Keywords

1 Introduction

Kamae and Irikura [1] estimated a broadband source model for the 1995 Hyogoken-Nanbu earthquake (Mw6.9) to fit near-field strong motion records by forward modeling using the empirical Green’s function method (e.g., Irikura [2]). Since Kamae and Irikura’s pioneering work, many researchers have estimated broadband source models in the period range from about 0.1 to 5 s for the other big earthquakes using the empirical Green’s function method. The broadband source models were composed of a few strong motion generation areas (SMGAs).

Irikura and Miyake [3] proposed the recipe for strong motion predictions. In the recipe, the source model was expressed by a few rectangular asperities and surrounding background areas. Here asperities were characterized from long-period heterogeneous kinematic slip models estimated by waveform inversion method using strong motion records in the period longer than about 2 s [4]. Asano and Iwata [5] studied on the relations between stress drops on asperities and depths of asperities for crustal earthquakes. Miyake et al. [6] showed that asperities coincide to SMGAs defined as areas that mainly generate strong ground motions. However, the period ranges to estimate asperities based on the waveform inversion results are relatively longer than those to estimate SMGAs by the empirical Green’s function method. Therefore, we develop empirical relations between stress drops on SMGAs and depths of SMGAs based on the previous broadband source models estimated by the empirical Green’s function method for crustal earthquakes in Japan to aim at the advancement of strong motion predictions.

2 Data

Data used in this study are shown in Table 6.1 [1, 6–34] and Fig. 6.1. A total of 22 articles on SMGAs [1, 6, 13, 16–34] for 13 crustal earthquakes of the moment magnitude Mw from 5.7 to 6.9 in 1995 to 2011 are used. The numbers of the strike-slip, reverse, and normal faults are six, six, and one, respectively. We independently treat each source model for the same earthquake, and so the total 25 source models are examined. We also independently treat each strong motion generation area. The stress drops on SMGAs estimated by Miyake et al. [6] are calculated assuming the single-asperity model. The others are calculated assuming single-crack models.

Table 6.1 List of earthquakes and references [1, 6–34]
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Fig. 6.1
figure 1

Epicenters by JMA and focal mechanisms by the Global CMT Project for No.1 earthquake and by F-net for the other earthquakes

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3 Results

Figure 6.2 shows the relations between stress drops on SMGAs and the top, center, and bottom depths of SMGAs for strike-slip faults and reverse faults. The number of SMGAs of each earthquake is one to three except for five by Hirai et al. [16]. The stress drops are less than 30 MPa except for those for the 2007 Noto Hanto earthquake estimated by Maeda et al. [27]. Although source models for the same earthquake are different among researchers, large SMGAs tend to be located at deep places.

Fig. 6.2
figure 2

Relations between stress drops and depths of SMGAs for strike-slip faults (left) and reverse faults (right). Large symbols denote the center depth of each SMGA. Small symbols denote the top and bottom depths of each SMGA

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Figure 6.3a shows the relations between the stress drops and the center depths of SMGAs for all types of earthquakes including a normal-fault earthquake and regression relations. The average stress drops are 21.2 MPa with standard error of 9.2 MPa, 13.3 MPa with standard error of 5.3 MPa, and 18.0 MPa with standard error of 8.6 MPa for reverse, strike-slip, and all types of faults, respectively. The average stress drop on SMGAs of 18.0 MPa in this study is larger than average stress drop on asperities of 10.5 MPa in the recipe by Irikura and Miyake [3]. The average center depths are 8.90 km, 8.65 km, and 8.65 km for reverse, strike-slip, and all types of faults, respectively. The relations between stress drops Δσ a [MPa] on SMGAs and the center depths h [km] are written as

Fig. 6.3
figure 3

(a) Relations between stress drops and the center depths of SMGAs. (b) Relations between Mo and total area of SMGAs

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$$ \Delta {\sigma}_{\mathrm{a}}=0.63\mathrm{h}+7.88\kern1em \left(\mathrm{standard}\ \mathrm{error}=5.26\right)\kern1em \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{strike}\hbox{-} \mathrm{slip}\ \mathrm{f}\mathrm{aults}, $$
(6.1)
$$ \Delta {\sigma}_{\mathrm{a}}=1.42\mathrm{h}+8.54\kern1em \left(\mathrm{standard}\ \mathrm{error}=8.39\right)\kern1em \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{r}\mathrm{everse}\ \mathrm{f}\mathrm{aults}, $$
(6.2)
$$ \Delta {\sigma}_{\mathrm{a}}=1.15\mathrm{h}+7.98\kern1em \left(\mathrm{standard}\ \mathrm{error}=8.05\right)\kern1em \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{all}\ \mathrm{types}\ \mathrm{o}\mathrm{f}\ \mathrm{f}\mathrm{aults} $$
(6.3)

The depth dependency for reverse faults is stronger than that for strike-slip faults. The stress drops for reverse faults is larger than those for strike-slip faults at the same depth. Although the standard errors [MPa] of the empirical relations are large, Eq. (6.3) means that stress drops increase by about 1 MPa every 1 km in depth. The empirical relations between stress drops on asperities Δσ asp and the center depths derived by Asano and Iwata [5] for crustal earthquakes in Japan shown in Fig. 6.3a is

$$ \Delta {\sigma}_{\mathrm{asp}}=1.10\mathrm{h}+4.2\kern1em \left(\mathrm{standard}\ \mathrm{error}=7.2\right)\kern1em \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{all}\ \mathrm{types}\ \mathrm{o}\mathrm{f}\ \mathrm{f}\mathrm{aults} $$
(6.4)

The depth dependency of the equations of (6.3) and (6.4) are similar, and the absolute value of the stress drop on SMGAs is about 4 MPa larger than the stress drop on asperities.

