ATTENUATION OF STRONG GROUND MOTION IN SHALLOW EARTHQUAKES

Transcription

1 3 th Wrld Cnference n Earthquake Engineering Vancuver, B.C., Canada August -6, 4 Paper N. 66 ATTENUATION OF STRONG GROUND MOTION IN SHALLOW EARTHQUAKES Símn ÓLAFSSON, Ragnar SIGBJÖRNSSON SUMMARY In this paper a theretical relatin is presented t mdel attenuatin f strng grund mtin. The relatin is based n the Brune spectra fr the near and far-field and uses parameters btained frm acceleratin recrds, such as seismic mment, spectral decay factr, duratin and stress drp. Attenuatin f hrizntal peak grund acceleratin as a functin f epicentral distance is presented fr bth hrizntal cmpnents. The theretical mdel is applied t PGA data frm tw Icelandic earthquakes (M w 6.6 and 6.5). Fr cmparisn the mdel is als applied t data frm Eurpean and Nrth-American earthquakes. The earthquakes are shallw (depth < 5 km) and with magnitude in the range M The recrds chsen are frm rck and stiff sil sites. The Icelandic earthquakes are strike-slip but the ther data cme frm nrmal and blique faults as well. The attenuatin curves fr the earthquakes are fund t have a similar slpe. The acceleratin levels are lwer by a factr f.7 fr the Icelandic data. Empirical attenuatin relatins fund in the literature are fund t prvide a pr fit t the data. INTRODUCTION Over the past decades many empirical attenuatin relatins, with the purpse f scaling strng-mtin acceleratin fr engineering purpses, have been presented. These attenuatin relatins are in mst cases similar in frm, with magnitude and distance frm surce t site as the independent variables. The parameters are estimated by fitting the relatins t the data (in mst cases PGA) by means f regressin analysis. When the different attenuatins relatins, that have been put frward in the literature, are examined, the disagreement between them is apparent (see, fr example, Duglas [] fr a recent review). This disagreement is partly due t the data sets frm which the mdels are derived and als due t different mdelling, prcessing and estimatin techniques. Senir scientist, University f Iceland, Earthquake Engineering Research Centre, Selfss, Iceland. E- mail: simn@afl.hi.is Prfessr, University f Iceland, Earthquake Engineering Research Centre, Selfss, Iceland. Ragnar.Sigbjrnssn@afl.hi.is

2 The attenuatin mdel used in this paper is derived theretically, althugh it has mdel parameters that can be determined empirically. It is based n Brune s near- and far-field mdels (see Brune [, 3]) and extended with an expnential term t accunt fr spectral decay at high frequencies (Olafssn and Sigbjörnssn [4, 5]). The theretical mdel is derived using Parseval s therem. A similar theretical apprach has als been derived fr respnse spectra (Snaebjörnssn et al. [6]). In this paper the theretical mdel, fr bth near- and far-field, is presented and then a cmparative study is made using data frm shallw strike-slip earthquakes in Iceland, June, f magnitude Mw 6.6 and 6.5 (see Sigbjörnssn et al. [7] and Thrarinssn et al. [8]). The mdel is als applied t PGA frm accelergrams btained in 7 earthquakes in Eurpe and Nrth-America. A cmparisn is then made f the results. STRONG-MOTION MODELLING The Brune mdel (Brune [, 3]) has been applied successfully t analyse Icelandic earthquakes and strng-mtin data (Ólafssn et al. [5, 9]; Ólafssn and Sigbjörnssn [4, ]; Ólafssn []). This mdel will als be used t derive grund mtin estimatin equatin in the fllwing and used t analyse strngmtin data and near surce effects. First sme preliminaries f the mdelling will be reviewed and the applied frmulas stated. The Brune mdel Seismic shear waves have been mdelled successfully by the s-called Brune mdel. It was was derived by cnsidering the effective stress needed t accelerate the sides f a circular causative fault n which a stress pulse is applied instantaneusly (Brune [, 3]). It is cmmnly used t btain fault dimensins frm spectra f shear waves fr small t mderate sized earthquakes (Udias []). The mdel describes nearand far-field displacement-time functins as well as spectra and includes the effect f fractinal stress drp. The near-field amplitude displacement spectrum is given as (Brune []): σβ D( ω ) = () µ ω ω + τ while the far-field rms displacement spectrum can be expressed as fllws (Brune []): σβ r D( ω) = R θφ () µ R ω + ω c Here, R θφ is the rms average f the radiatin pattern, ω dentes frequency in rad/s, β is the shear wave velcity, µ is the shear mdulus, r is the radius f the circular fault, R is the distance frm surce t site, ω c and τ are mdel parameters, respectively, the crner frequency and rise-time, given as (Brune [, 3]): ω c = 7π β 4 r (3) and

