## 3 Materials and Methods
### Stress Tensor Inversion  
Based on the theory of Coulomb failure stress changes, we need the stress tensor of the region where the source fault is located to calculate the \(\Delta CFS\) value of the receiver fault. The stress tensor is a tool used to generalize the state of a regional stress field. If we consider a point as an infinitesimal cube, then the traction force acting on this point can be split into three mutually perpendicular planes. By this method, the force acting on a point is split into nine components:  
\[\begin{bmatrix}\sigma_{xx}&\sigma_{xy}&\sigma_{xz}\\ \sigma_{yx}&\sigma_{yy}&\sigma_{yz}\\ \sigma_{zx}&\sigma_{zy}&\sigma_{zz}\\ \end{bmatrix}, \tag{5}\]  
where \(\sigma_{xx}\), \(\sigma_{yy}\), and \(\sigma_{zz}\) are normal stresses, and the other six components are shear stresses. Shear stress components with opposite subscripts are equal in magnitude and opposite in direction. Therefore, by knowing the six components, we can determine a stress tensor. During the calculation of \(\Delta CFS\), we need directions of the three principal stress axes (\(\sigma_{xx}\), \(\sigma_{yy}\), \(\sigma_{zz}\)) in the region where the source fault is located. Usually, we specify \(\sigma_{1}>\sigma_{2}>\sigma_{3}\) based on the magnitudes of the principal stresses. The stress ratio \(R\) defines the relationship between the magnitudes of the three principal stresses as follows:  
\[R=(\sigma_{1}-\sigma_{2})/(\sigma_{1}-\sigma_{3}), \tag{6}\]  
The prevailing stress tensor inversion techniques are performed using the focal mechanisms [53; 54; 55; 56; 57]. The fault plane tangential traction is often assumed to be parallel to the fault slide direction, and the research area's stress field is assumed to be uniform [58; 59]. In this study, we used STRESSINVERSE to calculate the stress tensor, which was proposed by [PERSON] [57]. This approach inverts jointly for stress and fault orientations and can eliminate errors due to an incorrect selection of fault planes. The accuracy of this method was tested by [PERSON] [57], which showed that the introduction of the fault instability criterion improves the accuracy of the linear inversion and removes the effect of fault uncertainty on the calculation results. The formula for determining the instability of a fault is as follows:  
\[I=\frac{\tau-\mu(\sigma-1)}{\mu+\sqrt{1+\mu^{2}}}, \tag{7}\]  
where  
\[\tau=\sqrt{n_{1}^{2}+(1-2R)n_{2}^{2}+n_{3}^{2}-\left[n_{1}^{2}+(1-2R)n_{2}^{2 }-n_{3}^{2}\right]^{2}}, \tag{8}\]  
\(n_{\rm i}\) are the components of the fault normal \(n\) (\(i=1,2,3\)), \(\mu\) is the fault friction, and \(R\) is the stress ratio defined in Equation (1).  
The value of fault instability \(I\) range from 0 to 1. The fault is considered most stable when \(I=0\) and most unstable when \(I=1\). The most unstable fault is the principal fault, the optimally oriented fault for shear faulting.  
This is how the fault instability criterion-based stress inversion operates. First, [PERSON]'s approach [56; 60] is used to construct the stress tensor, and fault planes are chosen at random from nodal planes supplied by focal mechanisms. Second, the nodal planes with the highest fault instability are known as fault planes based on the recoveredstress. Third, using the fault plane orientations determined by the fault instability criterion, the stress inversion is carried out once more. The procedure is continued until the final stress model is reached by the inversion.