FINAL TECHNICAL REPORT AWARD NUMBER: 08HQAG0115

FINAL TECHNICAL REPORT AWARD NUMBER: 08HQAG0115 Ground Moton Target Spectra for Structures Senstve to Multple Perods of Exctaton: Condtonal Mean Spectrum Computaton Usng Multple Ground Moton Predcton Models PI: Jack Baker Report co-author: Tng Ln (Stanford Unversty) Dept. of Cvl & Envronmental Engneerng Yang & Yamasak Envronment & Energy Buldng 473 Va Ortega, Room 283 Stanford CA 94305-4020 650-725-2573 (phone) 650-723-7514 (fax) bakerw@stanford.edu February 2009 Research supported by the U. S. Geologcal Survey (USGS), Department of Interor, under USGS award number 08HQAG0115. The vews and conclusons contaned n ths document are those of the authors and should not be nterpreted as necessarly representng the offcal polces, ether expressed or mpled, of the U. S. Government. 1

Abstract Probablstc sesmc hazard analyss (PSHA) combnes the probabltes of all earthquake scenaros wth dfferent magntudes and dstances wth predctons of the resultng ground moton ntensty, n order to compute sesmc hazard at a ste. PSHA also ncorporates uncertantes n those ground moton predctons, by consderng multple ground moton predcton ( attenuaton ) models (GMPMs). Current ground moton selecton uses the nformaton from earthquake scenaros wthout consderng multple GMPMs. Ths paper descrbes ways to ncorporate multple GMPMs, usng refnements to dsaggregaton and Condtonal Mean Spectrum (CMS). CMS, a new target spectrum proposed for ground moton selecton, utlzes the correlaton of spectral acceleraton (Sa) across perods to compute the expected or mean spectrum gven a target Sa at a sngle perod of nterest; we use a smplfed ste example to llustrate CMS computaton ncorporatng multple GMPMs. Dsaggregaton of GMPMs plays an mportant role n CMS computaton, n a smlar way as assgned weghts of GMPMs do n PSHA computaton. Just as the dsaggregaton of magntude and dstance dentfes the relatve contrbuton of each earthquake scenaro to exceedance of a gven Sa level, the dsaggregaton of GMPMs tells us the probablty that the exceedance of that Sa level was predcted by a specfc GMPM. We can further extend dsaggregaton to other ground moton parameters, such as earthquake fault types, to more accurately represent the parameters that contrbute most to Sa values of engneerng nterest. 2

Table of Contents Abstract... 2 1 Introducton... 4 2 Dsaggregaton Usng Multple Ground Moton Predcton Models... 5 2.1 Parameters... 5 2.2 Probablstc Sesmc Hazard Analyss... 6 2.3 Dsaggregaton of Magntude. Dstance, and Epslon... 7 2.4 Dsaggregaton of Ground Moton Predcton Models... 8 2.5 Dsaggregaton of Other Parameters... 9 3 Condtonal Mean Spectrum Computaton... 9 4 Calculaton Procedure and Ste Applcaton... 11 4.1 Descrpton of Ste and Events... 11 4.2 Dsaggregaton of Events... 12 4.3 Dsaggregaton of Ground Moton Predcton Models... 13 4.4 Dsaggregaton of Magntude, Dstance, and Epslon... 13 4.5 Condtonal Mean Spectrum Computaton Usng Two Approaches... 15 5 Dscusson and Conclusons... 18 Acknowledgments... 19 3

1 Introducton Performance of structures durng and after earthquakes s crtcal to publc safety and socetal functonalty. Performance-based earthquake engneerng ams to mprove sesmc desgn and analyss of structures through rsk analyss. In performance-based earthquake engneerng, we nput ground moton nto a structural model to predct structural response such as maxmum nterstory drft rato. Subsequently, we use structural response to categorze damage states and estmate losses n terms of dollars, down tme and fataltes, n order to quantfy performance of structures under earthquakes. To mtgate earthquake rsk, we must frst dentfy ground moton hazard, through probablstc sesmc hazard analyss (PSHA). PSHA combnes the probabltes of all earthquake scenaros wth dfferent magntudes and dstances n order to compute sesmc hazard at a ste. PSHA also ncorporates uncertantes n ground moton predcton, by consderng multple ground moton predcton models (GMPMs), formerly known as attenuaton equatons (e.g., Abrahamson and Slva 1997). GMPMs have nputs such as magntude and dstance, and outputs n terms of logarthmc mean and standard devaton of spectral acceleraton (Sa). When multple GMPMs are present, we typcally use a logc tree to assgn a weght to each GMPM. PSHA then estmates sesmc hazard at a ste ncorporatng uncertantes from earthquake scenaro and GMPMs. Ground moton selecton s a key step n defnng the sesmc load nput to structural analyss. Current ground moton selecton uses the nformaton from earthquake scenaros wthout consderng multple GMPMs. Whle PSHA computes the total sesmc hazard usng total probablty, ts reverse process--psha dsaggregaton--computes the relatve contrbuton of earthquake parameters to the total hazard usng condtonal probablty. Current ground moton selecton utlzes dsaggregaton results of magntude and dstance to dentfy causal events for a gven Sa value correspondng to a return perod. In ths paper we consder ways to ncorporate multple GMPMs nto ground moton selecton technques usng refnements to PSHA dsaggregaton. Ground moton selecton often nvolves specfcaton of a target spectrum. Baker (2005) proposed a target spectrum termed the condtonal mean spectrum (CMS), whch shows several mprovements over the commonly used unform hazard spectrum (UHS). For the UHS, the probablty of Sa exceedance s the same across all perods. But observed spectra rarely look lke 4

