Synthesis of Fractal Surfaces for Remote-Sensing Applications

A physical approach to synthesize fractal surfaces for a reliable and controllable use within remote-sensing applications is presented in this paper. In particular, the physical criteria to determine the minimum number of tones of the Weierstrass-Mandelbrot (WM) function needed to adequately synthesize realizations of fractional Brownian motion (fBm) processes are analytically derived. The presented rationale relies on considering, in an appropriate range of scales, the WM function as a spectral sampling of the fBm process, and linking the number of sampling functions to the width of the scale range and to the sampling rate; hence, it is shown how to set the lower and the upper scales according to a given remote-sensing problem. In this paper, the condition for determining the WM wavenumber sampling rate is analytically derived. The method is also applied to the efficient simulation of the synthetic aperture radar signal scattered by natural surfaces.