Diagnostics of linear and incremental approximations in 4D-Var

The validity of the tangent linear approximation and of the incremental 4D-Var formulation is evaluated in the operational context. In ECMWF's operational system, the linear and adjoint models are run at a lower resolution than the nonlinear model. Furthermore, the physics are simpler in the tangent linear and adjoint than in the nonlinear model. Comparisons are made between the output of the linear model and the finite difference obtained by running the nonlinear model twice, with and without adding the analysis increment. The accuracy of the linear model is assessed with respect to resolution, linearized physics and the length of the assimilation window. The most striking results are that linearization errors are larger than expected and that large errors appear very early in the assimilation window. It is also shown that higher resolution 4D-Var will require more accurate linear physics than currently available. A modification of the computation of the trajectory around which the problem is linearized is shown to improve the accuracy of the linearization. The results and diagnostic tools presented should provide guidance for further developments of the incremental 4D-Var, with respect to resolution changes and improvements in linearized physics and dynamics.