Abstract

A comparative analysis is presented of simple approaches to radiative transfer in plane-parallel layers, such as the self-consistent Haltrin approach, the Chandrasekhar–Klier exact solution for isotropic scatters, an extended version of two-flux radiative Kubelka–Munk theory, the neutron-diffuse Gate–Brinkworth theory, and different versions of the δ-Eddington theory. It is shown that the Haltrin approach is preferable to others and can be used for the solution of an inverse optical problem of the estimation of absorption and backscattering coefficients of aquatic environments from measured apparent optical properties. Two different methods of transformation from measured irradiance reflectance at combined illumination to irradiance reflectance induced by diffuse illumination only are developed. An analysis of the use of the different models for estimation of the effect of the bottom albedo is also presented.

© 2005 Optical Society of America

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  65. G. Dall’Olmo, A. A. Gitelson, D. C. Rundquist, “Towards a unified approach for remote estimation of chlorophyll-a in both terrestrial vegetation and turbid productive waters,” Geophys. Res. Lett. 30, 1-1–1-4 (2003).

2004 (1)

N. K. Højerslev, “Practical sea-water algorithms for fundamental bio-optical and remote sensing quantities,” Int. J. Remote Sensing 25, 1539–1543 (2004).
[Crossref]

2003 (6)

L. Sokoletsky, Z. Dubinsky, M. Shoshany, N. Stambler, “Estimation of phytoplankton pigment concentration in the Gulf of Aqaba (Eilat) by in situ and remote sensing single-wavelength algorithms,” Int. J. Remote Sensing 24, 5049–5073 (2003).
[Crossref]

G. Dall’Olmo, A. A. Gitelson, D. C. Rundquist, “Towards a unified approach for remote estimation of chlorophyll-a in both terrestrial vegetation and turbid productive waters,” Geophys. Res. Lett. 30, 1-1–1-4 (2003).

F. Fabbri, M. A. Franceschini, S. Fantini, “Characterization of spatial and temporal variations in the optical properties of tissuelike media with diffuse-reflectance imaging,” Appl. Opt. 42, 3063–3072 (2003).
[Crossref] [PubMed]

B. D. Piening, N. J. McCormick, “Asymptotic optical depths in source-free ocean waters,” Appl. Opt. 42, 5382–5387 (2003).
[Crossref] [PubMed]

P. J. Werdell, C. S. Roesler, “Remote assessment of benthic substrate composition in shallow waters using multispectral reflectance,” Limnol. Oceanogr. 48, 557–567 (2003).

C. S. Roesler, E. Boss, “Spectral beam attenuation coefficient retrieved from ocean color inversion,” Geophys. Res. Let. 30, 1468–1476 (2003).
[Crossref]

2002 (1)

2001 (1)

2000 (3)

M. Stramska, D. Stramski, B. G. Mitchell, C. D. Mobley, “Estimation of the absorption and backscattering coefficients from in-water radiometric measurements,” Limnol. Oceanogr. 45, 628–641 (2000).
[Crossref]

C. Rigollier, O. Bauer, L. Wald, “On the clear sky model of the ESRA—European Solar Radiation Atlas—with respect to the Heliosat method,” Solar Energy 68, 38–48 (2000).
[Crossref]

Y. Z. Yacobi, A. A. Gitelson, “Simultaneous remote measurement of chlorophyll and total seston in productive inland waters,” Verh. Int. Verein. Limnol. 27, 2983–2986 (2000).

1999 (5)

1998 (1)

1997 (1)

S. A. Garver, D. A. Siegel, “Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation. 1. Time series from the Sargasso Sea,” J. Geophys. Res. 102, 18607–18625 (1997).
[Crossref]

1995 (1)

C. S. Roesler, M. J. Perry, “In situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance,” J. Geophys. Res. 100, 13279–13294 (1995).
[Crossref]

1993 (2)

1991 (3)

A. Morel, “Light and marine photosynthesis: a model with geochemical and climatological implications,” Prog. Oceanogr. 26, 263–306 (1991).
[Crossref]

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

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[Crossref] [PubMed]

1989 (2)

H. R. Gordon, “Can the Lambert–Beer law be applied to the diffuse attenuation coefficient of oceanic water?,” Limnol. Oceanogr. 34, 1389–1409 (1989).

H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the Sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).

1988 (3)

1987 (1)

1985 (2)

S. Sugihara, M. Kishino, N. Okami, “Estimation of water quality parameters from irradiance reflectance using optical models,” J. Oceanogr. Soc. Jpn. 41, 399–406 (1985).
[Crossref]

V. I. Haltrin, “The self-consistent two-stream approximation in radiative transfer theory,” Atmos. Ocean. Phys. 21, 589–597 (1985).

1981 (2)

1979 (2)

V. A. Timofeeva, “Determination of light-field parameters in the depth regime from irradiance measurements,” Atmos. Ocean. Phys. 15, 774–776 (1979).

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

1978 (1)

1977 (2)

J. Texter, “Continuous K/S minimizing distributions in Kubelka–Munk systems,” J. Opt. Soc. Am. 67, 169–174 (1977).
[Crossref]

A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[Crossref]

1975 (1)

1974 (1)

1973 (1)

1972 (1)

1971 (1)

B. J. Brinkworth, “On the theory of reflection by scattering and absorbing media,” J. Phys. D 4, 1105–1106 (1971).
[Crossref]

1954 (1)

1948 (1)

1936 (1)

A. A. Gershun, “Transmission of light through a flat layer of light scattering medium,” Tr. Gos. Opt. Inst. (Proc. State Opt. Inst.), 11, 43–68 (1936).

