Abstract

Two factors influence the diffuse transmittance (t) of water-leaving radiance (Lw) to the top of the atmosphere: the angular distribution of upwelling radiance beneath the sea surface (Lu) and the concentration and optical properties of aerosols in the atmosphere. We examine these factors and (1) show that the error in Lw that is induced by assuming Lu is uniform (i.e., in treating the subsurface reflectance by the water body as Lambertian) is significant in comparison with the other errors expected in Lw only at low phytoplankton concentration and then only in the blue region of the spectrum; (2) show that when radiance ratios are used in biophysical algorithms the effect of the uniform- Lu approximation is even smaller; and (3) provide an avenue for introducing accurate computation of the uniform Lu diffuse transmittance into atmospheric correction algorithms. In an Appendix the reciprocity principle is derived for a medium in which the refractive index is a continuous function of position.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. A. Hovis, D. K. Clark, F. Anderson, R. W. Austin, W. H. Wilson, E. T. Baker, D. Ball, H. R. Gordon, J. L. Mueller, S. Y. E. Sayed, B. Strum, R. C. Wrigley, C. S. Yentsch, “Nimbus 7 coastal zone color scanner: system description and initial imagery,” Science 210k60–63 (1980).
    [CrossRef]
  2. H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments derived from the Nimbus-7 CZCS: initial comparisons with surface measurements,” Science 210, 63–66 (1980).
    [CrossRef] [PubMed]
  3. S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, “An overview of SeaWiFS and ocean color,” , Vol. 1, 1992 (NASA, Greenbelt, Md.).
  4. V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, H. Ostrow, “MODIS: advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sens. 27, 145–152 (1989).
    [CrossRef]
  5. H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, New York, 1983).
    [CrossRef]
  6. H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison between ship determinations and Coastal Zone Color Scanner estimates,” Appl. Opt. 22, 20–36 (1983).
    [CrossRef] [PubMed]
  7. H. R. Gordon, M. Wang, “Retrieval of water-leaving radiance and aerosol optical thickness over the oceans with SeaWiFS: a preliminary algorithm,” Appl. Opt. 33, 443–452 (1994).
    [CrossRef] [PubMed]
  8. H. R. Gordon, M. Wang, “Influence of oceanic whitecaps on atmospheric correction of SeaWiFS,” Appl. Opt. 33, 7754–7763 (1994).
    [CrossRef] [PubMed]
  9. H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytic radiance model of ocean color,” J. Geophys. Res. 93D, 10,909–10,924 (1988).
    [CrossRef]
  10. H. R. Gordon, “Atmospheric correction of ocean color imagery in the Earth Observing System era,” J. Geophys. Res. 102D, 17,081–17,106 (1997)
    [CrossRef]
  11. 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]
  12. A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters. II: bidirectional aspects,” Appl. Opt. 32, 6864–6879 (1993).
    [CrossRef] [PubMed]
  13. A. Morel, K. J. Voss, B. Gentili, “Bidirectional reflectance of oceanic waters: a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100C, 13,143–13,150 (1995).
    [CrossRef]
  14. D. Tanre, M. Herman, P. Y. Deschamps, A. de Leffe, “Atmospheric modeling for space measurements of ground reflectances, including bidirectional properties,” Appl. Opt. 18, 3587–3594 (1979).
    [CrossRef] [PubMed]
  15. Y. J. Kaufman, “Atmospheric effect on spatial resolution of surface imagery,” Appl. Opt. 23, 3400–3408 (1984).
    [CrossRef] [PubMed]
  16. P. Reinersman, K. L. Carder, “Monte Carlo simulation of the atmospheric point-spread function with an application to correction of the adjacency effect,” Appl. Opt. 34, 4453–4471 (1995).
    [CrossRef] [PubMed]
  17. H. R. Gordon, “Radiative transfer: a technique for simulating the ocean in satellite remote sensing calculations,” Appl. Opt. 15, 1974–1979 (1976).
    [CrossRef] [PubMed]
  18. H. R. Gordon, “Simple calculation of the diffuse reflectance of the ocean,” Appl. Opt. 12, 2803–2804 (1973).
    [CrossRef] [PubMed]
  19. H. R. Gordon, “Modeling and simulating radiative transfer in the ocean,” in Ocean Optics, R. W. Spinrad, K. L. Carder, M. J. Perry, eds. (Oxford U. Press, New York, 1994), pp. 3–39.
  20. T. J. Petzold, “Volume scattering functions for selected natural waters,” , 1972 (Scripps Institution of Oceanography, Visibility Laboratory, San Diego, Calif. 92152).
  21. J. T. O. Kirk, “Estimation of the scattering coefficient of natural waters using underwater irradiance measurements,” Aust. J. Mar. Freshwater Res. 32, 533–539 (1981).
    [CrossRef]
  22. J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (1989).
    [CrossRef]
  23. J. T. O. Kirk, “Estimation of the absorption and scattering coefficients of natural waters by the use of underwater irradiance measurements,” Appl. Opt. 33, 3276–3278 (1994).
    [CrossRef] [PubMed]
  24. E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” , 1979 (Air Force Geophysics Laboratory, Hanscomb AFB, Mass.).
  25. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).
  26. P. J. Reddy, F. W. Kreiner, J. J. Deluisi, Y. Kim, “Aerosol optical depths over the Atlantic derived from shipboard sunphotometer observations during the 1988 Global Change Expedition,” Global Biogeochem. Cycles 4, 225–240 (1990).
    [CrossRef]
  27. G. K. Korotaev, S. M. Sakerin, A. M. Ignatov, L. L. Stowe, E. P. McClain, “Sunphotometer observations of aerosol optical thickness over the North Atlantic from a Soviet research vessel for validation of satellite measurements,” J. Atmos. Oceanic Technol. 10, 725–735 (1993).
    [CrossRef]
  28. Y. V. Villevalde, A. V. Smirnov, N. T. O’Neill, S. P. Smyshlyaev, V. V. Yakovlev, “Measurement of aerosol optical depth in the Pacific Ocean and North Atlantic,” J. Geophys. Res. 99D, 20,983–20,988 (1994).
    [CrossRef]
  29. K. J. Voss, “Electro-optic camera system for measurement of the underwater radiance distribution,” Opt. Eng. 28, 241–247 (1989).
    [CrossRef]
  30. K. N. Liou, Radiation and Cloud Processes in the Atmosphere (Oxford U. Press, New York, 1992).
  31. A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters. III: implication of bidirectionality for the remote sensing problem,” Appl. Opt. 35, 4850–4862 (1996).
    [CrossRef] [PubMed]
  32. P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, G. Sèze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
    [CrossRef]
  33. K. M. Case, “Transfer problems and the reciprocity principle,” Rev. Mod. Phys. 29, 651–663 (1957).
    [CrossRef]
  34. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).
  35. S. Chandrasekhar, Radiative Transfer (Oxford U. Press, Oxford, UK, 1950).
  36. W. T. Welford, Aberrations in Optical Systems (Hilger, Bristol, UK, 1986).

