The problem of electromagnetic wave propagation over a horizontal, nonconstant immittance plane, whose immittance value is a function of incident grazing angle, is considered. An equivalent specification of the immittance surface is in terms of its angle-dependent reflection coefficient. Expressions are provided for the field on a vertical line given the field on a previous vertical line. The vertical line field is initialized at the plane containing the source where its aperture current distribution is specified. Both two-dimensional and three-dimensional fields are considered, and the expressions are valid for either polarization. The form of expressions is particularly suited for implementing with the Fourier split-step algorithm of the parabolic wave equation. Extension to inhomogeneous atmosphere to account for mild atmospheric inhomogeneities is presented. Several examples are considered where the immittance arises from small-scale and large-scale surface roughnesses. A numerical procedure is described wherein incomplete or approximate reflection coefficient data are made to conform to the assumptions made in the development of the expressions. This is demonstrated for a surface reflection coefficient which is governed by the Miller-Brown-Vegh roughness reduction factor. Numerical results are presented for propagation under ducting conditions over a rough surface for frequencies from HF through microwave.
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