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Characterized source model of the 2013 Lushan earthquake (M_{w} 6.6) by the empirical Green’s function method
Earth, Planets and Space volume 74, Article number: 50 (2022)
Abstract
Ground motions near the source area of the mainshock of the 2013 Lushan earthquake (M_{w} 6.6) in Sichuan Province in China were reproduced using the characterized source model and the empirical Green’s function method (EGFM). The bestfit characterized source model consisted of one strong motion generation area (SMGA) and a background area. The synthesized ground motions of the characterized source model were in fairly good agreement with the observed ground motions in the frequency range from 0.5 to 30.0 Hz at ten strong motion stations. For the 2013 Lushan earthquake (M_{w} 6.6), both the relationships between the SMGA and the seismic moment, and those between the flat amplitude of the acceleration source spectrum in the short period and the seismic moment almost followed the empirical scaling relationships of inner fault parameters developed for crustal earthquakes. The reasons for the largest peak ground acceleration (PGA) (> 1 g) in the strongmotion observation history of China recorded at the 51BXD strong motion station were investigated from the source and site effects. We found that the directivity effect did not contribute to the largest record by comparing the effect of different positions of the rupture starting point on the synthesized ground motions. The nonlinear effect of shallow layers was negligible, as indicated by the similarity of the earthquake H/V spectral ratios between the mainshock and EGF events. A large shearwave velocity contrast might not exist in the shallow layers as the station was situated on the slope of a small rock hill. Finally, we agreed with previous studies that the hangingwall effect and topographic effect might be the reasons for generating the largest record at Station 51BXD.
Graphical Abstract
Introduction
The 2013 Lushan earthquake (M_{w} 6.6) struck Lushan County, Ya’an City, Sichuan Province, China at 08:02 on April 20, 2013 (Beijing time, 00:02 on April 20, 2013, UTC). The mainshock caused considerable damage to buildings; 196 people died and 21 people were reported missing. The strong ground motions during the mainshock of the 2013 Lushan earthquake were recorded by as many as 114 strongmotion stations including some nearfault stations, as shown in Fig. 1. The largest PGA was 1005 cm/s^{2} calculated from the uncorrected (raw) acceleration record in the east–west (EW) component at Station 51BXD approximately 16 km from the epicenter, which was the first time that 1 g was exceeded in the strong motion observation history of China. It is important to clarify the source and site effects to interpret this large value.
To date, the source rupture process of this earthquake has been illuminated by many kinematic source models developed with the waveform inversion method using teleseismic data (Liu et al. 2013), joint inversion of strongmotion data and teleseismic data (Hao et al. 2013; Zhang et al. 2014), and joint inversion of strongmotion data, teleseismic data and GPS records (Li et al. 2017). Despite the different data sets and methods, these kinematic source models have much in common in that the mainshock ruptured a blind thrust fault in the southern part of the Longmen Shan fault belt and that large slip areas surrounded the hypocenter. The synthesized ground motions using these inverted source models agreed well with the longperiod (e.g., > 1 s) observed ground motions, whereas they could not match the observed ground motions at nearsource stations in the period range of engineering interests (e.g., 0.1–10.0 s). Then, the characterized source model consisting of several asperities and a background area in the seismogenic zone was applied to synthesize the ground motions. This model involves outer and inner fault parameters. The outer fault parameters are defined to outline the overall picture of the target earthquake, such as the entire source area and seismic moment, while the inner fault parameters are defined by asperities with large slips and stress drops. Strong ground motions of engineering interest are mostly generated from asperities. Therefore, the asperities are called strong motion generation areas (SMGAs). The outer and inner fault parameters follow specific scaling relationships for crustal earthquakes (Somerville et al. 1999; Dan et al. 2001; Irikura and Miyake 2001, 2011; Miyake et al. 2003; Miyakoshi et al. 2020). However, the empirical scaling relationships of inner fault parameters for crustal earthquakes in China have not been adequately investigated.
In this study, we elaborate the process to construct the characterized source model of the mainshock with the empirical Green’s function method (EGFM) (Irikura 1986) and then examine whether the inner fault parameters of the mainshock agree with the empirical scaling relationships thus far developed. Finally, we quantitatively discuss the source and site effects on the largest record (PGA > 1 g) at Station 51BXD.
