Abstract

We investigated the errors associated with the scalar approximation (i.e., an approach that neglects polarization effects) for simulating the intensity of the radiation reflected by an ice cloud. In a case for an optical thickness of τ=1, the relative errors of the scalar approximation are typically between -0.5% and 0.1%. We also investigated the effect of the order of scattering on the degree of linear polarization. It is shown that substantial errors can be introduced by the first-order scattering approximation, and thus, the multiple-scattering effect is essential to an accurate simulation of the polarization configuration of a radiation field. Furthermore, we investigate the effective Mueller matrix pertaining to multiple scattering of light by ice clouds. The effective Mueller matrix is a 4×4 matrix that relates the incident and scattered Stokes parameters. This matrix implicitly contains the effects of all orders of scattering and absorbing events in the entire radiation transfer process. The sensitivity of the (2,1) and (3,1) elements of the effective Mueller matrix to ice crystal shape and size indicates that polarimetric information may be useful for inferring the microphysical properties of ice crystals within ice clouds.

© 2006 Optical Society of America

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References

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    [Crossref]
  4. M. D. King, W.P Menzel, Y. J. Kaufman, D. Tanre, B. C. Gao, S. Platnick, S. A. Ackerman, L. A. Remer, R. Pincus, and P. A. Hubanks, “Cloud and aerosol properties, precipitable water, and profiles of temperature and water vapor from MODIS,” IEEE Trans. Geosci. Remote Sens. 41, 442–458 (2003).
    [Crossref]
  5. T. Nakajima and M. D. King, “Determination of the optical thickness and effective particle radius of clouds from reflected solar radiation measurements. Part I: Theory,” J. Atmos. Sci. 47, 1878–1893 (1990).
    [Crossref]
  6. J. E. Hansen, “Multiple scattering of polarized light in planetary atmospheres. Part 2: Sunlight reflected by terrestrial water clouds,” J. Atmos. Sci.,  28, 1400–1426 (1971).
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    [Crossref]
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    [Crossref]
  12. D. M. Winker, J. Pelon, and M. P. McCormickU. Singh, T. Itabe, and Z. Liu, “The CALIPSO mission: spaceborne lidar for observation of aerosols and clouds,”eds., Proc. SPIE 4893, 1–11 (2002).
    [Crossref]
  13. V. Grano, S. Ubhayakar, J. M. Haas, and C. Schueler, “Introduction, Overview, and Status of the NPOESS Aerosol Polarimetry Sensor (APS),” IEEE/IGARSS 2003 (Toulouse, France, 2003), pp. 374–376.
  14. M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
    [Crossref]
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    [Crossref]
  17. K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer 46, 413–423 (1991).
    [Crossref]
  18. G. W. Kattawar and G. N. Plass, “Radiance and polarization of multiple scattered light from haze and clouds,” Appl. Opt. 7, 1519–1527 (1968).
    [Crossref] [PubMed]
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  21. J. F. de Haan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987).
  22. P. Yang and K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
    [Crossref] [PubMed]
  23. Y. Takano and K. N. Liou, “Solar radiative transfer in cirrus clouds. I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
    [Crossref]
  24. A. Macke, J. Mueller, and E. Rascke, “Single-scattering properties of atmospheric crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
    [Crossref]
  25. C. N. Adams and G. W. Kattawar, “Solutions of the equations of radiative transfer by an invariant imbedding approach,” J. Quant. Spectrosc. Radiat. Transfer 10, 341–356 (1970).
    [Crossref]
  26. M. Hofer and O. Glatter, “Mueller matrix calculations for randomly oriented rotationally symmetric objects with low contrast,” Appl. Opt. 28, 2390–2400 (1989).
    [Crossref]
  27. J. W. Hovenier, “Structure of a general pure Mueller matrix,” Appl. Opt. 33, 8318–8324 (1994).
    [Crossref] [PubMed]
  28. P. Yang, H. Wei, G. W. Kattawar, Y. X. Hu, D. M. Winker, C. A. Hostetler, and B. A. Baum, “Sensitivity of the backscattering Mueller matrix to particle shape and thermodynamic phase,” Appl. Opt. 42, 4389–4395 (2003).
    [Crossref] [PubMed]
  29. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, Cambridge, UK, 2002).

