Supershear Transition Mechanism Induced by Step Over Geometry

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RESEARCH ARTICLE Supershear transition mechanism induced 10.1002/2016JB013333 by step over geometry Special Section: Feng Hu1,2,3, Jiankuan Xu1,2,3, Zhenguo Zhang1,2,3, and Xiaofei Chen1,2,3 Stress at Active Plate Boundaries - Measurement and 1School of Earth and Space Sciences, University of Science and Technology of China, Hefei, China, 2Mengcheng National Analysis, and Implications for Geophysical Observatory, University of Science and Technology of China, Hefei, China, 3Laboratory of Seismology and Seismic Hazard Physics of Earth Interior, University of Science and Technology of China, Hefei, China

Key Points: • Supershear ruptures can be induced Abstract Few studies have focused on the supershear transition mechanism induced by fault step overs, by a fault step over although seismic observations suggest that rupture speed transitions occur at geometrical complexities on • Stress waves radiated from the end of the primary fault control supershear faults. Based on dynamic rupture simulations on fault systems with step overs in a 3-D full space where the transitions on secondary segments initial stresses preclude a supershear transition on a single buried fault according to the Burridge-Andrews • Rapid rupture speed transitions at mechanism, we show that rupture speeds can transit from subshear on the primary fault to supershear on the step overs in a half-space secondary fault. The low normal stress zone and the high shear stress zone beyond the fault step, which radiate from the end of the primary fault if its rupture arrest is sudden, determine the supershear rupture occurrence on the secondary fault. However, a low shear stress zone traveling at the shear wave speed is also Correspondence to: radiated, making the rupture speed return to subshear in most cases. Sustained supershear ruptures are also X. Chen, [email protected] possible on compressional step overs under certain conditions. Self-arresting ruptures are observed in the overlap area on the secondary fault. In a half-space model where supershear rupture is induced by the free surface on the primary fault, the rupture speed on the secondary fault rapidly transits to subshear near the Citation: Hu, F., J. Xu, Z. Zhang, and X. Chen fault step if its width exceeds a critical value. (2016), Supershear transition mechanism induced by step over geometry, J. Geophys. Res. Solid Earth, 121, 1. Introduction 8738–8749, doi:10.1002/2016JB013333. Rupture speed, which is an essential factor of earthquake dynamics and seismic hazard assessments, may Received 3 JUL 2016 change from subshear to supershear in natural earthquakes. The rising interest in supershear rupture is Accepted 9 NOV 2016 Accepted article online 11 NOV 2016 mainly attributed to the potentially destructive Mach S wave it radiates [Bernard and Baumont, 2005; Published online 14 DEC 2016 Dunham and Bhat, 2008] and its close relationship with the rupture dynamics [Andrews, 1976]. Although many researchers have studied various supershear transition mechanisms on a planar fault [e.g., Andrews, 1985; Fukuyama and Olsen, 2002; Dunham et al., 2003; Dunham, 2007; Kaneko and Lapusta, 2010; Xia et al., 2005], few studies have extended their scope to the supershear transition mechanism caused by complex fault geometries [Oglesby et al., 2008; Ryan and Oglesby, 2014]. Studies of supershear rupture on a planar fault started with Burridge [1973], who ﬁrst found that a shear stress peak propagated at shear wave speed in front of a self-similar mode II crack. The supershear transition mechanism, known as the Burridge-Andrews mechanism, was later conﬁrmed by Andrews [1976] who noticed that a shear stress peak traveling at shear wave speed could initiate a daughter crack ahead of the main rupture if the initial stress exceeded a critical level based on dynamic simulations of a 2-D mode II crack. The initial shear

stress has been further quantified in terms of the nondimensional strength excess: S = (τu τ0)/(τ0 τd), where τu is the yielding strength, τ0 is the initial shear stress, and τd is the residual shear stress on the fault [Andrews, 1976; Das and Aki, 1977]. The maximum S value that allows the occurrence of a supershear transition on an unbounded planar fault is 1.77 in 2-D [Andrews, 1985], and 1.19 in 3-D [Dunham, 2007] and depends on fault width in bounded faults [Dunham, 2007; Xu et al., 2015]. In addition to the Burridge-Andrews mechanism, several different supershear transition mechanisms exist on a planar fault, including the Earth’s free surface [Zhang and Chen, 2006; Kaneko et al., 2008; Kaneko and Lapusta, 2010], favorable stress heterogeneities [Fukuyama and Olsen, 2002; Liu and Lapusta, 2008], bimaterial interfaces [Xia et al., 2005], a barrier on a planar fault [Dunham et al., 2003; Weng et al., 2015], the presence of a low-velocity damaged fault zone [Huang et al., 2016; Perrin et al., 2016], and fault roughness [Bruhat et al., 2016]. Supershear ruptures have also been observed in laboratory studies [e.g., Rosakis et al., 1999; Xia et al., 2004]. fi ©2016. American Geophysical Union. The rst earthquake proposed to have supershear rupture was the 1979 Imperial Valley earthquake (Mw = 6.5) All Rights Reserved. [Archuleta, 1984]. With the advance of strong motion seismograph networks in the near field, more evidence

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of supershear rupture has been found in natural earthquakes, such as, the 1999

