United States Office of Air Quality EPA-450/4-79-003

Environmental Protection Planning and Standards OAQPS No. 1.2-108

Agency Research Triangle Park, NC 27711 January 1979

Air

 Guideline Series

 Guideline for the

 Interpretation of

 Ozone Air Quality

 Standards
EPA-450/4-79-003

OAQPS No. 1.2-108

Guideline for the

Interpretation of Ozone

Air Quality Standards

Monitoring and Data Analysis Division

Prepared for

U.S. ENVIRONMENTAL PROTECTION AGENCY

Office of Air, Noise, and Radiation

Office of Air Quality Planning and Standards

Research Triangle Park, North Carolina 27711 

January 1979
ii

 OAQPS GUIDELINE SERIES

The guideline series of reports is being issued by the Office of Air Quality Planning

and Standards (OAQPS) to provide information to state and local air pollution

control agencies, for example, to provide guidance on the acquisition and

processing of air quality data and on the planning and analysis requisite for the

maintenance of air quality. Reports published in this series will be available - as

supplies permit - from the Library Services Office (MD-35), U.S. Environmental

Protection Agency, Research Triangle Park, North Carolina 27711; or, for a

nominal fee, from the National Technical Information Service, 5285 Port Royal

Road, Springfield, Virginia 22161.

Publication No. EPA-450/4-79-003

(OAQPS No . 1.2-108)
iii

Table of Contents

1. INTRODUCTION 1

1.1. Background 2

1.2. Terminology 3

1.3. Basic Premises 4

2. ASSESSING COMPLIANCE 7

2.1. Interpretation of "Expected Number" 7

2.2. Estimating Exceedances for a Year 8

2.3. Extension to Multiple Years 12

2.4. Example Calculation 14

3. ESTIMATING DESIGN VALUES 17

3.1. Discussion of Design Values 17

3.2. The Use of Statistical Distributions 18

3.3. Methodologies 20

3.4. Quick Test for Design Values 28

3.5. Discussion of Data Requirements 29

3.6. Example Design Value Computations 31

4. APPLICATIONS WITH LIMITED AMBIENT DATA 35

5. REFERENCES 37
1. INTRODUCTION

 The ozone National Ambient Air Quality Standards

(NAAQS) contain the phrase "expected number of days per

calendar year." [1] This differs from the previous NAAQS

for photochemical oxidants which simply state a particular

concentration "not to be exceeded more than once per year."

[2] The data analysis procedures to be used in computing the

expected number are specified in Appendix H to the ozone

standard. The purpose of this document is to amplify the

discussions contained in Appendix H dealing with compliance

assessment and to indicate the data analysis procedures

necessary to determine appropriate design values for use in

developing control strategies. Where possible, the

approaches discussed here are conceptually similar to the

procedures presented in the earlier "Guideline for

lnterpreting Air Quality Data With Respect to the

Standards" (OAQPS 1.2-008, revised February, 1977). [3]

However, the form of the ozone standards necessitates

certain modifications in two general areas: (1) accounting

for less than complete sampling and (2) incorporating data

from more than one year.

 Although the interpretation of the proposed standards may

initially appear complicated, the basic principle is relatively

straightforward. In general, the average number of days per

year above the level of the standard must be less than or

equal to 1. In its simplest form, the number of exceedances

each year would be recorded and then averaged over the past

three years to determine if this average is less than or equal

to 1. Most of the complications that arise are consequences

of accounting for incomplete sampling or changes in

emissions.

 Throughout the following discussion certain points are

assumed that are consistent with previous guidance [3] but

should be reiterated here for completeness. The terms hour

and day (daily) are interpreted respectively as clock hour

and calendar day. Air quality data are examined on a site by

site basis and each individual site must meet the standard. In

general, data from several different sites are not combined or

averaged when performing these analyses. These points are
2

discussed in more detail elsewhere. [3]

 This document is organized so that the remainder of this

introductory section presents the background of the problem,

terminology, and certain basic premises that were used in

developing this guidance. This is followed by a section

which examines methods for determining appropriate design

values. The final section discusses approaches that might be

employed in cases without ambient monitoring data. This

last section is brief and fairly general, because it treats an

aspect of the problem which would be expected to rapidly

evolve once these new forms of the NAAQS become

established. In several parts of this document the material is

developed in a conversational format in order to highlight

certain points.

1.1. Background

 The previous National Ambient Air Quality Standard

(NAAQS) for oxidant stated that no more than one hourly

value per year should exceed 160 micrograms per cubic

meter (.08ppm). [2] With this type of standard, the second

highest value for the year becomes the decision making value.

If it is above 160 micrograms per cubic meter then the

standard was exceeded. This would initially appear to be an

ideal type of standard. The wording is simple and the

interpretation is obvious, or is it? Suppose the second highest

value for the year is less than 160 micrograms per cubic

meter and the question asked is, "Does this site meet the

standard?" An experienced air pollution analyst would

almost automatically first ask, "How many observations

were there?" This response reflects the obvious fact that the

second highest measured value can depend upon how many

measurements were made in the year. Carried to the absurd,

if only one measurement is made for the year, it is

impossible to exceed this type of standard. Obviously, this

extreme case could be remedied by requiring some minimum

number of measurements per year. However, the basic point

is that the probability of detecting a violation would still be

expected to increase as the number of samples increased

from the specified minimum to the maximum possible number

of observations per year. Therefore, the present wording of

this type of standard inherently penalizes an area that performs
3

more than the minimum acceptable amount of monitoring.

Furthermore, the specification of a minimum data completeness criterion still does not solve the problem of what to

do with those data sets that fail to meet this criterion.

 A second problem with the current wording of the

standard is not as obvious but becomes more apparent when

considering what is involved in maintaining the standard

year after year. For example, suppose an area meets the

standard in the sense that only one value for the year is

above 160 micrograms per cubic meter. Because of the

variability associated with air quality data, the fact that one

value is above the standard level means that there is a chance

that two values could be above this standard level the next

year even though there is no change in emissions. In other

words, any area with emissions and meteorology that can

produce one oxidant value above the standard has a definite

risk of sometime having at least two such values occurring in

the same year and thereby violating the standard. This

situation may be viewed as analogous to the "10 year flood"

and "100 year flood" concepts used in hydrology; i.e., high

values may occur in the future but the likelihood of such

events is relatively low. However, with respect to air

pollution any rare violation poses distinct practical problems.

