Abstract

Quite recently, a semi-analytical approach to the sounding of multiply scattering media (clouds, seawaters) using multiple-field-of-view and CCD lidars with polarization devices was developed. The angular distributions of polarized components of the lidar returns from multiply scattering media computed on the basis of this theory using the small-angle approximation are presented and discussed. The semi-analytical nature of the solution makes the computation procedure faster. The obtained data are compared with results provided by the most advanced Monte Carlo algorithms for simulation of modern lidar performance. The good agreement between data provided by the semi-analytical approach and Monte Carlo computations assures one that these approaches can serve as a reliable theoretical base for interpretation and inversion of cloud lidar sounding data obtained with polarized lidars, including polarized multiple-field-of-view and CCD lidars.

© 2009 Optical Society of America

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References

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  1. G. Roy, L. R. Bissonnette, Ch. Bastille, and G. Vallee, “Retrieval of droplet-size density distribution from multiple-field-of-view cross-polarized lidar signals,” Appl. Opt. 38, 5202-5211 (1999).
    [CrossRef]
  2. N. Roy, G. Roy, L. R. Bissonnette, and J.-R. Simard, “Measurement of the azimuthal dependence of cross-polarized lidar returns and its relation to optical depth,” Appl. Opt. 43, 2777-2785 (2004).
    [CrossRef] [PubMed]
  3. N. Roy and G. Roy, “Influence of multiple scattering on lidar depolarization measurements with an ICCD camera,” in Proceedings of the Fourteenth International Workshop on Multiple Scattering Lidar Experiments (MUSCLE XIV) (Defence R&D Canada, 2006), pp. 17-26.
    [PubMed]
  4. D. C. Look, Jr., and Y. R. Chen, “Experimental demonstration of the effects of scattering on a linearly polarized laser beam by spherical particles,” Proc. SPIE 1779, 130-139 (1992).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  13. L. I. Chaikovskaya and E. P. Zege, “Theory of polarized lidar sounding including multiple scattering,” J. Quant. Spectrosc. Radiat. Transfer 88, 21-35 (2004).
    [CrossRef]
  14. U. G. Oppel, “Diffusion patterns of a pulsed laser beam seen by a monostatic and a bistatic CCD lidar,” Proc. SPIE 5829, 193-208 (2005).
    [CrossRef]
  15. U. G. Oppel and M. Wengenmayer, “A new approach to simulation of lidar multiple scattering returns and time-resolved diffusion patterns of a laser beam including polarization,” in Fourteenth International Workshop on Multiple Scattering Lidar Experiments (MUSCLE XIV) (Defence R&D Canada, 2006), pp. 53-68.
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    [CrossRef]
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  18. E. P. Zege and L. I. Chaikovskaya, “Polarization of multiply scattered lidar return from clouds and ocean water,” J. Opt. Soc. Am. A. 16, 1430-1438 (1999).
    [CrossRef]
  19. E. P. Zege and L. I. Chaikovskaya, “Approximate solutions of the polarized radiation transfer equation in media with highly anisotropic scattering,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 21, 1043-1049 (1985).
  20. E. P. Zege and L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transfer 55, 19-31 (1996).
    [CrossRef]
  21. E. P. Zege and L. I. Chaikovskaya, “Approximate theory of linearly polarized light propagation through a scattering medium,” J. Quant. Spectrosc. Radiat. Transfer 66, 413-435(2000).
    [CrossRef]
  22. L. I. Chaikovskaya, “Remote sensing of clouds using linearly and circularly polarized laser beams: techniques to compute signal polarization,” in Light Scattering Reviews 3, A. A. Kokhanovsky, ed., Springer Praxis Books (Springer, 2008), Part II, pp. 191-228.
    [CrossRef]
  23. I. L. Katsev, E. P. Zege, A. S. Prikhach, and I. N. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and oceanic sounding and imaging systems,” J. Opt. Soc. Am. A 14, 1338-1346 (1997).
    [CrossRef]
  24. U. G. Oppel, M. Hirschberger, M. Wengenmayer, N. Roy, G. Roy, L. R. Bissonnette, and J.-R. Simard, “Simulation of the azimuthal dependence of cross-polarized lidar returns and its relation to optical depth and a comparison with measurements,” in Fourteenth International Workshop on Multiple Scattering Lidar Experiments (MUSCLE XIV) (Defence R&D Canada, 2006), pp. 27-39.
  25. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
  26. S. Chandrasekhar , Radiative Transfer (Dover, 1960).
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    [CrossRef]

