Abstract

A method is evaluated for estimating the absorption coefficient a and the backscattering coefficient bb from measurements of the upward and downward irradiances Eu(z) and Ed(z). With this method, the reflectance ratio R(z) and the downward diffuse attenuation coefficient Kd(z) obtained from Eu(z) and Ed(z) are used to estimate the inherent optical properties R and K that are the asymptotic values of R(z) and Kd(z), respectively. For an assumed scattering phase function β˜, there are unique correlations between the values of R and K and those of a and bb that can be derived from the radiative transfer equation. Good estimates of a and the Gordon parameter G = bb/(a + bb) can be obtained from R and K if the true scattering phase function is not greatly different from the assumed function. The method works best in deep, homogeneous waters, but can be applied to some cases of stratified waters. To improve performance in shallow waters where bottom effects are important, the deep- and shallow-measurement reflectance models also are developed.

© 1997 Optical Society of America

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References

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  1. R. W. Preisendorfer, Hydrologic Optics, V.1. NTIS PB 259793/8ST (National Technical Information Service, Springfield, Va., 1976).
  2. D. A. Kiefer, B. G. Mitchell, “A simple, steady-state description of phytoplankton growth based on absorption cross section and quantum efficiency,” Limnol. Oceanogr. 28, 770–776 (1983).
    [CrossRef]
  3. M. Kishino, “Interrelationship between light and phytoplankton in the sea,” in Ocean Optics, R. W. Spinrad, K. L. Carder, M. J. Perry, eds. (Oxford U. Press, New York, 1994).
  4. A. A. Gershun, “The light field,” J. Math. Phys. (Cambridge, Mass.) 18, 51–151 (1939).
  5. K. L. Carder, D. J. Collins, M. J. Perry, H. L. Clark, J. M. Mesias, J. S. Cleveland, J Greenier, “The interaction of light with phytoplankton in the marine environment,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 42–55 (1986).
  6. L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
    [CrossRef]
  7. J. R. V. Zaneveld, “A reflecting tube absorption meter,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 124–136 (1990).
    [CrossRef]
  8. 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).
  9. H. R. Gordon, G. C. Boynton, “A radiance–irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: homogeneous waters,” Appl. Opt. 36, 2636–2641 (1997).
    [CrossRef] [PubMed]
  10. Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation using explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
    [CrossRef] [PubMed]
  11. S. Chandrasekhar, Radiative Transfer (Oxford U. Press, New York, 1950).
  12. M. Benassi, R. D. M. Garcia, A. H. J. Karp, C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” Astrophys. J. 280, 853–864 (1984).
    [CrossRef]
  13. N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
    [CrossRef]
  14. C. E. Siewert, “The FN method for solving radiative-transfer problems in plane geometry,” Astrophys. Space Sci. 58, 131–137 (1978).
    [CrossRef]
  15. L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  16. K. Stamnes, “The Chandrasekhar method and its applications to atmospheric radiative transfer,” Trans. Am. Nucl. Soc. 71, 213–214 (1994).
  17. Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media such as the atmosphere/ocean system,” Appl. Opt. 33, 431–442 (1994).
    [CrossRef] [PubMed]
  18. N. J. McCormick, “Analytical transport theory applications in optical oceanography,” Ann. Nucl. Energy 23, 381–395 (1996).
    [CrossRef]
  19. J. R. V. Zaneveld, “An asymptotic closure theory for irradiance in the sea and its inversion to obtain the inherent optical properties,” Limnol. Oceanogr. 34, 1442–1452 (1989).
    [CrossRef]
  20. N. J. McCormick, “Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient,” Limnol. Oceanogr. 40, 1013–1018 (1995).
    [CrossRef]
  21. 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]
  22. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989), pp. 274–286.
  23. T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography, Visibility Laboratory, San Diego, Calif., 1972).
  24. P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
    [CrossRef]
  25. G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).
    [CrossRef]
  26. V. I. Haltrin, “Algorithm for computing apparent optical properties of shallow waters under arbitrary surface illumination,” in Proceedings of the International Airborne Remote Sensing Conference and Exhibition (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), pp. 463–470.
  27. N. T. O’Neill, A. R. Kalinauskas, J. D. Dunlop, A. B. Hollinger, H. Edel, M. Casey, J. Gibson, “Bathymetric analysis of geometrically corrected imagery data collected using a two dimensional imager,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 191–202 (1986).

