Self‐affine (fractal) topography: Surface parameterization and radar scattering

Abstract

Starting with the assumption that planetary surfaces are self‐affine (fractal) over the scales applicable to radar scattering, we derive various surface parameters, e.g., rms slopes and autocorrelation functions, and examine the implications for radar scattering models. The results of this work provide several new insights of interest to planetary geologists and others using radar to study surface features. First, the unidirectional slope histograms of self‐affine surfaces are Gaussian, and the adirectional slope histograms are Rayleigh. Normalization of the adirectional histogram by solid angle results in a Gaussian adirectional slope density function and therefore a Gaussian quasi‐specular angular scattering function. Next, the wavelength dependent behavior of surface roughness inferred from lunar radar observations is consistent with self‐affine topography. Finally, surface rms height measurements are functions of profile length. Therefore, when determining the applicability of the small perturbation model to a surface based on those measurements, it is necessary to consider the length of the profile with respect to the sampling wavelength.