On the theory of the divergence method for quantifying source emissions from satellite observations
Erik Koene, Dominik Brunner, Gerrit Kuhlmann
Abstract
The divergence method, a lightweight approach for estimating emission fluxes from satellite images, relies on a number of tacit assumptions. This paper explicitly outlines these assumptions by deriving the method from first principles. The assumptions are: the enhanced mass flux is dominated by advection, normal fluxes vanish at the top and bottom of the atmosphere, steady-state conditions apply, sources are multiplications of temporal and spatial functions, sinks are described as first-order reactions, and effective wind fields are made by weighing the fields with the enhanced concentration profiles. No such assumptions have to be assumed for the background field. The commonly used ‘topography correction term’ does not follow from this analysis and likely corrects data artifacts. The cross-sectional flux method follows naturally from the derived theory, and the methods are compared. Effects of discrete pixels and finite-difference operations are explored, leading to recommendations, primarily the recommendation to work with small regions only to minimize the influence of noise. Numerical examples featuring Gaussian plume and COSMO-GHG simulated plumes are provided. The Gaussian plume example suggests that the divergence method might underestimate emissions when assuming only advection in the presence of cross-wind diffusion. Conversely, the cross-sectional flux method remains unaffected, provided fluxes are integrated across the entire plume. The COSMO-GHG example reveals frequent violations of steady-state assumption, although the assumption remains valid proximal to the source (<20 km in this example). It is the hope that this paper provides a solid theoretical foundation for the divergence and cross-sectional flux methods.