A SEMI-EMPIRICAL METHOD USING A HYBRID OF STOCHASTIC AND DETERMINISTIC FAULT MODELS: SIMULATION OF STRONG GROUND MOTIONS DURING LARGE EARTHQUAKES

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1 J. Phys. Earth, 36, , 1988 A SEMI-EMPIRICAL METHOD USING A HYBRID OF STOCHASTIC AND DETERMINISTIC FAULT MODELS: SIMULATION OF STRONG GROUND MOTIONS DURING LARGE EARTHQUAKES Masayuki TAKEMURA* and Tomonori IKEURA** *Kobori Research Complex, Kajima Corporation, Tokyo, Japan **Kajima Institute of Construction Technology, Kajima Corporation, Tokyo, Japan (Received March 2, 1988; Revised June 7, 1988) A new semi-empirical method is developed to estimate strong ground motions, which takes into account fault plane irregularities in the prediction of strong ground motions during large earthquakes. According to this method, the displacement on the fault plane of a large earthquake is divided into two parts. One is the average displacement Do over the fault plane and the other is the deviation ƒ Dij from Do. The seismic radiation SL(t) due to Do and the seismic radiation SS(t) due to ƒ Dij are synthesized separately using the observed seismograms for a small element earthquake. A parameter SD is introduced as the standard deviation of ƒ Dij divided by the displacement of the element earthquake, which shows the degree of inhomogeneity of displacement on the fault plane. The new method is applied to synthesis of strong ground motions at 18 stations during 8 large earthquakes with magnitude M of 6.7 to 7.9 occurring in and around Japan. The synthesized records are in good agreement with the observed ones except for the records obtained at the stations on soft soil deposits, if SD is chosen to be about 1.0. These results indicate the applicability of the new method for predicting strong ground motions on relatively stiff ground which does not show non-linear behavior during strong ground shaking. 1. Introduction Higher frequency strong ground motions are too complicated to simulate by a deterministic model, because of the sensitivity of high frequency waves to the details of fault plane irregularities and to heterogeneities of the earth's structure. To avoid the difficulty regarding the fault plane irregularities, stochastic models specified by a small number of statistical parameters have been proposed (e.g., HIRASAWA, 1979; IZUTANI, 1981; KOYAMA, 1983; PAPAGEORGIOU and Am, 1983). Their results indicated that the fault plane irregularity is an important factor in the excitation of 89

2 90M. TAKEMURA and T. IKEURA high frequency waves, while the excitation of low frequency waves can be described by the average displacement or the average stress drop over the fault plane. To estimate strong ground motions over a wide frequency range, therefore, a hybrid of deterministic and stochastic models is effective, in which gross features of rupture propagation are specified deterministically, but the details of the process are described by the stochastic model. Another approach was proposed by HARTZELL (1978) to synthesize strong ground motions, utilizing observed seismograms from small events as Green functions. This method is called a semi-empirical method and is useful for avoiding the difficulty concerning the heterogeneities of the earth's structure, because the Green functions include complex effects of the heterogeneous structure. IRIKURA (1983) adopted a similarity law among different size earthquakes on the kinematic source model of Haskell to improve the Hartzell's method and applied the method to the simulation of near-field velocity records for the 1980 Izu-toho-oki earthquake with magnitude M = 6.7. The results of the simulation can explain the observed records in the period range longer than about 1 s, but will underestimate the amplitudes in the period range shorter than about 1 s, because the spectrum of synthesized record has co high frequency fall-off due to the assumption of uniform displacement over the fault plane. Meanwhile, IMAGAWA and MIKUMO (1982) indicated through an analysis using another semi-empirical method that synthesized waveforms provide a satisfactory agreement to observed displacement records for the periods longer than 5s for the 1969 central Gifu earthquake (M= 6.6) and that a stochastic fault model will have to be applied for shorter period components. IRIKURA (1986) and DAN et al. (1987) proposed different methods to explain acceleration motions, which produce an co' high frequency spectral fall off. IRIKURA (1986) attempted to synthesize accelerograms for the 1983 Nihonkai-chubu earthquake (M=7.7) and its aftershock (M= 6.1). The synthesized accelerograms were in good agreement with observed ones for the case of the aftershock with the assumption of an ƒö-2 model, but the ƒö-2 model failed to fit the amplitudes of accelerograms for the main shock. IRIKURA (1986) indicated that the specific barrier model (PAPAGEORGIOU and AKI, 1983) should be introduced for the simulation of high frequency waves of the mainshock. In this study, a semi-empirical method for estimating strong ground motions over a wide frequency range is proposed with the assumption of hybrid deterministic and stochastic fault models. The deterministic model is based on the kinematic source model of Haskell and the similarity law of earthquakes, which are the same assumptions adopted by IRIKURA (1983). The stochastic model is described by a specific distribution function of displacement on the fault plane. To examine the applicability of the new method, strong ground motion records are simulated at 18 stations during 8 earthquakes of magnitude 6.7 to 7.9 in and around Japan and the results of the simulations are compared with observed records of acceleration, velocity, and displacement.

