The Doppler effect induced by earthquakes: A case study of the Wenchuan MS8.0 ea

Qiheng Li , Jingwen Sun , Guimei Xi , Jing Liu

a Mining College, Liaoning Technical University, Liaoning 123000, China

b Chaoyang Teachers College, Liaoning 122000, China

c Chaoyang Open University, Liaoning 122000, China

Keywords:Doppler effect First arrival frequency of P wave Fault rupture propagation velocity Wavelet transform The wenchuan earthquake

ABSTRACT Seismologists have found that the first arrival frequencies of P waves at different seismic stations have different widths, that is, different periods or frequencies, and they think that this phenomenon can be used to identify whether a Doppler effect is induced by earthquakes. However,the fault rupture process of a real earthquake is so complex that it is difficult to identify a frequency shift similar to the Doppler effect. A method to identify whether a Doppler effect is induced by an earthquake is proposed here. If a seismic station is in the direction of fault rupture propagation,this station could observe a Doppler effect induced by the earthquake.The Doppler effect causes the frequency of the seismic wave to shift from low frequency to high frequency,and the high frequency amplitudes become mutually superimposed.Under the combined influences of the absorption effect,geometric spreading effect and Doppler effect,the high frequency amplitude of the seismic wave will gradually become higher than the low frequency amplitude with increasing epicentral distance. If we find that the high frequency amplitude is higher than the low frequency amplitude with increasing epicentral distance in the direction of fault rupture propagation,then there is a Doppler effect. The fault that generated the Wenchuan earthquake is a reverse fault, and its horizontal rupture propagation velocity was low. To link fault rupture propagation velocity with the Doppler effect and identify the Doppler effect more easily,we decompose three-component records into two directions: the direction of fault rupture propagation and the direction perpendicular to the fault rupture propagation along the fault plane.The initial components of the two directions are processed by wavelet transform.Several seismic stations in the direction of fault rupture propagation of the Wenchuan earthquake were selected,and it was found that with increasing epicentral distance, the high frequency amplitudes of the wavelet spectra become obviously higher than the low frequency amplitudes.It can be concluded that due to the existence of the Doppler effect, high frequency amplitudes can overcome the influences of the absorption and geometric spreading effects on seismic waves in the fault rupture propagation process.

1. Introduction

The Doppler effect is widely observed in the fields of physics,optics, medicine, communication, etc. [1,2]. The Doppler effect is also caused by seismic activity [3-6]. Zhuo Yuru et al. [3] and Douglas A. et al. [4] analyzed the distributions of the first arrival frequencies of P waves around seismogenic faults and proved that a Doppler effect is induced by earthquakes. This method is concise and correct in theory, and when the first arrival frequencies of P waves vary greatly around a seismogenic fault,the Doppler effect is obvious, and it is easy to identify whether a Doppler effect is induced by an earthquake. However, when the first arrival frequencies of P waves vary slightly around a seismogenic fault, it is difficult to identify whether there is a Doppler effect.Furthermore,because picking the first arrival of the P wave from seismic records can result in considerable error, this method has important limitations in identifying whether a Doppler effect is induced by earthquakes.

Ge Jin et al.[5]studied records of the Wenchuan earthquake and determined whether there was a Doppler effect by using corner frequency.The main process of this method is as follows:they found that the corner frequency ahead and to the left of fault rupture propagation was 0.054 Hz,while the corner frequency behind and to the left of fault rupture propagation was 0.050 Hz.Thus,the corner frequency ahead and to the left of fault rupture propagation was slightly higher than the corner frequency behind and to the left of fault rupture propagation.Ge Jin et al.[5]also found that the corner frequency ahead and to the right of fault rupture propagation was 0.082 Hz,while the corner frequency behind and to the right of fault rupture propagation was 0.072 Hz.Thus,the corner frequency ahead and to the right of fault rupture propagation was slightly higher than the corner frequency behind and to the right of fault rupture propagation.Because the corner frequencies in front of fault rupture propagation were higher than the corner frequencies to the rear of fault rupture propagation,Ge Jin et al.[5]concluded that the above phenomena was caused by the Doppler effect. If the corner frequency is accurate, this method is feasible for identifying whether there is a Doppler effect,but a limitation of this method is that there is a human factor in picking the corner frequency.The sketch map of Fig. 1 shows how the corner frequency is picked, and the figure shows a seismic response spectrum.Point B is the intersection point of line AB and line BC, and the abscissa of point B is the corner frequency. It is obvious that the determination of point B has a human factor,introducing some degree of inaccuracy.

