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Automatic Identification of Rock Fracture Sets Using Finite Mixture Models

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Abstract

The clustering and classification of fracture orientation data are crucial tasks in geotechnical engineering and rock engineering design. The explicit simulation of fracture orientations is always applied to compensate for the lack of direct measurements over the entire rock mass. In this study, a single step approach based on the theory of finite mixture models, where the component distributions are Fisher distributions, is proposed for automatic clustering and simulation of fracture orientation data. In the proposed workflow, the spherical K-means algorithm is applied to select the initial cluster centers, and the component-wise expectation–maximization algorithm using the minimum message length criterion is used to automatically determine the optimal number of fracture sets. An additional advantage of the proposed method is the representation of orientation data using a full sphere, instead of the conventional hemispherical characterization. The use of a full spherical representation effectively solves the issue of clustering for fractures with high dip angles. In addition, the calculation process of the mean direction is also simplified. The effectiveness of the model-based clustering method is tested with a complicated artificial data set and two real world data sets. Cluster validity is introduced to evaluate the clustering results. In addition, two other clustering algorithms are also presented for comparison. The results demonstrate that the proposed method can successfully detect the optimal number of clusters, and the parameters of the distributions are well estimated. In addition, the proposed method also exhibits good computational performance.

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Acknowledgements

The authors gratefully acknowledge the support from the State Key Program of National Natural Science Fund of China (Grant No. 41330636), the National Natural Science Fund of China (Grant Nos. 41402242 and 41302218), and the Graduate Innovation Fund of Jilin University (Grant Nos. 2017137 and 2016208). We would like to thank the editors and two anonymous reviewers for their in depth comments and valuable suggestions.

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Authors and Affiliations

  1. College of Construction Engineering, Jilin University, Changchun, 130026, China

    Jiewei Zhan, Jianping Chen, Peihua Xu, Wen Zhang, Xudong Han & Xin Zhou

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  1. Jiewei Zhan
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  2. Jianping Chen
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  3. Peihua Xu
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  4. Wen Zhang
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Correspondence to Peihua Xu.

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Zhan, J., Chen, J., Xu, P. et al. Automatic Identification of Rock Fracture Sets Using Finite Mixture Models. Math Geosci 49, 1021–1056 (2017). https://doi.org/10.1007/s11004-017-9702-1

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