Engineering Geology 122 (2011) 34–42
Contents lists available at ScienceDirect
Engineering Geology
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e n g g e o
Empirical estimation of the Newmark displacement from the Arias intensity and
critical acceleration
Shang-Yu Hsieh, Chyi-Tyi Lee ⁎
Graduate Institute of Applied Geology, National Central University, No.300, Jungda Road, Jungli City, Taoyuan County, 32001, Taiwan
a r t i c l e
i n f o
Article history:
Accepted 20 December 2010
Available online 30 December 2010
Keywords:
Newmark displacement
Arias intensity
Critical acceleration
Empirical relation
a b s t r a c t
This study employs strong-motion data from the 1999 Chi-Chi earthquake, the 1999 Kocaeli earthquake, the 1999
Duzce earthquake, the 1995 Kobe earthquake, the 1994 Northridge earthquake and the 1989 Loma Prieta earthquake
to refine the relationship among critical acceleration (Ac), Arias intensity (Ia), and Newmark displacement (Dn). The
results reveal that, as expected, logDn is proportional to logIa when Ac is large. As Ac gets smaller however, the linearity
becomes less. We also found that logDn is proportional to Ac, and that the linearity is very stable through all Ia values.
These features are common to all six sets of data. Therefore, we add a third term in addition to the Jibson's form which
covers the aforementioned problem, and propose a new form for the relationship among Ia, Ac and Dn. Two
alternative forms were tested using each of the six data sets, before a final form was selected.
The final analyses grouped the data into a worldwide data set and a Taiwanese data set. Coefficients for the selected
form were derived from regression with the data, and two final empirical formulas, one for global, the other for local,
proposed. Site conditions are also considered in this study with empirical formulas being developed for soil and rock
sites, respectively. The estimation error is smaller and the goodness of fit is higher for both the local soil-site and
rock-site formulas. Since landslides are more likely to occur on hillsides, the rock site formula may be more
applicable for the landslide cases, whereas the soil site formula should be used for side slope of landfills.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
Landslides are one of the most damaging hazards from earthquakes.
Estimating where they are likely to occur and what kind of shaking
conditions will trigger them is an important topic in regional landslide
seismic hazard assessment. Newmark (1965) first developed a simple
model to estimate co-seismic slope displacement. Since then many
researchers have used the Newmark method and actual strong-motion
records to estimate the earthquake-induced slope displacement at
specific sites (e.g., Wilson and Keefer, 1983; Wieczorek et al., 1985;
Jibson and Keefer, 1993; Bray and Rathje, 1998; Pradel et al., 2005).
Others have empirically assessed the earthquake-triggered landslide
hazard using the Newmark displacements and a suitable formula
(Jibson et al., 1998; Mankelow and Murphy, 1998; Luzi and Pergalani,
1999; Miles and Ho, 1999; Jibson et al., 2000; Miles and Keefer, 2000,
2001; Del Gaudio et al., 2003; Haneberg, 2006; Saygili and Rathje, 2008).
In the Newmark method, a pseudo-static analysis is first used to
calculate a critical acceleration value. The critical acceleration (Ac) is
determined by interactively employing different horizontal earthquake accelerations in a static limit-equilibrium analysis until the
safety factor approaches 1.0.
Ac = ðFS−1Þgsin α;
ð1Þ
⁎ Corresponding author. Tel.: + 886 3 4253334; fax: +886 3 4223357.
E-mail addresses: [email protected] (S.-Y. Hsieh), [email protected] (C.-T. Lee).
0013-7952/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.enggeo.2010.12.006
where g is the acceleration of gravity, FS is the static factor of safety, and
α is the angle from the horizontal of the sliding surface or the thrust
angle (Newmark, 1965). The Newmark's cumulative displacement for a
sliding block is then calculated by double integration of the earthquake
acceleration time history data above the critical acceleration.
In a real landslide hazard case, however, it is not easy to find the
strong-motion record at a specific site for calculation of the Newmark
displacement. It is also complicated and time consuming to generate a
proper acceleration time history for a site for analysis. This is why
empirical relationships were developed. Ambraseys and Menu (1988)
were the first to propose various regression equations to estimate the
Newmark displacement (Dn) as a function of the critical acceleration
ratio (the ratio of critical acceleration to maximum acceleration)
based on analysis of 50 strong-motion records from 11 earthquakes.
They concluded that the following equation best characterizes the
results of their study:
log Dn = 0:90 + log
1−
Ac
A max
2:53 Ac
A max
−1:09 !
0:30;
ð2Þ
where Ac is critical acceleration, Amax is the maximum acceleration in a
strong-motion record, Dn is in centimeters, and the last term is the
estimation error of the model. Regression equations of different
forms have been proposed in other studies, with various additional
parameters included, to estimate Newmark displacement (Yegian
S.-Y. Hsieh, C.-T. Lee / Engineering Geology 122 (2011) 34–42
et al., 1991; Jibson, 1993; Ambraseys and Srbulov, 1995; Crespellani
et al., 1998; Jibson et al., 1998; Bray and Travasarou, 2007).
