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Statistics of concentrations due to single air pollution sources to be applied in numerical modelling of pollutant dispersion
Atmospheric Environment Vol. 24A, No. 5, pp. 1029 1035, 1990 Printed in Great Britain.
0004 6981,'90 $3.(1(1+0.(X) Pergamon Press pie
STATISTICS OF CONCENTRATIONS D U E TO SINGLE AIR POLLUTION SOURCES TO BE APPLIED IN NUMERICAL MODELLING OF POLLUTANT DISPERSION SERGIU TUMANOV Institute of Meteorology and Hydrology, 97 Bucure~ti Ploie~ti Highway, 71581 Bucharest, Romania (First received 25 November 1988 and received for publication 6 October 1989)
Abstraet A test of goodness of fit based on rank statistics was applied to prove the appl:'~abilityof the Eggenberger-Polya discrete probability law to hourly SO2-concentrationsmeasured in the vicinityof single sources. With this end in view, the pollutant concentration was considered an integral quantity which may be accepted if one properly chooses the unit of measurement(in this case Mgm- 3) and if account is taken of the limited accuracy of measurements. The results of the test being satisfactory,even in the range of upber quantiles, the Eggenberger-Polya law was used in association with numerical modelling to estimate statistical parameters, e.g. quantiles, cumulative probabilities of threshold concentrations to be exceeded, and so on, in the grid points of a network covering the area of interest. This only needs accurate estimations of means and variances of the concentration series which can readily be obtained through routine air pollution dispersion modelling. Key word index: Eggenberger-Polya distributions; dispersion modelling; frequency distribution.
INTRODUCTION
One of the most important applications of air pollution dispersion models lies in estimating, on account of the climatic features of an area, the statistical characteristics of the concentration series due to industrial facilities prior to their coming into operation. The statistical characteristics thus assessed are compared to air quality standards which helps in anticipating the degree of change in the purity of atmosphere over the area being investigated. This application is supported by the acceptance of the hypothesis that dispersion models are able to render with sufficient accuracy the required statistical characteristics. It may be stated that airborne pollutant dispersion modelling is in full swing as a consequence of the theoretical and experimental progress in studying the planetary boundary layer (PBL) (Briggs and Binkowski, 1985). In spite of the conceptual limitations of its background, the K-equation is largely used in dispersion modelling at different scales due to its possibilities to take into account BL inhomogeneities as well as systems of pollutants in chemical and phase transformation (see, for example Karamchandani et al., 1988). Other dispersion models to be developed are the high-order closure models, those based on numerical simulation through statistical approaches (Monte Carlo), nonlocal models, etc. (Hanna, 1982; Briggs and Binkowski, 1985). However, as Hanna (1982) noted, the Gaussian plume model is used in routine applications more than any other model. The same conclusion can be drawn from the international questionnaire study made by
Szepesi (1983) on dispersion modelling: from the total of 69 models used in 23 countries for operative and regulatory purposes, 59 were based in 1983 on the concept of a Gaussian plume. Mention should be made that, as the study of the convective BL and the dispersion of the pollutants in it using lidar techniques showed, the vertical distribution of the concentration in this case is not Gaussian, but the ground level concentration distribution can be estimated using the Gaussian model (Briggs and Binkowski, 1985). Finally, it should be added that most diffusion experiments were analyzed through the Gaussian model resulting in parametrizations for cry and crz utilized in various operative dispersion models (Hanna, 1982). The following discussion will be confined to the Gaussian model. There are some doubts as far as the performances of this model in the range of upper quantiles are concerned. Among other authors, Cats (19781 found out that the matching between measured and estimated values decreases towards the upper quantiles, i.e. in the very range of greatest interest in applications. The quoted author explained this limitation by regarding the Gaussian model as representing stationary homogeneous conditions while the high quantiles correspond to extreme weather conditions occurring only a few days in a year. The conclusion, and other authors' too, is that the Gaussian model is suited only for the estimation of long-term average concentrations. Ludwick et al. (1980), measuring SO2 concentrations around a coal-fired power plant in the U.S., noticed that the diffusion models used underestimate the frequencies and in-
1029
1030
SERGIU TUMANOV
tensities of exposure to the pollutants in points remote from the power plant in the surrounding rough terrain. Considering these findings the idea seems reasonable that the use of the Gaussian model should be limited to the estimation of those statistical characteristics for which accurate estimations can be obtained, i.e. the arithmetic mean and mean square deviation. Other statistical characteristics, quantiles included, having to be computed by means of a suitable probability law (Tumanov, 1983). THEORETICAL DISTRIBUTIONS APPLICABLE TO CONCENTRATIONS DUE TO SINGLE SOURCES