Izmit earthquake (Mw = 7.4) [Bouchon et al., 2000; Bouchon et al., 2001], the

2001 Kokoxili earthquake (Mw = 7.8) [Bouchon and Vallée, 2003], the 2002

Denali earthquake (Mw = 7.9) [Dunham and Archuleta, 2004], the 2010 Yushu

earthquake (Mw = 6.9) [Wang and Mori, 2012], and the 2013 Sea of Okhotsk

earthquake (Mw = 6.7) [Zhan et al., 2014]. Most of them are strike-slip earthquakes that occurred on complex fault systems. Step overs are a common geometrical Figure 1. Diagram of the step over model (an extensional case is shown complexity of fault systems. Although here), composed of two parallel vertical planar right-lateral strike-slip fault segments in a homogeneous full space. numerous studies have been performed on step overs to evaluate the rupture jump capability including both numerical [e.g., Harris et al., 1991; Harris and Day, 1993; Duan and Oglesby, 2006; Hu et al., 2016] and geological observation methods [e.g., Lettis et al., 2002; Wesnousky, 2006], few researchers have focused on the supershear transition mechanism induced by a step over geometry. Oglesby et al. [2008] ﬁrst noticed the supershear transition on a complex fault system with strike variations. Using an array of broadband stations deployed in Nepal, Vallée et al. [2008] tracked the rupture speed of the 2001 Kokoxili earthquake and showed that ruptures accelerated to supershear velocity and decelerated to the sub-Rayleigh regime in areas that were well correlated with geometrical fault complexities. With a linking fault model in 2-D simulations, Lozos et al. [2011] also noticed the occurrence of supershear rupture on an extensional step over. Based on 2-D numerical simulations, Ryan and Oglesby [2014] found that ruptures across step overs could lead to a supershear transition and showed that the minimum step width required for sustained supershear ruptures on compressional regimes obeying linear slip-weakening law was 0.6 km if the initial S value was 2.6 on the step overs. However, according to the critical jump distance study by

Hu et al. [2016], when S ≥ 1.5(Te = 1/(1 + S) ≤ 0.4) in a homogeneous 3-D full space or S ≥ 1.85 (Te ≤ 0.35) in a half-space, ruptures on both compressional and extensional step overs either die on the primary fault or stop soon after renucleation on the secondary fault. Thus, it is necessary to perform 3-D dynamic rupture simulations on step overs and investigate the supershear transition mechanism induced by step overs when ruptures on the primary fault travels at subshear speed in the full space. The rupture speeds on the secondary fault are also studied in the half-space where the supershear rupture is induced by the Earth’s free surface on the primary fault [Zhang and Chen, 2006; Kaneko et al., 2008; Kaneko and Lapusta, 2010].

2. Models and Methods We ﬁrst consider a simple 3-D step over model, which is composed of two parallel vertical right-lateral strike-slip fault segments, in a homogeneous, isotropic, linearly elastic full space (Figure 1). A fault segment parallel to the primary fault with a positive X coordinate represents an extensional segment. A compressional segment is a plane located at a negative X coordinate. Rupture starts from a circular nucleation patch with a radius of 3000 m centered at (yH, zH) = (6 km, 7.5 km). The primary fault is 40 km long, and the secondary fault is 30 km long with an 8 km overlap. Both fault widths are 15 km. The step width is ﬁxed at 1 km in the full space. The shear stresses on fault planes are bounded by the frictional strength [Day et al., 2005]. Spontaneous rupture propagates along fault planes obeying the linear slip-weakening friction

law [Ida, 1972; Andrews, 1976]. In that form, the frictional strength τc is

τc ¼τnμðÞℓ ; (1)

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Table 1. Medium and Friction Parameters which is the product of compressive normal

Parameters Value stress τn (positive in tension) and a coefficient of friction μ(ℓ) that depends on the slip S wave velocity, Vs 3.464 km/s ℓ fi μ ℓ P wave velocity, Vp 6.0 km/s path length . The coef cient of friction ( ) Density, ρ 2670 kg/m3 outside the nucleation patch is given by 8 Shear modulus, G 32038 MPa ðÞμ μ ℓ Normal stress, τ 120 MPa < s d n μ ; ℓ < Dc fi μ μðÞ¼ℓ s D ; Static friction coef cient, s 0.677 : c (2) fi μ μ ℓ > Dynamic friction coef cient, d 0.525 d; Dc The characteristic slip distance, Dc 0.4 m where μs and μd are coefficients of static and dynamic friction, respectively, and Dc is the critical slip-weakening distance. We assume no mechanism for the fault to heal and restrengthen to its static friction after a termination of a rupture event in the present paper. The medium and friction parameters are given in Table 1. Starting at time t = 0, the dynamic rupture on the primary fault is initiated by imposing a

time-varying coefficient of friction μn(ℓ, r, t) in the circular nucleation patch, where r is the distance from the hypocenter. The time-varying coefficient of friction μn(ℓ, r, t) is equal to the slip-weakening coefficient of friction μ(ℓ) when r ≥ 3000 m in the present paper. Descriptions of μn(ℓ, r, t) and the nucleation process are described in Appendix A. We simulate the dynamic rupture processes on step overs using the 3-D curved grid finite difference method developed by Zhang et al. [2014]. The spacing grid is chosen to be 50 m, and the time interval is 0.005 s to appropriately resolve the cohesive zone following the rationale introduced by Day et al. [2005]. In particular, we verified in our simulation outputs that the process zone (the region near the rupture front where slip- weakening occurs) is resolved by at least five fault nodes. As proposed by Bizzarri et al. [2010], the rupture

speed Vr(y, z) on each point on the faults is 1 VrðÞ¼y; z ; (3) jjjj∇trðÞy; z

where tr is the time when the slip velocity at fault point (y, z) ﬁrst exceeds 0.01 m/s. The gradient ∇tr(y, z)is calculated using equations (2) and (3) of Bizzarri and Das [2012].

3. Numerical Results We ﬁrst consider the dynamic rupture simulation results in the full space when the rupture speed on the primary fault is designed to stay subshear. Although the initial stresses on the primary fault preclude the inherent supershear transition on a planar fault based on the Burridge-Andrews mechanism, supershear rupture still occurs when the fault plane reaches the Earth’s free surface [Kaneko and Lapusta, 2010]. In section 3.4, dynamic rupture simulations and rupture speed distributions on step overs in the half-space with the same simulation parameters are presented for comparison. The supershear transition on a planar fault in a homogeneous 3-D full space is controlled by the initial S value [Andrews, 1985]. Rupture on an unbounded planar fault in the full space can only travel and remain at subshear speed if S > 1.19 [Dunham, 2007]. Thus, we simulate dynamic rupture processes in the present study with S ≥ 1.2 on step overs to guarantee that the occurrence of supershear rupture is not triggered by the Burridge-Andrews daughter crack mechanism at a low initial S value. Unlike the planar fault case, the initial S value on step overs cannot be much larger than 1.2, because rupture on a step over in a 3-D homogeneous full space with S ≥ 1.5 dies at the end of the primary fault or ceases soon on the secondary fault after renucleation based on previous results on the critical jump distance for step overs [Hu et al., 2016]. With the same fault length and width of the primary fault as assumed in the present paper, Hu et al. [2016] performed 3-D dynamic rupture simulations on step overs with different step widths in a half-space and analyzed the critical jump distance for step overs with different initial S values, background normal stresses, fault buried depths, and nucleation patch locations. Bai and Ampuero [2014] noticed the ratio between critical jump distance and fault width W depends on the nondimensional ratio

