Stress tensor similarity index based on Euclidean distance for numerical back analysis of in situ stress fields

Evaluating the similarity between measured and predicted stress states is critical for the numerical back analysis of three-dimensional in situ stress fields. However, conventional nontensor approaches process stress components or the magnitude and orientation of principal stresses separately, which may yield contradictory results among multiple indices. To address this issue, we propose the Euclidean distance similarity (EDS), which is a dimensionless single index based on spatial distance. As a case study, EDS was applied to a site with complex geological conditions for verification. EDS can be derived in the form of stress component or principal stress. EDS exhibited a positive correlation with the initial difference and magnitude error of principal stresses and changed periodically with the Euler angle of rotation around the principal stress. Moreover, three magnitude errors and three Euler angles had different effects on EDS. Finally, EDS can be classified into five levels (i.e., extremely high, high, normal, low, and very low) according to the magnitude errors of principal stresses and Euler angles, where the “normal” level is recommended as a general error standard. The thresholds of EDS differ according to the initial stress level. These results are a valuable contribution to the similarity evaluation of stress tensors.