The approximation of coefficients a
m , b
m and c
m in the range of parameter g
0.3 g 0.9 is presented in Tables 11.3 and 11.4.
The well-known relation of the rigorous theory is assumed for the isotropic and
conservative scattering (g ¼ 0, o 0 ¼ 1), namely:
r
0
ðm; m 0 Þ ¼
’ðmÞ’ðm 0 Þ
4ðm þ m 0 Þ
;
(11.14)
where ’(m) is Ambartsumyan’s function. In this case the following approximation
is correct: ’(m) ¼ 1.874 m þ 1.058 and it has been obtained that a
0
¼ 0.88,
b
0
¼ 0.47, and c
0
¼ 0.28. It is known that the reflection function for the isotropic
scattering does not differ very much from the anisotropic values of r
0
(m,m 0 ) if
m, m 0 > 0.25, so it is possible to improve this approach for the enlarged angle
ranges. The simple formula for the isotropic scattering could be corrected approximately with the linear dependence of the phase function parameter as follows:
r
0
ðm; m 0 Þ ¼
’ðmÞ’ðm 0 Þ þ g 4:8m 0 m À 3:0 m 0 þ m
ð
Þþ1:9
½
4 m 0 þ m
ð
Þ
:
(11.15)
In the case of Henyey-Greenstein phase function the high harmonics are close to
zero (r
m (m,m 0 ) % 0, m > 0) if either of zenith angle cosines m and m 0 are greater
than m limit . The values of m limit are different for different harmonics and they are
shown in Tables 2.3 and 2.4. The approximation by Eq. 11.13 with coefficients a
m
,
Table 11.3 Linear approximation for coefficients a
m , b
m и c
m in formula (7.11) for zero, first and
second azimuthal harmonics of the reflection function
m
a
m
b
m
c
m
m limit
0
2.051 g þ 0.508
À1.420 g þ0.831
0.930 g þ 0.023
–
1
1.821 gÀ0.558
À1.413 g þ0.387
1.150 gÀ0.239
0.80
2
2.227 gÀ0.669
À1.564 g þ0.481
1.042 gÀ0.293
0.55
Table 11.4 Power approximation for the coefficients a
m
, b
m и c
m in Eq. (2.14) for 3rd, 4th, 5th
and 6th azimuthal harmonics of the reflection function
m
0.3 g 0.9
a
m
b
m
c
m
limit
3 62.00 g
3 – 90.28 g
2 þ 42.42 g
À6.26
À15.24 g
3 þ 19.70 g
2
–8.73 g þ 1.25
2.75 g
2 –
2.03 g þ 0.39
0.50
4 105.26 g
3 – 155.06 g
2 þ 72.93 g
À10.76
À30.30 g
3 þ 43.04 g
2
–19.83 g þ 2.89
3.70 g
2 –
3.20 g þ 0.65
0.45
5 120.63 g
3 – 177.60 g
2 þ 83.48 g
À12.32
À25.84 g
3 þ 35.15 g
2
–15.61 g þ 2.22
3.23 g
2 –
2.75 g þ 0.55
0.35
6 144.92 g
3 – 202.16 g
2 þ 90.48 g
À12.85
À32.60 g
3 þ 43.88 g
2
–19.15 g + 2.67
3.90 g
2 –
3.41 g þ 0.70
0.35
11.3 The Analytical Presentation of the Reflection Function
111
m , b
m and c
m in the range of parameter g
0.3 g 0.9 is presented in Tables 11.3 and 11.4.
The well-known relation of the rigorous theory is assumed for the isotropic and
conservative scattering (g ¼ 0, o 0 ¼ 1), namely:
r
0
ðm; m 0 Þ ¼
’ðmÞ’ðm 0 Þ
4ðm þ m 0 Þ
;
(11.14)
where ’(m) is Ambartsumyan’s function. In this case the following approximation
is correct: ’(m) ¼ 1.874 m þ 1.058 and it has been obtained that a
0
¼ 0.88,
b
0
¼ 0.47, and c
0
¼ 0.28. It is known that the reflection function for the isotropic
scattering does not differ very much from the anisotropic values of r
0
(m,m 0 ) if
m, m 0 > 0.25, so it is possible to improve this approach for the enlarged angle
ranges. The simple formula for the isotropic scattering could be corrected approximately with the linear dependence of the phase function parameter as follows:
r
0
ðm; m 0 Þ ¼
’ðmÞ’ðm 0 Þ þ g 4:8m 0 m À 3:0 m 0 þ m
ð
Þþ1:9
½
4 m 0 þ m
ð
Þ
:
(11.15)
In the case of Henyey-Greenstein phase function the high harmonics are close to
zero (r
m (m,m 0 ) % 0, m > 0) if either of zenith angle cosines m and m 0 are greater
than m limit . The values of m limit are different for different harmonics and they are
shown in Tables 2.3 and 2.4. The approximation by Eq. 11.13 with coefficients a
m
,
Table 11.3 Linear approximation for coefficients a
m , b
m и c
m in formula (7.11) for zero, first and
second azimuthal harmonics of the reflection function
m
a
m
b
m
c
m
m limit
0
2.051 g þ 0.508
À1.420 g þ0.831
0.930 g þ 0.023
–
1
1.821 gÀ0.558
À1.413 g þ0.387
1.150 gÀ0.239
0.80
2
2.227 gÀ0.669
À1.564 g þ0.481
1.042 gÀ0.293
0.55
Table 11.4 Power approximation for the coefficients a
m
, b
m и c
m in Eq. (2.14) for 3rd, 4th, 5th
and 6th azimuthal harmonics of the reflection function
m
0.3 g 0.9
a
m
b
m
c
m
limit
3 62.00 g
3 – 90.28 g
2 þ 42.42 g
À6.26
À15.24 g
3 þ 19.70 g
2
–8.73 g þ 1.25
2.75 g
2 –
2.03 g þ 0.39
0.50
4 105.26 g
3 – 155.06 g
2 þ 72.93 g
À10.76
À30.30 g
3 þ 43.04 g
2
–19.83 g þ 2.89
3.70 g
2 –
3.20 g þ 0.65
0.45
5 120.63 g
3 – 177.60 g
2 þ 83.48 g
À12.32
À25.84 g
3 þ 35.15 g
2
–15.61 g þ 2.22
3.23 g
2 –
2.75 g þ 0.55
0.35
6 144.92 g
3 – 202.16 g
2 þ 90.48 g
À12.85
À32.60 g
3 þ 43.88 g
2
–19.15 g + 2.67
3.90 g
2 –
3.41 g þ 0.70
0.35
11.3 The Analytical Presentation of the Reflection Function
111