JOURNAL OF GEOPHYSICAL
RESEARCH, VOL. 104, NO. C4, PAGES 7991-8014, APRIL 15, 1999
Spatial regression and multiscale approximations for sequential data
assimilation
in ocean models
Toshio M. Chin, Arthur J. Mariano, and Eric P. Chassignet RosenstielSchoolof Marine and AtmosphericScience,Universityof Miami, Miami, Florida
Abstract. Effects of spatialregularity and locality assumptionsin the extendedKalman filter are examinedfor oceanicdata assimilationproblems.Biorthogonalwavelet basesare used to implementspatialregularitythrough multiscaleapproximations,while a Markov randomfield (MRF) is usedto imposelocalitythroughspatialregression.Both methods are shownto approximatethe optimal Kalman filter estimatesclosely,althoughthe stabilityof the estimatescan be dependenton the choiceof basisfunctionsin the wavelet case.The observedfilter performanceis nearly constantover a wide range of valuesfor the scalarweights(uncertaintyvariances)givento the model and data examinedhere. The MRF-based method,with its inhomogeneousand anisotropiccovarianceparameterization, has been shownto be particularlyeffective and stablein assimilationof simulated TOPEX/POSEIDON altimetry data into a reduced-gravity,shallow-waterequation model. 1.
Introduction
From a mathematicalperspective,data assimilationin physical oceanographyimplies solution of an overdeterminedsystem of equationsfor the prognosticvariablesof a circulation model. In numericalmodels,data are typicallyusedselectively to provide the necessaryvalues, including the initial and boundaryconditionsand forcingfields,to establishthe forward (time) recursionfor the prognosticvariables.In data assimilation the time trajectories of the model variables are further constrainedby extraneous data, usually inferred from ship, buoy, float, and satellite measurementsbut also possiblyobtained from measured or simulated specificationsof openboundaryconditionsand air-seainterfaces.A classicapproach to evaluatethe overconstrainedvariablesobjectivelyis to apply an optimality condition, such as a least squares criterion [Sasaki,1970;Thackerand Long, 1988;Bennett,1992; Wunsch, 1996], which is employed by most of the sophisticateddata assimilation methods under consideration today [Ghi! and Malanotte-Rizzoli,1991; Talagrand, 1997]. Although solution techniques for a least squares formulation are relatively straightforwardin theory, they presentseriouspracticalproblemswith computationalspeedand storageowingto the large number of variablesrequiredto representa prognosticstatein typical ocean circulationmodels.Approximate solution techniques are thus necessaryfor objectivedata assimilationat present.In this paper, suchapproximationtechniquesare examined in conjunctionwith the Kalman filter, a sequential algorithmbasedon the probabilitytheory, that allowsformulntic,n 9nctintprprptnticmc•fthp lpn•t qClllnr•q qc•l•ticmq in term• of the mean and covarianceof a normal (Gaussian)distribution [Andersonand Moore, 1979]. Assimilation of satellitebasedsea-surface heightdata into primitiveequationmodelsis then considered.
When applied to data assimilation,a Kalman filter performs time-sequentialstatisticalinterpolationthat incrementallycorrectsfor the discrepancybetweenthe observations(data) and
model predictions[Ghil et al., 1981]. The statisticalinterpolation is based on the prediction/analysiserror covariances, which are estimatedusinga forward recursionconsistentwith the model dynamics.An approximationtechniquefor Kalman filter in ocean data assimilationmust addresscomputational efficiencyof this forward recursion for the error covariance matrix, whose dimensionis the square of the number of the model prognosticvariables.Using both physicallyand algebraically motivated assumptions,various techniquesto parameterize the covariancematrix havebeen exploredand practiced to drasticallyreducethe matrix dimensions.Two qualitativeyet prevailing features of a geophysicalcovariance function are regularity(smoothness)and locality (decay or "e-folding") in space.The assumptionof regularity concernsmaintenance of consistency betweenthe spatialscalesof the dominantdynamic modesin the prognosticvariablesand the correlationscalcsof the correspondingerror process[Bennettand Budgell, 1987; Cane et al., 1996], and it tends to allow a coarselysampled representationof the covariancematrix. The assumptionof locality indicatesthat the correlation between a pair of variables decayssteeply (e.g., exponentially)with respectto the distancebetweenthem [Daley, 1991], and it facilitatesparameterization using a small number of regressioncoefficients. The structure of the error covariancematrix is partially dependenton the observationnetwork (the samplingpatterns associatedwith the extraneousdata) and henceis highlyvariable from caseto case,even with the same dynamicmodel. In practice, the regularity assumptionsare useful in situations where the data densityand analysisobjectiveare appropriate for long-distancecorrelationstructures.For data setssampled sparselyover space(e.g., sixtide gaugesfor the tropicalPacific [Miller and Cane, 1989]), only large-scalefeaturescan be sampled, for which casea highly "regular" (generallycontaining strongcorrelation over long distance)error covariancematrix would be desirablefor smooth interpolation uncontaminated with spuriousor artificial small-scalefeatures.A climatological analysisalsofocuseson large-scaleeventsto which smoothand
Copyright 1999 by the American GeophysicalUnion.
nonlocal
Paper number 1995JC900075.
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AND
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FOR
KALMAN
FILTER
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finer scales,the locality assumptionswould reduce oversmoothingand enhanceaccuracyin reproductionof features suchasmesoscale rings.Somedegreeof regularityin addition to localityis stillimportantin thiscaseto preventgenerationof destabilizinginertia-gravitywaves,as in the casewith initializationof data for meteorological models[Daley,1991]. Regularityin covariance canbe realizednumericallyby subspaceprojection,whichrestrictsthe dynamicbehaviorof the prediction/analysis error fieldsto a smallnumberof dominant modesrepresented bysmoothfunctions.Severalapproaches to Kalman filter approximationincludingthose by Ghil et al. [1981], Cohn and Parrish[1991],Fukumoriand MalanotteRizzoli[1995],and Caneet al. [1996]can be examinedin this unifyingmathematicalframework,as detailedin this paper.