2004 (1)

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[Crossref]

2003 (2)

M. D. King, W.P Menzel, Y. J. Kaufman, D. Tanre, B. C. Gao, S. Platnick, S. A. Ackerman, L. A. Remer, R. Pincus, and P. A. Hubanks, “Cloud and aerosol properties, precipitable water, and profiles of temperature and water vapor from MODIS,” IEEE Trans. Geosci. Remote Sens. 41, 442–458 (2003).
[Crossref]

P. Yang, H. Wei, G. W. Kattawar, Y. X. Hu, D. M. Winker, C. A. Hostetler, and B. A. Baum, “Sensitivity of the backscattering Mueller matrix to particle shape and thermodynamic phase,” Appl. Opt. 42, 4389–4395 (2003).
[Crossref] [PubMed]

2002 (2)

F. M. Breon, J. C. Buries, P. Couvert, P. Y. Deschamps, J. L. Deuze, M. Herman, P. Goloub, M. Leroy, A. Lifermann, C. Moulin, F. Parol, G. Seze, D. Tanre, C. Vanbauce, and M. Vesperini, “Scientific results from the polarization and directionality of the earth’s reflectances (POLDER),” Adv. Space. Res. 30, 2383–2386 (2002).
[Crossref]

D. M. Winker, J. Pelon, and M. P. McCormickU. Singh, T. Itabe, and Z. Liu, “The CALIPSO mission: spaceborne lidar for observation of aerosols and clouds,”eds., Proc. SPIE 4893, 1–11 (2002).
[Crossref]

2000 (1)

B. A. Baum, D. P. Kratz, P. Yang, S. C. Out, Y. Hu, P. F. Soulen, and S. C. Tsay, “Remote sensing of cloud properties using MODIS airborne simulator imagery during SUCCESS, 1,data and models,” J. Geophys. Res. 105, 11767–11780 (2000).
[Crossref]

1999 (1)

1998 (2)

1997 (3)

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J. J. Morcrette, “Second Simulation of the Satellite Signal in the Solar Spectrum, 6S: An overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[Crossref]

M. I. Mishchenko and L. D. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res. 102, 16,989–17,013 (1997).

A. Ben-David, “Mueller matrices and information derived from linear polarization lidar measurements: theory,” Appl. Opt. 37, 2448–2463 (1997).
[Crossref]

1996 (2)

P. Yang and K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
[Crossref] [PubMed]

A. Macke, J. Mueller, and E. Rascke, “Single-scattering properties of atmospheric crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[Crossref]

1994 (1)

1991 (1)

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer 46, 413–423 (1991).
[Crossref]

1990 (1)

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

1989 (2)

M. Hofer and O. Glatter, “Mueller matrix calculations for randomly oriented rotationally symmetric objects with low contrast,” Appl. Opt. 28, 2390–2400 (1989).
[Crossref]

Y. Takano and K. N. Liou, “Solar radiative transfer in cirrus clouds. I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[Crossref]

1987 (1)

J. F. de Haan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987).

1986 (1)

K. N. Liou, “Influence of cirrus clouds on weather and climate processes: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
[Crossref]

1974 (1)

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Science Rev. 16, 527–610 (1974).
[Crossref]

1971 (1)

J. E. Hansen, “Multiple scattering of polarized light in planetary atmospheres. Part 2: Sunlight reflected by terrestrial water clouds,” J. Atmos. Sci.,  28, 1400–1426 (1971).
[Crossref]

1970 (1)

C. N. Adams and G. W. Kattawar, “Solutions of the equations of radiative transfer by an invariant imbedding approach,” J. Quant. Spectrosc. Radiat. Transfer 10, 341–356 (1970).
[Crossref]

1968 (1)

Ackerman, S. A.

M. D. King, W.P Menzel, Y. J. Kaufman, D. Tanre, B. C. Gao, S. Platnick, S. A. Ackerman, L. A. Remer, R. Pincus, and P. A. Hubanks, “Cloud and aerosol properties, precipitable water, and profiles of temperature and water vapor from MODIS,” IEEE Trans. Geosci. Remote Sens. 41, 442–458 (2003).
[Crossref]

Adams, C. N.

C. N. Adams and G. W. Kattawar, “Solutions of the equations of radiative transfer by an invariant imbedding approach,” J. Quant. Spectrosc. Radiat. Transfer 10, 341–356 (1970).
[Crossref]

Baum, B. A.

