Recursive Two-Way Parabolic Equation Approach for Modeling Terrain Effects in Tropospheric Propagation

The Fourier split-step method is a one-way marching-type algorithm to efficiently solve the parabolic equation for modeling electromagnetic propagation in troposphere. The main drawback of this method is that it characterizes only forward-propagating waves, and neglects backward-propagating waves, which become important especially in the presence of irregular surfaces. Although ground reflecting boundaries are inherently incorporated into the split-step algorithm, irregular surfaces (such as sharp edges) introduce a formidable challenge. In this paper, a recursive two-way split-step algorithm is presented to model both forward and backward propagation in the presence of multiple knife-edges. The algorithm starts marching in the forward direction until the wave reaches a knife-edge. The wave arriving at the knife-edge is partially-reflected by imposing the boundary conditions at the edge, and is propagated in the backward direction by reversing the paraxial direction in the parabolic equation. In other words, the wave is split into two components, and the components travel in their corresponding directions. The reflected wave is added to the forward-wave in each range step to obtain the total wave. The wave-splitting is performed each time a wave is incident on one of the knife-edges. This procedure is repeated until convergence is achieved inside the entire domain.