DRAFT 1

Elastic Property of Damage Zones Inferred from In-situ Stresses: Changes in Seismic Wave Velocity Caused by Faulting

Kiyohiko Yamamoto, Namiko Sato, and Yasuo Yabe

Graduate School of Science, Tohoku University, Sendai 980-8578, Japan 1)e-mail: yama@aob.geophys.tohoku.ac.jp

Stresses have been measured at sites close to the Nojima fault just after the 1995 Kobe, Japan, earthquake (MJMA=7.3) by

deformation rate analysis and by hydraulic fracturing method. In this paper, coseismic changes in fracture density and crack density are estimated for the fault zone or the damage zone from these stress data using a fracturing process model. Further, coseismic changes in seismic wave velocities of the damage zone are calculated from the estimated change in crack density by means of the theory of effective elastic constants. Ignoring anisotropy due to preferred orientation of cracks, the decrease in seismic wave velocity is about 1.1 % of the host rock velocity for P-waves and about 2.7 % for S-waves. These amounts are nearly equal to or larger than those observed in the focal area of an earthquake. Taking anisotropy into consideration, the velocity of a mode of S-waves, which produce a change in crack volume, varies by about 50 % of the average velocity de-pending on the propagation direction and is the largest when the propagation direction is about 50° from the fault normal. This means that the first arrival of S-waves traveling in damage zones is not the direct S-waves but the S-waves reflecting at walls of a damage zone while traveling. The velocity-depth relationship and the P- to S-wave velocity ratio observed for some faults are consistently explained by the anisotropy of damage zones. Key words: weak fault, fault zone, in-situ stress measurement, anisotropy, fracture density, crack density

1. Introduction Frictional strength of faults is essential for modeling

deformation processes of the earth’s crust. The constitutive law for frictional sliding is substantial for modeling earth-quake generation process. Recently, Yamamoto et al. (2002) have shown the possibility that structure and mechanical properties of damage zones control the strength of faults, and probably affect the characteristics of slip propagation. Therefore, it is considered the most important requirement for the study of fault strength and slip charac-teristics to clarify the structure and the physical properties of damage zones or fault zones.

Stresses were measured near the Nojima fault ruptured during the 1995 Hyogo-ken Nanbu, Japan, earthquake (MJMA=7.3). Summarizing the data, Sato et al. (2003) have shown that the largest horizontal stress is almost perpen-dicular to the strike of this nearly vertical fault and that the shear stress is small in the vicinity of the fault core axis. The shear stress is larger outside of the zone while the de-formation is expected to be larger inside. This implies that the stresses equilibrate to the strength of the zone. On these results, Yamamoto et al. (2002) have inferred that a damage zone exists in a post-failure state under the compressive stress acting on the fault plane, of which a principal axis is in the fault normal direction. Based on this consideration, they have proposed a model of faults consisting of asperiti-es and apertures filled with damaged rocks and have shown that the elastic properties, that is large Young’s modulus and small rigidity, of the damaged rocks can make the strength of faults decrease. Here, damage zones mean aper-tures filled with damaged rocks.

However, the properties of damage zones have not be well understood. Yamamoto et al. (2002) have estimated the elastic constants of damage zones as follows. Yama-moto (1998) has proposed a model, called the fracture process model, to provide the relationship between fracture density or crack density and applied stress for rock speci-mens under tri-axial compression tests. This relationship can also be regarded as the relationship between strength

and fracture density for damaged rocks. Fracture density or crack density in damage zones can be estimated from measured stresses using this model by assuming that the stresses equilibrate to the fracture strength of rocks in damage zones. Further, they have calculated the elastic constants for these estimated crack density using the theo-ries of effective elastic constants. They have demonstrated by this calculation that the damage zone has large Young’s modulus and small rigidity to the applied stresses on the fault plane.

In order to verify the above model of damage zones, the first step is to confirm the theoretical seismic wave ve-locities. For the damage zone of the Nojima fault, Ito et al. (1996) obtained the seismic P-wave velocity from borehole logging and Kuwahara and Ito (1999), Li et al. (1998), and Nishigami (2000) estimated the S-wave velocity from the trapped waves. Yamamoto et al. (2002) estimated the frac-ture density of the damage zone to be about 0.8 from the measured stresses and they have calculated the seismic wave velocities of the damage zone at depths, assuming that the fracture density is constant in all depth and that pore pressure is hydrostatic. They have found that the cal-culated velocities approximately explain the observed ve-locities.

Feng and McEvily (1983) determined the P-wave ve-locity structure of the San Andreas Fault zone near Park field. This segment of the fault appears to be weak as far as inferred from the World Stress Map, Rel. 1997-1 on Web (Heidelberg Academy of Sciences and Humanities, Univer-sity of Karlsruhe/International Lithosphere Program). Ya-mamoto et al. (2002) calculated P-wave velocities and compared with the observed velocity structure, assuming that the fracture density is invariant throughout a damage zone and for every fault. The calculated velocity appears to explain the observed velocity to the depth of about 15 km. This implies that the model is realistic, and suggests that the fracture density and thus the stress in the damage zone of the San Andreas Fault are almost equal to those of the No-jima Fault and further that pressurized water is not neces-

DRAFT 2

sarily required to explain weak faults. Damaged rocks under axial stresses of compression

produce elastic anisotropy in damage zones. Nevertheless, the comparisons described above were performed neglect-ing this anisotropy. Therefore, it is necessary to prove that damage zones are anisotropic. Yamamoto et al. (2002) have mentioned that the S-wave velocity in damage zones de-termined by trapped waves is slightly larger than the esti-mated velocity, while the P-wave velocity appears to be reasonable as described above. Li et al. (1998) have shown that the travel times of P- and S-waves in a damage zone are shifting after an earthquake. Their analysis suggests that the ratio of the travel time shift of P-waves to that of S-waves cannot simply be explained by the theory of effec-tive elastic constants of an isotropic composite. One of the aims of this paper is to discuss the S-wave velocity in damage zones and the travel time shift ratio by taking the anisotropy into consideration.

Analyses of scattered waves or coda waves in seis-mograms suggest that seismic wave velocities change dur-ing an earthquake (e.g. Dodge and Beroza, 1997). Recently, it was reported that the seismic wave velocities decreased after the M6.1 earthquake near Iwate volcano in northeast Honshu, Japan (Nishimura et al., 2000; Matsumoto et al., 2001; Nakamura et al., 2002). Nishimura et al. (2000) showed that the velocity of S-coda waves decreased by about 0.3 to 1.0 %. Matsumoto et al., (2001) demonstrated that the phase arriving 2 sec after the first P-arrival was de-layed by about 30 ms after the earthquake. However, the traveling paths have not exactly been specified for these phases. Li et al. (1998, 2001) showed the gradual recovery of travel times of trapped waves after the 1992 M7.5 Land-ers, California, earthquake. The amount of these velocity decreases appears to be close to that of the recoveries. This implies that the coseismic velocity changes are caused by the crack density change in a damage zone due to stress change. Another purpose of this paper is to explain these velocity changes in terms of the stress change in damage zones caused by faulting to verify the damage zone model. 2. Stress change in the damage zone of the Nojima Fault

Stresses have been measured in holes drilled along the Nojima fault about 1 year after the 1995 Hyogo-ken Nanbu earthquake (MJMA=7.3). Ito et al. (1997), Ikeda et al. (2001), and Tsukahara et al. (2001) employed the hydraulic frac-turing method (HF) for the measurement and Sato et al. (2003) applied the deformation rate analysis to rock core sample recovered from the holes.