P. Yang, H. Wei, G. W. Kattawar, Y. X. Hu, D. M. Winker, C. A. Hostetler, and B. A. Baum, “Sensitivity of the backscattering Mueller matrix to particle shape and thermodynamic phase,” Appl. Opt. 42, 4389–4395 (2003).
[Crossref] [PubMed]

B. A. Baum, D. P. Kratz, P. Yang, S. C. Out, Y. Hu, P. F. Soulen, and S. C. Tsay, “Remote sensing of cloud properties using MODIS airborne simulator imagery during SUCCESS, 1,data and models,” J. Geophys. Res. 105, 11767–11780 (2000).
[Crossref]

Ben-David, A.

Bosma, P. B.

J. F. de Haan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987).

Breon, F. M.

F. M. Breon, J. C. Buries, P. Couvert, P. Y. Deschamps, J. L. Deuze, M. Herman, P. Goloub, M. Leroy, A. Lifermann, C. Moulin, F. Parol, G. Seze, D. Tanre, C. Vanbauce, and M. Vesperini, “Scientific results from the polarization and directionality of the earth’s reflectances (POLDER),” Adv. Space. Res. 30, 2383–2386 (2002).
[Crossref]

Burg, R.

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[Crossref]

Buries, J. C.

F. M. Breon, J. C. Buries, P. Couvert, P. Y. Deschamps, J. L. Deuze, M. Herman, P. Goloub, M. Leroy, A. Lifermann, C. Moulin, F. Parol, G. Seze, D. Tanre, C. Vanbauce, and M. Vesperini, “Scientific results from the polarization and directionality of the earth’s reflectances (POLDER),” Adv. Space. Res. 30, 2383–2386 (2002).
[Crossref]

Cairns, B.

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[Crossref]

Cameron, B. D.

Cote, G. L.

Couvert, P.

F. M. Breon, J. C. Buries, P. Couvert, P. Y. Deschamps, J. L. Deuze, M. Herman, P. Goloub, M. Leroy, A. Lifermann, C. Moulin, F. Parol, G. Seze, D. Tanre, C. Vanbauce, and M. Vesperini, “Scientific results from the polarization and directionality of the earth’s reflectances (POLDER),” Adv. Space. Res. 30, 2383–2386 (2002).
[Crossref]

de Haan, J. F.

J. F. de Haan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987).

Deschamps, P. Y.

F. M. Breon, J. C. Buries, P. Couvert, P. Y. Deschamps, J. L. Deuze, M. Herman, P. Goloub, M. Leroy, A. Lifermann, C. Moulin, F. Parol, G. Seze, D. Tanre, C. Vanbauce, and M. Vesperini, “Scientific results from the polarization and directionality of the earth’s reflectances (POLDER),” Adv. Space. Res. 30, 2383–2386 (2002).
[Crossref]

Deuze, J. L.

F. M. Breon, J. C. Buries, P. Couvert, P. Y. Deschamps, J. L. Deuze, M. Herman, P. Goloub, M. Leroy, A. Lifermann, C. Moulin, F. Parol, G. Seze, D. Tanre, C. Vanbauce, and M. Vesperini, “Scientific results from the polarization and directionality of the earth’s reflectances (POLDER),” Adv. Space. Res. 30, 2383–2386 (2002).
[Crossref]

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J. J. Morcrette, “Second Simulation of the Satellite Signal in the Solar Spectrum, 6S: An overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[Crossref]

Evans, K. F.

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer 46, 413–423 (1991).
[Crossref]

Gao, B. C.

M. D. King, W.P Menzel, Y. J. Kaufman, D. Tanre, B. C. Gao, S. Platnick, S. A. Ackerman, L. A. Remer, R. Pincus, and P. A. Hubanks, “Cloud and aerosol properties, precipitable water, and profiles of temperature and water vapor from MODIS,” IEEE Trans. Geosci. Remote Sens. 41, 442–458 (2003).
[Crossref]

Glatter, O.

M. Hofer and O. Glatter, “Mueller matrix calculations for randomly oriented rotationally symmetric objects with low contrast,” Appl. Opt. 28, 2390–2400 (1989).
[Crossref]

Goloub, P.

F. M. Breon, J. C. Buries, P. Couvert, P. Y. Deschamps, J. L. Deuze, M. Herman, P. Goloub, M. Leroy, A. Lifermann, C. Moulin, F. Parol, G. Seze, D. Tanre, C. Vanbauce, and M. Vesperini, “Scientific results from the polarization and directionality of the earth’s reflectances (POLDER),” Adv. Space. Res. 30, 2383–2386 (2002).
[Crossref]

Grano, V.

