Fractals, fractal dimensions and landscapes — a review

Mandelbrot's fractal geometry is a revolution in topological space theory and, for the first time, provides the possibility of simulating and describing landscapes precisely by using a mathematical model. Fractal analysis appears to capture some “new” information that traditional parameters do not contain. A landscape should be (or is at most) statistically self-similar or statistically self-affine if it possesses a fractal nature. Mandelbrot's fractional Brownian motion (fBm) is the most useful mathematical model for simulating landscape surfaces. The fractal dimensions for different landscapes and calculated by different methods are difficult to compare. The limited size of the regions surveyed and the spatial resolution of the digital elevation models (DEMs) limit the precision and stability of the computed fractal dimension. Interpolation artifacts of DEMs and anisotropy create additional difficulties in the computation of fractal dimensions. Fractal dimensions appear to be spatially variable over landscapes. The region-dependent spatial variation of the dimension has more practical significance than the scale-dependent spatial variation. However, it is very difficult to use the fractal dimension as a distributed geomorphic parameter with high “spatial resolution”. The application of fractals to landscape analysis is a developing and immature field and much of the theoretical rigour of fractal geometry has not yet been exploited. The physical significance of landscape fractal characteristics remains to be explained. Research in geographical information theory and fractal theory needs to be strengthened in order to improve the application of fractal geometry to the geosciences.