IMPROVEMENT OF THE SHAPE FUNCTION IN THE STOCHASTIC SIMULATION METHOD AND REPRODUCTION OF HIGH-FREQUENCY MOTION RECORDINGS OF THE KOBE EARTHQUAKE

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1 IMPROVEMENT OF THE SHAPE FUNCTION IN THE STOCHASTIC SIMULATION METHOD AND REPRODUCTION OF HIGH-FREQUENCY MOTION RECORDINGS OF THE KOBE EARTHQUAKE ABSTRACT: Masanori Horike 1 and Yoshihiro Onishi 1 Professor, Dept. of Architecture, Osaka Institute of Technoogy, Osaka, Japan Researcher, Geo-Research Institute, Osaka, Japan Emai: horike@archi.oit.ac.jp,onishi@geor.or.jp The stochastic method is idey used to simuate high-frequency motions. Hoever, it has a crucia defect that simuated high-frequency motions are underestimated in ampitude and duration except for near-faut zones. This is primariy caused by negecting the contributions from mutipy refected and transmitted aves and scattered aves. We remove this defect from the method in a simpe practica ay. We firsty introduce an epicentra-distance dependent shaping indo hich is inferred from borehoe data provided by KiK-net (Abbreviation of Kiban Kyoushin netork). We secondy introduce the radiation coefficients transiting from theoretica to isotropic vaues aong ith propagation. The vaidity of this improved stochastic method is demonstrated by reproducing three-component acceeration time histories of the 1995 Hyogo-ken Nanbu earthquake (Kobe) in ampitude, duration, and spectra at a the sites in the Osaka basin. KEY WORDS: Stochastic method, shaping indo, high-frequency motions, Kobe Earthquake 1. INTRODUCTION Wide-frequency strong ground motions are required for earthquake-hazard mitigation and earthquake-resistance design. Hoever, numerica simuation methods based on the easto-dynamic equations such as the finite-difference (FD) method, and the finite-eement method cannot generate high-frequency motions. The hybrid method (Kamae and Irikura, 1998) is, at present, a unique practica method for computing ide-frequency strong ground motions. In genera, the hybrid method sums o-frequency motions computed mosty by the FD method (For exampe, Graves, 1996) ith high-frequency motions computed by the stochastic method (Boore, 1983). Hoever, there are severa drabacks in the stochastic method. The first draback is that it generates ony a singe horizonta component. This draback has aready removed by Onish et a. (004). A crucia draback sti remained is that simuated motions rapidy decrease in ampitude and duration ith increasing the epicentra distance. This draback arises from negecting mutipy refected and transmitted aves and scattered aves generated aong ith propagation in inhomogeneous media.these aves have strong effects on high-frequency motions not ony by ay of reduction of ampitude decay (compared ith ampitude decay in homogeneous media), proongation of the duration but aso by ay of smear of the radiation coefficients. Therefore, the smear of the radiation coefficients is aso required to introduce into the method. The aim of this paper is to improve the stochastic method incorporating the effects of mutipy refected and transmitted aves and scattered aves and the smear of the radiation coefficients. The vaidity of this ne method is demonstrated by reproducing acceerations of the Kobe earthquake at 8 sites in the Osaka Basin.. IMPROVEMENT OF THE SHAPING WINDOW AND THE RADIATION COEFFICENTS. Inference of the parameters for the shaping indo It is difficut to numericay simuate high-frequency motions, because it requires a precise minute crusta mode. Thus, an aternative approach is taken in this study. Instead of the fixed shaping indo used in the conventiona method, e derive an epicentra-distance dependent shaping indo from the borehoe recordings provided by the KiK-net netork.

The shaping indo used in the stochastic method is expressed as s( t) γ α t / = λt e, (1) here parameters γ and α respectivey contro the shape of the initia and the ater portions of S aves, and γ + 1