V. Grano, S. Ubhayakar, J. M. Haas, and C. Schueler, “Introduction, Overview, and Status of the NPOESS Aerosol Polarimetry Sensor (APS),” IEEE/IGARSS 2003 (Toulouse, France, 2003), pp. 374–376.

Haas, J. M.

V. Grano, S. Ubhayakar, J. M. Haas, and C. Schueler, “Introduction, Overview, and Status of the NPOESS Aerosol Polarimetry Sensor (APS),” IEEE/IGARSS 2003 (Toulouse, France, 2003), pp. 374–376.

Hansen, J. E.

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[Crossref]

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Science Rev. 16, 527–610 (1974).
[Crossref]

J. E. Hansen, “Multiple scattering of polarized light in planetary atmospheres. Part 2: Sunlight reflected by terrestrial water clouds,” J. Atmos. Sci.,  28, 1400–1426 (1971).
[Crossref]

Herman, M.

F. M. Breon, J. C. Buries, P. Couvert, P. Y. Deschamps, J. L. Deuze, M. Herman, P. Goloub, M. Leroy, A. Lifermann, C. Moulin, F. Parol, G. Seze, D. Tanre, C. Vanbauce, and M. Vesperini, “Scientific results from the polarization and directionality of the earth’s reflectances (POLDER),” Adv. Space. Res. 30, 2383–2386 (2002).
[Crossref]

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J. J. Morcrette, “Second Simulation of the Satellite Signal in the Solar Spectrum, 6S: An overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[Crossref]

Hofer, M.

M. Hofer and O. Glatter, “Mueller matrix calculations for randomly oriented rotationally symmetric objects with low contrast,” Appl. Opt. 28, 2390–2400 (1989).
[Crossref]

Hostetler, C. A.

Hovenier, J. W.

J. W. Hovenier, “Structure of a general pure Mueller matrix,” Appl. Opt. 33, 8318–8324 (1994).
[Crossref] [PubMed]

J. F. de Haan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987).

Hu, Y.

B. A. Baum, D. P. Kratz, P. Yang, S. C. Out, Y. Hu, P. F. Soulen, and S. C. Tsay, “Remote sensing of cloud properties using MODIS airborne simulator imagery during SUCCESS, 1,data and models,” J. Geophys. Res. 105, 11767–11780 (2000).
[Crossref]

Hu, Y. X.

Hubanks, P. A.

M. D. King, W.P Menzel, Y. J. Kaufman, D. Tanre, B. C. Gao, S. Platnick, S. A. Ackerman, L. A. Remer, R. Pincus, and P. A. Hubanks, “Cloud and aerosol properties, precipitable water, and profiles of temperature and water vapor from MODIS,” IEEE Trans. Geosci. Remote Sens. 41, 442–458 (2003).
[Crossref]

Kattawar, G. W.

Kaufman, Y. J.

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[Crossref]

M. D. King, W.P Menzel, Y. J. Kaufman, D. Tanre, B. C. Gao, S. Platnick, S. A. Ackerman, L. A. Remer, R. Pincus, and P. A. Hubanks, “Cloud and aerosol properties, precipitable water, and profiles of temperature and water vapor from MODIS,” IEEE Trans. Geosci. Remote Sens. 41, 442–458 (2003).
[Crossref]

King, M. D.

M. D. King, W.P Menzel, Y. J. Kaufman, D. Tanre, B. C. Gao, S. Platnick, S. A. Ackerman, L. A. Remer, R. Pincus, and P. A. Hubanks, “Cloud and aerosol properties, precipitable water, and profiles of temperature and water vapor from MODIS,” IEEE Trans. Geosci. Remote Sens. 41, 442–458 (2003).
[Crossref]

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

Kratz, D. P.

B. A. Baum, D. P. Kratz, P. Yang, S. C. Out, Y. Hu, P. F. Soulen, and S. C. Tsay, “Remote sensing of cloud properties using MODIS airborne simulator imagery during SUCCESS, 1,data and models,” J. Geophys. Res. 105, 11767–11780 (2000).
[Crossref]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, Cambridge, UK, 2002).

Leroy, M.

