Statistics of approximately self-affine fractals: random corrugated surface and time series

Rough corrugated surfaces or time series are modeled as one-dimensional, stationary, Gaussian random processes with power-law power spectra over a limited range of frequencies and analyzed with the techniques of random processes. Surfaces with power-law spectra with small- and large-scale cutoffs exhibit approximate self-affine fractals behavior. There are two crossover scales, viz., a lower crossover scale and an upper crossover scale for nonfractal to fractal and fractal to nonfractal transition, respectively. We find the exact representation for the various statistical properties, viz., the mean square (MS) width, the MS slope, the MS curvature, the mean curve length (area), the mean zero crossing density the correlation function, and the structure function for these class of curves or corrugated surfaces. The importance of roughness exponent, upper and lower cutoff scales, and crossover of power-law spectra to non-power-law spectra to statistical properties of approximately self-affine fractal is emphasized. The scaling behavior of various statistical properties in the region between the two crossover scales is discussed. We suggest several methods for extracting fractal dimension. Finally, we apply our results to a granular flow experiment to characterize various time scales and gross statistical measure in this problem