# Study on the Vibration Variation of Rock Slope Based on Numerical Simulation and Fitting Analysis
## 3 Fitting Analysis of Blasting Vibration Prediction Formula Considering Elevation Value  
The prediction data of [PERSON]'s formula and previous research show that blasting vibration velocity will decrease with the increase of blasting distance [23]. However, a large number of foundation pits, slopes, and mine projects show that the vibration velocity will be amplified to a certain extent with the increase of elevation difference. In this regard, monitoring points were set up in the excavation blasting construction site to explore the response law of the foundation pit slope to blasting vibration during the blasting excavation at the bottom of the foundation pit. The existing formula reflecting the influence of elevation on blasting vibration velocity is as follows [24]:  
\[V=K\left(\frac{Q^{\frac{1}{3}}}{D}\right)^{\alpha}\left(\frac{Q^{\frac{1}{3}}}{ H}\right)^{\beta} \tag{1}\]  
where \(K\) and \(\alpha\) are the coefficient and attenuation index related to the topographic and geological conditions between the bursting points and the calculation of the protected object; \(Q\) is the maximum single section charge of blasting, kg; \(V\) is the designed vibration velocity for the ground particle where the protected object is located, cm/s; \(D\) is the horizontal distance from the bursting point to the measuring point, m; \(H\) is the height difference between the bursting point and the measuring point, m; \(\beta\) is the elevation coefficient. Equation (1) introduces new variables, height difference (\(H\)) and horizontal distance (\(D\)), and is a variation of Sadov's formula. It is only applicable to situations where there is a certain elevation difference between the burst center and the target mass point (the measurement point is higher than the burst center). Generally speaking, a smaller elevation coefficient \(\beta\) indicates a greater influence of elevation and a more significant amplification effect [25; 26]. Therefore, by fitting the blast vibration prediction formula, we can make a preliminary inference that the blast vibration is affected by the elevation. The TC-6850N blast vibration collector can collect blast vibration data in real time at a specific location in the X (vertical), Y (horizontal), and \(Z\) (tangential) directions. The horizontal distance (\(D\)) and height difference (\(H\)) need to be determined prior to blast vibration data collection. After being acquired, the data can be summarized in the form of an electronic file. It is important to note that the resultant velocity is not directly generated by the data collector but is calculated after data compilation. Therefore, the vertical and horizontal blast vibration velocities were selected for analysis. On-site monitoring data of vertical and horizontal vibration velocities were taken as fitting data, as shown in Table 1.  
According to the site monitoring data, the blasting vibration prediction formula was modified to obtain a new formula suitable for the geological conditions of the site. The blast velocity prediction equation in the vertical direction (\(V_{\rm V}\)), considering the elevation value, is shown below:  
\[V_{\rm V}=68.03\left(\frac{\mathrm{Q}^{\frac{1}{3}}}{D}\right)^{2.14}\left(\frac {\mathrm{Q}^{\frac{1}{3}}}{H}\right)^{-0.23} \tag{2}\]  
The prediction equation for blast vibration speed in the horizontal direction (\(V_{\rm L}\)), considering the elevation value, is shown below:  
\[V_{\rm L}=56.26\left(\frac{\mathrm{Q}^{\frac{1}{3}}}{D}\right)^{1.939}\left( \frac{\mathrm{Q}^{\frac{1}{3}}}{H}\right)^{-0.085} \tag{3}\]  
By fitting data, Equations (1)-(3) were obtained. The following conclusions can be obtained by comparing Equations (2) and (3). The correlation coefficients (R\({}^{2}\)) of Equations (2) and (3) are 0.946 and 0.93, indicating that there is a strong correlation between the fitting results and the field monitoring data and the research error is small. In Equation (2), the elevation coefficient is \(\beta=-0.23\), and in Equation (3), the elevation coefficient is \(\beta=-0.085\).