TITLE: Technical Guidance Manual for Performing Wasteload Allocations, 

Book VII: Permit Averaging Periods 

EPA DOCUMENT NUMBER: EPA-440/4-87.002 DATE: September 1984

ABSTRACT 

As part of ongoing efforts to keep EPA’s technical guidance readily accessible to water 

quality practitioners, selected publications on Water Quality Modeling and TMDL Guidance 

available at http://www.epa.gov/waterscience/pc/watqual.html have been enhanced for 

easier access. 

This document is part of a series of manuals that provides technical information related to 

the preparation of technically sound wasteload allocations (WLAs) that ensure that 

acceptable water quality conditions are achieved to support designated beneficial uses. 

The document presents a rational method for selecting the level of treatment required 

based on water quality considerations, and for incorporation of the water quality-based 

treatment requirements as permit limits. Conventional procedures for establishing a point 

source’s effluent limits using a WLA analysis do not quantify the degree to which a given 

limit protects against exceedances of acute toxicity water quality criteria. Also, the permit 

averaging period can have a substantial influence on the degree and cost of treatment 

required and on receiving water quality. 

The method presented in this document uses a probabilistic dilution model to evaluate the 

extent and frequency of acute criteria violations in the receiving water as computed with 

effluent concentrations based on daily, weekly, and monthly average permits. The model 

incorporates stream variability to develop probability distributions of daily stream 

concentrations for each permit limit, which can then be compared to water quality goals 

also expressed in terms of daily concentration frequencies. 

In addition to a detailed description of the methodology, the document presents an 

annotated example of the method performed first as a hand calculation and then using a 

computer program included in the manual. Several representatives applications are 

provided along with a discussion of suggested uses of the model. Appendices provide a 

review of log-normal distributions, a discussion of technical issues and assumptions, a 

listing of typical low flow characteristics for U.S. streams, and computer code for the 

model. 

KEYWORDS: Wasteload Allocations, Averaging Periods, Permit Limits, Lakes, 

Reservoirs, Water Quality Criteria, Water Quality Modeling

Click here for 

DISCLAIMER & 

NOTE TO BROWSER 

Document starts on next page
Technical Guidance Manual for 

Performing Waste Load Allocations 

Book VII: Permit Averaging Periods 

September 1984 

Final report 

for 

Office of Water Regulations and Standards 

Monitoring and Data Support Division, 

Monitoring Branch 

U.S. Environmental Protection Agency 

401 M Street, S.W. Washington, D.C. 20460
UNITED STATES ENVIRONMENTAL PROTECTION AGENCY 

WASHINGTON, D C 20460 

MEMORANDUM

SUBJECT: Technical Guidance Manual for Performing Waste Load 

Allocations Book VII, Permit Averaging Period 

TO: Regional Water Management Division Directors 

Regional Environmental Services Division Directors 

Regional wasteload Allocation Coordinators 

Attached, for national use, is the final version of the 

Technical Guidance Manual for Performing Waste Load Allocations, 

Book VII, Permit Averaging Periods. We are sending extra copies of 

this manual to the Regional Wasteload Allocation Coordinators for 

distribution to the States to use in conducting waste load 

allocations. 

Modifications to the February 1984 draft include: 

o The method to calculate the Reductions Factor in 

Chapter 2 has been elaborated to include the use of 95% 

cut-offs for frequency of permit violations. 

o The example calculation in Chapter 3 has been expanded. 

Step 7 has been added to the step-procedure to show how 

permit limits can be specified using 95% cut-offs for 

frequency of permit violations. 

o The document recommends that advanced treatment 

facilities should be built to meet the long-term 

average and the selected effluent variability. 

o A flow diagram and an IBM PC-compatible program have 

been added to Appendix D. 

If you have any questions or comments or desire additional 

information please contact Tim S. Stuart, Chief, Monitoring 

Branch, Monitoring and Data Support Division (WH-553) on (FTS) 

382-7074. 

Edwin L. Johnson, Director 

Office of Water Regulations 

and Standards (WH-551) 

Attachment 
TECHNICAL GUIDANCE MANUAL FOR 

PERFORMING WASTE LOAD ALLOCATIONS 

Book VII 

Permit Averaging Periods 

Contract Number 58-03-3131-WA9 

Project Officer 

Hiranmay Biswas 

Office of Water Regulations and Standards 

Monitoring and Data Support Division 

Monitoring Branch 

U.S. Environmental Protection Agency 

401 M Street, S.W. Washington, D.C. 20460 

September 1984 
FOREWORD 

This guidance document is a product of several years of 

research on many complex water quality issues. Although much 

progress has been made, some issues still remain. User 

participation will be needed to develop answers to these 

unresolved issues and will be key to future revisions of this 

document. 

Selection of permit averaging periods, as presented in this 

manual, is based on an assumed exceedance frequency of an acute 

violation in the stream no more than 1 day in 10 years. The EPA is 

currently considering the issue of allowable duration and 

frequency of exposure to acute as well as chronic toxicity. Based 

on this study, the choice of duration and frequency used in this 

document as examples may have to be changed. 

i
CONTENTS 

Chapter Page

FOREWORD ............................................................... i

LIST OF TABLES ........................................................ iv

LIST OF FIGURES ........................................................ v

LIST OF ABBREVIATIONS AND SYMBOLS .................................... vii

ACKNOWLEDGMENTS ....................................................... ix

EXECUTIVE SUMMARY ...................................................... 1

1 INTRODUCTION ..................................................... 1-1

1.1 Background ................................................... 1-1

1.2 Objectives ................................................... 1-2

1.3 Approach ..................................................... 1-3

1.4 Organization ................................................. 1-5

2 METHODOLOGY ...................................................... 2-1

2.1 Description of Probabilistic Dilution Model .................. 2-1

2.2 Choice of the Permit Averaging Period ........................ 2-9

3 EXAMPLE COMPUTATION .............................................. 3-1

3.1 Hypothetical Site-Specific Conditions ........................ 3-2

3.2 Example Computation - Hand Calculation ....................... 3-8

3.3 Example Computation - Computer Program ...................... 3-28

4 RANGE OF EXPECTED VALUES FOR STREAMS IN U.S. ..................... 4-1

4.1 Analysis for Conservative Substances ......................... 4-1

4.2 Use As a Screening Tool ..................................... 4-10

4.3 Preliminary Analysis for Dissolved Oxygen ................... 4-12

4.4 Analysis for Conservative Substances 1n Effluent-Dominated 

Streams ......................................................... 4-23

5 USES AND LIMITATIONS ............................................. 5-1

6 REFERENCES ....................................................... 6-1

ii
CONTENTS (Continued) 

Appendix Page

APPENDIX A STATISTICAL PROPERTIES OF LOG-NORMAL DISTRIBUTIONS ........ A-1

A-1. General Considerations ...................................... A-1

A-2. Probability Distributions ................................... A-3

A-3. Relationship Between Distributions .......................... A-6

A.4. Properties of Log-Normal Distributions ...................... A-6

A-5. Standard Normal Tables ..................................... A-10

A-6. References ................................................. A-12

APPENDIX B FIELD VALIDATION OF LOG-NORMAL DISTRIBUTION AND RELATED 

ASSUMPTIONS .......................................................... B-1

B-1. Use of the Log-Normal Distribution .......................... B-1

B-2. Verification of the Probabilistic Dilution Model ............ B-3

B-3. Appropriateness of Assumptions .............................. B-8

B-4. References ................................................. B-11

APPENDIX C CHARACTERISTIC VALUES FOR INPUT PARAMETERS ............... C-12

C-1. Treatment Plant Effluent Flows .............................. C-1

C-2. Treatment Plant Effluent Concentrations ..................... C-1

C-3. Stream Flow ................................................. C-3

C.4. References .................................................. C-6

APPENDIX D COMPUTER PROGRAM FOR THE PROBABILISTIC DILUTION MODEL – POINT 

SOURCE (PDM-PS) ..................................................... D-15

D-1. Formulation and Normalization ............................... D-1

D-2. Description of Program Use .................................. D-5

iii
LIST OF TABLES 

Table Page

2-1 Reduction factors for various coefficients of variation ........ 2-13 

4-1 Averaging period selection matrix for conservative substances: 

effluent dilution ratio - 7Q10/QE = 50 .............................. 4-5 

4-2 Averaging period selection for conservative substances: effluent 

dilution ratio - 7Q10/QE = b ........................................ 4-6 

4-3 Averaging period selection matrix for conservative substances: 

effluent dilution ratio - 7Q10/QE = 3 ............................... 4-7 

4-4 Averaging period selection matrix for conservative substances: 

effluent dilution ratio - 7Q10/QE = 1 ............................... 4-9 

4-5 Conditional moments for the low flow subpopulation (a = 16.75).. 4-17 

4-6 Permit averaging period selection matrix for MOD/DO: of fluent 

dilution ratio - 7Q10/OE = 5 ........................................ 4-20 

4-7 Permit averaging period selection matrix for BOD/DO: effluent 

dilution ratio - 7Q10/QE = 3 ....................................... 4-21 

4-8 Permit averaging period selection matrix for BOD/DO: effluent 

dilution ratio - 7Q10/QE = 1 ....................................... 4-22

4-9 Averaging period selection matrix for a fluent-dominated 

streams ............................................................. 4-26 

A-1 Probabilities for the standard normal distribution ............. A-11 

B-1 Comparison of observed and computed downstream 

concentrations(2

) .................................................... B-6 

B-2 Approximate overestimation of 10 year return period stream 

concentration by ignoring serial correlation ........................ B-10 

C-1 - Coefficient of variation of daily effluent flows, νQE........... C-2 

C-2 - Summary of secondary treatment plant performance - median 

coefficients of variation, νCE (from reference 1)..................... C-4 

C-3 - Effluent concentration variability for trickling filters (from 

reference 4) ........................................................ .C-6 

C-4 - Summary of stream flow characteristics ......................... C-9

iv
LIST OF FIGURES 

Figure Page

1-1 Schematic outline of probabilistic method ........................ 1-4 

2-1 Simple dilution model ............................................ 2-2 

2-2 Illustration of analysis results: stream concentration versus return 

period for three permit averaging periods ........................... 2-16 

3-1 Step procedure to select optimal permit averaging period ......... 3-1 

3-2 Sample stream concentration versus probability plot for 30-day 

averaging period .................................................... 3-15 

3-3 Sample stream concentration versus mean recurrence interval for 30-

day averaging period ................................................ 3-17 

3-4 Concentration versus probability plot for 1-, 7-, and 30-day 

averaging periods ................................................... 3-19 

3-5 Concentration versus mean recurrence interval plot for 1-, 7-, and 

30-day averaging periods ............................................ 3-19 

3-6 Concentration versus probability for PDM-PS computation ......... 3-33 

3-7 Concentration versus mean recurrence Interval for POM PS 

computation ......................................................... 3-33 

4-1 Effect of permit averaging period on stream concentrations for 

conservative substances: general analysis ............................ 4-4 

4-2 Effect of permit averaging period on stream, concentrations for 

BOD/DO .............................................................. 4-19 

4-3 Effect of permit averaging period on stream concentrations for 

conservative substances in effluent-dominated stream ................ 4-25 

A-1 Probability distribution ......................................... A-5 

A-2 Effect of coefficient of variation on frequency distribution ..... A-7 

A-3 Pertinent relationships for log-normal distributor ............... A-8 

A-4 Cumulative log-normal distribution ............................... A-9 

B-1: Evaluations of log-normal distribution for stream flows ......... B-2 

v
LIST OF FIGURES (continued) 

Figure Page

B-2 Probability distribution of treatment plant effluent concentrations – 

conventional pollutants .............................................. B-4 

B-3 Probability distribution of treatment plant effluent concentrations 

– heavy metals ....................................................... B-5 

C-1 - Typical low flow characteristics of U.S. streams ............... C-8 

D-1 CRT - displays ................................................... D-9 

D-2 - Example of printed output ..................................... D-10 

D-3 - Flow chart for PDM-PS program ................................. D-11 

D-4 - PDM-PS program listing - HP-85 compatible ..................... D-12 

D-5 - PDM-PS program listing - IBM-PC and MS-DOS compatible ......... D-15

vi
LIST OF ABBREVIATIONS AND SYMBOLS 

BASIC Computer language 

BOD Biochemical oxygen demand

BOD5 The amount of dissolved oxygen consumed in five days by 

biological oxidation of organic matter 

CE Treatment plant effluent concentration 

CFS Cubic feet per second, unit of flow 

CL Concentration equal to a water quality standard

CO Downstream concentration, after complete mixing

CRT Cathode ray tube 

Csat Saturation concentration of dissolved oxygen 

CS Stream concentration upstream of discharge 

D Flow ratio, equal to QS/QE

Dc Critical (or maximum) dissolved oxygen deficit

DO Dissolved oxygen 

EL Effluent limit. A maximum effluent concentration 

determined from a waste load allocation analysis, and 

specified by an NPDES permit 

FAV Final acute value 

FCV Final chronic value 

K Stream purification factor 

Ka Stream reaeration rate constant 

Kd BOD oxidation rate constant 

MRI Mean recurrence interval, expressed in years 

NPDES National pollutant discharge elimination system 

P Pollutant 

vii
LIST OF ABBREVIATIONS AND SYMBOLS (Continued) 

PDM-PS Probabilistic dilution model: point source 

POTW Publicly-owned treatment works 

Pr Probability 

7Q10 The lowest 7-day average stream flow with a recurrence 

interval of 10 years 

QE Treatment plant effluent flow 

QS Stream flow 

QT Total downstream flow, equal to QS + QE 

R Reduction factor, equal to the ratio of the mean CE for 

which a treatment plant is designed to the EL 

TSS Total suspended solids

WLA Waste load allocation 

WQ Water quality 

α Exceedence probability 

β Dimensionless unit of concentration equal to CO/CL 

μx Mean value of x 

σ Dilution factor 

σx Standard deviation of x 

νx Coefficient of variation of x 

Za Value of statistical parameter Z for a probability of a

viii
ACKNOWLEDGMENTS 

The contents of this section have been removed to comply with 

current EPA practice.

ix
EXECUTIVE SUMMARY 

Background

The conventional approach to developing Waste Load 

Allocations (WLAs) is based on a steady state analysis of stream 

conditions, using a design stream flow (usually the 7Q10) and a 

receiving water concentration (usually a water quality standard 

based on chronic criteria) for the pollutant to be allocated. An 

effluent concentration limit is computed for these conditions, and 

is used to establish the NPDES permit conditions. 

The water quality based permit conditions apply, in addition 

to technology based requirements (e.g., BAT, BCT, and secondary 

treatment). This effluent requirement may be incorporated into the 

permit as the daily maximum limit, the average limit over a week 

(for POTWs) or the average limit over a month (for industrial as 

well as municipal source)1. Typical practice for toxic pollutants 

is to incorporate the wasteload allocation result as the daily 

maximum permit limit. This document provides an innovative 

approach to determining which types of permit limits (daily 

maximum, weekly, or monthly average) should be specified for the 

steady-state model output based on the frequency of acute criteria 

violations. 

Approach

The method used to evaluate the effect of permit averaging 

periods is based on a probabilistic dilution model (PDM) in which 

it is assumed that the stream flows, effluent flows and 

concentration are log-normally distributed 

 1 See 40 C&R 122.45 (d)

1
and uncorrelated. The log-normal distribution is known to be 

representative of effluent behavior and to almost always underestimate the lowest stream flows somewhat. Thus, the analysis is 

generally conservative (overprotective) to some extent. However, a 

verification of the probabilistic dilution model indicates that, 

for the cases tested, it correctly estimates observed downstream 

concentration probability distributions to within the confidence 

limits of the data. 

The method applied in using this model to evaluate permit 

averaging period choices is based on the following observation. If 

chronic criteria and 7-day, 10-year low flow, or any other statespecified low flow, are used on the WLA analysis to develop the 

maximum effluent concentration, the use of monthly or weeklypermit limits for specifying this effluent requirement presents 

the possibility that simultaneous occurrences of high effluent 

concentrations and low stream flows may result in stream 

concentrations which exceed the acute criteria for a pollutant 

without violating maximum average discharge permit conditions. 

The analysis consists of computing the level of treatment 

required for the three averaging period options for specifying the 

WLA results as permit limits. The analysis computes the frequency 

at which acute stream criteria concentrations are violated under 

each of the permit averaging period options, taking into account 

the likely range of stream and effluent variability. Computation 

result are normalized so that summary results can be applied to a 

variety of pollutants based on their ratio of acute-to-chronic 

criteria concentrations. 

2
Uses

The primary use of this methodology will be specifying the 

required level of treatment and deriving permit limits based on 

water quality requirements. Care must be taken in the assumptions 

related to the permit limits and assumptions used in the 

methodology. For example, throughout this document, reference is 

made to 7-day and 30-day averages. These averages are equivalent 

to weekly and monthly permit limits where the assumption can be 

made that the monitoring data is adequate (i.e., that the data 

collected in a month adequately reflects the 30-day average). 

Where this requirement is not valid, alternative limits may be 

calculated which incorporate monitoring frequency, or monitoring 

frequency may be adjusted so that these conditions are met. 

In addition to the usefulness of this method for permit 

writers in selecting the averaging period for discharge permits, 

the method has been used to calculate suitable averaging periods 

for the range of stream and effluent conditions typified in the 

U.S. The results have been summarized in convenient graphic and 

tabular displays, and can be used as a "screening tool" that 

provides a guide for water quality decisions. These summaries 

show, for instance, that for toxic pollutants with acute-tochronic ratios of 10 or greater, 30-day permit averages will 

virtually always meet the criteria that have been adopted; that 

is, that acute criteria violations in the stream will recur with a 

frequency that averages less than 1 day in 10 years1

. 

 1 The EPA is presently considering the issue of allowable duration 

and frequency of exposure to toxicity. Based upon this work, 

duration and frequencies used as the decision criteria may change. 

This guidance does not recommend any particular minimum acceptable 

duration or frequency.

3
For pollutants with acute-to-chronic ratios of between 5 and 

10, monthly permit averages will be appropriate in most cases, 

although there will be some site-specific conditions that would 

call for the use of weekly averages. For pollutants with acute-tochronic ratios of less than 5, site specific conditions must be 

considered, and no general rule is possible. In these cases, sitespecific analyses of the effects of different permit averaging 

periods can be performed using the methods outlined in the text. 

Limitations

Several technical refinements to the probabilistic model 

would be required to more accurately reflect the deviation of 

lowest stream flow from log-normality, and to account for serial 

and cross-correlation of stream flows and effluent loads. For 

coupled reactions, such as BOD/00, the procedures would have to be 

extended to provider seasonal approach and results should be 

verified against field data. The analysis method would have to be 

extended to incorporate the variability of secondary water quality 

parameters such as pH, hardness and temperature, since these 

affect the toxicity of a number of pollutants. Finally, the 

chronic exposure event, as defined by the state design flow 

conditions, was used throughout the document to estimate the 

maximum effluent concentration. Further analyses to determine the 

possible underprotection or overprotection of chronic criteria 

based on the state design flow1

 were not done. 

 1

 The EPA is considering studying the Impact of uncertainties 

Involving the low flow estimating techniques on the selection of 

stream design flow. 

 

4
CHAPTER 1 

INTRODUCTION 

1.1 Background

The conventional procedure for establishing a point source 

effluent limit using a waste load allocation (WLA) analysis begins 

by specifying a target concentration of the pollutant in the 

stream, such as a state water Quality standard based on chronic 

criteria. This stream concentration is converted to a maximum 

effluent concentration using a mass balance calculation for 

conservative substances) or a steady-state analysis (for reactive 

substances). The inputs to these analyses are a design stream flow 

(representing low stream-flow conditions)1

 and a measure of the 

effluent flow, typically the mean effluent flow. Although this 

technique is presumed to provide adequate protection for receiving 

water quality, it fails to account for random and other 

fluctuations in the flow rate and concentration that naturally 

occur in both the stream and effluent. Thus, the degree to which a 

given limit protects against exceedances of acutely toxic 

concentrations is not quantified. 

Effluent permit limitations are currently specified as 

maximum concentrations for one day or averaged over a week or 

month. The number of observations from which the average is 

computed depends on the frequency of 

 1 The design stream flow most commonly used is the 7Q10 flow, which 

represents the low-flow condition with a recurrence interval of 10 

years based on a 7-day averaging period. Other flows, such as the 

30Q10 or 30Q5 are occasionally used as the design stream flow. 

Wherever the use of stream design flow is called for, these or 

other stream design flows can be substituted throughout this 

document

1-1 
monitoring. Although there is no generally accepted rational 

basis for selecting permit averaging periods, the effluent 

requirement derived from a WLA is typically expressed as a monthly 

average for conventional pollutants and as the daily maximum for 

toxic pollutants. A set of conversion factors is then used to 

convert these concentrations to other averaging periods. In this 

document the maximum daily, weekly, and monthly permit limits are 

referred to as 1-day, 7-day, and 30-day permit levels, 

respectively. 

The permit limit used to incorporate a WLA effluent 

requirement can have a substantial influence on the degree (and 

cost) of treatment required and on the quality of the receiving 

water. It is clear that a permit limit imposed as a daily maximum 

requirement is more restrictive than when the same permit limit is 

used as a 30-day average requirement, since in the latter case the 

effluent concentration can fluctuate above the effluent limit for 

days at a time and still meet the 30-day average requirement. Such 

fluctuations may or may not be significant in terms of receiving 

water quality. The appropriate choice of the averaging period, 

then, is one which ensures acceptable receiving water quality 

without imposing unnecessarily restrictive treatment requirements. 

1.2 Objectives

This guidance document is Intended to achieve the following: 

(1) Present a rational method for selecting the level of 

treatment required based on considerations of water 

quality; 

(2) Present a rational method to incorporate the water quality 

based treatment requirements as permit limits; 

1-2 
(3) Provide specific information, including detailed examples, 

so that the method can be applied to site-specific cases; 

(4) Use the method to provide an overall analysis of a broad 

range of conditions likely to be encountered, so as to 

provide a screening tool for the rapid assessment of a wide 

variety of cases; 

(5) Discuss the uses and limitations of the method. 

