Abstract

Lidar backscatter from clouds in the Delft University of Technology experiment is complicated by the fact that the transmitter has a narrow beam width, whereas the receiver has a much wider one. The issue here is whether reception of light scattered incoherently by cloud particles can contribute appreciably to the received power. The incoherent contribution can come from within as well as from outside the transmitter beam but in any case is due to at least two scattering processes in the cloud that are not included in the coherent forward scatter that leads to the usual exponentially attenuated contribution from single-particle backscatter. It is conceivable that a sizable fraction of the total received power within the receiver beam width is due to such incoherent-scattering processes. The ratio of this contribution to the direct (but attenuated) reflection from a single particle is estimated here by means of a distorted-Born approximation to the wave equation (with an incident cw monochromatic wave) and by comparison of the magnitude of the doubly scattered to that of the singly scattered flux. The same expressions are also obtained from a radiative-transfer formalism. The ratio underestimates incoherent multiple scattering when it is not small. Corrections that are due to changes in polarization are noted.

© 1999 Optical Society of America

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  1. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 3.
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 4.
  3. M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Sec. 2.4.2.
  4. The denominator of the Lorentz–Lorenz formula for the effective refractive index reduces to a factor of 3.
  5. A. Van Lammeren, H. Russchenberg, A. Apituley, H. Ten Brink, A. Feijt, “CLARA: a data set to study sensor synergy,” presented at the Workshop on Synergy of Radar and Lidar in Space, Geesthacht, Germany, 12–14 November 1997.
  6. Y. A. Kravtsov, L. A. Apresyan, “Radiative transfer: new aspects of the old theory,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1996), Vol. 36, pp. 200–212.
  7. A. Mannoni, C. Flesia, P. Bruscaglioni, A. Ismaeli, “Multiple scattering from Chebyshev particles: Monte Carlo simulations for backscattering in lidar geometry,” Appl. Opt. 36, 7151–7164 (1996).
    [CrossRef]
  8. F. Nicolas, L. R. Bissonnette, P. H. Flamant, “Lidar effective multiple scattering coefficients in cirrus clouds,” Appl. Opt. 36, 3458–3468 (1997).
    [CrossRef] [PubMed]
  9. L. S. Xu, G. T. Zhang, L. B. Cheng, “Parameterization of the shortwave radiative properties of water clouds for use in GCMS,” Theor. Appl. Climatol. 55, 211–219 (1996).
    [CrossRef]
  10. C. Flesia, A. V. Starkov, “Multiple scattering from clear atmosphere obscured by transparent crystal clouds in satellite borne lidar sensing,” Appl. Opt. 35, 2637–2641 (1996).
    [CrossRef] [PubMed]
  11. P. Bruscaglioni, “On the contribution of double scattering to the lidar returns from clouds,” Opt. Commun. 27, 9–12 (1978).
    [CrossRef]
  12. L. R. Bissonnette, “Multiple scattering of narrow lightbeams in aerosols,” Appl. Phys. B. 60, 315–323 (1995).
    [CrossRef]
  13. P. Bruscaglioni, A. Ismaeli, G. Zaccanti, “Monte-Carlo calculations of lidar returns: procedure and results,” Appl. Phys. B. 60, 325–329 (1995).
    [CrossRef]
  14. C. Flesia, P. Schwendimann, “Analytical multiple-scattering extension of the Mie theory: the lidar equation,” Appl. Phys. B. 60, 331–334 (1995).
    [CrossRef]
  15. A. V. Starkov, M. Noormohammadian, U. G. Oppel, “A stochastic model and a variance-reduction Monte-Carlo method for the calculation of light transport,” Appl. Phys. B. 60, 335–340 (1995).
    [CrossRef]
  16. D. M. Winker, L. R. Poole, “Monte-Carlo calculations of cloud returns for ground-based and space-based radars,” Appl. Phys. B. 60, 341–344 (1995).
    [CrossRef]
  17. E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B. 60, 345–353 (1995).
    [CrossRef]
  18. L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
    [CrossRef]
  19. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2, Chap. 14.
  20. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), Chap. 6.
  21. Ref. 19, Vol. I, Chaps. 7 and 8; see also Ref. 11, pp. 381–382.
  22. P. S. Ray, “Broadband complex refractive index of ice and water,” Appl. Opt. 2, 1836–1844 (1972).
    [CrossRef]

