Skip to main content
SpringerLink
Log in

Geostatistical Simulation of Geochemical Compositions in the Presence of Multiple Geological Units: Application to Mineral Resource Evaluation

  • Published:
Mathematical Geosciences Aims and scope Submit manuscript

Abstract

An accurate prediction of benefit in ore deposits with heterogeneous spatial variations requires the definition of geological domains that differentiate the types of mineralogy, alteration, and lithology, as well as the prediction of full mineral and geochemical compositions within each modeled domain and across boundaries between different domains. This paper proposes and compares various approaches (different combinations of log-ratio transformation, Gaussian and flow anamorphosis, and deterministic or probabilistic geological models) for geostatistical simulation of geochemical compositions in the presence of several geological domains. Different approaches are illustrated through an application to a nickel–cobalt laterite deposit located in Western Australia. Four rock types (ferruginous, smectite, saprolite, and ultramafic) are considered to define compositionally homogeneous domains. Geochemical compositions are comprised of six different components of interest (Fe, Al, Mg, Ni, Co, and Filler). The results suggest that the flow anamorphosis is a vital element for geostatistical modeling of geochemical composition due to its invariance properties and capability for reproducing complex patterns in input data, including: presence of outliers, presence of several populations (due to the presence of several geological domains), nonlinearity, and heteroscedasticity.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

References

  • Aitchison J (1982) The statistical analysis of compositional data (with discussion). J R Stat Soc B Methodol 44:139–177

    Google Scholar 

  • Aitchison J (1986) The statistical analysis of compositional data. Monographs on statistics and applied probability. Chapman & Hall, London. Reprinted in 2003 with additional material by The Blackburn Press

    Google Scholar 

  • Alabert F (1987a) The practice of fast conditional simulations through the LU decomposition of the covariance matrix. Math Geol 19(5):369–386. https://doi.org/10.1007/BF00897191

    Article  Google Scholar 

  • Alabert FG (1987b) Stochastic imaging of spatial distributions using hard and soft information. MSc thesis, Stanford University, Stanford, CA

  • Armstrong M, Galli A, Beucher H, Le Loc’h G, Renard D, Doligez B, Eschard R, Geffroy F (2011) Plurigaussian simulations in geosciences. Springer, Berlin

    Book  Google Scholar 

  • Bandarian EM, Bloom LM, Mueller UA (2008) Direct minimum/maximum autocorrelation factors within the framework of a two structure linear model of coregionalisation. Comput Geosci 34(3):190–200. https://doi.org/10.1016/j.cageo.2007.03.015

    Article  Google Scholar 

  • Boisvert JB, Rossi ME, Ehrig K, Deutsch CV (2013) Geometallurgical modeling at Olympic dam mine, South Australia. Math Geosci 45(8):901–925. https://doi.org/10.1007/s11004-013-9462-5

    Article  Google Scholar 

  • Buccianti A, Pawlowsky-Glahn V, Barceló-Vidal C, Jarauta-Bragulat E (1999) Visualization and modeling of natural trends in ternary diagrams: a geochemical case study. In: Lippard SJ, Naess A, Sinding-Larsen R (eds) IAMG’99: proceedings of the 5th annual conference of the International Association for Mathematical Geology, Trondheim, Norway, August 1999, pp 139–144

  • Camuti KS, Riel RG (1996) Mineralogy of the Murrin Murrin nickel laterites. In: Grimsey EJ, Neuss I (eds) Proceedings of Nickel ’96: mineral to market, Kalgoorlie, Western Australia, November 1996. AusIMM, Melbourne, pp 209–210

  • Chilès JP, Delfiner P (2012) Geostatistics: modeling spatial uncertainty. Wiley, New York

    Book  Google Scholar 

  • Desbarats AJ, Dimitrakopoulos R (2000) Geostatistical simulation of regionalized pore-size distributions using min/max autocorrelation factors. Math Geol 32(8):919–942

