Factors influencing soil surface seismic hazard curves

Soil Dynamics and Earthquake Engineering 83 (2016) 180–190

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Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn

Factors influencing soil surface seismic hazard curves Menzer Pehlivan a,n, Ellen M. Rathje b, Robert B. Gilbert b a

CH2M, 1100 112th Avenue NE, Suite 500, Bellevue, WA 98004, United States University of Texas at Austin, Department of Civil, Architectural, and Environmental Engineering, 301 E. Dean Keeton St, Stop C1700, Austin, TX 78712, United States

b

art ic l e i nf o

a b s t r a c t

Article history: Received 10 January 2015 Received in revised form 21 December 2015 Accepted 18 January 2016 Available online 12 February 2016

Performance-based seismic design of important structures requires design ground motions from probabilistic seismic hazard analysis (PSHA) that incorporate the effects of local site conditions. Seismic hazard curves incorporating site-specific soil conditions can be generated through the convolution of rock hazard curves with statistical models for site-specific ground motion amplification factors (AF). The AF relationships are developed from a series of site response analyses. The goal of this study is to evaluate how the AF relationships and the resulting surface hazard curves are influenced by different approaches in the site response analysis, specifically the time series (TS) vs. random vibration theory (RVT) approaches, and by different levels of shear wave velocity variability introduced in the site response analysis. The results show that the median AF relationships derived from TS and RVT analyses are similar, except at periods near the site period, where RVT analysis may predict larger AF. Including the effect of shear wave velocity variability reduces the median AF and increases the standard deviation associated with the AF relationship (σ lnAF ). Generally, the soil hazard curve derived by the AF relationship with the largest σ lnAF generates the largest ground motions, and this effect is most significant at small annual frequencies of exceedance. The effect of σ lnAF on soil hazard curves is larger than the effect of different median AF relationships. The value of σ lnAF is influenced significantly by the variability in the shear wave velocity and therefore proper specification of this variability is critical when developing soil hazard curves. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Convolution method Probabilistic site response analysis Random vibration theory Site property variability

1. Introduction Performance-based seismic design aims to ensure that a structure or facility under consideration can withstand a certain level of shaking defined by the design ground motion without excessive damage. The design ground motion is quantitatively estimated through probabilistic seismic hazard analysis (PSHA), which provides the annual frequency of exceedance (i.e. hazard level) for different ground motion levels. PSHA should incorporate the effects of the detailed site-specific soil conditions. The effect of local site conditions can be significant, with the site response changing the amplitude, frequency content, and duration of shaking. The influence of the local soil conditions on ground shaking is commonly estimated through dynamic site response analysis performed using site-specific soil properties. In practice this analysis is almost exclusively performed in a deterministic manner, in which the surface response is computed via dynamic n

Corresponding author: Tel.: þ 1 512 944 9354. E-mail addresses: [email protected] (M. Pehlivan), [email protected] (E.M. Rathje), [email protected] (R.B. Gilbert). http://dx.doi.org/10.1016/j.soildyn.2016.01.009 0267-7261/& 2016 Elsevier Ltd. All rights reserved.

site response analyses performed for a suite of rock motions selected to fit a single input uniform hazard response spectrum (UHS) for rock (Fig. 1). However, the surface response spectrum computed in this way does not correspond with the hazard level of the input UHS because it does not account for the additional variability in the site amplification [2]. The convolution approach is the most commonly used technique to incorporate the effects of site-specific soil conditions into a PSHA [15,2,21]. The convolution approach simply convolves a rock hazard curve with a statistical model for site amplification to generate a seismic hazard curve for the ground surface (Fig. 1). The statistical model for site amplification predicts the median amplification and its standard deviation for each spectral period as a function of the intensity of shaking on rock. These statistical models are derived from a large suite of site response analyses that span a broad range of input motion intensities and include statistical variations in the site properties. The required large suite of site response analyses makes the random vibration theory (RVT) approach to site response analysis [18,22] an attractive approach to use for this task, as RVT does not require selection of input time histories. However, recent research has indicated that RVT site response analysis may over-predict the median site amplification

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181

Fig. 1. Deterministic and probabilistic approaches incorporating site amplification in seismic hazard analysis.

