Abstract

A new method is proposed to analyze the effects of multiple scattering on simultaneously detected lidar returns for ground-based and space-borne lidars, and it is applied to a Monte Carlo-based simulation to test the feasibility of the new method. The experimental evidence of multiple scattering influences on both ground-based and space-borne lidar returns is presented. Monte Carlo-based evaluations of the multiple scattering parameters for the counter-looking lidar returns are separately obtained in order to correct the effective values of backscattering and extinction coefficients. Results show that for the typical cirrus cloud, the presence of the multiple scattering can lead to an underestimation of the extinction coefficient by as large as 70%, and the backscattering coefficient is overestimated by nearly 10%, which are retrieved by the Counter-propagating Elastic Signals Combination (CESC) technique in which the multiple scattering influences are neglected. Nevertheless, by the new method in which the multiple scattering effects are considered differently for the ground-based and space-borne lidar returns the extinction and backscattering coefficients can be more accurately obtained.

© 2009 Optical Society of America

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  2. D. M. Winker and L. R. Poole, "Monte-Carlo calculations of cloud returns for ground-based and space-based LIDARS," Appl. Phys. B: Lasers Opt. 60, 341-344 (1995).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  7. X. Wang, A. Boselli, L. D’Avino, R. Velotta, N. Spinelli, P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, "An algorithm to determine cirrus properties from analysis of multiple-scattering influence on lidar signals," Appl. Phys. B: Lasers Opt. 80, 609-615 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. L. R. Bissonnette, G. Roy, and N. Roy, "Multiple-scattering-based lidar retrieval: method and results of cloud probings," Appl. Opt. 44, 5565-5581 (2005).
    [CrossRef] [PubMed]
  14. L. R. Bissonnette, G. Roy, and G. Tremblay, "Lidar-Based Characterization of the Geometry and Structure of Water Clouds," J. Atmos. Oc. T. 24, 1364-1376 (2007).
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    [CrossRef]
  23. Y. Hu, Z. Liu, D. Winker, M. Vaughan, V. Noel, L. Bissonnette, G. Roy, and M. McGill, "Simple relation between lidar multiple scattering and depolarization for water clouds," Opt. Lett. 31, 1809-1811 (2006).
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  25. M. Hess, R. B. A. Koelemeijer, and P. Stammes, "Scattering matrices of imperfect hexagonal ice crystals," J. Quant. Spectrosc. Radiat. Transfer 60, 301-308 (1998), http://www.sciencedirect.com/science/article/B6TVR-3VJ3P1G-/2/07b1e14a2d9a9532a6d5c801084d6ea6.
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    [CrossRef] [PubMed]

2008 (1)

Z. Tao, M. P. McCormick, and D. Wu, "A comparison method for spaceborne and ground-based lidar and its application to the CALIPSO lidar," Appl. Phys. B: Lasers Opt. 91, 639-644 (2008).
[CrossRef]

2007 (2)

2006 (1)

2005 (2)

L. R. Bissonnette, G. Roy, and N. Roy, "Multiple-scattering-based lidar retrieval: method and results of cloud probings," Appl. Opt. 44, 5565-5581 (2005).
[CrossRef] [PubMed]

X. Wang, A. Boselli, L. D’Avino, R. Velotta, N. Spinelli, P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, "An algorithm to determine cirrus properties from analysis of multiple-scattering influence on lidar signals," Appl. Phys. B: Lasers Opt. 80, 609-615 (2005).
[CrossRef]

2001 (1)

Y.-X. Hu, D. Winker, P. Yang, B. Baum, L. Poole, and L. Vann, "Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study," J. Quant. Spectrosc. Radiat. Transfer 70, 569-579 (2001).
[CrossRef]

1998 (3)

U. Wandinger, "Multiple-Scattering Influence on Extinction-and Backscatter-Coefficient Measurements with Raman and High-Spectral-Resolution Lidars," Appl. Opt. 37, 417-427 (1998).
[CrossRef]