a UHS, for several reasons. Dfferent frequency regons of the UHS are often assocated wth dfferent earthquake events. The UHS s also typcally assocated wth above-average Sa values for the causal earthquake event, and due to mperfect correlaton of Sa values t s unlkely that ths above-average condton wll occur at all perods smultaneously. Thus, no sngle ground moton s lkely to produce a response spectrum as hgh as that of the UHS. To account for the varaton of causal earthquake events and Sa values for a gven causal event, the CMS makes use of the correlaton of Sa across dfferent perods. CMS answers the queston, Gven a Sa at the frst-mode perod of a structure, what wll be the expected Sa at other perods? The procedure for performng ths calculaton wll be descrbed later. The CMS calculatons utlzes the magntude, dstance and epslon values obtaned from dsaggregaton, but t also requres the user to choose a GMPM for the calculaton. Wth addtonal nsghts nto the relatve contrbuton of GMPMs from ths new dsaggregaton refnement, we can more effectvely choose approprate GMPMs for CMS calculatons, and thus overcome a practcal challenge to mplementng the CMS n performance based earthquake engneerng. 2 Dsaggregaton Usng Multple Ground Moton Predcton Models Computaton of the Condtonal Mean Spectrum requres dsaggregaton to dentfy the causal parameters. Snce multple Ground Moton Predcton Models are typcally used n practce for Probablstc Sesmc Hazard Analyss, PSHA dsaggregaton can be extended to nclude multple GMPMs. To demonstrate, we use the models and weghts specfed by the Unted States Geologcal Survey (USGS) for the Western US non-extensonal tectonc areas (Frankel et al. 2002): Abrahamson and Slva (1997), Boore et al.(1997), Campbell (1997) 1, and Sadgh et al.(1997) 2, wth equal weghts. 2.1 Parameters Dfferent GMPMs requre dfferent nput parameters. Ths varaton presents challenges for the dsaggregaton process. The parameters used n each model are shown n Table 1. The 1 Campbell (1997) was used n the proposed method nstead of Campbell and Bozorgna (2003) n the USGS method because the former was prevously programmed and thus readly avalable. We beleve the dfferences between the two models are mnor, and wll compare and verfy these dfferences n the future. 2 We have modfed the Sadgh et al. model to correct an error n the predcton equaton n Table 2 n the orgnal publcaton. The Abrahamson and Slva model has also been modfed to correct some errors n the orgnal publcaton. The ground moton predcton model scrpts are avalable at http://www.stanford.edu/~bakerw/attenuaton.html. 5

magntude defnton s dentcal for all models, but the models dffer n ther dstance defntons, as well as how they group and classfy ste condtons and rupture mechansms. When dfferent defntons or groupngs are used for the same ground moton parameter, we need to convert one defnton to another or re-group the nputs, n order to facltate consstent dsaggregaton. For nstance, Abrahamson and Slva (1997) and Sadgh et al. (1997) used the rupture dstance, r rup, the closest dstance from the recordng ste to the ruptured area, whle Boore et al. (1997) used the Joyner Boore dstance, r b, the shortest horzontal dstance from the recordng ste to the vertcal proecton of the rupture. For cases that nvolve dfferent defntons of dstance, Scherbaum et al. (2004) proposed converson approaches among the dfferent knds of dstance. Table 1: Parameters used for four Ground Moton Predcton Models Abrahamson and Slva Boore et al. Campbell Sadgh et al. Magntude M w M w M w M w Dstance r rup r b r ses r rup Fault Type RV, RO, others SS, RV, others SS, RV (TR, RO, TO) SS, RV(TR) Hangng Wall Yes No No No Ste Condton Sol, rock V s30 Sol, soft rock, hard rock Deep sol, rock M w = moment magntude r rup = the shortest dstance from the recordng ste to the ruptured area r b = the shortest horzontal dstance from the recordng ste to the vertcal proecton of the rupture r ses = the shortest dstance from the recordng ste to the sesmogenc porton of the ruptured area SS = strke-slp, RV = reverse, TR = thrust, RO = reverse oblque, TO = thrust oblque 2.2 Probablstc Sesmc Hazard Analyss PSHA ntegrates over all possble earthquake sources wth varous annual rate of occurrence, ν and aleatory uncertantes such as magntudes ( M ), dstances ( R ), and epslons (ε ) n order to compute the total annual rate of exceedance of a spectral acceleraton of nterest, ν (Sa > y). PSHA s usually done wth multple GMPMs, an epstemc source of uncertantes. We explctly consder the epstemc uncertanty n PSHA by ncorporatng weghts of GMPMs, P(GMPM ) nto Equaton (1), to compute the total hazard rate (Kramer 1996). 6

ν ( Sa > y) = ν f M, R, Ε ( m, r, ε ) P( Sa > y m, r, ε, GMPM) dmdrdε P( GMPM) (1) 2.3 Dsaggregaton of Magntude. Dstance, and Epslon Now we have computed the total hazard rate n Equaton (1), we can ask what wll be the dstrbuton of magntudes that cause Sa > y? Dsaggregaton of magntude answers ths queston. Snce all four models use moment magntudes as nputs, the dsaggregaton of magntudes wll be straghtforward and can be determned as follows. f 1 ν ( Sa > y) ν f M, R ( m, r, ε ) P( Sa > M Sa> y ( m, y) =, Ε y m, r, ε, GMPM) drdε P( GMPM) (2) The condtonal dstrbuton of magntude gven Sa, f M Sa> y ( m, y) s avalable on the USGS webste, as a dsaggregaton plot. Snce magntudes are usually dscretzed nto bns, the correspondng condtonal dstrbuton s expressed n terms of percentage contrbuton to Sa > y, P ( M = m Sa > y), nstead of f ( m, ) M Sa> y y. The resultng dsaggregated mean magntude, M Sa > y, whch s also provded by the USGS, can be calculated easly usng standard computaton for expected values, as follows: M Sa > y = E( M Sa > y) = mp( M = m Sa > y) (3) Ths dsaggregated mean value of magntude serves as an nput to the computaton of the CMS. The dsaggregaton of dstance s smlar n theory to the dsaggregaton of magntude, except for the complcaton of dfferng defntons of dstance n dfferent GMPMs as dscussed n 2.1. The dsaggregated dstrbuton of dstance, f R Sa> y ( r, y) can be found as follows, smlar to Equaton (2): f 1 ν ( Sa > y) ν f M, R ( m, r, ε ) P( Sa > R Sa> y ( r, y) =, Ε y m, r, ε, GMPM) dmdε P( GMPM) (4) The dsaggregaton of epslon, ε, s an mportant step for CMS computaton, snce CMS utlzes the correlaton between ε s across perods. Although t s smlar n concept to the dsaggregaton of magntude and dstance, we should pay addtonal attenton to the dfference between the approach of McGure (1995) and that of Bazzurro and Cornell (1999). McGure s dsaggregaton s condtoned on Sa = y, so there s a sngle value of epslon, ε that corresponds 7

