Skip to main content
SpringerLink
Log in

Fractals

  • Chapter
  • First Online:
Geomathematics: Theoretical Foundations, Applications and Future Developments

Part of the book series: Quantitative Geology and Geostatistics ((QGAG,volume 18))

  • 1837 Accesses

Abstract

As illustrated in previous chapters, many geological features display random characteristics that can be modeled by adopting methods of mathematical statistics. A question to which new answers are being sought is: Where does the randomness in Nature come from? Nonlinear process modeling is providing new clues to answers. Benoit Mandelbrot discovered about 50 years ago that many objects on Earth can be modeled as fractals with non-Euclidean dimensions. Spatial distribution of chemical elements in the Earth’s crust and features such as the Earth’s topography that traditionally were explained by using deterministic process models, now also are modeled as fractals or multifractals, which are spatially intertwined fractals. Which processes have produced phenomena that are random and often characterized by non-Euclidian dimensions? The physico-chemical processes that have resulted in the Earth’s present configuration were essentially deterministic and “linear” but globally as well as locally they may display random features that can only be modeled by adopting a non-linear approach that is increasingly successful in localized prediction. The situation is analogous to the relation between climate and weather. Longer term climate change can be modeled deterministically but short-term weather shows random characteristics that are best modeled by adopting the non-linear approach in addition to the use of conventional deterministic equations. This chapter reviews fractal modeling of solid Earth observations and processes with emphasis on topography, thickness measurements, geochemistry and hydrothermal processes. There is some overlap with multifractals that will be discussed in more detail in the next two chapters. Special attention is paid to improvements in goodness of fit and prediction obtained by non-linear modeling. The spatial distribution of ore deposits within large regions or within worldwide permissive tracts often is fractal. Illustrative examples to be discussed include lode gold deposits on the Canadian Shield and worldwide podiform Cr deposits, volcanogenic massive sulphide and porphyry copper deposits. The Pareto distribution is closely associated with fractal modeling of metal distribution within rocks or surficial cover in large regions. The Concentration-Area (C-A) method is a useful new tool for geochemical prospecting to help delineate subareas with anomalously high element concentration values that can be targets for further exploration with drilling.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Afzal P, Alghalandis YF, Khakzad A, Moarefvand P, Omran NR (2011) Delineation of mineralization zones in porphyry Cu deposits by fractal concentration-volume modeling. J Geochem Explor 108:220–232

    Google Scholar 

  • Agterberg FP (1980) Mineral resource estimation and statistical exploration. In: Miall AD (ed) Facts and principles of world oil occurrence, Canadian Society of Petroleum Geologists Memoir 6. Canadian Society of Petroleum Geologists, Calgary, pp 301–318

    Google Scholar 

  • Agterberg FP (1981) Geochemical crustal abundance models. Trans Soc Min Eng AIME 268:1823–1830

    Google Scholar 

  • Agterberg FP (1993) Calculation of the variance of mean values for blocks in mineral potential mapping. Nonrenew Resour 2:312–324

    Google Scholar 

  • Agterberg FP (1994) FORTRAN program for the analysis of point patterns with correction for edge effects. Comput Geosci 20(2):229–245

    Google Scholar 

  • Agterberg FP (1995a) Multifractal modeling of the sizes and grades of giant and supergiant deposits. Int Geol Rev 37:1–8

    Google Scholar 

  • Agterberg FP (1995b) Power laws versus lognormal models in mineral exploration. In: Proceedings of CAMI’95, 3rd Canadian conference on computer applications in the mineral industries, Montreal, pp 17–26

    Google Scholar 

  • Agterberg FP (2001) Multifractal simulation of geochemical map patterns. In: Merriam D, Davis JC (eds) Geologic modeling and simulation: sedimentary systems. Kluwer, New York, pp 327–346

    Google Scholar 

  • Agterberg FP (2007) Mixtures of multiplicative cascade models in geochemistry. Nonlinear Process Geophys 14:201–209