Methodology
The empirical Green’s function method (EGFM) was first proposed by Hartzell (1978) and then revised by Kanamori (1979) and Irikura (1983, 1986). The ground motions of a large event are expressed as a superposition of the ground motions of a small event in Eqs. (1)–(3) (Irikura 1986):
Where U(t) represents the ground motions of a large event, and u(t) is the ground motions of a small event regarded as the empirical Green’s functions (EGFs). The subscripts i and j are indices increasing along the strike and dip directions, respectively. The terms r, r_{ij}, and r_{0} are the respective distances from the station to the hypocenter of the small event, from the station to the (i,j) subfault, and from the station to the rupture starting point on the fault plane of the large event. In addition, ξ_{ij} is the distance between the rupture starting point and the (i,j) subfault. V_{s} and V_{r} are the Swave velocities in the source region and rupture velocity on the fault plane, respectively. F(t) is the correction function of slip velocity between large and small events which was modified by Irikura et al. (1997). The parameters, N and C, are the ratios of fault sizes (length and width) and stress drops between the large and small events, respectively. Furthermore, n’ is the division number to shift the artificial periodicity to a frequency higher than that of interests, e Euler’s number, and Γ the rise time of the large event.
The formulation of the EGFM in Eq. (1) implies that the synthesized ground motions consider the contribution of the Green’s functions from the farfield term only. Because the empirical Green’s functions contain the contributions of near, intermediate and farfield terms, the synthesized ground motions naturally contain the contributions from all terms. According to Nozu (2006), the Fourier spectral ratio of total waves corresponding to the near, intermediate and farfield terms to the farfield Swave term approaches 1 when the normalized frequency (= 2πfr/V_{s}, r is the hypocentral distance) is larger than 8. This means that the contributions of the near and intermediatefield terms can be neglected only if f > 8V_{s}/2πr. As V_{s} in the source region is 3.50 km/s (Han et al. 2014), and the shortest hypocentral distance is approximately 20 km, the farfield Swave term in the synthesized ground motions is comparable with that of the total terms without any corrections at frequencies larger than 0.2 Hz.
We adopted the method proposed by Miyake et al. (1999) to estimate the fault size of the EGF event, N and C by fitting the theoretical source spectral ratio to the observed one. The observed source spectral ratio is estimated from the ratio of the observed Fourier spectrum (vector summation of two horizontal components) removing the path effect between large and small events, while the theoretical source spectral ratio is expressed as the following equation:
where M_{o} and m_{o} are the seismic moments of large and small events, respectively, and f_{ca} and f_{cm} are the corner frequencies of large and small events, respectively.
Once the bestfit theoretical source spectral ratio is determined by searching the minimum of weighted least squares (Miyake et al. 2003), the corner frequencies f_{ca} and f_{cm} are directly obtained according to Eq. (4). The proportional parameters, N and C, are estimated from Eq. (5). The fault size r_{a} and stress drop ∆σ_{a} of the EGF event are estimated by applying Eq. (6) (Eshelby 1957; Brune 1970, 1971):
Except for the above parameters (i.e., N, C, r_{a}), we determine the position indices (i, j) of the rupture starting point within the SMGAs, V_{r} and Γ, by applying the adaptive simulated annealing (ASA) algorithm, which employs a more efficient sampling of the parameter space than the conventional simulated annealing algorithm. We gave a brief introduction to the ASA algorithm by referring to Satoh (2006).
Let x(t) be the vector composed of parameters x_{i}(t) (i.e., C, N, i, j, V_{r} and Γ). Here, x_{i}(t) is generated from the generating function by use of random variable ∆x_{i} as expressed in Eq. (7). ∆x_{i} is generated from the generating temperature T_{i,gen} and uniform random value u as expressed in Eq. (8):
where T_{i,gen} is calculated from the temperature reduction function as
where k is an annealing factor for each generating temperature, c is a constant defining the relative width of the generating distributions, and α is a factor associated with the speed of temperature decrease. The values of the parametersT_{i,gen}(0), c, and α are set to 1.0, 1, and 0.6, respectively, based on several numerical experiments.
Once new values for x(t) are generated, the difference of the misfit functions between new and previous is computed as
where E(x) is the misfit function defined by the residual between synthesized and observed records for acceleration envelopes and displacement waveforms (Miyake et al. 1999).
