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

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]

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

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.

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

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

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

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

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Figures (8)

Fig. 1.
Fig. 1.

Schematic picture for the case of plane transmittance. The Tp1(z) and t1(z) are the plane transmittance and spherical transmittance, respectively, after the first light passing through the slab of thickness z. Similarly, the Rp1(z) and r1(z) are the plane albedo and spherical albedo, respectively, before the first light passing through the slab of thickness z.

Fig. 2.
Fig. 2.

Scattering phase functions p(θ) used for modeling. The main selected p(θ) shown by solid lines, while corresponding Henyey–Greenstein p(θ) (with the same B values) shown by dash lines.

Fig. 3.
Fig. 3.

Plane transmittance as a function of optical depth τ, single-scattering albedo ω0, and backscattering ratio B computed by the MDOM method at θi=30.5°. The dependencies calculated by the BLB-a and BLB-c laws also shown for comparison.

Fig. 4.
Fig. 4.

Relative errors for the Tp(τ) computed by the (a) BLB-a, (b) BLB-c, (c) QSSA, (d) Gordon, (e) Kirk, (f) Cornet, (g) Ben-David, and (h) Lee models relative to Tp(τ) computed by the MDOM algorithm at θi=30.5° as a function of optical depth τ, single-scattering albedo ω0, and backscattering ratio B.

Fig. 5.
Fig. 5.

Plane transmittance Tp as a function of ω0 computed by the selected analytical methods at incidence angle θi=30.5°, optical depth τ=1, and three different phase functions with (a) g=0.00, (b) 0.50, and (c) 0.96.

Fig. 6.
Fig. 6.

Average diffuse attenuation coefficient K¯d(0z) normalized to the linear attenuation coefficient c as a function of optical depth τ, single-scattering albedo ω0 and backscattering ratio B computed by the MDOM method at θi=30.5°.

Fig. 7.
Fig. 7.

Same as Fig. 4, but for the K¯d(0z).

Fig. 8.
Fig. 8.

Dependence of K¯d(0z)μi/c on 1Fω0 computed by the MDOM for the stable light regime in the top of the layer (a) and at the asymptotic light regime (b). Note that a 11 line corresponds to the QSSA approximation.

Tables (3)

Tables Icon

Table 1. Parameters Used for p(θ) Generation

Tables Icon

Table 2. Accuracy of Selected Models for the Plane Transmittance Tp(θi,τ) at θi=30.5° and 0τ10a

Tables Icon

Table 3. Accuracy of Selected Models for the Diffuse Attenuation Coefficient Kd(θi,τ) at θi=30.5° and 0τ10a

Equations (11)

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

Tp(μi,τ)=1π02π01T(μi,μv,ϕ,τ)μvdμvdϕ=201T¯(μi,μv,τ)μvdμv,T¯(μi,μv,τ)=12π02πT(μi,μv,ϕ,τ)dϕ,
t(τ)=201Tp(μi,τ)μidμi=40101T¯(μi,μvi,τ)μiμvdμidμv.
It=Ii{1Rp1(z)}{1r1(z)}Tp1(z)[1+r12(z)t12(z)+r14(z)t14(z)+]=Ii{1Rp1(z)}{1r1(z)}Tp1(z)1r12(z)t12(z),
Tp(z)=It(z)Ii={1Rp1(z)}{1r1(z)}Tp1(z)1r12(z)t12(z).
t(z)={1r1(z)}2t1(z)1r12(z)t12(z).
MAPE(%)=100%i=1n|(x˜ixi)/x˜i|n,
NRMSE(%)=100%i=1n(x˜ixi)2n1x¯,
Kd(z)=ddzlnIt(z)Ii=1It(z)dIt(z)dz
Tp(μi,z)=It(μi,z)Ii(μi)=exp[K¯d(0z)z]=exp[K¯d(0z)cτ],
K¯d(0z)=1z0zKd(z)dz.
K¯d(0z)c=lnTp(μi,z)τ.

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