The idea put forward in the end of the previous section is not a new one. Venkatram (1979), for instance, quoted Tikvart and Freas who had suggested in 1977 the estimation of maximum and second highest concentrations making use of probability laws consistent with frequency distributions of the concentrations provided by diffusion models. In particular, the exponential distribution was found to describe with sufficient accuracy the statistical behaviour of the concentrations due to single sources. A similar idea was forwarded by Cats (1978) who, without recommending a probability law proper, suggested estimating the 0.98-quantile from the long term arithmetic mean on account of the finding that the ratio of Co.98 to the average is approximately constant. Interest in this field was shown by Camps and Bonometti (1974); they found that in points subject to the influence of single sources the fl-distribution is applicable. The use of experimental data provided by a diffusion experiment carried on at a thermal power station site in Romania (Molju et al., 1980) helped testing the applicability of the Eggenberger-Polya discrete distribution (Tumanov, 1979). This was found to approximate sufficiently well the empirical distribution of the SO2 hourly concentrations measured during the experiment described below. Denoting by C the pollutant concentration and assuming that it is an integer (an assumption which may be accepted if the concentration is expressed in conveniently chosen units and if account is being taken of the limited precision of the measuring method whatever this may be), the Eggenberger-Polya cumulative probability is given by the sum
P[C<<.c]= ~. p(C)
(1)
C=0
where p(C) =
m(m + d) (m + 2 d ) . . . (m + [ C - 1] d) CI (1 + d) m/a+c C=1,2 ....
p(O)
1 (1 +d) "/a
(2)
where d = s 2 / m - 1 >0, m being the arithmetic mean the variance of the concentration series (Brooks and Carruthers, 1953). A simple recurrence formula between the probabilities p(C) makes the electronic computation for the sum (1) not too lengthy even for large values of the concentration. Mention should be made, though, that with such an 'unorthodox' use of a discrete probability law, where values of the random quantity are considered instead of numbers of occurrences, the problem of the measure unit arises; the probabilities (2) are not absolutely invariant to the change of unit; it is, therefore, advisable to use #g m - 3, a unit for which the tests proved satisfactory. In order to test the matching of the EggenbergerPolya law, some series of hourly SO2 concentrations were used provided by a diffusion experiment performed in six meteorological and air quality measurement campaigns between 1975 and 1978 at the Mintia thermal power plant (840 MW) located not far from the town Deva, in the valley of the Mure~ river. The valley is 1-3 km wide at the site of the power station, confined by hills 100-200 m high. There are three stacks, each of 220 m. The air samples for measuring the SO2 concentration have been taken with AGI and Glass Developments semiautomated sequential systems (both English made), operating with programmable timers attached for 1 h sampling durations (intervals between samplings: 2 h); the flow rate: 90 ¢ h - 1. The samples were analyzed using p-rosanilin method (standardized in Romania under the code numbers STAS 10194/75). The SO2 concentration has been measured in 15 points located around the power station at distances varying from 0.5 to 16km (Moljiu et al., 1980; Tumanov, 1979). The results of the goodness of the fit tests concerning the applicability of the Eggenberger-Polya probability law to the hourly SO2 concentration series measured in two of the 15 points are shown in Figs 1 and 2 (on log-Weibull probability plot). The test is based on rank statistics; the empirical distribution function should keep within the confidence domain assessed with the aid of the theoretical distribution function to conclude that, at the chosen significance level, there is no reason for the second to be rejected (Dunin-Barkovsky and Smirnov, 1955; Sneyers, 1963). For comparison purposes, in Figs 3 and 4 the results of the test based on rank statistics concerning the applicability of the log-normal distribution are presented on log-Gauss probability plot. The empirical distribution curve lies almost entirely out of the confidence domain. Hence the statistical tests are clearly favourable to the Eggenberger-Polya distribution. The same test was applied with promising results to some short series of 3-h average SO 2 concentrations measured in the vicinity of another power plant, Turceni in the Gorj county, as well as to a relatively long series of average concentrations over the same interval in the town Baia Mare, in an area situated under the influence of a single SO 2 source. and s2
Numerical modelling of pollutant dispersion
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Fig. 2. Confidence domain (5% significance level) for the EggenbergerPolya probability law (continuous lines) adjusted to the empirical distribution function (dotted line) of the hourly SO z concentrations in a point located at 0.5 km from the source; number of values: 226.
