Hiller, Wolfgang

Janjic Pfander, Tijana

Nerger, Lars

Schröter, Jens

English

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Alfred Wegener Institutefor Polar and Marine ResearchOn the Relation of the SEIK and ETKF Assimilation MethodsLars Nerger, Tijana Janjic´, Wolfgang Hiller, and Jens Schro¨terAlfred Wegener Institute for Polar and Marine Research, Bremerhaven, GermanyContact: Lars.Nerger@awi.de · http://www.awi.deXY351During the recent years several ensemble-based Kalmanfilter algorithms have been developed that have beenclassified as ensemble square-root Kalman filters. Par-allel to these developments, the SEIK (Singular ’Evolu-tive’ Interpolated Kalman) filter [1] has been introduced.Some publications note that the SEIK filter is an en-semble Kalman filter or even an ensemble square-rootKalman filter.We discuss the relation of the SEIK filter to ensemblesquare-root Kalman filters in more detail. For this, wecompare the SEIK filter with the Ensemble TransformKalman Filter (ETKF) [2]. The comparison is conductedon the algorithmic formulations as well as in an applica-tion to the nonlinear Lorenz96 [6] model.The equations for the SEIK and ETKF algorithmsare displayed on the right hand side. Because theequations are very similar, one has to be careful whencomparing the algorithms.• ETKF uses the ensemble perturbation matrix Z torepresent the estimated error space. In contrast,SEIK uses the basis of the error space in matrix L,which has one column less than Z.• The ensemble transformation is computed in differ-ent spaces. Matrix A of the SEIK filter is smaller than˜A of ETKF by one row and one column. Nonethe-less, both contain the same information on the errorspace.• The ensemble in the SEIK filter is reduced by one tothe basis of the error space. Thus, the last memberof the analysis ensemble has to be re-generated fromthis information. This is performed by the matrix Ω.• SEIK and ETKF compute the analysis state xa usingthe same error space information. Due to this, theanalysis states are identical, if the same forecast en-semble and the same set of observations is used.• In addition, the analysis ensembles of both filter al-gorithms will be equal when a particular choice forthe matrix Ω is used. It is obtained when the House-holder reflection orthogonal to the vector (1, . . . ,1)Tis applied to the identity matrix.• When Ω is chosen to be a random matrix, it servesfor the randomization of the analysis ensemble whichis sometimes suggested to avoid a loss of rank in theanalysis ensemble.• The SEIK filter is an ensemble square-root filter likethe ETKF. ETKF uses all ensemble perturbations torepresent the error space, while SEIK directly uses abasis of it.• With deterministic transformations, SEIK and ETKFbecome equivalent. Then, they result in the sameanalysis ensemble. This is the case if both filters usethe symmetric square root of the transformation ma-trix (A, ˜A) and SEIK uses a matrix Ω that projectsfrom the error space to the ensemble space.• An assimilation experiment with the Lorenz96 modelshowed small differences in the estimated state forboth the SEIK and ETKF filters. They are caused bythe finite numerical precision of the computations, inparticular singular value decompositions.• These findings unify the separate developments thathave been performed for the ensemble square-rootKalman filters and the SEIK filter.SEIK ETKF(The equations mostly follow the notations of [4] and [5])Some definitionsState vector xa ∈ Rn equal to SEIKEnsemble of N members Xa =[xa(1), . . . ,xa(N)], Xa ∈ Rn×N equal to SEIKPerturbation matrix Za = Xa−Xa, Xa = [xa, . . . ,xa] equal to SEIKAnalysis covariance matrix Pa = 1N−1Za(Za)T equal to SEIKError subspace basis L f = X f T, L f ∈ Rn×(N−1) not used in ETKFT-matrix T =(I(N−1)×(N−1)01×(N−1))− 1N(1N×(N−1))not used in ETKFAnalysis covariance matrix Pa = L f A(L f )T Pa = Z f ˜A(Z f )Twith transformation matrix A ∈ R(N−1)×(N−1) ˜A ∈ RN×NA−1 = (N−1)TT T+(HL f )T R−1HL f ˜A−1 = (N−1)I+(HZ f )T R−1HZ fState analysisxa = x f +L f wSEIK xa = x f +Z f wET KFwith weight vector wSEIK = A(HL f )T R−1(yo−Hx f)wETKF = ˜A(HZ f )TR−1(yo−Hx f)Square-root of analysis covariance matrixZa = L f WSEIK Za = Z f WETKFwith weight matrix WSEIK =√N−1CΩT WETKF =√N−1 ˜CΓand square-roots C, ˜C CCT = A ˜C ˜CT = ˜AC, ˜C can be the symmetric square root C = US−1/2UT fromthe singular value decomposition USV = A−1. Alternativesquare-roots like a Cholesky factorization are possible.Matrices Ω and Γ Ω can be an arbitrary N× (N−1) matrix withorthogonal columns orthogonal to (1, . . . ,1)T .