Regularization and tempering for a moment-matching localized particle filter

Iterative ensemble filters and smoothers are now commonly used for geophysical models. Some of these methods rely on a factorization of the observation likelihood function to sample from a posterior density through a set of "tempered" transitions to ensemble members. For Gaussian-based data assimilation methods, tangent linear versions of nonlinear operators can be relinearized between iterations, thus leading to a solution that is less biased than a single-step approach. This study adopts similar iterative strategies for a localized particle filter (PF) that relies on the estimation of moments to adjust unobserved variables based on importance weights. This approach builds off a "regularization" of the local PF, which forces weights to be more uniform through heuristic means. The regularization then leads to an adaptive tempering, which can also be combined with filter updates from parametric methods, such as ensemble Kalman filters. The role of iterations is analyzed by deriving the localized posterior probability density assumed by current local PF formulations and then examining how single-step and tempered PFs sample from this density. From experiments performed with a low-dimensional nonlinear system, the iterative and hybrid strategies show the largest benefits in observation-sparse regimes, where only a few particles contain high likelihoods and prior errors are non-Gaussian. This regime mimics specific applications in numerical weather prediction, where small ensemble sizes, unresolved model error, and highly nonlinear dynamics lead to prior uncertainty that is larger than measurement uncertainty.