Some Distinctions Between Self-Similar and Self-Affine Estimates of Fractal Dimension with Case History

Compass, power-spectral, and roughness-length estimates of fractal dimension are widely used to evaluate the fractal characteristics of geological and geophysical variables. These techniques reveal self-similar or self-affine fractal characteristics and are uniquely suited for certain analysis. Compass measurements establish the self-similarity of profile and can be used to classify profiles based on variations of profile length with scale. Power spectral and roughness-length methods provide scaleinvariant self-affine measures of relief variation and are useful in the classification of profiles based on relative variation of profile relief with scale. Profile magnification can be employed to reduce differences between the compass and power-spectral dimensions; however, the process of magnification invalidates estimates of profile length or shortening made from the results. The power-spectral estimate of fractal dimension is invariant to magnification, but is generally subject to significant error from edge effects and nonstationarity. The roughness-length estimate is also invariant to magnification and in addition is less sensitive to edge effects and nonstationarity. Analysis of structural cross sections using these methods highlight differences between self-similar and self-affine evaluations. Shortening estimates can be made from the compass walk analysis that includes shortening contributions from predicted small-scale structure. Roughness-length analysis reveals systematic structural changes that, however, cannot be easily related to strain. Power-spectral analysis failed to extract useful structural information from the sections.