Incremental 4D-Var convergence study

Since its operational implementation at European Centre for Medium-Range Weather Forecasts, the incremental fourdimensional variational data assimilation system (4D-Var) has run with two outer loop iterations. It has been shown in the past that more outer loop iterations were leading to the divergence of the algorithm. We re-evaluate here the convergence of 4D-Var at outer loop level with the current system. Experimental results show that 4D-Var in its current implementation does diverge after four outer loop iterations. Various configurations are tested and show that convergence can be obtained when inner and outer loops are run at the same resolution, or at least with the same time-step. This is explained by the presence of gravity waves which propagate at different speeds in the linear and non-linear models. It is shown that these gravity waves are related to the shape of the leading eigenvector of the Hessian of the 4D-Var cost function which is determined by surface pressure observation and which controls the behaviour of the minimization algorithm. The influence of the choice of the inner loop minimization algorithm and preconditioner is also presented. Finally, some directions for possible future configurations of incremental 4D-Var are given