A New Implicit Equal-Weights Particle Filter for High Dimensional Non-linear Data Assimilation

Abstract:

Unlike traditional data assimilation approaches, particle filters make no assumptions about the linearity or Gaussianity of the problem; given a sufficient number of particles they will correctly sample the posterior distribution even when applied to non-Gaussian, highly nonlinear dynamical systems.

In the geosciences, we are typically interested in very high dimensional problems that involve assimilating a large number of independent observations, but the number of particles we can sample is restricted by computational constraints; this is a real stumbling block for particle filters. In such cases the standard particle filter suffers from filter degeneracy whereby all, or most, of the weight becomes concentrated on a single particle and all variability in the particles is lost.

In recent years, a new class of particle filters that avoid divergence has been developed; these filters use the freedom of proposal densities to change the model equations and draw the particles closer to the observations, under the assumption that the observations lie in the high-probability region of the posterior pdf. This alone is not enough to avoid degeneracy, but the proposal densities can be extended in a way that ensures the majority of particles have equal, or almost equal, weights by construction. The resulting so-called equal-weight particle filters do not solve the problem completely because of bias they introduce in the system. If the bias is smaller than the Monte-Carlo error this is not too problematic, but it restricts the method to very small ensemble sizes.

We have re-formulated the original implicit equal-weights particle filter (IEWPF) in order to address some of its deficiencies; specifically, to eliminate a gap in the proposal distribution that leads to a systematic bias and underestimation of the filter variance. Our new IEWPF is expressly designed for high dimensional systems, offers flexibility in the choice of target weight, and requires no additional parameter tuning or localisation. We will outline the formulation of this new filter and discuss its merits and weaknesses based on application to a high dimensional version of the chaotic Lorenz 96 system.