usgs award number: g17ap00103 project duration: 12 …
TRANSCRIPT
USGS AWARD NUMBER: G17AP00103
PROJECT DURATION: 12 MONTHS (09/01/2017 THROUGH 08/31/2018)
TECHNICAL FINAL REPORT
A New Model for Vertical to Horizontal Response Spectral Ratios for Central and Eastern
North America
Shahram Pezeshk1, Ali Farhadi2, and Alireza Haji-Soltani3
1Department of Civil Engineering
The University of Memphis
Memphis, TN 38152
Phone: (901) 678-4727
Fax: (901) 678-3026
2Department of Civil Engineering
The University of Memphis
Memphis, TN 38152
3Mueser Rutledge Consulting Engineers
14 Penn Plaza, 225 West 34th Street
New York, NY 10122
Phone: (646) 581-2232
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Abstract
It is a well-known fact that critical structures are required to be designed for the vertical effects of ground
motions as well as the horizontal effects. We developed a new model for the spectral ratio of vertical to
horizontal components of earthquakes (V/H ratio) for Central and Eastern North America (CENA). The
proposed V/H ratio model has the advantage of considering the earthquake magnitude, source to site
distance, and the shear-wave velocity of soil deposits in the upper 30m of the site for PGA and a wide
range of periods (0.001 to 10.0 seconds). The model evaluation is based on a comprehensive set of
regression analysis of the newly compiled Next Generation Attenuation (NGA-East) database of available
CENA recordings with M ≥ 3.4 and RRUP < 1000 km. The median value of the geometric mean of the
orthogonal horizontal motions rotated through all possible nonredundant rotation angles, known as the
GMRotD50 (Boore et al., 2006), is used along with the vertical component to perform regression using
the nonlinear mixed effect regression. We excluded the earthquakes and recording stations in the Gulf
Coast region due to their different ground-motion attenuation (Dreiling et al., 2014). To compute V/H
ratios for the Gulf Coast region, we refer the readers to the study of Haji-Soltani et al. (2017) as an
independent study performed on the Gulf Coast data. Moreover, we excluded the NEHRP site class E
(soft-soil) sites from consideration because of their complex site-response characteristics and their
potential for nonlinear site effects. Finally, we compared the predicted ratios from the proposed model
with recently published V/H ratio models. We suggest our model be used for developing the vertical
response spectra for CENA sites.
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Table of Contents
List of Figures…...…………………………………….……………………………….............. iii
List of Tables…...……………………………………….………………………………............. v
Introduction……………………………………………….……………………………….......... 1
Database…………………………………………………...……………………………….......... 5
Methodology……………………………………………………………………………….......... 6
Parametric Model………………………………………………...……………………….......... 6
Source Term……………………………………………………………………………….......... 7
Path Term………………………………………………………………………………….......... 8
Style of Faulting Term………………………………………………….………………............. 9
Site Term….……………………………………………………………………………............. 10
Residual Analysis…………………………………………………………………………........ 11
Comparison with Other Studies…………………………………………………………........ 12
Summary…….……………………………………………………………………………......... 15
References…….…………………………………………………………………………........... 15
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List of Figures
Figure 1. Proposed V/H model by McGuire et al. (2001) for hard rock spectrum and FEMA P-
750 (2009) model plotted for site class B, SS=0.5g, and S1=0.3g.………………........................ 19
Figure 2. (Left): CENA recording stations employed in the current study. The colors of the
symbols represent the NEHRP site class of the station. (Right): CENA earthquakes considered in
this study…………………………………………………….…………………………….......... 19
Figure 3. Number of usable records for each spectral period…………………………….......... 20
Figure 4. The magnitude–distance distribution of the selected CENA ground-motion recordings
for PGA………….………………………………………………………………………............ 21
Figure 5. The distribution of magnitude data versus VS30. Sites are classified into four groups
based on the NEHRP site classification scheme………………...………………………............ 22
Figure 6. The distribution V/H versus magnitude in four representative periods. The mean and
the standard deviation of error bars are calculated using magnitude bins of ±0.10 centered at the
marker ……………………………………………………………….……………………......... 23
Figure 7. Comparison of V/H for different styles of faulting. V/H ratios are plotted for M 5.0
and rupture distance of 50 km considering the VS30 equal to 270 m/s……………...………....... 24
Figure 8. The distribution of magnitude data versus VS30 for various faulting mechanisms........ 25
Figure 9. The distribution of between-event residuals versus magnitude for the Proposed Model
at four representative periods…………………………………………………………................ 26
Figure 10. The distribution of within-event residuals verses closest distance to the rupture area
for the Proposed Model at four representative periods. The mean and the standard deviation of
error bars are calculated by logarithmically spacing the distance range into 15 bins……........... 27
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Figure 11. The distribution of within-event residuals verses shear-wave velocity on the top 30
meters for the Proposed Model at four representative periods. The mean and the standard
deviation of error bars are calculated using VS30 bins of ±100 m/s centered at the marker.......... 28
Figure 12. V/H ratios computed from Proposed Model in comparison with the SP17 (Sedaghati
and Pezeshk, 2017), BAK11 (Bommer et al., 2011), GA11 (Gülerce and Abrahamson, 2011) and
HPMZ17 (Hahi-Soltani et al., 2017) models using M 5.0 and RRup = 30 km for soil (VS30 = 270
m/s) and rock (VS30 = 760 m/s) site conditions………………..………………..…………......... 28
Figure 12. V/H ratios computed from Proposed Model in comparison with the SP17 (Sedaghati
and Pezeshk, 2017), BAK11 (Bommer et al., 2011), GA11 (Gülerce and Abrahamson, 2011) and
HPMZ17 (Hahi-Soltani et al., 2017) models using M 6.0 and RRup = 30 km for soil (VS30 = 270
m/s) and rock (VS30 = 760 m/s) site conditions………………..………………..…………......... 29
v
List of Tables
Table 1. List of events used in the current study ...….………………………………...….......... 30
Table 2. Regression coefficients of the proposed vertical-to-horizontal (V/H) response spectral
ratio model….…………………………………………….……………………………….......... 32
Table 3. Distinctness table for the candidate models for peak ground acceleration (PGA).
