1 Introduction

The Longmen Shan fault (LMSF) is located in the boundary between the eastern Tibetan Plateau and the South China Block (Fig. 1). It is a complex fault zone formed as a result of collectively accommodating the crustal deformation of the eastern Tibetan Plateau induced by the Indo-Asian collision. The fault zone consists of three major thrust faults: the Wenchuan-Maoxian fault, the Yingxiu-Beichuan fault, and the Guanxian-Anxian fault (Deng et al., 1994). Each of these three faults exhibits distinct thrust characteristics, with a NW dip angle of 60–70° that gradually decreases with depth, ultimately converging into a low-dip shear zone (Feng et al., 2016; Royden et al., 2008).

Fig. 1
figure 1

Tectonic background of the eastern Tibetan Plateau. The blue dashed rectangles indicate the two seismic gaps (D: Dayi seismic gap; T: Tianquan-Kangding seismic gap) along the Longmen Shan fault. Red circles are the locations of instrumental earthquakes from 1970 to 2008; data are from the China Earthquake Network Center. Blue beach balls show the focal mechanism solution of earthquakes from GCMT (http://www.globalcmt.org). LMSF Longmen Shan fault; WC-MX Wenchuan-Maoxian fault; YX-BC Yingxiu-Beichuan fault; GX-AX Guanxian-Anxian fault; MJF Minjiang fault; XSHF Xianshuihe fault; LQSF Longquan Shan fault; FBHF Fubianhe fault; DKLF Dongkunlun fault; HYF Huya fault; QCF Qingchuan fault; ABF Aba fault; LRBF Longriba fault; DB-BLJF Diebu-Bailongjiang fault; HN-QSWF Hanan-Qingshanwan fault; XJHF Xiaojinhe fault; DLSF Daliang Shan fault; ANHF Anninghe fault

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The LMSF has been intensely seismically active recently, with the 2008 Mw7.9 Wenchuan earthquake and the 2013 Mw6.6 Lushan earthquake occurring successively. Surface ruptures caused by the Wenchuan and Lushan earthquakes are ~ 300 km of the central and northern LMSF, and ~ 30 km of the southern LMSF, respectively (Chen et al., 2013). However, there are still two seismic gaps on the LMSF that have not ruptured following these two events: the Dayi seismic gap, located between the rupture zones of the Wenchuan and Lushan earthquakes, and the Tianquan-Kangding seismic gap in the southwestern LMSF (Chen et al., 2013; Pei et al., 2014; Xu et al., 2013).

It is disputed whether these two seismic gaps have the potential to generate another major earthquake. One viewpoint suggests that the Dayi and Tianquan-Kangding seismic gaps pose a high risk of future earthquakes. For example, Chen et al. (2013) analyzed the seismic moment accumulation and release in the southwestern LMSF and indicated that both the Dayi and Tianquan-Kangding seismic gaps have a high potential for future earthquakes with magnitudes of Mw6.8 and Mw7.2, respectively. This viewpoint has been verified by in situ stress state (Li et al., 2022), fault locking (Li et al., 2023), and Coulomb stress changes (Guo et al., 2020; Liu et al., 2018; Toda et al., 2008). However, some other studies such as seismic tomography (Pei et al., 2014), global positioning system (GPS)-derived fault coupling (Li et al., 2018), and paleo-earthquakes (Dong et al., 2017) are to the contrary.

Determining the stress states in a region is crucial for evaluating seismic hazards. For example, following the 2008 Wenchuan earthquake, multiple scholars have calculated Coulomb stress changes (Guo et al., 2020; Liang et al., 2018; Liu et al., 2018; Parsons et al., 2008; Toda et al., 2008; Xu et al., 2013, 2020; Zhang et al., 2010) and found that the event has changed the regional stress field significantly. It is noteworthy that the 2013 Mw6.6 Lushan earthquake that occurred in the region with Coulomb stress changes surpassed 0.01 MPa. However, studies also show that some subsequent earthquakes occurred in regions where Coulomb stress changes caused by preceding earthquakes were negative (Gkarlaouni et al., 2008; Luo & Liu, 2018).

Compared with the background stress that has already accumulated on faults, Coulomb stress changes caused by earthquakes are a relatively small perturbation (Luo & Liu, 2018; Sun et al., 2020). Thus, a comprehensive analysis of full stress evolution on faults is crucial for further assessing the regional seismic hazards. However, determining the magnitude of regional stress remains challenging to date (Shi et al., 2018). Though Li et al. (2022) arranged two boreholes (1000 and 500 m deep) to carry out hydraulic fracturing in situ stress measurements in the Dayi seismic gap; the depth of the borehole is relatively small compared with the focal depth of regional earthquakes.

While it is still difficult to quantify the magnitude of regional stress, studies have shown that seismic b-values obtained from the Gutenberg–Richter (G–R) relation (Gutenberg & Richter, 1944) can provide insights into regional stress states. A lower b-value indicates a higher stress level, while a higher b-value indicates a lower stress level in a region (Dong et al., 2022; Scholz, 1968, 2015).

