Abstract

The reflected radiance and polarization are calculated for clouds with optical thicknesses from 10 to 100. The results are presented for both the haze C and nimbostratus model. The peak in the single scattered polarization at 140° for the nimbostratus model persists even with all the multiple scattering events that occur for the largest optical thicknesses considered here. The calculations are made by a Monte Carlo technique, which includes the effect of multiple scattering through all orders and a realistic anisotropic phase function for single scattering appropriate for the distribution of particle sizes in the cloud. The effect of the surface albedo is included in the calculations for the optical thickness of 10. The variation of the radiance and polarization with both the nadir and azimuthal angle is given for several solar zenith angles.

© 1971 Optical Society of America

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References

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  1. H. C. van de Hulst, K. Grossman, “Multiple Light Scattering in Planetary Atmospheres” in The Atmospheres of Venus and Mars, J. C. Brandt, M. B. McElroy, Eds. (Gordon and Breach, New York, 1968), pp. 35–55.
  2. J. E. Hansen, Astrophy. J. 155, 565 (1969).
    [Crossref]
  3. J. E. Hansen, J. Atmos. Sci. 26, 478 (1969).
    [Crossref]
  4. J. E. Hansen, Astrophy. J. 158, 337 (1969).
    [Crossref]
  5. R. E. Danielson, D. R. Moore, H. C. van de Hulst, J. Atmos. Sci. 26, 1078 (1969).
    [Crossref]
  6. A. Uesugi, W. M. Irvine, Astrophy. J. 159, 127 (1970).
    [Crossref]
  7. W. M. Irvine, Astrophys. J. 152, 823 (1968).
    [Crossref]
  8. S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
    [Crossref]
  9. M. Neiburger, J. Meteorol. 6, 98 (1949).
    [Crossref]
  10. I. Ruff, R. Koffler, S. Fritz, J. S. Winston, P. K. Rao, J. Atmos. Sci. 25, 323 (1968).
    [Crossref]
  11. V. V. Salomonson, “Anisotropy in Reflected Solar Radiation,” Atmospheric Science Paper No. 128, Colorado State U., Fort Collins, Colo. (1968).
  12. B. Brennan, W. R. Bandeen, Appl. Opt. 9, 405 (1970).
    [Crossref] [PubMed]
  13. G. N. Plass, G. W. Kattawar, Appl. Opt. 7, 361 (1968).
    [Crossref] [PubMed]
  14. G. N. Plass, G. W. Kattawar, Appl. Opt. 7, 415 (1968).
    [Crossref] [PubMed]
  15. G. N. Plass, G. W. Kattawar, Appl. Opt. 7, 699 (1968).
    [Crossref] [PubMed]
  16. G. W. Kattawar, G. N. Plass, Appl. Opt. 7, 869 (1968).
    [Crossref] [PubMed]
  17. G. N. Plass, G. W. Kattawar, Appl. Opt. 8, 2489 (1969).
    [Crossref] [PubMed]
  18. D. Deirmendjian, Appl. Opt. 3, 187 (1964).
    [Crossref]
  19. G. W. Kattawar, G. N. Plass, Appl. Opt. 6, 1377 (1967).
    [Crossref] [PubMed]
  20. G. W. Kattawar, G. N. Plass, Appl. Opt. 7, 1519 (1968).
    [Crossref] [PubMed]

1970 (2)

A. Uesugi, W. M. Irvine, Astrophy. J. 159, 127 (1970).
[Crossref]

B. Brennan, W. R. Bandeen, Appl. Opt. 9, 405 (1970).
[Crossref] [PubMed]

1969 (5)

G. N. Plass, G. W. Kattawar, Appl. Opt. 8, 2489 (1969).
[Crossref] [PubMed]

J. E. Hansen, Astrophy. J. 155, 565 (1969).
[Crossref]

J. E. Hansen, J. Atmos. Sci. 26, 478 (1969).
[Crossref]

J. E. Hansen, Astrophy. J. 158, 337 (1969).
[Crossref]

R. E. Danielson, D. R. Moore, H. C. van de Hulst, J. Atmos. Sci. 26, 1078 (1969).
[Crossref]

1968 (7)

1967 (2)

G. W. Kattawar, G. N. Plass, Appl. Opt. 6, 1377 (1967).
[Crossref] [PubMed]

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[Crossref]

1964 (1)

1949 (1)

M. Neiburger, J. Meteorol. 6, 98 (1949).
[Crossref]

Bandeen, W. R.

