Direct use of PGV for estimating peak nonlinear oscillator displacements
Sinan Akkar*,† and Bilge Küçükdoğan
Earthquake Engineering Research Center, Department of Civil Engineering, Middle East Technical
University 06531 Ankara, Turkey
SUMMARY
A predictive model is presented for estimating the peak inelastic oscillator displacements (Sd,ie) from
peak ground velocity (PGV). The proposed model accounts for the variation of Sd,ie for bilinear
hysteretic behavior under constant ductility (µ) and normalized lateral strength ratio (R) associated
with postyield stiffness ratios of α = 0% and α = 5%. The regression coefficients are based on a
ground-motion database that contains dense-to-stiff soil site recordings at distances of up to 30 km
from the causative fault. The moment magnitude (M) range of the database is 5.2 ≤ M ≤ 7.6 and the
ground motions do not exhibit pulse-dominant signals. Confined to the limitations imposed by the
ground-motion database, the model can estimate Sd,ie by employing the PGV predictions obtained from
the attenuation relationships (ground motion prediction equations). This way the influence of
important seismological parameters can be incorporated to the variation of Sd,ie in a fairly rationale
manner. This feature of the predictive model advocates its implementation in the probabilistic seismic
hazard analysis that employs scalar ground-motion intensity indices. Various case studies are
presented to show the consistent estimations of Sd,ie by the proposed model. The error propagation in
the Sd,ie estimations is also discussed when the proposed model is associated with various attenuation
relationships.
KEYWORDS: Peak ground velocity; Inelastic spectral displacement; Ground-motion predictive
models; Regression; Seismic design/performance assessment; Bilinear hysteretic model
*
Correspondence to: S. Akkar, Earthquake Engineering Research Center, Department of Civil Engineering,
Middle East Technical University 06531 Ankara, Turkey
†
Email: [email protected]
1
1. INTRODUCTION
Most approximate models that are developed for estimating the expected peak inelastic oscillator
displacements (inelastic spectral displacement, Sd,ie) have made use of elastic period (T) dependent
empirical relationships for a given displacement ductility (µ) or normalized lateral strength (R).
Concerning the historical development of these models, they are generally based on Rµ - µ - T
relationships whose ground breaking studies were conducted by Veletsos and Newmark [1] and
further improved by Newmark and Hall [2, 3] within the context of force-based design to approximate
the oscillator yield-strength (Fy) that would limit a predefined µ value. More recently various
researchers have provided useful period-dependent empirical regression equations for direct estimation
of Sd,ie either as a function of µ (e.g. References [4-7]) or R (e.g. References [8, 6]). Regardless of the
underlying approach, these studies used ground-motion datasets to establish the empirical relationships
based on the quantity and quality of the accumulated strong-motion records at the time when they
were conducted. Depending on the objective of the study the datasets were compiled to bring forward
various aspects of ground-motion parameters. The studies that followed the Veletsos and Newmark
approach have made use of peak ground-motion values to define the frequency content of the ground
motion. They then derived their empirical relationships to relate elastic to inelastic oscillator response
for the spectral period ranges described by the peak ground-motion ratios. These studies classified
their datasets according to ground-motion duration, moderate- to large-magnitude events, pulsedominant signals or severe events produced by a particular fault type (e.g. References [9-11]). Using
small- to large-size ground-motion datasets, studies conducted by Elghadamsi and Mohraz [12], Sewel
[13], Nassar and Krawinkler [14], Miranda [4, 15], Song and Pincheira [16], Ruiz-García and Miranda
[7, 8], Arroyo and Teran [17] and Peköz and Pincheira [18] mostly emphasized the influence of
different site classes on the estimation of Sd,ie. Chopra and Chintanapakdee [6, 19] classified their
ground motions according to different magnitude-distance bins as well as for different site classes.
Researchers like MacRae and Roeder [20], Baéz and Miranda [21], MacRae et al. [22] and Chopra and
Chintanapakdee [23] shaped their Sd,ie predictive expressions on the differences between near- and farfault ground-motion records. These studies revealed significant insight about the nonlinear oscillator
behavior under different load-deformation rules. The reader is referred to the above cited studies as
well as Miranda and Bertero [25] and Mahin and Bertero [26] for a detailed review on nonlinear
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2
oscillator response studies for the estimation of Sd,ie. In addition to these efforts few studies developed
empirical Rµ - µ - T relationships that are directly based on normalized spectrum [27-29] to account for
the influence of earthquake source in the estimation of inelastic oscillator response that is generally
overlooked by other studies.
Regardless of the approximate model, the major concept used in the estimation of Sd,ie is
S d ,ie =
µ
R
2
S d ,e =
µ⎛ T ⎞
⎜
⎟ S a ,e
R ⎝ 2π ⎠
(1)
In the case of Rµ - µ - T relationships, one would use the expected Rµ for a given µ - T pair to estimate
Sd,ie from its elastic counterpart Sd,e (or equivalently utilizing the more familiar elastic pseudo-spectral
acceleration, Sa,e that can be related to Sd,e through the constant (T/2π)2 as shown in Eq. (1)). For the
direct empirical relationships, the analyst relates Sd,ie to Sd,e by using the regression equations that
mimic the expected variation of µ/R either for a constant ductility level or normalized lateral strength
ratio. Thus, an alternative way of expressing Eq. (1) is
2
S d ,ie = C x S d ,e
⎛ T ⎞
= Cx ⎜
⎟ S a ,e
⎝ 2π ⎠
(2)
where Cx is the period-dependent modification factor either for constant µ or R. The expressions
presented in Eqs. (1) and (2) establish a linear relationship between inelastic and elastic spectral
displacements (or equivalently a linear variation between Sd,ie and Sa,e) for a given elastic period, T. As
a matter of fact this approach is one of the current methods used in the simplified nonlinear static
procedures (e.g. References [30-33]). Note that the ongoing research efforts for the estimation of Sd,ie
continuously result in new predictive models. In a recent study, Tothong and Cornell [34] proposed an
attenuation relationship for estimating Sd,ie as a function of oscillator yield-strength and magnitude.
Recent research that aims to associate structural deformation demands with alternative groundmotion intensities has conveyed valuable information about the merits of PGV. Various studies have
shown that PGV can be considered as a reasonable scalar ground-motion parameter that correlates well
with the peak nonlinear oscillator response (e.g. References [35-37]). Küçükdoğan [38] also
demonstrated that PGV reveals a good correlation with the global deformation demands (e.g.
maximum interstory drift ratio, MIDR) of mid-period, reinforced-concrete (RC) moment-resisting
frames. A suite of sample case studies conducted by Küçükdoğan [38] is presented in Figure 1 to
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illustrate the above conclusion. The scatter plots compare the correlation between PGV and MIDR
computed from the nonlinear response history analysis of a set of RC frame buildings with
fundamental periods between 0.40s < T1 ≤ 1.3s. The frame systems were subjected to the groundmotion dataset in Akkar and Özen [36]. The straight lines in each plot help to illustrate the variation of
MIDR with the scalar ground-motion index PGV. Spearman’s non-parametric coefficient (ρ) was used
to assess the correlation between these two variables because it does not require the assumption that
the relationship between PGV and MIDR is linear, nor does it require any assumption about the
frequency distribution of the variables [40]. The high ρ values presented in each plot advocates that
PGV reveals versatile information about the nonlinear multi-degree-of-freedom (MDOF) deformation
demands that has been observed previously for the peak nonlinear oscillator displacements. The
correlation between PGV and MIDR increases as the story number increases that is inherently related
to the shift in the building period towards longer spectral range. This observation was one of the
driving factors for using PGV as the ground-motion intensity measure to assess the seismic
performance of building stocks in the Istanbul metropolitan area [41, 42].
