Abstract

Calculations that use the Gauss–Seidel method are presented of the diffusely scattered light in a spherical atmosphere with polarization fully included. Comparisons are made between this method and the Monte Carlo calculations of other researchers for spherical geometry in a pure Rayleigh atmosphere. Comparisons with plane–parallel atmospheres are also presented. Single-scatter intensity comparisons with spherical geometry show excellent agreement. When all orders of scattering are included, comparisons of polarization parameters I, Q and U as well as the plane of polarization show good agreement when allowances are made for the statistical variability inherent in the Monte Carlo method.

© 1995 Optical Society of America

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References

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  1. B. M. Herman, A. Ben-David, K. J. Thome, “Numerical technique for solving the radiative transfer equation for a spherical-shell atmosphere,” Appl. Opt. 33, 1760–1770 (1994).
    [CrossRef] [PubMed]
  2. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), pp. 42, 262–265.
  3. R. E. Bellman, H. H. Kagiwada, R. E. Kalaba, “Diffuse reflection of solar rays by a spherical shell atmosphere,” Icarus 11, 417–423 (1969).
    [CrossRef]
  4. S. J. Wilson, K. K. Sen, “Light scattering by an optically thin inhomogeneous spherically symmetric planetary atmosphere,” Astrophys. Space Sci. 69, 107–113 (1980).
    [CrossRef]
  5. A. Dahlback, K. Stamnes, “A new spherical model for computing the radiation field available for photolysis and heating at twilight,” Planet. Space Sci. 39, 671–683 (1991).
    [CrossRef]
  6. D. E. Anderson, “The troposphere-stratosphere radiation field at twilight: a spherical model,” Planet. Space Sci. 31, 1517–1523 (1983).
    [CrossRef]
  7. G. I. Marchuk, G. A. Mikhailov, “The solution of problems of atmospheric optics by Monte Carlo method,” Izv. Acad. Sci. USSR Atmos. Ocean. Phys. 3, 147–155 (1967).
  8. D. G. Collins, M. B. Wells, “FLASH, a Monte Carlo procedure for use in calculating light scattering in a spherical-shell atmosphere,” Rep. AFCRL-70-0206 (Radiation Research Associates, Inc., Fort Worth, Tex., 1970).
  9. C. N. Adams, G. W. Kattawar, “Radiative transfer in spherical shell atmospheres: I. Rayleigh scattering,” Icarus 35, 139–151 (1978).
    [CrossRef]
  10. C. K. Whitney, “Implications of a quadratic stream definition in radiative transfer theory,” J. Atmos. Sci. 29, 1520–1530 (1972).
    [CrossRef]
  11. G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980), pp. 109–146.
  12. D. G. Collins, W. G. Blättner, M. B. Wells, H. G. Horak, “Backward Monte Carlo calculations of the polarization characteristics of radiation emerging from spherical shell atmospheres,” Appl. Opt. 11, 2684–2696 (1972).
    [CrossRef] [PubMed]
  13. W. G. Blättner, M. B. Wells, “Monte Carlo studies of sky radiation,” Rep. AFCRL-TR-73-0613 (Radiation Research Associates, Inc., Fort Worth, Tex., 1973).
  14. W. A. Asous, “Light scattering in spherical atmospheres,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1982), Chap. 3, pp. 16–60.
  15. K. J. Thome, “Radiative transfer model for a spherical atmosphere,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1990), Chap. 4, pp. 65–74.
  16. J. Lenoble, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (Deepak, Hampton, Va., 1985), Part 2, Chap. 3, pp. 247–255.
  17. Z. Sekera, “Scattering matrix for spherical particles and its transformation,” in Investigation of Skylight Polarization, con. AF 19(122)-239 (Department of Meteorology, University of California, Los Angeles, Calif., 1955), App. D.
  18. K. L. Coulson, J. V. Dave, Z. Sekera, Tables Relating to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (U. of California Press, Berkeley, Calif., 1960).
  19. W. G. Blättner, D. G. Collins, M. B. Wells, “Monte Carlo calculation in spherical-shell atmospheres,” Rep. AFCRL-71-0382 (Radiation Research Associates, Inc., Fort Worth, Tex., 1971).
  20. 1962 U. S. Standard Atmosphere (U.S. GPO, Washington, D.C., 1962), pp. 36–59.
  21. C. K. Whitney, H. L. Malchow, “Study of radiative transfer in scattering atmospheres,” Rep. AFGL-TR-78-0101 (The Charles Stark Draper Laboratory, Inc., Cambridge, Mass., 1977), pp. 25.
  22. W. G. Blättner, H. G. Horak, D. G. Collins, M. B. Wells, “Monte Carlo studies of the sky radiation at twilight,” Appl. Opt. 13, 534–547 (1974).
    [CrossRef] [PubMed]
  23. B. M. Herman, S. R. Browning, R. J. Curran, “The effect of atmospheric aerosols on scattered sunlight,” J. Atmos. Sci. 28, 419–428 (1971).
    [CrossRef]

