Abstract

Closed-form equations that describe the vector irradiance from an isotropic source embedded in the ocean are rigorously derived from the steady-state radiative transfer equation. The equations are exact for a homogeneous medium and are believed to be an excellent approximation along the vertical axis for a plane-parallel ocean. The equations are solved for the absorption coefficient as a function of distance from the source. For clear ocean water, it is shown that vector irradiance measurements alone provide sufficient information for an accurate calculation of the absorption coefficient. Measurements in Pacific water of the vector irradiance from an isotropic source are presented, and the absorption coefficient is computed. The estimated value of the absorption coefficient from a linear least-squares fit to the data has a standard error of ~ 1%.

© 1993 Optical Society of America

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References

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  1. J. R. V. Zaneveld, R. Bartz, J. C. Kitchen, “A reflective-tube absorption meter,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 124–136 (1990).
  2. E. S. Fry, G. W. Kattawar, R. M. Pope, “Integrating cavity absorption meter,” Appl. Opt. 31, 2055–2065 (1992).
    [Crossref] [PubMed]
  3. G. T. Bennett, E. S. Fry, F. M. Sogandares, “Photothermal measurement of the absorption coefficient of water at 590 nm,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.637, 172–180 (1986).
  4. C. C. Trees, K. J. Voss, “Optoacoustic spectroscopy and its application to molecular and particle absorption,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 149–156 (1990).
  5. N. Højerslev, “A spectral light absorption meter for measurements in the sea,” Limnol. Oceanogr. 20, 1024–1034 (1975).
    [Crossref]
  6. D. Spitzer, M. R. Wernand, “In situ measurements of absorption spectra in the sea,” Deep-Sea Res. 28A, 165–174, (1980).
  7. K. J. Voss, “Use of the radiance distribution to measure the optical absorption coefficient in the ocean,” Limnol. Oceanogr. 34, 1614–1622 (1989).
    [Crossref]
  8. W. Doss, W. Wells, “Undersea compound radiometer,” Appl. Opt. 31, 4268–4274 (1992).
    [Crossref] [PubMed]
  9. A. A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).
  10. R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Int. Union Geod. Geophys. Monogr. 10, 11–30 (1961).
  11. H. R. Gordon, “Ship perturbations of irradiance measurements at sea. 1: Monte Carlo simulations,” Appl. Opt. 24, 4172–4182 (1985).
    [Crossref] [PubMed]
  12. W. S. Helliwell, G. N. Sullivan, B. Macdonald, K. J. Voss, “Ship shadowing: model and data comparison,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 55–71 (1990).
  13. S. Q. Duntley, “Underwater lighting by submerged lasers and incandescent sources,” SIO Ref. 71-1 (Scripps Institution of Oceanography, University of California, San Diego, La Jolla, Calif, 1971).
  14. R. W. Preisendorfer, “Theoretical proof of the existence of characteristic diffuse light in natural waters,” J. Mar. Res. 18, 1–9 (1959).
  15. G. Sorenson, R. C. Honey, “Instrumentation for measuring visibility-limiting characteristics of sea water,” in Underwater Photo-Optical Instrumentation Applications II, C. N. DeMund, ed., Proc. Soc. Photo-Opt. Instrum. Eng.12, 115–122 (1968).
  16. G. Sorenson, R. C. Honey, “Underwater visibility meter concept study,” SRI Final Rep., Naval Undersea Warfare Center contract N60530-67-C-0891(SRI International, Menlo Park, Calif., 1968).
  17. G. D. Gilbert, R. C. Honey, R. E. Myers, G. P. Sorenson, “Optical absorption meter,” SRI Final Rep.Naval Undersea Research and Development Center contract N60530-68-C-1423 (SRI International, Menlo Park, Calif., 1969).
  18. R. A. Maffione, R. C. Honey, R. A. Brown, “Experiment for testing the closure property in ocean optics,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1537, 115–126 (1991).
  19. R. W. Preisendorfer, Radiative Transfer on Discrete Spaces (Pergamon, New York, 1965), Chap. X.
  20. See, for example, J. P. Elliott, “Milne’s problem with a point source,” Proc. R. Soc. London Ser. A 228, 424–433 (1955).
    [Crossref]
  21. H. R. Gordon, “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]
  22. G. Schellenberger, “Über Zusammenhänge zwischen optischen Parametern von Gewässern,” Acta Hydrophys. 10, 79–105 (1965).
  23. R. H. Stavn, “The three-parameter model of the submarine light field: radiant energy absorption and energy trapping in nepheloid layers,” J. Geophys. Res. 87, 2079–2082 (1982).
    [Crossref]
  24. N. K. Højerslev, J. R. V. Zaneveld, “A theoretical proof of the existence of the submarine asymptotic daylight field,” Rep. 34 (University of Copenhagen Institute of Physics and Oceanography, Copenhagen, 1977).
  25. J. R. V. Zaneveld, H. Pak, “Some aspects of the axially symmetric submarine daylight field,” J. Geophys. Res. 77, 2677–2680 (1972).
    [Crossref]
  26. K. J. Voss, A. L. Chapin, “Measurement of the point spread function in the ocean,” Appl. Opt. 29, 3638–3642 (1990).
    [Crossref] [PubMed]
  27. G. Kullenberg, “Observed and computed scattering functions,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemann Nielsen, eds. (Academic, New York, 1974), Chap 2.
  28. W. H. Wilson, “Spreading of light beams in ocean water,” in Ocean Optics VI, S. Q. Duntley, ed., Proc. Soc. Photo-Opt. Instrum. Eng.208, 64–72 (1979).
  29. K. J. Voss, “Electro-optic camera system for measurement of the underwater radiance distribution,” Opt. Eng. 28, 241–247 (1989).
  30. T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 72–78 (Scripps Institution of Oceanography, University of California, San Diego, La Jolla, Calif., 1972).
  31. R. A. Brown, R. C. Honey, R. A. Maffione, “Isotropic light source for underwater applications,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1537, 147–150 (1991).

