J. V. Dave and J. Gazdag, "A Modified Fourier Transform Method for Multiple Scattering Calculations in a Plane Parallel Mie Atmosphere," Appl. Opt. 9, 1457-1466 (1970)
A method for evaluating characteristics of the scattered radiation emerging from a plane parallel atmosphere containing large spherical particles is described. In this method, the normalized phase function for scattering is represented as a Fourier series whose maximum required number of terms depends upon the zenith angles of the directions of incident and of scattered radiation. Some results are presented to show that this method can be used to obtain reliable numerical values in a reasonable amount of computer time.
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Intensity of the Radiation Emerging at the Bottom of a Plane Parallel, Nonabsorbing, Rayleigh Atmosphere as Obtained Using Three Different Computational Proceduresa
θ
μ
Method
Chandrasekhar’s method
Modified fourier transform method
Phase matrix
Phase function
0.0°
1.00
0.01571
0.01755
0.01748
20.0
0.94
0.01785
0.01893
0.01886
36.0
0.81
0.02150
0.02127
0.02120
44.0
0.72
0.02360
0.02254
0.02246
48.0
0.67
0.02457
0.02308
0.02300
60.0
0.50
0.02632
0.02360
0.02350
64.0
0.44
0.02612
0.02307
0.02295
70.0
0.34
0.02436
0.02098
0.02094
72.0
0.31
0.02344
0.02002
0.01989
78.0
0.21
0.01922
0.01594
0.01576
82.0
0.14
0.01601
0.01305
0.01295
86.0°
0.07
0.01358
0.01103
0.01096
Note: Computations using Chandrasekhar’s method were carried out for the zenith variables μ and μ0, while those using the modified Fourier method were performed for the zenith variables θ and θ0. τ1 = 1.0, θ0 = 86.0°, μ0 = 0.07, φ0 − φ = 0°.
Table II
Difference Between the Maximum and Minimum Values of Φ(τ) as Observed Using Different Integration Increments in τ and θ′a
Δτ
Δθ in degrees
τ1
θ0 in degrees
Difference between maximum and minimum values of Φ(τ) for
Case A Rayleigh
Case B haze M, λ = 0.75 μ
0.01
2
0.1
0
0.007
0.0017
0.005
2
0.1
0
0.0006
0.0017
0.005
6
0.1
0
0.007
0.014
0.005
10
0.1
0
0.014
0.038
0.01
2
1.0
0
0.001
0.013
0.005
2
1.0
0
0.001
0.013
0.005
6
1.0
0
0.01
0.11
0.005
10
1.0
0
0.02
0.32
0.005
2
0.1
60
0.0008
0.0006
0.005
6
0.1
60
0.009
0.007
0.005
10
0.1
60
0.02
0.012
0.005
2
1.0
60
0.0014
0.0013
0.005
6
1.0
60
0.010
0.006
0.005
10
1.0
60
0.016
0.014
Maximum number iterations performed: 6 for τ1 = 0.1, 15 for τ1 = 1.0.
Table III
Intensity of the Radiation Emerging at the Top of a Plane Parallel Atmosphere as Obtained Using 2° and 10° Angular Increments for Integration over Zenith Angle. Solar Flux π Units per Unit Area Normal to the Direction of the Incident Radiationa
Nadir angle θ
τ1 = 0.1
τ1 = 1.0
Δθ = 2°
Δθ = 10°
Δθ = 2°
Δθ = 10°
0
0.00234
0.00225
0.0348
0.0358
10
0.00267
0.00259
0.0358
0.0360
20
0.00357
0.00349
0.0414
0.0414
30
0.00487
0.00478
0.0510
0.0509
40
0.00647
0.00634
0.0639
0.0638
50
0.00817
0.00797
0.0795
0.0797
60
0.0123
0.0120
0.105
0.105
70
0.0157
0.0149
0.126
0.127
80
0.0279
0.0257
0.145
0.145
Model: haze M, λ = 0.75 μ, m = 1.34 −0.0i, θ0 = 60°, φ0 − φ = 180°.
Table IV
Intensity of the Radiation Emerging at the Bottom of a Plane Parallel Atmosphere as Obtained Using 2° and 10° Angular Increments for Integration over Zenith Angle. Solar Flux π Units per Unit Area Normal to the Direction of the Incident Radiationa
Zenith angle θ
τ1 = 0.1
τ1 = 1.0
Δθ = 2°
Δθ = 10°
Δθ = 2°
Δθ = 10°
0
0.0107
0.0107
0.104
0.110
10
0.0192
0.0191
0.164
0.166
20
0.0383
0.0381
0.278
0.279
30
0.0849
0.0846
0.509
0.510
40
0.214
0.214
1.01
1.01
50
0.690
0.690
2.29
2.29
60
2.47
2.47
5.38
5.44
70
1.24
1.24
2.74
2.73
80
0.850
0.823
1.51
1.47
Model: haze M, λ = 0.75 μ, m = 1.34 −0.0i, θ0 = 60°, φ0 − φ = 0°.
