Abstract

Recent developments in the search for a practical method of exploiting the multiple-scattering contributions to lidar returns are consolidated in a robust retrieval algorithm. The theoretical basis is the small-angle diffusion approximation. This implies that the algorithm is limited to media of sufficient optical thickness to generate measurable multiple scattering and to geometries for which the receiver’s footprint diameter is less than the scattering mean free path. The primary retrieval products are the range-resolved extinction coefficient and the effective particle diameter from which secondary products such as the particle volume mixing ratio and the extinction at other wavelengths can be calculated. We recall briefly earlier validation tests and present new data and analysis that demonstrate and quantify the solutions’ accuracy. The results show that systematic lidar probings with the proposed multiple-scattering technique can provide valuable physical information on cloud formation and evolution.

© 2005 Optical Society of America

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  1. K. Sassen, “The polarization lidar technique: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866 (1991).
    [CrossRef]
  2. J. D. Klett, “Stable analytical inversion solutions for processing lidar returns,” Appl. Opt. 20, 211–220 (1981).
    [CrossRef] [PubMed]
  3. F. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652–653 (1984).
    [CrossRef] [PubMed]
  4. L. R. Bissonnette, “Sensitivity analysis of lidar inversion algorithms,” Appl. Opt. 25, 2122–2125 (1986).
    [CrossRef] [PubMed]
  5. S. T. Shipley, D. H. Tracy, E. W. Eloranta, J. T. Trauger, J. T. Stroga, 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]
  6. J. T. Stroga, E. W. Eloranta, S. T. Shipley, F. L. Roesler, P. J. Tryon, “High spectral resolution lidar to measure optical scattering properties of atmospheric aerosols. 2. Calibration and data analysis,” Appl. Opt. 22, 3725–3732 (1983).
    [CrossRef]
  7. A. Ansmann, M. Riebesell, C. Weitkamp, “Measurement of atmospheric aerosol extinction profiles with a Raman lidar,” Opt. Lett. 15, 746–748 (1990).
    [CrossRef] [PubMed]
  8. C. M. R. Platt, “Lidar and radiometer observations of cirrus clouds,” J. Atmos. Sci. 30, 1191–1204 (1973).
    [CrossRef]
  9. C. Flesia, P. Schwendimann, eds., “Topical feature on multiple-scattering lidar experiments,” Appl. Phys. B60, 315–362 (1995).
  10. L. R. Bissonnette, “Multiple-scattering lidar equation,” Appl. Opt. 35, 6449–6465 (1996).
    [CrossRef] [PubMed]
  11. 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).
    [CrossRef] [PubMed]
  12. L. R. Bissonnette, G. Roy, L. Poutier, S. G. Cober, G. A. Isaac, “Multiple-scattering lidar retrieval method: tests on Monte-Carlo simulations and comparisons with in situ measurements,” Appl. Opt. 41, 6307–6324 (2002).
    [CrossRef] [PubMed]
  13. L. R. Bissonnette, D. L. Hutt, “Multiple scattering lidar,” Appl. Opt. 29, 5045–5046 (1990).
    [CrossRef] [PubMed]
  14. D. L. Hutt, L. R. Bissonnette, L. Durand, “Multiple field of view lidar returns from atmospheric aerosols,” Appl. Opt. 33, 2338–2348 (1994).
    [CrossRef] [PubMed]
  15. G. Roy, L. R. Bissonnette, C. Bastille, “Efficient field-of-view control for multiple-field-of-view lidar receivers,” in Proceedings of Nineteenth International Laser Radar Conference, (U.S. Government Printing Office, 1998), pp. 767–770.
  16. N. Roy, G. Roy, L. R. Bissonnette, 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]
  17. L. R. Bissonnette, “Multiple scattering of narrow light beams in aerosols,” Appl. Phys. B 60, 315–323 (1995).
    [CrossRef]
  18. P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Monte-Carlo calculations of lidar returns: procedure and results,” Appl. Phys. B 60, 325–329 (1995).
    [CrossRef]
  19. 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]
  20. E. W. Eloranta, “Practical model for the calculation of multiply scattered lidar returns,” Appl. Opt. 37, 2464–2472 (1998).
    [CrossRef]
  21. J. A. Weinman, “Effects of multiple scattering on light pulses reflected by turbid atmospheres,” J. Atmos. Sci. 33, 1763–1771 (1976).
    [CrossRef]
  22. I. L. Katsev, E. P. Zege, A. S. Prikhach, 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]
  23. D. Deirmendjian, “Far-infrared and submillimeter wave attenuation by clouds and rain,” J. Appl. Meteorol. 14, 1584–1593 (1975).
    [CrossRef]
  24. E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their physical properties,” (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1979).
  25. G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.
  26. J. L. Lumley, H. A. Panofsky, The Structure of Atmospheric Turbulence (Wiley-Interscience, 1964).

2004

2002

1998

1997

1996

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).
[CrossRef] [PubMed]

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

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

1994

1991

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

1990

1986

1984

1983

1981

1976

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

1975

D. Deirmendjian, “Far-infrared and submillimeter wave attenuation by clouds and rain,” J. Appl. Meteorol. 14, 1584–1593 (1975).
[CrossRef]

1973

C. M. R. Platt, “Lidar and radiometer observations of cirrus clouds,” J. Atmos. Sci. 30, 1191–1204 (1973).
[CrossRef]

Ansmann, A.

Ayers, J. K.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Bailey, M.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Bastille, C.

G. Roy, L. R. Bissonnette, C. Bastille, “Efficient field-of-view control for multiple-field-of-view lidar receivers,” in Proceedings of Nineteenth International Laser Radar Conference, (U.S. Government Printing Office, 1998), pp. 767–770.

