-------
VII.2 METHOD OF MAXIMUM MARGINAL RETURN
The optimization method used to solve the resource allocation problems
is the method of maximum marginal return. It is particularly suited for
these problems since it solves all three problems in the same manner.
The features of the method of maximum marginal return are:
(1) It is very fast on the computer. The computation time grows
only proportionally with the size of the problem.
(2) If the function to be minimized is convex, this method will
yield the absolute minimum when the cost of resource quanta is
equal.
The cost C(s) can be easily shown to be convex—its second derivative is
strictly positive for s. < N (which is always the case) and p. < 1
(this is also satisfied, since p. is a probability). The only condi-
tion that is not satisfied for Problem 1 is the requirement that cost of
the quanta, r., be equal. However, the method will yield nearly the
optimum allocation if
max T± « B (7.5)
i.e., the largest cost of a sample is much smaller than the total budget
B. Then the difference between the solution obtained by this method and
the absolute minimum is negligible. Since (7.5) will be satisfied for
the monitoring resource allocation problem, the maximum marginal return
method is well suited for determining the sampling rates.
The method of maximum marginal return is basically a steepest descent
algorithm. It is based on the following intuitive idea: the best place
to allocate one unit of resource is where the marginal return (the
87
-------
decrease in cost - in our case undetected violation "cost" - accrued by
using that unit of. resource) is greatest. Therefore, by ordering the
marginal returns in descending order, one obtains a priority list with
the items having highest priority on top.
To be precise, the marginal return accrued when the sampling time on
the i source is increased from s.-l to s. is
C.(s.-l) - C.(s.)
( } . i i - 1_1_ (7< 6)
11 r1
In view of the convexity of C., these marginal returns are monotonlcally
decreasing with s., i.e.,
ui(si} > yi(8i + 1) (7<7)
The priorities of allocation are obtained by simply ordering these
marginal returns. If the ordering obtained is, for example,
U2(l) > y1(l) > y2(2) > M3(l) .... (7.8)
then effluent 2 is sampled with highest priority, then effluent 1, then
again effluent 2, then effluent 3, etc. Following this, a relation
between the minimized "cost" of undetected violations and the corresponding
resource cost is obtained. Therefore, this method solves simultaneously
the problem of minimizing the undetected violation "cost" subject to the
total budget and the minimization of the budget subject to a given
"cost" of undetected violations.
The problem of allocating an increment of resources to maximize the
improvement in an existing monitoring system is solved as follows: Set
up the priority list as described above, and remove from the list those
samples that have been allocated. The remaining items on the list are,
88
-------
in descending priority, the ones that should be monitored with an increase
in resources.
The above method will be illustrated via a simple numerical example.
Assume there are n =3 pollutant sources with "costs" of undetected
s
violation (for the period in consideration) as given in Table 7.1. It
is assumed, for the purpose of this example, that costs of monitoring
each of these effluents are the same (equal to one).
The "costs" C. of undetected violations are given in these tables as
functions of the corresponding number of samples. The maximum number of
samples per source is taken as 5. Also the marginal returns as defined
in (7.6) and the priority ordering according to (7.8) appear next to
each sample.
The priority list of the sources sampled appears in Table 7.2 together
with the "cost" of undetected violations as a function of the available
resources. This table shows immediately the necessary resources to
achieve a given "cost" of undetected violations and also the achievable
minimum "cost" of undetected violations for a given amount of resources
(number of samples).
As an example of Problem 1, consider the problem of finding the best
allocation of 6 samples. From column 2 of Table 7.2, one sees that the
6 samples should be taken from sources 3, 2, 1, 3, 1, and 2. The sampling
frequencies are then s, »2, s2'»2, s_»2. From column 3 of Table
7.2, the "cost" corresponding to these frequencies is 1.16.
As an example of Problem 2, consider the problem of finding the minimum
amount of resources required to bring the "cost" of undetected viola-
tions to 1.00 or less. From column 3 of Table 7.2, one sees that the
first time that the "cost" drops below 1.00 occurs for 7 samples, for
89
-------
Table 7.1 "COST" OF UNDETECTED VIOLATIONS AND PRIORITY ORDERING
Number
of
samples
Sl
0
1
2
3
4
5
S2
0
1
2
3
4
5
S3
0
1
2
3
4
5
"Cost"
undetected
violations
Cl
1.00
0.70
0.45
0.25
0.08
0.02
C2
1.00
0.65
0.42
0.27
0.15
0.08
C3
1.00
0.55
0.29
0.15
0.05
0.01
Marginal
return
yl
0.30
0.25
0.20
0.17
0.06
y2
0.35
0.23
0.15
0.12
0.07
M3
0.45
0.26
0.14
0.10
0.04
Priority
order
3
5
7
8
14
2
6
9
11
13
1
4
10
12
15
90
-------
Table 7.2 PRIORITY LIST OF SOURCES SAMPLED AND PERFORMANCE
AS FUNCTION OF TOTAL RESOURCES
Resources
accrued
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Source
number
sampled
none
3
2
1
3
1
2
1
1
2
3
2
3
2
1
3
Total "cost"
of
undetected
violations
3.00
2.55
2.20
1.90
1.64
1.39
1.16
0.96
0.79
0.64
0,50
0.38
0.28
0.21
0.15
0.11
91
-------
which the cost is 0.96. From column 2 of this table one sees that the
7 samples should be taken, in order, from sources 3, 2, 1, 3, 1, 2, and 1.
The corresponding sampling frequencies are thus s- = 3 (three samples at
source 1), s- = 2 (two samples at source 2), and s^ = 2 (two samples at
source 3).
As an illustration of how to use the information to allocate additional
resources to improve an existing monitoring system (Problem 3), assume
that the preassigned sampling frequencies are
s1 - 1, s2 - 2, s3 = 1
Consider the problem of optimally allocating four more samples. This is
solved as follows: Take the priority list and omit the first s.
samples on source i, as illustrated in Table 7.3. Then it is seen that
the priorities for the additional four samples are: first source #3,
then //I, again #1, and again //I. The resulting overall sampling fre-
quencies are
sl " 4» S2 * 2» 83
92
-------
Table 7.3 ALLOCATION OF ADDITIONAL INCREMENTS OF
RESOURCES TO A GIVEN MONITORING SYSTEM
Original
priority list
of sources
3
2
1
3
1
2
1
1
2
3
2
3
2
1
3
Priority list of sources
given the preassigned
samples
3
1
1
1
2
3
2
3
2
1
3
93
-------
SECTION VIII
RESOURCE ALLOCATION PROGRAM
Components of the allocation procedure were described in the previous
three sections. This section discusses how these components fit together
to form the Resource Allocation Program. Examples are also given showing
the operation of the Program.
VIII.1 GENERAL PROGRAM DESCRIPTION
A flowchart of the Resource Allocation Program is shown in Figure 8.1.
The following is a brief description of the function on the various
components.
(1) Initialize Statistical Description
Combine the raw self-monitoring and compliance monitoring data
to obtain an initial statistical description (distribution,
mean and standard deviation) for each pollutant of each source.
(2) Calculate Expected Damage and Probability of Violation
Use the statistical description of the effluent loads, the
effluent standards, and the stream parameters to obtain the
expected damage and probability of violation for each source.
(3) Determine Priorities
Use the method of maximum marginal return to obtain the
monitoring frequencies.
94
-------
RESOURCE ALLOCATION PROGRAM
INITIALIZE
STATISTICAL
DESCRIPTION
CALCULATE
EXPECTED DAMAGE AND
PROBABILITY OF
VIOLATION
1
DETERMINE
PRIORITIES
UPDATE
STATISTICS
MONITORING
SCHEDULE
K
MONITORING
PERIOD
I
Figure 8.1 Flow of Resource Allocation Program.
95
-------
(4) Monitoring Schedule
Take the sampling frequencies obtained in the previous component
and determine which day to sample which sources.
(5) Monitoring Period
This box represents the actual time spent monitoring the sources.
(6) Update Statistics
Combine new self-monitoring and compliance data with the initial
statistics to obtain an updated statistical description of the
effluents.
All the components except the "Monitoring Schedule" have been described
in detail in Sections V, VI, and VII. The scheduling of the sampling
depends on a number of factors which are difficult to quantify in an
optimization framework, such as: the spatial location of the various
effluent sources, the size of the monitoring agency's jurisdiction,
and the availability of personnel. This scheduling is beyond the scope
of this report.
Figure 8.2 gives a more detailed description of the Resource Allocation
Program. It describes in detail what data are needed by each component
of the Program. The basic output of the Program is the priorities and
the monitoring frequencies.
VIII.2 SIMPLIFIED EXAMPLE
The performance of the Resource Allocation Program is demonstrated in
this section, using a simplified example. Initially, it is assumed that
there are four sources to be monitored, each having four months of
self-monitoring data available from which to obtain the initial statistics.
96
-------
INITIAL MONITORING PERIOD
g
»-*
C
/ Data
For each aource:
• Description of known
•eatonal variation*
For each constituent:
• Normal or lognomal
• Past Bonthly meana
and maxima
• Number of monthly
•eaaureueata
• Feat compliance data
Obtain eatlmata of
Man and variance for
each conatltuent
SUBSEQUENT MONITORING FERIODS
/Data
For each
conatltuent:
• New aelf
monitoring
and compli-
ance data
I
Obtain new estimate of
the nean and variance
of each conatltuent
For each source:
* Mcen effluent flow
• Drought flow
e BOD-DO transfer
coefficient
For each conatltuent:
e Effluent standard
e Damage function
definition*
Calculate expected
damage and probability
of violation
f Data
• Reaourcaa needed to
monitor each aource
• Upper and lower
bound! on monitor-
ing frequenclea
* Budget or allowed
"coat" of undetect-
ed violation*
Uaing the method of
maximum marginal return
•obtain the monitoring
frequencies
Output:
• Prioritise
• Monitoring
frequenclea
e Reeource*
uaed
• Final
"coat" of
undetected
violations
Determine which day*
In monitoring period
to ample each source
Output:
Monitoring
schedule
Figure 8.2 Resource allocation program.
97
-------
The initial self monitoring data assumed are shown in Tables 8.3a through
8.3e. The data have been abstracted from real data that were used for
the demonstration case (Section IX). Using the procedure outlined in
Section V.3, Tables 8.4a through 8.4e present the initial statistics
obtained from the data. The estimated mean and estimated standard
deviation are the monthly estimates using the techniques developed in
Appendix A. For Source 4, the sample size of the effluent constituents
for a single month is 2; therefore, the data in months 1 and 2 and
months 3 and 4 have to be aggregated, as discussed in Section V.2.
Thus, only two estimates of the mean and two of the variance are given
in Table 8.4d and 8.4e. Tables 8.4a through 8.4e also show how the
estimates of the mean and standard deviation are sequentially updated as
the monthly estimates are combined to obtain the estimates to be used in
the Resource Allocation Program. For this case the design parameters k
n
and k^, which determine the degree of the discounting of past information,
have been set to 3.* The updated mean and variance for month 2 are
therefore the combined estimates derived from the 1st and 2nd monthly
estimates. The updated mean and variance for month 3 are the combination
of the updated estimates for month 2 and monthly estimate for month 3.
The same process is repeated for month 4, yielding the initial statistical
description to be used in the program.
The expected damage and probability of violation obtained from the data
are shown in Table 8.5, along with the estimated source flow and the
stream flow. For this case, the upstream concentration was assumed to
be at a level causing zero damage, and the distributions of the various
parameters were assumed uncorrelated. Certain of the entries deserve
some comment. Source 3 is a large sewage treatment plant. From the
table, the impact of BOD5 and phosphates is large; however, the standards
are also large and therefore the probability of violation for the parameter*
is small. Source 4 has a relatively small impact on the stream (i.e.,
small expected damage); however, the standards have been set so that the
probability of violation is very large. The resources required
* kQ and kv are discussed in Section V.2. The effect of changing k
and kv is shown in VIII.3.
98
-------
Table 8.3a SELF MONITORING DATA FOR SOURCE 1
Month
1
2
3
4
Mean
source
flow.
Ml/day
0.90
1.10
1.20
0.83
Parameter: pH Hue
at. standard: 9
Distribution: Normal
Mean
8.5
7.6
8.3
8.1
Max
10.6
9.0
9.8
9.3
w
20
22
22
20
Parameter: pH Mln
Eff. standard: 6
Distribution: Normal
Mean
8.5
7.6
8.3
8.1
Mln
6.0
5.4
6.4
6.4
Sample
20
22
22
20
Parameter: !•••«>
Eff. standard: 2 kg
Distribution: Normal
Mean,
kg
0.41
1.08
1.09
0.52
IT
1.0
1.7
6.3
1.8
Sample
six*
20
22
22
22
Table 8.3b SELF MONITORING DATA FOR SOURCE 2
Month
1
2
3
4
Mean
source
flow,
Ml/day
0.80
0.78
0.87
0.85
Parameter: Chronlum
Eff. standard: 0.45 kg
Distribution: Normal
Mean,
kg
0.216
0.313
0.214
0.132
Max,
kg
0.808
0.867
0.620
0.255
Sa.pl.
18
19
21
14
Parameter: Copper
Eff. standard: 1.5 kg
Distribution: Lognormal
Hun.
kg
0.524
0.374
0.364
0.110
Max,
**
1.89
1.87
1.25
0.42
Sample
18
19
22
14
Parameter: Fluoride
Eff. standard! 30 kg
Distribution: normal
Mean
kg
24.4
25.4
24.7
14.0
t5'
31.4
31.9
31.0
31.0
Sample
size
18
19
22
11
Table 8.3c SELF MONITORING DATA FOR SOURCE 3
Month
1
2
3
4
Mean
source
flow,
ML /day
105
110
109
108
Parameter: 8005
Eff. standard: 3500 kg
Distribution: Normal
Mean >
kg
1165
900
1395
1080
Maxi
kg
2115
2115
2880
2385
Sample
30
31
30
31
Parameter: Phosphate
Eff. standard: 500 kg
Distribution: Lognormal
Mean,
kg
178
171
171
88
Max,
kg
658
338
500
273
S«jpl.
30
31
30
31
Parameter! Sit*. Solid*
Eff. standard: 4050 kg
Distribution: Lognorms.1
Mean.
kg
2430
1663
3240
2160
Max,
kg
6030
5130
10935
4390
Sample
•iz*
30
31
30
31
Parameter:
Dissolved
oxygen
Mean,
1/1
3.9
3.8
4.2
4.1
Sample
site
30
31
30
31
99
-------
Table 8.3d SELF MONITORING DATA FOR SOURCE 4, FIFE 1
Month
1
2
3
4
Mean
source.
tlow
Ml/day
0.35
0.26
0.29
0.30
Parameter : Phosphates
Eff. standard: 0.6 kg
Diitribuclon: Normal
Mean.
kg
0.13
0.30
0.31
1.20
Hex,
kg
0.24
0.36
0.36
2.56
W
2
2
2
2
Parameter: SIM. Bolide
Eff. itand«rdi 25 kg
Distribution: Normal
Mean.
k|
12.0
14.6
16.4
11.0
Max,
k>
ia.»
18.9
ia.o
15.3
^i-
2
2
2
2
Table 8.3e SELF MONITORING DATA FOR SOURCE 4, PIPE 2
^••••-
1
2
3
4
Kaan
flow,
Ml/day
0.90
1.01
1.09
1.00
Parameter! Phosphate*
Eff. standard! 3.3 kg
Distribution: Normal
Mean.
kg
2.9
3.5
2.9
5.8
Max t
kg
3.2
3.9
3.1
9.8
w
2
2
2
2
Parameter: SIM. Solids
lit. standard! go kg
Distribution! Normal
Mean.
kg
158
ia
93
31
Max,
*
296
26
143
33
•as*
••^•••••mmmmmBBmmmBB*
2
2
2
2
100
-------
Table 8.4a INITIAL STATISTICS FOR SOURCE 1
Month
1
2
3
4
Parameter: pH Max
Distribution: Normal
Est.
mean
8.5
7.6
8.3
8.1
Est.
at. dev.
1.12
0.73
0.78
0.74
Updated
mean
_
8.03
8.12
8.12
Updated
at. dev.
—
1.06
0.98
0.92
Parameter: pH Hln
Distribution: Normal
Est.
mean
8.5
7.6
8.3
8.1
Est.
St. dev.
1.33
1.15
0.99
0.90
Updated
•can
8.03
8.12
8.12
Updated
st. dev.
1.33
1.22
1.14
Parameter: Lead
Distribution: Normal
Est.
0.41
1.08
1.09
0.515
Est. St.
dev., kg
0.31
0.32
2.72
0.67
Updated
0.76
0.87
0.78
Updated
0.51
1.62
1.45
Table 8.4b INITIAL STATISTICS FOR SOURCE 2
Month
1
2
3
4
Parameter: Chromium
Distribution! Normal
Est.
mean ,
kg
0.216
0.313
0.214
0.132
Est.
st. dev.,
kg
0.321
0.297
0.214
0.070
Updated
mean,
kg
_
0.266
0.247
0.218
Updated
st. dev.,
kg
_
0.308
0.277
0.246
Parameter: Copper
Distribution: Lognormal
Est.
mean ,
log kg
-0.437
-0.685
-0.570
-1.146
Est.
st . dev. ,
log kg
0.369
0.474
0.337
0.404
Updated
mean,
log kg
_
-0.565
-0.567
-0.711
Updated
st. dev.,
log kg
_
0.443
0.403
0.502
Parameter: Fluoride
Distribution: Normal
Est.
mean,
kg
24.4
25.4
24.7
24.0
Est.
st. dev.,
kg
3.79
3.49
3.29
4.17
Updated
mean,
kg
^
24.9
24.8
24.6
Updated
st. dev.,
kg
—
3.62
3.46
3.61
Table 8.4c INITIAL STATISTICS FOR SOURCE 3
Month
1
2
3
4
Parameter: BOD,
Distribution: Normal
Est.
mean,
kg
1165
900
1395
1080
Est.
st. dev. ,
kg
470
598
734
642
Updated
mean,
kg
...
1030
1150
1133
Updated
st. dev. ,
kg
...
555
648
643
Parameter: Phosphate
Distribution! Lognormal
Est.
mean,
log kg
2.12
2.20
2.12
1.85
Est.
st. dev. ,
log kg
0.339
0.157
0.268
0.2B6
Updated
mean,
log kg
_ _ „
2.16
2.16'
2.08
Updated
st. dev.,
log kg
...
0.265
0.264
0.313
Parameter! Suspended Solids
Distribution! Lognormal
Est.
mean,
log kg
3.33
3.13
3.40
3.30
Est.
st. dev. ,
log kg
0.218
0.282
0.312
0.175
Updated
mean,
log kg
...
3.23
3,29
3.29
Updated
at. dev.,
log kg
..w
0.277
0.302
0.274
Parameter:
Dissolved
oxygen
Est.
mean,
Bg/1
3.90
3.80
4.20
4.10
Updated
mean,
ng/1
^,»^
3.85
3.96
4.00
101
-------
Table 8.4d INITIAL STATISTICS FOR SOURCE 4, PIPE 1
Month
1
2
3
4
Parameter: Phosphates
Distribution: Normal
E*t.
. aean,
kg
_
0.225
-
0.755
Eat.
•t.dev. ,
kg
_
0.101
-
1.356
Updated
mean.
kg
.
-
-
0.490
Updated
•t.dev. ,
kg
-
-
-
0.925
Parameter: Suspended Solids
Distribution: Normal
Eat.
mean,
kg
-
13.3
-
13.7
Eat.
at.devi i
kg
-
4.21
-
3.23
Updated
Beaa,
kg
-
-
~
13.5
Updated
at. day.,
kg
-
-
V
3.38
Table 8.4e INITIAL STATISTICS FOR SOURCE 4, PIPE 2
Month
1
2
3
4
Parameter t Phosphate*
Distribution: Nornal
Eat.
•ean,
kg
.
3.20
-
4.35
Eat.
at.dev. ,
kg
_
0.526
-
4.096
Updated
•ean.
kg
_
-
-
3.78
Updated
•t.dev.,
kg
_
-
-
2.719
Parameter: Suspended Solids
Distribution: Normal
Eat.
nean,
kg
_
88.0
-
62.0
Eat.
•t.dev.,
kg
_
156.3
-
62.3
Updated
nean ,
kg
_
-
-
75.0
Updated
•t.dev.,
kg
_
-
-
108.2
102
-------
Table 8.5 EXPECTED DAMAGE AND PROBABILITY OF VIOLATION
o
1*3
Source
1
2
3
4
Pipe
1
1
1
1
2
Est. source
flow,
Ml/day
0.961
0.845
108
0.297
1.016
Stream
flow,
Ml/day
100
320
525
300
Parameter
PH
Lead
Chromium
Copper
Fluoride
BOD5
Phosphate
Suspended Solids
Phosphates
Suspended Solids
Phosphates
Suspended Solids
Expected
damage,
°ij
0.29
1.60
0.08
0.12
0.00
3.22
3.64
0.37
0.29
0.03
Probability
of no viola-
tion, pijtz
80.0
80.0
82.6
96.1
93.1
100.0
97.6
87.8
100.0
51.8
54.4
46.0
Expected
damage for
source, C.
1.60
0.12
3.64
0.29
Probability of
no violation
for source,
P^*.
64.0
74.0
85.6
13.0
-------
to sample the sources are given in Table 8.6, and the priority list is
given in Table 8.7. For the purposes of this example, it was assumed
that the sources could be sampled between 0 and 10 times. From the
table, one sees that Sources 1 and 3 should be sampled with higher
priority than Sources 2 and 4. This is due to the much larger expected
damage from the former sources. Source 4 appears relatively early in
the list, but most of the samples have low priority. This is because
the probability of violation is very large and therefore the chances are
that the source will be caught in violation after one or two visits.
Further sampling is therefore not necessary. Source 2 has a small
expected damage and a fairly large probability of no violation resulting
in a low sampling priority. Table 8.7 also gives the marginal return,
"cost" of undetected violations and resources used. The marginal
returns are decreasing (the list has been ordered in just this manner).
The "cost" of undetected violations is decreasing, and the resources
required are increasing as more sources are sampled.
If only, say, $10,000 were available for monitoring, then only the
sources with priority 1 through 18 would be monitored. The sampling
frequencies for this case are shown in Table 8.8. If, on the other
hand, a maximum allowed "cost" of undetected violations of, say, 1.0
were specified, then sources with priorities 1 through 19 would be
sampled. The sampling frequencies for this case are shown in Table 8.9.
The priority list in Table 8.7 also shows when the return from monitoring
(i.e. the marginal return) starts becoming negligible; the return, in
this case, for monitoring more than, say, 25 sources is very small.
