We study the statistical properties of time distribution of seimicity in California by means of a new method of analysis, the Diffusion Entropy. We find that the distribution of time intervals between a large earthquake (the main shock of a given seismic sequence) and the next one does not obey Poisson statistics, as assumed by the current models. We prove that this distribution is an inverse power law with an exponent µ = 2.06 ± 0.01. We propose the Long-Range model, reproducing the main properties of the diffusion entropy and describing the seismic triggering mechanisms induced by large earthquakes. The search for correlation in the space-time distribution of earthquakes is a major goal in geophysics. At the short-time and the short-space scale the existence of correlation is well established. Recent geophysical observations indicate that main fracture episodes can trigger long-range as well as short-range seismic effects [1,2,3,4]. However, a clear evidence in support of these geophysical indications has not yet been provided. This is probably the reason why one of the models adopted to describe the time distribution of earthquakes is still the Generalized Poisson (GP) model [5,6,7,8,9]. Basically the GP model assumes that the earthquakes are grouped into temporal clusters of events and these clusters are uncorrelated: in fact the clusters are distributed at random in time and therefore the time intervals between one cluster and the next one follow a Poisson distribution. On the other hand, the intra-cluster earthquakes are correlated in time as it is expressed by the Omori's law [10,11], an empirical law stating that the main shock, i.e. the highest magnitude earthquake of the cluster, occurring at time t 0 is followed by a swarm of correlated earthquakes (after shocks) whose number (or frequency) n(t) decays in time as a power law, n(t) ∝ (t − t 0 ) −p , with the exponent p being very close to 1. The Omori's law implies [12] that the distribution of the time intervals between one earthquake and the next, denoted by τ , is a power law ψ(τ ) ∝ τ −p . This property has been recently studied by the authors of Ref.[12] by means of a unified scaling law for ψ L,M (τ ), the probability of having a time interval τ between two seismic events with a magnitude * Corresponding author: vito.latora@ct.infn.it larger than M and occurring within a spatial distance L. This has the effect of taking into account also space and extending the correlation within a finite time range τ * , beyond which the authors of Ref.[12] recover Poisson statistics.In this letter, we provide evidence of inter-clusters correlation by studing a catalog of seismic events in California with a new technique of analysis called Diffusion Entropy (DE) [13,14]. This technique, scarcely sensitive to predictable events such as the Omori cascade of aftershocks, is instead very sensitive when the deviation from Poisson statistics generates Lévy diffusion [14,15]. This deviation, on the other hand, implies that the geophysical process generating clusters ha...

Mega et al. Reply: Any scaling measure, including the diffusion entropy (DE) method, when applied to the earthquake time series, may yield anomalous scaling for a variety of reasons, of stationary and nonstationary nature. Let us discuss first the stationary sources of anomalous diffusion: the non-Poisson distribution m / 1= m of the time distances between one earthquake swarm and the following (see [1]), and the Pareto's law, p n / n ÿ , with p n denoting the probability of a swarm of n earthquakes. The process with the smallest power index determines the asymptotic scaling , revealed by the DE method. In Ref.[1] we proposed a value of close to 2 so as to account for 0:94. [4], the scaling is 1= ÿ 1 if 2 < < 3, 0:5 for > 3 and 1 for < 2, yielding 0:8 in this case. This theoretical prediction is supported by Fig. 1, which also shows that to obtain 0:99 we should use 1:25 [5], which is even smaller than the value proposed in [2]. This proves that Pareto's law is not responsible for the anomalous diffusion generated by seismic fluctuations. Let us now discuss the nonstationary sources of anomalous diffusion. The recent work of our group shows that a drift on the diffusing variable x t with a derivative whose absolute value is larger (smaller) than 1, yields 1 (0). This is why in [1] we mentioned the possibility of relating 1 to a slow geological drift, which would make the main shocks predictable. However, we assigned to the GP m the form of an inverse-power law, while maintaining the assumption that these times are unpredictable. Another nonstationary source of anomalous diffusion is the Omori's law. According to the continuous random walk prescriptions [4], the diffusing variable should increase logarithmicaly in time (thereby producing localization) after an extended transition to scaling, with an index close to 1. We think that the discrepancy between Fig. 1 of [2], yielding 0:94, and Fig. 1 of this Reply, producing 0:8, is due to the adoption in Ref.[2] of an Omori's process with an extremely slow transient. This is confirmed by Fig. 2(a) of Ref.[2], which can be interpreted as a manifestation of the Omori's process with the time scale of months.In conclusion, the value 0:94 emerging from the real seismic data might be generated by an Omori's process with the time scale larger than a few months. Since Omori's law generally acts at shorter time scales, we consider the non-Poisson model of [1] a plausible way to account for the extended memory revealed by the DE analysis. We define the main shocks as those processes that cancel the memory of the earlier seismic activity. This yields no correlation among the m 's, while it predicts a strong time correlation among seismic events, if they are selected only on the basis of magnitude. Thus, the results of Fig. 2 of Ref.[2] reinforce our perspective rather than weakening it.

We show one possible dynamical approach to the study of the distribution of prime numbers. Our approach is based on two complexity methods, the Computable Information Content and the Entropy Information Gain, looking for analogies between the prime numbers and intermittency.

With the help of the Diffusion Entropy technique we show the non-Poisson statistics of the distances between consecutive Omori's swarms of earthquakes. We give an analytical proof of the numerical results of an earlier paper [Mega et al., Phys. Rev. Lett. 90 (2003) 188501].

We argue that the recent discovery of the non-Poissonian statistics of the seismic main-shocks is a special case of a more general approach to the detection of the distribution of the time increments between one crucial but invisible event and the next. We make the conjecture that the proposed approach can be applied to the analysis of terrorist network with significant benefits for the Inteligence Community.

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