An Efficient Implementation of the Ensemble Kalman Filter Based on an Iterative Sherman-Morrison Formula

TL;DR

This paper introduces a new efficient implementation of the ensemble Kalman filter using an iterative Sherman-Morrison formula, significantly reducing computational costs especially with large observation datasets.

Contribution

The paper presents a novel direct method for the EnKF that exploits matrix structure to improve efficiency without relying on matrix decompositions.

Findings

The new method matches existing implementations in accuracy.

It is significantly faster than traditional EnKF variants.

Performance remains robust even with large observation sets.

Abstract

We present a practical implementation of the ensemble Kalman (EnKF) filter based on an iterative Sherman-Morrison formula. The new direct method exploits the special structure of the ensemble-estimated error covariance matrices in order to efficiently solve the linear systems involved in the analysis step of the EnKF. The computational complexity of the proposed implementation is equivalent to that of the best EnKF implementations available in the literature when the number of observations is much larger than the number of ensemble members. Even when this conditions is not fulfilled, the proposed method is expected to perform well since it does not employ matrix decompositions. Computational experiments using the Lorenz 96 and the oceanic quasi-geostrophic models are performed in order to compare the proposed algorithm with EnKF implementations that use matrix decompositions. In terms of accuracy, the results of all implementations are similar. The proposed method is considerably faster than other EnKF variants, even when the number of observations is large relative to the number of ensemble members.