Deformation rate analysis (DRA) has been developed to measure the stresses from rock samples based on the rock property of stress memory (Yamamoto et al, 1990). It has been confirmed that the vertical stress measured by DRA is close to the overburden pressure. This implies that the measured stresses are those after the rocks have been placed at the depths, in other word, after the topography around the sites have been formed. It is also known that the stress can be measured from the rocks that have been re-trieved from depths several years before the measurement. Further, we can often obtain almost the same stress magni-tudes by repeating the measurement on a specimen. Rocks have been subjected to almost constant stresses at depths

for a long time. This stress memory is explained by the re-laxation of stress concentrations during the time, which is thought to result from plastic deformation of mineral grain boundaries (Yamamoto, 1995b). For these reasons, the measured stresses may be understood to be those averaged over more than several years before retrieving the rocks from depths. The stresses measured by DRA are thus ex-pected to be the average stress before the earthquake, while the stresses by HF are instantaneous stress after the earth-quake. If there are significant differences in the measured stress magnitudes between DRA and HF, the differences may be regarded as the stress variation due to faulting. 2.1 Definition of potential stress, r

In order to discuss stresses in relation to shear strength of the earth’s crust, a parameter r defined by

(1) is introduced, where !� and !� are the largest and the smallest principal stresses. In general, the shear strength of rocks increases proportionally to the normal stress on the shear plane. Thus, the parameter r may be thought as an index of the potential of the stress field for shear fracture. The value of r is calculated here on the assumption that one of the principal directions is vertical. If faulting occurs on a fault plane at r = rs, rs may be called the apparent friction coefficient of the fault. Hereafter, the parameter r is re-ferred to as r-value or potential stress. 2.2 Definition of damage zones

Sato et al. (2003) summarized the results of stress measurement performed at sites close to the Nojima fault. They have concluded that r-value does not decrease with an increase in depth, but is smaller near the fault core. Figure 1 shows the relationship between r-value and distance of measuring points from the fault core axis. The small r-value near the fault core axis can be confirmed in this figure. This property is probably one of the characteristics of the stress state near the fault core axis, which may be widely ob-served along a fault. Another characteristic is that the aver-

Fig. 1. Relationship between r-value and the distance from the fault

core axis. The difference in symbol denotes the difference in the method for measurement. HRBNIED (HF) are the data of hydraulic fracturing at HRB by Ikeda et al. (2001), HRBGSJ (DRA) are the data of DRA at HRBGSJ by Sato et al. (2003). TSM (DRA) and TSM (HF) respectively mean the data of DRA at TSM by Sato et al (2003) and those of hydraulic fracturing by Tsukahara et al. (2001). The r-values near the core axis scatter in the range indi-cated by arrows. This figure is modified from Sato et al. (2003).

DRAFT 3

average stress near the fault core is close to the lithostatic stress. The zone of small r-value is about 100 m in width from the core axis in the case of the Nojima fault as shown in Fig. 1.

We use the terminology by Chester et al. (1993) for fault zone structure. In the case of the Nojima fault, it has been pointed out that there are damaged rocks in a zone few tens meter wide along the fault core (Ito et al., 1996; Li et al., 1998). Yamamoto et al. (2002) have interpreted that the small r-value is a reflection of the small strength of the damaged rocks around the fault core. Although the source zone of small r-value cannot exactly be defined here, a damage zone means the source zone of the small r-value. The damage zone may be equivalent to the fault zone de-fined by Chester et al. without the fault core. 2.3 Stress change in damage zones caused by faulting

The stresses measured at sites close to the Nojima fault by DRA are expected to be those before the earthquake. If faulting had advanced damage, the shear stress obtained by HF should be smaller than that by DRA, since the strength of damage zones should decrease with an advance of dam-age. Whether the stresses are measured by HF or by DRA, the r-values at sites close to the fault core are small com-pared with those at distant sites. Roughly speaking, the r-values by HF and DRA in the zone appear to be almost identical to each other. This implies that the damage zone has developed with repeated faulting.

By inspecting Fig. 1 in more detail, it may be seen that the r-values of HF are smaller by about 0.06 on an average than those of DRA in the damage zone. However, it is dif-ficult to decide if this difference is significant for the fol-lowing two reasons: Firstly, the precision of the measure-ment is nearly equal to the difference. Secondly, their measurements have not been performed at the same points. Nevertheless, it may be said that if stress has dropped due to faulting, the amount may be about 0.06 in r at most. 3. Estimation of crack density change 3.1 Damage zone model and fracture process model

Figure 2a schematically illustrates a fault zone that is filled with damaged rocks. This simplification of fault zone

structure may be justifiable from the observational results by Chester et al. (1993) without the fault core, as described previously. This simplified fault zone is referred to as the damage zone model. The most important characteristic of the stress field in or near damage zones is that the largest principal stress lies almost perpendicular to the fault plane. Thus, for modeling damage zones, we assume as follows: 1) The largest principal stress lies in the direction perpen-dicular to the fault plane and the other principal stresses have the same magnitude. 2) The stresses in a damage zone equilibrate to the strength of the damaged rocks. In the damage zone under this stress field, cracks can be consid-ered to orient their surfaces parallel to the fault normal and distribute symmetrically around the fault normal.

The relationship of fracture density or crack density to strength of rocks at the stress state described above can be known from the relationship between elastic wave velociti-es and applied stress for rock specimens under tri-axial loading tests. Figure 2b illustrates the specimen, where the Cartesian coordinate is defined in order that x1 is in the di-rection parallel to the loading axis. Yamamoto (1995a, 1998) analyzed the data by Matsushima (1960) to propose an expression for the relationship between applied shear stress " and tensile crack density c in the specimens.

The relationship is given by , (2)

where C is a constant and !3 denotes the effective confining pressure. Here, the crack density c is defined by

and , respectively, are porosity and aspect ratio of

cracks, u is the applied shear stress normalized by the ulti-mate shear strength of the specimen. If the lithostatic pres-sure approximates the average stress at a depth and pore pressure is equal to the hydrostatic pressure, may be written as a function of depth by

, (3)

where !3 is expressed in GPa, , are densities (kg/m3) of overburden rocks and pore fluid, respectively. g and d are the gravitational acceleration (m/s2) and depth (m).

The function G(u) represents the density of shear mi-cro-fractures or fracture density. The fracture density is de-fined as the fraction of the volume elements in a specimen, which have lost their strength. The function G(u) is ex-pressed by

, (4) provided that the Coulomb criterion holds for microfrac-turing. Here, s0 is the normalization factor. The shear stress u may be approximately written by , where r is given by (1). rf is the ultimate shear strength of specimens and the equality holds when the cohesion can be neglected in the Coulomb criterion. The equation expressed by (4) is called fracture process function. The model for the equation is called the fracture process model. The derivation of

is reproduced in our previous paper (Yamamoto et al., 2002).

The behavior of G(u) for m = 5 is illustrated in Fig. 3. The stress u at the point B is the ultimate shear strength of

Fig. 2 a) Schematic illustration of the principal directions of com-

pressive stress acting to a damaged zone. The coordinate taken for the zone is shown together. One of the principal axes is taken in the fault normal direction. The principal stresses are denoted by !i (!1 > !2 > !3). The x1 axis is taken to be orthogonal to the fault plane and x3 axis to be in the depth direction. b) A rock specimen under tri-axial loading test and the coordinate defined for the specimen.

DRAFT 4

specimens under tri-axial compressive stresses. The value vc at B is the critical fracture density. When G is smaller than vc, G may increase along the curve AB with an in-crease in or r. When G is larger than vc, specimens have to deform until the applied stress decreases to the magnitude the specimens can sustain, that is the stress on the curve BC. The post-failure state is where G is larger than vc. The stress state on AB branch represents a quasi-stable state for rocks, where fracture density increases while sustaining an increasing applied stress. Here, we call this the ante-failure state. As seen in the explanation above, the function represents the fracture strength of rock speci-mens, of which fracture density is G. The stress and the fracture density in rocks should be in the hatched region enclosed by lines AB, BC, and CA. 3.2 Estimation of crack density change in damage zones

It is reasonable to consider that the small shear stress in damage zones is caused by the small shear strength of damaged rocks under post-failure state and the shear stress in damage zones equilibrates to the strength of the damaged rocks. Yamamoto et al. (2002) have assumed for analysis that the expression (3) is applicable throughout the range of fracture density G, although the expression has been con-firmed to hold only for the AB branch in laboratory ex-periment.

Figure 4 shows the function G(u) calculated for some values of parameter m. The value of m has been estimated to range from 5 to 10 for some types of rocks and the crust (Yamamoto, 1998). The largest shear strength of intact rocks, or the largest strength of the crust, may be given by rf = 0.6 from the stresses measured in the crust, referring to Zoback and Townend (2001). Assuming thus that rf = 0.6, the fracture density G(u) is estimated at a value between about 0.75 and 0.85 for r = 0.15 to 0.21 that has been ob-tained by the stress measurement around the damage zone as seen in Fig. 1. The density c of tensile cracks at a depth can be estimated by making use of (3) and (4), provided that r is invariable with depth.

