A Matrix-Free Posterior Ensemble Kalman Filter Implementation Based on a Modified Cholesky Decomposition

Elias Nino-Ruiz

doi:10.3390/atmos8070125

Abstract

In this paper, a matrix-free posterior ensemble Kalman filter implementation based on a modified Cholesky decomposition is proposed. The method works as follows: the precision matrix of the background error distribution is estimated based on a modified Cholesky decomposition. The resulting estimator can be expressed in terms of Cholesky factors which can be updated based on a series of rank-one matrices in order to approximate the precision matrix of the analysis distribution. By using this matrix, the posterior ensemble can be built by either sampling from the posterior distribution or using synthetic observations. Furthermore, the computational effort of the proposed method is linear with regard to the model dimension and the number of observed components from the model domain. Experimental tests are performed making use of the Lorenz-96 model. The results reveal that, the accuracy of the proposed implementation in terms of root-mean-square-error is similar, and in some cases better, to that of a well-known ensemble Kalman filter (EnKF) implementation: the local ensemble transform Kalman filter. In addition, the results are comparable to those obtained by the EnKF with large ensemble sizes.

Highlights

Data Assimilation is the process by which imperfect numerical forecasts and sparse observational networks are fused in order to estimate the state x∗ ∈ Rn×1 of a system [1] which evolves according to some model operator, x∗p = Mt p−1 →t p x∗p−1, for 1 ≤ p ≤ M, (1)
Where B ∈ Rn×n is the unknown background error covariance matrix, H : Rn×1 → Rm×1 is the observation operator, and R ∈ Rm×m stands for the data error covariance matrix
This paper proposes a posterior ensemble Kalman filter based on a modified Cholesky decomposition which works as follows: the precision matrix of the background error distribution is estimated in terms of Cholesky factors via a modified Cholesky decomposition, these factors are updated making use of a series of rank-one updates in order to estimate a precision matrix of the posterior distribution

Summary

Introduction

Data Assimilation is the process by which imperfect numerical forecasts and sparse observational networks are fused in order to estimate the state x∗ ∈ Rn×1 of a system [1] which (approximately) evolves according to some model operator, x∗p = Mt p−1 →t p x∗p−1 , for 1 ≤ p ≤ M , (1). In the context of sequential data assimilation, a forecast state xb ∈ Rn×1 is adjusted according to a real observation y ∈ Rm×1 , where m is the number of observed components from the numerical grid. X ∼ N xb , B , Atmosphere 2017, 8, 125; doi:10.3390/atmos8070125 (2). Where B ∈ Rn×n is the unknown background error covariance matrix, H : Rn×1 → Rm×1 is the observation operator, and R ∈ Rm×m stands for the data error covariance matrix. Covariance Estimation for Atmospheric CO2 Data Assimilation. Data Assimilation into a Primitive-Equation Model with a Parallel Ensemble Kalman Filter

Methods

Results

Conclusion