1997 (1)

H. R. Gordon, “Atmospheric correction of ocean color imagery in the Earth Observing System era,” J. Geophys. Res. 102D, 17,081–17,106 (1997)
[CrossRef]

1996 (1)

1995 (2)

P. Reinersman, K. L. Carder, “Monte Carlo simulation of the atmospheric point-spread function with an application to correction of the adjacency effect,” Appl. Opt. 34, 4453–4471 (1995).
[CrossRef] [PubMed]

A. Morel, K. J. Voss, B. Gentili, “Bidirectional reflectance of oceanic waters: a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100C, 13,143–13,150 (1995).
[CrossRef]

1994 (5)

Y. V. Villevalde, A. V. Smirnov, N. T. O’Neill, S. P. Smyshlyaev, V. V. Yakovlev, “Measurement of aerosol optical depth in the Pacific Ocean and North Atlantic,” J. Geophys. Res. 99D, 20,983–20,988 (1994).
[CrossRef]

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, G. Sèze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[CrossRef]

H. R. Gordon, M. Wang, “Retrieval of water-leaving radiance and aerosol optical thickness over the oceans with SeaWiFS: a preliminary algorithm,” Appl. Opt. 33, 443–452 (1994).
[CrossRef] [PubMed]

J. T. O. Kirk, “Estimation of the absorption and scattering coefficients of natural waters by the use of underwater irradiance measurements,” Appl. Opt. 33, 3276–3278 (1994).
[CrossRef] [PubMed]

H. R. Gordon, M. Wang, “Influence of oceanic whitecaps on atmospheric correction of SeaWiFS,” Appl. Opt. 33, 7754–7763 (1994).
[CrossRef] [PubMed]

1993 (2)

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters. II: bidirectional aspects,” Appl. Opt. 32, 6864–6879 (1993).
[CrossRef] [PubMed]

G. K. Korotaev, S. M. Sakerin, A. M. Ignatov, L. L. Stowe, E. P. McClain, “Sunphotometer observations of aerosol optical thickness over the North Atlantic from a Soviet research vessel for validation of satellite measurements,” J. Atmos. Oceanic Technol. 10, 725–735 (1993).
[CrossRef]

1991 (1)

1990 (1)

P. J. Reddy, F. W. Kreiner, J. J. Deluisi, Y. Kim, “Aerosol optical depths over the Atlantic derived from shipboard sunphotometer observations during the 1988 Global Change Expedition,” Global Biogeochem. Cycles 4, 225–240 (1990).
[CrossRef]

1989 (3)

J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (1989).
[CrossRef]

V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, H. Ostrow, “MODIS: advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sens. 27, 145–152 (1989).
[CrossRef]

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

1988 (1)

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytic radiance model of ocean color,” J. Geophys. Res. 93D, 10,909–10,924 (1988).
[CrossRef]

1984 (1)

1983 (1)

1981 (1)

