Technical Note

Evaluation of Seismic Slope Displacements Based on FullyCoupled Sliding Mass Analysis and NGA-West2 Database

Wenqi Du1; Gang Wang, M.ASCE2; and Duruo Huang3

Abstract: In this study, two predictive models for seismic slope displacements are developed based on an equivalent-linear fully coupledsliding mass model and 3,714 ground-motion recordings selected from the Next Generation Attenuation (NGA)-West2 database. Both pre-dictive models use the sliding system’s yield acceleration and initial fundamental period Ts as predictor variables, whereas they use differentsets of vector intensity measures (i.e., spectral acceleration at a degraded period of the system 1.5Ts and Arias intensity in one model; peakground acceleration and spectral acceleration at 2 s in another). The models are developed following the framework of model BT07, apredictive model for estimating the sliding displacement based on the fully coupled sliding mass analysis. The framework consists oftwo parts, namely, the probability of “zero” displacement (<1 cm) and the statistical distribution of “nonzero” displacement (≥1 cm).The proposed models in this study can be regarded as updates of the BT07 model, which can be used for estimating earthquake-induceddisplacements for a wide range of slope cases. DOI: 10.1061/(ASCE)GT.1943-5606.0001923.© 2018 American Society of Civil Engineers.

Author keywords: Seismic sliding displacement; Fully coupled analysis; Predictive models; Next Generation Attenuation (NGA)-West2database.

Introduction

Estimating earthquake-induced displacement of slopes is an impor-tant topic for evaluating and mitigating seismic hazards (Jibson2011). Since Newmark’s (1965) pioneering work on the rigid slid-ing block model, many researchers (e.g., Saygili and Rathje 2008;Du and Wang 2016) have proposed empirical equations to predictthe sliding displacement, which is regarded as a quantitative indexin various applications (e.g., Du and Wang 2014; Rathje et al.2014). Yet, the Newmark model is most suitable for modeling shal-low landslides of stiff materials that move along a well-defined slipsurface. Flexible/deep sliding masses generally deform internallywhen subjected to seismic shaking. Therefore, the dynamic behav-ior of the sliding mass should be considered in the analysis. For thispurpose, Rathje and Bray (2000) proposed a fully coupled slidingmass model to simulate stick-slip and deformation of the flexiblemass simultaneously.

Bray and Travasarou (2007) were the first to develop a predic-tive model for estimating the sliding displacement D based on thefully coupled sliding mass analysis. The model (called BT07 here-after) predicts D as a function of yield acceleration (ky), initial fun-damental period Ts of the slope, and the spectral acceleration of theground motion at a degraded period 1.5Ts [Sað1.5TsÞ]. The mo-ment magnitude (Mw) of the earthquake is also included in themodel to correct prediction bias. Detailed information of theexisting predictive models is summarized in Table 1.

Recent development of the Next Generation Attenuation(NGA)-West2 database (Ancheta et al. 2014) provides an opportu-nity to update these predictive models based on an extendedground-motion database. Therefore, this paper develops two pre-dictive models for seismic slope displacements based on the fullycoupled sliding mass analysis and 3,714 ground-motion recordingsselected from the NGA-West2 database. The predictive models fol-low a similar analysis framework of Bray and Travasarou (2007)using two sets of vector intensity measures (IMs) as predictor var-iables. They can be regarded as updates of the BT07 model.

Seismic Slope Displacement Analysis

Ground Motion Database

A subset of the Pacific Earthquake Engineering Research (PEER)Center’s NGA-West2 database is adopted for calculating the seis-mic slope displacements based on the fully coupled sliding massanalysis. The ground-motion selection criteria introduced byCampbell and Bozorgnia (2014) are used to exclude some low-quality recordings. To avoid any bias caused by low-amplitudemotions, we further exclude ground motions that fail to satisfythe following criteria: (1) Mw ≥ 4.5, (2) rupture distance ðRrupÞ ≤200 km, and (3) peak ground acceleration ðPGAÞ ≥ 0.01g. Thefinal database consists of 3,714 recordings from 102 earthquakes.The Mw-Rrup distribution of the selected ground motions is shownin Fig. 1. The two horizontal components of each recording areregarded as independent records. For each acceleration-time series,the maximum value computed from both the positive and the neg-ative directions is taken as the displacement D for this record.

