Abstract

Clear daytime skies persistently display a subtle local maximum of radiance near the astronomical horizon. Spectroradiometry and digital image analysis confirm this maximum's reality, and they show that its angular width and elevation vary with solar elevation, azimuth relative to the Sun, and aerosol optical depth. Many existing models of atmospheric scattering do not generate this near-horizon radiance maximum, but a simple second-order scattering model does, and it reproduces many of the maximum's details.

© 1994 Optical Society of America

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  1. We use radiance and brightness as synonyms in this paper, while recognizing that they are not linearly related. For example, see G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, New York, 1982), pp. 259 and 495.
  2. R. Perez, J. Michalsky, R. Seals, “Modeling sky luminance angular distribution for real sky conditions: experimental evaluation of existing algorithms,” J. Ilium. Eng. Soc. 21, 84–92 (1992).
  3. F. M. F. Siala, M. A. Rosen, F. C. Hooper, “Models for the directional distribution of the diffuse sky radiance,” J. Sol. Energy Eng. 112, 102–109 (1990).
    [CrossRef]
  4. C. R. Prasad, A. K. Inamdar, P. Venkatesh, “Computation of diffuse solar radiation,” Sol. Energy 39, 521–532 (1987).
    [CrossRef]
  5. J. V. Dave, “A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation. Part I: theory,” J. Atmos. Sci. 32, 790–798 (1975).
    [CrossRef]
  6. For basic discussions of optical path length's effect on sky radiance (in a purely molecular atmosphere) see C. F. Bohren, A. B. Fraser, “Colors of the sky,” Phys. Teach. 23, 267–272 (1985) and Ref. 7.
    [CrossRef]
  7. C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
    [CrossRef]
  8. G. Zibordi, K. J. Voss, “Geometrical and spectral distribution of sky radiance: comparison between simulations and field measurements,” Remote Sensing Environ. 27, 343–358 (1989).
    [CrossRef]
  9. R. L. Lee, “What are ‘all the colors of the rainbow’?” Appl. Opt. 30, 3401–3407, 3545 (1991).
    [CrossRef] [PubMed]
  10. D. K. Lynch, P. Schwartz, “Intensity profile of the 22° halo,” J. Opt. Soc. Am. A 2, 584–589 (1985).
    [CrossRef]
  11. A. Deepak, R. R. Adams, “Photography and photographic-photometry of the solar aureole,” Appl. Opt. 22, 1646–1654 (1983).
    [CrossRef] [PubMed]
  12. L. J. B. McArthur, J. E. Hay, “A technique for mapping the distribution of diffuse solar radiation over the sky hemisphere,” J. Appl. Meteorol. 20, 421–429 (1981).
    [CrossRef]
  13. M. A. Rosen, “The angular distribution of diffuse sky radiance: an assessment of the effects of haze,” J. Sol. Energy Eng. 113, 200–205 (1991).
    [CrossRef]
  14. A. W. Harrison, “Directional sky luminance versus cloud cover and solar position,” Sol. Energy 46, 13–19 (1991).
    [CrossRef]
  15. F. C. Hooper, A. P. Brunger, C. S. Chan, “A clear sky model of diffuse sky radiance,” J. Sol. Energy Eng. 109, 9–14 (1987).
    [CrossRef]
  16. C. G. Justus, M. V. Paris, “A model for solar spectral irradiance and radiance at the bottom and top of a cloudless atmosphere,” J. Cli. Appl. Meteorol. 24, 193–205 (1985).
    [CrossRef]
  17. J. V. Dave, “Extensive datasets of the diffuse radiation in realistic atmospheric models with aerosols and common absorbing gases,” Sol. Energy 21, 361–369 (1978).
    [CrossRef]
  18. R. L. Lee, “Colorimetric calibration of a video digitizing system: algorithm and applications,” Col. Res. Appl. 13, 180–186 (1988).
    [CrossRef]
  19. A Photo Research PR-704 spectroradiometer with a nominal 0.5° FOV was used.
  20. D. K. Lynch, “Step brightness changes of distant mountain ridges and their perception,” Appl. Opt. 30, 3508–3513 (1991).
    [CrossRef] [PubMed]
  21. C. F. Bohren, A. B. Fraser, “At what altitude does the horizon cease to be visible?” Am. J. Phys. 54, 222–227 (1986).
    [CrossRef]
  22. E. J. McCartney, Optics of the Atmosphere: Scattering by Molecules and Particles (Wiley, New York, 1976), pp. 136–138.
  23. As in the other models' simulations, all our radiance profiles are monochromatic. The wavelength λ used here is 475 nm, a typical dominant wavelength for the clear sky. This wavelength determines both the solar spectral irradiance and the angular scattering phase function for aerosols.
  24. R. L. Lee, “Twilight and daytime colors of the clear sky,” Appl. Opt. 33, 4629–4638 (1994).
    [CrossRef] [PubMed]
  25. G. E. Shaw, “Sun photometry,” Bull. Am. Meteorol. Soc. 64, 4–10 (1983).
    [CrossRef]

1994

1992

R. Perez, J. Michalsky, R. Seals, “Modeling sky luminance angular distribution for real sky conditions: experimental evaluation of existing algorithms,” J. Ilium. Eng. Soc. 21, 84–92 (1992).

