Abstract

We present an analysis of a number of different approximations for the plane transmittance Tp and diffuse attenuation coefficient Kd of a semi-infinite, unbounded, plane-parallel, and optically homogeneous layer. The maximally wide optical conditions (from the full absorption to the full scattering and from the fully forward to the fully backward scattering) were considered. The approximations were analyzed from the point of view of their physical limitations and closeness to the numerical solution of the radiative transfer equation for the plane transmittance. The main criterion for inclusion of the models for analysis was the possibility of practical use, i.e., approximations were well parameterized and included only easily measured or estimated parameters. A detailed analysis of errors for different Tp and Kd models showed that the two-stream radiative transfer Ben-David model yields the best results over all optical conditions and depths. However, the quasi-single-scattering and polynomial Gordon’s approximations proved to be the best for the depths close to zero.

© 2014 Optical Society of America

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  1. G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Math. Phys. Papers 4, 145–156 (1862).
  2. L. B. Tuckerman, “On the intensity of the light reflected from or transmitted through a pile of plates,” J. Opt. Soc. Am. 37, 818–825 (1947).
    [CrossRef]
  3. M. Gurevich, “Über eine rationelle klassifikation der lichtenstreuenden medien,” Physikalische Zeitschrift,” 31, 753–763 (1930).
  4. P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).
  5. P. Kubelka, “New contributions to the optics of intensely light-scattering material. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
    [CrossRef]
  6. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).
  7. L. G. Sokoletsky, A. A. Kokhanovsky, and F. Shen, “Comparative analysis of radiative transfer approaches for calculation of diffuse reflectance of plane-parallel light scattering layers,” Appl. Opt. 52, 8471–8483 (2013).
  8. J. M. Dlugach and E. G. Yanovitskij, “The optical properties of Venus and the Jovian planets. II. Methods and results of calculations of the intensity of radiation diffusely reflected from semi-infinite homogeneous atmospheres,” Icarus 22, 66–81 (1974).
    [CrossRef]
  9. M. King and Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
    [CrossRef]
  10. A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka-Munk theory,” J. Phys. D 40, 2210–2216 (2007).
    [CrossRef]
  11. A. Ben-David, “Multiple-scattering effects on differential absorption for the transmission of a plane-parallel beam in a homogeneous medium,” Appl. Opt. 36, 1386–1398 (1997).
    [CrossRef]
  12. L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectr. Rad. Transfer 110, 1132–1146 (2009).
    [CrossRef]
  13. A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
    [CrossRef]
  14. V. P. Budak, D. A. Klyuykov, and S. V. Korkin, “Complete matrix solution of radiative transfer equation for PILE of horizontally homogeneous slabs,” J. Quant. Spectrosc. Radiat. Trans. 112, 1141–1148 (2011).
    [CrossRef]
  15. V. P. Afanas’ev, D. S. Efremenko, and A. V. Lubenchenko, “On the application of the invariant embedding method and the radiative transfer equation codes for surface state analysis,” in Light Scattering Reviews 8: Radiative Transfer and Light Scattering, A. A. Kokhanovsky, ed. (Springer, 2013), pp. 363–423.
  16. H. C. van de Hulst, “Asymptotic fitting, a method for solving anisotropic transfer problems in thick layers,” J. Comput. Phys. 3, 291–306 (1968).
    [CrossRef]
  17. H. G. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water?” Limnol. Oceanogr. 34, 1389–1409 (1989).
    [CrossRef]
  18. H. R. Gordon, O. B. Brown, and M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975).
    [CrossRef]
  19. S. Sathyendranath and T. Platt, “The spectral irradiance field at the surface and in the interior of the ocean: a model for applications in oceanography and remote sensing,” J. Geophys. Res. 93, 9270–9280 (1988).
