Investigation of topographical effects on rupture dynamics and resultant ground motions

Investigation of topographical effects on rupture dynamics and resultant ground motions Abstract In this work, we investigate the effect of irregular topography on the dynamic rupture and resultant ground motions using the curved grid finite-difference method. The research is based on spontaneous dynamic rupture on vertical strike-slip faults by varying the shapes and relative locations of irregular topography to the critical supershear transition distance. The results show that seismic energy of a supershear earthquake can be transmitted farther with large amplitudes. However, its ground motion near the fault is weaker than that caused by a subshear (namely the sub-Rayleigh) rupture. Whether the irregular topography exhibits stronger ground motion overall depends on the irregular topography's ability to prevent the subshear-to-supershear transition. Finally, we also discuss the effects of the strength parameter S and a larger size of the irregular topography on the resultant ground motion. The modellings of San Andreas Fault with real and inverted topographical surfaces show the implications of the topographical effects from the real earthquake. Numerical solutions, Earthquake dynamics, Wave propagation, Wave scattering and diffraction, Dynamics and mechanics of faulting 1 INTRODUCTION The Earth's free surface acts as an important boundary in numerical simulation. The rupture with the flat surface has been investigated by many authors (e.g. Oglesby et al.2000a,b; Chen & Zhang 2006; Zhang & Chen 2006a,b; Kaneko & Lapusta 2010; Xu et al.2015). With numerical simulations, Kaneko & Lapusta (2010) pointed out that the free-surface-induced supershear rupture due to the phase conversion at the free surface. If a strike-slip rupture on a vertical plane in a homogeneous half-space grows continuously, a supershear rupture will always occur near the free surface (Xu et al.2015). Previous works treat the Earth's free surface as flat to simplify the numerical simulations. However, the Earth's surface is not always flat. Some numerical simulations have successfully taken the irregular topography into consideration and have shown that different rupture phases can develop (Ely et al.2010; Zhang et al.2016). The rupture speed of an earthquake is an important parameter that results in high frequencies and strong seismic radiation, particularly the transition from subshear rupture to supershear rupture (Madariaga 1983; Bernard & Madariaga 1984; Spudich & Cranswick 1984; Bizzarri & Spudich 2008; Bizzarri et al.2010). For the supershear rupture, the radiated S-wave can constructively form a Mach front that transports large seismic energy at farther distance away from the fault (Bernard & Baumon 2005; Dunham & Archuleta 2005). Zhang et al. (2016) showed that irregular topography can disrupt the critical conditions including the phase conversion and the rupture slip in some cases. However, they did not extend their research to the effect of irregular topography on resultant ground motions. Since the topographical surfaces can influence the rupture dynamics, the distribution of resultant ground motion should also be affected. In this work, we aim to study the effect of irregular topography on resultant seismic wave radiation. To investigate the effect of irregular topography on resultant near fault ground motion based on dynamic rupture models, we have done numerous simulations with different epicentral distances of the topographical irregularities. Then, we also discuss the initial shear stress and size of irregular topography on the distribution of resultant ground motions. At last, a real example of San Andreas Fault with different topographical surfaces is modelled and to discuss the topographical effects on rupture dynamics and ground motion. 2 METHODS AND MODEL SETTING In this work, we adopt the method of Curved Grid Finite Difference Rupture Dynamics Modeling (CGFD-RDM) developed by Zhang et al. (2014) to investigate the effect of irregular surface topography on resultant ground motion. We investigate free-surface topographical effects on rupture dynamics and resultant ground motions caused by that result from a dynamic strike-slip rupture on a vertical planar fault in a homogeneous half-space. All the simulations are divided into four groups (1–4) depending on the epicentre distances (7.5, 12.5, 17.5 and 22.5 km) of the topographic perturbations. Fig. 1 illustrates geometrical model discussed in this work. The grey rectangle indicates the nucleation area with a dimension of 3 km × 3 km, whose centre is 7.5 km from the left, bottom and free surface boundaries. The corresponding cases for the hill surfaces are not shown in Fig. 1 for clarity. The topographic surfaces are mathematically described by following Gaussian function Figure 1. View largeDownload slide Fault geometries of simulation models. The basic vertical fault extends 62 km and 15 km along the fault strike and dip directions, respectively. A small grey rectangle indicates a nucleation area with a dimension of 3 km × 3 km. The nucleation patch is the same in all comparison simulations. Only canyon-shaped topographic surfaces for Models 1 to 4 are shown, the corresponding hill surfaces are not shown for clarity Figure 1. View largeDownload slide Fault geometries of simulation models. The basic vertical fault extends 62 km and 15 km along the fault strike and dip directions, respectively. A small grey rectangle indicates a nucleation area with a dimension of 3 km × 3 km. The nucleation patch is the same in all comparison simulations. Only canyon-shaped topographic surfaces for Models 1 to 4 are shown, the corresponding hill surfaces are not shown for clarity  \begin{equation}z\ \left( r \right) = \ \pm 1000{\rm{exp}}\left( { - {r^2}/{{1500}^2}} \right), \end{equation} (1)where r is the horizontal distance in metres to the centre of the hill- (‘+’ sign) or canyon- (‘−’ sign) shaped topography. Note that the irregular topography is symmetrically distributed along the fault trace for all simulations in this work. In our simulations, the rupture criterion of faulting is governed by a linear slip-weakening friction law (Ida 1972) prescribed by   \begin{equation}{\mu _f} (l) = \left\{ \begin{array}{l@{\quad}c} {\mu _s} - l({\mu _s} - {\mu _d})/{d_0}, & l < {d_0}\\ {\mu _d}, & l \geq {d_0}\end{array} \right. \end{equation} (2) where l is the simulated slip and μs and μd represent the static and dynamic friction coefficients, respectively, d0 is the critical slip-weakening distance and set to be 0.4 m for all cases in this study. Moreover, the initial stress in the background medium is set to be homogeneous, with 120 MPa and 70.0 MPa as the normal and shear tractions. We apply a large initial shear stress (81.6 MPa) exceeding the failure strength within the nucleation patch to trigger the dynamic rupture. The subsequent rupture spontaneously propagates over the entire fault plane until it is stopped by surrounding artificial barriers with sufficiently high strength. The detailed parameters implemented in the simulations are listed in Table 1. Table 1. Parameters are used for simulations. Dynamic parameters  Value  Initial shear stress, τ0 (MPa)  70.0  Initial normal stress, −σn (Mpa)  120.0  Static friction coefficient, μs  0.677  Dynamic friction coefficient,  μd  0.525  Critical slip distance, d0 (m)  0.40  Media parameters  Value  P-wave velocity, Vp (m s−1)  6000  S-wave velocity, Vs (m s−1)  3464  Density, ρ (kg m − 3)  2670  Dynamic parameters  Value  Initial shear stress, τ0 (MPa)  70.0  Initial normal stress, −σn (Mpa)  120.0  Static friction coefficient, μs  0.677  Dynamic friction coefficient,  μd  0.525  Critical slip distance, d0 (m)  0.40  Media parameters  Value  P-wave velocity, Vp (m s−1)  6000  S-wave velocity, Vs (m s−1)  3464  Density, ρ (kg m − 3)  2670  View Large 3 RESULTS Previous studies have shown that the rupture dynamic on the strike-slip fault in half-space can generate a transition from subshear to supershear rupture at the free surface (Chen & Zhang 2006; Zhang & Chen 2006a,b; Xu et al.2015). However, such strong influence of free surface on rupturing doesn’t always accelerate the rupture speed and becomes complicated when the geometry of free surface is not flat. Zhang et al. (2016) have shown that the critical condition of supershear rupture transition near the free surface may be disrupted when the irregular topography is symmetrically distributed along the fault trace in some scenario cases. This is because that the topographical surfaces can influence the phase conversion and the rupture slip. In this study, we mainly focus on the features of seismic waves and resultant ground motion radiated from the complex rupturing controlled by various topographic surfaces. We carry out four groups (1–4) of simulations corresponding to epicentral distances (7.5, 12.5, 17.5, and 22.5 km) to the irregular topography. The parameters used in the simulation are the same except for the shapes of free surfaces. Each group has either a hill (H) or canyon (C) topographic feature. A