Abstract

To understand the connection between single-particle optics and the optics of a closely packed surface, controlled laboratory measurements of bidirectional reflectance distribution functions on layers of polymer and glass spheres are carried out. The measurements are compared with predictions from five radiative-transfer models; the Hapke’s models, the Lumme–Bowell model, the BRF algorithm of Mishchenko et al., and the discrete ordinate radiative transfer. It is found that models of strict numerical radiative-transfer equations (RTEs) predict measurements well in some regions but have errors in both forward- and backward-scattering directions. The improved Hapke’s model, although it has an anisotropic multiple-scattering term, still produces considerable errors compared with the strict RTE. The difference can be attributed to the exclusion of a diffraction contribution in the Hapke model.

© 2005 Optical Society of America

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  52. B. Hapke, “Bidirectional reflectance spectroscopy 3. Correction for macroscopic roughness,” Icarus 59, 41–59 (1984).
    [CrossRef]
  53. S. G. Warren, R. E. Brandt, P. O. Hinton, “Effect of surface roughness on bidirectional reflectance of Antarctic snow,” J. Geophys. Res. 103, 25,789–25,807 (1998).
    [CrossRef]

2004 (1)

2003 (3)

S. Kaasalainen, “Laboratory photometry of planetary regolith analogs. I. Effects of grain and packing properties on opposition effect,” Astron. Astrophys. 409, 765–769 (2003).
[CrossRef]

H. Zhang, K. J. Voss, R. P. Reid, E. M. Louchard, “Bidirectional reflectance measurements of sediments in the vicinity of Lee Stocking Island, Bahamas,” Limnol. Oceanogr. 48 (1, Part 2), 380–389 (2003).
[CrossRef]

X. Ma, J. Lu, R. S. Brock, K. M. Jacobs, P. Yang, X. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165–4172 (2003).
[CrossRef]

2002 (3)

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002).
[CrossRef]

Y. Shkuratov, A. Ovcharenko, E. Zubko, O. Miloslavskaya, K. Muinonen, J. Piironen, R. Nelson, W. Smythe, V. Rosenbush, P. Helfenstein, “The opposition effect and negative polarization of structural analogs for planetary regoliths,” Icarus 159, 396–416 (2002).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy 5. Coherent backscatter opposition effect and anisotropic scattering,” Icarus 157, 523–534 (2002).
[CrossRef]

2001 (1)

C. Bruegge, N. Chrien, D. Haner, “Spectralon BRF data for MISR calibration applications,” Remote Sens. Environ. 76, 354–366 (2001).
[CrossRef]

2000 (2)

K. J. Voss, A. L. Chapin, M. Monti, H. Zhang, “Instrument to measure the bidirectional reflectance distribution function of surfaces,” Appl. Opt. 39, 6197–6206 (2000).
[CrossRef]

A. F. Cheng, D. L. Domingue, “Radiative transfer models for light scattering from planetary surfaces,” J. Geophys. Res. 105, 9477–9482 (2000).
[CrossRef]

1999 (2)

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
[CrossRef]

B. Hapke, “Scattering and diffraction by particles in planetary regoliths,” J. Quant. Spectrosc. Radiat. Transfer 61, 565–581 (1999).
[CrossRef]

1998 (3)

A. A. Kokhanovsky, “On light scattering in random media with large densely packed particles,” J. Geophys. Res. 103, 6089–6096 (1998).
[CrossRef]

B. Hartman, D. Domingue, “Scattering of light by individual particles and the implications for models of planetary surfaces,” Icarus 131, 421–448 (1998).
[CrossRef]

S. G. Warren, R. E. Brandt, P. O. Hinton, “Effect of surface roughness on bidirectional reflectance of Antarctic snow,” J. Geophys. Res. 103, 25,789–25,807 (1998).
[CrossRef]

1997 (5)

D. Domingue, A. Verbiscer, “Reanalysis of the solar phase curves of the icy Galilean satellites,” Icarus 128, 49–74 (1997).
[CrossRef]

C. Leroux, J. Lenoble, G. Brogniez, J. W. Hovenier, J. F. de Haan, “A model for the bidirectional polarized reflectance of snow,” J. Quant. Spectrosc. Radiat. Transfer 61, 273–285 (1997).
[CrossRef]

J. K. Hillier, “Shadow-hiding opposition surge for a two-layer surface,” Icarus 128, 15–27 (1997).
[CrossRef]

S. Liang, “An investigation of remotely sensed soil depth in the optical region,” Int. J. Remote Sens. 18, 3395–3408 (1997).
[CrossRef]

M. I. Mishchenko, A. Macke, “Asymmetry parameters of the phase function for isolated and densely packed spherical particles with multiple internal inclusions in the geometric optics limit,” J. Quant. Spectrosc. Radiat. Transfer 57, 767–794 (1997).
[CrossRef]

1996 (2)

B. Hapke, “Are planetary regolith particles backscattering? Response to a paper by M. Mishchenko,” J. Quant. Spectrosc. Radiat. Transfer 55, 837–848 (1996).
[CrossRef]

M. I. Mishchenko, “Diffuse and coherent backscattering by discrete random media. 1. Radar reflectivity, polarization ratios, and enhancement factors for a half-space of polydisperse, nonabsorbing, and absorbing spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 56, 673–702 (1996).
[CrossRef]

1994 (1)

