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<p>Random fields provide a flexible framework for modelling spatial variability in environmental, mineral, and geological processes. In mineral exploration data, spatial continuity is often anisotropic because of formation processes such as sedimentary stratification. This type of structure can be represented through geometric anisotropy. Anisotropy parameters are commonly inferred from directional variograms. This study examines the applicability of the anisotropic Hybrid Spectral Ornstein–Uhlenbeck (HSOU) covariance model for multivariate geostatistical modelling under geometric anisotropy. The methodology is applied using collocated zinc (Zn) and lead (Pb) soil samples to support sustainable mining development. Gaussian anamorphosis based on Kernel Cumulative Distribution Estimation (KCDE) was applied to transform the data to an approximately standard Gaussian distribution. Directional empirical direct variograms and cross-variograms were used to characterise anisotropy in the spatial continuity structure. The estimated anisotropy ratios were <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(R_{Pb}=1.23\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(R_{Zn}=2.53\)</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(R_{Pb-Zn}=1.76\)</EquationSource> </InlineEquation>. Cokriging predictions obtained using isotropic and anisotropic HSOU models yielded comparable cross-validation performance. Uncertainty in the variogram parameters and predictions was evaluated through Monte Carlo simulations. The substantial uncertainty in the variogram parameters propagated to the cokriging predictions, resulting in considerable uncertainty in the estimates. However, the results indicate that the anisotropic HSOU model produces admissible covariance matrices and stable multivariate predictions under directional heterogeneity, supporting its use in mining geostatistical applications.</p>

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Multivariate geostatistical modelling of a bivariate natural resources dataset under geometric anisotropy

  • 期刊论文

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  • 发表日期：

    2026年7月20日

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  • Emmanouil A. Varouchakis,
  • Maria Chrysanthi,
  • Andrew Pavlides

摘要

Random fields provide a flexible framework for modelling spatial variability in environmental, mineral, and geological processes. In mineral exploration data, spatial continuity is often anisotropic because of formation processes such as sedimentary stratification. This type of structure can be represented through geometric anisotropy. Anisotropy parameters are commonly inferred from directional variograms. This study examines the applicability of the anisotropic Hybrid Spectral Ornstein–Uhlenbeck (HSOU) covariance model for multivariate geostatistical modelling under geometric anisotropy. The methodology is applied using collocated zinc (Zn) and lead (Pb) soil samples to support sustainable mining development. Gaussian anamorphosis based on Kernel Cumulative Distribution Estimation (KCDE) was applied to transform the data to an approximately standard Gaussian distribution. Directional empirical direct variograms and cross-variograms were used to characterise anisotropy in the spatial continuity structure. The estimated anisotropy ratios were \(R_{Pb}=1.23\) , \(R_{Zn}=2.53\) , and \(R_{Pb-Zn}=1.76\) . Cokriging predictions obtained using isotropic and anisotropic HSOU models yielded comparable cross-validation performance. Uncertainty in the variogram parameters and predictions was evaluated through Monte Carlo simulations. The substantial uncertainty in the variogram parameters propagated to the cokriging predictions, resulting in considerable uncertainty in the estimates. However, the results indicate that the anisotropic HSOU model produces admissible covariance matrices and stable multivariate predictions under directional heterogeneity, supporting its use in mining geostatistical applications.