Geology Reference
In-Depth Information
In the microwave region the expression of emissivity results
by combining equations (7.25) and (7.23). The emissivity
becomes simply the ratio between the brightness temper-
ature and the physical temperature of the medium (more
precisely the physical temperature of the radiating layer
of the medium):
forms: absorption and scattering. The wave extinction
per unit length is known as the extinction coefficient. It
has units of dB/length. According to the Beer‐Lambert
law [
Beer
, 1852], the intensity of an EM wave inside a
given substance decreases exponentially with depth.
Following this concept, if the transmitted power at a
point just below the surface is
P
(0+), then the power at
a depth
z
is given by the following equation [
Ulaby
et al.
, 1982]:
T
T
b
(7.28)
Therefore, while the constant of proportionality between
the brightness and physical temperatures of a gray body is
in the case of the microwave region, it is
1
4
in the case of
the TIR region. This makes the brightness temperature
more sensitive to changes in emissivity in the microwave
region than the TIR region. In fact, in the TIR region the
brightness temperature becomes much more sensitive to
variations in the physical temperature of the observed
object. This is particularly true if the emissivity of the
material does not virtually change in response to any
external factor. Water, ice, and snow represent this situa-
tion because their emissivities in the TIR region are
nearly constant and equal (around 0.96). That is how the
brightness temperature in the TIR observations can be easily
inverted to surface temperature (it is approximately equal
to the physical temperature). Discrimination between sea
ice and open water in cold seasons is usually performed
based on their TIR brightness temperature because ice
surface temperature is significantly lower than the water
temperature, while both have nearly equal emissivity.
On the contrary, in the microwave region the emissivity
of water is much smaller than that of ice and snow
(water is radiometrically colder in the microwave region).
Furthermore, the emissivity of ice or snow varies sub-
stantially in response to changes in atmospheric tempera-
ture or precipitation (especially when temperature changes
around the melting point). Therefore, the measured
microwave brightness temperature cannot be used to esti-
mate the physical temperature unless the emissivity
becomes known or assumed.
z
(7.29)
Pz P
() ( exp
0
k zdz
e
(
)
0
where
k
e
is the extinction coefficient. From equation
(7.29), the definition of penetration depth
δ
p
can be
implied in the following equation:
p
kzdz
e
() 1
(7.30)
0
If
k
e
is assumed to be approximately constant with depth,
then
δ
p
is given by
1
(7.31)
p
k
For large values of
k
e
, the signal penetration is very lim-
ited, hence any contribution to the observed signal in
remote sensing data is triggered by the surface compo-
sition; not the volume. According to the two extinction
processes mentioned above,
k
e
can be written as the
summation of two coefficients, one for the absorption
loss
k
a
and the other for scattering loss
k
s
:
kkk
e
s
(7.32)
a
The ratio
k
s
/
k
e
is known as the scattering albedo, which
varies between 0 and 1. Higher values imply that most of
the extinction is due to scattering. In practice, the scatter-
ing loss in the medium is difficult to compute and is there-
fore usually ignored. In this case, the penetration depth
can be calculated using equation (7.31) with
k
a
substituted
instead of
k
e
.
Penetration depth can be derived in terms of the com-
plex permittivity [equation (3.63)] from the solution of
the simplified equation of the EM propagation in a lossy
homogeneous medium when the wave is assumed to be
traveling along the positive
z
axis [
Ulaby et al.,
1981,
page 65]:
7.3.4. Penetration Depth
Penetration depth is defined as the depth over which
the radiative power is attenuated by a factor of 1/
e
(i.e.,
about 63% of the radiation is absorbed). Remote sensing
observations from any surface is triggered by its composi-
tion within the penetration depth. For optical and TIR
sensors the penetration depth is usually negligible as it
measures in submillimeters. Consequently, the theoretical
background presented in the following applies to the
microwave region only.
As mentioned above, EM wave propagation in matter
undergoes extinction that takes either one or two
EEe
jkz
0
(7.33)
where
k
is the wave number in free space (=2
π
/
λ
), and
λ
is
the wavelength in meters. In the case of a lossless material