    Article  Google Scholar 

  • Deutsch CV, Journel AG (1998) GSLIB: geostatistical software library and user’s guide. Oxford University Press, New York, NY

    Google Scholar 

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

    Article  Google Scholar 

  • Emery X (2007) Simulation of geological domains using the plurigaussian model: new developments and computer programs. Comput Geosci 33(9):1189–1201

    Article  Google Scholar 

  • Emery X (2008) A turning bands program for conditional co-simulation of cross-correlated Gaussian random fields. Comput Geosci 34(12):1850–1862

    Article  Google Scholar 

  • Emery X, González KE (2007a) Probabilistic modelling of lithological domains and it application to resource evaluation. J S Afr Inst Min Metall 107(12):803–809

    Google Scholar 

  • Emery X, González KE (2007b) Incorporating the uncertainty in geological boundaries into mineral resources evaluation. J Geol Soc India 69(1):29–38

    Google Scholar 

  • Emery X, Lantuéjoul C (2006) TBSIM: a computer program for conditional simulation of three-dimensional Gaussian random fields via the turning bands method. Comput Geosci 32(10):1615–1628. https://doi.org/10.1016/j.cageo.2006.03.001

    Article  Google Scholar 

  • Emery X, Arroyo D, Porcu E (2016) An improved spectral turning-bands algorithm for simulating stationary vector Gaussian random fields. Stoch Environ Res Risk Assess 30(7):1863–1873. https://doi.org/10.1007/s00477-015-1151-0

    Article  Google Scholar 

  • Korkmaz S, Goksuluk D, Zararsiz G (2014) MVN: an R package for assessing multivariate normality. R J 6:151–162

    Google Scholar 

  • Maleki M, Emery X, Cáceres A, Ribeiro D, Cunha E (2016) Quantifying the uncertainty in the spatial layout of rock type domains in an iron ore deposit. Comput Geosci 20(5):1013–1028. https://doi.org/10.1007/s10596-016-9574-3

    Article  Google Scholar 

  • Mardia KV (1970) Measures of multivariate skewness and kurtosis with applications. Biometrika 57(3):519–530. https://doi.org/10.1093/biomet/57.3.519

    Article  Google Scholar 

  • Mariethoz G, Caers J (2014) Multiple-point geostatistics: stochastic modeling with training images. Wiley, New York, NY

    Book  Google Scholar 

  • Markwell T (2001) Murrin Murrin Ni/Co resource estimation: MME resource modelling report. Anaconda Operations Pty Ltd, Melbourne, VIC

    Google Scholar 

  • McKinley JM, Hron K, Grunsky EC, Reimann C, de Caritat P, Filzmoser P, van den Boogaart KG, Tolosana-Delgado R (2016) The single component geochemical map: fact or fiction? J Geochem Explor 162:16–28

    Article  Google Scholar 

  • Mery N, Emery X, Cáceres A, Ribeiro D, Cunha E (2017) Geostatistical modeling of the geological uncertainty in an iron ore deposit. Ore Geol Rev 88:336–351. https://doi.org/10.1016/j.oregeorev.2017.05.011

    Article  Google Scholar 

  • Monti R, Fazakerley VW (1996) The Murrin Murrin nickel cobalt project. In: Grimsey EJ, Neuss I (eds) Proceedings of Nickel ’96: mineral to market, Kalgoorlie, WA, November 1996. AusIMM, Melbourne, VIC, pp 191–196

  • Montoya C, Emery X, Rubio E, Wiertz J (2012) Multivariate resource modelling for assessing uncertainty in mine design and mine planning. J S Afr Inst Min Metall 112:353–363

    Google Scholar 

  • Mueller U, Tolosana-Delgado R, van den Boogaart KG (2014) Approaches to the simulation of compositional data—a nickel-laterite comparative case study. In: Dimitrakopoulos R (ed) Proceedings of the orebody modelling and strategic mine planning symposium, Perth, WA, November 2014. AusIMM, Melbourne, VIC, pp 61–72

