Abstract

The scattering coefficient of water as a function of concentration of hydrosol particles is calculated. A new quantum-mechanical approach to calculate the multiple-scattering phenomenon in seawater is proposed. The approach is based on Maxwell’s equations for the light fields in stochastically scattering water with hydrosols. The water is modeled as a thermally fluctuating medium filled with the particles. It is found that at small concentrations of scatterers the scattering coefficient is linear in the concentration. At higher values of concentrations the dependence on the concentration may be approximated by a power law.

© 1999 Optical Society of America

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  1. Preliminary results of this paper have been partially presented at the 1997 International Geoscience and Remote Sensing Symposium, Denver, Colorado,2 and the 1998 Ocean Sciences Meeting, San Diego, California.3
  2. V. I. Haltrin, “Light scattering coefficient of water at concentrations of hydrosols typical for lakes and shallow marine waters,” in Proceedings of the Twelfth International Conference Applied Geologic Remote Sensing (ERIM International, Denver, Colo., 1997), Vol. I, pp. 417–424.
  3. V. I. Haltrin, “Nonlinear concentrational dependence of the water light scattering coefficient,” in Supplement to EOS Transactions (American Geophysical Union, Washington, D.C., 1998), Vol. 79, abstract OS12D-11.
  4. L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
    [CrossRef]
  5. D. K. Clark, E. T. Backer, A. E. Strong, “Upwelled spectral radiance distribution in relation to particular matter in water,” Boundary-Layer Meteorol. 18, 287–298 (1980).
    [CrossRef]
  6. U. Frish, “Wave propagation in random media,” in Probabilistic Methods in Applied Mathematics, A. T. Bharucha-Reid, ed. (Academic, New York, 1968), Vol. I, pp. 75–198.
  7. H. A. Gould, “Use of the Bethe–Salpeter equation in transport theory,” in Lectures in Theoretical Physics, Vol. IXC: Kinetic Theory, W. E. Brittin, ed. (Gordon and Breach, New York1967), pp. 651–691.
  8. M. Kaku, Quantum Field Theory: A Modern Introduction (Oxford U. Press, New York, 1993).
  9. J. B. Hartle, S. W. Hawking, “Wave function of the universe,” Phys. Rev. D 28, 2960–2975 (1983).
    [CrossRef]
  10. M. B. Green, J. H. Schwartz, E. Witten, Superstring Theory (Cambridge U. Press, Cambridge, UK, 1987), Vols. I and II.
  11. R. D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem (Dover, New York, 1976).
  12. A. J. Drummond, M. P. Thekaekara, eds. The Extraterrestrial Solar Spectrum (Institute of Environmental Science, Mount Prospect, Ill., 1973).
  13. K. S. Shifrin, Physical Optics of Ocean Water (American Institute of Physics, New York, 1988).
  14. 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. I. Case of absorption and elastic scattering,” Appl. Opt. 37, 3773–3784 (1998).
    [CrossRef]
  15. V. I. Haltrin, G. W. Kattawar, “Self-consistent solutions to the equation of transfer with elastic and inelastic scattering in oceanic optics. I. Model.” Appl. Opt. 32, 5356–5367 (1993).
    [CrossRef] [PubMed]
  16. V. I. Haltrin, A. D. Weidemann, “A method and algorithm of computing apparent optical properties of coastal sea waters,” in Remote Sensing for a Sustainable Future, Proceedings of the 1996 International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 305–309.
  17. V. I. Haltrin, “Algorithm for computing apparent optical properties of shallow waters under arbitrary surface illumination,” in Proceedings of the Third International Airborne Remote Sensing Conference (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), Vol. I, pp. 463–470.
  18. C. Acquista, J. L. Anderson, “A derivation of the radiative transfer equation for partially polarized light from quantum electrodynamics,” Ann. Phys. (N.Y.) 106, 435–443 (1977).
    [CrossRef]
  19. D. Bugniolo, “Transport equation for the spectral density of a multiple-scattered electromagnetic field,” J. Appl. Phys. 31, 1176–1182 (1960).
    [CrossRef]
  20. P. E. Scott, “A transport equation for the multiple scattering of electromagnetic waves,” J. Phys. A 1, 675–689 (1968).
    [CrossRef]
  21. K. M. Watson, “Quantum mechanical transport theory. I. Incoherent processes,” Phys. Rev. 118, 886–898 (1960).
    [CrossRef]
  22. S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 4: Wave Propagation through Random Media (Springer-Verlag, New York, 1989).
  23. V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, National Technical Information Service, Springfield, Va., 1971).
  24. The correct scalar equation for transfer that does not ignore polarization effects was recently proposed in Ref. 25.
  25. G. C. Pomraning, N. J. McCormick, “Approximate scalar equation for polarized radiative transfer,” J. Opt. Soc. Am. A 15, 1932–1939 (1998).
    [CrossRef]
  26. P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Parts 1 and 2 (McGraw-Hill, New York, 1953).
  27. A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 1963).
  28. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960).
  29. J. P. Boon, S. Yip, Molecular Hydrodynamics (Dover, New York, 1980).
  30. G. E. Uhlenbeck, L. S. Ornstein, “On the theory of Brownian motion,” Phys. Rev. 36, 823–841 (1930).
    [CrossRef]
  31. M. S. Wang, G. E. Uhlenbeck, “On the theory of Brownian motion. II,” Rev. Mod. Phys. 17, 323–342 (1945).
    [CrossRef]
  32. J. L. Lebowitz, E. Rubin, “Dynamical study of Brownian motion,” Phys. Rev. 131, 2381–2396 (1963).
    [CrossRef]
  33. G. W. Ford, M. Kac, P. Mazur, “Statistical mechanics of ensembles of coupled oscillators,” J. Math. Phys. (N.Y.) 6, 504–515 (1965).
    [CrossRef]
  34. A. Einstein, Investigations on the Theory of the Brownian Movement (Dover, New York, 1956).
  35. T. Matsubara, “A new approach to quantum-statistical mechanics,” Prog. Theor. Phys. 14, 351–367 (1955).
    [CrossRef]
  36. H. C. Van De Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  37. The root is regarded as physical if it behaves in accordance with physics, i.e., it is positive, is linear in concentrations when concentrations are small, and monotonically increases with the increase in concentration.