In this study, we establish a three-dimensional (3D) viscoelastic finite element model in the eastern Tibetan Plateau, and calculate the Coulomb stress changes caused by the Wenchuan and Lushan earthquakes. Additionally, we calculate the spatial distribution of the regional b-values based on the instrumental seismic catalog before the Wenchuan earthquake, and infer the regional background stress level according to the inverse correlation between the b-value and stress level. Finally, we analyze the regional seismic hazards by combining the results of Coulomb stress changes and regional b-values.

2 Finite Element Model

2.1 Model Setup

We establish a 3D finite element model of the eastern Tibetan Plateau. Figure 2 shows the mesh of the finite element model. The length and width of the model are 700 km, and the depth is 100 km. Based on the active tectonic block hypothesis (Deng et al., 2003; Shao et al., 2022), the model is horizontally divided into three blocks: South China Block, Songpan-Ganzi Block, and Sichuan-Yunnan Block (Fig. 2).

Fig. 2
figure 2

Three-dimensional finite element model of the eastern Tibetan Plateau

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The model consists of two rheological layers at depth: the elastic upper crust, and the viscoelastic middle-lower crust and upper mantle layer. The elastic upper crust (seismogenic layer) of the model is set to be 20 km since most earthquakes in this region occur within the depth of 20 km (Shen et al., 2009; Xu et al., 2009; Zhang et al., 2010). Results of receiver function analysis indicate that the crustal thickness of the eastern Tibetan Plateau is about 50–70 km, and the crustal thickness of the Sichuan Basin is about 35–50 km (Bao et al., 2020; Gao et al., 2022; Wang et al., 2023). Thus, the layered model of the South China Block consists of the upper crust (0–20 km), middle-lower crust (20–50 km), and upper mantle (50–100 km), while the Songpan-Ganzi and Sichuan-Yunnan Blocks consist of upper crust (0–20 km), middle-lower crust (20–60 km), and upper mantle (60–100 km) (Table 1).

Table 1 Material parameters of the finite element model
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According to the regional velocity and density structures from previous studies (https://igppweb.ucsd.edu/~gabi/crust1.html; Bao et al., 2020; Gao et al., 2022; Wang et al., 2023), we calculate Young’s modulus (E) and Poisson's ratio (\(\upsilon\)) of each block in the study area by using the following equations:

$$E = \frac{{V_{{\text{S}}}^{2} \rho (3V_{p}^{2} - 4V_{S}^{2} )}}{{V_{p}^{2} - V_{S}^{2} }},$$
(1)
$$\upsilon = \frac{{V_{p}^{2} - 2V_{S}^{2} }}{{2(V_{p}^{2} - V_{S}^{2} )}},$$
(2)

where \(\rho\) is the mean density, VS is the S-wave velocity, and VP is the P-wave velocity.

The four lateral boundaries of the model are fixed in the normal direction and free in the tangential direction. The bottom boundary is free to move horizontally and fixed vertically, and the upper surface of the model is free.

2.2 Co-seismic Slip Model of Earthquakes

After the Wenchuan earthquake, researchers have used teleseismic waveforms, high-resolution GPS observations, and other data to invert the co-seismic slip distribution of the earthquake. For example, Ji (2008) derived a single fault plane (strike: 229°; dip: 33°) with a length of 298 km and a width of 20 km through the analysis of Global Seismic Network broadband waveforms (Fig. 3a). Wang et al. (2008) constructed a double-listric finite fault model and reconstructed the source rupture process by combined inverting the teleseismic waveforms and near-field co-seismic displacements (Fig. 3b). Wang et al. (2011) inverted a refined co-seismic slip model (Fig. 3c) with a 60 km sub-horizontal décollement fault beneath the LMSF based on GPS, InSAR, and leveling data. Fielding et al. (2013) integrated teleseismic waveform data, geodetic data (GPS), and remote sensing data (InSAR and SAR) to invert the co-seismic slip model, and produced a five-stage model of the Wenchuan earthquake (Fig. 3d). These models share a common feature of showing two maximum slip regions beneath the Beichuan and Yingxiu cities, which were heavily damaged. However, due to variations in fault geometry, observation data, and inversion strategies, there are differences in the details of these models. Thus, this study utilizes these four co-seismic slip models (Fig. 3) to calculate the co-seismic deformation of the Wenchuan earthquake.

Fig. 3
figure 3

Co-seismic slip distributions of the rupture models for the Wenchuan earthquake. a–d Slip models developed by Ji (2008), Wang et al. (2008), Wang et al. (2011), and Fielding et al. (2013), respectively

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In contrast to the Wenchuan earthquake, the influence of different rupture models of the Lushan earthquake is not significant. Therefore, we have employed the co-seismic slip model of the Lushan earthquake as proposed by Liu et al. (2013) in this study.

We then incorporated the rupture geometries and the co-seismic slip distribution of the Wenchuan and Lushan earthquakes (Fig. 3) into the finite element model (Fig. 2) with the split-node technique (Melosh & Raefsky, 1981).

