- Email: [email protected]
Accounting for averaging time in air pollution modeling
Atmospheric Environment 36 (2002) 2165–2170
Accounting for averaging time in air pollution modeling Akula Venkatram* College of Engineering, University of California, Riverside, CA 92521, USA Received 28 August 2001; accepted 8 February 2002
Abstract This paper examines the validity of a commonly used expression to estimate concentrations for averaging times that are much smaller than that corresponding to the model estimate. These ‘‘peak’’ concentrations, which can be several times larger than the model estimate, are required in applications such as odor assessment. We show that this expression cannot be justified, and that information on short-term concentrations is only contained in the probability distribution of time-averaged concentrations. The paper proposes a simple model of concentration time series to examine the effect of averaging time on concentration probability distributions. The results from the model are compared with previous theoretical and experimental studies. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Averaging time; Odor assessment; Peak concentrations; Concentration distributions; Air pollution modeling
1. Motivation Several applications require estimates of concentrations averaged over shorter time periods that those estimated with models such as the Industrial Source Complex (ISC) model, used in the United States for regulatory applications. For example, predicting odor concentrations requires converting ISC 1 h estimates to values that correspond to averaging times of a few seconds to few minutes. Common practice (Engel et al., 1997; Scire et al., 2000) consists of converting model predicted estimates to shorter time periods using equations of the form: p T Cs ¼ Cm ; ð1Þ Ts where Cs is the required concentration for the short averaging time, Ts ; and Cm is the model estimated concentration corresponding to the averaging time, T; usually taken to be 1 h. The values of the exponent, p; used in the literature, range from 0.2 to 0.5. A second approach to modeling concentrations at short-time scales is based on Gifford’s (1959) fluctuating *Fax: +1-909-787-2899. E-mail address: [email protected] (A. Venkatram).
plume model. Mussio et al. (2001) have used the model to estimate ‘‘instantaneous’’ concentrations, which are then used to calculate odor impact frequencies around industrial facilities. Sykes (1984) has adapted Gifford’s model to examine the effect averaging time on the variance of time-averaged concentrations. The variance can, in principle, be used to estimate the occurrence of peak concentrations of concern. The focus of this paper is the use of modeled concentrations to estimate shortterm concentrations, and not the equally important problem of estimating instantaneous concentrations.
2. The problem In order to understand the meaning of equations such as Eq. (1), it is necessary to be clear about what the ideal model is designed to predict. Assume that the model uses a set of inputs, which will vary from model to model. Now imagine an infinite set of experiments in which these model inputs are kept constant, and make concentration observations during each one of these experiments, which we assume lasts for 1 h. Because the model inputs include only some of the variables that govern the concentrations, the average concentrations will differ from experiment to experiment. Most
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A. Venkatram / Atmospheric Environment 36 (2002) 2165–2170
dispersion models, such as ISC, are designed to estimate the average over the ensemble of concentration averages from these experiments. This means that any individual observation will always differ from the model predicted ensemble mean (see Venkatram, 1984). While it is possible to estimate the expected deviation of these observations from the model predicted ensemble mean, it is not possible to predict the actual observation during an individual hour. We can always string together 1 h concentration time series from each of the experiments to construct an infinitely long time series, which corresponds to a particular set of model inputs. The best that the model can do is to provide a time average of this time series. We see that breaking this time series into time intervals of less than an hour before deriving an overall time average is not going to change the value of the ensemble average, which is the average over the infinite time series. Thus, the ensemble average, which is what the model can predict, does not depend on the averaging time. On the other hand the distribution of concentrations will depend on the averaging time. We are more likely to see higher concentrations at smaller averaging times. It is this concept that equations like Eq. (1) are trying to capture. However, in the form that it is written, it implies that the ensemble average is a function of averaging time. The concentration, Cs ; in Eq. (1), has to be some sort of ensemble average. Otherwise, it has no meaning. The basic motivation behind using Eq. (1) is to incorporate the fact that the likelihood of observing a ‘‘high’’ concentration increases as the averaging time decreases. As formulated, all it does is to multiply the mean concentration by an arbitrary factor to yield a concentration that cannot be interpreted in a meaningful way. The next section uses a simple model to explain some of these concepts better.
