Metrics for assessing Linear Inverse Problems: a case study of a Trace Gas Inversion

Abstract. Multiple metrics have been proposed and utilized to assess the performance of linear Bayesian and geostatistical inverse problems. These metrics are mostly related to assessing reduction in prior uncertainties, comparing modeled observations to true observations, and checking distributional assumptions. These metrics though important should be augmented with sensitivity analysis to obtain a comprehensive understanding of the performance of inversions and critically improve confidence in the estimated fluxes. With this motivation, we derive analytical forms of the local sensitivities with respect to the number of inputs such as measurements, covariance parameters, covariates, and forward operator or jacobian. In addition to local sensitivity, we develop a framework for global sensitivity analysis that shows the apportionment of the uncertainty of different inputs to an inverse problem. The proposed framework is applicable to any other domain that employs linear Bayesian and geostatistical inverse methods. We show the application of our methodology in the context of an atmospheric inverse problem for estimating urban GHG emissions in Los Angeles. Within its context, we also propose a mathematical framework to construct correlation functions and components of uncertainty matrices from a pre-computed jacobian that encompasses non-stationary structures.