The University of Reading

A singular vector perspective of 4D-Var:The spatial structure and evolution of

baroclinic weather systems

C. JohnsonNational Center for Atmospheric Research, Boulder CO, USA

B.J. Hoskins, N.K.NicholsSchool of Mathematics, Meteorology and Physics, The University of

Reading, Whiteknights, Reading UK

and

S.P. BallardMet Office, JCMM, Earley Gate, Reading UK

NUMERICAL ANALYSIS REPORT 4/05

Department of Mathematics

Abstract

Theextentto whichFour-DimensionalVariationaldataassimilation(4D-Var) is able

to useinformationaboutthetime-evolutionof theatmosphereto infer theverticalspatial

structureof baroclinicweathersystemsis investigated.Generalresultsarederivedusing

the singularvaluedecompositionof the 4D-Var observability matricesin an idealized

Eadymodelsetting.Theseresultsareconfirmedwith 4D-Varanalyses.

Theresultsshow that4D-Varperformswell atcorrectingtheerrorsthatwouldother-

wiserapidlycorruptaforecast.However, in afew cases,4D-Varmayaddarapidlygrow-

ing errorinsteadof correctingadecayingerror. Theability to extractthetime-evolution

informationcanbe maximizedby placing the observationsas far apartaspossiblein

time, andthe specificationof thecase-dependentbackgrounderror variancesis crucial

in beingableto projecttheobservationalinformationontoanalysisincrementsthatlead

to theappropriategrowth rate.

2

1. Intr oduction

Thedynamicalinstability of theatmospheremeansthatsmallperturbationsthatareintroducedinto

theflow maygrow rapidly. For example,theflow at mid-latitudesis baroclinicallyunstabledueto

the vertical shearassociatedwith the meridionaltemperaturegradient. This wave-instabilitypro-

vides the dominantmechanismfor disturbancesto develop into mid-latitudeweathersystems.In

suchdevelopment,theverticalspatialstructureof thedisturbanceplaysa fundamentalrole in gov-

erningthedevelopment.For example,thenormal-modeanalysisof simplelinearmodels(Charney,

1947;Eady,1949)showedthat the fastestgrowing structureexhibits a westward tilt with heightin

the pressurefield. This vertical tilt leadsto a processknown asself-developmentwherethe upper

andlower level wavesact to intensifyeachother, leadingto exponentialmodalgrowth. In contrast,

thefastestdecayingstructureexhibits aneastwardtilt with heightsothatthecirculationsassociated

with theupperandlower level wavesact to weakeneachother. More recentstudies(Farrell, 1982,

1984)showedthatit is possiblefor thegrowth rateof adisturbanceto exceedtheexponentialgrowth

of thefastestgrowing normalmodeover a limited periodof time. Suchdisturbancesmaybefound

by computingthesingularvectorsof thelinearmodel(Farrell,1989;BuizzaandPalmer,1995).The

spatialstructuresof thesedisturbancesarecharacterizedby localizedtilted interior potentialvortic-

ity (PV) structureswhich separateanduntilt undertheactionof the shear, leadingto amplification

(BadgerandHoskins,2001).Again, theverticalstructureof thedisturbanceis fundamentalfor this

rapidnon-modalgrowth.

The rapid growth of small disturbancesin a baroclinicallyunstableflow hastwo consequences

for dataassimilation.The first consequenceis that dataassimilationalgorithmsneedto be ableto

generateanalyseswith theverticalstructuresthatarenecessaryfor bothmodalgrowth anddecayand

alsofor non-modalgrowth. Thesecondconsequenceis thatsmallerrorsin the initial conditionsof

anumericalweatherforecastmayrapidlydevelopinto largeforecasterrors.It is thereforeimportant

thatthedataassimilationalgorithmis ableto correctsuchrapidlygrowing errors.

Algorithms suchasThree-DimensionalVariationaldataassimilation(3D-Var) (Courtieret al.,

1998),usingspecifiederrorcovariancematrices,generateananalysisby blendingtogetherobserva-

tionsneartheanalysistime with a backgroundstate.AlgorithmssuchasFour-DimensionalVaria-

3

tional dataassimilation(4D-Var) (Rabieret al., 2000)andEnsembleFilters(Evensen,1994;Tippett

etal.,2003;Lorenc,2003a,b)extendthismethodby usinga forecastmodelto link togetherobserva-

tionsthataredistributedin time andalsoto evolve thebackgrounderrorcovariancematrix (Lorenc,

1986). This meansthat observationsarecombinedwith dynamicallyevolved covariancesso that

4D-Var is ableto generatetheverticalstructuresthatareneededfor baroclinicgrowth. For example,

singleobservationexperimentsby Thépautet al. (1996)showedthat3D-Var analysisincrementsdo

notexhibit any tilt with height,whereas4D-Varanalysisincrementsareanisotropicandalsoexhibit a

westwardtilt with height.Idealizedexperimentsby RabierandCourtier(1992)showedthat4D-Var

is ableto combinethe informationprovidedby the modeldynamicswith observationsof only the

eddypartof theflow to reconstructabaroclinicwave.

Thépautet al. (1996)demonstrateda stronglink betweenthe dominantsingularvectorsof the

tangentlinear model and 4D-Var analysisincrements. Theoreticalstudiesby Pireset al. (1996)

and Rabieret al. (1996) also showed that the 4D-Var cost function is most sensitive to analysis

incrementsgivenby the dominantsingularvectors. Hence4D-Var shouldprovide accuracy of the

unstablecomponentsof the flow by correctingthe componentsin the initial errorsthat arerapidly

growing.

Many of theprevious4D-Var studieshave consideredobservationsgivenat only theendof the

window. However, oneof themajoradvantagesof 4D-Var is thatit is ableto usethemodeldynamics

to link togethera time-sequenceof observations.Thisextra informationcanbeusedto build abetter

pictureof boththespatialandtemporalstructureof theatmosphereby ensuringconsistency between

the observationsandthe predictedevolution of the atmosphericstate. Hereandin the companion

paperJohnsonet al. (2005a),hereafterJHN, we usethesingularvaluedecomposition(SVD) of the

observability matrix to examinethespatialandtemporalinterpolationin 4D-Var thatresultsfrom the

interactionof a time-sequenceof observationswith themodeldynamics.Thetechniquewasusedin

JHNto investigatetheextentto which4D-Varcouldusethetimeevolutioninformationto reconstruct

thestatein anunobservedregion, whilst filtering theobservationalnoise. The techniqueis usedin

this paperto investigatethe extent to which 4D-Var is able to usethe information from a time-

sequenceof observationsto generatethe vertical spatialstructuresthat arenecessaryfor accurate

4

forecastsof baroclinicweathersystemsandalsoto correcterrorsin the initial conditionsthat result

in rapid growth. For a completeassessmentof whether4D-Var is ableto generatethe appropriate

vertical structures,we investigatewhether4D-Var is ableto generatetheappropriatestructuresfor

bothbaroclinicgrowth anddecay.

The2D Eadymodelis usedthroughoutthis paper. This linearmodelis oneof themostsimple

modelsof baroclinicinstability, andallows a clearunderstandingof the operationof 4D-Var. The

4D-Var algorithm,singularvectortechniqueandEadymodelaredescribedin section2. In section

3, we comparecaseswith growing anddecayingmodesandinvestigatethe impactof theaccuracy

of the observations,the temporalpositionof the observationsin the assimilationwindow and the

positionof theobservationsin thespatialdomain.Theexperimentsin section3 only considererrors

thatresultin modalgrowth or decaybut theexperimentsin section4 alsoconsidercaseswith errors

that result in non-modalgrowth. Thesefinal experimentsareusedto investigatethe impactof the

specifiederrorvariancesonthegrowth rateof thefollowing forecast.Themainconclusionsaregiven

in section5, andthepaperthenendswith adiscussion.Furtherstudiesassociatedwith thiswork may

befoundin Johnson(2003)andJohnsonet al. (2005b).

2. Description of the Eady modelexperiments

a. 4D-Var algorithm

4D-Var findstheoptimalstatetrajectorythat is closeto theobservedvaluesover a specifiedassim-

ilation time window andis closeto the backgroundstateat the beginningof the window. It is too

expensive to computethe optimal stateat every time level. Therefore,with the reductionof the

controlvariable(Le Dimet andTalagrand,1986),only theinitial conditionsareadjustedandthetra-

jectory is requiredto satisfythemodelequationsexactly. In thefollowing, we only considerlinear

models,but it is possibleto apply4D-Var to nonlinearmodelsby usingthenonlinearmodelin anon-

incrementalformulation,or usinga linearizedmodelin an incrementalformulation(Courtieret al.,

1994).

Mathematically, theanalysisat time���

, ��� , is givenby theinitial state,� � , which minimizesthe

5

costfunction,

� � �������� � ��� ��� ��������� � ��� ��� �! "#%$ �%& # �(' # � # ����)*���# %& # �+' # � # �-, (1a)

subjectto thelinearmodelconstraint,

� #%. � 0/ � #%. � ,1� # � � # for 2 435,7686768,79:�

observations, C& & �� , & � � ,767676=, & �"�

, to theinitial statevector, � � , andis denotedby

C'F ' �� ' � / � � ,>�����>� � 67676 ' "/ �

",>���8�G� � � 6 (2)

Thentheobservability matrix'

andits associatedsingularvaluedecompositionis givenby

'F C)*���ED�� C'*���ED��B HIG$ �J I-K�IGL �I 6 (3)

The singularvalues,left singularvectors(LSVs), right singularvectors(RSVs) and rank of the

observability matrixaredenotedbyJ I , KMI , L�I andN .

TheRSVsform anorthonormalbasisin theuncorrelatedstatespace,which meansthat the4D-

Varanalysisincrementscanbewrittenas:

� � � ��� HI-$ �O IQP1I ���ED�� L�I , (4a)

where

O I J � I� � J � I , (4b)

P1I K �I C) ���ED�� CR

J I 6 (4c)

andwhere CR C& � C' � � is the generalizedinnovation vector. The RSVsare independentof theobservedvaluesandthebackgroundstate.They give thepossiblespatialstructuresthatcanbeanal-

ysedby 4D-Var for the specifiedlinear modeldynamics,linearizationtrajectory, error covariances

andobservationlocationsin bothspaceandtime.

