# Structural Augmentation in Seismic Data for Fault Prediction
## 2 Methods
### Adding Virtual Folds  
The simulation of the virtual fold structures is divided into three parts in this study: large-scale folds \(S_{1}(X,Y,Z)\), local folds \(S_{2}(X,Y,Z)\), and dips \(S_{3}(X,Y,Z)\). Unlike conventional methods using standard Fourier series to model folds [95], elliptic equations are applied in Fourier series to increase the asymmetry in this study. Concretely, the standard Fourier series are diverse in the \(Z\)-direction, but they maintain symmetry in the \(X\)- or \(Y\)-direction (Figure 2a). As shown in Figure 2b, the generated large-scale folds can be asymmetric in the lateral direction through the following equation:  
\[ \begin{split} S_{1}(X,Y,Z)=A_{0}+\left(1-\left|\frac{a(X-X_{0})+ b(Y-Y_{0})}{X_{max}+Y_{max}}\right|\right)\sum_{k=1}^{N}\left(A_{k}\cos(\frac{k\pi E (X,Y)}{\lambda_{k}})+B_{k}\sin(\frac{k\pi E(X,Y)}{\lambda_{k}})\right)\\ E(X,Y)=\sqrt{\left(\frac{\cos\varphi_{k}(X-X_{0})-\sin\varphi_{1}( Y-Y_{0})}{L[x]_{k}}\right)^{2}+\left(\frac{\sin\varphi_{k}(X-X_{0})+\cos \varphi_{1}(Y-Y_{0})}{L[y]_{k}}\right)^{2}}\end{split} , \tag{1}\]  
where \(A_{k}\) and \(B_{k}\), and \(\lambda_{k}\) are the basis parameters in the standard Fourier series, controlling the amplitude and period of the trigonometric functions, respectively. The elliptic long axis \(L[x]_{k}\), elliptic short axis \(L[y]_{k}\), and rotation angle \(\varphi_{k}\) control the variation of the virtual folds in the lateral direction. By applying the linear decay function \(1-\left|\frac{a(X-X_{0})+b(Y-Y_{0})}{X_{max}+Y_{max}}\right|\), the amplitude of the large-scale folds gradually decreases from the center to the edges. In this study, we assume that large virtual folds belong to a single stratigraphic unit and  
Figure 1: The workflow of structural data augmentation. (**a**) Raw 3D seismic data. (**b**) Seismic data after adding virtual folds. (**c**) Seismic data after adding virtual faults and random noise. (**d**) Manual fault interpretation label data. (**e**) Label data after adding virtual folds. (**f**) Label data after adding virtual faults. have nearly uniform surfaces. If necessary, additional parameters can be applied in the \(Z\)-direction to increase diversity. Different from the symmetric standard Gaussian equation used by [PERSON] et al. [68], we use the following equation to generate asymmetric local virtual folds (Figure 2d):  
\[ \begin{array}{cc}S_{2}(X,Y,Z)=&c\frac{Z}{Z_{max}}\sum_{k=1}^{N}D_{k}\exp{ \left[-\left(\frac{\cos\phi_{k}(X-E_{k})-\sin\phi_{k}(Y-E_{k})}{2\sigma[y]_{k} }\right)\right)^{2}}\\ &-\left(\frac{\sin\phi_{k}(X-E_{k})+\cos\phi_{k}(Y-E_{k})}{2\sigma[y]_{k}} \right)^{2}\right],\end{array}  \tag{2}\]  
where the parameters \((E_{k},F_{k})\) and \(D_{k}\) control the center and amplitude of the local folds, respectively. As shown in Figure 2e, by controlling the parameters of long axis \(\sigma[x]_{k}\), short axis \(\sigma[y]_{k}\), and rotation angle \(\phi_{k}\), the generated local folds are not limited to symmetric circles. As shown in Figure 2g, the linear function that generates the dip structure uses parameters \(G\) and \(H\) to control the dip of the formation in the \(X\)- and \(Y\)-directions:  
\[S_{3}(X,Y,Z)=GX+HY+l. \tag{3}\]  
Figure 2: Demonstration of virtual folds. (**a**) A large-scale fold generated by the standard Fourier series. (**b**) A large-scale fold generated in this study. (**c**) A structure model with large-scale folds. (**d**) A local fold generated by the standard Gaussian function. (**e**) A local fold generated in this study. (**f**) A structure model with local folds. (**g**) Dips generated in this study. (**h**) A structure model with dips. (**i**) A structure model with multi-scale folds.