## 4 Establishment of a New Prediction Model for Blasting Vibration Attenuation
### Prediction Model for Blast Vibration Attenuation, Taking Elevation Effect into Account
Therefore, incorporating the bedding plane inclination into the attenuation model is necessary for a more accurate description of the blasting vibration attenuation pattern in interlayered sandstone-mudstone rock slopes. Prediction Model for Blast Vibration Attenuation Taking into Account the Angle of Inclination of the Layer  
The above analysis indicates that interlayered sandstone-mudstone rock slopes exhibit complex properties, and traditional predictive models fail to accurately predict the attenuation pattern of blasting vibrations. Predictive models that only consider the elevation effect show significant variability in the parameters \(k\) and \(\alpha\). Furthermore, existing blasting vibration attenuation models do not account for the influence of bedding plane inclination. Therefore, to incorporate the impact of bedding plane inclination, a blasting vibration prediction model that captures this effect is theoretically derived through dimensional analysis [25; 26; 27]. The specific steps are as follows: The attenuation of blast-induced seismic waves propagating within high rock slopes is influenced by multiple factors, including the characteristics of the blast source, the medium conditions of the slope body (such as lithology, joints, and geological structure), the distance from the blast source, and elevation differences. The summary of the main variables is presented in Table 5.  
Using the Buckingham \(\pi\) theorem from dimensional analysis, the peak particle velocity \(v\) in high rock slopes can be expressed as  
\[v=\varphi(Q,\mu,c,\rho,R,H,\theta,a,f,t) \tag{3}\]  
According to the \(\pi\) theorem, where the independent dimensional quantities are chosen as \(Q\), \(R\), and \(c\), the dimensionless terms can be represented by \(\pi\), yielding  
\[\left\{ \begin{array}{l}\pi=\frac{\rho}{Q^{R}R^{c}\tau},\pi_{1}=\frac{\rho} {Q^{R}R^{c}\tau},\pi_{2}=\frac{\rho}{Q^{R}R^{c}\tau}\\ \pi_{3}=\frac{\rho}{Q^{R}R^{c}\tau},\pi_{4}=\frac{a}{Q^{R}R^{c}\tau},\pi_{5}= \frac{f}{Q^{R}R^{c}\tau}\\ \pi_{6}=\frac{f}{Q^{R}R^{c}\tau},\pi_{7}=\frac{H}{Q^{R}R^{c}\tau}\end{array}  \right. \tag{4}\]  
In the formula, \(\alpha\), \(\beta\), and \(\gamma\) are undetermined coefficients. According to the principle of dimensional homogeneity, we have  
\[\left\{ \begin{array}{l}\pi=\frac{\rho}{c},\pi_{1}=\frac{\mu}{R},\pi_{2}= \frac{\rho}{Q^{R-3}},\pi_{3}=\frac{\theta}{1}\\ \pi_{4}=\frac{a}{Q^{R}R^{c}},\pi_{5}=\frac{f}{R^{-1}\tau},\pi_{6}=\frac{b}{Q ^{R-1}},\pi_{7}=\frac{H}{K}\end{array} \right. \tag{5}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable Type** & **Sign** & **Paraphrase** & **Dimension** \\ \hline \multirow{4}{*}{Dependent variable} & \(\mu\) & Displacement of particle vibration & L \\  & \(v\) & Peak velocity of particle vibration & LT\({}^{-1}\) \\  & \(a\) & Acceleration of particle vibration & LT\({}^{-2}\) \\  & \(f\) & Frequency of particle vibration & T\({}^{-1}\) \\ \hline \multirow{4}{*}{Independent variable} & \(Q\) & Explosive quality & M \\  & \(R\) & The distance between the source of the explosion and the particle & L \\  & \(H\) & Difference in elevation between the explosion source and the particles & L \\  & \(\rho\) & Density of rocks & ML\({}^{-3}\) \\  & \(c\) & Vibration wave speed of propagation & LT\({}^{-1}\) \\  & \(t\) & Detonation time & T \\  & \(\theta\) & Inclination angle & / \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Vital physical parameters for vibration in slope blasting. Substituting Equation (5) into Equation (3) yields Equation (6).