A New Efficient Joint Simulation Framework and Application in a Multivariable Deposit A Boucher 1 and R Dimitrakopoulos 2 ABSTRACT Ore mineralisations frequently contain more than one mineral or element of interest that are spatially related. As a result, they require the use of joint geostatistical simulation techniques that generate models conserving this correlation. Although joint simulation methods have long been available, they are impractical when it comes to more than two variables and mid to large size deposits. This paper presents a new framework for joint conditional simulations of non-Gaussian vectors of variables and stresses, in particular, the joint simulation directly at the block-support scale. The proposed framework is based on minimum/maximum autocorrelation factors (MAF) that decorrelate variables at all lags, thus allowing the simulation of independent variables. The MAF approach is combined with the direct block simulation framework presenting a new algorithm termed ‘DBMAFSIM’. It permits computationally efficient joint simulation of large, multivariable deposits. The proposed method is then applied at the Yandi iron ore deposit in Western Australia, with five major correlated attributes being successfully simulated and validated on block support. The application shows the efficiency and excellent performance of the method. INTRODUCTION Geostatistical simulation methods are used to quantify geological uncertainty in mineral deposits and assess risk in various aspects of mining project development and operation. Although methods for simulating individual attributes are generally efficient (see Benndorf and Dimitrakopoulos, 2007, this volume), existing methods for jointly modelling multivariable deposits are in practice limited, particularly when dealing with medium to large deposits having more than two attributes of interest. For example, a realistic model of an iron deposit must account for silica, alumina and phosphorus in addition to iron, and reproduce the joint local variability of the attributes of interest. Thus, there is a need to consider new efficient and practical joint simulation methods. Available approaches for joint simulation include early developments based on the model of linear coregionalisation and conditioning of simulated correlated fields (Chilès and Delfiner, 1999); extension of the conditional univariate LU decomposition method of Davis (1987) to joint simulation (Myers, 1988); and combination of the LU vector simulation and sequential simulation for large joint simulation of two variables (Verly, 1993), a method which becomes cumbersome for more than two variables. The major drawback with the above methods and their variations is that they require considerable computer processing capacity to solve the large systems of equations per simulated node, in addition to the inference of cross-variograms, and issues arising from data management. An alternative to the impractical common joint simulation methods is to factorise the variables involved to uncorrelated (orthogonal) factors that can be simulated independently of each other. Subsequent back- transformation of simulated factors to simulated realisations of variables aims to indirectly restore the histograms, variograms and cross-variograms of the data in the respective realisations. This type of a factorisation approach is introduced by David et al (1984) as a principal component analysis (PCA) data transformation (David, 1988; Suro-Perez and Journel, 1991). However, this transformation decorrelates variables only at lag zero, and is limited in practice (Wackernagel, Petitgas and Touffait, 1989; Goovaerts, 1993). Desbarats and Dimitrakopoulos (2000) and Dimitrakopoulos (in press) present a major improvement of the PCA approach to joint simulation of multiple variables by replacing PCA with the minimum/maximum autocorrelation factors (MAF), a factorisation method originally developed for remote-sensing applications (Switzer and Green, 1984). The advantage of MAF is that it produces uncorrelated factors at all lags, when the variogram model of the related variables follows the linear model of coregionalisation with two structures. The method gives access to a substantially wider range of variables that can be jointly simulated than is possible with PCA factorisation. Joint simulations of mineral deposits based on MAF are shown to be effective, relatively efficient, flexible and practical (eg Boucher, 2003; Dimitrakopoulos and Fonseca, 2003). The efficiency of joint simulation with MAF could be further enhanced if it were possible to simulate directly on a block-support scale. The block support on which an orebody is being numerically represented and modelled differs from the support size of the available data, thus requiring modelling and change of support. The current approach to change of support is safe but cumbersome. It consists of simulating points and then averaging them to the blocks needed, which has two computational drawbacks. First, the algorithm needs to process, store and manage large sets of data and files (several gigabytes). Second, the algorithm involves an additional operation (averaging) that can be time-consuming for large orebodies. An alternative simulation method is proposed by Godoy (2003), and it is termed ‘direct block simulation’. The method minimises the information stored in memory by retaining in memory only block values, a procedure that significantly speeds up the simulation process and also reduces the size of the output files, facilitating efficiency in data storage and management. Advantages of the method are that there is no assumption for change of support, and it is substantially more efficient than other existing methods (Benndorf and Dimitrakopoulos, 2007, this volume). The direct block simulation can be extended to the joint direct block simulation of multiple variables using MAF, an approach shown to be very efficient and effective (Boucher, 2003). This paper focuses on a new and efficient joint simulation framework. First, it outlines the MAF approach to joint geostatistical simulations at the conventional point-support scale and, subsequently, shows the extension of the approach to the direct joint simulation at the block-support scale. An application at the Yandi Central 1 iron ore deposit, Western Australia, follows and shows the joint simulation of iron content, silica, alumina, phosphorus and loss on ignition, directly at the block-support scale. Comments on the performance of the approach and conclusions follow. Orebody Modelling and Strategic Mine Planning Spectrum Series Volume 14 345 1. Department of Geological and Environmental Sciences, Stanford University, 450 Serra Mall, Braun Hall, Building 320, Stanford CA 94305-2115, USA. Email: aboucher@pangea.stanford.edu 2. MAusIMM, COSMO Laboratory, Department of Mining, Metals and Materials Engineering, McGill University, Frank Dawson Adams Building, Room 107, 3450 University Street, Montreal QC H3A 2A7, Canada. Email: roussos.dimitrakopoulos@mcgill.ca