Abstract

Abstract: A practical model for determining the time-dependent lidar attenuated backscattering coefficient β was developed for application to global lidar data. An analytical expression for the high-order phase function was introduced to reduce computational cost for simulating the angular distribution of the multiple scattering irradiance. The decay rate of the multiple scattering backscattered irradiance was expressed by incorporating the dependence on the scattering angle and the scattering order based on the path integral approach. The estimated β over time and the actual range showed good agreement with Monte Carlo simulations for vertically homogeneous and inhomogeneous cloud profiles, resulting in about 15% mean relative error corresponding to 4 times improved accuracy against the Ornstein–Fürth Gaussian approximation method.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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    [Crossref]
  18. R. J. Hogan, “Fast Lidar and Radar Multiple-Scattering Models. Part I: Small-Angle Scattering Using the Photon Variance–Covariance Method,” J. Atmos. Sci. 65(12), 3621–3635 (2008).
    [Crossref]
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    [Crossref]
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    [Crossref]
  21. H. W. Lewis, “Multiple scattering in an infinite medium,” Phys. Rev. 78(5), 526–529 (1950).
    [Crossref]
  22. L. T. Perelman, J. Wu, I. Itzkan, and M. S. Feld, “Photon Migration in Turbid Media Using Path Integrals,” Phys. Rev. Lett. 72(9), 1341–1344 (1994).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  27. H. Okamoto, S. Iwasaki, M. Yasui, H. Horie, H. Kuroiwa, and H. Kumagai, “An algorithm for retrieval of cloud microphysics using 95-GHz cloud radar and lidar,” J. Geophys. Res. 108(4226), D7 (2003).
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    [Crossref]
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  33. K. Sato, Research Institute for Applied Mechanics, Kyushu University, Kasuga, Fukuoka 816–8580, H. Okamoto, and H. Ishimoto are preparing a manuscript to be called “Modeling the depolarization of space-borne lidar signals.”

2016 (2)

2015 (1)

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

2011 (1)

K. Sato and H. Okamoto, “Refinement of global ice microphysics using spaceborne active sensors,” J. Geophys. Res. 116(D20), D20202 (2011).
[Crossref]

2010 (2)

R. Yoshida, H. Okamoto, Y. Hagihara, and H. Ishimoto, “Global analysis of cloud phase and ice crystal orientation from Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) data using attenuated backscattering and depolarization ratio,” J. Geophys. Res. 115, D00H32 (2010).
[Crossref]

H. Okamoto, K. Sato, and Y. Hagihara, “Global analysis of ice microphysics from CloudSat and CALIPSO: Incorporation of specular reflection in lidar signals,” J. Geophys. Res. 115(D22), D22209 (2010).
[Crossref]

2009 (2)

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

H. Iwabuchi and T. Suzuki, “Fast and accurate radiance calculations using truncation approximation for anisotropic scattering phase functions,” J. Quant. Spectrosc. Radiat. Transf. 110(17), 1926–1939 (2009).
[Crossref]

2008 (3)

A. B. Davis, “Multiple-scattering lidar from both sides of the clouds: Addressing internal structure,” J. Geophys. Res. 113(D14), D14S10 (2008).
[Crossref]

R. J. Hogan, “Fast Lidar and Radar Multiple-Scattering Models. Part I: Small-Angle Scattering Using the Photon Variance–Covariance Method,” J. Atmos. Sci. 65(12), 3621–3635 (2008).
[Crossref]

R. J. Hogan and A. Battaglia, “Fast Lidar and Radar Multiple-Scattering Models. Part II: Wide-Angle Scattering Using the Time-Dependent Two-Stream Approximation,” J. Atmos. Sci. 65(12), 3636–3651 (2008).
[Crossref]

2007 (1)

Y. Hu, “Depolarization ratio-effective lidar ratio relation: Theoretical basis for space lidar cloud phase discrimination,” Geophys. Res. Lett. 34(11), L11812 (2007).
[Crossref]

2005 (2)

S. Bony and J.-L. Dufresne, “Marine boundary layer clouds at the heart of tropical cloud feedback uncertainties in climate models,” Geophys. Res. Lett. 32(20), L20806 (2005).
[Crossref]

R. F. Cahalan, M. McGill, and J. Kolasinski, “THOR-cloud thickness from offbeam lidar returns,” J. Atmos. Ocean. Technol. 22(6), 605–627 (2005).
[Crossref]

2003 (1)

H. Okamoto, S. Iwasaki, M. Yasui, H. Horie, H. Kuroiwa, and H. Kumagai, “An algorithm for retrieval of cloud microphysics using 95-GHz cloud radar and lidar,” J. Geophys. Res. 108(4226), D7 (2003).

