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
=
|
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
s ( r ) dE s d
and substituting it to the relation dE s
dE d d
that
4
leads to the expression 1
=
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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