Stochastic versus Deterministic Approaches - Environmental Modelling: Finding Simplicity in Complexity - page 143

Doing so, one can obtain Darcy's law from the small-
scale equations (Neuman, 1977; Hassanizadeh,1986; de
Marsily, 1989).
For all those reasons, we argue that the traditional (and
controversial) question on the choice of a statistical, deter-
ministic or stochastic approach to model environmental
processes is obsolete. The world should not be considered
as a summary of statistical moments (statistical models),
either perfectly known (deterministic) or perfectly ran-
dom (stochastic). The three views of nature need to be
used altogether and need to be integrated. Certainly, this
integration requires an additional effort from scientists
and engineers who have to become familiar with the three
approaches. Still, we argue that there is no way nowadays
to make the economy of one or the other type of models.
Indeed, no one can reasonably defend the idea that a single
characterization of a natural system is sufficient to rep-
resent our current understanding. Neither does it make
more sense to argue that everything is known with 100%
accuracy than to argue that physical laws are irrelevant.
To conclude, it is necessary to emphasize that the meth-
ods discussed above are extremely efficient to describe
regularities and produce forecasts based on an assump-
tion of temporal or spatial stationarity: what is measured
in the past and at a given location has a high chance to
be reproduced in the future or in another similar loca-
tion. In this way, we can infer statistical models, derive
physical laws and build theories and models. That type
of reasoning is generally true and has allowed mankind
to make enormous progress but it is not always the case
and this is where the limits of these approaches lie. When
extreme events are not present in the data sets and in the
observations, there is almost no way to predict them with
reasonable accuracy. When the conceptual models on
which the deterministic models are based do not include
some crucial features that have not been observed, there is
very little chance that the forecasts will be correct and there
is little chance that the stochastic models will include such
exceptional features. Therefore, the modeller must always
remain very humble and remember that the probability
estimates, even if they were derived with a very rigorous
mathematical treatment, are only based on a subsample of
the real events. This argument is sometimes used to defend
the idea that stochastic methods are not useful and that
mini-max or scenario analyses are much more reasonable.
Our vision is that the choice of a model must be based
on the needs and constraints for a given project, as well
as on the current level of knowledge and resources. The
choice must not be dogmatic but driven by the principle
of maximum efficiency to solve a given problem.

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