F. M. Breon, J. C. Buries, P. Couvert, P. Y. Deschamps, J. L. Deuze, M. Herman, P. Goloub, M. Leroy, A. Lifermann, C. Moulin, F. Parol, G. Seze, D. Tanre, C. Vanbauce, and M. Vesperini, “Scientific results from the polarization and directionality of the earth’s reflectances (POLDER),” Adv. Space. Res. 30, 2383–2386 (2002).
[Crossref]

Lifermann, A.

F. M. Breon, J. C. Buries, P. Couvert, P. Y. Deschamps, J. L. Deuze, M. Herman, P. Goloub, M. Leroy, A. Lifermann, C. Moulin, F. Parol, G. Seze, D. Tanre, C. Vanbauce, and M. Vesperini, “Scientific results from the polarization and directionality of the earth’s reflectances (POLDER),” Adv. Space. Res. 30, 2383–2386 (2002).
[Crossref]

Liou, K. N.

P. Yang and K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
[Crossref] [PubMed]

Y. Takano and K. N. Liou, “Solar radiative transfer in cirrus clouds. I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[Crossref]

K. N. Liou, “Influence of cirrus clouds on weather and climate processes: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
[Crossref]

Lynch, D. K.

D. K. Lynch, K. Sassen, D. E. Starr, and G. L. Stephens, Cirrus (Oxford University Press, 2002).

Macke, A.

A. Macke, J. Mueller, and E. Rascke, “Single-scattering properties of atmospheric crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[Crossref]

Martins, V.

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[Crossref]

McCormick, M. P.

D. M. Winker, J. Pelon, and M. P. McCormickU. Singh, T. Itabe, and Z. Liu, “The CALIPSO mission: spaceborne lidar for observation of aerosols and clouds,”eds., Proc. SPIE 4893, 1–11 (2002).
[Crossref]

McCormick, N. J.

Mehrubeoglu, M.

Menzel, W.P

M. D. King, W.P Menzel, Y. J. Kaufman, D. Tanre, B. C. Gao, S. Platnick, S. A. Ackerman, L. A. Remer, R. Pincus, and P. A. Hubanks, “Cloud and aerosol properties, precipitable water, and profiles of temperature and water vapor from MODIS,” IEEE Trans. Geosci. Remote Sens. 41, 442–458 (2003).
[Crossref]

Mishchenko, M. I.

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[Crossref]

M. I. Mishchenko and L. D. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res. 102, 16,989–17,013 (1997).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, Cambridge, UK, 2002).

Morcrette, J. J.

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J. J. Morcrette, “Second Simulation of the Satellite Signal in the Solar Spectrum, 6S: An overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[Crossref]

Moulin, C.

F. M. Breon, J. C. Buries, P. Couvert, P. Y. Deschamps, J. L. Deuze, M. Herman, P. Goloub, M. Leroy, A. Lifermann, C. Moulin, F. Parol, G. Seze, D. Tanre, C. Vanbauce, and M. Vesperini, “Scientific results from the polarization and directionality of the earth’s reflectances (POLDER),” Adv. Space. Res. 30, 2383–2386 (2002).
[Crossref]

Mueller, J.

A. Macke, J. Mueller, and E. Rascke, “Single-scattering properties of atmospheric crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[Crossref]

Nakajima, T.

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

Out, S. C.

B. A. Baum, D. P. Kratz, P. Yang, S. C. Out, Y. Hu, P. F. Soulen, and S. C. Tsay, “Remote sensing of cloud properties using MODIS airborne simulator imagery during SUCCESS, 1,data and models,” J. Geophys. Res. 105, 11767–11780 (2000).
[Crossref]

Parol, F.

F. M. Breon, J. C. Buries, P. Couvert, P. Y. Deschamps, J. L. Deuze, M. Herman, P. Goloub, M. Leroy, A. Lifermann, C. Moulin, F. Parol, G. Seze, D. Tanre, C. Vanbauce, and M. Vesperini, “Scientific results from the polarization and directionality of the earth’s reflectances (POLDER),” Adv. Space. Res. 30, 2383–2386 (2002).
[Crossref]

Pelon, J.

D. M. Winker, J. Pelon, and M. P. McCormickU. Singh, T. Itabe, and Z. Liu, “The CALIPSO mission: spaceborne lidar for observation of aerosols and clouds,”eds., Proc. SPIE 4893, 1–11 (2002).
[Crossref]

Pincus, R.