1994 (1)

1991 (1)

A. Dahlback, K. Stamnes, “A new spherical model for computing the radiation field available for photolysis and heating at twilight,” Planet. Space Sci. 39, 671–683 (1991).
[CrossRef]

1983 (1)

D. E. Anderson, “The troposphere-stratosphere radiation field at twilight: a spherical model,” Planet. Space Sci. 31, 1517–1523 (1983).
[CrossRef]

1980 (1)

S. J. Wilson, K. K. Sen, “Light scattering by an optically thin inhomogeneous spherically symmetric planetary atmosphere,” Astrophys. Space Sci. 69, 107–113 (1980).
[CrossRef]

1978 (1)

C. N. Adams, G. W. Kattawar, “Radiative transfer in spherical shell atmospheres: I. Rayleigh scattering,” Icarus 35, 139–151 (1978).
[CrossRef]

1974 (1)

1972 (2)

1971 (1)

B. M. Herman, S. R. Browning, R. J. Curran, “The effect of atmospheric aerosols on scattered sunlight,” J. Atmos. Sci. 28, 419–428 (1971).
[CrossRef]

1969 (1)

R. E. Bellman, H. H. Kagiwada, R. E. Kalaba, “Diffuse reflection of solar rays by a spherical shell atmosphere,” Icarus 11, 417–423 (1969).
[CrossRef]

1967 (1)

G. I. Marchuk, G. A. Mikhailov, “The solution of problems of atmospheric optics by Monte Carlo method,” Izv. Acad. Sci. USSR Atmos. Ocean. Phys. 3, 147–155 (1967).

Adams, C. N.

C. N. Adams, G. W. Kattawar, “Radiative transfer in spherical shell atmospheres: I. Rayleigh scattering,” Icarus 35, 139–151 (1978).
[CrossRef]

Anderson, D. E.

D. E. Anderson, “The troposphere-stratosphere radiation field at twilight: a spherical model,” Planet. Space Sci. 31, 1517–1523 (1983).
[CrossRef]

Asous, W. A.

W. A. Asous, “Light scattering in spherical atmospheres,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1982), Chap. 3, pp. 16–60.

Bellman, R. E.

R. E. Bellman, H. H. Kagiwada, R. E. Kalaba, “Diffuse reflection of solar rays by a spherical shell atmosphere,” Icarus 11, 417–423 (1969).
[CrossRef]

Ben-David, A.

Blättner, W. G.

W. G. Blättner, H. G. Horak, D. G. Collins, M. B. Wells, “Monte Carlo studies of the sky radiation at twilight,” Appl. Opt. 13, 534–547 (1974).
[CrossRef] [PubMed]

D. G. Collins, W. G. Blättner, M. B. Wells, H. G. Horak, “Backward Monte Carlo calculations of the polarization characteristics of radiation emerging from spherical shell atmospheres,” Appl. Opt. 11, 2684–2696 (1972).
[CrossRef] [PubMed]

W. G. Blättner, M. B. Wells, “Monte Carlo studies of sky radiation,” Rep. AFCRL-TR-73-0613 (Radiation Research Associates, Inc., Fort Worth, Tex., 1973).

W. G. Blättner, D. G. Collins, M. B. Wells, “Monte Carlo calculation in spherical-shell atmospheres,” Rep. AFCRL-71-0382 (Radiation Research Associates, Inc., Fort Worth, Tex., 1971).

Browning, S. R.

B. M. Herman, S. R. Browning, R. J. Curran, “The effect of atmospheric aerosols on scattered sunlight,” J. Atmos. Sci. 28, 419–428 (1971).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), pp. 42, 262–265.

Collins, D. G.