2005 (2)

L. I. Chaikovskaya and E. P. Zege, “Backscattering patterns in the polarized lidar sounding of strongly scattering media,” Proc. SPIE 5829, 246-254 (2005).
[CrossRef]

U. G. Oppel, “Diffusion patterns of a pulsed laser beam seen by a monostatic and a bistatic CCD lidar,” Proc. SPIE 5829, 193-208 (2005).
[CrossRef]

2004 (3)

2000 (1)

E. P. Zege and L. I. Chaikovskaya, “Approximate theory of linearly polarized light propagation through a scattering medium,” J. Quant. Spectrosc. Radiat. Transfer 66, 413-435(2000).
[CrossRef]

1999 (2)

G. Roy, L. R. Bissonnette, Ch. Bastille, and G. Vallee, “Retrieval of droplet-size density distribution from multiple-field-of-view cross-polarized lidar signals,” Appl. Opt. 38, 5202-5211 (1999).
[CrossRef]

E. P. Zege and L. I. Chaikovskaya, “Polarization of multiply scattered lidar return from clouds and ocean water,” J. Opt. Soc. Am. A. 16, 1430-1438 (1999).
[CrossRef]

1998 (3)

1997 (2)

1996 (1)

E. P. Zege and L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transfer 55, 19-31 (1996).
[CrossRef]

1993 (1)

M. Dogariu and T. Asakura, “Polarization-dependent backscattering patterns from weakly scattering media,” J. Opt. 24, 271-278 (1993).
[CrossRef]

1992 (1)

D. C. Look, Jr., and Y. R. Chen, “Experimental demonstration of the effects of scattering on a linearly polarized laser beam by spherical particles,” Proc. SPIE 1779, 130-139 (1992).
[CrossRef]

1985 (2)

S. R. Pal and A. I. Carswell, “Polarization anisotropy in lidar multiple scattering from atmospheric clouds,” Appl. Opt. 24, 3464-3471 (1985).
[CrossRef] [PubMed]

E. P. Zege and L. I. Chaikovskaya, “Approximate solutions of the polarized radiation transfer equation in media with highly anisotropic scattering,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 21, 1043-1049 (1985).

1980 (1)

Asakura, T.

M. Dogariu and T. Asakura, “Polarization-dependent backscattering patterns from weakly scattering media,” J. Opt. 24, 271-278 (1993).
[CrossRef]

Bastille, Ch.

Bigio, I. J.

Bissonnette, L. R.

N. Roy, G. Roy, L. R. Bissonnette, and J.-R. Simard, “Measurement of the azimuthal dependence of cross-polarized lidar returns and its relation to optical depth,” Appl. Opt. 43, 2777-2785 (2004).
[CrossRef] [PubMed]

G. Roy, L. R. Bissonnette, Ch. Bastille, and G. Vallee, “Retrieval of droplet-size density distribution from multiple-field-of-view cross-polarized lidar signals,” Appl. Opt. 38, 5202-5211 (1999).
[CrossRef]

U. G. Oppel, M. Hirschberger, M. Wengenmayer, N. Roy, G. Roy, L. R. Bissonnette, and J.-R. Simard, “Simulation of the azimuthal dependence of cross-polarized lidar returns and its relation to optical depth and a comparison with measurements,” in Fourteenth International Workshop on Multiple Scattering Lidar Experiments (MUSCLE XIV) (Defence R&D Canada, 2006), pp. 27-39.

Cameron, B. D.

Carswell, A. I.

Chaikovskaya, L. I.

