Propagation modeling over terrain using the parabolic wave equation

We address the numerical solution of the parabolic wave equation over terrain using the Fourier/split-step approach. The method, referred to as a shift map, generalizes that of Beilis and Tappert (1979) who introduced a coordinate transformation technique to flatten the boundary. This technique is extended to a wide-angle form, allowing larger propagation angles with respect to the horizon. A new impedance boundary condition is derived for electromagnetic waves incident on a finitely conducting surface that enables solution of the parabolic wave (PWE) using the previously developed mixed Fourier transform. It is also shown by example that in many cases of interest, the boundary may be approximated by discrete piecewise linear segments without affecting the field solution. A more accurate shift map solution of the PWE for a piecewise linear boundary is, therefore, developed for modeling propagation over digitally sampled terrain data. The shift-map solution is applied to various surface types, including ramps, wedges, curved obstacles, and actual terrain. Where possible, comparisons are made between the numerical solution and an exact analytical form. The examples demonstrate that the shift map performs well for surface slopes as large as 10-15° and discontinuous slope changes on the order of 15-20°. To accommodate a larger range of slopes, it is suggested that the most viable solution for general terrain modeling is a hybrid of the shift map with the well-known terrain masking (knife-edge diffraction) approximation