simulation is labelled with H or C followed by a digit corresponding to a group 1–4. In the flat surface model, a subshear-to-supershear transition occurs at an epicentral distance of 7.5 km. 3.1 Rupture dynamics Fig. 2 illustrates the rupture time contours on fault planes for all the cases with different shapes and locations of topography. The conditions are the same for all simulations, except the topographical surfaces. For the reference model, the original subshear rupture is promoted into a free-surface-induced supershear rupture. In the presence of irregular free surfaces, this supershear mechanism can be modulated by the topographical surfaces (Zhang et al.2016). The rupture styles are significantly different for the C1, C2 and H2 models comparing with remaining models in Fig. 2. The hill- or canyon- shaped topography prevents the free-surface-induced supershear transition in some scenario C1, H2 and C2. Figure 2. View largeDownload slide Rupture time contours on fault planes for reference model (Flat surface in the uppermost row) and topographical models (hills on left panel; canyons on right panel) at four locations are contoured on the fault surface. The scenarios Hi and Ci (i = 1, …, 4) have been defined in the text. Figure 2. View largeDownload slide Rupture time contours on fault planes for reference model (Flat surface in the uppermost row) and topographical models (hills on left panel; canyons on right panel) at four locations are contoured on the fault surface. The scenarios Hi and Ci (i = 1, …, 4) have been defined in the text. The rupture front, which usually has the maximum slip velocity, contributes large-amplitude seismic radiation. The peak slip velocity (PSV) distributions on the fault are quite different between the supershear and subshear rupture cases (Fig. 3). For the supershear cases (Flat, H1, H3, C3, H4 and C4), the overlapping of the original rupture front radiated from the nucleation patch and the induced supershear front propagating from the free surface contributes to the large-amplitude PSV at the place where the two rupture fronts intersect. This causes a subduction-zone-shaped bend with a large PSV and a band of PSV (green area) that has smaller values than that in the same region of faults with subshear rupture. For the subshear cases (C1, H2 and C2), however, because no second rupture front is induced at the free surface, the rupture front on fault plane uniformly expands, producing nearly uniform PSV. Moreover, for the subshear cases, the area with a large PSV is concentrated near the free surface. The moment magnitudes of the simulation models do not differ substantially between the different scenarios. Figure 3. View largeDownload slide PSV distributions for reference model (Flat surface in the uppermost row) and topographical models (hills on left panel; canyons on right panel) at four locations are contoured on the fault surface. The scenarios Hi and Ci (i = 1, …, 4) have been defined in the text. The corresponding moment magnitude is noted on the bottom left of the figure. Supershear refers to the free-surface-induced supershear rupture cases, and subshear means subshear rupture ones. Figure 3. View largeDownload slide PSV distributions for reference model (Flat surface in the uppermost row) and topographical models (hills on left panel; canyons on right panel) at four locations are contoured on the fault surface. The scenarios Hi and Ci (i = 1, …, 4) have been defined in the text. The corresponding moment magnitude is noted on the bottom left of the figure. Supershear refers to the free-surface-induced supershear rupture cases, and subshear means subshear rupture ones. By comparing the PSV of the models with hill- and canyon-shaped topography in Group I (H1 and C1), we find their distribution characteristics are different. The topographic perturbations for H1 and C1 are located before the critical transition location. This finding indicates that the mechanism of the canyon-shaped topography is different from the hill-shaped topography. However, H1 results in a supershear rupture while C1 results in a subshear rupture. This difference may be attributed to the multi-reflection and scattering for the hill- and canyon-shaped topography on the incident wave, respectively. However, the distribution of the PSV in Group 2 concentrates near free surface in both the canyon- and hill-shaped topography model in which the topographic perturbations are beyond the subshear-to-supershear transition location. The PSV in Groups 3 and 4 is similar to that of the reference simulation (Flat model), because they all produce a free-surface induced supershear rupture. The only significant difference in PSV is locally near the topographic perturbation. 3.2 Seismic radiation The simulations show that the hill- or canyon-shaped topography can disrupt the supershear transition near free surface in some scenario cases. However, the effect of irregular topography on resultant ground motion is still unknown. Therefore, we make an investigation of its effect on the seismic radiation and resultant ground motions in this section. Fig. 4 shows snapshots at different times of the fault-parallel (FP) component of ground particle velocity (Vy). Note that the phenomenon of a Mach cone begins to emerge in the snapshot at 6.6 s for the F and H1 modes and becomes more obvious in later snapshots. However, the Mach cone does not occur in the C1 model. Clearly, the rupture front for the F and H1 models reaches the right boundary of the fault in the snapshots at 12.87 s in Fig. 4, showing that the rupture in the F and H1 modes propagates much faster than that in the C1 model. Additionally, when the Mach cone emerges, the Vy with relative large value (F and H1 modes) has wider range than the case without Mach cone (C1 model). The free surface FP particle velocities (Vy) of the topographical surfaces in Group 2 to 4 are not shown for clarity. Figure 4. View largeDownload slide Snapshots of fault-parallel component of velocity (Vy) on ground at different times for the left (Flat model), middle (H1 model) and right (C1 model). Those of the hill- and canyon-shaped topographical models in Groups 2 to 4 are not shown for clarity. Figure 4. View largeDownload slide Snapshots of fault-parallel component of velocity (Vy) on ground at different times for the left (Flat model), middle (H1 model) and right (C1 model). Those of the hill- and canyon-shaped topographical models in Groups 2 to 4 are not shown for clarity. Peak Ground Acceleration (PGA) is widely used to evaluate the seismic hazards. We investigate the effects of the hill- and canyon-shaped topography on the PGA distribution. Maps of the fault parallel (FP) and fault normal (FN) components of PGA (PGA-FP and PGA-FN, respectively) for the four groups (1 to 4) of simulations are shown in Figs 5 and 6, respectively. The simulated PGA displays many features related to the irregular topography. Topographical perturbations models C1, H2 and C2 prevent the subshear-to-supershear transition, resulting in a subshear rupture (Fig. 2). It is evident in Fig. 5 that the models (Flat, H1, H3, C3, H4 and C4) have larger PGA-FP than those with subshear rupture. As noted by Bernard & Baumon (2005) among others, supershear rupture can transport seismic energy a significant distance from the fault. However, the models with subshear rupture (C1, H2 and C2) have higher PGA-FN values on the free surface near the fault trace for distances beyond the irregular topography than the Flat, H1, H3, C3, H4 and C4 models. The effect of subshear rupture on the PGA-FN is well documented (Archuleta & Hartzell 1981; Somerville et al.1997). When the irregular topography prevents the supershear transition near the free surface, the distribution of the peak ground motion differs substantially from that of the irregular topography with the supershear. The irregular topography, which prevents the generation of supershear, results in strong shaking of the free surface near fault trace on the right side of the irregular topography. Figure 5. View largeDownload slide Distributions of the FP component of the PGA in map view. The reference simulation is labelled ‘Flat’ in the uppermost row, and the others correspond to the simulations with the hill- or canyon-shaped topography. All simulations, except for the C1, H2, and C2 models, produce a free-surface-induced supershear. The rupture styles (supershear or subshear) are marked on the upper left corner of each subfigure. Figure 5. View largeDownload slide Distributions of the FP component of the PGA in map view. The reference simulation is labelled ‘Flat’ in the uppermost row, and the others correspond to the simulations with the hill- or canyon-shaped topography. All simulations, except for the C1, H2, and C2 models, produce a free-surface-induced supershear. The rupture styles (supershear or subshear) are marked on the upper left corner of each subfigure. Figure 6. View largeDownload slide Distribution of the FN component of the PGA in map view. See Fig. 5 caption for explanation of the different panels. Figure 6. View largeDownload slide Distribution