P. Bruscaglioni, C. Flesia, A. Ismaelli, and P. Sansoni, "Multiple scattering and lidar returns," Pure Appl. Opt: J. European Opt. Soc. A 7, 1273-1287 (1998).
[CrossRef]

M. Hess, R. B. A. Koelemeijer, and P. Stammes, "Scattering matrices of imperfect hexagonal ice crystals," J. Quant. Spectrosc. Radiat. Transfer 60, 301-308 (1998), http://www.sciencedirect.com/science/article/B6TVR-3VJ3P1G-/2/07b1e14a2d9a9532a6d5c801084d6ea6.
[CrossRef]

1995 (4)

D. M. Winker and L. R. Poole, "Monte-Carlo calculations of cloud returns for ground-based and space-based LIDARS," Appl. Phys. B: Lasers Opt. 60, 341-344 (1995).
[CrossRef]

P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, "Monte-Carlo calculations of LIDAR returns: Procedure and results," Appl. Phys. B: Lasers Opt. 60, 325-329 (1995).
[CrossRef]

L. R. Bissonnette, "Multiple scattering of narrow light beams in aerosols," Appl. Phys. B: Lasers Opt. 60, 315-323 (1995).
[CrossRef]

C. Flesia and P. Schwendimann, "Analytical multiple-scattering extension of the Mie theory: The LIDAR equation," Appl. Phys. B: Lasers Opt. 60, 331-334 (1995).
[CrossRef]

1994 (1)

1987 (1)

1984 (1)

1981 (2)

J. D. Klett, "Stable analytical inversion solution for processing lidar returns," Appl. Opt. 20, 211-220 (1981).
[CrossRef] [PubMed]

C. M. R. Platt, "Remote Sounding of High Clouds. III: Monte Carlo Calculations of Multiple-Scattered Lidar Returns," J. Atmos. Sci 38, 156-167 (1981).
[CrossRef]

1976 (1)

K. E. Kunkel and J. A. Weinman, "Monte Carlo Analysis of Multiply Scattered Lidar Returns," J. Atmos. Sci 33, 1772-1781 (1976).
[CrossRef]

1973 (1)

C. M. R. Platt, "Lidar and Radioinetric Observations of Cirrus Clouds," J. Atmos. Sci. 30, 1191-1204 (1973).
[CrossRef]

1966 (1)

Baum, B.

Y.-X. Hu, D. Winker, P. Yang, B. Baum, L. Poole, and L. Vann, "Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study," J. Quant. Spectrosc. Radiat. Transfer 70, 569-579 (2001).
[CrossRef]

Bissonnette, L.

Bissonnette, L. R.

L. R. Bissonnette, G. Roy, and G. Tremblay, "Lidar-Based Characterization of the Geometry and Structure of Water Clouds," J. Atmos. Oc. T. 24, 1364-1376 (2007).
[CrossRef]

L. R. Bissonnette, G. Roy, and N. Roy, "Multiple-scattering-based lidar retrieval: method and results of cloud probings," Appl. Opt. 44, 5565-5581 (2005).
[CrossRef] [PubMed]

L. R. Bissonnette, "Multiple scattering of narrow light beams in aerosols," Appl. Phys. B: Lasers Opt. 60, 315-323 (1995).
[CrossRef]

Boselli, A.

X. Wang, A. Boselli, L. D’Avino, R. Velotta, N. Spinelli, P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, "An algorithm to determine cirrus properties from analysis of multiple-scattering influence on lidar signals," Appl. Phys. B: Lasers Opt. 80, 609-615 (2005).
[CrossRef]

Bruscaglioni, P.

X. Wang, A. Boselli, L. D’Avino, R. Velotta, N. Spinelli, P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, "An algorithm to determine cirrus properties from analysis of multiple-scattering influence on lidar signals," Appl. Phys. B: Lasers Opt. 80, 609-615 (2005).
[CrossRef]

P. Bruscaglioni, C. Flesia, A. Ismaelli, and P. Sansoni, "Multiple scattering and lidar returns," Pure Appl. Opt: J. European Opt. Soc. A 7, 1273-1287 (1998).
[CrossRef]

P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, "Monte-Carlo calculations of LIDAR returns: Procedure and results," Appl. Phys. B: Lasers Opt. 60, 325-329 (1995).
[CrossRef]

D’Avino, L.

