A PROCEDURE TO MODIFY THE FREQUENCY AND ENVELOPE CHARACTERISTICS OF EMPIRICAL GREEN'S FUNCTION. Lin LU 1 SUMMARY

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1 A POCEDUE TO MODIFY THE FEQUENCY AND ENVELOPE CHAACTEISTICS OF EMPIICAL GEEN'S FUNCTION Li LU SUMMAY Semi-empirical method, which divides the fault plae of large earthquake ito mets ad uses small groud motio ords as empirical Gree's fuctio, is ofte used to sythesize the groud motios of large earthquakes. The method adjusts the amplitude of the groud motio ords to satisfy the geometrical atteuatio for differet focal distaces betwee the ords ad the mets. The differet focal distace, however, chages ot oly the amplitude but also the frequecy cotets ad evelope shape of groud motio. A procedure that modifies the frequecy cotets ad evelope shape of the groud motio ords accordig to the relatio of the focal distaces of the ords ad the met is suggested. The Q factor ad the respose spectrum obtaied by atteuatio relatioship were used to modify the frequecy cotets. The groud motios of a sceario earthquake were sythesized as a example. The modificatio by the Q factor ad that by the respose spectrum gave similar results from the viewpoit of respose spectrum of the sythesized groud motio. Based o the cocept that the evelope ca be represeted by Fourier phase spectra give by the itegratio of group delay time, the stadard deviatio of the group delay time was employed to modify the evelope shape for differet focal distaces of the ords ad mets. With the modificatio of the frequecy cotets ad evelope shape, the frequecy cotets ad evelope shape are ot the same for each met ad a more realistic groud motio ca be sythesized INTODUCTION Groud motio time history is ecessary for the seismic desig of critical structures whe the dyamic respose of them is performed. The time history ca be strog groud motio ords if it fulfil the desig earthquake load or artificially sythesized groud motio if the required groud motio is ot available. For sythesizig of groud motio, stochastic process method ad semi-empirical method are ofte used. The stochastic process method, which is the summatio of a series of siusoidal waves, is simple but difficult to give a earthquakead site-specific solutio. The semi-empirical method, which divides the fault plae of a large earthquake ito small mets ad uses the groud motio ords as empirical Gree' fuctio to cosider the fault rupture process ad the path effects, are possible to icorporate the source, propagatio ad local site effects. The semi-empirical method origiated from Hartzell [978], where the groud displacemet of a maishock was sythesized by the ords of a aftershock, is ofte used to sythesize the groud motios of large earthquakes for egieerig purposes. There are several versios of semi-empirical method. I this paper, the procedure proposed by Irikura [986] was employed. Accordig to Irikura, the groud motio is sythesized by a sy ( ) ( ) ( t) a t t + a t tk () i j k Tokyo Electric Power Services Co., Ltd Higashi-Ueo, Taito-Ku, Tokyo 0-005, Japa, lu@aed.tepsco.co.jp