of the FN component of the PGA in map view. See Fig. 5 caption for explanation of the different panels. The above studies reveal that irregular topography obviously affects the rupture style, distribution of PSV on fault and PGA. With more details we also compare the seismograms from seismic Line 1 and Line 2 (Fig. 1) in the F, H2 and C2 models. Fig. 7 compares the seismograms from seismic Line 1 (parallel to the fault trace) in the F, H2 and C2 models. Note that the F model exhibits two types of rupture styles: the primary supershear and secondary subshear ruptures. However, the H2 and C2 models only have the subshear rupture. It can be seen that the amplitude of the FN component in the H2 and C2 models is approximately two to three times of that in the F model in Fig. 7(a). This difference corresponds to the PGA-FN distributions in Fig. 6, which show that the H2 and C2 models have larger PGA-FN on the free surface than the flat surface model (F), which can generate a transition from subshear to supershear, does. However, the amplitude of the FP component of the supershear rupture is slightly larger than that of the subshear rupture in the F model in Fig. 7(b). Fig. 8 compares the seismograms from seismic Line 2 (perpendicular to the fault trace) in the F, H2 and C2 models, clearly showing that as the perpendicular distance to the fault trace increases, the subshear rupture amplitude of the FP component decreases faster in the H2 and C2 models than the supershear rupture amplitude of the FP component in the F model in Fig. 8(b). This behaviour is consistent with the PGA-FP distribution in Fig. 5, which shows that PGA-FP in the F model has wider coverage areas than those in the H2 and C2 models (namely the Mach Cone). Figure 7. View largeDownload slide Seismograms along Line 1 (Fig. 1) paralleling the fault plane with a 0.5 km offset (Fig. 1) for the F, H2 and C2 models: (a) the FN component and (b) the FP component. The wave velocities Vp and Vs are inserted in each subfigure. Figure 7. View largeDownload slide Seismograms along Line 1 (Fig. 1) paralleling the fault plane with a 0.5 km offset (Fig. 1) for the F, H2 and C2 models: (a) the FN component and (b) the FP component. The wave velocities Vp and Vs are inserted in each subfigure. Figure 8. View largeDownload slide Seismograms along Line 2 (Fig. 1) which is normal to the fault at strike distance of 30 km for the F, H2 and C2 models: (a) the FN component and (b) the FP component. Figure 8. View largeDownload slide Seismograms along Line 2 (Fig. 1) which is normal to the fault at strike distance of 30 km for the F, H2 and C2 models: (a) the FN component and (b) the FP component. 4 DISCUSSIONS We have investigated the shapes and relative locations of irregular topography on resultant ground motion. The topographical surfaces can alter the distribution of the resultant ground motion in some cases. The effect of the canyon-shaped topography on resultant ground motion is stronger comparing with the hill-shaped topography. The initial shear stress and sizes of topographical surfaces are unknown in the real earthquakes. Therefore, we further discuss effects of the initial shear stress and the sizes of topographic features on the ground motion in this section. Note that we apply relative larger initial shear stress τ0 (small S value) to discuss the ground motion with the Burridge-Andrews (Burridge 1973; Andrews 1976; Das & Aki 1977) supershear and the free-surface-induced supershear simultaneously in some scenario cases. 4.1 The shear stress on ground motion The supershear discussed in the previous sections is induced by the free surface (hereafter referred to as the Super I). Dunham (2007) has shown that a subshear can transit into a supershear rupture on an unbound fault embedded in a homogenous medium when the value of the non-dimensional seismic ratio S = (τu − τ0)/(τ0 − τf) is small enough (S < 1.19 for 3-D). This intrinsic Burridge–Andrews supershear (hereafter referred to as the Super II) is caused by the high stress drop and has no relationship with the free surface. We set two groups of different S-values to investigate the effects of these two kinds of supershear rupture on ground motions. To gain insight into the ground motion of the Super I and the Super II, we simulate four models with different S values, which are given in Fig. 9, in full- and half-space respectively. All other parameters are identical with those shown in Table 1. Four models are labelled as follows: Hs1 and Hs2 denote the low- and high-stress cases in half-space, and Fs1 and Fs2 denote the low- and high-stress cases in full-space. For the models with half-space, fault is vertically aligned with zero buried depth. Figure 9. View largeDownload slide Strength parameter S for four models specified in full- and half-space. The other parameters are the same as in Table 1. Figure 9. View largeDownload slide Strength parameter S for four models specified in full- and half-space. The other parameters are the same as in Table 1. Fig. 10 shows the distributions of the PSV on fault planes in half- and full-space for different scenario cases. Comparing the PSV of the Hs1 and Fs1 models reveals that the distribution of the PSV in the Hs1 model displays a subduction-zone-shaped bend. The main discrepancy between Hs1 and Fs1 is caused by the free-surface-induced supershear rupture. For the ruptures occurring in high-stress cases (Fs2 and Hs2), larger PSV distributions and moment magnitudes are generated, relative to those in the low-stress cases (Fs1 and Hs1). Moreover, the effect of the free-surface-induced supershear rupture in high-stress cases is not as obvious as that in low-stress cases. The large stress drop and the high radiated seismic energy overlap the results from the free-surface-induced supershear rupture. Figure 10. View largeDownload slide Distributions of the PSV for the Hs1, Hs2, Fs1 and Fs2 models. The abbreviation for each model has been defined in the text. Rupture styles and corresponding moment magnitude for each model are noted on the top and bottom left of each subfigure, respectively. Figure 10. View largeDownload slide Distributions of the PSV for the Hs1, Hs2, Fs1 and Fs2 models. The abbreviation for each model has been defined in the text. Rupture styles and corresponding moment magnitude for each model are noted on the top and bottom left of each subfigure, respectively. Different rupture patterns result in different ground motion distributions. Figs 11 and 12 show the distributions of PGA-FP and PGA-FN, respectively. By comparing the distributions of the PGA for the Hs1 and Fs1 models, with the same initial shear stress applied in both models, the Hs1 model with the Super II produces ground motion over a larger area than does the Fs1 model with a subshear rupture. Figure 11. View largeDownload slide Distributions of the PGA-FP for the Hs1, Hs1, Fs1 and Fs2 models. The supershear rupture style in each subfigure is noted on the top left corner. The abbreviations in the illustrations refer to Fig. 9. Figure 11. View largeDownload slide Distributions of the PGA-FP for the Hs1, Hs1, Fs1 and Fs2 models. The supershear rupture style in each subfigure is noted on the top left corner. The abbreviations in the illustrations refer to Fig. 9. Figure 12. View largeDownload slide Distributions of the PGA-FN for the Hs1, Hs1, Fs1 and Fs2 models. The supershear rupture style in each subfigure is noted on the top left corner. The abbreviations in the illustrations refer to Fig. 9. Figure 12. View largeDownload slide Distributions of the PGA-FN for the Hs1, Hs1, Fs1 and Fs2 models. The supershear rupture style in each subfigure is noted on the top left corner. The abbreviations in the illustrations refer to Fig. 9. 4.2 The size of topography on ground motions To discuss the sizes of topographic features on subsequent ground motion, we rerun all simulations discussed above by increasing the size of irregular topography, which is defined by   \begin{equation} z( r) = \ \pm 2000{\rm{exp}}\left({ - {r^2}/{{2000}^2}}\right).