M. I. Mishchenko, “Asymmetry parameters of the phase function for densely packed scattering grains,” J. Quant. Spectrosc. Radiat. Transfer 52, 95–110 (1994).
[CrossRef]

1992 (2)

M. I. Mishchenko, “The angular width of the coherent backscatter opposition effect—an application to icy outer planet satellites,” Astrophys. Space Sci. 194, 327–333 (1992).
[CrossRef]

J. I. Peltoniemi, K. Lumme, “Light scattering by closely packed particulate media,” J. Opt. Soc. Am. A 9, 1320–1326 (1992).
[CrossRef]

1988 (2)

K. Stamnes, S. C. Tsay, W. Wiscombe, K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988).
[CrossRef] [PubMed]

T. Nakajima, M. Tanaka, “Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
[CrossRef]

1987 (1)

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

1984 (2)

A. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

B. Hapke, “Bidirectional reflectance spectroscopy 3. Correction for macroscopic roughness,” Icarus 59, 41–59 (1984).
[CrossRef]

1982 (3)

B. Hapke, “The Lumme–Bowell photometric parameters—reality or fantasy?” Bull. Am. Astron. Soc. 14, 726H (1982).

K. Lumme, E. Bowell, “A reply to Hapke’s criticism of the Lumme–Bowell photometric theory,” Bull. Am. Astron. Soc. 14, 726H (1982).

K. Lumme, W. M. Irvine, “Radiative transfer in the surfaces of atmosphereless bodies. III. Interpretation of lunar photometry,” Astron. J. 87, 1076–1082 (1982).
[CrossRef]

1981 (3)

B. Hapke, E. Wells, “Bidirectional reflectance spectroscopy 2. Experiments and observations,” J. Geophys. Res. 86, 3055–3060 (1981).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy 1. Theory,” J. Geophys. Res. 86, 3039–3054 (1981).
[CrossRef]

K. Lumme, E. Bowell, “Radiative transfer in the surfaces of atmosphereless bodies. I. Theory,” Astron. J. 86, 1694–1704 (1981).
[CrossRef]

1980 (1)

1977 (1)

W. J. Wiscombe, “The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

1975 (1)

1974 (1)

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

1960 (1)

G. D. Scott, “Packing of Spheres,” Nature (London) 188, 908–909 (1960).
[CrossRef]

Adam, J. A.

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002).
[CrossRef]

Bohren, C. F.

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

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
[CrossRef]

Bowell, E.

K. Lumme, E. Bowell, “A reply to Hapke’s criticism of the Lumme–Bowell photometric theory,” Bull. Am. Astron. Soc. 14, 726H (1982).

K. Lumme, E. Bowell, “Radiative transfer in the surfaces of atmosphereless bodies. I. Theory,” Astron. J. 86, 1694–1704 (1981).
[CrossRef]

E. Bowell, B. Hapke, K. Lumme, J. Peltoniemi, A. W. Harris, “Application of photometric models to asteroids,” in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, eds. (University of Arizona, Tucson, Ariz., 1989), pp. 524–556.

Brandt, R. E.

S. G. Warren, R. E. Brandt, P. O. Hinton, “Effect of surface roughness on bidirectional reflectance of Antarctic snow,” J. Geophys. Res. 103, 25,789–25,807 (1998).
[CrossRef]

Brock, R. S.

X. Ma, J. Lu, R. S. Brock, K. M. Jacobs, P. Yang, X. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165–4172 (2003).
[CrossRef]

Brogniez, G.

C. Leroux, J. Lenoble, G. Brogniez, J. W. Hovenier, J. F. de Haan, “A model for the bidirectional polarized reflectance of snow,” J. Quant. Spectrosc. Radiat. Transfer 61, 273–285 (1997).
[CrossRef]

Bruegge, C.

C. Bruegge, N. Chrien, D. Haner, “Spectralon BRF data for MISR calibration applications,” Remote Sens. Environ. 76, 354–366 (2001).
[CrossRef]

Chandrasekhar, S.

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

Chapin, A. L.

Cheng, A. F.

A. F. Cheng, D. L. Domingue, “Radiative transfer models for light scattering from planetary surfaces,” J. Geophys. Res. 105, 9477–9482 (2000).
[CrossRef]

Chrien, N.

C. Bruegge, N. Chrien, D. Haner, “Spectralon BRF data for MISR calibration applications,” Remote Sens. Environ. 76, 354–366 (2001).
[CrossRef]

de Haan, J. F.

C. Leroux, J. Lenoble, G. Brogniez, J. W. Hovenier, J. F. de Haan, “A model for the bidirectional polarized reflectance of snow,” J. Quant. Spectrosc. Radiat. Transfer 61, 273–285 (1997).
[CrossRef]

de Rooij, A. A.

A. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

Dlugach, J. M.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
[CrossRef]

Domingue, D.

B. Hartman, D. Domingue, “Scattering of light by individual particles and the implications for models of planetary surfaces,” Icarus 131, 421–448 (1998).
[CrossRef]

D. Domingue, A. Verbiscer, “Reanalysis of the solar phase curves of the icy Galilean satellites,” Icarus 128, 49–74 (1997).
[CrossRef]

Domingue, D. L.

A. F. Cheng, D. L. Domingue, “Radiative transfer models for light scattering from planetary surfaces,” J. Geophys. Res. 105, 9477–9482 (2000).
[CrossRef]

Haner, D.