  • Mueller U, van den Boogaart KG, Tolosana-Delgado R (2017) A truly multivariate normal score transform based on lagrangian flow. In: Gómez-Hernández JJ, Rodrigo-Ilarri J, Rodrigo-Clavero ME, Cassiraga E, Vargas-Guzmán JA (eds) Geostatistics Valencia 2016. Springer, Cham, pp 107–118. https://doi.org/10.1007/978-3-319-46819-8_7

    Chapter  Google Scholar 

  • Murphy M (2003) Geostatistical optimisation of sampling and estimation in a nickel laterite deposit. MSc thesis, Edith Cowan University (unpublished)

  • Ortiz JM, Emery X (2006) Geostatistical estimation of mineral resources with soft geological boundaries: a comparative study. J S Afr Inst Min Metall 106:577–584

    Google Scholar 

  • Pawlowsky-Glahn V, Egozcue JJ (2016) Spatial analysis of compositional data: a historical review. J Geochem Explor 164:28–32. https://doi.org/10.1016/j.gexplo.2015.12.010

    Article  Google Scholar 

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

    Google Scholar 

  • Pawlowsky-Glahn V, Egozcue JJ, Tolosana-Delgado R (2015) Modeling and analysis of compositional data. Statistics in practice. Wiley, Chichester

    Google Scholar 

  • Renard D, Beucher H (2012) 3D representations of a uranium roll-front deposit. Appl Earth Sci 121(2):84–88

    Article  Google Scholar 

  • Rondon O (2012) Teaching aid: minimum/maximum autocorrelation factors for joint simulation of attributes. Math Geosci 44:469–504

    Article  Google Scholar 

  • Soares A (1998) Sequential indicator simulation with correction for local probabilities. Math Geol 30(6):761–765

    Article  Google Scholar 

  • Switzer P, Green A (1984) Min/max autocorrelation factors for multivariate spatial imagery. Technical report no. 6, Department of Statistics, Stanford University, Stanford, CA

  • Székely GJ, Rizzo ML (2005) A new test for multivariate normality. J Multivar Anal 93(1):58–80. https://doi.org/10.1016/j.jmva.2003.12.002

    Article  Google Scholar 

  • Székely GJ, Rizzo ML (2013) Energy statistics: a class of statistics based on distances. J Stat Plan Inference 143:1249–1272. https://doi.org/10.1016/j.jspi.2013.03.018

    Article  Google Scholar 

  • Talebi H, Asghari O, Emery X (2013) Application of plurigaussian simulation to delineate the layout of alteration domains in Sungun copper deposit. Cent Eur J Geosci 5:514–522. https://doi.org/10.2478/s13533-012-0146-3

    Google Scholar 

  • Talebi H, Asghari O, Emery X (2014) Simulation of the lately injected dykes in an Iranian porphyry copper deposit using the plurigaussian model. Arab J Geosci 7:2771–2780. https://doi.org/10.1007/s12517-013-0911-8

    Article  Google Scholar 

  • Talebi H, Asghari O, Emery X (2015) Stochastic rock type modeling in a porphyry copper deposit and its application to copper grade evaluation. J Geochem Explor 157:162–168. https://doi.org/10.1016/j.gexplo.2015.06.010

    Article  Google Scholar 

  • Talebi H, Sabeti EH, Azadi M, Emery X (2016) Risk quantification with combined use of lithological and grade simulations: application to a porphyry copper deposit. Ore Geol Rev 75:42–51. https://doi.org/10.1016/j.oregeorev.2015.12.007

    Article  Google Scholar 

  • Talebi H, Lo J, Mueller U (2017) A hybrid model for joint simulation of high-dimensional continuous and categorical variables. In: Gómez-Hernández JJ, Rodrigo-Ilarri J, Rodrigo-Clavero ME, Cassiraga E, Vargas-Guzmán JA (eds) Geostatistics Valencia 2016. Springer, Cham, pp 415–430. https://doi.org/10.1007/978-3-319-46819-8_28

