Precise calculations of the total Rayleigh-scattering optical depth have been performed at 88 wavelengths ranging from 0.20 to 4.00 µm for the six well-known standard atmosphere models by integrating the volume Rayleigh-scattering coefficient along the vertical atmospheric path from sea level to a 120-km height. The coefficient was determined by use of an improved algorithm based on the Ciddor algorithm [
Appl. Opt. 35,
1566 (
1996)], extended by us over the 0.20–0.23-µm wavelength range to evaluate the moist air refractive index as a function of wavelength, air pressure, temperature, water-vapor partial pressure, and CO2 volume concentration. The King depolarization factor was also defined taking into account the moisture conditions of air. The results indicate that the influence of water vapor on Rayleigh scattering cannot be neglected at tropospheric altitudes: for standard atmospheric conditions represented in terms of the U.S. Standard Atmosphere (1976) model, the relative variations produced by water vapor in the Rayleigh scattering parameters at a 0.50-µm wavelength turn out to be equal to −0.10% in the moist air refractivity at sea level (where the water-vapor partial pressure is equal to ≈7.8 hPa), −0.04% in the sea-level King factor, −0.24% in the sea-level Rayleigh-scattering cross section, and −0.06% in the Rayleigh-scattering optical depth.
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Values of the King Factor F(λ) Calculated with Eq. (22) and Depolarization Ratio ρn(λ)a
Wavelength (µm)
King Factor F(λ)
Depolarization Ratio ρn(λ) × 102
0.200
1.07851
4.465
0.205
1.07610
4.335
0.210
1.07393
4.218
0.215
1.07199
4.112
0.220
1.07023
4.016
0.225
1.06864
3.930
0.230
1.06720
3.851
0.240
1.06470
3.714
0.250
1.06260
3.598
0.260
1.06084
3.501
0.270
1.05934
3.419
0.280
1.05806
3.348
0.290
1.05696
3.287
0.300
1.05601
3.234
0.310
1.05517
3.187
0.320
1.05444
3.147
0.330
1.05380
3.111
0.340
1.05323
3.079
0.350
1.05272
3.051
0.360
1.05227
3.026
0.370
1.05186
3.003
0.380
1.05150
2.983
0.390
1.05117
2.964
0.400
1.05087
2.947
0.450
1.04973
2.883
0.500
1.04898
2.841
0.550
1.04845
2.812
0.600
1.04808
2.791
0.650
1.04779
2.775
0.700
1.04758
2.763
0.750
1.04741
2.753
0.800
1.04727
2.745
0.850
1.04716
2.739
0.900
1.04707
2.734
0.950
1.04699
2.730
1.000
1.04693
2.726
1.500
1.04661
2.708
2.000
1.04650
2.702
2.500
1.04645
2.699
3.000
1.04642
2.698
3.500
1.04641
2.697
4.000
1.04640
2.696
The values are determined in terms of Eq. (5) for the sea-level atmospheric conditions given in Table 3 for Model 625 with the same accuracy adopted by Bodhaine5 and Bucholtz4, respectively.
Table 3
Values of Sea-Level Meteorological and Rayleigh-Scattering Parameters at Sea Level in the Six Standard Atmospheresa
Values of the Rayleigh-Scattering Cross Section σ0(λ) and Volume Rayleigh-Scattering Coefficient β0(λ) Calculated at the 88 Wavelengths Selected from 0.20 to 4.00 µm with the Present Rayleigh-Scattering Model for Sea-Level Atmospheric Conditions Defined in Model 6a
Wavelength λ (µm)
σ0(λ) (cm−2)
β0(λ) (km−1)
0.20
3.6064 (×10−25)
9.1813 (×10−1)
0.21
2.8349
7.2172
0.22
2.2686
5.7755
0.23
1.8422
4.6899
0.24
1.5146
3.8559
0.25
1.2586
3.2042
0.26
1.0558
2.6879
0.27
8.9311 (×10−26)