J. T. O. Kirk, “Estimation of the scattering coefficient of natural waters using underwater irradiance measurements,” Aust. J. Mar. Freshwater Res. 32, 533–539 (1981).
[CrossRef]

1980 (2)

W. A. Hovis, D. K. Clark, F. Anderson, R. W. Austin, W. H. Wilson, E. T. Baker, D. Ball, H. R. Gordon, J. L. Mueller, S. Y. E. Sayed, B. Strum, R. C. Wrigley, C. S. Yentsch, “Nimbus 7 coastal zone color scanner: system description and initial imagery,” Science 210k60–63 (1980).
[CrossRef]

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments derived from the Nimbus-7 CZCS: initial comparisons with surface measurements,” Science 210, 63–66 (1980).
[CrossRef] [PubMed]

1979 (1)

1976 (1)

1973 (1)

1957 (1)

K. M. Case, “Transfer problems and the reciprocity principle,” Rev. Mod. Phys. 29, 651–663 (1957).
[CrossRef]

Anderson, F.

W. A. Hovis, D. K. Clark, F. Anderson, R. W. Austin, W. H. Wilson, E. T. Baker, D. Ball, H. R. Gordon, J. L. Mueller, S. Y. E. Sayed, B. Strum, R. C. Wrigley, C. S. Yentsch, “Nimbus 7 coastal zone color scanner: system description and initial imagery,” Science 210k60–63 (1980).
[CrossRef]

Austin, R. W.

W. A. Hovis, D. K. Clark, F. Anderson, R. W. Austin, W. H. Wilson, E. T. Baker, D. Ball, H. R. Gordon, J. L. Mueller, S. Y. E. Sayed, B. Strum, R. C. Wrigley, C. S. Yentsch, “Nimbus 7 coastal zone color scanner: system description and initial imagery,” Science 210k60–63 (1980).
[CrossRef]

Baker, E. T.

W. A. Hovis, D. K. Clark, F. Anderson, R. W. Austin, W. H. Wilson, E. T. Baker, D. Ball, H. R. Gordon, J. L. Mueller, S. Y. E. Sayed, B. Strum, R. C. Wrigley, C. S. Yentsch, “Nimbus 7 coastal zone color scanner: system description and initial imagery,” Science 210k60–63 (1980).
[CrossRef]

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 semi-analytic radiance model of ocean color,” J. Geophys. Res. 93D, 10,909–10,924 (1988).
[CrossRef]

Ball, D.

W. A. Hovis, D. K. Clark, F. Anderson, R. W. Austin, W. H. Wilson, E. T. Baker, D. Ball, H. R. Gordon, J. L. Mueller, S. Y. E. Sayed, B. Strum, R. C. Wrigley, C. S. Yentsch, “Nimbus 7 coastal zone color scanner: system description and initial imagery,” Science 210k60–63 (1980).
[CrossRef]

Barnes, W. L.

V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, H. Ostrow, “MODIS: advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sens. 27, 145–152 (1989).
[CrossRef]

Bréon, F. M.

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, G. Sèze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[CrossRef]

Bricaud, A.

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, G. Sèze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[CrossRef]

Broenkow, W. W.

Brown, J. W.

Brown, O. B.

Buriez, J. C.

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, G. Sèze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[CrossRef]

Carder, K. L.

Case, K. M.

K. M. Case, “Transfer problems and the reciprocity principle,” Rev. Mod. Phys. 29, 651–663 (1957).
[CrossRef]

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, Oxford, UK, 1950).

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 semi-analytic radiance model of ocean color,” J. Geophys. Res. 93D, 10,909–10,924 (1988).
[CrossRef]

H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison between ship determinations and Coastal Zone Color Scanner estimates,” Appl. Opt. 22, 20–36 (1983).
[CrossRef] [PubMed]

W. A. Hovis, D. K. Clark, F. Anderson, R. W. Austin, W. H. Wilson, E. T. Baker, D. Ball, H. R. Gordon, J. L. Mueller, S. Y. E. Sayed, B. Strum, R. C. Wrigley, C. S. Yentsch, “Nimbus 7 coastal zone color scanner: system description and initial imagery,” Science 210k60–63 (1980).
[CrossRef]

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments derived from the Nimbus-7 CZCS: initial comparisons with surface measurements,” Science 210, 63–66 (1980).
[CrossRef] [PubMed]

de Leffe, A.

Deluisi, J. J.

P. J. Reddy, F. W. Kreiner, J. J. Deluisi, Y. Kim, “Aerosol optical depths over the Atlantic derived from shipboard sunphotometer observations during the 1988 Global Change Expedition,” Global Biogeochem. Cycles 4, 225–240 (1990).
[CrossRef]

Deschamps, P. Y.

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, G. Sèze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[CrossRef]

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

Esaias, W. E.

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, “An overview of SeaWiFS and ocean color,” , Vol. 1, 1992 (NASA, Greenbelt, Md.).

Evans, R. H.

Feldman, G. C.