This study presents alternative empirical regression equations as a function PGV for estimating
Sd,ie. The predictive equations were derived from a suite of dense-to-stiff soil ground motions at
relatively short source-to-site distances. The equations are devised for estimating 5%-damped Sd,ie for
bilinear oscillator response with postyield stiffness ratios of α = 0% and α = 5%. The estimations are
based on constant ductility and normalized lateral strength. The information revealed from the
constant ductility spectral displacements is useful for seismic design whereas Sd,ie obtained from the
normalized lateral strength yields direct information about the deformation demands on existing
structures. The bilinear hysteretic model and the associated α values are widely used to represent the
pushover curves of various MDOF systems in nonlinear static procedures. The paper also illustrates
the implementation of the predictive model by making use of PGV estimations computed from recent
ground motion prediction equations (GMPEs) under different scenarios. When implemented together
with the GMPEs, the predictive model can be useful for probabilistic seismic hazard analysis (PSHA)
methods that are based on a scalar ground-motion intensity measure.
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4
3 st. (T1=0.41-0.73s)
5 st. (T1=0.70-1.04s)
MIDR (%)
1
0.1
ρ = 0.687
ρ = 0.600
0.01
1
10
100
1
10
100
9 st. (T1=0.99-1.30 s)
7 st. (T1=0.87-1.15 s)
MIDR (%)
1
0.1
ρ = 0.680
ρ = 0.709
0.01
1
10
PGV (cm/s)
100
1
10
PGV (cm/s)
100
Figure 1. Correlation between PGV and MIDR for a set of moment-resisting frame systems subjected
to the ground motions in Akkar and Özen [36]. The plots present the MIDR vs. PGV scatters from 5
sets of building models with 3-, 5-, 7- and 9-stories. The nonlinear RHA were conducted by using
IDARC 2D [39]. All frame models have 3 bays and are regular in plan with 5m span width and 3m
story height. They comply with the modern seismic provisions such that they can undergo sufficient
nonlinear deformation without experiencing collapse when subjected to design earthquakes. Detailed
information about the models is presented in Küçükdoğan [38] that can be provided to the interested
reader upon request.
2. GROUND-MOTION DATASET
The ground-motion dataset comprises of 105 soil site records from 46 shallow events with a moment
magnitude range 5.2 ≤ M ≤ 7.6. The shortest horizontal distance from the surface projection of the
fault rupture (Rjb, [43]) of the accelerometric data is less than 30km that is of practical importance for
most engineering applications. The dataset was compiled from the studies by Akkar and Özen [36] and
Akkar and Bommer [44]. An arbitrary horizontal component was selected randomly from each
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5
accelerogram. Almost all pre-1999 data is analogue recordings and they constitute the majority in the
dataset. The records with forward directivity effects were excluded since such ground motions are
dominated by pulse-type signals that result in a distinct structural behavior depending on the pulse
period and the amplitude of ground velocity (e.g. References [45-48]). The main features of the
records used in the regression analysis are listed in Table A1 in Appendix A. The distribution of
records with respect to faulting style is uneven. The dataset is mainly dominated by strike-slip (S)
events (45 records). There are 31 records from reverse (R) faulting earthquakes and the number of
records from normal (N) faulting is 23. The style-of-faulting of the 6 records from the 1989 Loma
Prieta event is identified as reverse-oblique (RO) in the literature.
Figure 2 presents information about the magnitude-distance distribution and the usable period
range of the ground-motion dataset. The scatter plot presented in Figure 2.a indicates that the data has
a better resolution between 5.5 ≤ M ≤ 7.0 and Rjb ≤ 25km. The upper 30m average shear-wave velocity
(Vs,30) of each strong-motion station was used in the site classification. The 62% of the database is
dominated by NEHRP site class D records (180m/s ≤ Vs,30 < 360m/s) whereas the rest of the data is
recorded on NEHRP site class C (360m/s ≤ Vs,30 < 750m/s). The data number vs. oscillator period plot
displayed in Figure 2.b was obtained from the maximum usable elastic spectral period range criteria
established by Akkar and Bommer [49]. The low-cut (high-pass) filter frequency of each record was
used to define the corresponding spectral period range for which the filter influence is minimized in
spectral calculations. Figure 2.b shows that there is a significant reduction in the record number when
T > 3.5s. In this study the inelastic spectral displacements were calculated up to T = 2.0s as they are
expected to be influenced more by low-cut filtering due to the inherent period elongation in nonlinear
oscillator response. This way almost all data in the database could be used in the regression analyses
as can be depicted from Figure 2.b. It should be noted that the 2-seconds period limit is still a crude
assumption to minimize the low-cut filter influence on peak nonlinear oscillator displacements. To the
best of authors’ knowledge, there is no well-established criterion in the relevant literature for defining
usable period range of nonlinear spectral calculations to minimize the low-cut filter influence.
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6
8.0
NEHRP C
NEHRP D
100
Data Number
Magnitude (M)
7.5
7.0
6.5
6.0
80
60
40
5.5
(a)
5.0
0
5
10
15
20
25
30
(b)
20
0
1
Distance, Rjb (km)
2
3
4
5
6
7
8
9
10
Period (s)
Figure 2. Magnitude vs. distance (Rjb) scatter and useful spectral period range information.
3. REGRESSION ANALYSIS
3.1 Functional form
The general mathematical model for estimating Sd,ie for a given µ or R value is
S d ,ie
R,µ
= f (θ ) ⋅ PGV ⋅ T
(3)
The functional form f in Eq. (3) considers the influence of independent ground-motion parameters (θ)
such as magnitude, distance, site class etc. The regression analysis was conducted on the
dimensionless dependent variable Sd,ie/(PGV×T) because this parameter resulted in a simpler predictive
model as discussed next. Concerning the limited resolution of the actual ground-motion database, the
influence of independent ground-motion parameters on the predicted variable was investigated by
using some recent GMPEs. The behavior of Sd,e/(PGV×T) that is the elastic response version of the
predicted parameter was analyzed to achieve this objective. The main assumption in this sensitivity
analysis is that the general behavior of the predicted parameter will have a similar (but not the same)
pattern both for linear and nonlinear oscillator response. (This assumption is further discussed in
Figure 4). Figure 3 shows the comparative results from two recent GMPEs that are derived by Akkar
and Bommer [44, 50] and Boore and Atkinson [51]. These studies are abbreviated as AB07 and BA07,
respectively. The figures on the left display the results computed from AB07 whereas the pertaining
results of BA07 are presented on the right hand side. The AB07 prediction equations were derived
Revised Manuscript
7
from a recently compiled European ground-motion database. The BA07 GMPE uses a worldwide
ground-motion dataset that is compiled for Next Generation Attenuation (NGA) project. Both studies
estimate the spectral and peak ground-motion values with certain differences in their functional forms.
For example BA07 considers the nonlinear soil effects as a function of Vs,30 whereas AB07 does not
account for soil nonlinearity. Moreover AB07 introduces dummy variables to consider the site
conditions that are in accordance with NEHRP site class definitions. The BA07 model describes the
site influence through continuous Vs,30 values. The first row in Figure 3 shows the influence of distance
metric (Rjb) on Sd,e/(PGV×T) for the magnitude range of interest in this study. The figures on the left
and right panels display 3 sets of curves for the oscillator periods at T = 0.5, 1.0 and 2.0s. Each set
compares 3 distinct Rjb distances (i.e. Rjb = 10, 20 and 30 km) for a given oscillator period. The
discrete T and Rjb values presented fairly cover the period and distance ranges in this study. All plots
are produced for strike-slip events. The chosen site class is NEHRP C that is arbitrarily approximated
by using Vs,30 = 450m/s in AB07. The comparative plots show that for short-period oscillator response
(T = 0.5s) both GMPEs describe a slight departure for the Rjb = 10km curve with respect to the Rjb =
20km and Rjb = 30km curves in the small magnitude range. As far as the long-period oscillator
response (T = 2.0s) is concerned, the AB07 curves follow a trend similar to the one described for T =
0.5s whereas BA07 curves almost overlap each other for all Rjb distances. The observed differences in
the behavior of GMPEs for T = 2.0s may stem from their distinct magnitude scaling functional forms.
