Spudich, P.

We study how heterogeneous rupture propagation affects the coherence of shear– and Rayleigh–Mach wave fronts radiated by supershear earthquakes. We address this question using numerical simulations of ruptures on a planar, vertical strike–slip fault embedded in a three–dimensional, homogeneous, linear elastic half–space. Ruptures propagate spontaneously in accordance with a linear slip–weakening friction law through both homogeneous and heterogeneous initial shear stress fields. In the 3–D homogeneous case, rupture fronts are curved due to interactions with the free surface and the finite fault width; however, this curvature does not greatly diminish the coherence of Mach fronts relative to cases in which the rupture front is constrained to be straight, as studied by Dunham and Bhat (2008). Introducing heterogeneity in the initial shear stress distribution causes ruptures to propagate at speeds that locally fluctuate above and below the shear–wave speed. Calculations of the Fourier amplitude spectra (FAS) of ground velocity time histories corroborate the kinematic results of Bizzarri and Spudich (2008): 1) The ground motion of a supershear rupture is richer in high frequency with respect to a subshear one. 2) When a Mach pulse is present, its high frequency content overwhelms that arising from stress heterogeneity. Present numerical experiments indicate that a Mach pulse causes approximately an –1.7 high frequency falloff in the FAS of ground displacement. Moreover, within the context of the employed representation of heterogeneities and over the range of parameter space that is accessible with current computational resources, our simulations suggest that while heterogeneities reduce peak ground velocity and diminish the coherence of the Mach fronts, ground motion at stations experiencing Mach pulses should be richer in high frequencies compared to stations without Mach pulses. In contrast to the foregoing theoretical results, we find no average elevation of 5%–damped absolute response spectral accelerations (SA) in the period band 0.05–0.4 s observed at stations that presumably experienced Mach pulses during the 1979 Imperial Valley, 1999 Kocaeli, and 2002 Denali Fault earthquakes compared to SA observed at non–Mach pulse stations in the same earthquakes. A 20% amplification of short period SA is seen only at a few of the Imperial Valley stations closest to the fault. This lack of elevated SA suggests that either Mach pulses in real earthquakes are even more incoherent that in our simulations, or that Mach pulses are vulnerable to attenuation through nonlinear soil response. In any case, this result might imply that current engineering models of high frequency earthquake ground motions do not need to be modified by more than 20% close to the fault to account for Mach pulses, provided that the existing data are adequately representative of ground motions from supershear earthquakes.

We estimate fracture energy on extended faults for several recent earthquakes by retrieving dynamic traction evolution at each point on the fault plane from slip history imaged by inverting ground motion waveforms. We define the breakdown work (Wb) as the excess of work over some minimum traction level achieved during slip. Wb is equivalent to "seismological" fracture energy (G) in previous investigations. Our numerical approach uses slip velocity as a boundary condition on the fault. We employ a three-dimensional finite difference algorithm to compute the dynamic traction evolution in the time domain during the earthquake rupture. We estimate Wb by calculating the scalar product between dynamic traction and slip velocity vectors. This approach does not require specifying a constitutive law and assuming dynamic traction to be collinear with slip velocity. If these vectors are not collinear, the inferred breakdown work depends on the initial traction level. We show that breakdown work depends on the square of slip. The spatial distribution of breakdown work in a single earthquake is strongly correlated with the slip distribution. Breakdown work density and its integral over the fault, breakdown energy, scale with seismic moment according to a power law (with exponent 0.59 and 1.18, respectively). Our estimates of breakdown work range between 4e+5 and 2e+7 J/m2 for earthquakes having moment magnitudes between 5.6 and 7.2. We also compare our inferred values with geologic surface energies. This comparison might suggest that breakdown work for large earthquakes goes primarily into heat production.

We present a two-stage nonlinear technique to invert strong motions records and geodetic data to retrieve the rupture history of an earthquake on a finite fault. To account for the actual rupture complexity, the fault parameters are spatially variable peak slip velocity, slip direction, rupture time and risetime. The unknown parameters are given at the nodes of the subfaults, whereas the parameters within a subfault are allowed to vary through a bilinear interpolation of the nodal values. The forward modeling is performed with a discrete wave number technique, whose Green’s functions include the complete response of the vertically varying Earth structure. During the first stage, an algorithm based on the heat-bath simulated annealing generates an ensemble of models that efficiently sample the good data-fitting regions of parameter space. In the second stage (appraisal), the algorithm performs a statistical analysis of the model ensemble and computes a weighted mean model and its standard deviation. This technique, rather than simply looking at the best model, extracts the most stable features of the earthquake rupture that are consistent with the data and gives an estimate of the variability of each model parameter. We present some synthetic tests to show the effectiveness of the method and its robustness to uncertainty of the adopted crustal model. Finally, we apply this inverse technique to the well recorded 2000 western Tottori, Japan, earthquake (Mw 6.6); we confirm that the rupture process is characterized by large slip (3-4 m) at very shallow depths but, differently from previous studies, we imaged a new slip patch (2-2.5 m) located deeper, between 14 and 18 km depth.

