Abstract

Using three-dimensional Monte Carlo radiative transfer simulations, we examine the effect of beam transmissometer geometry on the relative error in the measurement of the beam-attenuation coefficient in an aquatic environment characterized by intense light scattering, especially within submerged bubble clouds entrained by surface-wave breaking. We discuss the forward-scattering error associated with the detection of photons scattered at small angles (<1°) and the multiple-scattering error associated with the detection of photons scattered more than once along the path length of the instrument. Several scattering phase functions describing bubble clouds at different bubble void fractions in the water are considered. Owing to forward-scattering error, a beam-attenuation meter (beam transmissometer) with a half-angle of receiver acceptance of 1.0° and a path length of 0.1 m can underestimate the true beam attenuation within the bubble cloud by more than 50%. For bubble clouds with a beam attenuation of as much as 100 m-1, the multiple-scattering error is no more than a few percent. These results are compared with simulations for some example phase functions that are representative of other scattering regimes found in natural waters. The forward-scattering error for the Petzold phase function of turbid waters is 16% for a typical instrument geometry, whereas for the Henyey-Greenstein phase function with the asymmetry parameter of 0.7 and 0.9 the error range is 8–28%.

© 2004 Optical Society of America

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References

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  1. R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Union Geod. Geophys. Inst. Monogr. 10, 11–30 (1961).
  2. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).
  3. N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).
  4. R. Bartz, J. R. V. Zaneveld, H. Pak, “A transmissometer for profiling and moored observations in water,” in Ocean Optics V, M. B. White, R. E. Stevenson, eds., Proc. SPIE160, 102–109 (1978).
    [CrossRef]
  5. C. Moore, E. J. Bruce, W. S. Pegau, A. Weidemann, “The WET Labs ac-9: field calibration protocol, deployment techniques, data processing and design improvements,” in Ocean Optics XIII, S. G. Ackleson, ed., Proc. SPIE2963, 725–730 (1996).
  6. J. T. O. Kirk, “Monte Carlo modeling of the performance of a reflective tube absorption meter,” Appl. Opt. 31, 6463–6468 (1992).
    [CrossRef] [PubMed]
  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. J. R. V. Zaneveld, R. Bartz, “Beam attenuation and absorption meters,” in Ocean Optics VII, M. A. Blizard, ed., Proc. SPIE489, 318–324 (1984).
    [CrossRef]
  9. K. J. Voss, R. W. Austin, “Beam attenuation measurement error due to small-angle scattering acceptance,” J. Atmos. Ocean. Technol. 10, 113–121 (1993).
    [CrossRef]
  10. D. Stramski, J. Tegowski, “Effects of intermittent entrainment of air bubbles by breaking wind waves on ocean reflectance and underwater light field,” J. Geophys. Res. 106, 31345–31360 (2001).
    [CrossRef]
  11. E. J. Terrill, W. K. Melville, D. Stramski, “Bubble entrainment by breaking waves and their influence on optical scattering in the upper ocean,” J. Geophys. Res. 106, 16815–16823 (2001).
    [CrossRef]
  12. X. Zhang, M. Lewis, B. Johnson, “Influence of bubbles on scattering of light in the ocean,” Appl. Opt. 37, 6525–6536 (1998).
    [CrossRef]
  13. J. Piskozub, A. R. Weeks, J. N. Schwarz, I. S. Robinson, “Self-shading of upwelling irradiance for an instrument with sensors on a sidearm,” Appl. Opt. 39, 872–1878 (2000).
    [CrossRef]
  14. J. Piskozub, P. J. Flatau, J. R. V. Zaneveld, “Monte Carlo study of the scattering error of a quartz reflective absorption tube,” J. Atmos. Oceanic Technol. 18, 438–445 (2001).
    [CrossRef]
  15. D. Stramski, J. Piskozub, “Estimation of scattering error in spectrophotometric measurements of light absorption by aquatic particles from three-dimensional radiative transfer simulations,” Appl. Opt. 42, 3634–3646 (2003).
    [CrossRef] [PubMed]
  16. X. Zhang, M. Lewis, M. Lee, B. Johnson, G. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47, 1273–1282 (2002).
    [CrossRef]
  17. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  18. E. J. Terrill, W. K. Melville, “A broad-band acoustic technique for measurement of bubble size distributions: laboratory and shallow water measurements,” J. Atmos. Ocean. Technol. 17, 220–239 (2000).
    [CrossRef]
  19. L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  20. T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 72–78 (Scripps Institution of Oceanography, La Jolla, Calif., 1972).
  21. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  22. C. D. Mobley, L. K. Sundman, E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41, 1035–1050 (2002).
    [CrossRef] [PubMed]
  23. G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).
    [CrossRef]
  24. V. I. Haltrin, “One-parameter two-term Henyey-Greenstein phase function for light scattering in seawater,” Appl. Opt. 41, 1022–1028 (2002).
    [CrossRef] [PubMed]
  25. E. J. Terrill, G. Lada, W. K. Melville, “Surf zone bubble populations,” Acoust. Oceanogr. (Part 2) 23, 212–219 (2001).
  26. R. M. Pope, E. S. Fry, “Absorption spectrum (380–700 nm) of pure water. 2. Integrating cavity measurements,” Appl. Opt. 36, 8710–8723 (1997).
    [CrossRef]
  27. W. S. Pegau, J. R. V. Zaneveld, K. J. Voss, “Toward closure of the inherent optical properties of natural waters,” J. Geophys. Res. 100, 13,193–13,199 (1995).
    [CrossRef]

