Skip to main content
SpringerLink
Log in

Selected Topics for Further Research

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

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

  • 1806 Accesses

Abstract

The case history studies described in the preceding eleven chapters leave some questions that could not be answered in full. New theoretical approaches in mathematical statistics and nonlinear physics provide new perspectives for the analysis of geoscience data. For example, bias due to incomplete information continues to be one of the most serious problems in 3-D mapping. How methods such as the jackknife and bootstrap can help to reduce this type of bias is briefly investigated and illustrated using volcanogenic massive copper deposits in the Abitibi area on the Canadian Shield. Compositional data analysis offers new ways to analyze multivariate data sets. Geochemical data from Fort à la Corne kimberlites in central Saskatchewan are used to illustrate the isometric logratio transformation for chemical data that form a closed number system. Three generalizations of the model of de Wijs are: (1) the 3-parameter model with finite number of iterations; (2) the random cut model in which the dispersion index d is replaced by a random variable D; and (3) the accelerated dispersion model in which d depends on concentration value during the cascade process. Universal multifractals constitute a useful generalization of multifractal modeling. As illustrated on the basis of the Pulacayo zinc values, new tools such as use of the first order structure function and double trace analysis generalize conventional variogram-autocorrelation fitting. Measurements on compositions of blocks of rocks generally depend on block size. For example, at microscopic scale chemical elements depend on frequencies of abundance of different minerals. On a regional basis, rock type composition depends on spatial distribution of contacts between different rock types. Frequency distribution modeling of compositional data can be useful in ore reserve estimation as well as regional mineral potential studies. During the 1970, Georges Matheron proposed the theory of permanent frequency distributions with shapes that are independent of block size. The lognormal is a well-known geostatistical example. The probnormal distribution is useful for the analysis of relative amounts of different rock types contained in cells of variable size. It arises when probits of percentage values are normally distributed. Its Q-Q plot has scales derived from the normal distribution along both axes. Parameters (mean and variance) of the probnormal distribution are related to the geometrical covariances of the objects of study. Practical examples are spatial distribution of acidic and mafic volcanics in the Bathurst area, New Brunswick, and in the Abitibi volcanic belt on the Canadian Shield in east-central Ontario and western Quebec.

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

  • Agterberg FP (1970) Autocorrelation functions in geology. In: Merriam DF (ed) Geostatistics. Plenum, New York, pp 113–142

    Chapter  Google Scholar 

  • Agterberg FP (1973) Probabilistic models to evaluate regional mineral potential. In: Proc Symp Mathematical Methods in the Geosciences, Přibram, Czechoslovakia, pp 3–38

    Google Scholar 

  • Agterberg FP (1974) Geomathematics. Elsevier, Amsterdam

    Google Scholar 

  • Agterberg FP (1977) Frequency distributions and spatial variability of geological variables. In: Ramani RV (ed) Application of computing methods in the mineral industry. American Institute of Mining, Metallurgical, and Petroleum Industry, New York, pp 287–298

    Google Scholar 

  • Agterberg FP (1978a) Use of spatial analysis in mineral resource evaluation. Math Geol 16(6):565–589

    Article  Google Scholar 

  • Agterberg FP (1978b) Quantification and statistical analysis of geological variables for mineral resource evaluation. Mém Bur Rech Géol Min 91:399–406

    Google Scholar 

  • Agterberg FP (1981) Cell value distribution models in spatial pattern analysis. In: Craig RG, Labovitz ML (eds) Future trends of geomathematics. Pion, London, pp 5–28

    Google Scholar 

  • Agterberg FP (1984) Use of spatial analysis in mineral resource evaluation. Math Geol 16:565–589

    Article  Google Scholar 

  • Agterberg FP (1994) Fractals, multifractals, and change of support. In: Dimitrakopoulos R (ed) Geostatistics for the next century. Kluwer, Dordrecht, pp 223–234

    Chapter  Google Scholar 

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

    Article  Google Scholar 

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

    Chapter  Google Scholar 

  • Agterberg FP (2001b) Aspects of multifractal modeling. In: Proceedings of the 7th conference of the International Association for Mathematical Geology (CD-ROM), Cancun, Mexico

    Google Scholar 

  • Agterberg FP (2005) Spatial analysis of cell composition data. In: Proceedings of CoDaWork’05, University of Girona, Spain (CD-ROM)

    Google Scholar 

  • Agterberg FP (2007a) New applications of the model of de Wijs in regional geochemistry. Math Geol 39(1):1–26

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Agterberg FP (2012b) Multifractals and geostatistics. J Geochem Explor 122:113–122

    Article  Google Scholar 

  • Agterberg FP, Fabbri AG (1978) Spatial correlation of stratigraphic units quantified from geological maps. Comput Geosci 4:284–294

    Google Scholar 

  • Aitchison J (1986) The statistical analysis of compositional data. Chapman and Hall, London