VIII.3 SENSITIVITY STUDIES
This subsection investigates the effect of various changes in the inputs
and design parameters of the example just discussed.
104
-------
Table 8.6. RESOURCES NEEDED TO SAMPLE
Source
1
2
3
4
Field and
office costs
$525
$525
$525
$525
Laboratory
costs
$10.50
$23.00
$38.00
$30.00
Total Cost
ri
$535.50
$548.00
$563.00
$555.00
105
-------
Table 8.7 PRIORITY LIST OF SAMPLES FOR SIMPLIFIED EXAMPLE
PRIORITY
1
2
3
4
5
6
7
a
9
10
11
12
13
1«
15
16
17
Ifl
19
20
21
22
23
2tt
25
26
27
28
29
30
31
32
33
31
35
36
37
3fl
39
40
SOURCE
SAMPLED
1
3
3
1
3
3
3
0
1
3
3
3
1
3
3
1
1
1
4
2
1
2
2
1
2
1
2
2
2
4
2
2
2
4
4
4
4
4
4
4
MARGINAL
RETURN X100
.10774492
.09326524
,07989130
. 068992/18
,06*43515
.05862177
,05021559
.OU526206
,04417806
,ft«301464
.03684665
.03156296
,0282«H6l
.OH703693
.02315992
.01*11409
.01159902
.00742722
,OOS9025«
.00556719
,00«75588
.00^12025
.00301938
.00304534
.00225683
.00195003
.00167027
,00123616
.00091488
,00076974
.00067710
.00050112
.00037087
.00010038
,00001309
.00000171
,00000022
.00000003
,00000000
.oooonooo
COST OF
UNDETECTED
VIOLATIONS
5,07571
4,5534?
4,10603
3,73658
3,35334
3.02506
2, 74385
2,49364
2,25607
2.01519
.80*55
,63209
,48061
.32920
.19951
.10251
1,04039
1,00062
,96786
,93735
.91138
,88931
.87260
.85629
.84392
,83348
.62432
,81755
.81254
,80826
,80455
.80181
.79978
.79932
.79915
.79914
,79914
.79911
,79914
.799J4
RESOURCES
REQUIRED
535.50
1095.50
1655.50
2191, CO
2751.00
3311,00
3871.00
4426,00
4961,50
5521,50
6061.50
66Ul,bO
7177.00
7737. On
8297.00
8832.50
9368.00
9903,50
10453.50
11006.50
11542,00
12090,00
12636,00
13173.50
13721.50
14257.00
11805.00
15353,00
15901.00
• 16456,00
17004,00
17552,00
18100,00
18655,00
19210.00
19765.00
20320.00
2.0675,00
21430,00
21985.00
106
-------
Table 8.8 FINAL ALLOCATION GIVEN MONETARY BUDGET
FIM&L. ALLOCATION
BUDGET 10000.00
SOURCE
KIN NO,
SAMPLES
REQUIRED
MAX NO.
SAMPLES
ALLOWED
TI*ES
SAMPLED
RESOURCES
USED
COST OF
UN.P6TECTEO
VIOLATIONS
1
2
3
4
0
0
0
0
10
10
10
10
7
0
10
1
3746.50
.00
5600.00
555.00
.07061
.11738
.77*76
.03767
TOTAL RESOURCES USED 9903.50
FINAL COST OF UNDETECTED VIOLATIONS 1.00062
Table 8.9 FINAL ALLOCATION GIVEN MAXIMUM ALLOWED COST OF
UNDETECTED VIOLATIONS
FINAL ALLOCATION
MAXIMUM ALLOWED COST OF UNDETECTED VIOLATIONS
1.00000
NP. MAX NO.
SAMPLES SAMPLES
SOURCE REQUIRED ALLOWED
COST OF
TIMES RESOURCES UNDETECTED
SAMPLED USED VIOLATIONS
1
2
3
4
0
0
0
0
10
JO
10
10
7
0
10
2
3746.50
.00
5600.00
1110.00
.07081
.11738
.77476
.00491
TOTAL RESOURCES USED 10458.50
FINAL COST OF UNDETECTED VIOLATIONS .96786
107
-------
Distribution
In order to check the sensitivity of the normal assumption versus log-
normal assumption in the distribution of pollutants, the loadings
of phosphate and suspended solids in Source 3 are now assumed normally
distributed. (The self monitoring data are given in Table 8.3c. They
are identical to the previous example.) The expected damage and probability
of no violation for phosphates are now 3.53 and 98.5% respectively, and
the expected damage and probability of no violation for suspended
solids are 0.41 and 76.0% respectively. These numbers can be compared
with the analagous values in Table 8.5. The major difference is in the
suspended solids where both the expected damage and probability of
violation changed by about 10%. The expected damage for the source is
now 3.54 (compared to 3.64), and the probability of no violation for the
source is 74.9% (compared to 85.6%). Table 8.10 gives the priority list
for this case. The priority ordering is slightly changed. It is therefore
seen that changing the distributional form will affect the sampling
frequencies by a small, but not negligible, amount.
Correlation
The effect of assuming that the constituents of a source were correlated
versus uncorrelated is investigated by first assuming that the constituents
of Source 2 are completely correlated. The constituents of the other
sources are assumed uncorrelated, as in the original example* The pro-
bability of no violation for source 2 is 82.6% as opposed to 74% for the
original example. The priority list for this case is given in Table
8.11. Comparing this table with Table 8.7 shows little change - the
priorities for source 2 have increased slightly.
Now assume that the constituents for all the sources are completely
correlated. The probabilities of no violation for sources 1,2,3 and 4
are 80.0%, 82.6%, 87.8% and 28.9% respectively.
108
-------
Table 8,10
PRIORITY LIST, CONSTITUENTS IN SOURCE 3
ALL NORMALLY DISTRIBUTED
PRIORITY LIST OF SAMPLES
PRIORITY
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
2 1
22
23
24
25
26
27
28
29
30
31
32
33
3ft
35
36
37
38
39
40
SOURCE
3
3
1
3
1
3
3
4
1
3
1
3
3
1
3
3
1
1
4
2
I
?
2
1
2
1
2
2
2
4
2
2
2
4
4
4
4
4
4
4
MARGINAL
RETURN X100
,15868572
.11880537
.10774492
,08894762
.06899248
.06659361
.04995753
.04526206
.04417606
.03732751
.02328*61
.0279^649
.02092307
.01811409
.01566476
.01172795
.01139902
.00742722
.00590254
.00556719
.00U75568
• PC-4J?0?5
,01)304938
.00304534
.OC225683
.00195003
.00167027
,001?3M6
. ooQ^Hae
,00076974
.0.7067710
.00.050112
.00037087
.00010038
,00001309
.00000171
,00000022
.00000003
,00000000
.ooooocoo
COST OF
UNDETECTED
VIOLATIONS
4,65774
3,99243
3.41546
?. 91735
2.54790
2.17497
1,89577
1.64456
1.40799
1 . 19696
1.04747
. 99097
.77380
,67660
.5B9C9
.5*340
• <*t>129
.42152
, 3«£76
.35325
.33278
,31020
.29349
.27718
.26482
.25437
.24522
•23*«5
.23343
«?29]6
.22545
.22270
.22067
.2201?
.22004
.22003
.22003
.22003
.22003
.22003
RESOURCES
560.00
1120. OC
| ^ejjjly.)
2215.50
2751,00
3311.00
3671,00
442**. 00
4961.50
S5?1.50
6J57.QO
6fc-J 7,00
7177.00
7712,50
6272,50
6632.50
9366.00
9903.50
10456.50
11006,50
11512,00
12090, O'O
1263'-., 00
13173.50
13721.50
14257. OP
1'4805.00
15353.00
15901,00
16'*56,00
1700<*,00
17552,00
1«100,00
18655,00
19210,00
19765,00
20320.00
cOt.75.00
21^30.00
21985.00
109
-------
Table 8.11 PRIORITY LIST, SOURCE 2 CONSTITUENTS CORRELATED
PRIORITY LIST OF SAMPLES
PRIORITY
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
2t
22
23
24
25
26
27
28
29
30
31
32
33
3^>
35
36
37
36
39
40
SOURCE
SA.'iPLEO
1
3
3
1
3
3
3
4
1
3
3
3
1
3
3
1
1
1
4
1
2
2
1
2
2
J
2
2
2
2
2
4
2
4
4
4
4
4
a
4
MARGINAL
RETURN X100
.10774492
.09326524
.07969130
.06899248
.06843515
.05662177
.05021559
.04526206
,04417806
.04301484
.03664665
.03156296
.0282*861
.02703693
.02315992
.01611409
.011599*2
.00742722
,0059025£i
.004755*8
.00371715
.00307210
.00304534
.00253898
.00209838
.00195003
.00173423
.00143328
.00116456
.00097899
,COOHf»9t 0
.00076974
,000o6870
.00010038
.00001309
.00000171
.00000022
.00000003
.00000000
.00000000
COST OF
UNDETECTED
VIOLATION'S
5.07571
4.55342
4.10603
3.73656
3.35*34
3.02506
2.74365
2.49264
2.25607
2.01519
,80865
.63209
.46061
.32920
.19951
.10251
.04039
1.00062
.967B6
,94?3<>
.92202
,90519
,88688
,67497
.86347
,85303
.34352
.83567
.82919
,62381
.81936
.81510
,8ii
ision.oo
16655.00
19210.00
19765,00
20320.00
20375.00
21430. CO
21965.00
110
-------
There is little change between the priority list for this case (Table 8.12)
the original priority list (Table 8.7).
No strong conclusions can be drawn from these examples. Cases can
clearly be devised where the priority list will be very sensitive to the
correlation assumption. However, from these examples it is seen that
*-n many cases the priorities will be insensitive to this assumption.
cjjjvlmizing Number of Undetected Violators
The objective of the Resource Allocation Problem can be changed to
Minimize the number of undetected violators (no "cost" due to environmental
damage) by setting all the expected damages in the priority procedure to
°ne. The statistics and the probability of not violating will be the
same as for the original problem. The new priority list is given in
Table 8.13. As would be expected, the priority list is very different
that for the case which considered damages.
Fast Data
past data are discounted by ensuring that the confidence parameters n
and v in the Bayesian update formula do not get too large. This is
accomplished by specifying that n <_ kn v' and v _< kv v1 where n1 and v1
are the confidence parameters for the month being used to update the
8tatistics. In the original example kn - kv - 3.0. Let us now assume
that k - k -1.5. The initial statistical description will therefore
, n v
QePend more strongly on the data in the months closer to the start of the
period.
8.14 compares the initial statistical description, at the start of
for the cases when k - k = 3.0 and k - k - 1.5. By
n v n v
comparing this table with the initial data (Tables 8.3a through 8.3e) it
is evident that the data for month 4 are more strongly felt for the
where k - k -1.5 than for the case where k - k - 3.0.
n v n v
111
-------
Table 8.12 PRIORITY LIST, SOURCES' CONSTITUENTS ALL
CORRELATED
PRIORITY LIST OF
PRIORITY
1
2
3
4
5
6
7
8
q
10
11
12
13
14
IS
16
17
18
19
20
21
22
23
21
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
SOURCE
S4^°L^f>
3
3
3
1
3
1
3
3
4
1
3
3
1
3
1
1
1
1
1
1
4
i
2
2
2
U
2
2
2
2
2
2
2
a
4
4
4
4
4
4
COST OF
MARGINAL UND&TECTFO
RFTURN *tOO VIOLATIONS
.07961626 5,20681
.06967035 4.81556
.06131743 4.47;?J8
.OS966003 4,15163
.05381148 3.85028
.04769870 3.59379
.04722435 3.32933
.04144355 3.09725
,03895?7a 2.88106
.03832751 ?. 67581
,03637039 2,47214
,03t«Mfl;?ii 2.29340
.03066384 2.12917
,02*01 109
.0245*2;??
.02454055
,01963632
.01571296
,01257317
.01006i)79
,00980149
.00805042
.00371715
,00307210
.00253898
.00246630
,0020«»e3*
.00173423
.00143326
,00119456
,00097899
,00080910
,00066«70
. 00162058
.00015615
,00003929
.97230
.83464
.70323
.59807
.51393
.44660
,39?73
.33833
.29522
.27485
.25801
,?"410
.23041
.21891
.20941
.20155
.19506
.18970
.16526
,18160
.17816
.17729
.17707
,OOOCO«*89 1.17702
.00000249 1,17700
.00000063 1.17700
.00000016 1.17700
RESOURCES
REQUIRED
560.00
1120.00
U80.00
2215.50
2775.50
3311.00
3871.00
4431.00
4986.00
5521.50
6081.50
6641.50
7177.00
7737.00
8297.00
8832.50
9368.00
9903.50
10439.00
10974.50
11529.50
12065.00
12613.00
13161.00
13709,00
14264,00
14812.00
15360.00
15908.00
16456,00
17004.00
17552,00
ISIOO.CO
1*655.00
19210,00
19765.00
20320.00
20875.00
21430.00
219*5.00
112
-------
Table 8.13 PRIORITY LIST, MINIMIZE
PRIORITY LIST OF SAMPLES
PHIO«TTY
1
2
3
a
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
36
39
40
SOURCE
SAMPLED
a
1
2
1
2
1
2
3
3
4
2
3
1
3
2
3
3
1
2
3
3
2
3
1
3
2
1
2
2
1
4
1
1
4
a
a
4
4
4
4
MARGINAL
.15668323
.'0671*497
.047427B1
.04300785
.03510111
,02753929
,025978! £
.02560657
.02193467
,02043?7fc
,01922634
.01878931
.01763427
.01609498
. 01"22933
,01378701
,01181000
.01129178
,01053106
,01011648
.00866581
,00779399
,007«?316
,00723049
,00635870
.00576630
,00462991
.00426909
.00315954
,00296468
,00266460
,00189836
.09121559
,00034748
.00004531
.OOOOC591
,0000«?077
.OCOOP01C
.00000001
,00000000
COST OF
UNDETECTED
VIOLATIONS
3,13041
2.77074
2.51084
2.26f*53
2.06617
1,94070
1,7983"
1,65494
1,53211
1.41871
1.31335
1.20*13
1.11370
1.02356
.94559
.86838
.80224
.74178
.66407
.62741
.57*09
.53617
,49*160
.'15589
.42C28
,38867
.36367
.34048
.32316
.30729
.29250
.28233
.27583
.27390
,27364
,27361
,27361
.27361
.27361
,?7361
RESOURCES
1090,50
163ft. 50
2174.00
27*2.00
3257.50
38C5.50
4365.50
49?5«50
5^60.50
6&P6.50
fei>38.50
7124,00
7664.00
8232.00
8792.00
9552.00
9t87.50
10435.50
10995.50
11555.50
1?1
-------
Table 8.14 EFFECT OF DISCOUNTING PAST DATA
Source
1
2
3
4
Pipe
1
1
1
1
2
Parameter
pH - Max
pH - Min
Lead
Chromium
Copper
Fluoride
BOD 5
Phosphate
Suspended Solids
Phosphate
Suspended Solids
Phosphate
Suspended Solids
k = k = 3
n v
Updated
mean
8.12
8.12
0.78
0.218
-0.711
24.6
1133
2.08
3.29
0.490
13.5
3.78
75.0
Updated
st. dev
0,92
1.14
1.45
0.246
0.502
3.61
643
0.313
0.274
0.925
3.38
2.72
108
k = k =1.5
n v
Updated
Fie fin
8.12
8.12
0.74
0.200
-0.798
24.5
1138
2.03
3.30
0.490
13.5
3.78
75.0
Updated
st . e\(*v
0.87
1.08
1.42
0.221
0.522
3.68
651
0.325
0.259
0.925
3.38
2.72
108
-------
Compliance Data
The effect of compliance data (effluent data obtained by the monitoring
agency) on the initial statistical descriptions of the source effluents
is investigated in this subsection. Suppose that Source 2 is monitored
twice in month 3. The compliance data for the two visits are given in
Table 8.15. Comparison of these data with the self-monitoring data for
Source 2, month 3 (Table 8.3b) shows that the compliance data for chromium
and copper are near the monthly maximum self-monitoring value. For
fluoride, one compliance value is near the maximum, the other is below
the mean.
Table 8.15 COMPLIANCE DATA - SOURCE 2, MONTH 3
Parameter
Chromium
Copper
Fluoride
Data Point
No, 1, kg.
0.53
1.80
28.0
Data Point
No. 2, kg
0.70
2.00
16.0
In the procedure that combines the self-monitoring and compliance monitor-
ing data, there is a design parameter, V, that specifies the relative
confidence one has in the self-monitoring as compared to the compliance
monitoring data. For example, a value of Y - 2 implies that one has
twice as much confidence in the compliance monitoring data as in the self-
monitoring data. In the examples that follow, Y will take on values 2 and A.
Tables 8.16a and 8.16b show the effect of the compliance data on the initial
statistical description; these tables are analogous to Table 8.4b. The
row opposite month 3 is the estimated mean and standard deviation for month
3 without the compliance data. The row opposite 3* includes the compliance
data. The tables show that the estimated mean and standard deviation for
the month is substantially increased for chromium and copper. For fluoride,
the mean is slightly decreased while the standard deviation is increased.
The effect of the compliance data on the estimates is clearly much greater
for Y « A than for Y - 2. By comparing the values of the updated mean and
standard deviation at the end of month 4 in Tables 8.4b, 8.16a, and 8.16b,
115
-------
Table 8.16a
INITIAL STATISTICS FOR SOURCE 2 WITH COMPLIANCE
MONITORING DATA: y - 2
Honth
1
2
3
3*
4
Parameter: Chromium
Distribution: Normal
Eat.
mean,
kg
0.216
0.313
0.214
0.280
0.132
Esc.
st.dev. ,
kg
0.321
0.297
0.214
0.261
0.070
Updated
mean,
kg
„_
0.266
0.271
0.236
Updated
st.dev. ,
kg
....
0.308
0.287
0.2S9
Parameter 1 Copper
Distribution: Lognormal
Est.
mean,
log kg
-0.437
-0.685
-0.570
-0.437
-1.146
Est.
st.dev.,
log kg
0.367
0.474
0.337
0.471
0.404
Updated
sean,
log kg
»_
-O.S65
-0.514
-0.672
Updated
st.dev. ,
log kg
0.443
0.455
0.551
Parameter: Fluoride
Dlatribution: Normal
Est.
mean,
kg
24.4
25.4
24.7
24.3
24.0
Est.
st.dev.,
kg
3.79
3.49
3.29
4.23
4.17
Updated
Bean,
kg
.
24.9
24.7
24.5
Updated
st.dev.,
kg
_—
3.62
—
3.84
3.88
*. Includes compliance monitoring data
Table 8.16b
INITIAL STATISTICS FOR SOURCE 2 WITH COMPLIANCE
MONITORING DATA: y = 4
Month
1
2
3
3*
4
Parameter) Chromium
Distribution! normal
Est.
mean,
kg
0.216
0.313
0.214
0.332
0.132
Est.
it.dev.,
kg
0.321
0.297
0.214
0.277
0.070
Updated
mean,
kg
___
0.266
—
0.291
0,251
Updated
st.devi ,
kg
—
0.308
0.295
0.268
Parana ter t Copper
Distribution: Lognormal
Bat.
mean,
log kg
-0.437
-0.685
-0.570
-0.333
-1.146
Est.
st.dev.,
log kg
0.369
0.474
0.337
0.515
-0.672
Updated
mean,
log kg
— —
-0.565
-0.473
-0.642
Updated
st.dev.,
log kg
-w~
0.443
___
0.486
0.583
Parameter! Fluoride
Dlatribution: Normal
Eat.
aean,
kg
24.4
25.4
24.7
23.8
24.0
Eat.
st.dev.,
kg
3.79
3.49
3.29
4.80
4.17
Updated
mean,
kg
•,„
24.9
— —
24.5
•24.4
Updated
st.dcv.,
kg
_.....
3.62
_
4.12
4.07
* Includes compliance monitoring data
116
-------
Table 8.17 EXPECTED DAMAGE AND PROBABILITY OF NO VIOLATION FOR SOURCE 2
Y
BCD*
2
4
Parameter
Chromium
Copper
Fluoride
Chromium
Copper
Fluoride
Chromium
Copper
Fluoride
Expected
damage
0.08
0.12
0.00
0.08
0.14
0.00
0.08
0.17
0.00
Probability of
no violation, Z
82.6
96.1
93.1
79.5
93.8
92.2
77.1
92.0
91.7
Expected daaage
for source
0.12
0.14
0.17
Probability of no violation
for source » Z
74.0
68.0
65.0
* Ho compliance data
-------
one can see the effect of the compliance monitoring data on the initial
statistical description. Again, the effect is substantial. Table 8.17
compares the value of the expected damage and probability of no violation
for source 2 for the three cases: no compliance data and compliance data
for y - 2 and y = 4. The compliance data, for this case, have increased
the expected damage and decreased the probability of no violation.
Upstream Concentration
The previous examples in this section have assumed that the concentra-
tion of each constituent, upstream from each source, has caused zero
environmental damage. In this subsection, we will investigate the
effect of changing the assumed upstream concentrations.
Five cases will be considered. Case I, for comparison purposes, cor-
responds to the zero upstream damage case described in Section VIII.2.
For Cases II and III the upstream concentration is set to cause damage
levels of 2 and 4 in the receiving waters (recall that "2" corresponds
to "excellent" water quality and "4" corresponds to "acceptable" water
quality). In Cases IV and V the upstream concentration is also set to
cause damages of 2 and 4; however, in this case, the expected damage
for each constituent that is calculated is the incremental damage, that
is, the expected damage due to the source's constituent minus the dam-
age in the receiving waters that exists if that constituent were not
present in the effluent. For reference, the five cases are described
in Table 8.18. Table 8.19 compares the expected damage for the five
cases. The table shows how the damage increases as the assumed upstream
concentration increases (Cases I, II and III). The incremental damage,
however, actually decreases for most cases (Cases I, IV and V). This
is because the damage functions are, for the most part, concave in shape.
The one exception, in this example, is the fluoride in Source 2« The
presence of fluoride in a stream does not cause any damage (it is actu-
ally beneficial) below a certain threshold. Above that threshold dam-
age increases rapidly. Thus, for fluoride, the incremental damage is
118
-------
zero under zero upstream concentration; It increases greatly for an up-
stream concentration causing a damage of 2; and it decreases for an
upstream concentration causing a damage of 4 (the damage curve is con-
cave for large values of concentration).
The priority lists for the five cases are compared in Table 8.20.
