Geoscience Reference
In-Depth Information
explain, because such fields can be neither periodic nor integrable, neither the usual
Fourier series nor integrals are formally applicable. However, according to
Lumley
and Panofsky
(
1964
), under a set of weak assumptions
u(t)
can be expanded in
another random, stochastic process
Z(ω)
:
+∞
e
iωt
dZ(ω).
u(t)
=
(15.6)
−∞
This is known as a stochastic Fourier-Stieltjes integral. The process
Z(ω)
has the
property
T
e
−
ibt
e
−
iat
1
2
π
−
=
−
lim
T
→∞
u(t) dt
Z(b)
Z(a).
(15.7)
−
it
−
T
The integrals are written in this way so that
Z(ω)
need not be differentiable.
We can perhaps make
Eqs. (15.6
)and
(15.7)
more familiar as follows. If we let
a
=
ω, b
=
ω
+
ω
, then we can write
e
−
ibt
e
−
iat
−
it
e
−
iωt
e
−
iωt
e
−
iωt
−
−
=
.
(15.8)
−
it
With the series expansion
e
−
iωt
1
−
iωt,
this becomes
e
−
ibt
e
−
iat
−
ωe
−
iωt
,
(15.9)
−
it
and
Eq. (15.7)
can be written, after dividing by
ω
and taking the limit as
ω
→
0
,
1
ω
T
u(t) dt
e
−
ibt
−
e
−
iat
−
1
2
π
lim
ω
lim
it
→
0
T
→∞
−
T
∞
1
2
π
dZ
dω
,
e
−
iωt
u(t) dt
=
=
(15.10)
−∞
assuming the derivative
dZ/dω
exists. Thus in that case
Eq. (15.10)
is the Fourier
transform of
u(t)
.From
Eq. (15.6)
the other half of this transform pair is
+∞
e
iωt
dZ
=
u(t)
dω
dω,
(15.11)
−∞
the inverse Fourier transform.
We'll see later in the chapter that the Fourier-Stieltjes representation is particu-
larly useful for finding exact statistical solutions to linear stochastic problems.
