Conditional Cosimulation of Multivariate Gaussian Random Fields with Unknown but Related Means

Source abstract

Abstract Multivariate geostatistical cosimulation is commonly used to reproduce spatial uncertainty and cross-variable relationships. However, conventional approaches based on simple or ordinary cokriging can become problematic under partially heterotopic sampling, particularly when an important variable is strongly undersampled or absent in part of the domain. This study proposes a two-step turning bands cosimulation framework that incorporates ordinary cokriging with related means. Non-conditional multivariate Gaussian realizations are first generated and subsequently conditioned to the available data by cokriging the residuals, while linear relationships between the unknown Gaussian means are incorporated into the conditioning system. The method is applied to an iron deposit where Fe, P, and S are widely sampled, whereas FeO is available only in the eastern part of the deposit. FeO is particularly important in this deposit because the Fe/FeO ratio helps distinguish magnetite-dominated preferred ore from hematite-dominated material and therefore contributes to mineralogical dilution assessment. To do so, the conditional realizations are generated at fine-scale discretization blocks, back-transformed to the original data units, and subsequently aggregated to selective mining-unit support for mineralogical dilution assessment. The proposed approach is compared with equivalent cosimulation frameworks based on simple and ordinary cokriging using spatial patterns, global statistics, distribution and correlation reproduction, trend analysis, cross-validation, and dilution maps. The related-mean approach maintained complete spatial coverage and provided competitive predictive performance, while ordinary cokriging became unstable in poorly sampled areas. This improved reproduction is transferred to the mineral dilution assessment, providing a more consistent characterization of mineralogical variability across the deposit. The proposed framework offers a practical approach for cosimulating undersampled mineralogical or geometallurgical variables when informative relationships with more densely sampled variables are available.