X. Wang, A. Boselli, L. D’Avino, R. Velotta, N. Spinelli, P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, "An algorithm to determine cirrus properties from analysis of multiple-scattering influence on lidar signals," Appl. Phys. B: Lasers Opt. 80, 609-615 (2005).
[CrossRef]

Elterman, L.

Fernald, F. G.

Flesia, C.

P. Bruscaglioni, C. Flesia, A. Ismaelli, and P. Sansoni, "Multiple scattering and lidar returns," Pure Appl. Opt: J. European Opt. Soc. A 7, 1273-1287 (1998).
[CrossRef]

C. Flesia and P. Schwendimann, "Analytical multiple-scattering extension of the Mie theory: The LIDAR equation," Appl. Phys. B: Lasers Opt. 60, 331-334 (1995).
[CrossRef]

Hess, M.

M. Hess, R. B. A. Koelemeijer, and P. Stammes, "Scattering matrices of imperfect hexagonal ice crystals," J. Quant. Spectrosc. Radiat. Transfer 60, 301-308 (1998), http://www.sciencedirect.com/science/article/B6TVR-3VJ3P1G-/2/07b1e14a2d9a9532a6d5c801084d6ea6.
[CrossRef]

M. Hess and M. Wiegner, "COP: a data library of optical properties of hexagonal ice crystals," Appl. Opt. 33, 7740-7746 (1994).
[CrossRef] [PubMed]

Hu, Y.

Hu, Y.-X.

Y.-X. Hu, D. Winker, P. Yang, B. Baum, L. Poole, and L. Vann, "Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study," J. Quant. Spectrosc. Radiat. Transfer 70, 569-579 (2001).
[CrossRef]

Ismaelli, A.

X. Wang, A. Boselli, L. D’Avino, R. Velotta, N. Spinelli, P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, "An algorithm to determine cirrus properties from analysis of multiple-scattering influence on lidar signals," Appl. Phys. B: Lasers Opt. 80, 609-615 (2005).
[CrossRef]

P. Bruscaglioni, C. Flesia, A. Ismaelli, and P. Sansoni, "Multiple scattering and lidar returns," Pure Appl. Opt: J. European Opt. Soc. A 7, 1273-1287 (1998).
[CrossRef]

P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, "Monte-Carlo calculations of LIDAR returns: Procedure and results," Appl. Phys. B: Lasers Opt. 60, 325-329 (1995).
[CrossRef]

Klett, J. D.

Koelemeijer, R. B. A.

M. Hess, R. B. A. Koelemeijer, and P. Stammes, "Scattering matrices of imperfect hexagonal ice crystals," J. Quant. Spectrosc. Radiat. Transfer 60, 301-308 (1998), http://www.sciencedirect.com/science/article/B6TVR-3VJ3P1G-/2/07b1e14a2d9a9532a6d5c801084d6ea6.
[CrossRef]

Kunkel, K. E.

K. E. Kunkel and J. A. Weinman, "Monte Carlo Analysis of Multiply Scattered Lidar Returns," J. Atmos. Sci 33, 1772-1781 (1976).
[CrossRef]

Kunz, G. J.

Liu, Z.

McCormick, M. P.

Z. Tao, M. P. McCormick, and D. Wu, "A comparison method for spaceborne and ground-based lidar and its application to the CALIPSO lidar," Appl. Phys. B: Lasers Opt. 91, 639-644 (2008).
[CrossRef]

McGill, M.

Noel, V.

Platt, C. M. R.

