A conjugate BFGS method for accurate estimation of a posterior error covariance matrix in a linear inverse problem
A conjugate BFGS method for accurate estimation of a posterior error covariance matrix in a linear inverse problem
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线性逆问题中后验误差协方差矩阵精确估计的共轭 BFGS 方法
DOI：
10.1002/qj.3838
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发表时间：
2020
影响因子：
8.9
通讯作者：
Fujii Yosuke
中科院分区：
文献类型：
--
作者：
Niwa Yosuke;Fujii Yosuke
一种有效的数据同化/反演方法是四维变分方法（4D-Var）。然而，对于传统的4D-Var来说，估计后验误差协方差矩阵是一项不平凡的任务。本研究提出一种估计后验误差协方差矩阵的方法，应用于大气成分的线性逆问题。该方法是在4D-Var框架内使用拟牛顿方法和Broyden-Fletcher-Goldfarb-Shanno（BFGS）算法构建的。所提出的方法被构造成使得增量向量对的集合之间的共轭性被确保。理论上证明，当这种共轭性质与预处理相结合时，可以从与观测相同数量的向量对中获得后验误差协方差矩阵的解析解。此外，为了加快收敛速度，该方法可以与集成方法相结合。通过一个简单的平流检验，证实了该方法可以在与观测值相同的迭代次数内获得后验误差协方差的分析矩阵。此外，该方法还使用大气CO2反问题进行了评估，这表明其实用性。评估表明，所提出的方法不仅可以提供后验误差协方差矩阵的对角元素的准确估计，而且还可以提供非对角元素的准确估计。虽然比最优状态估计昂贵得多，但计算效率对于实际使用来说是合理的，特别是与系综方法结合使用。该方法对后验误差协方差矩阵的精确估计可以为观测网的设计提供有关估计变量的不确定性以及观测影响的有价值的定量信息。此外，从估计的非对角元素导出的误差相关性可以有利于解释优化的参数变化。
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 CO2inverse 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.