2 where a sy (t) is the groud motio to be sythesized, a (t) is small groud motio ords, N is scalig factor, ad are the focal distaces of the groud motio ords ad met, respectively. Applig Fourier trasform to Equatio (), it becomes iφ ω iωt iωt ( ) k Fsy A e e + e () i j k where A ( ad φ ( are the Fourier amplitude ad phase agle, respectively. I Equatio (), the Fourier amplitude is modified by the ratio of the focal distace of the ords ad that of the met to satisfy the geometrical atteuatio of seismic wave ad the frequecy cotets ad evelope shape are idetical for all of the mets. O the other had, the differece of focal distace chages ot oly the amplitude but also the frequecy cotets ad evelope shape due to the physical absorptio of seismic waves, scatterig ad dispersio effects durig the propagatio of seismic wave. For a large earthquake, whose fault dimesio could be more tha a hudred kilometers, the differeces of the frequecy characteristics ad evelope shape caused by the differet focal distace amog mets might ot be egligible. The physical absorptio, which is caused by the imperfect elasticity of the medium, ca be measured by the Q factor. The scatterig caused by ihomogeeities is difficult to cosider with mathematical expressios. Izumi ad Katukura [983] have poited out that there was a close relatioship betwee the evelope shape ad the stadard deviatio of the group delay time of groud motio. I order to take ito accout the physical absorptio ad scatterig effects, a procedure, which modifies the frequecy cotets ad evelope shape for each met with the Q factor, the respose spectrum ad the stadard deviatio of group delay time, is suggested i this paper. MODIFICATION OF THE FEQUENCY CONTENTS Two methods were cosidered to modify the frequecy cotets of groud motio ords. Oe uses the Q factor ad aother makes use of the respose spectrum obtaied by atteuatio relatioship. While the modificatio by the Q factor is mathematically explicit ad simple, the modificatio by the respose spectrum is based o the observatio data. Modificatio by the Q factor The semi-empirical method requires a appropriate groud motio ords at the site to serve as the empirical Gree's fuctio. I order to estimate the differece of the frequecy cotets betwee the ords ad the Gree's fuctio of a met, a statistical model that gives the estimatio of the Fourier amplitude of accratio for a site with focal distace A CAS AL (3) was used. I Equatio (3), C reflects the effects of radiatio patter ad is a costat for a site, A S (, A P ( ad A L (ω ) deote the source spectrum, atteuatio effects ad local site amplificatio, respectively, ad the atteuatio effects is represeted by ω exp (4) QCs where C S is the velocity of shear wave. Although we kow its Fourier spectrum for the groud motio ords, it is difficult to separate it ito source spectrum, atteuatio effects ad local amplificatio clearly. Suppose the Fourier amplitude of the groud motio ords ca be represeted by Equatio (3), it is expressed as A C AS AL (5) I the same way, the Fourier amplitude for a met is give by A C AS AL (6) Sice the C, A S ( ad AL (ω ) ca be cosidered approximately the same for the ords ad the Gree's fuctio of met, the ratio of Equatio (6) to Equatio (5) becomes

3 A (7) A Substitutig Equatio (4) ito Equatio (7) results i ω( ) exp α( (8) QC S The ratio of the focal distaces i Equatio (8) has bee icluded i Equatio () ad α ( is eglected there. As we kow, the Q factor ad C S are strogly area depedet. I practical applicatio, the Q factor ad C S of that area should be used. A value of 3.0 km/sec for the shear wave velocity ad the results of the Q factor from Kiyoo [99] ω log Q 0.7 log( ) +. (9) (9) π were used here as a example. Equatio (9) is a average relatioship for Japa. Figure shows the chages of α ( with respect to ω for the cases of - beig equals to -50, 0 ad 50 km. From Figure, cosiderable Figure : Differece of the atteuatio of Fourier amplitude differeces amog α ( 's are for differet focal distace based o the Q factor observed i high frequecy rage. This differece ca be simply cosidered i the semi-empirical method by itroduce α( ito Equatio (), amely F sy iφ ω iωt iωt ( ) k α( A e e + e (0) i j k Modificatio with the shape of respose spectrum The modificatio by the Q factor takes ito accout the physical absorptio oly. It ca ot cosider the scatterig effects due to the ihomogeeities ad the local site effects if the ords ad the sythesizig site are ot the same site. I order to cosider the scatterig ad local site effects as well, the respose spectrum obtaied from atteuatio relatioship was used. There are a umber of atteuatio relatioships available ad a site specific atteuatio relatioship is preferred. I this paper, the atteuatio relatioship proposed by Aaka ad Nozawa [988] was used to explai how the frequecy cotets are modified. Their atteuatio relatioship is expressed i the form log S( T ) Cm ( T )( M ) Cd ( T )log( ) + Ch ( T ) H + C0 ( T ) () where T is atural period, M is magitude, H is focal depth, X exp(0.65M ) ad X is focal distace. Whe Equatio () is used for the evet whose ords are used as the empirical Gree's fuctio, we have log S ( T ) Cm ( T )( M ) Cd ( T )log( ) + Ch ( T ) H + C0 ( T ) () ad for a met, its respose spectrum is log S ( T ) Cm ( T )( M ) Cd ( T )log( ) + Ch ( T ) H + C0 ( T ) (3) The differece of the respose spectrum betwee the met ad the ords is give by S S ( C d ( T )log + C h ( T )( H H ) T ) 0 ( T ) ω β( ) π (4) 3

