  Ensemble Kalman filter techniques are widely used to assimilate observations
into dynamical models. The phase space dimension is typically much larger than
the number of ensemble members which leads to inaccurate results in the
computed covariance matrices. These inaccuracies can lead, among other things,
to spurious long range correlations which can be eliminated by
Schur-product-based localization techniques. In this paper, we propose a new
technique for implementing such localization techniques within the class of
ensemble transform/square root Kalman filters. Our approach relies on a
continuous embedding of the Kalman filter update for the ensemble members,
i.e., we state an ordinary differential equation (ODE) whose solutions, over a
unit time interval, are equivalent to the Kalman filter update. The ODE
formulation forms a gradient system with the observations as a cost functional.
Besides localization, the new ODE ensemble formulation should also find useful
applications in the context of nonlinear observation operators and observations
arriving continuously in time.