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, “An overview of SeaWiFS and ocean color,” , Vol. 1, 1992 (NASA, Greenbelt, Md.).

Fenn, R. W.

E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” , 1979 (Air Force Geophysics Laboratory, Hanscomb AFB, Mass.).

Gentili, B.

Gordon, H. R.

H. R. Gordon, “Atmospheric correction of ocean color imagery in the Earth Observing System era,” J. Geophys. Res. 102D, 17,081–17,106 (1997)
[CrossRef]

H. R. Gordon, M. Wang, “Influence of oceanic whitecaps on atmospheric correction of SeaWiFS,” Appl. Opt. 33, 7754–7763 (1994).
[CrossRef] [PubMed]

H. R. Gordon, M. Wang, “Retrieval of water-leaving radiance and aerosol optical thickness over the oceans with SeaWiFS: a preliminary algorithm,” Appl. Opt. 33, 443–452 (1994).
[CrossRef] [PubMed]

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytic radiance model of ocean color,” J. Geophys. Res. 93D, 10,909–10,924 (1988).
[CrossRef]

H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison between ship determinations and Coastal Zone Color Scanner estimates,” Appl. Opt. 22, 20–36 (1983).
[CrossRef] [PubMed]

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments derived from the Nimbus-7 CZCS: initial comparisons with surface measurements,” Science 210, 63–66 (1980).
[CrossRef] [PubMed]

W. A. Hovis, D. K. Clark, F. Anderson, R. W. Austin, W. H. Wilson, E. T. Baker, D. Ball, H. R. Gordon, J. L. Mueller, S. Y. E. Sayed, B. Strum, R. C. Wrigley, C. S. Yentsch, “Nimbus 7 coastal zone color scanner: system description and initial imagery,” Science 210k60–63 (1980).
[CrossRef]

H. R. Gordon, “Radiative transfer: a technique for simulating the ocean in satellite remote sensing calculations,” Appl. Opt. 15, 1974–1979 (1976).
[CrossRef] [PubMed]

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

H. R. Gordon, “Modeling and simulating radiative transfer in the ocean,” in Ocean Optics, R. W. Spinrad, K. L. Carder, M. J. Perry, eds. (Oxford U. Press, New York, 1994), pp. 3–39.

H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, New York, 1983).
[CrossRef]

Gregg, W. W.

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, “An overview of SeaWiFS and ocean color,” , Vol. 1, 1992 (NASA, Greenbelt, Md.).

Herman, M.

Hooker, S. B.

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, “An overview of SeaWiFS and ocean color,” , Vol. 1, 1992 (NASA, Greenbelt, Md.).

Hovis, W. A.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments derived from the Nimbus-7 CZCS: initial comparisons with surface measurements,” Science 210, 63–66 (1980).
[CrossRef] [PubMed]

W. A. Hovis, D. K. Clark, F. Anderson, R. W. Austin, W. H. Wilson, E. T. Baker, D. Ball, H. R. Gordon, J. L. Mueller, S. Y. E. Sayed, B. Strum, R. C. Wrigley, C. S. Yentsch, “Nimbus 7 coastal zone color scanner: system description and initial imagery,” Science 210k60–63 (1980).
[CrossRef]

Ignatov, A. M.

G. K. Korotaev, S. M. Sakerin, A. M. Ignatov, L. L. Stowe, E. P. McClain, “Sunphotometer observations of aerosol optical thickness over the North Atlantic from a Soviet research vessel for validation of satellite measurements,” J. Atmos. Oceanic Technol. 10, 725–735 (1993).
[CrossRef]

Kaufman, Y. J.

Kim, Y.

P. J. Reddy, F. W. Kreiner, J. J. Deluisi, Y. Kim, “Aerosol optical depths over the Atlantic derived from shipboard sunphotometer observations during the 1988 Global Change Expedition,” Global Biogeochem. Cycles 4, 225–240 (1990).
[CrossRef]

Kirk, J. T. O.

J. T. O. Kirk, “Estimation of the absorption and scattering coefficients of natural waters by the use of underwater irradiance measurements,” Appl. Opt. 33, 3276–3278 (1994).
[CrossRef] [PubMed]

J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (1989).
[CrossRef]

J. T. O. Kirk, “Estimation of the scattering coefficient of natural waters using underwater irradiance measurements,” Aust. J. Mar. Freshwater Res. 32, 533–539 (1981).
[CrossRef]

Korotaev, G. K.

G. K. Korotaev, S. M. Sakerin, A. M. Ignatov, L. L. Stowe, E. P. McClain, “Sunphotometer observations of aerosol optical thickness over the North Atlantic from a Soviet research vessel for validation of satellite measurements,” J. Atmos. Oceanic Technol. 10, 725–735 (1993).
[CrossRef]

Kreiner, F. W.

P. J. Reddy, F. W. Kreiner, J. J. Deluisi, Y. Kim, “Aerosol optical depths over the Atlantic derived from shipboard sunphotometer observations during the 1988 Global Change Expedition,” Global Biogeochem. Cycles 4, 225–240 (1990).
[CrossRef]

Leroy, M.

