An Efficient Implementation of the Ensemble Kalman Filter Based on an Iterative Sherman-Morrison Formula

We present a practical implementation of the ensemble Kalman (EnKF) filter 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. Even when this conditions is not fulfilled, the proposed method is expected to perform well since it does not employ matrix decompositions. Computational experiments using the Lorenz 96 and the oceanic quasi-geostrophic models are performed in order to compare the proposed algorithm with EnKF implementations that use matrix decompositions. In terms of accuracy, the results of all implementations are similar. The proposed method is considerably faster than other EnKF variants, even when the number of observations is large relative to the number of ensemble members

Numerical Linear Algebra in Data Assimilation

Data assimilation is a method that combines observations (that is, real world data) of a state of a system with model output for that system in order to improve the estimate of the state of the system and thereby the model output. The model is usually represented by a discretised partial differential equation. The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we will derive the most important variational and sequential data assimilation approaches, in particular three-dimensional and four-dimensional variational data assimilation (3D-Var and 4D-Var) and the Kalman filter. We will then consider more advanced methods which are extensions of the Kalman filter and variational data assimilation and pay particular attention to their advantages and disadvantages. The data assimilation problem usually results in a very large optimisation problem and/or a very large linear system to solve (due to inclusion of time and space dimensions). Therefore, the second part of this article aims to review advances and challenges, in particular from the numerical linear algebra perspective, within the various data assimilation approaches.Comment: 31 pages, 2 figure