Among those empirical estimations, Jibson (1993) proposed the
following regression equation based on 11 strong-motion records for
Ac values of 0.02, 0.05, 0.10, 0.20, 0.30, and 0.40 g:
log Dn = 1:460log Ia −6:642Ac + 1:546 0:409;
ð3Þ
where Ac is critical acceleration (in terms of g), Ia is the Arias intensity
(in meters per second), Dn is in centimeters, and the last term is the
estimation error of the model. This is called the Jibson93 model in the
present study. The Arias intensity (Arias, 1970) is given by
Ia =
π Td
∫ ðaðt ÞÞ2 dt;
2g 0
ð4Þ
where g is the acceleration due to gravity, a(t) is the recorded
acceleration time-history, Td is the duration of the ground motion.
The Jibson93 model has a fine goodness of fit of R2 = 0.87, but this
model is overly sensitive to small changes in Ac. Jibson et al. (1998)
modified the form of the Jibson93 model to make all terms logarithmic
and then performed rigorous analysis of 555 strong-motion records
from 13 earthquakes. They used the same Ac values as indicated for
Eq. (3) and generated the following regression equation:
log Dn = 1:521log Ia −1:993log Ac −1:546 0:375;
ð5Þ
it has a correlation coefficient R2 = 0.83 and is called the Jibson98
model in the present study.
In his review of previous regression models for finding the Newmark
displacement, Jibson (2007) concluded that factors included in the
models can be divided into 4 groups: (1) the critical acceleration ratio;
(2) the critical acceleration ratio and magnitude; (3) the Arias intensity
and critical acceleration; and (4) the Arias intensity and critical
acceleration ratio. These can be summarized as four basic factors:
Arias intensity, critical acceleration, maximum acceleration or PGA
(peak ground acceleration) and earthquake magnitude.
PGA is a common measure of ground-shaking intensity, but it just
measures a single point in an acceleration time history. The Arias intensity
is a measure of earthquake intensity given by the integration of squared
accelerations over time and it is related to the energy content of the
recorded signal. It has been demonstrated to be an effective predictor of
earthquake damage potential in relation to short-period structures,
liquefaction (Kayen and Mitchell, 1997), and seismic slope stability
(Wilson and Keefer, 1985; Harp and Wilson, 1995). Jibson (1993)
suggested using the Arias intensity, because it measures the total
acceleration of the record rather than just the peak value, so it provides
a more complete characterization of the shaking content of a strongmotion record than the PGA does. Moreover, the landslide ratio (landslide
pixels to total pixels ratio) also correlates well with the Arias intensity (Lee
et al., 2008). Therefore, the Arias intensity is considered to be superlative
among many seismic parameters in landslide hazard evaluation.
This study aims at developing a set of more effective empirical
equations than pre-existing ones for the estimation of the Newmark
displacement from the Arias intensity and other seismic parameters,
which can be used for seismic-induced landslide hazard analysis in
Taiwan and possibly worldwide. The critical acceleration or critical
acceleration ratio and/or earthquake magnitude may be used as
additional parameters in a regression model for the Newmark
displacement. In the present study, we consider critical acceleration as
an additional parameter in the model for the purpose of effectiveness
and practicality.
2. Data description
The main data set used in the present study encompasses strongmotion records from the 1999 Chi-Chi mainshock (Ma et al., 1999;
35
Kao and Chen, 2000) (Table 1). This includes 746 strong-motion
records of the two horizontal components from 373 recording
stations. The data are used for validation of previous empirical
formulas and analysis of the relationship among Dn, Ac and Ia to find
better regression models. They are also used to find local empirical
formulas suitable for use in Taiwan.
Other strong-motion data from well-recorded major earthquakes
worldwide were collected for the purpose of comparison and generation
of a common set of empirical equations. This includes strong-motion
data from the 1999 Duzce earthquake, the 1999 Kocaeli earthquake, the
1995 Kobe earthquake, the 1994 Northridge earthquake, and the 1989
Loma Prieta earthquake (Table 1). This additional data set includes 597
horizontal-component records.
All the strong-motion data were downloaded from the Pacific
Earthquake Research Center (PEER) website (http://peer.berkeley.edu/).
These data have been processed with baseline correction and band-pass
filtering by the PEER. Site conditions for each strong-motion station were
downloaded from the same website except for the Chi-Chi data set, for
which data from Lee et al. (2001) were used. The magnitude and distance
distribution of the ground-motion data set used in this study are 6.7–7.6
and 1–175 km, respectively (Figure 1).
3. Analysis
First, the Arias intensities are calculated for each strong-motion
record, according to the definition shown in Eq. (4). In the meantime,
the Newmark displacement is calculated by using the average of +a(t)
and −a(t) for different critical accelerations between 0.01 and 0.4 g for
each of the records. These derived data are stored in a plain text file for
visual checking and further use.
3.1. Verification of Jibson's formulas
This study verifies the applicability of the Jibson93 equation and the
Jibson98 equation in the Taiwanese geological setting by using the ChiChi data set. This data set is large and includes many strong-motion
records near the fault ruptures. The verification works by directly
inputting the data set into the Jibson93 and Jibson98 equations, and then
plotting the results for visual inspection. The estimation error is also
calculated for comparison.
