Dependence of earthquake recurrence times and independence of magnitudes on seismicity history

Abstract

The fulfillment of a scaling law for earthquake recurrence–time distributions is a clear indication of the importance of correlations in the structure of seismicity. In order to characterize these correlations we measure conditional recurrence–time and magnitude distributions for worldwide seismicity as well as for Southern California during stationary periods. Disregarding the spatial structure, we conclude that the relevant correlations in seismicity are those of the recurrence time with previous recurrence times and magnitudes; in the latter case, the conditional distribution verifies a scaling relation depending on the difference between the magnitudes of the two events defining the recurrence time. In contrast, with our present resolution, magnitude seems to be independent on the history contained in the seismic catalogs (except perhaps for Southern California for very short time scales, less than about 30 min for the magnitude ranges analyzed).

Introduction

Although seismology has intensively analyzed the characteristics of single earthquakes with great success, a clear description of the properties of seismicity (i.e., the structures formed by all earthquakes in space, time, magnitude, etc.) is still lacking. It is true that a number of empirical laws have been established since the end of the XIX century, but these laws are not enough to provide a coherent phenomenology for the overall picture of earthquake occurrence (Mulargia and Geller, 2003). In general, the approach to this phenomenon has to be statistical, as the Gutenberg–Richter law for the number of occurrences as a function of magnitude exemplifies. Moreover, for some studies it is necessary to assume that an earthquake can be reduced to a point event in space and time; despite this drastic simplification, the resulting point process for seismicity retains a high degree of complexity. In this context, the problem turns out to be very similar to those usually studied in statistical physics. Remarkably, in the last years the so-called ETAS model (which combines the Gutenberg–Richter law, the Omori law, and the law of productivity of aftershocks) has been widely studied (Ogata, 1988, Ogata, 1999, Helmstetter and Sornette, 2002); however, this model is still a first order approximation to real seismicity, useful mainly as a null-hypothesis model.

In contrast to the issues of the size of earthquakes or the temporal variations of the rate of occurrence, the timing of earthquakes (despite its potential interest for forecasting and risk assessment) has been the subject of relatively few empirical statistical studies, with the additional drawback that no unifying law for the time occurrence has emerged, at variance with the general acceptation of the Gutenberg–Richter law and the Omori law (Gardner and Knopoff, 1974, Udías and Rice, 1975, Koyama et al., 1995, Wang and Kuo, 1998, Ellsworth et al., 1999, Helmstetter and Sornette, 2004), see also the citations of Smalley et al. (1987) and Sornette and Knopoff (1997). Rather, results turn out to be in contradiction with each other, from the paradigm of regular cycles and characteristic earthquakes to the view of totally random occurrence (Bakun and Lindh, 1985, Sieh et al., 1989, Stein, 1995, Sieh, 1996, Murray and Segall, 2002, Stein, 2002, Kerr, 2004, Weldon et al., 2004, Weldon et al., 2005). In the opposite side of cycle regularity is the view of earthquake occurrence in the form of clusters, which has received more support from catalog analysis (Kagan and Jackson, 1991, Kagan and Jackson, 1995, Kagan, 1997).

A first step towards the solution of the time-occurrence problem and its insertion in a unifying description of seismicity was taken by Bak et al., who related, by means of a unified scaling law, the distribution of earthquake recurrence times with the Gutenberg–Richter law and the fractal distribution of epicenters (Bak et al., 2002, Christensen et al., 2002, Corral, 2003, Corral, 2004b, Corral and Christensen, 2006, Davidsen and Goltz, 2004). One of the goals of this line of research (apart from the applications in forecasting) should be to build a theory of seismicity that clarifies which kind of dynamical process we are dealing with. Simple behaviors such as periodicity, random occurrence, or even chaos seem too simplistic to account for the complexity of seismicity; in contrast, self-organized criticality fits better with this purpose (Bak, 1996, Jensen, 1998, Turcotte, 1999, Sornette, 2000, Hergarten, 2002), although different alternatives have been proposed more recently, see Rundle et al. (2003) and Sornette and Ouillon (2005).

Nevertheless, the recurrence–time distribution introduced by Bak et al. (2002) was defined in a somewhat complicated way through the mixing of recurrence times coming from (more or less) squared equally-sized regions with disparate seismic rates. A simpler, alternative perspective, in which each (arbitrary) region has its own recurrence–time distribution, has been developed (Corral, 2004a, Corral, 2005b). In this case, a universal scaling law involving the Gutenberg–Richter relation describes these distributions for a wide range of magnitudes and sizes of the regions, as long as seismicity shows a stationary behavior.

In this paper we extend this procedure to study conditional probability distributions. These distributions will provide important information about correlations in seismicity, which are fundamental for the existence of the scaling law for recurrence–time distributions. It turns out that recurrence times depend on previous recurrence times and magnitudes, in such a way that small values of the recurrence time and large magnitudes lead to small recurrence times, and vice versa. In contrast, the magnitude is fairly independent on history (although obviously we cannot reject the hypothesis of a dependence weaker than our uncertainty).