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Fig. 3. Confidence domain (5% significance level) for the log-normal probability law (continuous lines) adjusted to the same empirical distribution function as in Fig. 1.
1032
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Fig. 4. Confidence domain (5% significance level) for the log-normal probability law (continuous lines) adjusted to the same empirical distribution function as in Fig. 2.
"03
APPLICATIONS OF T H E E G G E N B E R G E R - P O L Y A DISTRIBUTION
To resume the idea of limiting the application of the Gaussian dispersion model to the assessment of statistical characteristics m and s, one can now specify that the Eggenberger-Polya probability could be an adequate tool in assessing the other statistical characteristics required in the case of single sources. In principle, there should be restrictions on the procedure according to the designed source siting in either an environment with pre-existing pollution background (urban environment) or an unpolluted environment. In the former case, the use of an adequate statistical law such as the log-normal distribution should be desirable instead of the probability (1) while in the latter case the Eggenberger-Polya probability should be resorted to. On another hand in many cases the influence of the desired source (or sources) upon larger areas is also of interest as they may cover urban--together with rural--type portions. Therefore the computation program should switch over from one probability law to another which would result not only in operative difficulties, but also in problems of continuity at the urban-rural boundary. In an attempt to avoid this type of difficulties the Eggenberger-Polya distribution was tested in an urban zone, where the log-normality of the measured SO2-concentrations (average daily values) was proved. It was found that the Eggenberger-Polya distribution approximates with sufficient accuracy the empirical distributions in such cases too. In other words, the distribution proved flexible equally fitting the frequency distributions with the mode situated at the end of the range (J-shaped) and the distributions with the mode within the range of the concentration. There have been so far few tests in the urban environment with the Eggenberger-Polya distribution and therefore they cannot yet represent the basis
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Fig. 5. Arithmetic mean of the concentrations, pg m- 3, estimated from the Gaussian model. of a final conclusion. It seems reasonable though that the procedure might possibly be applied without restrictions, i.e. in both cases of single and multiple source. The results of an application of the suggested procedure are presented in Figs 5-10. The NH 3 concentration due to 13 sources of 40-70 m high with emission rates of 1-12 g s -1 placed within a designed chemical works site are analyzed in 1 km- (0.5 kmwithin a smaller square centred on the works site) spaced points of a grid were computed through the Gaussian model.
Numerical modelling of pollutant dispersion
1033
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Fig. 6. Mean square deviation of the concentrations, /~g m-3, estimated from the Gaussian model.
Fig. 8. Frequency of equalling or exceeding 300 #g m 3, % , estimated from the Eggenberger-Polya probability law.
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Fig. 7. Frequency of equalling or exceeding 300 #g m - 3 %, estimated from the Gaussian model.
Fig. 9. The 0.999-quantile of the distribution (#g m 3} estimated from the Eggenberger-Polya probability law.
The long-term arithmetic mean and mean square deviation are shown in Figs 5 and 6, respectively. The expected probabilities of equalling or exceeding 300 p g m -3 concentration (the 30-min air quality
standard for N H 3 in Romania) estimated through the Gaussian model and computed from the Eggenberger-Polya probability law are shown in Figs 7 and 8, respectively. It should be noted that the
1034
SERGIU TUMANOV
1300#gm -3 while the concentrations computed directly from the Gaussian model do not exceed 750 ~g m - 3. Figure 9 shows the area where the 300/~gm -a concentration is exceeded, i.e. with a probability of 1-0.999=0.001 or 0.1% (cf. the 0.1% isoline in Fig. 8). As the computation programme supplies a set of 16 quantiles, there is a possibility to assess the areas where the critical concentration is expected to be equalled or exceeded with given probabilities.
CONCLUSIONS
Fig. 10. The 0.999943-quantile of the distribution (/~gm-3), corresponding to the frequency of one occurrence per year (annual maximum), estimated from the Eggenberger-Polya probability law.
