Mathematical Geosciences

, Volume 41, Issue 4, pp 447-474

First online:

Stationarity Scores on Training Images for Multipoint Geostatistics

  • Piotr W. MirowskiAffiliated withCourant Institute of Mathematical Sciences, New York University Email author 
  • , Daniel M. TetzlaffAffiliated withSchlumberger Information Solutions
  • , Roy C. DaviesAffiliated withGeology Department, University of Bergen
  • , David S. McCormickAffiliated withSchlumberger-Doll Research
  • , Nneka WilliamsAffiliated withDivision of Geological and Planetary Sciences, California Institute of Technology
  • , Claude SignerAffiliated withSchlumberger-Doll Research

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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