While localitycan be further imposedin this frameworkby choosingsmoothand localizedmode functionsfor the error subspace, the stabilityof the resultingdata assimilationsystem appearsto be dependenton the choicefor the modefunctions. Alternatively,a numericalrealizationof localityin covariance canbe achievedmore directlyby spatialregression, a multidimensionalgeneralizationof standardautoregression (e.g.,for time seriesanalysis),whichcan be formulatednaturallyusing a Markov randomfield (MRF). The MRF formalism[Kindermannand Snell,1980]usesa strictlylocal(in space)prescription of mathematicalexpressions to characterizestatisticaldependence over a wide range of spatial scales, finding applications in a varietyof fieldssuchas statisticalmechanics [Kindermannand Snell, 1980], population biology [Besag,
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1974],and imageprocessing [Gemanand Geman,1984]for dynamicallyupdatethe inhomogeneous MRF descriptionof statistical description of complexspatialpatterns.In our ap~ the covariancematrix in a mannerconsistent with a given proachto Kalmanfilter approximation the MRF essentially oceancirculationmodelis presented. Performance of the rereplacesthe covariance matrixwith a diagnostic (spatial)op- sultingapproximate Kalmanfilter is then assessed througha eratot, or a finite differenceoperatorwith a numericalcom- varietyof simulateddata assimilation experiments. plexitysimilarto that of a geostrophic balanceequation,operatingon the prediction/analysis errorprocess.The role of the diagnosticoperatorthen is to encodethe correlationstructure
intheerror process, andtheorder ofthisfinite difference 2. DataAssimilation WithKalman Filter operatordetermines the correlationscalesof the covariance We firstformulatethe dataassimilation taskasa weighted approximation. leastsquaresproblemand designate the Kalmanfilter as a The emphasis in thispaperis on realizationof localityasa meansto solvethisminimization problem.The readersfamilstrategy for assimilation of relativelydenseandwidelysampled iar with the Kalmanfilter mayskipto section3, after glancing satellite-based oceanic measurements. A numerical scheme to
at the mathematical notations in this section.
7994
CHIN
ET AL.'
SPATIAL
REGRESSION
AND
Let the equationsfor a numerical ocean circulationmodel be representedby the algebraicrecursion
Xk__fk(Xk_l)q_•odel
(1)
WAVELET
FOR
KALMAN
FILTER
computationof the filter gain K•, whichis obtainedthrougha separaterecursionof the error covariancematrix, defined as
P• = (e•e•), whereekistheestimation error,i.e.,4 = xf•- x• and e• =- x• - x•, and angle brackets denote ensemble average.To obtain an optimal recursionfor P•:consistentwith the leastsquares(3), eachinverseof the weightmatrices,P0 =
where the statevectorx/•, containingall the dynamicallyindependent prognosticvariables,representsthe synopticoceanin asthecovariance the model at time kAt, k = 0, 1, 2, ..., for a given model L•-I Q: =- m•-• andRi ---N•:• istreated time step At. The function f/• is a generic representationof matrix of an independentrandom vector representingthe reerrors in initialization (x0 - •), modeling (e•odel), computation of a single time step in a primitive equation spective and observation (e•t•). Such a statistical framework facilitates model, while e/• model is the aggregate of physical uncertainty model and numericerrorsin the model. Typically,•/• can only be determinationof the weightingparametersand interpretation characterizedstatistically.Let the vectory/•be a representation of the filter prediction/analysis. of the observable
variables
at time kAt
The optimaltime recursionfor the errorcovariance P• (ini-
such that
tializedas•
Yk= H•xk+ e•d•t•*
(2)
P• = FkP•_•F• , + Q•
ß The operator withan additivenoise/error term8kdata Ha sam-
plesthe synopticmap x•, by linear combinationsand thustends to be a diagonallystructuredand sparselybandedmatrix.The observedvalues("data") of Ykare usedto constrainthe prognosticvariablesin x/• to adjust the dynamictrajectory toward the observation.The general case where y/• is a nonlinear function of xa. can also be formulated in this framework throughlocal linearization[Cohnand Parrish,1991] but is not addressedin this paper, as practical effectivenessof such linearization is yet to be documented. The Kalman filter is intimately associatedwith the timedependentleastsquaresproblem(for a giventime indexk)
M•
(3)
= P0 = L•-•) canbe givenas
f
T
f
(6)
T
K• = PkHk(HkP•Hk + Rk)--1
(7)
P•: (I - K•Hk)P•(I- K•H•)r + K•R•K[
(8)
where the matrix F/• -- 0f/•/0x(•_•)
is a tangentlinear ap-
proximation of the model dynamicsabout the most recent estimate•__ •. The filter analysis• thus correspondsexactly
to the minimizingsolutionto the leastsquares(3) onlywhen the modeldynamicsf/•(x/•) is linear in x/•for all k. The tangent linear approximationof model dynamics(known as the extendedKalmanfilter) invokestheoreticalissuesinvolving"closure" of statisticalmoments [Cohn, 1993; Miller et al., 1994; Evensen,1994]; however,in practice,the approximationappearsto be suitablefor a typicaloceancirculationmodel that tends to have a small time step At relative to evolutiontimescalesof circulationfeatures[Menemenlis and Wunsch,1997]. The optimal filter gain K/• given in (7) would make the
where v • • vrWv denotesthe squarenormof a column data-updateequation(5) for a givenk equivalentto the optivector v weightedby a squarematrix W. The initial condition mal interpolation [Daley, 1991] and Gauss-Markovestimator • forthemodel (1)andthefullsetofobse•ations •, j = 1, [Wunsch,1996]appliedto a varietyof operationalgeophysical 2, -.., k, are providedexternallyand denotedwith the super- data analysis.The Kalman filter can thus be naturallyinterscriptd, indicating "data."Theweighting matrices Lo,Mi, and pretedasa time sequenceof spatialinterpolators,in whichthe N/ (forj - 1, 2, ..., k) aretheparameters of thisparticular optimality criteria evolve in a manner consistentwith the leastsquares formulation. The twotermsweighted byM• and modeldynamicsand data distribution(observationnetwork). N•, definingthe relativeconfidence in themodel(1) anddata Practicality of the extended Kalman filter in oceanic data (2), tend to dominatethe initial conditionterm weightedby L0 assimilationdependson two procedures'choice of the filter as more obse•ations becomeavailablewith increasingk. We parametersand reduction of the variable dimensions.The fildenotethe minimizingsolutionfor (3) as x•, •, .-., • (su- ter parametersto be specifiedare the matricesP0, Q/•, and perscripta for "analysis")and remind readersthat the opti- (equivalently,L0, M•, and N/•), k = 1, 2, .... Determination mality condition (3) is dependenton the time index k. The of realisticspatial correlationstructuresin Q/• for the model Kalman filter then is a time-recursivealgorithm to compute dynamicserror is still an openissue,althougha high degreeof only the latest member• of the optimal solutionfor (3) for regularity in Q/• has been determined crucial [Bennettand eachk = 1, 2,..-. Budge#,1987;Dee, 1991' Caneet al., 1996],especiallyfor asThe time recursionfor the Kalman filter (initialized as similationproblemswith sparselysampleddata. The need for
x; : •)is
a reduction
x• = f•(x•_,)
(4)
in the dimensions
of the filter
variables
is due
primarilyto demandsfor computationalresourcesto storeand time updatethe large predictionand analysiserror covariance
matrices, P• and•. If N denotes thedimension of theprognosticvariableset x/•, the correspondingerror covariancemawhere the superscriptf denotes"forecast."The last additive term in (5) computesthe correctionnecessa• to reducethe difference between the forecast and measuredvalues of y•. The matrix K•, known as the "filter gain," controlshow this correctionis incorporatedinto the prognosticvariables.Note
that if K• is null (zero), the recursion(4)-(5) is equivalentto the circulationmodel in (1). The essenceof the Kalman filter is in statisticallyoptimal
trix P• thencontains N 2 variables (elements), a tremendous storagerequirementfor a typicalvalueof N -- 104-10s. As the computationaldemandsare alreadyat a premiumfor the N-dimensionalrecursionof the modeldynamics(4) [Blecket al., 1995],the N-dimensional covariancerecursion(6)-(8) mustbe greatlysimplifiednumerically,in practice.A dimensionally reduced representation for the covariancematrix is intended to serve such a computational purpose, but it can
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alsoinfluencethe matrix structures(and hencecorrelation immediatelyapparentas the transformedcovariancematrix Pa structures)of the filter parameters to be determined. The is dimensionallyreduced to n x n. remainder of this paper thus focuseson approachesto reduce the covariancematrix Pk in dimension by parameterization.