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, G. Sèze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[CrossRef]

Liou, K. N.

K. N. Liou, Radiation and Cloud Processes in the Atmosphere (Oxford U. Press, New York, 1992).

Maymon, P. W.

V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, H. Ostrow, “MODIS: advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sens. 27, 145–152 (1989).
[CrossRef]

McClain, C. R.

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, “An overview of SeaWiFS and ocean color,” , Vol. 1, 1992 (NASA, Greenbelt, Md.).

McClain, E. P.

G. K. Korotaev, S. M. Sakerin, A. M. Ignatov, L. L. Stowe, E. P. McClain, “Sunphotometer observations of aerosol optical thickness over the North Atlantic from a Soviet research vessel for validation of satellite measurements,” J. Atmos. Oceanic Technol. 10, 725–735 (1993).
[CrossRef]

Montgomery, H. E.

V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, H. Ostrow, “MODIS: advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sens. 27, 145–152 (1989).
[CrossRef]

Morel, A.

Morel, A. Y.

H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, New York, 1983).
[CrossRef]

Mueller, J. L.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments derived from the Nimbus-7 CZCS: initial comparisons with surface measurements,” Science 210, 63–66 (1980).
[CrossRef] [PubMed]

W. A. Hovis, D. K. Clark, F. Anderson, R. W. Austin, W. H. Wilson, E. T. Baker, D. Ball, H. R. Gordon, J. L. Mueller, S. Y. E. Sayed, B. Strum, R. C. Wrigley, C. S. Yentsch, “Nimbus 7 coastal zone color scanner: system description and initial imagery,” Science 210k60–63 (1980).
[CrossRef]

O’Neill, N. T.

Y. V. Villevalde, A. V. Smirnov, N. T. O’Neill, S. P. Smyshlyaev, V. V. Yakovlev, “Measurement of aerosol optical depth in the Pacific Ocean and North Atlantic,” J. Geophys. Res. 99D, 20,983–20,988 (1994).
[CrossRef]

Ostrow, H.

V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, H. Ostrow, “MODIS: advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sens. 27, 145–152 (1989).
[CrossRef]

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected natural waters,” , 1972 (Scripps Institution of Oceanography, Visibility Laboratory, San Diego, Calif. 92152).

Podaire, A.

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, G. Sèze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[CrossRef]

Reddy, P. J.

P. J. Reddy, F. W. Kreiner, J. J. Deluisi, Y. Kim, “Aerosol optical depths over the Atlantic derived from shipboard sunphotometer observations during the 1988 Global Change Expedition,” Global Biogeochem. Cycles 4, 225–240 (1990).
[CrossRef]

Reinersman, P.

Sakerin, S. M.

G. K. Korotaev, S. M. Sakerin, A. M. Ignatov, L. L. Stowe, E. P. McClain, “Sunphotometer observations of aerosol optical thickness over the North Atlantic from a Soviet research vessel for validation of satellite measurements,” J. Atmos. Oceanic Technol. 10, 725–735 (1993).
[CrossRef]

Salomonson, V. V.

V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, H. Ostrow, “MODIS: advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sens. 27, 145–152 (1989).
[CrossRef]

Sayed, S. Y. E.

W. A. Hovis, D. K. Clark, F. Anderson, R. W. Austin, W. H. Wilson, E. T. Baker, D. Ball, H. R. Gordon, J. L. Mueller, S. Y. E. Sayed, B. Strum, R. C. Wrigley, C. S. Yentsch, “Nimbus 7 coastal zone color scanner: system description and initial imagery,” Science 210k60–63 (1980).
[CrossRef]

Sèze, G.

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, G. Sèze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[CrossRef]

Shettle, E. P.

E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” , 1979 (Air Force Geophysics Laboratory, Hanscomb AFB, Mass.).

Smirnov, A. V.

Y. V. Villevalde, A. V. Smirnov, N. T. O’Neill, S. P. Smyshlyaev, V. V. Yakovlev, “Measurement of aerosol optical depth in the Pacific Ocean and North Atlantic,” J. Geophys. Res. 99D, 20,983–20,988 (1994).
[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 semi-analytic radiance model of ocean color,” J. Geophys. Res. 93D, 10,909–10,924 (1988).
[CrossRef]

Smyshlyaev, S. P.

Y. V. Villevalde, A. V. Smirnov, N. T. O’Neill, S. P. Smyshlyaev, V. V. Yakovlev, “Measurement of aerosol optical depth in the Pacific Ocean and North Atlantic,” J. Geophys. Res. 99D, 20,983–20,988 (1994).
[CrossRef]

Stowe, L. L.

G. K. Korotaev, S. M. Sakerin, A. M. Ignatov, L. L. Stowe, E. P. McClain, “Sunphotometer observations of aerosol optical thickness over the North Atlantic from a Soviet research vessel for validation of satellite measurements,” J. Atmos. Oceanic Technol. 10, 725–735 (1993).
[CrossRef]

Strum, B.

W. A. Hovis, D. K. Clark, F. Anderson, R. W. Austin, W. H. Wilson, E. T. Baker, D. Ball, H. R. Gordon, J. L. Mueller, S. Y. E. Sayed, B. Strum, R. C. Wrigley, C. S. Yentsch, “Nimbus 7 coastal zone color scanner: system description and initial imagery,” Science 210k60–63 (1980).
[CrossRef]

Tanre, D.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

Villevalde, Y. V.

Y. V. Villevalde, A. V. Smirnov, N. T. O’Neill, S. P. Smyshlyaev, V. V. Yakovlev, “Measurement of aerosol optical depth in the Pacific Ocean and North Atlantic,” J. Geophys. Res. 99D, 20,983–20,988 (1994).
[CrossRef]

Voss, K. J.

A. Morel, K. J. Voss, B. Gentili, “Bidirectional reflectance of oceanic waters: a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100C, 13,143–13,150 (1995).
[CrossRef]

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

Wang, M.

Welford, W. T.

W. T. Welford, Aberrations in Optical Systems (Hilger, Bristol, UK, 1986).

Wilson, W. H.

W. A. Hovis, D. K. Clark, F. Anderson, R. W. Austin, W. H. Wilson, E. T. Baker, D. Ball, H. R. Gordon, J. L. Mueller, S. Y. E. Sayed, B. Strum, R. C. Wrigley, C. S. Yentsch, “Nimbus 7 coastal zone color scanner: system description and initial imagery,” Science 210k60–63 (1980).
[CrossRef]

Wrigley, R. C.

W. A. Hovis, D. K. Clark, F. Anderson, R. W. Austin, W. H. Wilson, E. T. Baker, D. Ball, H. R. Gordon, J. L. Mueller, S. Y. E. Sayed, B. Strum, R. C. Wrigley, C. S. Yentsch, “Nimbus 7 coastal zone color scanner: system description and initial imagery,” Science 210k60–63 (1980).
[CrossRef]

Yakovlev, V. V.

Y. V. Villevalde, A. V. Smirnov, N. T. O’Neill, S. P. Smyshlyaev, V. V. Yakovlev, “Measurement of aerosol optical depth in the Pacific Ocean and North Atlantic,” J. Geophys. Res. 99D, 20,983–20,988 (1994).
[CrossRef]

Yentsch, C. S.

W. A. Hovis, D. K. Clark, F. Anderson, R. W. Austin, W. H. Wilson, E. T. Baker, D. Ball, H. R. Gordon, J. L. Mueller, S. Y. E. Sayed, B. Strum, R. C. Wrigley, C. S. Yentsch, “Nimbus 7 coastal zone color scanner: system description and initial imagery,” Science 210k60–63 (1980).
[CrossRef]

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Appl. Opt. (12)

H. R. Gordon, “Radiative transfer: a technique for simulating the ocean in satellite remote sensing calculations,” Appl. Opt. 15, 1974–1979 (1976).
[CrossRef] [PubMed]

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

H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison between ship determinations and Coastal Zone Color Scanner estimates,” Appl. Opt. 22, 20–36 (1983).
[CrossRef] [PubMed]

Y. J. Kaufman, “Atmospheric effect on spatial resolution of surface imagery,” Appl. Opt. 23, 3400–3408 (1984).
[CrossRef] [PubMed]

H. R. Gordon, M. Wang, “Retrieval of water-leaving radiance and aerosol optical thickness over the oceans with SeaWiFS: a preliminary algorithm,” Appl. Opt. 33, 443–452 (1994).
[CrossRef] [PubMed]

J. T. O. Kirk, “Estimation of the absorption and scattering coefficients of natural waters by the use of underwater irradiance measurements,” Appl. Opt. 33, 3276–3278 (1994).
[CrossRef] [PubMed]

H. R. Gordon, M. Wang, “Influence of oceanic whitecaps on atmospheric correction of SeaWiFS,” Appl. Opt. 33, 7754–7763 (1994).
[CrossRef] [PubMed]

P. Reinersman, K. L. Carder, “Monte Carlo simulation of the atmospheric point-spread function with an application to correction of the adjacency effect,” Appl. Opt. 34, 4453–4471 (1995).
[CrossRef] [PubMed]

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

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters. II: bidirectional aspects,” Appl. Opt. 32, 6864–6879 (1993).
[CrossRef] [PubMed]

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

Aust. J. Mar. Freshwater Res. (1)

J. T. O. Kirk, “Estimation of the scattering coefficient of natural waters using underwater irradiance measurements,” Aust. J. Mar. Freshwater Res. 32, 533–539 (1981).
[CrossRef]

Global Biogeochem. Cycles (1)

P. J. Reddy, F. W. Kreiner, J. J. Deluisi, Y. Kim, “Aerosol optical depths over the Atlantic derived from shipboard sunphotometer observations during the 1988 Global Change Expedition,” Global Biogeochem. Cycles 4, 225–240 (1990).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (2)