1931 (1)

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

1930 (1)

M. Gurevič, “Über eine rationelle klassifikation der lichtstreuenden medien,” Phys. Z. 31, 753–763 (1930).

1924 (1)

G. A. Gamburtsev, “The question of sea color,” J. Russ. Phys.-Chem. Soc. 56, 226–234 (1924).

1906 (1)

K. Schwarzschild, “Über das gleichgewicht der sonnenatmosphere,” Göttingen Nachrichten 41, 1–24 (1906).

1905 (1)

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

Babin, M.

Baker, K. S.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

Barnard, A. H.

Bauer, O.

C. Rigollier, O. Bauer, L. Wald, “On the clear sky model of the ESRA—European Solar Radiation Atlas—with respect to the Heliosat method,” Solar Energy 68, 38–48 (2000).
[Crossref]

Boss, E.

C. S. Roesler, E. Boss, “Spectral beam attenuation coefficient retrieved from ocean color inversion,” Geophys. Res. Let. 30, 1468–1476 (2003).
[Crossref]

C. D. Mobley, L. K. Sundman, E. Boss, “Phase functions’ effects on oceanic light fields,” Appl. Opt. 41, 1035–1050 (2002).
[Crossref] [PubMed]

Brinkworth, B. J.

B. J. Brinkworth, “On the theory of reflection by scattering and absorbing media,” J. Phys. D 4, 1105–1106 (1971).
[Crossref]

Brown, J. W.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

Brown, O. B.

Carder, K. L.

K. L. Carder, F. R. Chen, Z. P. Lee, S. Hawes, “Semianalytic MODIS algorithms for chlorophyllI-a and absorption with bio-optical domains based on nitrate-depletion temperatures,” J. Geophys. Res. 104, 5403–5421 (1999).
[Crossref]

Chandrasekhar, S.

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

Chen, F. R.

K. L. Carder, F. R. Chen, Z. P. Lee, S. Hawes, “Semianalytic MODIS algorithms for chlorophyllI-a and absorption with bio-optical domains based on nitrate-depletion temperatures,” J. Geophys. Res. 104, 5403–5421 (1999).
[Crossref]

Clark, D. K.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

Dall’Olmo, G.

G. Dall’Olmo, A. A. Gitelson, D. C. Rundquist, “Towards a unified approach for remote estimation of chlorophyll-a in both terrestrial vegetation and turbid productive waters,” Geophys. Res. Lett. 30, 1-1–1-4 (2003).

Ding, K.

Dogniaux, R.

R. Dogniaux, M. Lemoine, “Classification of radiation sites in terms of different indices of atmospheric transparency,” in W. Palz, ed., Solar Energy Research and Development in the European Community, F. Volume 2, Solar Energy Data (D. Reidel Publ. Co., Dordecht, The Netherlands, 1983) pp. 94–107; see also http://www.helioclim.org/heliosat/csmodels_lib.c

Dubinsky, Z.

L. Sokoletsky, Z. Dubinsky, M. Shoshany, N. Stambler, “Estimation of phytoplankton pigment concentration in the Gulf of Aqaba (Eilat) by in situ and remote sensing single-wavelength algorithms,” Int. J. Remote Sensing 24, 5049–5073 (2003).
[Crossref]

Evans, R. H.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

Fabbri, F.

Fantini, S.

Fell, F.

Fournier-Sicre, V.

Franceschini, M. A.

Gamburtsev, G. A.

G. A. Gamburtsev, “The question of sea color,” J. Russ. Phys.-Chem. Soc. 56, 226–234 (1924).

Garver, S. A.

S. A. Garver, D. A. Siegel, “Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation. 1. Time series from the Sargasso Sea,” J. Geophys. Res. 102, 18607–18625 (1997).
[Crossref]

Gate, L. F.

Gentili, B.

Gershun, A. A.

A. A. Gershun, “Transmission of light through a flat layer of light scattering medium,” Tr. Gos. Opt. Inst. (Proc. State Opt. Inst.), 11, 43–68 (1936).

Gitelson, A. A.

G. Dall’Olmo, A. A. Gitelson, D. C. Rundquist, “Towards a unified approach for remote estimation of chlorophyll-a in both terrestrial vegetation and turbid productive waters,” Geophys. Res. Lett. 30, 1-1–1-4 (2003).

Y. Z. Yacobi, A. A. Gitelson, “Simultaneous remote measurement of chlorophyll and total seston in productive inland waters,” Verh. Int. Verein. Limnol. 27, 2983–2986 (2000).

Gong, W.

Gordon, H. R.

H. R. Gordon, K. Ding, W. Gong, “Radiative transfer in the ocean: computations relating to the asymptotic and near-asymptotic daylight field,” Appl. Opt. 32, 1606–1619 (1993).
[Crossref] [PubMed]

H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the Sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).

H. R. Gordon, “Can the Lambert–Beer law be applied to the diffuse attenuation coefficient of oceanic water?,” Limnol. Oceanogr. 34, 1389–1409 (1989).

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

H. R. Gordon, O. B. Brown, M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975).
[Crossref] [PubMed]

H. R. Gordon, O. B. Brown, “Irradiance reflectivity of a flat ocean as a function of its optical properties,” Appl. Opt. 12, 1549–1551 (1973).
[Crossref] [PubMed]

Gurevic, M.