C. Bruegge, N. Chrien, D. Haner, “Spectralon BRF data for MISR calibration applications,” Remote Sens. Environ. 76, 354–366 (2001).
[CrossRef]

Hansen, J. E.

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

Hapke, B.

B. Hapke, “Bidirectional reflectance spectroscopy 5. Coherent backscatter opposition effect and anisotropic scattering,” Icarus 157, 523–534 (2002).
[CrossRef]

B. Hapke, “Scattering and diffraction by particles in planetary regoliths,” J. Quant. Spectrosc. Radiat. Transfer 61, 565–581 (1999).
[CrossRef]

B. Hapke, “Are planetary regolith particles backscattering? Response to a paper by M. Mishchenko,” J. Quant. Spectrosc. Radiat. Transfer 55, 837–848 (1996).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy 3. Correction for macroscopic roughness,” Icarus 59, 41–59 (1984).
[CrossRef]

B. Hapke, “The Lumme–Bowell photometric parameters—reality or fantasy?” Bull. Am. Astron. Soc. 14, 726H (1982).

B. Hapke, E. Wells, “Bidirectional reflectance spectroscopy 2. Experiments and observations,” J. Geophys. Res. 86, 3055–3060 (1981).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy 1. Theory,” J. Geophys. Res. 86, 3039–3054 (1981).
[CrossRef]

B. Hapke, Theory of Reflectance and Emittance Spectroscopy (Cambridge University, New York, 1993).
[CrossRef]

E. Bowell, B. Hapke, K. Lumme, J. Peltoniemi, A. W. Harris, “Application of photometric models to asteroids,” in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, eds. (University of Arizona, Tucson, Ariz., 1989), pp. 524–556.

Harris, A. W.

E. Bowell, B. Hapke, K. Lumme, J. Peltoniemi, A. W. Harris, “Application of photometric models to asteroids,” in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, eds. (University of Arizona, Tucson, Ariz., 1989), pp. 524–556.

Hartman, B.

B. Hartman, D. Domingue, “Scattering of light by individual particles and the implications for models of planetary surfaces,” Icarus 131, 421–448 (1998).
[CrossRef]

Helfenstein, P.

Y. Shkuratov, A. Ovcharenko, E. Zubko, O. Miloslavskaya, K. Muinonen, J. Piironen, R. Nelson, W. Smythe, V. Rosenbush, P. Helfenstein, “The opposition effect and negative polarization of structural analogs for planetary regoliths,” Icarus 159, 396–416 (2002).
[CrossRef]

Hillier, J. K.

J. K. Hillier, “Shadow-hiding opposition surge for a two-layer surface,” Icarus 128, 15–27 (1997).
[CrossRef]

Hinton, P. O.

S. G. Warren, R. E. Brandt, P. O. Hinton, “Effect of surface roughness on bidirectional reflectance of Antarctic snow,” J. Geophys. Res. 103, 25,789–25,807 (1998).
[CrossRef]

Hovenier, J. W.

C. Leroux, J. Lenoble, G. Brogniez, J. W. Hovenier, J. F. de Haan, “A model for the bidirectional polarized reflectance of snow,” J. Quant. Spectrosc. Radiat. Transfer 61, 273–285 (1997).
[CrossRef]

Hu, X.

X. Ma, J. Lu, R. S. Brock, K. M. Jacobs, P. Yang, X. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165–4172 (2003).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
[CrossRef]

Irvine, W. M.

K. Lumme, W. M. Irvine, “Radiative transfer in the surfaces of atmosphereless bodies. III. Interpretation of lunar photometry,” Astron. J. 87, 1076–1082 (1982).
[CrossRef]

Jacobs, K. M.

X. Ma, J. Lu, R. S. Brock, K. M. Jacobs, P. Yang, X. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165–4172 (2003).
[CrossRef]

Jayaweera, K.

Kaasalainen, S.

S. Kaasalainen, “Laboratory photometry of planetary regolith analogs. I. Effects of grain and packing properties on opposition effect,” Astron. Astrophys. 409, 765–769 (2003).
[CrossRef]

Kokhanovsky, A. A.

A. A. Kokhanovsky, E. P. Zege, “Scattering optics of snow,” Appl. Opt. 43, 1589–1602 (2004).
[CrossRef] [PubMed]

A. A. Kokhanovsky, “On light scattering in random media with large densely packed particles,” J. Geophys. Res. 103, 6089–6096 (1998).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emissions by Small Particles (Cambridge University, New York, 2002).

Lenoble, J.

C. Leroux, J. Lenoble, G. Brogniez, J. W. Hovenier, J. F. de Haan, “A model for the bidirectional polarized reflectance of snow,” J. Quant. Spectrosc. Radiat. Transfer 61, 273–285 (1997).
[CrossRef]

Leroux, C.

C. Leroux, J. Lenoble, G. Brogniez, J. W. Hovenier, J. F. de Haan, “A model for the bidirectional polarized reflectance of snow,” J. Quant. Spectrosc. Radiat. Transfer 61, 273–285 (1997).
[CrossRef]

Liang, S.

S. Liang, “An investigation of remotely sensed soil depth in the optical region,” Int. J. Remote Sens. 18, 3395–3408 (1997).
[CrossRef]

Louchard, E. M.

H. Zhang, K. J. Voss, R. P. Reid, E. M. Louchard, “Bidirectional reflectance measurements of sediments in the vicinity of Lee Stocking Island, Bahamas,” Limnol. Oceanogr. 48 (1, Part 2), 380–389 (2003).
[CrossRef]

Lu, J.

