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Multifractal Modeling of Worldwide and Canadian Metal Size-Frequency Distributions

  • Frits AgterbergEmail author
Original Paper
  • 26 Downloads

Abstract

The Pareto-lognormal frequency distribution, which can result from multifractal cascade modeling, previously was shown to be useful to describe the worldwide size-frequency distributions of metals including copper, zinc, gold and silver in ore deposits. In this paper, it is shown that the model also can be used for the size-frequency distributions of these metals in Canada which covers 6.6% of the continental crust. Like their worldwide equivalents, these Canadian deposits show two significant departures from the Pareto-lognormal model: (1) there are too many small deposits, and (2) there are too few deposits in the transition zone between the central lognormal and the upper tail Pareto describing the size-frequency distribution of the largest deposits. Probable causes of these departures are: (1) historically, relatively many small ore deposits were mined before bulk mining methods became available in the twentieth century, and (2) economically, giant and supergiant deposits are preferred for mining and these have strongest geophysical and geochemical anomalies. It is shown that there probably exist many large deposits that have not been discovered or mined. Although overall the samples of the size-frequency distributions are very large, frequencies uncertainties associated with the largest deposits are relatively small and it remains difficult to estimate more precisely how many undiscovered mineral deposits there are in the upper tails of the size-frequency distributions of the metals considered.

Keywords

Multifractals Metal size-frequency distributions Base metals Precious metals Resources 

Notes

Acknowledgments

Thanks are due to Alberto Patiño-Douce and Renguang Zuo for discussion and helpful suggestions.

