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

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.The variations in metal grade relationships occur due to the transition into different geological domains. This raises two questions - how to define these geological domains and how the metal grade relationship is influenced by these geological domains. In this paper, linear relationship is assumed between Cu grade and other metal grades. We applies a Bayesian hierarchical (partial-pooling) model to quantify the linear relationships between Cu, Au, and Fe grades from geochemical bore core data. The hierarchical model was compared with two other models - 'complete-pooling' model and 'no-pooling' model. Mining blocks were split based on spatial domain to construct hierarchical model. Geochemical bore core data records metal grades measured from laboratory assay with spatial coordinates of sample location. Two case studies from different porphyry Cu deposits were used to evaluate the performance of the hierarchical model. Markov chain Monte Carlo (MCMC) was used to sample the posterior parameters. Our results show that the Bayesian hierarchical model dramatically reduced the posterior predictive variance for metal grades regression compared to the no-pooling model. In addition, the posterior inference in the hierarchical model is insensitive to the choice of prior. The data is well-represented in the posterior which indicates a robust model. The results show that the spatial domain can be successfully utilised for metal grade regression. Uncertainty in estimating the relationship between pay metals and both secondary and gangue elements is quantified and shown to be reduced with partial-pooling. Thus, the proposed Bayesian hierarchical model can offer a reliable and stable way to monitor the relationship between metal grades for ore sorting and other mineral processing options.