4. Coseismic Changes in seismic wave velocities

So far, attention has not been paid to the anisotropy of damage zones in either observational nor theoretical studies. In this section, firstly the averaged feature of seismic wave velocities and their coseismic changes will be discussed by calculating the elastic constants by neglecting anisotropy. Next, the velocities and their changes will be discussed for damage zones by taking the anisotropy due to preferred orientation of cracks into consideration. For the sake of calculating effective elastic constants, the matrix is assum-ed to be isotropic. The velocities and their changes calcu-lated for damage zones will be compared with the existing data obtained by the trapped wave observations. 4.1 Averaged behavior of velocity change 4.1.1 Theoretical procedure

Here we examine three causes for the velocity change due to faulting. One of them is the shear stress drop. An-other is the change in average stress or confining pressure. The other is the transition between water-saturation and dehydration of the crust.

P- and S-wave velocities of composites with spheroidal inclusions are generally expressed as follows;

. (5) Here, respectively are the incompressibility, rigid-ity and density. The variables with the superscripts h and i denote the properties of host rocks and inclusions, respec-tively. and respectively denote tensile crack density and crack orientation. The velocity changes by

with a crack density change by . Then, may be calculated by

. (6)

When the crack density change is caused by the change in r, may be calculated by

. (7)

When the crack density change is due to the change in the

Fig. 3 Fracture density G(u) as a function of applied shear stress u

normalized by the macroscopic strength. a) Explanation of G(u). The stress at the point B means the macroscopic fracture strength of a specimen. The path B to C through D means that all volume loses the strength at once or the macroscopic fracture. The paths AB and BC mean the ante-failure state and the post-failure state, respectively. b) Behavior of G(u) for some values of m. r is defined by (1) in the text.

Fig. 4 A change in G caused by faulting. A value of G, for example, is

estimated to change approximately from 0.80 to 0.85 at m = 5.

DRAFT 5

confining pressure, may be calculated by

. (8)

The rocks with water-saturated cracks stand for the satu-rated crust and those with void cracks stand for the dehy-drated crust. The differences in velocity between the satu-rated crust and the dehydrated crust mean the differences between the velocities for water-saturated cracks and those for void cracks at the same crack density.

The elastic constants of isotropic composites are cal-culated by the method of new self-consistent scheme (NSC) or differential scheme (DS) proposed by Yamamoto et al. (1981) and Norris (1985). The elastic constants of damage zones with cracks in a preferred orientation are calculated by an approximation method for weakly interacted inclu-sions (Weak-Interaction Approximation) by Yamamoto (1980, 1995). This method was reproduced in our previous paper (Yamamoto et al., 2002). Both the methods are based on the theory of Eshelby (1957), where cracks are theoreti-cally isolated from one another. This means that the pore fluid is pressurized or depressurized due to rapid deforma-tion of composites when cracks are saturated with fluid. This may actually occur for quick deformation such as that caused by incidence of seismic waves. 4.1.2 Parameter setting

The fracture density G has been estimated to be about 0.8 for the Nojima fault. Assuming that confining pressure is lithostatic and pore pressure is hydrostatic, the crack den-sity c is determined as a function of depth by substituting

and into (4). The corresponding seismic wave velocities are calculated for c. Here, the host rock is modeled on granite. Its Pois-son’s ratio and incompressibility respectively are taken to be 1/4 and 60 GPa. The aspect ratio " of cracks is taken to

be 10-3 in the present study, because it has been shown in our previous paper that the damage zone model approxi-mately elucidates the observed velocity structures of fault zones for this aspect ratio.

In order to discuss generally the observed velocity changes during an earthquake, the relationships between velocity and depth are calculated for r not only of damage zones but also of the earth’s crust. Figure 5 shows the ve-locities as the ratios to the velocities of host rocks. Since the fracture density G is two-valued function of r, r–values under the post-failure state are distinguished by the charac-ters ‘PF’ following after r-values. For convenience of dis-cussions, the crust is classified according to the potential stress level. The moderately stressed crust means the crust where r is smaller than about 0.4 under ante-failure state, nearly critical state means that r is more than 0.5, and the damage zone means the part of a fault zone where r is smaller than about 0.3 under post-failure state. 4.1.3 Velocity change with the change in potential stress r

If the stress in the crust equilibrates to the fracture strength of the crust, the stress is specified as a point on the curve in Fig. 3. This state of stress has been called the quasi-stable state. In the crust under the quasi-stable state, the fracture density increases with an increase in r under the ante-failure state, and r decreases with an increase in frac-ture density under the post-failure state. This decrease in r is caused because the fracture strength of the crust decreas-es with an increase in fracture density under the post-failure state, as seen in Fig. 3. The processes described above are irreversible for both the ante-failure and the post-failure states, since the fracturing process is generally irreversible. The Kaiser effect is one of the representatives for the ir-reversible process (e.g. Kuwahara et al., 1990).

The changes in the velocities of P- and S-waves with the change in r are calculated for some values of r under

Fig. 5 Seismic wave velocities of the water-saturated crust as the functions of depth (left; P-waves, right; S-waves). The velocity is shown as the ratio to the velocity of host rocks. The values of r marked (PF) are those of post-failure state. The value of G is taken to be 0.8. See text for other constants taken for the calculation.

DRAFT 6

both the ante-failure and the post-failure states by assuming that the crust is under a quasi-stable state. The calculated velocities are shown as the functions of depth in Fig. 5 and their changes with potential stress r are shown in Fig. 6. In damage zones under the post-failure state, the velocities decrease with a decrease in r. In an ordinary crust that is under the ante-failure state, the velocities decrease with an increase in r. These decrease and increase in the velocities are irreversible, as described above. Thus, their instantane-ous recovery with the recovery of r cannot be expected.

In damage zones, the value of r is about 0.21 before an earthquake. The value has been inferred to decrease by at most about 0.06 during an earthquake. This decrease in r corresponds to an increase by about 0.05 or 0.10 in fracture density, as seen in Fig. 4. The increase in fracture density makes the seismic wave velocities decrease. It is seen in Fig. 6 that the decreasing rates of velocities for a change in r slightly depend on depth. For discussing the changes in ve-locities of damage zones, we take r = 0.2 PF (under the post-failure state) and depths to be 2 and 15 km. At these depths, the decreasing rates, respectively, are about 0.19 and about 0.16 of the host rock velocity for P-waves and about 0.45 and about 0.35 of the host rock velocity for S-waves. Therefore, when r decreases by 0.06, the velociti-es decrease by 1.1 and 1.0 % of the host rock velocity for P-waves at the depths of 2 and 15 km, respectively, and by about 2.7 % and 2.1 % of the host-rock velocity for S-waves. Since the P-wave velocities of damage zones are about 0.65 and 0.80 of the host rock velocity and the S-wave velocities are about 0.25 and 0.60 of the host rock velocity at the depths, as seen in Fig. 5, the amounts of ve-locity reduction are about 1.7 and 1.3 % of the damage zone velocity for P-waves and about 11 and 3.5 % of damage zone velocity for S-waves.

When faulting occurs, shear stress may decrease not

only in the damage zone of the fault but also even in the ordinary crust at a distance from the fault. In the ordinary crust, however, the instantaneous increase in the seismic wave velocities with the stress-drop cannot be expected except for some limited regions in the vicinity of asperities along the fault. This is because fracturing is an irreversible phenomenon as described above in addition to the reason because the amount of stress drop is not so large as that in the damage zone. Although the velocity changes may be expected in some regions, it is probably difficult to actually observe the changes, because the asperity areas may be small if faults are weak. From these considerations, the changes in seismic wave velocities with the change in r are expected to be observable not for the moderately stressed crust but only for damage zones. 4.1.4 Velocity change with confining pressure and with saturation of fluid

When r is held constant, the fracture density does not change, while the density of tensile cracks increases and decreases in response to a decrease and an increase in ef-fective confining pressure, respectively. This behavior of crack density may be independent of the Kaiser effect or reversible. Figure 7 shows the velocity changes with con-fining pressure. It is seen in this figure that a large velocity change is restricted within the upper 5 km depth. It is as-sumed here that confining pressure changes by 1 MPa. In the moderately stressed crust, the change is estimated to be smaller than 0.1 % of the host rock velocity either for P-waves or for S-waves even at the depth of 2 km. If the confining pressure change means the change in the average stress due to faulting, the assumed amount of change of 1 MPa may be the upper limit. Also taking into consideration the small velocity changes at the larger depths together, it is difficult to observe the velocity changes for the moderately stressed crust. In damage zones, the change in S-wave ve-

Fig. 6 Velocity changes per unit change in potential stress (right; P-waves, left; S-waves). The values of r marked (PF) mean those of post-failure state. The constants taken for the calculation are the same as those for Fig. 5.

