Improved analysis‐error covariance matrix for high‐dimensional variational inversions: application to source estimation using a 3D atmospheric transport model

Abstract

Variational methods are widely used to solve geophysical inverse problems. Although gradient‐based minimization algorithms are available for high‐dimensional problems (dimension >106), they do not provide an estimate of the errors in the optimal solution. In this study, we assess the performance of several numerical methods to approximate the analysis‐error covariance matrix, assuming reasonably linear models. The evaluation is performed for a CO2 flux estimation problem using synthetic remote‐sensing observations of CO2 columns. A low‐dimensional experiment is considered in order to compare the analysis error approximations to a full‐rank finite‐difference inverse Hessian estimate, followed by a realistic high‐dimensional application. Two stochastic approaches, a Monte‐Carlo simulation and a method based on random gradients of the cost function, produced analysis error variances with a relative error <10%. The long‐distance error correlations due to sampling noise are significantly less pronounced for the gradient‐based randomization, which is also particularly attractive when implemented in parallel. Deterministic evaluations of the inverse Hessian using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm are also tested. While existing BFGS preconditioning techniques yield poor approximations of the error variances (relative error >120%), a new preconditioner that efficiently accumulates information on the diagonal of the inverse Hessian dramatically improves the results (relative error <50%). Furthermore, performing several cycles of the BFGS algorithm using the same gradient and vector pairs enhances its performance (relative error <30%) and is necessary to obtain convergence. Leveraging those findings, we proposed a BFGS hybrid approach which combines the new preconditioner with several BFGS cycles using information from a few (3–5) Monte‐Carlo simulations. Its performance is comparable to the stochastic approximations for the low‐dimensional case, while good scalability is obtained for the high‐dimensional experiment. Potential applications of these new BFGS methods range from characterizing the information content of high‐dimensional inverse problems to improving the convergence rate of current minimization algorithms.