Abstract

We present an analysis of a number of different approximations for the diffuse reflectance (spherical and plane albedo) of a semi-infinite, unbounded, plane-parallel, and optically homogeneous layer. The maximally wide optical conditions (from full absorption to full scattering and from fully forward to fully backward scattering) at collimated, diffuse, and combined illumination conditions were considered. The approximations were analyzed from the point of view of their physical limitations and compared to the numerical radiative transfer solutions, whenever it was possible. The main factors impacting the spherical and plane albedo were revealed for the known and unknown scattering phase functions. The main criterion for inclusion of the models in analysis was the possibility of practical use, i.e., approximations were well parameterized and only included easily measured or estimated parameters. We give a detailed analysis of errors for different models. An algorithm for recalculation of results under combined (direct and diffuse) illumination also has been developed.

© 2013 Optical Society of America

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2012 (1)

L. G. Sokoletsky, R. S. Lunetta, M. S. Wetz, and H. W. Paerl, “Assessment of the water quality components in turbid estuarine waters based on radiative transfer approximations,” Isr. J. Plant Sci. 60, 209–229 (2012).
[CrossRef]

2011 (1)

A. García-Valenzuela, F. L. S. Cuppo, and J. A. Olivares, “An assessment of Saunderson corrections to the diffuse reflectance of paint films,” J. Phys. Conf. Ser. 274, 012125 (2011).
[CrossRef]

2009 (2)

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. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[CrossRef]

D. Yudovsky and L. Pilon, “Simple and accurate expressions for diffuse reflectance of semi-infinite and two-layer absorbing and scattering media,” Appl. Opt. 48, 6670–6683 (2009).
[CrossRef]

2008 (2)

2007 (1)

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

2006 (3)

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 and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 2: plane albedo and reflection function,” Color Res. Appl. 31, 498–509 (2006).
[CrossRef]

G. Zonios and A. Dimou, “Modeling diffuse reflectance from semi-infinite turbid media: application to the study of skin optical properties,” Opt. Express 14, 8661–8674 (2006).
[CrossRef]

2005 (1)

2003 (1)

E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, and M. T. Vihena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
[CrossRef]

2002 (2)

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

V. I. Haltrin, “One-parameter two-term Henyey–Greenstein phase function for light scattering in seawater,” Appl. Opt. 41, 1022–1028 (2002).
[CrossRef]

2000 (1)

A. A. Kokhanovsky, “The determination of the effective radius of drops in water clouds from polarization measurements,” Phys. Chem. Earth B 25, 471–474 (2000).
[CrossRef]

1999 (2)

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, and 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. D. Mobley, “Estimation of the remote-sensing reflectance from above-surface measurements,” Appl. Opt. 38, 7442–7455 (1999).
[CrossRef]

1998 (2)

1997 (1)

P. E. Pierce and R. T. Marcus, “Radiative transfer theory solid color-matching calculations,” Color Res. Appl. 22, 72–87 (1997).
[CrossRef]

1996 (1)

1993 (2)

1992 (3)

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]

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef]

T. Nakajima and M. D. King, “Asymptotic theory for optically thick layers: application to the discrete ordinates method,” Appl. Opt. 31, 7669–7683 (1992).
[CrossRef]

1989 (2)

H. G. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[CrossRef]

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef]

1988 (1)

1987 (2)

R. F. Bonner, R. Nossal, S. Havlin, and G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
[CrossRef]

M. J. C. Gemert and W. M. Star, “Relations between the Kubelka–Munk and the transport equation models for anisotropic scattering,” Lasers Life Sci. 1, 287–298 (1987).

1986 (1)

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

1979 (1)

1977 (1)

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977).
[CrossRef]

1975 (1)

1974 (3)

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]

B. M. Golubitsky, I. M. Levin, and M. V. Tantashev, “Brightness coefficient of a semi-infinite layer of sea water,” Izv. Atmos. Ocean. Phys. 10, 1235–1238 (1974).

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

1973 (1)

1972 (1)

1971 (2)

O. V. Bushmakova, E. P. Zege, and I. L. Katsev, “On asymptotic equations for brightness coefficients of optically thick light scattering layers,” Doklady Acad. Sci. BSSR 15, 309–311 (1971).