L. I. Chaikovskaya and E. P. Zege, “Backscattering patterns in the polarized lidar sounding of strongly scattering media,” Proc. SPIE 5829, 246-254 (2005).
[CrossRef]

L. I. Chaikovskaya and E. P. Zege, “Theory of polarized lidar sounding including multiple scattering,” J. Quant. Spectrosc. Radiat. Transfer 88, 21-35 (2004).
[CrossRef]

E. P. Zege and L. I. Chaikovskaya, “Approximate theory of linearly polarized light propagation through a scattering medium,” J. Quant. Spectrosc. Radiat. Transfer 66, 413-435(2000).
[CrossRef]

E. P. Zege and L. I. Chaikovskaya, “Polarization of multiply scattered lidar return from clouds and ocean water,” J. Opt. Soc. Am. A. 16, 1430-1438 (1999).
[CrossRef]

E. P. Zege and L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transfer 55, 19-31 (1996).
[CrossRef]

E. P. Zege and L. I. Chaikovskaya, “Approximate solutions of the polarized radiation transfer equation in media with highly anisotropic scattering,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 21, 1043-1049 (1985).

L. I. Chaikovskaya and E. P. Zege, “Efficient technique to simulate angular patterns of polarized signals from multiply scattering media in MFOV and CCD lidars,” J. Quant. Spectrosc. Radiat. Transfer. (to be published).

L. I. Chaikovskaya, “Remote sensing of clouds using linearly and circularly polarized laser beams: techniques to compute signal polarization,” in Light Scattering Reviews 3, A. A. Kokhanovsky, ed., Springer Praxis Books (Springer, 2008), Part II, pp. 191-228.
[CrossRef]

Chandrasekhar, S.

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

Chen, Y. R.

D. C. Look, Jr., and Y. R. Chen, “Experimental demonstration of the effects of scattering on a linearly polarized laser beam by spherical particles,” Proc. SPIE 1779, 130-139 (1992).
[CrossRef]

Collet, C.

Cote, G. L.

Dogariu, M.

M. Dogariu and T. Asakura, “Polarization-dependent backscattering patterns from weakly scattering media,” J. Opt. 24, 271-278 (1993).
[CrossRef]

Eick, A. A.

Eloranta, E. W.

Freyer, J. P.

Hielscher, A. H.

Hirschberger, M.

U. G. Oppel, M. Hirschberger, M. Wengenmayer, N. Roy, G. Roy, L. R. Bissonnette, and J.-R. Simard, “Simulation of the azimuthal dependence of cross-polarized lidar returns and its relation to optical depth and a comparison with measurements,” in Fourteenth International Workshop on Multiple Scattering Lidar Experiments (MUSCLE XIV) (Defence R&D Canada, 2006), pp. 27-39.

Ivanov, A.

E. Zege, A. Ivanov, and I. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, 1991).
[CrossRef]

Katsev, I.

E. Zege, A. Ivanov, and I. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, 1991).
[CrossRef]

Katsev, I. L.

Kattawar, G. W.

Look, D. C.

D. C. Look, Jr., and Y. R. Chen, “Experimental demonstration of the effects of scattering on a linearly polarized laser beam by spherical particles,” Proc. SPIE 1779, 130-139 (1992).
[CrossRef]

Mehrubeoglu, M.

Mourant, J. R.

Oppel, U. G.

U. G. Oppel, “Diffusion patterns of a pulsed laser beam seen by a monostatic and a bistatic CCD lidar,” Proc. SPIE 5829, 193-208 (2005).
[CrossRef]

U. G. Oppel and M. Wengenmayer, “A new approach to simulation of lidar multiple scattering returns and time-resolved diffusion patterns of a laser beam including polarization,” in Fourteenth International Workshop on Multiple Scattering Lidar Experiments (MUSCLE XIV) (Defence R&D Canada, 2006), pp. 53-68.

