Block Kriging for geometallurgical optimization

Adpative processing is often related to the mineral composition of the mining block currently processed. The portions of waste, ore and secondary product minerals considered are always positive and typically sum to 100%, i.e. a composition. A typical solution to predict block averages of spatially dependent quantities would be block kriging. However, kriging is based on spatial correlations and it has been repeatedly shown that correlations of compositional data are spurious and blurred by the constant sum constraint.

Aitchison (1982) proposed a general solution for compositional problems, based on transforming the compositional data to a set of logarithms of ratios of components, and analysing the transformed scores. The transform is chosen to be invertible to ensure that no information is lost in the process. This methodology avoids spurious correlation problems and ensures coherence between results obtained with different subcompositions. Pawlowsky-Glahn and Olea (2004) already adapted the approach for point-support geostatistics, allowing the prediction of point compositions from spatially neighbouring data, without the artefacts induced by using standard multivariate cokriging. Unfortunately upscaling the results of this approach with block kriging is not straightforward, because of the nonlinearity of logratio transforms. Furthermore, we have to consider those nonlinearities created by the dependence on the block composition of extraction efficiency and processing costs of the possible processing choices.

This paper proposes a solution for predicting the conditional expectations of the benefit of processing the block with each processing choice using geostatistical simulations of the local values of the composition and the resulting block integrals. The computation for a large number of blocks can be done efficiently using Cholesky decomposition. The approach also allows the calculation of prediction errors for expected compositions and expected benefit at almost no additional computational cost. Such compositional approach is necessary, e.g.: when processing choices depend on proportions of certain minerals after other components have been removed; or if processing alters the composition in a multiplicative way, by partially removing a portion of some components.