## 3 Mathematical Model of Long Hole Blasting Vibration Velocity  
The primary safety control standards for blasting seismic effects are the peak particle vibration velocity and the main vibration frequency. Typically, the prediction model for blasting vibration velocity is a function of the maximum charge weight in a single section and the distance from the blasting working surface to the test point. Its general form is the following:  
\[V=kQ^{\kappa}R^{\beta} \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \begin{tabular}{c} **Parameters** \\ \end{tabular}  & \begin{tabular}{c} **Minimum** \\ **Value** \\ \end{tabular}  & \begin{tabular}{c} **Maximum** \\ **Value** \\ \end{tabular}  & \begin{tabular}{c} **Mean** \\ **Value** \\ \end{tabular}  & \begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular}  \\ \hline \ \begin{tabular}{c} The charge weight (kg) \\ Maximum charge per delay (kg) \\ The number of delays \\ The horizontal distance between the blasting point and \\ the measurement point (m) \\ The elevation difference between the blasting point and \\ the measurement point (m) \\ The peak particle velocity in the vertical direction (cm/s) \\ \end{tabular}  &
\begin{tabular}{c} 1500 \\ 319 \\ 519 \\ 519 \\ 519 \\ 519 \\ 519 \\ 519 \\ 519 \\  
where \(V\) is the blasting vibration intensity (velocity or acceleration); \(Q\) is the maximum charge per delay; \(R\) is the distance between the blasting working surface and the test point; and \(k\), \(\alpha\), and \(\beta\) are constants relating to geological terrain. Various situations necessitate different constants (\(\alpha\) and \(\beta\)), leading to the derivation of Formula (1) suitable for specific contexts, such as the Sadowsky formula, USBM formula, U-B formula, Indian formula, and Japanese formula. The Sadowsky empirical formula is commonly utilized for predicting the peak vibration velocity of particles. However, discrepancies often exist between the predicted values and the measured values. The literature [32; 33; 34] extensively compares different fitting formulas and indicates that the results obtained using the US Bureau of Mines (USBM) formula [6] closely align with reality. This comparison offers valuable insights when exploring the most accurate fitting formula for predicting vibration velocity. In this study, we employ the fitting equation based on the USBM formula for fitting analysis of blasting test data. The peak particle velocity (PPV) is primarily influenced by factors including the maximum charge per delay (\(Q\)), the distance from the blasting working face to the test point (\(R\)), the scaled distance (\(SD\)), and the site attenuation coefficients (\(k\) and \(\alpha\)). It can be expressed by the following:  
\[\text{PPV}=k(SD)^{\alpha} \tag{2}\]  
\[SD=\frac{R}{Q^{1/2}} \tag{3}\]  
Forty-six sets of vertical particle vibration velocity correlation monitoring data were employed to establish a relationship between the proportional distance and vertical particle vibration velocity. The site attenuation coefficients, which are related to terrain and geological conditions from the explosion center to the measuring point, were determined. A vertical particle vibration velocity model was formulated, as depicted in Figure 2(a). Utilizing this model, the predicted peak vibration velocity of the particle was calculated, resulting in a coefficient of determination (\(R^{2}\)) of 0.2587 between the predicted and measured values. The USBM fitting formula is \(V=72.28(\frac{R}{Q^{1/2}})^{-1.863}\), and the fitting equation is \(V=72.28\times SD^{-1.863}\). After excluding five sets of noisy data with significant accuracy errors, a vertical particle vibration velocity model within the monitoring range was established, as illustrated in Figure 2(b). This model was employed to obtain the predicted peak vibration velocity of the particle, yielding a coefficient of determination (\(R^{2}\)) of 0.6358.