X. Ma, J. Lu, R. S. Brock, K. M. Jacobs, P. Yang, X. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165–4172 (2003).
[CrossRef]

Lumme, K.

J. I. Peltoniemi, K. Lumme, “Light scattering by closely packed particulate media,” J. Opt. Soc. Am. A 9, 1320–1326 (1992).
[CrossRef]

K. Lumme, E. Bowell, “A reply to Hapke’s criticism of the Lumme–Bowell photometric theory,” Bull. Am. Astron. Soc. 14, 726H (1982).

K. Lumme, W. M. Irvine, “Radiative transfer in the surfaces of atmosphereless bodies. III. Interpretation of lunar photometry,” Astron. J. 87, 1076–1082 (1982).
[CrossRef]

K. Lumme, E. Bowell, “Radiative transfer in the surfaces of atmosphereless bodies. I. Theory,” Astron. J. 86, 1694–1704 (1981).
[CrossRef]

E. Bowell, B. Hapke, K. Lumme, J. Peltoniemi, A. W. Harris, “Application of photometric models to asteroids,” in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, eds. (University of Arizona, Tucson, Ariz., 1989), pp. 524–556.

Ma, X.

X. Ma, J. Lu, R. S. Brock, K. M. Jacobs, P. Yang, X. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165–4172 (2003).
[CrossRef]

Macke, A.

M. I. Mishchenko, A. Macke, “Asymmetry parameters of the phase function for isolated and densely packed spherical particles with multiple internal inclusions in the geometric optics limit,” J. Quant. Spectrosc. Radiat. Transfer 57, 767–794 (1997).
[CrossRef]

Miloslavskaya, O.

Y. Shkuratov, A. Ovcharenko, E. Zubko, O. Miloslavskaya, K. Muinonen, J. Piironen, R. Nelson, W. Smythe, V. Rosenbush, P. Helfenstein, “The opposition effect and negative polarization of structural analogs for planetary regoliths,” Icarus 159, 396–416 (2002).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
[CrossRef]

M. I. Mishchenko, A. Macke, “Asymmetry parameters of the phase function for isolated and densely packed spherical particles with multiple internal inclusions in the geometric optics limit,” J. Quant. Spectrosc. Radiat. Transfer 57, 767–794 (1997).
[CrossRef]

M. I. Mishchenko, “Diffuse and coherent backscattering by discrete random media. 1. Radar reflectivity, polarization ratios, and enhancement factors for a half-space of polydisperse, nonabsorbing, and absorbing spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 56, 673–702 (1996).
[CrossRef]

M. I. Mishchenko, “Asymmetry parameters of the phase function for densely packed scattering grains,” J. Quant. Spectrosc. Radiat. Transfer 52, 95–110 (1994).
[CrossRef]

M. I. Mishchenko, “The angular width of the coherent backscatter opposition effect—an application to icy outer planet satellites,” Astrophys. Space Sci. 194, 327–333 (1992).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emissions by Small Particles (Cambridge University, New York, 2002).

Mobley, C. D.

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, New York, 1994).

Monti, M.

Muinonen, K.

Y. Shkuratov, A. Ovcharenko, E. Zubko, O. Miloslavskaya, K. Muinonen, J. Piironen, R. Nelson, W. Smythe, V. Rosenbush, P. Helfenstein, “The opposition effect and negative polarization of structural analogs for planetary regoliths,” Icarus 159, 396–416 (2002).
[CrossRef]

Nakajima, T.

T. Nakajima, M. Tanaka, “Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
[CrossRef]

Nelson, R.

Y. Shkuratov, A. Ovcharenko, E. Zubko, O. Miloslavskaya, K. Muinonen, J. Piironen, R. Nelson, W. Smythe, V. Rosenbush, P. Helfenstein, “The opposition effect and negative polarization of structural analogs for planetary regoliths,” Icarus 159, 396–416 (2002).
[CrossRef]

Ovcharenko, A.

Y. Shkuratov, A. Ovcharenko, E. Zubko, O. Miloslavskaya, K. Muinonen, J. Piironen, R. Nelson, W. Smythe, V. Rosenbush, P. Helfenstein, “The opposition effect and negative polarization of structural analogs for planetary regoliths,” Icarus 159, 396–416 (2002).
[CrossRef]

Peltoniemi, J.

E. Bowell, B. Hapke, K. Lumme, J. Peltoniemi, A. W. Harris, “Application of photometric models to asteroids,” in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, eds. (University of Arizona, Tucson, Ariz., 1989), pp. 524–556.

Peltoniemi, J. I.

Piironen, J.

Y. Shkuratov, A. Ovcharenko, E. Zubko, O. Miloslavskaya, K. Muinonen, J. Piironen, R. Nelson, W. Smythe, V. Rosenbush, P. Helfenstein, “The opposition effect and negative polarization of structural analogs for planetary regoliths,” Icarus 159, 396–416 (2002).
[CrossRef]

Pogorzelski, R. J.

Reid, R. P.

H. Zhang, K. J. Voss, R. P. Reid, E. M. Louchard, “Bidirectional reflectance measurements of sediments in the vicinity of Lee Stocking Island, Bahamas,” Limnol. Oceanogr. 48 (1, Part 2), 380–389 (2003).
[CrossRef]

Rosenbush, V.

