Atmospheric methane is a potent greenhouse gas (GHG) and the second-largest contributor to anthropogenic climate forcing. After stabilizing in the early 2000s, the global methane concentration has sharply risen since 2007, mainly due to human-related activities. Curbing the rise of methane concentrations entails identifying and reducing methane emissions, which may otherwise significantly impact climate and air quality. Due to their near-continuous global coverage, satellite observations of methane are often combined with chemical transport models (CTMs) to improve model concentrations and emissions estimates. Previous methane studies are still faced with significant gaps and challenges such that considerable discrepancies among their results have been reported consistently. On the estimation side, most studies assumed that the model is perfect and characterization of uncertainties is already optimal. Obtaining information on methane uncertainties using conventional approaches requires extensive computational resources compared to model integration. Furthermore, there is a lack of independent and objective evaluation of those estimated uncertainties. The first thesis objective is to develop a novel cost-efficient data assimilation framework capable of estimating error statistics using a CTM. This method is referred to as parametric variance Kalman filter (PvKF), which relies on continuous formulation of error covariance propagation without making the perfect model assumption. We test the validity of our assumptions and the performance of the PvKF assimilation using simulated GOSAT observations. Our next goal is to conduct near-optimal assimilation to represent the true methane field. Cross-validation offers an objective manner to characterize the success of the method. We extend that method to the satellite observations and multiple covariance parameter estimations. Using estimated error statistics and GOSAT observations, we found that the quality of the analysis substantially depends on the optimality of those error covariances. Lastly, we evaluate the use of PvKF assimilation in a source inversion context in comparison with a traditional 4D-Var inversion. Using Observing System Simulation Experiments (OSSEs), we verify the ability of our new inversion framework to recover a distribution of known emissions. Our results indicate that both the analysis field and its error covariance exert a tangible influence in lowering the bias and variance of the recovered emissions.