Error Propagation in Air Dispersion Modeling
Air dispersion modeling has been evolving for most of this century. Over the last 15-25 years, strict environmental regulations and the availability of personal computers have fueled an immense growth in the use of mathematical models to predict the dispersion of air pollution plumes. My recently published book, "Fundamentals Of Stack Gas Dispersion," details the evolution of the widely used Gaussian air-dispersion models and their inherent assumptions and constraints.
Unfortunately, many users of such models are completely unaware of those assumptions and constraints and mistakenly believe that the precision achievable with computers equates to accuracy. This article discusses how the propagation of seemingly small errors in the Gaussian model parameters can cause very large variations in the model's predictions. Generally, the models over-predict (that is, provide a safety factor) of 2.5 or more.
In most dispersion models, determining the pollutant concentrations at ground-level receptors beneath an elevated, buoyant plume of dispersing, pollutant-containing gas involves two major steps:
- First, the height to which the plume rises at a given downwind distance from the plume source is calculated. The calculated plume rise is added to the height of the plume's source point to obtain the so-called "effective stack height", also known as the plume centerline height or simply the emission height.
Then, the ground-level pollutant concentration beneath the plume at the given downwind distance is predicted using the Gaussian dispersion equation. (1)
A host of assumptions and constraints are required to derive the Gaussian dispersion equation for modeling a continuous, buoyant plume from a single point-source in flat terrain . . . which is still a long way from the more sophisticated models now in use for multiple sources in complex terrain. The most important assumptions and constraints are related to:
- The accuracy of predicting the plume rise since that affects the emission height used in the Gaussian dispersion equation
- The accuracy of the dispersion coefficients (i.e., the vertical and horizontal standard deviations of the emission distribution) used in the Gaussian dispersion equation
- The assumption of the averaging time period represented by the calculated ground-level pollutant concentrations as determined by the dispersion coefficients used in the Gaussian equation. In other words, do the calculated ground-level concentrations represent a 5-minute, 10-minute, 15-minute, 30-minute or 1-hour average concentration?
- Obtaining the atmospheric stability classifications (which characterize the degree of turbulence available to enhance dispersion)
- Determining the profiles of windspeed versus emission height
- Converting ground-level short-term concentrations from one averaging time to another.
Deriving the Gaussian dispersion equation requires the assumption of constant conditions for the entire plume travel-distance from the emission source point to the downwind ground-level receptor. (1) Yet we cannot say with any reasonable certainty that the windspeed at the plume centerline height and the atmospheric stability class are known exactly, or that they are constant for the entire plume travel distance. Whether such homogeneity actually occurs is a matter of pure chance, particularly for large distances. Also, determining the exact windspeed and atmospheric stability class at the plume centerline height requires (a) the prediction of the exact plume rise and (b) the exact relation between windspeed and altitude-neither of which are achievable.
Plume Rise Uncertainty
Most Gaussian dispersion models use the Briggs plume rise equations (1,7, 8, 9) to predict buoyant plume rise. There are few knowledgeable dispersion modelers who would dispute that the Briggs equations could over- or underpredict actual plume rises by 20 percent.
Dispersion Coefficients
Most Gaussian models use modifications of the dispersion coefficients derived experimentally by Pasquill (10) in a rural area of fairly level, open terrain and for relatively moderate plume travel-distances. There are few knowledgeable dispersion modelers who would dispute that Pasquill's coefficients could be in error by +/-25%, especially when used for non-level, complex terrain and for large distances ranging up to 50 kilometers or more. Pasquill himself has proposed a re-examination of his coefficients (11) and has suggested they be revised.
Concentration Averaging Periods
As mentioned above, there is the question of what averaging time period the calculated ground-level concentration (i.e., C) represents when using Pasquill's dispersion coefficients. Turner (12) states that C is a 3- to 15-minute average. An American Petroleum Institute publication (13) believes C is a 10- to 30-minute average. An American Institute of Chemical Engineers publication, written by Hanna and Drivas (14), states that C represents a 10-minute average. The Tennessee Valley Authority (15) attributes a 5-minute average to their C values.