Brennan, B.

Danielson, R. E.

R. E. Danielson, D. R. Moore, H. C. van de Hulst, J. Atmos. Sci. 26, 1078 (1969).
[Crossref]

Deirmendjian, D.

Fritz, S.

I. Ruff, R. Koffler, S. Fritz, J. S. Winston, P. K. Rao, J. Atmos. Sci. 25, 323 (1968).
[Crossref]

Grossman, K.

H. C. van de Hulst, K. Grossman, “Multiple Light Scattering in Planetary Atmospheres” in The Atmospheres of Venus and Mars, J. C. Brandt, M. B. McElroy, Eds. (Gordon and Breach, New York, 1968), pp. 35–55.

Hansen, J. E.

J. E. Hansen, Astrophy. J. 155, 565 (1969).
[Crossref]

J. E. Hansen, J. Atmos. Sci. 26, 478 (1969).
[Crossref]

J. E. Hansen, Astrophy. J. 158, 337 (1969).
[Crossref]

Howell, H. B.

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[Crossref]

Irvine, W. M.

A. Uesugi, W. M. Irvine, Astrophy. J. 159, 127 (1970).
[Crossref]

W. M. Irvine, Astrophys. J. 152, 823 (1968).
[Crossref]

Jacobowitz, H.

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[Crossref]

Kattawar, G. W.

Koffler, R.

I. Ruff, R. Koffler, S. Fritz, J. S. Winston, P. K. Rao, J. Atmos. Sci. 25, 323 (1968).
[Crossref]

Moore, D. R.

R. E. Danielson, D. R. Moore, H. C. van de Hulst, J. Atmos. Sci. 26, 1078 (1969).
[Crossref]

Neiburger, M.

M. Neiburger, J. Meteorol. 6, 98 (1949).
[Crossref]

Plass, G. N.

Rao, P. K.

I. Ruff, R. Koffler, S. Fritz, J. S. Winston, P. K. Rao, J. Atmos. Sci. 25, 323 (1968).
[Crossref]

Ruff, I.

I. Ruff, R. Koffler, S. Fritz, J. S. Winston, P. K. Rao, J. Atmos. Sci. 25, 323 (1968).
[Crossref]

Salomonson, V. V.

V. V. Salomonson, “Anisotropy in Reflected Solar Radiation,” Atmospheric Science Paper No. 128, Colorado State U., Fort Collins, Colo. (1968).

Twomey, S.

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[Crossref]

Uesugi, A.

A. Uesugi, W. M. Irvine, Astrophy. J. 159, 127 (1970).
[Crossref]

van de Hulst, H. C.

R. E. Danielson, D. R. Moore, H. C. van de Hulst, J. Atmos. Sci. 26, 1078 (1969).
[Crossref]

H. C. van de Hulst, K. Grossman, “Multiple Light Scattering in Planetary Atmospheres” in The Atmospheres of Venus and Mars, J. C. Brandt, M. B. McElroy, Eds. (Gordon and Breach, New York, 1968), pp. 35–55.

Winston, J. S.