    Google Scholar 

  • Agterberg FP (2012) Sampling and analysis of element concentration distribution in rock units and orebodies. Nonlinear Process Geophys 19:23–44

    Google Scholar 

  • Agterberg FP (2013) Fractals and spatial statistics of point patterns. J Earth Sci 24(1):1–11

    Google Scholar 

  • Agterberg FP, Divi SR (1978) A statistical model for the distribution of copper, lead and zinc in the Canadian Appalachian region. Econ Geol 73:230–245

    Google Scholar 

  • Agterberg FP, Chung CF, Fabbri AG, Kelly AM, Springer JS (1972) Geomathematical evaluation of copper and zinc potential of the Abitibi area, Ontario and Quebec, Geological Survey of Canada Paper 71-41. Department of Energy, Mines and Resources, Ottawa

    Google Scholar 

  • Agterberg FP, Cheng Q, Wright DF (1993) Fractal modelling of mineral deposits. In: Proceedings of APCOM XX1V, international symposium on the applications of computers and operations research in the mineral industries, Canadian Institute of Mining, Metallurgy and Petroleum, Montreal, vol 1, pp 3–53

    Google Scholar 

  • Ahrens LH (1953) A fundamental law of geochemistry. Nature 712:1148

    Google Scholar 

  • Aitchison J, Brown JAC (1957) The lognormal distribution. Cambridge University Press, Cambridge

    Google Scholar 

  • Alldrick DJ (1987) Geology and mineral deposits of the Salmon River Valley, Stewart area, NTS 104A and 104B. British Columbia Ministry of Energy, Mines and Petroleum Resources, Geological Survey Branch, Open File Map 1987-22

    Google Scholar 

  • Allègre CJU, Lewin E (1995) Scaling laws and geochemical distributions. Earth Planet Sci Lett 132:1–3

    Google Scholar 

  • Anderson RG (1989) A stratigraphic, plutonic, and structural framework for the Iskut River map area, northwestern British Columbia. Geol Surv Can Pap 89-1E:145–154

    Google Scholar 

  • Anderson RG (1993) Mesozoic stratigraphic and plutonic framework for northwestern Stikinia (Iskut River area), northwestern British Columbia, Canada. In: Dunne C, McDougall KA (eds) Mesozoic paleogeography of the western United States, II, Pacific Section 71. Society of Economic Paleontologists and Mineralogists, Los Angeles, pp 477–494

    Google Scholar 

  • Baddeley A, Turner R (2012) Package ‘Spatstat’, manual, version 1.30.0 (Released 2012-12-23) PDF. http://140.247.115.171

  • Baddeley A, Bárány I, Scheicher R (2006) Spatial point processes and their applications. Lect Notes Math 192:1–75

    Google Scholar 

  • Baddeley A, Møller J, Poles AG (2008) Properties of residuals for spatial point processes. Ann Inst Stat Math 60(3):627–649

    Google Scholar 

  • Bak P (1996) The science of self-organised criticality. Copernicus Press, New York

    Google Scholar 

  • Barnsley M (1988) Fractals everywhere. Academic Press, Boston

    Google Scholar 

  • Barton CC, La Pointe PR (1995) Fractals in the earth sciences. Plenum, New York

    Google Scholar 

  • Blenkinsop TG (1995) Fractal measures for size and spatial distributions of gold mines: economic implications. In: Blenkinsop TG, Tromp PL (eds) Sub-Saharan economic geology, Special Publication Geological Society of Zimbabwe, 3. A.A. Balkema, Rotterdam, pp 177–186

    Google Scholar 

  • Brinck JW (1974) The geochemical distribution of uranium as a primary criterion for the formation of ore deposits. In: Chemical and physical mechanisms in the formation of uranium mineralization, geochronology, isotope geology and mineralization. International Atomic Energy Agency Proceedings Series STI/PUB/374, International Atomic Energy Agency, Vienna, pp 21–32

