  Ensemble methods, such as the ensemble Kalman filter (EnKF), the local
ensemble transform Kalman filter (LETKF), and the ensemble Kalman smoother
(EnKS) are widely used in sequential data assimilation, where state vectors are
of huge dimension. Little is known, however, about the asymptotic behavior of
ensemble methods. In this paper, we prove convergence in L^p of ensemble Kalman
smoother to the Kalman smoother in the large-ensemble limit, as well as the
convergence of EnKS-4DVAR, which is a Levenberg-Marquardt-like algorithm with
EnKS as the linear solver, to the classical Levenberg-Marquardt algorithm in
which the linearized problem is solved exactly.
