# Seismic Identification and Characterization of Deep Strike-Slip Faults in the Tarim Craton Basin
## 3 Seismic Resolution Improvement
### Fault-Focused Imaging  
Following the implementation of structure-oriented filtering, notable improvements in the fault resolution and clarity have been achieved, albeit with certain faults still exhibiting issues such as unclear boundaries and weak continuity. To address these limitations, applying fault-focused imaging technology has become essential for enhancing fault continuities and highlighting fault boundaries [42].  
To attain a more refined filtering effect on the features of geological structures and effectively reduce seismic noise, the pivotal step in this approach involves the derivation of the diffusion tensor [43]:  
\[\left\{\begin{matrix}\frac{\partial U}{\partial t}=div(D.\
abla U)\\ U|_{t=0}=U_{0}\end{matrix}\right. \tag{5}\]  
where \(t\) is the diffusion time; \(div\) is the divergence operator; \(D\) is the tensor diffusion coefficient of the diffusion filter; \(U\) is the result of tomography focusing imaging at time \(t\); and \(U_{0}\) is the original data at time \(t=0\).  
Moreover, \(|r|\) is the distance of the data point from the data centre.  
The gradient structure tensor can be constructed as follows:  
\[J_{p}(\
abla U)=G_{p}*J_{0}=G_{p}*\left(\
abla U(\
abla U)^{T}\right) \tag{6}\]  
where \(U\) is the original seismic data and \(J_{0}\) is the tensor product of the gradient vector.  
The Gaussian function with scale \(\rho\) is  
\[G_{\rho}(r)=\left(2\pi\rho^{2}\right)^{-\frac{3}{2}}\text{exp}\left(-\frac{|r |^{2}}{2\rho^{2}}\right) \tag{7}\]  
The eigenvector of the structure tensor can be obtained as follows:  
\[J_{\rho}(\
abla U)=(v_{1}\,v_{2}\,v_{3})\begin{bmatrix}\lambda_{1}&0&0\\ 0&\lambda_{2}&0\\ 0&0&\lambda_{3}\end{bmatrix}(v_{1}\,v_{2}\,v_{3})^{T} \tag{8}\]  
where \(v_{1}\), \(v_{2}\), and \(v_{3}\) are the three feature vectors of the gradient structure tensor, and they are regarded as the local orthogonal coordinate system. Then, \(v_{1}\) points to the gradient direction of the signal, and the plane composed of \(v_{2}\) and \(v_{3}\) is parallel to the local structural features of the signal. \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) represent the corresponding eigenvalues.  
Linear structure confidence metric:  
\[C_{line}=\frac{\lambda_{2}-\lambda_{3}}{\lambda_{2}+\lambda_{3}} \tag{9}\]  
Planar structure confidence metric:  
\[C_{plane}=\frac{\lambda_{1}-\lambda_{2}}{\lambda_{1}+\lambda_{2}} \tag{10}\]  
Focusing tensor:  
\[D=(v_{1}\,v_{2}\,v_{3})\begin{bmatrix}u_{1}&0&0\\ 0&u_{2}&0\\ 0&0&u_{3}\end{bmatrix}(v_{1}\,v_{2}\,v_{3})^{T}=\begin{bmatrix}D_{11}&D_{12}& D_{13}\\ D_{21}&D_{22}&D_{23}\\ D_{31}&D_{32}&D_{33}\end{bmatrix} \tag{11}\]  
where \(u_{1}\), \(u_{2}\), and \(u_{3}\) are the three non-negative eigenvalues of the focusing tensor. In the interval [0,1], they represent the filtering intensity of the fault-focused imaging along the three characteristic directions \(v_{1}\), \(v_{2}\), and \(v_{3}\).  
The fault-focused imaging algorithm effectively retains the distinctive \"featurely zone\" and \"flower-like structure\" associated with craton strike-slip faults while significantly enhancing the imaging capabilities of seismic data concerning branch faults (Figure 3c). Concurrently, the algorithm exhibits noise suppression properties, leading to an improvement in the transverse continuity of the in-phase axis and a higher signal-to-noise ratio in the seismic signals.