# Ground Motion and Site Characteristics Evaluation Using Hilbert-Huang Transform Method Based on Light-Moderate Earthquakes

Authors: Jinchang Chen; Lanmin Wang; Ailan Che; Hanxu Zhou
Journal: Shock and Vibration
 (2022)

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## Abstract

Compared with strong earthquakes (Ms ≥ 6.0), light-moderate earthquakes (Ms 4–5.9) occurred more frequently and distributed more widely, and rich ground motions have been collected around world. It provides a solid data base for the investigation on the ground motion and regional site characteristics which are closely related to seismic geological disaster. We chose two light-moderate earthquakes in China as study cases. Based on Hilbert-Huang transform, we analyzed the ground motion characteristics in time-frequency domain. Using a stable coda wave and marginal spectrum of it based on horizontal to vertical Fourier spectral ratio (HVSR) method, the site characteristics are clarified. Result shows that the amplitude (instantaneous frequency energy) of Hilbert-Huang spectrum is mainly concentrated within 20 Hz at the stations that are close to epicenter. The energy will change from concentration to divergence in time and frequency and decrease with the increase of epicenter distance under soil site condition. Two earthquakes present a low prominent frequency characteristic (1.075–2.45 Hz). Hilbert energy at soil layer is about 3 times greater than it at rock site when the epicenter distance is close and soil layer can also amplify the instantaneous energy. The predominant frequency of site at each station is within 3.6–6.2 Hz. The predominant frequencies of the earthquakes are lower than the predominant frequencies of the sites and would not produce resonance effects. The result can provide meaningful references for later study works about the characteristics and application of light-moderate earthquakes.


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## Body

## 1. Introduction

Ground motion and regional site characteristics are closely related to seismic geological disaster. The seismic response of slope will be significantly amplified, when the predominant frequency of ground motion approaches the natural frequency of slope [1]. The comparative study on the ground motion of bedrock site and soil site with close epicenter distance shows that the ground motion of bedrock site is much smaller than that of soil site, which is mainly due to the amplification effect of soil site on ground motion [2, 3]. Previous studies focused on the characteristics and effects of strong earthquakes (Ms ≥ 6.0), because large casualties and natural disasters were caused [4–6]. However, there are fewer large earthquakes with rich near-field waveform records on the mainland. Therefore, it is difficult to clarify the general law and site characteristics from these earthquakes. Correspondingly, there are many light-moderate earthquakes (Ms 4–5.9) with wide geographical distribution. With the development of observation technology, many light-moderate earthquakes have been recorded, and rich ground motion data have been accumulated all over the world. Using light-moderate earthquakes can comprehensively clarify the regional site characteristics and reveal the correlation between ground motion characteristics and disasters in a wider range. Meanwhile, light-moderate earthquakes also induced many disasters in history. For example, in 1995, an earthquake with magnitude of Ms 5.8 occurred at Yongdeng, Gansu province, China. Ten people died in the earthquake, which was mainly caused by the loess landslides [7]. A Ms 5.2 earthquake with shallow source depth occurred in Minxian-Zhangxian, Gansu province, China on November 13, 2003. In the earthquake, one person died and 26 people were injured, 3344 houses collapsed, and 3052 buildings became dangerous [8]. Relative studies to light-moderate earthquakes have been performed. A series of moderate earthquakes (M 5.8–6.5) in western Gulf greatly impacted the marine sedimentary [9]. The dynamical process of ionosphere following a moderate magnitude earthquake (M≈5.9) has been studied [10]. Ground motion prediction equation was proposed to assess the probabilistic seismic hazard in low-to-moderate seismic areas [11]. The characteristics of soil radon (Rn-222) during a series of moderate earthquakes (M 4.5–5.8) were clarified [12]. In low-to-moderate earthquake areas, non-uniform Timoshenko beam models have been used to assess the seismic performance of buildings [13]. As for the investigation about ground motion and regional site characteristics, and the relationship with seismic geological disasters using light-moderate earthquakes is insufficient.Seismic ground motion analysis enables us to know the natural characteristics of earthquakes and the stratum conditions. It is the base for further geotechnical disaster prevention, seismic design of structures, and seismological research. Time domain analysis of signals can reflect the characteristics of amplitude with time and the attenuation law and attenuation model of amplitude. However, frequency information of signal cannot be clarified. Fourier transform is widely used in seismic signal or ambient vibration [14, 15]; however, it is only suitable for linear and stationary data analysis in theory. The seismic wave triggered by an earthquake is a kind of sudden event, which can be regarded as a nonlinear and nonstationary process, which makes it difficult to apply the seismic signal processing technology based on the traditional linear stationary system theory. Hilbert-Huang transform (HHT) can be used for nonlinear and nonstationary data analysis, which developed rapidly in recent years and is used in many fields [16–20]. Especially in mining engineering, the structure and rock mass under different stress state show different spectral characteristics during failure process [21–25].For the record of microtremors, HVSR methods were often used to evaluate the site effects. The method is based on two assumptions [26]: 1. The vertical and horizontal values (amplitude and spectrum components) on the bedrock are equivalent; 2. The soil layer has a significant impact on the horizontal vibration, while it has little impact on the vertical vibration in the frequency components. It represents the transfer function of the surface horizontal component to the bedrock horizontal component, and the reference site is no longer needed. Studies [27] verified that the HVSR of light-moderate earthquake records can identify the amplification effect of the site, at least its resonance frequency domain. Later, this method was extended to analyze seismic response of nonlinear sites. Lermo & Chavez Garcia (1993) [28] analyzed the transfer functions of three sites in different cities in Mexico. The results of HVSR are very close to the transfer function obtained directly from the records. It shows that the predominant period and local amplification level of simple geological conditions can be obtained by the method. Bonilla, F. et al. [27] used direct S-wave, coda, and HVSR to estimate the site amplification effect in San Fernando Valley, California. The results show that the dominant period of site effect estimated by HVSR is the same as that calculated by direct S-wave and coda method, but the amplification effect is smaller. For microtremors or near-field vibration of light-moderate earthquakes, hypothesis 1 may be acceptable, but it is impossible for large earthquakes, especially for far-field nonlinear response in large earthquakes [29]. It reflects that site characteristics can be evaluated based on light-moderate earthquakes using HVSR method.We chose two light-moderate earthquakes with magnitude Ms 5.0 and Ms 4.9 that occurred in loess area of China as study cases and investigated the spectrum characteristics of them based on HHT method in the study. First, all the seismic waves recorded by different strong motion stations that in different site conditions and epicenter distances were performed baseline correction. Second, the seismic waves were decomposed into a series of Intrinsic Mode Functions (IMFs) through Empirical mode decomposition (EMD) and each of them satisfied the hypothesis of Hilbert transform. Third, the correlation coefficients between IMFs and original signal were calculated, and the seismic signals were reconstructed by eliminating the IMFs that have low correlation with the original signal. Forth, Hilbert transform was performed on the reconstructed signal. Then, the Hilbert-Huang spectrum, marginal spectrum, and instantaneous energy spectrum of each seismic wave were obtained. We analyzed the three types of spectrums of light-moderate earthquakes and clarified the characteristics of them from time-frequency domain. At last, the site characteristics are discussed based on the coda wave of the light-moderate earthquakes using HVSR method.


