Comparison of the Extended Kalman Filter and the Ensemble Kalman Filter Using the Barotropic General Circulation Model

Abstract

In this study, we compare assimilation techniques of the full-rank extended Kalman filter (EKF) and the ensemble Kalman filter (EnKF), using a barotropic general circulation model, called barotropic S-model, under the perfect model configuration. We investigate the accuracy of the EnKF in reference to the direct computation of the EKF and examine the influence of the localization for EnKF. The barotropic S-model is based on the primitive equations and predicts the vertical mean state of the atmosphere. Although it has the predictability comparable to the operational prediction models, the direct computation of the EKF is possible. Therefore, we can assess the accuracy of the EnKF as a function of the ensemble members. In this study, the convergence of the EnKF to the EKF is examined using various ensemble members of 20, 50, 100, 410, and 1000. The EKF and EnKF directly assimilate the observation in the spectral space, and the observational elements are model variables.According to the result of the root mean square error (RMSE), the EnKF converges to the full-rank EKF filter when the ensemble member is increased to more than 50. It is demonstrated that the 20 ensemble members are insufficient with respect to the convergence. An empirical orthogonal function (EOF) analysis is conducted using the covariance matrices of analysis error for both filters. The structure of the first EOF (EOF-1) indicates the characteristics of the baroclinic instability waves in mid-latitudes in both filters, showing the same geographical distributions when it has converged. Interestingly, another large analysis error is detected in the Arctic region. Furthermore, the influence of the localization is examined by introducing the local ensemble transform Kalman filter (LETKF), which assimilates the observations in the physical space. The observations which are assimilated by the LETKF are retrieved from the spectral space to the physical space. It is found that the analysis error of the non-localized EnKF in the spectral space is smaller than that of the LETKF in the physical space.It is concluded from the comparison of the RMSE that more than 50 ensemble members are required for the non-localized EnKF to converge to the full-rank EKF for the practical assimilation in the spectral space under the perfect model configuration of the barotropic general circulation model of the atmosphere.