Fractals and spatial statistics of point patterns

Abstract

The relationship between fractal point pattern modeling and statistical methods of parameter estimation in point-process modeling is reviewed. Statistical estimation of the cluster fractal dimension by using Ripley’s K-function has advantages in comparison with the more commonly used methods of box-counting and cluster fractal dimension estimation because it corrects for edge effects, not only for rectangular study areas but also for study areas with curved boundaries determined by regional geology. Application of box-counting to estimate the fractal dimension of point patterns has the disadvantage that, in general, it is subject to relatively strong “roll-off” effects for smaller boxes. Point patterns used for example in this paper are mainly for gold deposits in the Abitibi volcanic belt on the Canadian Shield. Additionally, it is proposed that, worldwide, the local point patterns of podiform Cr, volcanogenic massive sulphide and porphyry copper deposits, which are spatially distributed within irregularly shaped favorable tracts, satisfy the fractal clustering model with similar fractal dimensions. The problem of deposit size (metal tonnage) is also considered. Several examples are provided of cases in which the Pareto distribution provides good results for the largest deposits in metal size-frequency distribution modeling.

Keywords

fractal point pattern / spatial statistics / roll-off effect / cluster dimension / Pareto distribution

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