Environmental Engineering Reference
In-Depth Information
EXAMPLE 10.30
where
σ
V
=
. . m/s
. Hence, the average
velocity in the river has a mean and standard deviation
of 0.108 and 0.539 m/s, respectively. notably, the coef-
ficient of variation of
V
is much greater than that of
Q
and
A
, indicating that great care should be taken assess-
ing the uncertainty in the ratio of random variables.
0 290
=
0 539
The flow rate,
Q
, in an open channel, is sometimes esti-
mated using the relation
Q K S
=
0
where
K
is the conveyance of the channel and
S
0
is the
slope of the bottom of the channel. In a particular case,
estimates of
K
and
S
0
are uncertain such that
K
is char-
acterized by a mean and standard deviation of 3400 and
500 m
3
/s, respectively, and
S
0
is characterized by a mean
and standard deviation of 0.01 and 0.005, respectively.
Assuming that errors in estimating
K
and
S
0
are uncor-
related, estimate the mean and standard deviation of
Q
.
10.11.4 Other Functions
Many functions commonly used in water-quality analy-
sis involve more than addition, subtraction, multiplica-
tion, and division. Several of these functions and their
associated variances are given in Table 10.10. In the
general case where the random variables related by a
function of the form
Solution
y
= (
f x x
,
,
,
x
n
)
(10.159)
2
…
1
From the given data:
μ
K
= 3400 m
3
/s,
σ
K
= 500 m
3
/s,
μ
S
= 0.01, and
σ
S
= 0.005. The partial derivatives to be
used in estimating the variance of
Q
are as follows,
The mean and variance of
y
, denoted by
μ
y
and
σ
2
,
respectively, can be estimated by
µ
≈ (
µ µ
,
,
,
µ
)
(10.160)
2
…
y
1
n
∂
∂
Q
K
=
S
0
n
2
n
−
1
n
∂
∂
f
x
∂
∂
f
x
∂
∂
f
x
∑
∑
∑
σ
2
≈
σ
2
+
2
σ
(10.161)
y
i
ij
i
i
j
i
=
1
i
=
1
j
= +
i
1
1
2
∂
∂
Q
S
1
2
K
S
−
=
S K
=
0
2
where
μ
i
and
σ
i
2
are the mean and variance of
x
i
, respec-
tively, and σ
ij
is the covariance between
x
i
and
x
j
. The
approximations given by Equations (10.160) and
(10.161) are limited to cases in which the variances,
σ
i
2
,
are small. The variance estimation method given by
Equation (10.161) is sometimes called the
delta method
for estimating the variance of a dependent variable.
In cases where
x
i
's are independent (uncorrelated)
random variables, the mean and variance estimates sim-
plify to
0
0
Applying Equations (10.162) and (10.163) yields
m /s
µ
≈
µ µ
=
(
3400
)
0 01
.
=
340
Q
K
S
2
2
2
∂
∂
Q
K
∂
∂
Q
S
+
K
S
=
(
)
2
σ
2
≈
σ
2
+
σ
2
S
σ
2
σ
2
Q
K
S
0
K
S
2
0
0
2
3400
2 0 01
=
(
)
+
2
0 01
.
(
500
)
2
( .
0 005
)
2
=
9725
(m /s)
3
2
.
µ
≈ (
µ µ
,
,
,
µ
)
2
…
(10.162)
y
1
n
98 6
3
n
2
where
σ
Q
=
. m /s
. Hence, the estimated
flow rate in the channel is approximated as having a
mean and standard deviation of 340 and 98.6 m
3
/s,
respectively.
9725
=
∂
∂
f
x
∑
σ
2
≈
σ
2
(10.163)
y
i
i
i
=
1
TABLE 10.10. Variances of
Random Functions
10.12 KRIGING
y
σ
y
σ
µ
Kriging
, a data analysis procedure named in honor of
D.G. Krige* (Krige, 1951), is an optimal geostatistical
technique in which values of a random spatial function
(rSF) are estimated at a specified point, given measure-
x
x
ln
x
σ
µ
x
log
10
x
log
10
e
x
σ
µ
x
x
2
x
*
Danie G. Krige is a South African mining engineer who pioneered
the field of geostatistics.
a
exp(
a
x
µ
)
exp(
ax
)
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