Metrics for evaluating the quality in linear atmospheric inverse problems: a case study of a trace gas inversion

Abstract. Several metrics have been proposed and utilized to diagnose the performance of linear Bayesian and geostatistical atmospheric inverse problems. These metrics primarily assess the reductions in the prior uncertainties, compare modeled observations to true observations, and check distributional assumptions. Although important, these metrics should be augmented with a sensitivity analysis to obtain a comprehensive understanding of the atmospheric inversion performance and improve the quality and confidence in the inverse estimates. In this study, we derive closed-form expressions of local sensitivities for various input parameters, including measurements, covariance parameters, covariates, and a forward operator. To further enhance our understanding, we complement the local sensitivity analysis with a framework for a global sensitivity analysis that can apportion the uncertainty in input parameters to the uncertainty associated with inverse estimates. Additionally, we propose a mathematical framework to construct nonstationary correlation matrices from a precomputed forward operator, which is closely tied to the overall quality of inverse estimates. We demonstrate the application of our methodology in the context of an atmospheric inverse problem for estimating methane fluxes in Los Angeles, California.