W. G. Blättner, H. G. Horak, D. G. Collins, M. B. Wells, “Monte Carlo studies of the sky radiation at twilight,” Appl. Opt. 13, 534–547 (1974).
[CrossRef] [PubMed]

D. G. Collins, W. G. Blättner, M. B. Wells, H. G. Horak, “Backward Monte Carlo calculations of the polarization characteristics of radiation emerging from spherical shell atmospheres,” Appl. Opt. 11, 2684–2696 (1972).
[CrossRef] [PubMed]

D. G. Collins, M. B. Wells, “FLASH, a Monte Carlo procedure for use in calculating light scattering in a spherical-shell atmosphere,” Rep. AFCRL-70-0206 (Radiation Research Associates, Inc., Fort Worth, Tex., 1970).

W. G. Blättner, D. G. Collins, M. B. Wells, “Monte Carlo calculation in spherical-shell atmospheres,” Rep. AFCRL-71-0382 (Radiation Research Associates, Inc., Fort Worth, Tex., 1971).

Coulson, K. L.

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Relating to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (U. of California Press, Berkeley, Calif., 1960).

Curran, R. J.

B. M. Herman, S. R. Browning, R. J. Curran, “The effect of atmospheric aerosols on scattered sunlight,” J. Atmos. Sci. 28, 419–428 (1971).
[CrossRef]

Dahlback, A.

A. Dahlback, K. Stamnes, “A new spherical model for computing the radiation field available for photolysis and heating at twilight,” Planet. Space Sci. 39, 671–683 (1991).
[CrossRef]

Darbinjan, R. A.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980), pp. 109–146.

Dave, J. V.

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Relating to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (U. of California Press, Berkeley, Calif., 1960).

Elepov, B. S.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980), pp. 109–146.

Herman, B. M.

B. M. Herman, A. Ben-David, K. J. Thome, “Numerical technique for solving the radiative transfer equation for a spherical-shell atmosphere,” Appl. Opt. 33, 1760–1770 (1994).
[CrossRef] [PubMed]

B. M. Herman, S. R. Browning, R. J. Curran, “The effect of atmospheric aerosols on scattered sunlight,” J. Atmos. Sci. 28, 419–428 (1971).
[CrossRef]

Horak, H. G.

Kagiwada, H. H.

R. E. Bellman, H. H. Kagiwada, R. E. Kalaba, “Diffuse reflection of solar rays by a spherical shell atmosphere,” Icarus 11, 417–423 (1969).
[CrossRef]

Kalaba, R. E.

R. E. Bellman, H. H. Kagiwada, R. E. Kalaba, “Diffuse reflection of solar rays by a spherical shell atmosphere,” Icarus 11, 417–423 (1969).
[CrossRef]

Kargin, B. A.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980), pp. 109–146.

Kattawar, G. W.

C. N. Adams, G. W. Kattawar, “Radiative transfer in spherical shell atmospheres: I. Rayleigh scattering,” Icarus 35, 139–151 (1978).
[CrossRef]

Lenoble, J.

J. Lenoble, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (Deepak, Hampton, Va., 1985), Part 2, Chap. 3, pp. 247–255.

Malchow, H. L.

C. K. Whitney, H. L. Malchow, “Study of radiative transfer in scattering atmospheres,” Rep. AFGL-TR-78-0101 (The Charles Stark Draper Laboratory, Inc., Cambridge, Mass., 1977), pp. 25.

Marchuk, G. I.

G. I. Marchuk, G. A. Mikhailov, “The solution of problems of atmospheric optics by Monte Carlo method,” Izv. Acad. Sci. USSR Atmos. Ocean. Phys. 3, 147–155 (1967).

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980), pp. 109–146.

Mikhailov, G. A.

G. I. Marchuk, G. A. Mikhailov, “The solution of problems of atmospheric optics by Monte Carlo method,” Izv. Acad. Sci. USSR Atmos. Ocean. Phys. 3, 147–155 (1967).

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980), pp. 109–146.

Nazaraliev, M. A.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980), pp. 109–146.

Sekera, Z.

Z. Sekera, “Scattering matrix for spherical particles and its transformation,” in Investigation of Skylight Polarization, con. AF 19(122)-239 (Department of Meteorology, University of California, Los Angeles, Calif., 1955), App. D.

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Relating to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (U. of California Press, Berkeley, Calif., 1960).

Sen, K. K.

S. J. Wilson, K. K. Sen, “Light scattering by an optically thin inhomogeneous spherically symmetric planetary atmosphere,” Astrophys. Space Sci. 69, 107–113 (1980).
[CrossRef]

Stamnes, K.