    [CrossRef]
  20. R. H. Stavn, “Light attenuation in natural waters: Gershun’s law, Lambert-Beer law, and the mean light path,” Appl. Opt. 20, 2326–2327 (1981).
    [CrossRef]
  21. R. H. Stavn, “Lambert-Beer law in ocean waters: optical properties of water and of dissolved/suspended material, optical energy budgets,” Appl. Opt. 27, 222–231 (1988).
    [CrossRef]
  22. J. F. Cornet, C. G. Dussap, and G. Dubertret, “A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors: I. Coupling between light transfer and growth kinetics,” Biotechnol. Bioeng. 40, 817–825 (1992).
    [CrossRef]
  23. B. A. Bodhaine, N. B. Wood, E. G. Dutton, and R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean. Technol. 16, 1854–1861 (1999).
    [CrossRef]
  24. A. B. Kostinski, “On the extinction of radiation by a homogeneous but spatially correlated random medium,” J. Opt. Soc. Am. 18, 1929–1933 (2001).
    [CrossRef]
  25. H. R. Gordon and O. B. Brown, “Influence of bottom depth and albedo on the diffuse reflectance of a flat homogeneous ocean,” Appl. Opt. 13, 2153–2159 (1974).
    [CrossRef]
  26. J. T. O. Kirk, “Volume scattering function, average cosines, and the underwater light field,” Limnol. Oceanogr. 36, 455–467 (1991).
    [CrossRef]
  27. A. Ben-David, “Multiple-scattering transmission and an effective average photon path length of a plane-parallel beam in a homogeneous medium,” Appl. Opt. 34, 2802–2810 (1995).
    [CrossRef]
  28. Z. P. Lee, K. P. Du, and R. Arnone, “A model for the diffuse attenuation coefficient of downwelling irradiance,” J. Geophys. Res. 110, C02016 (2005).
    [CrossRef]
  29. A. A. Kokhanovsky, Light Scattering Media Optics (Springer-Praxis, 2004).
  30. A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 1: spherical albedo,” Color Res. Appl. 31, 491–497 (2006).
    [CrossRef]
  31. L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  32. V. P. Budak and S. V. Korkin, “On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering,” J. Quant. Spectr. Rad. Transfer. 109, 220–234 (2008).
    [CrossRef]
  33. C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 64, 227–254 (2000).
    [CrossRef]
  34. V. P. Budak and S. E. Sarmin, “Solution of the radiation transfer equation by the method of spherical harmonics in the small-angle modification,” Atmos. Opt. 3, 898–903 (1990).
  35. A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
    [CrossRef]
  36. H. C. van de Hulst, Multiple Light Scattering, Vol. 2 (Academic, 1980).
  37. H. C. van de Hulst, “The spherical albedo of a planet covered with a homogeneous cloud layer,” Astron. Astrophys. 35, 209–214 (1974).
  38. K.-N. Liou, An Introduction to Atmospheric Radiation (Academic, 1980).
  39. D. A. Siegel and T. D. Dickey, “Observations of the vertical structure of the diffuse attenuation coefficient spectrum,” Deep Sea Res. 34, 547–563 (1987).
    [CrossRef]
  40. D. R. Mishra, S. Narumalanu, D. Rundquist, and M. Lawson, “Characterizing the vertical diffuse attenuation coefficient for downwelling irradiance in coastal waters: implications for water penetration by high resolution satellite data,” ISPRS J. Photogrammetry Rem. Sens. 60, 48–64 (2005).
    [CrossRef]

2013 (1)

2011 (1)

V. P. Budak, D. A. Klyuykov, and S. V. Korkin, “Complete matrix solution of radiative transfer equation for PILE of horizontally homogeneous slabs,” J. Quant. Spectrosc. Radiat. Trans. 112, 1141–1148 (2011).
[CrossRef]

2010 (1)

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

2009 (1)

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectr. Rad. Transfer 110, 1132–1146 (2009).
[CrossRef]

2008 (1)

V. P. Budak and S. V. Korkin, “On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering,” J. Quant. Spectr. Rad. Transfer. 109, 220–234 (2008).
[CrossRef]

2007 (1)

A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka-Munk theory,” J. Phys. D 40, 2210–2216 (2007).
[CrossRef]

2006 (1)

A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 1: spherical albedo,” Color Res. Appl. 31, 491–497 (2006).
[CrossRef]