1998

1993

1983

J. B. Hartle, S. W. Hawking, “Wave function of the universe,” Phys. Rev. D 28, 2960–2975 (1983).
[CrossRef]

1981

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

1980

D. K. Clark, E. T. Backer, A. E. Strong, “Upwelled spectral radiance distribution in relation to particular matter in water,” Boundary-Layer Meteorol. 18, 287–298 (1980).
[CrossRef]

1977

C. Acquista, J. L. Anderson, “A derivation of the radiative transfer equation for partially polarized light from quantum electrodynamics,” Ann. Phys. (N.Y.) 106, 435–443 (1977).
[CrossRef]

1968

P. E. Scott, “A transport equation for the multiple scattering of electromagnetic waves,” J. Phys. A 1, 675–689 (1968).
[CrossRef]

1965

G. W. Ford, M. Kac, P. Mazur, “Statistical mechanics of ensembles of coupled oscillators,” J. Math. Phys. (N.Y.) 6, 504–515 (1965).
[CrossRef]

1963

J. L. Lebowitz, E. Rubin, “Dynamical study of Brownian motion,” Phys. Rev. 131, 2381–2396 (1963).
[CrossRef]

1960

K. M. Watson, “Quantum mechanical transport theory. I. Incoherent processes,” Phys. Rev. 118, 886–898 (1960).
[CrossRef]

D. Bugniolo, “Transport equation for the spectral density of a multiple-scattered electromagnetic field,” J. Appl. Phys. 31, 1176–1182 (1960).
[CrossRef]

1955

T. Matsubara, “A new approach to quantum-statistical mechanics,” Prog. Theor. Phys. 14, 351–367 (1955).
[CrossRef]

1945

M. S. Wang, G. E. Uhlenbeck, “On the theory of Brownian motion. II,” Rev. Mod. Phys. 17, 323–342 (1945).
[CrossRef]

1930

G. E. Uhlenbeck, L. S. Ornstein, “On the theory of Brownian motion,” Phys. Rev. 36, 823–841 (1930).
[CrossRef]

Abrikosov, A. A.

