Efficient and accurate algorithm for the evaluation of Kirchhoff scattering from fractal surfaces

Abstract

Scattering from a natural surface modeled by a fractional Brownian motion (fBm) two-dimensional process can be evaluated by using the Kirchhoff approximation if proper conditions are satisfied by surface parameters. This evaluation leads to a scattering integral that can be computed via two different asymptotic series expansions, whose behavior has been recently deeply investigated with the aim of finding suitable truncation criteria to compute, with a controlled absolute error, the field scattered by a fractal fBm surface. Based on those results, in this paper truncation criteria are used to compute aforementioned series with a controlled relative error instead of an absolute one. According to such an analysis, an algorithm is provided, which allows to automatically decide which of the two series, if any, can be used, and how it can be properly truncated for efficient and effective computation of the field scattered by natural surfaces. It turns out that by using the standard IEEE double-precision numbering format, a relative accuracy as high as 10^-5 can be achieved for most of allowable values of surface parameters. Finally, to illustrate its practical applicability, the proposed algorithm is employed to generate a Synthetic Aperture Radar (SAR) reflectivity map to be used within a SAR simulation scheme.