Abstract

Lidar measurements are often interpreted on the basis of two fundamental assumptions: absence of multiple scattering and sphericity of the particles that make up the diffusing medium. There are situations in which neither holds true. We focus our interest on multiply-scattered returns from homogeneous layers of monodisperse, randomly oriented, axisymmetric nonspherical particles. T 2 Chebyshev particles have been chosen and their single-scattering properties have been reviewed. A Monte Carlo procedure has been employed to calculate the backscattered signal for several fields of view. Comparisons with the case of scattering from equivalent (equal-volume) spheres have been carried out (narrow polydispersions have been used to smooth the phase functions’ oscillations). Our numerical effort highlights a considerable variability in the intensity of the multiply-scattered signal, which is a consequence of the strong dependence of the backscattering cross section on deformation of the particles. Even more striking effects have been noted for depolarization; peculiar behavior was observed at moderate optical depths when particles characterized by a large backscattering depolarization ratio were employed in our simulations. The sensitivity of depolarization to even small departures from sphericity, in spite of random orientation of the particles, has been confirmed. The results obtained with the Monte Carlo codes have been successfully checked with an analytical formula for double scattering.

© 1996 Optical Society of America

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References

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  1. J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1981).
  2. J. D. Klett, “Lidar inversion with variable backscatter/extinction ratio,” Appl. Opt. 24, 1638–1643 (1985).
  3. L. R. Bissonnette, “Sensitivity analysis of lidar inversion algorithms,” Appl. Opt. 25, 2122–2125 (1986).
  4. D. C. Woods, “Examples of realistic aerosol particles collected in a cascade impactor,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980).
  5. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).
  6. C. F. Bohren, S. B. Singham, “Backscattering by non-spherical particles: a review of methods and suggested new approaches,” J. Geophys. Res. 96, 5269–5277 (1991).
  7. K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866 (1991).
  8. M. I. Mishchenko, J. W. Hovenier, “Depolarization of light backscattered by randomly oriented nonspherical particles,” Opt. Lett. 20, 1356–1358 (1995).
  9. R. J. Allen, C. M. R. Platt, “Lidar for multiple backscattering and depolarization observation,” Appl. Opt. 16, 3193–3199 (1977).
  10. S. R. Pal, A. I. Carswell, “Polarization anisotropy in lidar multiple scattering from atmospheric clouds,” Appl. Opt. 24, 3464–3471 (1985).
  11. C. Werner, J. Streicher, H. Herrmann, H.-G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).
  12. L. R. Bissonnette, D. L. Hutt, “Multiply scattered aerosol lidar returns: inversion method and comparison with in situ measurements,” Appl. Opt. 34, 6959–6975 (1995).
  13. C. Flesia, P. Schwendimann, “Analytical multiple-scattering extension of the Mie theory: the LIDAR equation,” Appl. Phys. B 60, 331–334 (1995).
  14. 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).
  15. P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Monte-Carlo calculations of LIDAR returns: procedure and results,” Appl. Phys. B 60, 325–329 (1995).
  16. A. V. Starkov, M. Noormohammadian, U. G. Oppel, “A stochastic model for a variance-reduction Monte-Carlo method for the calculation of light transport,” Appl. Phys. B 60, 335–340 (1995).
  17. D. M. Winker, L. R. Poole, “Monte-Carlo calculations of cloud returns for ground-based and space-based LIDARS,” Appl. Phys. B 60, 341–344 (1995).
  18. A. Mugnai, W. J. Wiscombe, “Scattering from nonspherical Chebyshev particles. 1: Cross sections, single-scattering albedo, asymmetry factor, and backscattered fraction,” Appl. Opt. 25, 1235–1244 (1986).
  19. W. J. Wiscombe, A. Mugnai, “Scattering from nonspherical Chebyshev particles. 2: Means of angular scattering patterns,” Appl. Opt. 27, 2405–2421 (1988).
  20. A. Mugnai, W. J. Wiscombe, “Scattering from nonspherical Chebyshev particles. 3: Variability in angular scattering patterns,” Appl. Opt. 28, 3061–3073 (1989).
  21. W. J. Wiscombe, A. Mugnai, “Single scattering from non-spherical Chebyshev particles: a compendium of calculations,” (NASA Goddard Space Flight Center, Greenbelt, Md., 1986).
  22. P. C. Waterman, “Matrix formulation for electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
  23. P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
  24. P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric particles,” Appl. Opt. 14, 2864–2872 (1975).
  25. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  26. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1968).
  27. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  28. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
  29. P. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).
  30. P. Bruscaglioni, G. Zaccanti, “Multiple scattering in dense media,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, New York, 1990).
  31. P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Simple scaling relationships for calculation of lidar returns from turbid media in multiple scattering regime,” J. Mod. Opt. 39, 1003–1015 (1992).
  32. P. Bruscaglioni, “On the contribution of double scattering to the lidar returns from clouds,” Opt. Commun. 27, 9–12 (1978).

