This appendix describes some of the mathematical relationships and models used in exposure assessment.
Assessing exposure to a pollutant requires information on the pollutant concentration at a specific location (microenvironment) and the duration of contact with a person or population. If the concentration of a pollutant to which a person is exposed can be measured or modeled and the time spent in contact with the pollutant is known, exposure is determined from concentration and time. When concentration varies with time, the total exposure from time t1 to t2 is given by
where E is the exposure of a person to a pollutant at concentration C; C(t) represents the functional relationship of concentration with time t for an interval t1 through t2. The average ("time-weighted average") exposure during this interval is E/(t2–t1).
It is often assumed that the concentration is constant within a given microenvironment j for some finite interval, tj. Thus, any particular exposure within a given microenvironment ej is given by
which means that a person stays within the microenvironment with average con-
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Page 375 Appendix C Calculation and Modeling of Exposure This appendix describes some of the mathematical relationships and models used in exposure assessment. Calculation Of Exposure Assessing exposure to a pollutant requires information on the pollutant concentration at a specific location (microenvironment) and the duration of contact with a person or population. If the concentration of a pollutant to which a person is exposed can be measured or modeled and the time spent in contact with the pollutant is known, exposure is determined from concentration and time. When concentration varies with time, the total exposure from time t1 to t2 is given by where E is the exposure of a person to a pollutant at concentration C; C(t) represents the functional relationship of concentration with time t for an interval t1 through t2. The average ("time-weighted average") exposure during this interval is E/(t2–t1). It is often assumed that the concentration is constant within a given microenvironment j for some finite interval, tj. Thus, any particular exposure within a given microenvironment ej is given by which means that a person stays within the microenvironment with average con-
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Page 376 centration ¯Cj for the interval tj. A person's total exposure E to an airborne pollutant is the summation over all the microenvironments M in which the person is in contact with the pollutant: The latter equation includes the totality of all locations and activities that the person can occupy and engage in. To obtain the total exposure of a population Epop of N persons, it is necessary to sum the individual exposures Ei of all the persons in the population from i = 1 to N: Generally, the amount of time spent in each microenvironment is averaged over the exposed population, so that the average population exposure is given by Thus, it is necessary to estimate the atmospheric concentration of the pollutant to which people are exposed to obtain Cj and their activity patterns to obtain tj. Modeling Of Exposure It is often impossible or impractical to measure the exposures of individuals or populations directly, and instead mathematical models are used to estimate exposures. Microenvironmental concentrations are estimated with concentration models, which are based on the physics and chemistry of the environment. The time spend by an individual in a microenvironment with a pollutant is another important input to an exposure model. Population-exposure models combine data representing the time-activity patterns of an entire population with pollutant concentrations. Gaussian-Plume Models Gaussian-plume models are used by the Environmental Protection Agency (EPA) to estimate the concentration of a pollutant at locations some distance from an emission source. The models have this name because they represent the
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Page 377 plume of emissions from a stack as having a Gaussian, or normal, distribution, with a maximum at the center line, as shown in Figure C-1. The effect of boundaries (such as the ground or an atmospheric inversion cap), multiple emission sources, and deposition can alter the basic Gaussian distribution. Gaussian-plume models have been generalized to consider continuous and intermittent emissions, as well as emissions from points (e.g., concentrated emissions from a stack), areas (e.g., distributed emissions throughout a modeled region, such as home heating), and lines (e.g., roads). Gaussian-plume models have been further extended to complex topographic regions, such as valleys and bodies of water, and to industrial sources. They have also been designed for various temporal averaging periods. A number of Gaussian-plume models, with individual names, correspond to the various mathematical formulations used in the models. A few of the more commonly used Gaussian-plume models are the industrial-source complex long-term (ISCLT) and industrial-source complex short-term (ISCST) models, for long- and short-term averaging times, respectively; LONGZ (basic long-term model); Complex (for complex terrain); and Valley (for valleys). These are parts of the EPA UNAMAP modeling library (see Zannetti, 1990, for a brief description of each one and how to obtain the models). FIGURE C-1 Visualization of the dispersion of pollutants as described by a Gaussian-plume model. Source: Russell, 1988. Reprinted with permission; copyright 1988, Health Effects Institute, Cambridge, Mass.
