Source scaling relationships of small earthquakes estimated from the inversion method using stopping phases

We estimate source parameters of small earthquakes from stopping phases and investigate the scaling relationships between source parameters. The method we employed [Imanishi and Takeo, 2002] assumes an elliptical fault model proposed by Savage [1966]. In this model, two high-frequency stopping phases, Hilbert transformations of each other, are radiated and the difference in arrival times between the two stopping phases is dependent on the average value of rupture velocity, the source dimension, the aspect ratio of elliptical fault, the direction of rupture propagation and the orientation of the fault plane. These parameters can be estimated by a nonlinear least squares inversion method. Earthquakes studied occurred between May and August 1999 at the western Nagano prefecture, Japan, which is characterized by high levels of shallow earthquakes. The data consist of seismograms recorded by an 800 m deep borehole and a 46 surface seismic array whose spacing is a few km. In particular, the 800 m borehole data provide a wide frequency bandwidth and greatly reduce ground noise and coda wave amplitude compared to surface recordings. High-frequency stopping phases are readily detected on accelerograms recorded in the borehole. After correcting both borehole and surface data for attenuation, we also measure the rise time, which is defined as the time lag from the arrival time of the direct wave to the first slope change in the displacement pulse. Using these durations, we estimate source parameters of 25 earthquakes ranging in size from M1.2 to M2.6. The rupture aspect ratio is estimated to be about 0.8 on an average. This suggests that the assumption of a circular crack model is valid as a first order approximation for earthquakes analyzed in this study. Static stress drops range from approximately 0.1 to 5 MPa and do not vary with seismic moment. It seems that the breakdown seen in the previous studies by other authors using surface data is simply an artifact of attenuation in the crust. This is consistent with the conclusion by Stork et al. (2002) inferred from the spectral analysis using the 800m deep borehole data. The average values of rupture velocity do not depend on earthquake size, and are similar to those reported for moderate and large earthquakes. We then calculate the seismic energy following Sato and Hirasawa (1973). The magnitude scaling of the apparent stress is almost constant in the analyzed events, ranging from 0.05 to 1 MPa. Since most of apparent stresses for large earthquakes are in the range of 0.1 to 10 MPa, there may be small differences in apparent stress between large and small earthquakes. However, it is likely that earthquakes are self-similar over a wide range of earthquake size and the dynamics of small and large earthquakes are similar from a macroscopic viewpoint.