Skip to main content
SpringerLink
Log in

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

  • Published:
Mathematical Geosciences Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23

Similar content being viewed by others

References

  • Almeida AS, Journel AG (1994) Joint simulation of multiple variables with a Markov-type co-regionalization. Math Geol 26(5):565–588

    Article  Google Scholar 

  • Arpat B, Caers J (2007) Stochastic simulation with patterns. Math Geosci 39:177–203

    Google Scholar 

  • Bandarian E, Mueller U, Ferreira J, Richardson S (2010) Transformation methods for multivariate geostatistical simulation - minimum/maximum autocorrelation factors and alternating columns diagonal centres. In: Advances in Orebody Modelling and Strategic Mine Planning: Old and New Dimensions in a Changing World, The Australian Institute of Mining and Metallurgy, p 79–89

  • Barnett R, Manchuk JG, Deutsch CV (2014) Projection pursuit multivariate transform. Math Geosci 46(3):337–359

    Article  Google Scholar 

  • Boucher A, Dimitrakopoulos R (2008) Block simulation of multiple correlated variables. Math Geosci 41:215–237

    Article  Google Scholar 

  • Boucher A, Dimitrakopoulos R (2012) Multivariate block-support simulation of the Yandi iron ore deposit, Western Australia. Math Geosci 44:449–468

    Article  Google Scholar 

  • Boogaart KGvd, Tolosana-Delgado R, Lehmann M, Mueller U (2014) On the joint multipoint simulation of discrete and continuous geometallurgical parameters. In: Orebody Modelling and Strategic Mine Planning Symposium, Proceedings, Carlton. Australia, AusIMM, p 379–388

  • Boogaart KGvd, Mueller U, Tolosana-Delgado R (2016) An affine equivariant multivariate normal score transform for compositional data. Math Geosci, p 1–21. doi:10.1007/s11004-016-9645-y

  • Chatterjee S, Dimitrakopoulos R, Mustapha H (2012) Dimensional reduction of pattern-based simulation using wavelet analysis. Math Geosci 44:343–374

    Article  Google Scholar 

  • Chiles JP, Delfiner P (2012) Geostatistics: modeling spatial uncertainty, 2nd edn. Wiley, Hoboken

    Book  Google Scholar 

  • Chugunova TL, Hu LY (2008) Multiple-point simulations constrained by continuous auxiliary data. Math Geosci 40:133–146

    Article  Google Scholar 

  • Comon P (1994) Independent component analysis, a new concept? Signal Process 36:287–314

    Article  Google Scholar 

  • David M (1988) Handbook of applied advance geostatistical ore reserve estimation. Elsevier, Amsterdam

    Google Scholar 

  • De Laco S, Maggio S (2011) Validation techniques for geological patterns simulations based on variogram and multiple-point statistics. Math Geosci 43:483–500

    Article  Google Scholar 

  • De Vries LM, Carrera J, Falivene O, Gratacós O, Slooten LJ (2008) Application of multiple point geostatistics to non-stationary images. Math Geosci 41:29–42

    Article  Google Scholar 

  • Desbarats AJ, Dimitrakopoulos R (2000) Geostatistical simulation of regionalized pore-size distributions using min/max autocorrelation factors. Math Geol 32:919–942

    Article  Google Scholar 

  • Dimitrakopoulos R (2011) Stochastic optimization for strategic mine planning: a decade of developments. J Min Sci 47:138–150

    Article  Google Scholar 

  • Dimitrakopoulos R, Fonseca M (2003) Assessing risk in grade-tonnage curves in a complex copper deposit, Northern Brazil, based on an efficient joint simulation of multiple correlated variables. In: Camisani-Calzolari FA (ed) 31st International Symposium on Application of Computers and Operations Research in the Minerals Industries, Cape Town, South Africa, 14–16 May 2003

  • Dimitrakopoulos R, Mustapha H, Gloaguen E (2010) High-order statistics of spatial random fields: exploring spatial cumulants for modeling complex non-Gaussian and non-linear phenomena. Math Geosci 42:65–99