Theparticularcombinationof RSVsthatareincludedin theanalysisincrementis determinedby

thecoefficients, P>I . If thevector C) ���ED�� CR hasalargeprojectionontotheLSV K�I , thenthecorrespond-ing RSV is givena largeweight. Theobservationalnoisecanhave a largeprojectionontotheLSVs

with small singularvaluesso that thecorrespondingRSVswould dominatetheanalysisincrement.

However, thesearefilteredby thefilter factors,O I , whichdamptheRSVswith smallsingularvalues,

7

J � I ASA � � . Hencethealgorithmselectively filters unrealisticstructuresandtheanalysisincrementisdominatedby theRSVswith largesingularvalues.

c. Eady model

Thenon-dimensionalequationsfor the2D Eadymodel(Eady,1949)arenow described.Thebasic

stateis given by a linear zonalwind shearwith height in a domainbetweentwo rigid horizontal

boundaries. The domain is infinite in the meridionaldirection and the only dependenceon this

directionis theuniform meridionaltemperaturegradientwhich is in thermalwind balancewith the

zonalwind shear. Thedensity, staticstabilityandCoriolis parametersareall takento beconstants.

Theperturbationto thebasicstateis describedby thenon-dimensionalbuoyancy, T , on theupperandlowerboundariesandby thenon-dimensionalquasi-geostrophicpotentialvorticity (QGPV), U , intheinterior. Equivalently, theperturbationmayalsobedescribedby thenon-dimensionalgeostrophic

streamfunction,V , whichsatisfies:W � VWYX �

W � VW�Z � U , in Z@[ �;\ ,;\ , X][_^ 35,>`ba8, (5a)

W VW�Z T , on Z 4c;\ , X][_^ 35,>`ba8, (5b)

whereX

is thenon-dimensionaldistancein thezonaldirection,andZ

is thenon-dimensionalheight.

Thenon-dimensionaltimewill bedenotedby�. Theperturbationto thebasicstateis advectedzonally

by thebasicshearflow asdescribedby thenon-dimensionalQGthermodynamicequationandQGPV

equation:

WW � Z

WWYX T

W VWYX , on Z dc;\ , Xe[_^ 35,>`ba8, (6a)

WW � Z

WWYX U 435, in Z@[ �

;\ ,;\ , Xe[_^ 35,>`ba86 (6b)

Theperturbationis periodicin thehorizontalsothatthelateralboundaryconditionsare: T 35, Z ,1�f��T `e, Z ,1�f� and U 3Y, Z ,>�f�g U `e, Z ,1�f� . The model is discretizedusing11 vertical levels for QGPVwith 40 grid points in oneperiodic interval in

X. The advectionequationsarediscretizedusinga

8

leap-frogadvectionscheme.Thecomputationaldetailsto computethe4D-VaranalysisandtheSVD

areidenticalto thosein JHN.

DimensionalvaluesforX , Z

and�

will beusedto discusstheresults.Thesearebasedonadomain

heightof;=35hji

, buoyancy frequency9kl;=3 �m��no���

, CoriolisparameterO l;=3 �mp�no���

. Thedifference

in thebasicstatezonalwind shearovera heightof 10kmis q 38i no��� . Theimplied wavelengthof themostrapidly growing andmostrapidly decayingnormalmodesis aboutq 3Y3Y35hoi .

3. Resultsconsideringmodal growth and decay

In this section,we use the Eady model to investigatethe extent to which 4D-Var is able to use

the temporalevolution informationcontainedin the observationsto correctthe errorsin the initial

conditionsthatresultin eithermodalgrowth or decay.

a. Experiment Description

Thetruestateis givenby eitherthemostrapidlygrowing or themostrapidlydecayingnormalmode,

theequationsfor which aregiven in AppendixA. Thegrowing modeexhibits a westward tilt with

heightof thestreamfunctionfield, V , andaneastwardtilt with heightof thebuoyancy field, W Vsr W�Z .This vertical tilt is associatedwith the growth in amplitudeof the modewith time. In contrast,

the decayingmodeexhibits an eastward tilt in the streamfunctionfield anda westward tilt in the

buoyancy field.

Thebackgroundstateis givenby thetruestatebut with eitheranamplitudeor a phaseerror. For

the amplitudeerror, � � t3Y6vu ��w ; for the phaseerror, the true stateis shiftedwestward by 500km.Thus,in all theexperiments,thebackgroundstateerroris alsogivenby eitheragrowing or decaying

normalmode. Theseexperimentscanbe usedto comparecasesin which the errorsin the initial

conditionsareeithergrowing or decaying.

The interior QGPVis zerofor all the trueandbackgroundstates.Thecontrolvariablesusedin

the4D-Var algorithmaregivenby only theupperandlowerboundarybuoyancy at thebeginningof

thewindow. Thismeansthatanalysisincrementsmayonly beaddedto thebuoyancy field andnot to

9

theQGPVfield. It is assumedthatthereareno observationerrorcorrelations,)xdy

. Thespecified

backgrounderror correlations,�

, have a horizontalcorrelationlengthscale z {;=3Y3Y35hoi , andaredefinedin AppendixA.

Perfectsyntheticobservationsof only thelowerboundarybuoyancy aregivenattwo timesduring

a12hassimilationwindow. Theassumptionof perfectobservationsis not crucialhere.In anequiva-

lentexperimentwith noisyobservations,theRSVsthatcorrespondto noisewouldbedampedby the

filter factorsandhenceasimilaranalysiswouldbeobtained.Thus,in theseexperiments,varyingthe

sizeof theerrorvariance,� �

, hasthesameeffectasvaryingthesizeof theobservationalnoise.This

assumesthat in theassimilationof noisy observationstheappropriatevaluefor� �

is used.This is

describedin furtherdetail in AppendixB.

b. Preliminary Examples

Beforewediscussthemainresults,weshow someexample4D-Varanalysesfrom theseexperiments.

Theanalysesareshown at themiddleof the12hassimilationwindow. In Fig. 1a-b,thetruestateis

givenby agrowing normalmode,thebackgroundstatehasanamplitudeerrorandthespecifiederror

varianceratio is� � |;

. Observationsof the lower level buoyancy aregivenat the beginningand

the endof the 12h assimilationwindow. The 4D-Var algorithmdraws closeto the observationsso

that the analysisis closeto the true stateon the lower boundary. The algorithmalsoincreasesthe

amplitudeof theupperboundarysomewhatso that thegrowth rateof theanalysisthroughthe time

window is moreconsistentwith theobservations.Theequivalentanalysisfor thedecayingmodeis

shown in Fig. 1c-d. Thealgorithmagaindraws closeto theobservationson thelower boundary, but

theupperboundarywave is not adjustedasmuchasfor thegrowing mode. In particular, theupper

boundarywave is shiftedeastwardwith only aslight increasein theamplitude.

Figure2 shows similar experimentsbut with phaseerrorsinsteadof amplitudeerrors.Theanal-

ysis for thetruestategivenby a growing modeis shown in Fig. 2a-b. Theupperboundarywave is

movedeastward so that it is closerto the true state,but thereis alsoa reductionin the amplitude.

Theequivalentanalysisfor thetruestategivenby thedecayingmodeis shown in Fig. 2c-d. Again

the upperboundarywave hasbeenshiftedslightly but lessinformationaboutthe upperboundary

10

wave is inferredthanfor thegrowing mode.In boththeamplitudeandphaseerrorcases,4D-Var is

muchbetterableto correctthe unobservedupperboundarywave for the casewherethe true state,

andhencethebackgroundstateerrorresultsin growth ratherthandecay.

Theseexperimentsillustratethat4D-Var is ableto usethetime-evolution informationto correct

theupperlevel wave. However, it is not clearwhy 4D-Var is betterat correctingthegrowing errors

thanthedecayingerrors,andhow theanalysescanbeimproved.

In thefollowing experimentstheSVD framework is first employedto considerfurthertheextent

to which 4D-Var is able to correctthe upperlevel wave andproducean analysiswith the correct

growth rate. Theconceptsthatarelearnedfrom theSVD framework arethenconfirmedby 4D-Var

analyses.Theanalysesareverifiedby comparingthebehaviour, includingthemagnitudeandphase

error, over the following forecastinterval. The magnitudeis evaluatedusing the non-dimensional

kineticenergy (KE) norm:}e~ � X Z

(7)

where W Vsr W5X is the perturbationmeridionalwind. The phaseerror is evaluatedusing the

correlationbetweenthestreamfunctionfieldsof theanalysisandthetruestate.Theerror-correlation

takesavalueof onewhentheanalysisis completelyin phasewith thetruestateandtakesavalueof

minusonewhenthey arecompletelyoutof phase.

We first considerthe impactof theaccuracy of theobservationsby varyingthevalueof� �

; we

thenexaminetheimpactof thetemporalpositionof theobservationsby varyingthetime of thefirst

setof observations;andwe finally examinetheimpactof thespatialpositionof theobservationsby

observinghorizontallinesof thebuoyancy field at differentheights.

c. Accuracy of the observations

Theability of 4D-Var to reconstructtheupperlevel wave for differentvaluesof� �

is now examined.

Theobservationsareagainof thelower level buoyancy at thebeginningandtheendof thewindow,

but thevalueof� �

is varied.It is assumedthatin theappropriatevalueof� �

isusedin theassimilation

of realdata. A largevalueof� �

implies that relatively inaccurateobservationsareassimilatedand

sotheanalysisis closeto thebackgroundstate.A smallvalueof� �

impliesthatrelatively accurate

11

observationsareassimilatedandsotheanalysisis closeto the truestate.Hencevaryingthesizeof� �

allowsusto investigatetheimpactof theaccuracy of theobservations.

