The geometry of fault zones and its evolutionary trend were experimentally investigated for samples of calcareous siltstone nodule through a repetitive procedure of axial loading to grow faults a little, close-up photographing of faults that appeared on the side surface of the samples, and axial loading again. Irrespective of their size, fault zones are composed of three or four fault segments and compression-type fault jogs. Smaller segment-jog structures are nested in larger segments, forming a hierarchical fault zone structure as a whole, and they show self-similarities. Fault zones grow keeping this self-similar hierarchical structure. On the basis of this fault zone geometry, we successfully derived the Gutenberg-Richter's law as well as the previously known relationship of seismic nucleation sizes to seismic moments. These results suggest that fundamentally any seismic rupture nucleates at a smaller jog (asperity) of a lower hierarchical rank and terminates eventually at a larger jog (barrier) of a higher rank, mimicking the hierarchical fault zone geometry.
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 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.
 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].
Das and Aki  demonstrated first how the complexity of earthquake source processes and seismic radiations can be reproduced by “barrier model” of a fault plane. Aki  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].
 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 T1 of the “slow initial phase” or “nucleation phase” as M ∝ log T1. Ellsworth and Beroza  found an equivalent relation between seismic moment Mo of main shocks with the critical length of the nucleation zone Lc written as M0 ∝ Lc3. Ohnaka and Shen , Ohnaka [2000, 2003], and Ohnaka and Matsu'ura  found proportional relationships among the characteristic wave length λc of the fault plane topography, the critical length of the nucleation zone Lc and the breakdown displacement Dc, and Ohnaka  successfully derived the empirical relations of M ∝ log T1 and M0 ∝ Lc3.
 An important problem is how the fault segments and jogs are formed. Segall and Pollard  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  and Du and Aydin  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.
 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.  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.
 The previous studies referred above indicate how the geometry of fault segments and jogs is important for earthquake rupturing. Aydin and Nur  and Aydin and Schultz  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  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].
 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.
2. Experimental Method
 In order to understand the evolution of fault zone geometry, we performed fracturing experiments for the samples of calcareous siltstone nodule at the confining pressure of 100 MPa and a nominal strain rate of 10−4/s using a conventional triaxial apparatus. The samples were shaped into cylinders of 40 mm length and 20 mm diameter. The basal planes were finished parallel within an accuracy of 1/200. The axial load and displacement were measured by a load cell and a differential transformer set outside the pressure vessel. The successive growth of microfaults was traced by repeating the experimental procedure for a same sample; applying axial load to grow microfaults a little, retrieving carefully the sample from the pressure vessel after unloading, close-up photographing of the microfaults on the side surface of the sample, resetting carefully the sample in the pressure vessel and loading again. This procedure was repeated as many times as possible until a large-scale fault cut entirely through the sample.
 A technical key point of our experiments is whether we can stop axial loading at an appropriate time after microfaults grow and before large-scale faulting starts. This is not so easy for our machine without a servo-system, but we could determined the appropriate time to stop loading while watching the stress-strain curve, since at the confining pressure of 100 MPa the samples were associated with yielding just before large-scale faulting. The reason why calcareous siltstone nodule was used is its moderately ductile property as well as very homogeneous texture.
 Photographing is another technical key point. A marker grid was drawn on the side surface of the sample at an interval of 3.5 mm and 6.7 mm. A transparent vinyl jacket was used, and through it close-up photographs of the side surface of the sample were taken after every experimental cycle. The area that was covered effectively by one shot is 6.6 mm × 3.5 mm, and the whole area of the side surface was covered by 6 × 18 shots. The resolution limit of the photographs is within the range from 0.1 mm to 0.2 mm.
 The loading-photographing-reloading cycles were repeated successfully several times for two samples until they were eventually cut by a large-scale fault. The stress-strain curve for sample 1 is shown in Figure 1a. During the first-fifth experimental runs, we could stop axial loading successfully just after the yield point at a differential axial stress of about 320 MPa and before large-scale faulting. However, the sixth run was stopped just after the stress drop, and we found a large fault extending through one side of the sample. Loading during the seventh run was continued even after the small stress drop at which large-scale faulting occurred. The total slip of the large-scale fault was measured at 0.3 mm from the offset of the marker grid.
