Conducting Environmental Impact Assessment in Developing Countries (UNU, 1999, 375 pages)
 5. EIA tools
 5.1 Impact prediction
 (introduction...) 5.1.1 Application of methods to different levels of prediction 5.1.2 Informal modelling 5.1.3 Physical models 5.1.4 Mathematical models 5.1.5 Modelling procedure 5.1.6 Sensitivity analysis 5.1.7 Probabilistic modelling 5.1.8 Points to be considered when selecting a prediction model 5.1.9 Difficulties in prediction 5.1.10 Auditing of EIAs 5.1.11 Precision in prediction and decision resolution

### 5.1.4 Mathematical models

Box 5.1 Simple mathematical model

 A mathematical model might be illustrated as: Simple representation of a mathematical model The model consists of two rules imposing limits on variables A and B, and three equations. There are two kinds of inputs: A and B are external inputs to a system, while X and Y are parameters, which are actually internal to the system, but which can be changed to calibrate the model to fit observed data. C is an intermediate variable, which is not outputted. D and E are the outputs of interest to a user. The model could be implemented easily as a computer program, making it easy to process many inputs A and B, or to vary the parameters to tune the model. Suppose that measurements have been made for a particular event. A is 70, B is 100, and D has been recorded as 25 and E as 11. To match these values with the above model, we need to vary X and Y. This is easy as these parameters are simple multipliers and the required values can be found by algebra. These are X = 1.038 and Y = 1.729. In more complex models, parameters are fitted by trial and error or using an optimization technique

In a mathematical model the behaviour of an environmental system is represented by mathematical expressions of the relationships between variables. In general the output variable (x) is a function of one or more input variables (A,B,C,...):

X = f(A,B,C,...)

A model is a representation of the significant attributes of a real prototype, but is simpler and is easier to build, change, or operate.

The number of different variables in a model and the nature of the relationships between them are determined by the complexity of the system. The aim in mathematical modelling is to minimize the number of variables and keep the relationships as simple as possible, while retaining a sufficiently accurate and workable representation of the environment system.

When the structure and processes of ecosystems are understood, models such as Box 5.1 can be used to describe systems such as those below.

· The universal soil loss equation predicts erosion rates from knowledge about rainfall, slope, soil structure, vegetative cover, and management practices.

· The water budget is essentially a materials-balance equation involving rainfall, evapotranspiration, runoff, infiltration, and storage in a watershed or other hydrologic system.

· The salt budget accounts for all sources of salt, movement by solution, deposition (precipitation), uptake by plants, and export by leaching.

· Population dynamics predict the rise and fall of biological organisms and communities from knowledge of lifecycles, predator-prey relationships, food webs, and other factors affecting the lives of various species.

An example of a mathematical model is a simple form of the Gaussian plume dispersion equation for predicting air quality around a point source of emission:

where c= ground level concentration (mg/m3) at a distance of x meters in the wind direction; Q= rate of emission (mg/s); h= height of emission (= stack height + plume rise) (m); and sy and sz = lateral and vertical dispersion coefficients calculated for the required value of x from standard empirical formulae appropriate to the emission height, the roughness of the surrounding surface, and the atmospheric stability.

The exact shape of the relationship in a model is defined by establishing the model parameters, which may vary according to the circumstances in which the model is applied; for example, the dispersion coefficients in the example above must be defined according to the conditions of atmospheric stability, the surface roughness of the surrounding area, and the emission height. Various standard empirical formulae for these coefficients have been established by different workers for different types of emission and different meteorological and topographical conditions.