Comparing Cases II and III with Case I, it is seen that Sources 2 and 4
appear much higher on the list. Source 2 appears higher because of
the above large increase in expected damage due to fluoride. Source 4
appears earlier because it now has an expected damage comparable with
the other sources;-its expected damage in Case I was much smaller than
the expected damage for Sources 1 and 3. Comparing Cases IV and V
with the other cases, it is seen that Source 1 has lower sampling prior-
ity. Source 4 also appears lower on the lists. These phenomena both
reflect the lower expected incremental damage of Sources 1 and 4 as
compared to Sources 2 and 3.
Table 8.20 shows the large sensitivity of the priorities to changes in
assumed upstream concentration. It is preferable to use the incre-
mental expected damage over the "regular" expected damage since one is
basically interested in the damage caused by a source and not just by
the expected damage in the river (which will also depend on the up-
stream concentration). The value of assumed upstream concentration
used should reflect the average condition of the stream in a region
containing the source.
119
-------
Table 8.18 CASES CONSIDERED FOR SENSITIVITY STUDY
OF UPSTREAM CONCENTRATION
Case
I
II
III
IV
V
Assumed
upstream level
of damage
0
2
4
2
4
Incremental
damage
___
No
No
Yes
Yes
Table 8.19 COMPARISON OF EXPECTED DAMAGE FOR VARIOUS
ASSUMED UPSTREAM CONCENTRATIONS
Source
1
2
3
4
Constituent
pH
Lead
Chromium
Copper
Fluoride
BOD5
Phosphates
Suspended
Solids
Phosphates
Suspended
Solids
Expected Damage ,
Case I
0.29
1.60
0.08
0.12
0.00
3.22
3.64
0.37
0.29
0.03
Case II
2.13
2.45
2.05
2.03
3.49
4.29
4.59
2.03
2.28
2.02
Case III
4.02
4.40
4.00
4.00
4.49
5.20
5.19
3.67
4.09
4.00
Case IV
0.14
0.47
0.05
0.03
1.53
2.63
2.93
0.37
0.29
0.03
Case V
'
0.05
0.42
0,01
0.01
0.54
1.83
1.88
0.36
0.10
0.02
_-^
120
-------
Table 8.20 PRIORITY LISTS, VARIOUS ASSUMED UPSTREAM
CONCENTRATIONS
Priority
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Source Sampled
Case I
1
3
3
1
3
3
3
4
1
3
3
3
1
3
3
1
1
1
4
2
1
2
2
1
2
Case II
4
2
1
2
3
1
3
2
3
3
1
2
3
3
2
4
3
1
3
2
3
3
1
2
2
Case III
4
1
2
1
2
3
1
2
3
3
2
3
4
1
3
2
3
3
1
2
3
3
2
3
1
Case IV
3
2
3
3
2
3
4
3
2
3
1
3
2
3
2
3
1
3
2
Case V
3
3
3
3
1
3
2
3
2
3
1
4
3
2
3
3
1
2
2
•*• i ^
f
2
2
1
2
4
2
1
2
2
1
121
-------
SECTION IX
DEMONSTRATION PROJECT
The priority procedure will be demonstrated, in this section, using
data supplied by the State of Michigan, Department of Natural Resources.
The data, taken over a two year period, is from 30 industries and muni-
cipal treatment plants. Table 9.1 gives a brief description of the
various sources. As can be seen, a variety of pollutants and types of
plants have been included.
The purpose of the demonstration project is two-fold. First, it will
demonstrate the procedure on the types of data bases that will be avail-
able to the monitoring agencies. Second, it will compare the perform-
ance of the procedure with another, simpler, priority setting procedure.
IX.1 DESCRIPTION OF DATA AND ASSUMPTIONS
The quality of the data varied greatly from source to source. For
several sources, there were twenty four months of data; for others,
there was as little as six. Some sources sampled their effluent daily,
others weekly, and others monthly. Standards were not set for approxi-
matley 20% of the constituents reported. In order to test the priority
procedure with as many constituents as possible, reasonable hypotheti-
cal standards were established for these constituents. Also, most of
the standards were on the concentration of the constituent in the efflu-
ent. Since, in the future, standards will typically be on the mass
loading, it was decided to transform the given standards into mass
loading standards by multiplying them by the daily effluent flow of the
source, given on the permits.
The value of the upstream flow of the receiving waters was taken to be
the seven-day, ten-year low flow. This value will give a much smaller
flow than would be encountered in a typical month (it was used because
122
-------
Table 9.1 DESCRIPTION OF EFFLUENT SOURCES
Source
fiumber
1
2
3
4
5
6
7
8
Pipe
Humber
1
2
1
.2
1
1
2
1
1
2
1
1
Avg. daily
flow, MGD
0.07
0.0035
0.106
0.124
0.085
0.2
0.08
720.
4.436
8.07
0.75
0.14
Type of
plant
Chem
Porcelain
man.
Porcelain
man.
Auto parts
Power
Chem
Chem
Chem
Type of waste, %*
Proc
100
_ —
90
25
40
1
1
1
1
46
70
Cool
___
2
10
75
40
99
100
98
99
99
54
30
San
98
—
20
1
__.
Constituents
pH, chromium, nickel, chloroform extract
BOD, suspended solids, chloride
Phosphorus, pH, suspended solids, chloro-
form extract
Phosphorus, pH, suspended solids, chloro-
form extract
pH, suspended solids, phosphorus
pH, suspended solids, chloroform extract
pH, suspended solids, chloroform extract
pH, chloride
pH, oil-grease, phenol, COD
pH, oil-grease, phenol, COD
pH, suspended solids, phosphorus,
fluoride , copper , lead
pH, suspended solids, phosphorus, cyanide,
fluoride, chromium, copper, lead, chloro-
form extract
K>
U>
* "Proc", "Cool" and "San" denote processing, cooling and sanitary waste, respectively.
-------
Table 9.1 DESCRIPTION OF EFFLUENT SOURCES (Cont'd)
Source
number
9
10
11
12
13
14
15
16
17
Pipe
number
1
1
1
1
1
1
1
1
1
2
3
4
Avg. dally
flow, MGD
5.
0.35
0.69
1.1
0.129
0.38
0.223
0.184
0.53
0.123
0.137
0.828
Type of
plant
Auto
Auto
Auto body
Auto
Auto parts
Auto
Electronics
Metal
Type of waste, %*
Proc
40
100
100
24
14
57
100
20
100
Cool
60
76
86
43
80
100
100
100
___
San
___
Constituents
BOD, pH, suspended solids, chromium,
nickel, chloroform extract
pH, suspended solids, phosphorus, chloro-
form extract, oil-grease
pH, cyanide, chromium, copper, nickel
BOD, pH, suspended solids, chloroform
extract
BOD, pH
pH, suspended solids, cyanide, chromium,
copper, chloroform extract
pH, lead
pH, suspended solids, oil-grease, mercury
Chloroform extract
Chloroform extract
Chloroform extract
pH, suspended solids, phosphorus,
aluminum, chloroform extract
* "Proc", "Cool" and "San" denote processing, coolipg arid-sanitary, waste, respectively.
-------
Table 9.1 DESCRIPTION OF EFFLUENT SOURCES (Cont'd)
Source
number
18
19
20
21
22
23
24
25
26
27
Pipe
number
1
1
1 -
1
2
3
4
1
1
1
1
1
1
Avg. daily
flow, MGD
10.
1.3
0.527
Unknown
10.
0*114
0*718
43.6
1.91
1.54
Type of
plant
Chenrv
Glass
Refrig.
man.
Power
STP1"
STP
STP
STP
STP
STP
Type of waste, %*
Proc
86
Cool
100
14
100
100
100
100
San
100
100
100
100
100
100
Constituents
BOD, suspended solids, ammonia, dissolved
solids
Suspended solids, chloroform extract
pH, suspended solids, phosphorus
pH, chloride
BOD
Suspended solids
Suspended solids, BOD
DO, BOD, suspended solids, phosphorus
BOD, suspended solids, phosphorus
BOD, suspended solids
BOD, suspended solids
DO, BOD, suspended solids, phosphorus
BOD, suspended solids, phosphorus
to
Ul
* "Proc", "Cool" and "San" denote processing, cooling-and sanitary waste, respectively,
f Sewage treatment plant.
-------
Table 9.1 DESCRIPTION OF EFFLUENT SOURCES (Cont'd)
Source
number
28
29
30
Pipe
number
1
1
1
Avg. daily
flow, MGD
28.0
0.960
9.3
Type of
plant
STPt
STP
STP
Type of waste, %*
Proc
Cool
San
100
100
100
Constituents
DO, BOD, suspended solids, phosphorus
BOD, suspended solids
BOD, suspended solids
* "Proc", "Cool" and "San" denote processing, cooling and sanitary waste, respectively,
t Sewage treatment plant.
-------
it was readily available). In order to obtain better estimates of
the environmental damage that is likely to occur, it is suggested that
one use the minimum average monthly flow where the minimum is taken
°ver the months in the monitoring period.
The distributions used for the various constituents were obtained as
follows: The mean and standard deviation were first estimated for all
constituents under the normal distribution assumption. For those con-
8tituents whose standard deviation was greater than the mean, it was
inferred that the normal distribution did not give a good fit to the
data. The distribution assumption for these constituents was changed
to lognormal. This method of assigning distributions is based on the
Allowing considerations. Under the normal assumption,there is a fin-
ifce probability of having a negative discharge. Since this is almost
always impossible, this probability is interpreted as being the prob-
ability of having a zero discharge (i.e. the normal density function
8 changed so that all the area to the left of zero is put at zero).
*hus, the above method of assigning distributions, though somewhat
arbitrary, is based on the fact that if, under the normal distribution
assumption, the standard deviation is greater than the mean, then there
8 a large probability that the source will not produce that consti-
tuent. Since, typically, the constituent will be produced, a lognormal
Distribution is judged more appropriate.
assumptions made were:
The BOD-DO transfer coefficient, K^Q^rvQ* was assumed to be 0.5
for all sources.*
The saturation level of DO, DOSAT, was assumed to be 9 mg/1 for
all sources.*
? "—'
BOD-DO and DOSAT are defined in Section VI.1
127
-------
(3) The concentration of dissolved oxygen in an effluent was assumed
to be 0 mg/1 in the sources for which there was a standard for
BOD and which did not report their DO discharge.
(4) The design parameters k and k , which determine the degree of
discounting of past data, were set to 3. *
(5) The constituents of a source are assumed uncorrelated.
(6) The concentration of the pollutants upstream from the source (CU)
were assumed to be at a level to cause zero damage.
Table 9.2 lists the assumed monetary resources required to sample the
sources. The amounts are a function of two quantities: the number of
outfalls of the source and the number and types of pollutants sampled.
The exact method used to determine the resources is given in Appendix D.
* k and k, are defined in Section V.2.
n v
128
-------
Table 9.2 RESOURCES REQUIRED TO MONITOR
THE SOURCES
Source
1
2
3
4
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
22
23
24
25
26
27
28
29
30
Required Resources
$ 588.00
591.00
543.00
571.00
576.00
566.00
603.50
583.00
568.00
565.50
568.00
548.00
578.00
535.00
558.00
943.50
565.00
545.00
543.00
563.00
560.00
550.00
550.00
563.00
560.00
563.00
550.00
550.00
129
-------
IX.2 PERFORMANCE OF PROCEDURE
In this subsection the Resource Allocation Program is used to obtain
sampling frequencies based on the demonstration case data. Three
examples are considered. For each of the examples, the monitoring period
(i.e., the time period for which the allocation is based) is assumed to be
six months. The examples are:
Case I. Use the first twelve months of data to obtain the initial
source statistical descriptions. Determine the sampling
frequencies for the following monitoring period (i.e.,
months 13 through 18).
Case II. Use the data from months 13 through 18 to update the
statistical description of the sources used in Case I.
Determine the sampling frequencies for the following
monitoring period (i.e., months 19 through 24).
Case III. Obtain a revision of the sampling frequencies obtained in
Case II,under the assumption that the sampling has to be
interrupted in the middle of a sampling period due to a
measurement of very poor water quality in a given river
segment. (It is desired to sample two sources, which are
expected to cause the poor quality, twice in the remainder
of the monitoring period.)
This subsection is concluded with a comparison of the performance of
the priority procedure developed in this report with a procedure that
assigns sampling frequency on the basis of source flow.
Case I
The source expected damage and probability of no violation obtained
from the first 12 months of self-monitoring data is given in Table 9.3.*
The statistical description of the sources' constituents, and the
* Sources 5 and 21 are not included in this example due to insufficient
data.
130
-------
Table 9.3 DATA FOR CASE 1
SOURCE
PNV
EXP. DAMAGE
1
2
3
4
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
22
23
24
25
26
27
2A
29
30
.6^0361
.366900
•971J05
.434765
.966616
.111617
.006504
.089683
.072174
.814871
.138667
•921435
.964860
.000060
.981*22
•316116
•034052
.80926$
.696925
.117219
.716378
.992393
•050130
.396621
.000000
•663566
.883175
.256403
1,*24476
1.155373
.000615
3.428362
4.047932
3.517227
1.40«283
7.781987
4.489711
2.719460
5.660459
3.340151
2.432983
2.8J3442
4.072093
4.018275
8.9«?007
5.787722
.38B792
5.959835
4.?20618
I.fl9fl906
3.746962
.601895
6.574204
6.318164
6.091859
1.090917
Probability of no violation.
131
-------
expected damage and probability of no violation for each constituent
used to obtain the information in Table 9.3, is given in full in
Appendix G.I.
In this example, the upper and lower bounds on the sampling frequencies
are 0 and 3 respectively and the monitoring agency's budget is $25,000.
Table 9.4 gives the resulting priority list and Table 9.5 gives the
sampling frequencies. Comparing these tables with Table 9.3, it is
seen that those sources sampled most often and/or with highest priority
have high expected damage and low probability of no violation.
Case II
The assumptions and constraints for Case II are identical with those
for Case I. The new expected damage and probability of violation
for each source, based on the updated statistics is given in Table 9.6.
The large effect of the update procedure on this data can be determined
by comparing this table with Table 9.3. For example, the probability of no
violation for source 20 went from 69.7% for Case I to 33.3% for Case II
while the expected damage went from 0.387 to 0.737. The updated statistical
description, the expected damage and the probability of no violation, for
each constituent, is given in Appendix G.2.
Table 9.7 gives the priority list for this example, and Table 9.8 gives
the sampling frequencies. There are large differences in some of the
sampling frequencies for Cases I and II. For example, Source 6 was not
sampled in Case I (Table 9.5) but was sampled two times in Case II
(Table 9.8). Conversely, Source 11 was sampled three times in Case I
but was not sampled in Case II. These changes are due to changes in
the expected damage and probability of no violation for the sources.
132
-------
Table 9.4 PRIORITY LIST: DEMONSTRATION PROJECT, CASE I
PRIORITY
i
2
3
4
5
A
7
8
9
10
11
12
13
10
15
16
17
1*
19
20
21
22
23
2a
25
26
27
29
29
30
31
32
33
30
35
36
37
3*
39
40
41
42
43
44
45
46
47
46
49
5PURCE
SAMPLED
IB
9
27
2?
1?
10
25
7
15
28
a
17
28
8
23
19
28
19
23
a
30
2
19
29
12
29
1
23
22
9
29
17
11
11
1
26
4
7
11
2
10
1«
13
1
11
13
30
25
17
MARGINAL
RFTURN *100
1,52«76276
1.215107?6
1.17396499
.934U9891
,858379?!
.73339351
.646977S5
.55205745
.52535465
,37755728
,33<»37480
,29125971
.25053405
,23183490
.21376098
,20252984
,16624579
.16390496
.15313359
.14754831
.14709458
.14519254
.13264635
.12939626
.1 190?7?2
.114279S8
.11159047
.10970148
.10954107
.10897437
.10092891
,09207177
.08902778
.07254612
.07145823
.Q653M58
.064148*5
.06161888
,05«11570
.05327107
.05293204
.05205818
.04788692
.04575910
.04412466
.04065799
,03800971
.03256254
.02910533
COST OF
UNDETECTED
VIOLATIONS
95,09519
88.0111?
61.4369?
76.17569
71.30014
67.13447
63.57610
60.45145
57.63818
55,51253
53,57470
50.82667
49.41616
48.01704
46,81996
45.71619
44.78023
43.88694
43.02940
42.18690
41.37788
40.51979
39,79687
39.08519
38.40911
37.78058
37.12442
36.51010
35.89338
3S. 25806
34.70295
33.83425
33.33080
32.92055
32.50038
32.13240
31,76611
31,«1734
31.08304
30,76821
30.46756
30.17343
29.9U01
2«,64195
29,40014
29,17734
28.96628
28,78919
28.51456
RESOURCES
REOUIRfcO
565.00
1148.00
1708.00
2271.00
2839.00
3407.00
3957.00
4523.00
5058.50
5621.50
6102.50
7136,00
7699.00
8302.50
8862.50
9407.50
9970.50
1051S.50
11075.50
11646,50
12196.50
12787.50
13332.50
13882.50
14450.50
15000.50
15588,50
16148,50
16711.50
17294.50
178«4.50
16788.00
19353.50
19919,00
20507.00
21070.00
21641.00
22207.00
22772.50
23363.50
23931.50
24496.50
25044.50
25632.50
26180.50
26728,50
27278.50
27828.50
28772.00
133
-------
Table 9.5 SAMPLING FREQUENCIES: DEMONSTRATION PROJECT,
CASE I
BUDGET 25000.00
MIN NO. H4X NO.
SAMPLES SAMPLES
SOURCE REQUIRED ALLOWS
i
2
3
a
6
7
8
9
10
11
12
13
14
15
16
17
10
19
20
22
21
24
25
26
27
29
29
30
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
TIMES
SAMPLED
2
2
0
3
0
2
1
2
2
3
2
0
0
1
0
2
2
3
0
2
3
0
1
1
1
3
3
1
COST OF
RESOURCES UNDETECTED
USER VTOLATIONS
1176.00
1182.00
.00
1713.00
.00
1132.00
603.50
1166.00
1136.00
1696.50
1136.00
.00
.00
535.50
.00
1887.00
1130.00
1635.00
.00
1126.00
1680.00
.00
550.00
563.00
560.00
1689.00
1650.00
550.00
,7aei5
,18245
.00061
.28174
4.04793
.04382
.00916
.06259
,02339
1.47146
.10884
3.34015
2.43298
.00017
4.07209
.40154
.01037
3.06773
.38A79
.06189
1.55168
1,09891
.18859
.23391
.00000
1, 84605
4.19653
.28190
TOTAL RESOURCES USED 24496.50
FINAL COST OF UNDETECTED VIOLATIONS 30.17343
134
-------
Table 9.6 DATA FOR CASE II
SOURCE
1
2
3
4
6
T
a
9
10
U
1?.
13
14
15
U
17
10
H
20
22
23
24
25
2*
27
26
29
30
PNV
.696278
.147692
,979586
•531608
.308329
•111616
•000191
.032648
.075379
.938505
.172521
,743805
.991440
.000857
•994912
.316189
.021793
.974799
.333279
•232609
.394030
.922821
.107414
•484105
,00000ft
.615132
.792984
.163917
EXP. DAMAGE
2.747735
1.701061
.000607
3.106736
3, 76*632
3.517227
1.513851
8.724413
5.6230PO
?. 099448
5.397685
3.535545
2.267273
2.375780
4.120321
3.986394
9,106810
3,«2?610
.736836
5,389093
4.626968
.92^437
3.571862
.475020
6.675322
6,*03027
*• 596327
1.191331
* Probability of no violation.
135
-------
Table 9.7 PRIORITY LIST: DEMONSTRATION PROJECT, CASE II
PR'ORITY
1
2
3
4
5
6
7
a
9
10
11
12
13
14
15
16
17
16
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
4T
48
49
SOi'RCE
8AMPLFD
18
9
27
10
I*
22
25
7
23
A
15
28
17
28
u
fl
29
?
?.3
29
3ft
22
IS
28
?9
1
6
12
4
13
1
13
17
20
23
4
10
1
25
7
9
26
6
22
?
18
20
30
17
MARGINAL
,.«6M...
1 .14761255
1 * 19?05173
.915345/iB
,7863V>Otf 7
.73«5'5«fla
.579671^3
,55205768
.500896M
.45162335
.44327629
.42404060
,?8S91ftfm
,26084082
.2546/J597
,250797«1
,248230*4
.24531773
.19736782
.196682IS4
.181 10020
.17086395
.16528967
.160451/15
.15612404
.14192987
.13931040
.13566206
.13547814
.12294336
.09R82264
.09141594
,09135259
,090«7219
.07776874
.072021*6
.06899758
,06680803
.0622M93
,06161870
.04726118
,04352768
,04*95348
,03974448
.03623144
,03<13M29
,030152*50
,02968543
,02888465
COST or
VIOtATIOMS
94.8920?
B6,«5?«i4
79.7771?
74,57796
70, 1 1 1 49
A. 1C O 7 tC O 1C
OTJft' / v"»
62.787/rj
59.66311
56»fl£3f'9
5a,ii;5r.|j9
51 ,68t B'{
U0.49UUQ
46,76855
45.30002
43.84485
42.33? 29
40.9657fl
39.51591
38, «l 066
37.32780
36.33175
3S. 36979
3(l,46^no
33,56066
32,70197
31,86742
31.06500
30.29(144
29,52066
E8. 84713
28.26605
27,76493
26,90301
26,41175
25.97625
25.56500
25.17310
24.76851
24,42605
24,07729
23,80176
23.55669
23.30928
23.00552
22.87139
22.67725
22,51352
22.35026
22.07773
RESOURCES
REQUIRED
565.00
1148.00
1708,00
2276.00
2844.00
3407*00
3957.00
4523,00
5083.00
5659.00
6194.50
6757.50
7701.00
826(1.00
8835.00
9438,50
9988.50
10579. 50
11139.50
11689.50
12239.50
12802.50
13350.50
13913.50
JUU63.50
15051.50
15627.50
16195.50
16766.50
17314.50
17902.50
18450,50
19394,00
19937,00
20497,00
21068.00
21636.00
27224.00
22774.00
. 23310.00
23923.00
24486.00
25062.00
25625.00
26216.00
26781.00
27324,00
27874.00
28617.50
136
-------
Table 9.8 SAMPLING FREQUENCIES: DEMONSTRATION PROJECT, CASE II
BUDGET 25000.00
MIN NO. MAX NO.
