Stochastic versus Deterministic Approaches - Environmental Modelling: Finding Simplicity in Complexity - page 128

8

Stochastic versus Deterministic
Approaches

Philippe Renard 1 , Andres Alcolea 2 , and David Ginsbourger 3
1 Centre d'Hydrogeologie, Universite de Neuchatel, Switzerland
2 Geo-Energie Suisse, Basel, Switzerland
3 Department of Mathematics and Statistics, University of Bern, Switzerland

stage of development without incurring the full expense
of a full-sized prototype. Notwithstanding the use of these
types of models in other branches of science and engineer-
ing, the most popular models in environmental sciences
are mathematical. A mathematical model describes a sys-
tem by a set of state variables and a set of equations
that establish relationships between those variables and
the governing parameters. Mathematical models can be
analytical or numerical. Analytical models often require
many simplifications to render the equations amenable
to solution. Instead, numerical models are more versatile
and make use of computers to solve the equations.
Mathematical models (either analytical or numeri-
cal) can be deterministic or stochastic (from the Greek
τ

8.1 Introduction

In broad sense, modelling refers to the process of generat-
ing a simplified representation of a real system. A suitable
model must be able to explain past observations, inte-
grate present data and predict with reasonable accuracy
the response of the system to planned stresses (Carrera
et al ., 1987). Models have evolved together with science
and nowadays modelling is an essential and insepara-
ble part of scientific activity. In environmental sciences,
models are used to guarantee suitable conditions for sus-
tainable development and are a pillar for the design of
social and industrial policies.
Model types include analogue models, scale models
and mathematical models. Analogue models represent
the target system by another, more understandable or
analysable system. These models rely on Feynman's
principle (Feynman et al ., 1989, sec. 12-1): 'The same
equations have the same solutions.' For example, the
electric/hydraulic analogy (Figure 8.1a) establishes the
parallelism between voltage and water-pressure differ-
ence or between electric current and flow rate of water.
Scale models are representations of a system that is larger
or smaller (most often) than the actual size of the system
being modelled. Scale models (Figure 8.1b) are often built
to analyse physical processes in the laboratory or to test
the likely performance of a particular design at an early

o χ o ς for 'aim' or 'guess'). A deterministic model is
one in which state variables are uniquely determined by
parameters in the model and by sets of previous states of
these variables. Therefore, deterministic models perform
the same way for a given set of parameters and initial
conditions and their solution is unique. Nevertheless,
deterministic models are sometimes unstable - i.e., small
perturbations (often below the detection limits) of the
initial conditions or the parameters governing the prob-
lem lead to large variations of the final solution (Lorenz,
1963). Thus, despite the fact that the solution is unique,
one can obtain solutions that are dramatically different
by perturbing slightly a single governing parameter or
the initial condition at a single point of the domain.