2005 (2)

Z. P. Lee, K. P. Du, and R. Arnone, “A model for the diffuse attenuation coefficient of downwelling irradiance,” J. Geophys. Res. 110, C02016 (2005).
[CrossRef]

D. R. Mishra, S. Narumalanu, D. Rundquist, and M. Lawson, “Characterizing the vertical diffuse attenuation coefficient for downwelling irradiance in coastal waters: implications for water penetration by high resolution satellite data,” ISPRS J. Photogrammetry Rem. Sens. 60, 48–64 (2005).
[CrossRef]

2001 (1)

A. B. Kostinski, “On the extinction of radiation by a homogeneous but spatially correlated random medium,” J. Opt. Soc. Am. 18, 1929–1933 (2001).
[CrossRef]

2000 (1)

C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 64, 227–254 (2000).
[CrossRef]

1999 (1)

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean. Technol. 16, 1854–1861 (1999).
[CrossRef]

1997 (1)

1995 (1)

1992 (1)

J. F. Cornet, C. G. Dussap, and G. Dubertret, “A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors: I. Coupling between light transfer and growth kinetics,” Biotechnol. Bioeng. 40, 817–825 (1992).
[CrossRef]

1991 (1)

J. T. O. Kirk, “Volume scattering function, average cosines, and the underwater light field,” Limnol. Oceanogr. 36, 455–467 (1991).
[CrossRef]

1990 (1)

V. P. Budak and S. E. Sarmin, “Solution of the radiation transfer equation by the method of spherical harmonics in the small-angle modification,” Atmos. Opt. 3, 898–903 (1990).

1989 (1)

H. G. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water?” Limnol. Oceanogr. 34, 1389–1409 (1989).
[CrossRef]

1988 (2)

S. Sathyendranath and T. Platt, “The spectral irradiance field at the surface and in the interior of the ocean: a model for applications in oceanography and remote sensing,” J. Geophys. Res. 93, 9270–9280 (1988).
[CrossRef]

R. H. Stavn, “Lambert-Beer law in ocean waters: optical properties of water and of dissolved/suspended material, optical energy budgets,” Appl. Opt. 27, 222–231 (1988).
[CrossRef]

1987 (1)

D. A. Siegel and T. D. Dickey, “Observations of the vertical structure of the diffuse attenuation coefficient spectrum,” Deep Sea Res. 34, 547–563 (1987).
[CrossRef]

1986 (1)

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

1981 (1)

1975 (1)

1974 (3)

H. R. Gordon and O. B. Brown, “Influence of bottom depth and albedo on the diffuse reflectance of a flat homogeneous ocean,” Appl. Opt. 13, 2153–2159 (1974).
[CrossRef]

H. C. van de Hulst, “The spherical albedo of a planet covered with a homogeneous cloud layer,” Astron. Astrophys. 35, 209–214 (1974).

J. M. Dlugach and E. G. Yanovitskij, “The optical properties of Venus and the Jovian planets. II. Methods and results of calculations of the intensity of radiation diffusely reflected from semi-infinite homogeneous atmospheres,” Icarus 22, 66–81 (1974).
[CrossRef]

1968 (1)

H. C. van de Hulst, “Asymptotic fitting, a method for solving anisotropic transfer problems in thick layers,” J. Comput. Phys. 3, 291–306 (1968).
[CrossRef]

1948 (1)

1947 (1)

1941 (1)

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

1931 (1)

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

1930 (1)

M. Gurevich, “Über eine rationelle klassifikation der lichtenstreuenden medien,” Physikalische Zeitschrift,” 31, 753–763 (1930).

1905 (1)

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

1862 (1)

G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Math. Phys. Papers 4, 145–156 (1862).

Afanas’ev, V. P.

V. P. Afanas’ev, D. S. Efremenko, and A. V. Lubenchenko, “On the application of the invariant embedding method and the radiative transfer equation codes for surface state analysis,” in Light Scattering Reviews 8: Radiative Transfer and Light Scattering, A. A. Kokhanovsky, ed. (Springer, 2013), pp. 363–423.