U. G. Oppel, M. Hirschberger, M. Wengenmayer, N. Roy, G. Roy, L. R. Bissonnette, and J.-R. Simard, “Simulation of the azimuthal dependence of cross-polarized lidar returns and its relation to optical depth and a comparison with measurements,” in Fourteenth International Workshop on Multiple Scattering Lidar Experiments (MUSCLE XIV) (Defence R&D Canada, 2006), pp. 27-39.

Pal, S. R.

Polonsky, I. N.

Prikhach, A. S.

Rakovic, M. J.

Rastegar, S.

Roy, G.

N. Roy, G. Roy, L. R. Bissonnette, and J.-R. Simard, “Measurement of the azimuthal dependence of cross-polarized lidar returns and its relation to optical depth,” Appl. Opt. 43, 2777-2785 (2004).
[CrossRef] [PubMed]

G. Roy, L. R. Bissonnette, Ch. Bastille, and G. Vallee, “Retrieval of droplet-size density distribution from multiple-field-of-view cross-polarized lidar signals,” Appl. Opt. 38, 5202-5211 (1999).
[CrossRef]

N. Roy and G. Roy, “Influence of multiple scattering on lidar depolarization measurements with an ICCD camera,” in Proceedings of the Fourteenth International Workshop on Multiple Scattering Lidar Experiments (MUSCLE XIV) (Defence R&D Canada, 2006), pp. 17-26.
[PubMed]

U. G. Oppel, M. Hirschberger, M. Wengenmayer, N. Roy, G. Roy, L. R. Bissonnette, and J.-R. Simard, “Simulation of the azimuthal dependence of cross-polarized lidar returns and its relation to optical depth and a comparison with measurements,” in Fourteenth International Workshop on Multiple Scattering Lidar Experiments (MUSCLE XIV) (Defence R&D Canada, 2006), pp. 27-39.

Roy, N.

N. Roy, G. Roy, L. R. Bissonnette, and J.-R. Simard, “Measurement of the azimuthal dependence of cross-polarized lidar returns and its relation to optical depth,” Appl. Opt. 43, 2777-2785 (2004).
[CrossRef] [PubMed]

N. Roy and G. Roy, “Influence of multiple scattering on lidar depolarization measurements with an ICCD camera,” in Proceedings of the Fourteenth International Workshop on Multiple Scattering Lidar Experiments (MUSCLE XIV) (Defence R&D Canada, 2006), pp. 17-26.
[PubMed]

U. G. Oppel, M. Hirschberger, M. Wengenmayer, N. Roy, G. Roy, L. R. Bissonnette, and J.-R. Simard, “Simulation of the azimuthal dependence of cross-polarized lidar returns and its relation to optical depth and a comparison with measurements,” in Fourteenth International Workshop on Multiple Scattering Lidar Experiments (MUSCLE XIV) (Defence R&D Canada, 2006), pp. 27-39.

Shen, D.

Simard, J.-R.

N. Roy, G. Roy, L. R. Bissonnette, and J.-R. Simard, “Measurement of the azimuthal dependence of cross-polarized lidar returns and its relation to optical depth,” Appl. Opt. 43, 2777-2785 (2004).
[CrossRef] [PubMed]

U. G. Oppel, M. Hirschberger, M. Wengenmayer, N. Roy, G. Roy, L. R. Bissonnette, and J.-R. Simard, “Simulation of the azimuthal dependence of cross-polarized lidar returns and its relation to optical depth and a comparison with measurements,” in Fourteenth International Workshop on Multiple Scattering Lidar Experiments (MUSCLE XIV) (Defence R&D Canada, 2006), pp. 27-39.

Takakura, Y.

Vallee, G.

van de Hulst, H. C.

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

Wang, L. V.

Wengenmayer, M.

U. G. Oppel and M. Wengenmayer, “A new approach to simulation of lidar multiple scattering returns and time-resolved diffusion patterns of a laser beam including polarization,” in Fourteenth International Workshop on Multiple Scattering Lidar Experiments (MUSCLE XIV) (Defence R&D Canada, 2006), pp. 53-68.

Wengenmayer, , M.

