Data assimilation for the stochastic Camassa-Holm equation using particle filtering: a numerical investigation

TL;DR

This work tackles data assimilation for the stochastic Camassa–Holm equation with SALT transport noise and viscosity, casting the problem as Bayesian stochastic filtering and applying particle filters to estimate the posterior distribution of the signal given noisy observations. The authors develop and compare strategies—bootstrap, adaptive tempering, jittering, and Girsanov-based nudging—implemented with ensemble MPI parallelism and Firedrake adjoint capabilities to handle nonlinear optimization in the nudging step. Results from full-domain and half-domain observation scenarios show that bootstrap filters can diverge, while tempering with jittering stabilizes the filter and nudging can offer additional reductions in uncertainty, though with varying impact. The study demonstrates a scalable, high-fidelity data-assimilation framework for stochastic PDEs and outlines avenues for extending to higher dimensions, refining parameter choices, and improving parallel efficiency for practical, large-scale applications.

Abstract

In this study, we explore data assimilation for the Stochastic Camassa-Holm equation through the application of the particle filtering framework. Specifically, our approach integrates adaptive tempering, jittering, and nudging techniques to construct an advanced particle filtering system. All filtering processes are executed utilizing ensemble parallelism. We conduct extensive numerical experiments across various scenarios of the Stochastic Camassa-Holm model with transport noise and viscosity to examine the impact of different filtering procedures on the performance of the data assimilation process. Our analysis focuses on how observational data and the data assimilation step influence the accuracy and uncertainty of the obtained results.

Figures (11)

• Figure 1: Spatial and ensemble parallelism for an ensemble with 5 batches of particles, each executed in parallel over 5 processors, using 25 ranks in total. Spatial subcommunicators are used for domain decomposition algorithm for the iterative solvers involved in solving the forward equations for each particle, and ensemble subcommunicators are used to transfer particle states (and noises) during resampling.
• Figure 2: Initialization of all 150 particles and true state (the latter show in blue).
• Figure 3: Comparison of the evolution of the true state vs posterior ensemble and ensemble mean at data grids (weather stations). In order to assimilate data we use bootstrap partile filter and outcome is displayed for the mentioned assimilation steps.
• Figure 4: Evolution of the relative ensemble mean error (EMRE), relative bias (RB) and ensemble spread (RES) associated with bootstrap particle filter.
• Figure 5: Comparison of the evolution of the true state vs posterior ensemble and ensemble mean at data grids (weather stations). To assimilate data we use tempering and jittering and the outcome is displayed for the mentioned assimilation steps.