Y. Shkuratov, A. Ovcharenko, E. Zubko, O. Miloslavskaya, K. Muinonen, J. Piironen, R. Nelson, W. Smythe, V. Rosenbush, P. Helfenstein, “The opposition effect and negative polarization of structural analogs for planetary regoliths,” Icarus 159, 396–416 (2002).
[CrossRef]

Scott, G. D.

G. D. Scott, “Packing of Spheres,” Nature (London) 188, 908–909 (1960).
[CrossRef]

Shkuratov, Y.

Y. Shkuratov, A. Ovcharenko, E. Zubko, O. Miloslavskaya, K. Muinonen, J. Piironen, R. Nelson, W. Smythe, V. Rosenbush, P. Helfenstein, “The opposition effect and negative polarization of structural analogs for planetary regoliths,” Icarus 159, 396–416 (2002).
[CrossRef]

Smythe, W.

Y. Shkuratov, A. Ovcharenko, E. Zubko, O. Miloslavskaya, K. Muinonen, J. Piironen, R. Nelson, W. Smythe, V. Rosenbush, P. Helfenstein, “The opposition effect and negative polarization of structural analogs for planetary regoliths,” Icarus 159, 396–416 (2002).
[CrossRef]

Stamnes, K.

Tanaka, M.

T. Nakajima, M. Tanaka, “Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
[CrossRef]

Thomas, G. E.

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, New York, 2002).

Travis, L. D.

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

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emissions by Small Particles (Cambridge University, New York, 2002).

Tsai, W. C.

Tsay, S. C.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).

van der Stap, C. C. A. H.

A. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

Verbiscer, A.

D. Domingue, A. Verbiscer, “Reanalysis of the solar phase curves of the icy Galilean satellites,” Icarus 128, 49–74 (1997).
[CrossRef]

Voss, K. J.

H. Zhang, K. J. Voss, R. P. Reid, E. M. Louchard, “Bidirectional reflectance measurements of sediments in the vicinity of Lee Stocking Island, Bahamas,” Limnol. Oceanogr. 48 (1, Part 2), 380–389 (2003).
[CrossRef]

K. J. Voss, A. L. Chapin, M. Monti, H. Zhang, “Instrument to measure the bidirectional reflectance distribution function of surfaces,” Appl. Opt. 39, 6197–6206 (2000).
[CrossRef]

Warren, S. G.

S. G. Warren, R. E. Brandt, P. O. Hinton, “Effect of surface roughness on bidirectional reflectance of Antarctic snow,” J. Geophys. Res. 103, 25,789–25,807 (1998).
[CrossRef]

Wells, E.

B. Hapke, E. Wells, “Bidirectional reflectance spectroscopy 2. Experiments and observations,” J. Geophys. Res. 86, 3055–3060 (1981).
[CrossRef]

Wiscombe, W.

Wiscombe, W. J.

W. J. Wiscombe, “The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

W. J. Wiscombe, “Mie scattering calculations: Advances in technique and fast, vector-speed computer codes,” NCAR Tech. Note (National Center for Atmospheric Research, July1979, revised Aug.1996), ftp://climate.gsfc.nasa.gov/pub/wiscombe/Single_scatt/ .

Yang, P.

X. Ma, J. Lu, R. S. Brock, K. M. Jacobs, P. Yang, X. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165–4172 (2003).
[CrossRef]

Yanovitskij, E. G.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
[CrossRef]

E. G. Yanovitskij, Light Scattering in Inhomogeneous Atmospheres (Springer-Verlag, Berlin, 1997).
[CrossRef]

Zakharova, N. T.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
[CrossRef]

Zege, E. P.

Zhang, H.

H. Zhang, K. J. Voss, R. P. Reid, E. M. Louchard, “Bidirectional reflectance measurements of sediments in the vicinity of Lee Stocking Island, Bahamas,” Limnol. Oceanogr. 48 (1, Part 2), 380–389 (2003).
[CrossRef]

K. J. Voss, A. L. Chapin, M. Monti, H. Zhang, “Instrument to measure the bidirectional reflectance distribution function of surfaces,” Appl. Opt. 39, 6197–6206 (2000).
[CrossRef]

Zubko, E.

Y. Shkuratov, A. Ovcharenko, E. Zubko, O. Miloslavskaya, K. Muinonen, J. Piironen, R. Nelson, W. Smythe, V. Rosenbush, P. Helfenstein, “The opposition effect and negative polarization of structural analogs for planetary regoliths,” Icarus 159, 396–416 (2002).
[CrossRef]

Am. J. Phys. (1)

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

Appl. Opt. (4)

Astron. Astrophys. (2)

A. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

S. Kaasalainen, “Laboratory photometry of planetary regolith analogs. I. Effects of grain and packing properties on opposition effect,” Astron. Astrophys. 409, 765–769 (2003).
[CrossRef]

Astron. J. (2)

K. Lumme, E. Bowell, “Radiative transfer in the surfaces of atmosphereless bodies. I. Theory,” Astron. J. 86, 1694–1704 (1981).
[CrossRef]

K. Lumme, W. M. Irvine, “Radiative transfer in the surfaces of atmosphereless bodies. III. Interpretation of lunar photometry,” Astron. J. 87, 1076–1082 (1982).
[CrossRef]

Astrophys. Space Sci. (1)

M. I. Mishchenko, “The angular width of the coherent backscatter opposition effect—an application to icy outer planet satellites,” Astrophys. Space Sci. 194, 327–333 (1992).
[CrossRef]

Bull. Am. Astron. Soc. (2)

B. Hapke, “The Lumme–Bowell photometric parameters—reality or fantasy?” Bull. Am. Astron. Soc. 14, 726H (1982).

K. Lumme, E. Bowell, “A reply to Hapke’s criticism of the Lumme–Bowell photometric theory,” Bull. Am. Astron. Soc. 14, 726H (1982).

