EnKF radianceassimilation

JeffWhitaker1 andLiliLei21NOAA/ESRL/PSD

2CIRESandNanjingUniversity

Motivation(1)

• Hypothesis:4DEnVar(non-hybrid)andEnKFshouldperformsimilarlyifall‘extra’constraintsturnedoffinVar solver.

• Experiment:T254singleresolution4DEnVar(nostaticB,balanceconstraint)vs ‘pure’EnKF(80members,operationallocalizationsettings).

Motivation(2)4DEnVarvs EnKF,noradiances

RMSinnovationsforbackground

Virtuallynodifference,EnKF (red)perhapsslightlybetterforhumidity

Motivation(2)4DEnVarvs EnKF,includingradiances

RMSinnovationsforbackground

4DEnVar(blue)slightlybetter,esp inSH.Why?Hypothesis: Differenceinvertical localization

ObservationandModel-SpaceLocalization

Modelspace:• state-space covariances aretapered.• Involvesdistances betweenstatevariablesonly.• H applied after.• EnVar algorithmsusethisform.

Observationspace:• Computed afterH applied. • Involvesdistances betweenstateandobservationspacequantities. •Muchsimpler toimplement inEnKF systems.

Motivation(3)4DEnVarvs EnKF,radiancesonly

RMSinnovationsforbackground

4DEnVar(blue)nowsignificantly better(althoughusingdeeper localization(green)forEnKF helpsquiteabit)

Experimentswithasimplemodel

• 1-dKuramoto-Sivashinsky equation(https://www.encyclopediaofmath.org/index.php/Kuramoto-Sivashinsky_equation),oneofthesimplestPDEsthatexhibitsspatio-temporalchaos.

• u_t +u*u_x +u_xx +d*u_xxxx =0,periodicBCson[0,2*pi*L].– Energyentersthesystematlongwavelengths viau_xx (anunstable

diffusion term)– cascadestoshortwavelengthsduetothenonlinearity u*u_x– dissipates viad*u_xxxx– Solvedwithspectralmethod,usingNFouriercollocationpoints.

• Pythoncodeavailableathttps://github.com/jswhit/pyks.git

Naturerun:L=16,N=128,dt=0.5,d=1(semi-implicitRK3scheme)

Dataassimilationexperiments

• 10members,assimilationevery4timesteps(2timeunits).

• Forwardobservationoperator(H)includesanaveragingkernel,eitherGaussianorboxcar(runningaverage).R=0.1

• SerialEnSRF witheitherobservationspaceormodelspace localization.Observation‘location’assumedtobecenterofaveragingkernel.

• MultiplicativeinflationusingHodyss etal2016(dx.doi.org/10.1175/MWR-D-15-0329.1)algorithm.

‘Modulatedensemble’approach tomodel-space localizationintheEnKF

Modelspacelocalization isPloc =ρ ○ Psample (○ denoteselement-wiseproduct)

Let Psample = XXT,ρ =LLT,thenPloc =LLT○ XXT=ZZT,whereZ =L Δ XandΔdenotes ‘modulation product’(BishopandHodyss, 20081)

If X has N columns (ens.members) andL has M columns (eigenvectors), thenZ=[[X1 ○ L1, X2 ○ L1,…, XN ○ L1],[X1 ○ L2, X2 ○ L2,…, XN ○ L2],…, [X1 ○ LM, X2 ○ LM,…, XN ○ LM]]

1doi: 10.1111/j.1600-0870.2008.00372.x

RMSanalysiserrorasafunctionoflocalizationscale

•modelspacelocalizationworksbetterforBoxcarkernel.•ob spacelocalizationworksbetter(forshortlocalizationscales)forGaussiankernel.•Why?

Meancorrelationbetweenob priorsandstatepriors(usinglocalization=50)

•correlationbetweenstateandob spaceismaximumatedgeofBoxcarkernel(notcenter).

Whyisob spacelocalizationbetterinsomecircumstances?