M. D. King, W.P Menzel, Y. J. Kaufman, D. Tanre, B. C. Gao, S. Platnick, S. A. Ackerman, L. A. Remer, R. Pincus, and P. A. Hubanks, “Cloud and aerosol properties, precipitable water, and profiles of temperature and water vapor from MODIS,” IEEE Trans. Geosci. Remote Sens. 41, 442–458 (2003).
[Crossref]

Plass, G. N.

Platnick, S.

M. D. King, W.P Menzel, Y. J. Kaufman, D. Tanre, B. C. Gao, S. Platnick, S. A. Ackerman, L. A. Remer, R. Pincus, and P. A. Hubanks, “Cloud and aerosol properties, precipitable water, and profiles of temperature and water vapor from MODIS,” IEEE Trans. Geosci. Remote Sens. 41, 442–458 (2003).
[Crossref]

Pomraning, G. C.

Rakovic, M. J.

Rascke, E.

A. Macke, J. Mueller, and E. Rascke, “Single-scattering properties of atmospheric crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[Crossref]

Rastegar, S.

Remer, L. A.

M. D. King, W.P Menzel, Y. J. Kaufman, D. Tanre, B. C. Gao, S. Platnick, S. A. Ackerman, L. A. Remer, R. Pincus, and P. A. Hubanks, “Cloud and aerosol properties, precipitable water, and profiles of temperature and water vapor from MODIS,” IEEE Trans. Geosci. Remote Sens. 41, 442–458 (2003).
[Crossref]

Sassen, K.

D. K. Lynch, K. Sassen, D. E. Starr, and G. L. Stephens, Cirrus (Oxford University Press, 2002).

Schueler, C.

V. Grano, S. Ubhayakar, J. M. Haas, and C. Schueler, “Introduction, Overview, and Status of the NPOESS Aerosol Polarimetry Sensor (APS),” IEEE/IGARSS 2003 (Toulouse, France, 2003), pp. 374–376.

Seze, G.

F. M. Breon, J. C. Buries, P. Couvert, P. Y. Deschamps, J. L. Deuze, M. Herman, P. Goloub, M. Leroy, A. Lifermann, C. Moulin, F. Parol, G. Seze, D. Tanre, C. Vanbauce, and M. Vesperini, “Scientific results from the polarization and directionality of the earth’s reflectances (POLDER),” Adv. Space. Res. 30, 2383–2386 (2002).
[Crossref]

Shettle, E. P.

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[Crossref]

Soulen, P. F.

B. A. Baum, D. P. Kratz, P. Yang, S. C. Out, Y. Hu, P. F. Soulen, and S. C. Tsay, “Remote sensing of cloud properties using MODIS airborne simulator imagery during SUCCESS, 1,data and models,” J. Geophys. Res. 105, 11767–11780 (2000).
[Crossref]

Starr, D. E.

D. K. Lynch, K. Sassen, D. E. Starr, and G. L. Stephens, Cirrus (Oxford University Press, 2002).

Stephens, G. L.

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer 46, 413–423 (1991).
[Crossref]

D. K. Lynch, K. Sassen, D. E. Starr, and G. L. Stephens, Cirrus (Oxford University Press, 2002).

Takano, Y.

Y. Takano and K. N. Liou, “Solar radiative transfer in cirrus clouds. I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[Crossref]

Tanre, D.

M. D. King, W.P Menzel, Y. J. Kaufman, D. Tanre, B. C. Gao, S. Platnick, S. A. Ackerman, L. A. Remer, R. Pincus, and P. A. Hubanks, “Cloud and aerosol properties, precipitable water, and profiles of temperature and water vapor from MODIS,” IEEE Trans. Geosci. Remote Sens. 41, 442–458 (2003).
[Crossref]

F. M. Breon, J. C. Buries, P. Couvert, P. Y. Deschamps, J. L. Deuze, M. Herman, P. Goloub, M. Leroy, A. Lifermann, C. Moulin, F. Parol, G. Seze, D. Tanre, C. Vanbauce, and M. Vesperini, “Scientific results from the polarization and directionality of the earth’s reflectances (POLDER),” Adv. Space. Res. 30, 2383–2386 (2002).
[Crossref]

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J. J. Morcrette, “Second Simulation of the Satellite Signal in the Solar Spectrum, 6S: An overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[Crossref]

Travis, L. D.

