Scaling relationships between sizes of nucleation regions and eventual sizes of microearthquakes

We investigate the initial rupture process of microearthquakes to reveal relationships between nucleation region sizes and eventual earthquake sizes. In order to obtain high quality waveform data, we installed a trigger recording system with a sampling frequency of 10 kHz at the base of a deep borehole at the Nojima Fault, Japan. We analyze waveform data of 31 events around the borehole, with seismic moment ranging from 4.2 × 109 Nm to 7.1 × 1011 Nm. We use both a circular crack model with an accelerating rupture velocity (SK model) [Sato, T., Kanamori, H., 1999. Beginning of earthquakes modeled with the Griffith's fracture criterion, Bull. Seism. Soc. Am., 89, 80-93.], which generates a slow initial phase of velocity pulse, and a circular crack model with a constant rupture velocity (SH model) [Sato, T, Hirasawa, T., 1973. Body wave spectra from propagating shear cracks, J. Phys. Earth, 21, 415-431.], which generates a ramp-like velocity pulse. Source parameters of these two models are estimated by waveform inversion of the first half cycle of the observed velocity pulse applying both a grid search and a non-linear least squares method. 14 of 31 events are never reproduced by the SH model with a constant Q operator. But SK model with a constant Q operator provides a size of the pre-existing crack, corresponding to the size of the nucleation regions, and a size of the eventual crack. We recognize that (i) the eventual seismic moment is approximately scaled as the cube of the size of pre-existing cracks, (ii) the eventual seismic moment is scaled as the cube of the size of eventual cracks, and (iii) the size of eventual cracks is roughly proportional to the size of pre-existing cracks. We, thus, conclude that the size of eventual earthquakes is controlled by the size of the nucleation regions.