Eggenberger-Polya probabilities are greater than those estimated by means of the Gaussian model for locations remote from the source while the probabilities for locations close to the source are almost identical. This observation does not mean a validation of the procedure based on the EggenbergerPolya probability as there is not a comparison to experimental data but a possible index that the procedure suggested is appropriate to yield more realistic values, moreover if Ludwick's observation, that the Gaussian type models underestimate frequencies in points more distant from the source, were considered. At last, two quantiles computed through the Eggenberger-Polya distribution, i.e. those of 0.999 and 0.99993 ranks are shown in Figs 9 and 10. The latter rank was chosen according to the way of setting an air quality standard; a frequency of once a year is assigned to the 30-min annual maximum, which means a frequency of equalling or exceeding the maximum of 1/17520=0.000057 (there are 17,520 intervals of 30-min within a year), the complementary frequency being 0.999943. Following the 300 #g m-3 isoline in Fig. 10 one can easily notice that it has essentially the same configuration as the 0.01% isoline in Fig. 8 which comes out obviously as the 0.01% probability to equal or exceed the 300/~g m - 3 concentration is but the rounded value of the 0.000057 frequency transformed into the per cent of one occurrence a year. The same figure shows two nuclei of maximum Concentration out of which one reaches
As a result of the tests performed it may be stated that the Eggenberger-Polya probability law approximates with sufficient accuracy the empirical distributions of the concentration series due to air pollution single sources. This law will help compute some statistical characteristics of the concentrations after the long-term arithmetic mean and mean square deviation were estimated by using a Gaussian type dispersion model. The estimations thus obtained are plausible in magnitude and seem to be more realistic than those directly supplied by routine models. The results presented in this paper cannot stand for a validation of the procedure as the experimental data available were not sufficient. The objective of the paper was only to illustrate the capabilities of the Eggenberger-Polya statistical distribution for the concentrations due to single sources in view of comparing these concentrations to the air quality standards. Acknowledgements--The author is gratefulto his colleagueDr N. Romanof and to Mr V. Bo~caiu from the Bucharest Mathematical Statistics Centre for their helpful suggestions in working out the final version of the manuscript.
REFERENCES
Briggs G. A. and Binkowski F. S. (1985) Research on diffusionin atmospheric boundary layers:a position paper on status and needs. EPA report 600/3-85/072. Brooks C. E. P. and Carruthers N. (1953) Handbook of Statistical Methods in Meteorology, pp. 315-318. Her Majesty's Stationery Office, London. Camps M. and Bonometti G. (1974) Mod61e pour la repr6sentation de la pollution sur le site de LACQ. 11-brae Colloque international sur les atmosphbres pollubes, 8-10 Mai, Paris, pp 13.1-13.10. Cats G. J. (1978) A simple method for quick estimates of frequency distributions of air pollution concentrations from long-term average concentrations. Proc. WMO Syrup. on Boundary Layer Phys. Applied to Specific Problems of Air Pollution, 19-23 June, Norrkfping. WMO No. 510, pp 157-162. Dunin-BarkovskyI. V. and Smirnov N. V. (1955) The Theory of Probability and Technical Applications of the Mathematical Statistics (General Part). Gos. Izd. Tekhniko-Theoreticheskoy Literatury, Moscow (in Russian).
Numerical modelling of pollutant dispersion Hanna S. R. (1982) Review of atmospheric diffusion models for regulatory applications. WMO Technical Note No. 177. Karamchandani P., Venkatram A. and Misra P. K. (1988) Simulations of an acid deposition episode with a comprehensive long-range transport model. Proc. of the X V l t h NATO/CCMS International Technical Meeting on Air Pollution Modeling and its Application, April 1987, Lindau, F. R. G., pp. 403-413. Plenum Press, New York. Ludwick J. D., Weber D. B., Olsen K. B. and Garcia S. R. (1980) Air quality measurements in the coal fired power plant environment of Colstrip, Montana. Atmospheric Environment 14, 523-532. Mo[iu C., Romanof N. and Schmidt R, (1980) Studiul dispersiei SO2 in zona Centralei termoelectrice Mintia (SO 2 dispersion study in the thermal power station area of Mintia). Energetica 28, 493-495 (in Romanian).
1035
Sneyers R. (1963) Du test de validit~ d'un ajustement bas6 sur les fonctions de l'ordre des observations. Publications S6rie B No. 39, Institut Royal M&6orologique de Belgique. Szepesi D. (1983) Assessment on air quality simulation models--an international study. Proc. of the Vlth World Congress on Air Quality, May 1983, Paris, Vol. 3, pp. 263-269. Tumanov S. (1979) Frequency distributions of gas pollutant concentrations measured in the neighbourhood of a high single source. Met. Hydrol., Buch. 2, 11-16. Tumanov S. (1983) Aplica[ii ale metodelor statistice la evaluarea gradului de poluare a aerului (Applications of statistics in the assessment of air pollution level). Presented at the Annual Scientific Session of the Institute of Meteorology and Hydrology, Bucharest (manuscript in Romanian). Venkatram A. (1979) Applications of pollutant frequency distributions. J. Air Pollut. Control Ass. 29, 251 253.
