  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 developed specifically
developed to approximate high-dimensional covariance matrices using a small
number of samples. 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 initial members are propagated.
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. Numerical experiments carried out with a
quasi-geostrophic model show that the proposed implementations outperform
current methods such as the traditional EnKF formulation, square root filters,
and inflation-free EnKF implementations. The proposed implementations provide
good results with small ensemble sizes ($\sim 10$) and small percentages of
observed components from the vector state. These results are similar (and in
some cases better) to traditional methods using large ensemble sizes ($\sim
80$) and large percentages of observed components. The computational times of
the new implementations remain reasonably low.
