prediction of nonlinear soil response

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1609 Bulletin of the Seismological Society of America, Vol. 94, No. 5, pp. 1609–1629, October 2004 Prediction of Nonlinear Soil Effects by Stephen Hartzell, L. F. Bonilla, and Robert A. Williams Abstract Mathematical models of soil nonlinearity in common use and recently developed nonlinear codes are compared to investigate the range of their predictions. We consider equivalent linear formulations with and without frequency-dependent moduli and damping ratios and nonlinear formulations for total and effective stress. Average velocity profiles to 150 m depth with midrange National Earthquake Haz- ards Reduction Program site classifications (B, BC, C, D, and E) in the top 30 m are used to compare the response of a wide range of site conditions from rock to soft soil. Nonlinear soil models are compared using the amplification spectrum, calculated as the ratio of surface ground motion to the input motion at the base of the velocity profile. Peak input motions from 0.1g to 0.9g are considered. For site class B, no significant differences exist between the models considered in this article. For site classes BC and C, differences are small at low input motions (0.1g to 0.2g), but become significant at higher input levels. For site classes D and E the overdamping of frequencies above about 4 Hz by the equivalent linear solution with frequency- independent parameters is apparent for the entire range of input motions considered. The equivalent linear formulation with frequency-dependent moduli and damping ratios under damps relative to the nonlinear models considered for site class C with larger input motions and most input levels for site classes D and E. At larger input motions the underdamping for site classes D and E is not as severe as the over- damping with the frequency-independent formulation, but there are still significant differences in the time domain. A nonlinear formulation is recommended for site classes D and E and for site classes BC and C with input motions greater than a few tenths of the acceleration of gravity. The type of nonlinear formulation to use is driven by considerations of the importance of water content and the availability of laboratory soils data. Our average amplification curves from a nonlinear effective stress formulation compare favorably with observed spectral amplification at class D and E sites in the Seattle area for the 2001 Nisqually earthquake. Introduction Since the occurrence of the 1994 Northridge, 1995 Hyogoken-Nanbu (Kobe), and the 2001 Nisqually earth- quakes, soil nonlinearity has assumed a prominent role in the analysis of strong ground motion. After the Northridge earthquake, several studies reported widespread nonlinear sediment response (Field et al., 1997, 1998; Trifunac and Todorovska, 1996, 1998; Beresnev et al., 1998; Hartzell, 1998). Previous to this earthquake, clear measurements of soil nonlinearity in strong ground motion were generally limited to the extreme cases of liquefaction. Aguirre and Irikura (1997), Fukushima et al. (2000), and Frankel et al. (2002) have reported varying degrees of soil nonlinearity, dependent on the site conditions, for the Hyogoken-Nanbu and Nisqually earthquakes. Although there has been a gen- eral realization among seismologists that nonlinear effects are more common than previously believed, there is uncer- tainty concerning the appropriate mathematical model to use for predicting these effects. This article addresses the vari- ation that one can expect from one model to another and makes recommendations for their appropriate use. Soil nonlinearity and its correct prediction have major importance for seismic hazard assessment and building de- sign requirements. As an example, National Earthquake Hazards Reduction Program (NEHRP) (2000) recommen- dations for soil site amplification factors decrease to 1.0, reflecting expected nonlinear effects, for site classes C and D as effective peak acceleration increases to 0.5g at 5 Hz. However, these amplification factors are being continually reevaluated as new data become available (Crouse, 1995; Borcherdt, 1996; Abrahamson and Silva, 1997; Joyner and Boore, 2000). These studies generally indicate that the cur- rent amplification values may imply too much nonlinearity

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Page 1: Prediction of Nonlinear Soil Response

1609

Bulletin of the Seismological Society of America, Vol. 94, No. 5, pp. 1609–1629, October 2004

Prediction of Nonlinear Soil Effects

by Stephen Hartzell, L. F. Bonilla, and Robert A. Williams

Abstract Mathematical models of soil nonlinearity in common use and recentlydeveloped nonlinear codes are compared to investigate the range of their predictions.We consider equivalent linear formulations with and without frequency-dependentmoduli and damping ratios and nonlinear formulations for total and effective stress.Average velocity profiles to 150 m depth with midrange National Earthquake Haz-ards Reduction Program site classifications (B, BC, C, D, and E) in the top 30 m areused to compare the response of a wide range of site conditions from rock to softsoil. Nonlinear soil models are compared using the amplification spectrum, calculatedas the ratio of surface ground motion to the input motion at the base of the velocityprofile. Peak input motions from 0.1g to 0.9g are considered. For site class B, nosignificant differences exist between the models considered in this article. For siteclasses BC and C, differences are small at low input motions (0.1g to 0.2g), butbecome significant at higher input levels. For site classes D and E the overdampingof frequencies above about 4 Hz by the equivalent linear solution with frequency-independent parameters is apparent for the entire range of input motions considered.The equivalent linear formulation with frequency-dependent moduli and dampingratios under damps relative to the nonlinear models considered for site class C withlarger input motions and most input levels for site classes D and E. At larger inputmotions the underdamping for site classes D and E is not as severe as the over-damping with the frequency-independent formulation, but there are still significantdifferences in the time domain. A nonlinear formulation is recommended for siteclasses D and E and for site classes BC and C with input motions greater than a fewtenths of the acceleration of gravity. The type of nonlinear formulation to use isdriven by considerations of the importance of water content and the availability oflaboratory soils data. Our average amplification curves from a nonlinear effectivestress formulation compare favorably with observed spectral amplification at classD and E sites in the Seattle area for the 2001 Nisqually earthquake.

Introduction

Since the occurrence of the 1994 Northridge, 1995Hyogoken-Nanbu (Kobe), and the 2001 Nisqually earth-quakes, soil nonlinearity has assumed a prominent role inthe analysis of strong ground motion. After the Northridgeearthquake, several studies reported widespread nonlinearsediment response (Field et al., 1997, 1998; Trifunac andTodorovska, 1996, 1998; Beresnev et al., 1998; Hartzell,1998). Previous to this earthquake, clear measurements ofsoil nonlinearity in strong ground motion were generallylimited to the extreme cases of liquefaction. Aguirre andIrikura (1997), Fukushima et al. (2000), and Frankel et al.(2002) have reported varying degrees of soil nonlinearity,dependent on the site conditions, for the Hyogoken-Nanbuand Nisqually earthquakes. Although there has been a gen-eral realization among seismologists that nonlinear effectsare more common than previously believed, there is uncer-

tainty concerning the appropriate mathematical model to usefor predicting these effects. This article addresses the vari-ation that one can expect from one model to another andmakes recommendations for their appropriate use.