 The stress-strain curve for sample 2 shows nearly the same mechanical behavior as for sample 1. Large-scale faulting occurred as early as the third run with a stress drop of about 40 MPa (Figure 1b). The net slip was measured at 0.16 mm. The 4th run was continued up to a total axial strain of 12%, at which the slip of the large-scale fault attained 0.71 to 1.14 mm.
 Note here that our samples were fractured in a somewhat ductile manner associated with remarkable yielding and a slow stress drop.
3. Fault Zone Geometry in Sample 1
Figure 2 shows the mosaic photograph of the side surface of sample 1 after the final experimental run. Figure 2 covers only about 12% of the area of the whole side surface. A network pattern of conjugate faults is well developed, and the right-lateral largest fault (marked by F1 in Figure 2) corresponds to the large-scale fault with 0.3 mm net slip formed during the seventh run. Conjugate microfaults, appearing just like brushing trails in the photograph, are densely developed on the background of the network of conjugate larger faults. The former are at an angle of about 40° with the maximum compressional principal stress axis σ1, while about 35° for the latter. The densely developed microfaults are likely to be a cause of the ductile behavior of the analyzed samples.
 Fault F1 is composed of four segments, and only a part of it is shown in the figure. The central fault segment S2 (14 mm long) in the figure is connected with the adjacent segments S1 and S3 by jogs associated with colateral overlaps of several millimeters. The fault traces of F2 and F3 also are composed of two and three segments respectively of several mm lengths, and associated with jogs or step overs. The tips of segments curve apart from the maximum principle stress axis, and they tend to connect with the adjoining fault segments, forming compressional jogs. Such en echelon arrangement of fault segments with jogs is very common also in the smaller faults, and they appear to show a similar geometry. According to Du and Aydin , compressional jogs are formed if the spacing between two echelon faults is wide enough so that the fault interaction is weak, and if the angle between the maximum compressive stress and the echelon faults is smaller than 45°.
 As the experimental runs were repeated, the faults grew successively. The evolution of fault F3, showing a typical segment-jog structure, is shown in Figure 3. Fault F3 after the seventh run was composed of three segments (S1, S2 and S3) each of about 4 mm length (Figure 3f), and the inside of the jogs is whitish, indicating the development of microcracks due to the stress concentration.
 No forerunner of fault F3 can be found in the photographs after the first run and even after the second run (Figure 3a). After the third run, we found very small faults s1 and s2 (only 1.2 mm and 0.8 mm long, respectively) that are the embryos of S1 and S2 (compare Figures 3b and 3f). During the 4th run the segment s1 grew to form an en echelon array of s1-1, s1-2 (correlated with s1 at third run) and the embryonic s1-3 (Figure 3c). The fault s2 also became longer (s2-2), and a new segment s2-1 was formed on the lower left side of s2-2.
Figure 3d shows the fault F3 after the fifth run. A new segment s1-0 was formed on the lower left side of s1-1, and the segment s1-3 grew rightward. An embryonic segment of s2-3 also was formed on the right upper side of s2-2. Both the segment series s1-0 + s1-1 + s1-2 + s1-3 and s2-1 + s2-2 + s2-3 show left-step en echelon arrays, and they were connected by s1-3 that extended right-upward. A new small fault s3 also was formed ahead of the segment series s2-1 + s2-2 + s2-3. During the sixth run the fault F3 hardly grew at all (Figure 3e).
 The sample was cut into two pieces by fault F1 at the end of the seventh run (Figures 1 and 2), and F3 also grew drastically during the seventh run (Figures 2 and 3f). The en echelon segment arrays of s1-0 + s1-1 + s1-2 + s1-3 and s2-1 + s2-2 + s2-3 were integrated into the larger segments S1 and S2. On the upper right side of s3-1 (s3 during the previous runs) the segment s3-2 was newly formed, and they as a whole constitute the third large segment S3. The segments S1, S2 and S3 form an en echelon arrangement on a larger scale, being the new constituents of fault F3 that grew to 12 mm total length. The small segments preset before the seventh run are hardly distinguishable, but they were incorporated in the new large segments as internal structures of the thickened shear zones.