1995 (7)

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

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).

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

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

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

M. I. Mishchenko, J. W. Hovenier, “Depolarization of light backscattered by randomly oriented nonspherical particles,” Opt. Lett. 20, 1356–1358 (1995).

L. R. Bissonnette, D. L. Hutt, “Multiply scattered aerosol lidar returns: inversion method and comparison with in situ measurements,” Appl. Opt. 34, 6959–6975 (1995).

1992 (2)

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Simple scaling relationships for calculation of lidar returns from turbid media in multiple scattering regime,” J. Mod. Opt. 39, 1003–1015 (1992).

C. Werner, J. Streicher, H. Herrmann, H.-G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).

1991 (2)

C. F. Bohren, S. B. Singham, “Backscattering by non-spherical particles: a review of methods and suggested new approaches,” J. Geophys. Res. 96, 5269–5277 (1991).

K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866 (1991).

1989 (1)

1988 (1)

1986 (2)

A. Mugnai, W. J. Wiscombe, “Scattering from nonspherical Chebyshev particles. 1: Cross sections, single-scattering albedo, asymmetry factor, and backscattered fraction,” Appl. Opt. 25, 1235–1244 (1986).

L. R. Bissonnette, “Sensitivity analysis of lidar inversion algorithms,” Appl. Opt. 25, 2122–2125 (1986).

1985 (2)

S. R. Pal, A. I. Carswell, “Polarization anisotropy in lidar multiple scattering from atmospheric clouds,” Appl. Opt. 24, 3464–3471 (1985).

J. D. Klett, “Lidar inversion with variable backscatter/extinction ratio,” Appl. Opt. 24, 1638–1643 (1985).

1981 (1)

J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1981).

1978 (1)

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

1977 (1)

R. J. Allen, C. M. R. Platt, “Lidar for multiple backscattering and depolarization observation,” Appl. Opt. 16, 3193–3199 (1977).

1975 (2)

1971 (1)

P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).

1965 (1)

P. C. Waterman, “Matrix formulation for electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).

Allen, R. J.

R. J. Allen, C. M. R. Platt, “Lidar for multiple backscattering and depolarization observation,” Appl. Opt. 16, 3193–3199 (1977).

Asano, S.

Barber, P.

P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric particles,” Appl. Opt. 14, 2864–2872 (1975).

P. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Bissonnette, L. R.

Bohren, C. F.

C. F. Bohren, S. B. Singham, “Backscattering by non-spherical particles: a review of methods and suggested new approaches,” J. Geophys. Res. 96, 5269–5277 (1991).

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

Bruscaglioni, P.

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

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Simple scaling relationships for calculation of lidar returns from turbid media in multiple scattering regime,” J. Mod. Opt. 39, 1003–1015 (1992).

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

P. Bruscaglioni, G. Zaccanti, “Multiple scattering in dense media,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, New York, 1990).

Carswell, A. I.

Dahn, H.-G.

C. Werner, J. Streicher, H. Herrmann, H.-G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1968).

Flesia, C.

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

Herrmann, H.

C. Werner, J. Streicher, H. Herrmann, H.-G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).

Hill, S. C.

P. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Hovenier, J. W.

Huffman, D. R.

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

Hutt, D. L.

Ishimaru, A.

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

Ismaelli, A.

