Abstract

We propose the physically based parameterization of the radiative characteristics of liquid-water clouds as functions of the wavelength, effective radius, and refractive index of particles, liquid-water path, ground albedo, and solar and observation angles. The formulas obtained are based on the approximate analytical solutions of the radiative transfer equation for optically thick, weakly absorbing layers and the geometrical optics approximation for local optical characteristics of cloud media. The accuracy of the approximate formulas was studied with an exact radiative transfer code. The relative error of the approximate formula for the reflection function at nadir observations was less then 15% for an optical thickness larger than 10 and a single-scattering albedo larger than 0.95.

© 1998 Optical Society of America

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  1. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).
  2. I. N. Minin, Radiative Transfer in Planetary Atmospheres (Nauka, Moscow, 1988).
  3. K. N. Liou, Radiation and Cloud Processes in the Atmosphere (Oxford U. Press, New York, 1992).
  4. E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).
  5. A. Arking, J. D. Childs, “Retrieval of cloud cover parameters from multispectral satellite images,” J. Appl. Meteorol. 24, 322–333 (1985).
    [CrossRef]
  6. P. Damiano, P. Chylek, “Shortwave radiative properties of clouds: numerical study,” J. Atmos. Sci. 51, 1223–1233 (1994).
    [CrossRef]
  7. K. L. Coulson, Polarization and Intensity of Light in the Atmosphere (Deepak, Hampton, Va., 1988).
  8. T. Nakajima, T. Nakajima, “Wide-area determination of cloud microphysical properties from NOAA AVHRR measurements for FIRE and ASTEX regions,” J. Atmos. Sci. 52, 4043–4059 (1996).
    [CrossRef]
  9. S. Platnick, S. Twomey, “Determining the susceptibility of cloud albedo to changes in droplet concentration with the Advanced Very High Resolution Radiometer,” J. Appl. Meteorol. 33, 334–347 (1994).
    [CrossRef]
  10. G. V. Rozenberg, “Optical characteristics of thick weakly absorbing scattering layers,” Dokl. Akad. Nauk SSSR 145, 775–777 (1962).
  11. M. D. King, “Determination of the scaled optical thickness of clouds from reflected solar radiation measurements,” J. Atmos. Sci. 44, 1734–1751 (1987).
    [CrossRef]
  12. V. V. Sobolev, Light Scattering in Planetary Atmospheres (Nauka, Moscow, 1972).
  13. H. C. van de Hulst, Multiple Light Scattering (Tables, Formulas and Applications) (Academic, New York, 1980).
  14. T. Nakajima, M. D. King, “Asymptotic theory for optically thick layers: application to the discrete ordinates methods,” Appl. Opt. 31, 7669–7683 (1992).
    [CrossRef] [PubMed]
  15. M. D. King, “A method for determining the single scattering albedo of clouds through observation of the internal scattered radiation field,” J. Atmos. Sci. 38, 2031–2044 (1981).
    [CrossRef]
  16. M. D. King, R. Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
    [CrossRef]
  17. T. Nakajima, M. D. King, “Determination of the optical thickness and effective particle radius of clouds from reflected solar radiation measurements. I. Theory,” J. Atmos. Sci. 47, 1878–1893 (1990).
    [CrossRef]
  18. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  19. E. G. Yanovizkii, Light Scattering in Nonuniform Atmospheres (Springer-Verlag, New York, 1997).
  20. R. F. Calahan, W. Ridgway, W. J. Wiscombe, T. L. Bell, J. B. Snider, “The albedo of fractal stratocumulus clouds,” J. Atmos. Sci. 51, 2434–2455 (1994).
    [CrossRef]
  21. G. L. Stephens, “Radiation profiles in extended water clouds: parameterization schemes,” J. Atmos. Sci. 35, 2123–2132 (1978).
    [CrossRef]
  22. A. A. Kokhanovsky, E. P. Zege, “Local optical properties of spherical polydispersions: simple approximations,” Appl. Opt. 34, 5513–5519 (1995).
    [CrossRef] [PubMed]
  23. A. A. Kokhanovsky, E. P. Zege, “The determination of the effective radius of drops and liquid water path of water clouds from satellite measurements,” Earth Res. Space 2, 33–44 (1996).
  24. H. C. van de Hulst, Light scattering by Small Particles (Dover, New York, 1981).

