Radiative transfer solution using initial values in a scattering and absorbing atmosphere with surface reflection

The radiative transfer equation in a planar-stratified atmosphere with multiple scattering is solved by numerically integrating an ensemble of trial functions which are constructed so as to satisfy the boundary conditions (downward-propagating radiances) at the top of the atmosphere. The boundary conditions at the surface (reflection or scattering) are imposed after integration through the atmosphere. Opaque atmospheres constitute a special case of the latter boundary condition. The algorithm is very efficient because it requires solution, only once, of a set of linear equations of rank equal to half the number of radiation streams.