Greene second-order approximation of the wide-angle equation for terrain

The solution of a wide-angle parabolic wave equation solution using shift-map and finite-difference techniques is presented. The so-called Claerbout equation allows propagation angles up to 45° from the paraxial direction. The shift-map technique can be incorporated into this scheme allowing a varying terrain to be considered. The Claerbout one-order solution with respect to the terrain slope performs well for slopes up to about 15° and discontinuous slope changes up to about 30° and corresponds to the well-known split-step solution. The Claerbout second-order approximation is able to give better results than the split-step solution. Besides, under the condition of the phase error is less than 0.002, the acceptable propagation angle limits of the Claerbout and Greene are 36° and 47° respectively. This paper derives Greene second-order solution. To a certain extent, the Greene approximation can certainly give better results.