Mathematical Geosciences

, Volume 41, Issue 4, pp 447–474 | Cite as

Stationarity Scores on Training Images for Multipoint Geostatistics

  • Piotr W. MirowskiEmail author
  • Daniel M. Tetzlaff
  • Roy C. Davies
  • David S. McCormick
  • Nneka Williams
  • Claude Signer


This research introduces a novel method to assess the validity of training images used as an input for Multipoint Geostatistics, alternatively called Multiple Point Simulation (MPS). MPS are a family of spatial statistical interpolation algorithms that are used to generate conditional simulations of property fields such as geological facies. They are able to honor absolute “hard” constraints (e.g., borehole data) as well as “soft” constraints (e.g., probability fields derived from seismic data, and rotation and scale). These algorithms require 2D or 3D training images or analogs whose textures represent a spatial arrangement of geological properties that is presumed to be similar to that of a target volume to be modeled. To use the current generation of MPS algorithms, statistically valid training image are required as input. In this context, “statistical validity” includes a requirement of stationarity, so that one can derive from the training image an average template pattern. This research focuses on a practical method to assess stationarity requirements for MPS algorithms, i.e., that statistical density or probability distribution of the quantity shown on the image does not change spatially, and that the image shows repetitive shapes whose orientation and scale are spatially constant. This method employs image-processing techniques based on measures of stationarity of the category distribution, the directional (or orientation) property field and the scale property field of those images. It was successfully tested on a set of two-dimensional images representing geological features and its predictions were compared to actual realizations of MPS algorithms. An extension of the algorithms to 3D images is also proposed. As MPS algorithms are being used increasingly in hydrocarbon reservoir modeling, the methods described should facilitate screening and selection of the input training images.


Stationarity Orientation Multi-scale analysis Multiple point simulation Training image 


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  1. Caers J (2003) Geostatistical history matching under training-image based geological model constraints. SPE J 8(3):218–226 Google Scholar
  2. Caers J (2005) Petroleum geostatistics. SPE interdisciplinary primer series. Society of Petroleum Engineers, Richardson, 88 p Google Scholar
  3. Caers J et al (2002) Multiple-point geostatistics: a quantitative vehicle for integrating geologic analogs into multiple reservoir models. In: Grammer GM (ed) AAPG memoir, integration of outcrop and modern analogs in reservoir modeling Google Scholar
  4. Deng H, Clausi D (2004) Gaussian MRF rotation-invariant features for image classification. IEEE Trans Pattern Anal Mach Intell 26(7):951–955 CrossRefGoogle Scholar
  5. Feng X (2003) Analysis and approaches to image local orientation estimation. Unpublished master’s dissertation, University of California at Santa Cruz, 71 p Google Scholar
  6. Haralick RM, Shanmugham KS, Dinstein I (1973) Texture features for image classification. IEEE Trans Syst Man Cybernetics SMC-3(6):610–621 CrossRefGoogle Scholar
  7. Krzanowski WJ (1988) Principles of multivariate analysis: a user’s perspective. Oxford Statistical science series, vol. 3. Oxford University Press, London, 584 p Google Scholar
  8. Kurani AS, Xu DH, Furst J, Stan Raicu D (2004) Co-occurrence matrices for volumetric data. In: 7th IASTED international conference on computer graphics and imaging—CGIM, Kauai, Hawaii, August 16–18, 2004 Google Scholar
  9. Liu et al. (2004) Multiple-point simulation integrating wells, three-dimensional seismic data, and geology. Am Assoc Petr Geol Bull 88(7):905–921 Google Scholar
  10. Mardia KV (1972) Statistics of directional data. Academic Press, San Diego, 357 p Google Scholar
  11. Materka A, Strzelecki M (1998) Texture analysis methods—a review. Technical University of Lodz, Institute of Electronics, COST B11 report, Brussels, pp 9–11 Google Scholar
  12. NIST/SEMATECH (2005) e-Handbook of statistical methods.
  13. Randen T, Monsen E, Signer C, Abrahamsen A, Hansen JO, Sater T, Schlaf J, Sonneland L, Schlumberger Stavanger Research (2000) Three-dimensional texture attributes for seismic data analysis. In: Society of exploration geophysicists annual meeting, Calgary, Alberta, August 6–11, 2000 Google Scholar
  14. Rock NMS (1988) Lecture notes in earth sciences 18—Numerical geology. Springer, Berlin, pp 228–238 Google Scholar
  15. Strebelle S (2002) Conditional simulation of complex geological structures using multiple-point statistics. Math Geol 34(1):1–22 CrossRefGoogle Scholar
  16. Strebelle S, Payrazyan K, Caers J (2003) Modeling of a deepwater turbidite reservoir conditional to seismic data using principal component analysis and multiple-point geostatistics. SPE J 8(3):227–235 Google Scholar
  17. Tetzlaff D, Davies R, McCormick D, Signer C, Mirowski P, Williams N (2005) Application of multipoint geostatistics to honour multiple attribute constraints applied to a deepwater outcrop analog. In: Tanqua Karoo Basin, South Africa. Society of exploration geophysicists 75th annual meeting, Houston, Texas, November 6–11, 2005 Google Scholar

Copyright information

© International Association for Mathematical Geosciences 2008

Authors and Affiliations

  • Piotr W. Mirowski
    • 1
    Email author
  • Daniel M. Tetzlaff
    • 2
  • Roy C. Davies
    • 3
  • David S. McCormick
    • 4
  • Nneka Williams
    • 5
  • Claude Signer
    • 4
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Schlumberger Information SolutionsHoustonUSA
  3. 3.Geology DepartmentUniversity of BergenBergenNorway
  4. 4.Schlumberger-Doll ResearchCambridgeUSA
  5. 5.Division of Geological and Planetary SciencesCalifornia Institute of TechnologyPasadenaUSA

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