Conditional simulation of categorical spatial variables using Gibbs sampling of a truncated multivariate normal distribu
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ORIGINAL PAPER
Conditional simulation of categorical spatial variables using Gibbs sampling of a truncated multivariate normal distribution subject to linear inequality constraints Francky Fouedjio1
•
Celine Scheidt2 • Liang Yang1 • Yizheng Wang1 • Jef Caers1
Accepted: 23 October 2020 Ó Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper introduces a method to generate conditional categorical simulations, given an ensemble of partially conditioned (or unconditional) categorical simulations derived from any simulation process. The proposed conditioning method relies on implicit functions (signed distance functions) for representing the categorical spatial variable of interest. Thus, the conditioning problem is reformulated in terms of signed distance functions. The proposed approach combines aspects of principal component analysis and Gibbs sampling to achieve the conditioning of the unconditional categorical realizations to the data. It is applied to synthetic and real-world datasets and compared to the traditional sequential indicator simulation. It appears that the proposed simulation technique is an effective method to generate conditional categorical simulations from a set of unconditional categorical simulations. Keywords Categorical spatial variable Conditional simulation Gibbs sampler Implicit function Principal component analysis
1 Introduction Conditional simulations of categorical spatial variables in geostatistics are used to quantify spatial uncertainty relevant to variety of applications, such as environmental, groundwater, mineral, and oil/gas (Mariethoz and Caers 2014; Armstrong et al. 2011; Chiles and Delfiner 2012; Lantuejoul 2002; Deutsch 2002; Goovaerts 1998). Methods can be pixel-based, object-based or surface-based, although eventually all the results are rastered on a discrete mesh. In terms of pixel-based (or mesh-based) methods, one has variogram-based methods (Journel 1983; Deutsch 2006; Emery 2007), Markov-random field methods (Li 2007; Daly 2005; Elfeki and Dekking 2001; Tjelmeland and Besag 1998), and multiple-point-geostatistics methods
& Francky Fouedjio [email protected] 1
Department of Geological Sciences, Stanford University, 367 Panama Street, Stanford, CA 94305, USA
2
Department of Energy Resources Engineering, Stanford University, 367 Panama Street, Stanford, CA 94305, USA
(Strebelle 2002; Zhang et al. 2006; Arpat and Caers 2007; Mariethoz et al. 2010; Honarkhah and Caers 2010). Other works that deal with the problem of simulation of categorical spatial variables include methods based on spin models and maximum entropy (Zˇukovicˇ and Hristopulos 2009; Bogaert and Gengler 2018). Conditioning to exact observations (hard data) is often easily achieved, simply because the central value to be simulated on a mesh is taken to be the same support as the hard data. Conditioning in object-based or surface-based methods is more challenging, since an object or surface is represented as a shape or geometry rather than a regular mesh. A
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