M. Gurevič, “Über eine rationelle klassifikation der lichtstreuenden medien,” Phys. Z. 31, 753–763 (1930).

Haltrin, V. I.

V. I. Haltrin, “Diffuse reflection coefficient of a stratified sea,” Appl. Opt. 38, 932–936 (1999).
[Crossref]

V. I. Haltrin, “Self-consistent approach to the solution of 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]

V. I. Haltrin, G. W. Kattawar, “Self-consistent solutions to the equation of transfer with elastic and inelastic scattering in oceanic optics: I. Model,” Appl. Opt. 32, 5356–5367 (1993).
[Crossref] [PubMed]

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] [PubMed]

V. I. Haltrin, “The self-consistent two-stream approximation in radiative transfer theory,” Atmos. Ocean. Phys. 21, 589–597 (1985).

V. I. Haltrin, “Theoretical and empirical phase functions for Monte Carlo calculations of light scattering in seawater,” in Proceedings of the Fourth International Conference Remote Sensing for Marine and Coastal Environments (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), pp. 509–518.

V. I. Haltrin, “Diffuse reflectance of the optically deep sea under combined illumination of its surface,” in T. I. EditorIGARSS’97 Proceedings of the International Geoscience and Remote Sensing Symposium, 3–8 August 1997, Singapore.

V. I. Haltrin, “Modeling of sea optical signatures under natural illumination,” in Earth Surface Remote Sensing, Proc. SPIE3222, 538–549 (1997).
[Crossref]

Hawes, S.

K. L. Carder, F. R. Chen, Z. P. Lee, S. Hawes, “Semianalytic MODIS algorithms for chlorophyllI-a and absorption with bio-optical domains based on nitrate-depletion temperatures,” J. Geophys. Res. 104, 5403–5421 (1999).
[Crossref]

Højerslev, N. K.

N. K. Højerslev, “Practical sea-water algorithms for fundamental bio-optical and remote sensing quantities,” Int. J. Remote Sensing 25, 1539–1543 (2004).
[Crossref]

Humphreys, T. J.

Jacobs, M. M.

Judd, D. B.

D. B. Judd, G. Wyszecki, Color in Business, Science and Industry (Wiley, New York, 1975).

Kattawar, G. W.

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]

J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems, 2nd ed. (Cambridge U. Press, Cambridge, 1994).
[Crossref]

Kishino, M.

S. Sugihara, M. Kishino, N. Okami, “Estimation of water quality parameters from irradiance reflectance using optical models,” J. Oceanogr. Soc. Jpn. 41, 399–406 (1985).
[Crossref]

Klier, K. J.

Kortüm, G.

G. Kortüm, Reflectance Spectroscopy. Principles, Methods, Applications (Springer-Verlag, New York, 1969).
[Crossref]

Kubelka, P.

Latimer, P.

Leathers, R. A.

Lee, Z. P.

K. L. Carder, F. R. Chen, Z. P. Lee, S. Hawes, “Semianalytic MODIS algorithms for chlorophyllI-a and absorption with bio-optical domains based on nitrate-depletion temperatures,” J. Geophys. Res. 104, 5403–5421 (1999).
[Crossref]

Lemasle, B.

Lemoine, M.

R. Dogniaux, M. Lemoine, “Classification of radiation sites in terms of different indices of atmospheric transparency,” in W. Palz, ed., Solar Energy Research and Development in the European Community, F. Volume 2, Solar Energy Data (D. Reidel Publ. Co., Dordecht, The Netherlands, 1983) pp. 94–107; see also http://www.helioclim.org/heliosat/csmodels_lib.c

Loisel, H. D.

Lyzenda, D. R.

McCormic, N. J.

McCormick, N. J.

Meador, W. E.

Mitchell, B. G.

H. D. Loisel, D. Stramski, B. G. Mitchell, F. Fell, V. Fournier-Sicre, B. Lemasle, M. Babin, “Comparison of the ocean inherent opical properties obtained from measurements and inverse modeling,” Appl. Opt. 40, 2384–2397 (2001).
[Crossref]

M. Stramska, D. Stramski, B. G. Mitchell, C. D. Mobley, “Estimation of the absorption and backscattering coefficients from in-water radiometric measurements,” Limnol. Oceanogr. 45, 628–641 (2000).
[Crossref]

Mobley, C. D.

C. D. Mobley, L. K. Sundman, E. Boss, “Phase functions’ effects on oceanic light fields,” Appl. Opt. 41, 1035–1050 (2002).
[Crossref] [PubMed]

M. Stramska, D. Stramski, B. G. Mitchell, C. D. Mobley, “Estimation of the absorption and backscattering coefficients from in-water radiometric measurements,” Limnol. Oceanogr. 45, 628–641 (2000).
[Crossref]

C. D. Mobley, Light and Water—Radiative Transfer in Natural Waters (AcademicSan Diego, Calif., 1994).

Molenaar, R.

Morel, A.

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[Crossref] [PubMed]

A. Morel, “Light and marine photosynthesis: a model with geochemical and climatological implications,” Prog. Oceanogr. 26, 263–306 (1991).
[Crossref]

A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[Crossref]

Munk, F.

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

Noh, S. J.

Okami, N.

S. Sugihara, M. Kishino, N. Okami, “Estimation of water quality parameters from irradiance reflectance using optical models,” J. Oceanogr. Soc. Jpn. 41, 399–406 (1985).
[Crossref]

Pegau, W. S.

