Environmental Engineering Reference
In-Depth Information
Based on these results, a 51.9% reduction in FC load
under moist conditions will bring the stream into com-
pliance with both the geometric mean and 90-percentile
water-quality standards. Since the water-quality viola-
tions are occurring under moist conditions, this load
reduction should be applied to nonpoint sources. Such
load reductions will generally require improved storm-
water control measures in the contributing watershed.
where
A
i
represents the indicator for avail-
ability of the concentration data (1 if data
available, 0 if not). overbars denote sample
arithmetic means, and
L
s
is the resulting l
oa
d.
2.
In the second procedure, the mean flow
Q
is
replaced by the mean of all flow measure-
ments,
µ
Q
. Unlike the first procedure, this one
uses all the available data:
The statistical procedure of using the required reduc-
tion in concentration as a surrogate for the required
reduction in source loading is also commonly used in
cases where either there is no flow data, or the stream
flow rate is small or nonexistent.
∑
∑
N
∑
M
Ac
Q
i
i
i
i
−
1
i
=
1
L
w
=
N t
∆
=
c N t
µ
∆
Q
N
M
A
i
i
=
1
(4.172)
where
L
w
is the estimated contaminant load
and
4.7.2 Long-Term Contaminant Loads
Estimation of long-term contaminant loads generally
requires high-frequency contemporaneous measure-
ments of streamflow and contaminant concentration,
and the contaminant load,
L
T
(M), transported through
a river cross section during a time interval
T
(T) can be
estimated using the relation
∑
1
M
Q
i
(4.173)
µ
Q
=
i
M
where
M
is the number of flow measurements.
Walling and Webb (1981) and Ferguson (1987)
found that the estimators given by Equations
(4.171) and (4.172) are precise (i.e., give similar
results with different subsamplings from the
same data set) but biased and resulting in a
strong underestimation of the actual load. A
logical variation of Equation (4.172) is to take
µ
Q
as the average of the flow measurements
collected between concentrations measure-
ments (e.g., Cassidy and Jordan, 2011). In this
case, Equation (4.172) becomes
N
T
∑
=
∫
L
=
Q t c t dt
( ) ( )
≈
Qc t
∆
(4.170)
T
i
i
0
i
1
where
N
is the number of measurements, Δ
t
is the time
interval between measurements (T) (Δ
t
=
T
/
N
),
Q
i
is the
flow rate at time step
i
(L
3
T
−1
), and
c
i
is the concentra-
tion at time step
i
(ML
−3
). In reality, flow-rate measure-
ments are usually available at regular intervals, while
concentration measurements are seldom available at
regular intervals. Under these circumstances, other
more approximate methods must be used to estimate
the contaminant load,
L
. Existing methods for load esti-
mation can be classified into three classes: (1) averaging
estimators, (2) ratio estimators, and (3) regression
methods. These methods are described below.
∑
∑
N
A c
µ
i Qi
i
i
=
1
∆
(4.174)
L
w
=
N t
∆
=
c N t
µ
Q
N
A
i
i
=
1
where
µ
Qi
is the mean flow within the
i
th sam-
pling interval. This approach is similar to the
third procedure.
Averaging Estimators.
These estimators use the
means of concentrations and flows over a time
interval. The four principal averaging estimators
are described below.
1. The first procedure calculates separately the
mean flow and the mean concentration based
on data only from times when both variables
are measured:
3.
In the third procedure, the load is first calcu-
lated at times when both variables are mea-
sured, and then the contaminant load,
L
a
, is
given by
∑
N
AcQ
i
i
i
L
a
=
∑
1
i
=
N t
∆
=
cQN t
∆
(4.175)
N
∑
∑
N
∑
∑
N
A
Ac
AQ
i
i
i
i
i
i
=
1
L
s
=
i
=
1
i
=
1
N t
∆
=
cQN t
∆
N
N
A
A
i
i
This procedure produces a large bias when
concentration data are sparse, and to reduce
i
=
1
i
=
1
(4.171)
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