A spectral method for clustering of rock discontinuity sets

We demonstrate the use of a spectral clustering algorithm as a novel approach for the identification of rock discontinuity sets based on discontinuity orientations. We use the spectral clustering approach with a simple measure of similarity between normal unit vectors in spherical space that is specific to the clustering of rock discontinuity orientations. The performance of the algorithm is studied using benchmark test cases with data sets corresponding to real rock masses. The results show that the algorithm provides good clustering results, providing partitions that agree well with the results of several other clustering algorithms that are commonly used in rock engineering. Furthermore, we show an example case with data sets of discontinuity orientation compiled from the literature, in which the spectral clustering algorithm provides more natural partitions than the other algorithms considered. Additional advantages of the algorithm are that convergence is fast, and that it can be easily (and efficiently) implemented using popular software packages for numerical analysis. r 2006 Elsevier Ltd. All rights reserved.

[1]  R. D. Terzaghi.  Sources of Error in Joint Surveys , 1965 .

[2]  R. J. Shanley,et al.  Delineation and analysis of clusters in orientation data , 1976 .

[3]  M. R. Johnson.  Structural geology , 1977, Nature.

[4]  S. Priest,et al.  ESTIMATION OF DISCONTINUITY SPACING AND TRACE LENGTH USING SCANLINE SURVEYS , 1981 .

[5]  J. Kent.  The Fisher‐Bingham Distribution on the Sphere , 1982 .

[6]  Mahtab,et al.  A Rejection Criterion For Definition Of Clusters In Orientation Data , 1982 .

[7]  S. Priest,et al.  Hemispherical Projection Methods in Rock Mechanics , 1985 .

[8]  Pinnaduwa H.S.W. Kulatilake,et al.  Analysis of structural homogeneity of rock masses , 1990 .

[9]  H. H. Einstein,et al.  Problems with Representation of Rock-Fracture Clusters , 1991 .

[10]  S. Priest.  Discontinuity Analysis for Rock Engineering , 1992 .

[11]  Stephen D. Priest,et al.  The Collection and Analysis of Discontinuity Orientation Data for Engineering Design, with Examples , 1993 .

[12]  R. Goodman.  Block theory and its application , 1995 .

[13]  W. Dershowitz,et al.  A Stochastic Approach For Fracture Set Definition , 1996 .

[14]  P. Kulatilake,et al.  Box fractal dimension as a measure of statistical homogeneity of jointed rock masses , 1997 .

[15]  Jitendra Malik,et al.  Normalized cuts and image segmentation , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[16]  R. E. Hammah,et al.  Fuzzy cluster algorithm for the automatic identification of joint sets , 1998 .

[17]  Lianyang Zhang,et al.  Estimating the Mean Trace Length of Rock Discontinuities , 1998 .

[18]  R. Hammah,et al.  On Distance Measures for the Fuzzy K-means Algorithm for Joint Data , 1999 .

[19]  Yair Weiss,et al.  Segmentation using eigenvectors: a unifying view , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[20]  Reginald E. Hammah,et al.  Validity Measures for the Fuzzy Cluster Analysis of Orientations , 2000, IEEE Trans. Pattern Anal. Mach. Intell..

[21]  Michael I. Jordan,et al.  On Spectral Clustering: Analysis and an algorithm , 2001, NIPS.

[22]  Manhal Sirat,et al.  Application of artificial neural networks to fracture analysis at the Äspö HRL, Sweden : fracture sets classification , 2001 .

[24]  P. Starzec,et al.  Probabilistic predictions regarding key blocks using stochastic discrete fracture networks – example from a rock cavern in south-east Sweden , 2002 .

[25]  D. Marcotte,et al.  Automatic joint set clustering using a mixture of bivariate normal distributions , 2002 .

[26]  Wei Zhou,et al.  Implementation of multivariate clustering methods for characterizing discontinuities data from scanlines and oriented boreholes , 2002 .

[27]  P. Kulatilake,et al.  Stochastic fracture geometry modeling in 3-D including validations for a part of Arrowhead East Tunnel, California, USA , 2003 .

[28]  D. Tannant,et al.  A technique for identifying structural domain boundaries at the EKATI Diamond Mine , 2004 .

[29]  K. Obermayer,et al.  A new clustering approach for partitioning directional data , 2005 .

[30]  Hyuck-Jin Park,et al.  Probabilistic analysis of rock slope stability and random properties of discontinuity parameters, Interstate Highway 40, Western North Carolina, USA , 2005 .

[31]  A. Zille,et al.  The running-cone method for the interpretation of conical fold geometries: an example from the Badia Valley, Northern Dolomites (NE Italy) , 2005 .