Robust ensemble filtering and its relation to covariance inflation in the ensemble Kalman filter

TL;DR

This paper introduces a robust ensemble filtering approach based on $H_{}$ filtering theory, which enhances robustness in data assimilation and clarifies the connection between covariance inflation and the ensemble Kalman filter.

Contribution

It develops the time-local $H_{}$ filter and its ensemble version, showing covariance inflation in EnKF as a special case of the new framework.

Findings

EnTLHF offers increased robustness over EnKF.

Covariance inflation in EnKF aligns with EnTLHF.

Numerical examples demonstrate improved robustness.

Abstract

We propose a robust ensemble filtering scheme based on the $H_{\infty}$ filtering theory. The optimal $H_{\infty}$ filter is derived by minimizing the supremum (or maximum) of a predefined cost function, a criterion different from the minimum variance used in the Kalman filter. By design, the $H_{\infty}$ filter is more robust than the Kalman filter, in the sense that the estimation error in the $H_{\infty}$ filter in general has a finite growth rate with respect to the uncertainties in assimilation, except for a special case that corresponds to the Kalman filter. The original form of the $H_{\infty}$ filter contains global constraints in time, which may be inconvenient for sequential data assimilation problems. Therefore we introduce a variant that solves some time-local constraints instead, and hence we call it the time-local $H_{\infty}$ filter (TLHF). By analogy to the ensemble Kalman filter (EnKF), we also propose the concept of ensemble time-local $H_{\infty}$ filter (EnTLHF). We outline the general form of the EnTLHF, and discuss some of its special cases. In particular, we show that an EnKF with certain covariance inflation is essentially an EnTLHF. In this sense, the EnTLHF provides a general framework for conducting covariance inflation in the EnKF-based methods. We use some numerical examples to assess the relative robustness of the TLHF/EnTLHF in comparison with the corresponding KF/EnKF method.