Fractals, wavelets, and stochastic interface modeling

Abstract

The seafloor is one of many natural interfaces which, when viewed as a random process, exhibits a power law decay, i.e., the power spectral density (PSD) decays as A/fb, A>0, b>0 as f→∞. Power law spectra are usually taken as evidence that the stochastic process is multiscale or fractal. Multiscale interfaces exhibit features at all scales and their acoustical properties are distinct from those of single-scale interfaces, which contain features closely clustered about a mean size. Thus, determination of the scale structure, or size distribution of component features, is important in characterizing random interfaces for acoustical applications. However, because the PSD is only a second moment characterization, a power law PSD can allow both single-scale and multiscale processes. Only if the process is Gaussian is the PSD a complete description. Wavelet representations succeed where Fourier methods fail, as they are ideally suited for determining scale structure. In the case of fractal interfaces, the wavelet coefficients are Gaussian and independent across scale and space. Single-scale interfaces lead to a dependence of wavelet coefficients across scale for a given spatial location and vice-versa.