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

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Simple scaling relationships for calculation of lidar returns from turbid media in multiple scattering regime,” J. Mod. Opt. 39, 1003–1015 (1992).

Katsev, I. L.

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).

Klett, J. D.

J. D. Klett, “Lidar inversion with variable backscatter/extinction ratio,” Appl. Opt. 24, 1638–1643 (1985).

J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1981).

Mishchenko, M. I.

Mugnai, A.

A. Mugnai, W. J. Wiscombe, “Scattering from nonspherical Chebyshev particles. 3: Variability in angular scattering patterns,” Appl. Opt. 28, 3061–3073 (1989).

W. J. Wiscombe, A. Mugnai, “Scattering from nonspherical Chebyshev particles. 2: Means of angular scattering patterns,” Appl. Opt. 27, 2405–2421 (1988).

A. Mugnai, W. J. Wiscombe, “Scattering from nonspherical Chebyshev particles. 1: Cross sections, single-scattering albedo, asymmetry factor, and backscattered fraction,” Appl. Opt. 25, 1235–1244 (1986).

W. J. Wiscombe, A. Mugnai, “Single scattering from non-spherical Chebyshev particles: a compendium of calculations,” (NASA Goddard Space Flight Center, Greenbelt, Md., 1986).

Noormohammadian, M.

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

Oppel, U. G.

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

Pal, S. R.

Platt, C. M. R.

R. J. Allen, C. M. R. Platt, “Lidar for multiple backscattering and depolarization observation,” Appl. Opt. 16, 3193–3199 (1977).

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).

Poole, L. R.

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

Sassen, K.

K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866 (1991).

Schwendimann, P.

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

Singham, S. B.

C. F. Bohren, S. B. Singham, “Backscattering by non-spherical particles: a review of methods and suggested new approaches,” J. Geophys. Res. 96, 5269–5277 (1991).

Starkov, A. V.

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

Streicher, J.

C. Werner, J. Streicher, H. Herrmann, H.-G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).

van de Hulst, H. C.

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

Waterman, P. C.

P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).

P. C. Waterman, “Matrix formulation for electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).

Werner, C.

C. Werner, J. Streicher, H. Herrmann, H.-G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).

Winker, D. M.

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

Wiscombe, W. J.

A. Mugnai, W. J. Wiscombe, “Scattering from nonspherical Chebyshev particles. 3: Variability in angular scattering patterns,” Appl. Opt. 28, 3061–3073 (1989).

W. J. Wiscombe, A. Mugnai, “Scattering from nonspherical Chebyshev particles. 2: Means of angular scattering patterns,” Appl. Opt. 27, 2405–2421 (1988).

A. Mugnai, W. J. Wiscombe, “Scattering from nonspherical Chebyshev particles. 1: Cross sections, single-scattering albedo, asymmetry factor, and backscattered fraction,” Appl. Opt. 25, 1235–1244 (1986).

W. J. Wiscombe, A. Mugnai, “Single scattering from non-spherical Chebyshev particles: a compendium of calculations,” (NASA Goddard Space Flight Center, Greenbelt, Md., 1986).

Woods, D. C.

D. C. Woods, “Examples of realistic aerosol particles collected in a cascade impactor,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980).

Yamamoto, G.

Yeh, C.

Zaccanti, G.

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

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Simple scaling relationships for calculation of lidar returns from turbid media in multiple scattering regime,” J. Mod. Opt. 39, 1003–1015 (1992).

P. Bruscaglioni, G. Zaccanti, “Multiple scattering in dense media,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, New York, 1990).

Zege, E. P.

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).

Appl. Opt. (4)

J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1981).

J. D. Klett, “Lidar inversion with variable backscatter/extinction ratio,” Appl. Opt. 24, 1638–1643 (1985).

R. J. Allen, C. M. R. Platt, “Lidar for multiple backscattering and depolarization observation,” Appl. Opt. 16, 3193–3199 (1977).

A. Mugnai, W. J. Wiscombe, “Scattering from nonspherical Chebyshev particles. 1: Cross sections, single-scattering albedo, asymmetry factor, and backscattered fraction,” Appl. Opt. 25, 1235–1244 (1986).