From a control agency viewpoint, the question arises as to

what should be done about such a violation if it is highly

unlikely to reoccur in the next few years. If the decision is

made to ignore such a violation then the obvious implication

 is that the standard can occasionally be ignored. This is

not only undesirable but produces a state of ambiguity that

must be resolved to intelligently assess the risk of violating the

standard. In other words, some quantification is needed to

describe what it means to maintain the standard year after

year in view of the variation associated with air quality data.

The wording of the ozone standard is intended to alleviate

these problems.

1.2. Terminology

 The term "daily maximum value' refers to the maximum

hourly ozone value for a day. As defined in Appendix H,

a valid daily maximum means that at least 75 % of the hourly
4

values from 9:01 A.M. to 9:00 P.M. (LST) were measured

or at least one hourly value exceeded the level of the standard.

This criterion is intended to reflect adequate monitoring of

the daylight hours while allowing time for routine

instrument maintenance. The criterion also ensures that

high hourly values are not omitted merely because too few

values were measured. It should be noted that this is intended

as a minimal criterion for completeness and not as a

recommended monitoring schedule.

 A final point worth noting concerns terminology. The

term "exceedance" is used throughout this document to

describe a daily maximum ozone measurement that is above

the level of the standard. Therefore the phrase "expected

number of exceedances" is equivalent to "the expected

number of daily maximum ozone values above the level of the

standard."

1.3. Basic Premises

 By its very nature, the existence of a guideline document

implies several things: (1) that there is a problem, (2) that a

solution is provided, and (3) that there were several

alternatives considered in reaching the solution. Obviously, if

there is no problem then the guideline is of limited value,

and if there were not some alternative solutions then the

guidance is perhaps superfluous or at best educational. The

third point indicates that the "best" alternative, in some

sense, was selected. With this in mind, it is useful to briefly

discuss some of the key points that were considered in

judging the various options. The purpose of this section is

to briefly indicate the criteria used in developing this particular

guideline.

 The most obvious criterion is simplicity. This simplicity

extends to several aspects of the problem. When someone

asks if a particular area meets the standard they expect either

a "yes" or "no" as the answer or even an occasional "I don't

know". Secondly, this simplicity should extend to the reason

why the standard was met or violated. If a panel of experts is

required to debate the probability that an area is in compliance

then the general public may rightly feel confused about just

what is being done to protect their health. Also, the more 
5

clear-cut the status of an area is (and the reasons why) the

more likely it is that all groups involved can concentrate on

the real problem of maintaining clean air rather than

arguing over minor side issues.

 While simplicity is desirable, if the problem is complex the

solution cannot be oversimplified. In other words, the goal

is to develop a solution that is simple, and yet not

simple-minded. In order to do this, the approach taken in

this document is to recognize that there are two questions

involved in determining compliance: (1) was the standard

violated? and (2) if so, by how much? The first question is

the simpler of the two in that a "yes/no" answer is expected.

 The second question implies both a quantification and a determination of what to do about it. Therefore, it seems

reasonable to have a more complicated procedure for

determining the second answer.

 In addition to the trade-offs between simplicity and

complexity another problem is to allow a certain amount of

flexibility without being vague. There are several reasons

 for allowing some degree of flexibility. Not only do

available resources vary from one area to another but the

complexity of the air pollution problems vary. An area with

no pollution problem should not be required to do an

extensive analysis just because that level of detail is needed

someplace else. Conversely, an area with sufficient resources

to perform a detailed analysis of their pollution problem to

develop an optimum control strategy should not be

constrained from doing so simply because it is not warranted

elsewhere. Furthermore, a certain degree of flexibility is

essential to allow for modified monitoring schedules that are

used to make the best use of available resources.

 In addition to these points concerning simplicity and

flexibility, certain other considerations are of course

involved. In particular, the methodology employed cannot

merely ignore high values for a particular year simply because

they are unlikely to reoccur. The purpose of the standard is

to protect against high values in a manner consistent with the

likelihood of their occurrence.

 A final point is that the proposed interpretation
6

should involve a framework that could eventually be extended

to other pollutants, if necessary, and easily modified in the

future as our knowledge and understanding of air pollution

increases.

 It should be noted that no specific mention is made of

measurement error in the following discussions. While it

would be naive to assume that measurement errors do not

occur, at the present time it is difficult to allow for

measurement errors in a manner that is not tantamount to

re-defining the level of the standard. Obviously there is no

question that data values known to be grossly in error

should be corrected or eliminated. In fact the use of

multiple years of data for the ozone standards should facilitate

this process. The more serious practical problem is with the

level of uncertainty associated with every individual

measurement. The viewpoint taken here is that these

inherent accuracy limitations are accounted for in the choice

of the level of the standard and that equitable risk from one

area to another is assured by use of the reference (or an

equivalent) ambient monitoring method and adherence to a

required minimum quality assurance program. It should be

noted that the stated level of the standard is taken as

defining the number of significant figures to be used in

comparisons with the standard. For example, a standard

level of.12 ppm means that measurements are to be rounded

to two decimal places (.005 rounds up), and, therefore, .125

ppm is the smallest concentration value in excess of the level

of the standard.
7

2. ASSESSING COMPLIANCE

 This section examines the ozone standard with particular

attention given to the evaluation of compliance. This is

done in several steps. The first is a discussion of the term

"expected number." Once this is defined it is possible to

consider the interpretation when applied to several years of

data or to less than complete sampling data. An example

calculation is included at the end of this section to summarize

and illustrate the major points.

