Abstract

Seven models for computing underwater radiances and irradiances by numerical solution of the radiative transfer equation are compared. The models are applied to the solution of several problems drawn from optical oceanography. The problems include highly absorbing and highly scattering waters, scattering by molecules and by particulates, stratified water, atmospheric effects, surface-wave effects, bottom effects, and Raman scattering. The models provide consistent output, with errors (resulting from Monte Carlo statistical fluctuations) in computed irradiances that are seldom larger, and are usually smaller, than the experimental errors made in measuring irradiances when using current oceanographic instrumentation. Computed radiances display somewhat larger errors.

© 1993 Optical Society of America

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  26. G. Kattawar, C. Adams, “Errors induced when polarization is neglected in radiance calculations for an atmosphere–ocean system,” in Optics of the Air–Sea Interface: Theory and Measurement, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 2–22 (1992).
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    [CrossRef]
  45. H. Gordon, A. Morel, “Remote assessment of ocean color for interpretation of satellite visible imagers, a review,” in Lecture Notes on Coastal and Estuarine Studies (Springer Verlag, Berlin, 1983), Vol. 4.
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    [CrossRef] [PubMed]
  51. R. Stavn, R. Schiebe, C. Gallegos, “Optical controls on the radiant energy dynamics of the air/water interface: the average cosine and the absorption coefficient,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 62–67 (1984).
  52. G. Plass, T. Humphreys, G. Kattawar, “Ocean–atmosphere interface: its influence on radiation,” Appl. Opt. 20, 917–931 (1981).
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  56. H. Gordon, “Ship perturbation of irradiance measurements at sea. I: Monte Carlo simulations,” Appl. Opt. 24, 4172–4182 (1985).
    [CrossRef] [PubMed]
  57. C. Adams, G. Kattawar, “Radiative transfer in spherical shell atmospheres: Rayleigh scattering,” Icarus 35, 139–151 (1978).
    [CrossRef]
  58. G. Kattawar, C. Adams, “Radiative transfer in spherical shell atmospheres. II. asymmetric phase functions,” Icarus 35, 436–449 (1978).
    [CrossRef]
  59. G. Kattawar, “Radiative transfer in spherical shell atmospheres. III. applicationto Venus,” Icarus 40, 60–66 (1979).
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    [CrossRef]

1993 (2)

C. Adams, G. Kattawar, “Effect of volume scattering function on the errors induced when polarization is neglected in radiance calculations in an atmosphere–ocean system,” Appl. Opt. 20, 4610–4617 (1993).
[CrossRef]

Y. Ge, H. Gordon, K. Voss, “Simulation of inelastic-scattering contributions to the irradiance field in the oceanic variation in Fraunhofer line depths,” Appl. Opt. 32, 4028–4036 (1993).
[PubMed]

1992 (3)

1991 (1)

1989 (6)

G. Kattawar, C. Adams, “Stokes vector calculations of the submarine light field in an atmosphere–ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[CrossRef]

C. Mobley, “A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces,” Limnol. Oceanogr. 34, 1473–1483 (1989).
[CrossRef]

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

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

K. Voss, “Electro-optic camera system for measurement of the underwater radiance distribution,” Opt. Eng. 28, 241–247 (1989).

K. Voss, “Use of the radiance distribution to measure the optical absorption coefficient in the ocean,” Limnol. Oceanogr. 34, 1614–1622 (1989).
[CrossRef]

1988 (4)

1987 (3)

1986 (1)

R. Preisendorfer, C. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

1985 (1)

1983 (2)

M. Lewis, J. Cullen, T. Platt, “Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile,” J. Geophys. Res. 88, 2565–2570 (1983).
[CrossRef]

D. Brine, M. qbal, “Diffuse and global solar spectral irradiance under cloudless skies,” Sol. Energy 30, 447–453 (1983).
[CrossRef]

1981 (5)

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic water based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–3991981.
[CrossRef]

K. Stamnes, H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: Intensity computations,” J. Atmos. Sci. 38, 2696–2706 (1981).
[CrossRef]

R. Smith, K. Baker, “Optical properties of the clearest natural waters (200–800 nm),” Appl. Opt. 20, 177–184 (1981).
[CrossRef] [PubMed]

G. Plass, T. Humphreys, G. Kattawar, “Ocean–atmosphere interface: its influence on radiation,” Appl. Opt. 20, 917–931 (1981).
[CrossRef] [PubMed]

1979 (2)

1978 (2)

C. Adams, G. Kattawar, “Radiative transfer in spherical shell atmospheres: Rayleigh scattering,” Icarus 35, 139–151 (1978).
[CrossRef]

G. Kattawar, C. Adams, “Radiative transfer in spherical shell atmospheres. II. asymmetric phase functions,” Icarus 35, 436–449 (1978).
[CrossRef]

1976 (1)

1975 (2)

1974 (1)

1973 (1)

1972 (1)

G. Plass, G. Kattawar, “Monte-Carlo calculations of radiative transfer in the earth’s atmosphere ocean system: I. Flux in the atmosphere and ocean,” J. Phys. Oceanogr. 2, 139–145 (1972).
[CrossRef]

1971 (1)

L. Prieur, A. Morel, “Etude theorique du regime asymptotique: relations entre caracteristiques optiques et coefficient d’extinction relatif a la penetration de la lumiere du jour,” Cah. Oceanogr. 23, 35 (1971).

