Abstract

A formalism describing the influence of multiple scattering on cloud measurements with Raman and high-spectral-resolution lidars is presented. Model calculations including both particulate and molecular scattering processes are performed to describe the general effects of multiple scattering on both particulate and molecular lidar backscatter signals. It is found that, for typical measurement geometries of ground-based lidars, as many as five scattering orders contribute significantly to the backscattered light. The relative intensity of multiple-scattered light is generally larger in signals backscattered from molecules than in signals backscattered from particles. The multiple-scattering formalism is applied to measurements of water and ice clouds taken with a Raman lidar. Multiple-scattering errors of measured extinction coefficients are typically of the order of 50% at the bases of both water and ice clouds and decrease with increasing penetration depth to below 20%. In contrast, the multiple-scattering errors of backscatter coefficients are negligible in ice clouds and below 20% in water clouds.

© 1998 Optical Society of America

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    [CrossRef]
  2. A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, W. Michaelis, “Independent measurement of extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar,” Appl. Opt. 31, 7113–7131 (1992).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  18. P. Bruscaglioni, A. Ismaeli, G. Zaccanti, “Monte Carlo calculations of LIDAR returns: procedure and results,” Appl. Phys. B 60, 325–330 (1995).
    [CrossRef]
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    [CrossRef]
  20. 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]
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    [CrossRef]
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    [CrossRef] [PubMed]
  27. S. R. Pal, A. I. Carswell, “Multiple scattering in atmospheric clouds: lidar observations,” Appl. Opt. 15, 1990–1995 (1976).
    [CrossRef] [PubMed]
  28. C. M. R. Platt, “Lidar observations of a mixed-phase altostratus cloud,” J. Appl. Meteorol. 16, 339–345 (1977).
    [CrossRef]
  29. R. J. Allen, C. M. R. Platt, “Lidar for multiple backscattering and depolarization observations,” Appl. Opt. 16, 3193–3199 (1977).
    [CrossRef] [PubMed]
  30. A. Cohen, M. Kleiman, J. Cooney, “Lidar measurements of rotational Raman and double scattering,” Appl. Opt. 16, 1905–1910 (1978).
    [CrossRef]
  31. S. Egert, A. Cohen, M. Kleiman, N. Ben-Yosef, “Instantaneous integrated Raman scattering,” Appl. Opt. 22, 1592–1597 (1983).
    [CrossRef] [PubMed]

1995 (8)

L. R. Bissonnette, “Multiple scattering of narrow light beams in aerosols,” Appl. Phys. B 60, 315–324 (1995).
[CrossRef]

P. Bruscaglioni, A. Ismaeli, G. Zaccanti, “Monte Carlo calculations of LIDAR returns: procedure and results,” Appl. Phys. B 60, 325–330 (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 LIDARS,” 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, 345–354 (1995).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, 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]

U. Wandinger, A. Ansmann, J. Reichardt, T. Deshler, “Determination of stratospheric-aerosol microphysical properties from independent extinction and backscattering measurements with a Raman lidar,” Appl. Opt. 34, 8315–8329 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

A. Ansmann, J. Bösenberg, G. Brogniez, S. Elouragini, P. H. Flamant, K. H. Klapheck, H. Linné, L. Menenger, W. Michaelis, M. Riebesell, C. Senff, P.-Y. Thro, U. Wandinger, C. Weitkamp, “Lidar network observation of cirrus morphological and scattering properties during the International Cirrus Experiment 1989: The 18 October 1989 case study and statistical analysis,” J. Appl. Meteorol. 32, 1608–1622 (1993).
[CrossRef]

1992 (2)

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, W. Michaelis, “Independent measurement of extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar,” Appl. Opt. 31, 7113–7131 (1992).
[CrossRef] [PubMed]

1991 (1)

C. J. Grund, E. W. Eloranta, “University of Wisconsin high spectral resolution lidar,” Opt. Eng. 30, 6–12 (1991).
[CrossRef]

1990 (1)

1988 (1)

1985 (1)

1983 (2)

1981 (1)

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]

1978 (1)

1977 (2)

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

C. M. R. Platt, “Lidar observations of a mixed-phase altostratus cloud,” J. Appl. Meteorol. 16, 339–345 (1977).
[CrossRef]

