Review article :methods of fractal geometry used in the study of complex geomorphic networks

Abstract

Fractal geometry methods allow one to quantitatively describe self-similar or self-affined landscape shapes and facilitate the complex/ holistic study of natural objects in various scales. They also allow one to compare the values of analyses from different scales (Mandelbrot 1967; Burrough 1981). With respect to the hierarchical scale (Bendix 1994) and fractal self-similarity (Mandelbrot 1982; Stuwe 2007) of the fractal landscape shapes, suitable morphometric characteristics have to be used, and a suitable scale has to be selected, in order to evaluate them in a representative and objective manner. This review article defines and compares: 1) the basic terms in fractal geometry, i.e. fractal dimension, self-similar, self-affined and random fractals, hierarchical scale, fractal self-similarity and the physical limits of a system; 2) selected methods of determining the fractal dimension of complex geomorphic networks. From the fractal landscape shapes forming complex networks, emphasis is placed on drainage patterns and valley networks.

If the drainage patterns or valley networks are self-similar fractals at various scales, it is possible to determine the fractal dimension by using the method “fractal dimension of drainage patterns and valley networks according to Turcotte (1997)”. Conversely, if the river and valley networks are self-affined fractals, it is appropriate to determine fractal dimension by methods that use regular grids. When applying a regular grid method to determine the fractal dimension on valley schematic networks according to Howard (1967), it was found that the “fractal dimension of drainage patterns and valley networks according to Mandelbrot (1982)”, the “box-counting dimension according to Turcotte (2007a)” and the “capacity dimension according to Tichý (2012)” methods show values in the open interval (1, 2). In contrast, the value of the “box-counting dimensions according to Rodríguez-Iturbe & Rinaldo (2001) / Kolmogorov dimensions according to Zelinka & Včelař & Čandík (2006)” was greater than 2. Therefore, to achieve values in the open interval (1, 2) more steps are needed to be taken than in the case of other fractal dimensions.