Skip to main content
Log in

Assessment of Recoverable Resource Uncertainty in Multivariate Deposits Through a Simple Machine Learning Technique Trained Using Geostatistical Simulations

  • Original Paper
  • Published:
Natural Resources Research Aims and scope Submit manuscript


Mine design, mine production planning and the economic evaluation of multivariate ore deposits are based on the mineral resources that are recoverable after applying cutoffs to the grades of selective mining units. Being able to reliably assess the recoverable resource uncertainty of more than one element is key for the economic evaluation of mining projects. Multivariate geostatistical models are difficult to fit and strongly influence the resources reported. The proposed approach avoids the delicate modeling step by directly estimating multivariate tonnages and their associated confidence intervals from a reduced set of features extracted from data. The predictive models are obtained by training with conditional simulations. For each case considered in the training phase, the coregionalization parameters are drawn randomly from within specified intervals and multiple realizations make it possible to obtain deposits conditional to the simulated data sets. Multiple linear regression training is carried out using input–output data to generate predictive models that relate the input variables calculated from the simulated data sets and the output variables (i.e., mean tonnage and tonnage quantiles) computed from the simulated deposits. The results from a synthetic bivariate case indicate excellent tonnage prediction and credible uncertainty quantification. A lateritic nickel deposit subject to a constraint on the maximum silica/magnesia ratio shows the applicability of this approach in a real-world context. The resulting tonnage surface and the associated uncertainty quantification provide an essential tool to help assess the economic value of the mine project.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11

Similar content being viewed by others

Availability of data and materials

The datasets generated and analysed during the current study are not publicly available due to confidentiality reasons but are available from the corresponding author on reasonable request.

Code availability

There is no available public code.


  • Barnett, R. M., Manchuk, J. G., & Deutsch, C. V. (2014). Projection pursuit multivariate transform. Mathematical Geosciences, 46, 337–359.

    Article  Google Scholar 

  • Battalgazy, N., & Madani, N. (2019). Categorization of mineral resources based on different geostatistical simulation algorithms: A case study from an iron ore deposit. Natural Resources Research, 28, 1329–1351.

    Article  Google Scholar 

  • Bérubé, C. L., Olivo, G. R., Chouteau, M., Perrouty, S., Shamsipour, P., Enkin, R. J., Morris, W. A., Feltrin, L., & Thiémonge, R. (2018). Predicting rock type and detecting hydrothermal alteration using machine learning and petrophysical properties of the Canadian Malartic ore and host rocks, Pontiac Subprovince, Québec, Canada. Ore Geology Reviews, 96, 130–145.

    Article  Google Scholar 

  • Boucher, A., & Dimitrakopoulos, R. (2012). Multivariate block-support simulation of the Yandi iron ore deposit, Western Australia. Mathematical Geosciences, 44, 449–468.

    Article  Google Scholar 

  • Brooker, P. I. (1985). Two-dimensional simulation by turning bands. Mathematical Geology, 17, 81–90.

    Article  Google Scholar 

  • Desbarats, A. J., & Dimitrakopoulos, R. (2000). Geostatistical simulation of regionalized pore-size distributions using min/max autocorrelation factors. Mathematical Geology, 32, 919–942.

    Article  Google Scholar 

  • Dhaher, G. M., & Lee, M. H. (2013). A comparison between the performance of kriging and cokriging in spatial estimation with application. Matematika: Malaysian Journal of Industrial and Applied Mathematics, 33–41.

  • Dominy, S., Noppè, M., & Annels, A. (2002). Errors and uncertainty in mineral resource and ore reserve estimation: The importance of getting it right. Exploration and Mining Geology, 11, 77–98.

    Article  Google Scholar 

  • Emery, X., & Alegría, A. (2021). The Gauss hypergeometric covariance kernel for modeling second-order stationary random fields in euclidean spaces: Its compact support, properties and spectral representation. arXiv:2101.09558.

