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Relationship between concentrations of atmospheric pollutants and averaging time
Atmospheric Environment Pergamon Press 1972. Vol. 6, pp. 581-582. Printed in Great Britain.
LETTER TO THE EDITORS RELATIONSHIP
BETWEEN POLLUTANTS
CONCENTRATIONS AND AVERAGING
OF ATMOSPHERIC TIME*
THE CONCLUSIONS reported by McGuire and No11 seem to have been reached without regard for statistical principles. They state “although various frequencies of occurrence may be of interest, in most cases concern is directed to the maximum concentrations for various averaging times”. I submit, firstly, that knowledge about the “various frequencies of occurrence” of pollutant con~ntrations is not only of interest, it is vital to a proper interpretation of the results of air pollution sampling. Secondly, knowing the “maximum concentrations for various averaging times” far from being of concern in most cases is hardly ever of any practical value. The term “maximum concentration” can be applied properly only to the highest concentration amongst a finite number of samples. Then it is a statement of fact concerning a specific isolated experience in the past. It has therefore very little value in helping us predict quantitatively the possibihty of similar experiences occurring in the future. Nor, knowing its value, is our knowledge conceding the possible effects of air pollution enhanced since those effects receive contributions from all other concentrations above a threshold. The concentrations of a pollutant experienced at a given place are a consequence of the spatial and temporal distribution and strengths of sources as well as of the operation of certain natural processes such as atmospheric turbulence. The latter may be supposed to have operated at all times in the past and will continue to operate in the future although their presence becomes manifest only when tracers such as pollutants are available. However, because of the complexity of source distribution and our ignorance about the effect on concentrations at a given point in a large city of such source-variables as emission height and distance we must think in terms of the observed concentrations themselves instead of their underlying causes. If the distribution of sources around the point changed, the magnitudes of the observed concentrations would also change. The natural processes themselves would remain constant but the viewpoint from which they are seen would change. It is useful to think of the concentrations of a pollutant as existing during all past and future time and to constitute a “population” in the statistical sense. The atmosphere is sampled with the expectation of obtaining information about the “population”. For example, the mean, standard deviation and frequency distribution of the con~ntrations in the samples are estimates of the corresponding properties of that population. The confidence limits to be applied to those estimates vary with the number of samples collected and can, in principle be estimated using standard procedures. It is important to remember, however, that if the source distribution changes then another “population” is being sampled so that if observations are continued for a long time the samples may not be drawn from a homogeneous population. In the sequel, as in all attempts to analyze the concentrations of atmospheric pollutants at a fixed point, it will be assumed that no changes in source distribution have occurred during the observation period and that a homogeneous population is being sampled. The highest concentration observed amongst a finite number of samples is simply an estimate of the concentration which occurs in the population with a frequency implied by the reciprocal number of samples. For example, the highest concentration amongst a 100 samples is an estimate of the concentration having an occurrence frequency 0.01. Therefore the magnitude of the highest concentration observed will, on average, vary with the number of samples in a way that reflects the form of the frequency distribution for the population. For example, the highest concentration amongst a IO0 samples (frequency 0.01) will be lower than the highest amongst 1000 samples (frequency 0.001). Similarly the highest concentrations from sequences of the hourly-, daily-, monthly-, and yearlyaverage concentrations obtained during the course of a year will decrease in that order because they are estimates of concentrations having occurrence frequencies of 8760-l, 365-l, 12-’ and 1-r respectively. Therefore FIGS. 1 and 2 of McGuire and Noll’s paper show the effect of simultaneously varying the number of samples as well as the averaging times. If the purpose of the work was to investigate the effect of averaging time, the highest concentration should have been picked from sequences of hourly-, daily-, monthly-, and yearly-average concentrations each containing an equal number of samples. However, the vafues of the highest concentrations amongst different sets of samples each of equaI length will vary because of random statistical variations. For example, the highest hourly average SOZ concentrations for each of 7 yr in Chicago varied between 0.86 and 1.69 ppm (LARSEN,1970). Similar variations involving a factor of 2 or more are evident in Larsen’s results for other pollutants,
* MCGUIRE T. and NOLL K. E. Atmospheric Environment $291-298 581
(1971).