1969 (1)

1966 (1)

1954 (1)

C. Cox, M. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).

1908 (1)

G. Mie, “Beiträge zur Optik triber Medien, speziell Kolloidalen Metall-Lösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Abreu, L.

F. Kneizys, E. Shettle, L. Abreu, J. Chetwynd, G. Anderson, W. Gallery, J. Selby, S. Clough, “Users guide to lowtran-7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

Adams, C.

C. Adams, G. Kattawar, “Effect of volume scattering function on the errors induced when polarization is neglected in radiance calculations in an atmosphere–ocean system,” Appl. Opt. 20, 4610–4617 (1993).
[CrossRef]

G. Kattawar, C. Adams, “Stokes vector calculations of the submarine light field in an atmosphere–ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[CrossRef]

G. Kattawar, C. Adams, “Radiative transfer in spherical shell atmospheres. II. asymmetric phase functions,” Icarus 35, 436–449 (1978).
[CrossRef]

C. Adams, G. Kattawar, “Radiative transfer in spherical shell atmospheres: Rayleigh scattering,” Icarus 35, 139–151 (1978).
[CrossRef]

G. Kattawar, C. Adams, “Errors induced when polarization is neglected in radiance calculations for an atmosphere–ocean system,” in Optics of the Air–Sea Interface: Theory and Measurement, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 2–22 (1992).

G. Kattawar, C. Adams, “Errors in radiance calculations induced by using scalar rather than Stokes vector theory in a realistic atmosphere–ocean system,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 2–12 (1990).

Anderson, G.

F. Kneizys, E. Shettle, L. Abreu, J. Chetwynd, G. Anderson, W. Gallery, J. Selby, S. Clough, “Users guide to lowtran-7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

Austin, R.

J. Mueller, R. Austin “Ocean optics protocols for SeaWiFS validation,” SeaWiFS Project Tech. Rep. Vol. 5, S. B. Hooker, E. R. Firestone, series eds., NASA Tech. Memo. 104566 (National Aeronautics and Space Administration, Washington, D.C., July1992).

Baker, K.

K. Baker, R. Frouin, “Relation between photosynthetically available radiation and total insolation at the ocean surface under clear skies,” Limnol. Oceanogr. 32, 1370–1377 (1987).
[CrossRef]

R. Smith, K. Baker, “Optical properties of the clearest natural waters (200–800 nm),” Appl. Opt. 20, 177–184 (1981).
[CrossRef] [PubMed]

Blattern, W.

Brine, D.

D. Brine, M. qbal, “Diffuse and global solar spectral irradiance under cloudless skies,” Sol. Energy 30, 447–453 (1983).
[CrossRef]

Brown, O.

Castano, D.

Chandrasekhar, S.

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

Chetwynd, J.

F. Kneizys, E. Shettle, L. Abreu, J. Chetwynd, G. Anderson, W. Gallery, J. Selby, S. Clough, “Users guide to lowtran-7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

Clough, S.

F. Kneizys, E. Shettle, L. Abreu, J. Chetwynd, G. Anderson, W. Gallery, J. Selby, S. Clough, “Users guide to lowtran-7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

Collins, D.

Coombes, C.

A. Harrison, C. Coombes, “Angular distribution of clear sky short wavelength radiance,” Sol. Energy 40, 57–69 (1988).
[CrossRef]

Cox, C.

C. Cox, M. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).

Cullen, J.

M. Lewis, J. Cullen, T. Platt, “Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile,” J. Geophys. Res. 88, 2565–2570 (1983).
[CrossRef]

Dale, H.

K. Stamnes, H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: Intensity computations,” J. Atmos. Sci. 38, 2696–2706 (1981).
[CrossRef]

Deirmendjian, D.

D. Deirmendjian, “Scattering and polarization properties of polydisperse suspensions with partial absorption,” in Electromagnetic Scattering, M. Kerker, ed. (Pergamon, New York, 1963), pp. 171–189.

deLeffe, A.

Deschamps, R.

Elterman, L.

L. Elterman, “UV, visible, and IR attenuation for altitudes to 50 km,” Rep. AFCRL-68-0153 (U.S. Air Force Cambridge Research Laboratory, Bedford, Mass., 1968).

Frouin, R.

K. Baker, R. Frouin, “Relation between photosynthetically available radiation and total insolation at the ocean surface under clear skies,” Limnol. Oceanogr. 32, 1370–1377 (1987).
[CrossRef]

Gallegos, C.

R. Stavn, R. Schiebe, C. Gallegos, “Optical controls on the radiant energy dynamics of the air/water interface: the average cosine and the absorption coefficient,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 62–67 (1984).

Gallery, W.

F. Kneizys, E. Shettle, L. Abreu, J. Chetwynd, G. Anderson, W. Gallery, J. Selby, S. Clough, “Users guide to lowtran-7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

Ge, Y.

Gentili, B.

Gordon, H.

