Covariance Matrix Estimation for Ensemble-Based Kalman Filters with Multiple Ensembles

We consider the implementation of ensemble-based Kalman filters (EnKF) in the framework of ensembles of different accuracies and sizes that are smaller than the number of degrees of freedom. From a noise-free ensemble and a noisy one assuming Gaussian errors for which the hyper-parameters of the distribution are known, we suggest a maximum likelihood approach on reduced-dimension subspaces to estimate forecast error pseudo-anomaly matrices. We show that they can be introduced in most of the widely-used EnKF analyses scheme with little intrusion into the data assimilation system. The performances of this approach are assessed by twin experiments done in the Lorenz-96 system and a quasi-geostrophic model. It is shown that the estimation of the forecast covariance error matrices with this approach did not damage the performance of the EnKF, and even resulted in an improvement of the analysis root mean square error for the QG model, highlighting the benefits of using optimized forecast error covariance matrices.