    Google Scholar 

  • Burnett AI, Adams, Vienna, KC (1977) A geological, engineering and economic study of a portion of the Lloydminster Sparky Pool, Lloydminster, Alberta. Bull Can Petr Geol 25(2):341–366

    Google Scholar 

  • Cargill SM (1981) United States gold resource profile. Econ Geol 76:937–943

    Google Scholar 

  • Cargill SM, Root DH, Bailey EH (1980) Resources estimation from historical data: mercury, a test case. J Int Assoc Math Geol 12:489–522

    Google Scholar 

  • Cargill SM, Root DH, Bailey EH (1981) Estimating unstable resources from historical industrial data. Econ Geol 76:1081–1095

    Google Scholar 

  • Carlson A (1991) Spatial distribution of ore deposits. Geology 19(2):111–114

    Google Scholar 

  • Carranza EJM (2008) Geochemical anomaly and mineral prospectivity mapping in GIS, Handbook of Exploration and Environment Geochemistry 11. Elsevier, Amsterdam

    Google Scholar 

  • Cheng Q (1994) Multifractal modeling and spatial analysis with GIS: gold mineral potential estimation in the Mitchell-Sulphurets area, northwestern British Columbia. Unpublished doctoral dissertation, University of Ottawa

    Google Scholar 

  • Cheng Q (1999) Multifractality and spatial statistics. Comput Geosci 25:949–961

    Google Scholar 

  • Cheng Q (2001) Selection of multifractal scaling breaks and separation of geochemical and geophysical anomaly. J China Univ Geosci 12(1):54–59

    Google Scholar 

  • Cheng Q (2003) GeoData Analysis System (GeoDAS) for mineral exploration and environmental assessment, user’s guide. York University, Toronto

    Google Scholar 

  • Cheng Q (2005) A new model for incorporating spatial association and singularity in interpolation of exploratory data. In: Leuangthong D, Deutsch CV (eds) Geostatistics Banff 2004. Springer, Dordrecht, pp 1017–1025

    Google Scholar 

  • Cheng Q (2006) GIS based fractal and multifractal methods for mineral deposit prediction. In: Leuangthong D, Deutsch CV (eds) Geostatistics Banff 2004. Springer, New York, pp 1017–1026

    Google Scholar 

  • Cheng Q (2007) Mapping singularities with stream sediment geochemical data for prediction of undiscovered mineral deposits in Gejiu, Yunnan Province, China. Ore Geol Rev 32:314–324

    Google Scholar 

  • Cheng Q (2008) Non-linear theory and power-law models for information integration and mineral resources quantitative assessments. In: Bonham-Carter G, Cheng Q (eds) Progress in geomathematics. Springer, Heidelberg, pp 195–225

    Google Scholar 

  • Cheng Q (2012) Multiplicative cascade processes and information integration for predictive mapping. Nonlinear Process Geophys 19:1–12

    Google Scholar 

  • Cheng Q, Agterberg FP (1995) Multifactal modeling and spatial point processes. Math Geol 27(7):831–845

    Google Scholar 

  • Cheng Q, Agterberg FP (1996) Multifractal modelling and spatial statistics. Math Geol 28(1):1–16

    Google Scholar 

  • Cheng Q, Agterberg FP (2009) Singularity analysis of ore-mineral and toxic trace elements in stream sediments. Comput Geosci 35:234–244

    Google Scholar 

  • Cheng Q, Agterberg FP, Ballantyne SB (1994) The separation of geochemical anomalies from background by fractal methods. J Geochem Explor 51(2):109–130

    Google Scholar 

  • Cheng Q, Agterberg FP, Bonham-Carter GF (1996) A spatial analysis method for geochemical anomaly separation. J Geochem Explor 56:183–195

    Google Scholar 

  • Cheng Q, Xu Y, Grunsky EC (2001) Integrated spatial and spectrum analysis for geochemical anomaly separation. Nat Resour Res 9:43–51

    Google Scholar 

  • Cressie NAC (2001) Statistics for spatial data. Wiley, New York

    Google Scholar 

  • Crovelli RA (1995) The generalized 20/80 law using probabilistic fractals applied to petroleum field size. Nonrenew Resour 4(3):233–241