2003 (1)

2002 (3)

2001 (4)

E. J. Terrill, G. Lada, W. K. Melville, “Surf zone bubble populations,” Acoust. Oceanogr. (Part 2) 23, 212–219 (2001).

D. Stramski, J. Tegowski, “Effects of intermittent entrainment of air bubbles by breaking wind waves on ocean reflectance and underwater light field,” J. Geophys. Res. 106, 31345–31360 (2001).
[CrossRef]

E. J. Terrill, W. K. Melville, D. Stramski, “Bubble entrainment by breaking waves and their influence on optical scattering in the upper ocean,” J. Geophys. Res. 106, 16815–16823 (2001).
[CrossRef]

J. Piskozub, P. J. Flatau, J. R. V. Zaneveld, “Monte Carlo study of the scattering error of a quartz reflective absorption tube,” J. Atmos. Oceanic Technol. 18, 438–445 (2001).
[CrossRef]

2000 (2)

J. Piskozub, A. R. Weeks, J. N. Schwarz, I. S. Robinson, “Self-shading of upwelling irradiance for an instrument with sensors on a sidearm,” Appl. Opt. 39, 872–1878 (2000).
[CrossRef]

E. J. Terrill, W. K. Melville, “A broad-band acoustic technique for measurement of bubble size distributions: laboratory and shallow water measurements,” J. Atmos. Ocean. Technol. 17, 220–239 (2000).
[CrossRef]

1998 (1)

1997 (1)

1995 (1)

W. S. Pegau, J. R. V. Zaneveld, K. J. Voss, “Toward closure of the inherent optical properties of natural waters,” J. Geophys. Res. 100, 13,193–13,199 (1995).
[CrossRef]

1993 (1)

K. J. Voss, R. W. Austin, “Beam attenuation measurement error due to small-angle scattering acceptance,” J. Atmos. Ocean. Technol. 10, 113–121 (1993).
[CrossRef]

1992 (1)

1989 (1)

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

1975 (1)

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

1961 (1)

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

1941 (1)

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

Austin, R. W.

K. J. Voss, R. W. Austin, “Beam attenuation measurement error due to small-angle scattering acceptance,” J. Atmos. Ocean. Technol. 10, 113–121 (1993).
[CrossRef]

Bartz, R.

J. R. V. Zaneveld, R. Bartz, “Beam attenuation and absorption meters,” in Ocean Optics VII, M. A. Blizard, ed., Proc. SPIE489, 318–324 (1984).
[CrossRef]

R. Bartz, J. R. V. Zaneveld, H. Pak, “A transmissometer for profiling and moored observations in water,” in Ocean Optics V, M. B. White, R. E. Stevenson, eds., Proc. SPIE160, 102–109 (1978).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Boss, E.