as compared with traditional time series (TS) analysis (e.g. [14]), although both approaches use the same equivalent-linear model for the dynamic site response. Additionally RVT analysis does not include motion-to-motion variability and thus the variability for a site amplification model developed from a suite of RVT analyses may be different than that from a suite of TS analyses. This paper compares statistical site amplification relationships developed from equivalent-linear site response analyses performed using the traditional TS approach and the RVT approach, as well as the soil hazard curves derived from using the different site amplification relationships. The parametric uncertainty in local site conditions is introduced into the site response analysis through Monte Carlo simulation, which enables the quantification of the variability in the amplification model due to variability in the shear wave velocity. The goal of this study is to evaluate how different approaches of site response analysis and the level of shear wave velocity variability influence the resulting soil hazard curves. This paper builds off of the work of [16], which investigated the influence of different TS site response techniques (i.e., nonlinear and equivalent linear) and different constitutive model parameters on the resulting site amplification relationships and soil hazard curves. They found that both the site response technique and the model parameters significantly influenced the site amplification model and resulting soil hazard curves. The study presented here extends this previous work by considering RVT and TS site response techniques, including the effect of soil property variability through Monte Carlo simulation and usage of the convolution approach to develop the soil hazard curves.

2. Development of site-specific soil hazard curves PSHA provides a method for the consideration of uncertainties in the magnitude, location, and recurrence rate of earthquakes, together with the variability in the prediction of ground motions as a function of magnitude and distance. The results of a PSHA are seismic hazard curves for spectral acceleration, each of which plots the annual frequency of exceedance (AFE) as a function of ground shaking intensity for a different response spectral period. A uniform hazard spectrum can be defined from the hazard curves by identifying ground motion levels with the same annual frequency of exceedance for each period. PSHA commonly is performed using ground motion prediction equations that represent rock conditions or generic soil conditions. The details of the site-specific soil conditions are not taken into account. The convolution approach is one approach available to incorporate the site-specific soil conditions into the PSHA.

2.1. Convolution approach The convolution approach computes a soil hazard curve using the bedrock hazard curve at the period of interest and the sitespecific soil response. The soil response is quantified through an amplification factor, AF, defined as the ratio of the soil spectral acceleration (Sas ) to the rock spectral acceleration (Sar ) and soil nonlinearity is taken into account by the variation of AF with input intensity, Sar [3]. In the convolution approach, the annual frequency of exceedance of soil ground motion level z (AFEs ðzÞ) at spectral period T is

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0.25

Rock Hazard Curve

1e-1

0.20

1e-2

0.15

P(Sar)

Annual Frequency of Exceedance (1/yr)

1e+0

1e-3

0.10

1e-4

0.05

1e-5

0.00

0

1

2

3

1 1 1 2 5 7 1 .2 .3 .4 .5 .7 0 0 .0 0 0 .0 0 .0 0 .0 0 .0 0 . 0 0 0 0 0 0 .0 0

4

2

3

4

Sar (g)

Sar (g) 4.0

Amplification Factor, AF

1

Amplification Relationship Median AF +/- σlnAF

1.0

0.1 0.01

0.1

1

10

Rock Spectral Acceleration, Sar (g) Fig. 2. Requirements of convolution approach (a) rock hazard curve, (b) annual probability of occurrence of Sar , and (c) site amplification model.

computed as follows:   X    z AFEs ðzÞ ¼ P AF 4 xj ∙pSar xj xj x

ð1Þ

j

 than the where P AF Z xzjx is the probability that AF is greater  quantity xz given an Sar amplitude of xj and pSar xj is the annual probability of occurrence for Sar equal to xj . This probability is obtained by differencing a previously defined rock hazard curve   [19]. P AF Z xz j x can be computed by assuming AF is lognormally distributed and a function of x using Eq. (2): ! z  h z i ^ ln x  μlnAFjx ð2Þ P AF Z x ¼ Φ x σ ln AFjx where μlnAF jx is the mean value of lnAF given Sar ¼x and σ lnAF jx is ^ ð∙Þ is the the standard deviation of lnAF given Sa ¼x. Note that Φ r

standard Gaussian complementary cumulative distribution function. Parameters μlnAF jx and σ lnAF jx are obtained from site-specific AF relationships. Fig. 2 graphically illustrates the components of the convolution approach contained in Eq. (1). The first component is the rock hazard curve for Sar (Fig. 2a) and the annual probability of occurrence of each Sar (Fig. 2b), which is computed from the AFE at adjacent ground motion levels. The second component is the site-specific amplification relationship. The site-specific AF