0004 6981,'90 $3.(1(1+0.(X) Pergamon Press pie
STATISTICS OF CONCENTRATIONS D U E TO SINGLE AIR POLLUTION SOURCES TO BE APPLIED IN NUMERICAL MODELLING OF POLLUTANT DISPERSION SERGIU TUMANOV Institute of Meteorology and Hydrology, 97 Bucure~ti Ploie~ti Highway, 71581 Bucharest, Romania (First received 25 November 1988 and received for publication 6 October 1989)
Abstraet A test of goodness of fit based on rank statistics was applied to prove the appl:'~abilityof the Eggenberger-Polya discrete probability law to hourly SO2-concentrationsmeasured in the vicinityof single sources. With this end in view, the pollutant concentration was considered an integral quantity which may be accepted if one properly chooses the unit of measurement(in this case Mgm- 3) and if account is taken of the limited accuracy of measurements. The results of the test being satisfactory,even in the range of upber quantiles, the Eggenberger-Polya law was used in association with numerical modelling to estimate statistical parameters, e.g. quantiles, cumulative probabilities of threshold concentrations to be exceeded, and so on, in the grid points of a network covering the area of interest. This only needs accurate estimations of means and variances of the concentration series which can readily be obtained through routine air pollution dispersion modelling. Key word index: Eggenberger-Polya distributions; dispersion modelling; frequency distribution.
INTRODUCTION
One of the most important applications of air pollution dispersion models lies in estimating, on account of the climatic features of an area, the statistical characteristics of the concentration series due to industrial facilities prior to their coming into operation. The statistical characteristics thus assessed are compared to air quality standards which helps in anticipating the degree of change in the purity of atmosphere over the area being investigated. This application is supported by the acceptance of the hypothesis that dispersion models are able to render with sufficient accuracy the required statistical characteristics. It may be stated that airborne pollutant dispersion modelling is in full swing as a consequence of the theoretical and experimental progress in studying the planetary boundary layer (PBL) (Briggs and Binkowski, 1985). In spite of the conceptual limitations of its background, the K-equation is largely used in dispersion modelling at different scales due to its possibilities to take into account BL inhomogeneities as well as systems of pollutants in chemical and phase transformation (see, for example Karamchandani et al., 1988). Other dispersion models to be developed are the high-order closure models, those based on numerical simulation through statistical approaches (Monte Carlo), nonlocal models, etc. (Hanna, 1982; Briggs and Binkowski, 1985). However, as Hanna (1982) noted, the Gaussian plume model is used in routine applications more than any other model. The same conclusion can be drawn from the international questionnaire study made by
Szepesi (1983) on dispersion modelling: from the total of 69 models used in 23 countries for operative and regulatory purposes, 59 were based in 1983 on the concept of a Gaussian plume. Mention should be made that, as the study of the convective BL and the dispersion of the pollutants in it using lidar techniques showed, the vertical distribution of the concentration in this case is not Gaussian, but the ground level concentration distribution can be estimated using the Gaussian model (Briggs and Binkowski, 1985). Finally, it should be added that most diffusion experiments were analyzed through the Gaussian model resulting in parametrizations for cry and crz utilized in various operative dispersion models (Hanna, 1982). The following discussion will be confined to the Gaussian model. There are some doubts as far as the performances of this model in the range of upper quantiles are concerned. Among other authors, Cats (19781 found out that the matching between measured and estimated values decreases towards the upper quantiles, i.e. in the very range of greatest interest in applications. The quoted author explained this limitation by regarding the Gaussian model as representing stationary homogeneous conditions while the high quantiles correspond to extreme weather conditions occurring only a few days in a year. The conclusion, and other authors' too, is that the Gaussian model is suited only for the estimation of long-term average concentrations. Ludwick et al. (1980), measuring SO2 concentrations around a coal-fired power plant in the U.S., noticed that the diffusion models used underestimate the frequencies and in-
1029
1030
SERGIU TUMANOV
tensities of exposure to the pollutants in points remote from the power plant in the surrounding rough terrain. Considering these findings the idea seems reasonable that the use of the Gaussian model should be limited to the estimation of those statistical characteristics for which accurate estimations can be obtained, i.e. the arithmetic mean and mean square deviation. Other statistical characteristics, quantiles included, having to be computed by means of a suitable probability law (Tumanov, 1983). THEORETICAL DISTRIBUTIONS APPLICABLE TO CONCENTRATIONS DUE TO SINGLE SOURCES
The idea put forward in the end of the previous section is not a new one. Venkatram (1979), for instance, quoted Tikvart and Freas who had suggested in 1977 the estimation of maximum and second highest concentrations making use of probability laws consistent with frequency distributions of the concentrations provided by diffusion models. In particular, the exponential distribution was found to describe with sufficient accuracy the statistical behaviour of the concentrations due to single sources. A similar idea was forwarded by Cats (1978) who, without recommending a probability law proper, suggested estimating the 0.98-quantile from the long term arithmetic mean on account of the finding that the ratio of Co.98 to the average is approximately constant. Interest in this field was shown by Camps and Bonometti (1974); they found that in points subject to the influence of single sources the fl-distribution is applicable. The use of experimental data provided by a diffusion experiment carried on at a thermal power station site in Romania (Molju et al., 1980) helped testing the applicability of the Eggenberger-Polya discrete distribution (Tumanov, 1979). This was found to approximate sufficiently well the empirical distribution of the SO2 hourly concentrations measured during the experiment described below. Denoting by C the pollutant concentration and assuming that it is an integer (an assumption which may be accepted if the concentration is expressed in conveniently chosen units and if account is being taken of the limited precision of the measuring method whatever this may be), the Eggenberger-Polya cumulative probability is given by the sum
P[C<<.c]= ~. p(C)
(1)
C=0
where p(C) =
m(m + d) (m + 2 d ) . . . (m + [ C - 1] d) CI (1 + d) m/a+c C=1,2 ....