Y. Ge, H. Gordon, K. Voss, “Simulation of inelastic-scattering contributions to the irradiance field in the oceanic variation in Fraunhofer line depths,” Appl. Opt. 32, 4028–4036 (1993).
[PubMed]

H. Gordon, “Diffuse reflectance of the ocean: influence of nonuniform phytoplankton pigment profile,” Appl. Opt. 31, 2116–2129 (1992).
[CrossRef] [PubMed]

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

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

H. Gordon, D. Castano, “Coastal zone color scanner atmospheric correction algorithm: multiple scattering effects,” Appl. Opt. 26, 2111 (1987).
[CrossRef] [PubMed]

H. Gordon, “A bio-optical model describing the distribution of irradiance at the sea surface resulting from a point source embedded in the ocean,” Appl. Opt. 26, 4133–4148 (1987).
[CrossRef] [PubMed]

H. Gordon, “Ship perturbation of irradiance measurements at sea. I: Monte Carlo simulations,” Appl. Opt. 24, 4172–4182 (1985).
[CrossRef] [PubMed]

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

H. Gordon, A. Morel, “Remote assessment of ocean color for interpretation of satellite visible imagers, a review,” in Lecture Notes on Coastal and Estuarine Studies (Springer Verlag, Berlin, 1983), Vol. 4.

Guinn, J.

Harrison, A.

A. Harrison, C. Coombes, “Angular distribution of clear sky short wavelength radiance,” Sol. Energy 40, 57–69 (1988).
[CrossRef]

Herman, M.

Horak, H.

Humphreys, T.

Jacobs, M.

Jayaweera, K.

Jin, Z.

Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media such as the atmosphere–ocean system,” Appl. Opt. (to be published).

Kattawar, G.

C. Adams, G. Kattawar, “Effect of volume scattering function on the errors induced when polarization is neglected in radiance calculations in an atmosphere–ocean system,” Appl. Opt. 20, 4610–4617 (1993).
[CrossRef]

G. Kattawar, X. Xu, “Filling-in of Fraunhofer lines in the ocean by Raman scattering,” Appl. Opt. 31, 1055–1065 (1992).
[CrossRef]

G. Kattawar, C. Adams, “Stokes vector calculations of the submarine light field in an atmosphere–ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[CrossRef]

G. Plass, T. Humphreys, G. Kattawar, “Ocean–atmosphere interface: its influence on radiation,” Appl. Opt. 20, 917–931 (1981).
[CrossRef] [PubMed]

G. Kattawar, “Radiative transfer in spherical shell atmospheres. III. applicationto Venus,” Icarus 40, 60–66 (1979).
[CrossRef]

G. Kattawar, C. Adams, “Radiative transfer in spherical shell atmospheres. II. asymmetric phase functions,” Icarus 35, 436–449 (1978).
[CrossRef]

C. Adams, G. Kattawar, “Radiative transfer in spherical shell atmospheres: Rayleigh scattering,” Icarus 35, 139–151 (1978).
[CrossRef]

G. Kattawar, G. Plass, “Asymptotic radiance and polarization in optically thick media: ocean and clouds,” Appl. Opt. 15, 3166–3178 (1976).
[CrossRef] [PubMed]

G. Plass, G. Kattawar, J. Guinn, “Radiative transfer in the earth’s atmosphere and ocean: influence of ocean waves,” Appl. Opt. 14, 1924–1936 (1975).
[CrossRef] [PubMed]

G. Plass, G. Kattawar, “Monte-Carlo calculations of radiative transfer in the earth’s atmosphere ocean system: I. Flux in the atmosphere and ocean,” J. Phys. Oceanogr. 2, 139–145 (1972).
[CrossRef]

G. Plass, G. Kattawar, “Radiative transfer in an atmosphere–ocean system,” Appl. Opt. 8, 455–466 (1969).
[CrossRef] [PubMed]

G. Kattawar, C. Adams, “Errors in radiance calculations induced by using scalar rather than Stokes vector theory in a realistic atmosphere–ocean system,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 2–12 (1990).

G. Kattawar, C. Adams, “Errors induced when polarization is neglected in radiance calculations for an atmosphere–ocean system,” in Optics of the Air–Sea Interface: Theory and Measurement, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 2–22 (1992).

Kirk, J.

J. Kirk, “Monte Carlo procedure for simulating the penetration of light into natural waters,” Div. Plant Industry Tech. Paper 36 (Commonwealth Scientific and Industrial Research Organization, Canberra, Australia, 1981).

Kneizys, F.

F. Kneizys, E. Shettle, L. Abreu, J. Chetwynd, G. Anderson, W. Gallery, J. Selby, S. Clough, “Users guide to lowtran-7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

Lewis, M.

M. Lewis, J. Cullen, T. Platt, “Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile,” J. Geophys. Res. 88, 2565–2570 (1983).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur Optik triber Medien, speziell Kolloidalen Metall-Lösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Mobley, C.

C. Mobley, “A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces,” Limnol. Oceanogr. 34, 1473–1483 (1989).
[CrossRef]

R. Preisendorfer, C. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

C. Mobley, “A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces, part II: users’ guide and code listing,” NOAA Tech. Memo. ERL PMEL-81 (NTIS PB88-246871) (Pacific Marine Environmental Laboratory, Seattle, Wash., 1988).