Bruce, E. J.

C. Moore, E. J. Bruce, W. S. Pegau, A. Weidemann, “The WET Labs ac-9: field calibration protocol, deployment techniques, data processing and design improvements,” in Ocean Optics XIII, S. G. Ackleson, ed., Proc. SPIE2963, 725–730 (1996).

Flatau, P. J.

J. Piskozub, P. J. Flatau, J. R. V. Zaneveld, “Monte Carlo study of the scattering error of a quartz reflective absorption tube,” J. Atmos. Oceanic Technol. 18, 438–445 (2001).
[CrossRef]

Fry, E. S.

Greenstein, J. L.

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

Haltrin, V. I.

Henyey, L. C.

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

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Jerlov, N. G.

N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).

Johnson, B.

X. Zhang, M. Lewis, M. Lee, B. Johnson, G. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47, 1273–1282 (2002).
[CrossRef]

X. Zhang, M. Lewis, B. Johnson, “Influence of bubbles on scattering of light in the ocean,” Appl. Opt. 37, 6525–6536 (1998).
[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]

Kirk, J. T. O.

Korotaev, G.

X. Zhang, M. Lewis, M. Lee, B. Johnson, G. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47, 1273–1282 (2002).
[CrossRef]

Lada, G.

E. J. Terrill, G. Lada, W. K. Melville, “Surf zone bubble populations,” Acoust. Oceanogr. (Part 2) 23, 212–219 (2001).

Lee, M.

X. Zhang, M. Lewis, M. Lee, B. Johnson, G. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47, 1273–1282 (2002).
[CrossRef]

Lewis, M.

X. Zhang, M. Lewis, M. Lee, B. Johnson, G. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47, 1273–1282 (2002).
[CrossRef]

X. Zhang, M. Lewis, B. Johnson, “Influence of bubbles on scattering of light in the ocean,” Appl. Opt. 37, 6525–6536 (1998).
[CrossRef]

Melville, W. K.

E. J. Terrill, W. K. Melville, D. Stramski, “Bubble entrainment by breaking waves and their influence on optical scattering in the upper ocean,” J. Geophys. Res. 106, 16815–16823 (2001).
[CrossRef]

E. J. Terrill, G. Lada, W. K. Melville, “Surf zone bubble populations,” Acoust. Oceanogr. (Part 2) 23, 212–219 (2001).

E. J. Terrill, W. K. Melville, “A broad-band acoustic technique for measurement of bubble size distributions: laboratory and shallow water measurements,” J. Atmos. Ocean. Technol. 17, 220–239 (2000).
[CrossRef]

Mobley, C. D.

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

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).

Moore, C.

C. Moore, E. J. Bruce, W. S. Pegau, A. Weidemann, “The WET Labs ac-9: field calibration protocol, deployment techniques, data processing and design improvements,” in Ocean Optics XIII, S. G. Ackleson, ed., Proc. SPIE2963, 725–730 (1996).

Pak, H.

R. Bartz, J. R. V. Zaneveld, H. Pak, “A transmissometer for profiling and moored observations in water,” in Ocean Optics V, M. B. White, R. E. Stevenson, eds., Proc. SPIE160, 102–109 (1978).
[CrossRef]

Pegau, W. S.

W. S. Pegau, J. R. V. Zaneveld, K. J. Voss, “Toward closure of the inherent optical properties of natural waters,” J. Geophys. Res. 100, 13,193–13,199 (1995).
[CrossRef]

C. Moore, E. J. Bruce, W. S. Pegau, A. Weidemann, “The WET Labs ac-9: field calibration protocol, deployment techniques, data processing and design improvements,” in Ocean Optics XIII, S. G. Ackleson, ed., Proc. SPIE2963, 725–730 (1996).

Petzold, T. J.

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

Piskozub, J.

