Mathematical Geosciences

, Volume 50, Issue 1, pp 97–120 | Cite as

Which Path to Choose in Sequential Gaussian Simulation

  • Raphaël Nussbaumer
  • Grégoire Mariethoz
  • Erwan Gloaguen
  • Klaus Holliger


Sequential Gaussian Simulation is a commonly used geostatistical method for populating a grid with a Gaussian random field. The theoretical foundation of this method implies that all previously simulated nodes, referred to as neighbors, should be included in the kriging system of each newly simulated node. This would, however, require solving a large number of linear systems of increasing size as the simulation progresses, which, for computational reasons, is generally not feasible. Traditionally, this problem is addressed by limiting the number of neighbors to the ones closest to the simulated node. This does, however, result in artifacts in the realization. The simulation path, that is, the order in which nodes are visited, is known to influence the location and magnitude of these artifacts. So far, few rigorous studies linking the simulation path to the associated biases are available and, correspondingly, recommendations regarding the choice of the simulation path are largely based on empirical evidence. In this study, a comprehensive analysis of the influence of the path on the simulation errors is presented, based on which guidelines for choosing an optimal path were developed. The most common path types are systematically assessed based on the comparison of the simulation covariance matrices with the covariance of the underlying spatial model. Our analysis indicates that the optimal path is defined as the one minimizing the information lost by the omission of neighbors. Classification into clustering paths, that is, paths simulating consecutively close nodes, and declustering paths, that is, paths simulating consecutively distant nodes, was found to be an efficient way of determining path performance. Common examples of the latter are multi-grid, mid-point, and quasi-random paths, while the former include row-by-row and spiral paths. Indeed, clustering paths tend to inadequately approximate covariances at intermediate and large lag distances, because their neighborhood is only composed of nearby nodes. On the other hand, declustering paths minimize the correlation among nodes, thus ensuring that the neighbors are more diverse, and that only weakly correlated neighbors are omitted.


Visiting sequence Sequential simulation Artifact Covariance matrix Spiral path Random path Multi-grid path Quasi-random path 



This study has been supported by a Grant from the Swiss National Research Foundation.


