INTRODUCTION
Current studies of earthquake effects have tended to disregard the vertical
ground motion and concentrate mainly on the horizontal component. The
primary reason for this practice lies in the consideration that engineering
structures have adequate resistance to dynamic forces induced by the vertical
ground motion, which is generally much smaller than its horizontal components.
If the effect of vertical motion is explicitly included in design, it
is typically assumed that the ratio of vertical to horizontal (V/H) response
spectra will not exceed twothirds (UBC, 1997; GB500112001, EUROCODE8).
However, observations in recent earthquakes, such as the 1994 Northridge, California,
1989 Loma Prieta, California, 1999 ChiChi, Taiwan and the 1995 Kobe, Japan
have indicated that the ruleofthumb ratio of twothirds may not be a good
descriptor of vertical ground motion (Yang and Sato, 2000;
Bozorgnia and Campbell, 2004; Elgamal
and He, 2004). For example, the threedimensional records obtained at a
borehole array site in Kobe, Japan during the 1995 Kobe earthquake deserve particular
attention, which showed that the peak vertical acceleration at the surface was
twice as high as the peak horizontal acceleration. An integrated analysis of
this case history has suggested the key role of local site effects on the ground
motion records (Yang and Sato, 2000;
Yang et al., 2000).
On the other hand, large vertical component of nearfault earthquake records
has been considered and damage potentials of vertical ground motion on engineering
structures have been emphasized in recent years (Papazoglou
and Elnashai, 1996; Diotallevi and Landi, 2000; Button
et al., 2002).
In view of the fact that the effects of vertical ground motion are not
well understood, special attention of this study is hence placed on the
behavior of the vertical ground motion recorded during these recent large
earthquakes. In this study, it is presented equations for estimating peak
vertical acceleration and vertical absolute acceleration response spectra
in terms of magnitude, sourcedistance, site conditions and other parameters
for shallow earthquakes in the seismically active parts of the World.
Comparisons are also made between these empirical equations and equations
used widely in seismological researches. Some of the earlier conclusions
are important for the development of design codes, which does not consider
vertical earthquake spectra or limits for the nearfield ratio of vertical
to horizontal acceleration for damaging earthquakes.
MATERIALS AND METHODS
For these models, it is selected 751 freefield strongmotion records from
57 earthquakes following the freefield definition of Joyner
and Boore (1981) using the criteria; M_{w}≥5.2, R_{rup}≤100(km)
and h≤20(km).
Table 1: 
Summary of distribution of data in the set of records
used with respect to geographical location and faulting mechanism 