The new values x(t) are accepted if ∆E < 0. If ∆E > 0, x(t) is accepted if the uniform random value u is less than P, which is expressed as the following equation:
where the acceptance temperature is calculated as Eq. (9).
Determination of parameters required for the EGFM
The characterized source model in this study is composed of a SMGA consisting of N × N subfaults along the strike and dip directions, and a background area in the seismogenic zone on the fault plane. Because the background area mostly contributes to longperiod strong ground motions (Kurahashi and Irikura 2010), we estimated the strong ground motions only from the contribution of the SMGA neglecting the contributions from the background area. The number of SMGAs generally depends on the complexity of the source rupture process. For large magnitude earthquakes, multiple SMGAs seem to be distributed randomly (e. g. Kamae and Irikura 1998; Kurahashi and Irikura 2010, 2013). For the 2013 Lushan earthquake, several source–rupture–process studies based on waveform inversion analyses were common in that the large slip area was concentrated around the hypocenter. Therefore, it is reasonable to suppose that one SMGA located in the large slip area of the kinematic source models is suitable for synthesizing the ground motions.
First, we selected an aftershock with M_{w} 4.3 as an EGF. The aftershock occurred near the SMGA on 22:16, April 21, 2013 (Beijing time, 14:16, April 21, 2013, UTC) and had a similar source mechanism (Yi et al. 2016) as the mainshock. The positions of hypocenters and source mechanisms for the mainshock and this aftershock are listed in Table 1. The difference in the dip and rake angles between the mainshock and the aftershock are influential on the radiation pattern of ground motions from the source over a longer period range but less influential on those over a shorter period range, as the radiation characteristics over a shorter period are naturally smoothed due to diffraction and scattering along propagation paths (Kamae and Irikura 1992). Moreover, the site effects are more influential on the mainshock ground motions in a shorter period of engineering interest. Therefore, it is acceptable even if the dip and rake angles of the aftershock are not identical to those of the mainshock. Figure 1 shows the distribution of strong motion stations and beach ball diagrams of source mechanisms for the mainshock and the EGF event. Figure 1 also shows the projection of the kinematic source model with a large rectangle. The kinematic source model developed by the waveform inversion method (Li et al. 2017) is adopted for constructing the characterized source model, as the hypocenter is close to the position determined by highresolution relocation (Han et al. 2014). The ground motions of the M_{w} 4.3 aftershock at the stations, where the records of both the mainshock and the EGF event were available are regarded as EGFs. We synthesized ground motions at seven strong motion stations (e.g., 51BXD, 51BXM, 51BXY, 51YAM, 51QLY, 51YAD, 51PJD) surrounding the source region by applying the EGFM. The 51BXZ nearsource station is excluded, as a strong nonlinear effect is observed (see details in the “Discussion” section).
Second, we estimated the inner fault parameters related to the SMGA from the source spectral ratios of the mainshock ground motions to the EGF ones. We employed the Swave portion with a length of 40.96 s of the mainshock and EGF event to calculate the observed Fourier spectra. We followed the method of Miyake et al. (1999) to adopt the hypocenter distance for the mainshock and EGF events to remove the path effect (Q_{s}(f) = 155f^{0.6804}, Tao et al. 2016) from the Fourier spectra to obtain the observed source spectra. The average source spectral ratios and standard deviations for the target stations are shown with thick gray and dashed curves in Fig. 2. Three parameters, i.e., seismic moment ratio (M_{o}/m_{o}) between the mainshock and the EGF event, corner frequencies of the mainshock (f_{cm}) and the EGF event (f_{ca}), are required to estimate the theoretical source spectral ratio as expressed in Eq. (4). The ratio M_{o}/m_{o} is known from the moment magnitudes of the mainshock and the EGF event, while f_{cm} and f_{ca} are determined when the theoretical spectral ratio best fits the observed source spectral ratio in a specific frequency range. We set the lower frequency limit for fitting the observed source spectral ratio to be 0.5 Hz considering the low signaltonoise ratio of observed ground motions and the upper frequency limit to be 30.0 Hz, which is below the cutoff frequency of the antialiasing filter of the seismograph. We determined the best fitting of the theoretical source spectral ratio (red curves in Fig. 2) to the average observed spectral ratio (thick curves in Fig. 2) in the frequency range of 0.5 to 30.0 Hz when f_{ca} = 1.90 Hz and f_{cm} = 0.17 Hz. Subsequently, we applied Eqs. (5) and (6) to calculate N = 11, C = 2.12, r_{a} = 0.68 km, and ∆σ_{a} = 4.85 MPa. For small or moderate earthquakes, the fault length can be identical to the fault width; thus, the fault length or width of the EGF event is \(\sqrt{\pi }{r}_{a}\)=1.21 km. It should be noted that the ground motions from the SMGA are approximately expressed as omegasquared spectra with a corner frequency f_{cSMGA} (Boatwright 1988; Miyake et al. 2003). Then, f_{cm} and M_{o} in Eq. (5) can be expressed with f_{cSMGA} and M_{oSMGA} (seismic moment of the SMGA), respectively (Miyake et al. 2003; Somei et al. 2020). Therefore, N and C can be regarded as the ratio of the fault size between the SMGA and the EGF event, and the ratio of the stress drop between the SMGA and the EGF event, respectively.