Arnone, R.

Z. P. Lee, K. P. Du, and R. Arnone, “A model for the diffuse attenuation coefficient of downwelling irradiance,” J. Geophys. Res. 110, C02016 (2005).
[CrossRef]

Bass, L. P.

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectr. Rad. Transfer 110, 1132–1146 (2009).
[CrossRef]

Ben-David, A.

Bodhaine, B. A.

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean. Technol. 16, 1854–1861 (1999).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

Brown, O. B.

Budak, V. P.

V. P. Budak, D. A. Klyuykov, and S. V. Korkin, “Complete matrix solution of radiative transfer equation for PILE of horizontally homogeneous slabs,” J. Quant. Spectrosc. Radiat. Trans. 112, 1141–1148 (2011).
[CrossRef]

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectr. Rad. Transfer 110, 1132–1146 (2009).
[CrossRef]

V. P. Budak and S. V. Korkin, “On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering,” J. Quant. Spectr. Rad. Transfer. 109, 220–234 (2008).
[CrossRef]

V. P. Budak and S. E. Sarmin, “Solution of the radiation transfer equation by the method of spherical harmonics in the small-angle modification,” Atmos. Opt. 3, 898–903 (1990).

C-Labonnote, L.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

Cornet, C.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

Cornet, J. F.

J. F. Cornet, C. G. Dussap, and G. Dubertret, “A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors: I. Coupling between light transfer and growth kinetics,” Biotechnol. Bioeng. 40, 817–825 (1992).
[CrossRef]

Dickey, T. D.

D. A. Siegel and T. D. Dickey, “Observations of the vertical structure of the diffuse attenuation coefficient spectrum,” Deep Sea Res. 34, 547–563 (1987).
[CrossRef]

Dlugach, J. M.

J. M. Dlugach and E. G. Yanovitskij, “The optical properties of Venus and the Jovian planets. II. Methods and results of calculations of the intensity of radiation diffusely reflected from semi-infinite homogeneous atmospheres,” Icarus 22, 66–81 (1974).
[CrossRef]

Du, K. P.

Z. P. Lee, K. P. Du, and R. Arnone, “A model for the diffuse attenuation coefficient of downwelling irradiance,” J. Geophys. Res. 110, C02016 (2005).
[CrossRef]

Duan, M.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

Dubertret, G.

J. F. Cornet, C. G. Dussap, and G. Dubertret, “A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors: I. Coupling between light transfer and growth kinetics,” Biotechnol. Bioeng. 40, 817–825 (1992).
[CrossRef]

Dussap, C. G.

J. F. Cornet, C. G. Dussap, and G. Dubertret, “A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors: I. Coupling between light transfer and growth kinetics,” Biotechnol. Bioeng. 40, 817–825 (1992).
[CrossRef]

Dutton, E. G.

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean. Technol. 16, 1854–1861 (1999).
[CrossRef]

Efremenko, D. S.

V. P. Afanas’ev, D. S. Efremenko, and A. V. Lubenchenko, “On the application of the invariant embedding method and the radiative transfer equation codes for surface state analysis,” in Light Scattering Reviews 8: Radiative Transfer and Light Scattering, A. A. Kokhanovsky, ed. (Springer, 2013), pp. 363–423.

Emde, C.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

Gordon, H. G.

H. G. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water?” Limnol. Oceanogr. 34, 1389–1409 (1989).
[CrossRef]

Gordon, H. R.

Greenstein, J. L.

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Gurevich, M.

M. Gurevich, “Über eine rationelle klassifikation der lichtenstreuenden medien,” Physikalische Zeitschrift,” 31, 753–763 (1930).

Harshvardhan,

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

Henyey, L. C.

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

Jacobs, M. M.

Katsev, I. L.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

King, M.

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

Kirk, J. T. O.

J. T. O. Kirk, “Volume scattering function, average cosines, and the underwater light field,” Limnol. Oceanogr. 36, 455–467 (1991).
[CrossRef]

Klyukov, D. A.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

Klyuykov, D. A.

