On the Convergence of a Non-linear Ensemble Kalman Smoother

TL;DR

This paper proves the convergence of ensemble Kalman smoothers and related algorithms to their classical counterparts as the ensemble size grows large, providing theoretical insights into their asymptotic behavior in high-dimensional data assimilation.

Contribution

It establishes the first rigorous convergence results for ensemble Kalman smoothers and EnKS-4DVAR, enhancing understanding of their asymptotic properties in large-ensemble limits.

Findings

Ensemble Kalman smoother converges to the Kalman smoother in L^p as ensemble size increases.

EnKS-4DVAR converges to the classical Levenberg-Marquardt algorithm with large ensembles.

Provides theoretical foundation for the asymptotic behavior of ensemble-based data assimilation methods.

Abstract

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.