Abstract

We estimate the optical signal for an oceanographic lidar from the one-dimensional transient (time-dependent) radiative transfer equation using the discrete ordinates method. An oceanographic lidar directs a pulsed blue or green laser into the ocean and measures the time-dependent backscattered light. A large number of parameters affect the performance of such a system. Here the optical signal that is available to the receiver is calculated, rather than the receiver output, to reduce the number of parameters. The effects of albedo of a uniform water column are investigated. The effects of a school of fish in the water are also investigated for various school depths, thicknesses, and densities. The attenuation of a lidar signal is found to be greater than the diffuse attenuation coefficient at low albedo and close to it at higher albedo. The presence of fish in the water is found to have a significant effect on the signal at low to moderate albedo, but not at high albedo.

© 1999 Optical Society of America

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  1. C. D. Mobley, B. Gentili, H. Gordon, Z. Jin, G. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
    [CrossRef] [PubMed]
  2. M. M. Krekova, G. M. Krekov, I. V. Samokhvalov, V. S. Shamanaev, “Numerical evaluation of possibilities of remote laser sensing of fish schools,” Appl. Opt. 33, 5715–5720 (1994).
    [CrossRef] [PubMed]
  3. Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. 33, 431–442 (1994).
    [CrossRef] [PubMed]
  4. A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
    [CrossRef]
  5. Y. Yamada, “Light-tissue interaction and optical imaging in biomedicine,” Ann. Rev. Fluid Mech. Heat Transfer 6, 1–59 (1995).
  6. S. Kumar, K. Mitra, Y. Yamada, “Hyperbolic damped-wave models for transient light pulse propagation in scattering media,” Appl. Opt. 35, 3372–3378 (1996).
    [CrossRef] [PubMed]
  7. C. D. Mobley, “The optical properties of water,” in Handbook of Optics: Fundamentals, Techniques, Design (Academic, San Diego, Calif., 1995), Vol. 1, Chap. 43.
  8. J. H. Smart, K. H. Kwon, “Comparisons between in situ and remote sensing estimates of diffuse attenuation profiles,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, Proc. SPIE2964, 100–109 (1996).
  9. J. H. Churnside, J. R. Hunter, “Laser remote sensing of epipelagic fishes,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, Proc. SPIE2964, 38–53 (1996).
  10. J. H. Churnside, J. J. Wilson, V. V. Tatarskii, “Lidar profiles of fish schools,” Appl. Opt. 36, 6011–6020 (1997).
    [CrossRef] [PubMed]
  11. J. L. Squire, H. Krumboltz, “Profiling pelagic fish schools using airborne optical lasers and other remote sensing techniques,” Mar. Technol. Soc. J. 15, 443–448 (1981).
  12. D. L. Murphree, C. D. Taylor, R. W. McClendon, “Mathematical modeling for the detection of fish by an airborne laser,” AIAA J. 12, 1686–1692 (1974).
    [CrossRef]
  13. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1992).
  14. G. E. Hunt, “The generation of angular distribution coefficients for radiation scattered by a spherical particle,” J. Spectrosc. Radiat. Transfer 10, 857–864 (1970).
    [CrossRef]
  15. M. F. Modest, Radiative Heat Transfer (McGraw-Hill, New York, 1993).
  16. 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). Vol. 1, pp. I-509–I-518.
  17. G. R. Fournier, J. L. Forand, “Analytic phase function for ocean water,” in Ocean Optics XII, Proc. SPIE2258, 194–201 (1994).
    [CrossRef]
  18. E. Isaacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).
  19. M. P. Menguc, R. Viskanta, “Comparison of radiative transfer approximations for highly forward scattering planar medium,” J. Quant. Spectrosc. Rad. Transfer 29, 381–394 (1983).
    [CrossRef]
  20. S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
    [CrossRef]
  21. N. K. Madsen, R. F. Sincovec, “Algorithm 540 PDECOL, general collocation software for partial differential equations [D3],” ACM Trans. Math. Softwares 5, 326–361 (1979).
    [CrossRef]
  22. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters, (Academic, New York, 1994).
  23. H. R. Gordon, “Sensitivity of radiative transfer to small-angle scattering in the ocean: quantitative assessment,” Appl. Opt. 32, 7505–7511 (1993).
    [CrossRef] [PubMed]

1997 (1)

1996 (1)

1995 (2)

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

Y. Yamada, “Light-tissue interaction and optical imaging in biomedicine,” Ann. Rev. Fluid Mech. Heat Transfer 6, 1–59 (1995).

