A many-flux (discrete ordinate) radiative transfer calculation procedure is described with the goal of making the mathematics easy to learn and use. The major approximation is the neglect of polarization. Emission within the scattering medium is not included, and the formulas are restricted to a scattering medium bounded by parallel planes. The boundary conditions allow for a variety of kinds of illumination, and the surface reflection coefficients at the boundaries of the scattering medium are accurately determined. A comparison is made with the two-flux (Kubelka-Munk) and four-flux calculation methods, and this leads to empirical expressions for the scattering and absorption coefficients in these simple theories, which make them give nearly the same results as exact theories. These empirical expressions provide a very simple method for estimating the absolute reflectance and transmittance of turbid media and greatly increase the utility of the two-flux and four-flux calculation methods. The two-flux equations give excellent results provided the absorption is small compared to scattering and the optical thickness is greater than 5. A comparison with experimental data taken with collimated illumination shows that the four-flux equations give good results at any optical thickness even if the absorption is strong.
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Channel Divisions for Many-Flux Calculations with Twenty-Two Channelsa
Channel number
Angle of boundary with next channel
Diffuse flux Di
1
1°
0.00030
2
6°
0.01060
3
19°
0.09507
4
30°
0.14401
5
37°
0.11219
6
40°
0.05098
7
52°
0.20779
8
63°
0.17293
9
73°
0.12063
10
82°
0.06612
11
90°
0.01938
The diffuse flux enters each channel when the illumination is diffuse and there is no Fresnel reflection or refraction at the boundary.
Table II
Comparison of Reflectance Calculated Using the Many-Flux Equations with Those Reported by Orchard23 for Three Different Phase Functionsa
Collimated illumination
Diffuse illumination
Optical thickness
Exact
22-flux error
Exact
22-flux error
p(cosψ = 1.0 + 0.5P2(cosψ)
1
0.3404
−0.0000
0.4468
0.0014
2
0.5162
−0.0000
0.6101
0.0012
3
0.6211
0.0000
0.6985
0.0010
4
0.6897
0.0000
0.7541
0.0009
5
0.7377
0.0000
0.7924
0.0007
6
0.7729
0.0000
0.8204
0.0006
7
0.7998
0.0001
0.8417
0.0006
8
0.8211
0.0000
0.8585
0.0005
9
0.8382
0.0001
0.8721
0.0005
10
0.8524
0.0000
0.8833
0.0004
p(cosψ) = 1.0 + 1.0P1(cosψ) + 0.5P2(cosψ)
1
0.2346
0.0001
0.3580
0.0018
2
0.3990
0.0003
0.5157
0.0018
3
0.5104
0.0004
0.6104
0.0016
4
0.5886
0.0004
0.6739
0.0015
5
0.6458
0.0004
0.7197
0.0013
6
0.6892
0.0004
0.7541
0.0013
7
0.7231
0.0005
0.7810
0.0012
8
0.7504
0.0005
0.8026
0.0011
9
0.7729
0.0004
0.8204
0.0009
10
0.7916
0.0004
0.8352
0.0009
p(cosψ) = 1.0 + 1.5P1(cosψ) + 0.5P2(cosψ)
1
0.1678
0.0004
0.3020
0.0022
2
0.3163
0.0007
0.4490
0.0023
3
0.4266
0.0009
0.5437
0.0023
4
0.5085
0.0009
0.6104
0.0022
5
0.5705
0.0010
0.6601
0.0020
6
0.6189
0.0010
0.6985
0.0019
7
0.6575
0.0010
0.7292
0.0017
8
0.6891
0.0009
0.7541
0.0016
9
0.7153
0.0009
0.7749
0.0015
10
0.7375
0.0009
0.7924
0.0014
The error is the twenty-two-flux result minus the exact result.