3 Semi-Empirical Method Using a Hybrid Fault Model 91 Fig. 1. The model of displacement on the fault plane assumed for the synthesis of strong ground motions by the semi-empirical method. Explanations of parameters are described in the text. 2. Method Figure 1 shows the spatial distribution of displacement on the fault plane schematically for estimating strong ground motions by the semi-empirical method proposed here. The fault plane of a large earthquake is assumed to consist of small segments of a specific size. The displacement D1 of each segment is expressed as where Do is the average displacement over the fault plane and ƒ Dij is the deviation from Do. An element earthquake should be selected, whose hypocenter is near the focal region of the large earthquake. If the size of the fault area of the element earthquake is consistent with the size ƒ L of the segment, Dij can be rewritten as follows: where De is the displacement of the element earthquake, n is a scaling parameter determined from the cube root of seismic-moment ratio between the large and the element earthquakes (IRIKURA, 1983), and Kij is introduced as a ratio between the deviation ƒ Dij of each segment on the fault plane and the displacement De of the element earthquake. is assumed to be a probabilistic parameter of normal distribution with an average of 0 and with standard deviation of SD. SD shows the degree of heterogeneity of displacement on the fault plane. By using the observed seismogram Se(t) of the element earthquake, a seismic wave SL(t) due to the average displacement D, and a seismic wave Ss(t) due to the deviation ƒ Dij from Do are derived as follows:

4 M. TAKEMURA and T. IKEURA where Re is the source-to-site distance of element earthquake, Rij is the distance from each segment on the fault plane to the site, and Tijk is the total delay time due to the fault slip, the rupture propagation, and the travel time of seismic waves. According to IRIKURA (1983), the expression (3) possesses problems for practical applications in that the synthesized wave has an artificial periodicity of to/n, where to is the average rise time of displacement for the large earthquake. Therefore, we adopt the revised method proposed by IRIKURA (1983) to calculate SL(t). The final result of synthesized strong ground motion Ssyn(t) can be expressed as a sum of SL(t) and Sp). The semi-empirical method proposed here is characterized by the consideration of heterogeneous displacement over the fault plane. An inhomogeneity with a specific size on the fault plane was called a fault patch by KOYAMA (1983). According to KOYAMA (1983), the source spectrum has a second corner frequency due to the fault patch, which is inversely proportional to the average of the dimensions of the fault patches. IKEURA and TAKEMURA (1988) indicated theoretically that under the theory of the present method, the source spectrum also has a second corner frequency which is inversely proportional to the fault dimension of the element earthquake. Therefore, it is necessary to select the element earthquake such that its fault area is consistent with the fault patch area, based on the relation between the fault patch area ƒ S on the fault plane and magnitude M of the large earthquake. The relation between AS and M, and the value of SD will be discussed later. 3. Data Table 1 shows earthquakes and observation stations for strong motion records, which we attempt to simulate in the present study. M and M, are the magnitude of large and element earthquakes in the JMA (Japan Meteorological Agency) scale. Stations with parentheses are located on thick, soft, soil deposit, which behaves plastically during strong shaking due to a large earthquake. Stations with a star and with double stars provided us with displacement records and velocity records, respectively. The displacement records were obtained by strong motion seismographs of JMA type, of which natural period and static magnification are 6 s and 1.0. The velocity records were obtained by strong motion seismographs designed by MURAMATSU (1977) with the dynamic range from 100 to 0.01 cm/s over the period range from 0.05 to 50 s. Strong motion records at the other stations were obtained by SMAC-type and electromagnetic-type accelerographs. The records from element earthquakes were obtained by the same seismographs for the corresponding large events. Strong motion records from the 1980 Izu-toho-oki earthquake were simulated