1). SVD RESULTS

We first examinetheRSVsof theobservability matrix anddiscussthegeneralconclusionsthatare

implied from these. Section2 shows that the uncorrelatedanalysisincrementcanbe written asa

linear combinationof the RSVsof the observability matrix. For our purposes,only the RSVsthat

contributeto theanalysisincrementareof interest;thesearetheRSVsthathave non-zerovaluesof

P1I (4c). Thevaluesof P1I arelargewhenthegeneralizedinnovationvector CR is in thesamedirectionasthe correspondingLSVs. In thesecarefully constructedexperiments(with perfectobservations,

a truestategivenby a growing or decayingnormalmode,anda backgroundstatethathaseitheran

amplitudeor phaseerror) the innovations CR arealsoeithergivenby a growing or decayingnormalmode.Thismeansthatthereareonly four valuesof P1I thatarenon-zero,to machineprecision.Theseare for \ ,85,=5, and7. Further, the RSVsoccur in pairsso that RSVs2 and3 have the samesingularvalue(

J \ 6Y) andan identicalstructureexceptfor a phaseshift of

;=3Y3Y35hoi. Similarly

RSVs6 and7 bothhaveJ 356 q . Thus,it is only necessaryto examinethespatialstructureof the

secondandsixthRSVs.Theseareshown in Figs.3and4. NotethattheRSVs,definedin uncorrelated

statespace,arepremultipliedby thesquareroot of the backgrounderror correlationmatrix so that

they arein correlatedstatespaceandto beconsistentwith (4). Thestructureof thesamefieldsafter

they have beenevolvedby thelinearmodel/

arealsoshown. Thesegive theassociatedstructures

at theendof theassimilationwindow. It shouldbenotedthattheevolvedRSVat thefinal time is not

thesameastheLSV, astheseRSVsarethesingularvectorsof theobservability matrix'

andnotof

themodel/

.

RSV 2, from the first pair of RSVs, is shown in Fig. 3. It hasa large amplitudeon the lower

boundary, whichis theobservedregion. Thebuoyancy field tilts eastwardwith heightandthestream-

functionfield tilts westwardwith height. This tilt is characteristicof a growing solutionbut it does

not tilt asmuchasfor anormalmode.WhenthisRSVis evolvedby themodelto givethestructureat

thefinal timewefind thatthelowerboundarywaveincreasesslightly in amplitudeandthereis alarge

12

increasein theamplitudeof thewaveontheupperboundarywavesuchthatit is similar to thatonthe

lower boundary. Also theupperboundarybuoyancy wave moveswestwardandthelower boundary

wavemoveseastwardsothatthestreamfunctionfield exhibits a greatertilt with height,closeto that

of theunstablenormalmode.Thus,thestructureof theRSVat thefinal timemorecloselyresembles

thestructureof anormalmode.

RSV 6, from the secondpair of RSVs, is shown in Fig. 4. At the initial time, it hasa large

amplitudeon theunobservedupperboundary. Thebuoyancy field tilts westwardwith heightandthe

streamfunctionfield tilts eastwardwith height,characteristicof a decayingmode,but againit does

not tilt asmuchas for a normalmode. When the RSV is evolved to the endof the window, the

circulationassociatedwith the upperboundarywave actsto the weaken the lower boundarywave

until it becomeszeroandthenbeginsto grow againwith theoppositesignsothatat thefinal timethe

buoyancy field tilts eastwardwith heightandthestreamfunctionfield tilts westwardwith height. Its

structureat thefinal time is moresimilar to thatof agrowing wave thanadecayingwave.

The coefficients P1I have a large valuewhenthe generalizedinnovationvector CR is in the samedirectionasthecorrespondingLSVs. Thus,thestructureof theLSVs show how the observational

informationis projectedonto the RSVs. The LSVs exist in observation space,which is the lower

boundarybuoyancy at the initial andfinal time. The LSVs correspondingto the first pair of RSVs

have a similar structureto the lower boundarybuoyancy fields of the first pair of RSVs and are

thereforenot shown. Thereis very little changein the shapeandamplitudeof the wave with time

which impliesthatthefirst pair of RSVsgive thestructurecorrespondingto thegeneralshapeof the

observedwave. Thepairof LSVs correspondingto thesecondpairor RSVs(alsonotshown) havea

similar structureto thelowerboundarybuoyancy fieldsof thesecondpair of RSVs.Thewaveat the

final time hastheoppositesignto thatat the initial time. This impliesthat thesecondpair of RSVs

correspondto extractingthegrowth ratefrom theobservedwave.

FromtheTikhonov filter factors,O I (4b), thealgorithmdampstheRSVswith J � I � � so that

as the varianceratio,� �

, is increasedthe algorithmdampsmoreof the RSVswith small singular

values.In this case,thefirst pair of RSVswithJ \ 6Y

containthe informationthat is neededto

reconstructthestatein observedregionsandalsoto give a growing analysisincrement.In contrast,

13

thesecondpair of RSVs,withJ 356 q , containthe informationneededto reconstructthestatein

the unobserved regionsandalso to give a decayinganalysisincrement. In particular, theseRSVs

correspondto the informationdetectingthegrowth or decayof theobservedwave. This growth is

harderto detectthanthegeneralspatialstructureof thelowerboundarywaveandhencetheseRSVs

have a smallersingularvalue than the first pair of RSVsandaredampedby the algorithm if the

observationalnoiseis relatively largei.e� ? 356 q .

FromtheSVD results,we expectthatwhentheobservationsareextremelyaccurate,andhence�+l3

, thealgorithmis ableto addbothpairsof RSVsto thebackgroundstate.Whentheobserva-

tionalnoiseis largersothat�4356v q , thealgorithmwill filter out muchof thesecondpair of RSVs.

Theseare the vectorsthat are neededto reconstructthe upperlevel wave and to give a decaying

analysisincrement.Whentheobservationalnoiseis increasedfurther, so that�_ \ 6Y

, bothpairs

of RSVsarestronglyfiltered,andso thereis little differencebetweenthebackgroundstateandthe

analysis.This is now confirmedwith the4D-Var analysesandforecasts.

2). 4D-VAR ANALYSES

4D-Varanalysesusingobservationsof thelowerlevel buoyancy atthebeginningandtheendof a12h

assimilationwindow, but with differentspecifiedvarianceratiosarenow compared.Wefirst consider

thecaseswherethebackgroundstatehasanamplitudeerror. TheKE valuesfor the12hwindow and

following 36hforecastfor thegrowing modecaseareshown in Fig. 5a. Theanalysisis closeto the

true statewhen� �

is small (� � t356;

) but as� �

increases,the RSVsarefiltered moreso that the

analysisis closeto the backgroundstate. The equivalentdiagramfor the decayingmode(Fig. 5b)

is moreinteresting.When� � x356;

, theanalysisis closeto the true state.When� � ;

, thefirst

RSVscontributeto theanalysisincrement,but thesecondpair of RSVsarefiltered. It is thesecond

pair thatarerequiredto form a decayingincrement.Hence,theKE decaysthroughtheassimilation

window, but thenbegins to grow so that the 36h forecastis worsethanthat from the background

state.When� � ;Q3Y3

, theobservationsareassumedto besoinaccuratethatbothpairsof RSVsare

filtered,andtheanalysisremainscloseto thebackgroundstate.It shouldbenotedthat thediagram

for thedecayingmodehasadifferentscale,andsotheimpacton the36hforecastis smallcompared

14

with thegrowing mode.

Therearesimilar resultsfor the caseswherethe backgroundstatehasa phaseerror. For both

thegrowing anddecayingmodes,theKE valuesfor thebackgroundstateareidenticalto thosefor

the true state(Fig. 6). The error canonly be seenin the error correlationvalues(Fig. 7). For the

growing mode,when� � k356;

, both pairsof RSVscontribute to the analysisincrement,andboth

thephaseandKE arecloseto thetruestate.When� � ;

, only thefirst pair of RSVscontributeto

theanalysisincrement.This actsto correctthe lower level wave, but not theupperlevel wave and

hencethe growth rate is reducedbut the phaseerror is improved. When� � ;=3Y3

, both pairsof

RSVsarefilteredsothattheanalysisis closeto thebackgroundstate.For thedecayingmode,when� � d356;

, theanalysishasalmostthecorrectrateof decayandno phaseerrorfor themajority of the

forecast.When� � l;

, it is mostlyonly thefirst pairof RSVsthatareaddedto thebackgroundstate.

This meansthat thegrowing incrementeventuallydominatestheanalysisso that theKE grows and

theerror correlationeventuallyhasthewrongsign. This is becausethestreamfunctionfield of the

analysisbeginsto tilt westwardinsteadof eastward. When� � ;=3Y3

, bothRSVsarefilteredandso

theanalysisis closeto thebackgroundstate.Again, for thedecayingmode,the36hforecastwhen

observationsareassimilatedcanin factbeworsethantheforecastwithoutassimilatingobservations.

d. Temporal position of the observations

For the investigationof the impactof the temporalpositionof the observationsin the assimilation

window, observationsof the lower boundarybuoyancy areagaingiven at two timesduring a 12h

assimilationwindow. The first (initial) setof observationsaregivenat eitherthe beginning (T+0),

middle (T+6) or end(T+12) of thewindow whilst thesecond(final) setof observationsarealways

givenat theendof thewindow. Note that theexperimentswith the initial setof observationsat the

endof thewindow areequivalentto experimentswith a singlesetof observationsat only theendof

thewindow andwith thevarianceratio,� �

, halved.

15

1). SVD RESULTS

Singularvaluedecompositionsof thecorrespondingobservability matricesarefirst considered.For

a fixed assimilationwindow length,the temporalpositionof the initial observationshasvery little

impacton thespatialstructureof theRSVsandsotheir structuresarenot shown. Thereis, however,

a changein the singularvalues. Fig. 8 shows the singularvaluesof the first andsecondpairsof

RSVsthatcontributeto theanalysisincrementasselectedby thevaluesof P1I (notnecessarilyalwaysnumbers

\Y and

). Whenthe initial observationsareat T+0, thesingularvaluesare

J \ 6Yand

356 q . Thesecorrespondto theRSVsshown in Figs.3 and4. Whenthe initial observationsaremovedtowardstheendof thewindow, thesingularvalueof thefirst pair of RSVsincreaseswhilst

that of the secondpair decreases.For initial observationsat T+6, the singularvaluesare3.17and

0.34.Whentheinitial observationsareatT+12,thesingularvalueof thesecondpairof RSVsis zero;

whentheobservationsareonly theendof thewindow, only thefirst pair of RSVscancontributeto

theanalysisincrementasthereis noobservationalinformationaboutthegrowth of thewave.

