## 2 Materials and Methods
### Selection of the Related Factors  
\(R_{\rm n}\) is affected by the characteristics of falling rock, slope, and kinematics [29]. This study chose seven key factors related to \(R_{\rm n}\) based on previous research [7; 30; 31]. (1) Shape factor of falling rock  
In this study, the term \"shape factor\" refers specifically to the quantification of the rock's physical shape, particularly its degree of angularity or roundness [19]. \(\eta\) helps describe how the shape of the rock influences its kinematic behavior during the collision with the slope. \(\eta\) is used to classify rockfalls based on their shape characteristics, which are defined in terms of surface area distribution and geometric irregularity. \(\eta\) directly affects \(R_{\rm n}\), as it influences the amount of energy retained during impact. The four typical rock shapes selected in this study were based on different sphericity degrees, ranging from highly rounded blocks to more angular blocks, as shown in Figure 3. \(R_{\rm n}\) was then used to effectively characterize how these different \(\eta\) affect the rockfall's collision with the slope surface. The specific estimation formula is provided in Equation (1). \[\eta=\frac{1}{\psi\gamma}=\frac{3S_{0}}{S\times\left\{\frac{d_{\rm min}}{d_{ \rm max}}+\frac{d_{\rm min}}{d_{\rm mid}}+\frac{d_{\rm mid}}{d_{\rm max}} \right\}}, \tag{1}\]  
where \(\psi\) is the degree of true sphericity, \(\gamma\) is the abnormity index, \(S_{0}\) is the actual surface area of the falling rock, \(S\) is the surface area of a sphere of the same volume as the falling rock, \(d_{\rm max}\) is the longest dimension of the falling rock, \(d_{\rm min}\) is the smallest dimension of the falling rock perpendicular to \(d_{\rm max}\), and \(d_{\rm mid}\) is the dimension of the falling rock both perpendicular to \(d_{\rm min}\) and \(d_{\rm max}\). (2) Slope surface material  
The four slope surface materials selected for this study were concrete, wood, sand, and artificial turf, each representing a different category of slope surface types, as shown in Figure 4. These materials were chosen to simulate various natural slope conditions, including both hard and soft surfaces. Specifically, concrete was selected as a rigid, non-deformable surface that simulates highly compacted surfaces. Wood was used as a medium-rigidity, smooth-textured material, representing moderately soft and smooth surfaces often found in engineered slopes. Sand was chosen for its loose, granular texture, which simulates  
Figure 3: The four different falling rock shapes used in the tests. the frictional interaction in loose material environments. Turf was selected to mimic soft, energy-absorbing natural surfaces, such as those found in vegetated areas. Unlike rigid materials like plastic, turf was chosen because it provides better energy dissipation during collision, thereby more accurately simulating rockfall interactions with natural surfaces. To better quantify the differences in material behavior, a material factor (\(A\)) was introduced to reflect the mechanical properties of each surface. The formulation and corresponding parameters are provided in Equation (2) and Table 1.  
It is worth noting that the same material factor introduced in Table 1 is applied to both the slope surface material and the falling rock type. This shared parameter reflects the mechanical interaction between the rockfall and the slope during collision and is used to ensure consistency in evaluating surface-related effects across different experimental conditions. \[ \begin{split} A&=0.4173-0.0952\times\lambda+0.04 42\times SH,\\ \lambda&=vE\times\left[(1-2v)(1+v)\right]^{-1}\end{split}  \tag{2}\]  
where \(\lambda\) is the Lame constant, \(v\) is the Poisson's ratio, \(E\) is the elastic modulus, and \(SH\) is the Schmidt hardness.