Coseismic sediment deformation during the 1989 Loma Prieta

earthquake

Ronaldo I. Borja1 and WaiChing Sun1

Received 31 July 2007; revised 6 February 2008; accepted 14 April 2008; published 19 August 2008.

[1] Inelastic horizontal sediment deformation from the MS 7.1, 17 October 1989 LomaPrieta earthquake has been estimated to be on the order of 20 cm at a site inGilroy, California,about 32 km east of the epicenter. This estimate was based on combined deterministic-stochastic simulations using ground motion data and sediment properties measured from insitu seismic tests and laboratory tests on cored sediment samples of up to 170 m deep. Thecalculated deformation is comparable to measured horizontal coseismic displacements atvarious stations in the Santa Cruz network, suggesting that previously back-figured faultrupture mechanisms may have been influenced by the coseismic sediment deformation.

Citation: Borja, R. I., and W. C. Sun (2008), Coseismic sediment deformation during the 1989 Loma Prieta earthquake, J. Geophys.

Res., 113, B08314, doi:10.1029/2007JB005265.

1. Introduction

[2] Leveling surveys, geodolyte, Global PositioningSystem (GPS), and very long interferometry surveys hadbeen used to estimate the coseismic ground displacementsduring the MS 7.1, 17 October 1989 Loma Prieta earthquake[Arnadottir and Segall, 1994; Clark et al., 1990; Darraghand Shakal, 1991; Lisowski et al., 1990, 1991; Segall andLisowski, 1990; Snay et al., 1991; Williams and Segall,1996]. These surveys were used to infer the characteristicfault movement associated with this event with the assump-tion that measured surface displacements are bedrock dis-placements. Most stations used to determine coseismicmovements from the Loma Prieta earthquake are at or nearmountain tops (M. Lisowski, personal communication,2007). These stations were either former triangulation ortrilateration stations, and to provide a clear line of sightmountain tops (where accessible) were preferred. A moun-tain top, however, is not always a rock outcrop; sometimes itis formed by uplifted sediments. Where exposed outcrop isabsent the station tablets were set on rod driven to refusal, oron concrete piers buried 1 m or more into the ground. Forstations sitting on top of an uplifted sediment, the coseismicground surface movement is the sum of the bedrock dis-placement and the inelastic sediment deformation. To obtainthe bedrock movement it is necessary to subtract the inelasticsediment deformation from the measured surface movement.[3] There is no accurate technique for measuring the

coseismic inelastic sediment deformation [Ambraseys andMenu, 1988; Bray and Travasarou, 2007; Jibson, 1993;Kramer, 1996; Newmark, 1965]. In principle, one needs adownhole instrumentation array that extends all the waydown to the bedrock level, but even then these arrays are

equipped with accelerometers that only measure accelerationtime histories, not displacement time histories. To obtain thecorresponding displacement time history one must integratethe recorded acceleration time history twice, say, with a stableNewmark family algorithm. However, accelerograms con-tain noise and are not an accurate reproduction of the seismicevent. The records produced by the sensors are alwayscombinations of signals that represent the actual motionand the extraneous noise generated by insufficient decimalpoints in transcribing digitized data, physical external noisearound the seismograph, tilting of the seismograph base,uncertainty of the initial conditions, and other factors [Booreet al., 2002; Boore and Bommer, 2005; Bradner and Reichle,1973; Shakal and Petersen, 2001; Trifunac and Todorovska,2001]. Due to the arbitrary nature of the noise and thedifficulties to identify the source of errors, the residualdisplacements recovered from accelerograms are highlysensitive to the subjective choice of correction schemes andfiltering parameters (The process of selecting filter criteria isquite subjective: what may be noise to some could be animportant signal to others. In addition to the character of therecorded signal, factors considered in the selection of filtercorners include the magnitude of the event, source mecha-nism, wave propagation path, and the fundamental period ofthe structure if data were collected from a structure.). Thusthe absolute coseismic displacement produced from accelero-grams is often unreliable. Given the above difficulties, therehas been a serious lack of knowledge and understanding onthe magnitude of inelastic sediment deformation during anearthquake.[4] In this paper we use numerical simulations based on

published data on sediment properties to estimate thecoseismic sediment deformation at a specific site duringthe 1989 Loma Prieta earthquake. We construct a mechan-ical model for the sediment [Andrade and Borja, 2006;Borja and Amies, 1994; Borja et al., 1999, 2000, 2002;Borja and Sun, 2007] and subject this model to seismicexcitations recorded at this same site. The site investigated,Gilroy 2 (latitude 36.982N, longitude 121.556W, 12.7 km

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1Department of Civil and Environmental Engineering, StanfordUniversity, Stanford, California, USA.

