Viatcheslav B. Kisselev, Laura Roberti, and Giovanni Perona, "Finite-element algorithm for radiative transfer in vertically inhomogeneous media: numerical scheme and applications," Appl. Opt. 34, 8460-8471 (1995)
The recently developed finite-element method for solution of the radiative transfer equation has been extended to compute the full azimuthal dependence of the radiance in a vertically inhomogeneous plane-parallel medium. The physical processes that are included in the algorithm are multiple scattering and bottom boundary bidirectional reflectivity. The incident radiation is a parallel flux on the top boundary that is characteristic for illumination of the atmosphere by the Sun in the UV, visible, and near-infrared regions of the electromagnetic spectrum. The theoretical basis is presented together with a number of applications to realistic atmospheres. The method is shown to be accurate even with a low number of grid points for most of the considered situations. The fortran code for this algorithm is developed and is available for applications.
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Mean Deviations of the FEM and DOM Intensity Values for Different Numbers of Grid Points for Rural Aerosol with 23-km Visibility, λ= 0.55 μm, Λs = 0.5, and gs = 0.6a
Level Layer
Angle
Scheme
Grid Points
N = 2
N = 4
N = 8
N = 16
N = 32
z = 0 km
η > 0
FEM
0.457
0.224
0.093
0.031
0.001
DOM
37.421
18.071
10.642
4.410
0.617
η < 0
FEM
0.057
0.018
0.000
0.000
0.000
DOM
5.46
0.967
0.019
0.000
0.000
z = 2 km
η > 0
FEM
0.090
0.055
0.010
0.000
0.000
DOM
17.789
4.395
0.733
0.138
0.020
η < 0
FEM
0.229
0.091
0.000
0.000
0.000
DOM
3.847
0.713
0.077
0.034
0.022
z = 100 km
η < 0
FEM
0.182
0.039
0.007
0.001
0.000
DOM
0.673
0.570
0.042
0.008
0.009
The numbers in all the tables correspond to the real values multiplied by 100. In all the tables the deviations are computed with respect to the FEM solution with N = 64, which is considered as converged. The columns of the tables in which all the deviations are lower than 10−4 are removed.
Table 2
Mean and Maximum Deviations of the FEM Intensity Values for Different Numbers of Grid Points for Advective Fog with 0.2-km Visibility, λ= 0.55 μm, Λs = 0.8, and gs = 0
Grid Points
Layer Level
Angle
Deviation
N = 2
N = 4
N = 8
N = 16
z = 0 km
η > 0
Mean
0.059
0.013
0.002
0.000
Max
0.112
0.295
0.003
0.000
η < 0
Mean and max
0.014
0.007
0.001
0.000
z = 2 km
η > 0
Mean
0.064
0.056
0.018
0.001
Max
0.134
0.246
0.056
0.005
η < 0
Mean
1.751
0.622
0.014
0.002
Max
4.926
3.967
0.053
0.015
z = 100 km
η < 0
Mean
0.845
0.137
0.019
0.003
Max
1.637
0.474
0.045
0.012
Table 3
Mean and Maximum Deviations of the FEM Intensity Values for Different Numbers of Grid Points for Stratocumulus Clouds with λ = 0.55 μm, Λs = 0.7, and gs = 0.7
Grid Points
Layer Level
Angle
Deviation
N = 2
N = 4
N = 8
z = 0 km
η > 0
Mean
0.039
0.005
0.000
Max
0.067
0.007
0.001
η < 0
Mean and max
0.019
0.003
0.000
z = 2 km
η > 0