3. Simulating the concentration time series Most instantaneous concentration time series are highly intermittent with periods of spiky concentrations interspersed among periods of zero concentrations. We will model this as periods in which the concentration is constant at a level Cp separated by intervals of zero concentrations. This time series was used by Venkatram (1979) to examine the effects of time averaging on concentration variances. In the analysis, the time scale governing the concentration fluctuations was assumed to be proportional to a Eulerian time scale, which is a length scale of turbulent velocity fluctuations, divided by the mean wind speed. Sykes (1984) improved on Venkatram’s (1979) analysis by using Gifford’s (1959) fluctuating plume model to derive a relationship between the concentration time scale and
the Eulerian time scale. In the analysis to follow, we will compare our new results with those from Sykes (1984). The proposed model assumes that the concentration ‘‘windows’’ have a constant width Dt; but the spacing between these windows is a random variable with an average value of t: Fig. 1 depicts this simple model of concentrations. To make progress, we will assume that each time interval Dt is a Bernoulli trial in which the occurrence of a non-zero concentration has the probability g: Then, the number of non-zero concentration events in a number of trials is governed by a Binomial distribution. While this model is a major simplification of reality, it contains the essential features that allow us to investigate the effects of averaging time on concentrations. Consider a sampling interval of length T: This interval can contain a maximum of N ¼ T=Dt concentration ‘‘events’’, each of which can either record a zero concentration or Cp : Then, the ensemble mean concentration for this model is Cm ¼ Cp g:
ð2Þ
g is referred to as intermittency in the literature. The ensemble variance of the concentration time series is seen to be (Venkatram, 1979) /ðC Cm Þ2 S 1 ¼ 1: 2 Cm g
ð3Þ
Then, the average time interval between concentration events, t; can be defined through g¼
Dt : Dt þ t
ð4Þ
We can estimate the instantaneous plume time scale, Dt; as si ð5Þ DtB ; u where si is the dimension of the instantaneous plume, and u is the mean wind speed. The peak concentration
τ Concentration
∆t Cp
Time
Fig. 1. Idealized time series used to estimate the effect of averaging time on concentration distributions. The concentrations alternate between peak values, Cp ; and zero values. The width of the concentration event is constant at Dt; while the spacing between concentration events is a random variable with an average value of t:
A. Venkatram / Atmospheric Environment 36 (2002) 2165–2170
associated with this plume is then Cp ¼
Q ; pus2i
ð6Þ
where Q is the emission rate. By assuming that the mean concentration distribution is a symmetric Gaussian, the peak-to-mean ratio can be written as ! ! 2s2p Cp r2 exp ; ð7Þ ¼ Cm 2s2p s2i where sp is the standard deviation of the ensemble mean concentration distribution, and r is the distance of the receptor from the mean plume centerline. This expression neglects concentration fluctuations at Batchelor length scales within the instantaneous plume. Then, Eq. (4) yields " ! ! # 2s2p r2 t ¼ Dt exp 1 ð8Þ 2s2p s2i and on the plume centerline, tEDt
s2p s2i
¼
s2p : si u
ð9Þ
We see that t increases with the ratio of the total plume spread to the instantaneous spread, and with the distance from the plume centerline. Both the governing time scales, Dt and t; that govern the concentration distribution, are dependent on plume properties, and thus on the receptor location relative to the point of tracer release. The time scale, Dt; increases with distance from the source as the instantaneous plume spread increases. At short distances, t is much larger than Dt because the instantaneous plume spread is much smaller than the total plume spread; this difference becomes smaller with distance as the two plume spreads become comparable.
4. The effects of averaging time It is easy to estimate the effect of averaging time on the concentration variance by noting that the timeaveraged concentration when n non-zero values occur during the averaging interval, T; is C% T ðnÞ ¼
nCp Dt : T
ð10Þ
The time-averaged concentration, C% T ðnÞ; varies between averaging intervals because n varies. Then, the standard deviation, std, of C% T ðnÞ about the ensemble mean concentration is stdðC% T ðnÞÞ sTC ¼
stdðnÞCp Dt : T
ð11Þ
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For the binomial distribution, the standard deviation of n is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi stdðnÞ ¼ Ngð1 gÞ; ð12Þ where N ¼ T=Dt is the number of concentration events (Bernoulli trials) in the sampling interval. If we substitute Eq. (12) into Eq. (11) and use Eq. (4), we obtain the result t 1=2 sTC ¼ : ð13Þ T Cm This equation appears to tell us that standard deviation of the time-averaged concentrations is proportional to the ensemble mean, and is independent of the standard deviation of the instantaneous concentrations. Eq. (13) would seem to agree with the result obtained by Sykes (1984, Eq. (16)), who suggests that in a highly intermittent plume, the time-averaged concentration variance has ‘‘only a logarithmic dependence on the ensemble variance’’, and ‘‘is more nearly proportional to the square of the ensemble mean’’. But Eq. (13) does not support this result from Sykes (1984) because the concentration