A way of estimating the characteristic slip displacement

Abstract

During the ruptures of an earthquake, the strain energy, ΔE, will be transferred into, at least, three parts, i.e., the seismic radiation energy (E _s), fracture energy (E _g), and frictional energy (E _f), that is, $\Delta E = E_{\text{s}} + E_{\text{g}} + E_{\text{f}}$. Friction, which is represented by a velocity- and state-dependent friction law by some researchers, controls the three parts. One of the main parameters of the law is the characteristic slip displacement, D _c. It is significant and necessary to evaluate the reliable value of D _c from observed and inverted seismic data. Since D _c controls the radiation efficiency, $\eta_{\text{R}} = E_{\text{s}} /(E_{\text{s}} + E_{\text{g}} )$, the value of η _R is a good constraint of estimating D _c. Integrating observed data and inverted results of source parameters from recorded seismograms, the values of E _s and E _g of an earthquake can be measured, thus leading to the value of η _R. The constraint used to estimate the reliable value of D _c will be described in this work. An example of estimates of D _c based on the observed and inverted values of source parameters of the September 20, 1999 M _S 7.6 Chi-Chi (Ji-Ji), Taiwan region, earthquake will be presented.

1 Introduction

Friction controls the earthquake rupture processes. Experimental and theoretical studies showed two effects affecting the friction strength: the direct effect and the evolution one (Dieterich 1978, 1979; Rice 1983; Ruina 1983; Beeler et al. 1994; Marone 1998). The direct effect shows an instantaneous change of friction strength with a change in velocity, while the evolution effect evolves with slip following a change in velocity. Unstable slip in rock can result only when the evolution effect is larger than the direct one, which leads to the so-called velocity-weakening process. Bizzarri (2011) made a comprehensive description and review about the constitution laws and their intrinsic properties of friction, including time-weakening law, position-weakening law, slip-dependent law, rate-dependent law, and rate- and state-dependent law. This study, focus on the estimate of the characteristic slip displacement, D _c, of rate- and state-dependent law. D _c is denoted by L and named the slip-weakening distance by Bizzarri (2011), Ulutaş (2008, 2013, 2015), and Ulutaş et al. (2006, 2011). One of the weak points is that, as shown below, the law cannot be defined when the velocity is zero.

The one-state-variable slip and slowness (or aging) laws proposed by Dieterich (1978, 1979) and Ruina (1983) have been long used to represent the state-dependent evolution effect. Some authors compared the two laws based on the quasi-static model, which was often regarded as an acceptable approximation of the dynamic model (e.g., Rice and Ruina 1983; Gu et al. 1984; Rice and Tse 1986), especially for low velocities. From numerical simulations of earthquake nucleation on faults, Dieterich (1992) stressed that in most cases, the results using the two laws are similar. Rice and Ben-Zion (1996) stated that the slip law leads to periodically repeated events, but the slowness one allows apparently chaotic sequences of large events. Roy and Marone (1996) stated that pre-seismic slip is larger for the slowness law than for the slip law.

Ruina (1983) defined the rate- and state-dependent friction strength, μ, which employs state variables, $\theta$, in the following form: $\mu = \mu_{{0}} + a{ \ln }\left( {v/v_{{0}} } \right) + b{ \ln }\left( {\theta v_{{0}} /D_{\text{c}} } \right)$, where a and b are two constants, $v$ is the sliding velocity, $v_{{0}}$ is the reference velocity, and D _c is the characteristic slip distance. This expression leads to $\mu = \mu_{{0}}$ when $v = v_{0}$ and $\theta = D_{c} /v_{0}$. It is noted that the system is not at rest when $v = v_{0}$. The value of $v_{0}$ is usually considered to be a constant and not regarded as a significant parameter controlling the dynamic friction strength. Contributions to the total friction strength are scaled by a for the direct effect and b for the evolution effect. The three parameters a, b, and D _c for some rocks were determined experimentally (e.g., Dieterich 1979; Reinen and Weeks 1993), and the experimental values vary in large ranges.

Two one-state-variable friction laws are commonly used to describe the state-dependent evolution effect. One is the slip law given by ${\text{d}}\theta /{\text{d}}t = - \left( {\theta v /D_{\text{c}} } \right){ \ln }\left( {\theta v/D_{\text{c}} } \right)$. When $v = 0,\;{\text{d}}\theta /{\text{d}}t$ approximates to zero using the l’Hospital’s theorem. This implicates that no evolution occurs at zero sliding velocity. The other is slowness law given by ${\text{d}}\theta /{\text{d}}t = 1 - \theta v/D_{\text{c}}$. This equation shows that state is proportional to slowness, i.e., $\theta = D_{\text{c}} /v$ when ${\text{d}}\theta /{\text{d}}t \; = \;0$. A detailed description can be found elsewhere (Dieterich 1979; Ruina 1983; Marone 1998; Wang 2009a).

