The evaluation of the background error covariance matrices is an important aspect of any 3D-Var or 4D-Var data assimilation, as these matrices are a vehicle for spreading the innovation vectors spatially and to those model variables not explicitly used in the observation operators. It is generally recognized that the shortcomings in the existing operational systems result mainly from deficiencies in specifying the background error covariances, which are modeled through time invariant structure functions or the use of recursive filters. In this work, we propose a simple method to approximate the background error covariance matrices and compute their effect on the innovation vectors. Using an ensemble of perturbations as a representative sample for the background errors, we approximate the actual (unknown) covariance matrix with the ensemble covariance matrix. The square root of that matrix, needed in the 3D-Var and 4D-Var preconditioned incremental approach, is then approximated via a truncated matrix representation in terms of its orthonormal eigenvectors. The success of the method thus relies on a good choice for the ensemble members and the number of eigenvectors used to rewrite the ensemble covariance matrix. The advantage of the method is two fold: (1) the covariances are obtained naturally through vector inner products, and (2) the storage of real-size matrices in the computer memory is avoided. Preliminary results will be presented for mesoscale data assimilation.