    Book  Google Scholar 

  • Beckmann P (1973) Orthogonal polynomials for engineers and physicists. Golem Press, Boulder

    Google Scholar 

  • Beirlant J, Goegebeur Y, Segers J, Teugels J (2005) Statistics of extremes. Wiley, New York

    Google Scholar 

  • Blackman RB, Tukey JW (1958) The measurement of power spectra. Dover, New York

    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. In: International Atomic Energy Agency, Vienna, Proceedings Series STI/PUB/374, pp 21–32

    Google Scholar 

  • Chayes F (1971) Ratio correlation. University of Chicago Press, Chicago

    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, Agterberg FP (1996) Comparison between two types of multifractal modeling. Math Geol 28:1001–1015

    Article  Google Scholar 

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

    Article  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

    Article  Google Scholar 

  • Chung CF, Agterberg FP (1980) Regression models for estimating mineral resources from geological map data. Math Geol 12:473–488

    Article  Google Scholar 

  • Davis JC (2002) Statistics and data analysis in geology, 3rd edn. Wiley, New York

    Google Scholar 

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

    Google Scholar 

  • Dubois J, Cheminée JL (1991) Fractal analysis of eruptive activity of some basaltic volcanoes. J Volcan Geotherm Res 45:197–208

    Article  Google Scholar 

  • Efron B (1982) The jackknife, the bootstrap and other resampling plans. SIAM, Philadelphia

    Book  Google Scholar 

  • Egozcue JJ, Pawlowsky-Glahn V, Mateu-Figueras G, Barceló-Vidal C (2003) Isometric logratio transformations for compositional data analysis. Math Geol 35(3):279–300

    Article  Google Scholar 

  • Fabbri AG, Chung CF (2008) On blind tests and spatial prediction models. In: Bonham-Carter G, Cheng Q (eds) Progress in geomathematics. Springer, Heidelberg, pp 315–332

    Chapter  Google Scholar 

  • Filzmoser P, Hron K (2011) Robust statistical analysis. In: Pawlowsky-Glahn V, Buccianti A (eds) Compositional data analysis – theory and applications. Wiley, New York, pp 59–71

    Chapter  Google Scholar 

  • Filzmoser P, Hron K, Reimann C (2009) Univariate statistical analysis of environmental (compositional) data: problems and possibilities. Sci Total Environ 407(6):6100–6108

    Article  Google Scholar 

  • Garrett RG, Thorleifson LH (1995) Kimberlite indicator mineral and till geochemical reconnaissance, southern Saskatchewan. Geological Survey of Canada Open File 3119, pp 227–253

    Google Scholar 

  • Greenacre M (2009) Distributional equivalence and subcompositional coherence in gthe analysis of compositional data, contingency tables and ratio-scale measurements. J Classification 26(1):29–54

    Article  Google Scholar 

  • Grunsky EC, Kjarsgaard BA (2008) Classification of distinct eruptive phases of the diamondiferous Star kimberlite, Saskatchewan, Canada based on statistical treatment of whole rock geochemical analyses. Appl Geochem 23:3321–3336

    Article  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, Elsher JB (eds) Nonlinear dynamics in geosciences. Springer, New York, pp 121–150

    Chapter  Google Scholar 

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

    Google Scholar 

  • Hutchinson TP, Lai CD (1991) The engineering statistician’s guide to continuous bivariate distributions. Rumsby Scientific Publishing, Adelaide

    Google Scholar 

  • Jöreskog KE, Klovan JE, Reyment RA (1976) Geological factor analysis. Elsevier, Amsterdam

    Google Scholar 

  • Kotz S (1975) Multivariate distributions at a cross road. In: Patil GP, Kotz S (eds) A modern course on distributions in scientific work I – models and structures. Reidel, Dordrecht

    Google Scholar 

  • Krige DG (1951) A statistical approach to some basic valuation problems on the Witwatersrand. J S Afric Inst Mining Metallurgy 52:119–139

    Google Scholar 

  • Krige DG (1966) A study of gold and uranium distribution pattern in the Klerksdorp goldfield. Geoexploration 4:43–53

    Article  Google Scholar 

  • Lavallée D, Lovejoy S, Schertzer D, Schmitt F (1992) On the determination of universal multifractal parameters in turbulence. In: Moffatt HK, Zaslavsky GM, Conte P, Tabor M (eds) Topological aspects of the dynamics of fluids and plasmas. Kluwer, Dordrecht, pp 463–478

    Chapter  Google Scholar 

  • Lorenz EN (1963) Deterministic nonperiodic flow. J Atmos Sci 20:130–141

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Lovejoy S, Schertzer D (2010) On the simulation of continuous in scale universal multifractals, part I: spatially continuous processes. Comput Geosci 36:1393–1403

    Article  Google Scholar 

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

    Google Scholar 

  • Lovejoy S, Currie WJS, Tessier Y, Claereboudt MR, Bourget E, Roff JC, Schertzer D (2001) Universal multifractals and ocean patchiness: phytoplankton, physical fields and coastal heterogeneity. J Plankton Res 23:117–141