p(O)
1 (1 +d) "/a
(2)
where d = s 2 / m - 1 >0, m being the arithmetic mean the variance of the concentration series (Brooks and Carruthers, 1953). A simple recurrence formula between the probabilities p(C) makes the electronic computation for the sum (1) not too lengthy even for large values of the concentration. Mention should be made, though, that with such an 'unorthodox' use of a discrete probability law, where values of the random quantity are considered instead of numbers of occurrences, the problem of the measure unit arises; the probabilities (2) are not absolutely invariant to the change of unit; it is, therefore, advisable to use #g m - 3, a unit for which the tests proved satisfactory. In order to test the matching of the EggenbergerPolya law, some series of hourly SO2 concentrations were used provided by a diffusion experiment performed in six meteorological and air quality measurement campaigns between 1975 and 1978 at the Mintia thermal power plant (840 MW) located not far from the town Deva, in the valley of the Mure~ river. The valley is 1-3 km wide at the site of the power station, confined by hills 100-200 m high. There are three stacks, each of 220 m. The air samples for measuring the SO2 concentration have been taken with AGI and Glass Developments semiautomated sequential systems (both English made), operating with programmable timers attached for 1 h sampling durations (intervals between samplings: 2 h); the flow rate: 90 ¢ h - 1. The samples were analyzed using p-rosanilin method (standardized in Romania under the code numbers STAS 10194/75). The SO2 concentration has been measured in 15 points located around the power station at distances varying from 0.5 to 16km (Moljiu et al., 1980; Tumanov, 1979). The results of the goodness of the fit tests concerning the applicability of the Eggenberger-Polya probability law to the hourly SO2 concentration series measured in two of the 15 points are shown in Figs 1 and 2 (on log-Weibull probability plot). The test is based on rank statistics; the empirical distribution function should keep within the confidence domain assessed with the aid of the theoretical distribution function to conclude that, at the chosen significance level, there is no reason for the second to be rejected (Dunin-Barkovsky and Smirnov, 1955; Sneyers, 1963). For comparison purposes, in Figs 3 and 4 the results of the test based on rank statistics concerning the applicability of the log-normal distribution are presented on log-Gauss probability plot. The empirical distribution curve lies almost entirely out of the confidence domain. Hence the statistical tests are clearly favourable to the Eggenberger-Polya distribution. The same test was applied with promising results to some short series of 3-h average SO 2 concentrations measured in the vicinity of another power plant, Turceni in the Gorj county, as well as to a relatively long series of average concentrations over the same interval in the town Baia Mare, in an area situated under the influence of a single SO 2 source. and s2
Numerical modelling of pollutant dispersion
1031
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F~"ob~lity % Fig. 1. Confidence domain (5% significance level) for the EggenbergerPolya probability law (continuous lines) adjusted to the empirical distribution function (dotted line) of the hourly SO2 concentrations in a point located at 12.8 km from the source; number of values: 390.
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Fig. 2. Confidence domain (5% significance level) for the EggenbergerPolya probability law (continuous lines) adjusted to the empirical distribution function (dotted line) of the hourly SO z concentrations in a point located at 0.5 km from the source; number of values: 226.