2.3 Constitutive Equations

The constitutive equation of Maxwell viscoelasticity can be described as follows:

$$\dot{\varepsilon } = D^{ - 1} \dot{\sigma } + S\sigma ,$$
(3)

where \(\dot{\varepsilon } = \left[ {\dot{\varepsilon }_{xx} ,\dot{\varepsilon }_{yy} ,\dot{\varepsilon }_{zz} ,\dot{\gamma }_{yz} ,\dot{\gamma }_{xz} ,\dot{\gamma }_{xy} } \right]^{T}\), \(\sigma = \left[ {\sigma_{xx} ,\sigma_{yy} ,\sigma_{zz} ,\sigma_{yz} ,\sigma_{xz} ,\sigma_{xy} } \right]^{T}\). \(D^{ - 1}\), and \(S\) are the material matrices related to elastic and viscous deformation, respectively.

$${\text{D}}^{ - 1} = \left[ {\begin{array}{*{20}c} \frac{1}{E} & { - \frac{\nu }{E}} & { - \frac{\nu }{E}} & 0 & 0 & 0 \\ { - \frac{\nu }{E}} & \frac{1}{E} & { - \frac{\nu }{E}} & 0 & 0 & 0 \\ { - \frac{\nu }{E}} & { - \frac{\nu }{E}} & \frac{1}{E} & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{2(1 + \nu )}{E}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{2(1 + \nu )}{E}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {\frac{2(1 + \nu )}{E}} \\ \end{array} } \right],$$
(4)
$$S = \frac{1}{\eta }\left[ {\begin{array}{*{20}c} \frac{1}{3} & { - \frac{1}{6}} & { - \frac{1}{6}} & 0 & 0 & 0 \\ { - \frac{1}{6}} & \frac{1}{3} & { - \frac{1}{6}} & 0 & 0 & 0 \\ { - \frac{1}{6}} & { - \frac{1}{6}} & \frac{1}{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right],$$
(5)

where \(E\), \(\nu\), and \(\eta\) represent Young’s modulus, Poisson’s ratio, and viscosity, respectively.

To discretize the above constitutive equation, we have utilized the derivative formula to transform it into a differential form under the assumption of small time steps:

$$\dot{\sigma }^{t} = \frac{{\Delta \sigma^{t} }}{\Delta t} = \frac{{\sigma^{t} - \sigma^{t - \Delta t} }}{\Delta t},$$
(6)
$$\dot{\varepsilon }^{t} = \frac{{\Delta \varepsilon^{t} }}{\Delta t} = \frac{{\varepsilon^{t} - \varepsilon^{t - \Delta t} }}{\Delta t},$$
(7)

Combining Eqs. (3), (6), and (7), the viscoelastic constitutive relation can be expressed as:

$$\sigma^{t} = \tilde{D}\varepsilon^{t} + \tilde{\sigma },$$
(8)

where \(\tilde{D} = (D^{ - 1} + S\Delta t)^{ - 1}\) and \(\tilde{\sigma } = (D^{ - 1} + S\Delta t)^{ - 1} D^{ - 1} \sigma^{t - \Delta t} - (D^{ - 1} + S\Delta t)^{ - 1} \varepsilon^{t - \Delta t}\).

Based on the aforementioned theory, we independently developed the 3D viscoelastic finite element code to simulate co-seismic and post-seismic deformation based on the Abaqus secondary development platform. The reliability of this code has been verified (Gong et al., 2023; Sun et al., 2021).

3 Results

We calculate Coulomb stress changes caused by the Wenchuan and Lushan earthquakes and analyze stress transfer between major earthquakes. Then we calculate the spatial distribution of b-values based on the regional instrumental seismic catalog before the Wenchuan earthquake. Finally, we analyze regional seismic hazards by combining the stress transfer between major earthquakes and the spatial distribution of b-values.

3.1 Co-seismic Displacements Caused by Wenchuan Earthquake

The computation of co-seismic and post-seismic deformation is significantly influenced by the slip model of large earthquakes. Here, we employ four slip models of the Wenchuan earthquake (Fig. 3) to calculate the co-seismic surface displacements. To assess the accuracy of each model, we compare the simulated results from each model with GPS observations. The model that exhibits the closest match to the GPS observations is considered the most reliable and provides the best estimate of the actual co-seismic and post-seismic deformation.

The results demonstrate that modeled surface displacements generated by four slip models of the Wenchuan earthquake generally agree well with GPS observations in the far field (Fig. 4a–d). However, there are varying degrees of deviation near the rupture zone of the Wenchuan earthquake (Fig. 4). The difference may be attributed to various factors, including the complex deep structure of the earth and seismic rupture processes, as well as the relatively simplified model setting.

