Seismic coda due to non-linear elasticity

Non-linear elastic response of rocks has been widely observed in laboratory, but very few seismic studies are reported in the literature, even though it is the most natural environment where this feature could be observed. Analytic solutions to the non-linear wave propagation phenomena are not readily available, and there is a need to use approximated techniques. It is clear that when a seismic wave propagates through a homogeneous non-linear elastic media, it will be perturbed by the non-linearity. This perturbation can be treated as a source of scattering, spreading the energy of the primary wave in space and time, contributing to the seismic coda. This is in some sense similar to the effect of heterogeneities. The properties of the coda due to the non-linearity depend on the amount of non-linearity and the seismic moment. Using a perturbation approach we calculate the amplitude of the scattered waves, and show that it can describe reasonably well the main features of real seismic codas.     

Comment on 'Systematic determination of earthquake rupture directivity and fault planes from analysis of long-period P-wave spectra' by L. M. Warren and P. M. Shearer

Warren and Shearer describe a method of estimating the duration of P pulses radiated by earthquakes, on the assumption that the source is a unilateral fracture. The estimates are made in the frequency domain. The estimates obtained by Warren and Shearer for seven of the earthquakes are compared to durations estimated here in the time domain; the time-domain measurements being made on broad-band seismograms (∼0.1–4.0 Hz) derived by filtering from short-period recordings. Overall, the time-domain method indicates that the pulse duration of the earthquakes studied here range from 2.0 to 7.6 s, whereas the estimates from the results of Warren and Shearer range from 7.1 to 9.8 s. This suggests that the method of Warren and Shearer cannot resolve pulse lengths less than about 7 s. The suggestion is supported by the estimates of the duration of rupture—fault length/speed of rupture—implied by the results of Warren and Shearer. For, although the estimated fault-length ranges from 0.8 km to over 40 km most rupture durations are around 8 s.     

Size and duration of the high-frequency radiator in the source of the 2004 December 26 Sumatra earthquake

We recover the gross space–time characteristics of high-frequency (HF) radiator of the great Sumatra-Andaman islands earthquake of 2004 December 26 ( M _w= 9.1–9.3) using the time histories of the power of radiated HF P waves. To determine these time histories we process teleseismic P waves at 36 BB stations, using, in sequence: (1) bandpass filtering (four bands: 0.4–1.2, 1.2–2, 2–3 and 3–4 Hz); (2) squaring wave amplitudes, making 'power signals' for each band and (3) stripping the propagation-related distortion ( P coda, etc.) from the power signal and thus recovering source time function for HF power. In step (3) we employ an inverse filter constructed from an empirical Green's function, which is estimated as the power signal from an aftershock. For each ray we thus obtain signals with relatively well-defined end and no coda. From these signals we extract: total duration (joint estimate for all four bands) and temporal centroid of signal power for each band. Through linear inversion, the set of duration values for a set of rays delivers estimates of the rupture stopping point and stopping time. Similarly, the set of temporal centroids can be inverted to obtain the position of the space–time centroid of HF energy radiator. The quality of inversion for centroid is acceptable for lower-frequency bands but deteriorates for higher-frequency bands where only a fraction of stations provide useful data. For the source length and duration the following joint estimates were obtained: 1241 ± 224 km, 550 ± 10 s. The estimated stopping point position corresponds to the northern extremity of the aftershock zone. Spatial HF radiation centroids are located at distances 350–700 km from the epicentre, in a systematic way: the higher is the frequency, the farther is the centroid from the epicentre. Average rupture propagation velocity is estimated as 2.25 km s^–1.     

Simulation of the spontaneous growth of a dynamic crack without constraints on the crack tip path

The spontaneous growth of a dynamic in-plane shear crack is simulated using a newly developed method of analysis in which no a priori constraint is required for the crack tip path, unlike in other classical studies. We formulate the problem in terms of boundary integral equations; the hypersingularities of the integration kernels are removed by taking the finite parts. Our analysis shows that dynamic crack growth is spontaneously arrested soon after the bending of the crack tips, even in a uniformly stressed medium with homogeneously distributed fracture strengths. This shows that the dynamics of crack growth has a significant effect on forming the non-planar crack shape, and consequently plays an essential role in the arrest of earthquake rupturing.     