C. Mobley, R. Preisendorfer, “A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces,” NOAA Tech. Memo. ERL PMEL-75 (NTIS PB88-192703) (Pacific Marine Environmental Laboratory, Seattle, Wash., 1988).

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]

L. Prieur, A. Morel, “Etude theorique du regime asymptotique: relations entre caracteristiques optiques et coefficient d’extinction relatif a la penetration de la lumiere du jour,” Cah. Oceanogr. 23, 35 (1971).

H. Gordon, A. Morel, “Remote assessment of ocean color for interpretation of satellite visible imagers, a review,” in Lecture Notes on Coastal and Estuarine Studies (Springer Verlag, Berlin, 1983), Vol. 4.

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters. II. bidirectional aspects,” Appl. Opt. (to be published).

A. Morel, “Optical properties of pure water and pure sea water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, eds. (Academic, New York, 1974), pp. 1–24.

Mueller, J.

J. Mueller, R. Austin “Ocean optics protocols for SeaWiFS validation,” SeaWiFS Project Tech. Rep. Vol. 5, S. B. Hooker, E. R. Firestone, series eds., NASA Tech. Memo. 104566 (National Aeronautics and Space Administration, Washington, D.C., July1992).

Munk, M.

C. Cox, M. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).

Petzold, T.

T. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 72–78 (Scripps Institution of Oceanography, La Jolla, Calif., 1972).

Plass, G.

Platt, T.

T. Platt, S. Sathyendranath, “Oceanic primary production: estimation by remote sensing at local and regional scales,” Science 241, 1613–1620 (1988).
[CrossRef] [PubMed]

M. Lewis, J. Cullen, T. Platt, “Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile,” J. Geophys. Res. 88, 2565–2570 (1983).
[CrossRef]

Porto, S.

Preisendorfer, R.

R. Preisendorfer, C. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

R. Preisendorfer, “Eigenmatrix representations of radiance distributions in layered natural waters with wind-roughened surfaces,” NOAA Tech. Memo. ERL PMEL-76 (NTIS PB88-188701) (Pacific Marine Environmental Laboratory, Seattle, Wash., 1988).

R. Preisendorfer, Properties, Vol. V of Hydrologic Optics (U.S. Department of Commerce, Pacific Marine Environmental Laboratory, Seattle, Wash., 1976).

C. Mobley, R. Preisendorfer, “A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces,” NOAA Tech. Memo. ERL PMEL-75 (NTIS PB88-192703) (Pacific Marine Environmental Laboratory, Seattle, Wash., 1988).

Prieur, L.

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic water based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

L. Prieur, A. Morel, “Etude theorique du regime asymptotique: relations entre caracteristiques optiques et coefficient d’extinction relatif a la penetration de la lumiere du jour,” Cah. Oceanogr. 23, 35 (1971).

qbal, M.

D. Brine, M. qbal, “Diffuse and global solar spectral irradiance under cloudless skies,” Sol. Energy 30, 447–453 (1983).
[CrossRef]

Sathyendranath, S.

T. Platt, S. Sathyendranath, “Oceanic primary production: estimation by remote sensing at local and regional scales,” Science 241, 1613–1620 (1988).
[CrossRef] [PubMed]

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic water based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

Schiebe, R.

R. Stavn, R. Schiebe, C. Gallegos, “Optical controls on the radiant energy dynamics of the air/water interface: the average cosine and the absorption coefficient,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 62–67 (1984).

Selby, J.

F. Kneizys, E. Shettle, L. Abreu, J. Chetwynd, G. Anderson, W. Gallery, J. Selby, S. Clough, “Users guide to lowtran-7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

Shettle, E.

F. Kneizys, E. Shettle, L. Abreu, J. Chetwynd, G. Anderson, W. Gallery, J. Selby, S. Clough, “Users guide to lowtran-7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

Smith, R.

Stamnes, K.

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

K. Stamnes, H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: Intensity computations,” J. Atmos. Sci. 38, 2696–2706 (1981).
[CrossRef]

K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–3991981.
[CrossRef]

Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media such as the atmosphere–ocean system,” Appl. Opt. (to be published).

Stavn, R.

R. Stavn, A. Weidemann, “Raman scattering in ocean optics: quantitative assessment of internal radiant emission,” Appl. Opt. 31, 1294–1303 (1992).
[CrossRef] [PubMed]

R. Stavn, A. Weidemann, “Optical modeling of clear oceanlight fields: Raman scattering effects,” Appl. Opt. 27, 4002–4011 (1988).
[CrossRef] [PubMed]

R. Stavn, R. Schiebe, C. Gallegos, “Optical controls on the radiant energy dynamics of the air/water interface: the average cosine and the absorption coefficient,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 62–67 (1984).

Swanson, R. A.

K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–3991981.
[CrossRef]

Tanré, D.

Tsay, S. C.

van de Hulst, H.

H. van de Hulst, Multiple Light Scattering Tables, Formulas, and Applications (Academic, New York, 1980), Vol. 1.

Voss, K.

