Natural Resources Research

, Volume 26, Issue 2, pp 223–236 | Cite as

Spatial Pair-Copula Modeling of Grade in Ore Bodies: A Case Study

  • G. Nishani Musafer
  • M. Helen Thompson
  • E. Kozan
  • R. C. Wolff


A real-world mining application of pair-copulas is presented to model the spatial distribution of metal grade in an ore body. Inaccurate estimation of metal grade in an ore reserve can lead to failure of a mining project. Conventional kriged models are the most commonly used models for estimating grade and other spatial variables. However, kriged models use the variogram or covariance function, which produces a single average value to represent the spatial dependence for a given distance. Kriged models also assume linear spatial dependence. In the application, spatial pair-copulas are used to appropriately model the non-linear spatial dependence present in the data. The spatial pair-copula model is adopted over other copula-based spatial models since it is better able to capture complex spatial dependence structures. The performance of the pair-copula model is shown to be favorable compared to a conventional lognormal kriged model.


Geostatistical modeling Pair-copula Kriging Non-linear spatial dependence Mining 



This research was funded by the Australian Government’s Cooperative Research Centre for Optimising Resource Extraction Grant P3C-030. The authors thank the reviewers for their comments and guidance, which greatly improved discussion and practical aspects of the application.


  1. Aas, K., Czado, C., Frigessi, A., & Bakken, H. (2009). Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics, 44(2), 182–198.Google Scholar
  2. Bárdossy, A. (2006). Copula-based geostatistical models for groundwater quality parameters. Water Resources Research, 42(11), W11416.CrossRefGoogle Scholar
  3. Bárdossy, A., & Li, J. (2008). Geostatistical interpolation using copulas. Water Resources Research, 44(7), W07412.CrossRefGoogle Scholar
  4. Bedford, T., & Cooke, R. M. (2002). Vines: A new graphical model for dependent random variables. The Annals of Statistics, 30(4), 1031–1068.CrossRefGoogle Scholar
  5. Boardman, R. C., & Vann, J. E. (2011). A review of the application of copulas to improve modelling of non-bigaussian bivariate relationships (with an example using geological data). In F. Chan, D. Marinova, & R. S. Anderssen (Eds.), Proceedings of the 19th International Congress on Modelling and Simulation (MODSIM2011) (pp. 627–633). Perth: Modelling and Simulation Society of Australia and New Zealand (MSSANZ).
  6. Diggle, P. J., & Ribeiro, P. J. (2007). Classical parameter estimation. Model-based geostatistics (pp. 99–133). New York: Springer.Google Scholar
  7. Erhardt, T. M., Czado, C., & Schepsmeier, U. (2015a). R-vine models for spatial time series with an application to daily mean temperature. Biometrics, 71(2), 323–332.CrossRefGoogle Scholar
  8. Erhardt, T. M., Czado, C., & Schepsmeier, U. (2015b). Spatial composite likelihood inference using local C-vines. Journal of Multivariate Analysis, 138, 74–88.CrossRefGoogle Scholar
  9. Gaetan, C., & Guyon, X. (2010). Second-order spatial models and geostatistics. Spatial statistics and modeling (pp. 1–52). New York: Springer.CrossRefGoogle Scholar
  10. Genest, C., & Rivest, L. (1993). Statistical inference procedures for bivariate Archimedian copulas. Journal of the American Statistical Association, 88(423), 1034–1043.CrossRefGoogle Scholar
  11. Getis, A. (2007). Reflections on spatial autocorrelation. Regional Science and Urban Economics, 37(4), 491–496.CrossRefGoogle Scholar
  12. Gräler, B. (2014). Modelling skewed spatial random fields through the spatial vine copula. Spatial Statistics, 10, 87–102.CrossRefGoogle Scholar
  13. Gräler, B., & Appel, M. (2015). ‘spcopula’.
  14. Gräler, B., & Pebesma, E. (2011). The pair-copula construction for spatial data: A new approach to model spatial dependency. Procedia Environmental Sciences, 7, 206–211.CrossRefGoogle Scholar
  15. Haff, I. H., Aas, K., & Frigessi, A. (2009). On the simplified pair-copula construction-simply useful or too simplistic? Journal of Multivariate Analysis, 101(5), 1296–1310.CrossRefGoogle Scholar
  16. Haslauer, C. P., Li, J., & Bárdossy, A. (2010). Application of copulas in geostatistics. In P. M. Atkinson & C. D. Lloyd (Eds.), geoENV VII-geostatistics for environmental applications (pp. 395–404). Berlin: Springer.CrossRefGoogle Scholar
