Advancing Multivariate Simulations using Non-Euclidean Metrics

Multivariate data analysis in natural resources exploration can be beneficial for each variable investigated as the correlation between the variables increases the prediction accuracy and reduces the error variance. Geostatistical modeling of mineral deposits often encounters challenges in accurately representing spatial dependencies, particularly in complex geological formations and irregular sampling grids. While traditional Euclidean distances are commonly used, they may not adequately capture spatial relationships in such scenarios. Non-Euclidean distances, such as Manhattan and Chebyshev metrics, as well as geodesic distance (like a sphere manifold), offer alternative solutions that may better accommodate spatial fields with complex sampling grids. Such distances however may result in non-positive definite (thus not invertible) covariance matrices. This is further complicated when dealing with multivariate random fields as the resulting covariance-cross-covariance matrix may not be positive-definitive even in the Euclidean distance.This study builds on prior research to evaluate spatial dependencies for Aluminum (Al) and Zinc (Zn) concentrations in geochemical datasets under both Euclidean and non-Euclidean distance metrics. The data values have undergone Gaussian Anamorphosis with the previously introduced CDKC method. The recently introduced Harmonic Covariance Estimation (HCE) model is applied to generate covariance structures for co-kriging predictions, as well as multivariate simulations. Such simulations can assist in exploring the uncertainty of estimation (for example the 90% confidence interval) after the back-transform. The ability of HCE to maintain positive-definite cross-covariance matrices is a critical focus, particularly in multivariate simulations.In addition, this work investigates a separate dataset from a mine in Ireland, which includes Lead (Pb) and Zinc (Zn) concentrations. Here, the anisotropic form of the HCE model introduced and then applied in Euclidean space to account for directional dependencies. The performance of anisotropic HCE is then compared to kriging predictions using non-anisotropic HCE with non-Euclidean distances (Chebyshev, Manhattan, Spherical Manifold). This analysis aims to determine whether correcting for anisotropy or adopting non-Euclidean metrics yields better performance in this particular dataset, although more studies are required to reach a conclusion on the matter.The investigation results indicate that the HCE model results in invertible, positive-definite matrices that can be used for simulations and predictions with non-Euclidean distances, offering insights into optimizing spatial modeling for irregular datasets and complex deposit structures. The research project is implemented in the framework of H.F.R.I call “Basic research Financing (Horizontal support of all Sciences)” under the National Recovery and Resilience Plan “Greece 2.0” funded by the European Union – NextGenerationEU (H.F.R.I. Project Number: 16537)