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Nonlinear Site Ampliﬁcation Factors

for Constraining the NGA Models

Melanie Walling,a) M.EERI, Walter Silva,b) M. EERI, and

Norman Abrahamson,c) M.EERI

Ampliﬁcation factors computed from the equivalent-linear method using

the program RASCALS are used to develop constraints on the nonlinear soil

response for possible use by the NGA ground-motion model developers. The

site response computations covered site conditions with average VS30 values

ranging from 160 to 900 m/s, soil depths from 15 to 914 m, and peak

accelerations of the input rock motion 共VS30= 1100 m/ s兲between 0.01 g and

1.5 g. Four sets of nonlinear properties of the soils are used: EPRI, Peninsular

Range, Imperial Valley, and Bay Mud. The ﬁrst two soil models are used for

VS30ⱖ270 m /s and the later two are used for VS30ⱕ190 m / s. Simple

parametric models of the nonlinear ampliﬁcation factors that are functions of

the PGA on rock and VS30 are developed for the EPRI and Peninsula

models. 关DOI: 10.1193/1.2934350兴

INTRODUCTION

As part of the NGA project, analytical models of the site response were exercised to

help guide the development of the ground-motion models outside the range well con-

strained by the empirical data. The purpose of this paper is to document the nonlinear

site ampliﬁcation factors that were made available to the NGA developers for possible

use in their models. The decision on whether, and how, to use these results in the devel-

opers NGA model is the responsibility of each NGA developer and is described in their

model papers (Abrahamson and Silva 2008, Boore and Atkinson 2008, Campbell and

Bozorgnia 2008, Chiou and Youngs 2008, and Idriss 2008).

ANALYTICAL MODELING

The analytical model used for site response is the equivalent-linear method with ran-

dom vibration theory as implemented in the program RASCALS (Silva and Lee 1987).

Silva (2008) conducted site response simulations for a range of soil proﬁles parameter-

ized by the VS30 and the depth to VS=1000 m/ s. For each soil proﬁle, ampliﬁcation

factors were computed for a range of input rock PGA values 共0.001 to 1.5 g兲. For each

case, the ampliﬁcation with respect to VS30= 1100 m/ s was computed. For this study,

only the cases with the soil depth randomized over a broad range were used. The subset

of site response cases used in this study are summarized in Table 1.

a) Ph.D. candidate, Civil and Environmental Engineering Department, University of California, Berkeley, CA

94720

b) Paciﬁc Engineering and Analysis, 311 Pomona Ave, El Cerrito, CA 94546

c) Paciﬁc Gas and Electric Company, 245 Market Street, San Francisco, CA 94105

243

Earthquake Spectra, Volume 24, No. 1, pages 243–255, February 2008; © 2008, Earthquake Engineering Research Institute

VELOCITY PROFILE

The base shear-wave velocity proﬁles are shown in Figure 1 for the suite of VS30

values. For each base proﬁle, a range of soil depths are used. For each soil depth, the

base proﬁle is truncated at the soil depth with a jump to a velocity of 1000 m / s (Figure

2). At depths greater than the soil depth, the velocity proﬁle follows an average velocity

model for California.

The velocity proﬁle is randomized about the base proﬁle. For each base proﬁle, 30

realizations of the velocity proﬁle are generated. The proﬁle randomization scheme,

which varies both layer velocity and thickness, is based on a correlation model devel-

oped from an analysis of variance on about 500 measured shear-wave velocity proﬁles

(EPRI 1993, Silva et al. 1997). To accommodate variability in the modulus reduction

and damping curves on a generic basis, the curves were independently randomized about

Tabl e 1 . Subset of site response cases from Silva (2008)

VS30 (m/s)

Depth Range to Top of Rock

共VS=1000 m / s兲Nonlinear Material Properties

270 9– 305 m EPRI, Peninsular Range

400 9– 305 m EPRI, Peninsular Range

560 9– 152 m EPRI, Peninsular Range

760 27 m EPRI, Peninsular Range

900 12 m Peninsular Range

Figure 1. Base shear-wave velocity proﬁle.

244 WALLING, SILVA, AND ABRAHAMSON

the base case values. A truncated log normal distribution was assumed with a

ln of 0.35

at a cyclic shear strain of 3⫻10−2%. The distribution was truncated at ±2

to prevent

modulus reduction or damping models that are not physically possible. The random

curves are generated by sampling the modulus reduction or damping at 3⫻10−2%shear

strain, and applying the ratio of the sample value to the average value at all strains. The

random perturbation factor is reduced or tapered near the ends of the strain range to

preserve the general shape of the median curves (Silva 1992).

