On the construction of discrete distributions from training images

In reservoir characterisation is has become common practice to define a facies model by fitting a model to one or more training images. Particularly multi-point statistics models are popular. Then node values are simulated sequentially in a random order and conditional distributions for one node value given the previous ones are estimated from the training image(s). This is often combined with a multi-grid approach. In the present paper we focus on binary training images and consider a simplified variant of the multi-point statistics framework. We simulate the node values sequentially in a fixed order and do not include the multi-grid idea. The resulting model is a (higher-order) Markov chain and corresponding theory can be used to understand the properties of the fitted model. In particular the Markov chain is in general not stationary, from which it follows for example that the marginal probabilities vary spatially. This is clearly an unwanted property and we discuss several strategies for coping with the problem. One should note that by restricting us to a fixed simulation order we obtain explicit expressions for the probability distribution of the fitted model, which makes it easy to define a corresponding conditional distribution when observed data are available.