Y. Ge, H. Gordon, K. Voss, “Simulation of inelastic-scattering contributions to the irradiance field in the oceanic variation in Fraunhofer line depths,” Appl. Opt. 32, 4028–4036 (1993).
[PubMed]

K. Voss, “Electro-optic camera system for measurement of the underwater radiance distribution,” Opt. Eng. 28, 241–247 (1989).

K. Voss, “Use of the radiance distribution to measure the optical absorption coefficient in the ocean,” Limnol. Oceanogr. 34, 1614–1622 (1989).
[CrossRef]

Weidemann, A.

Wells, M.

Wiscombe, W.

Xu, X.

G. Kattawar, X. Xu, “Filling-in of Fraunhofer lines in the ocean by Raman scattering,” Appl. Opt. 31, 1055–1065 (1992).
[CrossRef]

Ann. Phys. (1)

G. Mie, “Beiträge zur Optik triber Medien, speziell Kolloidalen Metall-Lösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Appl. Opt. (20)

G. Kattawar, X. Xu, “Filling-in of Fraunhofer lines in the ocean by Raman scattering,” Appl. Opt. 31, 1055–1065 (1992).
[CrossRef]

C. Adams, G. Kattawar, “Effect of volume scattering function on the errors induced when polarization is neglected in radiance calculations in an atmosphere–ocean system,” Appl. Opt. 20, 4610–4617 (1993).
[CrossRef]

G. Plass, G. Kattawar, “Radiative transfer in an atmosphere–ocean system,” Appl. Opt. 8, 455–466 (1969).
[CrossRef] [PubMed]

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

W. Blattern, H. Horak, D. Collins, M. Wells, “Monte Carlo studies of the sky radiation at twilight,” Appl. Opt. 13, 534 (1974).
[CrossRef]

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

G. Plass, G. Kattawar, J. Guinn, “Radiative transfer in the earth’s atmosphere and ocean: influence of ocean waves,” Appl. Opt. 14, 1924–1936 (1975).
[CrossRef] [PubMed]

G. Kattawar, G. Plass, “Asymptotic radiance and polarization in optically thick media: ocean and clouds,” Appl. Opt. 15, 3166–3178 (1976).
[CrossRef] [PubMed]

D. Tanré, M. Herman, R. Deschamps, A. deLeffe, “Atmospheric modeling for space measurements of ground reflectances including bidirectional properties,” Appl. Opt. 18, 3587–3594 (1979).
[CrossRef] [PubMed]

R. Smith, K. Baker, “Optical properties of the clearest natural waters (200–800 nm),” Appl. Opt. 20, 177–184 (1981).
[CrossRef] [PubMed]

G. Plass, T. Humphreys, G. Kattawar, “Ocean–atmosphere interface: its influence on radiation,” Appl. Opt. 20, 917–931 (1981).
[CrossRef] [PubMed]

H. Gordon, “Ship perturbation of irradiance measurements at sea. I: Monte Carlo simulations,” Appl. Opt. 24, 4172–4182 (1985).
[CrossRef] [PubMed]

H. Gordon, “A bio-optical model describing the distribution of irradiance at the sea surface resulting from a point source embedded in the ocean,” Appl. Opt. 26, 4133–4148 (1987).
[CrossRef] [PubMed]

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

R. Stavn, A. Weidemann, “Optical modeling of clear oceanlight fields: Raman scattering effects,” Appl. Opt. 27, 4002–4011 (1988).
[CrossRef] [PubMed]

R. Stavn, A. Weidemann, “Raman scattering in ocean optics: quantitative assessment of internal radiant emission,” Appl. Opt. 31, 1294–1303 (1992).
[CrossRef] [PubMed]

H. Gordon, “Diffuse reflectance of the ocean: influence of nonuniform phytoplankton pigment profile,” Appl. Opt. 31, 2116–2129 (1992).
[CrossRef] [PubMed]

Y. Ge, H. Gordon, K. Voss, “Simulation of inelastic-scattering contributions to the irradiance field in the oceanic variation in Fraunhofer line depths,” Appl. Opt. 32, 4028–4036 (1993).
[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]

H. Gordon, D. Castano, “Coastal zone color scanner atmospheric correction algorithm: multiple scattering effects,” Appl. Opt. 26, 2111 (1987).
[CrossRef] [PubMed]

Cah. Oceanogr. (1)

L. Prieur, A. Morel, “Etude theorique du regime asymptotique: relations entre caracteristiques optiques et coefficient d’extinction relatif a la penetration de la lumiere du jour,” Cah. Oceanogr. 23, 35 (1971).

Icarus (3)

C. Adams, G. Kattawar, “Radiative transfer in spherical shell atmospheres: Rayleigh scattering,” Icarus 35, 139–151 (1978).
[CrossRef]

G. Kattawar, C. Adams, “Radiative transfer in spherical shell atmospheres. II. asymmetric phase functions,” Icarus 35, 436–449 (1978).
[CrossRef]

G. Kattawar, “Radiative transfer in spherical shell atmospheres. III. applicationto Venus,” Icarus 40, 60–66 (1979).
[CrossRef]

J. Atmos. Sci. (2)

K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–3991981.
[CrossRef]

K. Stamnes, H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: Intensity computations,” J. Atmos. Sci. 38, 2696–2706 (1981).
[CrossRef]

