Geoscience Reference
In-Depth Information
Equation (1.53) is the integral equation for the source function. Usually just this
equation is analyzed in the radiative transfer theory but not (1.47). The desired
radiance is linked with the solution of (1.53) with the simple expressions. It is
possible otherwise to substitute definition (1.50) to expressions (1.52) and to
obtain the integral equations for the radiance used in the numerical methods
of the radiative transfer theory.
It is possible to write the integral equation for the source function (1.53)
through the operator form (Hulst 1980; Lenoble 1985; Marchuk et al. 1980)
=
B
KB
+
q
,
(1.54)
=
τ
µ
µ
0
,
ϕ
where
B
)isthesourcefunction,
q
istheabsoluteterm,
K
is
the integral operator.
The operator kernel
and
theabsoluteterm
are expressed
according to (1.53) as:
B
(
,
,
)exp
−
τ
=
ω
0
(
τ
τ
)
−
=
K
(
τ
µ
µ
0
,
ϕ
τ
,
µ
,
ϕ
)
τ
χ
K
,
,
,
x
(
,
πη
µ
4
≤
τ
≤
τ
≤
µ
≤
1,
for
0
0
)exp
−
τ
−
ω
τ
τ
0
(
)
−
=
τµ
µ
0
,
ϕ
τ
,
µ
,
ϕ
)
=
τ
χ
K
K
(
,
,
x
(
,
πη
µ
4
(1.55)
τ
≤
τ
≤
τ
0
−1
≤
µ
≤
0,
for
=
K
0outofthepoined an ,
=
ω
0
(
τ
)
=
q
(
τ
µ
µ
0
,
ϕ
τ
χ
0
) exp(−
τ|µ
0
).
q
,
,
)
Sx
(
,
4
Remember that according to Kolmogorov and Fomin (1989) the operator
recording is:
b
K
(
x
,
x
)
y
(
x
)
dx
.
Ky
≡
a
Equation (1.54) is the Fredholmequation of the second kind. Themathematical
theory of these equations is perfectly developed, e. g. Kolmogorov and Fomin
(1989). The formal solution of the Fredholm equation of the second kind is
presented with the Neumann series:
q
+
Kq
+
K
2
q
+
K
3
q
+...
=
B
(1.56)
Expression (1.56) concerning the transfer theory is an expansionof the solution
(the source function) over powers of the scattering order. Actually, the item
q
isayieldofthefirstorderscatteringtothesourcefunction,theitem
Kq
is the
second order,
K
2
q
=
K
(
Kq
) is the third order etc. As kernel
K
is proportional
tothesinglescatteringalbedo,thevelocityoftheseriesconvergenceislinked
with this parameter: the higher
ω
0
(the scattering is greater) the higher order
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