## 2 EPSD Modeling of Explosive Ground Motions
### Modeling of Functional Relation between EPSD Parameters and Scaled Charge Quantity  
For the purpose of simulating the explosive ground motions reasonably and quantitatively, an exponential model is defined to describe the relationship between the EPSD parameters and the explosion mechanism. Consequently, Equation (6) is further expressed as:  
\[\lambda_{S}^{(i)}=F_{i}(\rho)=a_{i}\cdot\exp{(b_{i}\cdot\rho)}^{m_{i}} \tag{15}\]  
where \(a_{i}\), \(b_{i}\) and \(m_{i}\) are the \(i\)-th corresponding model parameters of \(F_{i}(\rho)\) which are also corresponding to the \(i\)-th EPSD parameters of \(\lambda_{S}^{(i)}\). It should be noted that the aforementioned proposed model is a typical pure mathematical model, and its model parameters have no definite physical meaning.  
In addition, in view of the fact that the identified errors are unavoidable during the identification process, the outliers ought to be eliminated from the EPSD parameters that have been determined in order to guarantee the dependability of the subsequent study. Moreover, the measured recordings of which of the PGA values are less than 5 cm/s\({}^{2}\) ought to be removed since their impacts on structures can be ignored. For that, the multivariate regression method [41] is utilized to concurrently obtain the parameters of the suggested exponential model.  
Figure 3 displays the fitting results, which reveals that the exponential model proposed in this study can roughly describe the changing trend of the EPSD parameters with the scaled charge quantity. It is shown in the figures that as the scaled charge quantity increases, \(\omega_{g}\), \(\omega_{t}\), \(c\), \(d\) and PGA grow while \(\xi_{g}\) and \(\xi_{t}\) decrease, which indicates that the frequency components, duration and PGA of the explosive ground motions become richer, longer and stronger, respectively. It should be mentioned that, currently, due to the limited amount of measured explosive ground motion data, the fitting results may not accurately reflect the trend of EPSD parameters describing explosive ground motions with changes in the explosion environment. Nevertheless, the fitting results can still roughly present the increasing trend of the duration, amplitude and frequency components of explosive ground motions as the scaled charge quantity increases. In subsequent research, it would be more feasible to achieve more accurate results as the number of measured explosive ground motion records increases. The regression parameters of the proposed exponential model are listed in Table 3.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameters** & \(c\) **(s)** & \(d\) & \(\omega_{g}\) **(rad/s)** & \(\xi_{g}\) & \(\omega_{t}\) **(rad/s)** & \(\xi_{t}\) \\ \hline Identified value & 0.40 & 1.65 & 81.74 & 0.14 & 67.37 & 0.29 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Identified parameters of the EPSD function for the typical example.  
Figure 3: The fitting results of EPSD parameters with scaled charge quantity: (**a**) regression result of \(\log(\omega_{\rm g})\); (**b**) regression result of \(\xi_{\rm g}\); (**c**) regression result of \(\log(\omega_{\rm f})\); (**d**) regression result of \(\xi_{\rm f}\); (**e**) regression result of \(c\); (**f**) regression result of \(d\); (**g**) regression result of PGA.  
Consequently, the corresponding EPSD parameter vector \(\mathbf{\lambda}_{S}\) of the explosive ground motions can be obtained by the given scaled charge quantity \(\rho\) according to Equation (15) and Table 3. Further, the EPSD function can be calculated by substituting \(\mathbf{\lambda}_{S}\) into Equation (1).  
In this section, the parameter identification method is proposed, of which the effectiveness is demonstrated by the determination coefficient. In addition, the EPSD parameter prediction formula is suggested based on the parameter identification results and the corresponding scaled charge quantity. This method realizes the prediction of seismic characteristics of explosive ground motions by giving the explosive environment conditions.