A New High-Order, Nonstationary, and Transformation Invariant Spatial Simulation Approach

  • Amir Abbas Haji AbolhassaniEmail author
  • Roussos Dimitrakopoulos
  • Frank P. Ferrie
Part of the Quantitative Geology and Geostatistics book series (QGAG, volume 19)


This paper presents a new high-order, nonstationary sequential simulation approach, aiming to deal with the typically complex, curvilinear structures and high-order spatial connectivity of the attributes of natural phenomena. Similar to multipoint methods, the proposed approach employs spatial templates and a group of training images (TI). A coarse template with a fixed number of data points and a missing value in the middle is used, where the missing value is simulated conditional to a data event found in the neighborhood of the middle point of the template, under a Markovian assumption. Sliding the template over the TI, a pattern database is extracted. The parameters of the conditional distributions needed for the sequential simulation are inferred from the pattern database considering a set of weights of contribution given for the patterns in the database. Weights are calculated based on the similarity of the high-order statistics of the data event of the hard data compared to those of the training image. The high-order similarity measure introduced herein is effectively invariant under all linear spatial transformations.

Following the sequential simulation paradigm, the template chosen is sequentially moved on a raster path until all missing points/nodes are simulated. The high-order similarity measure allows the approach to be fast as well as robust to all possible linear transformations of a training image. The approach respects the hard data and its spatial statistics, because it only considers TI replicate data events with similar high-order statistics. Results are promising.


Training Image Data Event Hard Data Sequential Simulation Conditioning Data 
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.



Funding was provided by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant 239019 and mining industry partners of the COSMO Laboratory (AngloGold Ashanti, Barrick Gold, BHP Billiton, De Beers Canada, Kinross Gold, Newmont Mining, and Vale) and the Group for Research in Decision Analysis (GERAD). Thanks are given to Prasun Lala for his assistance.


  1. Abdollahifard MJ, Faez K (2012) Stochastic simulation of patterns using Bayesian pattern modeling. Comput Geosci 17(1):99–116CrossRefGoogle Scholar
  2. Arpat GB, Caers J (2007) Conditional simulation with patterns. Math Geol 39(2):177–203CrossRefGoogle Scholar
  3. Chatterjee S, Dimitrakopoulos R (2012) Multi-scale stochastic simulation with wavelet-based approach. Comput Geosci 45(3):177–189CrossRefGoogle Scholar
  4. 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(1):65–99CrossRefGoogle Scholar
  5. Funkhouser HG (1930) A short account of the history of symmetric functions of roots of equations. Am Math Mon 37(7):357–365CrossRefGoogle Scholar
  6. Goovaerts P (1998) Geostatistics for natural resources evaluation. Cambridge University Press, CambridgeGoogle Scholar
  7. Guardiano FB, Srivastava RM (1993) Multivariate geostatistics: beyond bivariate moments. Geosatistics Tróia’92, Springer, pp 133–144Google Scholar
  8. Honarkhah M, Caers J (2010) Stochastic simulation of patterns using distance-based pattern modeling. Math Geosci 42(5):487–517CrossRefGoogle Scholar
  9. Mao S, Journel AG (1999) Generation of a reference petrophysical and seismic 3D data set: the Stanford V reservoir. Report—Stanford Center for Reservoir Forecasting Annual Meeting, Stanford. URL:
  10. Mariethoz G, Renard P, Straubhaar J (2010) The direct sampling method to perform multiple-point geostatistical simulations. Water Resour Res 46(11):W11536. doi:  10.1029/2008WR007621
  11. Minniakhmetov I, Dimitrakopoulos R (2016a) Joint high-order simulation of spatially correlated variables using high-order spatial statistics. Math Geosci, in pressGoogle Scholar
  12. Minniakhmetov I, Dimitrakopoulos R (2016b) A high-order, data-driven framework for joint simulation of categorical variables. GEOSTAT2016, in this volumeGoogle Scholar
  13. Mustapha H, Dimitrakopoulos R (2010a) A new approach for geological pattern recognition using high-order spatial cumulants. Comput Geosci 36(3):313–334CrossRefGoogle Scholar
  14. Mustapha H, Dimitrakopoulos R (2010b) High-order stochastic simulation of complex spatially distributed natural phenomena. Math Geosci 42(5):457–485CrossRefGoogle Scholar
  15. Mustapha H, Dimitrakopoulos R (2011) HOSIM: a high-order stochastic simulation algorithm for generating three-dimensional complex geological patterns. Comput Geosci 37(9):1242–1253CrossRefGoogle Scholar
  16. Mustapha H, Chatterjee S, Dimitrakopoulos R (2013) CDFSIM: efficient stochastic simulation through decomposition of cumulative distribution functions of transformed spatial patterns. Math Geosci 46(1):95–123CrossRefGoogle Scholar
  17. Stien M, Kolbjørnsen O (2011) Facies modeling using a Markov mesh model specification. Math Geosci 43(6):611–624CrossRefGoogle Scholar
  18. Strebelle S (2002) Conditional simulation of complex geological structures using multiple-point statistics. Math Geol 34(1):1–21CrossRefGoogle Scholar
  19. Wang Z, Bovik AC, Sheikh HR, Simoncelli EP (2004) Image quality assessment: from error visibility to structural similarity. IEEE Trans Image Process 13(4):600–612CrossRefGoogle Scholar
  20. Zhang T, Switzer P, Journel A (2006) Filter-based classification of training image patterns for spatial simulation. Math Geol 38(1):63–80CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Amir Abbas Haji Abolhassani
    • 1
    • 2
    Email author
  • Roussos Dimitrakopoulos
    • 2
  • Frank P. Ferrie
    • 1
  1. 1.Electrical and Computer Engineering DepartmentMcGill UniversityMontrealCanada
  2. 2.COSMO – Stochastic Mine PlanningMcGill UniversityMontrealCanada

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