Appl. Opt. (7)

Appl. Phys. B (1)

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

Appl. Phys. B (4)

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

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

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

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).

Bull. Am. Meteorol. Soc. (1)

K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866 (1991).

J. Geophys. Res. (1)

C. F. Bohren, S. B. Singham, “Backscattering by non-spherical particles: a review of methods and suggested new approaches,” J. Geophys. Res. 96, 5269–5277 (1991).

J. Mod. Opt. (1)

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Simple scaling relationships for calculation of lidar returns from turbid media in multiple scattering regime,” J. Mod. Opt. 39, 1003–1015 (1992).

Opt. Commun. (1)

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

Opt. Eng. (1)

C. Werner, J. Streicher, H. Herrmann, H.-G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).

Opt. Lett. (1)

Phys. Rev. D (1)

P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).

Proc. IEEE (1)

P. C. Waterman, “Matrix formulation for electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).

Other (8)

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

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1968).

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

P. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

P. Bruscaglioni, G. Zaccanti, “Multiple scattering in dense media,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, New York, 1990).

W. J. Wiscombe, A. Mugnai, “Single scattering from non-spherical Chebyshev particles: a compendium of calculations,” (NASA Goddard Space Flight Center, Greenbelt, Md., 1986).

D. C. Woods, “Examples of realistic aerosol particles collected in a cascade impactor,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980).

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

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Figures (9)

Fig. 1
Fig. 1

Shape of three T 2 Chebyshev particles with deformation parameters of (a) 0.05, (b) 0.15, (c) 0.25.

Fig. 2
Fig. 2

Phase functions of T 2 Chebyshev particles with size parameters of (a), (b) 6; (c), (d) 12; (e) 24; (f) 36. In (a), (c), (e), and (f) the phase function of a Gaussian polydispersion centered on the equal-volume sphere is drawn as a reference. Data for the 0–30-deg angular range, which is only slightly dependent on the deformation, are not shown.

Fig. 3
Fig. 3

Backscattering depolarization ratio δ L (π) [see Eq. (4) for a definition of this quantity] of T 2 Chebyshev particles with size parameters of (a) 6, (b) 12, (c) 24, (d) 36. Both random-2d and random-3d orientations are considered.

Fig. 4
Fig. 4

Total detected intensity as a function of photon path length inside the cloud. Data are reported for two different values of σext: (a), (c), (e) 3.33 × 10−3 m−1 (b), (d), (f) 0.02 m−1, corresponding to total optical thicknesses of 1 and 6, respectively. Fifteen scattering orders were taken into account in the Monte Carlo simulations. We used 15-mrad FOV for all the graphs.

Fig. 5
Fig. 5

Multiple-scattering depolarization curves for two Chebyshev particles with only (a) 2% and (b) 1% deformation compared with those of equal-volume polydisperse spheres. The number of scattering orders calculated and the FOV are as in Fig. 4. The extinction coefficient is 0.02 m−1.

Fig. 6
Fig. 6

Multiple-scattering depolarization of the detected signal as a function of photon path length inside the cloud. The extinction coefficient and the FOV are as in Fig. 5. All the curves for Chebyshev particles are shown as solid curves except where otherwise stated: (a) x = 6, random-3d orientation, 0.03–0.17 deformations (step 0.03) bottom to top; (b) x = 6, random-2d orientation, 0.08–0.16 deformations (step 0.02) bottom to top, 0.18–0.24 (step 0.02), dotted curves top to bottom; (c) x = 12, random-3d orientation, 0.05–0.11 deformations bottom to top; (d) x = 12, random-3d orientation, 0.01–0.05 deformations bottom to top. The data for spheres (dashed curves) are also shown for comparison.

Fig. 7
Fig. 7

Dependence of depolarization on the FOV of the receiver for two sample particles giving rise to the (a) anomalous and (b) spherelike behaviors described in the text. The extinction coefficient is as in Fig. 5. Deformation parameter (a) 0.11, (b) 0.05.

