An Ensemble-Based 4DVar Approach Based on SVD Technique

An ensemble-based four-dimensional variational data assimilation (4DVar) method is proposed to fit the model field to 4D observations in an increment form in the analysis step of data assimilation. The fitting is similar to that in the 4DVar but the analysis increment is expressed by a linear combination of the leading singular vectors extracted from an ensemble of 4D perturbation solutions, so the fitting is computationally very efficient and does not require any adjoint integration. In the costfunction used for the fitting, the background error covariance matrix is constructed implicitly by the perturbation solutions (through their representative singular vectors) similarly to that in the ensemble Kalman filter, but the perturbation solutions are not updated by the analysis into the next assimilation cycle, so the analysis is simpler and more efficient than that in the ensemble Kalman filter. The robustness and potential merits of the method are demonstrated by three sets of observing system simulation experiments (OSSEs) performed with a two-dimensional shallow water equation model. The method is shown to be robust even when the model is imperfect and the observations are incomplete. Furthermore, the method is not very sensitive to the SVD truncation as long as the truncation number is not small. And the method is moderately sensitive to observation density but not sensitive to the reduction of the analysis time window (from 12 to 6 hours) even for the imperfect-model case.