  1. Abdu H, Robinson DA, Seyfried M, Jones SB (2008) Geophysical imaging of watershed subsurface patterns and prediction of soil texture and water holding capacity. Water Resour Res 44(4):W00D18. doi: 10.1029/2008WR007043 CrossRefGoogle Scholar
  2. Barnsley MF, Devaney RL, Mandelbrot BB, Peitgen HO, Saupe D, Voss RF (1988) The science of fractal images. Springer, New York. doi: 10.1007/978-1-4612-3784-6 CrossRefGoogle Scholar
  3. Boulanger F (1990) Modélisation et simulation de variables régionalisées par des fonctions aléatoires stables. Ph.D. thesis, Ecole des Mines de Paris, FontainebleauGoogle Scholar
  4. Box GEP, Jenkins GM, Reinsel GC (2008) Time series analysis, vol 37. Wiley, Hoboken. doi: 10.1002/9781118619193 CrossRefGoogle Scholar
  5. Chilès JP, Delfiner P (1999) Geostatistics, Wiley series in probability and statistics, vol 497. Wiley, Hoboken. doi: 10.1002/9780470316993 Google Scholar
  6. Daly C (2005) Higher order models using entropy, Markov random fields and sequential simulation. In: Leuangthong O, Deutsch CV (eds) Geostatistics Banff 2004. Quantitative Geology and Geostatistics, vol 14. Springer, DordrechtGoogle Scholar
  7. Day-Lewis FD, Lane JW (2004) Assessing the resolution-dependent utility of tomograms for geostatistics. Geophys Res Lett 31(7):L07,503. doi: 10.1029/2004GL019617 CrossRefGoogle Scholar
  8. Delbari M, Afrasiab P, Loiskandl W (2009) Using sequential Gaussian simulation to assess the field-scale spatial uncertainty of soil water content. Catena 79(2):163–169. doi: 10.1016/j.catena.2009.08.001 CrossRefGoogle Scholar
  9. Deutsch CV, Journel AG (1992) GSLIB: Geostatistical software library and user’s guide. Technical Representative, New YorkGoogle Scholar
  10. Dimitrakopoulos R, Luo X (2004) Generalized sequential Gaussian simulation on group size and screen-effect approximations for large field simulations. Math Geol 36(5):567–591. doi: 10.1023/B:MATG.0000037737.11615.df CrossRefGoogle Scholar
  11. Dimitrakopoulos R, Farrelly CT, Godoy M (2002) Moving forward from traditional optimization: grade uncertainty and risk effects in open-pit design. Min Technol 111(1):82–88. doi: 10.1179/mnt.2002.111.1.82 CrossRefGoogle Scholar
  12. Emery X (2004) Testing the correctness of the sequential algorithm for simulating Gaussian random fields. Stoch Env Res Risk Assess 18(6):401–413. doi: 10.1007/s00477-004-0211-7 CrossRefGoogle Scholar
  13. Emery X, Peláez M (2011) Assessing the accuracy of sequential Gaussian simulation and cosimulation. Comput Geosci 15(4):673–689. doi: 10.1007/s10596-011-9235-5 CrossRefGoogle Scholar
  14. Fournier A, Fussell D, Carpenter L (1982) Computer rendering of stochastic models. Commun ACM 25(6):371–384. doi: 10.1145/358523.358553 CrossRefGoogle Scholar
  15. Gómez-Hernández JJ, Cassiraga EF (1994) Theory and practice of sequential simulation. Kluwer Academic Publishers, Dordrecht. doi: 10.1007/978-94-015-8267-4_10 CrossRefGoogle Scholar
  16. Gómez-Hernández JJ, Journel AG (1993) Geostatistics Tróia ’92, quantitative geology and geostatistics, vol 5. Springer, Dordrecht. doi: 10.1007/978-94-011-1739-5 Google Scholar
  17. Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, OxfordGoogle Scholar
  18. Goovaerts P (2001) Geostatistical modelling of uncertainty in soil science. Geoderma 103(1–2):3–26. doi: 10.1016/S0016-7061(01)00067-2 CrossRefGoogle Scholar
  19. Halton JH (1960) On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer Math 2(1):84–90. doi: 10.1007/BF01386213 CrossRefGoogle Scholar
  20. Hansen TM, Journel AG, Tarantola A, Mosegaard K (2006) Linear inverse Gaussian theory and geostatistics. Geophysics 71(6):R101–R111. doi: 10.1190/1.2345195 CrossRefGoogle Scholar
  21. Isaaks EH (1991) The application of Monte Carlo methods to the analysis of spatially correlated data. Ph.D. thesis, Stanford UniversityGoogle Scholar
  22. Isaaks EH, Srivastava RM (1989) An introduction to applied geostatistics. Oxford University Press, New YorkGoogle Scholar
  23. Johnson ME (1987) Multivariate statistical simulation. Wiley series in probability and statistics. Wiley, Hoboken. doi: 10.1002/9781118150740 CrossRefGoogle Scholar
