DART LAB Section02...DART_LAB Tutorial Section 2:How should observations impact an unobservedstate...

TheNationalCenterforAtmosphericResearchissponsoredbytheNationalScienceFoundation.Anyopinions,findingsandconclusionsorrecommendationsexpressedinthispublicationarethoseoftheauthor(s)anddonotnecessarilyreflecttheviewsoftheNationalScienceFoundation.

©UCAR2014

DART_LABTutorialSection2: Howshouldobservationsimpactanunobserved statevariable?

Multivariateassimilation.

Singleobservedvariable,singleunobservedvariable.

DART_LAB Section2:2 of45

Sofar,wehaveaknownobservationlikelihoodforasinglevariable.

Now,supposethepriorhasanadditionalvariable…

Wewillexaminehowensemblemembersupdatetheadditionalvariable.

Basicmethodgeneralizestoanynumberofadditionalvariables.

Ensemblefilters:Updatingadditionalpriorstatevariables

3

3.5

4

4.5

5

Unob

serv

ed S

tate

Var

iabl

e

−2 0 2 4Observed Variable

Assumethatallweknowisthepriorjointdistribution.

Onevariableisobserved.

Whatshouldhappentotheunobservedvariable?



33.5

44.5

5

Unob

s.




Updateobservedvariablewithoneofthepreviousmethods.






33.5

44.5

5

Unob

s.






Computeincrementsforpriorensemblemembersofobservedvariable.

33.5

44.5

5

Unob

s.


Increments






33.5

44.5

5

Unob

s.


Increments



Usingonlyincrementsguaranteesthatifobservationhadnoimpactonobservedvariable,theunobservedvariableisunchanged.

Highlydesirable!

33.5

44.5

5

Unob

s.


Increments




Howshouldtheunobservedvariablebeimpacted?

1st choice:leastsquares

Equivalenttolinearregression.

Sameasassumingbinormal prior.

3

3.5

4

4.5

5

Unob

serv

ed S

tate

Var

iabl

e


Increments



Havejointpriordistributionoftwovariables.

Howshouldtheunobservedvariablebeimpacted?

1st choice:leastsquares

Beginbyfindingleastsquaresfit.

3

3.5

4

4.5

5

Unob

serv

ed S

tate

Var

iabl

e


Increments




Next,regresstheobservedvariableincrementsontoincrementsfortheunobservedvariable.

Equivalenttofirstfindingimageofincrementinjointspace.

3

3.5

4

4.5

5

Unob

serv

ed S

tate

Var

iabl

e


Increments






3

3.5

4

4.5

5

Unob

serv

ed S

tate

Var

iabl

e


Increments




Regression:Equivalenttofirstfindingimageofincrementinjointspace.

Thenprojectingfromjointspaceontounobservedpriors.


3

3.5

4

4.5

5

Unob

serv

ed S

tate

Var

iabl

e


Increments






3

3.5

4

4.5

5

Unob

serv

ed S

tate

Var

iabl

e


Increments


Nowhaveanupdated(posterior)ensemblefortheunobservedvariable.

FittingGaussiansshowsthatmeanandvariancehavechanged.

Otherfeaturesofthepriordistributionmayalsohavechanged.3

3.5

4

4.5

5

Unob

serv

ed S

tate

Var

iabl

e

−2 0 2 4Obs.We’veexpandedthisplot.Same

informationaspreviousslides.Compressedthesetwo.Compressedthesetwo.






3.5

4

4.5

5

Unob

serv

ed S

tate

Var

iabl

e

Prior State Fit

−2 0 2 4Obs.






3.5

4

4.5

5

Unob

serv

ed S

tate

Var

iabl

e

Prior State Fit

Posterior Fit

−2 0 2 4Obs.



CRITICALPOINT:

Sinceimpactonunobservedvariableissimplyalinearregression,candothisINDEPENDENTLYforanynumberofunobservedvariables!

CouldalsodomanyatonceusingmatrixalgebraasintraditionalKalmanFilter.

3

3.5

4

4.5

5

Unob

serv

ed S

tate

Var

iabl

e

Prior State Fit

Posterior Fit

−2 0 2 4Obs.


Anderson,J.L.,2003:Alocalleastsquaresframeworkforensemblefiltering.Mon.Wea.Rev.,131,634-642

Matlab Hands-On:twod_ensemblePurpose:Explorehowanunobservedstatevariableisupdatedbyanobservationofanotherstatevariable.


Matlab Hands-On:twod_ensemble


Bivariateensembleplotwithprojectedmarginals forobserved,unobservedvariables.

Detailedplotforobservedvariable;sameasoned_ensemble.

Controlobservationvalueanderror;sameasoned_ensemble.

