A validation of the incremental formulation of 4D variational data assimilation in a nonlinear barotropic flow
In order to meet current operational limitations, the incremental approach is being used to reduce the computational cost of 4D variational data assimilation (4D-Var). In the incremental 4D-Var, the tangent linear (TLM) and adjoint of a simplified lower-resolution model are used to describe the time evolution of increments around a trajectory defined by a complete fullresolution model. For nonlinear problems, the trajectory needs to be updated regularly by integrating the full-resolution model during the minimization. These are referred to as outer iterations (or updates) by opposition to inner iterations done with the simpler TLM and adjoint models to minimize a local quadratic approximation to the actual cost function. In this study, the role of the inner and outer iterations is investigated in relation to the convergence properties as well as to the interactions between the large (resolved by both models) and small scale components of the flow. A 2D barotropic non-divergent model on a β-plane is used at two diverent resolutions to define the complete and simpler models. Our results show that it is necessary to have a minimal number of updates of the trajectory for the incremental 4D-Var to converge reasonably well. To assess the impact of restricting the gradient to its large scale components, experiments are carried out with a so-called truncated 4D-Var in which the complete model is used to compute the gradient which is truncated afterwards to retain only those components used in the incremental 4D-Var. A comparison between the truncated and incremental 4D-Var shows that the large-scale components of the gradient are well approximated by the lower resolution model. With frequent updates to the trajectory, the incremental 4D-Var converges to an analysis which is close to that obtained with the truncated 4D-Var. This conclusion is verified when perfect observations with a complete spatial and temporal coverage are used or when they are restricted to be available at a coarser resolution (in space and time) than that of the model. Finally, unbiased observational error was introduced and the results showed that at some point, the minimization is overfitting the observations and degrades the analysis. In this context, a criterion related to the level of observational noise is found to determine when to stop the minimization when the complete 4D-Var is used. This criterion does not hold however, for the incremental and truncated 4D-Var, thereby indicating that it may be very diycult to establish in a more realistic context when the error is biased and the model itself is introducing a biased error. The analysis and forecasts from the incremental 4D-Var compare well to those from a full-resolution 4D-Var and are more accurate than those obtained from a low-resolution 4D-Var that uses only the simplified model.
Document Type: Research Article
Publication date: October 1, 1998