    Chapter  Google Scholar 

  • Tercan AE (1999) Importance of orthogonalization algorithm in modeling conditional distributions by orthogonal transformed indicator methods. Math Geol 31:155–173. https://doi.org/10.1023/A:1007557701073

    Google Scholar 

  • Tolosana-Delgado R (2006) Geostatistics for constrained variables: positive data, compositions and probabilities—application to environmental hazard monitoring. Dissertation, Universitat de Girona, Girona

  • Tolosana-Delgado R, van den Boogaart KG (2013) Joint consistent mapping of high-dimensional geochemical surveys. Math Geosci 45:983–1004. https://doi.org/10.1007/s11004-013-9485-y

    Article  Google Scholar 

  • Tolosana-Delgado R, Mueller U, van den Boogaart KG, Ward C (2014) Compositional block cokriging. In: Pardo-Igúzquiza E, Guardiola-Albert C, Heredia J, Moreno-Merino L, Durán JJ, Vargas-Guzmán JA (eds) Mathematics of planet Earth: proceedings of the 15th annual conference of the International Association for Mathematical Geosciences, Madrid, September 2013. Springer, Berlin, pp 713–716. https://doi.org/10.1007/978-3-642-32408-6_154

  • Tolosana-Delgado R, Mueller U, van den Boogaart KG, Ward C, Gutzmer J (2015) Improving processing by adaption to conditional geostatistical simulation of block compositions. J S Afr Inst Min Metall 115:13–26

    Article  Google Scholar 

  • Tolosana-Delgado R, Mueller U, van den Boogaart KG (2016) Compositionally compliant contact analysis. In: Raju NJ (ed) Geostatistical and geospatial approaches for the characterization of natural resources in the environment: challenges, processes and strategies. Springer, Cham, pp 11–14. https://doi.org/10.1007/978-3-319-18663-4_2

    Chapter  Google Scholar 

  • van den Boogaart KG, Tolosana-Delgado R (2013) Analyzing compositional data with R. Springer, Berlin

    Book  Google Scholar 

  • van den Boogaart KG, Tolosana-Delgado R, Lehmann M, Mueller U (2014) On the joint multipoint simulation of discrete and continuous geometallurgical parameters. In: Dimitrakopoulos R (ed) Proceedings of the orebody modelling and strategic mine planning symposium, Perth, WA, November 2014. AusIMM, Melbourne, VIC, pp 379–388

  • van den Boogaart KG, Mueller U, Tolosana-Delgado R (2017) An affine equivariant multivariate normal score transform for compositional data. Math Geosci 49:231–251. https://doi.org/10.1007/s11004-016-9645-y

    Article  Google Scholar 

  • Wackernagel H (2003) Multivariate geostatistics: an introduction with applications. Springer, Berlin

    Book  Google Scholar 

Download references

Acknowledgements

The authors acknowledge financial support through DAAD-UA grant CodaBlockCoEstimation.

Author information

Authors and Affiliations

  1. School of Science, Edith Cowan University, 270 Joondalup Drive, Joondalup, WA, 6027, Australia

    Hassan Talebi & Ute Mueller

  2. Helmholtz Zentrum Dresden-Rossendorf, Helmholtz Institute Freiberg for Resource Technology, Chemnitzerstrasse 40, 09599, Freiberg, Saxony, Germany

    Raimon Tolosana-Delgado & K. Gerald van den Boogaart

Authors
  1. Hassan Talebi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ute Mueller
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Raimon Tolosana-Delgado
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. K. Gerald van den Boogaart
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hassan Talebi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Talebi, H., Mueller, U., Tolosana-Delgado, R. et al. Geostatistical Simulation of Geochemical Compositions in the Presence of Multiple Geological Units: Application to Mineral Resource Evaluation. Math Geosci 51, 129–153 (2019). https://doi.org/10.1007/s11004-018-9763-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11004-018-9763-9

Keywords

Search

Navigation