Pelevin, V. N.

V. N. Pelevin, V. V. Rostovtseva, “Sea water scattering and absorption models development using the classification of ocean waters on base contact measurement data,” in Proceedings of the First International Conference “Current Problems in Optics of Natural Waters” ONW-2001(St. Petersburg, Russia, 25–28 September 2001) pp. 377–382.

Perry, M. J.

C. S. Roesler, M. J. Perry, “In situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance,” J. Geophys. Res. 100, 13279–13294 (1995).
[Crossref]

Piening, B. D.

Plass, G. N.

Prieur, L.

A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[Crossref]

Rigollier, C.

C. Rigollier, O. Bauer, L. Wald, “On the clear sky model of the ESRA—European Solar Radiation Atlas—with respect to the Heliosat method,” Solar Energy 68, 38–48 (2000).
[Crossref]

Roesler, C. S.

P. J. Werdell, C. S. Roesler, “Remote assessment of benthic substrate composition in shallow waters using multispectral reflectance,” Limnol. Oceanogr. 48, 557–567 (2003).

C. S. Roesler, E. Boss, “Spectral beam attenuation coefficient retrieved from ocean color inversion,” Geophys. Res. Let. 30, 1468–1476 (2003).
[Crossref]

R. A. Leathers, C. S. Roesler, N. J. McCormic, “Ocean inherent optical property determination from in-water light field measurements,” Appl. Opt. 38, 5096–5103 (1999).
[Crossref]

C. S. Roesler, M. J. Perry, “In situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance,” J. Geophys. Res. 100, 13279–13294 (1995).
[Crossref]

Rostovtseva, V. V.

V. N. Pelevin, V. V. Rostovtseva, “Sea water scattering and absorption models development using the classification of ocean waters on base contact measurement data,” in Proceedings of the First International Conference “Current Problems in Optics of Natural Waters” ONW-2001(St. Petersburg, Russia, 25–28 September 2001) pp. 377–382.

Rundquist, D. C.

G. Dall’Olmo, A. A. Gitelson, D. C. Rundquist, “Towards a unified approach for remote estimation of chlorophyll-a in both terrestrial vegetation and turbid productive waters,” Geophys. Res. Lett. 30, 1-1–1-4 (2003).

Schuster, A.

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

Schwarzschild, K.

K. Schwarzschild, “Über das gleichgewicht der sonnenatmosphere,” Göttingen Nachrichten 41, 1–24 (1906).

Shoshany, M.

L. Sokoletsky, Z. Dubinsky, M. Shoshany, N. Stambler, “Estimation of phytoplankton pigment concentration in the Gulf of Aqaba (Eilat) by in situ and remote sensing single-wavelength algorithms,” Int. J. Remote Sensing 24, 5049–5073 (2003).
[Crossref]

Siegel, D. A.

S. A. Garver, D. A. Siegel, “Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation. 1. Time series from the Sargasso Sea,” J. Geophys. Res. 102, 18607–18625 (1997).
[Crossref]

Smith, R. C.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

Sokoletsky, L.

L. Sokoletsky, Z. Dubinsky, M. Shoshany, N. Stambler, “Estimation of phytoplankton pigment concentration in the Gulf of Aqaba (Eilat) by in situ and remote sensing single-wavelength algorithms,” Int. J. Remote Sensing 24, 5049–5073 (2003).
[Crossref]

Stambler, N.

L. Sokoletsky, Z. Dubinsky, M. Shoshany, N. Stambler, “Estimation of phytoplankton pigment concentration in the Gulf of Aqaba (Eilat) by in situ and remote sensing single-wavelength algorithms,” Int. J. Remote Sensing 24, 5049–5073 (2003).
[Crossref]

Stavn, R. H.

Stramska, M.

M. Stramska, D. Stramski, B. G. Mitchell, C. D. Mobley, “Estimation of the absorption and backscattering coefficients from in-water radiometric measurements,” Limnol. Oceanogr. 45, 628–641 (2000).
[Crossref]

Stramski, D.

H. D. Loisel, D. Stramski, B. G. Mitchell, F. Fell, V. Fournier-Sicre, B. Lemasle, M. Babin, “Comparison of the ocean inherent opical properties obtained from measurements and inverse modeling,” Appl. Opt. 40, 2384–2397 (2001).
[Crossref]

M. Stramska, D. Stramski, B. G. Mitchell, C. D. Mobley, “Estimation of the absorption and backscattering coefficients from in-water radiometric measurements,” Limnol. Oceanogr. 45, 628–641 (2000).
[Crossref]

Sugihara, S.

S. Sugihara, M. Kishino, N. Okami, “Estimation of water quality parameters from irradiance reflectance using optical models,” J. Oceanogr. Soc. Jpn. 41, 399–406 (1985).
[Crossref]

Sundman, L. K.

ten Bosch, J. J.

Texter, J.

Timofeeva, V. A.

V. A. Timofeeva, “Determination of light-field parameters in the depth regime from irradiance measurements,” Atmos. Ocean. Phys. 15, 774–776 (1979).

Wald, L.

C. Rigollier, O. Bauer, L. Wald, “On the clear sky model of the ESRA—European Solar Radiation Atlas—with respect to the Heliosat method,” Solar Energy 68, 38–48 (2000).
[Crossref]

Weaver, W. R.

Werdell, P. J.

P. J. Werdell, C. S. Roesler, “Remote assessment of benthic substrate composition in shallow waters using multispectral reflectance,” Limnol. Oceanogr. 48, 557–567 (2003).