2.1. Interpretation of "Expected Number'

 The wording of the ozone standard states that the

"expected number of days per calendar year" must be

"equal to or less than 1." The statistical term "expected

number" is basically an arithmetic average. Perhaps the

simplest way to explain the intent of this wording is to give an

example of what it would mean for an area to be in

compliance with this type of standard. Suppose an area has

relatively constant emissions year after year and its monitoring

station records an ozone value for every day of the year. At

the end of each year the number of daily values above the level

of the standard is determined and this is averaged with the

results of previous years. As long as this arithmetic average

remains "less than or equal to 1" the area is in compliance.

As far as rounding conventions are concerned, it suffices

to carry one decimal place when computing the average. For

example, the average of the three numbers 1,1,2 is 1.3 which

is greater than 1.

 Two features in this example warrant additional

discussion to clearly define how this proposal would be

implemented. The example assumes that a daily ozone

measurement is available for each day of the year so that the

number of exceedances for the year is known. On a practical

basis this is highly unlikely and, therefore, it will be

necessary to estimate this quantity. This is discussed in

section 2.2. In the example it is also assumed that several

years of data are available and there is relatively little change
8

in emissions. This is discussed in more detail in section 2.3.

 The key point in the example is that as data from additional

years are incorporated into the average this expected number

 of exceedances per year should stabilize. If unusual

meteorology contributes to a high number of exceedances

for a particular year then this will be averaged out by the

values for other "normal" years. It should be noted that

these high values would, therefore, not be ignored but

rather their relative contribution to the overall average is in

proportion to the likelihood of their occurrence. This use

of the average may be contrasted with an approach based

upon the median. If the median were used then the year

with the greatest number of exceedances could be ignored and

there would be no guarantee of protection against their

periodic reoccurrence.

2.2. Estimating Exceedances for a Year

 As discussed above, it is highly unlikely that an ozone

measurement will be available for each day of the year.

Therefore, it will be necessary to estimate the number of

exceedances in a year. The formula to be used for this

estimation is contained in Appendix H of the ozone standard.

The purpose of this section is to present the same basic

formula but to expand upon the rationale for choosing this

approach and to provide illustrations of certain points.

 Throughout this discussion the term "missing value" is

used in the general sense to describe all days that do not have

an associated ozone measurement. It is recognized that in

certain cases a so-called "missing value" occurs because the

sampling schedule did not require a measurement for that

particular day. Such missing values, which can be viewed as

"scheduled missing values," may be the result of planned

instrument maintenance or, for ozone, may be a consequence

of a seasonal monitoring program. In order to estimate the

number of exceedances in a particular year it is necessary to

account for the possible effect of missing values. Obviously,

allowance for missing values can only result in an

estimated number of exceedances at least as large as the

observed number. From a practical viewpoint, this means
9

that any site that is in violation of the standard based upon

the observed number of exceedances will not change status

after this adjustment. Thus, in a sense, this adjustment for

missing values is required to demonstrate attainment, but

may not be necessary to establish non-attainment.

 In estimating the number of exceedances in cases with

missing data, certain practical considerations are appropriate.

 In some areas, cold weather during the winter makes it very

unlikely that high ozone values would occur. Therefore it

is possible to discontinue ozone monitoring in some

localities for limited time periods with little risk of

incorrectly assessing the status of the area. As indicated in

Appendix H, the proposed monitoring regulations (CFR 58)

would permit the appropriate Regional Administrator to

waive any ozone monitoring requirements during certain

times of the year. Although data for such a time period

would be technically missing, the estimation formula is structured in terms of the required number of monitoring days

and therefore these missing days would not affect the

computations.

 Another point is that even though a daily ozone value is

missing, other data might indicate whether or not the missing

value would have been likely to exceed the standard level.

There are numerous ways additional information such as

solar radiation, temperature, or other pollutants could be used

but the final result should be relatively easy to implement

and not create an additional burden. An analysis of 258

site-yeas of ozone/oxidant data from the highest sites in the

90 largest Air Quality Control Regions showed that only 1%

of the time did the high value for a day exceed .12 ppm if the

adjacent daily values were less than .09 ppm. With this in

mind the following exclusion criterion may be used for ozone:

 A missing daily ozone value may be assumed to be less

than the level of the standard if the (daily maxima on both

the preceding day and the following, day do not exceed 75%

of the level of the standard.

 It should be noted that to invoke this exclusion criterion

data must be available from both adjacent days. Thus it does

not apply to consecutive missing daily values. Having defined
10

the set of missing values that may be assumed to be less than

the standard it is possible to present the computations required

to adjust for missing data.

 Let z denote the number of missing values that may be

assumed to be less than the standard. Then the following

formula shall be used to estimate the number of exceedances

for the year:

e=v+(v/n)*(N-n-z) (1) 

(* indicates multiplication)

Where N = the number of required monitoring days in the

year 

n = the number of valid daily maxima

v = the number of measured daily values above the level of the

standard

z - the number of days assumed to be less than the standard

level, and

e = the estimated number of exceedances for the year.

 This estimated number of exceedances shall be rounded

to one decimal place (fractional parts equal to .05 round up).

 Note that N is always equal to the number of days in the

year unless a monitoring waiver has been granted by the

appropriate Regional Administrator.

 The above equation may be interpreted intuitively in the

following manner. The estimated number of exceedances is

equal to the observed number plus an increment that

accounts for incomplete sampling. There were (N-n) missing

daily values for the year, but a certain number of these,

namely z, were assumed to be below the standard.

Therefore, (N-n-z) missing values are considered to be

potential exceedances. The fraction of measured values

that were above the level of
11

the standard was v/n and it is assumed that the same fraction

of these candidate missing values would also exceed the level

of the standard.

 The estimation procedures presented are computationally

simple. Some data processing complications result when

missing data are screened to ensure a representative data

base, but on a practical basis this effort is only required for

sites that are marginal with respect to compliance. Because

the exclusion criterion for missing values does not

differentiate between scheduled and non-scheduled missing

values it is possible to develop a computerized system to

perform the necessary calculations without requiring

additional information on why each particular value was

missing. In principle, if allowance is made for missing values

that are relatively certain to be less than the standard then it

would seem reasonable to also account for missing values

that are relatively certain to be above the standard. Although

this is a possibility, it will probably not be necessary initially

because such a situation would, of necessity, have at least two

values greater than the standard level. Therefore, it is quite

likely that this would be an unnecessary complication in that

it would not affect the assessment of compliance.

