## 4 Establishment of a New Prediction Model for Blasting Vibration Attenuation
### Prediction Model for Blast Vibration Attenuation, Taking Elevation Effect into Account
\begin{table}
\begin{tabular}{c c c c} \hline
**Blasting Vibration Attenuation** & \multicolumn{2}{c}{**Direction**} & \multicolumn{2}{c}{**The Fitting Results of**} & \multicolumn{2}{c}{**Coefficient of**} \\
**Prediction Model Equations** & **Direction** & **Equations (1) and (2)** & **Determination (\(r^{2}\))** \\ \hline \multirow{4}{*}{\
\begin{tabular}{c} Traditional blasting vibration attenuation prediction model \\ \end{tabular} } & X-direction & \(v=86.64\big{(}\frac{\sqrt{Q}}{R}\big{)}^{1.770}\) & 0.848 \\  & Y-direction & \(v=76.070\big{(}\frac{\sqrt{Q}}{R}\big{)}^{1.627}\) & 0.875 \\  & Z-direction & \(v=117.58\big{(}\frac{\sqrt{Q}}{R}\big{)}^{1.513}\) & 0.847 \\ \hline \multirow{4}{*}{
\begin{tabular}{c} Prediction model of blasting vibration attenuation considering \\ elevation effect \\ \end{tabular} } & X-direction & \(v=38.99\big{(}\frac{\sqrt{Q}}{R}\big{)}^{1.513}\big{(}\frac{H}{R}\big{)}^{-0.150}\) & 0.859 \\  & Y-direction & \(v=27.89\big{(}\frac{\sqrt{Q}}{R}\big{)}^{1.308}\big{(}\frac{H}{R}\big{)}^{-0.188}\) & 0.885 \\  & Z-direction & \(v=19.27\big{(}\frac{\sqrt{Q}}{R}\big{)}^{0.933}\big{(}\frac{H}{R}\big{)}^{-0.336}\) & 0.902 \\ \hline \end{tabular}
\end{table} 
Table 7: The fitting results of Equations (1) and (2) in each direction. blasting test data using Equations (1) and (2). This demonstrates that the blasting vibration speed in the three blasting tests is influenced by additional factors. This factor might be the layer's inclination angle, based on the analysis presented in the preceding section. By comparing the results in Tables 6 and 7, it is evident that the traditional blasting vibration attenuation prediction model and the blasting vibration attenuation prediction model that only takes the elevation effect into account have lower prediction accuracy than the model that takes the layer inclination angle into account. The latter has a fitting coefficient of determination greater than 0.93, whereas the first two have a fitting coefficient of determination between 0.8 and 0.9. Notably, for the Z-direction, the determination coefficient of the fit using Equation (14) reached 0.97. This indicates that bedding plane inclination has a certain impact on the propagation pattern of blasting vibration. In engineering applications involving interlayered rock slopes, such as sandstone-mudstone interlayers, considering the influence of bedding plane inclination on blasting vibration is fundamental for accurately predicting and controlling blasting vibrations. RMSE (Root Mean Square Error) and MAE (Mean Absolute Error) were computed, respectively, to further examine the prediction accuracy of the three prediction models. Figure 8 displays the results of the calculation. The patterns of RMSE and MAE are generally similar. The bar chart clearly shows that in the first model, the velocity error in the Z direction is the largest, followed by the X direction, with the smallest error observed in the Y direction. When compared to the second model, it can be seen that after accounting for the elevation effect, the velocity error in the Z direction is significantly reduced, while the errors in the X and Y directions also decrease, though less noticeably. This indicates that considering the elevation effect has the greatest impact on velocity in the Z direction. A comparison between the second and third models shows a substantial reduction in velocity errors across all three directions, demonstrating that the prediction model is most accurate when both elevation effect and bedding plane inclination are considered.