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 StandardsEPA-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 1979ii 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 371. INTRODUCTION

The ozone National Ambient Air Quality Standard s (NAAQS) contain the phrase "expected number of days pe r calen dar year." [1] This differs from the previous NAAQS for photochemical oxidants which simply state a particula r

concen tration "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 ozon e standard. The purpose of this document is to amplify the discussions contained in Appendix H dealing with compliance

assessme ntand to indicate the data analysis procedure s

necess ary to determine appropriate design values for use i n develo ping control strategies. Where possible, th e approa ches discussed here are conceptually similar to th e procedures presented in the earlier "Guideline fo r lnterpreting Air Quality Data With Respect to the Standards" (OAQPS 1.2-008, revised February, 1977). [3 ] However
, the form of the ozone standards necessitate s certain modifications in two general areas: (1) accounting for less than complete sampling and (2) incorporating dat a from more than one year.

Although the interpretat ion of the proposed standards may initially appear complicated, the basic principle is relativel y 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 t

he complications that arise are consequences of ac counting for incomplete sampling or changes i n emissions.

Throughout the following discussion certain points ar e assumed that are consistent with previous guidance [3] but shouldbe reiterated here for completeness. The terms hour and day (da ily) are interpreted respectively as clock hou r andcalendar day. Air quality data are examined on a site by site basis and eac h individual site must meet the standard. In general, data from several different sites are not combined or averaged when performing these analyses. These points are2

discussed in more detail elsewhere. [3]

This document is organized so that the remainder of this introductory section present s the background of the problem, terminology, and certain basic premises that were used i n develop ing this guidance. This is followed by a sectio n which examines metho ds for determining appropriate design values. The final section discusses approaches that might be empl oyed in cases without ambient monitoring data. Thi s last section is brief and fairly general, because it treats a n aspect of the problem wh ich would be expected to rapidly evolve once these new forms of the NAAQS becom e established. In several parts of this document the material is dev eloped in a conversational format in order to highligh t certain points.

1.1. Background

The previous National Ambient Air Quality Standar d (NAAQS) for oxidant stated that no more than one hourl y value per year should exceed 160 micrograms per cubi c mete r (.08ppm). [2] With this type of standard, the secon d highest value for the year becomes t he decision making value. If it is above 160 micrograms per cubic meter then th e standard was exceeded. This would initially appear to be an ideal type of standard. The wording is simple and th e interpretation is obvious, or is it? Suppose the second highest

va lue for the year is less than 160 micrograms per cubi c

me ter and the question asked is, "Does this site meet th e standar d?" An experienced air pollution analyst woul d almos t automatically first ask, "How many observation s were there?" This respon se reflects the obvious fact that the second highest measured value can depend upon how many measurements were made in t he year. Carried to the absurd, if only one measurement is made for the year, it i s impossib le

to exceed this type of standard. Obviously, thi s extreme case could be remedied by requiring some minimum number of measuremen

ts per year. However, the basic point is that the probability of detecting a violation would still b e expected to increase as the number of samples increase d from the specified min

imum to the maximum possible number of observation s per year. Therefore, the present wording of this type of standard inherently penaliz es an area that performs3

mor e than the minimum acceptable amount of monitoring

. Furt hermore, the specification of a minimum data com pleteness criterion sti ll 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 th e standard is not as obvious but becomes more apparent when considering what is involved in maintaining the standar d year after year. For example, suppose an area meets th e standard in the sense that only one value for the year i s ab ove 160 micrograms per cubic meter. Because of th e 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 i n emissions. In other words, any area with emissions and meteorology that ca n produce one oxidant value above the standard has a definite risk of s ometime having at least two such values occurring in the same year and thereby violating the standard. Thi s situ ation may be viewed as analogous to the "10 year flood" and "100 year flood" concepts used in hydrology; i.e., hig h valu es may occur in the future but the likelihood of suc h
events is relatively low. However, with respect to ai r polluti on any rare violation poses distinct practical problems. From a control agency viewpoint, the question arises as t o what sh ould be done about such a violation if it is highl y unlikely to reoccur in the next few years. If the decision i s made to ignore such a violation then the obvious implication is that the standard can occasionally be ignored. This is not only undesirable but pr oduces a state of ambiguity that must be resolved to intelligently asse ss the risk of violating the standard. In other words, some quantification is needed to desc ribe what it means to maintain the standard year afte r year in view of the variation associated with air quality data. The wording of the ozone standard is intended to alleviat e these problems.

