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Modeling directivity of heterogeneous earthquake ruptures

Département de Sismologie Institut de Physique du Globe de Paris, 75252 Paris Cedex, France

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 _max 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 -square-type radiation for frequencies below f_p = 1/_max and an ³ decay at higher frequencies. We instead introduce the concept of a scale-dependent rise-time (k_x), where k_x 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 _max 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 square distorted by frequency-dependent directivity effects. It presents holes at characteristic frequencies, the lowest being f_p. In the case of S waves and a narrow pulse model (L₀ << L), there is a strong frequency dependence of the directivity effect in the direction of rupture, with a larger amplification at low frequencies (f < f_p), controlled by C²_d, and a smaller amplification at higher frequencies, controlled by C_d, where C_d is the standard directivity coefficient 1/[1 – v/c cos()]. With broader pulses, f_p is shifted toward the lowest frequencies, masking the C²_d effect, thus reducing the directivity effects to a C_d amplification of the spectral level. More generally, any target spectrum of the far-field displacement—such as any power-law decay ⁿ—can be simulated: the far-field displacement spectrum is proportional to the spatial slip spectrum, and the corresponding directivity effects will be controlled by Cⁿ_d at low frequency and Cⁿ – 1_d 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 f_p, 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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