Stability of nonlinear filters - numerical explorations of particle and ensemble Kalman filters

Particle filters and ensemble Kalman filters are widely used in data assimilation but in the case of deterministic systems, which are quite commonly used in earth science applications, only a few theoretical results for their stability are available. Current numerical literature explores stability in terms of RMSE which, although practical, can not represent the distance between probability measures, convergence of which is what defines filter stability. In this study, we explore the distance between filtering distributions starting from different initial distributions as a function of time using Wasserstein metric, thus directly assessing the stability of these filters. These experiments are conducted on the chaotic Lorenz-63 and Lorenz-96 models for various initial distributions for particle and ensemble Kalman filters. We show that even in cases when both these filters are stable, the filtering distributions given by each of them may be distinct.