1976 (3)

J. A. Weinman, “Effects of multiple scattering on light pulses reflected by turbid atmospheres,” J. Atmos. Sci. 33, 1763–1771 (1976).
[CrossRef]

K. E. Kunkel, J. A. Weinman, “Monte Carlo analysis of multiply scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
[CrossRef]

S. R. Pal, A. I. Carswell, “Multiple scattering in atmospheric clouds: lidar observations,” Appl. Opt. 15, 1990–1995 (1976).
[CrossRef] [PubMed]

1972 (1)

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 77, 7123–7128 (1972).
[CrossRef]

1971 (2)

K.-N. Liou, R. M. Schotland, “Multiple backscattering and depolarization from water clouds for a pulsed lidar system,” J. Atmos. Sci. 28, 772–784 (1971).
[CrossRef]

G. N. Plass, G. W. Kattawar, “Reflection of light pulses from clouds,” Appl. Opt. 10, 2304–2310 (1971).
[CrossRef] [PubMed]

Allen, R. J.

Ansmann, A.

U. Wandinger, A. Ansmann, J. Reichardt, T. Deshler, “Determination of stratospheric-aerosol microphysical properties from independent extinction and backscattering measurements with a Raman lidar,” Appl. Opt. 34, 8315–8329 (1995).
[CrossRef] [PubMed]

A. Ansmann, J. Bösenberg, G. Brogniez, S. Elouragini, P. H. Flamant, K. H. Klapheck, H. Linné, L. Menenger, W. Michaelis, M. Riebesell, C. Senff, P.-Y. Thro, U. Wandinger, C. Weitkamp, “Lidar network observation of cirrus morphological and scattering properties during the International Cirrus Experiment 1989: The 18 October 1989 case study and statistical analysis,” J. Appl. Meteorol. 32, 1608–1622 (1993).
[CrossRef]

A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, W. Michaelis, “Independent measurement of extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar,” Appl. Opt. 31, 7113–7131 (1992).
[CrossRef] [PubMed]

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

A. Ansmann, M. Riebesell, C. Weitkamp, “Measurement of atmospheric aerosol extinction profiles with a Raman lidar,” Opt. Lett. 15, 746–748 (1990).
[CrossRef] [PubMed]

Benayahu, Y.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, 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]

Ben-Yosef, N.

Bissonnette, L. R.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, 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 light beams in aerosols,” Appl. Phys. B 60, 315–324 (1995).
[CrossRef]

L. R. Bissonnette, “Multiscattering model for propagation of narrow light beams in aerosol media,” Appl. Opt. 27, 2478–2484 (1988).
[CrossRef] [PubMed]

Bösenberg, J.

A. Ansmann, J. Bösenberg, G. Brogniez, S. Elouragini, P. H. Flamant, K. H. Klapheck, H. Linné, L. Menenger, W. Michaelis, M. Riebesell, C. Senff, P.-Y. Thro, U. Wandinger, C. Weitkamp, “Lidar network observation of cirrus morphological and scattering properties during the International Cirrus Experiment 1989: The 18 October 1989 case study and statistical analysis,” J. Appl. Meteorol. 32, 1608–1622 (1993).
[CrossRef]

Brogniez, G.

A. Ansmann, J. Bösenberg, G. Brogniez, S. Elouragini, P. H. Flamant, K. H. Klapheck, H. Linné, L. Menenger, W. Michaelis, M. Riebesell, C. Senff, P.-Y. Thro, U. Wandinger, C. Weitkamp, “Lidar network observation of cirrus morphological and scattering properties during the International Cirrus Experiment 1989: The 18 October 1989 case study and statistical analysis,” J. Appl. Meteorol. 32, 1608–1622 (1993).
[CrossRef]

Bruscaglioni, P.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, 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, 325–330 (1995).
[CrossRef]

Carswell, A. I.

Cohen, A.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, 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]

S. Egert, A. Cohen, M. Kleiman, N. Ben-Yosef, “Instantaneous integrated Raman scattering,” Appl. Opt. 22, 1592–1597 (1983).
[CrossRef] [PubMed]

A. Cohen, M. Kleiman, J. Cooney, “Lidar measurements of rotational Raman and double scattering,” Appl. Opt. 16, 1905–1910 (1978).
[CrossRef]

Cooney, J.

