Multivariate Categorical Modeling with Hierarchical Truncated Pluri-Gaussian Simulation

  • Diogo SilvaEmail author
  • Clayton Deutsch


Multiple categorical variables such as mineralization zones, alteration zones, and lithology are often available for geostatistical modeling. Each categorical variable has a number of possible categorical outcomes. The current approach for numerical modeling of categorical variables is to either combine the categorical variables or to model them independently. The collapse of multiple categorical variables into a single variable with all combinations is impractical due to the large number of combinations. In some cases, lumping categorical variables is justified in terms of stationary domains; however, this decision is often due to the limitations of existing techniques. The independent modeling of each categorical variable will fail to reproduce the collocated joint categorical relationships. A methodology for the multivariate modeling of categorical variables utilizing the hierarchical truncated pluri-Gaussian approach is developed and illustrated with the Swiss Jura data set. The multivariate approach allows for improved reproduction of multivariate relationships between categorical variables.


Simulation Geostatistics Geology Modeling 


  1. Acar S (2016) Process development metallurgical studies for gold cyanidation process. Miner Metall Process 33(4):161–171Google Scholar
  2. Angove J, Acar S (2016) Metallurgical test work: gold processingoptions, physical ore properties, and cyanide management. In: Adams MD (ed) Gold ore processing, 2nd edn. Elsevier, Amsterdam, pp 131–140. CrossRefGoogle Scholar
  3. Arroyo D, Emery X, Pelez M (2012) An enhanced Gibbs sampler algorithm for non-conditional simulation of Gaussian random vectors. Comput Geosci 46(Supplement C):138–148. CrossRefGoogle Scholar
  4. Astrakova A, Oliver DS, Lantuéjoul C (2015) Truncation map estimation based on bivariate probabilities and validation for the truncated pluriGaussian model. arXiv preprint arXiv:150801090
  5. Babak O, Deutsch CV (2008) Collocated cokriging based on merged secondary attributes. Math Geosci 41(8):921. CrossRefGoogle Scholar
  6. Barnett RM, Deutsch CV (2015) Multivariate imputation of unequally sampled geological variables. Math Geosci 47(7):791–817. CrossRefGoogle Scholar
  7. Barnett RM, Manchuk JG, Deutsch CV (2014) Projection pursuit multivariate transform. Math Geosci 46(3):337–359. CrossRefGoogle Scholar
  8. Black WE (2016) Multivariate geostatistical prediction of geochemical measurements for use in mineral prospectivity modeling. Ph.D. thesis, University of Alberta, Edmonton, Alberta.
  9. Bye A (2011) Case studies demonstrating value from geometallurgy initiatives. In: GeoMet 2011—1st AusIMM international geometallurgy conference 2011. AusIMM: Australasian Institute of Mining and Metallurgy, pp 9–30Google Scholar
  10. Deutsch CV, Journel AG (1998) Geostatistical software library and users guide, 2nd edn. Oxford University Press, OxfordGoogle Scholar
  11. Deutsch JL, Palmer K, Deutsch CV, Szymanski J, Etsell TH (2016) Spatial modeling of geometallurgical properties: techniques and a case study. Nat Resour Res 25(2):161–181. CrossRefGoogle Scholar
  12. Emery X, Cornejo J (2010) Truncated Gaussian simulation of discrete-valued, ordinal coregionalized variables. Comput Geosci 36(10):1325–1338. CrossRefGoogle Scholar
  13. Emery X, Arroyo D, Peláez M (2014) Simulating large Gaussian random vectors subject to inequality constraints by Gibbs sampling. Math Geosci 46(3):265–283. CrossRefGoogle Scholar
  14. Galli A, Gao H (2001) Rate of convergence of the Gibbs sampler in the Gaussian case. Math Geol 33(6):653–677. CrossRefGoogle Scholar
  15. Galli A, Beucher H, Le Loch G, Doligez B et al (1994) The pros and cons of the truncated Gaussian method. In: Armstrong M, Dowd PA (eds) Geostat Simul, vol 7. Springer, Dordrecht, pp 217–233. CrossRefGoogle Scholar
  16. Garza RAP, Titley SR, Pimentel BF (2001) Geology of the Escondida porphyry copper deposit, Antofagasta region, Chile. Econ Geol 96(2):307–324. CrossRefGoogle Scholar
  17. Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, OxfordGoogle Scholar
  18. Hunt JA, Berry RF (2017) Geological contributions to geometallurgy: a review. Geosci Can 44(3):103–118CrossRefGoogle Scholar
  19. Lantuéjoul C, Desassis N (2012) Simulation of a Gaussian random vector: a propagative version of the Gibbs sampler. In: The 9th international geostatistics congress, Oslo, NorwayGoogle Scholar
  20. León Carrera MF, Barbier M, Le Ravalec M (2018) Accounting for diagenesis overprint in carbonate reservoirs using parametrization technique and optimization workflow for production data matching. J Pet Explor Prod Technol.
  21. Renard D, Beucher H, Doligez B (2008) Heterotopic bi-categorical variables in pluri-Gaussian truncated simulation. In: Proceedings of the eighth international geostatistics congress geostats, Citeseer, pp 289–298Google Scholar
  22. Rossi ME, Deutsch CV (2014) Mineral resource estimation. Springer, Dordrecht. CrossRefGoogle Scholar
  23. Scheffel R, Guzman A, Dreier J (2016) Development metallurgy guidelines for copper heap leach. Miner Metall Process 33(4):187–199Google Scholar
  24. Silva DS (2018) Enhanced geologic modeling with data-driven training images for improved resources and recoverable reserves. Ph.D. thesis, University of Alberta, Edmonton, AlbertaGoogle Scholar
  25. Silva DSF, Deutsch CV (2017) Multiple imputation framework for data assignment in truncated pluri-Gaussian simulation. Stoch Environ Res Risk Assess 31(9):2251–2263. CrossRefGoogle Scholar
  26. Silva DS, Jewbali A, Boisvert JB, Deutsch CV (2018) Drillhole placement subject to constraints for improved resource classification. CIM J 9(1):21–32. CrossRefGoogle Scholar
  27. Spall JC (2005) Introduction to stochastic search and optimization: estimation, simulation, and control, vol 65. Wiley, New YorkGoogle Scholar
  28. Tonder EV, Deglon D, Napier-Munn T (2010) The effect of ore blends on the mineral processing of platinum ores. Miner Eng 23(8):621–626. CrossRefGoogle Scholar

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© International Association for Mathematical Geosciences 2019

Authors and Affiliations

  1. 1.5-052 Natural Resources Engineering FacilityUniversity of AlbertaEdmontonCanada
  2. 2.6-232 Donadeo Innovation Centre for EngineeringUniversity of AlbertaEdmontonCanada

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