DRAFT 7

locity is about 0.5 % and that in P-wave velocity is about 0.25%. These changes may be around the critical amount that is detectable.

Figure 8 shows the ratios of the velocities of dehydrat-ed rocks to that of saturated rocks. If crack density is as-sumed to be constant independently of a change in effective confining pressure, the following may be said from the fig-ure: The change in P-velocity is larger than the change in S-velocity for the transition between saturation and un-der-saturation. The change amounts to about 4 % at shallow depth for P-waves even in the moderately stressed crust, and about 1 % for S-waves. The change is very large in damage zones either for P-waves or for S-waves as seen in the figure. The velocity decreases to 10 # 40 % for P-waves and to 20 # 60 % for S-waves due to the transition from saturated to dehydrated rocks. These amounts of velocity change are larger than the resolution of observation even for the moderately stressed crust as well as for damage zones. 4.1.5 Interpretation of observational velocity changes

An M 6.1 earthquake occurred in September 3, 1998 near Iwate Volcano, northeast Honshu, Japan. The Re-search Group for Explosion Seismology (RGES) detonated explosives two times at the almost same sites just above the inferred fault plane. The first explosion was performed in August before the earthquake and the second in November after the earthquake (RGES, 1999). Some authors have been analyzed seismic waves from the two explosions to find evidences for the change in seismic wave velocities in the focal area, which are considered to be associated with the faulting (Nishimura et al., 2000; Matsumoto et al., 2001; Nakamura et al., 2002).

Nishimura et al. (2000) analyzed coda parts of seis-mograms produced by the explosions to show a velocity reduction of about 0.3 – 1.0 % in the focal region of the earthquake. This velocity reduction was determined from the increasing rate of travel time with an increase in lapse time for phases in coda waves. Their analysis has been per-formed on the assumption that the waves are scattered from

obstacles uniformly distributed in space. However the as-sumption seems not to be definite, since a fault zone may be especially strong heterogeneity. If the boundaries or walls at both sides of a damage zone are irregular in shape, the incident waves to the boundaries may be scattered at irregularities and some of the energy may be trapped in the fault zone. If the energy thus trapped is scattered outward from irregularities while propagating along the zone, the waves may be observed as the scattered waves from the zone, as illustrated in Fig. 9. Here, these scattered waves are called scattered trapped waves. Even if we interpret the delayed phases observed by Nishimura et al. (2000) as the scattered trapped waves, the observational results may not alter this interpretation.

Matsumoto et al. (2001) found an arrival time delay of about 30 ms for the phase arriving about 2 second after the first P-arrival. They determined the location of the anoma-lous region associated with this phase to be in the focal area, and have attributed the delay to the migration of the anomalous region associated with the earthquake. Accord-ing to their estimation, the anomalous region migrates in depth from about 3 km to about 12 km. However, there may be another possible interpretation as follows: This phase consists of the waves traveled along a ray path close to that for the first P-arrival. The path is longer than that of the first P-arrival by the length corresponding to the additional travel time of about 2 second. Since the delay was not ob-served for the first arrival, the delay should be produced in the additional path. The velocity decrease in the additional

Fig. 7 Velocity changes per unit change in confining pressure (right;

P-waves, left; S-waves). The values of r marked (PF) mean those of post-failure state. The constants taken for the calculation are the same as those for the previous figures.

Fig. 8 Ratio of seismic wave velocity of the dehydrated crust to that

of the water-saturated crust (right; P-waves, left; S-waves). The ratios near 100 % are shown with the horizontal axis magnified in the lower figures. The value of G and the aspect ratio of cracks are taken to be 0.8 and 10-3, respectively. The values of r marked (PF) mean those of post-failure state.

DRAFT 8

path is inferred thus to be about 1.5 %. Nakamura et al. (2002) have found that the first

P-arrival is delayed after the earthquake at some observa-tion stations in and around the focal area. They have esti-mated that the delays are caused by the reduction of P-wave velocity at shallow depths and the reduction amounts to at least 1 %. Further, they have found that the phases in coda parts abruptly begin to delay by about 20 ms at a certain lapse time after the earthquake. They have pointed out that the delay is caused in an anomalous region that lies at most about 13 km deep in the focal area, which is different from the region where the first P-arrival is delayed. 4.1.6 Comparison with observed velocity changes

Accepting the interpretations modified by us, the re-sults from the observations, except for the delay of the first P-arrival, may be summarized as follows. There is an as-semblage of obstacles or an anomalous region at depths between a few km and 13 km in the focal area of the M 6.1 earthquake near Iwate volcano. For the phases traveling along or going across the anomalous region, the travel time increases by about 20 or 30 ms after the earthquake. This increment of travel time corresponds to the reduction of the velocity of the anomalous region of between 0.3 and 1.5 %.

Firstly, it is tentatively assumed that the change in con-fining pressure is the cause of the delay. Figure 7 shows the change in the seismic wave velocities with a change in con-fining pressure. As seen in this figure, at all depths from 2 to 15 km in the ordinary crust, the P-wave velocity change is less than 0.05 % of the host rock velocity for the change of 1 MPa in confining pressure. From this, it is estimated that the variation of 1 % in the P-wave velocity requires the change by 20 MPa in confining pressure. Similar discussion is possible for the S-wave velocity. This confining pressure change is about ten times in magnitude as much as shear-stress drop due to faulting. Since it is not general to take the change in confining pressure larger than the shear stress-drop, it is reasonably concluded that the variation of 1 % in the velocity does not occur in the ordinary crust. For damage zones, it is seen from Fig. 7 that the variation of 1 % in P-wave velocity at 5 km depth requires the change in confining pressure by about 10 MPa. This amount of the change is also too large for the drop of the average stress. For the above reasons, it is concluded that the velocity

change is caused by the confining pressure change neither for the moderately stressed crust nor for the damage zone.

Another possible cause for the velocity change is the transition from the saturation to the under-saturation of rocks with pore-fluid. Figure 8 shows the ratio of the veloc-ity of dehydrated rocks to that of the saturated rocks for the fracture density held constant. Dehydration without a change in fracture density increases effective confining pressure to reduce crack density. The crack density reduc-tion may amount to at most about 25 %, as seen from (3). If this crack density reduction recover the under-saturation, seismic wave velocities may not decrease but rather in-crease and then decrease with an increase in crack density due to return back of water. Therefore, even if the velocities of P- and S-waves may decrease as inferred from the figure due to complete escape of water from the region, it is a dif-ficult problem to decide whether or not the cracks are still under-saturation. Since there is water at depths in general, under-saturation may be unlikely to occur. For this reason, the under-saturation of cracks is not the definitive cause of the velocity decrease at depths.