1992 (2)

1990 (1)

1989 (2)

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

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

1987 (1)

1985 (1)

1982 (1)

R. H. Stavn, “The three-parameter model of the submarine light field: radiant energy absorption and energy trapping in nepheloid layers,” J. Geophys. Res. 87, 2079–2082 (1982).
[Crossref]

1980 (1)

D. Spitzer, M. R. Wernand, “In situ measurements of absorption spectra in the sea,” Deep-Sea Res. 28A, 165–174, (1980).

1975 (1)

N. Højerslev, “A spectral light absorption meter for measurements in the sea,” Limnol. Oceanogr. 20, 1024–1034 (1975).
[Crossref]

1972 (1)

J. R. V. Zaneveld, H. Pak, “Some aspects of the axially symmetric submarine daylight field,” J. Geophys. Res. 77, 2677–2680 (1972).
[Crossref]

1965 (1)

G. Schellenberger, “Über Zusammenhänge zwischen optischen Parametern von Gewässern,” Acta Hydrophys. 10, 79–105 (1965).

1961 (1)

R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Int. Union Geod. Geophys. Monogr. 10, 11–30 (1961).

1959 (1)

R. W. Preisendorfer, “Theoretical proof of the existence of characteristic diffuse light in natural waters,” J. Mar. Res. 18, 1–9 (1959).

1955 (1)

See, for example, J. P. Elliott, “Milne’s problem with a point source,” Proc. R. Soc. London Ser. A 228, 424–433 (1955).
[Crossref]

1939 (1)

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

Bartz, R.

J. R. V. Zaneveld, R. Bartz, J. C. Kitchen, “A reflective-tube absorption meter,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 124–136 (1990).

Bennett, G. T.

G. T. Bennett, E. S. Fry, F. M. Sogandares, “Photothermal measurement of the absorption coefficient of water at 590 nm,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.637, 172–180 (1986).

Brown, R. A.

R. A. Maffione, R. C. Honey, R. A. Brown, “Experiment for testing the closure property in ocean optics,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1537, 115–126 (1991).

R. A. Brown, R. C. Honey, R. A. Maffione, “Isotropic light source for underwater applications,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1537, 147–150 (1991).

Chapin, A. L.

Doss, W.

Duntley, S. Q.

