A ‘‘separation of scales hypothesis’’ is an important component of both composite roughness and facet scattering theories applied to fractal (i.e., realistic) surfaces. It asserts that a scattering surface is separated by the acoustic wavelength into two scale regimes; features larger than the acoustically defined ‘‘separation scale’’ act to reflect rays specularly, whereas features smaller than the separation scale act to scatter acoustic energy diffusely. This paper explores the consequences of this hypothesis by formulating the relationship between an arbitrary local-scale scattering strength function, describing the diffuse component of scattering relative to a local-scale slope, and three global-scale scattering strength functions, which describe scattering over the entire surface relative to the horizontal. These include: mean and rms scattering strength, and grain ratio scattering strength (the ratio of scattering strength from two principal directions of an anisotropic surface). Each provides an important independent constraint on the relationship between local- and global-scale scattering properties: (1) Mean global-scale scattering strength is essentially a more diffuse version of local-scale scattering strength; (2) rms global-scale scattering strength exhibits great sensitivity at low grazing angles to the local-scale scattering strength model; and (3) grain ratio scattering strength is sensitive to the functional form of local-scale scattering strength, especially at low grazing angles. Examples demonstrate that sinusoidal local-scale scattering produces sinusoidal global-scale scattering, whereas Lambertian local-scale scattering does not result in Lambertian global-scale scattering.
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John A. Goff (1995) studied this question.