    Article  Google Scholar 

  • Lovejoy S, Gaonac’h H, Schertzer D (2008) Anisotropic scaling models of rock density and the earth’s surface gravity field. In: Bonham-Carter G, Cheng Q (eds) Progress in geomathematics. Springer, Heidelberg, pp 151–194

    Chapter  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

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

    Book  Google Scholar 

  • Matheron G (1962) Traité de géostatistique appliquée. Mém BRGM 14, Paris

    Google Scholar 

  • Matheron G (1974) Les fonctions de transfert des petits panneaux, Note Géostatistique 127. Centre de Morphologie Mathématique, Fontainebleau

    Google Scholar 

  • Matheron G (1976) Forecasting block grade distributions: the transfer functions. In: Guarascio M, David M, Huijbregts C (eds) Advanced geostatistics in the mining industry. Reidel, Dordrecht

    Google Scholar 

  • Matheron G (1980) Models isofactoriels pout l’effet zero, Note Géostatistique 659. Centre de Morphologie Mathématique, Fontainebleau

    Google Scholar 

  • May RM (1976) Simple mathematical models with very complicated dynamics. Nature 261:459–467

    Article  Google Scholar 

  • Monin AS, Yaglom AM (1975) Statistical fluid mechanics 2. MIT Press, Cambridge, MA

    Google Scholar 

  • Mosteller F, Tukey JW (1968) Data analysis including statistics. In: Lindzey G, Aronson E (eds) Handbook of social psychology, 2nd edn, vol 2. Addison-Wesley, Reading, pp 80–123

    Google Scholar 

  • Motter E, Campbell DK (2013) Chaos at fifty. Phys Today 66:27–33

    Article  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

    Article  Google Scholar 

  • Pawlowsky-Glahn V, Buccianti A (eds) (2011) Compositional data analysis – theory and applications. Wiley, New York

    Google Scholar 

  • Pawlowsky-Glahn V, Olea RA (2004) Geostatistical analysis of compositional data. Oxford University Press, New York

    Google Scholar 

  • Pearson K (1897) On a form of spurious correlation which may arise when indices are used in the measurements of organs. Proc R Soc Lond 60:489–498

    Article  Google Scholar 

  • Pilkington M, Todoeschuck J (1995) Scaling nature of crustal susceptibilities. Geophys Res Lett 22:779–782

    Article  Google Scholar 

  • Poincaré H (1899) Les methods nouvelles de la mécanique celeste. Gauthier-Villars, Paris

    Google Scholar 

  • Quenouille M (1949) Approximate tests of correlation in time series. J R Stat Soc Ser B 27:395–449

    Google Scholar 

  • Reyment RA, Mosoyana I, Oka M, Tanaka Y (2008) A note on seasonal variation in radiolarian abundance. In: Bonham-Carter G, Cheng Q (eds) Progress in geomathematics. Springer, Heidelberg, pp 413–430

    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

    Article  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

    Chapter  Google Scholar 

  • Schertzer D, Lovejoy S (eds) (1991a) Non-linear variability in geophysics. Kluwer, Dordrecht

    Google Scholar 

  • Schertzer D, Lovejoy S (1991b) 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

    Chapter  Google Scholar 

  • Schertzer D, Lovejoy S (1997) Universal multifractals do exist! Comments on “A statistical analysis of mesoscale rainfall as a random cascade”. J Appl Meteorol 36:1296–1303

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Schucany WR, Gray HL, Owen DB (1971) On bias reduction in estimation. J Am Stat Assoc 66:524–533

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Sornette A, Dubois J, Cheminée JL, Sornette D (1991) Are sequences of volcanic eruptions deterministically chaotic. J Geophys Res 96(11):931–945

    Google Scholar 

  • Sparrow C (1982) The Lorentz equation: bifurcations, chaos, and strange attractors. Springer, New York

    Google Scholar 

  • Szegö G (1975) Orthogonal polynomials, 9th edn, American Mathematical Society Colloquium Publications 23. American Mathematical Society, Providence

    Google Scholar 

  • Thompson RN, Esson J, Duncan AC (1972) Major element chemical variation in the Eocene lavas of the Isle of Skye, Scotland. J Petrol 13:219–253

    Article  Google Scholar 

  • Tukey JW (1970) Some further inputs. In: Merriam DF (ed) Geostatistics. Plenum, New York, pp 173–174

    Google Scholar 

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

    Book  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

    Article  Google Scholar 

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

    Google Scholar 

  • Wilson J, Schertzer D, Lovejoy S (1991) Continuous multiplicative cascade models of rain and clouds. In: Schertzer D, Lovejoy S (eds) Non-linear variability in geophysics. Kluwer, Dordrecht, pp 185–207

    Chapter  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). Selected Topics for Further Research. 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_12

Download citation

Publish with us

Policies and ethics

Search

Navigation