## 1. Introduction

[2] Earthquakes are the brittle response of the earth crust to stress–strain changes. These brittle seismic instabilities in the crust emerge as combined and complex effects of the response of heterogeneous media to small changes in loading rate which occur over a wide range of scales [e.g., *Bak and Tang*, 1989; *Sornette and Sornette*, 1989; *Main*, 1995; *Rundle et al.*, 2003]. These brittle deformations scale from dislocations and microcracks (∼1 *μ*m to 1 cm) to tectonic plate boundaries (10^{3}–10^{4} km), whereas time scales range from a few seconds during dynamic rupture to 10^{3}–10^{4} years (as the repeat times for the large *M* > 7 − 8 earthquakes) and to 10^{7}–10^{8} years (evolution of the plate boundaries) [e.g., *Rundle et al.*, 2003]. For earthquakes, *Gutenberg and Richter* [1944] suggested the frequency magnitude distribution as;

where *N* is the total number of earthquakes with magnitude *M* or greater, *a* and *b* are constants. Regional analyses [e.g., *Utsu*, 2002] suggest *b* – *values* in the 0.8–1.2 range, including for aftershock sequences. Variation in *b* – *value* across different stress regimes are suggested by *Schorlemmer et al.* [2005]. Aftershocks also are observed to obey Omori's law [*Utsu*, 1961]

where *N*(*t*) is the number of aftershocks per unit time, *t* is the elapsed time since the mainshock, *K*, *c* and *p* are constants. A median *p* – *value* of ∼1.1 is reported for the aftershock sequences in the various parts of the world, with a range of ∼0.6–2.5 [*Utsu et al.*, 1995]. *Narteau et al.* [2009] observed that *c* – *value* varied with mainshock faulting styles. As proposed by *Helmstetter and Sornette* [2003a]

*M*_{c} is threshold magnitude for catalog completeness, *α* is a parameter that controls the relative number of aftershocks triggered as a function of mainshock magnitude(*α* = 0.66 − 1.15, suggested by *Hainzl and Marsan* [2008] for the global catalog). Thirdly Båth's law for earthquake aftershocks is observed in many empirical and statistical studies. Initially reported by *Richter* [1958] as Båth's observation, it states the average magnitude difference (Δ*M*) between the mainshock and its largest aftershock is 1.2, regardless of mainshock magnitude [*Båth*, 1965]. A number of studies have been conducted for the physical interpretation of Båth's law [e.g., *Vere-Jones*, 1969; *Console et al.*, 2003; *Helmstetter and Sornette*, 2003b; *Shcherbakov et al.*, 2006; *Vere-Jones*, 2008]. Among them *Helmstetter and Sornette* [2003b] using ETAS (epidemic type aftershock sequence for seismicity model) simulations provide a comprehensive analysis of the empirical Båth's law.

[3] They suggest that Båth's law occurrence depends on both *α* – *value* of the mainshock and the relative difference, (*M*_{m} − *M*_{c}), between mainshock magnitude (*M*_{m}) and catalog completeness (*M*_{c}) value. When *M*_{m} − *M*_{c} ≥ 2 and *α* = 0.8 − 1.0, then Båth's law applies. In other cases, i.e., *M*_{m} − *M*_{c} < 2, *α*-value < 0.8, 〈Δ*M*〉 is smaller than 1.2 (i.e., ranging between 0 and 1.2) and it increases rapidly with *M*_{m}.

[4] In this latter case, the apparent increase in 〈Δ*M*〉 is correlated with a low *α* – *value*. A lower aftershock rate implies a lower picking rate in the Gutenberg-Richter law distribution, and thus a lower probability of a large magnitude occurrence [see*Helmstetter and Sornette*, 2003b, equation 1]. More recently *Saichev and Sornette* [2005] showed the relationship of Båth's Δ*M* = 1.2 value to the branching ratio (*n*) of the ETAS point process model of earthquake interactions. For high *n* (*n* ≥ 0.8), *α* (*α* ≥ 0.9) values, the ETAS model yields a constant value of Δ*M* = 1.2 (Båth's law) and for low *n* (*n* ≤ 0.6) and *α* (*α* ≤ 0.5) Båth's law does not apply.

[5] In this paper we extend Båth's law, (i) to space and time patterns of the largest aftershocks, and (ii) we consider the earthquake faulting style as a possible control parameter on size and location of the largest aftershock. To do this, we explore Δ*T* = *T*_{m} − *T*_{a} (*T*_{m} = mainshock time, *T*_{a} = largest aftershock time) and Δ*D** = *D*_{a}* is the normalized distance between the largest aftershock and the mainshock epicenter.

[6] Using the USGS global earthquake catalog, we verify that the Δ*M*, Δ*T* and Δ*D** values are independent of mainshock magnitude. Second, we investigate density distributions of size, time and space patterns of aftershocks. Third, we analyse Δ*M* and Δ*D** values as functions of earthquake faulting styles, as defined according mainshock rake angle [e.g., *Aki and Richards*, 2002].