1996 (2)

T. Nakajima, T. Nakajima, “Wide-area determination of cloud microphysical properties from NOAA AVHRR measurements for FIRE and ASTEX regions,” J. Atmos. Sci. 52, 4043–4059 (1996).
[CrossRef]

A. A. Kokhanovsky, E. P. Zege, “The determination of the effective radius of drops and liquid water path of water clouds from satellite measurements,” Earth Res. Space 2, 33–44 (1996).

1995 (1)

1994 (3)

R. F. Calahan, W. Ridgway, W. J. Wiscombe, T. L. Bell, J. B. Snider, “The albedo of fractal stratocumulus clouds,” J. Atmos. Sci. 51, 2434–2455 (1994).
[CrossRef]

S. Platnick, S. Twomey, “Determining the susceptibility of cloud albedo to changes in droplet concentration with the Advanced Very High Resolution Radiometer,” J. Appl. Meteorol. 33, 334–347 (1994).
[CrossRef]

P. Damiano, P. Chylek, “Shortwave radiative properties of clouds: numerical study,” J. Atmos. Sci. 51, 1223–1233 (1994).
[CrossRef]

1992 (1)

1990 (1)

T. Nakajima, M. D. King, “Determination of the optical thickness and effective particle radius of clouds from reflected solar radiation measurements. I. Theory,” J. Atmos. Sci. 47, 1878–1893 (1990).
[CrossRef]

1987 (1)

M. D. King, “Determination of the scaled optical thickness of clouds from reflected solar radiation measurements,” J. Atmos. Sci. 44, 1734–1751 (1987).
[CrossRef]

1986 (1)

M. D. King, R. Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
[CrossRef]

1985 (1)

A. Arking, J. D. Childs, “Retrieval of cloud cover parameters from multispectral satellite images,” J. Appl. Meteorol. 24, 322–333 (1985).
[CrossRef]

1981 (1)

M. D. King, “A method for determining the single scattering albedo of clouds through observation of the internal scattered radiation field,” J. Atmos. Sci. 38, 2031–2044 (1981).
[CrossRef]

1978 (1)

G. L. Stephens, “Radiation profiles in extended water clouds: parameterization schemes,” J. Atmos. Sci. 35, 2123–2132 (1978).
[CrossRef]

1962 (1)

G. V. Rozenberg, “Optical characteristics of thick weakly absorbing scattering layers,” Dokl. Akad. Nauk SSSR 145, 775–777 (1962).

Arking, A.

A. Arking, J. D. Childs, “Retrieval of cloud cover parameters from multispectral satellite images,” J. Appl. Meteorol. 24, 322–333 (1985).
[CrossRef]

Bell, T. L.

R. F. Calahan, W. Ridgway, W. J. Wiscombe, T. L. Bell, J. B. Snider, “The albedo of fractal stratocumulus clouds,” J. Atmos. Sci. 51, 2434–2455 (1994).
[CrossRef]

Calahan, R. F.

R. F. Calahan, W. Ridgway, W. J. Wiscombe, T. L. Bell, J. B. Snider, “The albedo of fractal stratocumulus clouds,” J. Atmos. Sci. 51, 2434–2455 (1994).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Childs, J. D.

A. Arking, J. D. Childs, “Retrieval of cloud cover parameters from multispectral satellite images,” J. Appl. Meteorol. 24, 322–333 (1985).
[CrossRef]

Chylek, P.

P. Damiano, P. Chylek, “Shortwave radiative properties of clouds: numerical study,” J. Atmos. Sci. 51, 1223–1233 (1994).
[CrossRef]

Coulson, K. L.

K. L. Coulson, Polarization and Intensity of Light in the Atmosphere (Deepak, Hampton, Va., 1988).

Damiano, P.

P. Damiano, P. Chylek, “Shortwave radiative properties of clouds: numerical study,” J. Atmos. Sci. 51, 1223–1233 (1994).
[CrossRef]

Deirmendjian, D.