1997 (1)

1996 (1)

N. J. McCormick, “Analytical transport theory applications in optical oceanography,” Ann. Nucl. Energy 23, 381–395 (1996).
[CrossRef]

1995 (1)

N. J. McCormick, “Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient,” Limnol. Oceanogr. 40, 1013–1018 (1995).
[CrossRef]

1994 (4)

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation using explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
[CrossRef] [PubMed]

K. Stamnes, “The Chandrasekhar method and its applications to atmospheric radiative transfer,” Trans. Am. Nucl. Soc. 71, 213–214 (1994).

Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media such as the atmosphere/ocean system,” Appl. Opt. 33, 431–442 (1994).
[CrossRef] [PubMed]

1992 (1)

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

1989 (1)

J. R. V. Zaneveld, “An asymptotic closure theory for irradiance in the sea and its inversion to obtain the inherent optical properties,” Limnol. Oceanogr. 34, 1442–1452 (1989).
[CrossRef]

1984 (1)

M. Benassi, R. D. M. Garcia, A. H. J. Karp, C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” Astrophys. J. 280, 853–864 (1984).
[CrossRef]

1983 (1)

D. A. Kiefer, B. G. Mitchell, “A simple, steady-state description of phytoplankton growth based on absorption cross section and quantum efficiency,” Limnol. Oceanogr. 28, 770–776 (1983).
[CrossRef]

1981 (1)

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

1978 (1)

C. E. Siewert, “The FN method for solving radiative-transfer problems in plane geometry,” Astrophys. Space Sci. 58, 131–137 (1978).
[CrossRef]

1975 (2)

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]

G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).
[CrossRef]

1941 (1)

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

Benassi, M.

M. Benassi, R. D. M. Garcia, A. H. J. Karp, C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” Astrophys. J. 280, 853–864 (1984).
[CrossRef]

Boynton, G. C.

Brown, O. B.

Carder, K. L.

K. L. Carder, D. J. Collins, M. J. Perry, H. L. Clark, J. M. Mesias, J. S. Cleveland, J Greenier, “The interaction of light with phytoplankton in the marine environment,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 42–55 (1986).

Casey, M.

N. T. O’Neill, A. R. Kalinauskas, J. D. Dunlop, A. B. Hollinger, H. Edel, M. Casey, J. Gibson, “Bathymetric analysis of geometrically corrected imagery data collected using a two dimensional imager,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 191–202 (1986).

Chandrasekhar, S.

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

Clark, H. L.

K. L. Carder, D. J. Collins, M. J. Perry, H. L. Clark, J. M. Mesias, J. S. Cleveland, J Greenier, “The interaction of light with phytoplankton in the marine environment,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 42–55 (1986).

Cleveland, J. S.

K. L. Carder, D. J. Collins, M. J. Perry, H. L. Clark, J. M. Mesias, J. S. Cleveland, J Greenier, “The interaction of light with phytoplankton in the marine environment,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 42–55 (1986).

Collins, D. J.

K. L. Carder, D. J. Collins, M. J. Perry, H. L. Clark, J. M. Mesias, J. S. Cleveland, J Greenier, “The interaction of light with phytoplankton in the marine environment,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 42–55 (1986).

Dunlop, J. D.

N. T. O’Neill, A. R. Kalinauskas, J. D. Dunlop, A. B. Hollinger, H. Edel, M. Casey, J. Gibson, “Bathymetric analysis of geometrically corrected imagery data collected using a two dimensional imager,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 191–202 (1986).

Edel, H.

N. T. O’Neill, A. R. Kalinauskas, J. D. Dunlop, A. B. Hollinger, H. Edel, M. Casey, J. Gibson, “Bathymetric analysis of geometrically corrected imagery data collected using a two dimensional imager,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 191–202 (1986).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989), pp. 274–286.

