On the Convergence of the Ensemble Kalman Filter

TL;DR

This paper proves that the ensemble Kalman filter converges to the classical Kalman filter as the ensemble size increases, using probabilistic methods and convergence theorems.

Contribution

It provides a rigorous proof of the convergence of the ensemble Kalman filter in the large ensemble limit.

Findings

Convergence of the ensemble sample covariance via a weak law of large numbers.

Convergence in probability of ensemble members using the continuous mapping theorem.

Establishment of $L^p$ convergence through bounds on the ensemble.

Abstract

Convergence of the ensemble Kalman filter in the limit for large ensembles to the Kalman filter is proved. In each step of the filter, convergence of the ensemble sample covariance follows from a weak law of large numbers for exchangeable random variables, the continuous mapping theorem gives convergence in probability of the ensemble members, and $L^p$ bounds on the ensemble then give $L^p$ convergence.