The GMPEs considered do not show a distance-wise sensitivity for T = 1.0s. Although it is crude,
these observations may lead to an assumption that the discrepancies on Sd,e/(PGV×T) emerging from
distance variation are secondary when compared to the magnitude influence. The second row in Figure
3 shows the significance of site class on Sd,e/(PGV×T). NEHRP site classes C and D are considered in
the comparative plots as the ground-motion dataset consists of records from these site categories. For
illustrative purposes Vs,30 = 450m/s and 270m/s are used in BA07 to represent NEHRP C and D site
classes, respectively. Similar to the plots in the first row, each panel displays 3 sets of Sd,e/(PGV×T) vs.
Rjb curves computed for T = 0.5, 1.0 and 2.0s. Each set corresponds to a particular oscillator period
and shows the variation in Sd,e/(PGV×T) for NEHRP C and D site classes, respectively. In order not to
Revised Manuscript
8
crowd the figures the AB07 predictions were used to display the results for M = 7 whereas BA07 was
used to illustrate the results from small magnitude events mimicked by M = 5. The style-of-faulting is
also chosen as strike-slip in these figures. Although there is a departure in Sd,e/(PGV×T) between
different site classes for large magnitude and mid-period values (T = 1.0s), the general picture from
these plots may also advocate that magnitude is a more prominent parameter than the site class in the
variation of Sd,e/(PGV×T). Plots similar to those presented in Figure 3 were also produced for normal
and reverse faulting. Although they are not shown in the paper, the observations are comparable to
those presented for strike-slip faults.
Akkar and Bommer (2007)
0.4
Boore and Atkinson (2007)
0.4
Vs,30 = 450 m/s
NEHRP C
0.3
0.3
T=0.5s
Sd,e/(PGVxT)
T=0.5s
0.2 T=1.0s
0.2
T=1.0s
T=0.5s, Rjb=10km
T=0.5s, Rjb=20km
T=0.5s, Rjb=30km
T=1.0s, Rjb=10km
T=1.0s, Rjb=20km
T=1.0s, Rjb=30km
T=2.0s, Rjb=10km
T=2.0s, Rjb=20km
T=2.0s, Rjb=30km
0.1
0.09
0.08 T=2.0s
0.07
0.06
T=2.0s
0.1
0.09
0.08
0.07
0.06
0.05
0.05
5
6
Magnitude (M)
5
7
0.30
Sd,e/(PGVxT)
T=0.5s, Rjb=10km
T=0.5s, Rjb=20km
T=0.5s, Rjb=30km
T=1.0s, Rjb=10km
T=1.0s, Rjb=20km
T=1.0s, Rjb=30km
T=2.0s, Rjb=10km
T=2.0s, Rjb=20km
T=2.0s, Rjb=30km
M=7
M=5
0.25
0.20
0.20
0.15
0.15
NEHRP C, T = 0.5s
NEHRP D, T = 0.5s
NEHRP C, T = 1.0s
NEHRP D, T = 1.0s
NEHRP C, T = 2.0s
NEHRP D, T = 2.0s
0.05
0.00
Vs,30 = 450m/s, T = 0.5s
Vs,30 = 270m/s, T = 0.5s
0.10
Vs,30 = 450m/s, T = 1.0s
Vs,30 = 270m/s, T = 1.0s
Vs,30 = 450m/s, T = 2.0s
Vs,30 = 270m/s, T = 2.0s
0.05
0.00
0
5
10
15
20
Distance, Rjb (km)
25
30
7
0.30
0.25
0.10
6
Magnitude (M)
0
5
10
15
20
25
30
Distance, Rjb (km)
Figure 3. Influence of certain ground-motion parameters on Sd,e/(PGV×T) using different GMPEs.
Revised Manuscript
9
Figure 4 exhibits the behavior of dependent variable obtained from the actual data as a function
of magnitude to rationalize the discussions presented in the above paragraph. The plots present the
variation of average Sd,ie/(PGV×T) points computed from the magnitude bins of half-unit intervals
starting from M = 5. The panels in the first row display this variation for constant ductility whereas the
second row plots describe the same relationships for constant R. Although the plots summarize the
results of elastoplastic (α = 0%) hysteretic behavior, similar trends also exist for the bilinear oscillator
response with α = 5%. The panels on the left show the magnitude dependent variation of Sd,ie/(PGV×T)
for µ = 1.5 (upper row) and R = 1.5 (lower row), respectively. The upper and lower row panels on the
right show the same variations for µ = 8 and R = 8, respectively. In order not to complicate the
information presented the average scatter points in each panel describe the magnitude dependent
variation at two periods: T = 0.5s and T = 2.0s. The data resolution allowed the computation of
average scatter points for two different distance intervals (i.e. Rjb < 10km and 10km < Rjb < 2 km). The
plots also show the quadratic curves that were fit to the variation of average scatter points computed
for each distance interval. The reason of choosing quadratic curves is their relatively high R2 values
that suggest a fairly good relation between the actual data trend and the fits. When the plots presented
in Figure 4 are compared with those in Figure 3, one would observe that the trends are fairly similar
except for the fact that the nonlinear oscillator response results in a shift in the vertical axis when the
level of inelasticity (i.e. µ or R) attains higher values. These limited observations verify the major
assumptions about the behavior of dependent parameter and fortify the general conclusions discussed
in Section 3.1 that are derived through the use of GMPEs.
Revised Manuscript
10
Sd,ie/(PGVxT)
0.6
0.5
0.4
0.4
0.3
0.3
0.2
0.1
0.09
0.08
0.07
0.06
0.05
T=0.5s
µ = 8.0
T=0.5s
0.2
0.1
0.09
0.08
0.07
0.06
0.05
T=2.0s
5
Sd,ie/(PGVxT)
0.6
0.5
µ = 1.5
6
7
Rjb < 10 km (T = 0.5s)
10 km < Rjb < 20 km (T = 0.5s)
T=2.0s
5
0.6
0.5 R = 1.5
0.6
0.5 R = 8.0
0.4
0.4
0.3
0.3
0.2
0.1
0.09
0.08
0.07
0.06
0.05
T=0.5s
6
7
T=0.5s
0.2
0.1
0.09
0.08
0.07
0.06
0.05
T=2.0s
5
Rjb < 10 km (T = 2.0s)
10 km < Rjb < 20 km (T = 2.0s)
6
Magnitude (M)
7
Rjb < 10 km (T = 0.5s)
10 km < Rjb < 20 km (T = 0.5s)
T=2.0s
Rjb < 10 km (T = 2.0s)
10 km < Rjb < 20 km (T = 2.0s)
5
6
7
Magnitude (M)
Figure 4. Variation of the observed data as a function of magnitude for different periods and distance
intervals. The upper row shows the variations for constant ductility for µ = 1.5 (left panel) and µ = 8.0
(right panel). Similar information is given in the lower row panels for normalized lateral strength.