Third, we applied the ASA algorithm to determine the optimal values of the parameters required by the EGFM, such as i, j, V_{r}, Γ as well as N and C determined by fitting the observed source spectral ratio which are not sufficiently accuracy considering the small signaltonoise ratio of the observed ground motions in the low frequency range. We fixed the rupture starting point at the hypocenter (Li et al. 2017) and kept the size of the fault length/width of the EGF event constant. Under the same acceptance temperature, the iteration of the ASA algorithm was carried out 5 times, and the search of indices (i, j) of the rupture starting point was carried out 10 times for each iteration, whereas the final value of k was set to 100. Therefore, the optimal parameters were determined when the misfit function reached the global minimum among the 5000 iterations. Table 2 lists the search range for each parameter in the second column. The search ranges for V_{r} and rise timeτ_{a} of the EGF event (Γ = Nτ_{a} for the mainshock) are approximately assigned, while the upper and lower limits for C and N are evaluated by the fitting of the theoretical source spectral ratio to the observed ones as shown in dashed curves lying one standard deviation above and below the average observed source spectral ratio. In accordance with the fitting frequency range of the observed source spectral ratio, we applied a bandpass filter of 0.5–30.0 Hz beforehand to the waveforms with a time window of 30 s, which contains the Swave portions at the seven target stations. Because steplike pulses were observed on the baseline of the records in the EW component at Station 51QLY for both the EGF event and the mainshock, we abandoned synthesizing the ground motions in the EW component to eliminate the special process of baseline correction.
Synthesized ground motions using the characterized source model
We obtained the bestfit SMGA of the characterized source model when the residuals between synthesized and observed records for the misfit function consisting of acceleration envelopes and displacement waveforms (Miyake et al. 1999) reached the global minimum throughout all iterations. The optimal values of the parameters, such as C, N, i, j, V_{r} and Γ, are listed in the third column in Table 2. The SMGA is 118.6 km^{2} (10.89 × 10.89 km), the stress drop is 12.9 MPa, and the seismic moment of SMGA is 6.85 × 10^{18} N·m. The location of the SMGA, as shown in Fig. 3, nearly coincides with the large slip area of the inverted kinematic source model (Li et al. 2017), which suggests that the SMGA can be regarded as equivalent to an asperity. Figure 4 shows the comparisons of waveforms between synthesized and observed acceleration, velocity and displacement in the NS and EW components at seven target stations. Figure 5 shows the comparisons of the acceleration Fourier spectra between synthesized and observed ground motions. The Fourier spectra are smoothed using a logarithmic window function with a band width parameter of b = 50 (Konno and Omachi 1998). Overall, not only the waveforms but also the Fourier spectra of the synthesized ground motions are in fairly good agreement with the counterparts of the observed ground motions at the target stations except for some stations, where synthesized acceleration waveforms are overestimated as nonlinearity of surface geology is not included in the EGFM. We also applied the same set of parameters of the bestfit SMGA to synthesize the ground motions at three nontarget stations, i.e., 51PJW, 51DJZ, and 51CDZ, where the observed ground motions are not considered in the above inversion analysis. The good consistency between the synthesized and observed ground motions is confirmed (Additional file 1: Fig. S1 and Additional file 2: Fig. S2), which further suggests that the characterized source model consisting of one SMGA is suitable for synthesizing the ground motions of the mainshock.
