Gaussian copulas for spatial estimation of ore grade in a copper deposit

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mining Engineering, Faculty of Engineering, National University of Trujillo, Trujillo, Peru.

2 Faculty of Chemical Engineering, National University of the Altiplano, Puno, Peru.

3 Department of Industrial Engineering, National University of Trujillo, Trujillo, Peru.

4 Mining Engineering School, Universidad Nacional Jorge Basadre Grohmann, Tacna, Peru.

5 Department of Metallurgy Engineering, National University of Trujillo, Trujillo, Peru.

10.22059/ijmge.2025.398485.595277

Abstract

This study evaluates the effectiveness of the Gaussian copula (GC) in estimating copper grades in a Peruvian copper deposit, comparing its performance with Ordinary Kriging (OK). The methodology was implemented in Python 3.11.7 and Jupyter Notebook 4.2.5. Model accuracy was assessed through 5-fold spatial cross-validation using metrics such as Mean Squared Error (MSE), Mean Bias Error (MBE), Mean Absolute Error (MAE), and Variance. The estimation was conducted using a database of 5,654 composites. The results demonstrate that GC outperforms OK, achieving an MSE of 0.0882, an MAE of 0.1956, an MBE of 0.0394, and a variance of 0.0369. These values indicate that GC provides more accurate and less biased estimates, capturing local grade variability more effectively than OK. Although GC shows slightly higher estimation variance (0.0369 vs. 0.027), it successfully captures the maximum copper grade observed in the real data (2.95%), unlike OK (1.62%), suggesting that GC mitigates the excessive smoothing of OK while still maintaining a centered and stable distribution. In conclusion, the GC method emerges as a robust alternative to OK, offering improved precision, better spatial representation, and enhanced reliability in mineral resource estimation. Its implementation in Python also promotes greater accessibility and reproducibility, reinforcing its value as a practical tool in geostatistics.