A. Dahlback, K. Stamnes, “A new spherical model for computing the radiation field available for photolysis and heating at twilight,” Planet. Space Sci. 39, 671–683 (1991).
[CrossRef]

Thome, K. J.

B. M. Herman, A. Ben-David, K. J. Thome, “Numerical technique for solving the radiative transfer equation for a spherical-shell atmosphere,” Appl. Opt. 33, 1760–1770 (1994).
[CrossRef] [PubMed]

K. J. Thome, “Radiative transfer model for a spherical atmosphere,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1990), Chap. 4, pp. 65–74.

Wells, M. B.

W. G. Blättner, H. G. Horak, D. G. Collins, M. B. Wells, “Monte Carlo studies of the sky radiation at twilight,” Appl. Opt. 13, 534–547 (1974).
[CrossRef] [PubMed]

D. G. Collins, W. G. Blättner, M. B. Wells, H. G. Horak, “Backward Monte Carlo calculations of the polarization characteristics of radiation emerging from spherical shell atmospheres,” Appl. Opt. 11, 2684–2696 (1972).
[CrossRef] [PubMed]

W. G. Blättner, M. B. Wells, “Monte Carlo studies of sky radiation,” Rep. AFCRL-TR-73-0613 (Radiation Research Associates, Inc., Fort Worth, Tex., 1973).

D. G. Collins, M. B. Wells, “FLASH, a Monte Carlo procedure for use in calculating light scattering in a spherical-shell atmosphere,” Rep. AFCRL-70-0206 (Radiation Research Associates, Inc., Fort Worth, Tex., 1970).

W. G. Blättner, D. G. Collins, M. B. Wells, “Monte Carlo calculation in spherical-shell atmospheres,” Rep. AFCRL-71-0382 (Radiation Research Associates, Inc., Fort Worth, Tex., 1971).

Whitney, C. K.

C. K. Whitney, “Implications of a quadratic stream definition in radiative transfer theory,” J. Atmos. Sci. 29, 1520–1530 (1972).
[CrossRef]

C. K. Whitney, H. L. Malchow, “Study of radiative transfer in scattering atmospheres,” Rep. AFGL-TR-78-0101 (The Charles Stark Draper Laboratory, Inc., Cambridge, Mass., 1977), pp. 25.

Wilson, S. J.

S. J. Wilson, K. K. Sen, “Light scattering by an optically thin inhomogeneous spherically symmetric planetary atmosphere,” Astrophys. Space Sci. 69, 107–113 (1980).
[CrossRef]

Appl. Opt. (3)

Astrophys. Space Sci. (1)

S. J. Wilson, K. K. Sen, “Light scattering by an optically thin inhomogeneous spherically symmetric planetary atmosphere,” Astrophys. Space Sci. 69, 107–113 (1980).
[CrossRef]

Icarus (2)

R. E. Bellman, H. H. Kagiwada, R. E. Kalaba, “Diffuse reflection of solar rays by a spherical shell atmosphere,” Icarus 11, 417–423 (1969).
[CrossRef]

C. N. Adams, G. W. Kattawar, “Radiative transfer in spherical shell atmospheres: I. Rayleigh scattering,” Icarus 35, 139–151 (1978).
[CrossRef]

Izv. Acad. Sci. USSR Atmos. Ocean. Phys. (1)

G. I. Marchuk, G. A. Mikhailov, “The solution of problems of atmospheric optics by Monte Carlo method,” Izv. Acad. Sci. USSR Atmos. Ocean. Phys. 3, 147–155 (1967).

J. Atmos. Sci. (2)

C. K. Whitney, “Implications of a quadratic stream definition in radiative transfer theory,” J. Atmos. Sci. 29, 1520–1530 (1972).
[CrossRef]

B. M. Herman, S. R. Browning, R. J. Curran, “The effect of atmospheric aerosols on scattered sunlight,” J. Atmos. Sci. 28, 419–428 (1971).
[CrossRef]

Planet. Space Sci. (2)

A. Dahlback, K. Stamnes, “A new spherical model for computing the radiation field available for photolysis and heating at twilight,” Planet. Space Sci. 39, 671–683 (1991).
[CrossRef]

D. E. Anderson, “The troposphere-stratosphere radiation field at twilight: a spherical model,” Planet. Space Sci. 31, 1517–1523 (1983).
[CrossRef]

Other (12)

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), pp. 42, 262–265.

