An Alternative Integral Equation for Propagation Over Irregular Terrain

Abstract

An elementary function that is closely related to the Sommerfeld flat‐earth attenuation function is used to derive an alternative integral equation for propagation over irregular terrain. This choice of elementary function is shown to satisfy a scalar ‘parabolic’ wave equation. For the special case of a spherical earth, the integral equation yields more accurate results than previous general formulations; in addition, the integral equation is numerically feasible for both vertical and horizontal polarization.

A Technique for Evaluating the Uncertainties in Path Loss Predictions Caused by Sparsely Sampled Terrain Data

“…Further, it shows an increase in signal level near the tops of hills facing the transmit side which is also noted by Ott [87]. As mentioned by Ott, this effect is not predicted by the typical diffraction-focused propagation techniques.…”

“…Because the goal of this study is to examine the uncertainties introduced by this undersampling, it is necessary to use a prediction technique which is as accurate as possible. Following the work of Hufford, Ott, Berry, Janaswamy, and Fernandez, a two dimensional integral equation approach is used [84,86,87,95,96,99]. The method used here, applying the method of ordered multiple interactions (MOMI) to the solution of the magnetic field integral equation (MFIE) [9], presented in chapter 4, includes fewer assumptions about the propagating wave than the above mentioned integral equation approaches.…”

“…Hufford's method was otherwise similar to performing a single (I − L) inversion on the source currents, or a half iteration of MOMI, to obtain the resulting surface currents. Following work by Ott and Berry also utilizes the antenna pattern in the source and scattering, though it also ignores backscatter effects [86,87,95]. Therefore, in a further attempt to reduce the error caused by the surface truncation, the source term will be modified with multiplication by the antenna pattern.…”

Direct Solution of Current Density Induced on a Rough Surface by Forward Propagating Waves

“…A Volterra integral equation of the second kind based on the parabolic approximation and different from (9) was also given in [24], [25] for the attenuation function over a lossy irregular terrain. Most previous works in radiowave and acoustic propagation, including [26], [27] and [25], were concerned with obtaining a numerical solution of the derived integral equations. Our goal, however, is to obtain a direct solution that will not involve matrix inversions.…”

“…Because the parabolic equation has only a first-order derivative along the range axis, the relevant integral equation will be of Volterra type as opposed to the Fredholm type [13] for the Helmholtz equation. Previous formulations of the integral equations for the parabolic equation [24], [25], [26], [27] or for low-grazing angle formulations [28] concentrated on obtaining the solution numerically using a variety of approaches. That the Volterra integral equation can be solved exactly appears to have escaped the attention of previous researchers and it is the purpose of the present paper to provide such a solution.…”