  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.