Francisco, P. W.

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

Garcia, R. D. M.

M. Benassi, R. D. M. Garcia, A. H. J. Karp, C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” Astrophys. J. 280, 853–864 (1984).
[CrossRef]

Gershun, A. A.

A. A. Gershun, “The light field,” J. Math. Phys. (Cambridge, Mass.) 18, 51–151 (1939).

Gibson, J.

N. T. O’Neill, A. R. Kalinauskas, J. D. Dunlop, A. B. Hollinger, H. Edel, M. Casey, J. Gibson, “Bathymetric analysis of geometrically corrected imagery data collected using a two dimensional imager,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 191–202 (1986).

Gordon, H. R.

Greenier, J

K. L. Carder, D. J. Collins, M. J. Perry, H. L. Clark, J. M. Mesias, J. S. Cleveland, J Greenier, “The interaction of light with phytoplankton in the marine environment,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 42–55 (1986).

Greenstein, J. L.

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

Haltrin, V. I.

V. I. Haltrin, “Algorithm for computing apparent optical properties of shallow waters under arbitrary surface illumination,” in Proceedings of the International Airborne Remote Sensing Conference and Exhibition (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), pp. 463–470.

Henyey, L. C.

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

Hollinger, A. B.

N. T. O’Neill, A. R. Kalinauskas, J. D. Dunlop, A. B. Hollinger, H. Edel, M. Casey, J. Gibson, “Bathymetric analysis of geometrically corrected imagery data collected using a two dimensional imager,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 191–202 (1986).

Jacobs, M. M.

Jin, Z.

Kalinauskas, A. R.

N. T. O’Neill, A. R. Kalinauskas, J. D. Dunlop, A. B. Hollinger, H. Edel, M. Casey, J. Gibson, “Bathymetric analysis of geometrically corrected imagery data collected using a two dimensional imager,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 191–202 (1986).

Karp, A. H. J.

M. Benassi, R. D. M. Garcia, A. H. J. Karp, C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” Astrophys. J. 280, 853–864 (1984).
[CrossRef]

Kattawar, G. W.

G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).
[CrossRef]

Kiefer, D. A.

D. A. Kiefer, B. G. Mitchell, “A simple, steady-state description of phytoplankton growth based on absorption cross section and quantum efficiency,” Limnol. Oceanogr. 28, 770–776 (1983).
[CrossRef]

Kishino, M.

M. Kishino, “Interrelationship between light and phytoplankton in the sea,” in Ocean Optics, R. W. Spinrad, K. L. Carder, M. J. Perry, eds. (Oxford U. Press, New York, 1994).

McCormick, N. J.

N. J. McCormick, “Analytical transport theory applications in optical oceanography,” Ann. Nucl. Energy 23, 381–395 (1996).
[CrossRef]

N. J. McCormick, “Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient,” Limnol. Oceanogr. 40, 1013–1018 (1995).
[CrossRef]

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation using explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
[CrossRef] [PubMed]

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

Mesias, J. M.

K. L. Carder, D. J. Collins, M. J. Perry, H. L. Clark, J. M. Mesias, J. S. Cleveland, J Greenier, “The interaction of light with phytoplankton in the marine environment,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 42–55 (1986).

Mitchell, B. G.

D. A. Kiefer, B. G. Mitchell, “A simple, steady-state description of phytoplankton growth based on absorption cross section and quantum efficiency,” Limnol. Oceanogr. 28, 770–776 (1983).
[CrossRef]

O’Neill, N. T.

N. T. O’Neill, A. R. Kalinauskas, J. D. Dunlop, A. B. Hollinger, H. Edel, M. Casey, J. Gibson, “Bathymetric analysis of geometrically corrected imagery data collected using a two dimensional imager,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 191–202 (1986).

Perry, M. J.

K. L. Carder, D. J. Collins, M. J. Perry, H. L. Clark, J. M. Mesias, J. S. Cleveland, J Greenier, “The interaction of light with phytoplankton in the marine environment,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 42–55 (1986).

Petzold, T. J.

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

Preisendorfer, R. W.

