Geoscience Reference
In-Depth Information
For such a function the theorem of Legendre Polynomials addition (Smirnov
1974; Korn and Korn 2000) is known. According to it the following is correct:
P
i
ϕ
)
µµ
+
(1 −
2
)
(1 −
µ
µ
2
) cos(
ϕ
−
(1.62)
i
(
i
−
m
)!
µ
µ
)+2
(
i
+
m
)!
P
i
(
µ
)
P
i
(
µ
) cos
m
(
ϕ
ϕ
)
+
P
i
(
)
P
i
(
−
=
m
1
where
P
i
(
z
) are associated Legendre Polynomials defined as:
2
d
m
P
i
(
z
)
dz
m
|
P
i
(
z
)
=
(1 −
z
)
m
and
P
i
(
z
)
=
P
i
(
z
).
(Letter
m
specifiesthesuperscriptandnotapowerhereandfurtherinthe
analogous relations). There are known recurrence relations for the practical
calculation of function
P
i
(
z
) (Korn G and Korn T 2000). Applying relation
(1.31) to expansion of the phase function (1.60) it is inferred:
N
χ
=
µ
µ
)
x
(
)
x
i
P
i
(
)
P
i
(
=
i
0
(1.63)
N
i
(
i
−
m
)!
(
i
+
m
)!
P
i
(
µ
)
P
i
(
µ
) cos
m
(
ϕ
ϕ
).
+2
x
i
−
=
=
i
1
m
1
After changing the summation order in the second item of (1.63) and account-
ing that for
m
=1 it is valid
i
=
=
=
1,...,
N
,and
m
2− is
i
2,...,
N
etc., we
finally obtain the following:
N
χ
=
p
0
(
µ
µ
)+2
p
m
(
µ
µ
) cos
m
(
ϕ
ϕ
),
x
(
)
,
,
−
=
i
1
(1.64)
N
x
i
(
i
−
m
)!
p
m
(
µ
µ
)
=
(
i
+
m
)!
P
i
(
µ
)
P
i
(
µ
).
,
=
i
m
Write the relations analogous to (1.64) for the radiance and source function:
N
τ
µ
µ
0
,
ϕ
=
I
0
(
τ
µ
µ
0
)+2
I
m
(
τ
µ
µ
0
) cos
m
ϕ
I
(
,
,
)
,
,
,
,
,
=
i
1
(1.65)
N
τ
µ
µ
0
,
ϕ
=
B
0
(
τ
µ
µ
0
)+2
B
m
(
τ
µ
µ
0
) cos
m
ϕ
B
(
,
,
)
,
,
,
,
,
=
i
1
τ
µ
µ
0
)and
B
m
(
τ
µ
µ
0
)for
m
=
where
I
m
(
0,...,
N
are certain unknown func-
tions. Substitute expansions (1.64) and (1.65) to expression for the source
,
,
,
,
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