## 2 Characterization of Random Fracture Networks in Broken Rock Masses
### Mathematical Description of Stochastic Model for Geometric Parameters of Structural Surfaces
#### 2.2.2 Occurrence of Structural Planes  
There are two main methods for describing the occurrence of structural planes [30]. One approach involves expressing the occurrence of structural planes in the normal direction. The other uses the strike and dip to describe the occurrence. The latter uses dip azimuth \(\varphi_{d}\) and dip angle \(\theta_{d}\) to represent the occurrence of structural plane, which is widely used in geological engineering, as shown in Figure 4.  
Figure 3: Density functions for log-normal and negative exponential distributions: (**a**) log-normal distribution; (**b**) negative exponential distribution.  
The commonly used probability distributions of structural plane occurrence (parameters including dip azimuth angle \(\varphi_{d}\) and dip angle \(\theta_{d}\)) include Bingham distribution, Fisher distribution, and bivariate normal distribution. Among these distributions, the first two regard the azimuth and inclination of the structural plane as two independent random variables, while the last one considers the correlation between the two parameters. Consider a set of two-dimensional random variables \(\varphi_{d}\) and \(\theta_{d}\) with sample means \(\varphi_{d}\) and \(\theta_{d}\), variances \(\sigma_{\varphi}{}^{2}\) and \(\sigma_{\theta}{}^{2}\), respectively, and a correlation coefficient \(\rho\) between the two random variables. The probability density function is as follows:  
\[f\left(\frac{\varphi_{d}-\overline{\varphi_{d}}}{\sigma_{\varphi}},\frac{ \theta_{d}-\overline{\theta_{d}}}{\sigma_{\theta}},\rho\right)=\frac{1}{2\pi \sigma_{\varphi}\varphi_{\theta}\sqrt{1-\rho^{2}}}\exp\left[\frac{\frac{2 \rho(\varphi_{d}-\overline{\varphi_{d}})\left(\theta_{d}-\overline{\theta_{d}} \right)}{\sigma_{\varphi}\varphi_{\theta}}-\left(\frac{\varphi_{d}-\overline {\varphi_{d}}}{\sigma_{\theta}}\right)^{2}-\left(\frac{\theta_{d}-\overline {\theta_{d}}}{\sigma_{\theta}}\right)^{2}}{2(1-\rho^{2})}\right] \tag{4}\]  
The joint probability distribution of the two variables can be expressed as follows:  
\[P(\varphi_{d}\leq u,\theta_{d}\leq v)=\int_{-\infty}^{\frac{v-\theta_{d}}{ \varphi_{\theta}}}\int_{-\infty}^{\frac{u-\varphi_{d}}{\varphi_{\theta}}}f \left(\frac{\varphi_{d}-\overline{\varphi_{d}}}{\sigma_{\varphi}},\frac{ \theta_{d}-\overline{\theta_{d}}}{\sigma_{\theta}},\rho\right)d\left(\frac{ \varphi_{d}-\overline{\varphi_{d}}}{\sigma_{\varphi}}\right)d\left(\frac{ \theta_{d}-\overline{\theta_{d}}}{\sigma_{\theta}}\right) \tag{5}\]  
When the dip azimuth angle \(\varphi_{d}\) and dip angle \(\theta_{d}\) are independent of each other, i.e., \(\rho=0\), the bivariate normal distribution in Equation (4) degenerates into a one-dimensional normal distribution. Substituting the mean and mean square deviation of the occurrence parameters \(\varphi_{d}\) and \(\theta_{d}\) into the one-dimensional normal distribution \(x_{i}=\sigma N_{i}+\mu\), the azimuth angle \(\varphi_{di}\) and dip angle \(\theta_{di}\) of the structural plane that satisfy the normal distribution can be obtained:  
\[\varphi_{di}=\sigma_{\varphi}N_{i}+\varphi_{d},\quad\theta_{di}=\sigma_{ \theta}N_{i}+\theta_{d} \tag{6}\]