C. M. R. Platt, "Remote Sounding of High Clouds. III: Monte Carlo Calculations of Multiple-Scattered Lidar Returns," J. Atmos. Sci 38, 156-167 (1981).
[CrossRef]

C. M. R. Platt, "Lidar and Radioinetric Observations of Cirrus Clouds," J. Atmos. Sci. 30, 1191-1204 (1973).
[CrossRef]

Poole, L.

Y.-X. Hu, D. Winker, P. Yang, B. Baum, L. Poole, and L. Vann, "Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study," J. Quant. Spectrosc. Radiat. Transfer 70, 569-579 (2001).
[CrossRef]

Poole, L. R.

D. M. Winker and L. R. Poole, "Monte-Carlo calculations of cloud returns for ground-based and space-based LIDARS," Appl. Phys. B: Lasers Opt. 60, 341-344 (1995).
[CrossRef]

Roy, G.

Roy, N.

Sansoni, P.

P. Bruscaglioni, C. Flesia, A. Ismaelli, and P. Sansoni, "Multiple scattering and lidar returns," Pure Appl. Opt: J. European Opt. Soc. A 7, 1273-1287 (1998).
[CrossRef]

Schwendimann, P.

C. Flesia and P. Schwendimann, "Analytical multiple-scattering extension of the Mie theory: The LIDAR equation," Appl. Phys. B: Lasers Opt. 60, 331-334 (1995).
[CrossRef]

Spinelli, N.

X. Wang, A. Boselli, L. D’Avino, R. Velotta, N. Spinelli, P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, "An algorithm to determine cirrus properties from analysis of multiple-scattering influence on lidar signals," Appl. Phys. B: Lasers Opt. 80, 609-615 (2005).
[CrossRef]

Stammes, P.

M. Hess, R. B. A. Koelemeijer, and P. Stammes, "Scattering matrices of imperfect hexagonal ice crystals," J. Quant. Spectrosc. Radiat. Transfer 60, 301-308 (1998), http://www.sciencedirect.com/science/article/B6TVR-3VJ3P1G-/2/07b1e14a2d9a9532a6d5c801084d6ea6.
[CrossRef]

Tao, Z.

Z. Tao, M. P. McCormick, and D. Wu, "A comparison method for spaceborne and ground-based lidar and its application to the CALIPSO lidar," Appl. Phys. B: Lasers Opt. 91, 639-644 (2008).
[CrossRef]

Tremblay, G.

L. R. Bissonnette, G. Roy, and G. Tremblay, "Lidar-Based Characterization of the Geometry and Structure of Water Clouds," J. Atmos. Oc. T. 24, 1364-1376 (2007).
[CrossRef]

Vann, L.

Y.-X. Hu, D. Winker, P. Yang, B. Baum, L. Poole, and L. Vann, "Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study," J. Quant. Spectrosc. Radiat. Transfer 70, 569-579 (2001).
[CrossRef]

Vaughan, M.

Velotta, R.

X. Wang, A. Boselli, L. D’Avino, R. Velotta, N. Spinelli, P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, "An algorithm to determine cirrus properties from analysis of multiple-scattering influence on lidar signals," Appl. Phys. B: Lasers Opt. 80, 609-615 (2005).
[CrossRef]

Wandinger, U.

Wang, X.

X. Wang, A. Boselli, L. D’Avino, R. Velotta, N. Spinelli, P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, "An algorithm to determine cirrus properties from analysis of multiple-scattering influence on lidar signals," Appl. Phys. B: Lasers Opt. 80, 609-615 (2005).
[CrossRef]

Weinman, J. A.

K. E. Kunkel and J. A. Weinman, "Monte Carlo Analysis of Multiply Scattered Lidar Returns," J. Atmos. Sci 33, 1772-1781 (1976).
[CrossRef]

Wiegner, M.

Winker, D.

