Further, some quantities vary over time, over space, or across individuals in a population; this is termed ``variability''. Variability is the heterogeneity between individual members of a population of some type, and is typically characterized through a frequency distribution. It is possible to interpret variability as uncertainty under certain conditions, since both can be addressed in terms of ``frequency'' distributions.
However, the implications of the differences in uncertainty and variability are relevant in decision making. For example, the knowledge of the frequency distribution for variability can guide the identification of significant subpopulations which merit more focused study. In contrast, the knowledge of uncertainty can aid in determining areas where additional research or alternative measurement techniques are needed to reduce uncertainty.
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Mathematical models are necessarily simplified representations of the
phenomena being studied and a key aspect of the modeling process is the
judicious choice of model assumptions. The optimal mechanistic model will
provide the greatest simplifications while providing an adequately accurate
representation of the processes affecting the phenomena of interest. Hence,
the structure of mathematical models employed to represent
transport-transformation systems is often a key source of uncertainty. In
addition to the significant approximations often inherent in modeling,
sometimes competing models may be available. Furthermore, the limited
spatial or temporal resolution (e.g., numerical grid cell size) of many
models is also a type of approximation that introduces uncertainty into
model results. In this context, different sources of model uncertainties are
summarized in
the following:
Model Structure: Uncertainty arises when there are alternative
sets of scientific or technical assumptions for developing a model. In such
cases, if the results from competing models result in similar conclusions,
then one can be confident that the decision is robust in the face of
uncertainty. If, however, alternative model formulations lead to different
conclusions, further model evaluation might be required.
Model Detail: Often, models are simplified for purposes of
tractability. These include assumptions to convert a complex nonlinear model
to a simpler linear model in a parameter space of interest. Uncertainty in
the predictions of simplified models can sometimes be characterized by
comparison of their predictions to those of more detailed, inclusive
models.
Extrapolation: Models that are validated for one portion of
input space may be completely inappropriate for making predictions in other
regions of the parameter space. For example, a dose-response model based on
high-dose, short duration animal tests may incur significant errors when
applied to study low-dose, long duration human exposures. Similarly, in
atmospheric modeling, models that are evaluated only for base case emission
levels, may introduce uncertainties when they are employed in the study of
future scenarios that involve significantly different
emissions levels (also known as 
E caveat).
Model Resolution: In the application of numerical
models, selection of a spatial and/or temporal grid size often
involves uncertainty. On one hand, there is a trade-off between the
computation time (hence cost) and prediction accuracy. On the other
hand, there is a trade-off between resolution and the validity of the
governing equations of the model at such scales. Very often, a coarse
grid resolution introduces approximations and uncertainties into model
results. Sometimes, a finer grid resolution need not necessarily
result in more accurate predictions. For example, in photochemical
grid modeling, the atmospheric diffusion equation (ADE) (see
Equation 6.1) is valid for grid cell sizes between 2 km and
20 km [137]. In this case, a coarse-grid model ignores
significant ``sub-grid'' detail, and a fine-grid model would require
more computer resources, and may even produce incorrect results, as
the governing equation may not be appropriate. This type of
uncertainty is sometimes dealt with through an appropriate selection
of model domain parameter values, or by comparing results based on
different grid sizes.
Model Boundaries: Any model may have limited boundaries in
terms of time, space, number of chemical species, types of pathways, and so
on. The selection of a model boundary may be a type of
simplification. Within the boundary, the model may be an accurate
representation, but other overlooked phenomenon not included in the model
may play a role in the scenario being modeled.
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Table 2.1 lists some of the sources of model and parametric uncertainty associated with the formulation and the application of transport-transformation models.