Theoretical description of the parabolic approximation/Fourier split‐step method of representing electromagnetic propagation in the troposphere

A theoretical foundation for the use of the parabolic wave equation/Fourier split‐step method for modeling electromagnetic tropospheric propagation is presented. New procedures are used to derive a scalar Helmholtz equation and to subsequently transform to a rectangular coordinate system without requiring approximations. The assumptions associated with reducing the resulting exact Helmholtz equation to the parabolic wave equation that is used for computations are then described. A similar discussion of the error sources associated with the Fourier split‐step solution technique is provided as well. These discussions provide an important indication of the applicability of the parabolic equation/split‐step method to electromagnetic tropospheric propagation problems. A rigorous method of incorporating an impedance boundary at the Earth's surface in the split‐step algorithm is also presented for the first time. Finally, a few example calculations which demonstrate agreement with other propagation models are provided.