Based on the above discussions, it was decided to consider the magnitude term as the only
explanatory variable in the predictive model. The quadratic variation of magnitude was selected for the
proposed model that seems to capture the variation of the dimensionless dependent variable
adequately for the overall magnitude range when the plots from AB07 (Figure 3) and the actual data
(Figure 4) are considered. The quadratic magnitude variation is also reasonable for the trends revealed
by BA07 for M ≤ 6.7. This GMPE shows a sharp linear decay for M > 6.7 (called as “hinging effect”
by the proponents) due to the magnitude scaling terms in the model that prevent oversaturation in the
predicted ground-motion variable. It should be noted that the sole consideration of magnitude
Revised Manuscript
11
influence is rough and at the expense of complexity a more complete model should contain the rest of
the independent ground-motion parameters that are omitted in this study. In their predictive model
Tothong and Cornell (2006) also considered magnitude as the only explanatory variable indicating that
other seismological independent parameters are not as influential as magnitude in the estimation of
Sd,ie. The final functional form used in the regression analysis is presented in Eq. (4).
⎛ S d ,ie R , µ
ln⎜
⎜ PGV × T
⎝
⎞
⎟ = b0 + b1 M + b2 M 2 + ε ⋅ σ
⎟
⎠
(4)
In the above expression b0, b1 and b2 are the regression coefficients to be determined from the
regression analysis. The last term is the random error term and it accounts for the variability in the
dependent parameter due to the unconsidered predictor parameters in the model. This term
corresponds to the difference between the estimated and observed dependent variable that is called as
the residual in the regression analysis. If the fitted model correctly accounts for the variation of the
observed data, the residual mean square is the unbiased estimator of the variance (σ2) about the
regression. However, if the model fails to explain the variation of the observed data, the residuals
contain both random and systematic errors due to the model inadequacy resulting in biased σ2
calculated from the residual mean square [52]. The term ε in Eq. (4) denotes the number of standard
deviations (σ) above or below the expected value of dependent variable.
3.2. Regression Technique
There are number of regression techniques to estimate the coefficients of predictive variables in a
functional form. In this study the least squares regression was used that would estimate the same
coefficients as of maximum likelihood regression method provided that the random error terms are
normally distributed with zero mean and σ2 [53]. This condition is satisfied here as discussed in the
succeeding paragraphs. The regressions were done period-by-period for 0.2s ≤ T ≤ 2.0s with
increments of 0.1s. The inelastic oscillator displacements were estimated at 8 distinct µ and R values
(i.e. µ or R = 1.5, 2, 3, 4, 5, 6, 7 and 8) for bilinear hysteretic model associated with 0% and 5%
postyield stiffness ratio (α). The initial damping was taken as 5% of critical in all nonlinear oscillator
responses. The regression coefficients as well as the associated σ values are presented in Küçükdoğan
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12
[38]. They are also posted on http://www.ce.metu.edu.tr/~sakkar/GMPE_coeffs.xls. The random error
term (σ) of the predictive model is independent of magnitude whereas some recent prediction
equations do consider the magnitude influence on σ. The magnitude-dependent standard deviations
might have been incorporated to the predictive model by implementing pure error analysis as outlined
in Douglas and Smit [54]. This was not done for the current study as the data resolution is limited to
partition the database into different magnitude-distance bins to observe the magnitude influence on the
standard deviations. Bommer et al. [55] showed that the magnitude dependence on the random
variability of the ground motion prediction models require further work because different magnitudedistance binning schemes may significantly influence the variation of σ as a function of magnitude.
Detailed analysis of variance (ANOVA) was carried out in order to judge the adequacy of the
predictive model for each set of T-µ-α (or T-R-α). The ANOVA calculations account for the random
variation of the repeated observations in the predicted variable for distinct T-M pairs [52]. F-test was
applied at the 5% significance level to examine the lack of model fit. Except for very short periods
there was no lack of model fit at the 5% significance level in the T-µ-α (or T-R-α) pairs indicating that
the predictive model can fairly represent the variation of the observed data. The corresponding R2
statistics also showed that the model can generally explain more than 50% of the data variation that
can be accepted as quite satisfactory for datasets containing repeated observations [52]. In essence,
considering the overall performance, the predictive model is accepted as adequate for descent
estimations of nonlinear peak oscillator displacements in terms of PGV.
3.3 Residual analysis and model evaluation
As noted previously the above calculations are only valid under the assumption that residuals are
normally distributed with zero mean and variance σ2. Figure 5 shows the sample normal probability
plots of residuals at T = 0.5, 1.0, 1.5 and 2.0s for µ and R equal to 4 when α = 0%. All plots indicate
that the normal distribution with zero mean assumption for residuals is reasonable since the residuals
fall near the solid line that connects different percentiles of normal distribution. Thus, the variances
computed from the residual mean squares can fairly account for the random error associated with the
Revised Manuscript
13
predictive model. The same observations also apply to the normal probability plots of the bilinear
hysteretic model with α = 5% [38]. These are not presented here due to the space limitations.
Figure 5. Normal probability plots of the residuals at T = 0.5, 1.0, 1.5 and 2.0 s for µ and R equal to 4
(upper and lower rows, respectively) when postyield stiffness is 0%.
The residuals were also examined to confirm that the predictions are unbiased due to the
omission of other explanatory parameters in the predictive model that are discussed in the previous
paragraphs. Figure 6 presents the residual plots against magnitude at T = 0.5 1.0, 1.5 and 2.0s for µ = 6
(first row) and for R = 6 (second row). Figure 7 shows the residuals vs. estimated dependent parameter
plots of the entire database for µ and R equal to 1.5, 3.0, 5.0 and 7.0. Similar to Figure 6, the upper and
lower rows display the relevant plots for constant ductility and normalized lateral strength ratio,
respectively. Both figures present residual scatters for α = 0% because the associated dispersion is
relatively higher when compared to the residuals of α = 5% case. The solid straight lines in these
figures show the trends fitted to the residuals; a significant slope in these trend lines would suggest the
biased estimations of the predictive model. The plots in Figures 6 and 7 do not exhibit a biased trend
in the residuals. Thus the proposed model satisfactorily accounts for the variation of the dependent
parameter regardless of the omission of other predictor parameters. Note that the residuals reported for
normalized lateral strength attain larger values than those of displacement ductility. This is expected
Revised Manuscript
14
since the peak oscillator displacements computed for constant R values are not limited to a predefined
value which is the case for constant µ peak oscillator displacements. The inherent difference between
the nonlinear oscillator responses imposed by constant µ and R results in increased dispersion about
the mean variation of peak oscillator displacements computed for normalized lateral strength.
Residual
2
T = 0.5 s., µ = 6.0
T = 1.0 s., µ = 6.0
2
T = 1.5 s., µ = 6.0
2
1
1
1
1
0
0
0
0
-1
-1
-1
-1
-2
5.0
5.5
6.0
2
Residual
2
6.5
7.0
7.5
T = 0.5 s., R = 6.0
-2
5.0
5.5
6.0
2
6.5
7.0
7.5
T = 1.0 s., R = 6.0
-2
5.0
5.5
6.0
2
6.5
7.0
7.5
T = 1.5 s., R = 6.0
-2
5.0
1
1
1
0
0
0
0
-1
-1
-1
-1
5.5
6.0
6.5
7.0
7.5
-2
5.0
5.5
Magnitude (M)
6.0
6.5
7.0
7.5
-2
5.0
Magnitude (M)
5.5
6.0
6.5
7.0
7.5
5.5
6.0
2
1
-2
5.0
T = 2.0 s., µ = 6.0
6.5
7.0
7.5
T = 2.0 s., R = 6.0
-2
5.0
Magnitude (M)
5.5
6.0
6.5
7.0
7.5
Magnitude (M)
Figure 6. Residual plots as a function of M for µ = 6 (upper row) and R = 6 (lower row) at T = 0.5, 1.0,
1.5 and 2s when α = 0%.
