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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union
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Volume 14, issue 3
Nonlin. Processes Geophys., 14, 201–209, 2007
https://doi.org/10.5194/npg-14-201-2007
© Author(s) 2007. This work is licensed under
the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 License.

Special issue: New developments in scaling, scale and nonlinearity in the...

Nonlin. Processes Geophys., 14, 201–209, 2007
https://doi.org/10.5194/npg-14-201-2007
© Author(s) 2007. This work is licensed under
the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 License.

  23 May 2007

23 May 2007

Mixtures of multiplicative cascade models in geochemistry

F. P. Agterberg F. P. Agterberg
  • Geological Survey of Canada, 601 Booth Street, Ottawa, K1A0E8, Canada

Abstract. Multifractal modeling of geochemical map data can help to explain the nature of frequency distributions of element concentration values for small rock samples and their spatial covariance structure. Useful frequency distribution models are the lognormal and Pareto distributions which plot as straight lines on logarithmic probability and log-log paper, respectively. The model of de Wijs is a simple multiplicative cascade resulting in discrete logbinomial distribution that closely approximates the lognormal. In this model, smaller blocks resulting from dividing larger blocks into parts have concentration values with constant ratios that are scale-independent. The approach can be modified by adopting random variables for these ratios. Other modifications include a single cascade model with ratio parameters that depend on magnitude of concentration value. The Turcotte model, which is another variant of the model of de Wijs, results in a Pareto distribution. Often a single straight line on logarithmic probability or log-log paper does not provide a good fit to observed data and two or more distributions should be fitted. For example, geochemical background and anomalies (extremely high values) have separate frequency distributions for concentration values and for local singularity coefficients. Mixtures of distributions can be simulated by adding the results of separate cascade models. Regardless of properties of background, an unbiased estimate can be obtained of the parameter of the Pareto distribution characterizing anomalies in the upper tail of the element concentration frequency distribution or lower tail of the local singularity distribution. Computer simulation experiments and practical examples are used to illustrate the approach.

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