After an earthquake rupture, the frictional stress on a fault plane decreases from an initial level, $\sigma_{{0}}$, to a dynamical one, $\sigma_{\text{d}}$, which is equal to or smaller than the final one, $\sigma_{\text{f}}$ (cf. Kanamori and Heaton 2000; Kanamori and Brodsky 2004). The process can be seen in Fig. 1. The static stress drop and the dynamic stress drop are, respectively, $\Delta \sigma_{\text{s}} = \sigma_{{0}} - \sigma_{\text{f}} \;{\text{and}}\;\Delta \sigma_{\text{d}} = \sigma_{0} - \sigma_{\text{d}}$. The friction law is complicated and depends upon either the slip or slip rate (cf. Marone 1998; Wang 2002). The friction law is displayed by a dashed line as shown in Fig. 1 and can be approximated by a piece-wise linear function. The strain energy, ΔE, can be transferred into, at least, three parts, i.e., the seismic radiation energy (E _s), the fracture energy (E _g), and the frictional energy (E _f), that is, $\Delta E = E_{\text{s}} + E_{\text{g}} + E_{\text{f}}$. According to the rate- and slip-weakening friction law, small E _g is associated with a short characteristic slip displacement, D _c. Venkataraman and Kanamori (2004) defined the radiation efficiency, $\eta_{\text{R}}$, to be the ratio of E _s to $\left( E_{\text{s}} + E_{\text{g}} \right)$, i.e., $\eta_{\text{R}} = E_{\text{s}} /\left( {E_{\text{s}} + E_{\text{g}} } \right)$. D _c related to $\eta_{\text{R}}$, which is a function of the ratio of rupture velocity over the S-wave velocity.

Fig. 1

The stress-slip function. The symbols are explained in the text (after Wang 2006a)

The D _c is the characteristic slip displacement, within which the frictional stress changes from a static one to a steady dynamic one (cf. Marone 1998; Wang 2002, 2009a). Laboratory results show D _c to be 10⁻⁵–10⁻³ m (Marone 1998). On the other hand, for real earthquakes, Mikumo et al. (2003) obtained D _c of 0.1 m to few meters. Their approach is based on the estimate of characteristic slip displacement at each point on the fault as the slip ($D_{\text{c}}^{\prime}$) at the time of peak slip velocity, supposing that the traction reaches its minimum value in that time. Fukuyama et al. (2003) have shown that the estimates of $D_{\text{c}}^{\prime}$ can be affected by an error of roughly 50 %. From theoretical analyses, Tinti et al. (2004) mentioned that the difference observed in this study between the D _c values and the inferred $D_{\text{c}}^{\prime}$ can range between few percent up to 50 %.

For the 1999 Chi-Chi (Ji-Ji), Taiwan region, earthquake, the values of D _c inferred by Zhang et al. (2003) and Ma and Mikumo (Wang 2006a, b) can be up to 10–12 m. Although there are high uncertainties in the estimate of D _c from earthquake data (Fukuyama et al. 2003), the value of D _c from real earthquakes is much longer than that from laboratory experiments.

Several methods have been suggested to estimate D _c (Ide and Takeo 1997; Guatteri and Spudish 2000; Pulido and Irikura 2000; Ohnaka 2000; Zhang et al. 2003; Mikumo et al. 2003; Tinti et al. 2004). Those methods are essentially based on the slip history inverted from recorded seismograms. Tinti et al. (2004) clearly explained the reasons to cause the difficulty of estimating D _c. According to the correlation between D _c and η _R, Wang (2006a) proposed a method to evaluate the value of D _c from the measures of E _s and E _g. In this study, his method will be discussed in detail based on theoretical consideration and calculations from the seismic data of the 1999 M _w 7.6 Chi-Chi (Ji-Ji), Taiwan region, earthquake (see Fig. 2) which is almost a unique event specified with a complete dataset, including different types of data. In addition, some values of D _c obtained by previous studies will be reviewed.

