dart lab section02...dart_lab tutorial section 2:how should observations impact an unobservedstate...
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![Page 1: DART LAB Section02...DART_LAB Tutorial Section 2:How should observations impact an unobservedstate variable? Multivariate assimilation. Single observed variable, single unobserved](https://reader035.vdocuments.mx/reader035/viewer/2022081614/5fbf757fc3dd4a51e966b317/html5/thumbnails/1.jpg)
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DART_LABTutorialSection2: Howshouldobservationsimpactanunobserved statevariable?
Multivariateassimilation.
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Singleobservedvariable,singleunobservedvariable.
DART_LAB Section2:2 of45
Sofar,wehaveaknownobservationlikelihoodforasinglevariable.
Now,supposethepriorhasanadditionalvariable…
Wewillexaminehowensemblemembersupdatetheadditionalvariable.
Basicmethodgeneralizestoanynumberofadditionalvariables.
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Ensemblefilters:Updatingadditionalpriorstatevariables
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−2 0 2 4Observed Variable
Assumethatallweknowisthepriorjointdistribution.
Onevariableisobserved.
Whatshouldhappentotheunobservedvariable?
DART_LAB Section2:3 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
33.5
44.5
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Unob
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−2 0 2 4Observed Variable
Assumethatallweknowisthepriorjointdistribution.
Onevariableisobserved.
Updateobservedvariablewithoneofthepreviousmethods.
DART_LAB Section2:4 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Assumethatallweknowisthepriorjointdistribution.
Onevariableisobserved.
Updateobservedvariablewithoneofthepreviousmethods.
33.5
44.5
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s.
−2 0 2 4Observed Variable
DART_LAB Section2:5 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Assumethatallweknowisthepriorjointdistribution.
Onevariableisobserved.
Updateobservedvariablewithoneofthepreviousmethods.
33.5
44.5
5
Unob
s.
−2 0 2 4Observed Variable
DART_LAB Section2:6 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Assumethatallweknowisthepriorjointdistribution.
Onevariableisobserved.
Computeincrementsforpriorensemblemembersofobservedvariable.
33.5
44.5
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Unob
s.
−2 0 2 4Observed Variable
Increments
DART_LAB Section2:7 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Assumethatallweknowisthepriorjointdistribution.
Onevariableisobserved.
Computeincrementsforpriorensemblemembersofobservedvariable.
33.5
44.5
5
Unob
s.
−2 0 2 4Observed Variable
Increments
DART_LAB Section2:8 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Assumethatallweknowisthepriorjointdistribution.
Onevariableisobserved.
Computeincrementsforpriorensemblemembersofobservedvariable.
33.5
44.5
5
Unob
s.
−2 0 2 4Observed Variable
Increments
DART_LAB Section2:9 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Assumethatallweknowisthepriorjointdistribution.
Onevariableisobserved.
Computeincrementsforpriorensemblemembersofobservedvariable.
33.5
44.5
5
Unob
s.
−2 0 2 4Observed Variable
Increments
DART_LAB Section2:10 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Assumethatallweknowisthepriorjointdistribution.
Onevariableisobserved.
Computeincrementsforpriorensemblemembersofobservedvariable.
33.5
44.5
5
Unob
s.
−2 0 2 4Observed Variable
Increments
DART_LAB Section2:11 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Usingonlyincrementsguaranteesthatifobservationhadnoimpactonobservedvariable,theunobservedvariableisunchanged.
Highlydesirable!
33.5
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DART_LAB Section2:12 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Assumethatallweknowisthepriorjointdistribution.
Howshouldtheunobservedvariablebeimpacted?
1st choice:leastsquares
Equivalenttolinearregression.
Sameasassumingbinormal prior.
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DART_LAB Section2:13 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Havejointpriordistributionoftwovariables.
Howshouldtheunobservedvariablebeimpacted?
1st choice:leastsquares
Beginbyfindingleastsquaresfit.
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DART_LAB Section2:14 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Havejointpriordistributionoftwovariables.
Next,regresstheobservedvariableincrementsontoincrementsfortheunobservedvariable.
Equivalenttofirstfindingimageofincrementinjointspace.
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DART_LAB Section2:15 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Havejointpriordistributionoftwovariables.
Next,regresstheobservedvariableincrementsontoincrementsfortheunobservedvariable.
Equivalenttofirstfindingimageofincrementinjointspace.
3
3.5
4
4.5
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Unob
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−2 0 2 4Observed Variable
Increments
DART_LAB Section2:16 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Havejointpriordistributionoftwovariables.
Next,regresstheobservedvariableincrementsontoincrementsfortheunobservedvariable.
Equivalenttofirstfindingimageofincrementinjointspace.
3
3.5
4
4.5
5
Unob
serv
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tate
Var
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−2 0 2 4Observed Variable
Increments
DART_LAB Section2:17 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Havejointpriordistributionoftwovariables.
Next,regresstheobservedvariableincrementsontoincrementsfortheunobservedvariable.
Equivalenttofirstfindingimageofincrementinjointspace.