relationship is defined as a function of Sar for each period (Fig. 2c). The two components are convolved together using Eq. (1) to obtain surface hazard curves. Another available approach to develop soil hazard curves involves modification of a rock ground motion prediction equation (GMPE) based on the developed site amplification function and its standard deviation [2], and this approach was used in the study by [16]. However, the GMPE modification approach is limited to specific functional forms of the site amplification relationship and these functional forms may not appropriately capture the variation of site amplification with Sar . Therefore, the study described here uses the more flexible convolution approach. 2.2. Development of site amplification relationships The site-specific AF relationships describe the change in AF with input motion intensity and quantifies the variability in AF. The AF relationships are developed through regression of an AF dataset derived from a suite of site response analyses. Onedimensional equivalent linear analysis using time series (TS) input motions or random vibration theory (RVT) can be used for the site response analysis. In TS analysis, rock acceleration-time histories are used as input and are obtained from available strong ground motion databases. In RVT site response analysis [18,14], the input motion is specified by its Fourier Amplitude Spectrum (FAS),

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60 m [11]. The nonlinear modulus reduction and damping curves are assigned for the four main velocity layers based on the empirical model of [10]. The small strain natural period of the site is about 0.8 s based on the quarter-wavelength method. In this study, multiple 1D shear wave velocity realizations are generated through the randomization tool implemented in the site response program Strata [13]. Strata uses the random field models of [23] to generate shear wave velocity profiles. These models assume that the VS is log-normally distributed at any given depth and correlated between adjacent layers. The statistical parameters required for the generation of the VS profiles are the standard deviation of the natural logarithm of the shear wave velocity (σ lnVs ) and the interlayer correlation (ρ). For this study, ρ is specified as 0.8 and the effect of different levels of variability is explored by taking σ lnVs as 0.1, 0.2, and 0.3. These values represent coefficients of variation of about 10%, 20%, and 30%, respectively. Minimum and maximum limits of the shear wave velocity assigned to any layer are specified at 7 2σ lnVs . Both TS and RVT site response analyses are performed with varied VS profiles to develop site amplification functions that include the effects of VS variability. Twenty VS profile realizations are generated for the TS and RVT analyses for each of the three different σ lnVs levels. Fig. 3 presents the varied VS profiles generated for σ lnVs of 0.1 and 0.3. Fig. 3 shows the increasing variability in the generated profiles with increasing σ lnVs , but in each case the median VS profile of the varied profiles agrees well with the baseline VS profile within about 5% at any given depth.

which can be specified directly from seismological theory [4] or derived from an acceleration response spectrum [20,13], and its duration. RVT analysis computes a response spectrum from the FAS and the specified duration using a peak factor derived from extreme value statistics [4]. Because the RVT input motion is specified only by its FAS, the site response must be modeled through frequency-domain transfer functions. Thus, only equivalent-linear analysis can be performed using RVT. Although nonlinear site response analysis may be more appropriate than equivalent-linear analysis for the largest input motions required to develop an AF relationship, only equivalent linear analysis was performed in this study because RVT site response analysis (commonly used for nuclear projects) was the focus of this study. An advantage of RVT-based site response analysis is that it does not require the selection of input time series. RVT site response predictions are comparable to the median response obtained through site response analysis performed using a suite of input TS, however there is the potential for RVT to predict a larger response at the modal frequencies of a site [14]. Additionally, RVT analysis does not capture motion-to-motion variability and thus the variability in the amplification factors predicted by RVT may be very different than by TS analysis, which will influence the soil hazard curves computed using the two approaches. An additional consideration in developing the site amplification relationships is the spatial variability and parametric uncertainty in the soil properties across a site. The shear wave velocity (VS) profile is the most critical parameter influencing the site response and the effect of the variability in VS can be taken into account by performing 1D analyses for multiple realizations of the VS profile. Multiple 1D profile realizations are generated by statistically varying the site VS profiles using Monte Carlo simulation.

3.2. Input motion characterization The rock hazard at the site in the San Fernando Valley is defined from the USGS Hazard Curve Application [17] for NEHRP B/C site class conditions (VS30 ¼ 760 m/s). The hazard curves for peak ground acceleration (PGA) and spectral acceleration (Sar ) at T¼ 1.0 s are shown in Fig. 4. At an AFE of 2.1  10–3 1/yr (i.e., 10% of probability of exceedance in 50 years), the PGA is 0.55 g and the Sar at T ¼1.0 s is 0.40 g, while at an AFE of 4  10–4 1/yr (i.e., 2% of probability of exceedance in 50 years), the PGA is 1.05 g and the Sar at T¼1.0 s is 0.80 g. Input motions for the site response analyses are required to span the range of input intensities indicated in the rock hazards curves.

