Michael A. Box and Claudia Sendra, "Retrieval of the albedo and phase function from exiting radiances with radiative perturbation theory," Appl. Opt. 38, 1636-1643 (1999)
We use radiative perturbation theory to develop a retrieval
technique for determining the radiative properties of a scattering
medium, such as the Earth’s atmosphere, based on measurements of the
radiation emerging at either the top or bottom of the medium. In a
previous paper [J. Quant. Spectrosc. Radiat. Transfer 54,
695 (1995)] we have shown the capacity of radiative perturbation
theory to describe variations in exiting intensity as a linear
combination of the parameters that characterize the scattering
medium. Here we show that it is possible to set up a matrix
relation such that the matrix inversion solves the inverse scattering
problem. Using simulated data, we observe that the quality of the
solution can be controlled by studying the singular values associated
with the kernel matrix, obtaining in this way a stable solution, even
in the presence of noise.
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Percentage Error in the Retrieved Scattering Coefficients
for Different γ Values and L = 7
Order
γ = 0
γ = 0.001
γ = 0.01
γ = 0.1
γ = 1.0
ω0
0.64
1.11
1.40
1.42
1.99
χ1
1.39
2.4
1.16
0.81
0.07
χ2
4.47
3.13
2.26
3.05
4.94
χ3
3.68
3.38
2.16
2.98
4.38
χ4
10.95
3.00
4.55
3.90
4.05
χ5
5.53
2.87
4.06
2.35
1.86
χ6
15.8
2.21
6.02
0.59
0.72
Table 2
Normalized Singular Values of the B Matrix
Calculated for L = 7 and Different γ Values
Order
γ = 0
γ = 0.001
γ = 0.01
γ = 0.1
γ = 1.0
1
1.0000
1.000
1.0000
1.0000
1.0000
2
0.2272
0.2515
0.8783
0.9658
0.7600
3
0.1645
0.1826
0.7276
0.8899
0.5781
4
0.1216
0.1351
0.5657
0.7397
0.4622
5
0.0647
0.0720
0.4311
0.6303
0.3505
6
0.0319
0.0355
0.2203
0.2053
0.2577
7
0.0094
0.0104
0.0656
0.1925
0.0595
Table 3
Normalized Singular Values of the Kernel Matrix Calculated
for γ = 0.1 and Different L Dimensions
Order
L = 7
L = 14
L = 19
L = 20
L = 22
1
1.0000
1.0000
1.0000
1.0000
1.0000
2
0.9658
0.9360
0.9294
0.9290
0.9307
3
0.8899
0.9297
0.9240
0.9220
0.9187
4
0.7297
0.7396
0.7399
0.7394
0.7448
5
0.6303
0.6284
0.6434
0.6430
0.6636
6
0.2053
0.6137
0.6086
0.6078
0.6068
7
0.1925
0.5277
0.5358
0.53645
0.5373
8
0.4143
0.4859
0.4864
0.4921
9
0.3098
0.3440
0.3746
0.3877
10
0.2887
0.3211
0.3217
0.3533
11
0.2471
0.3100
0.3198
0.3300
12
0.2003
0.2701
0.2733
0.2742
13
0.0283
0.2340
0.2479
0.2528
14
0.0130
0.1992
0.1989
0.1987
15
0.1578
0.1605
0.1787
16
0.0383
0.1122
0.1435
17
0.0175
0.0176
0.0964
18
0.0024
0.0098
0.0187
19
0.0007
0.0017
0.0131
20
0.0004
0.0028
21
0.0011
22
0.0000
Table 4
Percentage Error in the Expansion Coefficients and
Scattering Albedo Retrieved from γ = 0.1 and Different L
Dimensions of the Kernel Matrixa
Order
L = 7
L = 14
L = 19
L = 20
L = 22
ω0
1.42
1.44
1.44
1.45
1.45
1
0.81
0.78
0.76
0.76
0.75
2
3.05
2.98
2.94
2.94
2.92
3
2.98
2.87
2.82
2.82
2.79
4
3.90
3.70
3.65
3.65
3.60
5
2.35
2.08
2.02
2.01
1.97
6
0.59
0.21
0.15
0.14
0.09
7
1.81
1.86
1.88
1.95
8
4.69
4.74
4.76
4.87
9
6.43
6.49
6.51
6.62
10
8.76
8.83
8.87
8.95
11
8.50
8.59
8.61
8.69
12
9.37
9.46
9.49
9.59
13
7.97
8.09
8.13
8.25
14
7.67
7.72
7.85
15
6.10
6.14
6.28
16
3.41
3.44
3.58
17
0.77
0.71
0.55
18
6.87
6.70
6.42
19
12.01
11.55
20
19.81
21
25.05
Measured intensities at the bottom of the
atmosphere for τtotal = 1.0.
