Civil Engineering Reference
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tion and correlation structure as the theoretical 'target' distribution,
IM
|
IM
j
).
Therefore, for each realization, ideally a ground motion can be obtained
(either from databases of as-recorded motions (e.g. Chiou
et al.
, 2008) or
via ground motion simulation methodologies (e.g. Boore, 2003)) which has
the identical intensity measure values as the simulated vector
IM
nsim
.
However, because of several restrictions (e.g. the fi nite number of observed
ground motions available) fi nding such a ground motion will not generally
be possible. Therefore, for each random realization, it is instead attempted
to select the ground motion which has the minimum misfi t, or residual, as
compared to the randomly realized vector
IM
nsim
. If the misfi t between a
particular
m
th ground motions intensity measure vector,
IM
m
, and the real-
ized intensity measure vector,
IM
nsim
, is relatively small, then it is likely that
the empirical distribution
of the selected ground motion set,
IM
, will be
representative of
IM
|
IM
j
=
im
j
obtained from the GCIM approach (since
will be statistically similar to
IM
nsim
, and
IM
nsim
is drawn from
IM
|
IM
j
).
The residual of a
m
th prospective ground motion compared to the
nsim
th
random realization from the GCIM distributions,
r
m
,
nsim
, is given by:
IM
2
⎡
⎤
N
nsim
m
M
i
∑
I
ln
IM
−
ln
IM
i
i
r
=
w
[4.10]
⎢
⎢
⎥
⎥
mnsim
,
i
σ
⎣
⎦
nsim
i
=
1
ln
IM Rup
,
IM
i
j
where
IM
i
nsim
is the value of
IM
i
from the
nsim
th simulation;
IM
i
is the value
of
IM
i
of the
m
th prospective ground motion;
w
i
is the
i
th element of the
vector of normalized weights (i.e.
w
i
1
) which indicate the importance
of each of the
IM
i
s in ground motion selection;
∑=
σ
ln
IM
i
|
Rup
nsim
,
IM
j
is the condi-
im
j
for the randomly drawn
rupture of the
nsim
th
realization; and
N
IMi
is the number of
IM
i
s in the
vector
IM
. The functional form of the residual accounts for several factors
(Bradley, 2012b); the most important of which it is the hierarchy as to the
importance of each of the different intensity measures in the vector
IM
. For
example, it may be considered that, for a particular seismic response
problem, the cumulative effects of a ground motion are less important than
the ground motion amplitude. In such a case, it would be desired to place
a larger 'weight' on minimizing the misfi t between the intensity measures
indicative of ground motion amplitude than those indicative of ground
motion cumulative effects. Hence it can be seen that for each realization of
the GCIM distributions, the residual for all of the prospective ground
motions can be computed via Equation (4.10), and the ground motion with
the smallest residual value selected.
Because of the fact that ground motions are selected based on random
realizations of the vector
IM
, then the set of selected ground motions will
generally be different for different sets of realizations. For ground motion
sets of a reasonable size which are required for performance-based earth-
tional standard deviation of ln
IM
i
given
IM
j
=
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