A Dynamic Likelihood Approach to Filtering for Advection-Diffusion Dynamics

TL;DR

This paper extends the Dynamic Likelihood Filter to advection-diffusion problems, providing a Bayesian data assimilation method that improves estimates in advection-dominated systems with sparse, uncertain observations.

Contribution

The paper introduces a split step formulation of the Dynamic Likelihood Filter for advection-diffusion, enhancing phase-aware Bayesian filtering in such systems.

Findings

DLF outperforms other methods in advection-dominated scenarios

The approach handles sparse, low-uncertainty observations effectively

Computational cost is comparable to standard Kalman filtering

Abstract

A Bayesian data assimilation scheme is formulated for advection-dominated advective and diffusive evolutionary problems, based upon the Dynamic Likelihood (DLF) approach to filtering. The DLF was developed specifically for hyperbolic problems -waves-, and in this paper, it is extended via a split step formulation, to handle advection-diffusion problems. In the dynamic likelihood approach, observations and their statistics are used to propagate probabilities along characteristics, evolving the likelihood in time. The estimate posterior thus inherits phase information. For advection-diffusion the advective part of the time evolution is handled on the basis of observations alone, while the diffusive part is informed through the model as well as observations. We expect, and indeed show here, that in advection-dominated problems, the DLF approach produces better estimates than other assimilation approaches, particularly when the observations are sparse and have low uncertainty. The added computational expense of the method is cubic in the total number of observations over time, which is on the same order of magnitude as a standard Kalman filter and can be mitigated by bounding the number of forward propagated observations, discarding the least informative data.