V. P. Budak, D. A. Klyuykov, and S. V. Korkin, “Complete matrix solution of radiative transfer equation for PILE of horizontally homogeneous slabs,” J. Quant. Spectrosc. Radiat. Trans. 112, 1141–1148 (2011).
[CrossRef]

Kokhanovsky, A. A.

L. G. Sokoletsky, A. A. Kokhanovsky, and F. Shen, “Comparative analysis of radiative transfer approaches for calculation of diffuse reflectance of plane-parallel light scattering layers,” Appl. Opt. 52, 8471–8483 (2013).

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectr. Rad. Transfer 110, 1132–1146 (2009).
[CrossRef]

A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka-Munk theory,” J. Phys. D 40, 2210–2216 (2007).
[CrossRef]

A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 1: spherical albedo,” Color Res. Appl. 31, 491–497 (2006).
[CrossRef]

A. A. Kokhanovsky, Light Scattering Media Optics (Springer-Praxis, 2004).

Korkin, S. V.

V. P. Budak, D. A. Klyuykov, and S. V. Korkin, “Complete matrix solution of radiative transfer equation for PILE of horizontally homogeneous slabs,” J. Quant. Spectrosc. Radiat. Trans. 112, 1141–1148 (2011).
[CrossRef]

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

V. P. Budak and S. V. Korkin, “On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering,” J. Quant. Spectr. Rad. Transfer. 109, 220–234 (2008).
[CrossRef]

Kostinski, A. B.

A. B. Kostinski, “On the extinction of radiation by a homogeneous but spatially correlated random medium,” J. Opt. Soc. Am. 18, 1929–1933 (2001).
[CrossRef]

Kubelka, P.

P. Kubelka, “New contributions to the optics of intensely light-scattering material. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
[CrossRef]

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Kuznetsov, V. S.

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectr. Rad. Transfer 110, 1132–1146 (2009).
[CrossRef]

Lawson, M.

D. R. Mishra, S. Narumalanu, D. Rundquist, and M. Lawson, “Characterizing the vertical diffuse attenuation coefficient for downwelling irradiance in coastal waters: implications for water penetration by high resolution satellite data,” ISPRS J. Photogrammetry Rem. Sens. 60, 48–64 (2005).
[CrossRef]

Lee, Z. P.

Z. P. Lee, K. P. Du, and R. Arnone, “A model for the diffuse attenuation coefficient of downwelling irradiance,” J. Geophys. Res. 110, C02016 (2005).
[CrossRef]

Liou, K.-N.

K.-N. Liou, An Introduction to Atmospheric Radiation (Academic, 1980).

Lubenchenko, A. V.

V. P. Afanas’ev, D. S. Efremenko, and A. V. Lubenchenko, “On the application of the invariant embedding method and the radiative transfer equation codes for surface state analysis,” in Light Scattering Reviews 8: Radiative Transfer and Light Scattering, A. A. Kokhanovsky, ed. (Springer, 2013), pp. 363–423.

Lunetta, R. S.

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectr. Rad. Transfer 110, 1132–1146 (2009).
[CrossRef]

Mayer, B.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

Min, Q.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

Mishra, D. R.

D. R. Mishra, S. Narumalanu, D. Rundquist, and M. Lawson, “Characterizing the vertical diffuse attenuation coefficient for downwelling irradiance in coastal waters: implications for water penetration by high resolution satellite data,” ISPRS J. Photogrammetry Rem. Sens. 60, 48–64 (2005).
[CrossRef]

Munk, F.

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Nakajima, T.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

Narumalanu, S.

D. R. Mishra, S. Narumalanu, D. Rundquist, and M. Lawson, “Characterizing the vertical diffuse attenuation coefficient for downwelling irradiance in coastal waters: implications for water penetration by high resolution satellite data,” ISPRS J. Photogrammetry Rem. Sens. 60, 48–64 (2005).
[CrossRef]

Nikolaeva, O. V.

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectr. Rad. Transfer 110, 1132–1146 (2009).
[CrossRef]

Ota, Y.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

Platt, T.

