Geoscience Reference
In-Depth Information
Tab l e 2 . 3 .
Linear approximation for coefficients
a
m
,
b
m
,
c
m
in formula (2.37) for zero, first
and second azimuthal harmonics of the reflection function
a
m
b
m
c
m
µ
limit
m
0
2.051
g
+ 0.508
− 1.420
g
+ 0.831
0.930
g
+ 0.023
-
1
1.821
g
− 0.558
− 1.413
g
+ 0.387
1.150
g
− 0.239
0.80
2
2.227
g
− 0.669
− 1.564
g
+ 0.481
1.042
g
− 0.293
0.55
Tab l e 2 . 4 .
Power approximation for the coefficients
a
m
,
b
m
,
c
m
in (2.37) for 3rd, 4th, 5th and
6th azimuthal harmonics of the reflection function
0.3
≤
g
≤
0.9
a
m
b
m
c
m
µ
m
limit
62.00
g
3
− 90.28
g
2
+ 42.42
g
− 6.26
− 15.24
g
3
+ 19.70
g
2
−8.73
g
+1.25 2.75
g
2
−2.03
g
+ 0.39
3
0.50
105.26
g
3
− 155.06
g
2
+ 72.93
g
− 10.76
− 30.30
g
3
+ 43.04
g
2
− 19.83
g
+2.89 3.70
g
2
−3.20
g
+ 0.65
4
0.45
120.63
g
3
− 177.60
g
2
+ 83.48
g
− 12.32
− 25.84
g
3
+ 35.15
g
2
− 15.61
g
+2.22 3.23
g
2
−2.75
g
+ 0.55
5
0.35
144.92
g
3
− 202.16
g
2
+ 90.48
g
− 12.85
− 32.60
g
3
+ 43.88
g
2
− 19.15
g
+2.67 3.90
g
2
−3.41
g
+ 0.70
6
0.35
because it is also necessary to use a complicated model of the spherical atmo-
sphere and to take into account the refraction of solar rays for the small cosines
of zenith solar and viewing angles. These cases are not studied here.
The values of
ρ
m
(
µ
µ
0
)for
m
=
0,...,6havebeenanalyzed in the study
by Melnikova et al. (2000). The following expression, which is similar to the
formula for the zeroth harmonic in the topic by Sobolev (1972), is used for the
description of high harmonics
,
ρ
m
(
µ
µ
0
):
,
ρ
m
(
µ
µ
0
)
=
[
a
m
µµ
0
+
b
m
(
µ
µ
0
)+
c
m
]
|
µ
µ
0
) .
,
+
(
+
(2.37)
This presentation provides the reciprocity of the reflection function relative to
both zenith viewing and zenith solar angles.
The approximation of coefficients
a
m
,
b
m
and
c
m
in the range of parameter
g
0.3
0.9 is presented in Tables 2.3 and 2.4.
The well-known relation of the rigorous theory (Sobolev 1972; Minin 1988;
Yanovitskij 1997) is assumed for the isotropic and conservative scattering
(
g
≤
g
≤
=
ω
0
=
0,
1), namely:
=
ϕ
µ
ϕ
µ
0
)
(
)
(
ρ
0
(
µ
µ
0
)
,
,
(2.38)
µ
µ
0
)
4(
+
ϕ
µ
where
) is Ambartsumyan's function (Sobolev 1972). In this case the follow-
ing approximation is correct:
(
ϕ
µ
=
µ
(
)
1.874
+ 1.058 and it has been obtained
=
=
=
that
a
0
0.88,
b
0
0.47, and
c
0
0.28 (Melnikova 1992). It is known that
the reflection function for the isotropic scattering does not differ very much
from the anisotropic values of
µ
0
>
0.25 (Minin 1988; Melnikova
et al. 2000), so it is possible to improve this approach for the enlarged angle
ranges. The formula for the isotropic scattering (2.38) could be corrected ap-
proximately with the linear dependence upon the asymmetry parameter as
ρ
µ
µ
0
)if
µ
0
(
,
,
Search WWH ::
Custom Search