A conjugate BFGS method for accurate estimation of a posterior error covariance matrix in a linear inverse problem. (9th August 2020)

• Record Type: Journal Article
• Authors: Niwa, Yosuke; Fujii, Yosuke
• Is Part Of: Quarterly journal of the Royal Meteorological Society. Volume 146:Number 732(2020)
• Page Start: 3118
• Page End: 3143
• Publication Date: 2020-08-09
• Subjects: 4D‐Var -- posterior error covariance -- BFGS formula -- inverse analysis; Meteorology -- Periodicals; 551.5
• DOI: 10.1002/qj.3838
• Languages: English
• ISSNs: 0035-9009

Abstract

Abstract: One effective data assimilation/inversion method is the four‐dimensional variational method (4D‐Var). However, it is a non‐trivial task for a conventional 4D‐Var to estimate a posterior error covariance matrix. This study proposes a method to estimate a posterior error covariance matrix applied to the linear inverse problem of an atmospheric constituent. The method was constructed within a 4D‐Var framework using a quasi‐Newton method with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. The proposed method was constructed such that conjugacy among the set of increment vector pairs was ensured. It is theoretically demonstrated that, when this conjugate property is coupled with preconditioning, an analytical solution of a posterior error covariance matrix could be obtained from the same number of vector pairs as observations. Furthermore, to accelerate the speed of convergence, the method can be coupled with an ensemble approach. By performing a simple advection test, it was confirmed that the proposed method could obtain an analytical matrix of the posterior error covariance within the same number of iterations as the observations. Furthermore, the method was also evaluated using an atmospheric CO2 inverse problem, which demonstrated its practical utility. The evaluation revealed that the proposed method could provide accurate estimates not only of the diagonal but also of the off‐diagonal elements of the posterior error covariance matrix. Although far more expensive than optimal state estimation, the computational efficiency was found to be reasonable for practical use, especially in conjunction with an ensemble approach. The accurate estimation of a posterior error covariance matrix resulting from the proposed method could provide valuable quantitative information regarding the uncertainties of estimated variables as well as the observational impacts, which would be beneficial for designing observation networks. Furthermore, error correlations derived from the estimated off‐diagonal elements could benefit the interpretation of optimised parameter variations. Abstract : This study proposes a new method for accurate and efficient estimation of a posterior error covariance matrix, in which conjugacy among the set of increment vector pairs is ensured. The accurate estimation of a posterior error covariance matrix resulting from the proposed method could provide valuable quantitative information regarding the uncertainties of estimated variables as well as the observational impacts. Furthermore, error correlations derived from the estimated off‐diagonal elements could benefit the interpretation of optimised parameter variations.