Lc/W, (where Lc is deﬁned as GDc/Δτ, G is the shear modulus, and Δτ =|τn|(μs μd) is the breakdown stress drop). With the medium and friction parameters shown in Table 1, the ratio Lc/W is 0.047, the same value as used in the discussion of critical jump distance by Hu et al. [2016]. The critical jump distance for step overs

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in the full space is less than or equal to 1.0 km when the initial S value is between 1.2 and 1.5 [Hu et al., 2016]. Thus, dynamic rupture simulations on both compressional and extensional step overs are performed with a 1.0 km step width and three different initial S values of 1.2, 1.4, and 1.5 on secondary faults. The initial S value on the primary fault is always 1.2 to guarantee that ruptures can jump onto a fault step with a 1.0 km step width. The initial shear stresses corresponding to the initial S values of 1.2, 1.4, and 1.5 are 71.3, 70.6, and 70.3 MPa, respectively. Although different initial S values on the two fault segments of step overs used in the present study are arbitrarily designed to satisfy the subshear rupture speed conditions on the primary fault and successful rupture jump conditions on the secondary fault, the parameters are still reasonable in natural earthquakes, because heterogeneous fault stresses could be induced Figure 2. Distributions of NRS on the compressional step overs with a 1.0 km step width and three different initial S values 1.2, by previous earthquakes as proposed by Duan 1.4, and 1.5 on secondary fault segments. The initial S value on and Oglesby [2006]. the primary fault is always 1.2. The black and magenta lines represent rupture fronts with 1.0 s interval on the primary and 3.1. Compressional Step Overs secondary faults, respectively. Two rupture front contours of the secondary fault are labeled. Compressional step overs with a 1 km step width in the full space are considered ﬁrst. Figure 2 shows the distribution of nondimensional

rupture speed (NRS, deﬁned as the rupture speed Vr divided by the shear wave speed Vs) with three different S values on secondary faults. The NRS is convenient to indicate the zones of supershear rupture. The color

scales are capped at VP/Vs in all the figures showing the distribution of NRS in the present study. The rupture fronts at 1 s intervals are also plotted in Figure 2 (black lines on the primary fault, magenta lines on the secondary fault). On the primary fault, the NRS decreases near the rim of the prescribed nucleation region because of the nucleation method (Appendix A), which was designed to avoid local supershear induced by the nucleation area. Ruptures on the primary fault, which travel at subshear speed, arrive at the end of the primary fault between 12.0 and 13.0 s. The NRS distributions on the secondary fault segments are symmetric with respect to depth and depend on the assumed initial S values. On the secondary fault, when the initial S value is also 1.2, ruptures start at 14.0 s near the top edge, the bottom edge, and the middle depth. Renucleation occurs slightly beyond the end of the fault step, in consistency with the Coulomb friction analysis of compressional step overs provided by Harris et al. [1991], Harris and Day [1993], and Hu et al. [2016]. The rupture then propagates in both directions along strike. Supershear rupture is immediately observed in the forward rupture front beyond the fault step. However, the backward rupture in the overlap area of the secondary fault travels at subshear speed and gradually stops without reaching the end of the fault. Such phenomenon is defined as self-arresting. It is known that a crack-like rupture can stop spontaneously if its available elastic energy is insufficient to overcome fracture energy, which is expressed by a critical nucleation size for sustained rupture [e.g., Andrews, 1976; Galis et al., 2015; Xu et al., 2015]. Thus, our work shows that self-arresting rupture can also occur in the overlap area on the secondary fault of step overs. By contrast, the forward rupture beyond the fault step does not stop before it reaches the end of the secondary fault. Different from the Burridge-Andrews mechanism, in which rupture on a planar fault must propagate a critical distance before the supershear transition occurs, the supershear ruptures on the secondary fault induced by the step over geometry occur immediately after renucleation. At points near the top and bottom edges on the secondary fault located slightly beyond the end of the primary fault, supershear ruptures turn into subshear ruptures at 15.0–16.0 s. However, the supershear rupture at the middle depth propagates a longer

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distance before turning subshear at 17.0–18.0 s near 54 km in the Y direction. Thus, the rupture on the secondary fault of the compressional step over is an unsustained supershear rupture when S = 1.2. To understand the effect of different initial stresses on the secondary fault in determining the distribution of rupture speed, two more cases with initial S values of 1.4 and 1.5 on the secondary fault are also shown in Figure 2. Although supershear ruptures near the top and bottom edges still exist, they turn to subshear ruptures at 14.0–15.0 s, which is earlier than the transition time when S = 1.2. A sustained supershear rupture is observed when S = 1.4. An intersonic daughter crack starts in the middle depth, which runs ahead of the main rupture and jumps to an intersonic speed at 15.0–16.0 s. Concavities of the rupture fronts on the secondary fault are Figure 3. Same as Figure 2 but for the extensional step overs. noticed, which may contribute to the efﬁciency of the supershear transition in the middle depth. When S = 1.5, the rupture travels at subshear speed except for a small local supershear patch in the middle depth and local supershear rupture near the top and bottom edges of the secondary fault. Without the occurrence of the sustained supershear rupture in the middle depth, the rupture on the secondary fault is slower if the initial S value is larger, as expected from the lower available elastic energy.