Soil nonlinearity and its correct prediction have majorimportance for seismic hazard assessment and building de-sign requirements. As an example, National EarthquakeHazards Reduction Program (NEHRP) (2000) recommen-dations for soil site amplification factors decrease to 1.0,reflecting expected nonlinear effects, for site classes C andD as effective peak acceleration increases to 0.5g at 5 Hz.However, these amplification factors are being continuallyreevaluated as new data become available (Crouse, 1995;Borcherdt, 1996; Abrahamson and Silva, 1997; Joyner andBoore, 2000). These studies generally indicate that the cur-rent amplification values may imply too much nonlinearity

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1610 S. Hartzell, L. F. Bonilla, and R. A. Williams

Table 1NEHRP Site Classifications

NEHRP Site Class Vs30 (m/sec)

A �1500B 760–1500C 360–760D 180–360E �180

Figure 1. Average shear-wave velocity profiles to150-m depth for sites with midrange NEHRP site clas-sifications (Silva et al., 1998).

for stiff-soil sites, leading to an underprediction of groundmotion, and that nearly constant factors with increasing ef-fective peak acceleration are more consistent with the cur-rent data set. These amplification factors have been deter-mined by modeling data and making forward predictions;therefore, it is important to use the correct theoretical model.

Numerical schemes for predicting soil nonlinearity canbe divided into equivalent linear and nonlinear approaches.The equivalent linear method is more simplistic and assumesthat a damped linear model can approximate dynamic soilbehavior through the appropriate choice of material param-eters. Nonlinear formulations track a seismic load throughstress-strain space using different assumptions for the detailsof the stress-strain relationship. In addition, nonlinear meth-ods make a distinction between considering pore-water pres-sure (effective stress calculation) or assuming dry conditions(total stress calculation). Many studies have compared theperformance of subsets of these formulations (Constanto-poulos et al., 1973; Streeter et al., 1974; Finn et al., 1978;Yoshida, 1994; Yoshida et al., 1995; Yoshida and Iai, 1998;Hashash and Park, 2002; Kausel and Assimaki, 2002). Otherstudies have considered various aspects of the effects of non-linear soil response on ground motion (Yu et al., 1993; Niet al., 1997, 2000; Hartzell, 1998; Su et al., 1998; Silva etal., 1998; Hartzell et al., 2002). In this work we comparethe full range of methodologies from equivalent linear, withand without recently developed frequency-dependent mod-uli and damping ratios, to nonlinear formulations with totaland effective stress conditions. The nonlinear calculationsare based on newly developed finite difference codes byBonilla et al. (1998) and Bonilla (2000).

Method of Comparison

Nonlinear soil models are often compared in terms ofstress-strain plots or the variation in maximum stress orstrain with depth. We prefer to make our comparisons usinga more accessible quantity, namely, the Fourier spectral ratioof ground motion at the surface with respect to the inputmotion at depth. The NEHRP (1994) soil classificationscheme is used to consider the effects of different soil types.Site categories A, B, C, D, and E are based on the averageshear-wave velocity in the top 30 m, Vs30, as given in Table1. Silva et al. (1998) developed average shear-wave velocityprofiles for these site classes based on available shear-wavevelocity measurements. They selected velocity profiles withVs30 values closest to the midrange of the NEHRP site cate-gories. These profiles are averaged and shifted so that theirVs30 values agree with the midrange values for the differentNEHRP site categories. We have adopted these profiles forthis study (Fig. 1). Site class A is not considered here be-cause of its limited occurrence in the western United Statesand because the high surface velocities are expected to resultin approximately linear behavior. Class BC is midway be-tween classes B and C and fills the relatively large jump inmaterial properties between these two classes. The NEHRP

B/C boundary is also the reference site condition adoptedfor the 1996 National Seismic Hazard Maps (Frankel et al.,1996). Profiles BC, C, D, and E in Figure 1 range from softrock to soft soil with nonlinearity allowed to extend to adepth of 152.5 m (500 ft). Below this depth, linear wavepropagation is assumed. Class B is assumed to be nonlinearto a depth of 22.25 m (73 ft), where the shear-wave velocityreaches 1650 m/sec. For all the profiles except B, a smoothgradient extends from a depth of 152.5 m down to a depthof 305 m, bringing the velocities up to 1950 m/sec, the max-imum shear-wave velocity for profile B. The depth to whichnonlinearity persists influences the extent of nonlinear be-havior observed at the surface, with these effects becomingmore pronounced as the depth increases. We do not considervariations in the thickness of the nonlinear column; however,this topic has been investigated elsewhere (Yu et al., 1993;Ni et al., 1997; Silva et al., 1998; Hashash and Park, 2002).

In addition to shear-wave velocity information, we needto quantify the degree of nonlinearity within each profile.For this purpose we adopt the widely used modulus reduc-tion, G/Gmax, and damping ratio, n, curves as a function ofshear strain developed by the Electric Power Research In-stitute (EPRI) (1993) for rock and sand and the curves ofVucetic and Dobry (1991) for clay. These curves are shownin Figure 2. The EPRI (1993) curves are an improvementover those previously available in that they are a function of

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Prediction of Nonlinear Soil Effects 1611

Figure 2. EPRI (1993) modulus reduction and damping ratio curves for rock andsand, and the Vucetic and Dobry (1991) curves for clay. The EPRI curves incorporatereduced nonlinearity with depth, and the Vucetic and Dobry curves are for a range inplasticity indexes.

