Environmental Engineering Reference
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W t
12
10
8
6
4
2
t
200
400
600
800
1000
Figure 2.13. Example of a typical realization of the Wiener process. An enlargement
of a small portion of the path is shown in the inset to underline the fractal nature of
Brownian motion.
i.e., it can be mathematically represented as
ξ
gn ( t )
t )
gn ( t )
ξ
=
2 s gn
δ
( t
,
(2.74)
where s gn is the strength of the noise 4 ; (iii) all cumulants of
ξ gn ( t ) of orders higher
than the second vanish. The first two properties are valid for any white-noise process;
the third one is peculiar to white Gaussian noise. As for WSN, the problem with the
Gaussian white noise is that, even though the notion of white noise is commonly used,
it refers to a singular mathematical object. Thus Gaussian white noise is commonly
treated as the formal derivative of the Wiener process,
t
0 ξ
gn ( t )d t ,
=
W ( t )
(2.75)
a mathematical model for the displacement of the Brownian particle from some
arbitrary starting point at t
0 (Brownian motion). A typical path of the Wiener
process is represented in Fig. 2.13 .
The average and the covariance function of the Wiener process are ( Parzen , 1967 ,
p. 68)
=
W ( t )
=
0
,
(2.76)
W ( t ) W ( t )
2 min( t
t )
= σ
,
,
(2.77)
2 is the parameter of the Wiener process.
respectively, where
σ
4 The autocorrelation function of white Gaussian noise is often denoted as σ
2
δ ( t t ). However, we believe this
2 is commonly used to represent the
variance of a process, and the variance of white Gaussian noise diverges. Moreover, s gn is obtained as the limiting
value of the intensity of DMN, s dn =− 1 2 /
σ
notation may induce some confusion in the reader, because the symbol
( k 1 + k 2 ), as explained in Subsection 2.4.2 .
 
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