S. Q. Duntley, “Underwater lighting by submerged lasers and incandescent sources,” SIO Ref. 71-1 (Scripps Institution of Oceanography, University of California, San Diego, La Jolla, Calif, 1971).

Elliott, J. P.

See, for example, J. P. Elliott, “Milne’s problem with a point source,” Proc. R. Soc. London Ser. A 228, 424–433 (1955).
[Crossref]

Fry, E. S.

E. S. Fry, G. W. Kattawar, R. M. Pope, “Integrating cavity absorption meter,” Appl. Opt. 31, 2055–2065 (1992).
[Crossref] [PubMed]

G. T. Bennett, E. S. Fry, F. M. Sogandares, “Photothermal measurement of the absorption coefficient of water at 590 nm,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.637, 172–180 (1986).

Gershun, A. A.

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

Gilbert, G. D.

G. D. Gilbert, R. C. Honey, R. E. Myers, G. P. Sorenson, “Optical absorption meter,” SRI Final Rep.Naval Undersea Research and Development Center contract N60530-68-C-1423 (SRI International, Menlo Park, Calif., 1969).

Gordon, H. R.

Helliwell, W. S.

W. S. Helliwell, G. N. Sullivan, B. Macdonald, K. J. Voss, “Ship shadowing: model and data comparison,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 55–71 (1990).

Højerslev, N.

N. Højerslev, “A spectral light absorption meter for measurements in the sea,” Limnol. Oceanogr. 20, 1024–1034 (1975).
[Crossref]

Højerslev, N. K.

N. K. Højerslev, J. R. V. Zaneveld, “A theoretical proof of the existence of the submarine asymptotic daylight field,” Rep. 34 (University of Copenhagen Institute of Physics and Oceanography, Copenhagen, 1977).

Honey, R. C.

R. A. Maffione, R. C. Honey, R. A. Brown, “Experiment for testing the closure property in ocean optics,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1537, 115–126 (1991).

G. Sorenson, R. C. Honey, “Instrumentation for measuring visibility-limiting characteristics of sea water,” in Underwater Photo-Optical Instrumentation Applications II, C. N. DeMund, ed., Proc. Soc. Photo-Opt. Instrum. Eng.12, 115–122 (1968).

G. D. Gilbert, R. C. Honey, R. E. Myers, G. P. Sorenson, “Optical absorption meter,” SRI Final Rep.Naval Undersea Research and Development Center contract N60530-68-C-1423 (SRI International, Menlo Park, Calif., 1969).

G. Sorenson, R. C. Honey, “Underwater visibility meter concept study,” SRI Final Rep., Naval Undersea Warfare Center contract N60530-67-C-0891(SRI International, Menlo Park, Calif., 1968).

R. A. Brown, R. C. Honey, R. A. Maffione, “Isotropic light source for underwater applications,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1537, 147–150 (1991).

Kattawar, G. W.

Kitchen, J. C.

J. R. V. Zaneveld, R. Bartz, J. C. Kitchen, “A reflective-tube absorption meter,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 124–136 (1990).

Kullenberg, G.

G. Kullenberg, “Observed and computed scattering functions,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemann Nielsen, eds. (Academic, New York, 1974), Chap 2.

Macdonald, B.

W. S. Helliwell, G. N. Sullivan, B. Macdonald, K. J. Voss, “Ship shadowing: model and data comparison,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 55–71 (1990).

Maffione, R. A.

R. A. Maffione, R. C. Honey, R. A. Brown, “Experiment for testing the closure property in ocean optics,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1537, 115–126 (1991).

R. A. Brown, R. C. Honey, R. A. Maffione, “Isotropic light source for underwater applications,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1537, 147–150 (1991).

Myers, R. E.

G. D. Gilbert, R. C. Honey, R. E. Myers, G. P. Sorenson, “Optical absorption meter,” SRI Final Rep.Naval Undersea Research and Development Center contract N60530-68-C-1423 (SRI International, Menlo Park, Calif., 1969).

Pak, H.

J. R. V. Zaneveld, H. Pak, “Some aspects of the axially symmetric submarine daylight field,” J. Geophys. Res. 77, 2677–2680 (1972).
[Crossref]

Petzold, T. J.