R. W. Preisendorfer, Hydrologic Optics, V.1. NTIS PB 259793/8ST (National Technical Information Service, Springfield, Va., 1976).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989), pp. 274–286.

Prieur, L.

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

Sanchez, R.

Sathyendranath, S.

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

Siewert, C. E.

M. Benassi, R. D. M. Garcia, A. H. J. Karp, C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” Astrophys. J. 280, 853–864 (1984).
[CrossRef]

C. E. Siewert, “The FN method for solving radiative-transfer problems in plane geometry,” Astrophys. Space Sci. 58, 131–137 (1978).
[CrossRef]

Stamnes, K.

K. Stamnes, “The Chandrasekhar method and its applications to atmospheric radiative transfer,” Trans. Am. Nucl. Soc. 71, 213–214 (1994).

Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media such as the atmosphere/ocean system,” Appl. Opt. 33, 431–442 (1994).
[CrossRef] [PubMed]

Tao, Z.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989), pp. 274–286.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989), pp. 274–286.

Zaneveld, J. R. V.

J. R. V. Zaneveld, “An asymptotic closure theory for irradiance in the sea and its inversion to obtain the inherent optical properties,” Limnol. Oceanogr. 34, 1442–1452 (1989).
[CrossRef]

J. R. V. Zaneveld, “A reflecting tube absorption meter,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 124–136 (1990).
[CrossRef]

Ann. Nucl. Energy (1)

N. J. McCormick, “Analytical transport theory applications in optical oceanography,” Ann. Nucl. Energy 23, 381–395 (1996).
[CrossRef]

Appl. Opt. (4)

Astrophys. J. (2)

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

M. Benassi, R. D. M. Garcia, A. H. J. Karp, C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” Astrophys. J. 280, 853–864 (1984).
[CrossRef]

Astrophys. Space Sci. (1)

C. E. Siewert, “The FN method for solving radiative-transfer problems in plane geometry,” Astrophys. Space Sci. 58, 131–137 (1978).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).
[CrossRef]

Limnol. Oceanogr. (6)

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

J. R. V. Zaneveld, “An asymptotic closure theory for irradiance in the sea and its inversion to obtain the inherent optical properties,” Limnol. Oceanogr. 34, 1442–1452 (1989).
[CrossRef]

N. J. McCormick, “Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient,” Limnol. Oceanogr. 40, 1013–1018 (1995).
[CrossRef]

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

D. A. Kiefer, B. G. Mitchell, “A simple, steady-state description of phytoplankton growth based on absorption cross section and quantum efficiency,” Limnol. Oceanogr. 28, 770–776 (1983).
[CrossRef]

Trans. Am. Nucl. Soc. (1)

K. Stamnes, “The Chandrasekhar method and its applications to atmospheric radiative transfer,” Trans. Am. Nucl. Soc. 71, 213–214 (1994).

Other (11)

R. W. Preisendorfer, Hydrologic Optics, V.1. NTIS PB 259793/8ST (National Technical Information Service, Springfield, Va., 1976).

M. Kishino, “Interrelationship between light and phytoplankton in the sea,” in Ocean Optics, R. W. Spinrad, K. L. Carder, M. J. Perry, eds. (Oxford U. Press, New York, 1994).

A. A. Gershun, “The light field,” J. Math. Phys. (Cambridge, Mass.) 18, 51–151 (1939).

K. L. Carder, D. J. Collins, M. J. Perry, H. L. Clark, J. M. Mesias, J. S. Cleveland, J Greenier, “The interaction of light with phytoplankton in the marine environment,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 42–55 (1986).

J. R. V. Zaneveld, “A reflecting tube absorption meter,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 124–136 (1990).
[CrossRef]

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).

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

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989), pp. 274–286.

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

V. I. Haltrin, “Algorithm for computing apparent optical properties of shallow waters under arbitrary surface illumination,” in Proceedings of the International Airborne Remote Sensing Conference and Exhibition (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), pp. 463–470.

N. T. O’Neill, A. R. Kalinauskas, J. D. Dunlop, A. B. Hollinger, H. Edel, M. Casey, J. Gibson, “Bathymetric analysis of geometrically corrected imagery data collected using a two dimensional imager,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 191–202 (1986).