P. S. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
[CrossRef]

1970 (1)

L. W. Richards, “The calculation of the optical performance of paint films,” J. Paint Technol. 42, 276–286 (1970).

1962 (1)

G. V. Rozenberg, “Light characteristics of thick layers of a weakly absorbing scattering medium,” Doklady Acad. Sci. USSR 145, 775–777 (1962).

1948 (1)

1942 (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,” Zeitschrift für Technische Physik. 12, 593–601 (1931).

1930 (1)

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

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. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[CrossRef]

Bonner, R. F.

Brown, O. B.

Budak, V. 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. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[CrossRef]

Bushmakova, O. V.

O. V. Bushmakova, E. P. Zege, and I. L. Katsev, “On asymptotic equations for brightness coefficients of optically thick light scattering layers,” Doklady Acad. Sci. BSSR 15, 309–311 (1971).

Campos Velho, H. F.

E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, and M. T. Vihena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
[CrossRef]

Chalhoub, E. S.

E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, and M. T. Vihena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
[CrossRef]

Chandrasekhar, S.

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

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]

Cuppo, F. L. S.

A. García-Valenzuela, F. L. S. Cuppo, and J. A. Olivares, “An assessment of Saunderson corrections to the diffuse reflectance of paint films,” J. Phys. Conf. Ser. 274, 012125 (2011).
[CrossRef]

Dimou, A.

Dlugach, J. M.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, and 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]

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]

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]

Farrell, T. J.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef]

Field, M. S.

Flock, S. T.

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef]

Garcia, R. D. M.

E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, and M. T. Vihena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
[CrossRef]

García-Valenzuela, A.

A. García-Valenzuela, F. L. S. Cuppo, and J. A. Olivares, “An assessment of Saunderson corrections to the diffuse reflectance of paint films,” J. Phys. Conf. Ser. 274, 012125 (2011).
[CrossRef]

Gemert, M. J. C.

M. J. C. Gemert and W. M. Star, “Relations between the Kubelka–Munk and the transport equation models for anisotropic scattering,” Lasers Life Sci. 1, 287–298 (1987).

Gentili, B.

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, U.S. Department of Commerce, 1977).

Golubitsky, B. M.

B. M. Golubitsky, I. M. Levin, and M. V. Tantashev, “Brightness coefficient of a semi-infinite layer of sea water,” Izv. Atmos. Ocean. Phys. 10, 1235–1238 (1974).

Gordon, H. G.

H. G. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[CrossRef]

H. G. Gordon, “Simple calculation of the diffuse reflectance of ocean,” Appl. Opt. 12, 2803–2804 (1973).
[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).

Haltrin, V. I.

Hapke, B.

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

B. Hapke, Theory of Reflectance and Emittance Spectroscopy (University Press, 1993).

Harshvardhan,

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

Havlin, S.

Henyey, L. C.

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

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, U.S. Department of Commerce, 1977).

Jacobs, M. M.

Jin, Z.

Katsev, I. L.

O. V. Bushmakova, E. P. Zege, and I. L. Katsev, “On asymptotic equations for brightness coefficients of optically thick light scattering layers,” Doklady Acad. Sci. BSSR 15, 309–311 (1971).

Kattawar, G. W.

King, M.

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

King, M. D.

Klier, K.

Kokhanovsky, A. A.

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. Spectrosc. Radiat. 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 2: plane albedo and reflection function,” Color Res. Appl. 31, 498–509 (2006).
[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, “The determination of the effective radius of drops in water clouds from polarization measurements,” Phys. Chem. Earth B 25, 471–474 (2000).
[CrossRef]

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

L. G. Sokoletsky and A. A. Kokhanovsky, “Reflective characteristics of natural waters: The accuracy of selected approximations,” in Proceedings of the D. S. Rozhdestvensky Third International Conference: Current Problems in Optics of Natural Waters, I. Levin and G. Gilbert, eds. (Saint-Petersburg, Russia, 2005), pp. 56–63.

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,” Zeitschrift für Technische Physik. 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. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[CrossRef]

Levin, I. M.

B. M. Golubitsky, I. M. Levin, and M. V. Tantashev, “Brightness coefficient of a semi-infinite layer of sea water,” Izv. Atmos. Ocean. Phys. 10, 1235–1238 (1974).