## 2. Case Study

Light-moderate earthquakes (Ms 4–5.9) frequently occurred in Gansu Province, China. More than 119 light-moderate earthquakes occur in loess area of China from June 1, 1980 to October 28, 2019, which include 96 earthquakes with magnitude Ms 4–4.9 and 23 earthquakes with magnitude Ms 5–5.9. This study chose two typical light-moderate earthquake events in the area as study cases.The locations of epicenter of two earthquakes and the strong motion stations were shown in Figure1. The elevation of the area changed from 1224 m to 5218 m. Many faults distributed in this area such as Yumushan, Minle-Damaying, Huangcheng-Shuangta, Lenglongling, and Longshoushan. So, earthquakes frequently occur in this area. A Ms 5.0 earthquake with source depth about 11 km occurred on September 16, 2019. In the study area, two strong motion stations which are HZH and MLE recorded this event, as it is shown in Figure 1. Another Ms 4.9 earthquake with source depth about 16 km also occurred on May 11, 2012. In the area, four strong motion stations recorded the event, which are HJA, XCZ, YCH, and QSZ. The location, elevation, and epicenter distance of each strong motion station was shown in Table 1. It shows that HZH and HJA stations are about 2.4 km and 5.7 km away from the epicenter of Ms 5.0 and Ms 4.9, respectively, which can reflect many meaningful natural information of the two earthquake events. Among these stations, QSZ is on the rock site and with epicenter distance close to YCH and XCZ which are on soil sites. So, it can reflect the effect of site conditions on the ground motions.Figure 1
Study area and location of epicentral and strong motion stations.Table 1
Details about each strong motion station.
MagnitudeEpicenterStrong motion stationsLocationEpicentral distance (km)Elevation (m)Site conditionMs 5.0100.35 E, 38.6 NHZH100.4 E, 38.6 N2.42186SoilMLE100.8 E, 38.4 N38.422383SoilMs 4.9102 E, 37.75 NHJA102 E, 37.8 N5.72774SoilYCH102 E, 38.2 N50.252011SoilQSZ101.4 E, 37.5 N59.813085RockXCZ101.5 E, 38.2 N66.52511Soil


## 3. Method

HHT is composed of two parts which are EMD and Hilbert Transform [17]. First, the signal is processed by EMD to obtain a series of IMFs, and each order of IMF can satisfy the hypothesis of Hilbert transform; Second, each order of IMF is to perform Hilbert transform to obtain Hilbert-Huang spectrum, Hilbert marginal spectrum, and instantaneous energy spectrum.According to the theory of EMD, every time history of signal X(t) can be decomposed into n IMFs with frequency from high to low and a r(t), as it is shown in:(1)xt=∑i=1nIMFit+rt,where IMF1, IMF2, IMF3 … are the Intrinsic Mode Functions of different frequencies from high to low; r (t) is residual.Based on EMD, the intrinsic mode functions of seismic waves recorded by HZH-EW were obtained, as shown in Figure2. The duration of original signal is 76.87 s. Peak value 71.573 cm/s2 occurred at 23.56 s. The seismic wave was decomposed into 10 IMF with frequency from high to low and a residual. From IMF1 to IMF10, the amplitude gradually decreases, and IMF1-5 contain the most of amplitude of the original seismic wave.Figure 2
Intrinsic mode functions (IMFs) of HZH-EW.In the IMFs, some low frequency components are generated by numerical error and do not have any effects on the original signal. In order to acquire the useful components, the threshold of each signal was established [30]. The calculation method of threshold value ρ for extracting useful signals was shown in (2). When μi ≥ ρ, the former IMFi were retained, and the remaining IMF were eliminated [17].(2)ρ=maxμi10maxμi−3,i=1,2…n,where μi represents the correlation coefficient of the IMFi to the original signal; n is total number of IMFs.The correlation coefficients were shown in Figure3. The threshold of intrinsic mode functions of seismic waves recorded by HZH-EW is 0.16 according to the calculation method mentioned in (2). The correlation coefficients of IMF1-5 greater than threshold, which were retained. The correlation coefficients of IMF6-10 close to 0, which were eliminated. The IMF2 has the most correlation with original signal and the correlation coefficient up to 0.8. Then, the reconstructed signal composed by IMF1-5 was used to make Hilbert transfer.Figure 3
Correlation coefficient distribution.After the original signals were decomposed by EMD and extracted the useful components, the instantaneous frequency corresponding to each ofIMFi is calculated by Hilbert transform. Hibert-Huang spectrum is obtained. As it is shown in:(3)Ht,w=∑i=1nait,wi,where ωi (t) is the instantaneous frequency corresponding to each of IMFi; ait,wi is the amplitude corresponding to ωi (t) of the IMFi at certain a time t; H(t, ω) is time-frequency distribution of the amplitude of the whole signal.The Hilbert marginal spectrum is shown in:(4)hw=∫0THt,wdt.The instantaneous energy spectrum is acquired by integrating theH2t,w in frequency. As it shows in:(5)IEt=∫wH2t,wdw.All the seismic waves were performed baseline correction at first in this study. The calculation process of the HHT is implemented based on MATLAB software.