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, G. Sèze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[CrossRef]

V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, H. Ostrow, “MODIS: advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sens. 27, 145–152 (1989).
[CrossRef]

J. Atmos. Oceanic Technol. (1)

G. K. Korotaev, S. M. Sakerin, A. M. Ignatov, L. L. Stowe, E. P. McClain, “Sunphotometer observations of aerosol optical thickness over the North Atlantic from a Soviet research vessel for validation of satellite measurements,” J. Atmos. Oceanic Technol. 10, 725–735 (1993).
[CrossRef]

J. Geophys. Res. (4)

Y. V. Villevalde, A. V. Smirnov, N. T. O’Neill, S. P. Smyshlyaev, V. V. Yakovlev, “Measurement of aerosol optical depth in the Pacific Ocean and North Atlantic,” J. Geophys. Res. 99D, 20,983–20,988 (1994).
[CrossRef]

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytic radiance model of ocean color,” J. Geophys. Res. 93D, 10,909–10,924 (1988).
[CrossRef]

H. R. Gordon, “Atmospheric correction of ocean color imagery in the Earth Observing System era,” J. Geophys. Res. 102D, 17,081–17,106 (1997)
[CrossRef]

A. Morel, K. J. Voss, B. Gentili, “Bidirectional reflectance of oceanic waters: a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100C, 13,143–13,150 (1995).
[CrossRef]

Limnol. Oceanogr. (1)

J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (1989).
[CrossRef]

Opt. Eng. (1)

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

Rev. Mod. Phys. (1)

K. M. Case, “Transfer problems and the reciprocity principle,” Rev. Mod. Phys. 29, 651–663 (1957).
[CrossRef]

Science (2)

W. A. Hovis, D. K. Clark, F. Anderson, R. W. Austin, W. H. Wilson, E. T. Baker, D. Ball, H. R. Gordon, J. L. Mueller, S. Y. E. Sayed, B. Strum, R. C. Wrigley, C. S. Yentsch, “Nimbus 7 coastal zone color scanner: system description and initial imagery,” Science 210k60–63 (1980).
[CrossRef]

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments derived from the Nimbus-7 CZCS: initial comparisons with surface measurements,” Science 210, 63–66 (1980).
[CrossRef] [PubMed]

Other (10)

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, “An overview of SeaWiFS and ocean color,” , Vol. 1, 1992 (NASA, Greenbelt, Md.).

H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, New York, 1983).
[CrossRef]

E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” , 1979 (Air Force Geophysics Laboratory, Hanscomb AFB, Mass.).

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, Oxford, UK, 1950).

W. T. Welford, Aberrations in Optical Systems (Hilger, Bristol, UK, 1986).

K. N. Liou, Radiation and Cloud Processes in the Atmosphere (Oxford U. Press, New York, 1992).

H. R. Gordon, “Modeling and simulating radiative transfer in the ocean,” in Ocean Optics, R. W. Spinrad, K. L. Carder, M. J. Perry, eds. (Oxford U. Press, New York, 1994), pp. 3–39.

T. J. Petzold, “Volume scattering functions for selected natural waters,” , 1972 (Scripps Institution of Oceanography, Visibility Laboratory, San Diego, Calif. 92152).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Comparison of the scattering phase functions for Rayleigh scattering and Petzold’s20 turbid water measurements.

Fig. 2
Fig. 2

Subsurface upwelling radiance predicted with the Rayleigh and Petzold phase functions, with radiances normalized to their maximum value: (a) Rayleigh with θ0 = 40°; (b) Rayleigh with θ0 = 60°; (c) Petzold with θ0 = 40°; (d) Petzold with θ0 = 60°.

Fig. 3
Fig. 3

Error in t, induced by the assumption that Lu is uniform, as a function of wavelength and aerosol optical thickness at the SeaWiFS scan center and scan edge: (a) Rayleigh at the scan center, (b) Rayleigh at the scan edge, (c) Petzold at the scan center, (d) Petzold at the scan edge. Recall that we take the SeaWiFS scan edge to be θv ≈ 45°. The M90 aerosol model is used in all computations.

Fig. 4
Fig. 4

Error in t, induced by the assumption that Lu is uniform, as a function of wavelength, aerosol optical thickness, and viewing azimuth ϕv for θ0 = θv: (a) Rayleigh with θ0 = 40°, (b) Rayleigh with θ0 = 60°, (c) Petzold with θ0 = 40°, (d) Petzold with θ0 = 60°. The M90 aerosol model is used in all computations.

Fig. 5
Fig. 5

Error in t, induced by the assumption that Lu is uniform, as a function of wavelength, aerosol optical thickness, and viewing angle θv in the perpendicular plane of the Sun: (a) Rayleigh with θ0 = 40°, (b) Rayleigh with θ0 = 60°, (c) Petzold with θ0 = 40°, (d) Petzold with θ0 = 60°. Recall that for SeaWiFS we take the scan edge to be θv ≈ 45°. The M90 aerosol model is used in all computations.