I. Ruff, R. Koffler, S. Fritz, J. S. Winston, P. K. Rao, J. Atmos. Sci. 25, 323 (1968).
[Crossref]

Appl. Opt. (9)

Astrophy. J. (3)

J. E. Hansen, Astrophy. J. 155, 565 (1969).
[Crossref]

J. E. Hansen, Astrophy. J. 158, 337 (1969).
[Crossref]

A. Uesugi, W. M. Irvine, Astrophy. J. 159, 127 (1970).
[Crossref]

Astrophys. J. (1)

W. M. Irvine, Astrophys. J. 152, 823 (1968).
[Crossref]

J. Atmos. Sci. (4)

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[Crossref]

R. E. Danielson, D. R. Moore, H. C. van de Hulst, J. Atmos. Sci. 26, 1078 (1969).
[Crossref]

J. E. Hansen, J. Atmos. Sci. 26, 478 (1969).
[Crossref]

I. Ruff, R. Koffler, S. Fritz, J. S. Winston, P. K. Rao, J. Atmos. Sci. 25, 323 (1968).
[Crossref]

J. Meteorol. (1)

M. Neiburger, J. Meteorol. 6, 98 (1949).
[Crossref]

Other (2)

V. V. Salomonson, “Anisotropy in Reflected Solar Radiation,” Atmospheric Science Paper No. 128, Colorado State U., Fort Collins, Colo. (1968).

H. C. van de Hulst, K. Grossman, “Multiple Light Scattering in Planetary Atmospheres” in The Atmospheres of Venus and Mars, J. C. Brandt, M. B. McElroy, Eds. (Gordon and Breach, New York, 1968), pp. 35–55.

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Figures (10)

Fig. 1
Fig. 1

Reflected and transmitted radiance as a function of cosine (μ) of nadir or zenith angle for haze C and nimbostratus models for cloud optical thickness (τ) of 10, 30, 100 and for surface albedo (A) of 0, 0.2, 0.6, and 1. The cosine (μ0) of the solar zenith angle is −1 (sun at zenith). The incoming solar flux is normalized to unity.

Fig. 2
Fig. 2

Polarization of reflected radiation as a function of μ for haze C and nimbostratus models for τ = 10, 30, 100; A = 0; μ0 = −1.

Fig. 3
Fig. 3

Reflected radiance as a function of μ for μ0 = 0.15, τ = 10, 30, and 100, A = 0 and 1. The results have been averaged over the azimuthal angles in the range 0° < ϕ ≤ 30° or 150° < ϕ ≤ 180°. On all curves the solar horizon is on the left-hand side of the figure and the antisolar horizon is on the right-hand side.

Fig. 4
Fig. 4

Reflected radiance as a function of μ. Same as Fig. 3 except that the range of the azimuthal angle is 30° < ϕ ≤ 60° or 120° < θ ≤ 150°.

Fig. 5
Fig. 5

Reflected radiance as a function of μ. Same as Fig. 3 except that the range of the azimuthal angle is 60° < ϕ ≤ 120°.

Fig. 6
Fig. 6

Polarization of reflected radiation as a function of μ for μ0 = −0.15, τ = 10, 30, and 100, A = 0, and 0° ≤ ϕ < 30° or 150° < ϕ ≤ 180°.

Fig. 7
Fig. 7

Polarization of reflected radiation as a function of μ. Same as Fig. 6 except that the range of the azimuthal angle is 30° < ϕ ≤ 60° or 120° < ϕ ≤ 150°.

Fig. 8
Fig. 8

Polarization of reflected radiation as a function of μ. Same as Fig. 6 except that the range of the azimuthal angle is 60° < ϕ ≤ 120°.

Fig. 9
Fig. 9

Reflected radiance as a function of μ for nimbostratus model, μ0 = −0.55, τ = 30, A = 0, and various ranges of azimuthal angle.

Fig. 10
Fig. 10

Polarization of reflected radiation as a function of μ for nimbostratus model, μ0 = −0.55, τ = 30, A = 0, and various ranges of azimuthal angle.

Tables (1)

Tables Icon

Table I Mean Optical Path, Flux at Lower Boundary for A = 0, and Cloud Albedo for A = 0

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