Environmental Engineering Reference
In-Depth Information
uniformly varying from 0 to 1 for ϕ′,
B
, and
D
, respectively. These three uniform random
samples are then transformed into three standard Gaussian random samples using an Excel
built-in function “NORMSINV(),” which is the inverse function of the standard Gaussian
CDF. These three standard Gaussian random samples are generated in Cells B12- D12, and
their PDF values are calculated by invoking a built-in function “NORMSDIST()” in Cells
B13-D13. Then, these three standard Gaussian random samples are transferred into ran-
dom samples of the lognormal variable ϕ′ and discrete uniform random variables
B
and
D
.
The lognormal variable ϕ′ is expressed as (e.g., Ang and Tang, 2007; Au et al., 2010)
φ
′ =
exp
(
µ
+
σ
w
)
(7. 28)
N
1
in which μ
N
and σ
N
are the mean and standard deviation of ln(ϕ′ ) (i.e., Cells B7 and B8),
respectively, and
w
1
is the a standard Gaussian variable (i.e., Cell B12). For discrete uniform
random variables
B
and
D
, they are expressed as
BB
=
+
INTwNd
[( )
Φ
]
(7. 2 9)
min
2
B
B
DD
=
+
INT
[( )
ΦΝ
w
3
]
d
(7. 3 0)
min
D
D
in which
B
min
and
D
min
are the respective minimum possible values for
B
and
D
(i.e., Cells
E5 and F5 for
B
and
D
, respectively),
w
2
and
w
3
are the independent standard Gaussian
random variables;
d
B
and
d
D
are the respective increments of
B
and
D
(i.e., Cells E7 and
F7 for
B
and
D
, respectively),
N
B
and
N
D
are the respective numbers of possible values of
B
and
D
(i.e., Cells E8 and F8 for
B
and
D
, respectively), Φ() is the CDF of the standard
Gaussian variable, which is implemented by a built-in function “NORMSDIST()” in Excel,
and INT() is an Excel built-in function that rounds the number in the bracket down to the
nearest integer. In the uncertainty model worksheet,
Equations 7.28
through
7.30
are used
to generate, respectively, random samples of ϕ′,
B
, and
D
in Cells B14-D14 from random
into discrete integer random variables
INT
(Φ(
x
2
)
N
B
) and
INT
(Φ(
x
3
)
N
D
) ranging from 0
Then,
INT
(Φ(
x
2
)
N
B
) and
INT
(Φ(
x
3
)
N
D
) are further transformed into the discrete random
variables
B
and
D
in
Equations 7.29
and
7.30
using their respective minimum values (i.e.,
B
min
and
D
min
) and increments (i.e.,
d
B
and
d
D
).
7.6.3 Subset Simulation and rbD add-In
After the deterministic and uncertainty model worksheet are developed and linked together,
the Subset Simulation Add-In shown in
Figure 7.4a
is invoked for uncertainty propagation.
Using the Add-In, a Subset Simulation run with
N
= 10,000,
p
0
= 0.2, and
m
= 4 is executed,
and a total of
N
+
mN
(1 −
p
0
) = 42,000 samples are obtained from the simulation. Note that
the choice of
p
0
and
m
affects the efficiency of Subset Simulation. A reasonable range of
p
0
suggested in literature (e.g., Zuev et al., 2012) is from 0.1 to 0.3. For a given
p
0
value, the
number (i.e.,
m
) of Subset Simulation levels is selected to ensure that the target failure prob-
ability level is reached, that is,
p
+
<
1
.
In addition, the “Random variable(s),
X
” input field in
Figure 7.4a
records the cell refer-
ences of random variables ϕ′,
B
, and
D
or their respective equivalents in standard normal
m
p
0
T
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