## 1. Introduction

[2] The velocity at which a rupture propagates influences the amplitude and character of the radiated ground motion and stresses. A distinct manifestation of this occurs when ruptures exceed the S-wave speed and generate shear Mach waves that efficiently transmit ground motion and stresses away from the fault. Supershear speeds are also super-Rayleigh ones, and in an elastic half-space, we also expect Rayleigh Mach waves propagating along the free surface. Slip alters the component of normal stress parallel to the rupture front; for vertical strike-slip faults, negation of these stresses on the free surface generates Rayleigh waves. It follows from this line of reasoning that Rayleigh Mach fronts will be generated even if the rupture propagates at times the S-wave speed, the unique speed at which no shear Mach waves are produced [*Eshelby*, 1949]. The Rayleigh Mach front originates near the intersection of the rupture front with the free surface, and as one moves away from the fault along the free surface, the Rayleigh Mach front lags behind the shear Mach front. The expected pattern of shear and Rayleigh Mach fronts is illustrated in Figure 1.

[3] The objective of the current work is to quantify how the amplitude of radiated waves, specifically of those associated with the Mach waves, diminishes with distance from the fault. We further compare fields from supershear ruptures to those produced by sub-Rayleigh ruptures with the aim of contrasting the rate at which amplitudes decay with distance from the fault for both classes of ruptures.

[4] Supershear propagation, first suggested by analytical [*Burridge*, 1973] and numerical [*Andrews*, 1976; *Das and Aki*, 1977] studies, has since been confirmed in laboratory experiments [*Rosakis et al.*, 1999; *Xia et al.*, 2004]. Supershear speeds have been reported for a number of earthquakes, primarily from analyses of near-source records of the 1979 Imperial Valley [*Archuleta*, 1984; *Spudich and Cranswick*, 1984], 1999 Izmit and Düzce [*Bouchon et al.*, 2001, 2002], and 2002 Denali fault [*Ellsworth et al.*, 2004; *Dunham and Archuleta*, 2004] events. As *Savage* [1971] and *Ben-Menahem and Singh* [1981] have shown, there are also distinctive features of supershear ruptures at regional and teleseismic distances (e.g., changes in the radiation pattern), and an inversion of the 2001 Kokoxili (Kunlun) event [*Bouchon and Vallée*, 2003] using regional Love waves suggests supershear propagation. This conclusion is also supported by inversions of teleseismic body waves [*Robinson et al.*, 2006], though not all studies of such waves reach such a strong conclusion [*Antolik et al.*, 2004]. The unique characteristics of near-source records from supershear events, particularly of records at Pump Station 10 (located only 3 km from fault) in the 2002 Denali fault event, prompted a closer look at characteristics of radiated ground motion from supershear ruptures.

[5] We build on a number of previous studies that have examined the influence of rupture speed on near-source ground motion. *Ben-Menahem and Singh* [1987] studied the acceleration field generated by a point velocity dislocation (a singularity moving along a line and leaving in its wake a fixed moment per unit length) that travels a finite distance at a supershear speed before stopping. In addition to starting and stopping phases, shear Mach waves were implicated as carriers of large-amplitude accelerations. Their results further demonstrate how the Mach wave only passes through a particular region surrounding the fault. By considering propagation only along a line, their analysis applies only to observation points sufficiently removed from the fault (i.e., at distances much greater than the fault width). A number of other researchers have focused on the wavefield in the immediate vicinity of the fault (extending out to distances comparable to the fault width, but not much beyond that). By examining a sequence of kinematic models with various rupture speeds, *Aagaard and Heaton* [2004] demonstrated how the well-known two-sided fault-normal velocity waveform of sub-Rayleigh ruptures (the so-called “directivity pulse” that has been of primary concern in seismic hazard [*Somerville et al.*, 1997]) vanishes when ruptures exceed the S-wave speed. Instead, the largest amplitudes now occur at the Mach front. *Bernard and Baumont* [2005] combined kinematic models of supershear ruptures with an asymptotic isochrone-based analysis of fields near the Mach front to explore features of the Mach wave from supershear ruptures. Their asymptotic analysis, which did not include any corrections for a finite fault width, showed that for straight ruptures fronts, field amplitudes at the Mach front remain undiminished with distance from the fault; rupture-front curvature leads to an inverse square-root decay of amplitudes with distance due to a loss of coherence at the Mach front.