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

Harshvardhan, R.

M. D. King, R. Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
[CrossRef]

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).

Katsev, I. L.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).

King, M. D.

T. Nakajima, M. D. King, “Asymptotic theory for optically thick layers: application to the discrete ordinates methods,” Appl. Opt. 31, 7669–7683 (1992).
[CrossRef] [PubMed]

T. Nakajima, M. D. King, “Determination of the optical thickness and effective particle radius of clouds from reflected solar radiation measurements. I. Theory,” J. Atmos. Sci. 47, 1878–1893 (1990).
[CrossRef]

M. D. King, “Determination of the scaled optical thickness of clouds from reflected solar radiation measurements,” J. Atmos. Sci. 44, 1734–1751 (1987).
[CrossRef]

M. D. King, R. Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
[CrossRef]

M. D. King, “A method for determining the single scattering albedo of clouds through observation of the internal scattered radiation field,” J. Atmos. Sci. 38, 2031–2044 (1981).
[CrossRef]

Kokhanovsky, A. A.

A. A. Kokhanovsky, E. P. Zege, “The determination of the effective radius of drops and liquid water path of water clouds from satellite measurements,” Earth Res. Space 2, 33–44 (1996).

A. A. Kokhanovsky, E. P. Zege, “Local optical properties of spherical polydispersions: simple approximations,” Appl. Opt. 34, 5513–5519 (1995).
[CrossRef] [PubMed]

Liou, K. N.

K. N. Liou, Radiation and Cloud Processes in the Atmosphere (Oxford U. Press, New York, 1992).

Minin, I. N.

I. N. Minin, Radiative Transfer in Planetary Atmospheres (Nauka, Moscow, 1988).

Nakajima, T.

T. Nakajima, T. Nakajima, “Wide-area determination of cloud microphysical properties from NOAA AVHRR measurements for FIRE and ASTEX regions,” J. Atmos. Sci. 52, 4043–4059 (1996).
[CrossRef]

T. Nakajima, T. Nakajima, “Wide-area determination of cloud microphysical properties from NOAA AVHRR measurements for FIRE and ASTEX regions,” J. Atmos. Sci. 52, 4043–4059 (1996).
[CrossRef]

T. Nakajima, M. D. King, “Asymptotic theory for optically thick layers: application to the discrete ordinates methods,” Appl. Opt. 31, 7669–7683 (1992).
[CrossRef] [PubMed]

T. Nakajima, M. D. King, “Determination of the optical thickness and effective particle radius of clouds from reflected solar radiation measurements. I. Theory,” J. Atmos. Sci. 47, 1878–1893 (1990).
[CrossRef]

Platnick, S.

S. Platnick, S. Twomey, “Determining the susceptibility of cloud albedo to changes in droplet concentration with the Advanced Very High Resolution Radiometer,” J. Appl. Meteorol. 33, 334–347 (1994).
[CrossRef]

Ridgway, W.

R. F. Calahan, W. Ridgway, W. J. Wiscombe, T. L. Bell, J. B. Snider, “The albedo of fractal stratocumulus clouds,” J. Atmos. Sci. 51, 2434–2455 (1994).
[CrossRef]

Rozenberg, G. V.

G. V. Rozenberg, “Optical characteristics of thick weakly absorbing scattering layers,” Dokl. Akad. Nauk SSSR 145, 775–777 (1962).

Snider, J. B.

R. F. Calahan, W. Ridgway, W. J. Wiscombe, T. L. Bell, J. B. Snider, “The albedo of fractal stratocumulus clouds,” J. Atmos. Sci. 51, 2434–2455 (1994).
[CrossRef]

Sobolev, V. V.

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Nauka, Moscow, 1972).

Stephens, G. L.

G. L. Stephens, “Radiation profiles in extended water clouds: parameterization schemes,” J. Atmos. Sci. 35, 2123–2132 (1978).
[CrossRef]

Twomey, S.

S. Platnick, S. Twomey, “Determining the susceptibility of cloud albedo to changes in droplet concentration with the Advanced Very High Resolution Radiometer,” J. Appl. Meteorol. 33, 334–347 (1994).
[CrossRef]

van de Hulst, H. C.