Table III
Legendre Polynomial Coefficients for the Phase Functions.a
A
B
C
D
E
Geometric mean diamn. (Å)
1980
1980
1400
1980
Log-normal sigma
0.34
0.34
0.34
0.34
Vacuum wavelength (nm)
650
546
650
436
rayleigh scattering
a0
1.00000
1.00000
1.00000
1.00000
1.00000
a1
1.69440
1.62604
1.54660
1.32218
0.0
a2
1.53846
1.64414
1.11392
1.64140
0.50000
a3
1.01640
1.14862
0.52362
1.07084
—
a4
0.59130
0.78500
0.21394
1.06636
a5
0.25766
0.37350
0.04520
0.48348
a6
0.14220
0.26848
0.01366
0.55750
a7
0.03404
0.07154
0.00200
0.14176
a8
0.02386
0.07398
0.00024
0.21884
a9
0.00148
0.00770
—
0.02982
a10
0.00052
—
0.07372
a11
0.00006
0.00848
a12
—
0.01752
a13
−0.00032
a14
0.00570
a15
−0.00004
The particle size distributions and wavelengths used in calculating functions A through D from the Mie equations are indicated.
Table IV
Channel Divisions for Many-Flux Calculations with Twenty-six Channelsa
Channel number
Angle of boundary with next channel
Diffuse flux Di
Average Fresnel reflection coefficient ri
1
1°
0.00069
0.04520
2
6°
0.02404
0.04520
3
19°
0.21515
0.04574
4
30°
0.32348
0.05285
5
37°
0.24079
0.09495
6
40°
0.09027
0.25361
7
40.4°
0.00718
0.55989
8
40.5°
0.00077
0.79643
9
52°
0.0
1.0
10
63°
0.0
1.0
11
73°
0.0
1.0
12
82°
0.0
1.0
13
90°
0.0
1.0
The reflection coefficient and the flux entering each channel under conditions of diffuse illumination were calculated for a ratio of indices of refraction at the boundary of 1.54.
Table V
Values of S and K Required to Make the Two-Flux Calculation Give the Same Results As the Many-Flux Calculation
Phase Function
A
B
C
D
E
k/s
S/s
K/k
S/s
K/k
S/s
K/k
S/s
K/k
S/s
K/k
10−6
0.3270
1.990
0.3440
1.991
0.3643
1.992
0.4200
1.993
0.7514
1.996
10−5
0.3269
1.994
0.3439
1.994
0.3642
1.995
0.4199
1.996
0.7513
1.998
10−4
0.7503
0.3264
2.002
0.3434
2.003
0.3636
2.003
0.4192
2.003
2.003
10−3
0.3234
2.021
0.3403
2.020
0.3605
2.020
0.4158
2.018
0.7458
2.014
10−2
0.3120
2.053
0.3289
2.051
0.3486
2.051
0.4033
2.044
—
—
Table VI
Comparison of Four-Flux Calculations with the Data of Atkinsa
Sample
Calculated transmittance
Error
Calculated reflectance
Error
Surface reflection coefficients
r23
r32
16702.092
0.4172
−0.0014
0.4479
0.0017
0.6120
0.5701
16703.090
0.3685
−0.0029
0.3960
−0.0011
0.6133
0.5661
17407.096
0.1741
−0.0119
0.4215
0.0017
0.6090
0.5676
16704.093
0.2814
0.0133
0.3250
−0.0067
0.6151
0.5586
16802.096
0.6453
0.0195
0.2440
0.0005
0.6879
0.5886
17406.091
0.2875
−0.0202
0.3248
0.0094
0.6152
0.5585
16705.091
0.1937
−0.0110
0.2416
−0.0072
0.6189
0.5443
16803.091
0.6099
0.0224
0.2067
−0.0011
0.6895
0.5762
17405.093
0.4002
−0.0051
0.2410
0.0116
0.6320
0.5468
16706.093
0.0991
−0.0112
0.1659
−0.0054
0.6249
0.5193
16804.096
0.5216
0.0034
0.1664
0.0013
0.6763
0.5458
17404.093
0.5313
−0.0149
0.1655
0.0201
0.6795
0.5478
17506.096
0.0924
−0.0222
0.1656
0.0029
0.6250
0.5194
16707.095
0.0331
−0.0110
0.1115
0.0034
0.6324
0.4787
16805.097
0.4182
0.0037
0.1202
0.0000
0.6632
0.4975
17403.092
0.6426
−0.0022
0.1125
0.0132
0.7469
0.5690
17505.093
0.1899
−0.0203
0.1162
0.0056
0.6315
0.4837
17402.094
0.7110
−0.0062
0.0855
0.0097
0.8047
0.5957
16806.094
0.2980
−0.0148
0.0837
0.0001
0.6499
0.4552
17504.094
0.3002
−0.0186
0.0835
0.0037
0.6501
0.4440
16807.090
0.1688
−0.0142
0.0611
0.0010
0.6402
0.4000
17503.092
0.4005
−0.0229
0.0659
0.0035
0.6827
0.4436
17502.092
0.4639
−0.0169
0.0578
0.0036
0.7115
0.4394
Standard error
0.0147
0.0071
The error is the calculated minus the experimental value.
