An efficient implementation of the ensemble Kalman filter based on an iterative Sherman–Morrison formula

Abstract

We present a practical implementation of the ensemble Kalman filter (EnKF) based on an iterative Sherman–Morrison formula. The new direct method exploits the special structure of the ensemble-estimated error covariance matrices in order to efficiently solve the linear systems involved in the analysis step of the EnKF. The computational complexity of the proposed implementation is equivalent to that of the best EnKF implementations available in the literature when the number of observations is much larger than the number of ensemble members, as typically is case in practice. Moreover, the proposed method provides the best theoretical complexity when it is compared to generic formulations of matrix inversion based on the Sherman–Morrison formula. The stability analysis of the proposed method is carried out and a pivoting strategy is discussed in order to reduce the accumulation of round-off errors without increasing the computational effort. A parallel implementation is discussed as well. Computational experiments carried out using an oceanic quasi-geostrophic model reveal that the proposed algorithm yields the same accuracy as other EnKF implementations, but scales better with regard to the number of observations.