A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 1963).

Acquista, C.

C. Acquista, J. L. Anderson, “A derivation of the radiative transfer equation for partially polarized light from quantum electrodynamics,” Ann. Phys. (N.Y.) 106, 435–443 (1977).
[CrossRef]

Anderson, J. L.

C. Acquista, J. L. Anderson, “A derivation of the radiative transfer equation for partially polarized light from quantum electrodynamics,” Ann. Phys. (N.Y.) 106, 435–443 (1977).
[CrossRef]

Backer, E. T.

D. K. Clark, E. T. Backer, A. E. Strong, “Upwelled spectral radiance distribution in relation to particular matter in water,” Boundary-Layer Meteorol. 18, 287–298 (1980).
[CrossRef]

Boon, J. P.

J. P. Boon, S. Yip, Molecular Hydrodynamics (Dover, New York, 1980).

Bugniolo, D.

D. Bugniolo, “Transport equation for the spectral density of a multiple-scattered electromagnetic field,” J. Appl. Phys. 31, 1176–1182 (1960).
[CrossRef]

Clark, D. K.

D. K. Clark, E. T. Backer, A. E. Strong, “Upwelled spectral radiance distribution in relation to particular matter in water,” Boundary-Layer Meteorol. 18, 287–298 (1980).
[CrossRef]

Dzyaloshinski, I. E.

A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 1963).

Einstein, A.

A. Einstein, Investigations on the Theory of the Brownian Movement (Dover, New York, 1956).

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Parts 1 and 2 (McGraw-Hill, New York, 1953).

Ford, G. W.

G. W. Ford, M. Kac, P. Mazur, “Statistical mechanics of ensembles of coupled oscillators,” J. Math. Phys. (N.Y.) 6, 504–515 (1965).
[CrossRef]

Frish, U.

U. Frish, “Wave propagation in random media,” in Probabilistic Methods in Applied Mathematics, A. T. Bharucha-Reid, ed. (Academic, New York, 1968), Vol. I, pp. 75–198.

Gorkov, L. P.

A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 1963).

Gould, H. A.

H. A. Gould, “Use of the Bethe–Salpeter equation in transport theory,” in Lectures in Theoretical Physics, Vol. IXC: Kinetic Theory, W. E. Brittin, ed. (Gordon and Breach, New York1967), pp. 651–691.

Green, M. B.

M. B. Green, J. H. Schwartz, E. Witten, Superstring Theory (Cambridge U. Press, Cambridge, UK, 1987), Vols. I and II.

Haltrin, V. I.

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. I. Case of absorption and elastic scattering,” Appl. Opt. 37, 3773–3784 (1998).
[CrossRef]

V. I. Haltrin, G. W. Kattawar, “Self-consistent solutions to the equation of transfer with elastic and inelastic scattering in oceanic optics. I. Model.” Appl. Opt. 32, 5356–5367 (1993).
[CrossRef] [PubMed]

V. I. Haltrin, “Light scattering coefficient of water at concentrations of hydrosols typical for lakes and shallow marine waters,” in Proceedings of the Twelfth International Conference Applied Geologic Remote Sensing (ERIM International, Denver, Colo., 1997), Vol. I, pp. 417–424.

V. I. Haltrin, “Nonlinear concentrational dependence of the water light scattering coefficient,” in Supplement to EOS Transactions (American Geophysical Union, Washington, D.C., 1998), Vol. 79, abstract OS12D-11.

V. I. Haltrin, “Algorithm for computing apparent optical properties of shallow waters under arbitrary surface illumination,” in Proceedings of the Third International Airborne Remote Sensing Conference (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), Vol. I, pp. 463–470.

V. I. Haltrin, A. D. Weidemann, “A method and algorithm of computing apparent optical properties of coastal sea waters,” in Remote Sensing for a Sustainable Future, Proceedings of the 1996 International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 305–309.

Hartle, J. B.

J. B. Hartle, S. W. Hawking, “Wave function of the universe,” Phys. Rev. D 28, 2960–2975 (1983).
[CrossRef]

Hawking, S. W.

J. B. Hartle, S. W. Hawking, “Wave function of the universe,” Phys. Rev. D 28, 2960–2975 (1983).
[CrossRef]

Kac, M.