J. Geophys. Res. (1)

M. Lewis, J. Cullen, T. Platt, “Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile,” J. Geophys. Res. 88, 2565–2570 (1983).
[CrossRef]

J. Mar. Res. (1)

C. Cox, M. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).

J. Opt. Soc. Am. (1)

J. Phys. Oceanogr. (2)

R. Preisendorfer, C. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

G. Plass, G. Kattawar, “Monte-Carlo calculations of radiative transfer in the earth’s atmosphere ocean system: I. Flux in the atmosphere and ocean,” J. Phys. Oceanogr. 2, 139–145 (1972).
[CrossRef]

Limnol. Oceanogr. (7)

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

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

K. Baker, R. Frouin, “Relation between photosynthetically available radiation and total insolation at the ocean surface under clear skies,” Limnol. Oceanogr. 32, 1370–1377 (1987).
[CrossRef]

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic water based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

G. Kattawar, C. Adams, “Stokes vector calculations of the submarine light field in an atmosphere–ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[CrossRef]

C. Mobley, “A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces,” Limnol. Oceanogr. 34, 1473–1483 (1989).
[CrossRef]

K. Voss, “Use of the radiance distribution to measure the optical absorption coefficient in the ocean,” Limnol. Oceanogr. 34, 1614–1622 (1989).
[CrossRef]

Opt. Eng. (1)

K. Voss, “Electro-optic camera system for measurement of the underwater radiance distribution,” Opt. Eng. 28, 241–247 (1989).

Science (1)

T. Platt, S. Sathyendranath, “Oceanic primary production: estimation by remote sensing at local and regional scales,” Science 241, 1613–1620 (1988).
[CrossRef] [PubMed]

Sol. Energy (2)

D. Brine, M. qbal, “Diffuse and global solar spectral irradiance under cloudless skies,” Sol. Energy 30, 447–453 (1983).
[CrossRef]

A. Harrison, C. Coombes, “Angular distribution of clear sky short wavelength radiance,” Sol. Energy 40, 57–69 (1988).
[CrossRef]

Other (19)

R. Preisendorfer, “Eigenmatrix representations of radiance distributions in layered natural waters with wind-roughened surfaces,” NOAA Tech. Memo. ERL PMEL-76 (NTIS PB88-188701) (Pacific Marine Environmental Laboratory, Seattle, Wash., 1988).

C. Mobley, “A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces, part II: users’ guide and code listing,” NOAA Tech. Memo. ERL PMEL-81 (NTIS PB88-246871) (Pacific Marine Environmental Laboratory, Seattle, Wash., 1988).

Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media such as the atmosphere–ocean system,” Appl. Opt. (to be published).

L. Elterman, “UV, visible, and IR attenuation for altitudes to 50 km,” Rep. AFCRL-68-0153 (U.S. Air Force Cambridge Research Laboratory, Bedford, Mass., 1968).

C. Mobley, R. Preisendorfer, “A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces,” NOAA Tech. Memo. ERL PMEL-75 (NTIS PB88-192703) (Pacific Marine Environmental Laboratory, Seattle, Wash., 1988).

D. Deirmendjian, “Scattering and polarization properties of polydisperse suspensions with partial absorption,” in Electromagnetic Scattering, M. Kerker, ed. (Pergamon, New York, 1963), pp. 171–189.

G. Kattawar, C. Adams, “Errors in radiance calculations induced by using scalar rather than Stokes vector theory in a realistic atmosphere–ocean system,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 2–12 (1990).

G. Kattawar, C. Adams, “Errors induced when polarization is neglected in radiance calculations for an atmosphere–ocean system,” in Optics of the Air–Sea Interface: Theory and Measurement, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 2–22 (1992).

T. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 72–78 (Scripps Institution of Oceanography, La Jolla, Calif., 1972).

A. Morel, “Optical properties of pure water and pure sea water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, eds. (Academic, New York, 1974), pp. 1–24.

J. Kirk, “Monte Carlo procedure for simulating the penetration of light into natural waters,” Div. Plant Industry Tech. Paper 36 (Commonwealth Scientific and Industrial Research Organization, Canberra, Australia, 1981).

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters. II. bidirectional aspects,” Appl. Opt. (to be published).

R. Preisendorfer, Properties, Vol. V of Hydrologic Optics (U.S. Department of Commerce, Pacific Marine Environmental Laboratory, Seattle, Wash., 1976).

R. Stavn, R. Schiebe, C. Gallegos, “Optical controls on the radiant energy dynamics of the air/water interface: the average cosine and the absorption coefficient,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 62–67 (1984).

H. van de Hulst, Multiple Light Scattering Tables, Formulas, and Applications (Academic, New York, 1980), Vol. 1.

H. Gordon, A. Morel, “Remote assessment of ocean color for interpretation of satellite visible imagers, a review,” in Lecture Notes on Coastal and Estuarine Studies (Springer Verlag, Berlin, 1983), Vol. 4.

J. Mueller, R. Austin “Ocean optics protocols for SeaWiFS validation,” SeaWiFS Project Tech. Rep. Vol. 5, S. B. Hooker, E. R. Firestone, series eds., NASA Tech. Memo. 104566 (National Aeronautics and Space Administration, Washington, D.C., July1992).