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969), p. 78.

Deshler, T.

Egert, S.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, 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]

S. Egert, A. Cohen, M. Kleiman, N. Ben-Yosef, “Instantaneous integrated Raman scattering,” Appl. Opt. 22, 1592–1597 (1983).
[CrossRef] [PubMed]

Eloranta, E. W.

Elouragini, S.

A. Ansmann, J. Bösenberg, G. Brogniez, S. Elouragini, P. H. Flamant, K. H. Klapheck, H. Linné, L. Menenger, W. Michaelis, M. Riebesell, C. Senff, P.-Y. Thro, U. Wandinger, C. Weitkamp, “Lidar network observation of cirrus morphological and scattering properties during the International Cirrus Experiment 1989: The 18 October 1989 case study and statistical analysis,” J. Appl. Meteorol. 32, 1608–1622 (1993).
[CrossRef]

Flamant, P. H.

A. Ansmann, J. Bösenberg, G. Brogniez, S. Elouragini, P. H. Flamant, K. H. Klapheck, H. Linné, L. Menenger, W. Michaelis, M. Riebesell, C. Senff, P.-Y. Thro, U. Wandinger, C. Weitkamp, “Lidar network observation of cirrus morphological and scattering properties during the International Cirrus Experiment 1989: The 18 October 1989 case study and statistical analysis,” J. Appl. Meteorol. 32, 1608–1622 (1993).
[CrossRef]

Flesia, C.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, 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]

Grund, C. J.

C. J. Grund, E. W. Eloranta, “University of Wisconsin high spectral resolution lidar,” Opt. Eng. 30, 6–12 (1991).
[CrossRef]

Ismaeli, A.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, 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, 325–330 (1995).
[CrossRef]

Jayaweera, K.

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, 345–354 (1995).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, 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]

Kattawar, G. W.

Klapheck, K. H.

A. Ansmann, J. Bösenberg, G. Brogniez, S. Elouragini, P. H. Flamant, K. H. Klapheck, H. Linné, L. Menenger, W. Michaelis, M. Riebesell, C. Senff, P.-Y. Thro, U. Wandinger, C. Weitkamp, “Lidar network observation of cirrus morphological and scattering properties during the International Cirrus Experiment 1989: The 18 October 1989 case study and statistical analysis,” J. Appl. Meteorol. 32, 1608–1622 (1993).
[CrossRef]

Kleiman, M.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, 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]

S. Egert, A. Cohen, M. Kleiman, N. Ben-Yosef, “Instantaneous integrated Raman scattering,” Appl. Opt. 22, 1592–1597 (1983).
[CrossRef] [PubMed]

A. Cohen, M. Kleiman, J. Cooney, “Lidar measurements of rotational Raman and double scattering,” Appl. Opt. 16, 1905–1910 (1978).
[CrossRef]

Kunkel, K. E.

K. E. Kunkel, J. A. Weinman, “Monte Carlo analysis of multiply scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
[CrossRef]

Lahmann, W.

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

Linné, H.

A. Ansmann, J. Bösenberg, G. Brogniez, S. Elouragini, P. H. Flamant, K. H. Klapheck, H. Linné, L. Menenger, W. Michaelis, M. Riebesell, C. Senff, P.-Y. Thro, U. Wandinger, C. Weitkamp, “Lidar network observation of cirrus morphological and scattering properties during the International Cirrus Experiment 1989: The 18 October 1989 case study and statistical analysis,” J. Appl. Meteorol. 32, 1608–1622 (1993).
[CrossRef]

Liou, K.-N.

K.-N. Liou, R. M. Schotland, “Multiple backscattering and depolarization from water clouds for a pulsed lidar system,” J. Atmos. Sci. 28, 772–784 (1971).
[CrossRef]

Menenger, L.

A. Ansmann, J. Bösenberg, G. Brogniez, S. Elouragini, P. H. Flamant, K. H. Klapheck, H. Linné, L. Menenger, W. Michaelis, M. Riebesell, C. Senff, P.-Y. Thro, U. Wandinger, C. Weitkamp, “Lidar network observation of cirrus morphological and scattering properties during the International Cirrus Experiment 1989: The 18 October 1989 case study and statistical analysis,” J. Appl. Meteorol. 32, 1608–1622 (1993).
[CrossRef]

Michaelis, W.