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[Crossref]

M. I. Mishchenko and L. D. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res. 102, 16,989–17,013 (1997).

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Science Rev. 16, 527–610 (1974).
[Crossref]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, Cambridge, UK, 2002).

Tsay, S. C.

B. A. Baum, D. P. Kratz, P. Yang, S. C. Out, Y. Hu, P. F. Soulen, and S. C. Tsay, “Remote sensing of cloud properties using MODIS airborne simulator imagery during SUCCESS, 1,data and models,” J. Geophys. Res. 105, 11767–11780 (2000).
[Crossref]

Ubhayakar, S.

V. Grano, S. Ubhayakar, J. M. Haas, and C. Schueler, “Introduction, Overview, and Status of the NPOESS Aerosol Polarimetry Sensor (APS),” IEEE/IGARSS 2003 (Toulouse, France, 2003), pp. 374–376.

van de Hulst, H. C.

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

Vanbauce, C.

F. M. Breon, J. C. Buries, P. Couvert, P. Y. Deschamps, J. L. Deuze, M. Herman, P. Goloub, M. Leroy, A. Lifermann, C. Moulin, F. Parol, G. Seze, D. Tanre, C. Vanbauce, and M. Vesperini, “Scientific results from the polarization and directionality of the earth’s reflectances (POLDER),” Adv. Space. Res. 30, 2383–2386 (2002).
[Crossref]

Vermote, E. F.

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J. J. Morcrette, “Second Simulation of the Satellite Signal in the Solar Spectrum, 6S: An overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[Crossref]

Vesperini, M.

F. M. Breon, J. C. Buries, P. Couvert, P. Y. Deschamps, J. L. Deuze, M. Herman, P. Goloub, M. Leroy, A. Lifermann, C. Moulin, F. Parol, G. Seze, D. Tanre, C. Vanbauce, and M. Vesperini, “Scientific results from the polarization and directionality of the earth’s reflectances (POLDER),” Adv. Space. Res. 30, 2383–2386 (2002).
[Crossref]

Wang, L. V.

Wei, H.

Winker, D. M.

P. Yang, H. Wei, G. W. Kattawar, Y. X. Hu, D. M. Winker, C. A. Hostetler, and B. A. Baum, “Sensitivity of the backscattering Mueller matrix to particle shape and thermodynamic phase,” Appl. Opt. 42, 4389–4395 (2003).
[Crossref] [PubMed]

D. M. Winker, J. Pelon, and M. P. McCormickU. Singh, T. Itabe, and Z. Liu, “The CALIPSO mission: spaceborne lidar for observation of aerosols and clouds,”eds., Proc. SPIE 4893, 1–11 (2002).
[Crossref]

Yang, P.

Adv. Space. Res. (1)

F. M. Breon, J. C. Buries, P. Couvert, P. Y. Deschamps, J. L. Deuze, M. Herman, P. Goloub, M. Leroy, A. Lifermann, C. Moulin, F. Parol, G. Seze, D. Tanre, C. Vanbauce, and M. Vesperini, “Scientific results from the polarization and directionality of the earth’s reflectances (POLDER),” Adv. Space. Res. 30, 2383–2386 (2002).
[Crossref]

Appl. Opt. (8)

Astron. Astrophys. (1)

J. F. de Haan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987).

IEEE Trans. Geosci. Remote Sens. (2)

M. D. King, W.P Menzel, Y. J. Kaufman, D. Tanre, B. C. Gao, S. Platnick, S. A. Ackerman, L. A. Remer, R. Pincus, and P. A. Hubanks, “Cloud and aerosol properties, precipitable water, and profiles of temperature and water vapor from MODIS,” IEEE Trans. Geosci. Remote Sens. 41, 442–458 (2003).
[Crossref]

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J. J. Morcrette, “Second Simulation of the Satellite Signal in the Solar Spectrum, 6S: An overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[Crossref]

J. Atmos. Sci. (4)

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

J. E. Hansen, “Multiple scattering of polarized light in planetary atmospheres. Part 2: Sunlight reflected by terrestrial water clouds,” J. Atmos. Sci.,  28, 1400–1426 (1971).
[Crossref]

Y. Takano and K. N. Liou, “Solar radiative transfer in cirrus clouds. I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[Crossref]

A. Macke, J. Mueller, and E. Rascke, “Single-scattering properties of atmospheric crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[Crossref]