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, U.S. Department of Commerce, 1977).

Lunetta, R. S.

L. G. Sokoletsky, R. S. Lunetta, M. S. Wetz, and H. W. Paerl, “Assessment of the water quality components in turbid estuarine waters based on radiative transfer approximations,” Isr. J. Plant Sci. 60, 209–229 (2012).
[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. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[CrossRef]

Marcus, R. T.

P. E. Pierce and R. T. Marcus, “Radiative transfer theory solid color-matching calculations,” Color Res. Appl. 22, 72–87 (1997).
[CrossRef]

Masoliver, J.

G. H. Weiss, J. M. Porrà, and J. Masoliver, “The continuous-time random walk description of photon motion in an isotropic medium,” Opt. Commun. 146, 268–276 (1998).
[CrossRef]

Meador, W. E.

Mishchenko, M. I.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, and 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]

Mobley, C. D.

Morel, A.

Mudgett, P. S.

Munk, F.

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Zeitschrift für Technische Physik. 12, 593–601 (1931).

Nakajima, T.

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, U.S. Department of Commerce, 1977).

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. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[CrossRef]

Nossal, R.

Nussenzveig, H. M.

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977).
[CrossRef]

Olivares, J. A.

A. García-Valenzuela, F. L. S. Cuppo, and J. A. Olivares, “An assessment of Saunderson corrections to the diffuse reflectance of paint films,” J. Phys. Conf. Ser. 274, 012125 (2011).
[CrossRef]

Paerl, H. W.

L. G. Sokoletsky, R. S. Lunetta, M. S. Wetz, and H. W. Paerl, “Assessment of the water quality components in turbid estuarine waters based on radiative transfer approximations,” Isr. J. Plant Sci. 60, 209–229 (2012).
[CrossRef]

Partovi, F.

Patterson, M. S.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef]

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef]

Pierce, P. E.

P. E. Pierce and R. T. Marcus, “Radiative transfer theory solid color-matching calculations,” Color Res. Appl. 22, 72–87 (1997).
[CrossRef]

Pilon, L.

Porrà, J. M.

G. H. Weiss, J. M. Porrà, and J. Masoliver, “The continuous-time random walk description of photon motion in an isotropic medium,” Opt. Commun. 146, 268–276 (1998).
[CrossRef]

Rava, R. P.

Reinersman, P.

Richards, L. W.

P. S. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
[CrossRef]

L. W. Richards, “The calculation of the optical performance of paint films,” J. Paint Technol. 42, 276–286 (1970).

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, U.S. Department of Commerce, 1977).

Rogers, G. L.

Rozenberg, G. V.

G. V. Rozenberg, “Light characteristics of thick layers of a weakly absorbing scattering medium,” Doklady Acad. Sci. USSR 145, 775–777 (1962).

Saunderson, J. L.

Sobolev, V. V.

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, 1975).

Sokoletsky, L.

L. Sokoletsky, “A comparative analysis of selected radiative transfer approaches for aquatic environments,” Appl. Opt. 44, 136–148 (2005).
[CrossRef]

L. Sokoletsky, “A comparative analysis of simple radiative transfer approaches for aquatic environments,” in Proceedings of 2004 ENVISAT & ERS Symposium, H. Lacoste, ed. (Noordwijk, 2005), pp. 1–7.

Sokoletsky, L. G.

L. G. Sokoletsky, R. S. Lunetta, M. S. Wetz, and H. W. Paerl, “Assessment of the water quality components in turbid estuarine waters based on radiative transfer approximations,” Isr. J. Plant Sci. 60, 209–229 (2012).
[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. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[CrossRef]

A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 2: plane albedo and reflection function,” Color Res. Appl. 31, 498–509 (2006).
[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]

L. G. Sokoletsky and A. A. Kokhanovsky, “Reflective characteristics of natural waters: The accuracy of selected approximations,” in Proceedings of the D. S. Rozhdestvensky Third International Conference: Current Problems in Optics of Natural Waters, I. Levin and G. Gilbert, eds. (Saint-Petersburg, Russia, 2005), pp. 56–63.

Stamnes, K.

Star, W. M.