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1612 S. Hartzell, L. F. Bonilla, and R. A. Williams

Figure 3. Acceleration record for the 1994 North-ridge earthquake for the Stone Canyon Reservoir rocksite, north–south component. This record is scaledand used as input to the average velocity profiles inFigure 1.

depth, reflecting reduced nonlinearity with depth of burial.The EPRI (1993) rock curves are used for site classes B andBC, the sand curves for site classes C and D, and the Vuceticand Dobry (1991) curves for a plasticity index of 30 for siteclass E. By this association we consider C and D sites to becohesionless soils and E sites to be cohesive soils, such as,bay mud. In addition to these classifications we also considerthe Peninsular Range modulus reduction and damping ratiocurves developed by Silva et al. (1997) for site classes Cand D, referred to here as Cpen and Dpen. These curves wereempirically developed to reflect lower levels of nonlinearityfor the 1994 Northridge earthquake than predicted by theEPRI (1993) cohesionless soils curves. They are formed byusing the 15.5 to 36.5 m (51 to 120 ft) depth range EPRIcurve for depths 0 to 15.2 m (0 to 50 ft), and the 152.7 to304.8 m (501–1000 ft) depth range EPRI curve for depthsbelow 15.2 m (50 ft). Modulus reduction and damping ratiocurves suffice to quantify the nonlinear response in an equiv-alent linear formulation and in a nonlinear formulation thatis tied to a prescribed modulus reduction curve. However,other more general nonlinear formulations do not utilizemodulus reduction curves. We consider both types in thisarticle.

For the input motion to our soil profiles we use thenorth–south component of the rock site record from stationLA00 (Stone Canyon Reservoir, 34.106� N, 118.454� W) forthe 1994 Northridge earthquake (Fig. 3). This station is lo-cated in the Santa Monica Mountains on Mesozoic rock andhas a peak acceleration of 0.26g. We consider this record tobe typical of rock site ground motion for a magnitude 6 to7 earthquake, without any unusual behavior such as isolatedlarge spikes or pulses. To obtain peak input motions from0.1 to 0.9g, this record is scaled up or down. Other studiesof nonlinear soil behavior have used an impulse base exci-tation (Yu et al., 1993; Ni et al., 1997). Because we areinvestigating nonlinear processes where the solution is com-puted in the time domain, we consider an actual earthquakerecord to be a more appropriate input.

For the equivalent linear formulations, because they arelinear, we can calculate the transfer function of the soil col-umn directly, and thus the amplification values. For the non-linear methods we take the Fourier spectral ratio of the sur-face motion with respect to the input motion at depth. Thespectral ratios are then smoothed using a 0.25-Hz-wide run-ning mean filter to remove spikes caused by spectral holesin the denominator. In all cases we remove the factor-of-twofree-surface effect (Aki and Richards, 1980).

Soil Models

Equivalent Linear: SHAKE91

The first site response formulation considered is thecode SHAKE91 (Idriss and Sun, 1992), based on the originalcode by Schnabel et al. (1972). We modified the SHAKE91algorithm making parts double precision to improve stability

at high-input ground motions. Because of its ease of appli-cation, the equivalent linear approach, and particularly theSHAKE91 code, has been widely used. With the equivalentlinear approach the material properties of shear modulus, G,and damping, n, are constant within each layer and for theentire duration of the earthquake, regardless of whether thestrains are large or small at a given time. An iterative processis used to determine the values of G and n for each layerthat are consistent with the level of strain induced in thelayer. Because experimental soils data are based on simpleharmonic loading with constant peak shear strain, an effec-tive shear strain is used for earthquake records that is a frac-tion, �, of the maximum shear strain. A value of 0.65 istypically used for � and is assumed in this study. The effec-tive shear strain is then used to calculate G and n from em-pirical modulus reduction and damping ratio curves.

Yoshida and Iai (1998) have pointed out two weak-nesses of the equivalent linear method. First, damping isconstant, independent of frequency. For an unbounded, ho-mogeneous medium, the shear strain is given by the particlevelocity divided by the shear wave velocity of the medium.The spectrum of particle velocity, and therefore of shearstrain, decreases in amplitude approximately as 1/f at highfrequency. Then, the damping obtained from standard lab-oratory curves should also decrease at high frequencies.Viewed alternatively, the energy dissipated per loading cycleis proportional to the area enclosed by the hysteresis loop.Therefore, low-amplitude, high-frequency components ofstrain should have much less damping. The overattenuationof high frequencies by the equivalent linear model becomesmore pronounced at high levels of strain. Recently, Sugitoet al. (1994), Joyner and Boore (1998), and Joyner (1999)have formulated equivalent linear methods that use fre-quency-dependent damping. Second, equivalent linear anal-ysis can overestimate resonant-frequency amplitudes andshear stresses compared with nonlinear analysis. Becausesoil stiffness varies as a function of time in nonlinear anal-ysis, resonances are less likely to develop, compared with

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Prediction of Nonlinear Soil Effects 1613

the constant-layer-parameter approach of equivalent linearanalysis.

Equivalent Linear: TREMORKA

Recognizing the overdamping that occurs with the tra-ditional equivalent linear approach, Kausel and Assimaki(2002) have recently advanced a formulation with frequency-dependent moduli and damping. Except for the manner inwhich the frequency-dependent moduli and damping arechosen, their approach is the same as Schnabel et al. (1972).A two-parameter, smooth functional form is used to fit thestrain spectrum within each layer. This function is then usedto read off strain values as a function of frequency, whichin turn are used to specify the frequency-dependent moduliand damping from standard laboratory curves. The processis iterated in the same way as SHAKE91 to obtain internallyconsistent soil parameters. We have implemented this methodin the code TREMORKA.

There is value in obtaining an improved equivalent lin-ear code. The soil parameters needed by some nonlinearcodes can be more numerous and difficult to obtain than themodulus reduction and damping curves used by equivalentlinear analysis. The frequency-dependent equivalent linearformulation requires only two parameters, the shear modulusreduction and fraction of damping as a function of shearstrain. Both of these quantities are routinely measured inlaboratory studies of soils. Also, in some cases the equiva-lent linear method can be more numerically stable than anonlinear calculation.