Table V
Number of Iterations Needed for Obtaining a Four Significant Figure Convergence of the Diffuse Downward Flux at the Bottom of a Nonabsorbing, Mie Atmospherea
τ1
θ0 in degrees
Maximum number of iterations
Gauss-Seidel method
Successive scattering method
0.1
0
5
7
0.5
0
6
9
1.0
0
9
13
2.0
0
14
22
1.0
84
13
19
2.0
84
17
26
Model: haze M, λ = 0.75 μ, Δθ = 2°, Δτ = 0.01.
Tables (5)
Table I
Intensity of the Radiation Emerging at the Bottom of a Plane Parallel, Nonabsorbing, Rayleigh Atmosphere as Obtained Using Three Different Computational Proceduresa
θ
μ
Method
Chandrasekhar’s method
Modified fourier transform method
Phase matrix
Phase function
0.0°
1.00
0.01571
0.01755
0.01748
20.0
0.94
0.01785
0.01893
0.01886
36.0
0.81
0.02150
0.02127
0.02120
44.0
0.72
0.02360
0.02254
0.02246
48.0
0.67
0.02457
0.02308
0.02300
60.0
0.50
0.02632
0.02360
0.02350
64.0
0.44
0.02612
0.02307
0.02295
70.0
0.34
0.02436
0.02098
0.02094
72.0
0.31
0.02344
0.02002
0.01989
78.0
0.21
0.01922
0.01594
0.01576
82.0
0.14
0.01601
0.01305
0.01295
86.0°
0.07
0.01358
0.01103
0.01096
Note: Computations using Chandrasekhar’s method were carried out for the zenith variables μ and μ0, while those using the modified Fourier method were performed for the zenith variables θ and θ0. τ1 = 1.0, θ0 = 86.0°, μ0 = 0.07, φ0 − φ = 0°.
Table II
Difference Between the Maximum and Minimum Values of Φ(τ) as Observed Using Different Integration Increments in τ and θ′a
Δτ
Δθ in degrees
τ1
θ0 in degrees
Difference between maximum and minimum values of Φ(τ) for
Case A Rayleigh
Case B haze M, λ = 0.75 μ
0.01
2
0.1
0
0.007
0.0017
0.005
2
0.1
0
0.0006
0.0017
0.005
6
0.1
0
0.007
0.014
0.005
10
0.1
0
0.014
0.038
0.01
2
1.0
0
0.001
0.013
0.005
2
1.0
0
0.001
0.013
0.005
6
1.0
0
0.01
0.11
0.005
10
1.0
0
0.02
0.32
0.005
2
0.1
60
0.0008
0.0006
0.005
6
0.1
60
0.009
0.007
0.005
10
0.1
60
0.02
0.012
0.005
2
1.0
60
0.0014
0.0013
0.005
6
1.0
60
0.010
0.006
0.005
10
1.0
60
0.016
0.014
Maximum number iterations performed: 6 for τ1 = 0.1, 15 for τ1 = 1.0.
Table III
Intensity of the Radiation Emerging at the Top of a Plane Parallel Atmosphere as Obtained Using 2° and 10° Angular Increments for Integration over Zenith Angle. Solar Flux π Units per Unit Area Normal to the Direction of the Incident Radiationa
Nadir angle θ
τ1 = 0.1
τ1 = 1.0
Δθ = 2°
Δθ = 10°
Δθ = 2°
Δθ = 10°
0
0.00234
0.00225
0.0348
0.0358
10
0.00267
0.00259
0.0358
0.0360
20
0.00357
0.00349
0.0414
0.0414
30
0.00487
0.00478
0.0510
0.0509
40
0.00647
0.00634
0.0639
0.0638
50
0.00817
0.00797
0.0795
0.0797
60
0.0123
0.0120
0.105
0.105
70
0.0157
0.0149
0.126
0.127
80
0.0279
0.0257
0.145
0.145
Model: haze M, λ = 0.75 μ, m = 1.34 −0.0i, θ0 = 60°, φ0 − φ = 180°.
Table IV
Intensity of the Radiation Emerging at the Bottom of a Plane Parallel Atmosphere as Obtained Using 2° and 10° Angular Increments for Integration over Zenith Angle. Solar Flux π Units per Unit Area Normal to the Direction of the Incident Radiationa
Zenith angle θ
τ1 = 0.1
τ1 = 1.0
Δθ = 2°
Δθ = 10°
Δθ = 2°
Δθ = 10°
0
0.0107
0.0107
0.104
0.110
10
0.0192
0.0191
0.164
0.166
20
0.0383
0.0381
0.278
0.279
30
0.0849
0.0846
0.509
0.510
40
0.214
0.214
1.01
1.01
50
0.690
0.690
2.29
2.29
60
2.47
2.47
5.38
5.44
70
1.24
1.24
2.74
2.73
80
0.850
0.823
1.51
1.47
Model: haze M, λ = 0.75 μ, m = 1.34 −0.0i, θ0 = 60°, φ0 − φ = 0°.
Table V
Number of Iterations Needed for Obtaining a Four Significant Figure Convergence of the Diffuse Downward Flux at the Bottom of a Nonabsorbing, Mie Atmospherea