## 4 Establishment of a New Prediction Model for Blasting Vibration Attenuation
### Prediction Model for Blast Vibration Attenuation, Taking Elevation Effect into Account  
Considering the applicability conditions of the Sadowsky formula, and based on the relevant literature findings, it is shown that the vibration velocity of the rock slope is significantly affected by the altitude [22; 23; 24]. To more accurately capture the attenuation pattern of blasting vibration in interlayered sandstone-mudstone rock slopes and to improve prediction accuracy, a modified formula accounting for elevation effects was used to fit the experimental data (Equation (2)). Table 4 presents the fitting results. \[v=k\bigg{(}\frac{\sqrt[3]{Q}}{R}\bigg{)}^{\alpha}\bigg{(}\frac{H}{R}\bigg{)}^ {\beta} \tag{2}\]  
In the formula, \(v\) represents the blasting vibration velocity; \(Q\) is the explosive charge weight; \(R\) is the distance from the blast center, i.e., the distance from the measuring point to the center of the explosive charge; H is the elevation difference; \(k\) and \(\alpha\) are coefficients related to the blasting site and geological conditions; and \(\beta\) is the elevation influence factor. According to the fitting results in Table 4, the determination coefficients for each direction are above 0.9 when considering the elevation effect, indicating a good fit. Compared to the results without the elevation effect, the determination coefficients in all directions have improved, demonstrating that the elevation-adjusted fit aligns more closely with actual conditions and has higher accuracy. This improvement is particularly significant in the primary vibration direction, where the determination coefficient increased by up to 0.275. In practical engineering, to predict and control the impact of blasting vibration on slope stability more precisely, it is essential to consider the elevation effect when analyzing the attenuation pattern of blasting vibration in rock slopes. Additionally, as shown in Tables 3 and 4, the fitting parameters \(k\), \(\alpha\), and \(\beta\) vary among the three blasting tests. Engineering observations suggest that the primary difference among these tests lies in the inclination angles of the interlayered sandstone-mudstone planes along the vibration wave propagation path. This implies that the inclination angle \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test Sequence** & **Direction** & \begin{tabular}{c} **Fitting Results Considering** \\ **Elevation Effect Correction** \\ \end{tabular} & \begin{tabular}{c} **Coefficient of** \\ **Determination (\(r^{2}\))** \\ \end{tabular} \\ \hline I & \begin{tabular}{c} X-direction \\ Y-direction \\ Z-direction \\ \end{tabular} & \begin{tabular}{c} \(v=3.48\Big{(}\frac{\sqrt[3]{Q}}{R}\Big{)}^{0.703}\Big{(}\frac{H}{R}\Big{)}^{-0.673}\) \\ \(v=21.12\Big{(}\frac{\sqrt[3]{Q}}{R}\Big{)}^{1.069}\Big{(}\frac{H}{R}\Big{)}^{-0. 190}\) \\ \(v=4.24\Big{(}\frac{\sqrt[3]{Q}}{R}\Big{)}^{0.588}\Big{(}\frac{H}{R}\Big{)}^{-0. 590}\) \\ \end{tabular} & 0.974 \\ \hline II & \begin{tabular}{c} X-direction \\ Y-direction \\ Z-direction \\ \end{tabular} & \begin{tabular}{c} \(v=22.59\Big{(}\frac{\sqrt[3]{Q}}{R}\Big{)}^{1.242}\Big{(}\frac{H}{R}\Big{)}^{-0. 106}\) \\ \(v=27.20\Big{(}\frac{\sqrt[3]{Q}}{R}\Big{)}^{1.606}\Big{(}\frac{H}{R}\Big{)}^{-0. 269}\) \\ \(v=7.813\Big{(}\frac{\sqrt[3]{Q}}{R}\Big{)}^{0.749}\Big{(}\frac{H}{R}\Big{)}^{-0. 516}\) \\ \end{tabular} & 0.981 \\ \hline III & \
\begin{tabular}{c} X-direction \\ Y-direction \\ Z-direction \\ \end{tabular} &
\begin{tabular}{c} \(v=22.53\Big{(}\frac{\sqrt[3]{Q}}{R}\Big{)}^{0.846}\Big{(}\frac{H}{R}\Big{)}^{-0. 086}\) \\ \(v=19.88\Big{(}\frac{\sqrt[3]{Q}}{R}\Big{)}^{0.696}\Big{(}\frac{H}{R}\Big{)}^{-0. 137}\) \\ \(v=15.03\Big{(}\frac{\sqrt[3]{Q}}{R}\Big{)}^{0.542}\Big{(}\frac{H}{R}\Big{)}^{-0. 317}\) \\ \end{tabular} & 0.982 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fitting results considering elevation effect correction. may affect the blasting vibration attenuation pattern.