3
3.5
4
4.5
5
Unob
serv
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tate
Var
iabl
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−2 0 2 4Observed Variable
Increments
DART_LAB Section2:18 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Havejointpriordistributionoftwovariables.
Next,regresstheobservedvariableincrementsontoincrementsfortheunobservedvariable.
Equivalenttofirstfindingimageofincrementinjointspace.
3
3.5
4
4.5
5
Unob
serv
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tate
Var
iabl
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−2 0 2 4Observed Variable
Increments
DART_LAB Section2:19 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Havejointpriordistributionoftwovariables.
Regression:Equivalenttofirstfindingimageofincrementinjointspace.
Thenprojectingfromjointspaceontounobservedpriors.
DART_LAB Section2:20 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Havejointpriordistributionoftwovariables.
Regression:Equivalenttofirstfindingimageofincrementinjointspace.
Thenprojectingfromjointspaceontounobservedpriors.
DART_LAB Section2:21 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Havejointpriordistributionoftwovariables.
Regression:Equivalenttofirstfindingimageofincrementinjointspace.
Thenprojectingfromjointspaceontounobservedpriors.
DART_LAB Section2:22 of45
3
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Unob
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−2 0 2 4Observed Variable
Increments
![Page 23: DART LAB Section02...DART_LAB Tutorial Section 2:How should observations impact an unobservedstate variable? Multivariate assimilation. Single observed variable, single unobserved](https://reader035.vdocuments.mx/reader035/viewer/2022081614/5fbf757fc3dd4a51e966b317/html5/thumbnails/23.jpg)
Ensemblefilters:Updatingadditionalpriorstatevariables
Havejointpriordistributionoftwovariables.
Regression:Equivalenttofirstfindingimageofincrementinjointspace.
Thenprojectingfromjointspaceontounobservedpriors.
DART_LAB Section2:23 of45
3
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Unob
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−2 0 2 4Observed Variable
Increments
![Page 24: DART LAB Section02...DART_LAB Tutorial Section 2:How should observations impact an unobservedstate variable? Multivariate assimilation. Single observed variable, single unobserved](https://reader035.vdocuments.mx/reader035/viewer/2022081614/5fbf757fc3dd4a51e966b317/html5/thumbnails/24.jpg)
Ensemblefilters:Updatingadditionalpriorstatevariables
Havejointpriordistributionoftwovariables.
Regression:Equivalenttofirstfindingimageofincrementinjointspace.
Thenprojectingfromjointspaceontounobservedpriors.
DART_LAB Section2:24 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Nowhaveanupdated(posterior)ensemblefortheunobservedvariable.
FittingGaussiansshowsthatmeanandvariancehavechanged.
Otherfeaturesofthepriordistributionmayalsohavechanged.3
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−2 0 2 4Obs.We’veexpandedthisplot.Same
informationaspreviousslides.Compressedthesetwo.Compressedthesetwo.
DART_LAB Section2:25 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Nowhaveanupdated(posterior)ensemblefortheunobservedvariable.
FittingGaussiansshowsthatmeanandvariancehavechanged.
Otherfeaturesofthepriordistributionmayalsohavechanged.3
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Prior State Fit
−2 0 2 4Obs.
DART_LAB Section2:26 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
Nowhaveanupdated(posterior)ensemblefortheunobservedvariable.
FittingGaussiansshowsthatmeanandvariancehavechanged.
Otherfeaturesofthepriordistributionmayalsohavechanged.3
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4
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Prior State Fit
Posterior Fit
−2 0 2 4Obs.
DART_LAB Section2:27 of45
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Ensemblefilters:Updatingadditionalpriorstatevariables
CRITICALPOINT:
Sinceimpactonunobservedvariableissimplyalinearregression,candothisINDEPENDENTLYforanynumberofunobservedvariables!
CouldalsodomanyatonceusingmatrixalgebraasintraditionalKalmanFilter.
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DART_LAB Section2:28 of45
Anderson,J.L.,2003:Alocalleastsquaresframeworkforensemblefiltering.Mon.Wea.Rev.,131,634-642
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Matlab Hands-On:twod_ensemblePurpose:Explorehowanunobservedstatevariableisupdatedbyanobservationofanotherstatevariable.
DART_LAB Section2:29 of45
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Matlab Hands-On:twod_ensemble
DART_LAB Section2:30 of45
Bivariateensembleplotwithprojectedmarginals forobserved,unobservedvariables.
Detailedplotforobservedvariable;sameasoned_ensemble.
Controlobservationvalueanderror;sameasoned_ensemble.
Startcreatinganensemble.Seenextslide.
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Matlab Hands-On:twod_ensemble
DART_LAB Section2:31 of45
Startcreatinganensemble.Movecursorandclickinthisframetocreateensemblemembers.Clickoutsidethisframewhenallmembersarecreated.
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Matlab Hands-On:twod_ensemble
Explorations:• Createensemblemembersthatarenearlyonaline.Explorehowthe
unobservedvariableisupdated.
• Whathappensfornearlyuncorrelatedobservedandunobservedvariables?Createaroundishcloudofpointsfortheprior.