4 The ratio, otig as β (, is the fuctio of the focal distace ad focal depth. Figure shows the chages of β ( i the cases of / beig equals to 0.5, 0 ad.0 with H -H 0. For / equals to.0, β( is comparatively costat for differet frequecies. There is a cosiderable differece betwee / 0.5 ad.0 for the frequecy rage from 0.5 to 0 Hz. Figure 3 shows the chages of β ( i the cases of / beig equals to 0 ad H -H -0, 0 ad 0km. Comparig with the chages due to focal distace, the chages due to focal depth is mior. β ( does reflect the differece of the frequecy cotets betwee the ords ad met but, strictly speakig, ot ditly the differece of their Fourier amplitudes. Now we assume Equatio (4) ca be approximately applied to the Fourier amplitude. I this regard, β ( ca be used to modify the frequecy cotets of a met. Thus, Equatio () becomes iφ ω iωt iωt ( ) k Fsy β( A e e + e (5) i j k I Equatio (5), The modificatio by / is ot ecessary because the geometrical atteuatio is already icluded i the atteuatio relatioship. MODIFICATION OF THE ENVELOPE SHE OF GOUND MOTION Evelope shape or duratio of groud motio will chage as focal distace chages. I stead of ditly cosiderig the evelope shape, the cocept that the evelope ca be represeted by Fourier phase spectra give by the itegratio of group delay time,, was used. The group delay time is a differetial of Fourier phase spectra i frequecy domai. dφ( (6) dω The average ad the stadard deviatio of is defied as µ σ E E ω N 0 ωn 0 A ( dω A ( ( µ ) dω Izumi ad Katukura [983] poited out that the average ad the stadard deviatio of of strog motios for give frequecy widow i frequecy domai correspod to the cetroid ad the duratio of strog motios filtered with the same widow i time domai, respectively. Satoh et al. [997] foud that both the average ad the stadard deviatio have a close relatioship with the epicetral distace. The regressio models they suggested are µ bµ D (9) σ bσ D (0) where D is epicetral distace. Because the average of group delay time relates to the arrival of seismic wave ad it has bee cosidered i Equatio (), the relatioship betwee the stadard deviatio ad the epicetral distace was used to modify the evelope shape. Figure : Differece of the atteuatio of Fourier amplitude for differet focal distaces based o t h Figure 3: Differece of the atteuatio of Fourier amplitude for differet focal depth based o (7) (8) 4

5 With Equatio (0), the stadard deviatio of group delay time for the ords is estimated as σ b σ D () At the same time, the stadard deviatio of group delay time for a met is σ b σ D () Takig the ratio of Equatio () to Equatio (), oe obtais σ D γ (3) σ D The ratio depeds oly o the epicetral distace ad is ot the fuctio of frequecy, idicatig that the evelope shape chages simply as the epicetral distace chages. From Equatio (3), the stadard deviatio for a met is calculated by σ γσ (4) From statistical theory, Equatio (4) is satisfied if the group delay times of the ords ad met have the relatio: γ (5) I practical case, because FFT is used to perform the Fourier trasform, the group delay time is calculated by the differece istead of the differetial. φi+ ( φi i (6) ω From Equatios (6) ad (5), the phase agles of a met ca be obtaied by φ + ω γ ( ) (7) φ i + i i ω Figure 4: Differece of the evelope shape of time history for differet focal distaces The chage of the evelope shape, as a example, is show i Figure 4 for γ 0.5,.0 ad.0, where γ.0 is origial groud motio. I the case of γ 0.5, the duratio becomes shorter ad the peak value becomes larger comparig with the origial groud motio. O the cotrary, whe γ.0, the duratio becomes loger ad the peak value becomes smaller. This coicides with the physical pheomeo. 5

