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

摘要

Variational methods are widely used to solve geophysical inverse problems. Although gradient-based minimization algorithms are available for high-dimensional problems (dimension >10(6)), 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.