3.2. Extensional Step Overs Dynamic rupture simulations are also performed on extensional step overs with the three initial S values on the secondary faults. Figure 3 shows the resulting distribution of NRS and rupture fronts. Renucleation on the secondary fault always occurs after rupture fronts encounter the end of the primary fault [Harris et al., 1991; Harris and Day, 1993]. Thus, the distributions of NRS on the primary fault are the same as those on the compressional step overs (Figure 2). The rupture front also encounters the end of the primary fault between 12.0 and 13.0 s. However, the rupture on the secondary fault of the extensional step over starts at 13.0 s, earlier than the 14.0 s in the compressional cases. This differs from the 2-D simulation results shown by Harris et al. [1991], in which extensional steps are triggered later than the compressional steps with equal step width. Thus, the trigger time differences between extensional and compressional step overs may need further investigation. After renucleation, forward supershear ruptures can also be immediately observed on the secondary fault and eventually return to subshear speeds for all the three initial S values. Moreover, self-arrest of the backward front is also noticed in the overlap area on the secondary fault. The distributions of NRS on the extensional steps are also symmetric with respect to depth. When S = 1.2 on the secondary fault, supershear ruptures occur immediately near the top and bottom edges of the fault after renucleation. However, different from the result of compressional step overs, the rupture in the middle depth has to propagate a small distance before it turns into a supershear rupture. As shown in Figure 3, the transition from supershear to subshear on the secondary fault occurs gradually from the top and bottom edges to the middle depth. The supershear rupture in the middle depth returns to a subshear rupture at 16.0–17.0 s. If S is larger, the supershear patches near the top and bottom edges gradually disappear. When S = 1.4, the supershear rupture in the middle depth also returns to subshear speed at 16.0–17.0 s. When S = 1.5, the transition time from supershear to subshear is delayed to 17.0–18.0 s, and the rupture on the extensional segment propagates toward the top and bottom edges. Differences in rupture direction may affect the amplitudes of near-ﬁeld strong ground motion, because rupture directivity effects cause spatial variations in ground motion amplitude and differences between the strike-normal and strike-parallel components of horizontal ground motion [Somerville et al., 1997].

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Figure 4. (a–d) Slip rate snapshots at four different times with an initial S value of (a) 1.2, (b) 1.4, (c) 1.5 on the compressional segment, and (d) an initial S value of 1.2 on the extensional segment. The initial S value on the primary fault is always 1.2. (e–h) Normal stress and (i–l) shear stress snapshots on the segment shown in Figures 4a–4d, respectively.

3.3. Supershear Transition Mechanism To better understand the supershear transition mechanism on step overs, we depict the slip rate snapshots on the compressional segment at four different times with the three initial S values of 1.2, 1.4, and 1.5 in Figures 4a, 4b, and 4c, respectively. As a comparison, Figure 4d shows the slip rate distribution on the extensional segment with an initial S value of 1.2. Their initial S values on the primary fault are 1.2, the same as presented in Figures 2 and 3. Two phenomena are mainly addressed here. One is the occurrence of supershear on the secondary fault immediately after renucleation. The slip rates in the compressional regimes are shown at 13.5–14.0 s, just when renucleation occurs (Figures 4a–4c). Additionally, we select the slip rates on the extensional segment at 13.0–13.5 s (Figure 4d) because they are triggered earlier than compressional segments with the same step width in the full space here. When S = 1.2 on the compressional segment, renucleation patches are observed near the top and bottom edges at 13.5 s, and supershear ruptures occur at 14.0 s. Rupture also reinitiates in the middle depth at 14.0 s. The rupture speed is lower near the top and bottom edges at 14.0 s when the initial S value is larger. Ruptures in the extensional regime (Figure 4d), which reinitiate at 13.0 s, rapidly spread along the depth near 40 km in the Y direction at 13.5 s.

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The other important phenomenon is the transition from supershear to subshear on the secondary fault. As noticed in Figure 2, the transition from supershear to subshear in the middle depth of the compressional segment with an initial S value of 1.2 is shown at 17.0–18.0 s (Figure 4a). The sustained supershear on the secondary fault when S = 1.4 is observed in Figure 4b. The rupture mainly travels at subshear speed when S = 1.5 (Figure 4c). For the extensional segment, the transition from supershear to subshear occurs at 16.0–17.0 s (Figure 4d). The stress perturbation induced by the geometrical complexity is the key factor in determining the supershear transition mechanism on the secondary fault. Unlike a planar fault in the full space, normal stresses on step overs are not constant. Figures 4e–4h show the normal stress distribution on the secondary fault at the 4 times in the four cases presented in Figures 4a–4d, respectively. Close to the fault step, the normal stresses increase on compressional segments but decrease sharply on extensional segments. This confirms the dynamic clamping and unclamping of compressional and extensional step overs, respectively [Oglesby, 2005; Lozos et al., 2011]. An alternately low and high normal stress zone also exists on the secondary fault in front of the fault step. As shown in Figures 4e–4g, the locations of the high normal stress zone are identical despite the different initial S values on the secondary fault segments, indicating that the perturba- tions of normal stresses on the secondary fault are induced by the sudden stop of rupture fronts at the end of the primary fault. The low normal stress zone near 40 km in the strike direction on the extensional step over (Figure 4h) explains the rapid spreading of the rupture front along the depth close to the fault step at 13.5 s (Figure 4d). Moreover, the amplitude of the high normal stress zone in front of the fault step decreases as the rupture front propagates. Thus, the perturbation of normal stress may control the occurrence of supershear rupture on the secondary fault. The variation in normal stress changes the fault strength, and the rupture starts when the shear stress increases to the fault strength. As shown in Figures 4i–4k, the high shear stresses on the compressional segments concentrate at 13.5–14.0 s in front of the ruptured front near the top edge, the bottom edge, and the middle depth, where a low normal stress zone is already noticed (Figures 4e–4g). Thus, the high shear stresses and the low normal stresses there decrease the S value, which can cause a supershear transition if it is lower than the maximum S value [Andrews, 1976; Dunham, 2007]. However, the amplitudes of shear stress are lower if the initial S value is larger on the fault segments. Thus, the rupture at 14.0 s is slower for a larger initial S value (Figures 4a–4c). In the full space, the rupture front first encounters the end of the primary fault in the middle depth. Thus, the stress wave traveling at shear wave speed is radiated from the middle depth first and reaches the top and bottom edges on the primary fault last. This explains why the high shear stress zone (Figures 4e–4g) and the low normal stress zone (Figures 4i–4k) on the compressional segment at 14.0 s are not perpendicular to the top and bottom edges. The concavities of rupture fronts may contribute to the efficiency of supershear transition in the middle depth on the secondary fault. Moreover, as previously noticed in Figure 2, the supershear rupture on the compressional segment with an initial S value of 1.2 occurs immediately after renucleation. Festa and Vilotte [2006] and Lu et al. [2009] found that supershear rupture can occur immediately after nucleation on a single fault if the nucleation is vigorous enough, i.e., if it has a large nucleation area and/or high stress. Both conditions are likely met in our numerical simulations because of the dynamic stress carried by the coherent stopping phase from the end of the primary fault rupture. On the extensional segment, the high shear stress zone is also observed at 13.5 s (Figure 4l). In front of the high stress zone, there is also a low shear stress zone that controls the transition from supershear to subshear on the secondary fault in most cases simulated in the present paper. As shown in Figures 4i–4k, the locations of the low shear stress zones also are nearly identical despite different initial S values. Thus, they are also radiated from the abrupt arrest at the end of the primary fault. When the initial S value is 1.2 on the compressional segment (Figure 4i), the low shear stress zone travels