## 4. Results

### 4.1. Spectrum Characteristics of Strong Motion

#### 4.1.1. Hilbert-Huang Spectrum Characteristics

Hilbert-Huang spectrums of seismic waves recorded by station HZH, MLE, HJA, XCZ, YCH, and QSZ were shown in Figures4 and 5 respectively. Every station records the seismic wave with three directions and we just list the results of EW and NS directions for the amplitude of UD direction is much smaller than EW and NS directions. In the Hilbert-Huang spectrum, the relationship among time, frequency, and energy can be obtained. The amplitude of the spectrum represents the energy of instantaneous frequency.Figure 4
Hilbert-Huang spectrum of the reconstructed signal of HZH and MLE.Figure 5
Hilbert-Huang spectrum of the reconstructed signal of HJA, YCH, XCZ, and QSZ.
(a)(b)The Hilbert-Huang spectrums of seismic waves of Ms 5.0 recorded by two strong motion stations were shown in Figure4. Results show that the Hilbert-Huang spectrums of two stations with different epicenter distances show totally different characteristics and the distribution has certain similarity at the same station in different directions. The spectrum of HZH-EW shows that the energy concentrated at 20–30 s and instantaneous frequency concentrated within 20 Hz. Compared with HZH-EW, the energy of HZH-NS also concentrated at 20–30 s and instantaneous concentrated within 20 Hz, but peak value gradually shifts to relative high frequency (close to 20 Hz). The spectrum of MLE-EW shows that the energy widely distributed at 15–40 s and instantaneous frequency distributed within 40 Hz. Similarly, the energy of seismic wave which recorded by MLE-NS also distributed within 40 Hz. Except for the distribution characteristics, the amplitudes of two stations also have quite a difference. The peak energy of HZH-EW is about 62.3 cm/s2 and HZH-NS is about 41.2 cm/s2; however, the energy of MLE-EW is 4.6 cm/s2, and MLE-NS is 4.0 cm/s2. The peak amplitude of HZH-NS is about 10 times greater than MLE-NS and the HZH-EW is more than 10 times greater than MLE-EW, which reflect that the energy of seismic wave will changes from concentration to divergence in time and frequency and gradually decreases with the increase of epicenter distance.The Hilbert-Huang spectrums of Ms 4.9 were shown in Figure5. The spectrums of HJA-EW and HJA-NS show that the energy concentrated at 20–25 s and instantaneous frequency concentrated within 20 Hz. The peak amplitude of HJA-EW is about 30 cm/s2 and the peak value of HJA-NS is about 25 cm/s2. The location of YCH has the same longitude as HJA and lower elevation than HJA. The energy of seismic wave at YCH station is distributed within 10–40 s and with instantaneous frequency within 30 Hz. The peak values are 4.5 cm/s2 and 5.2 cm/s2 in the directions of EW and NS, respectively. Compared with the HJA, energy at YCH is more scattered and the peak value decreases about 5 times, which reflects that seismic energy attenuates with the epicenter distance increasing. The distribution of energy at station XCZ is more scattered compared with other stations and the peak amplitudes are 3.12 cm/s2 and 2.60 cm/s2, respectively, which are smaller than YCH, because the epicenter distance of XCZ is more far than YCH. The energy of seismic wave recorded by QSZ is concentrated at 20–25 s, the instantaneous frequency is concentrated within 20 Hz, and the peak amplitude are 1.5 cm/s2 and 1.58 cm/s2 in the directions of EW and NS, respectively. The epicenter distance of QSZ is between YCH and XCZ, whereas the energy of it is very concentrated compared with YCH and XCZ and the energy is smallest among them. This is because the site condition of QSZ station is rock, which is different from others. The results reflect that site conditions can significantly influence the distribution characteristics of instantaneous frequency energy. The energy of a seismic wave is more divergent in the soil than it is in the rock in the process of seismic wave propagation, the distribution characteristics of instantaneous frequency energy are closer to epicenter compared to the soil site condition, and the soil layer can greatly amplify the instantaneous frequency energy.


#### 4.1.2. Marginal Spectrum Characteristics

Marginal spectrum reflects the distribution law of Hilbert energy with frequency. Hilbert energy Marginal spectrum is different from Fourier spectrum. Marginal spectrum is based on the HHT and its frequency is a function of time. So, the frequency in marginal spectrum may not necessarily appear in the whole period, but at a certain time or several times [31].The marginal spectrums of Ms 5.0 and Ms 4.9 seismic waves at different strong motion stations were shown in Figures6(a) and 6(b), respectively. Result shows that the shapes of marginal spectrum in N-S and E-W directions are similar with each other at each station and Hilbert energy of frequency mainly distributed within 20 Hz. Peak Hilbert energy of HZH-NS is about 23.9 and the energy is about 4.0 at MLE-NS, which reflects that peak amplitude of HZH-NS is about 6.6 times greater than MLE-NS. Peak amplitudes of HZH-NS and MLE-NS occurred at 1.125 Hz. Peak amplitudes of HZH-EW and MLE-EW occurred at 1.625 Hz and 1.375 Hz, respectively, and peak amplitude of HZH-EW is about 6.03 times greater than MLE-EW. Hilbert energy of Ms 5.0 seismic waves shows a downward trend with the increase of epicenter distance and presents low prominent frequency characteristics. The energy of HJA, XCZ, YCH, and QSZ also mainly distributed within 20 Hz and the predominant frequency was within 1.075–2.45 Hz which shows low prominent frequency characteristic. The peak Hilbert energy of HJA, YCH, QSZ, and XCZ are 5.8, 4.7, 0.59, and 0.65, respectively, which shows a decreasing trend with the increase of epicenter distance except for QSZ. Result shows that the epicenter distance of XCZ is larger than QSZ, whereas the peak value is greater than QSZ. This result is related to the site conditions of two stations, which reflects that soil layer can greatly amplify the Hilbert energy of an earthquake. The result is consistent with the result of Hilbert-Huang spectrum.Figure 6
Marginal spectrum of each strong motion station. (a) Ms 5.0 (b) Ms 4.9.
(a)(b)Above results show that the prominent frequencies of the seismic waves of two earthquakes present low frequency (1.125–1.625 Hz and 1.075–2.45 Hz, respectively), the Hilbert energy shows a decreasing trend with the increase of epicenter distance, and the loess layer can amplify the Hilbert energy.