Tables (6)
Table I
Channel Divisions for Many-Flux Calculations with Twenty-Two Channelsa
Channel number
Angle of boundary with next channel
Diffuse flux Di
1
1°
0.00030
2
6°
0.01060
3
19°
0.09507
4
30°
0.14401
5
37°
0.11219
6
40°
0.05098
7
52°
0.20779
8
63°
0.17293
9
73°
0.12063
10
82°
0.06612
11
90°
0.01938
The diffuse flux enters each channel when the illumination is diffuse and there is no Fresnel reflection or refraction at the boundary.
Table II
Comparison of Reflectance Calculated Using the Many-Flux Equations with Those Reported by Orchard23 for Three Different Phase Functionsa
Collimated illumination
Diffuse illumination
Optical thickness
Exact
22-flux error
Exact
22-flux error
p(cosψ = 1.0 + 0.5P2(cosψ)
1
0.3404
−0.0000
0.4468
0.0014
2
0.5162
−0.0000
0.6101
0.0012
3
0.6211
0.0000
0.6985
0.0010
4
0.6897
0.0000
0.7541
0.0009
5
0.7377
0.0000
0.7924
0.0007
6
0.7729
0.0000
0.8204
0.0006
7
0.7998
0.0001
0.8417
0.0006
8
0.8211
0.0000
0.8585
0.0005
9
0.8382
0.0001
0.8721
0.0005
10
0.8524
0.0000
0.8833
0.0004
p(cosψ) = 1.0 + 1.0P1(cosψ) + 0.5P2(cosψ)
1
0.2346
0.0001
0.3580
0.0018
2
0.3990
0.0003
0.5157
0.0018
3
0.5104
0.0004
0.6104
0.0016
4
0.5886
0.0004
0.6739
0.0015
5
0.6458
0.0004
0.7197
0.0013
6
0.6892
0.0004
0.7541
0.0013
7
0.7231
0.0005
0.7810
0.0012
8
0.7504
0.0005
0.8026
0.0011
9
0.7729
0.0004
0.8204
0.0009
10
0.7916
0.0004
0.8352
0.0009
p(cosψ) = 1.0 + 1.5P1(cosψ) + 0.5P2(cosψ)
1
0.1678
0.0004
0.3020
0.0022
2
0.3163
0.0007
0.4490
0.0023
3
0.4266
0.0009
0.5437
0.0023
4
0.5085
0.0009
0.6104
0.0022
5
0.5705
0.0010
0.6601
0.0020
6
0.6189
0.0010
0.6985
0.0019
7
0.6575
0.0010
0.7292
0.0017
8
0.6891
0.0009
0.7541
0.0016
9
0.7153
0.0009
0.7749
0.0015
10
0.7375
0.0009
0.7924
0.0014
The error is the twenty-two-flux result minus the exact result.
Table III
Legendre Polynomial Coefficients for the Phase Functions.a
A
B
C
D
E
Geometric mean diamn. (Å)
1980
1980
1400
1980
Log-normal sigma
0.34
0.34
0.34
0.34
Vacuum wavelength (nm)
650
546
650
436
rayleigh scattering
a0
1.00000
1.00000
1.00000
1.00000
1.00000
a1
1.69440
1.62604
1.54660
1.32218
0.0
a2
1.53846
1.64414
1.11392
1.64140
0.50000
a3
1.01640
1.14862
0.52362
1.07084
—
a4
0.59130
0.78500
0.21394
1.06636
a5
0.25766
0.37350
0.04520
0.48348
a6
0.14220
0.26848
0.01366
0.55750
a7
0.03404
0.07154
0.00200
0.14176
a8
0.02386
0.07398
0.00024
0.21884
a9
0.00148
0.00770
—
0.02982
a10
0.00052
—
0.07372
a11
0.00006
0.00848
a12
—
0.01752
a13
−0.00032
a14
0.00570
a15
−0.00004
The particle size distributions and wavelengths used in calculating functions A through D from the Mie equations are indicated.
