Abstract

For measurement of aerosols over the ocean, the total radiance Lt backscattered from the top of a stratified atmosphere which contains both stratospheric and tropospheric aerosols of various types has been computed. A similar computation is carried out for an aerosol-free atmosphere yielding the Rayleigh scattered radiance Lr. The difference LtLr is shown to be linearly related to the radiance Las, which the aerosol would produce in the single scattering approximation. This greatly simplifies the application of aerosol models to aerosol analysis by satellite since adding to, or in some way changing, the aerosol model requires no additional multiple scattering computations. In fact, the only multiple computations required for aerosol analysis are those for determining Lr, which can be performed once and for all. The computations are explicitly applied to Band 4 of the CZCS, which, because of its high radiometric sensitivity and excellent calibration, is ideal for studying aerosols over the ocean. Specifically, the constant A in the relationship Las = A−1 (LtLr) is given as a function of position along the scan for four typical orbital–solar position scenarios. The computations show that Las can be retrieved from LtLr with an average error of no more than 5–7% except at the very edges of the scan.

© 1989 Optical Society of America

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References

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  1. “Earth Observing System: Science and Mission Requirements Working Group Report,” NASA Tech. Memo. 86129 (Aug.1984).
  2. M. Griggs, “Measurements of the Aerosol Optical Thickness Over Water Using ERTS-1 Data,” J. Air Pollut. Control Assoc. 25, 622 (1975).
    [CrossRef] [PubMed]
  3. Y. Mekler, H. Quenzel, G. Ohring, I. Marcus, “Relative Atmospheric Aerosol Content from Erts Observations,” J. Geophys. Res. 82, 967 (1977).
    [CrossRef]
  4. M. Griggs, “AVHRR Measurements of Atmospheric Aerosols Over Oceans,” NOAA National Environmental Satellite Service, Final Report contract MO-A01-78-00-4092 (Nov.1981).
  5. M. Griggs, “Satellite Measurements of Tropospheric Aerosols,” NASA Contract. Rep. 3459 (Aug.1981).
  6. M. Griggs, “AVHRR Aerosol Ground Truth Experiment,” NOAA National Environmental Satellite Service, Final Report contract NA-83-SAC-00106 (Jan.1984).
  7. R. S. Fraser, “Satellite Measurement of Mass of Sahara Dust in the Atmosphere,” Appl. Opt. 15, 2471 (1976).
    [CrossRef] [PubMed]
  8. P. Koepke, H. Quenzel, “Turbidity of the Atmosphere Determined From Satellite: Calculation of Optimum Viewing Geometry,” J. Geophys. Res. 84, 7847 (1979).
    [CrossRef]
  9. P. Koepke, H. Quenzel, “Turbidity of the Atmosphere Determined From Satellite: Calculation of Optimum Wavelength,” J. Geophys. Res. 86, 9801 (1981).
    [CrossRef]
  10. W. A. Hovis, “The Nimbus-7 Coastal Zone Color Scanner (CZCS) Program,” in Oceanography from Space, J. R. F. Gower, Ed. (Plenum, New York, 1981) pp. 213-225.
    [CrossRef]
  11. H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, New York, 1983).
  12. H. R. Gordon, “Removal of Atmospheric Effects from Satellite Imagery of the Oceans,” Appl. Opt. 17, 1631 (1978).
    [CrossRef] [PubMed]
  13. H. R. Gordon, D. J. Castaño, “The Coastal Zone Color Scanner Atmospheric Correction Algorithm: Multiple Scattering Effects,” Appl. Opt. 26, 2111 (1987).
    [CrossRef] [PubMed]
  14. H. R. Gordon, “Some Studies of Atmospheric Optical Variability in Relation to CZCS Atmospheric Correction,” NOAA National Environmental Satellite and Data Information Service, Final Report Contract NA-79-SAC-00714 (Feb.1984).
    [PubMed]
