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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