#### 4.1.3. Instantaneous Energy Spectrum Characteristics

Instantaneous energy is an energy corresponding to time, which can reflect the instantaneous energy fluctuation of earthquakes in the whole duration.Instantaneous energy spectrums of different strong motion stations were shown in Figure7. Result shows that the instantaneous energy at HZH is concentrated within 20–30 s and the amplitude of two directions up to maximum value at the same time. The instantaneous energy of MLE distributed within 15–40 s and the time that amplitude of MLE-EW up to maximum is before MLE-NS. The peak amplitudes of HZH and MLE in EW direction are 7.45 × 105 and 6.45 × 103, respectively, which are about 1.6–1.7 times greater than it in NS direction. The peak value of instantaneous energy of HZH is larger than that of MLE, which reflects that the instantaneous energy attenuates quickly with the increase of epicenter distance. The instantaneous energy of HJA concentrated within 20–25 s and the amplitude in two directions up to peak is also at the same time. The peak value of HJA-EW is 2.4 × 105, which is about 1.67 times larger than HJA-NS which is about 1.6 × 105. The instantaneous energy of XCZ is distributed within 10–40 s and the time that amplitude of two directions up to maximum is same. The peak value of XCZ-EW is about 1.36 times greater than XCZ-NS. The instantaneous energy of YCH is distributed within 20–50 s and the time that peak amplitude of YCH-EW and YCH-NS occurred is different. The results show that the peak amplitude of NS direction is larger than EW at YCH. The instantaneous energy of QSZ is concentrated within 20–30 s and the time that peak amplitude of QSZ-EW and QSZ-NS occurred is different. The peak value of QSZ-NS is about 2.3 times larger than QSZ-EW. The results reflect that distribution characteristics of instantaneous energy spectrum in two directions are not unified. The peak value of EW is larger than NS at HJA and XCZ, but the value has the opposite characteristics at YCH and QSZ. The instantaneous energy at QSZ is smaller than XCZ or YCH. This result shows that soil layer can amplify the instantaneous energy.Figure 7
Instantaneous energy spectrum of each strong motion station.


### 4.2. Site Characteristics

The HVSR method was often used to investigate the site effect [32]. Similarly, we used horizontal to vertical marginal spectrum ratio based on coda wave to study the site characteristics, because the marginal spectrum can reflect the frequency components more clearly than the Fourier spectrum. The calculation is shown in:(6)Rf=HfVf,where Hf is the marginal spectrum of horizontal ground motion, Vf is the marginal spectrum of vertical ground motion.The time history of seismic wave includes source, propagation path, and site information. Coda wave is a superposition of discontinuous scattered waves formed by the scattering of seismic waves on a nonuniform body in the Earth medium [33]. The generation of coda is related to the incompleteness elasticity, inhomogeneity, and anisotropy of the Earth medium, so coda contains a lot of information about the source and medium. So, coda wave analysis is better than the whole time history of ground motion to obtain the characteristics of the medium. Rautian [34] pointed out that t >> ts (ts is the arrival time of S wave from the time of earthquake occurrence) is the starting point of the coda wave. The starting point and length of coda wave do not have a unified selection method. We select the last 20 s of ground motion as the coda wave in this study, as shown in Figure 8.Figure 8
Coda wave.The HVSR results based on coda wave using marginal spectrum and the contour map of HVSR in study area are shown in Figures9 and 10 respectively. It reflects that the predominant frequency of site at each station is within 3.6–6.2 Hz. Although the data are limited, the site characteristics are obvious. The amplification factor (H/V) at rock site (QSZ) is 2, which is minimum among all stations. The factor reaches maximum at HZH which is soil site. The factors at other sites are within 3–5.2. Because of the lack of soil layer thickness data, relationship between the factor and soil layer thickness cannot be discussed further. However, it clearly reflects that soil layer can amplify the ground motion when compared with a rock site. So, HVSR based on coda wave of light-moderate earthquakes using marginal spectrum can be used to investigate the site effect.Figure 9
HVSR result based on coda wave using marginal spectrum.Figure 10
Contour map of HVSR in study area.


