## Chapter 16 On the Random Nature of Earthquake Sources and Ground Motions: a Unified Theory

Daniel Lavallee

Published 2008, SCEC Contribution #1136

The synthesis of fundamental principles of physics and of the theory of probability provides a coherent and unified picture of earthquake variability from its recording in the ground motions to its inference in source models. This theory, based on the representation theorem and the (generalized) Central Limit Theorem, stipulates that the random properties of the ground motions and the source for a single earthquake should be both (approximately) distributed according to a Lévy law. The Lévy laws are a special class of probability laws. According to the (generalized) Central Limit Theorem, a sum of Lévy random variables is simply a Lévy random variable. The Gauss and the Cauchy laws are special cases of the Lévy law.

Random models are best suited to describe the spatial heterogeneity embedded in earthquake source model of slip (or stress). For this purpose, we have developed a random model that can reproduce the variability in slip amplitude and the long-range correlation of the slip spatial distribution. Analysis of slip spatial distribution shows that a Non-Gaussian law, i.e. the Lévy law, is better suited to describe the distribution of slip amplitude values over the fault. Furthermore, a comparison of the random properties of the source and of the ground motion for the 1999 Chi-Chi and 2004 Parkfield earthquakes demonstrates that the slip distribution and the peak ground acceleration (PGA) can be described by Lévy laws. Additionally, the tails of the probability density functions (PDF) characterizing the slip and the |PGA| are controlled by a parameter, the Lévy index, with almost the same values as predicted by the (generalized) Central Limit Theorem. Thus, from the source to the ground motion, the Lévy index provides a universal law describing the tail of the PDF.

The PDF tail controls the frequency at which extreme large events occur. These large events correspond to the large stress drops -or asperities- distributed on the fault surface and to the large PGA observed in the ground motion. The theory and the results suggest that the frequency of these events is coupled: the PDF of the |PGA| is a direct consequence of the PDF of the asperities.

Key Words
earthquake, source model, ground motion, random model, Levy law

Citation
Lavallee, D. (2008). Chapter 16 On the Random Nature of Earthquake Sources and Ground Motions: a Unified Theory. , Netherlands: Elsevier. doi: 10.1016/S0065-2687(08)00016-2.