INCORPORATING PARAMETER UNCERTAINTY INTO ATTENUATION RELATIONSHIPS ABSTRACT

Transcription

1 INCORPORATING PARAMETER UNCERTAINTY INTO ATTENUATION RELATIONSHIPS Robb Erc S. Moss 1 and Armen Der Kureghan 2 ABSTRACT Strong ground moton attenuaton relatonshps estmate the mean and varance of ground shakng as t decreases wth dstance from an earthquake source. Current relatonshps use classcal regresson technques that treat the nput varables or parameters as exact, neglectng the uncertantes assocated wth the measurement of ground acceleraton, moment magntude, ste-to-source dstance, shear wave velocty, etc. Ths leads to a poorly constraned estmate of the uncertanty of strong ground motons. Ths paper dscusses the work n progress on; a) estmatng the statstcs of parameter uncertanty, and b) ncorporatng the parameter uncertanty nto the regresson of strong moton attenuaton data usng a Bayesan framework. The results are an mproved understandng of the uncertantes nherent n the phenomena of strong ground moton attenuaton, a reduced and better defned model varance, and better constraned estmates of rarer events assocated wth ground acceleratons towards the tal of the dstrbuton. Introducton Ths paper descrbes ongong research nto measurement error related to strong ground moton parameters and estmated varance related to strong ground moton attenuaton predctons. The current statstcal method for developng an attenuaton relatonshp s unvarate regresson on a database usng a fxed-effects or random-effects model (e.g., Boore et al., 1997; Abrahamson & Slva, 1997; Campbell & Bozorgna, 2003). Ths methodology assumes that the nput parameters are exact. There exsts, however, measurement error n the nput parameters. For nstance, the moment magntude of a partcular sesmc event s calculated usng a non-unque nverson process resultng n an unspecfed amount of uncertanty. Ths can be seen n the dfferences n reported moment magntudes by sesmology labs such as USGS and Harvard (Moss, 2003). Dfferences n nverson technques used over tme have also led to uncertanty n the moment magntude (Kagan, 2002). The geometrc mean of the peak ground acceleraton s an nput parameter that consders both horzontal drectons of ground shakng. Ths parameter has measurement error that s a functon of the orentaton of the strong moton sesmometer n relaton to the geometry of the fault 1 Asst. Prof., Dept. Cvl and Envronmental Engr., Calforna Polytechnc State Unversty, San Lus Obspo, CA 2 Prof., Dept. Cvl and Envronmental Engr., Unversty of Calforna Berkeley, CA

2 rupture. The horzontal motons can be numercally rotated provdng statstcal estmates of the medan and standard devaton as a functon of azmuth. Measurement errors also exst n the nput parameters that defne ste-to-source dstance, ste class as measured by shear wave velocty n the upper 30 meters, and other nput parameters. Treatng these nput parameters as nexact nstead of exact leads to a better understandng of the sources of uncertanty that propagate through the regresson analyss. Ths also results n reduced overall model varance. A Bayesan framework allows for the treatment of nput parameters as nexact, and provdes the mathematcal flexblty to use any type of functonal model form (Der Kureghan, 1999; Gardon et al, 2002; Moss et al., 2003). For ths study a Bayesan regresson methodology has been formulated for estmatng strong moton attenuaton usng exstng publshed attenuaton equatons. Ths paper uses Boore et al. (1997) for a feasblty study, comparng regresson results wth and wthout measurement error n the nput parameters. Future goals of ths research nclude; collectng more statstcal data on the nput parameters and evaluatng state-of-the-art attenuaton equatons usng a sngle database to measure relatve performance. Quantfyng Measurement Error The frst step n ncludng measurement error (.e., parameter uncertanty) nto a predctve model, n ths case a strong moton attenuaton relatonshp, s to evaluate and quantfy that uncertanty. There are two forms of uncertanty, epstemc and aleatory uncertanty. Aleatory uncertanty s the nherent randomness that s a functon of the phenomena that the model strves to predct. Aleatory uncertanty s nherent n nature and cannot be nfluenced by the observer or the manner of observaton. Ths type of uncertanty cannot be reduced. Epstemc uncertanty s a functon of our lack of knowledge, ncomplete descrpton of the phenomena n the model, measurement errors, and/or lack of suffcent measurements to fully capture the phenomena. Epstemc uncertanty s reducble. Ths study ams at reducng the epstemc uncertanty n attenuaton relatonshps by ncorporatng the uncertanty n the nput parameters, the measurement error, nto the regresson analyss. Peak Ground Acceleraton Uncertanty n the ground acceleraton can be observed n the varablty of the peak values n orthogonal drectons. Ths s a property of the orentaton of the rupture plane, complexty of the rupture plane, nature and geometry of the rupture, travel path complextes, surface topography, and other ste effects. Ground acceleraton measurements also contan uncertanty that s a functon of the orentaton of the strong moton sesmometer n relaton to the geometry of the fault rupture. To capture the uncertanty n strong ground motons, acceleraton tme hstores for dfferent events were evaluated. The motons were rotated through a sweep of 90 degrees, usng the method descrbed by Penzen & Watabe (1975), and the geometrc mean, medan, and coeffcent of varaton were measured. A 90 degree rotaton of orthogonal motons provdes a full sweep of the recorded moton n the horzontal drecton.