 One feature of these estimation procedures should be

noted. If an area does not record any values above the

standard, then the estimated number of exceedances for the

year is zero. An obvious consequence of this is that any area

that does not record a value above the standard level will be

in compliance. In most cases this confidence is warranted.

However, at least some qualification is necessary to indicate

that it is possible that the existing monitoring, data can be

deemed inadequate for use with these estimation formulas. In

general, data sets that are 75% complete for the peak

pollution potential seasons will be deemed adequate.

Although the general 75% completeness rule has been

traditionally used as an air quality validity criterion the key

point is to ensure reasonably complete monitoring of those

 time periods with high pollution potential. An additional

word of caution is probably required at this point concerning

attainment status determinations based upon limited data. If

a particular area has very limited data and shows no

exceedances of the standard it must be recognized that a

more intense monitoring program could possibly result in a
12

determination of non-attainment. Therefore, if it is critical to

immediately determine the status of a particular area and the

ambient data base is not very complete, the design value

computations presented in section 3 may be employed as a

guide to assess potential problems. The point is, that as

the monitoring data base increases, the additional data may

indicate nonattainment. Therefore some caution should be

used when viewing attainment status designations based

upon incomplete data.

2.3. Extension to Multiple Years

 As discussed earlier, the major change in the ozone

standard is the use of the term "expected number" rather than

just "the number." The rationale for this modification is to

allow events to be weighted by the probability of their

occurrence. Up to this point, only the estimation of the

number of exceedances for a single year has been discussed.

This section discusses the extension to multiple years.

 Ideally, the expected number of exceedances for a site

would be compared by knowing the probability that the site

would record 0,1,2,3,... exceedances in a year. Then each

possible outcome could be weighted according to its

likelihood of occurrence, and the appropriate expected value

or average could be computed. In practice, this type of

situation will not exist because ambient data will only be

available for a limited number of years.

 A period of three successive years is recommended as the

basis for determining attainment for two reasons. First,

increasing the number of years increases the stability of the

resulting average number of exceedances. Stated differently,

as more years are used, there is a greater chance of

minimizing the effects of an extreme year caused by

unusual weather conditions. The second factor is that

extending the number of successive years too far increases

the risk of averaging data during a period in which a real shift

in emissions and air quality has occurred. This would

penalize areas showing recent improvement and similarly

reward areas which are experiencing deteriorating ozone
13

air quality. Three years is thought by EPA to represent a

proper balance between these two considerations. This

specification of a three year time period for compliance

assessment also provides a firm basis for purposes of

decision-making. While additional flexibility is possible for

developing design values for control strategy purposes, a more

definitive framework seems essential when judging compliance

to eliminate possible ambiguity and to clearly identify the basis

for the decision.

 Consequently, the expected number of exceedances per

year at a site should be computed by averaging the estimated

number of exceedances for each year of data during the past

three calendar years. In other words, if the estimated number

of exceedances has been computed for 1974, 1975, and

1976, then the expected number of exceedances is estimated

by averaging those three numbers. If this estimate is greater

than 1, then the standard has been exceeded at this site. As

previously mentioned, it suffices to carry one decimal place

when computing this average. This averaging rule requires the

use of all ozone data collected at that site during the past three

calendar years . If no data are available for a particular year

then the average is computed on the basis of the remaining

years. If in the previous example no data were available for

1974, then the average of the estimated number of

exceedances for 1975 and 1976 would be used. In other

words, the general rule is to use data from the most recent

three years if available, but a single season of Inonitoring data

may still suffice to establish non-attainment. Thus, this three

year criterion does not mean that non-attainment decisions

must be delayed until three years of data are available. It

should be noted that to establish attainlnent by a particular

date, allowance will be permitted for emission reductions that

are known to have occurred.

 One point worth commenting on is the possibility that the

very first year is "unusual". While this could occur, in the case

of ozone rnost urbanized areas already have existing data

bases so that sorne measure of the normal number of

exceedances per year is available. Furthermore the nature of

the ozone problem makes it unlikely that areas currently well

above the standard would suddenly come into compliance.

Therefore, as these areas approach the standard additional
14

years of data would be available to determine the

expected number of exceedances for a year.

2.4. Example Calculation

 ln order to illustrate the key points that have been

discussed in this section it is convenient to consider the

following example for ozone.

Suppose a site has the following data history for

1978-1980:

1978: 365 daily values; 3 days above the standard

level.

1979: 285 daily values; 2 days above the standard

level; 21 missing days satisfying the exclusion criterion.

1980: 287 daily values; 1 day above the standard level;

7 missing days satisfying the exclusion criterion.

 Suppose further that in 1980 measurements were

not taken during the months of January and February (a

total of 60 days for a leap year) because the cold

weather minimizes any chance of recording

exceedances and a monitoring waiver had been granted

by the appropriate Regional Administrator.

 Because the three year average number of exceedances is clearly greater than 1, there is no computation required to determine that this site is not in

compliance. However, the expected number of exceedances may still be computed using equation 1 for

purposes of illustration.

 For 1978, there were no missing daily values and
15

therefore there is no need to use the estimated

exceedances formula. The number of exceedances for

1978 is 3.

 For 1979, equation 1 applies and the estimated

number of exceedances is:

 2 +(2/285)*(365 - 285 - 21) =

 2 + 0.4 = 2.4

 For 1980, the same extimation formula is used but

due to the monitoring waiver for January and February

the number of required monitoring days is 306 and

therefore the estimated number of exceedances is:

1 + (1/287)*(306 - 287 -7) = 

1 + (1/287)*(12) = 1.0

 Averaging these three numbers (3, 2.4, and 1.0)

gives 2.1 as the estimated expected number of

exceedances per year and completes the required

calculations.
16

Blank page.
17

3. ESTIMATING DESIGN VALUES

 The previous section addressed compliance with the

standard. As discussed, it suffices to treat questions

concerning compliance as requiring a "yes/no" type answer.