Distinctness index of each pairwise comparison (based on 100 bootstrap samples) given in the
intersecting box of a model pair. The whole dataset is considered to compute LLH scores in the
second last column.…………………………………………………. …………………............. 33
Table 4. Models ranking based on the LLH scores in four representative periods ……............. 33
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Introduction
Seismic codes require the effects of the vertical component of ground motions to be considered in the
analysis and design of critical structures. The effects of the vertical component of ground motions on
structures have been addressed in different studies, such as Kunnath et al. (2008) and Bozorgnia and
Campbell (2004). In the same way as the horizontal design, the vertical design of critical structures
requires a vertical design spectrum, which reflects the main seismological characteristic of the region.
There are basically two main approaches in developing a vertical design spectrum for a study site
(Gülerce and Abrahamson, 2011; Bommer et al., 2011). The first approach is to perform a Probabilistic
Seismic Hazard Analysis (PSHA) using vertical Ground Motion Prediction Equations (GMPEs) to
estimate the vertical ground motion hazard. The main disadvantage of this approach is the absence of
up-to-date vertical GMPEs in most regions such as Central and Eastern North America (CENA). Another
problem with this approach is a possible mismatch between the horizontal and vertical controlling
earthquakes. The second method is to use the V/H ratio of the ground motion to scale an available
horizontal design spectrum to a vertical spectrum.
The vertical and horizontal spectral ratios of ground motions have been studied in two methods: ratio of
Fourier spectra and ratio of response spectra of vertical to horizontal components of earthquakes.
Nakamura (1989) proposed the horizontal-to-vertical (H/V) Fourier spectral ratio technique to estimate
the dynamic properties of soil layers. He used the H/V Fourier spectral ratio of the micro tremors to
estimate the soil’s transfer function along about a 1500 km section of the Japan railway lines. Lermo and
Chávez-García (1993) extended the H/V ratio technique to strong motions. They estimated the empirical
transfer function without the reference station and concluded that if site effects are caused by geology, an
estimate of the first dominant period of the site and the local amplification can be obtained using the ratio
of horizontal to vertical spectral Fourier of records of only one station. Zandieh and Pezeshk (2011)
studied the Fourier domain H/V spectral ratios in the New Madrid seismic zone (NMSZ) to capture the
amplification effect of local soil sediments on earthquake ground motion. They used 500 broadband
seismograms from 63 events of magnitude Mw 2.5 to 5.2, recorded by broadband stations in the
Mississippi embayment. They compared the H/V ratios with the theoretical quarter-wavelength
approximation and showed that H/V ratios could be a first estimate of site amplifications. Niazi and
Bozorgnia (1992) studied a large number of V/H response spectra of the recorded earthquakes available at
the Taiwan strong motion array. They suggested the peak of the V/H ratio exceeds a value of 2/3 in the
near source regions. They also studied V/H ratio of Loma Prieta and the Northridge earthquakes for both
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soil and rock sites and suggested that the general pattern of the V/H ratio, such as having a distinct peak at
low spectral periods and the value of the peak, which they suggested to be 2/3, is universal. Their study
indicated that at longer periods, the V/H ratio increases slowly. Bozorgnia et al. (1995) observed that the
V/H ratio is a function of spectral period, distance to the fault, and earthquake magnitude. Furthermore,
they observed that the V/H ratio gets the largest value at short periods in near-field regions. In the near-
field region, the V/H ratio at the short period is larger than the V/H ratio at the peak ground accelerations.
In the short period range, the 2/3 value underestimates the V/H ratio, especially in near-field regions, and
at long periods the V/H ratio falls below 2/3.
Atkinson (1993) used small magnitude earthquakes recorded at distances beyond 20 km on rock site
conditions to develop an empirical model of the V/H ratio for the Central and Eastern United States
(CEUS). Atkinson (1993) studied the V/H ratio of the Fourier amplitude for the rock site conditions. For
the Saguenay earthquake the V/H ratio is between 0.7 and 1.0, suggesting a higher ratio in comparison
with the Western United States (WUS) at large distances. According to Atkinson (1993), the general
pattern of the V/H ratio in the 1.0 Hz to 10.0 Hz frequency range has an opposite trend compared to the
WUS. Atkinson (1993) also concluded that the magnitude dependency of the V/H ratio model for the
CEUS is smaller compared to the WUS.
Seismic codes suggested a variety of V/H models to obtain the vertical design spectrum. Regulatory
Guide 1.6 (1973) is among the first seismic codes that consider obtaining the vertical design spectrum
from the horizontal spectrum using the V/H ratio model. Regulatory Guide 1.6 assumes different values
of V/H for short and long periods. The vertical design response spectra values recommend by Regulatory
Guide 1.6 are 2/3 those of the horizontal design response spectra for frequencies less than 0.25 Hz. For
frequencies larger than 3.5 Hz they are the same, while the ratio varies between 2/3 and 1 for frequencies
between 0.25 Hz and 3.5 Hz. McGuire et al. (2001) studied the V/H ratio for rock site conditions in the
WUS and the CEUS to update the Regulatory Guide 1.6 values for the V/H ratio. For rock site conditions
in the WUS, they used three empirical GMPEs that included vertical motions: Abrahamson and Silva
(1997); Campbell (1997); and Sadigh et al., (1997; 1993). To develop V/H ratios for the WUS rock site
conditions, median V to median H ratios for strike slip mechanisms were produced for each relation and
averaged assuming equal weights. The V/H ratios were magnitude and distance dependent and were
recommended for a range of expected horizontal peak ground accelerations (PGAs) to accommodate the
magnitude and distance dependency. The range of expected horizontal peak accelerations were PGA ≤
0.2g; 0.2g < PGA ≤ 0.5g; and PGA ≥ 0.5g. For very hard rock conditions in CEUS, very few recordings
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were available for earthquakes with magnitudes greater than 5. They studied recordings of the Saguenay
event and the only three available recordings of the Nahanni and Gazli earthquakes. The very few data in
CEUS indicated very large V/H ratios may be likely at very close rupture distances to the CEUS
earthquakes. Data from more distant events suggested higher V/H ratios for the CEUS rock sites than the
WUS sites. Data showed that ratios at near-fault sites can be quite large at high frequencies. To develop
recommended values for applications for the CEUS, the simple point-source model was extended to
consider P-SV waves. Using the modified point-source model, they were able to predict the general trend
in V/H ratios at rock sites from the 1989 magnitude 6.9 Loma Prieta earthquake in WUS. The point-
source model predicted a peak in the ratios near 60 Hz, is associated with the vertical spectra and
corresponded to the peak in the WUS ratios but shifted from about 15 to 20 Hz to about 60 Hz. The
magnitude dependencies in the CEUS V/H ratios were smaller than for the WUS, probably because the
WUS models did not include magnitude saturation. The model showed higher ratios at low frequencies
< 3 Hz than the WUS ratios, consistent with available observations. Based on the trends model
predictions as well as the CEUS recordings, the recommended V/H ratios for rock sites in CEUS were
developed by shifting the WUS ratios to higher frequencies, so that the peaks correspond to about 60 Hz.