1994 (2)

1993 (2)

1990 (1)

S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
[CrossRef]

1983 (1)

M. P. Menguc, R. Viskanta, “Comparison of radiative transfer approximations for highly forward scattering planar medium,” J. Quant. Spectrosc. Rad. Transfer 29, 381–394 (1983).
[CrossRef]

1981 (1)

J. L. Squire, H. Krumboltz, “Profiling pelagic fish schools using airborne optical lasers and other remote sensing techniques,” Mar. Technol. Soc. J. 15, 443–448 (1981).

1979 (1)

N. K. Madsen, R. F. Sincovec, “Algorithm 540 PDECOL, general collocation software for partial differential equations [D3],” ACM Trans. Math. Softwares 5, 326–361 (1979).
[CrossRef]

1974 (1)

D. L. Murphree, C. D. Taylor, R. W. McClendon, “Mathematical modeling for the detection of fish by an airborne laser,” AIAA J. 12, 1686–1692 (1974).
[CrossRef]

1970 (1)

G. E. Hunt, “The generation of angular distribution coefficients for radiation scattered by a spherical particle,” J. Spectrosc. Radiat. Transfer 10, 857–864 (1970).
[CrossRef]

Chance, B.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

Churnside, J. H.

J. H. Churnside, J. J. Wilson, V. V. Tatarskii, “Lidar profiles of fish schools,” Appl. Opt. 36, 6011–6020 (1997).
[CrossRef] [PubMed]

J. H. Churnside, J. R. Hunter, “Laser remote sensing of epipelagic fishes,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, Proc. SPIE2964, 38–53 (1996).

Forand, J. L.

G. R. Fournier, J. L. Forand, “Analytic phase function for ocean water,” in Ocean Optics XII, Proc. SPIE2258, 194–201 (1994).
[CrossRef]

Fournier, G. R.

G. R. Fournier, J. L. Forand, “Analytic phase function for ocean water,” in Ocean Optics XII, Proc. SPIE2258, 194–201 (1994).
[CrossRef]

Gentili, B.

Gordon, H.

Gordon, H. R.

Haltrin, V. I.

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). Vol. 1, pp. I-509–I-518.

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1992).

Hunt, G. E.

G. E. Hunt, “The generation of angular distribution coefficients for radiation scattered by a spherical particle,” J. Spectrosc. Radiat. Transfer 10, 857–864 (1970).
[CrossRef]

Hunter, J. R.

J. H. Churnside, J. R. Hunter, “Laser remote sensing of epipelagic fishes,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, Proc. SPIE2964, 38–53 (1996).

Isaacson, E.

E. Isaacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

Jin, Z.

Kattawar, G.

Keller, H. B.

E. Isaacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

Krekov, G. M.

Krekova, M. M.

Krumboltz, H.

J. L. Squire, H. Krumboltz, “Profiling pelagic fish schools using airborne optical lasers and other remote sensing techniques,” Mar. Technol. Soc. J. 15, 443–448 (1981).

Kumar, S.

S. Kumar, K. Mitra, Y. Yamada, “Hyperbolic damped-wave models for transient light pulse propagation in scattering media,” Appl. Opt. 35, 3372–3378 (1996).
[CrossRef] [PubMed]

S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
[CrossRef]

Kwon, K. H.

J. H. Smart, K. H. Kwon, “Comparisons between in situ and remote sensing estimates of diffuse attenuation profiles,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, Proc. SPIE2964, 100–109 (1996).

