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Milton R. Beychok, Consulting Engineer
Newport Beach, California, USA

Air dispersion modeling has been evolving since before the 1930s.  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.  Beychok's 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.1

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.

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

Assumptions and Constraints

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?

Besides the assumptions and constraints in deriving the Gaussian equation, the methods for obtaining certain parameters used in the Gaussian models are also subject to many assumptions and constraints.  Those methods include: obtaining the atmospheric stability classifications (which characterize the degree of turbulence available to enhance dispersion), determining the profiles of windspeed versus emission height, and converting ground-level short-term concentrations from one averaging time to another.  This discussion of shortcomings in the Gaussian dispersion models is not unique.  The literature abounds with such discussions.2, 3, 4, 5, 6  Unfortunately, despite those discussions, there is a widespread belief that dispersion models can predict dispersed plume concentrations within a factor of two or three of the actual concentrations in the real world.  Indeed, there are some who believe the models are even more accurate than that.

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 equations1, 7, 8, 9 to predict buoyant plume rise.  There are few knowledgeable dispersion modelers who would dispute that the Briggs equations could over-or under-predict actual plume rises by 20 percent.

Dispersion Coefficients

Most Gaussian models use modifications of the dispersion coefficients derived experimentally by Pasquill10 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 plus or minus 25 percent, 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 coefficients11 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.  Turner12 states that C is a 3- to 15-minute average.  An American Petroleum Institute publication13 believes C is a 10- to 30-minute average.  An American Institute of Chemical Engineers publication, written by Hanna and Drivas14, states that C represents a 10-minute average.  The Tennessee Valley Authority15 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 shown1 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 direction are constant from source point to 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 the plume is conserved, meaning: no deposition or washout of the 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 accomodated 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 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 percent and the Pasquill vertical dispersion coefficients were decreased by 25 percent.

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.

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.

Figure 1 depicts the profiles, as predicted by Models A, B, C and D, of the pollutant ground-level concentrations versus downwind distance.

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 much more realistic to expect consistent predictions of real-world dispersing plume concentrations within a factor that may be as high as ten.  In fact, a recently released report by the Amarillo National Research Center16 states that field research by Texas A&M University has shown that the U.S. EPA's widely used ISCST model overpredicts downwind concentrations by a factor of 10 or more.

Although this article was first written in about 1988, the comments excerpted from the 1991 to 2002 references17, 18, 19, 20, 21, 22 presented in the Appendix to this article make it abundantly clear that very little has yet been done to quantify the uncertainty involved in air dispersion modeling.

Table 1: Calculated ground-level concentrations (micrograms/cubic meter)
at downwind distances ranging from 2 km to 10 km




Model Plus
Wind Shift
Model Plus
Wind Shift
and C10/C60



(1)   Beychok, M.R., "Fundamentals of Stack Gas Dispersion", published by author, Irvine, California, USA, Fourth Edition 2005

(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, "Gassian 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

(16)   "Tech Notes March/April 2000",  http://www.pu.org/main/technotes/tn200.html,  Amarillo National Research Center

Appendix:  Additional References and Comments Excerpted From Them

(17)   Stiggins, T.E., Parnell,C.B., Lacy, R.E., and Shaw, B.W., "Errors Associated With Time Average Concentrations in Gaussian Modeling", Department of Biological & Agricultural Engineering, Texas A&M University, College Station, Texas, presented at the 2002 Beltwide Cotton Conferences, National Cotton Council, Memphis, Tennessee

Excerpted comments:  "The 10-minute concentrations of the Gaussian model are assumed to be 60 minute concentrations in ISCT3.  It is demonstrated in this paper that this error in ISCT3 results in a significant over-estimate of downwind concentrations.  It is concluded that the downwind concentration estimates by regulatory agencies using ISCT3 over-predict by a factor of 2.5."

(18)  Britter, R. E., "The evaluation of technical models used for major-accident hazard installations", Report to Commission of the European Communities (DG X11), August 1991

Excerpted comments:  "In a broad sense, there is generally an over-confidence in the ability of mathematical models to provide accurate prediction.  In particular, there is significant over-confidence in the ability of very large numerical models to provide accurate prediction."

(19)  Chatwin, P.C., Lewis, D.M. and Mole, N., "Comments on the properties and uses of atmospheric dispersion datasets", School of Mathematics and Statistics, University of Sheffield, UK, EUROMECH Colloquium 338 and ERCOFTAC Workshop at Bologna, Italy, September 1995

Excerpted comments:  "Many, if not most, models of atmospheric dispersion and associated risk assessment now used by governments and regulatory authorities are, quite simply, wrong, both scientifically and practically, since they place undue emphasis on mean concentrations.  Little or no recognition is given to fluctuations."

(20)   Seibert, Petra, "Uncertainties in atmospheric dispersion modelling", Institute of Meteorology and Physics, University of Agricultural Sciences, Vienna, Proceedings Informal Workshop on Meteorological Modelling in Support of CTBT Verification, December 2000

Excerpted comments:  "However, the issue of uncertainties in atmospheric dispersion modelling has so far been addressed by the scientific community in rather general terms only, and we are still far from established practices."  and  "A variety of errors affect the results of transport calculations as well as source determination.  The biggest challenge in their quantification is the quantification of the transport errors ... In certain sensitive synoptic patterns, transport errors may lead to completely wrong calculated transport patterns."

(21)  "Summary of Short-Range Dispersion Modeling Workshop", Co-sponsored by the California Energy Commission and the California Air Resources Board, Sacramento, California, USA, January 2002

Excerpted comments:  "Methodologies for model performance evaluation have been relatively unsophisticated.  There are no standards to gauge model performance, and evaluation procedures are often ad hoc."  and  "A formal framework does not exist for assessing model uncertainty, and for assessing how model input errors propagate to-and manifest in-model outputs.  Methodologies for detecting the relative contributions of different sources of uncertainty are also lacking."

(22)  "Tracking and Predicting the Atmospheric Dispersion of Hazardous Material Releases: Implications for Homeland Security", National Research Council of the National Academies, National Academies Press, Washington, D.C., USA, 2003

Excerpted comments:  "... most models predict the average dispersion (over a large number of realizations of the given situation) and not the event-to-event variability about that average.  As a result, even a good atmospheric transport model may have single-event errors of more than a factor of ten."

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