Abstract
The divergence method, a lightweight approach for estimating emission fluxes from satellite images, rests on a few implicit 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 concentration-weighted wind fields. No such assumptions have to be made for the background field. A “topography correction term” does not follow from the theory, but is rather shown to be a practical correction for topography-dependent effective wind speed errors. 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 integrate over small regions only to minimize the influence of noise. Numerical examples featuring Gaussian plumes 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 the 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.
