
the above equations, as described in Appendix Section A3.2). Receiving water impacts would be
determined initially for existing conditions (no controls), and subsequently using modified pollutant
mass loadings that reflect the effect of the CSO control options being evaluated. Decisions on
control strategy could be guided by comparing each set of results with appropriate criteria and
standards, as discussed in Section 2.5 further below.
Streams and Rivers
Pollutant discharges resulting from individual storm events interact independently on water
quality in streams and rivers, because the loading pulse from one event will be transported some
distance downstream before loads from the next storm are discharged. In addition, the diluting flow
in the stream is also a variable element, and may be largely independent of the urban storm discharge
flows. CSO discharges can range from a minor to a substantial component of total stream flow
during specific storm events.
The analysis procedure is described in Appendix 3, Section A3.3, and operates as follows:
1. Stream concentrations of a pollutant are calculated from the combination of upstream
(background) flow and concentration, and CSO discharge flow and concentration. Variations in
stream pollutant concentrations below the discharge result from variations in each of these inputs.
The most significant source of variation results from whether or not it is raining; i.e., whether CSO
flows and loads are present.
Where a continuous point source load must also be considered, its impact would
be first computed using the same analysis method, and the result assigned as the
upstream condition for the CSO impact analysis.
Stream flows must be considered because of the major effect of dilution on the
resulting concentrations. However, upstream concentrations could be set at zero
for the computations, in which case the result obtained would be the exclusive
effect of the CSO discharge, and not the overall expected stream concentration.
2. The statistical properties of each input parameter are defined, either from analysis of
local data or using estimates guided by information presented in Appendix 2. The necessary
transforms of the input statistics for the log normal parameters used in the computations are made
using the information presented in Appendix 1. The four input parameters required for use in the
analysis are as listed below, together with the nomenclature used to designate the transforms.
11

INPUT ARITHMETIC
VALUE MEAN STD COEF
PARAMETER DEV VAR
(M) (S) (CV)
UPSTREAM
flow QS MQS SQS CVQS
concentration CS MCS SCS CVCS
CSO DISCHARGE
flow QR MQR SQR CVQR
concentration CR MCR SCR CVCR
LOGARITHMIC
MEAN STD
DEV
(U) (W)
UQS WQS
UCS WCS
UQR WQR
UCR WCR
3. A dilution factor (DF) is defined as the ratio of CSO discharge flow (QR) to total flow
(QS + QR):
QR 1
DF = =
QR + QS 1 + D
where D = QS / QR
The statistical parameters of the dilution factor (e.g., MDF, SDF, etc., per above
nomenclature) are computed from those for the stream and CSO flows, as described in Appendix 3.
4. The mean, standard deviation, and coefficient of variation of the variable stream
concentrations (CO) that result from the CSO discharges are then computed. During "wet" periods,
when overflows are being discharged to the stream, instream concentrations are computed by the
following equations:
MEAN stream concentration:
MCO = ( MCR * MDF) + (MCS * (1MDF))
STANDARD DEVIATION of stream concentrations:
SCO = SQR [ SDF2(MCRMCS)2 + SCR2(SDF2+MDF2) + SCS2(SDF2+(1MDF)2 ]
12

COEFFICIENT OF VARIATION of stream concentrations:
CVCO = SCO/MCO
From these values, the frequency with which specified criteria values, or other target
concentrations, will be exceeded can be determined as described in Appendix 1, and evaluated as
discussed in the next step in the procedure.
The above computation provides the stream characteristics for those periods of time during
which CSO events contribute pollutant loads. The fraction of the total time that runoff can occur is
estimated from the rainfall statistics as the ratio MDP / MTP.
MDP mean storm duration
= = fraction of time it rains
MTP mean interval between storms
During dry periods, there are no CSO discharges and hence CSO concentration and dilution
factor input parameters are zero. The frequency of exceeding a specified target concentration is
defined by the statistics of the upstream concentrations during these periods. Dry periods represent
a fraction of the total time equal to (1  MDP / MTP).
From probabilities of exceedance (PRd, PRw) computed for each condition as described in
Appendix 1, the overall probability (PRt) that stream concentration (CO) will exceed some target
(CT) is computed by:
PRt(CO>CT) = (MDP/MTP*PRw) + ((1MDP/MTP) * PRd)
2.5 COMPARE WATER QUALITY EFFECTS WITH CRITERIA  TASK D
Since it is not reasonable to expect 100 percent control to be achievable, the analysis should
be structured so that each control alternative and level of control, with its associated cost, can be
associated with a corresponding receiving water quality condition. Then, costeffectiveness can be
evaluated in terms of the projected degree of improvement in receiving water quality.
There are no hard and fast rules to describe how this should be done in a particular case. It
will vary with the local situation, the beneficial use that is influenced, and the way in which the CSO
pollutants affect that use. One possible approach is described below as an illustration.
For either estuaries or streams, the final computed outputs provide a value for mean,
standard deviation, and coefficient of variation of the stream concentrations resulting from the
CSOs. From this information, probabilities for any selected concentration value can be determined,
13

as described in Appendix 1. Specifically, the frequency and magnitude of exceeding some target
concentration representing a water quality criteria or standard can be computed using the equations
provided, for each of the conditions of interest.
The information for such a comparison is most conveniently summarized for the illustration
presented here by using the Appendix 1 procedure for converting results to a probability distribution
plot. This illustration assumes that the following results are produced by the analysis.
Existing CSO discharges result in the concentrations of the pollutant of interest
in the receiving water having a mean (MCO) of 100, and a coefficient of
variation (CVCO) of 1.25.
The water quality standard for this pollutant is 300.
Two control options are being considered. Option A results in a 25 percent
reduction in the longterm mean receiving water concentration; Option B
produces a 50 percent reduction. For both, it is assumed that the controls
reduce the mean, but do not change the variability.
From Appendix 1, Table A12 indicates that for a coefficient of variation of 1.25, the 50th,
90th, and 99th percentile values (expressed as multiples of the mean value) are 0.62, 2.16, and
5.97, respectively. Multiplying the mean concentrations of each of the three conditions being
examined by these factors provides concentration values for the three percentiles for each of the
control conditions.
The information may then be plotted as shown by Figure 21 and used as one basis for
evaluating the relative merits of the control alternatives being considered.
From the relationships plotted, the following information may be extracted, simply by
scaling from the plot.
Under existing conditions, the receiving water target is exceeded during 5
percent of the storm events. If the area averages 100 storms per year, there are
five violations per year. Once per year on average (one event per 100 or 1
percent exceedance), the receiving water concentration resulting from an
overflow event will be approximately 600, and exceed the target by a factor of
two.
For control Option A, which provides 25 percent reduction in the mean
receiving water concentration, target violations are reduced from five per year to
three per year.
For control Option B, which provides 50 percent reduction in the mean
receiving water concentration, the target is now exceeded on average during no
more than 1 percent of the storm events, or once per year when the area has 100

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storms per year. Higher concentrations are still expected, but less frequently.
For example, a receiving water concentration during a CSO event that exceeds
about 630 can be expected to occur during no more than 0.1 percent of the
events, or about an average of once during a 10year period.
When costs are associated with each of the control options being considered, the user is
provided with a basis for evaluating costeffectiveness on the basis of the relative water quality
improvements each option can be expected to produce. Other factors and considerations will
usually be involved in the decision process, but this analysis should provide a useful input to the
process. Because of the inherent variability of CSO discharges and their impacts, control actions
will not produce "eitheror" situations. Rather, as suggested by the illustration above, they must be
considered in terms of degrees of improvement (frequency and magnitude). Considerable value
judgement will, of necessity, enter the decision process.
16

3.0
PROCEDURE FOR ESTIMATING
PERFORMANCE OF CSO CONTROLS
3.1 GENERAL
Appendix 4 describes the methodology for making estimates of the longterm average
reductions produced by several control device types, under the intermittent and variable discharges
produced by CSOs. The computational procedure is described, but in addition, performance
characteristics are also summarized by a series of performance plots or design curves that
consolidate the basic computations using combinations of values for the input parameters that cover
the range of practical possibilities that will be encountered in CSO applications.
These design curves facilitate the application of the methodology. They describe the
performance characteristics of generic, rather than specific, control devices. That is, they represent
any device whose basic principle of operation corresponds with that on which the computation
procedure is based. For example, performance characteristics are described for device types whose
ability to reduce CSO flows and/or pollutant loads is influenced by applied rates of flow, by the
volumes of overflow generated, or by the effect of flow rate on the efficiency of a treatment process
(e.g., removal by sedimentation).
The user must select the appropriate design curve that applies for the specific device being
considered. Where the control system consists of different devices operating together (e.g., storage
and treatment), the overall removal will be determined by properly combining the efficiency of each
of the devices that make up the total control system.
This section illustrates the selection of the appropriate design curve from Appendix 4,
shows how it is used to estimate the effect of size on performance effectiveness, and how to
combine results for several different devices in an overall control system. Specific illustrations are
presented below to describe the use of the methodology to evaluate performance of CSO control
systems employing the following commonly used device types.
Regulators / interceptors
Offline storage
Sedimentation devices
17

From the basic information presented here and in Appendix 4, and the examples in this
section, it should be possible for some users to work out the appropriate procedures for assessing
the performance of devices that are not specifically covered by the following illustrations.
The use of the procedures and the illustration of comparisons that can be made to evaluate
design alternatives are most conveniently presented using specific numerical values. Accordingly,
the examples that follow are all based on an urban area served by a combined sewer collection
system that has the following site and climatic characteristics:
The urban catchment has an area of one (1) acre. It has a population density of 27
persons per acre, and a dry weather sanitary sewage flow (DWF) of 100 gcd. From
the percent impervious area, the runoff coefficient (Rv) is estimated to be 0.5.
Area rainfall has the following characteristics:
Mean storm event intensity (MIP) = 0.08 in/hr, coefficient of variation (CVIP) = 1.25.
Mean storm volume (MVP) = 0.40 inches; coefficient of variation (CWP) = 1.50
Mean interval between storm midpoints.(MTP) = 97 hours
Note, however, that for all the performance curves, the design size (volume or treatment
flow rate) of the device is defined as a ratio tothe mean of the runoff flows or volumes produced by
the catchment served by the device. Thus the design curves apply directly to any size urban
catchment.
3.2 PERFORMANCE OF REGULATOR / INTERCEPTOR SYSTEMS
Average dry weather flow for the 1acre urban area:
DWF = 27 * 100 = 2700 GPD = 15 CFH
Mean storm flow rate, from equation 22
MQR = 3630 * Rv * MIP * AURB
MQR = 3630 * 0.5 * 0.08 * 1 =145 CFH
Consider four alternate cases for the amount of interceptor capacity available in excess of
DWF. The amount of infiltration/inflow (VI) due to the condition of the collection system, the
settings and state of repair of the regulators, the size and presence of sediment deposits in the
interceptor, etc., can each or all determine the available capacity of the system to convey wet weather
flows away from overflow points.
18

CSO control measures being considered could include actions that address any or all of
these things. Sewer or regulator rehabilitation, interceptor flushing, or larger interceptors in critical
areas, are all possible approaches that can be addressed by this procedure.
If the existing and alternate conditions are estimated to result in conveyance capacities of 2,
3,4, and 5 times DWF for the system, then the "treatment rate" (QT) is computed as the capacity to
carry wet weather flows. It is the difference between the total hydraulic capacity of the system and
the amount used by the DWF. The ratio of treatment rate (available excess capacity), to the rate
produced by the mean storm is then defined, as QT/MQR.
Enter the flowcapture device performance curve (Appendix Figure A41) at the value for
this ratio. Extend a line vertically to the curve representing the CV of runoff flows (estimated to be
equal to CVIP, or 1.25 in this case). Then extend a line horizontally to read the expected longterm
average capture of wet weather flows provided by the regulator/interceptor system.
Figure 31 illustrates the procedure and Table 31 summarizes the results for the conditions
assumed for this example.
TABLE 31 INTERCEPTOR/REGULATOR PERFORMANCE
CASE CAPACITY EXCESS CAPACITY RATIO AVERAGE
*DWF >DWF CFH QT/MQR % CAPTURE
A 2 1 15 0.10 10%
B 3 2 30 0.21 18%
C 4 3 45 0.31 24%
D 5 4 60 0.41 30%
3.3 PERFORMANCE OF OFFLINE STORAGE DEVICE
Consider for the moment, a system in which the interceptor system capacity is just equal to
the dry weather flow. There is no excess capacity, and all storm flows will overflow.
Evaluate the performance of several alternative sizes of retention basin for an area with the
previously assigned characteristics. The storage device is designed to capture all overflows until it is
full, and then bypass all excess storm overflows. At the end of a storm event, any captured volume
19

0.5 0.75 1.00 1.25 1.50 1.75 2.00
From Appendix 4
Figure A41
0.2 0.3 0.4 0.6 0.8 1.0 2.0
TREATMENT CAPACITY
3.0 4.0 6.0 '8.0 10.0
«Ap,^
RAT'0:
MEAN RUNOFF FLOW
Figure 31. Interceptor/regulator performance
20

is pumped out of the basin at a specified constant rate and returned to the interceptor for conveyance
to the downstream treatment facility.
The rate at which captured overflow volume is emptied from the basin influences the actual
volume available for storage when the next storm occurs. The average volume available for storage
over all storm events is designated "effective volume" (VE), and will be less than the physical
volume of the basin (VB), by an amount that is influenced by the emptying rate.
For the purpose of clarity in illustrating the procedure, assume initially that the emptying
rate is very high, so that the effective volume is equal to the physical volume of the basin, that is,
(VE = VB). Analyze basin sizes that provide storage volumes equal to 0.5, 1.0, and 3.0 times the
volume of overflow discharged by the mean storm event
The volume of runoff (overflow) from the mean storm (MVR) is, per equation 21:
MVR = 3630 * Rv * MVP * AURB
MVR = 3630 * 0.5 * 0.40 * 1 =726 cu ft
In Appendix Figure A43, enter the horizontal axis at the selected values for the ratio of
effective basin volume to the above mean overflow volume (VE/MVR). Extend a vertical line
upward to intersect the curve for the CV of storm runoff volumes, CVVR = 1.5. On the vertical
axis, read the longterm average percent of overflow volumes that are captured by the storage
device.
Figure 32 illustrates the procedure and Table 32 summarizes the results for the conditions
assumed for this example.
TABLE 32 PERFORMANCE OF STORAGE DEVICE
CASE VOLUME VOLUME RATIO AVERAGE
VB (cu ft) VE/MVR % CAPTURE
A 363 0.5 34%
B 726 1.0 52%
C 2178 3.0 83%
However, in most cases extremely high emptying rates will not be feasible. Practical limits
will be imposed by the capacity of the conveyance system or the downstream treatment plant.
Appendix Figure A44 shows how the emptying rate may be used to estimate how much of the
physical basin volume (VB) will be available, on average, as effective volume (VE) for capture of
overflows.
21

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90
80
70
60
50
40
30
20
10
0.1
CoeHicient of Variation
of Runoff Volume CVVR
From
Appendix 4
Figure A43
0.2 0.3 0.40.50.6 0.81.0 2.0 3.0 4.05.06.0 8.010.0
EFFECTIVE BASIN VOLUME
RATIO:
MEAN RUNOFF VOLUME
(VE/VR)
Figure 32. Performance of storage device
22

