Parabolic equation-based propagation fields near surface irregularities including backscatter

Parabolic equations have become popular in describing electromagnetic propagation in a horizontally varying environment. The parabolic equation is, however, only an approximation of the governing elliptic differential equation in that it ignores backscatter and assumes that the fields vary slowly with range everywhere. Nevertheless, its solutions have been observed to agree well with other solutions and with measurements in regions in which the fields vary slowly with range even when regions of rapidly varying fields are present between the source and the observer. This is of little value, however, if it is the backscatter that is of specific interest, as is the case, for example, in radar clutter, or if it is desired to determine the solution in a region of rapid field variation such as near a terrain irregularity. In this article, Green\´s theorem is used to express the solution of the full elliptic differential equation (including backscatter) as an integral over the surface fields. A scheme is introduced which assumes that the parabolic solution represents an adequate approximation of the surface fields except near the ground irregularity where they generally vary rapidly. Near this irregularity, the surface fields are found from a numerical solution of the differential equation. Application of the method to a mixed path problem shows that the "exact" surface fields differ from the parabolic equation solution only near the ground-water interface.