History Matching Through a Smooth Formulation of Multiple-Point Statistics

Yulia Melnikova, Andrea Zunino, Katrine Lange, Knud Skou Cordua, Klaus Mosegaard

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We propose a smooth formulation of multiple-point statistics that enables us to solve inverse problems using gradient-based optimization techniques. We introduce a differentiable function that quantifies the mismatch between multiple-point statistics of a training image and of a given model. We show that, by minimizing this function, any continuous image can be gradually transformed into an image that honors the multiple-point statistics of the discrete training image. The solution to an inverse problem is then found by minimizing the sum of two mismatches: the mismatch with data and the mismatch with multiple-point statistics. As a result, in the framework of the Bayesian approach, such a solution belongs to a high posterior region. The methodology, while applicable to any inverse problem with a training-image-based prior, is especially beneficial for problems which require expensive forward simulations, as, for instance, history matching. We demonstrate the applicability of the method on a two-dimensional history matching problem. Starting from different initial models we obtain an ensemble of solutions fitting the data and prior information defined by the training image. At the end we propose a closed form expression for calculating the prior probabilities using the theory of multinomial distributions, that allows us to rank the history-matched models in accordance with their relative posterior probabilities. © 2014 The Author(s).
Original languageEnglish
JournalMathematical Geosciences
Issue number4
Pages (from-to)397-416
Publication statusPublished - 2014

Bibliographical note

This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


  • Earth and Planetary Sciences (all)
  • Mathematics (miscellaneous)
  • History matching
  • Inverse problems
  • Multiple-point statistics
  • Optimization


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