An Ensemble Kalman Filter Implementation Based on Modified Cholesky Decomposition for Inverse Covariance Matrix Estimation

TL;DR

This paper introduces EnKF-MC, an efficient ensemble Kalman filter implementation using modified Cholesky decomposition for sparse inverse covariance estimation, improving accuracy and reducing spurious correlations in data assimilation.

Contribution

It presents a novel EnKF implementation based on modified Cholesky decomposition, with convergence proof and superior performance over LETKF in various observation scenarios.

Findings

EnKF-MC reduces spurious correlations in covariance estimation.

The method outperforms LETKF in root mean square error.

Efficient sparse inverse covariance estimation enhances data assimilation accuracy.

Abstract

This paper develops an efficient implementation of the ensemble Kalman filter based on a modified Cholesky decomposition for inverse covariance matrix estimation. This implementation is named EnKF-MC. Background errors corresponding to distant model components with respect to some radius of influence are assumed to be conditionally independent. This allows to obtain sparse estimators of the inverse background error covariance matrix. The computational effort of the proposed method is discussed and different formulations based on various matrix identities are provided. Furthermore, an asymptotic proof of convergence with regard to the ensemble size is presented. In order to assess the performance and the accuracy of the proposed method, experiments are performed making use of the Atmospheric General Circulation Model SPEEDY. The results are compared against those obtained using the local ensemble transform Kalman filter (LETKF). Tests are performed for dense observations ($100\%$ and $50\%$ of the model components are observed) as well as for sparse observations (only $12\%$, $6\%$, and $4\%$ of model components are observed). The results reveal that the use of modified Cholesky for inverse covariance matrix estimation can reduce the impact of spurious correlations during the assimilation cycle, i.e., the results of the proposed method are of better quality than those obtained via the LETKF in terms of root mean square error.