at shear wave speed (Vs = 3.464km/s) in the middle depth of the secondary fault. When the supershear rupture is formed, it soon accelerates to the limiting velocity (P wave velocity) and becomes a subshear rupture when it reaches the low shear stress zone on the secondary fault. When S = 1.4 (Figure 4j), an intersonic daughter crack is observed at 17.0 s in the middle depth and develops into a sustained supershear rupture before it encounters the low shear stress zone. When S = 1.5, the initial shear stress is too small to generate a sustained supershear rupture (Figure 4k). On the extensional step over, the transition from supershear to subshear also occurs when S = 1.2 (Figure 4l). The shear stress in the overlap area on

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Figure 5. Distributions of NRS on (a) extensional and (b) compressional step overs with four different step widths. The black and magenta lines represent rupture fronts with 1.0 s interval on the primary and secondary faults, respectively. The timing of two rupture front contours of the secondary fault is labeled.

the secondary fault is relatively low because of the stopping phase at the strong boundary at the end of the primary fault. The stress shadow helps understanding the self-arresting phenomenon observed in the overlap area.

3.4. Distributions of Rupture Speed in a Half-Space The fault segments where supershear ruptures are observed in natural earthquakes usually reach the Earth’s free surface. Locally supershear ruptures near the free surface in strike-slip earthquakes can also exist because of the generalized Burridge-Andrews mechanism and the phase conversion of SV to P-diffracted waves at the free surface [Kaneko and Lapusta, 2010]. Moreover, Hu et al. [2016] have shown that the existence of the free surface also enhances the jump capability of ruptures on strike-slip step overs. The critical jump distance for ruptures without the Burridge-Andrews daughter crack is less than 4 km and further decreases to 2 km if we assume S ≥ 1.2. To verify the effect of step widths in determining the supershear transition, Figures 5a and 5b show the distributions of NRS on extensional and compressional step overs with four different step widths from 0.5 km to 2.0 km in the half-space and initial S values set to 1.2. Unlike the full space results shown in Figures 2 and 3, supershear ruptures are induced by the free surface on the primary fault. The rupture front encounters the end of the primary fault at 9.0–10.0 s, which is earlier than the full-space results. We examine whether supershear ruptures continue along the secondary fault. As shown in Figure 5a, when the step width is 0.5 km, supershear ruptures on the extensional segment immediately occur in front of the fault discontinuity after renucleation at 11.0 s and then merge with the supershear rupture induced by the free surface and propagate downward. If the step width is 1.0 km, the supershear rupture generated in the upper part propagates toward the free surface after renucleation at 12.0 s and turns into subshear earlier as the depth increases. A separate supershear patch is present near the bottom edge. However, when the step width is even larger, 1.5 km, renucleation occurs at 13.0 s close to the fault step near the bottom edge, where a small supershear patch exists. The rupture travels at subshear speed in the upper part beyond the fault step until the occurrence of the supershear rupture induced by the free surface at 18.0 s. If the step width is 2.0 km, the rupture dies soon on the extensional segment (Figure 5a) but not on the compressional segment (Figure 5b). This coincides with the critical jump distance studies by Hu et al. [2016].

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As shown in Figure 5b, when the step width is 0.5 km, the supershear rupture on the compressional segment occurs immediately after renucleation at 11.0 s beyond the fault step. The distribution of NRS seems to be nearly continuous across the two faults segments. As the step width increases to 1.5 km, the area of supershear rupture decreases in the upper part of the secondary fault and increases close to the fault step near the bottom edge. When the step width is 2.0 km, the supershear rupture, which returns to the subshear rupture at 15.0–16.0 s, concentrates at a small patch near the free surface after renucleation at 12.0 s. A supershear rupture is also generated from the small supershear patch close to the free surface and propagates upward. In the half-space, extensional steps are triggered later than compressional steps of the same step width (Figures 5a and 5b), which coincides with the results by Harris et al. [1991]. However, the results in the full space presented in Figures 2 and 3 are contrary. Thus, the free surface plays an important role in determining the trigger time differences between extensional and compressional steps. Moreover, because of the supershear rupture on the primary fault induced by the free surface, the subshear rupture speed close to the fault step on the secondary fault, including the extensional step over with a 1.5 km step width and the compressional step over with a 2.0 km step width, represents a sharp rupture speed transition that may increase the high-frequency radiation according to Madariaga [1977]. Vallée et al. [2008] tracked the rupture speed in the 2001 Kokoxili earthquake with an array of broadband stations deployed in Nepal and showed ruptures accelerated to supershear speed and decelerated to the sub-Rayleigh speed in areas well correlated with geometrical fault complexities. Our simulations provide numerical supports which may assist in understanding their observations.

