## 2 Characterization of Spatial Variability and Slope Stability Assessment Methods
### Generalized Hoek-Brown Criterion  
The generalized Hoek-Brown criterion is derived from extensive field tests and engineering practice. By incorporating empirical parameters that characterize rock mass properties, it establishes a quantitative correlation between the rock mass failure criterion and these parameters [26]. This criterion comprehensively accounts for both the geological characteristics and mechanical properties of rock masses, enabling the determination of equivalent mechanical parameters suitable for slope stability analysis. The mathematical expression of this criterion is as follows:  
\[\sigma_{1}=\sigma_{3}+\sigma_{ci}\bigg{(}m_{b}\frac{\sigma_{3}}{\sigma_{ci}}+s \bigg{)}^{a} \tag{5}\]  
where \(\sigma_{1}\) and \(\sigma_{3}\) denote the maximum and minimum principal stresses (MPa) at failure, respectively; \(\sigma_{ci}\) represents the uniaxial compressive strength (MPa) of the intact rock; \(m_{b}\) is an empirical parameter characterizing the rock mass, calculated using Equation (6); \(s\) is the integrity coefficient of the rock mass, determined by Equation (7); and \(a\) is a constant characterizing the jointed rock mass, as expressed in Equation (8).  
\[m_{b}=\exp\bigg{(}\frac{GSI-100}{28-14D}\bigg{)}m_{i} \tag{6}\]  
\[s=\exp\bigg{(}\frac{GSI-100}{9-3D}\bigg{)} \tag{7}\]  
\[a=0.5+\frac{1}{6}\bigg{[}\exp\bigg{(}-\frac{GSI}{15}\bigg{)}-\exp\bigg{(}- \frac{20}{3}\bigg{)}\bigg{]} \tag{8}\]  
in which \(D\in[0,1]\) denotes the excavation disturbance coefficient of the rock mass; \(m_{i}\) represents the rock hardness; and \(GSI\) is the geological strength index of the rock mass, with its engineering estimation formula expressed as follows:  
\[GSI=1.5J\text{{Cond}}_{89}+RQD/2 \tag{9}\]  
where \(J\text{{Cond}}_{89}\) denotes the structural plane condition coefficient, while \(RQD\) represents the rock quality designation, defined as the percentage of the cumulative length of intact rock core segments exceeding 10 cm in a core run relative to the total length of that core run.  
The tensile strength \(\sigma_{t}\) (MPa) of the rock mass is given by the following equation:  
\[\sigma_{t}=\frac{-s\sigma_{ci}}{m_{b}} \tag{10}\]  
When \(\sigma_{ci}<100\) MPa, the deformation modulus \(E_{m}\) of the rock mass can be approximated as follows:  
\[E_{m}(GPa)=(1-\frac{D}{2})\sqrt{\frac{\sigma_{ci}}{100}}10^{(\frac{GSI-10}{4 0})} \tag{11}\]  
Based on the area equivalence principle, the optimal fitting parameters for approximating the Hoek-Brown curve using the Mohr-Coulomb linear criterion in Equation (5) are determined. The equivalent internal friction angle \(\phi^{\prime}\) (\({}^{\circ}\)) and equivalent cohesion \(c^{\prime}\) (MPa) of the rock mass are derived as follows:  
\[\phi^{\prime}=\sin^{-1}\bigg{[}\frac{6\alpha m_{b}(s+m_{b}\sigma_{3n}^{ \prime})^{\alpha-1}}{2(1+\alpha)(2+\alpha)+6\alpha m_{b}(s+m_{b}\sigma_{3n}^{ \prime})^{\alpha-1}}\bigg{]} \tag{12}\]  
\[c^{\prime}=\frac{\sigma_{ci}[(1+2\alpha)s+(1-\alpha)m_{b}\sigma_{3n}^{\prime} ](s+m_{b}\sigma_{3n}^{\prime})^{\alpha-1}}{(1+\alpha)(2+\alpha)\sqrt{1+\frac{ 6\alpha m_{b}(s+m_{b}\sigma_{3n}^{\prime})^{\alpha-1}}{(1+\alpha)(2+\alpha)}}} \tag{13}\]  
Here, \(\sigma_{3n}^{\prime}\) represents the ratio of the maximum confining pressure \(\sigma_{3\text{max}}^{\prime}\) to the uniaxial compressive strength \(\sigma_{ci}\) of the rock block, and \(\sigma_{3\text{max}}^{\prime}\) is calculated as follows:  
\[\sigma_{3\text{max}}^{\prime}=0.72\sigma_{cm}\bigg{(}\frac{\sigma_{cnt}}{ \rho H}\bigg{)}^{-0.91} \tag{14}\]where \(\rho\) denotes the unit weight of the rock (MN/m\({}^{3}\)), \(H\) represents the engineering burial depth (m), and \(\sigma_{cm}\) corresponds to the uniaxial compressive strength of the rock mass (MPa). The analytical expression for \(\sigma_{cm}\) is as follows:  
\[\sigma_{cm}=\sigma_{ci}\frac{[m_{b}+4s-a(m_{b}-8s)]\left(\
icefrac{{m_{b}}}{{4} }+s\right)^{a-1}}{2(1+a)(2+a)} \tag{15}\]