Also, the low frequency WUS levels were scaled up by about 50%. The recommended V/H ratios for
CEUS were provided for the same horizontal PGA as for WUS ratios. The CEUS and WUS V/H ratios
recommend by McGuire et al. (2001) are shown in Figure 1.
Figure 1 also shows the resulting V/H model from the recommended vertical design spectrum by the
FEMA P-750 (2009). FEMA P-750 recommends a vertical design spectrum which is based on Site Class,
SDS (design earthquake spectral response acceleration parameter at short period), and SS (the mapped
maximum credible earthquake spectral response parameter at short period). The vertical design spectrum
suggested by FEMA P-750 is mainly based on the results of the study by Bozorgnia and Campbell
(2004).
Gülerce and Abrahamson (2011) developed GMPEs to predict the V/H ratio. They reviewed methods for
constructing the site-specific vertical design spectra from a Uniform Hazard Spectrum (UHS) and
Conditional Mean Spectrum (CMS) to be consistent with PSHA. The functional form of the GMPEs for
the V/H ratio is consistent with the horizontal GMPEs developed by Abrahamson and Silva (2008). They
used the Pacific Earthquake Engineering Research Center Next Generation of Ground Motion Attenuation
Phase 1 Project (PEER NGA-West1) database selected by Abrahamson and Silva (2008), which consists
of 2684 sets of recordings from 127 earthquakes. Their functional form to predict the V/H ratio is
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dependent on the earthquake magnitude, the source to site distance, and the type of faulting. They also
included the functional form developed by Walling et al. (2008) to predict the effects of the nonlinear soil
behavior in the V/H ratio model.
Bommer et al. (2011) reviewed V/H model ratios by different seismic codes and regulations such as the
Regulatory Guide 1.6, McGuire et al. (2001), the EC8 (Eurocode 2004), and the National Earthquake
Hazards Reduction Program (NEHRP, 2009). They also developed a model for the V/H ratio for Europe
and the Middle East. They found a simple functional form, expressing the V/H ratios as a function of
magnitude, the style of faulting (reverse, normal, and strike-slip), the source to site distance, and the site
class to appropriately describe the V/H model. Their proposed model is based on 1296 accelerograms
from 392 events occurring in Europe, the Middle East and surrounding regions, and predicts V/H ratios
for PGA and spectral accelerations from 0.02 to 3.0 seconds. Although their model predicts lower values
for V/H ratio, it has a general agreement with the model developed by Gülerce and Abrahamson (2011),
which is based on the data from Western North America. Their model can be used for a magnitude range
of 4.5 to 7.6 and distances up to 100 km.
The PEER NGA-West2 project was an extension of PEER NGA-West1 (Bozorgnia et al., 2014). A sub-
project in NGA-West2 was the development of GMPEs for vertical ground motions. A database of 5%
damped vertical PGAs and pseudo spectral accelerations (PSA) from shallow crustal earthquakes in
active tectonic regimes was compiled. Each GMPEs developer team, based on their selection criteria,
selected a subset of the vertical ground-motion data to develop their GMPEs for the vertical component.
The vertical models in PEER (2013) are developed by the teams of: Gülerce et al. (2013) [hereafter
GKAS13]; Stewart et al. (2013) [hereafter SSBA13]; Bozorgnia and Campbell (2013) [hereafter BC13];
and Chiou and Youngs (2013) [hereafter CY13]. Bozorgnia and Campbell (2016) studied the V/H ratios
obtained using the vertical model of Bozorgnia and Campbell (2015) and the horizontal GMPE of
Campbell and Bozorgnia (2014) developed in PEER NGA-West2 project. The functional forms used for
the vertical GMPEs are similar to those used by GMPE developers for the horizontal ground motions.
Akkar et al. (2014) [hereafter ASA14] developed a V/H ratios model for shallow active crustal regions in
Europe and the Middle East. Their V/H ratios model uses a functional form similar to that of Akkar et al.
(2013a, b) horizontal GMPEs; and therefore, produces compatible vertical ground motions. Their model
considers the style of faulting and includes a site response term, which uses VS30 parameter representing
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the site condition. The site response is evaluated with respect to a reference site condition of VS30 =
750 m/s.
In the present study, we developed a new model to estimate the V/H ratios for CENA. The resulting V/H
ratios model can be used in developing a site-specific vertical design spectrum for the study area from a
horizontal design spectrum. We selected several of trial functional forms to determine the best model. We
compared the V/H ratios based on this study with the V/H ratios proposed by other studies.
