S. K. Singh, M. Ordaz, T. Mikumo, J. Pacheco, C. Valdes, and P. Mandal
Implications of a composite source model and seismic-wave attenuation for the observed simplicity of small earthquakes and reported duration of earthquake initiation phase
Bulletin of the Seismological Society of America (October 1998), 88(5):1171-1181

Abstract:
An examination of P waves recorded on near-source, velocity seismograms generally shows that most small earthquakes (M _w <2 to 3) are simple. On the other hand, larger earthquakes (M _w > or =4) are most often complex. The simplicity of the seismograms of M _w <2 to 3 events may reflect the simplicity of the source (and, hence, may imply that smaller and larger earthquakes are not self-similar) or may be a consequence of attenuation of seismic waves. To test whether the attenuation is the cause, we generated synthetic P-wave seismograms from a composite circular source model in which subevent rupture areas are assumed to follow a power-law distribution. The rupture of an event is assumed to initiate at a random point on the fault and to propagate with a uniform speed. As the rupture front reaches the center of a subevent patch (all of which are circular), a P pulse is radiated that is calculated from the kinematic source model of Sato and Hirasawa (1973). Synthetic P-wave seismograms, which are all complex, are then convolved with an attenuation operator for different values of t (super *) . The results show that the observed simplicity of small events (M _w <2 to 3) may be entirely explained by attenuation if t (super *) > or =0.02 sec. The composite source model predicts that the average time delay between the initiation of the rupture and the rupture of the largest patch, tau , scales as M (super 1/3) ₀ , such that log tau = (1/3) log M ₀ -8.462. This relation is very similar to that reported by Umeda et al. (1996) between M ₀ and the observed time difference between the initiation of the rupture and the rupture of the "bright spot." It roughly agrees with the relation between M ₀ and the duration of the initiation phase reported by Ellsworth and Beroza (1995) and Beroza and Ellsworth (1996). The relation also fits surprisingly well the data on duration of slow initial phase, tsip, and M ₀ , reported by Iio (1995). One possible explanation of this agreement may be that the composite source model, which is essentially the "cascade" model, successfully captures the evolution of the earthquake source process and that the rupture initiation and the abrupt increase in the velocity amplitude observed on seismograms by previous researchers roughly corresponds to the rupture of the first subevent and the breaking of the largest sub-event in the composite source model.