1.2. Terminology

The ter m "daily maximum value' refers to the maximu m hourly ozone value for a day. As defined in Appendix H, a valid d aily maximum means that at least 75 % of the hourly4

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 dayligh t hours while allowing time for routin e instrume ntmaintenance. The criterion also ensures tha t 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 fi nal point worth noting concerns terminology. Th e te rm "exceedance" is used throughout this document t o describe a daily maximum ozone measurement that is above the level of the standard. Therefore the phrase "expecte d

num ber of exceedances" is equivalent to "the expecte d number of daily maximum o zone 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 severa l alternatives considere d 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 th e guidance is perhaps superfluous or at best educational. The third point indicates that the "best" alternative, in som e sense, was selected. With this in mind, it is useful to briefly discu ss some of the key points that were considered i n j u dging 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
ext ends to several aspects of the problem. When someon e 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 jus t what is being done to protect their health. Also, the more5

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

mor e likely it is that all groups involved can concentrate o n th e real problem of maintaining clean air rather tha n arguing over minor side issues.

While simplicity is desira ble, if the problem is complex the solution canno t be oversimplified. In other words, the goal is to de velop a solution that is simple, and yet no t simple-m inded. In order to do this, the approach taken in this do cument is to recognize that there are two question s in volved in determining compliance: (1) was the standar d 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 seem s reasonable to have a more complicated procedure fo r determining the second answer.

I n addition to the trade-offs between simplicity an d 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 d o availab le resources vary from one area to another but th e complexity of the air pollution problems vary. An area with

no p ollution problem should not be required to do a n extensiv e analysis just because that level of detail is needed someplace else. Con versely, an area with sufficient resources
to perfo rm a detailed analysis of their pollution problem t o develop an optimum control strategy should not b e

cons trained from doing so simply because it is not warranted elsew here. Furthermore, a certain degree of flexibility i s essential to allow for modified moni toring schedules that are used to make the best use of available resources.

In addition to these points concerning simplicity an d flexibility, certain other considerations are of cours e involved. In p articular, the methodology employed cannot merely ignore high val

ues for a particular year simply because they are unl ikely 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 interpretation6

should involve a framework tha t could eventually be extended
to other pollutants, if necessary, and easily modified in th e future as our knowledge and understanding of air pollutio n increases.

It should be noted that no specific mention is made o f
measurement

error in the following discussions. While i t would b e naive to assume that measurement errors do no t
occur, at the present time it is difficult to allow fo r

measurem ent

errors in a manner that is not tantamount t o re-def ining the level of the standard. Obviously there is n o question tha t data values known to be grossly in erro r should be corrected or eliminated. In fact the use o f multiple years of data for the ozone standar ds should facilitate th is process. The more serious practical problem is with th e level of uncerta inty associated with every individua l

measurem ent. The viewpoint taken here is that thes e inher ent accuracy limitations are accounted for in the choice of the level of the standard and that equitable risk from one

areato another is assured by use of the reference (or a n equival ent) ambient monitoring method and adherence to a requir ed minimum quality assurance program. It should b e note d that the stated level of the standard is taken a s defining th e number of significant figures to be used i n 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 p p m 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 ter m "expected number." Once this is defined it is possible to

cons ider the interpretation when applied to several years o f data or to less than complete sampling data. An exampl e calculation is included at the end of this section to summarize and illustrate the major points.

2.1. Interpretation of "Expected Number'

Th e wording of the ozone standard states that th e "expecte d number of days per calendar year" must b e "equal to or less than 1." The statistical term "expected

num ber" is basically an arithmetic average. Perhaps th e simplest way to explain the intent of this wording is to give an

ex ample of what it would mean for an area to be i n complia nce with this type of standard. Suppose an area ha s relatively constant emissions year a fter 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 va lues above the level of the standard is determined and this is averaged with the

res ults of previous years. As long as this arithmetic average remains "less than or equal to 1" the area is in compliance

. A s far as rounding conventions are concerned, it suffices to carry one d ecimal place when computing the average. For examp le, the average of the three numbers 1,1,2 is 1.3 which is greater than 1.

