This thesis consists of two parts. The first one is "Scattering of Waves in a Random Medium," and the second one is "Radiative Transfer in a Sphere Illuminated by a Parallel Beam: An Integral Equation Approach."
In the first part, a new formalism tor partially coherent wave scattering in a random medium is developed. In this formalism the coherent wave is the solution of a phenomenological wave equation, and the mutual coherence function of the wave field satisfies a simple integral equation. Using this formalism, the Peierls equation can be readily derived. Also, an improved version of the Peierls equation is derived in which the intensity of the wave field and the first order derivative of the mutual coherence function are calculated at the same time. A simple problem is solved to find the mutual coherence function produced by a laser beam in the atmosphere. The similarity between the mutual coherence function and the density matrix or quantum mechanics is explored and a measure of the randomness is defined for the partially coherent wave field.
In the second part of this work, the problem of multiple scattering of non-polarized light in a planetary body of arbitrary shape illuminated by a parallel beam is formulated using the integral equation approach. There exists a simple functional whose stationarity condition is equivalent to solving the equation of radiative transfer and whose value at the stationary point is proportional to the differential cross section. Our analysis reveals a direct relation between the microscopic symmetry of the. Phase function for each scattering event and the macroscopic symmetry of the differential cross section for the entire planetary body, and the intimate connection between these symmetry relations and the variational principle. The case of a homogeneous sphere containing isotropic scatterers is investigated in detail. It is shown that the solution can be expanded in a multipole series such that the general spherical problem is reduced to solving a set of decoupled integral equations in one dimension. Computations have been performed for a range of parameters of interest, and illustrative examples of applications to planetary problems are provided.
