Fitting model fields to observations by using singular value decomposition: An ensemble-based 4DVar approach

An ensemble-based four-dimensional variational data assimilation (4DVar) method is proposed to fit the model field to 4-D 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 4-D perturbation solutions, so the fitting is computationally very efficient and does not require any adjoint integration. In the cost function 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 potential merits of the method are demonstrated by three sets of observing system simulation experiments performed with a shallow-water equation model. The method is shown to be robust even when the model is imperfect and the observations are incomplete.