582
Letter to the Editors
other cities and other averaging times. If a large number of replicate experiments were carried out the highest concentrations observed would show considerable variations but the average of ali of them would tend to the ~on~en~a~on having the occurrence frequency implied by the number of samples in each experiment. For a given number of repfications standard statistical te&niques can be used to estimate, for a given probability level, the confidence limits within which the concentration corresponding to the implied occurrence frequency is expected. With the result of only one experiment the confidence limits cannot be estimated but would be so wide as to have no practical value. In FIG. 1 and 2 of McGuire and Noli’s paper the highest concentration for each of the four averaging times observed in a singIe experiment are plotted without any indication of the confidence limits. In spite of this, the slope (their parameter b) of the line plotted through the data points is given to three signiiicant figures. The use of a single observation, for example the highest, is, amongst the several statistical indices of sample variability, the least etIicient. For McGuire and NoI1 to suggest that their index b is analogous to a standard deviation is misleading. On the other hand, had all the data been used to obtain an estimate of the frequency distribution for the papulation, the concentration corresponding to any frequency within the range of observation could have been estimated with greater confidence. The question posed by McGuire and No11 should be restated in the following way: Wow does the ~n~e~trai~on corresponding to a given occmrence frequency change when the averaging time is varied? The answer can be inferred from the considerable data published by LARSEN (1970).For example, TABLE f &es the cumulative frequency distribution of SO2 concentrations reported by Larsen for Chicago from 1962 to 1968. From this it can be seen that the concentrations which are (ppm) AT CHICAGO BY AVERAGING TIME AND FREQUENCY 1962-1968 (LARSEN, 1970)
TABLE 1. SULFUR DIOXIDE C~~X~NTRATION
Averaging time 0.01 5 min lh 8h I day 1 month 1Yr
Per cent of samples in which concentration is exceeded 0.1 1 10 30 50 70
1.31 1.11
1.01 0.95 0.74
0.67 0.64 0.56 0‘49
0.32 0.32 0.31 0.30 0.27 0.12
O”l6 0.16 0.16 0.16
0.08 0.08 0.09 0.09
0.03 0.03 0.04 0.05
90 per cent 0.01 0.01 0.01 0.02
exceeded with frequencies in the range 10-50 per cent are very nearly constant. For frequencies less than 10 per cent, the concentration tends to decrease while for frequencies greater than 50 per cent, the concentration increases with averaging time, For example, the concentrations averaged over 5 min, 1 h, 8 h and daily intervals which are exceeded 1 per cent of the time are, respectively, 0.67, 0.64, 0.56 and 9.49 ppm= The 7-yr reporting period contains over ~~,~ S-min intervafs so that there are over 7ooO 5-min-average samples with concentrations exceeding 0.67 ppm. Similarly there are more than 500 hourly average samples having concentrations exceeding 0.64 ppm. In both cases the sampling statistics should be good. On the other hand only 77 g-h and 26 daily samples exceeding the stated concentrations are available at the 1 per cent occurrence frequency. For the 0.1 and 0.01 per cent occurrence frequencies the corresponding number of samples are smaller by a factor 10 and a factor 100, respectively. There are, for example, only S hourly-average samples having concentrations exceeding 1. f 1 ppm. IIow significant the 25 per cent decrease in con~ntration may be would have to be established before condusions could be drawn. However, Larsen% data cfearfy show the coneenfration corresponding to a given occurrence frequency is much less dependent on the averaging time than the equation of McGuire and No11 implies. More importantly, the relationship is more complex than they suggest, depending on such arbitrary variables, amongst others, as the number of samples available and range of averaging times considered. Properties introduced by the sampling and computation procedures used should not be attributed to the phenomenon under study. B&gy alpd He&& P&&a B~vi&m Ctafk R&M Nucfem Lulmafories Chufk R&r, Oniario Canada.