U. G. Oppel, M. Hirschberger, M. Wengenmayer, N. Roy, G. Roy, L. R. Bissonnette, and J.-R. Simard, “Simulation of the azimuthal dependence of cross-polarized lidar returns and its relation to optical depth and a comparison with measurements,” in Fourteenth International Workshop on Multiple Scattering Lidar Experiments (MUSCLE XIV) (Defence R&D Canada, 2006), pp. 27-39.

Zallat, J.

Zege, E.

E. Zege, A. Ivanov, and I. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, 1991).
[CrossRef]

Zege, E. P.

L. I. Chaikovskaya and E. P. Zege, “Backscattering patterns in the polarized lidar sounding of strongly scattering media,” Proc. SPIE 5829, 246-254 (2005).
[CrossRef]

L. I. Chaikovskaya and E. P. Zege, “Theory of polarized lidar sounding including multiple scattering,” J. Quant. Spectrosc. Radiat. Transfer 88, 21-35 (2004).
[CrossRef]

E. P. Zege and L. I. Chaikovskaya, “Approximate theory of linearly polarized light propagation through a scattering medium,” J. Quant. Spectrosc. Radiat. Transfer 66, 413-435(2000).
[CrossRef]

E. P. Zege and L. I. Chaikovskaya, “Polarization of multiply scattered lidar return from clouds and ocean water,” J. Opt. Soc. Am. A. 16, 1430-1438 (1999).
[CrossRef]

I. L. Katsev, E. P. Zege, A. S. Prikhach, and I. N. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and oceanic sounding and imaging systems,” J. Opt. Soc. Am. A 14, 1338-1346 (1997).
[CrossRef]

E. P. Zege and L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transfer 55, 19-31 (1996).
[CrossRef]

E. P. Zege and L. I. Chaikovskaya, “Approximate solutions of the polarized radiation transfer equation in media with highly anisotropic scattering,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 21, 1043-1049 (1985).

L. I. Chaikovskaya and E. P. Zege, “Efficient technique to simulate angular patterns of polarized signals from multiply scattering media in MFOV and CCD lidars,” J. Quant. Spectrosc. Radiat. Transfer. (to be published).

Appl. Opt. (6)

Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana (1)

E. P. Zege and L. I. Chaikovskaya, “Approximate solutions of the polarized radiation transfer equation in media with highly anisotropic scattering,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 21, 1043-1049 (1985).

J. Opt. (1)

M. Dogariu and T. Asakura, “Polarization-dependent backscattering patterns from weakly scattering media,” J. Opt. 24, 271-278 (1993).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. A. (1)

E. P. Zege and L. I. Chaikovskaya, “Polarization of multiply scattered lidar return from clouds and ocean water,” J. Opt. Soc. Am. A. 16, 1430-1438 (1999).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (3)

L. I. Chaikovskaya and E. P. Zege, “Theory of polarized lidar sounding including multiple scattering,” J. Quant. Spectrosc. Radiat. Transfer 88, 21-35 (2004).
[CrossRef]

E. P. Zege and L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transfer 55, 19-31 (1996).
[CrossRef]

E. P. Zege and L. I. Chaikovskaya, “Approximate theory of linearly polarized light propagation through a scattering medium,” J. Quant. Spectrosc. Radiat. Transfer 66, 413-435(2000).
[CrossRef]

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

L. I. Chaikovskaya and E. P. Zege, “Efficient technique to simulate angular patterns of polarized signals from multiply scattering media in MFOV and CCD lidars,” J. Quant. Spectrosc. Radiat. Transfer. (to be published).

Opt. Express (2)

Opt. Lett. (1)

Proc. SPIE (3)

D. C. Look, Jr., and Y. R. Chen, “Experimental demonstration of the effects of scattering on a linearly polarized laser beam by spherical particles,” Proc. SPIE 1779, 130-139 (1992).
[CrossRef]

U. G. Oppel, “Diffusion patterns of a pulsed laser beam seen by a monostatic and a bistatic CCD lidar,” Proc. SPIE 5829, 193-208 (2005).
[CrossRef]

L. I. Chaikovskaya and E. P. Zege, “Backscattering patterns in the polarized lidar sounding of strongly scattering media,” Proc. SPIE 5829, 246-254 (2005).
[CrossRef]

Other (7)

U. G. Oppel and M. Wengenmayer, “A new approach to simulation of lidar multiple scattering returns and time-resolved diffusion patterns of a laser beam including polarization,” in Fourteenth International Workshop on Multiple Scattering Lidar Experiments (MUSCLE XIV) (Defence R&D Canada, 2006), pp. 53-68.