    Google Scholar 

  • De Wijs HJ (1951) Statistics of ore distribution I. Geol Mijnbouw 13:365–375

    Google Scholar 

  • Diggle PJ (1983) Statistical analysis of spatial point patterns. Academic, London

    Google Scholar 

  • Drew LJ, Schuenemeyer JH, Bawiee WJ (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 

  • Evertsz CJG, Mandelbrot BB (1992) Multifractal measures. In: Peitgen H-O, Jurgens H, Saupe D (eds) Chaos and fractals. Springer, New York

    Google Scholar 

  • Falconer K (2003) Fractal geometry, 2nd edn. Wiley, Chichester

    Google Scholar 

  • Feder J (1988) Fractals. Plenum, New York

    Google Scholar 

  • Ford A, Blenkinsop TG (2009) An expanded de Wijs model for multifractal analysis of mineral production data. Miner Deposita 44(2):233–240

    Google Scholar 

  • Fowler A, L’Heureux I (1996) Self-organized banded sphalerite and banding galena. Can Mineral 34(6):1211–1222

    Google Scholar 

  • Freeden W, Schreiner M (2008) Spherical functions of mathematical geosciences. Springer, Heidelberg

    Google Scholar 

  • Freiling EC (1966) A comparison of the fallout mass-size distributions calculated by lognormal and power-law models, U.S. Naval Radiology Defense Laboratory Rep TR-1105. The Laboratory, San Francisco

    Google Scholar 

  • Garrett RG (1986) Geochemical abundance models: an update, 1975 to 1987, U.S. Geol Surv Circ 980:207–220

    Google Scholar 

  • Goehring L, Mahadevan L, Morris SW (2009) Nonequilibrium scale selection mechanism for columnar jointing. Proc Natl Acad Sci U S A 106(2):387–392

    Google Scholar 

  • Gupta VK, Troutman B, Dawdy D (2007) Towards a nonlinear geophysical theory of floods in river networks: an overview of 20 years of progress. In: Tsonis AA, Elsner JB (eds) Nonlinear dynamics in geosciences. Springer, New York, pp 121–150

    Google Scholar 

  • Harris DP (1984) Mineral resources appraisal. Clarendon, Oxford

    Google Scholar 

  • Heiskanen WA, Vening Meinesz FA (1958) The Earth and its gravity field. McGraw-Hill, New York

    Google Scholar 

  • Kaye BH (1989) A random walk through fractal dimensions. VCH Publishers, New York

    Google Scholar 

  • Korvin G (1992) Fractal models in the earth sciences. Elsevier, Amsterdam

    Google Scholar 

  • Li Q, Cheng Q (2004) Fractal singular value decomposition and anomaly reconstruction. Earth Sci 29(1):109–118 (in Chinese with English abstract)

    Google Scholar 

  • Lima A, Plant JA, De Vivo B, Tarvainen T, Albanese S, Cicchella D (2008) Interpolation methods for geochemical maps: a comparative study using arsenic data from European stream waters. Geochem Explor Environ Anal 8:41–48

    Google Scholar 

  • Lovejoy S, Schertzer D (1991) Multifractal analysis techniques and the rain and cloud fields from 10−3 to 106 m. In: Schertzer D, Lovejoy S (eds) Non-linear variability in geophysics. Kluwer, Dordrecht, pp 111–144

    Google Scholar 

  • Lovejoy S, Schertzer D (2007) Scaling and multifractal fields in the solid earth and topography. Nonlinear Process Geophys 14:465–502

    Google Scholar 

  • Lovejoy S, Schertzer D (2013) The weather and climate. Cambridge University Press, Cambridge

    Google Scholar 

  • Lovejoy S, Agterberg F, Carsteanu A, Cheng Q, Davidsen J, Gaonac’h H, Gupta V, L’Heureux I, Liu W, Morris SW, Sharma S, Shcherbakov R, Tarquis A, Turcotte D, Uritsky V (2010) Nonlinear geophysics: why we need it. EOS 90(48):455–456

