Multifractal Measures, Especially for the Geophysicist

Abstract

This text is addressed to both the beginner and the seasoned professional, geology being used as the main but not the sole illustration. The goal is to present an alternative approach to multifractals, extending and streamlining the original approach in Mandelbrot (1974). The generalization from fractal sets to multifractal measures involves the passage from geometric objects that are characterized primarily by one number, namely a fractal dimension, to geometric objects that are characterized primarily by a function. The best is to choose the function ρ(α), which is a limit probability distribution that has been plotted suitably, on double logarithmic scales. The quantity α is called Hölder exponent. In terms of the alternative function f(α) used in the approach of Frisch-Parisi and of Halsey et al., one has ρ(α) = f(α) − E for measures supported by the Euclidean space of dimension E. When f(α) ≥ 0, f(α) is a fractal dimension. However, one may have f(α) < 0, in which case α is called “latent.” One may even have α < 0, in which case α is called “virtual.” These anomalies’ implications are explored, and experiments are suggested. Of central concern in this paper is the study of low-dimensional cuts through high-dimensional multifractals. This introduces a quantity D q , which is shown for q > 1 to be a critical dimension for the cuts. An “enhanced multifractal diagram” is drawn, including f(α), a function called τ(q) and D q .