D. Stramski, J. Piskozub, “Estimation of scattering error in spectrophotometric measurements of light absorption by aquatic particles from three-dimensional radiative transfer simulations,” Appl. Opt. 42, 3634–3646 (2003).
[CrossRef] [PubMed]

J. Piskozub, P. J. Flatau, J. R. V. Zaneveld, “Monte Carlo study of the scattering error of a quartz reflective absorption tube,” J. Atmos. Oceanic Technol. 18, 438–445 (2001).
[CrossRef]

J. Piskozub, A. R. Weeks, J. N. Schwarz, I. S. Robinson, “Self-shading of upwelling irradiance for an instrument with sensors on a sidearm,” Appl. Opt. 39, 872–1878 (2000).
[CrossRef]

Pope, R. M.

Preisendorfer, R. W.

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

Robinson, I. S.

J. Piskozub, A. R. Weeks, J. N. Schwarz, I. S. Robinson, “Self-shading of upwelling irradiance for an instrument with sensors on a sidearm,” Appl. Opt. 39, 872–1878 (2000).
[CrossRef]

Schwarz, J. N.

J. Piskozub, A. R. Weeks, J. N. Schwarz, I. S. Robinson, “Self-shading of upwelling irradiance for an instrument with sensors on a sidearm,” Appl. Opt. 39, 872–1878 (2000).
[CrossRef]

Stramski, D.

D. Stramski, J. Piskozub, “Estimation of scattering error in spectrophotometric measurements of light absorption by aquatic particles from three-dimensional radiative transfer simulations,” Appl. Opt. 42, 3634–3646 (2003).
[CrossRef] [PubMed]

E. J. Terrill, W. K. Melville, D. Stramski, “Bubble entrainment by breaking waves and their influence on optical scattering in the upper ocean,” J. Geophys. Res. 106, 16815–16823 (2001).
[CrossRef]

D. Stramski, J. Tegowski, “Effects of intermittent entrainment of air bubbles by breaking wind waves on ocean reflectance and underwater light field,” J. Geophys. Res. 106, 31345–31360 (2001).
[CrossRef]

Sundman, L. K.

Tegowski, J.

D. Stramski, J. Tegowski, “Effects of intermittent entrainment of air bubbles by breaking wind waves on ocean reflectance and underwater light field,” J. Geophys. Res. 106, 31345–31360 (2001).
[CrossRef]

Terrill, E. J.

E. J. Terrill, W. K. Melville, D. Stramski, “Bubble entrainment by breaking waves and their influence on optical scattering in the upper ocean,” J. Geophys. Res. 106, 16815–16823 (2001).
[CrossRef]

E. J. Terrill, G. Lada, W. K. Melville, “Surf zone bubble populations,” Acoust. Oceanogr. (Part 2) 23, 212–219 (2001).

E. J. Terrill, W. K. Melville, “A broad-band acoustic technique for measurement of bubble size distributions: laboratory and shallow water measurements,” J. Atmos. Ocean. Technol. 17, 220–239 (2000).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Voss, K. J.

W. S. Pegau, J. R. V. Zaneveld, K. J. Voss, “Toward closure of the inherent optical properties of natural waters,” J. Geophys. Res. 100, 13,193–13,199 (1995).
[CrossRef]

K. J. Voss, R. W. Austin, “Beam attenuation measurement error due to small-angle scattering acceptance,” J. Atmos. Ocean. Technol. 10, 113–121 (1993).
[CrossRef]

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

Weeks, A. R.

J. Piskozub, A. R. Weeks, J. N. Schwarz, I. S. Robinson, “Self-shading of upwelling irradiance for an instrument with sensors on a sidearm,” Appl. Opt. 39, 872–1878 (2000).
[CrossRef]

Weidemann, A.

C. Moore, E. J. Bruce, W. S. Pegau, A. Weidemann, “The WET Labs ac-9: field calibration protocol, deployment techniques, data processing and design improvements,” in Ocean Optics XIII, S. G. Ackleson, ed., Proc. SPIE2963, 725–730 (1996).

Zaneveld, J. R. V.

