Direct Multivariate Simulation - A stepwise conditional transformation for multivariate geostatistical simulation

Abstract

Several applications in geoscience require the generation of multiple realizations of random fields of physical properties to mimic their spatial distribution and quantify the model uncertainty. Some modeling problems present complex multivariate distributions with heteroscedasticity and non-linear relations among the variables. We propose a new algorithm, namely Direct Multivariate Simulation, for the simulation of random fields of non-parametric multivariate joint distributions. The methodology is based on a generalization of the normal score and back transformation for multivariate distributions, also called Stepwise Conditional transformation. The sequential sampling of each variable is performed by decomposing the target joint distribution into a product of univariate marginal and conditional probability density functions. We propose numerical solutions to improve the algorithm efficiency for real case applications with a large number of variables. The method is demonstrated through an application where we sample a 6-variate joint distributions with strong nonlinear dependence among the variables. The results are validated by comparing the results to the Projection Pursuit Multivariate Transform and through the computation of the experimental semi-variograms, the marginal distributions, and the bi-histograms of the simulated variables.