Multivariate geostatistical simulation with sum and fraction constraints

Abstract

Geostatistical modelling of grades in mineral deposits often requires the simulation of multiple related variables that have sum and fraction constraints. Sum constraints occur when the sum of some variables may not exceed or must equal a given constant. Fraction constraints occur when a variable may not exceed another variable. In this case, the geostatistical simulations should reproduce the histograms, variograms, multivariate relationships and honour the constraints. We present a methodology for the geostatistical simulation of multiple related variables that considers sum and fraction constraints. The methodology is illustrated in a case study. The data were obtained from a bauxite deposit. First, the original variables were re-expressed as ratios. Second, the variables were transformed using the Projection Pursuit Multivariate Transform (PPMT). Then the PPMT transformed variables were simulated independently using sequential Gaussian simulation and back-transformed. The simulations reproduced the histograms, variograms and bivariate relationships and honoured the sum and fraction constraints.

Multivariate geostatistical simulation with PPMT: an application for uncertainty measurement

“…The limitation of the study is that compositional sum constraints were not considered. These constraints may be incorporated in the workflow by transforming the variables into ratios before applying the PPMT transformation (Bassani et al 2018).…”

“…To enable working with complex correlated data and any number of variables, Barnett et al (2014) proposed the Projection Pursuit Multivariate Transform (PPMT), which allows a complete decorrelation and multi-Gaussian transformation of the dataset. Several works demonstrating the applicability of PPMT for multivariate geostatistical simulation are reported in the literature (Bassani et al 2018; Battalgazy and Madani 2019).…”

Assessing heterotopic searching strategy in hierarchical cosimulation for modeling the variables with inequality constraints

“…However, these factorization methods have difficulty in reproducing geological inequality constraints present in bivariate relationships. Modeling of variables with inequality constraints has been attempted by using stepwise conditional transformation [Leuangthong and Deutsch, 2003], minimum/maximum autocorrelation factor [Desbarats andDimitrakopoulos, 2000, Vargas-Guzmán andDimitrakopoulos, 2003] with changing to new variables free of inequality constraint [Abildin et al, 2019], log-ratio transformation Olea, 2004, Pawlowsky-Glahn andEgozcue, 2006], projection pursuit multivariate transform on variables changed into ratios [Arcari Bassani et al, 2018] and stoichiometric relations of original variables [Mery et al, 2017, Adeli et al, 2018. One significant limitation of these factorization algorithms is that they can be applied mostly on isotopic data sets, wherever the sample observations of variables are required to share the exact locations.…”