Three-dimensional variational data assimilation for a limited area model Part I:: General formulation and the background error constraint

A 3-dimensional variational data assimilation (3D-Var) scheme for the HIgh Resolution Limited Area Model (HIRLAM) forecasting system is described. The HIRLAM 3D-Var is based on the minimization of a cost function that consists of one term J(b). which measures the distance between the resulting analysis and a background field, in general a short-range forecast. and another term J(o). which measures the distance between the analysis and the observations. This paper is concerned with the general formulation of the HIRLAM 3D-Var and with Jb. while the companion paper by Lindskog and co-workers is concerned with the handling of observations, including the J(o) term, and with validation of the 3D-Var through extended parallel assimilation and forecast experiments. The 3D-Var minimization requires a pre-conditioning that is achieved by a transformation of the minimization control variable. This change of variable is designed as an operator approximating an inverse square root of the forecast error covariance matrix in the model space. The main transformations are the Subtraction of the geostrophic wind increment, the bi-Fourier transform, and the projection on vertical eigenvectors. The spectral bi-Fourier approach allows one to derive non-separable structure functions in a limited area model. in the form of vertically dependent horizontal spectra and scale-dependent vertical correlations. Statistics have been accumulated from differences between +24 h and +48 h HIRLAM forecasts valid at the same time. Results from single observation impact studies as well as results from assimilation cycles using operational observations are presented. It is shown that the HIRLAM 3D-Var produces assimilation increments in accordance with the applied analysis structure functions, that the fit of the analysis to the observations is in agreement with the assumed error statistics. and that assimilation increments are well balanced. It is also shown that the particular problems associated with the limited area formulation have been solved. These results, together with the results of the companion paper, indicate that the 3D-Var scheme performs significantly better than the statistical interpolation scheme.