Stochastic versus Deterministic Approaches - Environmental Modelling: Finding Simplicity in Complexity - page 130

probability distributions from which statistical moments
can be evaluated such as the minimum travel time (i.e.
the maximum time for human beings to react to the
migration).
Despite the aforementioned advantages, the use of
stochastic models has not been excluded from debate.
Stochastic models are often surrounded with an aura
of esoterism and, in the end, they are often ignored by
most decision-makers, who prefer a single (deterministic)
solution (Carrera and Medina, 1999; Renard, 2007). One
might be tempted to give up and accept that stochastic
processes are not amenable to the quantitative and qual-
itative assessment of modelling. However, it is precisely
the large uncertainty associated with natural sciences
that makes stochastic models necessary. The goal of this
chapter is to propose a discussion of the strengths and
weaknesses of deterministic and stochastic models and
describe their applicability in environmental sciences.
The chapter starts by outlining some background con-
cepts and philosophical issues behind deterministic and
stochastic views of nature. We then present a summary of
the most widespread methods. The differences between
deterministic and stochastic modelling are illustrated by
means of a real-world application in Oman. The chapter
ends with a discussion and some recommendations about
the use of models in environmental sciences.

The motion of groundwater is then described by the
conservation principle, whose application leads to the
very well known groundwater-flow equation. It states
that the mass (or the volume if the fluid is assumed
uncompressible) of water that enters an elementary vol-
ume of porous medium per unit time must be equal to the
mass (or volume) of water that leaves that volume plus
the mass (or volume) stored in the elementary volume.
In terms of water volume and assuming constant density,
the groundwater flow equation can be expressed as:

S s ∂ h

∇ q =−

∂ t +

r ( x ) (8.2)

[L − 1 ] where t [T] represents time, S s is storativity,

q [ T − 1 ] represents the divergence of fluid flux (i.e.,
difference between incoming and outgoing volume of
water), and r [T − 1 ] is a sink/source term that may be
used to model, for example, the recharge to the aquifer
after rainfall. Note that all these parameters are, indeed,
heterogeneous in reality. Thus, they vary from one loca-
tion in space to another. K and S s can also vary in time
if the aquifer changes due to changes in porosity caused
by, e.g. clogging or precipitation processes. Yet, these are
often considered as constant in time. Instead, recharge
is a parameter that clearly depends on time. Finally, the
groundwater velocity is:

∇

v = q /φ (8.3)

8.2 A philosophical perspective φ −

where ] is the effective porosity of the aquifer (the

ratio of the volume of interconnected pores to the total
volume of the aquifer).
As one can see, this velocity can be obtained unequiv-
ocally from precise values (or spatial distributions if
heterogeneity is accounted for) of the physical param-
eters k , S s and φ , initial and boundary conditions and
sink/source terms (see also Chapter 5). Solving Equations
(8.1) to (8.3) twice with equal ingredients leads to two
identical solutions, without any room for randomness.
This approach is in line with the arguments of the Ger-
man mathematician and philosopher Leibniz, who quoted
the Greek philosopher Parmenides of Elea (fifth century
BCE), and stated the Principle of Sufficient Reason (Kab-
itz and Schepers, 2006): 'everything that is, has a sufficient
reason for being and being as it is, and not otherwise.'
In plain words, the same conditions lead to the same
consequences. This strong defence of determinism was
later on softened by the same Leibniz (Rescher, 1991).
As pointed out by Look (2008): 'most of the time these
reasons cannot be known to us.' This sentence plays a
crucial role in the remainder of this section.

[

The laws of motion expounded by Newton (1687) state
that the future of a system of bodies can be determined
uniquely, given the initial position and velocity of each
body and all acting forces. This radically deterministic
approach has been applied extensively to environmental
problems. For example, the flux of fluids (often ground-
water) through a porous medium is usually described by
Darcy's law (1856), which is analogous to Ohm's law in
electricity or Fourier's law in thermal energy. As with
most physical laws, it was first deduced from observa-
tions and later authenticated with a very large number
of experiments. In groundwater hydrology, Darcy's law
states that the flux of water q [L T − 1 ] through a unit sur-
face [L 2 ] is proportional to the gradient of hydraulic heads
h (a potential, if water density is constant, that depends
on water height and water pressure) and to a physical
parameter k [L T − 1 ], termed hydraulic conductivity, that
depends on the type of fluid and porous medium:

q =− k ∇ h (8.1)