RIVERS AND STREAMS - Water Quality Engineering in Natural Systems - page 126

p ( c )

required reduction
∆

distribution required
to meet water-quality
standard

distribution of
measurements

lognormal ( s log c , m log c )

log c std m log c log c b log c

Figure 4.20. Concentration reduction for given flow condition.

compliance with the water-quality standard under all
flow conditions for it not to be impaired, and so this
procedure must be applied to all flow conditions where
water-quality violations occur.
A limitation of the procedure described here is that
it neglects the uncertainty in estimating the β -percentile
of the observations, and so applying the Δ reduction to
the observed concentrations has a 50% likelihood of
successfully meeting the water-quality standard. If more
certainty in meeting the water-quality standard is
required, then a more sophisticated method, such as that
given by Chin (2009a), can be used.

c 90,std = 400 CFU/dL. The minimum required reduction
to meet each regulatory standard will be considered
separately.

Fifty-Percentile Standard. The 50-percentile con-
centration of the measurements is c

µ log

= 10 =

c

50

10 . Since this exceeds the 50-

percentile standard (200 CFU/dL), the stream is
impaired relative to the geometric mean (= 50-
percentile) standard. In accordance with Equation
(4.167), the required reduction, Δ, is given by

2 48 .

= 302 CFU/dL

∆ = log ( c ) − log ( c std ) = 2.482 − log (200) = 0.181

10 50 10 50, 10

and hence the reduction factor is 10 −Δ = 10 −0.181 = 0.659.
This is equivalent to a (1 − 0.659) × 100 = 34.1%
reduction in the FC load during moist conditions.
Ninety-Percentile Standard. The 90-percentile concen-
tration of the observations, c 90 , is given by

EXAMPLE 4.26

violations of the FC water-quality standard in a particu-
lar stream are found to occur mostly under “moist
conditions,” and instream measurements of FC concen-
trations under this condition are found to be adequately
described by a lognormal distribution. The mean and
standard deviation of the logarithm (to base 10) of the
FC data are 2.482 and 0.342, respectively, when the FC
concentrations are expressed in CFU/dL. Local water-
quality regulations require that FC concentration levels
have a geometric mean concentration less than or equal
to 200 CFU/dL and a 90-percentile concentration less
than or equal to 400 CFU/dL. Determine the required
FC load reduction under moist conditions for both
water-quality standards to be met.

log c = µ + z σ (4.169)

10 90 log c 90 log c

where z 90 = 1.282 is the 90-percentile standard normal
deviate. Substituting the statistics of the FC measure-
ments into Equation (4.169) gives

log 10 c = 2.482 (1.282)(0.342) + = 2.920

90

which corresponds to a 90-percentile value of
10 2.920 = 832 CFU/dL. Since this exceeds the regula-
tory 90-percentile value (400 CFU/dL), the stream is
impaired relative to the 90-percentile standard. In
accordance with Equation (4.167), the required
reduction, Δ, is given by

Solution

∆ = log ( c ) − log ( c ) = 2.920 − log (400) = 0.318

10 90 10 90, std 10

From the given data: µ logc = 2.482 and σ logc = 0.342. For
lognormal distributions, the geometric mean is equal to
the 50-percentile concentration, and so the regulatory
limits can be expressed as c 50,std = 200 CFU/dL and

and hence the reduction factor is 10 −Δ = 10 −0.318 = 0.481.
This is equivalent to a (1 − 0.481) × 100 = 51.9%
reduction in the FC load during moist conditions.