This approach facilitates the use of relatively simple

computational forMulas. It also makes it unnecessary to

define the type of statistical distribution that describes the

behavior of air quality data. The advantage of not invoking

a particular statistical distribution is that the key issue of

whether or not the standard is exceeded is not obscured by

which particular distribution best describes the data.

However, once it is established that an area exceeds the

standard, the next logical question is more quantitative and

requires an estimate of by how much the standard was

exceeded. This is done by first examining the definition of 

a design value for an "expected exceedances" standard

and then discussing various procedures that may be used to

estimate a design value. A variety of approaches are

considered such as fitting a statistical distribution, the use

of conditional probabilities, graphical estimation, and even

a table look-up procedure. In a sense each of these

approaches should be viewed as a means to an end, i.e.,

meeting the applicable air quality standard. As long as this

final goal is kept in mind any of these approaches are

satisfactory. As with the previous section discussing,

compliance, this section concludes with example calculations

illustrating the more important points.

3.1. Discussion of Design Values

 In order to determine the amount by which the standard

is exceeded it is necessary to discuss the interpretation of a

design value for the proposed standard. Conceptually tl1e

design value for a particular site is the value that should be

reduced to the standard level thereby ensuring that the site

will meet the standard. With the wording of the ozone

standard the appropriate design value is the concentration

with expected number of exceedances equal to 1. Although

this describes the design value in words it is useful to
18

introduce certain notations to precisely define this quantity.

 Let P(x < o) denote the probability that an observation x

is less than or equal to concentration c. This is also

denoted as F(c).

 Let e denote the nurnber of exceedances of the standard

level in the year, e.g., in the case of ozone this would be the

number of daily values above .12 ppm. Then the expected

value of e denoted as E(e) may be written as:

 E(e) = P(x > .12) * 365 = [1 - F( .12)] * 365

 For a site to be in compliance the expected number of

exceedances per year E(e), must be less than or equal to

1. From the above equation it follows that this is equivalent

to saying that the probability of an exceedance must be less

than or equal to 1/365.

 As indicated, the appropriate design value is that

concentration which is expected to be exceeded once per year.

Alternatively, the design value is chosen so that the

probability of exceeding this concentration is 1/365. If an

equation is known for F(c) then the design value may be

obtained by setting 1-F(c) equal to 1/365 and solving for c.

If a graph of F(c) is known then the design value may be

determined graphically by choosing the concentration value

that corresponds to a frequency of exceedance of 1/365.

Obviously in practice the distribution F(c) is not really

known. What is known is a set of air quality measurements

that may be approximated by a statistical distribution to determine a design value as discussed in the following section.

3.2. The Use of Statistical Distributions

 The use of a statistical distribution to approximately

describe the behavior of air quality data is certainly not new.

The initial work by Larsen [4] with the log-normal distribution

demonstrated how this type of statistical approximation could
19

be used. The proposed form of the ozone standard provides

a framework for the use of statistical distributions to assess

the probability that the standard will be met. An important

point in dealing with air pollution problems is that the rnain

area of interest is the high values. The National Ambient Air

Quality Standards are intended to limit exposure to high

concentrations. This has a direct impact on how statistical

distributions are chosen to describe the data. [5] If the

intended application is to approximate the data in the upper

concentration ranges then obviously it must be required that

any statistical distribution selected for this purpose has to fit

the data in these higher concentration ranges. Initially this

would appear to be an obvious truism but, in many cases,

a particular distribution may reasonably approximate" the

data in the sense that it fits fairly well for the middle 80% of

the values. This may be satisfactory for some applications

but if the top 10% of the data is the range of interest it may be

inappropriate.

 Over the years various statistical distributions have been

suggested for possible use in describing air quality data.

Example applications include the two-parameter 

Lognormal[4], the three-parameter 

lognormal[6], the Weibull[5,7], and the exponential

distribution[5,8]. Despite certain theoretical reservations

concerning factors such as interdependence of successive

values these approaches have been proven over time to be

useful tools in air quality data analysis. The appropriate

choice of a distribution is useful in determining the design

value. Viewed in perspective, however, the selection of the

appropriate statistical distribution is a secondary objective --

the primary objective is to determine the appropriate design

value. In other words, the question of interest is "what

concentration has an expected number of exceedances per

year equal to 1?" and not "which distribution perfectly

describes the data?" Therefore, it is not necessary to require

that any particular distribution be used. All that is necessary

is to indicate the characteristics that must be considered in

determining what is meant by a "reasonable fit". In fact it will

be seen later that a design value may be selected without even

knowing which particular distribution best describes the data.
20

 There are certain points that are implicit in the above

discussion which are worth commenting upon. One possible

approach in developing this type of guidance is to specify

a particular distribution to be used in determining a design

value. This approach is not taken here for a variety of

reasons. There is no guarantee that one family of

distributions would be adequate to describe ozone levels for

all areas of the country, for all weather conditions, etc. It

may well be that different distributions are needed for

different areas. Secondly, as control prog-rams take effect and

pollution levels are reduced the so-called "best" distribution

may change. Another point-that should be emphasized

involves the distinction between determining compliance and

deterMining a design value. Suppose, for example, that a

statistical distribution is selected and adequately describes

all but the highest five values each year. However, these five

values are always above the standard and consequently the

number of exceedances per year is always five. Such a site is

not in compliance even if the design value predicted from the

approximating distribution is below the standard level. In

such a case the expected number of exceedances per year is 5

(with complete sampling) and therefore the slte is in

violation. The design value is an aid in determining the

general reduction required, but it some cases it may be

necessary to further refine the estimate because of inadequate

fit for the high values.

3.3. Methodologies

 The purpose of this section is to present some acceptable

approaches to determine an appropriate design value, i.e., the

concentration with expected number of exceedances per year

equal to 1. As discussed, this may be alternatively viewed

as determining the concentration that will be exceeded 1 time

out of 365.