Fig. 4
figure 4

Comparison of co-seismic surface displacements caused by the Wenchuan earthquake between model results (red arrows) and GPS observations (blue arrows) (Wang et al., 2011). a–d Comparison between GPS observations and the model results using the co-seismic slip model from Ji (2008), Wang et al. (2008), Wang et al. (2011), and Fielding et al. (2013), respectively. e–h Residual displacements between model results of the corresponding models and GPS observations. Insets in e–f show the corresponding distribution of residual displacements (both northward displacements and eastward displacements)

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In order to further assess the differences between model results generated by the four slip models and GPS observations, we calculate the difference between model results and GPS observations. Figure 4e–h show the residual displacements (model displacements minus GPS displacements) for the four models. We also quantitatively count the residual of northward and eastward displacements between four model results and GPS observations (insets in Fig. 4e–h). The results indicate that the co-seismic slip model of the Wenchuan earthquake developed by Fielding et al. (2013) provides the best fit for horizontal surface displacements, with more than 80% samples of eastward (red line in subfigures of Fig. 4e–h illustrations) and northward (green line in subfigures of Fig. 4e–h illustrations) deviations stay within ±3 cm. Therefore, we then conducted a follow-up study based on the co-seismic slip model developed by Fielding et al. (2013).

3.2 Rheology of Songpan-Ganzi Block Derived from the Viscoelastic Model

Due to the viscoelasticity of the middle-lower crust and upper mantle, crustal deformation continues following a large earthquake. Previous studies have deployed a series of GPS stations to monitor the post-seismic deformation (Diao et al., 2018; Wang et al., 2021, 2022), and have used these data to constrain the viscosities of the middle-lower crust and upper mantle (Diao et al., 2018; Hu et al., 2016; Shao et al., 2011; Wang et al., 2021).

Given the sparsity of GPS stations in the South China Block and the Sichuan-Yunnan Block in the model region (Fig. 6) (Wang et al., 2021), we determined the viscosities of their middle-lower crust and upper mantle based on previous studies (Shi & Cao, 2008; Sun et al., 2013). Specifically, we set the value of 1 × 1019 Pa·s and 1 × 1020 Pa·s for the middle-lower crust, of 1 × 1021 Pa·s, and 1 × 1022 Pa·s for the upper mantle in the Sichuan-Yunnan Block and the South China Block, respectively.

We then use the Maxwell viscoelastic model to simulate the time-dependent displacement and constrain the viscosities of the middle-lower crust and upper mantle in Songpan-Ganzi Block based on the post-seismic deformation data for nine years after the Wenchuan earthquake (Wang et al., 2021). We test the viscosities range from 1 × 1018 to 1 × 1020 Pa s for the middle-lower crust and from 1 × 1019 to 1 × 1021 Pa s for the upper mantle. Using a grid-search approach, we carry out a series of cases to find the optimal viscosities that fit the GPS observations best. The misfit degree of model results to GPS observations is evaluated by using the formula (Chai & Draxler, 2014):

$${\text{RMSE}} = \sum\limits_{i = 1}^{N} {\sqrt {\left[ {\frac{{(x_{i} - x_{i}^{GPS} )^{2} }}{N}} \right]} } ,$$
(9)

where \({\text{RMSE}}\) is the root-mean-square error between GPS observations and model results, \(N\) is the number of stations, \(x_{i}^{GPS}\) and \(x_{i}\) are the GPS observations and model results at the particular station i, respectively.

Initially, we used the horizontal displacements of GPS data only to constrain the viscosities of the middle-lower crust and upper mantle in the Songpan-Ganzi Block. The test model indicates that when the viscosities for the middle-lower crust and upper mantle in the Songpan-Ganzi Block are 5.62 × 1018 Pa·s and 5.89 × 1020 Pa·s, and the RMSE is the minimum. However, there are still some discrepancies in the near-field stations if only horizontal displacements are considered (Hu et al., 2021).

Thus, we then constrain the viscosities of the middle-lower crust and upper mantle in the Songpan-Ganzi Block using both the horizontal and vertical displacements of GPS data. Figure 5 shows the data misfit as a function of viscosities for the middle-lower crust and upper mantle in the Songpan-Ganzi Block. The results show that the optimal viscosities for the middle-lower crust and upper mantle in the Songpan-Ganzi Block are 6.61 × 1018 Pa·s and 1.12 × 1020 Pa·s, respectively.

Fig. 5
figure 5

Data misfit map as function of viscosities of the Songpan-Ganzi Block. The red star represents the optimal fitting point

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The results show that the optimal viscosities for the middle-lower crust and upper mantle in the Songpan-Ganzi Block did not change much, and were within the same order of magnitude, between considering only horizontal displacements and considering both horizontal and vertical displacements. This consistency may be attributed to the relatively small magnitude and higher uncertainty associated with vertical displacements. The result is also consistent with previous studies (Hu et al., 2021; Zhang et al., 2021).

Figure 6 shows the comparison between the first 9-year surface displacements after the Wenchuan earthquake observed by GPS and that predicted in the model using the optimal viscosities (6.61 × 1018 Pa·s and 1.12 × 1020 Pa·s) of the middle-lower crust and upper mantle in the Songpan-Ganzi Block. The results show that the model can explain the observed post-seismic deformation well (both horizontal and vertical displacements). The results show both a continuous compression of the Songpan-Ganzi Block toward the South China Block overall, and an uplift of the Songpan-Ganzi Block.