4. Conclusions Previous studies have shown that a barrier can generate a supershear transition on a planar fault [Dunham et al., 2003; Weng et al., 2015]. Hu et al. [2014] stated that a step over model in a homogeneous half-space could produce similar ground motions than a barrier on a planar fault. This inspired us to investigate the supershear transition induced by a step over. Our simulation results indicate that step overs can play a key role in determining an earthquake’s rupture speed transition to supershear, especially near the fault step on the secondary fault segment. Supershear rupture can be observed on the secondary fault of both compressional and extensional step overs in 3-D full-space models immediately after renucleation, even if the initial shear stresses are low enough to preclude a supershear transition on a single buried fault according to the Burridge-Andrews mechanism. There is no need for the rupture to propagate a critical distance before the supershear transition as predicted on a planar fault [Andrews, 1976]. These results show that step overs, a common geometrical complexity on natural faults [Wesnousky, 2006], need to be carefully treated in earthquake hazard assessment. The low normal stress zone and the high shear stress zone on the secondary fault close to the fault step, induced by the sudden arrest of the rupture front at the end of the primary fault, determine the occurrence of supershear rupture on the secondary fault. Depending on the initial stress, the rupture speeds on the secondary fault beyond the fault step may remain supershear or return to subshear. The latter is mainly due to a low shear stress zone traveling at shear wave speed beyond the fault step on the secondary fault, which is radiated from the end of the primary fault and decreases the rupture speed when the supershear rupture front catches up with it. In the overlap area of the secondary fault of both compressional and extensional step overs, backward self-arresting ruptures are noticed. In half-space models, where supershear rupture is induced by the free surface on the primary fault [Zhang and Chen, 2006; Kaneko et al., 2008; Kaneko and Lapusta, 2010], we find that if the step over offset is short, the supershear rupture occurs immediately beyond the fault step. Thus, the distribution of rupture speed seems to be nearly continuous across the primary and secondary faults. However, on a wider extensional step over, the rupture on the secondary fault travels at subshear speed close to the free surface immediately after renucleation. Rapid rupture speed transitions can occur on sufficiently wide compressional step overs from supershear on the primary fault (induced by the free surface) to subshear beyond the fault step on the secondary fault, which may contribute to high frequency radiation at near field [Madariaga, 1977]. By contrast, Lozos et al. [2013] have shown that the presence of a step over reduces the maximum ground motion compared to a long planar fault. Further investigation is necessary to quantify the effect of step overs on near-field high-frequency strong ground motion.

HU ET AL. SUPERSHEAR BY STEP OVER 8746 Journal of Geophysical Research: Solid Earth 10.1002/2016JB013333

Appendix Rupture Initiation Procedure To smoothly nucleate dynamic rupture on the primary fault, we artificially reduce the friction coefficient beginning at a specified time T within a circular zone with a 3000 m radius surrounding the hypocenter exactly following the strategy used in the Southern California Earthquake Center (SCEC) benchmark TPV22

[Harris et al., 2009]. The time-varying coefﬁcient μn(ℓ, r, t) is given by the following formulas: μ ðÞ¼ℓ; ; μ þ ðÞμ μ ðÞ; ; n r t s d s max f 1 f 2 (A1) 8 < ℓ ; if ℓ < D ¼ c ; f 1 : Dc (A2) 1; if ℓ ≥ Dc 8 ; < > 0 if t T < ¼ t T; ≤ < þ ; f 2 > if T t T t0 (A3) :> t0 1; if t ≥ T þ t0 8 : > r 0 0 081rcrit 1 > þ ; if r < rcrit > 0:7Vs > B C < B 1 C T ¼ 0:7VsB 1C ; (A4) > @ r 2 A > 1 > > rcrit : 9 10 s; if r ≥ rcrit

where ℓ is the slip, T is the initial time of forced rupture, r is the distance from the hypocenter, and the

radius of nucleation zone rcrit is 3000 m. The coefﬁcient of friction decreases to its ﬁnal dynamic value at time t = T + t0. Thus, each fault point within the forced nucleation region begins to slide somewhere in the interval T < t < T + t0. In the present paper t0 is selected to be 0.5 s. The time T is computed so that the forced rupture expands at a variable speed. Near the hypocenter, the forced rupture expands at a

speed of 0.7 Vs. The rupture speed decreases as rupture front moves away from the hypocenter, ﬁnally reaching a speed of zero at a distance of 3000 m from the hypocenter. The variable speed allows for a smooth transition between forced rupture and spontaneous rupture, which guarantee that locally supershear ruptures induced by large nucleation area cannot exist. The time-varying coefﬁcient of

friction μn(ℓ, r, t) is equal to the slip-weakening coefﬁcient of friction μ(ℓ), which is described in section 2, when r ≥ rcrit. Further details can be found in the description of the SCEC benchmark TPV22 (http://scecdata.usc.edu/cvws).