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Page 378 Gaussian-plume models are among the simplest atmospheric-dispersion models, but they can still involve a number of complexities. For example, many sources emit their effluent at higher than ambient temperatures, so their pollutants tend to rise. The rise is a complex process to describe, requiring the simultaneous consideration of heat and mass transfer, atmospheric turbulence, and source characteristics. Conversely, a pollutant may be emitted without sufficient buoyancy or momentum to be lifted above the wake of turbulent air downwind of a building or a topographic feature. The pollutants can then be caught in the wake and downwashed, increasing the potential exposure. Specific Gaussian-plume models, such as the ISCLT and ISCST models, have been developed for that possibility (EPA, 1987). The ISCLT and ISCST models are often suggested for use in exposure assessment of air pollutants from industrial sources. The Human-Exposure Model (described below), which is used by EPA, also uses the industrial-source complex models. Multiple sources are treated by superimposing the calculated contributions of individual sources. It is possible to include the first-order chemical decay of pollutant species within the Gaussian-plume framework, as well as deposition of both gases and particles. Although Gaussian-plume models have been used for many years, their results are still subject to considerable error. In many cases, especially far from the source, they are biased to predict high concentrations. Applying Gaussian-plume models in complex terrain (such as hilly areas or areas with tall buildings) leads to even greater uncertainties and can result in significant overprediction and underprediction. Their rather simple formulation makes it difficult to handle complex terrain. Human-Exposure Model The HEM, one of the more commonly used models developed for EPA, incorporates a simple Gaussian-plume dispersion model with a fixed-location population model. Although EPA has developed several Gaussian-plume dispersion models for which validation studies have been conducted, the HEM was constructed with a model that incorporates an alternative approach to estimating the horizontal and vertical dispersion rates. The model was then compared with the standard UNIMAP models issued by the EPA Office of Air Quality, Planning, and Standards as part of the National Ambient Air Quality Standards (NAAQS) State Implementation Plan process, and it was found that they generally agreed to within a factor of 3. No comparison with field-measurement data was reported. In the most recent version of the program, the ISCLT model was incorporated as the default dispersion model, so that multiple emission points within the source area could be modeled, rather than aggregating all the emissions at a single point source within the source complex. It is possible to substitute concentration data from other dispersion models into the HEM. For example, LONGZ was used to model the dispersion of
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Page 379 arsenic from the ASARCO smelter in Tacoma, Wash. LONGZ is a complexterrain model that was optimized to reproduce the sulfur dioxide dispersion from this plant. However, it is not clear that it was adequately modified to take particle deposition into account, and it was found to overpredict the airborne concentration of arsenic by factors of 5-8 for distances of up to 3 km from the plant and factors of 1.6-1.8 for larger distances. Assays of arsenic in urine also suggested that the model substantially overestimated arsenic exposure. For distributed sources, such as perchloroethylene from dry cleaners, area sources were used with emission rates proportional to area population. The dispersion model was modified to incorporate the additional dispersion that comes from surface roughness and heat-island effects. The correction is included by making some of the parameters depend on the city geographic area. In the HEM, the population is based on data from the Bureau of the Census (enumeration district/block groups, ED/BGs). An ED/BG is the area containing on average about 800 people and can range from part of a single city block to several hundred square kilometers. The population of each ED/BG is assumed to be at the center of the population's geographic distribution (centroid). The pollutant concentration at that location is interpolated from the results of the dispersion model. The interpolation is logarithmic in the radial direction and linear in the azimuthal direction. The product of the population and the concentration summed over the total area is then the total annual population exposure. NAAQS Exposure Model The NAAQS exposure model (NEM) was developed to estimate exposure to the criteria pollutants (e.g., carbon dioxide, CO). In 1979, EPA began to develop this model by assembling a database of human activity patterns that could be used to estimate exposures to outdoor pollutants (Roddin et al., 1979). The data were then combined with measured outdoor concentrations in the NEM to estimate exposures to CO (Biller et al., 1981; Johnson and Paul, 1983). The NEM has recently been modified to include indoor exposures by incorporation of the Indoor Air Quality Model (IAQM) (Hayes and Lundberg, 1985). The IAQM is based on the recursive (stepwise) solution of a one-compartment mass-balance model and incorporates three basic indoor microenvironments: home, office or school, and transportation vehicle. It has been used to estimate distributions of ozone exposures (Hayes and Lundberg, 1985) and to evaluate mitigation strategies for indoor exposures to selected pollutants for five scenarios, such as exposure to CO from a gas boiler in a school (Eisinger and Austin, 1987). Simulation of Human Air Pollution Exposure (SHAPE) Model SHAPE (Ott, 1981) is a computerized simulation model that generates synthetic exposure profiles for a hypothetical sample of human subjects; the expo-