We also derive the relations between seismic moment Mo [Nm] and total area of SMGAs S a [km2] as shown in Fig. 6.3b in which S a is the average for each earthquake. The equations derived by constraining the slop to be 1/3 are written as

$$ {S}_{\mathrm{a}}=4.57\times {10}^{-16}{\left({\mathrm{M}}_{\mathrm{o}}\times {10}^7\right)}^{2/3}\kern0.5em \left(\mathrm{common}\ \mathrm{logarithm}\ \mathrm{o}\mathrm{f}\ \mathrm{standard}\ \mathrm{error}=0.18\right)\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{strike}\hbox{-} \mathrm{slip}\ \mathrm{f}\mathrm{aults} $$
(6.5)
$$ {S}_{\mathrm{a}}=3.64\times {10}^{-16}{\left({\mathrm{M}}_{\mathrm{o}}\times {10}^7\right)}^{2/3}\kern1em \left(\mathrm{common}\ \mathrm{logarithm}\ \mathrm{o}\mathrm{f}\ \mathrm{standard}\ \mathrm{error}=0.09\right)\kern1em \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{r}\mathrm{everse}\ \mathrm{f}\mathrm{aults} $$
(6.6)
$$ {S}_{\mathrm{a}}=4.02\times {10}^{-16}{\left({\mathrm{M}}_{\mathrm{o}}\times {10}^7\right)}^{2/3}\kern1em \left(\mathrm{common}\ \mathrm{logarithm}\ \mathrm{o}\mathrm{f}\ \mathrm{standard}\ \mathrm{error}=0.15\right)\kern1.25em \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{all}\ \mathrm{types}\ \mathrm{o}\mathrm{f}\ \mathrm{f}\mathrm{aults} $$
(6.7)

The scaling law for total area of asperities S asp by Somerville et al. [4] shown in Fig. 6.3b is written as

$$ {S}_{\mathrm{asp}}=5.00\times {10}^{-16}{\left({\mathrm{M}}_{\mathrm{o}}\times {10}^7\right)}^{2/3} $$
(6.8)

S a for strike-slip, reverse, and all types of faults are about 0.91, 0.73, and 0.80 times of S asp by Somerville et al. [4]. Although the standard error is large, S a for each reverse fault is smaller than S asp by Somerville et al. [4]. SMGAs are source models for strong motions in the period range from 0.1 to 5 s, while the asperities are source models for strong motions in the period longer than about 2 s. Therefore, the result that total area of SMGAs is smaller than total area of asperities is interpreted by frequency-dependent source radiations [35].

Short-period spectral level A which means the flat level of acceleration source spectrum [36] is proportional to stress drop and square root of total area of SMGAs (or asperities). Considering the equations of (6.1), (6.2), (6.5), and (6.6), A for reverse faults is larger than A for strike-slip faults. Satoh [35] showed the same results from strong motion records for big crustal earthquakes in Japan using the spectral inversion method. McGarr [37] showed that peak ground velocities PGVs normalized by Mo 1/3 and hypocentral distances depend on focal depths and are larger for reverse faults than normal faults. He pointed out that these results are expected from frictional laws. In addition he pointed out that data of strike-slip faults were insufficient in his analysis, but the normalized PGVs for strike-slip faults would lie between those for reverse and normal faults. Our results are qualitatively consistent with McGarr’s results, although site effects were not considered in McGarr’s results.

4 Conclusions

We developed empirical relations between stress drops on SMGAs and depths of SMGAs based on previous broadband source models estimated by the empirical Green’s function method. A total of 25 source models for 13 crustal earthquakes of Mw from 5.7 to 6.9 in Japan are used in this study. As a result it is found that stress drops on SMGAs for reverse faults are larger than those for strike-slip faults on average. The average stress drops are 21.2 MPa, 13.3 MPa, and 18.0 MPa for reverse, strike-slip, and all types of faults, respectively. In the derived empirical relation for all types of faults, the stress drops increase by about 1 MPa every 1 km in depth. The depth dependency of stress drops for reverse faults is stronger than that for strike-slip faults. We also showed that the total area of SMGAs is about 0.8 times of the total area of asperities by Somerville et al. [4]. This result can be interpreted by frequency-dependent source radiations, since asperities are estimated from longer-period strong motions (>2 s) than SMGAs. The empirical relations derived in this study would be useful for advancement of strong motion predictions for crustal earthquakes by considering together with regional differences and uncertainties.