SAMPLES SAMPLES TIMES
SOURCE REQUIRED ALLOWED SAMPLED
1
2
3
4
6
7
a
9
10
li
12
13
H
15
16
17
18
IP
20
22
23
24
25
26
27
28
29
30
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
1
0
3
2
2
1
2
2
0
2
3
0
1
0
2
1
0
1
2
3
0
2
1
1
3
3
1
COST OF
RESOURCES UNDETECTED
USED VIOLATIONS
1764.00
591,00
.00
1713.00
1152.00
1132,00
603.50
1166.00
1136.00
.00
1136.00
1644.00
.00
535.50
.00
1887,00
565.00
.00
543.00
1126.00
1660.00
.00
1100.00
563.00
560.00
1689.00
1650,00
550.00
.92752
.25123
.00061
.46674
.35770
.04382
.00029
.00930
.03195
2.09945
.16065
1,45490
2.26727
.00204
4.12032
.39854
.19S47
3,82261
,24557
.29159
.28319
.92144
,04121
.22996
.00000
1.44380
3.28924
,19528
TOTAL RESOURCES USED 24466.00
FINAL COST OF UNDETECTED VIOLATIONS
23.55669
137
-------
Case III
For Case III, it is assumed that sampling from Case II has been inter-
rupted in the middle of a monitoring period. It is expected that Source
14 or Source 19 is contributing to poor water quality. From Table 9.8,
it is seen that neither of these sources would normally be sampled
during this monitoring period.
Table 9.9 shows the number of times the sources were assumed
sampled before the interrupt and the optimal sampling frequencies after
the interrupt. Case III has shown how the priority procedure can be
used to respond to ambient monitoring reports.
Preliminary Performance Comparison
The performance of the Resource Allocation Program will be compared
with a simpler procedure that assigns sampling frequencies on the basis
of flow. The latter procedure, called the Allocation by Flow procedure,
assigns one sample to all the sources and then assigns the remaining samples,
within the budget, to the sources with largest flow.
The monitoring period used for this comparison will be the one corres-
ponding to Case II, (i.e., months 19 through 24) where the sampling
frequencies were based on data from months 1 through 18.
The performance criteria are (i) the observed "cost" of undetected
violations and (ii) the observed number of violators. These criteria
are observed values calculated for 14 sources for a month picked at
random from the monitoring period.*
*The number of sources considered for this comparison were reduced to 14
to reduce the amount of data handling required.
138
-------
Table 9.9 SAMPLING FREQUENCIES BEFORE AND AFTER INTERRUPT:
DEMONSTRATION PROJECT, CASE III
Source
1
2
3
4
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
22
23
24
25
26
27
28
29
30
Times sampled
before interrupt
2
1
0
1
1
2
0
0
1
0
1
2
0
0
0
0
0
0
1
1
2
0
0
1
0
1
2
Times sampled
after interrupt
0
0
0
1
0
0
1
2
0
i
i
0
2
I
0
2
1
3
0
1
1
0
1
0
1
2
1
'
139
-------
The observed "cost" of undetected ^violations fojj one month is
c - > c± P± i
sources
where
c = average damage due to source i
M
= i / (observed damage on day k)
M j~"i
k=l
1_ ^ = observed frequency of violation of source i
= (number of days in violation) * M
s = sampling frequency for source i.
and where M is the number of observed values of the effluent in the month.
The observed damage on day k is
max {d (CO (k))} (9'2)
constituents
j
where d is the damage function for constituent j and CO..(k) is the
J J
concentration of constituent j downstream from source i based on the
observed effluent value for constituent j on day k . (Note that the
assumed upstream concentration and stream parameters are the same as were
used in the priority procedure to determine the sampling frequencies.)
The observed number of violators in a month is simply
I
Sources
- P± *> (9.3)
140
-------
Table 9.10 shows the observed frequency of violation, 1-p , and the
average damage, C. , for the various sources along with the source flow.
These values were used to calculate the observed "cost" of undetected
violations and observed number of violators. Table 9.11 compares the
sampling frequencies obtained by the Allocation by Flow method and the
Resource Allocation Program (two lower bounds on sampling frequency were
chosen for the Resource Allocation Program: zero and one.) as well as
comparing the performance criteria. The budget was assumed to be $15,000.
Prom Table 9.11 it is seen that the Resource Allocation Program produces a
better allocation for this example than the Allocation by flow method.
The improvement is greater for the observed "cost" of undetected viola-
tions than for the observed number of violators.
It is recommended that more comparison studies be done in the future using
larger data bases. This study was hampered by the fact that only one month
of data was used. Since samples are highly correlated, day-to-day, for
many Industries, a small number of independent samples went into the calculation
°f the observed damage and the observed frequency of violation. (Note that
°ver half the sources were either always in violation or never in
violation.) It therefore is expected that much better performance of the
Resource Allocation Program would have been shown if more months of
data were used in the comparison.
141
-------
Table 9.10 OBSERVED FREQUENCY OF VIOLATION AND AVERAGE DAMAGE
Source
3
12
16
18
19
22
23
24
25
26
27
28
29
30
Source
flow, Ml/day
0.075
4.92
0.725
35.55
0.133
40.75
0.425
3.04
165.0
7.15
5.57
110.9
4.11
35.0
Observed frequency
of
violation,- %
0.0
41.4
22.2
50.0
0.0
100.0
13.3
0.0
100.0
100.0
100.0
100.0
5.0
87.1
Observed average
damage
0.00
5.73
3.01
3.90
1.32
6.85
4.01
0.98
3.70
0.43
2.61
4.15
5.79
1.13
142
-------
Table 9.11 PERFORMANCE COMPARISON
Source
3
12
16
18
19
22
23
24
25
26
27
28
29
30
Observed "cost" of undetected
violations
Observed number of violators
caught
Sampling Frequencies
Allocation
By
1
1
1
2
1
2
1
1
2
2
2
2
1
2
19.00
7.55
Optimal
Allocation
8i - 1
1
2
1
1
1
2
2
1
1
1
1
3
3
1
17.97
7.64
Optimal
Allocation
s > 0
0
2
0
2
0
3
3
0
2
1
1
3
3
1
17.23
7.77
143
-------
SECTION X
REFERENCES
1. Harm, Jr., R.W., et al., Evaluation of Factors Affecting Discharge Quality
Variation. Environmental Engineering Division, Civil Engineering Department,
Texas A & M University, September 1972.
2. Tarazi, D.S., et al., "Comparison of Waste Water Sampling Techniques",
J. Water Pollution Control Federation. 42, (5), 1970.
3. Kendall, M., and Stuart, A., The Advanced Theory of Statistics. Volume 2.
Hafner Publishing Company, New York, 1967.
4. System Control, Inc., "Palo Alto Waste Water Treatment Plant",
Automation Project Final Report (EPA Project R800356), May 1974.
5. "Development Document for Effluent Limitations , Guidelines and Standards
of Performance: Inorganic Chemicals, Alkali and Chlorine Industries",
General Technologies Corporation, June 1973.
6. "Development Document for Effluent Limitations, Guidelines and Standards
of Performance: Non-Fertilizer Phosphous Chemicals Industry", General
Technologies Corp., June 1973.
7. Raiffa, H., and Schlaiffer, R., Applied Statistical Decision Theory,
The M.I.T. Press, Cambridge Mass., 1961.
8, Beckers, C.V., et al., Quantitative Methods for Preliminary Design of
Water Quality SjmreJLllance Systems, Environmental Protection Agency,
Washington, D.C., Report No. EPA-R5-72-001, November 1972.
9. Hydroscience, Inc., Simplified Mathematical Modeling of Water Quality,
Environmental Protection Agency, Washington, D.C., March 1971.
144
-------
10. Environmental Protection Agency, Notice of Proposed Rulemaking; Effluent
Limitations Guidelines for Existing Sources and Standards of Performance
and Pretreatment Standards for New Sources, Federal Register, Vol 38,
No. 173, Washington, D.C., September 7, 1973.
11. Environmental Protection Agency, Proposed Rules; Effluent Limitations
Guidelines and Standards of Performance and Pretreatment Standards for
Electro-plating Point Source Category, Federal Register, Vol 38, No. 193,
Washington, D.C., October 5, 1973.
12. Environmental Protection Agency, Proposed Rules; Effluent Limitations
Guidelines and Standards of Performance and Pretreatment, Federal
Register, Vol 38, No. 196, Washington, D.C., October 11, 1973.
13. Environmental Protection Agency, Glass Manufacturing; Effluent Limitations
Guidelines, Federal Register, Vol 38, No. 200, Washington, D.C., October
17, 1973.
14. Environmental Protection Agency, Proposed Guidelines and Standards;
Ferroalloy Manufacturing Point Source Category, Federal Register,
Vol 38, No. 201, Washington, D.C., October 18, 1973.
15. Environmental Protection Agency, Proposed Effluent Limitations Guidelines
for Existing Sources and Standards for New Sources; Meat Products Point
Source Category, Federal Register, Vol 38, No. 207, Washington, D.C.,
October 29, 1973.
16. Environmental Protection Agency, Proposed Rules; Effluent Limitations
Guidelines for Asbestos Manufacturing Point Source Category, Federal
Register, Vol 38, No. 208, Washington, D.C., October 30, 1973.
145
-------
17. Environmental Protection Agency, Proposed Effluent Limitation Guidelines
for Existing Sources and Standards for New Sources; Canned and Preserved
Fruits and Vegetables Processing Industry Category, Federal Register, Vol__3§»
No. 216, Washington, D.C., November 9, 1973.
18. Environmental Protection Agency, Proposed Effluent Limitations Guidelines;
Nonferrous Metals Manufacturing Point Source Category, Federal Register,
Vol 38, No. 230, Washington, D.C., November 30, 1973.
19. Environmental Protection Agency, Grain Mills; Effluent Limitations Guide-
lines, Federal Register, Vol 38, No. 232, Washington, D.C., December
4, 1973.
20. Environmental Protection Agency, Fertilizer Industry Leather Tanning and
Finishing Industry Sugar Processing Industry; Effluent Limitations Guide-
lines and New Source Performance Standards. Federal Register, Vol 38,
No. 235, Washington, D.C., December 7, 1973.
21. Environmental Protection Agency, Proposal Regarding Minimizing Adverse
Environmental Impact; Cooling Water Intake Structures, Federal Register,
Vol 38, No. 239, Washington, D.C., December 13, 1973.
22. Environmental Protection Agency, Effluent Limitation Guidelines and
New Source Standards; Petroleum Refining Point Source Category, Federal
Register, Vol 38, No. 240, Washington, D.C., December 14, 1973.
23. Environmental Protection Agency, Organic Chemicals Manufacturing Industry;
Proposed Effluent Limitations Guidelines, Federal Register, Vol-38,
No. 241, Washington, D.C., December 17, 1973.
146
-------
24. Environmental Protection Agency, Dairy Products Processing Industry;
Proposed Effluent Limitations Guidelines, Federal Register, Vol 38.
No. 244, Washington, B.C., December 20, 1973,
25. Environmental Protection Agency, Proposed Effluent Limitation Guidelines
and New Source Standards; Soap and Detergent Manufacturing Point Source
Category, Federal Register, Vol 38. No. 246, Washington, D.C., December
26, 1973.
26. Environmental Protection Agency, Effluent Limitations Guidelines; Builders
Paper and Board Manufacturing Point Source Category, Federal Register.
Vol 39, No. 9, Washington, D.C., January 14, 1974.
27. Prati, L., et. al., "Assessment of Surface Water Quality by a Single
Index of Pollution", Water Research (GB), Vol. 5, pp. 741-751, 1971.
28. Horton, R. K., "An Index-Number System for Rating Water Quality",
Water Pollution Control Federation Journal. 37. pp. 300-306,
March, 1965.
29. McClelland, N. I., Water Quality Index Application in the Kansas
River Basin, Report No. EPA-907/9-74-001, U.S. Environmental Protection
Agency, Kansas City, Feb., 1974.
30. Kneese, A., and Bower, B. T., Managing Water Quality Economies Technology,
Institutions. John Hopkins Press, Baltimore, 1968.
31. Vermont Department of Water Resources, Development of a State Effluent
Charge System, Environmental Protection Agency Report No. 16110
GNT 02/72, February 1972.
147
-------
32. Dee, N., et. al., Environmental Evaluation System for Water Resource
Planning, Battelle Columbus Labs, Jan., 1972.
33. Mckee, J., and Wolf, H., (Eds.), Water Quality Criteria, Second Edition,
State Water Resources Control Board, California, Publication
No. 3-A, 1963.
34. Water Quality Criteria, Report of the National Technical Advisory
Committee, U.S. Dept. of Interior, Washington, D.C., 1968.
148
-------
SECTION XI
GLOSSARY
TERMINOLOGY
BOD - Biochemical oxygen demand.
COD - Chemical oxygen demand.
DO - Dissolved oxygen.
- BOD-dissolved oxygen transfer coefficient
Damage - A measure of effect of pollutants on water quality.
jSffluent Standard - A restriction on the quantities or concentrations of
constituents from an effluent source.
Monitor - The government agency having responsibility for enforcing laws
realting to the abatement of pollution.
Permit - A document or requirement regulating the discharge of pollutants.
Resources - Money required to obtain and process effluent samples obtained
during compliance monitoring.
Resource Allocation Program - Name given to procedure for setting compliance
monitoring priorities.
Source - A discharger or possible discharger of pollutants subject to effluent
atandards.
Water Quality Limited Segment - A segment of a river where it is known that
Water quality does not meet applicable water quality standards and which is not
expected to meet water quality standards even after the application of the
effluent limitations required by the Water Pollution Control Act.
MATHEMATICAL NOTATION
A - Maximum allowed cost of undetected violations.
01 - Level of significance of a statistical hypothesis test.
B - Monitoring agency's budget.
149
-------
C - Total "cost" of undetected violations.
C . (S . ) - "Cost" of undetected violations for source i.
c - Expected damage from all the constitutents of source i.
CO - Stream concentration at discharge point, constituent j, source i.
CU. - Upstream concentration, constituent j, source i.
CX. - Downstream concentration, constituent j, source i.
D . . - Expected damage due to constituent j, source i.
D - - Expected damage due to constituent j, from source i into
stream &.
d (T) - Damage function for constituent j.
DJ D^TN ~ Dissolved oxygen deficit due to BOD, source i.
1 , BUIJ
D - Dissolved oxygen deficit due to COD, source i.
DOMIN. Tj--. - Minimum DO level downstream from source i.
i , BUD
<)>..(•) - Density function of mass loading M . . .
Y - Parameter denoting relative weight given compliance data over self-
monitoring data.
h - Factor relating confidence in mean to number of measurements.
h - Factor relating confidence in variance to the number of measurements.
1 - Index denoting source.
j - Index denoting constituent.
k - Index denoting outfall.
i - Index denoting receiving water.
k - Constant for determining the confidence in mean.
k - Constant for determining the confidence in variance.
A. - Lower bound on sampling frequency for source i.
L. - Upper bound on the sampling frequency for source i.
150
-------
M - Effluent mass loading, constituent j. source i.
mg/1 - Milligrams per liter.
Ml - Megaliters.
m - Estimate of mean
y - Mean of a random process.
A
Vi - Estimate of mean, y.
y (•) ~ Marginal return function.
N - Number of sampling days in monitoring period.
n - Number of measurements or confidence in mean estimate.
n - Number of sources.
s
p(.) - Probability event occurs.
p - Probability of no violation for all constituents of source i.
p - Probability of no violation for constituent j, source i.
p - Probability of no violation due to constituent j, outfall k,
^ source i.
QS. - Effluent flow rate, source i.
QU - Upstream flow rate, source i.
QX. - Downstream flow rate, source i.
R(*) - Resources required to monitor all the sources.
r - Resources required to monitor source i once.
s - Number of times the ith source is sampled in monitoring period.
a - Standard deviation of a random process.
o - Estimate of standard deviation, c.
T.. - Standard for constituent j, source 1.
v - Estimate of variance.
V. - Event ith source is in violation.
151
-------
V. - Event ith source is not in violation •
v - Confidence in variance estimate.
x - Distance downstream from source or a random process.
y - Maximum of a set of data.
z - Compliance monitoring data.
152
-------
APPENDIX A
ESTIMATION OF DISTRIBUTION PARAMETERS
In this Appendix the estimation of the parameters of the normal and log-
normal probability density functions is discussed for the case where
the available data consist of the sample mean and maximum of a set of
observations. These two problems are treated in Sections A.I and A. 2*
Section A. 3 deals with the examination of the parameters when the avail-
able data consist of the maximum and the minimum value of a set of
observations.
A.I THE NORMAL CASE
In this case the process x is assumed normally distributed with mean p
and variance a2. The available data to estimate y and a is
z «• [m,£] (A. 1.1)
where m is the sample mean and
- max
(A. 1.2)
Approximate maximum likelihood estimates of y and o2 will now be
obtained .
The calculation of the likelihood function
requires the joint probability density function for m and £• This
density is not obtainable in closed form. Approximate maximum like-
lihood estimates can be obtained .by estimating u by m, the sample mean.
a, the estimate of a, is then that value of a that maximizes
- m,a2)
153
-------
The above density is obtained as follows:
Prob {max(X;L, ...,xn> £ £|y,o} » Fn (^) (A.I.5)
where
a
F(a) (2TT)'1/2 e"x2/2dx (A.I.6)
Therefore,
Fn
For convenience denote
x^ £ x± - y (A.1.8)
then
and, hence
Let
— 'y2/2
Denoting
(A. 1.10) becomes
154
f(y) - f(y) - --— e'y (A.i.ii)
I' » £lM (A.I.12)
a a
a) • nFn (C) f(5) — (A.1.13)
(7
-------
f (C) I
The likelihood equation is therefore
which is equivalent to
|*r - 0 (A.1.15)
Note that
||^- " Cf(£) (A.I.16)
Thus (A.1.15) can be written as follows
- 0
(A. 1.17)
or
*u 9 •« x «-i s — \
- n-1 (A.1.18)
The left hand side of (A. 1.18) is plotted in Figure A. 1.1. Using
>\
this figure, it is easy to determine a, the estimate of a, given
A
£» y and n. This is done by obtaining C for the given n from
Figure A. 1.1, then
(A.I. 19)
For example, suppose n - 31, p - 5 and C " 10. From the figure, (n-1)
- 30 implies C ~ 2.035. Thus
155
-------
I
c
2.5
Figure A.1.1 Plot of equation (A.1.18).
156
-------
A. 2 THE LOGNORMAL CASE
In the lognormal case*, if x. are the measurements, then
y1 = £n x± ~ e/P(y,a2 ) (A.2.1)
and 6 = [y,a] is the unknown parameter. Note that y and a are the
mean and standard deviation of the logs of the measurements rather than
of the measurements as in the normal case. Assume that the statistic
is, as before
z - IX £] (A.2.2)
i.e., the sample mean of the measurements, m, and the largest measure-
ment
£ - max{x1,...,xn} (A.2.3)
The estimate of the mean of x.^ is taken to be the sample mean m,
therefore
/N
,2.
m - E{x±} - exp j y + y
(A.2.4)
or
2
^ 0
y + y = £n m (A.2.5)
The maximum likelihood estimate of a is obtained by maximizing
pUlv - (*n m - f") • °) (A.2.6)
^Natural logarithms are used throughout the derivation. The final
results are given in terms of common logrithms.
157
-------
with respect to a. First, the distribution of £ is
P{max(x.,,...,x ) < 5} = P{max(y-,..;,y ) < Jin £} (A.2.7)
J- n "~ JL n —
e
- U
where F is the standard Gaussian distribution (A.1.6). Therefore,
denoting
p-f (A.2.8a)
A &n g - y An g - An m + (Q2/2)
™ — as — V.A.Z.ODJ
Jin p +(a2/2)
a
the density of £ is
Fn(n)
(A.2.9)
«s
From (A.2.8b) one has
(A.2.10)
***» V«,
Combining (A.2.9) and (A.2.10) yields
f(n)
158
-------
The likelihood equation is, therefore
n-2 2
fn(n-l)Fn'2(n) f (n)
- n Fn~ (n)n f(n)(a£) *| f£ (A.2.12)
- n F11"1^) f (n) a"2 T1 - 0
where use has been made of (A.1.16). Also
i - a
(A.2.13)
Inserting (A.2.13) into (A.2.12) yields the following equation for 0
[(n-i) f(n) - F(n)nl' (o - n) - P(n) - o (A.2.14)
where ri - r\(a) according to (A.2.8b).
The solution a of (A.2.14) for common logarithms is presented graphic-
ally in Figure A.2.1 as a function of the number of measurements n and
the ratio p between the maximum and the mean. For example, assume m -
10, £-30 and n - 30. Then p »3 and o » 0.27. The estimate y is
obtained using (A.2.5):
U - log m - AnlO - - 1 -(2.3) (0.0365) - 0,916
159
-------
0.6
0.5
< D
I
§O.A
M
s
0.3
10.2
0.1
I
0 12 3 4 5
RATIO OF MAXIMUM AMD MEAN - p
Figure A.2.1 Maximum likelihood estimate of standard deviation from mean
and maximum in lognormal case.
160
-------
A. 3 ESTIMATE OF MEAN AND STANDARD DEVIATION FROM MAXIMUM AND MINIMUM
Let x..,... ,x« be independent samples from a normal <^(y,O2) distribution
and let yn = min(xn,...,x ) and y = max(x, ,, .. ,x.T) . Then simple
J± 1 n N 1 N
estimates of y and a can be obtained from the midrange m m (y1 + y,.
and the range R = yN - y.^.
Estimate of Mean
The obvious estimate of the mean is the midrange. Kendall and Stuart
[Al] gives the relative efficiency of this estimate as compared
to the efficiency of the sample mean for several values of N (see
Table A.3.1).
Table A.3.1 RELATIVE EFFICIENCY OF MIDRANGE
AS AN ESTIMATE OF y
N
2
A
6
Relative efficiency
1.000
.915
.840
N
10
20
00
: Relative efficiency
.734
.591
0
jjist jmate of Standard Deviation
The estimate of the standard deviation from y- and yN has historically
[A2], [A3] been in the form
R/C
'N
(A. 3.1)
where R is the range and C » E(ft) where ft is the range of N samples for
a t/f (0,1) distribution. 0 is therefore an unbiased estimate of a, A
table of C versus N is given in Table A.3..2 [A3].
161
-------
Table A.3.2.
VERSUS N
N
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
CN
-
-
1.128
1.693
2.059
2.326
2.534
2.704
2.847
2.970
3.078
3.173
3.258
3.336
3.407
3.472
3.532
3.588
3.640
3.689
3,735
N
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
CN
3.778
3.819
3.858
3.895
3.930
3.964
3.997
4.027
4.057
4.086
4.113
4.139
4.165
4.189
4.213
4.236
4.259
4.280
4.301
4.322
In [A4], the relative efficiency of this estimate is given as compared
to the efficiency of the sample standard deviation. Several values
are shown in Table A.3.3.