M. J. C. Gemert and W. M. Star, “Relations between the Kubelka–Munk and the transport equation models for anisotropic scattering,” Lasers Life Sci. 1, 287–298 (1987).

Stavn, R. H.

Tantashev, M. V.

B. M. Golubitsky, I. M. Levin, and M. V. Tantashev, “Brightness coefficient of a semi-infinite layer of sea water,” Izv. Atmos. Ocean. Phys. 10, 1235–1238 (1974).

Thennadil, S. N.

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, Light Scattering by Small Particles (Dover, 1981).

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

Vihena, M. T.

E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, and M. T. Vihena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
[CrossRef]

Weaver, W. R.

Weiss, G. H.

G. H. Weiss, J. M. Porrà, and J. Masoliver, “The continuous-time random walk description of photon motion in an isotropic medium,” Opt. Commun. 146, 268–276 (1998).
[CrossRef]

R. F. Bonner, R. Nossal, S. Havlin, and G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
[CrossRef]

Wetz, M. S.

L. G. Sokoletsky, R. S. Lunetta, M. S. Wetz, and H. W. Paerl, “Assessment of the water quality components in turbid estuarine waters based on radiative transfer approximations,” Isr. J. Plant Sci. 60, 209–229 (2012).
[CrossRef]

Wilson, B.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef]

Wilson, B. C.

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef]

Wu, J.

Wyman, D. R.

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef]

Yanovitskij, E. G.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, and 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]

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]

Yudovsky, D.

Zakharova, N. T.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, and 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.

O. V. Bushmakova, E. P. Zege, and I. L. Katsev, “On asymptotic equations for brightness coefficients of optically thick light scattering layers,” Doklady Acad. Sci. BSSR 15, 309–311 (1971).

Zonios, G.

Appl. Opt. (14)

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
[CrossRef]

V. I. Haltrin, “Self-consistent approach to the solution of the light transfer problem for irradiances in marine waters with arbitrary turbidity, depth, and surface illumination. I. Case of absorption and elastic scattering,” Appl. Opt. 37, 3773–3784 (1998).
[CrossRef]

L. Sokoletsky, “A comparative analysis of selected radiative transfer approaches for aquatic environments,” Appl. Opt. 44, 136–148 (2005).
[CrossRef]

A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters. III. Implication of bidirectionality for the remote-sensing problem,” Appl. Opt. 35, 4850–4862 (1996).
[CrossRef]

C. D. Mobley, “Estimation of the remote-sensing reflectance from above-surface measurements,” Appl. Opt. 38, 7442–7455 (1999).
[CrossRef]

T. Nakajima and M. D. King, “Asymptotic theory for optically thick layers: application to the discrete ordinates method,” Appl. Opt. 31, 7669–7683 (1992).
[CrossRef]

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]

H. G. Gordon, “Simple calculation of the diffuse reflectance of ocean,” Appl. Opt. 12, 2803–2804 (1973).
[CrossRef]

V. I. Haltrin, “One-parameter two-term Henyey–Greenstein phase function for light scattering in seawater,” Appl. Opt. 41, 1022–1028 (2002).
[CrossRef]

P. S. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
[CrossRef]

W. E. Meador and W. R. Weaver, “Diffusion approximation for large absorption in radiative transfer,” Appl. Opt. 18, 1204–1208 (1979).
[CrossRef]

V. I. Haltrin, “Exact solution of the characteristic equation for transfer in the anisotropically scattering and absorbing medium,” Appl. Opt. 27, 599–602 (1988).
[CrossRef]

J. Wu, F. Partovi, M. S. Field, and R. P. Rava, “Diffuse reflectance from turbid media: an analytical model of photon migration,” Appl. Opt. 32, 1115–1121 (1993).
[CrossRef]

D. Yudovsky and L. Pilon, “Simple and accurate expressions for diffuse reflectance of semi-infinite and two-layer absorbing and scattering media,” Appl. Opt. 48, 6670–6683 (2009).
[CrossRef]

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. (1)

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

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. (3)

P. E. Pierce and R. T. Marcus, “Radiative transfer theory solid color-matching calculations,” Color Res. Appl. 22, 72–87 (1997).
[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 and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 2: plane albedo and reflection function,” Color Res. Appl. 31, 498–509 (2006).
[CrossRef]

Doklady Acad. Sci. BSSR (1)

O. V. Bushmakova, E. P. Zege, and I. L. Katsev, “On asymptotic equations for brightness coefficients of optically thick light scattering layers,” Doklady Acad. Sci. BSSR 15, 309–311 (1971).