return time scale, t; is not a flow variable, but depends on the intermittency (see Eq. (4)), and hence the instantaneous concentration variance. In fact, if we combine Eqs. (3), (4) and (13), we find that 1=2 sTC Dt 1 ¼ ¼ 1=2 : ð14Þ T N /ðC Cm Þ2 S1=2 This result, which is consistent with the assumption of independence of concentration events, says that the instantaneous plume time scale, Dt rather than the return time scale, t; governs the effect of averaging time on concentration variance. Eq. (14) implies that, unless the averaging time is comparable to the instantaneous plume time scale, Dt; measured variances are likely to be substantially smaller than the variance of the instantaneous concentrations. This suggests that unintentional time or space averaging induced by measuring instruments can substantially reduce measured concentration statistics. Eq. (13) allows us to relate the time-averaged standard deviations of concentrations at two different averaging times (at the same location) through the simple relationship 1=2 sTC1 T2 ¼ : ð15Þ T1 sTC2 This equation is very different from Eq. (1), which attempts to relate ensemble means at two different averaging times; the ensemble means are not functions of averaging times. Eq. (15) tells us that the likelihood of seeing high concentrations relative to the mean increases as the averaging time decreases. This can be seen more clearly by estimating the effects of averaging time on the
The concentration, C% T ðnÞ; is related to the number of events, n; that occur in T through Eq. (10). The probability that a specified concentration, C% T ðnÞ; occurs is equivalent to the probability that n events occurs in T: If we assume that these events follow a binomial distribution, this probability is ! N n PðC ¼ C% T ðnÞÞ ¼ g ð1 gÞNn ; ð16Þ n where the binomial coefficient is ! N N! ¼ n!ðN nÞ! n
ð17Þ
and N ¼ T=Dt: Figs. 2–5 show the behavior of the probability distribution for different averaging times, ranging from 5 times to 40 times the time scale, Dt: The intermittency is taken to be 0.1. The binomial distributions, shown in the figures, refer to discrete values of events, which have been converted into concentrations normalized by the mean. Although they do not mimic observed continuous distributions, they do illustrate the effects of averaging time on concentrations. In principle, the distribution can be converted approximately to a continuous distribution, but this does not add anything to the illustrative analysis of this paper. When T ¼ 5Dt; a maximum of only 5 concentration events can contribute to the time average. Thus, we see in Fig. 2 that the probability of observing a zero concentration is high at 60%. At the same time, the probability of seeing a concentration that is about 2 times the ensemble mean is over 30%, and that of
Probability (%)
70
45 40 35 30 25 20 15 10 5 0
0
1
4 2 3 Concentration/Mean
5
6
Fig. 3. Concentration distribution for averaging time equal to 10 times the instantaneous plume time scale. The plume intermittency is 0.1.
30 Probability (%)
5. Probability distribution of time-averaged concentrations
25 20 15 10 5 0
0
0.5
1 1.5 2 Concentration/Mean
2.5
3
Fig. 4. Concentration distribution for averaging time equal to 20 times the instantaneous plume time scale. The plume intermittency is 0.1. 25 Probability (%)
probability distributions of the time-averaged concentrations.
Probability (%)
A. Venkatram / Atmospheric Environment 36 (2002) 2165–2170
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20 15 10 5 0
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 Concentration/Mean
Fig. 5. Concentration distribution for averaging time equal to 40 times the instantaneous plume time scale. The plume intermittency is 0.1.
60 50 40 30 20 10 0
0
2 4 6 Concentration/Mean
8
Fig. 2. Concentration distribution for averaging time equal to 5 times the instantaneous plume time scale. The plume intermittency is 0.1.
observing a value 4 times the mean is about 7%. When the averaging time increases to 10Dt (Fig. 3), the probability of observing a zero concentration decreases to about 35%, and at the same time extreme concentrations become more rare: about 20% of the concentrations are twice the mean, while close to 2% of the observed events are 4 times the ensemble mean. When the averaging is 20Dt; about 12% of the values are expected to be zero. About 7% of the values are
A. Venkatram / Atmospheric Environment 36 (2002) 2165–2170
about twice the mean value, and large excursions from the mean, 3 times the mean, are close to 1%. Fig. 5 shows that as the averaging time becomes large (40Dt compared to the time scale Dt; the distribution approaches a normal distribution in the region surrounding the mean, as one expects from theory. However, note that concentrations cannot be less than zero, and the distribution does allow large positive excursions from the mean. While the binomial concentration distribution is a simplified representation of reality, it mimics some of the essential features of observed concentration distributions (Mylne and Thomson, 1991), which can be exponential, clipped normal, or normal depending on plume characteristics and receptor locations. We see that the binomial distribution can approximate these distributions depending on the ratio of averaging time, the instantaneous plume time scale, Dt; and plume intermittency, g: Thus, the model is capable of providing insight into the behavior of concentration time series.
