A robust formulation of the Ensemble Kalman Filter

The ensemble Kalman filter (EnKF) can be interpreted in the more general context of linear regression theory. The recursive filter equations are equivalent to the normal equations for a weighted least squares estimate that minimizes a quadratic functional. Solving the normal equations is numerically unstable and subject to large errors when the problem is ill-conditioned. These errors can act synergistically with other sources of error typically present in mesoscale ensemble data assimilation systems, such as sampling error, and make solutions more susceptible to filter divergence.

A numerically stable and efficient generalized least squares (GLS) algorithm is presented, based on the minimization of an alternative functional. The method relies on orthogonal rotations, is highly parallel and equivalent to a deterministic algorithm that does not `square' the covariance square root to compute the Kalman gain. Computation of eigenvalue and singular value decompositions is not required. We present numerical results for the Lorenz (2005, J. Atmos. Sci.) two-scale dynamics, using perfect and imperfect models. Compared to the Ensemble Transform Kalman Filter (ETKF), Monte-Carlo filters, and other square-root formulations, the GLS-EnKF algorithm results in lower mean square errors and is less susceptible to filter divergence. Although the vector form of the GLS algorithm and the two-step serial Ensemble Adjustment Kalman Filter (EAKF) described by Anderson (2003, Monthly Weather Review) are fundamentally different, they both lead to the same error levels.