Non-Gaussian measure in Gaussian Filtering Problem
The Ensemble Kalman Filter (EnKF) is known as an efficient Gaussian filter in high dimensional systems. However, when we apply EnKF to nonlinear chaotic systems, the PDF becomes non-Gaussian. The non-Gaussianity becomes severe when the observations are limited, i.e., sparse and infrequent, since the nonlinear dynamics plays a more significant role. Therefore, it is beneficial to find measures showing how non-Gaussian the regime is, so that we know how optimally EnKF is working in the given dynamics and observation. In this study, a 6-member Ensemble Transform Kalman Filter (ETKF) is applied to the 3-variable Lorenz63 model with several observing frequencies changing from 0.05 to 0.5 time units. We introduce a new measure of non-Gaussianity within the ETKF, or any other ensemble-based data assimilation methods. The method is summarized in the following 3 steps: (1) produce N independent ETKF analysis time series (or N replicas) with N independent identical twin experiments where a single nature run is used and observation noises are different, (2) average the N analysis time series to produce the replica mean, and (3) compute the difference between the nature run and the replica mean in each time step. The maximum value is used as the non-Gaussian measure. The replica mean should be closer to the nature run as N increases if EnKF is functioning optimally in the Gaussian regime. However, nonlinear dynamics and resulting non-Gaussian PDF cause the difference even if N goes to infinity. In this presentation, more details of the theory are discussed with the results from different observation frequencies.
Dr. Hideyuki Sakamoto RIKEN