Nonlinear: NOAHW

The NOAHW code is the first of three newly developed,second-order, staggered-grid finite difference codes consid-ered in this study. This code, and other nonlinear methods,operates in the time domain by tracking the earthquake loadthrough stress-strain space. In NOAHW, the Iwan (1967)model is implemented to take into account soil nonlinearity.Furthermore, in this model the stress-strain relation is spec-ified by a given modulus reduction curve. The main differ-ence between this formulation and the one implemented byJoyner and Chen (1975) is that real G/Gmax laboratory datacan be assigned to each layer. In addition, the implementedIwan (1967) model is one that uses a series of parallel elasticsprings that permits the computation of stress from the strain.For dynamic nonlinear wave propagation with a finite dif-ference formulation, it is easier to obtain the strain from thenode displacement gradient and then introduce it into theconstitutive equation to compute the stress. This type of for-mulation is advantageous when the only data available arethe G/Gmax curves. In our comparisons, the same modulusreduction curves are used as in the preceding equivalent lin-ear runs. Nonlinear damping comes from the hystereticloops, however, not from a damping ratio curve. The damp-ing ratio is given by DW/(4pW), where W is the maximumstored energy and DW is the energy dissipated per cycle(Ishihara, 1996; Kramer, 1996). The maximum stored en-

ergy at a strain level of ca is ca f (ca)/2 and the energy lossper cycle is obtained by integrating f (c) over a completecycle yielding (Ishihara, 1996)

caDW � 8(� f(c)dc � W),0

where f (c) is the functional relationship between the stressand the strain. For the Iwan (1967) model the hysteresisloops follow the so-called Masing criteria (Masing, 1926).If f (c) is the hyperbolic model, as we assume, the dampingapproaches 2/p � 0.63 at large strain (Ishihara, 1996).

Nonlinear: NOAHH

In the previous section we mentioned that the hysteresisloops follow the Masing (1926) criteria. In the original Mas-ing (1926) rules the initial loading or backbone curve is de-fined by a functional relationship between the stress, s, andthe strain, c, as s � f (c). Subsequent unloading and reload-ing cycles may be expressed as,

(1/j )(s � s ) � f [(1/j )(c � c )],H a H a

where (ca, sa) is the unload/reload reversal point and jH is ahysteresis scale factor. In Masing’s original formulation, thehysteresis scale factor is equal to 2. Subsequently, some ad-ditional constraints have been added to prevent the computedstress from exceeding the maximum strength of the material,smax. The ensemble of these criteria is known as the extendedMasing rules (Kramer, 1996). Several articles have dealtwith the implementation of the extended Masing rules in aphenomenological way (Pyke, 1979; Li and Liao, 1993;Finn, 1982; Vucetic, 1990; Iai et al., 1990a).

In comparison, the Iwan (1967) model fully complieswith these criteria by a mechanical model of a series of par-allel or serial elastic springs. Laboratory tests of soils undercyclic shear deformation show that the maximum dampingratio varies between 0.2 and 0.4 depending on the type ofmaterial. The Iwan (1967) model with a hyperbolic back-bone, on the other hand, may overdamp when large strainsare reached. Thus, we would like to find a compromise be-tween the shear modulus reduction and the damping ratio tobetter characterize the soil nonlinearity. Bonilla’s (2000) for-mulation utilizes a variable hysteresis scale factor that as-sures the stress-strain path during each loading/reloading isbounded by the maximum shear strength. In addition, sub-sequent unloading or reloading curves follow the backbonecurve if the previous maximum shear stress is exceeded. Thisphenomenological representation is called the generalizedMasing rules because its formulation contains Pyke’s (1979)and the original Masing jH � 2 models as special cases.Furthermore, this formulation allows, by controlling the hys-teresis scale factor, the reshaping of the backbone curve assuggested by Ishihara et al. (1985) so that the hysteresis pathfollows a prescribed damping ratio. Given the laboratory-measured soil damping, and since we know the damping asa function of the reference strain and hysteresis shape factor,

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1614 S. Hartzell, L. F. Bonilla, and R. A. Williams

Table 2Water Table Depths

NEHRP SiteClass

NOAHH Depth(m)

NOAHB Depth(m)

B Below profile Not consideredBC Below profile Not consideredC 15.0 Not consideredD 15.0 15.0 and 2.0E 5.0 5.0 and 2.5

we equate them to find a new reference strain compatiblewith the damping value. This reference strain is used to com-pute a new backbone curve. Thus, the hysteresis loop willcomply with the observed damping ratio. This procedure isfollowed at each time step. More details of this model aregiven by Lavallee et al. (2003).

The functional form of the backbone curve for NOAHHis the hyperbolic model given by (Kondner and Zelasko,1963),

G c G cmax maxs � � ,

G cmax1 � c 1 �s cmax ref

with,

G 1� .

G cmax 1 �cref

where Gmax is the low-strain shear modulus, smax is the max-imum shear stress that the material can support in the initialstate, and cref � smax/Gmax is the reference strain. The hy-perbolic model has been extensively used because it is rela-tively easy to implement and because of the low number ofparameters necessary for its description (Hardin and Drnev-ich, 1972; Pyke, 1979; Ishihara, 1996).

This code, and the following one, uses the multispringmodel plane strain formulation for the stress-strain relation-ship in each soil layer (Towhata and Ishihara, 1985; Iai etal., 1990a). Each spring follows the hyperbolic model andimplements hysteresis by the generalized Masing rules de-scribed previously. The multispring model was introducedby Towhata and Ishihara (1985) to model cyclic mobilityand dilatancy in sands. However, the NOAHH code is fortotal stress analysis; no pore-water pressure is considered.Under these conditions, the multispring model gives thesame results as the single hyperbolic element model custom-arily applied. NOAHH uses the multispring model for totalstress analysis because it is relatively easy to implement, ithas a two-dimensional representation (plane strain), and ittakes into account the anisotropy of soils (vertical and hor-izontal initial stresses may be different). In NOAHH we stillspecify typical water-table depths from the San FernandoValley, California (Gibbs et al., 1999) (Table 2), because thewater table influences the calculation of the over burden.

