Geoscience Reference
In-Depth Information
Fig. 1.5.
Interaction between radiation and elementary volume of the scattering medium
=
as
F
F
0
−
dF
after its penetrating the elementary volume (along the inci-
dent direction
r
0
). Take the relative change of incident energy as an extinction
characteristic:
dE
e
E
0
λ
(
F
0
−
F
)
dSd
dt
dF
F
0
=
=
.
λ
F
0
dSd
dt
As it is manifestly proportional to the length
dl
in the extenuating medium,
then it is possible to take the
volume extinction coefficient
α
as a characteristic
of radiation, attenuated by the elementary volume. This coefficient is equal to
a relative change of incident energy (measured in intervals [
λ
λ
λ
], [
t
,
t
+
dt
])
normalized to the length
dl
(i. e. reduced to the unit length) according to the
definition:
,
+
d
dE
e
dF
F
0
dl
α
=
E
0
dl
=
.
(1.19)
σ
κ
The analogous definitions of
thevolumescattering
co-
efficients
follow from the equality of extinction energy and the sum of the
scattering and absorption energies.
4
and absorption
dE
s
E
0
dl
dE
a
E
0
dl
σ
=
κ
=
α
=
σ
κ
,
,
+
.
(1.20)
Itwouldbepossibletointroduce
a volume coefficient of the directional scat-
tering
s
(
r
) considering energy
dE
d
(
r
) scattered along direction
r
in solid angle
d
Ω
=
|
Ω
dl
). However, it is not done
to use this characteristic. Actually, after accounting (1.20) we are obtaining
dE
d
(
r
)
analogously to (1.20):
s
(
r
)
dE
d
(
r
)
(
E
0
d
=
4
π
=
1
σ
Ω
Ω
s
(
r
)
dE
s
d
and substituting it to the relation
dE
s
dE
d
d
that
4
leads to the expression
1
Ω
=
1. It exactly corresponds to the normalizing
relation (1.17) for the phase function in the spherical coordinates (Figs. 1.4 and
1.5) after the setting
s
(
sd
σ
π
γ
ϕ
=
1
4
π
σ
γ
ϕ
γ
ϕ
)isthephasefunction
of the elementary volume. As has been mentioned above, we are considering
,
)
x
(
,
), where
x
(
,
4
Notice, that the introduced volume coefficients have the dimension of the inverse length (m
−1
,
km
−1
) and such values are usually called “linear” not “volume”. Further, we will substantiate this
terminological contradiction.
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