Civil Engineering Reference
In-Depth Information
3.2.4 Numerical simulation of spatially correlated
ground-motion parameters
In ground-motion models, the parameter of motion
Y
generated by earth-
quake
i
with magnitude
M
at a site
j
at distance
R
is estimated as a lognor-
mally distributed random variable as ln
Y
ij
N
(log
¯
ij
,
2
), where log
¯
i
,
j
f
(
e
i
,
p
i
,
j
,
s
i
,
j
). In addition to the mean value of ground motion
¯
ij
, we need to
generate the standard normal variates (errors) of
=
σ
=
ε
i
,
j
(see equation
3.1). The dependence structure among the variates is described by the cor-
relation matrix
η
i
and
.
For the generation of the
k
-site random fi eld of ground-motion values
that are spatially correlated, it is necessary to generate a Gaussian vector
of correlated standard normal variables (total residual term)
X
Σ
=
[X
1
,
X
2
, . . . , X
k
] with a symmetric correlation matrix
Σ
, or X
∼
N
k
(0,
Σ
). The cor-
relation matrix
Σ
is defi ned as follows:
1
…
…
ρ
ρ
⎛
⎞
12
1
k
⎜
⎜
⎜
⎜
⎟
⎟
⎟
⎟
ρ
1
ρ
21
2
k
Σ=
,
[3.21]
⎝
ρρ
1
⎠
k
1
k
2
where
ρ
kl
is the empirical correlation coeffi cient calculated for the sites
k
and
l
separated by a distance
. Descriptions of the procedure for genera-
tion of the
k
-site random fi eld of ground-motion error values that are
spatially correlated may be found in Johnson (1987) and Park
et al.
(2007).
The generation can be done in the following steps. First, a vector of inde-
pendent standard normal variates
U
Δ
=
[U
1
, U
2
, . . . , U
k
] with standard devia-
tion
is
constructed and a Cholesky decomposition is applied to represent the cor-
relation matrix
σ
T
, or
U
∼
N
k
(0,
σ
2
T
), is generated. Then, a correlation matrix
Σ
Σ
as the matrix product of matrix
B
and its transposition
B
T
, i.e.,
BU
. These
X
j
values are added to the median ground-motion term ln
¯
ij
to obtain a reali-
sation of spatially correlated ground motions. Figure 3.1 shows examples of
distribution of strong-motion residuals (peak ground acceleration (PGA),
ln g,
Σ
=
B B
T
. The required vector
X
is obtained as
X
=
0.5) generated for various correlation distances (25 and 5 km) and
for the case of spatially uncorrelated ground motion (all the variability is
within-earthquakes).
σ
T
=
2.3.5 Estimation of correlation models from empirical data
As can be seen from the next sections, the treatment of ground-motion
correlation is essential for the estimation of seismic hazard, damage and
loss for widely located building assets (portfolios) and spatially distributed
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