Subsequently, we examined the effectiveness of the inner fault parameters, such as the SMGA and acceleration level, in the bestfit source model. Here, the SMGA is considered equivalent to the combined area of an asperity. Figure 6 shows the empirical scaling relationship between the combined area of asperities with large slips estimated from waveform inversions for longperiod (> 1.0 s) motion and the seismic moment (M_{o}) proposed by Somerville et al. (1999) and Irikura and Miyake (2001, 2011) in the left panel, and the empirical scaling relationship between the flat amplitude of the acceleration source spectrum in the short period (A_{0}) (estimated according to Eq. (15), Dan et al. 2001) and M_{o} proposed by Dan et al. (2001) in the right panel. The relationship between the SMGA and M_{o} and the relationship between the A_{0} and M_{o} of the bestfit characterized source model during the 2013 Lushan earthquake (M_{w} 6.6) are found to be similar to these two empirical scaling relationships. Furthermore, we added the results of the SMGAs of the 2008 Wenchuan earthquake (M_{w} 7.9) (Kurahashi and Irikura 2010) and those of the 2014 Ludian earthquake (M_{w} 6.1) (Wang et al. 2017) and the 2017 Jiuzhaigou earthquake (M_{w} 6.5) (Wu et al. 2020), as shown with square, diamond and solid circle, respectively, in Fig. 6. It should be noted that Kurahashi and Irikura (2010) constructed the characterized source model only in the southern part of the fault model, and the seismic moment in the southern part was assumed to be half of the seismic moment for the entire fault. The relationships between the SMGAs and M_{o} for these crustal earthquakes in China agree well with the empirical scaling relationship (Somerville et al. 1999; Irikura and Miyake 2001, 2011; Miyake et al. 2003; Miyakoshi et al. 2020). The A_{0} values for the 2014 Ludian earthquake and the 2008 Wenchuan earthquake are slightly larger than those estimated from the empirical scaling relation, which is acceptable, as the values fall in the range indicating factors of 1/2 and 2 for the average (shown in dotted lines). Therefore, the relationships between A_{0} and M_{o} for these crustal earthquakes also follow an empirical scaling relationship (Dan et al. 2001). This implies that the empirical scaling relationships of inner fault parameters might be applicable to estimate the inner fault parameters for future crustal earthquakes in China. More inner fault parameters of the characterized source models for crustal earthquakes in China (if any) are required to investigate the applicability of the empirical scaling relationships.
Discussion
The EGFM is applicable for synthesizing the ground motions on the premise of linear superposition of the EGF, while the nonlinear effects on the ground motions are excluded for applying the EGFM. To examine the influence of nonlinearity on the surface geology of observed ground motions, we compare the earthquake H/V spectral ratio of the mainshock with that of the EGF event at seven target stations in Fig. 7. The acceleration Fourier spectra in three components are calculated from the Swave portions with a time length of 20.48 s, and then the H/V spectral ratios are evaluated following the method of Kawase et al. (2011). The degree of nonlinearity (DNL) (Noguchi and Sasatani 2008) at each station is evaluated to represent the extent of the nonlinear effect. A large DNL suggests a strong nonlinear effect, and vice versa. The similarity of the earthquake H/V spectral ratios between the mainshock and the EGF event and the small values of DNLs indicate that nonlinear effects are sufficiently weak at some target stations. Although the observed PGA of the mainshock is quite large at Station 51BXD, the smallest DNL value suggests that the nonlinear behavior of the site is negligible there. This is consistent with the fact that the site condition is rock, and the field survey also reported that the station was situated on the slope of a small rock hill (Wen and Ren 2014). The moderate values of DNL at Stations 51BXY and 51QLY suggest that the nonlinear behaviors of the sites are moderate. The synthesized acceleration ground motions are slightly larger than the observed ground motions at these stations due to the linear premise of the EGFM. An exception is the 51BXZ station, where the largest value of DNL suggests that nonlinear behavior is quite strong, which should be excluded when synthesizing the ground motions of the mainshock. The synthesis through the EGFM taking the nonlinear effect into account could be accomplished by introducing certain parameters (Nozu and Morikawa 2003), which is beyond the scope of this study.
