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, Λ_{s} = 0.5, and g_{s} = 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, Λ_{s} = 0.8, and g_{s} = 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, Λ_{s} = 0.7, and g_{s} = 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, Λ_{s} = 0.7, and g_{s} = 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, Λ_{s} = 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 η_{0} = 0.64

0.48

6.5E-2

2.2E-2

7.2E-3

1.5E-3

3.3E-4

0.00

Urban η_{0} = 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, Λ_{s} = 0.5, and g_{s} = 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, Λ_{s} = 0.8, and g_{s} = 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, Λ_{s} = 0.7, and g_{s} = 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, Λ_{s} = 0.7, and g_{s} = 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, Λ_{s} = 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