## 4.1. Spectrum Characteristics of Strong Motion

### 4.1.1. Hilbert-Huang Spectrum Characteristics

Hilbert-Huang spectrums of seismic waves recorded by station HZH, MLE, HJA, XCZ, YCH, and QSZ were shown in Figures4 and 5 respectively. Every station records the seismic wave with three directions and we just list the results of EW and NS directions for the amplitude of UD direction is much smaller than EW and NS directions. In the Hilbert-Huang spectrum, the relationship among time, frequency, and energy can be obtained. The amplitude of the spectrum represents the energy of instantaneous frequency.Figure 4
Hilbert-Huang spectrum of the reconstructed signal of HZH and MLE.Figure 5
Hilbert-Huang spectrum of the reconstructed signal of HJA, YCH, XCZ, and QSZ.
(a)(b)The Hilbert-Huang spectrums of seismic waves of Ms 5.0 recorded by two strong motion stations were shown in Figure4. Results show that the Hilbert-Huang spectrums of two stations with different epicenter distances show totally different characteristics and the distribution has certain similarity at the same station in different directions. The spectrum of HZH-EW shows that the energy concentrated at 20–30 s and instantaneous frequency concentrated within 20 Hz. Compared with HZH-EW, the energy of HZH-NS also concentrated at 20–30 s and instantaneous concentrated within 20 Hz, but peak value gradually shifts to relative high frequency (close to 20 Hz). The spectrum of MLE-EW shows that the energy widely distributed at 15–40 s and instantaneous frequency distributed within 40 Hz. Similarly, the energy of seismic wave which recorded by MLE-NS also distributed within 40 Hz. Except for the distribution characteristics, the amplitudes of two stations also have quite a difference. The peak energy of HZH-EW is about 62.3 cm/s2 and HZH-NS is about 41.2 cm/s2; however, the energy of MLE-EW is 4.6 cm/s2, and MLE-NS is 4.0 cm/s2. The peak amplitude of HZH-NS is about 10 times greater than MLE-NS and the HZH-EW is more than 10 times greater than MLE-EW, which reflect that the energy of seismic wave will changes from concentration to divergence in time and frequency and gradually decreases with the increase of epicenter distance.The Hilbert-Huang spectrums of Ms 4.9 were shown in Figure5. The spectrums of HJA-EW and HJA-NS show that the energy concentrated at 20–25 s and instantaneous frequency concentrated within 20 Hz. The peak amplitude of HJA-EW is about 30 cm/s2 and the peak value of HJA-NS is about 25 cm/s2. The location of YCH has the same longitude as HJA and lower elevation than HJA. The energy of seismic wave at YCH station is distributed within 10–40 s and with instantaneous frequency within 30 Hz. The peak values are 4.5 cm/s2 and 5.2 cm/s2 in the directions of EW and NS, respectively. Compared with the HJA, energy at YCH is more scattered and the peak value decreases about 5 times, which reflects that seismic energy attenuates with the epicenter distance increasing. The distribution of energy at station XCZ is more scattered compared with other stations and the peak amplitudes are 3.12 cm/s2 and 2.60 cm/s2, respectively, which are smaller than YCH, because the epicenter distance of XCZ is more far than YCH. The energy of seismic wave recorded by QSZ is concentrated at 20–25 s, the instantaneous frequency is concentrated within 20 Hz, and the peak amplitude are 1.5 cm/s2 and 1.58 cm/s2 in the directions of EW and NS, respectively. The epicenter distance of QSZ is between YCH and XCZ, whereas the energy of it is very concentrated compared with YCH and XCZ and the energy is smallest among them. This is because the site condition of QSZ station is rock, which is different from others. The results reflect that site conditions can significantly influence the distribution characteristics of instantaneous frequency energy. The energy of a seismic wave is more divergent in the soil than it is in the rock in the process of seismic wave propagation, the distribution characteristics of instantaneous frequency energy are closer to epicenter compared to the soil site condition, and the soil layer can greatly amplify the instantaneous frequency energy.


### 4.1.2. Marginal Spectrum Characteristics

Marginal spectrum reflects the distribution law of Hilbert energy with frequency. Hilbert energy Marginal spectrum is different from Fourier spectrum. Marginal spectrum is based on the HHT and its frequency is a function of time. So, the frequency in marginal spectrum may not necessarily appear in the whole period, but at a certain time or several times [31].The marginal spectrums of Ms 5.0 and Ms 4.9 seismic waves at different strong motion stations were shown in Figures6(a) and 6(b), respectively. Result shows that the shapes of marginal spectrum in N-S and E-W directions are similar with each other at each station and Hilbert energy of frequency mainly distributed within 20 Hz. Peak Hilbert energy of HZH-NS is about 23.9 and the energy is about 4.0 at MLE-NS, which reflects that peak amplitude of HZH-NS is about 6.6 times greater than MLE-NS. Peak amplitudes of HZH-NS and MLE-NS occurred at 1.125 Hz. Peak amplitudes of HZH-EW and MLE-EW occurred at 1.625 Hz and 1.375 Hz, respectively, and peak amplitude of HZH-EW is about 6.03 times greater than MLE-EW. Hilbert energy of Ms 5.0 seismic waves shows a downward trend with the increase of epicenter distance and presents low prominent frequency characteristics. The energy of HJA, XCZ, YCH, and QSZ also mainly distributed within 20 Hz and the predominant frequency was within 1.075–2.45 Hz which shows low prominent frequency characteristic. The peak Hilbert energy of HJA, YCH, QSZ, and XCZ are 5.8, 4.7, 0.59, and 0.65, respectively, which shows a decreasing trend with the increase of epicenter distance except for QSZ. Result shows that the epicenter distance of XCZ is larger than QSZ, whereas the peak value is greater than QSZ. This result is related to the site conditions of two stations, which reflects that soil layer can greatly amplify the Hilbert energy of an earthquake. The result is consistent with the result of Hilbert-Huang spectrum.Figure 6
Marginal spectrum of each strong motion station. (a) Ms 5.0 (b) Ms 4.9.
(a)(b)Above results show that the prominent frequencies of the seismic waves of two earthquakes present low frequency (1.125–1.625 Hz and 1.075–2.45 Hz, respectively), the Hilbert energy shows a decreasing trend with the increase of epicenter distance, and the loess layer can amplify the Hilbert energy.