In 1966, Aki K. [7] recorded the rupture directivity effect in the Parkfield earthquake in the United States for the first time. Later,additional rupture directivity effects were recorded in association with other earthquakes, such as the 1979 Imperial Valley earthquake in the United States[8],the 1992 Landers earthquake in the United States [9], the 1999 Chi-Chi earthquake in Taiwan, China[10], and many other earthquakes [11-15]. Many scholars have used the rupture directivity effect to prove that a Doppler effect is induced by earthquakes, and many others have used the Doppler effect to explain the rupture directivity effect.

Fig.1. Sketch to pick corner frequency.

In addition, some scholars believe that the fault rupture propagation velocity causes the Doppler effect when seismic activity occurs[3,16,17],and a small rupture surface can also be determined by the Doppler effect[3-6].The study of the Doppler effect induced by earthquakes is a basic aspect of earthquake engineering, which helps us to strengthen our understanding of earthquake engineering concepts and helps seismologists pay attention to the study of the spatial distributions of seismic wave frequencies during earthquakes. To find a clear method to identify whether there is a Doppler effect, this paper proposes using wavelet transform to identify whether a Doppler effect is induced by an earthquake.

2. Spatial distributions of seismic wave frequencies induced by the Doppler effect

When relative motion between the wave transmitter and wave receiver occurs,the frequency received by the wave receiver will be different from that transmitted by the wave transmitter; this phenomenon is called the Doppler effect.When an earthquake occurs,the relationship between the frequency received by the wave receiver and the frequency transmitted by the wave transmitter is given by Ref. [18].

where f is frequency received by the wave receiver, i.e., a seismic station in this paper;f0is frequency transmitted from the epicenter;u is the seismic shear wave velocity; v is the fault rupture propagation velocity; and θ (hereafter referred to as the azimuth) is the intersection angle between the line that links the epicenter and receiver and the line representing the direction of fault rupture propagation. Eq. (1) shows that the frequencies received by receivers at different azimuths will be different;if the seismic station is situated in of the direction of fault rupture propagation, the frequency received by seismic station (f) will be higher than the frequency transmitted at the epicenter(f0);if the seismic station is situated in the rear of fault rupture propagation, the frequency received by seismic station (f) will be lower than the frequency transmitted at the epicenter (f0); and if the seismic station is situated in the direction perpendicular to the direction of fault rupture propagation, the frequency received by seismic station (f) will be equal to the frequency transmitted at the epicenter (f0).

3. Observation of the Doppler effect during the Wenchuan earthquake

Douglas et al. [4] found that the first arrival frequencies of P waves at different seismic stations show different widths, that is,different periods or frequencies. Because seismic stations are situated around seismogenic faults and have different azimuths,Douglas et al. [4] concluded that the above phenomenon can be used to identify whether a Doppler effect is induced by earthquakes.However,the fault rupture process of a real earthquake is so complex that it is difficult to generate a frequency shift similar to the Doppler effect. Here, a frequency shift means that the first arrival frequencies of the P wave received by seismic stations at different azimuths are different,so the method that Douglas et al.[4] proposed to identify whether there is a Doppler effect is applicable only to the case in which the first arrival frequencies of the P wave recorded by seismic stations at different azimuths are obviously different. However, it is difficult to show differences in the first arrivals of P waves recorded by seismic stations in different azimuths due to the accuracy of the recording instruments.