Table 5
Percentage Error in the Expansion Coefficients and
Scattering Albedo Retrieved from Intensities at the Bottom of the
Atmosphere for Different Optical Thicknesses and γ = 0.1
Order
τtotal = 1.0
τtotal = 0.5
τtotal = 0.2
ω0
1.35
1.11
1.44
1
0.36
0.40
0.76
2
3.00
2.58
2.94
3
2.99
2.66
2.82
4
3.29
3.30
3.65
5
1.49
1.72
2.02
6
0.20
0.37
0.15
7
1.93
1.60
1.86
8
4.20
4.11
4.74
9
5.53
5.61
6.49
10
8.03
8.09
8.83
11
6.99
7.49
8.59
12
8.77
8.76
9.46
13
9.23
7.78
8.09
14
7.97
7.67
15
6.07
6.10
16
1.72
3.41
17
3.59
0.77
18
6.87
Table 6
Retrieved Coefficients from Intensities at the bottom of
the Atmosphere with 0% and 5% of Added
Noisea
Order
True Value
1st Guess
0% Noise
5% Noise
ω0
0.900
1.000
0.913
0.913
1
2.412
2.250
2.394
2.402
2
3.230
2.812
3.135
3.121
3
3.372
2.953
3.277
3.282
4
3.230
2.847
3.112
3.113
5
2.892
2.610
2.834
2.823
6
2.494
2.313
2.490
2.481
7
2.112
2.002
2.151
2.135
8
1.746
1.701
1.829
1.823
9
1.444
1.426
1.538
1.537
10
1.174
1.182
1.278
1.267
11
0.963
0.971
1.046
1.038
12
0.779
0.791
0.853
0.851
13
0.638
0.641
0.689
0.685
14
0.516
0.516
0.556
0.552
15
0.422
0.414
0.447
0.448
16
0.343
0.330
0.355
0.354
17
0.279
0.263
0.277
0.272
18
0.229
0.208
0.213
0.205
τtotal = 1.0.
Table 7
Expansion Coefficients and Scattering Albedo Retrieved
from Intensities at the Top of the Atmosphere for Different Optical
Thicknesses
Order
True Value
1st Guess
τtotal = 0.2
τtotal = 0.5
τtotal = 1.0
ω0
0.900
1.000
0.907
0.907
0.910
1
2.412
2.250
2.395
2.389
2.378
2
3.230
2.812
3.145
3.156
3.177
3
3.372
2.953
3.278
3.283
3.331
4
3.230
2.847
3.156
3.159
3.207
5
2.892
2.610
2.877
2.873
2.610
Table 8
Legendre Expansion Coefficients and Albedo Retrieved after
Successive Iterations from Intensities at the Top of the
Atmospherea
Order
True
1st Guess
1st Inversion
2nd Inversion
3rd Inversion
ω0
0.900
1.000
0.910
0.900
0.899
1
2.412
2.250
2.378
2.412
2.411
2
3.230
2.812
3.177
3.232
3.228
3
3.372
2.953
3.331
3.375
3.369
4
3.230
2.847
3.207
3.237
3.229
5
2.892
2.610
2.904
2.892
6
2.494
2.313
2.251
2.500
7
2.112
2.002
2.132
2.119
8
1.746
1.701
1.774
1.760
9
1.444
1.426
1.468
1.455
τtotal = 1.0.