Two features in this example warrant additiona l discussion to clearly define how this proposal would b e implement ed. The example assumes that a daily ozon e 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 b e

nece ssary to estimate this quantity. This is discussed i n section 2 .2. In the example it is also assumed that several
y e ars of data are available and there is relatively little change8

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 ave rage this expected number of excee dances per year should stabilize. If unusua l meteorology contributes to a high number of exceedances for a parti cular year then this will be averaged out by th e values for other "normal" years. It should be noted that these high values would, therefore, not be ignored bu t rather their relative contribution to the overall average is in propor tion to the likelihood of their occurrence. This us e of the average may be contrasted with an approach base d upon th e median. If the median were used then the yea r with the greatest number of exceedan ces could be ignored and there wo uld be no guarantee of protection against thei r periodic reoccurrence.

2.2. Estimating Exceedances for a Year

As discussed above, it is highly unlikely that an ozon e measurement will be available for each day of the year. Th erefore, it will be necessary to estimate the number o f exceed ances in a year. The formula to be used for thi s estimation i s contained in Appendix H of the ozone standard. The purpose of this section is to present the same basi c formula but to expand upon the rationale for choosing thi s 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 ass ociated 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 tha t part icular day. Such missing values, which can be viewed a s "sche duledmissing values," may be the result of planne d instrument maintenance or, for ozone, may be a consequence of a seasona

l monitoring program. In order to estimate the number of exceedances in a particular year it is necessary to

ac count for the possible effect of missing values. Obviously, allow ance for missing values can only result in a n estimated number of exceedances at least as large as th e observed number. From a practical viewpoint, this means9

that a ny site that is in violation of the standard based upo n 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, bu t may not be necessary to establish non-attainment.

In e stimating the number of exceedances in cases wit h missing data, certain practical considerations are appropriate. In some areas, cold we ather during the winter makes it very unlikely that high ozone values would occur. Therefore it is po ssible to discontinue ozone monitoring in som e loc alities for limited time periods with little risk o f inco rrectly assessing the status of the area. As indicated i n Appendix H, the proposed monitoring regulations (CFR 58) would permit the appropriate Regional Administrator t o waiv e any ozone monitoring requirements during certai n tim es of the year. Although data for such a time perio d would be technically missing, the es timation formula is structure d in terms of the required number of monitoring day s and th erefore these missing days would not affect th e computations.

Anoth er point is that even though a daily ozone value is missing, o ther data might indicate whether or not the missing value wo uld have been likely to exceed the standard level

. There are numerous ways additional information such a s solar radiation, temperature, or other pollutants could be used but the final result should be relatively easy to implement and n ot create an additional burden. An analysis of 25 8 site-yeas of oz one/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 i n 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 val ues. Having defined10

the set of missi ng 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 de note the number of missing values that may b e

assu med to be less than the standard. Then the followin g formula shal l 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 th e 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 assume d 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 th e year unless a monitoring waiver has been granted by th e appropriate Regional Administrator.

The above equation may be interpreted intuitively in th e following manner. The estimated number of exceedances i s equal to the observed number plus an increment tha t accounts for incomplete sampling. There were (N-n) missing daily val ues for the year, but a certain number of these
, namely z, were assumed to be below the standard

. Ther efore, (N-n-z) missing values are considered to b e potentia l exceedances. The fraction of measured value s that were above the level of11

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 procedu res presented are computationally simple. Some data processing complications result whe n mi ssing 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 no t diffe rentiate between scheduled and non-scheduled missin g values it is possible to develop a computerized system t o perform the necessary calculations without requirin g additional information on why each particular value wa s missing. In principle , if allowance is made for missing values that a re relatively certain to be less than the standard then it would seem reasonable to also account for missing value s that are relativ ely 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 u nnecessary complication in that it would not affect the assessment of compliance.

One feature of these estimation procedures should b e not ed. If an area does not record any values above th e st andard, then the estimated number of exceedances for the year is z ero. 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 indicat e that it is possible that the existing monitoring, data can b e deemed inadequate for use with these est imation formulas. In general, data sets that are 75% complete for the pea k pol lution potential seasons will be deemed adequate

. Although the general 75% completeness rule has bee n trad itionally used as an air quality validity criterion the ke y point is to ensu re reasonably complete monitoring of those time periods with high pollution potential. An additional word of caution is probably required at this point concerning attainme nt status determinations based upon limited data. If

a p articular area has very limited data and shows n o exceedances of the standard it must be recognized that a

more intense monitoring program could possibly result in a12

determination of non-att ainment. 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 valu e 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 ma y indicate nonattainment. Therefore some caution should b e used when viewing attainment status designations base d upon incomplete data.

