Advances in Geostatistical Modeling of Categorical Variables

Abstract

In the mining and petroleum industries, subsurface resources are modeled using data sets obtained from widely spaced drilling. It is crucial to optimize the use of available information from these data sets while reducing the amount required to adequately assess risks and uncertainties. Typically, these data sets include spatially correlated categorical and continuous variables. Geostatistics is commonly applied to model these types of spatial variables. Categorical variables are modeled first to establish stationary domains for continuous variables like ore grades and, therefore, are essential for the accurate modeling of continuous variables. Techniques such as object-based models (Lantuéjoul, 2002), sequential indicator simulation (A. G. Journel & Alabert, 1990), multiple-point statistics (S. Strebelle, 2002), truncated Gaussian and pluri- Gaussian simulation (Armstrong et al., 2011; Matheron et al., 1987), and hierarchical truncated pluri-Gaussian simulation (D. Silva, 2018) have been developed to model and characterize geological uncertainties. Modern approaches in multivariate modeling of continuous variables include principal component analysis (PCA) for dimension reduction and decorrelation. Suro (1988) applied PCA to decorrelate indicators and used kriging, followed by a back transformation, to derive the conditional cumulative distribution function. This research aims to obtain a greater understanding of the characteristics of categorical indicator variables. Studying the characteristics of categorical indicator variograms led to the demonstration that the nugget effect must be zero. Additionally, correlograms are shown to be a robust alternative in the presence of clustered data. This work also compares multiple-point statistics-based (MPS) conditional probabilities to variogram-based simple kriging (SK), ordinary kriging (OK), simple cokriging (SCK), and standardized ordinary cokriging (SOCK) estimates to evaluate which kriging variant comes closest to the benchmark MPS probabilities. SOCK stands out among the kriging variants when compared to the MPS probabilities. Subsequently, SOCK and OK estimates are compared to reference (training images) for several different cases. The difference in the root mean squared error (RMSE) gives a very small advantage to the SOCK method. The research work of this thesis also led to the implementation of a method to mitigate extreme weights that result from indefinite or ill-conditioned matrices in linear systems of equations. The method consists of inflating the diagonal of the left-hand side covariance matrix by a small constant. As the constant increases, all weights converge to 1 n, where n is the number of weights. This technique shows that the extreme weights are mitigated but does not significantly alter the overall estimation across an entire grid.