The ``scattering equation'' 1 describes the transport of light and is derived in [7][3],
where 
is the intensity of the light at the
position 
in the direction 
, 
is the scattering (phase) function, 
is the absorption
coefficient and 
is density of the material.
Previous attempts at approximating the solution of this equation have placed restrictions on the phase function [13], or the number of light paths between the light source and the observer, usually to one [9][6][5][2] or many single scattering events [8]. However, as pointed out in [7] the single scattering approximation is not sufficient in the case of clouds due to excessive self shadowing. A recent unpublished paper [12] brought to the authors attention approaches the problem in similar fashion to this paper, from more of a radiosity point of view.
In [7] an appoximate solution for the scattering equation was derived by expressing the intensities on a 3D grid as the sum of spherical harmonics
then solving for the coefficients 
through a relaxation-like
technique. Since Kajiya and Von Herzen [7] never
implemented the technique, it remains to be seen how well it would
perform.