Doklady Acad. Sci. USSR (1)

G. V. Rozenberg, “Light characteristics of thick layers of a weakly absorbing scattering medium,” Doklady Acad. Sci. USSR 145, 775–777 (1962).

Icarus (2)

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

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]

IEEE Trans. Biomed. Eng. (1)

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef]

Isr. J. Plant Sci. (1)

L. G. Sokoletsky, R. S. Lunetta, M. S. Wetz, and H. W. Paerl, “Assessment of the water quality components in turbid estuarine waters based on radiative transfer approximations,” Isr. J. Plant Sci. 60, 209–229 (2012).
[CrossRef]

Izv. Atmos. Ocean. Phys. (1)

B. M. Golubitsky, I. M. Levin, and M. V. Tantashev, “Brightness coefficient of a semi-infinite layer of sea water,” Izv. Atmos. Ocean. Phys. 10, 1235–1238 (1974).

J. Atmos. Sci. (1)

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

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

J. Paint Technol. (1)

L. W. Richards, “The calculation of the optical performance of paint films,” J. Paint Technol. 42, 276–286 (1970).

J. Phys. Conf. Ser. (1)

A. García-Valenzuela, F. L. S. Cuppo, and J. A. Olivares, “An assessment of Saunderson corrections to the diffuse reflectance of paint films,” J. Phys. Conf. Ser. 274, 012125 (2011).
[CrossRef]

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. Spectrosc. Radiat. Transfer (2)

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. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[CrossRef]

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, and 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]

Lasers Life Sci. (1)

M. J. C. Gemert and W. M. Star, “Relations between the Kubelka–Munk and the transport equation models for anisotropic scattering,” Lasers Life Sci. 1, 287–298 (1987).

Limnol. Oceanogr. (1)

H. G. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[CrossRef]

Med. Phys. (1)

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef]

Opt. Commun. (1)

G. H. Weiss, J. M. Porrà, and J. Masoliver, “The continuous-time random walk description of photon motion in an isotropic medium,” Opt. Commun. 146, 268–276 (1998).
[CrossRef]

Opt. Express (1)

Phys. Chem. Earth B (1)

A. A. Kokhanovsky, “The determination of the effective radius of drops in water clouds from polarization measurements,” Phys. Chem. Earth B 25, 471–474 (2000).
[CrossRef]

Physikalische Zeitschrift (1)

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

Sci. Am. (1)

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977).
[CrossRef]

Transp. Theory Stat. Phys. (1)

E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, and M. T. Vihena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
[CrossRef]

Zeitschrift für Technische Physik. (1)

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Zeitschrift für Technische Physik. 12, 593–601 (1931).

Other (11)

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

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, 1975).

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

B. Hapke, Theory of Reflectance and Emittance Spectroscopy (University Press, 1993).

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, U.S. Department of Commerce, 1977).

L. Sokoletsky, “A comparative analysis of simple radiative transfer approaches for aquatic environments,” in Proceedings of 2004 ENVISAT & ERS Symposium, H. Lacoste, ed. (Noordwijk, 2005), pp. 1–7.

, “Remote sensing of inherent optical properties: fundamentals, tests of algorithms, and applications”, Z. P. Lee, ed. (Naval Research Laboratory, Stennis Space Center, 2006).

S. L. Jacques, “Diffuse reflectance from a semi-infinite medium,” 1999, http://omlc.ogi.edu/news/may99/rd/index.html .

L. G. Sokoletsky and A. A. Kokhanovsky, “Reflective characteristics of natural waters: The accuracy of selected approximations,” in Proceedings of the D. S. Rozhdestvensky Third International Conference: Current Problems in Optics of Natural Waters, I. Levin and G. Gilbert, eds. (Saint-Petersburg, Russia, 2005), pp. 56–63.

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

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Figures (9)

Fig. 1.
Fig. 1.

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

Fig. 2.
Fig. 2.

