## Mathematical Models

The most sophisticated approach for impact prediction involves the use of mathematical models. Numerous types of mathematical models account for pollutant transport and fate within the environmental setting. Other models describe environmental features and the functioning of ecosystems. This review does not delineate the state-of-the-art of mathematical modeling with regard to environmental impact studies but discusses the availability of types of models which can be used in studies.

Stakhiv (1986) discusses several types of models used for forecasting in water resources planning. He notes the availability of predictive deterministic models for forecasting, including models addressing demographic, socioeconomic, and economic changes; and models for ecological, water quality, energy, hydraulics, hydrology, and land-use changes.

With regard to air quality dispersion, numerous models address point, line, and area sources of air pollution and the results of dispersion from these sources. In addition, within recent years, models are available for longrange transport of pollution and for atmospheric reactions leading to photochemical smog formation and acid rain. Many air quality models are available in software form for personal computers; they represent a usable technology for many studies.

Anderson and Burt (1985) note the following about hy-drologic modeling:

All models seek to simplify the complexity of the real world by selectively exaggerating the fundamental aspects of a system at the expense of incidental detail. In presenting an approximate view of reality, a model must remain simple enough to understand and use, yet complex enough to be representative of the system being studied.

Anderson and Burt (1985) classify hydrological models into three types:

1. Black-box models: These models contain no physically based transfer function to relate input to output; they depend upon establishing a statistical correspondence between input and output.

2. Conceptual models: These models occupy an intermediate position between the deterministic approach and empirical black-box analysis. They are formulated on the basis of a simple arrangement of a small number of components, each of which is a simplified representation of one process element in the system being modeled; each element of the model consists of a nonlinear reservoir in which the relationship between outflow (Q) and storage (S) is given by

where K and n represent constants.

3. Deterministic models: These models are based on complex physical theory. However, despite the simplifying assumptions to solve the flow equations, these models have huge demands in terms of computational time and data requirements and are therefore costly to develop and operate.

Figure 2.4.1 shows a generic method for selecting a mathematical hydrological model (Anderson and Burt 1985). The method emphasizes the dependence of the

modeling upon a clear definition of the problem to be solved, and upon the data base to describe the physical system.

Surface and groundwater quality and quantity models are also plentiful, with major research developments within the last decade occurring in solute transport in subsurface systems. Surface water quality and quantity models range from one-dimensional steady-state models to three-dimensional dynamic models which can be used for rivers, lakes, and estuarine systems (Anderson and Burt 1985). Groundwater flow models now include subsurface processes, such as adsorption and biological decomposition. The International Ground Water Modeling Center, along with the National Water Well Association, has hundreds of groundwater models in software form that address specific environmental consequences of projects or activities.

Noise impact prediction models are available for point, line, and area sources of noise generation. These models range in complexity from simple calculations involving the use of nomographs, to sophisticated computer modeling for airport operations. The technology for noise impact prediction is well developed as a result of numerous research studies related to highways and airports. Many noise impact prediction models are also available in software form for personal computers. Noise models can be used to address continuous or discontinuous noise sources, including instantaneous noise sources related to blasting.

Biological impact prediction models are characterized in the U.S. EIA practice as involving habitat approaches. Specifically, a widely used impact prediction approach involves the Habitat Evaluation Procedures developed by the U.S. Fish and Wildlife Service (1980). In addition, the Habitat Evaluation System developed by the U.S. Army Corps of Engineers (1980) has also been used in a number of impact studies. As noted earlier, these models involve calculating an index which incorporates both quality and quantity information. Prediction of impacts involves determining the index under baseline as well as future with- and without-project conditions. Other types of biological impact models include species population models which may be based on empirical approaches involving statistical correlations (Starfield and Bleloch 1986). The most sophisticated biological impact models are those involving energy system diagrams; these are used in some environmental impact studies.

Ad hoc models may be needed to address impact concerns associated with a proposed project, plan, program, or policy. Starfield and Bleloch (1986) address the building of models for conservation and wildlife management. Their topics include a simple single-species model, an exploratory stochastic model, a complex single-species model, and an ecosystem model, among others. Each topic begins with a management problem and describes how a model can be constructed to address that problem. They also describe modifications to the model in light of the available (or unavailable) data, how the model can be exercised, and what can be learned from it. They presume that the problem that needs to be modeled is often poorly defined; the processes and mechanisms are not well understood, and the data are often scant and difficult to obtain. Their work also includes information on the development of expert systems, as well as for resource management.

Model building for biological systems is also addressed by Armour and Williamson (1988) in their procedure for organizing and simplifying complex information into a cause-and-effect model through an interdisciplinary exercise. Information includes prerequisites to help ensure model completion, applications of model information, diagnosing and correcting modeling problems when users encounter difficulties, and technical limitations of the approach.

Predictive modeling is also possible for ascertaining the potential for archaeological resources in geographical study areas. Such modeling is primarily based upon evaluating factors that indicate the likelihood of archaeological resources being found, relating the factors to existing information, evaluating the likelihood for early occupations in the area, and other environmental and sociological factors. This type of modeling is often used to determine the need for planning and conducting archaeological field surveys.

Visual quality is also a subject in selected impact studies. Visual impact modeling approaches have been developed by several federal agencies, including the U.S. Forest Service, U.S. Bureau of Land Management, U.S. Soil Conservation Service, and U.S. Army Corps of Engineers (Smardon, Palmer, and Felleman 1986). These visual impact models typically involve evaluating a series of factors, in some cases quantitatively and in other cases descriptively, and assembling the information into an overall visual quality index for the study area. In this context, these models are similar to the environmental indices approach described earlier.

Impact prediction related to the socioeconomic environment is often associated with the use of human population and econometric models. Population forecasting can range from simple projections of historical trends to complicated cohort analysis models. Econometric models relate the population and economic characteristics of study areas so that interrelationships can be depicted between population changes and changes in economic features within given study areas. In addition to the econometric models and models related to population change, other impact predictions for the socioeconomic environment are addressed by the use of multiplier factors applied to population changes. In this regard, several input-output models can be used in environmental impact studies.

Table 2.4.3 provides a noninclusive listing of state-of-the-art books or reports related to quantitative models useful for impact forecasting. A common ingredient in the ap-

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