  This paper develops an efficient implementation of the ensemble Kalman filter
based on a modified Cholesky decomposition for inverse covariance matrix
estimation. This implementation is named EnKF-MC. Background errors
corresponding to distant model components with respect to some radius of
influence are assumed to be conditionally independent. This allows to obtain
sparse estimators of the inverse background error covariance matrix. The
computational effort of the proposed method is discussed and different
formulations based on various matrix identities are provided. Furthermore, an
asymptotic proof of convergence with regard to the ensemble size is presented.
In order to assess the performance and the accuracy of the proposed method,
experiments are performed making use of the Atmospheric General Circulation
Model SPEEDY. The results are compared against those obtained using the local
ensemble transform Kalman filter (LETKF). Tests are performed for dense
observations ($100\%$ and $50\%$ of the model components are observed) as well
as for sparse observations (only $12\%$, $6\%$, and $4\%$ of model components
are observed). The results reveal that the use of modified Cholesky for inverse
covariance matrix estimation can reduce the impact of spurious correlations
during the assimilation cycle, i.e., the results of the proposed method are of
better quality than those obtained via the LETKF in terms of root mean square
error.