J. Geophys. Res. (2)

B. A. Baum, D. P. Kratz, P. Yang, S. C. Out, Y. Hu, P. F. Soulen, and S. C. Tsay, “Remote sensing of cloud properties using MODIS airborne simulator imagery during SUCCESS, 1,data and models,” J. Geophys. Res. 105, 11767–11780 (2000).
[Crossref]

M. I. Mishchenko and L. D. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res. 102, 16,989–17,013 (1997).

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transfer (3)

C. N. Adams and G. W. Kattawar, “Solutions of the equations of radiative transfer by an invariant imbedding approach,” J. Quant. Spectrosc. Radiat. Transfer 10, 341–356 (1970).
[Crossref]

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer 46, 413–423 (1991).
[Crossref]

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[Crossref]

Mon. Weather Rev. (1)

K. N. Liou, “Influence of cirrus clouds on weather and climate processes: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
[Crossref]

Proc. SPIE (1)

D. M. Winker, J. Pelon, and M. P. McCormickU. Singh, T. Itabe, and Z. Liu, “The CALIPSO mission: spaceborne lidar for observation of aerosols and clouds,”eds., Proc. SPIE 4893, 1–11 (2002).
[Crossref]

Space Science Rev. (1)

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Science Rev. 16, 527–610 (1974).
[Crossref]

Other (4)

V. Grano, S. Ubhayakar, J. M. Haas, and C. Schueler, “Introduction, Overview, and Status of the NPOESS Aerosol Polarimetry Sensor (APS),” IEEE/IGARSS 2003 (Toulouse, France, 2003), pp. 374–376.

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

D. K. Lynch, K. Sassen, D. E. Starr, and G. L. Stephens, Cirrus (Oxford University Press, 2002).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, Cambridge, UK, 2002).

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Figures (6)

Fig. 1.
Fig. 1.

Six nonzero elements of the single-scattering phase matrix at a wavelength of λ=0.66 µm for randomly oriented hexagonal columns with aspect ratios of 2a/L=20µm/75µm and 2a/L=80µm/300µm and randomly oriented hexagonal plates with an aspect ratio of 2a/L=300µm/80µm. L and a are, respectively, the length and radius of the cylinder circumscribing an ice crystal.

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Fig. 2.
Fig. 2.

Same as Fig. 1 except for λ=2.13 µm.

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Fig. 3.
Fig. 3.

The vector intensity and relative error of intensity (REOI) of hexagonal columns with aspect ratio of 2a/L=80µm/300µm and hexagonal plates with aspect ratio of 2a/L=300µm/80µm for two wavelengths, 0.66 µm and 2.13 µm. The optical thickness of the ice cloud assumed in the computation is τ=1.

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Fig. 4.
Fig. 4.

The comparisons of the Degree of linear polarization (DLP) from the first-order scattering approximation with that including all orders of scattering for hexagonal columns and plates.

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Fig. 5.
Fig. 5.

Effective Mueller matrices for hexagonal columns with an aspect ratio of 2a/L=80µm/300µm and hexagonal plates with an aspect ratio of 2a/L=300µm/80µm at a wavelength of λ=0.66 µm.

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Fig. 6.
Fig. 6.

Effective Mueller matrix elements M11, M21/M11, M31/M11 and M41/M11 for hexagonal column with aspect ratios of 2a/L=80µm/300µm and 2a/L=20µm/75µm and hexagonal plates with an aspect ratio of 2a/L=300µm/80µm at a wavelength of λ=2.13 µm.

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Equations (7)

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

[ I Q U V ] = [ E l E l * + E r E r * E l E l * − E r E r * E l E r * + E r E l * i ( E l E r * − E r E l * ) ] .
[ I s Q s U s V s ] = M [ I 0 Q 0 U 0 V 0 ] ,
M = [ M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 ] .
M = [ I 1 I 2 − I 1 I 3 − I 1 I 4 − I 1 Q 1 Q 2 − Q 1 Q 3 − Q 1 Q 4 − Q 1 U 1 U 2 − U 1 U 3 − U 1 U 4 − U 1 V 1 V 2 − V 1 V 3 − V 1 V 4 − V 1 ] ,
P = [ P 11 P 12 0 0 P 12 P 22 0 0 0 0 P 33 − P 34 0 0 P 34 P 44 ]
REOI = I sca − I vec I vec × 100 % ,
DLP = − Q ⁄ I ,

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