The epicenter of the Wenchuan earthquake was located at 31.0°N, 103.4°E [19] in China. The fault rupture propagation occurred toward the northeast and southwest but mainly toward the northeast[19]. We study only the fault rupture propagation in the northeast direction. The Wenchuan earthquake fault is a reverse fault,and the component of horizontal rupture propagation velocity is small;therefore,according to Eq.(1),the Doppler effect of the Wenchuan earthquake will not be obvious.

A diagram of the seismic stations and faults in the Wenchuan earthquake is shown in Fig.2.The CDZ,MZD,MZQ,JYH,JYD and JYC seismic stations are selected to study the Doppler effect,and these seismic stations are situated in the direction of fault rupture propagation. The record of the Wenchuan earthquake in the E-W direction received by the CDZ seismic station is shown in Fig. 3a.The first arrival half period of the P wave picked by hand is 0.045 s,so the corresponding period is 0.09 s, and the corresponding frequency is 11.1 Hz. The record of the Wenchuan earthquake in the E-W direction received by the JYC seismic station is shown in Fig.3b.The first arrival half period of the P wave picked by hand is 0.04 s, the corresponding period is 0.08 s, and the corresponding frequency is 12.5 Hz. The record of Wenchuan earthquake in the E-W direction received by the MZQ seismic station is shown in Fig.3c,the record in the E-W direction received by the JYH seismic station is shown in Fig.3d,the record in the E-W direction received by the JYD seismic station is shown in Fig.3e,and the record in the E-W direction received by the MZD seismic station is shown in Fig. 3f. The first arrival frequencies of the P wave received by the MZQ,JYH and JYD stations are all 12.5 Hz,which is higher than that of the CDZ seismic station;the first arrival frequency of the P wave received by the MZD seismic station is 10 Hz, which is lower than that of the CDZ seismic station.

The above six seismic stations are situated in different azimuths,and most of the first arrival frequencies of the P wave at the above six seismic stations are not obviously different from each other,so it is difficult to identify whether there is a Doppler effect. To accurately identify whether a Doppler effect was induced by the earthquake, we need to find another method.

The greater the rupture propagation velocity of a seismogenic fault is, the more obvious the Doppler effect is, and it is easy to identify. The Wenchuan earthquake fault is a thrust fault, and its horizontal component of velocity is very small, which is why it is difficult to identify whether there is a Doppler effect using the first arrival frequency of the Wenchuan earthquake's P wave.

Fig. 2. Distributions of seismic stations in the Wenchuan earthquake.

4. Identification of the Doppler effect by using wavelet transform

Fourier transform is a valuable tool for processing stationary random signals, but it has difficulty processing nonstationary signals that are common in the real world, such as seismic signals.Furthermore, Fourier transform has difficulty capturing detailed changes when processing nonstationary signals because Fourier transform is a global transform of the whole signal and does not have the capacity to consider time-frequency information. Shorttime Fourier transform (STFT) can improve the capacity to consider time-frequency information, but its sliding window function is fixed in the operation course, so it does not have an adaptive capacity.

Wavelet transform was proposed for geophysical exploration data processing.It not only inherits and develops STFT localization but also overcomes its shortcomings, such as the window size not changing with frequency,which allows wavelet transform to carry out time-frequency localization analysis. By stretching, translation and other computing functions for multiscale refinement of signal analysis, wavelet transform can efficiently extract useful information from signal. Specifically, time refinement is performed in the wavelet transform at high signal frequencies, and frequency refinement is performed in the wavelet transform at low signal frequencies. Wavelet transform can gradually adapt to perform high-quality time-frequency signal analysis and highlight all the details of the signal.

The wavelet function is given by Ref. [20].

where t is time,T is the period of the seismic signal,At,T(c,d)is the wavelet spectrum of seismic signal f(t), and the independent variables are t and T.

The seismic record duration that is processed by the wavelet transform is from 0 to 5 s, whose advantage is that seismic waves received by seismic stations in the initial short duration of 0-5 s have no interference from reflected or refracted waves,and there is no problem of waveform conversion. In addition, the variation in fault rupture propagation velocity is a monotonic function over this duration. Suppose fault rupture propagation velocity at the beginning of an earthquake is v0,and the frequency range that is received by seismic stations in front of fault rupture propagation is from f0to fn. With the continuous increase in fault rupture propagation velocity, the frequency that is received by seismic stations in the direction of fault rupture propagation will increase due to the Doppler effect;see Eq. (1) for the specific reason.