to each Sa level (for a gven magntude and dstance). On the other hand, Bazzurro and Cornell s dsaggregaton s condtoned on Sa > y, the epslon value, ε that corresponds to Sa = y s the lower bound value that marks the begnnng of exceedance (Fgure 1). For each event (M = m, R = r), to get an equvalent mean value of epslon that corresponds to Sa > y, we can fnd a centrodal value of epslon, ε ntegrated from the lower bound value, ε to nfnty wth respect to epslon (Equatons (5) and (6)). Note that the tal of the ε dstrbuton does not contrbute sgnfcantly to ths mean, so we can truncate the dstrbuton at ε = 4 to 6, nstead of nfnty (Strasser et al. 2008). ε ( Sa > y, M = m, R = r) = xf Ε ( x) dx ( Sa > y, M = m, R = r) where f Ε (x) s the condtonal dstrbuton of ε gven Sa > y and M = m, R = r, as shown n Fgure 1 and defned by the followng equaton φ( x), x ε * f Ε ( x) ( Sa > y, M = m, R = r) = 1 Φ( ε*) ( Sa > y, M = m, R = r) (6) 0, x < ε * The dsaggregated dstrbuton of epslon, f Ε Sa> y ( ε, y) can be found as follows, smlar to Equatons (2) and (4): (5) f 1 ν ( Sa > y) ν f M, R ( m, r, ε ) P( Sa > Ε Sa> y ( ε, y) =, Ε y m, r, ε, GMPM) dmdrp( GMPM) (7) 2.4 Dsaggregaton of Ground Moton Predcton Models The dsaggregaton of GMPMs s smlar n concept to the dsaggregaton of magntude, dstance, and epslon; t tells us the probablty that the exceedance of a gven Sa level was predcted by a specfc GMPM, P( GMPM Sa > y). Whle equal weghts are often assgned to each GMPM at the begnnng of analyss, t turns out that the contrbuton of the GMPMs to predcton of a gven Sa > y s often unequal. Ths dscrepancy wll make a dfference for CMS computaton, snce new weghts of GMPMs wll offer addtonal nsghts nto the relatve contrbuton of GMPMs. The dsaggregated probablty of GMPM, P( GMPM Sa > y) can be found as follows, smlar to Equatons (2), (4), and (7): 8

P( GMPM 1 = ν ( Sa > y) Sa > y) ν f M R, Ε ( m, r, ε ) P( Sa >, y m, r, ε, GMPM) dmdrdε P( GMPM) (8) 2.5 Dsaggregaton of Other Parameters Smlarly, the total hazard, ν ( Sa > y) can be computed f other parameters, expressed as θ, are consdered: ν ( Sa > y) = ν f ( m, r, ε, θ ) P( Sa > M, R, Ε, Θ y m, r, ε, θ, GMPM) dmdrdεdθ P( GMPM) (9) Dsaggregaton can be extended to other parameters, f Θ Sa> y ( θ, y) n a smlar fashon to Equatons (2), (4), (7), and (8): f Θ Sa> y ( θ, y) 1 ν ( Sa > y) ν f M, R, Ε, ( m, r, ε, θ ) P( Sa > = Θ y m, r, ε, θ, GMPM) dmdrdεp( GMPM) (10) For nstance,θ could represent fault types. The current USGS dsaggregaton method assumes a random fault type. An alternatve to the USGS method mght use dsaggregaton based on avalable fault types n each ground moton predcton model. Fault types can be treated as dscrete random varables, sometmes wth several lumped nto one group. The relatve contrbuton of each fault type can be represented through hstograms smlar to those for dscretzed magntude and dstance. We could apply ths approach to other parameters, such as depth to top of rupture, n order to dentfy ther relatve contrbuton to exceedance of a gven Sa value. 3 Condtonal Mean Spectrum Computaton The Pacfc Earthquake Engneerng Research (PEER) Center s Ground Moton Selecton and Modfcaton Program (http://peer.berkeley.edu/gmsm/) has studed dfferent ground moton selecton methods. The CMS method (Baker 2005) s promsng, because t ams to match a realstc spectral shape, produces no bas n structural response when usng scaled 9

