Multivariate Simulation Using A Locally Varying Coregionalization Model

Multivariate spatial modeling is key to understanding the behavior of materials downstream in a mining operation. The ore recovery depends on the mineralogical composition, which needs to be properly captured by the model to allow for good predictions. Multivariate modeling must also capture the behavior of tailings and waste materials to understand the environmental risks involved in their disposal. However, multivariate spatial modeling is challenging when the variables show complex relationships, such as non-linear correlation, heteroscedastic behavior, or spatial trends. This contribution proposes a novel methodology for general multivariate contexts, with the idea of disaggregating the global non-linear behavior among variables into the spatial domain in a piece-wise linear fashion. We demonstrate that the complex multivariate behavior can be reproduced by looking at local correlations between variables at sample locations, inferred from a local neighborhood, and interpolating these local linear dependencies by using a non-stationary version of the Linear Model of Coregionalization. This mixture of locally varying linear correlations is combined to reproduce the global complex behavior seen in the multivariate distribution. The main challenge is to solve appropriately the interpolation of the known correlation matrices over the domain, as these local correlations defined at sample locations can be endowed with a manifold structure, on which the Euclidean distance is not a suitable metric for interpolation of such correlations. This is addressed by using tools from Riemannian geometry: correlation matrices are interpolated using a weighted Fr\'echet mean of the correlations inferred at sample locations. An application of the procedure is shown in a real case study with good results in terms of accuracy and reproduction of the reference multivariate distributions and semi-variograms.