Wyszecki, G.

D. B. Judd, G. Wyszecki, Color in Business, Science and Industry (Wiley, New York, 1975).

Yacobi, Y. Z.

Y. Z. Yacobi, A. A. Gitelson, “Simultaneous remote measurement of chlorophyll and total seston in productive inland waters,” Verh. Int. Verein. Limnol. 27, 2983–2986 (2000).

Zaneveld, J. R. V.

Zijp, J. R.

Appl. Opt. (22)

A. H. Barnard, J. R. V. Zaneveld, W. S. Pegau, “In situ determination of the remotely sensed reflectance and the absorption coefficient: closure and inversion,” Appl. Opt. 38, 5108–5117 (1999).
[Crossref]

R. A. Leathers, C. S. Roesler, N. J. McCormic, “Ocean inherent optical property determination from in-water light field measurements,” Appl. Opt. 38, 5096–5103 (1999).
[Crossref]

H. D. Loisel, D. Stramski, B. G. Mitchell, F. Fell, V. Fournier-Sicre, B. Lemasle, M. Babin, “Comparison of the ocean inherent opical properties obtained from measurements and inverse modeling,” Appl. Opt. 40, 2384–2397 (2001).
[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] [PubMed]

V. I. Haltrin, G. W. Kattawar, “Self-consistent solutions to the equation of transfer with elastic and inelastic scattering in oceanic optics: I. Model,” Appl. Opt. 32, 5356–5367 (1993).
[Crossref] [PubMed]

V. I. Haltrin, “Self-consistent approach to the solution of 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]

V. I. Haltrin, “Diffuse reflection coefficient of a stratified sea,” Appl. Opt. 38, 932–936 (1999).
[Crossref]

H. R. Gordon, O. B. Brown, M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975).
[Crossref] [PubMed]

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[Crossref] [PubMed]

L. F. Gate, “Comparison of the photon diffusion model and Kubelka–Munk equation with the exact solution of the radiative transport equation,” Appl. Opt. 13, 236–238 (1974).
[Crossref] [PubMed]

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

H. R. Gordon, O. B. Brown, “Irradiance reflectivity of a flat ocean as a function of its optical properties,” Appl. Opt. 12, 1549–1551 (1973).
[Crossref] [PubMed]

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]

G. N. Plass, T. J. Humphreys, G. W. Kattawar, “Ocean-atmospheric interface: its influence on radiation,” Appl. Opt. 20, 917–931 (1981).
[Crossref] [PubMed]

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] [PubMed]

H. R. Gordon, K. Ding, W. Gong, “Radiative transfer in the ocean: computations relating to the asymptotic and near-asymptotic daylight field,” Appl. Opt. 32, 1606–1619 (1993).
[Crossref] [PubMed]

P. Latimer, S. J. Noh, “Light propagation in moderately dense particle systems: a reexamination of the Kubelka–Munk theory,” Appl. Opt. 26, 514–523 (1987).
[Crossref] [PubMed]

R. Molenaar, J. J. ten Bosch, J. R. Zijp, “Determination of Kubelka–Munk scattering and absorption coefficients by diffuse illumination,” Appl. Opt. 38, 2068–2077 (1999).
[Crossref]

B. D. Piening, N. J. McCormick, “Asymptotic optical depths in source-free ocean waters,” Appl. Opt. 42, 5382–5387 (2003).
[Crossref] [PubMed]

D. R. Lyzenda, “Passive remote-sensing techniques for mapping water depth and bottom features,” Appl. Opt. 17, 379–383 (1978).
[Crossref]

C. D. Mobley, L. K. Sundman, E. Boss, “Phase functions’ effects on oceanic light fields,” Appl. Opt. 41, 1035–1050 (2002).
[Crossref] [PubMed]

F. Fabbri, M. A. Franceschini, S. Fantini, “Characterization of spatial and temporal variations in the optical properties of tissuelike media with diffuse-reflectance imaging,” Appl. Opt. 42, 3063–3072 (2003).
[Crossref] [PubMed]

Astrophys. J. (1)

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

Atmos. Ocean. Phys. (2)

V. I. Haltrin, “The self-consistent two-stream approximation in radiative transfer theory,” Atmos. Ocean. Phys. 21, 589–597 (1985).

V. A. Timofeeva, “Determination of light-field parameters in the depth regime from irradiance measurements,” Atmos. Ocean. Phys. 15, 774–776 (1979).

Geophys. Res. Let. (1)

C. S. Roesler, E. Boss, “Spectral beam attenuation coefficient retrieved from ocean color inversion,” Geophys. Res. Let. 30, 1468–1476 (2003).
[Crossref]

Geophys. Res. Lett. (1)

G. Dall’Olmo, A. A. Gitelson, D. C. Rundquist, “Towards a unified approach for remote estimation of chlorophyll-a in both terrestrial vegetation and turbid productive waters,” Geophys. Res. Lett. 30, 1-1–1-4 (2003).

