Noise-driven dynamical systems - Noise-induced Phenomena in the Environmental Sciences - page 40

We obtain the covariance function and first moments of the Gaussian white noise
as found in VanMarcke ( 1983 , pp. 110-111) by considering that Gaussian white noise
is the formal derivative of the Wiener process; as a consequence the mean ξ gn ( t ) =

gn ( t )

2

W ( t ) W ( t ) t ) 2

∂ W ( t ) /∂ t is zero and the covariance ξ gn ( t ) ξ =− ∂ /∂ ( t −

2

is reported in Eq. ( 2.74 ), where noise intensity s gn = σ / 2. The variance is κ = 2gn

σ

2

δ = δ

(0) 2 s gn (0), i.e., it diverges.

2.4.2 White Gaussian noise as a limiting case of DMN and WSN
We can obtain white Gaussian noise from DMN by taking the following parameter
values:

2
1

1 =− 2 →∞ , k 1 = k 2 =

2 s gn .

(2.78)

It is easily verified that we can obtain moments and autocorrelation function ( 2.74 )
fromEq. ( B2.1-5 ) by taking the limit for lim a → 0 e −| t / a | / (2 a )].

We can also obtain Gaussian white noise fromWSN by letting the frequency of the
shots go to infinity and the average intensity go to zero. This corresponds to taking,
for the zero-average WSN process

τ → 0 [note that δ ( t ) = c

ξ sn ,

s gn
λ

λ →∞ , α = . (2.79)

2.4.3 Relevance of Gaussian noise in the biogeosciences
In a large class of environmental processes, external noise varies at a much faster
time scale than the deterministic component of the system. This justifies the use of
a white-noise (memoryless) representation of the external forcing in a number of
environmental systems. Moreover, in many situations, fluctuations in the external
forcing are the result of several factors simultaneously acting on the environmental
system. When the number of these factors is large enough, the central-limit theorem
ensures that the fluctuations in the external parameter have approximatively a Gaussian
distribution. The quality of the approximation depends on the number of concurring
factors, the shape of their probability distributions, and the cross correlation among
the factors. The frequent simultaneous presence of these two conditions (memoryless
external forcing and numerous environmental factors) explains the common use of
Gaussian white noise to represent the external forcing in models of biological and
geophysical processes.
We refer again, as in Subsection 2.3.3 , to an example in which rainfall plays
the role of the external random force; we showed in Figure 2.10 that, considering
a daily resolution, rainfall can be approximatively modeled as a WSN process, in
which rainfall events occur at random times and are separated by relatively long dry