The family of curves that are used for the determination represent a series of "emptying rate
ratios" (ERR), the value of which is determined from the pumpout rate (QB), the mean time
between storms (MTP), and the mean overflow volume (MVR).
ERR = (QB * MTP) / MVR
The numerator reflects the volume that would be removed in the average time between storm
events. The denominator is the average volume introduced to the basin by a storm. The higher the
value of this emptying rate ratio, the smaller will be the basin volume occupied by carryover from
previous storms when a new event occurs.
Figure 33 illustrates the use of this plot to refine the performance estimates for Case C
above, which reflects the case of a basin volume that is 3 times the mean storm volume. Consider
the effect of emptying rates that are 33%, 50%, and 100% of the average DWF flow rate of 15
CFH. For the conditions assigned for this example, (MVR = 726 CF and MTP = 97 hours), the
selected emptying rates provide ERR ratios of 0.5, l.Oi and 2.0. Enter the plot on the horizontal
axis at the Case C value for VB/MVR (= 3.0), and extend a vertical line. At the intersection of the
appropriate value for ERR, extend a horizontal line to read the effective volume on the vertical axis.
Table 33 summarizes the results for the case considered. Longterm average removals are
computed as above (Figure 32), using the corrected values for effective volume (VE).
TABLE 33 EFFECT OF EMPTYING RATE ON EFFECTIVE VOLUME
ERR VB/MVR VE/MVR % REMOVAL
0.5 3.0 1.0 52%
1.0 3.0 2.2 75%
2.0 3.0 2.7 81%
3.4 COMBINED EFFECT OF INTERCEPTION AND STORAGE
The two preceding sections illustrated the procedure for estimating the longterm average
removal efficiency of interception and storage, when both are treated independently. In most CSO
control situations, however, the two devices will operate together as a system, with the storage
device processing the combined sewer flows that are not captured by the regulator/interceptor, and
hence overflow.
The fraction not captured by each device is
1  (% REMOVAL/100)
23

5.0
4.0
3.0
2.0
UJ IT
1.0
From Appendix 4
Figure A44
2.0 3.0 4.0
X§ _ rSTORAGE VOLUME (EMPTY!*]
VR ~ L MEAN RUNOFF VOLUME J
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0.75
0.5
0.1
5.0
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The combined fraction not removed by both devices operating in series is estimated by the
product of the two fractions. If Fi is this fraction for the interceptor, and Fv is the same for the
storage basin, then the overall fraction not removed by the combined system is:
Fiv = Fi * Fv
The overall percent removal from the combined system is
Combined % Removal = 100  (Fiv * 100)
The effect of a combination of interception and storage on the frequency at which overflows
occur can be estimated using Figure 34. It is a function of two ratios previously computed.
QT / MQR  This is the ratio of the excess interceptor capacity (QT), to
the runoff flow rate produced by the mean storm (MQR)
VE / MVR  This is the ratio of the effective basin volume (VE), and the
mean storm runoff volume (MVR)
For example, with no storage and no excess interceptor capacity, the probability of an
overflow is 1.0. An overflow will occur every time it rains. For a collection system with an
interceptor/regulator that provides an excess flow capacity that is 25% of the rate produced by the
mean storm, QT/MQR = 0.25, the probability of an overflow is reduced to 0.8. That is 20% of the
storm events are captured and produce no overflow.
When a storage device is added to the above system, and has an effective capacity equal to
the mean runoff volume, VE / MVR = 1.0, the probability of an overflow event is reduced to about
0.22. In this case, only 22% of the storm events result in an overflow to the receiving water.
3.5 PERFORMANCE OF SEDIMENTATION DEVICES
The analysis of performance for devices of this type requires information on the relationship
between removal efficiency for a pollutant and the hydraulic loading rate. Any usersupplied
relationship between flow rate and removal efficiency may be employed, provided it is defined (or
approximated) as an exponential function of the form:
% REMOVAL = constant * exp ( k * Q/A)
This is a commonly used format for general expressions of the performance of treatment
devices. Figure 35 shows this relationship for the removal of sanitary and combined sewage based
on results from several sources. Included in Figure 35 are the results of a study of the performance
of settling basins in Rochester, New York, for treating combined sewage at constant hydraulic
25

From Appendix 4. Figure A46
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THE CURVES REPORTED BY IMHOFF AND BUR.9 ASCE.10 AND SMITH11ARE
FOR SANITARY SEWAGE.
THE CURVE BY LAGER lal.8 IS FOR COMBINED SEWAGE.
THE COMPOSITE CURVE HAS THE FOLLOWING FUCTIONAL FORM'
IOOO
2OOO
3000
4OOO
SOOO
OVERFLOW RATE. q/Ay qal/ftday
Figure 35. Suspended solids removal by detention/sedimentation
27

loading rates. In this study, Lager et al. (8) give the following relationship for removal of TSS in
combined sewage by sedimentation:
% REMOVAL = 80 * exp ( k * Q/A)
where: k = 0.00064 when Q/A terms are... gal per day per sq ft
k = 0.115 when Q/A terms are... CFH per sq ft
Figure 36 is a semilog plot of the relationship, and is used in this example.
Two different modes of operation are possible for sedimentation basins treating urban
runoff. One mode is the case where the overflows are routed directly to the inlet of the device.
Applied flows, and hence overflow rates (Q/A) are different for each storm event, and event removal
efficiency also varies, in accordance with the relationship shown by Figure 36 (the operating point
moves back and forth along the curve). The longterm average reduction, considering this variable
performance, is estimated as illustrated by Figure 37 (which duplicates Appendix Figure A42).
For a selected basin surface area, compute the overflow rate at the mean storm flow rate.
Determine the removal efficiency at the mean storm rate either by scaling from Figure 36, or by
solving the performance equation.
FRM = 80 * exp(0.115*MQR/A)
where:
MQR = the mean CSO flow rate (145 CFH for this example)
A = the surface area of the basin (sq ft)
FRM = the fraction removed at the mean overflow rate
The longterm average removal is then either scaled from Figure 37 or computed by:
FRL = fmax*(r/(rln(FRM/fmax))r+1
where:
FRL = longterm average removal
FRM = removal for the mean storm
fmax = maximum removal at very low rates (80% this case)
r = 1/CVQR2 (CVQR for this example is 1.25)
If the performance estimate is to be scaled from Figure 37, the user must keep in mind that
this plot is for the case where fmax = 100%. (Both axes reflect the "removable fraction" of the
pollutant, and not the total concentration). Since the performance relationship being used indicates
that only 80% of the CSO suspended solids are removable by sedimentation, some simple
adjustments are required for proper use of this figure.
28

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5
10 15
Q/A(CFH/ft2)
20
25
1000 2000 3000
OVERFLOW RATE. Q/A (gal/day/ft2)
4000
5000
) = 80expk(Q/A)
k = 0.00064 for Q/A in GPD/ft2
k =0.115 for Q/A inCFH/ft2
Figure 36. Performance of sedimentation basin treating combined
sewer overflows
29

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From Appendix 4
Figure A42
of Runoff Flows CVq /
40 
20 
20
REMOVAL AT MEAN RUNOFF FLOW
MAXIMUM REMOVAL AT VERY LOW FLOW
(expressed as percent)
+ 69% 0.8 = 55%
86% =
68.6%
0.8
Figure 37. Performance of sedimentation device for CSO treatment
30

To illustrate, consider the case where a specified surface area results in a removal efficiency
of 68.6% (from Figure 36 or associated equation) at the mean storm flow rate. This value is based
on the total TSS. It represents:
68.6/0.80 = 86% of the "removable fraction"
Enter Figure 37 on the horizontal axis at 86% removal efficiency, and extend a vertical line
to the curve for the coefficient of variation of flow rates (1.25 for this example). A horizontal line
from this point indicates the longterm average removal on the vertical axis of 69%  based on the
removable fraction. The longterm average removal, based on the total amount of pollutant present
is:
69 * 0.80 = 55%
Table 34 summarizes the input data and projected performance for a series of design size
options that might be considered in an assessment of control alternatives. The values shown are
based on the conditions assigned for the example analyses, that is, a 1acre urban area with DWF =
15 CFH, runoff flow rate for the mean storm MQR = 145 CFH, with a coefficient of variation,
CVQR = 1.25. The initial screening of sedimentation basin sizes is defined in terms of the basin
surface area to be provided, expressed as a percent of the drainage area. In this way, the analysis is
valid for other sizes of drainage area as well as being specific for the 1acre example catchment.
TABLE 34 PERFORMANCE OF SEDIMENTATION DEVICE
SURFACE AREA OF BASIN
% of square feet
Drainage Area per Acre
OVERFLOW RATE
during MEAN storm
CFH/sqft GPD/sqft
% TSS REMOVAL
for longterm
MEAN storm AVERAGE
0.025
0.050
0.075
0.100
0.250
0.500
1.000
10.9
21.8
32.7
43.6
108.9
217.8
435.6
13.31
6.66
4.44
3.33
1.33
0.67
0.33
2390
1195
797
598
239
120
60
17.3
37.2
48.0
54.6
68.6
74.1
77.0
10.8
22.0
30.6
37.1
56.3
66.5
72.7
An alternate mode of operation would employ a sedimentation basin in a control system in
association with a storage device. As the captured overflows are emptied from the storage device,
they are routed at a constant rate through the sedimentation device, and then discharged to the
31

receiving water at the CSO overflow location. This is in contrast to the earlier example where such
captured flows were returned to the interceptor and completely eliminated from discharge at the CSO
location.
In this case, the removal efficiency is a simple function of the design overflow rate. The
overall removal for the storage/sedimentation basin system would be that computed for the storage
device with the selected emptying rate (as described earlier), but now factored by the removal
efficiency for the sedimentation device operating at a constant overflow rate (per the Figure 36
performance relationship).
Assume for this illustration that the storage device selected has a volume ratio (VB/MVR) of
3.0, and that an emptying rate corresponding to a ratio (ERR) of 1.0 is also chosen. Per Table 33,
this would capture 75% of overflows as a longterm average. This is also the reduction in pollutant
discharges at the CSO overflow location, if the CSO volumes emptied from the basin are completely
diverted from the discharge location by their return to the interceptor system.
For the example conditions, an ERR of 1.0 represents a flow rate of 50% of the DWF, or
7.5 CFH. Table 35 summarizes the performance of four alternate sizes of sedimentation basin that
would process the captured overflows emptied from the above storage device at the indicated rate.
Two removal efficiencies are listed.
The first is the removal effected by the sedimentation basin on the stormwater routed
through it.
The final column is the longterm average removal of all CSO suspended solids due to the
combined operation of the storage and treatment devices. Since the storage unit captures only 75%
of the overflow volumes, and the treatment unit removes the indicated percentage of solids from the
75% that has been "captured," the overall average removal is the product of the two efficiencies.
TABLE 35 STORAGE / TREATMENT PERFORMANCE
SURFACE AREA OVERFLOW RATE % REMOVAL
sq ft/Acre CFH/sqft GPD/sqft SED BASIN OVERALL
1.5 5.0 898 45 34
3.0 2.5 449 60 45
7.5 1.0 180 71 53
15.0 0.5 90 76 57
32

4.0
EXAMPLE COMPUTATIONS
4.1 ASSEMBLE AND SUMMARIZE SITE CHARACTERISTICS
(A) RAINFALL
The rainfall for the area has the following characteristics, determined either from a
"SYNOP" analysis of a local rain gage, or estimated from data in Appendix 2, Section A2.1.
STORM EVENT MEAN COEFofVAR
VOLUME (inch) MVP = 0.40 CVVP = 1.50
INTENSITY (in./hour) MIP = 0.80 CVIP = 1.25
DURATION (hours) MDP = 6.00 CVDP = 1.00
INTERVAL (hours) MTP = 97.0 CVTP = 1.00
(B) STUDY AREA PHYSICAL PROPERTIES
Assume that the study area is a 1acre urban catchment served by a combined sewer system
with one regulator/overflow point. It has the following characteristics.
RUNOFF COEFFICIENT (Rv) = 0.5
DRY WEATHER FLOW (DWF) = 15.0 CFH
INTERCEPTOR CAPACITY = 30.0 CFH (2* DWF)
EXCESS CAPACITY (QT) = 15.0 CFH (3015)
Assume that concentrations of the specific pollutant selected for analysis have the following
characteristics in combined sewage, and hence in the overflows.
MEAN CONCENTRATION (MCR) = 100
COEF of VARIATION (CVCR) = 0.75
33

(C) STREAM FLOW CHARACTERISTICS
The overflows discharge into a stream that has the following characteristics, determined
either from an analysis of the records of a local gage, or estimated using the information presented in
Appendix Section A2.4.
The stream segment of interest is ungaged, but from data on comparable streams in the area,
is estimated to have an average flow of 1.07 cfs per sq mile of drainage area. The total drainage area
upstream of the CSO discharge location is 10 times the contributing urban area, or 10 acres. Stream
flow statistics are as follows.
MEAN STREAM FLOW (MQS) = 60 CFH
COEF of VARIATION (CVQS) = 1.50
Pollutant concentrations in the receiving water upstream of the discharge location are
assumed to be "zero." Accordingly, the computations will reflect only the effect of the CSO
discharge.
4.2 COMPUTE RUNOFF / CSO FLOW PROPERTIES
(A) RUNOFF FROM MEAN STORM
The runoff generated by the mean storm event combines with the DWF to produce the
combined sewage flow rate during this mean event. However, since the DWF is assumed to be
constant, and always carried by the interceptor system, the overflow analysis can work with the
storm runoff itself and the "excess" interceptor capacity (MQR and QT). (See equations 21 and 22
in Section 2 for basis for volume and flow rate calculation.) The procedure assumes that the
variability of these runoff parameters is the same as that for the corresponding rainfall parameters.
RUNOFF FLOW RATE
MQR1 = 3630 * 0.5 * 0.08 * 1 =145 CFH
CVQR = CVIP = 1.25
RUNOFF VOLUME
MVR = 3630 * 0.5 * 0.40 * 1 = 726 CF
CVVR = CVVP = 1.50
34

The overflow rate during the mean storm will be less than the runoff flow rate (MQR) by
the amount of excess capacity available.
MEAN CSO FLOW RATE
MQR = MQR'QT = 145  15 = 130 CFH
The variability of CSO flow rates will be less than that for the surface runoff rates produced
by the storm events, because the interceptor will only slightly reduce the rate of flow for the larger
storms, but will completely eliminate the flows produced by a significant number of the smaller
storms. At the time of preparation of this document, there is no procedure available for estimating
the magnitude of the reduction in variability of flow rates. A conservative estimate is therefore used
by assuming that the coefficient of variation of the overflow rates is the same as that for the runoff
flows.
4.3 CALCULATE IMPACTS ON RECEIVING WATER
The statistical properties of the combined sewage flows and concentrations and the stream
flow characteristics developed by the steps above are now used to compute the receiving water
impact of the CSO. Specifically, the statistics of the stream concentrations downstream of the CSO
discharge are produced by the next calculation.
(A) COMPUTE STATISTICAL PARAMETERS OF INPUTS
The statistical properties of each of the input parameters that were established above, can be
computed from the mean (M) and coefficient of variation (CV) by using the equations presented in
Appendix Section A 1.4. For each of the input parameters, the following calculations are made.
Compute LOG SIGMA W = SQR(LN( 1+ CV2))
Compute LOG MEAN M = LN(M/SQR( 1+CV2))
Compute MEDIAN T = EXP(U)
Compute SIGMA S = M*CV
Results are summarized in the table below for the three input parameters used in the
analysis. Upstream concentration (CS) has been assumed to be zero, so that results reflect only the
impact of the CSO discharge. The table shows both the original irtput values for the arithmetic mean
and coefficient of variation, and the computed values for the other statistical parameters.
35