Mean
0.256
0.019
0.002
Max
0.803
0.073
0.005
η < 0
Mean
0.036
0.009
0.002
Max
0.140
0.016
0.002
z = 100 km
η < 0
Mean
0.356
0.037
0.009
Max
0.748
0.071
0.029
Table 4
Mean and Maximum Deviations of the FEM Intensity Values for Different Numbers of Grid Points for Rural Aerosol and Extreme Volcanic Profile with λ = 0.4 μm, Λs = 0.7, and gs = 0.7
Layer Level
Angle
Deviation
Grid Points
N = 2
N = 4
N = 8
N = 16
N = 32
z = 0 km
η > 0
Mean
15.708
10.079
2.842
0.825
0.041
Max
59.180
31.749
16.927
4.935
0.386
η < 0
Mean
0.183
0.012
0.000
0.000
0.000
Max
0.233
0.036
0.001
0.000
0.000
z = 10 km
η > 0
Mean
39.698
27.002
7.066
1.239
0.049
Max
90.909
63.881
24.177
7.915
0.384
η < 0
Mean
0.997
0.158
0.004
0.000
0.000
Max
1.823
0.538
0.014
0.001
0.000
z = 100 km
η < 0
Mean
2.287
0.644
0.012
0.002
0.000
Max
1.191
4.660
0.003
0.015
0.001
Table 5
Mean and Maximum Deviations of the FEM Intensity Values for Different Numbers of Grid Points for Rural Aerosol and Cirrus Clouds with λ = 0.4 μm, Λs = 0.8, and gs = 0.8
Layer Level
Angle
Deviation
Grid Points
N = 2
N = 4
N = 8
z = 0 km
η > 0
Mean
1.252
0.089
0.011
Max
2.139
0.171
0.037
η < 0
Mean
0.101
0.003
0.000
Max
0.144
0.015
0.000
z = 10 km
η > 0
Mean
0.663
0.145
0.026
Max
1.647
0.271
0.067
η < 0
Mean
1.114
0.044
0.004
Max
2.061
0.176
0.018
z = 100 km
η < 0
Mean
0.751
0.026
0.003
Max
1.440
0.142
0.014
Table 6
CPU Times on a VAX 7000-610 for Computation of One Harmonic for Different Values of N with Two Interpolation Angles and Levels
N
2
4
8
16
32
64
Seconds
0.14
0.30
1
4
15
85
Table 7
Maximum Value of Each Harmonic for Some Atmospheric Models that Were Measured over All the Interpolation Angles and Levels as a Function of Harmonic Number
Harmonic number (m)
0
4
10
25
50
75
100
Rural
0.34
5.1E-2
1.7E-2
5.3E-3
8.0E-4
1.8E-4
0.00
Urban η0 = 0.64
0.48
6.5E-2
2.2E-2
7.2E-3
1.5E-3
3.3E-4
0.00
Urban η0 = 0.9
1.11
0.42
0.35
0.22
5.2E-2
1.7E-2
4.0E-3
Stratocumulus
0.51
9.6E-3
0.00
0.00
0.00
0.00
0.00
Volcanic
0.61
0.29
0.17
5.5E-2
8.1E-3
1.8E-3
2.E-4
Tables (7)
Table 1
Mean Deviations of the FEM and DOM Intensity Values for Different Numbers of Grid Points for Rural Aerosol with 23-km Visibility, λ= 0.55 μm, Λs = 0.5, and gs = 0.6a
Level Layer
Angle
Scheme
Grid Points
N = 2
N = 4
N = 8
N = 16
N = 32
z = 0 km
η > 0
FEM
0.457
0.224
0.093
0.031
0.001
DOM
37.421
18.071
10.642
4.410
0.617
η < 0
FEM
0.057
0.018
0.000
0.000
0.000
DOM
5.46
0.967
0.019
0.000
0.000
z = 2 km
η > 0
FEM
0.090
0.055
0.010
0.000
0.000
DOM
17.789
4.395
0.733
0.138
0.020
η < 0
FEM
0.229
0.091
0.000
0.000
0.000
DOM
3.847
0.713
0.077
0.034
0.022
z = 100 km
η < 0
FEM
0.182
0.039
0.007
0.001
0.000
DOM
0.673
0.570
0.042
0.008
0.009
The numbers in all the tables correspond to the real values multiplied by 100. In all the tables the deviations are computed with respect to the FEM solution with N = 64, which is considered as converged. The columns of the tables in which all the deviations are lower than 10−4 are removed.
