On numerical properties of the ensemble Kalman filter for data assimilation

Abstract

Ensemble Kalman filter (EnKF) has been widely used as a sequential data assimilation method, primarily due to its ease of implementation resulting from replacing the covariance evolution in the traditional Kalman filter (KF) by an approximate Monte Carlo ensemble sampling. In this paper rigorous analysis on the numerical errors of the EnKF is conducted in a general setting. Error bounds are provided and convergence of the EnKF to the exact Kalman filter is established. The analysis reveals that the ensemble errors induced by the Monte Carlo sampling can be dominant, compared to other errors such as the numerical integration error of the underlying model equations. Methods to reduce sampling errors are discussed. In particular, we present a deterministic sampling strategy based on cubature rules (qEnKF) which offers much improved accuracy. The analysis also suggests a less obvious fact — more frequent data assimilation may lead to larger numerical errors of the EnKF. Numerical examples are provided to verify the theoretical findings and to demonstrate the improved performance of the qEnKF.