A hybrid approach to estimating error covariances in variational data assimilation

Data assimilation (DA) involves the combination of observational data with the underlying dynamical principles governing the system under observation. In this work we combine the advantages of the two prominent DA systems: the 4D-Var and the ensemble methods. The hybrid method described in this paper consists of identifying the subspace spanned by the major 4D-Var error reduction directions. These directions are then removed from the background covariance through a Galerkin-type projection, and are replaced by estimates of the analysis error obtained through a low-rank Hessian inverse approximation. The updated error covariance in one window can be used as the background covariance for the next window thus better capturing the 'error of the day'. The numerical results for a non-linear model demonstrate how the hybrid method leads to a good estimate of the true error covariance, and improves the 4D-Var analysis results.