Residual
2
µ = 1.5
2
µ = 3.0
µ = 5.0
2
1
1
1
0
0
0
0
-1
-1
-1
-1
-2
-2
-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -2.25
-2.00
-1.75
-1.50
-1.25
R = 3.0
R = 1.5
1
R = 5.0
2
1
1
1
1
0
0
0
0
-1
-1
-1
-1
-2
-2
-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -2.25
Predicted ln(Sd,ie/(PGVxT))
-2.00
-1.75
-1.50
Predicted ln(Sd,ie/(PGVxT))
-1.25
µ = 7.0
-2
-2
-2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00
2
2
2
Residual
2
R = 7.0
-2
-2
-2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -2.25-2.00-1.75-1.50-1.25-1.00-0.75-0.50
Predicted ln(Sd,ie/(PGVxT))
Predicted ln(Sd,ie/(PGVxT))
Figure 7.Residual plots in terms of the dependent parameter for distinct µ and R values considering the
entire database. The residuals pertain to the results obtained from the bilinear model with α = 0%.
Revised Manuscript
15
ln[Sd,ie/(PGVxT)]
0
0
-1
+σ
+σ
-1
−σ
-2
µ = 4 T = 0.5s
5.0 5.5 6.0 6.5 7.0 7.5 8.0
0
+σ
mean
-2
-3
-3
µ = 4, T = 1.0s
5.0 5.5 6.0 6.5 7.0 7.5 8.0
-3
µ = 4, T = 2.0s
5.0 5.5 6.0 6.5 7.0 7.5 8.0
0
+σ
+σ
-1
mean
-1
mean
−σ
-2
µ = 8, T = 0.5s
−σ
-2
−σ
-2
mean
−σ
0
-1
+σ
-1
mean
mean
-3
ln[Sd,ie/(PGVxT)]
0
-3
−σ
-2
µ = 8, T = 1.0s
-3
µ = 8, T = 2.0s
5.0 5.5 6.0 6.5 7.0 7.5 8.0
5.0 5.5 6.0 6.5 7.0 7.5 8.0
5.0 5.5 6.0 6.5 7.0 7.5 8.0
Magnitude (M)
Magnitude (M)
Magnitude (M)
Figure 8. Scatter plots of the actual variation of the dependent parameter together with the
corresponding mean ± σ estimations for µ = 4 (upper row) and µ = 8 (lower row). The plots represent
elastoplastic (α = 0%) behavior. The left, middle and the right panels compare the actual data and the
predictions at T = 0.5s, 1.0s and 2.0s, respectively. The diamonds designate the averages of the actual
data computed for the magnitude bins of half-unit intervals starting from M = 5.
Information about the predictive power of the proposed empirical model is reported in Figure 8.
The figure presents the magnitude-dependent scatter plots of the actual data for elastoplastic (α = 0%)
oscillator response superimposed with the mean ± σ estimations. The plots display the pertaining data
variation for µ = 4 (first raw) and µ = 8 (second row) that are computed at T = 0.5s (first column), 1.0s
(second column) and 2.0s (third column). The diamonds in all plots represent the average of the
observed data computed from the magnitude bins of half-unit intervals that start from M = 5. The
general picture depicted from these figures is that the proposed empirical model is able to capture the
general variation of the observed data fairly well. The exceptions are the short-period cases with large
inelasticity levels (e.g. the lower left corner panel). This observation is consistent with the conclusions
of Akkar and Özen [36] who observed poor correlation between PGV and Sd,ie at short-period
oscillator response. As stated in Section 3.2, this shortcoming of the model is tolerated by considering
Revised Manuscript
16
its overall performance for the entire spectral period range and inelastic levels covered in this study.
Though it is not reported here, similar conclusions are also valid for constant strength oscillator
response and bilinear hysteretic model with α = 5%.
4. APPLICATION OF THE PROPOSED MODEL
The predictive model can estimate the peak inelastic oscillator displacements for a PGV value that is
computed from a GMPE. Recalling the general functional form presented in Eq. (3) and applying
random variables theory under the assumption that both PGV and Sd,ie|R, µ are log-normal independent
varieties with negligible statistical correlation, one can incorporate the random error due to the
predicted PGV to the overall standard deviation of the Sd,ie estimation. This is given in Eq. (5).
[(
Var ln S d ,ie
R,µ
)] = Var[ln( f (θ ))] + Var[ln(PGV )]
(5)
The term on the left hand side of Eq. (5) is the total variance of the estimated Sd,ie that contains
random error terms due to the predictive model presented and the PGV estimated from a GMPE (first
and second terms on the right hand side, respectively). The predictive model presented here is derived
for random horizontal component definition and this requires a careful consideration of the GMPE
employed for the PGV estimation. If the chosen GMPE is not devised for the random components
effect, one must use a consistent scaling to bring the horizontal component definition of the chosen
GMPE in agreement with the random component definition used here. Beyer and Bommer [56, 57]
established empirical relationships between different horizontal component definitions of PGV for
their median estimations and for the associated random error terms. These relationships can be used
efficiently to obtain compatible and consistent results from the proposed predictive model when the
GMPE considered yields PGV estimations other than the random horizontal component definition.
Revised Manuscript
17
M = 5.5, PGV = 18.3 cm/s
10
10
Sd,ie (cm)
10
M = 7.0, PGV = 59.6 cm/s
M = 6.0, PGV = 29.7 cm/s
1
1
1
0.1
1
0.1
1
10
0.1
1
10
Sd,ie (cm)
10
µ = 1.0
µ = 4.0
µ = 6.0
µ = 8.0
1
1
0.1
Period (s)
1
R = 1.0
R = 4.0
R = 6.0
R = 8.0
1
0.1
1
Period (s)
0.1
1
Period (s)
Figure 9. Inelastic spectral displacement estimations of the proposed predictive model for constant µ
(upper row) and constant R (lower row) when α = 0%. The black solid curves that are designated by
either µ = 1 or R = 1 show the corresponding elastic spectral displacements computed from Akkar and
Bommer [50].
Figure 9 shows the variation in the expected Sd,ie for a set of constant µ and R values and for
reverse faulting events of increasing magnitude (M = 5.5, 6.0 and 7.0). For illustrative purposes the
site is assumed to be located 5 km from the surface projection of the fault rupture and its soil condition
is represented as NEHRP D. The plots present the Sd,ie estimations for α = 0%. The first row panels
display the constant ductility spectral displacement plots for M = 5.5, 6.0 and 7.0, respectively. The
second row presents the same information for normalized lateral strength spectral displacements. The
PGV values of the scenario events were computed from Akkar and Bommer [44] that uses geometric
mean component definition. The empirical relationships proposed by Beyer and Bommer [56, 57]
were used to adjust the differences between the random component and geometric mean definitions.
The figure also associates the elastic spectral displacements computed from Akkar and Bommer [50]
in order to verify the consistent behavior of the predicted Sd,ie with respect to its elastic counterpart.
Revised Manuscript
18
Figure 9 clearly displays the magnitude influence on the variation of Sd,ie. For small magnitude events
(M = 5.5), the inelastic spectral displacements start oscillating about a constant plateau after T > 1.0s.
For other magnitude values the inelastic spectral displacements follow a continuously increasing
pattern with increasing oscillator periods. This observation is consistent with the previous studies that
highlight the strong relationship between magnitude and spectral corner periods for defining the
commencement of constant spectral displacement plateau (e.g. References [32, 58, 59]. The PGV
dependent predictive model seems to capture this effect adequately underlining once again the strong
correlation between PGV and magnitude that has already been addressed by many studies (e.g.