  15. H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, W. W. Broenkow, “Phytoplankton Pigment Concentrations in the Middle Atlantic Bight: Comparison between Ship Determinations and Coastal Zone Color Scanner Estimates,” Appl. Opt. 22, 20 (1983).
    [CrossRef] [PubMed]
  16. H. R. Gordon, D. K. Clark, “Clear Water Radiances for Atmospheric Correction of Coastal Zone Color Scanner Imagery,” Appl. Opt. 20, 4175 (1981).
    [CrossRef] [PubMed]
  17. P. Y. Deschamps, M. Herman, D. Tanre, “Modeling of the Atmospheric Effects and its Application to the Remote Sensing of Ocean Color,” Appl. Opt. 22, 3751 (1983).
    [CrossRef] [PubMed]
  18. L. Elterman, “Vertical Attenuation Model with Eight Surface Meteorological Ranges 2 to 13 Kilometers,” AFCRL, Bedford, MA, Report AFCRL-70-0200 (Mar.1970).
  19. K. L. Davidson, C. W. Fairall, “Optical Properties of the Marine Atmospheric Boundary Layer: Aerosol Profiles,” Proc. Soc. Photo-Opt. Instrum. Eng. 637, 18 (1986).
  20. K. Bullrich, “Scattered Radiation in the Atmosphere and the Natural Aerosol,” in Advances in Geophysics, H. E. Landsberg, J. V. Mieghem, Eds. (Academic, New York, 1964), pp. 99–260.
    [CrossRef]
  21. H. Quenzel, M. Kastner, “Optical Properties of the Atmosphere: Calculated Variability and Application to Satellite Remote Sensing of Phytoplankton,” Appl. Opt. 19, 1338 (1980).
    [CrossRef] [PubMed]
  22. B. Sturm, “Ocean Color Remote Sensing and the Retrieval of Surface Chlorophyll in Coastal Waters Using the Nimbus-7 CZCS,” in Oceanography from Space, J. F. R. Gower, Ed. (Plenum, New York, 1981), pp. 267–280.
    [CrossRef]
  23. G. W. Kattawar, “A Three-Parameter Analytic Phase Function for Multiple Scattering Calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839 (1975).
    [CrossRef]
  24. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).
  25. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).
  26. M. Viollier, D. Tanre, P. Y. Deschamps, “An Algorithm for Remote Sensing of Water Color from Space,” Boundary Layer Meteorol. 18, 247 (1980).
    [CrossRef]
  27. One scan line of the CZCS contains 1968 pixels corresponding to a rotation of the scan mirror through a total angle of 78.72°. However, while the instrument field of view is 0.0495°, the pixel sample rate is 1 per 0.04°, so there is an ~25% overlap between adjacent pixels. Pixel 246 from the eastern edge of the scan corresponds to a position where the scan mirror has been rotated 30° from the subsatellite track toward the east.
  28. However, over the open ocean, where the pigment concentration is sufficiently low, the water-leaving radiance (known to be <1 DC in Band 4) is known in Bands 2 and 3.16 This may allow derivation of the above mentioned combination as a function of wavelength which could provide a means of proceeding with a reduced dependence on models or provide guidance in choosing models.
  29. H. R. Gordon, J. W. Brown, R. H. Evans, “Exact Rayleigh Scattering Calculations for use with the Nimbus-7 Coastal Zone Color Scanner,” Appl. Opt. 27, 862 (1988).
    [CrossRef] [PubMed]

1988 (1)

1987 (1)

1986 (1)

K. L. Davidson, C. W. Fairall, “Optical Properties of the Marine Atmospheric Boundary Layer: Aerosol Profiles,” Proc. Soc. Photo-Opt. Instrum. Eng. 637, 18 (1986).

1983 (2)

1981 (3)

H. R. Gordon, D. K. Clark, “Clear Water Radiances for Atmospheric Correction of Coastal Zone Color Scanner Imagery,” Appl. Opt. 20, 4175 (1981).
[CrossRef] [PubMed]

P. Koepke, H. Quenzel, “Turbidity of the Atmosphere Determined From Satellite: Calculation of Optimum Wavelength,” J. Geophys. Res. 86, 9801 (1981).
[CrossRef]

M. Griggs, “Satellite Measurements of Tropospheric Aerosols,” NASA Contract. Rep. 3459 (Aug.1981).

1980 (2)

