Nonlinear stability of the ensemble kalman filter with adaptive covariance inflation

Xin T. Tong, Andrew J. Majda, David Kelly

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)1283-1313
Number of pages31
JournalCommunications in Mathematical Sciences
Issue number5
StatePublished - 2016


  • Data assimilation
  • Ensemble kalman filter
  • Ergodicity of markov chains
  • Nonlinear stability

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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