### 4.1.3. Instantaneous Energy Spectrum Characteristics

Instantaneous energy is an energy corresponding to time, which can reflect the instantaneous energy fluctuation of earthquakes in the whole duration.Instantaneous energy spectrums of different strong motion stations were shown in Figure7. Result shows that the instantaneous energy at HZH is concentrated within 20–30 s and the amplitude of two directions up to maximum value at the same time. The instantaneous energy of MLE distributed within 15–40 s and the time that amplitude of MLE-EW up to maximum is before MLE-NS. The peak amplitudes of HZH and MLE in EW direction are 7.45 × 105 and 6.45 × 103, respectively, which are about 1.6–1.7 times greater than it in NS direction. The peak value of instantaneous energy of HZH is larger than that of MLE, which reflects that the instantaneous energy attenuates quickly with the increase of epicenter distance. The instantaneous energy of HJA concentrated within 20–25 s and the amplitude in two directions up to peak is also at the same time. The peak value of HJA-EW is 2.4 × 105, which is about 1.67 times larger than HJA-NS which is about 1.6 × 105. The instantaneous energy of XCZ is distributed within 10–40 s and the time that amplitude of two directions up to maximum is same. The peak value of XCZ-EW is about 1.36 times greater than XCZ-NS. The instantaneous energy of YCH is distributed within 20–50 s and the time that peak amplitude of YCH-EW and YCH-NS occurred is different. The results show that the peak amplitude of NS direction is larger than EW at YCH. The instantaneous energy of QSZ is concentrated within 20–30 s and the time that peak amplitude of QSZ-EW and QSZ-NS occurred is different. The peak value of QSZ-NS is about 2.3 times larger than QSZ-EW. The results reflect that distribution characteristics of instantaneous energy spectrum in two directions are not unified. The peak value of EW is larger than NS at HJA and XCZ, but the value has the opposite characteristics at YCH and QSZ. The instantaneous energy at QSZ is smaller than XCZ or YCH. This result shows that soil layer can amplify the instantaneous energy.Figure 7
Instantaneous energy spectrum of each strong motion station.


## 4.1.1. Hilbert-Huang Spectrum Characteristics

Hilbert-Huang spectrums of seismic waves recorded by station HZH, MLE, HJA, XCZ, YCH, and QSZ were shown in Figures4 and 5 respectively. Every station records the seismic wave with three directions and we just list the results of EW and NS directions for the amplitude of UD direction is much smaller than EW and NS directions. In the Hilbert-Huang spectrum, the relationship among time, frequency, and energy can be obtained. The amplitude of the spectrum represents the energy of instantaneous frequency.Figure 4
Hilbert-Huang spectrum of the reconstructed signal of HZH and MLE.Figure 5
Hilbert-Huang spectrum of the reconstructed signal of HJA, YCH, XCZ, and QSZ.
(a)(b)The Hilbert-Huang spectrums of seismic waves of Ms 5.0 recorded by two strong motion stations were shown in Figure4. Results show that the Hilbert-Huang spectrums of two stations with different epicenter distances show totally different characteristics and the distribution has certain similarity at the same station in different directions. The spectrum of HZH-EW shows that the energy concentrated at 20–30 s and instantaneous frequency concentrated within 20 Hz. Compared with HZH-EW, the energy of HZH-NS also concentrated at 20–30 s and instantaneous concentrated within 20 Hz, but peak value gradually shifts to relative high frequency (close to 20 Hz). The spectrum of MLE-EW shows that the energy widely distributed at 15–40 s and instantaneous frequency distributed within 40 Hz. Similarly, the energy of seismic wave which recorded by MLE-NS also distributed within 40 Hz. Except for the distribution characteristics, the amplitudes of two stations also have quite a difference. The peak energy of HZH-EW is about 62.3 cm/s2 and HZH-NS is about 41.2 cm/s2; however, the energy of MLE-EW is 4.6 cm/s2, and MLE-NS is 4.0 cm/s2. The peak amplitude of HZH-NS is about 10 times greater than MLE-NS and the HZH-EW is more than 10 times greater than MLE-EW, which reflect that the energy of seismic wave will changes from concentration to divergence in time and frequency and gradually decreases with the increase of epicenter distance.The Hilbert-Huang spectrums of Ms 4.9 were shown in Figure5. The spectrums of HJA-EW and HJA-NS show that the energy concentrated at 20–25 s and instantaneous frequency concentrated within 20 Hz. The peak amplitude of HJA-EW is about 30 cm/s2 and the peak value of HJA-NS is about 25 cm/s2. The location of YCH has the same longitude as HJA and lower elevation than HJA. The energy of seismic wave at YCH station is distributed within 10–40 s and with instantaneous frequency within 30 Hz. The peak values are 4.5 cm/s2 and 5.2 cm/s2 in the directions of EW and NS, respectively. Compared with the HJA, energy at YCH is more scattered and the peak value decreases about 5 times, which reflects that seismic energy attenuates with the epicenter distance increasing. The distribution of energy at station XCZ is more scattered compared with other stations and the peak amplitudes are 3.12 cm/s2 and 2.60 cm/s2, respectively, which are smaller than YCH, because the epicenter distance of XCZ is more far than YCH. The energy of seismic wave recorded by QSZ is concentrated at 20–25 s, the instantaneous frequency is concentrated within 20 Hz, and the peak amplitude are 1.5 cm/s2 and 1.58 cm/s2 in the directions of EW and NS, respectively. The epicenter distance of QSZ is between YCH and XCZ, whereas the energy of it is very concentrated compared with YCH and XCZ and the energy is smallest among them. This is because the site condition of QSZ station is rock, which is different from others. The results reflect that site conditions can significantly influence the distribution characteristics of instantaneous frequency energy. The energy of a seismic wave is more divergent in the soil than it is in the rock in the process of seismic wave propagation, the distribution characteristics of instantaneous frequency energy are closer to epicenter compared to the soil site condition, and the soil layer can greatly amplify the instantaneous frequency energy.