Fully Coupled Sliding Mass Analysis

The fully coupled sliding mass analysis follows Rathje and Bray(2000). The sliding mass is simplified as a generalized single-degree-of-freedom system governed by its first mode of vibra-tion. The nonlinear responses of the soil are modeled using an

1Senior Research Fellow, Institute of Catastrophe Risk Management,Nanyang Technological Univ., 50 Nanyang Ave., Singapore 639798.

2Associate Professor, Dept. of Civil and Environmental Engineering,Hong Kong Univ. of Science and Technology, Clear Water Bay, Hong Kong(corresponding author). Email: gwang@ust.hk

3Assistant Professor, Dept. of Hydraulic Engineering, Tsinghua Univ.,Beijing 100084, China.

Note. This manuscript was submitted on April 30, 2017; approved onFebruary 21, 2018; published online on May 31, 2018. Discussion periodopen until October 31, 2018; separate discussions must be submitted forindividual papers. This technical note is part of the Journal of Geotech-nical and Geoenvironmental Engineering, © ASCE, ISSN 1090-0241.

© ASCE 06018006-1 J. Geotech. Geoenviron. Eng.

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equivalent-linear approach, and permanent displacement wouldoccur if base acceleration exceeds ky (cf. Wang 2012).

The sliding mass was assigned with a constant unit weight of18 kN=m3 and a plasticity index of 30. The shear modulus reduc-tion and material damping ratio curves proposed by Darendeli(2001) were used to model the nonlinear properties of soils; theinput parameters for the Darendeli model were assigned as follows:overconsolidation ratio (OCR) is 1, loading frequency (f) is 1 Hz,number of cycles (N) is 10, and the mean confining pressure is101 kPa (1 atm).

The variables Ts and ky are commonly used to characterize thedynamic stiffness and strength of an earth slope. In this study, theearthquake-induced displacements were computed for slopes withspecified combinations of Ts (0–2 s) and ky (0.01–0.5g). Here, Tscan be estimated by Ts ¼ 4H=Vs, where H and Vs are the averageheight and shear-wave velocity of a sliding mass, respectively. Fornonzero Ts, H varied between 5 and 100 m, and Vs varied between200 and 400 m=s. The model reduces to the original Newmark’smodel for the rigid slope case (Ts ¼ 0).

Selection of Predictor Variables

When developing a new model, it is important to identify the op-timal IMs as predictor variables. Specifically, a set of IMs used inengineering applications should satisfy the efficiency and suffi-ciency criteria (Luco and Cornell 2007). Efficiency requires themodel has small variability in the prediction; sufficiency requiresthat the model would not significantly depend on other seismologi-cal parameters such as magnitude and distance of the earthquake.

Several ground-motion IMs were used to examine their corre-lations with D. These IMs include PGA, Arias intensity (Ia), spec-tral acceleration (Sa) at various periods, and Ts-dependentSa ordinates [e.g., Sað1.5TsÞ], etc. For each ðTs; kyÞ case, nonlinearregression analysis was performed by fitting a quadratic form for a

scalar IM or a vector IM. The calculated standard errors for variousslope conditions using the selected scalar and vector IMs are illus-trated in Figs. S1 and S2 in the Supplemental Data, respectively.The results indicate that using the [Sað1.5TsÞ, Ia] vector is an idealchoice to optimize the overall efficiency of the model. Another op-tion is to use the [PGA, Sað2 sÞ] vector, which also provides rea-sonably good efficiency in regressing D.

Proposed Models

Most of the existing models have adopted the mixed-random var-iable approach, which consists of estimating the probability of neg-ligible “zero” displacement and developing the conditionalprobability density function of nonzero displacement. In this study,displacements smaller than 1 cm are regarded as “zero” displace-ment following Bray and Travasarou (2007). Such truncation isnecessary to avoid prediction biases caused by small displacementsthat have little engineering significance. Yet, an inappropriatechoice of the truncation may affect the distribution of residuals.Several values were tested, and the truncation of 1 cm was foundto best preserve the normality of the residuals.

Two sets of functional forms using different IMs as predictorsare then proposed following the mixed-random variable model.Each predictive model was developed in two steps: (1) predictingthe probability of negligible “zero” displacement (D < 1 cm), and(2) estimating the distribution (median and standard deviation) ofnonzero displacement (D ≥ 1 cm). A probit regression analysis(Bray and Travasarou 2007) was used to derive the probabilityof zero displacement, while the nonzero displacements were ana-lyzed using the nonlinear mixed-effects regression analysis.

