# Ground-Borne Vibration Model in the near Field of Tunnel Blasting
## 4 Vibration near Blasting Area
### Vibration Model Based on New Approach  
Through the prediction formula of blasting vibration (15) obtained from dimensional analysis, the functional relationship between the blasting vibration velocity and delayed initiation time of each detonating period \(T\), the variable with three parameters \(\left(\rho_{r}/QR^{-3}\right)\), and the number of free surfaces of the blast holes \(n\) can be obtained, where \(K\), \(\gamma_{1}\), and \(\gamma_{2}\) are unknown parameters. The monitoring data in Table 2 were used and combined with the specific conditions of each group of monitoring data; all vibration data are brought in with the help of the Origin software, and the blasting vibration prediction formula was obtained.  
Figure 6: Vibration velocity in three directions of other blast holes behind the working face: (a) X direction; (b) Y direction; (c) Z direction.  
Since the number of fitting variables has changed from one to three, the fitting dimension has changed from two-dimensional to four-dimensional. Therefore, this problem has been considered more comprehensively by considering the impact of factors on blasting vibration, which will obviously improve the prediction accuracy of the obtained formula. Table 3 lists the parameters obtained by the software regression.  
The propagation model of blasting vibration velocity \(V_{X}\), \(V_{Y}\), and \(V_{Z}\) in three directions of \(X\), \(Y\), and \(Z\) can be expressed as:  
\[V_{X}=161.9T^{-1}(\frac{\rho_{r}}{QR^{-3}})^{-1.02}(n)^{16.5}\Big{(}R^{2}=0.79 \Big{)} \tag{28}\]  
\[V_{Y}=173.85T^{-1}(\frac{\rho_{r}}{QR^{-3}})^{-1}(n)^{15.90}\Big{(}R^{2}=0.75 \Big{)} \tag{29}\]  
\[V_{Z}=119.93T^{-1}(\frac{\rho_{r}}{QR^{-3}})^{-1.085}(n)^{17.84}\Big{(}R^{2}=0. 83\Big{)} \tag{30}\]  
When other factors remain unchanged, the change in vibration data in front of and behind the tunnel face is mainly affected by the density. The problem of density change was considered in Formula (15), so the laws in front of and behind the tunnel face do not need to be distinguished, and they will be displayed together in the calculation results of the formula. At the same time, the number of free surfaces of the blast holes was also introduced into the formula, and it is one of the main conditions to distinguish between cut hole blasting and other hole blasting. This implies that the prediction results of Formula (15) will not be deviated by the difference in the vibration law caused by the two types of blast holes initiated.  
These two variables are important factors affecting the vibration near the blasting area, and their introduction will greatly improve the prediction accuracy of the formula. The \(R^{2}\) values of the new method in the \(X\), \(Y\), and \(Z\) directions are 0.79, 0.75, and 0.83, respectively, which are greatly improved compared with the maximum \(R^{2}\) values of 0.77, 0.60, and 0.66 in the three directions of the Sadovsky formula. At the same time, the calculation complexity is greatly simplified because the new method only requires three formulas instead of twelve.  
In addition, the reason the \(R^{2}\) value is relatively low is that many blasting vibration data come from nearby blasting areas. When the measuring point is very close to the blasting source, the data will fluctuate greatly. In order to intuitively reflect the real situation of the nearby blasting area, the data were not processed in the above fitting process. If the data are denoised and fitted using 95% confidence interval data, the \(R^{2}\) value of Equations (28)-(30) can be increased to 0.86, 0.84, and 0.91.