An efficient matrix-free algorithm for the ensemble Kalman filter

In this work, we present an efficient matrix-free ensemble Kalman filter (EnKF) algorithm for the assimilation of large data sets. The EnKF has increasingly become an essential tool for data assimilation of numerical models. It is an attractive assimilation method because it can evolve the model covariance matrix for a non-linear model, through the use of an ensemble of model states, and it is easy to implement for any numerical model. Nevertheless, the computational cost of the EnKF can increase significantly for cases involving the assimilation of large data sets. As more data become available for assimilation, a potential bottleneck in most EnKF algorithms involves the operation of the Kalman gain matrix. To reduce the complexity and cost of assimilating large data sets, a matrix-free EnKF algorithm is proposed. The algorithm uses an efficient matrix-free linear solver, based on the Sherman–Morrison formulas, to solve the implicit linear system within the Kalman gain matrix and compute the analysis. Numerical experiments with a two-dimensional shallow water model on the sphere are presented, where results show the matrix-free implementation outperforming an singular value decomposition-based implementation in computational time.