Göttingen Nachrichten (1)

K. Schwarzschild, “Über das gleichgewicht der sonnenatmosphere,” Göttingen Nachrichten 41, 1–24 (1906).

Int. J. Remote Sensing (2)

L. Sokoletsky, Z. Dubinsky, M. Shoshany, N. Stambler, “Estimation of phytoplankton pigment concentration in the Gulf of Aqaba (Eilat) by in situ and remote sensing single-wavelength algorithms,” Int. J. Remote Sensing 24, 5049–5073 (2003).
[Crossref]

N. K. Højerslev, “Practical sea-water algorithms for fundamental bio-optical and remote sensing quantities,” Int. J. Remote Sensing 25, 1539–1543 (2004).
[Crossref]

J. Geophys. Res. (4)

K. L. Carder, F. R. Chen, Z. P. Lee, S. Hawes, “Semianalytic MODIS algorithms for chlorophyllI-a and absorption with bio-optical domains based on nitrate-depletion temperatures,” J. Geophys. Res. 104, 5403–5421 (1999).
[Crossref]

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

C. S. Roesler, M. J. Perry, “In situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance,” J. Geophys. Res. 100, 13279–13294 (1995).
[Crossref]

S. A. Garver, D. A. Siegel, “Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation. 1. Time series from the Sargasso Sea,” J. Geophys. Res. 102, 18607–18625 (1997).
[Crossref]

J. Oceanogr. Soc. Jpn. (1)

S. Sugihara, M. Kishino, N. Okami, “Estimation of water quality parameters from irradiance reflectance using optical models,” J. Oceanogr. Soc. Jpn. 41, 399–406 (1985).
[Crossref]

J. Opt. Soc. Am. (4)

J. Phys. D (1)

B. J. Brinkworth, “On the theory of reflection by scattering and absorbing media,” J. Phys. D 4, 1105–1106 (1971).
[Crossref]

J. Russ. Phys.-Chem. Soc. (1)

G. A. Gamburtsev, “The question of sea color,” J. Russ. Phys.-Chem. Soc. 56, 226–234 (1924).

Limnol. Oceanogr. (6)

H. R. Gordon, “Can the Lambert–Beer law be applied to the diffuse attenuation coefficient of oceanic water?,” Limnol. Oceanogr. 34, 1389–1409 (1989).

A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[Crossref]

M. Stramska, D. Stramski, B. G. Mitchell, C. D. Mobley, “Estimation of the absorption and backscattering coefficients from in-water radiometric measurements,” Limnol. Oceanogr. 45, 628–641 (2000).
[Crossref]

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

H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the Sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).

P. J. Werdell, C. S. Roesler, “Remote assessment of benthic substrate composition in shallow waters using multispectral reflectance,” Limnol. Oceanogr. 48, 557–567 (2003).

Phys. Z. (1)

M. Gurevič, “Über eine rationelle klassifikation der lichtstreuenden medien,” Phys. Z. 31, 753–763 (1930).

Prog. Oceanogr. (1)

A. Morel, “Light and marine photosynthesis: a model with geochemical and climatological implications,” Prog. Oceanogr. 26, 263–306 (1991).
[Crossref]

Solar Energy (1)

C. Rigollier, O. Bauer, L. Wald, “On the clear sky model of the ESRA—European Solar Radiation Atlas—with respect to the Heliosat method,” Solar Energy 68, 38–48 (2000).
[Crossref]

Tr. Gos. Opt. Inst. (Proc. State Opt. Inst.) (1)

A. A. Gershun, “Transmission of light through a flat layer of light scattering medium,” Tr. Gos. Opt. Inst. (Proc. State Opt. Inst.), 11, 43–68 (1936).

Verh. Int. Verein. Limnol. (1)

Y. Z. Yacobi, A. A. Gitelson, “Simultaneous remote measurement of chlorophyll and total seston in productive inland waters,” Verh. Int. Verein. Limnol. 27, 2983–2986 (2000).

Z. Tech. Phys. (1)

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

Other (12)

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

G. Kortüm, Reflectance Spectroscopy. Principles, Methods, Applications (Springer-Verlag, New York, 1969).
[Crossref]

D. B. Judd, G. Wyszecki, Color in Business, Science and Industry (Wiley, New York, 1975).

S. Sathyendranath, ed., Remote Sensing of Ocean Colour in Coastal, and Other Optically-Complex, Waters, , (International Ocean-Colour Coordinating Group, Dartmouth, Canada, 2000).

V. I. Haltrin, “Theoretical and empirical phase functions for Monte Carlo calculations of light scattering in seawater,” in Proceedings of the Fourth International Conference Remote Sensing for Marine and Coastal Environments (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), pp. 509–518.

R. Dogniaux, M. Lemoine, “Classification of radiation sites in terms of different indices of atmospheric transparency,” in W. Palz, ed., Solar Energy Research and Development in the European Community, F. Volume 2, Solar Energy Data (D. Reidel Publ. Co., Dordecht, The Netherlands, 1983) pp. 94–107; see also http://www.helioclim.org/heliosat/csmodels_lib.c

V. I. Haltrin, “Diffuse reflectance of the optically deep sea under combined illumination of its surface,” in T. I. EditorIGARSS’97 Proceedings of the International Geoscience and Remote Sensing Symposium, 3–8 August 1997, Singapore.

A. Albert, C. D. Mobley, “An analytical model for subsurface irradiance and remote sensing reflectance in deep and shallow case-2 waters,” Opt. Express11, 2873–2890 (2003), http://www.opticsexpress.org .
[Crossref]

C. D. Mobley, Light and Water—Radiative Transfer in Natural Waters (AcademicSan Diego, Calif., 1994).

V. I. Haltrin, “Modeling of sea optical signatures under natural illumination,” in Earth Surface Remote Sensing, Proc. SPIE3222, 538–549 (1997).
[Crossref]

J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems, 2nd ed. (Cambridge U. Press, Cambridge, 1994).
[Crossref]

V. N. Pelevin, V. V. Rostovtseva, “Sea water scattering and absorption models development using the classification of ocean waters on base contact measurement data,” in Proceedings of the First International Conference “Current Problems in Optics of Natural Waters” ONW-2001(St. Petersburg, Russia, 25–28 September 2001) pp. 377–382.

