In situ stress is an important parameter in many aspects of rock mechanics and often displays significant variability. Among many statistics, dispersion, which denotes how scatter or spread a data group is, is an effective tool to quantify the amount of variability. One common measure of dispersion is the standard deviation. However, this is only applicable to scalar data, and a robust approach to calculate a scalar-valued measure of stress dispersion is still not clear. This is mainly because of the tensorial nature of stress, which renders classical statistics inapplicable.
As multivariate statistics are widely used to analyse stress tensor components, here we investigate the use of three widely used scalar-valued scatter measures in multivariate statistics- namely, generalised variance, total variation and effective variance - to help identify a suitable scalar-valued measure to characterise stress variability. We compare these scatter measures with Euclidean dispersion, a tensorial statistic defined by us previously.
We show how the multivariate scatter measures are linked to Euclidean dispersion, and through the use of synthetic stress data illustrate that, in particular circumstances, all these measures can effectively characterise the variability of stress data. However, we have identified that there is significant potential for the multivariate total variation to generate erroneous results, due to a lack of coordinate invariance in the formulation of this statistic. Therefore, to remain loyal to the tensorial nature of stress, and avoid potential misuse, we recommend using the tensor-related Euclidean dispersion for characterisation of in situ stress variability.
In situ stress is an important parameter for a wide range of endeavours, including rock engineering design, hydraulic fracturing analysis, rock mass permeability and earthquake potential evaluation (Amadei & Stephansson, 1997; Zoback, 2010; Latham et al., 2013; Matsumoto et al., 2015). Because of the inherent complexity of fractured rock masses in terms of varying rock properties, the presence of discontinuities and unclear boundary conditions (Matsumoto et al., 2015), stress in rock often displays significant heterogeneity (Martin, 1990). For all rock engineering endeavours the characterisation and quantification of stress variability is therefore important.