4. Fault Zone Geometry in Sample 2
 The photograph of Figure 3 depicts realistically the process of fault growth, but the photo images are not as sharp. Therefore we present additional data for the evolutionary process of a fault formed in sample 2. Figures 4a and 4b show the geometry of a fault after the second run and the third run, respectively. The fault after the second run is composed of four large segments 2S1, 2S2, 2S3 and 2S4 with a length of about 5 mm. They show an en echelon array with jogs 1.4 to 2.7 mm long (Figure 4a). The intermediate two segments comprise four and five short segments with lengths of 1 to 3 mm, showing an en echelon array with jogs 0.3 to 1 mm long. As a whole, all of the short and long segments form a nested en echelon array.
 This fault grew dramatically as a large-scale fracture with a 40 MPa stress drop and 0.16 mm net slip during the third run (Figures 1b and 4b). The curved fault trace in this figure is simply due to the cylindrically curved sample surface. It is composed of three long segments 3S1, 3S2 and 3S3 (about 15 mm), arranged in an en echelon pattern with jogs 2.2 to 4.1 mm long. Each of these segments also is composed of several shorter segments; 3S2-1 to 3S2-4 for 3S2 and 3S3-1 to 3S3-6 for 3S3, showing en echelon arrays of a smaller scale. As a whole, the longer and shorter segments are systematically integrated into a nested en echelon array. Some of the short segments nested in the long segments of the second experimental run were combined into the short segments nested in the long segments of the third run, and others were inherited in the internal structures of the thickened shear zones of the long segments. The segments 2S1 and 2S4, the terminal parts of the fault at the second run, were abandoned, and the fault extended at a slightly smaller angle with σ1 axis.
 The connectivity configuration of fault segments observed in sample 1 and sample 2 is correlated consistently with the case of Du and Aydin  where the angle between maximum compressive stress axis and the orientation of segments is smaller than 45° and the friction coefficient is less than 0.6. It satisfies also the initial configuration for the coalescence of dynamically propagating segments of Ando et al. .
5. Evolution of Fault Zone Geometry
 As described in the preceding sections, the essential geometrical elements of fault zones are the length of segments, and the length and width of jogs. These structures are nested and show a hierarchical arrangement. Hereafter, we use “high (low)” ranks for larger (smaller) scale segment-jog structures. Irrespective of the hierarchical ranks, the number of segments in a given hierarchical rank is usually 3 (47%) or 4 (47%), and 5 (6%) at most. Since all fault zones were on the way to growth, some of them are composed of only two segments of the highest rank.
 After distinguishing the hierarchical ranks, excluding those of the incomplete highest rank, we measured the mean segment length LS, the mean jog length LJ and the mean jog width WJ as well as the total length of the segment series L0 for a given hierarchical rank of an individual fault zone. The relationships among LS, LJ, WJ and L0 are shown in Figure 5, and the data plots fit well to the power functions below.
All equations above demonstrate that the fault geometries are self-similar.
 Note here again that in some cases of our measurements L0 is for the total length of a fault zone, but in other cases it is for the total length of a segment series of a lower hierarchical rank. As described in sections 3 and 4, the sizes of the segments and jogs remain fundamentally unchanged, if once they are integrated into the segment-jog structure of a higher hierarchical rank. Therefore L0 even for the total length of a segment series can be interpreted as a total length of a fault zone at a growth stage. On the basis of this interpretation we conclude that equations (1) to (3) depict not only the self-similar fault zone geometry but also its evolutionary trend in which the self-similarities are kept at any evolutionary stage. Moreover, the fractal dimension (power value) of 0.64 in equation (3) and those close to 1 in equations (1) and (2) demonstrate that the growth of fault zones is allometric, that is, fault planes become smoother as fault zones grow.