Icarus (6)

B. Hartman, D. Domingue, “Scattering of light by individual particles and the implications for models of planetary surfaces,” Icarus 131, 421–448 (1998).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy 5. Coherent backscatter opposition effect and anisotropic scattering,” Icarus 157, 523–534 (2002).
[CrossRef]

Y. Shkuratov, A. Ovcharenko, E. Zubko, O. Miloslavskaya, K. Muinonen, J. Piironen, R. Nelson, W. Smythe, V. Rosenbush, P. Helfenstein, “The opposition effect and negative polarization of structural analogs for planetary regoliths,” Icarus 159, 396–416 (2002).
[CrossRef]

J. K. Hillier, “Shadow-hiding opposition surge for a two-layer surface,” Icarus 128, 15–27 (1997).
[CrossRef]

D. Domingue, A. Verbiscer, “Reanalysis of the solar phase curves of the icy Galilean satellites,” Icarus 128, 49–74 (1997).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy 3. Correction for macroscopic roughness,” Icarus 59, 41–59 (1984).
[CrossRef]

Int. J. Remote Sens. (1)

S. Liang, “An investigation of remotely sensed soil depth in the optical region,” Int. J. Remote Sens. 18, 3395–3408 (1997).
[CrossRef]

J. Atmos. Sci. (1)

W. J. Wiscombe, “The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

J. Geophys. Res. (5)

S. G. Warren, R. E. Brandt, P. O. Hinton, “Effect of surface roughness on bidirectional reflectance of Antarctic snow,” J. Geophys. Res. 103, 25,789–25,807 (1998).
[CrossRef]

A. F. Cheng, D. L. Domingue, “Radiative transfer models for light scattering from planetary surfaces,” J. Geophys. Res. 105, 9477–9482 (2000).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy 1. Theory,” J. Geophys. Res. 86, 3039–3054 (1981).
[CrossRef]

B. Hapke, E. Wells, “Bidirectional reflectance spectroscopy 2. Experiments and observations,” J. Geophys. Res. 86, 3055–3060 (1981).
[CrossRef]

A. A. Kokhanovsky, “On light scattering in random media with large densely packed particles,” J. Geophys. Res. 103, 6089–6096 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Quant. Spectrosc. Radiat. Transfer (8)

M. I. Mishchenko, “Diffuse and coherent backscattering by discrete random media. 1. Radar reflectivity, polarization ratios, and enhancement factors for a half-space of polydisperse, nonabsorbing, and absorbing spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 56, 673–702 (1996).
[CrossRef]

M. I. Mishchenko, “Asymmetry parameters of the phase function for densely packed scattering grains,” J. Quant. Spectrosc. Radiat. Transfer 52, 95–110 (1994).
[CrossRef]

B. Hapke, “Are planetary regolith particles backscattering? Response to a paper by M. Mishchenko,” J. Quant. Spectrosc. Radiat. Transfer 55, 837–848 (1996).
[CrossRef]

M. I. Mishchenko, A. Macke, “Asymmetry parameters of the phase function for isolated and densely packed spherical particles with multiple internal inclusions in the geometric optics limit,” J. Quant. Spectrosc. Radiat. Transfer 57, 767–794 (1997).
[CrossRef]

B. Hapke, “Scattering and diffraction by particles in planetary regoliths,” J. Quant. Spectrosc. Radiat. Transfer 61, 565–581 (1999).
[CrossRef]

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
[CrossRef]

C. Leroux, J. Lenoble, G. Brogniez, J. W. Hovenier, J. F. de Haan, “A model for the bidirectional polarized reflectance of snow,” J. Quant. Spectrosc. Radiat. Transfer 61, 273–285 (1997).
[CrossRef]

T. Nakajima, M. Tanaka, “Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
[CrossRef]

Limnol. Oceanogr. (1)

H. Zhang, K. J. Voss, R. P. Reid, E. M. Louchard, “Bidirectional reflectance measurements of sediments in the vicinity of Lee Stocking Island, Bahamas,” Limnol. Oceanogr. 48 (1, Part 2), 380–389 (2003).
[CrossRef]

Nature (London) (1)

G. D. Scott, “Packing of Spheres,” Nature (London) 188, 908–909 (1960).
[CrossRef]

Phys. Med. Biol. (1)

X. Ma, J. Lu, R. S. Brock, K. M. Jacobs, P. Yang, X. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165–4172 (2003).
[CrossRef]

Phys. Rep. (1)

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002).
[CrossRef]

Remote Sens. Environ. (1)

C. Bruegge, N. Chrien, D. Haner, “Spectralon BRF data for MISR calibration applications,” Remote Sens. Environ. 76, 354–366 (2001).
[CrossRef]

Space Sci. Rev. (1)

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

Other (12)

W. J. Wiscombe, “Mie scattering calculations: Advances in technique and fast, vector-speed computer codes,” NCAR Tech. Note (National Center for Atmospheric Research, July1979, revised Aug.1996), ftp://climate.gsfc.nasa.gov/pub/wiscombe/Single_scatt/ .