2002 (2)

H. Ishimoto and K. Masuda, “A Monte Carlo approach for the calculation of polarized light: application to an incident narrow beam,” J. Quant. Spectrosc. Radiat. Transf. 72(4), 467–483 (2002).
[Crossref]

L. R. Bissonnette, G. Roy, L. Poutier, S. G. Cober, and G. A. Isaac, “Multiple-scattering lidar retrieval method: tests on Monte Carlo simulations and comparisons with in situ measurements,” Appl. Opt. 41(30), 6307–6324 (2002).
[Crossref] [PubMed]

1999 (1)

1998 (1)

1997 (1)

1996 (2)

L. R. Bissonnette, “Multiple-scattering lidar equation,” Appl. Opt. 35(33), 6449–6465 (1996).
[Crossref] [PubMed]

E. P. Zege and L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transf. 55(1), 19–31 (1996).

1994 (1)

L. T. Perelman, J. Wu, I. Itzkan, and M. S. Feld, “Photon Migration in Turbid Media Using Path Integrals,” Phys. Rev. Lett. 72(9), 1341–1344 (1994).
[Crossref] [PubMed]

1993 (1)

J. M. Fernández-Varea, R. Mayol, J. Baró, and F. Salvat, “On the theory and simulation of multiple elastic scattering of electrons,” Nucl. Instrum. Methods Phys. Res. B 73(4), 447–473 (1993).
[Crossref]

1981 (1)

C. M. Platt, “Remote sounding of High Clouds. III: Monte Carlo Calculations of Multiple-Scattered Lidar Returns,” J. Atmos. Sci. 38(1), 156–167 (1981).
[Crossref]

1969 (1)

J. E. Hansen, “Absorption-line formation in a scattering planetary atmosphere: A test of Van de Hulst’s similarity relations,” Astrophys. J. 158, 337–349 (1969).
[Crossref]

1950 (1)

H. W. Lewis, “Multiple scattering in an infinite medium,” Phys. Rev. 78(5), 526–529 (1950).
[Crossref]

1949 (1)

R. P. Feynman, “Space-Time Approach to Quantum Electrodynamics,” Phys. Rev. 76(6), 769 (1949).
[Crossref]

1940 (1)

S. Goudsmit and J. L. Saunderson, “Multiple Scattering of Electrons,” Phys. Rev. 57(1), 24–29 (1940).
[Crossref]

Austin, R. T.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Bacmeister, J.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Barker, H. W.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Baró, J.

J. M. Fernández-Varea, R. Mayol, J. Baró, and F. Salvat, “On the theory and simulation of multiple elastic scattering of electrons,” Nucl. Instrum. Methods Phys. Res. B 73(4), 447–473 (1993).
[Crossref]

Bastille, C.

Battaglia, A.

R. J. Hogan and A. Battaglia, “Fast Lidar and Radar Multiple-Scattering Models. Part II: Wide-Angle Scattering Using the Time-Dependent Two-Stream Approximation,” J. Atmos. Sci. 65(12), 3636–3651 (2008).
[Crossref]

Beljaars, A.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Bissonnette, L.

Bissonnette, L. R.

Bony, S.

S. Bony and J.-L. Dufresne, “Marine boundary layer clouds at the heart of tropical cloud feedback uncertainties in climate models,” Geophys. Res. Lett. 32(20), L20806 (2005).
[Crossref]

Cahalan, R. F.

R. F. Cahalan, M. McGill, and J. Kolasinski, “THOR-cloud thickness from offbeam lidar returns,” J. Atmos. Ocean. Technol. 22(6), 605–627 (2005).
[Crossref]

Ceccaldi, M.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Cesana, G.

J. E. Kay, T. L’Ecuyer, H. Chepfer, N. Loeb, A. Morrison, and G. Cesana, “Recent Advances in Arctic Cloud and Climate Research,” Curr. Clim. Change Rep. 2(4), 159–169 (2016).
[Crossref]

Chaikovskaya, L. I.

E. P. Zege and L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transf. 55(1), 19–31 (1996).

Chepfer, H.

J. E. Kay, T. L’Ecuyer, H. Chepfer, N. Loeb, A. Morrison, and G. Cesana, “Recent Advances in Arctic Cloud and Climate Research,” Curr. Clim. Change Rep. 2(4), 159–169 (2016).
[Crossref]

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Chern, J.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Clerbaux, N.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Cober, S. G.

Cole, J.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Davis, A. B.

A. B. Davis, “Multiple-scattering lidar from both sides of the clouds: Addressing internal structure,” J. Geophys. Res. 113(D14), D14S10 (2008).
[Crossref]

Delanoë, J.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Domenech, C.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Donovan, D. P.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Dufresne, J.-L.

S. Bony and J.-L. Dufresne, “Marine boundary layer clouds at the heart of tropical cloud feedback uncertainties in climate models,” Geophys. Res. Lett. 32(20), L20806 (2005).
[Crossref]

Eloranta, E. W.

Feld, M. S.

