Probabilistic prediction on three-dimensional roughness of discontinuity based on two-dimensional traces under rock tunnel excavation based on Bayesian theory

Three-dimensional (3D) roughness of discontinuity affects the quality of the rock mass, but 3D roughness is hard to be measured due to that the discontinuity is invisible in the engineering. Two-dimensional (2D) roughness can be calculated from the visible traces, but it is difficult to obtain enough quantity of the traces to directly derive 3D roughness during the tunnel excavation. In this study, a new method using Bayesian theory is proposed to derive 3D roughness from the low quantity of 2D roughness samples. For more accurately calculating 3D roughness, a new regression formula of 2D roughness is established firstly based on wavelet analysis. The new JRC3D prediction model based on Bayesian theory is then developed, and Markov chain Monte Carlo (MCMC) sampling is adopted to process JRC3D prediction model. The discontinuity sample collected from the literature is used to verify the proposed method. Twenty groups with the sampling size of 2, 3, 4, and 5 of each group are randomly sampled from JRC2D values of 170 profiles of the discontinuity, respectively. The research results indicate that 100%, 90%, 85%, and 60% predicting JRC3D of the sample groups corresponding to the sampling size of 5, 4, 3, and 2 fall into the tolerance interval [JRCtrue–1, JRCtrue + 1]. It is validated that the sampling size of 5 is enough for predicting JRC3D. The sensitivities of sampling results are then analyzed on the influencing factors, which are the correlation function, the prior distribution, and the prior information. The discontinuity across the excavation face at ZK78 + 67.5 of Daxiagu tunnel is taken as the tunnel engineering application, and the results further verify that the predicting JRC3D with the sampling size of 5 is generally in good agreement with JRC3D true values.