6. Concluding remarks We have proposed a simple model of the concentration time series to examine the effect of averaging time on the distribution of concentrations. The results from the model show that the approach of Eq. (1), often used in regulatory practice, to estimate short-term ‘‘peak’’ concentrations from 1 h averaged model estimates cannot be justified. The approach is an attempt to estimate short-time averages that can be substantially larger than the hourly average that the model estimates. Using Eq. (1) to estimate a single short-term peak is not meaningful because short-term averages can vary substantially. These time-averaged concentrations need to be described in terms of a concentration distribution. We show that when the averaging time is decreased, the probability of exceeding the mean by a large factor increases. But the probability of seeing a zero concentration also increases at the same time, so that the ensemble mean concentration remains the same. The larger excursions from the mean at short averaging times are reflected in the standard deviation of the concentration distribution, which is a function of averaging time as seen in Eq. (14). On the other hand, the ensemble mean concentration does not depend on averaging time. Does this mean that the 1 h averaged concentration is the same as a 6 h averaged concentration? No, because the longer time average is likely to be affected by systematic turning of the wind, an effect which would be inconsistent with the assumption that the concentration time series is governed by only two scales, Dt; and t: In a laboratory, where the mean wind can be held steady, the ensemble mean is obviously not a function of averaging time,
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because the ensemble mean is the time-averaged mean. In real applications, the effect of wind turning is modeled by averaging over the ensemble averages corresponding to each hour of the total sampling period. The turning of the wind is in some ways equivalent to increasing the value of t; the average time interval between concentration events. The results obtained in this paper are consistent, in a qualitative sense, with the measurements made by Mylne and Mason (1991). They find that time averaging reduces the concentration variance much more at short distances than at larger distances. This supports the primacy of the role of the instantaneous plume time scale, Dt; in determining the decrease in concentration variance with averaging time, as seen in Eq. (14); N decreases as Dt increases with distance. Mylne and Thomson (1991) also find that as the averaging time increases, the probability density function evolves from an exponential to a clipped normal. This again agrees with the results presented in the previous section. It is useful to make suggestions on how the results of this paper can be used in air pollution modeling applications. It is clear that any meaningful analysis depends on an estimate of the standard deviation, sTC ; for a particular averaging time. This parameter can be empirically estimated as the standard deviation of the differences between concentration observations and the corresponding model estimates at a particular location, which can be taken to be the ensemble means. As an example, take this empirically determined standard deviation to be equal to the mean, and that we are interested in 5 min exposures. Eq. (15) can be used to estimate the 5 min averaged standard deviation, which works out to about 3.5 times the mean. This information can be used to calculate the probability that a specified concentration level is exceeded. For example, assume that this level is 4.5 times the model predicted mean. Then, if we assume that the time-averaged concentration is normally distributed, this ‘‘alarm’’ level will be exceeded 16% of the time. This probabilistic statement conveys substantially different information from an estimate of a 5 min averaged ‘‘peak’’ concentration, which cannot be meaningfully defined. If we want to use the Binomial distribution to make statements about the concentration distribution, we need to estimate Dt empirically. One way of doing so is to estimate the variance of the instantaneous concentrations using a top hat model for the instantaneous plume. Then the variance is given by /ðC Cm Þ2 S Cp ¼ 1; 2 Cm Cm
ð18Þ
where the ratio of the ‘‘peak’’ to mean concentration can be estimated by modeling the instantaneous spread of the plume. Then Eq. (14) allows us to estimate Dt from an empirical estimate of sTC : This information can, in
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A. Venkatram / Atmospheric Environment 36 (2002) 2165–2170
turn, be used to estimate the concentration distribution for any required averaging time using the methods described in this paper. Because of the simplicity of the proposed concentration time-series model, these suggestions on estimating short-term concentrations can only serve as a starting point for developing practical methods for predicting concentration distributions. Such methods. Will need to account for internal plume fluctuations, which are neglected in this analysis.
Acknowledgements I would like to thank Dr. Shuming Du, California Air Resources Board, for making several suggestions that improved the quality of this paper.