  17. Joe, H. (1996). Families of m-variate distributions with given margins and m(m-1)/2 bivariate dependence parameters. In L. Riischendorf, B. Schweizer, & M. D. Taylor (Eds.), Distributions with fixed marginal and related topics (pp. 120–141)., Lecture notes-monograph series Kolkata: Institute of Mathematical Statistics.CrossRefGoogle Scholar
  18. Journal, A. G., & Alabert, F. (2007). Non-Gaussian data expansion in the Earth Sciences. Terra Nova, 1(2), 123–134.CrossRefGoogle Scholar
  19. Kazianka, H., & Pilz, J. (2010). Copula-based geostatistical modeling of continuous and discrete data including covariates. Stochastic Environmental Research and Risk Assessment, 24(5), 661–673.CrossRefGoogle Scholar
  20. Kazianka, H., & Pilz, J. (2011). Bayesian spatial modeling and interpolation using copulas. Computers & Geosciences, 37(3), 310–319.CrossRefGoogle Scholar
  21. Khosrowshahi, S., & Shaw, W. (2001). Conditional simulation for resource characterization and grade control. In A. C. Edwards (Ed.), Mineral Resource and Ore Reserve Estimation - The AusIMM Guide to Good Practice. Australasian Institute of Mining and Metallurgy (AusIMM) (pp. 285–292).Google Scholar
  22. Kurowicka, D., & Cooke, R. (2006). Uncertainty analysis with high dimensional dependence modelling. Chichester: Wiley.CrossRefGoogle Scholar
  23. Li, J. (2010). Application of copulas as a new geostatistical tool. Ph.D dissertation. Institute for Water and Environmental System Modeling. University of Stuttgart. doi: 10.18419/opus-332
  24. Marchant, B. P., Saby, N. P. A., Jolivet, C. C., Arrouays, D., & Lark, R. M. (2011). Spatial prediction of soil properties with copulas. Geoderma, 162(3–4), 327–334.CrossRefGoogle Scholar
  25. Mclennan, J. A., & Deutsch, C. V. (2004). Conditional non-bias of geostatistical simulation for estimation of recoverable reserves. CIM Bulletin, 97(1080), 68–72.Google Scholar
  26. Musafer, G. N., & Thompson, M. H. (2016a). Non-linear optimal multivariate spatial design using pair-copulas. Stochastic Environmental Research and Risk Assessment. doi: 10.1007/s00477-016-1307-6.
  27. Musafer, G. N., & Thompson, M. H. (2016b). Optimal adaptive sequential spatial sampling of soil using pair-copulas. Geoderma, 271, 124–133.CrossRefGoogle Scholar
  28. Musafer, G. N., Thompson, M. H., Kozan, E., & Wolff, R. C. (2013). Copula-based spatial modelling of geometallurgical variables. In S. Dominy (Ed.), Proceedings of The Second AusIMM International Geometallurgy Conference (GeoMet2013) (pp. 239–246). Brisbane: Australasian Institute of Mining and Metallurgy (AusIMM).Google Scholar
  29. Nelsen, R. B. (2006). An introduction to copulas. New York: Springer.Google Scholar
  30. Peattie, R., & Dimitrakopoulos, R. (2013). Forecasting recoverable ore reserves and their uncertainty at Morila Gold Deposit, Mali: An efficient simulation approach and future grade control drilling. Mathematical Geosciences, 45(8), 1005–1020.CrossRefGoogle Scholar
  31. R Core Team. (2016). R: A language and environment for statistical computing. R Foundation for Statistical Computing. Vienna.
  32. Seo, D. J. (2013). Conditional bias-penalized kriging (CBPK). Stochastic Environmental Research and Risk Assessment, 27(1), 43–58.CrossRefGoogle Scholar
  33. Sklar, A. (1959). Fonctions de répartition á n dimensions et leurs marges. Publications de l’Institut de Statistique de L’Université de Paris, 8, 229–231.Google Scholar
  34. Trivedi, P. K., & Zimmer, D. M. (2007). Copula modeling: An introduction for practitioners. Boston: Now.Google Scholar
  35. Vann, J., & Guibal, D. (2001). Beyond ordinary kriging: An overview of non-linear estimation. In A. C. Edwards (Ed.), Mineral resource and ore reserve estimation-the AusIMM guide to good practice (pp. 249–256). Carlton: Australasian Institute of Mining and Metallurgy (AusIMM).Google Scholar

Copyright information

© International Association for Mathematical Geosciences 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesQueensland University of Technology (QUT)BrisbaneAustralia
  2. 2.Cooperative Research Centre for Optimising Resource Extraction (CRC ORE)PullenvaleAustralia
  3. 3.W.H. Bryan Mining and Geology Research CentreUniversity of Queensland (UQ)St LuciaAustralia

Personalised recommendations