Coherency of rupture and seismic spectrum

In considering the seismic spectrum, one of the methods to incorporate irregularity of fault motion statistically is to introduce the concept of coherency of fracture. In a classic paper,Aki (1967) investigated the scaling law of seismic spectrum on the basis of a statistical model in which an exponentially decaying function is fitted to the autocorrelation function of the dislocation velocity. It is found, however, thatAki's model does not necessarily express irregular fault motion, but corresponds to a smooth dislocation. We show that an analytical function of dislocation velocity gives the same autocorrelation function and the same seismic spectrum as those ofAki's model. In actual fault motion, there is considerable evidence which indicates that the dislocation is not continuous and smooth over the whole fault plane, but is often segmented in several parts. In order to take into consideration this feature we introduce a generalized autocorrelation function of the dislocation velocity in which many coherent fractures smaller than the size of the fault dimension are included. It is shown that the more small-scale coherent fractures, the larger the seismic wave energy in the high frequency range.Kanamori andAllen (1986) showed that a large ratio of seismic wave energy relative to the seismic moment means a large effective stress drop. On the other hand, it is well known that when a fault plane is segmented in several parts, stress drop becomes large (e.g.,Madariaga, 1979;Rudnicki andKanamori, 1981). These two results are fused in our model, because we find that large seismic wave energy is obtained when the fault motion includes small-scale fractures.Kanamori andAllen (1986) also showed that there is a tendency for earthquakes with long repeat times to have a large effective stress drop. Our model implies that a fracture corresponding to earthquakes with long recurrence intervals is more complex, and the strength is large, as also suggested byCao andAki (1986) using a numerical simulation. It should be noted that to the zeroth order, an approximate scaling relation is observed among earthquakes, which means that a large earthquake consists of a relatively large-scale coherent fracture. This fact seems to suggest that the condition of occurrence of a large earthquake is related to the maturing of a source region in which a large coherent fracture becomes feasible.