Γ is a random rotationmatrix or the identity.Ensemble transformationXa = Xa +L f WSEIK ˜Xa = Xa +Z f WETKFTwin experiments were conducted using the nonlinearLorenz96 model [6]. Synthetic observations of the fullstate were generated from a model run. The ETKF andthe SEIK filter were used to assimilate the observationsat each time step over 50000 time steps. For SEIK, theconfiguration was used that makes it equivalent to theETKF (see “Comparison of Filters” on the left) as well asa configuration with a square-root based on Cholesky de-compostion. The global formulations of SEIK and ETKFwere used. These were sufficient for the Lorenz96 model,but require larger ensembles than localized filters for com-parable performance.10 20 30 400.90.920.940.960.981ETKF, determininistic Γ (Γ=I)forgetting factorensemble size 0.170.175 0.180.185 0.190.195  0.20.205 0.21 0.22 0.23 0.24 0.25  0.3  0.4  0.5  0.6  0.8    110 20 30 400.90.920.940.960.981SEIK−symmetric SQRT, determininistic Ωforgetting factorensemble size 0.170.175 0.180.185 0.190.195  0.20.205 0.21 0.22 0.23 0.24 0.25  0.3  0.4  0.5  0.6  0.8    110 20 30 400.90.920.940.960.981SEIK−Cholesky SQRT, determininistic Ωforgetting factorensemble size 0.170.175 0.180.185 0.190.195  0.20.205 0.21 0.22 0.23 0.24 0.25  0.3  0.4  0.5  0.6  0.8    110 20 30 400.90.920.940.960.981ETKF, random Γforgetting factorensemble size 0.170.175 0.180.185 0.190.195  0.20.205 0.21 0.22 0.23 0.24 0.25  0.3  0.4  0.5  0.6  0.8    110 20 30 400.90.920.940.960.981SEIK−symmetric SQRT, random Ωforgetting factorensemble size 0.170.175 0.180.185 0.190.195  0.20.205 0.21 0.22 0.23 0.24 0.25  0.3  0.4  0.5  0.6  0.8    110 20 30 400.90.920.940.960.981SEIK−Cholesky SQRT, random Ωforgetting factorensemble size 0.170.175 0.180.185 0.190.195  0.20.205 0.21 0.22 0.23 0.24 0.25  0.3  0.4  0.5  0.6  0.8    1The mean RMS errors over 10 experiments dependingon ensemble size and forgetting factor (covariance infla-tion). The SEIK filter configured to be equivalent to ETKFprovides an almost identical result to the ETKF. The smalldifferences are statistically not significant and caused bysingular value decompositions of the matrices A and ˜A,which have different condition numbers. Errors fromthe SEIK filter using a Cholesky decomposition of thetransformation matrix A are larger. This is caused byan inferior ensemble quality in which a small number ofensemble members carry most of the variance. With ran-dom transformations, the error levels become equivalent.[1] DT Pham et al. (1998). Singu-lar evolutive Kalman filters for dataassimilation in oceanography. C. R.Acad. Sci. Series II 326: 255–260[2] CH Bishop et al.. (2001). Adap-tive Sampling with the EnsembleTransform Kalman Filter. Part I: The-oretical Aspects. Mon. Wea. Rev.129: 420–436[4] L Nerger and WW Gregg. (2007).Assimilation of SeaWiFS data into aglobal ocean-biogeochemical modelusing a local SEIK filter. J. Mar.Syst. 68: 237–254[5] SC Yang et al. (2009). Weightinterpolation for efficient data as-similation with the Local EnsembleTransform Kalman Filter. Q. J. Roy.Met. Soc. 135: 251–262[6] EN Lorenz. (1996). Predictabil-ity - a problem partly solved. Pro-ceedings Seminar on Predictability,ECMWF, Reading, UK, 1–18.IntroductionComparison of FiltersConclusionFilter Equations for SEIK and ETKFAssimilation ExperimentReferences

On the relation of the SEIK and ETKF filter methods

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