Electromagnetic wave scattering by power-law surfaces.

Abstract

A number of studies find that natural surfaces possess roughness spectra that have a power-law form; i.e., the magnitude of the roughness spectrum is inversely proportional to the spatial frequency raised to a power $\beta$ between 1.0 and 3.0. Such surfaces may be described by a fractal dimension. Synthetic fractal surfaces seem most realistic with a fractal dimension of about 2.2, which corresponds to a $\beta$ (spectral slope) of 2.6. However, several studies report that spectral slope values of 2.0 are characteristic of natural topography. This dissertation documents a study of power-law surfaces and their electro-magnetic scattering properties. Surface elevation profiles were collected on volcanic debris flow surfaces at Mount St. Helens. Spectral estimates derived from these measurements show that some of the debris flow surfaces possess power-law spectra. In determining these spectra, it was found that spectral leakage can cause.some spectral estimators to give an estimated spectral slope of 2.0 for profiles having slopes between 2.0 and 3.0. This slope insensitivity may explain the number of studies reporting a spectral slope of 2.0 for natural terrains. Spectral estimation using Capon's estimator was found to produce estimates with accurate spectral slopes and reduced variance. Verification of rough surface scattering models based on field measurements of radar backscatter is difficult because natural targets are often inhomogeneous and because volume scattering may be significant. To avoid these difficulties, power-law surface analogues were manufactured by milling surfaces with specified power-law roughness spectra on slabs of polyethylene. A 35 GHz scatterometer was used to measure the backscattering coefficient as a function of polarization, incidence angle, and surface roughness. Measured backscatter was compared to values predicted by classical rough surface scattering models and a recent model by Eftimiu. While none of the models was exceptionally accurate, the first-order small perturbation model results most closely resembled the measured values. This study is third in a series of projects on remote sensing of natural terrain by the author. Articles describing the first two projects are included as appendices.