserva'onscan‘blowup’.
ForinstanceunobservedregionsnearthetopofAGCMs.2. Magnitudeofλnormallyselectedbytrialanderror.
DARTTutorialSec'on9:Slide10
PhysicalSpaceVarianceInfla'oninLorenz63
−20
0
20 −20
0
2010
20
30
40
Observa'onoutsideprior:dangeroffilterdivergence.
Observa'oninred.Priorensembleingreen.
DARTTutorialSec'on9:Slide11
PhysicalSpaceVarianceInfla'oninLorenz63
−20
0
20 −20
0
2010
20
30
40
A\erinfla'ng,observa'onisinpriorcloud:filterdivergenceavoided.
Observa'oninred.Priorensembleingreen.
Inflatedensembleinmagenta.
DARTTutorialSec'on9:Slide12
PhysicalSpaceVarianceInfla'oninLorenz63
−20
0
20 −20
0
2010
20
30
40
Priordistribu'onissignificantly‘curved’.
Observa'oninred.Priorensembleingreen.
DARTTutorialSec'on9:Slide13
PhysicalSpaceVarianceInfla'oninLorenz63
−20
0
20 −20
0
2010
20
30
40
Inflatedprioroutsidealractor.Posteriorwillalsobeoffalractor.
Observa'oninred.Priorensembleingreen.
Inflatedensembleinmagenta.
Canleadtotransientoff-alractorbehavioror…Model‘blow-up’.
DARTTutorialSec'on9:Slide14
Basiccontrolofinfla'oninDARTisin&filter_nml
inf_flavor =0, 0,inf_ini'al_from_restart =.false., .false.,inf_sd_ini'al_from_restart =.false., .false.,inf_determinis'c =.true., .true.,inf_ini'al =1.0, 1.0,inf_sd_ini'al =0.0, 0.0,inf_damping =1.0, 1.0,inf_lower_bound =1.0, 1.0,inf_upper_bound =1000000.0, 1000000.0,inf_sd_lower_bound =0.0, 0.0,
BeforeAssimila'on
A\erAssimila'on
Flavor: 0=>NONE 1=>deprecated 2,3=>physicalspace
Infla'onValue
Ini'ally,we’llchangeinf_flavorandinf_ini*alinfirstcolumn.DARTTutorialSec'on9:Slide15
priorinfla'oncolumn
posteriorinfla'oncolumn
&assim_tools_nml filter_kind = 1 cutoff = 1000000.0 spread_restoration = .false. sampling_error_correction = .false. …&filter_nml ens_size = 20 perturb_from_single_instance = .false. … inf_flavor = 3, 0 inf_initial_from_restart = .false., .false. inf_sd_initial_from_restart = .false., .false. inf_initial = 1.0, 1.0 inf_sd_initial = 0.0, 0.0 inf_damping = 1.0, 1.0 inf_lower_bound = 1.0, 1.0 inf_upper_bound = 1000000.0, 1000000.0 inf_sd_lower_bound = 0.0, 0.0
Physicalspacevarianceinfla'oninLorenz96
Setinf_flavorto3tousestatespaceinfla'on.Inthefirstcolumn,setinf_ini*altovalueslike1.05,1.08,1.10
DARTTutorialSec'on9:Slide16
models/lorenz_96/work/Trysomevaluesandseewhathappenstoassimila'onswithLorenz96.
Varianceinfla'oninobserva'onspace
1.Forobservedvariable,havees'mateofprior-observedinconsistency.
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Prior PDF
S.D.
Obs. Likelihood
S.D.
NotcurrentlysupportedinDARTManhalan.
DARTTutorialSec'on9:Slide17
Varianceinfla'oninobserva'onspace
1.Forobservedvariable,havees'mateofprior-observedinconsistency.2.Expected(prior_mean–observa'on)=
Assumesthatpriorandobserva'onaresupposedtobeunbiased.Isitmodelerrororrandomchance?
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Prior PDF
S.D.
Obs. Likelihood
S.D.Expected Separation
Actual 4.714 SDs
σprior2 +σobs
2
DARTTutorialSec'on9:Slide18
Varianceinfla'oninobserva'onspace
1.Forobservedvariable,havees'mateofprior-observedinconsistency.2.Expected(prior_mean–observa'on)=3.Infla'ngincreasesexpectedsepara'on.
Increases‘apparent’consistencybetweenpriorandobserva'on.
−4 −2 0 2 40
0.2
0.4
0.6
0.8
Prob
abilit
y
Prior PDF Obs. Likelihood
S.D.Inflated S.D. Expected Separation
Actual 3.698 SDs
σprior2 +σobs
2
DARTTutorialSec'on9:Slide19
Varianceinfla'oninobserva'onspace:Lorenz96
Varianceinfla'oninobserva'onspacenotcurrentlysupported.TrysomevaluesandseewhathappenstoLorenz96assimila'on.
Setinf_flavor=1,observa'onspaceinfla'oninfirstcolumn.TrysomevaluesandseewhathappenstoLorenz96assimila'on.
Setinf_ini*altovalueslike1.05,1.08,1.10infirstcolumn.Makesurethatcutoff=10000000andens_size=20.
(Thesewereserngsthatdivergedwithoutinfla'on)
DARTTutorialSec'on9:Slide20
1. FilteringForaOneVariableSystem2. TheDARTDirectoryTree3. DARTRun>meControlandDocumenta>on4. Howshouldobserva>onsofastatevariableimpactanunobservedstatevariable?
Mul>variateassimila>on.5. ComprehensiveFilteringTheory:Non-Iden>tyObserva>onsandtheJointPhaseSpace6. OtherUpdatesforAnObservedVariable7. SomeAddi>onalLow-OrderModels8. DealingwithSamplingError9. MoreonDealingwithError;Infla>on10. RegressionandNonlinearEffects11. Crea>ngDARTExecutables12. Adap>veInfla>on13. HierarchicalGroupFiltersandLocaliza>on14. QualityControl15. DARTExperiments:ControlandDesign16. Diagnos>cOutput17. Crea>ngObserva>onSequences18. LostinPhaseSpace:TheChallengeofNotKnowingtheTruth19. DART-CompliantModelsandMakingModelsCompliant20. ModelParameterEs>ma>on21. Observa>onTypesandObservingSystemDesign22. ParallelAlgorithmImplementa>on23. Loca'onmoduledesign(notavailable)24. Fixedlagsmoother(notavailable)25. Asimple1Dadvec>onmodel:TracerDataAssimila>on
DARTTutorialIndextoSec'ons
DARTTutorialSec'on9:Slide21
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