P. J. BARRY
REFERENCE LARSEN R. I. (1970) Relating air pollutant effects to concentration
Ass. 20, 214-225.
and control. J. Air %Ilrri. Control
LETTER TO THE EDITORS RELATIONSHIP
BETWEEN POLLUTANTS
CONCENTRATIONS AND AVERAGING
OF ATMOSPHERIC TIME*
THE CONCLUSIONS reported by McGuire and No11 seem to have been reached without regard for statistical principles. They state “although various frequencies of occurrence may be of interest, in most cases concern is directed to the maximum concentrations for various averaging times”. I submit, firstly, that knowledge about the “various frequencies of occurrence” of pollutant con~ntrations is not only of interest, it is vital to a proper interpretation of the results of air pollution sampling. Secondly, knowing the “maximum concentrations for various averaging times” far from being of concern in most cases is hardly ever of any practical value. The term “maximum concentration” can be applied properly only to the highest concentration amongst a finite number of samples. Then it is a statement of fact concerning a specific isolated experience in the past. It has therefore very little value in helping us predict quantitatively the possibihty of similar experiences occurring in the future. Nor, knowing its value, is our knowledge conceding the possible effects of air pollution enhanced since those effects receive contributions from all other concentrations above a threshold. The concentrations of a pollutant experienced at a given place are a consequence of the spatial and temporal distribution and strengths of sources as well as of the operation of certain natural processes such as atmospheric turbulence. The latter may be supposed to have operated at all times in the past and will continue to operate in the future although their presence becomes manifest only when tracers such as pollutants are available. However, because of the complexity of source distribution and our ignorance about the effect on concentrations at a given point in a large city of such source-variables as emission height and distance we must think in terms of the observed concentrations themselves instead of their underlying causes. If the distribution of sources around the point changed, the magnitudes of the observed concentrations would also change. The natural processes themselves would remain constant but the viewpoint from which they are seen would change. It is useful to think of the concentrations of a pollutant as existing during all past and future time and to constitute a “population” in the statistical sense. The atmosphere is sampled with the expectation of obtaining information about the “population”. For example, the mean, standard deviation and frequency distribution of the con~ntrations in the samples are estimates of the corresponding properties of that population. The confidence limits to be applied to those estimates vary with the number of samples collected and can, in principle be estimated using standard procedures. It is important to remember, however, that if the source distribution changes then another “population” is being sampled so that if observations are continued for a long time the samples may not be drawn from a homogeneous population. In the sequel, as in all attempts to analyze the concentrations of atmospheric pollutants at a fixed point, it will be assumed that no changes in source distribution have occurred during the observation period and that a homogeneous population is being sampled. The highest concentration observed amongst a finite number of samples is simply an estimate of the concentration which occurs in the population with a frequency implied by the reciprocal number of samples. For example, the highest concentration amongst a 100 samples is an estimate of the concentration having an occurrence frequency 0.01. Therefore the magnitude of the highest concentration observed will, on average, vary with the number of samples in a way that reflects the form of the frequency distribution for the population. For example, the highest concentration amongst a IO0 samples (frequency 0.01) will be lower than the highest amongst 1000 samples (frequency 0.001). Similarly the highest concentrations from sequences of the hourly-, daily-, monthly-, and yearlyaverage concentrations obtained during the course of a year will decrease in that order because they are estimates of concentrations having occurrence frequencies of 8760-l, 365-l, 12-’ and 1-r respectively. Therefore FIGS. 1 and 2 of McGuire and Noll’s paper show the effect of simultaneously varying the number of samples as well as the averaging times. If the purpose of the work was to investigate the effect of averaging time, the highest concentration should have been picked from sequences of hourly-, daily-, monthly-, and yearly-average concentrations each containing an equal number of samples. However, the vafues of the highest concentrations amongst different sets of samples each of equaI length will vary because of random statistical variations. For example, the highest hourly average SOZ concentrations for each of 7 yr in Chicago varied between 0.86 and 1.69 ppm (LARSEN,1970). Similar variations involving a factor of 2 or more are evident in Larsen’s results for other pollutants,
* MCGUIRE T. and NOLL K. E. Atmospheric Environment $291-298 581
(1971).