A. Ansmann, J. Bösenberg, G. Brogniez, S. Elouragini, P. H. Flamant, K. H. Klapheck, H. Linné, L. Menenger, W. Michaelis, M. Riebesell, C. Senff, P.-Y. Thro, U. Wandinger, C. Weitkamp, “Lidar network observation of cirrus morphological and scattering properties during the International Cirrus Experiment 1989: The 18 October 1989 case study and statistical analysis,” J. Appl. Meteorol. 32, 1608–1622 (1993).
[CrossRef]

A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, W. Michaelis, “Independent measurement of extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar,” Appl. Opt. 31, 7113–7131 (1992).
[CrossRef] [PubMed]

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

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. Schwendimann, 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. Schwendimann, 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]

Pal, S. R.

Piironen, P.

Plass, G. N.

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 observations of a mixed-phase altostratus cloud,” J. Appl. Meteorol. 16, 339–345 (1977).
[CrossRef]

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

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, 345–354 (1995).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, 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 LIDARS,” Appl. Phys. B 60, 341–344 (1995).
[CrossRef]

Reichardt, J.

Riebesell, M.

A. Ansmann, J. Bösenberg, G. Brogniez, S. Elouragini, P. H. Flamant, K. H. Klapheck, H. Linné, L. Menenger, W. Michaelis, M. Riebesell, C. Senff, P.-Y. Thro, U. Wandinger, C. Weitkamp, “Lidar network observation of cirrus morphological and scattering properties during the International Cirrus Experiment 1989: The 18 October 1989 case study and statistical analysis,” J. Appl. Meteorol. 32, 1608–1622 (1993).
[CrossRef]

A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, W. Michaelis, “Independent measurement of extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar,” Appl. Opt. 31, 7113–7131 (1992).
[CrossRef] [PubMed]

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

A. Ansmann, M. Riebesell, C. Weitkamp, “Measurement of atmospheric aerosol extinction profiles with a Raman lidar,” Opt. Lett. 15, 746–748 (1990).
[CrossRef] [PubMed]

Roesler, F. L.

Schotland, R. M.

K.-N. Liou, R. M. Schotland, “Multiple backscattering and depolarization from water clouds for a pulsed lidar system,” J. Atmos. Sci. 28, 772–784 (1971).
[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]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, 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]

Senff, C.

A. Ansmann, J. Bösenberg, G. Brogniez, S. Elouragini, P. H. Flamant, K. H. Klapheck, H. Linné, L. Menenger, W. Michaelis, M. Riebesell, C. Senff, P.-Y. Thro, U. Wandinger, C. Weitkamp, “Lidar network observation of cirrus morphological and scattering properties during the International Cirrus Experiment 1989: The 18 October 1989 case study and statistical analysis,” J. Appl. Meteorol. 32, 1608–1622 (1993).
[CrossRef]

Shipley, S. T.

S. T. Shipley, D. H. Tracy, E. W. Eloranta, J. T. Trauger, J. T. Sroga, F. L. Roesler, J. A. Weinman, “High spectral resolution lidar to measure optical scattering properties of atmospheric aerosols. 1. Theory and instrumentation,” Appl. Opt. 22, 3716–3724 (1983).
[CrossRef] [PubMed]

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 77, 7123–7128 (1972).
[CrossRef]

S. T. Shipley, “The measurement of rainfall by lidar,” Ph.D. dissertation (University of Wisconsin, Madison, Wisconsin, 1978).

Sroga, J. T.

Starkov, A. V.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, 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]

Takano, Y.

Thro, P.-Y.

A. Ansmann, J. Bösenberg, G. Brogniez, S. Elouragini, P. H. Flamant, K. H. Klapheck, H. Linné, L. Menenger, W. Michaelis, M. Riebesell, C. Senff, P.-Y. Thro, U. Wandinger, C. Weitkamp, “Lidar network observation of cirrus morphological and scattering properties during the International Cirrus Experiment 1989: The 18 October 1989 case study and statistical analysis,” J. Appl. Meteorol. 32, 1608–1622 (1993).
[CrossRef]

Tracy, D. H.