F. Kneizys, E. Shettle, L. Abreu, J. Chetwynd, G. Anderson, W. Gallery, J. Selby, S. Clough, “Users guide to lowtran-7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1988).

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

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Figures (9)

Fig. 1
Fig. 1

(a) Inherent optical properties as a function of depth for problem 3. Coefficients a, b, and c have units of inverse meters; ω0 is dimensionless; (b) scattering-phase function for pure sea water, β ˜ w; for particles, β ˜ p; and for problem 3 at depths of z = 0, 17, and 60 m.

Fig. 2
Fig. 2

(a) Ed, Eou, and Lu as computed by the various models for problem 1, ω0 = 0.9; (b) the same quantities as computed for the case of ω0 = 0.2. The dotted line represents the air–water surface. Results from models II and DO are plotted with solid lines; models MC1–MC5 are plotted with dashed lines. Depth τ = 0 is in the water, just below the surface, and in air represents a point just above the surface.

Fig. 3
Fig. 3

Model predictions for problem 2, the base case: (a) ω0 = 0.9 and (b) ω0 = 0.2.

Fig. 4
Fig. 4

Model predictions for problem 3, the stratified-water case.

Fig. 5
Fig. 5

Ed near the surface for problem 4, the base case plus an atmosphere.

Fig. 6
Fig. 6

Model predictions near the surface for problem 5, the capillary-wave case. The wind speed is U = 7.23 m s−1, and the zenith angle of the sun is θsun = 80°.

Fig. 7
Fig. 7

Model predictions for problem 6, the finite-depth case. The bottom reflectance is 0.5.

Fig. 8
Fig. 8

Radiance distribution in the plane of the sun for problem 2, ω0 = 0.9. Angles (θv, ϕv) are viewing directions: θv = 180° − θ and ϕv = 180° + ϕ, where (θ, ϕ) are the directions of photon travel. The solid curves are L(τ, θv, ϕv) at selected depths within the water for models II and DO; models MC2–MC4 are shown by the dashed curves. The dotted curve is the asymptotic distribution Lv) normalized to the largest value of L at τ = 20.

Fig. 9
Fig. 9

Asymptotic radiance distributions Lv) for problems 1 and 2, as computed by various models (solid curves). The dotted curves give the exact analytic solution46 for the Rayleigh phase function of problem 1.

Tables (8)

Tables Icon

Table 1 Significant Symbols, Units, and Definitions

Tables Icon

Table 2 Phase Function Values Used in Defining the Particulate Phase Function β ˜ p (ψ)

Tables Icon

Table 3 Summary of the Canonical Problems

Tables Icon

Table 4 Representative Execution Times, and Numbers of Simulated Photons for Models MC1–MC5

Tables Icon

Table 5 Average Values of Ed, Eou, and Lu at Selected Depths for Problems 1–6a

Tables Icon

Table 6 Raman Scattering Contributions to Ed and Eu at ⊆ = 486 nm From an Excitation Wavelength of ⊆ex = 417 nma

Tables Icon

Table 7 Computed Values of k

Tables Icon

Table 8 Comparison of Percent Accuracies for Computing and Measuring Radiometric Variables

Equations (51)

