Abstract

A one-parameter two-term Henyey-Greenstein (TTHG) phase function of light scattering in seawater is proposed. The original three-parameter TTHG phase function was reduced to the one-parameter TTHG phase function by use of experimentally derived regression dependencies between integral parameters of the marine phase functions. An approach to calculate a diffuse attenuation coefficient in the depth of seawater is presented.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  2. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980) Vol. 1.
  3. C. D. Mobley, Light and Water (Academic, San Diego, Calif., 1994).
  4. N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).
  5. G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).
    [CrossRef]
  6. V. I. Haltrin, “Theoretical and empirical phase functions for Monte Carlo calculations of light scattering in seawater,” in Proceedings of the Fourth International Conference on Remote Sensing for Marine and Coastal Environments (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), pp. 509–518.
  7. T. J. Petzold, “Volume scattering functions for selected ocean waters,” Final Tech. Rep. SIO Ref. 72–78 (Scripps Institution of Oceanography Visibility Laboratory, San Diego, Calif., 1972), p. 79.
  8. V. A. Timofeyeva, “Relation between light-field parameters and between scattering phase function characteristics of turbid media, including seawater,” Izv. Acad. Sci. USSR Atmos. Ocean Phys. 14, 843–848 (1978).
  9. The regressions in Eqs. (11) and (12) are custom-made regressions. They were derived manually and nonformally by the trial and error method. The original regressions given in Ref. 8 are valid in the range of experimental measurements 0.05 ≤ B ≤ 0.25. The full range of variability of backscattering probability B = bb/b is between 0 (highly anisotropic delta-shaped in forward-direction scattering) and 0.5 (isotropic scattering). The original experimental data by Timofeyeva, who is regarded as one of the top experimentalists in hydrological optics of the 1960s and 1970s, were recovered from the scan of the figure published in Ref. 8 and checked with the original nonphysical regressions. I obtained the new regressions manually by forcing them to obey both the experimental points and the physical requirements listed above.
  10. V. I. Haltrin, “Chlorophyll-based model of seawater optical properties,” Appl. Opt. 38, 6826–6832 (1999).
    [CrossRef]
  11. V. I. Haltrin, “An algorithm to restore spectral signatures of all inherent optical properties of seawater using a value of one property at one wavelength,” in Proceedings of the Fourth International Airborne Remote Sensing Conference and Exhibition/21st Canadian Symposium on Remote Sensing (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1999), pp. I-680–I-687.
  12. V. A. Timofeyeva, “Optical characteristics of turbid media of the sea-water type,” Izv. Acad. Sci. USSR Atmos. Ocean Phys. 7, 1326–1329 (1971).
  13. V. I. Haltrin, “Empirical algorithms to restore a complete set of inherent optical properties of seawater using any two of these properties,” Can. J. Remote Sens. 26, 440–445 (2000).
  14. B. Bulgarelli, V. B. Kisselev, L. Roberti, “Radiative transfer in the atmosphere-ocean system: the finite-element method,” Appl. Opt. 38, 1530–1542 (1999).
    [CrossRef]
  15. V. I. Haltrin, “An analytic Fournier-Forand scattering phase function as an alternative to the Henyey-Greenstein phase function in hydrologic optics,” in IGARSS ’98: 1998 IEEE International Geoscience and Remote Sensing Symposium ProceedingsT. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), Vol. 2, pp. 910–912.
  16. V. I. Haltrin, “Self-consistent approach to the solution of the light transfer problem for irradiances in marine waters with arbitrary turbidity, depth, and surface illumination,” Appl. Opt. 37, 3773–3784 (1998).
    [CrossRef]
  17. V. I. Haltrin, “Apparent optical properties of the sea illuminated by Sun and sky,” Appl. Opt. 37, 8336–8340 (1998).
    [CrossRef]
  18. V. I. Haltrin, “Diffuse reflection coefficient of a stratified sea,” Appl. Opt. 38, 932–936 (1999).
    [CrossRef]
  19. V. I. Khalturin (a.k.a. V. I. Haltrin), “Propagation of light in the sea depth,” in Optical Remote Sensing of the Sea and the Influence of the Atmosphere, V. A. Urdenko, G. Zimmermann, eds. (German Democratic Republic Academy of Sciences Institute for Space Research, Berlin, 1985), Chap. 21, pp. 20–62 (in Russian).
  20. V. I. Haltrin, “Exact solution of the characteristic equation for transfer in the anisotropically scattering and absorbing medium,” Appl. Opt. 27, 599–602 (1988). This paper proposes a one-parameter analytic phase function that, if used with the scalar radiative transfer equation, produces in the asymptotic regime an analytical solution that has the form of the Henyey-Greenstein function.
  21. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  22. V. I. Haltrin, “Two-term Henyey-Greenstein light scattering phase function for seawater,” in IGARSS ’99: Proceeding of the International Geoscience and Remote Sensing SymposiumT. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 1423–1425.
  23. V. M. Loskutov, “Light regime in deep layers of turbid medium with strongly elongated phase function,” Vestn. Leningr. Univ. 13, 143–149 (1969); in Russian.