5 Semi-Empirical Method Using a Hybrid Fault Model 93 Table 1. Observation stations and magnitude for large and element earthquakes. ( ) Stations on thick, soft, soil deposit. * Displacement records. **Velocity records. by using various different methods and their results were compared (TAKEMURA and IKEURA, 1988). The present method was also used. In this paper, we will adopt the results of our method in that study pertaining to the Izu-toho-oki event. Locations of fault planes of large events and of epicenters of element earthquakes are summarized in Fig. 2. Arrows in the fault planes indicate directions of rupture propagations. Fault parameters used for the simulations are based on. other studies (FUKAO and FURUMOTO, 1975; IRIKURA, 1983; JMA, 1985; KANAMORI, 1971; MORI and SHIMAZAKI, 1984; SATO, 1985; SENO et al., 1980; SHIMAZAKI, 1974; SHIMAZAKI and SOMERVILLE, 1979; SHIONO and MIKUMO, 1980; TAKEMURA and IKEURA, 1987). Based on the results of FUKAO and FURUMOTO (1975), the Tokachi-oki event occurred in the form of multiple-shock activity which can be divided into two rupture processes. One has the rupture propagating from the initial break near the center of the eastern edge of the fault plane, and the second has the main faulting beginning from the southern edge at 40 to 45 s after the initial break. Results for long-period body waves in the far field (FUKAO and FURUMOTO, 1975) and those for displacement motions observed by JMA (MORI and SHIMAZAKI, 1984) are more consistent with the initial rupture process, while long-period surface waves in the far field can be explained by the process of the main faulting (KANAMORI, 1971). We may conclude that the excitation of short-period seismic waves is strongly related to the initial rupture process, while that of long-period seismic waves is mainly due to the main faulting. Therefore, the initial process is adopted for simulating strong ground motions, considering the period range of the analysis in the present study. The fault plane of the Nihonkai-chubu event is divided into three parts to explain the observed seismograms. SATO (1985) indicated that the fault plane of E3 weakly contributed to the generation of high frequency waves in comparison with

6 94 M. TAKEMURA and T. IKEURA Fig. 2. Locations of fault planes of large earthquakes and observation stations. Open circles indicate the epicenters of element earthquakes. Arrows show the direction of rupture propagation of each large event. the fault planes of El and E2. Therefore, we calculated the strong motion records using only rupture on the fault planes of El and E2 (TAKEMURA and IKEURA, 1987). Meanwhile, the rupture propagation of the 1968 Hyuga-nada event is assumed to begin at the center of the fault plane, though details of the rupture process have not yet been elucidated for this event. The element earthquakes, which are summarized in Table 1, are selected for the simulations based on the relation between magnitude M and fault patch area ƒ S for the large earthquakes. Figure 3 shows the relation between M and ƒ S, which is summarized based on the results of AKI (1983), IZUTANI (1984), and SATO (1985). AKI (1983) estimated barrier intervals from the average length of fault segments observed by geologists on the earth's surface after large earthquakes. These values are consistent with the barrier intervals obtained from acceleration power spectra using the specific barrier model (AKI, 1983). SATO (1985) estimated the barrier intervals for the 1983 Nihonkai-chubu earthquake by using the specific barrier model. IZUTANI (1984) obtained the second corner frequency fc* through the analysis of accelerograms for some Japanese earthquakes. According to KOYAMA (1983), fc* is inversely proportional to the average fault patch size. We calculated the fault patch areas from the barrier intervals by AKI (1983) and SATO (1985) and from fc*'s by IZUTANI (1984) on the assumption of a circular fault patch. The magnitude scale adopted in Fig. 3 is JMA magnitude or surface-wave magnitude. The fault areas Se of element earthquakes are also plotted in Fig. 3, which are