3. Study site and analyses performed 3.1. Site characteristics The Sylmar County Hospital (SCH) site in the San Fernando Valley of Southern California is used for the analyses in this study. The shear wave velocity profile of the SCH site is shown in Fig. 3a. The site has about 100 m of alluvium above bedrock, with the VS ranging from about 250 m/s at the surface and increasing to above 700 m/s at

VS (m/s)

VS (m/s) 0

250

500

0

750 1000 1250 1500

250

500

VS (m/s)

750 1000 1250 1500

0

0

lnVs

20

20

0

60

750 1000 1250 1500

σlnVs = 0.3

= 0.1

ρ = 0.8

Vs Profile Realizations

Vs Profile Realizations

20

Average Vs Profile

Depth (m)

Depth (m)

Depth (m)

60

40

500

ρ = 0.8

Average Vs Profile

40

250

0

σ

Baseline Vs Profile

183

40

60

80

80

80

100

100

100

Fig. 3. (a) Baseline VS profile of the site analyzed in this study and examples of VS realizations for σ lnVs of (b) 0.1 and (c) 0.3.

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For the TS analyses, input motions recorded at sites with an average shear wave velocity in the top 30 m (VS30) greater than 600 m/s are selected from the NGA strong motion database (peer. berkeley.edu/nga). Using this criterion 130 recorded ground motions from earthquakes with magnitudes ranging from 6.1 to 7.9 and distances ranging from 1.5 km to 90 km are selected for use in the TS analyses. The response spectra of the input motions for the TS analysis are shown in Fig. 5a. The PGA of the input motions range from about 0.005 g to 1.0 g, representing a wide range of bedrock input intensities. The input motions are not scaled from their recorded intensities. An important advantage of RVT site response analysis is that fewer analyses are required than for TS analysis. A set of 19 RVT motions are generated following the recommendations from NUREG/CR-6728 [15] and Regulatory Guide.1.208 [24]. The recommended approach defines controlling earthquake scenarios for the input motions from the deaggregation of the hazard curves

1

Annual Frequency of Exceedance (1/yr)

Rock Hazard Curves PGA 1.0 s 0.1

at low and high frequencies and at different hazard levels. For low frequencies (i.e., about 1 Hz) the controlling earthquake magnitude for the study site is about 6.8 and the associated distances are between 3.9 km and 37 km; for high frequencies (i.e., about 10 Hz) the controlling earthquake magnitude is about 6.7 and the associated distances are between 3.7 km and 29 km. The response spectra for the specified controlling earthquakes are computed as the average spectrum from four of the Next Generation Attenuation (NGA) ground motion prediction equations for rock conditions (VS30 ¼760 m/s). The NGA relationships by [1,5,7] and [9] are used. The computed response spectra obtained for low and high frequency controlling events for each hazard level are then scaled to match the rock uniform hazard spectrum (UHS) at that same hazard level. The smallest PGA for the resulting response spectra is only about 0.1 g. To ensure accurate amplification factors are obtained for lower levels of shaking, additional low intensity motions are specified by scaling the smallest response spectrum to PGA levels of 0.01 g and 0.03 g. Additionally, to represent the high intensity motions at long periods, a motion was added by scaling the response spectrum of the largest earthquake to 1.0 g at T¼ 1.5 s. Fig. 5b plots the final set of 19 input motions for the RVT analyses. The range of PGA values for the RVT input motions is 0.01 g to 2.4 g. The durations associated with each response spectrum for use in the RVT analysis are obtained from the Silva et al. [22] predictive relationship.

4. Results

0.01

4.1. Amplification relationships 0.001

0.0001 0.0

0.5

1.0

1.5

2.0

Rock Spectral Acceleration, Sar (g) Fig. 4. Rock hazard curves used in this study.

