Kalman filtering and smoothing for linear wave equations with model error

TL;DR

This paper investigates the impact of model error on Kalman filtering for linear wave equations, revealing conditions for consistent recovery and proposing methods to mitigate errors in high-dimensional wave propagation systems.

Contribution

It provides a theoretical analysis of how model errors affect Kalman filtering and smoothing in wave equations, including conditions for consistency and partial recovery, and introduces a computational approach to address these issues.

Findings

No model error leads to true signal recovery in large data limit.

Small model errors can cause inconsistent recovery, especially with constant velocity shifts.

Time-dependent model errors allow exact filtering but not smoothing recovery.

Abstract

Filtering is a widely used methodology for the incorporation of observed data into time-evolving systems. It provides an online approach to state estimation inverse problems when data is acquired sequentially. The Kalman filter plays a central role in many applications because it is exact for linear systems subject to Gaussian noise, and because it forms the basis for many approximate filters which are used in high dimensional systems. The aim of this paper is to study the effect of model error on the Kalman filter, in the context of linear wave propagation problems. A consistency result is proved when no model error is present, showing recovery of the true signal in the large data limit. This result, however, is not robust: it is also proved that arbitrarily small model error can lead to inconsistent recovery of the signal in the large data limit. If the model error is in the form of a constant shift to the velocity, the filtering and smoothing distributions only recover a partial Fourier expansion, a phenomenon related to aliasing. On the other hand, for a class of wave velocity model errors which are time-dependent, it is possible to recover the filtering distribution exactly, but not the smoothing distribution. Numerical results are presented which corroborate the theory, and also to propose a computational approach which overcomes the inconsistency in the presence of model error, by relaxing the model.