D. G. Collins, M. B. Wells, “FLASH, a Monte Carlo procedure for use in calculating light scattering in a spherical-shell atmosphere,” Rep. AFCRL-70-0206 (Radiation Research Associates, Inc., Fort Worth, Tex., 1970).

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980), pp. 109–146.

W. G. Blättner, M. B. Wells, “Monte Carlo studies of sky radiation,” Rep. AFCRL-TR-73-0613 (Radiation Research Associates, Inc., Fort Worth, Tex., 1973).

W. A. Asous, “Light scattering in spherical atmospheres,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1982), Chap. 3, pp. 16–60.

K. J. Thome, “Radiative transfer model for a spherical atmosphere,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1990), Chap. 4, pp. 65–74.

J. Lenoble, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (Deepak, Hampton, Va., 1985), Part 2, Chap. 3, pp. 247–255.

Z. Sekera, “Scattering matrix for spherical particles and its transformation,” in Investigation of Skylight Polarization, con. AF 19(122)-239 (Department of Meteorology, University of California, Los Angeles, Calif., 1955), App. D.

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Relating to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (U. of California Press, Berkeley, Calif., 1960).

W. G. Blättner, D. G. Collins, M. B. Wells, “Monte Carlo calculation in spherical-shell atmospheres,” Rep. AFCRL-71-0382 (Radiation Research Associates, Inc., Fort Worth, Tex., 1971).

1962 U. S. Standard Atmosphere (U.S. GPO, Washington, D.C., 1962), pp. 36–59.

C. K. Whitney, H. L. Malchow, “Study of radiative transfer in scattering atmospheres,” Rep. AFGL-TR-78-0101 (The Charles Stark Draper Laboratory, Inc., Cambridge, Mass., 1977), pp. 25.

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Figures (18)

Fig. 1
Fig. 1

Coordinate system that shows the variables used to define point s and a line of sight at point s.

Fig. 2
Fig. 2

Grid system that shows the zenith and conical boundary used to compute polarization parameters on the zenith. This two-dimensional cross section shows the boundary associated with the η and η + 180° lines. Labeled points are discussed in the text.

Fig. 3
Fig. 3

Comparison of reflected total intensity at the top of atmosphere calculated by three versions of the Gauss–Seidel code for optical depth τ = 1.0, solar zenith angle θ0 = 0.0°, and surface reflectivity A = 0.0.

Fig. 4
Fig. 4

Same as Fig. 3 except for θ0 = 80.0° and A = 0.5.

Fig. 5
Fig. 5

Reflected single-scatter intensity at the top of a homogeneous 100-km conservative Rayleigh atmosphere with τ = 0.25 and θ0 = 0.0°; the crosses are the spherical shell Monte Carlo calculations of Adams and Kattawar9 and the dashed curve is their plane–parallel solution.

Fig. 6
Fig. 6

Same as Fig. 5 except for τ = 1.0.

Fig. 7
Fig. 7

Same as Fig. 5 except for θ0 = 84.26°.

Fig. 8
Fig. 8

Same as Fig. 5 except for τ = 1.0 and θ0 = 84.26°.

Fig. 9
Fig. 9

Transmitted total intensity in the ϕ = 0 − 180° plane at the bottom of a 50-km homogeneous conservative Rayleigh atmosphere with τ = 0.25 and θ0 = 84.26°; the crosses are the spherical shell Monte Carlo calculations of Blattner et al.19 and the dashed curve is the Coulson et al.18 plane–parallel solution.

Fig. 10
Fig. 10

Same as Fig. 9 except for ϕ = 30 − 210° plane.

Fig. 11
Fig. 11

Reflected total intensity at top of the atmosphere for same conditions as in Fig. 9.

Fig. 12
Fig. 12

Transmitted Q parameter for the same conditions as in Fig. 9.

Fig. 13
Fig. 13

Transmitted Q parameter for the same conditions as in Fig. 9 except for ϕ = 30 − 210° plane.

Fig. 14
Fig. 14

Transmitted U parameter for the same conditions as in Fig. 13.

Fig. 15
Fig. 15

Transmitted plane of polarization for the same conditions as in Fig. 13.

Fig. 16
Fig. 16

Transmitted total intensity and single- and multiple-scatter intensity for 100-km vertically inhomogeneous Rayleigh atmosphere for τ = 0.25 and θ0 = 66.42°. The curves represent the results of the PSGS whereas the symbols are for the Blättner and Wells13 Monte Carlo calculations.