ground motons, and enlarges the pool of potental ground motons that can be selected and scaled for nonlnear dynamc analyss. The CMS makes use of the observed mult-varate normal dstrbuton of the logarthmc spectral acceleratons ( ln Sa ) at dfferent perods (T ). The mean of ln Sa at all perods T, s condtoned on ln Sa ( T1 ) = ln Sa( T1 )*, where Sa ( T 1 ) * s the target spectral acceleraton, and T 1 s the prmary perod of nterest. We use GMPMs to predct the logarthmc mean and standard devaton of Sa at a range of perods that pvot around the dsaggregated means M, R, ε gven ln Sa ( T1 ) = ln Sa( T1 ) *. Baker and Jayaram (2008) have provded the requred addtonal pece of nformaton, the correlaton coeffcent ρ of Sa across perods. The logarthmc mean and standard devaton of the CMS can be computed as follows: μ ln Sa( T ) ln Sa( T ) ln ( )* ln ( M, R, T) ln ( ),ln ( ) ln ( M, T ) ( T1 ) 1 = Sa T μ 1 Sa + ρ Sa T1 Sa T σ Sa ε (11) σ 2 ln Sa( T ) ln Sa( T1 ) ln Sa( T1 )* σ ln Sa ( M, T ) 1 ρ ln Sa( T1 ),ln Sa( T ) where ln Sa( T 1)*. = (12) μ = the mean lnsa at perod T, condtoned on ln Sa( T 1)*. ln Sa ( T ) ln Sa( T1 ) = ln Sa( T1 )* 1 σ = the standard devaton of lnsa at perod T, condtoned on ln Sa ( T ) ln Sa( T1 ) = ln Sa( T1 )* ρ = the correlaton coeffcent between the logarthmc spectral acceleratons ln Sa( T1 ),ln Sa( T ) at perods T and T 1. Sa ( T 1 )*. M, R, ε = the dsaggregated mean magntude, dstance, and epslon for the gven μ ( M, R, T ), σ ( M, T ) = the mean and standard devaton, respectvely, of the ln Sa ln Sa logarthmc spectral acceleraton at perod T, computed usng a ground moton predcton model. Whle ths approach appears to be advantageous n some stuatons (Baker 2005; Baker and Cornell 2006), challenges reman for ts mplementaton n practce. A prmary queston s, whch GMPMs should we use to evaluate the above equatons when multple GMPMs are used n a logc tree to perform the assocated PSHA? To address ths queston, we ntroduce two approaches and apply them to an example ste. 10

4 Calculaton Procedure and Ste Applcaton To demonstrate the concepts of PSHA dsaggregaton and CMS computaton usng multple GMPMs, we have appled two approaches of CMS computaton to an example ste as follows: 1. For Approach 1, frst we drectly apply one GMPM to compute the CMS, based on dsaggregated mean M/R/ε consderng all GMPMs. We could smply stop at ths pont and call the result a target CMS; we wll show ths ntermedate result below and label t Approach 0. Then we repeat Approach 0 for each GMPM used n the PSHA calculaton, and average the resultng Condtonal Mean Spectra (usng equal weghts for each) to obtan a weghted average CMS. 2. For Approach 2, frst we apply one GMPM to compute the CMS, usng the dsaggregated mean M/R/ε assocated wth that partcular GMPM. Then we repeat the same step for each GMPM used n the PSHA calculaton, and average the resultng Condtonal Mean Spectra (usng weghts for each obtaned usng the GMPM dsaggregaton), to compute a weghted average of CMS. 4.1 Descrpton of Ste and Events The example ste consdered s domnated by two earthquake events, as shown n Fgure 2. We selected ths ste because t has two events wth very dfferent magntudes and dstances, whch s helpful for llustratng PSHA, PSHA dsaggregaton, and CMS computaton. Event A, wth magntude, M = 6 and dstance, R = 10 km from the ste, has an annual occurrence rate of ν = 0.01; Event B, wth magntude, M = 8 and dstance, R = 25 km from the ste, has an annual occurrence rate of ν = 0.002. Both events have strke slp mechansms and no hangng walls. The ste has sol wth shear wave velocty V s30 = 310 m/s, correspondng to NEHRP Ste Class D. Assumng a vertcal fault that extends all the way to the ground surface (a reasonable assumpton for shallow crustal earthquakes n coastal Calforna), rupture dstance, r rup, s the same as r b. The earthquake events are assumed to rupture the whole of faults A and B, so the closest dstance to the ste for a gven earthquake wll be a known constant. We use four GMPMs to evaluate the annual rates of exceedng a target Sa level for both events. The logc tree that ncorporates four GMPMs s shown n Fgure 3. Accordng to USGS practce, we assgn equal weghts to each GMPM. The perod of nterest s 1 s. The probablty 11

of exceedng a target Sa level, gven an event wth ts assocated magntude and dstance, s computed usng each GMPM, and the results are plotted n Fgure 4. Accordng to Fgure 4, Event A has a lower probablty of Sa(1s) > y at all Sa levels than Event B does, but Event A occurs more frequently and thus stll contrbutes sgnfcantly to the ground moton hazard. Utlzng Equaton (1), the results from Fgure 4, and the fact that each event corresponds to a sngle magntude and dstance, Equaton (13) can be used to compute the mean annual rate of Sa > y due to these two events, evaluated usng four GMPMs. ν ( Sa > y) = 2 4 = 1 = 1 P( Sa > y Event, GMPM ) P( GMPM Event ) ν ( Event ) (13) as The weght of each GMPM s equal to 0.25 n ths case, so Equaton (13) can be rewrtten 2 4 ν ( Sa > y) = P( Sa > y Event, GMPM )(0.25) ν ( Event ) (14) = 1 = 1 The ground moton hazard curve obtaned from ths calculaton for ths smplfed ste s shown n Fgure 5. Ths calculaton s comparable to that used to produce the hazard curves avalable on the USGS webste (http://earthquake.usgs.gov/research/hazmaps/), for real stes n the Unted States. Two typcal desgn levels (2% and 10% n 50 years) along wth an addtonal llustratve desgn level (40% n 50 years) are marked n Fgure 5. Usng a Posson model for earthquake recurrence, a Sa exceedance probablty of 2% n 50 year s equvalent to ν(sa > y) = 0.0004 (a return perod of 2475 years), and hence a target Sa value, Sa(1s)* of 0.84g. Smlarly, Sa exceedance probabltes of 10% and 40% n 50 years are equvalent to return perods of 475 and 100 years, and have target Sa(1s)* values of 0.43g and 0.10g, respectvely. The followng plots of dsaggregaton results and CMS calculatons have these three desgn levels Sa levels marked for reference. 4.2 Dsaggregaton of Events P( Event where The condtonal probablty that each event caused Sa > y s gven by Equaton (15). ν ( Sa > y, Event ) Sa > y) = v( Sa > y) (15) 12

