## 4 Establishment of a New Prediction Model for Blasting Vibration Attenuation
### Prediction Model for Blast Vibration Attenuation, Taking Elevation Effect into Account
\[\frac{v}{c}=\varphi(\frac{\mu}{R},\frac{\rho}{QR^{-3}},\theta,\frac{a}{R^{-1}c^{2 }},\frac{f}{R^{-1}c},\frac{t}{Rc^{-1}},\frac{H}{R}) \tag{6}\]  
Since the product and power of different dimensionless numbers \(\pi\) remain dimensionless, we can combine \(\pi_{1}\), \(\pi_{2}\) and \(\pi_{7}\) as follows to obtain a new dimensionless number \(\pi_{8}\):  
\[\pi_{8}=(\pi_{2})^{\frac{1}{3}}\pi_{3}\pi_{7}=(\frac{\sqrt[3]{\rho}R}{\sqrt[3] {Q}})(\frac{H}{R})\theta \tag{7}\]  
For a given site, \(\rho\) and \(c\) can be approximated as constants. Thus, from Equation (7), \(v\) can be considered to have a functional relationship with \(\left(\frac{1}{Q^{1/3}R^{-1}}\right)\left(\frac{H}{R}\right)\theta\). Taking into account the attenuation relationship of vibration velocity with \(H/R\) and \(\theta\), the above functional relationship can be expressed as  
\[\ln v=\left[\alpha_{1}+\beta_{1}\ln(\frac{\sqrt[3]{Q}}{R})\right]+\left[\alpha _{1}{}^{\prime}-\beta_{1}{}^{\prime}\ln(\theta)\right]+\left[\alpha_{1}{}^{ \prime\prime}-\beta_{1}{}^{\prime\prime}\ln(\frac{H}{R})\right] \tag{8}\]  
If \(\ln v_{0}=\alpha_{1}+\beta_{1}\ln\left(\frac{\sqrt[3]{Q}}{R}\right)\), we obtain  
\[\ln v_{0}=\alpha_{1}+(\beta_{1}\ln Q)/3-\beta_{1}\ln R \tag{9}\]  
The attenuation index, \(\beta_{1}\), primarily reflects the influence of site medium conditions, while \(\alpha_{1}+(\beta_{1}\ln Q)/3\) comprehensively reflects the contribution of slope medium conditions and explosive amount to the slope rock mass particle vibration. \(-\beta_{1}\ln r\) in the formula represents the attenuation of blasting vibration speed with distance \(r\). If \(\alpha_{1}=\ln k_{1}\), we obtain  
\[v_{0}=k_{1}(\frac{\sqrt[3]{Q}}{R})^{\beta_{1}} \tag{10}\]  
The conventional Sadowsky formula is given by Equation (10). Equation (10) can be substituted into Equation (11) to obtain  
\[\ln v=\ln v_{0}+\left[\alpha_{1}{}^{\prime}-\beta_{1}{}^{\prime}\ln(\theta) \right]+\left[\alpha_{1}{}^{\prime\prime}-\beta_{1}{}^{\prime\prime}\ln( \frac{H}{R})\right] \tag{11}\]  
If \(\alpha_{1}{}^{\prime}=\ln k_{2},\alpha_{1}{}^{\prime\prime}=\ln k_{3},\beta_ {3}=-\beta_{1}{}^{\prime},\beta_{2}=-\beta_{1}{}^{\prime\prime}\), Equation (11) can be transformed into  
\[v=k_{1}k_{2}k_{3}\left(\frac{\sqrt[3]{Q}}{R}\right)^{\beta_{1}}\left(\frac{H} {R}\right)^{\beta_{2}}(\theta)^{\beta_{3}} \tag{12}\]  
If we assume \(k=k_{1}k_{2}k_{3}\), we obtain  
\[v=k\left(\frac{\sqrt[3]{Q}}{R}\right)^{\beta_{1}}\left(\frac{H}{R}\right)^{ \beta_{2}}(\theta)^{\beta_{3}} \tag{13}\]  
In the formula, \(k\) is the site influence coefficient; \(\beta_{1}\) is the attenuation coefficient; \(\beta_{2}\) is the parameter for the effect of elevation; and \(\beta_{3}\) is the coefficient representing the influence of bedding plane inclination. To maintain a unified and esthetically pleasing form of the equation, while also ensuring that the bedding plane inclination in the formula is expressed in radians, Equation (13) is further revised as follows:  
\[v=k\left(\frac{\sqrt[3]{Q}}{R}\right)^{\beta_{1}}\left(\frac{H}{R}\right)^{ \beta_{2}}\left(\frac{\theta}{\pi}\right)^{\beta_{3}} \tag{14}\]We can fit the field test results using Equation (14); During the fitting process, outliers were identified and removed using a MATLAB(R2019a)-based algorithm to ensure data reliability. Of note, the layer inclination angles of the test slope at the plane of the three blasting test holes were 60\({}^{\circ}\), 50\({}^{\circ}\), and 30\({}^{\circ}\), respectively, according to the on-site inspection of the layer occurrence. However, using Equation (14) to fit only the data from a single blast had no practical significance because the bedding inclination angles of the same test were the same.