Table IV
Channel Divisions for Many-Flux Calculations with Twenty-six Channelsa
Channel number
Angle of boundary with next channel
Diffuse flux Di
Average Fresnel reflection coefficient ri
1
1°
0.00069
0.04520
2
6°
0.02404
0.04520
3
19°
0.21515
0.04574
4
30°
0.32348
0.05285
5
37°
0.24079
0.09495
6
40°
0.09027
0.25361
7
40.4°
0.00718
0.55989
8
40.5°
0.00077
0.79643
9
52°
0.0
1.0
10
63°
0.0
1.0
11
73°
0.0
1.0
12
82°
0.0
1.0
13
90°
0.0
1.0
The reflection coefficient and the flux entering each channel under conditions of diffuse illumination were calculated for a ratio of indices of refraction at the boundary of 1.54.
Table V
Values of S and K Required to Make the Two-Flux Calculation Give the Same Results As the Many-Flux Calculation
Phase Function
A
B
C
D
E
k/s
S/s
K/k
S/s
K/k
S/s
K/k
S/s
K/k
S/s
K/k
10−6
0.3270
1.990
0.3440
1.991
0.3643
1.992
0.4200
1.993
0.7514
1.996
10−5
0.3269
1.994
0.3439
1.994
0.3642
1.995
0.4199
1.996
0.7513
1.998
10−4
0.7503
0.3264
2.002
0.3434
2.003
0.3636
2.003
0.4192
2.003
2.003
10−3
0.3234
2.021
0.3403
2.020
0.3605
2.020
0.4158
2.018
0.7458
2.014
10−2
0.3120
2.053
0.3289
2.051
0.3486
2.051
0.4033
2.044
—
—
Table VI
Comparison of Four-Flux Calculations with the Data of Atkinsa
Sample
Calculated transmittance
Error
Calculated reflectance
Error
Surface reflection coefficients
r23
r32
16702.092
0.4172
−0.0014
0.4479
0.0017
0.6120
0.5701
16703.090
0.3685
−0.0029
0.3960
−0.0011
0.6133
0.5661
17407.096
0.1741
−0.0119
0.4215
0.0017
0.6090
0.5676
16704.093
0.2814
0.0133
0.3250
−0.0067
0.6151
0.5586
16802.096
0.6453
0.0195
0.2440
0.0005
0.6879
0.5886
17406.091
0.2875
−0.0202
0.3248
0.0094
0.6152
0.5585
16705.091
0.1937
−0.0110
0.2416
−0.0072
0.6189
0.5443
16803.091
0.6099
0.0224
0.2067
−0.0011
0.6895
0.5762
17405.093
0.4002
−0.0051
0.2410
0.0116
0.6320
0.5468
16706.093
0.0991
−0.0112
0.1659
−0.0054
0.6249
0.5193
16804.096
0.5216
0.0034
0.1664
0.0013
0.6763
0.5458
17404.093
0.5313
−0.0149
0.1655
0.0201
0.6795
0.5478
17506.096
0.0924
−0.0222
0.1656
0.0029
0.6250
0.5194
16707.095
0.0331
−0.0110
0.1115
0.0034
0.6324
0.4787
16805.097
0.4182
0.0037
0.1202
0.0000
0.6632
0.4975
17403.092
0.6426
−0.0022
0.1125
0.0132
0.7469
0.5690
17505.093
0.1899
−0.0203
0.1162
0.0056
0.6315
0.4837
17402.094
0.7110
−0.0062
0.0855
0.0097
0.8047
0.5957
16806.094
0.2980
−0.0148
0.0837
0.0001
0.6499
0.4552
17504.094
0.3002
−0.0186
0.0835
0.0037
0.6501
0.4440
16807.090
0.1688
−0.0142
0.0611
0.0010
0.6402
0.4000
17503.092
0.4005
−0.0229
0.0659
0.0035
0.6827
0.4436
17502.092
0.4639
−0.0169
0.0578
0.0036
0.7115
0.4394
Standard error
0.0147
0.0071
The error is the calculated minus the experimental value.
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