Modelling Asymmetrical Facies Successions Using Pluri-Gaussian Simulations

  • Thomas Le BlévecEmail author
  • Olivier Dubrule
  • Cédric M. John
  • Gary J. Hampson
Part of the Quantitative Geology and Geostatistics book series (QGAG, volume 19)


An approach to model spatial asymmetrical relations between indicators is presented in a pluri-Gaussian framework. The underlying gaussian random functions are modelled using the linear model of co-regionalization, and a spatial shift is applied to them. Analytical relationships between the two underlying gaussian variograms and the indicator covariances are developed for a truncation rule with three facies and cut-off at 0. The application of this truncation rule demonstrates that the spatial shift on the underlying gaussian functions produces asymmetries in the modelled 1D facies sequences. For a general truncation rule, the indicator covariances can be computed numerically, and a sensitivity study shows that the spatial shift and the correlation coefficient between the gaussian functions provide flexibility to model the asymmetry between facies. Finally, a case study is presented of a Triassic vertical facies succession in the Latemar carbonate platform (Dolomites, Northern Italy) composed of shallowing-upward cycles. The model is flexible enough to capture the different transition probabilities between the environments of deposition and to generate realistic facies successions.


Gaussian Function Transition Rate Spatial Shift Asymmetrical Relation Sequence Indicator Simulation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors would like to thank the Earth Science and Engineering Department of Imperial College for a PhD studentship grant for T. Le Blévec and Total for funding O. Dubrule professorship at Imperial College.


  1. Alabert F (1989) Non‐Gaussian data expansion in the earth sciences. Terra Nova 1(2):123–134CrossRefGoogle Scholar
  2. Allard D, D’Or D, Froidevaux R (2011) An efficient maximum entropy approach for categorical variable prediction. Eur J Soil Sci 62(3):381–393CrossRefGoogle Scholar
  3. Apanasovich TV, Genton MG (2010) Cross-covariance functions for multivariate random fields based on latent dimensions. Biometrika 97(1):15–30CrossRefGoogle Scholar
  4. Armstrong M, Galli A, Beucher H, Loc’h G, Renard D, Doligez B, … Geffroy F (2011) Plurigaussian simulations in geosciences. Springer Science & Business Media, New YorkGoogle Scholar
  5. Burgess P, Wright V, Emery D (2001) Numerical forward modelling of peritidal carbonate parasequence development: implications for outcrop interpretation. Basin Res 13(1):1–16CrossRefGoogle Scholar
  6. Carle SF, Fogg GE (1996) Transition probability-based indicator geostatistics. Math Geol 28(4):453–476CrossRefGoogle Scholar
  7. Catuneanu O, Galloway WE, Kendall CGSC, Miall AD, Posamentier HW, Strasser A, Tucker ME (2011) Sequence stratigraphy: methodology and nomenclature. Newslett Stratigr 44(3):173–245CrossRefGoogle Scholar
  8. Chilès J-P, Delfiner P (2012) Geostatistics: modeling spatial uncertainty, vol 713. Wiley, HobokenGoogle Scholar
  9. D’Or D, Allard D, Biver P, Froidevaux R, Walgenwitz A (2008) Simulating categorical random fields using the multinomial regression approach. Paper presented at the Geostats 2008—proceedings of the eighth international geostatistics congressGoogle Scholar
  10. Dubrule O (2016) Indicator variogram models – do we have much choice?. Manuscript submitted for publicationGoogle Scholar
  11. Egenhoff SO, Peterhänsel A, Bechstädt T, Zühlke R, Grötsch J (1999) Facies architecture of an isolated carbonate platform: tracing the cycles of the Latemar (Middle Triassic, northern Italy). Sedimentology 46(5):893–912CrossRefGoogle Scholar
  12. Fischer AG (1964) The Lofer cyclothems of the alpine Triassic. Princeton University, PrincetonGoogle Scholar
  13. Genz A (1992) Numerical computation of multivariate normal probabilities. J Comput Graph Stat 1(2):141–149Google Scholar
  14. Genz A, Bretz F, Miwa T, Mi X, Leisch F, Scheipl F, Hothorn T (2009) mvtnorm: multivariate normal and t distributions. R package version 0.9-8. URL
  15. Gneiting T, Kleiber W, Schlather M (2012) Matérn cross-covariance functions for multivariate random fields. J Am Stat Assoc 105(491):1167–1177Google Scholar
  16. Goldhammer R, Dunn P, Hardie L (1990) Depositional cycles, composite sea-level changes, cycle stacking patterns, and the hierarchy of stratigraphic forcing: examples from Alpine Triassic platform carbonates. Geol Soc Am Bull 102(5):535–562CrossRefGoogle Scholar
  17. Grotzinger JP (1986) Cyclicity and paleoenvironmental dynamics, Rocknest platform, northwest Canada. Geol Soc Am Bull 97(10):1208–1231CrossRefGoogle Scholar
  18. Kendall M, Stuart A, Ord J (1994) Vol. 1: Distribution theory. Arnold, LondonGoogle Scholar
  19. Lantuéjoul C (2002) Geostatistical simulation: models and algorithms. Springer Science & Business Media, BerlinGoogle Scholar
  20. Li B, Zhang H (2011) An approach to modeling asymmetric multivariate spatial covariance structures. J Multivar Anal 102(10):1445–1453CrossRefGoogle Scholar
  21. Masetti D, Neri C, Bosellini A (1991) Deep-water asymmetric cycles and progradation of carbonate platforms governed by high-frequency eustatic oscillations (Triassic of the Dolomites, Italy). Geology 19(4):336–339CrossRefGoogle Scholar
  22. Oliver DS (2003) Gaussian cosimulation: modelling of the cross-covariance. Math Geol 35(6):681–698CrossRefGoogle Scholar
  23. Renard D, Bez N, Desassis N, Beucher H, Ors F, Laporte F (2015) RGeostats: the geostatistical package (Version 11.0.1). Retrieved from
  24. Sena CM, John CM (2013) Impact of dynamic sedimentation on facies heterogeneities in Lower Cretaceous peritidal deposits of central east Oman. Sedimentology 60(5):1156–1183CrossRefGoogle Scholar
  25. Sheppard W (1899) On the application of the theory of error to cases of normal distribution and normal correlation. Philos Trans R Soc London Ser A Containing Pap Math Phys Charact 192:101–531CrossRefGoogle Scholar
  26. Strasser A (1988) Shallowing‐upward sequences in Purbeckian peritidal carbonates (lowermost Cretaceous, Swiss and French Jura Mountains). Sedimentology 35(3):369–383CrossRefGoogle Scholar
  27. Tucker M (1985) Shallow-marine carbonate facies and facies models. Geol Soc London Spec Publ 18(1):147–169CrossRefGoogle Scholar
  28. Venables WN, Ripley BD (2002) Modern applied statistics with S, 4th edn. Springer, New YorkCrossRefGoogle Scholar
  29. Wackernagel H (2013) Multivariate geostatistics: an introduction with applications. Springer Science & Business Media, BerlinGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Thomas Le Blévec
    • 1
    Email author
  • Olivier Dubrule
    • 1
  • Cédric M. John
    • 1
  • Gary J. Hampson
    • 1
  1. 1.Imperial College, Royal School of MinesLondonUK

Personalised recommendations