Mathematical Geosciences

, Volume 41, Issue 4, pp 447–474

Stationarity Scores on Training Images for Multipoint Geostatistics


    • Courant Institute of Mathematical SciencesNew York University
  • Daniel M. Tetzlaff
    • Schlumberger Information Solutions
  • Roy C. Davies
    • Geology DepartmentUniversity of Bergen
  • David S. McCormick
    • Schlumberger-Doll Research
  • Nneka Williams
    • Division of Geological and Planetary SciencesCalifornia Institute of Technology
  • Claude Signer
    • Schlumberger-Doll Research

DOI: 10.1007/s11004-008-9194-0

Cite this article as:
Mirowski, P.W., Tetzlaff, D.M., Davies, R.C. et al. Math Geosci (2009) 41: 447. doi:10.1007/s11004-008-9194-0


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.


StationarityOrientationMulti-scale analysisMultiple point simulationTraining image
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© International Association for Mathematical Geosciences 2008