References [44, 60]). Note that the elastic displacement spectra plotted in these figures display a
compatible pattern with the magnitude-dependent inelastic spectral trends discussed above. This
observation provides further information about the trustable behavior of the proposed model. Another
common observation from these plots is that the spectral periods for the commencement of “equal
displacement rule” (i.e. inelastic spectral displacements practically attaining the same peak
displacements of the corresponding elastic oscillators) are sensitive to the level of PGV that is
essentially related to the magnitude. The increase in PGV (that is dictated by the increase in
magnitude) shifts the spectral regions towards longer periods where “equal displacement rule” starts to
hold. This observation was also noted by Tothong and Cornell [34] while deriving their inelastic
spectral displacement prediction equation. Note that the differences in the short-period peak
displacements between the displacement ductility and normalized lateral strength spectra are notable
since the latter spectrum type does not impose any limit on the computed inelastic oscillator
displacements. These observations have already been marked by various studies (e.g. References
[6,8]) and the proposed model can capture these prominent features of nonlinear oscillator response.
Revised Manuscript
19
Constant Strength
Constant Ductility
1.0
1.0
σ
Tot
0.9
Boore and Atkinson (2007)
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.0
0.5
1.0
1.5
2.0
1.0
0.9
0.9
Tot
σ
0.7
0.6
0.4
0.3
0.0
Akkar and Bommer (2007)
M = 5.5
0.5
1.0
1.5
2.0
1.5
2.0
Rµ = 1.5
Rµ = 3.0
Rµ = 5.0
Rµ = 8.0
0.7
0.6
0.5
0.4
0.3
0.0
Akkar and Bommer (2007)
M = 7.5
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.0
1.0
Akkar and Bommer (2007)
M = 5.5
0.5
1.0
1.5
2.0
1.0
1.0
0.9
0.5
0.8
µ = 1.5
µ = 3.0
µ = 5.0
µ = 8.0
0.5
Tot
0.3
0.0
1.0
0.8
σ
Boore and Atkinson (2007)
0.9
0.5
1.0
Period (s)
1.5
2.0
0.3
0.0
Akkar and Bommer (2007)
M = 7.5
0.5
1.0
1.5
2.0
Period (s)
Figure 10. Illustrative cases about the variation in the total standard deviation (σTot) of estimated Sd,ie
when the proposed model is associated with the scenario-based PGVs that are estimated from different
GMPEs. The plots on the left and on the right show the change in σTot for different µ and R values,
respectively. The gray shadows complete the entire picture about the variation of σTot by describing the
overall band in σTot for µ and R values ranging between 1.5 and 8.
Figure 10 illustrates the progress in the random error of estimated Sd,ie for various µ and R values
when the scenario PGV is described by a GMPE. The total variance expression presented in Eq. (5) is
used to address the change in the total standard deviation (σTot) when the scenario-based PGV values
are estimated through AB07 and BA07. One major difference between these predictive models is that
Revised Manuscript
20
AB07 takes into account of magnitude uncertainty in the random error term. This results in a
magnitude-dependent standard deviation that increases with decreasing magnitude. The predictive
model BA07 does not consider a magnitude influence on the variation of standard deviation. The
panels in the first row describe the period-dependent variation in σTot (the overall dispersion about the
Sd,ie estimations) when the scenario-based PGV values are computed from BA07. The left and right
panels show the change in σTot for constant µ and R, respectively. The empirical relationships
proposed by Beyer and Bommer [56, 57] were used once again to adjust the random error term in
BA07 for the differences between the horizontal component definitions. Owing to the magnitudeindependent standard deviation in BA07, the σTot values presented in these plots are not affected from
the changes in magnitude. The total standard deviation in BA07 for elastic spectral displacement
estimations is reported to vary between 0.6 and 0.7 for the period range of interest in this study. The
first row plots in Figure 10 show slightly higher standard deviations than those reported by BA07.
This observation suggests that the predictive model does not significantly amplify the uncertainty in
the estimated Sd,ie when it is associated with the PGV values computed from other GMPEs. The other
two rows in this figure show the variation in σTot when the scenario-based PGV is estimated by AB07.
Similar to the previous exercises, the empirical relationships of Beyer and Bommer [56, 57] were used
to fine-tune the standard deviations due to different horizontal component definitions. The second row
plots show the change in σTot for a small magnitude scenario event (M = 5.5). The third row plots
describe the same variation for a large magnitude event (M = 7.5). The comparisons of standard
deviations presented here and those reported in Akkar and Bommer [50] for elastic spectral
displacement predictions suggest once again that the proposed model does not severely amplify σTot of
Sd,ie estimations as a result of using the estimated PGV from AB07. A more important observation
depicted from these figures is that the magnitude dependency implemented in AB07 results in a
significant difference in the level of random error for the Sd,ie estimations. As far as the small
magnitude events (i.e. M = 5.5) are concerned, the use of AB07 for estimating the scenario PGV
results in about 30% increase in σTot when compared to the use of BA07. This ratio is reversed for
large magnitude events (i.e. M = 7.5) in the favor of AB07. The consideration of AB07 would
Revised Manuscript
21
approximately decrease σTot by 40% with respect to the use of BA07. Provided the fact that both AB07
and BA07 yield similar median PGV estimations [44] for a given scenario event large variations in the
standard deviations as a function of M may cast serious concerns about the description of random
error in the ground-motion model. Thus, as discussed by Bommer et al. [55], the magnitude influence
on the error propagation of the predicted variable should be studied further to clarify whether the
ground motion variability genuinely depends on magnitude.
5. SUMMARY AND CONCLUSIONS
The correlation between PGV and peak inelastic oscillator displacement is used to derive a simple
predictive model for estimating Sd,ie as a function of PGV. The model accounts for the 5%-damped
bilinear oscillator response between 0.2s ≤ T ≤ 2.0s associated with 0% and 5% postyield stiffness
ratios. It describes the variation of Sd,ie for constant ductility and normalized lateral strength ratios. The
regression analysis was conducted for a suite of dense-to-stiff soil site records with a magnitude
interval of 5.2 ≤ M ≤ 7.6. The records are selected from the close proximity of causative fault (Rjb < 30
km) and they do not exhibit pulse dominant signals. Confined to these limitations, this study presents
the results of regression analysis with a special emphasis on the model verification. The comparative
plots between the observed and estimated data as well as the residual analysis showed that, despite its
simplicity, the general performance of the model is appealing except for short periods where the model
may not fully explain the trends in the empirical data.
The case studies presented showed that the proposed model can properly address the effect of
important ground-motion parameters on the behavior of Sd,ie when the PGV values are associated
through a GMPE. The strong relationship between PGV and magnitude results in rationale Sd,ie
estimations that validate the magnitude influence on the spectral shapes. The estimated spectral
displacements start converging to constant values based on the variation of magnitude-dependent
spectral corner periods that reflect the genuine characteristic of the actual data. The empirical model
can also distinguish the differences in the nonlinear oscillator response imposed by constant ductility
and normalized lateral strength ratio. These observations suggest the adequacy of the proposed model
Revised Manuscript
22
in estimating Sd,ie. The case studies also illustrated that the total standard deviation about the estimated
Sd,ie is not amplified significantly when the proposed mode is associated with a GMPE. However, the
variation in σTot inherently depends on the random error description of the GMPE utilized (e.g.
magnitude dependency/independency of the standard deviation). In essence, the key features of the
model reported here not only make it practical for preliminary design and seismic performance
assessment of a broad class of building systems but also describe the model as a versatile tool for the
scalar PSHA methods.