Copyright 2008 by the American Geophysical Union.0148-0227/08/2007JB005265$09.00

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to fault rupture, 12.1 km to surface projection of rupture;see Figure 1), is part of the Gilroy instrumentation arrayconsisting of an alignment of six stations extending fromsandstone on the east, across the alluvial Santa Clara Valley,California, to sandstone on the west [Darragh and Shakal,1991]. The array is maintained by the California StrongMotion Instrumentation Program (CSMIP) in cooperationwith the United States Geological Survey (USGS). StationGilroy 1 (latitude 36.973N, longitude 121.572W, 11.2 kmto fault rupture, 10.5 km to surface projection of rupture) islocated about 1.5 km west of station Gilroy 2. Gilroy 1 isunderlain by moderately weathered sandstone at the sur-face, with thin beds of shale underneath, whereas Gilroy 2is underlain by alluvial-fan deposits, or stiff soil above thewater table, up to a depth of around 170 m. Below thesediment of Gilroy 2 is the same shale as that found inthe rock outcropping in Gilroy 1. The two stations aboveare the nearest rock-sediment pair of stations in the Gilroyinstrumentation array and have been selected precisely toinvestigate the dynamic local response of the alluvium.

2. Mechanical Model for Gilroy 2 Alluvium

[5] The mechanical model for Gilroy 2 alluvium con-sisted of horizontal layers, or slabs, of finite elements shown

in Figure 2. The model assumed horizontally stratifiedsediment layers resting on a horizontal bedrock, and bodywaves from the dynamic excitation of the bedrock propa-gated vertically through the model in a one-dimensionalfashion. Each slab was represented by a bounding surface-type elastoplastic constitutive law in which deformation wastaken as the sum of elastic, inelastic, and viscous parts[Borja and Amies, 1994; Borja et al., 1999, 2002]. Theelastic part of deformation is recoverable at the end ofseismic shaking, whereas the inelastic part is responsible forthe coseismic residual sediment deformation. The appropri-ate structural dynamics finite element code, called SPEC-TRA, has been described and documented in detail by Borjaet al. [1999, 2000].[6] The finite element code SPECTRA requires the

following soil parameters for input: mass density r; elasticshear modulus Ge; radius of the bounding surface R;coefficient h and exponent m of the exponential hardeningfunction; and coefficient of proportionality c relating theglobal viscous damping matrix D to the global elasticmoduli matrix Ce, i.e., D = cCe. We refer the readers toBorja and Amies [1994] and Borja et al. [1999] for somegeneral notations and background of the constitutive model.The elastic material parameters of the constitutive modelwere derived from results of seismic velocity tests; the

Figure 1. Loma Prieta region, California, showing horizontal coseismic displacements at stations in theSanta Cruz network (diamonds) and displacements estimated by the Stanford group (squares) from GPSmeasurements: 1, LP1; 2, Traill; 3, R57; 4, Crowell; 5, Cliff; 6, Porter; 7, R121; 8, Leon; 9, Pajaro 3.Error ellipses are 95% confidence interval. Station 10 is Gilroy 2 at which we computed the horizontalcoseismic sediment deformation. The shaded ellipse represents uncertainty associated with one standarddeviation away from the mean sediment properties. Figure adapted from Arnadottir and Segall [1994].