3 τ = O(r / β) (4) where the Landau symbl O indicates the prperties f the functinal relatinship. Finally, σ dentes the effective stress. If the effective stress des nt drp t zer but nly t a fractin f cmplete stress drp, the rise-time and high-frequency spectra are mdified especially in the lng-perid range (see Brune [, 3]). In the abve mentined studies f Icelandic earthquakes it is assumed that the effective stress equals the stress drp, i.e. σ = σ, where σ dentes the stress drp. Fr a duble cuple surce it can be shwn that the stress drp is related t the seismic mment, M, thrugh (see, fr instance, Udias []): 7 M σ = (5) 3 6 r Furthermre, it is assumed that the acceleratin spectrum can be derived frm the displacement spectrum by intrducing an expnential term t accunt fr spectral attenuatin at high frequencies (Andersn and Hugh [3]; Ólafssn []): A ( ω ) = ω D( ω) exp( κω) (6) where κ is the s-called spectral decay parameter. The spectral decay parameter is related t the quality factr Q thrugh the fllwing equatin: R κ = (7) βq The quality factr, Q, is in this cntext assumed t represent the average scattering and anelastic attenuatin ver the whle path. Studies f Icelandic strng-mtin data indicate that the spectral decay, κ, can be taken as cnstant (Ólafssn []), at least fr mderate epicentral distances, where the seismic wave field is dminated by shear waves and the Brune mdel is assumed t hld as an engineering apprximatin. The slw increase f κ with distance fund by Andersn and Hugh [3] is nt bserved in the available Icelandic data (Ólafssn []). This implies that the quality factr Q varies apprximately linearly with increasing distance frm the surce. This seems cnsistent with the fact that sites at great distance frm the surce are receiving shear waves that have penetrated thrugh lwer crustal layers with less attenuatin than the upper layers. Far-field apprximatin The acceleratin spectrum in the far-field can hence be expressed as fllws, accunting fr the freesurface effects and partitining f the wave energy int tw hrizntal cmpnents (Ólafssn []): CP R θφ M ( ω) = 3 4πβ ρr ω ( + ( ω/ ω A exp( c ) ) κω) (8a) Here, C P is the partitining factr, R θφ dentes the radiatin pattern and ρ is the material density f the crust. The fllwing expressin is suggested fr the gemetrical spreading functin (Ólafssn []):

4 D R = D n D n D D < D D < D D 3 (8b) where < n and R is a distance defined as: D + = d h (8c) Here, d is the epicentral distance and h is a depth parameter. The parameters D, D and D 3 are used t set the limits fr the different znes f the spreading functin. The first zne can be thught f as a crude apprximatin fr the intermediate field. Hence, the quantity D can be apprximated by h; D quantifies the size f the zne representing the intermediate field, which is related t the magnitude f the earthquake (as represented by the seismic mment) and the thickness f the seismgenic zne; while D 3 can be thught f as the distance where cylindrical waves begin t dminate the wave field. The time dmain prperties f acceleratin, a, can readily be derived frm the Furier spectrum, A, by applying the Parseval therem. This gives: T () ( ) π (9) I = a t dt = A ω dω The last integral can be evaluated after substituting Eq.(8). The result is: 3 /3 7 CP Rθφ σ I = 6 βρr Ψ M κ /3 () where Ψ dentes a dispersin functin given by the fllwing integral fr which a clsed frm slutin is readily btained: Ψ = λ 4 ϖ ( + ϖ ) e λϖ dϖ (a) Ψ = λ ci( λ) ( λ cs( λ) + 3sin( λ) ) λsi( λ) ( λ sin( λ) 3cs( λ) ) (b) Here, ci( ) and si( ) represent the csine and sine integrals, respectively, ϖ = ω/ ωc and: λ = κω c () where ωc is the crner frequency. The sine and csine integrals applied in Eq.() are given, respectively, as fllws:

5 π si( λ ) = + ci( λ ) = γ+ ln( λ ) + λ sin( t) dt t λ cs( t) dt t (3) where γ is the Euler cnstant (γ.577). Rms and PGA The rms grund acceleratin can be defined as fllws: T d arms = a () t dt T (6) d where T d is the duratin f shaking. A clsed frm attenuatin frmula can be derived using the Parseval therem and Eq. (6) and () abve. It is: 3 /3 7 CP Rθφ σ Ψ lg ( arms ) = lg + lg + lg ( M) lg ( R) 6 βρ κ Td 3 (7) The PGA can be related t the rms grund acceleratin by applying the thery f lcally statinary Gaussian prcesses (Vanmarcke and Lai [4]). The result is: a peak = pa (8) rms where p is the s-called peak functin, which depends n the strng mtin duratin, T d, and the predminant perid f the strng mtin phase f the acceleratin (Vanmarcke and Lai [4]; Hanks and McGuire [5]; Bre [6]). This relatin has been shwn t hld fr the available Icelandic strng-mtin data (see, fr instance, Ólafssn []). Hence, the same functinal frm as used fr the rms value can describe attenuatin f the peak grund acceleratin. Near-field apprximatin Rms and PGA The mdel described in the previus sectin is nt valid in the near-field and can, therefre, nt be expected t describe the peak grund acceleratin accurately clse t the fault. T be able t btain an apprximatin valid fr shear waves in the near-fault area, it is suggested that the Brune near-field mdel, Eq.(), is used. Hence, the near-field acceleratin spectrum can be apprximated as fllws, accunting fr the free surface and partitining f the energy int tw hrizntal cmpnents: A( ω) = 7 8 C p M ρβr 3 ω ω + τ exp( κ ω) (9) Here, κ is the spectral decay f the near-field spectra. Otherwise the same ntatin is used as abve. An apprximatin fr the rms and PGA is nw btained by applying the Parseval therem and, then, carrying ut the integratin. The result is:

6 7 C p Ψ lg ( arms ) = lg + lg 3 + lg ( M) 8 r π ρβ κ T () Here, the duratin is dented by T and Ψ is a dispersin functin given as: ϖ λϖ Ψ = λ e dϖ (a) + ϖ ( ci( λ) sin( λ) si( λ) cs( λ) ) Ψ = λ (b) where λ = κ / τ. It is seen that the PGA predicted by this equatin is independent f the epicentral distance and hence shuld give an estimate n the upper-bund f PGA. Anther result, which emerges when Eq.(5) is substituted int Eq.(), is that the RMS acceleratin is directly prprtinal t the stress drp. That is: a rms C p σ = π ρβ κ Ψ T () This indicates that assuming cnstant stress drp the grund acceleratin in terms f the rms value can decrease with increasing earthquake magnitude. NUMERICAL RESULTS In the fllwing the presented mdel is cmpared t tw different datasets: () Icelandic strng-mtin data frm Suth Iceland earthquakes and () shallw earthquakes frm Eurpe and Nrth-America, chse accrding t the cnditins f depth < 5 km and any magnitude in the range 6.4 t 6.6. The recrds chsen are frm rck r stiff sil sites. The data frm the Icelandic earthquakes is cmpsed f 98 hrizntal cmpnents f acceleratins frm tw earthquakes ccuring n June 7 th (M w 6.6) and June st (M w 6.5) (see Thrarinssn et al. [7]). The data frm the earthquakes are available in the ISESD database (Ambraseys et al. [6]). The data frm the Eurpean data are btained frm ISESD and the Nrth American data is btained frm the PEER database [7]. Icelandic strng-mtin data The data frm the earthquakes n 7 and f June are applied but have been scaled t fit the seismic mment f the 7 June earthquake. The far- and intermediate-field mdel is represented by the black curve while the near-field mdel is represented by the hrizntal dashed curve. Fllwing data are assumed: shear wave velcity, β = 3.5 km/s; density f rck, ρ =.8 g/cm 3 ; stress drp, σ = bar; average radiatin pattern, R θφ =.63; partitining parameter, C P = / ; peak factr, p =.94; spectral decay in the far-field, κ =.4 s, characteristic dimensin f the intermediate-field, R = 3 km; depth parameter, h = 9 km; expnent describing attenuatin in the intermediate-field, n = ; spectral decay in the near-field, κ =. s; characteristic fault dimensin (radius), r = 7. km; duratin used in near-field mdel, T = 3 r/β. These surce data give average slip equal t.5 m, which seems in fair accrdance with