\end{equation} (3) To make a distinction with the above groups of simulations, we append the letter L to the label in each model. The parameters in the H2L model are the same with those of the H2 model in addition to the size of topographical surfaces. Figs 13 and 14 illustrate the same distributions of the FP and FN components of the PGA as shown in Figs 5 and 6, respectively. By comparing with above results, it is evident that the FP and FN distributions of the PGA in the H2/C3/C4 (Figs 5 and 6) have distinct character with that of the H2L/C3L/C4L (Figs 13 and 14). This mainly because that the multireflection (hill) or diffracting (canyon) capability becomes strong as the size of hill- or canyon-shaped surface increases. Comparing the Figs 13 and 14 with Figs 5 and 6, the FN and FP distributions in the H2-H2L, C3-C3L and C4-C4L models have distinctly different patterns, and we can see that the size of irregular topography is also important to the distribution of the PGA for the canyon-shaped topography than the hill-shaped topography. Again, it is the difference between subshear and supershear rupture. The main point is that larger topographic perturbation has systematically changed the existence or non-existence of a supershear rupture. Figure 13. View largeDownload slide Similar to Fig. 5, except for the enlarged sizes of irregular topography as given by eq. (3). The definition of the label in each subfigure is the same with above statement, except that the last letter L in label representing larger topography. Figure 13. View largeDownload slide Similar to Fig. 5, except for the enlarged sizes of irregular topography as given by eq. (3). The definition of the label in each subfigure is the same with above statement, except that the last letter L in label representing larger topography. Figure 14. View largeDownload slide Similar to Fig. 6, except for the enlarged sizes of irregular topography as given by eq. (3). The definition of the label in each subfigure is the same with above statement, except that the last letter L in label representing larger topography. Figure 14. View largeDownload slide Similar to Fig. 6, except for the enlarged sizes of irregular topography as given by eq. (3). The definition of the label in each subfigure is the same with above statement, except that the last letter L in label representing larger topography. 5 EXAMPLES In previous simulations, we modelled and discussed rupture dynamics on simple fault models. We will make it closer to the reality by simulating dynamic rupture on the vertical San Andreas Fault (SAF) with real topographical surface. Fig. 15 marks the surface map of the target area. The white dashed line represents the SAF projected on the free surface, extending 170 and 30 km along the strike and dip directions, respectively. The epicentre we used to nucleate the dynamic modelling is represented by the red star. Figure 15. View largeDownload slide The white dashed line is our approximation to the surface trace of the SAF, The actual San Andreas Fault is not a critical line. The red star denotes the epicentre (47.5 km, 35 km). The length of the SAF along fault trace is about 170 km. Figure 15. View largeDownload slide The white dashed line is our approximation to the surface trace of the SAF, The actual San Andreas Fault is not a critical line. The red star denotes the epicentre (47.5 km, 35 km). The length of the SAF along fault trace is about 170 km. To show more information about the dynamic effect generated by the irregular free surfaces, here we run two comparing cases with different topographical models in our SAF modelling. The first simulation, the real topographical surface is implemented and the rupture time contours are presented in upper subfigure of Fig. 16. As it is clear that the supershear rupture is trigged at the right hand of nucleation patch when the rupture front reaches the free surface. In the second case, we invert the real topography. The rupture time is contoured in the bottom of Fig. 16. The significant difference between rupture times form different topographical surfaces is the rupture pattern at the right size of nucleation patch. The supershear rupture cannot be able to be triggered in the inverted topography model, comparing to the real topography case. Figure 16. View largeDownload slide Rupture time contours for (top) real topography and (bottom) inverted topography models of San Andreas Fault. Figure 16. View largeDownload slide Rupture time contours for (top) real topography and (bottom) inverted topography models of San Andreas Fault. Following the discussions as the simple fault model, the PGA for the real and inverted topography model of SAF are shown in Figs 17 and 18, respectively. We can see that inverted topography can prevent the free-surface supershear transition and the distributions of PGA mainly concentrate near the fault trace. Supershear rupture occurs in the real topography model, and the energy can radiate far away from the fault trace. Therefore, there is a consistent conclusion between the simple mathematical model and real complex examples of topographic features. Figure 17. View largeDownload slide Similar to Figs 5 and 6, except for topography with normal one as shown in Fig. 15. Figure 17. View largeDownload slide Similar to Figs 5 and 6, except for topography with normal one as shown in Fig. 15. Figure 18. View largeDownload slide Similar to Figs 5 and 6, except for topography with negative signal to one as shown in Fig. 15. Figure 18. View largeDownload slide Similar to Figs 5 and 6, except for topography with negative signal to one as shown in Fig. 15. 6 CONCLUSIONS By numerically modelling a reference model with flat surface and four groups of models with irregular topography, we study the effect of irregular topography on the near-fault ground motion using the CG-FDM. Our results show that the supershear rupture produces resultant ground motion over a larger area than the subshear rupture does. More importantly, the subshear rupture produces stronger ground motions near fault than the supershear rupture when the hill- or canyon-shaped topography (e.g. H2 and C1/2) prevents the generation of supershear. However, the moment magnitude does not differ substantially between the two cases. The detailed effects of different topography on the resultant ground motion depend on the size of the irregular topography and relative distance to the critical supershear transition distance. Moreover, we investigate and discuss the supershear transition caused by the free surface and high stress drop in full- and half-space, and gain insight into the effects of both the free surface and the initial shear stress on resultant ground motion in full- and half-space. The simulations show that the free surface is an important factor to the distribution of ground motion. Moreover, when two types of supershear (Super I and Super II) exit simultaneously, Super I has greater energy and more strongly affects the ground motion than Super II. We also discuss the size factor of the irregular topography on resultant ground motion to investigate the generality of the topographic phenomenon. The larger topographic perturbation has systematically changed the existence or non-existence of a supershear rupture. At last, we run the dynamic rupture simulations of SAF with real and inverted topographical surfaces to reveal the significance of topography on earthquake rupture and strong ground motion in the reality. Acknowledgements This work is supported by the National Natural Science Foundation of China (grants 41504040 and 41474037), the Science and Technology support plan of Sichuan province (2016SZ0067), and China Postdoctoral Science Foundation (grant 2016T90575). All data used in this paper are acquired from numerical simulations. REFERENCES Andrews D., 1976. Rupture velocity of plane strain shear cracks, J. geophys. Res , 81, 5679– 5687. Google Scholar CrossRef Search ADS   Archuleta R.J., Hartzell S.H., 1981. Effects of fault finiteness on near-source ground motion, Bull. seism. Soc. Am. , 71, 939– 957. Bernard, P. & Baumon, D., 2005. Shear Mach wave characterization for kinematic fault rupture models with constant supershear rupture velocity, Geophys. J. Int. , 162, 431– 447. CrossRef Search ADS   Bernard, P. & Madariaga, R., 1984. A new asymptotic method for the modeling of near-field accelerograms, Bull. seism. Soc. Am. , 74, 539– 557. Bizzarri A., Spudich P., 2008. Effects of supershear rupture speed on the high‐frequency content of S waves investigated using spontaneous dynamic rupture models and isochrone theory, J. geophys. Res. , 113, B05304, doi:10.1029/2007JB005146. Google Scholar CrossRef Search ADS   Bizzarri A., Dunham E.M., Spudich P., 2010. Coherence of Mach fronts during heterogeneous supershear earthquake rupture propagation: simulations and comparison with observations, J. geophys. Res ., 115, B08301, doi:10.1029/2009JB006819. Burridge R., 1973. Admissible speeds for plane-strain self-similar shear cracks with friction but lacking cohesion, Geophys. J. Int. , 35, 439– 455. Google Scholar CrossRef Search ADS   Chen X., Zhang H., 2006. Modelling rupture dynamics of a planar fault in 3-D half space by boundary integral equation method: an overview, Pure appl. Geophys. , 163, 267– 299. Google Scholar CrossRef Search ADS   Das S., Aki K., 1977. A numerical study of two-dimensional spontaneous rupture propagation, Geophys. J. Int. , 50, 643– 668. Google Scholar CrossRef Search ADS   Dunham E.M., 2007. Conditions governing the occurrence of supershear ruptures under slip-weakening friction, J. geophys. Res ., 112, B07302, doi:10.1029/2006JB004717. Google Scholar CrossRef Search ADS   Dunham E.M., Archuleta R.J., 2005. Near-source ground motion from steady state dynamic rupture pulses, Geophys. Res. Lett. , 32, L03302, doi:10.1029/2004GL021793. Google Scholar CrossRef Search ADS   Ely G.P., Day S.M., Minster J.