H. Quenzel, M. Kastner, “Optical Properties of the Atmosphere: Calculated Variability and Application to Satellite Remote Sensing of Phytoplankton,” Appl. Opt. 19, 1338 (1980).
[CrossRef] [PubMed]

M. Viollier, D. Tanre, P. Y. Deschamps, “An Algorithm for Remote Sensing of Water Color from Space,” Boundary Layer Meteorol. 18, 247 (1980).
[CrossRef]

1979 (1)

P. Koepke, H. Quenzel, “Turbidity of the Atmosphere Determined From Satellite: Calculation of Optimum Viewing Geometry,” J. Geophys. Res. 84, 7847 (1979).
[CrossRef]

1978 (1)

1977 (1)

Y. Mekler, H. Quenzel, G. Ohring, I. Marcus, “Relative Atmospheric Aerosol Content from Erts Observations,” J. Geophys. Res. 82, 967 (1977).
[CrossRef]

1976 (1)

1975 (2)

M. Griggs, “Measurements of the Aerosol Optical Thickness Over Water Using ERTS-1 Data,” J. Air Pollut. Control Assoc. 25, 622 (1975).
[CrossRef] [PubMed]

G. W. Kattawar, “A Three-Parameter Analytic Phase Function for Multiple Scattering Calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839 (1975).
[CrossRef]

Broenkow, W. W.

Brown, J. W.

Brown, O. B.

Bullrich, K.

K. Bullrich, “Scattered Radiation in the Atmosphere and the Natural Aerosol,” in Advances in Geophysics, H. E. Landsberg, J. V. Mieghem, Eds. (Academic, New York, 1964), pp. 99–260.
[CrossRef]

Castaño, D. J.

Clark, D. K.

Davidson, K. L.

K. L. Davidson, C. W. Fairall, “Optical Properties of the Marine Atmospheric Boundary Layer: Aerosol Profiles,” Proc. Soc. Photo-Opt. Instrum. Eng. 637, 18 (1986).

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

Deschamps, P. Y.

P. Y. Deschamps, M. Herman, D. Tanre, “Modeling of the Atmospheric Effects and its Application to the Remote Sensing of Ocean Color,” Appl. Opt. 22, 3751 (1983).
[CrossRef] [PubMed]

M. Viollier, D. Tanre, P. Y. Deschamps, “An Algorithm for Remote Sensing of Water Color from Space,” Boundary Layer Meteorol. 18, 247 (1980).
[CrossRef]

Elterman, L.

L. Elterman, “Vertical Attenuation Model with Eight Surface Meteorological Ranges 2 to 13 Kilometers,” AFCRL, Bedford, MA, Report AFCRL-70-0200 (Mar.1970).

Evans, R. H.

Fairall, C. W.

K. L. Davidson, C. W. Fairall, “Optical Properties of the Marine Atmospheric Boundary Layer: Aerosol Profiles,” Proc. Soc. Photo-Opt. Instrum. Eng. 637, 18 (1986).

Fraser, R. S.

Gordon, H. R.

Griggs, M.

M. Griggs, “Satellite Measurements of Tropospheric Aerosols,” NASA Contract. Rep. 3459 (Aug.1981).

M. Griggs, “Measurements of the Aerosol Optical Thickness Over Water Using ERTS-1 Data,” J. Air Pollut. Control Assoc. 25, 622 (1975).
[CrossRef] [PubMed]

M. Griggs, “AVHRR Measurements of Atmospheric Aerosols Over Oceans,” NOAA National Environmental Satellite Service, Final Report contract MO-A01-78-00-4092 (Nov.1981).

M. Griggs, “AVHRR Aerosol Ground Truth Experiment,” NOAA National Environmental Satellite Service, Final Report contract NA-83-SAC-00106 (Jan.1984).

Herman, M.

Hovis, W. A.

W. A. Hovis, “The Nimbus-7 Coastal Zone Color Scanner (CZCS) Program,” in Oceanography from Space, J. R. F. Gower, Ed. (Plenum, New York, 1981) pp. 213-225.
[CrossRef]

Kastner, M.

Kattawar, G. W.