1.3 Approach

The basis of the method is an evaluation of the extent and 

frequency of acute criteria violations to be expected in the 

stream receiving the Discharge as a result of imposing the 

effluent concentration, computed from a steady state wasteload 

allocation, as a daily, weekly, or monthly average permit. A 

probabilistic framework is adopted to account for the inherent 

variability of flows and concentrations. Acute criteria violations 

are assumed to be associated with random simultaneous occurrences 

of high effluent loadings and low stream flows.1

 The analysis is 

based on an examination of the probability distributions involved 

and how they combine to influence the concentration downstream. 

The probabilistic dilution model provides the analysis framework. 

The probabilistic dilution model is summarized in Figure 1-1. 

The inputs to the model include the flow and concentration 

histories (or projections) of both the effluent and the receiving 

stream. Each of these is 

 1 While it is apparent that effluent loadings and stream flows 

experience both random and nonrandom (e.g., seasonal) variations, 

the problem is analyzed here in purely random terms to limit the 

complexity of the analysis.

1-3 
1-4 

Figure 1.1 – Schematic outline of probabilistic 
expressed as a probability distribution; that is, in terms of the 

probability that a given value is exceeded. Next, the effluent and 

stream flows are combined to yield the probability distribution of 

the dilution factor; then the dilution factor and concentrations 

are combined to provide the probability distribution for the 

resulting stream concentration. The stream concentration 

probability distribution is then converted to a plot showing the 

recurrence interval to be associated with each stream 

concentration so that the frequency of occurrence of a given 

(high) stream concentration can be compared to water quality 

objectives. 

The probabilistic dilution model is used to guide the choice 

of the permit averaging period as follows. Given an effluent 

requirement from a WLA analysis, the mean effluent required to 

meet that WLA requirement is calculated for each of the three 

averaging periods, based on an assumed allowable frequency of 

effluent limit violation. This provides three levels of treatment 

for the plant in question. Each mean effluent concentration is 

then used, together with the parameters that characterize the 

stream variability, in the probabilistic dilution model. The 

result is a probability distribution of resulting stream 

concentration for each of the three treatment plant options, which 

can be compared to daily concentration/frequency water quality 

goals. The use of daily concentration frequencies allows the use 

of acute criteria in establishing water quality goals. 

1.4 Organization

This document is organized as follows. Chapter 2 provides a 

detailed description of the methodology for finding an optimum 

averaging 

1-5 
period based on a probabilistic dilution method. Chapter 3 

presents an annotated example of the method performed first as a 

hand calculation and then using the computer program provided in 

Appendix D. Chapter 4 uses the model in several representative 

applications, and Chapter 5 discusses the uses of the method. 

Several appendices to this document provide detailed additional 

material, including a review of relationships for log-normal 

distributions (Appendix A) and a discussion of technical issues 

and assumptions employed in the analysis (Appendix B). 

1-6 
2-1 

CHAPTER 2 

METHOD OF ANALYSIS 

This chapter lays the theoretical groundwork for the 

application of the probabilistic dilution model to the problem of 

permit averaging period selection. This discussion is presented in 

two parts. Section 2.1 describes the probabilistic dilution model. 

Section 2.2 develops the method whereby the probabilistic dilution 

model is employed to predict the water quality effects of the 

selection of different averaging periods. 

2.1 Description of the Probabilistic Dilution Model

The probabilistic dilution model is based on a simple stream 

dilution calculation. The complexity of the model arises from the 

probabi1istic framework that is superimposed upon the dilution 

equation. This section is intended to provide a description of the 

derivation of the model, and to reduce it to a manageable set of 

equations. While a strict mathematical derivation of the model is 

available [I], a rigorous treatment is considered beyond the scope 

of this manual. 

Figure 2-1 illustrates a treatment plant discharge entering a 

stream. The effluent discharge flow (QE), having a concentration 

(CE) of the pollutant of interest, mixes with the stream flow 

(QS), which may have a background concentration (CS). The 

receiving water concentration (CO) is the concentration that 

results after complete mixing of the effluent and stream flows. It 

is the cross-sectional average concentration downstream of the 

discharge, and is given by: 
Figure 2-1 - Simple dilution model 

2-2 
2-3 

 CO = (QE•CE) + (QS•CS) (2-1) 

 QE + QS 

If the dilution factor, φ, is defined as: 

φ = QE = 1 (2-2) 

 QE + QS 1 + D 

The calculated value of CO for a given day could be compared 

to a water quality standard (CL) or to any other stream 

concentration which relates water qua1ity to water use. This 

procedure could be repeated for a large number of days and the 

resulting set of values for CO could be subjected to standard 

statistical analysis procedures to obtain its probability 

distribution. If this were done, the total percentage of days on 

which the downstream concentration CO exceeded CL could be 

determined. 

The ability to perform this direct computation depends upon 

the availability of long time series of upstream and treatment 

plant flows and concentrations of each pollutant of interest. Such 

long data records are usually only available for stream flow, but 

estimates based on more limited data sets may be available for the 

other elements. An important objective of any modeling framework 

is to cast the problem into a manageable form while at the same 

time preserving its essential features. Therefore, it is necessary 

to characterize the fluctuating behavior of the upstream and 
 

effluent flows and concentrations in a concise and realistic 

fashion. 

The probabilistic dilution calculation procedure used in this 

report permits the probability distribution of downstream 

concentrations (CO) to be computed directly from the probability 

distributions of the flows and concentrations. 

The first step in the use of the probabilistic dilution model 

is to develop the statistics of the concentration and flow of both 

the stream and effluent.1

 These statistics include both the 

arithmetic and logarithmic forms of the mean (μ), standard 

deviation (σ), and coefficient of variation (ν). 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 effluent discharge, thus highlighting 

the comparative differences resulting from choice of permit 

averaging period. 

The amount of dilution at any time is a variable quantity and 

the dilution ratio (D=QS/QE) has a log-normal distribution when 

both stream flow (QS) and effluent flow (QE) are log-normal. The 

log standard deviation of the flow ratio QS/QE is designated as 

σlnD. This can be calculated from the log standard deviations of 

stream flow and effluent flow, assuming no cross-correlation 

between stream and effluent flows. 

 σlnD = √ σ2

lnQS + σ2lnQE (2-4)

1 Standard statistical procedures are used to compute the mean and 

standard deviation using the log transforms of the basic data. 

Conversion to the other statistical expressions used in the 

analysis is described in Appendix A.

2-4 
The probability distribution of the dilution factor, φ = 

1/(1+D) is not truly log-normal, even with log-normal runoff and 

stream flows. It has an upper bound of 1 and a lower bound of O, 

and where it approaches these values asymptotically, it deviates 

appreciably from a log-normal approximation. Deviations at values 

of approaching 0 are of no practical significance to the 

calculations being performed since they occur at high dilutions. 

For smaller streams relative to the size of the discharge, 

deviations from a log-normal approximation can be appreciable. 

They are large enough to introduce significant error into the 

calculated recurrence interval of higher stream concentrations. 

The error introduced is almost always conservative; that is, it 

projects high concentrations to recur more frequently than they 

actually would. The appropriateness of this assumption is 

discussed in detail in Appendix B. 

A procedure is provided in this report for accurately 

calculating the probability distribution of the dilution factor 

(φ ) and stream concentration (CO). This numerical method uses 

quadratures and would be prohibitively tedious to perform 

manually. It has, therefore, been provided in the form of a 

computer program which can be utilized on a microcomputer 

(Appendix D). 

For purposes of presenting the approach in a form which can 

be solved manually, and thereby better Illustrate the basic 

procedure employed, the methodology description which follows in 

this section develops a log-normal approximation for the dilution 

function φ and then proceeds with the calculations for stream 

concentration. Whether the log-normal approximation or the 

quadrature calculation is used, the subsequent steps in 

determining the appropriate averaging period are the same. 

2-5 
The manual procedure (moments method) estimates the mean 

and standard deviation of a log-normal approximation of dilution 

by first calculating, and then interpolating, between the 5% and 

95% probability values. The value of the dilution factor (φ ) for 

any probability percentile (a) is given by: 

 φa = QE ~ (2-5) 

 (QE + QS) exp (Z ~ ~ aσlnD) 

where the value of Zα is taken from a standard normal probability

table for the corresponding value of a (see Appendix A). 

For example, where 

a = 95%; Z95 = 1.65 

a = 5%; Z5 = -1.65 

a = 50%; Z50 = 0 

a = 84.13%; Z84 = 1.0 

The log mean dilution factor is estimated by interpolating 

between the 5% and 95% values, calculated above. 

 μlnφ = 1⁄2 [ln (φ95) + ln (φ5)] (2-6a) 

The log standard deviation is determined by tine following 

formula which, in effect, determines the slope of the straight 

line on the log-probability plot: 

 •ln• = [ln (•5) – ln (•95)] (2-6b) 

2 

From the log mean and log standard deviation of the dilution 

factor (φ ), the arithmetic statistics are computed using 

2-6 
μφ = exp (μlnφ + 1⁄2 σ2lnφ) 

σφ = μφ[exp(σ2lnφ)-1]1⁄2 (2-7) 

The arithmetic mean of the receiving water contaminant 

concentration (CO) downstream of the discharge after complete 

mixing, then, can be found by: 

μCO = [μCE (μφ)] + [μCS (1- μφ)] (2-8) 

The arithmetic standard deviation of stream concentration is: 

σCO = √ σφ

2

 (μCE - μCS)2 + σCE

2

 (σφ

2

 + μφ

2

) + σCS

2

 (σφ

2

 +[1-μφ]2) (2-9) 

The coefficient of variation of stream concentration (CO) is: 

ν = σCO/μCO (2-10) 

The arithmetic statistics used-to derive the log statistics 

will be used to develop the desired probability of exceedence. 

log standard deviation = σlnCO = √ ln(1+νCO

2

) (2-11) 

log mean = μlnCO = ln √1 + νμCO (2-12) 

CO

2 ( ) 

The probability (or expected frequency) at which a value of 

CO will occur is determined by constructing a probability 

distribution plot on log-probability paper. This is accomplished 

by computing the 50th percentile and 84th percentile 

concentrations and connecting them with a straight line: 

2-7 
50% concentration = CO = exp (μlnCO) 

84% concentration = exp (μlnCO + σlnCO) 

Using this procedure, any concentration of interest can be 

identified and its probability of occurrence scaled directly from 

the plot. 

Alternatively, the concentration that will not be exceeded at 

some specific frequency (or probability) can be calculated from: 

COa = exp (μlnCO + (Za σlnCO)) (2-13) 

where 

Za = the value of Z from a standard normal table which 

corresponds to the selected percentile a. 

To determine the probability of exceedence, (1 = a) is 

substituted in Equation 13. 

One can also work in the reverse direction; that is, given 

some target stream concentration (CL), the probability of CO 

exceeding that level can be determined by: 

Z = ln(CL) – μlnCO (2-14) 

σlnCO

A standard normal table will provide the probability for the 

calculated value of Z. 

Because of the way the standard normal table 1n Appendix A is 

organized, the probabilities calculated using this approach 

represent the fraction of time the target concentration (CL) is 

not exceeded. The 

2-8 
probability that the concentration will be exceeded is obtained 

by subtracting the value obtained from 1.0. 

2.2 Choice of the Permit Averaging Period

In order to examine the comparative effects of different 

choices of permit averaging periods on water quality, it is 

necessary to define the relationships between the established 

effluent limit (EL) from the steady state WLA, the permit 

averaging period, the treatment plant performance that results, in 

particular the mean effluent (CE ), the downstream concentration 

(CO), and a stream target concentration (CL). 

The objective of this section is to examine the relationships 

among these parameters in order to be able to predict the 

probability of an (adverse) water quality outcome based on known 

or estimated stream and effluent characteristics and the choice of 

permit averaging period. The approach is based on the assumption 

that the EL will be violated with a particular frequency. The mean 

effluent required to meet this level of compliance with EL is then 

calculated for each of the three permit averaging periods, and the 

probabilistic dilution model is then used to develop a probability 

distribution of the downstream concentration (CO) for the three 

cases. A level of acceptable adverse water quality (a decision 

expressed in terms of the probability or frequency of experiencing 

a selected high value of CO, such as the acute criteria 

concentration) is then compared with the probability distributions 

to determine the longest permit averaging period that meets the 

water quality goals. 

2-9 
The first step in this sequence is to establish the 

relationship between the mean effluent (CE ), the effluent limit 

(EL), and the permit averaging period. In fact, what is required 

is the relationship between the treatment plant performance

necessary to meet the effluent limit as either a daily, weekly, or 

monthly maximum permit. The reason for this is that the daily 

variation of stream quality is governed, not by the effluent limit 

which is a regulatory upper limit, but by the probability 

distribution of the daily effluent concentrations which results 

from the design of the treatment plant consistent with the 

effluent limit and the permit averaging period. For log-normally 

distributed random variables, this distribution is specified by 

the mean effluent concentration, CE , and its coefficient of 

variation, νCE. 

A particular effluent limit (say 30mg/l) established by 

permit as a maximum daily value would require a higher level of 

plant performance (a lower mean effluent concentration) to avoid 

permit violations than would the same limit specified as a maximum 

monthly average. In the latter case, excursions solve the effluent 

limit could be tolerated on individual days, without causing a 

violation of permit conditions. The reason for this is that a 

monthly average of 30 Individual dally effluent concentrations is 

less variable than the daily concentrations themselves. Occasional 

high daily concentrations are averaged together with lower 

concentrations to produce a less variable monthly average. Hence, 

treatment plant performance is directly related to the averaging 

period specified in the permit. 

In order to proceed with the analysis a quantification of 

this relationship is required. Daily treatment plant effluent 

concentration variations 

2-10
are well described by a log-normal distribution parameterized 

by a long term average concentration, CE , and a coefficient of 

variation, νCE. Thus, a relationship between these parameters and 

the permit effluent limit and averaging period is required. 

A method to be employed is based upon an interpretation of 

what is meant, in practice, by specifying permit effluent limits 

as maximum values which may never be exceeded for the specified 

averaging period without causing a violation. As Haugh, et al. [2] 

observe, fixed upper limits, which are never to be exceeded are 

conceptually inconsistent with the stochastic nature of wastewater 

treatment processes and the effluent concentrations they produce. 

Realistically, some exceedence frequency must be acknowledged, 

regardless of the averaging period assigned. For the present 

analysis, it will be assumed that the effluent limit specified by 

a permit is not to be exceeded more frequently than 5 percent or 1 

percent of the time. Of course, any other choice is possible. 

Once a specific choice is made, say 1 percent, then the 

probability of compliance is a = 99 percent and that establishes 

the fact that EL is the a-percentile effluent concentration: CEa. 

This procedure, then, gives a specific probabilistic 

interpretation to the effluent limit. It is the effluent 

concentration that 1s exceeded with no greater frequency than (1-a) 

percent of the time. If the permit is specified as a daily maximum 

value, then EL is the a-percentile of dally effluent 

concentrations. If the permit is specified as a weekly (or 

monthly) maximum value, then EL is the a-percentile of 7-day (or 

30-day) average effluent concentrations. 

2-11
In order to compute the long term average effluent 

concentration, CE , that would insure that CEa = EL as a daily, 

weekly, or monthly permit the coefficients of variation are 

required for 1-day and 7-day or 30-day averages of effluent 

concentrations. Table C-2 presents representative values. 

Thus, the requirement that: 

CEa = EL (2-15) 

and for a coefficient of variation νCE, the average effluent 

concentration CE can be computed from 

CE = Ra * EL (2-16) 

where the reduction factor relating CEa = EL to CE , that is, Ra = 

CE /CEa, is 

Ra = √1 + νCE

2

 exp [-Za √ln (1 + νCE

2

)] (2-17) 

the ratio of the arithmetic average to the a-percentile of a lognormal random variable with coefficient of variation, νCE. Table 2-

1 gives the values of Ra for various coefficients of variation. 

The derivation of this formula follows from the expression 

for the a-percentile of a log-normal random variable: 

CEa = exp (μlnCE + Za σlnCE) (2-18) 

and the arithmetic average of a log-normal random variable: 

2-12
TABLE 2-1 - Reduction factors for various coefficients of variation 

Reduction Factor 

Ra

Coefficient of 

Variation 

νCE a = 95% a = 99% 

0.1 0.853 0.797

0.2 0.736 0.643

0.3 0.644 0.527

0.4 0.571 0.439

0.5 0.514 0.372

0.6 0.468 0.321

0.7 0.432 0.281

0.8 0.403 0.249

0.9 0.379 0.224

1.0 0.360 0.204

1.1 0.344 0.187

1.2 0.330 0.173

1.3 0.319 0.162

1.4 0.310 0.152

1.5 0.302 0.144

2-13
 CE = exp (μlnCE + 1⁄2 σ2

lnCE) (2-19) 

Thus: Ra = CE/CEa = exp (1⁄2 σ2

lnCE – Za σlnCE) (2-20) 

and since exp (1⁄2 σ2

lnCE) = √1+ν2CE and σlnCE = √ln (1+ ν2CE) 

(appendix A, page A-8) equation (2-17) follows. 

At this point the effect of the choice of permit averaging 

period on treatment plant design can be illustrated. If the permit 

averaging period is 1-day, and the daily effluent coefficient of 

variation is νCE=0.7 (for example, extended aeration activated 

sludges, Table C-2), then for a 1 percent violation frequency a=99 

percent, Ra = 0.281, which indicates that the long term average 

effluent concentration must be 28.1 percent of the daily maximum 

permit limit. 

However, if the permit averaging period is 7 days, then the 

coefficient of variation of 7-day averages is νCE = 0.6 and Ra = 

0.321. Now the treatment plant can be designed to produce a long 

term average effluent concentration of 32.1 percent of the weekly 

permit limit. For a 30-day average permit limit νCE= 0.45 and Ra = 

0.404. Hence, if EL = 10 mg/l, the treatment plant average 

effluent concentration must be 2.81, 3.21, or 4.04 mg/l for a 

daily, weekly, or monthly permit specification, respectively. 

Hence the selection of the permit averaging period is related 

to the CE required for each of the three averaging periods in 

order to 

2-14
avoid exceeding the EL more often than the selected frequency. 

These average values are then used in the probabilistic dilution 

model (with other input parameters such as QS and QE ) to develop 

the probability distribution of CO for each of the three permit 

averaging periods. 

The value of CO in the probability distribution can be 

normalized in terms of a stream target concentration (such as the 

chronic criteria concentration, CL) so that the calculation can be 

used for a wide variety of pollutants. Stream concentration is 

therefore expressed in terms of β = CO/CI, β being a dimensionless 

unit of concentration. 

 

A convenient presentation of the resulting probability 

distribution makes use of the concept of return period. For daily 

stream concentrations the 1 percent exceedence value has an 

average recurrence rate of one day every 100 days so that its 

average return period is 100 days. Thus the return period for 

daily values is defined as: 

Return Period (days) = 1/Probability of Exceedence (2-28) 

The basic assumption in the use of return period as defined above 

is that the event whose probability is being examined has a 

characteristic time associated with it, in this case, one day for 

daily concentrations. Thus, it is assumed that daily stream 

concentrations are of concern, and each event corresponds to one 

day. 

Figure 2-2 illustrates how the results of such an analysis 

can be expressed in a plot of concentration versus return period. 

2-15
Figure 2-2 - Illustration of analysis results: stream 

concentration versus return period for three 

permit averaging periods

The stream target concentration (CL) for a typical WLA is the 

chronic criteria concentration of the pollutant under 

consideration. The use of the chrome criteria as the stream target 

concentration is convenient for the comparison of permit averaging 

periods because it represents a specific and frequently used 

procedure. The analysis that follows does not attempt to quantify 

the frequency with which chronic criteria concentrations are met 

by either the conventional ULA procedure or the guidance provided 

for selecting permit averaging period. Instead, the analysis is 

designed to relate the choice of the permit averaging period to 

the frequency with which severe, short term water quality impacts 

are expected as a result of an effluent limit. 

2-16
 

These snort-term impacts are perhaps most effectively 

evaluated with respect to acute criteria concentrations. If the 

stream concentration exceeds the acute criteria as a result of ah 

occasional high daily effluent loading, the result is presumed to 

be an undesirable impact. Hence, there is a direct connection 

between the permit averaging period and the probability of acute 

criteria violations. Specifying that the WLA requirement be met as 

a daily maximum permit limit significantly reduces the possibility 

of acute criteria violation since the effluent limit is specified 

using the chronic criteria, which is always a smaller 

concentration. 

The frequency with which daily stream concentrations are 

allowed to exceed acute criteria is a regulatory decision1

. The 

analyses presented herein employ a frequency that corresponds to a 

1-day in 10-year recurrence, on average. The choice of 10 years 

is, of course, used for example purpose only but it is consistent 

with the 10 year return period that is conventionally used for the 

design stream flow. 

The results of the permit averaging period analysis are 

presented in terms of CO/CL which is exceeded with a particular 

frequency, such as once in 10 years. This ratio can then be 

compared to the acute-to-chronic criteria concentration ratio for 

the pollutant of concern. For pollutants with large acute-tochronic ratios, occasional large daily fluctuations can be 

tolerated; and a 30-day permit averaging period provides 

protection from acute criteria violations. Conversely, pollutants 

with small acute-to-chronic ratios are more likely to require 

shorter day permit averaging periods. Site specific 

1 This is currently under EPA study

2-17
considerations, primarily the ratio of effluent to stream flow and 

stream flow variability, become significant in these cases. 

The final translation of the selected averaging period option 

to permit limits requires consideration of the monitoring 

frequency. The method assumes either daily monitoring or other 

monitoring adequate to describe the performance of the plant on a 

monthly basis. If such conditions are not met, alternate limits 

may be calculated which Incorporate monitoring frequency, or 

monitoring frequency may be adjusted so that these conditions are 

met. 

2-18
CHAPTER 3 

EXAMPLE COMPUTATION 

This chapter presents an example problem, showing step by 

step computations using the methodology described in the previous 

chapter. A set of hypothetical conditions that apply to a sitespecific situation is assumed, and an analysis is performed to 

determine the effect on receiving water quality-resulting from the 

assignment of different permit averaging periods to the steadystate model output. The steps used to conduct this analysis are 

summarized below in Figure 3-1. The format used in this chapter 

presents data and computations on the left-hand page, and 

pertinent commentary and supporting discussion on the facing page 

immediately opposite those computations. The manual computation 

using the moments approximation is described first, followed by an 

analysis using the computer program (PDM-PS) in Appendix D. Both 

examples use the same set of hypothetical site-specific 

conditions. 

GIVEN: 

• acute and chronic 

toxicity 

• design flows 

• flow and concentration 

variability for 

specific averaging 

periods 

STEP 1: 

Compute statistical 

parameters of stream and 

effluent flow and 

concentration

STEP 2: 

Compute statistical 

parameters of dilution 

factor

STEP 3: 

Compute statistical 

parameters of the 

resulting stream 

concentration

STEP 4: 

Compute frequency 

distributions 

STEP 7: 

Translate into daily, weekly, and monthly 

permit limits 

STEP 6: 

Select optimal permit coverage period

Step 5: 

Repeat steps 1 

through 4 for 

remaining averaging 

periods

Figure 3-1 - Step procedure to select optimal permit averaging 

period 

3-1 
3.1 HYPOTHETICAL SITE-SPECIFIC CONDITIONS

This section provides an example of the type and amount of 

information required to perform the analysis. It also establishes 

the basis for the example computations and assumes that pertinent 

site-specific conditions are as follows: 

A. Site-Specific Waste Load Allocation (WLA) Results

The pollutant (P) to be allocated has a chronic toxicity 

concentration (CL) of 2.5, and an acute toxicity

concentration of 6.25. 

WLA policy for the agency performing the analysis is to use 

7Q10 as stream assign flow, to use the design capacity of the 

treatment plant as the effluent flow, and to compute (e.g., 

using a water quality model) the effluent concentration of 

pollutant (P) that will result in a stream concentration 

after dilution less than or equal to the chronic value (2.5 = 

the stream target concentration, CL). For this example, it 

was assumed that: 

Design Effluent Flow (QE) = 5 MGD = 7.77 cfs 

Design Stream Flow (7Q10) = 23.3 cfs 

The stream target concentration (CL = 2.5) will be met under 

these design flow conditions, when the effluent concentration 

1s CE = 10. Therefore based on the WLA analysis, the effluent 

limit (EL) for pollutant (P) is specified by the permit as: 

EL = 10 

3-2 
COMMENTARY 

--- from EPA Criteria 

State water quality standards do not usually specify both 

values; they are usually based on chronic values. 

(Any concentration units may be assigned; stream 

concentrations will nave to be in the same units.) 

--- 77Q10 (the lowest 7-day average stream flow with a recurrence 

interval of 10 years) is the most common "design stream 

flow". Some states use other values (e.g., 30Q5). This 

analysis uses the numerical value of the "design flow". 

However, although the. example terminology uses "7Q10", it 

should be interpreted as "design stream flow" and the 

appropriate value substituted, regardless of the averaging 

period or the recurrence interval on which it is based. (For 

example, if design flow in a state were 30Q5, assume that 

30Q5 = 23.3 cfs). 

NOTE: The only exception to this is in Figure C1, in which the ratio of 7Q10 to average 

stream flow is used to estimate the 

variability of daily flows in the absence 

of a specific local analysis. The use of 

this figure is not requisite to either the 

analysis methodology or the computations. 

--- 

CL = (QE * CE) + (QS * CS) 

 QE + QS 

2.5 = (7.77 * CE) + (23.2 * 0) 

7.77 + 23.3 

CE = 10 = EL 

3-3 
HYPOTHETICAL SITE-SPECIFIC CONDITIONS 

(continued) 

3. Site-Specific Conditions

Stream Flow Mean Flow QS = 467 cfs

Coefficient of Variation (νQS) = 1.5 

Upstream Concentration Mean (CS ) = 0 

Coefficient of Variation (νCS) = 0 

Effluent Flow Mean (QE ) = 7.77 cfs 

Coefficient of Variation (νQE) = 0.20 

3-4 
COMMENTARY 

--- Stream flow data are obtained from analysis of flow 

gaging records for the stream in question; where the 

stream reach is engaged, it is obtained by extrapolation 

from an appropriate record. 

At present, records are not normally analyzed for the 

coefficient of variation, although the computation is 

straight forward and can be readily incorporated into a 

routine statistical analysis of daily stream flows. In 

the absence of specific analysis results, the coefficient 

of variation of daily stream flows can be estimated using 

the material presented in Figure C-1. 

--- Upstream concentration can be assumed to be zero if the 

stream concentration of the pollutant is very low 

compared to the discharge, or if the effect of the 

discharge only is to be examined. Site-specific values 

for upstream concentration statistics would be obtained 

from analysis of an appropriate STORET station, or from 

local monitoring records. If upstream concentrations are 

assigned, enter data here and in the equations when 

called for. 

--- The design effluent flow is assumed to be the mean 

effluent flow. The variability of daily effluent flows 

for a new facility must be estimated on the basis of 

available data for existing treatment facilities (such as 

Table C-1). For an existing facility being expanded, or 

simply re-permitted, variability could be based on an 

analysis of past plant records. For many industrial 

dischargers, this data will be available in Book VI 

(Design Conditions) of the waste load allocation 

technical guidance document series (specifically, in 

Chapter 4: Effluent Design Conditions). 

3-5 
HYPOTHETICAL SITE-SPECIFIC CONDITIONS 

(continued) 

Effluent Concentration Mean (CE ) = (*) 

Coefficient of Variation (νCE) = .7 

The mean concentration is a function of the permit averaging 

period and is that concentration required to avoid exceeding the 

effluent limit concentration (EL) more often than the compliance 

probability. 

The coefficient of variation for the hypothetical treatment plant 

is not known because the plant has yet to be constructed. Assuming 

that the plant will produce an effluent with a variability similar 

to the values given in Table C-2, the following values are used: 

Permit Averaging Coeff. of Var. 

Period (ν CE)

Daily 0.70 

7-Day 0.40 

30-Day 0.20 

Equation 2-17 is then used to determine the mean effluent 

concentration of (P) which is required to avoid a violation of EL 

more often than the compliance probability. For this example, 

assume that the exceedence probability is 1 percent. For a = 0.99 

percent, Za = 2.327. For νCE= 0.70, Ra= CE /EL is: 

Ra = √1 + ν2

CE exp [-Za √ln(1+ ν2CE)] 

= √1+ 0.49 exp [-2.327 √ln(1+0.49)] 

= 1.221 exp [-2.327 * 0.6315] 

=0.281 

The reduction factor for 7-day and 30-day averages are computed 

similarly with νCE (7-day) = 0.40 and νCE (30-day) = 0.20. The 

results are: 

Permit 

Averaging Period 

Coeff. Of Var. 

of Averaged 

Effluent 

Concentrations 

(νCE) 

Reduction Factor 

Ra = /EL 

Required Mean 

Effluent Conc. 

( = Ra EL)

Daily 0.70 0.281 0.281 * 10 = 2.81 

7-Day 0.40 0.439 0.439 * 10 = 4.39 

30-Day 0.20 0.643 0.643 * 10 = 6.43 

CE

CE 

3-6 
COMMENTARY 

--- The mean effluent concentration that a treatment 

facility is capable of producing is influenced 

significantly by process selection. For this example, it 

will be assumed that process selection will be made 

following the issuance of a permit, and influenced by its 

provisions. 

The mean effluent concentration that a facility is 

required to produce is influenced by the permit averaging 

period and the variability of effluent concentrations of 

the pollutant in question. 

The analysis employed here, which bases permit averaging 

period selection on receiving water impacts, is based on 

exceedance of the acute criteria on a daily basis. 

Therefore, all subsequent stream impact computations 

(Step 4) are based on the coefficient of variation of 

daily effluent concentrations, or 0.7, as shown. 

The mean concentration is shown by (*), because a 

different value is used for each permit averaging period. 

--- The recommended exceedence probability for the effluent 

limit is either 5 percent or 1 percent. For 5 percent, Za

would be Z95= 1.645. 

--- Longer averaging periods reduce the variability of 

effluent concentrations, and-allow permit exceedance 

limits to be mot with higher effluent means. Computation 

of the required mean (CE) uses the values of νCE for the 

corresponding permit averaging period. 

3-7 
3.2 EXAMPLE COMPUTATION - HAND CALCULATION

This section illustrates the hand computation using the moments 

approximation to evaluate the stream concentration probability 

distribution. 

STEP 1: Compute statistical parameters (arithmetic and 

logarithmic) of inputs using relationships for log-normal 

distributions (see notes on page 3-9 or Appendix A for 

equations). 

o For the mean effluent concentration (CE) for a 30-

day permit averaging period with X = CE, that is 

for the variable CE: 

ARITHMETIC

Mean (μx) = - - - - - - - (page 3-6)- - - - - - = 6.43 

Coef. Var. (νx) = - - - - - - - (page 3-6)- - - - - - = 0.70 

Std. Dev. (σx) = μx * νx = (6.43) * (0.70) - - - - - - = 4.50 

Median (x) = μx/ √1+ ν2

x = 6.43/ √1 + (0.7)2

 = 5.27 ~

LOGARTHMIC 

Log Mean (μlnx) = ln (x) = ln (5.27) = 1.662 ~

Log Std. Dev. (σlnx) = √ln (1 + νx

2

) = √ln (1 + (0.7)2) = 0.6315 

o These computations are repeated for each of the other 

input parameters. The results are tabulated below. 

Arithmetic Logarithmic

Mean Median Std Dev Coef Var Mean Std Dev

x μx x σx νx μlnx σlnx

Stream 

Flow: QS 467 259 701 1.50 5.5570 1.0857

Effluent 

Flow: QE 7.77 7.62 1.55 0.20 2.0307 0.1980

Upstream 

Concentration: 

CS 

0 0 0 0 0 0

Effluent 

Concentration: 

CE 

6.43 5.27 4.50 0.70 1.662 0.6315

~

3-8 
COMMENTARY 

The following parameters are used subsequently: 

Arithmetic Logarithmic

Input 

Parameter

Median Mean Std. 

Dev. 

Coef. 

Var. 

Log 

Mean 

Log 

S.D. 

x μx σx νx μ ln x σ ln x

Stream Flow QS QS μQS σQS νQS μlnQS σlnQS

Stream Conc. CS CS μCS σCS νCS μlnCS σlnCS

Effluent Flow QE QE μQE σQE νQE μlnQE σlnQE

Effluent Conc. CE CE μCE σCE νCE μlnCE σlnCE

~

~

~

~

~

The following definitions and equations summarize the 

relationships among the statistical parameters of log-normal 

random variables. 

Arithmetic Terms Logarithmic

x Random Variable ln x 

μx Mean μln x 

σ

2

x Variance σ

2

ln x 

σx Standard Deviation σln x 

νx Coefficient of Variation (not used)

 ~

x Median (not used)

μx = exp [μlnx + 1⁄2 σ2

lnx] μlnx = ln ( μx ) 

 ( √ 1 + νx

2

 ) 

x = exp [μlnx] ~

νx = √exp (σ2

lnx) – 1 σlnx = √ln (1-νx

2

) 

σx = μxνx

3-9 
EXAMPLE COMPUTATION - HAND CALCULATION 

(continued) 

STEP 2: (a) Compute the log standard deviation of the flow ratio 

QS/QE = 0. 

σlnD = √σ2

lnQS + σ2lnQE + 2p · σlnQS · σlnQE 

The first two terms are taken from the table in Step 1 

(and squared). Since, for this example, flows are not 

correlated (p=0), the third term drops out. Therefore, 

σlnD = √(1.0857)2

 + (0.1980)2 = 1.1036 

(b) Compute the 5th and 95th percentiles of the actual 

distribution of the dilution factor (φ ). 

φa = QE~

(QE + QS) · exp(ZaσlnD) ~ ~

where: 

QE , QS = median values for effluent and stream flows 

(from table in Step 1) 

Za = the standard normal Z score for selected 

percentiles(a) 

Z5 = 1.645; Z95 = 1.645 

σlnD = 1.1036 (computed in Step 2 (a)) 

Substituting the appropriate values gives: 

φ 95 = 0.004766 φ 5 = 0.1531 

(c) Compute the log mean and log standard deviation of the lognormal approximation of the distribution of the dilution factor 

(φ ). 

Log mean μlnφ = 1⁄2 [ln (φ95) + ln(φ5)] = -3.6115 

Log std dev σlnφ = 1 · (ln (φ5) – ln (φ95)) = 1.0546 

 1.645 2 

3-10
COMMENTARY 

--- This equation accounts for any correlation that may exist 

between stream flow and effluent flow; e.g., where higher 

effluent flows tend to occur during periods of high stream 

flow. 

--- Ordinarily, there is no reason to expect any such 

correlation; therefore ρ = 0, and the computation in step (a) 

is simplified as shown. 

--- φ95 = QE ~

(QE + QS)exp (ZaσlnD) ~ ~

= 7.62 

(7.62 + 259) exp [(1.645)(1.1036)] 

= 7.62 

7.62 +1591 

= 0.004766 

3-11
EXAMPLE COMPUTATION - HAND CALCULATION (continued) 

(d) Compute arithmetic statistical parameters (using equations 

on Page 3-9 and tabulate for convenience. 

Arithmetic Logarithmic

Mean Median Std Dev Coef Var Mean Std Dev

Dilution 

Factor 

(φ) 0.0471 0.0270 0.0673 1.43 -3.6115 1.0546

STEP 3: Compute the statistical parameters of the resulting instream concentration (CO). 

(a) Compute the arithmetic mean concentration using 

previously tabulated values, using Equation 2-8. 

μCO = [μCE · μφ] + [μCS · (1-μφ)] 

= [5.43 · 0.0471] + [0] = 0.303 

(b) Compute the standard deviation, using Equation 2-9. 

σCO = σ2

 · (μCE - μCS)2 (0.0673)2 · (6.43-0)2 0.187 

 + σ2

CE · (σφ

2

 + μφ

2

) = + (4.50)2 (0.06732 = + 0.137 + 0.04712

) 

 √ + σ2

CS · (σφ

2

 + (1 - μφ)2) √ + 0 √ + 0 

σCO = √0.324 = 0.569 

(c) Compute and tabulate for use in subsequent graphical 

or other summaries, the other statistical parameters 

of stream concentration. 

Arithmetic Logarithmic

Mean Median Std Dev Coef Var Mean Std Dev

Stream 

Concentration (CO)

0.303 0.142 0.569 1.88 -1.95 1.23

3-12
COMMENTARY

--- The equations are as follows: 

μφ = exp [μlnφ + 1⁄2σ2

lnφ] 

= exp [-3.6115 + 1⁄2 (1.0546)2

] 

= 0.0471 

νφ = √exp (σ2

lnφ) - 1 

= √exp [(1.0546)2

]-1 

= 1.429 

σφ = μφνφ

= (0.0471)(1.429) 

=0.06729 

--- When the manual ("moments" approximation) analysis presented 

here is used, the stream concentrations computed are assumed 

to be log-normally distributed. That is, the log-normal 

distribution computed is an approximate representation of the 

actual distribution that results. The degree of approximation 

is examined subsequently. 

--- The equations are: 

νCO = aCO/μCO = 0.569/0.303 

= 1.88 

μlnCO = ln (μCO) 

(√1 + νCO

2

) 

= ln (0.303) 

√(1 + (1.88)2 

= -1.95 

σlnCO = √ln (1 + ν2

CO) 

= √ln [1 + (1.88)2

] 

= 1.23 

3-13
EXAMPLE COMPUTATION - HAND CALCULATION 

 (continued) 

STEP 4: use the statistical parameters of stream concentration 

computed in the previous step to construct graphical OP 

tabular displays summarizing the frequency distribution. 

(a) To construct a probability plot using logprobability graph paper: 

• The median concentration is plotted at the 

50th percentile position. 

CO = CO50%

=exp (μlnCO) 

= exp (-1.95) 

=0.142 

~

• Any other plotting position is determined as 

follows: 

(1) From Table A-1, select a probability (a) and 

determine the corresponding value of Za. For 

example, 

Probability = 0.841 (84%) ..... Z84.1%=1.00 

Probability = 0.159 (16%) ..... Z15.9% = -1.00 

(2) Compute the concentration at probability (σ) 

from the log mean and log standard deviation of 

stream concentration (CO). 

Coa = exp (μlnCO + Za · σlnCO) 

84% plotting position 

CO84% = exp(-1.95 + 1.00 · 1.23) = 0.487 

16% plotting position 

CO16% = exp(-1.95 –1.00 · 1.23) = 0.0416 

(3) Plot these concentrations on log-normal 

probability paper and connect with a straight 

line. 

3-14
COMMENTARY 

Figure 3-2 - Sample stream concentration versus probability plot 

for 30-day averaging period

The probability plot indicates, for example, that the stream 

concentration of pollutant (P) will exceed a concentration of 1.0, 

at a frequency (probability) of about 5%. Since the analysis is 

based on daily values, this is interpreted as: 55 of all days will 

have stream concentrations greater than 1. 

3-15
EXAMPLE COMPUTATION - HAND CALCULATION

(continued) 

STEP 4 (continued) 

(b) To construct a recurrence interval (return period) 

plot using log-log graph paper: 

o the formula used in the previous Step 

Coa = EXP (μlnCO + Za · σlnCO) 

can be rearranged: 

Za = ln(COa) - μlnCO

σlnCO

The log mean and log standard deviation were determined 

in Step 3: 

μlnCO = -1.95 

σlnCO = 1.23 

o Plotting positions are determined as follows: 

(1) Select a series of values for stream 

concentration (CO) covering a range of interest, 

take the natural log (ln) and compute the value 

of 2. 

(2) From Table A-1 identify the probability (Pr) 

associated with each 2. 

(3) Compute the mean recurrence Interval (MRI) for 

each of the selected concentrations: 

MRI (years) = 1 · 1 

Pr 365 day/yr 

For example: 

Stream 

Concentration CO Z 

Probability 

Greater Than

Mean Recurrence 

Interval (years)

15 3.787 7.626 x 10-5 35.9 

10 3.457 2.732 x 10-4 10.0 

5 2.894 1.902 x 10-3 1.44 

1 1.585 5.648 x 10-2 0.0485 

Plot results. If necessary, compute additional values to 

assist in drawing a smooth curve. 

3-16
COMMENTARY

--- Probability results can be misleading for the water quality 

issues being considered here, unless interpreted very 

carefully. For example, a 1% probability of exceeding a 

significant stream concentration means that this occurs 

nearly 4, times in 1 year, and for more than a month of 

individual days over a 10 year period. Expressing results as 

recurrence intervals is believed to provide a more useful 

expression of analysis results. 

MEAN RECURRENCE INTERVAL - YEARS 

Figure 3-3 Sample stream concentration versus mean recurrence 

interval for 30-day averaging period 

Note that the acute concentration assumed for the 

pollutant (6.25) is exceeded an average of once every 

2.6 years. If the exceedance criteria to be met 1s an 

average of 1 acute toxicity exceedance every 10 years, 

then the assignment of a 30-day permit averaging period 

is insufficient; shorter averaging periods must be 

examined. 

However, if the pollutant had an acute concentration of 

12.5 (or an acute-to-chronic ratio of 5), the recurrence 

interval of 20 years would be sufficiently protective 

for acute events. 

3-17
EXAMPLE COMPUTATION - HAND CALCULATION 

(continued) 

STEP 5: Compute the receiving water quality impact that would 

result from assigning other permit averaging periods. 

Repeat Steps 1-4 using the values for CE that have been 

calculated for weekly and daily permit assignment. 

7-day permit average ........ CE = 4.39 

Daily maximum permit average....... CE = 2.81 

All other inputs remain unchanged. 

When the computations are repeated using these values, 

the statistical parameters for stream concentration 

(Step 3) that are developed are as follows: 

STREAM CONCENTRATION (CO) STATISTICS 

Permit 

Averaging 

Period 

Mean 

μCO

Median 

CO 

Std. Dev. 

σCO

Coef. Var. 

νCO

Mean 

μlnCO

Std. Dev. 

σlnCO

30-Day 0.303 0.142 0.570 1.88 -1.95 1.23

7-Day 0.207 0.0971 0.389 1.88 -2.33 1.23

1-Day 0.132 0.0622 0.248 1.88 -2.78 1.23

~

Probability and recurrence Interval plots are then 

constructed as described in Step 4 to provide a 

graphical comparison of the influence of alternative 

choices for averaging period on the frequency of 

exceeding acutely toxic concentrations of pollutant (P) 

in the receiving system. 

3-18
COMMENTARY

Figure 3-4 - Concentration versus probability plot for 1-, 7-, and 

30-day averaging periods

MEAN RECURRENCE INTERVAL - YEARS 

Figure 3-5 - Concentration versus mean recurrence interval plot 

 for 1-, 7-, and 30-day averaging periods 

3-19
EXAMPLE COMPUTATION - HAND CALCULATION

(continued) 

STEP 6: Select the appropriate permit averaging period. 

The appropriate permit averaging period is chosen to 

provide an acceptable level of receiving water quality. 

The decision is based on the assumption that an 

unacceptable exceedence of the acute criteria in the 

receiving stream is more than once every 10 years, on 

average. 

Therefore, the permit averaging period selected is the 

highest one that does not result in a mean recurrence 

interval for acute criteria violations that 1s less than 

10 years. For this example, recurrence intervals for a 

stream concentration of 6.25 are approximately 

30-day Avg. Period = 2.6 years 

7-Day Avg. Period = 7.7 years 

1-Day Avg. Period = 31 years 

For the site specific conditions assumed for this 

example, a 1-day permit averaging period could be 

assigned to the effluent limit of 10. However, as shown 

below using more exact calculations, a 7-day permit 

averaging period Is sufficiently protective for acute 

events. Thus a 7-day permit averaging period is assigned 

to the effluent limit of 10. 

3-20
COMMENTARY

--- For marginal cases, it should be recognized that the 

projections made using the moments approximation tend to be 

conservative. As shown below the more exact recurrence 

intervals are 6.4, 32, and 280 years". 

--- The acceptable frequency of acute criteria violation is, of 

course, a policy decision. Alternate levels are evaluated 

directly from Figures 3-3 and 3-4. 

--- The moments approximation used for the foregoing computations 

(because it approximates the distribution of dilution factor 

(φ ) with a log-normal distribution) provides an approximation 

of the probability distribution and recurrence interval of 

the stream concentrations. 

An exact computation that avoids the necessity of this 

approximation, is provided by use of the computer program 

detailed in the next section and in Appendix D. In this case, 

its use is warranted since a 7-day permit averaging period is 

sufficiently protective. 

Based on the selection of the 7-day permit averaging period, 

the maximum 7-day average permit limits = EL = 10 mg/l. This 

permit limit is equivalent to a long-term average effluent 

concentration CE = Ra EL = (0.439)(10) = 4.39, with 

coefficient of variation daily effluent concentration (νCE) = 

0.7. Thus, the design of the treatment facility and the 

selection of treatment process should be made to meet these 

specifications of CE = 4.39 mg/l with coefficient of 

variation of daily effluent concentrations νCE= 0.7. 

3-21
EXAMPLE COMPUTATION - HAND CALCULATION 

(continued) 

STEP 7: Compute permit limits for other averaging periods (daily 

maximum and monthly) and exceedence percentiles (1 

percent and 5 percent) that are consistent with the 

treatment performance level established in Step 6. 

At this point in the analysis, it has been determined 

that assigning the effluent limit of EL = 10 as a weekly 

permit, applicable to 7 day averages of the daily 

concentrations, is sufficiently protective. This choice 

is based upon an effluent limit violation frequency of 

one percent. The mean effluent concentration for these 

choices is CE = 4.39. 

If it is assumed that the same violation frequencies 

apply to the other permit concentrations, then they can 

be computed directly: 

Permit Limit = CE /Ra

Since Ra = CE /CEa and the permit limits are assumed to 

be the a-percentile concentrations for each averaging 

period. 

If other violation frequencies are desired, for example, 

5 percent, then permit limits of this frequency can also 

be calculated using the appropriate Ra for 1-a = 5 

percent. The table below presents the results for the 

example considered above. 

Reduction Factorsb

Ra Permit Limitsc

Permit 

Averaging 

Period 

Coeff. of Var. of 

Avg.’ed Effluent 

Concentrationa

νCE

1% 5% 1% 5% 

1-day 0.70 0.281 0.432 15.6 10.2

7-day 0.40 0.439 0.571 10.0 7.69 

30-day 0.20 0.643 0.736 6.83 5.96 

It should be pointed out that any or all of these 

permits are equivalent in the sense that a treatment 

plant meeting any of these requirements will also meet 

the desired water quality goal. Of course, this 1s true 

only 1f the actual coefficients of variation for daily 

values and 7 and 30 day average plant effluent 

concentrations are as specified. 

a

These are assumed to be representative of the treatment plant 

effluent behavior. 

b

Table 2-1, equation 2-17. 

c

Permit limit = CE /Ra; CE =4.39.

3-22
COMMENTARY

Permit Limits Daily Maximum Weekly Monthly 

Reduction factors (see p. 3-6) 0.281 0.439 0.643 

Choice of averaging period (from 

step 6) no yes no

Value for the selected averaging 

period (from step 6 – steady state 

model output) 

- 10 -

Permit limits using reduction 

factors, Ra 

10(0.439) = 15.6 

 0.281 10.0 10(0.439) = 6.8  0.643 

Long-term average effluent 

concentration, CE (see p. 3-6) 4.39 4.39 4.39

Coefficient of variation of daily, 

weekly, and monthly permit limits 

(see p. 3-6) 

0.7 0.4 0.2

The long term average effluent concentration for the 

required level of treatment is equal to 4.39 mg/l with 

the coefficient of variation of daily effluent 

concentrations equal to 0.7. To meet the water quality 

standard at the state specified design flow and to meet 

the acute criteria at all times except for 1 day once in 

10 years, the treatment facilities need to be built to 

meet the long term average concentration of 4.39 mg/l 

with coefficient of variation of daily effluent 

concentration νCE = 0.7. The permit limits derived above 

are based on daily, weekly, and monthly reporting 

procedures. If less than adequate monitoring is 

required, the appropriate permit limits must be derived 

using the long term average and equivalent coefficient 

of variation. 

3-23
EXAMPLE COMPUTATION - HAND CALCULATION 

(continued) 

Recapitulation 

In order to aid in the understanding of the suggested procedure, 

the sequence is reviewed below in outline form. 

1. Establish streamflow characteristics. 

QS νQS

2. Establish effluent flow characteristics. 

QE νQE

3. Establish effluent concentration variability characteristics 

(νCE) daily values and 7 and 30 day averages. 

Coefficient of Variation 

Averaging Period νCE

1-day 0.7 

7-day 0.4 

30-day 0.2 

4. Establish effluent limit from steady state wasteload 

allocation. 

EL = 10 

5. Establish violation frequency of EL. 

1-a = 1% 

a = 99% 

and assume CEa = EL 

3-24
COMMENTARY

1. These should be site specific since the computation is 

usually sensitive to the values. 

2. Mean effluent flow is important, but the coefficient of 

variation, since it is usually small, is usually not 

significant if νQE = νQS

3. These coefficients of variations specify the behavior of 

the daily values and temporal averages of effluent 

concentrations. More detailed evaluations for industry 

specific or pollutant specific situations are required 

to be more definitive. The values used are not suggested 

as universal. 

4. The analysis presented in this manual does not evaluate 

the degree of protection afforded by this choice. That 

is, the probability of violation of the chronic criteria 

is not calculated. It is assumed to be sufficiently 

protective. 

5. The choice of violation frequency is necessary in order 

to give a specific probabilistic meaning to EL. 

Reasonable values appear to be one or five percent. 

However, a problem may arise if too frequent a violation 

frequency is chosen. It may turn put that even 

specifying the permit as a daily maximum does not insure 

that acute criteria violations are sufficiently rare. In 

this case, a lower probability of violation must be 

specified. 

3-25
EXAMPLE COMPUTATION - HAND CALCULATION

(continued) 

6. For a (step 5) and coefficients of variation (step 3) compute 

ratio of mean effluent to effluent limit, Ra = CE /CEa and the 

resulting mean effluent concentration CE for each averaging 

period. 

Averaging Period 

Reduction Factor

Mean Effluent Concentration 

Ra CE 

1-day 0.281 2.81

7-day 0.439 4.39

30-day 0.643 6.43

7. Evaluate each mean effluent concentration using POM to 

compute me return period of acute criteria violation. Choose 

the appropriate averaging period. 

Return Period (years) for 

CO = 6.25 

Averaging 

Period CE 

Moments

Approximation

Quadrature 

Method 

1-day 2.81 31 281

7-day 4.39 7.7 31.8 > 10 years 

30-day 6.43 2.6 6.44

8. Establish appropriate permit limits for other averaging 

periods. CE = 4.39, 1-a = 1%. 

Averaging Period νCE Ra Permit Limita

1-day 0.70 0.281 15.6

7-day 0.40 0.439 10.0

30-day 0.20 0.643 6.83

a

Permit Limit = CE /Ra; 1% violation frequency. 

3-26
COMMENTARY

5. This calculation makes the connection between the 

effluent limit and the mean effluent concentration 

required to meet the effluent limit if it is assigned to 

daily values or 7 or 30 day averages. A treatment plant 

designed to produce CE and whose variability is as 

specified in (3) will meet the effluent limit with one 

percent violation frequency. 

7. The three treatment plant designs (the three mean 

effluent concentrations) and the daily effluent 

variability are used in PDM to compute the return period 

of an acute criteria violation. The moments approximation 

is sufficient if the return periods are significantly 

less than or greater than the 10 year criteria violation 

frequency being examined. In this case, the 7-day 

averaging period result is close to 10 years and the more 

accurate computer method is used to improve the accuracy 

of the calculation. The calculation indicates that a mean 

effluent concentration of CE = 4.39 and a daily νCE = 0.7 

is sufficiently protective for acute criteria violations. 

This, then, is the basis for the treatment plant design. 

8. The permit limits for the other averaging periods are now 

calculated to be consistent with the treatment plant 

design. That is, these permit limits are consistent with 

effluent mean and coefficients of variation as indicated, 

and specify the same performance. Thus, they are 

equivalent requirements. 

3-27
3.3 EXAMPLE COMPUTATION - COMPUTER PROGRAM

This section illustrates the use of the POM-PS computer program 

(included and described in Appendix D) to the solution of the 

example presented in the previous section. The site-specific 

conditions used to define input values in the previous section are 

used in this section as well. 

The PDM-PS is structured to accept inputs in the form of 

statistical parameters and ratios, determined readily from the 

data. The following ratios are entered for this example 

computation: 

Stream Flow Ratio 7Q10/QS = 23.3/467 = 0.05 

Effluent Dilution Ratio 7Q10/QE = 23.3/7.77 = 3.0 

Effluent Concentration CE /EL = (*) 

Reduction Factor 

(*) Reduction factor assigned depends on permit averaging period. 

As determined earlier. 

30 Day - - - - R = 0.643 

CE /EL 7 Day - - - - R = 0.439 

1 Day - - - R = 0.281 

The only other inputs called for are the coefficients of variation 

of stream flow, effluent flow, and effluent concentration, which 

have already been determined. 

The facing page illustrates the Input prompts that are displayed 

when the program is run, and the values entered in response to the 

prompts, in this case for evaluating the 30-day permit averaging 

period. 

3-28
COMMENTARY

DISPLAY AND PROMPTS RESPONSE ENTRIES

POINT SOURCE - RECEIVING WATER 

CONCENTRATION ANALYSIS 

+++++++++++++++++++++++++++++++++ 

INPUTS: COEF VAR OF QS,QE,CE 

RATIO...7Q10/avgQS 

RATIO...7Q10/avgQE 

RATIO... avgCE/EL 

BACKGROUND STREAM CONC (CS) 

IS ASSUMED TO BE ZERO 

+++++++++++++++++++++++++++++++++ 

ENTER COEF VAR OF QS,QE,CE?..........1.5, 0.2, 0.7 

ENTER FOLLOWING RATIOS: 

.......7Q10/avg QS?............. 0.05 

.......7Q10/avg QE?............... 3.0 

.......avgCE/EL?............. 0.643 

ENTER LOWEST, HIGHEST, AND 

INCREMENT OF MULT OF TARGET FOR 

WHICH% EXCEED IS DESIRED 

ENTER LOWEST, HIGHEST, AND 

INCREMENT OF MULT OF TARGET FOR 

WHICH% EXCEED IS DESIRED 

This prompt repeats after the 

selected range of values has 

been computed and displayed. It 

allows the user to be guided by 

output In selecting values and 

ranges for subsequent 

computations. 

0.01, 0.06, 0.01 

0.08, 0.36, 0.04 

0.40, 4.0, 0.2 

NOTE: The manual analysis presented earlier, computed the 

exceedance probability and recurrence interval for specific stream 

concentration values. The computerized computation generates these 

results for stream concentrations expressed as multiples of the 

target concentration (CL) that is explicitly assumed to result 

when 

Effluent Concentration CE = EL (the effluent limit) 

Effluent Flow QE = QE (average QE ) 

Stream Flow QS = 7Q10 (the design stream flow) 

3-29
EXAMPLE COMPUTATION - COMPUTER PROGRAM 

(continued) 

PROGRAM OUTPUT 

+++++++++++++++++++++ 

RECEIVING WATER CONC (CO) 

PROBABILITY DISTRIBUTION 

AND RETURN PERIOD 

FOR MULTIPLES OF TARGET CONC. 

DUE TO POINT SOURCE LOADS 

+++++++++++++++++++++ 

COEF VAR.....QS = 1.50 

COEF VAR.....QE = 0.20 

COEF VAR.....CE = 0.70 

7Q10/ave QS = 0.05 

7Q10/ave QE = 3.00 

ave CE/EL = 0.64 

+++++++++++++++++++++ 

STREAM CONCENTRATION (CO) 

MULT OF 

TARGET 

(CO/CL)

PERCENT 

OF TIME 

EXCEEDED

RETURN 

PERIOD 

(YEARS) 

0.01 92.699 0.003

0.02 80.916 0.003

0.03 71.039 0.004

0.04 62.788 0.004

0.05 55.862 0.005

0.08 40.808 0.007

0.12 28.659 0.010

0.16 21.170 0.013

0.20 16.201 0.017

0.24 12.728 0.022

0.28 10.206 0.027

0.32 8.320 0.033

0.36 6.875 0.040

0.40 5.746 0.048

0.60 2.650 0.103

0.80 1.399 0.190

1.00 0.804 0.341

1.20 0.490 0.559

1.40 0.312 0.878

1.60 0.206 1.331

1.80 0.140 1.961

2.00 0.097 2.821

3-30
COMMENTARY

--- This output is for a 30-day permit average period (Ra = 0.643) 

The range of values selected here is broad enough to 

facilitate construction of probability and recurrence 

interval plots. 

Stream concentrations listed are in terms of a ratio to the 

target concentration (CL). In this example, the target stream 

concentration is: 

CL = 2.5 

Actual stream concentration is this value multiplied by the 

listed value: e.g., the multiple of Target (CO/CL) = 0.4 

Corresponding stream concentration is: 

0.4 X 2.5=1.0 

Since the acute-to-chronic ratio for pollutant (P) is 

6.25/2.50 = 2.5, acute exceedences are reflected by multiple 

2.5. 

Probability or recurrence interval plots can be constructed, 

simply by plotting the values listed in the computer 

printout. 

Note that the probability distribution of stream 

concentrations deviates from-log-normal (a straight line) at 

the higher exceedance percentiles. 

3-31
EXAMPLE COMPUTATION - COMPUTER PROGRAM 

(continued} 

STREAM CONCENTRATION (CO) (cont.) 

MULT OF 

TARGET 

(CO/CL) 

PERCENT 

OF TIME 

EXCEEDED 

RETURN 

PERIOD 

(YEARS) 

2.20 0.069 3.977

2.40 0.050 5.507

2.60 0.036 7.509

2.80 0.027 10.098

3.00 0.020 13.411

3.20 0.016 17.612

3.40 0.012 22.894

3.60 0.009 29.482

3.80 0.007 37.640

4.00 0.006 47.674

3-32
COMMENTARY

% EQUAL OR LESS THAN 

Figure 3-6 - Concentration versus probability for PDM-PS 

 computation 

RETURN PERIOD-YEARS 

Figure 3-7 - Concentration versus mean recurrence Interval for POM 

 PS computation 

3-33
EXAMPLE COMPUTATION - COMPUTER PROGRAM 

(continued) 

To examine stream concentration effects for other permit averaging 

periods, repeat the analysis, substituting the appropriate value 

for the reduction factor (R = CE /EL) 

The return period curves provide a useful summary and perspective; 

however, the evaluation can be performed without constructing the 

graph. In this case, the range of concentrations specified might 

(as shown below) simply bracket those of principal interest. In 

this case, a range of CO/CL from 0.5 to 3 is selected, because the 

chronic limit (CL= 1), and the acute limit to be exceeded no more 

than once every 10 years 1s CO/CL = 2.5. 

The relevant portions of the output for the three permit averaging 

periods are shown below: 

STREAM CONCENTRATION (CO) 

MULT OF

TARGET 

(CO/CL) 

PERCENT

OF TIME 

EXCEEDED 

RETURN

PERIOD 

(YEARS) 

30-Day Average 0.50 3.818 0.072

1.00 0.804 0.341

CE/EL = 0.643 1.50 0.252 1.085

2.00 0.097 2.821

2.50 0.043 6.443

3.00 0.020 13.411

7-Day Average 0.50 1.717 0.160

1.00 0.272 1.008

CE/EL = 0.439 1.50 0.069 3.957

2.00 0.023 12.149

2.50 0.009 31.819

3.00 0.004 74.364

1-Day Average 0.50 0.560 0.489

1.00 0.060 4.601

CE/EL = 0.281 1.50 0.011 23.866

2.00 0.003 90.571

2.50 0.001 281.076

3.00 0.000 756.249

3-34
COMMENTARY

In this case a different averaging period would be selected than 

that based upon the manual computation. Acute criteria exceedences 

have a mean recurrence interval shorter than 10 years for a 30-day 

permit average, so it would be rejected in favor of a 7-day 

average, which meets the guideline. 

Note that the exact computation using the computer program 

indicates a 5.4 year return period for acute violations, compared 

with a 2.6 year return period estimated by the manual 

approximation. The manual approximation tends to give conservative 

projections for the longer return periods that are of interest, 

though differences vary depending on specific input conditions. 

Hence, there will be marginal cases where the approximate 

computation may reject a 30-day average inappropriately. 

On the other hand, wherever the manual approximation accepts a 30-

day permit average as appropriate, it is safe to assume that the 

more exact computation will not modify the choice. 

For the site specific conditions assumed for the example analysis: 

• Any pollutant with an acute-to-chronic 

ratio of 9.5 or greater would, based on the 

manual approximation, always be assigned a 

30-day permit average. 

• The POM-PS computation extends this to 

pollutants with acute-to-chronic ratios of 

3 or more. 

NOTE: EPA interprets any return period greater than 25 years as 

being highly improbable 

3-36
CHAPTER 4 

RANGE OF EXPECTED VALUES FOR STREAMS IN U.S. 

As illustrated in Chapter 3, the method can be applied to any 

site specific evaluation for which the relevant statistical 

parameters are available or can be estimated. The purpose of this 

section is to present a concise summary of the results of such 

computations for the range of site conditions that are likely to 

be encountered in practice. This chapter provides such a 

compilation along three lines. Section 4.1 describes the basis for 

the input values selected to provide a representative range of 

site conditions, and presents the results of an analysis using 

these typical ranges in the methodology described previously. The 

stream flow characteristics were determined from an analysis of 

180 streams and rivers; treatment plant effluent characteristics 

are based on analysis of data from over 400 POTWs. The results in 

this section apply for conservative (nonreacting) pollutants. 

Section 4.2 describes how the information provided by such an 

analysis can be used as a screening tool for selecting permit 

averaging periods. Section 4.3 presents results of a similar 

analysis, except that it is specific to oxygen depletion by 

biochemical oxygen demand (BOO) loadings. Section 4.4 extends the 

analysis for conservative pollutants to the special case of 

streams that are highly effluent dominated, including those with 

significant zero-flow Periods. 

4.1 Analysis for Conservative Substances

The review of stream flow and effluent statistics presented 

in Appendix 8 indicates that the following ranges are reasonable. 

Effluent 

4-1 
concentration variability, (νCE), is in the range of νCE = 0.3 - 

1.1. Effluent flow variability, (νQE), is generally small relative 

to stream flow variability and, therefore, does not greatly 

influence the computation. νQE = 0.2 is consistently used. Stream 

flow variability follows from the empirical relationship of νQS and 

7Q10/QS . For a specified ratio, the range of νQS, as indicated by 

the data discussed in Appendix B, is used. The ratio 7Q10/QS

varies considerably. A representative range is 7Q10/QS = 0.01 - 

0.25. Finally, the magnitude of the effluent flow relative to the 

stream flow is specified by the effluent dilution ratio: 7Q10/QE . 

A range from 7Q10/QE = 1 - 50 is chosen to represent effluent 

dominated streams and large streams with small discharges. A 10 

year return period has been selected as the acute criteria 

violation frequency. 

In order to compute the ratio of the mean effluent 

concentration to the effluent limit Ra = CE /EL, it is assumed that 

the permit violation frequency is one percent. The final 

specification required is the relationship of 7 and 30 day average 

effluent concentrations to the daily effluent concentration 

coefficient of variation, νCE. Based upon the data presented in 

Table C-2, it appears reasonable to expect that the 7-day averages 

have a coefficient of variation that 1s 0.8 of the dally values, 

and that 30 day averages have a coefficient of variation of 0.6 of 

the daily values. Thus, the reduction factors used are: 

Coefficient of Variation 

of Daily Values 

Reduction Factor, Ra

a = 99 percent 

νCE 1-day 7-day 30-day

0.3 0.527 0.593 0.671

0.7 0.281 0.340 0.425

1.1 0.187 0.229 0.296

4-2 
The results of these computations are summarized in Figure 

4-1 and given in detail in Tables 4-1 to 4-4. The three choices 

for permit average are shown. Each group of bars represents the 

range in effluent concentration variability, νCE. Each individual 

bar represents a particular effluent dilution, 7Q10/QE . Finally, 

the length of each bar represents the range that results from the 

range of stream flow variability (7Q10/QS = 0.01 - 0.25) and the 

associated coefficient of variation, νQS. The ordinate is the 

downstream concentration (in multiples of the chronic criteria) 

which has a 10 year return period. 

A number of features are immediately apparent. For pollutants 

with an acute to chronic ratio of greater than 10, no acute 

criteria violations are projected over the ranges Investigated, 

and 30-day average permit specifications appear to be sufficiently 

protective. For acute-to-chronic ratios of less than 10, site 

specific considerations are important. 

The results are most sensitive to the stream flow parameter 

7Q10/QS , as can be seen from the range covered by each bar. For 

example, the last bar in the figure, 30-day permit averaging 

period, 7Q10/QE = 50, νCE=1.1, covers the range from β= 0.9 to 4.6, 

corresponding to 7Q10/QS = 0.01 and νQS = 2-4. 

Following, in order of decreasing sensitivity, is the 

effluent dilution ratio: 7Q10/QE . A significant distinction can be 

found between 

1

The EPA is presently considering the issue of allowable duration 

and frequency of exposure to toxicity. Based upon this word, 

duration and frequencies used as the decision criteria may change. 

This guidance does not recommend any particular minimum acceptable 

duration or frequency. 

4-3 
*INDICATES THE STREAM CONCENTRATION (CO) WHICH WILL BE EXCEEDED WITH A 

FREQUENCY OF ONCE IN TEN YEARS, EXPRESSED AS A MULTIPLE OF THE CHRONIC 

CRITERIA (CL)

Figure 4-1 - Effect of permit averaging period on stream 

 concentrations for conservative substances: 

 general analysis 

4-4 
TABLE 4-1 - Averaging period selection matrix for conservative substanc

- 7Q10/QE = 50 

Stream Effluent 

νCE = 0.3 Effluent 

νCE = 0

Flow 

7Q10/QS 

Avg. Q 

Estim

a

t

e

of 

Variability 

Range 

νQS

30-Day Avg. 

7-Day Avg. 

1-Daily Max. 

30-Day Avg. 

7-Day Avg. 

L

O 2.00 1.0 0.9 0.8 0.9 0.7 

0.01 

P

R

O

B 3.00 3.0 2.7 2.4 2.5 2.0 

HI 4.00 6.1 5.4 4.8 4.9 3.9 L

O 1.00 0.9 0.8 0.7 0.9 0.7 

0.05 

P

R

O

B 1.50 2.3 2.0 1.8 2.2 1.7 

HI 2.00 4.7 4.2 3.7 4.2 3.4 L

O 0.75 1.0 0.9 0.8 1.1 0.8 

0.10 

P

R

O

B 1.00 1.7 1.5 1.4 1.8 1.4 

HI 1.50 4.4 3.9 3.4 4.2 3.3 L

O 0.60 1.0 0.9 0.8 1.2 0.9 

0.15 

P

R

O

B 0.90 2.0 1.8 1.6 2.2 1.7 

HI 1.25 4.1 3.7 3.3 4.1 3.3 L

O 0.50 1.3 1.1 1.0 1.6 1.3 

0.25 

P

R

O

B 0.75 2.4 2.1 1.0 2.6 2.1 

HI 1.00 4.1 3.6 3.2 4.2 3.4 

4-5 
TABLE 4-2 - Averaging period selection for conservative substances: effluen

= b 

Stream Effluent 

νCE = 0.3 Effluent 

νCE = 0

Flow 

7Q10/ 

Avg. Q 

Estim

a

t

e

of 

Variability 

Range 

νQS

30-Day Avg. 

7-Day Avg. 

1-Daily Max. 

30-Day Avg. 

7-Day Avg. 

L

O 2.00 1.0 0.9 0.8 0.9 0.7 

0.01 

P

R

O

B 3.00 2.2 1.9 1.7 2.0 1.6 

HI 4.00 3.1 2.8 2.5 3.1 2.5 L

O 1.00 0.9 0.8 0.7 0.9 0.8 

0.05 

P

R

O

B 1.50 1.9 1.7 1.5 1.9 1.5 

HI 2.00 2.9 2.6 2.3 3.0 2.4 L

O 0.75 1.0 0.9 0.8 1.1 0.9 

0.10 

P

R

O

B 1.00 1.5 1.4 1.2 1.7 1.4 

HI 1.50 2.8 2.5 2.2 3.1 2.5 L

O 0.60 1.0 0.9 0.8 1.2 1.0 

0.15 

P

R

O

B 0.90 1.8 1.6 1.4 2.0 1.6 

HI 1.25 2.8 2.5 2.2 3.2 2.5 L

O 0.50 1.3 1.1 1.0 1.6 1.3 

0.25 

P

R

O

B 0.75 2.0 1.8 1.6 2.4 1.9 

HI 1.00 2.8 2.5 2.2 3.4 2.7 

4-6 
TABLE 4-3 - Averaging period selection matrix for conservative substances: 

7Q10/QE = 3 

Stream Effluent 

νCE = 0.3 Effluent 

νCE = 0

Flow 

7Q10/ 

Avg. Q 

Estim

a

t

e

of 

Variability 

Range 

νQS

30-Day Avg. 

7-Day Avg. 

1-Daily Max. 

30-Day Avg. 

7-Day Avg. 

L

O 2.00 1.0 0.9 0.8 0.9 0.7 

0.01 

P

R

O

B 3.00 1.9 1.7 1.5 1.9 1.5 

HI 4.00 2.6 2.3 2.0 2.7 2.2 L

O 1.00 0.9 0.8 0.7 1.0 0.8 

0.05 

P

R

O

B 1.50 1.7 1.5 1.3 1.8 1.5 

HI 2.00 2.5 2.2 1.9 2.7 2.2 L

O 0.75 1.0 0.9 0.8 1.2 0.9 

0.10 

P

R

O

B 1.00 1.5 1.3 1.2 1.7 1.3 

HI 1.50 2.4 2.2 1.9 2.8 2.3 L

O 0.60 1.0 0.9 0.8 1.3 1.0 

0.15 

P

R

O

B 0.90 1.7 1.5 1.3 2.0 1.6 

HI 1.25 2.4 2.2 1.9 2.9 2.3 L

O 0.50 1.3 1.1 1.0 1.6 1.3 

0.25 

P

R

O

B 0.75 1.9 1.6 1.5 2.3 1.9 

HI 1.00 2.5 2.2 1.9 3.1 2.5 

4-7 
4-8 
TABLE 4-4 - Averaging period selection matrix for conservative substance

- 7Q10/QE = 1 

Stream Effluent 

νCE = 0.3 Effluent 

νCE = 0

Flow 

7Q10/ 

Avg. Q 

Estim

a

t

e

of 

Variability 

Range 

νQS

30-Day Avg. 

7-Day Avg. 

1-Daily Max. 

30-Day Avg. 

7-Day Avg. 

L

O 2.00 1.0 0.8 0.8 1.0 0.8 

0.01 

P

R

O

B 3.00 1.5 1.3 1.2 1.7 1.3 

HI 4.00 1.8 1.6 1.4 2.1 1.7 L

O 1.00 0.9 0.8 0.7 1.1 0.9 

0.05 

P

R

O

B 1.50 1.4 1.3 1.1 1.7 1.4 

HI 2.00 1.8 1.6 1.4 2.2 1.8 L

O 0.75 1.0 0.9 0.8 1.3 1.0 

0.10 

P

R

O

B 1.00 1.3 1.2 1.1 1.7 1.4 

HI 1.50 1.8 1.6 1.4 2.4 1.9 L

O 0.60 1.1 1.0 0.8 1.4 1.2 

0.15 

P

R

O

B 0.90 1.5 1.3 1.2 1.9 1.6 

HI 1.25 1.8 1.6 1.4 2.5 2.0 L

O 0.50 1.3 1.1 1.0 1.8 1.4 

0.25 

P

R

O

B 0.75 1.6 1.4 1.3 2.2 1.8 

HI 1.00 1.9 1.7 1.5 2.6 2.1 

4-9 
the effluent nominated streams, 7Q10/QE < 5, and the large stream 

case, QE = 50 for the latter cases, the stream flow variability is 

a more important determinant of the normalized downstream 

concentration. Finally, the effluent variability, νCE= affects the 

results by approximately a factor of 2, all other things being 

equal. 

4.2 Use As a Screening Tool

It is suggested that Figure 4-1 may be used as a screening 

tool to separate the cases which can be dealt with immediately 

from those for which more site specific information is required. 

For the latter cases, the flow ratios, 7Q10/QE and 7Q10/QS can 

usually be found quite easily so that a more specific answer can 

be found in Tables 4-1 to 4-4. The final determinant, νQS, requires 

a log-normal analysis of the stream flow record. Since this is 

reasonably straightforward, a more refined analysis is not 

excessively burdensome and would serve to reduce the range of 

possible values of β, from which the permit averaging decision can 

be made. 

As an example of such a screening analysis, consider the 

hypothetical case of a state establishing permit averaging periods 

for phenol. Phenol has an acute-to-chronic ratio of 4, so that 

stream concentrations which exceed a multiple of 4 times the 

chronic concentration will not be accepted (assuming that the 

acute criteria is not to be exceeded on a daily basis more often 

than once every 10 years). 

Comparing the bars on Figure 4-1 with the multiple of β = 4, the 

following conclusions relative to the permit averaging period can 

be 

4-10
drawn. For situations with an effluent dilution ratio of 5 or 

less (7Q10/QE ≤ 5): 

a. A 30-day permit averaging period will be selected whenever 

the νCE is 0.7 or less. 

b. Where νCE =1.1 a 7-day permit averaging period will meet 

requirements under all reasonable possibilities of stream 

flow variability (νQS). (The upper ends of the bars correspond 

to high values of νQS.) 

c. Even for effluent variability as high as νCE = 1.1, there will 

be many streams where it would be appropriate to select a 30-

day permit average, since only the upper end of the bars 

exceeds a multiple of 4. 

For an effluent dilution ratio 7Q10/QE = 5, the third column from 

the right (νCE =1.1; 30-day permit average) in Table 4-2 indicates 

that only the highly variable stream flows approach violations 

using a 30-day permit average. State records could be examined to 

determine if the set of streams under consideration (or a 

representative set from Appendix C) experiences νQS in this range. 

A conservative decision, then, would be to select a 7-day 

permit averaging period, although a site-specific assessment of 

stream flow variability or a restriction of νQS values could be 

expected (in most cases) to support selection of a 30-day permit 

averaging period. 

4-11
4.3 Preliminary Analysis for Dissolved Oxygen

The choice, of permit averaging periods for effluent limits 

of oxygen-consuming pollutants, such as BOD or ammonia, is a more 

complex problem than that addressed in the previous sections. The 

variations of the minimum or critical DO are caused not only by 

effluent concentration and dilution fluctuations, which are 

addressed by the probabilistic dilution model, but also by 

fluctuations in reaction rates and other sources and sinks of DO, 

such as algal production, respiration, and sediment oxygen demand. 

Stream flow and temperature variations affect these parameters, 

the latter also determining the DO saturation. A comprehensive 

probabilistic analysis that would include these effects as well is 

beyond the scope of this report. 

It is desirable, however, to provide at least a preliminary 

analysis for suitably restricted cases that are amenable to 

analysis using the probabilistic dilution model. The method to be 

employed makes use of the similarity of the formula for critical 

DO deficit for those streams for which the simple Streeter-Phelps 

formulation is adequate, and the dilution equation. The principal 

assumptions are (1) a single point source of BOD is the only DO 

sink; (2) the stream flow, geometry and reaction rates are 

spatially constant; and (3) the reaction rates are temporally 

constant. For this restrictive situation, the critical or maximum 

dissolved oxygen deficit (Dc) is a function of the reaeration rate 

(Ka), the BOD oxidation rate (Kd) and the ultimate-to-5-day BOD 

ratio. 

4-12
The Streeter-Phelps equation can be solved for the 

critical or dissolved oxygen deficit (Dc): 

 (4-1) 

where: 

CE = treatment plant effluent BOD5 concentration. 

F = ratio of ultimate/5-day BOD. Stream calculations are 

based on ultimate BOD; effluent criteria on 5-day BOD. 

φ = stream dilution factor QE/(QS = QE). 

P = stream purification factor; for a BOD oxidation rate (Ka) 

and stream reaeration rate (Ka) 

(Note that if the purification factor were constant then Equation 

4-1 would be formally equivalent to the dilution equation analyzed 

previously.) One remaining difficulty is that it is not the 

critical DO deficit (Dc) that is of concern but rather the critical 

dissolved oxygen (DOc) itself: 

(4-2) 

which is a function of stream temperature through the DO 

saturation concentration, Csat. Hence, the applicability of 

probabilistic dilution to the dissolved oxygen problem requires 

that the analysis be restricted to periods for which temperatures 

are essentially constant and fluctuations in the purification 

factor (P) are small. 

4-13
An evaluation of this latter effect can be made as 

follows. A relationship between P and stream depth, H, which 

follows from Ka and K<j versus depth relationships is [3]: 

 (4-3) 

and for many streams, depth is proportional to total stream flow, 

QT, to a power H = Qm

T with m = 0.4 - 0.6. Thus, 

(4-4) 

Consider Equation 4-1 for critical deficit. Taking natural logs 

and applying the formula for the variance of a sum of independent 

random variables yields: 

(4-5) 

wehere QT = QS + QE. This equation, of course, ignores the fact 

that and QT are correlated, but the point is that n2

 = 0.09 - 0.25 

so that if the log variance of QT is comparable to the effluent 

concentration log variance, then the n2 term is not a major 

contribution to critical deficit log variance; hence, it can be 

neglected. The fact that dilution (φ ) and total stream flow are 

negatively correlated would further reduce the effect. 

Hence, the key observation is that 1f it were possible to 

restrict consideration to those flows for which νQS = νCE, then 

purification factor fluctuations would not be very significant and 

probabilistic dilution can be applied. If these flows also 

correspond to periods of 

4-14
approximately constant temperature, then the two requirements 

for applying probabilistic dilution to critical dissolved oxygen 

have been met. For a site-specific analysis, the obvious solution 

is to seasonally analyze the stream flow and temperature data and 

apply probabilistic dilution, making any necessary corrections for 

purification factor variations. However, for the general case 

considered here, an alternate approach is required. 

Consider, instead, restricting consideration to that period 

of the year during, which flows are low. This period corresponds, 

presumably, to the period of time during which 7Q10 occurs, and 

includes the conditions for which the WLA was performed. 

Considering this period alone significantly reduces the 

variability of the stream flows to be considered. If, in addition, 

it can be argued that these low flows tend to occur during the 

same season each year, then the temperature variation is less than 

the annual variability and will be less significant as well. 

Hence, for these low flow periods, the assumption of constant P is 

much more realistic. 

The technical problem to be solved is to compute the 

reduction in the average stream flow and coefficient of variation 

when flows are restricted to the low values for this restricted 

period. We restrict consideration to the lowest one-sixth of the 

total population. This corresponds to an average of 2 out of 12 

months in each year, and the presumption is that this period 

recurs during the same months each year so that the temperaturevariation during this restricted period is small. This 

simplification also assumes that the lower one-sixth of the daily 

stream flows occur only in the two month period when temperature 

and reaction rates are assumed to be approximately constant. 

4-15
 As indicated earlier, a statistical analysis of actual stream 

data, stratified by month or critical season, could be performed 

to provide actual results and avoid the need for this type of 

estimate. However, data of this type are not presently available. 

The estimation described There is performed in order to allow a 

preliminary analysis for BOO/DO to be made. 

The computation of the required statistical parameters, the 

stream flow average and coefficient of variation for flows 

restricted to the lower a-quantil e of the total population, is 

straightforward! For log-normal random variables, it can be shown 

that these conditional moments, denoted by primes, are: 

QS = Q(σlnQS + Za)/Q(Za) (4-6) 

QS 

ν2

QS = exp(σ2lnQS) Q(2σ2lnQS + Za) Q(Za) - 1 (4-7) 

Q2

(σlnQS + Za) 

where Q(Z*) = Pr Z > Z* for Z, a standard normal random variable, 

and Za are the Z scores for the a-quantile which is the upper bound 

for the flows being considered. For a = 1/6, Za = 0.967. Table 4-5 

presents the results. These corrections, when applied to 7Q10/QS

and νQS in the first two columns of Tables 4-1 to 4-4 adjust these 

parameters to represent the low flow periods. For highly variable 

streams, νQS and therefore σlnQS are large and the corrections are 

quite substantial. 

Reduction factors for the mean range from 0.45 to 0.024 for 

the highly variable streams. The range in coefficient of variation 

is sharply 

4-16
TABLE 4-5 - Conditional moments for the low flow subpopulation 

(a = 16.75)

Coefficient of Variation 

for 

Entire Record 

νQS

Reduction 

in 

Mean 

QS’/QS 

Reduced 

Coefficient of 

Variation 

νQS’

0.50 0.450 0.188 

0.60 0.384 0.216 

0.75 0.306 0.254 

0.90 0.247 0.287 

1.00 0.216 0.306 

1.25 0.158 0.348 

1.50 0.120 0.381 

2.00 0.0761 0.431 

3.00 0.0389 0.500 

4.00 0.0241 0.547 

This table provides a basis for a preliminary estimate of the 

average stream flow and flow variability during critical low flow 

periods, relative to overall long-term characteristics. For sitespecific cases, the actual values can be determined readily from a 

statistical analysis of stream flows during the selected critical 

period of the year. 

4-17
compressed from νQS = 0.5 - 4.0 to νQS = 0.19 - 0.55, so that the 

sub-population of low flows fluctuates much less violently than 

the entire population, which includes the annual cyclical 

variation as well. 

A 10 year return period was selected for consistency with the 

general analysis, but since only one-sixth of the flow population 

is being considered, and we assume that no DO acute criteria 

violations occur during the remaining higher flows, the exceedence 

probability to be applied in the probabilistic dilution 

calculation is a 10/6 = 1.67 year return period. Figure 4-2 and 

Tables 4-6 to 4-8 present the results. 

In order to properly evaluate the computations, it is 

necessary to realize that they apply to 10 year return period 

critical deficit ratios. To convert critical DO concentrations to 

the deficit ratio (p) shown by the tables, the DO standard (CL) 

the DO saturation (Csat) used in the WLA, and the DO concentration 

taken to represent an acute criteria value are required. For most 

reasonable combinations of these values, the ratio will be between 

approximately 2.0 and 2.5. For example, if CS = 3, CL = 5, and 

acute DO = 2, then β=2.0. Alternatively, if these concentrations 

are CS = 9.0, CL = 6.0, acute DO = 1.5, then (the acute-to-chronic 

deficit ratio) β =2.5. 

Appropriate permit averaging periods are seen in Tables 4-6 

to 4-8 to be strongly influenced by local conditions of effluent 

load and stream flow variability. Because of this, a general 

statement on permit averaging period for effluent BOD/DO is not 

possible; it must be selected on the basis of site conditions. 

4-18
*INDICATES THE STREAK CONCENTRATION (co) WHICH WILL IE EXCEEDED 

WITH A FREQUENCY Of ONCE IN TEN YEARS, EXPRESSED AS A MULTIPLE OF 

THE CHRONIC CRITERIA (CL). 

Figure 4-2 - Effect of permit averaging period on stream, 

 concentrations for BOD/DO 

4-19
TABLE 4-6 - Permit averaging period selection matrix for MOD/DO: of 

7Q10/OE = 5 

Stream Flow Characteristics Effluent 

νCE = 0.3 Effluent 

νCE = 0

All Periods Low Flow Periods 7-

Day 

Avg.

7Q10/QS 

νQS 7Q10/QS’

νQS’

30-Day 

Avg. 

7-

Day 

Avg.

1-

Day 

Avg. 

30-Day 

Avg. 

2.00 0.13 0.43 0.5 0.4 0.4 0.5 0.4 

0.01 3.00 0.26 0.50 1.0 0.9 0.8 1.1 0.9 

4.00 0.41 0.55 1.6 1.4 1.3 1.8 1.4 1.00 0.23 0.31 0.6 0.6 0.5 0.8 0.6 

0.05 1.50 0.42 0.38 1.2 1.1 1.0 1.4 1.2 

2.00 0.66 0.43 1.9 1.7 1.5 2.2 1.8 0.75 0.33 0.25 0.8 0.7 0.6 1.0 0.8 

0.10 1.00 0.46 0.31 1.2 1.0 0.9 1.4 1.2 

1.50 0.83 0.38 2.0 1.8 1.6 2.5 2.0 0.60 0.39 0.22 0.9 0.8 0.7 1.1 0.9 

0.15 0.90 0.61 0.29 1.4 1.3 1.1 1.8 1.4 

1.25 0.95 0.35 2.2 1.9 1.7 2.7 2.2 0.50 0.55 0.19 1.2 1.0 0.9 1.5 1.2 

0.25 0.75 0.82 0.25 1.7 1.5 1.3 2.2 1.8 

1.00 1.16 0.31 2.4 2.1 1.9 3.0 2.4 

Critical DO deficit exceeded one day In 10 years as a multiple target defic

4-20
TABLE 4-7 - Permit averaging period selection matrix for BOD/DO: eff

7Q10/QE = 3 

Stream Flow Characteristics Effluent 

νCE = 0.3 Effluent 

νCE = 0

All Periods Low Flow Periods 7-

Day 

Avg.

7Q10/QS 

νQS 7Q10/QS’

νQS’

30-Day 

Avg. 

7-

Day 

Avg.

1-

Day 

Avg. 

30-Day 

Avg. 

2.00 0.13 0.43 0.5 0.4 0.4 0.6 0.5 

3.00

0.55 1.2 1.5 

0.01 0.26 0.50 1.0 0.9 0.8 1.2 0.9 

4.00 0.41 0.55 1.5 1.3 1.2 1.7 1.4 1.00 0.23 0.31 0.7 0.6 0.5 0.8 0.7 

0.05 1.50 0.42 0.38 1.2 1.1 1.0 1.5 1.2 

2.00 0.66 0.43 1.8 1.5 1.4 2.2 1.7 0.75 0.33 0.25 0.8 0.7 0.7 1.0 0.8 

0.10 1.00 0.46 0.31 1.2 1.0 0.9 1.5 1.2 

1.50 0.83 0.38 1.9 1.7 1.5 2.4 1.9 0.60 0.39 0.22 0.9 0.8 0.7 1.2 0.9 

0.15 0.90 0.61 0.29 1.4 1.2 1.1 1.8 1.4 

1.25 0.95 0.35 2.0 1.8 1.6 2.6 2.1 0.50 0.19 1.0 0.9 1.2 0.75 0.82 0.25 

0.31 

0.25 1.7 1.5 1.3 2.2 1.7 

1.00 1.16 2.2 1.9 1.7 2.9 2.3 

Critical DO deficit exceeded one day In 10 years as a Multiple target defic

4-21
TABLE 4-8 - Permit averaging period selection matrix for BOD/DO: eff

7Q10/QE = 1

Stream Flow Characteristics Effluent 

νCE

= 0.3 Effluent 

νCE

= 0

All Periods Low Flow Periods 

7Q10/QS 

7-

Day 

Avg.

0.7 

νQS 7Q10/QS’

νQS’

30-Day 

Avg. 

7-

Day 

Avg.

1-

Day 

Avg. 

30-Day 

Avg. 

2.00 0.13 0.43 0.6 0.5 0.5 0.6 

3.00 0.26 0.50 1.0 0.9 0.8 1.3 1.0 

0.55 

0.7 0.6 

1.0 

 1.3 

0.39 0.9 0.8 1.4 1.1 

1.1 1.5 

1.25 1.9

1.1 1.4 1.7 

0.01 

4.00 0.41 1.4 1.2 1.1 1.7 1.4 1.00 0.23

0.42 

0.310.38 

0.8 1.0 0.8 

1.50 1.2 1.1 1.6

2.1 

1.31.6

0.05 

2.00 0.66 0.43 1.5 1.4 1.2 0.75 0.33 0.25 0.9 0.8 0.7 1.2 1.0 1.00 0.46 0.31 1.2

1.6 

1.11.5 

0.91.3 

1.62.2 

0.10 

1.50 0.83 0.38 1.8 0.600.90 

0.22 1.0 

0.61 0.29

0.35 

1.4 1.2 1.8

2.3 

0.15 

0.95 1.7 1.5 1.3 

0.500.75 

0.550.82 

0.190.25 

1.21.5 

1.0 1.7 1.3 

0.25 1.2 2.1

1.00 1.16 0.31 1.8 1.6 1.4 2.5 2.0 

Critical DO deficit exceeded one day in 10 years as a multiple target defic

4-22
A table for the effluent dilution ratio (7Q10/QE ) equal to 

50 has not been prepared for BOD/DO. For small discharges entering 

larger streams, it is likely that an effluent BOD limit determined 

from a steady -state WLA analysis would be greater than the 

technology-based limit which would be used in the permit. The use 

of the standard matrix table* which would show a higher pattern of 

violations, would tend to be misleading, since the computations 

and the tables assume that the allowable effluent concentration 

determined from a WLA becomes the effluent limit (EL) specified by 

the permit. 

It should be emphasized at this point that the dissolved 

oxygen analysis presented in this section is meant only as a 

preliminary application. There are, as yet, no verification 

examples that support the applicability of a probabilistic 

dilution/critical deficit analysis, It has not been shown that 

actual stream 00 data conform to the probabilistic assumptions and 

simplifications used in this preliminary analysis. Further, it is 

well known that the DO distribution in streams cannot always be 

described by the simplest (Streeter-Phelps) model. Upstream 

sources of BOO and deficit are common, as are nitrification, algal 

effects, and sediment oxygen demand. A more comprehensive analysis 

would be required to Incorporate these effects into a calculation 

of the effect of selecting a permit averaging period.

4.4 Analysis for Conservative Substances 1n Effluent-Dominated 

Streams

An effluent -dominated stream 1s defined, for the purpose of 

this analysis, as one in which the effluent flow exceeds the 

design stream flow 

4-23
It should be recognized that as the degree of dilution 

decreases, a WLA-based EL becomes increasingly restrictive. When 

the design stream flow is zero, the effluent limit must equal the 

stream target concentration (CL). 

(e.g., the 7Q10). There are then two bounds to this analysis. 

The upper sound is the effluent dilution ratio 7Q10/avg QE = 1, 

which was the lowest dilution ratio examined in Section 4.1. The 

lower bound is provided by the case where the design stream flow 

is zero (7Q10 = 0). 

While the degree of effluent domination has a subsequent 

influence on the magnitude of an EL assigned in a permit, the 

screening analysis results presented below suggest that in most 

situations, a 30-day permit averaging period will be adequate for 

effluent dominated streams. 

The results of a broad hypothetical analysis of affluent 

dominated streams are summarized in Figure 4-3 and Table 4-9, 

using the format used earlier to illustrate the influence of 

permit averaging period, effluent variability and dilution ratio. 

o The bars on the provide the upper bound; i.e., the 

condition where 7Q10/avg QE = 1 (these results were also shown in 

Figure 4-1).

o The bars on the left represent an effluent dilution ratio 

of 7Q10/avg QE = 0.1, that is, where effluent How is ten times 

greater than design stream flow. High variability of daily flow is 

expected for such streams, together with a very small ratio of 

stream design flow to average stream flow. The screening analysis 

assumes that the coefficient of variation ranges between νQS =

4-24
Figure 4-3 - Effect of permit averaging period on stream 

 concentrations for conservative substances in 

 effluent-dominated stream

*INDICATES THE STREAM CONCENTRATION (co) WHICH WILL BE EXCEEDED 

WITH A FREQUENCY OF ONCE IN TEN YEARS, EXPRESSED AS A MULTIPLE OF 

THE CHRONIC CRITERIA (CL). 

4-25
TABLE 4-9 - Averaging period selection matrix for a fluent-do

Stream Effluent 

νCE = 0.3 Effluent 

νCE = 0.7 

Flow 

7Q10/ 

Avg. Q 

 Estimate 

of 

Variability 

Range 

νQS

LO 

30-Day Avg. 

7-Day Avg. 

1-Daily Max. 

0.5 

30-Day Avg. 

7-Day Avg. 

1-Daily Max. 

2.00 0.6 0.6 0.6 0.5 0.4

1.0 0.9 

.6 0.6 0.5 

1.0 

2.00 

1.2 

0.1 

B

HI P

R

O

B 4.00

5.00 

1.1 1.0 1.2 0.9 0.8 

HI 1.5 1.3 1.2 1.6 1.3 1.1 L

O 2.00 0.7

1.1 

0 0.8 0.6 

P

R

O

B 

HI 

4.00

5.00 

0.9 1.2

1.6 

1.0 0.8

1.0 

0.5 

1.3 1.2 1.1 1.3 

L

O 0.9 0.8 0.7 1.0

1.4 

 0.8 0.7 

P

R

O

B 4.00

5.00 

1.1 1.0 0.9 1.1 0.9 

0.2 

HI 1.3 1.1 1.0 1.7 1.3 1.1 L

O

P

R

O

2.00 1.0 0.9 0.8 1.0 0.8 

4.00

5.00 

1.2 1.1 0.9 1.6

1.8 

1.3 1.1 

1.3 1.2 1.0 1.5 1.2 

4-26
2 and νQS = 5, and estimates a stream flow ratio 7Q10/avg QS = 

0.005, for this condition near the lower bound for effluentdominated streams.

The conditions under which the design stream flow is greater 

than zero are listed in more detail in Table 4-9. Results for 

several additional intermediate effluent dilution ratios 

(7Q10/QE = 0.2 and 0.5) are also presented. A comparison of 

results for an effluent ratio of 1.0 presented here as an 

upper bound, and previously (Table 4-4 and Figure 4-1) as a 

lower bound will indicate that results are similar but not 

exactly the same. The differences are due to different 

assumed values for 7Q10/QS and the range of coefficients of 

variations used as inputs for the POM-PS model. 

For the case where-the design stream flow is zero, 7Q10 is 

zero and there appears to be a problem since 7Q10/QS and 

7Q10/QE are both zero. what actually matters is QS and QE

Thus, in order to evaluate these cases, the use of the actual 

QS, QE and a small 7Q10 suffices since the computation 

depends only on QS/QE and 7Q10 cancels out (Equation D-14). 

Finally, the use of a small 7Q10/QE correctly indicates that 

the WLA is done with QS = 0 (Equation D-15). Thus, no 

problems arise. 

Screening analysis results Indicate that 1n the case of 

effluent-dominated streams, a 30=day permit averaging period 

provides adequate protection for pollutants with the acuteto-chronic ratios summarized below: 

4-27
 When 

30-Day Permit Average 

Acute-to-Chronic Is Adequate for 

Ratio Acute Protection 

3 or more Always 

2 to 3 

Effluent variability is 

relatively high, but 

less than νCE = 1.1

 

4-28
USES AND LIMITATIONS 

CHAPTER 5 

The probabilistic dilution model has been demonstrated to be 

useful in selecting the appropriate averaging period for discharge 

permits. The method is easily adaptable to situations which vary 

widely in terms of stream and effluent characteristics, data 

availability, and policy-level assumptions used in the analysis. 

Although the example in Chapter 3 of how to use the method is 

based on the typical WLA assumptions of 7Q10 as the design flow 

and chronic criteria as the effluent limit, the method is easily 

adjusted to accommodate other assumptions. 

The method is intended to apply to pollutants for which the 

regulatory concern is at the point of complete mixing and for 

which the toxicity can be evaluated in terms of the total 

pollutant concentration. The method has been applied to a range of 

stream and effluent characteristics which typify the 

characteristics of streams and effluents in the United States. The 

results of this application are useful as a screening tool, by 

which the appropriate averaging period for many field situations 

can be readily; identified. However, pollutants whose toxicity is 

a function of pH, temperature, and harness require site-specific 

evaluations incorporating these parameters. 

There are also several limitations on the use of the method. 

One of the technical limitations is that the level of chronic 

protection is based on state-specified design flow, e.g., 7Q10, 

7Q2, etc., which may be overprotective or underprotective for many 

site-specific conditions. The EPA is presently considering the 

issue of allowable duration and frequency of 

5-1 
The effect of serial correlation on the return period 

specification would also need to be investigated, particularly 

with regard to the duration . of criteria violations. For example, 

a knowledge of the return period for n-day successive violations 

could be compared to the time scales of the criteria themselves. 

This would provide a direct link to the toxicity data. At a less 

sophisticated level of analysis, the tendency of criteria 

violations to cluster on successive days could be investigated to 

provide a basis for modifications to the method. 

exposure to acute as well as chronic toxicity. Users of this 

manual are advised to refer to Part A, Stream Design Flow, of Book 

VI, Selecting Design Conditions, when considering the choice of an 

appropriate chronic exposure event. Book VI is currently under 

peer review and will be issued by the Office of water Regulations 

and Standards once the peer review process is completed. 

Modifications are required to compute the probability 

distribution of 30-day average concentrations, as required for 

chronic criteria compliance; these would have to be investigated 

and verified in the field. 

The major shortcoming of the log-normal probabilistic 

dilution model is its misrepresentation of the lowest stream 

flows, thus tending to overestimate the probability of high stream 

concentrations. The use of a seasonally segmented approach could 

be investigated. 

For pollutants whose toxicity is a function of such secondary 

variables as pH, temperature and hardness, probabilistic methods 

are essential in that it is not possible to rationally choose 

"critical" or "sufficiently 

5-2 
5-3 

protective" values for these variables. Arbitrary choices cannot 

be defended in terms of the probability of criteria violations. 

Methods for analyzing these situations could be developed, 

following the logic of probabilistic dilution and incorporating 

the additional random variations of the variable. 

The application of this method to dissolved oxygen has 

indicated that the probabilistic method provides a useful approach 

to the problem of DO deficit. However this work has only been a 

first" step. Probabilistic methods can be further developed to 

assess the effects of DO fluctuations on resources and to provide 

a more rational approach to advanced waste treatment decisions. 
6-1 

CHAPTER 6 

REFERENCES 

1. DiToro, D. M., Probability Model of Stream Quality Due to 

Runoff. J. Environmental Engineering, American Society of Chemical 

Engineers, Vol. 110, No. 3, June 1984, p. 607-628. 

2. DiToro, 0. M. and Fitzpatriclc, J. J., Verification Analysis of 

the Probabilistic Dilution Model, report prepared for EPA Contract 

No. 68-01-6275, U. S. Environmental Protection Agency, Washington, 

D.C., 1982. 

3. Driscoll & Associates, Combined Sewer Overflow Analysis 

Handbook for Use in 201 Facility Planning, report prepared for EPA 

Contract No. 68-01-6148, U. S. Environmental Protection Agency, 

Washington, D.C. (1981) 

4. Hazen and Sawyer, Review of Performance of Secondary Municipal 

Treatment Works, Draft Final Report for Contract 68-01-6275, Work 

Assignment No. 5, U. S. Environmental Protection Agency, 

Washington, O.C., December 1982. 

5. Niku, Shroeder, and Samaniego, Performance of Activated Sludge 

Process and Reliability Related Design. JWPCF, Vol. 51, No. 12, 

December 1979. 

6. Niku, et al., Performance of Activated Sludge Processes; 

Reliability, Stability and Variability. EPA 600/52-81-227, 

December 1981. 
6-2 

7. Haugn, et al., Performance of Trickling Filter Plants:. 

Reliability Stability and Variability. EPA 600/52-81-228, December 

1981. 

8. Hydroscience, Inc., Simplified Mathematical Modeling of Water 

Quality, for u. S. Environmental Protection Agency, March 1971. 
APPENDIX A 

Statistical Properties of Log-Normal Distributions 
A-1

This appendix is intended to present a brief, simplified 

review of the statistical properties of log-normal distributions 

which characterize the important variables in the water quality 

analysis procedures used for 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 

relationships or equations used in the development of the methods 

employed. 

A-1. General Considerations

The factors which influence the concentration of a pollutant 

in a receiving water body are subject to a significant degree of 

variability. This variability results in fluctuations in the 

resulting stream concentration, which is compared with target 

concentrations such as criteria or standards, and which provides a 

basis for decisions on treatment requirements. The approach 

adopted in this report for examining the effects of different 

averaging periods on treatment plant discharges uses the concept 

"how much — how often" as a basis for such decisions. It is, 

therefore, essential that statistical aspects be Incorporated into 

the methodology even though they may add complexity. 

The standard statistical parameters of a population of values 

for a random variable which are used as a concise means of 

describing central tendency and spread are: 

Mean: (μx or x ) the arithmetic average. x defines the average of 

the available (usually limited) data set; 
A-2

μx denotes the true mean of the total population of variable x. x

will be an increasingly better approximation of μx as the size of 

the sample (the number of data points) Increases.

Variance: (σ2

x) by definition, the average of the square of the 

differences between individual values of x and the mean ( x ). The 

greater the variation in the data, the higher the variance:

σ2

x = (x1-x)2 + (x2-x)2 + .....(XN-x)2

N 

Standard 

Deviation: (σx) another measure of the spread of a population of 

random variables; by definition, the square root of the variance:

σx = √σ2

x

Coefficient of

Variation: (νx) is defined as the ratio of the standard deviation 

(σx) to the mean (μx):

νx = σx/μx

It is the principal measure of variation used in the 

analyses described in this report. The coefficient of 

variation is a dimensionless quantity and 1s thus freed 

from any dependence on 
A-3

the specific dimensions used to describe the variable 

(e.g., flow rate, concentrations, etc.). High 

coefficients of variation reflect greater variability in 

the random variable x. 

Median: ( x ) This is the value in a data set for which half the 

values are greater and half are lesser. 

Mode: The "most probable value" -- more of the individual data 

points are at this value (or are within this interval) 

than at other values or ranges. On a frequency histogram, 

this is the highest point on the graph. The mode has no 

real significance in the calculations in the methodology 

employed. 

Comparing the statistical properties of different data sets 

provides a convenient, concise way of recognizing similarities and 

differences. This could not be accomplished simply by "looking at 

the data" where reasonably large data sets are involved. These 

statistical properties convey no information concerning frequency, 

or the probability at which any particular value or range of 

values in the total population will occur. This essential item of 

information is provided by a knowledge of the type of 

distribution, technically, the probability distribution function 

(PDF). 

A-2. Probability Distributions

There are several different patterns which characterize the 

distribution of individual values in a large population of 

variable events. 
A-4

Most analysts are familiar with the normal distribution, in which 

a histogram of the frequency of occurrence of various values 

describes the familiar bell-shaped curve (Figure A-1(a)). When the 

cumulative frequency is plotted on probability paper, a straight 

line is generated as in Figure A-l(b) . 

Many variables, particularly those which are important in 

water quality applications, have been shown by a rapidly 

accumulating body of data to be represented by or adequately 

approximated by a log-normal distribution. A log-normal 

distribution has a skewed frequency histogram (Figure A-1(c)) 

which indicates an asymmetrical distribution of values about an 

axis defining the central tendency of the data set. There is a 

constraining limit to lower values (sometimes zero) and a 

relatively small number of rather large values but no upper 

constraint. Point source effluent concentrations [1,2]= and 

pollutant concentrations in combined sewer overflows and separate 

storm runoff [3,4], are parameters which are usually well 

characterized by log-normal distributions. In general, daily 

stream flows are satisfactorily approximated by log-normal 

distributions [5,6]. Scattered data from a number of unpublished 

sources suggest that receiving water concentrations are also lognormally distributed. Stream flows and concentrations are 

currently being examined from this perspective. A log-normal 

distribution appears as a straight line on log/probability paper 

(using cumulative frequency) as shown in Figure A-1(d). In this 

report natural (base "e") logs are used throughout. 

1

Cumulative frequency is the relative frequency (or probability) of 

values being less than or equal to a specific value. 
A-5

Figure A-1- Probability distribution 
A-6

A-3. Relationship Between Distributions

There are circumstances when two different types of 

distribution can begin to look similar -- so that either one will 

provide a reasonably good approximation of the probability 

distribution of a particular data set. For example, as the 

coefficient of variation becomes smaller and smaller, approaching 

zero, log-normal distributions begin to look more and more like a 

normal distribution. Figure A-2 shows a series of histograms for 

log-normally distributed populations, all having (arithmetic) 

population means of 100, but with different coefficients of 

variation (ν) as shown. As discussed above, smaller values of ν

approach a normal distribution. 

A.4. Properties of Log-Normal Distributions

Figure A-3 summarizes the pertinent statistical relationships 

for log-normal probability distributions. The mathematical 

formulas shown are based on statistical theory, and permit backand-forth conversions between arithmetic properties (in which 

concentrations, flows, and loads are reported) and the log of the 

variable (in which probability and frequency characteristics are 

defined). 

Normalized plots of probability versus the magnitude of a 

variable expressed as a multiple of the mean are presented 1n 

Figure A-4 for log-normal distributions. These plots present a 

family of curves reflecting the effect of coefficient of variation 

on probability of occurrence of events of specific magnitude. 

These plots can be used directly in the 
A-7

 distribution 

RANDOM VARIABLE 

Figure A-2 - Effect of coefficient of variation on frequency 
A-8

Frequency 

x is a random variable 

Definition of Terms

ln x 

μx .......... Mean...................... μln x

Variance...................

Standard Deviation......... σln x

νx .......... Coefficient of Variation... (not used) 

x .......... Median 

x .......... Random Variable.......... 

σ2

x .......... σ2ln x

σx ..........

Relationships Between Statistical Properties 

In Arithmetic and Log Space

μ = exp [μlnx + 1⁄2 σ2 ] μlnx = ln (μx) 

x = exp [μlnx] 

νx = √exp (σ2

lnx) – 1 σlnx = √ln (1 + νx

2

) 

x lnx

 (√1 + νx2) ~ 

σx = μxνx

Figure A-3 - Pertinent relationships for log-normal distributor 
A-9

Figure A-4 - Cumulative log-normal distribution 
A-10

A-5. Standard Normal Tables

analysis methodology and permit direct determination of frequency 

for events of any" specified magnitude with a known OP estimated 

coefficient of variation. 

FOP normal (or log-normal) distributions, probabilities can 

be defined in terms of the magnitude of a value, normalized by the 

standard deviation. This technique is used in the calculations of 

the probability of exceeding specified receiving water 

concentrations in this analysis. Standard normal tables can be 

obtained from any statistics textbook [8,9]. Table A-1 presents 

the standard normal table to provide a convenient source for the 

analyses used in this report. Table A-1 lists the probability fop 

the interval between 0 and the value of Z listed. Thus, it 

represents the probability that a value will be less than or equal 

to the selected value of Z. 
A-11

TABLE A-1 - Probabilities for the standard normal distribution 

Each 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 Z standard deviation units. 

Example: 88.69 percent of the area under a normal curve I1e$ to 

the left of a point 1.21 standard deviation units to the right of 

the mean. 

Z 0.0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359

0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5754

0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141

0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517

0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879

0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224

0.6 0.7258 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7518 0.7549

0.7 0.7580 0.7612 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852

0.8 0.7881 0.7910 0.7939 0.7967 0.7996 0.8023 0.8051 0.8079 0.8106 0.8133

0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389

1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621

1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830

1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015

1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177

1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319

1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9430 0.9441

1.6 0.9452 0.9463 0.9474 0.9485 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545

1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633

1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9700 0.9706

1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9762 0.9767

2.0 0.9773 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817

2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857

2.2 0.9861 0.9865 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890

2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916

2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936

2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952

2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964

2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974

2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9980 0.9980 0.9981

2.9 0.9981 0.9982 0.9983 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986

3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990

3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993

3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995

3.3 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997

3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 0.9998

3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998

3.6 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

3.8 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 1.0000 1.0000 1.0000

3.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
A-12

A-6. References

1. Niku, et al., "Performance of Activated Sludge Processes and 

Reliability Based Design." Journal WPCF, Vol. 51, No. 12, 

(December, 1979). 

2. McCarty, et al ., "Reliability of Advanced Wastewater 

Treatment." 

2. EPA Water Planning Division, "Preliminary Results of the 

Nationwide Uroan Runoff Program," (March 1982). 

4. Mancini, J. L., "Methods for Developing Wet Weather Water 

Quality Criteria." Progress Report, June 1981; EPA ORO Grant No. 

R8068280H Cincinnati. 

5. Chow, V.T. "Handbook of Applied Hydrology." Mc-Graw Hill, New 

York (1964). 

6. Linsley, et al., "Hydrology for Engineers." Mc-Graw Hill, 2nd 

Edition, (1975). 

7. Hydroscience, In., "A Statistical Method for the Assessment of 

Urban Stormwater." USEPA, EPA 440/3-79-023, (May 1979). 

8. Benjamin, J. R. and C. A. Cornell, "Probability, Statistics and 

Decision for Civil Engineers." McGraw-Hill, New York, (1970). 

9. Johnson, R. R., "Elementary Statistics." Duxbury Press, North 

Scituate, Massachusetts, (1980). 
APPENDIX B 

Field validation of Log-Normal Distribution and Related 

Assumptions 
B-1

This appendix presents a discussion of several technical 

issues and assumptions which are necessary to the use of the 

probabilistic dilution model to guide selection of permit 

averaging periods. This discussion is organized in two sections: 

the first provides- a justification for the use of the 

probabilistic dilution model in the method; the second provides a 

discussion of several key assumptions. 

B-1. Use of the Log-Normal Distribution

A relatively simple and straightforward analysis is made 

possible by the assumption that each of the input variables is 

log-normally distributed and independent. The appropriateness of 

these assumptions and their implications are discussed below. 