The results in both plots show a significant deviation of the data from
the median and large estimation errors equal to 1.052 and 0.925 for
Jibson93 and Jibson98, respectively. These estimation errors are much
larger than the values of 0.409 and 0.375, as shown in Jibson (1993) and
Jibson et al. (1998). These differences indicate that Jibson's formulas are
not suitable for the Taiwanese geological setting and need further study.
3.2. Test of the Jibson forms
To test the suitability of using Jibson's model in Taiwan, we used
the Jibson93 and Jibson98 forms, and the Chi-Chi data set, to perform
a new regression.
The new equation for the Jibson93 form is
log Dn = 1:782log Ia −12:104Ac + 1:764 0:671;
ð6Þ
Table 1
Strong-motion data used in the present study.
Earthquake
Magnitude
(MW)
Country
Year/month/
day (GMT)
Records #
Rock
sites #
Soil
sites #
Chi-Chi
Duzce
Kocaeli
Kobe
Northridge
Loma Prieta
7.6
7.1
7.4
6.9
6.7
6.9
Taiwan
Turkey
Turkey
Japan
U.S.A
U.S.A
1999/09/20
1999/11/12
1999/08/17
1995/01/16
1994/01/17
1989/10/18
746
20
41
24
376
136
326
16
20
8
172
74
420
4
21
16
204
62
36
S.-Y. Hsieh, C.-T. Lee / Engineering Geology 122 (2011) 34–42
Fig. 1. Magnitude and epicenter distance distribution of the ground motion data used in this study. (a) Rock site data, (b) soil site data.
Fig. 2. Ia − Dn relation diagrams when Ac = 0.05 g in the Chi-Chi data set. (a) Ia − Dn relation, (b) Ia − logDn relation, and (c) logIa − logDn relation.
where Dn is in centimeters, Ia is in meters per second, Ac is in terms of
g, R2 = 0.83 and σlog Dn = 0.671.
The new equation for Jibson98 form is
log Dn = 1:756log Ia −2:78logAc −2:728 0:658;
set tested, R2 is ranging from 0.70 to 0.98 for logIa − logDn relation. It
confirms that logDn being proportional to logIa is a worldwide
phenomenon.
ð7Þ
R2 = 0.87 and σlog Dn = 0.658.
The estimation errors have improved in the new regression from
those mentioned above, but they are still larger than those obtained in
Jibson (1993) and Jibson et al. (1998), although the goodness of fits are
similar to the original Jibson's results. These results are deemed less than
ideal and we judged that there was still room for improvement.
3.3. Relation between the Arias intensity and Newmark displacement
It has been proposed in Jibson (1993) and Jibson et al. (1998) that
the relation between logIa and logDn should be linear, when Ac is fixed.
We examine this relation. The Chi-Chi data are plotted and the
goodness of fit of a line for Ia − Dn, Ia − logDn and logIa − logDn
calculated with Ac being fixed at 0.05 g (Figure 2). It is clear from the
figure that the data are concentrated near the origin in the Ia − Dn plot
and R2 is only 0.38; the R2 in the Ia − logDn plot is only 0.26. However,
the logIa − logDn plot shows a better linear trend with R2 = 0.72. We
also tested a series of different Ac values between 0.01 g and 0.4 g. The
results show that R2 = 0.70–0.90; as Ac gets larger, the linearity
becomes better. This result reveals that logDn is proportional to logIa,
as expected, except when Ia is very small (b0.06 m/s).
We further tested the Ia and Dn relation using the worldwide data set.
They were plotted for visual observation and were calculated for
goodness of fit. The results are listed in Table 2. In all the calculations, the
correlation coefficient is larger than 0.77 for logIa − logDn plot, even
better than the value obtained for the Chi-Chi data set. In the entire data
3.4. Relation between the critical acceleration and Newmark displacement
Jibson (1993) first proposed the linear relation between Ac and
logDn after looking at strong-motion records from 11 earthquakes.
Jibson et al. (1998) modified their model to find the linear relation
between logAc and logDn using 555 strong-motion records from 13
earthquakes.
In the present study, we select some typical and large intensity data
(PGAN 0.3 g) from the Chi-Chi data set, plot and calculate the goodness
of fit of a line. The results show that the relation between Ac − logDn best
fits a line. A typical example from the Chi-Chi mainshock records from
station TCU072 (about 14 km from the fault rupture line) is selected for
exhibition. Ac − Dn, Ac − logDn, and logAc − logDn are plotted and the
goodness of fit calculated with Ia being fixed (Figure 3). The best linear
relation is shown in Fig. 3b for the Ac − logDn relation (R2 = 0.99). All the
Table 2
Goodness of fit to a line in different relation between Dn and Ia for different Acs
(0.01 ~ 0.4 g).