As seen from the above discussion, we can believe that only the potential stress change in damage zones is a rea-sonable cause for the observed travel time increase. The potential stress change in damage zones reduces the P-wave velocity by more than about 1.3 % and the S-wave velocity by more than about 3.5 %, as stated above. This suggests that the observed velocity reductions of 0.3 to 1.5 % are caused in damage zones. One plausible explanation is that the scattered trapped waves contribute to the phases in coda waves, although there might remain the problem if the scattered trapped waves are actually observable or not. 4.2 Change in wave velocities of anisotropic damage zones

Li et al. (1998, 1999, 2001) have observed and ana-lyzed the trapped waves for the Landers, California, fault zone. They have found that the travel time decreases with a lapse of time after the 1992 Landers earthquake, both for the trapped P-waves and for S-waves, and have shown that the ratio of the travel time reduction of P-waves to that of S-waves cannot simply be explained by the theory of elastic constants for isotropic composites. The damage zone model predicts that there is large anisotropy of P- and S-wave ve-locities in damage zones. Their results may imply that ani-sotropy has to be taken into consideration. Here, the seismic wave velocities of damage zones with elastic anisotropy are calculated using WIA. 4.2.1 Accuracy of the approximation method

Effective elastic constants calculated by WIA are ap-proximations. There is no well-established standard for the effective elastic constants of anisotropic composites in which inclusion geometry is arbitrary. Thus, the accuracy of the approximation has to be inferred by comparing the effective elastic constants in the case of isotropic compo-sites where inclusions are randomly oriented in isotropic matrix. For such isotropic composites, an established method, NSC or DS, is available, even if there is an implic-itly imposed condition on the inclusion geometry in the method (Yamamoto et al., 1981).

The accuracy of WIA is known to depend not only on a volume fraction and aspect ratio of inclusions but also on the properties of matrix and inclusions such as the Pois-

Fig. 9 Schematic illustration of scattered trapped waves. The arrows from a damaged zone indicate the seismic waves scattered from obstacles in a damaged zone.

DRAFT 9

son’s ratio and the incompressibility (Yamamoto, 1980, Yamamoto et al., 2001). In order to comprehend the accu-racy of WIA, the effective elastic constants of isotropic damage zones by WIA are compared with those by NSC, by assigning the same values as those adopted in the previ-ous paper to elastic constants and aspect ratio of cracks. These parameter values have been known to explain well the averaged elastic property of damage zones.

The relationships between seismic wave velocity and depth for a damage zone are calculated by means of WIA and NSC. Figure 10 shows these relationships. The veloci-

ties for G = 0.7 are calculated by WIA, while those for G = 0.7 and 0.8 are calculated by WIA and NSC. The aspect ra-tio of cracks is taken to be 10-3 both for NSC and for WIA. It is seen in this figure that the velocities for G = 0.8 by NSC lie between the velocities for G = 0.7 and G = 0.8 by WIA. If we want to estimate fracture density G from seis-mic wave velocity, WIA may determine G smaller than NSC by 10 % at most either from P- or S-waves. The dif-ference in velocity between WIA and NSC is about 5 % of the host rock velocity for S waves and a few percent for P-waves at the same fracture density. These differences may be insignificant when taking into consideration the differences between the model and the actual damage zone, for example, in geometry and spatial distribution of cracks, heterogeneity of materials, and so on. 4.2.2 Anisotropy of seismic wave velocities

In a damage zone, tensile cracks orient their surfaces in parallel with and symmetrically around the fault normal, as described in section 3. Seismic wave velocities of the zone are calculated for the propagation directions. Figure 11 il-lustrates the dependence of the seismic wave velocities on their propagation direction, and a) and b) are for the zone including water-saturated cracks and for the zone including void cracks, respectively. The x1-direction is taken parallel with the fault normal and thus the plane (x2, x3) is parallel with the fault plane. The planes, (x1, x2) and (x1, x3), are the cross-sectional planes of a fault, which are orthogonal to the fault plane. Here, P’-waves are the waves of which the propagation direction and the motion direction lie in the same cross-sectional planes. These waves correspond to P-waves in isotropic materials. S1-waves are the waves of which the motion direction and the propagation direction are also in the planes. These waves correspond to S-waves in isotropic materials. The motion directions and the propagation directions of P’-waves and S1-waves are thus in the same plane that is orthogonal to the fault plane. S2-waves correspond to S-waves. Their motion direction is parallel to the fault plane.

Vij denotes the velocity of the waves that propagate

Fig. 10 Comparison of the velocity calculated by WIA with that

by NSC in the case of isotropic orientation of cracks.

Fig. 11 Illustration for the dependence of P-, S1- and S2-wave velocities of damage zones on the propagation direction. The fault normal is in the x1-direction. The plane (x1, x2) indicates the horizontal crosssectional plane of a vertical fault. P-, S1- and S2-waves are the waves that propagate in the crosssectional plane. The motion directions of P- and S1-waves lie in the same horizontal plane as their propaga-tion directions, while the motion direction of S2-waves is vertical to the horizontal plane.

DRAFT 10

along xj or i with the motion direction in xi or j. The velocity V11 and V33 are the velocities of P’-waves. They are com-pared with VP in Fig. 12. The velocities V13 and V23 are the velocities of S1- and S2-waves, respectively. They are com-pared with the velocity VS in the figure. Vp and Vs are the velocities of P- and S-waves calculated by NSC for the same value of G on the assumption of isotropic damage zones. It is seen that V13 and V23 are very small compared with VS, while V33 is close to VP.

In order to qualitatively understand the relationship between anisotropy and crack orientation, we consider a two-phases composite consisting of matrix and a kind of inclusions and define the intersection angle as the angle between the propagation direction or the motion direction of waves and the crack surface. This angle can be regarded as an index of the amount of volumetric strain of cracks generated by incident waves. A larger index means the lar-ger volumetric strain when cracks are void. When waves are traveling in a direction of small intersection angles on

an average, the effect of cracks on their velocities is small. When the distributions of intersection angles are identical, the waves have the same velocity even for the waves of which propagation directions are different. For the S1-waves propagating in x1 direction, the intersection angle distribution is the same as that for the S2-waves propagating in the same direction. It is confirmed in Fig. 11 that the ve-locities of these waves are identical. The intersection angles for a half number of cracks are zero for the S1-waves propagating in x2 or x3 direction, while they are nonzero for the S2-waves propagating in the same direction. Therefore, the velocity of S1-waves is expected to be larger than that of S2-waves, when cracks are void. This is confirmed in Fig. 11b).

When aspect ratio of cracks is much larger than the ra-tio of incompressibility of pore materials to that of matrix, the cracks approximate void cracks. When the aspect ratio is much smaller than the ratio, the cracks look to be incom-pressible, that is, the cracks approximate pure shear-cracks. For S2-waves, the intersection angles are larger on an aver-age for the waves propagating in the x2 or x3 direction than those propagating in the x1-direction. Therefore, for void cracks, the velocity of S2-waves in the x2 or x3 direction re-duces by a larger amount than that of the waves in the x1 direction. For pure shear-cracks, the volumetric change of cracks is negligible. Therefore, the amount of the velocity reduction of S2-waves in the x2 or x3 direction lessens. This may be the reason why the velocity of S2-waves propagat-ing in the x2 or x3 direction is close to that of the waves in the x1 direction and that of the S1-waves in the x2 or x3 di-rection in Fig. 11a).

Figure 13 shows the dependence of the velocities on propagation direction calculated for P’- and S1-waves for some values of G. At a direction of about 50° from the fault normal, S1-waves propagate fastest while P’-waves do slowest. At this direction, the incidence of S1-waves pro-duces small volumetric strain of cracks and the incidence of P’-waves do shear strain of cracks relatively larger than volumetric strain. As seen in Fig. 13, the velocity of S1-waves to the direction of about 50° from the fault normal is approximately two times as fast as the waves to the di-rection parallel to the fault plane, when G = 0.8. This ani-sotropy suggests that the first arrival of S1-waves is not given by the direct S1-waves but the S1-waves reflecting at walls of a damage zone while traveling, when the source and the receivers for the waves are in the same damage zone.

Fig. 12 Seismic wave velocities in damaged zones calculated by

WIA. Vij means the velocity of the waves of which the mo-tion direction and the propagation direction are xi and xj. The velocities of the isotropic damaged zone by NSC are shown together for comparison.

DRAFT 11

4.2.3 Definition of ‘Effective Velocity’ A fault-damage zone is expected to be at most 1 km

wide and strongly anisotropic in elasticity. The P’- and S1-wave velocities in any propagation direction, respec-tively, are smaller than the P- and S-wave velocities of the host rock. The host rock means the rock outside the zone. When seismic waves are radiated from a source in a dam-age zone, a part of the wave energy may be trapped within the zone as fault-zone trapped waves. The wavelength of the trapped waves is about twice as long as the damage zone width in the case of the Landers fault as seen in the papers by Li et al. (1998, 2001). Even if the wave-theoretical approach or the Cagniard method (e.g. Fung Y. C., 1965) is preferable for discussing the trapped waves, the theoretical study has not been performed for such strongly anisotropic layers as damage zones. When the purposes is not to dis-cuss the frequency-amplitude relationships in relation to the damage zone width but to explain the elastic property of damage zones only, the ray-theoretical approach is expect-ed to be valid to a certain extent.