Despite that body of opinion, many of the dispersion models--whose use is mandated by most of our federal and state regulatory agencies-assume the Gaussian dispersion equation yields 1-hour average concentrations. It can be shown (1) that assuming the C values represent a 1-hour average, rather than a 10-minute average, constitutes a "built-in" over-prediction factor of as much as 2.5.
Other Assumptions and Constraints
Deriving the Gaussian dispersion equation also assumes the following: Windspeed and wind direction are constant from the source point to the receptor (for a windspeed of 2 m/s and a distance of 10 km, 80 minutes of constant conditions would be needed). Atmospheric turbulence is also constant throughout the plume travel distance. All of the plume is conserved, meaning: no deposition or washout of plume components; components reaching the ground are reflected back into the plume; no components are absorbed by bodies of water or by vegetation; and components are not chemically transformed. [Some of the more complex dispersion models do adjust for deposition and chemical transformation. However, such adjustments are separate from the basic Gaussian dispersion equation.] Only vertical and crosswind dispersion occurs (i.e., no downwind dispersion). The dispersion pattern is probabilistic and can be described exactly by Gaussian distribution. The plume expands in a conical fashion as it travels downward, whereas the ideal "coning plume" is only one of many observed plume behaviors. Terrain conditions can be accommodated by using one set of dispersion coefficients for rural terrain and another set for urban terrain. The basic Gaussian dispersion equation is not intended to handle terrain regimes such as valleys, mountains or shorelines.
In short, the Gaussian models assume an ideal steady-state of constant meteorological conditions over long distances, idealized plume geometry, uniform flat terrain, complete conservation of mass, and exact Gaussian distribution. Such ideal conditions rarely occur.
Sensitivity Study
A sensitivity study was performed by assuming reasonable degrees of error in some of the key variables used in the Gaussian models and determining the propagated end-result effect of those errors on the calculated, ground-level pollutant concentrations. Several comparative models were defined as follows:
- Base Model A--The base model uses Briggs' plume-rise equations, power-law conversion of surface windspeeds to obtain windspeeds at the source height (for use in the plume rise equations) and at the plume centerline height (for use in the Gaussian dispersion equations), and the calculated ground-level concentrations are taken to be 1-hour averages as per the U.S. EPA.
Adjusted Model B--Same as model A except that the calculated plume rises were increased by 20% and the Pasquill vertical dispersion coefficients were decreased by 25%.
Adjusted Model C--Same as model B except that the calculated ground-level concentrations reflect an assumed wind direction shift of 10 degrees.
Adjusted Model D Same as model C except that an over-prediction factor (C10/C60) of 2.5 was included to account for the EPA's assumption that the calculated ground-level concentrations represent 1-hour averages rather than 10-minute averages.
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16.1 |
24.3 |
20.5 |
15.8 |
12.0 |
9.4 |
7.4 |
6.0 |
5.0 |
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1.1 |
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13.8 |
13.7 |
12.0 |
10.1 |
8.4 |
7.0 |
5.9 |
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2.5 |
2.0 |
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0.2 |
1.7 |
2.4 |
2.3 |
1.9 |
1.5 |
1.2 |
1.0 |
0.8 |
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Table 1 presents the results of the sensitivity study. Comparing the ground-level concentrations calculated by the base model A to the concentrations calculated by the adjusted model D, it is seen that the base model A over-predicts model D by a factor ranging from 6 (at downwind distances of 6 to 10 km) to a factor of 80 (at a downwind distance of 2 km). Thus, seemingly minor changes in some of the key variables can result in a propagated over-prediction factor ranging from 6 to 80.
This study was not intended to downgrade the value of Gaussian dispersion models. They are very useful tools. However, we should be aware that they are merely tools and do not provide the ultimate truth. This study has shown that it is unrealistic to expect the Gaussian models to predict real-world dispersing plume concentrations consistently by a factor of two or three. It is probably more realistic to expect consistent predictions of real-world dispersing plume concentrations within a factor that may be as high as ten.
Given the limitations of the Gaussian models, what should the practicing engineer do? In the context of complying with regulatory requirements, there is very little that can be done, because the EPA mandates their use and the EPA considers them to be accurate within a factor of two or less. Until such time as the EPA is willing to admit that the models are not as accurate as they believe or until more of us are willing to contest the point, we have no choice but to comply.