References Engel, P.L., Williams, T.O., Muirhead, T., 1997. Utilizing ISCT to model composting facility odors. 90th Annual Air and
Waste Management Association Meeting and Exhibition, Toronto, Ont., Canada, 8–13 June. Gifford, F., 1959. Statistical properties of a fluctuating plume dispersal model. Advances in Geophysics 6, 117–137. Mylne, K.R., Mason, P.J., 1991. Concentration fluctuation measurements in a dispersing plume at a range of up to 1000 m. Q.J.R. Meteorol. Soc. 117, 177–206. Mussio, P., Gnyp, A.W., Henshaw, P.F., 2001. A fluctuating plume dispersion model for the prediction of odour-impact frequencies from continuous stationary sources. Atmospheric Environment 35, 2955–2962. Scire, J.S., Strimaitis, D.G., Yamartino, R.J., 2000. A users’s guide for the CALPUFF dispersion model (Version 5). Earth Tech, Inc., Concord, MA. Sykes, R.I., 1984. The variance in time-averaged samples from an intermittent plume. Atmospheric Environment 18, 121–123. Venkatram, A., 1979. The expected deviation of observed concentrations from predicted ensemble means. Atmospheric Environment 13, 1547–1549. Venkatram, A., 1984. The uncertainty in estimating dispersion in the convective boundary layer. Atmospheric Environment 18, 307–310.
Accounting for averaging time in air pollution modeling Akula Venkatram* College of Engineering, University of California, Riverside, CA 92521, USA Received 28 August 2001; accepted 8 February 2002
Abstract This paper examines the validity of a commonly used expression to estimate concentrations for averaging times that are much smaller than that corresponding to the model estimate. These ‘‘peak’’ concentrations, which can be several times larger than the model estimate, are required in applications such as odor assessment. We show that this expression cannot be justified, and that information on short-term concentrations is only contained in the probability distribution of time-averaged concentrations. The paper proposes a simple model of concentration time series to examine the effect of averaging time on concentration probability distributions. The results from the model are compared with previous theoretical and experimental studies. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Averaging time; Odor assessment; Peak concentrations; Concentration distributions; Air pollution modeling
1. Motivation Several applications require estimates of concentrations averaged over shorter time periods that those estimated with models such as the Industrial Source Complex (ISC) model, used in the United States for regulatory applications. For example, predicting odor concentrations requires converting ISC 1 h estimates to values that correspond to averaging times of a few seconds to few minutes. Common practice (Engel et al., 1997; Scire et al., 2000) consists of converting model predicted estimates to shorter time periods using equations of the form: p T Cs ¼ Cm ; ð1Þ Ts where Cs is the required concentration for the short averaging time, Ts ; and Cm is the model estimated concentration corresponding to the averaging time, T; usually taken to be 1 h. The values of the exponent, p; used in the literature, range from 0.2 to 0.5. A second approach to modeling concentrations at short-time scales is based on Gifford’s (1959) fluctuating *Fax: +1-909-787-2899. E-mail address: [email protected] (A. Venkatram).
plume model. Mussio et al. (2001) have used the model to estimate ‘‘instantaneous’’ concentrations, which are then used to calculate odor impact frequencies around industrial facilities. Sykes (1984) has adapted Gifford’s model to examine the effect averaging time on the variance of time-averaged concentrations. The variance can, in principle, be used to estimate the occurrence of peak concentrations of concern. The focus of this paper is the use of modeled concentrations to estimate shortterm concentrations, and not the equally important problem of estimating instantaneous concentrations.
2. The problem In order to understand the meaning of equations such as Eq. (1), it is necessary to be clear about what the ideal model is designed to predict. Assume that the model uses a set of inputs, which will vary from model to model. Now imagine an infinite set of experiments in which these model inputs are kept constant, and make concentration observations during each one of these experiments, which we assume lasts for 1 h. Because the model inputs include only some of the variables that govern the concentrations, the average concentrations will differ from experiment to experiment. Most
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A. Venkatram / Atmospheric Environment 36 (2002) 2165–2170
dispersion models, such as ISC, are designed to estimate the average over the ensemble of concentration averages from these experiments. This means that any individual observation will always differ from the model predicted ensemble mean (see Venkatram, 1984). While it is possible to estimate the expected deviation of these observations from the model predicted ensemble mean, it is not possible to predict the actual observation during an individual hour. We can always string together 1 h concentration time series from each of the experiments to construct an infinitely long time series, which corresponds to a particular set of model inputs. The best that the model can do is to provide a time average of this time series. We see that breaking this time series into time intervals of less than an hour before deriving an overall time average is not going to change the value of the ensemble average, which is the average over the infinite time series. Thus, the ensemble average, which is what the model can predict, does not depend on the averaging time. On the other hand the distribution of concentrations will depend on the averaging time. We are more likely to see higher concentrations at smaller averaging times. It is this concept that equations like Eq. (1) are trying to capture. However, in the form that it is written, it implies that the ensemble average is a function of averaging time. The concentration, Cs ; in Eq. (1), has to be some sort of ensemble average. Otherwise, it has no meaning. The basic motivation behind using Eq. (1) is to incorporate the fact that the likelihood of observing a ‘‘high’’ concentration increases as the averaging time decreases. As formulated, all it does is to multiply the mean concentration by an arbitrary factor to yield a concentration that cannot be interpreted in a meaningful way. The next section uses a simple model to explain some of these concepts better.