6. Implications of Fault Zone Geometry for Earthquakes
6.1. Formulation of Fault Zone Geometry
 Here we rewrite equations (1) to (3) as Ls = CSLL0, LJ = CJLL0 and WJ = CJWL0α, where L0 denotes the total length of a fault zone at an evolutionary stage. L0 has 1/CSL segments and 1/CSL − 1 jogs of the first hierarchical rank, and every segment of the first rank is composed of 1/CSL segments and 1/CSL − 1 jogs of the second rank. In general, every segment of the ith hierarchical rank is constructed by 1/CSL segments and 1/CSL − 1 jogs of i + 1th rank (see Figure 6). On the basis of this hierarchically nested structure, the length of segment LS(i) and its total number nS(i), the length of jog LJ(i), its width WJ(i) and its total number nJ(i) are expressed as follows:
Note that the segment length LS(i) was regarded above as CSLLS(i − 1) for simplicity, neglecting the half length of jogs.
 From the pairs of equations (4) and (5), (6) and (8), and (7) and (8), the size frequencies of segments and jogs are expressed as follows:
Equations (9) and (10) represent that the size frequencies of segment lengths, jog lengths and jog width are self-similar, and the fractal dimensions of the former two are 1 while 1/α for the last. The hierarchically nested segment-jog structures described in sections 3 and 4 and equations (1) and (2) demonstrate that the spatial distributions of fault segments and jogs also are self-similar with a fractal dimension of 1, irrespective of the hierarchical ranks.
6.2. Seismic Nucleation and Size of Main Shocks
 An empirical relation M0 ∝ Lc3 between the critical length of the nucleation zone Lc and the seismic moment Mo of main shocks [Ellsworth and Beroza, 1995] demonstrates as if seismic ruptures may predetermine their earthquake size already at their nucleation stages, but this is very unlikely. Considering both the hierarchical fault zone geometry and the field observations [King and Nabelek, 1985; Sibson, 1985, 1986; Barka and Kadinsky-Cade, 1988; Aki, 1989], it is likely that earthquake ruptures nucleate at a jog of a lower hierarchical rank and terminates at jogs of higher ranks, mimicking a part of the fault zone geometry. Assume here that the jog of the nucleation site is of ith rank. The propagating rupture will overcome the jogs of the ith rank and lower ranks, but will be arrested at jogs higher than (i − 1)th rank. If this is the case, the length of the activated part is equal to LS(i − 1). From equations (4) and (6), the equation below is obtained.
From the assumptions above, LS(i − 1) and LJ(i) are regarded as the length of seismic fault Lseis and the length of seismic nucleus Lnucle, respectively. If the experimental value of CJL = 0.0935 is adopted, Lnucle is as small as 1/10 of Lseis.
 Using the empirical relations among earthquake magnitude M, length Lseis (m), area A (m2) and mean net slip D (m) of seismic faults [Sato, 1979], as well as the definition of seismic moment Mo = μDA, we have Mo(Nm) = 105.84Lseis3. Substituting equation (11) into this equation, and if CJL = 0.0935 is used, the equation below is obtained.
This equation is a good match with the equation Mo = 109 (2Lc)3 of Ohnaka . Here 2Lc is defined by him as a critical nucleation size at the transition from the phase of unstably accelerating rupture propagation to the phase of steady propagation at the velocity approaching an elastic wave. Therefore it is very likely that Lnucle defined in this paper is correlated with 2Lc of Ohnaka .
 Our analytical result above was derived from the hierarchical self-similar fault zone geometry and a simple assumption that earthquake ruptures nucleate at a jog of a lower hierarchical rank and terminates at jogs of higher ranks, and we reached the conclusion consistent with the field observations by Umeda [1990, 1992], Iio [1992, 1995] and Ellsworth and Beroza . Therefore this assumption is fundamentally correct, even if lengths of fault segments and jogs in same rank are of a statistical deviation.
