Geoscience Reference
In-Depth Information
like:
3(1 −
g
)
s
1+
s
2
1. 5
g
−
+
O
(
s
3
),
1. 2
1+
g
=
k
8
s
1+
6−7.5
g
+
s
2
+
O
(
s
4
),
3. 6
1+
g
=
m
=
1−6
q
s
+18
q
2
s
2
+
O
(
s
3
),
l
(2.29)
1−4
s
+12
q
s
2
−
36
q
−6
g
−
1. 608
1+
g
s
3
+
O
(
s
4
),
a
∞
=
1−3
q
s
+
9
q
2
−3(1−
g
)−
s
2
+
O
(
s
3
).
2
1+
g
=
n
For the functions in (2.24)-(2.28) the followings expansions are correct ac-
cording to topics by Sobolev (1972), Minin (1988), and Yanovitskij (1997):
µ
=
µ
)(1 − 3
q
s
)+
K
2
(
µ
)
s
2
+
O
(
s
3
),
K
(
)
K
0
(
µ
=
µ
µ
)
s
2
+
a
3
(
µ
)
s
3
+
O
(
s
4
) ,
a
(
)
1−4
K
0
(
)
s
+
a
2
(
(2.30)
ρ
∞
(
µ
µ
0
)
=
ρ
0
(
µ
µ
0
)−4
K
0
(
µ
µ
0
)
s
+
ρ
2
(
µ
µ
0
)
s
2
+
ρ
3
(
µ
µ
0
)
s
3
+
O
(
s
4
),
,
,
)
K
0
(
,
,
where the nomination is introduced:
1
ζ
=
q
=
ζ
ζ
2
d
2
K
0
(
)
0. 714 .
0
ρ
0
(
µ
µ
0
)and
K
0
(
µ
ρ
∞
(
µ
µ
0
)
In these expansions functions
,
)arefunctions
,
µ
ω
0
=
and
K
(
)fortheconservativescattering(
1) correspondingly, functions
) are the coefficients by the item
s
2
. They are presented either in
analytical or in table form (Sobolev 1972; Hulst 1980; Minin 1988; Yanovitskij
1997). Asymptotic expansions (2.29) and (2.30) have been mathematically
rigorously derived, their errors are defined by items
µ
µ
a
2
(
)and
K
2
(
s
3
or
s
4
omitting in
∼
∼
the series.
The coefficients by items
s
2
and
s
3
in the expansion for reflection function
ρ
∞
(
µ
µ
0
) have been derived in the study by Melnikova (1992) and look like:
,
µ
µ
0
)
µ
µ
0
)
a
2
(
)
a
2
(
a
3
(
)
a
3
(
ρ
2
(
µ
µ
0
)
=
ρ
3
(
µ
µ
0
)
=
,
,
,
,
(2.31)
a
2
a
3
µ
µ
)arethecoefficientsby
s
2
and
s
3
in the series for
spherical
a
∞
albedo as per (2.29) and in series for plane
a
(
where
a
2
,
a
3
,
a
2
(
)and
a
3
(
µ
)albedoasper
(2.30) correspondingly.
According to the topic by Minin (1988), where it has been shown that it is
possible to neglect the dependence of escape function
K
0
(
µ
)uponthephase
function for the conservative scattering and values 0. 65
≤
g
≤
0. 9, we present
the following table:
Search WWH ::
Custom Search