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

Pope, R. M.

Preisendorfer, R. W.

R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Int. Union Geod. Geophys. Monogr. 10, 11–30 (1961).

R. W. Preisendorfer, “Theoretical proof of the existence of characteristic diffuse light in natural waters,” J. Mar. Res. 18, 1–9 (1959).

R. W. Preisendorfer, Radiative Transfer on Discrete Spaces (Pergamon, New York, 1965), Chap. X.

Schellenberger, G.

G. Schellenberger, “Über Zusammenhänge zwischen optischen Parametern von Gewässern,” Acta Hydrophys. 10, 79–105 (1965).

Sogandares, F. M.

G. T. Bennett, E. S. Fry, F. M. Sogandares, “Photothermal measurement of the absorption coefficient of water at 590 nm,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.637, 172–180 (1986).

Sorenson, G.

G. Sorenson, R. C. Honey, “Instrumentation for measuring visibility-limiting characteristics of sea water,” in Underwater Photo-Optical Instrumentation Applications II, C. N. DeMund, ed., Proc. Soc. Photo-Opt. Instrum. Eng.12, 115–122 (1968).

G. Sorenson, R. C. Honey, “Underwater visibility meter concept study,” SRI Final Rep., Naval Undersea Warfare Center contract N60530-67-C-0891(SRI International, Menlo Park, Calif., 1968).

Sorenson, G. P.

G. D. Gilbert, R. C. Honey, R. E. Myers, G. P. Sorenson, “Optical absorption meter,” SRI Final Rep.Naval Undersea Research and Development Center contract N60530-68-C-1423 (SRI International, Menlo Park, Calif., 1969).

Spitzer, D.

D. Spitzer, M. R. Wernand, “In situ measurements of absorption spectra in the sea,” Deep-Sea Res. 28A, 165–174, (1980).

Stavn, R. H.

R. H. Stavn, “The three-parameter model of the submarine light field: radiant energy absorption and energy trapping in nepheloid layers,” J. Geophys. Res. 87, 2079–2082 (1982).
[Crossref]

Sullivan, G. N.

W. S. Helliwell, G. N. Sullivan, B. Macdonald, K. J. Voss, “Ship shadowing: model and data comparison,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 55–71 (1990).

Trees, C. C.

C. C. Trees, K. J. Voss, “Optoacoustic spectroscopy and its application to molecular and particle absorption,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 149–156 (1990).

Voss, K. J.

K. J. Voss, A. L. Chapin, “Measurement of the point spread function in the ocean,” Appl. Opt. 29, 3638–3642 (1990).
[Crossref] [PubMed]

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

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

C. C. Trees, K. J. Voss, “Optoacoustic spectroscopy and its application to molecular and particle absorption,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 149–156 (1990).

W. S. Helliwell, G. N. Sullivan, B. Macdonald, K. J. Voss, “Ship shadowing: model and data comparison,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 55–71 (1990).

Wells, W.

Wernand, M. R.

D. Spitzer, M. R. Wernand, “In situ measurements of absorption spectra in the sea,” Deep-Sea Res. 28A, 165–174, (1980).

Wilson, W. H.

W. H. Wilson, “Spreading of light beams in ocean water,” in Ocean Optics VI, S. Q. Duntley, ed., Proc. Soc. Photo-Opt. Instrum. Eng.208, 64–72 (1979).

Zaneveld, J. R. V.

J. R. V. Zaneveld, H. Pak, “Some aspects of the axially symmetric submarine daylight field,” J. Geophys. Res. 77, 2677–2680 (1972).
[Crossref]

J. R. V. Zaneveld, R. Bartz, J. C. Kitchen, “A reflective-tube absorption meter,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 124–136 (1990).

N. K. Højerslev, J. R. V. Zaneveld, “A theoretical proof of the existence of the submarine asymptotic daylight field,” Rep. 34 (University of Copenhagen Institute of Physics and Oceanography, Copenhagen, 1977).