6.3. Size Frequency of Earthquakes
 On the basis of the assumption that earthquake ruptures mimic the preexisting fault zone, we can derive a simple model for the size frequency of earthquakes. Suppose a two dimensional fault zone with length L0 and width rL0, in which fault segments with same length/width ratio are hierarchically nested. In addition, presume that ith rank jogs are not ruptured until all of the ith rank segments are broken k times. If this is the case, the nS(1) segments will activate knS(1)2 times in total until any one of the first rank jogs is broken, and thereafter the fault zone activates over the whole area of rL02. Likewise the nS(2) segments of the second rank also will activate knS(2)2 times in total until any one of the second rank jogs is ruptured, and thereafter every nS(1) segments with LS(1) activate individually. In general, the nS(i) segments of the ith rank will activate knS(i)2 times in total until any one of the ith rank jogs is ruptured, and thereafter every nS(i − 1) segments with LS(i − 1) activate individually. This process is just like a reversed energy cascade.
 Denoting the activated fault length as L and its number as n, they are written as L = LS(i) and n = knS(i)2. Using equations (4) and (5), they are rewritten as L = CSLiL0 and n = kCSL−2i. From these equations, we have an equation below.
Using the well-known equation log L(m) = 0.5M + 1.12 [Sato, 1979], the equation above is expressed as
being consistent with Gutenberg-Richter's law, at least for the earthquakes on a given fault zone.
 We could derived the empirical relations Mo ∝ Lnucle3 and log n ∝ −M easily from the experimental fault zone geometry. This indicates that the self-similar hierarchical structure of fault segments and jogs is a deterministic parameter for nucleation, propagation and arresting of seismic ruptures. Why is this structure self-similar and hierarchical? A key observation was described in section 5. Irrespective of the evolutionary stages, any fault segment of a given hierarchical rank is composed usually of 3 or 4 segments of one lower rank. The second key observation is given in section 3. Once three or four segments of a given hierarchical rank coalesce, they as a whole behave as a new and longer segment of one higher rank. Third key point is given by Ando et al. . The initial condition for two segments to coalesce is the step over narrower than a critical distance normalized by the segment length. If these three are the cases, the fault zone embedded in an elastic medium is inevitably of a self-similar and hierarchical geometry at any scale of observation and at any evolutionary stage.
 According to the field observations of Aydin and Nur  and Aydin and Schultz , the length of fault jog is proportional to the jog width over several order of length scale. On the other hand, the number of steps per unit length along fault traces decreases nonlinearly as a cumulative geological offset increases [Wesnousky, 1988]. These two observations cannot be related directly, but appear to be inconsistent each other. Our observational result of equation (3) suggests that fault zones become smother as they grow, and is consistent with Wesnousky .
Kanamori and Allen  presented the observation that for a given length of the seismic fault, earthquakes with a longer repeat time tend to have a larger magnitude. They attributed it to the increase in static friction that increases as the time of stationary contact increases. Following our context in this paper, an alternative interpretation can be given as below. An active fault with a longer repeat time is in general of a shorter fault length, and it is regarded to be still at an early evolutional stage. Immature faults have jogs with a larger width/length ratio, and hence stronger than mature faults. Therefore the earthquake magnitude for the former tends to be larger than the latter, if the seismic faults are of same length.
 The following points summarize our conclusions.
 1. We presented examples of experimental fault zone structures. They are composed of essential structural elements: fault segments with an en echelon array and fault jogs. These elements are nested to form a hierarchical geometry as a whole.
 2. Nested segment-jog structures show a self-similar geometry at any hierarchical rank, and this holds at any evolutionary stage of fault zones. On the basis of the nested self-similar geometry, the size frequencies of segment length, the length of jogs and the width of jogs were formulated.
 3. The relation between the size of seismic nucleus and seismic moment of main shocks and the Gutenberg-Richter's law were successfully derived from the fault zone geometry.
 4. The conclusions above indicate that seismic ruptures nucleate at a smaller jog of a lower hierarchical rank and terminate at larger jogs of a higher hierarchical rank, mimicking the hierarchically nested fault zone structure.
 The authors thank B. S. Cramer for his kind help in polishing up English wording in the manuscript. They are also grateful to Yoshihisa Iio, Eiichi Fukuyama, and an anonymous associate editor for their critical reviewing and valuable suggestions.