In order to use the results of Sections 19.10—19.12 in various practical applications, one needs efficient techniques for solving the RTE in either the integral or the integro-differential form. Unfortunately, like many other integral and integro-differential equations, the RTE is difficult to solve analytically or numerically. In order to facilitate the solution, it is customary to make several simplifying assumptions. The most typical of them, which will be used throughout this chapter, are the assumptions that the particulate medium:

1. • is plane parallel;

2. • has an infinite horizontal extent; and

3. • is illuminated from above by a plane electromagnetic wave or a parallel polychromatic beam with quasi-monochromatic components.

These assumptions mean that all statistically averaged optical properties of the medium and all observable characteristics of the radiation field may vary only in the vertical direction and are independent of the horizontal coordinates. Taken together, these assumptions specify the so-called standard one-dimensional problem of the RTT and provide a model relevant to a great variety of applications in diverse fields of science and engineering.

To simplify the standard problem even further, we will also assume that the particulate medium is populated by statistically isotropic and mirror-symmetric random particles and use the extinction matrix given by Eq. (15.42) and the phase matrix given by Eqs. (15.20) and (15.21).

In this chapter we will derive several general equations describing the radiation field in the particular case of plane-parallel scattering geometry.