# Characterization of HVSR and VRSR in the Loess Plateau of China Based on Strong-Motion Data
## 3 Methods  
### Data Preprocessing Methods  
To meet the requirements for calculating HVSR and VRSR, the selected 477 strong-motion records were processed using the following four steps.  
(1) Baseline Correction: The polynomial fitting method [34] was applied to correct the baseline of the strong-motion records in the EW, NS, and UD directions individually in order to eliminate baseline drift effects.  
(2) S-Wave Identification: The fundamental assumption of HVSR analysis is that vertical ground motion remains unamplified within the frequency band where horizontal amplification occurs. Research has shown that, in some cases, vertical components can  
Figure 3: Distribution of PGA and epicentral distance in the Loess Plateau region of China.  
Figure 2: Distribution of \(M_{\text{wg}}\) magnitude and epicentral distance in the Loess Plateau region of China.  
exhibit amplification effects [35]. However, S-waves help reduce the influences of the source and propagation path [17]. Therefore, using S-wave spectral ratios has become a common practice for studying site dynamic characteristics [23]. In this study, the Husid function, which reflects the energy distribution throughout the entire acceleration time history, was used to identify the onset of the S-wave phase. The point where the Husid function indicates a sharp increase in energy is defined as the S-wave arrival time, t\({}_{\text{S1}}\). The Husid function [36] is determined by Equation (2). The end time of the S-wave phase was determined using the Cumulative Root Mean Square (CRMS) function [37], with the point where the CRMS function starts to decrease being defined as the end time, t\({}_{\text{S2}}\). The CRMS function is given by Equation (3).  
\[H_{n}(t)=\frac{\int_{0}^{t}{{{\left[{a(t)}\right]}^{2}}dt}}{\int_{0}^{\infty}{ {{\left[{a(t)}\right]}^{2}}dt}}, \tag{2}\]  
\[CRMS=\sqrt{\frac{1}{T}{\int_{0}^{T}{{{\left\|{a(t)}\right\|}^{2}}dt}}}, \tag{3}\]  
where \(a\) (\(t\)) denotes the acceleration time scale and \(T\) denotes the calculated holding time. The S-wave intercepts are shown in Figure 4:  
(3) Taper Preprocessing: After identifying the S-wave, it is common to observe non-zero, large values at the beginning and end of the data. This phenomenon could lead to side lobe effects when performing Fourier transforms, which would impact horizontal-to-vertical spectral ratio calculation. To mitigate this, a cosine taper window was applied to the beginning and end of the strong-motion records, each extending 5% of the S-wave duration beyond the S-wave window [38; 39]. Additionally, zero padding was added at both ends to effectively eliminate the influence of side lobe effects. Equation (4) represents the taper function at the start of the S-wave window, while Equation (5) represents the taper function at the end of the S-wave window.  
\[W_{i}=\frac{1}{2}\bigg{[}1+\cos(\pi\frac{n_{b}+i-1}{n_{b}})\bigg{]},i=1,n_{b}, \tag{4}\]  
\[W_{i}=\frac{1}{2}\bigg{[}1+\cos(\pi\frac{i-1}{n_{c}})\bigg{]},i=1,n_{c}, \tag{5}\]  
where \(n_{b}\) and \(n_{c}\) denote the number of data points contained in the first and last cosine window lengths in the Taper process, respectively.  
(4) Bandpass Filtering. A 4 th-order Butterworth bandpass filter was applied to the strong-motion data with a bandwidth of 0.1-25.00 Hz to remove the unwanted frequency components. Figure 5 shows the acceleration time series in the three directions of EW, NS, and UD for the Minxian station data after the process.  
Figure 4: Example of an S-wave recorded during a strong earthquake at Minxian Station (062 MXT), Loess Plateau region.