Y. Hu, Z. Liu, D. Winker, M. Vaughan, V. Noel, L. Bissonnette, G. Roy, and M. McGill, "Simple relation between lidar multiple scattering and depolarization for water clouds," Opt. Lett. 31, 1809-1811 (2006).
[CrossRef] [PubMed]

Y.-X. Hu, D. Winker, P. Yang, B. Baum, L. Poole, and L. Vann, "Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study," J. Quant. Spectrosc. Radiat. Transfer 70, 569-579 (2001).
[CrossRef]

Winker, D. M.

D. M. Winker and L. R. Poole, "Monte-Carlo calculations of cloud returns for ground-based and space-based LIDARS," Appl. Phys. B: Lasers Opt. 60, 341-344 (1995).
[CrossRef]

Wu, D.

Z. Tao, M. P. McCormick, and D. Wu, "A comparison method for spaceborne and ground-based lidar and its application to the CALIPSO lidar," Appl. Phys. B: Lasers Opt. 91, 639-644 (2008).
[CrossRef]

Yang, P.

Y.-X. Hu, D. Winker, P. Yang, B. Baum, L. Poole, and L. Vann, "Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study," J. Quant. Spectrosc. Radiat. Transfer 70, 569-579 (2001).
[CrossRef]

Zaccanti, G.

X. Wang, A. Boselli, L. D’Avino, R. Velotta, N. Spinelli, P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, "An algorithm to determine cirrus properties from analysis of multiple-scattering influence on lidar signals," Appl. Phys. B: Lasers Opt. 80, 609-615 (2005).
[CrossRef]

P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, "Monte-Carlo calculations of LIDAR returns: Procedure and results," Appl. Phys. B: Lasers Opt. 60, 325-329 (1995).
[CrossRef]

Appl. Opt. (7)

Appl. Phys. B: Lasers Opt. (6)

X. Wang, A. Boselli, L. D’Avino, R. Velotta, N. Spinelli, P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, "An algorithm to determine cirrus properties from analysis of multiple-scattering influence on lidar signals," Appl. Phys. B: Lasers Opt. 80, 609-615 (2005).
[CrossRef]

P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, "Monte-Carlo calculations of LIDAR returns: Procedure and results," Appl. Phys. B: Lasers Opt. 60, 325-329 (1995).
[CrossRef]

L. R. Bissonnette, "Multiple scattering of narrow light beams in aerosols," Appl. Phys. B: Lasers Opt. 60, 315-323 (1995).
[CrossRef]

C. Flesia and P. Schwendimann, "Analytical multiple-scattering extension of the Mie theory: The LIDAR equation," Appl. Phys. B: Lasers Opt. 60, 331-334 (1995).
[CrossRef]

D. M. Winker and L. R. Poole, "Monte-Carlo calculations of cloud returns for ground-based and space-based LIDARS," Appl. Phys. B: Lasers Opt. 60, 341-344 (1995).
[CrossRef]

Z. Tao, M. P. McCormick, and D. Wu, "A comparison method for spaceborne and ground-based lidar and its application to the CALIPSO lidar," Appl. Phys. B: Lasers Opt. 91, 639-644 (2008).
[CrossRef]

J. Atmos. Oc. T. (1)

L. R. Bissonnette, G. Roy, and G. Tremblay, "Lidar-Based Characterization of the Geometry and Structure of Water Clouds," J. Atmos. Oc. T. 24, 1364-1376 (2007).
[CrossRef]

J. Atmos. Sci (2)

C. M. R. Platt, "Remote Sounding of High Clouds. III: Monte Carlo Calculations of Multiple-Scattered Lidar Returns," J. Atmos. Sci 38, 156-167 (1981).
[CrossRef]

K. E. Kunkel and J. A. Weinman, "Monte Carlo Analysis of Multiply Scattered Lidar Returns," J. Atmos. Sci 33, 1772-1781 (1976).
[CrossRef]

J. Atmos. Sci. (1)

C. M. R. Platt, "Lidar and Radioinetric Observations of Cirrus Clouds," J. Atmos. Sci. 30, 1191-1204 (1973).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