7 Semi-Empirical Method Using a Hybrid Fault Model 95 Fig. 3. A relation between fault patch area AS (or fault area S e of element earthquake) and magnitude M of large earthquakes. estimated from magnitude using the relation between M and Se (SATO, 1979), It is found in Fig. 3 that ƒ S increases with M, though the relation between LIS and M cannot be accurately determined because of the large scattering of the data. We can also find that the fault area Se of element earthquakes selected in the present study are included within the distribution of the data of AS. SATO (1985) estimated radiuses of fault patches on the fault planes El and E2 of the Nihonkai-chubu event. The fault patch area ƒ S is about 190 km2 from the average of the radiuses. The magnitude of the element earthquake which we have to select is calculated to be M= 6.3 from LIS by using Eq. (5). This value is almost consistent with the magnitude Me of the element earthquake selected for the Nihonkai-chubu event shown in Table Synthesis of Strong Ground Motions Seismic moment Me, scaling parameters n, and the value of SD used for estimating strong ground motions are summarized in Table 2. n is obtained from the cube root of seismic moment ratio between large and element earthquakes. Except for the cases of the Nihonkai-chubu, Tokachi-oki, and Izu-toho-oki events, the seismic moment of the element earthquake is estimated from magnitude M using the empirical relation (SATO, 1979), Seismic moments estimated by the centroid-moment tensor method (DZIEWONSKI et al., 1983; DZIEWONSKI and WOODHOUSE, 1983) are adopted for element earthquakes

8 96 M. TAKEMURA and T. IKEURA Table 2. Source parameters for estimating strong ground motions. 1) KANAMORI ( 1971), 2) IZUTANI (1983), SATO (1985), 4) TAKEMURA and IKEURA (1987),5) SHIONO and Muumo (1980), 6) SHIMAZAKI (1974), 7)SENO et al. (1980), 8)HIRASAWA (1979), 9)DZIEWONSKI et al. (1985), 10)SHIMAZAKI and SOMERVILLE (1979). The value of n of the Miyagi-oki event is determined for the case of Miyako station. of the Nihonkai-chubu and the Tokachi-oki events. Seismic moment ratio is directly obtained from the spectral ratio of observed seismic waves in the long period range for the large and element earthquakes for the Izu-toho-oki event (IRIKURA, 1983). The value of SD is determined so as to match the spectral amplitude of the synthesized record with that of the observed one in the period range shorter than 1 s. We can find that the value of SD ranges from 0.7 to 1.5 irrespective of the magnitude and the seismic moment of the large earthquake. The available frequency range of the simulation depends on the S I N ratio of the observed seismograms for the element earthquake. Figure 4 shows observed and synthesized accelerograms at Akita of the Nihonkai-chubu event. The available frequency range of the accelerograms is 0.2 to 10 Hz. It is found that the amplitude of SS(t) is much larger than that of SL(t). This indicates that the heterogeneity of faulting is an important factor in the generation of high frequency waves. The synthesized record S syn(t) fails to fit the envelope with two peaks of observed accelerogram. SATO (1985) indicated that the rupture of the Nihonkai-chubu event stopped for 10 s at the boundary of the El and E2 fault planes. The reason for failure to fit the envelope is that the rupture is assumed to propagate with constant velocity in the calculation. Figure 5 shows response spectra with damping constant h = 0.05 of observed and synthesized records. The amplitude of SS(t) is almost equal to that of SL(t) at about 5 s, while SS(t) is about 5 to 10 times as large as SL(t) in the period range shorter than 1 s. We can easily guess that SL(t) becomes larger than 5(0 at longer periods, since the sum of the seismic moments for S3(t) waves is 0 over the fault plane. The spectrum of the synthesized record is in good agreement with that of the observed one. The SD value is determined to be 1.3 for the Nihonkai-chubu earthquake. From the displacement records of JMA, FUKUYAMA and IRIKURA (1986) estimated the distribution of displacement on the fault plane of the Nihonkai-chubu event, using a