[6] The starting point for our analysis is the two-dimensional (2D) steady state slip-pulse model developed by *Rice et al.* [2005] to examine stress fields near the rupture front of sub-Rayleigh ruptures. This model was extended to supershear speeds by *Dunham and Archuleta* [2005] and *Bhat et al.* [2007]. *Dunham and Archuleta* [2005] focused on ground motion (specifically, velocity records) from slip pulses in the context of models of the Denali fault event [*Ellsworth et al.*, 2004; *Dunham and Archuleta*, 2004]. *Bhat et al.* [2007] studied the off-fault damage pattern due to supershear ruptures and hypothesized that anomalous ground cracking observed at a few tens of kilometers from the fault during the 2001 Kokoxili (Kunlun) event resulted from the high stresses at the Mach front emanating from a supershear rupture. This observation raises the possibility that radiated stresses from a large supershear event might trigger slip on adjacent faults of the proper orientation. One objective of the current work is to quantify how the amplitude of Coulomb stresses on pre-existing structures is influenced by rupture speed and the finite fault width, and whether or not these amplitudes are sufficient to activate secondary faulting.

[7] The most distinctive feature of the 2D supershear slip-pulse models is the shear Mach wave. The combined assumptions of two dimensions (i.e., an infinite extent of the slipping region parallel to the rupture front), steady state propagation, and a homogeneous linear elastic medium cause the Mach waves to extend infinitely far from the fault and for the amplitude of fields at the Mach fronts to remain undiminished with distance from the fault. This study addresses the first of these assumptions by considering ruptures in three dimensions (3D), specifically right-lateral strike-slip ruptures on a finite-width vertical fault breaking the surface of an elastic half-space. The focus is on the wavefield after the rupture has propagated many times further than the fault width. In addition to the shear Mach waves found in 2D models, we also expect Rayleigh Mach waves emanating from the rupture front out along the free surface. The 2D plane-strain models feature large changes in the normal stress parallel to the rupture front. These changes appear only in the vicinity of the fault and not further away at the shear Mach front since fields there are nondilatational. When considering rupture in a half-space, the component of normal stress parallel to the rupture front is also normal to the free surface and must be negated there to satisfy the traction-free boundary condition. This can be accomplished by the superposition of normal loads on the free surface that negate the moving vertical-normal-stress pattern. These moving loads, which propagate at a super-Rayleigh speed, will then excite Rayleigh Mach waves [*Lansing*, 1966; *Georgiadis and Lykotrafitis*, 2001]. Since these waves are being excited by dilatational stresses, Rayleigh Mach waves will exist even at times the S-wave speed.

[8] To understand the effect of bounding the vertical extent of the slipping region, consider two limiting cases of rupture on a vertical surface-breaking fault of width *W* and half-length *L* (Figure 2a). At locations close to the fault and away from its edges and the free surface (specifically, at locations much closer that *W*), the fault width is unimportant and 2D models provide an accurate description of the fields, at least if the length of the slip-weakening zone, *R*, is much less than *W*. In this extremely near-source region, the shear Mach front assumes the form of a wedge (Figure 2b) and Mach-wave amplitudes will not diminish with distance from the fault. Of course, this region is further complicated by the presence of dilatational fields of comparable amplitude. At the opposite extreme, consider points far removed from the fault (specifically, at distances greatly exceeding *W*). From these distant points, the fault appears as a line source, and S-wave radiation now forms a Mach cone (Figure 2c). Since the cross-section of the cone is a circle, geometrical spreading dictates that Mach-wave amplitudes will decrease with the inverse square-root of radial distance from the fault. (Also, it is not clear that the Rayleigh Mach-wave amplitudes would attenuate at all, in the ideally elastic material considered.) It is of critical importance to hazard calculations to understand exactly how the transition between these two extremes occurs. Specifically, to what distances are large ground motion and stresses transported for realistic fault geometries? *Bhat et al.* [2007] hypothesized that the transition between the two limits occurs at distances comparable to *W*, and our results confirm this hypothesis, although the Rayleigh Mach fronts also contribute to the fields.