Fig. 8
Fig. 8

Schematic representation of the geometry of a generic double-scattering event. The analytical method for calculating the doubly-scattered received intensity described in Appendix A consists in one summing all the contributions to the double-scattering signal by integrating in α′ from 0 to α and in dx from cloud bottom to cloud top.

Fig. 9
Fig. 9

Results of Monte Carlo simulations relative to the first two scattering orders compared with the outcome of the analytical procedure for particles with (a) large or (b) small backscattering depolarization ratio. The extinction coefficient and FOV are as in Fig. 5.

Tables (1)

Tables Icon

Table 1 Asymmetry Factors for Spheres and a T 2 Particlesa

Equations (17)

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

r ( ϑ ) = r 0 [ 1 + ɛ T 2 ( ϑ ) ] = r 0 [ 1 + ɛ cos ( 2 ϑ ) ] ,
P ( ϑ ) = i + i 2 k 2 σ s ,
g = cos ( ϑ ) = 2 π 0 π P ( ϑ ) cos ( ϑ ) sin ( ϑ ) d ϑ ,
δ L ( π ) = I r ( π ) I r ( π ) + I l ( π ) ,
Q b ( π ) = P ( π ) π r 2 .
x = 2 π r eq λ = 2 π λ [ 3 V n ( ɛ ) 4 π ] 1 / 3 .
P ( ϑ ) = a 11 + 2 a 12 + a 22 2 k 2 = S 11 k 2 .
δ m s = I r I l + I r ,
S 1 ( r , α ) = σ c 2 r 2 exp ( - 2 σ d ) M ( π ) I 0 ,
S 2 ( r , α ) = σ 2 c exp ( - 2 σ d ) 0 α α d α 0 2 π d φ H b F ( x , t , α ) R ( φ ) M ( ϑ 2 ) R ( φ ) M ( ϑ 1 ) I 0 d x ,
S 1 , l ( r , α ) = σ 2 2 r 2 exp ( - 2 σ d ) a 22 ( π ) ,
S 1 , r ( r , α ) = σ 2 2 r 2 exp ( - 2 σ d ) a 12 ( π )
S 21 ( r , α ) = σ 2 c exp ( - 2 σ d ) 0 α α d α × H b F ( t , x , α ) G 1 ( ϑ 1 , ϑ 2 ) d x ,
S 2 r ( r , α ) = σ 2 c exp ( - 2 σ d ) 0 α α d α × H b F ( t , x , α ) G 2 ( ϑ 1 , ϑ 2 ) d x ,
G 1 ( ϑ 1 , ϑ 2 ) = a 12 ( ϑ 1 ) a 11 ( ϑ 2 ) + a 22 ( ϑ 1 ) a 12 ( ϑ 2 ) + a 11 ( ϑ 1 ) a 12 ( ϑ 2 ) + a 12 ( ϑ 1 ) a 22 ( ϑ 2 ) + 2 a 34 ( ϑ 1 ) a 34 ( ϑ 2 ) - 2 a 33 ( ϑ 1 ) a 33 ( ϑ 2 ) + 3 [ 2 a 12 ( ϑ 1 ) a 12 ( ϑ 2 ) + a 22 ( ϑ 1 ) a 22 ( ϑ 2 ) + a 11 ( ϑ 1 ) a 11 ( ϑ 2 ) ] ,
G 2 ( ϑ 1 , ϑ 2 ) = a 11 ( ϑ 1 ) a 11 ( ϑ 2 ) + 2 a 12 ( ϑ 1 ) a 12 ( ϑ 2 ) + a 22 ( ϑ 1 ) a 22 ( ϑ 2 ) + 2 a 33 ( ϑ 1 ) a 33 ( ϑ 2 ) - 2 a 34 ( ϑ 1 ) a 34 ( ϑ 2 ) + 3 [ a 12 ( ϑ 1 ) a 11 ( ϑ 2 ) + a 22 ( ϑ 1 ) a 12 ( ϑ 2 ) + a 11 ( ϑ 1 ) a 12 ( ϑ 2 ) + a 12 ( ϑ 1 ) a 22 ( ϑ 2 ) ] .
δ = S 1 , r + S 2 , r S 1 , l + S 1 , r + S 2 , l + S 2 , r .

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