A simple method to estimate the fractal dimension of a self‐affine series

Time/space series of natural variables (e.g., surface topography) are often self‐affine, i.e., measurements taken at different resolutions have the same statistical characteristics when rescaled by factors that are generally different for the horizontal and vertical coordinates. Self‐affinity implies that the standard deviation measured on a sample spanning a length w is proportional to w H = w 2 − D , where H is the Hurst exponent and D is the fractal dimension (1 ≤ D ≤2 for a fractal series). In this paper, a “roughness‐length” method based on this property of self‐affine series is presented. In practice, the root‐mean‐square roughness is computed in a number of windows of varying length w , and H is measured from the slope of a log‐log plot of roughness versus w . Montecarlo simulations show that the fractal dimension as measured by the roughness‐length method is approximately the same as that defined by the power spectrum. The roughness‐length method is closely related to the grid fractal dimension, is simple to implement, and can be applied to non‐uniformly spaced series.