Two-way fourier split step algorithm over variable terrain with narrow and wide angle propagators

Helmholtz´s wave equation can be approximated by means of two differential equations, corresponding to forward and backward propagating waves each of which is in parabolic wave equation (PWE) form. The standard PWE is very suitable for marching-type numerical solutions. The one-way Fourier split-step parabolic equation algorithm (SSPE) is highly effective in modeling electromagnetic (EM) wave propagation above the Earth´s irregular surface through inhomogeneous atmosphere. The two drawbacks of the standard PWE are: (i) It handles only the forward-propagating waves, and cannot account for the backscattered ones. The forward waves are usually adequate for typical longrange propagation scenarios. However, the backward waves become significant in the presence of obstacles that redirect the incoming wave. Hence, this necessitates the accurate estimation of the multipath effects to model the tropospheric wave propagation over terrain, (ii) It is a narrow-angle approximation, which consequently restricts the accuracy to propagation angles up to 10°-15° from the paraxial direction. To handle propagation angles beyond these values, wide-angle propagators have been introduced. Recently, a two-way SSPE algorithm was implemented to incorporate the backwardpropagating waves into the standard one-way SSPE, through a recursive forwardbackward scheme to model the tropospheric electromagnetic propagation over a staircase-approximated terrain. This algorithm has employed the standard narrowangle propagators in its implementation. The primary goal of this paper is to present the improved version of the algorithm based on wide-angle propagators, and to demonstrate the results of the comparison tests performed in some canonical scenarios, together with more complex scenarios involving variable terrains.