Two approaches to direct block-support conditional co-simulation $ Xavier Emery a,b,n , Julia ´ n M. Ortiz a,b a Department of Mining Engineering, University of Chile, Avenida Tupper 2069, Santiago 837 0451, Chile b ALGES Laboratory, Advanced Mining Technology Center, University of Chile, Beauchef 850, Santiago 837 0451, Chile article info Article history: Received 29 January 2010 Received in revised form 23 July 2010 Accepted 26 July 2010 Available online 7 November 2010 Keywords: Support effect Change of support Linear model of coregionalization Sequential Gaussian simulation Spectral simulation Fast Fourier transform Discrete Gaussian model abstract Change of support is a common issue in the geosciences when the volumetric support of the available data is smaller than that of the blocks on which numerical modeling is required. In this paper, we present two algorithms for the direct block-support simulation of cross-correlated random fields that are monotonic transforms of stationary Gaussian random fields. The first algorithm is a variation of sequential Gaussian co-simulation, in which each block value is simulated in turn, conditionally to the original data and to the previously simulated block values, while the second algorithm is based on spectral co-simulation in the framework of the discrete Gaussian change-of-support model. These two algorithms are implemented in computer programs and applied to a synthetic case study and to a mining case study. Their properties and performances are compared and discussed. & 2010 Elsevier Ltd. All rights reserved. 1. Introduction The numerical modeling of regionalized variables in the geos- ciences faces the problem that, most often, the volumetric support of the available samples is several orders of magnitude smaller than that at which modeling is required. In mining, drill hole samples are cylindrical volumes a few inches in diameter and several meters long, whereas selective blocks may be of thousands of cubic meters. Likewise, the volumetric support required for flow simulation or reservoir modeling is much larger than the cores obtained from wells. When passing from the quasi-point support of the samples to the block volume, a model is needed in order to account for the change in the statistical properties of the variable(s) under study. The traditional approach for constructing numerical models at a block support is to discretize each block into a finite set of points, perform point-support simulation, and average the point-support values simulated within each block (Verly, 1984; Journel and Kyriakidis 2004). The discretization must be at a sufficient resolution to approximate the average of the whole set of point-support values within the block support. Although simple and appealing, this approach is demanding in terms of CPU time when large domains (several millions of blocks) are considered and/or when a fine resolution is needed for block discretization (e.g., for large block supports). Memory storage problems also arise for simulation algorithms in which all the point-support values must be kept in memory, such as the LU decomposition of covariance matrix (Davis, 1987), the discrete spectral (Chil  es and Delfiner, 1997), and the sequential Gaussian algorithm with a random visiting path (Deutsch and Journel, 1992). In this context, the goals of this paper are threefold: (1) to propose algorithms for directly simulating one or more cross- correlated variables at a block support; (2) to illustrate their applicability, strengths, and weaknesses through examples and case studies; and (3) to provide practical considerations and computer codes for practitioners. Specifically we will consider a variation of the sequential Gaussian algorithm, as well as a discrete spectral algorithm in which an explicit change-of-support model (the discrete Gaussian model) is incorporated within the simula- tion procedure. So far, such a change-of-support model has been essentially associated with sequential and turning bands simula- tion in the univariate context (Marcotte, 1994; Emery, 2009); here it will be extended to the multivariate context and associated with another (discrete spectral) algorithm. The proposed approaches allow reducing CPU time and memory storage requirements when a block-support simulation is considered, with minimal loss of accuracy in comparison with traditional point-support simulation followed by block averaging. 2. Block-support sequential simulation 2.1. Principle Consider a regionalized variable, viewed as a realization of a real-valued random field Z that is a monotonic transform of Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/cageo Computers & Geosciences 0098-3004/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cageo.2010.07.012 $ Code available from server at http://www.iamg.org/CGEditor/index.htm. n Corresponding author at: Department of Mining Engineering, University of Chile, Avenida Tupper 2069, Santiago 837 0451, Chile. Tel.: +56 2 978 4498; fax: + 56 2 978 4985. E-mail address: xemery@ing.uchile.cl (X. Emery). Computers & Geosciences 37 (2011) 1015–1025