3.
Reduction
of Covariance
Dimension
Approximationof a covariancematrix through parameterization of its correlation structure is a common practice in geophysical datainterpolation[e.g.,MarianoandBrown,1992]. For the inhomogeneousand anisotropiccovariancematrix Pk in a Kalman filter, the regularity (smoothness)and locality (decay) assumptionsare realized mathematicallyusing "subspaceprojection" and "spatial regression"schemes,respectively.The mathematicalissuehere is to developa parameterized arithmetic recursion consistent with the optimal covariancedynamicequations(6)-(8).
3.1.1. Subspacesof dynamic modes. The key issuein the subspaceprojectionapproachis determinationof the approximationsubspace B. Choosinga smallsubspace (smalln) could make the approximatedcovariancerecursion(10)-(12) highly efficient;however,the danger of introducingspuriousspatial structures(estimationbiases)is higherwith a smallernumber of basis functions.
When the dynamicmodel is linear and time invariant;that is, f•(x•) = Fxk for all k, the range spaceof the dynamicsis spannedby the column vectorsof the matrix U given by the
singular valuedecomposition (SVD) F = USVr, whereU and
¾ are unitary matrices(matriceswhosecolumnsare normalized and mutually orthonormal) and S is a diagonalmatrix of singularvalues.Using linearized shallow-waterequation models, Ghil et al. [1981] and Cohn and Parrish[1991] have partitioned the dynamic modes as U - [U•ow, Uf•,t] into the numerically desirable"slow modes" U•ow and nondesirable"fast 3.1. SubspaceProjection for Spatial Regularity modes"Uf•st,respectively,representingthe Rossbywavesand A systematicapproach to impose regularity in the error inertial gravitywaves,leadingto a Kalman filter approximation covariance,essentiallytaken by Fukumori and Malanottewith the subspaceB - Us•ow. Rizzoli [1995] and Cane et al. [1996] amongothers,is to conFor nonlinear dynamics,direct modal decompositionis not strainevolutionof the error fieldseawithin a subspacedefined as straightforward.Alternatively, Cane et al. [1996] and Pham by a set of smooth, linearly independent,normalized basis et al. [1998] have used the empirical orthogonal functions ("mode") vectorsb•, as (EOFs) for the variability in the state x• as the mode vectors for the error processek (which is difficult to measureor simek• • •klbl- U•k (9) ulate directly). Typically,the total varianceof the model varil ability is contributedby a small number n of the EOFs, and using such EOFs as the error modes b, often results in an where •k is the columnvector of the modal coefficients•k• and efficient covariance recursion due to this small n. These staB is the N x n matrix whosecolumnsare the basisvectorsk, tistically dominant dynamic modes of the model variability i - 1, 2, .-., n. The idea is to constrainevolutionof e• using tend to have inherent large-scaleregularity. Constrainingthe a smallnumbern < N (whereN is the dimensionof the state error dynamicsusing these smooth mode functions has an x•) of carefullyselectedmodesb,, so that the resultingreduc- advantageof greatly reducing the chanceof exciting numerition in degreesof dynamicfreedommaintainsspatialregularity cally undesirable small-scale modes such as inertia-gravity in P• and accuracyin the estimatex• while decreasingcom- waves.An outstandingissuein the EOF subspaceapproachis putational costs. computation and selectionof the appropriate EOF modes. The error vector e• is transformedinto the modal subspace Both the choiceof the empiricalstatistics(e.g., designof simas • - Te•, by the pseudoinverse T of the mode matrix B, ulation experiments)and partitioning of the corresponding
givenasT • (BrB)- •Br. Substitution into(9) thenrevealsthe
EOFs
into "desirable"
and "undesirable"
modes can affect the
essenceof the approximationas e• • BTe•, in which the filter analysisto a large extent. A lack of sensitivityto smallermatrix operatorBT performsa spatiallow-passfiltering for a scale(e.g., mesoscale)featuresin data can alsobe expectedfor low-resolutionapproximationof the error fields[Fukumoriand typical EOF modes owing to their large-scaleand ensemble Malanotte-Rizzoli,1995]. The time recursionof the covariance (statistical)natures.
matrkPain themodalsubspace canbe derivedfrom(6)-(8)
by straightfo•ard linear transformations(note that TB : I, the identitymatrix) as
3.1.2.
Subspaces based on wavenumber and scale. To
prevent oversmoothingor large-scalebiasesthat might be introduced by statisticalmodes like EOFs, nonempirical mode : F•P•_,F•+ Q• (10) functionshavealsobeenconsidered.For example,the dynamic modes of linearized shallow-waterequationsover a periodic K• -•-r - -•R•) • domain are the Fourier basisfunctionswith which an approximation subspaceB can be defined in terms of wavenumber P•= (• - •aOa)P•(•- •aOa)r + KaRaKa spectra [Ghil et al., 1981]. Also, Fukumori and MalanotteRizzoli [1995] have useda cubicsplinefunction (a locally supwherethe transformed matricesare givenas Pa - TPaT•, Q• = TQaT•, Pa = TFaB,andOa = HaB.In principle, the ported cubic polynomial) for spatial smoothingand coarsegainmatrixcanbe computed asKa = B•, withwhichthe scale approximationof Kalman filter. In these approaches, data update (5) can then be performed.In practice,the data rudimentarygeometricnotionssuchaswavenumberand scale, update can be accomplishedmore efficientlywithout suchan respectively,are used to characterizethe approximationsubexplicitevaluationof the gain matrix Ka, i.e., by first comput- spaces. Specifyingthe error mode functionsin terms of a characteringthen-dimensional vector andthenmultiplyingthe resultingvector by B (inversetransform).For the istic scaleis especiallyappealing,as some notion of "locality" typical case of n 0.01. In particular, the RMS errors are, at most, 10% for q/r > 0.01, indicatingthat an estimationquality similarto one displayedin Figure 5 (correspondingto q/r = 1) can be expectedover a wide range of valuesfor q/r. The RMS error plots (Figure 6) displaydistinctcharacteristics for each of the four cases of data assimilation
scenarios.