The TS and RVT site response analyses are performed with the site response program Strata [13]. For the RVT analysis, the peak factor model of [8] is used along with the duration correction of [6]. For each input motion, surface response spectra and the associated AF values at each period are computed. Fig. 6 presents the computed AF results for both the TS and RVT analyses for the baseline VS profile. Both sets of analyses show peaks in AF for periods around 1.0 s, which corresponds with the first mode response of the site. The AF results in Fig. 6 are color-coded based on the input PGA, allowing one to observe the effects of soil

Fig. 5. Response spectra of rock input motions for (a) TS and (b) RVT analyses. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

M. Pehlivan et al. / Soil Dynamics and Earthquake Engineering 83 (2016) 180–190

nonlinearity on site amplification. There is a reduction in short period amplification and an elongation of the natural site period with increasing input intensity due to an increase in damping and

185

a reduction in stiffness associated with soil nonlinearity. These observed trends are clearer in the RVT results (Fig. 6b) because of the absence of motion-to-motion variability.

Fig. 6. Site amplification results for (a) TS and (b) RVT analyses. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 7. AF vs. Sar for PGA and T ¼1.0 s derived from TS and RVT analyses using the baseline VS profile.

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The AF values from Fig. 6 are plotted versus Sar in Fig. 7 for two selected spectral periods, PGA and T ¼1.0 s. For PGA, both the TS and RVT analyses show a reduction in AF at larger intensities due to soil nonlinearity. There is variability in the AF values from TS analysis due to motion-to-motion variability whereas the variability in the AF from RVT analysis is almost zero because motionto-motion variability is not taken into account. For T¼ 1.0 s, both the TS and RVT analyses show an initial increase in AF with increasing intensity (up to an Sar of about 0.4 g) followed by a decrease in AF. This shape in the AF relationship is also due to soil nonlinearity. The small-strain site period is about 0.8 s and as the site period elongates at moderate input intensities the AF at T ¼1.0 s increases. This behavior can be clearly observed in Fig. 6b by considering the AF values at T ¼1.0 s for PGA o0.06 g and PGA ¼0.24 to 0.48 g. At larger input intensities the site period continues to elongate past T ¼1.0 s and the AF at T¼ 1.0 s decreases. In Fig. 6b this trend is apparent in the small AF values at T ¼1.0 s for PGA 40.48 g. Statistical AF relationships are developed for the data in Fig. 7 by fitting a higher order polynomial to lnðAFÞ and lnðSar Þ using: lnðAFÞ ¼ a0 þ

n X

 k ak ∙ lnðSar Þ

ð3Þ

k¼1

When fitting Eq. (3) to the data, the order of the polynomial is selected to smoothly fit the data. The residuals between the relationship and data points are used to compute the standard deviation of lnðAFÞ, σ lnAF . Note that care must be taken when using higher order polynomials outside the range of Sar that is represented by

3.0

3.5

TS - PGA σlnVs=0.2

Amplification Factor , AF

Amplification Factor , AF

3.5

the available data (less than about 0.01 g and greater than about 1.5 g for the input motions used in this study). The AF relationship should be plotted over the entire range of Sar required for the convolution integral (Eq. 1) and, if necessary, modified at smaller or larger Sar to predict appropriate AF. This issue was only relevant for smaller Sar for this study, as the Sar values encompassed the full range of the values in the rock hazard curves. For Sar smaller than about 0.01 g, the AF was set to the AF predicted at 0.01 g. The median AFrelationships are shown in Fig. 7. For PGA, the relationships derived from TS and RVT analysis are very similar, but for T ¼1.0 s the RVT analysis shows larger amplification at Sar values between about 0.2 and 0.8 g. The maximum difference is about 25%. The larger amplification predicted by RVT at periods close to the site period is consistent with previous studies [14]. The results shown in Fig. 7 do not include the effects of the variability in VS. To account for this effect, site response analyses are performed with the statistically varied shear-wave velocity profiles generated by Monte Carlo simulation (Fig. 3). All the input motions are propagated through all of the statistically varied VS profiles. The AF data from the analyses using 20 statistically varied VS profiles with σ lnVs ¼ 0:2 are shown in Fig. 8 for PGA and T ¼1.0 s, along with the median AF relationships. Both the TS and RVT data now show significant variability, although the variability in the TS data includes both site-to-site variability and motion-to-motion variability while the variability in the RVT data only includes siteto-site variability. The AF relationships obtained from TS and RVT analyses for σ lnVs ¼ 0:0 (i.e., no VS variability) and for σ lnVs ¼ 0:2 are compared

2.5 2.0 1.5 1.0 0.5

3.0

TS - 1.0 s σlnVs=0.2

2.5 2.0 1.5 1.0 0.5 0.0

0.0 0.001

0.01

0.1

0.001

1

0.01

1

3.5

RVT - PGA σlnVs=0.2

Amplification Factor , AF

Amplification Factor , AF

3.5 3.0

0.1

Sar (g)

Sar (g)

2.5 2.0 1.5 1.0 0.5 0.0 0.001

3.0

RVT - 1.0 s σlnVs=0.2

2.5 2.0 1.5 1.0 0.5 0.0

0.01

0.1

Sar (g)

1

0.001

0.01

0.1

1

Sar (g)

Fig. 8. AF vs. Sar for PGA and T ¼ 1.0 s derived from TS and RVT analyses using shear wave velocity profiles statistically varied with σ lnVs ¼0.2.

