Geology Reference
In-Depth Information
Box 2.
{
k
(
j
+
1
)
}
=
∆
t h x t
{ ({ ( )},{ ( )},{ ( )})}
x t
z t
1
1
2
(
j
+
1
)
(
j
+
1
)
(
j
+
1
)
(
j
+
1
)
{
k
}
=
∆
t h x
{ ({
(
t
+
∆
t
/ )},{
2
x
(
t
+
∆
t
/ )},{ ( )}
2
z t
+
{
k
})}
(11)
2
1
1
2
(
j
+
1
)
(
j
+
1
)
(
j
+
1
)
(
j
+
1
)
{
k
}
=
∆
t h x
{ ({
(
t
+
∆
t
/ )},{
2
x
(
t
+
∆
t
/ )},{ ( )}
2
z t
+
{
k
})}
3
2
(
j
+
1
)
(
j
+
1
)
(
k
+
1
)
k
j
3
(
+
1
)
{
k
}
=
∆
t h x
{ ({
(
t
+
∆
t
)},{
x
(
t
+
∆
t
)},{ ( )
z t
}
+
{
})}
4
Box 3.
(
j
+
1
)
(
j
+
1
)
{ ( )}
x t
+
{
x
(
t
+
∆
t
)}
, {
{ ( )}
x t
+
{
x
(
t
+
∆
t
)}
(12)
{
x
(
j
+
1
)
(
t
+
∆
t
/ )}
2
=
x
(
j
+
1
)
(
t
+
∆
t
/ )}
2
=
2
2
e t M r
( , , )
and the ground acceleration spec-
trum
S f M r
1. Generate a discrete-time Gaussian
white noise sequence
ω
( , , )
are provided in the subse-
quent sections.
( )
t
=
I
/
∆
t
ξ
,
j
=
1
, ...,
n
,
where
j
j
T
=
1
, are independent, identi-
cally distributed standard Gaussian
random variables,
I
is the white noise
intensity,
∆
t
is the sampling interval, and
n
T
is the number of time instants equal to
the duration of the excitation
T
divided
by the sampling interval
2. The white noise sequence is modulated by
an envelope function
e t M r
ξ
j
,
j
, ...,
n
T
The probabilistic model for the seismic hazard
at the emplacement is complemented by consid-
ering that the moment magnitude
M
and epicen-
tral distance
r
are also uncertain. The uncer-
tainty in moment magnitude is modeled by the
Gutenberg-Richter relationship truncated on the
interval
M M
min
[ ]
which leads to the prob-
ability density function (Kramer, 2003)
,
,
max
( ,
, )
at the discrete
time instants
3. The modulated white noise sequence is
transformed to the frequency domain
4. The resulting spectrum is normalized by the
square root of the average square amplitude
spectrum
5. The normalized spectrum is multiplied by
a ground motion spectrum (or radiation
spectrum)
S f M r
b e
−
bM
p M
(
)
=
,
M
≤
M M
≤
6 0
.
b
8 0
.
b
min
max
e
−
−
e
−
(13)
where
b
is a regional seismicity factor. For the
uncertainty in the epicentral distance
r
,
a lognor-
mal distribution with mean value
r
(km) and
standard deviation
σ
r
(km) is used. The point
source stochastic model previously described is
well suited for generating the high-frequency
components of the ground motion (greater than
0.1Hz). Low-frequency components can also be
introduced in the analysis by combining the above
( ,
, )
at discrete frequencies
/ , , ..., 1 2
6. The modified spectrum is transformed back
to the time domain to yield the desired ground
acceleration time history. Details of the
characterization of the envelope function
f
=
l T l
=
n
l
T
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