## 1. Introduction

[2] The interaction between fault geometry and earthquake generation has been a main issue of seismology, because the problems where earthquake ruptures start and stop and what controls the variety of source processes are related intimately with the geometry of fault zones.

[3] As known well, a fault trace is not a straight line but segmented, and fault segments form an echelon arrangement associated with step overs or fault jogs. Well-constrained hypocenters of microearthquakes, fault zone trapped waves [*Li et al.*, 1994] and seismic wave scattering [*Nishigami*, 2000] delineated that segmented structures of fault zones extend deeper than 10 km in the earth's crust, indicating that they are fundamental structures throughout the seismogenic depth. On the basis of the knowledge in the 1980s, “characteristic earthquakes” tend to repeat in the same locations, and independent rupture segments can persist through several seismic cycles [e.g., *Schwartz and Coppersmith*, 1984]. Moreover, earthquake ruptures tend to initiate at the point below the smoother fault trace or a fault jog and terminate at fault jogs or fault bends [*King and Nabelek*, 1985; *Sibson*, 1985, 1986; *Barka and Kadinsky-Cade*, 1988; *Aki*, 1989]. Waveform inversion analyses now describe vividly the effects of the fault plane heterogeneity on the source processes; e.g., for the 1992 Landers earthquake [*Wald and Heaton*, 1994; *Bouchon et al.*, 1998], the 1995 Kobe earthquake [*Ide et al.*, 1996; *Wald*, 1996; *Yoshida et al.*, 1996], and the 1999 Chi-Chi earthquake [*Ma et al.*, 2001; *Wu et al.*, 2001].

[4] *Das and Aki* [1977] demonstrated first how the complexity of earthquake source processes and seismic radiations can be reproduced by “barrier model” of a fault plane. *Aki* [1979] suggested that the fracture energy of barriers is abnormally large and they can act not only a stopper of rupture but also as an initiator of rupture, as well as a stress concentrator causing the migration or propagation of major earthquakes. Thereafter, many computer simulations were performed for rupture propagation incorporated with the fault plane heterogeneity. A finite difference method demonstrated that strike-slip seismic ruptures can not jump both compressional and dilational fault steps wider than 5 km [*Harris and Day*, 1993]. The most advanced boundary integral equation method simulated successfully the spontaneous rupture transfer between nonplanar fault segments and its effects on ground motions during the 1992 Landers earthquake [*Aochi and Fukuyama*, 2002].

[5] As noted by previous workers, seismic nucleation is intimately related with the heterogeneous frictional properties of fault planes. *Umeda* [1990, 1992] and *Iio* [1992, 1995] found an empirical relation between the magnitude *M* of main shocks and the time duration *T*_{1} of the “slow initial phase” or “nucleation phase” as *M* ∝ log *T*_{1}. *Ellsworth and Beroza* [1995] found an equivalent relation between seismic moment *M*_{o} of main shocks with the critical length of the nucleation zone *L*_{c} written as *M*_{0} ∝ *L*_{c}^{3}. *Ohnaka and Shen* [1999], *Ohnaka* [2000, 2003], and *Ohnaka and Matsu'ura* [2002] found proportional relationships among the characteristic wave length λ_{c} of the fault plane topography, the critical length of the nucleation zone *L*_{c} and the breakdown displacement *D*_{c}, and *Ohnaka* [2000] successfully derived the empirical relations of *M* ∝ log *T*_{1} and *M*_{0} ∝ *L*_{c}^{3}.

[6] An important problem is how the fault segments and jogs are formed. *Segall and Pollard* [1980] studied this problem from the view point of static stress field in the area between tips of two nonplanar cracks. For a dilational jog both normal tractions on the overlapped crack ends and the mean compressive stress in the jog decrease to facilitate sliding. It tends to link the cracks and allow slip to be transferred through the discontinuity. In contrast, for a compressional jog they increase and inhibit frictional sliding. *Aydin and Schultz* [1989] and *Du and Aydin* [1995] studied the quasi-static growth of crack tips using a displacement discontinuity boundary element method. The fault interaction first enhances the growth of echelon faults as the inner tips pass each other and later impedes their growth after some degree of overlap. The shear fracture paths depend on the specific geometric configurations of echelon faults, the applied stress orientations and the coefficient of friction, producing a spectrum of the connectivity configurations.

[7] The growth of crack tips should be understood, of course, as a dynamic rupturing process. Using an elastodynamic boundary integral equation method, *Kame and Yamashita* [1999a, 1999b] predicted that a propagating crack spontaneously bends as a result of a stress wave concentrated near the extending crack tip, and that the crack tip growth is arrested soon after the onset of bending. Recently, *Ando et al.* [2004] simulated the growth of a planar fault segment approaching a preexisting noncoplanar segment. If the initial overlap of the two segments is smaller than the half length of the preexisting segment, the growing segment coalesces with the preexisting segment when the step over is narrower than about 1/4–1/2 the length of the preexisting segment but is repelled from the preexisting segment when the step over width is larger than this threshold distance. This mechanism explains the origin of fault jogs.

[8] The previous studies referred above indicate how the geometry of fault segments and jogs is important for earthquake rupturing. *Aydin and Nur* [1982] and *Aydin and Schultz* [1989] presented observations that the length/width ratio of fault jogs is constantly about 3 over the length range from tens of meters to tens of kilometers. *Wesnousky* [1988] found that the number of steps per unit length along the trace of major strike-slip fault zones is a smoothly decreasing function of cumulative geological offset, and suggested that faults may undergo a seismological evolution, whereby the size and frequency distribution of earthquakes is also a function of cumulative offset. His idea may be reasonable because the fractal size frequency and spatial distribution of fault populations also evolve [*Otsuki*, 1998; *Goto and Otsuki*, 2004].

[9] The main issue of this paper is to characterize the experimental fault zone geometry and to investigate its evolution law. This will offer a general constraint to the effects of the geometric heterogeneity of fault zone on earthquake source processes. Seismic slip events are regarded as a fractal phenomenon on the fractal backbone of preexisting faults. Therefore some rules for seismic events are expected to be derived only from fault zone geometry. The Gutenberg-Richter's law and the empirical relationship between the critical length of nucleation zones and seismic moment of main shocks are derived in this paper.