2.3. Extension to Multiple Years

As discussed earlier, the major change in the ozon e standard is the use of the term "expected number" rather than just "t he number." The rationale for this modification is t o allow events to be weighted by the probability of thei r

occurrenc e. Up to this point, only the estimation of th e

nu mber 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 sit e

wo uld be compared by knowing the probability that the sit e would record 0,1 ,2,3,... exceedances in a year. Then each poss ible outcome could be weighted according to it s li kelihood of occurrence, and the appropriate expected value

or av erage could be computed. In practice, this type o f situati on will not exist because ambient data will only b e available for a limited number of years.

A period of three successive years is recommended as the basis f or determining attainment for two reasons. First
, increasing the nu mber of years increases the stability of the resulting average number of excee dances. Stated differently,

as more years are used, there is a greater chance o f minimizing the effects of an extreme year caused b y unusual weather conditions. The second factor is tha t extending the number of successive years too far increases the risk of averagi ng data during a period in which a real shift in e missions and air quality has occurred. This woul d penalize areas showing recent improvement and similarl y

r e ward areas which are experiencing deteriorating ozone13

air quality. Three years is thought by EPA to represent a proper balance between these two considerations. Thi s sp ecification of a three year time period for complianc e assessment also provides a firm basis for purposes o f decision-making. While additional flexibility is possible fo r developing design values for control strategy purposes, a more definitive framework seems essential when judging compliance to eliminate possible ambiguity and t o clearly identify the basis for the decision.

Conse quently, the expected number of exceedances pe r 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, an d 1976, th en the expected number of exceedances is estimated by averaging those three numbers. If this estimate is greate r tha n 1, then the standard has been exceeded at this site. A s pr eviously mentioned, it suffices to carry one decimal plac e when computing this average. This averaging rule requires the use of all ozone data collected at that si te during the past three calendar years . If no data are available for a particular yea r then the average is computed on the basis of the remainin g years. If in the previous example no data were available fo r 19 74, then the average of the estimated number o f exceed ances for 1975 and 1976 would be used. In othe r words, the general rule is to use data from the most recen t three years if available , but a single season of Inonitoring data

ma y still suffice to establish non-attainment. Thus, this thre e year criterion does not mean that non-attainment decision s

mus t be delayed until three years of data are available. I t should be noted that to establish attainlnent by a particular date, allowance wil l be permitted for emission reductions that are known to have occurred. One point wor

th 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 dat a bases so that sorne measure of the normal number o f

ex ceedances per year is available. Furthermore the nature o f the o zone problem makes it unlikely that areas currently well above the standard would suddenly come into compliance

. There fore, as these areas approach the standard additional14

ye ars of data would be available to determine th e 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.

Supp ose a site has the following data history fo r 1978-1980:

197 8: 365 daily values; 3 days above the standar d level.

197 9: 285 daily values; 2 days above the standar d level; 21 missing days satisfying the ex clusion criterion.

1980: 287 daily values; 1 day above the standard level; 7 missing days satisfying the exclusion criterion.

Suppose further that in 1980 measurements wer e not taken during the mon ths of January and February (a total o f 60 days for a leap year) because the col d weather minimizes any chance of recordin g exceedances and a monitori ng 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 r equired to determine that this site is not i n compliance. However, th e expected number of exceedances may s till be computed using equation 1 for purposes of illustration.

For 1978, there were no missing daily values and15

there fore there is no need to use the estimate d exceedances formula. The number of exceedances fo r 1978 is 3.

For 1979, equation 1 applies and the estimate d 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 an d 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 o f

exc eedances per year and completes the require d calculations.16
Blank page.17

3. ESTIMATING DESIGN VALUES

T he previous section addressed compliance with th e stan dard. As discussed, it suffices to treat question s concerning compliance as requiring a "yes/ no" type answer. Thi s approach facilitates the use of relatively simpl e computational forMulas. It also makes it unnecessary t o def ine the type of statistical distribution that describes th e behavi or of air quality data. The advantage of not invoking

a pa rticular statistical distribution is that the key issue o f wheth er 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

re quires an estimate of by how much the standard wa s exceeded. This is done by first examining the definition of

a design value for an "expected exceedances" standar d and then discussing vari ous procedures that may be used to
est imate a design value. A variety of approaches ar e considered such as fitting a statistical distribution, the us e of conditional probabilities, graphical estimation, and even

a table look-up procedure. In a sense each of thes e 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 ar e satisfactory. As with the previous section discussing
, compliance, this se ction concludes with example calculations illustrating the more important points.

