### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Far-Field Surface Waves From Supershear Ruptures
- 3. Observation of Mach Waves From the Kokoxili Earthquake
- 4. Discussion
- Acknowledgments
- References

[1] Regional surface wave observations offer a powerful tool for determining source properties of large earthquakes, especially rupture velocity. Supershear ruptures, being faster than surface wave phase velocities, create far-field surface wave Mach cones along which waves from all sections of the fault arrive simultaneously and, over a sufficiently narrow frequency band, in phase. We present the first observation of far-field Mach waves from the major Kokoxili earthquake (Tibet, 2001/11/14,*M*_{w}7.9) and confirm that ground motion amplitudes are indeed enhanced on the Mach cone. Theory predicts that on the Mach cone, bandpassed surface wave seismograms from a large supershear rupture will be identical to those from much smaller events with similar focal mechanisms, with an amplitude ratio equal to the ratio of the seismic moments of the two events. Cross-correlation of 15–25 s Love waves from the Kokoxili event with those from a much smaller (*M*_{w}5) foreshock indicates a high degree of similarity (correlation coefficients ranging from 0.8 to 0.95) in waveforms recorded at stations near the far-field Mach cone. This similarity vanishes away from the Mach cone. These observations provide further evidence for supershear propagation of the Kokoxili rupture, and demonstrate how this simple waveform correlation procedure can be used to identify supershear ruptures.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Far-Field Surface Waves From Supershear Ruptures
- 3. Observation of Mach Waves From the Kokoxili Earthquake
- 4. Discussion
- Acknowledgments
- References

[2] The speed at which an earthquake rupture propagates influences the amplitude and character of the radiated wavefield. Rupture velocities less than the shear wave speed *β* are typically inferred by source inversions and seismic imaging studies. In fact, *β*is the limiting velocity in certain geometries, including along-strike propagation of megathrust ruptures in subduction zones. However, under mode II loading conditions, in which slip occurs parallel to the rupture propagation direction, rupture velocities in excess of*β* become possible [*Burridge*, 1973; *Andrews*, 1976; *Xia et al.*, 2004]. Seismic studies suggest supershear rupture velocities in several major strike-slip earthquakes (Izmit, Turkey, 1999; Kokoxili, Tibet, 2001; Denali, Alaska, 2002) [*Bouchon et al.*, 2001; *Bouchon and Vallée*, 2003; *Ellsworth et al.*, 2004; *Dunham and Archuleta*, 2004; *Aagaard and Heaton*, 2004; *Robinson et al.*, 2006; *Vallée et al.*, 2008; *Walker and Shearer*, 2009].

[3] The most distinctive features of supershear ruptures are Mach fronts. These sharp wavefronts occur whenever the source propagates faster than the speed of the waves it radiates. Supershear ruptures thus produce shear wave Mach fronts [*Freund*, 1979; *Ben-Menahem and Singh*, 1987], as well as surface wave Mach fronts for ruptures in a half-space [*Dunham and Bhat*, 2008]. These Mach fronts are predicted to transport extremely large particle velocities and stress perturbations out to distances comparable to the fault width [*Bernard and Baumont*, 2005; *Dunham and Bhat*, 2008], though this effect has not been substantiated observationally, possibly due to lack of Mach front coherence [*Bizzarri et al.*, 2010; *Andrews*, 2010].

[4] Thus far, almost all theoretical and numerical studies have focused on the wavefield in the near-source region (i.e., distances within a few source dimensions). In this work we explore properties of Mach waves in the far-field limit. Our focus is on surface waves, which carry the largest ground motion amplitudes outside the near-source region. In particular, we characterize how waves radiated by different sections of the fault interfere with each other, and how this leads to extreme amplification of surface wave motions at stations located along the far-field Mach cone. This directivity pattern is quite different from that of subshear ruptures, which features maximum amplification in the forward direction.

[5] We next prove that at stations along the far-field Mach cone, narrowband seismograms from a large supershear earthquake will be identical to those from a small earthquake of similar focal mechanism (except for an overall amplitude difference equal to the ratio of seismic moments). We test our theoretical predictions using regional Love wave records from the Kokoxili earthquake, and confirm that maximum directivity effects indeed occur at stations located along the far-field Mach cone.