Fig. 3. The first arrivals of the P wave at the CDZ, JYC, MQZ, JYH, JYD and MZD seismic stations in the E-W direction.

Fig. 4. Variation in seismic wave frequency caused by fault rupture propagation.

We know from Fig. 4 that when the fault rupture propagation velocity in the initial time is v0,the corresponding frequency range received by seismic stations in the direction of fault rupture propagation is from f0to fn,and the corresponding frequency range received is represented by the colorless rectangle in Fig. 4. When the fault rupture propagation velocity increases to vn, the corresponding frequency range is from f0+Δf to fn+Δf because of the Doppler effect,where Δf is the increase in frequency.The frequency range that is received by the seismic station is represented by the dark rectangle in Fig. 4. Because the fault rupture propagation velocity changes continuously, the overlapping part from f0+Δf to fnis the relatively high frequency range, and the amplitudes in this high frequency range will increase significantly after repeated superimposition. Even if the fault rupture propagation velocity changes from increasing to decreasing, the high frequency amplitude will still increase; in this case, the dark rectangle in Fig. 4 moves from right to left, that is, toward low frequency.

Crustal media absorb seismic waves, and geometric spreading occurs during the propagation process; hereafter, these phenomena are termed the absorption and geometric spreading effects. If there are only absorption and geometric spreading effects,both the high frequency and low frequency amplitudes of seismic wave will decrease with increasing epicentral distance, but the high frequency amplitude will decrease more rapidly than the low frequency amplitude.If a seismic station is situated in the direction of fault rupture propagation, there is a Doppler effect induced by the earthquake.The Doppler effect causes the frequency of the seismic wave to shift from low frequency to high frequency, and high frequency amplitudes become mutually superimposed. Under the combined influences of the absorption and geometric spreading effects and the Doppler effect,the high frequency amplitude of the seismic wave will gradually become higher than the low frequency amplitude with increasing epicentral distance. In the above process, the Doppler effect can shift energy only from high frequency to low frequency and can cause high frequency amplitudes to become mutually superimposed; under this circumstance, the Doppler effect can shift energy from low frequency to high frequency and cannot accumulate energy. Conversely, if we find that the high frequency amplitude is higher than the low frequency amplitude with increasing epicentral distance in the direction of fault rupture propagation,then there is a Doppler effect;this is our method to identify whether a Doppler effect is induced by an earthquake.If a seismic station is situated in the opposite direction of fault rupture propagation,there is also a Doppler effect induced by an earthquake. The Doppler effect causes the frequency of seismic waves to shift from high frequency to low frequency, and low frequency amplitudes become mutually superimposed. Under the combined influences of the absorption and geometric spreading effects and the Doppler effect, the low frequency amplitudes of seismic waves must be higher than the high frequency amplitudes. In the above process, the Doppler effect can shift energy from high frequency to low frequency but cannot make energy accumulate. If a seismic station is situated in a direction perpendicular to fault rupture propagation direction, there is no Doppler effect.The absorption and geometric spreading effects play primary roles in this direction, the amplitude of the seismic wave will attenuate during the propagation process, the high frequency amplitude of the seismic wave will attenuate faster than the low frequency amplitude,and the low frequency amplitude received by a seismic station in this direction will gradually become higher than the high frequency amplitude with increasing epicentral distance.

The variation in the amplitude of the wavelet spectrum with frequency depends on three situations. The first is the epicentral distance; the larger the epicentral distance is, the lower the amplitude of the wavelet spectrum received by the seismic station is, and amplitudes of the higher frequency of seismic waves will attenuate much faster than amplitudes of the lower frequency.The second is the amplification of the seismic wave at the engineering site,which depends on the specific engineering site.The third is the Doppler effect: if a seismic station is situated in the direction perpendicular to fault rupture propagation direction, there is no Doppler effect; if a seismic station is situated in the direction of fault rupture propagation, because of the Doppler effect, the high frequency amplitude of the wavelet spectrum may be higher than the low frequency amplitude;and if a seismic station is situated in the direction opposite the fault rupture propagation direction, the high frequency amplitude of the wavelet spectrum will be lower than the low frequency amplitude.