Bernstein, B. C.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Bissonnette, L. R.

N. Roy, G. Roy, L. R. Bissonnette, 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]

L. R. Bissonnette, G. Roy, L. Poutier, S. G. Cober, G. A. Isaac, “Multiple-scattering lidar retrieval method: tests on Monte-Carlo simulations and comparisons with in situ measurements,” Appl. Opt. 41, 6307–6324 (2002).
[CrossRef] [PubMed]

L. R. Bissonnette, “Multiple-scattering lidar equation,” Appl. Opt. 35, 6449–6465 (1996).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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

D. L. Hutt, L. R. Bissonnette, L. Durand, “Multiple field of view lidar returns from atmospheric aerosols,” Appl. Opt. 33, 2338–2348 (1994).
[CrossRef] [PubMed]

L. R. Bissonnette, D. L. Hutt, “Multiple scattering lidar,” Appl. Opt. 29, 5045–5046 (1990).
[CrossRef] [PubMed]

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

G. Roy, L. R. Bissonnette, C. Bastille, “Efficient field-of-view control for multiple-field-of-view lidar receivers,” in Proceedings of Nineteenth International Laser Radar Conference, (U.S. Government Printing Office, 1998), pp. 767–770.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Bruscaglioni, P.

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

Cober, S. G.

L. R. Bissonnette, G. Roy, L. Poutier, S. G. Cober, G. A. Isaac, “Multiple-scattering lidar retrieval method: tests on Monte-Carlo simulations and comparisons with in situ measurements,” Appl. Opt. 41, 6307–6324 (2002).
[CrossRef] [PubMed]

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Deirmendjian, D.

D. Deirmendjian, “Far-infrared and submillimeter wave attenuation by clouds and rain,” J. Appl. Meteorol. 14, 1584–1593 (1975).
[CrossRef]

Dreidger, N.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Durand, L.

Eloranta, E. W.

Evans, W. F. J.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Fabry, F.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Fenn, R. W.

E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their physical properties,” (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1979).

Fernald, F. G.

Glazer, A.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Gutelpe, I.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Hallett, J.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Hudak, D.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Hutt, D. L.

Isaac, G. A.

L. R. Bissonnette, G. Roy, L. Poutier, S. G. Cober, G. A. Isaac, “Multiple-scattering lidar retrieval method: tests on Monte-Carlo simulations and comparisons with in situ measurements,” Appl. Opt. 41, 6307–6324 (2002).
[CrossRef] [PubMed]

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Ismaelli, A.

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

Katsev, I. L.

Klett, J. D.

Korolev, A. V.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Lumley, J. L.

J. L. Lumley, H. A. Panofsky, The Structure of Atmospheric Turbulence (Wiley-Interscience, 1964).

Marcotte, D. L.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Minnis, P.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Murray, J.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Nguyen, L.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Panofsky, H. A.

J. L. Lumley, H. A. Panofsky, The Structure of Atmospheric Turbulence (Wiley-Interscience, 1964).

Platt, C. M. R.

C. M. R. Platt, “Lidar and radiometer observations of cirrus clouds,” J. Atmos. Sci. 30, 1191–1204 (1973).
[CrossRef]

Polonsky, I. N.

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]

Poutier, L.

Prikhach, A. S.

Ratvasky, T. P.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Reehorst, A.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Reid, J.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Riebesell, M.

Rodriguez, P.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Roesler, F. L.

Roy, G.

Roy, N.

Sassen, K.

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

Schneider, T.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Sheppard, B. E.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Shettle, E. P.

E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their physical properties,” (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1979).

Shipley, S. T.

Simard, J.-R.

Strapp, J. W.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Stroga, J. T.

Tracy, D. H.

Trauger, J. T.

Tryon, P. J.

Weinman, J. A.

Weitkamp, C.

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]

Wolde, M.

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

Zaccanti, G.

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

Zege, E. P.

Appl. Opt.

J. D. Klett, “Stable analytical inversion solutions for processing lidar returns,” Appl. Opt. 20, 211–220 (1981).
[CrossRef] [PubMed]

F. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652–653 (1984).
[CrossRef] [PubMed]

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

S. T. Shipley, D. H. Tracy, E. W. Eloranta, J. T. Trauger, J. T. Stroga, 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. T. Stroga, E. W. Eloranta, S. T. Shipley, F. L. Roesler, P. J. Tryon, “High spectral resolution lidar to measure optical scattering properties of atmospheric aerosols. 2. Calibration and data analysis,” Appl. Opt. 22, 3725–3732 (1983).
[CrossRef]

L. R. Bissonnette, “Multiple-scattering lidar equation,” Appl. Opt. 35, 6449–6465 (1996).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

L. R. Bissonnette, G. Roy, L. Poutier, S. G. Cober, G. A. Isaac, “Multiple-scattering lidar retrieval method: tests on Monte-Carlo simulations and comparisons with in situ measurements,” Appl. Opt. 41, 6307–6324 (2002).
[CrossRef] [PubMed]

L. R. Bissonnette, D. L. Hutt, “Multiple scattering lidar,” Appl. Opt. 29, 5045–5046 (1990).
[CrossRef] [PubMed]

D. L. Hutt, L. R. Bissonnette, L. Durand, “Multiple field of view lidar returns from atmospheric aerosols,” Appl. Opt. 33, 2338–2348 (1994).
[CrossRef] [PubMed]

N. Roy, G. Roy, L. R. Bissonnette, 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]

E. W. Eloranta, “Practical model for the calculation of multiply scattered lidar returns,” Appl. Opt. 37, 2464–2472 (1998).
[CrossRef]

Appl. Phys. B

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

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

Bull. Am. Meteorol. Soc.

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

J. