Geoscience Reference
In-Depth Information
1 m
2
I
λ
(0)
I
λ
(
s
)
s
s
= 0
Figure 4.13 Attenuation of a beam of radiation passing through
a medium with scattering particles (gray) and absorbing
molecules (black).
energy and reradiate at a different wavelength, and when molecules and par-
ticles scatter energy from the path of the beam. Spectral radiance can increase
due to emission from molecules and particles along the path, and when scatter-
ing directs energy into the path.
Consider the one-dimensional problem shown in
Figure 4.13,
in which
the spectral radiance,
I
l
(
s
), is changed by absorption, emission, and scattering
along a path length
s
. First, consider the attenuation of the beam due to the
absorption of energy by molecules acting as blackbodies. The
absorption coef-
ficient,
,
k
, is defined as the fraction of incident radiant energy absorbed per unit
path length, or
dI
I
dI
1
λ
λ
λ
k
/
− −
.
(4.23)
I
ds
ds
λ
Rearranging Eq. 4.23, we have the attenuation of the beam due to absorption:
J
dI
N
λ
K
O
=−
kI
.
(4.24)
λ
ds
L
P
ABS
Now, consider increases in the spectral radiance of the beam due to thermal
emission, which is governed by the Planck function,
B
l
(Eq. 4.3). According to
Kirchhoff's law (section 4.1), the emissivity is equal to the absorptivity, so
J
dI
N
λ
O
=+
kB
.
(4.25)
K
λ
ds
L
P
EMIS
The total change in spectral radiance for a beam traveling through a non-
scattering medium is then
dI
J
dI
N
J
dI
N
λ
λ
λ
K
O
K
O
=
+ =−
kB I
(
).
(4.26)
λ
λ
ds
ds
ds
L
P
L
P
ABS
EMIS
Equation 4.26 is known as
Schwarzschild's equation
. It states that a beam of
light will be attenuated in passing through a medium (with no scattering pres-
ent) if
I
l
>
B
l
and it will be enhanced if
I
l
<
B
l
.