References

  1. Agterberg, F. P. (1964). Statistical techniques for geological data. Tectonophysics, 1, 233–255.CrossRefGoogle Scholar
  2. Agterberg, F. P. (2014). Geomathematics: Theoretical foundations, applications and future developments. Quantitative geology and geostatistics (Vol. 18). Heidelberg: Springer.Google Scholar
  3. Agterberg, F. P. (2017a). Pareto-lognormal modeling of known and unknown metal resources. Natural Resources Research, 26(1), 3–20. (with erratum on p. 21).CrossRefGoogle Scholar
  4. Agterberg, F. P. (2017b). Pareto-lognormal modeling of known and unknown metal resources. II. Method refinement and further applications. Natural Resources Research, 26(3), 265–283.CrossRefGoogle Scholar
  5. Agterberg, F. P. (2018a). Can multifractals be used for mineral resource appraisal? Journal of Geochemical Exploration, 189, 59–63.CrossRefGoogle Scholar
  6. Agterberg, F. P. (2018b). Statistical modeling of regional and worldwide size-frequency distributions of metal deposits. In B. S. Daya Sagar, Q. Cheng, & F. Agterberg (Eds.), Handbook of mathematical geosciences: Fifty years of IAMG (pp. 505–527). Heidelberg: Springer.CrossRefGoogle Scholar
  7. Agterberg, F. P. (2018c). New method of fitting Pareto-lognormal size-frequency distributions to worldwide Cu and Zn deposit size data. Natural Resources Research, 27(4), 405–417.CrossRefGoogle Scholar
  8. Aitchison, J., & Brown, J. A. C. (1957). The lognormal distribution. Cambridge: Cambridge University Press.Google Scholar
  9. Billingsley, P. (1986). Probability and measure (2nd ed.). New York: Wiley.Google Scholar
  10. Cargill, S. M. (1981). United States gold resource profile. Economic Geology, 76, 937–943.CrossRefGoogle Scholar
  11. Cargill, S. M., Root, D. H., & Bailey, E. H. (1980). Resources estimation from historical data: Mercury, a test case. Mathematical Geology, 12, 489–522.CrossRefGoogle Scholar
  12. Cargill, S. M., Root, D. H., & Bailey, E. H. (1981). Estimating unstable resources from historical industrial data. Economic Geology, 76, 1081–1095.CrossRefGoogle Scholar
  13. Cheng, Q., & Agterberg, F. P. (1995). Multifractal modeling and spatial statistics. Mathematical Geology, 28, 1–16.CrossRefGoogle Scholar
  14. Cheng, Q., & Agterberg, F. P. (1996). Comparison between two types of multifractal modeling. Mathematical Geology, 29, 1001–1015.CrossRefGoogle Scholar
  15. Clauset, A., Shalizi, C. R., & Newman, M. E. J. (2009). Power-law distributions in empirical data. SIAM Review, 51, 661–703.CrossRefGoogle Scholar
  16. Crovelli, R. A. (1995). The generalized 20/80 law using probabilistic fractals applied to petroleum field size. Nonrenewable Resources, 4(3), 223–241.CrossRefGoogle Scholar
  17. Cuney, C., & Kyser, K. (2008). Recent and not-so-recent developments in uranium deposits and implications for exploration. Short Course Series 39. Quebec City, QC: Mineralogical Association of Canada.Google Scholar
  18. De Wijs, H. J. (1951). Statistics of ore distribution, I. Geologie en Mijnbouw, 30, 365–375.Google Scholar
  19. Drew, L. J., Schuenemeyer, J. H., & Bawlee, W. J. (1982). Estimation of the future rates of oil and gas discoveries in the western Gulf of Mexico. US Geological Survey Professional Paper 1252, U.S. GPO, Washington, DC.Google Scholar
  20. Frimmel, H. E., Groves, D. L., Kirk, J., Rutz, J., Chesley, J., & Minter, W. E. L. (2005). The formation and preservation of the Witwatersrand goldfields, the world’s largest gold province. Economic Geology, 100, 769–797.CrossRefGoogle Scholar
  21. Hald, A. (1952). Statistical theory and engineering applications. New York: Wiley.Google Scholar
  22. Kleiber, C., & Kotz, S. (2003). Statistical distributions in economics and actuarial sciences. Hoboken: Wiley.CrossRefGoogle Scholar
  23. Lévy, P. (1925). Calcul des Probabilités (Chapter 6). Paris: Gauthier-Villars.Google Scholar
  24. Lovejoy, S., & Schertzer, D. (2007). Scaling and multifractal fields in the solid Earth and topography. Nonlinear Processes in Geophysics, 14, 465–502.CrossRefGoogle Scholar
  25. Lydon, J. W. (2007). An overview of economic and geological contexts of Canada’s major mineral deposit types. In M. D. Goodfellow (Ed.), Mineral deposits in Canada (Vol. 5, pp. 3–48). ‎Montreal: Geological Survey of Canada, Special Publication.Google Scholar
  26. Mandelbrot, B. B. (1960). The Pareto-Lévy law and the distribution of income. International economic Review, 1, 79–106.CrossRefGoogle Scholar
  27. Mandelbrot, B. B. (1983). The fractal geometry of nature. San Francisco: Freeman.CrossRefGoogle Scholar
  28. Mandelbrot, B. B. (1998). Multifractals and 1/f noise. New York: Springer.Google Scholar
  29. Montroll, E. W., & Shlesinger, F. (1982). On 1/f noise and other distributions with long tails. Proceedings of the National Academy of Sciences of the United States of America, 70, 3380–3383.CrossRefGoogle Scholar
  30. Orris, G. J., & Bliss, J. D. (1985). Geologic and grade-volume data on 330 gold placer deposits. U.S. Geological Survey Open File Report.  https://doi.org/10.1007/s11053-015-9265-0.Google Scholar
  31. Pareto, V. (1895). La legge della domanda. Giornale degli Economisti, 10, 59–68. English translation in Rivista di Politica Economica, 87, 691–700 (1997).Google Scholar
  32. Patiño-Douce, A. E. (2016a). Metallic mineral resources in the twenty first century. I. Historical extraction trends and expected demand. Natural Resources Research, 25(1), 71–90.CrossRefGoogle Scholar
  33. Patiño-Douce, A. E. (2016b). Metallic mineral resources in the twenty first century. II. Constraints on future supply. Natural Resources Research, 25(1), 97–124.CrossRefGoogle Scholar
  34. Patiño-Douce, A. E. (2016c). Statistical distribution laws for metallic mineral deposit sizes. Natural Resources Research, 25(4), 365–387.CrossRefGoogle Scholar
  35. Patiño-Douce, A. E. (2017). Loss distribution model for metal discovery probabilities. Natural Resources Research, 25(3), 241–263.CrossRefGoogle Scholar
  36. Quandt, R. E. (1966). Old and new methods of estimation and the Pareto distribution. Metrica, 10, 55–82.CrossRefGoogle Scholar
  37. Reed, W. J. (2003). The Pareto law of incomes: An explanation and an extension. Physica A, 319, 579–597.CrossRefGoogle Scholar
  38. Reed, W. J., & Jorgensen, M. (2003). The double Pareto-lognormal distribution: A new parametric model for size distribution. Computational Statistics: Theory and Methods, 33(8), 1733–1753.Google Scholar
  39. Schuenemeyer, J. H., Drew, L. J., & Bliss, J. D. (2018). Predicting molybdenum deposit growth. In B. S. Daya Sagar, Q. Cheng, & F. Agterberg (Eds.), Handbook of mathematical geosciences: Fifty years of IAMG (pp. 395–409). Heidelberg: Springer.CrossRefGoogle Scholar
  40. Singer, D. A., & Menzie, W. D. (2010). Quantitative mineral resource assessments. New York: Oxford University Press.Google Scholar
  41. Turcotte, D. L. (1997). Fractals and chaos in geology and geophysics (2nd ed.). Cambridge: Cambridge University Press.CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2019

Authors and Affiliations

  1. 1.Geological Survey of CanadaOttawaCanada

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