Localization in the mapping particle filter

Data assimilation involves sequential inference in geophysical systems with nonlinear dynamics and observational operators. Non-parametric filters are a promising approach for data assimilation because they are able to represent non-Gaussian densities. The mapping particle filter is an iterative ensemble method that incorporates the Stein Variational Gradient Descent (SVGD) to produce a particle flow transforming state vectors from prior to posterior densities. At every pseudo-time step, the Kullback-Leibler divergence between the intermediate density and the target posterior is evaluated and minimized. However, for applications in geophysical systems, challenges persist in high dimensions, where sample covariance underestimation leads to filter divergence. This work proposes two localization methods, one in which a local kernel function is defined and the particle flow is global. The second method, given a localization radius, physically partitions the state vector and performs local mappings at each grid point. The performance of the proposed Local Mapping Particle Filters (LMPFs) is assessed in synthetic experiments. Observations are produced with a two-scale Lorenz system, while a one-scale Lorenz model is used as surrogate, introducing model error in the inference. The methods are evaluated with both full and partial observations, as well as with different linear and non-linear observational operators. The LMPFs with Gaussian mixtures in the prior density perform similarly to Gaussian filters such as the Ensemble Transform Kalman Filter (ETKF) and the Local Ensemble Transform Kalman Filter (LETKF) in most cases, and in some scenarios, they provide competitive performance in terms of analysis accuracy.