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A conjugate BFGS method for accurate estimation of a posterior error covariance matrix in a linear inverse problem
Author(s) -
Niwa Yosuke,
Fujii Yosuke
Publication year - 2020
Publication title -
quarterly journal of the royal meteorological society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.744
H-Index - 143
eISSN - 1477-870X
pISSN - 0035-9009
DOI - 10.1002/qj.3838
Subject(s) - broyden–fletcher–goldfarb–shanno algorithm , covariance , covariance matrix , mathematics , conjugate gradient method , estimation of covariance matrices , matrix (chemical analysis) , algorithm , diagonal , mathematical optimization , inverse , inversion (geology) , diagonal matrix , computer science , statistics , materials science , composite material , computer network , geometry , asynchronous communication , paleontology , structural basin , biology
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 CO 2 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.