LAUSR

Multiple variables that are correlated should be jointly simulated and the resulting realizations should reproduce the experimental data statistics (i.e. histogram, variogram, correlation coefficients). Multivariate transforms such as principal component analysis (PCA), minimum/maximum autocorrelation factors (MAF) and projection pursuit multivariate transform (PPMT) are commonly used to independently simulate correlated variables without the requirement of fitting a linear model of coregionalization to the direct and cross variograms of the variables. These transforms, however, operate at different spatial lags h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{h }}$$\end{document}; that is, while PCA and PPMT generate factors that are only pairwise (h=0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathbf{h }}=0)$$\end{document} uncorrelated, MAF generates factors that are both pairwise uncorrelated and have zero cross correlations at one chosen lag (h≠0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathbf{h }}\ne 0)$$\end{document}. In addition, PCA and MAF, due to being linear transforms, do not reproduce complex features (i.e. nonlinearity, heteroskedasticity, constraints) that exist in the multivariate distributions of the data; however, PPMT, being a multivariate Gaussian transform, accounts for these features. We show in a case study that these multivariate transforms reproduce the univariate and multivariate statistics of the experimental data. The correlated variables (Cd, Co, Cr, Cu, Ni, Pb and Zn) from the Jura data are transformed into uncorrelated factors using multivariate transforms. The factors are then independently simulated, and the performance of each multivariate transform is quantitatively assessed. The best reproduction of the experimental data statistics is obtained in the case where PPMT is used along with the MAF transform.