In the diffusion-perfectcase,lower valuesof the model-to-data ratio q/r are associatedwith more accurateestimates(lower widerangefrom 10-4 to 104, for eachfilter implementationRMS values) becauseof fidelity in the initial condition and
8004
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model numerics.For higher q/r values, implying increasing trust in data, the estimationaccuracydecreasesowing to the
WAVELET
FOR
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FILTER
8005
EOF Variances
SingularValues 1.1
ß
measurement noise. On the other hand, in the diffusion-flat
case,low q/r valueslead to high RMS errors and even filter failures,as the effectsof the inaccurate(flat) initial condition persistsowingto the implied overrelianceon the model. This suggests that the model-varianceparameterq, controllingthe retention time for information in the filter, needs to be rea-
1.0'
0.9
sonablyhigh relative to the data variancer to ensurethat the filter remainssensitiveto the data [Lewis,1986;Miller et al., 1994]. The same RMS error characteristicscan also be observed in the advection-flat
case for the same reason.
an unstable
The advection-flat
numerical
EOF-based
estimation
4
3
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I\•iii\\\ I \\ /i/' \\•IIt
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scheme.
case is the most difficult
basis index (n)
Unlike
the diffusion-perfectcasethe advection-perfectcasedoes not yield good estimateswhen the q/r ratio is small, implying inadequacyin the numericaladvectionscheme(an implementation of Lax-Wendroffscheme)for this particularconfiguration with relativelyhigh advectionspeedand undulatinginitial condition.For a higher q/r the noisydata have actually compensatedfor the numericalbiasesintroducedby the advection scheme.Miller [1986]reacheda similarconclusionthat Kalman filters can compensatefor imperfect numericsand can even stabilize
basis index (n)
-1
,-_
\/
assimilationproblems examined here owing to the combination of wrong initial condition and high spatialvariability. In this case the subspaceprojection approach (the waveletapproximatedfilter) is found dependenton the choiceof subspace basis functions. For example, a slightly different
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biorthogonal wavelet function •)4,4(Figure1,• = 2), aswellas the cubicB splineand severalother splinefunctions,haslead to an instableand inaccuratefilter. The performanceof subspaceprojectionfilter has been further examinedusingSVD and EOF subspaces (see section3.1.1). The EOFs used here have been obtained from the empirical covariancematrix of signalvariability in the analyticsolutionover 500 time steps.As depicted in Figure 7, all the singularvalues of the linear dynamicsF have comparablemagnitudes,as the advection dynamicshaveno mode selectivity.(Some singularvaluescorresponding to high-wavenumbermodes are clipped slightly owingto discretizationeffects.)The SVD subspaces are thus not expectedto offer compactapproximationsin this case.For example,a subspaceas large as n = 48 has been observed necessaryto produce estimatesthat begin to resemble the analytic solution qualitatively.Unlike the singularvalues the EOF variancesare dominatedby the first 10 modes,as shown in Figure 7. The subspaceusingthe corresponding10 EOFs leads to a stablefilter that producesestimateswith a constant bias(Figure7). This error hasresultedbecausenoneof the 10 EOF modesis able to represent(and hencecorrectfor) the constantbias in the flat initial condition(Figure 4). The fundamental difficulty is that the particular empirical statistics used to obtain the EOFs do not contain any constant bias owing to the method of ensemblesimulation.Such deficiency in the EOF subspacemay be difficult to assessin practice.In
,, ,, ,,
/
Figure 7. (top left) Singularvaluesof the 128 x 128 statetransitionmatrixF in the advectioncase.(top right) Variances (eigenvalues)of an empiricalvariabilitymatrixin the advection case.(bottom) Samplefinal estimate(solidline, at k = 200) for a subspace-projection filter using the subspaceof the 10 dominant empirical orthogonalfunctions(EOFs) (eigenvectors correspondingto the large variancesin Figure 7, top right). The correspondinganalytic solutionor the "truth" is shownby dashedline, while examplesof the noisyobservations (at k = 200) are shownby circles.Note that the set of 10 EOFs cannotcorrectfor the constantbias, despitethe information
available
in the observations.
reliable approximationof the second-orderstatistics;however, the correlationstructures(i.e., normalizedcovariance)are approximated well. Figure 3 depicts representativecovariance matricesfrom optimal Kalman filter and covarianceapproximation errorsby the MRF and wavelet filters. The MRF-based approachseemsto approximatelocal (near diagonal)correlation better, while the wavelet-based(and other subspaceprojection) approach tends to capture the far-field correlation structure
better
Linearized
than local correlation Shallow-Water
structure.
our idealized scenario, however, an ad hoc addition of an
4.2.
Model
eleventhbasisfunctionof a constant(a normalizedvectorof all ones)wouldadequatelyaugmentthe EOF subspace, leadingto an accurate,stable,and efficient(n = 11) approximatefilter for the advection-flatcase. This illustrates the potential for high computationalefficiencyaswell as strongdependencyon subspaceselectionfor the EOF-based method. In all approximatefiltering approachesthe absolutecovariance values are too sensitiveto operating conditionsfor a
A multivariate dynamicsystemis used to observethe performance of the optimal, wavelet-approximated,and MRFapproximatedKalman filters. The data assimilationproblem examinedhere is nearlyidentical(exceptfor somedifferences in discretizationand parameterization)to the configuration studied by Ghil et al. [1981]. The model here is given by a synoptic-scale atmosphericdynamicsexpressedby the linearized shallow-waterequation
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Figure 8. Examplesof final estimates(k = 1344) computedwith the optimalKalmanfilter (solidlines), wavelet-approximated filte• (dashedlines),and MRF-appmximatedfilte• (dash-dotted lines),alon• with the analyticsolutiontruth (dotted lines) and noisyobse•ations (circles)made eve• 4S time stepsfo• (top) •eopotential,(middle) zonalwind speed., and (bottom) mefidionalwind speed•.