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Figures (16)

Fig. 1
Fig. 1

Diffuse attenuation coefficient and irradiance reflectance profiles from simulated irradiance data (ω0 = 0.7, g = 0.85). The values of R are multiplied by 10.

Fig. 2
Fig. 2

Interdependence of R, ω0, and g for β ˜ HG.

Fig. 3
Fig. 3

Normalized sensitivity coefficient of ω0 with respect to R for g of β ˜ HG.

Fig. 4
Fig. 4

Normalized sensitivity coefficient of ω0 with respect to g of β ˜ HG.

Fig. 5
Fig. 5

Normalized sensitivity coefficient of c with respect to R for g of β ˜ HG.

Fig. 6
Fig. 6

Normalized sensitivity coefficient of c with respect to g of β ˜ HG.

Fig. 7
Fig. 7

Normalized sensitivity coefficient of a with respect to g of β ˜ HG.

Fig. 8
Fig. 8

Normalized sensitivity coefficient of a with respect to R for g of β ˜ HG.

Fig. 9
Fig. 9

Normalized sensitivity coefficient of bb with respect to g of β ˜ HG.

Fig. 10
Fig. 10

Normalized sensitivity coefficient of b or bb with respect to R for g of β ˜ HG.

Fig. 11
Fig. 11

Normalized sensitivity coefficient of G with respect to R for g of β ˜ HG.

Fig. 12
Fig. 12

Normalized sensitivity coefficient of G with respect to g of β ˜ HG.

Fig. 13
Fig. 13

Two-term Henyey–Greenstein phase function model for g = 0.85 and a, g1 = 0.85, α = 0; b, g1 = 0.88, g2 = −0.062, α = 0.97; c, g1 = 0.90, g2 = −0.26, α = 0.96; and d, g1 = 0.90, g2 = −0.64, α = 0.97.

Fig. 14
Fig. 14

Estimation of R in water of 5 optical depths with a purely absorbing bottom (Rb = 0). Shown are R from Eq. (13), the local irradiance ratio R(τ) that forms the asymptotic model (AM), and the depth-dependent estimates of R from the deepmeasurement reflectance model (DMRM) and the shallowmeasurement reflectance model (SMRM).

Fig. 15
Fig. 15

Estimation of R as in Fig. 14 for a Lambertian bottom reflectance of Rb = 0.2.

Fig. 16
Fig. 16

Estimation of R in three distinct water layers. From top to bottom, ω0 = 0.7, 0.65, and 0.60. Shown are R from Eq. (13), the local irradiance ratio R(τ), and the estimate of R(τ) from the DMRM.

Tables (5)

Tables Icon

Table 1 Predictions of K and R from the Simulated Kd(τ) and R(τ) Profiles of Fig. 1 versus the Depth of the Deepest Irradiance Measurement

Tables Icon

Table 2 Percent Errors in the Estimates of IOP’s for Selected Values of Percent Errors in R and K

Tables Icon

Table 3 Percent Errors in the Estimates of a and bb for Waters with Given Values of g and ω0a

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Table 4 Percent Errors in the Estimates of IOP’s for ω0 = 0.7 and the Two-Term Henyey–Greenstein Phase Functions Shown in Fig. 13a

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Table 5 Percent Errors in the Estimates of a and bb Obtained from Haltrin’s Model for c = 1 and the One-Term Henyey–Greenstein Phase Functiona

Equations (60)