G. W. Kattawar, “A Three-Parameter Analytic Phase Function for Multiple Scattering Calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839 (1975).
[CrossRef]

Koepke, P.

P. Koepke, H. Quenzel, “Turbidity of the Atmosphere Determined From Satellite: Calculation of Optimum Wavelength,” J. Geophys. Res. 86, 9801 (1981).
[CrossRef]

P. Koepke, H. Quenzel, “Turbidity of the Atmosphere Determined From Satellite: Calculation of Optimum Viewing Geometry,” J. Geophys. Res. 84, 7847 (1979).
[CrossRef]

Marcus, I.

Y. Mekler, H. Quenzel, G. Ohring, I. Marcus, “Relative Atmospheric Aerosol Content from Erts Observations,” J. Geophys. Res. 82, 967 (1977).
[CrossRef]

Mekler, Y.

Y. Mekler, H. Quenzel, G. Ohring, I. Marcus, “Relative Atmospheric Aerosol Content from Erts Observations,” J. Geophys. Res. 82, 967 (1977).
[CrossRef]

Morel, A. Y.

H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, New York, 1983).

Ohring, G.

Y. Mekler, H. Quenzel, G. Ohring, I. Marcus, “Relative Atmospheric Aerosol Content from Erts Observations,” J. Geophys. Res. 82, 967 (1977).
[CrossRef]

Quenzel, H.

P. Koepke, H. Quenzel, “Turbidity of the Atmosphere Determined From Satellite: Calculation of Optimum Wavelength,” J. Geophys. Res. 86, 9801 (1981).
[CrossRef]

H. Quenzel, M. Kastner, “Optical Properties of the Atmosphere: Calculated Variability and Application to Satellite Remote Sensing of Phytoplankton,” Appl. Opt. 19, 1338 (1980).
[CrossRef] [PubMed]

P. Koepke, H. Quenzel, “Turbidity of the Atmosphere Determined From Satellite: Calculation of Optimum Viewing Geometry,” J. Geophys. Res. 84, 7847 (1979).
[CrossRef]

Y. Mekler, H. Quenzel, G. Ohring, I. Marcus, “Relative Atmospheric Aerosol Content from Erts Observations,” J. Geophys. Res. 82, 967 (1977).
[CrossRef]

Sturm, B.

B. Sturm, “Ocean Color Remote Sensing and the Retrieval of Surface Chlorophyll in Coastal Waters Using the Nimbus-7 CZCS,” in Oceanography from Space, J. F. R. Gower, Ed. (Plenum, New York, 1981), pp. 267–280.
[CrossRef]

Tanre, D.

P. Y. Deschamps, M. Herman, D. Tanre, “Modeling of the Atmospheric Effects and its Application to the Remote Sensing of Ocean Color,” Appl. Opt. 22, 3751 (1983).
[CrossRef] [PubMed]

M. Viollier, D. Tanre, P. Y. Deschamps, “An Algorithm for Remote Sensing of Water Color from Space,” Boundary Layer Meteorol. 18, 247 (1980).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

Viollier, M.

M. Viollier, D. Tanre, P. Y. Deschamps, “An Algorithm for Remote Sensing of Water Color from Space,” Boundary Layer Meteorol. 18, 247 (1980).
[CrossRef]

Appl. Opt. (8)

Boundary Layer Meteorol. (1)

M. Viollier, D. Tanre, P. Y. Deschamps, “An Algorithm for Remote Sensing of Water Color from Space,” Boundary Layer Meteorol. 18, 247 (1980).
[CrossRef]

J. Air Pollut. Control Assoc. (1)

M. Griggs, “Measurements of the Aerosol Optical Thickness Over Water Using ERTS-1 Data,” J. Air Pollut. Control Assoc. 25, 622 (1975).
[CrossRef] [PubMed]

J. Geophys. Res. (3)

Y. Mekler, H. Quenzel, G. Ohring, I. Marcus, “Relative Atmospheric Aerosol Content from Erts Observations,” J. Geophys. Res. 82, 967 (1977).
[CrossRef]

