Civil Engineering Reference
In-Depth Information
3.2.3 Models of spatial correlation
The functions
) provide a set of experimental values for a
fi nite number of separation distances
ρ
ˆ
ε
(
Δ
) and
γ
ˆ
(
Δ
. A continuous function can be fi tted
based on these experimental values. The function
Δ
) may be represented
by an exponential decay function (e.g. Boore
et al.
, 2003; Wang and Takada,
2005; Goda and Hong, 2008a):
ρ
ε
(
Δ
(
)
()
=
ρ
ε
Δ
exp
a
Δ
b
[3.17]
where
a
and
b
are the model parameters. Again, if the PSA is considered,
the correlation function and parameters are the functions of
T
(Goda and
Hong, 2008a). Goda and Atkinson (2009) introduced an extended version
of equation (3.17), which is based on an assumption of uncorrelated ground-
motion parameters at well-separated sites
{
[
]
−
}
(
)
=
()
()
()
()
+
ρ
Δ
,
T
max
ϕ
T
exp
−
α
T
Δ
β
T
ϕ
T
10
,
[3.18]
ε
where
(
T
)
are the model parameters.
Wang and Takada (2005) suggested to characterize the function
α
(
T
),
β
(
T
), and
ϕ
) by
a single parameter - the so-called 'correlation distance'
R
C
. The correlation
distance shows the site-to-site distance for which the correlation coeffi cient
ρ
ε
(
ρ
ε
(
Δ
0.368.
Alternatively, an exponential model of the semi-variogram can be used
for modeling
Δ
) decreases to 1/
e
=
γ
(
Δ
) (and thus
ρ
ε
(
Δ
))
()
=− −
[
(
)
]
γ
Δ
a
1
exp
3
Δ
b
[3.19]
where
a
and
b
are the sill and the range of the semi-variogram function,
respectively. The sill of a semi-variogram equals to the variance of spatially
distributed random function; while the range is defi ned as the separation
distance
) equals 0.95 times the sill of the exponential semi-
variogram. The relation between semi-variogram and within-earthquake
correlation
Δ
at which
γ
(
Δ
ρ
ε
(
Δ
) is the following:
()
=−
()
[
]
γ
Δ
a
1
ρ
ε
Δ
[3.20]
Some models (e.g. Goda and Hong, 2008a; Goda and Atkinson, 2009)
interpolate empirical results obtained for moderate-to-long distances to
meet the theoretical requirement of perfect correlation at zero-separation
distance. The within-earthquake correlation model at relatively short sepa-
rate distances (less than 5 km), where available data are scarce, was ana-
lyzed by Goda and Atkinson (2010). The data from SK-net recording
stations, which are densely distributed in the Kanto (Japan) region, were
used.
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