    Google Scholar 

  • Malamud BD, Morein G, Turcotte DL (1998) Forest fires: an example of self-organized critical behavior. Science 281:1840–1842

    Google Scholar 

  • Mandelbrot BB (1975) Les objects fractals: forme, hazard et dimension. Flammarion, Paris

    Google Scholar 

  • Mandelbrot BB (1977) Fractals, form, chance and dimension. Freeman, San Francisco

    Google Scholar 

  • Mandelbrot BB (1983) The fractal geometry of nature. Freeman, San Francisco

    Google Scholar 

  • Mandelbrot BB (1989) Multifractal measures, especially for the geophysicist. Pure Appl Geophys 131:5–42

    Google Scholar 

  • Mandelbrot BB (1995) The statistics of natural resources and the law of Pareto. In: Barton CC, La Pointe PR (eds) Fractals in petroleum geology and the earth sciences. Plenum, New York, pp 1–12

    Google Scholar 

  • Mandelbrot BB (1999) Multifractals and 1/f noise. Springer, New York

    Google Scholar 

  • Over TM, Gupta VK (1996) A space-time theory of mesoscale rainfall using random cascades. J Geophys Res 101:26319–26331

    Google Scholar 

  • Park N-W, Chi K-H (2008) Quantitative assessment of landslide susceptibility using high-resolution remote sensing data and a generalized additive model. Int J Remote Sens 29(1):247–264

    Google Scholar 

  • Perrin J (1913) Les atomes. NRF-Gallimard, Paris

    Google Scholar 

  • Pickering G, Bul JM, Sanderson DJ (1995) Sampling power-law distributions. Tectonophysics 248:1–20

    Google Scholar 

  • Prey A (1922) Darstellung der Höhen- und Tiefen-Verhältnisse der Erde durch eine Entwicklung nach Kugelfunctionen bis zur 16. Ordnung. Nachr Ak Wiss Göttingen Math Phys Kl 11(1):1–29

    Google Scholar 

  • Quandt RE (1966) Old and new methods of estimation and the pareto distribution. Metrika 10:55–58

    Google Scholar 

  • Raines GL (2008) Are fractal dimensions of the spatial distribution of mineral deposits meaningful? Nat Resour Res 17:87–97

    Google Scholar 

  • Ripley BD (1976) The second-order analysis of stationary point processes. J Appl Probab 13:255–266

    Google Scholar 

  • Ripley BD (1981) Spatial statistics. Wiley-Interscience, New York

    Google Scholar 

  • Ripley BD (1987) Point processes for the earth sciences. In: Chung CF, Fabbri AG, Sinding-Larsen R (eds) Quantitative analysis of mineral and energy resources. Reidel, Dordrecht, pp 301–322

    Google Scholar 

  • Ripley BD (1988) Statistical inference for spatial processes. Cambridge University Press, Cambridge

    Google Scholar 

  • Rowlingson BS, Diggle PJ (1991) Estimating the K-function for a univariate point process on an arbitrary polygon, Lancaster University Math Dep Tech Rep MA91/58. Lancaster University, Lancaster, pp 1–15

    Google Scholar 

  • Rowlingson BS, Diggle PJ (1993) SPLANCS: spatial point pattern analysis code in S-Plus. Comput Geosci 19(5):627–655

    Google Scholar 

  • Rundle JB, Turcotte DL, Shcherbakov R, Klein W, Sammis C (2003) Statistical physics approach to understanding the multiscale dynamics of earthquake fault systems. Rev Geophys 41:1019

    Google Scholar 

  • Schertzer D, Lovejoy S (1985) The dimension and intermittency of atmospheric dynamics multifractal cascade dynamics and turbulent intermittency. In: Launder B (ed) Turbulent shear flow 4. Springer, New York, pp 7–33

