Environmental Engineering Reference
In-Depth Information
this bias,
c
i
can be replaced by a mean concen-
tration value over a defined period of time
(Donlan et al., 1981).
4. The fourth procedure, derived from Equation
(4.175), weighs the mean daily load by the
mean of all measured flows and estimates the
load,
L
c
, as follows
These unbiased estimators are well suited for
cases in which there are a large number of flow
data, but only a few concentration data are avail-
able. Preston et al. (1989) presented several of
these estimators, derived from Equation (4.177),
and the ratio estimator is mandated for use in the
Great Lakes region of the United States and
Canada by the International Joint Commission
(Richards and Holloway, 1987).
Regression methods.
These methods and their result-
ing rating curves define an empirical relationship
between streamflow and concentration. The most
common regression equation is the log-log linear
rating curve:
∑
∑
∑
∑
N
AcQ
i
i
i
i
=
1
N
∑
N
A
Q
µ
i
i
Q
i
=
1
i
=
1
L
c
=
N t
∆
=
cQ
Q
N t
∆
N
N
AQ
i
i
i
=
1
N
A
i
i
=
1
log
c
= +
a b
log
Q
(4.181)
(4.176)
10
10
where
a
and
b
are regression constants. once fitted
to available data by least square regression, Equa-
tion (4.181) can be used to generate concentration
values at the regular time intervals (Δ
t
) at which
flow is measured, and then the resulting contami-
nant load,
L
r
, is calculated by summing the product
of the concentration and streamflow using the
relation
The last two averaging procedures, Equations
(4.175) and (4.176), have been found to be less
biased than the first two, Equations (4.171)
and (4.172), but result in large variability in
load estimations (Walling and Webb, 1981).
The four averaging estimators described above
are widely used as a first approximation. However,
if the data set does not represent the entire range
of flows and concentration values, bias can be
important (Donlan et al., 1981; Ferguson, 1987).
These methods are defined for periodic concentra-
tion data; for aperiodic data, more suited methods
have been developed (e.g., Littlewood et al., 1998).
Ratio Estimators.
Ratio estimators multiply the pre-
vious estimator,
L
c
in Equation (4.176), by a ratio
that accounts for the covariance between load and
streamflow values. The contaminant load estimate,
L
re
, is given by
N
∑
L
r
=
c Q t
j
∆
(4.182)
j
j
=
1
The estimator
L
r
is precise, but has been shown to
produce a strong underestimation of the actual
load when using the log transformation. Ferguson,
1987 proposed corrections to get an unbiased esti-
mator,
L
cr
, given by
L
=
L
exp(2.651 )
s
2
(4.183)
cr
r
1
S
cQQ
cQ
where
s
indicates the estimated standard error of
the estimate of the rating curve in log
10
(mg/L)
units. The regression method does not require
extensive data, but the quality of the prediction
depends on the quality of the correlation between
the flows and the concentration. In practical appli-
cations, this requirement is often met for sedi-
ments, particulate and total phosphorus, and
pesticides, but less so for mobile chemicals, such as
nitrate and chlorides (Quilbé et al., 2006; vieux
and Moreda, 2003). Regression equations should
be used for interpolation but not extrapolation.
1
+
µ
N
Q
d
L
re
=
cQ
Q
N t
∆
(4.177)
S
Q
1
2
2
Q
1
+
N
d
where
N
∑
1
N
=
A
(4.178)
d
i
i
=
N
1
∑
S
=
AcQ N QcQ
−
(4.179)
cQ
i
i
i
d
N
−
1
d
i
=
1
In applying the above load estimators,
stratification
can be applied to improve the load estimations. The
stratification strategy classifies streamflow values into
N
1
∑
2
S
=
AQ N Q
−
(4.180)
2
i
i
d
Q
N
−
1
d
i
=
1
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