The persistence of directivity in small earthquakes

We derive a simple inversion of peak ground acceleration (PGA) or peak ground velocity (PGV) for rupture direction and rupture velocity and then test this inversion on the peak motions obtained from seven 3.5 <= M <= 4.1 earthquakes that occurred in two clusters in November 2002 and February 2003 near San Ramon, California. These clusters were located on two orthogonal strike-slip faults so that the events share the same approximate focal mechanism but not the same fault plane. Three earthquakes exhibit strong directivity, but the other four earthquakes exhibit relatively weak directivity. We use the residual PGAs and PGVs from the other six events to determine station corrections for each earthquake. The inferred rupture directions unambiguously identify the fault plane for the three earthquakes with strong directivity and for three of the four earthquakes with weak directivity. The events with strong directivity have fast rupture velocities (0.63 beta <= nu <= 0.87 beta); the events with weak directivity either rupture more slowly (0.17 beta <= nu <= 0.35 beta) or bilaterally. The simple unilateral inversion cannot distinguish between slow and bilateral ruptures: adding a bilateral rupture component degrades the fit of the rupture directions to the fault planes. By comparing PGAs from the events with strong and weak directivity, we show how an up-dip rupture in small events can distort the attenuation of peak ground motion with distance. When we compare the rupture directions of the earthquakes to the location of aftershocks in the two clusters, we find than almost all the aftershocks of the three earthquakes with strong directivity occur within 70 degrees of the direction of rupture.