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

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 120%), a new preconditioner that efficiently accumulates information on the diagonal of the inverse Hessian dramatically improves the results (relative error