2000

V. I. Haltrin, “Empirical algorithms to restore a complete set of inherent optical properties of seawater using any two of these properties,” Can. J. Remote Sens. 26, 440–445 (2000).

1999

1998

1988

1978

V. A. Timofeyeva, “Relation between light-field parameters and between scattering phase function characteristics of turbid media, including seawater,” Izv. Acad. Sci. USSR Atmos. Ocean Phys. 14, 843–848 (1978).

1975

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

1971

V. A. Timofeyeva, “Optical characteristics of turbid media of the sea-water type,” Izv. Acad. Sci. USSR Atmos. Ocean Phys. 7, 1326–1329 (1971).

1969

V. M. Loskutov, “Light regime in deep layers of turbid medium with strongly elongated phase function,” Vestn. Leningr. Univ. 13, 143–149 (1969); in Russian.

1941

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

Bulgarelli, B.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

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, “Empirical algorithms to restore a complete set of inherent optical properties of seawater using any two of these properties,” Can. J. Remote Sens. 26, 440–445 (2000).

V. I. Haltrin, “Chlorophyll-based model of seawater optical properties,” Appl. Opt. 38, 6826–6832 (1999).
[CrossRef]

V. I. Haltrin, “Diffuse reflection coefficient of a stratified sea,” Appl. Opt. 38, 932–936 (1999).
[CrossRef]

V. I. Haltrin, “Self-consistent approach to the solution of the light transfer problem for irradiances in marine waters with arbitrary turbidity, depth, and surface illumination,” Appl. Opt. 37, 3773–3784 (1998).
[CrossRef]

V. I. Haltrin, “Apparent optical properties of the sea illuminated by Sun and sky,” Appl. Opt. 37, 8336–8340 (1998).
[CrossRef]

V. I. Haltrin, “Exact solution of the characteristic equation for transfer in the anisotropically scattering and absorbing medium,” Appl. Opt. 27, 599–602 (1988). This paper proposes a one-parameter analytic phase function that, if used with the scalar radiative transfer equation, produces in the asymptotic regime an analytical solution that has the form of the Henyey-Greenstein function.

V. I. Haltrin, “Two-term Henyey-Greenstein light scattering phase function for seawater,” in IGARSS ’99: Proceeding of the International Geoscience and Remote Sensing SymposiumT. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 1423–1425.

V. I. Haltrin, “An analytic Fournier-Forand scattering phase function as an alternative to the Henyey-Greenstein phase function in hydrologic optics,” in IGARSS ’98: 1998 IEEE International Geoscience and Remote Sensing Symposium ProceedingsT. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), Vol. 2, pp. 910–912.

V. I. Haltrin, “An algorithm to restore spectral signatures of all inherent optical properties of seawater using a value of one property at one wavelength,” in Proceedings of the Fourth International Airborne Remote Sensing Conference and Exhibition/21st Canadian Symposium on Remote Sensing (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1999), pp. I-680–I-687.

V. I. Haltrin, “Theoretical and empirical phase functions for Monte Carlo calculations of light scattering in seawater,” in Proceedings of the Fourth International Conference on Remote Sensing for Marine and Coastal Environments (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), pp. 509–518.

Henyey, L. C.

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

Jerlov, N. G.

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

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]

Khalturin, V. I.

V. I. Khalturin (a.k.a. V. I. Haltrin), “Propagation of light in the sea depth,” in Optical Remote Sensing of the Sea and the Influence of the Atmosphere, V. A. Urdenko, G. Zimmermann, eds. (German Democratic Republic Academy of Sciences Institute for Space Research, Berlin, 1985), Chap. 21, pp. 20–62 (in Russian).