[Sa(1.5T s);I a] Model

The probability function of “zero” displacement is derived as

PðD¼ 0Þ ¼ 1−Φ

−2.282− 2.459 lnðkyÞ− 0.744Ts lnðkyÞ−2.057Ts þ 1.906 lnSað1.5TsÞ þ 0.57 ln Ia

!

ð1Þ

where PðD ¼ 0Þ = estimated probability of zero displacement(D < 1 cm); Φ = standard normal cumulative distribution function;Sað1.5TsÞ = spectral acceleration at 1.5Ts (g); and Ia = Arias in-tensity (m=s).

Fig. 2 shows the predicted zero probabilities associated with theempirical data selected from the present data set versus Sað1.5TsÞ,ky, and Ts, respectively. The proposed model generally captures thetrends of zero displacement. The zero probability increases asSað1.5TsÞ decreases or ky increases; with respect to Ts, the lowestzero probability occurs at Ts in the range of 0.2–0.4 s, possibly dueto the resonance effect.

Table 1. Summary of existing predictive models based on fully coupled approach

Predictor variables Earthquake category Number of earthquakes Number of records Reference

ky, Ts, Sað1.5TsÞ, Mw Shallow crustal 41 688 Bray and Travasarou (2007)ky, Ts, Sað2.0TsÞ, PGV Near-fault 46 243 Song and Rodriguez-Marek (2015)ky, Ts, Sað1.5TsÞ, Mw Subduction — 2,244 Macedo et al. (2017)ky, Ts, Sað1.5TsÞ, Ia, Mw Shallow crustal 102 3,714 This studyky, Ts, PGA, Sað2 sÞNote: Sað1.5TsÞ, Sað2.0TsÞ = spectral acceleration at a degraded period 1.5Ts and 2.0Ts, respectively; PGV = peak ground velocity; and Sað2 sÞ = spectralacceleration at 2 s.

Rupture Distance (km)

Mom

ent M

agni

tude

(M

w)

10−1

100

101

102

4.5

5

5.5

6

6.5

7

7.5

8

Fig. 1. Distribution of moment magnitude versus rupture distance forrecordings selected from the NGA-West2 database.

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The median nonzero displacement can be estimated by nonlin-ear mixed-effects regression analysis using the nonzero displace-ment data (D ≥ 1 cm) as

lnD ¼ −4.047 − 2.522 lnðkyÞ − 0.234ðlnðkyÞÞ2 þ 0.506Ts

− 0.651T2s − 0.286Ts lnðkyÞ þ 1.709 ln Sað1.5TsÞ

þ 0.204 lnðkyÞ ln Sað1.5TsÞ − 0.842maxðln Sað1.5TsÞ; 0Þþ 0.352Mw þ 0.486 ln Ia þ ε ð2Þ

where D = nonzero displacement in centimeters; and ε = normallydistributed (total) residuals with zero mean and standard deviationσlnD. All the coefficients yield very small p-values (commonlyused in statistical hypothesis testing), so they are statisticallysignificant. Note that Eq. (2) is only applicable to Ts ≥ 0.05 s,

because this functional form exhibits an underprediction biaswhen Ts ¼ 0 s. Therefore, for rigid slope cases of Ts < 0.05 s,the function form for the median nonzero displacement isrewritten as

lnD ¼ −3.707 − 2.522 lnðkyÞ − 0.234ðlnðkyÞÞ2 þ 1.709 ln PGA

þ 0.204 lnðkyÞ ln PGA − 0.842maxðln PGA; 0Þþ 0.352Mw þ 0.486 ln Ia þ ε ð3Þ

Note that Sað1.5TsÞ is replaced by PGA in Eq. (3) for practi-cal use.

Fig. 3 shows the total residuals of Eqs. (2) and (3) with respect toky, Ts, Sað1.5TsÞ, Ia, and Mw, respectively. In each plot, the resid-uals are partitioned into several bins; the open square symboldenotes the local mean of residuals in each bin. In general, no clear

Sa(1.5T

s) (g)

P(D

="0

")

ky=0.2 g, T

s=0.7 s

10−2

10−1

100

1010

0.2

0.4

0.6

0.8

1

ky (g)

P(D

="0

")

Ts=0.7 s

Sa(1.5T

s)=0.3 g

10−2

10−1

1000

0.2

0.4

0.6

0.8

1Empirical data

Eq. (2)

Ts (s)

P(D

="0

")

ky=0.2 g

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Fig. 2. Predicted probabilities of zero displacement for the ½Sað1.5TsÞ; Ia� model compared with empirical data versus Sað1.5TsÞ, ky, and Ts,respectively.