3.1. Discussion of Design Values

In orde r to determine the amount by which the standard is exc eeded it is necessary to discuss the interpretation of a design v alue for the proposed standard. Conceptually tl1e design value for a particular site is the value that should b e reduc ed to the standard level thereby ensuring that the sit e wi ll meet the standard. With the wording of the ozon e standard the appropriate design value is the concentratio n with expected number of exceedances equal to 1. Although this d escribes the design value in words it is useful to18
introduce certain notations to precisely define this quantity.

Let P(x < o) denote the probability that an observation x is le ss than or equal to concentration c. This is als o denoted as F(c).

Let e denote the nurnber of exceed ances of the standard le velin the year, e.g., in the case of ozone this would be th e

nu mber of daily values above .12 ppm. Then the expecte d value of e denoted as E(e) may be written as:

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

For a sit e to be in compliance the expected number o f exceedances per year E(e), must be less than or equal to 1. From the above equation it follows that this is equivalent
to sayi ng that the probability of an exceedance must be les s than or equal to 1/365.

As indicated, the appropriate design value is tha t concentration which is expected to be ex ceeded once per year. Alternatively, the design value is chosen so that th e probabi lityof exceeding this concentration is 1/365. If a n equati on is known for F(c) then the design value may b e obtained by setting 1-F (c) equal to 1/365 and solving for c. If a grap h of F(c) is known then the design value may b e det ermined graphically by choosing the concentration value that corresponds to a frequency of exceedance of 1/365

. Obvi ously in practice the distribution F(c) is not reall y known. W hat is known is a set of air quality measurement s that may be approximated by a statistical di stribution to determine a de sign value as discussed in the following section.

3.2. The Use of Statistical Distributions

The use of a statistical distribution to approximatel y describe the behavior of air qu ality data is certainly not new. The initial work by Larsen [4] with the log-normal distribution demonstrated how this type of statis tical approximation could19

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 o f interest is the high values. The National Ambient Air Quality Standards are intended to limit exposure to hig h
concent rations. This has a direct impact on how statistica l distributions are chosen to describe the data. [5] If th e intendedapplication is to approximate the data in the upper

co ncentration ranges then obviously it must be required tha t any statistical distribution selected for this purpose has to fit the dat a in these higher concentration ranges. Initially thi s would appear to be an obvious truism but, in many cases,

a particular distribution may reasonably approximate" th e data in the sense that it fits fairly well for the middle 80% of the value s. This may be satisfactory for some application s but if the top 10% of the data is t he 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 exponentia l distr ibution[5,8]. Despite certain theoretical reservation s concerning factors such as interdependence of successiv e val ues 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 desig n value. Viewed in pe

rspective, however, the selection of the appropriate statistica

l distribution is a secondary objective -- the pr imary objective is to determine the appropriate design value. In other words, the question of interest is "wha t concentration has an expected number of exceedances pe r year e qual to 1?" and not "which distribution perfectl y 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 i n determining what is meant by a "reasona ble fit". In fact it will be seen later that a desig n value may be selected without even knowing which pa

rticular distribution best describes the data.20
There are certain points that are implicit in the abov e di scussion which are worth commenting upon. One possible approach in developing this type of guidance is to specify

a p articular distribution to be used in determining a desig n valu e. This approach is not taken here for a variety o f

reasons. Th ere is no guarantee that one family o f di stributions would be adequate to describe ozone levels fo r all ar eas of the country, for all weather conditions, etc. I t may well be that different distributions are needed fo r 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 emphasize d inv olves the distinction between determining compliance and dete rMining a design value. Suppose, for example, that a statistical distribution is selected and adequately describes all but th e highest five values each year. However, these five values are always above the standard and consequently th e number of exceedances pe

r year is always five. Such a site is not in c ompliance even if the design value predicted from the approximating distribution is below the standard level. I n such a case the ex

pected number of exceedances per year is 5 (with complete sampling) and therefore the slte is i n violation

. The design value is an aid in determining th e gener alreduction required, but it some cases it may b e necessary to furth er refine the estimate because of inadequate fit for the high values.