L. T. Perelman, J. Wu, I. Itzkan, and M. S. Feld, “Photon Migration in Turbid Media Using Path Integrals,” Phys. Rev. Lett. 72(9), 1341–1344 (1994).
[Crossref] [PubMed]

Fernández-Varea, J. M.

J. M. Fernández-Varea, R. Mayol, J. Baró, and F. Salvat, “On the theory and simulation of multiple elastic scattering of electrons,” Nucl. Instrum. Methods Phys. Res. B 73(4), 447–473 (1993).
[Crossref]

Feynman, R. P.

R. P. Feynman, “Space-Time Approach to Quantum Electrodynamics,” Phys. Rev. 76(6), 769 (1949).
[Crossref]

Flamant, P. H.

Fujikawa, M.

Fukuda, S.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Genio, A. D.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Goudsmit, S.

S. Goudsmit and J. L. Saunderson, “Multiple Scattering of Electrons,” Phys. Rev. 57(1), 24–29 (1940).
[Crossref]

Hagihara, Y.

R. Yoshida, H. Okamoto, Y. Hagihara, and H. Ishimoto, “Global analysis of cloud phase and ice crystal orientation from Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) data using attenuated backscattering and depolarization ratio,” J. Geophys. Res. 115, D00H32 (2010).
[Crossref]

H. Okamoto, K. Sato, and Y. Hagihara, “Global analysis of ice microphysics from CloudSat and CALIPSO: Incorporation of specular reflection in lidar signals,” J. Geophys. Res. 115(D22), D22209 (2010).
[Crossref]

Hansen, J. E.

J. E. Hansen, “Absorption-line formation in a scattering planetary atmosphere: A test of Van de Hulst’s similarity relations,” Astrophys. J. 158, 337–349 (1969).
[Crossref]

Hirakata, M.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Hogan, R. J.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

R. J. Hogan, “Fast Lidar and Radar Multiple-Scattering Models. Part I: Small-Angle Scattering Using the Photon Variance–Covariance Method,” J. Atmos. Sci. 65(12), 3621–3635 (2008).
[Crossref]

R. J. Hogan and A. Battaglia, “Fast Lidar and Radar Multiple-Scattering Models. Part II: Wide-Angle Scattering Using the Time-Dependent Two-Stream Approximation,” J. Atmos. Sci. 65(12), 3636–3651 (2008).
[Crossref]

Horie, H.

H. Okamoto, S. Iwasaki, M. Yasui, H. Horie, H. Kuroiwa, and H. Kumagai, “An algorithm for retrieval of cloud microphysics using 95-GHz cloud radar and lidar,” J. Geophys. Res. 108(4226), D7 (2003).

Hu, Y.

Y. Hu, “Depolarization ratio-effective lidar ratio relation: Theoretical basis for space lidar cloud phase discrimination,” Geophys. Res. Lett. 34(11), L11812 (2007).
[Crossref]

Huenerbein, A.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Illingworth, A. J.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Isaac, G. A.

Ishimoto, H.

R. Yoshida, H. Okamoto, Y. Hagihara, and H. Ishimoto, “Global analysis of cloud phase and ice crystal orientation from Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) data using attenuated backscattering and depolarization ratio,” J. Geophys. Res. 115, D00H32 (2010).
[Crossref]

H. Ishimoto and K. Masuda, “A Monte Carlo approach for the calculation of polarized light: application to an incident narrow beam,” J. Quant. Spectrosc. Radiat. Transf. 72(4), 467–483 (2002).
[Crossref]

Itzkan, I.

L. T. Perelman, J. Wu, I. Itzkan, and M. S. Feld, “Photon Migration in Turbid Media Using Path Integrals,” Phys. Rev. Lett. 72(9), 1341–1344 (1994).
[Crossref] [PubMed]

Iwabuchi, H.

H. Iwabuchi and T. Suzuki, “Fast and accurate radiance calculations using truncation approximation for anisotropic scattering phase functions,” J. Quant. Spectrosc. Radiat. Transf. 110(17), 1926–1939 (2009).
[Crossref]

Iwasaki, S.

H. Okamoto, S. Iwasaki, M. Yasui, H. Horie, H. Kuroiwa, and H. Kumagai, “An algorithm for retrieval of cloud microphysics using 95-GHz cloud radar and lidar,” J. Geophys. Res. 108(4226), D7 (2003).

Jiang, J. H.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Jin, Y.

Kay, J. E.

J. E. Kay, T. L’Ecuyer, H. Chepfer, N. Loeb, A. Morrison, and G. Cesana, “Recent Advances in Arctic Cloud and Climate Research,” Curr. Clim. Change Rep. 2(4), 159–169 (2016).
[Crossref]

Kolasinski, J.

R. F. Cahalan, M. McGill, and J. Kolasinski, “THOR-cloud thickness from offbeam lidar returns,” J. Atmos. Ocean. Technol. 22(6), 605–627 (2005).
[Crossref]

Kollias, P.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Kuang, Z.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Kubota, T.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Kumagai, H.