Several seismic stations situated in the direction perpendicular to the fault rupture propagation direction of the Wenchuan earthquake are selected, and these sites have different epicentral distances.The seismic records from 0 to 5 s are processed by using the wavelet transform. If the amplitudes of the wavelet spectra decrease with increasing epicentral distance, especially at high frequencies, which indicates that the absorption and geometric spreading effects play primary roles,then there is no Doppler effect in this direction.Several seismic stations situated in the direction of the fault rupture propagation of the Wenchuan earthquake are selected, and these sites also have different epicentral distances.Their seismic records from 0 to 5 s are processed by using the wavelet transform. If the low frequency amplitudes of the wavelet spectra obviously decrease with increasing epicentral distance,then the high frequency amplitudes of the wavelet spectra are obviously higher than the low frequency amplitudes,and if the high frequency amplitudes of the wavelet spectra have no obvious relationship with the epicentral distance, then there is a Doppler effect in this direction.Seismic signals are recorded by three components,i.e.,the E-W,N-S,and U-D directions. To connect the Doppler effect with fault rupture propagation velocity and considering that fault rupture propagation occurs along the fault plane, we decompose the records of the three components into two directions:①the direction of fault rupture propagation (the H direction) and ②the direction perpendicular to fault rupture propagation along the fault plane(V direction). H and V are given by

where α is the acute angle between the fault strike and due north;β is the dip angle of the fault plane; and AEW, ANSand AUDare amplitudes in the east-west direction, north-south direction and direction perpendicular to the Earth's surface, respectively. According to inversion results [19], the Wenchuan earthquake occurred on a fault with a strike of 220°and a dip angle of 32°.

5. Identification of the Doppler effect by applying the wavelet transform to the Wenchuan earthquake

In this paper, wavelet spectra are calculated for the seismic record period from 0 to 5 s, the period range of the wavelet spectra that were extracted is limited from 0 to 1 s,and the corresponding frequency range is from ∞to 1 Hz. A period range of 0-1 s is adopted instead of the frequency range in the wavelet transform because all high frequencies are included in the period range of 0-1 s. In theory, no matter how high a frequency is, it cannot represent all high frequencies, and only if all high frequencies are covered cab the Doppler effect be identified. The real frequency range extracted from real seismic records is 10-1 Hz.

A diagram of the seismic stations and fault that generated the Wenchuan earthquake is shown in Fig. 2. The XJL, FSB and YBY seismic stations are roughly situated in the direction perpendicular to the fault rupture propagation direction, and the epicentral distances of these three seismic stations differ,as shown in Fig.2.We use Eqs.(4)and(5)to decompose the seismic records in the H and V directions, respectively, and then calculate the wavelet spectra of the two components.

The wavelet spectra of the XJL seismic station in the H and V directions are shown in Fig. 5a and Fig. 5b, respectively. The maximum amplitude of the wavelet spectrum in the H direction is 30 cm/s at 10 Hz, and the maximum amplitude of the wavelet spectrum in the V direction is 27 cm/s at 10 Hz.Because this seismic station is closest to the epicenter, its amplitudes of the wavelet spectra are higher than those of other seismic stations in the frequency range of 1-10 Hz,especially in the high frequency range.

The wavelet spectra of the FSB seismic station in the H and V directions are shown in Fig.5c and 5d,respectively.The epicentral distance of the FSB seismic station is greater than that of the XJL seismic station but shorter than that of the YBY seismic station.The maximum amplitude of the wavelet spectra in the H direction is 8 cm/s at a frequency of 10 Hz,and the maximum amplitude of the wavelet spectra in the V direction is 14 cm/s at a frequency of 10 Hz.As the epicentral distance of the FSB seismic station is greater than that of the XJL seismic stations,the high frequency amplitudes of its wavelet spectra are generally lower than those of the XJL seismic stations at a frequency of 10 Hz, which can be explained by the absorption and geometric spreading effects.