Figure 8 compares the observed PGAs at target stations with those predicted by the ground motion prediction equations (GMPEs) (Zhao et al. 2016). The site conditions of the target stations can be categorized into two classes, i.e., rock and hard soil, according to the average shearwave velocity within 30 m in depth (Vs30) (Xie et al. 2022), the predominant period of the H/V response spectral ratio or field survey, we employ the black curves to show GMPEs for the rock site and the red curves for the hard soil site. The observed PGAs are comparable with the predicted PGAs within one standard deviation at target stations except the 51BXD station, where the observed PGA is abnormally larger than the predicted PGA. The source effect (e.g., directivity effect), site effect (e.g., nonlinear behavior, large shearwave velocity contrast, topographic effect) and hangingwall effect could be considered the reasons. We investigated the directivity effect on the ground motions in the EW component at the 51BXD and 51YAM stations by comparing the synthesized ground motions for different rupturestarting points: hypocenter (bestfit SMGA), top of the SMGA (Model A), and bottom of the SMGA (Model B), as illustrated in Fig. 9. Figures 10 and 11 show the results of waveforms and acceleration Fourier spectra in the EW component for the above three models, as well as the observed results during the mainshock, at Stations 51BXD and 51YAM. The amplitude of the acceleration Fourier spectrum of the synthesized ground motion at Station 51BXD for Model A is similar to that for Model B at periods below 0.3 s. This suggests that the directivity has little effect on the shortperiod ground motions. On the other hand, the amplitude of the acceleration Fourier spectrum of the synthesized ground motion at Station 51BXD for Model A is larger than that for Model B at periods of 1.0 ~ 2.0 s. This indicates that the directivity affects the longperiod ground motions at Station 51BXD, which is in the forward direction of rupture propagation (Model A). Moreover, the synthesized waveforms of Model A are in better agreement with the observed waveforms than those of the bestfit characterized source model. However, if we compare the synthesized ground motions with the observed ones at other stations, e.g., 51YAM, as shown in Fig. 11, the synthesized ground motions of Model A are in poorer agreement with the observed ones than those of the bestfit SMGA model. In terms of the fitting of the synthesized ground motions at all target stations, the bestfit SMGA source model is preferred among the three models. Furthermore, the amplitude of the acceleration Fourier spectrum of Model B is quite larger at periods of 1.0 ~ 2.0 s than that of the other two models at Station 51YAM. This indicates that the directivity effect is noticeable at Station 51YAM, which is in the forward direction of rupture propagation (Model B). In contrast, the directivity effect is not significant at Station 51BXD even though the station is in the forward direction of rupture propagation (Model A). This is because the SMGA dips toward the northwest and the distances between each subfault of the SMGA and the 51BXD station are not as variant when the rupture propagates toward the 51BXD station. We concluded that the directivity effect could not be responsible for the largest PGA in the EW component at Station 51BXD station for the bestfit SMGA source model of which the rupture starting point is close to the center of the SMGA.
In addition, the synthesized ground motions for the bestfit SMGA of the characterized source model agree with the observed ground motions in the EW component at Station 51BXD during the mainshock, as shown in Fig. 4 and Fig. 5. The largest PGA can be attributed to the large peak of the Fourier spectrum at approximately 0.1 s which was observed during the mainshock and the EGF event, as shown in Fig. 12. On the other hand, if strong site amplification contributes to the largest record, it must be caused by the large shearwave velocity contrast between the soft and hard soils in the shallow layers. However, soft soil might not exist, as the station is situated on the slope of a small rock hill. It is appropriate to attribute the reason to the hangingwall effect and topographic effect which have been investigated in the previous studies (Dai and Li 2013; Xie et al. 2014; Wen and Ren 2014).
Conclusions
We successfully reproduced the ground motions for the mainshock during the 2013 Lushan earthquake (M_{w} 6.6) using the characterized source model and the empirical Green’s function method (EGFM). The source parameters of the bestfit source model are determined by applying the adaptive simulated annealing (ASA) algorithm (Satoh 2006) for the misfit function by Miyake et al. (1999) using the waveforms in a frequency range from 0.5 to 30.0 Hz at seven nearsource strong motion stations. The bestfit characterized source model consists of one strong motion generation area (SMGA) and a background area. We also confirmed the good consistency between the synthesized and observed ground motions at the other three strongmotion stations, where the observed ground motions were not used for the inversion analysis when the same set of parameters was employed to synthesize the ground motions there.
We found the relationships between the SMGA and M_{o} and the relationships between the flat amplitude of the acceleration source spectrum in the short period and M_{o}, for the characterized source model of the 2013 Lushan earthquake (M_{w} 6.6), together with the 2014 Ludian earthquake (M_{w} 6.1), the 2017 Jiuzhaigou earthquake (M_{w} 6.5) and the 2008 Wenchuan earthquake (M_{w} 7.9), followed the empirical scaling relationships of inner fault parameters (e.g., Somerville et al. 1999; Irikura and Miyake 2001, 2011; Dan et al. 2001). This implied that the empirical scaling relationships of inner fault parameters might be applicable for the prediction of strong ground motions for future crustal earthquakes in China.
The largest PGA during this earthquake was larger than 1 g at Station 51BXD, where large strong motions exceeding 1 g were obtained for the first time in the observation history of China. We explored the reason for the largest record from the source effect (e.g., directivity effect) and site effect (e.g., nonlinear behavior, large shearvelocity contrast, topographic effect). Our analyses suggested that the directivity effect cannot be the reason for comparing the effect of different positions of the rupturestarting point on the synthesized ground motions and that the nonlinear effect was not strong, as indicated by the similarity of the earthquake H/V spectral ratios between the mainshock and the EGF event. The Fourier spectra of the observed ground motions of both the mainshock and the EGF event dominated in the same shortperiod range (e.g., 0.1 ~ 0.2 s), which was responsible for the largest PGA. This result suggested that the strong site amplification, if present, was caused by the large shearwave velocity contrast between the soft and hard soils in the shallow layers. However, we considered this impossible, as the station was situated on the slope of a small rock hill. Finally, we agreed with the previous studies (Dai and Li 2013; Xie et al. 2014; Wen and Ren 2014) that the hangingwall effect and topographic effect could be the reasons for generating the largest record at Station 51BXD.
Availability of data and materials
The data sets analyzed during the current study are available from the corresponding author on reasonable request.
Abbreviations
 ASA:

Adaptive simulated annealing
 EGF:

Empirical Green’s function
 EGFM:

Empirical Green’s function method
 DNL:

Degree of nonlinearity
 PGA:

Peak ground acceleration
 SMGA:

Strong motion generation area
 Vs30:

Average shearwave velocity within 30 m in depth
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Acknowledgements
The China Strong Motion Network Center at the Institute of Engineering Mechanics, China Earthquake Administration is acknowledged for providing strong motion data. We are grateful to John Zhao and Hao Xing for discussions on the GMPE and Chengli Liu for kindly providing the slip data of the kinematic source model. The comments from editor Wataru Suzuki and two anonymous reviewers were helpful to improve the manuscript. Some of the calculations were implemented on the supercomputer system of Shandong Jianzhu University. All the figures were plotted by generic mapping tools (Wessel et al. 2013).
Funding
This work was supported by the Youth Innovation Project of Science and Technology of University in Shandong Province (2019KJG015) and special grant from Shandong Jianzhu University.
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HW designed the study, analyzed the data and drafted the manuscript. KI designed the study, participated in discussions, and interpreted the results. KM and KS contributed to discussions of the data processing and the results. All authors read and approved the final manuscript.
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Additional file 1:
Figure S1. Comparison of the observed ground motions (acceleration, velocity, displacement) (black) in the NS and EW components with the synthesized ground motions (red) at the 51PJW, 51DJZ and 51CDZ stations.
Additional file 2:
Figure S2. Comparison of the acceleration Fourier spectra (black) of the observed ground motions with those of the synthesized ground motions (red) at the 51PJW, 51DJZ and 51CDZ stations. The Fourier spectra are smoothed by use of a logarithmic window function (Konno and Omachi 1998).
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Wu, H., Irikura, K., Miyakoshi, K. et al. Characterized source model of the 2013 Lushan earthquake (M_{w} 6.6) by the empirical Green’s function method. Earth Planets Space 74, 50 (2022). https://doi.org/10.1186/s40623022016136
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DOI: https://doi.org/10.1186/s40623022016136
Keywords
 Strong motion generation area
 Empirical Green’s function method
 The 2013 Lushan earthquake (M _{w} 6.6)
 Empirical scaling relationships of inner fault parameters