The Coupling of the Random Properties of the Source and the Ground Motion for the 1999 Chi Chi Earthquake

Two of the most fundamental principles in science are founded on the similitude in properties between a single event and a sum of these events. In seismology, the principle of superposition stipulates that during an earthquake, the waveform observed at a given distance of the fault is essentially the sum of waves emitted by point sources distributed over the fault surface. On the other hand, the Central Limit theorem postulates that the sum of Levy random variables is also a Levy random variable. In studies of source models for several earthquakes-the 1979 Imperial Valley, the 1989 Loma Prieta, the 1994 Northridge and 1995 Hyogo-ken Nanbu (Kobe), we have found that the spatial distributions of slip and pre-stress are characterized by a Levy law. This result can be used to deduce statistical properties of the radiated field. During an earthquake the rupture front propagates over the fault surface; as the rupture front reaches different points on the fault, each point source will emit a wave with an amplitude proportional to the stress released or the "stress drop." Because the magnitude of the stress drop is distributed according to a Levy law, so will the point source wave amplitude. The signal observed at a given distance from the source will be the sum of the signals emitted by the point sources. Because the point source wave amplitudes are distributed according to a Levy law, the sum of these signal amplitudes observed at a given distance from the sources will also be distributed according to a Levy law. We show that both the slip distribution and the peak ground acceleration (PGA) for the 1999 Chi Chi earthquake can be described by Levy laws. Furthermore, we found that the tails of the probability density functions (PDF) characterizing the slip and the PGA are governed by a parameter, the Levy index, with almost the same values as predicted by the Central Limit theorem. The PDF tail controls the frequency at which extreme large events can occur. These events are the large stress drops -or asperities- distributed over the fault surface and the large PGA observed in the ground motion. Our results suggest that the frequency of these events is coupled, and the PDF of the PGA is a direct consequence of the PDF of the asperities.