The problem of rough surface scattering and propagation over rough terrain in ducting environments has been receiving considerable attention in the literature. One popular method of modeling this problem is the parabolic wave equation (PWE) method. In this method, the Helmholtz wave equation is replaced by a PWE under the assumption of predominant forward propagation and scattering. The resulting PWE subjected to the appropriate boundary condition(s) is then solved, given an initial field distribution, using marching techniques such as the split-step Fourier algorithm. As is obvious from the assumption on which it is based, the accuracy of the PWE approximation deteriorates in situations involving appreciable scattering away from the near-forward direction, i.e. when the terrain under consideration is considerably rough. The backscattered field is neglected in all PWE-based models.

An alternative and more rigorous method for modeling the problem under consideration is the boundary integral equation (BIE) method, which is formulated in two steps. The first step involves setting up an integral equation (the magnetic field integral equation, MFIE, or the electric field integral equation EFIE) governing currents induced on the rough surface by the incident field and solving for these currents numerically. The resulting currents are then used in the appropriate radiation integrals to calculate the field scattered by the surface everywhere in space. The BIE method accounts for all orders of multiple scattering on the rough surface and predicts the scattered field in all directions in space (including the backscattering direction) in an exact manner.

In homogeneous media, the implementation of the BIE approach is straightforward since the kernel (Green's function or its normal derivative) which appears in the integral equation and the radiation integrals is well known. This is not the case, however, in inhomogeneous media (ducting environments) where the Green's function is not readily known. Due to this fact, there has been no attempt, up to our knowledge, at using the BIE (except under the parabolic approximation) to model the problem under consideration prior to the work presented in this thesis.

In this thesis, a closed-form approximation of the Green's function for a two-dimensional ducting environment formed by the presence of a linear-square refractivity profile is derived using the asymptotic methods of stationary phase and steepest descents. This Green's function is then modified to more closely model the one associated with a physical ducting medium, in which the refractivity profile decreases up to a certain height, beyond which it becomes constant. This modified Green's function is then used in the BIE approach to study low grazing angle (LGA) propagation over rough surfaces in the aforementioned ducting environment. The numerical method used to solve the MFIE governing the surface currents is MOMI, which is a very robust and efficient method that does not require matrix storage or inversion.

The proposed method is meant as a benchmark for people studying forward propagation over rough surfaces using the parabolic wave equation (PWE). Rough surface scattering results obtained via the PWE/split-step approach are compared to those obtained via the BIE/MOMI approach in ducting environments. These comparisons clearly show the shortcomings of the PWE/split-step approach.