M. Pehlivan et al. / Soil Dynamics and Earthquake Engineering 83 (2016) 180–190

in Fig. 9. For PGA, all of the AF relationships are similar. For T ¼1.0 s, all of the AF relationships show a peak at about Sar ¼ 0.3 g but the peak is about 25% larger for RVT with σ lnVs ¼ 0:0. Note that this larger peak is suppressed when variability in the shear wave velocity is included in the RVT analysis, such that the AF relationship developed for RVT (σ lnVs ¼ 0:2) is similar to that developed for TS (σ lnVs ¼ 0:0). These small values of σ lnAF associated with RVT analysis without VS variability makes these amplification relationships inappropriate for the convolution method because they essentially represent deterministic conditions. The shear wave velocity variability also affects TS analysis such that the AF relationship for TS (σ lnVs ¼ 0:2) shows the smallest peak in Fig. 9. The incorporation of VS variability reduces the difference in median AF for RVT and TS analyses to about 10%. Fig. 10 presents the influence of σ lnVs on the median AF relationships for T ¼1.0 s for TS and RVT analyses. Introducing VS variability into the TS analysis reduces the observed peak at Sar  0.3 g, and the peak decreases more as σ lnVs increases (Fig. 10a). The TS analysis with σ lnVs ¼ 0.3 predicts AF values at Sar  0.3 g about 15% smaller than those predicted for σ lnVs ¼0.0. For RVT analysis (Fig. 10b), the effect of introducing VS variability is more significant compared to TS analysis. For σ lnVs values greater than 0.1, there is a 30% reduction in AF around Sar  0.3 g. The increased sensitivity to variations in VS for RVT analysis is caused

by the fact that motion-to-motion variability is not incorporated in RVT and thus changes in σ lnVs more strongly affect the computed AF relationship. The σ lnAF values for each AF relationship are summarized in Table 1 for PGA and T ¼1.0 s. Generally, the σ lnAF for TS and RVT are different, even for cases where the median AF are similar (Fig. 8). An increase in the σ lnVs from 0.0 to 0.3 results in an increase in σ lnAF by a factor of about 2 for the TS analysis and more than a Table 1 Comparison of of AF relationships.

3.0

Amplification Factor, AF

PGA

2.5 2.0 1.5 1.0

TS (σlnVs=0.0) TS (σlnVs=0.2) RVT (σlnVs=0.0) RVT (σlnVs=0.2)

0.5 0.0 0.001

0.01

0.1

2.0 1.5 1.0 0.5 0.0 0.001

1

1.0 s

2.5

TS (σlnVs=0.0) TS (σlnVs=0.2) RVT (σlnVs=0.0) RVT (σlnVs=0.2) 0.01

0.1

1

Sar (g)

Sar (g)

Fig. 9. Comparison of AF relationships from TS and RVT analyses: (a) PGA and (b) 1.0 s.

3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.01

3.0

T S - 1. 0 s Amplification Factor, AF

Amplification Factor, AF

Amplification Factor, AF

3.0

187

σlnVs = 0.0 σlnVs = 0.1 σlnVs = 0.2 σlnVs = 0.3 0.1

Sar (g)

1

RVT - 1.0 s

2.5 2.0 1.5 1.0 0.5 0.0 0.01

σlnVs = 0.0 σlnVs = 0.1 σlnVs = 0.2 σlnVs = 0.3 0.1

Sar (g)

Fig. 10. Influence of σ lnVs on AF relationships at T ¼ 1.0 s: (a) TS and (b) RVT analyses.