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, New York, 1994).

K. Stamnes, S. C. Tsay, W. Wiscombe, I. Laszlo, “DISORT, a general-purpose Fortran program for discrete-ordinate-method radiative transfer in scattering and emitting layered media: documentation of methodology,” Version 1.1, March2000, ftp://climate.gsfc.nasa.gov/pub/wiscombe/Multiple_scatt/ .

H. Zhang, K. J. Voss, R. P. Reid, “Determining the influential depth of sediment particles by BRDF measurements,” Opt. Express11, 2654–2665 (2003), http://www.opticsexpress.org .
[CrossRef] [PubMed]

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emissions by Small Particles (Cambridge University, New York, 2002).

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

E. G. Yanovitskij, Light Scattering in Inhomogeneous Atmospheres (Springer-Verlag, Berlin, 1997).
[CrossRef]

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, New York, 2002).

E. Bowell, B. Hapke, K. Lumme, J. Peltoniemi, A. W. Harris, “Application of photometric models to asteroids,” in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, eds. (University of Arizona, Tucson, Ariz., 1989), pp. 524–556.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
[CrossRef]

B. Hapke, Theory of Reflectance and Emittance Spectroscopy (Cambridge University, New York, 1993).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic of the simple goniometric scattering meter setup. Throughout this study the incident zenith angle θi is specified to be negative. Phase angle α is specified to be positive when opposite the incident beam.

Fig. 2
Fig. 2

Mie phase functions of (a) 200- and (b) 600-μm spheres: s-pol, p-pol, electric field of the incident irradiance perpendicular and parallel to the scattering plane, respectively. The unpolarized Mie phase function is their average.

Fig. 3
Fig. 3

Raw REFF of a 10-mm-thick, 200-μm-sphere layer for two orthogonal incident polarizations at −60° incidence. The viewing zenith angle in this configuration is a phase angle minus 60°.

Fig. 4
Fig. 4

Comparisons of MBRF and DISORT for a 200-μm-sphere layer with τ = 2000 and ϖ0 = 0.999. Incident zenith angles are (a) 0°, (b) −35°, and (c) −60°.

Fig. 5
Fig. 5

Comparisons of goniometric measurement, DISORT, LB model, HIMSA, and HAMSA for a 10-mm-thick (τ = 88.5), 200-μm-sphere layer. Incident zenith angles are (a) 0°, (b) −35°, and (c) −60°.

Fig. 6
Fig. 6

Comparisons of goniometric measurement, DISORT, LB model, HIMSA, and HAMSA for a 15-mm-thick (τ = 41.25), 600-μm-sphere layer. Incident zenith angles are (a) 0° and (b) −60°.

Fig. 7
Fig. 7

Contributions of single scattering [Eq. (42)] to total reflectance at (a) normal, (b) −35°, and (c) −60° incidence.

Fig. 8
Fig. 8

Comparison of δ-N truncated Mie phase function and the full phase function for the 200-μm spheres.

Fig. 9
Fig. 9

Comparisons of HAMSA, DISORT, and δN-DISORT for a 10-mm-thick (τ = 88.5), 200-μm layer. Incident zenith angles are (a) 0°, (b) −35°, and (c) −60°.

Fig. 10
Fig. 10

Comparisons of DISORT, roughness-corrected DISORT, and the measurements. Incident zenith angles are (a) 0°, (b) −35°, and (c) −60°.

Tables (1)

Tables Icon

Table 1 Parameters Used in the Modeling

Equations (57)