daily (every48 time steps).The measurements are artificially corruptedwith simulatedGaussianrandomnumberswith stan-
0
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- U•s
The filter parametermatrices(Po,Qk, andRk) usedhere are diagonal(uncorrelated)asbefore,and their variances(diagonal elements)are homogeneous for eachvariabletype (u, v, or qb).The variancesin R• are givenby the actualmeasurement
over a periodicspatialdomainalonga fixedlatitude,where the noisevariances,as above.The variancesin Poand Q• are setto of the spatialvariabilityin the state(prognostic)variablesare the variabilityin the zonaland be 1% (10% root-mean-square) meridionalwind speeds(u, v) and the geopotentialqb.The analyticsolutionfor each of the three variables.The resulting mean zonal flow, mean geopotential,and ambient Coriolis valuesfor the model-to-dataratio (q/r) are about0.02 for u and 1 for v and parameter aregivenasU = 2 m/s,q) = 30000m2/s 2,andf 0.0001 s-1, respectively. An analytic solution of thisequation, Like the advection-flatcase examinedpreviously,the nu-
alongwith other detailsof the experimentalsetup,are givenby Ghil et al. [1981].The spatialdomainis discretizedto N - 64 points.The spatialand temporalintervalsare As - 218.75 km and At = 15 min, respectively.The measurementsof the state variables u, v, qbare made by samplingthe analytic solutiononly at the 32 grid pointson the left sideof the domain (Figure 8, reflectingland-basedmeasurements),only twice
merical model here is marginally stable owing to the large dynamic-range mismatchamongthe threeprognostic variables (e.g., qboscillatesat an order of 1000 times larger amplitude
than u) combinedwith the spatialdifferentiationsin (21). Without data assimilationthe time trajectoriesof the variables do indeed deviate from the analytic solutionssystematically (Figure 9). Steadystateestimates(at k = 1344 or 14 days)
CHIN ET AL.' SPATIAL REGRESSION AND WAVELET FOR KALMAN FILTER
8007
RMS error: Geopotential 1000
,
,
,
,
,
,
,
I
i
i
i
i
i
I
i
i
i
,
,
,
,
,
,
i
i
i
i
i
i
i
i
i
i
i
I
8OO
6OO
400
2OO
RMS error: U 2
i
i
1.5
0.5
RMS error: V lO
i
i
I
6
..... 0 0
•
'
7
• / _• .__•. -C J_C
_'•L
• 9• 10• 1'11'2 1•3 1
8
day
Figure9. Timeseries oftheroot-mean-square errors between theanalytic solution andtheestimates bythe optimal Kalman filter(solidlines), wavelet-approximated (dashed lines),MRF-approximated (dash-dotted lines), andmodel runwithout dataassimilation (dotted lines)for(top)geopotential, (middle) zonalwind speedu, and(bottom)meridional windspeedv.
with data assimilation by the three filters are displayedin
4.3. Eddy-ResolvingShallow-WaterModel
Figure8, indicating againthatthe noisyandsparsedatacan The MRF-approximated filter is now appliedto a windcompensate for the imperfectmodelnumerics andthat the forced,nonlinear,reduced-gravity, shallow-water model conapproximate (wavelet andMRF) filtersarejustaseffective for figuredin a rectangular domain.The numericalimplementathispurpose asthe nonapproximated Kalmanfilter. tion of the dynamicmodelis a single-layer version(without The time seriesof RMS errors,displayedin Figure 9, show thermodynamics) of the Miami IsopycnalCoordinateOcean that the filter errorssaturatequickly,to valueslessthan the Model (MICOM) [Blecket al., 1992;Bleckand Chassignet, corresponding measurement errors(i.e.,lessthanthestandard 1994].Witha horizontal gridspacing of 20km andgridsizeof deviations givenabove).The RMS errorplotsare similarfor 100 x 100,thisparticularmodelconfiguration hasbeenused the MRF and optimalfilters,implyingnear-optimalperfor- to simulatemesoscalefeatureslike rings associatedwith the manceof the MRF-basedapproximation. The waveletfilter, free-jetportionof thewestern boundary currents [Chassignet et however,hasproducedestimates with systematically higher prognostic stateconsists of the RMS errors,as evidentfrom the time series(Figure 9) for v al., 1990].Sincethemultivariate and,lessobviously, O.Thebiorthogonal basis function I•4'4 horizontalcurrentvelocity(u, v) and layer thicknessh, the
(C= 2) hasbeenusedto construct thesubspace forthewavelet
state dimension is about N -- 30,000. This dimension is
filterexamined here.Thebasis function/b 3'3,whichhasbeenused relativelysmallfor a typicaloceangeneralcirculationmodel for thenonapproximated successfully in thediffusion/advection cases discussed previously,butisalreadytoolargetobepractical Kalmanfilter.The MRF-approximated filter usesthe secondis found to cause unstabilities in the multivariate estimates examshownin Figures2b and 2d, with inedhere.Theseagaindemonstrate sensitivity of the subspace order MRF configuration projection approach to theselection of thesubspace modes. whichthe informationmatrixis representedby approximately
8008
CHIN
ET AL.: SPATIAL
REGRESSION
AND
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FOR KALMAN
FILTER
Prediction:10-day update (day 180)
1400
1300
1200
1100
IOO0
Prediction:2-day update(day 180) 900
8OO
700
600 m
•
lm/s
Plate 5. Day-180predictionfieldsas in Plate 3, exceptusingthe data-updateintervalsof (top) 10 and (bottom) 2 days. 39N parameters.(This number is approximatedue to the staggered(C grid) discretization usedby MICOM. Also, 39N = 13 x 3 x N, where13 is the numberof parametersper grid pointin the second-order MRF (Figure2b) and3 is the number of variabletypes.)A similarlevel of compactness in covariance representationcan be achieved with the waveletapproximated filter (usingthelengthscale•e= 2); however,the waveletapproachis not examinedhereowingto the sensitivity to basisselectionas observedin previousexperiments.
lated measurements
from this truth. The tenth and eleventh
year estimatesof a spin-uprun are usedto initialize the prediction and truth, respectively.The initial conditionsfor the truth andpredictionare shownin Plate 1. To simulate(under the frameworkof the simplifiedmodelconfiguration)satellitebasedaltimetermeasurements, the layerthickness h (whichis directlyproportionalto the sea-surface heightin the reducedgravity model) have been sampledfrom the truth along a sequenceof the TOPEX-POSEIDON samplingswathswith a 4.3.1. Twin experiment with the TOPEX/POSEIDON sam- 10-dayrepeatcycle,asshownin Figure 10. Each satellitesampling pattern. We have conducteda set of "identical-twin" plingpointhasbeencollocated to the nearestmodelgridpoint. experiments, in whichone of two modelrunswith dynamically Sincethe RMS differencein h valuesbetweenadjacentgrid independentinitial conditionsis treated as the "truth," while pointsin the truth field is about 7.2 m and the full-field varithe other run, called"prediction"hereafter,assimilatessimu- ability of the samefield is 77 m, the collocationshave intro-
CHIN ET AL.: SPATIAL REGRESSION AND WAVELET FOR KALMAN FILTER
8009
Day 0
1300
1200
1 lOO
lOOO
Day 12
900
80O
7OO
m
•
lm/s
Plate6. (top)Initialand(bottom) finalstates ofageneralized dataassimilation problem inwhich the"data"
aretheweakconstrains simulating "land"alongtheleftedgeof thedomain(visibleasthetriangular shape in Plate6, bottom). Everytwovelocity vectors, in eachof thehorizontal andverticaldirections, areshown.
