Joint High-Order Simulation of Spatially Correlated Variables Using High-Order Spatial Statistics

Abstract

Joint geostatistical simulation techniques are used to quantify uncertainty for spatially correlated attributes, including mineral deposits, petroleum reservoirs, hydrogeological horizons, environmental contaminants. Existing joint simulation methods consider only second-order spatial statistics and Gaussian processes. Motivated by the presence of relatively large datasets for multiple correlated variables that typically are available from mineral deposits and the effects of complex spatial connectivity between grades on the subsequent use of simulated realizations, this paper presents a new approach for the joint high-order simulation of spatially correlated random fields. First, a vector random function is orthogonalized with a new decorrelation algorithm into independent factors using the so-termed diagonal domination condition of high-order cumulants. Each of the factors is then simulated independently using a high-order univariate simulation method on the basis of high-order spatial cumulants and Legendre polynomials. Finally, attributes of interest are reconstructed through the back-transformation of the simulated factors. In contrast to state-of-the-art methods, the decorrelation step of the proposed approach not only considers the covariance matrix, but also high-order statistics to obtain independent non-Gaussian factors. The intricacies of the application of the proposed method are shown with a dataset from a multi-element iron ore deposit. The application shows the reproduction of high-order spatial statistics of available data by the jointly simulated attributes.

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Acknowledgments

The work was funded from NSERC Collaborative Research and Development Grant CRDPJ 411270, entitled “Developing new global stochastic optimization and high-order stochastic models for optimizing mining complexes with uncertainty”, NSERC Discovery Grant 239019, and the COSMO consortium of mining companies (AngloGold Ashanti, Barrick Gold, BHP Billiton, De Beers, Kinross Gold, Newmont Mining and Vale).

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Correspondence to Ilnur Minniakhmetov.

Appendix: Statistics and Simulation Results

Appendix: Statistics and Simulation Results

The histograms of alumina, phosphorus, and LOI contents are shown in Figs. 23, 24, and 25, respectively. The scatter plots for Fe–SiO\(_{\mathrm {2 }}\) and SiO\(_{\mathrm {2}}\)–Al\(_{\mathrm {2}}\)O\(_{\mathrm {3}}\) are shown in Figs. 26 and 27, respectively. In Figs. 28, 29 and 30 simulation results for alumina, phosphorus, and LOI contents are shown.

Fig. 24
figure24

Histograms of phosphorus content: a drill-hole data; b training image

Fig. 25
figure25

Histograms of LOI content: a drill-hole data; b training image

Fig. 26
figure26

Scatter plot between Fe and SiO\(_{\mathrm {2}}\), for the data (a) and training image (b). Contents are shown in log-scale

Fig. 27
figure27

Scatter plot between SiO\(_{\mathrm {2}}\) and Al\(_{\mathrm {2}}\)O\(_{\mathrm {3}}\), for the data (a) and training image (b). Contents are shown in log-scale

Fig. 28
figure28

a Drill-hole data, b training image, and c, d two simulations of alumina content. Colors are shown in log-scale

Fig. 29
figure29

a Drill-hole data, b training image, and c, d two simulations of phosphorus content

Fig. 30
figure30

a Drill-hole data; b training image, and c, d two simulations of LOI content. Colors are shown in log-scale

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Minniakhmetov, I., Dimitrakopoulos, R. Joint High-Order Simulation of Spatially Correlated Variables Using High-Order Spatial Statistics. Math Geosci 49, 39–66 (2017). https://doi.org/10.1007/s11004-016-9662-x

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Keywords

  • High-order simulation
  • Joint simulation
  • Correlated variables
  • Non-Gaussian variables
  • Decorrelation
  • Diagonal domination
  • Multi-element mineral deposits