Fig. 6
figure 6

Comparison of post-seismic surface displacements 9 years after the Wenchuan earthquake between model results and GPS observations. a Comparison of post-seismic horizontal displacements. b Comparison of post-seismic vertical displacements. The red and black arrows show the model results and GPS observations (Wang et al., 2021), respectively

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3.3 Coulomb Stress Changes Caused by Wenchuan and Lushan Earthquakes

The 2008 Mw7.9 Wenchuan earthquake and 2013 Mw6.6 Lushan earthquake occurred on the LMSF, which has attracted considerable attention (Guo et al., 2020; Liang et al., 2018; Liu et al., 2018; Parsons et al., 2008; Toda et al., 2008; Xu et al., 2013, 2020; Zhang et al., 2010). These two earthquakes did not entirely rupture the LMSF and there are two distinct seismic gaps on the LMSF (Dayi seismic gap and Tianquan-Kangding seismic gap) (Liang et al., 2018; Liu et al., 2020). To analyze the effects of the Wenchuan and Lushan earthquakes on these seismic gaps and other regions, we calculate the co-seismic and post-seismic Coulomb stress changes caused by the Wenchuan and Lushan earthquakes. Coulomb stress changes (\(\Delta {\text{CFS}}\)) is defined as (King et al., 1994):

$$\Delta {\text{CFS}} = \Delta \tau + \mu ^{\prime}\Delta \sigma_{n} ,$$
(10)

where \(\Delta \tau\) is the shear stress on the fault plane, \(\Delta \sigma_{n}\) is the normal stress (tensile stress is positive) on the fault plane, and \(\mu ^{\prime}\) is the effective friction coefficient (\(\mu ^{\prime} = 0.4\) in this study). \(\Delta CFS{\kern 1pt} {\kern 1pt} > {\kern 1pt} {\kern 1pt} 0\) indicates that the fault plane tends to move toward the failure and the seismic hazard increases, while \(\Delta CFS{\kern 1pt} {\kern 1pt} < 0\) indicates that the fault plane tends to move away from the failure and the seismic hazard decreases (King et al., 1994; Stein, 1999).

Figure 7 illustrates the co-seismic Coulomb stress changes caused by the Wenchuan earthquake in four slip models (Fig. 3), which are projected to the plane (strike, 205°; dip, 33°; rake, 142°) of the Yingxiu-Beichuan fault (main seismogenic fault of the Wenchuan earthquake) (Shen et al., 2009; Zhou et al., 2006). The distribution of Coulomb stress changes exhibits differences among the different models, but also shows a certain degree of consistency. We further compared the correspondence between co-seismic Coulomb stress changes in different models and aftershocks of the Wenchuan earthquake. The spatial distribution of positive Coulomb stress changes calculated by the model proposed by Fielding et al. (2013) corresponds well with the aftershocks (Fig. 7d). These findings suggest that the model proposed by Fielding et al. (2013) provides a better explanation of the influence of co-seismic Coulomb stress changes on aftershocks. Therefore, it is reasonable for us to use the model of Fielding et al. (2013) to analyze Coulomb stress changes.

Fig. 7
figure 7

Co-seismic Coulomb stress changes caused by the Wenchuan earthquake calculated from different co-seismic slip models. a–d Results from the models of Ji (2008), Wang et al. (2008), Wang et al. (2011), and Fielding et al. (2013), respectively. The gray circles show the aftershocks of the Wenchuan earthquake

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Figure 8a illustrates that the Wenchuan earthquake resulted in significant stress adjustments in the surrounding area, especially in the southern and northern LMSF (+ 0.005 ~  + 1.175 MPa).

Fig. 8
figure 8

Coulomb stress changes caused by the Wenchuan and Lushan earthquakes. a Co-seismic Coulomb stress of the Wenchuan earthquake calculated using the model of Fielding et al. (2013). b Coulomb stress (co-seismic and post-seismic) generated by the Wenchuan earthquake prior to the Lushan earthquake. c Co-seismic Coulomb stress changes of Lushan earthquake. d Coulomb stress changes (co-seismic and post-seismic) caused by the Wenchuan and Lushan earthquakes (10 years after the Lushan earthquake)

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We also calculate the post-seismic Coulomb stress changes caused by the Wenchuan earthquake. The results (Fig. 8b), just before the 2013 Mw6.6 Lushan earthquake, show nearly the same spatial pattern with co-seismic Coulomb stress changes caused by the Wenchuan earthquake (Fig. 8a). The subsequent 2013 Mw6.6 Lushan earthquake occurred on the southern LMSF, where the Coulomb stress changes caused by the Wenchuan earthquake is nearly + 0.068 ~  + 0.144 MPa (co-seismic Coulomb stress changes, + 0.063 ~  + 0.132 MPa; post-seismic Coulomb stress changes, + 0.005 ~  + 0.012 MPa). It appears that the value has surpassed the triggering threshold of 0.01 MPa (Freed, 2005; King et al., 1994), implying a high probability that the Wenchuan earthquake triggered the Lushan earthquake.

