Environmental Engineering Reference
In-Depth Information
in which
P
(
x
i
|E
) is the occurrence probability of a toe debris with thickness
x
i
and
P
(
F|x
i
) is the probability of unsatisfactory performance of the pile with the given toe debris.
Similarly, if toe debris does exist and
x
is taken to be a continuous variable, its probability
distribution can be expressed as
f
(
x|t
) in which the parameter,
t
, is the mean of
x
and treated
as another random variable with a probability distribution
f
(
t
). The conditional probability
of unsatisfactory performance of the pile can be further given by
x
t
U
U
∫
∫
PF E
(|)
=
P Fx
( |)
fxtftt x
(|)()
dd
(14.17)
x
t
L
L
where
x
L
and
x
U
are the lower and upper bounds of
x
and
t
L
and
t
U
are the lower and upper
bounds of
t
, respectively.
For a pile that is constructed without any integrity testing, the empirical occurrence prob-
ability of toe debris can be considered as a prior distribution of
p
d
. The empirical perception
of toe debris thickness can also be used to establish the prior distribution of toe debris thick-
interface coring are available, the distributions of
p
d
and toe debris thickness can be updated
with the additional measurement information using the Bayesian approach (e.g., Ang and
Tang 2007), and
P
(
F
|
E
) of the pile with toe debris can be calculated using
Equation 14.16
or
14.4.2 updating occurrence probability of toe debris
The Bayesian approach provides a method to combine empirical information and field obser-
vations. Take the occurrence probability of toe debris,
p
d
, as an example. An initial PDF of
p
d
,
f
′(
p
d
), can be established based on theoretical models or expert judgments. When field
observations of
p
d
through on-site interface coring at a particular site are available,
f
′(
p
d
)
can be updated by combining the initial PDF with the field observations. The updated dis-
tribution can be regarded as a weighted average of the initial information and the additional
observations. More specifically, the updated PDF of
p
d
,
f
″(
p
d
), can be expressed as
fp
′′
()
=
KL Rp fp
(| )( )
′
(14.18)
d
d
d
where
K
is a normalization constant and
L
(
R
|
p
d
) is the likelihood of observing the outcome
R
.
Suppose
n
piles are selected randomly from a site for inspection, and
m
of them are found to
contain toe debris. Since a thick toe debris, if present in a pile, can be detected with certainty in
most cases by interface coring, it is reasonable to assume that the detection probability of inter-
face coring is equal to one. If statistical independence among pile inspections is further assumed,
then the likelihood that the assumed event occurs can be expressed as a binomial function
n
m
=
m
nm
−
LR p
(| )
p
(
1
−
p
)
(14.19)
d
d
d
f
′(
p
d
) can be formulated based on available test data on toe debris. Since
p
d
is bounded
between 0 and 1, it may be described by a beta distribution
1
(
q
−
1
)
(
r
−
1
)
fp
′
()
=
p
(
1
−
p
)
(14.20)
d
d
d
Bq r
(,)
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