Fractal surfaces and the Kirchhoff approximation for modeling quasi-specular surface scattering: Theory and applications

Surface scattering arises from scales greater than some fraction of the electromagnetic wavelength, lambda. The surface slope parameter relevant to scattering is estimated from height differences at points separated by a non-zero horizontal scale R. The RMS slope of actual surfaces varies with R.; The Kirchhoff approximation (KA), used to derive quasi-specular scattering laws such as Hagfors law, is valid when an effective radius of curvature, evaluated at the minimum horizontal scale significantly contributing to scattering, exceeds lambda. This minimum scale depends on lambda and surface parameters, and is estimated by method of moments.; Hagfors' C is not explicit in its scale dependence. The C-RMS slope relationship is indefinite due to the uncertainty in the minimum scale contributing to quasi-specular scattering. An infinite surface extent is assumed in the derivation of Hagfors and other KA-based laws. This may impact model accuracy depending on the relative radar resolution cell size to lambda and surface parameters.; A radar scattering law, based on KA and a generalized fractional Brownian surface model with a scale-dependent Hurst exponent, gives the surface RMS slope as a function of R. The fractal-based law subsumes the traditional Hagfors, Gaussian, exponential, and other laws, as well as linear combinations of these. Modeling of backscatter data by use of a linear superposition of scattering laws achieves statistically significant fits in case of complex data variations with angle. The superposition is equated to the fractal-based law, forming an integral equation which is solved via the Hankel transform to obtain surface reflectivity and RMS slope.; Fractal-based analysis of Magellan altimeter data for the surface of Venus (lambda = 12.6 cm) and Cassini scatterometer data for the surface of Titan (lambda = 2.18 cm) shows that the RMS slope estimate is reliable over an R-range from about lambda to several or tens of lambda depending on surface parameters. This analysis, in addition to that of Arecibo data for Titan's surface (lambda = 13 cm), demonstrates the increased ambiguity in the inferred surface parameters due to the existence of multiple statistically significant fits. The presence of considerable diffuse backscatter adds to the uncertainty in surface parameters.