Fractal analysis of topography and reflectance surfaces

Abstract

The definition of a fractal has been successfully deduced from constructing the Koch curve and the Cantor set. Principles of seven methods (the ruler, box-counting, spectral structure function, intersection methods, cube-counting, and triangular prism methods) for determining the fractal dimensions are illustrated and verified by the Koch curve, Cantor set, and the simulated 1-dimensional and 2-dimensional fBm samples by comparing the calculated with the theoretical D values of the theoretical fractal models. The application of appropriate methods to self-similar or self-affine fractals is essential due to different theoretical assumptions of the methodologies. The ruler dimension is different from the spectral dimension. The application of Hanning window to the synthetic fBm samples (Hanning window weighted) is important to obtain correct fractal dimensions for the spectral method and structure function methods. The multi-scaling behaviour of a fractal can be unveiled by revealing the difference between the 1st and 2nd order structure function methods. The zeroset theory is used to relate the D values of 1-d contour set with 2-d surface by analyzing the DEM data.The results of fractal analysing 132 topographic contours digitized from different scales (1:200,000, 1:50,000, 1:20,000) of maps of the border area between Spain and Portugal show that contours are self-similar, and have a fractal dimension of about D = 1.23 over length scales ranging from 30 m to 13 km scale (3 orders of magnitude). The thirteen filed and map profiles from Dorset area of southern England has a D value of 1.03 derived from the ruler method. The variations in D values are controlled by three geological factors: erosive processes, lithologies, and fractures. The dominant control is the erosive process and fractures, and lithologies can either result in significant difference or produce more subtle variation in D values of coastlines and contours. For example, the river down-cutting produces higher D value (1.1 ~ 1.5) than the wave action or cliff retreat erosive processes (1.01-1.10). The results of the fractal analysis of the five TM sub-image of Qatar have shown that D values of the TM images range from 2.10 to 2.96. The variations in D values are controlled by different types of surface, band variations, and methodologies. The study area B of a single rock type has the lowest D value (D is about 2.25) and is significant different from the other four study areas, whilst the urban area E yields the highest fractal dimension (about D = 2.6). Band 3 yields the highest fractal dimensions, followed by bands 4, 5, 1,and 6, and band 2 has the lowest D value. The difference between the D values derived from the 2nd and 1st order structure function methods for all the six bands of five study areas is D_2s(q=2) - D_2s(q=l) = 0.16 ± 0.13 (the uncertainty is the standard deviation), and suggests that the TM imagery has a multi-scaling property.

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