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.

翻译：多变量空间建模是理解采矿过程中下游物料行为的关键。矿石回收率取决于矿物组成，这需要通过模型准确捕捉以实现良好的预测。多变量建模还必须捕捉尾矿和废料的行为，以理解其处置所涉及的环境风险。然而，当变量呈现复杂关系（如非线性相关、异方差行为或空间趋势）时，多变量空间建模颇具挑战性。本文提出了一种适用于一般多变量背景的新方法，其核心思想是通过分段线性方式将变量间的全局非线性行为分解到空间域中。我们证明，通过考察样本位置处局部邻域推断的变量间局部相关性，并利用非平稳版线性协同区域化模型插值这些局部线性依赖关系，可以复现复杂的多变量行为。这种局部变化线性相关性的混合结构被组合起来，以再现多变量分布中观察到的全局复杂行为。主要挑战在于如何恰当地解决已知相关矩阵在空间域上的插值问题，因为这些在样本位置处定义的局部相关性具有流形结构，而欧氏距离并非适用于此类相关性插值的度量。为此，我们采用黎曼几何工具：利用样本位置处推断相关性的加权弗雷歇均值进行相关矩阵插值。通过在真实案例中的应用表明，该方法在精度、参考多变量分布及半变异函数再现方面均取得了良好效果。