Database
To perform V/H ratio analysis, earthquakes are selected from the NGA-East database (Goulet et al.,
2014). The NGA-East database includes ground-motion recordings from selected events with moment
magnitude M greater than 2.5 for distances up to 1,500 km recorded in CENA region since 1988. The
database contains records from 81 earthquake events and 1379 recording stations. In this study, we used
the NGA-East database of available CENA recordings with M ≥ 3.4 and RRUP < 1000 km. We included
both potentially induced events (PIEs) and tectonic earthquakes in our generating database. The
GMRotD50 of horizontal components of motions in the database is used in this study. We excluded
earthquakes and recording stations in the Gulf Coast region, which have been shown to exhibit
significantly different ground-motion attenuation because of the thick sediments in the region (Dreiling et
al., 2014). Haji-Soltani et al. (2017) performed an independent study on the Gulf Coast data and
suggested a new model for predicting V/H ratios in the Gulf Coast Region. Figure 2 (left) displays a map
of the recording stations with different colors representing their NEHRP site class and Figure 2 (right)
displays a map of the associated earthquakes. We excluded the NEHRP site class E (soft-soil) sites from
consideration because of their complex site-response characteristics and their potential for significant
nonlinear site effects. Therefore, the minimum VS30 in the database is 180 m/s. As can be observed from
Figure 2, the selected recordings were obtained on a variety of site conditions.
The number of ground motion records that can be used in regression analysis are different for various
spectral periods. This is because the NGA-East database developers have suggested a usable frequency
range for each of the vertical and the horizontal recordings. Following Gülerce and Abrahamson (2011),
we imposed the most restrictive criteria on the selection of the minimum usable frequency. In other
words, for each record, we considered the maximum of the minimum usable frequencies suggested for the
vertical and the horizontal components. Figure 3 shows the number of usable records versus period. As is
clear from Figure 3, the size of the generating database is reduced sharply for larger spectral periods.
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For PGA, the generating database is composed of 3829 ground motion records from 50 events with
magnitudes ranging from 3.42 to 5.85 and rupture distances up to 1000 km. These events were recorded
by 725 stations with VS30 between 215 and 2000 m/s. Table 1 gives a list of 50 earthquakes used for the
regression analysis. This table also provides some information about the date, the magnitude, the number
of records per event, and the geographical location of these events.
Figure 4 shows the distribution of magnitude data versus rupture distance for PGA. Based on Figure 4,
the database is sparse in short distances. We have relatively no data for moderate-to-large magnitude
events in short rupture distances. However, the database is sufficient for rupture distances more than 100
km. Figure 5 illustrates the distribution of the magnitude data versus VS30. In Figure 5, we grouped ground
motion records into four National Earthquake Hazard Reduction Program (NEHRP) site classes (class A:
VS30 ≥ 1500 m/s, class B: 760 ≤ VS30 ≤ 1500 m/s, class C: 360 ≤ VS30 ≤ 760 m/s, class D: 180 ≤ VS30 ≤ 360
m/s). According to this figure, the number of records from group A are less than other groups and there is
no record representing group E.
Methodology
The proposed model for the V/H ratio for CENA is developed for PGA and spectral periods of 0.01 to
10.0 seconds. The V/H values are determined using the same periods used by Pezeshk et al. (2018). We
calculated the 5% damped response spectra of vertical and GMRotD50 of horizontal components of
motions in the NGA-East database to develop a V/H ratios model. We used a two-step regression
approach. In the first step we performed the mixed-effects regression analysis to obtain the source and
path effects coefficients. Then, we obtained coefficients of the site-effects term by performing another
mixed-effects regression analysis considering the within-event residuals obtained from the first stage. To
arrive at a smooth model, we followed the procedure outlined by GKAS13. It is important to assure
smooth spectra and to constrain the model when data is limited. The sufficiency of the proposed V/H
ratio model is verified through the analysis of intra-event and inter-event residuals.
Parametric Model
Using the results of V/H ratio analysis from actual earthquakes, a parametric model is proposed to
estimate the V/H values within the study region. We used the moment magnitude, the rupture distance,
and the shear-wave velocity in the upper 30m as input parameters for the proposed functional form. We
explored several trial functional forms prior to the final regression analysis. We considered the ratios of
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vertical to horizontal component of motions (V/H) to be a function of four factors, including the source
term, the path term, the style of faulting, and the site effects. We made this assumption in accordance with
previous studies such as Haji-Soltani et al. (2017) and Pezeshk et al. (2018). Therefore, we considered the
median values of V/H ratios to be modeled as a function of these terms as indicated in Equation 1:
ln (V
H)= f
source+f
path+f
sof+f
site+Δes (1)
where fsourse, fpath, fsof, and fsite represent the source term, the path term, the style of faulting, and the site
effects, respectively. Moreover, Δes represents the variations around the median prediction of the
logarithm of the V/H ratios. We continue this section by describing each factor in more detail.
Source Term
To describe the source term, we investigated four functional forms from a complex model to a simple
form. We begin with a relatively complex functional form proposed by Haji-Soltani et al. (2017) for
predicting the V/H ratios in the Gulf Coast region. This model as our base functional form is presented as:
fsource
= { a1+a2(M-Mh)+a3(M-Mh)
2 for M≤Mh
a1+a4(M-Mh) for M>Mh
(2)
where Mh is hinge magnitude fixed at M 4.0. Coefficients a1 to a4 are fixed effect coefficients and were
determined from the regression. The model of Haji-Soltani et al. (2017) considers a second-order
magnitude term to account for the saturation effects, and divides the data into separate groups using a
single hinge magnitude. Based on our evaluations, the statistical p-values for the coefficients related to
hinge magnitude and the quadratic magnitude form were significantly large. In this step, we cannot mute
both coefficients at the same time since one of them may be statistically significant if we mute the other.
Therefore, we muted the second order magnitude term and evaluated another functional form with a term
for the hinge magnitude. The new form is:
fsource
= {a1+a2(M-Mh) for M≤Mh
a1+a4(M-Mh) for M>Mh (3)
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Again, the statistical p-values revealed that it is unnecessary to consider a separate term for the hinge
magnitude and to divide the magnitude data into separate groups. This is also confirmed in Figure 6
which represents the distribution of V/H ratios versus magnitude for various spectral periods. As shown
in Figure 6, the database is binned in various magnitude ranges, and error bars show the mean plus/minus
one standard deviation for each bin. According to Figure 6, we are unable to discern a specific trend from
the data. Therefore, it may not be necessary to perform the regression on different magnitude bins.