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μ L ( z , μ ) / z + c L ( z , μ ) = b - 1 1 β ˜ ( μ , μ ) L ( z , μ ) d μ ,
β ˜ ( μ , μ ) = 1 2 n = 0 M ( 2 n + 1 ) f n P n ( μ ) P n ( μ ) ,             f 0 = 1 ,
b b = b - 1 0 β ˜ ( μ , 1 ) d μ ,
b ˜ b = b b / b = ( 1 / 2 ) [ 1 - n odd ( 2 n + 1 ) f n 0 1 P n ( μ ) d μ ]
R ( z ) = E u ( z ) / E d ( z ) ,
E u ( z ) = - 1 0 μ L ( z , μ ) d μ ,
E d ( z ) = 0 1 μ L ( z , μ ) d μ .
K d ( z ) = - 1 E d ( z ) d E d ( z ) d z = - d ln [ E d ( z ) ] d z .
L ( z , μ ) = ϕ ( ± ν j , μ ) exp ( ± c z / ν j )
ϕ ( ± ν j , μ ) = ω 0 ν j 2 ( ν j μ ) × n = 0 M ( 2 n + 1 ) f n g n ( ± ν j ) P n ( μ ) ,             ν j > 1 ,
n g n ( v j ) = h n - 1 ν j g n - 1 ( ν j ) - ( n - 1 ) g n - 2 ( ν j ) ,
g N + 1 ( ν j ) = 0.
R = 0 1 ϕ ( - ν 1 , μ ) μ d μ 0 1 ϕ ( + ν 1 , μ ) μ d μ ,
K = c / ν 1 ,
L ( 0 - , μ ) = 0.7 δ ( μ - 0.866 ) + 0.3 ,             0 μ 1.
R m ( z ) = ( 1 + r u - r d 1 + r d ) R ( z ) ,
K d m ( z ) = K d ( z ) - d [ ln ( 1 + r d ) ] / d z .
R ( z ) R + [ R ( z r ) - R ] exp [ - P ( z - z r ) ] ,             z > z r ,
K d ( z ) K + [ K d ( z r ) - K ] exp [ - P ( z - z r ) ] ,             z > z r ,
R = R ( z 0 ) R ( z 2 ) - R 2 ( z 1 ) R ( z 0 ) + R ( z 2 ) - 2 R ( z 1 ) .
c = ν 1 ( R , g ) K ,
K c c K = K c ν 1 = 1 ,
R c c R = R c K ν 1 R = R ν 1 ν 1 R ,
g c c g = g K c v 1 g = g ν 1 ν 1 g .
a = c ( 1 - ω 0 ) ,             a / ω 0 = - c ,             a / c = ( 1 - ω 0 ) ,
b = c ω 0 ,             b / ω 0 = c ,             b / c = ω 0 ,
K a a K = K a a c c K = K a ( 1 - ω 0 ) ν 1 = 1 ,
R a a R = R a [ a c c R ( R c c R ) + a ω 0 ω 0 R ( R ω 0 ω 0 E ) ] = ( R c c R ) - ( ω 0 1 - ω 0 ) ( R ω 0 ω 0 R ) ,
g a a g = g a [ a c c g ( g c c g ) + a ω 0 ω 0 g ( g ω 0 ω 0 g ) ] = ( g c c g ) - ( ω 0 1 - ω 0 ) ( g ω 0 ω 0 g ) .
K b b K = K b ω 0 ν 1 = 1 ,
R b b R = R b [ b c c R ( R c c R ) + b ω 0 ω 0 R ( R ω 0 ω 0 R ) ] = ( R c c R ) + ( R ω 0 ω 0 R ) ,
g b b g = g b [ b c c g ( g c c g ) + b ω 0 ω 0 g ( g ω 0 ω 0 g ) ] = ( g c c g ) + ( g ω 0 ω 0 g ) .
g b b b b g = g b b g + g b ˜ b b ˜ b g .
g ( b b / a ) ( b b / a ) g = g b b b b g - g a a g = g b ˜ b b ˜ b g + 1 ( 1 - ω 0 ) g ω 0 ω 0 g ,
g G G g = a a + b b [ g ( b b / a ) ( b b / a ) g ] .
