Modifying Covariance Localization to Mitigate Sampling Errors from the Ensemble Data Assimilation

The ensemble-based Kalman filter requires at least a considerable ensemble (e.g., 10,000 members) to identify relevant error covariance at great distances for multidimensional geophysical systems. However, increasing numerous ensemble sizes will enlarge sampling errors. This study proposes a modified Cholesky decomposition based on the covariance localization (CL) scheme, namely a covariance localization scheme with modified Cholesky decomposition (CL-MC). Our main idea utilizes a modified Cholesky (MC) decomposition technique for estimating the background error covariance matrix; meanwhile, we employ the tunable singular value decomposition method on the background error covariance to improve the ensemble increment and avoid the imbalance of the system. To verify if the proposed method can effectively mitigate the sampling errors, numerical experiments are conducted on the Lorenz-96 model and large-scale model (SPEEDY model). The results show that the CL-MC method outperforms the CL method for different data assimilation parameters (ensemble sizes and localizations). Furthermore, by performing one year of assimilation experiments on the SPEEDY model, it is found that the 1-day forecast RMSEs obtained by the CL are approximately equal to the 5-day forecast RMSEs of CL-MC. So, the CL-MC method has potential advantages for long-term forecasting. Maybe the proposed CL-MC method achieves good prospects for widespread application in atmospheric general circulation models.