2.2737
0.28
7.6120
1.9379
0.29
6.5320
1.6629
0.30
5.6399
1.4358
0.31
4.8974
1.2468
0.32
4.2748
1.0883
0.33
3.7493
9.5451 (×10−2)
0.34
3.3031
8.4091
0.35
2.9221
7.4391
0.36
2.5950
6.6064
0.37
2.3129
5.8882
0.38
2.0684
5.2658
0.39
1.8557
4.7243
0.40
1.6698
4.2511
0.41
1.5069
3.8362
0.42
1.3634
3.4711
0.43
1.2368
3.1487
0.44
1.1246
2.8631
0.45
1.0249
2.6093
0.46
9.3614 (×10−27)
2.3833
0.47
8.5681
2.1813
0.48
7.8575
2.0004
0.49
7.2194
1.8379
0.50
6.6451
1.6917
0.51
6.1271
1.5599
0.52
5.6588
1.4406
0.53
5.2346
1.3326
0.54
4.8496
1.2346
0.55
4.4995
1.1455
0.56
4.1805
1.0643
0.57
3.8894
9.9017 (×10−3)
0.58
3.6233
9.2243
0.59
3.3797
8.6040
0.60
3.1562
8.0352
0.61
2.9510
7.5127
0.62
2.7622
7.0322
0.63
2.5884
6.5896
0.64
2.4280 (×10−27)
6.1814 (×10−3)
0.65
2.2799
5.8044
0.66
2.1430
5.4557
0.67
2.0162
5.1330
0.68
1.8987
4.8338
0.69
1.7896
4.5561
0.70
1.6883
4.2982
0.71
1.5941
4.0583
0.72
1.5064
3.8349
0.73
1.4246
3.6268
0.74
1.3483
3.4325
0.75
1.2771
3.2512
0.76
1.2105
3.0817
0.77
1.1482
2.9231
0.78
1.0899
2.7746
0.79
1.0352
2.6355
0.80
9.8394 (×10−28)
2.5049
0.82
8.9059
2.2673
0.84
8.0807
2.0572
0.86
7.3490
1.8709
0.88
6.6985
1.7053
0.90
6.1184
1.5576
0.92
5.5999
1.4256
0.94
5.1352
1.3073
0.96
4.7179
1.2011
0.98
4.3421
1.1054
1.00
4.0030
1.0191
1.10
2.7284
6.9461 (×10−4)
1.20
1.9234
4.8967
1.30
1.3947
3.5508
1.40
1.0359
2.6373
1.50
7.8549 (×10−29)
1.9997
1.60
6.0639
1.5438
1.70
4.7556
1.2107
1.80
3.7820
9.6282 (×10−5)
1.90
3.0453
7.7528
2.00
2.4796
6.3127
2.20
1.6927
4.3094
2.40
1.1947
3.0416
2.60
8.6714 (×10−30)
2.2076
2.80
6.4453
1.6409
3.00
4.8900
1.2449
3.50
2.6386
6.7173 (×10−6)
4.00
1.5463
3.9367
Air pressure p0 = 1013.00 hPa, temperature T0 = 288.20 K, water-vapor partial pressure e0 = 7.85075 hPa, and CO2 volume concentration C = 385 ppmv.
Table 5
Values of the Rayleigh-Scattering Optical Depth δ(λ) Obtained for the Six Standard Atmosphere Modelsa
WDCGG Stations, Their Latitude, Longitude, and Height and the Yearly Average and Monthly Mean Ground-Level Values of CO2 Volume Concentration C Measured in 2001
Values of the King Factor F(λ) Calculated with Eq. (22) and Depolarization Ratio ρn(λ)a
Wavelength (µm)
King Factor F(λ)
Depolarization Ratio ρn(λ) × 102
0.200
1.07851
4.465
0.205
1.07610
4.335
0.210
1.07393
4.218
0.215
1.07199
4.112
0.220
1.07023
4.016
0.225
1.06864
3.930
0.230
1.06720
3.851
0.240
1.06470
3.714
0.250
1.06260
3.598
0.260
1.06084
3.501
0.270
1.05934
3.419
0.280
1.05806
3.348
0.290
1.05696
3.287
0.300
1.05601
3.234
0.310
1.05517
3.187
0.320
1.05444
3.147
0.330
1.05380
3.111
0.340
1.05323
3.079
0.350
1.05272
3.051
0.360
1.05227
3.026
0.370
1.05186
3.003
0.380
1.05150
2.983
0.390
1.05117
2.964
0.400
1.05087
2.947
0.450
1.04973
2.883
0.500
1.04898
2.841
0.550
1.04845
2.812
0.600
1.04808
2.791
0.650
1.04779
2.775
0.700
1.04758
2.763
0.750
1.04741
2.753
0.800
1.04727
2.745
0.850
1.04716
2.739
0.900
1.04707
2.734
0.950
1.04699
2.730
1.000
1.04693
2.726
1.500
1.04661
2.708
2.000
1.04650
2.702
2.500
1.04645
2.699
3.000
1.04642
2.698
3.500
1.04641
2.697
4.000
1.04640
2.696
The values are determined in terms of Eq. (5) for the sea-level atmospheric conditions given in Table 3 for Model 625 with the same accuracy adopted by Bodhaine5 and Bucholtz4, respectively.
