Civil Engineering Reference
In-Depth Information
The experimental stationary semi-variogram
γ
ˆ
(
Δ
) is estimated from a
dataset as follows
()
∑
N
Δ
1
ˆ
()
=
{
}
2
γ
Δ
zz
u
−
[3.12]
u
+
Δ
()
2
N
Δ
α
α
α
=
1
where
z
u
denotes the data value at location
u
;
N
(
Δ
) denotes the number of
pairs of sites separated by
th pair. For a given
earthquake
i
at site
j
, the ground-motion residual is estimated as
Δ
; and {
z
u
α
,
z
u
α
+Δ
} denotes the
α
()
=
Zj
log
Y
−
log
Y
=
εη
+
[3.13]
j
j
j
The normalized residual
τ
˜
j
and the normalized within-earthquake resid-
ual
ε
˜
j
are computed as
εη
σ
+
ε
σ
j
j
τ
=
;
ε
=
[3.14]
j
j
ε
,
j
ε
,
j
σ
ε
,
j
denotes the standard deviation of the within-earthquake residual
at site
j
. Note that the between-earthquake residual
where
is a constant across
all the sites during a given earthquake. Thus, the experimental semi-
variogram function of
η
ε
˜
can be obtained as follows (see Baker and Jayaram,
2008):
()
∑
N
Δ
1
ˆ
()
=
{
}
2
γ
Δ
εε
−
u
u
+
Δ
()
α
α
2
N
Δ
α
=
1
(
)
−
(
)
−
2
(
)
−
(
)
−
()
∑
N
Δ
⎡
⎢
ln
Y
ln
Y
η
ln
Y
ln
Y
η
⎤
⎥
1
u
u
u
+
Δ
u
+
Δ
=
α
α
−
α
α
()
2
N
Δ
σ
σ
u
u
+
Δ
α
=
1
α
α
(
)
−
(
)
(
)
−
(
)
−
2
()
∑
N
Δ
ln
Y
ln
Y
ln
Y
ln
Y
1
⎡
⎢
⎤
⎥
u
u
u
+
Δ
u
+
Δ
α
α
α
α
=
()
2
N
Δ
σ
σ
u
u
+
Δ
α
=
1
α
α
()
∑
N
Δ
1
{
}
2
=
ττ
α
−
u
u
+
Δ
()
α
2
N
Δ
α
=
1
[3.15]
σ
u
α
is the standard deviation of the within-earthquake residual at
location
u
α
. The equation involves an approximation due to the mild
assumptions that
where
η
σ
η
=
.
σ
u
u
+Δ
α
α
For the case of normalized between-earthquake residuals, the relation
between semi-variogram
ˆ
ε
(
γ
ˆ
(
Δ
) and within-earthquake correlation
ρ
Δ
) is
the following:
ˆ
()
=−
()
ˆ
γ
Δ
1
ρ
ε
Δ
.
[3.16]
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