Geology Reference
In-Depth Information
equation can then be reduced to a form known as
Rayleigh‐Jean's equation:
10
10
10000 K
10
8
Rf
hkTf
c
B
()
2
2
6000 K (sun)
(7.23)
2
10
6
3000 K
(Tungsten filament)
The interesting feature of this equation is the linear rela-
tionship between the radiation and the physical tempera-
ture of the object. This is not the case in the TIR region
where the emitted energy is proportional to
T
4
as indicated
by equation (7.20). The relations between the emitted energy
and the physical temperature [equations (7.21) and (7.23)]
allow the conversion of energy measured by a radiometer
to temperature. This temperature would be equivalent to
the temperature of a blackbody that radiates the same
amount of the measured energy. It is called brightness
temperature
T
b
. Measurements of emitted radiation
R
B
by
any radiometer are usually provided as
T
b
. Radiation from
a TIR channel can be converted to brightness temperature
by the inversion of Planck's equation [
Gumley et al.,
1994]:
10
4
1000 K
500 K
100
300 K (Earth)
1
100 K
0.01
0.1
1
10
100
Wavelength (
μ
m)
Figure 7.19
Radiation from blackbody (Planck's equation)
against wavelength for different physical temperatures. Shaded
area marks the visible spectrum.
peak at a certain wavelength
λ
max
(in microns), which is
proportional to the physical temperature of the object.
This is known as Wien's displacement law (the hotter the
object the shorter the wavelengths of the maximum emit-
ted intensity).
5
6
(7.24)
Tc c R
b
/
ln( /(
10
)
1
2
1
B
where
c
1
= 1.1910439 × 10
− 16
W/m
2
,
c
2
= 1.4387686 ×
10
− 2
/m · K,
λ
is wavelength in meters, and
R
B
is radiance in
W/m
−2
· sr.
Most objects in nature radiate less energy compared to
a blackbody at the same temperature. Such an object
is known as a “gray” body and its radiation is denoted
(
R
G
). Emissivity is a physical property of material that
describes emitted radiation. It is defined as the ratio of
the emitted radiation from a given body to the radiation
from a blackbody at the same physical temperature (
R
B
):
aT
/
(7.22)
max
where
a
=2.879 × 10 m/K and
T
is in Kelvins.
Figure 7.19 is a plot of the radiation flux using
equation (7.21) for different physical temperatures. The
6000 K is the approximate average temperature of the
Sun. Although the Sun does not fit the definition of a
blackbody because it does not absorb energy, its radia-
tion follows the blackbody radiation and peaks at
approximately the 0.5
μ
m wavelength. The terrestrial
radiation resembles that of a blackbody at 300 K (aver-
age temperature of the Earth). The Earth‐atmosphere
system emits radiation, known as outgoing longwave
radiation, in the wavelength range from 3
μ
m to 100 mm.
It peaks around 10
μ
m, which corresponds to an atmos-
pheric window in the IR region (Figure 7.2). Earth surface
radiation in the microwave range is much weaker than
that in the TIR. Therefore, microwave sensors should
integrate the received radiation over a large IFOV (typically
a few kilometers width) in order to achieve a reasonable
signal‐to‐noise ratio.
In the microwave range of 1-300 GHz, the term
hf
/
kT
in equation (7.21) becomes very small (because of the
small frequency of the microwave signal). Then, in the limit,
the term in the exponential in equation (7.21) becomes
small and the exponential can then be well approximated
with the Taylor polynomials first‐order term. Planck's
R
R
G
(7.25)
B
The emissivity is always less than 1 but it approaches 1
as the object features more resemblance to blackbody
behavior. In terms of temperature ratio, emissivity takes
two different forms; one applies to the TIR and other to
the microwave emission. In the TIR region the emitted
radiation from a blackbody is proportional to the physi-
cal temperature raised to the fourth power [as shown by
equation (7.20)]. In this case, the expression of the emis-
sivity that follows from its definition in equation (7.25):
(7.26)
or
1
4
TT
b
(7.27)