ν ( Sa > y, Event ) = P( Sa > y GMPM, Event ) P( GMPM ) ν ( Event ) (16) The probabltes obtaned from Equaton (15) are plotted n Fgure 6. From Fgure 6, we can see that the smaller but more frequent Event A s most lkely to cause exceedance of small Sa ampltudes, whereas Event B s most lkely to cause exceedance of large Sa ampltudes. Ths s because the annual hazard rate nvolves two competng factors: annual rate of occurrence for an earthquake, and probablty of exceedng a Sa level gven that earthquake. Small earthquakes (for example, Event A) have a larger annual rate of occurrence than large earthquakes, whereas large earthquakes (for example, Event B) have a larger probablty of exceedng a Sa level. At a lower Sa level, the hazard s domnated by small frequent earthquakes; at a hgher Sa level, the hazard s domnated by large but nfrequent earthquakes because the small earthquakes very rarely generate such large ground moton ampltudes. The results n Fgure 6 are typcal of PSHA analyses for more realstc stes. 4.3 Dsaggregaton of Ground Moton Predcton Models Followng Equaton (8), the dsaggregaton of GMPMs s performed usng the followng equatons, P( GMPM where Sa > ν ( Sa > y, GMPM y) = v( Sa > y) ) (17) ν ( Sa > y, GMPM ) = P( Sa > y GMPM, Event ) P( GMPM ) ν ( Event ) (18) Note the smlarty between Equatons (16) and (18). The results of ths dsaggregaton calculaton are shown n Fgure 7. The dsaggregated GMPM contrbutons vary from 0.16 to 0.31, nstead of havng an equal weght of 0.25. The weghts are stll not too far from 0.25. GMPM dsaggregaton s useful for computaton of CMS, as the dsaggregated probabltes of GMPMs offer addtonal nsghts nto whch GMPM contrbutes most to the predcton of Sa values of nterest. 4.4 Dsaggregaton of Magntude, Dstance, and Epslon Snce only one magntude s assocated wth each event, the dsaggregated mean magntude can be found easly by determnng the sum of the products of the magntudes gven an event and the dsaggregated contrbuton of the event, as follows: 13

M Sa > y = ( M Event ) P( Event Sa > y) (19) where M s used to denote the mean value of M. Note that n ths example M Event s a determnstc relatonshp, snce there s only a sngle magntude assocated wth each event. Ths equaton s equvalent to Equaton (5). The dsaggregated mean magntude can be found usng a smlar method separately, for each GMPM, as shown n Equaton (20). The results are shown n Fgure 8. In ths fgure, the thn lnes ndcate the mean magntude, gven Sa > y and gven that the assocated GMPM was the model that predcted Sa > y, M GMPM, Sa > y. The heavy lne provdes the a weghted average (composte) of M GMPM, Sa > y over all GMPMs. The varaton of dsaggregated mean magntudes, s greater at hgher Sa values because that s where the GMPMs dffer more sgnfcantly due to lack of data to constran the predctons. M Sa > y = M GMPM, Sa > y = ( M Event, GMPM ) P( Event GMPM, Sa > y) (20) Snce only one dstance s assocated wth each event, the dsaggregated mean dstance can be found easly by determnng the sum of the products of the dstance gven an event and the dsaggregated contrbuton of the event, as follows: R Sa > y = ( R Event ) P( Event Sa > y) The dsaggregated mean dstance values are plotted n Fgure 9. In ths case the results are very smlar to the magntude dsaggregaton results due to the one-to-one correspondence between magntudes and dstances n ths smple example. For each event (M = m, R = r), epslon values change as Sa vares. The centrodal values of epslon gven Sa > y and an event, ε Sa > y, Event dsaggregated mean epslon can be computed as follows: (21) can be obtaned from Equaton (5). The 14

ε Sa > y = ( ε Sa > y, Event ) P( Event Sa > y) (22) The dsaggregated mean epslon values are plotted n Fgure 10. As Sa ncreases, the mean ε value ncreases, and ths can have a large mpact on the shape of the CMS that wll be computed n the next secton. 4.5 Condtonal Mean Spectrum Computaton Usng Two Approaches Wth the dsaggregaton nformaton of the prevous few sectons, t s now possble to compute the CMS usng Equaton (11). The frst-mode perod of nterest n ths example s 1s and our frst example calculaton s condtoned on Sa exceedng 0.84g, we use the dsaggregated mean values of M, R, and ε gven Sa(1s) > 0.84g. The mean predctons of lnsa at other perods, usng each GMPM, are then computed usng Equaton (11). There are two approaches to CMS computaton usng multple GMPMs. For Approach 1, we frst compute the mean M, R, and ε gven Sa > y usng all GMPMs, from Equatons (19), (21), and (22). Then we compute CMS, the CMS computed usng GMPM and the mean M, R, and ε gven all GMPMs, from Equatons (19), (21), and (22) as follows: CMS > = CMS ( M Sa > y, R Sa > y, ε Sa y) (23) The result from equaton (23) on ts own s also the Approach 0 mentoned above, as t can be used on ts own as a smple way to obtan a CMS. To use the results of equaton (23) n Approach 1, we compute a weghted sum of these CMS, usng the assgned weght of GMPM (0.25 n ths case), as follows: = CMS P( GMPM ) = CMS CMS (0.25) The results from equatons (23) and (24) are shown n Fgure 11. The result from equaton 24 s denoted CMS, Composte as t s a composte of the ndvdual CMS spectra from equaton (23). Smlarly for Approach 2, frst we compute the mean gven Sa > y usng each GMPM. Note that the dsaggregaton means used here here are condtonal on each GMPM (e.g., usng, Equaton (20)), nstead of on all GMPMs, as n Approach 1. (24) 15

