Environmental Engineering Reference
In-Depth Information
Table 4.2
Coordinates and measured values of
s
u
of the 30 samples
Sample no.
X (m)
Y (m)
c
u
(kPa)
Sample no.
X (m)
Y (m)
c
u
(kPa)
1
1.0
1.0
24.6
16
5.0
7.0
19.8
2
1.0
3.0
23.4
17
5.0
9.0
22.7
3
1.0
5.0
31.0
18
5.0
11.0
32.3
4
1.0
7.0
25.9
19
7.0
1.0
31.4
5
1.0
9.0
22.9
20
7.0
3.0
36.5
6
1.0
11.0
27.1
21
7.0
5.0
32.4
7
3.0
1.0
24.1
22
7.0
7.0
35.8
8
3.0
3.0
32.8
23
7.0
9.0
19.7
9
3.0
5.0
46.4
24
7.0
11.0
26.5
10
3.0
7.0
21.8
25
9.0
1.0
27.9
11
3.0
9.0
32.1
26
9.0
3.0
29.8
12
3.0
11.0
31.8
27
9.0
5.0
37.2
13
5.0
1.0
24.5
28
9.0
7.0
26.8
14
5.0
3.0
35.6
29
9.0
9.0
29.2
15
5.0
5.0
26.6
30
9.0
11.0
19.8
The correlation coefficient between ln(
c
ui
) and ln(
c
uj
) can be calculated based on the
correlation coefficient between
c
ui
and
c
uj
analytically (e.g., Law and Kelton 2000). In
general, these two correlation coefficients are not exactly the same. For simplicity, how-
ever, it is assumed in this example that the two correlation coefficients are equal to each
other.
The optimal values of θ are determined by maximizing the likelihood function in
Equation 4.9,
or equivalently, its log-likelihood function. When dealing with the spatial
variability model as shown above, the log-likelihood function is often quite flat. This
makes it difficult to maximize the likelihood with respect to μ, σ, and Ω simultane-
ously. To this end, Fenton (1999) suggested that θ can be estimated using the following
procedure:
1. Let Ω
i
(
i
= 1, 2, …,
m
) denote
m
possible values of Ω.
2. For
i
= 1, 2, …,
m
estimate the maximum likelihood of μ and σ for the case of
Ω = Ω
i
. Denote the maximum likelihood estimates as μ
i
and σ
i
. Thus, {μ
i
, σ
i
, Ω
i
} is
a local maximum likelihood point. Repeat this estimate for all
m
possible values
of Ω.
3. Among the
m
local maximum likelihood points, the one with the highest maxi-
mum likelihood value is the global maximum likelihood point.
The MATLAB code used to implement this concept is provided in
Figure 4.4
.
As an
example, the range of Ω is taken as [0.5 m, 10.0 m], and Ω values are assumed uniformly
distributed within this interval with a spacing of 0.1 m. Based on the MATLAB code
shown in
Figure 4.4,
the optimal values of μ, σ, and Ω are 28.48 kPa, 5.93 kPa, and
1.60 m, respectively.
4.2.3 Censored observations
The observation occasionally presents in such a way that its quantity is larger or smaller
than a threshold but the exact value is unknown. For instance, in a pile load test, a pile
may survive the prescribed applied load. In such a case, the capacity of the pile is only
known to be greater than the applied load. Such observations are known as censored data
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