Automatic joint set clustering using a mixture of bivariate normal distributions

Many engineering studies require an explicit Simulation of joints to compensate for the lack of direct measurements over the entire rock mass. For this simulation, a precise and concise description of the joint distribution is necessary. Due to the Usual multimodal nature of joint data, previous methods typically used a two-step procedure where homogenous. joint sets are first identified and then a parametric distribution is estimated separately for each set. Here we describe a single step approach based on the concept of finite mixture distributions where the component distributions of the mixture are Supposed bivariate normal. The truncation to the unit circle of the bivariate normal distribution implicitly defines a valid distribution on the sphere. The bivariate normality assumption enables to obtain maximum likelihood estimates of the mixture distribution by eigen-analysis of set orientation matrices. We develop an iterative and automatic procedure for set identification and parameter estimation. and present the equations for plotting the confidence ellipse of each set. Simulations indicate that: (a) the approximate test based on the increase of the log-likelihood is helpful in identifying the number of significant sets; (b) the orientation parameters used in the simulation are well retrieved with the proposed algorithm and (c) the dispersion parameters are more sensitive to the degree of overlap of the simulated joint sets. The method is applied to a real data set from the Murdochville area. (C) 2002 Elsevier Science Ltd. All rights reserved.