Bayesian Analysis Limitations for Determination of the Unconfined Compressive Strength from Direct and Correlated Indirect Tests

Abstract The unconfined compressive strength (UCS) (with test value σc) test of rock provides an important design parameter for rock engineering projects. Obtaining a random distribution of directly measured UCS test data are often expensive and spatially constrained by drill hole locations and testing costs. Indirect UCS testing methods, such as point load and Leeb hardness tests are cost-effective. UCS tests typically contain statistically small sample sizes and are prone to specimen selection bias toward stronger rock. Meanwhile, indirect tests can be taken more readily but are subject to potential correlation error. In this paper, the increasingly popular Bayesian analysis was explored as a rational basis to incorporate σc estimated by several testing methods to determine a suitable design value of σc. Bayesian statistics uses an initial assumption (prior distribution) and revised, or “new,” data to develop a likelihood function. The prior and the likelihood function are combined to determine a posterior estimate distribution and design strength value. Real case history data were analyzed to provide insight into practical application of Bayesian analysis, particularly the issues associated with incorporating real data. It was found that Bayesian analysis-based probability density function parameters were dominated by the number of tests from each method, making it difficult to rationally incorporate the data from different sources. The Bayesian approach was compared with more conventional statistical methods of σc design value determination, particularly the use of weighted average. It was revealed that careful judgment was required regarding the suitability of various indirect test statistical parameters, especially the standard deviation of the Leeb hardness test data. Ultimately, the Bayesian analysis alone was found limited as a means to provide an objective estimate of the design value of σc and that more sophisticated analyses (e.g., Markov Chain Monte Carlo) combined with Bayesian analysis are required.