Improving spatial localization in 4DEnVar

The four‐dimensional ensemble‐variational (4DEnVar) formulation is a credible alternative to the 4D‐Var formulation, especially for numerical weather prediction centres that have invested a lot in this latter formulation during the last decades. First implementations of this technique, however, rely on a simplified form for the localization of the 4D covariances inside the assimilation period. It is shown in this article that the use of a unique localization for all cross‐covariances between perturbations at different times can be a crude approximation, especially in areas where the mean flow speed is large. To overcome this problem, a Lagrangian advection of the localization is proposed. It is first tested in the simplified Burgers' model and then introduced in the real‐size system associated with the French global model Action de Recherche Petite Echelle Grande Echelle (ARPEGE). The test of this advection in both environments shows a significant positive impact in regions where the advection is large. The possibility of using such a Lagrangian advection to evolve the static initial covariance matrix in a flow‐dependent way inside the assimilation period, in a hybrid 4DEnVar formulation, is also investigated.