3 Fgure 1 shows three plots of the processed Hayward Bart Staton recordng from the Loma Preta earthquake. The frst shows a 90 degree sweep of the peak acceleraton n both horzontal drectons, the average, the geometrc mean, the medan geometrc mean, and the mnmum covarance angle. The second plot shows the frequency hstogram of the normalzed geometrc mean, that s the geometrc mean dvded by the medan of the geometrc mean. The thrd plot shows the cumulatve frequency dstrbuton of the normalzed geometrc mean. Fgure 1. Statstcal results of Loma Preta Hayward Bart Staton moton, typcal of the motons evaluated to date. It can be seen n the frequency hstogram that the randomness of the recorded moton throughout the rotated angles does not follow a common theoretcal probablty dstrbutons (e.g., normal or gaussan dstrbuton). Ths result s typcal of motons evaluated n ths study so far. Plotted for comparson s the unform frequency dstrbuton and the unform cumulatve dstrbuton, respectvely. For ths prelmnary analyss an average sample medan and standard devaton was calculated from the motons evaluated so far. Ths was used as the estmated measurement error n the subsequent regresson analyss.

4 Moment Magntude The uncertanty of the moment magntude can be attrbuted manly to the nverson process used to calculate the sesmc moment, and thus the moment magntude. Moment magntude s reported by sesmology laboratores followng an event, and terated on for a week or two untl the fnal revsed value s reported. Calculatng the moment magntude nvolves an nverse problem to determne the sesmc moment. The uncertanty n these calculatons comes from the non-unqueness of the nverson process. Uncertanty n moment magntude has also been shown to be a functon of tme. Kagan (2002) has estmated the standard devaton of the moment magntude as a functon of the nverson technque used to calculate the sesmc moment. The accuracy and compatblty of dfferent nverson technques has mproved over tme, thereby provdng a reduced standard devaton as we approach the present. Uncertanty n the moment magntude was quantfed for the NGA (Next Generaton Attenuaton) project funded by PEER (Pacfc Earthquake Engneerng Research). The standard devaton of moment magntude was estmated from multple reported magntudes for each event where they exsted. The standard devaton reported n the NGA dataset was based on the consderaton of statstcal standard devaton, tme, and qualty of the data and method used to derve magntude (Chou, 2005). Fgure 2. Standard devaton of moment magntude. Curves are regressed on the standard devatons reported n the NGA dataset. Fgure 2 shows magntude versus standard devaton as reported n the NGA dataset. There s a large amount of scatter n the data, but a decrease n uncertanty wth an ncrease n magntude can be observed. Ths trend was conjectured by Moss (2003) based on the logc that for the nverson of sesmc moment the dmensons of the fault plane and the amount of slp assocated wth larger magntude events tend to be easer to defne than wth smaller magntude events.