Trauger, J. T.

Voss, E.

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

Wandinger, U.

U. Wandinger, A. Ansmann, J. Reichardt, T. Deshler, “Determination of stratospheric-aerosol microphysical properties from independent extinction and backscattering measurements with a Raman lidar,” Appl. Opt. 34, 8315–8329 (1995).
[CrossRef] [PubMed]

A. Ansmann, J. Bösenberg, G. Brogniez, S. Elouragini, P. H. Flamant, K. H. Klapheck, H. Linné, L. Menenger, W. Michaelis, M. Riebesell, C. Senff, P.-Y. Thro, U. Wandinger, C. Weitkamp, “Lidar network observation of cirrus morphological and scattering properties during the International Cirrus Experiment 1989: The 18 October 1989 case study and statistical analysis,” J. Appl. Meteorol. 32, 1608–1622 (1993).
[CrossRef]

A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, W. Michaelis, “Independent measurement of extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar,” Appl. Opt. 31, 7113–7131 (1992).
[CrossRef] [PubMed]

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

Weinman, J. A.

S. T. Shipley, D. H. Tracy, E. W. Eloranta, J. T. Trauger, J. T. Sroga, F. L. Roesler, J. A. Weinman, “High spectral resolution lidar to measure optical scattering properties of atmospheric aerosols. 1. Theory and instrumentation,” Appl. Opt. 22, 3716–3724 (1983).
[CrossRef] [PubMed]

J. A. Weinman, “Effects of multiple scattering on light pulses reflected by turbid atmospheres,” J. Atmos. Sci. 33, 1763–1771 (1976).
[CrossRef]

K. E. Kunkel, J. A. Weinman, “Monte Carlo analysis of multiply scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
[CrossRef]

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 77, 7123–7128 (1972).
[CrossRef]

Weitkamp, C.

A. Ansmann, J. Bösenberg, G. Brogniez, S. Elouragini, P. H. Flamant, K. H. Klapheck, H. Linné, L. Menenger, W. Michaelis, M. Riebesell, C. Senff, P.-Y. Thro, U. Wandinger, C. Weitkamp, “Lidar network observation of cirrus morphological and scattering properties during the International Cirrus Experiment 1989: The 18 October 1989 case study and statistical analysis,” J. Appl. Meteorol. 32, 1608–1622 (1993).
[CrossRef]

A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, W. Michaelis, “Independent measurement of extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar,” Appl. Opt. 31, 7113–7131 (1992).
[CrossRef] [PubMed]

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

A. Ansmann, M. Riebesell, C. Weitkamp, “Measurement of atmospheric aerosol extinction profiles with a Raman lidar,” Opt. Lett. 15, 746–748 (1990).
[CrossRef] [PubMed]

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).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, 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]

Zaccanti, G.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, 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, 325–330 (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. Schwendimann, 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, 345–354 (1995).
[CrossRef]

Appl. Opt. (10)

S. R. Pal, A. I. Carswell, “Multiple scattering in atmospheric clouds: lidar observations,” Appl. Opt. 15, 1990–1995 (1976).
[CrossRef] [PubMed]

A. Cohen, M. Kleiman, J. Cooney, “Lidar measurements of rotational Raman and double scattering,” Appl. Opt. 16, 1905–1910 (1978).
[CrossRef]

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

S. Egert, A. Cohen, M. Kleiman, N. Ben-Yosef, “Instantaneous integrated Raman scattering,” Appl. Opt. 22, 1592–1597 (1983).
[CrossRef] [PubMed]

S. T. Shipley, D. H. Tracy, E. W. Eloranta, J. T. Trauger, J. T. Sroga, F. L. Roesler, J. A. Weinman, “High spectral resolution lidar to measure optical scattering properties of atmospheric aerosols. 1. Theory and instrumentation,” Appl. Opt. 22, 3716–3724 (1983).
[CrossRef] [PubMed]

Y. Takano, K. Jayaweera, “Scattering phase matrix for hexagonal ice crystals computed from ray optics,” Appl. Opt. 24, 3254–3263 (1985).
[CrossRef] [PubMed]