N. Roy and G. Roy, “Influence of multiple scattering on lidar depolarization measurements with an ICCD camera,” in Proceedings of the Fourteenth International Workshop on Multiple Scattering Lidar Experiments (MUSCLE XIV) (Defence R&D Canada, 2006), pp. 17-26.
[PubMed]

L. I. Chaikovskaya, “Remote sensing of clouds using linearly and circularly polarized laser beams: techniques to compute signal polarization,” in Light Scattering Reviews 3, A. A. Kokhanovsky, ed., Springer Praxis Books (Springer, 2008), Part II, pp. 191-228.
[CrossRef]

U. G. Oppel, M. Hirschberger, M. Wengenmayer, N. Roy, G. Roy, L. R. Bissonnette, and J.-R. Simard, “Simulation of the azimuthal dependence of cross-polarized lidar returns and its relation to optical depth and a comparison with measurements,” in Fourteenth International Workshop on Multiple Scattering Lidar Experiments (MUSCLE XIV) (Defence R&D Canada, 2006), pp. 27-39.

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

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

E. Zege, A. Ivanov, and I. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, 1991).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Lidar return and its components (a) from fog oil and (b) from water cloud versus sounding depth computed by the semi-analytical approach (solid curves) in comparison with Monte Carlo simulation (symbols). Triangles, circles, and diamonds denote the single-scattering component, multiple-scattering component, and total return, respectively. Extinction 0.1 m 1 , receiver FOV 10 mrad , distance 100 m .

Fig. 3
Fig. 3

Relative difference between Monte Carlo data and semi-analytical solution for multiple-scattered portions of returns (dashed curves) and total returns (solid curves) for (a) fog oil and (b) water cloud. Lidar parameters are the same as in Fig. 1.

Fig. 4
Fig. 4

Depolarization ratio versus sounding depth computed by the semi-analytical approach (curves) in comparison with Monte Carlo simulation (symbols) for (a) fog oil and (b) water cloud. Lidar parameters are the same as in Fig. 1.

Fig. 5
Fig. 5

Azimuth angular patterns for linearly polarized CCD return from fog oil for receiving (a) without analyzer and (b) with parallel and (c) cross analyzers computed by the semi-analytical approach (curves) in comparison with Monte Carlo simulation (symbols). Ring 1.7 2.3 mrad , distance 100 m , extinction 0.1 m 1 , range 9.5 12.5 m .

Fig. 6
Fig. 6

Same as in Fig. 4 for ring 3.7 4.3   mrad .

Fig. 7
Fig. 7

Same as in Fig. 4 except for return from water cloud for the range 0.5 3.5 m (curves are explained in text).

Fig. 8
Fig. 8

Same as in Fig. 6 for ring 3.7 4.3 mrad .

Equations (20)