Then we compute CMS, the CMS computed usng GMPM and the respectve mean M, R, and ε gven each GMPM, from procedures smlar to Equaton (20). > CMS ' = CMS ( M Sa > y, R Sa > y, ε Sa y) (25) Equaton (25) s dentcal to Equaton (23), except that the GMPM-specfc M/R/ε values of Equaton (25) are used n place of the overall M/R/ε dsaggregaton values. Fnally for Approach 2, we compute a weghted sum of these CMS (composte curve n Fgure 12), usng the dsaggregated contrbuton of GMPMs. Note that probablty of GMPM here s condtonal on Sa > y, and t dffers from the assgned equal probablty. Ths dsaggregated probablty of GMPM adds nsghts regardng the relatve contrbuton of GMPM to the predcton of the Sa level of nterest. The extenson of PSHA dsaggregaton to GMPMs offers a more complete soluton compared to the orgnal approach. CMS ' = CMS ' P( GMPM Sa > y) Fgure 13 shows that the approaches of Equatons (24) and (26) are not equvalent. Approach 0 s a raw attempt to compute the CMS usng a sngle GMPM wth avalable data. Approach 1 s a smplfed method to obtan a composte CMS by averagng the condtonal mean spectra obtaned from each of the GMPMs. Approach 2 computes a weghted average of the GMPM-specfc condtonal mean spectra, where the weghts come from GMPM dsaggregaton, and s thus probablstcally consstent. The drawback for Approach 2 s that we need addtonal data (.e., the dsaggregaton of GMPMs and the dsaggregaton of magntude, dstance, and epslon for each GMPM) whch are not readly avalable from current PSHA calculaton tools or the USGS webste. There s a large scatter among dfferent sngle-gmpm CMSs n Approach 0, showng the varablty among dfferent GMPMs. However, the dfference between Approach 1 and Approach 2 s not sgnfcant n ths example, whch suggests that we may be able to approxmate Approach 2 usng the smpler Approach 1. Further calculatons n more vared and more realstc condtons are needed to confrm ths possblty. (26) To nvestgate the effect of change n desgn level and perod of nterest on the accuracy of these approaches, four combnatons wth three desgn levels and two perods of nterest are shown: a) Sa(1s) > 0.84g, correspondng to 2% n 50 years Sa exceedance (Fgure 13); b) Sa(1s) > 0.43g, correspondng to 10% n 50 years Sa exceedance (Fgure 14); c) Sa(1s) > 0.10g, 16

correspondng to 40% n 50 years Sa exceedance (Fgure 15); d) Sa(0.2s) > 0.84g, correspondng to 10% n 50 years Sa exceedance but at a dfferent perod of nterest (Fgure 16). An examnaton of Equaton (11) reveals the components of CMS computaton at varous desgn levels,.e. target Sa(T 1 )*. As the target Sa(T 1 )* value changes, the dsaggregated mean M/R/ε change, whch change Sa predcton n terms of μ lnsa and σ lnsa. The correlaton coeffcent ρ, on the other hand, s not dependent on the target Sa(T 1 )* value. As ε ncreases, ερ ncreases, resultng n a larger dfference between Sa at the perod of nterest (usually the frst-mode perod) and Sa at perods further away from the perod of nterest and hence a sharper spectral shape. Ths s apparent n Fgure 17: the three CMSs usng the same approach dffer as a result of the varaton n the mean dsaggregated epslon values as shown n Fgure 10. For nstance, a decrease n target Sa from 0.84g to 0.43g results n a decrease n ε from 1.90 to 1.22 (as well as a decrease of M from 7.48 to 7.12 and a decrease of R from 21.1 to 18.4), whch results n a flatter CMS gven Sa(1s) > 0.43g. To evaluate the approprateness of approxmatng Approach 2 usng Approach 1 at a sngle Sa(T 1 )*, we can look at both the dsaggregated mean M/R/ε and dsaggregated GMPM contrbuton. If the dsaggregated mean M/R/ε s smlar for dfferent GMPMs (e.g., M Sa > y M Sa > y ) at Sa(T 1 )*, μ lnsa and σ lnsa wll be smlar,.e. CMS (Equaton (25)) s smlar to CMS (Equaton (23)). Approach 1 (Equaton (24)) and Approach 2 (Equaton (26)) are weghted averages of CMS / CMS respectvely and thus wll gve smlar results of CMS/CMS. In other words, the relatve contrbuton of GMPMs n Equaton (26) does not matter much snce CMS s almost equal to CMS. On the other hand, f the dsaggregated mean M/R/ε vary a great deal for varous GMPMs, the GMPM specfc CMS wll be substantally dfferent from the general CMS. The relatve contrbuton of GMPMs then accounts for the dfference n CMS and CMS. However, f the relatve contrbuton of GMPMs s about equal, then the average CMS s smlar to CMS, despte the dfference between CMS and CMS. The dfference between CMS and CMS for Sa(1s) > 0.10g (Fgure 15) s much smaller than that for Sa(1s) > 0.84g (Fgure 13), because the dsaggregated mean M/R/ε dffer less among dfferent GMPMs for Sa(1s) > 0.10g (from Fgure 7 to Fgure 9), resultng n a smaller dfference between CMS and CMS, and a less mportant relatve contrbuton of GMPMs whch s also closer to the equal value (0.25 n ths example). 17