The maximum shear stress for cohesionless soils is cal-culated by the expression,

s � ((1 � K )r sin u)/2,max o v

where Ko is the coefficient of lateral stress at rest, and � isthe angle of internal friction. Ko may be represented in termsof the angle of internal friction and the overconsolidationratio as (Kulhawy and Mayne, 1990),

sinuK � (1 � sin u)OCR .o

We assume a value for � of 30� and normal consolidation(OCR � 1), so that Ko is then 0.5. Gmax is calculated fromthe density, q, and the shear-wave velocity, Vs, by the ex-pression . rv is the vertical effective stress,2qVs

r � r � q g(z � z ),v t w w

where rt is the total vertical stress, qw is the density of water,g is the acceleration of gravity, z is the sample depth, andzw is the depth of the water table measured from the groundsurface.

Although the nonlinear formulation in NOAHH does notfollow a particular modulus reduction curve, as NOAHWdoes, we produce a soil column with the same strength im-plied by the modulus reduction curves used previouslythrough the addition of a cohesion term. With the additionof cohesion, we are able to model stiff soils, such as en-countered with site class B, and still have reasonable valuesof Ko and �. The Mohr–Coulomb relation is given bysmax � c � rn tan (�), where c is the cohesion and rn isthe normal stress, or alternatively (Jaeger and Cook, 1979),smax � rm sin � � c(cos �), where rm � (rv/3)(1 � 2Ko)is the effective mean stress. This expression for smax is thesame as the one given by Hardin and Drnevich (1972) withan axis transformation of the Mohr’s circle. Thus we havean expression for the cohesion in terms of the referencestrain, the coefficient of the earth at rest, and the angle ofinternal friction,

c � (c G � r sin u)/cos u.ref max m

Then, the procedure followed is to first determine cref fromthe desired modulus reduction curve as the value of strainwhen G/Gmax � 0.5. The cohesion is determined using thepreceding expression. Finally, smax is calculated using theexpression of Jaeger and Cook (1979).

For site classes B, BC, C, and D (rock and sand) a max-imum damping ratio of 0.30 is used. For site class E (clay)the maximum damping ratio is 0.20 (Fig. 2). Realistic in-trinsic attenuation with constant Q is incorporated by therheology of the generalized Maxwell body (Day, 1998). Inaddition, a small amount of viscous damping may be addedat low-strain levels for numerical stability (Li, 1990).

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Prediction of Nonlinear Soil Effects 1615

Table 3Dilatancy Parameters for NOAHB

�p 28.0Porosity 0.45

p1 0.5p2 0.6w1 5.0S1 0.01c1 1.0

Nonlinear: NOAHB

The NOAHB code is the third and final level of com-plexity in the nonlinear formulations we consider. NOAHBtreats undrained conditions of effective stress and also util-izes the Bonilla (2000) generalized Masing rules with thehyperbolic model discussed previously. The algorithm isbased on the multispring mechanism model first imple-mented by Towhata and Ishihara (1985) to simulate porepressure generation in sands under cyclic loading and un-drained conditions, and further developed by Iai et al.(1990a,b) to take into account the cyclic mobility and dila-tancy of sands. Under cyclic loading, as the pore pressureincreases, the effective stress decreases, and as a result, Gmax

and smax decrease. As Gmax and smax change, so does theshape of the backbone curve. An empirical approach devel-oped by Towhata and Ishihara (1985) and Iai et al. (1990a)is utilized to model the decrease of effective stress due tothe increase in pore pressure. Six parameters (�p, p1, p2, w1,S1, and c1), together with the porosity are used to describethe dilatancy of the material. Table 3 lists the values we haveadopted in this study. They are assumed to be the same forall layers. Iai et al. (1990b) give a more detailed descriptionof these model parameters. We consider two values for theangle of internal friction, � � 30 and 40 (corresponding toa medium-dense sand and a dense sand, respectively), tocover the range of typical values, and Ko � 0.5, and a max-imum damping ratio of 0.3.

With the introduction of pore pressure and effectivestress it is no longer possible to make a detailed comparisonwith the previous equivalent linear formulations or the totalstress nonlinear formulations. Significant buildups in pore-water pressure make a distinctly different problem. How-ever, for completeness we include this case. Because theeffects of pore pressure generation and liquefaction are sig-nificant only in cohesionless soils with low to mediumstrength, we limit our tests to site classes D and E. To thispoint class E has been considered a clay soil with the ap-propriate modulus reduction curves. For the NOAHB calcu-lations we drop this association because a clay soil is notexpected to undergo liquefaction. For NOAHB, site class Eis a cohesionless sandy soil similar to class D, only withlower shear-wave velocities. We consider two water-tabledepths (Table 2) to investigate the effect of the depth to thesaturated column.

Measurement of resistance to liquefaction of a sand isa standard laboratory test. In this procedure, an undrainedsample is subjected to cyclic loading. The pore-water pres-sure builds up steadily as the cyclic stress is applied, even-tually approaching the initial confining pressure, leading toan axial strain of about 5% peak-to-peak (or double ampli-tude, DA). This state is referred to as the start of liquefaction.Figure 4 shows the analytical liquefaction resistance curvesfor cyclic simple shear and soil classes D and E and ourassumed dilatancy parameters listed in Table 3. These curvesplot the number of cycles to produce 5% DA shear defor-

mation versus the cyclic stress ratio, sxy/rm0, where sxy isthe cyclic shear stress and rm0 is the initial effective confin-ing stress. Figure 4 shows the numerical response of themultispring model in NOAHB for our hypothetical profiles.All simulations are done with a reference effective meanstress, rma, equal to the initial effective confining stress, rm0,which is computed at the middle of each layer for soil mod-els D and E. The shaded area for a given angle of internalfriction represents the ranges of the liquefaction resistancecurves for all the layers within the profile. We can observethat the stronger the soil is, that is � � 40, the larger thecyclic stress ratio is needed to deform the material. In ad-dition, the effect of the water table is more evident when� � 40. Indeed, a deeper water table produces less buoy-ancy, and thus the material becomes stronger. Conversely,when the soil is relatively weak, � � 30, and combinedwith a shallow water table, we can observe that there is littledifference between the liquefaction resistance curves for theshallow and deeper layers. We recognize that the resultsstrongly depend on the chosen dilatancy parameters. How-ever, in the absence of specific geotechnical data on the liq-uefaction resistance curves, we show the effect of pore-waterpressure for these two soil classes.

