| Abstract: | Analysing terrain profiles of fields, roads, and other terrains, it was determined that terrain profiles are random and non-periodical. Mandelbrot has defined non-scaling, self-similar figures as fractals, and many investigators have tried to characterize natural forms and structures using fractal geometry. The work here investigates whether terrain profiles can be defined as fractals. Fractal dimensions of profiles were calculated. These were compared with a locus of Brownian motion further to investigate characteristics of terrain profiles. Fractals are defined to be self-similar and irregular. Measuring and analysing terrain profiles, it was established that the statistical characteristics of any part of a terrain profile are similar and that the statistical characteristics of profiles of any kind of terrain are similar irrespective of roughness. This means that terrain profiles are self-similar, and irregular. From these results, it was determined that terrain profiles are fractals. The fractal dimensions were calculated with a coarse-graining method and by Power Spectral Densities (PSD), and fractal dimensions by Scaling were between 1.1 and 1.8 and by PSD between 1.3 and 1.5. Using the locus of Brownian motion, fractal dimensions were 1.5 or slightly larger than those of the terrain profiles. Fractal dimensions for the locus of smoothed Brownian motion were nearly equal to terrain profiles. Therefore terrain profiles could be artificially generated from the locus of smoothed Brownian motion. It appears that terrain roughness is formed by random and non-periodical force. |
