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 > 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 CO 2 flux estimation problem using synthetic remote-sensing observations of CO 2 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 MonteCarlo 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.

Satellite observations of atmospheric methane and their value for quantifying methane emissions

“…A drawback is that error characterization is not included as part of the solution. Approximate methods are available at additional computational cost to estimate the posterior error covariance matrixŜ and from there the averaging kernel matrix A (Bousserez et al, 2015).…”

“…Given the prior estimate (x A , S A ) informed by the current observing system without the proposed instrument, we can quantify the information added by the proposed instrument by computingŜ from Eq. (9) or an adjoint-based approximation (Bousserez et al, 2015). From there we obtain the averaging kernel matrix A = I n −ŜS −1 A and the DOFS, and compare to the DOFS without the instrument to quantify the information to be gained.…”

“…This could enable separation between the local/regional methane enhancement near the surface and the higher-altitude methane background (Bousserez et al, 2015). Such multi-spectral retrievals are not yet produced operationally because of computational requirements and because of limitations in the quality and calibration of spectra across different detectors (Hervé Herbin, personal communication, 2016).…”

Constraints on methane emissions in North America from future geostationary remote-sensing measurements

“…In this study, following Bousserez et al (2015), the diagonal elements of P a (error variances) are computed using a randomization estimate of H T R −1 H. Here an ensemble of 500 random gradients of the cost function are used, based on the convergence of the uniform norm ( . ∞ ) of the inverse Hessian approximation.…”

“…Indeed, many perturbed inversions (typically about 50) are needed, each of them requiring numerous forward and adjoint model integrations (iterations) in case the problem is not well conditioned (about 50 iterations for our methane inversion). Alternatively, inverse Hessian approximations based on information from the minimization itself can be employed, but are usually of very low rank (e.g., Meirink et al, 2008;Bousserez et al, 2015). Therefore, most information content analyses in previous trace-gas Bayesian inversion studies have relied on explicit calculations of the inverse Hessian matrix, by either considering a regional domain (e.g., Wecht et al, 2014a) or performing a prior dimension reduction of the control vector (e.g., Wecht et al, 2014b;Turner and Jacob, 2015).…”