L. R. Bissonnette, “Multiscattering model for propagation of narrow light beams in aerosol media,” Appl. Opt. 27, 2478–2484 (1988).
[CrossRef] [PubMed]

A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, W. Michaelis, “Independent measurement of extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar,” Appl. Opt. 31, 7113–7131 (1992).
[CrossRef] [PubMed]

U. Wandinger, A. Ansmann, J. Reichardt, T. Deshler, “Determination of stratospheric-aerosol microphysical properties from independent extinction and backscattering measurements with a Raman lidar,” Appl. Opt. 34, 8315–8329 (1995).
[CrossRef] [PubMed]

G. N. Plass, G. W. Kattawar, “Reflection of light pulses from clouds,” Appl. Opt. 10, 2304–2310 (1971).
[CrossRef] [PubMed]

Appl. Phys. B (8)

L. R. Bissonnette, “Multiple scattering of narrow light beams in aerosols,” Appl. Phys. B 60, 315–324 (1995).
[CrossRef]

P. Bruscaglioni, A. Ismaeli, G. Zaccanti, “Monte Carlo calculations of LIDAR returns: procedure and results,” Appl. Phys. B 60, 325–330 (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 LIDARS,” 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, 345–354 (1995).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, 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. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

J. Appl. Meteorol. (2)

A. Ansmann, J. Bösenberg, G. Brogniez, S. Elouragini, P. H. Flamant, K. H. Klapheck, H. Linné, L. Menenger, W. Michaelis, M. Riebesell, C. Senff, P.-Y. Thro, U. Wandinger, C. Weitkamp, “Lidar network observation of cirrus morphological and scattering properties during the International Cirrus Experiment 1989: The 18 October 1989 case study and statistical analysis,” J. Appl. Meteorol. 32, 1608–1622 (1993).
[CrossRef]

C. M. R. Platt, “Lidar observations of a mixed-phase altostratus cloud,” J. Appl. Meteorol. 16, 339–345 (1977).
[CrossRef]

J. Atmos. Sci. (4)

K.-N. Liou, R. M. Schotland, “Multiple backscattering and depolarization from water clouds for a pulsed lidar system,” J. Atmos. Sci. 28, 772–784 (1971).
[CrossRef]

J. A. Weinman, “Effects of multiple scattering on light pulses reflected by turbid atmospheres,” J. Atmos. Sci. 33, 1763–1771 (1976).
[CrossRef]

K. E. Kunkel, J. A. Weinman, “Monte Carlo analysis of multiply scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
[CrossRef]

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]

J. Geophys. Res. (1)

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 77, 7123–7128 (1972).
[CrossRef]

Opt. Eng. (1)

C. J. Grund, E. W. Eloranta, “University of Wisconsin high spectral resolution lidar,” Opt. Eng. 30, 6–12 (1991).
[CrossRef]

Opt. Lett. (2)

Other (3)

S. T. Shipley, “The measurement of rainfall by lidar,” Ph.D. dissertation (University of Wisconsin, Madison, Wisconsin, 1978).

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969), p. 78.

E. W. Eloranta, “Calculation of doubly scattered lidar returns,” Ph.D. dissertation (University of Wisconsin, Madison, Wisconsin, 1972).

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Figures (9)

Fig. 1
Fig. 1

Scattering phase functions (a) of air molecules (Rayleigh scattering) and of droplets with the size distributions C1 (r eff = 6 μm) and Ns (r eff = 18 μm), (b) of N2 molecules for Raman scattering and of ice crystals having the shape of HP’s with 45-μm width and 18-μm length and of HC’s with 45-μm width and 112-μm length.

Fig. 2
Fig. 2

Geometry and parameters used in the Shipley model.