### 2. Far-Field Surface Waves From Supershear Ruptures

- Top of page
- Abstract
- 1. Introduction
- 2. Far-Field Surface Waves From Supershear Ruptures
- 3. Observation of Mach Waves From the Kokoxili Earthquake
- 4. Discussion
- Acknowledgments
- References

[6] In this section we discuss the relationship between far-field surface waves from a large supershear earthquake and a small earthquake located in the vicinity of the large one. Both earthquakes have identical focal mechanisms corresponding to horizontal slip on vertically dipping faults.

[7] First consider the small earthquake with seismic moment *m*_{0}. At sufficiently low frequencies, seismic wavelengths are larger than the source dimension and the earthquake can be described with the point source moment density *m*_{0}*δ*(**x**)*H*(*t*), where *δ*(⋅) and *H*(⋅) are the delta function and unit step function, respectively. Within the approximation of a layered medium (i.e., neglecting lateral heterogeneity in material properties), the far-field displacement spectrum corresponding to fundamental mode surface waves can be written in the form [*Aki and Richards*, 2002]

where *r*_{0} = |**x**| and *ϕ* are the distance and azimuthal angle between the source (at the origin) and the station, and *ω* is the natural frequency. The excitation function and wavenumber *k* = *k*(*ω*) are specific to the fundamental surface wave eigenmode, with the former also depending on the focal mechanism of the earthquake.

[8] Now consider a much larger earthquake, in the vicinity of the small one, involving unilateral rupture propagation at constant rupture velocity *v*_{r}. The seismic moment *M*_{0} is released over width *W* and length *L* (0 ≤ *x* ≤ *L*). At frequencies less than ∼ *β*/*W*, seismic wavelengths are larger than *W*and the source can be described in terms of the depth-averaged slip Δ*u*(*x*). The far-field surface wave displacement spectrum, in the far-field limit [*Aki and Richards*, 2002], is

where *μ* is the shear modulus. We have introduced the phase factors and *e*^{−ikxcosϕ}to account for variations in surface wave arrival times due to both the rupture time and source-receiver distance, respectively, for points along the length of the fault.

[9] Using (1) to eliminate the excitation function, we rewrite (2) as

where

captures the directivity effect involving the ratio of the rupture velocity *v*_{r} to the surface wave phase velocity *c*(*ω*) ≡ *ω*/*k*(*ω*).

[10] When *v*_{r} < *c*(*ω*), then receivers at any azimuth *ϕ* record wave arrivals in the chronological order in which they were emitted by the rupture; i.e., the first arrivals are from the hypocenter and the last are from the end of the fault. Maximum directivity effects occur at stations in the forward direction (*ϕ* ≈ 0). In contrast, for *v*_{r} > *c*(*ω*) there exist two distinct regions bounded by *ϕ* = ±*ϕ*_{M}(*ω*), where

is half the opening angle of the far-field Mach cone. Within the Mach cone (i.e., |*ϕ*| < *ϕ*_{M}(*ω*)), the first arriving waves come from the last section of the fault to rupture, and waves from the hypocenter arrive last. On the Mach cone itself, waves from all sections of the fault arrive simultaneously and interfere constructively. The resulting amplification of ground motion exceeds that caused by even the fastest subshear ruptures.

[11] The Mach angle *ϕ*_{M}(*ω*) is the value of *ϕ* for which *X*(*ϕ*, *ω*) = 0. Thus from (3) we see that on the Mach cone (and only on it), the displacement spectrum of the large earthquake is identical to that of the small earthquake:

a result that holds even for spatially variable slip in the large event since *M*_{0} ≡ *μW* ∫_{0}^{L} Δ*u*(*x*)*dx.* While a similar result holds for all *ϕ* at frequencies less than ∼ *β*/*L* (because |*X*(*ϕ*, *ω*)| ≪ 1), we emphasize that (6) applies at frequencies less than ∼ *β*/*W.*For large strike-slip earthquakes, this includes periods greater than about 5 s (considering*W* equal to 15 km and a shear wave speed of 3 km/s), rather than just those greater than ∼100 s.

[12] Since surface wave phase velocities *c*(*ω*) are slightly less than the shear wave speed *β*, then the surface wave Mach cone will exist for supershear earthquakes (for which *v*_{r} > *β*). Since the Mach angle (5) depends on frequency, observational confirmation of our theory is facilitated by working with a limited frequency band centered on *ω* = *ω*_{0} over which the average surface wave phase velocity is . The corresponding Mach angle is . For bandpassed signals recorded at stations along the Mach cone, we can inverse Fourier transform (6) to obtain the remarkable result

At these stations, the bandpassed seismogram from the large event is predicted to match that of the small event, up to an overall normalization factor that is the ratio of the moments of the two events.