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elastoplastic parameters were obtained from laboratory-derived modulus degradation and damping ratio curves.[7] Sediment deformations were calculated from the

mechanical properties of the alluvium at Gilroy 2. A seriesof geophysical surveys were performed to measure shearand compression wave velocities within the alluvium[EPRI, 1993]. Shear wave velocities vary from about200 m/s near the ground surface and reaches a value ofabout 700 m/s at 170 m depth; compressional wave veloc-ities are about 300 m/s to 2100 m/s at the correspondingdepths [see Figure 2]. Shear modulus reduction and damp-ing ratio increase with shear strain. They were established

from very detailed laboratory testing of cored samples atdifferent depths within the alluvium, and are relevant formodeling the inelastic deformation. Statistical variations ofmaterial properties also have been well documented for thealluvium [Andrade and Borja, 2006; Borja et al., 2000].[8] For the mechanical model to provide meaningful

solutions, the calculated sediment deformation should notbe significantly influenced by the noise and baseline offsetspresent in the input ground motion. Therefore we tested thesensitivity of the mechanical model to noise and baselineoffset corrections by subjecting the base of the alluvium tounprocessed and processed input ground motions from

Figure 3. Coseismic horizontal deformation of alluvium at Gilroy 2 calculated from the integratedaccelerograms at Gilroy 1 and Gilroy 2. Open circles denote start and end points of surface movementrelative to the bedrock, and the number next to the straight line denotes the inelastic sedimentdeformation. (left) Raw accelerograms produced a sediment deformation of 49.7 cm. (right) PEER-filtered accelerograms resulted in a sediment deformation of 0.05 cm.

Figure 2. Mechanical model for alluvium (‘‘stiff soil’’) at Gilroy 2. With a time shift, the ground motionmeasured at the rock outcropping Gilroy 1 was applied at the bottom of the finite element model and thecalculated response at the top was compared to the measured sediment response at Gilroy 2. Themechanical model utilized the elastic shear modulus inferred from S- and P-wave velocities and the shearmodulus reduction and damping ratio curves established from laboratory testing of cored samples. Inputmotion at the bedrock consisted of unprocessed (raw) and two filtered ground motion data from Gilroy 1.NS = North-South; EW = East-West; UD, Up-Down.

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Gilroy 1 and comparing the predicted responses. Theunprocessed (raw) data from Gilroy 1 contain no noisefiltering and no baseline or sensor offset corrections, andwere made available to the authors by the CaliforniaGeological Survey (CGS) for this research study (raw dataare not available from the CGS website). Two processedground motion data from Gilroy 1 were also considered inthe analyses: the first filtered accelerograms, downloadedfrom the CGS website, were processed by CGS/CSMIP,and have bandpass filtered with ramps at 0.080–0.160and 23.00–25.00 Hz. The second filtered accelerograms,downloaded from Pacific Earthquake Engineering ResearchCenter (PEER) website, have high pass and low-pass filtersof 0.2 Hz and 50 Hz, respectively.

3. Results: Coseismic Sediment Deformation atGilroy 2

[9] For purposes of definition, the coseismic sedimentdeformation is defined as the displacement of the ground

surface relative to the underlying bedrock. As noted earlier,it is not possible to calculate the absolute coseismic dis-placement of the underlying bedrock from the unprocessedaccelerograms alone because of noise and baseline offsetspresent in the ground motion data. This point is wellillustrated in Figure 3, which shows a comparison of thecoseismic deformations obtained by simply subtracting thetime-integrated accelerogram at Gilroy 1 from the time-integrated accelerogram at Gilroy 2 (The accelerograms atGilroy 1 and Gilroy 2 were generated by analog film-recording Kinemetric SMA-1 instruments.): the raw accel-erograms yielded a final coseismic deformation of 49.7 cm,whereas the PEER-filtered accelerograms resulted in nearlyzero deformation. Exactly how much of the 49.7 cmdeformation represent noise is unknown, so a time-integra-tion of the accelerograms alone cannot be used to estimatethe actual coseismic deformation.[10] In the following numerical simulations we resorted

instead to mechanical modeling to estimate the coseismicdeformations in the alluvium at Gilroy 2. Figure 4 shows the

Figure 4. Coseismic horizontal deformation of alluvium at Gilroy 2 from the 1989 Loma Prietaearthquake calculated by the proposed mechanical model. Open circles denote start and end points ofsurface movement relative to the bedrock, and the number next to the straight line denotes the inelasticsediment deformation. (lower right) Uncertainty ellipses for one-half (smallest ellipse), one, and two(largest ellipse) standard deviations from the mean sediment properties. Input functions are as follows:raw, unprocessed Gilroy 1 accelerograms; CGS/CSMIP and PEER, processed Gilroy 1 accelerograms.