9 Semi-Empirical Method Using a Hybrid Fault Model 97 Fig. 4 Fig. 5 Fig. 4. Observed and synthesized accelerograms in EW component at Akita. The observed ones are of the Nihonkai-chubu earthquake and the element event. The synthesized ones are of SL, Ss, and S+-1- Ss waves. Fig. 5. Response spectra of observed and synthesized accelerograms in EW component at Akita. Damping factor h is waveform inversion based on the semi-empirical method. We can calculate the SD value from their results, because the same element earthquake was used for the southern part of the fault by FUKUYAMA and IRIKURA (1986). The calculated value is 1.4, which is consistent with the value of SD in the present study. Figure 6 represents, synthesized and observed accelerograms and their spectra at Furofushi, which show a similar comparison between observed and synthesized records. The spectrum of the synthesized record is in good agreement with that of the observed one, while the wave form of the observed record is too complicated to be explained by the results of the simulation. The simulations for the displacement records are made at Tokyo, Sendai, and Hachinohe. The results are shown in Figs. 7 and 8. The period range of the analyses is 2 to 7 s. Figure 7 indicates that the amplitudes of Ss(t) is about twice as large as those of SL(t). This result is consistent with that of the comparison with spectral amplitudes of 5,(t) and SL(t) at Akita in Fig. 5. Synthetic waveforms in displacement provide a satisfactory agreement with the observed records. Figures 9, 10, 11, and 12 present accelerograms and their spectra at Kushiro for the Nemuro-oki event, at Hachinohe.for the Tokachi-oki event, at Miyako for the Miyagi-oki event, and at Shimizu-miho for the Oshima-kinkai event, respectively.

10 98 M. TAKEMURA and T. IKEURA Fig. 6. Observed and synthesized accelerograms in EW component at Furofushi during the Nihonkai-chubu earthquake and their response spectra with h= Fig. 7 Fig. 8 Fig. 7. Observed and synthesized displacement seismograms at Tokyo. The observed ones were obtained by the strong motion seismometer of the JMA network during the Nihonkai-chubu earthquake. Fig. 8. Observed and synthesized displacement seismograms at Sendai and at Hachinohe during the Nihonkai-chubu earthquake.

11 Semi-Empirical Method Using a Hybrid Fault Model 99 Fig. 9 Fig. 10 Fig. 9. Observed and synthesized accelerograms in EW component at Kushiro during the Nemuro-oki earthquake and their response spectra with h =0.05. Fig. 10. Observed and synthesized accelerograms in NS component at Hachinohe during the Tokachi-oki earthquake and their response spectra with h = The envelopes of synthesized waves are in relatively good agreement with those of observed waves for the Nemuro-oki, the Miyagi-oki, and the Oshima-kinkai events, compared with the results for the Tokachi-oki and the Nihonkai-chubu events. The synthetic waveforms are in good agreement with observed ones for the Izu-toho-oki event in the frequency range of 0.03 to 10 Hz (TAKEMURA and IKEURA, 1988) Magnitudes of the Tokachi-oki and the Nihonkai-chubu events are 7.9 and 7.^ which are larger than those of other events. These results suggest that the source process of a great earthquake with magnitude M larger than about 7.5 is too complicated to explain the high frequency waves by a statistical parameters such as SD. However, the spectra of synthesized records give us a satisfactory agreement with those of observed records in the period range shorter than 1 s not only for the earthquakes of M<7.5 but also for the great earthquakes of M> Discussions The semi-empirical method proposed here must face three problems for predicting strong ground motions due to disastrous earthquakes. The first problem is the estimation of strong ground motions on the soft soil deposits. Table 1 indicates that Aomori, Hosojima, and Shiogama are located on the thick, soft, soil

12 100 M. TAKEMURA and T. IKEURA Fig. 11 Fig. 12 Fig. 11. Observed and synthesized accelerograms in EW component at Miyako during the Miyagi-oki earthquake and their response spectra with h = Fig. 12. Observed and synthesized accelerograms in EW component at Shimizumiho during the Oshima-kinkai earthquake and their response spectra with h= deposits. Distribution of N-value obtained by the standard penetration test at each station (KURATA et al., 1970) is presented in Fig. 13. Figure 14 shows the results of simulating an accelerogram observed at Hosojima from the 1984 Hyuga-nada event. The value of SD is assumed to be 1.4 so that the peak acceleration of the synthesized waves is almost consistent with that of observed waves. However, we did not succeed in fitting the observed waveform, for which the dominant period changes during the strong shaking of acceleration. The period of the spectral peak for the observed waves is longer than that of the synthesized waves. These features may be caused by the non-linear behavior of soft soil deposit for strong ground motions. The same features are found in the results of the simulations at Aomori and Shiogama. Therefore, it is concluded that the semiempirical method based on the theory of linear system cannot be applied to the prediction of strong ground motions on thick, soft, soil deposits. The second problem is how to determine the magnitude of the element earthquake selected for the synthesis. The present method requires that the size of fault area of the element event should be consistent with the fault patch size of the

