Three-dimensional variational data assimilation for a limited area model - Part I : General formulation and the background error constraint
Assimilation variationnelle des données tri-dimensionnelle pour un modèle à domaine limité - Part I : Formulation générale et la contrainte des erreurs de départ
Gustafsson, N. ; Berre, L. ; Hörnquist, S. ; Huang, X.-Y. ; Lindskog, M. ; Navascués, B. ; K.S. Mogensen ; S. Thorsteinsson
Année de publication
2001
A 3-dimensional variational data assimilation (3D-Var) scheme for the <br>High Resolution Limited Area Model (HIRLAM) forecasting system is <br>described. The HIRLAM 3D-Var is based on the minimization of a cost <br>function that consists of one term <em>J</em><sub>b</sub>, which <br>measures the distance between the resulting analysis and a background <br>field, in general a short-range forecast, and another term <em>J</em><sub>0</sub>,<br> which measures the distance between the analysis and the observations. <br>This paper is concerned with the general formulation of the HIRLAM <br>3D-Var and with <em>J</em><sub>b</sub>, while the companion paper by Lindskog and co-workers is concerned with the handling of observations, including the <em>J</em><sub>o</sub><br> term, and with validation of the 3D-Var through extended parallel <br>assimilation and forecast experiments. The 3D-Var minimization requires a<br> pre-conditioning that is achieved by a transformation of the <br>minimization control variable. This change of variable is designed as an<br> operator approximating an inverse square root of the forecast error <br>covariance matrix in the model space. The main transformations are the <br>subtraction of the geostrophic wind increment, the bi-Fourier transform,<br> and the projection on vertical eigenvectors. The spectral bi-Fourier <br>approach allows one to derive non-separable structure functions in a <br>limited area model, in the form of vertically dependent horizontal <br>spectra and scale-dependent vertical correlations. Statistics have been <br>accumulated from differences between + 24 h and +48 h HIRLAM forecasts <br>valid at the same time. Results from single observation impact studies <br>as well as results from assimilation cycles using operational <br>observations are presented. It is shown that the HIRLAM 3D-Var produces <br>assimilation increments in accordance with the applied analysis <br>structure functions, that the fit of the analysis to the observations is<br> in agreement with the assumed error statistics, and that assimilation <br>increments are well balanced. It is also shown that the particular <br>problems associated with the limited area formulation have been solved. <br>These results, together with the results of the companion paper, <br>indicate that the 3D-Var scheme performs significantly better than the <br>statistical interpolation scheme.</div>
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