Mathematical Modeling and Computer-Aided Manufacturing of Rough Surfaces for Experimental Study of Seafloor Scattering

Abstract

Diverse aspects of stochastic processes, time-series analysis, fractal geometry, and manufacturing technology are brought together to provide a unified theoretical framework for the mathematical characterization and physical fabrication of scale-model representations of the rough ocean bottom. These scale models can be used to validate the predictions of interface-scattering theories, particularly those relating to the dependence of scattering strength on roughness parameters. In acoustical oceanography, rough surfaces are conventionally described in terms of power-spectral density (PSD) or fractal dimension. Here, recently developed concepts describing modified power-law PSD and approximately self-affine stochastic fractals are used to account for issues of finite sample size and finite manufacturing resolution. The large- and small- bandwidth restrictions that these issues inherently impose are related to their effects on the properties of surfaces as characterized both mathematically and by visual observation. This development provides the groundwork for numerical generation of surfaces, using spectral methods, and their physical manufacture, using computer-aided manufacturing (CAM) techniques. Effects of the spectral method of numerical generation on the statistics of generated topography are detailed. A manufacturing technique using a computer numerically controlled (CNC) milling machine is discussed and topography-specific guidelines for accurate manufacture of rough surfaces are prescribed.