4DEnVar: link with 4D state formulation of variational assimilation and different possible implementations

The four‐dimensional ensemble variational (4DEnVar) formulation is receiving increasing interest, especially in numerical weather prediction centres, which until now have mostly relied on the four‐dimensional variational (4D‐Var) formalism. It may indeed combine some of the best features of variational and ensemble methods. In this article, it is shown that the 4DEnVar formulation is linked with the 4D state formulation of variational assimilation, and that the 4DEnVar is relatively easy to precondition, in addition of being parallelizable. Practical implementations of the 4DEnVar are also investigated and two new preconditioned algorithms are proposed. The hybrid formulation of 4DEnVar, combining static and ensemble background‐error covariances, is discussed for the different possible algorithms. An application of the proposed implementations of 4DEnVar is shown with the Burgers model and compared to the use of 4D‐Var. Localized ensemble background error covariance between a point of a 1D domain and the neighbouring points distributed in space (x axis) and time (y axis). The paper shows that the 4DEnVar formulation is linked with the 4D state formulation of variational assimilation. Practical implementations of the 4DEnVar are investigated and two new preconditioned algorithms are proposed. The hybrid formulation of 4DEnVar, combining static and ensemble background error covariances is discussed. An application of the 4DEnVar is shown with the Burgers' model and compared to the use of 4D‐Var.