Table 2
Mean and Maximum Deviations of the FEM Intensity Values for Different Numbers of Grid Points for Advective Fog with 0.2-km Visibility, λ= 0.55 μm, Λs = 0.8, and gs = 0
Grid Points
Layer Level
Angle
Deviation
N = 2
N = 4
N = 8
N = 16
z = 0 km
η > 0
Mean
0.059
0.013
0.002
0.000
Max
0.112
0.295
0.003
0.000
η < 0
Mean and max
0.014
0.007
0.001
0.000
z = 2 km
η > 0
Mean
0.064
0.056
0.018
0.001
Max
0.134
0.246
0.056
0.005
η < 0
Mean
1.751
0.622
0.014
0.002
Max
4.926
3.967
0.053
0.015
z = 100 km
η < 0
Mean
0.845
0.137
0.019
0.003
Max
1.637
0.474
0.045
0.012
Table 3
Mean and Maximum Deviations of the FEM Intensity Values for Different Numbers of Grid Points for Stratocumulus Clouds with λ = 0.55 μm, Λs = 0.7, and gs = 0.7
Grid Points
Layer Level
Angle
Deviation
N = 2
N = 4
N = 8
z = 0 km
η > 0
Mean
0.039
0.005
0.000
Max
0.067
0.007
0.001
η < 0
Mean and max
0.019
0.003
0.000
z = 2 km
η > 0
Mean
0.256
0.019
0.002
Max
0.803
0.073
0.005
η < 0
Mean
0.036
0.009
0.002
Max
0.140
0.016
0.002
z = 100 km
η < 0
Mean
0.356
0.037
0.009
Max
0.748
0.071
0.029
Table 4
Mean and Maximum Deviations of the FEM Intensity Values for Different Numbers of Grid Points for Rural Aerosol and Extreme Volcanic Profile with λ = 0.4 μm, Λs = 0.7, and gs = 0.7
Layer Level
Angle
Deviation
Grid Points
N = 2
N = 4
N = 8
N = 16
N = 32
z = 0 km
η > 0
Mean
15.708
10.079
2.842
0.825
0.041
Max
59.180
31.749
16.927
4.935
0.386
η < 0
Mean
0.183
0.012
0.000
0.000
0.000
Max
0.233
0.036
0.001
0.000
0.000
z = 10 km
η > 0
Mean
39.698
27.002
7.066
1.239
0.049
Max
90.909
63.881
24.177
7.915
0.384
η < 0
Mean
0.997
0.158
0.004
0.000
0.000
Max
1.823
0.538
0.014
0.001
0.000
z = 100 km
η < 0
Mean
2.287
0.644
0.012
0.002
0.000
Max
1.191
4.660
0.003
0.015
0.001
Table 5
Mean and Maximum Deviations of the FEM Intensity Values for Different Numbers of Grid Points for Rural Aerosol and Cirrus Clouds with λ = 0.4 μm, Λs = 0.8, and gs = 0.8
Layer Level
Angle
Deviation
Grid Points
N = 2
N = 4
N = 8
z = 0 km
η > 0
Mean
1.252
0.089
0.011
Max
2.139
0.171
0.037
η < 0
Mean
0.101
0.003
0.000
Max
0.144
0.015
0.000
z = 10 km
η > 0
Mean
0.663
0.145
0.026
Max
1.647
0.271
0.067
η < 0
Mean
1.114
0.044
0.004
Max
2.061
0.176
0.018
z = 100 km
η < 0
Mean
0.751
0.026
0.003
Max
1.440
0.142
0.014
Table 6
CPU Times on a VAX 7000-610 for Computation of One Harmonic for Different Values of N with Two Interpolation Angles and Levels
N
2
4
8
16
32
64
Seconds
0.14
0.30
1
4
15
85
Table 7
Maximum Value of Each Harmonic for Some Atmospheric Models that Were Measured over All the Interpolation Angles and Levels as a Function of Harmonic Number