Startcreatinganensemble.Seenextslide.



Startcreatinganensemble.Movecursorandclickinthisframetocreateensemblemembers.Clickoutsidethisframewhenallmembersarecreated.


Explorations:• Createensemblemembersthatarenearlyonaline.Explorehowthe

unobservedvariableisupdated.

• Whathappensfornearlyuncorrelatedobservedandunobservedvariables?Createaroundishcloudofpointsfortheprior.

• Whathappenswithatwo-dimensionalbimodaldistribution?

• Trypriorensembleswithvarioustypesofoutliers.


SchematicofanEnsembleFilterforGeophysicalDataAssimilation

Ensemblestateestimateafterusingpreviousobservation(analysis)

Ensemblestateattimeofnextobservation(prior)

1. Usemodeltoadvanceensemble (3membershere)totimeatwhichnextobservationbecomesavailable.


HowanEnsembleFilterWorksforGeophysicalDataAssimilation

2. Getpriorensemblesampleofobservation,y = h(x),byapplyingforwardoperatorh toeachensemblemember.

Theory:observationsfrominstrumentswithuncorrelatederrorscanbedonesequentially.


Houtekamer,P.L.andH.L.Mitchell,2001:AsequentialensembleKalman filterforatmosphericdataassimilation.Mon.Wea.Rev.,129,123-137


3. Getobservedvalue andobservationalerrordistributionfromobservingsystem.



4. Findtheincrements forthepriorobservationensemble(thisisascalarproblemforuncorrelatedobservationerrors).

Note:Differencebetweenvariousensemblefiltermethodsisprimarilyinobservationincrementcalculation.



5. Useensemblesamplesofy andeachstatevariabletolinearlyregressobservationincrementsontostatevariableincrements.

Theory:impactofobservationincrementsoneachstatevariablecanbehandledindependently!



6. Whenallensemblemembersforeachstatevariableareupdated,thereisanewanalysis.Integratetotimeofnextobservation…


Matlab Hands-On:run_lorenz_63Purpose:ExplorebehaviorofensembleKalman filtersinalow-order,chaoticdynamicalsystem,the3-variableLorenz1963model.


‘Local’domain,localtimeframe

Fulldomain,fulltimeframe.

Thesecontrolsworkthesameasforoned_model.

Assimilationcanbeturnedoff,justdoesmodeladvances.

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

*


Observation

Matlab Hands-On:run_lorenz_63


Youcanusematlab toolstomodifyplots.

Here,theRotate3Dtoolshasbeenusedtochangetheangleofviewofboththelocalandglobalviewsoftheassimilation.

Matlab Hands-On:run_lorenz_63

Explorations:

• SelectStartAutoRunandwatchtheevolutionoftheensemble.Trytounderstandhowtheensemblespreadsout.

• RestarttheGUIandselectEAKF.Doindividualadvancesandassimilationsandobservethebehavior.

• Dosomeautorunswithassimilationturnedon.

• Explorehowdifferentareasoftheattractorhavedifferentassimilationbehavior.


Matlab Hands-On:run_lorenz_96Purpose:Explorethebehaviorofensemblefiltersina40-variablechaoticdynamicalsystem;theLorenz1996model.


Thesecontrolsworkthesameaslorenz_63.

Rootmeansquareerrorfromtruthandensemblespreadasfunctionoftime.

Priorandposteriorrankhistograms.

Parametersforensemblefilter.

Modelforcing.

Ensembleofmodelcontours(spaghettiplot).

Matlab Hands-On:run_lorenz_96StartaFreeRunoftheensemble(NoAssimilation).Aftersometime,theminuteperturbationsintheoriginalstatesleadtovisiblydifferentmodelstates.

Noposteriorinafreerun.

Takesawhileforthesmallinitialperturbationstogrow.

Ensemblesizeis20,sothereare21binsintherankhistogram.


Matlab Hands-On:run_lorenz_961) Stopthefreerunaftersometime.2) TurnontheEAKF3) Advancemodel,assimilate…

Yourfigureswillbedifferentdependingonyoursettings.That’sOK.

Note:All40statevariablesareobserved.Observationerrorstandarddeviationis4.0


Matlab Hands-On: run_lorenz_96

Explorations:• Doanextendedfreeruntoseeerrorgrowthintheensemble.

Howlongdoesittaketosaturate?• SelectEAKFandexplorehowtheassimilationworks.• Tryaddinginflation(maybe1.4)andrepeat.


DART LAB Section02...DART_LAB Tutorial Section 2:How should observations impact an unobservedstate...

Documents

Transcript of DART LAB Section02...DART_LAB Tutorial Section 2:How should observations impact an unobservedstate...