NONLINEAR DYNAMIC MATERIAL PROPERTIES

From the full set of results given by Silva (2008), a subset based on two sets of

G/Gmax and hysteretic damping curves are used: EPRI (1993) and Peninsular Range

(Silva et al. 1997). The G/Gmax and hysteric damping curves for EPRI and Peninsular

Range (PEN) models are shown in Figures 3a and 3b, respectively.

INPUT MOTIONS

The stochastic point-source model is used to compute the input motions at the sur-

face of the reference rock (Vs30 of 1100 m / s). Validations of the stochastic point-source

model (Hanks and McGuire 1981, Boore 1983, McGuire et al. 1984, Boore and Atkin-

son 1987, Silva and Lee 1987, Toro and McGuire 1987, EPRI 1993, Schneider et al.

1993, Silva and Darragh 1995, Silva et al. 1997) have demonstrated that it provides ac-

curate response spectral estimates. The earthquake magnitude and stress drop are ﬁxed

at M6.5 and 60 bars, respectively, since they have little affect on the ampliﬁcation fac-

tors. The model used for the anelastic Q(f) model is 176f0.6. For rock sites, the kappa is

0.04 sec. For soil sites, the rock kappa is adjusted to maintain a total small strain total

kappa value of 0.04 sec for deep soil sites as described in Silva (2008).

The point-source distance is selected such that the PGA on the rock 共VS30

Figure 2. Example of truncated base proﬁle at a depth of 76 m.

NONLINEAR SITE AMPLIFICATION FACTORS FOR CONSTRAINING THE NGA MODELS 245

=1100 m / s兲matches the desired level. In all, 11 distances corresponding to the follow-

ing 11 PGA values were used: 0.001 g,0.05 g,0.1 g,0.2 g,0.3 g,0.4 g,0.5 g,0.75 g,

1.0 g,1.25 g,1.5 g. For the soil sites, this distance is held ﬁxed and soil velocity proﬁle

is used in place of the rock velocity proﬁle. The ratio of the spectral acceleration on soil

to the spectral acceleration on rock is computed, resulting in a suite of ampliﬁcation fac-

tors as a function of reference rock outcrop peak acceleration values.

(

b)

(

a)

Figure 3. Generic G/Gmaz and hysteretic damping curves for the North Coast cohesionless soil

sites (EPRI 1993). Generic G/Gmaz and hysteretic damping curves for the Peninsular Range

cohesionless soil sites (EPRI 1993).

246 WALLING, SILVA, AND ABRAHAMSON

EXAMPLE OF ANALYTICAL RESULTS

As an example, the ampliﬁcation factors of the analytic model for T=0.2 sec. and

VS30 of 270, 398, 560, 750 and 850 m/ s are shown in Figure 4.

PARAMETERIZATION OF THE ANALYTICAL RESULTS

In the linear range, Boore et al. (1997) found that shear-wave velocity dependence of

the site ampliﬁcation can be modeled by the following:

ln共Amp兲=aln

冉

Vs30

VREF

冊

,共1兲

where VREF is a reference shear-wave velocity. To account for nonlinear site response,

Abrahamson and Silva (1997) developed an empirical model for site ampliﬁcation of

generic soil with respect to generic rock. They used the following model for the nonlin-

ear response:

ln共Amp兲=a+bln共PG

ˆArock +c兲,共2兲

where PG

ˆArock is the expected value of the peak acceleration on generic rock. Since this

model was only for a single site condition (generic soil), it did not include a VS30 de-

pendence. Choi and Stewart (2005) developed empirical ampliﬁcation factors for

NEHRP categories based on the VS30 and the PGA on rock using the following form:

Figure 4. Example of the ampliﬁcation factors for different VS30 values at T = 0.2 sec based on

the Peninsular Range soil model.

NONLINEAR SITE AMPLIFICATION FACTORS FOR CONSTRAINING THE NGA MODELS 247

ln共Amp兲=aln

冉

VS30

VREF

冊

+bln

冉

PGArock

0.1

冊

共3兲

In the Choi and Stewart model, the b coefﬁcients vary for each NEHRP category (e.g.,

b is a function of the Vs30).