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

H. C. van de Hulst, Light scattering by Small Particles (Dover, New York, 1981).

Wiscombe, W. J.

R. F. Calahan, W. Ridgway, W. J. Wiscombe, T. L. Bell, J. B. Snider, “The albedo of fractal stratocumulus clouds,” J. Atmos. Sci. 51, 2434–2455 (1994).
[CrossRef]

Yanovizkii, E. G.

E. G. Yanovizkii, Light Scattering in Nonuniform Atmospheres (Springer-Verlag, New York, 1997).

Zege, E. P.

A. A. Kokhanovsky, E. P. Zege, “The determination of the effective radius of drops and liquid water path of water clouds from satellite measurements,” Earth Res. Space 2, 33–44 (1996).

A. A. Kokhanovsky, E. P. Zege, “Local optical properties of spherical polydispersions: simple approximations,” Appl. Opt. 34, 5513–5519 (1995).
[CrossRef] [PubMed]

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).

Appl. Opt. (2)

Dokl. Akad. Nauk SSSR (1)

G. V. Rozenberg, “Optical characteristics of thick weakly absorbing scattering layers,” Dokl. Akad. Nauk SSSR 145, 775–777 (1962).

Earth Res. Space (1)

A. A. Kokhanovsky, E. P. Zege, “The determination of the effective radius of drops and liquid water path of water clouds from satellite measurements,” Earth Res. Space 2, 33–44 (1996).

J. Appl. Meteorol. (2)

A. Arking, J. D. Childs, “Retrieval of cloud cover parameters from multispectral satellite images,” J. Appl. Meteorol. 24, 322–333 (1985).
[CrossRef]

S. Platnick, S. Twomey, “Determining the susceptibility of cloud albedo to changes in droplet concentration with the Advanced Very High Resolution Radiometer,” J. Appl. Meteorol. 33, 334–347 (1994).
[CrossRef]

J. Atmos. Sci. (8)

T. Nakajima, T. Nakajima, “Wide-area determination of cloud microphysical properties from NOAA AVHRR measurements for FIRE and ASTEX regions,” J. Atmos. Sci. 52, 4043–4059 (1996).
[CrossRef]

P. Damiano, P. Chylek, “Shortwave radiative properties of clouds: numerical study,” J. Atmos. Sci. 51, 1223–1233 (1994).
[CrossRef]

M. D. King, “Determination of the scaled optical thickness of clouds from reflected solar radiation measurements,” J. Atmos. Sci. 44, 1734–1751 (1987).
[CrossRef]

M. D. King, “A method for determining the single scattering albedo of clouds through observation of the internal scattered radiation field,” J. Atmos. Sci. 38, 2031–2044 (1981).
[CrossRef]

M. D. King, R. Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
[CrossRef]

T. Nakajima, M. D. King, “Determination of the optical thickness and effective particle radius of clouds from reflected solar radiation measurements. I. Theory,” J. Atmos. Sci. 47, 1878–1893 (1990).
[CrossRef]

R. F. Calahan, W. Ridgway, W. J. Wiscombe, T. L. Bell, J. B. Snider, “The albedo of fractal stratocumulus clouds,” J. Atmos. Sci. 51, 2434–2455 (1994).
[CrossRef]

G. L. Stephens, “Radiation profiles in extended water clouds: parameterization schemes,” J. Atmos. Sci. 35, 2123–2132 (1978).
[CrossRef]

Other (10)

H. C. van de Hulst, Light scattering by Small Particles (Dover, New York, 1981).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

E. G. Yanovizkii, Light Scattering in Nonuniform Atmospheres (Springer-Verlag, New York, 1997).

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Nauka, Moscow, 1972).

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

K. L. Coulson, Polarization and Intensity of Light in the Atmosphere (Deepak, Hampton, Va., 1988).

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

I. N. Minin, Radiative Transfer in Planetary Atmospheres (Nauka, Moscow, 1988).

K. N. Liou, Radiation and Cloud Processes in the Atmosphere (Oxford U. Press, New York, 1992).

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).

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