Earthquake
Chi-Chi
Duzce & Kocaeli
Kobe
Loma Prieta
Northridge
D n − Ia
2
R = 0.3 ~ 0.5
R2 = 0.76 ~ 0.8
R2 = 0.7 ~ 0.89
R2 = 0.6 ~ 0.8
R2 = 0.6 ~ 0.8
logDn − Ia
2
R = 0.2 ~ 0.5
R2 = 0.8 ~ 0.87
R2 = 0.78 ~ 0.87
R2 = 0.32 ~ 0.5
R2 = 0.4 ~ 0.75
logDn − logIa
R2 = 0.7 ~ 0.9
R2 = 0.88 ~ 0.98
R2 = 0.9 ~ 0.98
R2 = 0.77 ~ 0.85
R2 = 0.77 ~ 0.85
S.-Y. Hsieh, C.-T. Lee / Engineering Geology 122 (2011) 34–42
37
Fig. 3. Ac −Dn relation diagrams for the north–south component of Chi-Chi earthquake at station TCU072; Ia =5.77 m/s. (a) Ac −Dn relation, (b) Ac −logDn relation, and (c) logAc −logDn relation.
Table 3
Goodness of fit to a line in different relation between Dn and Ac.
Earthquake
(Dn − Ac)
(logDn − Ac)
(logDn − logAc)
Chi-Chi
Duzce & Kocaeli
Kobe
Loma Prieta
Northridge
R2 = 0.6 ~ 0.7
R2 = 0.7 ~ 0.87
R2 = 0.67 ~ 0.82
R2 = 0.64 ~ 0.88
R2 = 0.61 ~ 0.88
R2 = 0.98 ~ 0.99
R2 = 0.98 ~ 0.99
R2 = 0.98 ~ 0.99
R2 = 0.98 ~ 0.99
R2 = 0.98 ~ 0.99
R2 = 0.89 ~ 0.97
R2 = 0.82 ~ 0.93
R2 = 0.89 ~ 0.96
R2 = 0.88 ~ 0.96
R2 = 0.81 ~ 0.97
regression with the Jibson93 and Jibson98 forms (Figure 4) in the first
step. The two plots in Fig. 4 give important information in that the data
show a steeper slope for a larger Ac, giving a critical hint that the form of
the equation can be modified. Therefore, we propose two new forms for
regression of the relationship of Newmark displacement to the Arias
intensity and critical acceleration.
To obtain new form I, we modified logIa to become AclogIa in the
first term of the Jibson93 form as follows:
log Dn = C1 Ac log Ia + C2 Ac + C3 ε;
R2 values that have been calculated for the Chi-Chi data set are listed in
Table 3.
We further test the Ac and Dn relation using the worldwide data
set. Some typical records were plotted for visual observation and the
goodness of fit calculated. The results are listed in Table 3. All
calculations for the worldwide data set show that the results are as
good as that obtained for the Chi-Chi data set with correlation
coefficients always larger than 0.98. This means that a very stable
linear relationship exists between Ac and logDn for all Ia values.
3.5. Regression models and regression method
We selected a subset data of 15 records from 8 strong-motion stations
(TCU065, TCU071, TCU072, TCU078, TCU079, TCU089, CHY074, and
ILA067) located at a proper fault distance (1 km–82 km) and site
condition (site class B, C, or D) (the 1997 UBC and 1997 NEHRP
classification scheme, ICBO, 1997; BSSC, 1998.) for studying the regression
model forms. We used this subset data to find the model coefficients by
ð8Þ
where C1, C2, and C3 are coefficients, Dn is in centimeters, Ia is in meters
per second, Ac is in terms of g, and the last term is the estimation error. In
new form II, we added a term logIa to the new form I to obtain
log Dn = C1 log Ia + C2 Ac + C3 Ac log Ia + C4 ε;
ð9Þ
where C1, C2, C3, and C4 are coefficients.
The subset data was used for regression of the coefficients and
estimation error with new form I and new form II. The results show
that a higher goodness of fit and a lower estimation error can be
obtained with the new forms than with the others (Figure 5, and
Table 4), especially for new form II.
Both new form I and new form II consist of linear equations and
our data distribution is rather uniform, making the regression analysis
using a least squares method via a common matrix inversion
technique easy and quick. The LU decomposition technique (Press
et al., 1992) was used in our computer code to solve the linear
equations and find coefficients for the equations.
Fig. 4. The Chi-Chi subset data fitting with (a) Jibson93 form, (b) Jibson98 form.
38
S.-Y. Hsieh, C.-T. Lee / Engineering Geology 122 (2011) 34–42
Fig. 5. The Chi-Chi subset data fitting with (a) new form I, (b) new form II.
and the equation for new form II is
Table 4
Estimation error and R2 of each equation for the Chi-Chi subset data.
σlog Dn
R2
Jibson93
equation
Jibson98
equation
Jibson93
form
Jibson98
form
New
form I
New
form II
0.853
–
0.745
–
0.467
0.831
0.466
0.831
0.321
0.920
0.305
0.928
Table 5
Coefficients, estimation error and R2 of each equation for the entire Chi-Chi data set.