PARAMETER CODE
STREAM FLOW
(QS)
CS OVERFLOW
(QR)
CONCENTRATION
(CR)
MEAN
COEFVAR
LOG SIGMA
LOG MEAN
MEDIAN
SIGMA
(M)
(CV)
(W)
(U~)
(T~)
60.0
1.50
1.08565878
3.50501706
33.3
90.0
130.0
1.25
0.97004296
4.39704277
81.2
162.5
100.0
0.75
0.66804723
4.38202663
80.0
75.0
(B) COMPUTE STATISTICAL PARAMETERS OF DILUTION FACTOR
A dilution factor (DF) has been defined as the ratio of CSO discharge flow (QR) to total
flow (QS + QR):
DF =
QR
1
QR + QS
where D = QS / QR
1+D
The statistical parameters of the dilution factor are computed from those for the stream and
CSO flows, as follows. For lognormal QR and QS, the ratio D is also lognormal, and for
uncorrelated flows the LOG SIGMA is:
WD = SQR (WQS2 + WQR2)
= SQR(1.0856587822 + 0.970042962) = 1.45589778
The 5th and 95th percentile values of the dilution factor (DF) are computed from this
value and the MEDIAN values developed in the preceding step.
DF95 = TQR / [ TQR + TQS * EXP (1.65 * WD) ]
= 81.2 / [ 81.2 + 33.3 * EXP ( 1.65 * 1.45589778) ]
= 0.18090832
DF5 = TQR / [ TQR + TQS * EXP (1.65 * WD) ]
= 81.2 / [ 81.2 + 33.3 * EXP (1.65 * 1.45589778) ]
= 0.96423127
36

The LOG MEAN and LOG SIGMA of the dilution factor are approximated by
interpolating between these values.
UDF = 0.5*[LN(DF95) + LN(DF5)]
= 0.5 * [ LN (0.18090832 ) + LN (0.96423127 ) ]
= 0.8730945
WDF = [ 1/1.65 ] * [ 0.5* ( LN ( DF5 )  LN (DF95 ) ]
= [ 1/1.65 ] * [ 0.5* ( LN ( 0.96423127 )  LN ( 0.18090832 ) ]
= 0.50707296
The remaining (arithmetic) statistics are then computed.
MDF = EXP ( UDF + 0.5 * WDF2 ) = 0.475
CVDF = SQR ( EXP (WDF2)  1 ) = 0.541
SDF = MDF* CVDF = 0.257
(C) COMPUTE STATISTICS OF STREAM CONCENTRATION
The mean, standard deviation, and coefficient of variation of the variable stream
concentrations (CO) that result from the CSO discharges are computed next. During "wet" periods,
when overflows are being discharged to the stream, instream concentrations are computed by the
following equations:
MEAN stream concentration:
MCO = (MCR * MDF) + ( MCS * (1MDF))
= (100.0 * 0.475) + 0
= 47.50
STANDARD DEVIATION of stream concentrations:
SCO = SQR [ SDF2(MCRMCS)2 + SCR2(SDF2+MDF2) + SCS2(SDF2+( 1MDF)2) ]
= SQR[ A + B + C ]
A = SDF2(MCRMCS)2 = 0.2572 * (100  O)2 = 660.49
B = SCR2(SDF2+MDF2) = 752 * (0.2572 + 0.4752) = 1640.67
C = SCS2(SDF2+( 1MDF)2) = O2 * (0.2572 +(10.475)2) = 0
SCO = SQR (660.49 + 1640.67 + 0 ) = 47.98
37

COEFFICIENT OF VARIATION of stream concentrations:
CVCO = SCO/MCO = 47.97/47.50 = 1.01
Then complete this step by computing the log transforms for the downstream concentration
of the pollutant.
LOG SIGMA W = SQR(LN(1+CVC02)) = 0.83869547
LOG MEAN U = LN (MCO / SQR (1+CVCO2 )) = 3.50893214
(D) COMPUTE PROBABILITY OF SPECIFIC CONCENTRATIONS
The frequency with which specified criteria values, or other target concentrations, will be
exceeded can be computed from the LOG MEAN and LOG SIGMA of the stream concentrations,
and the appropriate values of Z from the standard normal table (Appendix Table A11). The
concentration at any percentile (equal to or less than) is given by:
COa = EXP ( UCO + Za * WCO)
For example,
PERCENT PERCENT Z STREAM CONCENTRATION
EXCEEDING LESS THAN COa
10% 90% 1.28 EXP(3:50893214+1.28* 0.83869547) = 97.8
5% 95% 1.65 EXP(3.50893214+1.65* 0.83869547) = 133.3
1 % 99 % 2.33 EXP(3.50893214 + 2.33 * 0.83869547) = 235.8
The frequency at which a specified concentration will be exceeded can also be computed
directly. Assume that the criteria for the selected pollutant is a concentration of 80.
Z = (LN(80)UCO)/WCO
= " ( 4.3820266  3.50893214) / 0.83869547 = 1.04
From Appendix Table A11, this value of Z corresponds to a probability percentile of
0.851. Accordingly, 85.1% of the overflow events will result in stream concentrations that will be
less than 80, and 14.9% of CSO events (100  85.1) will produce stream concentrations that exceed
the criterion value of 80.
38

A graphical display of results may be prepared, so that conditions of interest can be scaled
from a plot on log probability paper. The distribution plot is constructed by plotting the median
concentration at the 50th percendle, and one or more of the above concentrations at its corresponding
percentile.
The above computation provides the stream characteristics for those periods of time during
which CSO events contribute pollutant loads. The fraction of the total time that runoff can occur is
defined from the rainfall statistics as the ratio MDP / MTP.
MDP mean storm duration
= = fraction of time it rains
MTP mean interval between
storm midpoints
With no control imposed, the mean duration of overflows (MDR) and interval between
overflows (MTR) are assumed to be the same as for the storm events. Thus:
MDR/MTR = 6/87 = 0.069 = fraction of time overflows discharge to stream
This amounts to (365 * 24) * 0.069 = 604 hours per year
During dry periods, there are no CSO discharges and hence the CSO concentration and
dilution factor input parameters are zero. The overall frequency of exceeding a specified target
concentration is defined by the statistics of the upstream concentrations during these periods. Dry
periods (i.e., periods when no overflows occur) represent a fraction of the total time equal
to (1  MDR/MTR).
From probabilities of exceedance (PRd, PRw) computed for each condition, the overall
probability (PRt) that stream concentration (CO) will exceed some target (CT), is computed by:
PRt(CO>CT) = (MDR/MTR * PRw ) + ((1MDR/MTR) * PRd)
PRw is the exceedance probability computed above (14.9 %). PRd will be zero in this
example because the upstream background concentration is assumed to be zero. Using the target
concentration of 80 for this example:
PRt ( CO > 80) = (0.069 * 0.149 ) + ((1 0.069) * 0) = 0.0103
Therefore the target of 80 is exceeded during about 15% of the overflow events, but during
about 1% of the hours during a year. The stream will be exposed to concentrations that exceed the
target of 80 for an average of:
0.0103 * (365*24) = 90 hours per year
39

4.4 EXAMINE THE EFFECT OF A CONTROL OPTION
Since it is not reasonable to expect 100% control to be achievable, the analysis is structured
so that each control alternative and level of control is associated with a corresponding receiving
water quality condition, specifically, the frequency at which specific concentration levels will be
exceeded.
This example shows numerical computation results for only one option, but tabulates results
for a range of storage basin sizes. In practice, other control techniques could be examined as well,
consisting of different design sizes for each control device.
(A) COMPUTE CHANGE IN CSO DISCHARGE PRODUCED BY CONTROL
Select an offline storage device that has a physical volume of 1000 cu ft. Assume it can be
pumped out during dry periods, at a rate equal to the excess capacity of the interceptor (QT = 15
CFH) so that the interceptor capacity (2* DWF) is not exceeded. Therefore:
Emptying Rate QB = QT = 15 CFH
Emptying Rate Ratio ERR = QB * MTR / MVR
= 15 * 87 / 726
= 1.80
Normalized Basin Size VB/MVR = 1000/726 = 1.38
Use Appendix Figure A44 to estimate the "effective" volume of the basin. Enter the
horizontal axis at the basin size (normalized by the mean storm volume) ratio of 1.38. Extend a
vertical line to the position where a curve representing the emptying rate ratio, ERR = 1.8, would be
intersected. Then, extend a horizontal line to the vertical axis and read the effective size ratio.
VE/MVR = 1.2
Use Appendix Figure A43 to estimate the longterm average reduction in overflow volume
and hence mass load produced by the basin. Enter the horizontal axis at the basin's effective size
(normalized by the mean storm volume) ratio of 1.2. Extend a vertical line to the position where a
curve representing the coefficient of variation of overflow volumes, CVVR = 1.5, is intersected.
Then, extend a horizontal line to the vertical axis and read the reduction efficiency.
% Reduction = 58 %

Use Appendix Figure A45 to estimate the change produced by the storage device in the
mean and coefficient of variation of CSO flows discharged to the stream. This figure also provides
an estimate of how the basin reduces the fraction of time during which overflows to the stream
occur. Enter each of the plots at the effective volume ratio (VE/MVR = 1.2) on the horizontal axis,
and scale off the value for the parameter.
(a) FLOW RATE RATIO  The mean flow rate of the overflows that escape the basin
increases, because the smaller events are captured and only the larger ones discharge. The plot
provides the ratio of the mean flow after treatment, to the mean flow rate without the device in place.
The discharge flow for this control option is then computed.
MQR(t)/MQR = 1.89
MQR = 130* 1.89 = 246 CFH
(b) FLOW COEFFICIENT OF VARIATION RATIO  Because of the capture of the smaller
flows, the variability of those that do overflow is reduced. The plot provides an estimate of the
treated to the untreated value of CVQR, and the new value for CVQR is computed.
CVQR(t)/CVQR = 0.62
CVQR = 1.25*0.62 = 0.78
(c) OVERFLOW TIME FRACTION  The fraction of time that discharges to the stream
occur is reduced, because small events are completely captured, and larger ones partly so. The
treated/untreated ratio of the time fraction of overflow discharges (DURATION/INTERVAL
BETWEEN EVENTS) is scaled from the plot, and the new value for this parameter is computed.
TIME RATIO = 0.25
MDR/MTR= 0.25*0.069 = 0.017
The storage device selected will reduce the fraction of time overflows enter the stream from
604 hours to 365*24*0.017 = 149 hours, a reduction of 75%.
(B) COMPUTE IMPACTS ON RECEIVING WATER
The sequence of computations described in Section 4.3 above is repeated using the same
values for stream flow and CSO concentration as inputs, but modifying the CSO flow inputs in
accordance with the computations performed immediately above. The new input values are:
MQR = 246 CVQR = 0.78
Following the computation procedure, the statistical properties of the stream concentrations
resulting from the overflows that escape the storage device are as follows:
41

Mean MCO = 65.71
CoefVar CVCO = 0.82
Median TCO = 50.78
Log Mean UCO = 3.92743847
Log Sigma WCO = 0.71801750
The stream concentrations during the resulting overflow events are higher but less variable.
They are higher because only the larger storm events cause overflows, but as indicated above, the
discharges occur during a much smaller fraction of the time.
The frequency at which the target concentration of 80 will be exceeded is computed for the
new condition.
Z = (LN (80)  UCO ) / WCO
= (4.38202663.92743847)7 0.71801750 = 0.633
From Appendix Table A11, this value of Z corresponds to a probability percentile of 0.73.
Accordingly, 73.7% of the overflow events will result in stream concentrations that will be less than
80, and 26.3% of CSO events (10073.7) will produce stream concentrations that exceed the criteria
value of 80.
The probability of exceeding the selected target concentration of 80 is computed to be:
PRt(CO>CT) = (MDR/MTR * PRw) + ((1MDP/MTP) * PRd)
PRt ( CO > 80) = (0.017 * 0.26.3 ) + ((! 0.017 ) * 0) = 0.0045
The target of 80 is exceeded during about 26% of the now reduced number of overflow
events, but during about 0.5 of 1% of the total time. The stream will be exposed to concentrations
that exceed the target of 80 for an average of:
0.0045 * (365*24) = 40 hours per year
4.5 COMPARE PERFORMANCE OF CONTROL OPTIONS
The comparison and evaluation of a set of alternative control options can be summarized for
decision purposes by tabulating the size, cost, and performance projected to apply for each of the
cases considered. Table 41 below summarizes the projected performance and receiving water
impact of a range of storage device sizes for the site conditions selected for the example.
Exceedance frequency is shown for a range of stream target concentrations to reflect other
possibilities for criteria values. Numerical details of the computations are not shown, but follow the

example presented, simply modifying the parameters for the overflow rate (QR) and the fraction of
time that overflows occur.
Slight differences in numerical output may result depending on how the intermediate
computation products are rounded. The stream impact computations shown in this section can be set
up quite conveniently on a spreadsheet that runs on a microcomputer, to facilitate comparison of
alternatives.
TABLE 41 SUMMARY RESULTS FOR ALTERNATE STORAGE VOLUMES
STORAGE VOLUME
CF/acre VB/MVR VE/MVR
OVERFLOW RATE
MEAN (MQR) COV (CVQR)
PROB O/F
(D/DELTA)
0
500
1000
2000
3000
0
0.69
1.38
2.75
4.13
0
0.65
1.20
2.25
3.40
130
212
246
298
342
1.25
0.86
0.78
0.68
0.63
0.0
0.026
0.017
0.009
0.005
STORAGE VOLUME
CF /acre
VOLUME/LOAD
REDUCTION (%)
HOURS/YEAR STREAM CONG IS GREATER THAN
CO=80 CO=100 CO=200 CO=400
0
500
1000
2000
3000
0
40
58
76
86
90
54
40
23
13
58
35
26
15
9
10
6
4
3
2
1
0
0
0
0
43

5.0
REFERENCES
1. Hydroscience, Inc. 1979. A Statistical Method for the Assessment of Urban
Stormwater. for USEPA NonPoint Sources Branch, EPA 440/379023, May.
2. DiToro, D.M. 1980. "Statistics of Advective Dispersive System Response to
Runoff." Proceedings "Urban Stormwater and Combined Sewer Overflow Impacts
on Receiving Water Bodies," Yousef et al. (eds.). EPA 600/98056, EPA
Municipal Environmental Research Laboratory, Cincinnati, Ohio, p. 437.
3. Ditoro, D.M. 1984. Probability Model of Stream Quality Due to Runoff. Journal of
the Environmental Engineering Division, ASCE, Vol. 110, No. 3, June, pp. 607627.
4. US EPA. 1984. "A Probabilistic Methodology for Analyzing Water Quality Effects
of Urban Runoff on Rivers and Streams," Draft Report, US EPA Office of Water,
Washington, D.C., February.
5. US EPA. 1984. "Technical Guidance Manual for Performing Waste Load
Allocations: Permit Averaging Periods," US EPA Monitoring and Data Support
Division, Monitoring Branch.
6. DiToro, D.M. and M.J. Small. 1979. Stormwater Interception and Storage. Journal
of the Environmental Engineering Division, ASCE, Vol. 105, No. EE1, Proc. Paper
14368, February.
7. Small, M.J. and D.M. DiToro. 1979. Stormwater Treatment Systems. Journal of the
Environmental Engineering Division, ASCE, Vol. 105, No. EE3, Proc. Paper 14617,
June.
8. Lager, J.A. et al. 1977. "Urban Stormwater Management and Technology: Update
and Users Guide," EPA 600/877014, US EPA, Cincinnati, Ohio, September.
9. Imhoff, K. and G.M. Fair. 1941. Sewage Treatment. John Wiley and Sons, Inc.,
New York.
10, American Society of Civil Engineers. 1960. "Sewage Treatment Plant Design,"
Manual of Practice No. 8.
11. Smith, R. 1968. "Preliminary Design and Simulation of Conventional Waste
Renovation Systems Using the Digital Computer," Report No. WP209, US Dept of
the Interior, Federal water Pollution Control Administration, Cincinnati, Ohio, March.
44

APPENDIX 1
STATISTICAL PROPERTIES
OF
LOGNORMAL DISTRIBUTIONS
A1.1 GENERAL CONSIDERATIONS
A 1.2 PROBABILITY DISTRIBUTIONS
A1.3 RELATIONSHIP BETWEEN DISTRIBUTIONS
A1.4 PROPERTIES OF LOGNORMAL DISTRIBUTIONS
All

Al.l GENERAL CONSIDERATIONS
This appendix presents a brief, simplified review of the statistical properties of lognormal
distributions that characterize the important variables in the water quality analysis procedures used in
this report. It is designed to help the user without a formal background in statistics to appreciate the
physical significance of the statistical properties employed. It is not the intent of this appendix to
present a theoretical discussion or to provide technical support for developing the relationships or
equations used in the development of the methods employed.
The standard statistical parameters of a population of values for a random variable that are
used as a concise means of describing central tendency and spread are:
MEAN : (pi x or X)  The arithmetic average, X defines the average of the
available data set, usually a limited sample of the total population.
ji x denotes the true mean of the total population of variable X.
X will be an increasingly better approximation of ]J x as the size of
the sample (the number of data points) increases.
VARIANCE : The average of the squares of the differences betveen individual
values of X and the mean, X . The greater the variability of the data
the higher the variance.
(xtx)2+ (x2x)2 +... (xnx)2
N
STANDARD DEVIATION : (

MEDIAN : ( X )  This is the value in a data set for vhich half the values
are greater, and half less than.
A1.2 PROBABILITY DISTRIBUTIONS
There are several different patterns that characterize the distribution of individual values in a
large population of variable events. Most users* are familiar with the normal distribution, in which a
histogram of the frequency of occurrence of various values describes the familiar bell shaped curve,
as in Figure A11 (a). When the cumulative frequency is plotted on probability paper, a straight line
is generated as in Figure All(b).
Many variables, particularly those that are important in water quality applications, are either
well represented, or adequately approximated by a lognormal distribution.
A lognormal distribution has a skewed frequency histogram, shown by Figure All(c), that
indicates an asymetrical distribution of values about an axis defining the central tendency of the data
set. There is a constraining limit to lower values (often zero) and a relatively small number of very
large values, but no upper limit.
A lognormally distributed data set appears as a bell shaped histogram and a straight line on
probability paper when the log transforms of the data are used, as shown by Figure All(d). The
log transforms of the values have a normal distribution. In this document, natural (base e)
logarithms are used throughout.
Rainfall events are usually best represented by a gamma distribution, which is characterized
by a large number of occurrences of very small values, and a small number of very large values, as
illustrated by Figure All(e).
A special case of the gamma distribution occurs when the coefficient of variation (CV) equals
1, which produces an exponential distribution. Values of CV greater than 1 result in a larger
percentage of values near the extremes (high and low), and fewer in the central range. As CVs
decrease below 1, the number of intermediate values increases and there are fewer occurrences of
very high and low values. As the CV approaches zero, the distribution approaches the bell shaped
curve of the normal distribution.
Al.3 RELATIONSHIP BETWEEN DISTRIBUTIONS
There are circumstances where two different types of distribution can begin to look quite
similar, so that either one can be used to provide a satisfactory approximation of the characteristics
of a particular data set. For example, as the coefficient of variation becomes smaller and smaller,
approaching zero, lognormal distributions begin to look more and more like a normal distribution.
Al3

HISTOGRAM
CUMULATIVE PROBABILITY
NORMAL
LOG NORMAL
o
oc
ui
CO
m
O
u.
O
E
ui
CO
O
H
V)
CD
O
u.
O
c
GAMMA
IA
O
I
c
I
o
c
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CO
(a)
V < 1
V> 1
MAGNITUDE OF VALUE (X)
(c)
V< 1
V> 1
MAGNITUDE OF VALUE (X)
le)
MAGNITUDE OF VALUE (X)
v = Coefficient of Variation
X
ui
10
60
90 99
PROBABILITY (K)
LESS THAN OR EQUAL
X
OJ
il
z>
Ou.
<0
10
SO
90 99
PROBABILITY (%)
LESS THAN OR EQUAL
O 3
u, _)
M
K U.
2°
O O
(f)
1 10
I
BO
90 99
PROBABILITY (%)
LESS THAN OR EQUAL
Figure A11. Probability distributions
Al4

Figure A12 shows a series of histograms for lognormally distributed populations, all having the
same arithmetic mean, but with different coefficients of variation, as shown. As discussed above,
as the variability decreases, the histogram approaches a normal distribution.
Similarly, at higher values for coefficient of variation, lognormal distributions become more
and more similar to gamma distributions. The tails at the high end will be different, but not greatly
so. At the very low end of the scale, differences will be substantial, but as a practical matter, errors
in the very small values will not significantly influence the accuracy of an overall analysis. Figures
A13 and A14 compare gamma and lognormal distributions having the same mean and coefficient
of variation.
Finally, when a number of different factors combine to produce some effect of interest, there
is a strong tendency for the combination to be lognormally distributed, regardless of the actual
probability distributions of the individual components. Accordingly, based on the above
considerations as well as the indications produced by analysis of available data for a variety of
parameters, the lognormal distribution has been selected as the basis for many of the probabilistic
computations described in this document. The fact that an extensive body of experience with
mathematical procedures for combining such distributions is available, permits the desired analyses
to be performed in a relatively simple and straightforward manner.
A1.4 PROPERTIES OF LOGNORMAL DISTRIBUTIONS
A few mathematical formulas based on statistical theory summarize the pertinent statistical
relationships for lognormal probability distributions, and provide the basis for back and forth
conversions between arithmetic properties (in which concentrations, flows, and loads are reported),
and logarithmic properties (in which probability and frequency characteristics are defined and
computed).
A convention using Roman (rather than Greek) letters for the statistical properties is adopted at
this point because it may make the presentation easier to follow for the user who does not use
mathematical formulas often enough to be comfortable working with them, because the type format
is simpler and clearer, and because this convention lends itself to convenient conversion to computer
programming. All of the computations required by the analyses described in this document will
operate quite rapidly on a micropersonal computer.
The statistical parameters are designated as follows:
ARITHMETIC LOGARITHMIC
MEAN M U
STD DEVIATION S W
COEF OF VARIATION CV
MEDIAN T
Al5

u
c
01
a
a
OJ
i_
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' Variation
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y
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Coefficient of
Variation
v a 2
X
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Figure A12. Effect of coefficient of variation on frequency distribution
Al6

PERCENT GREATER THAN OR EQUAL
J9 98 9'j 90 80 70 60 bO 40 30 ?0 10 b 2 1 OS 02 OIOOS 001
0.1
i : ;
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a.
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D
S
10 20 30 /0 SO 60 70 80 90 95 98 99
PERCENT LESS THAN/EQUAL TO
998 999 9999
Coef. of Variation: v = 1.0
Gamma _ _ _
Log Normal ^^^^^~
Figure A13. Comparison of gamma and lognormal probability distribution
for coefficient of variation = 1.0
Al7

PERCENT GREATER THAN/EQUAL TO
1 _
H _
7 _
ft _j
1
5 j
4
T
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PERCENT LESS THAN/EQUAL TO
Coef. of Variition: v  1.5
Gamma
Log Normal ^^
Figure A14. Comparison of gamma and lognormal probability distribution
for coefficient of variation = 1.5
Al8

For the lognormal distributions used in this document, the definition of any two of the above
statistical parameters automatically defines the values for all of the others. The following formulas
define the relationships that permit the other values to be computed.
T = exp (U) S = M * CV
M = exp (U + 0.5 * W2) W = SQR ( LN (1 + CV2) )
M = T * SQR (1 + CV2) U = LN (M/exp (0.5 * W2 ) )
V = SQR (exp (W2) 1) U = LN (M/ SQR (1 + CV2) )
Parameter designations are as above, and additionally:
LN(x) designates the base e log of the value x,
SQR(x) designates the square root of the value x,
exp(x) designates e to the power x.
The statistical parameters of a particular distribution may also be used to compute the
magnitude of the variable at any specified probability of exceedance, or conversely to compute the
probability of exceeding any specified value. The equations are:
Xa = exp ( U + Za * W)
LN(X)  U
Z =
W
For normal (or lognormal) distributions, probabilities can be defined in terms of the
magnitude of a value normalized by the value of the standard deviation. Cumulative probabilities
have a specific relationship to the normalized standard deviation, for which Z is the conventional
designation (e.g., Z = 1 is one standard deviation). Tables that summarize this relationship are
available in many texts.
Table Al1 presents the "standard normal table" in a form convenient for the analyses
described in this document. The probabilities listed are the total (equivalent to the area under the
curve) for the interval between zero and the specified value of Z. It therefore reflects the probability
that the magnitude of a variable will be equal to or less than the magnitude that corresponds with the
corresponding value of Z.
From this table, the value of Z used in the computation can be selected for any probability, or
viceversa. In the equations, the subscript "a" represents the percentile (probability) of interest. Xa
is the value of the variable having that probability, and Za is the value of Z that corresponds with
percentile "a".
Al9

TABLE Al1 PROBABILITIES FOR THE STANDARD NORMAL DISTRIBUTION
Caen entry In the table Indicates the proportion of the total area under the
normal curve to the left of a perpendicular raised at a distance of 21
standard deviation units.
Example: 68.69 percent of the area under a normal curve ties to the left
of a point 1.21 standard deviation units to the right of the mean.
z
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
.0
.1
.2
.3
.4
2.5
2.6
2.7
2. B
2.9
.0
.1
.2
.3
.4
.5
.6
.7
.0
.9
0.00
0.5000
0.5398
0.5793
0.6179
0.6554
0.6915
0.7257
0.7580
0.7881
0.8159
0.8113
0.8613
0.8819
0.9032
0.9192
0.9332
0.9152
0.9551
0.9611
0.9713
0.9772
0.9621
0.9861
0.9893
0.9918
0.9938
0.9953
0.9965
0.9974
0.9981
0.9986
0.9990
0.9993
0.9995
0.9997
0.9998
0.9998
0.9999
0.9999
1.0000
0.01
0.5010
0.5138
0.5832
0.6217
0.6591
0.6950
0.7291
0.7612
0.7910
0.8186
0.8138
0.8665
0.8869
0.9019
0.9207
0.9315
0.9463
0.9564
0.9649
0.9719
0.9778
0.9826
0.9864
0.9896
0.9920
0.9910
0.9955
0.9966
0.99/5
0.9982
0.9987
0.9991
0.9993
0.9995
0.9997
0.9998
0.9998
0.9999
0.9999
1.0000
0.02
0.5080
0.5176
0.5871
0.6255
0.6628
0.6985
0.7324
0.7612
0.7939
0.8212
0.8461
0.8686
0.8888
0.9066
0.9222
0.9357
0.9474
0.9573
0.9656
0.9726
0.9783
0.9830
0.9868
0.9898
0.9922
0.9911
0.9956
0.9967
0.9976
0.9982
0.9987
0.9991
0.9994
0.9995
0.9997
0. 9998
0.9999
0.9999
0.9999
1.0000
0.03
0.5120
0.5517
0.5910
0.6293
0.6664
0.7019
0.7357
0.7673
0.7967
0.8238
0.8485
0.8708
0.8907
0.9082
0.9236
0.9370
0.9484
0.9582
0.9664
0.9732
0.9768
0.9834
0.9871
0.9901
0.9925
0.9943
0.9957
0.9968
0.9977
0.9983
0.9988
0.9991
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
0.04
0.5160
0.5557
0.5948
0.6331
0.6700
0.7051
0.7389
0.7704
0.7995
0.8264
0.8508
0.8729
0.8925
0.9099
0.9251
0.9382
0.9495
0.9591
0.9671
0.9738
0.9793
0.9838
0.9875
0.9904
0.9927
0.9915
0.9919
0.9969
0.9977
0.9984
0.9988
0.9992
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
l.OOCO
C
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.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
1.
1.05
5199
5596
5987
6368
6736
7008
7422
7734
8023
8289
8531
8719
8944
9115
9265
9391
9505
9599
9678
9744
9798
9842
9678
9906
9929
9916
9960
9970
9978
9984
9969
9992
9994
9996
9997
9998
9999
9999
9999
0000
O.Ofi
0.5239
0.5636
0.6026
0.6406
0.6772
0.7123
0.7454
0.7764
0.8051
0.8315
0.8554
0.8770
0.8962
0.9131
0.9279
0.9406
0.9515
0.9608
0.9686
0.9750
0.9803
0.9846
0.9881
0.9909
0.9931
0.9948
0.9961
0.9971
0.9979
0.9985
0.9989
0.9992
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
0.07
0.5279
0.5675
0.6064
0.6443
0.6808
0.7157
0.7486
0.7794
0.8078
0.8340
0.8577
0.8790
0.8980
0.9147
0.9292
0.9418
0.9525
0.9616
0.9693
0.9756
0.9808
0.9850
0.9884
0.9911
0.9932
0.9949
0.9962
0.9972
0.9979
0.9985
0.9989
0.9992
0.9995
0.9996
0.9997
0.9998
0.9999
0.9999
1.0000
1.0000
0.08
0.5319
0.5714
0.6103
0.6480
0.6844
0.7190
0.7518
0.7823
0.8106
0.8365
0.8599
0.8810
0.8997
0.9162
0.9306
0.9429
0.9535
0.9625
0.9699
0.9761
0.9812
0.98S4
0.9887
0.9913
0.9934
0.9951
0.9963
0.9973
0.9980
0.9986
0.9990
0.9993
0.9995
0.9996
0.9998
0.9998
0. 9999
0.9999
1.0000
1.0000
0.09
0.5359
0.5753
0.6141
0.6517
0.6879
0.7224
0.7549
0.7852
0.8133
0.8389
0.8621
0.8830
0.9015
0.9177
0.9319
0.9441
0.9545
0.9633
0.9706
0.9767
0.9817
0.9857
0.9890
0.9916
0.9936
0.9952
0.9964
0.9974
0.9981
0.9986
0.9990
0.9993
0.9995
0.9997
0.9998
0.9998
0.9999
0.9999
1.0000
1.0000
Al10