-B., 2010. Dynamic rupture models for the southern San Andreas fault, Bull. seism. Soc. Am. , 100, 131– 150. Google Scholar CrossRef Search ADS   Ida Y., 1972. Cohesive force across the tip of a longitudinal-shear crack and Griffith's specific surface energy, J. geophys. Res. , 77, 3796– 3805. Google Scholar CrossRef Search ADS   Kaneko Y., Lapusta N., 2010. Supershear transition due to a free surface in 3-D simulations of spontaneous dynamic rupture on vertical strike-slip faults, Tectonophysics , 493, 272– 284. Google Scholar CrossRef Search ADS   Madariaga R., 1983. High frequency radiation from dynamic earthquake fault models, Ann. Geophys. , 1, 17– 23. Oglesby D.D., Archuleta R.J., Nielsen S.B., 2000a. Dynamics of dip-slip faulting: explorations in two dimensions, J. geophys. Res. , 105, 13 643–13 653. Oglesby D.D., Archuleta R.J., Nielsen S.B., 2000b. The three-dimensional dynamics of dipping faults, Bull. seism. Soc. Am. , 90, 616– 628. Google Scholar CrossRef Search ADS   Somerville P.G., Smith N.F., Graves R.W., Abrahamson N.A., 1997. Modification of empirical strong ground motion attenuation relations to include the amplitude and duration effects of rupture directivity, Seismol. Res. Lett. , 68, 199– 222. Google Scholar CrossRef Search ADS   Spudich P., Cranswick E., 1984. Direct observation of rupture propagation during the 1979 Imperial Valley earthquake using a short baseline accelerometer array, Bull. seism. Soc. Am. , 74, 2083– 2114. Xu J., Zhang H., Chen X., 2015. Rupture phase diagrams for a planar fault in 3-D full-space and half-space, Geophys. J. Int. , 202, 2194– 2206. Google Scholar CrossRef Search ADS   Zhang H., Chen X., 2006a. Dynamic rupture on a planar fault in three-dimensional half-space–II. Validations and numerical experiments, Geophys. J. Int. , 167, 917– 932. Google Scholar CrossRef Search ADS   Zhang H., Chen X., 2006b. Dynamic rupture on a planar fault in three-dimensional half space—I. Theory, Geophys. J. Int. , 164, 633– 652. Google Scholar CrossRef Search ADS   Zhang Z., Zhang W., Chen X., 2014. Three-dimensional curved grid finite-difference modelling for non-planar rupture dynamics, Geophys. J. Int. , 199, 860– 879. Google Scholar CrossRef Search ADS   Zhang Z., Xu J., Chen X., 2016. The supershear effect of topography on rupture dynamics, Geophys. Res. Lett. , 43, 1457– 1463. Google Scholar CrossRef Search ADS   © The Authors 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Geophysical Journal International Oxford University Press

Investigation of topographical effects on rupture dynamics and resultant ground motions

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Abstract

Abstract In this work, we investigate the effect of irregular topography on the dynamic rupture and resultant ground motions using the curved grid finite-difference method. The research is based on spontaneous dynamic rupture on vertical strike-slip faults by varying the shapes and relative locations of irregular topography to the critical supershear transition distance. The results show that seismic energy of a supershear earthquake can be transmitted farther with large amplitudes. However, its ground motion near the fault is weaker than that caused by a subshear (namely the sub-Rayleigh) rupture. Whether the irregular topography exhibits stronger ground motion overall depends on the irregular topography's ability to prevent the subshear-to-supershear transition. Finally, we also discuss the effects of the strength parameter S and a larger size of the irregular topography on the resultant ground motion. The modellings of San Andreas Fault with real and inverted topographical surfaces show the implications of the topographical effects from the real earthquake. Numerical solutions, Earthquake dynamics, Wave propagation, Wave scattering and diffraction, Dynamics and mechanics of faulting 1 INTRODUCTION The Earth's free surface acts as an important boundary in numerical simulation. The rupture with the flat surface has been investigated by many authors (e.g. Oglesby et al.2000a,b; Chen & Zhang 2006; Zhang & Chen 2006a,b; Kaneko & Lapusta 2010; Xu et al.2015). With numerical simulations, Kaneko & Lapusta (2010) pointed out that the free-surface-induced supershear rupture due to the phase conversion at the free surface. If a strike-slip rupture on a vertical plane in a homogeneous half-space grows continuously, a supershear rupture will always occur near the free surface (Xu et al.2015). Previous works treat the Earth's free surface as flat to simplify the numerical simulations. However, the Earth's surface is not always flat. Some numerical simulations have successfully taken the irregular topography into consideration and have shown that different rupture phases can develop (Ely et al.2010; Zhang et al.2016). The rupture speed of an earthquake is an important parameter that results in high frequencies and strong seismic radiation, particularly the transition from subshear rupture to supershear rupture (Madariaga 1983; Bernard & Madariaga 1984; Spudich & Cranswick 1984; Bizzarri & Spudich 2008; Bizzarri et al.2010). For the supershear rupture, the radiated S-wave can constructively form a Mach front that transports large seismic energy at farther distance away from the fault (Bernard & Baumon 2005; Dunham & Archuleta 2005). Zhang et al. (2016) showed that irregular topography can disrupt the critical conditions including the phase conversion and the rupture slip in some cases. However, they did not extend their research to the effect of irregular topography on resultant ground motions. Since the topographical surfaces can influence the rupture dynamics, the distribution of resultant ground motion should also be affected. In this work, we aim to study the effect of irregular topography on resultant seismic wave radiation. To investigate the effect of irregular topography on resultant near fault ground motion based on dynamic rupture models, we have done numerous simulations with different epicentral distances of the topographical irregularities. Then, we also discuss the initial shear stress and size of irregular topography on the distribution of resultant ground motions. At last, a real example of San Andreas Fault with different topographical surfaces is modelled and to discuss the topographical effects on rupture dynamics and ground motion. 2 METHODS AND MODEL SETTING In this work, we adopt the method of Curved Grid Finite Difference Rupture Dynamics Modeling (CGFD-RDM) developed by Zhang et al. (2014) to investigate the effect of irregular surface topography on resultant ground motion. We investigate free-surface topographical effects on rupture dynamics and resultant ground motions caused by that result from a dynamic strike-slip rupture on a vertical planar fault in a homogeneous half-space. All the simulations are divided into four groups (1–4) depending on the epicentre distances (7.5, 12.5, 17.5 and 22.5 km) of the topographic perturbations. Fig. 1 illustrates geometrical model discussed in this work. The grey rectangle indicates the nucleation area with a dimension of 3 km × 3 km, whose centre is 7.5 km from the left, bottom and free surface boundaries. The corresponding cases for the hill surfaces are not shown in Fig. 1 for clarity. The topographic surfaces are mathematically described by following Gaussian function Figure 1. View largeDownload slide Fault geometries of simulation models. The basic vertical fault extends 62 km and 15 km along the fault strike and dip directions, respectively. A small grey rectangle indicates a nucleation area with a dimension of 3 km × 3 km. The nucleation patch is the same in all comparison simulations. Only canyon-shaped topographic surfaces for Models 1 to 4 are shown, the corresponding hill surfaces are not shown for clarity Figure 1. View largeDownload slide Fault geometries of simulation models. The basic vertical fault extends 62 km and 15 km along the fault strike and dip directions, respectively. A small grey rectangle indicates a nucleation area with a dimension of 3 km × 3 km. The nucleation patch is the same in all comparison simulations. Only canyon-shaped topographic surfaces for Models 1 to 4 are shown, the corresponding hill surfaces are not shown for clarity  \begin{equation}z\ \left( r \right) = \ \pm 1000{\rm{exp}}\left( { - {r^2}/{{1500}^2}} \right), \end{equation} (1)where r is the horizontal distance in metres to the centre of the hill- (‘+’ sign) or canyon- (‘−’ sign) shaped topography. Note that the irregular topography is symmetrically distributed along the fault trace for all simulations in this work. In our simulations, the rupture criterion of faulting is governed by a linear slip-weakening friction law (Ida 1972) prescribed by   \begin{equation}{\mu _f} (l) = \left\{ \begin{array}{l@{\quad}c} {\mu _s} - l({\mu _s} - {\mu _d})/{d_0}, & l < {d_0}\\ {\mu _d}, & l \geq {d_0}\end{array} \right. \end{equation} (2) where l is the simulated slip and μs and μd represent the static and dynamic friction coefficients, respectively, d0 is the critical slip-weakening distance and set to be 0.4 m for all cases in this study. Moreover, the initial stress in the background medium is set to be homogeneous, with 120 MPa and 70.0 MPa as the normal and shear tractions. We apply a large initial shear stress (81.6 MPa) exceeding the failure strength within the nucleation patch to trigger the dynamic rupture. The subsequent rupture