The seismic P- and S-waves radiated from a point in linear-elastic media may generally be expressed as an inte-gral of the Fourier component waves. If damage zones are assumed to be isotropic, the major phases expected to be observed in damage zones may be of direct waves, dif-fracted waves or head waves and reflected waves, as shown in Fig. 14. Direct waves travel in damage zones with the velocities of the zones. Diffracted waves propagate along the boundaries of a fault zone with the velocities near or equal to those of the host rocks. Their approximate ampli-tudes can be evaluated by integrating the component waves around the critical incident angle. The waves have small amplitude at the onset.

Here, we infer the behavior of the waves in anisotropic media from that of the waves in isotropic media as de-scribed above. Trapped waves are formed by interference of the waves of which incident angles are around and larger than the critical angle. They arrive with small amplitude at the time when the diffracted waves arrive and then grow up. The near-field term can probably be neglected, because the epicentral distances are large compared with the wave-length. As far as we inspect the seismograms recorded for the Landers fault zone by Li et al. (1998, 2001), the waves traveling with the velocities comparable to the host rock velocities cannot clearly be identified. Therefore, we may neglect the diffracted waves for the following discussions.

Thus, the major phases observed in damage zones are the direct waves, the reflected waves and the trapped waves.

In roughly speaking, the condition for the waves to be trapped in a fault zone is that the apparent velocity in the directions parallel to the fault surface is smaller than that of S-waves of the host rocks. The critical incident angle to the walls that gives the apparent velocity may be calculated by

(9) where Vd(#c) and VHOST are the velocity of a damage zone at the critical incident angle, #c, and that of the host rocks, re-spectively. The dot-lines in Fig. 13 indicate the relationship of (9), where # stands for #c. The line for P-waves repre-sents the critical angle for the incidence of P’-waves.

The critical angle is about 50° for P’-waves as seen in Fig. 13, when G is about 0.8. For P’-waves, the apparent velocity parallel to the fault surface is larger than the S-wave velocity of host rocks for any propagation direction. This implies that all the P’-waves except for the direct waves may decay while traveling in a damage zone. Thus, the clear arrival of waves may be perceived by the direct P’-waves. For S1-waves, the critical incident angle is about 56° for about 0.8 of G. The waves with the incident angle larger than the critical angle may be supposed to totally re-flect at the walls as inferred from the waves in isotropic media. It is noticed here that the velocity of S1 waves to the direction of # nearly equal to 50° is twice as much as the velocity of the waves to the direction of # equal to 90°. This suggests that the direct S1-waves do not travel with the fastest speed along a fault surface but the reflected waves. It is seen by a rough calculation that the reflecting waves of their incident angles near 60° have the largest velocity

Fig. 13 Dependence of P- and S1-wave velocities on propa-gation direction in damaged zones. The definitions of P- and S1-waves are the same as those in Fig. 10. The propagation direction # is measured clockwise from the fault normal.

Fig. 14 Illustration of the trapped waves reflecting while propagating along a fault plane with the largest velocity.

DRAFT 12

along the fault. This angle is slightly larger than the critical angle. Therefore, the amplitude of the waves may be com-parable to the direct S1-waves. This suggests that the first clear onset of S-waves is given by S1-waves in a damage zone.

It is assumed that the clear onset of S-waves in a dam-age zone is given by the reflected S1-waves described as above. The path for the waves is shown in Fig. 14. When the waves progress a distance l along the fault surface, the travel time may be written by

. (10) is the velocity of the waves propagating to the direc-

tion # from the fault normal. The ‘Effective Velocity’ is defined to be

. (11) A change in fracture density G may shift the travel time by

for P-waves and by for S-waves. Their ratio, or the travel time shift ratio, may be written by

, (12) where

, (13)

provided that .

Figure 15 shows the ‘Effective Velocities’ and the travel time shift ratios calculated for dG equal to 0.6, 0.7 and 0.8. The ‘Effective Velocity’ of P’-waves is actually identical to the velocity of the waves propagating directly to the direc-tion # equal to 90°. 4.2.4 Comparison with observational data

Yamamoto et al. (2002) have calculated the P-wave

velocity of a damage zone for G = 0.8, on the assumption that the damage zone is isotropic. They have shown that the calculated velocity explains well the velocity of the San Andreas Fault zone by Feng and McEvilly (1983) up to the depth of about 15 km. The velocity has been determined mainly by analyzing the reflected waves from depths. The ray paths of the waves are considered to be nearly vertical. Thus, we may understand that the velocity corresponds to the velocity V33, which is close to Vp, as shown in Fig. 12. Therefore, it is unnecessary to modify the conclusion of our previous paper (Yamamoto et al., 2002), even if the anisot-ropy of the damage zone is taken into consideration.

Li et al. (2000, 2001, 2002b) analyzed trapped waves at several sites by assuming isotropic fault zones. The S-wave velocities they have determined are summarized in Table 1. Further, Li et al. (1998, 2001) performed explosion experiments to study the strength recovery of a fault after the 1992 M7.5 Landers, California, Earthquake. The recov-ery was estimated from the travel time shifts and the travel-time shift ratios of the waves in the fault zone with time. They obtained the ratios from the trapped waves pro-duced by near-surface explosions, too. Since the travel times were measured for at most several kilometer of epi-central distance, the results may reflect the changes in velocities at shallow depths of at most a few kilometers. The travel time shift ratio has been obtained for the Hector Mine Fault zone, too (Li et al, 2002a). The ratios are ap-proximately in the range from about 0.6 to about 0.8 both for the S- and the trapped waves. The shadowed region in Fig. 15 indicates the range of the travel time shift ratio ob-tained by their experiments.

Figure 16 shows the velocities of P- and S-waves and the travel time shift ratio for isotropic damage zones of G = 0.8. Here, the velocities are calculated for the cases of satu-rated cracks and of void cracks. In the case of saturated

Fig. 15 ‘Group velocities’ and travel time shift ratio for the trapped waves in water-saturated damaged zones. Vg

p and Vgs

are the ‘group velocities’ of P’- and S1-waves of trapped waves along a fault plane. dtp/dts denotes the travel time shift ratio. dtp and dts are the travel time shifts of the P’-waves and that of the S1-waves calculated from the ‘group velocities’. The travel time shift ratios have been determined in the shaded range by observations (Li et al., 2001).

Fig. 16 Seismic wave velocities and travel time shift ratio for the waves in water-saturated damaged zone (wt) and those for the waves in dehydrated damaged zones (vd), when ori-entation of cracks is random or isotropic. A shaded zone in-dicates the range of travel time shift ratios determined by observations.

DRAFT 13

cracks, the S-wave velocity is about 40 % of the host rock velocity and the travel time shift ratio is about 0.13 at depth of 5 km. Although the calculated velocity is close to the observed one, the time shift ratio is seen to be quite differ-ent. In the case of void cracks, S-wave velocity decreases to a value smaller than 20 % of the host rock velocity. This velocity is too small compared with the observed one, while the ratio is near the observed one. Thus we can confirm that there is no solution that simultaneously satisfies both the observed velocity and the observed time shift ratio, when anisotropy is neglected.

In our previous paper, the calculated velocity of S-waves appears to be somewhat smaller than the observed one. Here, the observed S-wave velocity is discussed again taking account of the anisotropy of damage zones. The ‘Ef-fective Velocity’ and the travel time shift ratio calculated for G = 0.8 are compared with the S-wave velocity and the ratio observed by Li et al. (1998, 2001, 2002a, 2002b) in Fig. 17. It is seen in this figure that the S-wave velocity ob-tained from the trapped waves is close to the ‘Effective Velocity’ at all depths except for the velocities at small depths of the Landers Fault zone. Although the calculated ratios appear to not exactly lie in the observed range of the ratio, they are close to the calculated ones compared with those calculated for isotropic damage zones. It is clear that the conformity of the theoretical results with the observed ones has been greatly improved by taking anisotropy into account.