1997 (1)

1996 (3)

A. Mannoni, C. Flesia, P. Bruscaglioni, A. Ismaeli, “Multiple scattering from Chebyshev particles: Monte Carlo simulations for backscattering in lidar geometry,” Appl. Opt. 36, 7151–7164 (1996).
[CrossRef]

L. S. Xu, G. T. Zhang, L. B. Cheng, “Parameterization of the shortwave radiative properties of water clouds for use in GCMS,” Theor. Appl. Climatol. 55, 211–219 (1996).
[CrossRef]

C. Flesia, A. V. Starkov, “Multiple scattering from clear atmosphere obscured by transparent crystal clouds in satellite borne lidar sensing,” Appl. Opt. 35, 2637–2641 (1996).
[CrossRef] [PubMed]

1995 (7)

L. R. Bissonnette, “Multiple scattering of narrow lightbeams in aerosols,” Appl. Phys. B. 60, 315–323 (1995).
[CrossRef]

P. Bruscaglioni, A. Ismaeli, G. Zaccanti, “Monte-Carlo calculations of lidar returns: procedure and results,” Appl. Phys. B. 60, 325–329 (1995).
[CrossRef]

C. Flesia, P. Schwendimann, “Analytical multiple-scattering extension of the Mie theory: the lidar equation,” Appl. Phys. B. 60, 331–334 (1995).
[CrossRef]

A. V. Starkov, M. Noormohammadian, U. G. Oppel, “A stochastic model and a variance-reduction Monte-Carlo method for the calculation of light transport,” Appl. Phys. B. 60, 335–340 (1995).
[CrossRef]

D. M. Winker, L. R. Poole, “Monte-Carlo calculations of cloud returns for ground-based and space-based radars,” Appl. Phys. B. 60, 341–344 (1995).
[CrossRef]

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B. 60, 345–353 (1995).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

1978 (1)

P. Bruscaglioni, “On the contribution of double scattering to the lidar returns from clouds,” Opt. Commun. 27, 9–12 (1978).
[CrossRef]

1972 (1)

P. S. Ray, “Broadband complex refractive index of ice and water,” Appl. Opt. 2, 1836–1844 (1972).
[CrossRef]

Apituley, A.

A. Van Lammeren, H. Russchenberg, A. Apituley, H. Ten Brink, A. Feijt, “CLARA: a data set to study sensor synergy,” presented at the Workshop on Synergy of Radar and Lidar in Space, Geesthacht, Germany, 12–14 November 1997.

Apresyan, L. A.

Y. A. Kravtsov, L. A. Apresyan, “Radiative transfer: new aspects of the old theory,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1996), Vol. 36, pp. 200–212.

Benayahu, Y.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

Bissonnette, L. R.

F. Nicolas, L. R. Bissonnette, P. H. Flamant, “Lidar effective multiple scattering coefficients in cirrus clouds,” Appl. Opt. 36, 3458–3468 (1997).
[CrossRef] [PubMed]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

L. R. Bissonnette, “Multiple scattering of narrow lightbeams in aerosols,” Appl. Phys. B. 60, 315–323 (1995).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 3.

Born, M.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Sec. 2.4.2.

Bruscaglioni, P.

A. Mannoni, C. Flesia, P. Bruscaglioni, A. Ismaeli, “Multiple scattering from Chebyshev particles: Monte Carlo simulations for backscattering in lidar geometry,” Appl. Opt. 36, 7151–7164 (1996).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

P. Bruscaglioni, A. Ismaeli, G. Zaccanti, “Monte-Carlo calculations of lidar returns: procedure and results,” Appl. Phys. B. 60, 325–329 (1995).
[CrossRef]

P. Bruscaglioni, “On the contribution of double scattering to the lidar returns from clouds,” Opt. Commun. 27, 9–12 (1978).
[CrossRef]

Cheng, L. B.

L. S. Xu, G. T. Zhang, L. B. Cheng, “Parameterization of the shortwave radiative properties of water clouds for use in GCMS,” Theor. Appl. Climatol. 55, 211–219 (1996).
[CrossRef]

Cohen, A.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

Egert, S.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

Feijt, A.