P. Koepke, H. Quenzel, “Turbidity of the Atmosphere Determined From Satellite: Calculation of Optimum Viewing Geometry,” J. Geophys. Res. 84, 7847 (1979).
[CrossRef]

P. Koepke, H. Quenzel, “Turbidity of the Atmosphere Determined From Satellite: Calculation of Optimum Wavelength,” J. Geophys. Res. 86, 9801 (1981).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

G. W. Kattawar, “A Three-Parameter Analytic Phase Function for Multiple Scattering Calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839 (1975).
[CrossRef]

NASA Contract. Rep. (1)

M. Griggs, “Satellite Measurements of Tropospheric Aerosols,” NASA Contract. Rep. 3459 (Aug.1981).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

K. L. Davidson, C. W. Fairall, “Optical Properties of the Marine Atmospheric Boundary Layer: Aerosol Profiles,” Proc. Soc. Photo-Opt. Instrum. Eng. 637, 18 (1986).

Other (13)

K. Bullrich, “Scattered Radiation in the Atmosphere and the Natural Aerosol,” in Advances in Geophysics, H. E. Landsberg, J. V. Mieghem, Eds. (Academic, New York, 1964), pp. 99–260.
[CrossRef]

H. R. Gordon, “Some Studies of Atmospheric Optical Variability in Relation to CZCS Atmospheric Correction,” NOAA National Environmental Satellite and Data Information Service, Final Report Contract NA-79-SAC-00714 (Feb.1984).
[PubMed]

L. Elterman, “Vertical Attenuation Model with Eight Surface Meteorological Ranges 2 to 13 Kilometers,” AFCRL, Bedford, MA, Report AFCRL-70-0200 (Mar.1970).

M. Griggs, “AVHRR Aerosol Ground Truth Experiment,” NOAA National Environmental Satellite Service, Final Report contract NA-83-SAC-00106 (Jan.1984).

M. Griggs, “AVHRR Measurements of Atmospheric Aerosols Over Oceans,” NOAA National Environmental Satellite Service, Final Report contract MO-A01-78-00-4092 (Nov.1981).

W. A. Hovis, “The Nimbus-7 Coastal Zone Color Scanner (CZCS) Program,” in Oceanography from Space, J. R. F. Gower, Ed. (Plenum, New York, 1981) pp. 213-225.
[CrossRef]

H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, New York, 1983).

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

B. Sturm, “Ocean Color Remote Sensing and the Retrieval of Surface Chlorophyll in Coastal Waters Using the Nimbus-7 CZCS,” in Oceanography from Space, J. F. R. Gower, Ed. (Plenum, New York, 1981), pp. 267–280.
[CrossRef]

One scan line of the CZCS contains 1968 pixels corresponding to a rotation of the scan mirror through a total angle of 78.72°. However, while the instrument field of view is 0.0495°, the pixel sample rate is 1 per 0.04°, so there is an ~25% overlap between adjacent pixels. Pixel 246 from the eastern edge of the scan corresponds to a position where the scan mirror has been rotated 30° from the subsatellite track toward the east.

However, over the open ocean, where the pigment concentration is sufficiently low, the water-leaving radiance (known to be <1 DC in Band 4) is known in Bands 2 and 3.16 This may allow derivation of the above mentioned combination as a function of wavelength which could provide a means of proceeding with a reduced dependence on models or provide guidance in choosing models.

“Earth Observing System: Science and Mission Requirements Working Group Report,” NASA Tech. Memo. 86129 (Aug.1984).

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Figures (9)

Fig. 1
Fig. 1

Aerosol phase functions used in the study: ▼, Haze L; ▲, Haze C; ●, marine aerosol model.

Fig. 2
Fig. 2

Las as a function of LtLr at the center of the scan for the nontilted orbits with the aerosol restricted to the troposphere. ●, ▼, and ▲ refer, respectively, to the marine aerosol, Haze L, and Haze C.

Fig. 3
Fig. 3

Las as a function of LtLr at the center of the scan for the tilted orbits (tilt = 20°) with the aerosol restricted to the troposphere. ●, ▼, and ▲ refer, respectively, to the marine aerosol, Haze L, and Haze C.

