Constraining distance-based multipoint simulations to proportions and trends Gregoire Mariethoz a, b, c, * , Julien Straubhaar a , Philippe Renard a , Tatiana Chugunova d , Pierre Biver d a University of Neuch^ atel, Centre of Hydrogeology and Geothermics, Neuchatel, Switzerland b University of Lausanne, Institute of Earth Surface Dynamics, Switzerland c The University of New South Wales, School of Civil and Environmental Engineering, Sydney, Australia d TOTAL SA, CTJS Avenue Larribau, 64000, Pau, France Keywords: Training image Probability Non-stationarity Spatial modeling Geostatistics abstract In the last years, the use of training images to represent spatial variability has emerged as a viable concept. Among the possible algorithms dealing with training images, those using distances between patterns have been successful for applications to subsurface modeling and earth surface observation. However, one limitation of these algorithms is that they do not provide a precise control on the local proportion of each category in the output simulations. We present a distance perturbation strategy that addresses this issue. During the simulation, the distance to a candidate value is penalized if it does not result in proportions that tend to a target given by the user. The method is illustrated on applications to remote sensing and pore-scale modeling. These examples show that the approach offers increased user control on the simulation by allowing to easily impose trends or proportions that differ from the pro- portions in the training image. 1. Introduction Multiple-point geostatistics (MPS) has emerged in the last years as a family of stochastic simulation tools that can be used in various areas of earth systems imaging. Although the original application domain focused on modeling the internal structures of subsurface deposits ranging from pore-scale (El Ouassini et al., 2008; Okabe and Blunt, 2007; Tahmasebi and Sahimi, 2013; Zhang et al., 2006a) to reservoir-scale (Huysmans and Dassargues, 2012; Ronayne et al., 2008; Yin, 2013), it has thereafter been extended to very different fields such as mining (Goodfellow et al., 2012; Rezaee et al., 2014), soil science (Meerschman et al., 2014), remote sensing (Boucher, 2009; Ge and Bai, 2011; Jha et al., 2013; Stisen et al., 2011; Vannametee et al., 2014), for modeling the occurrence of rainfall (Oriani et al., 2014; Wojcik et al., 2009), and even in medical imaging (Pham, 2012). Several multiple-point simulation methods have been devel- oped over the last decade. MPS algorithms can be classified in two categories, based on their underlying statistical approach. The first category comprises methods that are in the continuity of classical variogram-based geostatistics such as SGS (Deutsch and Journel, 1998) where a probability distribution is inferred for each grid node, which is subsequently sampled for a simulated value. This is the case of the first available MPS method, SNESIM (Strebelle, 2002), as well as its memory-efficient implementation, IMPALA (Straubhaar et al., 2011). One important advantage of computing local probability density functions is that it is possible to perturb them in order to influence the sampling. SNESIM and IMPALA allow using probability aggregation methods (Allard et al., 2012; Krishnan, 2008) to update, on-the-fly, the local probability den- sity functions (pdfs) for each simulated node. In the context of the patchwork simulation (El Ouassini et al., 2008), a method based on the perturbation of the transition probabilities between patches has been proposed by Faucher et al. (2014). Approaches based on the perturbation of a probability allow the user to impose two specific constraints on the resulting simulations: 1) To control the global proportions of a categorical variable (also known as servo-system). 2) To increase or decrease locally the proportion of a given facies, therefore allowing to generate non-stationary models. * Corresponding author. University of Lausanne, Institute of Earth Surface Dynamics, UNIL-Mouline, Geopolis, 1015 Lausanne, Switzerland E-mail address: gregoire.mariethoz@unil.ch (G. Mariethoz). Published in Environmental Modelling & Software 72, 184-197, 2015 which should be used for any reference to this work 1