7 mre refined estimates. The necessity f accunting fr the abve-mentined near-surce effects, which are significant fr distances shrter than km, is bvius The results btained fr the hrizntal PGA are displayed in Figure including data frm the tw abvementined earthquakes. The mdel fr the far- and intermediate-fields, represented by the slid black curves, is seen t fit the data reasnably well. Furthermre, the near-field mdel, given by the slid hrizntal line, appears t give sensible values fr the near-fault acceleratins. The errr r the standard deviatin, a measure f the uncertainty, has a value σ =.83. The mean value f the attenuatin given by the theretical mdel +/- ne standard deviatin is indicated n Figure as tw dashed lines. PEAK GROUND ACCELERATION (g) DISTANCE (km) Figure : PGA values frm 98 cmpnents f data frm the June earthquakes in Iceland. The slid black curve represent the mean value f the theretical attenuatin mdel. The dashed lines represent +/- ne standard deviatin. Strng-mtin data btained in shallw earthquakes in Eurpe and Nrth-America The theretical mdel is applied t the data frm the Eurpean and Nrth-American earthquakes. The selected data includes 7 earthquakes and recrds f hrizntal acceleratin (tw cmpnents fr each site). The result are seen is Figure. The standard deviatin is σ =.9 and is indicated with tw dashed lines n Figure. It is seen that the rate f attenuatin is similar as fr the Icelandic earthquakes. The same mdel parameters are used as fr the mdel except the spectral decay parameter f σ?=. s instead f σ?=.4 s. This is rughly speaking equivalent t lifting the curve in Figure by a factr.

8 PEAK GROUND ACCELERATION (g) DISTANCE (km) Figure : PGA values frm cmpnents f data frm seven Eurpean and Nrth-America earthquakes. The slid black curve represent the mean value f the theretical attenuatin mdel. The dashed lines represent +/- ne standard deviatin. DISCUSSION AND CONCLUSION The theretical mdel is seen t fit the PGA data frm the Eurpean and Nrth-American data equally well as the data frm the Icelandic earthquakes, as can see by cmparing Figures and. The parameters are the same as except κ =. instead f κ =.4 s as already mentined. This is seen the lift the curve by a factr apprximately equal t cmpared with the curve seen in Figure. The same frm f the attenuatin curve is seen t fit bth data sets equally well. There is hwever trade-ff between the mdel parameters σ and κ (see Bre et al. [8]). Different pair f σ and κ can therefre give the same results. Instead f lwering the κ values the radius culd be decreased, which increased the stress drp. The surce parameters have been estimated fr the tw Icelandic earthquakes using the acceleratin data with an estimatin prcedure similar t the prcedure described in Ólafssn et al. [5] and Ólafssn []. They agree well with the parameters used here as mdel parameters fr the attenuatin mdel. The parameters frm the ther data have nt been cmputed by the same prcedure. We knw, hwever, that values f κ =.3 s t.4 s and σ = 7 bar are fund t give a gd fit t Nrth-American data (see Bre [9]). The standard deviatin was smaller fr the Icelandic earthquake σ =.83 cmpared with σ =.9 fr the Nrth-American and Eurpean data-set. The nn-icelandic data is als seen t have a few utliers. The utliers are seen t bee arund epicentral distances f 4 and km. The utliers can pssibly be due t Mh bunce (see Smerville et al. [], Duglas []) Several f the empirical attenuatin relatins that have been presented in the literature have been applied t the Icelandic data frm the June earthquakes, and have been fund t give a pr fit t the data