H. Okamoto, S. Iwasaki, M. Yasui, H. Horie, H. Kuroiwa, and H. Kumagai, “An algorithm for retrieval of cloud microphysics using 95-GHz cloud radar and lidar,” J. Geophys. Res. 108(4226), D7 (2003).

Kuroiwa, H.

H. Okamoto, S. Iwasaki, M. Yasui, H. Horie, H. Kuroiwa, and H. Kumagai, “An algorithm for retrieval of cloud microphysics using 95-GHz cloud radar and lidar,” J. Geophys. Res. 108(4226), D7 (2003).

L’Ecuyer, T.

J. E. Kay, T. L’Ecuyer, H. Chepfer, N. Loeb, A. Morrison, and G. Cesana, “Recent Advances in Arctic Cloud and Climate Research,” Curr. Clim. Change Rep. 2(4), 159–169 (2016).
[Crossref]

Lewis, H. W.

H. W. Lewis, “Multiple scattering in an infinite medium,” Phys. Rev. 78(5), 526–529 (1950).
[Crossref]

Li, J.-L. F.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Loeb, N.

J. E. Kay, T. L’Ecuyer, H. Chepfer, N. Loeb, A. Morrison, and G. Cesana, “Recent Advances in Arctic Cloud and Climate Research,” Curr. Clim. Change Rep. 2(4), 159–169 (2016).
[Crossref]

Makino, T.

Masuda, K.

H. Ishimoto and K. Masuda, “A Monte Carlo approach for the calculation of polarized light: application to an incident narrow beam,” J. Quant. Spectrosc. Radiat. Transf. 72(4), 467–483 (2002).
[Crossref]

Mayol, R.

J. M. Fernández-Varea, R. Mayol, J. Baró, and F. Salvat, “On the theory and simulation of multiple elastic scattering of electrons,” Nucl. Instrum. Methods Phys. Res. B 73(4), 447–473 (1993).
[Crossref]

McGill, M.

R. F. Cahalan, M. McGill, and J. Kolasinski, “THOR-cloud thickness from offbeam lidar returns,” J. Atmos. Ocean. Technol. 22(6), 605–627 (2005).
[Crossref]

Meng, H.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Minnis, P.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Morrison, A.

J. E. Kay, T. L’Ecuyer, H. Chepfer, N. Loeb, A. Morrison, and G. Cesana, “Recent Advances in Arctic Cloud and Climate Research,” Curr. Clim. Change Rep. 2(4), 159–169 (2016).
[Crossref]

Nakajima, T.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Nakajima, T. Y.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Nicolas, F. O.

Nishizawa, T.

H. Okamoto, K. Sato, T. Nishizawa, N. Sugimoto, T. Makino, Y. Jin, A. Shimizu, T. Takano, and M. Fujikawa, “Development of a multiple-field-of-view multiple-scattering polarization lidar: comparison with cloud radar,” Opt. Express 24(26), 30053–30067 (2016).
[Crossref] [PubMed]

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Ohno, Y.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Okamoto, H.

H. Okamoto, K. Sato, T. Nishizawa, N. Sugimoto, T. Makino, Y. Jin, A. Shimizu, T. Takano, and M. Fujikawa, “Development of a multiple-field-of-view multiple-scattering polarization lidar: comparison with cloud radar,” Opt. Express 24(26), 30053–30067 (2016).
[Crossref] [PubMed]

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

K. Sato and H. Okamoto, “Refinement of global ice microphysics using spaceborne active sensors,” J. Geophys. Res. 116(D20), D20202 (2011).
[Crossref]

R. Yoshida, H. Okamoto, Y. Hagihara, and H. Ishimoto, “Global analysis of cloud phase and ice crystal orientation from Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) data using attenuated backscattering and depolarization ratio,” J. Geophys. Res. 115, D00H32 (2010).
[Crossref]

H. Okamoto, K. Sato, and Y. Hagihara, “Global analysis of ice microphysics from CloudSat and CALIPSO: Incorporation of specular reflection in lidar signals,” J. Geophys. Res. 115(D22), D22209 (2010).
[Crossref]

H. Okamoto, S. Iwasaki, M. Yasui, H. Horie, H. Kuroiwa, and H. Kumagai, “An algorithm for retrieval of cloud microphysics using 95-GHz cloud radar and lidar,” J. Geophys. Res. 108(4226), D7 (2003).

Oki, R.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Perelman, L. T.

L. T. Perelman, J. Wu, I. Itzkan, and M. S. Feld, “Photon Migration in Turbid Media Using Path Integrals,” Phys. Rev. Lett. 72(9), 1341–1344 (1994).
[Crossref] [PubMed]

Platnick, S.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Platt, C. M.

C. M. Platt, “Remote sounding of High Clouds. III: Monte Carlo Calculations of Multiple-Scattered Lidar Returns,” J. Atmos. Sci. 38(1), 156–167 (1981).
[Crossref]

Poutier, L.