A. Van Lammeren, H. Russchenberg, A. Apituley, H. Ten Brink, A. Feijt, “CLARA: a data set to study sensor synergy,” presented at the Workshop on Synergy of Radar and Lidar in Space, Geesthacht, Germany, 12–14 November 1997.

Flamant, P. H.

Flesia, C.

C. Flesia, A. V. Starkov, “Multiple scattering from clear atmosphere obscured by transparent crystal clouds in satellite borne lidar sensing,” Appl. Opt. 35, 2637–2641 (1996).
[CrossRef] [PubMed]

A. Mannoni, C. Flesia, P. Bruscaglioni, A. Ismaeli, “Multiple scattering from Chebyshev particles: Monte Carlo simulations for backscattering in lidar geometry,” Appl. Opt. 36, 7151–7164 (1996).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

C. Flesia, P. Schwendimann, “Analytical multiple-scattering extension of the Mie theory: the lidar equation,” Appl. Phys. B. 60, 331–334 (1995).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 3.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2, Chap. 14.

Ismaeli, A.

A. Mannoni, C. Flesia, P. Bruscaglioni, A. Ismaeli, “Multiple scattering from Chebyshev particles: Monte Carlo simulations for backscattering in lidar geometry,” Appl. Opt. 36, 7151–7164 (1996).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

P. Bruscaglioni, A. Ismaeli, G. Zaccanti, “Monte-Carlo calculations of lidar returns: procedure and results,” Appl. Phys. B. 60, 325–329 (1995).
[CrossRef]

Katsev, I. L.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B. 60, 345–353 (1995).
[CrossRef]

Kleiman, M.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), Chap. 6.

Kravtsov, Y. A.

Y. A. Kravtsov, L. A. Apresyan, “Radiative transfer: new aspects of the old theory,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1996), Vol. 36, pp. 200–212.

Mannoni, A.

A. Mannoni, C. Flesia, P. Bruscaglioni, A. Ismaeli, “Multiple scattering from Chebyshev particles: Monte Carlo simulations for backscattering in lidar geometry,” Appl. Opt. 36, 7151–7164 (1996).
[CrossRef]

Nicolas, F.

Noormohammadian, M.

A. V. Starkov, M. Noormohammadian, U. G. Oppel, “A stochastic model and a variance-reduction Monte-Carlo method for the calculation of light transport,” Appl. Phys. B. 60, 335–340 (1995).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

Oppel, U. G.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

A. V. Starkov, M. Noormohammadian, U. G. Oppel, “A stochastic model and a variance-reduction Monte-Carlo method for the calculation of light transport,” Appl. Phys. B. 60, 335–340 (1995).
[CrossRef]

Polonsky, I. N.

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B. 60, 345–353 (1995).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

Poole, L. R.

D. M. Winker, L. R. Poole, “Monte-Carlo calculations of cloud returns for ground-based and space-based radars,” Appl. Phys. B. 60, 341–344 (1995).
[CrossRef]

Ray, P. S.

P. S. Ray, “Broadband complex refractive index of ice and water,” Appl. Opt. 2, 1836–1844 (1972).
[CrossRef]

Russchenberg, H.

A. Van Lammeren, H. Russchenberg, A. Apituley, H. Ten Brink, A. Feijt, “CLARA: a data set to study sensor synergy,” presented at the Workshop on Synergy of Radar and Lidar in Space, Geesthacht, Germany, 12–14 November 1997.

Schendimann, P.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

Schwendimann, P.

C. Flesia, P. Schwendimann, “Analytical multiple-scattering extension of the Mie theory: the lidar equation,” Appl. Phys. B. 60, 331–334 (1995).
[CrossRef]

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), Chap. 6.

Starkov, A. V.

C. Flesia, A. V. Starkov, “Multiple scattering from clear atmosphere obscured by transparent crystal clouds in satellite borne lidar sensing,” Appl. Opt. 35, 2637–2641 (1996).
[CrossRef] [PubMed]

A. V. Starkov, M. Noormohammadian, U. G. Oppel, “A stochastic model and a variance-reduction Monte-Carlo method for the calculation of light transport,” Appl. Phys. B. 60, 335–340 (1995).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

Ten Brink, H.

A. Van Lammeren, H. Russchenberg, A. Apituley, H. Ten Brink, A. Feijt, “CLARA: a data set to study sensor synergy,” presented at the Workshop on Synergy of Radar and Lidar in Space, Geesthacht, Germany, 12–14 November 1997.