High-resolution structures of the Landers fault zone inferred from aftershock waveform data

High-frequency body waves recorded by a temporary seismic array across the surface rupture trace of the 1992 Landers, California, earthquake were used to determine fault-zone structures down to the seismogenic depth. We first developed a technique to use generalized ray theory to compute synthetic seismograms for arbitrarily oriented tabular low-velocity fault-zone models. We then generated synthetic waveform record sections of a linear array across a vertical fault zone. They show that both arrival times and waveforms of P and S waves vary systematically across the fault due to transmissions and reflections from boundaries of the low-velocity fault zone. The waveform characteristics and arrival-time patterns in the record sections allow us to locate the boundaries of the fault zone and to determine its P - and S -wave velocities independently as well as its depth extent. Therefore, the trade-off between the fault-zone width and velocities can be avoided. Applying the method to the Landers waveform data reveals a low-velocity zone with a width of 270–360 m and a 35–60 per cent reduction in P and S velocities relative to the host rock. The analysis suggests that the low-velocity zone extends to a depth of ∼7 km. The western boundary of the low-velocity zone coincides with the observed main surface rupture trace.     

Dynamic nucleation process of shallow earthquake faulting in a fault zone

Two distinct phases are commonly observed at the initial part of seismograms of large shallow earthquakes: low-frequency and low-amplitude waves following the onset of a P wave ( P ₁) are interrupted by the arrival of the second impulsive phase P₂ enriched with high-frequency components. This observation suggests that a large shallow earthquake involves two qualitatively different stages of rupture at its nucleation.
We propose a theoretical model that can naturally explain the above nucleation behaviour. The model is 2-D and the deformation is assumed to be anti-plane. A key clement in our model is the assumption of a zone in which numbers of pre-existing cracks are densely distributed; this cracked zone is a model for the fault zone. Dynamic crack growth nucleated in such a zone is intensely affected by the crack interactions, which exert two conflicting effects: one tends to accelerate the crack growth, and the other tends to decelerate it. The accelerating and decelerating effects are generally ascribable to coplanar and non-coplanar crack interactions, respectively. We rigorously treat the multiple interactions among the cracks, using the boundary integral equation method (BIEM), and assume the critical stress fracture criterion for the analysis of spontaneous crack propagation.
Our analysis shows that a dynamic rupture nucleated in the cracked zone begins to grow slowly due to the relative predominance of non-coplanar interactions. This process radiates the P₁ phase. If the crack continues to grow, coalescence with adjacent coplanar cracks occurs after a short time. Then, coplanar interactions suddenly begin to prevail and crack growth is accelerated; the P₂ phase is emitted in this process. It is interpreted that the two distinct phases appear in the process of the transition from non-coplanar to coplanar interaction predominance.     

Scaling relations of earthquakes and aseismic deformation in a damage rheology model

We perform analytical and numerical studies of scaling relations of earthquakes and partition of elastic strain energy between seismic and aseismic components using a thermodynamically based continuum damage model. Brittle instabilities occur in the model at critical damage level associated with loss of convexity of the strain energy function. A new procedure is developed for calculating stress drop and plastic strain in regions sustaining brittle instabilities. The formulation connects the damage rheology parameters with dynamic friction of simpler frameworks, and the plastic strain accumulation is governed by a procedure that is equivalent to Drucker–Prager plasticity. The numerical simulations use variable boundary forces proportional to the slip-deficit between the assumed far field plate motion and displacement of the boundary nodes. These boundary conditions account for the evolution of elastic properties and plastic strain in the model region. 3-D simulations of earthquakes in a model with a large strike-slip fault produce scaling relations between the scalar seismic potency, rupture area, and stress drop values that are in good agreement with observations and other theoretical studies. The area and potency of the simulated earthquakes generally follow a linear log–log relation with a slope of 2/3, and are associated with stress drop values between 1 and 10 MPa. A parameter-space study shows that the area-potency scaling is shifted to higher stress drops in simulations with parameters corresponding to lower dynamic friction, more efficient healing, and higher degree of seismic coupling.