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A semi-empirical basis for a modified Lambert's law for low-grazing bottom backscattering

Recently developed bottom scattering models require an array of parameters to describe low-grazing backscattering. In stark contrast to this is the simplicity of Lambert's law which requires only a single empirical coefficient. Lambert's law is currently used for most sonar applications, but it has shortcomings. It has a cosine-squared angular functional form which sometimes fails to fit data. What little theoretical justification there is for Lambert's law is based on surface roughness scattering. Today, it is commonly thought that, at low-grazing, volume inhomogeneities are mainly responsible for backscatter. An angular functional form for volume-induced backscattering referenced to the bottom surface plane would, in general, be different from cosine-squared. The authors discuss a development of volume-induced backscattering referenced to a flat bottom which provides a theoretical basis for a modified angular functional form. This functional form involves a Lambert-like coefficient times an angular function. The coefficient may have values close to those used in Lambert's law (e.g., Mackenzie's value of 27 dB) for various sediments and, through part of the angular range, the angular form can be Lambert like. Thus, Lambert's law is a subset, and has been successful in some angular regions where it applies. While a theoretical form for the coefficient and the angular dependency has been derived some of the parameters needed would probably be established empirically, leaving the possibility that one might as well determine the single coefficient empirically. From this viewpoint, the extensive work already done with Lambert's law could be extended to this new angular form.