3. Simulating the concentration time series Most instantaneous concentration time series are highly intermittent with periods of spiky concentrations interspersed among periods of zero concentrations. We will model this as periods in which the concentration is constant at a level Cp separated by intervals of zero concentrations. This time series was used by Venkatram (1979) to examine the effects of time averaging on concentration variances. In the analysis, the time scale governing the concentration fluctuations was assumed to be proportional to a Eulerian time scale, which is a length scale of turbulent velocity fluctuations, divided by the mean wind speed. Sykes (1984) improved on Venkatram’s (1979) analysis by using Gifford’s (1959) fluctuating plume model to derive a relationship between the concentration time scale and
the Eulerian time scale. In the analysis to follow, we will compare our new results with those from Sykes (1984). The proposed model assumes that the concentration ‘‘windows’’ have a constant width Dt; but the spacing between these windows is a random variable with an average value of t: Fig. 1 depicts this simple model of concentrations. To make progress, we will assume that each time interval Dt is a Bernoulli trial in which the occurrence of a non-zero concentration has the probability g: Then, the number of non-zero concentration events in a number of trials is governed by a Binomial distribution. While this model is a major simplification of reality, it contains the essential features that allow us to investigate the effects of averaging time on concentrations. Consider a sampling interval of length T: This interval can contain a maximum of N ¼ T=Dt concentration ‘‘events’’, each of which can either record a zero concentration or Cp : Then, the ensemble mean concentration for this model is Cm ¼ Cp g:
ð2Þ
g is referred to as intermittency in the literature. The ensemble variance of the concentration time series is seen to be (Venkatram, 1979) /ðC Cm Þ2 S 1 ¼ 1: 2 Cm g
ð3Þ
Then, the average time interval between concentration events, t; can be defined through g¼
Dt : Dt þ t
ð4Þ
We can estimate the instantaneous plume time scale, Dt; as si ð5Þ DtB ; u where si is the dimension of the instantaneous plume, and u is the mean wind speed. The peak concentration
τ Concentration
∆t Cp
Time
Fig. 1. Idealized time series used to estimate the effect of averaging time on concentration distributions. The concentrations alternate between peak values, Cp ; and zero values. The width of the concentration event is constant at Dt; while the spacing between concentration events is a random variable with an average value of t:
A. Venkatram / Atmospheric Environment 36 (2002) 2165–2170
associated with this plume is then Cp ¼
Q ; pus2i
ð6Þ
where Q is the emission rate. By assuming that the mean concentration distribution is a symmetric Gaussian, the peak-to-mean ratio can be written as ! ! 2s2p Cp r2 exp ; ð7Þ ¼ Cm 2s2p s2i where sp is the standard deviation of the ensemble mean concentration distribution, and r is the distance of the receptor from the mean plume centerline. This expression neglects concentration fluctuations at Batchelor length scales within the instantaneous plume. Then, Eq. (4) yields " ! ! # 2s2p r2 t ¼ Dt exp 1 ð8Þ 2s2p s2i and on the plume centerline, tEDt
s2p s2i
¼
s2p : si u
ð9Þ
We see that t increases with the ratio of the total plume spread to the instantaneous spread, and with the distance from the plume centerline. Both the governing time scales, Dt and t; that govern the concentration distribution, are dependent on plume properties, and thus on the receptor location relative to the point of tracer release. The time scale, Dt; increases with distance from the source as the instantaneous plume spread increases. At short distances, t is much larger than Dt because the instantaneous plume spread is much smaller than the total plume spread; this difference becomes smaller with distance as the two plume spreads become comparable.