Fig. 4
Fig. 4

Las as a function of LtLr at a position located 246 pixels from the eastern edge of the scan (a scan mirror rotation angle of 30°). The sensor is in the untilted mode, and the aerosol is restricted to the troposphere. ●, ▼, and ▲ refer, respectively, to the marine aerosol, Haze L, and Haze C.

Fig. 5
Fig. 5

Las as a function of LtLr at a position located 246 pixels from the eastern edge of the scan (a scan mirror rotation angle of 30°). The sensor is in the tilted mode (tilt = 20°), and the aerosol is restricted to the troposphere. ●, ▼, and ▲ refer, respectively, to the marine aerosol, Haze L, and Haze C.

Fig. 6
Fig. 6

Las as a function of LtLr at the center of the scan for the nontilted orbits. Aerosols are located in both the troposphere and the stratosphere. ●, ▼, and ▲ refer, respectively, to the marine aerosol, Haze L, and Haze C, restricted to the troposphere. ♦ refers to cases with aerosols in both the troposphere and the stratosphere, i.e., τs ≠ 0.

Fig. 7
Fig. 7

Las as a function of LtLr at the center of the scan for the tilted orbits (tilt = 20°). Aerosols are located in both the troposphere and stratosphere. ●, ▼, and ▲ refer, respectively, to the marine aerosol, Haze L, and Haze C, restricted to the troposphere. ♦ refers to cases with aerosols in both the troposphere and stratosphere, i.e., τs ≠ 0, and + refers to cases for which the tropospheric aerosol has ω0 = 0.9.

Fig. 8
Fig. 8

Las as a function of LtLr at a position located 246 pixels from the eastern edge of the scan (a scan mirror rotation angle of 30°). The sensor is in the untilted mode. Aerosols are located in both the troposphere and stratosphere. ●, ▼, and ▲ refer, respectively, to the marine aerosol, Haze L, and Haze C restricted to the troposphere. ♦ refers to cases with aerosols in both the troposphere and stratosphere, i.e. τs ≠ 0.

Fig. 9
Fig. 9

Las as a function of LtLr at a position located 246 pixels from the eastern edge of the scan (a scan mirror rotation angle of 30°). The sensor is in the tilted mode (tilt = 20°). Aerosols are located in both the troposphere and stratosphere. ●, ▼, and ▲ refer, respectively, to the marine aerosol, Haze L, and Haze C restricted to the troposphere. ♦ refers to cases with aerosols in both the troposphere and stratosphere, i.e., τs ≠ 0, and + refers to cases for which the tropospheric aerosol has ω0 = 0.9.

Tables (2)

Tables Icon

Table I Scenes Examined In the Present Studya

Tables Icon

Table II Linear Regression Estimates of the Coefficient A in Eq. (8) as a Function of the Mirror Rotation Angle α. ΔLas is the Average Error in Las in Percent and N is the Number of Points Used in the Individual Regressions

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

L t = L r s + L a s .
L x s = ω x τ x F 0 p x   ( θ , θ 0 ) / 4 π   cos θ ,
p x   ( θ , θ 0 ) = P x   ( θ _ , ) + [ ρ ( θ ) + ρ ( θ 0 ) ] P x   ( θ + ) , cos θ ± = ±  cos θ 0  cos θ  sin θ 0  sin θ  cos  ( ϕ ϕ 0 ) .
F 0 = F 0  exp [ τ 0 z   ( 1 / cos θ + 1 / cos θ 0 ) ] ,
L t = L r + L a + C R , P ,
b x = b x 0  exp ( z / H x ) ,
τ x = b x 0 H x .
P x ( θ ) = α f ( θ , g 1 ) + ( 1 α ) f ( θ , g 2 ) ,
f ( θ , g ) = ( 1   g 2 ) ( 1   +   g 2   2 g  cos θ ) 3 / 2 ,
L a s = [ ω a τ a p a ( θ , θ 0 ) + ω s τ s p s ( θ , θ 0 ) ] F 0 4 π   cos θ .
A L a s = ( L t L r )
Δ L a s = | L a s   ( true ) L a s   ( fit ) L a s   ( true ) | × 100 % ,

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