162
-------
TABLE A.3.3 RELATIVE EFFICIENCY
OF THE ESTIMATE 0
N
2
4
6
Relative efficiency
1.000
0.975
0.933
N
10
20
50
Relative efficiency
0.850
0.700
0.490
REFERENCES
Al. Kendall, M. and Stuart, A., The Advanced Theory of Statistics,
Volume 2, Hafner Publishing Company, New York, 1967.
A2. Hosteller, G., "On Some Useful 'Inefficient1 Statistics," Ann.
Math. Stat.. Vol. 17, 1946, pp. 377-408.
A3. Tippett, L. H. C., "On the Extreme Individuals and the Range of
Samples Taken from a Normal Distribution," Biometrika, Vol. 17,
1925, pp. 364-387.
A4. Snedecor, G. W. and Cochran, W. G., Statistical Analysis,
University of Iowa Press, 1972.
163
-------
APPENDIX B
INVESTIGATION OF THE CORRELATION
BETWEEN EFFLUENT CONSTITUENTS
In this appendix a procedure is presented to test for the uncorrelatedness
of normal random variables with unknown mean and unknown variance. Sub-
sequently, it is applied to data from the Palo Alto Municipal Waste Treat-
ment Plant.
B.I THE UNCORRELATEDNESS TEST
Consider two normal random variables x, and y, from which n independent
samples x., i - 1, ..., n and y±, i = 1, ..., n are available. The
true means and variances are unknown and can be estimated by the well-
known equations
- 1 v - 1 V
x'n 5, xi» y "a ^yi
(B.I.I)
We want to test whether their correlation
A E [(x - Ex) (v - Ey)]
" [E (, - E,)2 E
-------
Define the sample correlation as
(y± - y)
(B.I.5)
It has been pointed out in Kendall and Stuart [Bl] that the distribution
of this sample correlation converges very slowly to the normal and thus
a test based on the normality assumption is not accurate. The exact
test is presented next. As shown in [Bl]
|(n-2)r
7*r
(B.I.6)
has a t-distribution with v - n-2 degrees of freedom. Thus the above
simple transformation enables one to test HQ against H. using readily
available tables.
To illustrate the procedure, consider for example n » 30. The t
values corresponding to various values of the sample correlation r are
presented in Table B.I.I. Also, the significance levels above which
H would be rejected (and E^ accepted) for these values of r are given.
Table B.I.I UNCORRELATEDNESS TEST FOR N-30 SAMPLES
r
0.5
0.4
0.35
0.3
0.25
t
3.06
2.31
1.99
1.66
1.37
c&
<1
3
6
11
18
165
-------
If the observed value is r = 0.'35, then at 5% level of significance
(probability of error of type I) H would be accepted.
B.2 EXAMPLE OF UNCORRELATEDNESS TESTS FOR EFFLUENT CONSTITUENTS
Tests were run on a number of constituents from the Palo Alto
Municipal Waste Treatment Plant. The data consisted of daily composite
samples of the following
1. Flow
2. Suspended Solids
3. BOD (Biological Oxygen Demand)
4. TOC (Total Organic Carbon)
5. COD (Chemical Oxygen Demand)
Data was obtained from a dry month (July 1973) and a wet month
(November 1973) each-with 30 samples. The correlation coefficients were
computed for the actual measurements, under the normal assumption and for
the logarithms of the measurements, under the-logaormal assumption. (The
goodness of these assumptions was examined in Section V.I).
The resulting correlation coefficients are presented in Tables B.2.1
and B.2.2. An examination of these tables reveals that the sample
correlations are such that only at relatively low significance levels
(a * 1%) would the hypothesis of uncorrelatedness be accepted in some
cases. This can be seen from the uncorrelatedness test illustrated in
Table B.2.1. However, the variation of the correlation coefficients
seems to be large from season to season and no clear pattern seems to
emerge. For example, the r2_ term (SS vs. BOD) is positive in a dry
month while in a wet month it can become negative. Also notice that
there is no appreciable difference in the correlation tests when done
under normal or lognormal assumption. The hypothesis that the effluent
constituents are highly (near unity) correlated is even less likely than
their being uncorrelated.
166
-------
Table B.2.1 SAMPLE CORRELATIONS OF THE MEASUREMENTS
Month
Dry
Wet
Variable
sampled
1
2
3
4
1
2
3
4
2
0.28
-0.24
3
0.33
0.46
0.45
-0.19
4
0.55
0.39
0.43
0,29
0.27
0.35
5
0.58
0.62
0.50
0.47
0.22
0.25
0.13
0.51
Table B.2.2 SAMPLE CORRELATIONS OF LOGS OF THE MEASUREMENTS
Month
Dry
Wet
Variable
sampled
1
2
3
4
1
2
3
4
2
0.30
-0.24
3
0.32
0.45
0.50
-0.18
4
0.58
0.40
0.51
0.28
0.37
0.30
5
0.60
0.68
0.45
0.49
0.23
0.33
o.io
0.59
167
-------
REFERENCE
Bl. Kendall, M. and Stuart, A., The Advanced Theory of Statistics,
Volume 2, Hafner Publishing Company, New York, 1967.
168
-------
APPENDIX C
EXPECTED DAMAGE AND PROBABILITY OF VIOLATION CALCULATIONS
C.I INTRODUCTION
The sampling frequencies to choose, in determining whom to monitork mini-
mize the "cost" of undetected violations. This "cost" was derived in
Section VI to be:
ct pj1 (C.I.I)
sources
i
where c. is the expected damage caused by the i source, p. is the
th
probability that the i source will riot violate any effluent standard,
and s. is the number of times the i source is to be monitored, c
equals the maximum of the expected damages due to the various constituents
of the ±t source, or
c. • max D.. (C.I.2)
1 J 3
where D is the expected damage due to the j constituent of the i
source, p , assuming independence between the various constituents, is
p - n p , (C.I.3)
j J
where p is the probability 'the standard on the j constituent is not
violated. If the constituents are completely correlated, then
169
-------
p - min p (C.I.A)
j ^
This appendix describes in detail both the calculation of the ex-
pected damage, DJJ» due to constituent j from source i and p..,
the probability that constituent j, source i, does not violate its
standard. It is organized as follows: Section C.2 calculates D.. and
p.. under the assumption that only one set of effluent standards is
given for the source. This corresponds to the case where there is only
one outfall or the permits are written for the combined discharge from
several outfalls. Section C.3 describes how these calculations are
generalized to the case when standards are set for many outfalls from a
single industry or municipal treatment plant. Section C.4 evaluates
certain integrals that arise often in the expected damage and probability
of violation calculation.
C.2 EXPECTED DAMAGE AND PROBABILITY OF VIOLATION DERIVATION:
ONE SET OF STANDARDS
This section describes the derivation of the expected damage from a
source and the probability of violation when there is either a single
outfall from the source or there are several outfalls, all to the same
river, and there is one set of standards for the total discharge from the
source. When there are several outfalls but only one set of standards
for the total effluent, the monthly self-monitoring reports are on the
total effluent, and so the several outfalls can be treated as one.
The section is divided into four subsections. The first subsection con-
siders the majority of constituents. All the calculations needed to
determine the expected damage and probability of violation for this set
of constituents are the same. pH, BOD, and temperature require slightly
different calculations, and they will be treated separately in the re-
maining subsections.
170
-------
C.2.1 Noncoupled Constituents
This subsection derives expected damage and probability of violation
for all the indicators listed in Table 6.1 except pH, temperature, and
dissolved oxygen.
Inputs
The data needed to calculate expected damage and probability of violation
are:
For source i:
P » index set of pollutants
y = mean of mass loading of j pollutant (kg)
a . « standard deviation of mass loading of j pollutant (kg)
y • distribution of j pollutant—normal or lognormal
QU. - flow of stream above source (Ml/day)
QS. - effluent flow (Ml/day)
CU ** concentration of the pollutant upstream from source fag/l)
EFST.. » effluent standard for j pollutant (kg)
For each pollutant j:
d. (k) » concentration of pollutant when damage equals 2(k-l),
k - 1, 2, ..., 6.
d.(k) is the value of the abscissa of the damage function at the k
J .
breakpoint. The damage function breakpoints for the constituents of
interest were given in Table 6.1. The damage function of the j pollutant
is then
171
-------
«- d (10)
(d.(fcfl) -
'
+ 10 *(d(6), », a) (C.2.1)
where a is the concentration of pollutant and $ is the characteris-
tic function:
1 ; x _< a < 9
*(x,y,a) - (C.2.2)
) ; otherwise.
Maximum Downstream Concentration
The maximum downstream concentration for the j pollutant - 1 source
is given by the conservation law:
M±1 + CU QU
cc-2-3)
where M. . is the mass loading of the J pollutant - i source
(M.. is a random variable with mean y. ., standard deviation o. . and
3-J 1J 1J I.L
distributional form y..) and CU.. is the concentration of the j
th "
pollutant upstream from the i source. (C.2.3) can be rewritten to
yield
C0±j - a^ + b±j (C.2.4a)
where
a± - 1/(QU± + QS^ (C.2.4b)
and
172
-------
CIL
QU± + QS±
(C.2.4c)
Expected Damage
The expected damage due to pollutant j from the 1 source Is
then
/
:(M) dM
(C.2.5)
where £(•) is the expectation operator and $.. is the probability
density function of the mass loading M... Using (C.2.4),
(C.2.6)
Combining (C.2.1) and (C.2.6),
^ / / I
^ J hjkM + fijk; *u°° ** (c-2-7a)
a
ijk
where
aijk
a,
d (k+1)
k™ 1, 2, •••» 6
(C.2.7b)
; k - 1, 2 5 (C.2.7c)
; k - 6
173
-------
J
2a1/(dj(k+l) - dj(k)) ; k - 1,2 5
(C.2.7d)
; k - 6
2(bJ4 - d4(k))
(C.2.7e)
10 ; k - 6
If 01... or 3j.ii, are less than 0, set them to 0. (C.2.7) can be re-
written
6
where
/
, y, a) - /
a
, f , a, 3, y, a) - / (ex + f) ^(x) dx (C.2.9)
and where Is the normal density function with mean y and variance
2 Y
o if y « Normal, and Is lognormal, with mean and variance of corres-
2
ponding normal distribution being y and a , respectively. If
y - Lognormal. (C.2.9) Is evaluated In Section C.4 for the normal and
lognormal cases.
Probability of a Violation
The probability that a standard for the j pollutant 1 source
is not violated is
(M) dM
(0, 1, -», EFSTi;J, Uy, Ojj) (C.2.10)
174
-------
where Iy is defined in (£.2.90
C.2.2 5-Day Biochemical Oxygen Demand - BOD5
The presence of BOD,- in the water causes a depletion in the dissolved
oxygen (DO). The difference between the saturated level of dissolved
oxygen, DOSAT, in the water and the actual level is called the dis-
solved oxygen deficit or DO-deficit. The degree of depletion caused
by a given amount of BOD_ from a source depends on several stream para-
meters such as stream depth, flow rate, temperature, and the distance
downstream from the source. The relationship between BOD. and DO-defi-
cit can be expressed (see Section VI.1) in the form
Dmax " "BOD-DO0" (C.2.11)
where CO is the concentration of BOD- at the source, D is the
D max
maximum DO deficit downstream from the source, and K^D-DO is the
BOD.-DO transfer coefficient.
Inputs
The data needed to calculate the expected damage and probability of vio-
lation due to BOD5 is:
For source i:
V* T>rtn " n6811 of mass loading of BOD- (kg)
i,BOD J
o4 urtn " standard deviation of mass loading of BOD. (kg)
1 , BOD • J
Vj -onn " distribution of mass loading of BOD.
i,BOD -J
CS . - mean of DO concentration of the source (mg/1)
1 ,DO
K_0_ _o - BOD5-DO transfer coefficient
175
-------
QU. - flow of stream above source (Ml/day)
QS. - effluent flow (Ml/day)
DOSAT1 = saturation level of DO in the stream (mg/1)
EFST. ___ = effluent standard for BODC (kg)
1, bu.u j
CU. B_D = upstream concentration of BOD- (mg/1).
Maximum Downstream Concentration
The concentration of BOD at the point where the outfall empties into
the stream is given by
= Mi.BOD + CUi.BOD QUi 2
C°i,BOD QU± + QS± (C.2.12)
The concentration of DO is similarly
CSi DO QSi + CUi DO QUi
j s 1»IJU •*• lillU 1 ff, -
i,DO QU+QS (C.2.
and CU. are unknown and must be chosen in a way that is
consistent with determining the sampling frequencies. It was suggested
in Section VI that they be chosen so that a given level of damage will
occur when the loading M is zero and the concentration of DO in
the source, CS. , is equal to DOSAT.. Since there are two unknowns
1,LMJ 1
and only one requirement, the upstream DO concentration, CU. n_, shall
be arbitrarily set equal to DOSAT.. The value of CU. Brtp. will be set
1 1) a(JD
to give the desired downstream DO concentration under zero load.
The minimum concentration of DO downstream from the source can be ap-
proximated by (see Section VI.1):
(C.2.14)
176
-------
Using (C.2.12) and (C.2.13) and noting that Cl^ DQ » DOSAT± we obtain
CSi,DO
QU± + QS±
(C.2.15)
or
C°i,DOMIN " ai,BOD Mi,BOD + bi,BOD (C.2.16a)
where
ai BOD * ~ ^OD-DO /(QUi + QSi^ (C.2.16b)
and
CSi,DO<>Si + (DOSAVKBOD,D01CUi,BOD) ^±
ui,BOD (QU±
(C.2.16c)
Probability of Violation
The probability that the BOD5 effluent standard will not be violated is
given by (C.2.10) with j - BOD.
C.2.3
pH is a measure of the acidity (alkalinity) of a solution. It is
defined as the negative of the log of the concentration^* of H+ ions.
pOH is defined to be the negative of the log of the concentration of OH
ions. pOH and pH are related by the equation
t The concentration is in moles/liter.
177
-------
pOH + pH - 14 (C.2.17)
For pure water (H20), pOH - pH - 7. pH < 7 implies an acidic solu-
tion and pH > 7 implies a basic or alkaline solution. If two acidic
solutions are combined, then the number of H ions is equal to the sum of
the H ions from the two original solutions.* Similarly, if two basic
solutions are combined, the number of OH~ ions add. Therefore, if, for
example, we combine X liters of an acid with pH » p^ and Y liters
of an acid with pH » p2» then the concentration of H ions is
-Pi -P2
X 10 A + Y 10 /„ , i ON
xTY (c*2'18)
and the pH of the resulting solution is the negative log of this quantity.
So, as long as both the effluent and the receiving waters are both acidic
or both basic, the concentration of ions can be considered as a conserva-
tive constituent.
The standards for pH require that pH lie between two values: one above
7, the other below. The damage as measured by pH and the distri-
butions of effluent pH can also be divided into two parts: one for
pH > 7, the other for pH < 7. Similarly, to consider the worse case
problem, the receiving waters will be assumed to have the same quality
(acidic or basic) as the effluent.
The self-monitoring data for pH will either be (1) a monthly maximum and
minimum or (2) a monthly maximum, minimum, and mean. If the data are the
former, then the mean and standard deviation can be estimated using the
midrange and the range respectively. If they are the latter, then two sta-
tistical descriptions can be obtained, one using the mean and maximum, the
other the mean and minimum. Two standard deviations would be estimated
* We are assuming that no chemical reaction or buffering takes place.
178
-------
using the estimation technique described in Appendix A.I. The proba-
bility density function for pH would have the shape shown in Figure C.2.1,
DENSITY FUNCTION
pH
Figure C.2.1 Example of probability density function of pH.
Inputs
The data needed to calculate the expected damage and probability of vio-
lation are given below. The subscript J denotes either H or OH.
The distribution of pH or pOH is assumed normal.
For source i:
iJ
EFST
QU±
QS±
iJ
mean of pJ(y±OH " 14 - y±H)
standard deviation of pj
upstream concentration of J ions (Moles/1)
effluent standard for pj
flow of stream upstream from source (Ml/day)
effluent flow (Ml/day).
The damage function for pH was given in Table 6.1. It is much
easier to obtain expressions for the expected damage if the damage
function is given in units of concentration of ions. The damage
179
-------
function is therefore redefined as shown in Table C.2.1 (the damage
function is assumed linear, in concentration, between the given values)
and it is plotted in Figure C.2.2. Therefore, for J » H or OH, the
following is defined:
d (k) = concentration of J ions when damage equals k-1
J
where k-1, 2, ...,11.
The damage function DT(a)
•J
Z10 ( (a - d (k))
(dT(k+l) - dT(
k=l I J J
(k» - > d* <*
(C.2.19)
+ 10 $(d(ll), ~, a)
where $ is defined in (C.2.2).
Maximum Downstream Concentration
The maximum downstream concentration of H or OH ions is
CS QS + CU QU
COiJ - QS.+QU
where CS.j is the concentration of J ions in the effluent. Note
that
CS^ = 10~pJ (C.2.21)
where pj is the pH or pOH of the effluent and is a normal random
variable. The upstream concentration is set to give the desired level
of damage under zero source load.
180
-------
TABLE C.2.1 DAMAGE FUNCTION BREAKPOINTS
Damage
function
value
0
1
2
3
4
5
6
7
8
9
10
H+ ions
pH
7.00
6.75
6.50
6.25
6.00
5.50
5.00
4.50
4.00
3.95
3.90
Cone
1.00 x 10"7
1.78 x 10"7
3.16 x io~7
5.62 x 10~7
1.00 x 10~6
3.16 x 10~6
1.00 x 10~5
3.16 x 10~5
1.00 x 10~4
1.12 x 10~4
1.26 x 10~4
OH~ ions
pOH
7.00
6.50
6.00
5.80
5.60
5.30
5.00
4.50
4.00
3.95
3.90
Cone
1.00 x 10"7
3.16 x 10~7
1.00 x 10"6
1.58 x 10"6
2.51 x 10~6
5.01 x 10"6
1.00 x 10"5
3.16 x 10"5
1.00 x 10~4
1.12 x 10~4
1.26 x 10"4
181
-------
10"7 3*10~7 10"' 3*10~5 10~5 3*10~5
CONC. OP IONS H+ OR OH"
3x10"*
I I I 1 I I 1 L
I . I I L
6.5
5.5
4.5
pH OR pOH
Figure C.2.2 Damage function for pH and pOH
3.5
182
-------
Therefore
(C.2.22a)
where
a. - QSi/(QU± + QS±)
(C.2.22b)
iJ
Expected Damage
QUi \
i + ^Si/
(C.2.22c)
The expected damage can be separated into two parts—-the damage due to H
ions and the damage due to OH~ ions. Let q. be the density function
tli
for pH for the i source. Define
q±(x)
*iOH(x) "
X £ 7
x >_ 7
x < 7
0 ; x >. 7
Then the damage due to J ions (where J - OH~ or H ) is
(C.2.23)
'iJ
(C.2.24)
and the total damage associated with pH is
As discussed earlier, the density function can, depending on the in-
put data, be described by a mean value and either one or two values
183
-------
of standard deviation (see Figure C.2.1 for an example of the latter
distribution) . First consider the case when the distribution is des-
cribed by a single standard deviation. Using (C.2.22a), (C.2.24)
becomes
J
10
"pJ
where ^j is a normal random density function with mean p.- and
standard deviation a^. Making a change of variables, let w - 10"pJ;
then
iJ
• / Vaijw +
"'lO-?
dw
(C.2.26)
where g^ is a lognormal density function with corresponding normal
distribution having mean -y^- and standard deviation o^. Analogous
to the derivation of (C.2.7) we obtain
LJ
11 fBiJk
"S i {e±jkW
iJk
dw
where
a
iJk
dj(k> * bu
- a -
(dj(k+l)
3
iJk
k - 1, 2 ..... 11
(C.2.27a)
k - 1, 2, ..., 10
k - 11
(C.2.27b)
If aUk < 10~' then reset aiJk *
6iJk > lf then reset 0 " ! aad
and
' If
184
-------
'Uk
ai/(dJ(W-l)
; k-1, 2 10
k-11
(C.2.27c)
LiJk
' (biJ " dJ(k))
10
+ (k-1) ; k-1, 2, ..., 10
; k-11 (C.2.27d)
(C.2.27) can be rewritten
11
'u ^ i/-ijk' fuk» auk» 6ijk' "^ij» au)
k-1
(C.2.28)
where IT Is defined In Section C.4.
L
Now consider the case when the density function is of the form
shown in Figure C.2.1 (i.e., the density is defined in terms of a
mean y,.T and two standard deviations a. and cr.rtll). Th.en the
^ f^ it* i^rfl
density function for pH can be written
- q±(x)
, x)
qiH(x) + q1QH(x)
(C.2.29)
where $ is the characteristic function defined in (C.2.2). q.1T(x) is
in
the result when a normal density function with mean y and standard
in
deviation <7.u is restricted to the range x < y and set to zero for
In •— xH
x > y4IT> q^nu ^s similarly defined.
In 3.UH
There are two cases to consider: y.., £ 7 or y.H > 7. First sup-
pose y.u > 7, then using (C.2.23)
iti
185
-------
and
Wx>
qiH(x)
0
q10H(14 - x)
q1R(14 - x)
x < 7
x > 7
(C.2.30a)
(C.2.30b)
x > 7
For this case, the formula for D... is given by (G.2.27). D10H»
however, is now (analogous to (C.2.25))
JiOH
D
iOH
- pJ)dpJ
+ ' VaiOH10~PJ + W
biOH)
biOH>
/• lo'^iOH
' / 7 VaiOHW
10~7
f1
+ J DJ(aiOHW'
(C.2.31)
where ip... and iKOH are lognormal density functions whose corresponding
normal distributions have mean -U.nu and standard deviations 0... and
l(Jn Xu
a.-., respectively. Using (C.2.19), (C.2.31) becomes
lOn
186
-------
DiOH
"iOHk
10
/**4 ATI 0
iv/nx>>
KoH*» + b
10 UIOH £ 10IU*
r "IOHk
J {eiOHkW+biOHk} ^10H(w)dw
aiOHk
^^iOHk* fiOHk* aiOHk* ^iOHk' "^"n»
k-1
~U10H
' aiOHA » 10
"UiOH
' fiOHA±» 10 '^iOH^' ~yiOH' °iOH)
11
' fiOHk* aiOHk» eiOHk» ~P10H» C710H)
(C.2.32)
k-Jl±+l
where £. e' {1, 2, ..., 11} is defined so that a.nvo < 10 1UH < e,
l lUnX. ~~ .
