Wide-Angle Shift-Map PE for a Piecewise Linear Terrain

A wide-angle parabolic wave equation solution, using shift-map and finite-difference techniques, is presented. The corresponding split-step Fourier solution is well known. The solution using finite-difference technique, where the standard parabolic wave equation is modified into the so-called Claerbout equation allowing propagation angles up to 45 from the paraxial direction, is also well known. Here, we present an extension to that solution in which the shift-map technique is incorporated into the finite-difference scheme allowing a varying terrain to be considered. The result is a solution that corresponds to the well known split-step solution, which is believed to perform well for terrain slopes up to 10 - 15, and discontinuous slope changes on the order of 15 - 20. This solution is of first order with respect to the terrain slope. The resulting difference scheme is for solving a tridiagonal system. When using the finite-difference technique, it is also possible to find a second order solution with respect to the terrain slope. The resulting difference scheme in this case is for solving a pentadiagonal system. This new solution performs well for slopes up to about 15 - 20 and discontinuous slope changes up to about 30 - 40, which is an improvement.