Results

Figures 5 through 9 plot amplification as a function offrequency for the site classes discussed previously (B, BC,C, Cpen, D, Dpen, and E) and peak base input motions of0.1g to 0.9g in 0.1g increments. The linear response curves,designated by an L, are also shown for each site classifica-tion. Consider first the SHAKE91 results in Figure 5. As ex-pected, the low input acceleration curves most closely followthe linear curve, but they diverge more as the stiffness of thesoil decreases, from class B (rock) to E (clay). Most of theequivalent linear curves lie below the linear response be-cause of higher damping at higher strain levels. However,resonant peaks at lower frequencies (below about 3 Hz) canrise above the linear response caused by the familiar down-shift in energy from the reduction of the shear modulus.Also, for site class B, higher input motions yield higher am-plifications than the linear case. This result comes from thecompeting effects of reduction in the shear modulus and in-creased damping. For class B the reduction of the shear mod-ulus leads to an increase in amplification that is greater than

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1616 S. Hartzell, L. F. Bonilla, and R. A. Williams

Figure 4. Liquefaction resistance curves for soilclasses D and E. The strength of the material is relatedto the angle of internal friction, �, shown in each plot.For all models the reference effective mean stress,rma, is equal to the initial effective confining stress,rm0. The dilatancy parameters are listed in Table 3.The shaded area represents the range of liquefactionresistance curves for all the layers within the profile.

the attenuation caused by the increased damping. However,the most striking feature of the SHAKE91 curves is the severeattenuation for softer soils and larger input motions. For siteclasses D and E, amplification values are in the range from10�3 to 10�4 for frequencies greater than 6 Hz and inputmotions above 0.5g. This response has the effect of remov-ing nearly all high frequencies in areas with deep soil profilesand runs counter to observations (Durward et al., 1996).

As discussed previously, and shown in Figure 6, theTREMORKA amplification curves have significantly less at-tenuation for soft soils and large strains than the SHAKE91results. The moduli and damping ratios are now a functionof frequency based on the spectrum of the strain induced ineach layer by the earthquake motion. For site class B thedifference between SHAKE91 and TREMORKA is negligible,indicating that constant moduli and damping ratios are agood approximation for this site class. For all the other siteclasses, however, there are significant differences. Siteclasses C and Cpen have nearly constant amplification atfrequencies above 4 Hz for different input motions. The am-plification for class D never goes much below 1.0 and isabout 0.1 for class E at high frequencies and large inputmotions.

Figure 7 shows amplification spectra for NOAHW, anonlinear formulation that utilizes the same modulus reduc-tion curves as the previous equivalent linear models to de-scribe the backbone stress-strain curve. These amplificationcurves are more complicated because they are calculatedfrom the spectral ratios of the output to the input motionsand because of the more complicated response produced bytracking the load through stress-strain space. Smoothing outthe peaks in the curves and considering only the generalamplification levels, the NOAHW results for class B are verysimilar to the TREMORKA results. For the softer site classes,C and softer, the NOAHW results give amplification valuesup to a factor of 2 less than those from TREMORKA at fre-quencies above about 2 Hz. This difference is due to thedifferent way in which damping is incorporated into the twocodes. TREMORKA uses damping values picked from an em-pirical curve. NOAHW uses the dissipation of energy in eachhysteresis loop following the Masing criteria and the hyper-bolic model, thus at large strains the maximum dampingis 2/p.

The NOAHH amplification curves are shown in Figure8. This code uses a hyperbolic backbone curve and addscohesion to obtain the same strength as implied by the em-pirical modulus reduction curves. For the stiffer profiles (B,BC, C, and Cpen) there is very little difference between theNOAHW and NOAHH results for both small- and large-inputmotions. For site classes D and E there is a divergence be-tween the two solutions, with NOAHH yielding lower am-plification values above 2 Hz by 50% at low input motionsand up to a factor of 2 at large input motions. Recall thatNOAHH constrains the maximum damping ratio to 20% forclays and 30% for sands. This difference is discussed furthernext when we consider time domain records.

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Prediction of Nonlinear Soil Effects 1617

Figure 5. Equivalent linear SHAKE91 spectral amplification curves as a function offrequency for the average NEHRP velocity profiles in Figure 1 and for peak inputmotions from 0.1g to 0.9g. An L indicates the linear response curve. Cpen and Dpenuse the site class C and D velocity profiles, respectively, but modulus reduction anddamping curves with reduced nonlinearity (see text and Silva et al., 1997). The resultsfor class E are plotted on both linear and log scales to reveal details of the curves.

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1618 S. Hartzell, L. F. Bonilla, and R. A. Williams

Figure 6. Same as Figure 5 for the equivalent linear code TREMORKA with fre-quency-dependent moduli and damping ratios.

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Prediction of Nonlinear Soil Effects 1619

Figure 7. Same as Figure 5 for the nonlinear total stress code NOAHW based onthe Iwan (1967) model and hysteresis loops according to the Masing (1926) criteriawith prescribed modulus reduction curves.

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1620 S. Hartzell, L. F. Bonilla, and R. A. Williams

Figure 8. Same as Figure 5 for the nonlinear total stress code NOAHH with thehyperbolic model backbone curve and hysteresis loops following generalized Masingrules and added cohesion to match given modulus reduction curves.

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Prediction of Nonlinear Soil Effects 1621

Figure 9. Same as Figure 5 for the nonlinear effective stress code NOAHB with twoassumptions for the depth of the water table and two different values for the angle ofinternal friction, phi.

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1622 S. Hartzell, L. F. Bonilla, and R. A. Williams

In Figure 9 we show the amplification curves forNOAHB, an effective stress calculation including the effectsof pore-water pressure. We consider only soft soils withclass D and E velocity profiles. A deeper and a shallowerwater table are considered for both site classes (Table 2);15 m and 2 m for class D, and 5 m and 2.5 m for class E.We also consider two angles of internal friction, � � 30and 40, which covers the range of typical values. As thewater table shallows the trend is for greater attenuation athigher frequencies above 1 to 2 Hz, and somewhat largerresonant peaks below 1 Hz. Decreasing the angle of internalfriction has the effect of increasing the attenuation at nearlyall frequencies. This effect is expected because decreasingthe angle of internal friction means a reduction of the initialstrength as well.

