## 2 EPSD Modeling of Explosive Ground Motions  
### EPSD Function of Explosive Ground Motions  
Generally, explosive ground motion can be regarded as a stochastic process \(U(t)\), which is non-stationary in time domain, and its evolutionary power spectrum density (EPSD) function can be defined as follows [33]:  
\[S(\omega,t;\lambda_{S})=\Big{|}f(t;\lambda_{f})\Big{|}^{2}\overline{S}(\omega \ ;\lambda_{\overline{S}}) \tag{1}\]  
where \(S(\omega,t;\lambda_{S})\) denotes the EPSD function, \(f(t;\lambda_{f})\) indicates the intensity modulation function, and \(\overline{S}(\omega\ ;\lambda_{\overline{S}})\) indicates the corresponding one-sided power spectrum density (PSD) function, respectively. The intensity modulation function \(f(t;\lambda_{f})\), adopting [PERSON]'s model with double parameters capable of reflecting the non-stationary characteristics of intensity, is given by [34]:  
\[f(t;\lambda_{f})=\big{[}\frac{t}{c}\exp(1-\frac{t}{c})\big{]}^{d} \tag{2}\]  
where \(c\) and \(d\), respectively, control the peak arrival time and the decay speed of the explosive ground motions. Thus, the parameter vector of the intensity modulation function \(f(t;\lambda_{f})\) can be expressed as \(\lambda_{f}=(c,d)\). For the PSD function \(\overline{S}(\omega\ ;\lambda_{\overline{S}})\) of the corresponding stationary ground motions, the Clough-Penzien model is adopted in this paper, followed by [35]:  
\[\overline{S}(\omega\ ;\lambda_{\overline{S}})=2\frac{\omega_{\mathrm{g}}^{4}+4 \xi_{\mathrm{g}}^{2}\omega_{\mathrm{g}}^{2}\omega^{2}}{\left(\omega^{2}-\omega _{\mathrm{g}}^{2}\right)^{2}+4\xi_{\mathrm{g}}^{2}\omega_{\mathrm{g}}^{2} \omega^{2}}\cdot\frac{\omega^{4}}{\left(\omega^{2}-\omega_{\mathrm{f}}^{2} \right)^{2}+4\xi_{\mathrm{f}}^{2}\omega_{\mathrm{f}}^{2}\omega^{2}}\cdot S_{0} \tag{3}\]  
where \(\omega_{\mathrm{g}}\) and \(\xi_{\mathrm{g}}\) denote the dominant frequency and critical damping of the soil layer, respectively. \(\omega_{\mathrm{f}}\) and \(\xi_{\mathrm{f}}\) denote the dominant frequency and critical damping of the rock bed, respectively. \(S_{0}\) is the spectrum intensity factor, which represents the energy intensity of the explosive ground motions, defined as follows [36]:  
\[S_{0}=\frac{A^{2}}{r^{2}\omega_{\mathrm{e}}};\quad\omega_{\mathrm{e}}=\frac{1 }{S_{0}}\int_{0}^{\infty}\overline{S}(\omega\ ;\lambda_{\overline{S}})\mathrm{d}\omega \tag{4}\]  
where \(r\) and \(A\), respectively, denote the peak factor and peak ground acceleration (PGA) of the explosive ground motions. In light of the fact that the peak factor is insensitive to the changes in the seismic environment, it can thus be considered as a fixed constant valued by \(r=3\) in this study. Therefore, the parameter vector of the PSD function \(\overline{S}(\omega\ ;\lambda_{\overline{S}})\) can be expressed as \(\lambda_{\overline{S}}=(\omega_{\mathrm{g}},\xi_{\mathrm{g}},\omega_{\mathrm{ f}},\xi_{\mathrm{f}},A)\). As previously mentioned, there are a total of seven parameters in the EPSD function which need to be determined. Specifically, the parameters involved in the intensity modulation function control the seismic duration, and the parameters included in the PSD function control the seismic amplitude and the shape of the spectrum, respectively. As a result, the overall parameter vector of the EPSD function \(S(\omega,t;\lambda_{S})\) can be written as:  
\[\lambda_{S}=(\lambda_{f},\lambda_{\overline{S}})=(c,d,\omega_{\mathrm{g}}, \xi_{\mathrm{g}},\ \omega_{\mathrm{f}},\ \xi_{\mathrm{f}},A) \tag{5}\]  
Prior research has demonstrated that the seismic characteristics of explosive ground motions are mostly influenced by charge quantity and explosion source distance. To this end, the scaled charge quantity, which can comprehensively reflect charge quantity and distance to the explosion source, is extensively employed herein to describe the seismic characteristics.