The results of co-seismic Coulomb stress changes caused by the Lushan earthquake are relatively small, and its impact was limited (Fig. 8c).

We found that stress states in the Dayi and Tianquan-Kangding seismic gaps are significantly influenced by the Wenchuan and Lushan earthquakes (Fig. 8d). Though there are negative co-seismic Coulomb stress changes caused by the Lushan earthquake observed in certain parts of these two seismic gaps, the value is relatively small (Fig. 8c). Coulomb stress changes caused by the Wenchuan and Lushan earthquakes are + 0.216 ~  + 2.607 MPa and + 0.021 ~  + 0.211 MPa in Dayi seismic gap and Tianquan-Kangding seismic gap, respectively (Fig. 8d).

To further investigate the effects of the Wenchuan and Lushan earthquakes on the regional fault system and analyze regional seismic hazards, we calculate the cumulative Coulomb stress changes (including both co-seismic and post-seismic Coulomb stress changes caused by the Wenchuan and Lushan earthquakes) projected onto each fault plane on the regional fault system (Fig. 9). Detailed fault parameters are shown in Table 2 (Clarence et al., 1991; He & Chéry, 2008; He et al., 2008; Zhou et al., 1999), with median value selected for the floating parameters. Our results demonstrate that Coulomb stress changes caused by the Wenchuan and Lushan earthquakes have significant effects on regional fault system (Fig. 9). In general, Coulomb stress changes increase predominantly in the southern LMSF, southern XSHF (encompassing the Kangding-Shimian segment and Daofu-Kangding segment), XJHF, central LQSF, western QCF, HN-QSWF, western DB-BLJF, northern MJF, DKLF, and northern LRBF (Fig. 9). Across most of above fault segments, Coulomb stress changes exceed 0.01 MPa, indicating a heightened risk of seismic hazards in these regions. Coulomb stress changes decrease in the northern XSHF, northern LMSF, ANHF, DLSF, southern MJF, southern LRBF, FBHF, ABF, eastern DB-BLJF, and HYF (Fig. 9).

Fig. 9
figure 9

Cumulative Coulomb stress changes caused by the Wenchuan and Lushan earthquakes on major faults in the eastern Tibetan Plateau. Labels are explained in Fig. 1

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Table 2 Parameters of main active faults in the study area
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We note that several earthquakes in the vicinity (2014 Mw6.1 Kangding earthquake, 2017 Mw6.5 Jiuzhaigou earthquake, and 2022 Mw6.6 Luding earthquake) occurred in regions with positive Coulomb stress changes (Fig. 9) caused by the Wenchuan and Lushan earthquakes. These findings provide evidence of triggering between these earthquakes.

While it is widely accepted that large earthquakes can trigger subsequent earthquakes in the surrounding area through positive Coulomb stress changes (King et al., 1994; Parsons et al., 2008; Stein, 1999), it should be noted that these changes represent second-order perturbations of stress on nearby faults (King et al., 1994; Tormann et al., 2015). Many studies have also found that some subsequent earthquakes occur in areas with negative Coulomb stress changes caused by preceding earthquakes (Gkarlaouni et al., 2008; Luo & Liu, 2018). In fact, Coulomb stress changes triggered by earthquakes on surrounding faults is a relatively small perturbation (Luo & Liu, 2018; Sun et al., 2020), compared to the background stress already accumulated on those faults. The background stress plays a critical role in determining whether an earthquake will occur. Thus, to better understand regional seismic hazards, it is crucial to conduct a comprehensive analysis of total stress evolution on faults.

3.4 Spatial Distribution of b-Values in the Eastern Tibetan Plateau

The magnitude–frequency relationship (also called the G–R relationship) is one of the most important statistical relationships for describing seismic activity (Gutenberg & Richter, 1944). The relationship between magnitude and frequency can be expressed as follows:

$$LgN = a - bM,$$
(11)

where M is the magnitude of the earthquake (M ≥ Mc), Mc is the minimum complete magnitude, and N is the cumulative frequency of earthquakes. The parameter \(a\) is the constant reflecting the regional seismicity, and the parameter b is the relative proportion of large and small earthquakes in a region.

The b-value has attracted considerable attention (Kun et al., 2013; Liu & Pei, 2017; Main et al., 1992; Scholz, 2015; Schorlemmer et al., 2005). Rock fracture experiments (Main et al., 1992; Scholz, 1968), numerical simulation (Gao et al., 2020), and statistical results of regional seismicity (Kun et al., 2013; Schorlemmer et al., 2005; Zhang & Zhou, 2016) all indicate that the b-value has a clear physical meaning. A lower b-value indicates a higher stress level in a region, while a higher b-value indicates a lower stress level in a region. Thus, we then calculate the regional b-value to further estimate the stress level in the eastern Tibetan Plateau before the 2008 Mw7.9 Wenchuan earthquake.

