Maximum likelihood estimation of covariance parameters for Bayesian atmospheric trace gas surface flux inversions

This paper introduces a Maximum Likelihood (ML) approach for estimating the statistical parameters required for the covariance matrices used in the solution of Bayesian inverse problems aimed at estimating surface fluxes of atmospheric trace gases. The method offers an objective methodology for populating the covariance matrices required in Bayesian inversions, thereby resulting in better estimates of the uncertainty associated with derived fluxes and minimizing the risk of inversions being biased by unrealistic covariance parameters. In addition, a method is presented for estimating the uncertainty associated with these covariance parameters. The ML method is demonstrated using a typical inversion setup with 22 flux regions and 75 observation stations from the National Oceanic and Atmospheric Administration-Climate Monitoring and Diagnostics Laboratory (NOAA-CMDL) Cooperative Air Sampling Network with available monthly averaged carbon dioxide data. Flux regions and observation locations are binned according to various characteristics, and the variances of the model-data mismatch and of the errors associated with the a priori flux distribution are estimated from the available data.