spontaneously propagates over the entire fault plane until it is stopped by surrounding artificial barriers with sufficiently high strength. The detailed parameters implemented in the simulations are listed in Table 1. Table 1. Parameters are used for simulations. Dynamic parameters  Value  Initial shear stress, τ0 (MPa)  70.0  Initial normal stress, −σn (Mpa)  120.0  Static friction coefficient, μs  0.677  Dynamic friction coefficient,  μd  0.525  Critical slip distance, d0 (m)  0.40  Media parameters  Value  P-wave velocity, Vp (m s−1)  6000  S-wave velocity, Vs (m s−1)  3464  Density, ρ (kg m − 3)  2670  Dynamic parameters  Value  Initial shear stress, τ0 (MPa)  70.0  Initial normal stress, −σn (Mpa)  120.0  Static friction coefficient, μs  0.677  Dynamic friction coefficient,  μd  0.525  Critical slip distance, d0 (m)  0.40  Media parameters  Value  P-wave velocity, Vp (m s−1)  6000  S-wave velocity, Vs (m s−1)  3464  Density, ρ (kg m − 3)  2670  View Large 3 RESULTS Previous studies have shown that the rupture dynamic on the strike-slip fault in half-space can generate a transition from subshear to supershear rupture at the free surface (Chen & Zhang 2006; Zhang & Chen 2006a,b; Xu et al.2015). However, such strong influence of free surface on rupturing doesn’t always accelerate the rupture speed and becomes complicated when the geometry of free surface is not flat. Zhang et al. (2016) have shown that the critical condition of supershear rupture transition near the free surface may be disrupted when the irregular topography is symmetrically distributed along the fault trace in some scenario cases. This is because that the topographical surfaces can influence the phase conversion and the rupture slip. In this study, we mainly focus on the features of seismic waves and resultant ground motion radiated from the complex rupturing controlled by various topographic surfaces. We carry out four groups (1–4) of simulations corresponding to epicentral distances (7.5, 12.5, 17.5, and 22.5 km) to the irregular topography. The parameters used in the simulation are the same except for the shapes of free surfaces. Each group has either a hill (H) or canyon (C) topographic feature. A simulation is labelled with H or C followed by a digit corresponding to a group 1–4. In the flat surface model, a subshear-to-supershear transition occurs at an epicentral distance of 7.5 km. 3.1 Rupture dynamics Fig. 2 illustrates the rupture time contours on fault planes for all the cases with different shapes and locations of topography. The conditions are the same for all simulations, except the topographical surfaces. For the reference model, the original subshear rupture is promoted into a free-surface-induced supershear rupture. In the presence of irregular free surfaces, this supershear mechanism can be modulated by the topographical surfaces (Zhang et al.2016). The rupture styles are significantly different for the C1, C2 and H2 models comparing with remaining models in Fig. 2. The hill- or canyon- shaped topography prevents the free-surface-induced supershear transition in some scenario C1, H2 and C2. Figure 2. View largeDownload slide Rupture time contours on fault planes for reference model (Flat surface in the uppermost row) and topographical models (hills on left panel; canyons on right panel) at four locations are contoured on the fault surface. The scenarios Hi and Ci (i = 1, …, 4) have been defined in the text. Figure 2. View largeDownload slide Rupture time contours on fault planes for reference model (Flat surface in the uppermost row) and topographical models (hills on left panel; canyons on right panel) at four locations are contoured on the fault surface. The scenarios Hi and Ci (i = 1, …, 4) have been defined in the text. The rupture front, which usually has the maximum slip velocity, contributes large-amplitude seismic radiation. The peak slip velocity (PSV) distributions on the fault are quite different between the supershear and subshear rupture cases (Fig. 3). For the supershear cases (Flat, H1, H3, C3, H4 and C4), the overlapping of the original rupture front radiated from the nucleation patch and the induced supershear front propagating from the free surface contributes to the large-amplitude PSV at the place where the two rupture fronts intersect. This causes a subduction-zone-shaped bend with a large PSV and a band of PSV (green area) that has smaller values than that in the same region of faults with subshear rupture. For the subshear cases (C1, H2 and C2), however, because no second rupture front is induced at the free surface, the rupture front on fault plane uniformly expands, producing nearly uniform PSV. Moreover, for the subshear cases, the area with a large PSV is concentrated near the free surface. The moment magnitudes of the simulation models do not differ substantially between the different scenarios. Figure 3. View largeDownload slide PSV distributions for reference model (Flat surface in the uppermost row) and topographical models (hills on left panel; canyons on right panel) at four locations are contoured on the fault surface. The scenarios Hi and Ci (i = 1, …, 4) have been defined in the text. The corresponding moment magnitude is noted on the bottom left of the figure. Supershear refers to the free-surface-induced supershear rupture cases, and subshear means subshear rupture ones. Figure 3. View largeDownload slide PSV distributions for reference model (Flat surface in the uppermost row) and topographical models (hills on left panel; canyons on right panel) at four locations are contoured on the fault surface. The scenarios Hi and Ci (i = 1, …, 4) have been defined in the text. The corresponding moment magnitude is noted on the bottom left of the figure. Supershear refers to the free-surface-induced supershear rupture cases, and subshear means subshear rupture ones. By comparing the PSV of the models with hill- and canyon-shaped topography in Group I (H1 and C1), we find their distribution characteristics are different. The topographic perturbations for H1 and C1 are located before the critical transition location. This finding indicates that the mechanism of the canyon-shaped topography is different from the hill-shaped topography. However, H1 results in a supershear rupture while C1 results in a subshear rupture. This difference may be attributed to the multi-reflection and scattering for the hill- and canyon-shaped topography on the incident wave, respectively. However, the distribution of the PSV in Group 2 concentrates near free surface in both the canyon- and hill-shaped topography model in which the topographic perturbations are beyond the subshear-to-supershear transition location. The PSV in Groups 3 and 4 is similar to that of the reference simulation (Flat model), because they all produce a free-surface induced supershear rupture. The only significant difference in PSV is locally near the topographic perturbation. 3.2 Seismic radiation The simulations show that the hill- or canyon-shaped topography can disrupt the supershear transition near free surface in some scenario cases. However, the effect of irregular topography on resultant ground motion is still unknown. Therefore, we make an investigation of its effect on the seismic radiation and resultant ground motions in this section. Fig. 4 shows snapshots at different times of the fault-parallel (FP) component of ground particle velocity (Vy). Note that the phenomenon of a Mach cone begins to emerge in the snapshot at 6.6 s for the F and H1 modes and becomes more obvious in later snapshots. However, the Mach cone does not occur in the C1 model. Clearly, the rupture front for the F and H1 models reaches the right boundary of the fault in the snapshots at 12.87 s in Fig. 4, showing that the rupture in the F and H1 modes propagates much faster than that in the C1 model. Additionally, when the Mach cone emerges, the Vy with relative large value (F and H1 modes) has wider range than the case without Mach cone (C1 model). The free surface FP particle velocities (Vy) of the topographical surfaces in Group 2 to 4 are not shown for clarity. Figure 4. View largeDownload slide Snapshots of fault-parallel component of velocity (Vy) on ground at different times for the left (Flat model), middle (H1 model) and right (C1 model). Those of the hill- and canyon-shaped topographical models in Groups 2 to 4 are not shown for clarity. Figure 4. View largeDownload slide Snapshots of fault-parallel component of velocity (Vy) on ground at different times for the left (Flat model), middle (H1 model) and right (C1 model). Those of the hill- and canyon-shaped topographical models in Groups 2 to 4 are not shown for clarity. Peak Ground Acceleration (PGA) is widely used to evaluate the seismic hazards. We investigate the effects of the hill- and canyon-shaped topography on the PGA distribution. Maps of the fault parallel (FP) and fault normal (FN) components of PGA (PGA-FP and PGA-FN, respectively) for the four groups (1 to 4) of simulations are shown in Figs 5 and 6, respectively. The simulated PGA displays many features related to the irregular topography. Topographical