Figure 10 summarizes the preceding comparisons byplotting a low-strain curve (0.2g) and a high-strain curve(0.7g) from each of the five different nonlinear soil models.For site class B differences between the different formula-tions are not significant. For this comparison we considermean levels and average over higher-frequency oscillations.All the curves are also similar for site class C for an inputmotion of 0.2g. However, there are significant differencesbetween the equivalent linear solutions and the nonlinearsolutions for an input motion of 0.7g. The two equivalentlinear solutions overestimate the amplitude of the resonantpeak at 1.0 Hz. This tendency has been noted by other au-thors (Yoshida and Iai, 1998). For the softer soil profiles(classes D and E), the SHAKE91 solution becomes more di-vergent from the others, particularly at larger strains. Theother solutions group more closely together, with the TRE-MORKA curves generally lying above the nonlinear solu-tions. The NOAHB curves for an angle of internal friction of30 are also included for site classes D and E. An angle ofinternal friction of 30 was also used for the total stress codeNOAHH. NOAHW does not use this parameter. However, aswe mentioned previously, when significant pore-water pres-sures develop, results from the effective stress calculation ofNOAHB will depart from the total stress analysis. We includeit here to see the effect of pore-water pressure. The NOAHBcurves in Figure 10 are for the deeper water-table values(15 m for class D and 5 m for class E). For these parametersthe NOAHB results are generally intermediate to the two totalstress calculations.

Figure 11a,b shows the same comparisons in the timedomain as in Figure 10. As noted previously, there is verylittle difference between the different solutions for site classB at both low and high input motions. The same can be saidfor class C at lower input motions (0.2g). However, startingwith class C at high input motions (0.7g), differences de-velop and grow with the reduction in the strength of the soilcolumn. For classes D and E, the SHAKE91 solutions clearlylack high frequencies that are present in the TREMORKA andnonlinear solutions. The TREMORKA and SHAKE91 solu-tions yield similar peak accelerations for all site classes andinput motions, even for classes D and E. This observation is

explained by the fact that the peak acceleration is controlledby the frequency band 1 to 3 Hz, where the two solutionshave similar amplitudes. The nonlinear solutions, on theother hand, all have significantly lower amplitudes in thisfrequency band, and thus lower peak accelerations. The peakaccelerations for the nonlinear formulations are all similar.The increased attenuation of higher frequencies, above about2 Hz, for the NOAHH solution compared with the NOAHWsolution is apparent for site classes D and E.

The input record (Fig. 3) shows a progression fromlower-amplitude, higher-frequency motion at the beginningto larger-amplitude, lower-frequency motion with the arrivalof the main shear-wave energy. This character is typical ofearthquake ground motion. The nonlinear soil responses tendto amplify the pattern of higher frequencies early and lowerfrequencies late, because the larger, late motions causegreater nonlinearity with resulting lower rigidity and higherdamping. This pattern of ground motion has also been ob-served in earthquake records on soft soils (Satoh et al., 1995;Frankel et al., 2002). However, large input motions with softsoils can result in cyclic mobility with late, high-frequency,cusped waveforms (Archuleta, 1998). Both characteristicsare seen in Figure 11a,b.

The fact that there is damping control in NOAHH doesnot necessarily mean that the result will be less damped thanthe Iwan (1967) model. Indeed, what happens, especially atshallow depths or where there is a low-strength layer, is thatlarge strains may develop. This can be understood because,to keep the same energy (hysteresis loop area) in a dampedcontrolled system, the material has to develop larger strainsthan in one following the traditional Masing criteria. Thesedeformations, typically obtained at shallow depths, result inreduced rigidity that consequently damps the higher fre-quencies in the ground motion. This effect has been pointedout by Ishihara et al. (1985).

Nisqually

The 28 February 2001 M 6.8 Nisqually earthquake pro-duced a ground motion data set that shows significant soilnonlinearity (Frankel et al., 2002), manifested as shifts inresonant spectral peaks at lower frequencies, increased at-tenuation at higher frequencies, and cusped waveforms afterthe arrival of the S wave (Archuleta, 1998). An ML 3.4 af-tershock 14 hr after the mainshock and recorded at the samestations provides a linear response reference point. We usethese records for comparison with our model predictions.Figure 12 shows the stations we consider. In terms of thesurficial geology, the stations fall into the two general cate-gories of young, soft sediments, consisting of artificial filland Holocene alluvium (BOE, HAR, KDK, and SDS), andolder, stiffer sediments of Pleistocene age (BHD, LAP, andTHO). The soft sediment sites had peak accelerations from0.19g to 0.22g and the Pleistocene sediments had peak ac-celerations from 0.11g to 0.16g. Following Frankel et al.(2002), we estimate the amplification at each site by calcu-

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Prediction of Nonlinear Soil Effects 1623

Figure 10. Comparison of amplification spectra for peak input motions of 0.2g and0.7g for the different equivalent linear and nonlinear soil response formulations con-sidered in this article. Black, SHAKE91; red, TREMORKA; green, NOAHW; blue,NOAHH; magenta, NOAHB. The NOAHB curves assume the deep-water table and anangle of internal friction of 30.

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1624 S. Hartzell, L. F. Bonilla, and R. A. Williams

Figure 11. Comparison of time domain surface recordsfor the spectral amplification curves in Figure 10. (a) peakinput motion of 0.2g. (b) peak input motion of 0.7g.