G. W. Ford, M. Kac, P. Mazur, “Statistical mechanics of ensembles of coupled oscillators,” J. Math. Phys. (N.Y.) 6, 504–515 (1965).
[CrossRef]

Kaku, M.

M. Kaku, Quantum Field Theory: A Modern Introduction (Oxford U. Press, New York, 1993).

Kattawar, G. W.

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 4: Wave Propagation through Random Media (Springer-Verlag, New York, 1989).

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960).

Lebowitz, J. L.

J. L. Lebowitz, E. Rubin, “Dynamical study of Brownian motion,” Phys. Rev. 131, 2381–2396 (1963).
[CrossRef]

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960).

Matsubara, T.

T. Matsubara, “A new approach to quantum-statistical mechanics,” Prog. Theor. Phys. 14, 351–367 (1955).
[CrossRef]

Mattuck, R. D.

R. D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem (Dover, New York, 1976).

Mazur, P.

G. W. Ford, M. Kac, P. Mazur, “Statistical mechanics of ensembles of coupled oscillators,” J. Math. Phys. (N.Y.) 6, 504–515 (1965).
[CrossRef]

McCormick, N. J.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Parts 1 and 2 (McGraw-Hill, New York, 1953).

Ornstein, L. S.

G. E. Uhlenbeck, L. S. Ornstein, “On the theory of Brownian motion,” Phys. Rev. 36, 823–841 (1930).
[CrossRef]

Pomraning, G. C.

Prieur, L.

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

Rubin, E.

J. L. Lebowitz, E. Rubin, “Dynamical study of Brownian motion,” Phys. Rev. 131, 2381–2396 (1963).
[CrossRef]

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 4: Wave Propagation through Random Media (Springer-Verlag, New York, 1989).

Sathyendranath, S.

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

Schwartz, J. H.

M. B. Green, J. H. Schwartz, E. Witten, Superstring Theory (Cambridge U. Press, Cambridge, UK, 1987), Vols. I and II.

Scott, P. E.

P. E. Scott, “A transport equation for the multiple scattering of electromagnetic waves,” J. Phys. A 1, 675–689 (1968).
[CrossRef]

Shifrin, K. S.

K. S. Shifrin, Physical Optics of Ocean Water (American Institute of Physics, New York, 1988).

Strong, A. E.

D. K. Clark, E. T. Backer, A. E. Strong, “Upwelled spectral radiance distribution in relation to particular matter in water,” Boundary-Layer Meteorol. 18, 287–298 (1980).
[CrossRef]

Tatarskii, V. I.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 4: Wave Propagation through Random Media (Springer-Verlag, New York, 1989).

V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, National Technical Information Service, Springfield, Va., 1971).

Uhlenbeck, G. E.

M. S. Wang, G. E. Uhlenbeck, “On the theory of Brownian motion. II,” Rev. Mod. Phys. 17, 323–342 (1945).
[CrossRef]

G. E. Uhlenbeck, L. S. Ornstein, “On the theory of Brownian motion,” Phys. Rev. 36, 823–841 (1930).
[CrossRef]

Van De Hulst, H. C.

H. C. Van De Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Wang, M. S.

M. S. Wang, G. E. Uhlenbeck, “On the theory of Brownian motion. II,” Rev. Mod. Phys. 17, 323–342 (1945).
[CrossRef]

Watson, K. M.

K. M. Watson, “Quantum mechanical transport theory. I. Incoherent processes,” Phys. Rev. 118, 886–898 (1960).
[CrossRef]

Weidemann, A. D.

V. I. Haltrin, A. D. Weidemann, “A method and algorithm of computing apparent optical properties of coastal sea waters,” in Remote Sensing for a Sustainable Future, Proceedings of the 1996 International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 305–309.

Witten, E.

M. B. Green, J. H. Schwartz, E. Witten, Superstring Theory (Cambridge U. Press, Cambridge, UK, 1987), Vols. I and II.

Yip, S.

J. P. Boon, S. Yip, Molecular Hydrodynamics (Dover, New York, 1980).

Ann. Phys. (N.Y.)