New form I
New form II
C1
C2
C3
C4
σlog Dn
R2
18.388
0.766
− 21.536
−19.945
2.344
13.744
–
2.196
0.503
0.458
0.804
0.837
4. Results and evaluations
4.1. Empirical formula for all site conditions
Regressions were performed using the entire Chi-Chi data set with
new form I and new form II. The results are shown in Fig. 6 and
Table 5. The best-fit equation for new form I is
log Dn = 18:388Ac log Ia −21:536Ac + 2:344 0:503;
ð10Þ
log Dn = 0:766log Ia −19:945Ac + 13:744Ac log Ia + 2:196 0:458:
ð11Þ
The goodness of fit for new form I and new form II are R2 = 0.804
and R2 = 0.837, respectively.
A comparison of the goodness of fit and estimation error between
new form I and new form II shows the goodness of fit and deduction of
estimation error to be better for new form II. An examination of the plots
in Fig. 6 shows that new form I does not appear to function as well when
Ia is larger than about 2 m/s, especially for lower Ac. Therefore, new form
II is much superior to new form I and we suggested that new form II is
better suited for use in Taiwan.
Before developing a global empirical formula, we made a
regression analysis for records from each earthquake and checked
the superiority between new form I and new form II. It confirms that
new form II always has better goodness of fit and less estimation error.
We also checked the homogeneity of data from different regions or
different earthquakes by comparing the regression lines for each
subset data at different Ac levels. Results of different subset data fitting
to new form II are shown in Fig. 7. It is clear that the best-fit lines of 4
worldwide subsets have clustered tightly, whereas the Chi-Chi set is
apart from the cluster except when Ac is small. Because of the
Fig. 6. Results of the Chi-Chi data set fitting to (a) new form I, (b) new form II.
S.-Y. Hsieh, C.-T. Lee / Engineering Geology 122 (2011) 34–42
39
Fig. 7. Results of different data set fitting to new form II, (a) Ac = 0.02 g, (b) Ac = 0.1 g, (c) Ac = 0.2 g.
Fig. 8. Results of worldwide data set fitting to (a) new form I, (b) new form II.
inhomogeneity phenomenon and because the Chi-Chi set contains
more than 50% of the whole data set, we decided that global analyses
should not include the Chi-Chi data set. Only the 4 worldwide subsets
were combined to form a worldwide data set for further analyses.
The worldwide data set were then adopted for regression using
new form I and new form II. The results are shown in Fig. 8 and
Table 6. In this case, the equation for new form I is
log Dn = 11:287Ac log Ia −11:485Ac + 1:948 0:357;
ð12Þ
the equation for new form II is
log Dn = 0:847log Ia −10:62Ac + 6:587Ac log Ia + 1:84 0:295: ð13Þ
The goodness of fit for new form I and new form II are R2 = 0.84
and R2 = 0.89, respectively.
A comparison of the goodness of fit and estimation error between
new form I and new form II shows that new form II has better
Table 6
Coefficients, estimation error and R2 of each equation for the worldwide data set.
New form I
New form II
C1
C2
C3
C4
σlog Dn
R2
11.287
0.847
−11.485
− 10.62
1.948
6.587
–
1.84
0.357
0.295
0.84
0.89
goodness of fit and less estimation error. The plots in Fig. 8 show that
the fit of new form I to the data is not as good when Ia is large.
Therefore, new form II is reconfirmed to be superior to new form I and
is suggested for use worldwide.
4.2. Empirical formula for different site conditions
It is necessary to develop an empirical formula for rock sites
because most natural slope problems occur at rock sites. Even when
the slope failure appears to be a debris soil slide, the sloped terrain
beneath the debris and soil is most likely the rock or soft rock type.
We first divided the Chi-Chi strong-motion data into rock-site
data and soil-site data according to Lee et al. (2001). The site
conditions for the Duzce earthquake and Kocaeli earthquake data
were adopted from Rathje (2001). The site conditions for the
other three earthquake data sets were assigned according to United
States Geological Survey (USGS) site classifications. Site classes A(Vs N
1500 m/s), B (1500 N Vs N 760 m/s) and C (760 N Vs N 360 m/s) were
grouped into the rock site category, site classes D (360N Vs N 180 m/s)
and E (Vs b 180 m/s) were grouped into soil site category; site class E was
not included in the present study.
The Chi-Chi rock-site subset data were put into a regression using
new form II. We obtained a local rock-site empirical formula
log Dn = 0:555log Ia −20:488Ac + 14:555Ac log Ia + 2:295 0:414:
ð14Þ
40
S.-Y. Hsieh, C.-T. Lee / Engineering Geology 122 (2011) 34–42
Fig. 9. Results of the Chi-Chi data set fitting to new form II, (a) rock site, (b) soil site.
Table 7
Coefficients, estimation error and R2 of each site condition for the Chi-Chi data set.
Chi-Chi data
C1
C2
C3
C4
σlog Dn
R2
Rock site
Soil site
0.555
0.802
− 20.488
− 19.246
14.555
12.757
2.295
2.153
0.414
0.445
0.875
0.843
Note: Using new form II.
The goodness of fit is R2 = 0.843. These two empirical equations and
their data fits are plotted in Fig. 9; coefficients are listed in Table 7. The
results reveal that with both the rock-site formula and the soil-site
formula we obtain better goodness of fit than that of all-sites formula
as well as reduction of estimation error. This indicates that
differentiation of site conditions does improve the results.
