Geology Reference
In-Depth Information
and
n
t
(definitions of the complex permittivity are intro-
duced in section 3.6). Since a transmitted signal from
radar sensors is polarized, the two Fresnel equations can
be applied to determine the reflected radar signal with the
same or orthogonal polarization with respect to the
polarization of the incident wave. For example, for the
incident wave with horizontal polarization
h
, one of two
Fresnel equations determines the horizontally polarized
reflection coefficient, denoted
R
hh
, and the other deter-
mines
R
hv
. For an EM wave traveling in air (
i
= 1) and
incident with an angle
θ
on an interface of a medium that
has complex permittivity
t
and the relative permeability
equals 1, the two Fresnel equations can be obtained by
replacing
n
t
in equation (7.4) and (7.5) with
n
from equa-
tion (7.8) and assuming
n
i
=1. The resulting equations are
1. 0
0.8
6.0
Ґ
h
0.4
Brestwer's
angle
Ґ
υ
2
cos
sin
0.2
t
R
hh
(, )
(7.9)
t
2
cos
sin
t
2
sin
cos
0.0
t
t
R
vv
(,)
(7.10)
0.0
20
40
60
80
t
2
sin
cos
t
t
Incidence angle
Figure 7.17
Reflectivity calculated from Fresnel equations for
the radar signal as a function of incidence angle for a smooth
surface of lossless medium (e.g., freshwater ice) with a dielec-
tric constant
= 3.0 − j0.
These are complex quantities since
t
is a complex num-
ber. Their magnitude varies between 0 and 1 (with 1
indicating total reflection). The reflectivity in the power
domain can be obtained from the reflection coefficients
using the following equations:
In using equations (7.11) and (7.12) for sea ice, it should
be noted that its dielectric constant in the microwave
region varies over a relatively wide range depending on the
frequency of the EM wave. The typical value for FY ice is
3.5 −
j
2.0. For seawater it takes much higher values.
Ellison
et al.
[2003] measured the permittivity of seawater in the
spectral microwave range 30-105 GHz over the tempera-
ture. At −2 °C, they found that complex permittivity was
10.66 −
j
19.46 and 7.43 −
j
9.38 for 37 and 89 GHz, respec-
tively. It is obvious that the loss component of seawater is
significantly higher than that of sea ice.
2
RR R
(7.11)
hh
hh
hh
hh
2
RR R
(7.12)
vv
vv
vv
vv
Equations (7.11) and (7.12
)
are plotted in Figure 7.17
for a smooth surface of a lossless medium with permittiv-
ity (3.2 −
j
0), which represents freshwater ice. Although
the plots are generated for the radar signal, the trend
shown is also valid for an optical signal. In general, the
horizontally polarized waves are reflected more readily
than vertically polarized waves. It increases monotoni-
cally with increasing incidence angle and reaches a peak
of 1 at the grazing angle (close to 90°). On the other hand,
reflection of the vertically polarized waves decreases as
the incidence angle increases until it reaches zero at the
Brewster angle. Beyond this angle, the reflection increases
as shown in Figure 7.17 and reaches its peak also at the
grazing angle. As indicated in section 6.1.1 when an unpo-
larized monochromatic light beam strikes a transparent
plate, the reflected beam becomes plane polarized if the
incident angle is equal to the Brewster angle for the given
material and the wavelength.
7.3.2.3. Transmission
The part of the EM wave that is not reflected at the
surface penetrates the medium. Depending on the
manner of the energy interaction with the atoms and
electrons, the medium can be fully transparent medium,
fully absorbing (opaque medium), or anything in‐between
these two limits. Signal transmission is a function of the
optical or the electrical properties of the material as well
as the wavelength
λ
of the signal. The transmittance of a
signal
T
λ
is defined as the ratio of the intensity of the
transmitted to the incident radiation.
TII
/
0
(7.13)
