Nonlinear stability of the ensemble Kalman filter with adaptive covariance inflation

TL;DR

This paper introduces an adaptive covariance inflation technique for ensemble Kalman filters that guarantees nonlinear stability, prevents catastrophic divergence, and improves accuracy in high-dimensional data assimilation tasks.

Contribution

It proposes a novel adaptive inflation method with a complete nonlinear stability theory, addressing stability issues in ensemble Kalman filters.

Findings

Adaptive inflation eliminates catastrophic divergence.

Enhanced stability allows use of unstable forecast integrators.

Numerical results show improved accuracy over standard methods.

Abstract

The Ensemble Kalman filter and Ensemble square root filters are data assimilation methods used to combine high dimensional nonlinear models with observed data. These methods have proved to be indispensable tools in science and engineering as they allow computationally cheap, low dimensional ensemble state approximation for extremely high dimensional turbulent forecast models. From a theoretical perspective, these methods are poorly understood, with the exception of a recently established but still incomplete nonlinear stability theory. Moreover, recent numerical and theoretical studies of catastrophic filter divergence have indicated that stability is a genuine mathematical concern and can not be taken for granted in implementation. In this article we propose a simple modification of ensemble based methods which resolves these stability issues entirely. The method involves a new type of adaptive covariance inflation, which comes with minimal additional cost. We develop a complete nonlinear stability theory for the adaptive method, yielding Lyapunov functions and geometric ergodicity under weak assumptions. We present numerical evidence which suggests the adaptive methods have improved accuracy over standard methods and completely eliminate catastrophic filter divergence. This enhanced stability allows for the use of extremely cheap, unstable forecast integrators, which would otherwise lead to widespread filter malfunction.