The worldwide rock-site subset data were adopted for regression
using new form II. We obtained the global rock-site empirical formula
log Dn = 0:788log Ia −10:166Ac + 5:95Ac log Ia + 1:779 0:294: ð16Þ
Table 8
Coefficients, estimation error and R2 of each site condition for the worldwide data set.
Worldwide Data
C1
C2
C3
C4
σlog Dn
R2
Rock site
Soil site
0.788
0.802
− 10.166
− 10.981
5.95
7.377
1.779
1.914
0.294
0.274
0.884
0.905
The goodness of fit is R2 = 0.884. Similarly, we determined the
global soil-site empirical formula
log Dn = 0:802log Ia −10:981Ac + 7:377Ac log Ia + 1:914 0:274:
Note: Using new form II.
ð17Þ
The goodness of fit is R2 = 0.875. For the Chi-Chi soil-site subset
data we obtained a local soil-site empirical formula
log Dn = 0:802log Ia −19:246Ac + 12:757Ac log Ia + 2:153 0:445:
ð15Þ
The goodness of fit is R2 = 0.905. Coefficients for these two
empirical formulas are listed in Table 8, and their data fits are plotted
in Fig. 10. There is further improvement in the goodness of fit and
estimation error with either the rock-site formula or the soil-site
formula confirming that differentiation of site conditions does
improve the results.
Fig. 10. Results of worldwide data set fitting to new form II, (a) rock site, (b) soil site.
S.-Y. Hsieh, C.-T. Lee / Engineering Geology 122 (2011) 34–42
41
Fig. 11. Comparison among suggested empirical formulas. Rock site, soil site, and all site formulas are compared at Ac = 0.02 g, 0.1 g, and 0.2 g. (a) local formulas, (b) global formulas.
4.3. Comparison among suggested empirical formulas
We now compare the 6 suggested empirical formulas (local all-sites,
local rock-site, local soil-site, global all-sites, global rock-site, and global
soil-site). We now plot the 6 equations with different Ac values on one
figure (Figure 11). The results reveal that the formulas are rather similar
among soil-site, rock-site, and all-site ones, either for local formulas or
global formulas. However, as demonstrated above, the soil-site and
rock-site formulas are more exact than the all-site formulas, and could
be used in a more strict sense, whereas the all-site formulas could be
applied to general seismic slope stability cases.
Although the local formulas are different from the global formulas,
however, they predict similar Dn values when Ia is large. This is very
important in practical application, because most seismic slope failures
occur when Ia is large. This difference becomes larger, when Ia gets
smaller and Ac becomes larger.
The local formulas predict smaller Dn values than that of the global
formulas when Ac gets larger. This phenomenon is here interpreted as a
local feature in Taiwan and could be attributable to similar active
orogens, like New Zealand. Our experience and some local studies (e.g.,
Loh et al., 2000; Lin et al., 2010) show that seismic ground-motion
attenuates faster in Taiwan as compared with other worldwide
attenuation equations, like NGA models (Next Generation Attenuation
project) attenuation equations (Abrahamson and Silva, 2008; Boore and
Atkinson, 2008; Campbell and Bozorgnia, 2008; Chiou and Youngs,
2008; Idriss, 2008), especially for short-period motion. Therefore,
strong-motion records at larger fault distance will contain less short-
period motions and show less peaks in the acceleration time history, so
that the double integration above a certain Ac value will show lower Dn
values especially for records farther from fault rupture.
4.4. Comparison with pre-existing empirical equations
We compare the new formulas only with those formulas using Ac
and Ia as variables; those formulas using critical acceleration ratio or
maximum acceleration as variables were difficult to compare and
were not considered. Therefore, we now compare the new formulas
with the Jibson93 and Jibson98 formulas. The Jibson93 formula and
the new formula for worldwide and all-site data are plotted in
Fig. 12a, and the Jibson98 formula and the new formulas for
worldwide and all-site data are plotted in Fig. 12b. The main
difference between the new worldwide and all-site formula and
Jibson's formulas is the Ia − Dn pattern in the plots; the former has a
radiating pattern (as a consequence of the correlating term AclogIa)
whereas the latter exhibits a parallel pattern. When the Ia value is
about median to large, the new formula is quite similar to Jibson's
formulas thus predicting similar Dn values. However, they significantly diverge when Ia gets smaller. When Ia is less than 1 m/s, Jibson's
formulas over-predict the Dn value and are actually too conservative.
In summary, the new formulas possess higher goodness of fits (R2)
and lower estimation errors than that of previous ones (R2 = 0.87 and
0.83 for Jibson93 and Jibson98, respectively, whereas R2 = 0.89 for
new global all-site formula; σlog Dn = 0.409 and 0.375 for Jibson93
equation and Jibson98 equation, respectively, whereas σlog Dn = 0.295
Fig. 12. Comparison of new global formulas (estimation error = 0.295) with Jibson's formulas, (a) comparison with Jibson93 formula (estimation error = 0.409), (b) comparison
with Jibson98 formula (estimation error = 0.375).