by the MRF-approximated filterhasremainedstaducedabout 10% (•7.2/77) signaluncertainty,on average. computed The layer-thickness sampleshavethusbeen artificiallycor- ble throughout.As before,diagonalparametermatricesare ruptedwitha 10%-RMSadditiveGaussian noiseto simulate usedfor the filter, with the model-to-dataratio q/r = 10 for the collocationeffects.This noiselevel is also in rough agree-
the variableh. Plate 2 showsan instanceof predictionat day
truth,displaying a transitionperiod mentwiththe datanoise(4/40cm)expected across mesoscale 30 andthe corresponding to trackthe truth,given features in areas such as the free-jet portion of the Gulf duringwhichtheMRF filterattempts the sparseandnoisyobservations of theh field.Plate3 shows Stream.We note, however,that realisticTOPEX/POSEIDON data noisecanbe higher(4/9 cm) in lesstopologically active that by day 180 the predictionhasbeen able to reproduce all the mesoscale featuresin the truth.The RMS (lowsignal)oceanbasins andthatthealong-track correlation essentially (prediction error)betweenthetruthandprediction in datanoiseis ignoredwhilelumpingthesemeasurement and difference (Plate4, blacklines)displays an exponential decayin time, collocationerrors together. At a timestepAt of 20 min, the twinexperiment hasbeen with ane-foldingscaleof about30 days.Theseresultsindicate of the TOPEX/POSEIDON performed for360simulated days,duringwhichtheprediction that a half-yearto yearsequence
8010
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ET AL.: SPATIAL
REGRESSION
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Day 1
FOR KALMAN
FILTER
Day 2
_
Day 3
Day 4
ß•
_
Day 5
Day 6
Day 7
Day8
Day 9
Day 10
_
Figure 10. TOPEX-POSEIDONsatellitetracksoverthe 100 x 100grid (at 20 km gridsize)usedfor the Miami Isopycnal CoordinateOceanModel (MICOM) shallow-water model.Circlesrepresentthe observed gridpointsfor a givenday.Stippledlinesdenoteall the tracksfrom the previousdays.
altimetrydata can be usedto estimatethe entire prognostic intervalscan undersample dynamicmesoscale featuressuchas stateof a reduced-gravity, shallow-water dynamicsfor repro- advecting rings[Chassignet et al., 1992],whichdesirably should ductionof mesoscaleand possiblylarger-scalefeatures. not be compromised for computational convenience. 4.3.2. Data-update frequency. The duration of each Data-updateintervalsof 1, 2, 5, and 10 dayshavebeenused TOPEX/POSEIDON
satellite swath over the model domain is
well within the modeltime stepof 20 min. The data updates are thus performedonly two or three timesper day, correspondingto the numberof swathsper day (Figure 10). For computationaleconomy,as well as for possibledynamicbenefitsof increaseddata density,we haveconsidered performing data update(5) at an intervalsignificantly largerthan At, by accumulating the measurements duringthat intervalandtreating the accumulated data set as "synoptic." An updateinterval equalingthe repeatcycleof 10 daysis particularly attractive, not onlyfor the near-regular coverageduringthe assumed synoptic interval(Figure10) but alsofor possiblecomputational saving due to a steadystatefilter approximation[e.g.,Fukumoriand Malanotte-Rizzoli, 1995]enabledby the time independence in datapatternoverthisinterval.On the otherhand,largeupdate
to repeat the identical-twinexperiments.Plate 4 showsthe
resultingRMS predictionerrorsin time for eachupdateinterval. In general,the RMS error at a giventime increasesmonotonicallywiththe lengthof the updateintervals,indicatingthat systematic biaseshavebeenintroducedto the estimates owing to delaysin data update.For the longerintervalsof 5 and 10 days,more pronouncedoscillations in RMS error are observed after eachdata updateowingto larger data densities.These oscillations are especially noticeablein the error plotsfor the velocitycomponents (u, v). The steadystateestimatesare, nevertheless, stablefor eachupdate-interval, asexemplified by Plate 5, displaying the day-180predictions for the 2-dayand 10-day update intervals,where the mesoscalefeatures have been reproducedwell in both cases.The differencesbetween the two predictionin Plate 5 are in the detailsof the features,
CHIN
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8011
suchas the locationsand intensitiesof the rings.Such differencesin qualitativefeaturesare significantlysmallerbetween the casesof 2-dayand 20-minintervals(Plate 3, top), in accord with the RMS error values(Plate4). Theseimplythat the data can be updatedat an interval larger than the model time step without degradingfeature reproductionand filter stability,if the increasein the intervalis moderate,i.e., up to 2 daysin this particulartwin experiment. 4.3.3. Assimilation of generalizeddata. The generic observationequation(2) in the Kalmanfilter leadsto the weak
covariances. A highlycompactparameterizationis not only a computationalnecessitybut also inevitableby virtue, as error characterizations availableto data assimilationtend to be only qualitativeand/orempirical.For example,if a physicalor systematic descriptionof prediction errors were available, it is much more likely that this knowledgeis used to improve the prognosticequations(4) rather than the data assimilationprocedure.The two numericalapproachesexaminedin thispaper, subspaceprojectionand MRF-based regression,can effectively
Algebraically,it makesno differencewhether or not this data term containsphysicalmeasurements. Any extraneous constraint on the prognosticvariablesat a given time, e.g., a discretized diagnosticequation,can be expressed as a quadraticnorm and treatedas a data term. With thisperspective of data in Kalman filter, a data assimilation technology can,in principle,be applied for numericalincorporationof suchexternalconstraints as open boundaryconditions,air-seainterfacesin coupledmodels,and multiscale(e.g.,coastaland openocean)coupling. Plate 6 illustratesthat an arbitrarydiagnosticconstraintcanbe assimilatedas a conceptuallygeneralizedobservationusingthe MRF-approximatedfilter. The constraintin this caseis intended to form an intrusiveboundary,referredto as "coastline"hereafter, whichcanbe seenin Plate 6, bottom,asthe slantedright side of the vectorless triangularregion,referredto as"land,"near the left edgeof the domain.The equationsof the constraint,
projection,n canhavea wide rangeof valuesdependingon the subspacedimension.In particular, an EOF subspacetends to
represent theN x N errorcovariance matrixusingonlyn2 and constraint termI• - I-Ikxk 2in theleastsquares formulation (3). nN parameters,respectively,where n < N. For subspace have a very small n, e.g., n • 10 for casesin section 4.1. The
scale-dependent subspaceof waveletsusuallyrequiresa larger n (n/N is constrainedby the scaleconstant),while the spectral (wavenumber)mode subspaceis still larger. For the MRFbasedapproach,n is determinedby the order of approximation, e.g., n = 13 for the second-orderstructure(Figure 2b) used in our numerical experiments,all of which have been performed on a desktopworkstation. Our calculationswith simple but geophysicallyrelevant models of fluid flow have demonstratedthat the second-orderMRF parameterization (inhomogeneous and anisotropic)is both efficientand effective in producingnear-optimalestimatesof the prognosticvariables. The sequentialassimilationschemesapproximatingthe Kalman filter basedon the two parameterizationapproacheshave consistentlyproducedestimates(analyses)nearly identicalto 4u - • = 0 (22) thoseof nonapproximated Kalmanfilter, under a wide rangeof h- • (23) dynamicequationsand filter parametersexaminedin this pawhere• - 1000m isthemeanlayer-thickness, areimposed per. The numericalexperiments,conductedwith uncorrelated on the prognosticvariablesin the "land" area. The first equa- parametermatricesQ•,.and R/•,havepresenteda stringenttestof tion (22) is designedto constrainthe ocean current to flow regularityin the covariancestructureand hence stabilityof the parallelto the coastline.Note that the left-handsideof (22) is filter. It is then noted that stabilityof the subspaceprojection an inner product betweenthe currentvector (u, •) and the approachcan be dependenton choiceof the modal functions. normalvector(4, -1) that definesthe orientationof the coast- Both the empirical(e.g.,EOFs) and prescribed(e.g.,wavelets) while the MRF approach line. The secondequation,(23), is designedto null the fluid modeshaveshownsuchdependency, flow over land by smoothing.The algebraicconstraints(22) and (23) canbe organizedasH/•xk -- yk and then be incorporated into the shallow-watermodel dynamicsusinga Kalman filter. The initial and final estimatesproduced by the MRF filter, appliedover 12 days(864 time steps)to a 64 x 64 grid version of the model equation used in the twin experiment describedpreviously,are shownin Plate 6. The final estimate at day 12 displaysintended patterns of velocity and layerthicknessfields.Note that the modelperformsdynamicevolution of the statevariablesover the land throughoutasif it were "ocean";the appearanceof the triangularland regionis strictly
has been stable under all numerical
situations examined here.