 Throughout this discussion it is important to recognize that

the number of rneasurements must be treated properly. In

particular, missing values that are known to be less than the

standard level should be accounted for so that they do not

incorrectly affect the empirical frequency distribution. For

example, if an area does not monitor ozone in December,

January, and February, because no values even approaching
21

the standard level have ever been reported in these months

then these observations should not be considered missing but

should be assigned some value less than the standard. The

exact choice of the value is arbitary and is not really important

because the primary purpose is to fit the upper tail of the

distribution.

 In discussing the various acceptable approaches several

different cases are presented. This is intended to illustrate the

general principles that should be applied in determining the

design value. Throughout these discussions it is generally

assumed that more than one year of data is available. The

difficulty with using a single year of data is that any effect due

to year to year variations in meteorology is obviously not

accounted for. Therefore, any results based upon only one

year of data should be Yiewed as a guide that may be sub ject

to revision.

 (1) Fitting One Statistical Distribution to Several Years of

Data

 One of the simplest cases is when several years of fairly

complete data are available during a time of relatively constant

emissions. In this situation the data can be plotted to

determine an empirical frequency distribution. For example, all

data for a site from a 3-5 year period could be ranked from

smallest to largest and the empirical frequency distribtion

plotted on semi-log paper. This type of plot emphasizes the

behavior of the upper tail of the data as shown in Figure 1. A

discussion of this plotting is contained elsewhere. [5] Figure

2 illustrates how different types of distrihutions would appear

on such a plot. The data may also be plotted on other types of

graph paper, such as log-normal or Weibull. The ideal situation is when the data points lie approximately on a straight

line. The next step is to choose a statistical distribution that

approximately describes the data and to fit the distribution to

the data. This may be done by least squares, maximum

likeihood estimation, or any method that gives a reasonable fit

to the top 10% of the data. An obvious quesl;ion is "what

constitutes a reasonable fit?" This can be judged visually by

plotting the fitted distribution on the same graph as the data

points. Because of the intended use of the distribution the

degree of approximation for the top 10%, 5%, 1%, and even

23

.5% of the data must be examined. The most obvious

check is to examine departures of the actual data points

from the fitted distribution. As a general rule there should

be no obvious pattern to the lack of fit in terms of under- or 

over-prediction ,or trend. For example, if the fitted

distribution underestimates all of the last eight data points

by more than 5%, then it must be established that the

fitted distribution is reasonable. Such an argument might involve showing that the majority of these data points all

occurred in the same period and that the meteorology for 

these particular days was extremely unusual. The claim

that this meteorology was unusual would also have to be 

substantiated by examining historical meteorological data. 

It should be noted that this extra effort is not routinely

required and would onLy be necessary when the fit appears 

inadequate. The design value corresponds to a frequency

of l/365 and in some cases the empirical frequency 

distribution function will be plotted in this range. In such

cases, the fitted distribution should be consistent with the

empirical distribution in this range. This can be examined

graphically by locating the concentration on the empirical

frequency distribution function corresponding to a 

frequency of 1/365. By construction, there will be

measured data points on either side of this value. The two 

measured concentrations below this value and the two

measured concentrations above this value will be used as a

constraint in fitting a distribution. If the fitted distribution

results in a design value that differs by more than 5% from

all four of these measured concentrations, some 

explanation should be presented indicating the reasons for

this discrepancy. It should be noted that in some cases 

there may be only one, rather than two, measured values

on the empirical frequency distribution with frequencies less 

than l/365. In these cases the upper constraint would

consist of one rather than two data points.

 (2) Using the Empirical Frequency Distribution of Several

Years of Data (Graphical Estimation)

 It should be noted that if several years of fairly cornplete

data are available it is not necessary to even fit a statistical

distribution. The concentration value corresponding to a

frequency of l/365 may be read directly off the graph of the

25

empirical distribution function and used as the design value.

 If the data records are not sufficiently complete then the

empirical distribution function will not be plotted for the

1/365 frequency and it will be necessary to fit a distribution

to estimate the design value. However, whenever

sufficient data are available, this technique provides a

convenient means of graphically estimating the design value.

 (3) Table Look-up

 An obvious point that can initially be overlooked in the

discussion of these techniques is that the final choice of a

design value is primarily influenced by the few highest values

in the data set. With this in mind, it is possible to construct

a simple table look-up procedure to determine a design value.

 Again, it is important to treat the number of values properly

to ensure that the data adequately reflects all portions of the

year.

 To use this tabular approach it is only necessary to know

the total number of daily values, and then determine a few

of the highest data values. For example, if there are 1,017

daily values then the ranks of the lower and upper bounds

obtained from Table 1 are 3 and 2. This means that an

appropriate design value would be between the third-highest

and second-highest observed values. In using this table the

higher of the two concentrations may be used as the design

value. Therefore in this particular case, it suffices to know the

three highest measured values during the time period.

 This look-up procedure is basically a tabular technique

for determining what point on the empirical frequency

distribution corresponds to a frequency of 1/365. By

construction, the table look-up procedure overestimates the

design value. For instance, in the example with 1,017

values an acceptable design value would lie closer to the lower

bound. This could be handled by interpolation between the

second and third highest values. However, rather than

introduce interpolation formulas it would be simpler to merely

use the previously discussed graphical procedure.
26

 For the cases that are 75% complete but still have less

than 365 days the maximum observed concentration may

be used as a tentative design value as long as the data set

was 75% complete during the peak times of the year. In

this case it must be recognized that the design value is

quite likely to require future revision. In principle, if

statistical independence applied, this maximum observed

concentration would egual or exceed the 1/365

concentration about half the time. However, the failure

to adequately account for yearly variations in

meteorology makes any estimate based on a single year

of data very tentative.

 (4) Fitting a Separate Distribution for Each Year of

Data (Conditional Probability Approach)

 The previous method required grouping data from

several years into a single frequency distribution. In some

cases data processing constraints may make this

cumbersome. Therefore, an alternate approach may be

used that allows each year to be treated individually. In

considering this alternate approach it is useful to briefly

indicate the underlying framework. This particular

approach uses conditional probabilities and in most

cases it would probably be more convenient to use one

of the previous methods. However, the underlying

framework of this method has sufficient flexibility to

warrant its inclusion.

