Scalar-valued measures of stress dispersion

In situ stress is an important parameter in rock mechanics, but localised measurements often display significant variability; for meaningful analyses it is essential that such variability is appropriately quantified. Among many statistics, dispersion, which denotes how scattered or spread out a data group is, is an effective tool to quantify the amount of variability. However, dispersion measures are commonly only used for scalar and vector data, and it is not yet clear what robust scalar-valued measures of stress dispersion - i.e. measures that are faithful to the tensorial nature of stress - are available. Here, using stress tensors referred to a common Cartesian coordinate system, we consider several dispersion measures, namely, Euclidean dispersion (a tensor version of standard deviation), and the three widely used multivariate dispersions of total variation, generalised variance and effective variance, for scalar-valued quantification of stress variability and to improve the existing related work. We compare these measures, show how they are linked to the covariance matrix of tensor components, and derive their invariance with respect to change of coordinate system. Through the use of synthetic two-dimensional stress data we demonstrate that these measures can effectively characterise the dispersion of stress data. Further analysis of randomly generated three-dimensional stress data reveals that generalised variance and effective variance, which consider both variances of, and covariances between, tensor components, are more effective than Euclidean dispersion and total variation which ignore covariances. The transformational invariance of generalised variance and effective variance allows these measures to be applied in any convenient coordinate system.