A conjugate BFGS method for accurate estimation of a posterior error covariance matrix in a linear inverse problem

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.

Carbon Monitoring System Flux Net Biosphere Exchange 2020 (CMS-Flux NBE 2020)

“…The data sources for the prior fluxes are listed in Table 7 and provided in the gridded fluxes. Methods to generate prior ocean carbon fluxes and fossil fuel emissions are documented in Brix et al (2015), Caroll et al (2020, and Oda et al (2018). The focus of this dataset is optimized terrestrial biosphere fluxes, so we briefly describe the prior terrestrial biosphere fluxes and their uncertainties.…”

A global CO<sub>2</sub> flux dataset (2015–2019) inferred from OCO-2 retrievals using the Tan-Tracker inversion system

“…where , +1 is the prior perturbations of the (r + 1)th window and , is the prior perturbations of the rth window. After obtaining the prior and posterior uncertainties of the scaling factors, the prior and posterior total flux uncertainties ( and ) can be calculated according to the correlation between fluxes and scaling factors as follows (Niwa and Fujii, 2020):…”

Estimation of fire-induced carbon emissions from Equatorial Asia in 2015 using in situ aircraft and ship observations

“…To link the atmospheric observations to surface carbon fluxes, we performed an inverse analysis of atmospheric CO 2 using the Nonhydrostatic Icosahedral Atmospheric Model (NICAM; Tomita and Satoh, 2004;Satoh et al, 2008Satoh et al, , 2014based Inverse Simulation for Monitoring CO 2 (NISMON-CO 2 ) (formerly NICAM-TM 4D-Var; Niwa et al, 2017a, b). The inversion system uses the NICAM-based transport model (NICAM-TM; Niwa et al, 2011). Using the same atmospheric transport model, Niwa et al (2012) performed a CO 2 inverse analysis and demonstrated a strong constraint of the CONTRAIL data for Equatorial Asia.…”