Robust Ensemble Filtering and Its Relation to Covariance Inflation in the Ensemble Kalman Filter
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 timelocal constraints instead, and hence we call it the timelocal $H_{\infty}$ filter (TLHF). By analogy to the ensemble Kalman filter (EnKF), we also propose the concept of ensemble timelocal $H_{\infty}$ 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 EnKFbased methods. We use some numerical examples to assess the relative robustness of the TLHF/EnTLHF in comparison with the corresponding KF/EnKF method.
 Publication:

Monthly Weather Review
 Pub Date:
 December 2011
 DOI:
 10.1175/MWRD1005068.1
 arXiv:
 arXiv:1108.0158
 Bibcode:
 2011MWRv..139.3938L
 Keywords:

 Physics  Data Analysis;
 Statistics and Probability;
 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 Accepted manuscript, to appear in Monthly Weather Review (Early online release available from the URL http://journals.ametsoc.org/doi/pdf/10.1175/MWRD1005068.1)