S. Sathyendranath and T. Platt, “The spectral irradiance field at the surface and in the interior of the ocean: a model for applications in oceanography and remote sensing,” J. Geophys. Res. 93, 9270–9280 (1988).
[CrossRef]

Prikhach, A. S.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

Rozanov, V. V.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

Rundquist, D.

D. R. Mishra, S. Narumalanu, D. Rundquist, and M. Lawson, “Characterizing the vertical diffuse attenuation coefficient for downwelling irradiance in coastal waters: implications for water penetration by high resolution satellite data,” ISPRS J. Photogrammetry Rem. Sens. 60, 48–64 (2005).
[CrossRef]

Sarmin, S. E.

V. P. Budak and S. E. Sarmin, “Solution of the radiation transfer equation by the method of spherical harmonics in the small-angle modification,” Atmos. Opt. 3, 898–903 (1990).

Sathyendranath, S.

S. Sathyendranath and T. Platt, “The spectral irradiance field at the surface and in the interior of the ocean: a model for applications in oceanography and remote sensing,” J. Geophys. Res. 93, 9270–9280 (1988).
[CrossRef]

Schuster, A.

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Shen, F.

Siegel, D. A.

D. A. Siegel and T. D. Dickey, “Observations of the vertical structure of the diffuse attenuation coefficient spectrum,” Deep Sea Res. 34, 547–563 (1987).
[CrossRef]

Siewert, C. E.

C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 64, 227–254 (2000).
[CrossRef]

Slusser, R.

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean. Technol. 16, 1854–1861 (1999).
[CrossRef]

Sokoletsky, L. G.

L. G. Sokoletsky, A. A. Kokhanovsky, and F. Shen, “Comparative analysis of radiative transfer approaches for calculation of diffuse reflectance of plane-parallel light scattering layers,” Appl. Opt. 52, 8471–8483 (2013).

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectr. Rad. Transfer 110, 1132–1146 (2009).
[CrossRef]

A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 1: spherical albedo,” Color Res. Appl. 31, 491–497 (2006).
[CrossRef]

Stavn, R. H.

Stokes, G. G.

G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Math. Phys. Papers 4, 145–156 (1862).

Tuckerman, L. B.

van de Hulst, H. C.

H. C. van de Hulst, “The spherical albedo of a planet covered with a homogeneous cloud layer,” Astron. Astrophys. 35, 209–214 (1974).

H. C. van de Hulst, “Asymptotic fitting, a method for solving anisotropic transfer problems in thick layers,” J. Comput. Phys. 3, 291–306 (1968).
[CrossRef]

H. C. van de Hulst, Multiple Light Scattering, Vol. 2 (Academic, 1980).

Wood, N. B.

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean. Technol. 16, 1854–1861 (1999).
[CrossRef]

Yanovitskij, E. G.

J. M. Dlugach and E. G. Yanovitskij, “The optical properties of Venus and the Jovian planets. II. Methods and results of calculations of the intensity of radiation diffusely reflected from semi-infinite homogeneous atmospheres,” Icarus 22, 66–81 (1974).
[CrossRef]

Yokota, T.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

Zege, E. P.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

Appl. Opt. (7)

Astron. Astrophys. (1)

H. C. van de Hulst, “The spherical albedo of a planet covered with a homogeneous cloud layer,” Astron. Astrophys. 35, 209–214 (1974).

Astrophys. J. (2)

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Atmos. Opt. (1)

V. P. Budak and S. E. Sarmin, “Solution of the radiation transfer equation by the method of spherical harmonics in the small-angle modification,” Atmos. Opt. 3, 898–903 (1990).