A more general form of the ampliﬁcation factor can be written as:

ln共Amp兲=aln

冉

VS30

VLIN

冊

+fb共VS30,VLIN兲ln共PGArock +fc共VS30,VLIN兲兲 +d,共4兲

where VLIN is a reference velocity above which the site response is linear. The d term is

needed because the VLIN may not be the reference velocity, VREF. We assume that in the

linear range, the site response should reduce to the form used by Boore et al. (1997).

That is, as PGArock goes to zero or as VS30 goes to VLIN, the ln(Amp) becomes propor-

tional to ln共VS30兲.

There are two simple forms of Equation 4 that satisfy these constraints. In the ﬁrst

form, the fbterm includes a Vs30 dependence and the fcis constant. This results in the

following model:

ln共Amp兲=

冦

aln

冉

VS30

VLIN

冊

+bln

冉

VS30

VLIN

冊

ln共PGArock +c兲+dfor VS30 ⬍VLIN

aln

冉

VS30

VLIN

冊

+dfor VS30 艌VLIN

冧

共5兲

While this form meets the constraints, it has the undesired consequence that there exists

aPGArock value at which the ampliﬁcation factor becomes independent of the VS30 for

VS30⬍VLIN. In particular, if the PGArock= exp共−a/b兲−c, then the ampliﬁcation is con-

stant for all VS30 values (Walling and Abrahamson 2006). This feature of the model is

considered to be unrealistically restrictive.

As an alternative, we considered the case where fbis independent of VS30 and fcis

dependent on VS30. The following model meets the constraints described above:

ln共Amp兲=

冦

aln

冉

VS30

VLIN

冊

−bln共PGArock +c1兲

+bln

冉

PGArock +c

冉

VS30

VLIN

冊

n

冊

+dfor VS30 ⬍VLIN

共a+bn兲ln

冉

VS30

VLIN

冊

+dfor VS30 艌VLIN

冧

共6兲

This form avoids the feature of the ampliﬁcation being independent of Vs30 atagiven

PGA level.

248 WALLING, SILVA, AND ABRAHAMSON

EXAMPLE FITTING

The ampliﬁcation factors are modeled using Equation 6 and the parameters n, c,

VLIN, and b were estimated using ordinary least squares. The n and c coefﬁcients were

constrained to be independent of frequency and the parameters VLIN and b were esti-

mated at each frequency. An example of the parametric ﬁt of the analytic model for

VS30= 270 m/ s and 560 m/ s at T = 0.2 sec is seen in Figure 5.

SMOOTHING OF PARAMETERS

A smoothed model of the period dependence of the b and VLIN terms was developed.

The b and VLIN term are correlated parameters. This correlation is addressed by ﬁrst

smoothing the VLIN values and using the smoothed VLIN values to recompute the b

terms. These resulting b terms were then smoothed. The functional form of the model

used for smoothing the b and VLIN term is given by

x=

冦

2T艌T2

␣

0+兺

i=1

6

␣

i关ln共T/T0兲兴iT1⬍T⬍T2

1T艋T1

冧

,共7兲

where xis either ln共VLIN兲or band Tis the spectral period. The resulting coefﬁcients for

the smoothed b and VLIN models listed in Table 2. The period dependence of the b and

VLIN terms is shown in Figures 6 and 7 for both the EPRI and Peninsular Range models.

Figure 5. Example of the parametric ﬁt of the analytic model for VS30 = 270 and 560 m /s using

Eq. 6 at T= 0.2 s based on the PEN model.

NONLINEAR SITE AMPLIFICATION FACTORS FOR CONSTRAINING THE NGA MODELS 249

COMPARISON OF RESULTS

The smoothed nonlinear ampliﬁcation model from this study is compared to the lin-

ear ampliﬁcation model from Boore et al. (1997) and the nonlinear ampliﬁcation model

from Choi and Stewart (2005). The Choi and Stewart (2005) model was derived using

the empirical models of Abrahamson and Silva (1997), Sadigh et al. (1997), and Camp-

bell and Bozorgnia (2003) for generic rock. The Choi and Stewart (2005) model refer-

ence VS30 derived with the Abrahamson and Silva (1997) model is 532±93 m/s for a

Tabl e 2 . Parameterization of the nonlinear site response from the Silva simulations

EPRI Model Peninsular Model

VLIN bVLIN b

T00.0133 0.02 0.025 1.00

T10.020 0.025 0.025 0.0125

T21.1 2.5 1.25 2.5

␣

07.244 −1.1050 6.763 −1.9546

␣

1−1.6411 −0.42439 0.20784 1.9097

␣

22.7107 1.482073 0.400139 1.16744

␣

3−1.42332 −1.329229 −0.5196731 −0.3716778

␣

40.294717 0.45954657 0.1566076 −0.3893755

␣

5−0.0216321 −0.0705797 −0.0144830 −0.0931755

␣

60.0 0.00418515 0.0 −0.007279

16.9431 −1.139 6.7628 −1.190

26.0380 −0.650 5.9964 0.1504

n 1.30 1.18

c 1.38 1.88

Figure 6. Comparison of the estimated VLIN terms with the smoothed models.