The seismic catalog from 1970 to 2008 used in this study is provided by the China Earthquake Network Center. Figure 10 shows the statistical results of seismic activity over time based on the regional catalog. The results show that the cumulative and annual frequency of earthquakes is roughly bounded by the year 2000 (red dashed line in Fig. 10). Specifically, the cumulative rate and annual frequency of earthquakes prior to 2000 is notably lower than that observed after 2000. We note that the Regional Telemetered Digital Seismograph Network of China underwent its second update around the year 2000 (Liu et al., 2003), resulting in a significant improvement in its monitoring capabilities. It is generally considered that the seismic catalog after the year 2000 is more complete. Therefore, we selected the catalog from 2000 to 2008 for our calculations.

Fig. 10
figure 10

Statistics of instrumental earthquakes in the eastern Tibetan Plateau. a and b Cumulative and annual frequency of earthquakes from 1970 to 2008, respectively

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Figure 11 shows the magnitude–frequency distribution based on the seismic catalog from 2000 to 2008. We use the maximum curvature method (Wiemer & Wyss, 2000) to calculate the minimum completed magnitude \(M_{c}\) and the maximum likelihood method (Aki, 1965) to calculate the regional b-values. The results show that \(M_{c}\) = 1.8 and b = 1.03 ± 0.01. The b-value is also approximately similar to the global average of 1.0 (El-Isa & Eaton, 2014).

Fig. 11
figure 11

Earthquake magnitude–frequency distribution. The non-cumulative and cumulative magnitude–frequency distributions are represented by the red triangles and black squares, respectively

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To further analyze the spatial distribution of regional b-values, the study area is subdivided into 0.1° × 0.1° grids. We calculate the Mc and b-value at each grid node with an initial statistical radius R = 20 km. We prescribed the number of earthquakes N ≥ 50 within the statistical radius and the magnitude span Mspan ≥ 2.5 (Kutliroff, 2017). If the prescribed number of earthquakes and magnitude span were not satisfied, we increase the statistical radius by 10 km. However, we also impose a condition that the statistical radius should not exceed 60 km. If seismic data within a grid node do not satisfy the preset conditions, Mc and b-values in this grid node were not calculated.

Figure 12a shows the spatial distribution of Mc. The results show that Mc across the eastern Tibetan Plateau generally ranges from 1.2  to  2.0. Figure 12b shows the spatial distribution of the b-values, with notably lower values found near the faults, including the middle segment of LMSF, XSHF, XJHF, FBHF, HN-QSWF, and DKLF. It is noteworthy that the study area has recently experienced several large earthquakes, including the 2008 Mw7.9 Wenchuan, 2013 Mw6.6 Lushan, 2014 Mw6.1 Kangding, 2017 Mw6.5 Jiuzhaigou, and 2022 Mw6.6 Luding earthquakes, all of which occurred in regions with low b-values. Since the b value is negatively correlated with stress (Kun et al., 2013; Main et al., 1992; Nuannin et al., 2012; Scholz, 1968, 2015; Schorlemmer et al., 2004, 2005), high stress has already accumulated in these regions prior to these earthquakes. Thus, the b-value may be an important indicator of regional seismic hazards.

Fig. 12
figure 12

Spatial distribution of Mc (a) and b-values (b) in the eastern Tibetan Plateau before the Wenchuan earthquake

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4 Discussion

In this study, we set up a 3D viscoelastic finite element model in the eastern Tibetan Plateau, and calculate Coulomb stress changes caused by 2008 Mw7.9 Wenchuan and 2013 Mw6.6 Lushan earthquakes. The computation of Coulomb stress changes is significantly influenced by the slip model of large earthquakes and viscosities of the middle-lower crust and upper mantle. Thus, we analyze four slip models of the Wenchuan earthquake and choose the viscoelastic model that produced the closest match to the GPS data. Based on the chosen slip model of the Wenchuan earthquake, we also invert the viscosities of the middle-lower crust and upper mantle for the Songpan-Ganzi Block.

The results show that Coulomb stress changes caused by the Wenchuan earthquake near the epicenter of 2013 Mw6.6 Lushan earthquake is positive (+ 0.063 ~  + 0.132 MPa and + 0.005 ~  + 0.012 MPa for co-seismic and post-seismic Coulomb stress changes, respectively) (Fig. 8b), and the total value exceeds the threshold of 0.01 MPa (Freed, 2005; King et al., 1994). This suggests that the Wenchuan earthquake indeed triggered the occurrence of the Lushan earthquake, which is generally consistent with previous studies (Guo et al., 2020; Nalbant & McCloskey, 2011; Parsons et al., 2008; Toda et al., 2008).