Now, we try another functional form with the term related to the hinge magnitude muted. This functional
form was used by Soghrat and Ziyaeifar (2016) to describe the source term for estimating V/H ratios
within the Iranian plateau. The third trial functional form for describing the source term is presented as:
fsource
= a1+a2M+a2M 2 (4)
By repeating our evaluations, we reached the conclusion that the term related to the quadratic magnitude
form is not statistically significant either. Therefore, it is unnecessary to make the source term relatively
complex by adding a term related to the hinge magnitude or considering a quadratic form. In other words,
the following simple linear model can sufficiently describe the V/H ratios with respect to magnitude.
fsource
= a1+a2M (5)
Equation (5) is the form used in Bommer et al. (2011) to describe the source term for the prediction of
V/H ratios in Europe and the Middle East.
Path Term
It is well known that ground motion diminishes in amplitude with distance from the seismic source. The
attenuation of ground motion is influenced by the large-scale crustal velocity structure of earth, anelastic
absorption of seismic energy, and scattering of seismic waves. The path effect is divided into two parts,
geometric spreading and anelastic attenuation (Zandieh and Pezeshk, 2010; Haji-Soltani et al., 2017;
Sedaghati and Pezeshk, 2017; Nazemi et al., 2017). The geometric spreading involves amplitude loss due
to the expanding surface area of the wave front as the distance from the source increases. The anelastic
attenuation term involves the conversion of the elastic wave energy to heat and is frequency-dependent.
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Regarding the path term, we initially considered a model with a term for the anelastic attenuation.
However, we found the coefficient related to anelastic attenuation statistically insignificant in all spectral
periods and decided to eliminate this term and represent the path term as follows:
fpath
= a3M ln(√RRup 2 +h
2) (6)
The form considered for describing the path term was also adopted by Haji-Soltani et al. (2017) to
describe the path effect for the V/H ratios in the Gulf Coast region. In Equation (6), the regression
coefficient a3 is a fixed effect coefficient and h is the hypothetical depth coefficient, assumed to be 6.0 km
following Pezeshk et al. (2018). Moreover, RRup is the closest distance to the fault ruptured area.
Style of Faulting Term
Style of faulting can be modeled using dummy variables in the regression analysis. According to the
database, events with normal mechanisms and unknown style of faulting are too scarce to be modeled
independently. We may consider these events as either strike-slip or reverse faulting earthquakes. We
considered them as earthquakes with reverse faulting mechanisms to increase the number of dip slip
events. This assumption will not significantly affect the result. In fact, we will get the same model if we
categorize events with normal mechanisms and unknown style of faulting as strike-slip earthquakes.
Finally, we modeled the style of faulting by a single term, FS. FS is set to be 1.0 for strike-slip events and
zero for other faulting mechanisms.
fsof
= a5FS (7)
Figure 7 shows the V/H ratios versus period for the two styles of faulting. According to this figure, V/H
ratios for reverse mechanisms are significantly larger than those of strike-slip faults. The difference is
unreasonably large, especially for periods greater than 1.0 second. The reason for this might be the
reduced number of records with reverse mechanisms at longer periods. However, we noticed that only a
small number of the records with strike-slip mechanism were recorded on rock. In other words, by
dividing the data into two groups of mechanisms we categorized them as events recorded either on rock
or soil. This is obvious from Figure 8 which shows the distribution of magnitude data versus VS30 for
various styles of faulting. As can be observed from Figure 2, the majority of strike-slip events were
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recorded by stations with the VS30 up to 760 m/s. In fact, the difference between the predicted values of
V/H ratios for these mechanisms is not really because of the style of faulting but is due to site effects.
Moreover, the style of faulting in the NGA-East database does not seem to be reliable and exact.
Therefore, we decided not to include a term for the style of faulting in our final functional form.
Site Term
The parameter VS30 is the most commonly used parameter for describing the site effects for soil deposits.
During recent years, ground motion model developers used this parameter extensively as an important
seismological factor to describe the local site effects (Sedaghati et al., 2018). Moreover, recently
published building codes classify soil types mainly based on the value of VS30. The site term for the V/H
ratio depends on the amplification of the horizontal component and the amplification of the vertical
component. The largest event in our database has M 5.8, and we have few observations at close distances
(≤50 km). Consequently, the number of records that might potentially include any nonlinear effects is
negligible. Due to this fact, Haji-Soltani et al. (2017) assumed a linear model for the site effects of V/H
ratios in the Gulf Coast region. In this study, we followed the procedure of previous studies and
considered the following simple linear functional form to define the site term:
fsite
= a4l n(VS30
VRef
)+a5 (8)
We used a two-step regression approach. In the first step we found coefficients a1, a2, and a3. Then, in the
second step, we determined a4 and a5. The considered site term is also similar to the site term used by
Sedaghati and Pezeshk (2107).
To sum up, we suggest the following functional form as an optimal model for predicting V/H ratios in
CENA:
ln (V
H)= a1+a2M+a3M ln(√RRup
2 +h2)+a4l n(
VS30
VRef
)+a5+Δes (9)
Coefficients a1 to a5 are given in Table 2. Clearly, the proposed model has a relatively simple functional
form compared to those developed for the horizontal component. However, this simple functional form
for predicting V/H ratios is well justified. Bommer et al. (2011) have also reported the same result for
11
Europe and Middle East. They started with a complex form compatible with models developed for the
horizontal component but ended up with a simple form for prediction of V/H ratios. The term Δes
represents the variations around the median predictions of the V/H ratios and can be decomposed into
between-event variability (ΔBe) and within-event variability (ΔWes). The between-event and within-event
variabilities have standard deviations of τ and φ, respectively. Furthermore, the within-event standard
deviation can be decomposed into a standard deviation of the site-to-site residuals (φS2S) and single-station
within-event standard deviation (φSS). Therefore, the standard deviation of the total variation (Δes) can be
obtained by combining τ and ϕ as follows:
σ=√τ 2+φS2S 2 +φ
SS
2 (10)
The standard deviation represented in Equation (10) is known as ergodic sigma (Al Atik et al., 2010).