Fig. 3. Distribution of residuals for the ½Sað1.5TsÞ; Ia� model with respect to predicator variables.

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biases with respect to these predictor variables can be observed.There are some biased local means [e.g., Sað1.5TsÞ < 0.05g],which is not a critical issue since the displacements for these casesare less significant.

The variability coefficients of the proposed model are listed inTable 2, and σlnD obtained from the mixed-effects regression is0.69. It is tempting to examine the relationship between σlnDand ky=Sað1.5TsÞ values, which is shown in Fig. 4. The standarddeviations become larger for larger ky=Sað1.5TsÞ bins, so a bilinearrelationship is proposed as

σlnD ¼

8>>><>>>:

0.66 forky

Sað1.5TsÞ< 0.65

0.36þ 0.46 ·ky

Sað1.5TsÞfor

kySað1.5TsÞ

≥ 0.65

ð4Þ

As demonstrated in Fig. 4, the bilinear model appropriately fitsthe empirical points.

In summary, the probability of zero displacement is specified viaEq. (1); the nonzero displacement is assumed to follow a lognormaldistribution with median displacementD provided in Eqs. (2) and (3)and σlnD specified in Eq. (4). The probability of seismic displacementexceeding a given displacement value d can be estimated as

PðD > dÞ ¼ ½1 − PðD ¼ 0Þ� ·�1 − Φ

�lnd − lnD

σlnD

��ð5Þ

The predicted displacement according to a specified percentile p(in decimal form) can then be estimated as

lnDp ¼ ln D̄þ σlnD · Φ−1�p − PðD ¼ 0Þ1 − PðD ¼ 0Þ

�ð6Þ

where Φ−1 = inverse standard normal cumulative distributionfunction.

[ PGA;Sa(2 s)] Model

Using PGA and Sað2 sÞ as predictors, the probability of zero displacement can be evaluated asFor Ts ≤ 0.2 s

PðD ¼ 0Þ ¼ 1 − Φ

−1.521 − 3.783 ln ky − 0.152ðln kyÞ2 þ 18.26Ts − 36.30T2s

þ3.255 ln PGAþ 0.533 lnðSað2sÞÞ

!ð7aÞ

For Ts ≥ 0.3 s

PðD ¼ 0Þ ¼ 1 − Φ

�−1.00 − 3.837 ln ky − 0.299ðln kyÞ2 − 3.423Ts þ 0.77T2s

þ0.804 ln PGAþ 1.145 lnðSað2sÞÞ − 0.491 lnTs lnðPGA=kyÞ

�ð7bÞ

where ky, Ts, PGA, Sað2 sÞ, and Φ are as defined previously.A notable feature of this model is that it calculates the prob-ability of zero displacement for short periods (Ts ≤ 0.2 s) andmoderate-to-long periods (Ts ≥ 0.3 s) separately. Linear interpo-lation can be used for obtaining the zero probability for Tsbetween 0.2 and 0.3 s. This two-part statistical analysis allowsmore flexibility in capturing the zero probability versus Ts.Fig. 5 shows the predicted zero probabilities via Eq. (7) comparedwith the empirical data with respect to ky and Ts, respectively.The predictive curves are in reasonable agreement with empiricaldata.

The functional form of the predicted median nonzero displace-ment is written as follows:

lnD ¼ b0ðky;TsÞ − 2.209 lnðkyÞ − 0.141ðlnðkyÞÞ2þ ð1.414þ 0.359 lnðkyÞÞ ln PGA − 0.135ðln PGAÞ2− 0.294Ts lnðkyÞ þ ð0.653 − 0.307 lnðkyÞÞ ln Sað2sÞþ 0.135ðln Sað2sÞÞ2 þ ε ð8Þ

where b0ðky;TsÞ is a piecewise function as

σ lnD

ky /S

a(1.5T

s)

0 0.2 0.4 0.6 0.8 10.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 4. Calculated binned standard deviations versus ky=Sað1.5TsÞ forthe ½Sað1.5TsÞ; Ia� model.