•Localizationapplied toPbremovessomeofnegativesidelobes.•H operatorapplied tolocalizedcovariancethenproducestoolargeavaluefor K..•SeeLeiandWhitaker2015:DOI:dx.doi.org/10.1175/MWR-D-14-00413.1•Symptom(seenwithsomesatellite obs):single-obexperimentwith/withoutlocalization, increment islargerwith localizationthanwithout.

Conclusions(simplemodelexpts)

• Modelspacelocalizationperformsbetterwhencorrelationbetweenob priorandmodelpriorsisnotmaximumatcenterofaveragingkernel(nominal’oblocation’).AdvantageisevenlargerthanshownbyCampell etal2009(dx.doi.org/10.1175/2009MWR3017.1).– Obspacelocalizationwith‘empiricallocalizationfunctions’(ELFs)canworkinthiscircumstance…

ExampleofEmpiricalLocalizationFunction(ELF)withBoxcarKernelH (Lorenz40variablemodel)fromAndersonandLei2013:

http://dx.doi.org/10.1175/MWR-D-12-00330.1

Conclusions(simplemodelexpts)

• Modelspacelocalizationperformsbetterwhencorrelationbetweenob priorandmodelpriorsisnotmaximumatcenterofaveragingkernel(nominal’ob location’).

• Ifcorrelationismaximumatcenterofaveragingkernel,ob spacelocalizationcanperformbetteriftherearenegativesidelobesincorrelationandlocalizationscaleisshort(LeiandWhitaker2015:DOI:dx.doi.org/10.1175/MWR-D-14-00413.1).

EnKF radianceassimilationwithmodel-spacelocalization

• Asbefore,withGFST25480memberensemble,butusingmodelspacelocalization

• ‘Modulatedensemble’approachusedtoimplementmodel-spacelocalization intheverticalonly.

‘Modulatedensemble’approachtomodel-spaceverticallocalization• Assume localization isseparable. Performhorizontal localization inob

space,vertical inmodel space.• Truncatethevertical localizationmatrix,retainingtheM eigenvectors

thatexplain90-95%ofthevariance (M isO(10)forthecurrentoperational configuration,N=80).The‘modulated ensemble’ thencontains MN members. - Horizontal localizationstillcomputed inobservation space.- TheEnKF algorithm isunchanged,exceptthatverticallocalizationisturned

off,and‘modulated’ensemble (inmodelandobservation space) isingestedinsteadof‘raw’ensemble.

- OnlytheoriginalNensemblemembers areupdatedintheDA,usingcovariances derived fromthefullMN membermodulatedensemble.

- Notstraightforward toimplementinLETKF,sinceanalysisweightsapplytofullMNmember ensemble (notoriginalNmemberensemble).

Results(radiance-onlyassimilation)

• UsingmodelspacelocalizationintheEnKF improvestheuseofradiancedata.

• Performancesimilarto4DEnVar.

Howmanymembersareneededtoturnoffverticallocalization?

• Since12eigenvectorsofverticallocalizationmatrixexplain99%ofvariance,thissuggeststhat80*12=960membersshouldbesufficient.

• Wehaveruna960memberLETKFensemblewithoutverticallocalization(veryefficientinLETKF,sinceanalysisweightscanbecomputedforentirecolumnatonce).

4DEnVar(80members)vsLETKF(960members):radiancesonly

• Significantimprovementfromincreaseensemblesize/eliminationofverticallocalization.

• Performancesuperiorto4DEnVarwith80members.

Conclusions• CaremustbetakenwhenassimilatingradianceobservationsintheEnKF withobservation-spacelocalization.

• O(1000)membersshouldbeenoughtoobviatetheneedforverticallocalization.

• Modelspacelocalizationimprovestheassimilationofradianceobservations.CanbeimplementedinEnKFusing‘modulatedensembles’,butwithasignificantincreaseincost.– Alternately,empiricallyderivedlocalizationfunctions(ELFs,Leietal2016:dx.doi.org/10.1002/2016MS000627) foreachinstrument/channel.

ExamplefromLeietal2016:AMSU-AChannel9