Environmental Engineering Reference
In-Depth Information
10.34.
The flow rate in a drainage channel has a mean
and standard deviation of 5.21 and 3.89 m
3
/s,
respectively, and the total nitrogen concentration
has a mean and standard deviation of 2.86 and
1.59 kg/m
3
, respectively. Assuming that the total
nitrogen concentration is independent of the
flow rate, estimate the mean and standard devia-
tion of the total nitrogen mass flux in the river.
10.38.
Prove Equation (10.172).
10.39.
Prove Equation (10.177).
10.40.
use the definition of the semivariogram,
γ
(
h
),
and the covariance function,
C
(
h
), to show that
they are related by
γ( )
h
=
C
( )
0
−
C
( )
h
10.35.
A flow rate is characterized by a mean and stan-
dard deviation of 8.86 and 1.80 m
3
/s, respectively,
and separate measurements indicate that the
flow area has a mean of 35 m
2
and standard
deviation of 4 m
2
. Assuming that errors in the
flow rate and flow area are uncorrelated, what is
the mean and standard deviation of the average
velocity?
10.41.
Analysis of rainfall measurements indicate that
the monthly rainfall in a region has a mean of
13.2 cm, and the spatial correlation is adequately
described by the Gaussian covariance function
2
h
C h
( )
=
13 6
. exp
8 5
.
where
h
is in kilometers and
C
(
h
) is in cm
2
. What
is the range of the covariance function? Estimate
the rainfall at a location,
X
, based on the mea-
sured rainfall at four nearby stations, where the
coordinates and rainfall measurements at the
stations are given by
10.36.
The flow rate,
Q
(m
3
/s), over a suppressed rect-
angular weir can be estimated using
the
relation
3
2
Q
= 1 83
.
LH
where
L
is the length of the weir (m), and
H
is
the head of water on the weir (m). In a particular
case, estimates of
L
and
H
are uncertain, such
that
L
is characterized by a mean and standard
deviation of 3 and 0.5 m, respectively, and
H
is
characterized by a mean and standard deviation
of 0.31 and 0.03 m, respectively. Assuming that
errors in estimating
L
and
H
are uncorrelated,
estimate the mean and standard deviation of
Q
.
10.37.
The stormwater runoff rate,
Q
(m
3
/s), from a
catchment is frequently estimated using the
relation
Station
Coordinate (km, km)
Measured rainfall (cm)
X
(0, 0)
-
1
(0.96, 1.08)
8.1
2
9.3
(−1.20, 1.43)
3
(−1.50, −0.98)
6.9
4
(1.23, −1.89)
10.4
What is the error variance of the estimated
rainfall?
10.42.
Measurements of the transmissivity in an aquifer
suggest a semivariogram of the form
γ( )
h
= 43 1
.
h
1 5
.
Q CIA
=
where
h
is in meters and
γ
(
h
) is in m
4
/d
2
. The
mean transmissivity in the region is estimated to
be 1500 m
2
/d. Estimate the transmissivity,
T
, at
location
X
, based on the measured transmissivity
at four nearby stations, where the coordinates
and transmissivity measurements are given by
where
C
is a runoff coefficient (dimensionless),
I
is the rainfall intensity (m/s), and
A
is the area of
the catchment (m
2
). The variables
C
,
I
, and
A
, are
random variables subject to measurement errors
with means and standard deviations given as:
Station
Coordinates (km, km)
Transmissivity (m
2
/d)
Variable
Mean
Standard Deviation
X
(0, 0)
-
C
0.85
0.12
1
(120, 105)
2150
I
25 mm/h
5 mm/h
2
(155, −180)
2390
A
10 ha
0.70 ha
3
(−133, −148)
2280
4
(−192, 106)
2060
use a first-order uncertainty analysis to estimate
the mean and standard deviation of the esti-
mated runoff,
Q
.
What is the error variance of the estimated trans-
missivity at
X
?
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