Type of Document Dissertation Author Awadallah, Ra'id S. M.S. Author's Email Address rawadall@vt.edu URN etd-4598-0751 Title Rough Surface Scattering and Propagation over Rough Terrain in Ducting Environments Degree PhD Department Electrical Engineering Advisory Committee

Advisor Name Title Brown, Gary S. Committee Chair Besieris, Ioannis M. Committee Member Kohler, Werner E. Committee Member Mili, Lamine M. Committee Member Nayfeh, Ali H. Committee Member Keywords

- ducting environments
- rough surfaces
- Electromagnetic propagation
- numerical methods
- Asymptotic techniques
Date of Defense 1998-04-30 Availability unrestricted AbstractThe problem of rough surface scattering and propagation over rough terrain inducting 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.

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