Long-time stability and accuracy of the ensemble Kalman-Bucy filter for fully observed processes and small measurement noise

TL;DR

This paper analyzes the long-term stability and accuracy of the ensemble Kalman-Bucy filter in continuous-time data assimilation, providing theoretical guarantees and empirical validation for fully observed systems with small measurement noise.

Contribution

It offers the first theoretical analysis of the ensemble Kalman-Bucy filter's stability and accuracy, including mean field limits and consistency with classical filters.

Findings

Uniform-in-time accuracy and stability results for finite ensemble sizes.

Consistency with the classical Kalman-Bucy filter for linear Gaussian systems.

Empirical validation on the Lorenz-63 system.

Abstract

The ensemble Kalman filter has become a popular data assimilation technique in the geosciences. However, little is known theoretically about its long term stability and accuracy. In this paper, we investigate the behavior of an ensemble Kalman-Bucy filter applied to continuous-time filtering problems. We derive mean field limiting equations as the ensemble size goes to infinity as well as uniform-in-time accuracy and stability results for finite ensemble sizes. The later results require that the process is fully observed and that the measurement noise is small. We also demonstrate that our ensemble Kalman-Bucy filter is consistent with the classic Kalman-Bucy filter for linear systems and Gaussian processes. We finally verify our theoretical findings for the Lorenz-63 system.