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Page 380 sure profiles can be summarized into exposure measuressay, integrated exposuresto estimate the distribution for the exposure measure of interest. The bulk of the model estimates the exposure profile for pollutants attributable to local sources; the contribution of remote sources is assumed to be the same as that of a background site where there is no local source. The total exposure is therefore estimated as the exposure due to local sources plus the ambient concentration at the background site. For each person in the hypothetical sample, the model generates a profile of activities and pollutant concentrations attributable to local sources over a given period, such as a 24-hour period. The activity profiles are generated by a modified Markov model. A later version of SHAPE can accept given profiles of activities, instead of using simulation to generate the activity profiles. At the beginning of the profile, an initial microenvironment is generated according to a probability distribution with the time spent in it, generated according to a microenvironment-specific probability distribution: each microenvironment has a specific probability distribution for its duration. At the end of the duration, a transition into another microenvironment is generated according to a transition probability distribution with another duration. The procedure is repeated until the end of the given period. For each time unit, such as a minute, spent in a given microenvironment, a pollutant concentration is generated according to a microenvironment-specific probability distribution, and each microenvironment has a specific probability distribution for its pollutant concentration. All random values are generated independently of each other. Convolution Model Duan (1981) originally developed the convolution model for the integrated exposure attributable to local sources and later (1987) expanded it for a broader context. In this model, distributions of exposure are calculated from the distributions of concentrations observed in each defined microenvironment and the distribution of time spent in those microenvironments. Thus, distributions of exposure are calculated for a population by assuming that values of concentration and time can be independently drawn from the exposure distributions and combined to yield a series of individual exposures. The exposures can then be summed over time to yield a time-integrated exposure for an individual in the population. Enough cases are drawn to provide a distribution of exposures for the entire population. Variance-Component Model The variance-component model assumes that short-term pollutant concentrations comprise two components, a time-varying component and a time-invariant component. If neither the time-varying component nor the time-invariant
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Page 381 component is negligible, SHAPE and the convolution method can no longer be used; it is necessary to use the variance-component model, which can incorporate both the time-variant and the time-invariant components. Depending on the needs of the analyst, the two components can be either summed or multiplied to estimate the modeled concentration value. Contaminant concentrations are usually more variable at higher values, so the multiplicative form may often be more realistic. It is first necessary to determine the distributions of the two components. If random samples of locales belonging to the same microenvironment are available and if there are continuous monitoring data for at least a random sample of locales, it is possible to estimate the distributions of time-varying and time-invariant components of the concentration directly. If integrated personal monitoring data are available, the methods described by Duan (1987) can be applied. Once those distributions are available, exposure distributions are estimated with a computer simulation similar to that in SHAPE. However, instead of generating a contaminant concentration for each time unit independently, as in SHAPE, values of the time-invariant and time-varying components for each time unit are generated and then combined to determine 1-minute average concentrations. The remainder of the simulation is identical with SHAPE. References Biller, W.F., T.B. Feagans, T.R. Johnson, G.M. Duggan, R.A. Paul, T. McCurdy, and H.C. Thomas. 1981. A general model for estimating exposure associated with alternative NAAQS. Paper No. 81-18.4 in Proceedings of the 74th Annual Meeting of the Air Pollution Control Association, Philadelphia, Pa. Duan, N. 1981. Micro-Environment Types: A Model for Human Exposure to Air Pollution. SIMS Technical Report No. 47. Department of Statistics, Stanford University, Stanford, Calif. Duan, N. 1987. Cartesianized Sample Mean: Imposing Known Independence Structures on Observed Data. WD-3602-SIMS/RC. The RAND Corporation, Santa Monica, Calif. Eisinger, D.S., and B.S. Austin. 1987. Indoor Air Quality: Problem Characterization and Computer Simulation of Indoor Scenarios and Mitigation Strategies. Report No. SYSAPP-87/170. Systems Applications, Inc., San Rafael, Calif. EPA (U.S. Environmental Protection Agency). 1987. Industrial Source Complex (ISC) User's Guide, 2nd ed., revised, Vols. 1 and 2. EPA 450/4-88-002a,b. U.S. Environmental Protection Agency, Research Triangle Park, N.C. Hayes, S.R., and G.W. Lundberg. 1985. Further Improvement and Sensitivity Analysis of an Ozone Population Exposure Model. Draft final report to the American Petroleum Institute. Report No. SYSAPP-85/061. Systems Applications, Inc., San Rafael, Calif. Johnson, T.R., and R.A. Paul. 1983. The NAAQS Exposure Model (NEM) Applied to Carbon Monoxide. EPA-450/5-83-003. Prepared for the U.S. Environmental Agency by PEDCo Environmental Inc., Durham, N.C. under Contract No. 68-02-3390. U.S. Environmental Protection Agency, Research Triangle Park, N.C. Ott, W. 1981. Computer Simulation of Human Air Pollution Exposures to Carbon Monoxide. Paper 81-57.6. Paper presented at 74th Annual Meeting of the Air Pollution Control Association, Philadelphia, Pa.
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Page 382 Roddin, M.F., H.T. Ellis, and W.M. Siddiqee. 1979. Background Data for Human Activity Patterns, Vols. 1, 2. Draft Final Report prepared for Strategies and Air Standards Division, Office of Air Quality Planning and Standards, U.S. Environmental Protection Agency, Research Triangle Park, N.C. Russell, A.G. 1988. Mathematical modeling of the effect of emission sources on atmospheric pollutant concentrations. Pp. 161-205 in Air Pollution, the Automobile, and Public Health, A.Y. Watson, R.R. Bates, and D. Kennedy, eds. Washington, D.C.: National Academy Press. Zannetti, P. 1990. Air Pollution Modeling: Theories, Computational Methods, and Available Software. New York: Van Nostrand Reinhold.