The secondpair of RSVscorrespondto the informationaboutthe growth rateof thewave. In-

tuitively, we know thatmoreinformationaboutthegrowth ratecanbeextractedif thetime between

the initial andfinal time observationsis longer. Whenthe observationsarefar apartin time, there

is a large differencein theamplitudeof theobservedwave at the initial andfinal times. Whenthe

observationsaremoved closertogetherin time, thereis a small differencein the amplitudeat the

initial andfinal timesandthereforeit is muchmoredifficult to infer thegrowth rateaccuratelywhen

assimilatingrelatively noisyobservations. It is for this reasonthat thesingularvalueof thesecond

pair of RSVsincreasesastheobservationsaremovedfurtherapartin time.

To beableto includethesecondpair of RSVsin theanalysisincrement,we needa valueof� �

that is smallerthanabout0.5. Further, the singularvaluesshow that the secondpair of RSVswill

have a larger contribution to the incrementif the initial observationsarecloserto the beginningof

thewindow. This is now confirmedby the4D-Varanalyses.

16

2). 4D-VAR ANALYSES

4D-Var analysesusingobservationsof thelower level buoyancy but at differenttimesarenow com-

pared.Thefirst setof observationsaregivenat thebeginning,middleor endof thewindow andthe

secondsetof observationsarealwaysgivenat the endof the window. For all the cases,we select

a varianceratio of� � 356;

. This is closeto thesingularvalueof thesecondpair of RSVsandso

allows theeffectof thepositionof theobservationsto beillustratedclearly.

The resultsfor the caseswith an amplitudeerror areshown in Fig. 9. For the growing mode,

thereis little differencein the KE valuesfor the differentcases,andall arecloseto the truth. For

thedecayingmode,wherethesecondpairof RSVsaremoreimportant,theanalysisis indeedcloser

to the true statewhenthe initial observationsareat thebeginningof thewindow. Whenthe initial

observationsareat theendof thewindow, theanalysisincrementleadsto growth ratherthandecay

becausethesecondpair of RSVsarefilteredcompletely.

The resultsfor the caseswith a phaseerror (Figs. 10 and 11) are somewhat similar. For the

growing mode, the phaseerrorsare significantly improved for all threecasesas the lower level

wavehasbeenshiftedto thecorrectpositionandtheKE valuesareslightly worseastheupperlevel

wave is not shiftedcompletely. Further, both the phaseerrorandKE valuesfor thegrowing mode

areworsewhenthe initial observationsareat the endof the window ratherthanat the beginning.

For the decayingmode,the differencesareagainmorepronounced.With observationsat the end

of the window, only the first pair of RSVs are includedand the KE valuesbegin to grow whilst

thestreamfunctionfield eventuallytilts westwardinsteadof eastward,giving a negativecorrelation.

With observationsat thebeginningof thewindow, theRSVsneededto give decayarefiltered less,

improving boththeKE valuesandthephaseerror.

Fromtheequivalencebetween4D-Var andtheKalmanFilter, 4D-Var implicitly propagatesthe

backgrounderror covariancematrix throughtime. This meansthat the error covarianceat the end

of the interval is more sensitive to the flow regime (more flow-dependent)than at the beginning

(Thépautet al., 1996),which suggeststhat you would expectobservationsto bemoreusefulat the

endof thewindow. However, theseexperimentsshow that the4D-Var analysesareimprovedwhen

the initial andfinal observationsarefurtherapartin time. Thus,in thesecases,theobservationsat

17

thebeginningof thewindow arealsoimportant.

e. Spatial position of the observations

The impactof thespatialpositionof the observationsis consideredby repeatingthe SVD analysis

and4D-Var experimentswith the horizontalline of observationsgivenat differentheights.This is

achievedby observingtheinterior non-dimensionalbuoyancy, which is theverticalderivativeof the

non-dimensionalstreamfunction.The observationsin eachcasearegivenat the beginningandthe

endof a 12hassimilationwindow.

1). SVD RESULTS

Thesingularvaluesof thefirst andsecondpairsof RSVs,asa functionof theheightof thebuoyancy

observations,areshown in Fig. 12. Dueto thesymmetryof theof theEadymodel,theexperiments

usingobservationson the lower boundaryareequivalentto thoseusingobservationson the upper

boundary. Hence,the curvesaresymmetricalabout5km. As the heightof the observationsis in-

creasedfrom 0 to 4.5km,thesingularvaluesof boththefirst andsecondRSVsdecrease.Theformer

decreasesfrom 2.99to 1.64andthelatterfrom 0.74to 0.57.

The SVD shows that we expectthe 4D-Var analysesto be closestto the truth whenthe obser-

vationsarenearestto the upperor lower boundaries,asthesegive the largestsingularvalues.The

normalmodesaredrivenfrom theboundariesandtheirbuoyancy fieldshavetheir largestamplitudes

at theboundaries.Henceobservingtheboundarybuoyancy couldbeexpectedto beoptimumandthe

differencebetweentheobservationsandthebackgroundstateis largestat theboundaries.Therefore,

moreaccurateinformationcanbeextractedin thepresenceof relatively noisyobservationswhenthe

observationsareclosestto therelevantboundary, herethelowerone.This is now confirmedwith the

actual4D-Varanalyses.

2). 4D-VAR ANALYSES

Whenthetruestateis givenby thegrowing mode,thevarianceratio is specifiedas� � ;=3

. With

this value,the positionof the observationshasthe most impacton the filtering of the first pair of

18

RSVs. Whenthe heightof the observationsis increased,the first pair of RSVsarefiltered more.

TheKE values(Fig. 13a)show that theanalysisis indeedclosestto thebackgroundstatewhenthe

observationsarenearto thecentreof thedomainandcloseto thetruestatewhentheobservationsare

on thelowerboundary.

Whenthe true stateis givenby the decayingmode,thevarianceratio is specifiedas� � x356;

.

With this value,thepositionof the observationshasthe mostimpacton the filtering of the second

pair of RSVs.Whentheheightof theobservationsis increased,thesecondpair of RSVsarefiltered

moresothat thefirst pair dominatethesolution,leadingto growth insteadof decay. TheKE values

(Fig. 13b) show that the bestanalysisis indeedobtainedwhen the observationsare on the lower

boundary.

f. Summary

Caseswherethetruestateis givenby eitheragrowing or decayingnormalmode,andwheretheback-

groundstatehaseitheranamplitudeor phaseerrorhavebeenconsidered.Theright singularvectors

(RSVs)of the4D-Var observability matrix provide a usefultool to examinehow the observational

informationis projectedinto theanalysisincrement.In thesecases,thereareonly two pairsof RSVs

thatareneededto form theanalysisincrement.Thefirst pairof RSVsbothhavelargesingularvalues,

andleadto a growing analysisincrement.Thesecondpair of RSVshave smallsingularvaluesand

areneededto reconstructtheupperlevel waveandto giveadecayinganalysisincrement.

The4D-Var algorithmfilters theRSVswith relatively smallsingularvalues,comparedwith the

sizeof the observationalnoise. This implies that4D-Var preferentiallyaddsan analysisincrement

that resultsin growth. Suchbehaviour meansthat 4D-Var is extremelyefficient in correctingthe

errorsin the initial conditionsthat would otherwiselead to large forecasterrors. However, if the

errorsin the backgroundstateleadto decay, addingan analysisincrementthat leadsto growth is

detrimentalto theforecast.

4D-Var is ableto extract the informationaboutthe growth rateof the true statefrom the time-

sequenceof observations. This informationis containedin thesecondpair of RSVsandis filtered

lesswheneither� �

decreasesor whentheir singularvalueincreases.We have examinedthreeways

19

in whichthiscanoccur. First,astheaccuracy of theobservationsincreases,thevalueof� �

decreases.

Second,asthetime betweentheinitial andfinal setsof observationsincreases,thesingularvalueof

the secondpair of RSVsincreases.Third, asthe line of buoyancy observationsis movedfrom the

centreof thedomainto eithertheupperor lower boundary, thesingularvaluesof boththefirst and

secondpairsof RSVsincreases.

4. Resultsconsideringmodal and non-modal growth

Theexperimentsin section3 arebasedon backgroundstateerrorsthataregivenby eithera grow-

ing or decayingnormalmode.Suchexperimentsareusefulfor understandingthedifferencesin the

behaviour of 4D-Var in thepresenceof growing anddecayingerrors.However it is possiblefor an

error in the initial conditions,typically associatedwith interior QGPVanomalies,to exceedtheex-

ponentialgrowth of themostrapidlygrowing normalmode.In thisfinal sectionof results,theextent

to which4D-Var is capableof correctingsuchanerroris investigatedby comparingthebehaviour of

4D-Var in caseswherethe truestateis givenby eitherthemostrapidly growing normalmodeor a

PV-dipoleperturbationthatexhibitsnon-modalrapidgrowth.

a. Experiment Description

Theexperimentsin section3 only allowedtheanalysisincrementsto beaddedto theboundarybuoy-

ancy. Theexperimentsin thissectionallow theanalysisincrementsto beaddedto boththeboundary

buoyancy andthe interior QGPV. The backgrounderror variancefor buoyancy andQGPV canbe

expectedto be different,andthis is allowed for. The ratio betweenthe observed andbackground

buoyancy errorvarianceis denotedby� �� and

� � is theratiobetweenthebuoyancy observationerror

varianceand the backgroundQGPV error variance. Horizontalbackgrounderror correlationsare

specifiedby thematrices��%� and

� , both with lengthscalesz xuY3m35hji . � �%� is block diagonal,

with two horizontalcorrelationmatriceson thediagonal,which correspondto thebuoyancy on the

upperandlower levels.� �

is alsoblockdiagonal,but with 11horizontalcorrelationmatricesonthe

diagonalthatcorrespondto the11 verticallevelsfor QGPV. Thedetailsof thehorizontalcorrelation

20

matricesaregivenin AppendixA. To simplify theexperiments,we considerthecasewhereall the

backgroundstatevaluesarezero, � � 3 . Thecostfunctionis thengivenby:

� � �7� � ��� ����%�

33 � � � �

�� C& � C' � ��C& � C' � � (8)

The true stateis given by either the most rapidly growing normal modeor a PV-dipole that

exhibitsnon-modalgrowth, theequationsfor whicharegivenin AppendixA. Thehorizontaldomain

is increasedto 8000km,with 80grid pointsin thehorizontal.Horizontallinesof perfectobservations

of thenon-dimensionalbuoyancy aregivenat thebeginningandtheendof a 6h window at a height

of 4.5km. Providing observationsin the middle of the spatialdomainallows an interior buoyancy

anomalyto bewell observed.