M. Hess, R. B. A. Koelemeijer, and P. Stammes, "Scattering matrices of imperfect hexagonal ice crystals," J. Quant. Spectrosc. Radiat. Transfer 60, 301-308 (1998), http://www.sciencedirect.com/science/article/B6TVR-3VJ3P1G-/2/07b1e14a2d9a9532a6d5c801084d6ea6.
[CrossRef]

Y.-X. Hu, D. Winker, P. Yang, B. Baum, L. Poole, and L. Vann, "Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study," J. Quant. Spectrosc. Radiat. Transfer 70, 569-579 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Pure Appl. Opt: J. European Opt. Soc. A (1)

P. Bruscaglioni, C. Flesia, A. Ismaelli, and P. Sansoni, "Multiple scattering and lidar returns," Pure Appl. Opt: J. European Opt. Soc. A 7, 1273-1287 (1998).
[CrossRef]

Other (4)

D. Winker, "Accounting for Multiple Scattering in Retrievals from Space Lidar," Proc. SPIE 5059, 128-139. (12th international conference on Lidar Multiple Scattering Experiments, Oberfaffenhofen, Germany, 10-12 September 2002)
[CrossRef]

V. E. Zuev, "Laser-Light Transmission Through the Atmosphere," in Laser Monitoring of the Atmosphere, E.D.Hinkley, ed. (Springer, New York, 1976), pp. 29-69.

http://www.lrz-muenchen.de/~uh234an/www/mitarb/mhess.html.

A. Boselli, M. Armenante, L. D'Avino, G. Pisani, N. Spinelli, and X. Wang, "Characterization of atmospheric aerosol in the urban area of Napoli in the framework of EARLINET Project," in Remote Sensing of Clouds and the Atmosphere Viii(SPIE, Barcelona, Spain, 2004), pp. 643-650.

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Figures (4)

Fig. 1.
Fig. 1.

Relation between Racc and δacc for three different altitude ranges (a) from 0 to 2km, (b) from 2 to 5km, (c) from 8 to 15km. The black square represents values for water clouds, F1 is the continuous curve obtained by Eq. (3) and the red circle stands for the actual cases from the year 2006 to 2008.

Fig. 2.
Fig. 2.

The simulated RCSg and RCSS of three layers system-cirrus cloud (9-9.5km) and pure molecule Rayleigh scattering (7-9km, 9.5-10.5km), where ‘1’ and ‘t’ correspond to single and total scattering signal

Fig. 3.
Fig. 3.

The supposed, single, effective and corrected profiles of the extinction and backscattering coefficients, which are represented by ‘Supposed’, ‘Single’, ‘Effect’ and numbers, respectively. The number corresponds to the iteration step

Fig. 4.
Fig. 4.

Multiple scattering parameters FG and FS for the ground-based and space-borne lidar returns at the four iteration steps. The number corresponds to the iteration step

Tables (1)

Tables Icon

Table 1. Monte Carlo parameters used in the simulation

Equations (11)

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Racc(z)=z0zβG(z')dz'z0zβS(z')dz'
δacc(z)=z0zβS,(z')dz'z0zβS,(z')dz'
As=0.9993.906δacc+6.263δacc23.554δacc3
RCSg(z)=R0βexp(20z((1FG)αp(z')+αm(z')dz'))
RCSs(z)=A0βexp(20zs((1FS)αp(z')+αm(z'))dz')
R(z)=k'exp[40z(αp(z')+αm(z'))dz'2zszFSαp(z')dz'20zFGαp(z')dz']
αp(z)=14ddz(InR(z))αm(1FG+FS2)=αpeff(z)(1FG+FS2)
P(z)=kβ2exp(20zFGαp(z')dz'+2zzsFSαp(z')dz')
β(z)=βm(z*)[P(z)P(z*)]1/2exp(zz*(FGFS)αp(z')dz')
=βeffexp(zz*(FGFS)αp(z')dz')
Fi=ddzIn(Pi(tot)(z)Pi(1)(z))2αpeff(z)+ddzIn(Pi(tot)(z)Pi(1)(z))

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