A Bayesian hierarchical model for the inference between metal grade with reduced variance: Case studies in porphyry Cu deposits

Yufu Niu, Mark Lindsay, Peter Coghill, Richard Scalzo, Lequn Zhang

Geoscience Frontiers ›› 2024, Vol. 15 ›› Issue (2) : 101767.

Geoscience Frontiers ›› 2024, Vol. 15 ›› Issue (2) : 101767. DOI: 10.1016/j.gsf.2023.101767

A Bayesian hierarchical model for the inference between metal grade with reduced variance: Case studies in porphyry Cu deposits

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Abstract

Ore sorting is a preconcentration technology and can dramatically reduce energy and water usage to improve the sustainability and profitability of a mining operation. In porphyry Cu deposits, Cu is the primary target, with ores usually containing secondary ‘pay’ metals such as Au, Mo and gangue elements such as Fe and As. Due to sensing technology limitations, secondary and deleterious materials vary in correlation type and strength with Cu but cannot be detected simultaneously via magnetic resonance (MR) ore sorting. Inferring the relationships between Cu and other elemental abundances is particularly critical for mineral processing.

Keywords

Bayesian hierarchical model / Porphyry Cu deposit / Ore sorting / Metal grade / Linear regression

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Yufu Niu, Mark Lindsay, Peter Coghill, Richard Scalzo, Lequn Zhang. A Bayesian hierarchical model for the inference between metal grade with reduced variance: Case studies in porphyry Cu deposits. Geoscience Frontiers, 2024, 15(2): 101767 https://doi.org/10.1016/j.gsf.2023.101767