3.3. Methodologies

The purpose of this section is to present some acceptabl e approaches to determine an appropriate desi gn value, i.e., the
concentrat ion with expected number of exceedances per year equ al 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 imp ortant to recognize that the number of rneasurements must be treated properly. I n particular, missing values that are known to be less than the standard level should be accounted for so that they do no t incorrectly affect the empirical frequency distribution. For example, if an area does not monitor ozone in December, January, and February, because no values even approaching21

the standard level have ever been reported in these month s then thes e observations should not be considered missing but should be assigned some value less than the standard. Th e exact choice of the value is arbit ary and is not really important because th e primary purpose is to fit the upper tail of th e distribution.

In discussing the various acceptable approaches severa l different ca ses are presented. This is intended to illustrate the ge neral principles that should be applied in determining th e de sign value. Throughout these discussions it is generall y

assu med that more than one year of data is available. Th e difficulty with using a sing le year of data is that any effect due
to year to year variations in meteorology is obviously no t

accoun ted for. Therefore, any results based upon only on e year of data should be Yiewed as a guide that may be sub ject to revision.

(1) Fitting One Statist

ical Distribution to Several Years of Data

O ne of the simplest cases is when several years of fairl y complete data are available during a tim e of relatively constant emissi ons. In this situation the data can be plotted t o determine an empirical frequency distribu tion. For example, all data fo r a site from a 3-5 year period could be ranked fro m smalle st to largest and the empirical frequency distribtio n plotte d on semi-log paper. This type of plot emphasizes th e behavior o f the upper tail of the data as shown in Figure 1. A discussion of this plotting is contained elsewhere. [5] Figur e 2 illustrates how d

ifferent 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 straigh t line

. The next step is to choose a statistical distribution tha t approxim ately describes the data and to fit the distribution to the data. This may be done by least squares, maximu m likeihood estimation, or any me thod that gives a reasonable fit
to the top 10% of the data. An obvious quesl;ion is "wha t consti tutes a reasonable fit?" This can be judged visually b y plott ing the fitted distribution on the same graph as the dat a poin ts. Because of the intended use of the distribution th e degr ee of approximation for the top 10%, 5%, 1%, and even23

.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 thefitted 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 shoul d be noted that if several years of fairly cornplet e data are available it is not necessary to even fit a statistica l distribution. The concentration value corresponding to a frequency of l/365 may be read directly off the graph of the25
empirical distribution function and used as the design value.

If the data records are not sufficiently complete then the empirica l distribution function will not be plotted for th e 1/365 frequency and it will be necessar y to fit a distribution
to estimate the design value. However, whenever

su fficient 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 th e 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-upprocedure 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 fe w of the highest da

ta values. For example, if there are 1,017 daily values then the ranks of the lower and upper bound s obtain ed from Table 1 are 3 and 2. This means that a n appropriate design value would be between the third-highest and second-highest observed values. In using this table th e higher of the two conc entrations may be used as the design value. Therefore in this particular cas e, 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 frequenc y distributi on corresponds to a frequency of 1/365. B y construction, the table look-up procedure overestimates the design valu e. For instance, in the example with 1,01 7 values an acceptable design value would lie closer to the lower bound. This could be handled by in terpolation between the

secon d and third highest values. However, rather tha n introduce interpolation formulas it would be simpler to merely use the previously discussed graphical procedure.26

For the cas es that are 75% complete but still have less than 365 days the maximum observed concen tration may be used as a tentat

ive design value as long as the data set was 75% complete during the peak time s of the year. In this case it must be recognized that the design value i s quit e likely to require future revision. In principle, i f statistical independe nce applied, this maximum observed

co ncentration would egual or exceed the 1/36 5

conce ntration about half the time. However, the failure
to adequately account for yearly variations i n meteorology makes any estimate based on a single year of data very tentative.

(4) Fitt ing a Separate Distribution for Each Year o f Data (Conditional Probability Approach)

The p revious method required grouping data fro m several years into a single frequency di stribution. In some

cases data processing constraints may make thi s cumbersome. Therefor

e, an alternate approach may be used that allows each year to be treated individually. In

c o nsidering this alternate approach it is useful to briefly indicate the underlying framework. This particula r approach uses conditional probabilities and in mos t

cases it would probably be more convenient to use on e of the previous methods. However, the underlyin g fra mework

of this method has sufficient flexibility t o warrant its inclusion.