4. The effects of averaging time It is easy to estimate the effect of averaging time on the concentration variance by noting that the timeaveraged concentration when n non-zero values occur during the averaging interval, T; is C% T ðnÞ ¼
nCp Dt : T
ð10Þ
The time-averaged concentration, C% T ðnÞ; varies between averaging intervals because n varies. Then, the standard deviation, std, of C% T ðnÞ about the ensemble mean concentration is stdðC% T ðnÞÞ sTC ¼
stdðnÞCp Dt : T
ð11Þ
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For the binomial distribution, the standard deviation of n is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi stdðnÞ ¼ Ngð1 gÞ; ð12Þ where N ¼ T=Dt is the number of concentration events (Bernoulli trials) in the sampling interval. If we substitute Eq. (12) into Eq. (11) and use Eq. (4), we obtain the result t 1=2 sTC ¼ : ð13Þ T Cm This equation appears to tell us that standard deviation of the time-averaged concentrations is proportional to the ensemble mean, and is independent of the standard deviation of the instantaneous concentrations. Eq. (13) would seem to agree with the result obtained by Sykes (1984, Eq. (16)), who suggests that in a highly intermittent plume, the time-averaged concentration variance has ‘‘only a logarithmic dependence on the ensemble variance’’, and ‘‘is more nearly proportional to the square of the ensemble mean’’. But Eq. (13) does not support this result from Sykes (1984) because the concentration return time scale, t; is not a flow variable, but depends on the intermittency (see Eq. (4)), and hence the instantaneous concentration variance. In fact, if we combine Eqs. (3), (4) and (13), we find that 1=2 sTC Dt 1 ¼ ¼ 1=2 : ð14Þ T N /ðC Cm Þ2 S1=2 This result, which is consistent with the assumption of independence of concentration events, says that the instantaneous plume time scale, Dt rather than the return time scale, t; governs the effect of averaging time on concentration variance. Eq. (14) implies that, unless the averaging time is comparable to the instantaneous plume time scale, Dt; measured variances are likely to be substantially smaller than the variance of the instantaneous concentrations. This suggests that unintentional time or space averaging induced by measuring instruments can substantially reduce measured concentration statistics. Eq. (13) allows us to relate the time-averaged standard deviations of concentrations at two different averaging times (at the same location) through the simple relationship 1=2 sTC1 T2 ¼ : ð15Þ T1 sTC2 This equation is very different from Eq. (1), which attempts to relate ensemble means at two different averaging times; the ensemble means are not functions of averaging times. Eq. (15) tells us that the likelihood of seeing high concentrations relative to the mean increases as the averaging time decreases. This can be seen more clearly by estimating the effects of averaging time on the
The concentration, C% T ðnÞ; is related to the number of events, n; that occur in T through Eq. (10). The probability that a specified concentration, C% T ðnÞ; occurs is equivalent to the probability that n events occurs in T: If we assume that these events follow a binomial distribution, this probability is ! N n PðC ¼ C% T ðnÞÞ ¼ g ð1 gÞNn ; ð16Þ n where the binomial coefficient is ! N N! ¼ n!ðN nÞ! n
ð17Þ
and N ¼ T=Dt: Figs. 2–5 show the behavior of the probability distribution for different averaging times, ranging from 5 times to 40 times the time scale, Dt: The intermittency is taken to be 0.1. The binomial distributions, shown in the figures, refer to discrete values of events, which have been converted into concentrations normalized by the mean. Although they do not mimic observed continuous distributions, they do illustrate the effects of averaging time on concentrations. In principle, the distribution can be converted approximately to a continuous distribution, but this does not add anything to the illustrative analysis of this paper. When T ¼ 5Dt; a maximum of only 5 concentration events can contribute to the time average. Thus, we see in Fig. 2 that the probability of observing a zero concentration is high at 60%. At the same time, the probability of seeing a concentration that is about 2 times the ensemble mean is over 30%, and that of
Probability (%)
70
45 40 35 30 25 20 15 10 5 0
0
1
4 2 3 Concentration/Mean
5
6
Fig. 3. Concentration distribution for averaging time equal to 10 times the instantaneous plume time scale. The plume intermittency is 0.1.
30 Probability (%)
5. Probability distribution of time-averaged concentrations
25 20 15 10 5 0
0
0.5
1 1.5 2 Concentration/Mean
2.5
3
Fig. 4. Concentration distribution for averaging time equal to 20 times the instantaneous plume time scale. The plume intermittency is 0.1. 25 Probability (%)
probability distributions of the time-averaged concentrations.
Probability (%)
A. Venkatram / Atmospheric Environment 36 (2002) 2165–2170
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20 15 10 5 0
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 Concentration/Mean
Fig. 5. Concentration distribution for averaging time equal to 40 times the instantaneous plume time scale. The plume intermittency is 0.1.
60 50 40 30 20 10 0
0
2 4 6 Concentration/Mean
8
Fig. 2. Concentration distribution for averaging time equal to 5 times the instantaneous plume time scale. The plume intermittency is 0.1.
observing a value 4 times the mean is about 7%. When the averaging time increases to 10Dt (Fig. 3), the probability of observing a zero concentration decreases to about 35%, and at the same time extreme concentrations become more rare: about 20% of the concentrations are twice the mean, while close to 2% of the observed events are 4 times the ensemble mean. When the averaging is 20Dt; about 12% of the values are expected to be zero. About 7% of the values are
A. Venkatram / Atmospheric Environment 36 (2002) 2165–2170
about twice the mean value, and large excursions from the mean, 3 times the mean, are close to 1%. Fig. 5 shows that as the averaging time becomes large (40Dt compared to the time scale Dt; the distribution approaches a normal distribution in the region surrounding the mean, as one expects from theory. However, note that concentrations cannot be less than zero, and the distribution does allow large positive excursions from the mean. While the binomial concentration distribution is a simplified representation of reality, it mimics some of the essential features of observed concentration distributions (Mylne and Thomson, 1991), which can be exponential, clipped normal, or normal depending on plume characteristics and receptor locations. We see that the binomial distribution can approximate these distributions depending on the ratio of averaging time, the instantaneous plume time scale, Dt; and plume intermittency, g: Thus, the model is capable of providing insight into the behavior of concentration time series.
