# Characterization of HVSR and VRSR in the Loess Plateau of China Based on Strong-Motion Data
## 4 Results and Discussion
### Characterization of the Power Spectrum  
In the context of random processes, the power spectrum represents the average spectral characteristics of the intensity of each component of the random process as it varies with frequency. It also represents the spectral characteristics of the process's variance. In engineering seismology, the power spectrum is used to describe the relationship between the intensity of the frequency (or period) components contained in the seismic time history and the corresponding frequency (or period). This is expressed using the amplitude spectrum of the square of the seismic motion time series. It characterizes the energy distribution in the frequency domain of the seismic motion time history, highlighting the impact of each frequency (or period) component on the structure [40].  
In engineering applications, the power spectrum is typically calculated using the following method:  
In engineering applications, the power spectrum is usually calculated using the following methods. For a given seismic time history x(t) with a duration T, the principle of energy conservation states that energy remains invariant in both the time and frequency domains under Fourier transformation. According to [PERSON]'s theorem for Fourier transforms,  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**\(R\) (km)**} & \multicolumn{2}{c}{**EW**} & \multicolumn{2}{c}{**NS**} & \multicolumn{2}{c}{**UD**} \\ \cline{2-7}  & \(F_{max}\) **(gal)** & \(T_{f}\) **(s)** & \(F_{max}\) **(gal)** & \(F_{max}\) **(gal)** & \(T_{f}\) **(s)** & \(F_{max}\) **(gal)** \\ \hline
0–20 & 8.22 & 0.17 & 9.44 & 0.15 & 3.81 & 0.16 \\
20–50 & 4.10 & 0.18 & 3.69 & 0.20 & 2.21 & 0.12 \\
50–200 & 3.86 & 0.14 & 3.99 & 0.12 & 1.77 & 0.10 \\ \(\geq\)200 & 32.10 & 3.00 & 32.18 & 3.00 & 15.92 & 2.78 \\ \hline \end{tabular}
\end{table}
Table 2: List of maximum and predominant periods of the mean Fourier amplitude spectrum for different epicenter distance subgroups.  
Figure 8: Mean Fourier amplitude spectra for different epicentral distance groupings in the Loess Plateau region.  
as expressed in Equation (8), Equation (9) can be derived. The left-hand side of Equation (9) represents the energy per unit time. For a specific seismic time history, if the power is calculated based on this equation, the right-hand side can be reformulated as Equation (10). In Equation (10), S(\(\omega\)) denotes the power spectrum of the given seismic time history, also referred to as the energy spectral density function, which is defined in Equation (11).  
\[\int_{-\infty}^{+\infty}x^{2}(t)dt=\frac{1}{2\pi}\int_{-\infty}^{+\infty}|F( \omega)|^{2}d\omega \tag{8}\]  
\[\frac{1}{T}\int_{-\infty}^{+\infty}[x(t)]^{2}dt=\frac{1}{2\pi T}\int_{-\infty }^{+\infty}|F(\omega)|^{2}d\omega \tag{9}\]  
\[\frac{1}{2\pi T}\int_{-\infty}^{+\infty}|F(\omega)|^{2}d\omega=\frac{1}{2\pi} \int_{-\infty}^{+\infty}S(\omega)d\omega \tag{10}\]  
\[S(\omega)=\frac{1}{T}|F(\omega)|^{2} \tag{11}\]  
where: \(x(t)\) represents the original function, \(|F(\omega)|\) is the Fourier amplitude spectrum, and \(\left|F(\omega)\right|^{2}\) is the energy spectral density, also referred to as the energy spectrum.