5 Uncertanty also stems from dfferent nverson technques used: partal or complete waveforms, regonal or telesesmc recordngs, and dfferent Green s functons. Bgger magntude events also have more statons recordng the event (bgger sample sze), generally have a hgher sgnal to nose rato, and dfferent sesmology labs may be usng some of the same statons resultng n correlated results. Shown n Fgure 2 are a lnear regresson lne, logarthmc regresson lne, and the equaton from Moss (2003). All three curves exhbt a smlar slope, although the ntercepts of the regresson lnes are lower. For ths prelmnary analyss the logarthmc regresson lne was used to estmate the uncertanty assocated wth moment magntude for the subsequent regresson analyss. Other Parameters The measurement errors assocated wth other nput parameters have not been evaluated yet, as we are stll n the prelmnary stages of ths research. In partcular, uncertanty n the ste class as measured by V S30 (the shear wave velocty n the upper 30 meters) appears to have some mpact on the model varance. Also, measurement errors assocated wth the ste-to-source dstance, and the rake angle of the rupture plane may be quantfed for future analyses. For acceleraton, not just the peak acceleraton but spectral acceleraton values throughout the frequency range need to be evaluated. These are topcs that wll be covered n subsequent stages of ths research. Regresson Analyss Predctng strong ground attenuaton uses a unvarate-type model. It s unvarate because only one quantty of nterest s to be predcted from a set of measurable varables x=(x 1,x 2, x n ). The quantty of nterest n ths case s the spectral acceleraton. The general unvarate model can be wrtten as, Z = Z (x,θ) (1) where Θ denotes a set of model parameters used to ft the model to the observed data. In ths study varous models, based on attenuaton relatonshps proposed prevously n the lterature, wll be used. The generalzed unvarate model can then be wrtten as, Z(x,Θ) = ẑ(x,θ) + ε (2) where ẑ(x,θ) s the selected attenuaton relatonshp and ε s a random normal varate wth zero mean and unknown standard devaton that s the model error term. Aleatory uncertanty s found n the measured varables x and partly n the error term ε. Epstemc uncertanty s found n the model parameters Θ and partly n the model error term ε. Model Uncertanty In ths model formulaton the error term ε captures the mperfect ft of the model to the measurements. The mperfect ft may be due to nexact model form or due to mssng varables. The mssng varables can be consdered nherently random and that porton of the model error term s aleatory uncertanty. The porton of the model error term that s from the nexact model form s epstemc uncertanty.

6 Measurement Error Measurement error tends to comprse a large porton of the epstemc uncertanty n geoscence problems. Ths uncertanty comes from mprecse measurement of the varables x=(x 1,x 2, x n ). These measurement errors are treated as statstcally ndependent normally dstrbuted random varables wth zero mean (assumng unbased measurement errors) and measurable standard devaton. The errors are ncorporated as x = xˆ + e x where xˆ x s the measured value and e x s the measurement error. Statstcal Uncertanty The sze of the sample n wll nfluence the accuracy of the model parameters Θ. The larger the sample sze the less epstemc uncertanty ntroduced nto the model parameters. In ths case, there s a lmted amount of strong moton recordngs for model fttng. Parameter Estmaton through Bayesan Updatng A Bayesan framework s used to estmate the unknown model parameters (.e., regresson). The Bayesan approach s useful because t ncorporates all forms of uncertanty related to the problem of strong ground moton attenuaton nto the regresson analyss. Bayes rule s derved from smple rules of condtonal probablty, yet the smplcty portends lttle of the power of the Bayesan technque. Bayes rule can be wrtten as (Box & Tao, 1992), f (Θ) = c L(Θ) p(θ) (3) where; f (Θ) s the posteror dstrbuton representng the updated state of knowledge about Θ, L(Θ) s the lkelhood functon contanng the nformaton ganed from the observatons of x, p(θ) s the pror dstrbuton contanng our apror knowledge about Θ, and c = [ L(Θ) p(θ) d(θ)] 1 s the normalzng constant. The lkelhood functon s proportonal to the condtonal probablty of the observed events, gven the value of Θ. The lkelhood functon ncorporates the objectve nformaton that, n ths case, are the measurements assocated wth strong ground moton attenuaton. The pror dstrbuton can nclude subjectve nformaton known about the dstrbutons of Θ. The posteror dstrbuton ncorporates both the objectve and subjectve nformaton nto the dstrbutons of the model parameters. The process of performng Bayesan updatng nvolves formulatng the lkelhood functon, selectng a pror, calculatng the normalzng constant, and then calculatng the posteror statstcs. The pror dstrbuton tends to be the most controversal ssue for detractors of Bayesan methods. Box & Tao (1992) have shown that the use of a non-nformatve pror can lead to an unbased, data-drven estmate of the model parameters. A non-nformatve pror allows the data, through the lkelhood functon, to domnate the posteror dstrbuton, thereby mnmzng the role of the subjectve nformaton. A non-nformatve pror, by defnton, has no effect on the shape of the posteror dstrbuton and s used when no pror nformaton about the parameters s avalable. Gardon et al., (2002) have shown that for a unvarate model where the unknown parameters Θ are the coeffcents n a lnear expresson and the standard devatons σ