Table 2. Variability coefficients of proposed models

Model τ σ σlnD

½Sað1.5TsÞ; Ia� 0.26 0.64 0.69½PGA; Sað2 sÞ� 0.21 0.71 0.74

Note: Sað1.5TsÞ = spectral acceleration at a degraded period 1.5Ts;Sað2 sÞ = spectral acceleration at 2 s; and τ , σ, and σlnD = standarddeviations of between-event, within-event, and total residuals, respectively.

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b0ðky;TsÞ ¼

8>>>>>>>>>>><>>>>>>>>>>>:

0.641 − 1.257Ts ln ky if Ts ≤ 0.05 s

1.818þ 0.073 ln ky þ ð0.393þ 0.045 ln kyÞ lnðTsÞ if 0.05 s < Ts ≤ 0.2 s

0.979 − 0.128 lnðTsÞ if 0.2 s < Ts ≤ 0.4 s

0.231 − 0.944 lnðTsÞ if 0.4 s < Ts ≤ 0.8 s

−0.064 − 2.267 lnðTsÞ if 0.8 s < Ts ≤ 1.4 s

0.331 − 3.442 lnðTsÞ if Ts > 1.4 s

ð9Þ

Eq. (8) includes a quadratic function of ln(PGA) and ln½Sað2sÞ�and a piecewise function b0ðky;TsÞ. All coefficients in Eqs. (8) and(9) are statistically significant.

Fig. 6 illustrates the total residuals with respect to ky, Ts, PGA,Sað2 sÞ, and Mw, respectively. Again, the residuals are partitionedinto several bins, and the local means are shown in each plot.

Fig. 6. Distribution of total residuals for the [PGA, Sað2 sÞ] model with respect to predicator variables.

ky (g)

P(D

="0

")

Ts=0.6 s

PGA=0.37 g

Ia=0.19 m/s

10−2

10−1

100

0

0.2

0.4

0.6

0.8

1Empirical data

Eq. (8)

Ts (s)

P(D

="0

")k

y=0.2 g

PGA=0.37 g

Ia=0.19 m/s

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Fig. 5. Predicted probabilities of zero displacement for the [PGA, Sað2 sÞ] model compared with empirical data versus ky and Ts, respectively.

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The residuals are generally unbiased with respect to the predictorvariables used. Specifically, the residuals versus Mw are onlyslightly biased, implying that the [PGA, Sað2 sÞ] model can satisfythe sufficiency requirement.

As listed in Table 2, the standard deviation of the total residualsbased on Eqs. (8) and (9) is 0.74. Fig. 7 shows the computed σlnDfor bins of ky=PGA values. The σlnD values do not show a cleardependence on ky=PGA, and thus an average σlnD value (0.72) canbe used for all cases.

The proposed models are based on the assumption that the non-zero D are log-normally distributed. Fig. 8 shows the normalizedQ-Q plots of the displacement residuals for the two models. Theresiduals generally match the 1∶1 line, especially in the range of−2 to 2, which justifies the choice of truncation value and the re-gression approaches.

Model Comparisons

In this section, the performance of the proposed models is com-pared with the BT07 model. An earthquake scenario with Mw ¼ 7that occurred on a strike-slip fault is considered. Permanentdisplacement analysis is performed for hypothetic slopes locatedat a soft soil site and a Rrup of 10 km. The ground-motion predic-tion equations proposed by Campbell and Bozorgnia (2014) andTravasarou et al. (2003) are adopted to estimate PGA, Sa, andIa, respectively.

The predicted distributions of slope displacements for threeky cases using three predictive models (i.e., ½Sað1.5TsÞ; Ia�,½PGA; Sað2 sÞ�, and BT07) are shown in Fig. 9. The predictedmedian nonzero displacements and the probabilities of zero dis-placement with respect to Ts are plotted in Figs. 9(a and b), respec-tively. Fig. 9(c) shows the predicted median (50th percentile)displacements computed via Eq. (6) for ky ¼ 0.05, 0.1, and0.2g, respectively. All these plots indicate a similar trend, exceptthat the new model predictions tend to die down faster with an in-creasing Ts at Ts > 1 s. This difference might be due to the over-estimation of the BT07 model at large Ts. As was reported in Bray

Observed Data Quantiles

Sta

ndar

d N

orm

al Q

uant

iles

−6(a) (b)

−4 −2 0 2 4 6−6

−4

−2

0

2

4

6

Observed Data Quantiles

Sta

ndar

d N

orm

al Q

uant

iles

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

Fig. 8. Q-Q plots of the displacement residuals: (a) the ½Sað1.5TsÞ; Ia� model; and (b) the [PGA, Sað2 sÞ] model.

