Grade Estimation Through the Gaussian Copulas: A Case Study | ||
| Journal of Mining and Environment | ||
| مقاله 1، دوره 16، شماره 1، فروردین 2025، صفحه 1-13 اصل مقاله (2.01 M) | ||
| نوع مقاله: Original Research Paper | ||
| شناسه دیجیتال (DOI): 10.22044/jme.2024.14287.2668 | ||
| نویسندگان | ||
| Babak Sohrabian* 1؛ Abdullah Erhan Tercan2 | ||
| 1Associate Professor, Department of Mining Engineering, Faculty of Environment, Urmia University of Technology, Urmia, Iran | ||
| 2Professor, Department of Mining Engineering, Hacettepe University, Ankara, Turkey | ||
| چکیده | ||
| Mineral Resources have commonly been estimated through the kriging method that assigns weights to the samples based on variogram distance to the estimation point without considering their values. More robust estimators such as spatial copulas are promising tools because they consider both distance and sample values in determining weights. The purpose of this study is to demonstrate the effectiveness of the Gaussian copulas (GC) by estimating the copper grade values in the Sungun porphyry copper deposit located in Iran. Performance of the method was compared to ordinary kriging (OK) and indicator kriging (IK) by running the Jackknife test of cross-validation. The metrics used in measuring performance of the methods are global accuracy and precision of the distribution of the estimates, error statistics, and variability for globally accurate and precise estimates. The case study shows advantages of GC over OK and IK by producing globally accurate and precise estimates with acceptable error statistics and variability. | ||
| کلیدواژهها | ||
| Gaussian copula؛ Indicator kriging؛ Jackknife test؛ Ordinary kriging؛ Porphyry copper deposit | ||
| مراجع | ||
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[1]. Bastante, F.G., Taboada, J., Alejano, L.R., & Ordóñez, C. (2005). Evaluation of the resources of a slate deposit using indicator kriging. Engineering Geology, 81(4), 407-418.
[2]. 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, 76(3), 217-226.
[3]. Vargas-Guzmán, J. (2008). Unbiased Resource Evaluations with Kriging and Stochastic Models of Heterogeneous Rock Properties. Nat Resour Res, 17, 245–254.
[4]. Daya, A.A. (2012). Reserve estimation of central part of Choghart north anomaly iron ore deposit through ordinary kriging method. International Journal of Mining Science and Technology, 22(4), 573-577.
[5]. Moosavi, E., & Gholamnejad, J. (2015). Long-term production scheduling modeling for the open pit mines considering tonnage uncertainty via indicator kriging. J Min Sci, 51, 1226–1234.
[6]. Monteiro da Rocha, 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. Nat Resour Res, 9,223–235.
[7]. Lloyd, C., & Atkinson, P. (2001). Assessing uncertainty in estimates with ordinary and indicator kriging. Computational Geosciences, 27(8), 929–937.
[8]. Carr, J., & Mao, N. (1993). A general-form of probability kriging for estimation of the indicator and uniform transforms. Math Geol,25(4), 425–438.
[9]. Carle, S., & Fogg, G. (1996). Transition probability-based indicator geostatistics. Math Geol, 28(4), 453–476.
[10]. Bogaert, P. (1999). On the optimal estimation of the cumulative distribution function in presence of spatial dependence. Math Geol,3(2), 213–239.
[11]. Pardo-Igúzquiza, E., & Dowd, P. (2005). Multiple indicator cokriging with application to optimal sampling for environmental monitoring. Computational Geoscience, 31(1),1–13.
[12]. Bardossy, A. (2006). Copula-based geostatistical models for groundwater quality parameters. Water Resour. Res, 42.
[13]. Bardossy, A., & Li, J. (2008). Geostatistical interpolation using copulas. Water Resour. Res, 44.
[14]. Kazianka, H., & Pilz, J. (2010). Copula-based geostatistical modeling of continuous and discrete data including covariates. Stoch. Env. Res. Risk A., 24, 661–673.
[15]. Atalay, F., & Tercan, A.E. (2017). Coal resource estimation using Gaussian copula, International Journal of Coal Geology, 175.
[16]. Käärik, E., & Käärik, M. (2009). Modeling dropouts by conditional distribution, a copula-based approach. Journal of Statistical Planning and Inference, 139(11), 3830-3835.
[17]. Parsa, R.A., & Klugman, S.A. (2011). Copula regression. Var. Adv. Sci. Risk, 5, 45-54.
[18]. Kwak, M. (2017). Estimation and inference on the joint conditional distribution for bivariate longitudinal data using Gaussian copula, Journal of the Korean Statistical Society, 46(3), 349-364.
[19]. Chang, B., & Joe, H. (2019). Prediction based on conditional distributions of vine copulas, Computational Statistics & Data Analysis, 139, 45-63.