Fig. 3
Fig. 3

(a) Multiple-to-single-scattering ratio in a C1 cloud with an extinction coefficient of 17.25 km-1 for a laser beam divergence of 0.1 mrad, a RFOV of 1 mrad (full angles), and a wavelength of 1064 nm. The result of the Shipley model (◇) is compared with results from Monte Carlo [Cohen et al.30 (▼), Bruscaglioni et al.18 (●), and Winker and Poole21 (△)], stochastic [Starkov et al.20 (◆)], radiative transfer [Zege et al.22 (□) and Bissonnette et al.17 (▲)], and recursive Mie approaches [Flesia and Schwendimann17 (○)].23 (b) Ratios P 0 2 / P 0 1 (dashed–dotted curve), P 0 3 / P 0 1 (dashed curve), P 0 4 / P 0 1 (dotted curve), and P 0 ms / P 0 1 = P 0 2 / P 0 1 + P 0 3 / P 0 1 + P 0 4 / P 0 1 (thick solid curve) calculated with the Shipley model for the same case as in (a) in comparison with the ratios P 0 2 / P 0 1 and P 0 ms / P 0 1 from the stochastic model (thin solid curve).

Fig. 4
Fig. 4

(a) Ratio P 0 2 / P 0 1 for a C1 cloud with an extinction coefficient of 16.8 km-1, a wavelength of 700 nm, and a laser beam divergence and a RFOV of 10 mrad (full angles) obtained from the Shipley model (solid curve), from the analytical solution for double scattering by Eloranta10 (dashed curve) and from Platt’s Monte Carlo model15 (dotted curve). (b) Ratios P 0 2 / P 0 1 (dotted curve), P 0 ( 2 + 3 ) / P 0 1 (dashed curve), P 0 ( 2 + 3 + 4 ) / P 0 1 (dashed–dotted curve), and P 0 ( 2 + 3 + 4 + 5 ) / P 0 1 (solid curve) calculated with the Shipley model for a Ns cloud with an extinction coefficient of 10 km-1, a wavelength of 700 nm, a beam divergence of 0.2 mrad, and a RFOV of 10 mrad (full angles) in comparison with the ratios P 0 ms / P 0 1 from Monte Carlo simulations by Platt15 (○) and Kunkel and Weinman13 (●).

Fig. 5
Fig. 5

Ratios P i 2 / P i 1 (solid curve), P i 3 / P i 1 (dashed curve), P i 4 / P i 1 (dashed–dotted curve), and P i 5 / P i 1 (dotted curve) for backscattering by particles (left, i = 0) and by molecules (middle, i = R) and corresponding parameters F 0, F R , and F 0/F R (right) for (a) a C1 cloud with an extinction coefficient of 10 km-1 and (b) a HP cloud with an extinction coefficient of 1 km-1. The laser beam divergence is 0.1 mrad, and the RFOV is 0.4 mrad (full angles).

Fig. 6
Fig. 6

Parameters F 0 (solid curve), F R (dashed curve), and F 0/F R (dotted curve) for different clouds obtained with a laser beam divergence of 0.1 mrad and a RFOV of 0.4 mrad (full angles).

Fig. 7
Fig. 7

Fig. 7. P i tot / P i 1 (solid curve), ln [ P i tot / P i 1 ] (dashed curve), and exponential fit to P i tot / P i 1 (dotted curve) for particulate (left, i = 0) and molecular (right, i = R) backscattering from a Ns cloud with an extinction coefficient of 20 km-1 obtained with a laser beam divergence of 0.1 mrad and a RFOV of 0.4 mrad (full angles).

Fig. 8
Fig. 8

Profiles of the extinction coefficient, backscatter coefficient, and lidar ratio obtained with a Raman lidar with a laser beam divergence of 0.1 mrad and a RFOV of 0.4 mrad (full angles) in a water cloud (upper, solid curve). The profiles were corrected for multiple scattering by use of phase functions of the C1 size distribution (dotted curve) and of the Ns size distribution (dashed curve). The lower graphs show the corresponding relative multiple-scattering errors.

Fig. 9
Fig. 9

Profiles of the extinction coefficient, backscatter coefficient, and lidar ratio obtained with a Raman lidar with a laser beam divergence of 0.1 mrad and a RFOV of 0.4 mrad (full angles) in a cirrus cloud (upper, solid curve). The profiles were corrected for multiple scattering by use of phase functions of HP’s (dotted curve) and HC’s (dashed curve). The lower graphs show the corresponding relative multiple-scattering errors.

