Scenario Reduction Applied to Geostatistical Simulations
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Scenario Reduction Applied to Geostatistical Simulations Margaret Armstrong · Aziz Ndiaye · Rija Razanatsimba · Alain Galli
Received: 28 August 2010 / Accepted: 23 August 2012 / Published online: 15 September 2012 © International Association for Mathematical Geosciences 2012
Abstract Having a large number of geostatistical simulations of a mineral or petroleum deposit provides a better idea of its upside potential and downside risk; however, large numbers of simulated realizations of a deposit may pose computational difficulties in subsequent decision-making phases. Hence, depending on the specific case, there can be a need to select a representative subset of conditionally simulated deposit realizations. This paper examines and extends an approach developed by the stochastic optimization community based on stochastic mathematical programming with recourse and is discussed here in the context of mineral deposits while it is possibly suitable for other earth science applications. The approach is based on measuring the “distance” between simulations and the introduced distance measure between simulated realizations of a mineral deposit is based on the metal above a given set of cutoff grades while a pre-existing mine design is available. The approach is tested on 100 simulations of the Walker Lake data with promising results. Keywords Stochastic optimization · Multi-stage programming with recourse 1 Introduction Over the past two decades, computer power has increased enormously and it is now possible to generate many conditional simulations of orebodies. For open-pit mines, the next steps in evaluating an orebody are to optimize the pit contour and determine the optimum schedule then to calculate the net present value (NPV). The advantage of having a large number of simulations is that it provides a better idea of the upside potential and the downside risk. Unfortunately, there is no way of post-processing all of the simulations that can be generated. Therefore, our M. Armstrong () · A. Ndiaye · R. Razanatsimba · A. Galli Cerna, Mines-Paris, 60 Boulevarde Saint-Michel, 75272 Paris, France e-mail: [email protected]
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Math Geosci (2013) 45:165–182
Fig. 1 Binomial tree with three stages (time periods)
objective is to reduce the set of simulations to a more manageable size by selecting a representative subset of them and of associating probabilities. Different approaches to this problem exist in the literature. On the petroleum geosciences side, Scheidt and Caers (2009a, 2009b) propose an approach based on kernel clustering that is designed for the oil industry and with fluid flow problems in mind. Amongst the stochastic optimization community, a very similar problem arises when multistage programming with recourse is used (Sahinidis 2004; van der Vlerk 2007; Beasley 2010). In this method revisited in Appendix A, a distinction is made between the state variables or those outside the control of the decision-makers, and the control variables. The evolution of the state variables is represented b
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