  • Emery, X., Arroyo, D., & Mery, N. (2021). Twenty-two families of multivariate covariance kernels on spheres, with their spectral representations and sufficient validity conditions. Stochastic Environmental Research and Risk Assessment.

  • Emery, X., & Lantuéjoul, C. (2006). TBSIM: A computer program for conditional simulation of three-dimensional gaussian random fields via the turning bands method. Computers & Geosciences, 32, 1615–1628.

    Article  Google Scholar 

  • Emery, X., Porcu, E., & White, P. (2021). Flexible validity conditions for the multivariate Matérn covariance in any spatial dimension and for any number of components. arXiv e-prints arXiv:2101.04235.

  • Eze, P. N., Madani, N., & Adoko, A. C. (2019). Multivariate mapping of heavy metals spatial contamination in a Cu-Ni exploration field (Botswana) using turning bands co-simulation algorithm. Natural Resources Research, 28, 109–124.

    Article  Google Scholar 

  • Faria, P. H., Costa, J. F. C. L., & Bassani, M. A. A. (2021). Multivariate geostatistical simulation with PPMT: an application for uncertainty measurement. Applied Earth Science, 130, 174–184.

    Google Scholar 

  • Ghezelbash, R., Maghsoudi, A., Bigdeli, A., & Carranza, E. J. M. (2021). Regional-scale mineral prospectivity mapping: Support vector machines and an improved data-driven multi-criteria decision-making technique. Natural Resources Research, 30, 1977–2005.

    Article  Google Scholar 

  • Goovaerts, P. (1993). Spatial orthogonality of the principal components computed from coregionalized variables. Mathematical Geology, 25, 281–302.

    Article  Google Scholar 

  • Granian, H., Tabatabaei, S. H., Asadi, H. H., & Carranza, E. J. M. (2015). Multivariate regression analysis of lithogeochemical data to model subsurface mineralization: a case study from the Sari Gunay epithermal gold deposit, NW Iran. Journal of Geochemical Exploration, 148, 249–258.

    Article  Google Scholar 

  • Hosseini, S., Asghari, O., & Emery, X. (2017). Direct block-support simulation of grades in multi-element deposits: application to recoverable mineral resource estimation at Sungun porphyry copper-molybdenum deposit. Journal of the Southern African Institute of Mining and Metallurgy, 117, 577–585.

    Article  Google Scholar 

  • Kalam, S., Khan, R. A., Khan, S., Faizan, M., Amin, M., Ajaib, R., & Abu-Khamsin, S. A. (2021). Data-driven modeling approach to predict the recovery performance of low-salinity waterfloods. Natural Resources Research, 30, 1697–1717.

    Article  Google Scholar 

  • Karbalaei Ramezanali, A., Feizi, F., Jafarirad, A., & Lotfi, M. (2020). Geochemical anomaly and mineral prospectivity mapping for vein-type copper mineralization, Kuhsiah-e-Urmak area, Iran: Application of sequential gaussian simulation and multivariate regression analysis. Natural Resources Research, 29, 41–70.

    Article  Google Scholar 

  • Keskinkilic, E. (2019). Nickel laterite smelting processes and some examples of recent possible modifications to the conventional route. Metals, 9.

  • Leuangthong, O., & Deutsch, C. V. (2003). Stepwise conditional transformation for simulation of multiple variables. Mathematical Geology, 35, 155–173.

    Article  Google Scholar 

  • McKay, G., & Harris, J. (2015). Comparison of the data-driven random forests model and a knowledge-driven method for mineral prospectivity mapping: A case study for gold deposits around the Huritz Group and Nueltin Suite, Nunavut, Canada. Natural Resources Research, 25.

  • Mery, N., & Marcotte, D. (2021). Quantifying mineral resources and their uncertainty using two existing machine learning methods. Mathematical Geosciences.

  • Mery, N., Marcotte, D., & Dutaut, R. (2020). Constrained kriging: An alternative to predict global recoverable resources. Natural Resources Research, 29, 2275–2289.

