Multivariate distribution model for stress variability characterisation

In situ stress is an important parameter in rock mechanics, but localised measurements of stress often display significant variability. For improved understanding of in situ matrices that satisfy both Eq. stress it is important that this variability be correctly characterised, and for this a robust statistical distribution model - one that is faithful to the tensorial nature of stress - is essential. Currently, variability in stress measurements is customarily characterised using separate scalar and vector distributions for principal stress magnitude and orientation respectively. These customary scalar/vector approaches, which violate the tensorial nature of stress, together with other quasi-tensorial applications found in the literature that consider the tensor components as statistically independent variables, may yield biased results. To overcome this, we propose using a multivariate distribution model of distinct tensor components to characterise the variability of stress tensors referred to a common Cartesian coordinate system. We discuss why stress tensor variability can be sufficiently and appropriately characterised by its distinct tensor components in a multivariate manner, and demonstrate that the proposed statistical model gives consistent results under coordinate system transformation. Transformational invariance of the probability density function (PDF) is also demonstrated, and shows that stress state probability is independent of the coordinate system. Thus, stress variability can be characterised in any convenient coordinate system. Finally, actual in situ stress results are used to confirm the multivariate characteristics of stress data and the derived properties of the proposed multivariate distribution model, as well as to demonstrate how the quasitensorial approach may give biased results. The proposed statistical distribution model not only provides a robust approach to characterising the variability of stress in fractured rock mass, but is also expected to be useful in risk- and reliability-based rock engineering design.