Plane albedo as a function of transport single-scattering albedo ωtr. (a) Computations performed by the numerical (IIM). (b)–(d) Selected analytical methods at incidence angle θi=30.5° shown for three different phase functions with g=0.00 [(a), (b)]; 0.50 [(a), (c)]; and 0.96 [(a), (d)].

Fig. 3.
Fig. 3.

Errors of selected plane albedo approximations compared to the IIM-derived values at θi=30.5° for different phase functions versus ωtr.

Fig. 4.
Fig. 4.

Plane albedo as a function of Gordon’s parameter G computed by numerical and selected analytical methods at θi=30.5° for different phase functions.

Fig. 5.
Fig. 5.

Same as Fig. 3, but errors versus G.

Fig. 6.
Fig. 6.

Errors of selected plane albedo approximations compared to the IIM-derived values at θi=30.5° for different phase functions versus G in situations when the scattering phase function is unknown.

Fig. 7.
Fig. 7.

Spherical albedo as a function of ωtr computed by numerical and selected analytical methods for different phase functions.

Fig. 8.
Fig. 8.

Errors of selected spherical albedo approximations compared to the IIM-derived values for different phase functions versus ωtr.

Fig. 9.
Fig. 9.

Errors of selected spherical albedo approximations compared to the IIM-derived values for different phase functions versus G in situations when the scattering phase function is unknown. The values of backscattering ratio B are (a) 0.4986, (b) 0.1559, and (c) 0.0087.

Tables (4)

Tables Icon

Table 1. Compendium of IOPs Used in the Work and Their Mathematical Definitions

Tables Icon

Table 2. Parameters Used for the p(θ) Generation

Tables Icon

Table 3. Accuracy of Selected Models for the Plane Albedo Rp(θi) at θi=30.5°a

Tables Icon

Table 4. Accuracy of Selected Models for the Spherical Albedo ra

Equations (25)

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

Rp(μi)=1π02π01R(μi,μv,φ)μvdμvdφ,
Rp(μi)=201R¯(μi,μv)μvdμv,R¯(μi,μv)=12π02πR(μi,μv,φ)dφ.
r=201Rp(μi)μidμi=40101R¯(μi,μv)μiμvdμidμ.
MAPE(%)=100%i=1n|(x˜ixi)/x˜i|n,
NRMSE(%)=100%i=1n(x˜ixi)2n1x¯,
Rp(μi)=rRp(μi)/Rp(1)Rp(μef)/Rp(1),
θef=48+14.12s22.77s2+19.24s3(deg),
Rp(μi)Rp(1)=exp{[3.599ln(1s)0.550ln2(1s)+0.0416ln(1g)×ln(1s)](1μi)2}
r=(10.139s)(1s)1+1.170s.
R¯(μi,μv)=0.25ω0H(μi)H(μv)μi+μv,
H(μ)=1+2μ1+2μ1ω0.
ω0=ba+bωtr=btra+btr=1s2.
R¯(μi,μv)=0.25(1s2)H(μi)H(μv)μi+μv
H(μ)=1+2μ1+2μs.
Rp(μi)=0.5(1s2)(1+2μi)s(1+2μis)×{1+(1s)ln(1+2s)2s(2μis1)+μis(2μi1)[lnμiln(1+μi)]2μis1}.
R¯(μi,μv)=0.25ω0[p(θ)+H(μi)H(μv)1]μi+μv
H(μ)={1μω0[r0+(0.5μr0)ln(1+1/μ)]}1,r0=11ω01+1ω0.
Rp(μi)=0.5ω0{[p(θ)1][1+μilnμiln(1+μi)]+H(μi)01H(μv)μvμi+μvdμv}.
p(θ)=1g2(12gcosθ+g2)1.5.
B=1g2g(1+g1+g21).
g=14.440B+12.11B223.87B3+23.52B49.317B5,
xj=(2j+1)gj.
r=1+KSKS(KS+2),r=ln(1+ξ)ξln(1ξ)+ξ,ωtr=2ξln[(1+ξ)/(1ξ)],
η=a/K=1atωtr=0,η=0.5atωtr=1,andχ=btr/S=4/3atωtr=1.
Rc(μi)=IrIi=Ii,dir(μi)Rp(μi)+Ii,difrIi=(1dE)Rp(μi)+dEr.

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