Fig. 5. Wavelet spectra of seismic stations situated in the direction perpendicular to the fault rupture propagation direction.

The wavelet spectra of the YBY seismic station in the H and V directions are shown in Fig. 5e and 5f, respectively. The epicentral distance of the YBY seismic station is greater than those of the XJL and FSB seismic stations. We can see that the amplitudes of the wavelet spectra of the YBY seismic station in the H and V directions are relatively low at a high frequency of 1 Hz, and the maximum amplitude appears at a low frequency range, which indicates that the absorption and geometric spreading effects play primary roles,especially at high frequencies, and the Doppler effect is not observed in this direction, which is perpendicular to the fault rupture propagation direction. The frequencies corresponding to the maximum amplitudes of the above three stations are listed in Table 1.

The positions of the seismic stations and the fault that generated the Wenchuan earthquake are shown in Fig.2.The wavelet spectra of the FSB,CDZ,JYH,JYC,JYD and GYZ seismic stations in the H and V directions are shown in Fig.6.These seismic stations are situated in the direction of fault rupture propagation, are close to the fault,and feature increasing epicentral distances.

Table 1 The frequencies corresponding to the maximum amplitudes at the three stations.

Fig. 6. Wavelet spectra of seismic stations on the southeastern side of ruptured fault.

The wavelet spectra of the FSB seismic station in the H and V directions are shown in Fig.6a and 6b,respectively.The epicentral distance of the FSB seismic station is relatively small, so the amplitudes of the wavelet spectra are relatively high in both the high frequency range and the low frequency range. The high frequency amplitudes of the wavelet spectra at a period of 0.1 s, that is, at a frequency of 10 Hz, are obviously higher than those at low frequencies. Why are the high frequency amplitudes of the wavelet spectra higher than the low frequency amplitudes? This is the Doppler effect. Because the FSB seismic station is situated in the direction of fault rupture propagation and is closer to the fault than the CDZ seismic station, the Doppler effect is more obvious, as it superimposes seismic waves at high frequencies to form large amplitudes.

The wavelet spectra of the CDZ seismic station in the H and V directions are shown in Fig. 6c and 6d, respectively. The epicentral distance of the CDZ seismic station is relatively large,so amplitudes are relatively low in both the high-frequency range and the lowfrequency range at the CDZ seismic station. However, the high frequency amplitudes are higher than the low frequency amplitudes,especially in the direction of H.Because the position of seismic station CDZ deviates from the direction of fault rupture propagation,the Doppler effect is not obvious.

The wavelet spectra of the JYH seismic station in the H and V directions are shown in Fig. 6e and 6f, respectively; the wavelet spectra of the JYC seismic station in the H and V directions are shown in Fig.6g and 6h,respectively;the wavelet spectra of the JYD seismic station in the H and V directions are shown in Fig.6i and 6j,respectively;and the wavelet spectra of the GYZ seismic station in the H and V directions are shown in Fig.6k and 6l,respectively.The high frequency amplitudes of the wavelet spectra in the period of 0.1 s,that is,at a frequency of 10 Hz,of the abovementioned seismic stations are obviously higher than the low frequency amplitudes,which can also be explained by the Doppler effect.In particular,the epicentral distance of the GYZ station is the greatest, and the high frequency amplitudes of its wavelet spectra at approximately 10 Hz are obviously higher than the low frequency amplitudes. If we consider only absorption and geometric spreading effects,the high frequency amplitudes of its wavelet spectra cannot be significantly higher than the low frequency amplitudes.If we consider only the amplification effect of the site on the seismic wave,it is impossible for only the high-frequency component to be amplified. The Doppler effect can shift the frequency of the seismic wave from low frequency to high frequency and superimpose the amplitudes,making the high frequency amplitudes of the wavelet spectra obviously higher than the low frequency amplitudes.