In the context of performing pollution dispersion studies (unrelated to any regulatory issues), the practicing engineer should realize that the models are only tools and not the ultimate truth. The models can be used to qualitatively compare alternative new plant sites or new air pollution control systems. In other words, will site A (or system A) create higher or lower ground-level concentrations than site B (or system B). But don't place too much confidence in using the models to quantitatively predict the correct or actual real-world concentrations.
One reassuring note on this point is that the over-conservative results obtained by present-day Gaussian modeling adequately protect the general public-and then some. The regulatory Gaussian models have a built-in over-prediction factor of 2.5 as discussed in my article. The dense gas (non-Gaussian) models used for many of the accidental hazardous gas release scenarios may or may not be more accurate than the Gaussian models (the jury is still out on that point). However, EPA mandates (a) a so-called "worst case" release scenario which could hardly be more pessimistic and (b) their concentration levels of concern are equally conservative.
For example, the National Institute of Occupational Safety and Health (NIOSH) publishes the so-called IDLH values for a great many hazardous chemicals. NIOSH defines the IDLH levels as being concentrations to which one could be subjected "for 30 minutes without experiencing any escape-impairing or irreversible health effects". The EPA then divides the IDLH values by a safety factor of ten and uses those IDLH/10 values as their mandated levels of concern. In other words, in addition to the mandated very pessimistic "worst case" releases, the concentrations within the evacuation zone must not exceed one-tenth of the IDLH values published by NIOSH. I do believe quite strongly that we need to take every precaution against accidental releases of hazardous gases. My comments here are not meant to be a diatribe against the EPA, but simply to point out that the EPA already has one large safety factor upon another large safety factor.
References:
(1) Beychok, M.R., Fundamentals of Stack Gas Dispersion, published by author, Irvine, Calif., 1994
(2) "Atmospheric dispersion modeling, a critical review," JAPCA, Sept. 1979
(3) Ellis, H.M. et al, "Comparision of predicted and measured concentrations for 58 alternative models of plume transport in complex terrain," JAPCA, June 1980
(4) American Petroleum Institute, An evaluation of short-term air quality models using tracer study data, API Report No. 4333, Oct. 1980
(5) Bowne, N.J. et al, Overview, results and conclusions for the EPRI plume model validation and development project: Plains Site, Electric Power Research Institute, Final Report 1616-1 for Project EA-3704, 1983
(6) Benarie, M.M., "Editorial: The limits of air pollution modeling," Atmospheric Environment, 21:1-5, 1987
(7) Briggs, G.A., "Plume rise," USAEC Critical Review Series, 1969
(8) Briggs, G.A., "Some recent analyses of plume rise observation," Proc. Second Internat'l. Clean Air Congress, Academic Press, New York, 1971
(9) Briggs, G.A., "Discussion: Chimney plumes in neutral and stable surroundings", Atmospheric Environment, 6:507-510, 1972
(10) Pasquill, F., "The estimation of the dispersion of windborne material", Meteorology Magazine, Feb. 1961
(11) Pasquill, F., "Atmospheric dispersion parameters in Gaussian plume modeling. Part II: Possible requirements for change in Turner Workbook values," U.S. EPA Publication 600/4-76-030b, June 1976
(12) Turner, D.B., Workbook of atmospheric dispersion estimates, U.S. EPA Publication AP-26, revised 1970
(13) American Petroleum Institute, Gaussian dispersion models applicable to refinery emissions, API Publication 52, Oct. 1977
(14) Hanna, S.R. and Drivas, P.J., Guidelines For The Use Of Vapor Cloud Dispersion Models, Center For Process Safety, American Institute of Chemical Engineers, 1987
(15) Montgomery, T.C. and Coleman, J.H., "Empirical relationship between time-averaged SO2 concentrations," Environmental Science & Technology, Oct. 1975
The author may be contacted at:
2233 Martin St. #205, Irvine, CA 92612. Phone & Fax: 714-833-8871. E-mail: mbeychok@air-dispersion.com. or mbeychok@deltanet.com.
There are copies available of the author's book, Fundamentals of Stack Gas Dispersion.