Following non-ergodic assumption, repeatable components of ground motions can be identified and
removed from aleatory variability of a specific model (Rodriguez-Marek et al., 2014). We separated the
site-to-site component δS2S from the total residual. Accordingly, the single station standard deviation can
be obtained from (Rodriguez-Marek et al., 2014; Kotha et al., 2016):
σSS=√τ 2+φSS2 (11)
In the following section we will evaluate the residuals of this model with respect to magnitude, distance,
and shear wave velocity at four spectral periods. Moreover, we will compare the derived model with some
of the existing models for regions other than the study region.
Residual Analysis
Here, we provide plots of residuals versus predictor variables including magnitude, distance, and VS30 at
four representative periods. We selected various spectral ordinates to represent a large frequency range.
The total (rij) residuals shown by Equation (12) are generally defined as the differences between the
natural logarithm of the observed and the predicted values based on our model:
rij= ln (Yij) - ln (Yij) (12)
12
where Yij is the observed V/H ratio from the jth records of the ith event, and 𝑌𝑖�� is the predicted value of
lnYij. The between-event residual can be defined as the mean of residuals for the ith event with Ni records
as follows:
ηi=
1
Ni
∑ rij
Ni
j=1
(13)
The within-event (intra-event) and between-event (inter-event) residuals are defined, respectively, as:
rij intra=rij-ηi
(14)
ri inter=η
i-
1
Nevent
∑ ηi
Nevent
i=1
(15)
Plots of residuals are provided for the PGA and spectral periods of 0.5, 1.0, and 2.0 s. Figure 9 shows the
distribution of between-event residuals versus magnitude, while Figure 10 and Figure 11 show the
distribution of within-event residuals versus distance and VS30, respectively. In Figure 10 and Figure 11
error bars are plotted only for bins with at least three observations. According to these figures, we observe
no discernable residuals trend versus the mentioned predictor variables. This confirms the sufficiency of
the derived equation in describing the V/H ratios in all representative periods.
Comparison with Other Studies
In this section, we compared proposed model with models derived by various researchers for other
regions. We compared our model with models of Sedaghati and Pezeshk (2017) [hereafter SP17], Haji-
Soltani et al. (2107) [hereafter HPMZ17], Gülerce and Abrahamson (2011) [hereafter GA11], and
Bommer et al. (2011) [hereafter BAK11]. We performed the comparison in two ways. First, we made
visual comparisons by providing plots of V/H ratios versus period for various models and two earthquake
scenarios. Then, we quantitatively compared models developed for other regions with the proposed model
using the popular loglikelihood (LLH) method of Scherbaum et al. (2009).
Figure 12 provides the comparison plots for a scenario earthquake of M 5 at a rupture distance of 30 km
for both soil (VS30 = 270 m/s) and rock sites (VS30 = 760 m/s). As can be seen from Figure 12, predictions
13
of the suggested model are compatible with those of others specifically in short periods and rock site
conditions. For median to large periods, this study is more similar to the study of SP17 and GA11 than
others. In this period range, our model proposes relatively larger V/H ratios than other competing models.
The ratio of the vertical to horizontal component is crucial in moderate-to-large magnitudes and short
distances. Therefore, we provided Figure 13 for a scenario earthquake larger than that used to plot Figure
12. Figure 13 illustrates the comparison plots for scenario earthquake of M 6 at the same rupture distance
of 30 km for both soil (VS30 = 270 m/s) and rock sites (VS30 = 760 m/s). For the earthquake scenario of
Figure 13, we have no similar record in our database. In fact, our database is insufficient in moderate-to-
large magnitudes and short distances, and we extrapolated the derived empirical model to predict V/H
ratios for such a magnitude-distance range. As is clear from Figure 13, our model is in good agreement
with those of others despite being used outside its generating dataset.
Now we directly compare some of the candidate ground motion models for predicting V/H ratios with our
proposed model. We implemented the popular loglikelihood (LLH) method of Scherbaum et al. (2009) to
examine relative performance of the candidate models. This method has been used by various researchers
for the purpose of selecting suitable ground motion models in regional scale (see Mak et al., 2017;
Zafarani and Farhadi, 2017). In the LLH method, one needs to compute the probability density function
(PDF) of a given observation by assuming a normal distribution for the logarithmic predictions of each
candidate GMM. This normal distribution is characterized by the mean and the standard deviation equal
to the logarithmic median prediction and the standard deviation in log unit of the model. In other words,
for an individual observation, 𝑥𝑖, we first compute the log-likelihood value from log2 [g (xi)], where g (xi)
is the PDF of the candidate GMM. Then, the LLH score is the average of the log-likelihood values
computed for all observed data from Equation (16), with N representing the total number of observations:
LLH= -1
N∑ log
2[g(𝑥𝑖)]
N
i=1
(16)
We did not rank the candidate models based on the absolute values of LLH scores. In fact, we addressed
score variability by following the procedures of Mak et al. (2017) and Farhadi et al. (2018). We used the
cluster bootstrap technique to examine score variability by resampling 100 datasets from the generating
database. The cluster bootstrap technique can be summarized as the distinctness index (DI) that shows if
the two models are truly different given the score variability. DI ranges from -1.0 to 1.0 and a positive
14
value of DI indicates that the model scores better more often than another one, given the variability of the
evaluation data. In this study, we used DI values to rank candidate models instead of ranking them based
on the absolute value of their final score. A model having all positive DIs is the best model and, more
often than not, scores better than the rest of the GMMs. However, the second-best model should have a
single negative DI with respect to the best model. One may compute the DI from the following equation:
DIij=1
Nbs
∑ 1 (si(k)
,sj(k))
Nbs
k
1 (si(k)
,sj(k))=
{
1 when si
(k)<sj
(k)
-1 when si(k)
>sj(k)
0 when si(k)
=sj(k)
(17)
where 𝐷𝐼𝑖𝑗 is the DI of model i with respect to model j. 𝑁𝑏𝑠 represents the number of bootstrap samples
and 𝑠𝑖(𝑘)
is the score of model i for the kth bootstrap sample. A model with better performance has a
smaller score. ��(. ) is the modified indicator function.