The incident angle of the reflected waves traveling with the ‘effective velocity’ is close to the critical incident

angle as stated above. The wave number kr along the radial direction r, that is taken parallel to the fault plane, may be written by

. (14) and

. (15)

In order to evaluate the amplitude of the waves at a point in a damage zone, it may be necessary to integrate the com-ponent waves with respect to kr. The trapped waves may be mainly composed of the waves of their incident angles of #c < # < 90°. If the seismic energy is uniformly radiated from a point in a damage zone, the integral with respect to kr may be replaced with the integral with respect to # through (15). Referring to the critical angle in Fig. 13, we may infer that the amplitude becomes large near the critical angle that is around 60°. The observed velocity of trapped S-waves is seen to be almost equivalent to the ‘Effective Velocity’ of S1-waves from about 2 km depth to a depth greater than 15 km. This may be explained by considering that the trapped waves consist of the waves whose incident angles are close to that for the ‘effective velocity’. The travel-time shift ra-tio for the trapped waves is also close to that for the S-waves. This seems to be consistent with the above infer-ence. Thus, the observed elastic properties of damage zones are better predicted when the anisotropy of damage zones is taken into consideration. This is one piece of evidence of the validity of the present damage zone model.

Table 1 S-wave velocity in damaged zones. The velocity is represented as the ratio to the velocity of host rocks.

Fault/Section Layers Authors Depth, km 1.0 1.5 4.0 7.0 10.0 Landers Vs/Vs

h 0.56 0.56 0.62 0.66 0.69 Li et al. (2000)

Depth, km 1.5 5.0 10.0 17.0 20.0

Vs/Vsh 0.62 0.63 0.67 0.71 0.75

Jacinto /BRF, CVF

/CCF Vs/Vsh 0.65 0.67 0.70 0.71 0.75

Li et al. (2002b)

Depth, km - Nojima Vs/Vs

h 0.40 Nishigami (2000)

Depth, km Nojima Vs/Vs

h 5.0 < 0.66

Kuwahara & Ito (1999)

Depth, km Mozumi-

Sukenobu Vs/Vsh

0.5 < 0.30 Nishigami (2000)

DRAFT 14

For comparing the results from the present model with the results of observation, it has been assumed that the fracture density G is equal to 0.8 for damage zones. This value has been obtained from the stresses measured in the vicinity of the damage zone of the Nojima fault. Neverthe-less, the value seems to explain the observed elastic proper-ties of damage zones for the San Andreas Fault and for faults around it. This may imply that the largest fracture density of a damage zone is almost equal for every fault. 5. Summary

For the damage zone of the Nojima fault, the stresses have been measured by DRA and hydraulic fracturing method (HF) just after faulting. The measured stresses are in the range from 0.15 to 0.21 in the potential stress that is defined as the maximum shear stress divided by the normal stress on the shear plane, and the potential stress by HF are slightly smaller than those by DRA. Although this differ-ence may not be significant, it is inferred from this that the upper limit on the stress drop due to faulting amounts to about 0.06 in potential stress, provided that the stress drop has occurred in the zone.

A damage zone model has been already proposed to explain the strength of faults. For the model, fracture den-sity, crack density and the elastic constants can be calcu-lated from the stresses in damage zones using the fracture process model and the theories of effective elastic constants. Further, the coseismic change in seismic wave velocity can be estimated from the change in stress in the damage zone. The fracture density in the damage zone has been estimated to be about 0.8 for the Nojima fault. Making use of the in-ferred stress drop together with this fracture density, the coseismic decrease in seismic wave velocity in damage zones is estimated at 1.4 % for P-waves and at 5.5 % for

S-wave. It has been reported that the arrival times of some

phases in seismic coda waves are delayed after an earth-quake. It has been shown that this delay is caused in the fo-cal region. If it is assumed that these phases are of the wav-es traveling partly in the damage zones, the amount of the delay corresponds to a decrease of 0.3 to 1.5 % in the ve-locities of the zones. This velocity decrease is consistent with the estimated velocity decrease in damage zones due to faulting. This is one piece of evidence to verify the valid-ity of the present damage zone model.

The gradual shifts in the travel times with time after an earthquake have been observed for trapped P- and S-waves in the Landers Fault zone. It has been shown that the ratio of the travel time shift for P-waves to that for S-wave is not simply explained by the theory of effective elastic constants for isotropic composites. It is expected from the damage zone model that three modes of seismic waves, P’-waves, S1-waves, and S2-waves, propagate in damage zones. S1-waves are the S-waves that couple with the P-waves. S1-waves propagate with the largest velocity to the direc-tion of about 50° from the fault normal. This study theo-retically shows that the first arrival of trapped S-waves is not given by the direct S1-waves but by the S1-waves re-flecting at walls of a damage zone. The velocity of the re-flecting S-waves traveling along the fault axis is called ‘ef-fective velocity’ of S-waves. The present paper shows that the S-wave velocity distribution for depth in damage zones and the observed ratio of travel time shift can be properly explained by taking account of the anisotropy expected from the damage zone model. This is another piece of evi-dence to justify the validity of the present damage zone model.

Fig. 17 Comparison of ‘group velocity’ and travel time shift ratio with the observational data. The ‘group veloc-ity’ and the travel time shift ratio are calculated for G = 0.8. The velocity data are for the Landers fault and the San Jacinto fault by Li et al. (2001, 2002b). The data of travel time shift ratio is for the Landers fault by Li et al. (2001, 2002a). The velocity data are presented in Table 1.

DRAFT 15

Acknowledgements: The stress measurements and the acquisition of stress data are owed a lot to Prof. M. Ando, Nagoya University, Dr. H. Ito and Dr. Y. Kuwahara, Geological Survey of Japan, Prof. R. Ikeda, Hokkaido Univ., Prof. H. Tsukahara, Shinsyu Univ.. Further, the discussions with them and the suggestions from them are very useful to progress this study. Discussions on the interpre-tation of the field data with Drs. Nishimura, Matsumoto, and Na-kamura were very useful to provide this manuscript. We would like to express very much thanks to all of them. I have to express special thanks to Dr. S. Maxwell and an anonymous reviewer for their comments based on their careful reading of this manuscript. Their comments are very much helpful for revising this manu-script. References Brune, J. N., T. L. Henyey, and R. F. Roy, Heat flow, stress and

rate of slip along the San Andreas fault, California, J. Geo-phys. Res., 74, 3821-3827, 1969.

Byerlee, J., Friction of rocks. PAGEOPH, 116, 615-626, 1978. Chester F. M., J. P. Evans, and R. L. Biegel, Internal structure and

weakening mechanisms of the San Andreas fault, J. Geo-phys. Res., 98, 771-786, 1993.

Dodge D. A., and G. C. Beroza, Source array analysis of coda waves near the 1989 Loma Prieta, California, maishock: Implications for the mechanism of coseismic velocity changes, Journ. Geophys. Res., 102, 24,437-24,458, 1997.

Eshelby, J. D., The determination of the elastic field of an ellip-soidal inclusions, and related problems, Proc. Roy. Soc., Ser. A, 241, 376-396, 1957.

Fung, Y. C., ‘Foundation of Solid Mechanics’, Prentice Hall, Englewood Cliffs, New Jersey, U.S.A., pp. 524, 1965.

Feng R., and T. V. McEvilly, Interpretation of seismic reflection profiling data for the structure of the San Andreas fault zone, Bull. Seism. Soc. Am., 73, 1701-1720, 1983.

Iio, Y., Frictional coefficient on faults in a seismogenic region inferred from earthquake mechanism solutions. Journ. Geo-phys. Res., 102, 5,403-5,412, 1997.

Ikeda, R., Y. Iio, and K. Omura, In-situ stress measurements in NIED boreholes in and around the fault zone near the 1995 Hyogoken-Nanbu earthquake, Japan, The Island Arc, 10, 252-260, 2001.