C. Acquista, J. L. Anderson, “A derivation of the radiative transfer equation for partially polarized light from quantum electrodynamics,” Ann. Phys. (N.Y.) 106, 435–443 (1977).
[CrossRef]

Appl. Opt.

Boundary-Layer Meteorol.

D. K. Clark, E. T. Backer, A. E. Strong, “Upwelled spectral radiance distribution in relation to particular matter in water,” Boundary-Layer Meteorol. 18, 287–298 (1980).
[CrossRef]

J. Appl. Phys.

D. Bugniolo, “Transport equation for the spectral density of a multiple-scattered electromagnetic field,” J. Appl. Phys. 31, 1176–1182 (1960).
[CrossRef]

J. Math. Phys. (N.Y.)

G. W. Ford, M. Kac, P. Mazur, “Statistical mechanics of ensembles of coupled oscillators,” J. Math. Phys. (N.Y.) 6, 504–515 (1965).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. A

P. E. Scott, “A transport equation for the multiple scattering of electromagnetic waves,” J. Phys. A 1, 675–689 (1968).
[CrossRef]

Limnol. Oceanogr.

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

Phys. Rev.

J. L. Lebowitz, E. Rubin, “Dynamical study of Brownian motion,” Phys. Rev. 131, 2381–2396 (1963).
[CrossRef]

G. E. Uhlenbeck, L. S. Ornstein, “On the theory of Brownian motion,” Phys. Rev. 36, 823–841 (1930).
[CrossRef]

K. M. Watson, “Quantum mechanical transport theory. I. Incoherent processes,” Phys. Rev. 118, 886–898 (1960).
[CrossRef]

Phys. Rev. D

J. B. Hartle, S. W. Hawking, “Wave function of the universe,” Phys. Rev. D 28, 2960–2975 (1983).
[CrossRef]

Prog. Theor. Phys.

T. Matsubara, “A new approach to quantum-statistical mechanics,” Prog. Theor. Phys. 14, 351–367 (1955).
[CrossRef]

Rev. Mod. Phys.

M. S. Wang, G. E. Uhlenbeck, “On the theory of Brownian motion. II,” Rev. Mod. Phys. 17, 323–342 (1945).
[CrossRef]

Other

Preliminary results of this paper have been partially presented at the 1997 International Geoscience and Remote Sensing Symposium, Denver, Colorado,2 and the 1998 Ocean Sciences Meeting, San Diego, California.3

V. I. Haltrin, “Light scattering coefficient of water at concentrations of hydrosols typical for lakes and shallow marine waters,” in Proceedings of the Twelfth International Conference Applied Geologic Remote Sensing (ERIM International, Denver, Colo., 1997), Vol. I, pp. 417–424.

V. I. Haltrin, “Nonlinear concentrational dependence of the water light scattering coefficient,” in Supplement to EOS Transactions (American Geophysical Union, Washington, D.C., 1998), Vol. 79, abstract OS12D-11.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Parts 1 and 2 (McGraw-Hill, New York, 1953).

A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 1963).

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960).

J. P. Boon, S. Yip, Molecular Hydrodynamics (Dover, New York, 1980).

H. C. Van De Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

The root is regarded as physical if it behaves in accordance with physics, i.e., it is positive, is linear in concentrations when concentrations are small, and monotonically increases with the increase in concentration.

A. Einstein, Investigations on the Theory of the Brownian Movement (Dover, New York, 1956).

M. B. Green, J. H. Schwartz, E. Witten, Superstring Theory (Cambridge U. Press, Cambridge, UK, 1987), Vols. I and II.

R. D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem (Dover, New York, 1976).

A. J. Drummond, M. P. Thekaekara, eds. The Extraterrestrial Solar Spectrum (Institute of Environmental Science, Mount Prospect, Ill., 1973).

K. S. Shifrin, Physical Optics of Ocean Water (American Institute of Physics, New York, 1988).

U. Frish, “Wave propagation in random media,” in Probabilistic Methods in Applied Mathematics, A. T. Bharucha-Reid, ed. (Academic, New York, 1968), Vol. I, pp. 75–198.

