Skip to main content

Plurigaussian modeling of geological domains based on the truncation of non-stationary Gaussian random fields

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

The plurigaussian model is used in mining engineering, oil reservoir characterization, hydrology and environmental sciences to simulate the layout of geological domains in the subsurface, while reproducing their spatial continuity and dependence relationships. However, this model is well-established only in the stationary case, when the spatial distribution of the domains is homogeneous in space, and suffers from theoretical and practical impediments in the non-stationary case. To overcome these limitations, this paper proposes extending the model to the truncation of intrinsic random fields of order k with Gaussian generalized increments, which allows reproducing spatial trends in the distribution of the geological domains. Methodological tools and algorithms are presented to infer the model parameters and to construct realizations of the geological domains conditioned to existing data. The proposal is illustrated with the simulation of rock type domains in an ore deposit in order to demonstrate its applicability. Despite the limited number of conditioning data, the results show a remarkable agreement between the simulated domains and the lithological model interpreted by geologists, while the conventional stationary plurigaussian model turns out to be unsuccessful.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

References

  1. Armstrong M, Galli A, Beucher H, Le Loc’h G, Renard D, Renard B, Eschard R, Geffroy F (2011) Plurigaussian simulations in geosciences. Springer, Berlin

    Book  Google Scholar 

  2. Arroyo D, Emery X (2015) Simulation of intrinsic random fields of order k with Gaussian generalized increments by Gibbs sampling. Math Geosci 47(8):955–974

    Article  Google Scholar 

  3. Arroyo D, Emery X (2016) Spectral simulation of vector random fields with stationary Gaussian increments in d-dimensional Euclidean spaces. Stoch Environ Res Risk Assess. doi:10.1007/s00477-016-1225-7

    Google Scholar 

  4. Arroyo D, Emery X, Peláez M (2012) An enhanced Gibbs sampler algorithm for non-conditional simulation of Gaussian random vectors. Comput Geosci 46:138–148

    Article  Google Scholar 

  5. Bernstein DS (2009) Matrix mathematics. Princeton University Press, Princeton

    Book  Google Scholar 

  6. Beucher H, Galli A, Le Loc’h G, Ravenne C (1993) Including a regional trend in reservoir modeling using the truncated Gaussian method. In: Soares A (ed) Geostatistics Tróia’92. Kluwer Academic, Dordrecht, pp 555–566

    Chapter  Google Scholar 

  7. Biver P, Haas A, Bacquet C (2002) Uncertainties in facies proportion estimation II: application to geostatistical simulation of facies and assessment of volumetric uncertainties. Math Geol 34(6):703–714

    CAS  Article  Google Scholar 

  8. Chilès JP, Delfiner P (2012) Geostatistics: modeling spatial uncertainty. Wiley, New York

    Book  Google Scholar 

  9. Christakos G (1992) Random field models in earth sciences. Academic Press, San Diego

    Google Scholar 

  10. Delfiner P (1976) Linear estimation of nonstationary spatial phenomena. In: Guarascio M, David M, Huijbregts CJ (eds) Advanced geostatistics in the mining industry. Reidel, Dordrecht, pp 49–68

    Chapter  Google Scholar 

  11. Emery X (2007) Simulation of geological domains using the plurigaussian model: new developments and computer programs. Comput Geosci 33(9):1189–1201

    Article  Google Scholar 

  12. Emery X (2008) Uncertainty modeling and spatial prediction by multi-Gaussian kriging: accounting for an unknown mean value. Comput Geosci 34(11):1431–1442

    Article  Google Scholar 

  13. Emery X (2010) Iterative algorithms for fitting a linear model of coregionalization. Comput Geosci 36(9):1150–1160

    Article  Google Scholar 

  14. Emery X, Cornejo J (2010) Truncated Gaussian simulation of discrete-valued, ordinal coregionalized variables. Comput Geosci 36(10):1325–1338

    Article  Google Scholar 

  15. Emery X, Lantuéjoul C (2006) TBSIM: a computer program for conditional simulation of three-dimensional Gaussian random fields via the turning bands method. Comput Geosci 32(10):1615–1628

    Article  Google Scholar 

  16. Emery X, Lantuéjoul C (2008) A spectral approach to simulating intrinsic random fields with power and spline generalized covariances. Comput Geosci 12(1):121–132

    Article  Google Scholar 

  17. Emery X, Arroyo D, Peláez M (2014) Simulating large Gaussian vectors subject to inequality constraints by Gibbs sampling. Math Geosci 46:265–283