Tsang, L.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), Chap. 6.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 4.

Van Lammeren, A.

A. Van Lammeren, H. Russchenberg, A. Apituley, H. Ten Brink, A. Feijt, “CLARA: a data set to study sensor synergy,” presented at the Workshop on Synergy of Radar and Lidar in Space, Geesthacht, Germany, 12–14 November 1997.

Winker, D. M.

D. M. Winker, L. R. Poole, “Monte-Carlo calculations of cloud returns for ground-based and space-based radars,” Appl. Phys. B. 60, 341–344 (1995).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Sec. 2.4.2.

Xu, L. S.

L. S. Xu, G. T. Zhang, L. B. Cheng, “Parameterization of the shortwave radiative properties of water clouds for use in GCMS,” Theor. Appl. Climatol. 55, 211–219 (1996).
[CrossRef]

Zaccanti, G.

P. Bruscaglioni, A. Ismaeli, G. Zaccanti, “Monte-Carlo calculations of lidar returns: procedure and results,” Appl. Phys. B. 60, 325–329 (1995).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

Zege, E. P.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B. 60, 345–353 (1995).
[CrossRef]

Zhang, G. T.

L. S. Xu, G. T. Zhang, L. B. Cheng, “Parameterization of the shortwave radiative properties of water clouds for use in GCMS,” Theor. Appl. Climatol. 55, 211–219 (1996).
[CrossRef]

Appl. Opt. (4)

A. Mannoni, C. Flesia, P. Bruscaglioni, A. Ismaeli, “Multiple scattering from Chebyshev particles: Monte Carlo simulations for backscattering in lidar geometry,” Appl. Opt. 36, 7151–7164 (1996).
[CrossRef]

P. S. Ray, “Broadband complex refractive index of ice and water,” Appl. Opt. 2, 1836–1844 (1972).
[CrossRef]

C. Flesia, A. V. Starkov, “Multiple scattering from clear atmosphere obscured by transparent crystal clouds in satellite borne lidar sensing,” Appl. Opt. 35, 2637–2641 (1996).
[CrossRef] [PubMed]

F. Nicolas, L. R. Bissonnette, P. H. Flamant, “Lidar effective multiple scattering coefficients in cirrus clouds,” Appl. Opt. 36, 3458–3468 (1997).
[CrossRef] [PubMed]

Appl. Phys. B. (7)

L. R. Bissonnette, “Multiple scattering of narrow lightbeams in aerosols,” Appl. Phys. B. 60, 315–323 (1995).
[CrossRef]

P. Bruscaglioni, A. Ismaeli, G. Zaccanti, “Monte-Carlo calculations of lidar returns: procedure and results,” Appl. Phys. B. 60, 325–329 (1995).
[CrossRef]

C. Flesia, P. Schwendimann, “Analytical multiple-scattering extension of the Mie theory: the lidar equation,” Appl. Phys. B. 60, 331–334 (1995).
[CrossRef]

A. V. Starkov, M. Noormohammadian, U. G. Oppel, “A stochastic model and a variance-reduction Monte-Carlo method for the calculation of light transport,” Appl. Phys. B. 60, 335–340 (1995).
[CrossRef]

D. M. Winker, L. R. Poole, “Monte-Carlo calculations of cloud returns for ground-based and space-based radars,” Appl. Phys. B. 60, 341–344 (1995).
[CrossRef]

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B. 60, 345–353 (1995).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60, 355–362 (1995).
[CrossRef]

Opt. Commun. (1)

P. Bruscaglioni, “On the contribution of double scattering to the lidar returns from clouds,” Opt. Commun. 27, 9–12 (1978).
[CrossRef]

Theor. Appl. Climatol. (1)

L. S. Xu, G. T. Zhang, L. B. Cheng, “Parameterization of the shortwave radiative properties of water clouds for use in GCMS,” Theor. Appl. Climatol. 55, 211–219 (1996).
[CrossRef]

Other (9)

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 3.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 4.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Sec. 2.4.2.

The denominator of the Lorentz–Lorenz formula for the effective refractive index reduces to a factor of 3.

