Surprisingly, the solution involves the diagonally predominant property of the unique positive symmetric ETM Ts, 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 SST 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 Ts. Experiments using real observations show that initial perturbations obtained from λI produce ensemble forecasts better than the ones obtained from the conventional ETM Ts 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
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