Clustering of fracture orientations using a mixed Bingham distribution and its application to paleostress analysis from dike or vein orientations

The clustering and classification of fracture orientations are important in rock mechanics and in brittle tectonics, the latter of which includes the paleostress analysis of extension fractures hosting dikes or mineral veins. Here, we present an unsupervised clustering method for the orientations of extension fractures using mixed Bingham distributions. The method not only detects the elliptical clusters and girdles made by the poles to such planar features, but also determines the appropriate number of those groups by means of Bayesian information criterion (BIC) without a priori information. The method was tested with artificial data sets, and successfully detected the assumed groups, when the clusters had little overlaps. However, clusters with the common maximum concentration orientation and large aspect ratios were distinguished, provided that their minimum concentration orientations were separated by a large angle. Our method separated two stress states from natural data from a Miocene dike swarm in SW Japan. The method also evaluated the probabilities of the stresses to form each of the dike.