Dynamical structure functions in a four-dimensional variational assimilation: A case study

This paper contributes to the understanding of the structure functions used implicitly in the four-dimensional variational assimilation (4D-Var) developed at the European Centre for Medium-Range Weather Forecasts in the last few years. The theoretical equivalence between 4D-Var and the Kalman filter allows us to interpret (after normalization by the error standard deviations) the analysis increments produced by one single observation as the structure functions used implicitly in 4D-Var. The shape of the analysis increments provides a three-dimensional picture of the covariances of the background errors, modified by the dynamics. We study a baroclinic situation and observations have been regularly distributed along a latitude circle crossing the baroclinic wave. Eight standard pressure levels have been considered to sample the vertical. The forecast error standard deviations and the structure functions implied in 4D-Var may differ considerably from those used in the 3D-Var analysis. Unlike 3D-Var, the structure functions are flow dependent: the effective background error standard deviation can be four times larger and the correlation length scale twice as short in the vicinity of a low. A meridional extension of the experimentation at the surface shows that the effective background error standard deviations at 1000 hPa are largest in the areas of strong pressure gradient. We quantify the link between the analysis increments produced by 4D-Var and the fastest growing perturbations over the same time interval. In the depression, the explained variance of the analysis increments by the first 13 singular vectors reaches 30%. The impact of the temporal dimension is assessed. A period of 24 hours seems a minimum for the increments to develop fully baroclinic structures.