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.

This paper presents the formulation and preliminary results of a 3D ensemble‐variational data assimilation algorithm (3DEnVar) for the AROME‐France model at 3.8 km horizontal resolution. This algorithm is a deterministic, variational data assimilation scheme that uses background‐error covariances sampled from an ensemble. Our ensemble is an ensemble of data assimilation (EDA) at convective scale, based on the same system with the same spatial resolutions. In ensemble schemes, localization of the covariances is necessary to filter sampling noise. Two different localization schemes have been implemented, one in spectral space and one in grid‐point space. We also evaluate hybrid formulations, where the background‐error covariances are a weighted linear combination of the sampled covariances with the climatological ones. Cycled experiments are performed over a five‐week time period with 3 h updates. The 3DEnVar scheme largely outperforms standard 3D‐Var in terms of forecast scores. The best experiment is the one with the grid‐point localization scheme. A diagnostic of objective localization can provide guidance about the horizontal and vertical localization lengths that give best performance. The hybrid configuration with 80% of ensemble covariances and 20% of climatological ones performs also significantly better than the 3D‐Var, but to a lesser extent than the best 3DEnVar configuration, although it has better balanced initial fields.

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