Theratio� � r � � � givesthevarianceof thenon-dimensionalboundarybuoyancy errorsdividedby

thevarianceof thenon-dimensionalinterior QGPVerrorsin thebackgroundstate. In practice,the

sizeof this ratio will dependon the forecastmodelusedto generatethebackgroundstate,previous

analysiserrorsandtheparticularflow situation. The following experimentscomparethe resultsof

the SVD of the observability matricesand4D-Var analyseswith differentspecificationsfor the a

priori errorvarianceratios.

Extremevaluesarechosento illustrateclearly thebehaviour of 4D-Var. Thesearesummarized

in Table1. Specification1 usesa large valueof� � r � � � (

;=38). This assumesthat we think that the

backgroundstateerrorsin the boundarybuoyancy field aremuch larger than thosein the interior

QGPV field. Specification3 usesa small valueof� � r � � � (

;=3 � ). This assumesthat we think that

the backgroundstateerrorsin the buoyancy field aremuchsmallerthanthosein the QGPV field.

Specification2 usesvaluesthatareacompromiseof thetwo extremes(� � r � � �

u�;=3 �m�).

21

b. SVD results

Theobservability matrix correspondingto thecostfunctiongivenby (8) canbewrittenas:

'F ' � �� �ED���%�

33 � � �ED��� (9)

We now examinethe RSVs of the observability matricesfor different specificationsof the error

varianceratios. We only examinetheRSVsthathave largevaluesof P1I whenthebackgroundstateerrorsaregivenby a growing modeandtheobservationsareperfect.It shouldbenotedthat for the

non-modalcase,therearemany valuesof P1I thatarenon-zeroandhencetherearemany RSVswitha varietyof spatialscalesthat contribute to the analysisincrements.However, for our purposes,it

sufficesto examinethestructuresof only a few of theseRSVs.

TheRSVsfor thecasewith specification1 (� � r � � �

;Q37) areshown in Fig. 14a-b. Theinterior

QGPVhasa relatively smallmagnitudewhilst thebuoyancy field hasa relatively largemagnitude.

The buoyancy field for RSV4 exhibits an eastward tilt with heightand that for RSV10exhibits a

westwardtilt, similar to theRSVsin section3. Suchstructuresleadto modalgrowth andarethere-

fore likely to beappropriatefor correctingsucherrors. TheRSVsfor thecasewith specification3

(� � r � � �

l;=3 � ) areshown in Fig. 14c-d.TheinteriorQGPVnow hasarelatively largemagnitudein

comparisonto thebuoyancy field. Suchstructuresleadto non-modalgrowth throughPV-unshielding

andarethereforelikely to beappropriatefor correctingsucherrors.

c. 4D-var analyses

We now considerthe impactof differenterror variancespecificationson analyseswherethe back-

groundstateerrorsareeithergivenby modalgrowth or non-modalgrowth. Theanalysesareverified

by comparingthespatialstructureat thebeginningof thewindow, andalsotheexponentialgrowth

rate,which is definedas:

5� �f� ;\Y � z¡ }e~

w .m¢ w}e~w � ¢ w

(10)

where �

is thetimestep.

22

Thespatialstructuresof theanalysesareshown at thebeginningof thewindow. Wefirst consider

caseswherethe truestateis givenby thegrowing normalmode.The truestate,shown in Fig. 15a,

is given by buoyancy waves and zero interior QGPV. This givesan exponentialKE growth rate,

shown in Fig. 16a, that is constantthroughoutthe forecastperiod. The analysesusing the three

different varianceratio specificationsare shown in Fig. 15. For specification1, a large analysis

increment(amplitude56vu

) is addedto thebuoyancy fieldsandasmallanalysisincrement(amplitude;Q3 �mp

) is addedto theinterior QGPV. This givesananalysisthat lookssimilar to thetruestateat the

initial time, andthe KE growth ratevalues(shown in Fig. 16a)arealsosimilar. For specification

2, a smalleranalysisincrement(amplitude\ 6 q ) is addedto thebuoyancy field anda largeranalysis

increment(amplitudeq 6; ) is addedto theQGPV. As theanalysishasa largeamplitudein theinterior,the incrementgrows by thePV-unshieldingmechanism,leadingto a largergrowth ratethanthatof

thetruestate.For specification3, thestructureof theRSVsaresuchthat4D-Varaddsanevenlarger

analysisincrement(amplitude£ 6u ) to theinterior QGPV, leadingto anevenlargergrowth rate.Finally, we considercaseswherethetruestateis givenby a PV-dipoleperturbationthatexhibits

rapid finite-time non-modalgrowth. The true state,shown in Fig. 17a,correspondsto a positive

temperatureanomalywhich mayhave resultedfrom, for example,diabaticheating.As discussedby

BadgerandHoskins(2001), the anomalyevolvesby the PV-unshieldingmechanismto give rapid

finite time growth which peaksat 6h, asshown in Fig. 16b. The analysesusingthe threedifferent

varianceratio specificationsareshown in Fig. 17. For specification1, a large analysisincrement

(amplitude56

) is addedto thebuoyancy fieldsandasmallanalysisincrement(amplitude¤;=3 �mp

)

is addedto theQGPVfields. This resultsin bothanextremelyunrealisticanalysisandgrowth rate.

Although the time-evolution information is containedin the observations,it is projectedonto the

inappropriateRSV structures. For specification2, a larger analysisincrement(amplitude\ 6v

) is

addedto the interior QGPV field, giving an analysisthat hasa similar structureto the true state.

The growth ratealsohasa similar behaviour to that of the true state,but never exceedsthat of the

most unstablenormal mode. For specification3, the observational information can be projected

onto the appropriateRSV structures.A larger analysisincrement(amplitudeuY6v

) is addedto the

interior QGPV anda smalleranalysisincrementis addedto the boundariesso that in the analysis,

23

thebuoyancy anomalyhasa small verticalscale,similar to the trueanomaly. Thegrowth ratenow

achievesa valuethat exceedsthat of the fastestgrowing normalmode. However, it still doesnot

reachthemaximumvalueof thetruestateandit is likely thatmany moreaccurateobservationsare

requiredto produceagrowth ratethatis closeto thatof thetruestate.

d. Summary

Wehavecomparedcaseswherethebackgroundstateerrorsexhibit eithermodalor non-modalgrowth

andwheretherearedifferentspecificationsfor thebackgrounderrorvariancesfor theinteriorQGPV

andtheboundarybuoyancy.

The RSVs of the 4D-Var observability matrix illustratedthat the possibleanalysisincrement

structuresarepre-determinedby themodel,observation locations,andthecovariances,without the

knowledgeof the observed values. When the specifiedboundarybuoyancy varianceis relatively

large,theRSVshave maximaon theboundaries.Suchstructuresresultin modalgrowth. Whenthe

specifiedinterior QGPV varianceis relatively large, the RSVshave maximain the interior. Such

structuresresultin non-modalgrowth.

With an inappropriatespecificationof theerrorvarianceratios,theobservationalinformationis

projectedontotheinappropriatestructures.Althoughtheobservationsmaycontaininformationthat

canbeusedto infer thatthetruestateis for examplea PV-dipole,this maybemistakenlyfilteredby

the4D-Var algorithmif theerror covariancesarespecifiedinappropriately. Thegrowth ratesshow

clearly that this canleadto poor forecasts,especiallyfor thecasewherethetruestateexhibits non-

modalgrowth. Hencethe specificationof the backgrounderror covarianceis crucial to enablethe

benefitsof 4D-Var to bemaximized.

5. Conclusions

In this paperwe have usedthe singularvalue decomposition(SVD) of the 4D-Var observability

matrix to investigatethe extent to which 4D-Var is ableto usethe informationaboutthe temporal

developmentof idealizedbaroclinicweathersystemsto infer the vertical spatialstructure.Simple

24

casesusingthe2D Eadymodelwith a singlehorizontalline of buoyancy observationsat two times

during the assimilationperiod have beeninvestigated. The SVD provides a generalframework,

withouttheneedfor repeatingnumerous4D-Var identicaltwin experiments.However, to confirmthe

anticipatedresultsfrom theSVD, we have alsoexaminedmany 4D-Var analysesandcomparedthe

evolutionof theKE valuesandthecorrelationof thestreamfunctionfield throughouttheassimilation

window andfollowing forecasts.

Threemainconclusionsfrom thiswork canbedrawn. First,wehaveshown that4D-Varpreferen-

tially generatesananalysisincrementthatleadsto growth, but adecayinganalysisincrementcanbe

generatedprovidedthat theinformationabouttheevolution canbeextractedfrom theobservations.

Second,theability of 4D-Var to extract thevaluabletime-evolution informationcanbemaximized

by adjustingthe locationsof theobservationsin spaceandtime. Third, thespecificationof theap-

propriatebackgrounderrorcovariancesis crucial in beingableto generateanalysisincrementsthat

leadto theappropriategrowth rate.

