Estimating model error covariance matrix parameters in extended Kalman filtering

The extended Kalman filter (EKF) is a popular state estimation method for nonlinear dynamical models. The model error covariance matrix is often seen as a tuning parameter in EKF, which is often simply postulated by the user. In this paper, we study the filter likelihood technique for estimating the parameters of the model error covariance matrix. The approach is based on computing the likelihood of the covariance matrix parameters using the filtering output. We show that (a) the importance of the model error covariance matrix calibration depends on the quality of the observations, and that (b) the estimation approach yields a welltuned EKF in terms of the accuracy of the state estimates and model predictions. For our numerical experiments, we use the two-layer quasi-geostrophic model that is often used as a benchmark model for numerical weather prediction.