Phase error analysis of several parabolic equation forms

The intrinsic phase error of the parabolic equation (PE) is mainly caused by the paraxial approximation. In order to select the optimal PE form in the target scenario, we analyze the accuracy of the existing six PE forms in the typical tropospheric electromagnetic wave propagation, underwater acoustic propagation, and forest electromagnetic wave propagation scenarios through studying their phase error properties. The dispersion relation of each PE is derived. It is found that in tropospheric electromagnetic wave propagation scenario, the Lin-Duda PE is the most accurate, the Tappert and Feit-Fleck PEs are almost identical, and the Padé PE is closely followed. In underwater acoustic propagation scenario, the Padé and Lin-Duda PEs get the highest accuracy. For forest electromagnetic wave propagation scenario, Lin-Duda PE still maintains the highest accuracy. Furthermore, the acceptable propagation angle limit of each PE is given for a fixed phase error limit.