Environmental Engineering Reference
In-Depth Information
where
z
is a random variable characterizing the modeling error of the factor of safety. With
Equation 4.24
,
the liquefaction probability can be expressed as follows:
F
z
=
S
PPF
=
(
<
1
)
=
P
<
1
Pz
(
> =−
F
)
1
F F
()
(4.25)
L
Sa
S
S
where
F
() is the cumulative density function of
z
. Let μ and δ denote the mean and coef-
ficient of variation (COV) of
z
, respectively. The task of calibrating the
P
L
-
F
S
relationship,
which is in the form of
P
L
=
f
(
F
S
), is then reduced to the task of calibrating μ and δ, or in the
notation in this chapter, θ = {μ, δ}.
The shape of the assumed probability distribution of
z
may affect the results of the max-
imum-likelihood analysis. To derive the best
P
L
−
F
S
relationship, the following cumulative
distribution functions (denoted as Q
1
, Q
2
, Q
3
, and Q
4
) are examined and compared:
Q
1
—Gaussian (normal):
[
]
Fz
()
=
Φµ µδ
(
z
−
) (
⋅
)
(4.26)
Q
2
—Lognormal:
(
(
)
)
2
2
Fz
()
=
Φ
ln()
z
−
ln
µ
1
+
δ
ln(
1
+
δ
)
(4.27)
Q
3
—Minimum Gumbel:
‡
Ž
(
z
−
µπ
)(
−
0 5772 6
.
⋅
δ
)
‰
‰
‰
‰
Fz
()
=− −
1
exp xp
(4.28)
6
⋅
µδ
⋅
Q
4
—Maximum Gumbel:
‡
Ž
−− −
(
z
µπ
)(
0 5772 6
.
⋅
δ
)
‰
‰
‰
‰
Fz
()
=
exp xp
−
(4.29)
6
⋅
µδ
⋅
Based on the above assumptions about the cumulative density function of
z
, four
P
L
−
F
S
relationships for
z
are generated based on
Equation 4.25
,
as summarized in
Table 4.6
.
Note
that in these derivations, COV of
z
is treated as a constant.
4.4.2 Calibration and ranking of P
L
-F
s
relationships
The four probability models as shown in
Table 4.6
can be calibrated using the database
as shown in
Table 4.5
(Ku et al. 2012). As an example, the analysis is conducted with the
assumption that the variable
z
follows the lognormal distribution which is designated as
Q
2
.
The weighted likelihood function can be expressed as follows:
{
}
(
)
n
L
(
)
∑
1
L
(| )
θ
d
=
w
ln
−
Φ
ln(
F
)ln
−
µ
1
+
δ
2
ln(
1
+
δ
2
)
L
si
i
=
1
(
)
n
NL
(
)
∑
+
w
ln
Φ
ln() ln
F
−
µ
1
+
δ
2
ln(
1
+
δ
2
)
(4.30)
N
L
sj
j
=
1
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