    Google Scholar 

  • Schertzer D, Lovejoy S (1987) Physical modeling and analysis of rain and clouds by anisotropic scaling of multiplicative processes. J Geophys Res 92:9693–9714

    Google Scholar 

  • Schertzer D, Lovejoy S (1991) Non-linear geodynamical variability: multiple singularities, universality and observables. In: Schertzer D, Lovejoy S (eds) Non-linear variability in geophysics. Kluwer, Dordrecht, pp 41–82

    Google Scholar 

  • Schertzer D, Lovejoy S, Schmitt F, Chigirinskaya Y, Marsan D (1997) Multifractal cascade dynamics and turbulent intermittency. Fractals 5:427–471

    Google Scholar 

  • Sharma AS (1995) Assessing the magnetosphere’s nonlinear behavior: its dimension is low, its predictability high. Rev Geophys 33:645

    Google Scholar 

  • Sinclair AJ (1991) A fundamental approach to threshold estimation in exploration geochemistry: probability plots revisited. J Geochem Explor 41(1):1–22

    Google Scholar 

  • Singer DA, Menzie WD (2008) Map scale effects on estimating the number of undiscovered mineral deposits. In: Bonham-Carter G, Cheng Q (eds) Progress in geomathematics. Springer, Heidelberg, pp 271–283

    Google Scholar 

  • Singer DA, Menzie WD (2010) Quantitative mineral resource assessments. Oxford University Press, New York

    Google Scholar 

  • Stanley H, Meakin P (1988) Multifractal phenomena in physics and chemistry. Nature 335:405–409

    Google Scholar 

  • Steinhaus H (1954) Length, shape and area. Colloq Math 3:1–13

    Google Scholar 

  • Stoyan D, Kendall WS, Mecke J (1987) Stochastic geometry and its applications. Wiley, Chichester

    Google Scholar 

  • Turcotte DL (1997) Fractals and chaos in geology and geophysics, 2nd edn. Cambridge University Press, Cambridge

    Google Scholar 

  • Turcotte DL (2002) Fractals in petrology. Lithos 65(3–4):261–271

    Google Scholar 

  • Uritsky VM, Donovan E, Klimas AJ (2008) Scale-free and scale-dependent modes of energy release dynamics in the night time magnetosphere. Geophys Res Lett 35(21):L21101, 1–5

    Google Scholar 

  • Veneziano D, Furcolo P (2003) Marginal distribution of stationary multifractal measures and their Haar wavelet coefficients. Fractals 11(3):253–270

    Google Scholar 

  • Veneziano D, Langousis A (2005) The maximum of multifractal cascades: exact distribution and approximations. Fractals 13(4):311–324

    Google Scholar 

  • Vening Meinesz FA (1951) A remarkable feature of the earth’s topography. Proc K Ned Akad Wet Ser B Phys Sci 54:212–228

    Google Scholar 

  • Vening Meinesz FA (1964) The earth’s crust and mantle. Elsevier, Amsterdam

    Google Scholar 

  • Vistelius AB (1960) The skew frequency distribution and the fundamental law of the geochemical processes. J Geol 68:1–22

    Google Scholar 

  • Walsh J, Watterson J, Yielding G (1991) The importance of small-scale faulting in regional extension. Nature 351:391–393

    Google Scholar 

  • Wang Q, Deng J, Zhao J, Li N, Wan L (2013) The fractal relationship between orebody tonnage and thickness. J Geochem Explor 122:4–8

    Google Scholar 

  • Zuo RG, Cheng QM, Xia QL, Agterberg FP (2009) Application of fractal models to distinguish between different mineral phases. Math Geosci 41:71–80

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ministry of Natural Resources Canada, Geological Survey of Canada (NRCan-GSC), Ottawa, ON, Canada

    Frits Agterberg

Authors
  1. Frits Agterberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Agterberg, F. (2014). Fractals. In: Geomathematics: Theoretical Foundations, Applications and Future Developments. Quantitative Geology and Geostatistics, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-319-06874-9_10

Download citation

Publish with us

Policies and ethics

Search

Navigation