Fractally Invariant Distributions and an Application in Geochemical Exploration
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Fractally Invariant Distributions and an Application in Geochemical Exploration1 Wei Shen2,3 and D. R. Cohen3 Fractal modelling has been applied extensively as a means of characterizing the spatial distribution of geological phenomena that display self-similarity at differing scales of measurement. A fractal distribution exists where the number of objects exhibiting values larger than a specified magnitude displays a power-law dependence on that magnitude, and where this relationship is scale-invariant. This paper shows that a number of distributions, including power-function, Pareto, log-normal and Zipf, display fractal properties under certain conditions and that this may be used as the mathematical basis for developing fractal models for data exhibiting such distributions. Population limits, derived from fractal modelling using a summation method, are compared with those derived from more conventional probability plot modelling of stream sediment geochemical data from north-eastern New South Wales. Despite some degree of subjectivity in determining the number of populations to use in the models, both the fractal and probability plot modelling have assisted in isolating anomalous observations in the geochemical data related to the occurrence of mineralisation or lithological differences between sub-catchments. Thresholds for the main background populations determined by the fractal model are similar to those established using probability plot modelling, however the summation method displays less capacity to separate out anomalous populations, especially where such populations display extensive overlap. This suggests, in the geochemical data example provided, that subtle differences in the population parameters may not significantly alter the fractal dimension. KEY WORDS: fractal modelling, anomaly detection, geochemistry, summation method.
INTRODUCTION Isolating geochemical patterns or populations associated with particular domains or geological processes, including separating background from anomalous element concentrations, is a common objective in geochemical data processing (Cheng, 1999; Garrett, 1989; Govett and others, 1975; Miesch, 1981; Sinclair, 1991). Whereas recent attention has been directed toward multivariate pattern 1Received
3 September 2004; accepted 21 February 2005. of Higher and New Techniques Applied to Land Resources, State Key Laboratory of GeoProcesses and Mineral Resources, China University of Geosciences, Xueyuan Road 29, Beijing 100083, P. R. China; e-mail: [email protected] or [email protected] 3School of Biological, Earth and Environmental Sciences, University of New South Wales, Sydney, NSW 2052, Australia. 2Institute
895 C 2005 International Association for Mathematical Geology 0882-8121/05/1100-0895/1
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recognition, there is still a place for characterizing and modelling individual variables. Most geochemical anomaly detection approaches emphasize frequency distribution, correlation and cross-correlations between element concentrations while ignoring s
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