Ito, H., Y. Kuwahara, T. Miyazaki, O. Nishizawa, T. Kiguchi, K. Fujimoto, T. Ohtani, H. Tanaka, T. Higuchi, S. Agar, A. Brie, H. Yamamoto, Structure and physical properties of the Nojima fault, by the active fault drilling, Butsuri-Tansa, 49, 522-535, 1996 (in Japanese).

Jones, L. M., Focal mechanisms and the state of stress on the San Andreas Fault in southern California, Journ. Geophys. Res., 93, 8,869-8,891, 1988.

Kuwahara Y., K. Yamamoto, and T. Hirasawa, An experimental and theoretical study of inelastic deformation of brittle rocks under cyclic uniaxial loading, Tohoku Geophys. Journ. (Sci. Rep. Tohoku Univ., Ser. 5), 33, 1-21, 1990.

Kuwahara, Y., and H. Ito, Deep structure of the Nojima fault by trapped wave analysis, Proc. Int. W/S on the Nojima fault core and borehole data analysis, Nov. 22-23, Tsukuba, Ja-pan, (GSJ Interim Rep. No. EQ/00/1: USGS Open-file Rep. 00-129), edited by H. Ito, K. Fujimoto, H. Tanaka, and D. Lockner, pp. 283-289, 1999.

Li, Yong-Gang, K Aki, J. E. Vidale, and M. G. Alvarez, A de-lineation of the Nojima fault ruptured in the m 7.2 Kobe, Japan, earthquake of 1995 using fault zone trapped waves, Jorn.Geophys.Res.,103, 7247-7263.

Li, Yong-Gang, K Aki, J. E. Vidale, and F. Xu, Shallow structure of the Landers fault zone from explosion-generated trapped waves, Journ. Geophys. Res., 104, 20,257-20,275, 1999.

Li, Yong-Gang, K. Aki, J. E. Vidale, S. M. Day, and D. D Oglesby, Characterization of spatial and Temporal Varia-tions of Landers and Hector Mine rupture zones by

fault-zone trapped waves, Proc. Int. W/S on Physics of ac-tive faults (Technical. Note of NIED, No. 234), Feb. 26-27, 2002, Tsukuba, Japan, edited by E. Fukuyama and R. Ikeda, 130-137, 2002a.

Li, Yong-Gang, J. E. Vidale, Healing of the shallow fault zone from 1994-1998 after the 1992 m7.5 Landers, California, earthquake, Geophys. Res. Let.,. 28, 2999-3002, 2001.

Li, Yong-Gang, J. E. Vidale, K. Aki, and Fei Xu, Depth-dependent structure of the Landers fault zone from trapped waves generated by aftershocks, Journ. Geophys. Res., 105, 6237-6254, 2000.

Li, Yong-Gang, J. E. Vidale, K. Aki, and Fei Xu, and T. Burdette, Evidence of shallow fault zone strengthening after the 1992 M7.4 Landers, California, Earthquake, Science, 279, 217-219, 1998.

Li, Yong-Gang, J. E. Vidale, S. M. Day, D. D Oglesby, and the SCEC Field Workin Team, Study of the 1999 M 7.1 Hector Mine, California, Earthquake fault plane by trapped waves, Bull. Seism. Soc. Am, 92, 1318-1332, 2002b.

Matsumoto S., K. Obara, K. Yoshimaoto, T. Saito, A. Ito, and A. Hasegawa, Temporal change in P-wave scatterer distribu-tion associated with the M 6.1 earthquake near Iwate vol-cano, northeastern Japan, Geophys. J. Int., 145, 48-58, 2001.

Matsushima, S., Variation of the elastic wave velocities of rocks in the process of deformation and fracture under high pres-sure, Disas. Prevention Res. Inst. Kyoto Univ., Bull., 32, 2-8, 1960.

Nakamura, A., A. Hasegawa, N. Hirata, T. Iwasaki, and H. Ha-maguchi, Temporal variations of seismic wave velocity of associated with 1998 M 6.1 Shizukuishi earthquake, PAGEOPH, 159, 1183-1204, 2002.

Nishigami, K., Investigation of deep structure of active faults us-ing scattered waves and trapped waves, in Seismogenic Process Monitoring, edited by H. Ogasawara, T. Yanagida-ni and M. Ando, Balkema, Rotterdam, pp. 245-256, 2000.

Nishimura T., N. Uchida, H. Sato, M. Ohtake, S. Tanaka, and H. Hamaguchi, Temporal changes of the crustal structure as-sociated with the M 6.1 earthquake on September 3, 1998, and the volcanic activity of Mount Iwate, Japan, Geophys. Res. Let., 27, 269-272, 2000.

Norris, A. N., 1985, A differential scheme for the effective moduli of composites, Mechanics of Materials, 4, 1-16.

Oppenheimer, D. H., P. Reasenberg, and R. W. Simpson, Fault plane solutions for the 1984 Morgan Hill, California, earthquake sequence: Evidence for the state of stress on the Calaveras Fault, Journ. Geophys. Res., 93, 9007-9027, 1988.

Rice, J. R, Fault stress states, pore pressure distributions, and the weakness of the San Andreas Fault, in Fault mechanics and transport properties of rocks: A festschrift in honor of W. F. Brace, edited by B. Evans and W. Teng-Fong, 475-503, academic press, New York, 1992.

Research Group for Explosion Seismology, Seismic refrac-tion/wide-angle reflection experiments in the backbone range of northern Honshu, Japan, Abstr. Japan Earth Planet. Sci. Joint Mtng., sk013, 1999 (in Japanese).

Sato, N., Y. Yabe, K. Yamamoto, and H. Ito, Estimation of stresses in the vicinity of the Nojima fault, Awaji Is., Hyogo Pref. Japan, from core samples, submitted to Zisin 2, 56, 157-169, 2003.

Sibson, R. H., F. Robert and K. H. Poulsen, 1988, High-angle re-verse faults, fluid-pressure cycling, and mesothermal gold-quartz deposits, Geology, 16, 551-555.

Tsukahara, H., R. Ikeda, and K. Yamamoto, 2001. In situ stress measurements around 1500 m depths in a borehole close to the earthquake fault of the 1995 Kobe earthquake, The Is-land Arc, 10, 261-265, 2001.

Yamamoto, K., Theoretical determination of effective elastic con-

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stants of composite and its application to seismology, Doc-toral thesis, Tohoku Univ., pp. 199, 1980.

Yamamoto, K., Strength distribution of microfracture elements in granites under compression test, Proc. 3rd SEGJ/SEG Symp., 327-334, 1995a.

Yamamoto, K., The rock property of in-situ stress memory: Dis-cussions on its mechanism. In Matsuki & Sugawara eds. Proceeding of International Work Shop on Rock Stress Measurement at Great Depth, Tokyo, 1995, pp. 46-51, 1995b.

Yamamoto, K, Estimation of fracture stress for intact rocks and possibility of long-term earthquake prediction, Zisin 2, 50, Sup. 169-180, 1998 (in Japanese with English abstract).

Yamamoto, K., M. Kosuga, and T. Hirasawa, A theoretical method for determination of effective elastic constants of isotropic composites, Sci. Rep. Tohoku Univ., Ser. 5, Geo-physiscs, 28, 47-67, 1981.

Yamamoto, K., N. Sato and Y. Yabe, Strength of fault as inferred

from the stresses measured in the vicinity of the Nojima fault (Extended abstract), Tohoku Geophys. Journ. (Sci. Rep. Tohoku Univ., Ser. 5), 36, 272-290, 2001.

Yamamoto K., Y. Kuwahara, N. Kato and T. Hirasawa, Deforma-tion rate analysis: A new method for in situ stress estima-tion from inelastic deformation of rock samples under uni-axial compressions. Tohoku Geophys. Journ. (Sci. Rep. Tohoku Univ., Ser 5), 33: 127-147, 1990.

Yamamoto K., N. Sato and Y. Yabe, Elastic property of damaged zone inferred from in-situ stresses and its role on the shear strength of fault, Earth Planets Space, 54, 1181-1194, 2002.

Zoback M.. D. et al., New evidence on the state of stress of the San Andreas fault system, Science, 238, 1,105-1,111, 1987.

Zoback, M. D., and Townend, J., Implication of hydrostatic pore pressures and high crustal strength for the deformation of intraplate lithosphere, Tectonophysics, 336, 19-30, 2001.