perturbations models C1, H2 and C2 prevent the subshear-to-supershear transition, resulting in a subshear rupture (Fig. 2). It is evident in Fig. 5 that the models (Flat, H1, H3, C3, H4 and C4) have larger PGA-FP than those with subshear rupture. As noted by Bernard & Baumon (2005) among others, supershear rupture can transport seismic energy a significant distance from the fault. However, the models with subshear rupture (C1, H2 and C2) have higher PGA-FN values on the free surface near the fault trace for distances beyond the irregular topography than the Flat, H1, H3, C3, H4 and C4 models. The effect of subshear rupture on the PGA-FN is well documented (Archuleta & Hartzell 1981; Somerville et al.1997). When the irregular topography prevents the supershear transition near the free surface, the distribution of the peak ground motion differs substantially from that of the irregular topography with the supershear. The irregular topography, which prevents the generation of supershear, results in strong shaking of the free surface near fault trace on the right side of the irregular topography. Figure 5. View largeDownload slide Distributions of the FP component of the PGA in map view. The reference simulation is labelled ‘Flat’ in the uppermost row, and the others correspond to the simulations with the hill- or canyon-shaped topography. All simulations, except for the C1, H2, and C2 models, produce a free-surface-induced supershear. The rupture styles (supershear or subshear) are marked on the upper left corner of each subfigure. Figure 5. View largeDownload slide Distributions of the FP component of the PGA in map view. The reference simulation is labelled ‘Flat’ in the uppermost row, and the others correspond to the simulations with the hill- or canyon-shaped topography. All simulations, except for the C1, H2, and C2 models, produce a free-surface-induced supershear. The rupture styles (supershear or subshear) are marked on the upper left corner of each subfigure. Figure 6. View largeDownload slide Distribution of the FN component of the PGA in map view. See Fig. 5 caption for explanation of the different panels. Figure 6. View largeDownload slide Distribution of the FN component of the PGA in map view. See Fig. 5 caption for explanation of the different panels. The above studies reveal that irregular topography obviously affects the rupture style, distribution of PSV on fault and PGA. With more details we also compare the seismograms from seismic Line 1 and Line 2 (Fig. 1) in the F, H2 and C2 models. Fig. 7 compares the seismograms from seismic Line 1 (parallel to the fault trace) in the F, H2 and C2 models. Note that the F model exhibits two types of rupture styles: the primary supershear and secondary subshear ruptures. However, the H2 and C2 models only have the subshear rupture. It can be seen that the amplitude of the FN component in the H2 and C2 models is approximately two to three times of that in the F model in Fig. 7(a). This difference corresponds to the PGA-FN distributions in Fig. 6, which show that the H2 and C2 models have larger PGA-FN on the free surface than the flat surface model (F), which can generate a transition from subshear to supershear, does. However, the amplitude of the FP component of the supershear rupture is slightly larger than that of the subshear rupture in the F model in Fig. 7(b). Fig. 8 compares the seismograms from seismic Line 2 (perpendicular to the fault trace) in the F, H2 and C2 models, clearly showing that as the perpendicular distance to the fault trace increases, the subshear rupture amplitude of the FP component decreases faster in the H2 and C2 models than the supershear rupture amplitude of the FP component in the F model in Fig. 8(b). This behaviour is consistent with the PGA-FP distribution in Fig. 5, which shows that PGA-FP in the F model has wider coverage areas than those in the H2 and C2 models (namely the Mach Cone). Figure 7. View largeDownload slide Seismograms along Line 1 (Fig. 1) paralleling the fault plane with a 0.5 km offset (Fig. 1) for the F, H2 and C2 models: (a) the FN component and (b) the FP component. The wave velocities Vp and Vs are inserted in each subfigure. Figure 7. View largeDownload slide Seismograms along Line 1 (Fig. 1) paralleling the fault plane with a 0.5 km offset (Fig. 1) for the F, H2 and C2 models: (a) the FN component and (b) the FP component. The wave velocities Vp and Vs are inserted in each subfigure. Figure 8. View largeDownload slide Seismograms along Line 2 (Fig. 1) which is normal to the fault at strike distance of 30 km for the F, H2 and C2 models: (a) the FN component and (b) the FP component. Figure 8. View largeDownload slide Seismograms along Line 2 (Fig. 1) which is normal to the fault at strike distance of 30 km for the F, H2 and C2 models: (a) the FN component and (b) the FP component. 4 DISCUSSIONS We have investigated the shapes and relative locations of irregular topography on resultant ground motion. The topographical surfaces can alter the distribution of the resultant ground motion in some cases. The effect of the canyon-shaped topography on resultant ground motion is stronger comparing with the hill-shaped topography. The initial shear stress and sizes of topographical surfaces are unknown in the real earthquakes. Therefore, we further discuss effects of the initial shear stress and the sizes of topographic features on the ground motion in this section. Note that we apply relative larger initial shear stress τ0 (small S value) to discuss the ground motion with the Burridge-Andrews (Burridge 1973; Andrews 1976; Das & Aki 1977) supershear and the free-surface-induced supershear simultaneously in some scenario cases. 4.1 The shear stress on ground motion The supershear discussed in the previous sections is induced by the free surface (hereafter referred to as the Super I). Dunham (2007) has shown that a subshear can transit into a supershear rupture on an unbound fault embedded in a homogenous medium when the value of the non-dimensional seismic ratio S = (τu − τ0)/(τ0 − τf) is small enough (S < 1.19 for 3-D). This intrinsic Burridge–Andrews supershear (hereafter referred to as the Super II) is caused by the high stress drop and has no relationship with the free surface. We set two groups of different S-values to investigate the effects of these two kinds of supershear rupture on ground motions. To gain insight into the ground motion of the Super I and the Super II, we simulate four models with different S values, which are given in Fig. 9, in full- and half-space respectively. All other parameters are identical with those shown in Table 1. Four models are labelled as follows: Hs1 and Hs2 denote the low- and high-stress cases in half-space, and Fs1 and Fs2 denote the low- and high-stress cases in full-space. For the models with half-space, fault is vertically aligned with zero buried depth. Figure 9. View largeDownload slide Strength parameter S for four models specified in full- and half-space. The other parameters are the same as in Table 1. Figure 9. View largeDownload slide Strength parameter S for four models specified in full- and half-space. The other parameters are the same as in Table 1. Fig. 10 shows the distributions of the PSV on fault planes in half- and full-space for different scenario cases. Comparing the PSV of the Hs1 and Fs1 models reveals that the distribution of the PSV in the Hs1 model displays a subduction-zone-shaped bend. The main discrepancy between Hs1 and Fs1 is caused by the free-surface-induced supershear rupture. For the ruptures occurring in high-stress cases (Fs2 and Hs2), larger PSV distributions and moment magnitudes are generated, relative to those in the low-stress cases (Fs1 and Hs1). Moreover, the effect of the free-surface-induced supershear rupture in high-stress cases is not as obvious as that in low-stress cases. The large stress drop and the high radiated seismic energy overlap the results from the free-surface-induced supershear rupture. Figure 10. View largeDownload slide Distributions of the PSV for the Hs1, Hs2, Fs1 and Fs2 models. The abbreviation for each model has been defined in the text. Rupture styles and corresponding moment magnitude for each model are noted on the top and bottom left of each subfigure, respectively. Figure 10. View largeDownload slide Distributions of the PSV for the Hs1, Hs2, Fs1 and Fs2 models. The abbreviation for each model has been defined in the text. Rupture styles and corresponding moment magnitude for each model are noted on the top and bottom left of each subfigure, respectively. Different rupture patterns result in different ground motion distributions. Figs 11 and 12 show the distributions of PGA-FP and PGA-FN, respectively. By comparing the distributions of the PGA for the Hs1 and Fs1 models, with the same initial shear stress applied in both models, the Hs1 model with the Super II produces ground motion over a larger area than does the Fs1 model with a subshear rupture. Figure 11. View largeDownload slide Distributions of the PGA-FP for the Hs1, Hs1, Fs1 and Fs2 models. The supershear rupture style in each subfigure is noted on the top left corner. The abbreviations in the illustrations refer to Fig. 9. Figure 11. View largeDownload slide Distributions of the PGA-FP for the Hs1, Hs1, Fs1 and Fs2 models. The supershear rupture style in each subfigure is noted on the top left corner. The abbreviations in the illustrations refer to Fig. 9. Figure 12. View largeDownload slide Distributions of the PGA-FN for the Hs1, Hs1, Fs1 and Fs2 models. The supershear rupture style in each subfigure is noted on the top left corner. The abbreviations in the illustrations refer to Fig. 9. Figure 12. View largeDownload slide Distributions of the PGA-FN for the Hs1, Hs1, Fs1 and Fs2 models. The supershear rupture style in each subfigure is noted on the top left corner. The abbreviations in the illustrations refer to Fig. 9. 