|Title||Incremental 4D-Var convergence study|
|Year of Publication||2005|
|Secondary Title||Technical Memorandum|
Since its operational implementation at ECMWF, incremental 4DVar 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 reevaluate here the convergence of 4DVar at outer loop level with the current system. Experimental results show that 4DVar 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 timestep. This is explained by the presence of gravity waves which propagate at different speeds in the linear and nonlinear models. It is shown that these gravity waves are related to the shape of the leading eigenvector of the Hessian of the 4DVar cost function which is determined by surface pressure observation and which controls the behaviour of the minimisation algorithm. The influence of the choice of the inner loop minimisation algorithm and preconditioner is also presented. Finally, some directions for possible future configurations of incremental 4DVar are given.