Keywords

Main Subjects

References

[1] Sohrabian, B., & Tercan, A. E. (2014). Introducing minimum spatial cross-correlation kriging as a new estimation method of heavy metal contents in soils. Geoderma. doi:https://doi.org/10.1016/j.geoderma.2014.02.014.
[2] Sohrabian, B., Ozcelik, Y., & Hasanpour, R. (2017). Estimating major elemental oxides of an andesite quarry using compositional kriging. International Journal of Mining, Reclamation and Environment. doi:https://doi.org/10.1080/17480930.2016.1168019.
[3] Jeuken, R., Xu, C., & Dowd, P. (2020). Improving coal quality estimations with geostatistics and geophysical logs. Natural Resources Research. doi:https://doi.org/10.1007/s11053-019-09609-y.
[4] Rahimi, H., Asghari, O., & Afshar, A. (2018). A geostatistical investigation of 3D magnetic inversion results using multi-Gaussian kriging and sequential Gaussian co-simulation. Journal of Applied Geophysics. doi:https://doi.org/10.1016/j.jappgeo.2018.05.003.
[5]  Lloyd, C. D., & Atkinson, P. M. (2001). Assessing uncertainty in estimates with ordinary and indicator kriging. Computers & Geosciences. doi:https://doi.org/10.1016/S0098-3004(00)00132-1.
[6]  Gräler, B., & Pebesma, E. (2011). The pair-copula construction for spatial data: A new approach to model spatial dependency. Procedia Environmental Sciences. doi:https://doi.org/10.1016/j.proenv.2011.07.036.
[7]  Sohrabian, B., Soltani-Mohammadi, S., Pourmirzaee, R., & Carranza, E. J. M. (2023). Geostatistical evaluation of a porphyry copper deposit using copulas. Minerals, 13(6), Article 732. doi:https://doi.org/10.3390/min13060732.
[8]  Sohrabian, B., & Tercan, A. E. (2024). Copula-based data-driven multiple-point simulation method. Spatial Statistics, 58, 100802. doi:https://doi.org/10.1016/j.spasta.2023.100802.
[9]  Sohrabian, B., & Tercan, A. (2025). Grade estimation through the Gaussian copulas: A case study. Journal of Mining and Environment, 16, 1–13.
[10] Gräler, B. (2014). Modelling skewed spatial random fields through the spatial vine copula. Spatial Statistics, 8, 1–14. doi:https://doi.org/10.1016/j.spasta.2014.01.001.
[11]  Shaked, M., & Joe, H. (1998). Multivariate models and dependence concepts. Journal of the American Statistical Association. doi:https://doi.org/10.2307/2669872.
[12] Frahm, G., Junker, M., & Szimayer, A. (2003). Elliptical copulas: Applicability and limitations. Statistics & Probability Letters. doi:https://doi.org/10.1016/S0167-7152(03)00092-0.
[13] Bárdossy, A. (2006). Copula-based geostatistical models for groundwater quality parameters. Water Resources Research. doi:https://doi.org/10.1029/2005WR004754.
[14] Van de Vyver, H., & Van den Bergh, J. (2018). The Gaussian copula model for the joint deficit index for droughts. Journal of Hydrology. doi:https://doi.org/10.1016/j.jhydrol.2018.03.064.
[15] Li, F., Zhou, J., & Liu, C. (2018). Statistical modelling of extreme storms using copulas: A comparison study. Coastal Engineering. doi:https://doi.org/10.1016/j.coastaleng.2018.09.007.
[16] Lourme, A., & Maurer, F. (2017). Testing the Gaussian and Student’s t copulas in a risk management framework. Economic Modelling. doi:https://doi.org/10.1016/j.econmod.2016.12.014.
[17] Marchant, B. P., Saby, N. P. A., Jolivet, C. C., Arrouays, D., & Lark, R. M. (2011). Spatial prediction of soil properties with copulas. Geoderma. doi:https://doi.org/10.1016/j.geoderma.2011.03.005.
[18] Quessy, J. F., Rivest, L. P., & Toupin, M. H. (2019). Goodness-of-fit tests for the family of multivariate chi-square copulas. Computational Statistics & Data Analysis. doi:https://doi.org/10.1016/j.csda.2019.04.008.
[19] Musafer, G. N., Thompson, M. H., Wolff, R. C., & Kozan, E. (2017). Nonlinear multivariate spatial modeling using NLPCA and pair-copulas. Geographical Analysis. doi:https://doi.org/10.1111/gean.12126.
[20] Pardo-Igúzquiza, E., & Dowd, P. A. (2005). EMLK2D: A computer program for spatial estimation using empirical maximum likelihood kriging. Computers & Geosciences. doi:https://doi.org/10.1016/j.cageo.2004.09.020.
[21] Heriawan, M. N., & Koike, K. (2008). Uncertainty assessment of coal tonnage by spatial modeling of seam distribution and coal quality. International Journal of Coal Geology. doi:https://doi.org/10.1016/j.coal.2008.07.014.
[22] Vargas-Guzmán, J. A. (2008). Unbiased resource evaluations with kriging and stochastic models of heterogeneous rock properties. Natural Resources Research. doi:https://doi.org/10.1007/s11053-008-9082-9