  24. Journel AG (1989) Fundamentals of geostatistics in five lessons, vol 16. American Geophysical Union, Washington. doi: 10.1029/SC008 CrossRefGoogle Scholar
  25. Juang KW, Chen YS, Lee DY (2004) Using sequential indicator simulation to assess the uncertainty of delineating heavy-metal contaminated soils. Environ Pollut 127(2):229–238. doi: 10.1016/j.envpol.2003.07.001 CrossRefGoogle Scholar
  26. Kocis L, Whiten WJ (1997) Computational investigations of low-discrepancy sequences. ACM Trans Math Softw 23(2):266–294. doi: 10.1145/264029.264064 CrossRefGoogle Scholar
  27. Lantuéjoul C (2002) Geostatistical simulation. Springer, Berlin. doi: 10.1007/978-3-662-04808-5 CrossRefGoogle Scholar
  28. Lee SY, Carle SF, Fogg GE (2007) Geologic heterogeneity and a comparison of two geostatistical models: sequential Gaussian and transition probability-based geostatistical simulation. Adv Water Resour 30(9):1914–1932. doi: 10.1016/j.advwatres.2007.03.005 CrossRefGoogle Scholar
  29. Leuangthong O, McLennan JA, Deutsch CV (2004) Minimum acceptance criteria for geostatistical realizations. Nat Resour Res 13(3):131–141. doi: 10.1023/ CrossRefGoogle Scholar
  30. Lin YP, Chang TK, Teng TP (2001) Characterization of soil lead by comparing sequential Gaussian simulation, simulated annealing simulation and kriging methods. Environ Geol 41(1–2):189–199. doi: 10.1007/s002540100382 CrossRefGoogle Scholar
  31. McLennan J (2002) The effect of the simulation path in sequential gaussian simulation. Technical Representative, University of AlbertaGoogle Scholar
  32. Meyer TH (2004) The discontinuous nature of kriging interpolation for digital terrain modeling. Cartogr Geogr Inf Sci 31(4):209–216. doi: 10.1559/1523040042742385 CrossRefGoogle Scholar
  33. Mowrer H (1997) Propagating uncertainty through spatial estimation processes for old-growth subalpine forests using sequential Gaussian simulation in GIS. Ecol Model 98(1):73–86. doi: 10.1016/S0304-3800(96)01938-2 CrossRefGoogle Scholar
  34. Omre H, Sølna K, Tjelmeland H (1993) Simulation of random functions on large lattices. In: Soares A (ed) Geostatistics Tròia ’92. Kluwer Academic Publishers, Dordrecht, pp 179–199. doi: 10.1007/978-94-011-1739-5_16 CrossRefGoogle Scholar
  35. Rivoirard J (1984) Le comportement des poids de krigeage. Ph.D. thesis, Ecole des Mines de Paris, FontainebleauGoogle Scholar
  36. Safikhani M, Asghari O, Emery X (2017) Assessing the accuracy of sequential gaussian simulation through statistical testing. Stoch Env Res Risk Assess 31(2):523–533. doi: 10.1007/s00477-016-1255-1 CrossRefGoogle Scholar
  37. Srinivasan BV, Duraiswami R, Murtugudde R (2008) Efficient kriging for real-time spatio-temporal interpolation Linear kriging. In: 20th conference on probablility and statistics in atmospheric sciences, pp 228–235Google Scholar
  38. Tran TT (1994) Improving variogram reproduction on dense simulation grids. Comput Geosci 20(7–8):1161–1168. doi: 10.1016/0098-3004(94)90069-8 CrossRefGoogle Scholar
  39. Trefethen LN, Bau D III (1997) Numerical linear algebra, vol 50. SIAM, PhiladelphiaCrossRefGoogle Scholar
  40. Verly GW (1993) Sequential Gaussian cosimulation: a simulation method integrating several types of information. In: Soares A (ed) Geostatistics Tròia ’92. Kluwer Academic Publishers, Dordrecht, pp 543–554. doi: 10.1007/978-94-011-1739-5_42 CrossRefGoogle Scholar
  41. Zhao Y, Xu X, Huang B, Sun W, Shao X, Shi X, Ruan X (2007) Using robust kriging and sequential Gaussian simulation to delineate the copper- and lead-contaminated areas of a rapidly industrialized city in Yangtze River Delta, China. Environ Geol 52(7):1423–1433. doi: 10.1007/s00254-007-0667-0 CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2017

Authors and Affiliations

  1. 1.Institute of Earth SciencesUniversity de LausanneLausanneSwitzerland
  2. 2.Institute of Earth Surface DynamicsUniversity de LausanneLausanneSwitzerland
  3. 3.Centre Eau Terre EnvironnementInstitut National de la Recherche ScientifiqueQuébecCanada

Personalised recommendations