Stochastic versus Deterministic Approaches - Environmental Modelling: Finding Simplicity in Complexity - page 133

skewness and the kurtosis (third and fourth moments,
respectively) are often used as well. Yet, describing a
random variable with just two, three or four statistical
moments is inherently limited because it does not take
into account any knowledge of the physical process that
generated the observed values. Estimating only the first
few moments of X can therefore lead to an incomplete
picture of the statistical distribution of the variable of
interest. Suitable parametric models help to alleviate this
problem.
In this framework, and coming back to the example, the
classical approach to model the lifetime of radionuclides
consists of using the following parametric exponential
pdf distribution:

above. Still, the same concepts and methodology apply
to multivariate/multiparametric problems. This method
is especially useful to address the correlations between
different types of observations (multivariate statistics),
or between the same type of observations measured at
different locations (spatial statistics).

8.3.2 Deterministicmodels

While descriptive statistical models have the ability to cap-
ture and describe repetitive patterns observed in nature
and make forecasts, they usually do not integrate physi-
cal concepts. Thus, forecasts made with those techniques
might be incoherent with the underlying physics gov-
erning the process under study. For example, it is a
current practice in groundwater modelling to interpolate
the hydraulic head measurements h(x,t) in space and time
in a reservoir using geostatistical techniques to produce
global maps (Rivest et al ., 2008). Those maps are compat-
ible with observations and known trends. They agree with
the observed spatial variability but they are most often
incompatible with basic physical principles, such as the
conservation of the mass of water in Equation (8.2). For
example, nothing guarantees that the interpolated heads
in the domain would not be lower than the lowest dis-
charge point. As such, flow directions might not make any
sense. This lack of physical coherency limits the applica-
tion of purely statistical methods. Instead, deterministic
models are rather preferred by scientists and engineers.
To illustrate the use of a deterministic model, let us
imagine the following problem: a company is planning to
construct a new building on an alluvial plain containing
an aquifer (Figure 8.2a). The problem is then to evaluate
whether the basement of the building will be below or
above the groundwater table. To that end, we want to
predict the hydraulic head along a flow line connecting
the recharge area of the aquifer (in grey; Figure 8.2a) to a
river that acts as a discharge area. We simplify the problem
drastically to obtain an analytical solution defining the
hydraulic head distribution. First, we assume that it is
sufficient to consider one-dimensional flow along a single
flow line. In addition, we consider that the aquifer is
confined (just to keep the equations amenable to didactic
use), and we assume a constant recharge r due caused
by rainfall all over the domain. By combining Equations
(8.1) and (8.2), the simplified one-dimensional problem
is expressed as follows:

f λ ( x ) = λ e − λ x for x ≥ 0, and 0 otherwise,

(8.7)

λ > where 0 the positive decay coefficient (also termed

rate parameter) is the unique parameter )inplay.

When such a parametric expression is available, instead of
directly estimating the mean, variance, or other moments
of X, it is preferable to estimate directly the parameters
controlling the distribution from the available data. In
the example above, the integration of the exponential
distribution in Equations (8.4) and (8.6) shows that

ε ( λ

1
λ

) 2 are the mean and the variance, respectively
(see standard references on calculus such as Bostock and
Chandler, 1981). Thus given,

/λ

and (1

ε λ

( ), one can use these

relations to estimate ˆ

λ from available measurements. The
main advantage of this approach is that, once the estimate
of λ is known, it allows the evaluation of the probability
of any event of practical interest. For example, one can
use the estimated pdf f λ (Equation 8.7) to compute the
probability for a radionuclide to decay during a time
interval [a,b]:

b

P ( X ∈ [ a , b ]) = f λ ( x ) dx (8.8)

a

Applications of this principle in environmental sci-
ences are widespread. For instance, it is very common
to evaluate the probability of flood events or volcanic
eruptions by first identifying a suitable parametric sta-
tistical distribution, inferring its parameters and then
making forecasts (Jaquet and Carniel, 2001). Several
difficulties may arise in this process, such as the low
occurrence of extreme events in the data set, which makes
the inference of an appropriate law and its parame-
ters an arduous task. For the sake of brevity, only the
case of a single variable with corresponding pdf fully
characterized by a single parameter has been described

∂
∂

k ∂

h

− = r (8.9)

x ∂ x