Are Fractal Dimensions of the Spatial Distribution of Mineral Deposits Meaningful?



It has been proposed that the spatial distribution of mineral deposits is bifractal. An implication of this property is that the number of deposits in a permissive area is a function of the shape of the area. This is because the fractal density functions of deposits are dependent on the distance from known deposits. A long thin permissive area with most of the deposits in one end, such as the Alaskan porphyry permissive area, has a major portion of the area far from known deposits and consequently a low density of deposits associated with most of the permissive area. On the other hand, a more equi-dimensioned permissive area, such as the Arizona porphyry permissive area, has a more uniform density of deposits. Another implication of the fractal distribution is that the Poisson assumption typically used for estimating deposit numbers is invalid. Based on datasets of mineral deposits classified by type as inputs, the distributions of many different deposit types are found to have characteristically two fractal dimensions over separate non-overlapping spatial scales in the range of 5–1000 km. In particular, one typically observes a local dimension at spatial scales less than 30–60 km, and a regional dimension at larger spatial scales. The deposit type, geologic setting, and sample size influence the fractal dimensions. The consequence of the geologic setting can be diminished by using deposits classified by type. The crossover point between the two fractal domains is proportional to the median size of the deposit type. A plot of the crossover points for porphyry copper deposits from different geologic domains against median deposit sizes defines linear relationships and identifies regions that are significantly underexplored. Plots of the fractal dimension can also be used to define density functions from which the number of undiscovered deposits can be estimated. This density function is only dependent on the distribution of deposits and is independent of the definition of the permissive area. Density functions for porphyry copper deposits appear to be significantly different for regions in the Andes, Mexico, United States, and western Canada. Consequently, depending on which regional density function is used, quite different estimates of numbers of undiscovered deposits can be obtained. These fractal properties suggest that geologic studies based on mapping at scales of 1:24,000 to 1:100,000 may not recognize processes that are important in the formation of mineral deposits at scales larger than the crossover points at 30–60 km.


Fractal dimensions mineral deposits deposit density 



I wish to acknowledge Neal Fordyce of Placer Dome who felt that Carl Carlson’s papers raised interesting questions and asked if fractal dimensions might provide insights on where to explore. Steve Ludington, Mike Zientek, Vic Mossotti, and Don Singer have provided useful discussions and suggestions during the course of the research for this article. Vic Mossotti, Frits Agterberg, Quiming Cheng, and P. Gumeil provided helpful reviews of this article. I would also like to acknowledge the work of Berger, Bookstom, Ludington, Moring, Mutschler, Long, DeYoung, and Singer who compiled the databases of mineral deposits that made this analysis possible.


  1. Agterberg, F. P., Cheng, Q., and Wright, D. F., 1993, Fractal modeling of mineral deposits. Proceedings 24th APCOM Symposium, Canadian Inst. Mining, Metallurgy, and Petroleum Engineers, v. 1, p. 43–53.Google Scholar
  2. Barnsley M. F. (1988) Fractals everywhere. Academic Press, Boston, MA, 394pGoogle Scholar
  3. Blenkinsop T. G., Sanderson D. J. (1999) Are gold deposits in the crust fractals? A study of gold mines in the Zimbabwe craton. In McCaffrey K. J. W., Lonergan L., Wilkinson J. J. (Eds.), Fractures, fluid flow and mineralization. Geological Society, London, Special Publication 155, pp. 141–151Google Scholar
  4. Carlson C. A. (1991) Spatial distributions of ore deposits. Geology 19:111–114 CrossRefGoogle Scholar
  5. Cheng Q., Agterberg F. P. (1995) Multifractal modeling and spatial point processes. Math Geol 27(7): 831–845CrossRefGoogle Scholar
  6. Herzfeld U. C. (1993) Fractals in geosciences – challenges and concerns. In Davis J. C., Herzfeld U. C. (eds.), Computers in geology – 25 years of progress. Oxford University Press, New York, pp. 217–230Google Scholar
  7. Johnston J. D., McCaffrey K. J. W. (1996) Fractal properties of vein systems and the variation of scaling relationships with mechanism. Jour Struct Geol 18:349–358CrossRefGoogle Scholar
  8. King, P. B., and Beikman, H. M., 1974, Geologic map of United States. US Geological Survey, 3 sheets, scale 1:2,500,000.Google Scholar
  9. Long, K. R., DeYoung, J., and Ludington, S. D., 1998, Database of significant deposits of gold, silver, copper, lead, and zinc in the United States. US Geological Survey Open-File Report 98–206, Online version 1.0 (–206/).
  10. Mandelbrot B. B. (1985) Self-affine fractals and fractal dimension. Physica Scripta 32:257–260CrossRefGoogle Scholar
  11. McCaffrey K. J. W., Johnston J. D. (1996) Fractal analysis of mineralized vein deposit. Curraghinalt gold deposit, County Tyrone. Mineralium Deposita 31:52–58CrossRefGoogle Scholar
  12. Mutschler, F. E., Ludington, S., and Bookstrom, A. A., 1999, Giant porphyry-related metal camps of the world—a database: and database.
  13. Raines, G. L., Sawatzky, D. L., and Connors, K. A., 1996, Great Basin geoscience data base. US Geological Survey Digital Data Series 41.Google Scholar
  14. Roberts S., Sanderson D. J., Gumiel P. (1998) Fractal analysis of Sn-W mineralization from Central Iberia-insights into the role of fracture connectivity in the formation of an ore deposit. Economic Geology 93(3):360–365CrossRefGoogle Scholar
  15. Roberts S., Sanderson D. J., Gumiel P. (1999) Fractal analysis and percolation properties of veins. In McCaffrey K. J. W., Lonergan L., Wilkinson J. J. (Eds.), Fractures, fluid flow and mineralization. Geological Society, London, Special Publication 155, pp. 7–16Google Scholar
  16. Sanderson D. J., Roberts S., Gumiel P. (1994) A fractal relationship between vein thickness and gold grade in drill-core from La Codosera, Spain. Economic Geology 89:68–173Google Scholar
  17. Singer D. A. (1994) Conditional estimates of the number of podiform chromite deposits. Nonrenewable Resources 2(2):69–81CrossRefGoogle Scholar
  18. Singer, D. A., Berger, V. I., and Moring, B. C., 2005, Porphyry copper deposits of the world: database, map, and grade and tonnage models. US Geological Survey Open File Report 2005-1060, p. 9 and database,
  19. Singer D. A., Berger V. I., Menzie W. D., Berger B. R. (2005) Porphyry copper deposit density. Economic Geology 100:491–514CrossRefGoogle Scholar
  20. Singer, D. A., and Menzie, W. D., 2005, Statistical guides to estimating the number of undiscovered mineral deposits: an example with porphyry copper deposits. Proceeding of IAMG’05, GIS and Spatial Analysis, v. 2, p. 1028–1029.Google Scholar
  21. Stewart, J. H., and Carlson, J. E., 1978, Geologic map of Nevada. US Geological Survey, 2 sheets, scale 1:500,000.Google Scholar
  22. U.S.Geological Survey National Mineral Assessment Team, 1998, Assessment of undiscovered deposits of gold, silver, copper, lead, and zinc in the United States. US Geological Survey Circular 1178, p. 29 and database.Google Scholar

Copyright information

© International Association for Mathematical Geology 2008

Authors and Affiliations

  1. 1.U.S. Geological Survey (retired)c/o Mackay School of Earth Sciences, UNRRenoUSA

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