42
S.-Y. Hsieh, C.-T. Lee / Engineering Geology 122 (2011) 34–42
for new global all-site formula). Therefore, the new formulas predict
more accurate Dn values throughout the entire Ia range and are thus
more suitable for the use in seismic slope stability evaluation.
5. Conclusions
After testing a large amount of strong-motion data, we conclude
that the logarithmic Newmark displacement logDn is proportional to
the critical acceleration Ac, and the linearity is very stable through all
Arias intensity Ia values. This relation has been used in Jibson (1993)
and is adopted in the present study. We proposed and tested two new
forms for the regression model for Newmark displacement. Our
results suggest that the new form II, which adds a third term — AclogIa
to the form used by Jibson (1993), is more appropriate for use.
One global empirical formula and one local empirical formula for
all site conditions are suggested for use in evaluation of earthquakeinduced slope failures. The local formula could be used in Taiwan and
in a similar tectonic environment (active orogen) outside Taiwan. The
global formula could be used worldwide except active orogen. This
study further suggests empirical formulas for soil site and for rock
sites. Because most natural slope failures occur on hillsides, which
usually lay atop bedrock, a rock-site formula may be more valid for
use in regional earthquake-induced landslide hazard evaluation. The
soil-site formula, however, should be used for slopes of landfills.
Acknowledgments
We extend our deepest thanks to the Pacific Earthquake Engineering
Research Center (PEER) for providing the processed strong-motion data.
The manuscript was greatly improved based on the comments and
suggestions made by the editors, Prof. Roberto W. Romeo, and one
anonymous reviewer. This research was partially supported by the
Taiwan Earthquake Research Center (TEC) funded through the National
Science Council (NSC) under Grant Number NSC98-2116-M-008-009.
References
Abrahamson, N.A., Silva, W.J., 2008. Summary of the Abrahamson & Silva NGA GroundMotion Relations. Earthquake Spectra 24 (1), 67–97.
Ambraseys, N.N., Menu, J.M., 1988. Earthquake-induced ground displacements.
Earthquake Engineering and Structural Dynamics 16, 985–1006.
Ambraseys, N.N., Srbulov, M., 1995. Earthquake induced displacements of slope. Soil
Dynamics and Earthquake Engineering 14, 59–71.
Arias, A., 1970. A measure of earthquake intensity. In: Hansen, R.J. (Ed.), Seismic Design
for Nuclear Power Plants. Massachusetts Institute of Technology Press, Cambridge,
MA, pp. 438–483.
Boore, D.M., Atkinson, G.M., 2008. Ground-motion prediction equations for the average
horizontal component of PGA, PGV, and 5%-damped PSA at spectral periods between
0.01s and 10.0s. Earthquake Spectra 24, 99–138.
Bray, J.D., Rathje, E.M., 1998. Earthquake-induced displacements of solid-waste
landfills. Journal of Geotechnical and Geoenvironmental Engineering 124, 242–253.
Bray, J.D., Travasarou, T., 2007. Simplified procedure for estimating earthquake-induced
deviatoric slope displacements. Journal of Geotechnical and Geoenvironmental
Engineering 133 (4), 381–392.
Building Seismic Safety Council, 1998. 1997 Edition NEHRP Recommended Provisions
for Seismic Regulations for New Buildings and Other Structures. FEMA 302/303,
Part 1 (Provisions) and Part 2 (Commentary), developed for the Federal Emergency
Management Agency, Washington, D. C. 337 pp.
Campbell, K.W., Bozorgnia, Y., 2008. NGA ground motion model for the geometric mean
horizontal component of PGA, PGV, PGD and 5% damped linear elastic response
spectra for periods ranging from 0.01 to 10s. Earthquake Spectra 24, 139–171.
Chiou, B.S., Youngs, R.R., 2008. An NGA model for the average horizontal component of
peak ground motion and response spectra. Earthquake Spectra 24, 173–215.
Crespellani, T., Madiai, C., Vannucchi, G., 1998. Earthquake destructiveness potential
factor and slope stability. Geotechnique 48 (3), 411–419.
Del Gaudio, V., Pierri, P., Wasowski, J., 2003. An approach to time-probabilistic
evaluation of seismically induced landslide hazard. Bulletin. Seismological Society
of America 93, 557–569.
Haneberg, W.C., 2006. Effects of digital elevation model errors on spatially distributed
seismic slope stability calculations: an example from Seattle, Washington.
Environmental and Engineering Geoscience 12, 247–260.
Harp, E.L., Wilson, R.C., 1995. Shaking intensity for rock falls and slides: evidence from
1987 Whittier Narrows and Superstition Hills earthquake strong-motion records.
Bulletin of the Seismological Society of American 85 (6), 1739–1757.
Idriss, I.M., 2008. An NGA empirical model for estimating the horizontal spectral values
generated by shallow crustal earthquakes. Earthquake Spectra 24, 217–242.
International Conference of Building Officials, 1997. Uniform Building Code, Whittier,
California. 1050 pp.
Jibson, R.W., 1993. Predicting earthquake-induced landslide displacements using
Newmark's sliding block analysis. Transportation Research Record 1411, 9–17.
