# Simulation of Cross-Correlated Random Fields for Transversely Anisotropic Soil Slope by Copulas
## 2 Random Field Theory
### Transversely Anisotropic Random Field  
A stationary random field can be divided into two types: isotropic random field and anisotropic random field [35]. In the former, the material properties and SOF are independent of the direction. In contrast, the latter is characterized by the direction-dependency of SOF, which is represented by the main SOF \(\delta_{1}\), the secondary SOF \(\delta_{2}\) and the direction angle \(\varphi\). The spatial variation of random variables is the smoothest along the direction of \(\delta_{1}\), and the roughest in the direction of \(\delta_{2}\). A transversely anisotropic random field is a special case of the anisotropic random field. Soil parameters change smoothly in the direction of \(\delta_{1}\), and faster in the direction of \(\delta_{2}\).  
To characterize isotropic and transversely anisotropic random fields, SOF and ACF must be formulated. As for ACF, theoretical ACFs are generally employed to approximate the autocorrelation. Notably, exponential ACF is the most commonly used in geotechnical engineering. In this paper, soil parameters are also assumed to obey an exponential auto-correlation structure.  
The formulation of SOF in the isotropic case is circular and direction-independent, namely, \(\delta_{1}=\delta_{2}\). Then, the theoretical formulation of SOF \(\delta_{\varphi}\) at directional angle \(\varphi\) can be expressed as below [35]:  
\[\delta_{\varphi}=\delta_{1}=\delta_{2} \tag{1}\]  
The corresponding exponential ACF is:  
\[\rho=\exp\left[-2\frac{\sqrt{\Delta x^{2}+\Delta y^{2}}}{\delta_{\varphi}}\right] \tag{2}\]  
where \(\Delta x\) and \(\Delta y\), respectively, represent absolute horizontal and vertical distances between two spatial locations.  
As for anisotropic random fields, the formulation of SOF is generally assumed as an ellipse and is parameterized by using \(\delta_{\varphi}\), \(\delta_{1}\) and \(\delta_{2}\).  
\[\frac{\left(\delta_{\varphi}\cos\varphi\right)^{2}}{\delta_{1}^{2}}+\frac{ \left(\delta_{\varphi}\sin\varphi\right)^{2}}{\delta_{2}^{2}}=1 \tag{3}\]Via trigonometric transformation, Equation (3) can be transformed as  
\[\frac{\delta_{\varphi}^{2}\frac{1}{1+\tan^{2}\varphi}}{\delta_{1}^{2}}+\frac{ \delta_{\varphi}^{2}\frac{\tan^{2}\varphi}{1+\tan^{2}\varphi}}{\delta_{2}^{2}}=1 \tag{4}\]  
Let  
\[\tan\varphi=\frac{\Delta y}{\Delta x} \tag{5}\]  
Substituting Equation (5) into Equation (4) yields  
\[\sqrt{\left(\frac{\Delta x}{\delta_{1}}\right)^{2}+\left(\frac{\Delta y}{ \delta_{2}}\right)^{2}}=\frac{\sqrt{\Delta x^{2}+\Delta y^{2}}}{\delta_{ \varphi}} \tag{6}\]  
According to the definition of a transversely anisotropic random field, the spatial variation of random variables is the smoothest in the direction of \(\delta_{1}\), and the roughest in the direction of \(\delta_{2}\). Assuming that the formulation of SOF is elliptical, \(\delta_{\varphi}\) can be expressed as  
\[\delta_{\varphi}=\sqrt{\frac{\delta_{1}^{2}\delta_{2}^{2}\left(1+\tan^{2} \varphi\right)}{\delta_{2}^{2}+\delta_{1}^{2}\tan^{2}\varphi}} \tag{7}\]  
The corresponding exponential ACF is:  
\[\rho=\exp\left[-2\sqrt{\frac{\Delta x^{2}}{\delta_{1}^{2}}+\frac{\Delta y^{2}} {\delta_{2}^{2}}}\right] \tag{8}\]