Abstract

A theoretical model for constructing the -squared model is proposed by modifying the k-squared model of Bernard et al. (1996). The k-squared model provides a theoretical basis for the empirical -squared model under the assumptions that (1) the spatial wavenumber spectrum of the slip distribution falls off as the inverse of the wavenumber squared (k-squared), (2) the Fourier amplitudes of the slip velocity are independent of at high frequencies, and (3) the rupture velocity is constant. In this study, a more realistic model is proposed by modifying the last two assumptions. First, a Kostrov-type slip velocity model is proposed by superposing equilateral triangles, in which a source-controlled fmax is imposed by the minimum duration among the triangles. The Fourier amplitude of our slip velocity model falls off as the inverse of at high frequencies less than fmax. Next, in order to model variable rupture velocities, the incoherent rupture time (t_r), namely, the difference between the actual rupture time and the coherent (average) rupture time, is introduced. After checking various models for t_r distributions, the k-squared model for t_r, similar to that for the slip distributions of the k-squared model, is found to be the most plausible. Finally, it is confirmed that the proposed source model (we call it as the -inverse-squared model), which consists of the combination of the slip velocity proposed here and the k-squared distributions for both slip and t_r, not only is consistent with the empirical -squared model, but also provides the theoretical basis for constructing realistic source models at broadband frequencies.

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