# Structural Augmentation in Seismic Data for Fault Prediction
## 2 Methods
### Adding Virtual Faults
#### 2.2.2 Fault Surface  
As shown in Figure 5, the generation of fault surfaces is mainly divided into two parts in this study: ellipsoidal surface \(f_{1}(\bar{X},\bar{Y})\) and random perturbation \(f_{2}(\bar{X},\bar{Y})\).  
\[f(\bar{X},\bar{Y})=f_{1}(\bar{X},\bar{Y})+f_{2}(\bar{X},\bar{Y}). \tag{7}\]  
Unlike the conventional methods that use only the random simulation algorithms to generate faults [104; 105; 68], which has a low probability of generating listric faults, we apply an ellipsoidal surface equation to create listric faults with different shapes:  
\[\begin{array}{c}f_{1}(\bar{X},\bar{Y})=\left[1-\left(\frac{X_{R}}{\Gamma_{ MB}}\right)^{2}-\left(\frac{Y_{R}}{\Gamma_{MB}}\right)^{2}-\left(\frac{Z_{R}}{ \Gamma_{MB}}\right)^{2}\right]Amp_{z}\\ \left[\begin{array}{c}X_{R}\\ Y_{R}\\ Z_{R}\end{array}\right]=R_{x}R_{y}R_{z}\left[\begin{array}{c}\bar{X}-\bar{X} _{0}\\ \bar{Y}-\bar{X}_{0}\\ Z_{0}\end{array}\right]\\ R_{x}(\theta_{x})=\left[\begin{array}{cc}1&0&0\\ 0&\cos\theta_{x}&-\sin\theta_{x}\\ 0&\sin\theta_{x}&\cos\theta_{z}\end{array}\right].R_{y}(\theta_{y})=\left[ \begin{array}{cc}\cos\theta_{y}&0&\sin\theta_{y}\\ 0&1&0\\ -\sin\theta_{y}&0&\cos\theta_{y}\end{array}\right],R_{z}(\theta_{z})=\left[ \begin{array}{cc}\cos\theta_{z}&-\sin\theta_{z}&0\\ \sin\theta_{z}&\cos\theta_{z}&0\\ 0&0&1\end{array}\right]\end{array}, \tag{8}\]  
where \((\bar{X}_{0},\bar{Y}_{0},\bar{Z}_{0})\), \((L[x_{R}],L[y_{R}],L[z_{R}])\), and \((\theta_{x},\theta_{y},\theta_{z})\) are the classical parameters of the standard ellipsoidal surface equation, controlling the center, radius, and rotation angle of the ellipsoidal surface, respectively. In some fault examples (Figure 6a,b), there are local  
Figure 4: Demonstration of fault displacement distribution. (**a**) Near-field displacement with displacement rotation angle \(\Psi_{dis}\). (**b**) Near-field displacement with attenuation rotation angle \(\Psi\). The dashed arrow indicates the direction of displacement, and the dotted line indicates the direction of the short axis of the attenuation ellipse.  
Figure 5: Generation of fault surfaces. (**a**) Ellipsoidal surface. (**b**) Random perturbation.  
structures with uniform orientation, such as ridge-and-groove morphology [106; 107; 108], so we control the random perturbation of the fault surface by the following equation:  
\[f_{2}(\bar{X},\bar{Y})=\sum_{k=1}^{N}Amp_{k}\exp\left[-\left(\frac{\cos\Psi_{k}( \bar{X}-I_{k})-\sin\Psi_{k}(\bar{Y}-K_{k})}{2\sigma[\bar{x}]_{k}}\right)^{2}- \left(\frac{\sin\Psi_{k}(\bar{X}-I_{k})+\cos\Psi_{k}(\bar{Y}-K_{k})}{2\sigma[ \bar{y}]_{k}}\right)^{2}\right], \tag{9}\]  
where the parameters \((I_{k},K_{k})\), \((\sigma[\bar{x}]_{k},\sigma[\bar{y}]_{k})\) and \(Amp_{k}\) control the center, radius, and magnitude of the local perturbation, respectively. The random perturbation of the fault can be made consistent with the fault displacement direction by controlling the rotation angle \(\Psi_{k}\) (Figure 6c).  
With the calculation of \(f_{1}(\bar{X},\bar{Y})\) and \(f_{2}(\bar{X},\bar{Y})\), numerous curved fault surfaces can be generated, and the insufficiency of random seed methods in listric fault simulation can be avoided.