1991

R. L. Lee, “What are ‘all the colors of the rainbow’?” Appl. Opt. 30, 3401–3407, 3545 (1991).
[CrossRef] [PubMed]

M. A. Rosen, “The angular distribution of diffuse sky radiance: an assessment of the effects of haze,” J. Sol. Energy Eng. 113, 200–205 (1991).
[CrossRef]

A. W. Harrison, “Directional sky luminance versus cloud cover and solar position,” Sol. Energy 46, 13–19 (1991).
[CrossRef]

D. K. Lynch, “Step brightness changes of distant mountain ridges and their perception,” Appl. Opt. 30, 3508–3513 (1991).
[CrossRef] [PubMed]

1990

F. M. F. Siala, M. A. Rosen, F. C. Hooper, “Models for the directional distribution of the diffuse sky radiance,” J. Sol. Energy Eng. 112, 102–109 (1990).
[CrossRef]

1989

G. Zibordi, K. J. Voss, “Geometrical and spectral distribution of sky radiance: comparison between simulations and field measurements,” Remote Sensing Environ. 27, 343–358 (1989).
[CrossRef]

1988

R. L. Lee, “Colorimetric calibration of a video digitizing system: algorithm and applications,” Col. Res. Appl. 13, 180–186 (1988).
[CrossRef]

1987

F. C. Hooper, A. P. Brunger, C. S. Chan, “A clear sky model of diffuse sky radiance,” J. Sol. Energy Eng. 109, 9–14 (1987).
[CrossRef]

C. R. Prasad, A. K. Inamdar, P. Venkatesh, “Computation of diffuse solar radiation,” Sol. Energy 39, 521–532 (1987).
[CrossRef]

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

1986

C. F. Bohren, A. B. Fraser, “At what altitude does the horizon cease to be visible?” Am. J. Phys. 54, 222–227 (1986).
[CrossRef]

1985

C. G. Justus, M. V. Paris, “A model for solar spectral irradiance and radiance at the bottom and top of a cloudless atmosphere,” J. Cli. Appl. Meteorol. 24, 193–205 (1985).
[CrossRef]

For basic discussions of optical path length's effect on sky radiance (in a purely molecular atmosphere) see C. F. Bohren, A. B. Fraser, “Colors of the sky,” Phys. Teach. 23, 267–272 (1985) and Ref. 7.
[CrossRef]

D. K. Lynch, P. Schwartz, “Intensity profile of the 22° halo,” J. Opt. Soc. Am. A 2, 584–589 (1985).
[CrossRef]

1983

1981

L. J. B. McArthur, J. E. Hay, “A technique for mapping the distribution of diffuse solar radiation over the sky hemisphere,” J. Appl. Meteorol. 20, 421–429 (1981).
[CrossRef]

1978

J. V. Dave, “Extensive datasets of the diffuse radiation in realistic atmospheric models with aerosols and common absorbing gases,” Sol. Energy 21, 361–369 (1978).
[CrossRef]

1975

J. V. Dave, “A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation. Part I: theory,” J. Atmos. Sci. 32, 790–798 (1975).
[CrossRef]

Adams, R. R.

Bohren, C. F.

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

C. F. Bohren, A. B. Fraser, “At what altitude does the horizon cease to be visible?” Am. J. Phys. 54, 222–227 (1986).
[CrossRef]

For basic discussions of optical path length's effect on sky radiance (in a purely molecular atmosphere) see C. F. Bohren, A. B. Fraser, “Colors of the sky,” Phys. Teach. 23, 267–272 (1985) and Ref. 7.
[CrossRef]

Brunger, A. P.

F. C. Hooper, A. P. Brunger, C. S. Chan, “A clear sky model of diffuse sky radiance,” J. Sol. Energy Eng. 109, 9–14 (1987).
[CrossRef]

Chan, C. S.

F. C. Hooper, A. P. Brunger, C. S. Chan, “A clear sky model of diffuse sky radiance,” J. Sol. Energy Eng. 109, 9–14 (1987).
[CrossRef]

Dave, J. V.

J. V. Dave, “Extensive datasets of the diffuse radiation in realistic atmospheric models with aerosols and common absorbing gases,” Sol. Energy 21, 361–369 (1978).
[CrossRef]

J. V. Dave, “A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation. Part I: theory,” J. Atmos. Sci. 32, 790–798 (1975).
[CrossRef]

Deepak, A.