6. Concluding remarks We have proposed a simple model of the concentration time series to examine the effect of averaging time on the distribution of concentrations. The results from the model show that the approach of Eq. (1), often used in regulatory practice, to estimate short-term ‘‘peak’’ concentrations from 1 h averaged model estimates cannot be justified. The approach is an attempt to estimate short-time averages that can be substantially larger than the hourly average that the model estimates. Using Eq. (1) to estimate a single short-term peak is not meaningful because short-term averages can vary substantially. These time-averaged concentrations need to be described in terms of a concentration distribution. We show that when the averaging time is decreased, the probability of exceeding the mean by a large factor increases. But the probability of seeing a zero concentration also increases at the same time, so that the ensemble mean concentration remains the same. The larger excursions from the mean at short averaging times are reflected in the standard deviation of the concentration distribution, which is a function of averaging time as seen in Eq. (14). On the other hand, the ensemble mean concentration does not depend on averaging time. Does this mean that the 1 h averaged concentration is the same as a 6 h averaged concentration? No, because the longer time average is likely to be affected by systematic turning of the wind, an effect which would be inconsistent with the assumption that the concentration time series is governed by only two scales, Dt; and t: In a laboratory, where the mean wind can be held steady, the ensemble mean is obviously not a function of averaging time,
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because the ensemble mean is the time-averaged mean. In real applications, the effect of wind turning is modeled by averaging over the ensemble averages corresponding to each hour of the total sampling period. The turning of the wind is in some ways equivalent to increasing the value of t; the average time interval between concentration events. The results obtained in this paper are consistent, in a qualitative sense, with the measurements made by Mylne and Mason (1991). They find that time averaging reduces the concentration variance much more at short distances than at larger distances. This supports the primacy of the role of the instantaneous plume time scale, Dt; in determining the decrease in concentration variance with averaging time, as seen in Eq. (14); N decreases as Dt increases with distance. Mylne and Thomson (1991) also find that as the averaging time increases, the probability density function evolves from an exponential to a clipped normal. This again agrees with the results presented in the previous section. It is useful to make suggestions on how the results of this paper can be used in air pollution modeling applications. It is clear that any meaningful analysis depends on an estimate of the standard deviation, sTC ; for a particular averaging time. This parameter can be empirically estimated as the standard deviation of the differences between concentration observations and the corresponding model estimates at a particular location, which can be taken to be the ensemble means. As an example, take this empirically determined standard deviation to be equal to the mean, and that we are interested in 5 min exposures. Eq. (15) can be used to estimate the 5 min averaged standard deviation, which works out to about 3.5 times the mean. This information can be used to calculate the probability that a specified concentration level is exceeded. For example, assume that this level is 4.5 times the model predicted mean. Then, if we assume that the time-averaged concentration is normally distributed, this ‘‘alarm’’ level will be exceeded 16% of the time. This probabilistic statement conveys substantially different information from an estimate of a 5 min averaged ‘‘peak’’ concentration, which cannot be meaningfully defined. If we want to use the Binomial distribution to make statements about the concentration distribution, we need to estimate Dt empirically. One way of doing so is to estimate the variance of the instantaneous concentrations using a top hat model for the instantaneous plume. Then the variance is given by /ðC Cm Þ2 S Cp ¼ 1; 2 Cm Cm
ð18Þ
where the ratio of the ‘‘peak’’ to mean concentration can be estimated by modeling the instantaneous spread of the plume. Then Eq. (14) allows us to estimate Dt from an empirical estimate of sTC : This information can, in
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turn, be used to estimate the concentration distribution for any required averaging time using the methods described in this paper. Because of the simplicity of the proposed concentration time-series model, these suggestions on estimating short-term concentrations can only serve as a starting point for developing practical methods for predicting concentration distributions. Such methods. Will need to account for internal plume fluctuations, which are neglected in this analysis.
Acknowledgements I would like to thank Dr. Shuming Du, California Air Resources Board, for making several suggestions that improved the quality of this paper.
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