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

μ L ( τ ; μ , ϕ ) τ = - L ( τ ; μ , ϕ ) + ω 0 ( τ ) ( μ , ϕ ) Ξ L ( τ ; μ , ϕ ) × β ˜ ( τ ; μ , ϕ μ , ϕ ) d μ d ϕ + S ( τ ; μ , ϕ ) .
L ( τ ; u , v ) 1 Ω u v ( μ , ϕ ) Q u v L ( τ ; μ , ϕ ) d μ d ϕ .
β ˜ ( τ ; μ , ϕ μ , ϕ ) β ˜ ( τ ; cos ψ ) = l = 0 2 N - 1 ( 2 l + 1 ) g l ( τ ) P l ( cos ψ ) ,
μ d L m ( τ , μ ) d τ = - L m ( τ , μ ) + ω 0 ( τ ) - 1 1 L m ( τ , μ ) × β m ( τ ; μ , μ ) d μ + S m ( τ , μ ) ,
β m ( τ ; μ , μ ) = 1 2 l = m 2 N - 1 ( 2 l + 1 ) g l ( τ ) × ( l - m ) ! ( l + m ) ! P l m ( μ ) P l m ( μ ) .
p ( z x , z y ) d z x d z y = 1 π σ 2 exp ( - z x 2 + z y 2 σ 2 ) d z x d z y ,
p ( θ n , ϕ n ) d θ n d ϕ n = 1 π σ 2 exp ( - tan 2 ϕ n σ 2 ) × tan ϕ n sec 2 ϕ n d θ n d ϕ n .
σ 2 = 0.003 + 0.00512 U .
θ n = 2 π ρ θ n , ρ ϕ n = 1 2 π 2 σ 2 0 ϕ n exp ( - tan 2 ϕ n σ 2 ) tan ϕ n sec 2 ϕ n d ϕ n .
W = cos ω sec ϕ n cos ω > 0 p ( z x , z y ) cos ω sec ϕ n d z x d z y ,
d n ( r ) d r 1 r v + 1 ,
f = f ( x ) p ( x ) d x .
f = f ( x ) p ( x ) p ˜ ( x ) p ˜ ( x ) d x = f ( x ) w ( x ) p ˜ ( x ) d x ,
σ 2 [ f ( x ) w ( x ) ] = [ f ( x ) w ( x ) - f ] 2 p ˜ ( x ) d x .
p ˜ ( τ ) d τ = exp ( - τ ) d τ 1 - exp ( - τ b ) ,             0 τ τ b .
β ˜ w ( μ , ϕ μ , ϕ ) β ˜ w ( ψ ) = 3 16 π ( 1 + cos 2 ψ )
ψ = cos - 1 [ μ μ + ( 1 - μ 2 ) 1 / 2 ( 1 - μ 2 ) 1 / 2 cos ( ϕ - ϕ ) ] .
2 π 0 π β ˜ ( ψ ) sin ψ d ψ = 1.
β = β w + β p ,
β ˜ = b w b β ˜ w + b p b β ˜ p .
a p = 0.04 C 0.602 ,
b p = 0.33 C 0.620 ,
C ( z ) = C 0 + h s 2 π exp [ - 1 2 ( z - z max s ) 2 ] .
C 0 = 0.2 mg m - 3 ,
s = 9 m ,
z max = 17 m ,
h = 144 mg m - 2 ,
a w = 0.0257 m - 1
b w = 0.0029 m - 1 .
ω 0 b c = b w + b p ( z ) a w + a p ( z ) + b w + b p ( z ) .
τ aerosol = 0.264 , τ Rayleigh = 0.145.
σ 2 = 0.003 + 0.00512 U ,
β ˜ Ram ( ψ ) = 3 16 π 1 + 3 ρ 1 + 2 ρ ( 1 + 1 - ρ 1 + 3 ρ cos 2 ψ ) ,
a w ( 417 ) = 0.0156 m - 1 , b w ( 417 ) = 0.0063 m - 1 , a w ( 486 ) = 0.0188 m - 1 , b w ( 486 ) = 0.0032 m - 1 .
L ( θ ) = L 0 1 + k cos θ .
E d = - 2 π L 0 k 2 [ k + ln ( 1 - k ) ] , E o u = 2 π L 0 k ln ( 1 + k ) .
s x ¯ = [ 1 N - 1 i = 1 N ( x i - x ¯ ) 2 ] 1 / 2 1 N i = 1 N x i ,
( 1 - k μ ) L ( μ ) = ω 0 0 2 π - 1 1 L ( μ ) β ˜ ( ψ ) d μ d ϕ .
S in ( z , θ , λ ) = 1 4 π l = 0 N b in ( l ) ( z , λ ex λ ) P l ( cos θ ) E l ( z , λ ex ) d λ ex ,
E l ( z , λ ex ) = 2 π 0 π P l ( cos θ ) L ( 0 ) ( z , θ , λ ex ) × sin θ d θ , β in ( z , ψ , λ ex λ ) = 1 4 π l = 0 N b in ( l ) ( z , λ ex λ ) P l ( cos ψ ) .
b in ( 0 ) ( z , λ ex λ ) b in ( z , λ ex λ ) = Ξ β in ( z ; θ , ϕ θ , ϕ ; λ ex λ ) d Ω = 2 π 0 π β in ( z , ψ , λ ex λ ) sin ψ d ψ .
β Ram ( z , ψ , λ ex λ ) = β ˜ Ram ( ψ ) b Ram ( z , λ ex λ ) ,
β Ram = 3 16 π ( 1 + 3 ρ 1 + 2 ρ ) [ 1 + 1 3 γ + 2 3 γ P 2 ( cos ψ ) ] × b Ram ( z , λ ex λ ) .
b Ram ( 0 ) ( z , λ ex λ ) = b Ram ( z , λ ex λ ) , b Ram ( 2 ) = 1 2 ( 1 - ρ 1 + 2 ρ ) b Ram ( z , λ ex λ ) ,
S Ram ( z , θ , λ ) = 1 4 π b Ram ( z , λ ex λ ) E 0 ( z , λ ex ) × [ 1 + 1 2 ( 1 - ρ 1 + 2 ρ ) E 2 ( z , λ ex ) E 0 ( z , λ ex ) P 2 ( cos θ ) ] d λ ex .
S in ( z , θ , λ ) = 1 4 π b in ( z , λ ex λ ) Δ λ ex E 0 ( z , λ ex ) × l = 0 N b in ( l ) ( z , λ ex λ ) E l ( z , λ ex ) b in ( z , λ ex λ ) E 0 ( z , λ ex ) P l ( cos θ ) .
p ( z ) = E 0 ( z , λ ex ) 0 E 0 ( z , λ ex ) d z .
ρ j = 0 z p ( z ) d z .
p ( θ z ) = 1 4 π l = 0 N b in ( l ) ( z , λ ex λ ) E l ( z , λ ex ) b in ( z , λ ex λ ) E 0 ( z , λ ex ) P l ( cos θ ) ,
ρ j + 1 = 0 θ p ( θ z ) d θ .
W = b in ( z , λ ex λ ) Δ λ ex 0 E 0 ( z , λ ex ) d z ,

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