Geoscience Reference
In-Depth Information
is characteristic of the incoherent scatter process for probing directions perpendicular to
B
and for heights where electron temperature exceeds the ion temperature (i.e.,
T
e
>
T
i
)as
described before. At even higher altitudes, scattered signals become weaker and weaker as
the ionospheric electron density vanishes.
To model incoherent scatter radar measurements using the propagation model presented
in this section, an extra level of complexity has to be considered because, within the
range of aspect angles illuminated by the antenna beams, propagation and scattering effects
vary quite rapidly. For this reason, the measured backscattered radar signals need to be
carefully modeled taking into account the shapes of the antenna beams. A model for the
beam-weighted incoherent scatter spectrum that considers magnetoionic propagation and
collisional effects is formulated next.
9.3 Soft-target radar equation and magnetoionic propagation
In this section, the soft-target radar equation is reformulated using the wave propagation
model described above. Consider a radar system composed of a set of antenna arrays (located
in the same area) with matched filter receivers connected to the antennas used in reception.
The mean square voltage at the output of the
i
-th receiver can be expressed as
E
t
K
i
ˆ
2
r
c
2
2
2
|T
(
r
)
|
|R
i
(
r
)
|
Γ
i
(
r
)
|
χ
(
t
−
,
ω
)
|
d
2
2
|
v
i
(
)
|
=
(
k
,
ω
)
t
dr d
Ω
σ
v
,
(78)
k
2
r
2
π
4
π
where
t
is the radar delay,
E
t
is the total energy of the transmitted radar pulse, and
K
i
is
the
i
-th calibration constant (a proportionality factor that accounts for the gains and losses
along the
i
-th signal path).
Integrals are taken over range
r
, solid angle
Ω
, and Doppler
frequency
2
k
o
r
denotes the relevant Bragg vector for a radar with a
carrier wavenumber
k
o
and associated wavelength
ω
/2
π
. In addition,
k
=
−
λ
. Above,
T
(
r
)
and
R
i
(
r
)
are the antenna
2
2
factors of the arrays used in transmission and reception.
Note that
|T
(
r
)
|
and
|R
i
(
r
)
|
2
2
are antenna gain patterns and the product
|T
(
r
)
|
|R
i
(
r
)
|
is the corresponding two-way
radiation pattern. The polarization coefficient
Γ
i
(
r
)
is defined as
p
i
Π
(
r
)
p
t
2
Γ
i
(
r
)=
,
(79)
where
p
t
and
p
i
are the polarization unit vectors of the transmitting and receiving antennas,
and
¯
Π
(
r
)
is the two-way propagator matrix for the wave field components propagating along
r
(incident on and backscattered from the range
r
). Note that
p
t
and
p
i
are normal to
r
because propagating fields are represented as TEM waves. In addition,
χ
(
t
,
ω
)
is the radar
ambiguity function and
is the volumetric RCS spectrum, functions that have been
defined before. Similarly, the cross-correlation of the voltages at the outputs of the
i
-th and
j
-th receivers can be expressed as
σ
v
(
k
,
ω
)
E
t
K
i
,
j
ˆ
R
i
(
r
)
R
j
(
r
)
k
2
2
|T
(
r
)
|
2
c
2
2
Γ
i
,
j
(
r
)
|
χ
(
t
−
,
ω
)
|
d
v
j
(
v
i
(
)
)
=
(
k
,
ω
)
t
t
dr d
Ω
σ
v
,
(80)
r
2
π
4
π
where
K
i
,
j
is a cross-calibration constant (dependent on gains and losses along the
i
-th and
j
-th signal paths), and
Γ
i
,
j
(
r
)
is a cross-polarization coefficient defined as
p
i
Π
(
r
)
p
t
p
j
Π
(
r
)
p
t
∗
.
Γ
i
,
j
(
r
)=
(81)
