Some distinctions between self-similar and self-affine estimates of fractal dimension with case history

Compass. power-spectral, and roughness-length estimates of fractal dimensio n 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. Comp ass measurements establish the self-similarity of profile and cart be used to classify profiles based on variations of profile length with scale. Powe r spectral and roughness-length methods provide scale-invariant self-affine measures of relief variation and are useful in the classification of profi les based on relative variation of profile relief with scale. Profile magni fication can be employed to reduce differences between the compass and powe r-spectral dimensions; however, the process of magnification invalidates es timates of profile length or shortening made from the results. The power-sp ectral estimate of fractal dimension is invariant to magnification, but is generally subject to significant error from edge effects and nonstationarit y. 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 betwee n self-similar and self-affine evaluations. Shortening estimates can be mad e from the compass walk analysis that includes shortening contributions fro m predicted small-scale structure. Roughness-length analysis reveals system atic structural changes that, however; cannot be easily related to strain. Power-spectral analysis failed to extract useful structural information fro m the sections.