The performance of the approximate filters, if stable, is generallyinsensitiveto the relativeweights(i.e., the ratio q/r) givento the model and data, especiallywhen the weightfor the data is at least that of the model (q/r >_ 1). This parameter insensitivityis encouragingwith respect to robustnessof the filter outputs,suchas analysesof satellitealtimetric data. Assimilation of simulated TOPEX/POSEIDON
data into a mul-
tilayer versionof MICOM has resultedin asymptoticallydiminishingRMS trackingerrors (to be presented),similar to those in the single-layercase discussedin section 4.3. For a muchmore sparselysampleddata set, assimilationoutputscan the effect of data assimilation. The numerical treatment of the coastlineboundaryconditionexaminedhere is applicable(in become sensitiveto the filter parameters,particularlyto the term of the algorithmic structure) to time-dependent con- correlationstructureof model error in Q/•, as noted by Bennett straintslike an open-boundarycondition as well. A realistic andBudgell[1987]and others.This impliesthat the correlation simulation of basin-scaleocean often requires a continuous scaleof the error covariancesdependspartly on the resolving incorporationof both measurements and extraneousboundary power of the observationnetwork at hand. The particularmethod of MRF parameterizationhas addiconditionsinto the model equations.The illustrationhere imtional possibilitiesthat remain to be investigated.First, the plies the possibilitythat a data assimilationschemelike the MRF framework has been used for statistical characterization MRF filter can be used as a single consistentframework for numericaltreatment of both typesof extraneousinformation. of geometricalpatternslike discontinuitycontoursand estimation of them usinga stochasticrelaxationprocedurelike simulated annealing[Gemanand Geman, 1984].This may lead to 5. Discussion and Summary an assimilationprocedurefor contour data suchas fronts and Successin data assimilationdependsheavily on character- rings detected on sea-surfacetemperature maps (satellite ization and parameterizationof the prediction/analysis error based). Second,the MRF parametersshould be able to be
8012
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dynamicallyupdatedby simulationof error ensemble[Evensen, 1994], as an alternative to the tangent linear approximation schemeusedin thispaper.This alternativeappearsto be useful for a predictionof MRF over multiple time steps.Finally, the use of diagnosticequations as "generalized data" (section 4.3.3) needsto be examinedfor practicalcasessuchas "assimilation" of open boundaryconditions.In summary,the Kalman filter generallyregardsthe prognosticequations(4) as a firstorder autoregression in time, on which the covarianceprediction equation (6) is based.It is then conceptuallyconsistent that the covariancestructurebe parameterizedin spaceby a regressionprocedureas well, and this has been achievedsuccessfullyhere with an MRF. Spatialregressionwith an MRF is implicative of a differential/differencediagnosticequation, such as the geostrophicbalance,which is a classicconstraint for the error fields in data analysis/assimilation.
WAVELET
FOR
KALMAN
FILTER
Further (diadic) reductionsin the subspacedimensioncan
beachieved bycascading. BylettingB("') denotethe2m x m subspace matrixwith the exactsamelocal structureas givenin (25), an/'th scale-levelsubspace is obtainedas
B = B(N/2)B(N/4)... B(N/2()
(26)
resulting in a N x N/2e subspace B for Kalmanfilterapproximation.
An advantageof this particularbiorthogonalwavelet transformation is that the basis functionsfor the pseudoinverse subspaceT (wavelet decomposition)are locally supported, promotingcomputationalefficiency.In particular,the columns
of Tr canbe obtainedin exactlythe samemannerascolumns of thecorresponding B, startingfromthescaling functions •b3'3 and (D4'4, ß
ß
0
Appendix A: Wavelet Subspaces
0
0.0378 0.0663
Considera discreteone-dimensionalperiodic domain consistingof N points. Compactlysupportedbasisfunctionsin a multiresolutionframeworkcanbe givenasthe so-calledscaling functionsof orthonormaland biorthogonalwavelettransforms [Daubechies,1992]. In particular,the symmetricbiorthogonal basisfunctionsgiven by Daubechies[1992, section8.3] have compactlysupportedinversesas well. For example,the two
-0.0238 -0.1989 -0.1106
-0.1547 0.3774 0.9944
0.8527
0.9944
(27)
0.3774
-0.1547 -0.1106 -0.1989
scalingfunctions, referredto as •b3'3and I• 4'4,usedin this
-0.0238 0.0663
paper are given by
0.0378 0 0 .
0
ß
-0.0645 -
whicharethecounterparts of i•)3'3andI•)4'4,respectively. Fig-
0.0407
ure 1 showsthe basisfunctionsassociated with the four scaling
0.1768
functions •b3'3,•4,4, (D3,3, and (D4'4,for the firstthreescale
0.4181
•3,3=0.5303 0.5303 '
0.7885
(24)
levels.
0.4181
0.1768 -0.0407
Appendix B: Time Recursion of Information
-
Matrix
0
0.0645
The Kalman
0
The basisvectorsbi in (9) are periodicallyshiftedversionsof these scalingfunctions.In the standardmultiresolutionanalysiswith diadic scaling,the scalingfunctionsare shiftedby two
spatialindices. For example, usingthe scaling function•b3'3,
0 0
0
0.5303
0
0
0.5303
0.1768
0.1768
0.5303
0 0
0
0.5303
0.1768
0
0.1768
0.5303
that results from
time recursion
O/• = (FkM•F• r + L•_•) " -• F/½Mk r
0
0.1768
filter
(25)
of the
information matrix Lk instead of the covariancePk is often referred to as the informationfilter [Lewis,1986].The MRFconformingrecursion(see (16)-(18)) is an approximationof suchfiltering equations.The nonapproximatedversionof the equationscan be retrievedby replacing(16) with
the subspaceB (whosecolumnsare bi) would look like
0
ß
(28)
A systematic wayto numericallyapproximatethisis to perform the matrix inverseon the right-hand side with the Jacobiiterations [Chin et al., 1992, 1994], where (16) representsthe simplestof suchapproximations. Since(17) and (18) are sums of quadratic terms, Lk can remain as positive semidefinite throughoutthe recursionfor any matrix Ok, i.e., for an arbitrary approximationof (28). If Ok has a full rank for eachk, then Lk would retain positivedefiniteness. This appendixpresentsa derivationof the exact recursion for the informationmatrix; Lk is not approximatedby trunca-
0
0
0.5303
tion here. While
0
0
0.1768
volvesexplicitcomputationof inverse(time reverse)dynamics
.
