Irina N. Melnikova,
Zhanna M. Dlugach,
Teruyuki Nakajima,
and Kazuaki Kawamoto
When this research was performed, I. N. Melnikova, T. Nakajima, and K. Kawamoto were with the Center for Climate Research, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan.
I. N. Melnikova (irina.melnikova@pobox.spbu.ru) has returned to the Laboratory for Global Climate Change, Research Center for Ecological Safety, Russian Academy of Sciences, Korpusnaya St. 18, St. Petersburg 197119, Russia.
Z. M. Dlugach is with the Main Astronomical Observatory, Ukrainian National Academy of Sciences, Golosiiv 252650, Kyiv-22, Ukraine.
Irina N. Melnikova, Zhanna M. Dlugach, Teruyuki Nakajima, and Kazuaki Kawamoto, "Calculation of the reflection function of an optically thick scattering layer for a Henyey–Greenstein phase function," Appl. Opt. 39, 4195-4204 (2000)
Simple analytical methods are proposed for calculating the
reflection function of a semi-infinite and conservative scattered
layer, the value of which is needed to solve many atmospheric optics
problems. The methods are based on approximations of the exact
values obtained with a strict numerical method. For a
Henyey–Greenstein phase function, knowledge of the zeroth and sixth
higher harmonics appears to be sufficient for a quite accurate
approximation of the angle range, which is acceptable for solution of
direct and inverse problems in atmospheric optics when a plane
atmosphere is assumed. An error estimation and a comparison with
the exact solution are presented.
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LA’s for Coefficients am, bm, and cm of the
Zeroth, First, and Second Harmonics of the Analytical Presentation of
the Reflection Function
m
am
bm
cm
μlimit
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
Table 2
Exponential Approximations of Coefficients am, bm, and cm of the Third through Sixth Harmonics of the Reflection Function (PF Method)
m
0.3 ≤ g ≤ 0.9
am
bm
cm
μlimit
3
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
0.50
4
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
0.45
5
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
0.35
6
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
0.35
Table 3
Deviation of the Approximation of Zeroth Harmonic
ρ0(0.67, 0.67) from 1
Variable
Value of g
0.3
0.5
0.75
0.8
0.85
0.9
Deviation of |1 - ρ0(0.67, 0.67)|
0.0037
0.024
0.021
0.0059
0.013
0.0046
Table 4
Contributions of the First through Fourth Harmonics
Relative to the Zeroth Harmonic to the First and Second Scattering
Orders for Phase Function Parameter g = 0.5
Variable
Parameter δ
Parameter Δ (i = 1)
Parameter Δ (i = 2)
Case I
Case II
Case III
Case I
Case II
Case III
Case I
Case II
Case III
μ; μ0
0; 0.1
0; 0.9
0.9; 0.9
0; 0.1
0; 0.9
0.9; 0.9
0; 0.1
0; 0.9
0.9; 0.9
m = 1
0.600
0.089
0.0078
0.048
0.008
0.0055
0.009
0.007
0.003
m = 2
0.345
0.018
2.9 × 10-4
0.014
0.001
1.5 × 10-4
0.001
3.2 × 10-4
5.7 × 10-5
m = 3
0.192
0.0035
1.2 × 10-5
0.004
1.1 × 10-4
5.1 × 10-6
2.3 × 10-4
1.6 × 10-5
1.8 × 10-6
m = 4
0.105
6.9 × 10-4
6.0 × 10-6
0.001
1.3 × 10-5
1.6 × 10-6
0
0
2.4 × 10-7
Table 5
Contributions of the First through the Seventh Harmonics
Relative to the Zeroth Harmonic to the First and Second Scattering
Orders for Phase Function Parameter g = 0.85
Variable
Parameter δ
Parameter Δ (i = 1)
Parameter Δ (i = 2)
Case I
Case II
Case III
Case I
Case II
Case III
Case I
Case II
Case III
μ; μ0
0; 0.1
0; 0.9
0.9; 0.9
0; 0.1
0; 0.9
0.9; 0.9
0; 0.1
0; 0.9
0.9; 0.9
m = 1
0.905
0.086
0.016
0.158
0.050
0.016
0.062
0.040
0.015
m = 2
0.805
0.017
5.8 × 10-4
0.121
0.007
5.9 × 10-4
0.041
0.005
5.5 × 10-4
m = 3
0.705
0.0038
2.5 × 10-5
0.088
0.001
2.3 × 10-5
0.026
8.8 × 10-4
2.1 × 10-5
m = 4
0.611
9.1 × 10-4
1.1 × 10-6
0.064
4.0 × 10-4
9.2 × 10-6
0.016
1.2 × 10-4
7.9 × 10-6
m = 5
0.528
2.3 × 10-4
5.1 × 10-7
0.047
3.8 × 10-5
4.5 × 10-7
0.010
1.8 × 10-5
3.5 × 10-7
m = 6
0.454
5.1 × 10-5
2.8 × 108
0.034
6.9 × 10-6
2.6 × 10-8
0.006
3.1 × 10-6
2.1 × 10-8
m = 7
0.389
1.1 × 10-5
1.0 × 10-9
0.024
1.0 × 10-6
7.8 × 10-9
0.003
4.3 × 10-7
5.9 × 10-9
Table 6
Results of Approximate and Numerical Method Calculations
of the Reflection Function Zeroth
Harmonica
The CI method calculates the isotropic
zeroth harmonic and adds the item that is linearly dependent on phase
function parameter g for correction; the LA method
calculates the zeroth harmonic with the linear approximation on the
parameter g according to Table 1.
Table 7
Results of Approximations and Exact Methods of the
Reflection Function Calculationsa
The LS method takes into account all the
scattering orders for the zeroth harmonic and only the first scattering
order for high harmonics; the PF method fits the power regression on
parameter g for harmonics with numbers higher than 2.
Tables (7)
Table 1
LA’s for Coefficients am, bm, and cm of the
Zeroth, First, and Second Harmonics of the Analytical Presentation of
the Reflection Function
m
am
bm
cm
μlimit
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
Table 2
Exponential Approximations of Coefficients am, bm, and cm of the Third through Sixth Harmonics of the Reflection Function (PF Method)
m
0.3 ≤ g ≤ 0.9
am
bm
cm
μlimit
3
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
0.50
4
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
0.45
5
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
0.35
6
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
0.35
Table 3
Deviation of the Approximation of Zeroth Harmonic
ρ0(0.67, 0.67) from 1
Variable
Value of g
0.3
0.5
0.75
0.8
0.85
0.9
Deviation of |1 - ρ0(0.67, 0.67)|
0.0037
0.024
0.021
0.0059
0.013
0.0046
Table 4
Contributions of the First through Fourth Harmonics
Relative to the Zeroth Harmonic to the First and Second Scattering
Orders for Phase Function Parameter g = 0.5
Variable
Parameter δ
Parameter Δ (i = 1)
Parameter Δ (i = 2)
Case I
Case II
Case III
Case I
Case II
Case III
Case I
Case II
Case III
μ; μ0
0; 0.1
0; 0.9
0.9; 0.9
0; 0.1
0; 0.9
0.9; 0.9
0; 0.1
0; 0.9
0.9; 0.9
m = 1
0.600
0.089
0.0078
0.048
0.008
0.0055
0.009
0.007
0.003
m = 2
0.345
0.018
2.9 × 10-4
0.014
0.001
1.5 × 10-4
0.001
3.2 × 10-4
5.7 × 10-5
m = 3
0.192
0.0035
1.2 × 10-5
0.004
1.1 × 10-4
5.1 × 10-6
2.3 × 10-4
1.6 × 10-5
1.8 × 10-6
m = 4
0.105
6.9 × 10-4
6.0 × 10-6
0.001
1.3 × 10-5
1.6 × 10-6
0
0
2.4 × 10-7
Table 5
Contributions of the First through the Seventh Harmonics
Relative to the Zeroth Harmonic to the First and Second Scattering
Orders for Phase Function Parameter g = 0.85
Variable
Parameter δ
Parameter Δ (i = 1)
Parameter Δ (i = 2)
Case I
Case II
Case III
Case I
Case II
Case III
Case I
Case II
Case III
μ; μ0
0; 0.1
0; 0.9
0.9; 0.9
0; 0.1
0; 0.9
0.9; 0.9
0; 0.1
0; 0.9
0.9; 0.9
m = 1
0.905
0.086
0.016
0.158
0.050
0.016
0.062
0.040
0.015
m = 2
0.805
0.017
5.8 × 10-4
0.121
0.007
5.9 × 10-4
0.041
0.005
5.5 × 10-4
m = 3
0.705
0.0038
2.5 × 10-5
0.088
0.001
2.3 × 10-5
0.026
8.8 × 10-4
2.1 × 10-5
m = 4
0.611
9.1 × 10-4
1.1 × 10-6
0.064
4.0 × 10-4
9.2 × 10-6
0.016
1.2 × 10-4
7.9 × 10-6
m = 5
0.528
2.3 × 10-4
5.1 × 10-7
0.047
3.8 × 10-5
4.5 × 10-7
0.010
1.8 × 10-5
3.5 × 10-7
m = 6
0.454
5.1 × 10-5
2.8 × 108
0.034
6.9 × 10-6
2.6 × 10-8
0.006
3.1 × 10-6
2.1 × 10-8
m = 7
0.389
1.1 × 10-5
1.0 × 10-9
0.024
1.0 × 10-6
7.8 × 10-9
0.003
4.3 × 10-7
5.9 × 10-9
Table 6
Results of Approximate and Numerical Method Calculations
of the Reflection Function Zeroth
Harmonica
The CI method calculates the isotropic
zeroth harmonic and adds the item that is linearly dependent on phase
function parameter g for correction; the LA method
calculates the zeroth harmonic with the linear approximation on the
parameter g according to Table 1.
Table 7
Results of Approximations and Exact Methods of the
Reflection Function Calculationsa
The LS method takes into account all the
scattering orders for the zeroth harmonic and only the first scattering
order for high harmonics; the PF method fits the power regression on
parameter g for harmonics with numbers higher than 2.