Acknowledgments References We thank Steve Day, Julian Lozos, and – Jean Paul Ampuero for their insightful Andrews, D. J. (1976), Rupture velocity of plane strain shear cracks, J. Geophys. Res., 81(32), 5679 5687, doi:10.1029/JB081i032p05679. comments that helped improve the Andrews, D. J. (1985), Dynamic plane-strain shear rupture with a slip-weakening friction law calculated by a boundary integral method, Bull. – manuscript. No data were used in this Seismol. Soc. Am., 75(1), 1 21. – study. Simulations were done using the Archuleta, R. J. (1984), A faulting model for the 1979 Imperial Valley earthquake, J. Geophys. Res., 89(6), 4559 4585, doi:10.1029/ ﬁnite difference code CG-FDM, cited in JB089iB06p04559. the references, and numerical data are Bai, K., and J. P. Ampuero (2014), The effect of seismogenic zone depth on the likelihood of fault stepover jump, AGU Fall Meeting, available to anyone upon request Abstracts 1, 4446. ([email protected]). This work was Bernard, P., and D. Baumont (2005), Shear Mach wave characterization for kinematic fault rupture models with constant supershear rupture – supported by the National Natural velocity, Geophys. J. Int., 162, 431 447, doi:10.1111/j.1365-246X.2005.02611.x. Science Foundation of China (grants Bizzarri, A., and S. Das (2012), Mechanics of 3-D shear cracks between Rayleigh and shear wave rupture speeds, Earth Planet. Sc. Lett., – – 41504039 and 41274053) and partially 357 358, 397 404, doi:10.1016/j.epsl.2012.09.053. by China Postdoctoral Science Bizzarri, A., E. M. Dunham, and P. Spudich (2010), Coherence of Mach fronts during heterogeneous supershear earthquake rupture Foundation (grant 2016 M590574), the propagation: Simulations and comparison with observations, J. Geophys. Res., 115, B08301, doi:10.1029/2009JB006819. Fundamental Research Funds for the Bouchon, M., and M. Vallée (2003), Observation of long supershear rupture during the magnitude 8.1 Kunlunshan earthquake, Science, – Central Universities (grant 301(5634), 824 826, doi:10.1126/science.1086832. WK2080000072), and the Science and Bouchon, M., N. Toksöz, H. Karabulut, M. P. Bouin, M. Dietrich, M. Aktar, and M. Edie (2000), Seismic imaging of the 1999 Izmit (Turkey) rupture – Technology Support Plan of Sichuan inferred from the near-fault recordings, Geophys. Res. Lett., 27(18), 3013 3016, doi:10.1029/2000GL011761. Province (grant 2016SZ0067). Bouchon, M., M. P. Bouin, H. Karabulut, N. Toksöz, M. N. Dietrich, and A. J. Rosakis (2001), How fast is rupture during an earthquake? New insights from the 1999 Turkey earthquakes, Geophys. Res. Lett., 28(14), 2723–2736, doi:10.1029/2001GL013112. Bruhat, L., Z. Fang, and E. M. Dunham (2016), Rupture complexity and the supershear transition on rough faults, J. Geophys. Res. Solid Earth, 121, 210–224, doi:10.1002/2015JB012512.

HU ET AL. SUPERSHEAR BY STEP OVER 8747 Journal of Geophysical Research: Solid Earth 10.1002/2016JB013333

