# Structural Augmentation in Seismic Data for Fault Prediction
## 2 Methods
### Adding Virtual Faults
#### 2.2.3 Fault Drag  
As shown in Figure 7, the fault drag is described in two types in this study: reverse drag and normal drag. For the reverse drag simulation, the following nonlinear equation is popularly used [110]:  
\[\alpha(\bar{X},\bar{Y},\bar{Z})=\left(1-\frac{|\bar{Z}-f(\bar{X},\bar{Y})|^{2} }{R}\right)\qquad f(\bar{X},\bar{Y})-R\leq\bar{Z}\leq f(\bar{X},\bar{Y})+R, \tag{10}\]  
where \(f(\bar{X},\bar{Y})\) is the fault surface defined in Equation (7), and \(R\) is the drag radius.  
Inspired by the sigmoid function [111], the normal drag is simulated using the following truncated sigmoid function:  
\[\alpha(\bar{X},\bar{Y},\bar{Z})=\begin{cases}\frac{2}{1+\exp\left(\frac{-\mu \bar{Z}-f(\bar{X},\bar{Y})-\delta}{\delta}\right)}-1&f(\bar{X},\bar{Y})-R\leq \bar{Z}\leq f(\bar{X},\bar{Y})\\ \frac{2}{1+\exp\left(\frac{-\mu\bar{Z}-f(\bar{X},\bar{Y})+\delta}{\delta} \right)}-1&f(\bar{X},\bar{Y})\leq Z\leq f(\bar{X},\bar{Y})+R\,,\\ 1&|\bar{Z}-f(\bar{X},\bar{Y})|\geq R\end{cases} \tag{11}\]  
where the truncation distance \(\delta\) controls the displacement at the drag center, and the scaling factor \(\mu\) controls the shape of the drag.  
Different from the conventional methods [68; 96], we apply the drag rotation angle \(\Psi_{drag}\) to control the generation of fault drags. In nature, fault drag exists not only with dip-slip faults [112; 113; 114] but also with strike-slip faults [115; 116; 117]. Therefore, different strike-slip and dip-slip drags can be combined to form arbitrary drag directions on a fault surface (Figure 7):  
Figure 6: Demonstrations of ridge-and-groove morphology. (**a**) Normal fault surface modified after [108]. (**b**) Strike-slip fault surface modified after [109]. (**c**) Ridge-and-groove morphology generated in this paper. The red arrows represent the fault displacement direction. Reprinted with permission from [109]. 2022, John Wiley and Sons.  
To ensure the rationality of the hanging wall and foot wall on both sides of the fault surface, the displacement in the normal direction of the fault surface is defined by the following formula:  
\[D(\bar{z})(\bar{X},\bar{Y},\bar{Z})=f(\bar{X}+D[\bar{x}](\bar{X},\bar{Y},\bar{Z}),\bar{Y}+D[g](\bar{X},\bar{Y},\bar{Z}))-f(\bar{X},\bar{Y}). \tag{13}\]  
Finally, after calculating the three components of the displacement field, we can generate virtual faults by the following equation:  
\[\left[\begin{array}{c}\bar{X}\\ \bar{Y}\\ \bar{Z}\end{array}\right]=\mathbb{R}^{-1}\left(\left[\begin{array}{c}\bar{X }\\ \bar{Y}\\ \bar{Z}\end{array}\right]+\left[\begin{array}{c}D[\bar{x}](\bar{X},\bar{Y}, \bar{Z})\\ D[\bar{y}](\bar{X},\bar{Y},\bar{Z})\\ D[\bar{z}](\bar{X},\bar{Y},\bar{Z})\end{array}\right]\right)+\left[\begin{array} []{c}X_{0}\\ Y_{0}\\ Z_{0}\end{array}\right], \tag{14}\]  
where the rotation factor \(\mathbb{R}\) is mentioned in Equation (5). By arbitrarily combining the parameters of near-field displacement \(d(\bar{X},\bar{Y})\), fault surface \(f(\bar{X},\bar{Y})\), and fault drag \(\alpha(\bar{X},\bar{Y},\bar{Z})\), many unique virtual faults can be generated. As shown in Figure 1c, numerous virtual faults including linear faults, listric faults, oblique-slip faults, and normal and reverse drag faults can be generated through a series of calculations. By adding a variety of virtual faults and folds, a highly diverse field-like dataset can be obtained.