Geoscience Reference
In-Depth Information
V
will produce a scattered field
2
filling a small radar volume
Δ
N
o
Δ
V
p
=
1
N
o
Δ
V
p
=
1
r
e
r
e
r
E
o
r
p
E
op
e
−
j
2
k
o
r
p
e
j
k
·
r
p
.
E
s
=
−
≈−
(3)
Here
N
o
is the mean density of free electrons within
V
and the rightmost expression amounts
to invoking a plane wave approximation
3
of the incident and scattered fields in terms of
scatterer position vector
r
p
and a Bragg wave vector
k
=
−
Δ
2
k
o
r
pointing from the center
Δ
of subvolume
V
to the location of the radar antenna (assuming a mono-static backscatter
radar geometry).
With electrons in (non-relativistic) motion, scattered field phasor (3) turns into
N
o
Δ
V
p
=
1
r
e
r
E
o
r
c
e
j
k
·
r
p
(
t
−
)
E
s
(
t
)=
−
(4)
including a propagation time delay
r
/
c
of the scattered field from the center of volume
4
Δ
V
.
It then follows that the auto-correlation function (ACF) of the scattered field is
2
N
o
Δ
V
p
=
1
N
o
Δ
V
q
=
1
e
j
k
·
[
r
q
(
t
+
τ
−
c
)
−
r
p
(
t
−
c
)]
,
r
e
r
2
|
E
s
(
t
)
E
s
(
t
+
τ
)
=
E
i
|
(5)
where angular brackets denote an expected value (ensemble average) operation.
Using
e
j
k
·
[
r
q
−
r
p
=
q
]
=
e
j
k
·
r
q
e
−
j
k
·
r
p
=
=
)
0 for statistically independent electrons (
p
q
, this
reduces to
r
e
r
2
|
r
e
r
2
|
r
c
r
c
E
s
(
2
N
o
Δ
V
e
j
k
·
[
r
q
(
t
+
τ−
)
−
r
q
(
t
−
)]
=
2
N
o
Δ
V
e
j
k
·
Δ
r
t
)
E
s
(
t
+
τ
)
=
E
i
|
E
i
|
(6)
r
c
)
−
r
q
(
r
c
)
with
Δ
r
≡
r
q
(
t
+
τ
−
t
−
denoting particle displacements over time intervals
τ
.
Only with (6), i.e., only under a strict incoherent scatter scenario, we can obtain
r
e
r
2
|
2
2
N
o
Δ
|
E
s
(
t
)
|
=
E
i
|
V
,
(7)
a result that implies a total scattered power which is a simple sum over all scatterers
individually described by (1).
Collective effects in general invalidate the results (6) and (7) from being directly applicable.
Nevertheless the desired spectral model for ionospheric incoherent scatter can be expressed
in terms of (6) and (7) after suitable corrections and transformations. To obtain the model let
us first re-express (4) as
r
e
r
E
o
n
e
(
k
,
t
r
c
)
E
s
(
t
)=
−
−
(8)
2
We assume here that
ω
o
is sufficiently large so that dispersion effects due to plasma density
N
o
can be
neglected (or treated as perturbation effects). Also, multiple scattering is neglected.
3
V
2/3
/
V
1/3
.
Justified for
r
>
2
k
o
Δ
π
, the far-field condition for an antenna of size
Δ
4
Δ
V
is sufficiently small for electrons to move only an insignificant fraction of the radar wavelength
during an interval for light to propagate across
Δ
V
.
