Random Fractals Geometry in Surface Roughness Modeling of Geological Formations Using Synthetic Aperture Radar Images

Some geological formations are more subject of weathering and alteration, so they are physically softer; in contrast, some other formations are more resistant against physical and chemical alterations, so they are harder and rocky on the ground surface. According to lack of geometric texture detection capability in usual optical images, the radar data must be undertaken. Because of the surface description capability of microwave remote sensing data, they are so much efficiently useful for morphological studies. Surface morphological modeling using Synthetic Aperture Radar (SAR) data requires topographic and micro-topographic surface model. Utilizing the ability of geometric pattern discrimination needs to relate dielectric and surface roughness parameters to radar signals back-scattering. Euclidean geometry is less capable in comparison to fractal geometry to describe natural phenomena. So far, some efforts are made to use fractal parameters in order to improve back-scattering model, but in none of them Euclidean geometry is not completely replaced by fractal geometry.
Because of irregular nature of earth surface, geometry of the radar signal incidence on the earth surface is not Euclidean geometry and experimentally, fractal geometry describes it much better. Roughness of geological surfaces is an example of such natural phenomena. In literature, using fractal geometry in IEM has been performed by changing the computation process of some factors but, in all of such methods Euclidean geometry is the obvious rule of computation in IEM. In this paper, it is desired to replace the Euclidean geometry, basically by fractal geometry. Therefore, instead of conventional procedure of correlation length ( ) and rms-height ( ) calculation, two following equations are utilized: Where and are fractal surface parameters. According to adaptability of fractal theory with natural phenomena, it is supposed to generally have better results in IEM after having applied such improvements on the model input parameters.
The methodology is implemented for PALSAR (ALOS satellite sensor) data analysis results of Anaran (between Dehloran and Ilam cities in Iran) geological formations. Field measurement of surface roughness using a total station and the data gathering performed on a grid of points. Thus, the digital elevation model (DEM) of the surface with sampling intervals smaller than the correlation length is formed.
Surface roughness has been computed by IEM and get compared with field measurements on 20 selected pixels which show the most obvious improvement. In general, the behavior of the correlation function of the two polarization parameters are very close to each other. Although at sites 1 and 2 in some cases, lower standard deviation can be seen for Euclidean geometry, but mean standard deviation for fractal input parameters in IEM is considerably lower. Exponential ACF shows better for Site 1, and in contrary Gaussian ACF for Site 2 is more efficient; which confirms the fact that exponential ACF is suitable for soft surfaces and Gaussian one for rough ones.
Due to irregular and fractal nature of the surface roughness, electromagnetic backscattering modeling of radar signals using fractal geometry calculates surface parameters closer to actual values. Gaussian correlation function is suitable for smooth and exponential correlation function is more appropriate for rough surfaces. The mean improvement in the use of fractal geometry for both polarizations hh and vv is about 50%. Comparison of three different sites with different levels of roughness provides similar results, and in particular, results improvement in areas with larger roughness parameters is more pronounced.