6. Discussion

The fact that 4D-Var is efficient in correctingrapidly growing errorsis not a new result. Previous

studiesby Thépautet al. (1996);Pireset al. (1996)andRabieret al. (1996)demonstratedthata 4D-

Varalgorithmwith nobackgroundtermis ableto useobservationsgivenattheendof theassimilation

window to correctsucherrors.However, thispaperhasextendedthisconclusionto show that4D-Var

is moreefficient in correctingsucherrorswheninformationaboutthetime-evolutionof theobserved

systemcanbeinferred.Further, we haveshown thatin thecasewheretheactualerrorsaredecaying

with time, it is possiblefor theobservationsto beprojectedontoanalysisincrementstructuresthat

leadto growth insteadof decay;suchincrementsaredetrimentalto theforecast.An incrementthat

leadsto decaycanbegeneratedprovidedthereis accurateinformationaboutthegrowth rateof the

system,but otherwise4D-Var generatesan incrementthat leadsto growth. So not only is 4D-Var

efficient in correctinggrowing errors,it is alsoefficient in addinggrowing errors.Thecaseswhere

4D-Varcorrectssucherrorsmostlikelyoutweighthecaseswhere4D-Varaddstheseerrors.However,

therearelikely to bea few occasionswhere4D-Var addsrapidly growing errorsto thebackground

25

state,resultingin aworseforecastthanif observationshadnot beenassimilated.

The benefitsof 4D-Var canbe maximizedby placingthe observationsin the optimal locations

in spaceandtime. Suchadaptive observation strategieshave previously beenconsidered(Snyder,

1996),andincludeplacingtheobservationsin regionsthataresensitive to errorgrowth (Buizzaand

Montani, 1999)or wherethe observationshave a large impacton a measureof the forecasterror

(BishopandToth,1999).Thispapersuggeststhatanalternativestrategy couldbebasedonoptimiz-

ing theinformationthatcanbeextractedby the4D-Varalgorithm.Further, the4D-Varobservability

matrix enablesus to considertheoptimal locationfor observationsin both time andspace(Daescu

andNavon,2004),andtakesinto accountboththemodeldynamicsanderrorcovariances.

This work hasalsohighlightedthe importanceof usingcase-dependentbackgrounderror vari-

ances,sothatthemaximumamountof informationcanbeextractedfrom theobservationsandsothat

thealgorithmcangeneratethemodalandnon-modalgrowth structures.Inappropriateerrorvariances

canleadto usefulinformationbeingmistakenlyfilteredfrom thesolution,or informationcanbepro-

jectedontotheinappropriatestructures.Case-dependentcovariancesmaybeachievedusingmethods

suchaserror-breeding(TothandKalnay,1997),or alternatively by computingtheappropriatevalues

from thedata(Dee,1995;Gonget al., 1998;DesroziersandIvanov, 2001).

Acknowledgements

The authorsthank A. Lawless from the University of Readingfor the many fruitful discussions

throughoutthis project. The authorsalso thank A. Hollingsworth from the EuropeanCentrefor

MediumRangeWeatherForecastsfor stimulatingoneof us(BJH) into trying to understand4D-Var

andfor hismany usefulcommentsandinterpretationof thiswork. Theresearchof CJwassupported

by a NERCCASEstudentshipat theUniversityof Readingwith theMet Office asthecooperating

organization.Revisionswerecompletedwhile CJwassupportedby a post-doctoralfellowshipfrom

theAdvancedStudyProgramat theNationalCenterfor AtmosphericResearch.TheNationalCenter

for AtmosphericResearchis supportedby theNationalScienceFoundation.

26

APPENDIX A

Definitions of the true stateinitial conditions and background error

correlation matrices

Thenormalmodesaredefined(e.g.HoskinsandBretherton,1972)as:

V X , Z ��0¥8¦Y§>¨ h Z ��¥8¦Y§ h X �!�_©§>ª¬«¨ h Z ��§>ª¬« h X �-, Growing Mode (A.1a)V X , Z ��0¥8¦Y§>¨ h Z ��¥8¦Y§ h X �� _©§>ª¬«®¨ h Z ��§>ª¬« h X �-, DecayingMode (A.1b)

wherehb¯;=6v

,©0 \ 6uY

andthe interior QGPV is zero. The interior QGPV-dipole perturbation

consistsof two spatiallyconfinedvorticesby defining U X , Z �� O X �±° Z � , where

O X ���² p �d³� 4´¬µ·¶�¸ ��¹ ³ºp�¹ �² � A X A �g² p� ² » �¼¾½ ´�¸ ¹ ³º¹ � ² p A X A ² p�² p 4³� � ´¬µ·¶�¸ ��¹ ³ºp�¹ ² p A X A ² �

(A.2a)

° Z ��4¥8¦Y§� i Z ��§>ª¬« i Z � \ A Z A £ hji (A.2b)

with ¿ £ 3Y35hji , h( \8À rY¿ , i À r 356 £ andboth O X � and ° Z � arezerooutsidethe specifiedranges.

Thebackgrounderrorcorrelationmatricesaredefinedby assumingthatthebackgroundstateer-

rors arecorrelatedin the horizontal,but that thereareno correlationsbetweenvertical levels. The

matrix�

is block-diagonalwith matricesÁ on the diagonal,which definethe horizontalcorrela-tion betweenvariableson onevertical level. The correlationsarespecifiedby definingthe inverse

correlationmatrix,asin JHN:

Á ���ày zp\ %Ä ³Å³ �� 6 (A.3)

Ä ³Å³ is a finite differencesecondderivative matrix in the X direction which incorporatesperiodicboundaryconditions,z is thehorizontalcorrelationlength-scale,and  is a scalarparameterthat isspecifiedsothatdiagonalelementsof Á haveavalueof one.

27

APPENDIX B

On the useof perfect observations

Theexperimentsin this paperuseperfectobservationsratherthanobservationswith errors,but

still includethe effect of filtering provided by the backgroundterm in the cost function. We now

describe,first theoreticallyandthenwith experiments,why theseresultsarerelevantto assimilation

with real data. The effectsof observationalnoiseandfiltering by 4D-Var aredescribedin further

detail in JHN.

Following from (4), whenimperfectobservations C&ÇÆ C' ��w _È areassimilated,the analysisincrementscanbe written in termsof componentsfrom the true signal C' ��w andthe observationalnoise

È.

� ���ED�� � � � � � �� IO IfP wI L�I I

O IQP ÆI L�I (B.1a)

where

P wI K �I )*��� C' � w � C' ��� � r J I (B.1b)P ÆI K �I )*���È r J I (B.1c)

Typically, the observationalnoisehasa large projection, P ÆI , onto the RSVswith small spatialscalesandassociatedwith smallsingularvalues,whilst thetruesignal, P wI , hasa largeprojectionontothe RSVswith large spatialscalesandassociatedwith large singularvalues. This is illustratedin

Fig.B.1.Theroleof thefilter factor,O I , is to filter thecontributionfrom thenoisewhilst retainingthe

contribution from thetruesignal. Whentheappropriatevaluefor thevarianceratio is specified,the

contribution from thenoise, I O IfP ÆI L�I (thesecondterm in equationB.1a),shouldbecloseto zeroso that the equationsfor perfectand imperfectobservationsarealmostidentical. This meansthat

the analysiswith perfectobservationsshouldbe closeto the analysiswith imperfectobservations.

Thusthe relevanceof perfectobservationsrelieson the assumptionthat the observationalnoiseis

projectedonto theRSVswith small singularvaluesandthereforethat thenoiseis filtered from the

28

solution.It is only with this sameassumptionthatvariationalassimilationmethodscanbeof use.

The analysiswith perfectobservationsdoesaccountfor the fact that the observationalnoiseis

filtered, becausethe appropriateamountof filtering is alsoappliedto the true signal. This is very

differentto a similar analysiswith perfectobservationsbut with no backgroundtermandhenceno

filtering.

Theremaybesomesmalldifferencesbetweentheanalysesfor perfectandimperfectobservations

as it is possiblefor the observational noiseto project onto the RSVs with large singularvalues.

Thesesmall differenceswill dependon the actualobservational errorsand are likely to become

importantfor a seriesof assimilationwindows in which theforecastsbecomethebackgroundstates

for thenext analysis.However, for a singleassimilationwindow it is unlikely that thesedifferences

have a significantimpacton theresults.Further, asthesedifferencesdependon thestructureof the

observationalnoise,anensembleof experiments,with differentvaluesfor therandomnoise,would

beneededto deduceconcreteconclusions.Theuseof perfectobservationseliminatesthis need.

To finally confirm that the resultswith perfectobservationsarealmostequivalentto thosewith

imperfectobservations,we compareananalysisthatusesperfectobservationswith ananalysisthat

usesnoisyobservations.The truestateis givenby a growing modeandthebackgroundstatehasa

phaseerror. The casewith noisy observationshasrandomnoiseaddedto the perfectobservations

thathasa Gaussiandistribution with standarddeviation 1.5. Theappropriatevalueof� �

, andhence

the appropriateamountof filtering, is found by repeatingthe analyseswith differentvaluesof� �

andselectingthe casewherethe valueof the costfunction at theminimum is equalto the number

of observations(Talagrand,1998). This gives an appropriatevalue of� � 56u

. The analysed

wave, shown at the beginningof the window in Fig. B.2a-b,is closeto the true stateon the lower

boundaryandmid-waybetweenthebackgroundstateandtruestateon theupperboundary. Thecase

for perfectobservationsagainuses� � x56u

. The analysis(Fig. B.2c-d) is almostidenticalto that

for theperfectobservations,with theanalysedwavecloseto thetruestateon thelowerboundaryand

mid-way on theupperboundary. Therearesomeslight differences,which dependon thevaluesof

theobservationalnoise.

29

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32

List of Tables

1 Threedifferentspecificationsof thevarianceratiosfor theinteriorQGPV,� �

, andfor

thebuoyancy on theboundaries,� �� , for usein 4D-Varexperiments. . . . . . . . . . 34

33

Specification� � � �

�� � r � � �

1 1;Q3 � ;=37

2u�;=3 �mp ;Q3 �m� u¤;=3 �m�

3;=3 �

1;=3 �

Table 1: Threedifferentspecificationsof thevarianceratiosfor the interior QGPV,� �

, andfor the

buoyancy on theboundaries,� �� , for usein 4D-Varexperiments.

