A practical model to compute shallow-water boundary reverberation is described. Normal modes are used to calculate the acoustic energy propagating from the source to the scattering area, and from the scattering area to the receiver. At the scattering patch each mode is decomposed into up- and down-going waves, then ray-mode analogies and empirical scattering functions can be used to compute the scattered energy. The method was first described by Bucker and Morris [J. Acoust. Soc. Am. 44, 827–828 (1968)], and papers which first appeared in the Chinese literature. Their work is extended here by using group velocities to obtain the travel times for each mode pair, and by further developing the ray-mode analogy. The effects of summing the modes coherently or incoherently and of including the time spreading due to the modal group velocities are examined. This paper deals with the range-independent monostatic case, although the technique is extendible to bistatic geometries and range-dependent environments. Calculations show excellent agreement with some ray-based models, and, using the Lambert bottom-scattering coefficient as the only adjustable parameter, good agreement is obtained with some measured shallow-water reverberation.
The ocean bottom scattering function depends, in general, on the grazing angles and the azimuthal angles of the incident and scattered energy. However, most measurements are for backscatter only. The few general measurements that are available indicate strong forward scattering near the angle of the specularly reflected ray and weaker, azimuthally isotropic, diffuse scattering away from the specular angle. By combining Lambert’s law scattering with a surface scattering function based on the Kirchhoff approximation, Ellis and Haller [J. Acoust. Soc. Am. Suppl. 1 82, S124 (1987)] proposed a function that incorporated these features. The function is quite simple, and depends on three parameters that can be fitted to backscatter measurements. The functional form thus allows a reasonable extension from backscatter to the general three-dimensional scattering function, which can then be used in bistatic reverberation calculations. It is an improvement over two commonly used methods (which do not include azimuthal dependence) for extrapolating backscattering to general scattering: the separable approximation, and the half-angle approximation. This paper discusses the three-dimensional function in more detail, and presents some comparisons between model predictions and measured bistatic reverberation.
Shallow-water seabeds are often varied and complex, and are known to have a strong effect on acoustic propagation. Some of these seabeds can be modeled successfully as fluid or solid half-spaces. However, unexpectedly high propagation loss with respect to these models has been measured in several regions with rough, partially exposed, hard-rock seabeds. It is shown that the high propagation loss in these areas can be modeled successfully by introducing a thin layer of elastic–solid sediment over the hard-rock substrate. Propagation loss predictions using the safari fast-field program exhibit bands of high loss at regularly spaced frequencies. Normal-mode calculations show resonance phenomena, with large peaks in the modal attenuation coefficients at these same frequencies, and with rapid changes in the mode wavenumbers. Bottom reflection loss calculations indicate that the high propagation loss is due to absorption of shear waves in the sediment layer.
Weston’s [J. Acoust. Soc. Am. 32, 647–654 (1960)] concept of the effective depth of a Pekeris-type shallow water waveguide has been extended to admit seabeds that support shear waves. The amended effective depth formula leads to estimates of normal mode phase speeds that are surprisingly accurate, without numerical iteration. Results for six hypothetical, though realistic, bottom types are compared with an ‘‘exact’’ normal mode model that includes shear wave effects and another normal mode model that does not.
In shallow water environments where the uppermost sediment layer is a fine-grained fabric (e.g. clay or silty-clay), the observed reverberation may be dominated by scattering from the subbottom. Here, reverberation predictions from normal mode and energy flux models are compared for the case where the scattering arises from a sub-bottom half-space under a fine-grained sediment layer. It is shown that in such an environment, the position of the angle of intromission, in addition to the angular dependence of the scattering kernel, is a factor controlling the reverberation and its vertical angle distribution. It is also shown that the reverberation from a sub-bottom horizon is typically governed by higher grazing angles than the case where the scattering occurs at the watersediment interface. There was generally very close agreement between the models as a function of frequency (200-1600 Hz), layer thickness (0-8 m), and range (1-15 km). The model comparisons, showing some differences, illuminate the result of different approximations in the two approaches.
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