Two Localization Strategies for Sequential MCMC Data Assimilation with Applications to Nonlinear Non-Gaussian Geophysical Models

Abstract

We present a localized data assimilation (DA) scheme based on the sequential Markov Chain Monte Carlo (SMCMC) technique [Ruzayqat et al., 2024], a provably convergent method for filtering high-dimensional, nonlinear, and potentially non-Gaussian state-space models. Unlike particle filters, which are exact methods for nonlinear non-Gaussian models, SMCMC does not assign weights to samples and therefore avoids weight degeneracy in small-ensemble regimes. We design two localization approaches within the SMCMC framework that exploit spatial sparsity of observations to reduce the effective degrees of freedom and improve efficiency. The first variant collects observed blocks into a single reduced domain and runs parallel MCMC chains over this combined region. The second variant further reduces the per-chain state dimension by decomposing the observed region into independent blocks, each augmented with a compact halo, and applying Gaspari--Cohn observation-noise tapering to smoothly down-weight distant observations. When the observation model is linear and Gaussian, we show that our approximate filtering density reduces to a Gaussian mixture from which independent samples can be drawn exactly. For nonlinear or non-Gaussian observation models, we employ an MCMC kernel. We test on high-dimensional ($d \sim 10^4 - 10^5$) state-space models, including a linear Gaussian model and a nonlinear multilayer shallow water equation with both linear and nonlinear observation operators. We consider Gaussian and non-Gaussian (Student-$t$) observation noise, showing that LSMCMC naturally handles heavy-tailed errors that cause ensemble Kalman methods to diverge. Observations include synthetic and real data from the Surface Water and Ocean Topography (SWOT) mission (NASA) and ocean drifter data (NOAA). We compare the two variants against each other and the local ensemble transform Kalman filter (LETKF).