Fig. 2

A figure to show the epicenter (in a solid star), the Chelungpu fault (in a line), and the sites of two shallow boreholes (in solid circles) (after Wang 2006a)

2 Basic principle

Usually, it is not easy to estimate ΔE and E _f. Wang (2004, 2006a) suggested a way to estimate the value of ΔE, from which $E_{\text{f}} = \sigma_{\text{d}} {\text{\it DA }}\left( {D = {\text{slip}}} \right)$ can be obtained from $E_{\text{f}} = \Delta E - \left( {E_{\text{s}} + E_{\text{g}} } \right)$. Of course, the values of ΔE and E _f are not necessary in this study. The value of E _s can be directly measured from seismograms (cf. Wang 2004, 2006a). E _g can be evaluated indirectly from seismograms. Kanamori and Heaton (2000) suggested the following formula to evaluate the fracture energy, E _g:

$$E_{\text{g}} = \left[\frac{( 1- v_{\text{R}} /\beta )}{( 1+ v_{\text{R}} /\beta )}\right]^{ 1/ 2} \Delta \sigma_{\text{d}} D_{ \hbox{max} } A/ 2 ,$$

(1)

where β, $v_{\text{R}}$, $\Delta \sigma_{\text{d}}$, and A are, respectively, the S-wave velocity of source area, rupture velocity, dynamic stress drop, and fault area. On a simple fault plane, D _max is just the final displacement. On a complicated fault plane, the final displacement varies from place to place, and thus, D _max is the average of final displacements over the plane. Of course, the rupture velocity also varies from place to place. Equation (1) is obtained based on a crack-like rupture model (Tinti et al. 2005), and E _g computed from Eq. (1) is an average global value because $\Delta \sigma_{\text{d}}$ and D _maxare both average values on the fault plane. All parameters in Eq. (1) can be evaluated from recorded seismograms and geological surveys. Define $G = E_{\text{g}} /A$ to be the specific fracture energy, i.e., the fracture energy per unit area. From the dislocation theory, the static-specific fracture energy is expressed by $G = \lambda K^{2} /Y$ (Scholz 1990), where λ is a geometry factor, K is the stress intensity factor, and Y is the Young modulus.

Based on a symmetrical circular crack model proposed by Sato and Hirasawa (1973), Ide (2002) stated that (1) E _s increases with $v_{R} /\beta$; (2) E _g decreases with increasing $v_{R} /\beta$; (3) $E_{\text{s}} + E_{\text{g}}$ increases with $v_{R} /\beta$; and (4) η _R increases with $v_{\text{R}} /\beta$, and η _R is larger than 0.6 when $v_{\text{R}} /\beta \; > \;0.7$. Using a different source model, Dong and Papageorgiou (2002) also stated that η _R is an increasing function of $v_{R} /\beta$ and larger than 0.6 when $v_{R} /\beta \; > \; 0.4$. Clearly, η _R approaches 1 when $v_{\text{R}}$ is close to β. Faster v_R results in higher E _s. E _g is generally much smaller than ΔE, because $v_{R} /\beta$ is usually in the range of 0.75–0.85 (Kanamori and Heaton 2000). From theoretical analyses based on friction caused by thermal pressurization, Wang (2009b) obtained a relationship between η _R and slip. The relationship suggests that η _R decreases with increasing slip and depends on model parameters of thermal pressurization.

Bizzarri (2010) theoretically explored the relationships between the fracture energy density, e.g., and the source parameters, such as the rupture velocity, $v_{\text{R}}$, the total fault slip, and the dynamic stress drop. He performed numerical simulations of three dimensional, spontaneous, fully dynamic ruptures developing on planar faults of finite width, obeying different governing laws and accounting for both homogeneous and heterogeneous friction. His results indicate that E _g depends on the adopted governing law and mainly on the rupture mode (pulse-like or crack-like, sub or supershear regime). For subshear, homogeneous ruptures, ${\text{\it e}}_{\text{g}} \; = \;[ {1 - \left( {v_{\text{R}} /v_{\text{S}} } \right)^{2} }]^{1/2},$ where $v_{\text{S}}$ is the shear-wave velocity, like the theoretical prediction. But for ruptures that accelerate up to supershear speeds, it is difficult to infer a clear dependence of E _g on $v_{\text{R}}$, especially in heterogeneous configurations. Hence, Eq. (1) holds only in the case of subshear events.

