A Theoretical Omega-Square Model Considering the Spatial Variation in Slip and Rupture Velocity

A theoretical model for constructing the x-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 x-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 x 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 superpos- ing equilateral triangles, in which a source-controlled fmax is imposed by the mini- mum duration among the triangles. The Fourier amplitude of our slip velocity model falls off as the inverse of x at high frequencies less than fmax. Next, in order to model variable rupture velocities, the incoherent rupture time ( Dtr), namely, the dif- ference between the actual rupture time and the coherent (average) rupture time, is introduced. After checking various models for Dtr distributions, the k-squared model for Dtr, 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 x-inverse-squared model), which consists of the combination of the slip velocity proposed here and the k-squared distributions for both slip and Dtr, not only is consistent with the empirical x-squared model, but also provides the theoretical basis for constructing realistic source models at broadband frequencies.