Truncation Trees in Hierarchical Truncated PluriGaussian Simulation

Abstract

Geological uncertainty is a major source of risk in resource projects and must be characterized. Extensive research has been undertaken in the development of sophisticated techniques given the economic impact of this uncertainty. The Hierarchical Truncated PluriGaussian (HTPG) simulation is well‑known among existing categorical modeling techniques for its ability to portray realistic features. HTPG uses latent Gaussian variables to simulate categories. The rules to map continuous variables to categories, and to introduce juxtaposition constraints are key and known as truncation trees. The number of truncation trees is large, and their structures are very flexible, however, choosing the correct one is a daunting task. This work focuses on developing tools to choose the truncation tree that leads to the optimal model. The process starts with the inference of possible trees from any source of limited data including drillholes, point samples, or images. Then, a tree is chosen based on measures of goodness. This research has made several contributions. First, it presented tools to enumerate all possible trees in friendly plots given the number of categories. Second, it introduced interval probabilities to quantify the geometrical associations of categories and proposed a dissimilarity matrix based on this concept to summarize the associations and to be used in inference algorithms. Third, it implemented Single Linkage Cluster Analysis (SLCA), and spectral partitioning from graph theory for the inference of trees. Lastly, it showed an optimization framework in a synthetic example that used all trees to recommend transition probabilities as an appropriate measure of goodness. The tools and methodologies were used in a case study where the chosen tree obtained good metrics and respected the geological understanding.