9 (see Ólafssn and Sigbjörnssn []). This has als been the bservatin fr Icelandic earthquakes with lesser magnitudes (see Sigbjörnssn []). The empirical attenuatin relatins are als fund t be dependent n gegraphical regins, where the data riginates, which they are based n. In additin t this the empirical equatins d ften nt fit well t specific earthquakes. This is nt surprising cnsidering the fact that the regressin parameters are ften estimated frm earthquakes ver a wide magnitude range. Mre research is needed t determine if the theretical mdelling f attenuatin as is presented here can be useful fr hazard assessment. The mdel has t be applied t mre accelergrams earthquakes which are gruped accrding t type f mechanism, sil cnditins etc., in rder t estimate mdel parameters and determine attenuatin f grund mtin. ACKNOWLEDGEMENTS The wrk presented herein was supprted by a grant frm the University f Iceland Research Fund and EU cntract EVG-CT--73 (PREPARED). REFERENCES. Duglas J. Earthquake grund mtin estimatin using strng-mtin recrds: A review f equatins fr the estimatin f peak grund acceleratin and respnse spectral rdinates. Earth- Science Reviews 3; 6(-): Brune J. Tectnic stress and the spectra f shear waves frm earthquakes. Jurnal f Gephysical Research 97; 75: Brune J. Crrectin. Jurnal f Gephysical Research 97; 76: Ólafssn S and Sigbjörnssn R. A theretical attenuatin mdel fr earthquake-induced grund mtin. Jurnal f earthquake engineering 999; 3: Ólafssn S, Remseth S and Sigbjörnssn R. Stchastic mdels fr simulatin f strng grund mtin in Iceland. Earthquake engineering and structural dynamics ; 3; Snaebjörnssn, JTh, Sigbjörnssn R and Ólafssn S Mdelling f Earthquake Respnse Spectra fr Strike-slip Earthquakes in the Near- and Far-Field Prceeding f the 3 th Wrld Cnference n Earthquake Engineering, Paper N. 366, Vancuver, B. C., Canada, Thrarinssn Ó, Bessasn B, Snaebjrnssn J, Ólafssn S, Baldvinssn GI, Sigbjörnssn R. The Suth Iceland Earthquake in : Strng mtin measurements Prceedings f the Twelfth Eurpean Cnference n Earthquake Engineering, Lndn UK, Paper N. 3, Oxfrd: Elesvier Science,. 8. Sigbjörnssn R, Ólafssn S and Thrarinssn Ó. Strng-Mtin Recrdings in Iceland. Prceedings f the 3 th Wrld Cnference n Earthquake Engineering, Paper N. 37, Vancuver, Canada, Ólafssn, S, Sigbjörnssn R and Einarssn P. Estimatin f surce parameters and Q frm acceleratin recrded in the Vatnafjöll Earthquake in Suth Iceland. Bulletin f the Seismlgical Sciety f America 998; 88: Ólafssn S and Sigbjörnssn R. Attenuatin f strng-mtin in the Suth Iceland earthquakes. Prceedings f the th Eurpean Cnference n Earthquake Engineering, Lndn, UK. Paper n. 3. Elsevier Science,.. Ólafssn S. Estimatin f earthquake-induced respnse. Department f Structural Engineering, Nrwegian University f Science and Technlgy, Trndheim, Udias A. Principles f seismlgy. Cambridge University Press, Cambridge, 475, Andersn JG and Hugh SE. A mdel fr the shape f the Furier amplitude spectrum f acceleratin at high frequencies. Bulletin f the Seismlgical Sciety f America 984; 74:

10 4. Vanmarcke EH and Lai SP. Strng mtin duratin and rms amplitude f earthquake recrds. Bulletin f the Seismlgical Sciety f America 98; 7: Hanks TC and McGuire RK The character f high-frequency strng grund mtin. Bulletin f the Seismlgical Sciety f America 98; 7: Ambraseys NN et al. ISESD Internet Site fr Eurpean Strng-Mtin Data. Eurpean Cmmissin, Research-Directrate General, Envirnment and Climate Prgramme, PEER (Pacific Eearthquake Engineering Research Centre) Strng Mtin Database, Bre DM, Jyner WB and Wennerberg, L. Fitting the Stchastic ω surce mdel t bserved respnse spectra in western nrth America: Trade-ffs between σ and κ. Bulletin f the Seimlgical Sciety f America 99; 8(4): Atkinsn GM and Bre DM. Grund-Mtin Relatins fr Eastern Nrth America. Bulletin f the Seismlgical Sciety f America 995; 85(): Smerville P, & Yshimura J. The influence f critical Mh reflectins n strng grund mtins recrded in San Francisc and Oakland during the 989 Lma Prieta earthquake. Gephysical Research Letters 99; 7(8): Duglas J. A critical reappraisal f sme prblems in engineering seismlgy. PhD thesis, University f Lndn,.. Sigbjörnssn R, 99. Strng mtin measurements in Iceland and seismic risk assessment. Prceedings f Ninth Eurpean Cnference n Earthquake Engineering, vl. -A, pp. 5.