Civil Engineering Reference
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2
σ
σ
()
=−
d
ρ
1
[3.3]
T
2
2
T
where
σ
d
denotes the variance of (
η
i
+
ε
i
,
j
)
−
(
η
i
+
ε
i
,
k
) (Boore
et al.
, 2003).
The within-earthquake correlation coeffi cient
ρ
ε
(
Δ
), which by defi nition is
given by
[
]
COV
εε
,
()
=
ij
ik
ρ
[3.4]
ε
σ
2
ε
where
COV
(
A
,
B
) denotes the covariance of
A
and
B
, can be expressed as
σ
σ
2
()
=−
d
ρ
1
[3.5]
ε
2
2
ε
Note that the derivation of the correlation coeffi cients shown in equa-
tions (3.2) to (3.5) are based on the consideration that residuals are related
to the statistics of the logarithm of a single randomly oriented horizontal
component
Y
ij
. If the correlation coeffi cients are derived by considering the
statistics of the geometric mean of two orthogonal horizontal components
of the ground motion parameter, one has the following relationship:
()
+
1
ρ
()
=
C
ρ
Δ
ρ
Δ
[3.6]
ε
gm
2
where
) is the correlation coeffi cient based on geometric mean of two
orthogonal horizontal components;
ρ
gm
(
Δ
ρ
C
is the correlation coeffi cient between
ground-motion parameters of two orthogonal horizontal components.
Baker and Cornell (2006) provided the following empirical expression for
ρ
C
, which depends on
T
:
()
ρ
C
=−
0 79
.
0 023
.
ln
T
[3.7]
3.2.2 Procedures of analysis
Let
Z
(
x
) denote the total residuals between the log of observed ground
motion parameter at a site
x
and that of the value predicted from the
attenuation relationship, i.e.
Z
(
x
)
log
¯
ij
. Several methods, which
can be adopted to estimate within-earthquake correlation of ground motion
residuals, have been described in the literature. Here we briefl y describe
three of them.
In
Procedure 1
, Boore
et al.
(2003) performed the following analysis to
investigate the within-earthquake correlation of ground motion residuals:
(1) select the data for a selected earthquake
i
; (2) compute separation dis-
tance
=
log
Y
ij
−
for all pairs of stations; (3) compute the differences
Z
(
x
); (4) divide
the range of
Δ
Δ
into bins so that the separation distance in the same bin is
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