Wide-Angle Shift-Map PE for a Piecewise Linear Terrain—A Finite-Difference Approach

A wide-angle parabolic wave equation solution is presented using shift-map and finite-difference techniques. 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 45deg 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 10deg-15deg and discontinuous slope changes on the order of 15deg-20deg. This solution is a first-order one with respect to the terrain slope. However, when using the finite-difference technique, it is also possible to find a second-order solution with respect to the terrain slope. This new solution performs well for slopes up to about 15deg and discontinuous slope changes up to about 30deg, which is an improvement.