Notes and correspondence an explanation for the diagonally predominant property of the positive symmetric ensemble transform matrix

Abstract

In the ensemble transform Kalman filter (ETKF), an ensemble transform matrix (ETM) is a matrix that maps background perturbations to analysis perturbations. All valid ETMs are shown to be the square roots of the analysis error covariance in ensemble space that preserve the analysis ensemble mean. ETKF chooses the positive symmetric square root Ts as its ETM, which is justified by the fact that Ts is the closest matrix to the identity I in the sense of the Frobenius norm. Besides this minimum norm property, Ts is observed to have the diagonally predominant property (DPP), i.e., the diagonal terms are at least an order of magnitude larger than the offdiagonal terms. To explain the DPP, first, the minimum norm property has been proved. Although ETKF relies on this property to choose its ETM, this property has never been proved in the data assimilation literature. The extension of this proof to the scalar multiple of I reveals that Ts is a sum of a diagonal matrix D and a full matrix P whose Frobenius norms are proportional, respectively, to the mean and the standard deviation of the spectrum of Ts. In general cases, these norms are not much different but the fact that the number of non-zero elements of P is the square of the ensemble size whereas that of D is the ensemble size causes the large difference in the orders of elements of P and D. However, the DPP is only an empirical fact and not an inherently mathematical property of Ts. There exist certain spectra of Ts that break the DPP but such spectra are rarely observed in practice since their occurrences require an unrealistic situation where background errors are larger than observation errors by at least two orders of magnitude in all modes in observation space.