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 toreduce the computational cost of 4D variational data assimilation (4D-Var). In the incremental4D-Var, the tangent linear (TLM) and adjoint of a simplified lower-resolution model are usedto describe the time evolution of increments around a trajectory defined by a complete fullresolutionmodel. For nonlinear problems, the trajectory needs to be updated regularly byintegrating the full-resolution model during the minimization. These are referred to as outeriterations (or updates) by opposition to inner iterations done with the simpler TLM and adjointmodels 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 propertiesas well as to the interactions between the large (resolved by both models) and small scalecomponents of the flow. A 2D barotropic non-divergent model on a b-plane is used at twodifferent resolutions to define the complete and simpler models. Our results show that it isnecessary to have a minimal number of updates of the trajectory for the incremental 4D-Varto converge reasonably well. To assess the impact of restricting the gradient to its large scalecomponents, experiments are carried out with a so-called truncated 4D-Var in which the completemodel is used to compute the gradient which is truncated afterwards to retain only thosecomponents 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 approximatedby the lower resolution model. With frequent updates to the trajectory, the incremental 4D-Varconverges to an analysis which is close to that obtained with the truncated 4D-Var. Thisconclusion is verified when perfect observations with a complete spatial and temporal coverageare 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 resultsshowed that at some point, the minimization is overfitting the observations and degrades theanalysis. In this context, a criterion related to the level of observational noise is found todetermine when to stop the minimization when the complete 4D-Var is used. This criteriondoes not hold however for the incremental and truncated 4D-Var, thereby indicating that itmay be very difficult to establish in a more realistic context when the error is biased and themodel itself is introducing a biased error. The analysis and forecasts from the incremental4D-Var compare well to those from a full-resolution 4D-Var and are more accurate than thoseobtained from a low-resolution 4D-Var that uses only the simplified model.