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Fig. 3. Confidence domain (5% significance level) for the log-normal probability law (continuous lines) adjusted to the same empirical distribution function as in Fig. 1.
1032
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Fig. 4. Confidence domain (5% significance level) for the log-normal probability law (continuous lines) adjusted to the same empirical distribution function as in Fig. 2.
"03
APPLICATIONS OF T H E E G G E N B E R G E R - P O L Y A DISTRIBUTION
To resume the idea of limiting the application of the Gaussian dispersion model to the assessment of statistical characteristics m and s, one can now specify that the Eggenberger-Polya probability could be an adequate tool in assessing the other statistical characteristics required in the case of single sources. In principle, there should be restrictions on the procedure according to the designed source siting in either an environment with pre-existing pollution background (urban environment) or an unpolluted environment. In the former case, the use of an adequate statistical law such as the log-normal distribution should be desirable instead of the probability (1) while in the latter case the Eggenberger-Polya probability should be resorted to. On another hand in many cases the influence of the desired source (or sources) upon larger areas is also of interest as they may cover urban--together with rural--type portions. Therefore the computation program should switch over from one probability law to another which would result not only in operative difficulties, but also in problems of continuity at the urban-rural boundary. In an attempt to avoid this type of difficulties the Eggenberger-Polya distribution was tested in an urban zone, where the log-normality of the measured SO2-concentrations (average daily values) was proved. It was found that the Eggenberger-Polya distribution approximates with sufficient accuracy the empirical distributions in such cases too. In other words, the distribution proved flexible equally fitting the frequency distributions with the mode situated at the end of the range (J-shaped) and the distributions with the mode within the range of the concentration. There have been so far few tests in the urban environment with the Eggenberger-Polya distribution and therefore they cannot yet represent the basis
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l\
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o,.
Fig. 5. Arithmetic mean of the concentrations, pg m- 3, estimated from the Gaussian model. of a final conclusion. It seems reasonable though that the procedure might possibly be applied without restrictions, i.e. in both cases of single and multiple source. The results of an application of the suggested procedure are presented in Figs 5-10. The NH 3 concentration due to 13 sources of 40-70 m high with emission rates of 1-12 g s -1 placed within a designed chemical works site are analyzed in 1 km- (0.5 kmwithin a smaller square centred on the works site) spaced points of a grid were computed through the Gaussian model.
Numerical modelling of pollutant dispersion
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101 o~ ~
~5
Fig. 6. Mean square deviation of the concentrations, /~g m-3, estimated from the Gaussian model.
Fig. 8. Frequency of equalling or exceeding 300 #g m 3, % , estimated from the Eggenberger-Polya probability law.
N
5O
\
0
1
2
sO
3kin
Fig. 7. Frequency of equalling or exceeding 300 #g m - 3 %, estimated from the Gaussian model.
Fig. 9. The 0.999-quantile of the distribution (#g m 3} estimated from the Eggenberger-Polya probability law.
The long-term arithmetic mean and mean square deviation are shown in Figs 5 and 6, respectively. The expected probabilities of equalling or exceeding 300 p g m -3 concentration (the 30-min air quality
standard for N H 3 in Romania) estimated through the Gaussian model and computed from the Eggenberger-Polya probability law are shown in Figs 7 and 8, respectively. It should be noted that the
1034
SERGIU TUMANOV
1300#gm -3 while the concentrations computed directly from the Gaussian model do not exceed 750 ~g m - 3. Figure 9 shows the area where the 300/~gm -a concentration is exceeded, i.e. with a probability of 1-0.999=0.001 or 0.1% (cf. the 0.1% isoline in Fig. 8). As the computation programme supplies a set of 16 quantiles, there is a possibility to assess the areas where the critical concentration is expected to be equalled or exceeded with given probabilities.
CONCLUSIONS
Fig. 10. The 0.999943-quantile of the distribution (/~gm-3), corresponding to the frequency of one occurrence per year (annual maximum), estimated from the Eggenberger-Polya probability law.
