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where
f
EDP
(
EDP
>
y
|
IM
1
,
IM
2
) is the PSDM for a particular EDP and two
IMs,
f
IM
1
(
IM
1
|
IM
2
) is the probability density function for the fi rst IM condi-
tioned on the second and d
(
IM
2
) is the annual rate of exceeding a specifi ed
value of IM
2
(within some small increment). For EDPs depending on a
vector of ground motion parameters, a vector-valued PSHA is required
(Bazzurro and Cornell, 2002). The vector-valued seismic hazard integral for
calculating the structural demand is (Gülerce and Abrahamson, 2010):
ν
⎧
∫
∫
∫
∫
()(
)
(
)
(
)
⎫
N
⋅
f
M
f
M R f
,
ε
f
ε
ε
min
M
R
ε
IM
1
ε
IM
2
IM
1
IM
1
IM
2
⎪
⎪
M
R
ε
ε
IM
1
IM
2
⎨
⎬
ˆ
(
)
=
(
[
(
)
(
)
]
ν
EDP
>
y
P EDP
>
y EDP IM
M R
,,
ε
,
IM
M R
,,
ε
,
⎪
⎪
⎪
⎪
1
IM
1
2
IM
2
)
σ
d
MMR
IM
dd
ε
d
ε
⎩
⎭
ln
EDP
1
IM
2
[20.10]
where
ε
IM
1
and
ε
IM
2
are the epsilons for IM
1
and IM
2
,
f
ε
IM
1
(
ε
IM
1
) is the prob-
ability density function for
ε
IM
1
, and
f
ε
IM
2
(
ε
IM
2
|
ε
IM
1
) is the probability density
function for
ε
IM
1
. The form in Equation 20.10 differs
from the formulation given by Bazzurro and Cornell (2002) such that the
double integral in Equation 20.10 is carried out over
ε
IM
2
conditioned by
ε
IM
2
, rather
than IM
1
and IM
2
. The modifi ed formulation clearly shows that the correla-
tion of the variability of the intensity measures needs to be considered
(Gülerce and Abrahamson, 2010).
f
ε
IM
1
and
ε
IM
1
(
ε
IM
1
) is given by the standard normal
distribution, whereas
f
ε
IM
2
(
ε
IM
2
|
ε
IM
1
) is conditioned on the value of
ε
IM
1
and
depends on the correlation of
ε
IM
1
and
ε
IM
2
. The covariance of
ε
IM
2
with
respect to
ε
IM
1
is computed from the correlation of the normalized residuals
from the GMPE regression analysis. The probability density function for
ε
IM
2
can be defi ned as a function of
ε
IM
1
as:
()
=×
(
)
±
ε
T
ρ
ε
T
=
T
σ
ε
[20.11]
IM
22
IM
1
1
ε
IM
2
IM
1
where
σ
ε
IM
2
|
ε
IM
1
is the standard deviation
of the correlation. A vector-IM-based analysis facilitates the record selec-
tion process, assuming that no dependence upon any other parameters is
remained (Baker and Cornell, 2005). Using a vector-valued IM also
decreases the variability in the PSDM and thus the number of required
nonlinear analyses to develop the PSDM model. However, to perform
vector-valued PSHA, the correlation of residuals of the ground motion
model (or models) across the periods need to be known. Only a few of the
recent GMPEs provide these correlation coeffi cients to be used in vector-
valued analysis (e.g., Abrahamson and Silva, 2008; Gülerce and Abraham-
son, 2011). Another drawback of the vector-valued IMs is the inability of
the standard PSHA software packages to run vector-valued calculations.
ρ
is the correlation coeffi cient and
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