Venkataraman and Kanamori (2004) defined the radiation efficiency, η _R, to be the ratio of E _s to (E _s + E _g). For a Model-III crack, η _R is a function of $v_{R} /\beta$ in the following form (cf. Kanamori and Heaton 2000; Kanamori 2004; Kanamori and Brodsky 2004):

$$\eta_{\text{R}} = 1- \left[ \frac {( 1- v_{\text{R}} /\beta )}{( 1+ v_{\text{R}} /\beta )} \right]^{ 1/ 2}$$

(2)

The first derivative of Eq. (2) is ${{\text d}}\eta_{R} \text d\left( {v_{R} /\beta } \right) = \left( {1 - v_{\text R} /\beta } \right)^{ - 1/2} \left( {1 + v_{\text R} /\beta } \right)^{ - 3/2} /2$. Since $v_{\text R} /\beta$ is smaller than 1, ${\text{d}}\eta_{\text{R}} /{\text{d}}\left( {v_{\text{R}} /\beta } \right)$ must be positive. Obviously, there is a singularity at $v_{\text{R}} = \beta$. Figure 3 shows the variation of η _R with $v_{\text R} /\beta$ from Eq. (2) when $v_{\text R} /\beta$ ranges from 0.5 to 1.0. Clearly, η _R increases with $v_{\text R} /\beta$ and the increasing rate becomes larger when $v_{\text{R}} /\beta \; > \;0.90$.

Fig. 3

The curve shows the variation of the radiation efficiency, η _R, with ${\text{v}}_{\text{R}} /\beta$ calculated from Eq. (2) in the text when ${\text{v}}_{\text{R}} /\beta = 0.5{-}1.0$. Two crosses denote η _R at ${\text{v}}_{\text{R}} /\beta = 0.75{\text{ and }}0.80$ (Ma et al. 2001). ‘J’ displays η _R = 0.65 at ${\text{v}}_{\text{R}} /\beta = 0.66$ by Ji et al. (2003). ‘V’ displays η _R = 0.8 by Venkataraman and Kanamori (2004). ‘O’ shows both η _RS = 0.61 at v_R/β = 0.75 and η _RN = 0.81 at v_R/β = 0.80 from Wang (2006a, b). ‘S₁’ denotes η _RS = 0.69 at v_R/β = 0.75 for D _cS = 1 m. The symbols ‘N’ with numbers represent η _RN at ${\text{v}}_{\text{R}} /\beta = 0.80$ when D _cN = 1, 3.7, 6, and 10 m, which are, respectively, displayed with an integer near the symbols (after Wang 2006a).

The definition of $\eta_{\text{R}} = E_{\text{s}} /\left( {E_{\text{s}} + E_{\text{g}} } \right)$ leads to $E_{\text{g}} /E_{\text{s}} = \left( {1 - \eta_{\text{R}} } \right)/\eta_{\text{R}}$. The variation in friction with slip, which is displayed by a dashed line in Fig. 1, controls the values of both E _s and E _g. The ratio E _g/E _s is also controlled by such a variation. Although E _g is a function of D _max from Eq. (1), D _c must be a major factor in influencing E _g because the value of E _g between D _c and D _max under the line CD is very small. Hence, E _g/E _s relates to D _c. In other words, D _c, which is associated with E _g/E _s, also relates to $\left( {1 - \eta_{\text{R}} } \right)/\eta_{\text{R}}$. hence, the value of η _R can be a constraint on the estimate of D _c.

3 Basic data for estimating D _c: an example

On September 20, 1999, the M _S 7.6 Chi-Chi (Ji-Ji), Taiwan region, earthquake ruptured the Chelungpu fault, which is a ~100-km-long and east-dipping thrust fault in central Taiwan (cf. Ma et al. 1999; Shin and Teng 2001). The epicenter and fault trace are displayed in Fig. 2. The source parameters of the earthquake were estimated by several groups of researchers from near-field and/or teleseismic seismograms with or without GPS data. The observed data and inversed results of source parameters are not uniform and vary from place to place on the fault plane. Wang (2003, 2004, 2006a, b) summarized the observed and inferred results of related source parameters from different source materials. He also pointed out the existence of differences in source parameters between the Northern and Southern segments, which are separated almost at the middle point of the fault (see Fig. 2).

Ma and Mikumo (see Wang 2006a, b) inferred the spatial distribution of characteristic slip distance, D _c. Their values of D _c vary from 0 to 4 m, with an average of ~1 m, in the south and from 0 to 12 m, with an average of ~10 m, in the north. Ma et al. (2001) observed v_R = 2.28 km/s and $v_{\text R} /\beta \; = \;0.75$ in the southern segment and $v_{\text{R}} = 2.69\, {\text{km/s}}$ and $v_{\text{R}} /\beta \; = \;0.80$ in the northern one. Huang et al. (2001) evaluated the values of $\Delta \sigma_{\text{s}}$ for the two segments from near-field seismograms. Hwang et al. (2001) measured the values of E _s from near-fault seismograms. Wang (2004) correlated their values by eliminating the effect due to finite frequency bandwidth limitation. The correlated results are E _sN = 3.981 × 10¹⁶ J for the northern segment, $E_{\text{sS}} = 0.326 \; \times \;10^{16}\, {\rm J}$ for the southern one, and E _s = E _sS + E _sN = 4.307 × 10¹⁶ J for the whole fault.