Coulomb stress evolutions caused by the Wenchuan and Lushan earthquakes have a significant impact on regional fault system (Fig. 9). Overall, Coulomb stress changes increase mainly in the southern LMSF, southern XSHF (encompassing the Kangding-Shimian segment and Daofu-Kangding segment), XJHF, central LQSF, western QCF, HN-QSWF, western DB-BLJF, northern MJF, DKLF, and northern LRBF. We found that recent earthquakes (2014 Mw6.1 Kangding earthquake, 2017 Mw6.5 Jiuzhaigou earthquake, and 2022 Mw6.6 Luding earthquake) occurred in regions where Coulomb stress changes caused by the Wenchuan and Lushan earthquakes are positive. However, there are still many regions with highly positive Coulomb stress changes that have not experienced major earthquakes yet (Fig. 9), and these regions are deemed high potential for seismic hazards in the future. We also noticed that Coulomb stress changes between the Dayi and Tianquan-Kangding seismic gaps are far greater than 0.01 MPa (Fig. 9).

Compared with background stress already accumulated on faults, Coulomb stress changes caused by earthquakes is a relatively small perturbation (Luo & Liu, 2018; Sun et al., 2020). However, when the background stress (stress already accumulated on faults) is relatively high and in the critical state of failure on faults, the increased Coulomb stress changes may trigger the subsequent earthquakes.

Previous studies have indicated that seismic b-values decrease linearly with increasing stress (Kun et al., 2013; Main et al., 1992; Nuannin et al., 2012; Scholz, 1968; Schorlemmer et al., 2004). To better understand the background stress state in the eastern Tibetan Plateau, we then analyze the spatial distribution of b-values using the regional catalog from 2000 to 2008. The findings suggest that numerous earthquakes, including the 2008 Mw7.9 Wenchuan earthquake, 2013 Mw6.6 Lushan earthquake, 2014 Mw6.1 Kangding earthquake, 2017 Mw6.5 Jiuzhaigou earthquake, and 2022 Mw6.6 Luding earthquake, occurred in areas with low b-values (Fig. 12), which is consistent with other studies (Zhang & Zhou, 2016; Zhao et al., 2019). This implies that these regions already have a high background stress level, increases of Coulomb stress changes (Fig. 9) could trigger subsequent earthquakes. Therefore, an assessment of seismic hazards utilizing both Coulomb stress changes and seismic b-values proves a more comprehensive approach for regional seismic analysis.

By integrating the Coulomb stress changes resulting from earthquakes and the b-value distribution, we identified four fault segments (southern XSHF, northern XJHF, DKLF, and HN-QSWF) with both low b-values and positive Coulomb stress changes (black dashed circles in Fig. 12b). Among these areas, only two have witnessed subsequent large earthquakes, namely the 2014 Mw6.1 Kangding earthquake, 2017 Mw6.5 Jiuzhaigou earthquake, and 2022 Mw6.6 Luding earthquake. In contrast, no major earthquakes have occurred in other regions yet, indicating an increased potential for seismic hazards in the future.

Our finite element model used in the calculations is relatively simplified and did not take into account historical earthquakes, the dynamic propagation of seismic waves, and tectonic loading. Therefore, future research could incorporate these factors to provide more comprehensive references for regional seismic hazard analysis.

5 Conclusion

In this study, we constructed a 3D viscoelastic finite element model to calculate Coulomb stress changes caused by the Wenchuan and Lushan earthquakes. We also calculated the spatial distribution of regional b-values. We then combined the stress transfer between earthquakes with the spatial distribution of b-values to assess seismic hazards in the area. Our main conclusions from this study are as follows:

  • 1. Constrained by 9-year post-seismic deformation due to the 2008 Mw7.9 Wenchuan earthquake, we estimated the optimal viscosities for the middle-lower crust and upper mantle in the Songpan-Ganzi Block as 6.61×1018 Pa·s and 1.12×1020 Pa·s, respectively.

  • 2. Regional earthquakes (including the 2008 Mw7.9 Wenchuan earthquake, 2013 Mw6.6 Lushan earthquake, 2014 Mw6.1 Kangding earthquake, 2017 Mw6.5 Jiuzhaigou earthquake, and 2022 Mw6.6 Luding earthquake) occurred in regions with low b-values. Meanwhile, subsequent earthquakes occurred in regions where Coulomb stress changes caused by the Wenchuan and Lushan earthquakes were positive. This suggests that regions with both low b-values and positive Coulomb stress changes may pose higher seismic hazards.

  • 3. After the Wenchuan and Lushan earthquakes, there are four regions (southern Xianshuihe fault, Dongkunlun fault, northern Xiaojinhe fault, and Hanan-Qingshanwan fault) with both highly positive Coulomb stress changes and low b-values, which may indicate high-stress accumulation and high seismic hazard in the future.

  • 4. Coulomb stress changes caused by the Wenchuan and Lushan earthquakes increased significantly in the Dayi seismic gap (+0.216 ~ +2.607 MP), and Tianquan-Kangding seismic gap (+0.021 ~ +0.211 MPa), while the result of high b-values for both seismic gap indicates relatively low accumulation of background stress. However, with continued tectonic loading, seismic hazards on both seismic gaps should cause our attention.