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calculated coseismic horizontal deformations of the alluviumat Gilroy 2 from performing site response analyses using theraw and processed Gilroy 1 accelerograms as input groundmotions. The processed accelerograms resulted in a 10%reduction of the calculated sediment deformation relative tothe unprocessed bedrock motion. Considering the complex-ity of earthquake records, this difference is nearly inconse-quential and suggests that the mechanical model is notsignificantly affected by the choice of bandpass filteringcriteria. That the noise and baseline offsets in the inputground motion did not significantly affect the calculatedsediment deformation could be explained from the facts thatthey only produced rigid body motions for a constantresidual velocity, and when the residual velocity was notconstant the baseline offset was too small to produce anysignificant inelastic deformation.[11] We also tested the sensitivity of the mechanical

model to statistical variations of sediment properties byperforming a combined stochastic-deterministic analysissimilar to that presented by Andrade and Borja [2006].The rationale for conducting this study is that values of thesediment properties used in the numerical simulations havetheir own uncertainties, and so the calculated coseismicsediment deformations have probability distributions.Uncertainties in the elastic sediment properties were quan-tified from data available for the alluvium [EPRI, 1993]; inthe absence of sufficient statistical data for the modulus

Figure 5. Comparison of unprocessed recorded (red/thin)and calculated (blue/thick) responses at Gilroy 2, EWcomponent: (top) acceleration time history, (middle) accel-eration response spectra at 5% damping, (bottom) Fourieracceleration amplitude spectra.

Figure 6. Comparison of unprocessed recorded (red/thin)and calculated (blue/thick) responses at Gilroy 2, NScomponent: (top) acceleration time history, (middle) accel-eration response spectra at 5% damping, (bottom) Fourieracceleration amplitude spectra.

Figure 7. Time history of maximum resolved shear straindeveloped in the sediment column at Gilroy 2 site usingunprocessed Gilroy 1 ground motion as bedrock forcingfunction. Shaded square, bottom of slab i; open square, topof slab.

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reduction and damping ratio curves, uncertainties in thesecurves were estimated from generic soil data [Andrade andBorja, 2006]. For the combined stochastic-deterministicsimulations we utilized the unprocessed (raw) data atGilroy 1 deterministically as input into the mechanical

model and treated the uncertain sediment properties asrandom variables. Statistical bands of random variables forthe Gilroy 2 alluvium were reported by Andrade and Borja[2006], which we used in the stochastic (Monte Carlo-type)simulations. Empirical cumulative distributions functions

Figure 8. Final inelastic shear strain profile developed in the Gilroy 2 sediment column: (a) magnitudeof resolved shear strain (Di); (b) orientation of plane of resolved shear strain (qi).

Figure 9. Displacement amplitude response spectra at 5% damping. Red (thin) curves are spectraobtained from the CGS/CSMIP-processed accelerograms at Gilroy 2. Blue (thick) curves are spectraobtained from the calculated site response at Gilroy 2, in which the CGS/CSMIP-processedaccelerograms at Gilroy 1 was used as input forcing function.

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were generated for East-West (EW) and North-South (NS)coseismic sediment deformations, and were combined toform uncertainty ellipses as shown in Figure 4. It can beobserved that the uncertainty ellipses did not inscribe theorigin of the displacement axes up to two standard deviationsbeyond the mean values, implying that the relative displace-ments did not reverse in sign (that is, it continued to exhibit anortheasterly trend) even though the sediment properties hadbeen perturbed by up to two standard deviations away fromthe mean values.[12] Figure 4 suggests that the coseismic horizontal

deformation of the alluvium at Gilroy 2 is on the order of20 cm (mean value) with a standard deviation of around4 cm. This deformation is comparable in magnitude to thecoseismic horizontal displacements measured at variousstations in the Santa Cruz network. This horizontal defor-mation is plotted as a vector at Station 10 in Figure 1,along with the coseismic displacements obtained from

leveling surveys, geodolyte, GPS, electronic distancemeasurements (EDM), and very long interferometry sur-veys at various stations in the Santa Cruz network. Notethat error ellipses for the other stations in Figure 1represent 95% confidence interval associated with survey-ing errors, and do not have the same meaning as theuncertainty ellipse used for Station 10.[13] The present analysis assumed the ground to be flat,

so evidently the sediment deformation must be adjusted fordifferent sites to account for spatially varying sedimentproperties, bedrock elevation, basin effects, water table,and sloping grounds. Because a sediment column represen-tation tends to constrain the kinematics of deformation, andtherefore underestimate deformation, three-dimensional ba-sin effects and sloping grounds, among others, are expectedto further amplify the ground displacements. Nevertheless,the results from the vertical sediment column representationsuggest that the coseismic sediment deformation that was

