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through an envelope function
e
(
t
;
M,r
), which
again depends on
M
and
r
. These frequency and
time domain functions,
A
(
f
;
M,r
) and
e
(
t
;
M,r
),
completely describe the model and their charac-
teristics are provided by predictive relationships
that relate them directly to the seismic hazard, i.e.,
to
M
and
r
. More details on them are provided in
(Boore, 2003; Taflanidis et al., 2008). Particularly,
the two-corner point-source model by Atkinson
& Silva (2000) can be selected for the source
spectrum because of its equivalence to finite fault
models. This equivalence is important because
of the desire to realistically describe near-fault
motions and adaptation of a point-source model
might not efficiently address the proximity of the
site to the source (Mavroeidis & Papageorgiou,
2003). The spectrum developed by Atkinson &
Silva (2000) has been reported in their studies to
efficiently address this characteristic.
The time-history (output) for a specific event
magnitude,
M
, and source distance,
r
, is obtained
according to this model by modulating a white-
noise sequence
Z
w
=[
Z
w
(
iΔt
):
i
=1,2,…,
N
T
] by
e
(
t
;
M
,
r
) and subsequently by
A
(
f
;
M
,
r
) through
the following steps: the sequence
Z
w
is multiplied
by the envelope function
e
(
t
;
M
,
r
) and then trans-
formed to the frequency domain; it is normalized
by the square root of the mean square of the ampli-
tude spectrum and then multiplied by the radiation
spectrum
A
(
f
;
M
,
r
); finally it is transformed back
to the time domain to yield the desired accelera-
tion time history. The model parameters consist of
the seismological parameters,
M
and
r
, describing
the seismic hazard, the white-noise sequence
Z
w
and the predictive relationships for
A
(
f
;
M
,
r
) and
e
(
t
;
M
,
r
). Figure 4 shows functions
A
(
f
;
M
,
r
) and
e
(
t
;
M
,
r
) for different values of
M
and
r
. It can be
seen that as the moment magnitude increases the
duration of the envelope function also increases
and the spectral amplitude becomes larger at all
frequencies with a shift of dominant frequency
content towards the lower-frequency regime.
Reduction of
r
primarily contributes to an overall
increase of the spectral amplitude. Alternatively, a
“record-based” stochastic ground motion could be
used for modelling the high frequency component
of the ground motion. An example of such a model
is the one developed recently by Rezaeian & Der
Kiureghian (2008), which has the advantage that
additionally addresses spectral nonstationarities.
Long Period Pulse
For describing the pulse characteristic of near-
fault ground motions, the simple analytical model
developed by Mavroeidis & Papageorgiou (2003)
is selected. This model is based on an empirical
description of near-fault ground motions and has
been calibrated using actual near-field ground mo-
tion records from all over the world. According
to it, the pulse component of near-fault motions
is described through the following expression for
the ground motion velocity pulse:
A
2
f
π
γ
γ
γ
p
p
p
p
p
p
V t
( )
=
1
+
cos
(
t
−
t
) cos
2
π
f
(
t
−
t
)
+
ν
if
t
∈ −
t
,
t
+
o
p
o
p
o
o
2
2
f
2
f
p
= otherwise
0
(18)
where
A
p
,
f
p
,
ν
p
,
γ
p
, and
t
o
describe the signal ampli-
tude, prevailing frequency, phase angle, oscillatory
character (i.e., number of half cycles), and time
shift to specify the envelope's peak, respectively.
Note that all parameters have an unambiguous
physical meaning. A number of studies (Alavi
& Krawinkler, 2000; Bray & Rodriguez-Marek,
2004; Mavroeidis & Papageorgiou, 2003) have
been directed towards developing predictive rela-
tionships that connect these pulse characteristics
to the seismic hazard of a site. These studies link
the amplitude and frequency of near-fault pulse
to the moment magnitude and epicentral distance
of seismic events, but they also acknowledge that
significant uncertainty exists in such relationships.
This indicates that a probabilistic characterization
is more appropriate. For example, according to
the study by Bray and Rodriguez-Marek (2004)
the following relationships hold for the pulse
frequency
f
p
and the peak ground velocity
A
v
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