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Figures (9)

Fig. 1
Fig. 1

Direct Rdir and diffuse Rdif constituents of the total irradiance reflectance coefficient calculated within the framework of the Haltrin approach plotted as functions of the cosine of the refracted solar zenith angle just below the surface μw and the average cosine of the asymptotic light field μ ¯ . The values of μw are represented by numbers in the inset of the legend.

Fig. 2
Fig. 2

Dependence of the reflectance coefficient for a plane-parallel layer illuminated by diffuse light Rdif on the optical depth for the diffuse attenuation coefficient τk at selected values of the diffuse reflectance coefficient for an infinite layer Rdif (solid curves, squares, and triangles for 2%, 10%, and 20%, respectively) and the bottom albedo AB (solid, long-dashed, and short-dashed curves for 0%, 20%, and 60%, respectively): (a) Haltrin model, (b) KM model, (c) CK model, (d) Lyzenda model, and (e) Albert—Mobley model.

Fig. 3
Fig. 3

Dependence of the scattering power pS = (ΔZ)S on the optical depth for the diffuse attenuation coefficient τk = kΔZ and Rdif computed according to the extended KM theory. The curves correspond to values of Rdif of 0.1, 0.2, …, 0.9 (from left to right).

Fig. 4
Fig. 4

Dependence of the absorption power pK = (ΔZ)K on the optical depth for the diffuse attenuation coefficient τk = kΔZ and Rdif computed according to the extended KM theory. The curves correspond to values of Rdif of 0.01, 0.1, 0.2, …, 0.9 (from right to left).

Fig. 5
Fig. 5

Relations between μ ¯ a/k, a/bb, bb/kb, and Rdif calculated within the framework of the self-consistent Haltrin approach.

Fig. 6
Fig. 6

Dependence of the ratio σ = S/bb on the parameters τk and Rdif. S was computed within the framework of the KM theory, whereas bb was computed based on the Haltrin self-consistent approach.

Fig. 7
Fig. 7

Dependence of the ratio κ = K/a on the parameters τk and Rdif. K was computed within the framework of the KM theory, whereas a was computed based on the Haltrin self-consistent approach.

Fig. 8
Fig. 8

Dependence of the ratios σ = S/bb and κ = K/a on Rdif for different approximations. S and K were computed within the framework of the KM theory, whereas a and bb were computed based on the approaches of Haltrin (H), Chandrasekhar and Klier (CK), Gate and Brinkwort (GB), and Meador and Weaver (MW-2 and MW-4).

Fig. 9
Fig. 9

Dependence of the normalized reflectance coefficient of a plane-parallel infinite layer illuminated by direct light Rdir on the cos μ0 of the incoming radiation angle (in air): (a) Morel and Gentili model (MGM). The numbers in parentheses in the legend are values of bb/a (numbers to the left) and ηb (numbers to the right); (b) KM model. The numbers in parentheses in the legend are also values of bb/a: (c) Haltrin model (HM). Numbers in parentheses in the legend are values of bb/a.

Tables (1)

Tables Icon

Table 1 Estimated Accuracy (NRMSE for the Estimation of Rdif and μ ¯ by Use of Eqs. (4) and (3), Respectively

Equations (52)