Eggenberger-Polya probabilities are greater than those estimated by means of the Gaussian model for locations remote from the source while the probabilities for locations close to the source are almost identical. This observation does not mean a validation of the procedure based on the EggenbergerPolya probability as there is not a comparison to experimental data but a possible index that the procedure suggested is appropriate to yield more realistic values, moreover if Ludwick's observation, that the Gaussian type models underestimate frequencies in points more distant from the source, were considered. At last, two quantiles computed through the Eggenberger-Polya distribution, i.e. those of 0.999 and 0.99993 ranks are shown in Figs 9 and 10. The latter rank was chosen according to the way of setting an air quality standard; a frequency of once a year is assigned to the 30-min annual maximum, which means a frequency of equalling or exceeding the maximum of 1/17520=0.000057 (there are 17,520 intervals of 30-min within a year), the complementary frequency being 0.999943. Following the 300 #g m-3 isoline in Fig. 10 one can easily notice that it has essentially the same configuration as the 0.01% isoline in Fig. 8 which comes out obviously as the 0.01% probability to equal or exceed the 300/~g m - 3 concentration is but the rounded value of the 0.000057 frequency transformed into the per cent of one occurrence a year. The same figure shows two nuclei of maximum Concentration out of which one reaches
As a result of the tests performed it may be stated that the Eggenberger-Polya probability law approximates with sufficient accuracy the empirical distributions of the concentration series due to air pollution single sources. This law will help compute some statistical characteristics of the concentrations after the long-term arithmetic mean and mean square deviation were estimated by using a Gaussian type dispersion model. The estimations thus obtained are plausible in magnitude and seem to be more realistic than those directly supplied by routine models. The results presented in this paper cannot stand for a validation of the procedure as the experimental data available were not sufficient. The objective of the paper was only to illustrate the capabilities of the Eggenberger-Polya statistical distribution for the concentrations due to single sources in view of comparing these concentrations to the air quality standards. Acknowledgements--The author is gratefulto his colleagueDr N. Romanof and to Mr V. Bo~caiu from the Bucharest Mathematical Statistics Centre for their helpful suggestions in working out the final version of the manuscript.
REFERENCES
Briggs G. A. and Binkowski F. S. (1985) Research on diffusionin atmospheric boundary layers:a position paper on status and needs. EPA report 600/3-85/072. Brooks C. E. P. and Carruthers N. (1953) Handbook of Statistical Methods in Meteorology, pp. 315-318. Her Majesty's Stationery Office, London. Camps M. and Bonometti G. (1974) Mod61e pour la repr6sentation de la pollution sur le site de LACQ. 11-brae Colloque international sur les atmosphbres pollubes, 8-10 Mai, Paris, pp 13.1-13.10. Cats G. J. (1978) A simple method for quick estimates of frequency distributions of air pollution concentrations from long-term average concentrations. Proc. WMO Syrup. on Boundary Layer Phys. Applied to Specific Problems of Air Pollution, 19-23 June, Norrkfping. WMO No. 510, pp 157-162. Dunin-BarkovskyI. V. and Smirnov N. V. (1955) The Theory of Probability and Technical Applications of the Mathematical Statistics (General Part). Gos. Izd. Tekhniko-Theoreticheskoy Literatury, Moscow (in Russian).
Numerical modelling of pollutant dispersion Hanna S. R. (1982) Review of atmospheric diffusion models for regulatory applications. WMO Technical Note No. 177. Karamchandani P., Venkatram A. and Misra P. K. (1988) Simulations of an acid deposition episode with a comprehensive long-range transport model. Proc. of the X V l t h NATO/CCMS International Technical Meeting on Air Pollution Modeling and its Application, April 1987, Lindau, F. R. G., pp. 403-413. Plenum Press, New York. Ludwick J. D., Weber D. B., Olsen K. B. and Garcia S. R. (1980) Air quality measurements in the coal fired power plant environment of Colstrip, Montana. Atmospheric Environment 14, 523-532. Mo[iu C., Romanof N. and Schmidt R, (1980) Studiul dispersiei SO2 in zona Centralei termoelectrice Mintia (SO 2 dispersion study in the thermal power station area of Mintia). Energetica 28, 493-495 (in Romanian).
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Sneyers R. (1963) Du test de validit~ d'un ajustement bas6 sur les fonctions de l'ordre des observations. Publications S6rie B No. 39, Institut Royal M&6orologique de Belgique. Szepesi D. (1983) Assessment on air quality simulation models--an international study. Proc. of the Vlth World Congress on Air Quality, May 1983, Paris, Vol. 3, pp. 263-269. Tumanov S. (1979) Frequency distributions of gas pollutant concentrations measured in the neighbourhood of a high single source. Met. Hydrol., Buch. 2, 11-16. Tumanov S. (1983) Aplica[ii ale metodelor statistice la evaluarea gradului de poluare a aerului (Applications of statistics in the assessment of air pollution level). Presented at the Annual Scientific Session of the Institute of Meteorology and Hydrology, Bucharest (manuscript in Romanian). Venkatram A. (1979) Applications of pollutant frequency distributions. J. Air Pollut. Control Ass. 29, 251 253.