• Whathappenswithatwo-dimensionalbimodaldistribution?
• Trypriorensembleswithvarioustypesofoutliers.
DART_LAB Section2:32 of45
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SchematicofanEnsembleFilterforGeophysicalDataAssimilation
Ensemblestateestimateafterusingpreviousobservation(analysis)
Ensemblestateattimeofnextobservation(prior)
1. Usemodeltoadvanceensemble (3membershere)totimeatwhichnextobservationbecomesavailable.
DART_LAB Section2:33 of45
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HowanEnsembleFilterWorksforGeophysicalDataAssimilation
2. Getpriorensemblesampleofobservation,y = h(x),byapplyingforwardoperatorh toeachensemblemember.
Theory:observationsfrominstrumentswithuncorrelatederrorscanbedonesequentially.
DART_LAB Section2:34 of45
Houtekamer,P.L.andH.L.Mitchell,2001:AsequentialensembleKalman filterforatmosphericdataassimilation.Mon.Wea.Rev.,129,123-137
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HowanEnsembleFilterWorksforGeophysicalDataAssimilation
3. Getobservedvalue andobservationalerrordistributionfromobservingsystem.
DART_LAB Section2:35 of45
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HowanEnsembleFilterWorksforGeophysicalDataAssimilation
4. Findtheincrements forthepriorobservationensemble(thisisascalarproblemforuncorrelatedobservationerrors).
Note:Differencebetweenvariousensemblefiltermethodsisprimarilyinobservationincrementcalculation.
DART_LAB Section2:36 of45
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HowanEnsembleFilterWorksforGeophysicalDataAssimilation
5. Useensemblesamplesofy andeachstatevariabletolinearlyregressobservationincrementsontostatevariableincrements.
Theory:impactofobservationincrementsoneachstatevariablecanbehandledindependently!
DART_LAB Section2:37 of45
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HowanEnsembleFilterWorksforGeophysicalDataAssimilation
6. Whenallensemblemembersforeachstatevariableareupdated,thereisanewanalysis.Integratetotimeofnextobservation…
DART_LAB Section2:38 of45
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Matlab Hands-On:run_lorenz_63Purpose:ExplorebehaviorofensembleKalman filtersinalow-order,chaoticdynamicalsystem,the3-variableLorenz1963model.
DART_LAB Section2:39 of45
‘Local’domain,localtimeframe
Fulldomain,fulltimeframe.
Thesecontrolsworkthesameasforoned_model.
Assimilationcanbeturnedoff,justdoesmodeladvances.
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Matlab Hands-On:run_lorenz_63Bothpanelsshowtimeevolutionoftruestate(black).20ensemblemembersareshowningreenintopwindow.
At each observation time, the three components of the truth are 'observed’ by adding a random draw from a standard normal distribution to the true value.
Increments
*
DART_LAB Section2:40 of45
Observation
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Matlab Hands-On:run_lorenz_63
DART_LAB Section2:41 of45
Youcanusematlab toolstomodifyplots.
Here,theRotate3Dtoolshasbeenusedtochangetheangleofviewofboththelocalandglobalviewsoftheassimilation.
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Matlab Hands-On:run_lorenz_63
Explorations:
• SelectStartAutoRunandwatchtheevolutionoftheensemble.Trytounderstandhowtheensemblespreadsout.
• RestarttheGUIandselectEAKF.Doindividualadvancesandassimilationsandobservethebehavior.
• Dosomeautorunswithassimilationturnedon.
• Explorehowdifferentareasoftheattractorhavedifferentassimilationbehavior.
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Matlab Hands-On:run_lorenz_96Purpose:Explorethebehaviorofensemblefiltersina40-variablechaoticdynamicalsystem;theLorenz1996model.
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Thesecontrolsworkthesameaslorenz_63.
Rootmeansquareerrorfromtruthandensemblespreadasfunctionoftime.
Priorandposteriorrankhistograms.
Parametersforensemblefilter.
Modelforcing.
Ensembleofmodelcontours(spaghettiplot).
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Matlab Hands-On:run_lorenz_96StartaFreeRunoftheensemble(NoAssimilation).Aftersometime,theminuteperturbationsintheoriginalstatesleadtovisiblydifferentmodelstates.
Noposteriorinafreerun.
Takesawhileforthesmallinitialperturbationstogrow.
Ensemblesizeis20,sothereare21binsintherankhistogram.
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Matlab Hands-On:run_lorenz_961) Stopthefreerunaftersometime.2) TurnontheEAKF3) Advancemodel,assimilate…
Yourfigureswillbedifferentdependingonyoursettings.That’sOK.
Note:All40statevariablesareobserved.Observationerrorstandarddeviationis4.0
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Matlab Hands-On: run_lorenz_96
Explorations:• Doanextendedfreeruntoseeerrorgrowthintheensemble.
Howlongdoesittaketosaturate?• SelectEAKFandexplorehowtheassimilationworks.• Tryaddinginflation(maybe1.4)andrepeat.
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