4.2 The size of topography on ground motions To discuss the sizes of topographic features on subsequent ground motion, we rerun all simulations discussed above by increasing the size of irregular topography, which is defined by   \begin{equation} z( r) = \ \pm 2000{\rm{exp}}\left({ - {r^2}/{{2000}^2}}\right).\end{equation} (3) To make a distinction with the above groups of simulations, we append the letter L to the label in each model. The parameters in the H2L model are the same with those of the H2 model in addition to the size of topographical surfaces. Figs 13 and 14 illustrate the same distributions of the FP and FN components of the PGA as shown in Figs 5 and 6, respectively. By comparing with above results, it is evident that the FP and FN distributions of the PGA in the H2/C3/C4 (Figs 5 and 6) have distinct character with that of the H2L/C3L/C4L (Figs 13 and 14). This mainly because that the multireflection (hill) or diffracting (canyon) capability becomes strong as the size of hill- or canyon-shaped surface increases. Comparing the Figs 13 and 14 with Figs 5 and 6, the FN and FP distributions in the H2-H2L, C3-C3L and C4-C4L models have distinctly different patterns, and we can see that the size of irregular topography is also important to the distribution of the PGA for the canyon-shaped topography than the hill-shaped topography. Again, it is the difference between subshear and supershear rupture. The main point is that larger topographic perturbation has systematically changed the existence or non-existence of a supershear rupture. Figure 13. View largeDownload slide Similar to Fig. 5, except for the enlarged sizes of irregular topography as given by eq. (3). The definition of the label in each subfigure is the same with above statement, except that the last letter L in label representing larger topography. Figure 13. View largeDownload slide Similar to Fig. 5, except for the enlarged sizes of irregular topography as given by eq. (3). The definition of the label in each subfigure is the same with above statement, except that the last letter L in label representing larger topography. Figure 14. View largeDownload slide Similar to Fig. 6, except for the enlarged sizes of irregular topography as given by eq. (3). The definition of the label in each subfigure is the same with above statement, except that the last letter L in label representing larger topography. Figure 14. View largeDownload slide Similar to Fig. 6, except for the enlarged sizes of irregular topography as given by eq. (3). The definition of the label in each subfigure is the same with above statement, except that the last letter L in label representing larger topography. 5 EXAMPLES In previous simulations, we modelled and discussed rupture dynamics on simple fault models. We will make it closer to the reality by simulating dynamic rupture on the vertical San Andreas Fault (SAF) with real topographical surface. Fig. 15 marks the surface map of the target area. The white dashed line represents the SAF projected on the free surface, extending 170 and 30 km along the strike and dip directions, respectively. The epicentre we used to nucleate the dynamic modelling is represented by the red star. Figure 15. View largeDownload slide The white dashed line is our approximation to the surface trace of the SAF, The actual San Andreas Fault is not a critical line. The red star denotes the epicentre (47.5 km, 35 km). The length of the SAF along fault trace is about 170 km. Figure 15. View largeDownload slide The white dashed line is our approximation to the surface trace of the SAF, The actual San Andreas Fault is not a critical line. The red star denotes the epicentre (47.5 km, 35 km). The length of the SAF along fault trace is about 170 km. To show more information about the dynamic effect generated by the irregular free surfaces, here we run two comparing cases with different topographical models in our SAF modelling. The first simulation, the real topographical surface is implemented and the rupture time contours are presented in upper subfigure of Fig. 16. As it is clear that the supershear rupture is trigged at the right hand of nucleation patch when the rupture front reaches the free surface. In the second case, we invert the real topography. The rupture time is contoured in the bottom of Fig. 16. The significant difference between rupture times form different topographical surfaces is the rupture pattern at the right size of nucleation patch. The supershear rupture cannot be able to be triggered in the inverted topography model, comparing to the real topography case. Figure 16. View largeDownload slide Rupture time contours for (top) real topography and (bottom) inverted topography models of San Andreas Fault. Figure 16. View largeDownload slide Rupture time contours for (top) real topography and (bottom) inverted topography models of San Andreas Fault. Following the discussions as the simple fault model, the PGA for the real and inverted topography model of SAF are shown in Figs 17 and 18, respectively. We can see that inverted topography can prevent the free-surface supershear transition and the distributions of PGA mainly concentrate near the fault trace. Supershear rupture occurs in the real topography model, and the energy can radiate far away from the fault trace. Therefore, there is a consistent conclusion between the simple mathematical model and real complex examples of topographic features. Figure 17. View largeDownload slide Similar to Figs 5 and 6, except for topography with normal one as shown in Fig. 15. Figure 17. View largeDownload slide Similar to Figs 5 and 6, except for topography with normal one as shown in Fig. 15. Figure 18. View largeDownload slide Similar to Figs 5 and 6, except for topography with negative signal to one as shown in Fig. 15. Figure 18. View largeDownload slide Similar to Figs 5 and 6, except for topography with negative signal to one as shown in Fig. 15. 6 CONCLUSIONS By numerically modelling a reference model with flat surface and four groups of models with irregular topography, we study the effect of irregular topography on the near-fault ground motion using the CG-FDM. Our results show that the supershear rupture produces resultant ground motion over a larger area than the subshear rupture does. More importantly, the subshear rupture produces stronger ground motions near fault than the supershear rupture when the hill- or canyon-shaped topography (e.g. H2 and C1/2) prevents the generation of supershear. However, the moment magnitude does not differ substantially between the two cases. The detailed effects of different topography on the resultant ground motion depend on the size of the irregular topography and relative distance to the critical supershear transition distance. Moreover, we investigate and discuss the supershear transition caused by the free surface and high stress drop in full- and half-space, and gain insight into the effects of both the free surface and the initial shear stress on resultant ground motion in full- and half-space. The simulations show that the free surface is an important factor to the distribution of ground motion. Moreover, when two types of supershear (Super I and Super II) exit simultaneously, Super I has greater energy and more strongly affects the ground motion than Super II. We also discuss the size factor of the irregular topography on resultant ground motion to investigate the generality of the topographic phenomenon. The larger topographic perturbation has systematically changed the existence or non-existence of a supershear rupture. At last, we run the dynamic rupture simulations of SAF with real and inverted topographical surfaces to reveal the significance of topography on earthquake rupture and strong ground motion in the reality. Acknowledgements This work is supported by the National Natural Science Foundation of China (grants 41504040 and 41474037), the Science and Technology support plan of Sichuan province (2016SZ0067), and China Postdoctoral Science Foundation (grant 2016T90575). All data used in this paper are acquired from numerical simulations. REFERENCES Andrews D., 1976. 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Geophysical Journal InternationalOxford University Press

Published: Jan 1, 2018

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