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

R = E u dir + E u dif E d dir + E d dif = R dir E d dir + R dif E d dif E d dir + E d dif = s R dir + R dif 1 + s , s = E d dir E d dif ,
R dir = ( 1 - μ ¯ ) 2 1 + μ w μ ¯ ( 4 - μ ¯ 2 ) .
R dif = ( 1 - μ ¯ 1 + μ ¯ ) 2 .
R dif = R { 1 + 2 R [ μ w ( 1 - R ) ] 3 } .
R dif ( Z ) = R dif + ξ ( Z ) 1 + R 0 dif ξ ( Z ) ,
R 0 dif = 2 + μ ¯ 2 - μ ¯ R dif ,
ξ ( Z ) = A B - R dif 1 - R 0 dif A B exp [ - k ( Z B - Z ) ( 7 + 2 μ ¯ 2 - μ ¯ 4 ) 3 - μ ¯ 2 ] ,
T b = E 0 ( Z B ) E 0 ( Z ) = exp [ - a μ ¯ ( Z B - Z ) ] = exp [ - k ( Z B - Z ) ] ,
R dif = 2 A R b A B - R b - A B R b A B - 1 ,
R b = 1 + K / S - [ K / S ( K / S + 2 ) + T b 2 ] , T b = exp ( - τ k ) .
A = 0.5 ( R dif + 1 / R dif )
K / S = A - 1 = 0.5 ( R dif - 1 ) 2 / R dif .
R dif = 1 - A B [ A - B coth ( B S Δ Z ) ] A + B coth ( B S Δ Z ) - A B ,
B S Δ Z = arcoth ( 1 - A R b B R b ) ,
B = ( A 2 - 1 ) 1 / 2 = 0.5 ( R dif - 1 / R dif ) .
R dif = 1 - A B ( A - B coth τ k ) A + B coth τ k - A B ,
R dif = R dif [ 1 - exp ( - 2 τ k ) ] + A B exp ( - 2 τ k ) .
R dif = R dif [ 1 - 1.0546 × exp ( - 2 τ k ) ] + 0.9755 A B exp ( - 2 τ k ) ,
R b , = R dif ( 1 - T b 2 ) = R b dif [ 1 - exp ( - 2 τ k ) ] ,
p S , = τ k / B , S = k / B .
K = k ( 1 - R dif ) / ( 1 + R dif ) .
k = a / μ ¯ ,
a = k 1 - ( R dif ) 1 / 2 1 + ( R dif ) 1 / 2 .
a b b = [ 1 - ( R dif ) 1 / 2 ] 2 [ 1 + 4 ( R dif ) 1 / 2 + R dif ] 4 R dif ,
b b = 4 k ( 1 / R dif - 1 ) [ 1 + 4 ( R dif ) 1 / 2 + R dif ] .
σ ( H ) = arcoth [ ( 1 - A R b ) / B R b ] 2 τ k 1 + 4 ( R dif ) 1 / 2 + R dif 1 + R dif ,
κ ( H ) = arcoth [ ( 1 - A R b ) / B R b ] τ k 1 + 2 ( R dif ) 1 / 2 + R dif 1 + R dif .
σ ( H ) = 1 + 4 ( R dif ) 1 / 2 + R dif 2 ( 1 + R dif ) ,
κ ( H ) = 1 + 2 ( R dif ) 1 / 2 + R dif 1 + R dif ,
lim σ ( H ) = 0.5 R dif 0 , lim κ ( H ) = 1 R dif 0 , lim σ ( H ) = 1.5 R dif 1 , lim κ ( H ) = 2 R dif 1 .
σ ( CK ) = 4 ξ ω 0 ( 1 / R dif - R dif ) ,
κ ( CK ) = ξ ( 1 - R dif ) ( 1 - ω 0 ) ( 1 + R dif ) ,
R dif = ln ( 1 + ξ ) - ξ ln ( 1 + ξ ) + ξ ,
ω 0 = 2 ξ ln [ ( 1 + ξ ) / ( 1 - ξ ) ] .
ξ = 1 + 0.1104 R dif - 4.705 ( R dif ) 2 + 6.114 ( R dif ) 3 - 2.533 ( R dif ) 4 ,
ω 0 = 5.949 R dif - 21.59 ( R dif ) 2 + 51.99 ( R dif ) 3 - 73.64 ( R dif ) 4 + 54.23 ( R dif ) 5 - 15.94 ( R dif ) 6 ,
σ ( CK ) = 0.6137 + 3.846 R dif - 7.107 ( R dif ) 2 + 6.400 ( R dif ) 3 - 2.259 ( R dif ) 4 ,
κ ( CK ) = 1 + 4.651 R dif - 9.324 ( R dif ) 2 + 9.019 ( R dif ) 3 - 3.360 ( R dif ) 4 ,
lim σ ( CK ) = 2 ( 1 - ln 2 ) 0.6137 R dif 0 , lim κ ( CK ) = 1 R dif 0 , lim σ ( CK ) = 1.5 R dif 1 , lim κ ( CK ) = 2 R dif 1 .
σ ( GB ) = 7 ω 0 - 4 2 ω 0 , κ ( GB ) = 2 ,
σ ( MW - 2 ) = 4 ω 0 - 1 2 ω 0 , κ ( MW - 2 ) = 2 ;
σ ( MW - 4 ) = 15 - ( 1 - ω 0 ) κ ( MW - 4 ) ( 16 - 3 κ ( MW - 4 ) ) ω 0 ( 16 - 3 κ ( MW - 4 ) ) , κ ( MW - 4 ) = 224 132 - 55 ω 0 + 35 [ 1 + 2 ( 1 - ω 0 ) / 35 + 121 ( 1 - ω 0 ) 2 / 49 ] 1 / 2 ,
κ ( CK ) = { σ ( CK ) + ( 0.449 ± 0.027 ) ,             R dif ( 0 , 0.3 ] σ ( CK ) + ( 0.464 ± 0.031 ) ,             R dif ( 0 , 1 ]
κ ( MV - 4 ) = { σ ( CK ) + ( 0.500 ± 0.018 ) , R dif ( 0 , 0.3 ) σ ( CK ) + ( 0.499 ± 0.015 ) , R dif ( 0 , 1 )
κ ( H ) = σ ( H ) + 0.5 ,             for any values of R dif .
R dif = [ ( 0.6279 - 0.2227 η b - 0.0513 η b 2 ) + ( 0.2465 η b - 0.3119 ) μ 0 ] ( b b / a ) ,
R dir = [ 0.31 + ( 2.181 μ ¯ s - 1.654 ) ( μ 0 - 1 - 1 ) ] ( b b / a ) ,
R dir = { 0.31 + [ 2.126 exp ( - 1.297 b b / a ) - 1.654 ] μ 0 - 1 ) } ( b b / a ) .
R dir = ( 0.6279 - 0.3119 μ 0 ) ( b b / a ) .
μ ¯ = { a / b b 3 + a / b b + [ 9 + 4 ( a / b b ) ] 1 / 2 } 1 / 2 ,
b b a = 0.01439 + 2.279 R dif + 10.54 ( R dif ) 2
10.54 q ( R dif ) 2 + ( 2.279 q + 1 ) R dif + [ 0.01439 q - ( 1 + s ) R ] = 0 , q = ( 0.6279 - 0.3119 μ 0 ) s .

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