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

r = d L R d E i ,
BRDF = r cos θ i ,
REFF = π BRDF = π r cos θ i ,
REFF HIMSA ( μ 0 , μ , α ) = ϖ 0 4 1 μ 0 + μ { [ 1 + B ( α ) ] P ( α ) + H ( μ ) H ( μ 0 ) - 1 } × S ( μ 0 , μ , ϕ , θ ¯ ) ,
H ( x ) 1 1 - ϖ 0 x ( r 0 + 1 - 2 r 0 x 2 ln 1 + x x ) ,
r 0 = 1 - ( 1 - ϖ 0 ) 1 / 2 1 + ( 1 - ϖ 0 ) 1 / 2 .
B ( α ) = B 0 1 + 1 h tan α 2 ,
REFF HAMSA ( μ 0 , μ , α ) = ϖ 0 4 1 μ 0 + μ × { [ 1 + B ( α ) ] P ( α ) + M ( μ 0 , μ ) } ,
M ( μ 0 , μ ) = Π ( μ 0 ) [ H ( μ ) - 1 ] + Π ( μ ) [ H ( μ 0 ] - 1 ] + Δ [ H ( μ ) - 1 ] [ H ( μ 0 ) - 1 ] ,
Π ( x ) = 1 + n = 1 A n b n P n ( x ) ,
Δ = 1 + n = 1 A n 2 b n ,
A n = 0             n even ,
A n = ( - 1 ) n + 1 2 n 1 × 3 × 5 × × n 2 × 4 × 6 × × ( n + 1 )             n odd ,
P ( α ) = 1 + n = 1 b n P n ( cos α ) .
REFF LB ( μ 0 , μ , α ) = R S + R M = ϖ 0 4 1 μ + μ 0 × [ 2 Φ R Φ S P ( α ) + H ( μ , ϖ 0 * ) × H ( μ 0 , ϖ 0 * ) - 1 ] ,
ϖ 0 * = 1 - g 1 - g ϖ 0 ϖ 0 ,
g = cos α = 1 2 - 1 1 P ( cos α ) d ( cos α ) .
Φ R = 1 + ( 1 - q ) ρ ξ 1 + ρ ξ ,
ρ = H R = tan ( θ ¯ ) ,
ξ = ( μ 2 + μ 0 2 - 2 μ μ 0 cos α ) 1 / 2 μ μ 0 .
Φ S y + 3 / 4 y + 3 / 2 ,
y = D 2.38 μ + μ 0 ( μ 2 + μ 0 2 - 2 μ μ 0 cos α ) 1 / 2 ,
P ( μ , μ 0 , ϕ ) = P 0 ( μ , μ 0 ) + m = 1 m max P m ( μ , μ 0 ) cos ( m ϕ ) ,
P m ( μ , μ 0 ) = ( - 1 ) m s = m S max β s P m 0 s ( μ ) P m 0 s ( μ 0 ) ,
P ( Θ ) = s = 0 s max β s P s ( cos Θ ) ,
Θ = π - α .
L ( - μ , ϕ ) = μ 0 R ( μ , μ 0 , ϕ ) F ,
R ( μ , μ 0 , ϕ ) = R 0 ( μ , μ 0 ) + 2 m = 1 m max R m ( μ , μ 0 ) cos m ϕ ,
( μ + μ 0 ) R m ( μ , μ 0 ) = ϖ 0 4 P m ( - μ , μ 0 ) + ϖ 0 2 μ 0 × 0 1 P m ( μ , μ ) R m ( μ , μ 0 ) d μ + ϖ 0 2 μ 0 1 R m ( μ , μ ) P m ( μ , μ 0 ) d μ + ϖ 0 μ μ 0 0 1 0 1 R m ( μ , μ ) P m ( - μ , μ ) × R m ( μ , μ 0 ) d μ d μ .
REFF BMRF = π L π F μ 0 = R .
μ d L ( τ , μ , ϕ ) d τ = L ( τ , μ , ϕ ) - ϖ 0 4 π 0 2 π d ϕ × - 1 1 d μ P ( μ , ϕ ; μ , ϕ ) L ( τ , μ , ϕ ) ,
L ( τ , μ , ϕ ) = m = 0 2 M - 1 L m ( τ , μ ) cos m ϕ ,
P ( μ , ϕ ; μ , ϕ ) P ( cos Θ ) = l = 0 2 M - 1 ( 2 l + 1 ) g l P l ( cos Θ ) ,
P ( cos Θ ) = l = 0 2 M - 1 ( 2 l + 1 ) g l [ P l ( μ ) P l ( μ ) + 2 m = 1 l Λ l m ( μ ) Λ l m ( μ ) cos m ( ϕ - ϕ ) ] ,
Λ l m ( μ ) = [ ( l - m ) ! ( l + m ) ! ] 1 / 2 P l m ( μ )
μ d L m ( τ , μ ) d τ = L m ( τ , μ ) - ϖ 0 2 - 1 1 [ l = 0 2 M - 1 ( 2 l + 1 ) g l Λ l m ( μ ) Λ l m ( μ ) ] × L m ( τ , μ ) d μ .
μ i d L m ( τ , μ i ) d τ = L m ( τ , μ i ) - j = - N j 0 N w j D m ( τ , μ i , μ j ) L m ( τ , μ j ) × ( ι = ± 1 , ± 2 ± N ) ,
REFF DISORT = L DISORT ,
REFF Smooth ( μ 0 , μ , α ) = ϖ 0 4 1 μ 0 + μ [ P ( α ) + H H - 1 ] ,
B ( α ) 0 , Φ R 1 , Φ S 1 / 2.
REFF Single = ϖ 0 4 1 μ + μ 0 P ( α ) .
REFF Single = ϖ 0 4 1 μ + μ 0 P ( α ) × { 1 - exp [ - τ ( 1 μ + 1 μ 0 ) ] } .
REFF raw = 1 2 ( L l s L c s + L l p L c p ) ,
Corr = 1 2 ( L c s + L c p ) cos ( θ v ) f ( θ v 0 ) ,
REFF Gonio = REFF raw × C o r r .
f ( D ) = 1 ( 2 π ) 1 / 2 σ exp [ - ( D - D 0 ) 2 2 σ 2 ] ,
f ( D ) = 1 w 0 D exp [ - ( ln D D 0 ) 2 ( ln σ ) 2 ] ,
τ = N σ ext z ,
τ = 0 S N ( z ) σ ext ( z ) d z N σ ext s ,
N = n V = f 4 3 π r eff 3
τ 3 f s 2 r eff ,
τ 200 1 - g = τ .
P ( cos Θ ) 2 f δ ( 1 - cos Θ ) + ( 1 - f ) × l = 0 2 N - 1 ( 2 l + 1 ) g l * P l ( cos Θ ) ,
g l * = g l - f 1 - f ( l = 0 , , 2 N - 1 ) ,
f = g 2 N .
P trunc ( cos Θ ) l = 0 99 ( 2 l + 1 ) ( g l - g 100 ) P l ( cos Θ )
DISORT Rough = DISORT - ( 1 - Φ R ) ϖ 0 4 1 μ + μ 0 P ( α ) ,

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