J. Piskozub, P. J. Flatau, J. R. V. Zaneveld, “Monte Carlo study of the scattering error of a quartz reflective absorption tube,” J. Atmos. Oceanic Technol. 18, 438–445 (2001).
[CrossRef]

W. S. Pegau, J. R. V. Zaneveld, K. J. Voss, “Toward closure of the inherent optical properties of natural waters,” J. Geophys. Res. 100, 13,193–13,199 (1995).
[CrossRef]

R. Bartz, J. R. V. Zaneveld, H. Pak, “A transmissometer for profiling and moored observations in water,” in Ocean Optics V, M. B. White, R. E. Stevenson, eds., Proc. SPIE160, 102–109 (1978).
[CrossRef]

J. R. V. Zaneveld, R. Bartz, “Beam attenuation and absorption meters,” in Ocean Optics VII, M. A. Blizard, ed., Proc. SPIE489, 318–324 (1984).
[CrossRef]

Zhang, X.

X. Zhang, M. Lewis, M. Lee, B. Johnson, G. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47, 1273–1282 (2002).
[CrossRef]

X. Zhang, M. Lewis, B. Johnson, “Influence of bubbles on scattering of light in the ocean,” Appl. Opt. 37, 6525–6536 (1998).
[CrossRef]

Acoust. Oceanogr. (Part 2) (1)

E. J. Terrill, G. Lada, W. K. Melville, “Surf zone bubble populations,” Acoust. Oceanogr. (Part 2) 23, 212–219 (2001).

Appl. Opt. (7)

Astrophys. J. (1)

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

J. Atmos. Ocean. Technol. (2)

K. J. Voss, R. W. Austin, “Beam attenuation measurement error due to small-angle scattering acceptance,” J. Atmos. Ocean. Technol. 10, 113–121 (1993).
[CrossRef]

E. J. Terrill, W. K. Melville, “A broad-band acoustic technique for measurement of bubble size distributions: laboratory and shallow water measurements,” J. Atmos. Ocean. Technol. 17, 220–239 (2000).
[CrossRef]

J. Atmos. Oceanic Technol. (1)

J. Piskozub, P. J. Flatau, J. R. V. Zaneveld, “Monte Carlo study of the scattering error of a quartz reflective absorption tube,” J. Atmos. Oceanic Technol. 18, 438–445 (2001).
[CrossRef]

J. Geophys. Res. (3)

D. Stramski, J. Tegowski, “Effects of intermittent entrainment of air bubbles by breaking wind waves on ocean reflectance and underwater light field,” J. Geophys. Res. 106, 31345–31360 (2001).
[CrossRef]

E. J. Terrill, W. K. Melville, D. Stramski, “Bubble entrainment by breaking waves and their influence on optical scattering in the upper ocean,” J. Geophys. Res. 106, 16815–16823 (2001).
[CrossRef]

W. S. Pegau, J. R. V. Zaneveld, K. J. Voss, “Toward closure of the inherent optical properties of natural waters,” J. Geophys. Res. 100, 13,193–13,199 (1995).
[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. (2)

X. Zhang, M. Lewis, M. Lee, B. Johnson, G. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47, 1273–1282 (2002).
[CrossRef]

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

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

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

Other (8)

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).

N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).

R. Bartz, J. R. V. Zaneveld, H. Pak, “A transmissometer for profiling and moored observations in water,” in Ocean Optics V, M. B. White, R. E. Stevenson, eds., Proc. SPIE160, 102–109 (1978).
[CrossRef]

C. Moore, E. J. Bruce, W. S. Pegau, A. Weidemann, “The WET Labs ac-9: field calibration protocol, deployment techniques, data processing and design improvements,” in Ocean Optics XIII, S. G. Ackleson, ed., Proc. SPIE2963, 725–730 (1996).

J. R. V. Zaneveld, R. Bartz, “Beam attenuation and absorption meters,” in Ocean Optics VII, M. A. Blizard, ed., Proc. SPIE489, 318–324 (1984).
[CrossRef]

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

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Figures (10)

Fig. 1
Fig. 1

Scattering phase functions for bubble populations calculated from Mie theory by use of bubble size distributions measured acoustically under breaking waves at several void fractions. The bubble phase functions for various void fraction ranges are written here and below as bubxy, where x and y define the void fraction range from 10- x to 10- y . The Mie calculations were made for λ = 550 nm.

