Communications in Mathematical Sciences

Volume 14 (2016)

Number 5

Nonlinear stability of the ensemble Kalman filter with adaptive covariance inflation

Pages: 1283 – 1313

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n5.a5

Authors

Xin T. Tong (Centre for Atmosphere Ocean Science and Courant Institute of Mathematical Sciences, New York University, New York, N.Y., U.S.A.)

Andrew J. Majda (Centre for Atmosphere Ocean Science and Courant Institute of Mathematical Sciences, New York University, New York, N.Y., U.S.A.)

David Kelly (Courant Institute of Mathematical Sciences, New York University, New York, N.Y., 10012, U.S.A.)

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.

Keywords

ensemble Kalman filter, nonlinear stability, ergodicity of Markov chains

2010 Mathematics Subject Classification

37A25, 60G25, 62L12, 65C20, 93E11

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Published 18 May 2016