Acta Hydrophys. (1)

G. Schellenberger, “Über Zusammenhänge zwischen optischen Parametern von Gewässern,” Acta Hydrophys. 10, 79–105 (1965).

Appl. Opt. (5)

Deep-Sea Res. (1)

D. Spitzer, M. R. Wernand, “In situ measurements of absorption spectra in the sea,” Deep-Sea Res. 28A, 165–174, (1980).

Int. Union Geod. Geophys. Monogr. (1)

R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Int. Union Geod. Geophys. Monogr. 10, 11–30 (1961).

J. Geophys. Res. (2)

J. R. V. Zaneveld, H. Pak, “Some aspects of the axially symmetric submarine daylight field,” J. Geophys. Res. 77, 2677–2680 (1972).
[Crossref]

R. H. Stavn, “The three-parameter model of the submarine light field: radiant energy absorption and energy trapping in nepheloid layers,” J. Geophys. Res. 87, 2079–2082 (1982).
[Crossref]

J. Mar. Res. (1)

R. W. Preisendorfer, “Theoretical proof of the existence of characteristic diffuse light in natural waters,” J. Mar. Res. 18, 1–9 (1959).

J. Math. Phys. (1)

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

Limnol. Oceanogr. (2)

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

N. Højerslev, “A spectral light absorption meter for measurements in the sea,” Limnol. Oceanogr. 20, 1024–1034 (1975).
[Crossref]

Opt. Eng. (1)

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

Proc. R. Soc. London Ser. A (1)

See, for example, J. P. Elliott, “Milne’s problem with a point source,” Proc. R. Soc. London Ser. A 228, 424–433 (1955).
[Crossref]

Other (15)

W. S. Helliwell, G. N. Sullivan, B. Macdonald, K. J. Voss, “Ship shadowing: model and data comparison,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 55–71 (1990).

S. Q. Duntley, “Underwater lighting by submerged lasers and incandescent sources,” SIO Ref. 71-1 (Scripps Institution of Oceanography, University of California, San Diego, La Jolla, Calif, 1971).

G. Sorenson, R. C. Honey, “Instrumentation for measuring visibility-limiting characteristics of sea water,” in Underwater Photo-Optical Instrumentation Applications II, C. N. DeMund, ed., Proc. Soc. Photo-Opt. Instrum. Eng.12, 115–122 (1968).

G. Sorenson, R. C. Honey, “Underwater visibility meter concept study,” SRI Final Rep., Naval Undersea Warfare Center contract N60530-67-C-0891(SRI International, Menlo Park, Calif., 1968).

G. D. Gilbert, R. C. Honey, R. E. Myers, G. P. Sorenson, “Optical absorption meter,” SRI Final Rep.Naval Undersea Research and Development Center contract N60530-68-C-1423 (SRI International, Menlo Park, Calif., 1969).

R. A. Maffione, R. C. Honey, R. A. Brown, “Experiment for testing the closure property in ocean optics,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1537, 115–126 (1991).

R. W. Preisendorfer, Radiative Transfer on Discrete Spaces (Pergamon, New York, 1965), Chap. X.

J. R. V. Zaneveld, R. Bartz, J. C. Kitchen, “A reflective-tube absorption meter,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 124–136 (1990).

G. T. Bennett, E. S. Fry, F. M. Sogandares, “Photothermal measurement of the absorption coefficient of water at 590 nm,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.637, 172–180 (1986).

C. C. Trees, K. J. Voss, “Optoacoustic spectroscopy and its application to molecular and particle absorption,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 149–156 (1990).

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

R. A. Brown, R. C. Honey, R. A. Maffione, “Isotropic light source for underwater applications,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1537, 147–150 (1991).

G. Kullenberg, “Observed and computed scattering functions,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemann Nielsen, eds. (Academic, New York, 1974), Chap 2.

W. H. Wilson, “Spreading of light beams in ocean water,” in Ocean Optics VI, S. Q. Duntley, ed., Proc. Soc. Photo-Opt. Instrum. Eng.208, 64–72 (1979).

N. K. Højerslev, J. R. V. Zaneveld, “A theoretical proof of the existence of the submarine asymptotic daylight field,” Rep. 34 (University of Copenhagen Institute of Physics and Oceanography, Copenhagen, 1977).