We compared the proposed model with models of GA11, HPMZ17, and BAK11. We did not directly
compare our model with the SP17 model because this model provides separate equations for vertical and
horizontal components rather than a single equation for direct evaluation of V/H ratios. Results of
comparisons are provided for the PGA in Table 3. This table gives the DI computed for each model.
According to this table, the proposed model is the best fitting model for the PGA. The proposed model
outperforms the second-best model (HPNZ17) in 65% of resampled datasets. The proposed model also
scores better than the GA11 and BAK11 models in nearly all resampled datasets.
For remaining representative periods, we did not provide the distinctness tables since the model ranking
based on these tables is similar to the ranking based on the absolute values of scores. We summarized the
results for the four representative periods in Table 4. Table 4 shows the LLH scores for the four candidate
models in PGA and spectral periods including 0.5, 1.0, and 2.0 s. According to this table, the proposed
model is the best fitting model for the majority of the representative periods. Better performance of this
model compared to the other models is not surprising since we are testing this model against its
generating dataset. For the short period of 0.5 s, the score of the proposed model is very close to the score
of the GA11 model, and GA11 marginally outperforms the proposed model. In general, the proposed
15
model shows a stable performance over the entire frequency range compared to the other candidate
models, and can be considered as an appropriate model for predicting V/H ratios in CENA.
Summary
We developed a new model to predict V/H ratios for CENA using the newly developed NGA-East
database. The developed model has the advantage of considering the dominant magnitude, source-to-site
distance, and shear-wave velocity of soil deposits in the upper 30m of the site in a wide period range of
0.0 (PGA) to 10.0 seconds. We thoroughly analyzed the developed model considering residuals and
compared our model with the V/H ratios of other studies. Our proposed model describes the data well and
its performance is better than models developed for other study regions over the entire frequency range.
Therefore, we suggest our model be considered as the most appropriate model for predicting V/H ratios in
CENA.
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19
Figure 1. Proposed V/H model by McGuire et al. (2001) for hard rock spectrum and FEMA P-750
(2009) model plotted for site class B, SS=0.5g, and S1=0.3g.
Figure 2. (Left): CENA recording stations employed in the current study.The colors of the symbols
represent the NEHRP site class of the station. (Right): CENA earthquakes considered in this study.
10-2
10-1
100
101
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Period-S
V/H
CEUS PGA <=0.2g
0.2g=<CEUS PGA <=0.5g
CEUS PGA >=0.5g
WUS PGA <=0.2g
0.2g=<WUS PGA <=0.5g
WUS PGA >=0.5g
FEMA P-750 (NEHRP)
20
Figure 3. Number of usable records for each spectral period.
21
Figure 4. The magnitude–distance distribution of the selected CENA ground-motion recordings for PGA.
22
Figure 5. The distribution of magnitude data versus VS30. Sites are classified into four groups based on
the NEHRP site classification scheme.
23
Figure 6. The distribution V/H versus magnitude in four representative periods. The mean and the
standard deviation of error bars are calculated using magnitude bins of ±0.10 centered at the marker.
24
Figure 7. Comparison of V/H for different styles of faulting. V/H ratios are plotted for M 5.0 and rupture
distance of 50 km considering the VS30 equal to 270 m/s.
25
Figure 8. The distribution of magnitude data versus VS30 for various faulting mechanisms.
26
Figure 9. The distribution of between-event residuals versus magnitude for the Proposed Model at four
representative periods.
27
Figure 10. The distribution of within-event residuals verses closest distance to the rupture area for the
Proposed Model at four representative periods. The mean and the standard deviation of error bars are
calculated by logarithmically spacing the distance range into 15 bins.
28
Figure 11. The distribution of within-event residuals verses shear-wave velocity on the top 30 meters for
the Proposed Model at four representative periods. The mean and the standard deviation of error bars
are calculated using VS30 bins of ±100 m/s centered at the marker.
Figure 12. V/H ratios computed from Proposed Model in comparison with the SP17 (Sedaghati and
Pezeshk, 2017), BAK11 (Bommer et al., 2011), GA11 (Gülerce and Abrahamson, 2011) and HPMZ17
(Hahi-Soltani et al., 2017) models using M 5.0 and RRup = 30 km for soil (VS30 = 270 m/s) and rock (VS30
= 760 m/s) site conditions.
29
Figure 13. V/H ratios computed from Proposed Model in comparison with the SP17 (Sedaghati and
Pezeshk, 2017), BAK11 (Bommer et al., 2011), GA11 (Gülerce and Abrahamson, 2011) and HPMZ17
(Hahi-Soltani et al., 2017) models using M 6.0 and RRup = 30 km for soil (VS30 = 270 m/s) and rock (VS30
= 760 m/s) site conditions.
30
Table 5. List of events used in the current study.
