Bayesian analysis for uncertainty quantification of in situ stress data

Abstract Estimates of in situ stress state may be unreliable due to the inherent variation of stresses in rock masses coupled with the usual lack of sufficient stress data. This renders uncertainty quantification critically important for stress estimation, as it both permits quantitative assessment of the reliability of estimated stresses and facilitates application of reliability-based design in rock engineering. This paper presents a Bayesian approach that can probabilistically quantify uncertainty in both mean stress estimation and predicted stresses. We show that the quantified uncertainty supports our general understanding in that (i) more stress data tend to result in more reliable estimates, and (ii) the usual case of small numbers of stress data is liable to yield highly uncertain estimates of the mean stress. The results reveal that large uncertainty may exist in estimates of both the magnitudes and orientations of the principal mean stress, and this suggests that in practice the estimation reliability should be considered for not only stress magnitudes but also stress orientations. The results also show that the three principal mean stress components may display different degrees of uncertainty, thereby highlighting the importance of identifying which mean stress components are of most interest for the specific engineering objective. Finally, we demonstrate how, within the Bayesian framework, the large uncertainty in mean stress estimates arising from limited stress data can be effectively reduced by means of informative priors.