13 Semi-Empirical Method Using a Hybrid Fault Model 101 Fig. 13 Fig. 14 Fig. 13. N-Values of the grounds at Aomori, Shiogama, and Hosojima. Fig. 14. Observed and synthesized accelerograms in EW component at Hosojima during the 1984 Hyuga-nada earthquake and their response spectra with h= large earthquake. Figure 15 shows the relation between fault areas Se of element earthquakes used in the present study and seismic moments Ma of large earthquakes. The fault patch areas ƒ S for large earthquakes are also plotted in Fig. 15, which are calculated from the data of barrier intervals summarized by An (1983). KANAMORI and ANDERSON (1975) and SATO (1979) indicated that seismic moment M, is proportional to the 1.5 power of fault area S under the similarity law of earthquakes. The relation of SATO (1979) is also shown by a solid line in Fig. 15. It can be written as follows: Comparing the relation of SATO (1979) and the data of Se and LIS, we find that Se's and ƒ S's almost satisfy the relation of log M0 `1.5 log Se (or ƒ S). Under this relation, the scaling parameter n becomes constant irrespective of the magnitude of the large earthquake. The dotted line in Fig. 15 corresponds to the Me-Se relation when n = 7. The relation of log M0 `1.5 log ƒ S indicates the similarity law between

14 (8) 102 M. TAKEMURA and T. IKEURA Fig. 15. A relation between fault patch area AS (or fault area Se of element earthquake) and seismic moment Mo. Solid line indicates the similarity relation between Mo and fault area of large earthquake and dotted line the Mo-Se relation for the scaling parameter n= 7. seismic moment and fault patch area. The same results were obtained for the second corner frequency ft.* by KOYAMA (1983) and IZUTANI (1983). KOYAMA (1983) and IZUTANI (1983) indicated that the relation of M0 `fc*-3 is applicable to the Matsushiro earthquake swarm. On the other hand, IRIKURA (1984), and IZUTANI (1984) pointed out that the second corner frequency fc* is almost constant or independent of the earthquake magnitude for aftershocks of the Nihonkai-chubu event and for some moderate earthquakes near Japan, respectively. IZUTANI (1984) suggested a regional difference in Mo-f.* relations. The validity of the similarity condition between seismic moment and fault patch area for estimating strong ground motions is an important problem which remains to be elucidated. The third problem is how to determine the value of SD. To resolve this problem, we have to discuss the physical meaning of SD in relation to the stochastic source model. HIRASAWA (1979) and IZUTANI (1981) proposed a stochastic source model in the high frequency range. According to this model, the acceleration due to S-waves radiated from a large number of element faults consisting of a circular fault of an earthquake can be approximated by a random pulse sequence. The power spectral density of an acceleration is proportional to the root-mean-square (rms) of stress drop E(ƒÑ2), which is a determinant factor of amplitudes of high frequency waves. We assume that the average stress drop E(Tij) of element faults is equal to global stress drop ƒ ƒð and that the segment on the fault plane shown in Fig. 1 corresponds to the area of the element fault. Based on the similarity law of earthquakes, the global stress drop ƒ ƒð can be written using the displacement De of the element earthquake as follows: where C is a constant determined by elastic constants and AL is a characteristic