582
Letter to the Editors
other cities and other averaging times. If a large number of replicate experiments were carried out the highest concentrations observed would show considerable variations but the average of ali of them would tend to the ~on~en~a~on having the occurrence frequency implied by the number of samples in each experiment. For a given number of repfications standard statistical te&niques can be used to estimate, for a given probability level, the confidence limits within which the concentration corresponding to the implied occurrence frequency is expected. With the result of only one experiment the confidence limits cannot be estimated but would be so wide as to have no practical value. In FIG. 1 and 2 of McGuire and Noli’s paper the highest concentration for each of the four averaging times observed in a singIe experiment are plotted without any indication of the confidence limits. In spite of this, the slope (their parameter b) of the line plotted through the data points is given to three signiiicant figures. The use of a single observation, for example the highest, is, amongst the several statistical indices of sample variability, the least etIicient. For McGuire and NoI1 to suggest that their index b is analogous to a standard deviation is misleading. On the other hand, had all the data been used to obtain an estimate of the frequency distribution for the papulation, the concentration corresponding to any frequency within the range of observation could have been estimated with greater confidence. The question posed by McGuire and No11 should be restated in the following way: Wow does the ~n~e~trai~on corresponding to a given occmrence frequency change when the averaging time is varied? The answer can be inferred from the considerable data published by LARSEN (1970).For example, TABLE f &es the cumulative frequency distribution of SO2 concentrations reported by Larsen for Chicago from 1962 to 1968. From this it can be seen that the concentrations which are (ppm) AT CHICAGO BY AVERAGING TIME AND FREQUENCY 1962-1968 (LARSEN, 1970)
TABLE 1. SULFUR DIOXIDE C~~X~NTRATION
Averaging time 0.01 5 min lh 8h I day 1 month 1Yr
Per cent of samples in which concentration is exceeded 0.1 1 10 30 50 70
1.31 1.11
1.01 0.95 0.74
0.67 0.64 0.56 0‘49
0.32 0.32 0.31 0.30 0.27 0.12
O”l6 0.16 0.16 0.16
0.08 0.08 0.09 0.09
0.03 0.03 0.04 0.05
90 per cent 0.01 0.01 0.01 0.02
exceeded with frequencies in the range 10-50 per cent are very nearly constant. For frequencies less than 10 per cent, the concentration tends to decrease while for frequencies greater than 50 per cent, the concentration increases with averaging time, For example, the concentrations averaged over 5 min, 1 h, 8 h and daily intervals which are exceeded 1 per cent of the time are, respectively, 0.67, 0.64, 0.56 and 9.49 ppm= The 7-yr reporting period contains over ~~,~ S-min intervafs so that there are over 7ooO 5-min-average samples with concentrations exceeding 0.67 ppm. Similarly there are more than 500 hourly average samples having concentrations exceeding 0.64 ppm. In both cases the sampling statistics should be good. On the other hand only 77 g-h and 26 daily samples exceeding the stated concentrations are available at the 1 per cent occurrence frequency. For the 0.1 and 0.01 per cent occurrence frequencies the corresponding number of samples are smaller by a factor 10 and a factor 100, respectively. There are, for example, only S hourly-average samples having concentrations exceeding 1. f 1 ppm. IIow significant the 25 per cent decrease in con~ntration may be would have to be established before condusions could be drawn. However, Larsen% data cfearfy show the coneenfration corresponding to a given occurrence frequency is much less dependent on the averaging time than the equation of McGuire and No11 implies. More importantly, the relationship is more complex than they suggest, depending on such arbitrary variables, amongst others, as the number of samples available and range of averaging times considered. Properties introduced by the sampling and computation procedures used should not be attributed to the phenomenon under study. B&gy alpd He&& P&&a B~vi&m Ctafk R&M Nucfem Lulmafories Chufk R&r, Oniario Canada.
P. J. BARRY
REFERENCE LARSEN R. I. (1970) Relating air pollutant effects to concentration
Ass. 20, 214-225.
and control. J. Air %Ilrri. Control