H. A. Gould, “Use of the Bethe–Salpeter equation in transport theory,” in Lectures in Theoretical Physics, Vol. IXC: Kinetic Theory, W. E. Brittin, ed. (Gordon and Breach, New York1967), pp. 651–691.

M. Kaku, Quantum Field Theory: A Modern Introduction (Oxford U. Press, New York, 1993).

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 4: Wave Propagation through Random Media (Springer-Verlag, New York, 1989).

V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, National Technical Information Service, Springfield, Va., 1971).

The correct scalar equation for transfer that does not ignore polarization effects was recently proposed in Ref. 25.

V. I. Haltrin, A. D. Weidemann, “A method and algorithm of computing apparent optical properties of coastal sea waters,” in Remote Sensing for a Sustainable Future, Proceedings of the 1996 International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 305–309.

V. I. Haltrin, “Algorithm for computing apparent optical properties of shallow waters under arbitrary surface illumination,” in Proceedings of the Third International Airborne Remote Sensing Conference (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), Vol. I, pp. 463–470.

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Figures (7)

Fig. 1
Fig. 1

Graphical form of Dyson’s equation (48).

Fig. 2
Fig. 2

Perturbation series for the vertex part Γαβ.

Fig. 3
Fig. 3

Graphical form of the approximate integral equation (50) for the vertex part Γαβ.

Fig. 4
Fig. 4

Concentration dependence of the relative scattering coefficient Re(z) for size parameter x=(2πr¯)/λ=20.

Fig. 5
Fig. 5

Same as Fig. 4, but for x=200.

Fig. 6
Fig. 6

Same as Fig. 4, but for x=2000.

Fig. 7
Fig. 7

Same as Fig. 4, but for x=20,000.

Tables (2)

Tables Icon

Table 1 Values of the Parameter Re(z) for Different Size Parameters x=(2πr¯)/λ and Concentrations

Tables Icon

Table 2 Regression Coefficients for Equation (63)

Equations (80)