    Article  Google Scholar 

  18. Galli A, Beucher H, Le Loc’h G, Doligez B (1994) The pros and cons of the truncated Gaussian method. In: Armstrong M, Dowd PA (eds) Geostatistical simulations. Kluwer, Dordrecht, pp 217–233

    Chapter  Google Scholar 

  19. Geman S, Geman D (1984) Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans Pattern Anal Mach Intell 6:721–741

    CAS  Article  Google Scholar 

  20. Guilbert JM, Park CF (1986) The geology of ore deposits. Freeman, New York

    Google Scholar 

  21. Langlais V, Beucher H, Renard D (2008) In the shade of truncated Gaussian simulation. In: Ortiz JM, Emery X (eds) Proceedings of the eighth international geostatistics congress. Gecamin Ltda, Santiago, pp 799–808

    Google Scholar 

  22. Lantuéjoul C (2002) Geostatistical simulation, Models and Algorithms. Springer, New York

    Book  Google Scholar 

  23. Lantuéjoul C, Desassis N (2012) Simulation of a Gaussian random vector: a propagative version of the Gibbs sampler. In: Ninth international geostatistics congress, Oslo

  24. Le Loc’h G, Galli A (1997) Truncated plurigaussian method: theoretical and practical points of view. In: Baafi EY, Schofield NA (eds) Geostatistics Wollongong’96. Kluwer Academic, Dordrecht, pp 211–222

    Google Scholar 

  25. Lowell JD, Guilbert JM (1970) Lateral and vertical alteration-mineralization zoning in porphyry ore deposits. Econ Geol 65:373–408

    Article  Google Scholar 

  26. Madani N, Emery X (2015) Simulation of geo-domains accounting for chronology and contact relationships: application to the Río Blanco copper deposit. Stoch Environ Res Risk Assess 29:2173–2191

    Article  Google Scholar 

  27. Matheron G (1971) The theory of regionalized variables and its applications. Ecole Nationale Supérieure des Mines de Paris, Fontainebleau, p 212

    Google Scholar 

  28. Matheron G (1973) The intrinsic random functions and their applications. Adv Appl Probab 5(3):439–468

    Article  Google Scholar 

  29. Ravenne C, Galli A, Doligez B, Beucher H, Eschard R (2002) Quantification of facies relationships via proportion curves. In: Armstrong M, Bettini C, Champigny N, Galli A, Remacre A (eds) Geostatistics Rio 2000. Kluwer Academic, Dordrecht, pp 19–40

    Chapter  Google Scholar 

  30. Serrano L, Vargas R, Stambuk V, Aguilar C, Galeb M, Holmgren C, Contreras A, Godoy S, Vela I, Skewes MA, Stern CR (1996) The late Miocene to early Pliocene Río Blanco-Los Bronces copper deposit, Central Chilean Andes. In: Camus F, Sillitoe RH, Petersen R (eds) Andean copper deposits: new discoveries, mineralization, styles and metallogeny. Special publication No. 5. Society of Economic Geologists, Littleton, p 119

    Google Scholar 

  31. Skewes MA, Stern CR (1995) Genesis of the late Miocene to Pliocene copper deposits of central Chile in the context of Andean magmatic and tectonic evolution. Int Geol Rev 37(10):893–909

    Article  Google Scholar 

  32. Stambuk V, Aguilar C, Blondel J, Galeb M, Serrano L, Vargas R (1988) Geología del yacimiento Río Blanco. Technical Report. Codelco-Chile, División Andina

  33. Stein ML (2002) Fast and exact simulation of fractional Brownian surfaces. J Comput Graph Stat 11(3):587–599

    Article  Google Scholar 

  34. Xu C, Dowd PA, Mardia KV, Fowell RJ (2006) A flexible true plurigaussian code for spatial facies simulations. Comput Geosci 32(10):1629–1645

    Article  Google Scholar 

Download references

Acknowledgements

This research was funded by the Chilean Commission for Scientific and Technological Research, through projects CONICYT/FONDECYT/REGULAR/No 1130085 and CONICYT PIA Anillo ACT1407. The authors also acknowledge the support from Mr. Claudio Martínez from Codelco-Chile (Andina Division), who provided the data set used in this work.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xavier Emery.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Madani, N., Emery, X. Plurigaussian modeling of geological domains based on the truncation of non-stationary Gaussian random fields. Stoch Environ Res Risk Assess 31, 893–913 (2017). https://doi.org/10.1007/s00477-016-1365-9

Download citation

Keywords

  • Geological domaining
  • Subsurface heterogeneity
  • Intrinsic random fields of order k
  • Generalized covariance function