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We use 2010-2015 observations of atmospheric methane columns from the GOSAT satellite instrument in a global inverse analysis to improve estimates of methane emissions and their trends over the period, as well as the global concentration of tropospheric OH (the hydroxyl radical, methane's main sink) and its trend. Our inversion solves the Bayesian optimization problem analytically including closed-form characterization of errors. This allows us to (1) quantify the information content from the inversion towards optimizing methane emissions and its trends, (2) diagnose error correlations between constraints on emissions and OH concentrations, and (3) generate a large ensemble of solutions testing different assumptions in the inversion. We show how the analytical approach can be used, even when prior error standard deviation distributions are lognormal. Inversion results show large overestimates of Chinese coal emissions and Middle East oil and gas emissions in the EDGAR v4.3.2 inventory but little error in the United States where we use a new gridded version of the EPA national greenhouse gas inventory as prior estimate. Oil and gas emissions in the EDGAR v4.3.2 inventory show large differences with national totals reported to the United Nations Framework Convention on Climate Change (UNFCCC), and our inversion is generally more consistent with the UNFCCC data. The observed 2010-2015 growth in atmospheric methane is attributed mostly to an increase in emissions from India, China, and areas with large tropical wetlands. The contribution from OH trends is small in comparison. We find that the inversion provides strong independent constraints on global methane emissions (546 Tg a(-1)) and global mean OH concentrations (atmospheric methane lifetime against oxidation by tropospheric OH of 10.8 +/- 0.4 years), indicating that satellite observations of atmospheric methane could provide a proxy for OH concentrations in the future.