Table 1 shown summary of distribution of
data in the set of records used with respect to geographical location and faulting
mechanism.
The records are uniformly distributed with respect to magnitude and distance
and consequently the equations obtained based on this set of data are well constrained
and representative of the entire data space: 0≤R_{rup}≤100 (km)
and M_{W}≥5.2. The strong motion parameters include corrected vertical
peak ground acceleration (PGA) and 5% damped pseudo response spectral acceleration
(PSA) at natural periods range from 0.01 to 10.0 sec. Definition of the size
of an earthquake is in terms of M_{W} (Kanamori,
1997). The sourcetosite distance is defined as the shortest distance between
the recording site and the rupture plane of earthquakes. The faulting mechanism
of an earthquake is classified based on rake angle into strikeslip, normal
and reverse groups.
DEVELOPMENT OF ATTENUATION RELATIONS
A number of different regression methods exist for deriving attenuation relations,
which affect in different ways the predictive equations. For this regression
analysis, it is used random effects model. In this method the error is assumed
to consist of two parts: an earthquaketoearthquake component η, which
is the same for all records from the same earthquake and a recordtorecord
component ε, which expresses the variability between each record not expressed
by the earthquaketoearthquake component (Abrahamson and
Youngs, 1992; Joyner and Boore, 1993). The standard
deviation of these two errors is found along with the coefficients. This method
is thought to take better account of the fact that each record from the same
earthquake is not strictly independent. This algorithm uses an iterative approach
to find the maximum likelihood solution. There are two terms defining the standard
deviation for the model: an interevent term τ which is the standard deviation
of the η and an intraevent term σ which is the standard deviation
of the ε.
GROUND MOTION ATTENUATION MODELS
The median estimate of ground motion for the vertical component of PGA
and PSA is given by:
Scaling of magnitude: The magnitude scaling characteristics f_{1}(M_{w})
and f_{2}(M_{w}) are given by follow:
Coefficients C_{3}, C_{4}, C_{5} and C_{6}
are held to be fix during regression analysis and supposed to be 1.323,
0.423, 1.842, 0.724 for vertical PGA and PSA.
Scaling of distance: The distance scaling characteristics are
given by:
where, R_{rup} is the shortest distance between the recording
site and the rupture plane of earthquake and the coefficient c_{7}(T)
is constrained to varying with period.
Scaling of faulting mechanism: The period dependence for the style
of faulting mechanism is given by:
Scaling of hanging wall: The HW model is developed to including
three tapers on JoynerBoore distance R_{JB}, magnitude and dip
as follow:
For the regression analysis, only a single HW parameter c_{12}(T)
is estimated. The factor HW is set to 1.0 for sites affected by hanging
wall.
Scaling of site response: Based on the Youngs’s approach (Youngs,
1993), the linear and nonlinear soil response is modeled by:
Table 2: 
Coefficients and statistical parameters from the regression
analysis of vertical PGA and PSA 

The linear site amplification equation is given by:
The nonlinear term is given by:
The coefficients, V_{lin}, c_{13}(T) and c_{14}(T)
are depend on period and V_{S30}. These coefficients are prescribed
based on the on the study of Choi and Stewart (2005).
PGA_{nonlin} is the predicted PGA in (g) for rock site, as given in
Eq. 1.
Scaling of standard deviation: The total standard deviation is
considered to this bellow form:
Regression analysis: This model includes a large number of coefficients.
To arrive at a smooth model, the coefficients were smoothed in a series of steps.
The final smoothed coefficients for the median ground motion are shown in Table
2. It should be noted that the quality of selected data is not sufficient
at long periods (T>3 sec) to provide reliable spectral (Ambraseys
et al., 2005; Akkar and Bommer, 2006).
RESULTS OF THE EMPIRICAL MODELS
The median response spectra for the current model are shown in Fig.
1 for reverse and strikeslip faulting. In general, the predicted
response spectra increase with magnitude. In the magnitude condition,
large amplitude in median periods is observed when magnitude is increases.
The effects of style of faulting mechanism and local site condition are
plotted in Fig. 2. Predicted response spectra plotted
for reverse event shows higher amplitude in median and long periods compared
to strikeslip and normal events. As it is shown in Fig.
2, the behavior of spectral acceleration is significantly different
with respect to site conditions for reverse faulting. At short periods,
the amplitudes of soil site is greater than rock site for PSA and indicates
the soil site may increase the amplitude of strong groundmotions in short
to median periods. The amplitudes for soft soil and soft rock are somewhat
higher at short periods and the amplitudes for firm rock and hard rock
are significantly lower at long periods for the specific values of magnitude
(M_{W} = 7.0) and distance (R_{rup} = 10 km) used in the
evaluation. The shortperiod amplitudes for soft clays are found to be
much higher than the other site categories in short to median periods,
consistent with reduced nonlinear site effects.