    Article  Google Scholar 

  • Goodfellow R, Consuegra FA, Dimitrakopoulos R, Lloyd T (2012) Quantifying multi-element and volumetric uncertainty, Coleman McCreedy deposit, Ontario, Canada. Comput Geosci 42:71–78

    Article  Google Scholar 

  • Honarkhah M (2011) Stochastic simulation of patterns using distance-based pattern modeling. Ph.D. dissertation, Stanford University, Stanford, USA

  • Hyvärinen A (1999) Independent component analysis for time-dependent stochastic processes. ICANN 98 Perspectives in Neural Computing, p 135–140

  • Journel AG (2005) Beyond covariance: the advent of multiple-point geostatistics. In: Geostatistics Banff 2004. Springer, The Netherlands

  • Journel AG (2007) Roadblocks to the evaluation of ore reserves—the simulation overpass and putting more geology into numerical models of deposits. In: Orebody Modelling and Strategic Mine Planning Uncertainty and Risk Management, 2nd Edn. The AusIMM, Spectrum Series 14

  • Journel AG, Huijbregts C (1978) Min Geostat. Academic Press, New York

    Google Scholar 

  • Kiers HAL (2000) Towards a standardized notation and terminology in multiway analysis. J Chemom 14:105–122

    Article  Google Scholar 

  • Lebedev NN (1965) Special functions and their applications. Prentice-Hall Inc., New York

    Google Scholar 

  • Leuangthong O, Deutsch CV (2003) Stepwise conditional transformation for simulation of multiple variables. Math Geol 35:155–173

    Article  Google Scholar 

  • Li L, Srinivasan S, Zhou H, Gómez-Hernández JJ (2013) Simultaneous estimation of geologic and reservoir state variables within an ensemble-based multiple-point statistic framework. Math Geosci 46:597–623

    Article  Google Scholar 

  • Lochbühler T, Pirot G, Straubhaar J, Linde N (2013) Conditioning of multiple-point statistics facies simulations to tomographic images. Math Geosci 46:625–645

    Article  Google Scholar 

  • Maleki M, Emery X (2015) Joint simulation of grade and rock type in a stratabound copper deposit. Math Geosci 47:471–495

    Article  Google Scholar 

  • Mariethoz G, Renard P (2010) Reconstruction of incomplete data sets or images using direct sampling. Math Geosci 42:245–268

    Article  Google Scholar 

  • Mariethoz G, Renard P, Straubhaar J (2010) The direct sampling method to perform multiple-point geostatistical simulations. Water Resour Res 46

  • Morton J (2010) Scalable implicit symmetric tensor approximation, preprint. Pennsylvania State University, University Park

    Google Scholar 

  • Morton J, Lim LH (2010) Principal cumulant components analysis, preprint. University of Chicago, Chicago

    Google Scholar 

  • Mueller UA, Ferreira J (2012) The U-WEDGE transformation method for multivariate geostatistical simulation. Math Geosci 44:427–448

    Article  Google Scholar 

  • Mueller UA, Tolosana-Delgado R, van den Boogaart KG (2014) Approaches to the simulation of compositional data—a nickel-laterite comparative case study. In: Proceedings of Orebody Modelling and Strategic Mine Planning Symposium, AusIMM, Melbourne, p 61–72

  • Mustapha H, Chatterjee S, Dimitrakopoulos R (2013) CDFSIM: efficient stochastic simulation through decomposition of cumulative distribution functions of transformed spatial patterns. Math Geosci 46:95–123

    Article  Google Scholar 

  • Mustapha H, Dimitrakopoulos R (2010) High-order stochastic simulations for complex non-Gaussian and non-linear geological patterns. Math Geosci 42:455–473

    Article  Google Scholar 

  • Mustapha H, Dimitrakopoulos R, Chatterjee S (2011) Geologic heterogeneity representation using high-order spatial cumulants for subsurface flow and transport simulations. Water Resour Res 47(8). doi:10.1029/2010WR009515

