Multifractals and Local Singularity Analysis

Abstract

Multifractals are spatially intertwined fractals. For example, a chemical concentration value obtained from rock samples in a study area may be a fractal with fractal dimension different from those of other concentration values for the same element, but together the fractal dimensions may form a multifractal spectrum f (α) that is a continuous function of the singularity α, which depends on the concentration value. Self-similar patterns produce multifractals of this type. If a block of rock with chemical concentration value X is divided into equal parts, the halves have concentration values (1 + d) · X and (1 − d) · X where d is a constant. The model of de Wijs assumes that the dispersion index d is independent of block size. This cascade process produces a multifractal. The properties of a multifractal can be estimated by the method of moments or by the histogram method. The four-step method of moments has the advantage that the assumption of multifractality is being tested during its first step because this should produce an array of straight lines on a log-log plot of the spatial mass-partition function χ (ϵ,q) against measure of block size (ϵ) used. It should be kept in mind, however, that the multifractal spectrum estimated by the method of moments is primarily determined by the majority of measurements that are clustered around the mean or median. Very large or very small observed values are rare; because of this, the low-singularity and high-singularity tails of a multifractal spectrum generally cannot be estimated with sufficient precision. Because of strong local autocorrelations effects, singularity analysis can provide better estimates of singularities associated with the very large or very small observed values. Practical examples of multifractal modeling include the distribution of gold in the Mitchell-Sulphurets area, northwestern British Columbia, uranium resources in the U.S. and worldwide, lengths of surface fractures in the Lac du Bonnet Batholith, eastern Manitoba, geographic distribution of gold deposits in the Iskut map area, British Columbia. Local singularity mapping is useful for the detection of geochemical anomalies characterized by local enrichment even if contour maps for representing average variability are not constructed. Examples include singularity maps based on various element concentration values from stream sediment samples and their relation to tin deposits in the Gejiu area, Yunnan Province, as well as Ag and Pb-Zn deposits in northwestern Zhejiang Province, China. The iterative Chen algorithm for space series of element concentration values offers a new way to separate local singularities from regional trends. This technique is applied to the Pulacayo zinc and KTB copper values.