Environmental Engineering Reference
In-Depth Information
were randomly selected from a site and were interface cored. Two piles were found to con-
tain toe debris, whose thicknesses were x 1 = 100 mm and x 2 = 50 mm, respectively. This
observation corresponds to case 2 described earlier. For comparison purposes, imaginary
scenarios corresponding to case 1 (i.e., no toe debris is found in the 67 piles) and case 3 (i.e.,
all the toe debris at the site are detected and repaired) are also studied.
The prior distributions of the occurrence probability and the toe debris thickness are updated
for the three cases. Figure 14.7 compares the updated PDFs of occurrence probability of toe
debris for cases 1 and 2 for a diffuse prior and a prior beta distribution. The four updated
PDFs in Figure 14.7 are sharper than the respective prior PDFs. The updated standard devia-
tions of the occurrence probability, calculated by
( ) ( 1 based on the
beta prior distribution, are 1.3 and 1.9% for cases 1 and 2, respectively. All these updated
standard deviations are significantly smaller than the prior standard deviation of 3.7% found
earlier. The uncertainties in the occurrence probability are substantially reduced through the
tests. For the prior beta distribution, the Bayesian estimators of occurrence probability, calcu-
lated by ( q + m )/( q + r + n ), for the three cases are 1.9, 4.0, and 0%, respectively. For the diffuse
prior distribution, the Bayesian estimators, calculated by ( m + 1)/( n + 2), for the three cases are
1.5, 4.3, and 0%, respectively, which will be used for p d in Equation 14.15 .
Similarly, the mean thickness of toe debris can also be updated based on the test results.
Figure 14.8 shows the prior and updated PDFs of mean toe debris thickness for case 2. The
updated PDF is sharper than the prior PDF. The corresponding standard deviation of the
mean thickness of toe debris after the tests, calculated by ′′
qr
′′
′′
/
qr
′′ + ′′ +
/
qr
′′ + ′′
/( ) 3 is 45 mm, which is
smaller than the prior standard deviation of 58 mm found in the previous section. Hence,
the uncertainty in the mean thickness is substantially reduced through the coring tests. The
Bayesian estimators of the mean thickness, calculated by ˆ
νκ
′′
t
′′ =+ +
=
(
ν
Σ
m
x
) (
/
κ
m
),
for the
i
1
i
three cases are 145, 128, and 0 mm, respectively.
There is no observation of toe debris thickness in case 1 and all toe debris, if any, is
repaired in case 3. Therefore, no updating exercise for toe debris thickness is conducted for
the two cases.
Table 14.7 summarizes the assumptions in the three cases and whether the PDFs of f ( p d )
and f ( t ) are updated or not in each case. Note that all piles would be tested and repaired in
case 3 and hence no toe debris would be present.
70
Diffuse prior distribution
Prior beta distribution
Updated distribution with diffused prior, case 1
Updated distribution with diffused prior, case 2
Updated beta distribution, case 1
Updated beta distribution, case 2
60
50
40
30
20
10
0
0.00
0.03
0.06
0.09
0.12
0.15
0.18
0.21
Occurrence probability of toe debris
Figure 14.7 A comparison between updated and prior PDFs of occurrence probability of toe debris.
 
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