In order to calculate the values of E _g for the 1999 Chi-Chi earthquake from Eq. (1), the basic values of parameters in use (Wang 2006a, b) are (1) $v_{\text{RS}} = 0.75\beta ,\;\Delta \sigma_{{\rm dS}}= 6.52 \,{\text{MPa}}, \, D_{\text{cS}} = 1 {\text{m}}$, and A _S = 4.551 × 10⁸ m² for the southern segment; and (2) v_RN = 0.80β, Δσ _dN = 2.97 MPa, D _cN = 10 m, and A _N = 3.615 × 10⁸ m² for the northern one. The estimated values of G and E _g are (1) for the southern segment, G _S = 9.32 × 10⁵ J/m² and E _gS = 4.24 × 10¹⁴ J, which is 13 % of E _sS; (2) for the northern segment, G _N = 3.68 × 10⁷ J/m² and E _gN = 1.33 × 10¹⁶ J, which is 33 % of E _sN; and (3) for the whole fault, G = 1.68 × 10⁷ J/m² and E _g = E _gS + E _gN = 1.37 × 10¹⁵ J, which is 32 % of E _s. The values of η _R are 0.88 for the Southern segment, 0.75 for the Northern one, and 0.76 for the whole fault. Based on the inverted results of source rupture processes from teleseismic data, Ma and Mikumo (see Wang 2006a, b) obtained G = (1–3) × 10⁸ J/m², with the largest value in the grid having the greatest displacement. Their values are about one-order-of-magnitude larger than those by Wang (2006a, b). The difference might be due to the use of different kinds of data by respective studies. They also observed an increase in G from south to north. Venkataraman and Kanamori (2004) evaluated the radiation efficiency of η _R = 0.8 for the earthquake from teleseismic data, thus giving $E_{\text{g}} \approx 0.25E_{\text{s}}$. Obviously, the value of E _g measured from near-fault seismograms is similar to that from teleseismic data. Hence, the results are reliable. The data to be used in this study, i.e., E _s, D (average displacement), D _c, E _g, η _R, and G, are given in Table 1. In the table, E _g is evaluated through two ways: (1) from Eq. (1) with D _S = 4.88 m and D _N = 7.15 m; and (2) from Eq. (3) as mentioned below with D _cS = 1 m and D _cN = 1, 3.7, 6, and 10 m. Meanwhile, in the table, the subscripts “S” and “N” are, respectively, applied to show the southern and northern segments of the Chelungpu fault.

Table 1 Related values in use

In 2000, two shallow boreholes near the Chelungpu fault (see Fig. 2) were drilled under a Taiwan-Japan Collaborative Project (cf. Tanaka et al. 2002). The possible fracture zones associated with the Chi-Chi (Ji-Ji) earthquake are in the range of 225–330 m at the northern borehole and 177–180 m at the southern one. Different depths of the inferred fracture zones are mainly due to the difference in the distances of the boreholes to the fault trace. The values of related physical parameters obtained from the two holes can be seen in Wang (2010). From the core samples, Otsuki et al. (2001) and Tanikawa et al. (2004) obtained D _c = ~1 m at the northern borehole. This value of D _c is close to the average of D _c in the southern segment inferred by Ma and Mikumo (See Wang 2006a, b) from teleseismic data and much shorter than those inferred by Zhang et al. (2003) and Ma and Mikumo (see Wang 2006a, b) for the northern segment from teleseismic data.

4 Estimates of D _c and discussion

Numerous factors can influence D _c. From laboratory experiments, Wong (1982) found that temperature, pressure, rock type etc., can all change the shear fracture energy up to an order of magnitude. Marone (1998) observed that these factors are also able to influence the value of D _c. Kostrov and Das (1988) obtained $G \; = \; 1 - 10^{2} \;{\text{J/m}}^{ 2}$ for a single crystal. Scholz (1990) reported $G = 0.27 - 10^{2} {\text{J/m}}^{ 2}$ for some geological materials and $G \; = \;10^{6} - 10^{7} {\text{J}}/{\text{m}}^{ 2}$ for earthquakes. In addition, $G \; = \;1.1 \; \times \;10^{3} \;{\text{J/m}}^{ 2}$ when the materials melt and $G \; = \;1.1 \; \times \;10^{5} \;{\text{J/m}}^{ 2}$ in the condition of dissociation. Pittarello et al. (2008) estimated the surface energy from microcrack density inside clast (i.e., cracked grains) entrapped in the pseudotachylyte and in the fault wall rock and the values range between 0.10 and 0.85 × 10³ J/m². Their estimates for the studied fault segments suggest that ~97 %–99 % of the energy was dissipated as heat during seismic slip. They concluded that at 10 km depth, less than 3 % of the total mechanical work density is adsorbed as surface energy on the fault plane during earthquake rupture.

