## 3 Generation of Explosive Ground Motions Based on Dimension-Reduction Method
### Numerical Examples  
In this study, three cases are, respectively, assumed as the scaled charge quantity \(\rho\): 0.02, 0.03 and 0.04, which are typical situations in engineering. In this end, the corresponding EPSD parameters used for simulation are listed in Table 4, which can be calculated through substituting the scaled charge quantity \(\rho\) into the prediction formula, i.e., Equation (15), by referring to Table 3. The parameters of the original spectral representation formula, i.e., Equation (16), are presented in Table 5.  
Figure 4 displays the generated representative samples of explosive ground motions with different scaled charge quantities, which have different seismic characteristics under the conditions of the three cases. Obviously, as the scaled charge quantity \(\rho\) increases, namely, the charge quantity \(Q\) increases or the distance to the explosion source \(R\) decreases, the effective intensity duration, PGA and frequency components of the simulated explosive ground motions get longer, stronger and richer, respectively, which is in line with the understanding of the impact of explosion mechanisms on the seismic characteristics of explosive ground motions. Figure 5 depicts comparisons of the mean and standard deviation (Std. D) between the simulated time-histories with the corresponding target ones under the three cases, which reveals that the mean and Std. D are both well fitted to the corresponding targets. Specifically, it can be acknowledged by inspection of Figure 5a that the simulated explosive ground motions are zero-mean stochastic processes, and the information obtained by analyzing Figure 5b is that as \(\rho\) increases, the effective intensity duration and response Std. D of the explosive ground motions, respectively, get longer and stronger. Therefore, the effectiveness of the dimension-reduction method for generating explosive ground motions can be further verified. The average relative errors (AREs) upon the mean and Std. D are calculated under the cases of the number of representative samples--which are 144, 377 and 610, respectively--generated by the dimension-reduction method and the Monte Carlo method to further verify the accuracy and superiority of the former. The results in detail are listed in Table 6. Obviously, the AREs upon the mean of the dimension-reduction method are practically \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Cases**} & \multicolumn{6}{c}{**EPSD Parameters**} \\ \cline{2-9}  & \(\mathbf{c}\) **(s)** & \(\mathbf{d}\) & \(\mathbf{\omega_{\text{g}}}\) **(rad/s)** & \(\mathbf{\xi_{\text{g}}}\) & \(\mathbf{\omega_{\text{f}}}\) **(rad/s)** & \(\mathbf{\xi_{\text{f}}}\) & **PGA** \\ \hline \(\rho=0.02\) & 0.41 & 0.87 & 103.42 & 0.26 & 63.16 & 0.26 & 94.44 \\ \(\rho=0.03\) & 0.61 & 1.09 & 123.84 & 0.25 & 71.80 & 0.16 & 121.60 \\ \(\rho=0.04\) & 0.91 & 1.36 & 148.28 & 0.25 & 81.62 & 0.09 & 156.56 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The predicted EPSD parameters used for generating explosive ground motions. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Parameter**} & **Value** \\ \hline Upper cut-off frequency (rad/s) & \(\omega_{\text{u}}=240\) \\ Frequency increasement (rad/s) & \(\Delta\omega=0.15\) \\ Simulated duration (s) & \(T=5\) \\ Time increasement (s) & \(\Delta t=0.001\) \\ Number of representative samples & \(n_{\text{sel}}=144\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters for original spectral representation formula. equal to zero, which fully indicates the simulated stochastic explosive ground motions are zero-mean processes. The AREs upon the Std. D of the dimension-reduction method are smaller than 5%, which is acceptable for engineering purposes. The table also shows that as the number of the representative samples rises, the AREs upon both the mean and Std. D of the dimension-reduction method decline.