7 of ε, the nonformatve pror smplfes to, 1 p(σ ) (4) σ The mean vector M Θ and covarance matrx Σ ΘΘ can be calculated from the posteror dstrbuton of Θ. Computaton of these statstcs and the normalzng constant s non-trval, requrng multfold ntegraton over the Bayesan kernel. Importance samplng, a samplng algorthm as descrbed n Gardon (2002), was used to effcently perform these calculatons. Lkelhood Functon As defned above the lkelhood functon s proportonal to the condtonal probablty of observng a partcular event gven a value of Θ. In order to formulate the lkelhood functon a lmt-state must be defned to provde a threshold for defnng the probablty of observaton. For ths feasblty study the attenuaton relatonshp from Boore et al. (1997), s used as a bass for the lkelhood functon. Boore et al. (1997) was chosen because the database used n the regresson was provded n the paper. The functon form of ths attenuaton relatonshp s, log(y ) = θ +θ (M 6) +θ (M 6) 2 θ ln( R w 3 w 4 jb +θ 5 ) θ 6 ln(v s /θ 7 ) (5) where Y represents the spectral acceleraton value, M w s the moment magntude, R jb s the Joyner-Boore dstance, V S s the shear wave velocty n the upper 30 meters, and the θ s are the model parameters. Boore et al., (1997) determned the parameters of ths model by usng classcal regresson wth a two step procedure. To present ths attenuaton relatonshp as a lmt-state functon, the equaton s rearranged to descrbe the most lkely locaton of a threshold gven a value of Θ. Ths lmt-state would be where the threshold les at the zero mean of the error term at a value of Z for a gven x. Ths thereby mnmzes the error on ether sde of the threshold at that pont. From Equaton 2, Z = ẑ(x,θ ) + ε or ε = g (θ ) where g (θ ) = Z ẑ(x,θ ) and ε s the model error term at the th observaton. The attenuaton relatonshp of Campbell et al., (1997), shown n Equaton 5, then becomes, g(θ) = log(y ) [θ +θ (M 6) +θ (M 6) 2 θ ln( R w 3 w 4 jb +θ 5 ) θ 6 ln(v s /θ 7 )] (6) The lkelhood functon for the problem of strong ground moton attenuaton s the product of the probabltes of observng n values wth the lmt-state co-located wth the zero mean of the error term. Gven exact measurements and statcally ndependent observatons, the lkelhood can be wrtten as, n L(θ,σ ε ) P I{g ( θ ) = ε } (7) =1 where σ ε s the standard devaton of the error term ε. Gven that ε s a standard normal varate, Equaton 7 can be wrtten as, n 1 g (θ ) L(θ,σ ε ) ϕ (8) =1 σ ε σ ε