Madsen, N. K.

N. K. Madsen, R. F. Sincovec, “Algorithm 540 PDECOL, general collocation software for partial differential equations [D3],” ACM Trans. Math. Softwares 5, 326–361 (1979).
[CrossRef]

Majumdar, A.

S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
[CrossRef]

McClendon, R. W.

D. L. Murphree, C. D. Taylor, R. W. McClendon, “Mathematical modeling for the detection of fish by an airborne laser,” AIAA J. 12, 1686–1692 (1974).
[CrossRef]

Menguc, M. P.

M. P. Menguc, R. Viskanta, “Comparison of radiative transfer approximations for highly forward scattering planar medium,” J. Quant. Spectrosc. Rad. Transfer 29, 381–394 (1983).
[CrossRef]

Mitra, K.

Mobley, C. D.

C. D. Mobley, B. Gentili, H. Gordon, Z. Jin, G. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
[CrossRef] [PubMed]

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

C. D. Mobley, “The optical properties of water,” in Handbook of Optics: Fundamentals, Techniques, Design (Academic, San Diego, Calif., 1995), Vol. 1, Chap. 43.

Modest, M. F.

M. F. Modest, Radiative Heat Transfer (McGraw-Hill, New York, 1993).

Morel, A.

Murphree, D. L.

D. L. Murphree, C. D. Taylor, R. W. McClendon, “Mathematical modeling for the detection of fish by an airborne laser,” AIAA J. 12, 1686–1692 (1974).
[CrossRef]

Reinersman, P.

Samokhvalov, I. V.

Shamanaev, V. S.

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1992).

Sincovec, R. F.

N. K. Madsen, R. F. Sincovec, “Algorithm 540 PDECOL, general collocation software for partial differential equations [D3],” ACM Trans. Math. Softwares 5, 326–361 (1979).
[CrossRef]

Smart, J. H.

J. H. Smart, K. H. Kwon, “Comparisons between in situ and remote sensing estimates of diffuse attenuation profiles,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, Proc. SPIE2964, 100–109 (1996).

Squire, J. L.

J. L. Squire, H. Krumboltz, “Profiling pelagic fish schools using airborne optical lasers and other remote sensing techniques,” Mar. Technol. Soc. J. 15, 443–448 (1981).

Stamnes, K.

Stavn, R.

Tatarskii, V. V.

Taylor, C. D.

D. L. Murphree, C. D. Taylor, R. W. McClendon, “Mathematical modeling for the detection of fish by an airborne laser,” AIAA J. 12, 1686–1692 (1974).
[CrossRef]

Tien, C. L.

S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
[CrossRef]

Viskanta, R.

M. P. Menguc, R. Viskanta, “Comparison of radiative transfer approximations for highly forward scattering planar medium,” J. Quant. Spectrosc. Rad. Transfer 29, 381–394 (1983).
[CrossRef]

Wilson, J. J.

Yamada, Y.

S. Kumar, K. Mitra, Y. Yamada, “Hyperbolic damped-wave models for transient light pulse propagation in scattering media,” Appl. Opt. 35, 3372–3378 (1996).
[CrossRef] [PubMed]

Y. Yamada, “Light-tissue interaction and optical imaging in biomedicine,” Ann. Rev. Fluid Mech. Heat Transfer 6, 1–59 (1995).

Yodh, A.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

ACM Trans. Math. Softwares (1)

N. K. Madsen, R. F. Sincovec, “Algorithm 540 PDECOL, general collocation software for partial differential equations [D3],” ACM Trans. Math. Softwares 5, 326–361 (1979).
[CrossRef]

AIAA J. (1)

D. L. Murphree, C. D. Taylor, R. W. McClendon, “Mathematical modeling for the detection of fish by an airborne laser,” AIAA J. 12, 1686–1692 (1974).
[CrossRef]

Ann. Rev. Fluid Mech. Heat Transfer (1)

Y. Yamada, “Light-tissue interaction and optical imaging in biomedicine,” Ann. Rev. Fluid Mech. Heat Transfer 6, 1–59 (1995).

