Fractal Planets: a Generalized Surface Roughness Model for Remote Sensing

Remote sensing plays an enormous role in our understanding of the geology of solid bodies in the solar system. Inferences based on remote sensing of planetary surfaces require the use of two types of models: one for the scattering and emission of electromagnetic radiation from matter; and one which quantitatively describes the surface composition and roughness of geologic surfaces. Here, we use the term roughness to indicate any deviations, statistical or otherwise, from a planar surface. In order for our geologic inferences to be soundly based, each of these models must be realistic, i.e., based upon physical principles and/or empirically observed behavior, and general enough to encompass most or all types of surfaces one is likely to encounter. While the interaction of electromagnetic waves and matter has been reasonably well characterized for over a century, realistic quantitative models of geologic surface roughness have only recently been developed [1,2,3]. In this abstract, we advocate the use of fractals as a simple, quantitative, and general model for geologic surface roughness. Fractals implicitly assume that the surface is “noise” and therefore do not model deterministic structures well. However, in areas of homogeneous surface structure, i.e., areas where a single geologic “theme” is randomized to form the landscape, they provide a robust estimator of the scaledependent surface roughness. We briefly describe fractal statistics, and then illustrate their utility for remote sensing by computing a shadowing function for fractal surfaces observed at nadir. To demonstrate the generality of fractal surface models, we compare the shadowing behavior of fractal surfaces to the shadowing behavior of three published surface roughness models, including the well known Hapke [4] model. The term “fractal” is often taken to be synonymous with “scale-invariance”, the well known property that many geologic phenomenon look the same at all scales. However, for topography and surface roughness, scale-invariance is rare. More commonly, the vertical and horizontal axes of a surface scale at different rates. Quantitatively, this behavior, called “self-affinity”, can be described by two relations: