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75
n * The effective number of days per year.
If ph is given by (3-47) it follows that:
/ VyW
>\ E
(3-51)
Substituting Eq. (3-51) into (3-50) yields:
HSTD
In
1
- (1-
nd/y
1/nh/d"
n -1 Vk
In
h/y
(3-52)
Equation (3-52) connects the parameters of the two standards when both are
connected by the same distribution of hourly average concentrations. It
allows comparisons to be made between the two kinds of standards when both
standards are simultaneously met. Table 3-23 shows the corresponding
levels of the standards when the expected exceedances for each are the same.
In the one case the hourly average standard level is held constant at 0.08
ppm for each value of the exceedance. In the other, the daily maximum
hourly average concentration level is held constant. In this table the
effective hours per year are the full 8760 and the effective hours per day
are 24. The calculation is done for three levels of k.
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76
It is seen that in the range of 1 to 20 exceedances, the difference
between the two standards is negligible for complete independence of hours
and a stationary stochastic process.
To partially remove the effect of nonstationarity, the calculation
was repeated for effective hours per year of 600 and effective hours per
day of 5 (Table 3-24). There is still not much effect up to 10 exceedances.
Thus, for the case of independent hours the two standards are essen-
tially identical for expected exceedances in the range of interest. That
also means the risks of exceeding thresholds would be the same. That is,
either one could be used for the same concentration level and expected
number of exceedances within reasonable limits and give the same protection.
The analysis says that a relatively large number of exceedances would
be required before there was sufficient clustering within a day (in this
case by a purely random process) to cause the standards to begin to
separate.
Unfortunately, the analysis cannot be stopped here because while we
have accounted for nonstationarity to some extent we probably have not
in this case, gone far enough. For example, suppose a simple, but not
far fetched, coupling were introduced, such as: If one exceedance occurs
the next hour will almost always exceed the standard, while the probability
that the third hour will exceed the standard is about the same as the first
hour. In this case the exceedances will cluster very strongly into pairs. The
above analysis of the independence case shows that beyond that little additional
random clustering will occur if the total exceedances are not too large ( < 5).
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77
Now in this case it can be concluded that if the same concentration level
is set for the two types of standard the following very nearly holds:
Ed/y * 1/2 (3-53)
In other words, the parameters of the two standards would now have to be
significantly different to provide the same protection.
The analysis in the proceeding paragraphs shows that for complete inde-
pendence the standards are nearly the same over a broad range of E values.
For ozone concentrations, however, the nonstationarity is extensive. To go
further requires working directly with ambient air data since at the present
time there is no mathematical model for the nonstationary stochastic proces-
ses generating hourly average ozone concentrations.
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78
Table 3-23 COMPARISON OF DAILY MAXIMUM HOURLY AVERAGE AND HOURLY AVERAGE STANDARDS
(1)
Expected
Exceedances^ '
1
5
10
15
20
50
Level of Dally Maximum Standard When
Hourly Standard Held at 0.08 ppm
0.5
0.0800
0.0799
0.0797
0.0795
0.0793
0.0779
1.0
0.0800
0.0799
0.0798
0.0797
0.0796
0.0789
2.0
0.0800
0.0800
0.0799
0.0799
0.0798
0.0795
1
5
10
15
20
50
Level of Hourly Standard When
Dally Maximum Standard Held at 0.08 ppm
0.5
0.0800
0.0801
0.0803
0.0805
0.0807
0.0822
1.0
0.0800
0.0801
0.0802
0.0803
0.0804
0.0811
2.0
0.0800
0.0800
0.0801
0.0801
0.0802
0.0805
(1) Number of hours per year = 8760
(2) Number of hours per day = 24
Expected exceedances the same for both standards.
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79
Table3-24 COMPARISON OF DAILY MAXIMUM AVERAGE AND HOURLY AVERAGE STANDARDS WITH
PARTIAL CORRECTION FOR NONSTATIONARITY IN HOURLY AVERAGE CONCENTRATIONS^1
Expected Level of Daily Maximum Standard When
Exceedances(2) Hourly Standard Held at 0.08 ppm
1
5
10
15
20
50
k = 0.5
0.0799
0.0794
0.0787
0.0777
0.0767
0.0674
1.0
0.0800
0.0797
0.0793
0.0789
0.0783
0.0734
2.0
0.0800
0.0799
0.0797
0.0794
0.0792
0.0766
Level of Hourly Standard When
Daily Maximum Standard Held at 0.08 ppm
1
5
10
15
20
50
k - 0.5
0.0801
0.0806
0.0814
0.0823
0.0835
0.0950
1.0
0.0800
0.0803
0.0807
0.0812
0.0817
0.0872
2.0
0.0800
0.0801
0.0803
0.0806
0.0809
0.0835
(1) Effective number of hours per year = 600
Effective number of hours per day = 5
(Z) Expected exceedances the same for both standards.
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80
3.5.2 Comparison of Hourly and Dally Maximum Standards When
Hourly Average Concentrations Exhibit Dependence
To determine the effect of dependence of hourly average concentrations
use was made of an analysis performed by the Monitoring and Data Analysis
Division of OAQPS, at the Pollutants Strategies Branch request on data from
93 urbanized areas with populations greater than 200,000. Data were assem-
bled on the three highest ozone readings for selected sites in each area
and the dates on which they occurred. In a number of cases the highest
readings would duplicate and thus lead in some cases to having, in effect,
as many as the six highest readings. The dates of each of these readings
were also recorded. From these data it could be determined whether or not
the second highest day and the second highest hour tend to correspond to
the same reading. The MDAD data are shown in Table 3-25 grouped according to
air quality region. The right hand column shows that in 55% of the site-
years the second highest hour and the second highest daily maximum hour are
the same. That is, in 552 of the cases the highest hourly average concen-
tration and the second highest concentration occur on different days. In
45% of the cases the 3rd, or 4th, or 5th, etc. highest hourly average
corresponds to the second high day.
It is seen in Table 3-25 that the data for each region are shown in
two groupings. This is necessary because in many cases insufficient data
were available to determine which hourly average concentration corresponded
to the second high daily maximum hourly average. For example, if the three
highest concentrations were all different, thru data points would have been
recorded for that site that year. If these all occurred on the same day
then all that would be known is that the second high day corresponded to a
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concentration less than the third highest hour. It is possible to redis-
tribute these cases into the upper group if it is assumed the distribution
of the upper group is approximately correct. For example, looking at the
far right column of Table 3-25, 11.49% indefinite cases have the second
high day corresponding to less than the third highest hour. These cases
can be distributed among the 4th, 5th, 6th highest, etc. in the grouping
above roughly in proportion to the current distribution.
Thus, if the 1U69 is assumed to distribute entirely between the 4th,
and 5th high, 11.49 x 4.56/5.91 - 8.9% should go to the 4th and 11.49 x
1.35/5.91 = 2.6% should go to the 5th high. The 2.96% less than the
fourth high (Table 3-25, far right column ) can be distributed to the 5th
high. The 1.2% greater than the 5th high can be distributed to the sixth
high. The 1% remaining is attributed to the seventh and higher values.
The calculation could be repeated using the newly obtained distribution.
Table 3-26 shows the result for a single iteration. Using these data to
weight the ranking it can be calculated (Table 3-26) that the second
highest daily maximum concentration corresponds on the average to the 2.8
highest hourly concentration in a given year.
Table 3-26 can be used to obtain an approximate comparison of at least
one statistical form of the hourly and daily maximum hour forms. However,
it is first noted that if the forms of the standards were the expected values
of the annual maximum hourly average concentration and daily maximum hourly
average concentration, both standards would be identical. This follows from
the fact that the maximum hour in any year is also the maximum daily maximum
hour. This would not be true of the expected value of the second highest
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83
Table3-26 Daily Maximum Hourly Average Standard versus Hourly Average Standard
Per Cent
Second Highest Dally Maximum Concentration the Same as: of Years
Second highest hourly concentration 55
Third highest hourly concentration 23
Fourth highest hourly concentration 13
Fifth highest hourly concentration 7
Sixth highest hourly concentration 1
<, Seventh highest hourly concentration 1
100
Weighted Average Rank =
0.55x2 + 0.23x3 + 0.13x4 + 0.07x5 + 0.01x6 + 0.01x7 = 2.8
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84
hour versus the expected value of the second highest dally maximum hour.
The equivalent daily maximum standard would have a lower expected second
high concentration.
Where Table 3-26 can be used is in the case of a standard of the form,
X ppm with an expected number of exceedances per year less than or equal
to one. At any given number of hourly exceedances of a given concentration
level the data in Table 3-26 can be used to estimate the distribution of
exceedances of the same concentration level by the daily maximum hour concentra-
tions. When the annual hourly exceedances are 0 or 1, the connection is straight-
forward. If there are no hourly exceedances, there can be no exceedance
of a daily maximum hour standard with the same specified concentration
level. If there is a single hourly exceedance there will be a single daily
maximum hour exceedance.
In the case of two hourly exceedances, the data in Table 3-26 indicate
that in 45% of the years there will be a single exceedance of a daily maxi-
mum standard. In the remaining 55% of the years there will be two exceed-
ances. Unfortunately, the data in Table 3-26 are not sufficient to fill
out the table for more than two exceedances of the hourly average standard.
The values shown in Table 3-27 for 3 and 4 hourly exceedances are coarse
estimates. The numbers in the single daily exceedance column, however, do
follow from Table 3-26. For example, if there are three hourly exceedances
but only one daily maximum hour exceedance, this corresponds in Table 3-26
to cases in which the second highest day corresponds to the 4th, the 5th,
etc. highest hour. These add to 22% of the years. Similarily, for four
hourly exceedances the number is 9%. These have been rounded to 20% and
10% in Table 3-27.
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85
Table 3-27 Distribution of Exceedances per Year for Daily Maximum Hour
Standard Versus Exceedances per Year for Hourly Standard' '
Exceedances
per Year of
Hourly
Standard
0
1
2
3
> 4
Estimated Distribution of Exceedances for
Daily Maximum Hour Standard
dances
ear = 0
1.00
0.00
0.00
0.00
0.00
1234
1.00
0.45 0.55
0.20 0.35 0.45
0.10 0.30 0.30 0.30
Weighted^
Average
Exceedances
Daily Std.
0
1.00
1.55
2.25
2.71
Relative^
Frequencey of
Exceedances
Hourly Std.
0.37
0.37
0.18
0.06
0.02
(1) Both standards have the same specified concentration level.
(2) For two hourly exceedances the weighted average daily maximum
hour exceedance rate is 0.00 x 0 + 0.45 x 1 + 0.55 x 2 = 1.55.
(3) For expected exceedances/yr = 1, assuming independence of hours.
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86
From this distribution the weighted average exceedances of the daily
maximum hour standard can be calculated for each hourly exceedance. A
sample calculation is shown in Table 3-27.
To compare the two standards, the next step is to assume a probability
distribution of the hourly exceedances. The last column of Table 3-27
shows how they are distributed for an expected exceedance rate of one if
all hours of the year are independent (calculated from the binomial distri-
bution). The corresponding expected expected exceedance rate for the daily
maximum hourly average is obtained by combining the last two columns of
Table 3-27:
0.37 x 0 + 0.37 x 1.00 * 0.18 x 1.55 + 0.06 x 2.25 + 0.02 x 2.71 = 0.84
Thus, the daily standard is only somewhat more restrictive than the hourly
standard when the hourly expected exceedance rate is one.
Without much loss of accuracy it can be said that a daily maximum
hour standard with an expected exceedance of once per year corresponds to
an hourly standard at the same specified concentration level with an
expected rate of 1/0.84 = 1.2 exceedances per year.
A sensitivity analysis of the assumptions in Table 3-27 with regard
to the distribution corresponding to hourly exceedances of 3 and 4 and the
relative frequencies of exceedances of the hourly standard shows that the
last figure given is most probably in the range 1.2 to 1.4. If it is
assumed 1.3 expected exceedances per year is the most likely value, Eq.
(3-14) can be used to calculate the equivalent concentration level of the
daily maximum hour standard if both standards have an expected exceedance
rate of one. The comparison is shown in Table 3-28.
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87
Table 3-28 Approximate Comparison of Statistical Daily Maximum Hour and Hourly
Standards When Both Have Expected Exceedances per Year of One '
Level of Level of
Hourly Standard Daily Max. Hour Standard
(ppm) (ppm)
0.06 0.061
0.08 0.082
0.10 0.102
0.12 0.123
0.14 0.143
(1) Assumes distribution of hourly average concentrations
is represented by a Wei bull distribution for 8760
hours whose shape factor is: k = 1.25.
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88
While the above calculation is crude the conclusion that the two
forms of the standard do not differ much in their relative severities
and that the daily maximum hour standard is the less severe of the two
can be made with reasonable confidence.
3.5.3 Comparison of Deterministic Hourly and Daily Maximum Standards
The preceeding sections compared statistical forms of hourly and daily
maximum standards. For a standard in which only one exceedance per year of
specified concentration level is permitted the results are different and
depend upon how strictly the standard is enforced. If the restriction to
no more than one exceedance a year is strictly applied then both standards
are identical in their end result since both (following arguments presented
in Section 3.1) will lead to an air quality in which the standard level is
never exceeded.
In getting to this air quality level, the "design values" for the daily
maximum hour standard will initially average lower than those for an hourly
standard. This follows from the data and analysis presented in Section 3.4
and 3.5.2. In application of a "not to be exceeded more than once" standard
the design value for calculating required levels of emission control is the
highest second high over a period of several base years. If only one base
year is available,the second high for the area for that year is used. In
this case the highest second highest daily maximum hourly average concen-
tration will be equal to less than that of the highest second highest hourly
average concentration. This situation is seen in Table 3-29 which was
obtained from the MDAD analysis referred to in the previous section and which
provided the basis for Tables 3-25 through 3-27. In this case the highest
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89
Table 3-29 Comparison of Percent Hydrocarbon Control Required by Daily
Maximum Hourly Average and Hourly Average Standards
Highest % Control Hourly Max. - % Daily Max.
Hourly Average
Concentrati on
(ppm)
0.115
0.144
0.174
0.203
0.233
0.262
0.292
0.321
0.351
0.380
Average
(«)
1.0
4.0
7.0
4.5
4.0
2.8
1.5
1.4
1.4
1.4
Maximum
(%)
2.8
11.2
16.0
13.0
10.0
8.0
6.0
5.0
4.0
3.5
Minimum
(%)
0
0
0
0
0
0
0
0
0
0
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90
second high for the hourly and daily maximum hour concentration for each
of the 93 urbanized areas was used to calculate the percent control of
hydrocarbon emissions needed to meet the standard. Linear rollback was
used ( Eq. (3-38)). The results were summarized to give the average differ-
ence in calculated control level as well as the highest and lowest individual
value over a range of air quality levels (as exemplified by the highest
hourly average concentration).
It is seen in Table 3-29 that at each level of air quality the minimum
difference is zero. And indeed, for many of the regions the second highest
hour and the second highest daily maximum hour are the same. This follows
directly from the results in Table3-25which show that this situation occurs
on the average in 552 of the site years of data. Thus, for any region as
data are collected over an increasing number of years the design values
for the two standards will eventually come together.
Thus, in the short term a deterministic daily maximum will either
require the same level or less control than an hourly standard. In the
long run both standards will require the same level of control.
Notice the situation is different in statistical standards and depends
on the form. They are the same for a standard which specifies a maximum
value for the expected value of the annual highest concentration. However,
for an expected second high standard or standard that sets an expected
exceedance rate of a specified level the daily maximum hour standard will
be less than the hourly standard if the hourly average concentrations are
autocorrelated. In the case of ozone there is a relatively small difference
between the standards based on expected exceedances.
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91
3.6 Conclusions and Recommendations
The present deterministic form of the photochemical oxidants standard
does not take into account the statistical behavior of hourly average ozone
concentrations. As a result the only way in which strict compliance with
the standard can be achieved is by controlling precursor emissions to the
degree that the standard concentration level is never exceeded. The standard
is, therefore, significantly more stringent than apparent from its seeming
allowance of at least one exceedance of the standard level per year. The
uses of the second highest ozone concentration as an indicator of how far
a given Air Quality Control Region is out of compliance and as a design
value to calculate required degree of control of precursor emissions are also
misleading. Because second high concentrations are statistical variables
they do not provide a measure of the overall air quality for a region and
will lead to an under estimation of the degree of control of precursor emis-
sions required to gain compliance. Use of the highest second high for
several years of data only partially corrects for this problem.
Statistical standards provide a way out of these difficulties by fully
taking into account the statistical behavior of air pollutant concentrations.
They provide a clearly defined target for determining compliance and a mean-
ingful measure of existing air quality.
The statistical forms unlike deterministic forms lend themselves to
quantitative or mathematical treatment. The terms used in statistical
standards such as expected exceedance rate or expected annual maximum
hourly average concentration have precise meanings and are relatable to the
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92
cumulative frequency or probability distributions of the time average con-
centrations. With some statistical forms, such as expected annual maximum
concentration, it is necessary to take into account autocorrelation between
concentrations close to each other in time and nonstationarity in the under-
lying stochastic processes giving rise to the random changes in concentrations.
A simple approximate method has been presented dealing with this lack of
independence.
Comparison of statistical and deterministic standards which are set at
the same concentration level indicates the deterministic standards to be
more stringent if the deterministic standard is strictly enforced. An AQCR
just meeting a statistical standard allowing an expected exceedance rate
of one will violate on an average of once every four years a deterministic
standard allowing not more than one exceedance per year of the same concen-
tration level. A statistical standard based on an expected annual maximum
concentration will lead to violations of the above deterministic standard
on an average of once every eight years when the statistical standard is
just met.
When the comparison is made in terms of degree of precursor emission
control, in the short term the deterministic standard could appear on the
average to be somewhat more lenient. In the long term and if the determin-
istic standard is strictly enforced it will require greater control of
precursor emissions than comparable statistical standards. (In the case of
the present deterministic oxidant standard the comparable statistical stan-
dard is based on an expected exceedance rate of one. A standard based on
one expected annual maximum hourly average concentration would be somewhat,
though not strictly, comparable.)
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93
Comparisons between oxidant standards based on hourly and daily maximum
concentrations have led to the following conclusions. If both standards are
deterministic in the short term it will appear the daily maximum standard
is somewhat more lenient than the hourly standard. The difference in
apparent control of precursor emissions required for individual Air Quality
Control Regions (AQCR), in which the second high design value is based on
one to several years of monitoring data, will vary from zero to somewhat
over 15%. In the long run both standards are equally stringent and will
require the same level of control.
In the case of statistical forms there would be no difference in the
two forms if the hourly average concentrations were not autocorrelated.
The autocorrelation in the case of oxidant causes the daily maximum standard
to be slightly less stringent.
It is recommended that EPA consider changing all its short-term
national ambient air quality standards from the present deterministic to
suitable statistical forms. Its standards based on annual mean concentra-
tions could be considered as already being in a statistical form.
Initial guidlines have been developed by the Office of Air Quality
Planning and Standards on how to determine compliance and degree of depar-
ture from compliance with the proposed statistical standard for oxidant.
These guidlines are a good first step, but more study is needed on these
questions to fully utilize the benefits of the statistical form.
It would also be desirable to perform a more complete investigation
of the applicability of Weibull and other distribution functions to ozone
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94
and other pollutants. The question of within year and year to year (or
longer term) variations in air pollutants requires further study.
Another major area needing investigation is the autocorrelation of
air pollutant concentrations and nonstationarity in the underlying stochas-
tic processes. Improved understanding in this area will allow consideration
of a broader range of statistical forms and more accurate treatment or
aerometric data.
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95
3.7 References
1. U.S. Code of Federal Regulations, 40 CFR, Part 51.14, 1975.
2. "Uses, Limitations, and Technical Basis for Procedures for Quanti-
fying Relationships between Photochemical Oxidants and Precursors",
U.S. Environmental Protection Agency, Research Triangle Park, N.C.,
EPA 450/2-77-021 a, November, 1977.
3. "Investigation of Alternative Methods of Expressing the National
Ambient Air Quality Standard for Photochemical Oxidants", Final
Report on Task #4 of U.S. Environmental Protection Agency Contract
No. 68-02-2393, William F. Biller, East Brunswick, N.J., in press.
4. R. I. Larsen, "A Mathematical Model for Relating Air Quality
Measurements to Air Quality Standards", U.S. Environmental Protection
Agency, Research Triangle Park, N.C., Publication No. AP-89, November,
1971.
5. T.C. Curran and N.H. Frank, "Assessing the Validity of the Lognormal
Model When Predicting Maximum Air Pollution Concentrations", Presented
at 68th Annual Meeting of the Air Pollution Control Association,
Boston, June 15-20, 1975.
6. "The Validity of the Weibull Distribution as a Model for the Analysis
of Ambient Ozone Data", Report, U.S. Environmental Protection Agency,
PEDCo Environmental, Inc., Cincinatti, Ohio, in press.
7. J. Horowitz and S. Barakatz, "Statistical Analysis of the Maximum
Concentration of an Air Pollutant: Effects of Autocorrelation and
Nonstationarity", submitted for publication to Atmospheric Environment.
8. J. V. Uspensky, Introduction to Mathematical Probability. McGraw-Hill,
1937, p.46.
9. Private Communication, T. Johnson, C. Nelson, PEDCo Environmental, Inc.
Durham, N. C.
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96
4.0 ASSESSMENT OF RISK ASSOCIATED WITH POSSIBLE CHOICES OF A PRIMARY
NATIONAL AMBIENT AIR STANDARD
4.1 Background
If National Ambient Air Standards are expressed in statistical forms it
is now possible as discussed in the proceeding sections to associate standard
levels in a meaningful way with the general levels of air quality which will
meet the standard. The way is then paved for beginning to assess the risks
of adverse health effects to the most susceptible members of the population
associated with alternative levels of the standard.
A quantitative assessment of these risks can be of major value to the
EPA standard setting process. This section present a mathematical formalism
for risk assessment. It is based on underlying concepts developed by Mr.
Thomas Feagans of the Office of Air Quality Planning and Standards. This
report will deal only with the mathematical treatment which was the contractor's
contribution. A complete discussion of the method may be found elsewhere1- .
4.2 Underlying Principles of Risk Assessment Method
A broad perscription for setting primary National Ambient Air Quality
Standards is given by the following passage from the Clean Air Act.
National primary ambient air quality standards... shall be ambient
air quality standards, the attainment and maintenance of which in
the judgment of the Administrator, based on [air quality] criteria
and allowing an adequate margin of safety, are requisite to protect
the public health'-1-'.
Support material to the Act clarifies this perscription somewhat:
An ambient air quality standard therefore, should be the maximum
permissible ambient air level of an air pollution agent or class
of such agents (related to a perido of time) which will protect
the health of any group of the population
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97
A fundamental concept implicit in the above passages is the concept of
a health effects threshold; i.e., a concentration C_ such that if the con-
centration of pollutant, averaged over a suitable time period, is less than
CT, the pollutant will neither cause nor contribute to effects which are
detrimental to human health - even in people who are most susceptible to the
health effects at higher concentration levels.
The provisions in the Act for an adequate margin of safety implicitly
acknowledges that the concentration of the health effects threshold will be
uncertain. That we do not have precise knowledge on the location of health
effect thresholds is amply demonstrated in the EPA criteria documents which
discuss in detail the health effects research and which are intended to help
decide the pollutant concentrations at which health effects occur. In the
case of photochemical oxidants, for example, numerous toxicological, clinical,
and epidermiological studies are cited which show effects at various levels
but do not directly allow a determination of the lowest concentration level of
oxidants in ambient air at which the most susceptible portion of the population
would suffer adverse health effects. The data from different studies are
often in conflict and in cases of effects shown in animals it is not certain
the effects observed will be caused to occur in man exposed to oxidants in
ambient air.
Fortunately, through the use of techniques developed in the field of
decision analysis, the criteria document can be used to quantify the uncer-
tainty in the location of health effect thresholds. Roughly the procedure
would be to have highly qualified medical people study the criteria document
in detail. After they have absorbed all the pertinent health data, the
technique of probability encoding would be used to quantify their subjective
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98
judgments by eliciting a probability distribution for the location of individ-
ual health effects. That is, they would define functions:
DT = Pr (CT < C) (1)
Ti Ti ~
Where the quantity D is the probability that the health effects thres-
TI
hold of an effect i is less than or equal to the concentration C. The
method of probability encoding by which the probability distribution repre-
sented by Eq. (4-1) is determined is discussed in detail in a comprehensive
f3l
review article J and its application to photochemical oxidants is discussed
T41
in the report referred to above . The following discussion will take
Eq. (4-1) as its starting point and not discuss how it is obtained nor how
the concept of health effects threshold is given precise meaning so that it
may be successfully encoded. Nor will it be concerned with the concept of
subjective probability and its utility in decision making processes. These
matters are sufficiently treated in the references cited above.
There is also a second source of uncertainty that must be accounted
for in setting an adequate margin of safety. This is the uncertainty in
the peak levels that will be reached over a time period of one or more years
when the standard is met. As discussed in Section 3, setting National Ambient
Air Quality Standards cannot place an absolute lid on the highest pollutant
concentrations which may be observed in a given time period. This follows
from the stochastic behavior of pollutant concentrations, and,as also discussed
in Section 3, leads to the conclusion that standards should be expressed in
a way that acknowledges this fact. That is, they should be expressed in suit-
able statistical forms. Since it is not known what the highest concentration
values will be in any given period, the margin of safety again must be adequate
-------
99
to give assurance that these highest concentrations will not be likely to
exceed a health effects threshold for the pollutant. The uncertainty in
the peak concentration levels can be expressed as:
PC = Pr (CMax 1
where the quantity Pp is the probability that no concentration will exceed
the concentration level C in a given time period (e.g., 5 years). The concen-
tration CM, is the maximum time averaged concentration observed during the
naX
given time period.
It should be noted here that the concentration C appearing in Eqs.
(4-1) and (4-2) is a time averaged concentration. The averaging time is set
by health effect considerations. If short-term peaks cause the onset of
health effects,then averaging times of one hour are used. If exposure to
a concentration level over a more extended period of time is necessary, the
averaging time could be several hours, one or more days or even a year. For
photochemical oxidants the averaging time is, as mentioned earlier, one hour.
To assure that there is an adequate margin of safety is the same as
setting the primary standard so that the risk is adequtely small that the most
susceptible members of the population will be exposed to concentration levels
exceeding a health effect threshold in some given time period. If the expres-
sions (4-1) and (4-2) are known either as mathematical expressions or in tabular
form this risk can be calculated for each alternative standard setting. The
essential judgment to be made by EPA decision makers is, therefore, deciding
on the level of risk that will be tolerated, taking into account the seriousness
of the health effects involved. Choosing the acceptable risk level will then
-------
TOO
lead to choosing the corresponding alternative standard, and this standard
will, by definition, have an adequate margin of safety.
A decision also needs to be made regarding the time period. This
subject is discussed more completely elsewhere*- •". In brief, it appears
reasonable that the time period minimally be keyed into the five year
standard review cycle.
4.3 Mathematical Description of the Method
In the proceeding section it was pointed out that the risk of exceeding
a true health effects threshold in a given period of time is determined by
the uncertainty in the location of the health effects threshold (or in the
case of multiple thresholds, the uncertainty in the location of the lowest
threshold) and the uncertainty in the maximum oxidant concentration over the
given time period. It can be shown, by application of the theory of proba-
bility, that the following equation gives the relationship between the
risks and the above uncertainties:
= 1 - Pc(C)pT(C)dC (4-3)
where:
R = Probability (Risk) that a true health effect threshold,
or the lowest of a multiple number of thresholds, is ex-
ceeded one or more times in a given time period (e.g. one
year, five years, etc.).
PC(C) = Probability that the highest observed time averaged (e.g.
one hour, two hour, etc.) pollutant concentration does not
exceed the concentration C in the given time period.
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101
P,(C) = The probability density function for the health effect
threshold or in the case of multiple health effects the
function for the lowest effect (the composite density
function).
The derivation of Equation (4-3) is given in Appendix B.
Equation (4-3) also holds for a more general case in which R is defined
as the risk of m or more exceedances of a threshold in a given time period,
where m may have the integral values 1, 2, 3, etc. In this case PC is
redefined as the probability that the mth highest time averaged pollutant
concentration does not exceed the concentration C in the given time period.
In other words, PC is the cumulative distribution of the mth highest time
averaged pollutant concentration in the time period of interest. Specifying
the national ambient air quality standard for a pollutant limits the range
of PC(C) functions which will satisfy the air quality requirements of the
standard. If this limitation can be expressed quantitatively, and the
health effect threshold density functions have been determined, then for
any given specification of the standard, a range of risks associated with
that specification can be calculated from Eq. (4-3).
In practice, it is only convenient to use Eq. (4-3) directly to
calculate risk when a single health effect with an existence probability
of one is involved. When the risk that lowest of n health effects will
be exceeded is to be calculated and when there is uncertainty as to whether
one or more of the effects actually occur in the sensitive population at
any attainable pollutant concentration, the following expanded version of
Eq. (4-3) is used. (See Appendix B):
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102
R • 1 - J Pc(C)p° dC - (1 - ei)(l - e2)----(l - en) (4-4)
where:
p = The probability density function for the location of the
lowest of n thresholds over the possible range of concen-
trations of the pollutant.
e.j = The probability that the ith health effect actually occurs
in the most sensitive population in the possible range of
concentrations of the pollutant.
The function p" is calculated from:
<«>
where:
p? = The probability density function for the threshold of the
ith health effect assuming its e. = 1. (That is,
CO
r o
J pi >'
In practice p! is obtained by asking the subject health expert in an
encoding session to first give his best judgment of the value of e-j and
then encoding him as to the location of the threshold assuming that the
effect actually occurs in the sensitive population at an attainable pollu-
tant concentration. The encoding procedure for the location of health
effects threshold gives the cumulative distributions for the health effects
-------
103
thresholds. These functions are differentiated by numerical methods to
obtain the p^.
The terms Q. and Q in Eq.. (4-5) are calculated from:
C
Q. = 1 - e. J pt dC (4-6)
Q - Q,(J& •- Qn
The Q^ are seen to be functions of the cumulative distribution of the
o
probability density functions p^.
The approach used above in treating uncertainty about the existence
of a causal relationship between the pollutant and a given health effect
is to define a density function PT assuming the effect exists and a proba-
bility e that the effect is caused by the pollutant to occur in humans.
The area under the probability density functions p° is one. The approach
multiplies this function by e, thus reducing the area to e. To bring the
total area to one the remaining area (1-e) is assumed to be accounted for
in a concentration range very much above the concentration range which
effectively contains p,-. This situation is illustrated in Fig. 4-1. The
distribution on the left is ep . The area under this function is, therefore,
e. The function to the right can have any shape. However, the area under
the function must be (1-e). A more detailed treatment of the approach is
given in Appendix B.
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104
Figure 4-1 Hypothetical Threshold Probability Density Function
for a Health Effect with a 20% Chance of Being Real
0.15
C a 100,000 ppm
Concentrations, ppm
-------
105
When more than one independent health effect is involved, p,. is the
distribution of the threshold for the lowest effect. This distribution is
a composite of the distributions of the individual thresholds and is formed
from them in accordance with Eq. (4-5). Figure 4-2(a) shows two threshold
distributions for two hypothetical, independent health effects. Figure
4-2(b) shows three different composite distributions formed by assigning
different probabilities of existence to the two health effects. A further
discussion of compositing and the effect of assigning existence probabili-
ties is given in Appendix C.
Thus, given the Pr(C) functions corresponding to different levels
w
of the standard, the probability density functions p! for n independent
health effects and their corresponding e. > Eqs. (4-4) through (4-7) can
be used to calculate the range of risks associated with alternate
specifications of the ambient air quality standard. The risks can be
calculated for the individual health effects and for composites of two
or more of the effects in any combination. Calculating risks for individ-
ual health effects and various combinations can be of value where the
effects differ significantly in their seriousness.
The calculations involved in Eqs. (4-4) through (4-7) can be most
conveniently carried out with a computer. The function p° can also
be calcualted by differentiating its cumulative distribution function.
The cumulative distribution is a function of the existence probabilities
e- and the cumulative distribution functions of the p?. (See Appendix B.)
Less computational labor is involved with this method if there is no
o
interest in knowing the density functions p .
i
-------
106
Figure 4-2 Variation in Composite Health Effect Threshold Probability
Density Functions of Two Independent Density Functions as Different
Probabilities of Existence are Assigned to the Lower Threshold Function
= 0.20 ppm
a. = 0.04 ppm
0.2 0.3 0.4
Concentration, ppm
(a) Individual Health Effect Threshold Probability Density Functions
0.4
0.5
0.2 0.3
Concentration, ppm
(b) Composite Probability Density Functions for Various Probabilities
A
of Existence are Assigned to Function with T = 0.2 ppm
-------
107
4.4 Obtaining the PC(C) Distributions
As indicated in earlier sections the Pp(C) function is the cumulative
distribution function for highest time averaged pollutant concentration
for a specified period of time. If the risk is calculated for m or more
exceedances, it is the cumulative distribution for the mth highest concen-
tration. To simplify the following discussion it will be assumed that the
concentration averaging time is one hour.
The Pr(C) function for a pollutant in the air over a given region is
u
a measure of the air quality for that region with respect to the pollutant.
Specifying a National Ambient Air Quality Standard places a limitation on
the range of PC(C) functions corresponding to air quality just meeting the
standard. For example, if the ambient air quality standard specified an
expected (average) value of maximum hourly average concentration for a one
year period was not to exceed a given level, this would immediately locate
the mean value of the distribution of maximum values and thus define the
concentration region in which the preponderance of maximum values must
occur. However, depending upon the area and the control methods used to
meet the standard, the distribution of maximum values about the mean could
be relatively narrow or spread out. It is expected, however, that there
would be practical limitations on the degree of spread of the distribution.
Therefore, specifying the expected maximum concentration limits the P_(C)
0
distributions just satisfying the standard.
In applying the risk assessment method it is necessary to determine
the range of PC(C) functions just meeting each alternate specification of
the standard. While, in principle, this should be possible for almost any
-------
108
type of standard it is more readily done for standards with statistical
forms than for standards with deterministic forms.
The PC distribution function is related to the distribution of hourly
average concentrations. However, as discussed in Section 3 the presence
of correlation between hourly average concentration observed in different
hours and dependence of concentrations on time of day or period of the
year can strongly affect this relationship. Air pollutants commonly show
this correlation and time dependence and these effects must, therefore,
be taken into account in developing suitable P~ functions. Again, as
discussed in Section 3, taking these effects into account may make use of
the case in which independence of hours and no time dependence of hours
are assumed. This case will be discussed first.
If no correlation or time dependence of hours exists, Eq. (3-22)
applies, namely:
Pc - (1 - G(C)) (4-8)
where P- is the distribution of the highest concentration for n hours.
(n= 8760 hrs. for one year or 43,800 hrs. for 5 years.) The function G(C)
is defined by Eq. (3-4):
fi(C) - Pr [CQbs > C] (4-9)
That is, G(C) is the probability that an observed hourly average concentra-
tion is greater than C.
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109
If the broader definition of Pr is used, Eq. (3-26) applies, where
\*
G(C) is substituted for p:
v=0
where p is the distribution of the mth highest hourly average concentra-
\s
tion for n hours.
Thus, if the distribution function G(C) is known, the desired P-
function can be obtained from application of Eq. (4-8) or (4-10). Studies
have found that the distributions of short-term time averaged concentra-
tions of air pollutants can usually be represented by lognormal, Wei bull,
T51
or gamma distribution functions . Of these the Weibull function provides
a good fit to photochemical oxidant air monitoring data^- •* and is convenient
to use since its G(C) function can be stated explicitly as in Eq. (3-6).
G(C) = e" (C/6) (4-11)
The parameter 6 is referred to as the scale factor. It is the concentra-
tion corresponding to G(C) - 0.368. It establishes the approximate position
of the mid-concentration values of the distribution. The parameter k is
called the shape factor. It tends to be a measure of the spread of the
distribution. The larger k the more compact the distribution. If the
-------
values of k and 6 have been determined for a given geographic area the
corresponding PC functions can then be obtained by use of Eqs. (4-8) and
(4-11) or Eqs. (4-10) and (4-11) at the given level of air quality.
For the risk assessment it is necessary to connect alternative
levels of the ambient air quality standard with the corresponding PC
function. This is easily done through the Weibull distribution, Eq. (4-11).
The proposed form for the ozone standard is:
CSTD ppm hourly average concentration with an expected
number of exceedances per year less than or equal to E.
It is shown by Eq. (3-12) that for any region to which the Weibull function
applies and just meets the standard:
6(C) - e - E ST (4-12)
where:
C<-Tn - level of ambient air standard.
E = expected number of exceedances in n_ hours.
The term n£ is customarily the number of hours in one year or 8760 hours
The expected exceedance rate would normally be one for an air standard.
In this case Eq. (4-12) becomes:
8(C) - e-9'078
-------
Ill
It should be pointed out that from the point of view of the risk assessment
method developed in this report, the designation of the expected number of
exceedances is a relatively arbitrary matter and could be set at any value
that gives a convenient level for C-TD so long as the risk is the same
(m)
(the value of m in P~ of Eq. (4-10) has a more direct impact on health
since it directly bears on the number of exceedances of a true threshold).
For example, if it is decided that it is undesirable to have any exceedances
of a true threshold over a given time period then the Pr used in the calcu-
(1) C
lation is PC (see Eq. (4-10)) and once an acceptable level of risk is
chosen any combination of E and C-TD values which yields this risk value
in a given area give the same level of protection.
The relationship between the G(C) and P~ distribution for a Weibull
distribution of hourly averages is shown in Fig. 4-3. The combination C-,-
and E determine the general location of P~ while k determines its spread.
Figure 4-4 shows the P functions for a one year period for a series of
u
alternate levels of C with E = 1 and for the Wei bull shape factor k = 1.
Figure 4-5 shows the effect of changing the shape factor at CSTQ = 0.1 ppm
and E = 1. Figure 4-6 shows the effect of changing k for a PC covering a
5 year period. It is seen form the three figures that changing CSTD
displaces the PC function over a wide range while having a relatively small
effect on its shape. Changing k causes little actual displacement of the
PC for a one year period but has a large effect on its shape. The changing
k causes the P- function in Fig. 4-5 to pivot about the point (0.10, 0.368).
The effect of k on the PC for a five year period (Fig. 4-6) is still largely
in the shape of the function, but there also seems to be more displacement.
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112
Figure 4-3 Hypothetical Distribution of Hourly Average Concentration and the
Corresponding Distribution of the Annual Maximum Hourly Average Concentration
•2 -4 .6 .8 1.0 1.2 1.4 1.6
Concentration, C/C<-Tn
(a) Distribution of Hourly Average Concentration
0 -2 .4 ,6 .8 1.0 1.2 1.4 1.6
Concentration, (C/C$TD)
(b) Distribution of Annual Maximum Hourly Average Concentration
-------
113
O
O_
dJ
O
II
O
II
UD
O
•
O
II
O
CO
O
01
3
cr
Q-
o.
O
c
O
+J
(O
Ol
u
CO
C3
UD
c
CM
O
O
a.
-------
114
s_
o
u
OJ
O
(/I
VI
QJ
c
o>
OJ
M-
5-
(D
Q.
O)
O)
c
o
(J
c
3
in
i
o>
3
CD
LO
O
CC
•
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(U
o
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o
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«
c
o
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00
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00
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o
-------
115
Figure 4-6 P~ Function for 5 Year Period for Different
Values of Wei bull Scale Factor, k
1.01-
0.8
0.6
£
D-
0.4
0.2
0.09
k = 2.0
Concentration, ppm
-------
116
This results from the fact that the pivot point for the distribution is
now very close to the concentration axis (0.10, 0.007). In general,
calculated risk values will be more sensitive to changing values of C^jD
than to changes in k over the usual ranges of these parameters.
From the proceeding discussion it is seen that the assumption of
independence of hours and the use of the Wei bull function to represent
the distribution of hourly average concentrations readily yield Pr and
(m) L
PC functions. The Weibull can be used with little loss in accuracy
even where other distributions such as the lognormal distribution provides
a better fit to the concentration data. The primary concern in estimating
the appropriate P function is to have it placed properly along the con-
v*
concentration axis and have the correct degree of spread. The parameters
6 and k in the Weibull function provide wide flexibility in this regard.
As shown above, the standard level essentially places the P- function.
The appropriate values of k can be obtained by fitting a Weibull distri-
bution to hourly average concentration data obtained from air monitoring
sites. The range of applicable k values for standard setting purposes
should be determined by examining aerometric data in areas that are close
to the concentration ranges of the alternate levels of the standard under
consideration. Where this is not possible, the k values can be determined
at existing levels of air quality and the results extrapolated to potential
standard levels.
While using a Weibull distribution where a loqnormal or gamma distri-
bution might be more appropriate does not appear to lead to serious errors
in the risk estimates, ignoring the possible dependence of hourly average
-------
117
concentrations can lead to significant error. As discussed in Section 3,
internal EPA study*- -* showed that dependence of one hourly concentration
on the value of another did not lead to serious errors if independence of
hours was assumed. It also showed that dependence of concentrations
on time of day or year can lead to PC functions which were placed lower on
the concentration axis than would be obtained assuming no time dependence.
When this time dependence can be modeled it should be possible to generate
the corresponding PC tables of functions and express them mathematically.
This was done in the EPA study for the daily maximum hourly average ozone
concentration.
Another approach can be taken as discussed in Section 3, if the time
dependence is such that the maximum concentration tends to occur only
within some determinable period and the probability distribution of con-
centrations is approximately the same for all hours within the time period.
In this case all hours outside the time period can be excluded and inde-
pendence assumed for the hours within the period. A Weibull distribution
then could be fit to the hours within the time period to determine the
appropriate k values and Eqs. (4-8) and (4-12) used to calculate PC . To
the extent that the period under consideration is also likely to contain
the mth highest hourly average concentration, P_ could be obtained
w
using Eq. (4-10). The term n£ in Eq. (4-12) would be set equal to the
hours in one calendar year of the time period. The term n in Eqs. (4-8)
and (4-12) would be n- times the number of years for which the risk is to
be estimated. In the following section, this procedure is applied to
ozone.
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118
4.5 Determination of PC Functions for Ozone
As suggested in the proceeding sections estimating the appropriate
P- function involves: 1. determination of mathematical function or functions
which best describe the distribution of time averaged ambient concentrations
of the pollutant; 2. given the distribution of time averaged concentration,
the standard, and the nature of the correlation between concentrations in
neighboring time periods and dependence upon time of day and year, derive
sutiable P- functions and 3. estimate the range of values of parameters
appearing in the P~ function.
A study performed under contract for EPA involving 14 sites scattered
around the United States and involving 22 site years of data showed that
the Weibull distribution provides an excellent fit to hourly ozone concen-
trations . In only two cases did a lognormal distribution give a
superior fit.
Ozone hourly average concentrations exhibit strong correlation and
strong dependence on time of day and time of year. The day of the week
also has some effect. As indicated in Section 4-3, the correlation
between neighboring concentrations does not appear to have an important
effect on the PC function; however, the time dependence does.
The method of dealing with the time dependence, as discussed, was to
find a time period in which the time dependence was relatively constant
and which was highly likely to contain the maximum hourly average concen-
tration for the time period. If such a time period existed and a distri-
bution could be fit to the time-averaged concentrations within this period,
then the appropriate PC function could be derived by assuming complete
-------
119
independence of the time averaged concentrations. For ozone this period
is during the midday hours of the warm months of the year for urban areas.
In a continuation of the above mention study*- -" it was found that
Weibull distributions could be fit to data obtained between 11 AM and 6 PM
both from May through September and July through August. It was further
shown the maximum ozone concentration had significantly more chance of
occurring during the longer of the two periods. Therefore, this time period,
which contains 1071 hours, was used in the derivation of the P/. function.
The form of the National Ambient Air Quality Standard proposed for
ozone is: C<.Tn ppm hourly average concentration with an expected number of
exceedances per year less than or equal to one. For a region whose air
quality just meets this standard the Weibull distribution of hourly averages
for the hours 11 AM to 6 PM, May through September, would be according to
Eq. (4-12):
G(c)=e-(lnl071)(C/CSTD)k (4.u)
And the Pr function for a period of n, years would be from Eq. (4-8)
*" y
"" ny
By substituting alternate levels, C-T_, of the standard into Eq. (4-15)
the Pr function for n years needed for the risk assessment can be obtained.
** y
However, before this can be done it is necessary to estimate the range of
values of the parameter k (Weibull shape factor) over different regions in
-------
120
the U.S. Table 3-11 shows measured k values for the 1071 hour time period
as well as other time periods. The range for the values based on 1071
hours is 1.31 to 2.04.
For the risk assessment best estimates were made of the lower bound and
upper bound values of k. This was done by probability encoding the contractor
and a PEDCo researcher involved in the development of the Weibull distribu-
tions and the PC functions. The information base was the data in Table 3-11,
plus the data developed during the Weibull distribution studies. The data
shown in Tabel 3-11 are, by and large, for geographic regions above the range
of alternative ozone standards considered. The Weibull studies suggest that
the k factors at the standard levels would be somewhat higher than those
shown in Table 3-11. This factor was taken into account in the encoding.
The contractor first encoded Mr. T. Johnson of PEDCo and then himself. The
results of the encoding are shown in Tables 4-1 and 4-2. In encoding Mr.
Johnson the upper and lower bounds of the variable were first encoded
[3]
followed by the use of the interval method . The subject was then allowed
to make slight adjustments based on viewing the plotted probability function.
When the contractor was his own subject the same approach was used but the
final adjustments were made directly to the plotted curve. In this way a
more completely defined curve was obtained.
The distributions were combined by Mr. T. Feagans of EPA into a single
distribution for the lower bound and a single distribution for the upper
bound (Table 4-3). The median values for the location of the lower and
upper bound shape factors were 1.36 and 2.54 respectively. Since the
range of k values varies somewhat with the standard level, the range,
strictly speaking, should be estimated for each alternative level of the
standard. In the case of ozone the difference is not likely to be large
enough to seriously affect the risk estimates.
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121
Table 4-1 Subjective Probability Distributions of Lower and Upper Bounds
of Weibull Shape Factor k For n at Approximately 1000 Hours
and Maximum Concentration at Approximately 0.1 ppm
Subject: Ted Johnson
PEDCo
Lower Bound k
Cum.
k Probability
1.30 0.001
1.38 0.25
1.43 0.50
1.48 0.75
1.50 0.999
Interviewer
Date:
Upper
k
2.30
2.37
2.50
2.70
3.00
: W.F. Biller
1/26/78
Bound k
Cum.
Probability
0.001
0.25
0.50
0.75
0.999
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122
Table 4-2 Subjective Probability Distributions of Lower and Upper Bounds
of Weibull Shape Factor k For n at Approximately 1000 hours
and Maximum Concentration at Approximately 0.1 ppm
Subject: W.F. Biller Interviewer: W.F. Biller
Date: 1/31/78
Lower
k
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
Bound k
Cum.
Probability
0.01
0.02
0.05
0.10
0.16
0.27
0.60
0.86
0.95
0.99
Upper
k
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
Bound k
Cum.
Probability
0.01
0.09
0.30
0.60
0.75
0.83
0.88
0.91
0.94
0.96
0.97
0.98
0.99
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123
Table 4-3 Subjective Probability Distributions for Upper and Lower
Bound Weibull Shape Factors (k) for Distributions of Hourly
Average Ozone Concentrations
Probability
that k is
below specified
value
0.10
0.30
0.50
0.60
0,90
Lower
Bound
k
1.15
1.30
1.36
1.41
1.47
Upper
Bound
k
2.41
2.50
2.54
2.66
2.98
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124
4.6 Example Application of Method
Suppose that three health effects have been identified for a given
pollutant and that there is some uncertainty as to the precise location
of each of the effects. Suppose further that the evidence for one of
the effects is very weak at the present time and there is considerable
doubt the effect exists. However, if it does exist, it is a serious
effect and,therefore,it should have some bearing on the setting of the
standard. Suppose there is also some doubt about the existence of one
of the other effects, but in this case the effect is not considered to
be too serious. However, it also should be taken into account. It will
be assumed there is no question about the existence of the third effect
which is considered to be serious. A thorough review of the information
available on each health effect by a group of medical experts establishes
that the uncertainties in the locations of the effects are best described
by normal distributions. The findings of the group are summarized as
follows:
Health
Effect
1
2
3
Most Probable
Location of
Threshold
(ppm)
0.15
0.20
0.30
Standard
Deviation of
Density Function
(ppm)
0.05
0.04
0.08
Probability
Effect
is Real
15
80
100
Seriousness
of
Effect
Serious
Mild
Serious
In a real situation much more information would be supplied concerning the
seriousness of the effect. It also would not be possible to describe the
probability density function for the location of the threshold so easily.
A tabulation of the function versus concentration for each effect would
have to be developed.
-------
125
Simple inspection of the table does not provide a clear indication
as to where the level of the national ambient air standard should be
set to provide an adequate margin of safety against all three effects
simultaneously. However, application of the procedure outlined in the
previous section gives an assessment of the medical risks involved at
each potential level of the standard and can be used, therefore, in
judging where the standard can be set to give an adequate margin of safety.
The margin of safety requirement means that an exceedance of the
standard concentration will not, if the margin is adequate, generally
result in an exceedance of a true health effect threshold. To apply
the approach it is first necessary to adopt a form of the standard that
provides protection appropriate to the medical effects under consideration.
Suppose the effects are such that any exposure for a short period (such
as one hour) over the thresholds are to be avoided. In this case one
appropriate form would be:
C^ pom hourly average with expected exceedances per year
less than or equal to one.
This form is close to that of the current oxidant standard but differs from
that form in that its statistical nature allows it to be directly related
to yearly distributions of hourly average concentrations.
It is assumed that a Wei bull distribution with k = 1 fits the
ambient concentrations and there is no time dependence (i.e., n,- = 8760 hrs.)
Also assume the risk is to be calculated for a one year period. It follows
from the discussion in Section 4, 3 that the P~ function is:
. (,.e-0n 87«0)(C/CSTD),
87M
-------
126
Equation (4-16) plus the information supplied in the above table on the
health effects thresholds can then be used to calculate for each level
of the standard: 1. The risk, R, that no hourly average concentration
will exceed any of the true health effect thresholds in a given year
and 2. The individual risks, R^, that no hourly average concentration
will exceed the i-th true threshold in a given year. For the three
thresholds the risk R. . that no hourly average concentration exceeds
• »J
either the i-th or j-th true threshold can also be calculated. The
calculation of these risks for each standard level is achieved by:
1. Developing the appropriate Pr function for each level at
w
which the standard may be set, in this case Eq.(4-]6).
2. Forming the composite distribution function p° (C) for
all three thresholds from Eq.(4-5).
3. Determining R by numerically integrating the product of
P (C) and p° (C) as indicated by Eq.(4-4).
4. Determining each R. by carrying out the process in Eq.(4-4)
for Pr(C) and pT (normal distributions in this example).
t i.j
5. Repeating ^3 and *4 for any composite of two threshold
functions.
When this procedure is carried out for the above three health effects the
tabulation shown in Table 4-3 is obtained (Details of the calculations are
given in Appendix D).
It is seen that if the standard is set at 0.10 ppm, the margin of
safety is such that there is a 0.05 risk (or one chance in 20) that a
health effect threshold will be exceeded in a given year. At the same
-------
127
Table 4-4 Risks of an Hourly Average Concentration Exceeding True
Health Effect Thresholds in a Given Year for Different
Values of a National Ambient Air Quality Standard
Standard
(ppm)
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
RMT
0.009
0.013
0.022
0.033
0.050
0.072
0.102
0.141
0.190
0.248
0.315
0.383
0.457
R1
0.007
0.010
0.015
0.022
0.030
0.040
0.050
0.062
0.073
0.085
0.096
0.106
0.115
R2
0.001
0.001
0.002
0.005
0.011
0.022
0.040
0.066
0.101
0.147
0.202
0.265
0.334
R3
0.002
0.002
0.004
0.006
0.008
0.012
0.017
0.025
0.034
0.045
0.060
0.077
0.098
Rl,3
0.008
0.013
0.019
0.028
0.038
0.052
0.067
0.085
0.104
0.126
0.149
0.175
0.202
-------
128
time the risks of exceeding the individual thresholds are 0.03 (1 in 33),
0.011 (1 in 91), and 0.008 (1 in 125). Since the first and third effects
are serious, this level of protection might be considered as the minimal
acceptable value by the EPA decision makers. If so, the standrard would
be set at 0.10 ppm. Considering, however, that the second effect is mild,
it is of interest to determine the probability of not exceeding either the
first or third threshold. This probability is shown in Column 6 of the
table. If the .05% level is still considered acceptable the standard could
be set at 0.11. This level would still give a 96.0% probability of not
exceeding the mild health effect.
The above example is greatly simplified since it is primarily intended
to show how the equations developed in the previous section are used. In
actual application of the method other considerations would enter. Further-
more, considerable power can be added to the method through a sensitivity
analysis of the uncertainties involved in the determination of the PC and
T41
P functions. These added considerations are treated elsewhere1- J.
-------
129
4.7 Conclusions and Recommendations
It is concluded that the method of risk assessment presented in this
section is a practical method for developing national ambient air quality
standards which are in accord with the intent and spirit of the Clean Air
Act as presently written. The method leads naturally to the use of standards
expressed in statistical forms. It gives a proper perspective to viewing
exceedances of a standard level. The real concern is to protect against
pollutant concentrations exceeding health effects thresholds and to set
the standard at a level which reduces the risk of such exceedances to an
acceptable level. Exceedances of the standard level do not necessarily
mean a health effects threshold has been exceeded and the risk that they do
can be set as low as the EPA Administrator deems necessary to protect the
public health.
The risk assessment method also is a logical starting point for:
1. estimating the expected number of people in a population who will suffer
health effects at given air quality levels and the uncertainties in such
estimates and 2. Cost-benefit analyses.
In those aspects of the method to which the contractor contributed there
is a need to: determine P£ functions accurately, determine the extent to which more
accurate functions can be developed,and to obtain more complete information
on the range of P- functions experienced in different parts of the United
States. In investigating the question of improved P- functions it will be
necessary to further study the interdependence of time averaged concentra-
tions and the nonstationarity of the underlying stochastic processes which
are responsible for observed pollutant levels and to determine how these
factors affect the PC functions.
-------
130
There is also a need to better define the behavior of aerometric
data and, therefore,the P functions at the standard level.
V
It is recommended that the above mentioned problem areas be further
investigated. In addition it would be worthwhile to extend the basic
method to estimating expected numbers of people affected at air pollution
levels above the health effects thresholds and the attendant uncertainties
and risks.
-------
131
4.8 References
1. Clean Air Act, Section 109, 42 U.S.C.
2. "A Legislative History of the Clean Air Amendments of 1970", U.S.
Senate Committee on Public Works, 1970.
3. C.S. Spetzler and C.A.S. Stael von Holstein, "Probability Encoding
in Decision Analysis", Management Science, Vol.22, No. 3, November 1975.
4. "A Method for Assessing Health Risks Associated With Alternative Air
Quality Standards for Photochemical Oxidants", Report by U.S. Environ-
mental Protection Agency, Office of Air Quality Planning and Standards,
in press.
5. T.C. Curran and N.H. Frank, "Assessing the Validity of the Lognormal
Model When Predicting Maximum Air Pollution Concentrations", Presented
at 68th Annual Meeting of the Air Pollution Control Association,
Boston, June 15-20, 1975.
6. "The Validity of the Weibull Distribution as a Model for the Analysis
of Ambient Ozone Data". Draft Report to EPA by PEDCO Environmental,
November 17, 1977.
7. Private Communication, J. Horowitz, Office of Air and Waste Management,
U.S. Environmental Protection Agency, Washington, D.C.
-------
APPENDIX A
SUPPORT MATERIAL FOR SECTION 3
1. Derivation of Approximate Relationship (3-18)
Starting with Equation (3-16):
n . n! ,E »m ,, E ,n-m
The following approximately holds for n » m:
(n-m!)
« nm (A-2
If define a constant X such that:
X » !|- (A-3)
Eq. (A-l) is approximated by:
mm p n-m
Pm - x E (1 - I-) (A-4)
If E « n the term in brackets can be expanded as a binomial series to
yield:
2! 3!
132
-------
133
The approximation series expression is the expansion of the exponential function.
Therefore:
c n-m _^£
it p
Substituting into the approximation (A-4) yields:
which is Expression (3-18) appearing in Section 3.
2. Demonstration That the Concentration Corresponding to the Most Probable
Value of the mth Highest Concentration in n Hours is the Concentration
With an Expected Exceedance Rate of m.
This relationship is true if the distribution of hourly average con-
centrations is described by a Weibull function with k = 1. To find the
concentration corresponding to the most probable value of the mth highest
concentration in n hours the derivative of Eq. (3-27) is set equal to zero.
To simplify the expression use the 6 notation for the Weibull and set k = 1.
Thus:
- (A-6)
-------
134
If the derivative of Eq. (A-6) is taken with respect to C and all
terms cleared that are not zero at the maximum value, the following expres-
sion is obtained.
* = e" C/6 (A-7)
Equation (A-7) has the same form as Eq. (3-8):
I - .- (A-B,
Therefore:
C° = CE (A-9)
E = m (A-10)
3. Demonstration of Expression (3-37)
Using the following series expansion:
(1 - X)" = 1 - nX + HlHi
2!
-------
135
the right hand side of Eq. (3-35) expands to:
= 1 - nQp for p very small
Similarity the right side of Eq. (3-36) also becomes approximately (1 - n p)
for small p. Since we are concerned with the highest observations, the
value of p is much less than one and the approximation should hold closely.
Note also,this result is independent of the value k.
-------
136
APPENDIX B
DERIVATION OF BASIC EQUATIONS FOR ASSESSING HEALTH RISKS
ASSOCIATED WITH ALTERNATIVE AIR QUALITY STANDARDS
1.0 Risk of Exceeding a True Health Effect Threshold or the Lowest of
Several Thresholds
First determine probability P where:
P = Probability that np_ hourly average concentration exceeds
the true health effect threshold or the lowest of several
thresholds in a given period.
Let:
P-(C) = Probability that no time averaged concentration exceeds
the concentration C in the given period.
Py(C) = Probability density function expressing the uncertainty
in the location of the true threshold or the lowest of
several thresholds.
Note:
fb
j PT(C)dC = Probability that a true threshold or the lowest of
a several thresholds is in the interval a,b.
It follows that:
Pr(C)pT(C)dC = Probability that the true threshold or the lowest of
w 1
several thresholds is contained in the interval dC
and this value is not exceeded by any hourly average
concentration in the given period.
-------
137
By integrating over all values of C the quantity P, defined above, is
determined:
oo
P = Pr(C)PT(C)dC (B-l)
i C T
If R is the probability or risk of exceeding a health effects threshold or
the lowest of several thresholds one or more times in the given period, then:
R = 1 - P
R a 1 - Pc(C)pT(C)dC (B-2)
Note that if the function PC is defined as the probability that no time
averaged concentration exceeds the concentration C it must also be the pro-
bability that the highest time averaged concentration observed in the time
period does not exceed C (therefore, is < C). In other words, Pr is the
~~~ L
cumulative distribution of the highest time averaged concentration occurring
in the given time period.
It is further noted that if it is desired to estimate the risk of
exceeding a health effects threshold or the lowest of several threshold
"m" or more times in a given time period, then P- becomes the probability
that a threshold will not be exceeded more than m-1 times in the given time
period. In this case P~ is the probability distribution of the mth highest
time averaged concentration.
2.0 Determination of Probability Density Function for the Lowest of Several
Health Effect Threshold
Assume n different health effects and that a probability density function
has been obtained for each. If the function Pp(C) is known, Eq. (B-l) can
then be used to evaluate the individual probabilities for each health effect.
-------
138
P. = Pr(C)PT (C)dC (B-3)
' J u ' •{
o '
Let:
P ~ The probability that no hourly average concentration ex-
ceed any of the n true thresholds in the given time period.
To evaluate this probability for all possible configurations it is assumed
that each threshold in turn is the lowest threshold.
Differential elements of the following type can be formed:
*•** ww OQ
Pr(C)PT (C)dC • f PT (x)dx ' f PT (x)dx---- f PT (x)dx
L 'l J T2 J T3 J Tn
C C C
This term is the probability that: 1. no hourly average concentration
exceeds the value C; 2. the threshold T-j is in the interval dC; and
3. all the other thresholds are above this value.
If:
CO
Q.(C) = J PT (x)dx
C
And:
Q(C) = TTQ(C)
1=1 1
-------
139
The above expression simplifies to:
pr 2- p_ dc
L WT 11
Integrating over the concentration range gives:
This integral is the probability that no hourly average concentration exceeds
any of the n thresholds in the time period and T, is the lowest threshold.
In order to enumerate all cases, it is necessary to form the sum of integrals
of this type in which each threshold in turn is assumed to be the lowest.
Therefore:
P = V t P 2- P, dC
or:
oo n
P - f Pr [Q
J • i
0 1=1
or:
P = ! P. PT dC
w I
This equation is, as would be expected, identical to Eq. (B-l) except that
P is now a composite probability density function such that:
p, = Q n pT /Qi (B-4)
1 1-1 Ti
-------
140
Where :
00
Q.(C) = f pT> dC (B-5)
C 1
and
n
Q » n
and:
D = Pr (T, < C)
1
It follows from probability theory that:
n
1-0T = n 0-D- ) (B-7)
1 i=l '1
The composite probability density function is by definition:
dD
p = -1 (B-8)
' dC
Differentiating (B-7) yields:
dDT n n dD
-------
141
Since by definition:
CO
1-D- = f PT dC = Q. (B-10)
TI J T,
Q = n (I-DT )
i Ti
pT = dDT /dC (B-ll)
1 i i
Equation (B-9) is equivalent to Equation (B-4).
Equations (B-7) and (B-8) provide an alternate route to obtaining
the composite density function, p_, which can then be used in Eq. (B-2)
to estimate risk.
3.0 Inclusion of Uncertainty that One or More Health Effects Exist
For some health effects there may be uncertainty that the effect
actually occurs in humans. It would be desirable to include this consid-
eration when considering the uncertainty in the location of the threshold
on the concentration axis. Given uncertainty only in the location of the
threshold it has been shown that Eq. (B-2) gives the risk that the true
threshold T^ is exceeded one or more times in a given period.
If there is uncertainty as to whether the ith effect occurs in humans,
assign the probability e. that the effect does exist. Then choose a value
of C = u such that u is many times larger than any concentration likely to
be encountered. In other words, u is many times beyond the concentration
range of interest. In this case, if pi (C) is defined as the probability
i-j
density function for the location of the ith effect if it does exist.
A new overall probability density function can be written:
PT>(C) = e.p° + (i-e.) 6(C-u) i = 1 , n (B-12)
-------
142
where 5(C-u) is the Dirac delta function. It has the property:
= 1 if a <_ u <_ b
(B-13)
[ 6(C-u)dC
0 if a > u or b < u
In other words, if the interval of integration includes C = u the
value of the integral is unity. If it does not include C = u the integral
is zero. The Dirac delta function itself may be considered as zero for
all values of C except u. At u it has an infinite value. It is also
assumed that pZ (C) is essentially zero in the vicinity of C = u and above.
'i
Equation (B-12) is based on the premise that saying an effect does not
exist is mathematically equivalent to saying that the true threshold is at
a very high concentration which is above any concentration likely to be
encountered. Thus, if the probability e can be assigned to the certainty
that it does exist, the fraction e of the total area under the probability
density curve can be assigned to the concentration range in which the effect
is thought to be located if it does occur. The rest of the area, (1-e), can
be assigned to a range above any concentration likely to occur.
Note also, the use of the Dirac delta function is only a matter of
convenience since it leads to a simple form of Eq. (B-12). Any function
can be used in this outer region as long as it has the value zero in the
region of interest and has the area (1-e). The Dirac function is simply
a convenient form for -including the desired property in the probability
density function.
Substituting Eq. (B-12) into (B-3) gives:
P1 = e. PpdC + (1-e.) P S(C-u)dC
Since Pr will be essentially 1 at C = u, the above equation yields:
v»
GO
P. = e. f PrpT dC + (1-e.)
i i j L \. i
(B-14)
-------
143
To find the probability that no hourly average concentration exceeds any
of n thresholds, it is necessary to derive the appropriate form of the
composite threshold probability density function. Its general form is
given by Eq. (B-4)
Substituting Eq. (B-12) into Eq. (B-4) and rearranging gives:
n e " 1-e-
PT • Q E ITPT. + [Q Eor1^0-"*
The first term on the right side of the Eq. (B-15) is evaluated in the
ambient concentration range of the pollutant well below the value C = u.
In this region the functions Q. have a simple interpretation. Substitut-
ing Eq. (B-12) into Eq. (B-5).
Q. = e. I PT dC + (1-e^ 6(C-u)dC (B-16)
c 1 c
For all C < u
00
Q. = e. j p^dC + (1-e.) (B-17)
ft 1
The second term on the right side of the Eq. (B-15) is evaluated in
the vicinity of C = u, far above the ambient concentration range. In this
region the behavior of the Q. functions needs to be more carefully consid-
ered. The problem is the behavior of the second term on the right side of
Eq. (B-16) as C passes through u.
Note that in the vicinity of C = u the first term on the right side
of Eq. (B-16) is zero by definition.
-------
144
Thus:
oo
Qi » 0-e..) I fi(C-u)dC for C -\, u (B-18)
C
Expanding the second term of Eq. (B-15) gives:
[ Q2Q3*'"Qn(1"el) * QlQ3%"Qn(1"e2) * '" + QlQ2*"Qn-l(1"en) ] 6(C"u)
which on substituting Eq. (B-18) yields:
oo
n fi(C-u) [ f 6(C-u)dC ]"" J U (1-e^
The portion of this term in large brackets can be shown to have the proper-
ties of a Dirac delta function (Eq. (B-13)) as follows:
Let:
00
) = n 6(C-u) [ f 6(C-u)dC ]"" (B-19)
and:
<|>(C) = f 6(C-u)dC (B-20)
then:
Substituting into the bracketed terms:
= - n
dC
-------
145
or
d*
(B-22)
To show that the derivative has the properties of the Dirac delta function,
integrate between the limits a and b.
dn = $(a)n - (b)n
6(C-u)dC ]" - [ f (C-u)dC (B-24)
a
= 0 if a > uorb< u
where b > a.
Therefore, i|>{C-u) is also a Dirac delta function.
We can, therefore, write:
e.
PT = (Q
-------
146
where:
CO
o
= A ' *
Q. = en. p° dC + (1-e.)
and therefore:
or alternatively:
C
PT dC
and
Q » 0^2 Qn (B-27)
Substituting Eq. (B-25) into Eq. (B-l) gives:
00
P = I Pcp°dC + II (1-e.) (B-28)
= 1 - j PcP°dC + II (l-ei) (B-29)
The Eq. (B-25) through (B-29) are the basic working equations for estimating
risk when a multiple number of health effects thresholds are involved, and
where one or more of the health effects may not occur in humans.
-------
147
APPENDIX C
EXAMPLES OF COMPOSITE HEALTH EFFECT THRESHOLD DISTRIBUTIONS FORMED
FROM SIMPLE HYPOTHETICAL DISTRIBUTIONS FOR INDIVIDUAL THRESHOLDS
The following simple examples will help illustrate the relationship
of individual health effect threshold probability density functions and
their composite functions. Assume two health effects with probability
density distributions. The composite density function is:
PT = e!Q2pl + 62Q1P2 + O-e^O-e^ 6 (C-u) (C-l)
(The superscripts on the p. are omitted for convenience.)
Assume: 1. p, and p? are box-like functions; 2. P2 lies beyond p, on
the concentration axis with no overlap; 3. both effects are certain to
exist. Eq. (C-l) then becomes:
°
P = Q2P + Q.^2 (C-2)
or
°° f
p = P1 J p2dC + p2 J
P]dC (C-3)
C
-------
148
This situation is shown in Figure C-l(a). The area under each box is 1.
In the interval [0,a], p, and p« are both 0, so that the composite
distribution PT = 0. In the interval [a,b], p-j = p , Q? = 1, p = 0.
Therefore, pT = p . Similarily it can be shown that above point b,
PT = 0. Thus, the plot of the composite interval (Fig. C-l(b))is identi-
cal to the distribution p, alone.
This result is expected because in this example it is known with
certainty that Threshold #1 is below Threshold #2. Thus, the probability
that the lowest threshold is in a given interval is determined only by
the density function for Threshold #1.
Figure C-l(c) shows the composite probability density function for a
different situation in which the density functions for Threshold #1 and
#2 are both box-like functions and are identical. That is, both have the
same distribution as shown for p, in Fig. C-l(c). Since the density functions
for the individual thresholds are uniform over the interval [a,b], the most
probable point at which the lowest of the two will be encountered is at the
lowest end of the interval [a,b]. If there are several identical, fully
overlapping box-like threshold distributions, the composite distribution
becomes curvilinear (Fig. C-l(d))with more of its area displaced toward the
lower end of the interval. The area under the composite function is, of
course, one in both Figs. C-2(c) and (d).
Figure C-2(a) depicts a situation similar to that shown in Fig. C-l(a)
in all respects except that there is only a 30% certainty that the first
health effect exists. In this case the probability density function for
Threshold #1 is divided between the region in which the threshold would
be located if it does exist and a region of very high concentration above
any concentration that could be encountered in the ambient atmosphere.
-------
149
Figure C-l
Composite Threshold Probability Densities
When Health Effects Are Certain to Exist
Ti
(a)
abcdOab
C+ C
(b)
n = 2
n = 3
0
(O
(d)
-------
150
Figure C-2
Composite Threshold Probability Densities When the Existence of
One Or More Health Effects is Not Certain
Ti
0.3p0
0 a be d7X e f
C * u
(a)
0.7pr
0.3pQ-
c d
C f
(b)
-------
151
For graphical purposes a simple box function is shown in this region
instead of the Dirac delta function. Both yield the same result.
The composite function can be written:
PT = Q2e1p1 + Q1P2 (C-4)
where:
(C-5)
Q2 - P2dC (C-6)
Note the absence of a third term in Eq, (C-4) which would give the value
of PJ in the region of C = u. This term is zero because e? = 1 and,
therefore, (l-e-|)(l-e-) 6(C-u) = 0. The graph of the composite function
(Fig. C-2(b)) is obtained by evaluating (C-4) at each point on the concen-
tration axis.
In the interval [0,a] both p-j and p_ are zero and, therefore, p- - 0.
In the interval [a,b] p, = p , p? = 0. Thus, only the first term on
the right side of the equation contributes. Since in this interval (L = 1
and e] » 0.3, PMJ « 0.3 pQ.
In the interval [b,c] again both p- and p are zero and, therefore,
PT = 0.
-------
152
In the interval [c,d] the p-j term in Eq. (C-4) is zero, but the second
term, Q^p2 has the value (1-0.3 x 1) x pQ = 0.7 pQ. Therefore, PT = 0.7 pQ.
In the interval above point d, p, * 0, p~ = 0 and the
-------
153
APPENDIX D
DETAILS OF CALCULATIONS PERFORMED IN SECTION 4.5
EXAMPLE APPLICATION OF METHOD
The table of risks in Section 4.5 was obtained as follows:
D.I Develop the Appropriate P- Function for Each Alternate Standard Level
Rearranging Eq. (4-16) of Section 4, the constant in the exponential
term can be calculated for each level of the standard:
-In R7fin 876°
Pf(C) = [1 - exp ( 7 876° C)l (D-l)
u LSTD
Thus:
CSTD
0.06
0.07
0.08
0.09
0.10
0.11
0.12
In 8760/CSTD
151.30
129.69
113.47
100.87
90.78
82.53
75.65
CSTD
0.13
0.14
0.15
0.16
0.17
0.18
In 8760/C$TD
69.83
64.84
60.52
56.74
53.40
50.433
Substituting a constant from this table into Eq. (D-l) above gives the
P~(C) function corresponding to each level of the standard. Figure D-l
shows plots of PC(C) for various levels of the standard generated from the
constants in the table.
-------
c
o
3
JO
4J
Q
«
o
o
CJ
Q.
O
C
3
OJ
O!
re
1
Q
OJ
rs
en
tn
evi
Q.
-------
155
D.2 Form the Composite Distribution Function, PJ(C), For the Three Thresholds
Equation (4-5), Section 4.2 is used to calculate the composite distri-
bution function. In the example, theree health effects are considered.
Expanding Eq. (4-5) for three thresholds yields:
PT(0 - eQQP + ep + eQ (D-2)
Note from Eq. (4-6) that:
o
Qi = "eiQi (D-3)
where:
.c o
Q! = j p! dC (D-4)
0
Thus, in addition to requiring tabulations of the threshold probability
density functions p? versus concentration, tabulations of the cumulative
distribution is also needed. In the example all three p? are normal distri-
butions.
From the constants provided in the small table on the first page of
Section 4.5 one can write:
Pl° = 2x0.05 ^ 6XP HC-0.15)2 / 2 x (0.05)2)
p° = 3.989 exp (-200 (C-0.15)2) (D-5)
The functions for p2 and p° can be developed in the same manner.
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156
TABLE D-l
INDIVIDUAL AND COMPOSITE HEALTH EFFECTS THRESHOLD PROBABILITY DENSITY
FUNCTIONS VS. CONCENTRATION
STANDARD
(ppm)
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20
0.21
0.22
0.23
0.24
0.25
0.26
0
0.09
0.16
0.27
0.45
0.71
1.08
1.58
2.22
2.99
3.88
4.84
5.79
6.66
7.37
7.82
7.98
7.82
7.37
6.66
5.79
4.84
3.88
2.99
2.22
1.58
1.08
0.71
o
PT
T2
0.00
0.00
0.00
0.00
0,00
0.01
0.02
0.05
0.11
0.23
0.44
0.79
1.35
2.16
3.24
4.57
6.05
7.53
8.80
9.67
9.97
9.67
8.80
7.53
6.05
4.57
3.24
0
PT3
0.00
0.01
0.01
0.02
0.03
0.04
0.06
0.08
0.11
0.16
0.22
0.30
0.40
0.52
0.67
0.86
1.08
1.33
1.62
1.94
2.28
2.65
3.02
3.40
3.76
4.10
4.40
o;
0.001
0.003
0.005
0.008
0.014
0.023
0.036
0.055
0,081
0.115
0.159
0.212
0.274
0.345
0.421
0.500
0.579
0.655
0.726
0.788
0.841
0.885
0.919
0.945
0.964
0.977
0.986
«;
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.001
0.003
0.006
0.012
0.023
0.040
0.067
0.106
0.159
0.227
0.308
0.401
0.500
0.599
0.692
0.773
0.841
0.894
0.933
«;
0.000
0.000
0.000
0.000
0.001
0.001
0.002
0.002
0.003
0.004
0.006
0.009
0.012
0.017
0.023
0.030
0.040
0.052
0.067
0.100
0.106
0.129
0.159
0.189
0.227
0.264
0.309
PMT
0.01
0.03
0.05
0.09
0.14
0.21
0.31
0.45
0.64
0.92
1.27
1.77
2.37
3.14
4.05
5.07
6.09
6.99
7.64
7.83
7.82
7.30
6.44
5.41
4.31
3.33
2.50
o
P1,3
0.01
0.03
0.05
0.09
0.14
0.20
0.30
0.41
0.56
0.74
0.94
1.15
1.37
1.58
1.77
1.96
2.11
2.25
2.38
2.49
2.64
2.81
2.98
3.19
3.40
3.62
3.82
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157
Table D-l (CONTINUED)
STANDARD
(ppm) p°
0.27 0.45
0.28 0.27
0.29 0.16
0.30 0.09
0.31 0.05
0.32 0.02
0.33 0.01
0.34 0.01
0.35 0.00
0.36 0.00
0.37 0.00
0.38 0.00
0.39 0.00
0.40 0.00
0.41
0.42
0.43
0.44
0.45
0.46
0.47
0.48
0.49
0.50
0.51
0.52
0.53
0.54
0.55
0.56
0.57
0.58
0.59
0.60
o o
PT PT
'2 '3
2.16 4.65
1.35 4.83
0.79 4.95
0.44 4.99
0.23 4.95
0.11 4.83
0.05 4.65
0.02 4.40
0.01 4.10
0.00 3.76
0.00 3.40
0.00 3.02
0.00 2.65
0.00 2.28
1.94
1.62
1.33
1.08
0.86
0.67
0.52
0.40
0.30
0.22
0.16
0.11
0.09
0.06
0.04
0.03
0.02
0.01
0.01
0.00
0; ,; Q;
0,992 0,956 0.352
0,995 0.977 0.401
0.997 0.988 0.448
0.999 0.994 0.500
1.000 0.997 0.552
0.999 0.599
1.000 0.648
0.691
0.736
0.773
0.811
0.841
0.871
0.900
0.948
0.970
0.983
0.991
0.996
0.998
0.999
1.000
p°
1.89
1.45
1.18
1.02
0.92
0.85
0.80
0.75
0.70
0.64
0.58
0.51
0.45
0.39
0.33
0.28
0.23
0.18
0.15
0.11
0.09
0.07
0.05
0.04
0.03
0.02
0.01
0.01
0.01
0.01
0.00
o
P1,3
4.00
4.13
4.22
4.25
4.21
4.11
3.95
3.74
3.49
3.20
2.89
2.57
2.25
1.94
1.65
1.38
1.13
0.92
0.73
0.57
0.44
0.34
0.26
0.19
0.14
0,09
0.07
0.05
0.03
0.03
0.02
0.01
0.01
0.00
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158
Since the distributions are normal, their integrals as a function of
(C-C/cr) can be obtained from tables generally available in statistical text-
books. The p. functions and integrals, Q% are tabulated for various values
of C in Table D-l of this appendix. The composite distribution p-j- and the
semicomposite distribution p, - are also given in the table. The function
i >o
p-r is calculated from the individual p^ as follows: Take as an example
C = 0.20 ppm. By substitution of the value at this concentration calcualte
the Q1 (0.20) from Eq. (D-3).
Q1 (0.20) = 1 - 0.15 x 0.841 = 0.874
Q2 (0.20) « 1 - 0.80 x 0.500 = 0.600
Q3 (0.20) = 1 - 1.0 x 0.106 = 0.894
Note the e, are from the table on the first page of Section 4.5. Subs ti tut-
in these values and the p? (0.20) from the table into Eq. (0-2) gives:
P°T (0.20) = 0.15 x 0.6 x 0.894 x 4.84 + 0.8 x 0.874 0.894 x
9.97 + 1.0 x 0.874 x 0.600 x 2.28 = 7.82
The function p, ~ is calculated in a similar manner.
i ,j
D.3 Determine the Risk, R, That no True Threshold Will be Exceeded
The risk is calculated from Eq. (4-2). Note that the product of
probabilities terms in Eq. (4-4) is zero because e^ = 1 . For this case,
therefore, one can write:
R = 1 Pc (C,CSTD) p° (C)dC. (D-5)
-------
159
Given any potential value of the standard, the appropriate constant can
be calculated for PC as was done in the Table in Step #1. Substituting
this constant into Eq. (C-l) a value of P to be calculated for every
L>
value of Py in Table D-l of this appendix. The corresponding product is
then calculated and numerically integrated over the range of p°.
For example, for a level of the standard CSTD = 0.14 ppm have:
PC(C) = [1 -exp (-64.84C)]8760
Evaluating this expression at C * 0.20 ppm gives:
Pr(0.2) * [1 -exp (-64.84 x 0.2)]8760 = 0.980
That is, there is a 0.980 chance that the hourly average 0.2 ppm will not
be exceeded in a given calendar year. The corresponding p° was calculated
in Step #3 to be 7.82. The product is:
Pr (0.2) p° (0.2) » 7.66
w I
This product is tabulated as concentration over the range of p°. The product
is then numerically integrated over the range of p° using methods such as
Simpsons' rule or trapezoidal integration. The procedure is repeated for
each level of the standard shown in Table 4-4, Section 4.5.
Figure D-2 shows the procedure graphically for a standard at 0.14 ppm.
Figure D-2(a) shows the PC function. Figure D-2(b) shows the composite
probability density function p°, and Figure D-2(c) shows their product.
-------
160
Figure D-2 Graphical Illustration of PT Calculation
8
6
4
2
0
1.0
.8
.6
.4
.2
0
(a)
0.2 0.3
Concentration, ppm
(b)
0.5
0.1
0.2 0.3
Concentration, ppm
0.4
0.5
88
6
4
2
0
PCPTdC
0.2 0.3
Concentration, ppm
0.4
0.5
-------
161
The dashed line in Figure D-2(c) sketches in the original composite density
function. The area under the solid curve is PT, the probability of not ex-
ceeding the threshold. The area between the dashed line and the full line
is the probability of exceeding the true threshold. The sum of the areas,
is of course, the total area under the composite density function alone.
D.4 Determine R.
The risk, R.., of exceeding any individual threshold is determined
proceeding in the same manner as described in Step #3.
D.5 Determine any R,- ^
* »j
The risk of exceeding the lower of two thresholds, R. . can be
i »J
calculated by forming the composite in a manner similar to Step #2 and
then proceeding to Step #3.
-------
TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing}
1. REPORT NO.
3. RECIPIENT'S ACCESSION-NO.
4. TITLE AND SUBTITLE
STUDIES IN THE REVIEW OF THE PHOTOCHEMICAL
OXIDANT STANDARD
5. REPORT DATE
January 1979. Date of Issue
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
William F. Biller
8. PERFORMING ORGANIZATION REPORT NO,
9. PERFORMING ORGANIZATION NAME AND ADDRESS
William F. Biller
68 Yorktown Road
East Brunswick, N.J. 08816
10. PROGRAM ELEMENT NO.
11. CONTRACT/GRANT NO.
68-02-2589
12. SPONSORING AGENCY NAME AND ADDRESS
Environmental Protection Agency
Research Triangle Park
North Carolina 27711
13. TYPE OF REPORT AND PERIOD COVERED
Final Report
14. SPONSORING AGENCY CODE
15. SUPPLEMENTARY NOTES
16. ABSTRACT
Studies were performed in three areas for the Office of Air Quality Planning
and Standards to support its review of the photochemical oxidants standard: 1. the
desirability of changing from a photochemical oxidants to an ozone standard, 2. the
suitability of alternate ways of expressing the standard and, 3. the assessment of
health risks at alternate levels of the standard. From the first study it was
concluded that a change to an ozone standard could lead to benefits in administering
the standard with little or no loss in protection of the public health and welfare.
The second study concluded that national ambient air standards should be expressed
in a statistical form. A statistical form is more in accord with the actual
behavior of pollutant concentrations in the atmosphere. In the third area the
mathematical methodology was developed for estimating the risk that health effect
thresholds for sensitive populations will be exceeded during a given period of time
at alternate levels of the ambient air quality standard. The method quantitatively
treats the uncertainties in health effects thresholds and the extreme concentration
levels of pollutant to be experienced in the given time period.
1 7.
j
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
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Air Pollution
Oxidant
Ozone
Stantdard
Risk
DISTRIBUTION STATEMENT
Release unlimited
19. SECURITY CLASS (This Report)
NA
21. NO. OF PAGES
161
20. SECURITY CLASS (This page)
NA
22. PRICE
EPA Form 2220-1 (9-73)
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