2 The shaping indo used in the stochastic method is expressed as s( t) γ α t / = λt e, (1) here parameters γ and α respectivey contro the shape of the initia and the ater portions of S aves, and γ λ = ( α Γ ( γ + 1)). We inferred to parameters γ and α from acceeration recordings acquired at the borehoe sites of the KiK-net netork. At these sites three-component acceerometers ere instaed on ground surface and in the borehoe at a depth deeper than 100m. We seected the borehoe sites, here the S ave veocity is above 1.7km/s, to reduce the contamination of surface aves as much as possibe. The earthquake events and the KiK-net sites used in this study are shon in Figure 1. The Osaka Basin is surrounded ith these events and sites. The foca depths of these events are beteen 8km and 70km and the moment magnitudes are beteen 3.7 and 5.6. The method for inferring the to parameters is the same as the method in Saragoni and Hart (1974). The estimates of parameter γ are not dependent on the epicenta distance and the magnitude, but scatter in the range beteen 0.1 and.5. The estimates of parameter α decrease ith increasing the epicentra distance, suggesting that the duration increases ith increasing the epicentra distance. Figure 1: Earthquakes (Circes) and Kik-net sites (Trianges) used for the inference of the parameters for the shaping indo. These events occurred in southest Japan incuding the Osaka Basin (Rectange). We have aready obtained the estimates for the to parametersγ and α. Hoever, e derive the regression formua not for the to parameters γ and α but for ne parameters ε and T, because the ne parameters are obvious in physica meanings; specificay, T denotes the duration of the shaping indo, and parameter ε denotes the ratio of time for the peak of the shaping indo to the duration (Figure ). Figure Schematic diagram of the shaping indo for interpretation of parameters η, ε, and T.

3 To do this, e convert parameters γ and α into parameters ε and T using the reations (Boore, 1983) { 1+ ε (n( ) 1) } γ = ε n( η) ε ( γ ) ( ) α = εt (3) We empoy the same vaue of 0.05 for parameter η as that in Boore (1983). Equation means that parameter ε is aso a fixed vaue because parameter γ is a fixed vaue. Hoever, parameter ε is not determined as a specific vaue, but is confined in the range beteen 0.01 and 0.35 hich corresponds to the range for parameter γ beteen 0.1 and.5. We derive the regression formua for parametert. For this purpose the estimates of parameter α are converted into durationt, using equation (3). For the case of ε = 0., correspondingy γ =. 5, converted durations T are shon in Figure 3. As noted, durations T amost ineary increase ith the epicentra distance, though they scatter idey. This hods true for the other earthquakes. T/ (s) M=4.0 EW NS UD M= M= M=5.6 T/ (s) Distance (km) Distance (km) Figure 3 Parameter T converted from parameter α on an assumption of η = and ε = 0.. Assuming that duration T is a inear function for the epicentra distance r, it is specified as T = c + Kr. (4) We estimated proportiona constants K for a the earthquake by the east square fitting. Because the estimates of K appear to decrease ineary ith the magnitude and, ess ikey, the foca depth, e derive the regression formua of K from their estimates as a inear function of the foca depth Z in unit of km and the moment magnitude M. The fooing formua is obtained, 3 K = ( Z ). (5) M

4 Since duration T is a sum of duration of radiated puse increment aong ith propagation, it is expressed as, fc, here f c is the corner frequency, and an T = f + fct K ( r r ) (6) c off here term fct, a factor dependent on parameter ε, is specified as /(1 + ε (n( ε ) 1)). Considering that the conventiona method can reproduce earthquake motions in cose-faut zones (Onishi et a., 004), e introduce r off and set it at 10 km in this study. Putting a specific vaue of parameters ε in the range beteen 0.01 and 0.35 and duration T cacuated from the regression formua (6) into equations () and (3), parameters γ and α are determined. We obtain the epicenta-distance-dependent shaping indo by putting these ne parameters into equation (1). This ne shaping indo can proong the duration but sti underestimate ampitude. Thus, e introduce another term into equation (1) in compensation for ampitude underestimation. The shaping indo is expressed in a form of n γ T α s( t) = λ t exp( t) f, (7) c here n does not exceed unity because a peak ampitude of the ne shaping indo, hich is shaved off due to scattering, is ess than that of the conventiona (or the fixed) shaping indo. Numerica experiments suggest that n is appropriate in the range beteen 0.5 and Radiation coefficients The inhomogeneity of the propagating media affects the radiation coefficients as e. Aong ith propagation, SH SV P the radiation coefficients, F, F, and F change from theoretica vaues dependent on the take-off ange to isotropic vaues expressed in equations (8) and (9) as SV P ( F ) = 4 15 (8) SH ( F ) + ( F ) = / 5. (9) The manner of the transition is controed by the statistica properties of the inhomogeneity. Hoever, at present, e ack sufficient knoedge of these properties. Thus, e assume that the radiation coefficients change as foos if if if D < LW t LW < D 1 D > L W t 1 t < L W theoretica vaue Interpoation beteen theoretica and isotropic vaues isotropic vaue, (10) here Dt and W denote the trave distance and the aveength. In the computations, e assume that the factors L1 and L are set at 1 and 5 for P and S aves. The isotropic radiation coefficients for SH and SV aves are not the individua isotropic vaues, but share the same isotropic vaue of 1 5 derived from equation (9). We incorporate the path effects such as geometric spreading and damping due to absorption into the method. Specificay, the geometrica spreading factors, R p and R s, for P and S aves are evauated by the dynamic ray theory (Červený et a. 1977) and the damping is evauated aong the ray paths using frequency- and depth-

