# Predictive Modelling for Blasting-Induced Vibrations from Open-Pit Excavations
## 4 Prediction Models
### Empirical Formula  
Each empirical formula of the 50 construction sites was generated using Equation (6) with the site factors, K and m. For instance, Equation (13) represents the empirical formula of Site 1 with K and m values of 67.4 and \(-\)1.59, respectively. The site factors of each site are represented in Figure 2. Through this method, 50 empirical formulas were generated and defined as EF-1. Each of the formulas included in EF-1 can only be applied to the PPV prediction at the site where it was generated. The K of Site 2, which is far higher than the rest, seems to be noise. In geotechnical engineering, some noise could have happened due to uncertainties. Thus, datasets obtained from Site 2 should also be analyzed with other datasets.  
\[\text{V}=67.4(\text{SD})^{-1.59} \tag{13}\]  
Test blasting is required to obtain site factors K and m, used in empirical formulas such as EF-1. However, it is difficult to perform test blastings at the preliminary design stage, and representative values of K and m are needed to compensate for this weakness. Representative K and m values of 200 and \(-\)1.6 were proposed based on Design and Construction Guidelines for Open-pit blasting in Road construction published by the Ministry of Land, Infrastructure, and Transport in South Korea [33]. We defined Equation (14) as EF-2 using the K and m. Many engineers have designed preliminary blasting patterns, applying Equation (14).  
\[\text{PPV}=200(\text{SD})^{-1.6} \tag{14}\]  
To derive one representative empirical formula for the 50 sites, we calculated K and m values of 74.9 and \(-\)1.535 using datasets of 50 open-pit blasting construction sites. Equation (15) expresses the representative empirical formula and it was defined as EF-3. Since this is a representative equation of 50 sites, it will show lower prediction accuracy than EF-1. However, it could be used at the preliminary design stage like EF-2. Figure 5 shows EF-3 (solid line) and the 893 datasets (circles) on a log-log plot where the vertical axis is PPV and the horizontal axis is SD. As mentioned in Section 2.2, EF-3 was obtained from the linear regression of the 893 blasting datasets.  
\[\text{PPV}=74.9(\text{SD})^{-1.535} \tag{15}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Characteristics** & **Details** \\ \hline \multirow{4}{*}{Datasets} & Total datasets & 1191 \\  & Training and validation datasets & 714, 179 \\  & Test datasets & 298 \\ \hline \multirow{4}{*}{Structure} & Number of input parameters & 7 \\  & Number of output parameter & 1 \\  & Number of hidden layers & 2 \\  & Number of nodes & 21, 28 \\ \hline \multirow{4}{*}{Training parameters} & Activation function & ReLU \\  & Optimization & Adam \\ \cline{1-1}  & Normalization & Standard scaling \\ \cline{1-1}  & Regularization & Early-stopping \\ \cline{1-1}  & Number of epochs & 4208 \\ \cline{1-1}  & Training algorithm & Back-propagation \\ \hline \hline \end{tabular}
\end{table}
Table 5: Characteristics of the ANN model.  
Equations (16) and (17) are prediction models proposed by the International Society of Explosives Engineers (ISEE) [34] and USBM [35], respectively. These two equations have been widely used to predict PPVs. We defined Equations (16) and (17) as the ISEE model and USBM model, respectively.  
\[\text{PPV}=172.5(\text{SD})^{-1.6} \tag{16}\]  
\[\text{PPV}=71.4(\text{SD})^{-1.6} \tag{17}\]