15 Semi-Empirical Method Using a Hybrid Fault Model 103 length of the element earthquake which is consistent with the length of a segment on the fault plane (Fig. 1). Based on the assumption that E(ƒÑij)= ƒ ƒð and relation (8), a stress drop ti; of each element fault can be given as, where Kij is the ratio of the deviation ƒ Dij from the average displacement Do to the displacement Do of the element earthquake (Fig. 1). Since E(Kij)=0 and E (9) as (Kij2)= SD, the rms stress drop E(ƒÑ2) can be easily obtained from Eq. (9) follows: The value of SD and the global stress drop du obtained for each earthquake are summarized in Table 2. The values of E(ƒÑ2)* are rms of stress drop calculated from SD and du using Eq. (10). For the 1968 and 1984 Hyuga-nada events, the SD value has large uncertainties, because the simulation of strong ground motions was not successful due to the problem of soft soil deposit as described above. Therefore, E(ƒÑ2)* is not calculated for these events and SD values for these events will not be used for the discussions below. HIRASAWA (1979), IZUTANI (1983), and SATO (1985) estimated the rms of stress drops E(ƒÑ2) from peak acceleration and power spectral density of observed accelerograms for some earthquakes. We find that the rms of stress drop E(ƒÑ2)* calculated from SD and ƒ ƒð are consistent with E(ƒÑ2) from observed accelerograms. This result indicates the validity of Eq. (10). According to Table 2, SD value shows 0.7 to 1.5 for 6 events, while d c ranges from 32 to 130 bar. The variations of VE(-ƒÑ2)* with SD and Au are defined as follows: (10) (11) (12) where f(ƒ ƒð, SD) =.E(ƒÑ2)*; and SD, sp, max, and SD min are the average, maximum, and minimum of SD respectively; and Au,.do-max, and ƒ ƒðmin, are the average, maximum, minimum of Au, respectively. Based on the values of SD and Au, the variation of rms of stress drop E(ƒÑ2)* is about 60% with SD value, and about 160% with the global stress drop ƒ ƒð using Eqs. (11) and (12). These results suggest that the value of SD may be fixed to the average value SD = 1.0 for predicting strong ground motions, because the variation of rms of stress drop is affected mainly by ƒ ƒð. 6. Conclusion A semi-empirical method is proposed, which is characterized by a con-

16 104 M. TAKEMURA and T. IKEURA sideration of heterogeneous faulting. This method is applied to the simulations of strong ground motion records at 18 stations during 8 earthquakes in and around Japan with magnitude M from 6.7 to 7.9. The results obtained can be summarized as follows. (1) The spectra obtained by the simulation provided us with satisfactory agreement for all events in the frequency range from 0.1 to 10 Hz. However, the wave envelope forms of the events of M> 7.5 are not consistent with the observed ones because of the complicated source processes of great earthquakes, while those of the events of M< 7.5 can be simulated satisfactorily. (2) By the present method, we failed to simulate the strong ground motions on thick, soft, soil deposits, which behave plastically during large earthquakes. This is because the present method is based on the theory of a linear system. (3) The amplitude of Ss(t) waves due to the deviation 4.Dij from the average displacement Do over the fault plane is much larger than that of SL(t) waves due to Do in the high frequency range. This indicates that fault plane irregularities are an important factor in the excitation of high frequency waves. (4) The present method requires that the size of the fault area of the element event be consistent with the size of the fault patch area of the large earthquake. According to the relation between the fault area of selected element events and the seismic moment of large earthquakes, we find that the synthetics in the present study are based on the assumption of the similarity law between fault patch area and seismic moment. (5) The value of SD is the standard deviation of ƒ Dij divided by the displacement De of the element earthquake, which describes the degree of heterogeneity of the displacement on the fault plane. SD values are obtained from 0.7 to 1.5 for 6 earthquakes. According to HIRASAWA (1979) and IZUTANI (1983), peak acceleration and power spectral density of acceleration is proportional to the rms of stress drop E(ƒÑ2) of the element fault (fault patch). The relation among E(ƒÑ2), SD, and global stress drop ƒ ƒð can be derived based on the assumption of the similarity law of earthquakes. Based on this relation, the variation of VE(ƒÑ2) is about 60% with SD value, while it is about 160% with ƒ ƒð. This indicates that the value of.\// E(ƒÑ2) is mainly determined by the value of du. Therefore, SD value may be fixed to the average Sp= 1.0 provisionally for predicting strong ground motions during large earthquakes. The authors wish to express their hearty thanks to Emeritus Prof. I. Muramatsu of Gifu University, Dr. K. Irikura of the Disaster Prevention Research Institute, Kyoto University, and Dr. T. Ohta of Kajima Institute of Construction Technology, Kajima Corporation, for their useful suggestions during the early stage of the present study. The authors also express their sincere gratitude to Dr. S. Noda of the Department of Civil Engineering, Tottori University, Mr. Y. Sawada of the Central Research Institute of Electric Power Industry, and Mr. E. Kurata of the Port Harbour Research Institute, Ministry of Transport, for allowing the use of magnetic tapes of digital records of displacement and acceleration.

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