Comparative analysis of radiative transfer approaches for calculation of plane transmittance and diffuse attenuation coefficient of plane-parallel light scattering layers
Leonid G. Sokoletsky, Vladimir P. Budak, Fang Shen, and Alexander A. Kokhanovsky
Leonid G. Sokoletsky, Vladimir P. Budak, Fang Shen, and Alexander A. Kokhanovsky, "Comparative analysis of radiative transfer approaches for calculation of plane transmittance and diffuse attenuation coefficient of plane-parallel light scattering layers," Appl. Opt. 53, 459-468 (2014)
We present an analysis of a number of different approximations for the plane transmittance and diffuse attenuation coefficient of a semi-infinite, unbounded, plane-parallel, and optically homogeneous layer. The maximally wide optical conditions (from the full absorption to the full scattering and from the fully forward to the fully backward scattering) were considered. The approximations were analyzed from the point of view of their physical limitations and closeness to the numerical solution of the radiative transfer equation for the plane transmittance. The main criterion for inclusion of the models for analysis was the possibility of practical use, i.e., approximations were well parameterized and included only easily measured or estimated parameters. A detailed analysis of errors for different and models showed that the two-stream radiative transfer Ben-David model yields the best results over all optical conditions and depths. However, the quasi-single-scattering and polynomial Gordon’s approximations proved to be the best for the depths close to zero.
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The errors values (MAPE and NRMSE) derived for the are shown in the upper, middle, and bottom rows for the with , 0.5033, and 0.9583, respectively, while the errors values derived for computed for the ranges of , , and are shown in the left, middle, and right positions, respectively. Note that . Errors more than 100% noted by “-.”
Table 3.
Accuracy of Selected Models for the Diffuse Attenuation Coefficient at and a
Model
MAPE (%)
NRMSE (%)
BLB-a
24; 62; 93
26; 61; 95
14; 44; 88
15; 43; 86
1.3; 6.2; 11
1.5; 5.8; 29
BLB-c
19; 62; -
21; 61; -
39; -; -
39; -; -
63; -; -
58; -; -
QSSA
2.5; 2.8; 21
3.9; 4.9; 24
5.8; 14; 17
7.6; 17; 19
0.8; 2.9; 4.2
1.1; 3.9; 13
Gordon
2.3; 4.7; 28
3.2; 5.8; 27
3.8; 7.1; 6.7
5.9; 11; 9.3
3.0; 13; 16
2.8; 11; 26
Kirk
-; -; -
-; -; -
17; 33; 50
17; 34; 53
2.3; 13; 22.4
2.2; 10; 44
Cornet
3.5; 6.3; 3.5
5.6; 10; 6.3
11; 44; -
12; 44; -
29; -; -
27; -; -
Ben-David
3.5; 6.4; 3.5
5.6; 10; 6.3
4.2; 12; 19
5.4; 13; 22
0.7; 3.7; 6.6
0.8; 3.2; 20
Lee
40; -; -;
43; -; -
10; 46; -;
2.1; 6.9; 17
0.9; 4.5; 8.9
0.2; 0.6; 2.6
The errors values (MAPE and NRMSE) derived for the are shown in the upper, middle, and bottom rows for the with , 0.5033, and 0.9583, respectively, while the error values derived for computed for the ranges of , , and are shown in the left, middle, and right positions, respectively. Errors more than 100% noted by “-.”
Tables (3)
Table 1.
Parameters Used for Generation
0.0019
0.4986
0.006
1.2
0
0.5033
0.1559
0.116
1.25
0.001
0.9583
0.0087
5
1.2
0.01
Table 2.
Accuracy of Selected Models for the Plane Transmittance at and a
The errors values (MAPE and NRMSE) derived for the are shown in the upper, middle, and bottom rows for the with , 0.5033, and 0.9583, respectively, while the errors values derived for computed for the ranges of , , and are shown in the left, middle, and right positions, respectively. Note that . Errors more than 100% noted by “-.”
Table 3.
Accuracy of Selected Models for the Diffuse Attenuation Coefficient at and a
Model
MAPE (%)
NRMSE (%)
BLB-a
24; 62; 93
26; 61; 95
14; 44; 88
15; 43; 86
1.3; 6.2; 11
1.5; 5.8; 29
BLB-c
19; 62; -
21; 61; -
39; -; -
39; -; -
63; -; -
58; -; -
QSSA
2.5; 2.8; 21
3.9; 4.9; 24
5.8; 14; 17
7.6; 17; 19
0.8; 2.9; 4.2
1.1; 3.9; 13
Gordon
2.3; 4.7; 28
3.2; 5.8; 27
3.8; 7.1; 6.7
5.9; 11; 9.3
3.0; 13; 16
2.8; 11; 26
Kirk
-; -; -
-; -; -
17; 33; 50
17; 34; 53
2.3; 13; 22.4
2.2; 10; 44
Cornet
3.5; 6.3; 3.5
5.6; 10; 6.3
11; 44; -
12; 44; -
29; -; -
27; -; -
Ben-David
3.5; 6.4; 3.5
5.6; 10; 6.3
4.2; 12; 19
5.4; 13; 22
0.7; 3.7; 6.6
0.8; 3.2; 20
Lee
40; -; -;
43; -; -
10; 46; -;
2.1; 6.9; 17
0.9; 4.5; 8.9
0.2; 0.6; 2.6
The errors values (MAPE and NRMSE) derived for the are shown in the upper, middle, and bottom rows for the with , 0.5033, and 0.9583, respectively, while the error values derived for computed for the ranges of , , and are shown in the left, middle, and right positions, respectively. Errors more than 100% noted by “-.”