Grid-point response to the incremental strategy for variational applications

The incremental approach is an efficient way to reduce the computational cost of four-dimensional variational assimilation. In this approach, the discrepancy between the forecast fields and the observations is calculated with a full physics model at high resolution, but the minimization is performed at a lower resolution with a simplified physics model. The incremental approach is also used in sensitivity-analysis applications where errors in the initial conditions are estimated a posteriori by replacing observations by a subsequent analysis. The incremental strategy as used in typical sensitivity analyses with observations corresponding to a full model state is re-examined in the context of a grid-point model by establishing how the grid-transfer operations in grid-point space influence the convergence properties of the whole incremental process. We show that two incremental grid-point formulations are possible. In the first formulation, the tangent-linear perturbations are interpolated at high resolution to be compared with the forecast errors. In the second one, the forecast errors are interpolated to low resolution before the minimization. Using a one-dimensional study and identical twin experiments in three dimensions, we study the impact of aliasing associated with the grid-transfer operations on the two incremental formulations. They are both affected by the misinterpretation of unresolved forecast errors leading to a contribution to resolved forecast errors. However, the first formulation has the property of damping this aliasing contribution and this effect is stronger with higher-order grid-transfer operators.