Radiative transport theory, at least in its present form,
began early this century. Since then the field has been greatly
advanced, with many excellent treatises on the subject now available
including works by Chandrasekhar (1960), Mihalas (1978) and
van de Hulst (1981).
Over the years it has been applied to many types of problems
in engineering and the
physical sciences including solar atmospheres, terrestrial atmospheres,
underwater radiation fields, reflective properties of many types
of surfaces, biophysical studies such as scattering and reflection by
blood cells and tissue, and in the study of high-temperature machinery
(Flatau and Stephens, 1988) to name a few. Examples of other applications
can be found in the
*Journal of Quantitative Spectroscopy and Radiative Transfer*,
**36**(1), 1986.

The transfer of solar radiation in terrestrial atmospheres
is governed by a
single equation:
the radiative transfer equation. Its solution results in a description
of the radiation field at each point in the atmosphere.
Much of the
progress over recent years deals with how this equation can be solved in
an efficient and accurate manner. As a result many different solution
techniques exist, each with its own advantages and disadvantages.
Among the more common are the doubling and adding method (*e.g.*: Plass *et al*.,
1973), the discrete ordinates method (*e.g.*: Stamnes and Conklin, 1984),
spherical harmonic method
(*e.g.*: Karp *et al*., 1980), Monte-Carlo solutions (*e.g.*: Collins *et al*., 1972),
invariant imbedding (*e.g.*: Bellman *et al*., 1963), the method of *X* and *Y*
functions (*e.g.*: Chandrasekhar, 1960), and finally the successive
orders of scattering method (*e.g.*: Irvine, 1975). It is the last method
that will be used in this study.

This chapter first presents the underlying physics describing how solar radiation interacts with atmospheric matter. Following this is a detailed outline to the solution of the radiative transfer equation and the assumptions made in arriving at it.