A Comparison of Error Subspace Kalman Filters

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Nerger, L. , Hiller, W. and Schröter, J. (2005): A Comparison of Error Subspace Kalman Filters , Tellus series a-dynamic meteorology and oceanographyA(5), 57 , pp. 715-735 . doi: 10.1111/j.1600-0870.2005.00141.x
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Three advanced filter algorithms based on the Kalman filter arereviewed and presented in a unified notation. They are the wellknown Ensemble Kalman filter (EnKF), the Singular EvolutiveExtended Kalman (SEEK) filter, and the less common SingularEvolutive Interpolated Kalman (SEIK) filter.For comparison, the mathematical formulations of the filters arereviewed in relation to the extended Kalman filter as errorsubspace Kalman filters. The algorithms are presented in theiroriginal form and possible variations are discussed. The comparisonof the algorithms shows their theoretical capabilities forefficient data assimilation with large-scale nonlinear systems. Inparticular, problems of the analysis equations are apparent in theoriginal EnKF algorithm due to the Monte Carlo sampling ofensembles. Theoretically, the SEIK filter appears to be anumerically very efficient algorithm with high potential for usewith nonlinear models.The superiority of the SEIK filter is demonstrated on the basis ofidentical twin experiments using a shallow water model with nonlinearevolution. Identical initial conditions for all three filters allowfor a consistent comparison of the data assimilation results. Theseshow how choices of particular state ensembles and assimilationschemes lead to significant variations of the filterperformance. This is related to different qualities of thepredicted error subspaces as is demonstrated in a examination ofthe predicted state covariance matrices.

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