CMS employs the same overall dsaggregated mean M/R/ε (e.g., M Sa > y ) whereas CMS utlzes GMPM-specfc dsaggregated mean M/R/ε (e.g., M Sa > y ). Snce the same M/R/ε are used for CMS, the scatter n CMS n Approach 0 (Fgure 11) reflects the varablty n Sa predcton by dfferent GMPMs. However, CMS (Fgure 12) shows a larger scatter than CMS (Fgure 11), wth varablty from both Sa predcton and dsaggregaton of M/R/ε. If the dsaggregaton of M/R/ε dffer very lttle among dfferent GMPMs, there wll be a smaller dfference between CMS and CMS. When the desgn level or perod of nterest changes, the event that causes Sa(T 1 ) > Sa(T 1 )* (shown by dsaggregated mean M/R/ε ) vares. Wth the same perod of nterest, when the desgn level changes, the spectral shape changes, as shown n Fgure 17. Wth the same desgn level, when the perod of nterest changes, the spectral shape changes, as shown n Fgure 18. The former can be used to select ground motons for the same structure at varous desgn levels, potentally used n ncremental dynamc analyss. The latter can be used to select ground motons for dfferent structures at the same desgn level, or alternatvely, select ground motons for the same structure wth varous perods of nterest (e.g. frst-mode perod and hgher mode perod for a tall buldng) at the same desgn level, potentally usng multple CMSs (Baker 2005). 5 Dscusson and Conclusons The applcaton of condtonal mean spectrum (CMS) computaton to multple ground moton predcton models (GMPMs) requres the dsaggregaton of GMPMs. Ths approach s consstent wth the probablstc treatment of random varables n tradtonal probablstc sesmc hazard analyss (PSHA), and s an extenson from the currently avalable method of dsaggregaton of events. The CMS method appled drectly to the dsaggregated mean M, R, and ε consderng all GMPMs, may produce dfferent results from that appled to the dsaggregated mean M, R, and ε consderng each GMPM separately, along wth ts assocated dsaggregated contrbuton. The spectral shape of CMS, as well as the causal event, vares as the desgn level or the perod of nterest changes. CMS usng multple GMPMs has practcal sgnfcance snce multple GMPMs are usually used to obtan an aggregate hazard rate. It wll be probablstcally consstent to consder the condtonal relatve contrbuton of GMPMs. The 18

dsaggregaton of GMPMs provdes addtonal nsghts nto whch GMPM contrbutes most to predcton of Sa values of engneerng nterest. Currently, GMPM dsaggregaton s typcally not avalable (e.g., from USGS hazard maps). CMS appled drectly to the dsaggregated mean M, R, and ε consderng all GMPMs wthout the dsaggregaton of GMPMs and the dsaggregaton of M, R, and ε wth respect to each GMPM, may produce naccurate results, especally for cases where dsaggregaton of M, R, and ε dffer sgnfcantly usng dfferent GMPMs. Although the computaton of CMS usng multple GMPM s slghtly more complcated, the mproved accuracy may be worthwhle n some cases. Performng GMPM dsaggregaton n typcal PSHA calculatons would be very benefcal n facltatng the mproved CMS calculatons presented n ths paper. We can further extend dsaggregaton to other ground moton parameters, such as earthquake fault types, to more accurately represent the parameters that contrbute most to Sa values of engneerng nterest. Future collaboratons wth USGS are essental to overcome challenges n practcal mplementaton of CMS usng multple GMPMs. Such mplementaton can be ncorporated nto future USGS hazard maps and made avalable to the publc, whch would then facltate the CMS calculaton procedure above and ad ground moton selecton efforts. Acknowledgments Ths work was supported by the U.S. Geologcal Survey, under Award Number 07HQAG0129. Gudance, assstance, and tmely feedback provded by the P.I., Dr. Jack Baker, are gratefully apprecated and acknowledged here. Any opnons, fndngs, and conclusons or recommendatons presented n ths materal are those of the author and do not necessarly reflect those of the fundng agency. 19

f Ε (ε) 1 PDF of ε PDF of ε, gven Sa>y 0 0 ε * ε ε Fgure 1: Probablty densty functon of epslons demonstratng the dfference between McGure dsaggregaton (ε ) and Bazzurro and Cornell dsaggregaton ( ε ). 20

Fgure 2: Layout of an example ste domnated by two earthquake events A and B. 21

Abrahamson and Slva p = 0.25 Event A Boore et al. p = 0.25 ν A Campbell p = 0.25 Sadgh et al. p = 0.25 Abrahamson and Slva p = 0.25 Event B Boore et al. p = 0.25 ν B Campbell p = 0.25 Sadgh et al. p = 0.25 Fgure 3: Logc tree of GMPMs used to calculate hazard n the example ste. 22

P(Sa(1s)>y Event, GMPM ) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Event A, Abrahamson and Slva Event A, Boore et al. Event A, Campbell Event A, Sadgh et al. Event B, Abrahamson and Slva Event B, Boore et al. Event B, Campbell Event B, Sadgh et al. 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Sa(1s) [g] Fgure 4: Predcton of ground moton exceedance for two events usng four dfferent GMPMs n the example ste. 23

0.012 0.01 Sesmc Hazard Curve 40% n 50 years 10% n 50 years 2% n 50 years 0.008 ν(sa(1s)>y) 0.006 0.004 0.002 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Sa(1s) [g] Fgure 5: Probablstc Sesmc Hazard Analyss curve for the example ste. Three Sa levels of later nterest are also noted. 24

1 0.9 0.8 P(Event Sa(1s)>y) 0.7 0.6 0.5 0.4 0.3 Event A Event B 40% n 50 years 10% n 50 years 2% n 50 years 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Sa(1s) [g] Fgure 6: Dsaggregaton of events, gven Sa(1s) > y, for the example ste. Dsaggregaton results correspondng to Sa exceedance probabltes of 40%, 10%, and 2% n 50 years are marked on the fgure. 25