250 WALLING, SILVA, AND ABRAHAMSON

frequency= 0.3 Hz and 519 ±69 m / s for a frequency=1.0 Hz (Choi and Stewart 2005).

This is consistent with other estimates of the reference velocity for Abrahamson and

Silva (1997) rock as approximately 550 m / s.

To allow for comparisons of the analytical model results with the Choi and Stewart

(2005) model, the ampliﬁcation factors from the analytical model were normalized by

the ampliﬁcation for Vs30= 550 m/ s and the PGA1100 was converted to PGA550.Asan

example, the ampliﬁcation for VS30= 274 m/ s, corresponding to generic soil is shown as

a function of the PGA550 in Figures 8–10 for spectral periods of 0.01, 0.2, and 1.0 sec,

Figure 7. Comparison of the estimated b terms with the smoothed models.

Figure 8. Comparison of site ampliﬁcation models for VS30 = 274 m / s for PGA.

NONLINEAR SITE AMPLIFICATION FACTORS FOR CONSTRAINING THE NGA MODELS 251

respectively. The linear ampliﬁcation from the Boore et al. (1997) model is also shown

for comparison.

As seen in each ﬁgure, the three models are in general agreement at the low levels of

shaking, becoming approximately equal at PGA550= 0.1 g.ForPGA550 values greater

than 0.2 g, the Choi and Stewart (2005) model shows less nonlinearity than the models

derived in this paper. These trends are more noticeable at shorter periods where soil re-

sponse is more nonlinear, as seen in Figure 9 for T= 0.2 s, and but less at the longer

periods T⬎1.0 s, where soil response is primarily linear as seen in Figure 10 for T

=1.0 s.

Figure 9. Comparison of site ampliﬁcation models for VS30 = 274 m / s for T = 0.2.

Figure 10. Comparison of site ampliﬁcation models for Vs30 = 274 m / s for T = 1.0 s.

252 WALLING, SILVA, AND ABRAHAMSON

USE OF RESULTS BY NGA DEVELOPERS

Of the ﬁve NGA developers, two used the analytical site response model results de-

scribed in this paper to constrain the nonlinear site response. Abrahamson and Silva

(2008) and Campbell and Bozorgnia (2008) used the form of the ampliﬁcation given in

Equation 6 and constrained the b, VLIN,c1, and n terms to be given by the values from

the PEN model (Table 2). The linear site response (a term) was derived from the em-

pirical data. They also constrained the impact of the nonlinear response on the standard

deviation of the ground motion using the PEN model. The Boore and Atkinson (2008)

model constrained the nonlinear ampliﬁcation in their model using the Choi and Stewart

(2005) model results. Chiou and Youngs derived the nonlinear response based on the em-

pirical data and checked these results against the analytical results shown in this paper.

The ﬁfth model, Idriss (2008), is for rock only, so it does not include a nonlinear site

response term.

CONCLUSIONS

The analytical site response results based on the equivalent-linear method with RVT

were provided to the NGA developers as one possible approach for constraining the non-

linear site response in their empirical ground-motion models. Two of the four developers

that included site response used the analytical results based on the PEN soil to constrain

their ground-motion models. With these constraints, the site ampliﬁcation from these

two ground motion-models will, in general, be consistent with site-speciﬁc site response

approaches commonly used for engineering projects. In particular, hazard studies con-

ducted using the empirical soil ground-motion models are expected to be generally con-

sistent with site-speciﬁc studies using empirical rock ground motions models as input to

site-speciﬁc site response analyses.

ACKNOWLEDGMENTS

This study was sponsored by the Paciﬁc Earthquake Engineering Research Center’s

Program of Applied Earthquake Engineering Research of Lifelines Systems supported

by the California Department of Transportation, the California Energy Commission, and

the Paciﬁc Gas and Electric Company.

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(Received 7 August 2007; accepted 21 April 2008兲

NONLINEAR SITE AMPLIFICATION FACTORS FOR CONSTRAINING THE NGA MODELS 255