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

sph=ω¯c10-10W m
ω¯=2πcnwλ¯,λ¯=500nm5×10-7m,
ρph=Lω¯c1013m-3(50μm)-3.
Vr=Srτrc=NphVph.
Nph=SrτrLω¯=L Srτrλ¯nw2πc.
S=Tˆ exp-i0H^int(t)dt.
|0=S|Tˆ exp-i0H^int(t)dt|.
δE=(4π)-1d3r E(r, t)δD(r, t),
Dm(r, t)=E(r, t)+0fm(t)E(r, t-t)dt,
Um(r-rm)=1,rVm0,rVm,
δD(r, t)=0δf(r, t)E(r, t-t)dt,
δf(r, t)=mUm(r-rm)m[fm(t)-fw(t)],
F(r, t)α1ρ(r, t)+α2ρ2(r, t)+,
F(r, t)d3r=1,ρ(r, t)=ρ0+δρ(r, t),
ρ01g/cm3,
F(r, t)ρ(r, t)V0ρ0=1V0+δρ(r, t)V0ρ0,
Um(r-rm)m=F(r, t)Um(r-r)d3r
=Cm+1V0ρ0 δρ(r, t)Um(r-r)d3r,
δf(r, t)=mCm[fm(t)-fw(t)]+1ρ0 δρ(r, t)v(r-r, t)d3r,
v(r, t)=1V0 mUm(r)[fm(t)-fw(t)].
δD(r, t)=δD0(r, t)+δDs(r, t),
δD0(r, t)=mCm0dt[fm(t)-fw(t)]E(r, t-t),
δDs(r, t)=ρ0-10dtd3r δρ(r, t)×v(r-r, t-t)E(r, t-t).
fm(t)-fw(t)(m-0w)δ(t+0),
v(r, t)=δhCVΔ(r)δ(t+0),
δh=CV-1m(m-0w)Cm=CV-1mmCm-0w=¯h-0w,
Δ(r)=1δhCVV0 m(m-0w)Um(r),
Δ(r)d3r=1.
ρ(r, t)t+div[q˙(r, t)ρ(r, t)]=0,
δρ(r, t)ρ0=-q(r, t)=-ϕ(r, t)u0ρ0,
δE=-δhCV4πu0ρ0 d3r E(r, t)×d3r Δ(r-r)ϕ(r, t)E(r, t),
Δ(r)=rf(r)dr4π0r2 drrf(r)dr.
E(r, t)=-1c A(r, t)t-grad[Φ(r, t)],
ψ(r, t)=14π r¯3T1/2d3r Δ(r-r)ϕ(r, t)-r¯4π ru¯0(T)ρ0T1/2Δ(r-r)δρ(r, t)d3r,
g=kgCV,kg=δhTr¯3ρu02(T)1/2103105,
H^int(t)=-gc2 d3r A^α(r, t)t ψˆ(r, t) A^α(r, t)t,
α=1,2,3,
Gαβ(τ1-τ2, r1, r2)
=-Tr{exp[(F-H^int)/T]exp(H^int|τ1-τ2|)×Aα(r1)exp(-H^int|τ1-τ2|)Aβ(r2)},
[(ω, r)ω2δαβ-curlαγ curlγδ]Gδβ(ω, r, r)
=4πδαβδ(r1-r2).
[0(ω, r)ω2δαβ-curlαγ curlγδ]Gδβ(0)(ω, r, r)
=4πδαβδ(r1-r2).
Gαβ(0)(ω, k)=G0tr(ω, k)(δαβ-nαnβ)+G0l(ω, k)nαnβ,
G0tr(ω, k)=4π0ω2/c2-k2,G0l(ω, k)=4πc20ω2,
Gαβ(ω, k)=Gtr(ω, k)(δαβ-nαnβ)+Gl(ω, k)nαnβ,
Gtr(ω, k)=4πtrω2/c2-k2,Gl(ω, k)=4πc2lω2,
αβ(ω, k)=tr(ω, k)(δαβ-nαnβ)+l(ω, k)nαnβ.
αβ0(ω)=0(ω)δαβ0(ω)(δαβ-nαnβ)+0(ω)nαnβ.
tr(ω, k)=0(ω)+δtr(ω, k),
l(ω, k)=0(ω)+δl(ω, k),
0(ω)=0(ω)+i0(ω),
±iωnGαβ(0)(ωn, k),
Gαβ(0)(ωn, k)=-4π(δαβ-nαnβ)0ωn2+k2-4πnαnβ0ωn2.
ωn2Gαβ(0)(ωn, k).
r¯34π2T Δ2(q)D(0)(0, q)-r¯34π2T Δ2(q),
D(0)(ω, q)=ω02(q)ω2-ω02(q)+iδ,
ω0(q)=u0|q|,Δ(q)d3r exp(-iqr)Δ(r)
-g2T(2π)3n.
Gαβ(ωn, k)=Gαβ(0)(ωn, k)+Gαγ(0)(ωn, k)×πγμ(ωn, k)Gμβ(ωn, k),
παβ(ωn, k)=-ghd3q Δ2(q)Gαγ(ωn, k-q)×Γγβ(ωn, k-q, k),
h=r¯3ωn4/(2π)5,
δαβ(ωn, k)=4πωn2 παβ(ωn, k).
Γαβ(k-p, k)
=gδαβ+hd3q Δ2(q)Γαμ(k-p, k-p-q)×Gμη(k-p-q)Γην(k-p-q, k-q)×Gνκ(k-q)Γκβ(k-q, k).
Γtr=g+r¯2ω42π2k2c4 (Γtr)3,Γl=g+23π2(l)2 (Γl)3.
tr=0+ig rω¯|ω|16πkc2 Γtr,l=0-g6π2l Γl.
k2=trω2/c2.
l=0-g6π2l Γl,Γl=g+23π2(l)2 (Γl)3,
k2=tr ω2c2,tr=0+igx16πtr Γtr,
Γtr=g+x22π2tr (Γtr)3,
g<420/3,CV<2×10-3(orCp<2 g/m3),
l=01-η3 sin13 sin-1(3η),
η=CVδhπ0 T2r¯3ρu021/2,
k=k0+i(a+b)/2;
k=ωtrc.
z=itr
b=4πλ Re(z)4πλ Im(tr).
16πz(0+z2)[g2+128(0+z2)2]-g4x=0.
b=4πλ β1g+β2g2+β3g3,g<g0αgγ,gg0,r2>0.999.

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