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Figures (3)

Fig. 1
Fig. 1

Spherical components of vector irradiance. The three components of the vector irradiance are the differences of the irradiances incident upon the opposing faces.

Fig. 2
Fig. 2

Vector irradiance from an isotropic source embedded in a homogeneous scattering and absorbing medium.

Fig. 3
Fig. 3

Vector irradiance measurements from an isotropic source taken on 26 August 1990 off the coast of southern California.

Equations (30)

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

( ξ ^ · ) L ( r , ξ ^ ) = - c ( r ) L ( r , ξ ^ ) + L * ( r , ξ ^ ) ,
( ξ ^ · ) = ( cos θ x , cos θ y , cos θ z ) · ( x , y , z )
L * ( r , ξ ^ ) = 4 π L ( r , ξ ^ ) β ( r , ξ ^ · ξ ^ ) d Ω ( ξ ^ ) .
- · E ( r ) = a ( r ) E 0 ( r ) .
E 0 ( r ) = 4 π L ( r , ξ ^ ) d Ω ( ξ ^ ) .
E x = 2 π 0 π L ( r , ξ ^ ) cos θ x sin θ x d θ x = 2 π [ 0 π / 2 L ( r . ξ ^ ) cos θ x sin θ x d θ x - π / 2 π L ( r , ξ ^ ) cos θ x sin θ x d θ x ] = E + x - E - x ,             etc .
- E z ( z ) z = a E 0 ( z ) .
- a E 0 = 1 r 2 ( r 2 E r ) r + 1 r sin θ ( sin θ E θ ) θ + 1 r sin θ E ϕ ϕ .
- a E 0 = 1 r 2 ( r 2 E r ) r + 1 r E θ θ = E r r + 1 r ( 2 E r + E θ θ ) .
- a E 0 = E r r + 2 E r r ,
μ ¯ r ( r ) = E r ( r ) E 0 ( r ) ,
d E r ( r ) E r ( r ) = - a ( r ) μ ¯ r ( r ) d r - 2 r d r .
E r ( r ) = E r ( r 0 ) exp [ - r 0 r a ( r ) μ ¯ r d r ] ( r / r 0 ) 2 .
E r ( r 0 ) = Φ r 0 4 π r 0 2 ,
E r ( z ) = Φ z 0 4 π z 2 exp [ - z 0 z a ( z ) μ ¯ r ( z ) d z ] .
E r ( z ) = Φ z 0 4 π z 2 exp ( - a Δ z ) ,
E r ( z ) = Φ z 0 4 π z 2 exp [ - a z 0 z d z μ ¯ r ( z ) ] = Φ z 0 4 π z 2 exp ( - a z ¯ ) ,
z ¯ = z 0 z d z μ ¯ ( z ) ,
E r ( z ) = Φ z 0 4 π z 2 exp [ - a ( Δ z + δ z ) ] ,
a ( z ) = μ ¯ r ( z ) [ - 1 E r ( z ) d E r ( z ) d z - 2 z ] = μ ¯ r ( z ) [ K E r ( z ) - 2 z ] ,
K E r ( z ) - 1 E r ( z ) d E r ( z ) d z .
E r ( z ) = E r ( z 0 ) exp [ - z 0 z K E r ( z ) d z ] .
K E r = a ( z ) μ ¯ r ( z ) + 2 z .
a ( z ) μ ¯ r ( z ) K E r ( z ) ,
K E r ( z n ) = ln [ E r ( z n - 1 ) E r ( z n + 1 ) ] z n + 1 - z n - 1 ,
a ( z n ) = μ ¯ r ( z n ) { ln [ E r ( z n - 1 ) E r ( z n + 1 ) ] z n + 1 - z n - 1 - 2 z n } .
μ ¯ r ( z ) 0 θ max L ( z , θ ) cos θ sin θ d θ 0 θ max L ( z , θ ) sin θ d θ .
μ ¯ = 1 - ω 0 1 - ω 0 ,
E ˜ = k e Φ 0 exp ( - a z ) 4 π z 2 ,
ln [ z 2 E ˜ ( z ) ] = ln ( k e Φ 0 4 π ) - a z .

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