34

List of Figures

1 4D-Varanalysesfor thecaseswherethebackgroundstatehasanamplitudeerrorand

thetruestateis givenby themostrapidly (a)-(b)growing and(c)-(d)decayingmode.

The analysis( � � , solid), backgroundstate( � � , dashed)and true state( ��w , dotted)fieldsareall shown at themiddleof thetime window. Theupperpanels(a) and(c)

show thebuoyancy on theupperboundaryandthelowerpanels(b) and(d) show the

buoyancy on thelowerboundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2 4D-Var analysesfor thecaseswherethebackgroundstatehasa phaseerrorandthe

true stateis given by the most rapidly (a)-(b) growing and(c)-(d) decayingmode.

Thedetailsareasfor Fig. 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Thetop panelsshow thesecondRSV, L � , of theobservability matrix, premultipliedby thesquarerootof thebackgrounderrorcorrelationmatrix,

� �ED��. Thelowerpanels

show theresultof integratingthesefieldsby theEadymodelovera12hinterval. The

numbersatthetopleft of eachplot indicatesthemaximummagnitudeof eachplotted

field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Thestructureof thesixth RSV, (� �ED�� L�É ). Thedetailsareasfor Fig. 3. . . . . . . . . 41

5 The evolution of the kinetic energy (KE) for the caseswherethe backgroundstate

hasanamplitudeerrorandthetruestateis givenby themostrapidly (a)growing and

(b) decayingmode. The true stateKE is shown by the thick solid line (T) andthe

backgroundstateKE is shown by thethin solid line (B). Theanalyseshavespecified

errorvarianceratios� �

of356;

(dash),;

(dot-dash),;Q3

(dot-dot-dash)and;=3m3

(dot). . 42

6 Theevolution of theKE for thecaseswherethebackgroundstatehasa phaseerror

andthe true stateis givenby the mostrapidly (a) growing and(b) decayingmode.

Thedetailsareasfor Fig. 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

35

7 The evolution of the error correlationfor thecaseswherethe backgroundstatehas

a phaseerror and the true stateis given by the most rapidly (a) growing and (b)

decayingmode. The backgroundstatecorrelationis shown by the solid line (B).

The analyseshave specifiederror varianceratios� �

of356;

(dash),;

(dot-dash),;Q3

(dot-dot-dash)and;=3Y3

(dot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

8 The singularvaluesof the observability matrix that correspondto the first (solid)

andsecond(dashed)pairsof RSVsthatcontributeto theanalysisincrement,plotted

againstthetime of theinitial observations.In all cases,thefinal setof observations

aregivenat T+12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

9 Theevolution of theKE for thecaseswherethebackgroundstatehasanamplitude

error andthe true stateis given by the most rapidly (a) growing and(b) decaying

mode.Thefirst setof observationsareeitherat thebeginning(T+0,dash),themiddle

(T+6,dot-dash)or theend(T+12,dot)andthesecondsetarealwaysat theendof the

window. ThetruestateKE is shown by thethick solid line (T) andthebackground

stateKE is shown by thethin solid line (B). . . . . . . . . . . . . . . . . . . . . . . 46

10 Theevolution of theKE for thecaseswherethebackgroundstatehasa phaseerror

andthe true stateis givenby the mostrapidly (a) growing and(b) decayingmode.

Thedetailsareasfor Fig. 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

11 The evolution of the error correlationfor thecaseswherethe backgroundstatehas

a phaseerror and the true stateis given by the most rapidly (a) growing and (b)

decayingmode.Thefirst setof observationsareeitherat thebeginning(T+0, dash),

themiddle (T+6, dot-dash)or theend(T+12, dot) andthesecondsetarealwaysat

theendof thewindow. Thebackgroundstateerrorcorrelationis shown by thesolid

line (B). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

12 The singularvaluesof the observability matrix that correspondto the first (solid)

andsecond(dashed)pairsof RSVsthatcontributeto theanalysisincrement,plotted

againsttheheightof thehorizontalline of observations. . . . . . . . . . . . . . . . 49

36

13 Theevolution of theKE for thecaseswherethebackgroundstatehasanamplitude

errorand(a) thetruestateis givenby thegrowing modeandthespecifiederrorvari-

ance� �

is;=3

, and(b) thetruestateis givenby thedecayingmodeandthespecified

error variance� �

is356;

. The horizontalline of observationsis given at heightsof3km (dash),

;86ukm (dot-dash)and

56ukm (dot). . . . . . . . . . . . . . . . . . . . . 50

14 The structureof the non-dimensionalQGPV (upperpanels)and buoyancy (lower

panels)for selectedRSVs� �ED�� L of theobservability matriceswherethevariancera-

tiosare(a)-(b)Specification1, and(c)-(d)Specification3. Thesehavecorresponding

singularvalues,(a)J p £ 3 \ , (b) J � �Ê \ ;= , (c)J p Ë;= q and(d)J �Ì� kuY . The

valuesat thetop left of eachpaneldenotethemaximummagnitudeof thefield. . . . 51

15 4D-Varanalysesnon-dimensionalQGPV(upper)andbuoyancy (lower) for thecases

wherethetruestateis givenby themostrapidly growing mode(shown in (a)). The

specifiederrorvarianceratiosare:(b) Specification1, (c) Specification2 (d) Specifi-

cation3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

16 Theevolutionof theKE growth ratesfor thecaseswherethebackgroundstatevalues

areall zeroandthetruestateis givenby (a) themostrapidly growing modeand(b)

aPV-dipoleperturbation.Thetruestateis shown by thesolid line andtheotherlines

show theanalysesusingthedifferentspecificationsfor thevarianceratios. . . . . . . 53

17 4D-Varanalysesnon-dimensionalQGPV(upper)andbuoyancy (lower) for thecases

wherethetruestateis givenby aninterior QGPVdipoleperturbation(shown in (a))

thatexhibits rapidfinite-timenon-modalgrowth. Thedetailsareasfor Fig. 15. . . . 54

B.1 Illustration of typical Picardratios, z¾Í °�Î P>I Î , for the true signaland the noise. ThePicardratio for the true signal, z¾Í °ÏÎ P wI Î , decreaseswith increasingsingularvectorindex, j. ThePicardratio for theerror, z¾Í °ÏÎ P ÆI Î , increaseswith j. . . . . . . . . . . . . 55

B.2 4D-Var analysescreatedusingobservationswith (a-b)noisewith standarddeviation

1.5and(c)-(d) no noise.Both experimentsuse� � d56u

andtheanalysesareshown

at thebeginningof a12hourassimilationwindow. . . . . . . . . . . . . . . . . . . 56

37

1000 10002000 20003000 30004000 4000-5-5

00

55

Lower Buoyancy

zonal direction (km)zonal direction (km)

-5 -5

00

5 5

(d)(b)

Growing Mode(c)(a)

Decaying Mode

Upper Buoyancy

tbx

x

xa

Figure 1: 4D-Var analysesfor thecaseswherethebackgroundstatehasanamplitudeerrorandthe

truestateis givenby themostrapidly (a)-(b)growing and(c)-(d) decayingmode.Theanalysis( � � ,solid), backgroundstate( � � , dashed)andtruestate( � w , dotted)fieldsareall shown at themiddleofthe time window. The upperpanels(a) and(c) show the buoyancy on the upperboundaryandthe

lowerpanels(b) and(d) show thebuoyancy on thelowerboundary.

38

1000 10002000 2000 30003000 40004000-5-5

00

55

Lower Buoyancy

zonal direction (km) zonal direction (km)

-5-5

00

5 5(a)

(d)

(c)Decaying ModeGrowing Mode

(b)

Upper Buoyancy

tbx

x

xa

Figure 2: 4D-Var analysesfor thecaseswherethebackgroundstatehasa phaseerrorandthe true

stateis givenby themostrapidly (a)-(b)growing and(c)-(d) decayingmode.Thedetailsareasfor

Fig. 1.

39

100010001000 200020002000 30003000 300040004000 4000

0

0

2

2

4

4

6

6

8

8

10

10

heig

ht (k

m)

0.354

zonal direction (km) zonal direction (km) zonal direction (km)

heig

ht (k

m)

0.468-1

-0.5

-1

-0.5

0

0

0.5

1

0.5

1

0.194

0.603-1

-0.5

-1

-0.5

0

0.5

0

0.5

1

1

0.532

0.780

Initial

Final

Streamfunction Upper Buoyancy Lower Buoyancy

Figure3: Thetoppanelsshow thesecondRSV, L � , of theobservability matrix,premultipliedby thesquareroot of the backgrounderror correlationmatrix,

� �ED��. The lower panelsshow the resultof

integratingthesefieldsby theEadymodelover a 12h interval. Thenumbersat the top left of each

plot indicatesthemaximummagnitudeof eachplottedfield.

40

1000 1000 100020002000 2000 30003000 30004000 40004000

0

0

2

2

4

4

6

6

8

8

10

10

0.317

0.325 0.532 0.194

0.1320.513

heig

ht (k

m)

zonal direction (km)zonal direction (km)zonal direction (km)

heig

ht (k

m)

-1

-1

-0.5

0

-0.5

0.5

0

1

0.5

1

-1

-1

-0.5

0

-0.5

0

0.5

1

0.5

1

Final

Streamfunction Upper Buoyancy Lower Buoyancy

Initial

Figure4: The structure of the sixth RSV, (� �ED�� L�É ). The details are as for Fig. 3.

41

00 1212 24 24 3636 48 4800

0.22

40.4

6

0.6

0.8

8 1

10

12

1.2

14

1.4

16

1.6

1.8

2

KE

Time (hours) Time (hours)

Decaying Mode

Ð Ñ Ñ Ò Ó Ò Ô Õ Ö Ò × ØÙ Ò Ø Ú × Û(b)

T

Growing Mode

1

0.1

Ù Ò Ø Ú × Û(a)

Ð Ñ Ñ Ò Ó Ò Ô Õ Ö Ò × Ø

1000.1

10

1

B

10

T

1

0.1

10

100B

Figure 5: Theevolution of thekinetic energy (KE) for thecaseswherethebackgroundstatehasan

amplitudeerrorandthe truestateis givenby themostrapidly (a) growing and(b) decayingmode.