5 dependent Q vaues..4 Computation procedure The moment rate function for the ne stochastic method is derived by convoving the correction function (Onish et. a., 004) ith the ne shaping indo specified by equation (7) Using this moment rate function, e obtain acceeration time histories for the three types of body aves propagating from each sub-faut to the top of the crust. Their Fourier spectra for SH aves are expressed as A ~ 0.5 SH M ( ) c ( f, r) exp( iπft s ) ρ = 0β 0 F π f r, f ) exp R dz, ρ β r 4πρ β ( ) s 1 1 S β Q 0 0 s SH ( 5 ~ here M c ( f, r) is the Fourier transform of the moment rate function of a subfaut of a arge earthquake event 3 3 d M c ( t, r) dt. Parameters ρ 0, α 0, and β 0 are the density, the P-ave veocity, and the S-ave veocity in the source region, and ρ 1, α 1, and β 1 are the density, the P-ave veocity, and the S-ave veocity at the uppermost part of the crust. Q p and Q s denote quaity factors for P and S aves aong the seismic ray paths rp and r s. Simiar expression for SV and P aves are obtained by repacing severa arguments ith suitabe ones. The effects of the sediments are introduced into the method as the same ay described in Onishi et a. (004). 3. SIMULATION OF KOBE EARTHQUAKE In this section e examine the vaidation of the method by comparing simuated motions ith strong motion recordings of the 1995 Hyogo-ken Nanbu (Kobe) earthquake at a the sites in the Osaka basin depoyed by the Committee of Earthquake Observation and Research in the Kansai Area (Abbreviated as CEORKA). 3.1 Sites and Source Mode We compute acceerations at 8 sites in the Osaka sedimentary basin (Figure 3) ith the ne stochastic method. As described in the previous section, a one-dimensiona (1-D) subsurface structure mode is required at each site. Fortunatey, a three-dimensiona (3-D) subsurface structure mode of the basin as proposed by Yamada and Horike (007). We pick out 1-D modes for the 8 sites from this 3-D mode. We adopt the earthquake source mode in Yamada et a. (000). This source mode has 4 asperities and their geometries are shon in Figure 4. The rupture as initiated from the bottom of the boundary beteen asperities 1 and 3, being denoted by the star symbo in the Figure, and after 8s the rupture of asperity 4 as initiated from the eft oer corner. Figure 3: Map of the faut surface projection of the Kobe Earthquake (shaded area), and the 8 sites (circes).

$Figure 4: Source mode. The four rectanges are the asperities. The star is initiation point of the fracture.$

6 Figure 4: Source mode. The four rectanges are the asperities. The star is initiation point of the fracture. Figure 5: Comparison of the simuated (upper three traces) acceeration traces ith the recorded (oer three traces) acceeration traces at the 8 sites in the Osaka Basin. 3. Comparison Figure 5 shos the comparison of acceeration time histories simuated by the improved stochastic method for ε = 0.15 ith strong motion recordings at 8 sites (See Figure 3). Simuated motions are generay simiar to recorded motions in ampitude and duration for the three components at each site. Refecting the agreement in the time histories, the acceeration spectra are aso simiar to each other at every site. This indicates that the improved method is a practica usefu too for simuation of high-frequency earthquake motions.

7 4. DISCUSSION We shoed that the improved method fairy e reproduced three-component acceeration recordings of the Kobe Earthquake at a the sites in the Osaka Basin. Hoever, a coser ook of Figure 5 reveas that the simuated vertica motions at cose faut site Mot are significanty sma in ampitude, compared ith recorded motions. The spectra anayses sho that the cause for this is the deficiency of the vertica spectra beo 0.8 Hz. Thus, e discuss the deficiency of the vertica o-frequency spectra in the fooings. We at first began ith adding an extra asperity at various ocations on the faut surface. Hoever, these trias did not yied any improvement in the vertica spectra at site Mot, suggesting that faut modes are not responsibe for the deficiency of o-frequency vertica spectra. Besides, the radiated spectra may not be a cause either, because the to horizonta motions are reproduced e at a the sites incuding site Mot. As a resut, e suppose that a convincing cause is the negect of the near-fied and the intermediate-fied terms of the eastic ave Green s function. We sho that this cause can accounts for the deficiency of the o-frequency vertica spectra at sites cose to a faut such as site Mot Figure 6: Comparison of the contributions from the terms of the eastic ave Green s function radiated from a point source. The four eft panes are for the radia component and the four right panes are for the transverse component. The abbreviations are as foos; N: Near-fied term motion, IP: Intermediate-fied term P motion, IS: intermediate-fied term S motion, FP: Far-fied term P motion, FS: Far-fied term S motion, and Tota: Sum of near-fied, intermediate-fied, and Far-fied term motions. Using equations (4.3) in Aki and Richards (1980), e computed the veocity aveform for the individua term of the eastic ave Green s function at an epicentra distance of 5km in the homogeneous media (Figure 6). The