0.32 0.3 P(GMPM Sa(1s)>y) 0.28 0.26 0.24 0.22 Abrahamson and Slva Boore et al. Campbell Sadgh et al. 40% n 50 years 10% n 50 years 2% n 50 years 0.2 0.18 0.16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Sa(1s) [g] Fgure 7: Dsaggregaton of GMPMs, gven Sa(1s) > y, for the example ste. 26

8 7.8 7.6 7.4 E(M Sa(1s)>y) 7.2 7 6.8 Composte Abrahamson and Slva Boore et al. 6.6 Campbell Sadgh et al. 6.4 40% n 50 years 10% n 50 years 2% n 50 years 6.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Sa(1s) [g] Fgure 8: Dsaggregaton of magntude M, gven Sa(1s) > y, for the example ste. 27

26 24 22 E(R Sa(1s)>y) 20 18 Composte 16 Abrahamson and Slva Boore et al. Campbell 14 Sadgh et al. 40% n 50 years 10% n 50 years 2% n 50 years 12 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Sa(1s) [g] Fgure 9: Dsaggregaton of dstance R, gven Sa(1s) > y, for the example ste. 28

2.5 2 E(ε Sa(1s)>y) 1.5 1 Composte Abrahamson and Slva Boore et al. 0.5 Campbell Sadgh et al. 40% n 50 years 10% n 50 years 2% n 50 years 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Sa(1s) [g] Fgure 10: Dsaggregaton of epslon ε, for the example ste. 29

10 0 Sa(T) [g] 10-1 CMS, Composte CMS, Abrahamson and Slva CMS, Boore et al. CMS, Campbell CMS, Sadgh et al. Target Sa(T 1 )* 10-1 10 0 T [s] Fgure 11: CMS computaton usng Approach 1 condtonal on Sa(T 1 = 1s) > 0.84g (2% n 50 years), for the example ste. 30

10 0 Sa(T) [g] 10-1 CMS, Composte CMS, Abrahamson and Slva CMS, Boore et al. CMS, Campbell CMS, Sadgh et al. Target Sa(T 1 )* 10-1 10 0 T [s] Fgure 12: CMS computaton usng Approach 2 condtonal on Sa(T 1 = 1s) > 0.84g (2% n 50 years), for the example ste. 31

10 0 Sa(T) [g] 10-1 CMS, Approach 0, Abrahamson and Slva CMS, Approach 0, Boore et al. CMS, Approach 0, Campbell CMS, Approach 0, Sadgh et al. CMS, Approach 1 CMS, Approach 2 Target Sa(T 1 )* 10-1 10 0 T [s] Fgure 13: Comparson of CMS computaton usng Approaches 0, 1, and 2 condtonal on Sa(T 1 = 1s) > 0.84g (2% n 50 years), for the example ste. 32

10 0 Sa(T) [g] 10-1 CMS, Approach 0, Abrahamson and Slva CMS, Approach 0, Boore et al. CMS, Approach 0, Campbell CMS, Approach 0, Sadgh et al. CMS, Approach 1 CMS, Approach 2 Target Sa(T 1 )* 10-1 10 0 T [s] Fgure 14: Comparson of CMS computaton usng Approaches 0, 1, and 2 condtonal on Sa(T 1 = 1s) > 0.43g (10% n 50 years), for the example ste. 33

10 0 CMS, Approach 0, Abrahamson and Slva CMS, Approach 0, Boore et al. CMS, Approach 0, Campbell CMS, Approach 0, Sadgh et al. CMS, Approach 1 CMS, Approach 2 Target Sa(T 1 )* Sa(T) [g] 10-1 10-1 10 0 T [s] Fgure 15: Comparson of CMS computaton usng Approaches 0, 1, and 2 condtonal on Sa(T 1 = 1s) > 0.10g (40% n 50 years), for the example ste. 34

10 0 Sa(T) [g] 10-1 CMS, Approach 0, Abrahamson and Slva CMS, Approach 0, Boore et al. CMS, Approach 0, Campbell CMS, Approach 0, Sadgh et al. CMS, Approach 1 CMS, Approach 2 Target Sa(T 1 )* 10-1 10 0 T [s] Fgure 16: Comparson of CMS computaton usng Approaches 0, 1, and 2 condtonal on Sa(T 1 = 0.2s) > 0.84g (10% n 50 years). 35

10 0 Sa(T) [g] 10-1 CMS, Approach 1, Sa(1s) > 0.84g CMS, Approach 2, Sa(1s) > 0.84g Target Sa(T 1 )*, Sa(1s) > 0.84g CMS, Approach 1, Sa(1s) > 0.43g CMS, Approach 2, Sa(1s) > 0.43g Target Sa(T 1 )*, Sa(1s) > 0.43g CMS, Approach 1, Sa(1s) > 0.10g CMS, Approach 2, Sa(1s) > 0.10g Target Sa(T 1 )*, Sa(1s) > 0.10g 10-1 10 0 T [s] Fgure 17: CMS condtonal on Sa(1s) > Sa(T 1 )* for 2% n 50 years (Sa(T 1 )* = 0.84g), 10% n 50 years (Sa(T 1 )* = 0.43g), and 40% n 50 years (Sa(T 1 )* = 0.10g) desgn levels. 36

10 0 Sa(T) [g] 10-1 CMS, Approach 1, Sa(1s) > 0.43g CMS, Approach 2, Sa(1s) > 0.43g Target Sa(T 1 )*, Sa(1s) > 0.43g CMS, Approach 1, Sa(0.2s) > 0.84g CMS, Approach 2, Sa(0.2s) > 0.84g Target Sa(T 1 )*, Sa(0.2s) > 0.84g 10-1 10 0 T [s] Fgure 18: CMS condtonal on Sa(T 1 ) > Sa(T 1 )* for 10% n 50 years desgn level (Sa(1s)* = 0.43g and Sa(0.2s)* = 0.84g). 37

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