Kisselev, V. B.

Loskutov, V. M.

V. M. Loskutov, “Light regime in deep layers of turbid medium with strongly elongated phase function,” Vestn. Leningr. Univ. 13, 143–149 (1969); in Russian.

Mobley, C. D.

C. D. Mobley, Light and Water (Academic, San Diego, Calif., 1994).

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” Final Tech. Rep. SIO Ref. 72–78 (Scripps Institution of Oceanography Visibility Laboratory, San Diego, Calif., 1972), p. 79.

Roberti, L.

Timofeyeva, V. A.

V. A. Timofeyeva, “Relation between light-field parameters and between scattering phase function characteristics of turbid media, including seawater,” Izv. Acad. Sci. USSR Atmos. Ocean Phys. 14, 843–848 (1978).

V. A. Timofeyeva, “Optical characteristics of turbid media of the sea-water type,” Izv. Acad. Sci. USSR Atmos. Ocean Phys. 7, 1326–1329 (1971).

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980) Vol. 1.

Appl. Opt.

Astrophys. J.

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

Can. J. Remote Sens.

V. I. Haltrin, “Empirical algorithms to restore a complete set of inherent optical properties of seawater using any two of these properties,” Can. J. Remote Sens. 26, 440–445 (2000).

Izv. Acad. Sci. USSR Atmos. Ocean Phys.

V. A. Timofeyeva, “Optical characteristics of turbid media of the sea-water type,” Izv. Acad. Sci. USSR Atmos. Ocean Phys. 7, 1326–1329 (1971).

V. A. Timofeyeva, “Relation between light-field parameters and between scattering phase function characteristics of turbid media, including seawater,” Izv. Acad. Sci. USSR Atmos. Ocean Phys. 14, 843–848 (1978).

J. Quant. Spectrosc. Radiat. Transfer

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

Vestn. Leningr. Univ.

V. M. Loskutov, “Light regime in deep layers of turbid medium with strongly elongated phase function,” Vestn. Leningr. Univ. 13, 143–149 (1969); in Russian.

Other

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

V. I. Haltrin, “Two-term Henyey-Greenstein light scattering phase function for seawater,” in IGARSS ’99: Proceeding of the International Geoscience and Remote Sensing SymposiumT. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 1423–1425.

V. I. Haltrin, “An analytic Fournier-Forand scattering phase function as an alternative to the Henyey-Greenstein phase function in hydrologic optics,” in IGARSS ’98: 1998 IEEE International Geoscience and Remote Sensing Symposium ProceedingsT. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), Vol. 2, pp. 910–912.

V. I. Haltrin, “Theoretical and empirical phase functions for Monte Carlo calculations of light scattering in seawater,” in Proceedings of the Fourth International Conference on Remote Sensing for Marine and Coastal Environments (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), pp. 509–518.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” Final Tech. Rep. SIO Ref. 72–78 (Scripps Institution of Oceanography Visibility Laboratory, San Diego, Calif., 1972), p. 79.

The regressions in Eqs. (11) and (12) are custom-made regressions. They were derived manually and nonformally by the trial and error method. The original regressions given in Ref. 8 are valid in the range of experimental measurements 0.05 ≤ B ≤ 0.25. The full range of variability of backscattering probability B = bb/b is between 0 (highly anisotropic delta-shaped in forward-direction scattering) and 0.5 (isotropic scattering). The original experimental data by Timofeyeva, who is regarded as one of the top experimentalists in hydrological optics of the 1960s and 1970s, were recovered from the scan of the figure published in Ref. 8 and checked with the original nonphysical regressions. I obtained the new regressions manually by forcing them to obey both the experimental points and the physical requirements listed above.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980) Vol. 1.

C. D. Mobley, Light and Water (Academic, San Diego, Calif., 1994).

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

V. I. Haltrin, “An algorithm to restore spectral signatures of all inherent optical properties of seawater using a value of one property at one wavelength,” in Proceedings of the Fourth International Airborne Remote Sensing Conference and Exhibition/21st Canadian Symposium on Remote Sensing (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1999), pp. I-680–I-687.