[20]. Musafer, G. N., Thompson, M. H., Kozan, E., & Wolff, R. C. (2016). Spatial Pair-Copula Modeling of Grade in Ore Bodies: A Case Study. Natural Resources Research, 26(2), 223–236.
[21]. Addo, E., Chanda, E.K., & Metcalfe, A.V. (2017). Estimation of direction of increase of gold mineralization using pair-copulas, International Congress on Modelling and Simulation.
[22]. Sohrabian, B., Soltani-Mohammadi, S., Pourmirzaee, R., & Carranza, E.J.M. (2023). Geostatistical Evaluation of a Porphyry Copper Deposit Using Copulas. Minerals, 13(6), 732.
[23]. Bárdossy, A., & Hörning, S. (2023). Definition of Spatial Copula Based Dependence Using a Family of Non‐Gaussian Spatial Random Fields. Water Resources Research, 59(7).
[24]. Sklar, A. (1959). Fonctions de Répartition à n Dimensions et Leurs Marges. Publications de l’Institut Statistique de l’Université de Paris, 8, 229-231.
[25]. Addo, E., Chanda, E.K. & Metcalfe, A.V. (2018). Spatial Pair-Copula Model of Grade for an Anisotropic Gold Deposit. Mathematical Geosciences, 51(5), 553–578.
[26]. Isaaks, E.H., & Srivastava, R.M. (1989). An Introduction to Applied Geostatistics. Oxford University Press, New York, 413.
[27]. Journel, A.G. (1983). Nonparametric estimation of spatial distributions. Math. Geol., 15, 445–468.
[28]. Tercan, A.E., & Dowd, P.A. (1995). Approximate Local Confidence Intervals under Change of Support. Mathematical Geology, 27(1), 149–72.
[29]. Deutsch, C.V. (1996). Direct Assessment of Local Accuracy and Precision. Geostatistics Wollongong 96, Vol 1, Baafi and Schofield, editors, Kluwer Academic Publishers, pp 115-125.
[30]. Pyrcz, M.J., & Deutsch, C.V. (2014). Geostatistical Reservoir Modeling, 2nd Edition, Oxford University Press, New York, p. 448.
[31]. Berberian, M., & King, G.C.P. (1981). Towards a paleogeography and tectonic evolution of Iran, Canadian Journal of Earth Sciences, 18, 210–265.
[32]. Alavi, M. (1991). Sedimentary and structural characteristics of the Paleo-Tethys remnants in northeastern Iran, Geological Society of America Bulletin, 103, 983–992.
[33]. Kou, G.Y., Xu, B., Zhou, Y., Zheng, Y.C., Hou, Z.Q., Zhou, L.M., Zhang, L.M., & Yu, J.X. (2021). Geology and petrogenesis of the Sungun deposits: Implications for the genesis of porphyry-type mineralisation in the NW Urumieh–Dokhtar magmatic Arc, Iran. Ore Geology Reviews [Internet].
[34]. Lescuyer, J.L., Riou, R., Babakhani, A., Alavi Tehrani, N., Nogol, M.A., Dido, J., & Gemain, Y.M. (1978). Geological map of the Ahar area. Geological Survey of Iran.
[35]. Mehrpartou, M. (1993). Contributions to the geology, geochemistry, ore genesis and fluid inclusion investigations on Sungun Cu–Mo porphyry deposit, (North-West of Iran). PhD Thesis, Hamburg University 1993. Hamburg, 245 pp.
[36]. Simmonds, V., Moazzen, M., & Mathur, R. (2017). Constraining the timing of porphyry mineralization in northwest Iran in relation to Lesser Caucasus and Central Iran; Re–Os age data for Sungun porphyry Cu–Mo deposit. International Geology Review, 59(12), 1561–1574.
[37]. Calagari, A.A., Pattrick, R.A.D., & Polya, D.A. (2001). Veinlets and micro-veinlets studies in Sungun porphyry copper deposit, East Azarbaidjan, Iran. Iranian Journal of Geoscience, 39–40, 70–79.
[38]. Soltani, F., Afzal, P., & Asghari, O. (2014). Delineation of Alteration Zones Based on Sequential Gaussian Simulation and Concentration–Volume Fractal Modeling in the Hypogene Zone of Sungun Copper Deposit, NW Iran. Journal of Geochemical Exploration, 140, 64–76.
[39]. Sullivan, J. (1984). Non-parametric estimation of spatial distributions. Unpublished PhD dissertation, Stanford University, 367 pp.
[40]. Myers, J.C. (1997). Geostatistical Error Management: Quantifying Uncertainty for Environmental Sampling and Mapping, John Wiley and Sons, 596 pp.
[41]. Sohrabian, B. (2021a). Geostatistical prediction through convex combination of Archimedean copulas. Spatial Statistics, 41, 100488.
[42]. Sohrabian, B. (2021b). Capacity assessment of Gumbel-Clayton copula for geostatistical estimation. Iran J Min Eng, 16(51), 52–67.
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