Parallel Direct Solution of the Ensemble Square Root Kalman Filter Equations with Observation Principal Components

The ensemble square root Kalman filter (ESRF) is a variant of the ensemble Kalman filter used with deterministic observations that includes a matrix square root to account for the uncertainty of the unperturbed ensemble observations. Because of the difficulties in solving this equation, a serial approach is often used where observations are assimilated sequentially one after another. As previously demonstrated, in implementations to date the serial approach for the ESRF is suboptimal when used in conjunction with covariance localization, as the Schur product used in the localization does not commute with assimilation. In this work, a new algorithm is presented for the direct solution of the ESRF equations based on finding the eigenvalues and eigenvectors of a sparse, square, and symmetric positive semidefinite matrix with dimensions of the number of observations to be assimilated. This is amenable to direct computation using dedicated, massively parallel, and mature libraries. These libraries make it relatively simple to assemble and compute the observation principal components and to solve the ESRF without using the serial approach. They also provide the eigenspectrum of the forward observation covariance matrix. The parallel direct approach described in this paper neglects the near-zero eigenvalues, which regularizes the ESRF problem. Numerical results show this approach is a highly scalable parallel method.