# Improving the Pattern Reproducibility of Multiple-Point-Based Prior Models Using Frequency Matching

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## Abstract

Some multiple-point-based sampling algorithms, such as the *snesim* algorithm, rely on sequential simulation. The conditional probability distributions that are used for the simulation are based on statistics of multiple-point data events obtained from a training image. During the simulation, data events with zero probability in the training image statistics may occur. This is handled by pruning the set of conditioning data until an event with non-zero probability is found. The resulting probability distribution sampled by such algorithms is a pruned mixture model. The pruning strategy leads to a probability distribution that lacks some of the information provided by the multiple-point statistics from the training image, which reduces the reproducibility of the training image patterns in the outcome realizations. When pruned mixture models are used as prior models for inverse problems, local re-simulations are performed to obtain perturbed realizations. Consequently, these local re-simulations lead to additional pruning in the set of conditioning data, which further deteriorates the pattern reproduction. To mitigate this problem, it is here suggested to combine the pruned mixture model with a frequency matching model. The multiple-point statistics of outcome realizations from this combined model has improved degree of match with the statistics from the training image. An efficient algorithm that samples this combined model is suggested. Finally, a tomographic cross-borehole inverse problem with prior information expressed by the combined (prior) model is used to demonstrate the effect of pattern reproducibility on the resolution of an inverse problem.

## Keywords

Markov model Cross-borehole tomography Probabilistic inverse problem Multiple-point statistics Training image Sequential simulation Frequency matching Metropolis algorithm## Notes

### Acknowledgments

We thank DONG E&P for financial support. Moreover, we are very grateful for the comments and suggestions that were provided by associated editor Odd Kolbjørnsen and two anonymous reviewers, which we believe improved this paper considerably.

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