We describe a system for constraining the spatial distribution of fossil fuel emissions of CO2. The system is based on a modified Kaya identity which expresses emissions as a product of areal population density, per capita economic activity, energy intensity of the economy, and carbon intensity of energy. We apply the methodology of data assimilation to constrain such a model with various observations, notably, the statistics of national emissions and data on the distribution of nightlights and population. We hence produce a global, annual emission field at 0.25° resolution. Our distribution of emissions is smoother than that of the population downscaling traditionally used to describe emissions. Comparison with the Vulcan inventory suggests that the assimilated product performs better than downscaling for distributions of either population or nightlights alone for describing the spatial structure of emissions over the United States. We describe the complex structure of uncertainty that arises from combining pointwise and area-integrated constraints. Uncertainties can be as high as 50% at the pixel level and are not spatially independent. We describe the use of 14CO2 measurements to further constrain national emissions. Their value is greatest over large countries with heterogeneous emissions. Generated fields may be found online (http://ffdas.org/).