Fig. 2
Fig. 2

Comparison of the cumulative scattering phase functions for (a) bubble populations at a number of void fraction ranges (see text and Fig. 1 for explanation) with (b) the Petzold cumulative phase function from San Diego Harbor and the HG cumulative phase functions for asymmetry parameter g = 0.7, 0.8, 0.9.

Fig. 3
Fig. 3

Measurement error expressed as 1 - c meas/c true as a function of true beam attenuation c true. The measured beam attenuation coefficient, c meas, is calculated from the Monte Carlo radiative transfer model of the beam transmissometer, and the true beam attenuation, c true, is used as input to the Monte Carlo model. The modeled instrument has a path length of 0.1 m, an initial beam radius of 0.001 m, a half-angle of beam divergence of 0.8°, a half-angle of receiver acceptance of 1°, and a receiver radius of 0.0095 m. The Monte Carlo simulations were made for the HG phase functions with asymmetry parameter g = 0.7, 0.8, 0.9.

Fig. 4
Fig. 4

Same as Fig. 3 but for instrument path length d = 0.25 m.

Fig. 5
Fig. 5

Mean cosine of the light field relative to the beam axis produced by photons within the plane containing the collecting area of the receiver. The modeled instrument has a path length of 0.1 m, an initial beam radius of 0.001 m, and a half-angle of beam divergence of 0.8°. The parameters of the receiver itself are irrelevant to the results presented, as all the photons (both detected and undetected) reaching the plane of the receiver contribute to the mean cosine. The calculations were made for the HG scattering phase function with g = 0.8.

Fig. 6
Fig. 6

Measurement error of the beam attenuation for several receiver apertures as indicated by the receiver acceptance radius. The modeled instrument has a path length of 0.1 m, an initial beam radius of 0.001 m, a half-angle of beam divergence of 0.8°, and a half-angle of receiver acceptance of 1.0°. The HG scattering phase function with g = 0.8 was used in these calculations.

Fig. 7
Fig. 7

Measurement error of the beam attenuation for several acceptance angles of the receiver as indicated. The modeled instrument has a path length of 0.1 m, an initial beam radius of 0.001 m, a half-angle of beam divergence of 0.8°, and a receiver radius of 0.0095 m. The HG scattering phase function with g = 0.8 was used in these calculations. The curve for the acceptance angle of 0.5° (which is smaller than the beam divergence) is negative for the relatively small values of c true that are due to losses caused by part of the beam’s not being accepted by the receiver.

Fig. 8
Fig. 8

Comparison of the measurement error of the beam attenuation for all the scattering phase functions considered in this study. The modeled instrument has a path length of 0.1 m, an initial beam radius of 0.001 m, a half-angle of beam divergence of 0.8°, a half-angle of receiver acceptance of 1°, and a receiver radius of 0.0095 m.

Fig. 9
Fig. 9

Multiple-scattering error expressed as a ratio of 1 - c meas/c true to c meas/c true for c true = 1 m-1 plotted as a function of c true multiplied by c meas/c true for c true = 1 m-1. Both variables are subject to multiple scattering after the effects of forward scattering at angles of <1.0° are removed. The relationship is shown for all the phase functions examined in this study. The modeled instrument has a path length of 0.1 m, an initial beam radius of 0.001 m, a half-angle of beam divergence of 0.8°, a half-angle of receiver acceptance of 1°, and a receiver radius of 0.0095 m.

Fig. 10
Fig. 10

Measurement error expressed as 1 - c meas/c true as a function of true scattering coefficient b true for several values of absorption coefficient a, as indicated. An absorption value of as much as a = 20 m-1 was added with no significant effect on the calculated measurement error. The modeled instrument has a path length of 0.1 m, an initial beam radius of 0.001 m, a half-angle of beam divergence of 0.8°, a half-angle of receiver acceptance of 1°, and a receiver radius of 0.0095 m. The HG scattering phase function with g = 0.8 was used in these calculations.

Tables (1)

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Table 1 Relative Forward-Scattering Error Expressed as 1 - c meas/c true for Various Scattering Phase Functionsa

Equations (3)

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cmeas= 1dlnΦ0Φ,
β˜λ, ψ= βψ, λbλ,
β˜HGg, ψ= 14π1-g21+g2-2g cos ψ3/2.

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