Seismic radiation from circular cracks growing at variable rupture velocity

We have investigated the far-field body waves emitted by a circular fault growing at variable rupture velocity. The slip motion on the fault is constructed by assuming that the self-similar slip distribution holds at every successive instant of rupture formation for a circular crack. The present method permits us to evaluate not only the high-frequency but also the low-frequency radiation. It is also possible to handle both abrupt and continuous change in rupture velocity. The resultant expression for the radiation does not involve an integral. It is expressed in a closed form that contains information about the isochrone properties of the two extreme points having the minimum and maximum distances from the observer. Because of its simplicity and high computational speed, the present method provides us with a basic tool for simulating the radiation from seismic sources with a multitude of irregular rupture growth. The property of acceleration pulse radiated by continuous change in rupture velocity has been investigated. The time duration of the velocity change is assumed to be short but to take a finite time. The shape of the rupture front that is effective in radiating the high-frequency radiation is presumed to be semicircular. The acceleration pulse demonstrates a directivity with respect to both the pulse width and amplitude. The pulse width is, on average, given by the duration of the velocity change, and it is modulated by the directivity factor, which depends on the rupture velocity averaged over the change in rupture velocity. The pulse width radiated toward the growth direction of the semi-circular rupture front is shorter than that radiated toward the opposite side. The spectral amplitude of the acceleration pulse depends linearly on the strain, the radius of the rupture front at which the rupture velocity starts to change, the magnitude of the change in rupture velocity, and the generalized radiation pattern coefficient. The directivity of the radiation pattern coefficient is stronger than that for the case of an abrupt change in rupture velocity. In the case where the rupture stops completely, the radiation pattern coefficient is simply controlled by the magnitude of the change in rupture velocity. If we deal with a partial drop of the rupture velocity, however, we must consider the average rupture velocity as well to fully describe the radiation pattern. A procedure is presented for retrieving the model parameters from acceleration pulse data. The present results with regard to the acceleration pulse do not require the coherent movement of the rupture front over the entire circle. The results are applicable to more general cases where the rupture front moves coherently over a certain restricted segment, though the range of the segment has to become wider as the duration of the change in rupture velocity increases.