The theoretical values of G are usually calculated for a homogeneous model, with constant K and Y, under a normal temperature and a constant pressure. Whereas the temperature, lithostatic pressure, and rock properties on the fault plane change with depth, and, thus, K and Y must also be a function of depth. This suggests depth-dependence of G. Of course, these properties also vary from place to place, and, thus, K and Y must also be a function of locality. The observation of G _N > G _S by Ma and Mikumo (see Wang 2006a, b) as mentioned above indicates the differences on the temperature, lithostatic pressure, and rock properties between the southern and northern fault segments. Huang and Wang (2002) observed the changes of scaling of displacement spectra of the 1999 Chi-Chi (Ji-Ji), Taiwan region, earthquake from near-fault seismograms from south to north. From the observed and inferred results, Wang (2003, 2004, 2006a, b) pointed out the differences in source parameters between the northern and southern segments. Wang (2006a) found the difference of pore pressures between the two fault segments. From seismic reflection experiments, Wang et al. (2004) also observed a difference in sub-surface fault geometry between the two segments. Consequently, different physical properties between the northern and southern segments lead to G _N  > G _S. Bizzarri (2011) stressed that a large number of chemical and physical mechanisms can also affect D _c.

Since the measured value of D _c is usually in the range of 0.1 m to few meters from earthquake data (Mikumo et al. 2003), the value of D _cS = 1 m for the southern fault segment inferred by Zhang et al. (2003) and Ma and Mikumo (see Wang 2006a, b) is reasonable. On the other hand, their values of D _cN = ~10 m for the northern fault segment sound to be unusual, even though it is less than the maximum displacement of 15 m at the depth of 8 km on the northern fault plane as inferred by Ma et al. (2001). There are two possible reasons for causing large D _cN. The first one is an under-shot of the northern segment during faulting, thus resulting in a fact that D _cN was almost the final offset. This does not seem possible, because there was a long slip displacement of 15 m on the fault plane as mentioned above. The second reason is that the value of D _cN = 10 m was over-estimated, thus resulting in an over-estimate of E _gN. The second reason seems reasonable. However, it is actually difficult to ensure an exact value of D _c based on the inferred stress-slip function from seismograms as pointed out by Tinti et al. (2004). Fukuyama et al. (2001) stressed that there is high uncertainty in the estimate of D _c. Here, it is necessary to explore the acceptable range of D _c for the Northern segment.

Equation (2) shows that the ratio of the rupture velocity to shear-wave velocity, i.e., $v_{\text{R}} /\beta$, is the main parameter causing an overall effect on η _R. Hence, the effect on η _R due to this ratio must be considered in advance. In the following, the value of η _R calculated from Eq. (2) is denoted by η _RC. Ma et al. (2001) observed v_R/β = 0.75 in the southern segment and v_R/β = 0.80 in the northern one. The related values of η _R calculated from Eq. (2) are η _RCS = 0.62 at v_R/β = 0.75 for the southern segment and η _RCN = 0.67 at v _R /β = 0.80 for the northern one, and the data points are denoted by two crosses in Fig. 3. Based on Eq. (1), Venkataraman and Kanamori (2004) evaluated the value of E _g from v_R/β = 0.66 by Ji et al. (2003), and the values of Δσ _d, D _max, and A by Ma et al. (2001). Hence, their value of η _R is 0.8. The related data point is denoted by ‘V’ in Fig. 3. For η _R = 0.8, from Eq. (2), the related value of v_R/β is 0.92 which is larger than 0.66 inferred by Ji et al. (2003). On the other hand, the value of η _RC calculated from v_R/β = 0.66 inferred by Ji et al. (2003) based on Eq. (2) is 0.55 and is shown by a symbol ‘J’ in Fig. 3. Since v _R/β = 0.66 by Ji et al. (2003) is the smallest in comparison with those inferred by others, η _R = 0.55 must be the smallest one based on the increase in η _R with v_R/β as displayed in Fig. 3. Clearly, the value of η _R = 0.8 is larger than η _R = 0.55. For the southern segment, the measured and calculated values are, respectively, η _RS = 0.61 (displayed by a symbol ‘O’ in Fig. 3) and η _RCS = 0.62. The two values are similar to each other. For the northern segment, the two values are, respectively, η _RN = 0.81 (demonstrated also by a symbol ‘O’ related to v_R/β = 0.80 in Fig. 3) and η _RCN = 0.66. The former is 0.14 larger than the latter. This inconsistency might be due to either an over-estimate of E _s or an under-estimate of η _R using Eq. (2). Since Wang (2004) eliminated possible effects on the estimate of E _s due to several factors, including the finite frequency bandwidth limitation, site effect, path effect, and radiation pattern, so his measured values should be reliable. Hence, it is assumed that this inconsistency is due to an under-estimate of η _R using Eq. (2), which was obtained based on dry rocks. Indeed the mechanisms due to fluids, for example, lubrication (Brodsky and Kanamori 2001) or thermal fluid pressurization (Sibson 1973; Mase and Smith 1984/1985), also play a significant role on faulting. Since the ratio v_R/β in Eq. (2) cannot represent such mechanisms, an advanced model including such mechanisms is needed to construct.

