On Two Localized Particle Filter Methods for Lorenz 1963 and 1996 Models

Nonlinear data assimilation methods like particle filters aim to improve the numerical weather prediction (NWP) in non-Gaussian setting. In this manuscript, two recent versions of particle filters, namely the Localized Adaptive Particle Filter (LAPF) and the Localized Mixture Coefficient Particle Filter (LMCPF) are studied in comparison with the Ensemble Kalman Filter when applied to the popular Lorenz 1963 and 1996 models. As these particle filters showed mixed results in the global NWP system at the German meteorological service (DWD), the goal of this work is to show that the LMCPF is able to outperform the LETKF within an experimental design reflecting a standard NWP setup and standard NWP scores. We focus on the root-mean-square-error (RMSE) of truth minus background, respectively, analysis ensemble mean to measure the filter performance. To simulate a standard NWP setup, the methods are studied in the realistic situation where the numerical model is different from the true model or the nature run, respectively. In this study, an improved version of the LMCPF with exact Gaussian mixture particle weights instead of approximate weights is derived and used for the comparison to the Localized Ensemble Transform Kalman Filter (LETKF). The advantages of the LMCPF with exact weights are discovered and the two versions are compared. As in complex NWP systems the individual steps of data assimilation methods are overlaid by a multitude of other processes, the ingredients of the LMCPF are illustrated in a single assimilation step with respect to the three-dimensional Lorenz 1963 model.