Practical Implementation of Non-linear Transforms for Modeling Geometallurgical Variables

Abstract

Evaluation of process performance within mining operations requires geostatistical modeling of many related variables. These variables are a combination of grades and other rock properties, which together provide a characterization of the deposit that is necessary for optimizing plant design, blending and stockpile planning. Complex multivariate relationships such as stoichiometric constraints, non-linearity and heteroscedasticity are often present. Conventional covariance-based techniques do not capture these multivariate features; nevertheless, these complexities influence decision making and should be reproduced in geostatistical models. There are non-linear transforms that help bridge the gap between complex geologic relationships and practical geostatistical modeling tools. Logratios, Min./Max. Autocorrelation Factors, Normal Scores, and Stepwise Conditional transformation are a few of the available transforms. In many circumstances these transforms are used in sequence to model the variables for a given deposit. As each technique possesses its own limitations, challenges may arise in choosing the appropriate transforms and the order in which they are applied. These practical challenges will be examined, with a new technique named conditional standardization introduced as a potential solution to address non-linear and heteroscedastic multivariate features. A generalized workflow is proposed to aid in the selection and ordering of multiple transformations. Common problems such as bias and poor reproduction of spatial correlation are illustrated in a geometallurgic case study, along with a demonstration of the corrective measures. Although these transforms are presented within a mining context, they are equally suited to any petroleum or environmental application where multiple variables are being considered.