Copula-based spatial modelling of geometallurgical variables

The most important aspect of modelling a geological variable, such as metal grade, is the spatial correlation. Spatial correlation describes the relationship between realisations of a geological variable sampled at different locations. Any method for spatially modelling such a variable should be capable of accurately estimating the true spatial correlation. Conventional kriged models are the most commonly used in mining for estimating grade or other variables at unsampled locations, and these models use the variogram or covariance function to model the spatial correlations in the process of estimation. However, this usage assumes the relationships of the observations of the variable of interest at nearby locations are only influenced by the vector distance between the locations. This means that these models assume linear spatial correlation of grade. In reality, the relationship with an observation of grade at a nearby location may be influenced by both distance between the locations and the value of the observations (ie non-linear spatial correlation, such as may exist for variables of interest in geometallurgy). Hence this may lead to inaccurate estimation of the ore reserve if a kriged model is used for estimating grade of unsampled locations when nonlinear spatial correlation is present. Copula-based methods, which are widely used in financial and actuarial modelling to quantify the non-linear dependence structures, may offer a solution. This method was introduced by Bárdossy and Li (2008) to geostatistical modelling to quantify the non-linear spatial dependence structure in a groundwater quality measurement network. Their copula-based spatial modelling is applied in this research paper to estimate the grade of 3D blocks. Furthermore, real-world mining data is used to validate this model. These copula-based grade estimates are compared with the results of conventional ordinary and lognormal kriging to present the reliability of this method.