A. Van Lammeren, H. Russchenberg, A. Apituley, H. Ten Brink, A. Feijt, “CLARA: a data set to study sensor synergy,” presented at the Workshop on Synergy of Radar and Lidar in Space, Geesthacht, Germany, 12–14 November 1997.

Y. A. Kravtsov, L. A. Apresyan, “Radiative transfer: new aspects of the old theory,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1996), Vol. 36, pp. 200–212.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2, Chap. 14.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), Chap. 6.

Ref. 19, Vol. I, Chaps. 7 and 8; see also Ref. 11, pp. 381–382.

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Figures (10)

Fig. 1
Fig. 1

Sketch of geometry with one intermediate off-axis scattering.

Fig. 2
Fig. 2

Absolute value of the Mie-scattering coefficient as a function of scattering angle at λ = 1.064 µm: (a) small angles, (b) all angles.

Fig. 3
Fig. 3

Absolute value of the Mie-scattering coefficient as a function of scattering angle at λ = 0.532 µm: (a) small angles, (b) all angles.

Fig. 4
Fig. 4

Ratio of double-scattering to single-scattering power as a function of attenuation 8.686γ(R - R 0) at λ = 1.064 µm: ground observation.

Fig. 5
Fig. 5

Ratio of double-scattering to single-scattering power as a function of attenuation 8.686γ(R - R 0) at λ = 0.532 µm: ground observation.

Fig. 6
Fig. 6

Ratio of double-scattering to single-scattering power as a function of attenuation 8.686γ(R - R 0) at f = 94 GHz: ground observation.

Fig. 7
Fig. 7

Ratio of double-scattering to single-scattering power as a function of attenuation 8.686γ(R - R 0) at λ = 1.064 µm: satellite observation.

Fig. 8
Fig. 8

Ratio of double-scattering to single-scattering power as a function of attenuation 8.686γ(R - R 0) at λ = 0.532 µm: satellite observation.

Fig. 9
Fig. 9

Wave number and field vectors with relevant geometric scattering angles.

Fig. 10
Fig. 10

Geometry relevant to the radiative-transfer calculation.

Equations (43)

