Geoscience Reference
In-Depth Information
which means that there is exactly one pair of values which is taken on by any
particular
u(t)
at
t
1
and
t
2
. Further, we see that (from
(13.16)
and
(13.17)
)
+∞
β
2
(u
1
,u
2
;
t
1
,t
2
)du
2
=
β(u
1
;
t
1
),
(13.20)
−∞
and correspondingly for the integral over
u
1
. Its Fourier transform, called a
characteristic function, is
+∞
e
iκ
1
u
1
+
iκ
2
u
2
β
2
(u
1
,u
2
;
f
2
(κ
1
,κ
2
;
t
1
,t
2
)
=
t
1
,t
2
)du
1
du
2
−∞
e
iκ
1
u
1
+
iκ
2
u
2
.
=
(13.21)
Th
rough the property
(13.20)
all the moments at
t
1
or
t
2
only can be obtained, e.g.,
u
n
(t
1
)
. There are also joint moments (also called cross moments):
+∞
u
1
u
2
β
2
(u
1
,u
2
;
u
n
(t
1
)u
m
(t
2
)
=
t
1
,t
2
)du
1
du
2
.
(13.22)
−∞
The complete set of these moments formally represents
f
2
,
∞
1
n
!
m
!
u
n
(t
1
)u
m
(t
2
)(iκ
1
)
n
(iκ
2
)
m
,
f
2
=
(13.23)
n,m
=
0
and hence
β
2
.
Rather than considering
u(t)
at two different times, we could have considered
two different processes, say
u(t)
and
v(t)
, at the same time - two components of
velocity, for example, or velocity and a scalar. We can also consider products similar
to
(13.16)
of any number of
φ
's, giving simultaneous information at arbitrarilymany
times, for several different processes, if desired. For a single process, for example,
we could define
1
,
2
,...,
(13.24)
with the interpretation paralleling that given before. The entire set of such densities,
for all
n
, gives all statistical information about
u(t)
.
β
n
(u
1
, ..., u
n
;
t
1
, ..., t
n
),
n
=
13.2.4 Stationarity
A random process is
stationary
when
β
is not a function of time, and
β
n
,n
≥
2is
a function only of the time differences. That is,
β(u
;
t)
=
β(u),
β
2
(u
1
,u
2
;
t
1
,t
2
)
=
β
2
(u
1
,u
2
;
t
1
−
t
2
),
(13.25)
