Fractal dimensions of landscapes and other environmental data

Abstract

Mandelbrot¹ has introduced the term ‘fractal’ specifically for temporal or spatial phenomena that are continuous but not difierentiable, and that exhibit partial correlations over many scales. The term fractal strictly defined refers to a series in which the Hausdorf–Besicovitch dimension exceeds the topological dimension. A continuous series, such as a polynomial, is differentiable because it can be split up into an infinite number of absolutely smooth straight lines. A non-differentiable continuous series cannot be so resolved. Every attempt to split it up into smaller parts results in the resolution of still more structure or roughness. For a linear fractal function, the Hausdorf–Besicovitch dimension D may vary between 1 (completely differentiate) and 2 (so rough and irregular that it effectively takes up the whole of a two-dimensional topological space). For surfaces, the corresponding range for D lies between 2 (absolutely smooth) and 3 (infinitely crumpled). Because the degree of roughness of spatial data is important when trying to make interpolations from point data such as by least-squares fitting or kriging², it is worth examining them beforehand to see if the data contain evidence of variation over different scales, and how important these scales might be. Mandelbrot's work¹ suggests that the fractal dimensions of coastlines and other linear natural phenomena are of the order of D = 1.2–1.3, implying that long range effects dominate. I show here that published data on many environmental variables suggest that not only are they fractals, but that they may have a wide range of fractal dimensions, including values that imply that interpolation mapping may not be appropriate in certain cases.