ACKNOWLEDGEMENTS
The authors express their sincere gratitude to Prof. M.T. Yılmaz who gave valuable insight about the
regression and ANOVA analyses. Dr. David M. Boore kindly ran many simulations based on different
source models to discuss the behavior of regression coefficients with the first author. The valuable
comments of two anonymous reviewers improved the technical quality of the paper significantly. This
study is partially funded by the Scientific and Technological Research Council of Turkey under Award
No. 105G016.
Revised Manuscript
23
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Revised Manuscript
27
APPENDIX – A GROUND MOTION DATA SET
Table A.1. List of ground motions used and their important features
Earthquake
CO1
Aigion
Aigion
Alkion
Alkion
Alkion
Ano Liosia
Ano Liosia
Ano Liosia
Ano Liosia
Basso Tirreno
Campano Lucano
Campano Lucano
Cape Mendocino
Cape Mendocino
Cerkes
Chi-Chi
Chi-Chi
Chi-Chi
Coyote Lake
Coyote Lake
Coyote Lake
Dinar
Duzce
Duzce
Faial
Firuzabad
Friuli
Friuli
Friuli
Friuli
Friuli
Gazli
Imperial Valley
Imperial Valley
Imperial Valley
Imperial Valley
Imperial Valley
Imperial Valley
Imperial Valley
Imperial Valley
Imperial Valley
Imperial Valley
Imperial Valley
Imperial Valley
Ionian
Izmit
Izmit
Izmit
Izmit
Kalamata
Kalamata
Komilion
Komilion
Kozani
Landers
GR
GR
GR
GR
GR
GR
GR
GR
GR
IT
IT
IT
USA
USA
TR
TA
TA
TA
USA
USA
USA
TR
TR
TR
PO
IR
IT
IT
IT
IT
IT
UZ
USA
USA
USA
USA
USA
USA
USA
USA
USA
USA
USA
USA
GR
TR
TR
TR
TR
GR
GR
GR
GR
GR
USA
Revised Manuscript
Date
15/06/1995
15/06/1995
24/02/1981
24/02/1981
25/02/1981
07/09/1999
07/09/1999
07/09/1999
07/09/1999
15/04/1978
23/11/1980
23/11/1980
25/04/1992
25/04/1992
14/08/1996
20/09/1999
20/09/1999
20/09/1999
06/08/1979
06/08/1979
06/08/1979
01/10/1995
12/11/1999
12/11/1999
09/07/1998
20/06/1994
11/09/1976
15/09/1976
15/09/1976
15/09/1976
15/09/1976
17/05/1976
15/10/1979
15/10/1979
15/10/1979
15/10/1979
15/10/1979
15/10/1979
15/10/1979
15/10/1979
15/10/1979
15/10/1979
15/10/1979
15/10/1979
04/11/1973
17/08/1999
17/08/1999
13/09/1999
13/09/1999
13/09/1986
13/09/1986
25/02/1994
25/02/1994
19/05/1995
28/06/1992
Time
(UTC)
00:15:51
00:15:51
20:53:39
20:53:39
20:53:39
11:56:51
11:56:51
11:56:51
11:56:51
23:33:48
18:34:52
18:34:52
18:06:11
18:06.11
02:59:41
17:47:35
17:47:35
17:47:35
17:05:28
17:05:28
17:05:28
15:57:13
16:57:20
16:57:20
05:19:07
09:09.03
16:35:03
09:21:19
09:21:19
09:21:19
09:21:19
02:58:42
23:17:00
23:17:00
23:17:00
23:17:00
23:17:00
23:17:00
23:17:00
23:17:00
23:17:00
23:17:00
23:17:00
23:17:00
15:52:12
00:01:40
00:01:40
11:55:30
11:55:30
17:24:34
17:24:34
02:30:50
02:30:50
06:48:49
11:57:53
Station
PGV
(cm/s)
M
Rjb
(km)
Site2
Aigio-OTE Building
Amfissa-OTE Building
Korinthos-OTE Building
Xilokastro-OTE Building
Korinthos-OTE Building
Athens-Syntagma 1st lower level
Athens 3 Kallithea District
Athens-Sepolia Metro Station
Athens-Sepolia Garage
Patti-Cabina Prima
Calitri
Brienza
Rio Dell - 101/Painter St. Overseas
Petrolia
Merzifon Meteorology Station
TCU051
TCU082
CHY006
Gilroy Array #3 Sewage Treatment
SJB Overpass, Bent 3
Gilroy Array #2
Dinar Meteorology Station
LDEO Station No. C1062 FI
Bolu
Horta
Zanjiran
Buia
Breginj Fabrika IGLI
Forgaria Cornio
San Rocco
Buia
Karakyr Point
Parachute Test Facility,El Centro
Calexico Fire Station
Casa Flores, Mexicali
El Centro Array #10
Aeropuerto Mexicali
El Centro Array #2
El Centro Array #4
Dogwood Rd., Diff. Array, El Centro
Bonds Corner
McCabe School, El Centro Array #11
James Rd., El Centro Array #5
Borchard Ranch, El Centro Array #1
Lefkada OTE Building
Duzce Meteorology Station
Iznik Highway Patrol
Adapazari Kadin D. Cocuk B. Evi
Yarimca-Petkim
Kalamata Prefecture
Kalamata OTE Building
Lefkada OTE Building
Lefkada Hospital
Karpero Town Hall
Joshua Tree Fire Station
52.36
9.72
23.62
28.04
13.67
12.99
15.70
17.84
21.32
15.21
29.36
11.50
42.63
48.30
5.02
40.58
41.03
42.09
16.89
4.74
31.88
43.99
18.25
55.17
34.37
40.44
21.71
27.74
23.97
19.40
12.53
54.75
17.27
18.95
31.51
45.96
42.03
32.71
38.10
41.15
44.33
45.24
49.71
10.36
56.90
50.70
26.82
7.02
8.07
33.10
34.61
14.53
12.08
14.85
42.71
6.5
6.5
6.6
6.6
6.3
6.0
6.0
6.0
6.0
6.0
6.9
6.9
7.0
7.0
5.6
7.6
7.6
7.6
5.7
5.7
5.7
6.4
7.2
7.2
6.1
5.9
5.5
6.0
6.0
6.0
6.0
6.7
6.5
6.5
6.5
6.5
6.5
6.5
6.5
6.5
6.5
6.5
6.5
6.5
5.8
7.6
7.6
5.8
5.8
5.9
5.9
5.4
5.4
5.2
7.3
7.0
22.0
10.0
8.0
19.0
8.0
8.0
5.0
5.0
13.0
13.0
23.0
7.9
0.0
13.0
7.7
5.2
9.8
6.8
20.4
8.5
0.0
14.0
12.0
11.0
7.0
7.0
14.0
9.0
9.0
9.0
4.0
12.7
10.5
9.8
6.2
0.0
13.3
4.9
5.1
0.5
12.5
1.8
19.8
11.0
13.6
29.0
27.0
27.0
0.0
0.0
16.0
15.0
16.0
11.0
C
D
D
D
D
C
C
C
C
D
C
C
C
C
D
D
D
D
D
C
D
D
D
D
D
C
D
C
C
C
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
C
C
D
D
C
C
F3