Table 9
Differences between the Measured and Retrieved Intensities
at the Top of the Atmosphere after Each
Iterationa
(θobs, φobs)
E - Ebc
E - Eretr1
E - Eretr2
E - Eretr3
(87°, 180°)
-0.0194
-0.00675
-0.000731
-0.0000710
(59°, 180°)
-0.0184
-0.0294
-0.00124
-0.0000681
(31°, 180°)
-0.0166
-0.00243
-0.000260
-0.0000127
(4°, 180°)
-0.0258
-0.00144
-0.000303
-0.00000378
(4°, 0°)
-0.0296
-0.00604
-0.000126
-0.0000155
(31°, 0°)
-0.0500
-0.00767
-0.000381
-0.0000643
(59°, 0°)
-0.0930
-0.00338
-0.000169
-0.0000626
(87°, 0°)
-0.898
-0.0560
-0.00163
-0.0000365
τtotal = 1.0.
Tables (9)
Table 1
Percentage Error in the Retrieved Scattering Coefficients
for Different γ Values and L = 7
Order
γ = 0
γ = 0.001
γ = 0.01
γ = 0.1
γ = 1.0
ω0
0.64
1.11
1.40
1.42
1.99
χ1
1.39
2.4
1.16
0.81
0.07
χ2
4.47
3.13
2.26
3.05
4.94
χ3
3.68
3.38
2.16
2.98
4.38
χ4
10.95
3.00
4.55
3.90
4.05
χ5
5.53
2.87
4.06
2.35
1.86
χ6
15.8
2.21
6.02
0.59
0.72
Table 2
Normalized Singular Values of the B Matrix
Calculated for L = 7 and Different γ Values
Order
γ = 0
γ = 0.001
γ = 0.01
γ = 0.1
γ = 1.0
1
1.0000
1.000
1.0000
1.0000
1.0000
2
0.2272
0.2515
0.8783
0.9658
0.7600
3
0.1645
0.1826
0.7276
0.8899
0.5781
4
0.1216
0.1351
0.5657
0.7397
0.4622
5
0.0647
0.0720
0.4311
0.6303
0.3505
6
0.0319
0.0355
0.2203
0.2053
0.2577
7
0.0094
0.0104
0.0656
0.1925
0.0595
Table 3
Normalized Singular Values of the Kernel Matrix Calculated
for γ = 0.1 and Different L Dimensions
Order
L = 7
L = 14
L = 19
L = 20
L = 22
1
1.0000
1.0000
1.0000
1.0000
1.0000
2
0.9658
0.9360
0.9294
0.9290
0.9307
3
0.8899
0.9297
0.9240
0.9220
0.9187
4
0.7297
0.7396
0.7399
0.7394
0.7448
5
0.6303
0.6284
0.6434
0.6430
0.6636
6
0.2053
0.6137
0.6086
0.6078
0.6068
7
0.1925
0.5277
0.5358
0.53645
0.5373
8
0.4143
0.4859
0.4864
0.4921
9
0.3098
0.3440
0.3746
0.3877
10
0.2887
0.3211
0.3217
0.3533
11
0.2471
0.3100
0.3198
0.3300
12
0.2003
0.2701
0.2733
0.2742
13
0.0283
0.2340
0.2479
0.2528
14
0.0130
0.1992
0.1989
0.1987
15
0.1578
0.1605
0.1787
16
0.0383
0.1122
0.1435
17
0.0175
0.0176
0.0964
18
0.0024
0.0098
0.0187
19
0.0007
0.0017
0.0131
20
0.0004
0.0028
21
0.0011
22
0.0000
Table 4
Percentage Error in the Expansion Coefficients and
Scattering Albedo Retrieved from γ = 0.1 and Different L
Dimensions of the Kernel Matrixa
Order
L = 7
L = 14
L = 19
L = 20
L = 22
ω0
1.42
1.44
1.44
1.45
1.45
1
0.81
0.78
0.76
0.76
0.75
2
3.05
2.98
2.94
2.94
2.92
3
2.98
2.87
2.82
2.82
2.79
4
3.90
3.70
3.65
3.65
3.60
5
2.35
2.08
2.02
2.01
1.97
6
0.59
0.21
0.15
0.14
0.09
7
1.81
1.86
1.88
1.95
8
4.69
4.74
4.76
4.87
9
6.43
6.49
6.51
6.62
10
8.76
8.83
8.87
8.95
11
8.50
8.59
8.61
8.69
12
9.37
9.46
9.49
9.59
13
7.97
8.09
8.13
8.25
14
7.67
7.72
7.85
15
6.10
6.14
6.28
16
3.41
3.44
3.58
17
0.77
0.71
0.55
18
6.87
6.70
6.42
19
12.01
11.55
20
19.81
21
25.05
Measured intensities at the bottom of the
atmosphere for τtotal = 1.0.
