Maximum likelihood estimation of error covariances in ensemble-based filters and its application to a coupled atmosphere-ocean model

We propose a method for estimating optimal error covariances in the context of sequential assimilation, including the case where both the system equation and the observation equation are nonlinear. When the system equation is nonlinear, ensemble-based filtering methods such as the ensemble Kalman filter (EnKF) are widely used to deal directly with the nonlinearity. The present approach for covariance optimization is a maximum likelihood estimation carried out by approximating the likelihood with the ensemblemean. Specifically, the likelihood is approximated as the samplemean of the likelihood of each member of the ensemble. To evaluate the sampling error of the proposed ensemble-approximated likelihood, we construct a method for examining the statistical significance using the bootstrap method without extra ensemble computation. We apply the proposed methods to an EnKF experiment where TOPEX/POSEIDON altimetry observations are assimilated into an intermediate coupled model, which is nonlinear, and estimate the optimal parameters that specify the covariances of the system noise and observation noise. Using these optimal covariance parameters, we examine the estimates by the EnKF and the ensemble Kalman smoother (EnKS). The effect of smoothing decreases by 1/e approximately one year after the filtering step. One of the properties of the smoothed estimate is that westerly wind anomalies over the western Pacific are not reproduced around the period of an El Niño event, while those over the central Pacific are strengthened. From additional experiments, we find that (1) the westerly winds in the western Pacific are phenomena outside of the coupled model and are not necessary tomodel ElNiño, (2) themodel El Niño is maintained by thewesterlies over the central Pacific, and (3) the modelled evolution process of the sea-surface temperature (SST) requires improvement to reproduce the westerly winds over the western Pacific.