## 4.1.2. Marginal Spectrum Characteristics

Marginal spectrum reflects the distribution law of Hilbert energy with frequency. Hilbert energy Marginal spectrum is different from Fourier spectrum. Marginal spectrum is based on the HHT and its frequency is a function of time. So, the frequency in marginal spectrum may not necessarily appear in the whole period, but at a certain time or several times [31].The marginal spectrums of Ms 5.0 and Ms 4.9 seismic waves at different strong motion stations were shown in Figures6(a) and 6(b), respectively. Result shows that the shapes of marginal spectrum in N-S and E-W directions are similar with each other at each station and Hilbert energy of frequency mainly distributed within 20 Hz. Peak Hilbert energy of HZH-NS is about 23.9 and the energy is about 4.0 at MLE-NS, which reflects that peak amplitude of HZH-NS is about 6.6 times greater than MLE-NS. Peak amplitudes of HZH-NS and MLE-NS occurred at 1.125 Hz. Peak amplitudes of HZH-EW and MLE-EW occurred at 1.625 Hz and 1.375 Hz, respectively, and peak amplitude of HZH-EW is about 6.03 times greater than MLE-EW. Hilbert energy of Ms 5.0 seismic waves shows a downward trend with the increase of epicenter distance and presents low prominent frequency characteristics. The energy of HJA, XCZ, YCH, and QSZ also mainly distributed within 20 Hz and the predominant frequency was within 1.075–2.45 Hz which shows low prominent frequency characteristic. The peak Hilbert energy of HJA, YCH, QSZ, and XCZ are 5.8, 4.7, 0.59, and 0.65, respectively, which shows a decreasing trend with the increase of epicenter distance except for QSZ. Result shows that the epicenter distance of XCZ is larger than QSZ, whereas the peak value is greater than QSZ. This result is related to the site conditions of two stations, which reflects that soil layer can greatly amplify the Hilbert energy of an earthquake. The result is consistent with the result of Hilbert-Huang spectrum.Figure 6
Marginal spectrum of each strong motion station. (a) Ms 5.0 (b) Ms 4.9.
(a)(b)Above results show that the prominent frequencies of the seismic waves of two earthquakes present low frequency (1.125–1.625 Hz and 1.075–2.45 Hz, respectively), the Hilbert energy shows a decreasing trend with the increase of epicenter distance, and the loess layer can amplify the Hilbert energy.


## 4.1.3. Instantaneous Energy Spectrum Characteristics

Instantaneous energy is an energy corresponding to time, which can reflect the instantaneous energy fluctuation of earthquakes in the whole duration.Instantaneous energy spectrums of different strong motion stations were shown in Figure7. Result shows that the instantaneous energy at HZH is concentrated within 20–30 s and the amplitude of two directions up to maximum value at the same time. The instantaneous energy of MLE distributed within 15–40 s and the time that amplitude of MLE-EW up to maximum is before MLE-NS. The peak amplitudes of HZH and MLE in EW direction are 7.45 × 105 and 6.45 × 103, respectively, which are about 1.6–1.7 times greater than it in NS direction. The peak value of instantaneous energy of HZH is larger than that of MLE, which reflects that the instantaneous energy attenuates quickly with the increase of epicenter distance. The instantaneous energy of HJA concentrated within 20–25 s and the amplitude in two directions up to peak is also at the same time. The peak value of HJA-EW is 2.4 × 105, which is about 1.67 times larger than HJA-NS which is about 1.6 × 105. The instantaneous energy of XCZ is distributed within 10–40 s and the time that amplitude of two directions up to maximum is same. The peak value of XCZ-EW is about 1.36 times greater than XCZ-NS. The instantaneous energy of YCH is distributed within 20–50 s and the time that peak amplitude of YCH-EW and YCH-NS occurred is different. The results show that the peak amplitude of NS direction is larger than EW at YCH. The instantaneous energy of QSZ is concentrated within 20–30 s and the time that peak amplitude of QSZ-EW and QSZ-NS occurred is different. The peak value of QSZ-NS is about 2.3 times larger than QSZ-EW. The results reflect that distribution characteristics of instantaneous energy spectrum in two directions are not unified. The peak value of EW is larger than NS at HJA and XCZ, but the value has the opposite characteristics at YCH and QSZ. The instantaneous energy at QSZ is smaller than XCZ or YCH. This result shows that soil layer can amplify the instantaneous energy.Figure 7
Instantaneous energy spectrum of each strong motion station.


## 4.2. Site Characteristics

The HVSR method was often used to investigate the site effect [32]. Similarly, we used horizontal to vertical marginal spectrum ratio based on coda wave to study the site characteristics, because the marginal spectrum can reflect the frequency components more clearly than the Fourier spectrum. The calculation is shown in:(6)Rf=HfVf,where Hf is the marginal spectrum of horizontal ground motion, Vf is the marginal spectrum of vertical ground motion.The time history of seismic wave includes source, propagation path, and site information. Coda wave is a superposition of discontinuous scattered waves formed by the scattering of seismic waves on a nonuniform body in the Earth medium [33]. The generation of coda is related to the incompleteness elasticity, inhomogeneity, and anisotropy of the Earth medium, so coda contains a lot of information about the source and medium. So, coda wave analysis is better than the whole time history of ground motion to obtain the characteristics of the medium. Rautian [34] pointed out that t >> ts (ts is the arrival time of S wave from the time of earthquake occurrence) is the starting point of the coda wave. The starting point and length of coda wave do not have a unified selection method. We select the last 20 s of ground motion as the coda wave in this study, as shown in Figure 8.Figure 8
Coda wave.The HVSR results based on coda wave using marginal spectrum and the contour map of HVSR in study area are shown in Figures9 and 10 respectively. It reflects that the predominant frequency of site at each station is within 3.6–6.2 Hz. Although the data are limited, the site characteristics are obvious. The amplification factor (H/V) at rock site (QSZ) is 2, which is minimum among all stations. The factor reaches maximum at HZH which is soil site. The factors at other sites are within 3–5.2. Because of the lack of soil layer thickness data, relationship between the factor and soil layer thickness cannot be discussed further. However, it clearly reflects that soil layer can amplify the ground motion when compared with a rock site. So, HVSR based on coda wave of light-moderate earthquakes using marginal spectrum can be used to investigate the site effect.Figure 9
HVSR result based on coda wave using marginal spectrum.Figure 10
Contour map of HVSR in study area.


