# Numerical Modeling of 3D Slopes with Weak Zones by Random Field and Finite Elements
## 2 Methodology
### Method for Characterization of Anisotropic Correlation Structure  
To make the simulation more similar to the actual slope, a 3D model was established, taking into account rotational anisotropy. The coordinates of the Gaussian random field  
Figure 1: Reconstructions of two-phase random media on the basis of the volume fraction of weak zones and rock. Figure 2: Schematic diagram of the distribution of weak zones (red represents rocks, blue represents weak zones). (**a**) Typical sample of slope with homogenous weak zones (correlation length along all directions is 5 m); (**b**) typical sample of slope with layered weak zones (correlation length along x directions is 20 m; correlation length along y direction is 5 m; correlation length along z direction is 1000 m). were rotated around the \(X\)-axis in accordance with the given inclination angle. The rotation process can be expressed as in Equation (9). \[\left[ \begin{array}{c}x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{array} \right]=\left[ \begin{array}{ccc}1&0&0\\ 0&\cos\alpha&\sin\alpha\\ 0&-\sin\alpha&\cos\alpha\end{array} \right]\left[ \begin{array}{c}x\\ y\\ z\end{array} \right] \tag{9}\]  
where \(x^{\prime},y^{\prime}\) and \(z^{\prime}\) are the corresponding coordinates of \(x,y\) and \(z\) after rotation of the \(\alpha\) angle about the \(X\)-axis. The distance after rotation can be obtained using Equations (10)-(12), as shown below. \[\Delta x^{\prime}=\Delta x \tag{10}\]  
\[\Delta y^{\prime}=\Delta y\cos\alpha+\Delta z\sin\alpha \tag{11}\]  
\[\Delta z^{\prime}=\Delta z\cos\alpha-\Delta y\sin\alpha \tag{12}\]  
The expression of the squared exponential autocorrelation function of rotational anisotropy can be obtained as:  
\[R^{\prime}_{x}\big{(}x^{\prime},y^{\prime},z\big{)}=\exp\Biggl{\{}-\pi\biggl{(} \frac{\Delta x}{\theta_{x}}\biggr{)}^{2}-\pi\biggl{(}\frac{\Delta y\cos\alpha +\Delta z\sin\alpha}{\theta_{y}}\biggr{)}^{2}-\pi\biggl{(}\frac{\Delta z\cos \alpha-\Delta y\sin\alpha}{\theta_{z}}\biggr{)}^{2}\Biggr{\}} \tag{13}\]  
Figure 3 illustrates the rotationally anisotropic after rotation of the \(\alpha\) angle about the \(X\)-axis, the transversely anisotropic weak zone structure, and the correlation lengths. It can be clearly found from the figure that transverse anisotropy is a special case of rotational anisotropy. In addition, the autocorrelation function can well characterize the weak zones inside the slope. Figure 3: Correlation structure of different related structures. (**a**) Horizontal rock slope; (**b**) schematic diagram of the correlation length of the \(x=0\); (**c**) rotational anisotropic slope; (**d**) correlation length of rotational anisotropy after rotation of the \(\alpha\) angle about the \(X\)-axis. For rotated anisotropy around the \(Y\)-axis, the correlation distance after rotation can be obtained using Equations (14)-(16), as shown below. \[\Delta x^{\prime}=\Delta x\cos\alpha+\Delta z\sin\alpha \tag{14}\]  
\[\Delta y^{\prime}=\Delta y \tag{15}\]  
\[\Delta z^{\prime}=\Delta z\cos\alpha-\Delta x\sin\alpha \tag{16}\]  
The expression of the squared exponential autocorrelation function of rotational anisotropy can be obtained as:  
\[R_{y}^{\prime}(x^{\prime},y^{\prime},z)=\exp\Biggl{\{}-\pi\biggl{(}\frac{ \Delta x\cos\alpha+\Delta z\sin\alpha}{\theta_{x}}\biggr{)}^{2}-\pi\biggl{(} \frac{\Delta y}{\theta_{y}}\biggr{)}^{2}-\pi\biggl{(}\frac{\Delta z\cos\alpha -\Delta x\sin\alpha}{\theta_{z}}\biggr{)}^{2}\Biggr{\}} \tag{17}\]  
Figure 4 illustrates the rotationally anisotropic after rotation of \(\alpha\) angle about the \(Y\)-axis, the transversely anisotropic weak zone structure, and the corresponding correlation lengths. For rotated anisotropy around the \(Z\)-axis, the correlation distance after rotation can be obtained by Equations (18)-(20), as shown below.