Analogously if p^ <. 7, then the equation for DIQH is given by (C.2.27),
and D .„ becomes
187
-------
Difl " 2~l ^^iHk* fiHk» "iHk' ^iHk'
k-1
• fiitt • aiiu • 10 ' -*<*• is given by (C.2.23).
For the case where the probability distribution is defined by a mean
^iH^iOH " 14 " ^iH^ and a sin8le standard deviation (JIH - cri()H,
p._ is given by
PiJ " IM(0» 1» " °°» EFST^, Wjj. cr±J) (C.2.35)
where L. is defined in Section C.4. The probability of no violation
+ -
due to both H and OH ions is 1 - (p41I + p.nu).
Itl lUn
188
-------
Now consider the case where the density function is defined in terms
of a mean and two standard deviations o and C4rk1] (see Figure
In iOH
C.2.1). Suppose V^ > 7, then P^ is given by (C.2.34). If
yiOH > EFSTiOH then PiOH is also glven bv (C.2.35), otherwise
PiOH
/
/
J I
EFSTiOH
iOH
V
Similarly, if ^£7, then P±()H is given by (C.2.36). If
V1R > EFST^j, then p^ is also given by (C.2.35), otherwise
0.5 + ^(0, 1. UIH, EFST, W, a) (C.2.37)
The probability of no violation is then 1 - (p + p ).
1H iOH
C.2.4 Temperature
The damage due to heat from an effluent is a function of the change in
temperature from its ambient value.
Inputs
The data needed to calculate expected damage and the probability of a
violation are:
For the i source:
y._ « mean temperature change (°C)
189
-------
a.- « standard deviation of the temperature change ( C)
QU. » flow of stream above source (Ml/day)
QS. - effluent flow (Ml/day)
The damage function which is a function of the temperature change from
ambient is of the same form as (C.2.1) with the breakpoints dT(k)
given. in Table 6.1.
Temperature Change
The temperature change or temperature difference between the influent
and effluent is measured for various industries. This section speci-
fies the calculations needed to determine expected damage and probabil-
ity of a violation.
The temperature downstream from the source is
TU, QU + TS, QS ,
TOi ' qp'+Qs"
where TS. is the temperature of the effluent and TU. is the upstream
temperature. The change in the temperature of the river, AT., is
(TS. - TU.) QS.
AT± - TO, - TU± - V, * QS, (C'2'39)
Letting ATS. • TS. - TU. be the change between the influent and
effluent temperature, we have
/
ATi " ATSi (QU,
aiT ATSi
where
190
-------
QS
"IT ' QU, +V (C-2-*°b>
Expected Damage
The expected damage due to temperature change is
DiT " E(DT(ATi))
- / DT(AT1)<|)1T(ATS)d(ATS) (C.2.41)
where .T is the probability density function of the change in effluent
temperature. Combining (C.2.40) and (C.2.41)
DiT " / VaiATS)*iT(ATS)d(ATS) (C.2.42)
Since the damage function is in the same form as in Section C.2.1 (with
b. » 0), the expression for D._ is the same as given by (C.2.7),
(C.2.8) with j - T«
Probability of a Violation
The probability that the standard for temperature is not violated for
source i is given by (C.2.10) with J - T.
C.3 EXPECTED DAMAGE AND PROBABILITY OF VIOLATION DERIVATION
—SEVERAL SETS OF STANDARDS
This section describes the derivation of the expected damage from a
source and the probability of violation when there are several out-
falls, each with its own effluent standards. The most complicated
case treated occurs when the outfalls floW into different bodies of
water.
191
-------
C.3.1 Inputs
The data needed to calculate expected damage and probability of viola-
tion are:
For source i:
R „ » index set of outfalls flowing into stream x.
IX
For source i, stream x:
QU.g « flow of stream above source (Ml/day)
DOSAT . « saturation level of DO in stream (mg/1)
KBOD_DQ - BOD5 - DO transfer coefficient
For source i, outfall k:
P - index set of pollutants
y • mean of mass loading of j pollutant (kg)
ijk
a... » standard deviation of mass loading of j pollutant (kg)
y . - distribution of j* pollutant
QS.. - effluent flow (Ml/day)
» effluent standard for j pollutant (kg)
CU... B upstream concentration of j pollutant (mg/1)
y . - mean of pJ, J - H or OH (pH or pOH)
a.., • standard deviation of pJ, J - H or OH (pH or pOH)
EFST.T - effluent standard of pJ, J • H or OH (pH or pOH)
192
-------
C.3.2 Expected Damage
Expected Damage on Stream & - All Variables Except pH
The expected damage to a single stream depends on the total mass load
of pollutants flowing into it. It is assumed here that the outfalls
are located close enough together, as far as damage is concerned, for
the effluents to be considered as coming from a single outfall. The
following development is for all constituents except pH (pH will be
treated later). It is assumed that the distribution of the jth pollu
tant is the same for all the pipes, that is, -^ , « constant for
k e R.. The effluent flows add, therefore
QS' - total effluent flow from source i into stream £
2-r
QS±k (C.3.1)
Normal Case - For the case where the probability distribution of the
mass loadings is normal, the distribution of the total loading into
stream i is normal with mean and variance equal to the sum of the indivi-
dual means and variances. (This is true under the assumption that the
loadings are independent, which will most likely be the case, since dif-
ferent outfalls are almost always connected to different processes.)
Therefore, for source i, stream A:
y * m mean of mass loading of J pollutant
ijx
(C.3.2)
'. 0) - variance of mass loading of j pollutant
ij*
(C.3.3)
193
-------
t,ognonnal Case - In the lognormal case, the sum of lognormal random
variables is no longer lognormal. In fact, the distribution is, in
general, very complicated, and to use it would be untractable for our
purposes. Since the mean and variance of the distribution of the total
mass loading are equal to the sum of the means and variances, respective-
ly, of the individual loadings, an approximation of the distribution can
be obtained by assuming that the resulting distribution is lognormal with
mean and variance equal to the sum of the means and variances respective-
ly. This will be a very good approximation for the cases of interest.
2
Recalling that P and a. ., are the mean and variance of the corres-
ponding normal distribution, the mean, n»..., an^ variance, v... , of the
lognormal distribution are
=
mijk j (C.3.4)
and
where y * ^n 10 • The mean and variance of the total mass loading are
then
CC.3.7)
Assuming that the resulting distribution is lognormal, the corresponding
normal distribution has variance and mean
194
-------
iog
Expected Damage on Stream A - pH
Since the distribution of pH (or pOH) is normal, the distribution of H
(or OH~) ions is lognormal. Since the loadings of ions add, the dis-
tribution of pH for the total effluent into stream S, is very similar
to the lognormal case just discussed. The major difference arises from
the fact that pH is defined as the negative of the log of the concentra
tion of ions. Thus, equations (C.3.4) through (C.3.9) hold with
then
Total Expected Damage
All the data have now been combined in terms of the total loading due to
source i in stream H. The formulas of Section C.2 can now be used to
obtain the expected damage where all the variables have an extra subscript
denoting the stream into which the outfall flows. So letting D..« be
th
the expected damage due to the flow of the j constituent from source i
into stream &, the expected damage due to the i source can then be
written (analogous to (C.I. 2))
c. - max D , „ (C.3.10)
1 J,A J
C . 3 . 3 Probability of Violation
The calculation of the probability of violation is not complicated
since, if we assume that the effluents from the various outfalls are In-
dependent, the probability of no violation from all the outfalls is the
product of the probability of no violation in each of the outfalls. To
be precise, let
p . - probability of no violation due to pollutant j,
outfall k, source 1.
195
-------
The calculation of P.* .it. is discussed in Section C.2. Using (C.I.3)
and (C.I.4), the probability of no violation of any standard from out-
fall k, source i is
fll p. , ; uncorrelated constituents
1 eP •*
min p... ; correlated constituents
J£Pik
(C.3.11)
The probability of no violation from any pollutant of any outfall for
the source i is then
P± - n p . (C.3.12)
i .fc IK
where we have assumed that the pollutant loadings in the outfalls are
independent.
C.4 CALCULATION OF IMPORTANT INTEGRALS
C.4.1 Normal Case
f8
IN - IN(a, b, a, g, y,a) - / (ax + b) f(x)dx (C.4.1)
f^ *y
where f is normal with mean y and variance or . Therefore
•8
* f ^27raA ( 2a'
a x
•e / 2)
a(x - y) + ya + b ) (x - y) ( . . , 0.
—» J-i—w- ' exp < 2 ( dx (C.4.2)
Let x - (x - y)/a, a - (a - y) /a and § - (3 - y)/o, then
196
-------
/
-x 12
ax + -——-^ I e
a
- aa [f(a) - f(3)] + (ya + b) [F(3) - F(a)] (C.4.3)
where F and f are, respectively, the standard normal distribution and
density function with mean 0 and standard deviation 1.
C.4.2 Lognormal Case
f*
IL - IL(a, b, a, 3, y, a) = / (ay + b) g.(y)dy
a
where g is a lognormal density function whose corresponding normal
2 —
density function has mean y and variance a . Let a « log a,
"3 - log 3 and x * log y, then
H
3
(ae1" + b) 7(x)dx (C.4.4)
f\
where f is normal with mean y, variance a , and where we used the
identity y - 10X - *** » ^ » In 10. Thus
I w
t
«
\
-------
•/:
J a
a
exp
r
a v 2ira
-2'21 ' (*-(u*Av)2
exp
2a
2a
dx
dx
a exp
+ b
(P(ei) -
(C.4.7)
where
Jl
*2
!.=.
a
. 3 - (u + a k)
- ak
a - V
a
a - (u + cTk)
a
^ a, - ak
and F is defined In Section C.4.1.
198
-------
APPENDIX D
RESOURCES REQUIRED TO MONITOR A SOURCE
The monitoring resources, r., required to monitor source i include
field, laboratory, office, and transportation costs. The field and
laboratory costs contain costs due to manpower and equipment. Each
monitoring agency should examine its costs to determine r., since these
costs will vary due to differences in agency structure, size of regions
that are in agencies1 jurisdiction, etc. The purpose of this Appendix
is to develop reasonable values for r^ to be used in the demonstration
part of the project. The transportation costs to travel to the various
sites are assumed small and will be neglected.
D.I FIELD AND OFFICE COSTS
Estimates of manpower requirements for compliance monitoring are given
in [Dl]. It is estimated that it will take 8 man-days to travel to
plant, set up equipment, take measurements, remove equipment, and return
to point of origin. If more than 5 outfalls are to be sampled, the
manpower requirements must be increased. Also, there may be some savings
if additional surveys are conducted in the same vicinity. Mr. R. Christiansen
of the Michigan Water Resources Commission estimated that a two man
crew can make one 24-hour composite measurement in two days (including
set-up and removal) and that the crew can handle four closely spaced
outfalls in this period. Combining these estimates, we shall assume
that it takes 2 men 2-1/2 days (or 5 man-days) to monitor 4 outfalls.
We shall assume that the two man team can be divided between two sources
Private communication.
199
-------
if, at most, two outfalls are located at each source. The office cost
to process the compliance monitoring data is estimated, in [1], to be 3
man-days. In addition, Mr. Christiansen, estimated the equipment costs
to monitor the sources at about $2,500/year. Based on these assumptions,
Table D.I gives the total cost of monitoring a source (not including
laboratory costs) based on a man day costing $64.
Table D.I TOTAL FIELD AND OFFICE COSTS
No. of
outfalls
1 or 2
3 or 4
5 or 6
Manpower
field costs
$320
$640
$960
Manpower
office costs
$192
$192
$192
*
Equipment
costs
$13
$25
$38
Total
cost
$ 525
$ 857
$1190
D.2 LABORATORY COST
The laboratory costs must include both the cost of making the analyses
and the costs of report writing. If a private laboratory is used, then
overhead costs will also be included. If the analyses are done by the
monitoring agency or another government agency, then the capital equipment
costs need not be included, since costs will exist regardless of the
analyses made. For this project, a price list from a private laboratory
was used to estimate the laboratory costs (see Table D.2).
REFERENCE
Dl. "Model Water Monitoring Program", Environmental Protection Agency,
Office of Water Enforcement, 1974.
200
-------
Table D.2 LABORATORY COSTS
Analysis
Cost
Analysis
Cost
Aluminum $ 8.50
Ammonia 10.00
BOD5 20.00
Carbon 10.00
COD 10.00
Chloride 5.00
Chloroform Extract 15.00
Chromium 7.50
Coliforms-Total 15.00
Coliforms-Fecal 15.00
Copper 7.50
Cyanide 15.00
Dissolved Oxygen 3.00
Fluoride 8.00
Iron $ 7.50
Lead 7.50
Manganese 7.50
Mercury 15.00
Nickel 7.50
Nitrogen 10.00
Oil-Grease 10.00
pH 3.00
Phenol 12.50
Phosphorus 10.00
Dissolved Solids 10.00
Suspended Solids 5.00
Tin 8.50
Zinc 7.50
201
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APPENDIX E
BAYESIAN UPDATE FORMULA
Consider the case when both parameters of an independent normal process
are estimated. Using the Bayesian approach, the parameters of the
process, the mean y and precision h, (the precision is equal l/o
where a is the standard deviation) are themselves treated as random
variables. The most convenient [El] joint distribution of the parameters
called the natural conjugate prior - is defined by
f (y,h|m,n,v,V)
(E.I)
x exp < - j
This distribution is known as the normal-gamma distribution and is
uniquely defined by the parameters m,n,v, and v. m is the estimated mean of
the process, v is the estimated variance of the process, n is a constant
expressing the confidence (or uncertainty) in the estimated mean, and V
is a constant expressing the confidence in the estimated variance. For
the case where the estimated mean, m, and variance, v, were obtained from
N identically distributed, independent, normal random variables,{x.},
using the sample mean and sample variance, that is
N
202
-------
and (E.2)
N
then n is equal N and v is equal N-l. Thus n and v express the degrees
of freedom used to obtain the estimates m and v.
Suppose that estimates of the mean and variance, m and v, of a random
process are available with confidence parameters n and v respectively.
The prior distribution is normal-gamma with parameters (m,n,v,v). If a
new sample from the random process is made (independent from the process
which yielded m and v) yielding a sufficient statistic (m1 , n1, v1 , v'),
then the posterior distribution is again normal-gamma with parameters [El]
n,, m n'm* + nm (E.3a)
n' + n
n" - n' + n (E-3b)
i. Tv'v' + n'm'2] + [vv + nm2] - (n' + n)m"2 fv . ,
V « -* - > - \Ci.3C)
v' + v + 1
V" - V' + V + 1 (E.3d)
m" and v" are Bayesian posterior estimates of the mean and the variance
and n" and v" are the corresponding confidence parameters. The formulas
in (E.3) describe how to update old estimates (m,n,v,v) as new estimates
(m' , n1 , v' , v1) become available. If the new estimates are from a
single data point z, then mf «• z, n1 » 1 and v' • v? =0.
203
-------
REFERENCE
El. Raiffa, H. and Schlaiffer, R., Applied Statistical Decision Theory,
The M.I.T. Press, Cambridge, Mass., 1961.
204
-------
APPENDIX F
ESTIMATION OF THE BOD-DO AND COD-DO TRANSFER COEFFICIENTS
AND THE SATURATION LEVEL OF DO*
On streams, rivers, and vertically well-mixed reservoirs the maximum
dissolved oxygen deficit (D. BQ_) due to a BOD effluent is related to
the BOD stream concentration at the effluent source (CO. ) , the BOD
1 y iSUJJ
decay rate (K,), the stream reaeration rate (K ), and the waste dispersion
u a
rate. The initial BOD stream concentration is given by
CO - "i.BOP * CUi.BOD
i.BOD QU± + QS '
The relationship between D^BQD and C.^ BQD can be estimated using a transfer
coefficient as
Di,BOD = (KBOD-DO) (COifB(lD) (F*2>
K can be obtained using Figure F.I along with Figure F.2.
Figure F.I shows that in streams, K^^^ varies primarily with \/^d>
Also, as rivers become more tidally influenced and broad, KBOJ^DQ
increases. Values of K&/Kd can be found for various applications using
Figure F.2.
The damage due to COD loadings is difficult to quantify since there are
many different kinds of chemicals, each with their own reaction
Information derived from Simplified Mathematical Modeling of Water Quality [Fl]
205
-------
figure F.I Dissolved oxygen response as a function of water body type and .
(Note: 4>-Ka/Ka)
-------
H
Q
N.
Creeks &
Shallow
Streams
10-20
1-10
Upstream
Feeders
2-5
10-100
Interme-
diate
Channels
5-10
100-1000
Main
Drainage
Rivers
10-20
1000-
iopoo
Large
Rivers
20-30
IO,OOC
Impounded
Rivers
30 — «-
10.0
e
o
o-i
1.0
0.1
Probable Range
Probable Limits
Figure F.2
10.
DEPTH IN FEET
(K/K) as a function of depth.
100.
207
-------
characteristics, which may demand oxygen. Furthermore, the lab tests
for COD are performed by heating the sample, which probably would not
indicate actual stream damages* For these reasons COD is generally not
a modeled constituent and little is known about its stream characteristics.
However, Prati et al. [F2] found that the damages due to COD are proportional
to those from an equivalent concentration of BOD. The maximum DO uptake
due to COD is related to the initial stream COD concentration (C )
o
through a transfer function (KCon_no^» wnlcn can be estimated as
KCOP-DO * °-15 ''-
Therefore the COD transfer coefficient can be estimated using Figures
F.I and F.2 along with equation (F.3).
DOSAT, the saturation level of dissolved oxygen in the stream, can be
found for various temperatures and salinities using Figure F.3.
REFERENCES
Fl. Hydroscience, Inc., Simplified Mathematical Modeling of Water Quality.
Environmental Protection Agency, Washington, D.C., March 1971.
F2. Prati, L., et al., "Assessment of Surface Water Quality by a
Single Index of Pollution", Water Research (GB) . Vol. 5, pp. 741-751, 1971.
208
-------
to
55
o
Ed
1
8
M
H
I
tn
a
s
o
CO
CO
15. Or-
14.0
13.0
12.0
11.0
10.0
9.0
8.0
7.0
6.0
5.0
4.0
3.0
2.0
1.0
CHLORIDES
A
10
Figure F.3
15
30
35
40
20 25
TEMPERATURE-, "C
Dissolved oxygen saturation versus temperature ant) chlorides.
45
50
-------
APPENDIX G
DATA FOR DEMONSTRATION PROJECT
This appendix contains the statistical description, expected damage and
probability of no violation for each constituent of each source for Case I
and Case II of the demonstration project. The following notation is used:
DIST - Distribution
N - Normal
L - Lognormal
EST. MEAN - Estimated Mean
EST. SIGMA - Estimated Standard Deviation
The units for the standards for the various constituents are in kilograms,
except for pH where the units are in pH. The units for estimated mean and
estimated standard deviations are in kilograms, if the distribution is
normal, and log kilograms if the distribution is lognormal (recall, in the
lognormal case, the mean and standard deviation are of the logs of the data)
For pH the units are in pH,
In the case of pH-max and pH-min only one value of expected damage and
probability of violation is given since only one value is calculated (see
Appendix C). Also note that the expected damage only appears once for
each constituent of a source for the cases where there are two or more
pipes from the source flowing into the stream.
