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Large sample asymptotics for the ensemble Kalman filter

The ensemble Kalman filter (EnKF) has been proposed as a Monte Carlo, derivative-free, alternative to the extended Kalman filter, and is now widely used in sequential data assimilation, where state vectors of huge dimension (e.g. resulting from the discretization of pressure and velocity fields over a continent, as considered in meteorology) should be estimated from noisy measurements (e.g. collected at sparse in-situ stations). Even if the state and measurement equations are linear with additive Gaussian white noise, computing and storing the error covariance matrices involved in the Kalman filter is practically impossible, and it has been proposed to represent the filtering distribution with a sample (ensemble) of a few elements and to think of the corresponding empirical covariance matrix as an approximation of the intractable error covariance matrix. Extensions to nonlinear state equations have also been proposed. Surprisingly, very little is known about the asymptotic behaviour of the EnKF, whereas on the other hand, the asymptotic behaviour of many different classes of particle filters is well understood, as the number of particles goes to infinity. Interpreting the ensemble elements as a population of particles with mean-field interactions (and not merely as an instrumental device producing the ensemble mean value as an estimate of the hidden state), we prove the convergence of the EnKF, with the classical rate 1/\sqrt{N}, as the number N of ensemble elements increases to infinity. In the linear case, the limit of the empirical distribution of the ensemble elements is the usual (Gaussian distribution associated with the) Kalman filter, as expected, but in the more general case of a nonlinear state equation with linear observations, this limit differs from the usual Bayesian filter. To get the correct limit in this case, the mechanism that generates the elements in the EnKF should be interpreted as a proposal importance distribution, and appropriate importance weights should be assigned to the ensemble elements.