he M 7.4 Landers earthquake triggered widespread seismicity in the Western U.S. Because the transient dynamic stresses induced at regional distances by the Landers surface waves are much larger than the expected static stresses, the magnitude and the characteristics of the dynamic stresses may bear upon the earthquake triggering mechanism. The Landers earthquake was recorded on the UPSAR array, a group of 14 triaxial accelerometers located within a 1-square-km region 10 km southwest of the town of Parkfield, California, 412 km northwest of the Landers epicenter. We used a standard geodetic inversion procedure to determine the surface strain and stress tensors as functions of time from the observed dynamic displacements. Peak dynamic strains and stresses at the Earth's surface are about 7 microstrain and 0.035 MPa, respectively, and they have a flat amplitude spectrum between 2 s and 15 s period. These stresses agree well with stresses predicted from a simple rule of thumb based upon the ground velocity spectrum observed at a single station. Peak stresses ranged from about 0.035 MPa at the surface to about 0.12 MPa between 2 and 14 km depth, with the sharp increase of stress away from the surface resulting from the rapid increase of rigidity with depth and from the influence of surface wave mode shapes. Comparison of Landers-induced static and dynamic stresses at the hypocenter of the Big Bear aftershock provides a clear example that faults are stronger on time scales of tens of seconds than on time scales of hours or longer.

We use a two-stage nonlinear technique to invert strong motions records and geodetic data to retrieve the rupture history of an earthquake on a finite fault. The unknown model parameters, spatially variable peak slip velocity, slip direction, rupture time and rise time, are given at the vertices of subfaults, whereas the parameters within a subfault can vary through a bilinear interpolation of the vertex values. The forward modeling is performed with a discrete wavenumber technique, whose Green's functions include the complete response of the vertically varying non-attenuating Earth structure. The GPS coseismic data are compared with the synthetic displacements using a L2 norm, while the recorded and modeled waveforms are compared in the frequency domain, using a cost function that is a hybrid representation between L1 and L2 norms. During the first stage (search), an algorithm based on heat-bath simulated annealing generates an ensemble of models that efficiently sample the good data-fitting regions of the parameter space. During this stage multiple Earth structures can be used to allow for uncertainty in the true structure. In the second stage (appraisal), the algorithm performs a statistical analysis of the model ensemble and computes a weighted mean model and its standard deviation by weighting all models by the inverse of the cost function values. We do not use any smoothing operator. This technique, rather than simply looking at the best model, extracts the most stable features of the earthquake rupture that are consistent with the data and gives an estimate of the variability of each model parameter. We present some applications to recent earthquakes such as the 2000 western Tottori (Mw 6.7) and the 2007 Niigata (Mw 6.6) (Japan) earthquakes in order to test and show the effectiveness of the method. Our methodology allows the use of different slip velocity time functions and we emphasize the relevance of adopting source time functions in kinematic inversions compatible with earthquake dynamics. We have verified that the choice of source time function affects ground motion time histories within the frequency band commonly used in waveform inversions and has a clear impact on the inferred peak slip velocity and rise time and, consequently, on the dynamic traction evolution inferred from kinematic models. Furthermore, the assessment of model uncertainty could be useful to predict ground motion time histories for seismic hazard assessment.

In this paper we attempt to reconcile a theoretical understanding of the earthquake energy balance with current geologic understanding of fault zones, with seismological estimates of fracture energy on faults, and with geological measurements of surface energy in fault gouges. In particular, we discuss the mechanical work absorbed on the fault plane during the propagation of a dynamic earthquake rupture. We show that, for realistic fault zone models, all the mechanical work is converted in frictional work defined as the irreversible work against frictional stresses. We note that the eff γ of Kostrov and Das (1988) is zero for cracks lacking stress singularities, and thus does not contribute to the work done on real faults. Fault shear tractions and slip velocities inferred seismologically are phenomenological variables at the macroscopic scale. We define the macroscopic frictional work and we discuss how it is partitioned into surface energy and heat (the latter includes real heat as well as plastic deformation and the radiation damping of Kostrov and Das). Tinti et al. (2005) defined and measured breakdown work for recent earthquakes, which is the excess of work over some minimum stress level associated with the dynamic fault weakening. The comparison between geologic measurements of surface energy and breakdown work revealed that 1-10% of breakdown work went into the creation of fresh fracture surfaces (surface energy) in large earthquakes, and the remainder went into heat. We also point out that in a realistic fault zone model the transition between heat and surface energy can lie anywhere below the slip weakening curve.

In this paper we achieve three goals: (1) We demonstrate that crack tips governed by friction laws, including slip weakening, rate- and state-dependent laws, and thermal pressurization of pore fluids, propagating at supershear speed have slip velocity functions with reduced high-frequency content compared to crack tips traveling at subshear speeds. This is demonstrated using a fully dynamic, spontaneous, three-dimensional earthquake model, in which we calculate fault slip velocity at nine points (locations) distributed along a quarter circle on the fault where the rupture is traveling at supershear speed in the inplane direction and subshear speed in the antiplane direction. This holds for a fault governed by the linear slip-weakening constitutive equation, by slip weakening with thermal pressurization of pore fluid, and by rate- and state-dependent laws with thermal pressurization. The same is also true even assuming a highly heterogeneous initial shear stress field on the fault. (2) Using isochrone theory, we derive a general expression for the spectral characteristics and geometric spreading of two pulses arising from supershear rupture, the well-known Mach wave, and a second lesser known pulse caused by rupture acceleration. (3) We demonstrate that the Mach cone amplification of high frequencies overwhelms the de-amplification of high-frequency content in the slip velocity functions in supershear ruptures. Consequently, when earthquake ruptures travel at supershear speed, a net enhancement of high-frequency radiation is expected, and the alleged ‘‘low’’ peak accelerations observed for the 2002 Denali and other large earthquakes are probably not caused by diminished high-frequency content in the slip velocity function, as has been speculated.