Figure 10. Calculated coseismic horizontal deformation of alluvium at Gilroy 2 from the 1989 LomaPrieta earthquake with EW and NS ground motions applied separately. Open circles denote start and endpoints of surface movement relative to the bedrock, and the number next to the straight line denotes theinelastic sediment deformation. (lower right) The coseismic deformation predicted by a coupledsimulation in which the EW and NS motions were applied simultaneously. Input functions are as follows:RAW, unprocessed Gilroy 1 accelerograms; CGS/CSMIP and PEER, processed Gilroy 1 accelerograms.

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initially thought to be small for a ‘‘stiff soil’’ at Gilroy 2 wasin fact comparable to the coseismic bedrock displacementsin the Loma Prieta region. This result could have importantimplications for studies of fault rupture mechanisms partic-ularly when surveying stations are located on top of upliftedsediments.[14] To validate the results of Figure 3 the mechanical

model was used to reproduce the acceleration time historyresponses at the top of the alluvium for comparison to therecorded Gilroy 2 accelerograms. Figures 5 and 6 showgenerally good agreement between the EW and NS (raw)ground motions at Gilroy 2 with the calculated responses atthe top of the alluvium. In the time-history plot the calculatedresponses were shifted by 0.3 s to make the predicted peakground accelerations coincide with the recorded peak valueand thus account for the delay in the arrival of seismic waves.Note that in Figures 5 and 6 it would be impossible for thepredicted and recorded ground motions to be one on top ofthe other due to the following reasons: (1) Gilroy 2 is on theedge of a steeply dipping bedrock interface where two-dimensional basin effects could be pronounced [Silva,1991]; (2) there could be some alteration of signal as theseismic waves travelled from Gilroy 1 to the bedrock atGilroy 2; and (3) there could be some inaccuracies in themathematical representation of the alluvium. However,

Figures 5 and 6 show that the agreement between therecorded and predicted responses is generally good, lendingcredibility to the estimated coseismic horizontal deformationof the alluvium at Gilroy 2.[15] Figures 7 and 8 describe the extent of inelastic

behavior occurring within the various layers at Gilroy 2site as the sediments responded to the unprocessed Gilroy 1excitation applied at the base of the sediment column.Figure 7 shows the time histories of the maximum resolvedshear strain developed over the 170 m-thick sedimentdeposit. The maximum resolved shear strain was calculatedas follows. For any slab i and at any time t, let dNS and dEWrepresent, respectively, the NS and EW total displacementsat the top, and let �dNS and �dEW denote the correspondingdisplacements at the bottom of the slab. The relativehorizontal displacements in the two directions are dNS =dNS � �dNS and dEW = dEW � �dEW, so the resolved shearstrain in the slab is

gi ¼ Di=h; Di ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2NS þ d2EW

q; ð1Þ

where h = 2 m is the slab thickness. For any time t themaximum resolved shear strain is equal to max(gi) for all i,and is plotted in Figure 7. The time-history plot shown inFigure 7 indicates a peak shear strain value of 1.45%coinciding with the arrival of the strongest seismic pulse,and tapering off to a value of 1.30%. These strains areslightly higher than the 0.8% maximum shear strainreported in Figure 20 of Borja et al. [2000], which wascalculated using the CGS/CSMIP-processed Gilroy 1ground motion as input forcing function.[16] Figure 8a reveals that the maximum resolved shear

strain actually occurred approximately at a depth of 40 mwithin a softer sediment layer just before it transitioned intoa stiffer sediment layer at the bottom (see the shear modulusprofile shown in Figure 2). Secondary peak strain values onthe order of 0.3 to 0.7% developed at depths of 80 to 100 m,where, according to Figure 2, softer sediments were alsoencountered. The remainder of the plastic shear strain wason the order of approximately 0.1%. As can be gleanedfrom the modulus reduction curves of Figure 2, these plasticshear strains represent significant modulus degradation, inwhich the secant stiffnesses had been reduced to as much as10 to 50% of their initial elastic values.[17] For any slab i and time t the orientation of the plane