    Article  Google Scholar 

  • Minnitt, R., & Deutsch, C. (2014). Cokriging for optimal mineral resource estimates in mining operations. Journal of the Southern African Institute of Mining and Metallurgy, 114, 189–203.

    Google Scholar 

  • Montoya, C., Emery, X., Rubio, E., & Wiertz, J. (2012). Multivariate resource modelling for assessing uncertainty in mine design and mine planning. Journal of the Southern African Institute of Mining and Metallurgy, 112, 353–363.

    Google Scholar 

  • O’Brien, J. J., Spry, P. G., Nettleton, D., Xu, R., & Teale, G. S. (2015). Using random forests to distinguish gahnite compositions as an exploration guide to Broken Hill-type Pb-Zn-Ag deposits in the Broken Hill domain, Australia. Journal of Geochemical Exploration, 149, 74–86.

    Article  Google Scholar 

  • Pan, G., Gaard, D., Moss, K., & Heiner, T. (1993). A comparison between cokriging and ordinary kriging: Case study with a polymetallic deposit. Mathematical Geology, 25, 377–398.

    Article  Google Scholar 

  • Sagadin, C., Luidold, S., Wagner, C., & Wenzl, C. (2016). Melting behaviour of ferronickel slags. JOM, 68, 3022–3028.

    Article  Google Scholar 

  • Tercan, A., & Sohrabian, B. (2013). Multivariate geostatistical simulation of coal quality data by independent components. International Journal of Coal Geology, 112, 53–66. Special issue on geostatistical and spatiotemporal modeling of coal resources.

  • van den Boogaart, K. G., Mueller, U., & Tolosana-Delgado, R. (2017). An affine equivariant multivariate normal score transform for compositional data. Mathematical Geosciences, 49, 231–251.

    Article  Google Scholar 

  • Vergara, D., & Emery, X. (2013). Conditional bias for multivariate resources estimation. In J. Ambrus, J. Beniscelli, F. Brunner, J. Cabello, & F. Ibarra (Eds.), 3rd International seminar on geology for the mining industry (pp. 27–33).

  • Yalçin, E. (2005). Cokriging and its effect on the estimation precision. Journal of The South African Institute of Mining and Metallurgy, 105, 223–228.

    Google Scholar 

  • Zhang, S., Carranza, E. J. M., Xiao, K., Wei, H., Yang, F., Chen, Z., Li, N., & Xiang, J. (2021a). Mineral prospectivity mapping based on isolation forest and random forest: Implication for the existence of spatial signature of mineralization in outliers. Natural Resources Research.

  • Zhang, S. E., Nwaila, G. T., Tolmay, L., Frimmel, H. E., & Bourdeau, J. E. (2021b). Integration of machine learning algorithms with Gompertz curves and Kriging to estimate resources in gold deposits. Natural Resources Research, 30, 39–56.

Download references


This research was funded by the National Agency for Research and Development of Chile (ANID) through the Doctorado Becas Chile program (Grant Number 72180581), the National Research Council of Canada (Grant RGPIN-2015-06653), Quebec’s Merit Scholarship Program for Foreign Students (PBEEE, Grant Number 0000274857) and Polytechnique Montréal’s Doctoral Scholarship Program.

Author information

Authors and Affiliations



Nadia Mery: Study conception, outlining and carrying out the experiments and analysis. Denis Marcotte: Study conception and supervise the research. The first draft was written by Nadia Mery, and subsequent versions were reviewed by Denis Marcotte. Both authors read and approved the final manuscript.

Corresponding author

Correspondence to Nadia Mery.

Ethics declarations

Conflict of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work presented in this manuscript.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mery, N., Marcotte, D. Assessment of Recoverable Resource Uncertainty in Multivariate Deposits Through a Simple Machine Learning Technique Trained Using Geostatistical Simulations. Nat Resour Res 31, 767–783 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: