Simulation of Ground Motion From Finite‐Fault Modeling Incorporating the Influence of Duration

On 21 May 2021 (local time), an Mw 6.1 earthquake struck the Yangbi County, Yunnan Province, China. The finite fault stochastic simulation approach is usually used to simulate ground motions in high‐frequency band (f > 1 Hz). Model parameters needed for earthquake ground motion simulation mainly include source, path, and site. In the high‐frequency ground motion simulation program widely used in earthquake engineering, the reciprocal of corner frequency is typically used to define the source rise time; however, the complete step‐wise derivation process is unavailable. In deriving the static corner frequency f0, source rise time is typically estimated to be the time required for the rupture to reach 50% of the final location, and the source rise time then can be obtained as 0.27/f0, which is consistent with the hypothesis of corner frequency and rupture velocity, and the correlation and integrity between parameters are established. This study also focuses on the influence of different source rise times and path durations on the simulation results, such as Fourier acceleration spectrum, pseudo‐spectral acceleration, and peak ground acceleration. The results show that the source rise time recommended in this simulation can improve the accuracy of near‐fault ground motion simulations. This study provides suggestions for a reasonable selection of path duration in different engineering applications.


Introduction
The stochastic simulation method of ground motion comprises two parts: the ground motion amplitude spectrum fitted based on a statistical model and the ground motion phase spectrum based on random vibration theory.These are typically described as a function of magnitude, path, and site and as the phase corresponding to the limited bandwidth Gaussian white noise obtained from the intensity envelop related to duration (Boore, 2003).The stochastic finite-fault technique based on the static corner frequency (FINSIM) was first proposed by Atkinson and Berenev (Beresnev & Atkinson, 1998).Because the rupture fault size cannot be neglected in the near-fault ground motion simulation, Beresnev and Atkinson (1998) divided the entire rupture surface into subfaults, and each subfault was approximated as a point source (Aki, 1967;Brune, 1970).A rupture triggers from the source and spreads to the surrounding subfaults.When the rupture propagates to the center of each subfault, the subfault starts to rupture, propagating to the edge of the subfault.In FINSIM, the source rise time is defined as the time required for the rupture to propagate from the center of the subfault to the edge of the subfault.
Subsequent studies showed that different subfault sizes result in significantly different synthetic results.In 2005, Motazedian and Atkinson (2005) developed a stochastic finite-fault approach based on dynamic corner frequency (EXSIM) and then developed a scaling factor based on the velocity spectrum to conserve the far-field received energy under different subfault sizes.In this case, the synthetic results depend less on the subfault size.To further weaken the influence of subfault size on the synthetic results, Boore (2009) proposed calculating the scaling factor based on the square of the acceleration spectrum, and Boore (2009) defined source rise time as the reciprocal of the static corner frequency f 0 ; however, Boore (2009) did not present a straightforward derivation process.The improved programs have been widely adopted worldwide for generating synthetic accelerograms for different engineering applications (Ameri et al., 2011;Chopra et al., 2012;Dang et al., 2022;Ghasemi et al., 2010;Ghofrani et al., 2013;Mittal & Kumar, 2015;Mittal et al., 2016;Shen et al., 2014;Sutar et al., 2020;Yalcinkaya et al., 2012;Zengin & Cakti, 2014).Compared with the point-source modeling, finite fault modeling is advantageous as it simultaneously considers the geometry of the fault and the effect of rupture direction of the source; as a result, finite fault modeling can more accurately synthesize the near-field ground motion of large earthquakes.
• The stochastic finite-fault method was applied to simulate the near-field ground motion of the Yangbi earthquake in China • The input parameters were selected and the simulated and observed results were compared • The effects of path duration and source rise time on the synthetized results were analyzed

Supporting Information:
Supporting Information may be found in the online version of this article.
DANG ET AL.
10.1029/2023EA002871 2 of 13 EXSIM, which is widely used, is not suitable for simulating ground motion in specific areas.For example, in the EXSIM program developed by Boore (2009), the path duration model used is based on the rich ground motion recordings in North America and thus may not apply to other regions (Sun et al., 2015).Moreover, the source rise time is expressed as 1/f 0 without providing the derivation process.The expression of source rise time varies across different studies (Tang, 2022), owing to these differences, it is challenging to estimate source rise time in large earthquakes.In actual ground motion simulation, the lack of clear assumptions and ambiguous definition of source rise time can introduce unnecessary uncertainty to the simulation results.
The Yangbi earthquake in Yunnan Province is taken as an example, and the influence of commonly used path duration and several different source rise times on the simulation results are discussed; the findings of this study provide insights for guiding the use of EXSIM in practical engineering applications in future.In this study, the subfault is estimated to be a circular fault.

Stochastic Method
In a simulation, a large fault is always divided into smaller subfaults, each of which is regarded as a point source (Boore, 1983); Stochastic point-source modeling was applied to synthetized the waveforms generated by each subsource (Boore, 1983(Boore, , 2003)); synthetic results of large rupture faults can be calculated, in the time domain, by summarized all acceleration time series (Motazedian & Atkinson, 2005): in which △t ij indicates the relative time delay, nl and nw denote the number of subfaults along the fault strike and dip direction, respectively.
In EXSIM, M 0ij is defined as follows (Motazedian & Atkinson, 2005): where M 0 represents the seismic moment of the large fault.D ij indicates the slip of the ijth subfault.
In EXSIM, the dynamic corner frequency f 0ij of the ijth subfault is expressed as a function of time t (Motazedian & Atkinson, 2005) and can be given by 0() = 4.9 × 10 6  ( Δ 0ave in which M 0ave , the average seismic moment in the unit of dyne-cm, is given as M 0ave = M 0 /N, and N(t) denotes the total number of ruptured subfaults on the large fault at time t.
In addition, a high-frequency scaling factor is adopted to conserve the spectral level of the subfault at the high frequencies (Motazedian & Atkinson, 2005): in which N denotes the total number of subfaults on the main fault and can be defined as N = nl × nw.f 0 represents the static corner frequency and can be obtained by the expression f 0 = 4.9 × 10 6 β(△σ/M 0 ) 1/3 and f 0ij indicates the dynamic corner frequency of the ijth subfault defined by Equation 4.

Recordings of the Yangbi Earthquake
An Mw 6.1 earthquake struck the Yangbi County, Yunnan Province, China at 21:48 on 21 May 2021 local time.The epicenter was located at 25.67°N and 99.87°E, which is approximately 9.0 km away from Yangbi County, and the focal depth was about 8 km.The results of the seismic macro intensity map released by the Yunnan Earthquake Agency showed that the highest intensity of this earthquake was VIII defined by the Chinese Seismic Intensity Scale.In this case, most houses are moderately damaged and the structures are damaged and require repair (The Chinese seismic intensity scale, GB/T 17742-2020, 2020).The focal mechanism solution given by the Global Centroid Moment Tensor showed that the magnitude of the Yangbi mainshock was Mw 6.1, and it was a strike-slip earthquake.Studies have shown that the Yangbi earthquake occurred in an NW-SE right rotation strike-slip secondary fault on the west side of the Weixi-Qiaohou Fault in the north section of the Honghe Fault (Qiang et al., 2021).The Honghe Fault is the southwest boundary of the Sichuan-Yunnan rhombic block in the southeast of the Qinghai-Tibet Plateau.This area is the boundary zone of collision and compression between the Indian plate and the Eurasian plates.The Sichuan-Yunnan rhombic block is considered to be the most active block at the southeast edge of the Qinghai-Tibet Plateau.The complex geological structure and strong fault zone activity make it one of the areas with the most frequent seismic activity in China (Kan et al., 1977;Wang et al., 2003).As of May 22, the earthquake killed 3 people, injured 28 and collapsed 192 houses.The maximum peak ground acceleration (PGA) was recorded by the Yangbi strong-motion station 053YBX, approximately 8 km away from the epicenter.The PGAs of east-west (EW), north-south (NS), and vertical (UD) components were 379.9, 720.3, and 448.4 cm/s 2 , respectively.
After the Yangbi earthquake, the Yunnan strong motion network obtained 25 sets of strong motion records, with only 5 stations within 100 km from the epicenter.The relevant parameters of the selected ground-motion stations are presented in Table 1. Figure 1 shows the locations of all stations and faults triggered during this earthquake.In this study, the method proposed by Boore ( 2001) is adopted to process the near-field recordings.The baseline correction and the cosine window are added at the beginning and end of the recording waveform to eliminate the recording error, and then the Butterworth non-causal band-pass filter (0.08-30 Hz) is applied to process the records.As one of the essential input parameters, and the effective frequency band of the recordings after filter process is 0.1-24 Hz (Qiang et al., 2021).
The source model of the Yangbi earthquake was established using the scaling law based on global and local source parameters and truncated normal distribution density function (Dang, Cui, Liu, & Ji, 2023;Wang, 2004), as shown in Figure 2. The results show that the fault size is 10 × 9 km 2 and the subfaults are sized at 1 × 1 km 2 (Dang, Cui, Liu, & Ji, 2023).

Ground-Motion Duration
In EXSIM, the ground-motion duration plays a vital role in the synthetic results.In EXSIM, the ground motion duration T is composed of path duration T path (R) and source rise time T rise (f).

Source Rise Time
Source rise time is also known as duration.To capture the ground motion caused by a large fault of a large earthquake, it is necessary to discretize a large fault into several small subfaults.When a rupture propagates from the starting point to the center of a subsource, the subsource begins to rupture and ends when the rupture reaches the edge of the subsource (Beresnev & Atkinson, 1998;Sun, 2010).Therefore, the original source rise time can be defined as:  Different definitions of T rise will result in different expressions of T rise .In earthquake engineering, T rise is typically estimated to be the time required for the rupture to reach 50% of the final location (Beresnev & Atkinson, 1997): where τ denotes a characteristic time parameter controlling the rate of the displacement increase.
Solving Equation 6gives the solution T rise /τ = 1.68.Beresnev and Atkinson (1997) introduced a notation ω c = 1/τ, then the corner frequency is defined as: Other scholars have also provided different definitions of T rise .For example, Boore (2009) defined T rise as the reciprocal of corner frequency: Boatwright and Choy (1992) defined T rise as: and Hough and Dreger (1995) defined T rise as: In Equations 9-11, f 0 represents the corner frequency.To weaken the influence of subfault dimension on synthetic results, in EXSIM, the source rise time was usually expressed as 1/f 0 (Boore, 2009).

Path Duration
In EXSIM, the original path duration is typically defined as (Akinson & Boore, 1995): Other studies also use the following formula to define the path duration of ground motion (Ghofrani et al., 2013): where a and b represent the model coefficient, and generally taken a as 0.5 or 0.1.R indicates epicentral distance in km.
The significant duration is determined based on the energy accumulation process of earthquake ground motion, and it is expressed by the integral of the square of site acceleration, velocity, or displacement (Bora et al., 2014).
In engineering practices, acceleration is usually used to calculate the significant duration, and the normalized Arias intensity can be defined as: In Equation 14, a (t) denotes the acceleration time series, g indicates the gravitational acceleration (9.8 m/s 2 ), and t indicates the ground motion duration.The time when the standard energy accumulates to 5%-95% is defined as 90% significant duration expressed as D 5-95 .To highlight the body wave energy, the time when the energy accumulates to 5%-75% is used to define 70% significant duration, which is expressed as D 5-75 (Sun & Li, 2019).Boore and Thompson (2014) suggested using 20%-80% energy accumulation time to emphasize S-wave energy.
Based on different definitions of energy accumulation time, the following formula can be used to directly fit the path duration of earthquake ground motion (Dang et al., 2022;Ghofrani et al., 2013;Sun et al., 2015;Wang et al., 2022): where m and n indicate regression coefficients and R represents epicentral distance.
In light of the abovementioned earthquake duration models, reasonably selecting these in-ground motion simulations is the next biggest concern among researchers.In this study, we compare different effective durations (Figure 3) and use 90% effective duration (Sun et al., 2015;Tang, 2022) to fit the path duration of the Yangbi earthquake (Figure 4).T path adopted in this study can be expressed in T path = 0.2705R + 4.928 for EW and T path = 0.238R + 3.787 for NS. Figure 4 shows that the difference of 90% duration recorded and calculated by other stations is small in the two horizontal components, except for station 053BCJ.Therefore, the path duration in the North-South component with a high goodness of fit is selected as the path duration of the Yangbi earthquake.

Results and Discussion
As one of the most important input parameters, site amplification considerably influences the synthetic results.
For bedrock sites, typically, only crustal site amplification is considered, while for soft soil sites, local site amplification has to be taken into account (Dang et al., 2022;Yalcinkaya et al., 2012).The crustal amplification function applied in this study adopted the crustal amplification factors of class C and class A sites established by Boore and Joyner (1997) and Atkinson and Boore (2006), respectively (Figure 5).The single station technique has been adopted to obtain the H/V spectral ratio (Meneisy et al., 2020).A forward calculation and inversion of the H/V ratio could be performed, however, the upper and lower limits of material elastic parameters, such as density, S-wave velocity and P-wave velocity needed a preliminary model regarding the characteristics of near surface layers.These profiles could be obtained from boreholes, geological and geophysical data.Due to there are no detailed shear wave velocity profiles in this region, herein, in the current study, the local site amplification factors of the five selected stations are approximately estimated by the H/V spectral ratio given by Nakamura (1989), then the local site effect is obtained from the resulted H/V curves (Meneisy et al., 2020;Toni, 2017), which is plotted in Figure 6.The figure indicates that the H/V curves at stations 053BCJ, 053YBX, 053DLY, and 053YPX shows multiple peak characteristics, this may be due to the existence of multiple impedance interfaces under the surface where these stations are located or the resonance of S-wave (Lu & Wang, 2018).Application of H/V ratio method in different site types, such as Class A, B, C, and D sites, has been detailed discussed by previous studies (Dang, Cui, & Liu, 2023;Dang et al., 2022) suggesting that the crustal and local site amplification functions  should be used for soft soil sites.In this simulation, the stress drop was determined by the trail-and-error method to yield the minimum error between the predicted and observed values.In our simulation, the relationships between epicentral distance and high-frequency attenuation kappa of EW and NS components can be obtained: κ = 0.00153R + 0.01569 for EW component, and κ = 0.00151R + 0.01709 for NS component (Figure S1 in Supporting Information S1), then the best fitting value of 0.017 s was applied in this study.The quality factor Q was given by Xu et al. (2010) and can be defined as Q = 180f 0.5 , which obtained from the recordings of medium, small, and strong earthquakes in Sichuan and Yunnan.Other model parameters needed for the ground motion simulation are listed in Table 2. Based on these parameters, EXSIM is used to simulate the acceleration records of five near-fault stations of the Yangbi earthquake, and the corresponding pseudo-spectral acceleration (PSA) and Fourier spectrum are obtained.The effects of different source rise times and path durations on simulated results in terms of PGA, PSA, and Fourier acceleration spectrum (FAS) are discussed in the following sections.

Effect of Source Rise Time
In the simulation, path duration is defined as T path = 0.238R + 3.787.Figure 7 shows the PSA of five near-field stations of the Yangbi earthquake simulated by different source rise times defined by Equations 5 and 8-11 and recorded values.Figure 7 shows that the synthetic results of stations 053BCJ, 053BTH, and 053YPX are consistent with the recoded values in the short term (T < 1 s).The simulated PSAs of stations 053DLY and 053YBX agree with the recorded values in the medium and long term (T > 0.4 s).However, the simulated values of station 053YPX overestimate the observed values in the long term, which may be due to the slow high-frequency attenuation caused by the small kappa parameter selected in this study or because the local site amplification roughly obtained from H/V cannot adequately reflect the local site characteristics of the local site.Figure 8 shows the simulated FAS and observed FAS of five selected stations during the Yangbi, China, earthquake, indicating that the simulated FAS of stations 053BCJ, 053BTH, 053DLY, 053YBX, and 053YPX are consistent with the recorded values in the high frequency band (f > 1 Hz), which also shows that EXSIM has significant advantages in high-frequency ground motion simulation.For further analyses, the model deviation of five near-field stations simulated by the different source rise times (Figure 9) is calculated as follows (Mena et al., 2010): where  PSA sim i ( ) and  PSA obs i ( ) represent the simulated and recorded PSA of the ith station, respectively, and n indicates the total number of stations.Original: Equation 12Akinson and Boore (1995) Source rise time 0.27/f 0 Beresnev and Atkinson (1997) 0.37/f 0 Hough and Dreger (1995) 0.5/f 0 Boatwright and Choy (1992) 1.0/f 0 Boore (1983Boore ( , 2003) ) Original:  √  × ∕∕rup Beresnev and Atkinson (1997) Rupture velocity 0.69β, 0.8β This study

Crustal amplification
Class A for station 053BTH Atkinson and Boore (2006) Class C for the other stations Boore and Joyner (1997) Local site amplification H/V ratios This study

Table 2
Model Parameters Required for the Yangbi Earthquake The total deviation of all five stations is plotted in Figure S2 in Supporting Information S1; the simulated value calculated by the source rise time defined in Equation 8 is closer to the recorded value in the short term with T less than 1 s, while the model deviation obtained by other models is small.Figure 10 shows the simulated and observed PGAs of five near-field stations; the PGA synthesized by the source rise time defined  in Equation 9 differs slightly from the observed PGA at stations 053BCJ, 053BTH, 053DLY, and 053YPX.Equation 8 adopts the basic assumptions generally accepted in earthquake engineering, such as slip reaching 50% of the final displacement, which is consistent with the assumption used in deriving the static corner frequency.Therefore, to ensure the integrity of the total corner frequency, rupture velocity, and source rise time, the source rise time defined in Equation 8 is recommended as the input parameter of source time.

Effect of Path Duration
Another factor that affects the duration of earthquakes is path duration.Similar to the source rise time, the PSA, FAS, model deviation, averaged model deviation, and PGA of five near-field stations simulated by different path durations are compared with the observed values, and these are presented in Figures 11-14, respectively.In the simulation, the source rise time is defined by Equation 5. Figure 11 reveals that the synthetic PSAs of stations 053BCJ, 053BTH, 053DLY, and 053YPX are well matched with the observed values in the short term (T < 1 s).At station 053YPX, the difference between synthetized and recorded values is evident in the short term.Figure 12 indicates that the simulated FAS of stations 053BCJ, 053BTH, 053DLY, 053YBX, and 053YPX are consistent with the observed values at high frequencies.
As EXSIM is typically adopted to simulate high-frequency ground motions, this focus was on the high-frequency range in this study.Figure 13 shows that the model deviation obtained by the original path duration (Equation 12) is closer to 0 in the short term (T < 1 s) than that obtained by the other two path durations.The path duration fitted by vital motion records performs well in the long term and is between the path duration defined by Equations 12 and 13 in the short term.The model deviation obtained by the three path durations is nearly consistent with the recorded values; the maximum value of the model deviation in the whole period is no more than 0.41.This is consistent with the conclusion of Dang and Liu (2020), who simulated the 2013 Mw 6.7 Lushan, China earthquake and Dang et al. (2021), who simulated the 2016 Mw 6.2 Tottori, Japan earthquake. Figure 14 shows   In summary, in ground motion simulation, this study recommends using the source rise time determined in Equation 8 as the input parameter, which is consistent with the assumption used in deriving the static corner frequency, in theory, forming a unified whole, but also verified in the actual simulation.In terms of path duration, this study presents three duration models, all of which obtain ground motion characteristics that agree with the observed values.When using EXSIM to predict the ground motion field in an area with few ground motion records or using the generated acceleration records as the ground motion input for analyzing the seismic analysis of buildings, the path duration is defined by Equations 12 and 13 are recommended.

Conclusions
In this simulation, the slip model of the Yangbi, China, earthquake is established by using the scaling laws of the global and local parameters and the truncated normal distribution density function.The local site effects of five near-field stations are approximately calculated by the H/V method.
Considering different source rise times and path durations, the acceleration observation records of five near-fault stations of the Yangbi earthquake are simulated using EXSIM, and the corresponding PSA and FAS are calculated.
In addition, based on PSA, the model deviation and its average value of five near-field stations are calculated.The main conclusions are as follows.
1.When EXSIM is used in practical engineering, the source rise time defined by 0.27/f 0 is recommended.Because the assumption used in this rise time is consistent with that used in the derivation of corner frequency.However, the assumptions used in the definition of other rise time are inconsistent with the assumptions used in the commonly used expression of static corner frequency.In addition, the simulated and observed values of different source rise times are compared from the aspects of PSA, FAS, PGA, and model deviation.The results indicate that the synthetic  results of source rise time defined as 0.27/f 0 are well-matched with the observed values.2. By comparing the differences between the synthesized values and the recorded values obtained by different path durations, this study shows that when guiding the post-disaster rescue and reconstruction, the results synthesized using the path durations fitted by ground motion records can provide ideal results that are agreement with the observed values in terms of PGA, PSA, and FAS.The path duration is defined by Equations 12 and 13 is the fastest and ideal choice when EXSIM is used to predict the seismic field in areas with few seismic records or the acceleration records generated by EXSIM are used as the ground motion input for analyzing the seismic analysis of buildings.

Figure 1 .
Figure 1.Distribution of stations and faults during the Yangbi earthquake.Red triangle: near-field acceleration observation stations selected in this paper, blue triangle: remaining stations during the Yangbi earthquake, yellow five-pointed-star: the source.

Figure 4 .
Figure 4.The 90% effective path duration of the two horizontal directions of the fitted Yangbi earthquake.Blue ball: 90% effective duration calculated for each record, red solid line: the path duration model of linear fitting, magenta-filled area: 95% confidence interval.

Figure 3 .
Figure 3. Several commonly used significant duration models.053BCJ station in the Yangbi earthquake is taken as an example.

Figure 5 .
Figure 5. Crustal amplification factors of class A and class C sites used in the simulation.Red ball: class C site, blue ball: class A site.

Figure 6 .
Figure 6.Local site amplification factors of the near-field stations roughly estimated by the H/V method.

Figure 7 .
Figure 7.Comparison between pseudo-spectral acceleration and observed values of the selected stations simulated by different source rise times.Red solid line: the observed value, solid lines of other colors: the simulated value when the source rise time is defined by T rise = 0.27/f 0 , T rise = 0.37/f 0 , T rise = 0.5/f 0 , T rise = 1.0/f 0 and the original one.

Figure 8 .
Figure 8.Comparison between Fourier acceleration spectrum and observed values of the selected stations simulated by different source rise times.Red solid line: the observed value, solid lines of other colors: the simulated value when the source rise time is defined by T rise = 0.27/f 0 , T rise = 0.37/f 0 , T rise = 0.5/f 0 , T rise = 1.0/f 0 and the original one.

Figure 9 .
Figure 9. Model deviations and average values of five near-field stations calculated by different source rise times.Gray solid line: the synthetic value of each station, red solid line: the average value of five stations, filled area: the range of standard deviation.

Figure 10 .
Figure 10.Comparison between peak ground acceleration (PGA) and observed values of simulated near-field stations with different source rise times.One PGA value is the average of 30 simulated values.

Figure 11 .
Figure 11.Comparison between pseudo-spectral acceleration and observed values of selected stations for different path durations.Black and orange solid lines: the observed values, yellow, green, and blue solid lines: the simulated values when the path duration is defined by the original one, T path = 0.5R, and T path = 0.238R + 3.787, respectively.
the simulated PGAs obtained by the path duration defined by Equations 13 and 15 are between the two horizontal recorded values at five stations, while the simulated PGAs obtained by the original path duration defined by Equation 12 significantly overestimate the observed values at stations 053BCJ and 053YPX.

Figure 12 .
Figure 12.Comparison between Fourier acceleration spectrum and observed values of selected stations simulated by different path durations.Black and orange solid lines: the observed values, yellow, green, and blue solid lines: the simulated values when the path duration is defined by the original one, T path = 0.5R, and T path = 0.238R + 3.787, respectively.

Figure 13 .
Figure 13.Average model deviation of all stations obtained by different path durations.Blue solid line, red dashes line, and magenta dotted line: the model deviation when the path duration is defined by the original one, T path = 0.5R, and T path = 0.238R + 3.787, respectively.

Figure 14 .
Figure 14.Comparison between peak ground acceleration (PGA) and observed values of simulated near-field stations with different path durations.One PGA value is the average of 30 simulated values.

Validation: Pengfei Dang, Yi Li, Wenbo Wang Visualization: Pengfei Dang Writing -original draft: Pengfei Dang Writing -review & editing: Pengfei Dang
Unit: PGA, cm/s 2 ; R, km; Latitude, °N; Longitude, °E.The site conditions are obtained from the header file of the acceleration recordings.Details of Near-Fault Stations Triggered by the 2021 Mw 6.1 Yangbi Earthquake in China