Mathematical Geosciences

, Volume 51, Issue 2, pp 155–186 | Cite as

Enhanced Multiple-Point Statistical Simulation with Backtracking, Forward Checking and Conflict-Directed Backjumping

  • Mohammad ShahraeeniEmail author


During a conventional multiple-point statistics simulation, the algorithm may not find a matched neighborhood in the training image for some unsimulated pixels. These pixels are referred to as the dead-end pixels; the existence of the dead-end pixels means that multiple-point statistics simulation is not a simple sequential simulation. In this paper, the multiple-point statistics simulation is cast as a combinatorial optimization problem, and the efficient backtracking algorithm is developed to solve this optimization problem. The efficient backtracking consists of backtracking, forward checking, and conflict-directed backjumping algorithms that are introduced and discussed in this paper. This algorithm is applied to simulate multiple-point statistics properties of some synthetic training images; the results show that no anomalies occurred in any of the produced realizations as opposed to previously published methods for solving the dead-end pixels. In particular, in simulating a channel system, all the channels generated by this method are continuous, which is of paramount importance in fluid flow simulation applications. The results also show that the presence of hard data does not degrade the quality of the generated realizations. The presented method provides a robust algorithmic framework for performing MPS simulation.


Multiple-point statistics Constraint satisfaction problem Backtracking Generalized arc consistency Conflict-directed backjumping 



The author would like to thank the Geoscience Research Centre (GRC) of TOTAL EP UK for funding and supporting this research. The author also would like to thank Tatiana Chugunova (TOTAL SA) for providing one of the training images, and her constructive comments and suggestions during the development of this work.

Supplementary material

11004_2018_9761_MOESM1_ESM.pdf (2 mb)
Supplementary material 1 (PDF 2075 kb)


  1. Arpat GB, Caers J (2007) Conditional simulation with patterns. Math Geol 39:177–203CrossRefGoogle Scholar
  2. Beldiceanu N, Carlsson M, Rampon JX (2010) Working version of SICS. Accessed 27 July 2018
  3. Brélaz D (1979) New methods to color the vertices of a graph. Commun ACM 22:251–256. CrossRefGoogle Scholar
  4. Caers J, Hoffman T (2006) The probability perturbation method: a new look at Bayesian inverse modeling. Math Geol 38:81–100. CrossRefGoogle Scholar
  5. Comunian A, Renard P, Straubhaar J (2012) 3D multiple-point statistics simulation using 2D training images. Comput Geosci 40:49–65CrossRefGoogle Scholar
  6. Hu LY, Chugunova T (2008) Multiple-point geostatistics for modeling subsurface heterogeneity: a comprehensive review. Water Resour Res. Google Scholar
  7. Jeavons P, Krokhin A, Živný S (2014) The complexity of valued constraint satisfaction. Bull EATCS 113:21–55Google Scholar
  8. Krząkała F, Zdeborová L (2008) Phase transitions and computational difficulty in random constraint satisfaction problems. J Phys Conf Ser 95:012012CrossRefGoogle Scholar
  9. Lhomme O (1993) Consistency techniques for numeric CSPs. IJCAI 93:232–238Google Scholar
  10. Mackworth AK (1977) Consistency in networks of relations. Artif Intell 8:99–118. CrossRefGoogle Scholar
  11. Mariethoz G, Renard P, Caers J (2010a) Bayesian inverse problem and optimization with iterative spatial resampling. Water Resour Res. Google Scholar
  12. Mariethoz G, Renard P, Straubhaar J (2010b) The direct sampling method to perform multiple-point geostatistical simulations. Water Resour Res. Google Scholar
  13. Minton S, Johnston MD, Philips AB, Laird P (1992) Minimizing conflicts: a heuristic repair method for constraint satisfaction and scheduling problems. Artif Intell 58:161–205. CrossRefGoogle Scholar
  14. Poole DL, Mackworth AK (2010) Artificial intelligence: foundations of computational agents. Cambridge University Press, New York, p 122CrossRefGoogle Scholar
  15. Prosser P (1993) Hybrid algorithms for the constraint satisfaction problem. Comput Intell 9:268–299. CrossRefGoogle Scholar
  16. Russell SJ, Norvig P (2010) Artificial intelligence: a modern approach. Pearson Education Limited, Kuala Lumpur, pp 202–233Google Scholar
  17. Strebelle S (2002) Conditional simulation of complex geological structures using multiple-point statistics. Math Geol 34:1–21. CrossRefGoogle Scholar
  18. Strebelle S (2003) New multiple-point statistics simulation implementation to reduce memory and cpu-time demand. In: Proceedings to the IAMG 2003Google Scholar
  19. Suzuki S, Strebelle S (2007) Real-time post-processing method to enhance multiple-point statistics simulation. In: EAGE. Petroleum Geostatistics 2007, Cascais, Portugal.
  20. Wu J, Zhang T, Journel A (2010) Fast filtersim simulation with score-based distance. Math Geosci 40:773–788CrossRefGoogle Scholar
  21. Yokoo M, Hirayama K (2000) Algorithms for distributed constraint satisfaction: a review. Auton Agent Multi-Agent Syst 3:185–207. CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2018
corrected publication September 2018

Authors and Affiliations

  1. 1.Geoscience Research CentreTOTAL EP UKAberdeenUK

Personalised recommendations