Geoscience Reference
In-Depth Information
The variance of
f
is determined by squaring and ensemble averaging
Eq. (2.47)
.
Since
f
is real,
f
is equal to its complex conjugate
f
∗
and we can write
∞
∞
f(κ
n
) f
∗
(κ
m
)e
i(κ
n
−
κ
m
)x
.
f
2
ff
∗
=
=
(2.48)
n
=−∞
m
=−∞
It is a property of the Fourier-series representation of a homogeneous function that
Fourier coefficients of different wavenumbers are uncorrelated - that is,
f(κ
n
) f
∗
(κ
m
)
=
0
,κ
n
=
κ
m
.
(2.49)
With
Eq. (2.
48)
we
ca
n make this plausible as follows. The homogeneity of
f(x)
implies that
f
2
f
2
(x)
, and therefore the lhs of
Eq. (2.48)
is inde
pendent of
x.
On the rhs, for
κ
n
=
=
κ
m
the exponential is a nonzero function of
x
,so
f(κ
n
) f
∗
(κ
m
)
must vanish to make the rhs independent of
x
.Thisis
Eq. (2.49)
.
Given the constraint
(2.49)
, the variance in
Eq. (2.48)
is given by a single sum:
∞
f(κ
n
) f
∗
(κ
n
).
f
2
=
(2.50)
n
=−∞
We
define
φ(κ
n
)
,the
power spectral density
of
f
, as the contribution to the variance
f
2
per unit interval of wavenumber,
f(κ
n
) f
∗
(κ
n
)
κ
2
π
L
,
φ(κ
n
)
=
,κ
=
(2.51)
so that
∞
f
2
=
φ(κ
n
)κ.
(2.52)
n
=−∞
Thus,
φ(κ)
is the density of contributions to the variance. In the limit as
L
and
N
approach infinity
Eq. (2.52)
becomes an integral:
∞
f
2
=
φ(κ)dκ.
(2.53)
−∞
2.5.2 Extension to three dimensions
Next we generalize
Eq. (2.47)
to represent a real, homogeneous, three-dimensional,
random, conserved scalar field
c(x
1
,x
2
,x
3
;
α)
=
c(
x
;
α)
in a cube of side
L
.Now
the wavenumber is a vector
κ
=
(κ
1
,κ
2
,κ
3
)
so we write
α)e
i(
κ
·
x
)
.
c(
x
;
α)
=
c(
κ
;
ˆ
(2.54)
κ