Table 5
Percentage Error in the Expansion Coefficients and
Scattering Albedo Retrieved from Intensities at the Bottom of the
Atmosphere for Different Optical Thicknesses and γ = 0.1
Order
τtotal = 1.0
τtotal = 0.5
τtotal = 0.2
ω0
1.35
1.11
1.44
1
0.36
0.40
0.76
2
3.00
2.58
2.94
3
2.99
2.66
2.82
4
3.29
3.30
3.65
5
1.49
1.72
2.02
6
0.20
0.37
0.15
7
1.93
1.60
1.86
8
4.20
4.11
4.74
9
5.53
5.61
6.49
10
8.03
8.09
8.83
11
6.99
7.49
8.59
12
8.77
8.76
9.46
13
9.23
7.78
8.09
14
7.97
7.67
15
6.07
6.10
16
1.72
3.41
17
3.59
0.77
18
6.87
Table 6
Retrieved Coefficients from Intensities at the bottom of
the Atmosphere with 0% and 5% of Added
Noisea
Order
True Value
1st Guess
0% Noise
5% Noise
ω0
0.900
1.000
0.913
0.913
1
2.412
2.250
2.394
2.402
2
3.230
2.812
3.135
3.121
3
3.372
2.953
3.277
3.282
4
3.230
2.847
3.112
3.113
5
2.892
2.610
2.834
2.823
6
2.494
2.313
2.490
2.481
7
2.112
2.002
2.151
2.135
8
1.746
1.701
1.829
1.823
9
1.444
1.426
1.538
1.537
10
1.174
1.182
1.278
1.267
11
0.963
0.971
1.046
1.038
12
0.779
0.791
0.853
0.851
13
0.638
0.641
0.689
0.685
14
0.516
0.516
0.556
0.552
15
0.422
0.414
0.447
0.448
16
0.343
0.330
0.355
0.354
17
0.279
0.263
0.277
0.272
18
0.229
0.208
0.213
0.205
τtotal = 1.0.
Table 7
Expansion Coefficients and Scattering Albedo Retrieved
from Intensities at the Top of the Atmosphere for Different Optical
Thicknesses
Order
True Value
1st Guess
τtotal = 0.2
τtotal = 0.5
τtotal = 1.0
ω0
0.900
1.000
0.907
0.907
0.910
1
2.412
2.250
2.395
2.389
2.378
2
3.230
2.812
3.145
3.156
3.177
3
3.372
2.953
3.278
3.283
3.331
4
3.230
2.847
3.156
3.159
3.207
5
2.892
2.610
2.877
2.873
2.610
Table 8
Legendre Expansion Coefficients and Albedo Retrieved after
Successive Iterations from Intensities at the Top of the
Atmospherea
Order
True
1st Guess
1st Inversion
2nd Inversion
3rd Inversion
ω0
0.900
1.000
0.910
0.900
0.899
1
2.412
2.250
2.378
2.412
2.411
2
3.230
2.812
3.177
3.232
3.228
3
3.372
2.953
3.331
3.375
3.369
4
3.230
2.847
3.207
3.237
3.229
5
2.892
2.610
2.904
2.892
6
2.494
2.313
2.251
2.500
7
2.112
2.002
2.132
2.119
8
1.746
1.701
1.774
1.760
9
1.444
1.426
1.468
1.455
τtotal = 1.0.
Table 9
Differences between the Measured and Retrieved Intensities
at the Top of the Atmosphere after Each
Iterationa