Multiple-scatter and shadowing effects are included in an extended theory of high-frequency scattering from a surface rough in one dimension. The single-scatter probability of slopes relation, corrected for shadowing, is an immediate consequence for any stationary random process. The double-scatter contribution (shadow corrected) is derived as well, and it provides a significant correction for surfaces with appreciable rms slope. The total power scattered by a perfectly reflecting rough surface is numerically evaluated as a test of energy conservation; the results show that the double-scatter formulation is substantially more accurate than the conventional single-scatter, unshadowed theory, particularly in the cases of large angles of incidence or very rough surfaces.
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Lynch et al. (1970) studied this question.
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