The positions of the MXT,MXN,MXD and PWM seismic stations are shown in Fig. 2. The above four seismic stations are situated northwest of the ruptured fault,near the fault,and in the direction of fault rupture propagation,and their epicentral distances increase successively. The wavelet spectra of the above seismic stations in the H and V directions are shown in Fig. 7. The high frequency amplitudes of their wavelet spectra in the H and V directions are much higher than the low frequency amplitudes;in general,we can see that the high frequency amplitudes of the wavelet spectra are much higher than the low frequency amplitudes. The high frequency amplitudes of the MXN seismic station are higher than the low-frequency amplitudes at 2.5 Hz, where 2.5 Hz is a relatively high frequency. The amplitudes of the MXT and MXD seismic stations at 5 Hz are obviously higher than the amplitudes at low frequencies, and the high frequency amplitude of the PWM seismic station at 4 Hz is obviously higher than the low frequency amplitude. The high frequency amplitudes of the wavelet spectra are obviously higher than the low frequency amplitudes,which can be explained by the Doppler effect.

6. Discussion and conclusions

Because previously proposed methods for determining whether a Doppler effect is induced by an earthquake are not mature, we propose to use the wavelet transform method to determine whether there is a Doppler effect. Specifically, we select the initial short periods of seismic records to be processed using wavelet transform.The advantage of selecting the initial short period is that there is no interference from reflected or refracted waves,and there is no interference from waveform conversion.Seismic fault rupture propagation will cause the Doppler effect in the direction of fault rupture propagation, and the Doppler effect will cause the frequency to shift from low frequency to high frequency. Seismic waves are also superimposed in the high frequency range. Finally,the high frequency amplitudes of wavelet spectra will become higher than the low frequency amplitudes. In contrast, under certain conditions, the increase in high frequency amplitudes of wavelet spectra is evidence of the existence of a Doppler effect.Specifically, several seismic stations situated in the direction perpendicular to the Wenchuan earthquake fault are selected,and seismic records with durations of 0-5 s are processed by using wavelet transform. It is found that with the increase in epicentral distance,the amplitudes in the high frequency range of the wavelet spectra decrease rapidly, and the low frequency amplitudes of the wavelet spectra are significantly higher than the high frequency amplitudes. When the epicentral distance is large enough, the absorption and geometric spreading effects play primary roles in this direction, and we can conclude that there is no Doppler effect in this direction. Several seismic stations that are situated in the direction of fault rupture propagation during the Wenchuan earthquake are selected,and seismic records with durations of 0-5 s are also processed by using wavelet transform.It is found that with the increase in epicentral distance, the high frequency amplitudes of the wavelet spectra are significantly higher than the low frequency amplitudes, which indicates that although the seismic waves are influenced by the absorption and geometric spreading effects and site-specific amplification effects,the high frequency amplitudes of the wavelet spectra still increase because of the Doppler effect induced by the earthquake.

Assuming that some seismic stations are situated in the direction of fault rupture propagation,if the high frequency amplitudes of their wavelet spectra are significantly higher than the low frequency amplitudes and if the high frequency amplitudes of the wavelet spectra are still significantly higher than the low frequency amplitudes with increasing epicentral distance, then there is a Doppler effect; this is our method to identify the Doppler effect induced by earthquakes.

The Doppler effect induced by earthquakes is of practical value;for example,Frez J.et al.[13]used the Doppler effect to determine the rupture surface of a small earthquake. It can be seen from Eq.(1) that if the relevant parameters are known, the fault rupture propagation velocity can be determined. This is a meaningful method to determine the fault rupture propagation velocity in theory, which is of great significance to the seismic fortification of buildings.

Fig. 7. Wavelet spectra of seismic stations on the northwestern side of the ruptured fault.

Credit author statement

Qicheng Li：determine the topic of the article, determine the research methods, and write the article.

Jingwen Sun:collect data，process some data and complete the first draft.Guimei Xi: collect data, draw pictures and polish articles.Jing Liu: collect information and polish articles.

Conflicts of interest

The authors declare that there is no conflicts of interest.

Acknowledgments

Seismic data are from the Institute of Engineering Mechanics,China Earthquake Administration.