1

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factor of 8 to 10 for RVT analysis. Of course, this effect is larger for RVT because there is almost no variability in AF when σ lnVs ¼ 0.0. For both PGA and T¼ 1.0 s, the σ lnAF from TS analysis is larger than that of RVT because TS includes motion-to-motion variability and site-to-site variability. The exception is σ lnVs ¼0.3, where the σ lnAF from RVT analysis is larger than that of TS. This trend indicates that there may exist some negative correlation between the effects of motion-to-motion variability and site-to-site variability. Close inspection of the AF data presented in Figs. 7 and 8 indicates that the variability in the AF data varies with Sar . Generally, the scatter is larger at larger Sar because the different motions and different velocity profiles induce different levels of nonlinearity at these larger levels of shakings and these differences influence the computed AF. To investigate the relationship between σ lnAF and Sar , σ lnAF is computed within different bins of Sar for the AF data from TS and RVT analyses and the resulting values are shown in Fig. 11. Fig. 11 clearly shows that the variability in AF is significantly larger at larger input intensities for both PGA and T¼1.0 s. In some cases, the value of σ lnAF increases by a factor of more than 3 as Sar increases. At larger Sar , which often drive the design motions, the variability in AF is generally larger than the values represented by a constant σ lnAF (Table 1). The increased σ lnAF at larger Sar is an important feature and the influence of incorporating this feature is investigated subsequently in this paper. 4.2. Soil hazard curves Soil hazard curves are computed using the convolution approach (Eq. 1). For this calculation, the rock hazard curves in Fig. 4 are differenced to obtain the annual probability of occur  rence, pX xj , for each Sar , and μlnAF jx and σ lnAF jx for each Sar were computed from the developed AF relationships (Figs. 9 and 10, Table 1). Note that the AF relationships are only defined within the range of available input Sar and if needed these higher order polynomials can be carefully linearly extrapolated to larger Sar in AF-ln(Sar ) space for use in the convolution integral. Fig. 12 presents site-specific soil hazard curves for PGA and T¼1.0 s obtained using the AF relationships and the associated σ lnAF derived from the TS and RVT analyses for σ lnVs of 0.0 and 0.2 (Table 1). These intensity-independent values of σ lnAF are used to more clearly compare the results for the different analyses, although the intensity-dependent σ lnAF more accurately capture the variability in AF. For PGA (Fig. 12a), the four curves are similar except for some slight deviations at AFE smaller than about 0.001 1/

0.4

yr. The RVT (σ lnVs ¼0.0) curve predicts the smallest ground motions because σ lnAF is essentially zero for this AF relationship. This level of variability essentially indicates that the site amplification is perfectly known, which is unrealistic. Therefore, RVT analysis that does not consider VS variability should not be used to develop AF relationships for use in the convolution approach. The other three curves in Fig. 12a are very similar with each other, but each predicts larger ground motions than RVT (σ lnVs ¼0.0), because σ lnAF is considerably larger (Table 1). This difference is only observed at smaller AFE because the variability in AF has a stronger influence on the predicted ground motion at low AFE. For T¼1.0 s (Fig. 12b), there are larger differences in the hazard curves because there are more significant differences between the median AF relationships and σ lnAF values. The RVT (σ lnVs ¼ 0:0) hazard curve has a distinctly different shape than the others due to the strong peak in the AF relationship and the very small variability in AF (σ lnAF ¼ 0:01). The shape of this AF relationship results in a maximum Sas of about 1.2 g for Sar values greater than 0.5 g. When coupled with essentially zero σ lnAF , this AF relationship generates a hazard curve that has an asymptote at about 1.2 g. However, as noted earlier, the very small σ lnAF is unrealistic and therefore should not be used to develop site-specific soil hazard curves. The other three hazard curves in Fig. 12b have similar shapes and predict similar levels of shaking at AFE larger than around 0.002 1/yr. A smaller AFE, the predicted ground motions increase as the σ lnAF of the model increases. Although RVT (σ lnVs ¼ 0.2) and TS (σ lnVs ¼0.2) have similar σ lnAF (Table 1), RVT predicts larger motions due to larger median AF (Fig. 9b). Thus, it is important to note the combined effect of the median amplification relationship and σ lnAF on the soil hazard curves shown in Fig. 12. To demonstrate the independent effects of the median AF relationship and σ lnAF , consider the hazard curves for T¼ 1.0 s shown in Fig. 13. The soil hazard curves in Fig. 13a are calculated using a constant σ lnAF of 0.2 coupled with four different median AF relationships developed by RVT analyses with different σ lnVs . These median AF relationships were shown in Fig. 10b. The differences observed in the soil hazard curves in Fig. 13a are therefore solely due to differences in median AF relationships. For smaller AFE, the σ lnVs ¼0.0 model predicts the largest ground motion because this AF relationship contains the largest peak (Fig. 10b). The motions generally get smaller as σ lnVs increase and the peak in the AF relationship decreases. At AFE equal to 0.0004 1/yr, the difference in the predicted ground motions between σ lnVs ¼0.0 and σ lnVs ¼0.3 is about 20%. Fig. 13b shows soil hazard curves computed using the

