Exploring Positive-Definiteness in Multivariate Geostatistics with Non-Euclidean Distances

In complex geographic environments, spatial relationships are often distorted by natural barriers and irregular terrain as well as irregular sampling. These sub-optimal conditions often present challenges for geostatistical modeling, especially in multivariate data. Traditional covariance and variogram models relying on Euclidean distance may fail to capture such complexities, necessitating the use of alternative approaches. Building on previous research, this study investigates the performance of various covariance and variogram models applied to multivariate geostatistical data from a mine in Ireland. The dataset, consisting of published concentrations of co-located metals, provides a good opportunity to explore the utility of advanced spatial modeling techniques in continuation of our previous (univariate) studies.The analysis employs Gaussian anamorphosis with the Kernel Cumulative Density Estimator (KCDE) to normalize the Multivariate data, with the use of a look-up Table for the back-transform of the predicted grid values. The resulting data follow the Normal Distribution N(0,1) and thus the transformed data are gaussian and the values are of the same range.  A range of theoretical variogram models, (for example the Exponential, Gaussian, Spherical) as well as the previously introduced Harmonic Covariance Estimation (HCE) model, is applied to assess their suitability for co-kriging in a multivariate context. Emphasis is placed on ensuring positive-definiteness of the resulting covariance matrices through Eigenvalue analysis [1]. With more than two variables, the invertibility of the augmented covariance matrix is not ensured, even for Euclidean distances. Positive definiteness is further complicated with the utilization of non-Euclidean distance metrics such as Manhattan, Minkowski, or Chebyshev distances.The primary goal of this study is to evaluate the comparative performance of the HCE model against traditional variogram models in modeling multivariate spatial relationships and the positive definiteness of the multivariate covariance matrices. Furthermore, the accuracy of co-kriging predictions with the various established models and HCE, both before and after the back-transform will be tested and discussed. This research extends the understanding of non-Euclidean geostatistical modeling in multivariate contexts, with potential applications in other regions with complex terrains or spatiotemporal phenomena.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) 1) Curriero, F.C.: On the use of non-euclidean distance measures in geostatistics. Mathematical Geology 38, 907–926 (2006)