of resolved shear strain is given by the angle

qi ¼ tan�1dNS=dEW: ð2Þ

[18] Figure 8b shows that at large values of t (steadystate) most values of this angle are between zero and 90�,implying that the entire sediment deposit has been shearednearly in the same northeasterly direction (compare themovement of Station 10 in Figure 1, for example). Thetotal relative displacement of the ground is 20 cm represent-ing the cumulative plastic deformation of the sedimentcolumn (see Figure 4). This corresponds to an averageshear strain of (0.20/170) � 100% = 0.12%, or an averagemodulus reduction of 25 to 50%, according to Figure 2.[19] Figure 9 compares the recorded and calculated

displacement amplitude response spectra at 5% dampingfor Gilroy 2 ground motions. For purposes of definition,the recorded ground motions (red/thin) are the CGS/

Figure 11. Comparison of unprocessed recorded (red/thin)and calculated (blue/thick) responses at Gilroy 2, EWcomponent: (top) acceleration time history, (middle) accel-eration response spectra at 5% damping, (bottom) Fourieracceleration amplitude spectra (cf. Figure 5). Note: NScomponent of forcing function was assumed 0.

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CSMIP-processed accelerograms, while the calculatedground motions (blue/thick) were those obtained from thesite response analysis utilizing the CGS/CSMIP-processedground motions at Gilroy 1 as input forcing function on thesediment column. Observe that the maximum displacementamplitudes occurred at higher periods (between 3 and 5 s)compared with the acceleration response spectra where themaximum amplitudes occurred at lower periods (between0.5 and 1.0 s; see Figures 5 and 6). The PEER-processedground motions produced very similar displacement am-plitude spectra that are not anymore reported in thispaper.

4. Some Aspects of Three-Dimensional InelasticDynamic Analysis

[20] This section is devoted to some aspects of inelasticfinite element analysis in the dynamic regime and theirimplications for inelastic deformation calculation. Webelieve that some discussions on this topic are in ordersince there are very few inelastic numerical models avail-able in the literature that can address the problem ofinelastic deformation for earthquake engineering applica-tions. In site response studies, particularly those that utilizean equivalent linear analysis procedure [Schnabel et al.,1972; Yoshida et al., 2002], it is customary to calculate the

total seismic response of horizontally layered sedimentsfrom the sum of the individual responses to two orthog-onal excitations applied separately. Such a decoupledmethod of analysis is not meaningful in nonlinear analysissince the principle of superposition does not hold when theresponse is inelastic. In the following discussion weexamine the importance of the coupling effect on thecalculated ground response of the alluvium at Gilroy 2.[21] We conducted hypothetical simulations to calculate

the coseismic horizontal deformation of the Gilroy 2 allu-vium by applying the EW and NS Gilroy 1 ground motionsseparately. Figure 10 shows that the coseismic sedimentdeformations, obtained by simply adding the two horizontalcomponents of displacements, are much smaller than thosepredicted by the solution that assumed full kinematicalcoupling. This result is to be expected since inelasticdeformations do not add up linearly but instead combinein a nonlinear fashion to amplify the displacements. Thisexample thus affirms that full kinematical coupling isessential for a meaningful capture of inelastic sedimentdeformation. (With the present constitutive model, whichonly accounts for plastic yielding in shear (i.e., deviatoricplasticity), the vertical component of ground motion is notcoupled with the horizontal components for waves propa-gating vertically on flat grounds. For plasticity models thataccount for combined volumetric and deviatoric yielding,

Figure 13. Comparison of unprocessed recorded (red/thin)and calculated (blue/thick) hypothetical ‘‘elastic’’ responsesat Gilroy 2, EW component. The elastic simulationssuppressed plastic hysteretic damping through a very largebounding surface. Top: acceleration-time history; middle:acceleration response spectra at 5% damping; bottom:Fourier acceleration amplitude spectra (cf. Figure 6).

Figure 12. Comparison of unprocessed recorded (red/thin)and calculated (blue/thick) responses at Gilroy 2, NScomponent: (top) acceleration time history, (middle) accel-eration response spectra at 5% damping, (bottom) Fourieracceleration amplitude spectra (cf. Figure 6). Note: EWcomponent of forcing function was assumed 0.

