Ensemble Kalman filter implementations based on shrinkage covariance matrix estimation

This paper develops efficient ensemble Kalman filter (EnKF) implementations based on shrinkage covariance estimation. The forecast ensemble members at each step are used to estimate the background error covariance matrix via the Rao-Blackwell Ledoit and Wolf estimator, which has been specifically developed to approximate high-dimensional covariance matrices using a small number of samples. Two implementations are considered: in the EnKF full-space (EnKF-FS) approach, the assimilation process is performed in the model space, while the EnKF reduce-space (EnKF-RS) formulation performs the analysis in the subspace spanned by the ensemble members. In the context of EnKF-RS, additional samples are taken from the normal distribution described by the background ensemble mean and the estimated background covariance matrix, in order to increase the size of the ensemble and reduce the sampling error of the filter. This increase in the size of the ensemble is obtained without running the forward model. After the assimilation step, the additional samples are discarded and only the model-based ensemble members are propagated further. Methodologies to reduce the impact of spurious correlations and under-estimation of sample variances in the context of the EnKF-FS and EnKF-RS implementations are discussed. An adjoint-free four-dimensional extension of EnKF-RS is also discussed. Numerical experiments carried out with the Lorenz-96 model and a quasi-geostrophic model show that the use of shrinkage covariance matrix estimation can mitigate the impact of spurious correlations during the assimilation process.