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

F ^ ( x ; z ) = ( a 1 ( x ; z ) b 1 ( x ; z ) 0 0 b 1 ( x ; z ) a 2 ( x ; z ) 0 0 0 0 a 3 ( x ; z ) b 2 ( x ; z ) 0 0 b 2 ( x ; z ) a 4 ( x ; z ) )
F ^ ( x , z ) = F ^ 0 ( x , z ) + F ^ 1 ( x , z ) + F ^ 2 ( x , z ) , F ^ 0 = ( a 1 0 0 0 0 a 0 0 0 0 a 0 0 0 0 a 4 ) , F ^ 1 = ( 0 b 1 0 0 b 1 0 0 0 0 0 0 b 2 0 0 b 2 0 ) , F ^ 2 = ( 0 0 0 0 0 a + 0 0 0 0 a + 0 0 0 0 0 ) ,
W ( n r ) = A T [ L ^ ( ψ r ) B ^ ( n r ) L ^ ( ψ r ) ] P ,
L ^ ( ψ ) = ( 1 0 0 0 0 cos 2 ψ sin 2 ψ 0 0 sin 2 ψ cos 2 ψ 0 0 0 0 1 ) .
[ L ^ ( ψ r ) B ^ ( n r ) L ^ ( ψ r ) ] = B ^ 0 ( n r ) + L ^ ( ψ r ) B ^ 1 ( n r ) L ^ ( ψ r ) + B ^ 2 ( n r ) L ^ ( 2 ψ r ) , B ^ 0 = ( B 11 0 0 0 0 B 0 0 0 0 B 0 0 0 0 B 44 ) , B ^ 1 = ( 0 B 12 0 0 B 21 0 0 0 0 0 0 B 34 0 0 B 43 0 ) , B ^ 2 = ( 0 0 0 0 0 B + 0 0 0 0 B + 0 0 0 0 0 ) ,
B i k m ( n r ) = σ s ( z ) d z d n ˜ P i k m ( n ˜ ; z ) 4 π J i k eff ( z , r = 0 , n ˜ ; n r ) , m = 0 , 1 , 2 ; i k = 11 , 22 , 33 , 44 if     m = 0 , i k = 12 , 21 , 34 , 43 if     m = 1 , i k = 22 , 33 if     m = 2.
P i k m ( n ˜ ; z ) = F i k m ( | n ˜ | ; z ) cos 2 m ( ψ ˜ ψ r ) ,
J i k eff ( z , r = 0 , n ˜ ; n r ) = d r 0 d n 0 Φ src eff ( r 0 , n 0 ; n r ) G i k eff ( z , r = 0 , n ˜ ; 0 , r 0 , n 0 ) ,
Φ src eff ( r , n ; n r ) = d r d n Φ src ( 0 , r , n ) Φ rec ( 0 , r + r , n + n ; n r ) ,
D { G i k eff ( z , r , n ˜ ; 0 , r 0 , n 0 ) } = σ s eff ( z ) 4 π d n K i k ( x ; z ) G i k eff ( z , r , n ; 0 , r 0 , n 0 ) + δ ( z ) δ ( r r 0 ) δ ( n ˜ n 0 ) ,
D = n R + σ e eff ( z ) ,
σ s eff ( z ) = 2 σ s ( z ) ,
σ e eff ( z ) = 2 σ e ( z ) ,
K i i ( x ; z ) = { a i ( x ; z ) at i = 1 , 4 a + ( x ; z ) at i = 2 , 3 , K i k ( x ; z ) = { 0.5 [ a 1 ( x ; z ) + a + ( x ; z ) ] at   i k = 12 , 21 , 0.5 [ a 4 ( x ; z ) + a + ( x ; z ) ] at   i k = 43 , 34 .
W ( t = 2 z v ; n r ) = A T [ L ^ ( ψ r ) B ^ ( t = 2 z v , n r ) L ^ ( ψ r ) ] P
B i k m ( t = 2 z v , n r ) = v 2 σ s ( z ) d n ˜ P i k m ( n ˜ ; z ) 4 π , J i k eff ( z , r = 0 , n ˜ ) , m = 0 , 1 , 2 ; i k = 11 , 22 , 33 , 44 if     m = 0 , i k = 12 , 21 , 34 , 43 if     m = 1 , i k = 22 , 33 if m = 2.
r = R MC R SA R MC
W ( ϑ r , ψ r ) = B 11 ( ϑ r ) + B 12 ( ϑ r ) cos 2 ψ r ,
W ( ϑ r , ψ r ) = B ¯ ( ϑ r ) + B 2 ( ϑ r ) cos 2 ψ r + B + ( ϑ r ) cos 4 ψ r ,
W ( ϑ r , ψ r ) = B ¯ ( ϑ r ) B + ( ϑ r ) cos 4 ψ r ,

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