The sequential spectral turning band simulator as an alternative to Gibbs sampler in large truncated- or pluri- Gaussian
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ORIGINAL PAPER
The sequential spectral turning band simulator as an alternative to Gibbs sampler in large truncated- or pluri- Gaussian simulations Dany Lauzon1
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Denis Marcotte1
Ó Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The Sequential Spectral Turning Bands Method (S-STBM) builds Gaussian random fields (GRF) calibrated to desired response functions. An interesting application of S-STBM concerns the simulation of GRF subject to inequality constraints. S-STBM works by choosing the phase of each cosine function of the STBM algorithm instead of perturbating nodes of the GRF many thousand times using conditional distributions as in Gibbs sampler. Each chosen phase increasingly constrains the nodes to the desired inequalities. A method based on the sequential Gaussian simulation is introduced to accelerate convergence at the end of the process. Examples shown compare S-STBM approach to Gibbs sampler. Orders of magnitude reduction in computation time is achieved with our spectral method. Furthermore, examples show that the phase selection has no significant influence on the spatial correlation. Our approach is easily generalized to pluriGaussian simulations. Compared to Gibbs sampler, S-STBM is not limited to small systems (no memory limitation) and its complexity of O(n) makes it an efficient tool to simulate large GRF subject to inequality constraints. Keywords Truncated Gaussian random field Spectral simulation Sequential Gaussian simulation Gibbs Sampler PluriGaussian simulation Inequality constraints
1 Introduction Geostatistical simulations of categorical fields are used to assess uncertainties of geological facies for petroleum reservoir or mineral deposits modelling. The facies identification is often the most influential step in the evaluation of the uncertainty of a reservoir or mining deposit. As example, facies connectivity is often the major control for flow simulation in reservoirs. Similarly, size, morphology and spatial relations of geological units are directly influential in determining optimal cut-off grade (Armstrong et al. 2011; Refsgaard et al. 2012; Talebi et al. 2013; Zhang 2015; Jakeman et al. 2016). The calibration of categorical fields to dynamic data typically requires to associate one or many latent Gaussian variables to the categorical one (Ravalec-Dupin and Hu & Dany Lauzon [email protected] 1
Civil, Geological and Mining Department, Polytechnique Montre´al, C.P. 6079 Succ. Centre-ville, Montre´al, Qc H3C 3A7, Canada
2005; Hu et al. 2013; Rezaee and Marcotte 2018) as it is much easier to directly perturb continuous variables than discrete ones. The latent GRFs are the basis of methods like the truncated Gaussian and the PluriGaussian methods (Allard 1994; Galli et al. 1994; Armstrong et al. 2011; Allard et al. 2012; Deutsch and Deutsch 2014; Madani and Emery 2014; Desassis et al. 2019). They were also used for calibration of categorical fields generated by multipoints (Rezaee and Marcotte 2018). The bottleneck of these methods is the genera
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