V. I. Khalturin (a.k.a. V. I. Haltrin), “Propagation of light in the sea depth,” in Optical Remote Sensing of the Sea and the Influence of the Atmosphere, V. A. Urdenko, G. Zimmermann, eds. (German Democratic Republic Academy of Sciences Institute for Space Research, Berlin, 1985), Chap. 21, pp. 20–62 (in Russian).

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 (4)

Fig. 1
Fig. 1

Dependence of integral parameters (B, cos ϑ¯, cos2 ϑ¯) of the marine TTHG phase function on asymmetry parameter g.

Fig. 2
Fig. 2

Samples of marine TTHG phase functions for different values of asymmetry parameter g.

Fig. 3
Fig. 3

Comparison of the HG phase function, the one-parameter seawater TTHG phase function, and the experimental Petzold phase function7 with the same value of backscattering probability B = 0.119.

Fig. 4
Fig. 4

Comparison of average cosines of the asymptotic radiance distribution computed for all possible combinations of B and ω0 with the experimental average cosines plotted against Gordon’s parameter g x = Bω 0/(1 - ω0 + Bω 0).

Tables (2)

Tables Icon

Table 1 Eigenvalues γ of the Asymptotic Equation for Transfer for the One-Parameter Seawater TTHG Scattering Phase Function for Different Values of the Single-Scattering Albedo ω0 and the Probability of Backscattering B

Tables Icon

Table 2 Coefficients γ n 0) for Eq. (A15)

Equations (33)

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

pHGμ, g=1-g21-2gμ+g23/2, μ=cos ϑ,
cos ϑ¯=12-11 pμμdμ, 12-11 pμdμ=1.
pHGμ, g=n=02n+1gnPnμ,0g1.
cos2 ϑ¯=12-11 pHGμμ2dμ=13+23 g2,
BHG=12-10 pHGμ, gdμ1201 pHG-μ, gdμ=1-g2g1+g1+g21/2-1.
pTTHGμ, α, g, h=αpHGμ, g+1-α×pHGμ, -h, 0α, g, h1.
pTTHGμ, α, g, h=n=02n+1αgn+1-α×-hnPnμ.
B12-10 pTTHGμ, gdμ=α1-g2g1+g1+g21/2-1+1-α1+h2h1-1-h1+h21/2,
cos ϑ¯=12-11 pTTHGμμdμ=αg+1-α-hαg+h-h,
cos2 ϑ¯=12-11 pTTHGμμ2dμ=13+23αg2-h2+h2.
cos ϑ¯=21-2B2+B,r20.99999,
cos2 ϑ¯=6-7B32+B,r20.99999.
α=h1+hg+h1+h-g,
h=-0.3061446+1.000568g-0.01826332g2+0.03643748g3,0.30664<g1.
k=cγB, ω0,
bb=0.000977+0.01b-0.0019540.962+0.532b1+0.01064b, r2=0.993,
Bbbb=0.000977b+0.01×1-0.001954b0.962+0.532b1+0.01064b, λ=515 nm.
g=1-0.001247 m-1b515 nm, 0.001954 m-1b515 nm2 m-1,r2=0.985.
Lτ, μ=L0Ψμexp-γτ,
1-γμΨμ=ω02-11 Ψμp¯μ, μdμ,
p¯μ, μ=12π02π pcos ϑdϕ, cos ϑ=μμ+1-μ21-μ21/2cosϕ-ϕ
120π pcos ϑsin ϑdϑ=1.
pcos ϑ=n=0 snPncos ϑ,s0=1,
p¯μ, μ=n=0 snPnμPnμ.
Ψμ=n=02n+1ΨnPnμ.
2n+1μPnμ=nPn-1μ+n+1Pn+1μ,
ψ0-γψ1=ω0ψ0s0,γn+1ψn+1-2n+1-ω0snψn+γnψn-1=0,n=1,, .
Δ=1-ω0-γ00-γ3-ω0s1-2γ00-2γ5-ω0s2-3γ00-3γ7-ω0s3=0.
1-ω0=γ23-ω0s1-4γ25-ω0s2-9γ27-ω0s3-,
1-ω0=Δ1,
Δn=nγ22n+1-ω0sn-Δn+1,n=1,, .
ψ0=1,ψ1=1-ω0γ,ψn=2-1+ω0sn-1nψn-1γ-1-1n ψn-2,n=2,, 
γB, ω0=γ0ω0-n=15 γnω0-1nBn.

Metrics