  • Osterholt V, Dimitrakopoulos R (2007) Simulation of wireframes and geometric features with multiple-point techniques: application at Yandi iron ore deposit. In: Orebody modelling and strategic mine planning, 2nd edn. AusIMM, Spectrum Series, 14, p 95–124

  • Perry JM (1977) A class of conjugate gradient algorithms with a two-step variable-metric memory. Center for Mathematical Studies in Economics and Management Science, Northwestern University, Evanston (discussion paper 269)

    Google Scholar 

  • Rezaee H, Mariethoz G, Koneshloo M, Asghari O (2013) Multiple-point geostatistical simulation using the bunch-pasting direct sampling method. Comput Geosci 54:293–308

    Article  Google Scholar 

  • Rondon O (2012) Teaching aid: minimum/maximum autocorrelation factors for joint simulation of attributes. Math Geosci 44:469–504

    Article  Google Scholar 

  • Rosenblatt M (1985) Stationary sequences and random fields. Birkhauser, Boston

    Book  Google Scholar 

  • Savas B, Lim LH (2009) Quasi-Newton methods on Grassmannians and multilinear approximations of tensors. SIAM J Sci Comput 32:3352–3393

    Article  Google Scholar 

  • Soares A (2001) Direct sequential simulation and co-simulation. Math Geol 33:911–926

    Article  Google Scholar 

  • Straubhaar J, Renard P, Mariethoz G, Froidevaux R, Besson O (2011) An improved parallel multiple-point algorithm using a list approach. Math Geosci 43:305–328

    Article  Google Scholar 

  • Strebelle S, Cavelius C (2013) Solving speed and memory issues in multiple-point statistics simulation program SNESIM. Math Geosci 46:171–186

    Article  Google Scholar 

  • Switzer P, Green AA (1984) Min/max autocorrelation factors for multivariate spatial imaging. Technical report no. 6, Department of Statistics, Stanford University

  • Tahmasebi P, Hezarkhani A, Sahimi M (2012) Multiple-point geostatistical modeling based on the cross-correlation functions. Comput Geosci 16:779–797

    Article  Google Scholar 

  • Toftaker H, Tjelmeland H (2013) Construction of binary multi-grid Markov random field prior models from training images. Math Geosci 45:383–409

    Article  Google Scholar 

  • Verly GW (1993) Sequential Gaussian co-simulation: a simulation method integrating several types of information. Quant Geol Geostat. In: Geostatistics Tróia 92, Springer, The Netherlands

  • Wackernagel H (1998) Principal component analysis for autocorrelated data: a geostatistical perspective. In: Technical report no. 22/98/G, Centre de Géostatistique, Ecole des Mines de Paris, Paris, France

  • Welling M (2005) Robust higher-order statistics. In: Proc. 10th Int Workshop Artif Intell Statist, p 405–412

  • Zhang T, Switzer P, Journel A (2006) Filter-based classification of training image patterns for spatial simulation. Math Geol 38:63–80

    Article  Google Scholar 

  • Zhang T (2015) MPS-driven digital rock modeling and upscaling. Math Geol 47:937–954

    Google Scholar 

Download references

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).

Author information

Authors and Affiliations

  1. Department of Mining and Materials Engineering, McGill University, Montreal, H3A 0E8, Canada

    Ilnur Minniakhmetov

  2. COSMO-Stochastic Mine Planning Laboratory, McGill University, Montreal, QC, H3A 0E8, Canada

    Ilnur Minniakhmetov & Roussos Dimitrakopoulos

Authors
  1. Ilnur Minniakhmetov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roussos Dimitrakopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

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
figure 24

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

Full size image
Fig. 25
figure 25

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

Full size image
Fig. 26
figure 26

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

Full size image
Fig. 27
figure 27

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

Full size image
Fig. 28
figure 28

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

Full size image
Fig. 29
figure 29

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

Full size image
Fig. 30
figure 30

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

Full size image

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11004-016-9662-x

Keywords

Search

Navigation