and have a dimension of N x N/2.
the usual form
of the information
filter in-
F•-1, the recursion steps(28), (17), and (18) avoidsuchan inversion.Duality with the standardrecursionof the covariancematrix Pk can also be recognized,when comparing(28)
CHIN ET AL.: SPATIALREGRESSIONAND WAVELET FOR KALMAN FILTER
8013
formulasandblockmatrix with(7),(17)with(8),and(18)with(6).Theoriginal recursionmumlikelihood(ML) estimation inversion techniques. (6)-(8)of thecovariance matrixin Kalman filtercorresponds Assumefor the momentthatthe regression operatorF•_ • is
to the followingdynamicsystemfor the error processek:
ek= Fke•_•+ w•
(29)
0 = H•e• + vk
(30)
available, sothat F•_ •ek_• = 5. Augmenting with (29) will yieldanML formulation fora jointestimation of e•_• ande• as
where% andv/• are zero-mean processes representing the modelanddataerrorprocesses, respectively, with respective
Because e•_ • isuncorrelated with% (a fundamental assumption in the standardKalmanfilter), thisML problemis equivderiveanequivalent timerecursion fortheinformation matrix alentto the optimalsingle-step forecasting for the estimation covariance matricesof Q/• and R/•. Our interesthere is to
Lk, defined asthesquare L/•-- F•F/• of theregression oper- error. The standardformula for the 2N x 2N posteriorestiator F/• satisfying(15). B1.
mation error covariance9• for the ML problemis [e.g.,Lewis, fiO?l
Information Matrices for Model and Data Errors
The information matrices associatedwith the model and
dataerrorprocesses % andv/•canbe obtained simplyasthe
inverses ofthecorresponding covariance matrices, M/•-=Q/• (34) andN/• -= R• •, yieldingtheweighting matrices in the least squares formulation (3). Fortheseerrorprocesses theregressionoperator canusually betranslated fromlineardiagnosticin whichthesquareof theregression operatorF•_ • hasbeen balance equations thataregivenor assumed. Thecorrespond-replaced by the information matrixL•_ • in the laststep.By inginformation matrixis thenobtained by squaring. For ex- partitioning thiscovariance matrixbyN x N blocksP,sas ample,JiangandGhil [1993](amongothers)haveassumed
that the model errors/residuals(u', v', h') in a twodimensional,shallow-waterdynamicsystemsatisfya geostro-
PllP•:]
•5: P21P22'
(35)
phicbalance, whichcanbewritten(assuming a proportionalitywecanidentifyfrom(33)thattheposterior covariance fore• is betweenthe dynamicpressure andlayerdepth)as P2:.The desired information matrixcanthusbe obtained as 0
u
c c')
•
fOy
(31)
fox h
- M; - M•F•(F/•MkF/, + L•_•) r , -1F/•M• r
(36)
The laststepis dueto a blockmatrixinversion formula,i.e.,
for a proportionality constant c andCoriolisparameter f. Whenthisequality isaccepted withsomeformof uncertainty
denoting(34) conciselyas
•:
(e.g.,replacing the right-hand sidewitha zero-mean uncorre-
L21L12] L22/ ' [L• -1
latedprocess withavariance ofw-t), theleftmatrixoperator it can be verifiedby substitution that Pl• = L• • + represents a regression operator fortheerrorvector[u', v', Lll-IL I 2P22L2 - 1, P22 -- (L22 - L21Lll -•L 12)-t ' P1-'> = 1L 11 h'] T.Thecorresponding information matrixMgcostrophy would -L(i•L 12P22, and P2 1 -P22L21L• • ' Equation (36) canbc be a finite difference version of shownto be equivalent to the recursion steps(28) and(17),
1
0 Mgcøstrøphy :W C 0
0
I C 0
c
0 1
fox
B3. Data Update of Information Matrix
To verifythedata-update step(18),we assume initiallythat
0
theregression operator Ff•isavailable, sothatF•e/•= 5.The
fOy c
=w 0
(2S) into (17).
roy C 0
y Oy• 0• 0 1
1
whichcanbeverifiedbyelimination of termsaftersubstituting
1 0 c0
O
fox
(32)
following ML estimation problemcanthenbe formulated by augmentation withtheobservation equation(30):
c2
cOcO f2V
f Oy fox
posterior covariance matrixisthedesired analysis error whereV is the Laplacianoperator.The differenceoperators whose
Usingthestandard ML formulaasbefore,wethen tendto maketheinformation matrixlikeMgeostrophy singular, covariance. have preventing anexplicit expression of thecorresponding covariancematrixQgeostrophy'
L• =- (P•) -•
B2. Forecastingof Information Matrix
We consider computing theinformation matrixL/•associated with the forecasterror, giventhe nearest-past analysis errorinformation L•_ • andthe forecastequation(29). The
keyalgebraic stepsinvolve application of thestandard maxi- which is (18).
r = L• + H•N•H•
(38)
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FILTER
Acknowledgments. We thank the Office of Naval Research for Evensen, G., Sequential data assimilationwith a nonlinear quasigeostrophicmodel using Monte Carlo methodsto forecast error funding this study through grants N00014-95-1-0257 (T.M.C. and statistics,J. Geophys.Res., 99, 10,143-10,162, 1994. A.J.M.) and N00014-93-1-0404(E.P.C.), as well as NASA through grant NAGW-4329. We are gratefulto Ichiro Fukumori(Jet Propul- Fukumori, I., and P. Malanotte-Rizzoli, An approximateKalman filter for ocean data assimilation:An example with an idealized Gulf sionLaboratory)for the TOPEX/POSEIDON satellitetrackdata.We Stream model,J. Geophys.Res., 100, 6777-6793, 1995. also appreciatehelpful commentson the manuscriptby Bob Miller, Geman, S., and D. Geman, Stochasticrelaxation, Gibbs distributions, Dimitris Menemenlis, and an anonymousreviewer. and the Bayesianrestorationof images,IEEE Trans.PatternAnal.
References Anderson,B. D. O., and J. B. Moore, OptimalFiltering,Prentice-Hall, Englewood Cliffs, N.J., 1979. Bennett, A. F., InverseMethodsin PhysicalOceanography, Cambridge Univ. Press, New York, 1992.
Bennett, A. F., and W. P. Budgell, Ocean data assimilationand the Kalman filter: Spatial regularity,J. Phys.Oceanogr.,17, 1583-1601, 1987.
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(ReceivedNovember 10, 1997;revisedNovember 10, 1998; acceptedNovember19, 1998.)