Table 3
Values of Sea-Level Meteorological and Rayleigh-Scattering Parameters at Sea Level in the Six Standard Atmospheresa
Values of the Rayleigh-Scattering Cross Section σ0(λ) and Volume Rayleigh-Scattering Coefficient β0(λ) Calculated at the 88 Wavelengths Selected from 0.20 to 4.00 µm with the Present Rayleigh-Scattering Model for Sea-Level Atmospheric Conditions Defined in Model 6a
Wavelength λ (µm)
σ0(λ) (cm−2)
β0(λ) (km−1)
0.20
3.6064 (×10−25)
9.1813 (×10−1)
0.21
2.8349
7.2172
0.22
2.2686
5.7755
0.23
1.8422
4.6899
0.24
1.5146
3.8559
0.25
1.2586
3.2042
0.26
1.0558
2.6879
0.27
8.9311 (×10−26)
2.2737
0.28
7.6120
1.9379
0.29
6.5320
1.6629
0.30
5.6399
1.4358
0.31
4.8974
1.2468
0.32
4.2748
1.0883
0.33
3.7493
9.5451 (×10−2)
0.34
3.3031
8.4091
0.35
2.9221
7.4391
0.36
2.5950
6.6064
0.37
2.3129
5.8882
0.38
2.0684
5.2658
0.39
1.8557
4.7243
0.40
1.6698
4.2511
0.41
1.5069
3.8362
0.42
1.3634
3.4711
0.43
1.2368
3.1487
0.44
1.1246
2.8631
0.45
1.0249
2.6093
0.46
9.3614 (×10−27)
2.3833
0.47
8.5681
2.1813
0.48
7.8575
2.0004
0.49
7.2194
1.8379
0.50
6.6451
1.6917
0.51
6.1271
1.5599
0.52
5.6588
1.4406
0.53
5.2346
1.3326
0.54
4.8496
1.2346
0.55
4.4995
1.1455
0.56
4.1805
1.0643
0.57
3.8894
9.9017 (×10−3)
0.58
3.6233
9.2243
0.59
3.3797
8.6040
0.60
3.1562
8.0352
0.61
2.9510
7.5127
0.62
2.7622
7.0322
0.63
2.5884
6.5896
0.64
2.4280 (×10−27)
6.1814 (×10−3)
0.65
2.2799
5.8044
0.66
2.1430
5.4557
0.67
2.0162
5.1330
0.68
1.8987
4.8338
0.69
1.7896
4.5561
0.70
1.6883
4.2982
0.71
1.5941
4.0583
0.72
1.5064
3.8349
0.73
1.4246
3.6268
0.74
1.3483
3.4325
0.75
1.2771
3.2512
0.76
1.2105
3.0817
0.77
1.1482
2.9231
0.78
1.0899
2.7746
0.79
1.0352
2.6355
0.80
9.8394 (×10−28)
2.5049
0.82
8.9059
2.2673
0.84
8.0807
2.0572
0.86
7.3490
1.8709
0.88
6.6985
1.7053
0.90
6.1184
1.5576
0.92
5.5999
1.4256
0.94
5.1352
1.3073
0.96
4.7179
1.2011
0.98
4.3421
1.1054
1.00
4.0030
1.0191
1.10
2.7284
6.9461 (×10−4)
1.20
1.9234
4.8967
1.30
1.3947
3.5508
1.40
1.0359
2.6373
1.50
7.8549 (×10−29)
1.9997
1.60
6.0639
1.5438
1.70
4.7556
1.2107
1.80
3.7820
9.6282 (×10−5)
1.90
3.0453
7.7528
2.00
2.4796
6.3127
2.20
1.6927
4.3094
2.40
1.1947
3.0416
2.60
8.6714 (×10−30)
2.2076
2.80
6.4453
1.6409
3.00
4.8900
1.2449
3.50
2.6386
6.7173 (×10−6)
4.00
1.5463
3.9367
Air pressure p0 = 1013.00 hPa, temperature T0 = 288.20 K, water-vapor partial pressure e0 = 7.85075 hPa, and CO2 volume concentration C = 385 ppmv.
Table 5
Values of the Rayleigh-Scattering Optical Depth δ(λ) Obtained for the Six Standard Atmosphere Modelsa
WDCGG Stations, Their Latitude, Longitude, and Height and the Yearly Average and Monthly Mean Ground-Level Values of CO2 Volume Concentration C Measured in 2001