A Shallow-Water Reverberation Model Based on Perturbation Theory

A shallow-water reverberation model is developed based on Bass perturbation theory. The key component for shallow-water reverberation modeling, the modal backscattering matrix (MBSM), has been investigated and the explicit analytic expressions of the MBSM both for the scattering due to interface roughness and due to sediment volume inhomogeneities are derived. For rough interface scattering, it is shown that the angular dependence and the frequency dependence of the MBSM can be separated: the angular dependence is given by sin
²
(?
_m
P/2)sin
²
(?
_n
P/2), where ?
_m
is the modal grazing angle and P is a bottom parameter related to the bottom reflection phase shift, and the frequency dependence is given by k
₀

^4-n
, where n depends on the power spectrum of the roughness, for instance, n = 3 for Goff-Jordan spectrum. The difference between the scattering due to sediment volume inhomogeneities and the scattering due to rough interface is that there is an extra factor contributed by the vertical correlation and the exponential attenuation of the modal function weighting on the vertical direction in the sediment medium. This extra factor has some important impacts: (1) there will be, in general, a coupled angular-frequency behavior, (2) the angular pattern will decrease more rapidly at small grazing angle area, and (3) the angular pattern is no longer separable. It is shown that only for a ?thin? layer (or low frequency), the scattering due to volume inhomogeneities will have a similar behavior as the scattering due to rough interface. The significant feature of the angular pattern for both kinds of scattering is a marked departure from Lambert's law at small grazing angle area. The explicit analytic expression of the MBSM, and the differences between the two kinds of bottom scattering, given in this paper, provide the opportunity for a comprehensive model-data comparison and a better understanding of the scattering mechanism.