Insights into the statistical relationship between focal mechanisms and stress from synthetic experiments

Abstract: [Objective] The extent to which the spatial distribution patterns (particularly the clustering characteristics) of fault nodal planes or the P, B, and T axes of a set of focal mechanism data can provide information about background stress causing earthquakes has long been a controversial academic topic. [Methods] This study systematically investigates this issue through synthetic experiments designed on the basis of the stress–fault slip relationship, with stress parameters including the orientations of the three principal stresses and the stress shape ratio (R). [Results] The experimental results demonstrate that the spatial distribution patterns of both fault nodal planes and PBT axes are jointly controlled by the stress shape ratio and the fault failure conditions. In most cases, the two nodal planes exhibit widely scattered spatial distributions. Only when the shape ratio is close to 0.5 and the contact area between the Mohr-Coulomb failure envelope and Mohr’s circle is minimized do the distributions of both the actual and the auxiliary planes become relatively concentrated. Under these specific conditions, the fault nodal planes (their normals) gain statistical significance for estimating stress orientations. Identifying the actual fault plane among the two nodal planes in focal mechanisms would enhance the determination of principal stress directions. Notably, the spatial distribution of PBT axes effectively captures both the principal stress orientations and the shape ratio. Key findings include: ① Due to the influence of fault failure conditions and the shape ratio, the P, B, and T axes may not cluster or disperse simultaneously. However, when clustering occurs, they converge near the σ1 , σ2 , or σ3  axes, respectively. ② A ring-shaped (toroidal) distribution of T axes indicates a high R-value. ③ P and T axes never exhibit fully random scattering; if such disorder is observed in real data, it suggests that the focal mechanisms may not share a common stress regime. [Conclusion] This study provides critical constraints for evaluating whether focal mechanism data used in stress inversion belong to a unified stress regime and for predicting stress parameters from the distribution of PBT axes. [Significance] These results offer significant implications for developing and applying stress inversion methodology using focal mechanisms.