Rossow, W. B.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Roy, G.

Salvat, F.

J. M. Fernández-Varea, R. Mayol, J. Baró, and F. Salvat, “On the theory and simulation of multiple elastic scattering of electrons,” Nucl. Instrum. Methods Phys. Res. B 73(4), 447–473 (1993).
[Crossref]

Sato, K.

H. Okamoto, K. Sato, T. Nishizawa, N. Sugimoto, T. Makino, Y. Jin, A. Shimizu, T. Takano, and M. Fujikawa, “Development of a multiple-field-of-view multiple-scattering polarization lidar: comparison with cloud radar,” Opt. Express 24(26), 30053–30067 (2016).
[Crossref] [PubMed]

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

K. Sato and H. Okamoto, “Refinement of global ice microphysics using spaceborne active sensors,” J. Geophys. Res. 116(D20), D20202 (2011).
[Crossref]

H. Okamoto, K. Sato, and Y. Hagihara, “Global analysis of ice microphysics from CloudSat and CALIPSO: Incorporation of specular reflection in lidar signals,” J. Geophys. Res. 115(D22), D22209 (2010).
[Crossref]

Satoh, M.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Saunderson, J. L.

S. Goudsmit and J. L. Saunderson, “Multiple Scattering of Electrons,” Phys. Rev. 57(1), 24–29 (1940).
[Crossref]

Shephard, M. W.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Shimizu, A.

Stephens, G. L.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Sugimoto, N.

Sun-Mack, S.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Suzuki, T.

H. Iwabuchi and T. Suzuki, “Fast and accurate radiance calculations using truncation approximation for anisotropic scattering phase functions,” J. Quant. Spectrosc. Radiat. Transf. 110(17), 1926–1939 (2009).
[Crossref]

Szedung, T.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Takano, T.

Tao, W. K.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Tompkins, A. M.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Vallée, G.

van Zadelhoff, G.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Vane, D. G.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Velázquez-Blázquez, A.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Waliser, D. E.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Walker, C.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Wandinger, U.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Wehr, T.

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Woods, C. P.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Wu, D.

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

Wu, J.

L. T. Perelman, J. Wu, I. Itzkan, and M. S. Feld, “Photon Migration in Turbid Media Using Path Integrals,” Phys. Rev. Lett. 72(9), 1341–1344 (1994).
[Crossref] [PubMed]

Yasui, M.

H. Okamoto, S. Iwasaki, M. Yasui, H. Horie, H. Kuroiwa, and H. Kumagai, “An algorithm for retrieval of cloud microphysics using 95-GHz cloud radar and lidar,” J. Geophys. Res. 108(4226), D7 (2003).

Yoshida, R.

R. Yoshida, H. Okamoto, Y. Hagihara, and H. Ishimoto, “Global analysis of cloud phase and ice crystal orientation from Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) data using attenuated backscattering and depolarization ratio,” J. Geophys. Res. 115, D00H32 (2010).
[Crossref]

Zege, E. P.

E. P. Zege and L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transf. 55(1), 19–31 (1996).

Appl. Opt. (5)

Astrophys. J. (1)

J. E. Hansen, “Absorption-line formation in a scattering planetary atmosphere: A test of Van de Hulst’s similarity relations,” Astrophys. J. 158, 337–349 (1969).
[Crossref]

Bull. Am. Meteorol. Soc. (1)

A. J. Illingworth, H. W. Barker, A. Beljaars, M. Ceccaldi, H. Chepfer, N. Clerbaux, J. Cole, J. Delanoë, C. Domenech, D. P. Donovan, S. Fukuda, M. Hirakata, R. J. Hogan, A. Huenerbein, P. Kollias, T. Kubota, T. Nakajima, T. Y. Nakajima, T. Nishizawa, Y. Ohno, H. Okamoto, R. Oki, K. Sato, M. Satoh, M. W. Shephard, A. Velázquez-Blázquez, U. Wandinger, T. Wehr, and G. van Zadelhoff, “The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation,” Bull. Am. Meteorol. Soc. 96, 1311–1332 (2015).

Curr. Clim. Change Rep. (1)

J. E. Kay, T. L’Ecuyer, H. Chepfer, N. Loeb, A. Morrison, and G. Cesana, “Recent Advances in Arctic Cloud and Climate Research,” Curr. Clim. Change Rep. 2(4), 159–169 (2016).
[Crossref]

Geophys. Res. Lett. (2)

Y. Hu, “Depolarization ratio-effective lidar ratio relation: Theoretical basis for space lidar cloud phase discrimination,” Geophys. Res. Lett. 34(11), L11812 (2007).
[Crossref]

S. Bony and J.-L. Dufresne, “Marine boundary layer clouds at the heart of tropical cloud feedback uncertainties in climate models,” Geophys. Res. Lett. 32(20), L20806 (2005).
[Crossref]

J. Atmos. Ocean. Technol. (1)

R. F. Cahalan, M. McGill, and J. Kolasinski, “THOR-cloud thickness from offbeam lidar returns,” J. Atmos. Ocean. Technol. 22(6), 605–627 (2005).
[Crossref]

J. Atmos. Sci. (3)

R. J. Hogan, “Fast Lidar and Radar Multiple-Scattering Models. Part I: Small-Angle Scattering Using the Photon Variance–Covariance Method,” J. Atmos. Sci. 65(12), 3621–3635 (2008).
[Crossref]

R. J. Hogan and A. Battaglia, “Fast Lidar and Radar Multiple-Scattering Models. Part II: Wide-Angle Scattering Using the Time-Dependent Two-Stream Approximation,” J. Atmos. Sci. 65(12), 3636–3651 (2008).
[Crossref]

C. M. Platt, “Remote sounding of High Clouds. III: Monte Carlo Calculations of Multiple-Scattered Lidar Returns,” J. Atmos. Sci. 38(1), 156–167 (1981).
[Crossref]

J. Geophys. Res. (6)

R. Yoshida, H. Okamoto, Y. Hagihara, and H. Ishimoto, “Global analysis of cloud phase and ice crystal orientation from Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) data using attenuated backscattering and depolarization ratio,” J. Geophys. Res. 115, D00H32 (2010).
[Crossref]

H. Okamoto, K. Sato, and Y. Hagihara, “Global analysis of ice microphysics from CloudSat and CALIPSO: Incorporation of specular reflection in lidar signals,” J. Geophys. Res. 115(D22), D22209 (2010).
[Crossref]

K. Sato and H. Okamoto, “Refinement of global ice microphysics using spaceborne active sensors,” J. Geophys. Res. 116(D20), D20202 (2011).
[Crossref]

A. B. Davis, “Multiple-scattering lidar from both sides of the clouds: Addressing internal structure,” J. Geophys. Res. 113(D14), D14S10 (2008).
[Crossref]

D. E. Waliser, J.-L. F. Li, C. P. Woods, R. T. Austin, J. Bacmeister, J. Chern, A. D. Genio, J. H. Jiang, Z. Kuang, H. Meng, P. Minnis, S. Platnick, W. B. Rossow, G. L. Stephens, S. Sun-Mack, T. Szedung, W. K. Tao, A. M. Tompkins, D. G. Vane, C. Walker, and D. Wu, “Cloud ice: A climate model challenge with signs and expectations of progress,” J. Geophys. Res. 114, D00A21 (2009).

H. Okamoto, S. Iwasaki, M. Yasui, H. Horie, H. Kuroiwa, and H. Kumagai, “An algorithm for retrieval of cloud microphysics using 95-GHz cloud radar and lidar,” J. Geophys. Res. 108(4226), D7 (2003).

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

H. Ishimoto and K. Masuda, “A Monte Carlo approach for the calculation of polarized light: application to an incident narrow beam,” J. Quant. Spectrosc. Radiat. Transf. 72(4), 467–483 (2002).
[Crossref]

H. Iwabuchi and T. Suzuki, “Fast and accurate radiance calculations using truncation approximation for anisotropic scattering phase functions,” J. Quant. Spectrosc. Radiat. Transf. 110(17), 1926–1939 (2009).
[Crossref]

E. P. Zege and L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transf. 55(1), 19–31 (1996).

Nucl. Instrum. Methods Phys. Res. B (1)

J. M. Fernández-Varea, R. Mayol, J. Baró, and F. Salvat, “On the theory and simulation of multiple elastic scattering of electrons,” Nucl. Instrum. Methods Phys. Res. B 73(4), 447–473 (1993).
[Crossref]

Opt. Express (1)

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[Crossref]

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[Crossref]

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[Crossref]

Phys. Rev. Lett. (1)

L. T. Perelman, J. Wu, I. Itzkan, and M. S. Feld, “Photon Migration in Turbid Media Using Path Integrals,” Phys. Rev. Lett. 72(9), 1341–1344 (1994).
[Crossref] [PubMed]

Other (4)

K. N. Liou, “An Introduction to Atmospheric Radiation,” (Academic Press, 2002).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

K. Sato, Research Institute for Applied Mechanics, Kyushu University, Kasuga, Fukuoka 816–8580, H. Okamoto, and H. Ishimoto are preparing a manuscript to be called “Modeling the depolarization of space-borne lidar signals.”