G.I DATA FOR CASE I
210
-------
S3
PIPE* 1 MEAN
CONSTITUENT
PH-MAX
PN.MIN
CHROMIUM
NICKEL
CHLOROFORM EXTRACT
PIPE* 2 MEAN
CONSTITUENT
BODS
SUSPENDED SOLIDS
CHLORIDE
DISCHARGE (ML/DAY)
STANDARD
9.5000
6.5000
.5299
2.6497
3.97«6
DISCHARGE tML/DAY)
STANDARD
.1987
.2650
2.6500
•
DIST
N
N
L
L
L
e
DIST
N
N
N
SOURCE 1
***********
.3407
EST. MEAN
7.5895
7.5895
-U4006
•1.7560
.0151
EST. MEAN
.1184
.2010
.0306
UPSTREAM FLOW
EST. SIGMA
.6730
.4395
.8876
.6635
1.6513
UPSTREAM FLOW
EST. SIGMA
.0484
.0896
.0044
CML/DAY)«
EXPECTED
DAMAGE
********
.2860
.R077
1.3251
1.82^5
(ML/DAY)*
EXPECTED
DAMAGE
1.2426
.0038
.0005
4. £937
PRO1?, OF NO
VIOLATION
***********
.9911
.9673
.9970
,9233
4.8937
PRO". OF NO
VIOLATION
.9515
.76?*
1.0000
**************************************************
SOURCE EXPECTED DAMAGE 1.8245
SOURCE PROBABILITY OF NO VIOLATION ,6*100
**************************************************
-------
PXPE«
MEAN- DISCHARGE (ML/OAY)«
***********
SOURCE 2
***********
.5779
UPSTREAM FLOW (ML/OAY)i
185.9600
CONSTITUENT
PHOSPHORUS
PH.HAX
PH-MIN
SUSPENDED SOLIDS
CHLOROFORM EXTRACT
STANDARD
.8025
9.5000
6.SOOO
24.9476
6.0196
DIST
L
N
N
L
L
EST, MEAN
•1.3866
7.1500
7.1500
.7711
.0432
EST. SIGMA
.saia
.5119
,4994
.3048
.2269
EXPECTED
DAMAGE
.0931
********
.0104
.0502
1,3550
PROR. OF NO
VIOLATION
.99U
***********
.9035
.9000
.999a
ro
PIPE* 2
CONSTITUENT
PHOSPHORUS
PH«MAX
PH-MIN
SUSPENDED SOLIDS
CHLOROFORM EXTRACT
CHARGE (ML/DAY)"
STANDARD
.9388
9.5000
6.5000
58.9670
7.0407
DIST
L
N
N
L
L
.7621
EST. MEAN
-,25?0
7.9333
7.9200
1.5876
.5393
UPSTREAM FLOW
EST. SIGMA
,3547
.3064
,2460
,5508
,2396
(ML/DAY)«
EXPECTED
DAMAGE
********
********
********
********
********
185.9600
PROB. OF NO
VIOLATION
.7367
***********
1.0000
.6302
.9009
**************************************************
SOURCE EXPECTED DAMAGE 1.3554
SOURCE PROBABILITY OF NO VIOLATION .3669
**************************************************
-------
***********
SOURCE 3
***********
PIPE* i MEAN DISCHARGE (MU/OAYJ- .0750 UPSTREAM FLOH (ML/DAY)* is«i.aooo
EXPECTED PROP. OF "0
CONSTITUENT STANDARD DIST E8T. MEAN EST. SIGMA DAMAGE
PH-HAX 9.5000 N 7.7258 .7828 ******** ***********
Ph-MlN 6.5000 N 7.7258 .5798 .0005 .9711
SUSPENDED SOLIDS 16.0875 N ,5?63 .4465 .0000 1.0000
PHOSPHORUS 3.2175 N .0477 .0704 .0006 1.0000
**************************************************
SOURCE EXPECTED DAMAGE .0006
SOURCE PROBABILITY OF NO VIOLATION .9711
**************************************************
to
H
u>
-------
to
PIPE* 1 MEAN
CONSTITUENT
PH-MAX
PH-MIN
SUSPENDED SOLIDS
CHLOROFORM EXTRACT
PIPE* 2 MEAN
CONSTITUENT
PH-MAX
PH-MIN
SUSPENDED SOLIDS
CHLOROFORM EXTRACT
DISCHARGE (ML/OAY)i
STANDARD
9.0000
6.0000
18.9265
7.5706
DISCHARGE (ML/DAY)>
STANDARD
9.0000
6.0000
7.5705
3.0282
i
DIST
N
N
L
L
i
DIST
N
N
L
L
***********
SOURCE 4
***********
.8026
EST. MEAN
8.0385
8.0385
.7329
.5695
.3762
E8T. MEAN
7.9083
7.9833
.8516
.2921
UPSTREAM FLOM
EST, SIGMA
.1256
.1234
.1648
.1684
UPSTREAM FLOW
EST. SIGMA
.2509
.2342
.2116
.2107
(ML/OAY)*
EXPECTED
DAMAGE
********
.0451
.02*3
3.4284
(ML/DAY)*
EXPECTED
DAMAGE
********
********
********
********
51.3840
PRGR. OF NQ
VIOLATION
***********
1.0000
.9995
.9670
51.3040
PROP. OF NO
VIOL*TIQN
***********
1.0000
.5517
.8153
**************************************************
SOURCE EXPECTED DAMAGE 3.428«
SOURCE PROBABILITY OF NO VIOLATION .4348
**************************************************
-------
Ul
PIPE* i
CONSTITUENT
PH-HAK
PM-MIN
OIL-GREASE
PHENOL
PIPE- 2
CONSTITUENT
PH-HAX
PH.MIN
OIL-GREASE
PHENOL
MEAN .DISCHARGE (ML/DAY)*
STANDARD
10.3000
5.8000
251.8800
.9072
MEAN DISCHARGE (ML/DAY)*
STANDARD
10.3000
5.8000
059.2050
1.3608
DIST
N
N
N
L
DIST
Kl
N
L
***********
SOURCE 6
***********
18.7919
EST. MEAN
7.6797
7.6797
140.9276
•.9000
34.2069
EST. MEAN
7.8392
7.8392
165.8726
-.6423
UPSTREAM FLOW
EST. SIGMA
.4289
.3663
128.6569
.3921
UPSTREAM FLOW
EST. SIGMA
.2907
.3918
68.0088
.3921
fML/DAY}«
EXPECTED
DAMAGE
********
.0416
4.0479
1.4198
CML/DAY)»
EXPECTED
DAMAGE
********
********
********
********
1358.0000
PROD, OF NO
VIOLATION
***********
1.0000
1.0000
,9904
1358.0000
PRO*. OF NO
VIOLATION
***********
1.0000
1.0000
.9762
**************************************************
SOURCE EXPECTED DAMAGE 4.0479
SOURCE PROBABILITY OF NO VIOLATION .9668
**************************************************
-------
PIPE* i
MEAN DISCHARGE (ML/DAY)"
***********
SOURCE 7
***********
2.B95S
UPSTREAM FLOW CML/ruY)«
28.3890
CONSTITUENT
PH-MAX
PH-MIN
SUSPENDED SOLIDS
PHOSPHORUS
FLUORIDE
COPPER
LEAD
STANDARD
9.5000
6.5000
42.5835
5.6778
2.83*9
1.4194
.8517
DIST
N
N
N
N
N
N
L
EST, MEAN
6.9933
6.9933
12.5510
1.4573
5.667tt
.2221
•.0866
EST. SIGMA
.4946
.6821
6.6731
.4256
4.9940
• 11U
.4972
EXPECTED
DAMAGE
********
.2903
.0401
.9342
.0004
.7131
3.5172
PRO*. OF NO
VIOLATION
***********
.7652
i.oooo
1.0000
.2840
1.0000-
.5136
N>
**************************************************
SOURCE EXPECTED DAMAGE 3.5172
SOURCE PROBABILITY OF NO VIOLATION .1116
**************************************************
PIPE* 1 MEAN
CONSTITUENT
PH-MAX
PH-MIN
SUSPENDED SOLIDS
PHOSPHORUS
CYANIDE
FLUORIDE
CHROMIUM
COPPER
LEAD
CHLOROFORM EXTRACT
DISCHARGE (ML/OAY)
STANDARD
9.5000
6.5000
15.9982
1.0599
.1325
9.53*9
.2650
.5299
.0530
7.9491
8
DIST
N
N
N
L
L
N
L
L
L
N
***********
SOURCE 8
***********
.0005
EST. MEAN
8.6090
8.6Q90
3. 7636
•1.7588
•1.2287
15.6843
-.8677
•1.1041
•.2816
.2653
UPSTREAM FLOW
EST. SIGMA
.4199
.5094
2.3490
.4756
.1800
6.3081
.5863
.6289
.7189
.2743
(ML/DAY)*
EXPECTED
DAMAGE
********
.0001
.9020
,0032
.1099
.0000
.1679
.1108
1.0083
.0740
195.7400
PROP. OF NO
VIOLATION
***********
.9631
1.0000
.9999
.7697
.1650
.6901
.9061
.0833
1.0000
*«************«*******>***************************
-------
PIPE* 1
MEAN DISCHARGE CML/DAY)i
SOURCE 9
***********
5.5546
UPSTREAM FlOW (ML/DAY)»
78.2990
CONSTITUENT
BOOS
PH-MAX
PH-MIN
SUSPENDED SOLIDS
CHROMIUM
NICKEL
CHLOROFORM EXTRACT
STANDARD
180,2700
9. "5000
6.5000
473.1750
5.6781
5.6781
283.9050
OIST
N
N
N
L
L
L
N
EST, MEAN
435.3778
8.1921
8.1921
1.3117
1.0355
-1.8793
43.5777
EST. SIGMA
390.7530
1.0480
,9570
• Z691
1.0919
1.0136
33.3680
EXPECTED
DAMAGE
4.6630
********
1.2080
.0296
3.9607
.2167
7.7820
PROB. OP MO
VIOLATION
.2644
***********
.8555
1.0000
».398«
.9953
1.0000
**************************************************
SOURCE EXPECTED DAMAGE 7.7820
SOURCE PROBABILITY OF NO VIOLATION ,0897
**************************************************
PIPE* 1
MEAN DISCHARGE (ML/DAY)«
***********
SOURCE 10
***********
1.2846
UPSTREAM FLOW
-------
MEAN DISCHARGE (ML/DAY)*
***********
SOURCE tl
***********
2.6909
UPSTREAM FLOW (ML/DAY>«
26.1180
CONSTITUENT
PH«MAX
PH*MIN
CYANIDE
CHROMIUM
COPPER
NICKEL
STANDARD
10.5000
6.5000
• 6530
5. 2238
2.6119
13.0595
DIST
N
N
L
L
L
L
EST. MEAN
a. 1815
8. IMS
..9011
•1.0685
.1026
••2420
EST. SIGMA
.6907
.6776
.3946
.3222
.3071
.5054
EXPECTED
DAMAGE
********
.9760
1.1064
.3893
2,7195
1.A169
PROS. OF NO
VIOLATION
***********
.9931
.9723
1.0000
.6470
,9964
**************************************************
SOURCE EXPECTED DAMAGE 2.7195
SOURCE PR08ABTLITY OF NO VIOHTtQN .8149
**************************************************
N3
H
oo
***********
SOURCE 12
***********
PIPE* i
CONSTITUENT
BODS
PK»MAX
PH-MIN
SUSPENDED SOLIDS
CHLOROFORM EXTRACT
CHARGE (ML/DAY)
STANDARD
41.6380
9.0000
6.0000
104.0950
41.6380
*
DIST
N
N
N
L
N
4.9209
EST. MEAN
60.5332
7,5779
7,5779
1.55*7
61.0981
UPSTREAM FLOW
EST. SIGMA
52.8607
.2A8D
.2910
.4165
140.8109
(ML/DAYJ*
EXPECTED
DAMAGE
.9089
********
.0452
.0304
5.6605
1B3.5200
PROR. OF NO
VIOLATION
.3604
***********
1.0000
.8646
.4050
**************************************************
SOURCE EXPECTED DAMAGF 5.6605
SOURCE PRQBABTLITY OF NO VIOLATION . ,iU7
**************************************************
-------
PIPE* i
MEAN DISCHARGE (ML/DAY)"
***********
SOURCE 13
***********
,4044
UPSTREAM FLOW (ML/OAY)i
,0466
CONSTITUENT
BOOS
PH-HAX
PH-MIK
STANDARD
4.0630
10.5000
6.0000
OIST
N
N
N
EST. HEAN
a. 4000
7.8927
7.8927
EST. SIGMA
1.7550
.3508
.2699
EXPECTED
DAMAGE
3.3402
********
.6080
PRQR. PF NO
VIOLATION
.9214
***********
1.0000
**************************************************
SOURCE EXPECTED DAMAGE 3.3402
SOURCE PROBABILITY OF NO VIOLATION .9214
**************************************************
to
***********
SOURCE 14
***********
PIPE* i
CONSTITUENT
PH-MAX
PH-HIN
SUSPENDED SOLIDS
CYANIDE
CHROMIUM
COPPER
CHLOROFORM EXTRACT
CHARGE (HL/OAY)'
STANDARD
9.5000
6. 500Q
50.3440
.3596
«.3l52
2.8768
21.5760
•
DIST
N
N
N
N
L
L
N
.1514
EST. MEAN
7.8002
7.8002
2.4662
.0140
•2.2696
-1.0677
UPSTREAM FLOW
EST. SIGMA
.4945
.4487
1.4728
.0241
1.5801
.2796
.7672
(ML/DAY)*
EXPECTED
DAMAGE
********
.0398
.0126
.1842
.6237
.5301
2.4330
19.5750
PROB. OF NO
VIOLATTON
***********
.9978
1.0000
t.QOOO
.9670
i.oooo
1.0000
**************************************************
SOURCF EXPECTED DAMAGE 2.4330
SOURCE PROBABILITY OF NO VIOLATION .9649
**************************************************
-------
***********
SOURCE 15
***********
PIPE* t MEAN DISCHARGE (ML/DAY)* .9024 UPSTREAM FLOW (HI/DAY)* 66.0650
EXPECTED PRO*. OF NO
CONSTITUENT STANDARD DIST EST. MEAN EST. SIGMA DAMAGE VIOLATION
PH-MAX
PH-MIN
IEAO
9.0000
6. 0000
.0084
N
N
L
7.3853
7.3653
-.0143
1.4051
1.7557
.5497
********
.7313
2.6134
***********
.6530
.0001
**************************************************
SOURCE EXPECTED DAMAGE 2.8134
SOURCE PROBABILITY OF NO VIOLATION .0001
**************************************************
N>
10
O
***********
SOURCE 16
***********
PIPE* 1 MEAN DISCHARGE (ML/DAY)* .7253 UPSTREAM FLOW (ML/DAY)« 6.9649
CONSTITUENT
PH-MAX
PH-MIN
SUSPENDED SOLIDS
OIL-GREASE
MERCURY
**************************************************
SOURCE EXPECTED DAMAGE 4.0721
SOURCE PROBABILITY OF NO VIOLATION .9618
**************************************************
STANDARD
9.5000
6.5000
24.3771
10.4473
.0035
DIST
N
N
N
N
L
EST. MEAN
7.9391
7.9J91
4.A520
4.9166
•3.1565
EST. SIGMA
.1695
.1738
2.6631
2.6286
.2107
EXPECTED
DAMAGE
********
.3645
.0636
4.0721
.2040
PRO*. OF NO
VIOLATION
***********
1.0000
1.0000
.9623
.9995
-------
to
PIPE" 1 MEAN
CONSTITUENT
CHLOROFORM EXTRACT
PIPE" 2 MEAN
CONSTITUENT
CHLOROFORM EXTRACT
PIPE* 3 MEAN
CONSTITUENT
CHLOROFORM EXTRACT
PIPE* 4 MEAN
CONSTITUENT
PH-MAX
PH»MIN
SUSPENDED SOLIDS
PHOSPHORUS
ALUMINUM
CHLOROFORM EXTRACT
DISCHARGE (ML/DAY)*
STANDARD DIST
40.12^0 L
DISCHARGE (ML/DAY)*
STANDARD DIST
9.3118 L
DISCHARGE (ML/DAY)*
STANDARD DIST
10.3718 L
DISCHARGE (ML/DAY)*
STANDARD DIST
9.5000 N
6.5000 N
156.7100 L
31.3420 N
62.6840 N
62.6840 L
*«*#***#**«
SOURCE 17
***********
1.6632
EST. .MEAN
.8641
.4707
ESI. MEAN
.3052
.6279
EST. MEAN
.1147
.0035
EST. MEAN
6.1343
6.8303
1.0723
25.6556
2.0438
1.1427
UPSTREAM FLON (ML/DAY)*
EXPECTED
EST. SIGMA DAMAGE
.5312 4.0183
UPSTREAM FLOH (ML/DAY)*
EXPECTED
EST. SIGMA DAMAGE
.3105 ********
UPSTREAM FLOW (HL/DAY)*
EXPECTED
EST. SIGMA DAMAGE
.4631 ********
UPSTREAM FLO* tML/DAYJ*
EXPECTED
EST. SIGMA DAMAGE
.7875 ********
.9770 .1516
.5123 .0080
14.5367 1.7431
1.2657 1.3103
.5517 ********
293,6200
PROS., OF NO
VIOLATION
.9180
293.6200
PROB, OF NO
VIOLATION
.9837
293.6200
PRO". OF NO
VIOLATTON
.9742
293.6?00
PROB. OF NQ
VIOLATION
***********
.6335
.9858
.6522
1.0000
.8822
**************************************************
SOURCE EXPECTED DAMAGE 4.0183
SOURCE PROBABILITY OF NO VIOLATION .3161
«*********************************»*»*******»,,fM*
-------
***********
SOURCE 16
***********
PIPE* i
MEAN DISCHARGE fML/OAY)«
35.5510
UPSTREAM FLOW
-------
PIPE* i
MEAN DISCHARGE (ML/HAY)*
***********
SOURCE 20
***********
.6176
UPSTREAM FLOW CML/DAY)(
195.7500
CONSTITUENT
PH-MAX
PH«MIN
SUSPENDED SQUIDS
PHOSPHORUS
STANDARD
9.5000
6.5000
49.8725
9.9745
DIST
N
N
N
N
EST. MEAN
6.7000
6.7000
28.1263
3,5321
EST. SIGMA
.3809
.2137
16.3831
3.5105
EXPECTED
DAMAGE
********
,0073
.0105
.3888
PROB, OF NO
VIOLATION
***********
.7941
.9078
.9668
*********************************************,****
SOURCE EXPECTED DAMAGE .388*
SOURCE PROBABILITY OF NO VIOLATION ,6969
**************************************************
10
CO
***********
SOURCE 22
***********
PIPE« 1 MEAN DISCHARGE
-------
N>
10
PIPES 1 MFAN
CONSTITUENT
BODS
SUSPENDED SOLIDS
PHOSPHORUS '
DISCHARGE (ML/DAYJ
STANDARD
184.1583
104.7798
4.3152
m
DIST
N
N
N
***********
. SOURCE 23
***********
.4251
EST. MEAN
39.9745
30.4148
3,7928
UPSTREAM FLOW
EST. SIGMA
17.9516
28.3043
.8987
(HL/DAY)«
EXPECirD
DAMAGE
2.8746
.2114
4.2206
14,9260
PROB. -OF NO
VIOLATION
1.0000
.9957
.7195
SOURCE EXPECTED DAMAGE 4.2206
SOURCE PROBABILITY OF NO VIOLATION ,7164
PIPE- 1
MEAN DISCHARGE (ML/DAY)*
CONSTITUENT
STANDARD
DIST
SOURCE 24
***********
3.0492
EST. MEAN
UPSTREAM FLOW (ML/DAY)"
EST. SIGMA
EXPECTED
DAMAGE
269.160ft
PROS. OP NO
VIOLATION
BODS
SUSPENDED SOLIDS
408.2328
272.1552
N
N
244.2449
146.0120
67*0962
37.1962
1.0989
.0536
.9927
.9997
**************************************************
SOURCE EXPECTED DAM»GE 1.0989
SOURCE PROBABILITY OF NO VIOLATION .9924
**************************************************
-------
SOURCE 25
***********
i *EAN DISCHARGE (ML/DAY). 1*4.9721 UPSTREAM FLOM (ML/OAY)« 1827.8000
EXPECTED PROB.'OF MO
CONSTITUENT STANDARD DIST EST. MEAN EST. SIGMA DAMAGE VKH.ATTON
M «*.«,. 4535.9200 N 5622.2911 2094.1224 5.7470 .3316
SUSPENDED SOLIDS 3628.7360 I 3.7494 .1844 .3084 !l518
**************************************************
SOURCE EXPECTFD DAMAGE / 3.7470
SOURCP PR08ABTLITY OF NO VIOLATION .0503
I*************************************************
N>
10
Ul
***********
SOURCE 26
***********
MEAN DISCHARGE (ML/OAYJa 7.1535 UPSTREAM FLOW (ML/DAY)«
MEAN DO CONCENTRATION (MG/L)» 5.7838
STANDARD DIST EST. MEAN EST. SIGMA DAMAGE VIOLATION^
BODS
SUSPENDED SOLIDS
PHOSPHORUS
278.9591
302.0923
72.2990
L
L
N
2.0125
2.2752
43.2767
.4327
.3726
74.5506
.1150
.0146
.6019
.8416
.7086
.6515
**************************************************
SOURCE EXPECTED DAMAGE .6019
SOURCE PROBABTLITY OF NO VIOLATION .3886
*»*****»****************»*******»**»*»*****»*»***,
-------
PIPE* 1
CONSTITUENT
BODS
SUSPENDED SOLIDS
PHOSPHORUS
MEAN DISCHARGE (ML/DAY)«
STANDARD
272.1552
272.1552
58.2940
•
DIST
M
N
N
***********
SOURCE 27
***********
5.5699
EST. MEAN
3543.7104
2791.1669
291.9943
UPSTREAM FLOW
EST. SIGMA
1179.1592
1131.6056
61.2851
(ML/DAY)«
EXPECTED
DAMAGE
6.4625
.7859
6,5742
349.9000
PROS*- OF NO
VIOLATION
.0028
inooi
**************************************************
SOURCE EXPECTED QAM4GE 6.5742
SOURCE PROBABILITY OF NO VIOLATION .0000
V*************************************************
S3
***********
SOURCE 28
***********
1 MEAN DISCHARGE CML/OAY)* 110.8503
MEAN DO CONCENTRATION CHS/D* 4.8750
UPSTREAM FLOW CML/DAYJI
266.7100
CONSTITUENT
BOD5
SUSPENDED SOLIDS
PHOSPHORUS
STANDARD
4980.5120
4082.3210
529.9SOO
DIST
N
N
N
EST. MEAN
1228,5462
2549.2877
333.7J07
EST. SIGMA
1460.4089
2072.4836
177.2986
EXPECTED P(
DAMAGE \
4.5018
.7041
6.3182
JOB. OF NO
VIOLATION
.9950
.7703
.8658
**************************************************
SOURCE EXPECTED DAM4GE 6.3182
SOURCE PROBABILITY OF NO VIOLATION ,6636
**************************************************
-------
PIPE- i
CONSTITUENT
BODS
SUSPENDED SOLIDS
MEAN DISCHARGE (ML/DAY)
STANDARD
170.0970
170.0970
•
DIST
N
L
SOURCE 29
***********
4.1106
EST. *EAN
73.5866
1.6046
UPSTREAM FLOW
EST. SIGMA
67.2152
.3664
(ML/OAY)>
EXPECTFO
DAMAGE
6.0919
.3525
12.2340
PROP. 'OP NO
VIOLATION
.9245
,«55J
M***********************************!^, „»„,„,,
SOURCE EXPECTED DAMAGE 6.0919
SOURCE PROBABILITY OF NO VIOLATION .8832
*************************«*********,**,„„, ******
K>
NS
***********
SOURCE SO
***********
PIPE* i
MEAN DISCHARGE (ML/DAY)'
35.0425
UPSTREAM FLOW tML/OAY)* 1862.1000
CONSTITUENT
STANDARD
DIST
BODS
SUSPENDED SOLIDS
EST. MEAN
EXPECTED
EST. SIGMA DAMAGE
PROW, OF NO
VIOLATION
1567.5720
1360.7760
N
N
1437.6229
1460.9076
570.2724
552.7676
1.0909
.0770
.6036
.42*1
SOURCE EXPECTED DAMAGE
SOURCE PROBABILITY OF NfJ VIOLATION
**
1.0«0«
.2584
-------
00
PIPE" i MEAN
CONSTITUENT
PH-MAX
PH-HIN
CHROMIUM
NICKEL
CHLOROFORM EXTRACT
PIPE* 2 MEAN
CONSTITUENT
BODS
SUSPENDED SOLIDS
CHLORIDE
***********
SOURCE 1
***********
DISCHARGE (ML/DAY)s .3407
STANDARD
9.5000
6.5000
.5299
2.6497
3.9746
DISCHARGE (ML/DAY)
STANDARD
.1987
.2650
2.6500
DIST
. N
N
L
L
L
•
DIST
N
N
N
EST. MEAN
7,5679
7.5766
•1.8839
-1,5674
•1.0848
.0151
EST. MEAN
.1371
.1914
.0287
UPSTREAM FLOW
EST. SIGMA
.6912
.3141
.6766
.6141
1.4235
UPSTREAM FLOW
EST. SIGMA
.0447
.0646
.0032
(ML/RAY)»
EXPECTED
DAMAGE
********
.2891
.6285
1.1391
2.7477
(ML/DAY)"
EXPECTED
DAMAGE
1.2462
.0036
.00.04
4,8937
PROB, OF NO
VIOLATION
***********
.9971
.9913
.9994
.8816
4.8937
PPOB. OF NO
VIOLATION
.9161
.9727
1.0000
**************************************************
SOURCE EXPECTED DAMAGE 2.7477
SOURCE PROBABILITY OF NO VIOLATION .6963
**************************************************
-------
***********
SOURCE 2
***********
PIPES
CONSTITUENT
PHOSPHORUS
PH.MAX
PH-MIN
SUSPENDED SOLins
CHLOROFORM EXTRACT
CHARGE (ML/DAY)
STANDARD
.8025
9.5000
6.5000
24.9476
6.0186
y
DIST
L
N
K
L
L
.3779
EST. MEAN
•.8619
7.1062
7.1062
.8869
.1969
UPSTREAM FLOW
EST. SIGMA
.7768
1.0055
.6984
.3668
.3496
(ML/RAY)*
EXPECTED
DAMAGE
.1993
********
.0530
.0645
1.7011
185.9600
PROP. OF' NO
VIOLATION
.83«1
***********
.7987
.9179
.9522
N>
PIPES 2
CONSTITUENT
PHOSPHORUS
PH.MAX
PH*MIN
SUSPENDED SOLIDS
CHLOROFORM EXTRACT
CHARGE (ML/DAY)
STANDARD
.9388
9.5000
6.5000
58.9670
7.0407
s
DIST
L
N
N
L
L
.7621
EST. MEAN
-.1358
7.8937
7.8862
1.7498
.6289
UPSTREAM FLOW
EST. SIGMA
.4329
.2363
.1998
.5023
.2428
(ML/DAY)-
EXPECTED
DAMAGE
********
********
********
********
********
lflS.9600
PROP. OF NO
VIOLATION
.5989
***********
1.0000
.5165
.8161
**************************************************
SOURCE EXPECTED DAMAGE 1.7011
SOURCE PROBABILITY OF NO VIOLATION .1477
**************************************************
-------
***********
SOURCE 3
***********
PIPE* 1
CONSTITUENT
PH-MAX
PH«MIN
SUSPENDED SOLIDS
PHOSPHORUS
CHARGE (ML/DAY)
STANDARD
9.5000
6.5000
16.0875
3.2175
•
OIST
N
N
N
N
.0750
EST. MEAN
7.7466
7.7466
.6242
.0466
UPSTREAM FLOW
EST. SIGMA
.6625
.5844
,6757
.0661
-------
ro
u>
I-1
PIPE" 1 MEAN
CONSTITUENT
PH-MAX
PH-MIN
SUSPENDED SOLIDS
CHLOROFORM EXTRACT
PIPE" 2 MEAN
CONSTITUENT
PH-HAX
PH-MIN
SUSPENDED SOLIDS
CHLOROFORM EXTRACT
DISCHARGE (ML/DAY)*
STANDARD
9.0000
6.0600
16.9265
7.5786
DISCHARGE (MU/DAYJ*
STANDARD
9.0000
6.0000
7.5705
3,0282
DIST
N
N
L
L
DIST
N
N
L
L
***********
SOURCE . 4
.8026
EST, MEAN
8,0437
8.0437
.6157
.3023
.3762
EST. MEAN
7.6273
7.9273
.6035
.0320
UPSTREAM FLOW
EST. SIGMA
.1003
.1087-
.2950
.5246
UPSTREAM FLOW
EST. SIGMA
.2418
.2991
.4231
.4540
(ML/DAY)"
EXPECTED
DAMAGE
********
.0400
.0222
3.1067
(ML/OAY)«
EXPECTED
DAMAGE
********
********
********
********
51.3840
PROB. OF NO
VIOLATION -
***********
i.oooo
.9875
.8643
51.3840
PROS. OF NO
VIOLATION
***********
i.oooo
.7426
.8)88
**************************************************
SOURCE EXPECTED DAMAGE 3.1667
SOURCE PROBABILITY OF NO VIOLATION *53t6
**************************************************
-------
***********
SOURCE 4
***********
PIPE- 1
MEAN DISCHARGE (ML/DAY)*
18.7919
UPSTREAM FLOW (ML/DAY)* 1356.0000
CONSTITUENT
PH-HAX
PH.MIN
OIL-GREASE
PHENOL
STANDARD
10.3000
5.8000
251.8800
.9072
DI8T
N
N
M
L
EST. MEAN
7,6891
7.6891
95.7457
-.0200
EST. SIGMA
.3800
.3377
100.3298
,6685
EXPECTED
DAMAGE
********
.0378
3.7626
3.4322
PROS, OF NO
VIOLATION
***********
1.0000
.9402
.4867
10
PIPE* 2
MEAN DISCHARGE (ML/DAY)«
34.2089
UPSTREAM FLOH (ML/DAY)* 1358.0000
CONSTITUENT
PHoMAX
PHoMIN
OIL-GREASE
PHENOL
STANDARD
10.3000
5.8000
458.2050
1.3608
DI8T
N
N
N
L
EST. MEAN
7,7908
7.7908
137.1951
••0996
EST. SIGMA
.3865
.3283
63.3132
.5166
EXPECTED
DAMAGE
********
********
********
********
PROP. OF NO
VIOLATION
***********
l.OOAO
1.0000
.6738
**************************************************
SOURCE EXPECTED DAMAGE 3.7626
SOURCE PROBABILITY OF NO VIOLATION ,3063
CM***********************************************
-------
PIPE* 1
MEAN DISCHARGE (ML/DAY)a
CONSTITUENT
STANDARD
PH-HAX
PHwMIN
SUSPENDED SOLIDS
PHOSPHORUS
FtUORlOE
COPPER
LEAD
DIST
9.5000
6.5000
42.5835
5.6778
2.8389
1.4194
.6517
N
N
N
N
N
L
***********
SOURCE 7
***********
2.895*
EST. MEAN
»•••»•»•§»•*••
6.9933
6.9933
12.5510
1.4573
5.6674
.2221
•.0666
UPSTREAM FLOW (HL/DAYJi
EST. SIGMA
imm*mmmmmmm+
.4946
.6821
6.6731
.4256
4.9540
.1116
.4972
EXPECTED
DAMAGE
********
.2903
,0404
.9317
.0004
.7131
3.5172
28.3690
PROB. QF NO
VIOLATION
mmmmmmmmmfm
***********
.76*2
1.0000
i.oooo
.2640
1.0000
.5136
*!i!!S*!*!!!**************»***********»**»****»***»
SOURCE EXPECTED DAMAGE 3.517?