[23] Ali Akbar, D. (2012). Reserve estimation of central part of Choghart north anomaly iron ore deposit through ordinary kriging method. International Journal of Mining Science and Technology. doi:https://doi.org/10.1016/j.ijmst.2012.01.022.
[24] Rohma, N. N. (2022). Estimation of ordinary kriging method with jackknife technique on rainfall data in Malang Raya. International Journal on Information and Communication Technology (IJoICT). doi:https://doi.org/10.21108/ijoict.v8i2.678.
[25] Lamamra, A., Neguritsa, D. L., & Mazari, M. (2019). Geostatistical modeling by the ordinary kriging in the estimation of mineral resources on the Kieselguhr mine, Algeria. IOP Conference Series: Earth and Environmental Science, 362(1), 012051. doi:https://doi.org/10.1088/1755-1315/362/1/012051.
[26] Da Rocha, M. M., & Yamamoto, J. K. (2000). Comparison between kriging variance and interpolation variance as uncertainty measurements in the Capanema iron mine, State of Minas Gerais-Brazil. Natural Resources Research. doi:https://doi.org/10.1023/a:1010195701968.
[27] Fogg, G. E. (1996). Transition probability-based indicator geostatistics. Mathematical Geology. doi:https://doi.org/10.1007/bf02083656.
[28] Carr, J. R., & Mao, N. (1993). A general form of probability kriging for estimation of the indicator and uniform transforms. Mathematical Geology. doi:https://doi.org/10.1007/BF00894777.
[29] Bárdossy, A., & Li, J. (2008). Geostatistical interpolation using copulas. Water Resources Research. doi:https://doi.org/10.1029/2007WR006115.
[30] Kazianka, H., & Pilz, J. (2010). Copula-based geostatistical modeling of continuous and discrete data including covariates. Stochastic Environmental Research and Risk Assessment. doi:https://doi.org/10.1007/s00477-009-0353-8.
[31] Atalay, F., & Tercan, A. E. (2017). Coal resource estimation using Gaussian copula. International Journal of Coal Geology. doi:https://doi.org/10.1016/j.coal.2017.03.010.
[32] Käärik, E., & Käärik, M. (2009). Modeling dropouts by conditional distribution, a copula-based approach. Journal of Statistical Planning and Inference. doi:https://doi.org/10.1016/j.jspi.2009.05.020.
[33] Klugman, S. A. (2011). Copula regression. Variance, 5.
[34] Kwak, M. (2017). Estimation and inference on the joint conditional distribution for bivariate longitudinal data using Gaussian copula. Journal of the Korean Statistical Society. doi:https://doi.org/10.1016/j.jkss.2016.11.005.
[35] Chang, B., & Joe, H. (2019). Prediction based on conditional distributions of vine copulas. Computational Statistics & Data Analysis. doi:https://doi.org/10.1016/j.csda.2019.04.015.
[36] Addo, E., Chanda, E. K., & Metcalfe, A. V. (2017). Estimation of direction of increase of gold mineralisation using pair-copulas. Proceedings of the 22nd International Congress on Modelling and Simulation (MODSIM 2017). doi:https://doi.org/10.36334/modsim.2017.a2.addo.
[37] Musafer, G. N., Thompson, M. H., Kozan, E., & Wolff, R. C. (2017). Spatial pair-copula modeling of grade in ore bodies: A case study. Natural Resources Research. doi:https://doi.org/10.1007/s11053-016-9314-3.
[38] Bárdossy, A., & Hörning, S. (2023). Definition of spatial copula based dependence using a family of non-Gaussian spatial random fields. Water Resources Research. doi:https://doi.org/10.1029/2023WR034446.
[39] Agarwal, G., Sun, Y., & Wang, H. J. (2021). Copula-based multiple indicator kriging for non-Gaussian random fields. Spatial Statistics. doi:https://doi.org/10.1016/j.spasta.2021.100524.
[40] Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges (Distribution functions of n dimensions and their marginals). Publications de l’Institut Statistique de l’Université de Paris, 8.
[41] Hohn, M. E. (1991). An introduction to applied geostatistics. Computers & Geosciences. doi:https://doi.org/10.1016/0098-3004(91)90055-i.
[42] Deutsch, C. V. (1996). Correcting for negative weights in ordinary kriging. Computers & Geosciences. doi:https://doi.org/10.1016/0098-3004(96)00005-2.
[43] Pesquer, L., Cortés, A., & Pons, X. (2011). Parallel ordinary kriging interpolation incorporating automatic variogram fitting. Computers & Geosciences. doi:https://doi.org/10.1016/j.cageo.2010.10.010.
[44] Munyati, C., & Sinthumule, N. I. (2021). Comparative suitability of ordinary kriging and inverse distance weighted interpolation for indicating intactness gradients on threatened savannah woodland and forest stands. Environmental and Sustainability Indicators. doi:https://doi.org/10.1016/j.indic.2021.100151.