Since D _c is usually in the range of 0.1 m to few meters from earthquake data (Ide and Takeo 1997; Mikumo et al. 2003), the maximum values of D _cN, i.e., 10 and 12 m, and the average ones, i.e., 6 and 10 m, inferred, respectively, by Zhang et al. (2003) and Ma and Mikumo (see Wang 2006a, b) seem to be unusually large. In principle, at any locality on the fault plane, D _max must be larger than D _c, and, thus, on the whole fault plane average, D _max should be also larger than average D _c. The average value of D _c (= 10 m) inferred by Ma and Mikumo (see Wang 2006a, b) is larger than the average displacement D _N = 7.15 m, even though it is smaller than D _maxN = 15 m at H = 8 km inferred by Ma et al. (2001). Scholz (1988) stressed that the fault would be stable and could not generate large earthquakes if D _c is too long. This means that D _c cannot be larger than average D _N. Hence, D _cN = 10 m was over-estimated. Of course, D _cN = 6 m could also be over-estimated, even though it is smaller than the average D _N = 7.15 m. Fukuyama et al. (2003) stressed that there is high uncertainty in estimates of D _c from the inferred stress-slip function. Hence, it is necessary to explore the range of D _cN.

Wang (2006a) proposed a way to explore the acceptable value or range of D _c. The way is discussed in detail here. From Fig. 1, E _g can be approximated by

$$E_{\text{g}} \approx\frac {\Delta \sigma_{\text{d}} D_{\text{c}} {\text{A}}} {2}.$$

(3)

It is noted that since the variation of stress versus slip must be described by the dashed line in Fig. 1, Eq. (3), of course, gives an approximation and upper bound estimate of E _g. E _g estimated from Eq. (3) is slightly smaller than that done from Eq. (1). The difference between the values of E _g evaluated from the two equations depends on the stress drops, frictional strengths, frictional law, and the nonlinear decrease of stress with increasing slip. From Eq. (3), we can calculate the value of η _R from $E_{\text{s}} /\left( {E_{\text{s}} + E_{\text{g}} } \right)$, where E _s was taken from Wang (2004). A comparison between Eqs. (1) and (3) suggests an approximated relation of D _c versus D _max:

$$D_{\text{c}} \approx \left[ \frac{( 1- v_{\text{R}} /\beta )}{( 1+ v_{\text{R}} /\beta )} \right]^{ 1/ 2} D_{ \hbox{max} }.$$

(4)

It is noted that Eq. (4) does not hold for supershear earthquakes. Like E _g, D _c estimated from Eq. (4) is slightly different than that obtained from the inferred stress-slip function. Nevertheless, Eq. (4) is still acceptable because only the comparison of the values of η _R calculated from several values of D _c is taken into account.

Based on Eq. (3), D _cS = 1 m gives E _gS = 0.15 × 10¹⁶ J and thus η _RS = 0.69, which is only slightly larger η _RCS = 0.62 as mentioned above and represented by ‘S₁’ in Fig. 3 for the southern segment. From Eq. (4), we have D _cS = 1.4 m from D _maxS = 4.88 m inferred by Ma and Mikumo (see Wang 2006a, b). The two values of D _cS are close to each other, thus suggesting that D _cS = 1 m is acceptable.