6 Whe calculatig the phase agle with Equatio (7) for each met rather tha usig the origial phase agles for all the mets i the sythesizig of strog groud motio, the chage of evelope shape due to the differet focal distaces amog the mets ca be cosidered. SYNTHESIZING OF GOUND MOTION WITH MODIFIED GEEN'S FUNCTION The strog groud motios of a sceario earthquake were sythesized with modifyig the frequecy cotets of empirical Gree's fuctio by the Q factor ad the respose spectrum. The sythesized groud motio time histories ad their respose spectra were compared with those without the modificatio. The parameters of the earthquake are listed i Table. The groud motio ords from a earthquake with magitude of 6.6 ad focal distace of 86 km appeared i Figure 4 was used as the empirical Gree's fuctio. Accordig to the scalig law, the fault plae of the sceario earthquake was divided ito 4 by 4 mets. The locatio of the site ad fault plae is show i Figure 5. Table : Parameters of sceario earthquake Magitude Fault legth Fault width Fault depth Slip ise time Shear velocity upture velocity km 50 km 0 km 3. m 3. sec 3.0 km/sec. km/sec The Sythesized groud motio time histories with the modificatio of the frequecy cotets by the Q factor ad the respose spectrum ad that without the modificatio are show i Figure 6. The peak values of the time history with modificatio are about 30% lager tha that without modificatio. The respose spectra of the sythesized groud are compared i Figure 7. The modificatios by the Q factor ad the respose spectrum have close results. Comparig with the respose spectrum without modificatio, the differece is small for log periods ad large for short periods. This is because the focal distace of the groud motio ords is loger tha that of the almost mets. I this case, whe the groud motio ords is used as it is, the seismic waves are overatteuated for the mets who has the shorter focal distace tha the ords. The over-atteuated effects are corted whe the modificatio of the frequecy cotets is performed. It is cosidered that the modificatios of the frequecy cotets ad the evelope shape ca result i more realistic respose spectrum ad duratio for sythesized groud motio. CONCLUSIONS Figure 5: Locatio of the site ad fault plae A procedure that modifies the frequecy cotets ad evelope shape of the Gree's fuctio by the Q factor, the respose spectrum obtaied by atteuatio relatioship ad the stadard deviatio of group delay time accordig to the relatio of the focal distace betwee the ords ad the met is suggested. Followig this procedure, the frequecy cotets ad evelope shape of the Gree's fuctio are ot the same for each met, which, comparig with the covetioal procedure, ca result i more realistic groud motio. Site 6deg. 50km km Cross sectio of fault plae 87km Projectio of fault plae 0km 00km 5deg. 6

7 EFEENCES. Aaka, T, ad Y. Nozawa (988), "A probabilistic model for seismic hazard estimatio i the Kato district", Proceedigs of 9 th world Coferece o Earthquake Egieerig, Vol.II, pp Hartzell, S.H. (978), "Earthquake aftershocks as Gree's fuctio", Geophysical esearch Letters, Vol.5, pp Irikura, K. (986), "Predictio of strog accratio motios usig empirical Gree's fuctio", Proceedigs of the 7 th Japa Earthquake Egieerig Symposium, pp Figure 6: Sythesized groud motios with ad without frequecy cotets modificatio 4. Izumi, M ad K, Katukura (983), "A fudametal study o the phase properties of seismic waves", Joural of Structure Costructio Egieerig, Vol.37, pp Kiyoo, J. (99), Idetificatio ad sythesis of seismic groud motio i structural respose aalysis, PhD Thesis, Dept. of Civil Egieerig, Kyoto Uiversity. 6. Satoh, T. et al. (996), "A study o evelope characteristics of strog motios i a period rage of to 5 secods by usig group delay time", Proceedigs of 9 th world Coferece o Earthquake Egieerig, Paper No. 49. Figure 7: espose spectra of the ords ad 7