## 2 Theory and Method  
### Derivation of the Reflection Coefficient Equation  
In the investigation of the stiffness matrix for anisotropic media, we consider a specific case involving a group of horizontal, rotationally invariant fractures within a vertically transversely isotropic (VTI) medium. The stiffness matrix for this configuration is derived and expressed as follows:  
\[C^{VTI}=C^{Iso}+C^{H}_{f} \tag{1}\]  
Figure 1: Classical fracture model ([PERSON], 2015) [9]. (**a**) [PERSON]’s penny-shaped crack model and (**b**) [PERSON]’s linear-slip model. where \(C^{Iso}\) represents the stiffness coefficient matrix of the background isotropic medium while \(C^{H}_{f}\) corresponds to the stiffness coefficient matrix induced by a single group of horizontal fractures. Similarly, the stiffness coefficient matrix of HTI media, which is caused by a set of vertical, rotationally invariant fractures, can be expressed as the sum of the background isotropic medium's stiffness matrix and the matrix resulting from the vertical fractures. Building on this principle of superposition, the stiffness matrix for TTI media can be formulated as the sum of the stiffness matrix for the background isotropic medium and the stiffness matrix induced by a set of inclined fractures. Furthermore, the stiffness matrix resulting from a single group of inclined fractures in TTI media can be decomposed into two components: one from the vertical fractures and the other from the horizontal fractures. Therefore, the total stiffness coefficient matrix for the TTI medium is given by the following:  
\[C^{TTI}=C^{Iso}+C^{T}_{f}=C^{Iso}+C^{V}_{f}\cos\xi+C^{H}_{f}\sin\xi \tag{2}\]  
where \(\xi\) represents the angle between the fracture and the vertical direction in the TTI medium. \(C^{V}_{f},\cos\xi\) refers to the difference in the stiffness matrix caused by the inclined fracture in the vertical direction, and the influence of this component on the seismic amplitude is solely dependent on the incident angle. On the other hand, \(C^{H}_{f}\sin\xi\) represents the difference in the stiffness matrix due to the inclined fracture in the horizontal direction, where its impact on the seismic amplitude is influenced by both the incident angle and the azimuth angle. Therefore, the stiffness coefficient matrix of the TTI medium, induced by a single group of inclined fractures, is given by the following:  
\[C= \begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\ C_{12}&C_{22}&C_{23}&0&0&0\\ C_{13}&C_{23}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{55}&0\\ 0&0&0&0&0&C_{66}\end{bmatrix}  \tag{3}\]  
Its specific forms are as follows:  
\[ \begin{array}{l}C_{11}=M\big{[}1-(1-2g)^{2}\delta_{N}\sin\xi-\delta_{N}\cos \xi\big{]},\\ C_{12}=\lambda[1-(1-2g)\delta_{N}\sin\xi-\delta_{N}\cos\xi],\\ C_{13}=\lambda(1-\delta_{N}\sin\xi-\delta_{N}\cos\xi),\\ C_{22}=M\big{[}1-(1-2g)^{2}\delta_{N}\sin\xi-(1-2g)^{2}\delta_{N}\cos\xi\big{]},\\ C_{23}=\lambda[1-\delta_{N}\sin\xi-(1-2g)\delta_{N}\cos\xi],\\ C_{33}=M\big{[}1-\delta_{N}\sin\xi-(1-2g)^{2}\delta_{N}\cos\xi\big{]},\\ C_{44}=\mu(1-\delta_{T}\sin\xi),\\ C_{55}=\mu(1-\delta_{T}\sin\xi-\delta_{T}\cos\xi),\\ C_{66}=\mu(1-\delta_{T}\cos\xi).\end{array} \]  
where \(\lambda\) and \(\mu\) refer to the Lame coefficient of the background isotropic medium, while \(M\) denotes the longitudinal wave modulus, and its unit is Pa. \(g\) refers to the square of the ratio between the transverse and longitudinal wave velocities. \(\delta_{N}\) represents the normal weakness of the fracture, which quantifies the difference in elastic parameters across the fracture surface and along its normal direction. \(\delta_{T}\) refers to the tangential weakness of the fracture, capturing the difference in elastic parameters along the tangential direction of the fracture. For \(g\) and \(\delta_{N}\), \(\delta_{T}\), they are all dimensionless parameters.