The Diagonally Predominant Property of the Positive Symmetric Ensemble Transform Matrix and Its Application in Ensemble Forecast.

An ensemble generation method based on the ensemble transform Kalman filter (ETKF) has been developed with the following properties: (1) the dimension of error subspace is maximized given a number of ensemble members; (2) analysis perturbations represent analysis error covariances; and (3) analysis perturbations run independently from each other. While the two first properties are inherited from ETKF, the third property is inspired from the breeding method. These properties together dictate a special form for the ensemble transform matrix (ETM) in ETKF: that is a scalar multiple of the identity I. However, all valid ETMs in ETKF do not support such form of ETM and the best we can do is to find the best approximation under this form.

Surprisingly, the solution involves the diagonally predominant property of the unique positive symmetric ETM T^s, i.e. the diagonal elements are at least an order of magnitude larger than the off-diagonal elements. This property follows from an important theorem for factorization SS^T of a symmetric positive-definite matrix M: among all factorizations the positive symmetric square root yields the closest matrix to a scalar multiple of I. This theorem points out that the best approximation is λI where λ is the average of eigenvalues of T^s. Experiments using real observations show that initial perturbations obtained from λI produce ensemble forecasts better than the ones obtained from the conventional ETM T^s and the breeding method. This outperformance arises from two factors: independent run of perturbations and the built-in inflation of the ETM λI.

Keywords: ensemble transform Kalman filter, ensemble transform matrix, positive symmetric square root, diagonally predominant property, inflation