Abstract

Equations are obtained for determining the Legendre polynomial expansion coefficients of the phase function from angular or angular-spatial integrals of the radiance measured either in water or at the surface.

© 2002 Optical Society of America

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References

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  1. J. R. V. Zaneveld, H. Pak, “Some aspects of the axially symmetric submarine daylight field,” J. Geophys. Res. 77, 2677–2680 (1972).
    [CrossRef]
  2. C. D. Mobley, Light and Water Radiative Transfer in Natural Waters (Academic, New York, 1994), Secs. 5.10 and 10.2.
  3. L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  4. G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).
    [CrossRef]
  5. V. Haltrin, “Two-term Henyey-Greenstein light scattering phase function for seawater,” in Proceedings of the International Geoscience and Remote Sensing Symposium: IGARSS ’99 (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 1423–1425.
  6. C. D. Mobley, L. K. Sundman, E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41, 1035–1050 (2002).
    [CrossRef] [PubMed]
  7. W. H. Wells, “Techniques for measuring radiance in sea and air,” Appl. Opt. 22, 2313–2321 (1983).
    [CrossRef] [PubMed]
  8. W. Doss, W. H. Wells, “Undersea compound radiometer,” Appl. Opt. 31, 4268–4274 (1992).
    [CrossRef] [PubMed]
  9. L. J. Holl, N. J. McCormick, “Ocean optical-property estimation with the Zaneveld–Wells algorithm,” Appl. Opt. 34, 5433–5441 (1995).
    [CrossRef] [PubMed]
  10. N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
    [CrossRef]
  11. A. H. Hakim, N. J. McCormick, “Ocean optics estimation for absorption, backscattering, and phase function parameters,” submitted for publication.
  12. J. C. Oelund, N. J. McCormick, “Sensitivity of multiple-scattering inverse transport methods to measurement errors,” J. Opt. Soc. Am. A 2, 1972–1978 (1985).
    [CrossRef]
  13. J. T. O. Kirk, “Multiple scattering of a photon flux: implications for the integral average cosine of the underwater light field,” Appl. Opt. 38, 3134–3140 (1999).
    [CrossRef]
  14. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Sec. 2.4.
  15. 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).
  16. J. T. O. Kirk, “The vertical attenuation of irradiance as a function of the optical properties of the water,” Limnol. Oceanogr. (to be published).

2002 (1)

1999 (1)

1995 (1)

1992 (1)

1985 (1)

1983 (1)

1979 (1)

N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
[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]

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]

1941 (1)

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

Boss, E.

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Sec. 2.4.

Doss, W.

Greenstein, J. L.

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

Hakim, A. H.

A. H. Hakim, N. J. McCormick, “Ocean optics estimation for absorption, backscattering, and phase function parameters,” submitted for publication.

Haltrin, V.

V. Haltrin, “Two-term Henyey-Greenstein light scattering phase function for seawater,” in Proceedings of the International Geoscience and Remote Sensing Symposium: IGARSS ’99 (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 1423–1425.

Henyey, L. C.

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

Holl, L. J.

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.

J. T. O. Kirk, “Multiple scattering of a photon flux: implications for the integral average cosine of the underwater light field,” Appl. Opt. 38, 3134–3140 (1999).
[CrossRef]

J. T. O. Kirk, “The vertical attenuation of irradiance as a function of the optical properties of the water,” Limnol. Oceanogr. (to be published).

McCormick, N. J.

L. J. Holl, N. J. McCormick, “Ocean optical-property estimation with the Zaneveld–Wells algorithm,” Appl. Opt. 34, 5433–5441 (1995).
[CrossRef] [PubMed]

J. C. Oelund, N. J. McCormick, “Sensitivity of multiple-scattering inverse transport methods to measurement errors,” J. Opt. Soc. Am. A 2, 1972–1978 (1985).
[CrossRef]

N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
[CrossRef]

A. H. Hakim, N. J. McCormick, “Ocean optics estimation for absorption, backscattering, and phase function parameters,” submitted for publication.

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, New York, 1994), Secs. 5.10 and 10.2.

Oelund, J. C.

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).

Sundman, L. K.

Wells, W. H.

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]

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Sec. 2.4.

Appl. Opt. (5)

Astrophys. J. (1)

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

J. Geophys. Res. (1)

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

J. Math. Phys. (1)

N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
[CrossRef]

J. Opt. Soc. Am. A (1)

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]

Other (6)

V. Haltrin, “Two-term Henyey-Greenstein light scattering phase function for seawater,” in Proceedings of the International Geoscience and Remote Sensing Symposium: IGARSS ’99 (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 1423–1425.

C. D. Mobley, Light and Water Radiative Transfer in Natural Waters (Academic, New York, 1994), Secs. 5.10 and 10.2.

A. H. Hakim, N. J. McCormick, “Ocean optics estimation for absorption, backscattering, and phase function parameters,” submitted for publication.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Sec. 2.4.

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).

J. T. O. Kirk, “The vertical attenuation of irradiance as a function of the optical properties of the water,” Limnol. Oceanogr. (to be published).

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Equations (22)

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β˜Ψ=4π-1n=02n+1β˜nPncos Ψ,
β˜n=2π 0π β˜ΨPncos Ψsin ΨdΨ,
β˜Ψ=4π-1n=02n+1gnPncos Ψ  HG=4π-1n=02n+1αgfn+1-αgbnPncos Ψ  TTHG
μ Lz, μz+cLz, μ=b2n=02n+1β˜nPnμ×-11 PnμLz, μdμ, z0,
k+1dEk+1zdz+k dEk-1zdz+2k+1a/σk×Ekz=0, k0,
Ekz=-11 PkμLz, μdμ, k0,
σk=1+b/a1-β˜k-1, k0.
β˜k=1-a1-σkbσk.
aμ¯z=-1E1zdE1zdz.
σk/a=-2k+1z1z2 Ekzdzk+1Ek+1zz1z2+kEk-1zz1z2, k0, z1, z2>0,
b/ck=0K-1k2k+1β˜kEkz22-Ekz1]2=4Sz2-Sz1,
b/ck=0k-1k2k+1γ˜kE˜kz22-E˜kz12=4S˜z2-S˜z1,
Sz=01 Lz, μLz, -μdμ,
E˜kz=-11 μPkμLz, μdμ,
S˜z=01 μ2Lz, μLz, -μdμ,
γ˜k=β˜k/1-b/cβ˜k.
μLz, μz+cLz, μ=b2n=02n+1β˜nPnμ×-11 PnμLz, μdμ+LsμΘμδz, z0,
c-bfkLk,0=L¯sPkμ0¯,
Lk,0=0-11 PkμLz, μdμdz=0 Ekzdz, L¯s=01 Lsμ0dμ0,
Pkμ0¯=L¯s-101 Pkμ0Lsμ0dμ0.
σk=Lk,0L0,0=Pkμ0¯-10 Ekzdz0 E0zdz.
σ1=0 E1zdzμ¯00 E0zdz=0μ¯zE0zdzμ¯00 E0zdz=μ¯cμ¯0.

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