Geoscience Reference
In-Depth Information
ionosphere in slabs of equal width (see Figure 13) perpendicular to the propagation direction
such that within each slab the physical plasma parameters (as electron density, electron
and ion temperatures, and magnetic field) can be considered constants.
9
The transverse
component of the wave electric field propagates from the bottom to the top of the
i
-th slab
according to (66), that is
e
−
jk
o
n
i
Δ
r
e
−
jk
o
Δ
n
i
Δ
r
p
O
p
O
+
e
jk
o
Δ
n
i
Δ
r
p
X
p
X
E
i
=
E
i
−
1
,
(68)
T
i
which is the superposition of the O- and X-modes of magnetoionic propagation detailed in
the previous section. Above,
T
i
denotes the
i
-th propagator matrix (expressed in cartesian
coordinates), where
k
o
≡
2
π
/
λ
o
is the free-space wavenumber,
Δ
r
is the width of the slab,
n
X
,
i
2
are the mean and half difference between the
refractive indices of the propagation modes in the
i
-th layer. The polarization vectors of the
O- and X-modes are
n
O
,
i
+
n
X
,
i
n
O
,
i
−
and where
n
i
≡
n
i
≡
and
Δ
2
ˆ
ˆ
ˆ
ˆ
θ
−
ja
i
φ
=
−
ja
i
θ
+
φ
p
O
=
1
and
p
X
1
(69)
a
i
a
i
+
+
F
O
,
i
Y
L
,
i
Y
L
,
i
F
X
,
i
ˆ
ˆ
where
a
i
φ
i
are a pair of
mutually orthogonal unit vectors perpendicular to
k
whose directions depend on the relative
orientation of the propagation vector
k
and the magnetic field
B
i
(see Figure 12). Neglecting
reflection from the interfaces between slabs, the field components of an upgoing plane wave
propagating in the
≡
=
−
is the polarization parameter,
and
θ
i
and
k
direction (at a distance
r
i
=
r
from the origin) can be computed by
the successive application of the propagator matrices; that is,
+
i
Δ
E
i
=
T
i
···
T
2
T
1
E
o
,
(70)
where
E
o
is the wave field at the origin (perpendicular to
k
), and
T
1
...
T
i
are the propagator
matrices from the bottom layer to the
i
-th layer. Similarly, taking advantage of the
bidirectionality of the propagator matrices, the field components of a downgoing plane wave
propagating in the
k
direction (from the
i
-th layer to the ground) can be written as
−
E
o
=
T
1
T
2
···
T
i
E
i
,
(71)
where
E
i
is the field at the top of the
i
-th layer.
In radar experiments, the transverse field component of the signal backscattered from a radar
range
r
i
=
i
Δ
r
can be modeled as
E
o
∝
κ
i
T
1
T
2
···
T
i
T
i
···
T
2
T
1
E
o
,
(72)
Π
i
where
E
o
and
E
o
are the fields transmitted and received by the radar antenna in the
k
direction.
Above,
Π
i
denotes a two-way propagator matrix that accounts for the polarization effects on
9
In the ionosphere, electron density and plasma temperatures can be considered to be functions of
altitude
f
. Thus, the values of these physical parameters at any position
r
from a radar placed at
the origin are given by
f
(
z
)
(
ψ
)
r
cos
where
r
is the radar range and
ψ
is the zenith angle.