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median AF relationship obtained for RVT (σ lnVs ¼ 0.2) analysis and σ lnAF values ranging from 0.0 to 0.5. As can be clearly seen from Fig. 13b, the predicted ground motions increase with increasing level of σ lnAF , especially at smaller AFE. At AFE equal to 0.0004 1/yr, the difference in the predicted ground motions between σ lnAF ¼0.0 and σ lnAF ¼0.5 can be as large as 45%. Another consideration is the variation of σ lnAF with input motion intensity (Fig. 11) and its influence on the soil surface hazard curve. Fig. 14 compares the soil hazard curves obtained for the TS (σ lnVs ¼0.2) AF relationships at T ¼1.0 s when using a constant σ lnAF (Table 1) and variable σ lnAF (Fig. 11). For these models, the variable σ lnAF is almost 50% larger than the constant σ lnAF at high input intensities (Sar 40.3 g). The soil hazard curve computed with variable σ lnAF predicts larger ground motions at small AFE due to the larger variability associated with the higher input intensities. At an AFE of 0.0004 1/yr the ground motion is about 10% larger when the variable σ lnAF is taken into account. The variation in σ lnAF with Sar should be modeled when calculating the soil hazard curves using the convolution approach.

5. Conclusions Design of important structures requires estimation of design ground motions corresponding to a specific hazard level at the soil surface based on probabilistic seismic hazard analysis. The effects of the local soil conditions can be taken into account through the convolution approach, which incorporates site response analysis into the probabilistic seismic hazard analysis through convolution of the rock hazard curve and a site amplification relationship developed from site response analyses. This paper focused on evaluating how site amplification relationships and the resulting soil surface hazard curves are influenced by different approaches to site response analysis and different levels of variability in the shear wave velocity. A suite of site-response analysis were performed using equivalent-linear TS and RVT approaches to develop site amplification relationships. The median AF relationships derived from TS and RVT analyses were similar, except at periods close to the site period where RVT analysis predicted site amplification about 25% larger than TS analysis at the peak in the AF relationship.

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Acknowledgments

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TS - 1.0 s The research presented in this paper was supported by funding from the U.S. Nuclear Regulatory Commission under Grant NRC04–09-134. This support is gratefully acknowledged.

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References

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Introducing variability in the shear wave velocity via Monte Carlo simulation reduced the difference between the RVT and TS site amplification relationships to about 10%. RVT analysis with no shear wave velocity variability produces an AF relationship with essentially zero σ lnAF because RVT analysis does not include the effects of motion-to-motion variability. This level of variability in AF is unrealistic, and thus RVT analysis without consideration of shear wave velocity variability should not be used to develop site amplification relationships for the convolution method. When variability in the shear wave velocity is introduced into the site response analyses, the σ lnAF associated with RVT and TS analysis are comparable. Both the median AF relationship and the associated σ lnAF influence the computed soil surface seismic hazard curves, particularly at small AFE. Generally, the AF relationship with the largest σ lnAF generates the largest ground motions, most significantly at small AFE. The effect of σ lnAF is more significant than the effect of the median AF relationships derived from the different site response approaches. The σ lnAF increases with increasing input intensity and this effect should be modeled in the convolution analysis. Incorporating the variation in σ lnAF with the input intensity in the convolution analysis leads to larger ground motions, predominantly at lower AFE. This study illustrates the importance of the statistical amplification models developed for use in convolution approach to estimate the site-specific soil hazard curves. These amplification models require a large suite of site response analyses that span a wide range of input intensities and incorporate the variability in the site properties. RVT site response analysis is an efficient approach to perform the required analyses because input time series are not needed, but the variability in the site properties must be taken into account to properly quantify the site amplification variability. Whether TS or RVT analysis is performed, the variability in the shear wave velocity profile (as well as other soil properties) must be quantified. An increase in site property variability may result in a smaller median amplification relationship but it also leads to larger variability in the amplification, which results in larger ground motions in most cases. This result provides incentive to collect additional site property data to reduce this variability, if possible.

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