[13] To summarize, in the case of a long unilateral rupture (*L* ≫ *W*) observed in the far field (*r*_{0} ≫ *L*), three key observations provide evidence for Mach waves and thus proof that an earthquake is supershear: (1) Bandpassed waveforms from the large and small events are proportional at stations in particular azimuthal directions (which define the far-field Mach cone). (2) On the Mach cone, the amplitude ratio of these waveforms (or their spectral amplitude) is equal to the moment ratio. (3) In all other directions, the waveforms of the large earthquake are more complex than those of the small one. The amplitude ratio also decreases because signals from the large event are spread in time and waves from different parts of the fault are subject to more destructive interference. This is substantially different than the directivity pattern for subshear ruptures, for which directivity is maximized in the forward direction (*ϕ* = 0) and decreases monotonically as |*ϕ*| is increased to 180°.

### 3. Observation of Mach Waves From the Kokoxili Earthquake

- Top of page
- Abstract
- 1. Introduction
- 2. Far-Field Surface Waves From Supershear Ruptures
- 3. Observation of Mach Waves From the Kokoxili Earthquake
- 4. Discussion
- Acknowledgments
- References

[15] The rupture propagated unilaterally from west to east over about 100 s. After 130 km of subshear propagation, the rupture jumped into the supershear regime [*Vallée et al.*, 2008; *Walker and Shearer*, 2009; *Robinson et al.*, 2006]. The average rupture velocity over the following 170-km-long segment (bounded by points*P*_{1} and *P*_{2} in Figure 1) has been determined to be between 5 and 6.5 km/s, a value clearly higher the 3.5 km/s crustal shear wave velocity. The rupture velocity in the last part of the earthquake is less well known, but appears to be subshear.

[16] The Kokoxili earthquake, as well as a similar but much smaller foreshock (2000/11/26, *M*_{w} 5) located nearby, were recorded by several regional broadband seismometers belonging to the Federation of Digital Seismometers Network (FDSN; Figure 2). Because of the strike-slip character of the two earthquakes, the dominant waves are dispersive Love waves. We focus on 15–25 s Love waves to limit the effects of dispersion (see previous section). The average phase velocity in this period range can be estimated from the recent regional group velocity maps derived from earthquakes [*Chen et al.*, 2010] or seismic noise [*Li et al.*, 2011]. This estimation can be made using the average value of 3.5 km/s for the phase velocity of the 25 s Love waves in the Kunlun fault area (GDM52 model of *Ekström* [2011]), and the relation between group and phase velocities. Also taking into account the variability of group velocity in the *Li et al.* [2011] model around the Kunlun fault, km/s. Acceptable values of , along with the possible values of rupture velocity *v*_{r}in the supershear regime, enable us to predict the geometry of the far-field Love wave Mach cone (Figure 2).

[17] Three stations (ULN, HIA, and KMI) are on the far-field Mach cone. InFigure 3we show that at these stations, waveforms from the main shock are very similar to those of the small foreshock, as theoretically predicted. After aligning in time the Love wave arrivals, the normalized cross-correlation coefficient for the entire Love wave train exceeds 0.8 at these three stations, and even reaches 0.95 at station ULN. Moreover, when taking into account the amplifying factor applied inFigure 3, we observe that the amplitude ratio on the Mach cone is approximately 13,000–16,000. The predicted moment ratio between the events is 22,000, which is larger than the observed ratio. This discrepancy can be explained by the fact that the first segment of the Kokoxili event ruptured at a subshear speed. Generalizing the theory presented in the previous section to a compound rupture containing segments with different rupture speeds is straightforward, and for this specific earthquake we find that 20 s Love waves from the first (subshear) segment interfere destructively at stations along the Mach cone and contribute little to the overall waveform. Thus, only the supershear segment needs to be considered, and it likely released 60–80% of the overall moment [*Lasserre et al.*, 2005; *Robinson et al.*, 2006]. This reduces the expected moment ratio to values close to those observed.

[18] We also find that the cross-correlation values and the amplitude ratios are quite small for stations away from the Mach cone (either inside or outside it). As expected, both of these values reach a minimum when the stations are the furthest from the Mach cone (here for stations ENH and XAN, which are in the forward direction). Taken together, these results provide direct evidence of far-field Mach waves and thus supershear rupture speeds over a large section of the Kunlun fault.