Fig. 17
Fig. 17

Same as Fig. 16 except for θ0 = 84.26°.

Fig. 18
Fig. 18

Comparison of the transmitted total intensity for 100-km vertically inhomogeneous and homogeneous Rayleigh atmospheres for τ = 0.25 and θ0 = 84.26°. The dashed curve represents the results of Ref. 18.

Equations (18)

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I = ( I l , I r , U , V ) = ( I 1 , I 2 , I 3 , I 4 ) .
I p ( s , θ , ϕ ) = I p ( s 0 , θ , ϕ ) exp [ - τ ( s , s 0 ) ] + s 0 s J p ( s , θ , ϕ ) exp [ - τ ( s , s ) ] κ ρ d s .
τ ( s , s ) = s s κ ρ d s .
J p ( s , θ , ϕ ) = P p q ( s , θ , ϕ , θ 0 , ϕ 0 ) F q ( s , θ 0 , ϕ 0 ) + 0 2 π 0 π P p q ( s , θ , ϕ , θ , ϕ ) I q ( s , θ , ϕ ) × sin θ d θ d ϕ ,
P p q ( s , θ , ϕ , θ , ϕ ) = 3 8 π [ cos 2 ψ μ 2 sin 2 Δ ϕ - μ cos ψ sin Δ ϕ μ 2 sin 2 Δ ϕ cos 2 Δ ϕ μ sin Δ ϕ cos ϕ 2 μ cos ψ sin Δ ϕ - 2 μ sin Δ ϕ cos Δ ϕ - μ μ sin Δ ϕ + cos ψ cos Δ ϕ ] ,
I p ss ( s i , θ , ϕ ) = 0 s i P p q ( s , θ , ϕ , θ 0 , ϕ 0 ) F q ( s , θ 0 , ϕ 0 ) × exp [ - τ ( s i , s ) ] κ ρ d s .
I p ms ( s i , θ , ϕ ) = 0 s i 0 2 π 0 π P p q ( s , θ , ϕ , θ , ϕ ) I q ( s , θ , ϕ ) × sin θ d θ d ϕ exp [ - τ ( s i , s ) ] κ ρ d s .
I p ms ( s i , θ , ϕ ) = I p ms ( a , θ , ϕ ) exp [ - τ ( s i , a ) ] + a s i J p ms ( s , θ , ϕ ) exp [ - τ ( s i , s ) ] κ ρ d s ,
I p ms ( a , θ , ϕ ) = I p ss ( a , θ , ϕ ) R ( a , θ , ϕ ) .
J ¯ p q m s ( θ , ϕ ) = a s i 0 2 π 0 π P p q ( s , θ , ϕ , θ , ϕ ) I q ( s , θ , ϕ ) sin θ d θ d ϕ exp [ - τ ( s i , s ) ] κ ρ d s a s i exp [ - τ ( s i , s ) ] κ ρ d s .
P ¯ p q ( θ , ϕ ) = J ¯ p q ms ( θ , ϕ ) 0 2 π 0 π I q ( s ¯ , θ , ϕ ) sin θ d θ d ϕ ,
b s i ( Ψ 0 , η ) J p ms ( s , θ , ϕ ) exp { - τ [ s i ( Ψ 0 , η ) , s ] } κ ρ d s = P ¯ p 1 ( θ , ϕ ) 0 2 π 0 π I 1 [ s ¯ ( Ψ 0 ) , θ , ϕ ] sin θ d θ d ϕ × b s i ( Ψ 0 , η ) exp { - τ [ s i ( Ψ 0 , η ) , s ] } κ ρ d s + P ¯ p 2 ( θ , ϕ ) 0 2 π 0 π I 2 [ s ¯ ( Ψ 0 ) , θ , ϕ ] sin θ d θ d ϕ × b s i ( Ψ 0 , η ) exp { - τ [ s i ( Ψ 0 , η ) , s ] } κ ρ d s + J ¯ p 3 ms ( θ , ϕ ) b s i ( Ψ 0 , η ) exp { - τ [ s i ( Ψ 0 , η ) , s ] } κ ρ d s .
I = I 1 + I 2 ,
Q = I 1 - I 2 ,
U = I 3 ,
V = I 4 ,
P = ( Q 2 + U 2 ) 1 / 2 I × 100 ,
tan 2 χ = U Q .

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