SOURCE PROBABILITY OF NO VIOLATION .1116
N>
to
U>
Wt* I
MEAN DISCHARGE (ML/DAY)"
***********
SOURCE 8
***********
.0005
UPSTREAM FLOW (ML/DAY)« 195.7400
CONSTITUENT
PH«MAX
PH-HIN
SUSPENDED SOLIDS
PHOSPHORUS
CYANIDE
FLUORIDE
CHROMIUM
COPPER
LEAD
CHLOROFORM EXTRACT
STANDARD
9.5000
6.5000
15.6982
1,0599
• 1325
9.5369
.2650
.5299
.0530
7.9491
DIST
N
N
N
L
L
N
L
L
L
N
EST. MEAN
8.6556
6.6556
3.6416
•1.6940
•1.2287
23,5479
-.6769
•.7528
••1176
.2652
EST. SIGMA
.3375
.3978
2.5
-------
PIPED 1 MEAN (
CONSTITUENT
BODS
PH.MAX
PH-MIN
SUSPENDED SOLIDS
CHROMIUM
NICKEL
CHLOROFORM EXTRACT
JISCHARGE (ML/DAY!
STANDARD
189.2700
9,5000
6.5000
473.1750
5.6761
5.6781
283.9050
la
DIST
N
N
N
L
L
L
N
SOURCE 9
***********
5.5546
EST. MEAN
426.1661
8.9937
8.9937
1.2750
1.3116
•1.6607
83.6446
UPSTREAM FLOk
EST. SIGMA
269.2156
1.1545
1.1431
.3221
.8630
.6699
58.0347
1 fML/PAY)s
EXPECTED
DAMAGE
4.6290
********
2.7035
.0296
4.5256
.?358
6.7244
78.2990
PROS, OF NO
VIOLATION
.189U
***********
.6549
1.0000
.2639
.9972
.9997
**************************************************
SOURCE EXPECTED DAMAGE 6.7244
SOURCE PROBABILITY OF NO VIOLATION .0326
**************************************************
CO
DISCHARGE (ML/OAY)«
***********
SOURCE 10
***********
1.2648
UPSTREAM FLOW (ML/DAY)*
112.5600
CONSTITUENT
PH.MAX
PH.MIN
SUSPENDED SOLfDS
PHOSPHORUS
CHLOROFORM EXTRACT
OIL-CREASE
STANDARD
10.5000
6.5000
46.3715
1.3249
19.8735
19,8735
DIST
N
N
L
L
L
L
EST. MEAN
6.2339
6.2339
1.4912
.0772
1.3489
1.3559
EST. SIGMA
.7128
.8945
.2836
.2934
.3416
.3163
EXPECTED
DAMAGE
********
.2517
.0337
.2637
5.6230
3.9SS2
PROB. OF MO
VIOLATION
***********
.9730
.7314
,5609
.4011
.0262
**************************************************
SOURCE EXPECTED DAMAGE 5.6?30
SOURCE PROBABILITY OF NO VIOLATION ,075ft
»*******%<*************%**************************
-------
NJ
<*>
Ot
PIPE- 1
MEAN DISCHARGE (ML/DAV)!
***********
SOURCE 11
***********
2.6900
UPSTREAM FLOW
-------
PIPE* i
CONSTITUENT
BOD5
PH-MAX
PH-MIN
CHARGE (ML/DAY)*
STANDARD DIST
4.6830 N
10.5000 M
6.0000 N
***********
SOURCE 13
***********
.4444
EST. MEAN
3.1851
7.7968
7.7968
UPSTREAM FLOK
EST. SIGMA
2.5917
.2909
.2994
1 (ML/OAY)*
EXPECTED
DAMAGE
3.5355
********
.4554
2.4468
PROB. OF NO
VIOLATION
.7438
***********
1.0000
**************************************************
SOURCE EXPECTED DAMAGE 3.535*
SOURCE PROBABILITY OF MO VIOLATION .7438
**************************************************
to
PIPE* i
MEAN DISCHARGE (ML/OAY)"
***********
SOURCE 14
***********
• 1514
UPSTREAM FLOM (ML/DAY))
19.5750
CONSTITUENT
PH-MAX
PH-MIN
SUSPENDED SOLIDS
CYANIDE
CHROMIUM
COPPER
CHLOROFORM EXTRACT
STANDARD
9.5000
6.5000
50.3440
.3596
4.3152
2.8768
21.5760
DI8T
N
N
N
N
L
L
N
EST. MEAN
7.8169
7.8169
2.4780
.0177
-1.9036
-1.0361
1.2122
EST. SIGMA
.3460
.3163
1.5581
.0284
1.0644
.2755
•5944
EXPECTED
DAMAGE
********
.0287
.0127
.2265
.4666
.5661
2.2673
PROP. OF NO
VIOLATION
***********
1.0000
1.0000
1.0000
.9915
1.0000
1.0000
**************************************************
SOURCE EXPECTED DAMAGE 2.2673
SOURCE PROBABILITY IF NO VIOLATION .9914
**************************************************
-------
PIPE* i
MEAN DISCHARGE (HL/DAY}«
CONSTITUENT
STANDARD
DIST
***********
SOURCE t^
***********
.9024
EST. MEAN
UPSTREAM FLOW (MU/OAY)»
EST, SIGMA
EXPECTED
DAMAGE
66.0650
PROS.' OF. NO
VIOLATION
PH«HAX
PH-MXN
LEAD
9.0000
6.0000
.0084
N
N
L
8.4552
8.4552
••2231
1.8041
1.6507
.6257
********
1.5457
2.3758
***********
.5526
.0016
**************************************************
SOURCE EXPECTED DAMAGE 2.3758
SOURCE PROBABILITY OF NO VIOLATION .0009
**************************************************
v*>
PIPE- i
MEAN DISCHARGE (ML/DAY)*
***********
SOURCE 16
***********
.7251
UPSTREAM FLOW (ML/DAY)•
6.9649
CONSTITUENT
PH«MAX
SUSPENDED SOLIDS
OIL-GREASE
MERCURY
STANDARD
9.5000
6.5000
24i3771
10.4473
.0035
DIST
N
N
N
N
L
EST. MEAN
7.9801
7.9001
4.6869
4.3342
-3.0912
EST. SIGMA
.2045
.2307
2.8447
1.9318
.2410
EXPECTED
DAMAGE
********
.4128
.0617
4.1203
.2459
PROB. OF NO
VmL*TION
***********
1.0000
1.0000
.9992
.9957
**************************************************
SOURCE EXPECTED DAMAGE 4.1203
SOURCE PROBABILITY OF NO VIOLATION .994*
**************************************************
-------
NJ
U>
00
PIPE* 1 MEAN
CONSTITUENT
CHLOROFORM EXTRACT
PIPE« 2 MEAN
CONSTITUENT
CHLOROFORM EXTRACT
PIPE* s MEAN
CONSTITUENT
CHLOROFORM EXTRACT
PjPEs 4 MEAN
CONSTITUENT
PH-MAX
PH.MIN
SUSPENDED SOLIDS
PHPSPHORUS
ALUMINUM
CHLOROFORM EXTRACT
DISCHARGE (ML/DAY)*
STANDARD DIST
40.1240 L
DISCHARGE
-------
***********
SOURCE IB
***********
PIPE* 1
MEAN DISCHARGE (ML/BAY)»
35.5519
UPSTREAM FLOW
-------
PIPE" 1
MEAN DISCHARGE (ML/DAY)a
***********
SOURCE 20
***********
8176
UPSTREAM FLOW (ML/DAY)
195.7500
CONSTITUENT
PH-MAX
PH-MIN
SUSPENDED SOLIDS
PHOSPHORUS
STANDARD
0.5000
6.5000
49.8725
9.9745
DIST
N
N
N
N
EST. MEAN.
6,6901
6.6613
33.5698
6.4116
EST. SIGMA
.533«
.4555
22.4706
7.5453
EXPECTED
DAMAGE
********
.0161
.0174
.7368
PROB. OF NO
VIOLATION
***********
.6364
.7659
.6616
**************************************************
SOURCE EXPECTED DAMAGE .7368
SOURCE PROBABILITY OF NO VIOLATION .3333
**************************************************
***********
SOURCE 22
***********
PIPES 1 MEAN DISCHARGE (ML/DAY)s
MEAN DO CONCENTRATION (MG/L)» 4.3690
CONSTITUENT
40.7535
UPSTREAM FLO* (ML/DAY)*
203.0900
BODS
SUSPENDED SOLIDS
PHOSPHORUS
STANDARD
1360.7760
907,1640
378.5300
DIST
N
L
L
EST. MEAN
1537. 6694
2.9206
1.6472
EST. SIGH*
1462.6501
.5022
.4856
EXPECTED
DAMAGE
5.3891
.6357
3.6269
PROP. OF NO
VIOLATION
.4519
.5294
.9723
**************************************************
SOURCE EXPECTED DAMAGE 5.3891
SOURCE PROBABILITY OF NO VIOLATION .2326
**************************************************
-------
MEAN DISCHARGE (ML/DAY)i
***********
SOURCE 21
***********
.4251
UPSTREAM FLOW (ML/DAY)m
14.9260
CONSTITUENT
BODS
SUSPENDED SOLIDS
PHOSPHORUS
STANDARD
184.1503
104.7798
4.3152
DIST
N
N
N
EST. MEAN
64.2895
02.5845
4.6613
EST. SIGMA
29.6657
22.5484
1,3021
EXPECTED
DAMAGE
3.8687
.2791
4.6290
PROB. OF MO
VIOLATIO"
1.0000
.9971
.3952
**************************************************
SOURCE EXPECTED DAMAGE 4.6290
SOURCE PROBABILITY OF NO VIOLATION .3940
**************************************************
CO
MEAN DISCHARGE
-------
SOURCE 25
***********
1 MEAN DISCHARGE (ML/DAV)e 164.9721 UPSTREAM FLOW CML/DAY)« 1827.8000
EXPECTED PRO*. .OF NO
CONSTITUENT STANDARD DIST EST. MEAN EST. SIGMA DAMAGE VIOLATION
BODS 4535.9200 N 5085.6584 2481,7701 3,5719
SUSPENDED SOLIDS 3628,7360 L 3.7497 t2959 ,3556 ,2605
**************************************************
SOURCE EXPECTED DAMAGE 3.5719
SOURCE PROBABILITY OF NO VIOLATION ,107*
I*************************************************
&
***********
SOURCE 26
***********
PIPE* i MEAN DISCHARGE (ML/OAY>« 7.1535 UPSTREAM FLOW « 1862.1000
MEAN DO CONCENTRATION fMG/D* 5.9653
CONSTITUENT
BOOS
SUSPENDED SOLIDS
PHOSPHORUS
STANDARD
278.9591
302,0923
72.2990
DI8T
L
L
N
EST. MEAN
2.0957
2.2366
43.0084
EST, SIGMA
.4399
.3561
32.3824
EXPECTED PROP. 0? NO
DAMAGE VIOLATION
.1345
.0129
.4750
.7868
,7530
,8171
**************************************************
SOURCE EXPECTED DAMAGE .4750
SOURCE PROBA8ILITV OF NO VIOLATION ,4841
**************************************************
-------
PIPE" 1 MEAN
CONSTITUENT
BODS
SUSPENDED SOLIDS
PHOSPHORUS
DISCHARGE CHL/DAY)*
STANDARD OIST
272.1552 N
272.1552 N
58.2940 N
SOURCE 27
***********
5,5699
EST. MEAN
3603.6948
3382.8377
311.0573
UPSTREAM FLOW
EST. SIGMA
817.3354
1500,1360
64.9729
(ML/OAYJa
EXPECTED
DAMAGE
6.5243
.9534
6.6753
349.9000
PROS. .OF NO
VIOLATION
.0096
.0191
.0001
**************************************************
SOURCE EXPECTED DAMAGE 6.6753
SOURCE PROBA8ILW OF NO VIOLATION .0000
**************************************************
to
***********
SOURCE 28
***********
PIPE* 1 MEAN DISCHARGE (ML/DAY)*
MEAN 00 CONCENTRATION (MG/L>« 4.8551
CONSTITUENT
110.8503
UPSTREAM FLOW CML/DAYJs
266.7100
BODS
SUSPENDED SOLIDS
PHOSPHORUS
STANDARD
4989.5120
4082.3280
529.9500
DIST
N
N
N
EST. MEAN
1413.0719
3151.0186
301.2218
EST. SIGMA
845.8808
2321.0495
148.7913
EXPECTED
DAMAGE
4.6933
.8593
6,2030
PRQB. OF NO
VIOLATION
1.0000
.6559
.9379
I*************************************************
SOURCE EXPECTED 04:uSf 6,2030
SOURCE PROBABILITY rtf NO VIOLATION .6151
I*************************************************
-------
PIPE- i
MEAN DISCHARGE (ML/DAY)"
SOURCE 29
***********
4.1106
UPSTREAM FLOW (ML/DAY)s
CONSTITUENT
STANDARD
DIST
E8T. MEAN EST. StGMA
EXPECTED
DAMAGE
12.2340
PROP* OF NO
VIOLATION
BODS
SUSPENDED .SOLIDS
170,0970
170.0970
N
L
93.4627
1.7007
77.9805
.3274
6.5963
.4079
.8371
.9473
**************************************************
SOURCE EXPECTED DAMAGE 6.5961
SOURCE PROBABILITY OF NO VIOLATION .7930
**************************************************
***********
SOURCE 30
***********
PIPE- i
MEAN DISCHARGE (ML/DAY)'
CONSTITUENT
STANDARD
35.0425
OIST EST. MEAN
UPSTREAM FLOW CML/OAY)» 1862.1000
EST. SIGMA
EXPECTED
DAMAGE
PROB. OF NO
VIOLATION
BODS
SUSPENDED SOLIDS
1587.5720
1360.7760
N
N
1625.9873
1654.7131
759.3214
720.3438
1.1913
.0874
.4798
.3416
**************************************************
SOURCE EXPECTED DAMAGE 1.1913
SOURCE PROBABILTTY OF NO VIOLATION .1639
**************************************************
-------
TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/5-75-015
3. RECIPIENT S ACCESSION1 NO.
4. TITLE AND SUBTITLE
A QUANTITATIVE METHOD FOR EFFLUENT COMPLIANCE
MONITORING RESOURCE ALLOCATION
5. REPORT DATE
September 1975
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
Arthur I. Cohen, Yaakov Bar-Shalom,
Wendy Winkler, G.Pauj^ Grimsrud
8. PERFORMING ORGANIZATION REPORT NO,
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Systems Control, Inc.
1801 Page Mill Road
Palo Alto, California 94304
10. PROGRAM ELEMENT NO.
1HC619
11. CONTRACT/GRANT NO.
68-01-2232
12. SPONSORING AGENCY NAME AND ADDRESS
Office of Research and Development
U.S. Environmental Protection Agency
Washington, D.C. 20460
13. TYPE OF REPORT AND PERIOD COVERED
Final
14. SPONSORING AGENCY CODE
15. SUPPLEMENTARY NOTES
16. ABSTRACT •
rhis report develops and demonstrates a quantitative method for the preliminary design
of effluent standard surveillance systems. The principal output of the report is a
procedure to be used in the state or EPA water quality programs to determine the fre-
quency of effluent compliance monitoring visits. The procedure allocates compliance
monitoring budgetary resources so as to minimize environmental damage. It utilizes a
statistical model of the effluents that is obtained from self-monitoring and compliance
monitoring data. The procedure is demonstrated on an example river basin using data
supplied by the State of Michigan.
This report is submitted in fulfillment of Contract Number 68-01-2232 by Systems
Control, Inc., under the sponsorship of the Office of Research and Development Environ-
mental Protection Agency. Work was completed as of January 1975.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS
c. COSATI Field/Group
Wastewater Monitoring, Wastewater Stan-
dards, Effluent Monitoring, Water Quality
Control, Effluent Compliance Monitoring,
Resource Allocation, Statistical Analysis,
Michigan Water Resources, Cost Effectiveness
Resource Allocation
Program, Effluent
Standards Compliance
Monitoring
14A
Methods and
Equipment/
Cost Effective-
ness
18. DISTRIBUTION STATEMENT
UNLIMITED
19. SECURITY CLASS (ThisReport)
UNCLASSIFIED
21. NO. OF PAGES
20. SECURITY CLA
pat*)
22. PRICE
KPA Form 2220-1 (t-73)
245
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