Tables (1)

Tables Icon

Table 1 Parameters for the Approximation of Phase Functions with Sums of Gaussian Functionsa

Equations (20)

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

P 0 1 z = 1 z 2 β mol λ 0 ,   z + β par λ 0 ,   z × exp - 2   0 z α mol λ 0 ,   ζ + α par λ 0 ,   ζ d ζ .
P R 1 z = 1 z 2   β R λ 0 ,   z exp - 0 z α mol λ 0 ,   ζ + α par λ 0 ,   ζ + α mol λ R ,   ζ + α par λ R ,   ζ d ζ .
β R λ 0 ,   z = N R z d σ R d Ω λ 0 , π .
α par λ 0 ,   z = d / d z ln N R z / P R z z 2 - α mol λ 0 ,   z - α mol λ R ,   z 1 + λ 0 / λ R k .
β par λ 0 ,   z = - β mol λ 0 ,   z + β par λ 0 ,   z 0 + β mol λ 0 ,   z 0 × P R z 0 P 0 z P 0 z 0 P R z N R z N R z 0 × exp - z 0 z α par λ R ,   ζ + α mol λ R ,   ζ d ζ exp - z 0 z α par λ 0 ,   ζ + α mol λ 0 ,   ζ d ζ .
S par λ 0 ,   z = α par λ 0 ,   z β par λ 0 ,   z .
α par eff = 1 - F α par .
P 0 tot z = 1 z 2 β mol λ 0 ,   z + β par λ 0 ,   z × exp - 2   0 z 1 - F 0 λ 0 ,   ζ α par λ 0 ,   ζ + α mol λ 0 ,   ζ d ζ ,
P R tot z = 1 z 2   β R λ 0 ,   z × exp - 0 z 1 - F R λ 0 ,   ζ α par λ 0 ,   ζ + α par λ R ,   ζ + α mol λ 0 ,   ζ + α mol λ R ,   ζ d ζ ,
α par eff λ 0 ,   z = 1 - F R λ 0 ,   z α par λ 0 ,   z .
β par eff λ 0 ,   z = β par λ 0 ,   z × exp - z 0 z   F R λ 0 ,   ζ α par λ 0 ,   ζ + α par λ R ,   ζ d ζ exp - z 0 z   2 F 0 λ 0 ,   ζ α par λ 0 ,   ζ d ζ .
S par eff λ 0 ,   z = α par eff λ 0 ,   z β par eff λ 0 ,   z = S par λ 0 ,   z 1 - F R λ 0 ,   z × exp - z 0 z   α par λ 0 ,   ζ 2 F 0 λ 0 ,   ζ - F R λ 0 ,   ζ 1 + λ 0 λ R k d ζ .
P i tot z = P i 1 z exp 0 z   F i λ 0 ,   ζ × α par λ 0 ,   ζ + α par λ i ,   ζ d ζ ,
F i λ 0 ,   z = 1 α par λ 0 ,   z + α par λ i ,   z d d z ln P i tot z P i 1 z .
I L t ,   ϑ ,   ϕ = I 0 π ϑ t 2 exp - ϑ / ϑ t 2 ,
p F ϑ = f = 1 f max   p f ϑ = f = 1 f max a f π ϑ f 2   exp - ϑ / ϑ f 2 , 0 ϑ π / 2   forward ,
p B ϑ = b = 1 b max   p b ϑ = b = 1 b max a b π ϑ b 2 exp - π - ϑ / ϑ b 2 , π / 2 ϑ π backward .
P n P 1 z = R = 2 τ n - 1 b = 1 b max   p b π 0 1 d u 1 0 u 1 d u 2   × 0 u n - 1 d u n - 1 b = 1 b max f = 1 f max   p b π J n ,
J n = i = 1 n - 1   a f u i ρ u i D n 1 - exp - D n ψ 2 C n + ϑ t 2 D n , C n = R - R c R 2 i = 1 n - 1   ϑ f 2 u i u i 2 + j = 1 n - 2 k = j + 1 n - 1 ϑ f 2 u j ϑ f 2 u k ϑ b 2 u n u j - u k 2 , D n = 1 + i = 1 n - 1 ϑ f 2 u i ϑ b 2 u n 1 - R - R c R   u i 2 .
P i tot P i 1 = 1 + P i 2 P i 1 + P i 3 P i 1 +   .

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