Insights into the statistical relationship between focal mechanisms and stress from synthetic experiments
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摘要: 文章依据应力与断层滑动的关系设计合成实验，系统研究了在同一应力体系下的不同断层上发生地震，这组震源机制的断层节面或P轴、B轴和T轴的空间分布形式能否反映出应力信息的问题。应力信息包括3个主应力的空间方位以及应力形因子（R）。实验结果表明，在大多数情况下，震源机制的2个节面在空间中的分布范围很广，不能有效地反映出主应力的方向信息，而P轴、B轴和T轴的空间分布可以很好地反映主应力的方向和R值的信息，具体为：受断层破裂条件和本身应力R值的影响，P轴、B轴和T轴的分布可能不是同时丛集或离散的，但不论哪个轴丛集分布，都会分别丛集于主应力σ1轴、σ2轴和σ3轴的周围；若T轴呈现环形分布，可以说明应力的R值较大；P轴和T轴不会出现无规律混乱分布的现象，若实际数据中观察到这种现象，预示着这些数据不适于同一应力背景。研究成果可以用来评估反演应力所使用的震源机制数据是否属于同一应力体系，以及根据P轴、B轴和T轴的分布预测应力结果，对基于震源机制反演应力的方法和应用研究都具有重要的参考意义。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 $ {\sigma }_{1} $, $ {\sigma }_{2} $, or $ {\sigma }_{3} $ 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. -
Key words:
- focal mechanism /
- fault nodal planes /
- PBT axes /
- stress tensor
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图 1 失稳系数的定义以及不同失稳系数的断层在摩尔圆中的位置
横纵坐标分别为正应力和剪应力的大小，单位Pa；图中红色圆点为最易失稳的断层；蓝色圆点代表任意断层在摩尔圆中的投影位置；蓝色线段的长度代表了其失稳的难易程度；用蓝色线段与红色线段长度之比计算失稳性归一化，定义为失稳系数（I）；σ1、σ2和σ3分别表示最大、中间、最小主应力，下图同
Figure 1. Definition of the instability coefficient and positions of faults with different instability coefficients on the Mohr's circle
The horizontal and vertical axes represent the magnitudes of normal stress and shear stress, respectively(Pa). In the diagram, the red dot represents the most unstable fault, and the blue dot indicates the projected position of an arbitrary fault on the Mohr circle. The length of the blue line segment reflects the difficulty of failure, and instability is normalised by its ratio to the length of the red line segment, which is defined as the instability coefficient (I). σ1，σ2 and σ3 indicate the magnitudes of three principal stresses, respectively， same as the figure below.
图 2 不同的应力形因子（R）下模拟的发震断层面法向的分布
圆圈的颜色代表了断层的失稳性大小
Figure 2. Distributions of simulated fault-plane normals under different shape ratios (R), with circle colors indicating fault instability.
图 3 不同的应力形因子（R）下模拟的辅助断层面法向的分布
圆圈的颜色代表了相应发震断层的失稳性大小
Figure 3. Distributions of simulated auxiliary fault-plane normals under different shape ratios (R), with circle colors indicating the instability of the corresponding seismogenic faults
The subplots are significant in the same way as in Fig. 2. The only difference is that the colour of each normal is plotted according to the instability of the seismogenic fault plane.
图 4 走滑型应力体系下的震源机制断层节面
实验的应力模型中 $ {\sigma }_{1} $ 轴的走向/倾角为30°/0°；$ {\sigma }_{3} $ 轴的走向/倾角为120°/0°； R=0.5，临界失稳系数为0.9a—2组满足失稳条件的断层在摩尔圆中的位置（红色和蓝色区域为 $ {\sigma }_{1} $ 轴两侧的2组断层面）；b—2组真实断层面（红色、蓝色）以及相应的辅助断层面（绿色）
Figure 4. Fault nodal planes of focal mechanisms under a strike-slip stress regime
The stress model employed in this experiment is characterized by a $ {\sigma }_{1} $ axis oriented at 30° (trend) / 0° (plunge), a $ {\sigma }_{3} $ axis at 120° (trend) / 0° (plunge), an R value of 0.5, and a critical instability coefficient of 0.9. (a) Positions of two fault sets satisfying the instability criterion on the Mohr circle (Colors in the upper and lower semicircles distinguish sets on opposite sides of the $ {\sigma }_{1} $ axis); (b) The two actual fault sets (red and blue) and their corresponding auxiliary fault planes (green).
图 5 不同应力形因子（R）的情况下震源机制P轴的空间分布
Figure 5. Spatial distributions of P axes for focal mechanisms under different shape ratios (R)
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