One additional normalization permits a full generalization of the characteristics of lognormal
distributions, and a convenient approach for frequent use. This involves describing the magnitude
of the variable at any percentile as its ratio to the mean.
The equations summarizing the relationships for a lognormal distribution presented earlier can
be combined and reduced to directly compute the value of x at any percentile "a", expressed as a
multiple of the mean (Xa / M):
where:
Xa/M = EXP (Za* SQR (A)  0.5 * A)
A = LN (1 + CV2)
Thus, when magnitude is expressed as a multiple of the mean, the value at any probability is
seen to depend only on the coefficient of variation (CV). This permits the construction of standard
tables or graphs that can be used in place of the computations.
Table A12 presents a tabulation of the relationships for lognormal distributions for a range of
CVs. It was constructed using the equations immediately above, and the information in Table A11.
The user can quite easily modify this table to include other probabilities or values for CV, where this
might be desired to support local analyses. Figure A15 converts the tabulated results to a
generalized probability plot, from which probabilities and magnitudes of interest may be scaled
directly.
TABLE Al2 LOGNORMAL RELATIONSHIPS
COEF
OF
VAR
VALUE OF X AS MULTIPLE OF MEAN
MEDIAN 80% 84.13% 90% 95% 99% 99.9% 99.99%
Z = 0.000 0.842 1.000 1.280 1.645 2.327 3.090 3.750
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
0.06
0.22
0.45
0.69
0.94
1.18
1.40
1.61
1.80
1.98
0.97
0.89
0.80
0.71
0.62
0.55
0.50
0.45
0.41
0.37
1.19
1.33
1.40
1.43
1.41
1.38
1.34
1.30
1.26
1.21
1.24
1.43
1.56
1.63
1.65
1.64
1.62
1.59
1.55
1.52
1.33
1.64
1.88
2.05
2.16
2.23
2.26
2.27
2.26
2.25
1.45
1.95
2.40
2.78
3.08
3.31
3.48
3.60
3.70
3.76
1.72
2.68
3.79
4.91
5.97
6.94
7.80
8.56
9.23
9.82
2.08
3.85
6.30
9.26
12.52
15.88
19.25
22.54
25.71
28.75
2.44
5.26
9.80
16.05
23.74
32.52
42.06
52.07
62.37
72.79
Al11

tt«» t»t
100
PERCENT OF VALUES EQUAL OR GREATER THAN
ti W (0 70 40 SO JO X) 10 10 s > I 05 01 01 001 001
IX
LU
2
o
HI
_i
Q.
I
_l
3
COEFFICIENT OF
VARIATION
0.5
1.5
2.0
111
100
10
1.0
_LJ
001 DOT 01 01 OJ '0 10 50 IO 10 JO JO SO 60 70 »0 »0 »S » »» »ti «»
PERCENT OF VALUES EQUAL OR LESS THAN
0.1
Figure A15. Log normal distributions

APPENDIX 2
CHARACTERISTIC VALUES
FOR PARAMETERS
USED IN THE ANALYSIS
A2.1 RAINFALL STATISTICS
A2.2 RUNOFF COEFFICIENT
A2.3 POLLUTANT CONCENTRATIONS
A2.4 STREAM FLOW VARIABILITY
A2.5 DISPERSION COEFFICIENT
A21

This appendix presents information on representative values for various parameters used in
the computations. It is intended to serve as a reference that will permit the user to make preliminary
estimates for use in a screening analysis, and for comparing local values against those developed
from a broader data base.
A2.1 RAINFALL STATISTICS
Longterm rainfall patterns for an area are recorded in the hourly precipitation records of rain
gages maintained by the U.S. Weather Service (USWS). The analysis procedures used in this
manual are based on the statistical characteristics of storm "events." As illustrated by Figure A21,
the hourly record may be converted to an "event" record by the specification of a minimum number
of dry hours that defines the separation of storm events. Routine statistical procedures are then used
to compute the statistical parameters (mean, standard deviation, coefficient of variation) of all events
in the record, for the rainfall properties of interest.
A computer program, SYNOP, documented in a publication of EPA's Nationwide Urban
Runoff Program (NURP), computes the desired statistics from rainfall data tapes obtainable from
USWS. It generates outputs based on the entire record, and also on a stratification of the record by
month, which is convenient for evaluating seasonal differences.
Table A21 summarizes the statistics for storm event parameters for rain gages in selected
cities distributed throughout the country. These data may be used to guide local estimates, pending
analysis of specific data based on a site specific rain gage. The tabulations provide values for mean
and coefficient of variation for storm event volumes, average intensities, durations, and intervals
between storm midpoints. The cities for which results have been tabulated are grouped by region of
the country. Results are presented for both the longterm average of all storms, and for the June
through September period that is commonly the critical period for many receiving water impacts.
Figure A22 provides initial estimates of storm event characteristics for broad regions of the
country based on data in the foregoing table.
From the statistics of the storm event parameters, other values of interest may be determined.
The ratio of mean storm duration (D) to the mean interval between storms (A) reflects the
percent of the time that storm events are in progress.
% time that it is raining = D
A
The average number of storms during any period of time is defined by the ratio between the
total number of hours in the selected period and the average interval between storms (A). For
example, on an annual basis:
A22

() HOURLY RAINFALL VARIATION
1
Time
(b) STORM EVENT VARIATION
V% ft
d
1
n
^^^^ V
H
I
V77A
Volume
Duration
Average intensity
Interval between
event midpoints
PARAMETER
For each
V
d
i
6
storm event
(inches)
(hours)
(inch/hour)
(hours)
For all
Mean
V
0
1
A
storm events
Coef Var
»V
"d
v\
"6
Figure A21. Characterization of a rainfall record
A23

3949CI
Table A2I. RAINFALL EVENT CHARACTERISTICS FOR SELECTED CITIES
Annual
Mean
Location
Great Lakes
Champa IgnUrbana, IL
Chicago, IL (3)
Chicago, IL (5)
Davenport, IA
Detroit, Ml
Louisville, KY
Minneapolis, MN
Steubenvllle, OH
Toledo, OH
Zanesville, OH
Lansing, Ml (5) (30 yr)
Lansing, Ml (9X21 yr)
Ann Arbor, Ml (5)
Lower Mississippi Valley
Memphis, TN
New Orleans, LA (8)
Shreveport, LA (9) (17 yr)
Lake Charles, LA (10)
Average
Texas
Abilene, TX
Austin, TX
Brownsvi 1 le, TX
Dallas, TX
Waco, TX
V
0.35
0.27
0.27
0.38
0.21
0.38
0.24
0.31
0.22
0.30
0.21
0.26
0.52
0.61
0.54
0.66
0.58
0.32
0.33
0.27
0.39
0.36
1
.063
.053
.053
.077
.050
.064
.043
.057
.048
.061
.041
.047
.086
.113
.080
.108
.097
.083
.078
.072
.079
.086
0
6.1
4.4
5.7
6.6
4.4
6.7
6.0
7.0
5.0
6.1
5.6
6.2
6.9
6.9
7.8
7.7
7.3
4.2
4.0
3.5
4.2
4.2
A
80
62
72
98
57
76
87
79
62
77
62
87
89
89
110
109
99
128
96
1.09
100
1.06
Coefficient
%
.47
.44
.59
.37
.59
.45
.48
.28
.52
.24
.56
.42
1.36
1.46
1.39
1.64
1.46
1.52
1.88
2.02
1.64
1.66
vi
.37
.58
.54
.24
.16
.42
.22
.03
.16
.01
.55
.42
1.31
1.40
1.27
1.40
1.35
.24
.53
.43
.23
.40
of Variation
vd
.02
.06
.08
.40
.02
.08
.08
.39
0.99
0.93
1.10
0.95
1.07
1.24
1.09
1.26
1.17
.01
.06
.20
.00
.08
WA
.02
.12
.00
.01
.07
.00
0.98
.00
.03
.03
.02
.00
1.01
1.02
0.99
0.99
1.00
.45
.44
.50
.32
.36
V
0.45
0.33
0.37
0.49
0.27
0.36
0.34
0.39
0.29
0.36
0.29
0.34
0.44
0.53
0.49
0.63
9.52
0.42
0.38
0.33
0.38
0.40
June to
Mean
1
.102
.091
.090
.112
.095
.094
.075
.094
.083
.100
.073
.078
.112
.142
.105
.130
.122
.121
.106
.104
.100
.117
D
4.6
6.2
4.5
5.3
3.1
4.5
4.5
5.9
3.7
4.3
4.2
5.1
4.7
5.0
5.3
5.9
5.2
3.3
3.3
2.8
3.2
3.3
A
87
67
76
91
64
78
74
68
69
60
71
89
68
65
109
66
67
114
108
101
III
124
September
Coefficient
Utf Vj
.44
.49
.42
.32
.43
.40
.34
.28
.43
.23
.39
.25
1.35
1.40
1.50
1.90
1.54
.56
.82
.94
.65
.60
.22
.37
.37
.14
.32
.31
.26
.27
.37
.11
.25
.13
1.28
1.42
1.27
1.41
1.35
.32
.71
.33
.24
.34
of Variation
vd UA
.01
.00
.04
.22
0.82
.01
.00
.76
.93
0.95
0.98
0.90
1.12
1.34
1.28
1.43
1.29
0.98
1.02
1.30
1.01
1.07
.05
.13
.02
0.94
.14
.04
0.92
0.95
1.06
1.06
1.00
0.98
1.06
1.08
1.09
0.99
1.06
.46
.49
.67
.44
.39
irage
0.33 0.080 4.0 108
1.74 I
1.07 1.41 0.38
.110
3.2 112 1.71 1.39 I/
1.49

39491
Table A2I. RAINFALL EVENT CHARACTERISTICS FOR SELECTED CITIES (continued)
to
Ul
Mean
Location
Northeast
Caribou, ME
Boston, MA
Lake George, NY
Kingston, NY
Poughkeepsie, NY
New York City, NY
Mineola LI, NY
Upton LI, NY
Wantagh LI, NY (2 YR)
Long Island, NY
Washington, D.C.
Baltimore, MD (3)
Southeast
Greensboro, NC
Columbia, SC
Atlanta, GA
Birmingham, ALA
Gainesville, FLA
Tampa, FLA
Average
V
0.21
0.33
0.23
0.37
0.35
0.37
0.43
0.43
0.40
0.41
0.36
0.40
0.32
0.38
0.50
0.53
0.64
0.40
0.49
1
.034
.044
.067
.052
.052
.053
.088
.076
.075
.126
.067
.069
.067
.102
.074
.086
.139
.110
.102
0
5.8
6.1
5.4
7.0
6.9
6.7
5.8
6.3
5.6
4.2
5.9
6.0
5.0
4.5
8.0
7.2
7.6
3.6
6.2
Annual
June to September
Coefficient of Variation Mean Coefficient of Variation
A «u
Vj vd VA V 1 0 A « Vj vd «A
55 .58 0.97 1.03 1.03 0.24 .054 4.4 55 .64 .15 1.00 1.01
68 .67
76 .26
80 .35
81 .31 (
77 .37
89 .34
81 .42
83 .54
93 .35
80 .45
82 .48
67 .40
68 .55
94 .37
85 .44
106 .35
93 .63
89 1.47
.02 1.03 1.06 0.30 .063 4.2 73 .60 .20 1.12 1.12
.98 0.91 1.48 0.27 .076 4.5 72 .25 .61 0.86 1.44
.01 0.91 0.98 0.35 .073 5.0 79 .46 .27 1.00 1.08
).95 0.87 0.95 0.36 .081 4.9 82 .48 .16 0.96 1.00
.04 0.93 0.89 0.30 .076 .8 75 .51 .28 1.03 0.95
.14 .30 0.99 0.41 .114 .5 88 .42 .17 1.48 1.03
.06 .09 0.99 0.42 .101 .6 88 .56 .10 1.23 1.02
.24 .03 1.03 0.34 .091 .0 74 .59 .08 1.28 0.99
.30 .12 1.72 0.41 .127 .4 99 .52 .15 1.21 1.57
.18 .03 1.00 0.41 .107 4.1 78 .67 .38 1.10 1.06
.21 .01 1.03 0.43 .107 4.2 79 .66 .49 1.08 1.08
.44 .11 1.18 0.34 .093 3.6 62 .67 .43 .20 1.19
.59 .13 1.18 0.41 .153 3.4 58 .59 .68 .25 1.13
.16 .11 0.93 0.45 .100 6.2 87 .43 .27 .31 0.97
.31 .09 1.00 0.45 .III 5.0 76 .47 .33 .18 1.01
.14 .66 1.06 0.65 .161 6.6 70 .41 .13 .65 0.92
.21 .11 1.10 0.44 .138 3.1 49 .70 .28 .28 1.01
.28 1.22 1.05 0.48 .133 4.9 68 1.52 1.34 1.33 1.01

3949C3
Table A2I. RAINFALL EVENT CHARACTERISTICS FOR SELECTED CITIES (concluded)
Annual
Mean
Location
Rocky Mountains
Denver, CO (3) 8 YRS
Denver, CO (3) 25 YRS
Denver, CO (13) 24 YRS
Rapid City, SO (3)
Rapid City. SD (12)
Salt Lake City, UT (3)
Salt Lake City, UT (3)
(2 GAGES)
Average (2)
California
Oakland, CA
San Francisco, CA (75)
Southwest
El Paso, TX
Phoenix, AZ
Average'
Northwest
Portland, OR (3) 25 YRS
Portland, OR (10) 10 YRS
Eugene, OR (6)
Eugene, OR (15)
Eugene, OR (20)
Seattle, WA (15)
Average
V
0.15
0.15
0.22
0.15
0.20
0.14
0.18
0.15
0.19
0.78
0.15
0.17
0.17
0.17
0.36
0.39
0.63
0.72
0.46
0.48
1
.033
.033
.032
.039
.033
.031
.025
.036
.033
.017
.047
.055
.045
.017
.023
.030
.026
.025
.023
.024
0
4.3
4.8
9.1
4.0
8.0
4.5
7.8
4.4
4.3
59
3.3
3.2
3.6
5.4
15.5
10.9
23.1
29.2
21.5
20.0
A
97
101
144
86
127
94
133
94
320
515
226
286
277
60
83
73
118
136
101
101
Coefficient
*
2.00
.73
.49
.81
.46
.42
.32
1.77
1.62
1.45
1.54
1.38
1.51
.60
.51
.85
.88
.85
.45
1.61
wi
.58
.07
.13
.63
.09
0.91
1.06
1.35
0.74
0.89
1.12
1.26
1.04
0.85
0.79
0.87
0.88
0.91
0.86
0.84
of Variation
wd
.24
.20
.15
.21
.24
0.92
0.85
1.20
1.03
1.37
1.07
0.97
1.02
.00
.09
.25
.35
.34
.26
1.23
*
1.25
1.15
0.92
1.33
0.95
1.39
0.97
1.24
1.60
0.72
1.43
1.42
1.48
.47
.32
.74
.30
.19
.02
1.21
V
0.18
0.15
0.22
0.20
0.25
0.14
0.16
0.18
0.11
0.14
0.19
0.21
0.17
0.15
0.22
0.21
0.28
0.31
0.29
0.26
June to
Mean
1
.053
.055
.053
.063
.059
.041
.031
.059
.020
.017
.069
.090
.060
.019
.027
.033
.029
.027
.024
.027
D
3.2
3.2
4.4
3.0
6.1
2.8
6.8
3.1
2.9
11.2
2.6
2.4
2.6
4.5
9.4
6.3
12.0
15.0
12.7
11.4
A
82
80
101
75
101
125
164
78
756
830
142
379
425
109
179
167
226
250
159
188
September
Coefficient
*
.90
.85
.78
.63
.50
.51
.43
1.74
1.63
1.46
1.68
1.51
1.61
.45
.32
.32
.28
.24
.45
1.35
,
.44
.51
.53
.36
.46
.13
.06
1.44
0.56
0.70
1.28
1.64
1.16
0.99
1.33
1.01
1.07
1.15
0.92
1.11
of Variation
vd
.20
.20
.35
.08
.39
0.80
1.01
1.14
1.00
1.67
1.20
0.84
1.01
0.95
.13
.05
.22
.19
.24
1.20
VA
1.26
1.05
0.23
1.20
0.94
1.41
0.98
1.13
1.09
0.75
1.44
1.25
1.26
.64
.20
.49
.20
.11
.04
1.15

MM
)
1
3
4
5
1
7
1
1
KMOO
MMJAl
MMCK
MMUAl
SUMMCA
MNUAl
SUMMER
AJMUAl
SUMMf*
MWA1
$UM«K«
AJMUA1
SUMMER
ANNUAL
SUMMER
ANNUAL
SUMMER
ANNUAl
SUMMER
HAWAII STATISTICS
HXUMf Ml
MUM
Ut
OJ2
OJt
0.40
0.41
0.41
on
tat
on
Ul
017
0.17
0.41
OJt
0.14
0.14
0.11
0.11
tv.
1.41
1JI
1.4S
157
147
Ii2
1.4t
I.M
1.74
1.71
Ml
1J1
1.11
US
142
lit
1.77
174
riENSin IWIMRI
MUD
CLM1
O.M7
0.066
0.101
.102
.)U
fl7
.1C
.OM
.110
IMS
00
ton
ton
431
Ml
jue
M
C.«.
1J1
1JI
UJ
U7
in
1 J4
US
us
1.37
U»
\M
1.11
OJ4
1.11
0.11
1.13
US
1.44
OUKATm IM«i
MUD
SJ
4.4
iJ
4.]
1.2
4.9
7J
1.2
40
12
3.S
2.1
20.0
11.4
4.S
2J
4.4
1.1
c.«.
105
1.14
1.0S
109
1.22
1.33
1.17
129
107
1.01
1.02
141
123
1JO
012
OJO
1.20
114
WTE*VAI mm
HUH
n
71
77
77
n
u
n
17
lOt
112
J77
42S
101
in
M
12S
N
71
CV.
1.07
107
1.0S
1M
IDS
1.01
1.00
1.06
141
1.49
1.41
126
1.21
115
1.31
141
1.24
1.19
Figure A22. Representative regional rainfall statistics for preliminary estimates
A27

Avg. number of storms per year = 365 * 24
A
The storm event parameters of interest have been shown to be well represented by a gamma
distribution, and the results listed in Table A21 indicate that the coefficient of variation of the event
parameters generally falls between 1.0 and 1.5. Figure A23 plots the probability distribution of
gamma distributed variables with coefficient of variation of 1.0, 1.25, and 1.5, in terms of
probability of occurrence as a function of the magnitude expressed as a multiple of the mean. This
plot can be used to approximate the magnitude of an event with a specified frequency of occurrence.
For example, at a location where storm events have volume statistics for MEAN and CV of
0.4 inch, and 1.5 respectively, Figure A22 can be used to estimate that 1 percent of all storm events
have volumes that exceed about 7.5 times the mean (or 7.5 * 0.4 = 3 inches). If the same location
has an average interval between storms (A) of 87.5 hours, there will be an average of:
(365 * 24) / 87.5 = 100 events/year
and the 1 percentile event (3 inches) reflects a storm volume exceeded, on average, once per year.
A2.2 RUNOFF COEFFICIENT
Runoff coefficient (Rv) is defined as the fraction of rainfall that appears as surface runoff.
The substantial data base developed under EPA's NURP program indicated that Rv varied from
event to event at any site, but that the median value for a site was best estimated by the percent of
impervious surface in the drainage area.
Figure A24 illustrates the relationship between the median runoff coefficient observed at an
urban site and the percent of impervious area in the catchment.
This information may be used to guide estimates of the surface runoff entering a combined
sewer system during storm events.
A2.3 POLLUTANT CONCENTRATIONS
The data summaries presented below may be used to guide local estimates of pollutant
concentrations in CSOs and in discharges from separate storm sewers (URO). The available data
bases indicate that there are appreciable sitetosite differences in strength for both categories.
Accordingly, the summary tables list three sets of concentrations, plus an estimate of the
eventtoevent variability.
The concentration listings in these tables are site median concentrations for:
A28

PERCENT EQUAL OR GREATER
90 80 70 60 50 40 30 20 10 5 21 0.5 0,2 0.1
10
20
30 40 50 60 70 80 90 95 96 98 M* MJ Mi
PERCENT EQUAL OR LESS
Figure A23. Probability distribution for a variable with a
gamma distribution
A29

I.U
.9
7
ml
.5
.4
.
i
n
o
o
*0°
1
0
o r
0 On
i
O
1
°o °
D G
**0
W
O
o
D
(
O
O
o
0
o
o
)
1
c
D
C
i
0 10 20 30 40 SO 60 70 80 90 100
% IMPERVIOUS
Figure A24. Relationship between percent impervious area and
median runoff coefficient
A210

LOW concentration sites  This corresponds approximately with the site
median concentration of the pollutant for the 10th percentile
site. About 10 percent of all sites in the data base have a
median EMC equal to or less than this concentration.
MEDIAN site  This is the median of all sites in the data base.
HIGH concentration sites  This corresponds approximately with the site
median concentration of the pollutant for the 90th percentile
site. About 90 percent of all sites in the data base have a
median EMC equal to or less than this concentration.
From the available data, flows and concentrations for each monitored event were analyzed to
compute an event mean concentration (EMC). The EMCs of all events at a particular site were
analyzed statistically and determined to be adequately approximated by a lognormal distribution.
Using the procedures described in Appendix 1, the median EMC for the site (the "site median") and
the coefficient of variation of all EMCs at the site, were computed.
The examination of all such site characteristics (site median EMC and coef. of variation of
EMCs) indicates that sites may differ significantly in their site medians for all of the pollutants, but
that the eventtoevent variability falls into a consistent range and is unrelated to the magnitude of the
site median.
Table A22 summarizes quality characteristics for CSOs, based on the EPA sponsored data
base developed and maintained by the University of Florida. CSO data from a total of 164 overflow
events, at 18 sites in 7 cities, form the basis for this summary table.
Table A23 summarizes quality characteristics for URO, based on the EPA sponsored
Nationwide Urban Runoff Program (NURP). Monitoring data from 85 sites in 25 cities are
represented in this data base.
A2.4 STREAM FLOW VARIABILITY
The analysis procedure for computing the impact of stormwater discharges in advective
systems (rivers and streams), requires that values be assigned for the mean and coefficient of
variation of upstream flows. The mean flow for a particular stream will often be available from
hydrologic data summaries from the USGS stream gage network. In some cases it will be necessary
to estimate the mean flow from the data from nearby gages, on the basis of CFS/sq mi of drainage
area. In either case, good local estimates of mean flow will usually be available for the segment of 
the stream into which the CSOs discharge.
Although the coefficient of variation of daily flows could be computed from the flow record
using standard statistical procedures, this is not routine practice and hence is not normally available.
However, the coefficient of variation of daily flows for perennial streams (7Q10 > 0) exhibits a well
defined relationship with the ratio of 7Q10/Average Flow. It may be estimated using the relationship
A211

TABLE A22 CSO QUALITY CHARACTERISTICS
POLLUTANT
TSS .
BOD
COD
PO4P
TOTAL P
TKN
AMMONIA N
NO2N
NO3N
CHROME
COPPER
LEAD
ZINC
FECAL COLIFORM
SITE MEDIAN CONCENTRATION
(mg/1)
10th 90th
peicentile MEDIAN percentile
SITE SITE SITE
121
32
62
0.5
0.8
4.1
0.9
0.1
0.2
0.008
0.039
0.158
0.223
0.5
184
53
132
0.8
2.4
6.5
1.9
0.1
1.0
0.090
0.102
0.346
0.348
279
86
283
1.1
7.9
10.3
3.9
0.1
4.5
0.957
0.271
0.755
0.544
(1,000,000 MPN/lOOml)
1 5
COEFofVAR
of
EVENT MEAN
CONCENTRATIONS
0.7
0.7
0.8
0.4
0.7
0.6
0.8
0.6
0.5
0.6
0.5
0.6
0.6
1.5
TABLE A23 URO QUALITY CHARACTERISTICS
POLLUTANT
TSS
BOD
COD
TOTAL P
SOLUBLE P
TKN
NO2 + NO3 N
COPPER
LEAD
ZINC
FECAL COLIFORM
SUMMER
WINTER
SITE MEDIAN CONCENTRATION
(mg/1)
10th 90th
percentile MEDIAN percentile
SITE SITE SITE
33
5
30
0.16
0.07
0.68
0.26
0.012
0.060
0.050
10
0.4
100
9
65
0.33
0.12
1.50
0.68
0.034
0.144
0.160
300
15
140
0.70
0.21
3.30
1.75
0.093
0.350
0.500
(1,000 MPN/lOOml)
20
1
50
3
COEFofVAR
of
EVENT MEAN
CONCENTRATIONS
1.5
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
1.5
1.5
A212

presented in Figure A25, which is based on the analysis of about 150 streams in different areas of
the country.
A2.5 DISPERSION COEFFICIENT
For CSO discharges to tidal rivers and estuaries, the analysis procedure requires the
assignment of a value for a dispersion coefficient (E). This is a lumped parameter that reflects all
transport factors other than advection and pollutant decay. It is influenced by a number of factors,
including strength of tides, fresh water flow and velocity, salinity gradients and geomorphology.
For a given tidal river/estuary system, it varies with freshwater flow and distance from the mouth.
Typical values reported range from about 0.3 to 30 sq mi/day.
The most reliable basis for assigning a value of E for the segment receiving the CSO
discharges, is from the value used in a previously calibrated mathematical model of the system.
However, it may also be estimated from a salinity profile that spans the area of interest, using the
following relationship.
Cu = Cd exp (UX/E)
E = Ux/LN(Cu/Cd)
where:
E = dispersion coefficient (sq mi/day)
Cu = concentration at upstream location (mg/1, parts per thousand, etc.)
Cd = concentration at downstream location (mg/1, parts/thousand, etc.)
X = distance between locations used for Cy and Cd (miles)
U = freshwater flow velocity (miles/day)
Approximate ranges for values of dispersion coefficient (E) are as follows:
E = 0.52
more upstream segments of tidal rivers; no depth stratification; small
vertical salinity gradients.
E = 812
estuarine segments; appreciable tidal effects; some stratification;
relatively strong vertical salinity gradients.
E = 2030
locations near the mouth of estuaries; strong tidal effects,
stratification, and high vertical salinity gradients.
A213

10
i
o
CO
111
o
cc
UJ
I
o
o
<
oc
10
2
10
3
i r i i i i i i
I I I I I I I L
i i i i i
i i i i i i i i
10
,1
10
COEFFICIENT OF VARIATION DAILY STREAM FLOWS
Figure A25. Guide for estimating coefficient of variation of daily stream flows
A214

APPENDIX 3
ANALYSIS METHODOLOGY
FOR
RECEIVING WATER IMPACTS
A3.1 INTRODUCTION
A3.2 ESTUARIES AND TIDAL RIVERS
A3.3 RIVERS AND STREAMS
A31

A3.1 INTRODUCTION
Procedures for calculating water quality effects of continuous waste loads under steady state
conditions are comparatively straightforward. However, a detailed analysis, employing
mathematical receiving water models, can in some cases be a relatively complex exercise, even for a
steady state, continuous point source load. Complexity is introduced by the type of effects to be
modeled, complexities in receiving water geometry, multiple load sources, inflow from downstream
tributaries or groundwater accretion, and the influences these factors may have on the reaction rates
of various constituents. Simplified analyses can be used in many cases to provide an approximation
of receiving water impacts (where the necessary simplifying assumptions can be accepted, and the
implications are not lost sight of).
For the "time variable" situations that are characteristic of CSO discharges, the basic approach
required is necessarily more complex. A procedure for estimating water quality effects must account
for variability in pollutant loadings with respect to occurrence, magnitude, and duration. The
approach in this document is less concerned with what happens during any particular individual
storm than with the overall effect from all storms.
There will be a continuous spectrum covering very small to very large storms, and any
rational control strategy will stop short of attempting complete control of the very largest storm
conceivable. The frequency of occurrence of water quality impacts of various magnitudes and
severity, with and without controls, provides a basis for evaluating alternate control strategies.
The analysis methods presented in this document provide a direct computation of the statistical
characteristics of receiving water responses using the statistical properties of pollutant loads and
other pertinent factors. These techniques use simplifying assumptions and the results are therefore
approximations. They are convenient to use and permit a quantitative evaluation, even where time
and budget limitations exist The theoretical basis is complex, but execution is relatively simple and
straightforward, consisting of solving a few equations or scaling from graphs. The equations
employed can be solved on hand calculators.
This appendix presents a description of procedures for calculating the probability or frequency
distribution of water quality effects resulting from intermittent, variable storm loads. Two different
water body types are addressed: estuaries and tidal rivers, and streams and rivers.
A3.2 ESTUARIES AND TIDAL RIVERS
Estuaries tend to provide relatively high degrees of dilution and relatively strong mixing of the
waste loads that enter. The analysis employed was developed by Hydroscience for EPA (1), and the
theoretical development has been described by DiToro (2). The procedure takes advantage of these
natural characteristics of dispersive receiving water systems.
The high degree of mixing in estuarine systems makes the separate analysis of individual
storms inappropriate. The residual effects of previous storms will often still be present when the
A32

current storm discharge occurs, and the impacts of each of the storms must be superimposed to
determine the total stormwater response. The mean and variability of pollutant concentrations in an
estuary can be computed directly, using equations derived from the response shape of a single
loading pulse to an advectivedispersive system and the assumption that these pulses occur as a
Poisson process (2).
The equations account for the dilution, tidal mixing and dispersion in the system, but not for
tidal translation. The computation, in effect, assumes that the load is on a moving reference system,
always entering the estuary at the same point in the tidal excursion. However, since the computation
is a longterm average, with loads expected to enter at all points in the range, the estimate is believed
to be satisfactory, particularly at stream stations relatively close to the discharge location. At more
remote locations, the estimate of the mean will be reasonable, but the computation would be
expected to underestimate the variability to some extent.
The mean and standard deviation of concentrations of a pollutant in an estuary, caused by
intermittent storm event loadings, is estimated by the following equations:
Mean concentration (MCO) in estuary
MLR
MCO = * exp(a * [l±m]) * 3.04
AEST * vel * m * MTP
Standard deviation of concentrations (SCO) in estuary
SLR2 + MLR2 exp (a)
SCO = SQR ( * K0(b)) * * 3.04
2 * IT * E * MTP AEST
where:
MCO = mean concentration of pollutant in estuary (mg/1)
SCO = standard deviation of estuary concentrations (mg/1)
MLR = mass load from mean storm (pounds)
SLR = standard deviation of storm loads (pounds)
= ML * CVVP
(where CVVP is the coef of variation of storm volumes)
AEST = crosssectional area of estuary (square feet)
m = SQR (1 + (4 * K * E / vel2) ) (dimensionless)
vel = freshwater velocity (miles/day)
= (FLOW(cfs)/AEST(sqft))* 16.36
A33

