Geoscience Reference
In-Depth Information
in terms of 3D spatial Fourier transform
N
o
Δ
V
p
=
1
e
−
j
k
·
r
p
t
n
e
(
k
,
t
)
≡
(9)
of the electrons
5
of the
microscopic
density function
n
e
(
r
,
t
)=∑
p
δ
(
r
−
r
p
(
t
))
in volume
Δ
V
.
V
can then be expressed
6
The scattered field spectrum for volume
Δ
as
r
e
r
2
|
2
2
2
|
(
ω
)
|
=
E
i
|
|
(
k
,
ω
)
|
Δ
E
s
n
e
V
(11)
in terms of the electron density frequency spectrum
ˆ
∞
1
Δ
r
c
)
r
c
+
τ
)
2
e
−
j
ωτ
n
e
(
k
,
t
|
(
k
,
ω
)
|
≡
V
−
(
k
,
t
−
n
e
d
τ
n
e
(12)
−
∞
which simplifies as
N
o
ˆ
∞
−
∞
2
e
−
j
ωτ
e
j
k
·
Δ
r
2
|
n
e
(
k
,
ω
)
|
=
d
τ
≡|
n
te
(
k
,
ω
)
|
(13)
2
for independent electrons. We also have an identical expression
describing the
density spectrum independent ions in the same volume in terms of ion displacements
|
n
ti
(
k
,
ω
)
|
Δ
r
.
2
2
While neither
are accurate representations of the density
spectra of electrons and ions in a real ionosphere (because of the neglect of collective effects),
it turns out that an accurate model for
|
n
te
(
k
,
ω
)
|
nor
|
n
ti
(
k
,
ω
)
|
2
|
n
e
(
k
,
ω
)
|
can be expressed as a linear combination
2
2
of
|
n
te
(
k
,
ω
)
|
and
|
n
ti
(
k
,
ω
)
|
given by
2
2
2
2
=
|
j
ω
o
+
σ
i
|
|
n
te
(
k
,
ω
)
|
+
|
σ
e
|
|
n
ti
(
k
,
ω
)
|
2
|
n
e
(
k
,
ω
)
|
,
(14)
|
ω
o
+
σ
e
+
σ
i
|
2
|
ω
o
+
σ
e
+
σ
i
|
2
j
j
where
σ
e
,
i
denote the AC conductivities of electrons and ions in the medium. This result
can be derived (e.g., Kudeki & Milla, 2011) by enforcing charge conservation (i.e., continuity
equation) in a plasma carrying quasi-static
macroscopic
currents
σ
e
,
i
E
forced by longitudinal
polarization fields
7
E
produced by the mismatch of thermally driven electron and ion density
fluctuations
n
te
(
k
,
t
)
and
n
ti
(
k
,
t
)
. Furthermore, Nyquist noise theorem (e.g., Callen & Welton,
1951) stipulates that the required conductivities are related to the thermal density spectra via
relations
2
k
2
e
2
ω
2
|
n
te
,
i
(
k
,
ω
)
|
=
2
KT
e
,
i
Re
{
σ
e
,
i
(
k
,
ω
)
}
.
(15)
5
Here
δ
(
·
)
's denote Dirac's deltas utilized to highlight the trajectories
r
p
(
t
)
of individual electrons.
6
This expression can be generalized as a soft-target radar equation
ˆˆ
|
2
/2
E
i
|
η
o
d
2
r
e
|
2
P
r
=
A
r
4
π
n
e
(
k
,
ω
)
|
dV
(10)
r
2
4
π
π
for backscatter ISR's having a scattering volume defined by the beam pattern associated with the
effective area function
A
r
(
r
)
.
7
Note that it is the response of individual particles to the quasi-static
E
that produces the mutual
correlations in their motions.