Quantification of geological uncertainty and mine planning risk using metric spaces

Abstract

Major sources of financial risk for mining projects include geological uncertainty and uncertainty in future commodity prices, costs, demand levels and interest rates. Geological uncertainty is difficult to model as there are complicated spatial considerations which are not present in the other sources of uncertainty. Stochastic simulations are now the common approach to assessing geological uncertainty, and one of the most common practical methods of producing realisations is conditional sequential simulation. Conditional sequential simulation algorithms can create multiple realisations that honour the original histogram and covariance matrix. One of the shortcomings of the conditional simulation algorithms is that there is no parameter that can provide further information about high order statistics for generated realisations. By visually comparing the colour realisation images (if they are 2D), we can easily see uncaptured spatial differences; therefore, any possible dissimilarity or similarity between the realisations cannot be captured by descriptive geostatistics. Distance computation as a technique to measure dissimilarity or similarity between images, objects, and models has received attention in recent years. This thesis presents a formal measure of dissimilarity for generated realisations by adapting the Kantorovich metric to the geostatistics context. We propose a new methodology for mapping the space of uncertainty by a distance function that is based upon a physically meaningful notion of dissimilarity between pairs of realisations. We are able to quantify the dissimilarity of different realisations. In this framework, the pairwise dissimilarities between realisations can be used to make a relation or a precise mathematical structure between them, which can describe the variability of parameters on interest (for example, grade) inside the space of uncertainty. This method provides a powerful tool to address how realisations are connected to each other and how this connection (structure) can answer some controversial questions in geostatistical simulations.

Furthermore, the mining processes such as mine optimisation, open pit design and long term scheduling are only able to handle relatively modest numbers of realisations. It is difficult to say how many realisations are required to achieve a prescribed level of accuracy based on a very large number of possible realisations. This method has the ability to construct a collection of schedules (coming from generated realisations) so that the overall uncertainty is captured in a way prescribed by the user. We argue that this small set of candidate schedules produce more robust outcomes than schedules selected by other existing risk-based approaches.