United States
Environmental Protection
Agency
Industrial Environmental Research
Laboratory
Research Triangle Park NC 27711
EPA-600/2-78-004u
August 1978
Research and Development
Source Assessment:
Analysis of
Uncertainty -
Principles and
Applications
-------
RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into nine series. These nine broad cate-
gories were established to facilitate further development and application of en-
vironmental technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The nine series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
6. Scientific and Technical Assessment Reports (STAR)
7. Interagency Energy-Environment Research and Development
8. "Special" Reports
9. Miscellaneous Reports
This report has been assigned to the ENVIRONMENTAL PROTECTION TECH-
NOLOGY series. This series describes research performed to develop and dem-
onstrate instrumentation, equipment, and methodology to repair or prevent en-
vironmental degradation from point and non-point sources of pollution. This work
provides the new or improved technology required for the control and treatment
of pollution sources to meet environmental quality standards.
EPA REVIEW NOTICE
This report has been reviewed by the U.S. Environmental Protection Agency, and
approved for publication. Approval does not signify that the contents necessarily
reflect the views and policy of the Agency, nor does mention of trade names or
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This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
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EPA-600/2-78-004u
August 1978
Source Assessment:
Analysis of Uncertainty -
Principles and Applications
by
R.W. Serth, T.W. Hughes, R.E. Opferkuch, and E.G. Eimutis
Monsanto Research Corporation
1515 Nicholas Road
Dayton, Ohio 45407
Contract No. 68-02-1874
ROAPNo. 21AXM-071
Program Element No. 1AB015
EPA Project Officer: Dale A. Denny
Industrial Environmental Research Laboratory
Office of Energy, Minerals, and Industry
Research Triangle Park, NC 27711
Prepared for
U.S. ENVIRONMENTAL PROTECTION AGENCY
Office of Research and Development
Washington, DC 20460
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PREFACE
The Industrial Environmental Research Laboratory (IERL) of EPA
has the responsibility for insuring that pollution control tech-
nology is available for stationary sources to meet the require-
ments of the Clean Air Act, the Federal Water Pollution Control
Act and solid waste legislation. If control technology is
unavailable, inadequate, or uneconomical, then financial support
is provided for the development of the needed control techniques
for industrial and extractive process industries. The Chemical
Processes Branch of the Industrial Processes Division of IERL
has the responsibility for investing tax dollars in programs to
develop control technology for a large number of operations (more
than 500) in the chemical industries.
Monsanto Research Corporation (MRC) has contracted with EPA to
investigate the environmental impact of various industries which
represent sources of pollution in accordance with EPA's respon-
sibility as outlined above. Dr. Robert C. Binning serves as MRC
Program Manager in this overall program entitled "Source Assess-
ment," which includes the investigation of sources in each of
four categories: combustion, organic materials, inorganic
materials, and open sources. Dr. Dale A. Denny of the Industrial
Processes Division at Research Triangle Park serves as EPA Pro-
ject Officer. Reports prepared in this program are of three
types: Source Assessment Documents, State-of-the-Art Reports
and Special Project Reports.
Source Assessment Documents contain data on emissions from spe-
cific industries. Such data are gathered from the literature,
government agencies and cooperating companies. Sampling and
analysis are also performed by the contractor when the available
information does not adequately characterize the source emis-
sions. These documents contain the information necessary for
IERL to decide whether a reduction in emissions is required.
State-of-the-Art Reports include data on emissions from specific
industries which are also gathered from the literature, govern-
ment agencies and cooperating companies. However, no extensive
sampling is conducted by the contractor for such industries.
Results from such studies are published as State-of-the-Art
Reports for potential utility by the government, industry, and
others have specific needs and interests.
111
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Special projects provide specific information or services which
are applicable to a number of source types or have special
utility to EPA but are not part of a particular source assess-
ment study. This special project report, "Source Assessment:
Analysis of Uncertainty, Principles and Applications," was pre-
pared to describe a procedure for analyzing the uncertainty
associated with data and other information contained in source
assessment studies. The general principles and procedures
used in the analysis are illustrated by means of four applica-
tions to decision making in the area of environmental control.
The application of these principles to the study of air emissions
in the Source Assessment Program is described in detail.
IV
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ABSTRACT
This report provides the results of a study that was conducted
to analyze the uncertainties involved in the calculation of the
decision parameters used in the Source Assessment Program and
to determine the effect of these uncertainties on the decision-
making procedure.
A general procedure for performing an analysis of uncertainty
is developed based on the principles of error propagation and
statistical inference. It is shown that this simple and
straightforward method represents an approximation to standard
statistical techniques. The approximate method is illustrated
by application to four problems in the area of environmental
control.
The general procedure is used to establish guidelines for
conducting air emissions studies in the Source Assessment
Program. In particular, guidelines are established for preci-
sion in field sampling and analytical work, and for setting
critical values of decision parameters.
This report was submitted in partial fulfillment of Contract
No. 68-02-1874 by Monsanto Research Corporation under the
sponsorship of the U.S. Environmental Protection Agency. The
study described in this report covers the period November
1976 to March 1978.
v
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CONTENTS
Preface ill
Abstract v
Figures ix
Tables x
Abbreviations and Symbols xi
1. Introduction 1
2. Summary 4
Background 4
Source severity 4
National emission burden 5
Offset calculations 5
Reasonably available control technology 6
3. Statistical Basis for the Procedure 8
4. Source Assessment Methodology 13
Definition of indices 13
Expanded discussion of source severity 15
Representative source 17
Source severity distributions 19
Stochastic approach to source severity 19
5. Source Assessment: Source Severity 22
Introduction 22
Uncertainties in parameters 23
Deterministic decision approach 25
Stochastic decision approach 28
Tests of hypothesis 30
Comparison of alternate decision approaches .... 35
Guidelines for source assessment program 35
Allowable random uncertainty in emission factor . . 39
Effect of random uncertainty in emission factor . . 40
Operating characteristics of the test 42
Summary 46
6. Source Assessment: Emissions Burden 47
Introduction 47
Uncertainty in Ng 48
Statistical test of hypothesis 50
Summary 51
7. Offset Calculations: Plant Expansion Problem 52
Introduction 52
Working equations 52
Uncertainty in (Q2 - Qt) 53
Test of hypothesis 55
Numerical Example 56
Summary 57
vii
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CONTENTS (continued)
8. Comparison of Alternative Controls: Reasonably
Available Control Technology 59
Introduction 59
Governing equations . 59
Test of hypothesis 61
Numerical example 62
Summary 63
9. Conclusions and Recommendations 64
Conclusions 64
Recommendations 66
References 68
Appendices
A. Accuracy, error, and uncertainty 74
B. Use and interpretation of error propagation formulas . . 77
C. Derivation of source severity equations 98
D. Plume rise 110
E. Alternative methods for estimating "acceptable"
concentration for noncriteria pollutants 116
F. Construction of operating characteristic curves .... 120
G. Relationship of sampling and analysis bias to
systematic errors 127
H. Source severity simulation and probabilistic
sensitivity analysis 129
I. Quantification of uncertainty in source severity . . . .139
J. Example calculations for source assessment: carbon
black manufacture 151
Glossary 154
Conversion Factors and Metric Prefixes 156
Vlll
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FIGURES
Figure Page
1. Deterministic source severity distribution for
carbon monoxide emissions from carbon black
plants 20
2. Impact of alternative hypothesis test approaches
on S* 36
3. Variation in critical source severity as a function
of emission factor data quality and the accept-
able number of days for severity to exceed 1.0
when treating the true source severity as a
random parameter 36
4. Schematic representation of the fiducial statis-
tical approach 42
5. Operating characteristic curves for Case A with
br = 0.10 . . . 43
6. Effective operating characteristic curve for
Case A with b = 0.10 44
7. Effective operating characteristic curves for
Case A as a function of random uncertainty in
emission factor 45
8. Operating characteristic curve for the stochastic
decision test approach 45
9. Effective operating characteristic curve for
example problem 57
IX
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TABLES
Number
1. Measure/Analysis Aspects of Source Severity 5
2. Measure/Analysis Aspects of the National
Emissions Burden 6
3. Generalized Error Bounds for Source Assessment .... 37
4. Contribution of Individual Uncertainties to Total
Uncertainty in Source Severity (Deterministic
Decision Approach) 40
5. Contribution of Individual Uncertainties to Error
Bounds on Source Severity (Deterministic
Decision Approach) 41
x
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ABBREVIATIONS AND SYMBOLS
SECTIONS 1 THROUGH 5
A estimated value of average production capacity, kg/s
a random uncertainty associated with A, kg/s
ar -- ar/A
B measured value of emission factor, g/kg
b bias
b,,, b bounds on bias
JG LI
b random uncertainty associated with B, g/kg
b uncertainty in emission factor due to imprecision in
ri
r2
sampling, g/kg
b uncertainty in emission factor due to imprecision in
analysis, g/kg
b uncertainty in emission factor due to process
variation, g/kg
b uncertainty in emission factor due to imprecision in
measurement of production rate, g/kg
si
su'
bsu
c
CAP
CO
Cr
ฃr
br/B
" bs/B
. upper and lower systematic uncertainties associated
s with B, g/kg
b /B
su'
estimated value of effective emission height,
production capacity, kg/s
carbon monoxide
- random uncertainty associated with C, m
-- cr/C
m
c , c . upper and lower systematic uncertainties associated
su s* with C, m
c c /C
su su7
D estimated value of "acceptable" concentration, g/m3
xi
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ABBREVIATIONS AND SYMBOLS (continued)
ds* - ds/D
d , d . upper and lower systematic uncertainties associated
su siL with D, g/m3
d d /D
su su7 ^
e natural base logarithm, = 2.72
E (x) expected value of % (underbar denotes random variable)
EF emission factor, g/kg
e random uncertainty associated with Sr
*l ~ er/SC
/s
e , e . upper and lower systematic uncertainties associated
su SJ6 with Sc
> A " esu/SC
e , e. upper and lower total relative uncertainties associ-
u ated with SG
F "acceptable" pollutant concentration, g/m3
flf f2 weighting factors
G factor which converts TLV to "equivalent" PAAQS
h physical stack height, m
H effective emission height, m
HQ null hypothesis
HA alternate hypothesis
I calculated value of decision index
I true value of decision index
k constant, s/m
m 97.5% point of distribution of y
M annual mass emissions of given criteria pollutant
n from all stationary sources nationwide, metric tons
M annual mass emissions of given criteria pollutant
from given source type, metric tons
M annual mass emissions of given criteria pollutant
s from all stationary sources within a given state,
metric tons
n, nj, n2 sample size
n fraction of a year
NO nitrogen oxides
PAAQS primary ambient air quality standard, g/m3
xii
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ABBREVIATIONS AND SYMBOLS (continued)
Q mass emission rate, g/s
r number of days per year
s, BI, 82 sample standard deviation
S source severity
S_, "calculated" source severity
S* critical test value of calculated source severity
w
SO sulfur oxides
X
S "true" source severity
t averaging time, min; or student t value
t weighted average Student t value
ave
TLV threshold limit value, g/m3
t short-term averaging time, 3 min
V(x) variance of x
u wind speed, m/s
a level of test
Y correction factor for Gaussian plume equation
AH plume rise, m
y, yi, y2 population mean
a population standard deviation
X estimate of population mean
x"' x"i i X2 sample mean
Xmax maximum time-averaged ground level concentration, g/m3
v
Ameas,
X_red measured and predicted values of ground level concen-
p tration, g/m3
Z number of days per year
SECTION 6
A measured value of CAP , kg/yr
* +
a relative random uncertainty associated with A
B measured value of EF^, g/kg
*
b relative random uncertainty associated with B
j-, f b upper and lower systematic uncertainties associated
su ai with B, g/kg
CAP total production capacity of given source type, kg/yr
D measured value of M , kg
/s ^
d relative random uncertainty associated with D
xiii
-------
ABBREVIATIONS AND SYMBOLS (continued)
d , d 0 upper and lower systematic uncertainties associated
su s* .with D, kg
EF representative emission factor for given source type,
R
e relative random uncertainty associated with (NB)
XT . ^f
e , e upper and lower systematic uncertainties associated
su sJi with (NB)C
e , e\ relative upper and lower total uncertainty in N
H null hypothesis
M annual mass emissions of a given criteria pollutant
n from all stationary sources nationwide, kg
MTTTP^O estimate of M obtained from 1972 National Emissions
1*EDS Report, kg n
M annual mass emissions of a given criteria pollutant
" from a given source type, kg
N_. national emissions burden
13
(N,,)-, calculated value of N,,
B C- o
(NB)T ~ true value of NB
(N*)_ critical test value of (N_)n
D t- D L.
a level of test
SECTION 7
b random uncertainty associated with B , g/s
b N- random uncertainty in Q . , g/s
b random uncertainty associated with BI , g/s
b i random uncertainty associated with BI ' , g/s
b . random uncertainty in Ql . , g/s
b , . random uncertainty in Q , . , g/s
b random uncertainty associated with B^, g/s
b , b upper and lower systematic uncertainties
un associated with BN/ g/s
b ., bOM. upper and lower systematic uncertainties
UNI AMI associated with Q ^, g/s
b , bff upper and lower systematic uncertainties
u associated with BI , g/s
b , , b. , upper and lower systematic uncertainties
ul associated with Bj1, g/s
xiv
-------
ABBREVIATIONS AND SYMBOLS (continued)
b ., ba,. upper and lower systematic uncertainties in Q. . ,
ui] *ID g/s ID
b ,., b,, ,. upper and lower systematic uncertainties in Q ,.,
3 D g/s l D
b ., bป. upper and lower systematic uncertainties
u associated with B., g/s
B measured value of Q , g/s
BI measured value of Ch > g/s
BI, measured value of Qi,, g/s
B. measured value of A i
e random uncertainty associated with difference
BN - BA, g/s
e , e ., upper and lower systematic uncertainties
su S associated with B - B. , g/s
e , e upper and lower total uncertainties associated
U . with BN - V g/S
Ha alternative hypothesis
A
HO null hypothesis
Q emission rate from new unit, g/s
Q . emission rate from ith emission point in new
unit, g/s
Qi original emission rate, g/s
Qi, emission rate from original unit after expansion,
g/s
Q . . emission rate before expansion from jth emission
^ point in original unit, g/s
Q ,. emission rate after expansion from jth emission
* point in original unit, g/s
Q2 emission rate after expansion, g/s
(Q2 - QI),-, calculated value of Q2 - QI = B - B., g/s
(Q2 - QI)T true value of Q2 - QI/ g/s
AQi Q2 - QIii - reduction in emission rate from
original unit, g/s
(AQj). difference in emission rate before and after
^ expansion from jth emission point in original
unit, g/s
a level of test
b random uncertainty in Qj, g/s
XV
-------
SECTION 8
ABBREVIATIONS AND SYMBOLS (continued)
rl n!
etc.
br2
bui' bฃi
bui = bui/Bi
etc.
B
esu'
V e*
HA
HO
Qz
a
(ฃ2)c
(ฃ2 -
SECTION 9
/\
br
TLV
APPENDIX B
a
A
relative random uncertainties
-- random uncertainty in Q , g/s
upper and lower systematic uncertainties in Q , g/s
-- relative systematic uncertainties
upper and lower systematic uncertainties in Q , g/s
measured value of Ql , g/s
-- measured value of Q , g/s
upper and lower systematic uncertainties in
(ฃ2 - ei)
upper and lower total uncertainties in (ฃ2 ~ e1)
alternative hypothesis
null hypothesis
uncontrolled emission rate, g/s
controlled emission rate with presently installed
device, g/s
level of test
lower and upper systematic uncertainties in e2
-- random uncertainty in e2
-- control efficiency of presently installed device
control efficiency of RACT
nominal value of RACT control efficiency
nominal value of (e2 - e^)
true value of (e2 - e^
relative random uncertainty in emission factor
threshold limit value
uncertainty associated with A
nominal value of X or x
xvi
-------
ABBREVIATIONS AND SYMBOLS (continued)
ai, a2 lower and upper systematic uncertainties associ-
ated with A
b uncertainty associated with B
B nominal value of X2 or x2
bj, b2 lower and upper systematic uncertainties associ-
ated with B
c uncertainty associated with C
C nominal value of y or y
GI, C2...C constants
dn-;i Behrens-Fisher statistic
f effective degrees of freedom
f'(x) first derivative of f(x)
f(xj, x2) arbitrary function of xi and x2
k, k constants
n , n sample sizes
!1 if x > 0
-1 if x < 0
0 if x = 0
s pooled estimate of standard deviation
s sample standard deviation of x
J\.
tj_ , (1 - a/2) percentage point of t-distribution
i-a/2, v with v degrees of freedom
V , V, variance factors
xi, x2...x independent variables
xj, x2 mean values of xj and x2
y dependent variable
y mean value of y
Z, , (1 - a/2) percentage point of standard normal
~a/2 distribution
a one minus confidence level
04, a2ป
3lป 32 factors defined in Table A-3
eA, EB...
errors associated with A, B. . -N, and AB
maximum and minimum values of error in product AB
mm r
xvii
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ABBREVIATIONS AND SYMBOLS (continued)
a2 variance of x
H
APPENDIX C
A area of annular region, m2
A' area of sector, m2
a, b parameters in relationship for a
c, d, f parameters in relationship for a
z
Dp population density, persons/m2
F "acceptable" concentration, g/m3
H effective emission height, m
P affected population, persons
PAAQS primary ambient air quality standard, g/m3
Q mass emission rate, g/s
S source severity
t averaging time, min
TLV threshold limit value, g/m3
t short-term averaging time, 3 min
u wind speed, m/s
x downwind distance from emission source, m
xi, X2 roots of equation x/F = 1-0, m
y horizontal distance from plume centerline, m
a lateral dispersion coefficient, m
a vertical dispersion coefficient, m
z
X short-term (3 min) time-averaged ground level concen-
tration, g/m3
X short-term maximum ground level concentration, g/m3
max
X ~~ maximum time-averaged ground level concentration, g/m3
max
APPENDIX D
C estimated value of effective emission height, m
C-. heat capacity at constant pressure of effluent,
'P
kcal/g-ฐK
AHcalc>
XVlll
-------
ABBREVIATIONS AND SYMBOLS (continued)
c , c upper and lower systematic error bounds for effective
emission height, m
Csu - Csu/(Hs + AHcalc>
D stack diameter, m
H physical stack height, m
s
M molecular weight of effluent, g/g mole
P atmospheric pressure
R gas constant = 8.314 x 105 dyne-m/g mole-ฐK
T ambient air temperature, ฐK
a
T stack gas temperature, ฐK
S
u wind speed, m/s
V stack gas exit velocity, m/s
s
AH plume rise, m
AH , calculated value of plume rise, m
calc
AH, -- true value of plume rise, m
APPENDIX E
b - exponent in averaging time correction factor
C pollutant exposure concentration, g/m3
D estimated value of "acceptable" concentration, g/m3
d., d lower and upper uncertainties associated with D, g/m3
A/ U
F "acceptable" concentration, g/m3
G conversion factor
LD50 dosage which results in mortality to 50% of exposed
population, g/m3 or yg/m3
P probability of a lifetime response to pollutant
exposure
p* "acceptable" probability of a lifetime response to
pollutant exposure
PAAQS primary ambient air quality standard, g/m3 or yg/m3
PAAQS2i+ estimated 24-hr primary ambient air quality standard,
g/m3 or yg/m3
t averging time for PAAQS, hr
3. V CJ
TLV threshold limit value, g/m3 or yg/m3
xix
-------
ABBREVIATIONS AND SYMBOLS (continued)
PAAQS primary ambient air quality standard, g/m3 or yg/m3
PAAQS2it estimated 24-hr primary ambient air quality standard,
g/m3 or yg/m3
t averging time for PAAQS, hr
clVCJ
TLV threshold limit value, g/m3 or yg/m3
APPENDIX F
/\
b relative random uncertainty associated with emission
factor
s*
e relative random uncertainty associated with S
e , e 0 upper and lower systematic uncertainties associated
su s* with s_
/s ^ w
e , e . upper and lower relative systematic uncertainties
su s associated with S_
A A V^
e , e. upper and lower total relative uncertainties associ-
u * ated with S,,
\*
F(z) area under normal curve between minus infinity and z
P probability
Sc calculated source severity
S true source severity
z standard normal deviate
a standard deviation
y mean
APPENDIX G
B , B. measured and true values of emission factor, g/kg
meas true
A A
b , b . upper and lower relative systematic uncertainties
su s in emission factor
B , 3~ bounds on positive and negative bias in sampling
and analysis
APPENDIX H
A coefficient in expression for a
parameters in expression for a
% Be fraction of beryllium in coal
CONS coal consumption, g/s
xx
-------
ABBREVIATIONS AND SYMBOLS (continued)
d stack diameter, m
EF emission factor, g/kg
h physical emission height, m
H h + Ah = effective emission height, m
P
p, barometric pressure, mb
Q mass emission rate, g/s
S source severity
S~ mean source severity
J\,
So.gs 95% point of source severity distribution
T ambient temperature, ฐK
a
TLV threshold limit value, g/m3
T stack gas temperature, ฐK
S
u wind speed, m/s
V stack gas exit velocity, m/s
S
x downwind distance, m
Ah plume rise, m
a lateral dispersion coefficient, m
a vertical dispersion coefficient, m
z
Y Y , short-term average ground level concentration, g/m3
, peaK
X2. . -- 24-hr average ground level concentration, g/m3
APPENDIX I
S*
a relative random uncertainty in average production
r capacity
^ป
b relative random uncertainty in emission factor
A. 27 /\
b p , b lower and upper relative systematic uncertainties
s su in emission factor
CO carbon monoxide
s\
c relative random uncertainty in average emission
*r * height
c . , c lower and upper relative systematic uncertainties
in average emission height
/\ /\
d ., d lower and upper relative systematic uncertainties
s su in "acceptable" concentration
xxi
-------
ABBREVIATIONS AND SYMBOLS (continued)
(F ). "calculated" value of "acceptable" concentration
for pollutant i, g/m3
(F ). "true" value of "acceptable" concentration for
1 pollutant i, g/m3
G factor which converts TLV to "equivalent" PAAQS
G average TLV conversion factor
G. TLV conversion factor for pollutant i
m 97.5% point of distribution of y
NO nitrogen oxides
H
PAAQS primary ambient air quality standard, g/m3
s sample standard deviation
SO sulfur oxides
X
t averaging time, min
TLV threshold limit value, g/m3
(TLV). threshold limit value of pollutant i, g/m3
t short-term averaging time, min
x sample mean
I _^
3,3 bounds on positive and negative bias in sampling
and analytical procedures
Y correction factor for Gaussian plume equation
y population mean
a population standard deviation
Y maximum time-averaged ground level concentration,
max g/m3
X ~x a measured and predicted values of ground level
meas, pred concentration, g/m3
APPENDIX J
/s
a relative random uncertainty in average production
capacity
/\
b relative random uncertainty in emission factor
/\ JL s\
b ., b lower and upper relative systematic uncertainties in
emission factor
/\
c relative random uncertainty in emission height
e , e lower and upper uncertainties in source severity
POM polycyclic organic material
xxii
-------
ABBREVIATIONS AND SYMBOLS (continued)
S calculated source severity
S* critical test value of calculated source severity
S true source severity
TLV threshold limit value
XXlll
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SECTION 1
INTRODUCTION
The Industrial Environmental Research Laboratory (IERL) of EPA
uses emissions data to generate information essential to environ-
mental decision making. These data are always subject to a
degree of uncertainty because of inherent errors in the methods by
which they are obtained. When projections based on the data are
employed in decision making, this uncertainty, which extends into
the projections, must be considered.
This report develops a decision analysis procedure for dealing
with the uncertainty in source assessment information in a system-
atic and logical fashion. The procedure, as developed, makes
allowance, on a mathematically sound basis, for the degree of
uncertainty introduced into environmental decision making by
uncertainty in the data.
The procedure described in this report applies specifically to
the evaluation of technical information used in determining the
need to reduce emissions from a source. It does not, however,
take into account value judgments, bias in projecting biological
tests from test animals to humans, or unpredictable variations in
magnitude of emissions due to process upsets, spills, or accidents.
The decision analysis procedure developed in this report consti-
tutes an approximation to standard statistical methods. Such an
approximation is necessary, in general, due to the complexity of
the types of problems considered. The procedure is simple and
straightforward, requiring no detailed knowledge of statistical
distribution theory. Application of the decision analysis method
requires only the manipulation of a few simple formulas (tabu-
lated in Appendix B) with which most engineers and scientists
have some familiarity. Therefore, lack of training in statis-
tical methods should not prove a deterrent to utilization of
this procedure.
The decision analysis procedure is illustrated by means of four
examples related to environmental control. The first two exam-
ples relate to decision problems associated with lERL's Source
Assessment Program involving air emissions. The third example
involves compliance with a given emissions offset policy concern-
ing industrial plant expansions. The fourth example involves a
comparison between two alternative emission control techniques.
-------
The general decision analysis procedure is outlined in Section 3
of the report. The approximations involved in the procedure and
its relationship to standard statistical methods are described.
In Section 4, the decision indices employed in the Source Assess-
ment Program are defined and their use in decision analysis is
discussed.
Sections 5 and 6 present applications of the general decision
analysis procedure to two indices used in the Source Assessment
Program, namely: source severity and national emissions burden.
Source severity combines emissions data, dispersion modeling, and
health effects information to yield a measure of the pollution
hazard represented by a given source of emissions.
Section 7 addresses a problem involving compliance with local air
pollution regulations. The situation considered is that in which
an industrial plant expansion must be accomplished without an
increase in total pollutant emissions in order to prevent deteri-
oration of ambient air quality. Section 8 considers a problem
associated with environmental improvement by means of reasonably
available control technology (RACT). The decision to be made is
whether to require adoption of RACT in preference to an existing,
installed control device.
The major results presented in the report are summarized in
Section 2. Conclusions and recommendations are presented in
Section 9.
The decision analysis procedure utilizes standard concepts of
statistical hypothesis testing. However, there are several areas
where the report used a robust application of statistical theory
in order to make the decision analysis procedure useful to de-
cision makers who are unfamiliar with detailed statistical
methods. Two aspects of the test of hypothesis as used in
this report that are not commonly employed in statistical analy-
ses involve: 1) treatment of systematic errors separately from
random errors, and 2) estimation of the risks of making incorrect
decisions. Those who rigorously apply statistical methods in a
puristic fashion may take exception to the approximate statis-
tical methods employed in the report. The discussion presented
below is designed to identify areas where approximate statis-
tical methods have been employed.
Treatment of Systematic Errors
Systematic errors in data are treated separately from random
errors rather than being combined with them. This treatment is
designed to make the systematic errors (bias) more visible to
the decision maker so that he can deal with the uncertainty
attributed to systematic errors in a conscious fashion. This
treatment leads to a family of operating characteristic (OC)
-------
curves for the test of hypothesis and to the associated concept
of an "effective" OC curve for the test. As explained in
Section 5, the "effective" OC curve represents an entire family
of OC curves and contains a horizontal section, or plateau, in
the center. While such an OC curve may be anathematic to
statisticians, this procedure permits systematic errors to be
treated in a logical and straightforward manner without recourse
to arguments involving statistical distributions. It permits
the distinction between bias and imprecision to be maintained
throughout the analysis. In addition, the authors believe that
this procedure is at least as valid as one which treats system-
atic and random errors as nonseparable entities.
Estimating the Risk of an Incorrect Decision
In developing the estimated risks of making an incorrect de-
cision, the decision alternatives were handled in a conservative
fashion. Normal decision analysis methods divide the decision
into three regions: the acceptance region; the indifference
region; and the rejection region. Further, usual practice in
decision analysis methods is to treat the indifference region
as though it is in the acceptance region. The conservative
approach taken in the report treats the indifference region as
though it is in the rejection region. This approach provides
the decision maker opportunity to ensure that a decision is not
counter to his charter. Specifically, it enables IERL the
opportunity to ensure that pollution control technology is avail-
able to meet the requirements of environmental legislation. The
decision analysis technique does not and is not intended to
ensure that pollution control technology is applied. The
decision to apply control technology requires input for trade-
off (economic, social, political, etc) considerations to be
made and those considerations are beyond the scope of lERL's
source assessment program.
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SECTION 2
SUMMARY
BACKGROUND
The objective of completing a source assessment in the Source
Assessment Program is to provide sufficient information to aid
IERL in determining the need for reducing pollution from station-
ary sources. A source is an entire industrial, commercial, or
municipal operation which is national in scope. An assessment is
the evaluation of a source's pollution based on all available
information on process emissions, discharges, and pollution con-
trol. The product of a source assessment is a Source Assessment
Document. The ultimate result is an EPA decision regarding the
need for further study of the source.
This report describes a decision analysis procedure that can be
used as an aid in evaluating the information contained in Source
Assessment Documents. This procedure, which constitutes an
approximate statistical analysis, employs the concepts of error
propagation, confidence intervals, and hypothesis testing. The
statistical concepts cannot be applied in a rigorous manner due to
both the complexity of the problems considered and the subjective
nature of environmental decision making.
The general decision analysis principles are illustrated by four
examples in the areas of emissions assessment and control describ-
described in the following subsections.
SOURCE SEVERITY
Source severity is defined as the ratio of two concentrations:
1) concentration of pollutant to which the human population may
be exposed and 2) concentration of pollutant which represents an
"acceptable" concentration. The exposure concentration is the
time-averaged maximum ground level pollutant concentration as
determined by Gaussian plume dispersion methodology. The "accept-
able" concentration is estimated in two ways. It is the primary
ambient air quality standard for criteria pollutants, or it is a
surrogate primary ambient air quality standard for noncriteria
pollutants.
The values of source severity which IERL uses as a guide for
screening sources are shown in Table 1. The value of S = 0.05 as
-------
TABLE 1. MEASURE/ANALYSIS ASPECTS OF SOURCE SEVERITY
Measure
a'b'c Analysis
P
S E > 0.05 There is sufficient cause to consider a source as
a candidate for further study.
P
S = < 0.05 There is not sufficient cause to consider a source
as a candidate for further study.
S = calculated source severity.
C = concentration to which the population is exposed.
F = potentially hazardous concentration.
a milepost was obtained by evaluating the uncertainties in sampl-
ing and analysis results, atmospheric dispersion models, and
health effects data.
NATIONAL EMISSIONS BURDEN
The national emissions burden is the mass of each criteria pol-
lutant emission from a source divided by the national mass of
each criteria pollutant emission. This index uses engineering
and emissions data for a source to develop an emissions inventory
for that source. The values which IERL uses as a guide for
screening sources are shown in Table 2. The value of BN = 0.05
as a milepost was obtained by evaluating the uncertainties in
sampling and analysis data and the uncertainties in national
emission inventories such as those generated by EPA's National
Emissions Data System (NEDS). In short, if emissions from a
source amount to more than 0.05% of the U.S. total emission rate
for a given criteria pollutant, the source is considered as a
candidate for further study.
OFFSET CALCULATIONS
As part of EPA's overall program to achieve and maintain national
ambient air quality standards, the states are required (through
state implementation plans) to determine the amount of emissions
reduction necessary to offset the probable impact of increased
population, industrial activity, motor vehicle traffic, and other
growth factors. Of particular concern is the policy regarding
increased emissions due to increased industrial activity through
expansions of existing plants.
One way to offset emissions from a plant expansion is to require
the plant to make a corresponding emissions reduction from its
existing unit. Such a requirement entails the associated problem
of determining a posteriori whether the plant is complying with
-------
TABLE 2. MEASURE/ANALYSIS ASPECTS OF
THE NATIONAL EMISSIONS BURDEN
Measure * * Analysis
BN = Trrr-(lOO) > 0.05 There is sufficient cause to consider a
N source as a candidate for further study.
IMp
BN = y-ri(100) < 0.05 There is not sufficient cause to consider
N a source as a candidate for further
study.
B = national criteria pollutant emissions burden, percent.
Mp = mass of criteria pollutant emissions from a source.
M^ = national mass of criteria pollutant emissions.
the offset policy. Such a determination is not entirely straight-
forward because the emission rates are not known exactly. This
uncertainty should be taken into account in the decision analysis
process.
To quantify the uncertainty in the emission rates before and
after plant expansion, error bounds on the following quantities
are required:
Emission rates from each emission point in the new unit.
Emission rates before and after expansion from each
emission point in the original unit which has different
emissions after expansion.
A simple, approximate statistical test is used to determine
whether the plant may be in compliance with the offset policy.
In order to ensure that air quality is not degraded, the pro-
cedure requires that emissions from the original plant be reduced
by an amount sufficient to compensate not only for the emissions
from added capacity, but also for the uncertainty in emissions
data.
REASONABLY AVAILABLE CONTROL TECHNOLOGY
In order to meet environmental regulations, a plant may be
required to install reasonable available control technology (RACT)
based on a comparison of two alternative control technologies.
For this analysis, it is assumed that a plant currently employs a
given control technology having a nominal (measured) control effi-
ciency which is less than that of RACT for a given application.
In order that the plant be required to adopt RACT, it should be
established that RACT is indeed superior to the installed control
technology. Since neither of the two control efficiencies can be
-------
known with certainty, this determination should not be based on
the nominal efficiencies alone.
A simple, approximate statistical test can be employed to compare
the effectiveness of RACT with that of the installed technology.
The essence of the procedure is to ensure that the incremental
emissions reduction obtained by RACT more than compensates for
the uncertainties in the control efficiencies for the existing
control device and RACT.
-------
SECTION 3
STATISTICAL BASIS FOR THE PROCEDURE
The general procedure set forth in this report is based on the
statistical concepts of confidence intervals and hypothesis test-
ing. In this initial effort, no attempt has been made to apply
alternative decision-theoretic precepts such as the minimax prin-
ciple or the principle of maximum expected utility (1, 2) .
The general procedure can be summarized as follows:
Define a numerical index to be used as an aid in decision
making. Source severity, defined in Section 4, is such
an index. The value of the index may depend upon experi-
mental data, a mathematical model, and/or a semiempirical
relationship, each of which introduces an element of
uncertainty into the problem.
Perform an error analysis to determine the uncertainty in
the value of the index due to uncertainties in the data,
the mathematical model, etc.
Based on the results of the error analysis, formulate a
statistical test of hypothesis as the basis for using the
index in the decision-making process.
Set the values of controllable parameters (e.g., the
level of the test, the critical test value, the accept-
able random uncertainty in the data) to keep the proba-
bility of making an incorrect evaluation of the index
within acceptable limits.
The above procedure may be an iterative one. For example, after
completing the above steps, the decision index may be redefined
in order to reduce the risk of incorrect decisions. In this
manner, the entire decisionmaking process, from initial program
planning through data collection to the final decision, is
focused toward and guided by the needs of the decision maker.
Specifically, the decisionmaking procedure is based on the con-
cept of controlling the risk of making an incorrect decision.
(1) Savage, L. J. The Foundations of Statistics. John Wiley &
Sons, Inc., New York, New York, 1954.
(2) Lindley, D. V. Making Decisions. John Wiley & Sons, Ltd.,
London, 1971.
-------
At the heart of the above procedure is the second steperror
analysis. This analysis makes use of formulas for the propaga-
tion of random and systematic errors presented in Appendix B.
The error analysis procedure can be summarized as follows:
Use formulas for propagation of random errors to obtain
the random component of uncertainty in the index.
Use formulas for the propagation of systematic errors to
obtain the systematic component of uncertainty in the
index.
Combine the results of the above two steps to obtain the
total uncertainty in the index; i.e., the interval of
uncertainty associated with the index.
The above error analysis procedure represents an approximate sta-
tistical technique, as shown in Appendix B. The last two steps
of the procedure, involving the test of hypothesis, employ stand-
ard statistical procedures. However, since the test of hypothe-
sis is based on the results of the error analysis, the entire
procedure constitutes an approximate statistical technique.
The nature of the above approximation and the circumstances which
necessitate it are discussed in the following paragraphs. The
discussion involves the calculation of confidence intervals using
the "Student t" statistic. This subject is covered in most books
on elementary statistics, such as Reference 3.
Let IG denote the ("calculated") value of the decision index
obtained from experimental data, and let IT denote the ("true")
value which would be obtained if there were no errors in the data,
the mathematical model employed, etc. Ic can be regarded as a
value assumed by a random variable, I^/ which is an estimator or
the parameter I-p.3 The value, Ic / isHref erred to as a point esti-
mate of IT/ and the simplest approach to decision making is to
base the decision on this value. However, in order to take
account of the uncertainty in the data and thereby control the
risk of making an incorrect decision, it is desirable to obtain
an interval estimate (i.e., a confidence interval) for IT-
The standard statistical approach to this problem is to determine
the probability distribution of the random variable IG_, and to
use this distribution to construct a confidence interval for IT
and/or to set up a statistical test of hypothesis concerning IT-
The confidence interval thus obtained or, alternatively, the out-
come of the statistical test, then determines the decision to be
made.
Underbars denote random variables.
(3) Freund, J. E. Mathematical Statistics. Prentice-Hall, Inc.,
Englewood Cliffs, New Jersey, 1962. 390 pp.
-------
The above statistical approach is not generally applicable to the
type of problem considered in this report for the following
reasons:
The estimator, IQ, is in general biased due to biases in
the experimental methods, and the value of the bias in
I_C is not known.
lฃ_ is generally a complex function of other random vari-
ables, so that its distribution function cannot be
determined analytically.
The approach used in this report utilizes the principles of error
propagation to obtain an approximate confidence interval for IT.
Although an approximation, the resulting confidence interval is
generally conservative, thereby permitting the establishment of
an upper bound on the risk of an incorrect decision.
To aid in clarifying the relationship between the approximate pro-
cedure used in this report and the rigorous statistical approach,
two simple examples are given below. Both examples involve varia-
tions of the fundamental problem of estimating the mean, y, of a
normally distributed population having variance, a2, given a
random sample of size n_from that population. In the usual proce-
dure, the sample mean, \, and sample standard deviation, s, are
used to obtain a confidence interval for y:
y = x ฑ (1)
/n
where t represents the appropriate value of Student's statistic.
Example A
Suppose now that instead of )(, we use a different statistic, X'
to estimate the mean, y. Further, suppose the random variable
ฃ has the same distribution as )(, except that ฃ is biased by an
amount, b. That is, the expected value of ฃ is given by:
E(ฃ) = y - b (2)
We would then obtain a confidence interval for y as follows:
y - b = ฃ ฑ (3)
/n
y = x + b ฑ (4)
/n
X + b -
-------
Suppose, however, that we do not know the value of b, but that we
have bounds on it as follows:
(6)
Combining Inequalities 5 and 6 gives :
Inequality 7 does not represent a confidence interval in the
usual sense, because we cannot associate a specific confidence
level with it. However, if Inequality 6 is strictly true, then
Interval 7 contains Interval 5. Hence, the confidence level
associated with Interval 7 must be at least as great as that
associated with Interval 5. For example, if the latter repre-
sents a 95% confidence interval, the confidence level associated
with the former must be greater than or equal to 95%. Further-
more, Inequality 7 is the smallest interval that could be con-
structed with the given information for which the confidence
level is at least 95%.
In the approximate procedure used in this report, formulas for
the propagation of systematic errors are used to obtain bounds on
the bias as in Inequality 6. These bounds are then used to
construct an approximate confidence interval as in Inequality 7.
Example B
Consider next another variation of the standard problem in which
it is desired to estimate the sum of means from several normal
populations based on independent random samples from each of
these populations. (For simplicity, we consider the sum of two
means only.) Thus, we wish to estimate (y^ + y2) based on
samples of size nj and n2 from two normal populations whose
variances, aj2 and aj2, are unknown. In this case, there is no
generally applicable statistic corresponding to the t statistic
which can be used to obtain a confidence interval for y^ + y2.
The method reported here uses formulas for the propagation of
random errors to generate an approximate confidence interval
which is based on a weighted average of the two individual
t-values. The (approximate) interval corresponding to Interval 1
is :
ฑ t s (8)
where s = I - + - J (9)
Vn, n, /
11
-------
tave
fl = - (11)
Si2/r\i + s22/n2
s22/n2
f2 = - (12)
Sl2/nl + s22/n2
In using the error propagation formulas, Interval 8 is obtained
directly without the necessity of computing either t or s.
a, V G
In the general case, the problems illustrated in examples A and B
both occur, and it is necessary to use both random and systematic
error propagation formulas. The two calculations are made
separately, and the results are then combined to obtain an approx-
imate confidence interval. This interval can be used directly or,
alternatively, it can be used to set up an approximate statisti-
cal test for decision making.
12
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SECTION 4
SOURCE ASSESSMENT METHODOLOGY
DEFINITION OF INDICES
The study of air emissions in the Source Assessment Program com-
prises a screening procedure designed to identify the need for
application of or development of new control technology for sta-
tionary sources of air pollution. Five numerical indices are
used in the program to identify those industries, or source types,
which are considered to be potential candidates for emissions
reduction.
Source Severity
Source severity, S , is defined as
where Xmax ^s tne maximum time-averaged ground level concentra-
tion of a given pollutant emitted from the source type in ques-
tion, and F represents an "acceptable" concentration of that
pollutant. The source type is considered to be a potential can-
didate for emissions reduction if S ฃ 1.0. Since the true
value of severity is unknown, however, the decision must be
based on an estimated, or "calculated," value of source severity,
SG. In recognition of the uncertainty associated with the esti-
mated severity, the cut points for decision making were original-
ly (June, 1975) set up as follows:
If S > 1.0: Source type is a potential candidate
for emissions reduction.
If 0.1 < Sc < 1.0: Source type may be a potential candidate
for emissions reduction.
If S < 0.1: Source type is not a potential candidate
for emissions reduction.
National Emissions Burden
The national emissions burden is defined as the ratio M /M_
where Mp denotes the total annual mass of emissions of each
13
-------
criteria pollutant from a given source type, and Mn denotes the
total annual mass of emissions of that pollutant from all station-
ary sources nationwide. When the national emissions burden from
a specific source type exceeds 0.001 (0.1%), the source type is
considered to be a potential candidate for emissions reduction.
State Emissions Burden
The state emissions burden is defined as the ratio Mp/Ms where Mp
now denotes the annual mass of emissions of each criteria pollut-
ant from a given source type within a given state, and Ms is the
annual mass of emissions of that pollutant from all stationary
sources within that state. When the state emissions burden
exceeds 0.01 (1%), the source type is considered to be a poten-
tial candidate for emissions reduction.
The preceding are the three dominant screening criteria and com-
prise the basic components of the decisionmaking process. Since
the first index is directly proportional to the emission rate
from a given source, a discrepancy arises when comparing a source
type having a few large plants with a source type having many
small plants. The second and third criteria, which deal with the
total mass of emissions, are designed to overcome this problem.
The two remaining decision indices are used to support or modify
decisions based on the first three indices.
Affected Population
Affected population designates the average number of persons
exposed to high concentrations (e.g., those for which )(/F>1.0) of
a given emission from a given source. This quantity is useful in
characterizing emissions because a given source may exceed the
first three criteria; yet, if it is located in a sparsely popu-
lated area, it may have a relatively small impact on human health
compared with a source located in a densely populated area. In
addition, a source may have a large value of source severity due
to a low emission height. Again, its impact on human health may
be small because the low emission height results in pollutants
being dispersed over a very small area in the immediate vicinity
of the source. The calculation of affected population is
described in Appendix C.
Growth Factor
The growth factor is determined from the ratio of known to pro-
jected emissions from a source type. For example,
Growth factor = Projected emissions in 1978
Emissions in 1973 v '
Other 5-yr periods (e.g., 1975 and 1980) could also be used
depending on available data. The main purpose of this index is
14
-------
to eliminate from consideration those sources whose emissions are
expected to decrease greatly in the near future; e.g., due to the
implementation of new emission controls or to a process being
phased out of use.
In the subsequent analysis of uncertainty, the discussion centers
mainly on the first decision index, source severity. The nation-
al emissions burden is also considered in more detail in Section 6.
EXPANDED DISCUSSION OF SOURCE SEVERITY
Basic Definition
As noted previously in Equation 13, the source severity, SC' fฐr a
given pollutant emitted from a given source is defined as
C - -f
where Xmax equals the maximum time-averaged ground level concen-
tration of pollutant which would result from the source emitting
into a standard receiving atmosphere, and where F equals an
"acceptable" concentration. The standard receiving atmosphere is
defined as one having Class C stability and a mean wind speed of
4.5 m/s. The averaging time for the mean concentration is the
same as that used in the specification of F. For the Source
Assessment Program, the "acceptable" concentration is specified
as follows:
F = PAAQS for criteria pollutants
= (TLV)G for noncriteria pollutants (4)
where G is a factor which converts the TLV into an "equivalent"
PAAQS. The averaging time used in the PAAQS for each criteria
pollutant is shown in Table C-4 of Appendix C. The averaging
time used for noncriteria pollutants is 24 hr.
Appendix E develops a correlation between the Primary Ambient
Air Quality Standard for criteria pollutants and their corre-
sponding TLV's. The analysis indicates that "G" for criteria
pollutant TLV's ranges from 0.0260 to 0.158 with an average value
of 0.0467.
PAAQS represents primary ambient air quality standard.
(4) TLVsฎ Threshold Limit Values for Chemical Substances and
Physical Agents in the Workroom Environment with Intended
Changes for 1976. American Conference of Governmental
Industrial Hygienists, Cincinnati, Ohio, 1976. 94 pp.
15
-------
Calculation of Source Severity
Ideally, SQ would be determined by placing the source in the
standard receiving atmosphere and measuring the resulting maximum
ground level concentration. If the measurement could be made
with no error, the measured value would yield the true source
severity. Obviously, such a procedure is impossible in practice.
Therefore, S^ must be estimated by measuring the source emission
rate and by using dispersion modeling to relate emission rate to
Ymax- In the Source Assessment Program, the simple Gaussian
plume equation is used to calculate source severity as follows
(5, 6) :
S = 2 Q /3\ฐ-17 = 2 CAP EF/3\ฐ-17 (15)
C TreuFH2\t/ ueuFH2 \t/
where Q = mass emission rate, g/s
H = effective emission height, m
u = average wind speed (4.5 m/s)
e = natural base logarithm = 2.72
CAP = production capacity, kg/s
EF = emission factor; i.e., mass of emissions generated
per unit of product produced, g/kg
t = averaging time for mean concentration, min
The concentration obtained from the Gaussian plume equation cor-
responds to an averaging time of approximately 3 min (5). The
factor (3/t)ฐ-17 in Equation 15 corrects this value for averaging
times between 3 min and 24 hr (5, 7). Equation 15 is used for
all pollutants with the exception of nitrogen oxide (NOX), for
which the primary standard averaging time is 1 yr. Since the
above correction factor is not valid for averaging times of this
magnitude, a modified approach, which is described in Appendix C
is necessitated.
(5) Turner, D. B. Workbook of Atmospheric Dispersion Estimates.
Public Health Service Publication No. 999-AP-26, U.S. Depart-
ment of Health, Education, and Welfare, Cincinnati, Ohio,
1969. 84 pp.
(6) Pasquill, F. Atmospheric Diffusion, Second Edition. John
Wiley & Sons, Inc. (Halsted Press), New York, New York, 1974,
429 pp.
(7) Cheremisinoff, P. N., and A. C. Morresi. Predicting Trans-
port and Dispersion of Air Pollutants from Stacks. Pollu-
tion Engineering, 9 (3): 3, 26, 1977.
16
-------
The conversion factor, G, for converting TLV to an equivalent
PAAQS is specified as follows:
G = 300" = 2? 100" = ฐ-0033
Theoretically, the factor 8/24 adjusts the TLV from an 8-hr work
day to continuous (24-hr) exposure, and the factor of 1/100 is
designed to account for the fact that the general population
constitutes a higher risk group than healthy workers. Alterna-
tive methods of estimating "acceptable" concentrations for non-
criteria pollutants are considered in Appendix E.
REPRESENTATIVE SOURCE
In assessing the potential pollution problems associated with an
industry or source type, it is necessary to take into account
plant-to-plant variations in physical parameters such as stack
height, production capacity, or process technology. In the
Source Assessment Program, this problem is handled in two ways:
1) through the concept of a representative source and 2) through
the use of source severity distributions. The representative
source concept is discussed in this subsection; source severity
distributions are discussed in the following subsection.
The concept of a representative source constitutes a simplified
approach to plant-to-plant variability in which an industry or
source type is characterized by means of a "typical" (or repre-
sentative) source or sources. In general, the representative
source is a hypothetical plant having physical parameters which
are typical or average values for the source type in question.
The methodology employed in defining a representative source is
described in the following discussion (8).
Simple Case
The representative source, for a simple case, can be defined as
one having an average plant capacity, average stack height, aver-
age county population density, and an average emission factor.
The following definitions can be used in calculating required
average parameters.
Average plant capacity is the total capacity of the
industry divided by the number of plants.
Average stack height is the sum of all known stack heights
for that industry divided by that number of stacks.
(8) Hughes, T. W., R. B. Reznik, R. W. Serth, and Z. S. Kahn.
Source Assessment Methodology. Contract 68-02-1874, U.S.
Environmental Protection Agency, Research Triangle Park,
North Carolina, January 7, 1976. 33 pp.
17
-------
Average county population density is the sum of the
population densities of all counties within which
plants are located divided by the number of sites.
Average emission factor is the sum of the known emis-
sion factors divided by the number of factors known.
These definitions are valid when there is no correlation between
plant size, emission factor, stack height, and population density
and when emission factor data are taken from a random sample of
the population. One simple way to detect possible correlations
is to graphically depict the variables in question; e.g., plant
size versus population density.
Special Cases
Small Number (Fewer than 10) of Sources
When a small number (fewer than 10) of sources comprises the
source type being studies, a representative source is defined as
the largest source within the group. It may also be desirable to
treat each particular source individually.
Large Number of Sources
For source types with a large number of sites, it is impractical
to determine all of the average parameters. This problem can be
handled in several ways depending on the data base available.
Some possibilities include
Consideration of 30 plants at random. If the parameters
are normally distributed, this is sufficient to find an
average. Random election of 30 plants will give fairly
accurate estimates of the industry's mean parameters while
describing the variation within the industry.
Use of a simulation technique.
Use of state average population density.
Nonhomogeneous Sources--
When qualitative differences in subgroups exist, a representative
source is defined by subdividing into homogeneous subgroups and
considering each group separately.
Log-Normal Distributions
In some sources, the plant capacity is log-normally distributed.
Most of the plants are small, but the few large plants account
for a major portion of the production. Under these circum-
stances, it would be better to subdivide into size groups or to
treat only the larger plants as a worst case condition.
18
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Correlation of Averaging Parameters
Sources which exhibit a correlation between the averaging param-
eters (emission factor, plant size, stack height, population den-
sity) should be treated on a case-by-case basis. If the emission
factor or population density depends on plant size, a weighted
average should be calculated. For other situations, calculating
the distribution of severities using a simulation technique may
be appropriate.
SOURCE SEVERITY DISTRIBUTIONS
A source severity distribution depicts the distribution of source
severities among individual plants within an industry. It thus
provides more detailed information about potential air pollution
problems associated with a given industry than does the repre-
sentative source severity. When sufficient information is avail-
able to compute a source severity for each plant in an industry,
a deterministic source severity distribution can be constructed
for the industry. An example of a deterministic severity distri-
bution is shown in Figure 1 (9). When such detailed information
is not available, a simulation technique can be employed to
generate an approximate (simulated) source severity distribution
based on data from a sample of plants. The methodology used
to simulate source severity distributions is described in
Reference 10.
STOCHASTIC APPROACH TO SOURCE SEVERITY
In the Source Assessment Program, a deterministic approach to
calculating source severity has been adopted by employing the con-
cept of a fixed receiving atmosphere. The main advantage of this
approach is its simplicity: It permits a wide variety of source
types to be analyzed and compared on a consistent basis with a
minimum of experimental and computational effort.
An alternative approach is to allow meteorological parameters
(i.e., stability class or wind speed) to vary stochastically.
This approach generates a frequency distribution of source sever-
ities rather than a single value, as is obtained in the
(9) Serth, R. W., and T. W. Hughes. Source Assessment: Carbon
Black Manufacture. Contract 68-02-1874, U.S. Environmental
Protection Agency, Research Triangle Park, North Carolina.
(Preliminary document submitted to the EPA by Monsanto
Research Corporation, December 1975.) 145 pp.
(10) Eimutis, E. C., B. J. Holmes, and L. B. Mote. Source Assess-
ment: Severity of Stationary Air Pollution Souces--A Simu-
lation Approach. EPA-600/2-76-032e, U.S. Environmental
Protection Agency, Research Triangle Park, North Carolina,
July 1976. 119 pp.
19
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10.0
9.0
8.0
7.0
6.0
5.0
4.0
3.0
2.0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
s
l/l
0.1
0.09
0.08
0.06
0.05
0.04
0.03
0.02
0.01
0.009
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
CO BOILER AND THERMAL INCINERATOR
i i i
ii
0.01 C.05'0.1 0.2 0.5 1 2 5 20 30 40 50 60 70 BO 90 95 98 " W-6
PERCENT OF PLANTS HAVING SOURCE SEVERITY LESS THAN OR EQUAL TO OROINATE
Figure 1. Deterministic source severity distribution for carbon
monoxide emissions from carbon black plants (9).
20
-------
deterministic approach. For making decisions concerning the need
for emissions reduction, a value of source severity corresponding
to a relatively high (e.g., 99 or 99.9) percentage point of this
distribution would be appropriate. For example, for severities
based on 24-hr mean concentrations, values higher than the 99%
point would be expected to occur about four times per year. Thus,
a low value of severity at the 99% point would indicate a low risk
of potentially hazardous conditions arising from the source in
question. The same would not be true of a low value of the mean
severity, for example, since severities much higher than the mean
might occur many times over the course of a year.
The stochastic approach to the calculation of source severity is
investigated in Appendix H. A Monte Carlo simulation and proba-
bilistic sensitivity analysis are carried out in which wind speed,
stability class, lateral and vertical dispersion coefficients,
peak-to-mean concentration ratio-, etc., are allowed to vary sto-
chastically. The output from the simulation is a probability dis-
tribution of values of source severity for beryllium emissions
from a coal-fired powerplant. In this example, the upper 95%
point of the severity distribution is approximately 1.9, and the
upper 99% point is approximately 3.0.
By comparison, the deterministic value of severity (corrected for
plume rise) for the above example is 3.3. This value corresponds
roughly to the 99% point of the stochastic severity distribution.
Thus, on the basis of this example, it appears that the determin-
istic approach is consistent with the stochastic approach.
The simulation described in Appendix H is based on the general-
ized Gaussian dispersion equation (Equation H-7). Hence, the
results are restricted to meteorological conditions under which
the Gaussian model is valid. A complete solution to the problem
would have to include dispersion models applicable to other mete-
orological conditions, such as fumigation (inversion breakup) and
trapping, and the frequency with which the various meteorological
conditions occur.
Although ground level concentrations under trapping and fumiga-
tion conditions can be considerably higher than those predicted
by the Gaussian model (coning conditions), these conditions gener-
ally persist for relatively short periods (30 min to 4 hr).
Therefore, these conditions may have a relatively small effect on
24-hr mean concentrations, which are of primary interest in rela-
tion to source severity. The above reasoning provides justifica-
tion for the exclusive use of the Gaussian model as an approxima-
tion in the simulation procedure.
The example given in Appendix H also indicates that the determin-
istic approach to source severity yields a worst case value in
the sense that higher values would be expected to occur only
infrequently, on the order of once per year.
21
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SECTION 5
SOURCE ASSESSMENT: SOURCE SEVERITY
INTRODUCTION
In this section, the general procedure outlined in Section 3 is
applied to the source severity decision index. It was noted in
the previous section that the true value of source severity is
always unknown and that decisions must be based on an estimated,
or "calculated," value of severity. The uncertainty associated
with the estimated value of severity is related to uncertainties
in individual parameters (i.e., production capacity or emission
factor) used to calculate severity. In the following subsections,
a mathematical expression for this relationship is obtained and
used to establish overall guidelines for decision making in the
Source Assessment Program.
Only the source severity of a representative source will be con-
sidered. In addition, it will be assumed that the source type in
question is homogeneous with respect to emission factor, so that
the representative source emission factor can be obtained by sam-
pling at a single plant.9 This assumption may appear to pose a
severe restriction on the applicability of the analysis; in prac-
tice, however, economic constraints often dictate that sampling
be conducted at a single plant. Use of the measured emission
factor to calculate the severity of the representative source
then involves the implicit assumption of homogeneity with respect
to emission factor.
For the purpose of this analysis, it is assumed that the true
source severity, S , is given by the following equation:
0.17
where y is a correction factor for the simple Gaussian plume
equation. Since we are dealing with the representative source,
CAP and H now represent arithmetic mean production capacity and
effective emission height, respectively, for the source type in
This assumption is made to simplify the analysis and does not
represent an inherent limitation of the general procedure. .
22
-------
question. The emission factor, EF, in Equation 16 is the emis-
sion factor for the plant at which sampling was performed.
The following discussion will analyze Equation 16 by defining
the uncertainties contained in each of the variables, by showing
how the uncertainties are propagated, by developing alternate
decision tests (tests of hypothesis), by quantifying and compar-
ing the alternate decision tests, and by developing guidelines
for the Source Assessment program. Two alternate decision tests
will be developed analytically - a deterministic approach and a
stochastic approach. The deterministic approach assumes that
the true value of source severity is a fixed parameter and
treats large uncertainties in Equation 16 as systematic errors.
The stochastic approach assumes that the true value of source
severity is a random parameter (i.e., changes daily) and treats
all uncertainties as random errors.
UNCERTAINTIES IN PARAMETERS
The uncertainties associated with each of the individual param-
eters appearing in Equation 16 are discussed below. The concepts
of random and systematic uncertainties are discussed in Appendices
A and B.
CAP (Average Production Capacity)
If CAP is determined from a sample of plants in the industry or
source type, there is a random sampling uncertainty which shall
be equated with the 95% confidence interval about the sample mean
plant capacity. If CAP is determined from a knowledge of all
plants in the industry, then this uncertainty is zero.
H (Average Effective Emission Height)
The average effective emission height, H, is comprised of two
components, H = h+AH, where h is the physical stack height and
AH is the plume rise. A statement similar to the one made for
CAP applies to h as well. In addition, there is uncertainty in
estimating plume rise, AH. The error in plume rise can be
assumed to be a randomly distributed uncertainty over the popu-
lation of all sources and atmospheric conditions, provided AH is
indeed included in H.
In the development of the deterministic decision test later,
source severity is computed for a fixed source and a fixed re-
ceiving atmosphere. Thus, the estimated severity, S_, can be
\*,
The receiving atmosphere was defined by specifying stability
class and wind speed. These two parameters do not suffice to
define a unique state of the atmosphere. However, it may be
assumed that the additional meteorological parameters necessary
to define a unique atmospheric state are maintained at fixed,
albeit unspecified, values.
23
-------
considered as a sample value from a population of severities in
which both source type and receiving atmosphere are fixed. In
this use, the error in plume rise is treated as a bias in Sr since
the error is amplified by squaring H in Equation 16. In the
development of the stochastic decision test developed later,
source severity is computed for a fixed source and a variable
receiving atmosphere. In this use, the error in plume rise
represents a random uncertainty in S_,.
EF (Emission Factor)
There is an uncertainty in EF which is composed of process varia-
tion, sampling uncertainty, analysis uncertainty, and uncertainty
in production rate during sampling. The total uncertainty has
both a random and a systematic component, the latter resulting
from possible bias in the sampling and analytical procedures.
The random component is given by:
a
ซ_ 9 i 9 . i_9 . i_9 , i_ 9
D = D T D T D ~r D
where b = random uncertainty in emission factor, g/kg
b = uncertainty in emission factor due to imprecision
in sampling, g/kg
b = uncertainty in emission factor due to imprecision
in analysis, g/kg
b = uncertainty in emission factor due to process
variation, g/kg
b = uncertainty in emission factor due to imprecision
in measurement of production rate, g/kg
Y (Correction Factor)
There is an uncertainty in the correction factor, y/ which is a
reflection of the uncertainty inherent in the Gaussian dispersion
equation itself. For the purpose of this analysis, values of y
are assumed to be log-normally distributed over source types and
atmospheric conditions with mean of 1.0 and a variance of 0.264
(standard deviation = 0.514). The 2.5% and 97.5% points of the
distribution are taken to be 1/m and m (where m = 3).
In the deterministic decision test developed later, the uncer-
tainty in Y is treated as a systematic uncertainty with m = 3.
In the stochastic decision test, the uncertainty in Y is treated
as a randam uncertainty with a variance of 0.264.
If the source type is not assumed to be homogeneous with respect
to emission factor, then the random uncertainty contains an addi-
tional term due to plant-to-plant variability of emission factor,
In general, the plant-to-plant distribution of emission factors
is not a normal distribution, which further complicates the
analysis.
24
-------
(t /t)ฐป17 (Averaging Time Correction Factor)
For the purpose of the present analysis, the uncertainty associ-
ated with the averaging time correction factor (t /t)ฐ-^7 is
assumed to be included in the uncertainty associated with y
This assumption is discussed further in the following section.
F (Acceptable Pollutant Concentration)
There may be an uncertainty associated with the "acceptable"
pollutant concentration, F, depending upon interpretation. It
can be argued that there is no pollutant concentration, however
small, which is entirely innocuous. Therefore, specification of
an acceptable concentration represents a subjective judgment on
the part of the decision maker. Hence, the "true" value of F may
be regarded as whatever value the decision maker specifies. In
this case then, there is no uncertainty in F since the "true"
value equals the "calculated" (specified) value.
Alternatively, it can be argued that the PAAQS represents a pol-
lutant concentration at which no adverse effects on human health
can be detected. Therefore, the "true" value of F can be equated
with the PAAQS. In this case, there is no uncertainty in F for
criteria pollutants. For noncriteria pollutants, however, there
is uncertainty associated with the conversion factor, G, which
converts the TLV into an equivalent PAAQS. This uncertainty can
be assumed to be randomly distributed over the population of all
pollutants. However, since source severity is computed for a
given pollutant, this uncertainty should be treated in the analy-
sis as a systematic uncertainty.
DETERMINISTIC DECISION APPROACH
In developing a deterministic decision test it is assumed that
the true value of source severity is a fixed parameter. The test
is developed for a fixed source in a fixed atmosphere. It treats
small uncertainties as random errors and large uncertainties as
systematic errors. The approach employs the development of formu-
lae for the propagation of errors.
Propagation of Random Uncertainties
The total uncertainty in source severity is determined by the
propagation of uncertainties in CAP, H, EF, y, and F through
Equation 16. For convenience, let A, B, C, and D represent
estimated or measured (as opposed to true) values as follows:
A = CAP
B = EF
C = H
D = F
25
-------
The "measured" value of y is taken as unity. Furthermore, let
k = -Mir1) = constant (18)
ireuyt /
Considering the random uncertainty first,
s
C
where ar, br, cr, and er are the random uncertainties in A, B, C,
and SQ, respectively, and Sc is the nominal, or "calculated,"
source severity; i.e.,
S_ = . (20)
C DC2
Thus, if we are working at the 95% confidence level, (A ฑ ar)
represents a 95% confidence interval for the industry mean produc-
tion capacity, CAP.
Now let
ar = arA
= brB
- crC
e = e S_
r r C
i.e., ar, br, cr, and er are the relative random uncertainties
associated with A, B, C, and Sc- Using the error propagation
formula for multiplication (see Appendix B),
(A ฑ aWB ฑ
= AB ฑ
= AB ฑ WB2A2ar2 + A2B2br2
= AB ฑ AB ar2 + br2 (21)
Using the error propagation formula for exponentation,
i cj 2 = C2 ฑ 2 Ccr = C2 ฑ 2 C2cr (22)
26
-------
Using the division formula,
AB ฑ AB Ja 2 + b 2 fln 1fc*B2(ฃ 2 + ฃ2)
* JiT JL _ t\D , ป ^ i
"
c 2
C2 ฑ 2 C28 C2 C1* C8
= AB ฑ AB /j 2 + ฃ 2 + 4 ฃ 2
c2 c2 T r r r
Thus,
S ฑ e = ฑ a 2 + ฃ 2 + 4 ฃ2 (24)
C r C2D C2D f r r r
Therefore.
and
= e = Ja 2 + b 2 + 4 3 2 (26)
kAB/C2D r ' r r r
Propagation of Systematic Uncertainties
In general, the systematic components of uncertainty are unsymmet-
rical. Therefore, let
+b +c +d +e
B_bSU, C_CSU, D_dSU, and S- SU
C 0 C 0 O 0 n
be systematic error bounds on EF, H, F, and S-p, and let bsu, bsฃ,
etc., denote the corresponding relative uncertainties. The upper
and lower bounds on y are taken as m and 1/m, respectively.
Since all variables in Equation 16 are non-negative, the upper
bound on source severity can be obtained by substituting upper
bounds in the numerator and lower bounds in the denominator.
Thus,
m(B
(E - O
q j. Q = flrA\ i ' O~l\
O . T C I JX ฃ\ I I Z / )
C SU ' " ' ^ - '
Rearranging this equation yields
27
-------
su
- 1
(28)
The lower bound is obtained in a similar manner.
m l
(29)
Uncertainty Interval for Source Severity
The total uncertainty associated with source severity is the sum
of the random and systematic components (see Appendix B). Symbol
ically,
A A A
e = e + e
u r su
s\ /\ s\
e* = er + esฃ
Using Equations 26, 28, and 29 for er, esu, and eSฃ yields:
- 1
/A ', T
u = Var +
ฃsu)
-V
a_> + b..2 + 4 cj- + 1 -
su
(30)
(31)
The uncertainty interval for source severity is then given by the
following inequality:
(32)
STOCHASTIC DECISION APPROACH
In developing the stochastic decision test approach the true value
of source severity is assumed to be a random parameter. The test
is developed for a fixed source in a random receiving atmosphere.
It treats all uncertainties as random errors. The approach differs
from the deterministic decision approach in that the deterministic
approach fixes the receiving atmosphere. The approach also differs
from the one presented in Appendix H in that the latter uses vari-
able emission sources as well as a random receiving atmosphere.
28
-------
Uncertainties in the Stochastic Decision Approach
In the deterministic decision approach, S is treated as a fixed
parameter by treating y as a constant. It is intuitively reason-
able to attempt to describe ST in a stochastic manner since
meteorological change day to day. The calculated source severity
is:
/t \ ฐ l 7
_ _ 2 (CAP) (EF) To] { .
SC eyFH' \~] (33)
which may be rewritten as.
q _ v (CAP) (EF)
sc - K -Z - (34)
where K is a constant. Recalling Equation 16, we have
/t \ ฐ . * 7
_ 2 (CAP) (EF) ^o)
ST - Y eyFH^ - (/
which may be rewritten as
(CAP) (EF) ,
ST ~ YK ^ (35)
Based on the discussions presented earlier, ST can be treated as a
log-normally distributed random parameter. S_ is the product of a
constant and a log-normally distributed variable (y) Treatment
of the above equations is best accomplished by taking logarithms.
In S_ = In K = In (CAP) + In (EF) - In y - In F - 2 In H
\
The expected value of Sc (E(log S_) ) is
E(log Sr) = log
.(CAPHEF! . ,
The variance of log S_ (V(log S )) is
V- V,
Vdog Sc) = i
S
Uncertainty Interval for Source Severity
The uncertainty interval for ST is derived as follows:
In Sc - ZaVVdog SG) < log ST < In SG + Zaป/V(log SG) (38)
29
-------
TESTS OF HYPOTHESIS
The above discussion leads to the development of two approaches to
hypothesis testing for source severity. The deterministic decision
approach treats S as a fixed parameter and tests whether S > 1.0.
The stochastic approach treats S_ as a random parameter; it tests
the number of days for which S_ 2 1 againsts an "acceptable" number
of such occurances. The following discussion developes the hypo-
thesis testing procedure for both decision approaches.
Hypothesis Testing for Deterministic Decision Approach
Inequality 32 can be used to set up a formal test of hypothesis.
The hypothesis that we wish to test (the null hypothesis, H0) is
that the true severity is greater than or equal to unity; i.e.,
H0: ST > 1.0
This hypothesis is tested against the alternative hypothesis, HA>
HA: ST < 1.0
The statistic which is used to make the test is Sc- If SQ
exceeds a specified critical value, Sฃ, H0 is accepted and it is
concluded that ST > 1.0; if SQ is less than Sฃ, H0 is rejected
and it is concluded that ST < 1.0.
The basic idea behind the test is that it should be concluded
that ST > 1.0 whenever the upper end point of the uncertainty
interval (Interval 32) exceeds unity. Thus, the critical value,
Sฃ, is given by
S*l + e = 1.0
S* = -~- (39)
1 + eu
With any such statistical test, two types of errors can occur.
These errors, usually designated Type I and Type II, are defined
as follows:
Type I Error; Reject H0 when it is, in fact, true. That is, we
conclude that Sm < 1.0 when it is, in fact,
greater than unity.
Type II Error; Accept Hg when it is, in fact, false. That is,
we conclude that ST > 1.0 when it is, in fact,
less than unity.
30
-------
These possible errors can be represented schematically as
follows:
HO is true
HO is false
Accept HO
Reject HO
Correct decision
Type I error
Type II error
Correct decision
Within the context of the Source Assessment Program, the effects
of the two types of incorrect decisions are as follows: a
Type I error may results in a needed emissions reduction not
being accomplished; a Type II eror may result in an emissions
reduction which is not really needed.
In order to see how the risk of an incorrect decision can be
controlled, it is convenient to use Inequality 32: as repeated
here.
(32)
Now assuming that 95% confidence intervals (i.e., A ฑ ar, etc.)
were used to construct the approximate Interval (32), and further
assuming that the confidence level associated with Interval 32 is
greater than or equal to the nominal level, the probability that
the true value of ST will fall in Interval 32 is greater than or
equal to 95%. In other words, the probability of obtaining a
value of ST which falls on either side of Interval 32 is less
than or equal to 2.5%.
Thus,
fe, <
1.0.
Then we have T ~
K < rh; s> HHS< < r?r \-^
2.5%
Using Equation 39, we obtain:
P|SC < Sc'ST - 1'
2'5%
(41)
31
-------
But Sc < 5$ is precisely the condition for rejecting H0. There-
fore, the probability of rejecting H0 when it is true, i.e., of
making a Type I error, is less than or equal to 2.5%. In general,
if we work at the (1 - 2a)(100%) confidence level in constructing
the uncertainty interval (32) , then the probability of making a
Type I error will be less than or equal to 100 1/d - eฃ) , the probabil
ity of a Type II error will be less^than 2.5%. However, H0 is
accepted whenever Sc > Sฃ = 1/(1 + eu). Thus, for values of Sc
in the interval
(42)
there will be a (relatively) large chance of a Type II error.3
Therefore, in order to minimize the number of Type II errors in
repeated applications of the test procedure, the Interval 36
should be made as small as possible; i.e., the smaller the un-
certainty in source severity, the fewer the number of incorrect
decisions that are expected to be made.
Within the context of the Source Assessment Program, the width of
the Interval 36 can be controlled primarily through the random
uncertainty in emission factor, br. This uncertainty can (theo-
retically) be made as small as desired by analyzing a sufficient-
ly large number of samples. The other uncertainties which
contribute to Interval 36 cannot readily be controlled in this
program.
In order to Specify an allowable value for b , a width must be
specified for Interval 36. Although this laiter specification is
arbitrary, an upper bound on br can be obtained by setting the
upper limit in Interval 36 equal to infinity. Roughly speaking,
An alternative procedure isAto reject H0 when G
Sc < 1/(1 + eu) ,
accept HO when S > 1/(1 - eป) , and make no decision when SG
falls in the Interval 36. This procedure is considered in more
detail below. For the present, it is assumed that a decision
(accept or reject H0) is always required.
32
-------
this condition means that we can never be sure that S > 0, no
matter how large the calculated severity, SQ, may be. This can
be seen by setting (1 - ep) = 0 in Interval 32. Stated another
way, when this condition prevails, there will always be a large
probability of a Type II error, regardless of how large the value
of SQ may be. From Interval 36, this upper bound on 6r is found
by solving the equation
ej, = 1.0 (43)
Hypothesis Testing for Stochastic Decision Approach
The hypothesis that we wish to test is not whether S > 1.0 but
whether the number of times per year S > 1.0 is more than some
acceptable number of times; i.e., Ho: ST > 1 on more than r days
per year. This hypothesis is tested against the alternative
hypothesis, HA,
H : S > on less than r days per year
ฃ\ J.
Obviously this hypothesis would be most relevant when the "measure
of harmful effect" is proportional to the number of days of
threshhold exceedance. If the harm done depends in a more compli-
cated way on S_, for example on the actual amounts by which S_
exceeds 1.0, tnen other hypotheses may be more appropriate.
The hypothesis H may be rephrased as:
Ho: The expected number of days on which S^ > 1, exceeds r.
or
H0: E(Z) > r
where E denotes the mean (expected value), but, E(Z) = 365 P\S >l|
so that l 1 '
H0:
where n = r/365.
The hypothesis becomes,
Ho:
H0: p|lny>lnS >>n
The hypothesis becomes,
Ho: E(ln Sc)>V(lnY)Z-
33
-------
where V(lny) is the variance of log y (V(lny) = 0.3136 from
Appendix I) and Z is the n-percentage point of the standard
normal distribution. The level a test for H0 is to reject H0 if
In S>, < In S* where
Pjln Sc < In S*|W(ln Y)Z- = aj
so that
or
in S* - fV(ln Y)Z-= z (44)
V(ln Sc)a
log S* = (Z-)[v(log Y)]*5 + (Za) [v(log SG)] h (45)
The critical value of S- for an a-level test is
S* = e(ZE)tv(log Y)^ + (Za} CV(lo9 SC^h (46)
As an example, assume that r = 1 day per year (current basis for
several of the criteria pollutants), and that a = 0.05 is desired,
We have :
r = 1
n = 1/365 = 0.00274
Z- = -2.786
n
Z = Zo .05 = -1.645
Hence, we would reject Ho if,
c ._..,,. x^y Y) + 1.645/V(log S_)) .._.
Of-, *- C \ * / \^ I I
Remembering that
V(log Y) = 0.3134
or
[V(log v)]35 = 0.560
we would reject H0 if
< e-/1.555 + 1.645 V{log
(48)
34
-------
COMPARISON OF ALTERNATE DECISION APPROACHES
The above approaches to hypothesis testing utilize different start-
ing assumptions (S_ is a "fixed" versus a "random" parameter) and
they treat uncertainties in different manners (large uncertainties
treated as systematic errors versus all uncertainties treated as
random errors). Yet, the critical values of source severity for
decision making are quite similar. This is best illustrated in
Figure 2 where S* is plotted as a function of the alternative
decision appraches and uncertainty in emission factor. The most
frequently occuring range of b (emission factor uncertainties)
is 0.6 to 1.0. The bias (b )rin available sampling and analysis
technologies ranges from 0.6uto 0.5. The values S* vary from 0.5
to 0.8 for the general cases displayed in Figure 2 regardless of
the decision approach used.
The following observations of Figure 2 are readily apparent.
Given the current (1979) capabilities in measurement technologies,
both decision approaches give the same S* values. The determin-
istic decision approach is insensitive to improvements in measure-
ment technologies. The stochastic decision approach is quite
sensitive to measurement technology improvements.
Figure 3 shows how S* changes as r increases for the stochastic
decision approach.
GUIDELINES FOR SOURCE ASSESSMENT PROGRAM
The above test of hypothesis can be applied to the decisionmaking
process by computing the critical severity, Sฃ, for each pollut-
ant emitted from each source type; i.e., the test can be perform-
ed for each individual pollutant and each source type studied.
An example of this approach is given in Appendix J. Alternative-
ly, the test procedure can be used to derive general guidelines
for decision making in the Source Assessment Program. The latter
approach is followed in this section.
In order to establish overall guidelines for the Source Assessment
Program, generalized error bounds have been estimated as shown in
Table 3. These estimates are derived in Appendix I. The values
listed for^ar and cr represent 95% confidence limits. In this
analysis, br is considered to be an independent variable. The
values given for br in Table 3 indicate the range in which br can
be expected to fall in most cases. Three sets of values are
listed for the uncertainty in "acceptable" concentration, F. In
Case A, there is no uncertainty associated with F (cf. discussion
on page 30). In Case B\, the "acceptable" concentration is
considered to be uncertain, and the TLV conversion factor, G, is
equal to the mean value (0.047) for criteria pollutants (see
Appendices E and I). Case B2 is the same as B\ except that
G = 0.0033, the value currently used in the Source Assessment
Program.
35
-------
o
QC
O
in
-0.2
^x ASSUMING ST IS PERMITTED TO
xv EXCEED 1.0 ONLY 1 DAY PER YEAR
ST IS A "FIXED PARAMETER"
ST IS A "RANDOM PARAMETER"
MOST FREQUENTLY OCCURING RANGE
OF br ON SOURCE ASSESSMENT PROGRAM
,7
OBSERVED RANGE OF BIAS IN AVAILABLE
SAMPLING AND ANALYSIS TECHNOLOGIES.
0.1 0.2 0.3 0.4 0.5 0.6 Q.I 0.8 0.9
UNCERTAINTY IN EMISSION FACTOR, br
Figure 2. Impact of alternative hypothesis
test approaches on S*
i.o
-------
TABLE 3. GENERALIZED ERROR BOUNDS FOR SOURCE ASSESSMENT
Uncertainty Value
A
a
A
C
A
b
b
b
A
C
A
C
A
d
A
d
m
0.05
r
0.05
r
0 to 1.0
r
0.5
su
0.1
O A/
0.1
su
~ o 0.5
O A/
0 4.2 71.6
A Bl B2
sฃ 0 0.81 0
3.0
For the Deterministic Decision Approach
Substituting values from Table 3 into Equations 30 and 31 yields
the following error bounds for source severity:
Case A
ea =
eu = ^tปr2 + 0.01 + 17
+0.01 +0.75
(49)
Case
+ 0.01 + 94
= Jbr2 +0.01 + 0.95
(50)
37
-------
Case B?_
" * + 0.01 + 17
(51)
The corresponding uncertainty interval for source severity is
obtained by substituting Equation 49 , 50 / or 51 into Inequal-
ity 32. For example, taking br = 1.0 as a worst case,3
Case A 0 < ST < 19 Sc (52)
Case Bi 0 < ST < 96 Sc (53)
Case B? 0 < S < 19 S (54)
c
At the other extreme, when b = 0,
Case A 0.15 SG < ST < 18 SG (55)
Case B! 0 < S^ < 95 Sn (56)
~- "" ฑ "* \*
Case B? 0 < S < 18 S (57)
T ~n --rn"-'~- ~ ^
The critical value of the "calculated" severity is given by Equa-
tion 39. Substituting for eu from Equations 49, 50, and 51
yields the following results:
Case A . S* = 1 (58)
18 + Vbr2 +0.01
Case BT S* = l (59)
95 + Vbr2 +0.01
Case B7 S* = (60)
18 + Vb 2 + 0.01
Note that source severity cannot be negative, so the lower bound
cannot be less than zero.
38
-------
As br varies from 0 to 1.0, SJ varies from 0.055 to 0.053 in
Cases A and B2 and from 0.0105 to 0.0104 in Case BX. It follows
that the critical test value should be set at 0.05 for Cases A
and B2 and at 0.01 for Case BX.
For the Stochastic Decision Approach
Substituting values from Table 3 into Equation 26 yields the
following value for V(log S ), regardless of how F is treated.
V(log Sp) = Vb^ + 0.01
\~r I.
The corresponding uncertainty interval for source severity is
obtained by substituting Equation 61 into Equation 48. For ex-
ample, taking b = 1.0 as a worst case, we get,
0.0404 Sc < ST < 24.7 SG (62)
The corresponding uncertainty interval for source severity is ob-
tained by substituting by taking b =0.0.
0.179 SG < ST < 5.58 Sc (63)
ALLOWABLE RANDOM UNCERTAINTY IN EMISSION FACTOR
An upper bound on the allowable value of br is obtained by solv-
ing Equation 44. Substituting for e from Equations 49, 50, and
51 gives the following results:
Deterministic Stochastic
Approach Approach
Case A br < 0.23 (59)
Case BI No solution
Case B2 No solution
infinity
In Cases BI and B2 , the restriction imposed by Equation 44 can-
not be met even with 6r = 0. Hence, in these cases, there will
always be a large risk of a Type II error in the screening
procedure.
Strictly speaking, Equations 58 through 61 are valid for a test
level, a, of 0.025. However, these results are nearly independ-
ent of a. This is a reflection of the fact that the total
uncertainty in source severity is dominated by the systematic
component of uncertainty (see below).
39
-------
In principle, then, for Case A (no uncertainty in "acceptable"
concentration), Inequality 61 can be used with standard proce-
dures to estimate the number of samples required in a sampling
program. In most situations involving environmental sampling,
however, the restriction imposed by Equation 61 will result in an
impractically large number of samples. In practice then, there
is generally little that can be done to control the risk of a
Type II error when using the present screening procedures. This
is again a reflection of the fact that the random uncertainty in
emission factor makes a relatively small contribution to the
total uncertainty in source severity.
EFFECT OF RANDOM UNCERTAINTY IN EMISSION FACTOR
Comparison of Inequalities 52 through 53 with Inequalities 55
through 57 shows that the random uncertainty in emission factor
has a relatively small effect on the total uncertainty in source
severity. The total uncertainty is dominated by the systematic
uncertainties, as illustrated in Table 4. This table lists the
reduction in the width of the uncertainty interval (Inequality 41,
42, or 43) which would be obtained if each of the individual
uncertainties was eliminated while leaving the others unchanged.
The table was constructed using the values of the individual
uncertainties listed in^Table 3, together with br = 1.0, to cal-
culate the width (eu + ejj,) of the Interval 32. Each of the
individual uncertainties was then set to zero while leaving the
others unchanged, and the resulting reduction in the width of the
interval was computed.
TABLE 4. CONTRIBUTION OF INDIVIDUAL UNCERTAINTIES
TO TOTAL UNCERTAINTY IN SOURCE SEVERITY
(DETERMINISTIC DECISION APPROACH) .
Reduction in width of uncertainty
interval for source severity, %^
Uncertainty eliminated Case A Case BI Case 62
1.
2.
3.
4.
5.
6.
7.
Random uncertainty in emission factor.
Systematic uncertainty in emission factor.
Total uncertainty in emission factor (1 + 2) .
Uncertainty in dispersion equation.
Uncertainty in plume rise.
Total uncertainty in dispersion modeling (4 + 5) .
Uncertainty in "acceptable" concentration.
10
30
40
63
69
87
0
2
33
35
65
74
90
80
10
30
40
60
68
83
1.
3
Uncertainty interval computed assuming b = 1.0.
K
No uncertainty in acceptable concentration.
Acceptable concentration uncertain; TLV conversion factor equal to geometric mean value.
Acceptable concentration uncertain; TLV conversion factor equal to 1/300.
Contrary to what might be expected, Case B2, in which the accept-
able concentration is "intentionally underestimated," has a nar-
rower uncertainty interval than does Case Bj. In general,
underestimating the acceptable concentration decreases both the
40
-------
upper bound (which tends to decrease the total uncertainty) and
the lower bound (which tends to increase total uncertainty) on
source severity. However, in Case BI the lower bound is already
zero, so it cannot be decreased further. Therefore, the only
effect of underestimating the acceptable concentration is to
decrease the upper bound, which reduces the total uncertainty in
source severity.
In addition, since eu can vary from zero to infinity while ej,
can vary from zero to one, the above calculation tends to give
more weight to the upper bound than to the lower bound. Although
underestimating the "acceptable" concentration does not reduce
the uncertainty associated with acceptable concentration, it does
shift it to the lower bound where it is less conspicuous. There-
fore, in Table 4, Case Bj is more indicative of the relative
magnitude of the uncertainty associated with acceptable concen-
tration than is Case B2.
The upper and lower error bounds on source severity are tabulated
in Table 5 for each of the cases listed in Table 4. These error
bounds correspond to the situation in which ar = cr = 0, so that
the random component of uncertainty becomes
er = War^ + br2 + 4 c^ = br (64)
TABLE 5. CONTRIBUTION OF INDIVIDUAL UNCERTAINTIES TO ERROR BOUNDS
ON SOURCE SEVERITY3 (DETERMINISTIC DECISION APPROACH)
Lower bound;
1
2.
3.
4.
5.
6.
7.
8.
'Uncertainty eliminated
None.
Random uncertainty in emission factor.
Systematic uncertainty in emission factor.
Total uncertainty in emission factor
(2 + 3).
Uncertainty in dispersion equation.
Uncertainty in plume rise.
Total uncertainty in dispersion modeling
(5+6).
Uncertainty in "acceptable" concentration.
b
Case A
0.25 -
0.25
0.28 -
0.28
0.74 -
0.30 -
0.90 -
0.25 -
h
r
br
b
br
h
r
br
, 1
Case BI
0.
0.
0.
0.
0.
0.
05 -
0.05
05 -
0.05
14 -
06 -
17 -
25 -
h
r
br
b
br
h
r
br
' "l
Case 82 '
0.00
0.00
0.00
0.00
0.01 - b
0.00
0.01 - b
r
0.25 - br
Upper bound , 1 H
Case A Case BiC
18 +
18
12 +
12
6 +
4.5 +
1.5 +
18 +
b 95 + b
r r
95
b 63 + b
63
b 32 + b
r r
b 24 + b
r r
b 7.9 + b
r r
b 18 + b
h e
u
A
Case 82
18 + b
r
18
12 + b
12
6 + b
r
4.5 + b
1.5 + b
r
18 + b
* - T c u
No uncertainty in acceptable concentration.
Acceptable concentration uncertain; TLV conversion factor equal to geometric mean value.
d
Acceptable concentration uncertain; TLV conversion factor equal to 1/300.
Q
Since source severity is nonnegative, the lower bound cannot be less than zero.
From Table 5, it can be seen that in Case B2 the uncertainty
associated with acceptable concentration has no effect on the
upper bound since the values for Case B2 are identical to those
for Case A (no uncertainty in acceptable concentration). On the
41
-------
other hand, the lower bound is dominated by the uncertainty in
acceptable concentration in Case B2. In fact, the lower bound is
essentially zero for Case Bฃ in all instances except Item 8,
which corresponds to no uncertainty in acceptable concentration.
Table 5 also shows how each of the individual uncertainties con-
tributes to the departure from the ideal situation in which both
the upper and lower bounds are equal to unity. (In this situa-
tion, Inequality 32 becomes
< ST <
.e.
= SQ.) The
ideal situation is approached only in Item 7, Case A (no uncer-
tainty in dispersion modeling or acceptable concentration) ^and
then only when the random uncertainty in emission factor, br,
is small.
OPERATING CHARACTERISTICS OF THE TEST
In order to fully describe the risk of a Type II error in a test
of hypothesis such as that described in this section, it is neces-
sary to construct an operating characteristic curve for the test.
The probability of a Type II error can be expressed as a function
either of the "true" value (ST in this case) or the estimated
value (S^;) of the parameter in question.
In the Neyman-Pearson theory of statistical inference, the true
value of the unknown parameter, ST, is regarded as a fixed entity,
and the measurements, SG, are visualized as distributed about
this value. This situation is illustrated in Figure 4.
to
5
o
CO
i
0.
TRUE VALUE
-MEASURED VALUES -
Figure 4.
Schematic representation of the
fiducial statistical approach.
Deterministic Decision Approach
Operating characteristic (OC) curves are given in Figure 5 for
Case A (no uncertainty in "acceptable" concentration) with
br = 0.1. a The curves give the probability that ST < 1 . 0 as a
function of the true severity, ST. For ST < Scd - e^) , the
curves give the probability of not making a Type I error; for
ST > Sc(l + eu) , they give the probability of making a Type II
error. A family of OC curves is obtained with the systematic
error in source severity as a parameter. In Figure 5, three
Construction of these curves is discussed in Appendix F.
42
-------
members of this family of curves are shown corresponding to no
systematic error, systematic error equal to esu, and systematic
error equal to esฃ. The latter two curves represent the extreme
cases; i.e., all OC curves for this problem lie between these two
curves.
Note that in Figure 5 the left-most curve (systematic error
equal to esu) represents the worst case with respect to Type I
errors, while the right-most curve (systematic error equal to es&)
represents the worst case with respect to Type II errors. Hence,
a convenient means of representing the family of OC curves is a
combination of the upper half of the left-most curve and the
lower half of the right-most curve, as shown in Figure 6. (The
other halves of the curves are of lesser interest since they
correspond to error probabilities of 50% or greater.) The curve
in Figure 6 is termed an "effective" OC curve since it corre-
sponds to treating the Interval 32 as an exact 95% confidence
interval.
1.0
0.9
0.8
5-0.7
en
-rr'-c 0.6
O
-------
consequence whether the worst case probability of a Type II error
is, for example, 0.5 or 0.9. The important point is that there
is a large risk of a Type II error in this region, and this is
clearly indicated by the "effective" curve.) Thus, the graph
shows that the worst case probability of a Type II error cannot
be reduced to less than 0.5 in this region by reducing the random
uncertainty alone. This result can only be achieved by reducing
the systematic component of uncertainty.
1.0
0.9
_ 0.8
V o
B
a 0.4
0.3
0.2 -
0.1 -
0
0.01
0.1
1.0
100
Figure 6. "Effective" operating characteristic
curve for Case A with ฃ>r = 0.10.
In Figure 7, "effective" OC curves are given for Case A (no
uncertainty in "acceptable" concentration) as a function of br,
the random uncertainty in emission factor.3 (The curves for
Cases BI and B2 are similar except that the horizontal portions
of the curves extend to infinity for all values of br. ) The
curves show that when br > 0.23, the worst case probability of a
Type II error is greater than or equal to 0.5 for all values of
the estimated severity, Sc (cf. Equation 48, inequality 59, and
the related discussions).
The left-hand branch of the curve is insensitive to the value of
br due to the dominant effect of the systematic component of
uncertainty.
44
-------
V__
n
1.U
0.8
|ฐ0.7
|fl.6
ฐ 0.5
|aซ
ฃ 0.3
0.2
0.1
0
0
.
-
NO RANDOM ERROR. ar = t>r = Cr = 0
IN ALL OTHER CASES. ฃ. = ^=0.05
r"
i <
i b iO.23
\V b -0-0
\\^^
\^^____
01 0.1 1.0 10 10
ST _
Figure 7. "Effective operating characteristic
curves for Case A as a function of
random uncertainty in emission factor.
Stochastic Decision Approach
The operating characteristic curve for the stochastic decision
approach is given in Figure 7. OC curves for r = 1 and r = 183
are shown; corresponding curves for other values of r are parallel
to those shown in Figure 8. Since the stochastic decision test
approach treats all uncertainties as random errors, there is no
plateau in the curves as found in the previous OC curves.
Figure 8. Operating characteristic curve
for the stochastic decision test
approach .
45
-------
SUMMARY
The results of the above analysis can be summarized as follows:
The stochastic decision test approach should be used
in preference to the deterministic decision test
approach.
The critical value, Sฃ, for the calculated source
severity should be set at 0.05 to 0.08.
In general, there is no justification for taking a
large number of samples to obtain a very precise esti-
mate for the emission factor, since precision in source
severity is limited by the precision in available
measurement technologies.
The last conclusion should not be construed as implying that
sampling and analytical procedures are unimportant in the Source
Assessment Program. In the first place, bias in sampling and
analysis does have a significant effect on the total uncertainty
as shown in Table 4. Secondly, in many cases the effect of
random uncertainty in emission factor will be considerably
greater than indicated in Table 4. For sources having negligible
plume rise, for example, the total uncertainty will be reduced by
69% to 74% over that used in Table 4, and the contribution from
the random uncertainty in emission factor will be correspondingly
greater. Thirdly, reliable emissions data are of value in their
own right, above and beyond their utility in estimating source
severity.
46
-------
SECTION 6
SOURCE ASSESSMENT: EMISSIONS BURDEN
INTRODUCTION
For criteria pollutants, the source assessment screening process
makes use of a second decision index, the national emissions
burden, in addition to source severity. For a given source type
and given criteria pollutant, the national emissions burden, NB/
is defined as follows:
M
NB = gB (65)
n
where M = annual mass emissions of given criteria pollutant
" from the given source type, kg
M = annual mass emissions of given pollutant from all
n
stationary sources nationwide, kg
Equation 65 defines the "true" value of N . In practice, it is
calculated as follows:
CAP E.F )
R; (66)
where CAP^, = total production capacity of source type, kg/yr
EF = representative emission factor for source type,
g/kg
M^ = estimate of Mn obtained from 1972 National
Emissions Report (11) ,a kg
The National Emissions Report contains data on emissions from
both stationary and mobile sources. Only the data for station-
ary sources are used in estimating Mn.
(11) 1972 National Emissions Report; National Emissions Data
System (NEDS) of the Aerometric and Emissions Reporting
System (AEROS). EPA-450/2-74-012, U.S. Environmental
Protection Agency, Research Triangle Park, North Carolina,
June 1974. 422 pp.
47
-------
Note that the emission factor in Equation 53 is not the same as
that used to calculate source severity. The latter emission
factor is for a single emission point within a representative
plant, while the emission factor in Equation 53 is the total for
all emission points within a representative plant.
As noted in Section 5, the national emissions burden is presently
used for screening a source type as follows: the source type is
considered to be a potential candidate for control technology
development only if ND > 0.001.
D ~~
The objective of this section is to perform an analysis for NB
similar to that given in the previous section for source severity,
UNCERTAINTY IN ND
B
The uncertainty in N_ arises from the following components:
D
There is a random uncertainty in production capacity
as discussed in the previous section.
There is uncertainty in the emission factor which has
both random and systematic components.
There is uncertainty in nationwide annual emissions,
due to the use of the NEDS data base, which has both
random and systematic components.
For convenience, let A, B, and D denote the "measured" values of
the variables as follows:
A = CAP
T
B = EF
R
D = M
n
Let ar, br, and dr be the relative random uncertainties associ-
ated with A, B, and D. Using the error propagation formulas from
Table B-l, the random uncertainty in N_, is found to be:
_,
a
er =
V - V
(67)
Considering next the systematic uncertainty, let
-b -d
48
-------
be systematic error bounds on EF^ and Mn, and let bsu, bsฃ, etc.,
denote the corresponding relative error bounds. Using the formu-
las from Table B-8 yields the following systematic error bounds
on N :
b + d
= - - (68)
su
b + d
sฃ ^ su (69)
1 + d
su
The total uncertainty in N is the sum of the random and system-
atic components:
/N //\ /N /\ ,_ on Q V
... - ปK2 + br2 + dr2 + ฐ" , (70)
/\ /\
b_n + d.
su
Thus, we have the following uncertainty interval for the true
value of N_.:
c
-------
1 December 1976). There are approximately 100 categories which
contribute to the national totals. Hence, the variability in the
totals is estimated (assuming all categories contribute equally
to the total) to be:
Vioo(ฐ.Q5)2 = fo.oos
100
The 95% confidence limits are approximately twice this value, or
ฑ0.01.
The systematic error bounds on Mn were obtained by assuming that
the NEDS totals are systematically low due to incomplete inven-
tories. Assuming the emissions inventories are at least 70%
complete gives ds& = 0 and dsu = 0.3.
With the above values, Equations 70 and 71 become:
^\ *\i /\
eu = br + 0.5
/v f\j /v
e, = br + 0.3
The uncertainty interval for N_, is then:
c
(NB)C (0.7 - br) < (NB)T < (NB)C (br + 1.5) (74)
STATISTICAL TEST OF HYPOTHESIS
Inequality 74 can be used as in the previous section to formally
set up a statistical test of hypothesis. The null hypothesis in
this case is:
H0: (NB)T > 0.001
This hypothesis will be accepted if the upper limit in Inequality
61 exceeds 0.001. Therefore, the critical value for the test,
(N*)c, is given by:
= o.ooi {75)
br + 1.5
Inverting Inequality 74 yields:
(NB)T (N )
~ 5-2- < (NB)C < ^-V (76)
br+1.5 " ^ 0.7-br
50
-------
The upper end point of this interval becomes infinite for br
= 0.7, which places an upper bound on thง acceptable uncertainty
in emission factor. With this value of br, Equation 62 becomes
(N*)c 3* 0.0005.
As br ranges between zero and one, (Ng)c varies from 0.00067 to
0.00042. Sin^e the level of the test, a, enters the calculation
only through br, the critical test value is again nearly indepen-
dent of the test level.3
SUMMARY
The above analysis leads to the following guidelines for use of
the second source assessment decision index:
1. The critical value of national emissions burden should be
set at 0.0005 = 0.05%.
2. The random uncertainty in emission factor should be less
than ฑ70%.
The emission factor referred to here is the total emission factor
for the representative plant. However, the above restriction can
be met if the uncertainty in the emission factor for each emis-
sion point in the representative plant is required to be less
than ฑ70%.
The analysis of the first source assessment decision index result-
ed in an upper bound on the random uncertainty in emission factor
for criteria pollutants of ฑ23%. Hence, the present value of
ฑ70% theoretically places no additional restriction on sampling
and analytical procedures. In practice, however, just the oppo-
site may be true. The value ฑ23% may be impractical and, there-
fore, may be ignored entirely, while the value ฑ70% may be
reasonable, and, therefore, represent a real restriction.
ar has been neglected compared to br.
Criteria pollutants are covered by Case A.
51
-------
SECTION 7
OFFSET CALCULATIONS: PLANT EXPANSION PROBLEM
INTRODUCTION
As part of EPA's overall program to achieve and maintain National
Ambient Air Quality Standards, the individual states are required
(through State Implementation Plans) to determine the amount of
emissions reduction necessary to offset the probable impact of
increased population, industrial activity, motor vehicle traffic,
and other growth factors. Of particular concern in this section
is the policy regarding increased emissions due to increased in-
dustrial activity through expansions of existing plants.
One way to offset emissions from a plant expansion is to require
the company to make a corresponding reduction in emissions from
its existing unit. Such a policy entails the associated problem
of determining a posteriori whether the company is complying with
the policy. Such a determination is not entirely straightforward
because the emission rates are not known exactly. This uncer-
tainty should be taken into account in the decision-analysis
process .
The objective of this section is to analyze the uncertainty
associated with the above problem and construct an approximate
statistical test as the basis for making a decision regarding
compliance with the offset policy.
WORKING EQUATIONS
The difference in emission rates of a given pollutant before and
after plant expansion can be expressed as follows:
Q2 = Qi - AQi + QN (77 )
Q2 - Ql = QN -
where Q2 = emission rate after expansion, g/s
QI = original emission rate, g/s
QN = emission rate from new unit, g/s
AQj = reduction in emission rate from original unit, g/s
5-2
-------
Both QN and AQi are composed of terms corresponding to the indi-
vidual emission points within the plant. Thus,
Q = ฃQNi (sum over all emission (73)
i points in new unit)
and
AQj = I(AQi). (sum over all emission points in
j original unit which have different
emissions after expansion)
= SQjj - ZQ1(j (79)
In these equations Q^ and Qj'j denote emission rates from the
jth. emission point in the original unit before and after expan-
sion, respectively.
Equations 77, 78, and 79 are the basic relationships governing
the problem. Note that the difference (Q2 - Qj) depends only on
emissions from the new unit and those emissions from the original
unit which are changed in the expansion.
UNCERTAINTY IN (Q2 - Ql)
Uncertainty in (Q2 - QI) is due to uncertainties in the individ-
ual emission rates QNI/ QI-J' an<^ QI'-J* Once the random and sys-
tematic uncertainties in these values have been determined, the
uncertainty in (Q2 - QI) can be obtained using error propagation
formulas for addition and subtraction as outlined below.
Consider first the random uncertainty. Let BN and B^ denote the
measured values of QN and AQlf and let brn and brA be the corre-
sponding random uncertainties. Then the random uncertainty in
(Q2 - QI) is, according to the subtraction formula in Table B-l,
Appendix B:
e =
where
b2
D rN "" rNi
and
rlj ' - ri'j - \" rij
53
-------
Therefore, we have:
er ' b + + b2' (80)
where brNi/ ^riv anc^ ^ri'j are t'ie ran<3om uncertainties asso-
were rNi/ riv anc ri'j e ie ran OU
V "XT " A /
IN , a , N A
JIN ฃA Sfc
be systematic error bounds for Q , AQlf and (Q2 - QI). Then,
using the addition and subtraction formulas from Table B-8, we
obtain:
esu = buN + bฃA = Z'D--"- + ^ib"'-! + b.--i-;i (81)
= bฃN + buA = + Eb + b' (82)
where bujj-^, b ^, bu i ^ , etc., are systematic bounds correspond-
ing to QNi, Q* and
The total uncertainty in (Qa - QI) is the sum of the random and
systematic components. Combining Equations 80, 81, and 82, we
obtain:
e = M- T ,
u L rNi \ rl] rl
- H/2 + 2
D/ i
= Zt>2 XT- + Z^b2 , + fa2 , . M /2 + ฃ*
I . rNi . y rl] rl'j/l .
. - . bnil .\ (84)
ฃNi .1 ul] ฃ1 '
The uncertainty interval for (Q2 - QI) is given by:
BN - BA - 6ฃ - (Q2 - Ql} ~ BN - BA + 6U (85)
Alternatively, Inequality 72 may be written as:
(Q2 - Qi)c - eฃ < (Q2 - Q!)T < (Q2 - Qi)c + eu (86)
where (Q2 - Qi)ip denotes the true value of the emission rate dif-
ference and (Q2 - Qi)c = BN - BA is the calculated difference.
54
-------
TEST OF HYPOTHESIS
Inequality 86 forms the basis for a formal test of hypothesis.
The hypothesis to be tested is:
H0: (Qa ~ Ql)T > ฐ' i.e., Q2 > Q!
This is tested against the alternative hypothesis:
HA: (Q2 - Qi)T < 0; i.e., Q2 < Qj
The following simple procedure is proposed for carrying out the
test:
1. Specify level of test, a (a = probability of Type I error)
2. Determine (Q2 - QI),, = B - B
\^f IN ฃA
3. Calculate e according to Equation 83. [Note: the b 's
should be (1 - 2a) x 100% confidence limits, where a is the
specified level of the test.]
4. If (Q2 - Q!)c + eu is ^ 0, reject H0; otherwise, accept H0.
(Note: If H0 is accepted, the plant is considered out of
compliance.)
Notice that compliance requires that the emissions from the orig-
inal plant be reduced by the amount of emissions from the new
capacity plus an additional amount, eu, which compensates for the
uncertainty in the data.
The two kinds of incorrect decisions that can be made using the
above procedure are:
Type I Error; Reject H0 when it is, in fact, true; i.e., decide
that the plant is in compliance when it is, in
fact, in violation.
Type II Error; Accept H0 when it is, in fact, false; i.e.,
decide the plant is in violation when it is, in
fact, in compliance.
The probabilities of making the two types of errors in decision
making are as follows:
1. Probability of Type I error is 1 a.
2. Probability of Type II error is a function of a, (Q2 - QI)Q,
eu, and e^. It must be determined individually for each
specific problem.
55
-------
Theoretically, the risk of a Type II error can be controlled by
specifying the sample size to be used in obtaining emissions data,
However, the effect of a Type II error is to require the plant to
make an additional reduction in emissions. In practice, there-
fore, specification of the sample size involves an economic trade-
off between the cost of reducing uncertainty in the data and the
potential cost of reducing emissions.
NUMERICAL EXAMPLE
In order to illustrate the test procedure, numerical values have
been assigned to the variables involved in the test. These val-
ues are for illustration only, and do not correspond to actual
field data.
1. Take a = 0.05 [Note:
confidence limits.]
the br's should be (1 - 2a) x 100 = 90%
Let
brN = 6 g/s
brA = 8 g/s
e =
rN
= 10
3. Let
buN = 4
= 2
= 3
uA
bฃA = 6 g/s
6su = buN + bฃA = 10 9/S
buA = 5
Then
e = e + e =20 g/s
u r su ^
= e
esฃ = 15 g/S
Let
BN = 50 g/s
BA = 65 g/s
(Q2 - QI)C =
= -15 g/s
56
-------
6. Uncertainty interval for (Q2 - QI)T is then
+e
+20
Since (Q2 - QI)Q + eu = +5 9/s' the nul1 hypothesis is
accepted; i.e., it is concluded that (Q2 - QI)T > 0 and the
plant is out of compliance. Notice that if the uncertainty
in the data were not taken into account, the opposite con-
clusion would be reached since the nominal value of (Q2 - QI)
is -15 g/s .
An "effective" operating characteristic curve for the test is
illustrated in Figure 9. This curve is obtained by treating
the uncertainty interval (86) as an exact 90% confidence in-
terval. This approximate treatment indicates a relatively
large risk of a Type II error in this particular example
(PType II = ฐ'79)-
1.0
0.8
V 0.6
j_>
o*
| 0.4
O_
0.2
0-4
~
-15 .
-5 0 5
'VVr
10
15
20
Figure 9. "Effective" operating characteristic
curve for example problem.
SUMMARY
The data required to quantify the uncertainty in the emission
rate difference before and after plant expansion consist of ran-
dom and systematic error bounds on the following quantities:
Emission rates from each emission point in the new unit.
57
-------
Emission rates before and after expansion from each emis-
sion point in the original unit which has different
emissions after expansion.
The above data can be used to perform a simple, approximate sta-
tistical test to determine whether the plant is in compliance
with offset policy. The essence of this procedure is the follow-
ing: In order to ensure that air quality is not degraded, emis-
sions from the original plant must be reduced by an amount
sufficient to compensate not only for the emissions from the
added capacity, but also for the uncertainty in emissions data.
58
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SECTION 8
COMPARISON OF ALTERNATIVE CONTROLS:
REASONABLY AVAILABLE CONTROL TECHNOLOGY
INTRODUCTION
The problem to be addressed in this section involves comparison
of the control efficiencies of two alternative control technolo-
gies. It is assumed that a plant currently employs a given con-
trol technique with a nominal (measured) control efficiency which
is less than that of reasonably available control technology
(RACT) for the given application. In order that the plant be
required to adopt RACT, it should be established that RACT is
indeed superior to the installed control technology. Since
neither of the two efficiencies can be known with certainty, this
determination should not be based on the nominal efficiencies
alone.
The purpose of this section is to formulate a statistical test to
serve as the basis for deciding whether RACT is superior to a
given installed control technique. Only the simplest case, a
single pollutant emitted from a single source, is considered.
GOVERNING EQUATIONS
The control efficiency, ej, of the installed device is given by
Qi - Q.2 Q2
(87)
where Qj and Q2 are the uncontrolled and controlled emission
rates, respectively. Letting e2 denote the control efficiency
of RACT, we have
Q2
e2 - ej = e2 + QY - 1 (88)
For the purpose of this example, it is assumed that EJ is meas-
ured by measuring QI and Q2 . The value of e2 is assumed to have
been determined previously on a similar but different installa-
tion. Therefore, the measured value of e2, together with its
59
-------
associated random and systematic uncertainties, is assumed to be
given.
We now wish to develop equations for the total uncertainty in the
difference (ฃ2 - EI). Let BI and B2 be the measured values of
Ql and Q2 with associated random uncertainties brj and br . Let
(e2)c ke the given nominal value of e2 with associated random
uncertainty 3r- Then the nominal value (e2 - EI)C of tne control
efficiency difference is given b
(E2 - EI)C = (e2)c + 3 -- 1
(89)
The associated random uncertainty, e , is obtained using error
propagation formulas for addition and division:
where
V
and
(90)
Considering next the systematic uncertainty, let
+b
+b
B
' B2
+e
be systematic error bounds for Qj, Q2 , e2 an^ (E2 ~ EI)ซ Using
addition and division formulas for propagation of systematic
errors yields:
esu = 3u + B -
where
"/s"
again denotes relative uncertainty.
The total uncertainty in (e2 - ei) is the sum of the random and
systematic components. From Equations 79, 80, and 81 we obtain
60
-------
,
ri
r2
(93)
V 2 /B
-r + hr
ri
+ b
(94)
The uncertainty interval for (e2 ~
(e2 - ฃi) - e < (e2 - E
is given by:
< (e2 - ฃi)
(95)
where (e2 - EI)T represents the true value of the control effi-
ciency difference, (e2 - ฃi)c is given by Equation 89, and e and
e are given by Equations 93 and 94.
J6
TEST OF HYPOTHESIS
Inequality 95 forms the basis for a statistical test of hypothe-
sis. The idea behind the test is the following: if the lower
uncertainty limit (e2 - ฃi)c - e exceeds zero, then we can con-
clude that (e2 - EI)T is greater than zero, i.e., that RACT has
a higher efficiency than the installed device. If the lower
uncertainty limit is less than or equal to zero, then we cannot
conclude that RACT is better.
The null hypothesis for this case is that RACT is no better than
the installed control device, i.e.,
H0: (E2 - EI)T < 0
The alternative hypothesis is that RACT is better; i.e.,
HA: (ฃ2 - EI)T > 0
The test is formally carried out as follows:
1. Specify level of test, a (a = probability of Type I error).
2. Calculate (e2 - ฃi)c from Equation 89.
3. Calculate e from Equation 94. [Note that the random uncer-
tanties should be (1 - 2a) x 100% confidence limits.]
If (e2 - e1)c - e > 0, reject H0 (conclude RACT i;
otherwise accept H0 (conclude RACT is not better).
61
-------
The two types of incorrect decisions that can be made using the
above test are:
Type I Error; Conclude RACT is better when in fact it is not
better.
Type II Error; Conclude RACT is not better when, in fact, it
is better.
NUMERICAL EXAMPLE
In order to illustrate the test procedure, numerical values have
been assigned to the variables involved in the test. These val-
ues are for illustration only, and do not represent actual field
data.
1. Take a = 0.05 [Note: the random uncertainties should be
(1 - 2a) x 100 = 90% confidence limits.]
2. Let
4.
= 0.10; i.e., the nominal efficiency of the
c
installed device is (ฃ1)^, = 0.90.
3. Let the nominal efficiency of RACT be (ฃ2),-, = 0.95. Then
(ฃ2 - El) = 0.05. c
= 0.20
ui
er = o.oi
= b
u2
Then,
V 2
=>3T. +
B,
l
B~
ri
+ b.
T2
-
= 0.048
5. (ฃ2 - EI)
= 0.05 - 0.048 = 0.002 > 0
Therefore, we reject the null hypothesis and conclude that RACT
has a higher control efficiency than the installed device.
62
-------
SUMMARY
Before a plant is required to install RACT in preference to a
previously installed alternative control method, the superiority
of RACT should be demonstrated in light of uncertainties in emis-
sions data. A simple approximate statistical test can be employ-
ed for this purpose. The essence of the test is the following:
in order to demonstrate superiority, the RACT control efficiency
is required to be sufficiently high that the difference in
control efficiencies is statistically significant for the given
level of uncertainty in the data.
The information required to perform the test consists of the ran-
dom and systematic components of uncertainty in the two control
efficiencies to be compared. Usually, this information will be
derived from corresponding uncertainties in controlled and uncon-
trolled emission rates.
63
-------
SECTION 9
CONCLUSIONS AND RECOMMENDATIONS
The principles of decision analysis can be used to form a sound
basis for evaluating the uncertainties in environmental data as
they relate to decision making. These principles have been
applied to two numerical indices (source severity and national
emissions burden) to develop guideposts for use as an aid in
environmental decision making. However, these principles cannot
be used in a rigorous fashion due to the complexity of the prob-
lems to be considered and due to the subjective nature of environ-
mental decision making. Seldom, if ever, will an environmental
decision be made only on the basis of a given numerical index
(e.g., source severity). In general, other experimental data;
social, political, and economic considerations; and human judg-
ment all enter into the decisionmaking process. The index is
only one piece of information which the decision maker may util-
ize in arriving at a decision. The index provides only technical
information; value judgments must be made by the decision maker
(in this case, IERL). This fact does not alter the basic
approach used to analyze the uncertainty associated with the
index. Therefore, in this report, it is convenient to think of
the index alone as determining the outcome of the decision-making
process. In practice, the index may not be used as the sole
basis for decision making; IERL may consider other factors.
CONCLUSIONS
The following conclusions pertain to the Source Assessment
Program:
The stochastic decision approach is more than the
deterministic approach, to uncertainties in emission
factors, hence, it is the preferred approach.
The critical value of the calculated source severity
should be set at 0.05 or 0.08.
The critical value for national emissions burden should
be set at 0.05%.
The following guidelines for planning sampling and
analytical procedures should be observed:
64
-------
Criteria pollutantsThe random uncertainty in emission
factor should be less than ฑ23%. If this restriction
requires an impractically large number of samples, suffi-
cient samples should be collected to maintain the random
uncertainty below ฑ70%.
Noncriteria pollutantsThe random uncertainty in emission
factor should be less than ฑ23%. If this restriction
requires an impractically large number of samples,
the guideline for Cases B! and B2 should be followed.
In general, there will be a large risk of a Type II error
(see Section 6) in the screening procedure no matter how
large the value of the calculated source severity. This
situation results from the dominant effect of systematic
uncertainties associated with sampling and analytical
procedures, dispersion modeling, and health effects
information. This situation can be ameliorated only by
reducing these uncertainties (e.g., by using more
detailed modeling techniques or through a program of
fundamental health effects research) or by devising a
decision index other than source severity which will
circumvent these uncertainties.
Eliminating the uncertainty associated with dispersion
modeling may be possible by appropriately defining the
true (deterministic) source severity, ST. Experimental
data presented in Appendix I indicate that the simple
Gaussian dispersion model correctly predicts the ensemble
median ground level concentration under Class C stability
conditions. Thus, by using this ensemble median value in
the definition of ST, the simple Gaussian equation could
be used with correction factor (y) equal to unity. In
effect, the condition y = 1.0 would be specified by the
definition of ST.
Results of the source severity simulation presented in
Appendix H indicate that the definition suggested in the
preceeding conclusion for the deterministic source sever-
ity would be consistent with the stochastic approach to
source severity. In particular, results indicate that the
deterministic severity so defined would yield a worst
case value in the sense that higher severities would be
expected to occur only infrequently, on the order of
once per year.
Based on experimental data cited in Appendix I, the dis-
tribution of ground level concentration under Class C
stability and fixed source conditions is approximately
log-normally distributed, with 98% of the values falling
within a factor of three of the median value. As noted
above, the data indicate that the median concentration
is correctly predicted by the simple Gaussian dispersion
model.
65
-------
RECOMMENDATIONS
The following recommendations are made for conducting the Source
Assessment Program:
The critical value of source severity for criteria pollut-
ants should be set at 0.05.
For noncriteria pollutants, the most consistent and
logically defensible approach would be to estimate
"acceptable" concentration using the mean TLV conversion
factor for criteria pollutants (i.e., take F = TLV/21.4)
and set the critical value of source severity at 0.01.
The net effect would be to reduce the source severity
(as originally specified for the Source Assessment Pro-
gram) by a factor of about 15, and to reduce the
original lower cut point by a factor of 10 from 0.1 to
0.01.
For criteria pollutants, sampling and analytical strate-
gies should be planned so as to maintain the random
uncertainty in emission factor below ฑ23% if practicable;
otherwise, below ฑ70%.
For noncriteria pollutants, the minimum number (generally
three) of samples required to obtain a valid estimate of
the random uncertainty in emission factor should be col-
lected and analyzed. This procedure is consistent with
the second recommendation above, concerning interpreta-
tion and estimation of "acceptable" concentration.
The following recommendations are made for additional related
work:
The present study has served to underscore the need for
additional information in the following areas:
- Dispersion modelinga simple method is needed
to accurately predict the dispersion of airborne
pollutants.
- A vastly expanded data base is needed relating
pollutant exposure to human health effects (extrap-
olation of animal data to humans).
- Much more information is needed concerning the
possible bias associated with individual sampling
and analytical techniques used to measure pollut-
ant emissions.
It is therefore recommended that additional research be under-
taken in each of the above areas.
66
-------
The present study, including the above recommendations,
should be considered as a first step toward establishing
a coherent procedure for environmental decision making.
It is recommended that a followup study be undertaken to
investigate the applicability of other principles of
decision theory, such as the principle of maximum expect-
ed utility, to the types of problems considered in this
report. The present study is not the ultimate answer
to analysis of the uncertainties in environmental data.
67
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73
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APPENDIX A
ACCURACY, ERROR, AND UNCERTAINTY
In the analysis of errors in experimental data, it is important
to distinguish the following concepts: 1) accuracy of the meas-
urement process, 2) error in the measured value, and 3) uncertain-
ty in our knowledge of the true value which is measured. The
relationship between these concepts is indicated schematically in
Figure A-l.
MEASUREMENT PROCESS MEASURED VALUE KNOWLEDGE OF TRUE VALUE
ACCURACY ^- (TOTAL) ERROR ^- (TOTAL) UNCERTAINTY
PRECISION *- RANDOM ERROR *- RANDOM UNCERTAINTY
BIAS ^- SYSTEMATIC ERROR ^- SYSTEMATIC UNCERTAINTY
Figure A-l. Schematic representative of relationship
between accuracy, error, and uncertainty
and between the components of each term.
The terms accuracy, precision, and bias refer to the measurement
process itself (12). Accuracy refers to the closeness between
true and measured values which is characteristic of a given mea-
surement process. Accuracy (strictly speaking, inaccuracy) is
composed of two parts: precision (strictly speaking, imprecision)
and bias. Precision is a measure of the closeness together, or
lack of scatter, in successive independent measurements when all
controllable variables are held fixed. Scatter in the measure-
ments is due to fluctuations in values of variables which are not
controlled in the measurement process. Bias is the magnitude and
direction of the tendency of the measurement process to measure
something other than what is intended. Failure to maintain
isokinetic conditions during particulate sampling is a typical
example of bias in a measurement process.
The inaccuracy of the measurement process results in errors in
the measured values. The total error in a measurement is the
magnitude and sign of the deviation of the measured value from
the true value.
(12) Eisenhart, C. Expression of the Uncertainties of Final
Results. Science, 160(3833):1201-1204 , 1968.
74
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The total error can be decomposed into a random component and a
systematic component. The random error in the measured value
results from the imprecision of the measurement process, while
the systematic error results from the bias in the measurement
process. The decomposition of total error into random and
systematic components is illustrated in Figure A-2. When the
mean of the population of all possible measured values coincides
with the true value in Figure A-2, the measurement process is
said to be unbiased.
on
OL
O
O
O
U-
o
>-
s
on
TRUE
VALUE
MEAN OF ALL POSSIBLE
MEASURED VALUES
-SYSTEMATIC ERROR-
VALUE OF A SINGLE
MEASUREMENT
RANDOM
" ERROR "
-TOTAL ERROR
-MEASURED VALUES-
Figure A-2.
Decomposition of total error into
random and systematic components.
In practice, the magnitude of the error in the measured value is
not known. If it were known, the measured value could simply be
corrected by the amount of the error to obtain the true value.
The best that can usually be done is to place reasonable bounds
on the possible error. In the case of random errors, the bounds
can be estimated statistically, and the results expressed in the
form of a confidence interval about the measured value. In the
case of systematic errors, bounds must be estimated based on a
knowledge of the measurement process. For example, in field
sampling work, the measurement method can be tested on standard
samples to determine the bias under varying conditions. The
largest value of the bias so obtained may then be used to esti-
mate a bound on the systematic error under field conditions.
Again, these results can be expressed in the form of an interval
about the measured value.
Thus, the estimation of bounds for the error in a measured value
results in an interval about the measured value within which the
true value can reasonably be expected to lie; i.e., an interval
of uncertainty. Uncertainty, then, refers to our lack of know-
ledge of the true value of a quantity due to imperfect data;
75
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i.e., due to error in the measured value. The total uncertainty
can be decomposed into a random component, which is due to the
random error in the measured value, and a systematic component,
which is due to the systematic error in the measured value. The
uncertainty is quantified by estimating bounds on the random and
systematic errors.
When measured values are used in a calculation, the corresponding
uncertainties result in an uncertainty of the calculated quantity
in relation to the true value. The manner in which uncertainties
propagate through a calculation depends upon the functional rela-
tionship involved. This topic is discussed in Appendix B and is
generally referred to as error propagation. Strictly speaking,
however, it should be called propagation of uncertainty. In this
report, then, we are concerned with uncertainty and its effect on
decision making.
76
-------
APPENDIX B
USE AND INTERPRETATION OF ERROR PROPAGATION FORMULAS
Any experimentally determined quantity has associated with it
some degree of error for which bounds must be determined in order
for the measurement to be meaningful. Indeed, a measured value
for which no error bounds can be determined is essentially use-
less. When the measured value is subsequently used in a calcula-
tion, its associated error results in an error in the computed
value. One is then faced with the problem of determining error
bounds on the computed value in terms of errors associated with
input values.
The manner in which errors (strictly speaking, uncertainties9)
propagate in a calculation depends upon the functional relation-
ship between variables and upon the type (random or systematic9)
of error involved. In this appendix, the basic formulas for
propagating the two types of errors are reviewed, and a number
of example problems are presented which illustrate their use and
limitations. Some brief derivations are included to point out
underlying assumptions and to indicate the extension to arbitrary
functional relationships.
As pointed out in Appendix A, the subject of the present section
should more properly be termed "propagation of uncertainty"
rather than "propagation of error." However, in order to adhere
to convention as well as for conciseness of expression, the term
"error" is used in this section to denote both exact error (which
is useful in theoretical discussions) and uncertainty (which is
useed in practice). The intended meaning will usually be clear
from the context. When it is not clear, the meaning will be
indicated parenthetically.
RANDOM ERRORS
Error propagation formulas for random errors are listed in Table
B-l. for the four basic arithmetic operations. These formulas are
special cases of the general formula also yiven in Table B-l.
The formulas give the confidence interval for the appropriate
function of two independent variables in terms of the confidence
intervals for the two independent variables. The extension to
See discussion in Appendix A.
77
-------
cases involving more than two independent variables should be
evident from the form of the general equation listed in the table.
These formulas as well as others are given in various forms in
the literature (see, for example, References 13, 14, and 15). All
of the formulas are based on the following equation for the
variance of a linear function of n statistically independent
random variables (3):
02y = (cJVx! + (c2)202x2 + . . . + (cJ2o2xn (B-l)
where y = CiXi + C?xo + . . . + C x , and the C. are constants.
j i i *. *. nn i
If the X-L are not statistically independent, then the equation
for o2 contains the covariances of the x variables and hence the
equations in Table B-l are not valid. A derivation of the error
propagation formula for the sum of two variables is first given,
and then its extension to nonlinear relationships is indicated.
TABLE B-l. ERROR PROPAGATION FORMULAS FOR RANDOM ERRORS
Operation Error propagation formula
Addition, xi + x2 A + B ฑ \a2 + b2
Subtraction, KI - x2 A - B ฑ \a2 + b2
Multiplication, XiX2 AB ฑ \B2a2 + A2b2
Division, X!/x2 (A/B) ฑ y g^ + fir fa2
General case: f(x1,x2) f(A,B) ฑ^pg;B)] 2 a2 + [8f^B)]2 b2
NOTE. A ฑ a and B ฑ b are confidence intervals for xl and x2 .
The formulas give confidence intervals for the various
mathematical operations performed with Xi and x2 . The
formulas are valid only when A and B are statistically
independent.
(13) Braddick, H. J. J. The Physics of Experimental Method.
John Wiley & Sons, Inc., New York, New York, 1954. 404 pp.
(14) Volk, W. Applied Statistics for Engineers. McGraw-Hill
Book Company, New York, New York, 1958. 354 pp.
(15) Beers, Y. Introduction to the Theory of Error. Second Edi-
tion. Addison-Wesley Publishing Co., Inc., Reading, Massa-
chusetts, 1957.
78
-------
Derivation of Addition Formula
Let y = xj + x2 , where xฑ and x2 are independent variables whose
measurements are subject to random errors. The (exact) errors
are assumed to be normally distributed with zero means and vari-
ances o2Xi an^ 02X . Let A and B denote the estimated means of
xi and X2 determined from n^ measurements of Xj and ng measure-
ments of X2 . Then (i_ - a) x 100% confidence intervals for the
true average values x^ and x2 are given by
a
xi
xi = A ฑ Z . a, = A ฑ Z , , - = A ฑ a (B-2)
1 l-a/2 A l-a/2
a
= B +~ Zi-a/2 aB = B +- Zi-a/2
where Zj_a/2 denotes the (1 - a/2) percentage point of the stand
ard normal distribution, where a2^ = a2x ,/n^ and a2g = o2x /ng
are the variances of the two sample means A and B, and where
b = Z , -j= (B-5)
l-a/2 n
A (i_- a) x 1(K)% confidence interval for y is required. Since
y = xj +_x2, y is normally distributed (3), and the sample esti-
mate of y is C = A + B. Using Equation B-l,
ฃ3
(B_6)
Thus, the (i - a) x 100% confidence interval for y is
= C * Zl-a/2 ฐc = A + B ฑ Z.
= A
+ B ฑ Va2 + b2 (B-7)
When the variances of a2Xl and o2X2 are not known, they must be
estimated by the sample variances
79
-------
, nA . nB
2 A^ / A \ 2 2 ^- \T^ / r> \ 2
k
The confidence intervals for xj and x2 are then
s.
A ฑ t, ,
and
'
1Q + 4-
where tl_GL/f v denotes the (1 - a/2) percentage point of the
Student t distribution with v degrees of freedom. In this case,
the above derivation is no longer valid because although
s2 s2
xi X2
s2 = - + - (B-9)
C nA nB
is an unbiased estimate of o2Q, it is not a_sum-of-square esti-
mate. Therefore, the random variable (C - y) /SQ will not follow
the t distribution. (The special case when a2Xi = a2x2 an<^
nA = nB is an exception. See Example 3 below.) Thus in this
case, the error propagation formulas are approximations which are
accurate for large sample sizes n& and ng, for which the esti-
mates s2x, and s*X2 approach the true variances cr2x and 02X2-
However, it might be expected that the formulas will not be
greatly in error except for sample sizes smaller than about 5,
for which the percentage points of the t distribution deviate
markedly from those of the normal distribution. This point is
examined further in the examples given below.
Nonlinear Relations
When y = f(xi,x2) and f is nonlinear, the above derivation does
not apply because Equation B-l no longer holds. However, when
errors in measurements are small (or more precisely, when stand-
ard deviations aXl and aX2 are small compared with mean values
Xi and x2), the problem can be linearized. Thus, neglecting
second-order terms ,
C E f(A,B) % f(Xi,x2) + (A - Xl) + ^ (B - x2)
3f(x1/x2) _ 3f(x1,x2)
(B -
(B-10)
eo
-------
Since this expression is linear in A and B and since to this
degre_e of approximation the partial derivatives evaluated at
(xi,x2) and (A,B) are equal, Equation B-l yields
2 _ pf(A,B)12 , pf(A,B)-|2 ,
ฐ C ~ L 9*1 J ฐ A + L 9x2 J ฐ B (E
For example, when y = f(x!,x2) = X}X2, then 9f(A,B)/9x1 = B and
9f(A,B)/9x2 = A, so that
The confidence interval for y is then
a_ = AB ฑ Z, , 4/B2a2 + A2o2
C l-a/2 HA B
= AB ฑ N/B2a2 + A2b2 (B-13)
which is the multiplication formula given in Table B-l. In this
manner, error propagation formulas can be derived for any func-
tional relationship. However, the formulas will be accurate for
only small errors.
Non-Normally Distributed Errors
When error distributions deviate significantly from normality,
Chebyshev's theorem (3) can be used to estimate confidence inter-
vals. This theorem states that for a random variable, x, which
has a mean, y, and standard deviation, a, the probability that
|x - y| > ka is less than 1/k2; i.e.,
P(|x - y| > ka) < (B-14)
k2
Thus, a (i - a) x 100% confidence interval is given by
y ฑ ka
where ka = ->
a
Actually, the confidence level will be greater than (i - a) x 100%
since the above probability is strictly less than 1/k2. The
important point is that this confidence interval is independent
of the distribution of the random variable, x.
If Chebyshev's theorem is used to compute all confidence inter-
vals, the derivations of error propagation formulas given above
remain valid if ka is substituted for z . . The formulas will
be only approximately correct if sample variances are used to
81
-------
estimate error variances or if normal statistics are used to com-
pute one confidence interval and Chebyshev's theorem is used to
compute the other.
For a 95% confidence interval,
l - a = 0.95
a = 0.05
k = 4.47
a
This value of ka is more than twice the corresponding value of
z, / =1.96 for a normally distributed variate. For this
reason, Chebyshev's theorem is seldom used to compute confidence
intervals. It is recommended that it be used only when error
distributions are known to be highly non-normal.
Examples; Random Errors
Example 1
The following data on atmospheric emission factors for ammonia
emitted from ammoniation and granulation plants were obtained
from various published sources.
TABLE B-2. AMMONIA EMISSIONS FROM AMMONIUM NITRATE PLANTS
Ammoniator-
granulator
Dryer and
cooler
Number of measurements, n 7 16
Mean emission factor, g/kg 0.503 0.316
s, g/kg 0.564 0.262
s//n 0.213 0.065
to.975/ n - 1 2,447 2.131
95% Confidence interval for mean ฑ0.521 (ฑ104%) ฑ0.139 (ฑ44%)
The emission factor for the entire plant is the sum of the values
for the ammoniator-granulator and the dryer and cooler. We wish
to compute the confidence interval for the emission factor for
the entire plant and to compare the result obtained using the
error propagation formula with that obtained by rigorous statis-
tical methods.
Using the error propagation formula,
A = 0.503, a = 0.521, B = 0.316, b = 0.139
Hence,
82
-------
C = A + B = 0.819
c = \a2 + b2 = V0.271 + 0.019 = 0.539
The 95% confidence interval is thus 0.819 ฑ 0.539, or
0.819 ฑ 66%.
The confidence interval can be approximated using
the method of Welch (16, 17). Actually, an approxi-
mation to Welch's method is used which is valid when
sample sizes are not too small (18). The following
quantities are computed:
VA E
= 0.2132 = 0.0454
B ~ n
B
= 0.0652 = 0.0042
f =
V_, = 0.2228 = "effective" standard deviation of mean
13
(V
V 2 V 2
-X-+ B
"
B
- 2 = "effective" degrees of freedom
f = 8.37 - 2 = 6.37
The confidence interval is computed using the t distribu-
tion with f ' degrees of freedom, f being the nearest
integer to f, or f = 6 in this case. Thus,
= t0. 975, 6
= 2.447(0.2228) = 0.545
(16) Pearson, E. S., and H. 0. Hartley. Biometrika Tables for
Statisticians, Third Edition, Volume 1. Cambridge Univer-
sity Press, New York, New York, 1966. 264 pp.
(17) Welch, B. L. The Generalization of "Student's" Problem When
Several Different Population Variances Are Involved. Bio-
metrika, 34:28-35, 1947.
(18) Natrella, M^ G. Experimental Statistics. National Bureau
of Standards Handbook 91, U.S. Department of Commerce,
Bureau of Standards, Washington, D.C., 1963. 504 pp.
83
-------
The 95% confidence interval is then 0.819 ฑ 0.545 or
0.819 ฑ 67%. The agreement with the previous calcula-
tion is excellent.
Suppose that it is known a priori that the variances
o2Xl and o2x? of the two error distributions are equal,
(Actually, using the F-test, the difference in the two
sample variances is not significant at the 0.05 level,
but is significant at the 0.10 level.) In this case,
the confidence interval can be computed rigorously
using the t distribution. The method is essentially
the same as that employed for comparing two means (21),
The pooled estimate of the unknown variance is com-
puted as follows:
(
nA -
x2
- 2
- 0.1399
The estimated standard deviation of the mean is then
nA + l/nfi = 0.3741J1/7 + 1/16 = 0.1695
The confidence limit is given by
10 . 9 7 5 / nA + n~ ~ 2
B
1/nB = t 0 . 9 7 5 , 2 1 (0 . 1695)
= 2.080(0.1695)
= 0.353
Hence, the confidence interval is 0.819 ฑ 0.353 or
0.819 ฑ 43%. It is seen that the additional information
about the error variances can be used to obtain a
smaller confidence interval.
Example 2
The data are similar to those of Example 1, but the emission
factors are for particulate matter rather than ammonia.
TABLE B-3. PARTICULATE EMISSIONS FROM AMMONIUM NITRATE PLANTS
Aminoniator- Dryer and
granulator cooler
Number of measurement, n
Mean emission factor, g/kg
s, g/kg
s//n
^0. 975' n ~ 1
95% Confidence limit for mean
2
0.175
0.070
0.049
12.706
0.623(356%)
12
0.230
0.173
0.050
2.201
0.110(48%)
84
-------
This example represents an extreme case since only two measure-
ments are available for the ammoniator-granulator.
Using the error propagation formula,
A = 0.175, a = 0.623, B = 0.230, b = 0.110
Hence,
C = A + B = 0.405
c = Va2 + b2 = 0.632
The 95% confidence interval is thus 0.405 ฑ 0.632 or
0,405 ฑ 156%.
The approximation to Welch's method used in Example 1 is
not valid in this case due to the small sample size
involved (nA = 2). On the other hand, the exact version
of the method cannot be used because the percentage
points of the appropriate distribution function have not
been tabulated for degrees of freedom less than 8 (19).
The approximate calculation is included here for
completeness.
Va = = 0.00245
** n
VB =
- 0.00250
f =
'A + VB = 0.0704
(VA + VB J2
V
LnA + 1 + nfi -f 1
Therefore,
- 2 = 9.88 - 2 = 7.88
f = 8
to.975,8\/VA + VB = 2.306(0.0704) = 0.162
85
-------
Hence, the confidence interval is 0.405 ฑ 0.162 or
0.405 ฑ 40%. The agreement here is quite poor, but no
conclusion can be made since the accuracy of the present
calculation is unknown.
It is assumed that a2x = o2X2. (Using the F test, the
difference in the sample variances is not significant
even at the 0.25 significance level.) Then
- ls2 + n - l\s2
}s2 + (n^ - l\
) xj ( B )
- 2
x2
= ฐ-0278
+ l/nB = 0.0162
to.975' nA + nB - 2 sp^l/nA + l/nfi = 2.179(0.0162)
= 0.277
Hence, the confidence interval is 0.405 ฑ 0.277 or
0.405 ฑ 68%. Since the difference in the sample vari-
ances is not significant at the 0.25 level, the present
calculation is probably the most accurate of the three
methods in this case. Thus, it appears that error prop-
agation formulas can yield very conservative error
estimates when small sample sizes are involved.
Example 3
It is desirable to investigate further the behavior of the
formulas for small sample sizes since repeating an experiment
more than once or twice is often impractical. As previously
noted, when y = xj + x2, the random variable
C - y
SC
follows the t distribution with 2(n - 1) degrees of freedom in
the special case when o2x = a2X2 an^ n^ = nB = n. Hence, the
confidence limits for y are given by
* tl-a/2, 2(n-i) SC
On the other hand, the addition formula yields the confidence
limits
* tl-a/2, n-i SC
The percent difference is thus
86
-------
Vl SC -
t2(n-l) sc t2(n-l)
where the confidence level subscripts have been dropped for con-
venience. This factor is tabulated below for the 95% and 99%
(two-sided) confidence levels (i - a/2 = 0.975 and 0.995).
TABLE B-4. PERCENT DIFFERENCES IN SIZES OF APPROXIMATE
AND EXACT CONFIDENCE INTERVALS
n
2
3
4
5
10
t , - t_ /.
n-l 2 0
t2(n-i)
95% Level
191
55
30
25
8
n~ ( too )
99% Level
541
116
58
37
13
It is clear from these values that error propagation formulas can
yield very conservative results for smaller sample sizes. How-
ever, at the 95% level the one most often used in practice the
difference does not exceed 25% for n = 5. A conservative error
of this magnitude in the confidence limits should be acceptable
for most purposes. In effect, the calculated confidence interval
simply corresponds to a slightly higher significance level than
the nominal (95%) level.
When a2X! * a?-X2, the accuracy of the error propagation formulas
can be estimated by means of the Behrens-Fisher confidence inter-
val, dn_, sc (19). Aside from philosophical considerations, the
Behrens-Fisher method may be disputed on the grounds that the
actual confidence level is not exactly equal to the nominal level,
but varies with the ratio ฐx\/ax2 (16). However, the method
should give a good indication of the accuracy of the error formu-
las. The percent error in the addition formula is given by
Vl " dn-i
n-i
where dn_ l is tabulated in Table VI of Reference 19. Since
depends on SA/SB = tan 0, the extreme value of dn-1 for
(19) Fisher, R. A., and F. Yates. Statistical Tables for
Biological, Agricultural, and Medical Research. Oliver and
Boyd, London, 1963. 356 pp.
87
-------
0ฐ < 0 < 90ฐ is used to estimate the maximum error. The values
are~listed below for the 95% and 99% levels.
TABLE B-5. MAXIMUM PERCENT DIFFERENCES IN SIZES OF
APPROXIMATE AND EXACT CONFIDENCE
INTERVALS WHEN ERROR VARIANCES ARE UNEQUAL
n
2
4
6
t , - d
n-i r
d
n-i
95% Level
-29
-1.8
+0.35
11 / -I nn\
\ J.UU )
99% Level
-29
+4.3
+5.8
Significant errors occur for only the smallest sample sizes;
however, the errors need not be conservative.
Example 4
Derive error propagation formulas for the exponential and loga-
rithmic operations. For the exponential operation,
y = f(x) = xn (B-15)
f (x) = nxn-1 (B-16)
C = f (A) = An (B-17)
and
x = A ฑ a (B-18)
where a = z, , a.
l-a/2 A
Following the linearization procedure described above under
"Nonlinear Relations,"
ac2 = [f'(A)]2(aA2) (B-19)
ac = f ' (A)aA (B-20)
ฑ Zl-a/2(ac)
88
-------
Y = A" ฑ Z!-a/2 *** ^A) (B-23)
y = An ฑ nAn~l(a) (B-24)
Symbolically, this relationship may be written as
(A ฑ a)n = An ฑ nhn~l(a) (B-25)
For the logarithmic operation,
y = f (x) = ln(x) (B-26)
f ' (x) = 1/x (B-27)
C = f(A) = In (A) (B-28)
Linearizing as before,
ac2 = [f (A)]2(aA2) (B-29)
ac = f (A)aA = io (B-30)
Therefore,
^ =C +- Zi-a/2(ac) (B
= in (A) ฑ Zi_a/2 1 (aA) (B-32)
y = In (A) ฑ | (B-33)
Symbolically, this result may be written
In (A ฑ a) = ln(AJฑ | (B-34)
The above formulas may also be obtained as special cases of the
general error propagation formula given in Table B-l.
SYSTEMATIC ERRORS
Error propagation formulas for systematic errors are given in
Table B-6 for the four basic arithmetic operations. The first
two formulas are valid in general, while the latter two are
restricted by the conditions a < |A| and b < |B|. In all cases,
the two variables as well as the corresponding errors are assumed
to be functionally independent. The first two formulas together
with the linearized versions of the multiplication and division
89
-------
formulas have been reported in the literature (20). A derivation
of the multiplication formula is given here to demonstrate the
method.
TABLE B-6. ERROR PROPAGATION FORMULAS FOR SYSTEMATIC ERRORS
Operation Lower bound Upper bound
Addition A + B - (a + b) A + B + (a + b)
Subtraction A - B - (a + b) A - B + (a + b)
Multiplication AB + sgn(AB)ab - (a|B| +b|A|) AB+ sgn(AB)ab + (aJB| + b|A|)
... A atsl + b|A| A alel + bJAl
Division '' '' +
B _o . ,._,, i_ i B
B2 + sgn(AB)b|B| B B2 - sgn(AB)b|B|
NOTE.A ฑ a and Bib are error bounds for x\ and x2 The formulas give
upper and lower error bounds for the four basic mathematical operations
performed with xi and Xฃ The formulas are valid only when xj and X2 are
functionally independent variables.
Let y = xiX2 where KI and Xฃ are independent variables; let A
and B represent the measured values of KI and X2; let e^ and eg
signify the (exact) errors associated with A and B; and let a
and b stand for upper bounds on |e,| and |eBl- Then
y = Xlx2 = (A + EA)(B + eB) (B-35)
y = AB + eAB + EBA + EAEB (B-36)
Letting y = AB + c,u, the (exact) error in the product enn is
i f\D f\D
given by
ฃAB = ฃAB + ฃBA + ฃAฃB (B-37)
Since in this case there is no statistical basis for a partial
cancellation of errors, a worst-case analysis is the only
recourse. Hence, the maximum and minimum values of eaR for
-a < e, < a and -b < ฃn < b must be determined.
A * D
Assuming a < |A| and b < |B|, the results are readily found to be
as illustrated in Table A-7.
The four cases can be combined into the single formula
sgn(AB)ab ฑ (a|B| + b|A|), where sgn(AB) denotes the algebraic
sign of the product AB.
(20) Jenson, V. G., and G. V. Jeffreys. Mathematical Methods in
Chemical Engineering. Academic Press, New York, New York,
1963, pp. 356-360.
90
-------
TABLE B-7. MAXIMUM AND MINIMUM ERRORS IN THE PRODUCT AB
I I ฃ,._ Maximum ฃ,._. Minimum
sgn A sgn B AB AB
+ + ab + aB + bA ab - aB - bA
ab - aB - bA ab + aB + bA
+ - -ab - aB + bA -ab + aB - bA
+ -ab + aB - bA -ab - aB + bA
When a and b are small relative to |A| and |B|, the product ab
can be neglected and the formula reduced to
|eAB|max = a|B| + b|A| (B-38)
Dividing by |AB| gives
AB
(B-39,
which states that the relative absolute errors are additive.
This is the form given in Reference 20. It can be derived more
easily by linearizing the problem at the outset. Thus, to terms
of first order,
y = f (Xl, x2) f(Af B) + tr-(xl - A) + x^ * B>
(B-40)
y - f (A/ B) - 8f(B)(x1 - A) + ^(x2 - B) (B-41)
e = 9f(A, B) 3f(A, B)
EAB 3xi A 9x2 ฃB (B-42)
Taking f(xi, x2) =
ฃAB = BฃA + AฃB
Thus,
(ฃAB)max = lBla + !A'b (B-44)
(ฃAB)min = -'Bla - 'Alb (B-45)
or
= alBl + b'Al (B-46)
91
-------
From Equation B-42, the general formula for computing linearized
systematic error bounds is
f(A, B)
ฑ[
9f(A, B)
1
a +
3f(A, B)
9x2
For operations more complex than those given in Table B-6, error
bounds are best determined by the above linearization procedure
if errors are small or by direct substitution if errors are large.
In the latter case, however, the bounds on y cannot generally be
obtained by substituting the corresponding bounds on the x's into
the function f. In fact, determining the bounds on y rigor-
ously requires solving two nonlinear programming problems, namely,
Max
and
Min
eA' V '
subject to
N
Y =
B
eB,...,N
(B-47)
N
= f (
-a < e < a, -b <
Examples; Systematic Errors
A + eA, B + eB,...,N +
< b,...,-n < e < n
(B-48)
(B-49)
Example 5
For comparison, Example 1 is reworked assuming that the errors
are systematic rather than random. Taking A = 0.503, B = 0.316,
a = 0.521, and b = 0.139, the bounds on the sum are given by
A + B ฑ (a + b) = 0.819 ฑ 0.660
= 0.819 ฑ 81%
Of course, the systematic error bounds will always be larger than
those for random errors because
Va2 + b2 < a + b
(B-50)
according to the triangle inequality. A similar relationship can
be demonstrated for the (linearized) multiplication and division
formulas.
92
-------
Example 6
The specific reaction rate of a first order chemical reaction is
3.0 x 10~7 ฑ 50% sec"1, and the initial reactant concentration is
0.1 ฑ 20% kg-mole/m3. It is desired to compute error bounds on
the initial reaction rate YO = ^co assuming that the errors are
systematic. For this problem, A = 3 x 10~', a = 1.5 x 10~7,
B = 0.1, and b = 0.02.
Using the multiplication formulas in Table B-6,
Lower bound
(3 x 10~8) + (3 x 10~9) - (1.5 x 10~8 + 6 x 10~9)
= (3.3 x 10~8) - (2.1 x 10~8)
= 1.2 x 10~8
Upper bound
(3.3 x 10~8) + (2.1 x 10~8) = 5.4 x 10~8
Thus,
1.2 x 10~8 < (Y = 3 x 10~8) < 5.4 x 10~8
or
YQ = 3 x 10-8 ft 2-4 x 10Z8 kg_mole/m3/s
Using the linearized version of the formulas yields
0.9 x 10~8 < (Y = 3 x 10~8) < 5.1 x 10~8
or
YQ = 3 x 10~8 ฑ 2.1 x 10~8 kg-mole/m3/s
This represents an error of 25% in the lower bound and 6% in the
upper bound. However, the estimate for the lower bound is con-
servative, while that for the upper bound is not.
Unsymmetrical Systematic Errors
The formulas given in Table B-6 assume that the upper and lower
error bounds are symmetrical about the nominal values of the
variables. This is often an unrealistic assumption for system-
atic errors. For example, the algebraic sign of a systematic
error may be known even though the magnitude can only be esti-
mated. Unsymmetrical error bounds can also arise from physical
93
-------
restrictions on the variables; e.g., the variables may be
restricted to non-negative values. In addition, use of the
multiplication or division formulas in Table B-6 will result in
unsymmetrical bounds. Thus, if a sequence of calculations is to
be performed, formulas for variables having unsymmetrical error
bounds are required.
Error propagation formulas are given in Table B-8 for the case in
which
~ ai 1 xi < A
a2
(B-51)
where
and
where
a2 > 0
B -
< x2 < B + b2
(B-52)
bj, b2 > 0
TABLE B-8.
ERROR PROPAGATION FORMULAS FOR
UNSYMMETRICAL SYSTEMATIC ERRORS
Operation
Addition
Subtraction
Multiplication
A
A
AB
B
Lower bound
+ B - (ai + bj)
- B - (aj + b2)
+ sgn(AB)a]Bi - (aiJB| + BI 1
-------
Examples; Unsymmetrical Systematic Errors
Example 7
In Example 5, A = 0.503, a = 0.521, B = 0.316, and b = 0.139.
Thus, the lower bound on KI was 0.503 - 0.521 = -0.018. However,
xj and X2 represent real emission factors and hence cannot be
negative. Thus, it is more realistic to take HI = 0.503,
a2 = 0.521, and bi = b2 = 0.139. Using the addition formula
from Table B-8 yields
Lower bound
A + B - (a! + b!) = 0.819 - 0.642
= 0.177
Upper bound
A + B + (a2 + b2) = 0.819 + 0.660
= 1.479
In Example 5, a lower bound of 0.159 was obtained for the sum.
However, the present value of 0.177 is still more conservative
than the value of 0.280 obtained in Example 1 using the formulas
for random errors.
Example 8
Obtain an error propagation formula for unsymmetrical systematic
errors for a functional relationship of the form.
y = f(Xl, x2) = - (B-53)
where Xj, x2 > 0
Using the linearization procedure described previously, start
with Equation B-42,
_ 3f(A, B) , . 3f(Af B)
R 'e-^ + s \ BI
(B-55)
ea/ป2 = "^'V^ (ฃA) + "XV:."/ (ej (B-54)
1 / ' v 2 A
B-
where -aj < e < a2 and -bi < e < b2 . It is assumed that the
measured values, A and B, are both positive. Hence, it is
readily seen that
(ฃA/B2)max = 7+ T(bi> (B-56)
95
-------
B2 B3
These equations can be expressed symbolically as follows:
(B-57)
a2
2 A
B3
(bi)
b2
B2
(B-58)
B
al o A
+ (b2)
B2 B3
Since both variables are positive, it is a simple matter to
obtain the exact error bounds (as opposed to the linearized
bounds) by direct substitution. Thus, the upper bound is
obtained by substituting the largest value for the numerator and
the smallest value for the denominator.
Upper bound--
A +
(B - 1
Lower
ai A(l + a2/A) f 1 + a2/A
A i L n
Dl)2 B2
A
B2
A
B2
bound
(1 - bi/B)2 B2f (1 - bi/B)2 "
(1 + a2/A) - (1 - 2 bi/B + bi2/B2)
-\ ฑ
X T
a2/A + 2 bi/B - bi2/B2"
T a
A -
A(l - ai/A) A
(B + b2)2 B2(l + b2/B)2 B2
- ai/A
- 1
A
B2
A
B2
1 +
1 -
- ai/A) -
+ b2/B)2
2 b2/B + b22/B2)
(1 + b2/B)2
2 b2/B + b22/B2)
+ b2/B)2
(B-59)
(B-60)
(B-61)
(B-62)
(B-63)
(B-64)
Thus,
96
-------
"A/B2 max 2
:A/B2 min
a2/A + 2
~ai/A + 2 b2/B + b22/B
2/n2
b2/B)
(B-65)
(B-66)
These results reduce to those obtained using the linearization
procedure when the error in the denominator is small; i.e., when
L/B
ซ 1 and b
2/B
ซ 1.
CONCLUSIONS
The assumptions underlying the error propagation formulas for
random errors can be summarized as follows:
The errors are normally distributed. (This assumption
is required if normal statistics are used to compute
confidence intervals.)
The errors are statistically independent.
When the error variances are estimated from the measured
data, the formulas are valid for large sample sizes. In
practice, a sample size of five should be adequate for
most purposes.
For nonlinear relations, the formulas are valid for
small errors only; i.e., a ซ |A| and b ซ |B|.
For systematic errors, the only assumptions are that the vari-
ables are functionally independent and, in the case of the multi-
plication and division formulas, a < |A| and b < |B|. The
linearized versions of these equations are accurate for small
errors; i.e., a ซ |A| and b ซ |B|.
When both types of errors are important in a calculation, they
should be treated separately. The total error in the calculated
value is then the sum of the random and systematic errors (15).
When the type of error is not known, as is sometimes the case
with data obtained from the literature, the formulas for system-
atic errors should be used since they yield more conservative
error bounds. The systematic error formulas can also be used as
a conservative approximation for propagation of random errors
when the assumptions underlying the formulas for random errors
are invalid. For the addition and subtraction of quantities sub-
ject to random errors, sharper bounds can be obtained by using
the appropriate statistical method when the error variances are
known to be equal.
97
-------
APPENDIX C
DERIVATION OF SOURCE SEVERITY EQUATIONS
SUMMARY OF EQUATIONS
The source severity of pollutants can be calculated using the
mass emission rate, Q, the effective height of the emissions,
H, and the primary ambient air quality standard, PAAQS, or the
threshold limit value, TLV. The severity equations shown in
Table C-l are derived in this appendix.9
TABLE C-l. SOURCE SEVERITY EQUATIONS FOR ELEVATED SOURCES
Pollutant _ Severity equation
Particulate matter S = -
H2
SO S =
x
NO
Hydrocarbons S = 162 Q
H2
CO S =
H
Others S = 5'5 Q
TLV(H2)
DERIVATION OF EQUATION FOR x =
luclX
The most widely accepted formula for predicting time-averaged
ground level concentrations downwind from a point source is the
generalized Gaussian dispersion equation (5, 6).
For convenience, the subscript "C" for "calculated" severity is
omitted in this appendix.
98
-------
x = -FTT ฃXPh ^'"i ISXP|- Tf^l I
-------
2 = 0 (C-7)
V
or
Oy . Oz . _s_
Substituting this result back into Equation C-2 yields the
desired result,
The validity of the assumption ay = az can be ascertained by
referring to Tables C-2 and C-3 121-24) . These tables give the
functional dependence of a and oz on downwind distance. It can
be seen that for stability^Class C, Equation C-5 is approximately
satisfied.
The averaging (i.e., sampling) time associated with Equation C-9
is approximately 3 min.a For averaging times between 3 min and
24 hr, a semiempirical correction factor given by Turner (5) is
applied to Equation c-9.b
The two sets of values in Tables C-2 and C-3 are not entirely
consistent with respect to averaging time. The values for ay in
Table C-2 correspond to an averaging time of 3 min. The values
for oz in Table C-3 correspond to averaging times in excess of
some limiting value, rm, which is approximately proportional to
emission height for emission heights up to 100 m, at which
height and above rm is approximately 10 min (24). Thus, Turner
(5) states that concentrations calculated using these values for
Oy and az correspond to an averaging time between 3 min and
ID min.
The correction factor is applicable to situations in which the
mean wind direction remains constant during the period of
interest. That is, as the averaging time increases, the width
of the plume increases, but the position of the plume centerline
does not change. Since the mean wind direction does not, in
general, remain constant over an extended period of time (e.g.,
24 hr), this correction factor corresponds to a worst case
situation with respect to pollutant concentration.
(21) Eimutis, E. C., and M. G. Kcnicek. Derivations of Continu-
ous Functions for the Lateral and Vertical Atmospheric
Dispersion Coefficients. Atmospheric Environment, 6(11):
859-863, 1972.
(22) Tadmor, J., and Y. Gur. Analytical Expressions for the
(continued)
100
-------
TABLE C-2. VALUES OF a FOR THE COMPUTATION OF a a (21)
Stability class
A
B
C
D
E
F
a
0.3658
0.2751
0.2089
0.1471
0.1046
0.0722
For the equation
where x = downwind distance
b = 0.9031 (from Reference 40)
TABLE C-3.
VALUES OF THE CONSTANTS USED TO
ESTIMATE VERTICAL DISPERSION3 (23)
Usable range, Stability
m class
>1,000 A
B
C
D
E
F
100 to 1,000 A
B
C
D
E
F
<100 A
B
C
D
E
F
0.
0.
0.
1.
6.
18.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
Coefficient
00024
055
113
26
73
05
C2
0015
028
113
222
211
086
192
156
116
079
063
053
2
1
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
.094
.098
.911
.516
.305
.18
d2
.941
.149
.911
.725
.678
.74
.936
.922
.905
.881
.871
.814
-9
2
0
-13
-34
-48
f
9
3
0
-1
-1
-0
0
0
0
0
0
0
.6
.0
.0
.6
2
.27
.3
.0
.7
.3
.35
3
For the equation
a
= cxd +
f
101
-------
- /t \ฐ17
y _ X I O I
max max\t /
or
,0.17
(C-ll)
where t = "short-term" averaging time; i.e., 3 min
t = averaging time of interest
X = maximum ground level concentration corresponding to
averaging time, t
DEVELOPMENT OF SOURCE SEVERITY EQUATIONS
Source severity, S, has been defined as follows:
S = ^|K (C-12)
where x is given by Equation C-ll with u = 4.5 m/s
ITlclX
and where F = PAAQS for criteria pollutants
= TLV/300 for noncriteria pollutants
The averaging time associated with )^max is 24 hr for noncriteria
pollutants; for criteria pollutants, it is the value specified in
the corresponding PAAQS, as shown in Table C-4 (25).
(continued)
Vertical and Lateral Dispersion Coefficients in Atmospheric
Diffusion. Atmospheric Environment, 3:688-689, 1969.
(23) Martin, D. 0., and J. A. Tikvart. A General Atmospheric
Diffusion Model for Estimating the Effects on Air Quality of
One or More Sources. Presented at the 61st Annual Meeting
of the Air Pollution Control Association, St. Paul, Minne-
sota, June 23-27, 1968. 18 pp.
(24) Pasquill, F. Atmospheric Dispersion Parameters in Gaussian
Plume Modelling, Part II, Possible Requirements for Change
in the Turner Workbook Values. EPA-600/4-76-030b (PB 258
036), U.S. Environmental Protection Agency, Research Tri-
angle Park, North Carolina, June 1976. 43 pp.
(25) Code of Federal Regulations, Title 42Public Health,
Chapter IVEnvironmental Protection Agency, Part 410
National Primary and Secondary Ambient Air Quality Stand-
ards, April 28, 1971. 16 pp.
102
-------
TABLE C-4. SUMMARY OF NATIONAL AMBIENT AIR QUALITY STANDARDS (25)
Pollutant
Particulate
SO
X
CO
Nitrogen dioxide
Photochemical oxidants
d
Hydrocarbons (nonmethane)
Averaging time,
hr
Annual
(geometric mean)
24b
Annual
(arithmetic mean)
24
3
8b
1
Annual
(arithmetic mean)
h
1ฐ
3
(6 a.m. to 9 a.m.)
Primary
standards
75
260
80
24
365
10,000
40,000
100
160
160
yg/m3
(0.03)
(0.14)
None.
(9)
(35)
(0.05)
(0.08)
(0.24)
Secondary
standards
(ppm)
60a
150
60
365
c
260
1,300
40,000
100
160
160
(0.02)
(0.14)
(0.1)
(0.5)
None.
(35)
(0.05)
(0.08)
(0.24)
a 0
The secondary annual standard (60 yg/m3) is a guide for assessing implementa-
tion plans to achieve the 24-hr secondary standard.
Not to be exceeded more than once per year.
C 3
The secondary annual standard (260 yg/m3) is a guide for assessing implemen-
tation plans to achieve the annual standard.
d
Recommended guideline for meeting the primary ambient air quality standard
for photochemical oxidants.
CO Severity
The primary standard for CO is reported for a 1-hr averaging time.
Therefore,
t = 60 min
t = 3 min
x
max
xmax\60/
= 2 Q
ireuH2
(-)0'1'
\60/
2 Q
(3.14) (2.72) (4.5)H2
-(0.6)
(C-13)
(C-14)
(C-15)
103
-------
- (3'12 *210'2)Q
Setting F equal to the primary standard for CO, i.e., 0.04 g/m3 ,
yields
c _ Xmax _ (3.12 x 10"2)Q ,_ 1Q>
S = -- - (C-18)
* 0.04 H2
or
H2
Hydrocarbon Severity
For nonmethane hydrocarbons, a 3-hr averaging time is used.
t = 180 min
- _ / 3 \ฐ- 17
xmax ~ Xmaxll86/ (C-20)
= (0.5) (0.052)Q (C-22)
H2
^max 2
For nonmethane hydrocarbons, the concentration of 1.6 x 10~k g/m3
has been established as a guide for achieving oxidant standards.
Therefore,
S = ฐ'ฐ26 Q (C-24)
(1.6 x 10~U)H2
or
s _ (c_25)
H2
104
-------
Particulate Severity
The primary standard for particulate matter is reported for a
24-hr averaging time. Therefore,
- _ (3 \ฐ-17
xmax " xmax\l,440/ (C-26)
= (0.052)Q(0.35) (C-27)
H2
The primary standard for particulate matter is 2.6 x 10 ** g/m3 .
Thus,
S = ฐ-ฐ182 Q (C-29)
(2.6 x 10~1+)H2
or
S - (C-30)
H2
SO Severity
X
The primary standard for SOX is 3.65 x 10"4 g/m3 for an averaging
time of 24 hr. Thus, proceeding as before,
max
max 2
Xmax 0.0182 Q
(c_32)
(3.65 x
or
S (c_33)
H2
N02 Severity
Since NO 2 has a primary standard with a 1-yr averaging time,
Equation C-10 cannot be used to calculate Xmax- Hence, the
105
-------
following equation is used which gives the annual mean ground
level concentration (2):
(C-34)
To obtain the equation for Xmax' Equation C-34 is differentiated
with respect to x, and the derivative is substituted into Equa-
tion C-3. These operations result in the following equation:
(c-35)
From Table C-3, for stability Class C, a has the form
= ax
(C-36)
Substituting Equation C-36 into C-35 and solving for x yields
(C-37)
Using the values a = 0.113 and b = 0.911 from Table C-3 yields
H1 -098
max
0.137
(C-38)
and
K)
max
= ฐ'691 H
Substituting these values for x and o back into Equation C-34
and setting u = 4.5 m/s gives z
= ฐ-ฐ314 Q
H2.098
(C-40)
Equation C-34 is based on a 16-point wind rose; i.e., the 360ฐ
of the compass are divided into 16 sectors of 22.5ฐ each. Equa-
tion C-34 represents a worst case situation in that it assumes
that the wind always blows from the same sector during the
entire year. Thus, for example, if the wind directions were
distributed equally over the 16 sectors during the year, the
corresponding concentration would be one-sixteenth of the value
given by Equation C-34.
106
-------
Since the N02 standard is 1.0 x 10"1* g/m3 , the N02 severity
equation is
S = - ฐ
(1.0 x
or
S (c_42)
H2.1
Noncriteria Emissions
For noncriteria pollutant concentrations, the averaging time is
24 hr. Thus,
/ 3 \Q.l7
Xmax ~ Xmax\l,440/ (C-43)
(0>35) (c_44)
H2
Since F = TLV/300, the equation for source severity is
0.0182 Q
(TLV/300) H2
or
s s
TLV(H2)
AFFECTED POPULATION CALCULATION
The affected population is calculated using Equation C-34 to pre-
dict average concentration, )(, as a function of downwind distance.
The value of az corresponding to stability Class C is used in
Equation C-34; i.e.,
a = 0.113 x0-911 (C-48)
z
107
-------
The downwind distances at which x/F = 1-0 are determined by an
iterative technique;9 i.e., the roots of the following equation
are determined:
a uxF
exp - ฑ MM - 1.0 = 0
(C-49)
where a is given by Equation C-48.
A typical plot of "x/F as a function of downwind distance appears
as follows:
where
and x2 are the roots of Equation C-49
As previously noted, Equation C-34 gives the concentration for a
worst-case situation in which the wind blows continuously from
one sector of width 22.5ฐ (one-sixteenth of a circle). Hence,
the area in that sector over which x/F exceeds 1.0 is given by
(see following sketch) .
A' =
TT (x2
2 _
:)
(C-50)
The population density, Dp, in the vicinity of the representative
source is obtained by averaging the county population densities
for each plant in the source type. The worst case arises when
most of the population is concentrated in the sector downwind
from the source; e.g., in a metropolitan area.b This situation
This calculation is not entirely consistent in that different
averaging times are used for x and F. _Equation c-34 gives an
annual or long-term average value for x- The averaging times
for F vary from 1 hr to 1 yr. This discrepancy is considered
acceptable in view of the very crude overall approach used to
estimate the exposed population.
The worst case arises when the wind blows from one sector most
of the time and most of the population is contained in the down-
wind sector. The present approximation can be considered as a
limiting case in which the wind always blows from the same sector
and the entire population is contained in the downwind sector.
108
-------
is approximated by assuming that the entire population in the
vicinity of the representative source is distributed uniformly
in the downwind sector.
The population, P, in the area over which x/F>l-0 is then given
by
P = Dp(x22 - Xl2) (C-51)
or
P = DA (persons) (C-52)
where
A = TT(x22 - Xi2) (C-53)
is the area contained in an annular region surrounding the source.
The quantity, P, is designated the effected population.
109
-------
APPENDIX D
PLUME RISE
The problem of determining plume rise from an elevated point
source is discussed in this appendix. Equations are recommended
for calculating plume rise in the Source Assessment Program, and
error bounds for the calculated plume rise are estimated. These
error bounds are employed in Section 5 and Appendix I.
RECOMMENDED PROCEDURE
According to Briggs (26), as of 1969 there were over 30 plume
rise formulas in the literature with new ones appearing at a rate
of about two per year. This multitude of formulas can be clas-
sified into two broad categories: 1) those which give the plume
rise as a function of downwind distance and 2) those which yield
a single value for the plume rise. Although the former equations
show better agreement with observed plume rise, they are incom-
patible with the simple Gaussian dispersion equation used in
Source Assessment. Of the equations in the second category, a
modified Holland equation appears (on the basis of extensive data
presented in References 26 and 27) to be about as good as any.
The original Holland equation is (27)
AH =(- 1.5 V D + 0.04 QTT) (D-l)
where AH = plume rise, m
u = wind speed, m/s
V = stack gas exit velocity, m/s
D = stack diameter, m
Qu = heat emission rate, kcal/s
n
(26) Briggs, G. A. Plume Rise. AEC Critical Review Series.
U.S. Atomic Energy Commission, Division of Technical Infor-
mation Extension, Oak Ridge, Tennessee, 1969. 81 pp.
(27) Moses, H., and M. R. Kraimer. Plume Rise DeterminationA
New Technique Without Equations. Journal of the Air Pollu-
tion Control Association, 22(8):621, 1972.
110
-------
The heat emission rate is calculated as follows (30) :
H
where P = atmospheric pressure, dyne/m2
R = gas constant = 8.314 x 10 5 dyne-m/gmole-ฐK
M = molecular weight of effluent, g/gmole
Cp = heat capacity at constant pressure of effluent,
kcal/g-ฐK
T = stack gas temperature, ฐK
s
T = ambient air temperature, ฐK
a,
The values calculated from Equation D-l are generally too low.
For example, values of AHcaic/ ^observed for 22 stacks given in
Reference 26 range from a low of 0.04 to a high of 1.18. Exclud-
ing the lowest and highest values, the range is 0.18 to 0.66.
The mean is 0.45. The stack heights range from 60 m to 180 m.
Extensive data are also summarized in Reference 30. For small
stacks (30 m to 40 m) , the average value of AHcaic/AHo^g is 0.17.
For medium-size stacks (60 m to 120 m) , the average value is 0.43,
For the largest stacks (150 m to 180 m) , the average is 1.0. For
another set of data (Bringfelt data from Sweden) covering 36
stacks ranging from 20 m to 150 m in height, the average value of
AHcale/AHobs is 0.5. The average for all four sets of data in
Reference 27 is 0.33.
Based on the value of 0.33 for AHcaic/AH0kS, the plume rise calcu-
lated from Holland's equation should be multiplied by a factor of
3. This value is in good agreement with a value of 2.92 recom-
mended by Stumke (26) based on independent observations. A cor-
rection factor of 2.2 results from the data given by Briggs (29)
(2.2 = 1/0.45) .
For electric generating stations, an equation is available which
has been optimized for best fit to existing data (28) .
4,000 QH\ฐ.'""' QO-^"
AH = 0.414| - 2- = (8.892) - (D-3)
(28) Thomas, F. W., S. B. Carpenter, and W. C. Colbaugh. Plume
Rise estimates for electric Generating Stations. Journal
of the Air Pollution Control Association, 20(3):170, 1970.
Ill
-------
The following plume rise equations are recommended for source
assessment:
For general sources,
AH =(- 1.5 V D + 0.04 Q__) (D-4)
\U S rl/
AH = (8.892) - (D-3)
For electric utility combustion sources,
Q
U0.694
ESTIMATED ERROR BOUNDS FOR PLUME RISE
Based on data in References 26 and 27, error bounds for plume
rise when using Equation D-4 are estimated to be
AH ,
A ,- calc 0 _
0.5 < 77j - < 3.0
An .
true
A H
0.33 < Atjtrue < 2.0
' AHcalc -
ฐ'33 AHcalc ฑ AHtrue ฑ 2 AHcalc
AHcalc(1-ฐ - ฐ-67) 1 AHtrue ^ AHcalc(1'0
Therefore,
csu , n ฐsa
= 1.0; -r^ = 0.67
AHcalc ' AHcalc
For the error analysis, values of c and c are required where
SX1 S jL
- ฐsu - ฐSU -
and
112
-------
Since
A Csu / AHcalc \
CSU = AiTT-lH + AH . I (ฐ-7)
calc \ s calc/
values of the term in parenthesis are needed. The following data
were obtained from Reference 27:
StaCk AH /H
size Plant true7 s
Small Argonne I 0.17) - n >
Small Argonne II 0.25) g ~
Medium Harwell 0.87J
Medium Gernsheim 0.38> Avg = 0.5
Medium Duisberg 0.31*
Large Lakeview 1.34
Large Widows Creek 1.14 -10
Large Gallatin 1.49 Avg ~ L"L
Large Paradise 0.89
For small stacks,
AHtrue = ฐ'2 Hs and Ali = ฐ-5 - AHcalc = Q'1 Hs
true
Therefore,
AH
calc _ 0.1
H + AH . 1.1 w'w'
s calc
For large stacks,
AHcalc
AH,. = 1.2 H and . =3.0 ->- AH , =3.6H
true s AH calc s
Therefore,
AH
calc 3.6
Hs + AHcalc
Note that csu is a result of errors for small stacks and cSฃ for
large stacks. Thus, error bounds for plume rise calculated from
Equation D-4 are
= 1.0(0.09) = 0.1 (D-8)
113
-------
c . = 0.67(0.78) = 0.5 (D-9)
S A/
Error bounds for plume rise calculated from Equation D-3 are
estimated from data given in Reference 28 as follows:
0.6 AH ,
-------
Thus, error bounds for plume rise calculated from Equation D-3
are
csu
= ฐ'23
115
-------
APPENDIX E
ALTERNATIVE METHODS FOR ESTIMATING "ACCEPTABLE"
CONCENTRATION FOR NONCRITERIA POLLUTANTS
METHOD ORIGINALLY ADOPTED FOR SOURCE ASSESSMENT PROGRAM
For noncriteria pollutants, the "acceptable" concentration, F, is
specified as follows:
The conversion factor, G = 1/300, converts the TLV to an "equiva-
lent" primary ambient air quality standard. The factor 8/24
adjusts the TLV from an 8-hr work day to continuous (24-hr) expo-
sure, and the factor of 1/100 is designed to account for the fact
that the general population constitutes a higher risk group than
healthy workers* This latter factor was chosen after consulta-
tion with EPA health effects experts.
METHOD BASED ON CRITERIA POLLUTANTS
An alternative method for estimating "acceptable" concentration
for noncriteria pollutants is to determine the relationship
between TLV's and primary ambient air quality standards (PAAQS)
based on the criteria pollutants. This relationship is shown in
Table E-l. The (geometric) mean conversion factor (i.e.,
PAAQS2it/TLV) for the four pollutants listed in the table is
0.0467. Thus,
F = 0.0467 TLV = ~-^ (E-2)
The "acceptable" concentration based on data for criteria pollut-
ants is therefore higher than that adopted for the Source Assess-
ment Program by a factor of about 15.
Assuming that criteria pollutants constitute a random sample of
size 4 from the population of all pollutants, and further assum-
ing that the conversion factors are log-normally distributed over
this population, a 95% confidence interval for the geometric mean
conversion factor is found to be (0.0123, 0.179). The correspond-
ing confidence interval for the inverse of the mean conversion
factor is (5.6, 81.3). Thus, the value of 300 obtained by
116
-------
TABLE E-l. RELATIONSHIP BETWEEN TLV'S AND PRIMARY
AMBIENT AIR QUALITY STANDARDS
Criteria pollutant
Particulate matter
Sulfur dioxide
Carbon monoxide
Nitrogen dioxide
PAAQS,
pg/m3
260
365
10,000
100
Averaging
time,
(fcavg) ป
hr
24
24
8
8,760
Estimated
24-hr
PAAQS2k,
pg/m3
260
365
8,670
370
TLV,
yg/m3
10,000
13,000
55,000
9,000
Conversion
factor,
PAAQS ?u /TLV
0.0260
0.0281
0.158
0.0411
Calculated using the method given in Reference 29. The equation used was
PAAQS24 = PAAQS(tavg/24)b
where b = 0.13 for CO and 0.22 for N0ฃ. These exponents are averages of
. the values given in Table 2 of Reference 32.
Method 1 lies well beyond the upper end point, 81.3, of this
interval.
METHOD BASED ON LD50 DATA
The problem of estimating permissible concentrations of pollut-
ants from available health effects data has been studied by Handy
and Schindler (30). Their results yield the following two esti-
mates of "acceptable" concentration in terms of animal LD50
values:9
F = 1.07 x 10-1* (LD50)
F = 8.1 x 10~5
(E-3)
Equation E-3 is based on a statistical correlation between TLV's
and animal LDso values for some 240 different compounds (30).
LD5o is the dosage which results in mortality to 50% of exposed
population.
(29) McGuire, T., and K. C. Knoll. Relationship Between Concen-
trations of Atmospheric Pollutants and Averaging Times.
Atmospheric Environment, 5(5):291, 1971.
(30) Handy, R., and A. Schindler. Estimation of Permissible
Concentrations of Pollutants for Continuous Exposure. EPA-
600/2-76-155 (PB 253 959), U.S. Environmental Protection
Agency, Washington, D.C., June 1976. 136 pp.
117
-------
Equation E-4 is based on an assumed permissible body concentra-
tion of pollutant together with an assumed biological half-life
of 30 days (30). Thus, nearly identical results are obtained by
two entirely different approaches.
In order to compare Equations E-3 and E-4 with Equations E-l and
E-2, the least squares fit to the relationship between TLV and
LD50 data obtained in Reference 30 is utilized:
LD50 = 34.5 (TLV) (E-5)
With this relationship, Equations E-3 and E-4 become
F = 0.0037 (TLV) (E-6)
F = 0.0028 (TLV) (E-7)
For comparison, Equation E-l is rewritten as follows:
F = 0.0033 (TLV) (E-8)
Clearly, Equations E-l, E-3, and E-4 are essentially equivalent
definitions for F. In fact, the coefficient 0.0033 in Equation
E-8 is the mean of the two coefficients in Equations E-6 and E-7.
It should be noted, however, that for a particular compound, the
value of F based on TLV may differ by more than an order of magni-
tude from the value based on LDsg. This variability is due to
the large deviations of individual data points from the least
squares fit to the TLV versus LD50 data (30).
METHOD BASED ON DOSE-RESPONSE RELATIONSHIPS
The three previous methods have dealt with the estimation of an
"acceptable" concentration using currently available health
effects data. The present subsection considers the hypothetical
situation in which a dose-response relationship for the human
population has been established for a particular pollutant. Such
a relationship is illustrated in Figure E-l.
In this figure, probability, P, of a lifetime response to pollu-
tant exposure is given as a function of the concentration, C, to
which exposure occurs. The solid line represents the curve
fitted to experimental data obtained at high concentrations.
The dotted line represents a statistical extrapolation (31) of
(31) Cornfield, J. Carcinogenic Risk Assessment. Science,
198(4318):693-699, 1977.
118
-------
Figure E-l.
Schematic representation of a dose-response
relationship, with a positive no-response
level.
the experimental curve to the no-response level (P = 0). The
dashed lines represent 95% confidence limits for the experimental
curve.
The estimated value, D, of "acceptable" concentration in this
case would be the concentration at which the extrapolated experi-
mental curve intersects the C-axis. The associated uncertain-
ties, d^ and du, would be obtained from the 95% confidence limits
as indicated in Figure E-l.
The positive no-response level exhibited by the dose-response
relationship shown in Figure E-l is not the only type of behavior
that can occur at low concentrations. The extrapolated experi-
mental curve may pass through the origin as illustrated in Figure
E-2, or it may intersect the P-axis as illustrated in Figure E-3.
In order to determine an "acceptable concentration in the latter
two cases, an "acceptable" probability, P*, of a response to
pollutant exposure would have to be arbitrarily chosen. The
estimated value, D, of "acceptable" concentration and the asso-
iated uncertainties, dฃ and du, would then be obtained as shown
in Figures E-2 and E-3.
119
-------
APPENDIX F
CONSTRUCTION OF OPERATING CHARACTERISTIC CURVES
This appendix explains the construction of the operating charac
teristic (OC) curves presented in Section 5. Attention is
focused specifically on construction of the curves presented in
Figure 4. For convenience, this figure is reproduced here as
Figure F-l, to which the plotted points have been added.
i.o
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
NO SYSTEMATIC
ERROR
SYSTEMATIC
ERROR = e,..
0.01
0.1
1.0
100
Figure F-l. Operating characteristic curves
for Case A with b =0.10.
GOVERNING RELATIONSHIPS
s\
From Equation 38 (Section 5) (taking br = 0.10)
e = e + e =17.14
u r su
e + e = 0.89
r sฃ
(F-l)
(F-2)
-V
b 2 + 0.01 = 0.14
e =17
su
(F-3)
(F-4)
120
-------
esฃ = ฐ'75
(F-5)
The uncertainty interval for source severity is therefore
ScC1 - ^) ^ ST < Sc(l + eu) (F-6)
or
0.11
18.14
(F-7)
OC CURVE FOR SYSTEMATIC ERROR EQUAL TO -e
sฃ
The OC curve gives the (fiducial) probability that ST < 1.0 as a
function of Sc The fiducial probability distribution of Sm is
shown schematically in Figure F-2 for the case in which the
systematic error is -esฃ = -S^ esฃ. The distribution is centered
at the point Sc - eSฃ = 3^(1 - esฃ). Since er represents a 95%
confidence limit in this case, the 2.5% and 97.5% points of the
distribution are located at a distance of er = SQ er on either
side of the center point.
Figure F-2. Schematic representation of fiducial distribution
of S for systematic error equal to ~esn-
From Figure F-2, when the central point of the distribution is at
1.0, P(S < 1.0} = 0.5. This situation occurs when
c/'i. o^ in ( "F R ^
sc =
1 - e
0.25
= 4.0
(F-9:
The point SG = 4.0, P = 0.5 is shown plotted in Figure F-l.
121
-------
Similarly, the 0.975 probability point is found from the equation
SC = 0.25 + 0.14 = 2'56 (
The point SG = 2.56, P = 0.975 is also plotted in Figure F-l
The 0.025% point is found in a similar manner.
S =
C 0.25 - 0.14 '
The point S = 9.1, P = 0.025 is shown plotted in Figure F-l.
Additional points on the OC curve are obtained by assuming that
the probability distribution in Figure F-2 is a normal distribu-
tion.9 For illustration, take S- = 7.0. The mean of the distri-
bution of S is then
Sc(l - egฃ) = 7.0(0.25) = 1.75 (F-14)
The 97.5% point is
Sc(l - egฃ + er) = 7.0(0.39) = 2.73 . (F-15)
The 2.5% point is
Sc(l - egฃ - er) = 7.0(0.11) = 0.77 (F-16)
The distribution is shown schematically in Figure F-3. The prob-
ability that ST < 1.0 is equal to the area under the normal curve
to the left of 1.0. In order to compute this area, the standard
deviation of the distribution is needed. Since the 97.5% point
of a normal distribution occurs at 1.96 standard deviations from
the mean,
1.75 + 1.96 a = 2.73 (F-17)
or
a = 0.50 (F-18)
Note that the three points previously determined do not depend
on this assumption. Thus, only the precise shape of the OC
curve between these points is affected by this assumption. The
precise shape of the curve is of little interest in the present
study.
122
-------
t/1
s
CO
o
Qi
Q.
Figure F-3. Schematic representation of the distribution of
S for systematic error equal to -e . and SG = 7.0.
Next, the standard normal deviate, z, corresponding to S~ = 1.0
is calculated.
sm - y
z =
_ 1.0 ~ 1.75 _ , ,n
z -- 1.50
(F-19)
(F-20)
The area, F(z), under the standard normal curve to the left of z
is given in Table F-l (17). Because of the symmetry of the
normal distribution,
F(-z) = 1 - F(z) (F-21)
Thus, entering Table F-l with z = 1.50,
F(1.50) = 0.9332 (F-22)
F(-1.50) = 1 - F(1.50) = 0.0668 (F-23)
The point Sc = 7.0, P = 0.0668 is shown plotted in Figure F-l.
The complete OC curve is constructed by repeating the above calcu-
lation for other values of S .
OC CURVE FOR SYSTEMATIC ERROR EQUAL TO e
su
The fiducial distribution of S-p is shown schematically in
ure F-4 for the case in which the systematic error is e
The 0.5 probability point is obtained from
su
)
SC 1+17
= 0.0556
^
S,,e
(F-24)
(F-25)
123
-------
TABLE F-l. AREA UNDER THE NORMAL CURVE, F(z) (14)
">
,,=/
oo
z
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
0.00
0.5000
0.5398
0.5793
0.6179
0.6554
0.6915
0.7257
0.7580
0.7881
0.8159
0.8413
0.8643
0.8849
0.9032
0.9192
0.9332
0.9452
0.9554
0.9641
0.9713
0.9772
0.9821
0.9861
0.9893
0.9918
0.9938
0.9953
0.9965
0.9974
0.9981
0.9987
0.9990
0.9993
0.9995
0.9997
0.01
0.5040
0.5438
0.5832
0.6217
0.6591
0.6950
0.7291
0.7611
0.7910
0.8186
0.8438
0.8665
0.8869
0.9049
0.9207
0.9345
0.9463
0.9564
0.9649
0.9719
0.9778
0.9826
0.9864
0.9896
0.9920
0.9940
0.9955
0.9966
0.9975
0.9982
0.9987
0.9991
0.9993
0.9995
0.9997
0.02
0.5080
0.5478.
0.5871
0.6255
0.6628
0.6985
0.7324
0.7642
0.7939
0.8212
0.8461
0.8686
0.8888
0.9066
0.9222
0.9356
0.9474
0.9573
0.9656
0.9726
0.9783
0.9830
0.9868
0.9898
0.9922
0.9941
0.9956
0.9967
0.9976
0.9982
0.9987
0.9991
0. 9994
0.9995
0.9997
0.03
0.5120
0.5517
0.5910
0.6293
0.6664
0.7019
0.7357
0.7673
0.7967
0.8238
0.8485
0.8708
0.8907
0.9082
0.9236
0.9370
0.9484
0.9582
0.9664
0.9732
0.9788
0.9834
0.9871
0.9901
0.9925
0.9943
0.9957
0.9968
0.9977
0.9983
0.9988
0.9991
0.9994
0.9996
0.9997
0.04
0.5160
0.5557
0.5948
0.6331
0.6700
0.7054
0.7389
0.7704
0.7995
0.8264
0.8508
0.8729
0.8925
0.9099
0.9251
0.9382
0.9495
0.9591
0.9671
0.9738
0.9793
0.9838
0.9875
0.9904
0.9927
0.9945
0.9959
0.9969
0.9977
0.9984
0.9988
0.9992
0.9994
0.9996
0.9997
0.05
0.5199
0.5596
0.5987
0.6368
0.6736
0.7088
0.7422
0.7734
0.8023
0.8289
0.8531
0.8749
0.8944
0.9115
0.9265
0.9394
0.9505
0.9599
0.9678
0.9744
0.9798
0.9842
0.9878
0.9906
0.9929
0.9946
0.9960
0.9970
0.9978
0.9984
0.9989
0.9992
0.9994
0.9996
0.9997
0.06
0.5239
0.5636
0.6026
0.6406
0.6772
0.7123
0.7454
0.7764
0.8051
0.8315
0.8554
0.8770
0.8962
0.9131
0.9279
0.9406
0.9515
0.9608
0.9686
0.9750
0.9803
0.9846
0.9881
0.9909
0.9931
0.9948
0.9961
0.9971
0.9979
0.9985
0.9989
0.9992
0.9994
0.9996
0.9997
0.07
0.5279
0.5675
0.6064
0.6443
0.6808
0.7157
0.7486
0.7794
0.8078
0.8340
0.8577
0.8790
0.8980
0.9147
0.9292
0.9418
0.9525
0.9616
0.9693
0.9756
0.9808
0.9850
0.9884
0.9911
0.9932
0.9949
0.9962
0.9972
0.9979
0.9985
0.9989
0.9992
0.9995
0.9996
0.9997
0.08
0.5319
0.5714
0.6103
0.6480
0.6844
0.7190
0.7517
0.7823
0.8106
0.8365
0.8599
0.8810
0.8997
0.9162
0.9306
0.9429
0.9535
0.9625
0.9699
0.9761
0.9812
0.9854
0.9887
0.9913
0.9934
0.9951
0.9963
0.9973
0.9980
0.9986
0.9990
0.9993
0.9995
0.9996
0.9997
0.09
0.5359
0.5753
0.6141
0.6517
0.6879
0.7224
0.7549
0.7852
0.8133
0.8389
0.8621
0.8830
0.9015
0.9177
0.9319
0.9441
0.9545
0.9633
0.9706
0.9767
0.9817
0.9857
0.9890
0.9916
0.9936
0.9952
0.9964
0.9974
0.9981
0.9986
0.9990
0.9993
0.9995
0.9997
0.9998
Figure F-4 . Schematic representation of fiducial distribution
of
for systematic error equal to e
124
-------
The 0.975 probability point is given by
SC = 1 + ll 0.14 = ฐ-0551
Similarly, the 0.025 probability point is given by
SC = 1 + 17- 0.14 = ฐ-0560
These three points are shown plotted in Figure F-l . Since these
points are nearly colinear on this plot, the remainder of the OC
curve is approximated by drawing a straight line between the
points. Of course, the OC curve must eventually bend over at the
ends and approach the horizontal lines P = 1.0 and P = 0 asymp-
totically. However, these portions of the curve are of little
interest here.
"EFFECTIVE" OC CURVES
The "effective" operating characteristic curve shown in Figure 5
(Section 5) can be constructed using the appropriate branches of
the curves in Figure F-l. The upper branch of the curve for
systematic error equal to esu and the lower branch of the curve
for systematic error equal to -esฃ are connected by a horizontal
line .
Alternatively, Figure 5 can be constructed by working directly
with an "effective" probability distribution. The "effective"
fiducial distribution of ST is obtained by treating Interval F-6
as an exact 95% confidence interval for ST- The resulting dis-
tribution is shown schematically in Figure F-5. The area under
each branch of the density function is 0.5. The 2.5% point
of the distribution is Sr(l - e_n - er) , and the 97.5% point is
" * \ ^ป Sx, *-
scd + esu + erj '
By following the procedure described in the previous subsections,
it can be verified that the density function shown in Figure F-5
generates the "effective" OC curve of Figure 5.
125
-------
l/l
s
Q
CD
-------
APPENDIX G
RELATIONSHIP OF SAMPLING AND ANALYSIS BIAS TO SYSTEMATIC ERRORS
In this appendix, the relationship of sampling and analysis bias
to systematic errors is developed.
Let 6 = upper bound on positive bias in sampling and analysis
$ = upper bound on negative bias in sampling and analysis
where 3 , B~ > 0
Then, letting Bmeas and B^rue represent the measured and true
values of the emission factor,
B - B.
a+ . meas true
p > - -
B. - B
true meas
-
B
true
(1 -J- Q 1 *ป "R
X i D J ^ D
M / - meas
B
meas
true - 1 + 6+
B \
meas ^
P 5
true
B > B (l - B
meas - true ^
B
^ meas
Btrue S , -
/
^ meas
By definition of b and b
SV1 S X- /
B
meas
Therefore,
B
1-6"
^ Btrue ^ Bmeas
= B (l + b V
meas \ su/
(G-l)
and
127
-------
and
meas
1 + 6
or
1
T=Bmeas(1 ' O (G'2)
= 1 + b (G-3)
1-6
or
1
1 + 6n
Therefore,
= 1 - b (G-4)
b = _ (G_5)
su 1-6
1 + 6
128
-------
APPENDIX H
SOURCE SEVERITY SIMULATION AND PROBABILISTIC SENSITIVITY ANALYSIS
BACKGROUND
In the beginning of the Source Assessment Program, a decision
index, "source severity," was developed. For noncriteria pol-
lutants, i.e., pollutants other than particulate matter, sulfur
dioxide, nitrogen dioxide, hydrocarbons, and carbon monoxide, sta-
tionary source severity was defined as
S = -S^I-Q- (H-l)
TLV(h2)
where S = stationary source severity
Q = air pollutant emission rate, g/s
TLV = threshold limit value, g/m3
h = physical emission height, m (typically, chimney or
stack height)
This simple and usable expression was derived after a number of
conservative, simplifying assumptions had been made. For screen-
ing and evaluative purposes, this form of the decision index was
deemed appropriate at the time.
During the course of the Source Assessment Program, questions
arose regarding the effects of uncertainty in various parameters.
Addressed here in one combined analysis are a more complete formu-
lation of the severity model, physical distributions of param-
eters, and uncertainties in several parameters. In this simula-
tion study are included effects of varying atmospheric stability;
variations in wind speed; uncertainties in the ratio of peak con-
centrations, Xpeafc, to 24-hr maxima, X2i+; uncertainties in the
emission factor; uncertainties in lateral dispersion coefficients,
Oy; uncertainties in vertical dispersion coefficients, az; varia-
tions in temperature; and resulting variations in plume rise.
D. B. Turner's well-documented air dispersion model (5) has been
used. The analytical effort to rigorously calculate the effects
of all uncertainties is formidable if not impossible. A Monte
Carlo simulation was used to investigate effects of uncertainties
in input parameters.
129
-------
DESCRIPTION OF REPRESENTATIVE PLANT
The simulated representative plant was a coal-fired, steam elec-
tric utility assumed to be emitting beryllium from one virtual
point or stack. Simulated plant constants and fuel parameters
are shown in Table H-l. Meteorological data for atmospheric sta-
bility and wind speed correspond to central Alabama and were used
because they were readily available in a form suitable for this
simulation. The meteorological data were the joint frequency of
occurrence of atmospheric stability and wind speed. Frequency-
distributed variables and their parameters are shown in Table H-2.
Since there were only six wind speed ranges and six stability
classes, these parameters were not distributed but were used by
means of table lookup procedures.
Equations that were used in the severity simulation, ordered in
calculational sequence, may be found in Example 1 at the end of
this appendix. The source severity calculation for this particu-
lar source, emitting beryllium, is shown in Example 2. This
severity, subsequently referred to as deterministic severity, was
calculated to be 18.6. If plume rise is included in the calcula-
tion, the resulting deterministic severity is 3.3. The latter
calculation is given in Example 3.
SIMULATION RESULTS
The ground level concentration of a pollutant emitted by an
elevated point source is a function of several variables, one of
which is the downwind distance. For this initial analysis, a
distance was used where the mean of the ground level concentra-
tion distribution is a maximum. This distance could not be calcu-
lated directly and was approximated by a binary search. It was
found to lie between 2 km and 4 km. A distance of 3 km was thus
used for all simulations. Results of the baseline simulation,
where all variables were distributed, and of probabilistic sensi-
tivity analyses are presented in Table H-3.
GENERAL COMMENTS ON RESULTS
In a Monte Carlo simulation, since distributions are used as
inputs and not single deterministic values, the result is also a
distribution of valuesin this case, a distribution of values
for severity. When a probabilistic sensitivity analysis was per-
formed, the variables were fixed one at a time at either a high
level or low level. However, since the remaining variables are
still distributions of values, the resulting severity will again
be a distribution of values.
The sensitivity analyses were performed by fixing each of the
variables that were normally distributed at the +2o level and the
-2a level. Thus, in the second row of Table G-3, the notation
SpO.05 designates that the value of P (the ratio of peak to 24-hr
.-1-30
-------
TABLE H-l. PLANT AND FUEL PARAMETERS USED IN SIMULATION
Parameter Value
Coal consumption rate, g/s 34,532,
Stack height, m 100
Barometric pressure, mb 1,013,
Stack gas temperature, ฐK ^08,
Fraction of beryllium in coal, ppm ^.3,
Stack I.D., m 4>23h
Stack gas velocity, m/s 14.1
TLV of beryllium, g/m3 2 x 10~6
1,092,000 metric tons/yr (nominal value for a
500-MW powerplant). (1 metric ton equals 106
grams; conversion factors and metric system
prefixes are presented at the end of this
report).
Nominal value for a 500-MW powerplant.
c
Reference 4.
TABLE H-2. ASSUMED DISTRIBUTION PARAMETERS
Variable
xpeak _ p
x24-hr
Emission factor
Ambient temperature, ฐ
az
ay
Atmospheric stability
Wind speed
Assumed
distribution
Normal.
Normal .
K Normal.
Normal.
Normal .
None (table lookup)
None (table lookup)
Mean
6.2a
0.85
286.1
1.0
1.0
Standard
deviation
0.93a
0.347
4.9
0.255
0.051
Lower
limit
4.3
0.17
276.5
0.5
0.9
Upper
limit
8.1
1.53
295.7
1.5
1.1
NOTE. Blanks indicate
numerical value is not
applicable.
(32) Montgomery, T. L., and J. H. Coleman. Empirical Relation-
ships Between Time-Averaged S0? Concentrations. Environ-
mental Science and Technology, 9 (10) :953-957, 1975.
131
-------
TABLE H-3. SIMULATION RESULTS OF PROBABILISTIC
SENSITIVITY ANALYSES
Baseline
So
SP0.05
Po.05 = 4.3
SP0.95
PO . 95 = 6-1
SEF0.05
EFo.05 = 0.17
SEF0.95
EFfl.95 = 1-53
Tao.05
Tao.os = 276.5
STa0.95
Ta0.95 = 295.7
ZO . 05
ฐz = ฐz - ฐ'5 ฐz
'Sa
Z0.95
az = az + 0.5 az
Sa
yo.os
ฐy = ฐy - O'1 ฐy
ayo.9s
ฐy = ฐy + O'1 ฐy
Stability Class A
Stability Class F
.Stability D
'Uo.05 = 2.05
Stability D
U0.95 = 6.99
SP0.5
Pfl.5 = 6.2
SEF0.5
EF0<5 = 0.85
Mean
Sx
0.0828
0.122
0.065
0.0163
0.148
0.076
0.086
0.027
0.139
0.097
0.08
0.0242
9.3 x 10~8
0.0153
0.112
0.085
0.812
Standard
deviation
0.
1.
0.
0.
1.
0.
0.
0.
1.
0.
0.-
0.
4.75
0.
0.
0.
0.
788
11
592
153
38
749
832
367
06
842
689
168
x 10~5
154
72
773
776
S
1
2
1
0
3
1
2
0
2
2
1
0
<0
0
1
1
1
0.95
.85
.70
.44
.372
.41
.89
.05
.97
.28
.05
.70
.34
.01
.65
.48
.62
.91
P(S
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
< 0.1)
.585
.561
.597
.688
.541
.586
.568
.806
.451
.576
.586
.513
.0
.764
.147
.581
.565
P(S
0
0
0
1
0
0
0
0
0
0
0
1
1
0
0
0
0
< 1.0)
.841
.771
.887
.00
.703
.853
.827
.954
.675
.8?j
.859
.00
.0
.999
.832
.838
.824
P(S
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
> 1.0)
159
229
113
0
297
147
173
046
325
177
141
0
0
001
168
162
176
132
-------
maximum concentrations) was fixed at the -2a level where P is
equal to 4.3. The third row shows that for this sensitivity
analysis, P was fixed at the +2a level (or 8.1). Since results
of the probabilistic simulation are also distributions, several
parameters are available for making comparisons. One of the obvi-
ous parameters is the mean of the resulting distributions; other
parameters are listed in Table H-3 as column headings and include
the standard deviation, the severity at the 95% cumulative level,
and the probability that the severity is <0.1, <1.0, and >1.
All of the resulting distributions are nonnormal. Looking at the
equations in Example 1, the mean severity values shift in the
appropriate direction. Thus, when the emission factor is low,
the mean severity is low; and when the emission factor is high,
the mean severity is high with respect to the baseline case.
When the ambient temperature is low, the buoyancy factor in the
plume rise equation (Equation H-4) is high; therefore, the plume
rise is higher and thus the severity is less than the baseline
case. The opposite is true when the ambient temperature is high.
The effect of oz is more complex, since az is both in the exponen-
tial term as well as in the denominator of the premultiplying
factor in the Gaussian plume equation (Equation H-7). The effect
of the exponential term overshadows the premultiplier term, as
can be seen from the lower value of az yielding a lower value of
severity when compared to the baseline case. The opposite is
true for higher values of az.
In the case of ay, the effect is straightforward since ay is used
in only the denominator of the premultiplying factor and thus the
low values of ay give higher values of severity compared to the
baseline case. The opposite is true for higher values of ay.
For stability Class A, a lower value of mean severity is obtained
since values of ay and az are large in this stability class. The
exponential term thus becomes unity, and only the premultiplying
term becomes significant. Since ay and az are in the denomina-
tor, a lower value of mean severity is obtained compared to the
baseline case. For stability Class F, values of ay and az are
very small, smaller than the plume rise by a significant factor.
Thus, for example, if H = 300 m and a = 30 m,
Z
exp [-0.5(!ฃ?_)2] = 1.9 x 10-22 {H-2)
and the severity is much lower than the baseline case.
Stability D was fixed to evaluate the effect of fixed wind speed.
Using a low value of wind speed results in a higher plume rise;
this in turn gives a lower value of severity. The wind speed
term in the premultiplier in the Gaussian plume equation is
133
-------
overriden by the plume rise term. The converse is true when the
wind speed is higher.
The last two rows of Table H-3 show results when mean values of
the parameters were used. When P was set equal to its mean value
of 6.2, a mean severity approximately equal to the baseline case
was obtained. The same was true when a mean emission factor of
0.85 was used. No subsequent simulations using mean values of
the parameters were run.
SPECIFIC RESULTS
One indicator of sensitivity is the value of S at the 95% cumula-
tive level. These results are in Column 3 of Table H-3. If the
true emission factor is higher than the value being used, the
highest severity at the 95% cumulative level is the value of 3.4.
Low values of P and high values of az are also sensitive vari-
ables. The uncertainties in az and ay are the main reason for
saying that the dispersion model is no better than a "factor of
2 or 3." However, in this simulation, separation of the individ-
ual effects and the overall result should be a good approximation
of the "true severity" for this source.
The column labeled "P(S > 1.0)" is another indicator of sensitiv-
ity and is a way of looking at the cumulative severity distribu-
tion from a different perspective. In one case, if concern
relates to the magnitude of the severity at the 95% probability
level, Column 3 would be used for sensitivity. If, on the other
hand, interest involved the magnitude of_ the probability that
S > 1.0, Column 6 would be used. In either case, P, a , and the
emission factor are the most sensitive variables.
LARGE SYSTEMATIC ERROR IN EMISSION FACTOR
One final question that was addressed involves the effect of a
large systematic error in the emission factor. First, plant and
fuel parameters are assumed to be known exactly. While this is
not strictly true (e.g., stack gas flow rate and temperature may
vary), uncertainties are assumed to be small. The emission
factor in this analysis is expressed as a fraction of beryllium
that is emitted based on the amount of beryllium in the coal. To
preserve the uncertainty distribution on the emission factor and
to facilitate computation, the fraction of beryllium is set in
coal (2.3 ppm) at 5 times the baseline value (11.5 ppm). Results
of this simulation are shown in Table H-4. ~
CONCLUSIONS DERIVED FROM THE SIMULATION ANALYSIS
If this simulation approach is assumed to yield a "truer" picture
of the actual severity at this representative plant (the true
severity distribution being obtainable only by long-term sampling
for 24-hr beryllium maxima in a comprehensive sampling network),
the following observations can be made:
134
-------
TABLE H-4. SIMULATION OF A LARGE SYSTEMATIC
ERROR IN THE EMISSION FACTOR
SD
0.95
0.1) P(S < 1.0) P(S > 1.0)
0.414 3.94
9.5
0.523
0.62
0.38
The deterministic severity is conservative by a factor
of 225 when compared to the mean simulated severity for
the baseline case (18.6/0.0828).
The largest value of S at the 95% probability is 3.41
for a high emission factor when looking only at emission
factor precision. This represents a conservative factor
of 5.5 compared to the deterministic severity.
Even if the systematic error in emissions is a factor of
5, the deterministic severity is conservative by a
factor of 2, (18.6/9.5).
Thus, one may conclude that the first decision index, the
deterministic source severity, is indeed a worst case estimate
and can be used as one of several tools to aid the EPA decision
maker in the screening process.
Example 1. Equations Used in the Severity Simulation
Q = (CONS)(EF)(%Be)
where Q = emission rate, g/s
CONS = coal consumption, g/s
EF = emission factor for beryllium, g/kg
%Be = fraction of beryllium in coal
'V
Ah =
U
1.5 + (2.68 x 10~3)p
(H-3)
(H-4)
where
Ah
V
s
d
u =
T =
plume rise, m
stack gas exit velocity, m/s
inside stack diameter, m
wind speed, m/s
barometric pressure, mb
stack gas temperature, ฐK
Ordered in calculational sequence,
135
-------
T = air temperature, ฐK
cl
2.8 x 10~3 = constant with units of (mb)"1 m"1
a = Ax0-9031
(H-5)
where a = lateral dispersion coefficient, m
x = downwind distance, m
A = function of stability class as shown below
Stability class
A
B
C
D
E
F
0.3658
0.2751
0.2089
0.1471
0.1046
0.0722
= A!X
(H-6)
where a = horizontal dispersion coefficient, m
AI, BI, and Cj
(for x > 1,000 m) = functions of stability as shown below
where
Stability class
Ai
BI
Ci
X =
H =
h =
A
B
C
D
E
F
0.00024
0.055
0.113
1.26
6.73
18.05
Q f ^ r
\f ^^W*"t 1 lit
x yฐzu exp[ ฐ-
2.094
1.098
0.911
0.516
0.305
0.18
-/ H\21
* l ll
u
\ z/ J
-9.
2.
0.
-13
-34
-48.
6
0
0
6
short-term peak concentration, g/m3
h + Ah =
physical
effective emission
stack height, m
Y = X-
*21t p
height,
m
(H-7)
(H-8)
where
X2i+
P
maximum 24-hr concentration, g/m3
empirically derived peak to 24-hr maximum ratio
S =
(TLV) (8/24) (1/100)
(H-9)
136
-------
where S = source severity
TLV = threshold limit value, g/m3
Example 2. Source Assessment Deterministic Severity Calculation
. Q = (CONS) (EF) (%Be) (H-10)
where Q = emission rate, g/s
CONS = coal consumption rate, g/s
= 3.4532 x 1(T g/s
EF = emission factor for beryllium
= 0.85
%Be = fraction of beryllium in coal
= 0.23 x 10~5 g Be/g coal
Q = (3.4532 x 104) (0.85) (0.23 x 10~5)
= 6.75 x 10~2 g/s
S = 5'5 Q (H-ll)
TLV(h2)
(5.5) (6.75 x 10"2)
(2 x 10~6) (100) 2
= 18.6
Example 3. Deterministic Severity Using Plume Rise Correction
The plume rise is calculated using Equation D-3.
Q 0 .kkk
Ah = (8.892) - (H-3)
where QH ^ 67 x 10" 3 YS b (H.12)
h = 100 m
u = 4 . 5 m/s
T = 407.95 ฐK
S
T = 286.1 ฐK
3.
V =14.1 m/s
o
d = 4.23 m
Equation D-l (uncorrected Holland equation) gives a plume rise
of 65 m and a source severity of 6.8. Equation D-4 (corrected
Holland equation) gives a plume rise of 195 m and a source
severity of 2.1.
137
-------
Thus,
QH = (67 x ID'3) (14.1) (4.23)2(1,013) (4084~8286j
QR = 5,137 cal/s
.. 8.892(5,137)ฐ-l+1+lt
An =
(4.5)0.69^
Ah = 139 m
H = h + Ah = 239 m
S = (5.5) (6.75 x IP"2)
(2 x 10~6) (239)2
S S 3.3
138
-------
APPENDIX I
QUANTIFICATION OF UNCERTAINTY IN SOURCE SEVERITY
Inequality 32, derived in Section 5, can be used directly in the
decision-making process by computing values of eฃ and eu corre-
sponding to each individual calculated severity, Sc. Alternative-
ly, Inequality 32 can be used to derive general guidelines for
decision making in the^Source Assessment Program by computing
generalized values of e& and eu. The latter approach is followed
in this report. In this appendix, generalized values are derived
for each of the individual uncertainties required in the calcula-
tion of e^ and eu.
RANDOM UNCERTAINTIES IN AVERAGE CAPACITY AND EMISSION HEIGHT
(ar AND cr)
For organic and inorganic chemicals source types having small
numbers of plants, ar and cr are usually zero since data for all
plants are usually available. For source types having large num-
bers of sources, survey data are generally obtained from a suffi-
cient number of plants to make these terms small. For example,
in a source assessment of asphalt hot-mix plants, 600 plants were
surveyed from a population of approximately 4,000 plants to yield
95% confidence limits on mean capacity and mean stack height of
ฑ4% and ฑ3%, respectively, Thus, for most source types, ฑ5%
appears to be a conservative value for these terms. Then
ar2 + 4 cr2 = 0.01 (1-1)
SYSTEMATIC UNCERTAINTY IN EMISSION FACTOR (b AND b )
S Xf o U.
Estimates of systematic and random errors involved in sampling
and analytical procedures have been determined for a number of
standard EPA methods via collaborative testing. The available
.results are summarized in Table 1-1 (33-43). This table shows
that average biases for standard EPA methods range rom -35% to +10%,
(33) Hamil, H. F., and R. E. Thomas. Collaborative Study of
Method for Determination of Stack Gas Velocity and Volumet-
ric Flow Rate in Conjunction with EPA Method 5. EPA-650/4-
74-033 (PB 240 342), U.S. Environmental Protection Agency,
Research Triangle Park, North Carolina, 1974. 40 pp.
(continued)
139
-------
TABLE 1-1. COLLABORATIVE TEST RESULTS FOR EPA SAMPLING AND ANALYTICAL METHODS
EPA
High volume.
(Ambient air.
Method
2
3
S
6
7
8
9
10
104
b
)
Chemiluminescent
(NO ), ambient air.)
X
Chemiluminescent (photochemical
oxidants, ambient air) .
NDIR (ambient
air) .
Bias,
* of standard
concentration
0
0
-2
(Analysis only.)
+1.4% Opacity.
-2 to +10
-20
0
0
-35 to -15
(From 0.05 ppm
to 0.50 ppm.)
+2.5
Precision (standard deviation) ,
% of mean concentration
Within lab
5.5
10 to 30
10 to 30
4
7
60
2% Opacity.
2 to 7
44
3
2
(At 250 pg/m3)
1.3
(At 1,000 pg/rn3)
7 to 8
(At 100 yg/m3)
9.2
(At 0.05 ppm)
5.9
(At 0. 1 ppm)
3.2
(At 0. 5 ppm)
20
(At 3 mg/m3)
Between lab
5.6
15 to 35
20 to 40
5.5
10
65
2.5% Opacity.
4 to 13
58
3.7
4
(At 250 pg/m3)
2.6
(At 1,000 ug/m3)
30
(At 3 mg/m3)
95% (20) confidence
limits, % of mean
Within lab
11
20 to 60
20 to 60
8
14
120
4% Opacity.
4 to 14
88
6
4
(At 250 pg/m3)
2.6
(At 1,000 pg/m3)
14 to 16
(At 100 ug/m3)
18
(At 0.1 ppm)
12
(At 0.1 ppm)
6.5
(At 0.5 ppm)
40
(At 3 mg/m3)
Rptwppn 1 ab
11.2
30 to 70
40 to 80
11
20
130
5% Opacity.
8 to 26
116
7.4
8
5.2
60
(At 3 mg/m3)
Minimum
detectable
limit Reference
36
37
38-41
3 ppm 38,42
38,43,44
38
38
20 ppm 38,45
38,46
3 mg 38
25 pg/m3 38
10 pg/m3 38
0.3 mg/m3 ^8
0.3 mg/m3 38
NOTE.Blanks indicate no information available.
Values listed are for a single measurement. For n measurements, the values shouJa be divided by >^n.
-------
Biases were determined by measuring a gas of known composition
(standard) under controlled conditions. Values reported are aver-
ages of a number of runs by various collaborators at different
levels of pollutant concentration. Since bias can vary greatly
among collaborators and with pollutant concentration, the range
of the individual measurements would be more appropriate for
determining systematic error bounds. For example, in the measure-
ment of carbon monoxide by EPA Method 10 (44), bias in individual
tests ranged from -12% to +18%, and there was a definite trend
from positive to negative bias with increasing CO concentration.
Hence, more realistic error bounds would be obtained by using the
limits -12% and +18% for bias rather than the average values of
-2% to +10%. For the purpose of the present analysis, however,
the average values will be used.
The value of -35% for negative bias is probably a conservative
estimate for source assessment anyway, since it corresponds to a
method for sampling ambient air rather than a stack sampling
method.
(continued)
(34) Hamil, H. F., and R. E. Thomas. Collaborative Study of
Method for Stack Gas Analysis and Determination of Moisture
Fraction with Use of Method 5. EPA-650/4-73-026 (PB 236
929), U.S. Environmental Protection Agency, Research Tri-
angle Park, North Carolnia, 1974.
(35) Smith, F., and J. Buchanan. IERL-RTP Data Quality Manual.
EPA-600/2-76-159, U.S. Environmental Protection Agency,
Research Triangle Park, North Carolina, 1976.
(36) Hamil, H. F., and D. E. Camann. Collaborative Study of
Method for the Determination of Particulate Matter Emissions
from Stationry Sources (Portland Cement Plants). EPA-650/
4-74-029 (PB 237 346), U.S. Environmental Protection Agency,
Research Triangle Park, North Carolina, 1974. 54 pp.
(37) Hamil, H. F., and R. E. Thomas. Collaborative Study of
Method for the Determination of Particulate Matter Emissions
from Stationary Sources (Fossil Fuel-Fired Steam Generators).
EPA-650/4-74-021 (PB 234 150), U.S. Environmental Protection
Agency, Research Triangle Park, North Carolina, 1974. 36 pp.
(38) Hamil, H. F., and R. E. Thomas. Collaborative Study of
Method for the Determination of Particulate Matter Emissions
from Stationary Sources (Municipal Incinerators). EPA-650/
4-74-022 (PB 234 151), U.S. Environmental Protection Agency,
Research Triangle Park, North Carolina, 1974. 37 pp.
(39) Hamil, H. F., and D. E. Camann. Collaborative Study of
Method for the Determination of Sulfur Dioxide Emissions
from Stationary Sources (Fossil Fuel-Fired Steam Generators),
EPA-650/4-74-024 (PB 238 293), U.S. Environmental Protection
Agency, Research Triangle Park, North Carolina, 1973. 64 pp.
(continued)
141
-------
From Appendix F,
b
su
su 1-3
s -
SJt 1 + 3
where $~ and 3 are bounds on_the negative and positive bias,
respectively. Substituting 3 =0.35 and 3+ = 0.10 yields
b = 0.5
su
t> = 0-1
sฃ
A
, RANDOM UNCERTAINTY IN EMISSION FACTOR (b )
The random uncertainty in emission factor will be treated as an
independent variable since it can (theoretically) be controlled
by specifying the number of samples to be collected and analyzed
(continued)
(40) Hamil, H. F., and D. E. Camann. Collaborative Study of
Method for the Determination of Nitrogen Oxide Emissions
from Stationary Sources (Fossil Fuel-Fired Steam Generators).
EPA-650/4-74-025 (PB 238 555), U.S. Environmental Protection
Agency, Research Triangle Park, North Carolina, 1973. 102 pp.
(41) Hamil, H. F., and R. E. Thomas. Collaborative Study of
Method for the Determination of Nitrogen Oxide Emissions
from Stationary Sources (Nitric Acid Plants). EPA-650/4-74-
028 (PB 236 930), U.S. Environmental Protection Agency,
Research Triangle Park, North Carolina, 1974. 41 pp.
(42) Constant, P. C., G. Scheil, and M. C. Sharp. Collaborative
Study of Method 10Reference Method for Determination of
Carbon Monoxide Emissions from Stationary SourcesReport of
Testing. EPA-650/4-75-001 (PB 241 284), U.S. Environmental
Protection Agency, Research Triangle Park, North Carolina,
1975.
(43) Constant, P. C., and M. C. Sharp. Collaborative Study of
Method 104Reference Method for Determination of Beryllium
Emissions from Stationary Sources. EPA-650/4-74-023 (PB 245
Oil), U.S. Environmental Protection Agency, Research Tri-
angle Park, North Carolina, 1974. 94 pp.
(44) Title 40Protection of Environment. Chapter 1Environ-
mental Protection AgencyPart 60Standards of Performance
for New Stationary Sources. Subchapter CAir Programs.
Method 10Determination of Carbon Monoxide Emissions from
Stationary Sources. Federal Register, 39:9319-9321, 1974.
142
-------
The range over which b can be expected to vary can be determined
by reference to Table I-l, which lists the precision of standard
EPA methods.
The precision of each measurement method was determined by an
analysis of variance (ANOVA) of collaborative test results
obtained under field conditions. The "within-laboratory" stand-
ard deviation is obtained from the residual, or error, variance
in the standard ANOVA procedure. It is an estimate of the impre-
cision inherent in the sampling and analytical procedures and is
the imprecision which would be obtained if repeat measurements
were performed under identical conditions by the same personnel
using the same equipment. The "between-laboratory" standard
deviation is obtained from the total variability in the test
data, and is a measure of the total imprecision due to 1) the
inherent imprecision in the method, 2) differences in personnel
and equipment among laboratories, and 3) any process variations
which occurred during the tests. In the absence of process
variation, the difference of the between-laboratory and within-
laboratory variances is a measure of the variability due to
differences in factors such as personnel or equipment and is
designated "laboratory bias." (Note that laboratory bias is a
component of imprecision rather than bias.) In this case,
a2 = a2 + a2 (1-4)
between labs within lab laboratory bias
Assuming that errors are normally distributed and that estimates
of standard deviations are based on a large number of degrees of
freedom (The latter assumption is not justified in some cases due
either to poor experimental design or to a limited number of
usable data points.), the 95% confidence limits are approximately
equal to twice the standard deviation. These values are listed
in Table I-l for both the within-laboratory and between-laboratory
components.
Usually, at least three samples are collected and analyzed. Thus,
dividing the 95% confidence limits by /3, the largest random
uncertainty in emission factor should be about 75%. Larger
values could result from large process variations, however.
Measurement of trace elements and polycyclic organic material
(POM) may also result in larger uncertainties due to the rela-
tively large imprecision of analytical methods used for these
species. For the purpose of this analysis, it will be assumed
that the random uncertainty in emission factor varies between 0
and 100%; i.e.,
/\
0 < br < 1.0 (1-5)
SYSTEMATIC UNCERTAINTY IN EFFECTIVE EMISSION HEIGHT (c AND c )
S Xf S U
Bounds on the systematic error in effective emission height due
to use of semiempirical equations to estimate plume rise are
143
-------
derived in Appendix C. The bounds corresponding to the modified
Holland equation (Equation C-4) are used in the present analysis:
c =05
ฐsa ฐ'b
UNCERTAINTY IN CORRECTION FACTOR FOR DISPERSION EQUATION (m)
There are no experimental data which can be compared with disper-
sion calculations under the conditions of interest in this report;
i.e., fixed-source parameters, wind speed of 4 . 5 m/s, and Class C
stability. However, recent measurements by Guzewich and Pringle
(45) permit such a comparison for (approximately) fixed-source
parameters and Class C stability. A total of 33 measurements was
made under Class C stability conditions with wind speeds ranging
from 1.6 m/s to 13.2 m/s. Emission rates varied from 8.4 g/s to
9.8 g/s;a other source parameters remained constant during the
measurements.
The distribution of the ratio of measured to calculated concentra-
tion is shown in Figure I-l.b Calculated values were obtained
using simple Gaussian dispersion theory together with Briggs1
plume rise formula (26) . Data are approximately log-normally
distributed with a mean of 1.0 and a variance of 0.3136. The 1%
and 99% points are approximately equal to 1/3 and 3, respectively.
Data variability can be assigned to variability in the following
factors :
Measured values
Wind speed
Emission rate
Estimated plume rise
Other atmospheric parameters (other than wind speed and
stability class)
Only the last of these contributions is pertinent to the present
analysis. That is, error bounds for the correction factor, y,
based on Figure 1-1, should be on the conservative side.
Emission rates were accurately known since the "pollutant" was a
tracer substance injected into the stack gas stream.
Raw data were supplied by R. L. Hanson, Office of the Project
Manager for Chemical Demilitarization and Installation Restora-
tion, U.S. Department of the Army, Aberdeen Proving Ground,
Maryland.
(45) Guzewich, D. C., and W. J. B. Pringle. Validation of the
EPA-PTMTP Short-Term Gaussian Dispersion Model. Journal of
the Air Pollution Control Association, 27 (6) :540-542, 1977.
144
-------
is 1.0 -
S 0.8 -
^ 0.6 -
0.4 -
0.2 -
0.010.050.10.20.51 2 5 10 20 3040506070
90 95 98 99 99.8 99.9 99.99
Figure 1-1. Comparison of measured and predicted
values of ground level concentration.
Therefore, the value m = 3 will be used in the analysis as a
conservative estimate.
The above "factor of three" is consistent with the best case
estimate given by Turner (5). This factor corresponds to the
estimation of the pollutant concentration at a given point down-
wind of the source rather than estimation of the maximum concen-
tration, Xmax- Theoretical and empirical analyses indicate that
a "factor of two" may be more appropriate for estimation of
Xmax (46-47). However, the "factor of three" will be used here
in order to take into account uncertainty in the averaging time
correction (see following subsection).
Data in Figure 1-1 indicate that the uncertainty in the disper-
sion equation could be eliminated by appropriately modifying the
(46) Scriven, R. A. Variability and Upper Bounds for Maximum
Ground Level Concentrations. Philosophical Transactions of
the Royal Society of London, Series A, 265:209-220, 1969.
(47) Weber, A. H. Atmospheric Dispersion Parameters in Gaussian
Plume Modling, Part I, Review of Current Systems and Pos-
sible Future Developments. EPA-600/4-76-030a (PB 257 893),
U.S. Environmental Protection Agency, Research Triangle
Park, North Carolina, July 1976. 58 pp.
145
-------
definition of source severity. The value of Xmax was defined as
the value which would result from the source emitting into an
atmosphere with a wind speed of 4.5 m/s and Class C stability.
If a large number of tests were made holding source parameters,
wind speed, and stability class fixed, but allowing otheฃ atmos-
pheric parameters to vary,3 a distribution of values of Xmax
rather than a single value would be obtained. Figure 1-1 indi-
cates that the Gaussian dispersion model correctly predicts the
median of this distribution (median correction factor equals
l.Q). Therefore, by specifying the value of Xmax to be used in
the definition of source severity as the above ensemble median,
inherent uncertainty associated with the dispersion model would
be eliminated.
Of course, in interpreting the above modified version of source
severity, it would have to be kept in mind that actual ground
level concentrations under conditions of Class C stability and
wind speed of 4.5 m/s could be "a factor of three" higher or
lower than the median value. Hence, "actual" severities could be
higher or lower than the nominal value by a "factor of three"
under these atmospheric conditions.
From the standpoint of decision making, the modified version of
source severity would result in a decision based on the median
value of Xmax fฐr Class C stability and wind speed of 4.5 m/s
rather than on the upper 97.5% value of Xmax-
UNCERTAINTY IN AVERAGING TIME CORRECTION FACTOR (tQ/t)0-17
The averaging time correction factor given by Turner (5),
(to/t)0*1 , is a semiempirical relationship derived from data on
lateral and vertical diffusion coefficients in steady winds. It
applies to situations in which the mean wind direction remains
constant over the period of interest. Thus, it corresponds to a
worst case situation with respect to pollutant concentration.
A number of studies have been conducted to determine the relation-
ship between measured concentration and averaging time for ambi-
ent air monitors located at fixed sites (29, 32, 48). Results of
these studies tend to predict considerably lower mean concentra-
tions than does the above correction factor, which is to be
expected since the mean wijd direction does not, in general,
remain constant over an extended period of time. The diversity
For example, stability Class C defines a range of atmospheric
stability conditions rather than a single unique condition.
Previously, it was assumed that other atmospheric parameters
were held fixed so that a unique atmospheric state (and a cor-
responding unique value of x^ ) was obtained.
(48) Shiruaikar, V. V., and P. R. Patel. Long Term Statistics
of Peak/Mean Concentrations From A Point Source. Atmos-
pheric Environment, 11 (4):387-389, 1977.
146
-------
of results obtained in the three studies cited above also empha-
sizes the fact that the relationship between concentration and
sampling time is strongly dependent upon local meteorological
conditions; e.g., wind direction statistics.
For the purpose of the present analysis, it is assumed that the
mean wind direction remains constant during the period of inter-
est. This assumption is consistent with the concept of a fixed-
state receiving atmosphere assumed previously. The uncertainty
associated with the averaging time correction factor under these
conditions is then required. In lieu of other information, this
uncertainty is assumed to be included in the uncertainty associ-
ated with the dispersion equation, and the latter uncertainty is
conservatively taken to be a "factor of three" (see preceding
subsection) .
The effect on source severity of relaxing the assumption of con-
stant mean wind direction is considered in Appendix G.
^ /\
UNCERTAINTY IN ACCEPTABLE CONCENTRATION (d AND d )
SV1 S J6
It is assumed that for each noncriteria pollutant, i, the "true"
value, FT, and calculated value, FC, of the "acceptable" concen-
tration are given by
= (TLV)i Gi (1-6)
(Fc)i = (TLV)i
where (TLV). = TLV of pollutant i as listed in Reference 4
G. = conversion factor which converts (TLV). to an
equivalent PAAQS 1
G = an "average" conversion factor
Since (TLV). is taken as the nominal value listed in Reference 4,
there is no1uncertainty associated with this value in the context
of the present analysis. All variability due to the methodology
used in establishing TLV's is associated with the conversion
factors, G..
From Equations 1-6 and 1-7 are obtained
(FT)i fi
Wh~ G
F ) .
In -7j-i- = In GL - In G (1-8)
147
-------
It is assumed that the In Gi are normally distributed (i.e., the
G^ are log-normally distributed) over the population of pollut-
ants with mean y and variance a2. It then follows from Equa-
tion I-8_ that ln(FT/Fc) is normally distributed w^th mean
y - In G and variance a2. If it is assumed that: G is the geo-
metric mean of the distribution of Gj_, then In G = y, and the
distribution of ln(FT/Fc) nas a mean of zero.
Now it is assumed that criteria pollutants constitute a random
sample of size four from the population of all pollutants. From
Table E-l, the values shown in Table 1-2 are obtained.
TABLE 1-2. CONVERSION FACTORS FOR CRITERIA POLLUTANTS
Pollutant
Particulate matter
Sulfur oxides (SOX)
Carbon monoxide (CO)
Nitrogen oxides (NO )
J\.
Gi
0.0260
0.0281
0.158
0.0411
In G.
-3.650
-3.573
-1.845
-3.191
Using these values, the estimated mean and standard deviation of
the In G. are
x = -3.064
s = 0.8395
Thus, the estimate for G is
G = ey = ex = 0.0467 (1-9)
The estimated 95% confidence interval for ln(FT/Fc) is ฑ1.96 s;
i.e. ,
/M
-1.645 < ln(=i)< 1.645
\ C/
FT
0.193 < =rฑ. < 5.18
" C ~
0.193 FC < FT < 5.18 FC (1-10)
/\ /\
By definition of d and d ,
su sx,
^ FT <
-------
From Equations 1-10 and 1-11,
d = 4.2
su
In the Source Assessment Program, G = 0.0033, so that In G = -5.70
In this case, the estimated 95% confidence interval for F^/FQ is
given by
[-3.064 - (-5.70) ] ฑ 1.96 s
That is,
FT
2.64 - 1.645 < In =ฑ- < 2.64 + 1.645
FC "
FT
2.70 < =ฑ- < 72.6
~ FC ~
2.70 FC < FT < 72.6 FC (1-12)
Comparing Equations 1-11 and 1-12 yields
/\
dsu = 71'6
Thus, there are three sets of values for the uncertainty in
"acceptable" concentration.
Case A. No uncertainty in acceptable concentration.
s\ s\
d = d =0
si su
Case B! . Acceptable concentration uncertain; TLV conversion
factor equal to geometric mean value.
Case 62 . Acceptable concentration uncertain; TLV conversion
factor equal to 1/300.
3Note that d ฃ > 0 by definition.
149
-------
dsu = 71'6
150
-------
APPENDIX J
EXAMPLE CALCULATIONS FOR SOURCE ASSESSMENT:
CARBON BLACK MANUFACTURE
In this appendix, the principles developed in the body of the
report are applied to a source assessment study of carbon black
manufacturing operations .(49). In this study, emission factors
for the main process vent were measured at a typical carbon black
plant. Mean emission factors and 95% confidence limits for the
mean values (computed from the experimental data) are listed in
Table J-l. These values were taken directly from Reference 52.
Also listed in Table J-l are the random and systematic uncertain-
ties associated with the measured emission factors. The values
of br are simply the 95% confidence limits for the mean emission
factors. The systematic uncertainties for nitrogen oxides and
carbon monoxide were obtained by applying Equations G-5 and G-6
to the biases listed in Table 1-1 for EPA Methods 7 and 10,
respectively. The systematic uncertainties for the other species
in Table J-l were equated with the generalized values given in
Table 3 (Section 5), since no information is available concerning
biases in the methods used to measure emissions of these
compounds.
TREATMENT OF ST AS A FIXED PARAMETER
The random uncertainties^in average plant capacity and average
emission height (ar and cr) were set equal to zero since data
were available for all plants in the industry. Values given in
Table 3 (Section 5) were used for the uncertainties in plume
rise, dispersion equation, and "acceptable" concentration.
For each pollutant listed in Table J-l, the following quantities
were computed:
Estimated severity, S (including plume rise).
Lower and upper bounds for the true severity, S .
The critical test value, S*.
\^
(49) Serth, R. W,, and T. W. Hughes. Source Assessment: Carbon
Black Manufacture. EPA-600/2-77-107k, U. S. Environmental
Protection Agency, Research Triangle Park, North Carolina,
October 1977.244 pp.
151
-------
Results for criteria pollutants are presented in Table J-2. Cri-
teria pollutants correspond to Case A of Section 5; i.e., no
uncertainty in "acceptable" concentration. The critical test
value derived in Section 5 by the generalized analysis was 0.05.
The values for nitrogen oxides and carbon monoxide are signifi-
cantly higher than the generalized value because of the smaller
systematic uncertainties in emission factors for these two species
TABLE J-l.
EMISSION FACTORS AND ASSOCIATED UNCERTAINTIES
FOR CARBON BLACK PROCESS VENT
Material
emitted
Particulate matter
Nitrogen oxides
Hydrocarbons
Carbon monoxide
Hydrogen sulfide
Carbon disulfide
Carbonyl sulfide
Isobutane
n-Butane
POM (total)
Measured emission
factor , g/kg
0.11 ฑ 70%
0.28 ฑ 15%
50 ฑ 48%
1,400 ฑ 19%
30 ฑ 82%
30 ฑ 76%
10 ฑ 99%
0.1 ฑ 80%
0.27 ฑ 57%
0.002 ฑ 52%
b
r
0.70
0.15
0.48
0.19
0.82
0.76
0.99
0.80
0.57
0.52
*sฃ
0.1
0
0.1
0.09
0.1
0.1
0.1
0.1
0.1
0.1
b
su
0.5
0
0.5
0.02
0.5
0.5
0.5
0.5
0.5
0.5
TABLE J-2.
EXAMPLE CALCULATIONS FOR CRITERIA POLLUTANTS
(CASE A: NO UNCERTAINTY IN ACCEPTABLE CONCENTRATION)
Material
emitted
Particulate matter
Nitrogen oxides
Hydrocarbons
Carbon monoxide
Calculated
severity, S
0.004
0.032
4.2
0.36
Uncertainty interval
S e < S S^- for these species), but not
with regard to nitrogen oxides and particulate emissions (since
SG < S* for these species).
152
-------
For noncriteria pollutants, calculations were made for both Case
BI (TLV conversion factor of 0.0467) and Case B2 (TLV conversion
factor of 0.0033). Results of these calculations are given in
Tables J-3 and J-4, respectively. The critical test values
obtained by the generalized analysis in Section 5 were 0.01 for
Case BI and 0.05 for Case B2. The critical test values listed in
Tables J-3 and J-4 are essentially the same as the generalized
results since the same uncertainty values were used in both
cases.
TABLE J-3.
EXAMPLE CALCULATION FOR NONCRITERIA POLLUTANTS
(CASE BI: TLV CONVERSION FACTOR = 0.0467)
Material
emitted
Hydrogen sulfide
Carbon disulfide
Carbonyl sulfide
Isobutane
n-Butane
POM (total)
Calculated
severity, S
ซu>
0.28
0.11
0.030
0.000014
0.000042
0.030
Uncertainty interval
0
0
0
0
0
0
<_ ST 1
- ST 1
-------
GLOSSARY
accuracy: Closeness between true and measured values which is
characteristic of a given measurement process.
atmospheric stability class: Class used to designate degree of
turbulent mixing in the atmosphere.
bias: Magnitude and direction of the tendency of a given mea-
surement process to measure a quantity other than the
nominal or intended quantity.
criteria pollutant: Emission species for which an ambient air
quality standard has been established.
decision index: Numerical index used as an aid in decision
making.
emission factor: Weight of material emitted to the atmosphere
per unit weight of product produced.
expected value: First moment about the origin (i.e., the mean)
of the probability distribution of a random variable.
Monte Carlo simulation: Simulation in which variables are
represented by probability distributions.
national emissions burden: Decision index defined as the ratio
of emissions of a criteria pollutant from a given source
type to emissions of the same pollutant from all stationary
sources nationwide.
Neyman-Pearson theory: Most widely accepted theory of statis-
tical inference.in which population parameters are re-
garded as fixed.
noncriteria pollutant: Emission species for which no ambient
air quality standard has been established.
operating characteristic curve: Curve which gives the (fiducial)
probability that the true severity is less than one as a
function of the calculated severity.
plume rise: Distance above point of emission where plume rises
due to its momentum and buoyancy.
154
-------
precision: Closeness together, or lack of scatter, in replicate
measurements which is characteristic of a given measure-
ment process.
random error: Component of the total error which results from
imprecision of the measurement process.
random uncertainty: Component of the total uncertainty which
results from random error in the measured value.
reasonably available control technology (RACT): Available,
economically feasible control technology required for
attainment of national ambient air quality standards (as
specified in Code of Federal Regulations, Title 40,
Chapter I, Part 51. Requirements for Preparation, Adopt-
ion, and Submittal of Implementation Plans).
representative source: Hypothetical source having typical values
of parameters which is used to characterize an industry
or source type.
source severity: Decision index defined as the ratio of maxi-
mum time-averaged ground level pollutant concentration to
an acceptable pollutant concentration.
systematic error: Component of the total error which results
from bias in the measurement process.
systematic uncertainty: Component of the total uncertainty
which results from the systematic error in the measured
value.
Type I error: Error in decision making which results from
rejecting the null hypothesis when it is true.
Type II error: Error in decision making which results from
accepting the null hypothesis when it is false.
uncertainty: Measure of the lack of knowledge of the true
value of a quantity due to imperfect data; i.e., due to
errors in the measured values of the quantity.
155
-------
CONVERSION FACTORS AND METRIC PREFIXES
(50)
CONVERSION FACTORS
To convert from
Degree Celsius (ฐC)
Degree Kelvin (ฐK)
Gram/second (g/s)
Joules (J)
Kilogram (kg)
Kilometer (km)
Meter (m)
To
Meter'1
(m2)
Meter3 (m3)
Meter/second
Metric ton
Metric ton
Metric ton
Newtons (n)
Pascal (Pa)
Pascal (Pa)
Watts (W)
(m/s)
Degree Fahrenheit
Degree Celsius
Pound/hr
Calories
Pound-mass (avoirdupois)
Mile
Foot
Foot2
Foot3
Mile/hr
Pound-mass
Kilogram
Ton (short, 2,000 pound mass)
Dyne
Millibars
r\
Pound-force/inch'' (psi)
Calories/min
Multiply by
^
t*
= 1.8 tฐ + 32
= tฐ - 273.15
7.937
2.388 x 10"1
2.205
6.214 x 10"1
3.281
1.076 x 101
3.531 x 101
2.237 x 103
2.205 x 103
1.000 x 103
1.102
1.000 x 10
1.450 x I0~k
1.434 x 101
METRIC PREFIXES
Prefix Symbol Multiplication factor
Kilo
Mega
Micro
k
M
P
10
3
-6
Example
1 kPa = 1 x 103 pascals
1 MJ ' ""
1 pg
= i x lu pascal'
= 1 x 166 joules
1 x 10~6 gram
(50) Standard for Metric Practice. ANSI/ASTM Designation
E 380-76 , IEEE Std 268-1976, American Society for Testing
and Materials, Philadelphia, Pennsylvania, February 1976.
37 pp.
156
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA~600/2~78~004u
2.
3. RECIPIENT'S ACCESSION NO.
4. TITLE AND SUBTITLE
SOURCE ASSESSMENT: Analysis of
5. REPORT DATE
August 1978
6. PERFORMING ORGANIZATION CODE
and E.C.Eimutis
, T. W. Hughes, R. E. Opferkuch,
8. PERFORMING ORGANIZATION REPORT NO.
MRC-DA-632
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Monsanto Research Corporation
1515 Nicholas Road
Dayton, Ohio 45407
10. PROGRAM ELEMENT NO.
1AB015; ROAP 21AXM-071
11. CONTRACT/GRANT NO.
68-02-1874
12. SPONSORING AGENCY NAME AND ADDRESS
EPA, Office of Research and Development
Industrial Environmental Research Laboratory
Research Triangle Park, NC 27711
13. TYPE OF REPORT AND PERIOD COVERED
Task Final; 11/76-3/78
14. SPONSORING AGENCY CODE
EPA/600/13
is. SUPPLEMENTARY NOTES IERL-RTP project officer is Dale A.. Benny, Mail Drop 62, 919/
541=2547. Similar previous reports are in the EPA-600/2~76~u32 and -77-107 series.
is. ABSTRACT r^ rgp0rj. gjves results of: an analysis of the uncertainties involved in cal-
culating the decision parameters used in the Source Assessment Program; and a
determination of the effect of these uncertainties on the decision-making procedure.
A general procedure for performing an analysis of uncertainty is developed, based on
the principles of error propagation and statistical inference. It is shown that this sim-
ple and straightforward method represents and approximation to standard statistical
techniques. The approximate method is illustrated by application to four problems in
the area of environmental control. The general procedure Is used to establish guide-
lines for conducting air emissions studies in the Source Assessment program. In
particular, guidelines are established for precision in field sampling and analytical '
work, and for setting critical values of decision parameters.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS
COSATI Field/Group
Pollution
Uncertainty Principle
Statistical Inference
Statistical Distributions
Sampling
Pollution Control
Stationary Sources
Source Assessment
Error Propagation
13B
20J
12A
05A
Field Tests
13. DISTRIBUTION STATEMENT
Unlimited
19. SECURITY CLASS (This Report)
Unclassified
21. NO. OF PAGES
180
20. SECURITY CLASS (This page)
Unclassified
22. PRICE
EPA Form 2320-t (S-73)
157
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