R ( b b / a ) ( b b / a ) R = R b ˜ b b ˜ b R + 1 ( 1 - ω 0 ) R ω 0 ω 0 R ,
R G G R = a a + b b [ R ( b b / a ) ( b b / a ) R ] .
β ˜ ( g 1 , g 2 , α ) = α β ˜ ( g 1 ) + ( 1 - α ) β ˜ ( g 2 ) ,
a b b = ( 1 - R ) 2 ( 1 + 4 R + R ) 4 R ,
K / c = ( 1 - ω 0 ) { 1 + G 1 + 2 G - [ G ( 4 + 5 G ) ] 1 / 2 } 1 / 2 .
a = K { 1 + G 1 + 2 G - [ G ( 4 + 5 G ) ] 1 / 2 } - 1 / 2 ,
( 1 - R 1 + R ) 2 = [ E d ( z ) - E u ( z ) 2 ] 2 z 1 z 2 [ E d ( z ) + E u ( z ) 2 ] 2 z 1 z 2 .
R = { R ( z ) - R b exp [ - 2 K d ( z ) ( z b - z ) ] } / { 1 - exp [ - 2 K d ( z ) ( z b - z ) ] } .
E ± ( z ) = j = 1 J [ C ( ν j ) g ˜ 1 ( ± ν j ) exp ( - c z / ν j ) + C ( - ν j ) g ˜ 1 ( ± ν j ) exp ( c z / ν j ) ] ,
g ˜ 1 ( ν 1 ) = 0 1 ϕ ( ν 1 , μ ) μ d μ , g ˜ 1 ( - ν 1 ) = - 1 0 ϕ ( ν 1 , μ ) μ d μ = 0 1 ϕ ( - ν 1 , μ ) μ d μ ,
E ± ( z ) [ C ( ν 1 ) g ˜ 1 ( ± ν 1 ) exp ( - c z / ν 1 ) + C ( - ν 1 ) g ˜ 1 ( ± ν 1 ) exp ( c z / ν 1 ) ] ,
R = g ˜ 1 ( - ν 1 ) / g ˜ 1 ( ν 1 ) ,
ω 0 R = 1 R f / ω 0 ,
ω 0 g = 1 R f / ω 0 R f g ,
ν 1 R = ν 1 f ω 0 ω 0 R = ν 1 f ω 0 1 R f / ω 0 ,
ν 1 g = ν 1 f g + ν 1 f ω 0 ω 0 g = ν 1 f g - ν 1 f ω 0 1 R f / ω 0 R f g .
E + ( z ) ± E - ( z ) g ˜ 1 ( ν 1 ) ( 1 ± R ) [ C ( ν 1 ) exp ( - c z / ν 1 ) ± C ( - ν 1 ) exp ( c z / ν 1 ) ] .
[ E + ( z ) ± E - ( z ) ] 2 g ˜ 1 2 ( ν 1 ) ( 1 ± R ) 2 [ C 2 ( ν 1 ) × exp ( - 2 c z / ν 1 ) + C 2 ( - ν 1 ) × exp ( 2 c z / ν 1 ) ± 2 C ( ν 1 ) C ( - ν 1 ) ] .
R ( z ) = E - ( z ) E + ( z ) = R + C ( - ν 1 ) exp ( 2 c z ν 1 ) / C ( ν 1 ) 1 + C ( - ν 1 ) R exp ( 2 c z / ν 1 ) / C ( ν 1 ) ,
R ( z ) - R = C ( - ν 1 ) ( 1 - R ) exp ( 2 c z / ν 1 ) / C ( ν 1 ) 1 + R C ( - ν 1 ) exp ( 2 c z / ν 1 ) / C ( ν 1 ) .
R ( 0 + ) - R = C ( - ν 1 ) ( 1 - R ) / C ( ν 1 ) 1 + R C ( - ν 1 ) / C ( ν 1 ) ,
R b - R = C ( - ν 1 ) ( 1 - R ) exp ( 2 c z / ν 1 ) / C ( ν 1 ) 1 + R C ( - ν 1 ) exp ( 2 c z b / ν 1 ) / C ( ν 1 ) .
( R b - R ) exp ( - 2 c z b / ν 1 ) = [ R ( 0 + ) - R ] [ 1 + C ( - ν 1 ) R / C ( ν 1 ) ] 1 + R C ( - ν 1 ) exp ( 2 c z b / ν 1 ) / C ( ν 1 ) .
( R b - R ) exp ( - 2 c z b / ν 1 ) R ( 0 + ) - R .
R = [ R ( 0 + ) - R b exp ( - 2 K z b ) ] / [ 1 - exp ( - 2 K z b ) ] .

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