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E=E0+αtα·E0+αβαtα·tβ·E0+αβαγβtα·tβ·tγ·E0+.
Esc2r=k4αβαg0r-rα·fαkˆfα, kˆαβ·g0rα-rβ·fβkˆαβ, kˆβ0·E0r.
E=E0 exp-γ|r-r0|,  γ=2π/kr0rdz 0dDn4z, DImfMie0, D.
g0r-rαgr-rαexp-γ|r-rα|g0r-rα,  g0rα-r0grα-r0exp-γ|rα-r0|g0rα-r0,
Ebs1=gr-rpf-kˆ0, kˆ0grp-rE0,
Pbs1=14πR4 f-kˆ0, kˆ02 exp-4γΔR,  ΔR|r-r0|,  kˆ0=kzˆ
Pbs1=14πR4 |fMieπ, D|2 exp-4γΔR.
Ebs2=α gr-rαfkˆα0, kˆrαgrα-rp×fkˆrα, kˆ0gRE0.
Pbs2=14π6  dvα0dDn4D×exp-2γl1+l2+R-R0l32l22R2×|fMieπ-θrα, D|2|fMieθα, D|2|fbwθ0α|2,  l1|r1-rα|,  l2|rα-r|,  l3|rα-rp|, R0|r-r0|,  R=|r-rp|.
|fbw(θ|2=exp-2.77×106θ2
Pbs2=2πn34π60π/2dθra sin θra×0lfdlaexp-2γRl1+l2+R-R0l32R2×|f¯Mieπ-θrα|2|f¯Mieθα|2|fbwθ0α|2,
Pbs2Pbs1=n3R28π0π/2d θra sin θra×0l¯fdlaexp-2γRl¯1+l¯2-1+ζ0l¯32×|f¯Mieπ-θrα|2|f¯Mieθα|2|fbwθ0α|2|fMieπ, D|2,
γ=n32πk ImfMie0, D,  ImfMie0, D=0dDn4DImfMie0, D0dDn4D.
Pbs2Pbs1=k2γR32π20π/2dθra sin θra×0l¯fdlaexp-2γRl¯1+l¯2-1+ζ0l¯32×|f¯Mieπ-θrα|2|f¯Mieθα|2|fbwθ0α|2ImfMie0|fMieπ, D|2,  sin θ0α=l¯2 sin θral¯3,  tan θα=tanθra+θ0α,  l¯3=l¯2 sin θra2+1-l¯2 cos θra21/2,  l¯1=1-ζ01-l¯2 cos θral¯3.
γ=2πk n3 ImfMie0
|f¯Mieθ|8.5×10-5|sin0.4θ+0.6||0.4θ+0.6|.
ln|f¯Mieθ|-10.2-0.0878θ+0.000618θ2-1.282×10-6θ3.
|f¯Mieθ|1.8×10-4|sin0.78θ+0.6||0.78θ+0.6|,
ln|f¯Mieθ|-9.92-0.1076θ+0.0008756θ2-2.257×10-6θ3.
E1θE1φ=fθcos θ00fθcos φsin φ-sin φcos φE0xE0y,
E1θE1φ=f1 cos θ00f1E0ρE0φ,
E2θE2φ=f2 cosπ-θ00f2cosπ/2sinπ/2-sinπ/2cosπ/2×E1xE1y.
E2θE2φ=2 cos θ00f2E1θE1φ.
E2θE2φ=-cos φ-sin φ-sin φcos φE2xE2y
E2xE2y=f1f2-cos φ-sin φ-sin φcos φcos θ001×cos θ001-cos φ-sin φ-sin φcos φE0xE0y=f1f2sin2 θ cos2 φ-cos 2φ-½1+cos2 θsin 2φ-½1+cos2 θsin 2φsin2 θ cos2 φ+cos 2φE0xE0y.
E2x=f1f2a-bE0x-cE0y,  E2y=f1f2-cE0x+a+bE0y,
|E2|2=|f1f2|2a2+b2+c2|E0x|2+|E0y|2-2ab|E0x|2-|E0y|2-4ac ReE0xE0y*.
02πdφ 0π/2dθ sin θ|E2|2=1.6167π|f1f2|2×|E0x|2+|E0y|2.
dIτ, Ωˆdτ=-Iτ, Ωˆ+1σt  d2Ω|fΩˆ, Ωˆ|2Iτ, Ω.
dIl, Ωˆdl=-n3σtIl, Ωˆ+n3  d2Ω|fΩˆ, Ωˆ|2Il, Ω.
n3σt=4πk n3 ImfΩˆ, Ωˆ=4πk  dDn4DImfΩˆ, Ω, Dˆ2γ.
dIl, Ωˆdl=-2γIl, Ωˆ+n3  d2Ω|fΩˆ, Ωˆ|2Il, Ω.
IrP, Ωˆ=exp-2γlQPIrQ, Ωˆ+n3QPdl exp-2γlP-l× d2Ω|fΩˆ, Ωˆ|2Il, Ω.
S1R= d2Ωexp-2γlQRIrQ, Ωˆ|fbwΩ|2.
S1R=exp-2γRR2  dSIrP, -kˆ0.
S1R=exp-2γRR2 f-kˆ0, kˆ02S0P.
S2R=n3  d2Ω|fbwΩ|2αPdlα×exp-2γlR-lα× d2Ω|fΩˆ, Ωˆ|2Irα, Ω.
Irα, Ωˆ=exp-2γlβαIrβ, Ωˆ+n3×γαdl exp-2γlα-l
 d2Ω|fΩˆ, Ωˆ|2 exp-2γlβαIrβ, Ω|fkˆf, kˆα|2 exp-2γlPα  d2ΩIrP, Ωˆ|fkˆf, kˆα|2exp-2γlPαlPα2  dSIrP, Ωˆ,
 dSIrP, Ωˆ=|fkˆα, kˆ0|2S0P
 d2Ω|fΩˆ, Ωˆ|2 exp-2γlβαIrβ, Ωn3exp-2γlPαlPα2 |fkˆf, kˆα|2|fkˆα, kˆ0|2S0P.
S2R=n3  d2Ω  dlαexp-2γlR-lα-2γ(lPα)lPα2×|fbwΩ|2|fkˆf, kˆα|2|fkˆα, kˆ0|2S0P.
S2R=n3  dvαexp-2γlαR+lPαlαR2lPα2×|fbwΩ|2|fkˆf, kˆα|2|fkˆα, kˆ0|2S0P.

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