N
N
N
N
N
N
N
N
N
S
N
N
R
R
S
R
R
R
S
S
S
N
S
S
S
S
R
R
R
R
R
R
S
S
S
S
S
S
S
S
S
S
S
S
R
S
S
S
S
N
N
S
S
N
S
f c4
(Hz)
0.08
0.09
0.10
0.07
0.06
0.10
0.14
0.16
0.15
0.15
0.10
0.15
0.07
0.07
0.16
0.04
0.04
0.04
0.25
0.20
0.25
0.09
0.05
0.05
0.18
0.13
0.20
0.14
0.15
0.15
0.15
0.20
0.07
0.07
0.10
0.07
0.15
0.07
0.07
0.07
0.10
0.07
0.07
0.07
0.10
0.10
0.04
0.08
0.08
0.15
0.05
0.20
0.20
0.13
0.07
28
Lazio Abruzzo
Livermore
Livermore
Loma Prieta
Loma Prieta
Loma Prieta
Loma Prieta
Loma Prieta
Loma Prieta
Manesion
Montenegro
Montenegro
Montenegro
Montenegro
Montenegro
Morgan Hill
Morgan Hill
Morgan Hill
Morgan Hill
Morgan Hill
Morgan Hill
North Palm Springs
Northridge
Northridge
Northridge
Northridge
Northridge
Northridge
Northridge
Parkfield
Preveza
Pyrgos
Racha
Racha
Sicilia-Orientale
South Iceland
South Iceland
South Iceland
South Iceland
Spitak
Umbria Marche
Umbria Marche
Umbria Marche
Umbria Marche
Umbria Marche
Umbria Marche
Volvi
Whittier Narrows
Whittier Narrows
Whittier Narrows
IT
USA
USA
USA
USA
USA
USA
USA
USA
GR
MN
MN
MN
MN
MN
USA
USA
USA
USA
USA
USA
USA
USA
USA
USA
USA
USA
USA
USA
USA
GR
GR
GRG
GRG
IT
IC
IC
IC
IC
AR
IT
IT
IT
IT
IT
IT
GR
USA
USA
USA
07/05/1984
01/24/1980
01/24/1980
18/10/1989
18/10/1989
18/10/1989
18/10/1989
18/10/1989
18/10/1989
07/06/1989
15/04/1979
15/04/1979
15/04/1979
24/05/1979
24/05/1979
24/04/1984
24/04/1984
24/04/1984
24/04/1984
24/04/1984
24/04/1984
07/08/1986
17/01/1994
17/01/1994
17/01/1994
17/01/1994
17/01/1994
17/01/1994
17/01/1994
27/06/1966
10/03/1981
26/03/1993
03/05/1991
03/05/1991
13/12/1990
17/06/2000
17/06/2000
21/06/2000
21/06/2000
07/12/1988
26/09/1997
26/09/1997
26/09/1997
26/09/1997
12/10/1997
06/10/1997
20/06/1978
10/01/1987
10/01/1987
10/01/1987
17:49:42
19:00
02:33
00:04:21
00:04:21
00:04:21
00:04:21
00:04:21
00:05:21
19:45:54
06:19:41
06:19:41
06:19:41
06:19:41
06:19:41
21:15:28
21:15:28
21:15:28
21:15:28
21:15:28
21:15:28
09:20
12:31:03
12:31:03
12:31:03
12:31:03
12:31:03
12:31:03
12:31:03
04:26
15:16:20
11:58:15
20:19:39
20:19:39
00:24:26
00:51:48
00:51:48
00:51:48
00:51:48
07:41:24
00:13:16
00:13:16
09:40:30
09:40:30
11:08:36
23:24:00
20:03:22
14:42
14:42
14:42
Cassino Sant Elia
Livermore VA Hospital
Morgan Territory Park
Gilroy #6 San Ysidoro
Gilroy Gavilan Coll
Gilroy #2 – Hwy 101/Bolsa Rd
Gilroy #3 - Gilroy Sewage Plant
Saratoga 1-Story School Gym
Corralitos Eureka Canyon Rd
Patra OTE Building
Petrovac Hotel Oliva
Bar Skupstina Opstine
Ulcinj Hotel Olimpic
Bar Skupstina Opstine
Budva PTT
Gilroy Gavilan College
Gilroy #2
Halls Valley
Gilroy #7
Gilroy #6
Gilroy #3
Fun Valley
Los Angeles - UCLA Grounds
6850 Coldwater Canyon Ave., N. Hollywood
Brentwood V.A. Hosp.
Pacoima Kagel Canyon
14145 Mulholland Dr., Beverly Hills
17645 Saticoy St.
7769 Topanga Canyon Blvd.,Canoga Park
Cholame, Shandon, Array #5
Lefkada OTE Building
Pyrgos Agriculture Bank
Ambrolauri
Oni Base Camp
Catania Piana
Hella
Selsund
Solheimar
Kaldarholt
Gukasian
Colfiorito
Castelnuovo Assisi
Castelnuovo Assisi
Gubbio Piana
Foligno Santa Maria Infraportas Base
Castelnuovo Assisi
Thessaloniki City Hotel
200 S. Flower, Brea
7420 Jaboneria,Bell Gardens
Los Angeles Obregon Park
11.12
17.39
11.04
13.92
28.93
39.23
34.48
37.19
55.20
2.34
39.96
52.81
51.70
16.54
27.73
3.39
4.99
39.57
5.76
11.26
11.88
6.12
21.88
23.07
24.01
50.88
57.94
59.82
59.84
25.44
5.94
19.02
25.10
1.97
10.78
55.26
22.06
40.95
26.62
30.09
23.01
6.44
13.06
17.72
1.07
7.55
16.08
7.07
28.00
21.78
5.9
5.5
5.8
6.9
6.9
6.9
6.9
6.9
6.9
5.2
6.9
6.9
6.9
6.2
6.2
6.1
6.1
6.1
6.1
6.1
6.1
6.2
6.7
6.7
6.7
6.7
6.7
6.7
6.7
6.1
5.4
5.4
5.6
5.6
5.6
6.5
6.5
6.4
6.4
6.7
5.7
5.7
6.0
6.0
5.2
5.5
6.2
6.1
6.1
6.1
18.0
NI
10.3
17.9
9.2
10.4
12.2
8.5
0.2
24.0
3.0
3.0
13.0
15.0
10.0
14.8
13.7
3.5
12.1
9.9
13.0
12.8
13.8
7.9
12.9
5.3
9.4
0.0
0.0
9.6
21.0
10.0
11.0
17.0
24.0
5.0
20.0
4.0
12.0
20.0
3.0
24.0
23.0
30.0
20.0
20.0
13.0
18.4
10.3
4.5
D
D
C
C
C
D
D
C
C
D
C
C
C
C
C
C
D
D
D
C
D
D
C
C
C
C
D
D
D
D
D
D
D
D
D
C
C
C
C
D
C
D
D
D
D
D
D
D
D
D
N
S
S
RO
RO
RO
RO
RO
RO
S
R
R
R
R
R
S
S
S
S
S
S
S
R
R
R
R
R
R
R
S
R
S
R
R
S
S
S
S
S
R
N
N
N
N
N
N
N
R
R
R
0.17
0.20
0.60
0.16
0.16
0.16
0.16
0.50
0.10
0.17
0.11
0.10
0.10
0.09
0.07
0.40
0.16
0.16
0.30
0.20
0.20
0.20
0.16
0.09
0.10
0.14
0.13
0.07
0.16
0.07
0.14
0.15
0.15
0.22
0.18
0.08
0.17
0.09
0.10
0.06
0.17
0.15
0.10
0.10
0.13
0.12
0.13
0.16
0.10
0.40
1
CO is used to abbreviate countries Armenia (AR), Georgia (GRG), Greece (GR), Iceland (IC), Iran (IR), Italy (IT), Montenegro (MN), Portugal (PO), Turkey
(TR), the United States of America (USA), Uzbekistan (UZ) and Taiwan (TA). 2 S designates the faulting style. 3 F designates the style-of-faulting. 4 fc abbreviates
the low-cut filter frequency.
Revised Manuscript
29