In order to explore the acceptable values of D _cN for the northern segment, η _RN is calculated for five particular values of D _cN when Δσ _d = 2.97 MPa and v_R/β = 0.80. For the first case, we take the value of D _cN = 1.8 m calculated from the average displacement on the northern fault plane of average D _maxN = 7.15 m inferred from seismograms (cf. Wang 2006b) based on Eq. (4). This leads to η _RN = 0.81. Its data point is the same as that denoted by ‘O’ related to v_R/β = 0.80 in Fig. 3. For the second case 2, we take D _cN = 1 m inferred by Tanikawa et al. (2004) from the core samples obtained at the depth range of 225–330 m in a shallow northern borehole by using numerical simulations based on the thermal pressurization model. This leads to E _gN = 0.54 × 10¹⁶ J and η _RN = 0.99, which is represented by ‘N₁’ in Fig. 3. For the third case, considering E _sN = 3.98 × 10¹⁶ J with Δσ _dN = 2.97 × 10⁷ N/m² and A _N = 3.929 × 10⁸ m² measured from seismograms (Wang 2004, 2006a), we have D _cN = 3.7 m. This gives η _RN = 0.67 at v _R/β = 0.80. Its data point is denoted by a symbol ‘N₃’. For the fourth case, considering D _cN = 6 m, which was inferred by Zhang et al. (2003), we have E _gN = 3.22 × 10¹⁶ J. This leads to η _RN = 0.55, and the related data point is shown by ‘N₆’ in Fig. 3. For the fifth case, considering D _cN = 10 m, which was inferred by Ma and Mikumo (see Wang 2006a, b), we have E _gN = 5.37 × 10¹⁷ J. This leads to η _RN = 0.43, and the related data point is shown by ‘N₁₀’ in Fig. 3. Obviously, η _RN decreases with increasing D _cN. Related values of E _sN, E _gN, G _N, and η _RN of the five cases are listed in Table 1.

Since the value of D _cN = 1 m was inferred by Tanikawa et al. (2004) from the core samples at a shallow borehole, it can only represent the value of D _cN at shallow depths and thus cannot be the average of D _cN over the northern segment. The value of D _cN = 1.8 could be the lower bound of D _cN, because it was calculated from the average D _maxN = 7.15 m inferred from seismograms and larger than 1 m for the shallow depths. The data point denoted by ‘N₃’ related to D _cN = 3.7 m calculated from E _sN = 3.98 × 10¹⁶ J with Δσ _dN = 2.97 × 10⁷ N/m² and A _N = 3.929 × 10⁸ m² measured from seismograms is much close to the data point denoted by a cross on the theoretical curve in Fig. 3. Hence this D _cN = 3.7 m could be the upper bound of D _cN. The two values of the fourth and fifth cases are both smaller than those (0.75–0.95) of normal earthquakes (Kanamori and Brodsky 2004), and their data points are below the theoretical curve. Hence, D _cN = 6 and 10 m could be over-estimated. In addition, η _RN = 0.81 of case 2 and 0.67 of case 3, respectively, related to D _cN = 1.8 and 3.7 m are both larger than η _RS = 0.61. Hence, it is reasonable to consider D _cN to be in between 1.8 and 3.7 m. The mean value of the two values is 2.25 m, which is close to D _cN = 2.3 m inferred by Mori (2005) from near-fault seismograms. Conclusively, the acceptable value of D _cN must be in between 1.8 and 3.7 m, with the mean value of 2.25 m to be most acceptable one.

5 Conclusions

The characteristic slip displacement (or slip-weakening distance), D _c, of the velocity- and state-dependent friction law is an important parameter of friction law and its value varies in a large range. Meanwhile, the values measured from laboratory experiments are much smaller than those estimated from seismic data. Essentially, it is difficult to estimate an acceptable value of D _c. Nevertheless, seismologists need its value to understand the source processes. In this study, a simple way is proposed to estimate D _c based on the constraint from the radiation efficiency, i.e., $\eta_{\text{R}} = E_{\text{s}} /\left( {E_{\text{s}} + E_{\text{g}} } \right)$. In order to describe the way clearly, an example of estimate of D _c is made for the 1999 M _w 7.6 Chi-Chi (Ji-Ji) earthquake, which ruptured along the Chelungpu fault in central Taiwan region, because there are fruitful data generated by this event. The values of D _c for the two segments of the fault were estimated by Zhang et al. (2003) and Ma and Mikumo (see Wang 2006a, b). Results are, respectively, D _cS = 1 m (for the southern segment) and D _cN = 10 m (for the northern segment). D _cS seems to be reasonable, while D _cN is too long to fit the theoretical requirement. From the measured and calculated values of η _R for five values of D _cN through two approaches, the acceptable value of D _cN is in between 1.8 and 3.7 m. The mean value of 2.25 m is considered to be the most acceptable one.