Abstract

The k-square broadband kinematic model of earthquake rupture proposed by Herrero and Bernard (1994) is reconsidered in order to relax the assumption of an instantaneous slip. The basic k−2 decay of the final slip spectrum is preserved, as well as the constant rupture velocity v. The rise-time τmax is finite, equal to L0/v, where L0 is the width of the slip pulse propagating at the velocity v along the fault (length L). The slip velocity is not constant during the slipping phase; otherwise, the model would produce an ω-square-type radiation for frequencies below fp = 1/τmax and an ω3 decay at higher frequencies. We instead introduce the concept of a scale-dependent rise-time τ(kx), where kx is the wavenumber in the direction of propagation: at a scale smaller than L0, the partial slip at a given wavelength is assumed to be completed in a time proportional to this wavelength after the arrival of the rupture front. At a larger scale, the rise-time τmax is constant, being limited by the arrival of the stopping phases, or by the “self-healing” of the slip pulse. The resulting radiation spectrum is an ω square distorted by frequency-dependent directivity effects. It presents holes at characteristic frequencies, the lowest being fp. In the case of S waves and a narrow pulse model (L0L), there is a strong frequency dependence of the directivity effect in the direction of rupture, with a larger amplification at low frequencies (f < fp), controlled by C2d, and a smaller amplification at higher frequencies, controlled by Cd, where Cd is the standard directivity coefficient 1/[1 − v/c cos(ϑ)]. With broader pulses, fp is shifted toward the lowest frequencies, masking the C2d effect, thus reducing the directivity effects to a Cd amplification of the spectral level. More generally, any target spectrum of the far-field displacement—such as any power-law decay ωn—can be simulated: the far-field displacement spectrum is proportional to the spatial slip spectrum, and the corresponding directivity effects will be controlled by Cnd at low frequency and Cn − 1d at high frequencies. In light of this simple model, we analyzed the spectral shapes of the Landers 1992 regional broadband records. The dominant spectral hole observed at about 0.3 Hz is associated to fp, leading to a pulse width of about 10 km—thus a “broad” pulse—and the absence of strong directive amplification suggests a rather slow mean rupture velocity (v/c ≦ 0.8), which is in agreement with published results of the Landers 1992 waveform inversions.

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