Fast Non-Localized Equal-Weight Particle Filters for Nonlinear Data Assimilation

Recent developments in nonlinear data assimilation for the geosciences have concentrated on particle filters with localization. Localization is a technique borrowed from Ensemble Kalman Filtering, in which observations are only allowed to influence the ocean area close to them. The need for this restriction is that, because of the relatively small ensemble size, spurious unphysical ensemble correlations can lead to artificial updates in remote ocean areas. In particle filtering the technique is used to limit the number of independent observations in an update area. This is because standard particle filters are degenerate when the number of independent observations is too large. However, even with localization the local areas contain too many observations to avoid degeneracy, and creating smooth posterior particles via gluing local particles together remains troublesome.

We propose another route that avoids localization and all of its problems. We explore the proposal-density freedom of particle filters, which allows one to change the model equations to get closer to the observations, at the expense of extra proposal weights. Although it has been shown that proposal densities cannot avoid degeneracy we realized that the family of proposal densities can be extended in a way that degeneracy can be avoided. The resulting so-called equal-weight particle filters do not solve the problem completely because of a bias they introduce in the system. As long as 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 recently managed to develop equal-weight particle filters that have unbiased first and second moments, with the outlook that more moments can be made unbiased by extending the number of proposal steps. We will outline the methodology and discuss its merits and drawbacks based on application to a highly nonlinear high-dimensional shallow-water system.