E = dispersion coefficient (square miles/day)
K = reaction rate of pollutant (I/day)
X = distance along estuary from discharge point (miles)
(+ for downstream;  for upstream direction)
A = mean time between storms (days)
a = (vel * X) / (2 * E)
KQ = modified Bessel function (from tables or Figure A31)
NOTE: K0(b) = KQ(b)
b = (vel * X * m/E) the argument of the Bessel function
3.04 = conversion factor for units
Note that the exponential term in the equation for the mean must be negative, because the term
reflects decay in pollutant concentration with distance from the point of load addition to the estuary.
Since the convention adopted for location X is such that negative values (X) are assigned for
estuary locations upstream of a loading point, and positive values (+X) for downstream directions,
then:
for upstream locations (X), select the equation form that uses the
term (1 + m)
for downstream locations (+X), select the equation form that uses
the term (1  m)
The concentrations calculated are those due only to the storm loads. They will reflect actual
conditions only to the extent that loads from other sources (upstream or downstream in an estuary)
are negligible. The analysis results will be sensitive to the accuracy of the initial estimates of the
input parameters. It is, however, a relatively simple exercise to examine how adjustments in the
input values will modify the resulting receiving water concentrations.
A3.3 RIVERS AND STREAMS
Basic Approach
Figure A32 schematically illustrates a situation wherea series of CSOs are discharged to a
stream. The discharges usually enter the stream at several locations, but are aggregated into an
equivalent discharge flow which enters the system at a single location. This consolidated discharge
is used to estimate the water quality impacts from the sum of the individual CSO overflows. The
equivalent discharge flow (QR) is the sum of the individual discharges, and the equivalent
concentration (CR) is the site mean concentration for the constituent of concern. If the mass
discharge from each individual CSO of concern is known, the average concentration is the total mass
divided by total flow.
A34

10
I I I
Note: K0(0) = 0.0
10
11
~
10
3
10
4
10
,51   I II I I 'I I
0123456789 10
borb
Figure A31. Bessel fu notion
A35

STREAM
CSO DISCHARGE
BLEND
O J
z
in
o
z
o
o
QS
QR
TIME
UQS
4
PROBABILITY
CS
CR
TIME
LnCS
rfi
TIME
PROBABILITY
TIME
LnCR
PROBABILITY
PROBABILITY
Note: 0 Indicates computation step as described in
procedure description.
DILUTION
PROBABILfTY
STREAM CONCENTRATION
LnCO
Strewn
Cone. CO
RECURRENCE
INTERVAL
Figure A32. Schematic outline of probabilistic method for computing impact
of CSO discharge on a stream
A36

The receiving water concentration that results from mixing the CSO discharge with stream
flow is influenced by the upstream flow (QS) and the upstream concentration (CS) during a runoff
event The receiving water concentraton (CO) is the resulting concentration after complete mixing of
the overflow and stream flows, and should be interpreted as the average concentration just
downstream of all of the CSO discharges resulting from the overflow event
As illustrated by Figure A32, the elements that determine the stream concentration (CO) are
all variable and may have any of a range of values during any storm event. The elements that
determine the stream concentration resulting from CSO discharges are:
1. CSO discharge flow (QR)
2. CSO concentration (CR)
3. Stream flow (QS)
4. Stream concentration (CS)
For an individual combined sewer overflow event, it is possible, in principle, to measure a
value for each of these variables. The average stream concentration (CO), during this event, could
be calculated:
(QR*CR) + (QS*CS)
CO =
(QR + QS)
If a dilution factor, DF, is defined as:
QR 1
DF =
QR + QS 1 + D
CO may be defined in terms of DF by:
CO = (DF*CR) + ([1DF]* CS )
The calculated value (CO) for an individual event could be compared to some concentration
limit selected as a target (CT), such as a water quality standard, or to any other stream concentration
which relates water quality to protection or impairment of water use. When CO is less than CT then
water quality is satisfactory and it will be assumed that the individual event would not impair the
beneficial water use. By contrast, if the comparison of CO and CT indicates that during this event
receiving water concentratons of the constituent in question exceed the limit, the relative
contributions of CSO runoff and upstream sources to the violaton could be ascertained as discussed
later in the section.
A37

In principle, this procedure could be repeated for a large number of overflow events. The
set of variable stream concentration values that were produced could then be subjected to standard
statistical analysis procedures. If this were done, the total percentage of the CSO events during
which stream concentration, CO, exceeded target limits, CT, could be determined. The relative
effectiveness of CSO treatment alternatives could be defined in terms of the differences in the
percentage of overflow events that cause the stream concentration (CO) to exceed the selected target
concentration (CT).
This section of the report describes a probabilistic dilution calculation that permits the
statistics and probability distribution of downstream concentrations (CO) to be computed directly
from the probability distributions of the flows and concentrations. More comprehensive and detailed
discussions of the procedure are available in the technical literature (3), and in other EPA documents
(4,5).
The first step in the use of this probabilistic dilution model (PDM) is to develop the statistics
of the concentrations and flows for both the stream and the CSO discharges. These statistics include
both the arithmetric and logarithmic forms of the mean (M), standard deviation (S), and coefficient
of variation (CV). The analysis is simplified here by specifying an upstream concentration of zero
(CS = 0) so that the results reflect only those effects on the receiving water due to the CSO
discharge, thus highlighting the comparative differences resulting from control actions.
Table A31 summarizes the four basic steps in the calculation of the impact of variable CSO
discharges (or other storm runoff) on water quality of a river or stream. Each of the elements is
described and discussed below.
STEP1
The statistical parameters of each of the four basic inputs parameters are computed.
Data on the upstream flow and concentration (QS and CS) may be extracted from available
records (STORET, USGS) or obtained during monitoring efforts, that should cover both wet and
dry periods. Where such records are available, standard statistical analyses will produce the desired
data, and permit the user to check the assumption used in the procedure that these inputs are
adequately approximated by an appropriate lognormal distribution.
CSO flow and concentration characteristics (QR and CR) may be developed using similar
procedures applied to appropriate local monitoring data. However, there will normally be no
longterm record, as in the case for the stream characteristics. For these overflow parameters
particularly, but also for the stream parameters, preliminary estimates of appropriate input values
may be developed using information presented in Appendix 2.
Input parameters may be tabulated as shown below for convenience, using the information
provided in Appendix 1 for computing the transforms.
A38

INPUT
VALUE
PARAMETER
UPSTREAM
flow QS
concentration CS
CSO DISCHARGE
flow QR
concentration CR
ARITHMETIC
MEAN STD COEF
DEV VAR
(M) (S) (CV)
MQS
MCS
MQR
MCR
SQS CVQS
SCS CVCS
SQR CVQR
SCR CVCR
LOGARITHMIC
MEAN STD
DEV
(U) (W)
UQS WQS
UCS WCS
UQR WQR
UCR WCR
TABLE A31. PROCEDURE FOR DEFINING WATER
QUALITY IMPACTS OF CSO DISCHARGES
A. Develop a table summarizing the pertinent statistical properties of the four input parameters used
in the methodology.
Input Parameters
Stream How (QS)
Upstream Cone. (CS)
Runoff Flow (QR)
Runoff Cone. (CR)
Statistics Required
Arithmetic Logarithmic (base e)
mean (M)
standard
deviation (S)
coefficient of
variation (CV)
median (T)
log mean (U)
log standard
deviation (W)
B. Estimate the same statistical properties of the dilution factor (DF).
C. From the values derived in steps A and B, calculate the mean and standard deviation of stream
concentrations.
D. Using the statistical properties of stream concentrations from step C, determine the probability
(percent of CSO events) during which stream concentrations will violate any selected
concentration limit Appendix 1 describes how to do this using either:
a probability plot prepared from the statistics, or
direct calculations using the equations provided.
A39

STEP 2  Dilution Factor CDF)
A dilution factor is defined as:
QR 1
DF =
QR + QS 1 + QS/QR 1 + D
The statistical properties of the dilution factor that are required for the analysis are calculated
from the statistics of the CSO discharge flow and stream flow, specifically their log standard
deviations (WQR and WQS). One additional element in the formula is the correlation coefficient
between the two flows. This could be calculated from the analysis of paired data on stream flow and
rainfall (converted to overflow) derived from analyzing stream gage and rain gage values at
corresponding times.
It is, however, appropriate to assume that there is no significant correlation between CSO
discharge flows and stream flows, especially in the case of a preliminary screening analysis.
Assuming a correlation coefficient of zero provides a conservative estimate for the results, but a
sensitivity analysis (4) indicates the overestimate of stream concentrations to be no more than 10 to
15%, even in cases where flows may be rather highly correlated.
The amount of dilution at any time is a variable quantity and the flow ratio (D = QS/QR) has a
lognormal distribution when both stream flow (QS) and discharge flow (QR) are lognormal. The
log standard deviation of the flow ratio QS/QR is designated as WD. This can be calculated from the
log standard deviations of effluent flow and stream flow. Thus, assuming no crosscorrelation
between stream and effluent flows:
WD = SQR (WQS2 + WQR2 )
The probability distribution of the dilution factor is not truly lognormal, even with lognormal
CSO and stream flows. It has an upper bound of 1 and lower bound of 0, and in the region of the
plot where it approaches these values asymptotically, it deviates appreciably from lognormal
approximation. Deviations at values of DF approaching 0 are of no practical significance to the
analysis being performed since they occur at high dilutions. For smaller streams relative to the size
of the discharge, deviations from a lognormal approximation can be appreciable. They are large
enough to introduce significant error into the calculated recurrence of higher stream concentrations.
The error introduced is almost always conservative; that is, it projects high concentrations to occur
more frequently than they actually would.
A procedure is available for accurately calculating the probability distribution of dilution (DF)
and stream concentraton (CO). This numerical method uses quadratures and would be prohibitively
tedious to perform manually. A computer program listing which can be utilized on a microcomputer
is available and can be obtained from reference 4.
A310

For the purposes of presenting the approach in a form that can be solved manually, and
thereby better illustrate the basic procedure employed, the methodology description which follows in
this section develops a lognormal approximation for the dilution function DF and then proceeds with
the calculations for stream concentration. Whether the lognormal approximation or the quadrature
calculation is used, the subsequent steps in determining the statistical properties of the resulting
stream concentration are the same.
The manual procedure (using the method of moments), estimates the mean and standard
deviation of a lognormal approximation of dilution by first calculating, and then interpolating
between, the 5% and 95% probability values. The value of the dilution factor (DF) for any
probability percentile (a) is defined by:
TQR
(TQR + TQS) * exp(Za*WD)
where the value of Za is taken from any standard normal probability table for the corresponding
value of percentile "a" (see Appendix 1, Table Al1).
For example, where a = 95%; Z95 = 1.65
a = 5%; Z5 = 1.65
a = 50%; Z50 = 0
a = 84.13; Z^ = 1.0 (+1 standard deviation)
The log mean dilution factor (UDF) is estimated by interpolating between the 5% and 95%
values, calcuated above.
UDF = 0.5 * (In [DF95] + In [DF5])
The log standard deviation (WDF) is determined by the following formula, which in effect
determines the slope of the straight line on the logprobability plot, recognizing that Zg4 (1 standard
deviation) = 1.0:
1 (In [DF5]  In [DF95])
WDF = *
Z95 2
From the log mean and log standard deviation of the dilution factor (DF), the arithmetic statistics are
computed using the transform equations presented in Appendix 1.
A311

STEP 3  Stream Concentration
The arithmetic mean of the receiving water contaminant concentration (MCO) downstream of
the discharge, after complete mixing, is computed from the arithmetic mean values of CSO
concentrations (MCR), upstream concentrations (MCS), and the dilution factor (MDF).
MCO = ( MCR * MDF ) + ( MCS * [1  MDF])
The arithmetic standard deviation of stream concentration (SCO) is computed from the arithmetic
means and standard deviations of the same factors.
SCO = SQR ( SDF2*[MCR  MCS]2 + SCR2*[SDF2+MDF2] + SCS2*[SDF2+{ 1MDF}2] )
The coefficient of variation (CVCO) is :
CVCO = SCO/MCO
The arithmetic statistics are now used to derive the log transforms which will be used to
develop the desired information on probability.
log standard deviation WCO = SQR (LN [1 + CVCO2])
log mean UCO = LN [ MCO / SQR (1 + CVCO2) ]
STEP 4  Probability of Specific Stream Concentrations
The probability (or expected frequency) at which a value of CO will occur may be determined
by using the procedures described in Appendix 1. The concentration that will not be exceeded at
some specific frequency (or probability) can be calculated from:
COa = exp ( UCO + Za * WCO)
where
Za = the value of Z from a standard normal table which
corresponds to the selected percentile, a.
To determine the probability of exceedance, replace Za with Z(j . ay
One can also work in the reverse direction; that is, given some target stream concentration
(CT), the probability of CO exceeding that level can be determined by:
ln[CT]  UCO
Z =
WCO
A standard normal table will provide the probability for the calculated value of Z.
A312

Because of the way the standard normal table in Appendix 1 is organized, the probabilities
calculated using this approach represent the fraction of time the target concentration (CT) is not
exceeded. The probability that the concentration will be exceeded is obtained by subtracting the
value obtained from 1.0.
In some cases, the user may prefer to plot results as a probability distribution plot on
logprobability paper as a useful way to summarize and compare the effects of different control
alternatives. This is accomplished by computing concentrations for any two percentiles, plotting the
positions and connecting them with a straight line. In this case it is convenient to select the 50th
percentile (median, where Z=0), and the 84th percentile (+1 standard deviation, where Z=l).
50% concentration = CO = exp (UCO)
84% concentration = CO = exp (UCO + WCO)
Using this procedure, any concentration of interest can be identified and its probability of occurrence
scaled directly from the plot.
The probability relationships described by the above computations represent the distribution of
stream concentrations during "wet" periods, i.e., when CSOs take place. The concentration shown
is the result of both upstream contributions and CSOs.
In many cases, CSO control decisions can be made on the basis of differences in stream
effects during these "wet" overflow periods only. However, from the information available, it is
also possible to calculate the overall percentage of the time that a particular condition will occur,
considering both wet and dry periods. The overall probability that stream concentration CO will
exceed the target concentration CT is defined by:
PrT { CO > CT } = (MDP/MTP) * (Prwet) + (1MDP/MTP) * (
where:
Pry = overall probability of a violation, or that a selected concentration will
occur  (% of time, considering total time including both wet and dry
periods)
Prwet = probability of occurrence during wet periods
MDP/MTP = probability that it is raining.
MDP = average duration of storms
MTP = avg. interval between storms
MDP/MTP*Prwet = % of total time (or probability) that a violation occurs due to runoff events
A313