ThetruestateKE is shown by thethick solid line (T) andthebackgroundstateKE is shown by the

thin solid line (B). Theanalyseshavespecifiederrorvarianceratios� �

of356;

(dash),;

(dot-dash),;=3

(dot-dot-dash)and;=3Y3

(dot).

42

00 1212 24 24 3636 484800

20.2

40.4

0.6

6

8

0.8

10

1

12

1.2

14

1.4

16

1.6

1.8

2

KE

Time (hours) Time (hours)

Decaying Mode

(b)(a)

Growing Mode

0.1

Ü Ý Þ ß à áâ ã ã Ý ä Ý å æ ç Ý à Þ â ã ã Ý ä Ý å æ ç Ý à ÞÜ Ý Þ ß à á0.1,100

1,10

T,B

100

1

10

T,B

Figure 6: Theevolution of theKE for thecaseswherethebackgroundstatehasa phaseerrorand

thetruestateis givenby themostrapidly (a) growing and(b) decayingmode.Thedetailsareasfor

Fig. 5.

43

00 1212

Assimilation

Correlation

Window WindowAssimilation

(b)

Decaying ModeGrowing Mode

Time (Hours)

(a)

Time (hours)2424 3636 4848

-1 -1

-0.5-0.5

0 0

0.5 0.5

1 110

B

0.1

1001

B

1 10 0.1

100

Figure 7: The evolution of the error correlationfor the caseswherethe backgroundstatehasa

phaseerrorandthetruestateis givenby themostrapidly (a) growing and(b) decayingmode.The

backgroundstatecorrelationis shownby thesolidline (B). Theanalyseshavespecifiederrorvariance

ratios� �

of356;

(dash),;

(dot-dash),;=3

(dot-dot-dash)and;=3Y3

(dot).

44

120 60

1

2

3

4

Time of the Initial Set Of Observations

SingularValue

Second Pair of RSVs

First Pair of RSVs

Figure 8: The singularvaluesof the observability matrix that correspondto the first (solid) and

second(dashed)pairsof RSVsthatcontributeto theanalysisincrement,plottedagainstthetime of

theinitial observations.In all cases,thefinal setof observationsaregivenat T+12.

45

00 1212 2424 36 36 484800

20.2

40.4

6

0.6

8

0.8

10

1

12

1.2

14

1.4

1.6

16

1.8

2

KE

Time (hours) Time (hours)

Decaying ModeGrowing Mode

(a) (b)

è é é ê ë ê ì í î ê ï ðñ ê ð ò ï óè é é ê ë ê ì í î ê ï ðñ ê ð ò ï ó

T+12

T+12

T

B

T+6T+0

B

T+12

T+6

T+0

T

T+0

T+6

Figure 9: Theevolution of theKE for thecaseswherethebackgroundstatehasanamplitudeerror

andthe true stateis givenby the mostrapidly (a) growing and(b) decayingmode. Thefirst setof

observationsareeitherat thebeginning (T+0, dash),the middle (T+6, dot-dash)or the end(T+12,

dot)andthesecondsetarealwaysat theendof thewindow. ThetruestateKE is shown by thethick

solid line (T) andthebackgroundstateKE is shown by thethin solid line (B).

46

00 12 1224 2436 36 48480 0

2 0.2

0.44

6

0.6

0.8

8 1

10 1.2

12

14

1.4

16

1.6

1.8

2

KE

Time (hours) Time (hours)

T+0

T,B

T+6

T+12

(b)(a)

Growing Mode Decaying Mode

T,B

T+12

T+6

T+0

T+0

T+6

T+12

ô õ õ ö ÷ ö ø ù ú ö û üý ö ü þ û ÿô õ õ ö ÷ ö ø ù ú ö û üý ö ü þ û ÿ

Figure 10: Theevolution of theKE for thecaseswherethebackgroundstatehasa phaseerrorand

thetruestateis givenby themostrapidly (a) growing and(b) decayingmode.Thedetailsareasfor

Fig. 9.

47

0 0

Correlation

Time (hours)12

Time (hours)

Decaying ModeGrowing Mode

(a)

12

(b)

2424 3636 48 48

-1-1

-0.5 -0.5

0 0

0.50.5

1 1

BT+0

T+12T+12 T+6

B

T+6T+0

Figure 11: The evolution of the error correlationfor the caseswherethe backgroundstatehasa

phaseerrorandthetruestateis givenby themostrapidly (a) growing and(b) decayingmode.The

first setof observationsareeitherat the beginning (T+0, dash),the middle (T+6, dot-dash)or the

end(T+12,dot) andthesecondsetarealwaysat theendof thewindow. Thebackgroundstateerror

correlationis shown by thesolid line (B).

48

0 2 4 6 8 100

1

2

3

Height of the Observations

Singular Value

First pair of RSVs

Second pair of RSVs

Figure 12: The singularvaluesof the observability matrix that correspondto the first (solid) and

second(dashed)pairsof RSVsthatcontributeto theanalysisincrement,plottedagainsttheheightof

thehorizontalline of observations.

49

00 1212 24 2436 36 48480 0

2 0.2

0.44

0.6

6 0.8

8 1

10 1.2

12

14

1.4

16

1.6

1.8

2(a) (b)

KE

Time (hours) Time (hours)

µ Decaying Mode, =0.12Growing Mode, =10

T

µ2

� � � � � �

0

3.5

1.51.5

0

3.5

� � � � � � � � � �� � � � � �0

� � � � � � � � � �1.5

3.5

T

B

B

Figure13: Theevolutionof theKE for thecaseswherethebackgroundstatehasanamplitudeerror

and(a) thetruestateis givenby thegrowing modeandthespecifiederrorvariance�� is ��� , and(b)thetruestateis givenby thedecayingmodeandthespecifiederrorvariance� is ����� . Thehorizontalline of observationsis givenatheightsof � km (dash),����� km (dot-dash)and ����� km (dot).

50

0

2

4

6

8

100.0001495

4000 8000

0.5

2.5

4.5

6.5

8.5

10123.1

0.0001898

4000 8000

122.9

0

2

4

6

8

1061.21

4000 8000

0.5

2.5

4.5

6.5

8.5

1018.91

63.49

4000 8000

6.938

(a)RSV 4 (b)RSV 10 (c)RSV 4 (d)RSV 12

µq2=1 µ

b2=10−5 µ

q2=10−5 µ

b2=1Specification 1 Specification 3

Figure14: Thestructureof thenon-dimensionalQGPV(upperpanels)andbuoyancy (lowerpanels)

for selectedRSVs ����� ��! of the observability matriceswherethe varianceratiosare(a)-(b) Spec-ification 1, and(c)-(d) Specification3. Thesehave correspondingsingularvalues,(a) "$#&%('�)+* ,(b) " �-, %.*0/�1 , (c)"$#2%(/�3�4 and(d)" �-� %65�7 . The valuesat the top left of eachpaneldenotethemaximummagnitudeof thefield.

51

0

2

4

6

8

100.0000

4000 8000

0.5

2.5

4.5

6.5

8.5

103.53

0.0001199

4000 8000

3.53

4.128

4000 8000

2.422

8.489

4000 8000

2.424

(a) Truth (b) µq2=1 µ

b2=10−5 (c) µ

q2=5×10−4 µ

b2=10−2 (d) µ

q2=10−5 µ

b2=1

Buoyancy

QGPV

Specification 1 Specification 2 Specification 3

Figure 15: 4D-Var analysesnon-dimensionalQGPV (upper)andbuoyancy (lower) for the cases

wherethe truestateis givenby themostrapidly growing mode(shown in (a)). Thespecifiederror

varianceratiosare:(b) Specification1, (c) Specification2 (d) Specification3.

52

6 12 18 24 30 36 42 48−0.5

0

1

2

3

4x 10

−5

Time (hours)

Growth Rateσ

KE (s−1)

6 12 18 24 30 36 42 48−0.5

0

1

2

3

4x 10

−5

Time (hours)

(a)Truth(b)(c)(d)

(a) (b)

Specification 1Truth

Specification 2Specification 3

Figure 16: The evolution of the KE growth ratesfor the caseswherethe backgroundstatevalues

areall zeroandthe true stateis given by (a) the most rapidly growing modeand(b) a PV-dipole

perturbation.Thetruestateis shown by thesolid line andtheotherlinesshow theanalysesusingthe

differentspecificationsfor thevarianceratios.

53

0

2

4

6

8

107.89

4000 8000

0.5

2.5

4.5

6.5

8.5

100.9932

0.0007027

4000 8000

3.65

2.935

4000 8000

0.8651

5.94

4000 8000

0.9837

(a) Truth (b) µq2=1 µ

b2=10−5 (c) µ

q2=5×10−4 µ

b2=10−2 (d) µ

q2=10−5 µ

b2=1

Buoyancy

QGPV

Specification 1 Specification 2 Specification 3

Figure 17: 4D-Var analysesnon-dimensionalQGPV (upper)andbuoyancy (lower) for the cases

wherethe true stateis given by an interior QGPV dipole perturbation(shown in (a)) that exhibits

rapidfinite-timenon-modalgrowth. Thedetailsareasfor Fig. 15.

54

noiseperfect

εc

ct

singular vector index, j

clog

FigureB.1: Illustrationof typicalPicardratios, 8:9=@?BAC= , for thetruesignalandthenoise.ThePicardratio for thetruesignal, 8:9=E?GIA = , increaseswith j.

55

1000 1000 20002000 30003000 40004000−5−5

00

55−5

0

5

−5

0

5

y

x

(c)

zonal direction (km)

bx

Buoyancy

a

Upper

(d)(b)

xt

(a)

zonal direction (km)

BuoyancyLower

FigureB.2: 4D-Varanalysescreatedusingobservationswith (a-b)noisewith standarddeviation1.5

and(c)-(d) no noise.Both experimentsuseJ�KMLON�PRQ andtheanalysesareshown at thebeginningofa12hourassimilationwindow.

56