 Suppose that the air quality data at a particular site may be

approximated by some statistical distribution F(xl0), where 0

denotes the fitted parameters. Suppose further that the

values of the fitted parameters differ from year to year, but

that the data may still be approximated by the same type of

distribution. Intuitively this would mean that while the same

type of distribution describes each year of data, the values of

the parameters would change from year to year reflecting

the prevailing meteorology for the year. In theory it could be

possible to define a set of meteorological classes, say m(i),

so that the distribution function of the air quality data could be

defined for each one of these meteorological classes. Then

for each meteorological class, m(i), there would be an

associated air quality distribution function denoted as

F(x|m(i)), the distribution function for x given the

meteorological class m(i). Using the standard rules of
27

conditional probability the distribution function F (x) may be

written as:

F(x) = 3 {F(x|m(i))} P[m(i)]

 \

where P[m(i)] is the probability of meteorological class m(i)

occurring.

Continuing this approach the expected number of

exceedances may be written as:

E(e) = 3 P[x > s | m (i) ] * P[m(i)]

 \

where s denotes the standard level.

Initally the above framework may seem to be too

theoretical to have much pratical use. However, it will be

seen in Section 4 that this approach may afford a convenient

means of determining the expected number of exceedances per

year when limited historical data is available. For the present

discussion it suffices to indicate how this approach may be

used when ambient data sets are available.

Suppose that five years of ambient measurements are

available. An approximating statistical distribution may be

determined as discussed previously for each year, denoted as

Fi (x). This would be analogous to the F (x|m(i)) in the above

discussion. Then the distribution function of F(x) may be

written as:

 5

F(x) = 3 Fi (x) * 1/5

 \=1

where Fi is analogous to F [x|m(i)] and P[m(i)] is assumed to

be 1/5. The design value may then be determined by setting

1-F(d) = 1/365 and solving for d, the design value. This is

equivalent to determining the concentration d so that:

5

3 [1-Fi (d) ] * 1/5 = 1/365.

 \=1
28

 In general it may not be possible to explicity solve this

equation for d, but the answer may be obtained iteratively by

first guessing an appropriate design value.

 The use of this equation can perhaps best be illustrated

by a simple example with two years of data. Suppose the data

for each year may be approximated by an exponential

distribution although the paralneter is different for the two

years. In particular let

F1(x) = 1 -EXP(-43.4x) and 

F2(x) = 1 -EXP(-37.6x).

Using the previous equation, the design value (d) must be

determined so that

1/2 EXP(-43.4d) + 1/2 EXP(-37.6d) = 1/365 or 

365 * {1/2 EXP(-43.4d) + 1/2 EXP(-37.6d)} = 1.

 If .15 is used as an initial guess for d this equation

gives a value of .92 rather than 1. If .145 is used the resulting

value is 1.12 indicating that the design value is between .145

and .15. Guessing .148 gives a value of .99, i.e.

365{1/2 EXP(-43.4 * .148) + 1/2 EXP(-37.6 * .148)}=

 .99

 This is sufficiently close to 1 and is a reasonable stopping

place in determining the design value.

 3.4. Quick Test for Design Values

 All of the approaches in the previous section have one thing

in common; namely, their purpose. Each technique is

intended to select an appropriate design value, i.e., a

concentration with expected number of yearly exceedances
29

equal to 1. With this in mind a quick check may be made to

determine how reasonable the selected design value is. This

may be done by counting the number of observed daily values

that exceed the selected design value and computing the

average number of exceedances per year. For example, if the

selected design value was exceeded 4 times in 3 years, then

the average number of exceedances per year is 1.3.

Ideally, this average should be less than or equal to 1, but for

a variety of reasons somewhat higher values may occur.

However, if this average is greater than 2.0 the design value

is questionable. In such cases the design value should

either be changed or, if not changed, careful examination

should be performed to substantiate this choice of a design

value.

3.5. Discussion of Data Requirements

 The use of the previous approaches presupposes the

existence of an adequate data base. Both approaches were

presented in the context Or having several years of arnbient

data. In many practical cases the available data base may not

be so extensive. Although these statistical approaches rnay be

used with less data, some caution is still required to ensure

a minirnally acceptable data set. In general, statistical

procedures permit inferences to be made from limited data

sets. Nevertheless, the initial iata set must be

representative. For example, if no data is available from the

peak season, ttlen any extrapolations would require more

than merely statistical procedures. Therefore, the input

data sets should be at least 50% complete for the peak season

with no systernatic pattern of missing potential peak hours.

This 50% completeness criterion should be viewed in the

context of the type of monitoring performed. A

continuous monitor that fails to produce data sets meeting this

criteria has in effect a down-time of more than 50%. With

such a high percentage of down-time for the instrument even

the recorded values should be viewed with caution.

 In employing approaches that group data from all years

into one frequency distribution, it should be verified that all

years have approximately the same pattern of missing

values. Furthermore, if the number of measurements during

the oxidant season differs by more than 20% from one year to
30

another, then the conditional probability approach should

be used. The reason for this constraint is to ensure that

variations in sample sizes do not result in

disproportionate weighting of data from different years.

 Another point of concern is how many years of data

should be used. Intuitively it would be reasonable to use

as many years of data as possible as long as emissions have

not changed "appreciably". Obviously this suggests that

some guidance be provided on what percent change in

emissions is perrnissible. To some degree any such

specification is arbitrary. However, the more relevant

point is that the specified percentage be reasonable.

The reason for a cut-off is to ensure that the impact of

increased emissions is not masked by the use of air quality

data occurring prior to these emission increases. If an

area is in violation of the standard, then emission changes

should be expected as control programs take effect.

Also, the design value serves as a guide to achieving the

standard and is, in a sense, merely the means to an end

rather than an end in itself. Therefore, no more than a 20%

variation between the lowest and highest years is

recommended. It should be noted that a total variation

of 20% may translate into a + or - 10% variation around

the average.

