On the theory of the divergence method for quantifying source emissions
from satellite observations
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.