= probability of violation occurring during dry periods (no runoff)
1  MDP/MTP = probability that it is not raining, i.e., that there is no urban runoff
(1MDP/MTP^Pr^y = % of total time (or probability) that a violation occurs from causes other
than urban runoff
The values for MDP and MTP are provided by the rainfall statistics. The exceedance
probability during "wet" periods is provided by the computations just described. The probability of
exceedance during "dry" periods is computed using the same procedure, but with overflow values
(QR and CR) set at zero, so that only upstream concentration statistics are used in the determination.
A314

APPENDIX 4
ANALYSIS METHODOLOGY
FOR ESTIMATING
PERFORMANCE OF CSO CONTROLS
A4.1 GENERAL
A4.2 RAINFALL
A4.3 FLOWCAPTURE DEVICES
A4.4 FLOWTREATMENT DEVICES
A4.5 VOLUMECAPTURE DEVICES
A4.6 REDUCTION IN OVERFLOWS BY INTERCEPTION PLUS STORAGE
A41

A4.1 GENERAL
Performance estimates for the stormwater control devices addressed in this report are
computed using probabilistic analysis procedures conceived and formulated by DiToro, and
developed by DiToro and Small (6,7). These procedures provide a direct solution for the longterm
average removal of stormwater and pollutants for several different modes of operation of a control
technique. The variable nature of storm runoff is treated by specifying the rainfall and the runoff it
produces in probabilistic terms, established by an appropriate analysis of a longterm precepitation
record for an area.
Longterm average reduction is considered an appropriate measure of performance for several
reasons. It recognizes the highly variable nature of storm runoff which, for a drainage area of fixed
size, will result in higher removal efficiencies during some storm events and lower efficiencies in
others. In addition, characterizing control device performance in this manner provides a direct tiein
with the methods described in the previous section for characterizing the water quality impacts of
CSOs.
The specification of the size or design capacity of a control device is often ambiguous,
because the rate and volume of individual storm runoff events vary so greatly. This is influenced by
regional differences in rainfall patterns, by the size of the drainage area the device serves, and by the
land use distribution of this area, which determines the degree of impervious cover, and the amount
of runoff that any particular storm generates. For the procedures used in this report, variable
rainfall/runoff rates, volumes, durations, and intensities are specified as a mean and coefficient of
variation. Device size or capacity is specified in terms of its ratio to the mean of the appropriate
storm runoff parameter. This permits a convenient generalization of the analyses performed, and
allows results to be readily applied to various combinations of local conditions.
Analysis procedures for computing size/performance relationships for three operational modes
are presented in this section (flowcapture, flowtreatment, and volumecapture). A particular
stormwater control device may incorporate one or more of these modes. Performance estimates for
specific devices, presented in Section 3 of the report, then reduce to selecting and combining the
procedures for the modes that are appropriate, or adapting them to the specific circumstances dictated
by the nature of the device and the way it is incorporated in the collection system.
A4.2 RAINFALL
A longterm record of hourly precipitation data, available from the U.S. Weather Service for
many locations, may be separated into a sequence of discrete storm "events," for each of which
volume, duration, average intensity, and interval since the preceding event can be readily
determined.
The full set of values for each of these parameters may then be subjected to standard statistical
analysis procedures, and the mean and standard deviation determined, as well as the probability
distribution of the set of all values for a parameter. A NURP publication (1) documents a computer
A42

program (SYNOP) that computes these statistics (and other information) from a USWS hourly
precipitation record.
Appendix 2 presents a tabulated summary of storm statistics for gages in various parts of the
country, and information for estimating runoff coefficient This information is provided to assist the
user in estimating appropriate values for local analyses.
Analysis of a number of rainfall records, indicates that the storm parameters that are used in
the analyses described in this report are well represented by a gamma distribution. This distribution
has accordingly been incorporated in the probabilistic analysis procedures described in this section
of the document.
A4.3 FLOWCAPTURE DEVICES
This procedure addresses the condition where a device captures 100% of all applied flows, up
to its capacity QT, and bypasses all flows in excess of this. No consideration is given to what
happens to the "capture" fraction, other than that it no longer discharges with the uncontrolled
fraction. For example, in a combined sewer overflow situation, the amount of the total wet weather
flow that is carried away from the overflow point by a regulator/interceptor, and conveyed to a
downstream sewage treatment plant, can be considered to have been "captured," or removed from
the overflows that would otherwise occur.
The further consideration that must be given to the storm runoff so captured, e.g., removals
by the sewage treatment plant under wet weather conditions, is not addressed here. The technique
simply determines the longterm average reduction (or capture) in stormwater volumes and loads
processed by the device.
For storm flows that are gamma distributed, and a device that captures all inflows up to a rate,
QT, the longterm fraction not captured is given by (3):
.. n . °°
6 1 r nr Tl ' , _ .
J expCr, E )dE
6(r,) J
where:
fFC = fraction not removed by FlowCapture device
r i = 1/CV2 (reciprocal of square of CV of runoff flows)
G(rj) = Gamma function for rj
E = q/QR  QT/QR
q = runoff flow rate for an event
A43

QR = runoff flow rate for mean storm
QT = flow rate  capacity of device
Transformed for numerical integration by Laguerre quadrature, this performance equation
becomes:
r rl  2 n (QT/QR) n
6(r,)  2> W]f(Xj)
where:
f(xj) = Xj (Xj/q + QT/QR) rl'l
xj, wj = abcissas and weights for Laguerre quadrature
This equation has been solved for a range of values for device capacity and variability of
storm runoff flows (CVQR). The treatment capacity of the device (QT) is defined as its ratio to the
flow rate produced by the mean storm (QR). By summarizing performance in terms of this
normalized device treatment capacity (QT/QR), the solution is general and can be applied to any site.
Results are presented in Figure A41 which illustrates the effect of the above variables on
longterm control efficiency of a device with this mode of operation.
A4.4 FLOW  TREATMENT DEVICES
This procedure addresses the performance of a device under variable input flows, when the
treatment or removal efficiency for a pollutant varies with the rate of applied flow. It differs from
the previous case in that all flow directed to the device is processed. This may be the entire runoff
flow, or the fraction of the total that is captured and diverted to the device. An example would be a
sedimentation basin which has less efficient pollutant removals at higher flowthrough rates, than it
does at lower ones.
For variable runoff flows entering a treatment device, that are gamma distributed and
characterized by a mean flow and coefficient of variation (CV), the longterm average removal is:
r+
FRL = fmax
r 
A44

100
0.5 0.75 1.00 1.251.50 175 2.00
0.2 0.3 0.4 0.6 0.8 1.0 2.0 3.0 4.0 6.0 8.0 10.0
TREATMENT CAPACITY
RATIO:
MEAN RUNOFF FLOW

where:
FRL = fraction removed as a longterm average
FRM = fraction removed at mean flow rate
r = 1/CV2 (reciprocal of square of CV of runoff flows)
fmax = maximum removal at very low rates
A graphical solution to this equation is presented by Figure A42 and illustrates the effect on
longterm performance caused by variability of stormwater flows. The plot summarizes the case
where fmax = 1; that is, 100% removal of the pollutant can be achieved at very low flows. As a
practical matter, the relationship shown can be considered to relate to the "removable fraction," or
that portion of the total pollutant concentration that is "removable" by the physical processes
employed by the device. For example, for a sedimentation or filtration device, only the pollutant
fraction that is not soluble would constitute the removable fraction. In addition, the analysis
assumes that removal is an exponential function of flow, thus:
% REMOVED = exp(k*QR)
While not exact, this relationship appears to approximate many removal relationships
adequately, and certainly for the purpose of a planning level analysis.
A4.5 VOLUME  CAPTURE DEVICES
This procedure addresses devices whose effectiveness is a function of the storage volume
provided. The mode of operation it reflects is illustrated by a basin that captures runoff flows until it
is filled and thereafter passes (untreated) all additional stormwater. The captured stormwater runoff
is then removed in some manner from the basin once runoff ceases, in preparation for the next
event.
The analysis does not consider what happens to the captured volume; it simply assumes it to
be removed from the total discharge processed by the device. Offline detention basins for CSOs,
which pump back to the sewer system for processing at the treatment facility provide one example of
this mode of operation.
For storm volumes that are gamma distributed, the fraction not captured, over all storms, is:
A46

N
CO
z
LU
LU
CC
LU
O
LU
CC
LU
O
CC
LU
LU
0>
D.
es
O a
5 S
LU ""
cc
of Runoff Flows CVq /
100
REMOVAL AT MEAN RUNOFF FLOW
MAXIMUM REMOVAL AT VERY LOW FLOW
(expressed as percent)
Figure A42. Average long term performance: flowtreatment device
A47

f., =
oo
G(r,)G(r2)
exp(r,q)
00
A0
rz1
[VI
A + QJ exp (r2A) dA dq
where:
TI = 1 / CVq2 (reciprocal of square of CV of runoff flows)
f2 = 1 / CV

Xj, Xk, Wj, Wk = abcissas and weights for Laguerre integration (from any
handbook of mathematical functions)
n = number of orders used in computation
This integral has been solved for a range of values of V (= VE/VR) and values for coefficient
of variation in a range typically observed for rainfall/runoff. Results are plotted in Figure A43,
which may be used instead of the computation.
From this figure, the average longterm performance of a volume device may be estimated
based on the basin volume relative to the mean storm volume, and the variability of individual event
volumes being processed. However, the relationship is based on "effective" basin volume (VE)
which may be quite different from the physical storage volume of the basin (VB). In the original
CSO application, DiToro and Small present a procedure for approximating the effective volume,
based on an emptying rate ratio (ERR):
MTP * QB
ERR =
VR
where:
MTP = average interval between storms (t)
QB = rate at which basin empties (L3/0
MTP*QB = volume removed between storms, on average (L3)
VR = volume from mean storm
The effect of emptying rate ratio on the fraction of physical basin volume which is effective is
described by Figure A44. As indicated, in cases where the volume that can be removed in the
average interval between storms is small relative to the storm volume which enters on average, much
of the available volume may be occupied with carryover from prior storms each time it rains. In
such cases, effective volume may be considerably smaller than the physical storage volume
provided.
The expression MTP*QB may be thought of as the volume emptied from the basin during the
average interval between storm events. The smaller this quantity is relative to VR, the average
volume entering the basin during storms, the more likely it is that the basin will still contain leftover
runoff when a storm begins, and the smaller the effective volume. When this ratio, ERR, is less
than about 2, the effective volume becomes quite small compared with the physical volume
provided, especially for the larger basins.
The installation of a storage device, in addition to the capture of overflow volumes and loads,
also results in substantial changes in the statistical characteristics of the flows that are not captured
A49

o
D
eo
tc
LU
(E
U
cc
LU
Q.
Coefficient of variation
of Runoff Volume CVVR
0.3 0.4 0.50.6 0.8 1.0
3.0 4.0 5.0 6.0 8.0 10.0
RATIO EFFECTIVE BASIN VOLUME
MEAN RUNOFF VOLUME

5
X
oo
O

and discharge to the receiving water. The smaller storms are completely captured; only the larger
ones escape. As a result, the mean value of the flows that do discharge is greater, and the variability
of these flows lower, than is the case when there is no storage provided. In addition, the number of
storms that produce overflows to the receiving water is reduced, increasing the interval between
discharge events. The ratio of duration (D) to interval between events (DELTA), which defines the
fraction of time that overflows occur, becomes smaller.
Because of the important effect that these parameters have on receiving water impacts, it is
important to adjust them when storage devices are used. Figure A45 shows the adjustment factors
for mean overflow flow rate (MQR), coefficient of variation of flows (CVQR), and the D/DELTA
ratio (MDR/MTR).
The values tabulated and plotted represent the ratio of the value for the parameter with the
storage device in place to the value of the parameter with no control. Multiplying the original value
of the parameter by this factor provides the corrected value for use in the computation of receiving
water impacts.
A4.6 REDUCTION IN OVERFLOWS BY INTERCEPTION PLUS STORAGE
The effect of a combination of interception and storage on the frequency at which overflows
occur can be estimated using Figure A46. It is a function of two ratios previously computed:
QT / MQR  This is the ratio of the excess interceptor capacity (QT) to the
runoff flow rate produced by the mean storm (MQR).
VE / MVR  This is the ratio of the effective basin volume (VE) and the
mean storm runoff volume (MVR).
For example, with no storage and no excess interceptor capacity, the probability of an
overflow is 1.0. An overflow will occur every time it rains. For a collection system with an
interceptor/regulator that provides an excess flow capacity that is 25% of the rate produced by the
mean storm, QT/MQR = 0.25, the probability of an overflow is reduced to 0.8. That is, 20% of
the storm events are captured and produce no overflow.
When a storage device is added to the above system and has an effective capacity equal to the
mean runoff volume, VE / MVR = 1.0, the probability of an overflow event is reduced to about
0.22. In this case, only 22% of the storm events result in an overflow to the receiving water.
The vertical axis represents the probability that an overflow will occur, given the condition
that a storm occurs. The overall probability of an overflow is the product of the probability that it is
raining, and the probability that a storm will produce an overflow. For an area with the above
controls, and precipitation statistics that provide a D/DELTA ratio (MDP/MTP) of 0.07, the
probability of an overflow (fraction of total time overflows occur) is:
PRT = 0.07 * 0.22 = 0.015
A412

VE /MVR
001
002
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
020
0.30
0.40
0.30
060
070
080
0.90
1 00
EFFECT OF A STORAGE DEVICE ON
OVERFLOW CHARACTERISTICS
RATIO OF TREATED TO UNTREATED PARAMETER VALUE
D/DELTA MTJR CVQR
300
4.00
500
600
7.00
8.00
9.00
0.955
0.924
0.897
0.874
0.852
0833
0815
0.798
0.782
0.767
0.647
0.563
0.497
0.444
0.401
0.363
0.332
03O4
0280
0.140
0.080
0.050
0.033
0.022
0015
0.011
0.008
1.037
1.062
1.083
1.102
1.119
1.136
1.151
1.065
1.179
1.192
1.303
1.392
1.469
1.538
1.601
1 659
1.714
1.765
1.814
2.214
2.525
2.788
3.021
3.232
3.426
3.607
3.777
0.969
0949
0934
0921
0.909
OB99
0.889
0.880
0872
0865
0808
0.769
0739
0.713
0695
0.678
0.663
0649
0.637
0.558
0513
0.482
0.458
0440
0425
0.412
0400
in
O
4
3
2
<
OC
Mean Flow, MQR
I I
4 5
VE/MVR
I
6
I
8
I
9
UJ
O
OC
I
UJ
O
oc
I
<
oc
Coef. of Variation of
Flow Rates. CVQR
Ratio. Duration/Interval between Overflows
0.4 
0.2 
VE/MVR
Figure A45. Effect of a storage device on overflow characteristics


Overflows will occur on the average of 1.5% of the time, or about 135 hours per year. If the
average duration of a storm is 6 hours, and it is assumed that the control system has a minor effect
on average duration compared with its effect on interval between overflow events, then the system
will reduce overflows to about:
135/6 = 22 events per year
A415