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for sloping grounds, or for waves traveling in arbitrarydirections, all three components of motion must be appliedsimultaneously.)[22] On the other hand, Figures 11 and 12 show, respec-

tively, the EWand NS acceleration responses of the Gilroy 2alluvium obtained by applying the two horizontal compo-nents of bedrock motion separately. The calculated acceler-ation responses are very similar to the solutions obtainedwith full kinematical coupling (cf. Figures 5 and 6, respec-tively). The predictions also seem to agree quite well withthe recorded ground motions, suggesting that the lack ofkinematical coupling does not have much impact on theacceleration responses. Of course, the previous exampleshowed that this is not the case with displacements. It thusappears that comparing the acceleration responses alone is aweaker test of the robustness of a model, and that oneshould also devise an experimental program to validate themodel with the stronger deformation test.[23] Finally, we illustrate the impact of inelastic defor-

mation on the acceleration responses. We recall that theconstitutive model defines two distinct types of damping:plastic hysteretic and viscous. The former pertains todamping generated by the nonlinear soil behavior, whereasthe latter pertains to rate effects. Typically, one of theeffects of nonlinear soil behavior is to increase the naturalperiods of the soil deposit. To illustrate this point, wesuppressed the plastic hysteretic damping in the constitu-

tive model on another hypothetical simulation (that is, weneglected plasticity altogether). The results are shown inFigures 13 and 14. Synthetic accelerograms (blue/thickcurves) are now much higher compared to the recordedaccelerograms (red/thin). This is to be expected since thesynthetic accelerograms neglected plastic hysteretic damp-ing, and so overall the system is now ‘‘underdamped.’’Comparing with Figures 5 and 6, with plastic hystereticdamping the acceleration amplitudes have decreased whilethe natural periods at peak resonances have increased,from about 0.4 s for the elastic case to about 0.6 s forthe full elastoplastic case.

5. Closure

[24] The coseismic horizontal sediment deformation dur-ing the 1989 Loma Prieta earthquake has been estimated at asite in Gilroy, California to be around 20 ± 4 cm. Thisestimate accounts for the constitutive properties and thick-ness of the alluvium, the statistical variations of sedimentconstitutive properties, the ground motion data, and theeffects of baseline corrections on the input ground motion.The mechanical model assumes a flat ground surface; for asloping ground the coseismic horizontal sediment deforma-tion is expected to be greater. To extrapolate this estimate toother sites it is necessary to incorporate the local sedimentthicknesses and the local constitutive properties, as well asthe spatial variation of the input bedrock excitation.[25] Coseismic sediment deformation is difficult to mea-

sure in the field since the bedrock on which the sedimentrests is not exposed for geodetic surveying or satelliteimaging. Even if a sediment site contains downhole arraysequipped with accelerometers, the coseismic displacementscannot be readily calculated from integrating the accelera-tion time history data due to the presence of noise andbaseline offsets that could lead to meaningless estimates ofthe residual displacements. The procedure adopted in thispaper thus calculates the coseismic sediment deformationbased on the constitutive properties of the sediment. Wehave shown that this procedure is not significantly affectedby baseline corrections on the input ground motion.

[26] Acknowledgments. We thank Dr. H. Haddadi for providing rawground motion data for Gilroy 1 and Gilroy 2 stations, Dr. M. Lisowski forassistance with interpreting the coseismic movements in the Loma Prietaregion, Dr. Charles Menun for allowing us to use his structural reliabilitycode, Dr. Paul Spudich for suggesting relevant references, and the reviewersand editor of the journal for their truly constructive reviews of themanuscript. This research was funded by NSF grant CMS-0201317.

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Figure 14. Comparison of unprocessed recorded (red/thin)and calculated (blue/thick) hypothetical elastic responses atGilroy 2, NS component. The elastic simulations suppressedplastic hysteretic damping through a very large boundingsurface Top: acceleration-time history; middle: accelerationresponse spectra at 5% damping; bottom: Fourier accelera-tion amplitude spectra (cf. Figure 6).

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�����������������������R. I. Borja and W. C. Sun, Department of Civil and Environmental

Engineering, Stanford University, Stanford, CA 94305, USA. (borja@stanford.edu)

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