8 where ϕ s the standard normal dstrbuton functon. When measurement errors are consdered the lkelhood functon becomes, n 1 gˆ (θ ) L(θ,σ ε ) ϕ (9) =1 σˆ ε (θ,σ ε ) σˆ ε (θ,σ ε ) The above formulaton was used to estmate the statstcs of the model parameters, Θ, and the model error, ε, for the gven functonal form of the attenuaton relatonshp and the gven database. These estmated terms are analogous to the coeffcents solved for usng classcal regresson n Boore et al., (1997). The means and standard devatons of the coeffcents are used to defne the predctve model. The model varance s found usng a second order Taylor seres expanson about the mean pont. Feasblty Study Results The results of the feasblty study, usng the functonal form of the attenuaton relatonshp and the database from Boore et al., (1997), are shown n Fgure 3. Ths fgure shows a comparson of Boore et al., versus prelmnary results from ths study. M w =7.5 V S =750m/s Mechansm: unspecfed Fgure 3. Comparson plot of attenuaton relatonshp estmated usng classcal regresson wth exact parameters versus Bayesan regresson that ncorporates parameter uncertanty. The black curves are from Boore et al. (1997), the red curves from ths study. Plus/mnus one standard devaton curves are shown as dashed lnes.

9 There s a slght dfference n the lmt-states or mean regresson curves found usng classc versus Bayesan regresson. Ths s due to the nfluence of ncludng nexact parameters n the Bayesan regresson analyss. More mportant s the reduced model standard devaton found usng Bayesan regresson. The standard devaton s reduced because the parameter uncertanty, or measurement error of the nput parameters, s quantfed and ncorporated n the analyss. By ncludng the addtonal nformaton of parameter uncertanty we acheve an mproved (.e., reduced) estmate of the model standard devaton. For ths prelmnary study, Bayesan regresson was ntally performed wthout parameter uncertanty to confrm that a smlar standard devaton was calculated as Boore et al. Then the Bayesan regresson was performed ncludng parameter uncertanty, wth a coeffcent of varaton (standard devaton dvded by the mean) of ~0.10 for moment magntude and ~0.30 for peak acceleraton. As shown n Fgure 3 the total model standard devaton (the square root of the model varance) of the natural log of peak ground acceleraton,σ lny, s 0.386, compared to from Boore et al. The earthquake to earthquake component of the standard devaton, σ e, s the same for the two studes. Summary Presented here s a method to ncorporate parameter uncertanty, or the measurement error assocated wth the nput parameters, nto strong ground moton attenuaton relatonshps. Ths method uses Bayesan regresson for ncorporatng nexact parameters nto the regresson analyss. A feasblty study was carred out usng the functonal form of the attenuaton relatonshp and the database from Boore et al., (1997). The results of ths feasblty study demonstrate that a reduced or better-constraned model varance s one beneft of the presented methodology. As part of ths ongong research study; further analyss of the statstcs of strong ground moton attenuaton, exploraton of other benefts of usng a Bayesan approach such as model optmzaton and correlaton analyss, and analyss usng other attenuaton models wll be carred out n the future. Acknowledgements Components of ths study are supported by the PEER (Pacfc Earthquake Engneerng Research center) Lfelnes Program n conjuncton wth the NGA (next generaton attenuaton) study. Ths support s gratefully acknowledged. Thanks to Dr. Alson Fars for her assstance n troubleshootng the Bayesan regresson algorthm. References Ang, A.H-S. & Tang, W. H. (1975) Probablty Concepts n Engneerng Plannng and Desgn, Vol. 1. John Wley and Sons, Inc. Abrahamson, N.A., & Slva, W.J. (1997). Emprcal Response Spectral Attenuaton Relatons for Shallow Crustal Earthquakes. Sesmologcal Research Letters, Vol. 68, No. 1. Boore, D.M., Joyner, W.B., and Fumal, T.E. (1997). Equatons for Estmatng Horzontal Response Spectra and Peak Acceleraton from Western North Amercan Earthquakes: A Summary of Recent Work. Sesmologcal Research Letters, Vol. 68, No. 1.

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