[45] Daya, A. A., & Bejari, H. (2015). A comparative study between simple kriging and ordinary kriging for estimating and modeling the Cu concentration in Chehlkureh deposit, SE Iran. Arabian Journal of Geosciences. doi:https://doi.org/10.1007/s12517-014-1618-1.
[46] Paoli, J. N., Tisseyre, B., Strauss, O., & Roger, J. M. (2024). Methods to define confidence intervals for kriged values: Application to precision viticulture data. Precision Agriculture. doi:https://doi.org/10.3920/9789086865147_079.
[47] Virtanen, P., Gommers, R., Oliphant, T. E., et al. (2020). SciPy 1.0: Fundamental algorithms for scientific computing in Python. Nature Methods. doi:https://doi.org/10.1038/s41592-019-0686-2.
[48] Abraj, M., Wang, Y. G., & Thompson, M. H. (2022). A new mixture copula model for spatially correlated multiple variables with an environmental application. Scientific Reports. doi:https://doi.org/10.1038/s41598-022-18007-z.
[49] Bevilacqua, M., Alvarado, E., & Caamaño-Carrillo, C. (2024). A flexible Clayton-like spatial copula with application to bounded support data. Journal of Multivariate Analysis. doi:https://doi.org/10.1016/j.jmva.2023.105277.
[50] Cristianini, N. (2004). Cross-validation (K-fold cross-validation, leave-one-out, jackknife, bootstrap). Dictionary of Bioinformatics and Computational Biology. doi:https://doi.org/10.1002/9780471650126.dob0148.pub2.
[51] Jandaghian, Z., & Berardi, U. (2021). The coupling of the Weather Research and Forecasting model with the Urban Canopy Models for climate simulations. In Urban Microclimate Modelling for Comfort and Energy Studies. doi:https://doi.org/10.1007/978-3-030-65421-4_11.
[52] Kazianka, H., & Pilz, J. (2011). Bayesian spatial modeling and interpolation using copulas. Computers & Geosciences. doi:https://doi.org/10.1016/j.cageo.2010.06.005.
[53] Hodson, T. O., Over, T. M., & Foks, S. S. (2021). Mean squared error, deconstructed. Journal of Advances in Modeling Earth Systems. doi:https://doi.org/10.1029/2021MS002681.
[54] Cotrina, M., Marquina, J., Noriega, E., Mamani, J., Ccatamayo, J., Gonzalez, J., & Arango, S. (2024). Predicting open pit mine production using machine learning techniques: A case study in Peru. Journal of Mining and Environment, 15, 1345–1355.
[55] Marquina, J., Cotrina, M., Mamani, J., Noriega, E., & Vega, J., Cruz, J. (2024). Copper ore grade prediction using machine learning techniques in a copper deposit. Journal of Mining and Environment, 15, 1011–1027.
[56] Marquina-Araujo, J. J., Cotrina-Teatino, M. A., Cruz-Galvez, J. A., Noriega-Vidal, E. M., & Vega-Gonzalez, J. A. (2024). Application of autoencoders neural network and K-means clustering for the definition of geostatistical estimation domains. Mathematical Modelling of Engineering Problems, 11, 1207–1218.
[57]      Zhang, M., Zhang, Y., & Yu, G. (2017). Applied geostatistics analysis for reservoir characterization based on the SGeMS (Stanford Geostatistical Modeling Software). Open Journal of Yangtze Oil and Gas. doi:https://doi.org/10.4236/ojogas.2017.21004.
[58] Marwanza, I., Nas, C., Azizi, M. A., & Simamora, J. H. (2019). Comparison between moving windows statistical method and kriging method in coal resource estimation. Journal of Physics: Conference Series, 1402(3), 033016. doi:https://doi.org/10.1088/1742-6596/1402/3/033016
[59] Lemenkova, P. (2019). Computing and plotting correlograms by Python and R libraries for correlation analysis of the environmental data in marine geomorphology. Journal of Geomorphological Researches, 1.
[60]      Cotrina-Teatino, M. A., Marquina-Araujo, J. J., & Riquelme, Á. I. (2025). Comparison of machine learning techniques for mineral resource categorization in a copper deposit in Peru. Natural Resources Research. doi:https://doi.org/10.1007/s11053-025-10505-x.
[61] Cotrina-Teatino, M. A., Riquelme, Á. I., Marquina-Araujo, J. J., Mamani-Quispe, J. N., Arango-Retamozo, S. M., Ccatamayo-Barrios, J. H., Donaires-Flores, T., Calla-Huayapa, M. A., & Gonzalez-Vasquez, J. A. (2025). KMeans-Riemannian model for classification of mineral resources in a copper deposit in Peru. International Journal of Mining, Reclamation and Environment. doi:https://doi.org/10.1080/17480930.2025.2518987.
International Journal of Mining and Geo-Engineering

Articles in Press, Accepted Manuscript
Available Online from 01 November 2025

Files

Share

How to cite

Statistics