Burridge, R. (1973), Admissible speeds for plane-strain self-similar shear cracks with friction but lacking cohesion, Geophys. J. Int., 35(4), 439–455, doi:10.1111/j.1365-246X.1973.tb00608.x. Das, S., and K. Aki (1977), Fault plane with barriers: A versatile earthquake model, J. Geophys. Res., 82(36), 5658–5670, doi:10.1029/ JB082i036p05658. Day, S. M., L. A. Dalguer, N. Lapusta, and Y. Liu (2005), Comparison of finite difference and boundary integral solutions to three-dimensional spontaneous rupture, J. Geophys. Res., 110, B12307, doi:10.1029/2005JB003813. Duan, B., and D. D. Oglesby (2006), Heterogeneous fault stresses from previous earthquakes and the effect on dynamics of parallel strike-slip faults, J. Geophys. Res., 111, B05309, doi:10.1029/2005JB004138. Dunham, E. M. (2007), Conditions governing the occurrence of supershear ruptures under slip-weakening friction, J. Geophys. Res., 112, B07302, doi:10.1029/2006JB004717. Dunham, E. M., and H. S. Bhat (2008), Attenuation of radiated ground motion and stresses from three-dimensional supershear ruptures, J. Geophys. Res., 113, B08319, doi:10.1029/2007JB005182. Dunham, E. M., and R. J. Archuleta (2004), Evidence for a supershear transient during the 2002 Denali fault earthquake, Bull. Seismol. Soc. Am., 94(6B), S256–S268, doi:10.1785/0120040616. Dunham, E. M., P. Favreau, and J. M. Carlson (2003), A supershear transition mechanism for cracks, Science, 299, 1557–1559, doi:10.1126/ science.1080650. Festa, G., and J. P. Vilotte (2006), Influence of the rupture initiation on the intersonic transition: Crack-like versus pulse-like modes, Geophys. Res. Lett., 33, L15320, doi:10.1029/2006GL026378. Fukuyama, E., and K. B. Olsen (2002), A condition for super-shear rupture propagation in a heterogeneous stress field, Pure Appl. Geophys., 159, 2047–2056, doi:10.1007/978-3-0348-8203-3_9. Galis, M., C. Pelties, J. Kristek, P. Moczo, J. P. Ampuero, and P. M. Mai (2015), On the initiation of sustained slip-weakening ruptures by localized stresses, Geophys. J. Int., 200(2), 890–909, doi:10.1093/gji/ggu436. Harris, R. A., and S. M. Day (1993), Dynamics of fault interaction: Parallel strike-slip faults, J. Geophys. Res., 98(B3), 4461–4472, doi:10.1029/ 92JB02272. Harris, R. A., R. J. Archuleta, and S. M. Day (1991), Fault steps and the dynamic rupture process: 2-D numerical simulations of a spontaneously propagating shear fracture, Geophys. Res. Lett., 18(5), 893–896, doi:10.1029/91GL01061. Harris, R. A., et al. (2009), The SCEC/USGS dynamic earthquake-rupture code verification exercise, Seismol. Res. Letts., 80(1), 119–126. Hu, F., J. Xu, Z. Zhang, W. Zhang, and X. Chen (2014), Construction of equivalent single planar fault model for strike-slip stepovers, Tectonophysics, 632, 244–249, doi:10.1016/j.tecto.2014.06.025. Hu, F., Z. Zhang, and X. Chen (2016), Investigation of earthquake jump distance for strike-slip step overs based on 3-D dynamic rupture simulations in an elastic half-space, J. Geophys. Res. Solid Earth, 121, 994–1006, doi:10.1002/2015JB012696. Huang, Y., J. P. Ampuero, and D. V. Helmberger (2016), The potential for supershear earthquakes in damaged fault zones—Theory and observations, Earth Planet. Sci. Lett., 433, 109–115, doi:10.1016/j.epsl.2015.10.046. 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, doi:10.1029/JB077i020p03796. Kaneko, Y., and N. Lapusta (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, doi:10.1016/j.tecto.2010.06.015. Kaneko, Y., N. Lapusta, and J. P. Ampuero (2008), Spectral element modeling of spontaneous earthquake rupture on rate and state faults: Effect of velocity-strengthening friction at shallow depths, J. Geophys. Res., 113, B09317, doi:10.1029/2007JB005553. Lettis, W., J. Bachhuber, R. Witter, C. Brankman, C. E. Randolph, A. Barka, W. D. Page, and A. Kaya (2002), Influence of releasing step-overs on surface fault rupture and fault segmentation: Examples from the 17 August 1999 Izmit earthquake on the North Anatolian Fault, Turkey, Bull. Seismol. Soc. Am., 92,19–42. Liu, Y., and N. Lapusta (2008), Transition of mode II cracks from sub-Rayleigh to intersonic speeds in the presence of favorable heterogeneity, J. Mech. Phys. Solids, 56(1), 25–50, doi:10.1016/j.jmps.2007.06.005. Lozos, J. C., D. D. Oglesby, B. Duan, and S. G. Wesnousky (2011), The effects of double fault bends on rupture propagation: A geometrical parameter study, Bull. Seismol. Soc. Am., 101(1), 385–398. Lozos, J. C., D. D. Oglesby, and J. N. Brune (2013), The effects of fault stepovers on ground motion, Bull. Seismol. Soc. Am., 103(3), 1922–1934. Lu, X., N. Lapusta, and A. J. Rosakis (2009), Analysis of supershear transition regimes in rupture experiments: The effect of nucleation conditions and friction parameters, Geophys. J. Int., 177(2), 717–732. Madariaga, R. (1977), High-frequency radiation from crack (stress drop) models of earthquake faulting, Geophys. J. R. Astron. Soc., 51, 625–651. Oglesby, D. D. (2005), The dynamics of strike-slip step-overs with linking dip-slip faults, Bull. Seismol. Soc. Am., 95(5), 1604–1622. Oglesby, D. D., P. M. Mai, K. Atakan, and S. Pucci (2008), Dynamic models of earthquakes on the North Anatolian fault zone under the Sea of Marmara: Effect of hypocenter location, Geophys. Res. Lett., 35, L18302, doi:10.1029/2008GL035037. Perrin, C., I. Manighetti, J. P. Ampuero, F. Cappa, and Y. Gaudemer (2016), Location of largest earthquake slip and fast rupture controlled by along-strike change in fault structural maturity due to fault growth, J. Geophys. Res. Solid Earth, 121, 3666–3685, doi:10.1002/ 2015JB012671. Rosakis, A. J., O. Samudrala, and D. Coker (1999), Cracks faster than the shear wave speed, Science, 284(5418), 1337–1340, doi:10.1126/ science.284.5418.1337. Ryan, K. J., and D. D. Oglesby (2014), Dynamically modeling fault step overs using various friction laws: Modeling stepovers with various friction, J. Geophys. Res. Solid Earth, 119, 5814–5829, doi:10.1002/2014JB011151. Somerville, P. G., N. F. Smith, R. W. Graves, and N. A. Abrahamson (1997), Modification of empirical strong ground motion attenuation relations to include the amplitude and duration effects of rupture directivity, Seismol. Res. Lett., 68(1), 199–222. Vallée, M., M. Landès, N. M. Shapiro, and Y. Klinger (2008), The 14 November 2001 Kokoxili (Tibet) earthquake: High-frequency seismic radiation originating from the transitions between sub-Rayleigh and supershear rupture velocity regimes, J. Geophys. Res., 113, B07305, doi:10.1029/2007JB005520. Wang, D., and J. Mori (2012), The 2010 Qinghai, China, earthquake: A moderate earthquake with supershear rupture, Bull. Seismol. Soc. Am., 102(1), 301–308, doi:10.1785/0120110034. Weng, H., J. Huang, and H. Yang (2015), Barrier-induced supershear ruptures on a slip-weakening fault: Barrier-induced supershear ruptures, Geophys. Res. Lett., 42, 4824–4832, doi:10.1002/2015GL064281. Wesnousky, S. G. (2006), Predicting the endpoints of earthquake ruptures, Nature, 444, 358–360, doi:10.1038/nature05275. Xia, K., A. J. Rosakis, and H. Kanamori (2004), Laboratory earthquakes: The sub-Rayleigh-to-supershear rupture transition, Science, 303(5665), 1859–1861, doi:10.1126/science.1094022.

HU ET AL. SUPERSHEAR BY STEP OVER 8748 Journal of Geophysical Research: Solid Earth 10.1002/2016JB013333

Xia, K., A. J. Rosakis, H. Kanamori, and J. R. Rice (2005), Laboratory earthquakes along inhomogeneous faults: Directionality and supershear, Science, 308, 681–684, doi:10.1126/science.1108193. Xu, J., H. Zhang, and X. Chen (2015), Rupture phase diagrams for a planar fault in 3-D full-space and half-space, Geophys. J. Int., 202, 2194–2206, doi:10.1093/gji/ggv284. Zhan, Z., D. V. Helmberger, H. Kanamori, and P. M. Shearer (2014), Supershear rupture in a Mw 6.7 aftershock of the 2013 Sea of Okhotsk earthquake, Science, 345, 204–207, doi:10.1126/science.1252717. Zhang, H., and X. Chen (2006), Dynamic rupture on a planar fault in three-dimensional half-space—II. Validations and numerical experiments, Geophys. J. Int., 167, 917–932, doi:10.1111/j.1365-246X.2006.03102.x. Zhang, Z., W. Zhang, and X. Chen (2014), Three-dimensional curved grid ﬁnite-difference modelling for non-planar rupture dynamics, Geophys. J. Int., 199, 860–879, doi:10.1093/gji/ggu308.

HU ET AL. SUPERSHEAR BY STEP OVER 8749