Event Name Date (Month/Day/Year) Latitude (°) Longitude (°) Magnitude No. Records
Charleston 11/11/2002 32.40 -79.94 4.03 28
FtPayne 4/29/2003 34.49 -85.63 4.62 26
Slaughterville 10/13/2010 35.20 -97.31 4.36 242
Enola 5/4/2001 35.21 -92.19 4.37 8
Greenbrier 2/28/2011 35.27 -92.34 4.68 227
Guy 10/15/2010 35.28 -92.32 3.86 202
Guy 11/20/2010 35.32 -92.32 3.9 198
Sparks 11/6/2011 35.54 -96.75 5.68 189
Lincoln 2/27/2010 35.55 -96.75 4.18 203
Sparks 11/5/2011 35.57 -96.70 4.73 210
Jones 1/15/2010 35.59 -97.26 3.84 203
Arcadia 11/24/2010 35.63 -97.25 3.96 126
Jefferson 12/9/2003 37.77 -78.10 4.25 52
Mineral 8/23/2011 37.91 -77.98 5.74 130
Mineral 8/25/2011 37.94 -77.90 3.97 115
Caborn 6/18/2002 37.98 -87.80 4.55 36
Sullivan 6/7/2011 38.12 -90.93 3.89 253
MtCarmel 4/18/2008 38.45 -87.89 5.3 61
MtCarmel 4/25/2008 38.45 -87.87 3.75 49
MtCarmel 4/21/2008 38.47 -87.82 4.03 51
MtCarmel 4/18/2008 38.48 -87.89 4.64 45
Montgomery 7/16/2010 39.17 -77.25 3.42 40
Greentown 12/30/2010 40.43 -85.89 3.85 186
PrairieCntr 6/28/2004 41.44 -88.94 4.18 50
Ashtabula 1/26/2001 41.87 -80.80 3.85 19
Boyd 11/3/2002 42.77 -98.90 4.18 8
Acadia 10/3/2006 44.35 -68.15 3.87 59
AuSableForks 4/20/2002 44.51 -73.70 4.99 51
Hawkesbury 3/16/2011 45.58 -74.55 3.59 76
Thurso 2/25/2006 45.65 -75.23 3.7 76
31
ValDesBois 6/23/2010 45.90 -75.50 5.1 108
MontLaurier 10/19/1990 46.47 -75.59 4.47 9
StFlavien 7/23/2010 46.58 -71.67 3.51 38
CapRouge 11/6/1997 46.80 -71.42 4.45 21
Kipawa 1/1/2000 46.84 -78.93 4.62 9
EagleLake 7/14/2006 46.92 -68.68 3.46 67
Miramichi 3/31/1982 47.00 -66.57 4.46 1
BarkLake 10/12/2003 47.01 -76.36 3.82 54
BaieStPaul 4/7/2006 47.37 -70.48 3.72 53
Laurentide 7/12/2000 47.55 -71.08 3.65 12
Charlevoix 5/22/2001 47.65 -69.92 3.6 8
LaMalbaie 10/28/1997 47.67 -69.91 4.29 11
LaMalbaie 6/13/2003 47.70 -70.09 3.53 34
RiviereDuLoup 11/15/2008 47.74 -69.74 3.57 56
RiviereDuLoup 3/6/2005 47.75 -69.73 4.65 72
Saguenay 11/25/1988 48.12 -71.18 5.85 14
Saguenay 11/23/1988 48.13 -71.20 4.19 6
Saguenay 11/26/1988 48.14 -71.30 3.53 6
CoteNord 3/16/1999 49.62 -66.34 4.43 13
LacLaratelle 6/5/2002 52.85 -74.35 3.81 18
32
Table 6. Regression coefficients of the proposed vertical-to-horizontal (V/H) response spectral ratio model.
Period a1 a2 a3 a4 a5 τ φS2S
φSS
σ
PGA -0.520 0.004 -0.002 0.104 -0.684 0.169 0.303 0.286 0.450
0.01 -0.516 0.007 -0.002 0.107 -0.701 0.170 0.304 0.286 0.451
0.02 -0.488 0.017 -0.005 0.099 -0.642 0.173 0.309 0.286 0.456
0.03 -0.478 0.027 -0.007 0.094 -0.609 0.164 0.293 0.303 0.452
0.04 -0.472 0.032 -0.008 0.085 -0.553 0.158 0.308 0.307 0.463
0.05 -0.485 0.018 -0.005 0.076 -0.495 0.164 0.314 0.303 0.466
0.075 -0.464 -0.025 0.001 0.061 -0.399 0.129 0.297 0.343 0.472
0.1 -0.434 -0.073 0.008 0.062 -0.405 0.148 0.346 0.317 0.492
0.15 -0.423 -0.095 0.011 0.088 -0.576 0.183 0.363 0.288 0.499
0.2 -0.419 -0.085 0.010 0.124 -0.815 0.187 0.382 0.273 0.505
0.25 -0.448 -0.057 0.006 0.147 -0.967 0.192 0.396 0.277 0.521
0.3 -0.463 -0.024 0.001 0.156 -1.027 0.209 0.381 0.295 0.525
0.4 -0.479 0.001 -0.001 0.163 -1.075 0.222 0.388 0.299 0.538
0.5 -0.475 0.015 -0.003 0.171 -1.123 0.239 0.387 0.316 0.554
0.75 -0.491 0.020 -0.003 0.167 -1.092 0.265 0.377 0.346 0.576
1 -0.457 0.017 -0.003 0.156 -1.011 0.262 0.368 0.354 0.574
1.5 -0.376 0.010 -0.005 0.131 -0.850 0.256 0.324 0.383 0.563
2 -0.090 -0.032 -0.007 0.129 -0.835 0.276 0.304 0.402 0.575
3 0.265 -0.117 -0.004 0.132 -0.856 0.325 0.269 0.423 0.597
4 0.673 -0.221 -0.001 0.151 -0.983 0.402 0.276 0.427 0.648
5 0.870 -0.302 0.006 0.160 -1.042 0.440 0.284 0.430 0.678
7.5 0.590 -0.221 0.003 0.174 -1.125 0.429 0.223 0.485 0.685
10 -0.136 -0.003 -0.007 0.192 -1.237 0.402 0.198 0.568 0.724
33
Table 7. Distinctness table for the candidate models for peak ground acceleration (PGA). Distinctness
index of each pairwise comparison (based on 100 bootstrap samples) given in the intersecting box of a
model pair. The whole dataset is considered to compute LLH scores in the second last column.
Model i
Model j Th
is s
tud
y
HP
MZ
17
GA
11
BA
K1
1
LL
H
Ran
k
This study 0.00 0.30 1.00 1.00 0.87 1
HPMZ17 -0.30 0.00 1.00 0.96 0.88 2
GA11 -1.00 -1.00 0.00 -1.00 1.76 4
BAK11 -1.00 -0.96 1.00 0.00 0.99 3
Table 8. Models ranking based on the LLH scores in four representative periods.
V/H Model PGA T = 0.5 T = 1 T = 2
LLH Rank LLH Rank LLH Rank LLH Rank
Proposed Model 0.87 1 1.13 2 1.24 1 1.25 1
HMPZ17 0.88 2 1.45 4 1.32 2 1.35 3
GA11 1.76 4 1.12 1 1.37 3 1.34 2
BAK11 0.99 3 1.24 3 1.43 4 1.42 4