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Figure 12. Seattle area map showing surficial geology and locations of acceler-ometer stations that recorded the 2001 Nisqually earthquake and its large aftershock,and for which there is also a measure of the shallow shear-wave velocity.

lating the Fourier spectral ratios with respect to site SEW onTertiary sedimentary rock (Fig. 12). The spectra are basedon the rms of the two horizontal components of accelerationand 40 sec of the S-wave record. This duration ensures thatwe have the average spectral levels over all significant-amplitude shear-wave energy. The ratios are smoothed withthe same 0.25-Hz running mean filter as our nonlinear theo-retical amplification spectra. Because both the mainshockand aftershock were deep (52 km and 54 km, respectively)and within a few kilometers of each other, differences in thesource-to-station pathlengths are small, and no propagationpath corrections are considered necessary. Figure 13 showsmainshock and aftershock spectral ratios for the soft-soil andstiff-soil sites. For the stiff-soil sites on Pleistocene sedi-ments there is little difference between the mainshock andaftershock spectral ratios. For the soft-soil sites, however,frequency downshifts of resonant spectral peaks at lowerfrequencies are apparent, as well as greater attenuation athigher frequencies. Williams et al. (1999) have determinedVs30 values (average shear-wave velocity in the top 30 m)for the seismic stations (Table 4). These values allow us tospecify the NEHRP site class for each station. In Figure 14

we use this information to compare the observed mainshockspectral ratios with the nonlinear predictions. Our objectivein this comparison is to see how well the average NEHRPsite class velocity profiles perform without detailed velocityinformation at the site. For site classes D and E, we use theeffective stress results from NOAHB and the shallow-watertable profile with � � 40. These sites all lie within theDuwamish River Valley, which is known from P-wave seis-mic refraction data to have a water table in the 2- to 4-mdepth range (Williams et al., 1999). The site class C stationsare on Pleistocene sediments with a deeper water table. Forthese stations we use the total stress NOAHH results. Recall,however, that all the formulations considered give similarresults for class C with lower input motions. Assuming thatthe Tertiary sedimentary rock site SEW is site class B, wecalculate the theoretical amplification spectra by dividing bythe site class B response. Two input ground motion levelsare considered in Figure 14, 0.1g and 0.2g, which bracketthe range of values in the Seattle area for the Nisqually earth-quake. At lower frequencies, below 2 Hz, there are differ-ences between the frequencies and amplitudes of resonantspectral peaks that are due to the details of the velocity struc-

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1626 S. Hartzell, L. F. Bonilla, and R. A. Williams

Figure 13. Nisqually mainshock (solid curves)and aftershock (dashed curves) shear-wave spectralratios relative to the Tertiary rock site SEW. Spectraare based on 40 sec of the rms of the two horizontalcomponents of acceleration.

Figure 14. Comparison of observed mainshockspectral ratios from Figure 13 (dashed and dottedcurves) with model predictions for site classes C, D,and E for two peak input motions of 0.1g and 0.2g(solid curves). The model predictions for class C arefrom the nonlinear total stress code NOAHH. Themodel predictions for classes D and E are from thenonlinear effective stress code NOAHB with the shal-low water table and angle of internal friction of 40.

Table 4Vs30 for Ground Motion Recording Stations in Seattle,

Washington

Station Site ClassVs30

(m/sec)Distance of Vs30 from

Station (m)

HAR E 145 0SDS E 148 100BOE E 166 0KDK D 245 200LAP C 367 0THO C 425 100BHD C 673 0

ture at each station, not contained in our smooth, averagevelocity profiles. Using high-resolution S-wave seismic re-flection data near stations BOE, SDS, and KDK, Williamset al. (2000) and Frankel et al. (2002) found that the resonantspectral peaks below 2 Hz are consistently observed duringweak and strong motion and are likely caused by fill/allu-

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Prediction of Nonlinear Soil Effects 1627

vium or young alluvium/bedrock impedance boundaries lo-cated in the 16- to 50-m depth range. Because these impe-dance boundaries are not captured in the average velocityprofiles used in this study, the nonlinear models underpredictthe observed peak amplifications by up to a factor of 2 atthese resonant peaks for site classes E and C. However, theamplification levels over broader frequency bands are gen-erally well represented, especially considering that no iter-ations were performed to fine tune the theoretical results tothe Nisqually data. There is a trend for lower observed am-plification at the class C sites. Two recent boreholes in thePleistocene sediments on the hills east of downtown Seattleshow nearly uniform shear-wave velocities of 500 m/sec toa depth of at least 100 m (Odum et al., 2003). These veloc-ities are significantly lower than our average site class Cprofile and would explain the greater attenuation.

Conclusions

Nonlinear soil effects have been examined as measuredby Fourier amplification spectra for equivalent linear for-mulations with and without frequency-dependent moduliand damping ratios and nonlinear formulations with andwithout pore-water pressure. No significant differences existbetween these solutions for NEHRP site class B and peakinput motions from 0.1g to 0.9g. Amplification spectrawithin the frequency band 1 to 2 Hz and average spectrallevels above 2 Hz are essentially the same for site classesBC and C at lower peak input motions of 0.1g to 0.2g. Asthe input motion increases, differences develop between thesolutions. For the softer soils (C, D, and E), the equivalentlinear solutions can have up to 50% larger amplitude reso-nant peaks below 2 Hz than nonlinear formulations becausethey include no time dependence in the solution. For siteclasses D and E, the overdamping of the SHAKE91 solutionabove about 4 Hz is apparent for the entire range of inputmotions considered. The equivalent linear formulation withfrequency-dependent moduli and damping ratios underdamps relative to the nonlinear models considered for siteclass C with larger input motions and most input levels forsite classes D and E. At larger input motions the under-damping for site classes D and E is not as severe as theoverdamping with the frequency-independent formulation,but there are still significant differences in the time domain.Based on these comparisons, a nonlinear approach is rec-ommended for site classes D and E, and for site class C forinput motions greater than a few tenths of the accelerationof gravity.

Our average site class D and E amplification spectrabased on a nonlinear effective stress calculation and shallowwater table are consistent with observed soil-to-rock spectrafor the Nisqually earthquake, excluding low-frequency res-onances. Class C sites for the Nisqually earthquake showgreater attenuation at frequencies above 2 Hz than the pre-dictions of a total stress nonlinear calculation. This differ-ence is attributed to velocities lower than the average site

class C velocity profile to depths of 100 m or more in theSeattle Pleistocene glacial deposits.

Acknowledgments

We thank Walt Silva for supplying his average velocity profiles forthe NEHRP site categories. S.H. benefited from discussions with PengchengLiu and Alena Leeds. This article was improved by reviews from ChrisCramer, Charles Mueller, and two anonymous reviewers.

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U.S. Geological SurveyDenver Federal CenterBox 25046 MS 966Denver, Colorado 80225

(S.H., R.A.W.)

Institute de Radioprotection et de Surete NucleaireIRSN/DEI/SARG/BERSSINBP 17-92262 Fontenay-aux-RosesCedex, France

(L.F.B.)

Manuscript received 18 June 2004.