Biotechnol. Bioeng. (1)

J. F. Cornet, C. G. Dussap, and G. Dubertret, “A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors: I. Coupling between light transfer and growth kinetics,” Biotechnol. Bioeng. 40, 817–825 (1992).
[CrossRef]

Color Res. Appl. (1)

A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 1: spherical albedo,” Color Res. Appl. 31, 491–497 (2006).
[CrossRef]

Deep Sea Res. (1)

D. A. Siegel and T. D. Dickey, “Observations of the vertical structure of the diffuse attenuation coefficient spectrum,” Deep Sea Res. 34, 547–563 (1987).
[CrossRef]

Icarus (1)

J. M. Dlugach and E. G. Yanovitskij, “The optical properties of Venus and the Jovian planets. II. Methods and results of calculations of the intensity of radiation diffusely reflected from semi-infinite homogeneous atmospheres,” Icarus 22, 66–81 (1974).
[CrossRef]

ISPRS J. Photogrammetry Rem. Sens. (1)

D. R. Mishra, S. Narumalanu, D. Rundquist, and M. Lawson, “Characterizing the vertical diffuse attenuation coefficient for downwelling irradiance in coastal waters: implications for water penetration by high resolution satellite data,” ISPRS J. Photogrammetry Rem. Sens. 60, 48–64 (2005).
[CrossRef]

J. Atmos. Ocean. Technol. (1)

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean. Technol. 16, 1854–1861 (1999).
[CrossRef]

J. Atmos. Sci. (1)

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

J. Comput. Phys. (1)

H. C. van de Hulst, “Asymptotic fitting, a method for solving anisotropic transfer problems in thick layers,” J. Comput. Phys. 3, 291–306 (1968).
[CrossRef]

J. Geophys. Res. (2)

S. Sathyendranath and T. Platt, “The spectral irradiance field at the surface and in the interior of the ocean: a model for applications in oceanography and remote sensing,” J. Geophys. Res. 93, 9270–9280 (1988).
[CrossRef]

Z. P. Lee, K. P. Du, and R. Arnone, “A model for the diffuse attenuation coefficient of downwelling irradiance,” J. Geophys. Res. 110, C02016 (2005).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Phys. D (1)

A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka-Munk theory,” J. Phys. D 40, 2210–2216 (2007).
[CrossRef]

J. Quant. Spectr. Rad. Transfer (1)

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectr. Rad. Transfer 110, 1132–1146 (2009).
[CrossRef]

J. Quant. Spectr. Rad. Transfer. (1)

V. P. Budak and S. V. Korkin, “On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering,” J. Quant. Spectr. Rad. Transfer. 109, 220–234 (2008).
[CrossRef]

J. Quant. Spectrosc. Radiat. Trans. (1)

V. P. Budak, D. A. Klyuykov, and S. V. Korkin, “Complete matrix solution of radiative transfer equation for PILE of horizontally homogeneous slabs,” J. Quant. Spectrosc. Radiat. Trans. 112, 1141–1148 (2011).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111, 1931–1946 (2010).
[CrossRef]

C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 64, 227–254 (2000).
[CrossRef]

Limnol. Oceanogr. (2)

J. T. O. Kirk, “Volume scattering function, average cosines, and the underwater light field,” Limnol. Oceanogr. 36, 455–467 (1991).
[CrossRef]

H. G. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water?” Limnol. Oceanogr. 34, 1389–1409 (1989).
[CrossRef]

Math. Phys. Papers (1)

G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Math. Phys. Papers 4, 145–156 (1862).

Physikalische Zeitschrift (1)

M. Gurevich, “Über eine rationelle klassifikation der lichtenstreuenden medien,” Physikalische Zeitschrift,” 31, 753–763 (1930).

Z. Tech. Phys. (1)

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Other (5)

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

V. P. Afanas’ev, D. S. Efremenko, and A. V. Lubenchenko, “On the application of the invariant embedding method and the radiative transfer equation codes for surface state analysis,” in Light Scattering Reviews 8: Radiative Transfer and Light Scattering, A. A. Kokhanovsky, ed. (Springer, 2013), pp. 363–423.

H. C. van de Hulst, Multiple Light Scattering, Vol. 2 (Academic, 1980).

K.-N. Liou, An Introduction to Atmospheric Radiation (Academic, 1980).

A. A. Kokhanovsky, Light Scattering Media Optics (Springer-Praxis, 2004).

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