Stochastic versus Deterministic Approaches - Environmental Modelling: Finding Simplicity in Complexity - page 129

clay lenses

coarse sand coarse sand

Coarser
sand

Medium-coarse sand
(semi-confined layer)

(a) (b)

Figure 8.1 Types of models: (a) Electrical analogue model of the groundwater flow in the Areuse catchment in Switzerland (Device
built by J. Tripet), (b) scale model of an aquifer (Courtesy of F. Cornation).

Conversely, stochastic model parameters are described by
random variables or distributions rather than by a single
value. Correspondingly, state variables are also described
by probability distributions. Thus, a stochastic model
yields a manifold of equally likely solutions, which allow
the modeller to evaluate the inherent uncertainty of the
natural system being modelled.
Mathematical models (either analytical or numerical,
deterministic or stochastic) can also be classified as direct
or inverse. Direct or forward modelling consists of obtain-
ing the value of the state variables given a model structure
and values or distributions of the parameters governing
the state equations. Instead, inverse modelling refers to
the process of gathering information about the model
and its parameters from measurements of what is being
modelled (Carrera et al ., 2005). In practice, the governing
parameters and the model structure are highly uncertain.
Thus, direct modelling is restricted mainly to academic
purposes. On the other hand, inverse modelling corre-
sponds to the quotidian situation, where measurements
(either of parameters or state variables or both) are col-
lected at a few selected locations in space and time and a
model structure and parameter distributions are inferred
from those measurements.
Either deterministic or stochastic, direct or inverse,
modelling is a crucial step in environmental sciences.
Just to mention one example, the disposal of nuclear
wastes in deep geological formations requires the esti-
mation of the potential environmental impact in the
biosphere caused by a possible release of hazardous
radionuclides. This problem requires detailed studies of
their migration through the subsurface, including the use

of numerical models to predict travel times and trajecto-
ries. A deterministic model assumes a certain geometry of
the geological bodies, fractures, and so forth, and a deter-
ministic (unique) spatial distribution of the parameters
governing the model equations - for example, hydraulic
conductivity and storativity. Thus, a deterministic model
yields a unique prediction of the migration. As such, a
radionuclide migrates (with probability one) to the bio-
sphere following a 'single deterministic' trajectory and
after a 'single deterministic' travel time. Unfortunately,
it is impossible to get 'the perfect' characterization of
geology, hydraulic conductivity, and so forth, because
they are scarcely measured and therefore, our knowl-
edge is inherently uncertain. Even being omnipotent
and gathering the required information at every point
in space and time, the model would still be uncertain
due to the presence of measurement errors. Stochas-
tic models acknowledge model uncertainties, including
(1) conceptual uncertainties, such as lack of knowledge
about the dominant processes driving the modelled phe-
nomenon; (2) measurement uncertainties due to the
limited accuracy of instruments; and (3) uncertainties
due to the scarcity or the lack of measurements in space
and time. For instance, one can simulate the migration of
the radionuclide using many different geological scenar-
ios accounting for, presence or absence of fractures for
example. These simulations are a set of different predic-
tions of the migration under different conditions, from
which the modeller or the policy-maker can evaluate
probabilities of occurrence of a given event (such as the
probability that the radionuclide reaches the biosphere in
less than 10 000 years). These events are characterized by