## 5. Discussion

### 5.1. Comparison of Predominant Frequency of Ground Motion and Site

The severity of disasters caused by earthquakes is not only related to earthquake intensity but also closely related to ground motion and site characteristics. Site characteristics are related to landform, soil type, soil thickness, stratum structure, and so on. The predominant frequency of site is the comprehensive reflection of these factors. When the predominant frequency of site is close to the seismic ground motion, the resonance effect would be produced and the disaster would also be more serious. Figure11 presents the HVSR results of HZH and HJA stations and the marginal spectrum of two light-moderate earthquakes in the study. It reflects that the predominant frequencies of the earthquakes are in the low frequency range, which are lower than the predominant frequencies of sites. So, the disasters induced by the two earthquakes are minor.Figure 11
Marginal spectrum and HVSR results.


### 5.2. Comparison of Fourier Spectrum and Marginal Spectrum

The frequency characteristics of two light-moderate earthquakes in this study are clearly according to the Hilbert-Huang spectrum and marginal spectrum based on HHT. Generally, we use FFT to clarify the frequency characteristics. However, FFT applicable to linear and stationary signal analysis, which cannot present the frequency features of seismic waves for the earthquake signal is nonlinear and nonstationary. In order to clear that, we performed FFT on the seismic waves of Ms 5.0, as it is shown in Figure12. The prominent frequency of HZH is within 0–8 Hz and the prominent frequency of MLE is within 0–20 Hz. Compared with marginal spectrum that is shown in Figure 6 Fourier spectrum cannot reflect the predominant frequency clearly and the low frequency energy is not well presented. So, HHT is better than FFT in the frequency analysis of seismic waves, which gives a direction for future relative studies.Figure 12
Fourier spectrum of Ms 5.0 earthquake. (a) HZH (b) MLE.
(a)(b)


## 5.1. Comparison of Predominant Frequency of Ground Motion and Site

The severity of disasters caused by earthquakes is not only related to earthquake intensity but also closely related to ground motion and site characteristics. Site characteristics are related to landform, soil type, soil thickness, stratum structure, and so on. The predominant frequency of site is the comprehensive reflection of these factors. When the predominant frequency of site is close to the seismic ground motion, the resonance effect would be produced and the disaster would also be more serious. Figure11 presents the HVSR results of HZH and HJA stations and the marginal spectrum of two light-moderate earthquakes in the study. It reflects that the predominant frequencies of the earthquakes are in the low frequency range, which are lower than the predominant frequencies of sites. So, the disasters induced by the two earthquakes are minor.Figure 11
Marginal spectrum and HVSR results.


## 5.2. Comparison of Fourier Spectrum and Marginal Spectrum

The frequency characteristics of two light-moderate earthquakes in this study are clearly according to the Hilbert-Huang spectrum and marginal spectrum based on HHT. Generally, we use FFT to clarify the frequency characteristics. However, FFT applicable to linear and stationary signal analysis, which cannot present the frequency features of seismic waves for the earthquake signal is nonlinear and nonstationary. In order to clear that, we performed FFT on the seismic waves of Ms 5.0, as it is shown in Figure12. The prominent frequency of HZH is within 0–8 Hz and the prominent frequency of MLE is within 0–20 Hz. Compared with marginal spectrum that is shown in Figure 6 Fourier spectrum cannot reflect the predominant frequency clearly and the low frequency energy is not well presented. So, HHT is better than FFT in the frequency analysis of seismic waves, which gives a direction for future relative studies.Figure 12
Fourier spectrum of Ms 5.0 earthquake. (a) HZH (b) MLE.
(a)(b)


## 6. Conclusions

In the study, we analyzed the spectrum characteristics of two light-moderate earthquakes that occurred in loess area of China from time-frequency domain based on HHT and the site characteristics based on the coda wave of the earthquakes using HVSR method. The main conclusions are shown as follows:(1)
Hilbert-Huang spectrum based on HHT can clearly reflect the relationship among time, frequency, and amplitude (instantaneous frequency energy) of seismic waves of light-moderate earthquakes. The energy of seismic wave at strong motion station that close to epicenter is mainly concentrated within 20 Hz. With the increase of epicenter distance, the energy will change from concentration to divergence in time and frequency and gradually decrease at same site condition. In the process of propagation, the energy of a seismic wave is more concentrated under the rock site condition than it under the soil site situation and a loess layer can amplify the energy.(2)
The marginal spectrum distinctly shows the accumulation of energy of light-moderate earthquake in time. Two earthquakes present low prominent frequency characteristics. Peak amplitude of Ms 5.0 earthquake in two directions occurred within 1.125–1.625 Hz, and the value of Ms 4.9 earthquake is within 1.075–2.45 Hz, respectively. With the increase of epicenter distance, Hilbert energy shows a downward trend. The Hilbert energy at soil site is about 3 times greater than at rock site when the epicenter distance is close, which suggests that soil layer can greatly amplify the Hilbert energy of an earthquake.(3)
Instantaneous energy spectrum shows the change law of instantaneous energy of light-moderate earthquakes with time. Instantaneous energy attenuates quickly with the increase of epicenter distance. Site conditions greatly impact on the characteristics of instantaneous energy. Instantaneous energy at the rock site is much smaller than at soil site, which reflects that soil layer can significantly amplify the instantaneous energy of an earthquake.(4)
HVSR based on coda wave of light-moderate earthquakes using marginal spectrum can be used to investigate the site characteristics. The predominant frequency of site at each station is within 3.6–6.2 Hz. Compared with a rock site, soil layer can amplify the ground motion. The amplification factor at rock site is minimum with value 2 and the factor reaches 30.2 at HZH which is soil site.



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*Source: 5241257-2022-08-12.xml*