A basic feature of any random time series of numerical values 

is its probability distribution function, which specifies the 

distribution of values and their frequency of occurrence. More 

detailed characterizations which account for seasonal trends and 

day-to-day correlations are also possible, but at minimum the 

univariate probability density function is required. An 

examination of flow data from a number of streams indicates that 

the data can be reasonably well represented by a log-normal 

distribution. Figure B-l summarizes an examination of the adequacy 

of a log-normal distribution for dally flows of 60 streams with 

long periods of record. The actually observed 10th and 1st 

percent, ie low flows are compared with the flow estimated by a 

log-normal distribution. The major important discrepancy occurs at 

the lowest flows where the predicted distribution is lower than 

that actually observed. The most likely cause 
B-2

Figure B-1: Evaluations of log-normal distribution for stream 

 flows 
B-3

is the presence of a base stream flow which does not vary 

appreciably. THUS, the log-normal representation is generally a 

lower bound characterization of this distribution of the very 

lowest flows, which will tend to provide upper bound estimates of 

stream concentrations if these misrepresented-low flows are 

important. For the analysis results in this report, therefore, the 

calculations may be overprotective in some cases. 

Log probability plots of treatment plant effluent flows and 

concentrations are illustrated in Figure B-2 for conventional 

pollutants and figure B-3 for heavy metals. Essentially, all data 

examined to date indicate that a log-normal characterization is 

representative. 

B-2. Verification of the Probabilistic Dilution Model

In addition, detailed analysis of actual discharges into 

streams, (11 data sets for 5 streams) has been performed [2]. 

Observed data were available for upstream and effluent flows and 

concentrations, as well as for downstream concentrations. The lognormal probability dilution model was used to predict the 

probability distribution of downstream concentrations. Table 8-1 

compares the observed and computed median and 95tn percentiles 

values for selected water quality parameters. The 95% confidence 

limits of these observed quantities, computed from the known 

sampling 

The probabilistic dilution model itself has been subjected to 

a number of tests in order to check its validity and realism. 

Detailed simulation studies using Monte Carlo methods [1] have 

verified the calculated downstream concentration probability 

distribution when the upstream and effluent flows and 

concentrations are exactly log-normal. 
Figure B-2 – Probability distribution of treatment plant effluent 

 concentrations – conventional pollutants

B-4
Figure B-3 – Probability distribution of treatment plant effluent 

 concentrations – heavy metals

B-5
B-6

TABLE B-1 - Comparison of observed and computed downstream 

concentrations (2

) 

Median (50th Percentile) Concentrations

Location Variable Model 

Prediction

Observed 

Quantile 

Confidence 

Limit of 

Observed 

Quantile 

North Buffalo Creek, NC BOD (mg/l) 9.7 10.0 8.5 – 11.0

COD (mg/l) 51.0 59.0 47.0 – 66.0

TSS (mg/l) 16.0 15.0 12.0 – 22.0

Jackson River, VA BOD (mg/l) 6.0 5.3 4.2 – 6.0

TSS (mg/l) 15.8 13.6 10.0 – 17.0

Color (PCU) 110.0 100.0 90.0 – 130.0

Haw River, NC BOD (mg/l) 2.0 1.7 1.5 – 1.7

COD (mg/l) 23.8 22.0 19.0 – 26.0

Pigeon River, NC BOD (mg/l) 3.7 3.8 3.0 – 5.1

COD (mg/l) 85.0 78.0 65.0 – 87.0

Mississippi River, MN NH3 (mg/l) 1.0 1.1 1.0 – 1.2

95th Percentile Concentrations

North Buffalo Creek, NC BOD (mg/l) 31.0 22.0 20.0 – 33.0

COD (mg/l)

TSS (mg/l)

120.0 97.0 

13.6 

82.0 – 129.0

15.8 10.0 – 17.0

BOD (mg/l) 15.6 

TSS (mg/l) 32.0 

Jackson River, VA 18.1 13.0 – 20.0

41.6 30.0 – 40.0

Color (PCU) 324.0 330.0 300.0 – 410.0

Haw River, NC BOD (mg/l) 4.5 4.7 3.2 – 5.6

COD (mg/l) 43.0 46.0 33.0 - 53.0

Pigeon River, NC BOD (mg/l) 8.7 7.6 6.4 – 9.4

COD (mg/l) 186.0 229.0 188.0 – 233.0

Mississippi River, MN NH3 (mg/l) 3.5 43 3.2 – 5.0
B-7

The influence of. possible deviations from the assumed lognormality of the upstream and effluent flows and concentrations 

upon more extreme quantiles is unknown at present due to lack of 

larger data sets that encompass these extreme quantiles. However, 

the quality of the alternatives to and the simplicity of this 

model argue strongly for Its use in the present context of 

describing comparative differences in water quality impacts. 

distribution of quantiles, are also listed. In all but one case, 

the computed quantiles are within the confidence limits. 

Thus, there is no statistical evidence, to reject the 

computed quantiles as not being the true quantiles of the observed 

concentration distribution. This is strong statistical evidence 

that indeed the log-normal probabilistic dilution model is 

representative of actually observed downstream concentration 

distributions for the 95th percentile at least. 

The 11 data sets used in the verification analysis were 

examined for cress correlations between effluent flows and 

concentrations. The observed ranges in correlation coefficients 

have no significant impact on the computation. Correlations 

between stream flow and effluent load for a point source are not 

expected. Upstream concentrations are not employee in the 

comparison of permit averaging period effects, so that any 

correlation between stream flow and concentration is not relevant 

to this analysis. Modifications to the probabilistic dilution 

model computations are available for use in situations where cross 

correlations must be considered [1]. 
B-8

B-3. Appropriateness of Assumptions

We have chosen to ignore the seasonal and day-to-day 

correlation structure of both stream flow and effluent behavior in 

order to simplify the characterization of each variable. The 

consequences of this simplification are discussed below in more 

detail, but it should be pointed out that trends and correlations 

do not invalidate the use of the log-normal probability 

distribution function to characterize the frequency of occurrence 

of flows and concentrations. Trends and day-to-day correlations 

affect the time sequences with which certain values occur, but not 

their long term frequency of occurrence. This is judged to be an 

acceptable penalty to be endured when compared to the 

simplification achieved. If a more refined, site specific analysis 

is required, then a seasonal breakdown of the data, with the 

appropriate means and standard deviations for each time period, 

can be generated and the analysis performed as described below. 

The consequence of a possible serial correlation can be 

approximately quantified as follows. If, in fact, the serial 

correlation is such that 10 consecutive daily violations always 

occur when one violation occurs, then the proper percentile to 

consider 1s not 0.0274 (10 years) but rather 0.274 (1 year return 

period). The degree to which the 10 year return period 

concentration is overestimated can be estimated by comparing the 

ratio of the 10 year to the 1 year stream concentrations which are 

compiled without regard to serial correlation. 

The ratio of the 10 year return period concentration to that 

for 
B-9

some other return period can be computed for log-normally 

distribute random concentrations by: 

C 10 yr = EXP [(Z10 yr – Zx yr) σlnC] 

Cx yr

Where 

σlnC = log standard deviation of stream concentrations (C) 

Z10 yr, C10 yr = Z score and concentration corresponding to a 10 year 

return period 

Zx yr, Cx yr = I score and concentration corresponding to an x year 

return period 

Table 3-2 summarizes results for a range of values for 

coefficient of variation of stream concentrations. Clustering 

tendencies of 5 and 10 are examined as approximations of the 

degree of serial correlation which might exist. If clusters of 10 

occur, the comparison is between 10 and 1 year return periods as 

discussed above; for clusters of 5, the comparison is between 10 

and 2 year return periods. On the basis of this analysis, the 

water quality effects presented in Chapter 4 for various permit 

averaging periods may overstate the 10 year stream concentrations 

by approximately a factor of 1.5 to 2.0. 

Until stream and effluent data can be analyzed to define the 

serial correlation structure and the methodology modified to 

incorporate it, the results presented in Table B-2 should be 

interpreted to indicate with the following possibilities: 
B-10

TABLE B-2 - Approximate overestimation of 10 year return period 

stream concentration by ignoring serial correlation 

 Variability of Ratio of Stream Concentration

Stream Concentration At Indicated Average Return Periods

Coefficient 

of Variation

(vC)

Log 

Sigma

(σlnc)

10 Year

to 1 Year

(C10/C1) 

to 2 Year 

10 Year 

(C10/C2)

.5 .4724 1.4 1.25 

1.0 .8326

1.5 1.0857 1.65 

1.8 1.50

2.1

2.0 1.2686 2.4 1.80

C10= EXP[(Z10 - Z1,2) σlnc] 

Z1 (1 year Return Period) = 2.778 

Z1 (2 year Return Period) = 2.778 

C1,2

Z10 (10 year Return Period) = 3.456
B-11

o Stream concentrations indicated by the methodology used 

in the report to recur on average for 1 day every 10 

years would, if they actually never occur except in 

clusters of 5 to 10 days, have return periods of 50 to 

100 years. 

o Conversely, for the same clustering assumptions, the 

stream concentrations that occur at 10-year intervals 

should be 50 to 70% (1/2 to 1/1.5) of the 10-year 

concentrations projected by the report methodology. 

B-4. References

1. DiToro, D.M., "Probability Model of Stream Quality Due to 

Runoff." J. Environmental Engr. ASCE, Vol. 110., #3, June 1984 p. 

607-628. 

2. DiToro, D.M. and Fitzpatrick, J.J., "Verification Analysis of 

the Probabilistic Dilution Model" Report prepared for EPA Contract 

No. 68-01-6275, U.S. Environmental Protection Agency, Washington, 

D.C., (1982). 
APPENDIX C 

Characteristic Values for Input Parameters 
C-1

The results reported here represent an attempt to develop 

characteristic values and ranges for stream flow and effluent 

variability. These values and ranges have been extracted from the 

results of published analyses, and are used in Chapter 4 to 

evaluate the influence of the permit averaging period on typical 

receiving water conditions. These values are provided for effluent 

flows (Section 1), effluent concentrations (Section 2), and stream 

flow (Section 3). 

C-1. Treatment Plant Effluent Flows

A recent study [1] analyzed several years of performance data 

from approximately 400 secondary treatment plants in 8 different 

process categories. Average plant effluent flows ranged from 0.002 

to 82 MGD. Table C-1 summarizes the coefficient of variation of 

treatment plant effluent flows. 

C-2. Treatment Plant Effluent Concentrations

Data on the variability of effluent BOD5 and total suspended 

solids (TSS) from municipal biological treatment plants are 

available from several sources. Niku, et al . [2] provide analysis 

results for 37 activated sludge plants which show the coefficient 

of variation of effluent 8005 concentrations to range between 0.34 

and 1.11 for individual plants. The median of the individual 

plant- values was 0.635. The EPA research report [3] on which the 

foregoing was based1 reported a mean coefficient of variation for 

43 activated sludge plants using a variety of processes. Daily 

effluent concentrations were found to be well represented 
C-2

Process Category Individual Plants 

Median of 

All Plants 

TABLE C-1 - Coefficient of variation of daily effluent flows, νQE

Number of 

Plants 

Range For 

Trickling Filter 

Rock 

64 0.06 - 0.97 0.27 

Trickling Filter 

Plastic 

17 0.16 - 0.88 0.38 

Conventional Activated 

Sludge 

66 0.04 - 1.04 0.24 

Contact Stabilization 

Activated Sludge 

57 0.06 - 1.35 0.34 

Extended Aeration 

Activated Sludge 

28 0.11 – 1.32 

Contact 

Oxidation Ditch 

0.34 

Rotating Biological 27 0.12 – 1.19 0.31 

28 0.09 – 1.16 0.31 

Stabilization Pond 37 0.00 – 0.83 0.31 
C-3

by a log-normal distribution. The mean of all plants analyzed had 

coefficients of variation of 0.7 for BOD5 and 0.84 for TSS. 

Two recent studies have extended the analysis of effluent 

concentration variability, and report coefficients of variation of 

BOD5 and TSS for 7- and 30-day averages as well as for daily 

values. Results reported by Hazen and Sawyer [1] provide the basis 

for the summary presented in Table C-2 as well as the two other 

sources cited in the table. An analysis of the performance of 11 

trickling filter plants by Haugh, et al. [4] produced me results 

summarized by Table C-3. 

Based on available data, a single representative value for 

coefficient of variation of effluent concentrations cannot be 

defined. The most appropriate characteristic value will be 

influenced by process category, effluent concentration averaging 

period, and the pollutant in question (e.g., BOD, TSS, etc.), as 

well as individual plant differences. The computations in this 

report are performed using a range of values estimated to 

encompass most of the conditions of interest. 

C-3. Stream Flow

Figure C-1 provides a basis for estimating the coefficient of 

variation of daily stream flows on the basis of the ratio of 7Q10 

to average (QS) stream flow. These flow values are usually readily 

available. The relationship shown is derived from a set of flow 

measurements and statistics which has been developed for a sample 

of 130 streams in various areas of the country [5] and is 

summarized in Table C-4, along with additional details on the 

location of the stream gages used. The ranges 
C-4

Process Category 

TABLE C-2 - Summary of secondary treatment plant performance - median coef

(from reference 1)

Number 

of 

Plants 

Effluent BOD (mg/l) 

Coefficient of 

 Mean Variation* 

 Daily 7-day 30-Day 

 Values Avgs. Avgs 

E

 Mean 

 

 

28 

Trickling Filter Rock 64 26.0 0.40 0.30 0.25 25.3 Trickling Filter Plastic 17 19.0 0.50 0.35 0.30 19.4 Conventional Activated Sludge 

66 14.8 0.65 0.55 0.40 14.3 

Contact Stabilization Activated Sludge 

57 12.6 0.60 0.50 0.40 13.8 

Extended Aeration Activated Sludge 

28 7.2 0.70 0.60 0.45 9.8 

Rotating Biological Contacter 

27 17.0 0.60 0.45 0.35 15.2 

Oxidation Ditch Stabilization Pond 

8.4 0.60 0.55 0.40 

22.7 0.50 0.45 0.40 

12.3 

37 39.5 

Values shown are rounded to nearest 0.05 for v(CE)

*Basis: vCE = Standard Deviation of Median Plant

 Mean of Median Plant 
C-5

Chemical Precipitation/Settling1

Pollutant Coefficient of Variation 

Cr .99 

Cu .60 

Mn .34 

Plant Number

Fe .57 

Ni .81 

Zn .84 

Tss .66 

Pharmaceutical Industry2

___________Coefficient of Variation_______

 BOD (n) TSS (n)

12015 1.01 46 .85 195 

12072 .97 392 .63 395 

12026 

 .74 366 1.12 364 

12160 

12161 

 .71 54 .50 54 

12187 .21 12 .26 12 

12136 

 .58 50 .55 52 

 1.55 39 1.34 38 

 .95 44 .49 53 

12036 

12097 1.08 222 1.21 249 

12098 1.37 24 1.52 25 

12117 .70 39 .81 51 

 .92 34 1.11 32 

 .55 249 .99 355 

12186 

 1.02 110 1.16 111 

12248 

12257 .64 56 .92 56 

12294 .93 56 1.25 50 

12307 

1

From Table 3, page 14 of 10-18-83 memorandum from H. Kahn to E. 

Hall titled, "Revisions to Data and Analysis of the Combined 

Metals Data Base." 

2

From preliminary descriptive statistics generated on 

pharmaceutical data by SRI International, 11-12-82. 
C-6

BOD

TABLE C-3 - Effluent concentration variability for trickling 

filters (from reference 4) 

 5 TSS

Mean for 11 plants (mg/l) 29.6 29.3 

Coefficient of Variation (median of 

Individual plant values): 

 Daily Values 0.39 0.55 

 7-Day Averages 0.35 0.31 

 30-Day Averages 0.31 0.26 

shown reflect the bulk of the data in the sample of stream records 

which were used. However, a relatively small percentage of streams 

will have coefficients of variation which fall outside the 

indicated ranges. The statistical analysis was performed for the 

entire period of record. Results in some cases may be distorted, 

if flow regulation works were installed on the stream sometime 

during the period of record. 

C.4. References

3. Niku, et al., "Performance of Activated Sludge Processes: 

Reliability, Stability and Availability." EPA 600/52-81-227, 

(December 1981). 

1. Hazen and Sawyer, "Review of Performance of Secondary Municipal 

Treatment Works." Draft Final Report for Contract 68-01-6275, Work 

Assignment No. 5, U.S. Environmental Protection Agency, 

Washington, D.C., (December 1982). 

2. Niku, Shroeder, and Samaniego, "Performance of Activated Sludge 

Process and Reliability Related Design." JWPCF, Vol. 51, No. 12, 

(December 1979). 

4. Haugh, et al. "Performance of Trickling Filter Plants: 

Reliability, Stability and Variability." EPA 600/52-81-228. 

(December 1981). 

5. Driscoll & Associates, "Combined Sewer Overflow Analysis 

Handbook for Use in 201 Facility Planning." Report prepared for 
C-7

EPA Contract No. 68-01-6148, U.S. Environmental Protection Agency, 

Washington, D.C. (1981). 
C-8

Figure C-1 - Typical low flow characteristics of U.S. streams 
C-9

TABLE C-4 - Summary of stream flow characteristics 
C-10

TABLE C-4 (Cont.) 
C-11

TABLE C-4 (Cont.) 
C-12

TABLE C-4 (Cont.) 
C-13

TABLE C-4 (Cont.) 
C-14

TABLE C-4 (Cont.) 
Probabilistic Dilution Model - Point Source 

APPENDIX D 

Computer Program for the 

(PDM-PS) 
D-1

This appendix describes a computer program (PDM-PS) which 

performs the computations of the Probabilistic Dilution Model for 

Point Source discharges using numerical methods based on 

quadratures. The program is written 1n BASIC for the HP-85 and the 

IBM-PC, and should be readily applicable to other personal 

computers with perhaps minor modifications to reflect individual 

machine characteristics. 

The program is structured around slightly different Input 

format than that used for the manual calculation using the moments 

approximation. A series of normalizations (ratios) of certain of 

the input data items is used to provide a computation framework 

that provides a more generalized perspective 

The appendix is organized as follows. Section 1 describes the 

basis for the formulation and normalization of the input data, as 

used in program. Section 2 provides an annotated description of 

the CRT and functions, as well as the nature of the user's 

response. Figures and D-2 provide the results of running the PDMPS through the example described in Section 3.2 of this report. 

Finally, Figure D-3 provides a of the POM-PS program for entry 

into a personal computer. 

D-1. Formulation and Normalization

The analysis can be made more useful in a general way if the 

normalization described below is applied to reduce certain of the 

inputs recognized ratios, and to express-results (stream 

concentration as a multiple or fraction of the target stream 

concentration (CL). 
D-2

- The stream target concentration (CL) is produced when the 

discharge flow is the mean effluent flow (

The explicit assumptions in the normalization scheme that is 

used are that: 

QE ), the discharge 

pollutant concentration is equal to the permit effluent limit 

(EL), and the stream flow is equal to the design value (here 

designated 7Q10 - though any other basis may be used for 

designating the numerical value of stream design flow, e.g., 

30Q5, 30Q10, etc.). 

- The reduction factor (R = CE /EL) determines the mean effluent 

concentration of the pollutant being evaluated. It, could be 

selected arbitrarily; however, as applied in this manual for 

evaluating the permit averaging period, the value selected 

will be dictated by the variability of effluent 

concentrations and the permit averaging period. 

In the usual case, where the stream target concentration (CL) is 

set at the chronic toxicity level, the multiples of the target - 

in which stream concentrations are expressed (CO/CL) - correspond 

with the acute toxicity level. The basis for the normalization 

scheme adopted is as follows. 

The downstream concentration, CO, is given by the dilution 

equation: 

CO = _CE QE_ = φCE

 QS + QE 

 (D-1) 

For a chronic criteria concentration, CL, the effluent limit 

concentration, 
D-3

EL, is computed using QS * 7Q10 and an average effluent flow, QE :

__

 CL = _ EL QE_ = ELφSTD

 __ 

7Q10 + QE 

(D-2) 

where φ STD is the effluent dilution factor at the standard 

conditions, φ STD = QE /(7Q10 + QE ). Thus:

(D-3)

EL = CL/φSTD

However, the choice of permit averaging period forces a reduction 

of CE of magnitude, R, so that permit violations occur only 5 

percent of the time. Thus the actual long term average effluent 

concentration is: 

__

 CE = R EL = R CL/φSTD

CO/CE = (CE/CE) __QE_

(D-4) 

The problem is to compute the probability that the downstream 

concentration exceeds a multiple, β, of the chronic concentration, 

CL. In particular, if the acute criteria concentration is 

selected, then β is the acute to chronic criteria ratio for the 

pollutant being regulated. Hence it is necessary to compute: 

__

Pr [CO > βCL] = PR [CO > βφSTD CE/R]

 

 (D-5) 

where Equation D-4 has been substituted for CL. Dividing both 

sides of the inequality by CE provides the first normalization 

site 

__

_ 

 QS + QE 

(D-6) 
D-4

and CE/CE is the normalized effluent concentration. The 

probability distribution of this random variable no longer depends 

upon the mean effluent concentration, but only on the coefficient 

of variation, νCE This is easily seen from the following 

representation of a log-normal random variable: 

lnCE = lnCE + ZσlnCE

~

(D-7) 

 

where CE is the median, σlnCE is the log standard deviation, and Z 

is a standard normal random variable with zero mean and unit 

standard deviation. For log-normal random variables,

 ⎯ ~

 CE = CE √(1 + vCE

2

)

(D-8) 

and 

σ2

lnCE = ln(1 + vCE

2

)

(D-9) 

so that Equation D-7 becomes 

__

 ln(CE/CE) = -1/2σ2

lnCE + ZσlnCE 

(D-10) 

Thus, it is seen that CE/CE is log-normal with log mean = -

1/2σ2

lnCE and only the coefficient of variation, which specifies 

σlnCE through equation 0-9, is required to completely specify the 

behavior of CE/CE .

