An Analysis of Narrow-Angle and Wide-Angle Shift-Map Parabolic Equations

The capabilities of the narrow-angle (NA) and wide-angle (WA) parabolic equations (PEs) in the physical coordinate system are well known. For use on the irregular surface, a shift-map (SM) PE (SMPE) form was formulated in terms of the transformed coordinates. In sharp contrast, little is known of the accuracy of the NA and WA-SMPEs in the transformed coordinates. It was thought that the WA-SMPE is entirely superior to the NA-SMPE according to the behavior in the NA and WA propagators. In this article, we analyze the accuracy of the WA and NA-SMPEs in the transformed coordinates through studying their artificial dispersion properties. It is found that the artificial dispersion of the NA-SMPE is independent of the terrain parameters, whereas for the WA-SMPE, the dependence of the dispersion characteristics on the terrain slope is very strong. Also, as the terrain slope increases, its WA capability deteriorates dramatically. Particularly, when the shady terrain slope is larger than about 5°, the accuracy of the WA-SMPE is even worse than the NA-SMPE. To substantiate the accuracy analysis, numerical experiments involving wave propagation over irregular terrain are also presented.