Ts (s)

Med

ian

nonz

ero

disp

(cm

)

ky=0.05 g

ky=0.1 g

ky=0.2 g

0

(a) (b) (c)0.5 1 1.5 2

0

10

20

30

40

50

60

70S

a(1.5T

s)+I

a model

PGA+Sa(2s) model

BT07 model

Ts (s)

P(D

="0

")

ky=0.1 g

ky=0.2 g

ky=0.05 g

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Sa(1.5T

s)+I

a

PGA+Sa(2s)

BT07 model

Ts (s)

Med

ian

disp

(cm

)

ky=0.05 g

ky=0.1 g

ky=0.2 g

0 0.5 1 1.5 20

10

20

30

40

50

60 Sa(1.5T

s)+I

a model

PGA+Sa(2s) model

BT07 model

Fig. 9. Comparisons of the predictions of three models: (a) predicted median nonzero displacements; (b) probability of zero displacements; and(c) predicted median displacements (50th percentile).

σ lnD

ky/PGA

0 0.2 0.4 0.6 0.8 10.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 7. Calculated binned standard deviations versus ky=PGA for the[PGA, Sað2 sÞ] model.

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and Travasarou (2007), the residuals of the BT07 model exhibiteda negative bias for Ts > 1 s, indicating an overestimation ofdisplacement.

Discussions

Two predictive models for seismic displacements are developedfollowing the analytical framework of the BT07 model. Comparedwith the BT07 model, several distinguishing features are incorpo-rated in the new models. First, a much expanded ground-motiondatabase is used in this study. Second, both models developeduse two IMs as predictor variables. Many studies have indicatedthe advantage of considering a vector IM on engineering applica-tions (e.g., Saygili and Rathje 2008; Wang 2012). Furthermore, theaverage sliding displacements from two horizontal components ofground motions were used for the regression analysis in the BT07model, while the horizontal components of each recording are re-garded as separate empirical data in this study.

The developed models could be regarded as updates of the BT07model. Specifically, the [PGA, Sað2 sÞ] model has some desirablefeatures. First, the model does not need to incorporate an Mw termin the functional form, indicating that it satisfies the sufficiencyrequirement. Second, this model uses Ts-independent IMs as pre-dictors, making it convenient to utilize existing PGA and Sað2 sÞhazard maps for a regional-scale slope displacement analysis(e.g., Du and Wang 2014). The application of the vector IM modelsrequires the study of correlation between IMs, which has been ex-tensively studied recently (e.g., Baker and Jayaram 2008; Wangand Du 2013; Du and Wang 2013; Huang and Wang 2015).

Conclusions

Two predictive models (i.e., ½Sað1.5TsÞ; Ia� and [PGA, Sað2 sÞ])are developed for estimating the earthquake-induced slope dis-placements based on equivalent-linear fully coupled analysis and3,714 ground-motion records selected from the NGA-West2 data-base. Both models use vector IMs as predictor variables, whichprovide better representation of ground-motion characteristicscompared with a scalar IM model. The proposed models, togetherwith existing ones, can be used in a logic-tree framework to quan-tify the epistemic uncertainty of seismic displacements in engineer-ing applications.

Acknowledgments

This work was supported by Hong Kong Research GrantsCouncil through General Research Fund No. 16213615 andDAG11EG03G. The authors thank the associate editor and twoanonymous reviewers for their helpful comments.

Supplemental Data

Further information about the selection of predictor variables forregressing analysis, including Eqs. (S1) and (S2) and Figs. S1 andS2, is available online in the ASCE Library (www.ascelibrary.org).

References

Ancheta, T. D., et al. 2014. “NGA-West2 database.” Earthquake Spectra30 (3): 989–1005. https://doi.org/10.1193/070913EQS197M.

Baker, J. W., and N. Jayaram. 2008. “Correlation of spectral accelerationvalues from NGA ground motion models.” Earthquake Spectra 24 (1):299–317. https://doi.org/10.1193/1.2857544.

Bray, J. D., and T. Travasarou. 2007. “Simplified procedure for estimatingearthquake-induced deviatoric slope displacements.” J. Geotech. Geo-environ. Eng. 133 (4): 381–392. https://doi.org/10.1061/(ASCE)1090-0241(2007)133:4(381).