K. Sato, H. Okamoto, and H. Ishimoto, “Modeling Lidar Multiple Scattering,” EPJ Web of Conferences 119, 21005 (2016).
[Crossref]

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Figures (15)

Fig. 1
Fig. 1 Schematic of the cloud area, the satellite position and the foot print sizes of the beam and field-of-view at cloud position.
Fig. 2
Fig. 2 Relation between Iss (black), Ims,tot,on (blue) and Ims,off (green). The direction of the arrows indicates the sources and sinks. Since all of the backscattered irradiances are defined by the maximum z-axis value of the photon trajectory, the arrows are all looking downstream. The z-t dependences of I are indicated by the numbers in parentheses.
Fig. 3
Fig. 3 (a.) Schematic of the components of Itot(t3):① Iss, ② Ims,tot,on and ③,④,⑤ Ims,off from different cloud layers, but observed at the same time 2t3. (b.) The components ①-⑤ shown in (a) are modeled by introducing the geometrical scattering angles Θon and Θoff that represent the photon trajectories. The returning paths of I are abbreviated by simple arrows.
Fig. 4
Fig. 4 Example of Θon and Θoff in Eqs. (9,10 and 11). Θon and Θoff are determined by the geometrical lengths,①, ②and,③,④,⑤, respectively. The fraction of Iss(t1) scattered at z1 in the direction 0<Θ<Θon(t3) becomes the source for Ims,tot,on(t3). Ims,off (z4, t6) and Ims,off (z4, t7) are the time delayed returns from z4 at different times t6 and t7, respectively. The fraction of Ims,tot,on(t3) scattered at z3 within the angle Θoff,3 →4(t5)<Θ<Θoff,3 →4(t6) and Θoff,3 →4(t6)<Θ<Θoff,3 →4(t7) becomes the source for Ims,off (z4, t6) and Ims,off (z4, t7), respectively. The returning paths of I are abbreviated by simple arrows.
Fig. 5
Fig. 5 Normalized n-th order scattering phase function at 532nm for a liquid particle with an effective radius of 10µm. The colors of the lines indicate the scattering order n = 1,3,5,7,10,20,30; a transition in the scattering phase function from a forward-peaked shape to a more isotropic shape was observed. Enlarged figure of the scattering phase functions for n = 20 and 30 is shown in the upper right corner.
Fig. 6
Fig. 6 (a) Geometry for defining θds,j in Eq. (22). (b) Schematic of the dependence of θd,j on scattering order j when θmax = 〈θj in Eq. (23). For satellites, θds,j is approximately constant for homogenous profiles, but varies for inhomogeneous profiles.
Fig. 7
Fig. 7 Vertical profile of (a.) cloud reff and (b.) σext for Cases 1, 2, and 3.
Fig. 8
Fig. 8 (a, b) Comparison between the physical model (PM) and the Monte Carlo (MC) simulation for Case 1 for β(z,t). In (a) and (b), the colors indicate β(z,t) from cloud layers 1~7 and 8~17, respectively. (c.) The contributions of ss, ss + ms,on and ms,off to βtot. (d) Same as (a,b) but with reduced number of lines.
Fig. 9
Fig. 9 Comparison between βtot obtained by the physical model (PM: red solid and dotted lines,) and the Monte Carlo simulation (MC: solid black and gray lines) for (a) Case 1 as well as simulation with a 10-times larger field of view angle (10θfov) and for (b) water cloud cases with the same reff = 10 µm as (a) but with extinction coefficient of 3km−1 and 40km−1 at the 532nm CALIPSO lidar specification. In (a) and (b), βtot simulated by the OFGA method [19] (blue solid and dotted lines) are also shown.
Fig. 10
Fig. 10 Comparison between log10βtot,MC and log10βtot,PM in Fig. 9(a) for Case1. Black and red lines correspond to a 1:1 ratio and the slope (s) of the linear regression line forced through the origin with correlation coefficient r2, respectively.
Fig. 11
Fig. 11 Relative error of βtot,PM and βtot,OFGA corresponding to (a,b) Fig. 9(a) and (c,d) Fig. 9(b) as functions of the optical thickness τ. In (c), τ range larger than that corresponding to Fig. 9(b) as indicated by the black dotted line is shown. The mean relative errors <ERR> and standard deviations (sd) in percentage for each case at 4 different τ ranges are shown in each figure.
Fig. 12
Fig. 12 (a,b,d) Same as Fig. 8(a,b,d) but for Case 2. (c) Same as Fig. 9 but for Case2. (e) Same as Fig. 11 but the relative errors of βtot corresponding to (c) are shown.
Fig. 13
Fig. 13 (a,b,c,d,e) Same as Fig. 12(a,b,c,d,e) but for Case 3.
Fig. 14
Fig. 14 Same as Fig. 13(a), but for a thick molecular layer.
Fig. 15
Fig. 15 Relative error of βtot againtst Monte Calro simulation for (a-d) the PM and for (e-h) the OFGA method for 14 experiments summarized in Table 1 and categorized according to the optical thickness, i.e., (a,e), 0≤τ (b,f) τ≤2, (c,g) 2<τ<8, (d,h) 8≤τ. Relative errors out of the vertical axis range were indicated in parentheses. Different shaded areas denote simulations performed with different σext profiles. The maximum τ considered are about 48 for the σext = 40km−1 cases and about 18 for the others. The overall mean relative errors 〈ERR〉 and standard deviations [%] listed in the upper right corner of each figure are estimated from all 12 homogeneous profiles.