References

J.P. Albers, F.J. Kleinhampl. Spatial relation of mineral deposits to tertiary volcanic centers in Nevada. US Geol. Surv. Prof. Pap., 700 (1970), pp. C1-C10
C. Andrieu, N. De Freitas, A. Doucet, M.I. Jordan. An introduction to MCMC for machine learning. Mach. Learn., 50 (2003), pp. 5-43
A. Arora, R. Fitch, S. Sukkarieh. An approach to autonomous science by modeling geological knowledge in a Bayesian framework. 2017 IEEE Int. Conf. Intell. Robots Syst. (2017), pp. 3803-3810
L. Bauwens, M. Lubrano, J.-F. Richard. Bayesian Inference in Dynamic Econometric Models. Oxford University Press, Oxford (2000)
D. Bennett, D. Miljak, J. Khachan. The measurement of chalcopyrite content in rocks and slurries using magnetic resonance. Miner. Eng., 22 (9–10) (2009), pp. 821-825
M. Betancourt. A conceptual introduction to Hamiltonian Monte Carlo. ArXiv Preprint (2017)
N. Bozorgzadeh, R.J. Bathurst. Hierarchical Bayesian approaches to statistical modelling of geotechnical data. Georisk., 16 (3) (2022), pp. 452-469
S. Brooks. Markov chain Monte Carlo method and its application. Journal of the Royal Statistical Society, Series D (The Statistician), 47 (1) (1998), pp. 69-100
S. Chib. Markov chain Monte Carlo methods: computation and inference. Handb. Econom., 5 (2001), pp. 3569-3649
S. Chib, E. Greenberg. Understanding the metropolis-hastings algorithm. The Am. Stat., 49 (4) (1995), pp. 327-335
P. Coghill, D. Miljak, E. Williams. Consequences of fractal grade distribution for bulk sorting of a copper porphyry deposit. Geosci. Front., 6 (4) (2015), pp. 477-480
M. Cubillos, J.N. Wulff, S. Wøhlk. A multilevel Bayesian framework for predicting municipal waste generation rates. Waste Manage., 127 (2021), pp. 90-100
M. de la Varga, J.F. Wellmann. Structural geologic modeling as an inference problem: a Bayesian perspective. Interpretation, 4 (3) (2016), pp. SM1-SM16
S. Duane, A.D. Kennedy, B.J. Pendleton, D. Roweth. Hybrid monte carlo. Phys. Lett. B., 195 (2) (1987), pp. 216-222
B. Efron. Bayes’ theorem in the 21st century. Science, 340 (6137) (2013), pp. 1177-1178
S. Elias, D. Alderton. Encyclopedia of Geology. Academic Press (2020)
A.M. Ellison. An introduction to Bayesian inference for ecological research and environmental decision-making. Ecol. Appl., 6 (4) (1996), pp. 1036-1046
A.M. Ellison. Bayesian inference in ecology. Ecol. Lett., 7 (6) (2004), pp. 509-520
X. Emery, J.M. Ortiz. Estimation of mineral resources using grade domains: critical analysis and a suggested methodology. J. South. Afr. Inst. Min. Metall., 105 (4) (2005), pp. 247-255
Y. Feng, K. Gao, A. Mignan, J. Li. Improving local mean stress estimation using Bayesian hierarchical modelling. Int. J. Rock Mech. Min. Sci., 148 (2021), Article 104924
M.F. Gazley, C.M. Tutt, L.A. Fisher, A.R. Latham, G. Duclaux, M.D. Taylor, S.J. de Beer. Objective geological logging using portable XRF geochemical multi-element data at Plutonic Gold Mine, Marymia Inlier, Western Australia. J. Geochem. Explor., 143 (2014), pp. 74-83
A. Gelman. Multilevel (hierarchical) modeling: what it can and cannot do. Technometrics, 48 (3) (2006), pp. 432-435
A. Gelman, J. Hill. Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press (2006)
A. Gelman, D.B. Rubin. Inference from iterative simulation using multiple sequences. Stat. Sci., 7 (4) (1992), pp. 457-472
A. Gelman, K. Shirley. Inference from simulations and monitoring convergence. Handbook of Markov Chain Monte Carlo., 6 (2011), pp. 163-174
J. Geweke. Bayesian inference in econometric models using Monte Carlo integration. Econometrica: Journal of the Econometric Society (1989), pp. 1317-1339
J. Geweke. Using simulation methods for Bayesian econometric models: inference, development, and communication. Econom. Rev., 18 (1) (1999), pp. 1-73
W.K. Hastings. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57 (1) (1970), pp. 97-109
R. Herrington. Road map to mineral supply. Nat. Geosci., 6 (11) (2013), pp. 892-894
M.D. Hoffman, A. Gelman. The No-U-Turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. J. Mach. Learn. Res., 15 (1) (2014), pp. 1593-1623
A.A. Johnson, M.Q. Ott, M. Dogucu. Bayes Rules!: An Introduction to Applied Bayesian Modeling. CRC Press (2022)
T.-J. Kim, H.-H. Kwon, C. Lima. A Bayesian partial pooling approach to mean field bias correction of weather radar rainfall estimates: Application to Osungsan weather radar in South Korea. J. Hydrol., 565 (2018), pp. 14-26
D. Maulud, A.M. Abdulazeez. A review on linear regression comprehensive in machine learning. Journal of Applied Science and Technology Trends, 1 (4) (2020), pp. 140-147
R. McElreath. Statistical Rethinking: A Bayesian Course with Examples in R and STAN. CRC Press (2020)
N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller. Equation of state calculations by fast computing machines. J. Chem. Phys., 21 (6) (1953), pp. 1087-1092
C.C. Monnahan, J.T. Thorson, T.A. Branch. Faster estimation of Bayesian models in ecology using Hamiltonian Monte Carlo. Methods in Ecol. Evol., 8 (3) (2017), pp. 339-348
J.L. Peugh. A practical guide to multilevel modeling. J. Sch. Psychol., 48 (1) (2010), pp. 85-112
S.S. Qian, T.F. Cuffney, I. Alameddine, G. McMahon, K.H. Reckhow. On the application of multilevel modeling in environmental and ecological studies. Ecology, 91 (2) (2010), pp. 355-361
C. Robben, H. Wotruba. Sensor-based ore sorting technology in mining—past, present and future. Minerals, 9 (9) (2019), p. 523
J. Salvatier, T.V. Wiecki, C. Fonnesbeck. Probabilistic programming in Python using PyMC3. PeerJ Comput. Sci., 2 (2016), p. e55
S. Sharma. Markov chain Monte Carlo methods for Bayesian data analysis in astronomy. Annu. Rev. Astron. Astrophys., 55 (2017), pp. 213-259
R.H. Sillitoe. Porphyry copper systems. Econ. Geol., 105 (1) (2010), pp. 3-41
H. Talebi, L.J.M. Peeters, A. Otto, R. Tolosana-Delgado. A truly spatial random forests algorithm for geoscience data analysis and modelling. Math. Geosci., 54 (2022), pp. 1-22
E. Thrane, C. Talbot. An introduction to Bayesian inference in gravitational-wave astronomy: parameter estimation, model selection, and hierarchical models. Publ. Astron. Soc. Aust., 36 (2019), p. e010
E. Thrane, C. Talbot. An introduction to Bayesian inference in gravitational-wave astronomy: parameter estimation, model selection, and hierarchical models—Corrigendum. Publ. Astron. Soc. Aust., 37 (2020), p. e036
Tirumala, S.S., Narayanan, A., 2015. Hierarchical data classification using deep neural networks. Neural Information Processing: 22nd International Conference., ICONIP 2015, Istanbul, Turkey, November 9-12, 2015, Proceedings, Part I 22, 492–500.
R. Van de Schoot, D. Kaplan, J. Denissen, J.B. Asendorpf, F.J. Neyer, M.A.G. Van Aken. A gentle introduction to Bayesian analysis: applications to developmental research. Child. Dev., 85 (3) (2014), pp. 842-860
D. Van Ravenzwaaij, P. Cassey, S.D. Brown. A simple introduction to Markov Chain Monte-Carlo sampling. Psychon. Bull. Rev., 25 (1) (2018), pp. 143-154
Q.-H. Vuong, V.-P. La, M.-H. Nguyen, M.-T. Ho, T. Tran, M.-T. Ho. Bayesian analysis for social data: a step-by-step protocol and interpretation. MethodsX., 7 (2020), Article 100924
S. Wang, X. Sun, U. Lall. A hierarchical Bayesian regression model for predicting summer residential electricity demand across the USA. Energy, 140 (2017), pp. 601-611
J.F. Wellmann, M. De La Varga, R.E. Murdie, K. Gessner, M. Jessell. Uncertainty estimation for a geological model of the Sandstone greenstone belt, Western Australia–insights from integrated geological and geophysical inversion in a Bayesian inference framework. Geol. Soc., London, Spec. Publ., 453 (1) (2018), pp. 41-56
Y. Xiong, R. Zuo, E.J.M. Carranza. Mapping mineral prospectivity through big data analytics and a deep learning algorithm. Ore Geol. Rev., 102 (2018), pp. 811-817
B. Yin, R. Zuo, Y. Xiong. Mineral prospectivity mapping via gated recurrent unit model. Nat. Resour. Res. (2022), pp. 1-15
Young, C.A., 2019. SME Mineral Processing and Extractive Metallurgy Handbook. Society for Mining, Metallurgy & Exploration.
C. Youngflesh. MCMCvis: tools to visualize, manipulate, and summarize MCMC output. J. Open. Source. Softw., 3 (24) (2018), p. 640

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