Suppose that the air quality data at a particular site may be approx imated by some statistical distribution F(xl0), where 0 denot es the fitted parameters. Suppose further that th e value s of the fitted parameters differ from year to year, bu t that t he data may still be approximated by the same type of distri bution. Intuitively this would mean that while the sam e type of distribution describes each year of data, the values of the parameter s would change from year to year reflecting the prevailing meteorology for the year. In theory it could be pos sible 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 e ach meteorological class, m(i), there would be a n associated air quality distribution function denoted as F(x |m(i)), the distribution function for x given th e meteorological class m(i). Using the standard rules of27

conditional probability the distribution function F (x) may be written as:

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

wher e P[m(i)] is the probability of meteorological class m(i ) occurring.

Conti nuing this approach the expected number o f exceedances may be written as:

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

where s denotes the standard level.

Inita lly the above framework may seem to be to o the oretical to have much pratical use. However, it will b e

seen in Section 4 that this approach may afford a convenient means of determining the expected number o f exceedances per year when l imited historical data is available. For the present discus sion it suffices to indicate how this approach may b e used when ambient data sets are available.

Su ppose that five years of ambient measurements are avail able. An approximating statistical distribution may b e dete rmined as discussed previously for each year, denoted as Fi (x). This would be analogou s to the F (x|m(i)) in the above discussi on. Then the distribution function of F(x) may b e 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 settin g 1-F(d) = 1/365 and solving for d, the design value. This i s equivalent to determining the concentration d so that:

5 3 [1-Fi (d) ] * 1/5 = 1/365. \ =128
In general i t 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.

Th e use of this equation can perhaps best be illustrate d by a simple example with two years of data. Suppose the data for e ach year may be approximated by an exponentia l dist ribution although the paralneter is different for the tw o years. In particular let

F1(x) = 1 -EXP(-43.4x) and F2(x) = 1 -EXP(-37.6x).

Usin g the previous equation, the design value (d) must b e 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 us ed as an initial guess for d this equatio n 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 s ufficiently 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 previ ous section have one thing in common; namely, their purpose. Each technique i s intended to select an appropriate design value, i.e., a concentration with e xpected number of yearly exceedances29
equal to 1. With this in mind a quic k check may be made to determine how reasonable the selected design value is. Thi s may be done by counting

the number of observed daily values that exceed the selected design value and computing th e average number o f exceedances per year. For example, if the selected desi gn value was exceeded 4 times in 3 years, then the a verage number of exceedances per year is 1.3

. Ideally, this average should be le ss than or equal to 1, but for

a variety of r easons somewhat higher values may occur. Ho wever, if this average is greater than 2.0 the design value is question able. In such cases the design value shoul d ei ther be changed or, if not changed, careful examinatio n sh ould be performed to substantiate this choice of a design value.

3.5. Discussion of Data Requirements

The use of the previous approaches presupposes th e existe nce of an adequate data base. Both approaches were presented in the context Or having several years of arnbient data. In m any practical cases the available data base may not be so extensive. Although these s tatistical approaches rnay be

us ed with less data, some caution is still required to ensure

a minir nally acceptable data set. In general, statistica l procedures permit inferences to b e made from limited data
sets. Nevertheless, the initial iata set must b e 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 l east 50% complete for the peak season with no systernatic pattern of missing potential peak hours

. This 50 % completeness criterion should be viewed in th e cont ext of the type of monitoring performed. A continuous monitor that fails to prod uce data sets meeting this criteri a 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 i n to one frequency distribution, it should be verified that all years have approximately the same pattern of missin g

va lues. Furthermore, if the number of measurements durin g the oxidant season differs by more than 20% from one year to30

another, then the conditional probability approach shoul d 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 dat a shou ld be used. Intuitively it would be reasonable to us e as many years of data as poss ible as long as emissions have

n o t changed "appreciably". Obviously this suggests that

some guidance be provided on what percent change i n emis sions is perrnissible. To some degree any suc h sp ecification is arbitrary. However, the more relevan t point i s that the specified percentage be reasonable

. Th e reason for a cut-off is to ensure that the impact o f increased e missions is not masked by the use of air quality dat a occurring prior to these emission increases. If a n area is in violation of the standard, then emission changes should be expected as control programs take effect

. Also, th e design value serves as a guide to achieving th e standard and is, in a sense, merely the means to an en d rather than an end in itself. Therefor e, no more than a 20% varia tion between the lowest and highest years i s recommended. It should be noted that a total variation of 20% may translate into a + or - 10% variation around the average.