Fig. 1: 
PSA predicted from the groundmotion relations showing
the effects of (a) magnitude for the rock site and (b) magnitude for
the soil site 

Fig. 2: 
PSA predicted from the groundmotion relations showing
the effects of faulting mechanism (a) reverse, (b) strikeslip and
(c) normal for different local soil conditions for the vertical component
of groundmotion 
Hard rock and firm rock have similar amplitudes at all periods. Figure
2 also shows the spectra tend to have higher amplitudes for reverse faulting
at short to mid periods respected to other events.
The possibility of sites to be located over the hanging wall of reverse
and thrust faults is higher than strikeslip and normal faults, which
increases their groundmotion at close distances. The influence of local
site condition is in general limited for strikeslip and normal faulting,
although moderate effects are observed for soft rock at shorter periods
and soft soil at longer periods.
DISCUSSION
These new groundmotion relations are compared to five groundmotion relations
that are widely used to estimate horizontal response spectra for seismological
and engineering analyses in nonextensional regions (Abrahamson
and Silva, 1997; Campbell, 1997; Sadigh
et al., 1997; Ambraseys and Douglas, 2003;
Campbell and Bozorgnia, 2003). Almost all of the relations
address the vertical component for both soil and rock.
They all define the faulting mechanism to strikeslip and reverse categories,
where this study includes both reverse and strikeslip faulting as defined normal
faulting. These relations use different definitions for local site conditions.
Figures 3 and 4 show the predicted spectral
acceleration from this groundmotion relation compared to predicted from five
selected groundmotion relations for a site located 10 km from a reverse earthquake
of M_{W} = 7.0. This distance corresponds to R_{JB} = R_{rup}
= 10 km and Rseis = 10.4 km. The R_{seis} is defined as the shortest
distance between the recording site and the zone of the seismogenic energy release
on the causative fault (Campbell, 1997).

Fig. 3: 
Comparison of predicted PSA for generic soil from groundmotion
relation in this study and five groundmotion relations widely used
in seismology and engineering 

Fig. 4: 
Comparison of predicted PSA for generic rock from groundmotion
relation in this study and five groundmotion relations widely used
in seismology and engineering 

Fig. 5: 
Comparison of predicted standard deviations of spectral
acceleration (σ_{lnY}) from this study and four groundmotion
relations widely used in seismology and engineering (a) magnitude
M = 5.5 and (b) magnitude M = 7.5 
Comparisons
are shown for the two most common site conditions used in engineering analysis,
namely: generic soil, V_{S30} = 310 m sec^{1} and generic rock,
V_{S30} = 620 m sec^{1} (Boore et al.,
1997). In generally, the comparisons in Fig. 3 and 4 show that the relations predict spectral accelerations similar to those of the
other five groundmotion relations when evaluated for generic soil and generic
rock. This groundmotion relatively predicts higher amplitudes at short and
median periods on generic soil. The prediction of these groundmotion relations
relatively shows the same amplitudes at short periods for the vertical component
on generic rock. Only this relation addressed these differences by site parameter,
V_{S30} and the relation of Abrahamson and Silva
(1997) and this relation incorporate nonlinear site effects. Figure
5 shows a comparison of the standard deviations (natural log) predicted
from the groundmotion relations evaluated for two magnitudes. The standard
deviation is important because it contributes significantly to deterministic
and probabilistic estimates of groundmotion. Figure 5 indicates
that the study vertical standard deviations are belonging to the lowest values.
CONCLUSIONS
The groundmotion relations developed in this study are considered valid
to estimating vertical PGA and PSA for earthquakes from shallow crustal
earthquakes. The study refines some new parameters in the model. Local
site conditions do not significantly influence vertical spectra of strikeslip
and normal faulting. However, we find that at long period vertical spectra
for soft sites show systematically larger amplification than for stiff
sites. This trend is reversed at short and intermediate periods. If this
trend is genuine, it may suggest that decay of the high frequency motion
in thin superficial soft soil deposits is due to high straining. The study
shows that the medianpredicted spectral accelerations among strikeslip,
reverse and normal faulting earthquakes are different. These differences
were found to become negligible at long periods, where dynamic stress
drop is expected to have little effect on the amplitude of strong groundmotion.
The standard deviation is magnitude dependent with smaller magnitudes
leading to larger standard deviations.
ACKNOWLEDGMENT
This study was supported by the Institute of Earthquake Engineering and
Seismology Research (IEESR) of Amirkabir University of Technology with
grant No. 1061384.