Fraser, A. B.

C. F. Bohren, A. B. Fraser, “At what altitude does the horizon cease to be visible?” Am. J. Phys. 54, 222–227 (1986).
[CrossRef]

For basic discussions of optical path length's effect on sky radiance (in a purely molecular atmosphere) see C. F. Bohren, A. B. Fraser, “Colors of the sky,” Phys. Teach. 23, 267–272 (1985) and Ref. 7.
[CrossRef]

Harrison, A. W.

A. W. Harrison, “Directional sky luminance versus cloud cover and solar position,” Sol. Energy 46, 13–19 (1991).
[CrossRef]

Hay, J. E.

L. J. B. McArthur, J. E. Hay, “A technique for mapping the distribution of diffuse solar radiation over the sky hemisphere,” J. Appl. Meteorol. 20, 421–429 (1981).
[CrossRef]

Hooper, F. C.

F. M. F. Siala, M. A. Rosen, F. C. Hooper, “Models for the directional distribution of the diffuse sky radiance,” J. Sol. Energy Eng. 112, 102–109 (1990).
[CrossRef]

F. C. Hooper, A. P. Brunger, C. S. Chan, “A clear sky model of diffuse sky radiance,” J. Sol. Energy Eng. 109, 9–14 (1987).
[CrossRef]

Inamdar, A. K.

C. R. Prasad, A. K. Inamdar, P. Venkatesh, “Computation of diffuse solar radiation,” Sol. Energy 39, 521–532 (1987).
[CrossRef]

Justus, C. G.

C. G. Justus, M. V. Paris, “A model for solar spectral irradiance and radiance at the bottom and top of a cloudless atmosphere,” J. Cli. Appl. Meteorol. 24, 193–205 (1985).
[CrossRef]

Lee, R. L.

R. L. Lee, “Twilight and daytime colors of the clear sky,” Appl. Opt. 33, 4629–4638 (1994).
[CrossRef] [PubMed]

R. L. Lee, “What are ‘all the colors of the rainbow’?” Appl. Opt. 30, 3401–3407, 3545 (1991).
[CrossRef] [PubMed]

R. L. Lee, “Colorimetric calibration of a video digitizing system: algorithm and applications,” Col. Res. Appl. 13, 180–186 (1988).
[CrossRef]

Lynch, D. K.

McArthur, L. J. B.

L. J. B. McArthur, J. E. Hay, “A technique for mapping the distribution of diffuse solar radiation over the sky hemisphere,” J. Appl. Meteorol. 20, 421–429 (1981).
[CrossRef]

McCartney, E. J.

E. J. McCartney, Optics of the Atmosphere: Scattering by Molecules and Particles (Wiley, New York, 1976), pp. 136–138.

Michalsky, J.

R. Perez, J. Michalsky, R. Seals, “Modeling sky luminance angular distribution for real sky conditions: experimental evaluation of existing algorithms,” J. Ilium. Eng. Soc. 21, 84–92 (1992).

Paris, M. V.

C. G. Justus, M. V. Paris, “A model for solar spectral irradiance and radiance at the bottom and top of a cloudless atmosphere,” J. Cli. Appl. Meteorol. 24, 193–205 (1985).
[CrossRef]

Perez, R.

R. Perez, J. Michalsky, R. Seals, “Modeling sky luminance angular distribution for real sky conditions: experimental evaluation of existing algorithms,” J. Ilium. Eng. Soc. 21, 84–92 (1992).

Prasad, C. R.

C. R. Prasad, A. K. Inamdar, P. Venkatesh, “Computation of diffuse solar radiation,” Sol. Energy 39, 521–532 (1987).
[CrossRef]

Rosen, M. A.

M. A. Rosen, “The angular distribution of diffuse sky radiance: an assessment of the effects of haze,” J. Sol. Energy Eng. 113, 200–205 (1991).
[CrossRef]

F. M. F. Siala, M. A. Rosen, F. C. Hooper, “Models for the directional distribution of the diffuse sky radiance,” J. Sol. Energy Eng. 112, 102–109 (1990).
[CrossRef]

Schwartz, P.

Seals, R.

R. Perez, J. Michalsky, R. Seals, “Modeling sky luminance angular distribution for real sky conditions: experimental evaluation of existing algorithms,” J. Ilium. Eng. Soc. 21, 84–92 (1992).

Shaw, G. E.

G. E. Shaw, “Sun photometry,” Bull. Am. Meteorol. Soc. 64, 4–10 (1983).
[CrossRef]

Siala, F. M. F.

F. M. F. Siala, M. A. Rosen, F. C. Hooper, “Models for the directional distribution of the diffuse sky radiance,” J. Sol. Energy Eng. 112, 102–109 (1990).
[CrossRef]

Stiles, W. S.