Fig. 6
Fig. 6

Subsurface upwelling radiance distributions measured by Voss with the radiance distribution system (RADS)13,29: (a) λ = 450 nm, θ0 = 58.2°; (b) λ = 500 nm, θ0 = 59.7°.

Fig. 7
Fig. 7

Error in t, induced by the assumption that Lu is uniform, as a function of wavelength and aerosol optical thickness, with the Voss subsurface upwelling radiance distribution: (a) θ0 = 60° viewing in the perpendicular plane to the Sun, (b) θv = θ0 = 60° with viewing azimuth ϕv. The M90 aerosol model is used in all computations.

Tables (1)

Tables Icon

Table 1 Water-Leaving Radiance Lw at 443 and 550 nm as a Function of Pigment Concentration C

Equations (31)

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

Ltλi=Lotherλi+Lwaλi,
tξˆvLwaξˆvLwξˆv,
tξˆ=LwaξˆLwξˆ
Lwξˆ=Tfξˆmw2 Luξˆ.
t-ξˆ0=1F0ξˆ0·nˆ0Tfξˆ0Ωdξˆ·nˆLRξˆLu-ξˆLu-ξˆ0dΩξˆ,
t*-ξˆ0=1F0ξˆ0·nˆ0Tfξˆ0Ωdξˆ·nˆLRξˆdΩξˆ=ERξˆ0F0ξˆ0·nˆ0Tfξˆ0,
ΔLw-ξˆ0Lw-ξˆ0Lw*-ξˆ0-Lw-ξˆ0Lw-ξˆ0=t-ξˆ0-t*-ξˆ0t*-ξˆ0,
ΔLw-ξˆ0Lw-ξˆ0=1ERξˆ0Ωdξˆ·nˆLRξˆLu-ξˆLu-ξˆ0-1dΩξˆ.
Luθ, ϕ=Cω0, PF0Tθ0cos θ0PΘcos θ0-cos θ,
cos Θ=cos θ cos θ0+sin θ sin θ0 cosϕ-ϕ0.
Cω0,Pξˆ·nˆ>oLdξˆcos θPΘcos θ-cos θdΩξˆ,
cos Θ=cos θ cos θ+sin θ sin θ cosϕ-ϕ
PΘ=brPRΘ+bpPpΘbr+bp,
ddL, ξˆm2=-cL, ξˆm2+4πβ; ξˆξˆL, ξˆm2dΩξˆ,
ddL1, ξˆm2=-cL1, ξˆm2+4πβ; ξˆξˆL1, ξˆm2dΩξˆ,
-ddL2, -ξˆm2=-cL2, -ξˆm2+4πβ; ξˆ-ξˆL2, ξˆm2dΩξˆ,
4πβ; ξˆξˆL2, -ξˆm2dΩξˆ.
4πdΩξˆV m2ddL1, ξˆm2L2, -ξˆm2dV,
VdV dΩξˆm2  β; ξˆξˆL1, ξˆm2L2, -ξˆm2×dΩξˆ-VdV dΩξˆm2× β; ξˆξˆL1, ξˆm2L2, -ξˆm2dΩξˆ.
4πdΩξˆV m2ddL1, ξˆm2L2, -ξˆm2dV=0
12ddA4πL1, ξˆm2L2, -ξˆm2m2dΩξˆdAξˆ×d=0,
A4πL1, ξˆL2, -ξˆm2dΩξˆdAξˆ12=0,
SdS4π ξˆ·nˆL1ρ, ξˆL2ρ, -ξˆm2ρdΩξˆ=0,
SdSξˆ·nˆ<0ξˆ·nˆL1ρ, ξˆL2ρ, -ξˆm2ρ-L1ρ, -ξˆL2ρ, ξˆm2ρdΩξˆ=0,
4πdΩξˆVdVQ2r, -ξˆL1r, ξˆm2r-Q1r, ξˆL2r, -ξˆm2r.
L2ρT, -ξˆ0=1F0ξˆ0·nˆΩdξˆ·nˆ×L1ρB, ξˆL2ρB, -ξˆmw2dΩξˆ,
L2ρT, -ξˆ0L2ρB, -ξˆ0=1F0ξˆ0·nˆmw2Ωdξˆ·nˆ×L1ρB, ξˆL2ρB, -ξˆL2ρB, -ξˆ0dΩξˆ.
L1ρB, ξˆ=Tfξˆ0F0ξˆ0·nˆξˆ0·nˆδξˆ-ξˆ0,
L2ρT, -ξˆ0=Tfξˆ0mw2 L2ρB, -ξˆ0,
L2ρB, -ξˆ0=mw2Tfξˆ0LwρB, -ξˆ0,
L2ρT, -ξˆ0LwρB, -ξˆ0=1F0ξˆ0·nˆTfξˆ0Ωdξˆ·nˆ×L1ρB, ξˆL2ρB, -ξˆL2ρB, -ξˆ0dΩξˆt-ξˆ0.

Metrics