Complexity and scale in geomorphology: Statistical self-similarity vs. characteristic scales

Abstract

Two models of the relationship between complexity and scale of geomorphic lines are compared, one based on statistical self-similarity (in which complexity is invariant for some range of scale), and the other on the concept of characteristic scales (in which complexity changes continuously with scale). Two corresponding techniques are used in the comparison, fractal analysis utilizing the divider method, and an angle measure technique. These techniques are applied to three types of coastlines: fiord, volcanic, and tectonic, in order to ascertain which model, statistical self-similarity or characteristic scales, is more useful in understanding variations in coastline complexity for scale. Apparently linear log-log plots of number of steps against steplength produced by fractal analysis display slight but significant curvature. Upon closer examination, it is determined that using fractal dimension to compare even the same types of features is unreliable because of the dependency of fractal dimension on scale of measurement, even if the same steplengths are used throughout the study. These results are corroborated by the use of the angle measure technique, a method based on measuring angles between points along a digitized line. It is concluded that the coastlines examined display no evidence of statistical self-similarity and that the characteristic scales model is more useful in investigating complexity and scale in geomorphology.