We use radiance and brightness as synonyms in this paper, while recognizing that they are not linearly related. For example, see G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, New York, 1982), pp. 259 and 495.

Venkatesh, P.

C. R. Prasad, A. K. Inamdar, P. Venkatesh, “Computation of diffuse solar radiation,” Sol. Energy 39, 521–532 (1987).
[CrossRef]

Voss, K. J.

G. Zibordi, K. J. Voss, “Geometrical and spectral distribution of sky radiance: comparison between simulations and field measurements,” Remote Sensing Environ. 27, 343–358 (1989).
[CrossRef]

Wyszecki, G.

We use radiance and brightness as synonyms in this paper, while recognizing that they are not linearly related. For example, see G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, New York, 1982), pp. 259 and 495.

Zibordi, G.

G. Zibordi, K. J. Voss, “Geometrical and spectral distribution of sky radiance: comparison between simulations and field measurements,” Remote Sensing Environ. 27, 343–358 (1989).
[CrossRef]

Am. J. Phys.

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

C. F. Bohren, A. B. Fraser, “At what altitude does the horizon cease to be visible?” Am. J. Phys. 54, 222–227 (1986).
[CrossRef]

Appl. Opt.

Bull. Am. Meteorol. Soc.

G. E. Shaw, “Sun photometry,” Bull. Am. Meteorol. Soc. 64, 4–10 (1983).
[CrossRef]

Col. Res. Appl.

R. L. Lee, “Colorimetric calibration of a video digitizing system: algorithm and applications,” Col. Res. Appl. 13, 180–186 (1988).
[CrossRef]

J. Appl. Meteorol.

L. J. B. McArthur, J. E. Hay, “A technique for mapping the distribution of diffuse solar radiation over the sky hemisphere,” J. Appl. Meteorol. 20, 421–429 (1981).
[CrossRef]

J. Atmos. Sci.

J. V. Dave, “A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation. Part I: theory,” J. Atmos. Sci. 32, 790–798 (1975).
[CrossRef]

J. Cli. Appl. Meteorol.

C. G. Justus, M. V. Paris, “A model for solar spectral irradiance and radiance at the bottom and top of a cloudless atmosphere,” J. Cli. Appl. Meteorol. 24, 193–205 (1985).
[CrossRef]

J. Ilium. Eng. Soc.

R. Perez, J. Michalsky, R. Seals, “Modeling sky luminance angular distribution for real sky conditions: experimental evaluation of existing algorithms,” J. Ilium. Eng. Soc. 21, 84–92 (1992).

J. Opt. Soc. Am. A

J. Sol. Energy Eng.

F. M. F. Siala, M. A. Rosen, F. C. Hooper, “Models for the directional distribution of the diffuse sky radiance,” J. Sol. Energy Eng. 112, 102–109 (1990).
[CrossRef]

M. A. Rosen, “The angular distribution of diffuse sky radiance: an assessment of the effects of haze,” J. Sol. Energy Eng. 113, 200–205 (1991).
[CrossRef]

F. C. Hooper, A. P. Brunger, C. S. Chan, “A clear sky model of diffuse sky radiance,” J. Sol. Energy Eng. 109, 9–14 (1987).
[CrossRef]

Phys. Teach.

For basic discussions of optical path length's effect on sky radiance (in a purely molecular atmosphere) see C. F. Bohren, A. B. Fraser, “Colors of the sky,” Phys. Teach. 23, 267–272 (1985) and Ref. 7.
[CrossRef]

Remote Sensing Environ.

G. Zibordi, K. J. Voss, “Geometrical and spectral distribution of sky radiance: comparison between simulations and field measurements,” Remote Sensing Environ. 27, 343–358 (1989).
[CrossRef]

Sol. Energy

J. V. Dave, “Extensive datasets of the diffuse radiation in realistic atmospheric models with aerosols and common absorbing gases,” Sol. Energy 21, 361–369 (1978).
[CrossRef]

C. R. Prasad, A. K. Inamdar, P. Venkatesh, “Computation of diffuse solar radiation,” Sol. Energy 39, 521–532 (1987).
[CrossRef]

A. W. Harrison, “Directional sky luminance versus cloud cover and solar position,” Sol. Energy 46, 13–19 (1991).
[CrossRef]

Other

A Photo Research PR-704 spectroradiometer with a nominal 0.5° FOV was used.

We use radiance and brightness as synonyms in this paper, while recognizing that they are not linearly related. For example, see G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, New York, 1982), pp. 259 and 495.

E. J. McCartney, Optics of the Atmosphere: Scattering by Molecules and Particles (Wiley, New York, 1976), pp. 136–138.

As in the other models' simulations, all our radiance profiles are monochromatic. The wavelength λ used here is 475 nm, a typical dominant wavelength for the clear sky. This wavelength determines both the solar spectral irradiance and the angular scattering phase function for aerosols.

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