Appl. Opt. (6)

J. Heat Transfer (1)

S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
[CrossRef]

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

M. P. Menguc, R. Viskanta, “Comparison of radiative transfer approximations for highly forward scattering planar medium,” J. Quant. Spectrosc. Rad. Transfer 29, 381–394 (1983).
[CrossRef]

J. Spectrosc. Radiat. Transfer (1)

G. E. Hunt, “The generation of angular distribution coefficients for radiation scattered by a spherical particle,” J. Spectrosc. Radiat. Transfer 10, 857–864 (1970).
[CrossRef]

Mar. Technol. Soc. J. (1)

J. L. Squire, H. Krumboltz, “Profiling pelagic fish schools using airborne optical lasers and other remote sensing techniques,” Mar. Technol. Soc. J. 15, 443–448 (1981).

Phys. Today (1)

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

Other (9)

C. D. Mobley, “The optical properties of water,” in Handbook of Optics: Fundamentals, Techniques, Design (Academic, San Diego, Calif., 1995), Vol. 1, Chap. 43.

J. H. Smart, K. H. Kwon, “Comparisons between in situ and remote sensing estimates of diffuse attenuation profiles,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, Proc. SPIE2964, 100–109 (1996).

J. H. Churnside, J. R. Hunter, “Laser remote sensing of epipelagic fishes,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, Proc. SPIE2964, 38–53 (1996).

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1992).

M. F. Modest, Radiative Heat Transfer (McGraw-Hill, New York, 1993).

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). Vol. 1, pp. I-509–I-518.

G. R. Fournier, J. L. Forand, “Analytic phase function for ocean water,” in Ocean Optics XII, Proc. SPIE2258, 194–201 (1994).
[CrossRef]

E. Isaacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

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

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Figures (6)

Fig. 1
Fig. 1

Schematic of the problem under consideration.

Fig. 2
Fig. 2

Comparison of the phase function obtained by Mobley to its Delta–Eddington approximation.

Fig. 3
Fig. 3

Comparison of the backscattered signal as a function of depth for various types of ocean water using the discrete ordinates method.

Fig. 4
Fig. 4

Comparison of the backscattered signal as a function of depth for different number density of fish using the discrete ordinates method.

Fig. 5
Fig. 5

Comparison of the backscattered signal as a function of depth for different thicknesses of schools of fish using the discrete ordinates method.

Fig. 6
Fig. 6

Comparison of the backscattered signal as a function of depth for different depths from the ocean surface where schools of fish are present using the discrete ordinates method.

Tables (1)

Tables Icon

Table 1 Optical Properties of Different Types of Water for Green Light

Equations (15)

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

1cLz, μ, tt+μ Lz, μ, tz=-keLz, μ, t+ks2-11 Lz, μ, tpμμdμ+S(z, μ, t),
120π pθsin θ d θ=1.
pΘ=m=0M amPmcos Θ,
12π02π Pmcos Θdψ=PmμPmμ,cos Θ=μμ+1-μ21-μ2 cosψ-ψ,
ppΘ=2fδ1-cos Θ+1-fp*Θ,
ppμ, ϕμ, ϕ=2fδμ-μδϕ-ϕ+1-fp*μ, ϕμ, ϕ,
120π p*θsin θ d  θ=1.
pfθ=83πsin Θ-Θ cos Θ.
pcos Θ=kspks ppΘ+ksfks pfΘ,
1cLiz, tt+μiLiz, tz=-keLiz, t+ks2j=-KK wjLjz, tpμjμi+Sz, μi, t,  i, j0,
Lcz, μ, t=Lincident exp-kezt-z/cexp×-t-z/c/τHt-z/cδμ-1,
Sz, μ, t=ks2-11 Lcz, μ, tpμμμ.
Lz=0, μ>0, t=Lz=D, μ<0, t=0.
ksf=2RAN,
kaf=21-RAN,

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