A split-step Padé solution for the parabolic equation method

A split-step Padé solution is derived for the parabolic equation (PE) method. Higher-order Padé approximations are used to reduce both numerical errors and asymptotic errors (e.g., phase errors due to wide-angle propagation). This approach is approximately two orders of magnitude faster than solutions based on Padé approximations that account for asymptotic errors but not numerical errors. In contrast to the split-step Fourier solution, which achieves similar efficiency for some problems, the split-step Padé solution is valid for problems involving very wide propagation angles, large depth variations in the properties of the waveguide, and elastic ocean bottoms. The split-step Padé solution is practical for global-scale problems.