It is well understood that planet Earth is undergoing a period of severe climate change, brought on by decades of anthropogenic emission of greenhouse gases into our atmosphere. The increased abundance of greenhouse gases like carbon dioxide and methane has warmed our planet dramatically above pre-industrial levels. Rising global temperatures lead to increasingly frequent severe weather patterns and habitat destruction, and so we now find ourselves faced with the daunting task of stemming the tide of greenhouse gas emissions as quickly as possible. Although it is desirable that we decrease anthropogenic emissions of all greenhouse gases, a case can be made that reductions in methane emissions should be prioritised in order to mitigate the worst near-term effects of global warming. Methane is a much stronger greenhouse gas than carbon dioxide, capable of trapping 80 times more energy in our atmosphere over a 20-year timescale after emission. Methane also has a shorter atmospheric lifetime than carbon dioxide (just over a decade for methane compared to centuries for carbon dioxide), and thus reductions in methane emissions today will lead to a reduction in the global atmospheric abundance of methane in the near future. Over the past two decades, satellites have begun to be used to monitor greenhouse gases, and are crucial for maintaining accountability as nations commit to reductions in emissions. The latest such satellite is the TROPOspheric Monitoring Instrument (TROPOMI), capable of observing methane globally on a daily basis. However, TROPOMI observations of methane are often spatially disrupted due to cloud cover and other factors that prevent accurate retrievals of methane abundances. In this thesis, I present a Bayesian model capable of learning the extent to which TROPOMI observations of methane are spatially correlated with observations of nitrogen dioxide, and we use this model to spatially augment TROPOMI methane observations over the Permian basin in Texas. We then explore the efficacy of this model when used with TROPOMI observations of a variety of fossil fuel producing regions around the globe. Additionally in this thesis, I explore the effect of spatially disrupted TROPOMI observations on regional methane emission rate estimation. Regional methane emission rate estimates are crucial for providing timely updates on progress made towards national reductions in methane emissions. We find that spatially disrupted data may result in underestimated methane emission rates, and develop an optimised methodology for producing non-negative spatial maps of regional methane emission.