8 radiation coefficients are negected from these aveforms because equation (4.33) in Aki and Richards (1980) ensures that the radiation coefficients are common to the terms. The source parameters for the point source ere 16 as foos; the seismic moment of 5 10 Nm, the stress drop of 8 MPa, and the high frequency imit of 10 Hz. The radia motions due to the far fied P (FP) term in the eft third pane are appreciaby sma compared ith the tota radia motions in the eft bottom pane due to the four terms of the near-fied (N) term, the intermediate-fied P (IP) term, the intermediate-fied S (IS) term, and the far-fied P (FP) term. The tota radia motions are rather simiar to the summation motions in the second pane due to the N term, the IP term, the IS term This accounts for the sma vertica ampitude of the simuated time history at site Mot, noting that the vertica motions in cose-faut zones are amost the same as the radia motions. More specificay, since the contributions from the N, IP and IS terms are not taken into account in the stochastic method, it underestimates vertica motions in cose-faut zones The spectra of the far-fied term are appreciaby sma beo 0.8Hz, compared ith the spectra of the tota motions. In contrast, the transverse motions due to the far-fied S (FS) aone in the right third pane are quite simiar in ampitude and aveform to the tota transverse motions in the right second pane due to the NS, IP, IS, and FS terms, because the contribution from the N, IP, and IS term are significanty sma, compared to the contribution from the FS term (See and compare the right second and the right third panes). Noting that horizonta motions in cose-faut zones is amost the same as the transverse motions, the above resuts account for the fact that the stochastic method reproduces the horizonta motions at cose-faut site Mot fairy e, despite the underestimation of the vertica motions. 5. CONCLUSION We incorporated the effects of inhomogeneity of the propagating media such as the reduction of the ampitude decay, the proongation of the duration, and the smear of the radiation coefficients into the method. This ne method successfuy reproduced the acceeration time histories and the spectra for the three components of the Kobe Earthquake at a the sites depoyed by CEORKA in the Osaka basin. We think that this ne stochastic method becomes a practica usefu too to generate high-frequency earthquake motions not ony in cose-faut areas but aso ide areas encompassing earthquake fauts. REFERENCES Aki, K., and P. G. Richards (1980). Quantitative Seismoogy: Theory and Method, Vo. 1, W. H. Freeman, San Francisco. Boore, D. M. (1983). Stochastic simuation of high-frequency ground motions based on seismoogica modes of the radiated spectra, Bu. Seism. Soc. Am., 173, Červený, V., I. A. Mootokov, and I. Pšenčík (1977). Ray method in seismoogy, Univerzita Karova Praha. Graves, R.W. (1996). Simuating seismic ave propagation in 3D eastic media using staggered-grid finite difference, Bu. Seism. Soc. Am., 86, Kamae, K. and K. Irikura (1998). Source mode of the 1995 Hyogo-ken Nanbu Earthquake and simuation of near-source ground motion, Bu. Seism. Soc. Am., 88, Onishi, Y. M. Horike, and Y. Kaamoto.(004). A method for simuating three-component, near-faut, strong ground motions using stochastic Green s function, DVD in 13th Word Conference on Earthquake Engineering, Paper No Saragoni R., and G. Hart (1974). Simuation of artificia earthquakes, Earthquake Engineerings and Structura Dynamics,, Yamada, K., and Horike, M. (007). Inference of Q-Vaues beo 1 Hz from borehoe and surface data in the Osaka basin by three-component aveform fitting, Bu. Seism. Soc. Am., 97, No.4, Yamada, M., T. Hirai, T. Iashita, K. Kamae, K. Irikura (000). Revision of the Hyogo-ken Nanbu earthquake source mode, Abstract and Programme of 000 Fa Meeting of the Seismoogica Society of Japan, A46.

RISKY HIGH-RISE BUILDINGS RESONATING WITH THE LONG-PERIOD STRONG GROUND MOTIONS IN THE OSAKA BASIN, JAPAN

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