## 2 EPSD Modeling of Explosive Ground Motions
### Modeling of EPSD Function
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Test Sequence**} & \multirow{2}{*}{**No. of Instrument Tag**} & **Distance to Explosion** & \multirow{2}{*}{**PGA (cm/s\({}^{2}\))**} \\  & & **Source (m)** & & \\ \hline \multirow{6}{*}{\(1\)} & L20-N24677 & 146 & 41.88 \\  & L20-N24783 & 219 & 6.33 \\  & L2015-100 & 389 & 99.36 \\  & L20-N24972 & 387 & 52.98 \\  & L20-N25512 & 132 & 136.61 \\  & L20-N25128 & 372 & 114.47 \\ \hline \multirow{6}{*}{\(2\)} & L20-N24677 & 437 & 7.08 \\  & L20-N24783 & 353 & 81.18 \\  & L2015-100 & 194 & 34.30 \\  & L20-N24972 & 958 & 422.74 \\  & L20-N25512 & 464 & 3.95 \\  & L20-N25128 & 445 & 27.57 \\ \hline \multirow{6}{*}{\(3\)} & L20-N24677 & 272 & 288.24 \\  & L20-N24783 & 220 & 55.64 \\  & L2015-100 & 162 & 101.50 \\  & L20-N24972 & 1531 & 3.53 \\  & L2015-100 & 160 & 0.41 \\  & L20-N24972 & 1534 & 0.67 \\ \hline \multirow{6}{*}{\(4\)} & L20-N24677 & 270 & 0.32 \\  & L20-N24783 & 220 & 0.32 \\  & L2015-100 & 162 & 26.37 \\  & L20-N24972 & 1531 & 2.18 \\  & L20-N25512 & 1273 & 1.28 \\  & L20-N25128 & 437 & 0.58 \\ \hline \multirow{6}{*}{\(5\)} & L20-N24677 & 213 & 1.11 \\  & L20-N24783 & 163 & 1.46 \\ \cline{1-1}  & L2015-100 & 115 & 4.52 \\ \cline{1-1}  & L20-N24972 & 1472 & 28.16 \\ \cline{1-1}  & L20-N25512 & 1218 & 5.10 \\ \cline{1-1}  & L20-N25128 & 413 & 6.65 \\ \hline \multicolumn{2}{c}{Remarks} & \multicolumn{2}{c}{Charge quantity: \(Q=75\)kg} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The information of the measured explosive ground motion records. It should be noted that for the purpose of eliminating the influence of the PGA and peak factor of the spectrum, the amplitudes of all the selected measured records are uniformly scaled to 200 cm/s\({}^{2}\). Figures 1 and 2 compare the NTED and FTED between the measured records obtained by L2015-100 and identified models, respectively. Figures 1a and 2a demonstrate that the identified NTED and FTED both fit well with the corresponding target measured records, which can describe the cumulative distribution of seismic energy with time and frequency, respectively. Figure 1b reveals that the identified intensity modulation function can roughly describe the non-stationary characteristics of explosive ground motions, and Figure 2b demonstrates that the identified PSD function can precisely capture the energy distribution characteristics on the frequency domain of explosive ground motions. Specifically, the energy of the explosive ground motions is mainly concentrated in the range of 50 rad/s-100 rad/s, indicating that the explosive ground motions are provided with more abundant high-frequency components. Obviously, the aforementioned results illustrate the efficacy of the parameter identification method presented in this research. Figure 1: The fitting results of NTED between the measured records and the identified model: (**a**) NTED; (**b**) acceleration. Figure 2: The fitting results of FTED between the measured records and the identified model: (**a**) FTED; (**b**) PSD function. Further, to quantify the effectiveness of the parameter identification method proposed in this paper, the determination coefficient is employed as follows [40]:  
\[R^{2}=1-\frac{\sum\limits_{m=1}^{M}{(\hat{y}_{m}-y_{m})^{2}}}{\sum\limits_{m=1}^{ M}{(\overline{y}-y_{m})^{2}}} \tag{14}\]  
where \(M\) is the total number of the data to be fitted by \(y\). \(\hat{y}\) and \(\overline{y}\), respectively, indicate the mean of the data to be fitted and the fitted data. \(R^{2}\) is the determination coefficient over the interval \([0,1]\). As previously mentioned in the typical example, the determination coefficient of NTED and FTED are, respectively, 0.95 and 0.99, demonstrating the effectiveness of the proposed parameter identification method. Table 2 displays the identified parameters of this typical example.