 If emissions have increased by more than 20% then

additional years should not be incorporated unless the air

quality values can be adjusted for the change in

emissions. For cases in which emissions have decreased

by more than 20% the earlier data may be used after adjustment or used without change knowing that the design

value will consequently be conservative. Although this

document does not discuss methods for perforrning this

adjustment, it is useful to mention the basic principle

involved. The selection of a design value inherently

implies the existence of an acceptable model for taking an

air quality value and determining the emission reduction

required to reduce this value to the standard. In

principle, then, this same rnodel may be used in reverse to

take the emission change known to have occurred and

use the model to scale the previous data sets. Attempting

to adjust older historical data may initially seem to be an

unnecessary complication but the more data that can be

used to estimate the design value the more likely it is that

a proper design value is selected. Because considerable 
31

effort could be expended in revising a control strategy

this additional effort may be warranted.

3.6. Example Design Value Computations

 As in the previous discussion of compliance

assessment, it is convenient to conclude this section with

examples illustrating the main point involved in applying

these various techniques. For purposes of illustration all

four techniques are used on the salne data set. Figures

3,4, and 5 display semi-log plots of daily ozone values for

1974, 1975 and 1976 at a sample site. These data are

plotted using previously discussed conventions. [5] The

horizontal axis is concentration (in ppm) and the vertical

axis is the fraction of values exceeding this concentration.

A horizontal dotted line is shown at a frequency of

1/365 and the dotted line represents a Weibull distribution

approximating the data. This particular fit was done

by "eye-balling" the data, but suffices for the purposes

of illustration. Figure 6 is a similar plot for all three

years of data grouped together. The high and second

high values for the three years are: (.13 and .12), (.16

and .16), and (.15 and .14).

Method 1: Fitting a single distribution to data from all

three years.

 The Weibull distribution plotted in Figure 6 for the

three years of data is described by the equation:

2.011

 F(x) = 1 - EXP[-(x/.0609) ].

 Setting, F(x) = 1 - 1/365 and solving for x gives .147

which is the design value because it corresponds to 3

frequency of exceedance of 1/365. Using this quick

check, there are three values above .147 so the average

number of yearly exceedances is 1.


34

Method 2: Graphical estimation

Referring to Figure 6 it may be seen that the empirical

frequency distribution function crosses the line plotted at

1/365 at a concentration of .15 and, therefore, this is the

design value selected by this method.

Using the quick check there are only two data values

above .15 and, therefore, the average number of yearly

exceedances of the design value is .67 which is acceptable.

Method 3: Table look-up

 A total of 1,017 data values were recorded during the three

year period. Using Table 1, this method says that the second

highest value may be used as the design value. Therefore this

method yields .16 as the design value. The quick check gives

O as the average number of yearly exceedances of the design

value although there are two values exactly equal to this

estimated design value. As indicated earlier, this procedure

is somewhat conservative in that it tends to overestimate the

design value.

Method 4: Conditional probabiIities

 Separate two parameter Weibull distributions were fitted

to each yearly data set as shown in the graphs. Using the form

of equation 5 gives the equation:

 1.835

1/365 = 1/3 EXP{-(d/.0467) } + 

 2.139

 1/3 EXP{-(d/.0705) } +

2.180

 1/3 EXP{-(d/.0629) }

Solving for d (by successive guesses) gives .15 as the design

value. Using the quick check gives two values above the

design value and therefore an average yearly exceedance rate

of 2/3.
35

4. APPLICATIONS WITH LIMITED AMBIENT DATA

 Virtually all of this discussion has focused upon the use

of ambient data. Historically, air quality models have been

quite useful in providing estimates of air quality levels in the

absence of ambient data. The proposed wording of the

standard does not preclude the use of such models. As

models that provide frequency distributions of air quality are

developed their use with the proposed standard will be

convenient.

 Another potential means of estimating air quality data

involves the use of conditional probabilities. While the use

of conditional probabilities was discussed earlier in terms of

combining different years of data, a more promising use of

this technique would involve the construction of historical air

quality data sets from relatively short monitoring studies.

Very limited ambient data or air quality models may be used

to develop frequency distributions for certain types of days or

meteorological conditions. Then past historical

meteorological data may be used to determine the frequency

of occurrence associated with these meteorological conditions.

This information may then be combined using conditional

probabilities to obtain a general air quality distribution. This

particular approach could even be expanded to allow for

changes in emissions.

 No matter what approach is chosen the two quantities of

interest are: (1) the expected number of exceedances per year

and (2) the design value, i.e., that concentration with

expected number of yearly exceedances equal to 1.

However, these modelling and conditional probability

constructions may make it possible to assess the risk of

violating the standard in the future based upon limited

historical data.
36

Blank page
37

5. REFERENCES

1. 40CFR50.9

2. Fed. Reg.,36(84):8186(April 30,1971)

3. "Guidelines for Interpretation of Air Quality Standards,"

 Office of Air Quality Planning and Standards Publ.

 1.2-008 U.S. Environmental Protection Agency, 

 Research Triangle Park, North Carolina, February, 

 1977.

4. Larsen, R . I. A Mathematical Model for Relating Air

 Quality Measurements to Air Quality Standards. U.S.

 Environmental Protection Agency Research Triangle 

 Park, North Carolina Publication Number AP-89, 

 1971

5. Curran, T.C. and N.H. Frank. Assessing the Validity of

 the Lognormal Model When Predicting Maximum 

 Air Pollutant Concentrations. Paper No. 75-51.3, 

 68th Annual Meeting of the Air Pollution Control 

Association, Boston, Massachusetts, 1975.

6. Mage, D.T. and W. R . Ott An Improved Statistical

 Model for Analyzing Air Pollution Concentration Data. 

 Paper No. 75-51.4, 68th Annual Meeting of the Air 

 Pollution Control Association, Boston, Massachusetts, 

 1975.

7. Johnson,T. A Comparison of the Two-Parameter Weibull

 and Lognornal Distributions Fitted to Ambient Ozone 

 Data, Quality Assurance in Air Pollution Measurement 

 Conference, New Orleans, Louisiana, March, 1979.

8. Breiman,L. et al. Statistical Analysis and Interpretation 

 of Peak Air Pollution measurement. Technology 

 Service Corporation, Santa Monica,California. 1978.