Campbell, K. W., and Y. Bozorgnia. 2014. “NGA-West2 ground motionmodel for the average horizontal components of PGA, PGV, and 5%damped linear acceleration response spectra.” Earthquake Spectra30 (3): 1087–1115. https://doi.org/10.1193/062913EQS175M.

Darendeli, M. B. 2001. “Development of a new family of normalizedmodulus reduction and material damping curves.” Ph.D. thesis, Dept.of Civil, Architectural and Environmental Engineering, Univ. of Texasat Austin.

Du, W., and G. Wang. 2013. “Intra-event spatial correlations for cumulativeabsolute velocity, Arias intensity and spectral accelerations based onregional site conditions.” Bull. Seismol. Soc. Am. 103 (2): 1117–1129. https://doi.org/10.1785/0120120185.

Du, W., and G. Wang. 2014. “Fully probabilistic seismic displacementanalysis of spatially distributed slopes using spatially correlated vectorintensity measures.” Earthquake Eng. Struct. Dyn. 43 (5): 661–679.https://doi.org/10.1002/eqe.2365.

Du, W., and G. Wang. 2016. “A one-step Newmark displacement model forprobabilistic seismic slope displacement hazard analysis.” Eng. Geol.205: 12–23. https://doi.org/10.1016/j.enggeo.2016.02.011.

Huang, D., and G. Wang. 2015. “Region-specific spatial cross-correlationmodel for stochastic simulation of regionalized ground-motion time his-tories.” Bull. Seismol. Soc. Am. 105 (1): 272–284. https://doi.org/10.1785/0120140198.

Jibson, R. W. 2011. “Methods for assessing the stability of slopes duringearthquakes—A retrospective.” Eng. Geol. 122 (1): 43–50. https://doi.org/10.1016/j.enggeo.2010.09.017.

Luco, N., and C. A. Cornell. 2007. “Structure-specific scalar intensitymeasures for near-source and ordinary earthquake ground motions.”Earthquake Spectra 23 (2): 357–392. https://doi.org/10.1193/1.2723158.

Macedo, J., J. Bray, and T. Travasarou. 2017. “Simplified procedure forestimating seismic slope displacements in subduction zones.” In Proc.,16th World Conf. on Earthquake Engineering, Santiago, Chile.

Newmark, N. M. 1965. “Effects of earthquakes on dams and embank-ments.” Géotechnique 15 (2): 139–160. https://doi.org/10.1680/geot.1965.15.2.139.

Rathje, E. M., and J. D. Bray. 2000. “Nonlinear coupled seismic slidinganalysis of earth structures.” J. Geotech. Geoenviron. Eng. 126 (11):1002–1014. https://doi.org/10.1061/(ASCE)1090-0241(2000)126:11(1002).

Rathje, E. M., Y. Wang, P. J. Stafford, G. Antonakos, and G. Saygili. 2014.“Probabilistic assessment of the seismic performance of earth slopes.”Bull. Earthquake Eng. 12 (3): 1071–1090. https://doi.org/10.1007/s10518-013-9485-9.

Saygili, G., and E. M. Rathje. 2008. “Empirical predictive models forearthquake-induced sliding displacements of slopes.” J. Geotech. Geo-environ. Eng. 134 (6): 790–803. https://doi.org/10.1061/(ASCE)1090-0241(2008)134:6(790).

Song, J., and A. Rodriguez-Marek. 2015. “Sliding displacement of flexibleearth slopes subject to near-fault ground motions.” J. Geotech. Geoen-viron. Eng. 141 (3): 04014110. https://doi.org/10.1061/(ASCE)GT.1943-5606.0001233.

Travasarou, T., J. D. Bray, and N. A. Abrahamson. 2003. “Empirical attenu-ation relationship for Arias intensity.” Earthquake Eng. Struct. Dyn.32 (7): 1133–1155. https://doi.org/10.1002/eqe.270.

Wang, G. 2012. “Efficiency of scalar and vector intensity measures for seis-mic slope displacements.” Front. Struct. Civ. Eng. 6 (1): 44–52.

Wang, G., and W. Du. 2013. “Spatial cross-correlation models for vectorintensity measures (PGA, Ia, PGV, and SAs) considering regional siteconditions.” Bull. Seismol. Soc. Am. 103 (6): 3189–3204. https://doi.org/10.1785/0120130061.

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