Tables (1)

Tables Icon

Table 1 Experiment setting

Equations (35)

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I ¯ i,j = 1 t j+1/2 t j1/2 1 z i+1/2 z i1/2 t j1/2 t j+1/2 z i1/2 z i+1/2 I( z,t ) dz dt
I ¯ i,j I( z i , t j )
I tot ( t j )= i=1 j I( z i , t j )
I tot ( t j )= I ss ( t j )+ I ms,tot ( t j )
I ms,tot ( t j )= I ms,tot,on ( t j )+ I ms,tot,off ( t j )
I ms,tot, off ( t j )= i=1 j I ms,off ( z i , t j )
I tot ( t j )= I ss ( t j )+ I ms,tot,on ( t j )+ i=1 j I ms,off ( z i , t j )
I( z i , t j )={ I ss ( t j )+ I ms,tot,on ( t j )+ I ms,off ( z j , t j ),( i=j ) I ms,off ( z i , t j ),( ij )
Θ on ( t j )=( cos 1 [ ( t j t 1 )c/ { ( t j+1/2 t 1 )c } ] )
Θ off,ki ( t j )= cos 1 [ ( t i t k )c/ { ( t j+1/2 t k )c } ]
β w = I w / ( I 0 C s )
C s = Ac τ s / ( 2 R s 2 )
β tot ( t j )= β ss ( t j )+ β ms,tot ( t j )
p n ( Θ )= l=0 2l+1 4π ( A l ) n P l ( cosΘ )
I ss ( t j )= C s I o exp[ 2τ( z j1/2 ) ][ exp{ 2 σ ext ( Z j ) l j }1 2 σ ext ( Z j ) l j ] σ sca ( z j ) p ss ( z j ,π )/( 4π )
Δ I ss = I o [ 1 { exp( 2 σ ext ( z 1 ) l 1 )1 }/ { [ 2 σ ext ( z 1 ) l 1 ] } ]
I ms,tot,on ( t j )= B on ( t j )[ exp{ 2 σ ext,on ( Z j ) l j }1 2 σ ext,on ( Z j ) l j ] σ sca ( z j ) p ss ( z j ,π )/( 4π )
B on ( t j )= C s Δ I ss F on ( Θ on ( t j ) )exp[ 2 i=1 j1 σ ext,on ( z i ) l i ]
F on ( Θ on ( t j ) )= 1 4π 0 2π 0 Θ on ( t j ) p 1 ( z 1 ,Θ )sinΘdΘdϕ
σ ext,on ( z j )= σ ext ( z j )( 1 0 2π 0 θ d,j p j ( θ ) sinθdθdϕ )
θ d,j = θ ds,j =ta n ( r fov ( z j )/ l j ),( j2 )
θ d,j = θ ds,j 2 +( θ max 2 θ ds,j 2 )tanh( (j1) θ ds,3 / θ 3 ) ,( 3j ) θ max =max( θ ds,j , θ j )
I ms,off ( z i , t j )= k=1 i [ B on ( t k ) F off ( Θ off,ki ( t j ) ) R( z i ) ×exp[ 2 q=k i σ ext,off ( z q , z k , z i , t j )( l q +Δd( z q , z k , z i , t j ) ) ] σ sca ( z i ) p i ( π )/( 4π )
F off ( Θ off,ki ( t j ) )= C f 1 4π 0 2π Θ off,ki ( t j1 ) Θ off,ki ( t j ) p k ( Θ )sinΘdΘdϕ
σ ext,off ( z q , z k , z i , t j )= σ ext,on ( z q )/ N q ,( kqi )
N q = n q / [ C 0,i cos 2 Θ off,ki ( t j1/2 )+ C 1,i cos Θ off,ki ( t j1/2 )+ C 2,i ]
n q = h=1 q g h1
C 0,i = C 1,i =1 1/τ i
C 2,i =1+1/ τ i 1/ h=1 i g h1
Δd( z q , z k , z i , t j )=( t j t i )c g q ik+1
R( z i )= 1 4π 0 2π π θ d,i π p i ( θ ) sinθdθdϕ
I tot ( t j )= I ss ( t j )+ I ms,tot ( t j ) = I ss ( t j )+ I ms,tot,on ( t j )+ i=1 j I ms,off ( z i , t j )
β tot ( t j )= β ss ( t j )+ β ms,tot,on ( t j )+ i=1 j β ms,off ( z i , t j )
ERR PMorOFGA = ( β PMorOFGA β MC )/ β MC
p n ( Θ )= l=0 2l+1 4π { j=1 n ( A l ( z j ) ) n j } P l ( cosΘ )

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