Jibson, R.W., Keefer, D.K., 1993. Analysis of the seismic origin of landslides: examples
from the NewMadrid seismic zone. Geological Society of America Bulletin 105,
521–536.
Jibson, R.W., Harp, E.L., Michael, J.M., 1998. A method for producing digital probabilistic
seismic landslide hazard maps: an example from the Los Angeles, California area.
US Geological Survey Open-File Report 98-113. . 17 pp.
Jibson, R.W., Harp, E.L., Michael, J.M., 2000. A method for producing digital probabilistic
seismic landslide hazard maps. Engineering Geology 58, 271–289.
Jibson, R.W., 2007. Regression models for estimating coseismic landslide displacement.
Engineering Geology 91, 209–218.
Kao, H., Chen, W.P., 2000. The Chi-Chi earthquake sequence: active out-of-sequence
thrust faulting in Taiwan. Science 288, 2346–2349.
Kayen, R.E., Mitchell, J.K., 1997. Assessment of liquefaction potential during earthquakes by
Arias intensity. Journal of Geotechnical and Geoenvironmental Engineering 123 (12),
1162–1174.
Lee, C.T., Cheng, C.T., Liao, C.W., Tsai, Y.B., 2001. Site classifications of Taiwan free-field
strong-motion stations. Bulletin. Seismological Society of America 91 (5), 1283–1297.
Lee, C.T., Huang, C.C., Lee, J.F., Pan, K.L., Lin, M.L., Dong, J.J., 2008. Statistical approach to
earthquake-induced landslide susceptibility. Engineering Geology 100, 43–58.
Lin, P.S., Lee, C.T., Cheng, C.T., 2010. Response Spectral Attenuation Relations for Shallow
Crustal Earthquakes in Taiwan. Engineering Geology, in review.
Loh, C.H., Lee, Z.K., Wu, T.C., Peng, S.Y., 2000. Ground motion characteristics of the ChiChi earthquake of 21 September 1999. Earthquake Engineering and Structural
Dynamics 29 (6), 867–897.
Luzi, L., Pergalani, F., 1999. Slope instability in static and dynamic conditions for urban
planning: the “Oltre Po Pavese” case history (Regione Lombardia — Italy). Natural
Hazards 20, 57–82.
Ma, K.F., Lee, C.T., Tsai, Y.B., 1999. The Chi-Chi, Taiwan earthquake: large surface
displacements on an inland fault. EOS. Transactions of the American Geophysical
Union 80 (50), 605–611.
Mankelow, J.M., Murphy, W., 1998. Using GIS in the probabilistic assessment of earthquake
triggered landslide hazards. Journal of Earthquake Engineering 2, 593–623.
Miles, S.B., Ho, C.L., 1999. Rigorous landslide hazard zonation using Newmark's method
and stochastic ground motion simulation. Soil Dynamics and Earthquake
Engineering 18, 305–323.
Miles, S.B., Keefer, D.K., 2000. Evaluation of seismic slope performance models using a
regional case study. Environmental and Engineering Geoscience 6, 25–39.
Miles, S.B., Keefer, D.K., 2001. Seismic Landslide Hazard for the City of Berkeley, California.
US Geological Survey Miscellaneous Field Studies Map MF-2378.
Newmark, N.M., 1965. Effects of earthquakes on dams and embankments. Geotechnique 15, 139–159.
Pradel, D., Smith, P.M., Stewart, J.P., Raad, G., 2005. Case history of landslide movement
during the Northridge earthquake. Journal of Geotechnical and Geoenvironmental
Engineering 131, 1360–1369.
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1992. Numerical Recipes in
Fortran — The Art of Scientific Computing, 2nd Ed. Cambridge University Press. 963 pp.
Rathje, E.M., 2001. Processing of strong motion data from the 1999 Kocaeli and Duzce
earthquakes. Final Report to Pacific Earthquake Engineering Center. . November, 11 pp.
Saygili, G., Rathje, E.M., 2008. Empirical predictive models for earthquake-induced
sliding displacements of slopes. Journal of Geotechnical and Geoenviromental
Engineering 134 (6), 790–803.
Wieczorek, G.F., Wilson, R.C., Harp, E.L., 1985. Map showing slope stability during
earthquakes in San Mateo County California: US Geological Survey Miscellaneous
Investigations Map I-1257-E, scale 1:62,500.
Wilson, R.C., Keefer, D.K., 1983. Dynamic analysis of a slope failure from the 6 August
1979 Coyote Lake, California, earthquake. Bulletin. Seismological Society of America
73, 863–877.
Wilson, R.C., Keefer, D.K., 1985. Predicting areal limits of earthquake-induced
landsliding. In: Ziony, J.I. (Ed.), Evaluating Earthquake Hazards in the Los Angeles
Region An Earth-science Perspective: USGS Professional Paper, 1360, pp. 316–345.
Yegian, M.K., Marciano, E.A., Ghahraman, V.G., 1991. Earthquake induced permanent
deformations: probabilistic approach. Journal of Geotechnical Engineering 117, 35–50.