Geoscience Reference
In-Depth Information
15.2.3 The power spectral density, or spectrum
The differences or increments
dZ(ω)
may be thought of as the complex amplitudes
of the Fourier modes of frequency
ω
.Theyare
orthogonal
; that is, nonoverlapping
members are uncorrelated:
dZ(ω
1
)dZ
∗
(ω
2
)
=
1
=
0
,
ω
2
,
(15.12)
with
dZ
∗
the complex conjugate. The variance of the differences is the differential
of the power spectral distribution function
F
,
ω
φ(ω
)dω,
dZ(ω)dZ
∗
(ω)
=
dF(ω)
=
φ(ω)dω,
F(ω)
=
(15.13)
−∞
with
φ(ω)
the power spectral density or, loosely, the spectrum. The autocorrelation
and power spectral distribution functions are related through
+∞
+∞
e
iωt
dF(ω)
e
iωt
φ(ω)dω,
u
2
ρ(t)
=
=
(15.14)
−∞
−∞
which indicates that the autocorrelation function
u
2
ρ(t)
is the Fourier transform of
the spectrum
φ
. The inverse transform relation is
+∞
1
2
π
e
−
iωt
ρ(t)u
2
dt.
φ(ω)
=
(15.15)
−∞
Since
ρ
is an even function,
φ
is also an even function.
The adjective
power
is used here because when
u(t)
is a voltage signal, its
square is proportional to power
.
F
is a
distribution
function because
F(ω)
gives
the contribution to the variance
u
2
from frequencies below
ω
(from
(15.13)
).
According to
Lumley and Panofsky
(
1964
) this set of theorems is called the
Wiener-Khintchine theorem, and in effect restores everything that was lost due to
the lack of integrability or periodicity of our stochastic function
u(t)
.
15.2.4 Cross correlations and cross spectra
The foregoing analysis can be extended to two different stationary random
functions, say
u(t)
and
v(t)
. Their cross covariance is
+∞
e
iωt
1
−
iω
t
2
dZ
u
(ω) dZ
v
(ω
).
u(t
1
)v(t
2
)
=
C
uv
(t
1
−
t
2
)
=
(15.16)
−∞
