Ensemble Kalman filter with the unscented transform

Abstract

A modification scheme to the ensemble Kalman filter (EnKF) is introduced based on the concept of the unscented transform [S. Julier, J. Uhlmann, H. Durrant-Whyte, A new method for the nonlinear transformation of means and covariances in filters and estimators, IEEE Trans. Automat. Control. 45 (2000) 477–482; S.J. Julier, J.K. Uhlmann, Unscented filtering and nonlinear estimation, Proc. IEEE 92 (2004) 401–422], which therefore will be called the ensemble unscented Kalman filter (EnUKF) in this work. When the error distribution of the analysis is symmetric (not necessarily Gaussian), it can be shown that, compared with the ordinary EnKF, the EnUKF has more accurate estimations of the ensemble mean and covariance of the background by examining the multidimensional Taylor series expansion term by term. This implies that, the EnUKF may have better performance in state estimation than the ordinary EnKF in the sense that the deviations from the true states are smaller. For verification, some numerical experiments are conducted on a 40-dimensional system due to Lorenz and Emanuel [E.N. Lorenz, K.A. Emanuel, Optimal sites for supplementary weather observations: Simulation with a small model, J. Atmos. Sci. 55 (1998) 399–414]. Simulation results support our argument.