## 1. Introduction

[2] Statistical models for seismic hazard evaluation are generally based on two well established empirical laws: the Gutenberg-Richter (GR) law [*Gutenberg and Richter*, 1944] and the Omori law [*Omori*, 1894]. The GR law states that the distribution *P*(*M*) of magnitude *M* follows an exponential behaviour *P*(*M*) ∼ 10^{−bM}, where usually *b* ≃ 1. The Omori law predicts that the number of earthquakes, after the occurrence of a large seismic event, decays in time as a power law, *n*(*t*) ∼ (*t* + *c*)^{−p}, where *p* ≃ 1. A third scaling relation is the productivity law stating that the number of aftershocks of an *M*_{M} mainshock is proportional to with *α* ≃ *b* [*Felzer et al.*, 2004; *Helmstetter et al.*, 2005]. Combining the above three laws one has that the probability to have an aftershock with magnitude *M* > *M*_{I} at the time *t* after a *M*_{M} mainshock occurred at the time *t*_{M} is given by

[3] A further relation is the Båth's law [*Båth*, 1965] stating that the magnitude difference between the main shock and the maximum detected aftershock, Δ*M* ≃ 1.2, is approximately a constant independent of the main shock magnitude.

[4] A topic of wide interest is whether the parameter *c* has a physical meaning or it is an artifact of instrumental deficiency [*Narteau et al.*, 2002; *Shcherbakov et al.*, 2004; *Kagan*, 2004; *Lise et al.*, 2004; *Lolli and Gasperini*, 2006; *Peng et al.*, 2006, 2007]. *Shcherbakov et al.* [2004, 2005a] show that *c* scales with the lower magnitude cut off *M*_{I}. More precisely, combining the GR, the Omori and the Båth laws, they obtain for *p* > 1

where *β* is a constant and *c** represents a lower cut off for *c*'s. Using the above dependence of *c* on *M*_{I} and *M*_{M}, they obtain the so-called generalized Omori law that fits very well experimental data. This law, inserted in a non-homogenous Poisson process, reproduces the experimental scaling behavior of interoccurrence time distribution [*Shcherbakov et al.*, 2005a, 2005b, 2006].

[5] A result similar to equation (2) was obtained by *Kagan* [2004] who investigated the magnitude and temporal distribution during the early stage of aftershock sequences for several strong earthquakes. He observed that the effective sequence starting time *c* depended on the magnitudes of the mainshock and aftershocks as

where *M*_{1} = 4, *δ* = 1 are experimental constants and time is measured in days. Kagan proposed a Short Term Aftershocks Incompleteness (STAI) hypothesis stating that many small aftershocks are not reported in the earthquake catalogues during the first days after the main event. According to this hypothesis the dependence of *c* on *M*_{I} and *M*_{M} is only an artifact and catalogs result complete for *M* ≥ *M*_{I} only for times larger than *c*. This implies a threshold magnitude *M*_{C} depending on the time from the main-shock, by setting *t* − *t*_{M} = *c* in equation (3)

[6] For *M* > *M*_{C}, the GR law is satisfied whereas for *M* < *M*_{C} an almost flat behavior of *P*(*M*) is expected. This is consistent with experimental data [*Kagan*, 2004; *Shcherbakov et al.*, 2006; *Enescu et al.*, 2007]. According to equation (4) for a *M*_{M} = 7 mainshock, the catalog can be considered complete above *M*_{I} > 2 only after 10 days from the main event. A direct inspection [*Helmstetter et al.*, 2005] of the time-magnitude distribution for several main-aftershock sequences gives a time dependent threshold magnitude that follows equation (4) but with *δ* = 0.75 and *M*_{1} = 4.5. Notice that equations (2) and (3) are equivalent if *δ* = and *M*_{1} = Δ*M* − *δ*log_{10}(*c**). Furthermore, the dependence of *c* on the main magnitude has also been observed by *Lise et al.* [2004] in their analysis of aftershock sequences.

[7] Recently, *Lippiello et al.* [2007] have considered subsets of the Southern California Catalog, each containing 125 earthquakes with magnitude *M* ≥ 3, and have computed the average magnitude inside each subset. The averaging procedure reduces statistical fluctuations and allows to detect a significant non-null magnitude correlation not related to STAI. They propose a dynamical scaling (DS) hypothesis relating magnitude to time differences to explain the observed magnitude correlations. This idea originates from a critical dynamics approach in which energy and time do not behave as independent quantities, but one observes a self-similar behaviour if energy is rescaled by an appropriate power of time. Lippiello et al. are also able to obtain the GR and the Omori law and to generate synthetic catalogues reproducing the statistical features of experimental ones.

[8] In this paper we apply the DS hypothesis to the analysis of aftershock sequences and discuss the physical meaning of the constant *c*. In particular we show that its dependence on *M*_{M} and *M*_{I} can be a direct consequence of the DS hypothesis. In the Discussions we compare our prediction with the recent findings of *Peng et al.* [2007].