The final normalization results from expressing Equation D-6 

as 

__ __

 CO/CE = _CE/CE__

 1 + QS/QE 

(D-11) 
D-5

F1 = 7Q10/QS 

 __ 

 (D-13) 

Then 

Note mat QS/QE is log-normally distributed since both QS and QE 

are assumed to be log-normal. Thus, only the ratio of the average 

flows, QS/QE, is required. A convenient normalization using ratios 

that are more readily available results if the average effluent 

and stream flows are standardized relative to design stream flow 

(here designated by 7Q10). Defining 

 __ 

 (D-12) 

F2 = 7Q10/QE 

 __ __ (D-14) 

 QS/QE = F2/F1 

And 

φSTD = __1__

 1 + F2 

 (D-15) 

These ratios, F1 and F2, together with the coefficients of 

variation, νQS, νQE, and νCE, completely specify the characteristics 

of the random variables in the normalized dilution Equation D-11. 

R specifies the effect of permit averaging period and β, the acute 

to chronic criteria ratio, specifies the toxicity behavior of the 

substance being considered. This completes the normalization. 

D-2. Description of Program Use

The program is easy to use. The values of the input variables 

are sequentially requested on the CRT. Once the input values are 

entered, a summary of the input data is printed out, as is a 

tabular listing of the 
D-6

results of the calculations. The user should be thoroughly 

familiar with the theoretical and practical bases for the PDM-PS 

as described in Chapters 2 and 3 before attempting to use the PDMPS. 

CRT: Displays title and general descriptive material shown in 

Figure D-1. 

USER: Initiates program execution. 

PRINTER: Writes title. 

CRT: Question #1 is displayed: "Enter coefficient of variation 

of QS, QE, and CE. 

USER: Enters the values of νQS, νQS and νCE= separated by commas.

CRT: Question #2 is displayed: "7Q10/avg QS?" 

USER: Enters the ratio of the 7Q10 flow to the average stream 

flow (QS). 

CRT: Question #3 1s displayed: "7Q10/avg QE?" 

USER: Enters the design dilution ratio, i.e., the ratio of 7Q10 

flow rate to the average effluent flow rate (QE ). 

CRT: Question #4 1s displayed: "avg CE/EL?" 

USER: Enters the ratio of the average effluent concentration 

which the treatment plant will be designed to produce 

(avg CE), to the effluent concentration derived from the 
D-7

WLA analysis (EL). This latter value is that concentration in 

the effluent which will result in the stream target 

concentration being met, when the following flow conditions 

prevail: 

Stream flow (QS) is at the 7Q10 flow rate. 

Effluent flow (QE) is at the average discharge rate of flow. 

PRINTER: Prints a tabular summary of the input data selected. 

CR: Question #5 is displayed: "Enter lowest, highest and 

increment of multiple of target for which % exceedence is 

desired." 

USER: Decides on a range of stream concentrations (expressed as 

multiples of the target concentration, CL) for which the 

probability of occurrence and the recurrence interval are 

desired. The user enters (1) the lowest value, (2) the 

highest value and (3) the incremental step desired for 

values between the highest and lowest. 

PRINTER: Prints tabular listing of results. For each multiple of 

CL, the exceedence frequency and return period are 

listed. When the printing is completed, a tone sounds and 

Question 5 is repeated. 

USER: Enters a new set of values for multiples of CL, if 
D-8

desired. This allows the user to conveniently search out 

the ranges of interest and select the most appropriate 

levels of incremental detail. When the desired amount of 

output has been obtained, the program is interrupted, and 

begun again at Question #1 to examine another set of 

conditions. The user can formally "end" the program by 

entering 0,0,0 in response to Question 5. 
D-9

POINT SOURCE – RECEIVING WATER 

 RATIO...avg CE/EL 

.05 

3 

? 

 CONCENTRATION ANALYSIS 

******************************** 

INPUTS: COEF. VAR OF QS, QE, CE 

 RATIO...7Q10/avgQS GENERAL DESCRIPTIVE MATERIAL 

 RATIO...7Q10/avgQE 

 BACKGROUND STREAM CONC (CS) 

 IS ASSUMED TO BE ZERO 

******************************** 

ENTER COEF VAR OF QS, QE, CE? QUESTION #1 

1.6, .2, .7 

ENTER FOLLOWING RATIOS: 

...... 7Q10avg/ QS ? QUESION #2 

...... 7Q10avg/ QE ? QUESTION #3 

...... avg CE/ EL ? QUESTION #4 

.57 

ENTER LOWEST, HIGHEST, AND INCREM- QUESTION #5(CONTINUES TO 

ENT OF MULT OF TARGET FOR WHICH REPEAT AS 

% EXCEED IS DESIRED NEEDED) 

ENTER LOWEST, HIGHEST, AND INCREMENT OF MULT OF TARGET FOR WHICH 

% EXCEED IS DESIRED 

? 

2.5, 3, .05 

Figure D-1 CRT - displays 
D-10

 RECEIVING WATER CONC (CO) 

 COEF VAR .....QS = 1.50 

 TARGET EXCEEDED (YEARS) 

 2.00 0.112 2.4 

 3.00 0.024 11.3 

 4.00 0.007 39.4 

 2.60 0.043 6.4 

 2.85 0.030 9.2 

 2.95 0.026 10.6 

****************************** TITLE 

 PROBABILITY DISTRIBUTION 

 AND RETURN PERIOD 

 FOR MULTIPLES OF TARGET CONC 

 DUE TO POINT SOURCE LOADS 

****************************** 

 COEF VAR .....QE = 0.20 

 COEF VAR .....CE = 0.70 

 7Q10/avg QS = 0.05 SUMMARY OF INPUT DATA 

 7Q10/avg QE = 3.00 

 avg CE/ EL = 0.05 

VIOLATION PERCENT RETURN 

 MULT OF OF TIME PERIOD 

--------- -------- ------- 

 1.00 0.894 0.3 

 5.00 0.002 114.4 

 2.50 0.050 5.5 

 2.55 0.046 5.9 

 2.65 0.040 6.9 CALCULATED RESULTS 

 2.70 0.037 7.4 

 2.75 0.034 8.0 

 2.80 0.032 8.6 

 2.90 0.028 9.9 

 3.00 0.024 11.3 

Figure D-2 - Example of printed output 
D-11

 

 yes

Start 

Clear screen 

no 

Print header 

messages

Prompt for and 

input coef. of 

variations for 

QS, QE, CE 

 

Prompt for and Iterate on CO/CL values 

input ratios 

 __ 

of 7Q10/QS, 

 __ 

 7Q10/QE, 

and avg. CE/CL 

Compute normal 

and reverse normal 

coefficients 

 Next CO/CL 

Prompt for and input 

lowest, highest, and 

delta increment of 

multiples of CO/CL 

To use 

All 

CO/CL values 

= φ?

Stop

Clear screen 

Print input values 

and table header 

Evaluate Q(x) 

Compute return 

period 

Print CO/CL, % of time 

exceeded, and 

return period 

end

Figure D-3 - Flow chart for PDM-PS program 
D-12

Figure D-4 - PDM-PS program listing - HP-85 compatible 
D-13

Figure D-4 (cont'd.) 
D-14

Figure D-4 (cont'd.) 
D-15

Figure D-5 - PDM-PS program listing - IBM-PC and MS-DOS compatible
D-16

Figure D-5 (cont'd.) 
D-17

Figure D-5 (cont'd.) 
D-18

Figure D-5 (cont'd.) 
D-19

RECEIVING WATER CONC (CO) PROBABILITY DISTRIBUTION 

FOR MULTIPLES OF TARGET CONC 

++++++++++++++++++++++++++++ 

POINT SOURCE - RECEIVING WATER 

CONCENTRATION ANALYSIS 

++++++++++++++++++++++++++++ 

INPUT COEF OF VAR OF QS,QE,CE 

RATIO...7Q10/AVG QS 

RATIO...7Q10/AVG QE 

RATIO...AVG CE/CL 

++++++++++++++++++++++++++++ 

? 1.5, .2,.7 

ENTER THE FOLLOWING RATIOS: 

.......7Q10/AVG QE ? 3.0 

COEF OF VAR.....QE .2 

++++++++++++++++++++++++++++ 

AND RETURN PERIOD 

DUE TO POINT SOURCE LOADS 

BACKGROUND STREAM CONC (CS) IS ASSUMED TO BE 

ZERO 

ENTER COEF OF VAR OF QS, QE, CE 

.......7Q10/AVG QS ? .05 

.........AVG CE/EL ? .67 

COEF OF VAR.....QS 1.5 

COEF OF VAR.... CE .7 

7Q10/AVG QS = .05 

7Q10/AVG QE = 3 

AVG CE/EL = .67 

++++++++++++++++++++++++++++ 

ENTER LOWEST, HIGHEST, AMD INCREMENT OF MULT OF TARGET FOR WHICH % 

EXCEED IS DESIRES? 1,5,1 

COEF OF VAR.....QS 1.5 

COEF OF VAR.....QE .2 

COEF OF VAR.....CE .7 

7Q10/AVG QS = .05 

7Q10/AVG QE = 3 

AVG CE/EL = .67 

Figure D-5 (cont'd.) 
PERCENT 

--------------------------------------- 

 STREAM CONC (CO) 

MULT OF 

TARGET 

(CO/CL) 

OF TIME 

EXCEEDED 

RETURN 

PERIOD 

(YEARS) 

1.000 0.894 0.306 

2.000 0.112 2.443 

3.000 0.024 

114.356 

11.313 

4.000 0.007 39.429 

5.000 0.002 

ENTER <CR> TO CONTINUE, OR ‘STOP’? 

 COEF OF VAR..... QE = .2 

 7Q10/AVG QS = .05 

 AVG CE/EL = .67 

ENTER LOWEST, HIGHEST, AJJD INCREMENT OF MJLT OF TARGET FOR WHICH 

% EXCEED IS DESIRED? 2.5, 3, .1 

COEF OF VAR.....QE = .2 

7Q10/AVG QE = 3 

AVG CE/EL = .67 

------------------------------------ 

MULT OF 

(CO/CL) 

RETURN 

PERIOD 

(YEARS) 

 COEF OF VAR..... QS = 1.5 

 COEF OF VAR..... CE = .7 

 7Q10/AVG QE = 3 

-------------------------------------- 

COEF OF VAR.....QS = 1.5 

COEF OF VAP......CS = .7 

7Q10/AVG QS = .05 

 STREAM CONC (CO) 

TARGET 

PERCENT 

OF TIME 

EXCEEDED 

2.500 0.050 5.501 

2.600 0.043 6.395 

2.700 0.037 7.410 

2.800 0.032 8.558 

2.900 0.028 9.854 

3.000 0.024 11.313 

ENTER <CR> TO CONTINUE, OR STOP? STOP 

Figure D-5 (cont'd.) 
DISCLAIMER 

We have made efforts to ensure that this electronic document is an accurate reproduction 

of the original paper document. However, this document does not substitute for EPA 

regulations; nor is it a regulation itself. Thus, it does not and cannot impose legally 

binding requirements on EPA, the states, tribes or the regulated community, and may not 

apply to a particular situation based on the circumstances. If there are any differences 

between this web document and the statute or regulations related to this document, or 

the original (paper) document, the statute, regulations, and original document govern. We 

may change this guidance in the future. 

Supplemental material such as this disclaimer, a document abstract and glossary entries 

may have been added to the electronic document. 

NOTE TO THE BROWSER 

These original guidance documents - enhanced for easier access in 2006/2007 – still 

contain much of EPA’s current thinking with regards to water quality modeling and TMDLs. 

However, the reader may discover that some of the referenced tools and materials have 

been superseded or are no longer in general use. Information on the latest EPA-supported 

and other models is available at the EPA Center for Exposure Assessment Modeling 

(CEAM), currently located online at http://www.epa.gov/ceampubl/. 
GLOSSARY 

Ammonia - Inorganic form of nitrogen; product of hydrolysis of organic nitrogen and 

denitrification. Ammonia is preferentially used by phytoplankton over nitrate for uptake of 

inorganic nitrogen. 

Chronic Toxicity - Toxicity, marked by a long duration, that produces an adverse effect 

on organisms. The end result of chronic toxicity can be death although the usual effects 

are sublethal; e.g., inhibits reproduction, reduces growth, etc. These effects are reflected 

by changes in the productivity and population structure of the community. 

Complete mixing - No significant difference in concentration of a pollutant exists across 

the transect of the waterbody. 

Activated sludge - A secondary wastewater treatment process that removes organic 

matter by mixing air and recycled sludge bacteria with sewage to promote decomposition. 

Acute toxicity - A chemical stimulus severe enough to rapidly induce an effect; in aquatic 

toxicity tests, an effect observed within 96 hours or less is considered acute. When 

referring to aquatic toxicology or human health, an acute effect is not always measured in 

terms of lethality. 

Advanced waste treatment (AWT) - Wastewater treatment process that includes 

combinations of physical and chemical operation units designed to remove nutrients, toxic 

substances, or other pollutants. Advanced, or tertiary, treatment processes treat effluent 

from secondary treatment facilities using processes such as nutrient removal (nitrification, 

denitrification), filtration, or carbon adsorption. Tertiary treatment plants typically achieve 

about 95% removal of solids and BOD in addition to removal of nutrients or other 

materials. 

Biochemical oxygen demand (BOD) - The amount of oxygen per unit volume of water 

required to bacterially or chemically oxidize (stabilize) the oxidizable matter in water. 

Biochemical oxygen demand measurements are usually conducted over specific time 

intervals (5,10,20,30 days). The term BOD generally refers to standard 5 day BOD test. 

Combined sewer overflows (CSOs) - A combined sewer carries both wastewater and 

stormwater runoff. CSOs discharged to receiving water can result in contamination 

problems that may prevent the attainment of water quality standards. 

Concentration - Amount of a substance or material in a given unit volume of solution. 

Usually measured in milligrams per liter (mg/l) or parts per million (ppm). 

Conservative substance - Substance that does not undergo any chemical or biological 

transformation or degradation in a given ecosystem. 

Conventional pollutants - As specified under the Clean Water Act, conventional 

contaminants include suspended solids, coliform bacteria, biochemical oxygen demand, 

pH, and oil and grease. 

Design stream flow - The stream flow used to conduct steady state wasteload allocation 

modeling. 

Dilution - Addition of less concentrated liquid (water) that results in a decrease in the 

original concentration. 

Discharge permits (NPDES) - A permit issued by the U.S. EPA or a State regulatory 

agency that sets specific limits on the type and amount of pollutants that a municipality or 

industry can discharge to a receiving water; it also includes a compliance schedule for 
Dissolved oxygen (DO) - The amount of oxygen that is dissolved in water. It also refers to 

a measure of the amount of oxygen available for biochemical activity in water body, and as 

indicator of the quality of that water. 

Heavy Metals - Metals that can be precipitated by hydrogen sulfide in acid solution, for 

example, lead, silver, gold, mercury, bismuth, copper. 

Nitrification - The oxidation of ammonium salts to nitrites (via Nitrosomonas bacteria) and 

the further oxidation of nitrite to nitrate via Nitrobacter bacteria. 

Organic matter - The organic fraction that includes plant and animal residue at various 

stages of decomposition, cells and tissues of soil organisms, and substance synthesized 

by the soil population. Commonly determined as the amount of organic material contained 

in a soil or water sample. 

achieving those limits. It is called the NPDES because the permit process was established 

under the National Pollutant Discharge Elimination System, under provisions of the 

Federal Clean Water Act. 

Effluent - Municipal sewage or industrial liquid waste (untreated, partially treated, or 

completely treated) that flows out of a treatment plant, septic system, pipe, etc. 

In situ - In place; in situ measurements consist of measurement of component or 

processes in a full scale system or a field rather than in a laboratory. 

Load allocation (LA) - The portion of a receiving water's total maximum daily load that is 

attributed either to one of its existing or future nonpoint sources of pollution or to natural 

background sources. 

Low flow (7Q10) - Low flow (7Q10) is the 7 day average low flow occurring once in 10 

years; this probability based statistic is used in determining stream design flow conditions 

and for evaluating the water quality impact of effluent discharge limits. 

Mass balance - An equation that accounts for the flux of mass going into a defined area 

and the flux of mass leaving the defined area. The flux in must equal the flux out. 

Mathematical model - A system of mathematical expressions that describe the spatial 

and temporal distribution of water quality constituents resulting from fluid transport and the 

one, or more, individual processes and interactions within some prototype aquatic 

ecosystem. A mathematical water quality model is used as the basis for waste load 

allocation evaluations. 

Modeling - The simulation of some physical or abstract phenomenon or system with 

another system believed to obey the same physical laws or abstract rules of logic, in order 

to predict the behavior of the former (main system) by experimenting with latter (analogous 

system). 

Monitoring - Routine observation, sampling and testing of designated locations or 

parameters to determine efficiency of treatment or compliance with standards or 

requirements. 

Organic - Refers to volatile, combustible, and sometimes biodegradable chemical 

compounds containing carbon atoms (carbonaceous) bonded together and with other 

elements. The principal groups of organic substances found in wastewater are proteins, 

carbohydrates, and fats and oils. 

Oxidation - The chemical union of oxygen with metals or organic compounds 

accompanied by a removal of hydrogen or another atom. It is an important factor for soil 

formation and permits the release of energy from cellular fuels. 
Oxygen Deficit - The difference between observed oxygen concentration and the amount 

that would theoretically be present at 100% saturation for existing conditions of 

temperature and pressure. 

Oxygen depletion - Deficit of dissolved oxygen in a water system due to oxidation of 

organic matter. 

Secondary treatment plant - Waste treatment process where oxygen demanding organic 

materials (BOD) are removed by bacterial oxidation of the waste to carbon dioxide and 

water. Bacterial synthesis of wastewater is enhanced by injection of oxygen. 

Sediment oxygen demand (SOD) - The solids discharged to a receiving water are partly 

organics, and upon settling to the bottom, they decompose anaerobically as well as 

aerobically, depending on conditions. The oxygen consumed in aerobic decomposition 

represents another dissolved oxygen sink for the waterbody. 

Oxygen demand - Measure of the dissolved oxygen used by a system (microorganisms) 

in the oxidation of organic matter. See also biochemical oxygen demand. 

Partition coefficients - Chemicals in solution are partitioned into dissolved and particulate 

adsorbed phase based on their corresponding sediment to water partitioning coefficient. 

Point source - Pollutant loads discharged at a specific location from pipes, outfalls, and 

conveyance channels from either municipal wastewater treatment plants or industrial 

waste treatment facilities. Point sources can also include pollutant loads contributed by 

tributaries to the main receiving water stream or river. 

Pollutant - A contaminant in a concentration or amount that adversely alters the physical, 

chemical, or biological properties of a natural environment. The term include pathogens, 

toxic metals, carcinogens, oxygen demanding substances, or other harmful substances. 

Examples of pollutant sources include dredged spoil, solid waste, incinerator residue, 

sewage, garbage, sewage sludge, munitions, chemical waste, biological material, 

radioactive materials, heat, wrecked or discharged equipment, sediment, cellar dirt, 

hydrocarbons, oil, and municipal, industrial, and agricultural waste discharged into surface 

water or groundwater. 

Quality - A term to describe the composite chemical, physical, and biological 

characteristics of a water with respect to it’s suitability for a particular use. 

Reaeration - The absorption of oxygen into water under conditions of oxygen deficiency. 

Respiration - Biochemical process by means of which cellular fuels are oxidized with the 

aid of oxygen to permit the release of the energy required to sustain life; during respiration 

oxygen is consumed and carbon dioxide is released. 

Sediment - Particulate organic and inorganic matter that accumulates in a loose, 

unconsolidated form on the bottom of natural waters. 

Simulation - Refers to the use of mathematical models to approximate the observed 

behavior of a natural water system in response to a specific known set of input and forcing 

conditions. Models that have been validated, or verified, are then used to predict the 

response of a natural water system to changes in the input or forcing conditions. 

Stabilization pond - Large earthen basins that are used for the treatment of wastewater 

by natural processes involving the use of both algae and bacteria. 

Steady state model - Mathematical model of fate and transport that uses constant values 

of input variables to predict constant values of receiving water quality concentrations. 
Storm runoff - Rainfall that does not evaporate or infiltrate the ground because of 

impervious land surfaces or a soil infiltration rate lower than rainfall intensity, but instead 

flows onto adjacent land or waterbodies or is routed into a drain or sewer system. 

Wastewater treatment - Chemical, biological, and mechanical procedures applied to an 

industrial or municipal discharge or to any other sources of contaminated water in order to 

remove, reduce, or neutralize contaminants. 

STORET - U.S. Environmental Protection Agency (EPA) national water quality database 

for STORage and RETrieval (STORET). Mainframe water quality database that includes 

physical, chemical, and biological data measured in waterbodies throughout the United 

States. 

Streamflow - Discharge that occurs in a natural channel. Although the term "discharge" 

can be applied to the flow of a canal, the word "streamflow" uniquely describes the 

discharge in a surface stream course. The term streamflow is more general than "runoff" 

as streamflow may be applied to discharge whether or not it is affected by diversion or 

regulation. 

Suspended solids or load - Organic and inorganic particles (sediment) suspended in and 

carried by a fluid (water). The suspension is governed by the upward components of 

turbulence, currents, or colloidal suspension. 

Trickling filter - A wastewater treatment process consisting of a bed of highly permeable 

medium to which microorganisms are attached and through which wastewater is 

percolated or trickled. 

Verification (of a model) - Subsequent testing of a precalibrated model to additional field 

data usually under different external conditions to further examine model validity (also 

called validation). 

Waste load allocation (WLA) - The portion of a receiving water's total maximum daily 

load that is allocated to one of its existing or future point sources of pollution. 

Wastewater - Usually refers to effluent from a sewage treatment plant. See also domestic 

wastewater. 

Water quality criteria (WQC) - Water quality criteria comprised numeric and narrative 

criteria. Numeric criteria are scientifically derived ambient concentrations developed by 

EPA or States for various pollutants of concern to protect human health and aquatic life. 

Narrative criteria are statements that describe the desired water quality goal. 

Water quality standard (WQS) - A water quality standard is a law or regulation that 

consists of the beneficial designated use or uses of a waterbody, the numeric and 

narrative water quality criteria that are necessary to protect the use or uses of that 

particular waterbody, and an antidegradation statement.