If em issions have increased by more than 20% the n add itional years should not be incorporated unless the air quality values can be adjusted for the change i n emi ssions. For cases in which emissions have decrease d by mor e than 20% the earlier data may be used after ad

justment or used without change knowing that the design value will consequently be conservative. Although thi s document does not discuss methods for perforrning thi s adj ustment, it is useful to mention the basic principl e involve d. The selection of a design value inherentl y imp lies the existence of an acceptable model for taking an air quality value and determ ining the emission reduction required to reduce this value to the standard. I n princip le, then, this same rnodel may be used in reverse to take the emission change known to have occurred and
use t he 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 b e used to estimate the design value the more likely it is that a proper design value is selected. Because considerable31

e f fort 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 complianc e assessment, it is convenient to conclu de this section with examples illustrating the main point involved in applying these various technique s. For purposes of illustration all f o ur techniques are used on the salne data set. Figures 3,4, and 5 display semi-log plots of daily ozone values for 1974, 197 5 and 1976 at a sample site. These data are plotted using previously discussed conventions. [5] Th e horizontal axis is concentration (in ppm) and the vertical axis is the fraction of values ex ceeding this concentration. A horizontal dotted line is shown at a frequency o f 1/365 and the dotted line represents a We ibull distribution approxi mating the data. This particular fit was done by "eye-balling" the data, but suffices for the purpose s of illustration. Figure 6 is a similar plot for all thre e years of data grouped together. The high and secon d 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 Wei bull distribution plotted in Figure 6 for th e three years of data is described by the equation:

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

Setting, F(x) = 1 - 1/365 a nd solving for x gives .147 which is the design value because it corresponds to 3 fr equency of exceedance of 1/365. Using this quic k 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 fre quency distribution function crosses the line plotted a t 1/3 65 at a concentration of .15 and, therefore, this is th e design value selected by this method.

Using the quick check there are only two data values ab ove .15 and, therefore, the average number of yearl y exceedances of the design value is .67 which is acceptable.

Method 3: Table look-up

A total of 1,017 data values were record ed during the three year peri od. Using Table 1, this method says that the second highest value may be used as the d esign value. Therefore this metho d yields .16 as the design value. The quick check gives O as t he average number of yearly exceedances of the design value a lthough there are two values exactly equal to thi s estim ated design value. As indicated earlier, this procedur e is somewhat conservat

ive 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 gra phs. 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 f or d (by successive guesses) gives .15 as the design value. Using the quick check gives two values above th e design value and there fore an average yearly exceedance rate of 2/3.35

4. APPLICATIONS WITH LIMITED AMBIENT DATA

Virtua lly all of this discussion has focused upon the us e of ambient data. Historically, air quality models have bee n quite useful in providing estimates of air quality levels in th e absenc e of ambient data. The proposed wording of th e stan dard does not preclude the use of such models. A s models that provide frequency distributions of air quality are developed their use with the proposed standard will b e convenient.
Another potential means of estimating air quality dat a involves the use of conditional probabilities. While the use of conditional proba

bilities was discussed earlier in terms of combinin g different years of data, a more promising use o f 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 f or certain types of days or met eorological conditions. Then past historica l meteorological data may be used to determine the frequency of occurrence associated with these meteorolo gical conditions. This information may then be combined using conditiona l probabilities to obtain a

general air quality distribution. This particul ar approach could even be expanded to allow fo r 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 wit h expected numbe r of yearly exceedances equal to 1. Howe ver, these modelling and conditional probabilit y construc tions may make it possible to assess the risk o f viol ating the standard in the future based upon limite d historical data.36
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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 Qu ality Planning and Standards Publ. 1.2-008 U.S. Environmental Protection Agency, Research Triangle Park, North Carolina, February, 1977. 4. Larsen, R . I. A Ma

thematical 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. A ssessing the Validity of the Lognormal Model When Predicting Maximum Air Pollutant Concentrations. Paper No. 75-51.3, 68th A nnual 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 o f 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.