Natural Resources Research

, Volume 25, Issue 4, pp 365–387 | Cite as

Statistical Distribution Laws for Metallic Mineral Deposit Sizes

  • Alberto E. Patiño Douce


Given that expanding industrial societies rely on the continuing supply of metallic raw materials, it is of interest to be able to generate estimates of the likely magnitude of undiscovered metal resources. Statistical predictions of this kind require that one knows the frequency distribution of the metal content (or tonnage) of mineral deposits. The distribution of metal content among deposits of different sizes may be treated empirically (i.e., without explicit formulation of a density function), or formally, starting from a density function that is known to represent the distribution of metal in known deposits. The empirical approach has been applied to the prediction of undiscovered resources of some metals, most notably copper. The focus here is on the formal approach, which may be simpler and more general than the empirical one. It has long been suspected that the distribution of metal tonnages is heavy-tailed, in the sense that a relatively few very large deposits are likely to contain much of the metal endowment. There is, however, uncertainty about the precise nature of the distribution of metal tonnages. Lognormal and power law models have been proposed. Under some conditions, which are examined here, the choice of one or the other of these two models may result in widely diverging estimates of the likely magnitude and distribution of undiscovered resources. However, a detailed statistical analysis of the available deposit tonnage data for 20 metals reveals that it is not possible to decide between the two models. Using conservative significance levels, both models fit the data equally well. Two different procedures are used to attempt to decide whether either of the two models is more plausible: likelihood ratio tests and power law fits to lognormally distributed synthetic data sets that mimic the natural data sets. Neither of the two procedures can detect any preference for one or the other of the two models, for any of the metals discussed here. This uncertainty may arise from the relatively small sizes of the samples available, which are limited to known and reasonably well-characterized metallic mineral deposits. There is at present no statistical justification to choose between a lognormal and a power law description of metal distribution in mineral deposits. This uncertainty should be kept in mind when preparing, and interpreting, estimates of the magnitude of undiscovered mineral resources.


Resource distribution statistics power law lognormal deposit size Pareto 



Reviews by Frits Agterberg and an anonymous reviewer were most helpful in improving this manuscript and are gratefully acknowledged.


  1. Agterberg, F. P. (1995). Multifractal modeling of the size and grade of giant and supergiant deposits. International Geology Review, 37, 1–8.CrossRefGoogle Scholar
  2. Agterberg, F. P. (2007). Mixtures of multiplicative cascade models in geochemistry. Nonlinear Processes in Geophysics, 14, 201–209.CrossRefGoogle Scholar
  3. Agterberg, F. P. (2011). Principles of probabilistic regional mineral resource estimation. Journal of China University of Geosciences, 36, 189–199. doi: 10.3799/dqkx.2011.020.Google Scholar
  4. Ahrens, L. H. (1954). The lognormal distribution of the elements (a fundamental law of geochemistry and its subsidiary). Geochimica et Cosmochimica Acta, 5, 49–73.CrossRefGoogle Scholar
  5. Allègre, C. J., & Lewin, E. (1995). Scaling laws and geochemical distributions. Earth and Planetary Science Letters, 132, 1–13.CrossRefGoogle Scholar
  6. Cargill, S. M., Root, D. H., & Bailey, E. H. (1981). Estimating usable resources from historical industry data. Economic Geology, 76, 1081–1095.CrossRefGoogle Scholar
  7. Clauset, A., Shalizi, C. R., & Newman, M. E. J. (2009). Power-law distributions in empirical data. SIAM Review, 51, 661–703. doi: 10.1137/070710111.CrossRefGoogle Scholar
  8. Fazio, G., & Modica, M. (2015). Pareto or log-normal? Best fit and truncation in the distribution of all cities. Journal of Regional Science, 55, 736–756. doi: 10.1111/jors.12205.CrossRefGoogle Scholar
  9. Johnson, K. M., Hammarstrom, J. M., Zientek, M. L., & Dicken, C. L. (2014). Estimate of undiscovered copper resources of the world, 2013. USGS Fact sheet 20143004.Google Scholar
  10. Mitzenmacher, M. (2004). A brief history of generative models for power law and lognormal distributions. Internet Mathematics, 1, 226–251.CrossRefGoogle Scholar
  11. Patiño Douce, A. E. (2016a). Metallic mineral resources in the twenty first century: I. Historical extraction trends and expected demand. Natural Resources Research, 25, 71–90.CrossRefGoogle Scholar
  12. Patiño Douce, A. E. (2016b). Metallic mineral resources in the twenty first century: II. Constraints on future supply. Natural Resources Research, 25, 97–124.CrossRefGoogle Scholar
  13. Root, D. H., Menzie, W. D., & Scott, W. A. (1992). Computer Monte Carlo simulations in quantitative resource estimation. Nonrenewable Resources, 1, 125–138.CrossRefGoogle Scholar
  14. Singer, D. A. (1995). World class base and precious metal deposits. A quantitative analysis. Economic Geology, 90, 88–104.CrossRefGoogle Scholar
  15. Singer, D. A. (2013). The lognormal distribution of metal resources in mineral deposits. Ore Geology Reviews, 55, 80–86.CrossRefGoogle Scholar
  16. Singer, D. A., & Menzie, W. D. (2010). Quantitative mineral resource assessments—An integrated approach. New York, NY: Oxford University Press.Google Scholar
  17. Turcotte, D. L. (1986). A fractal approach to the relationship between ore grade and tonnage. Economic Geology, 81, 1528–1532.CrossRefGoogle Scholar
  18. Turcotte, D. L. (2002). Fractals in petrology. Lithos, 65, 261–271.CrossRefGoogle Scholar
  19. Wasserman, L. (2005). All of statistics. A concise course in Statistical Inference. Springer: New York.Google Scholar

Copyright information

© International Association for Mathematical Geosciences 2016

Authors and Affiliations

  1. 1.Department of GeologyUniversity of GeorgiaAthensUSA

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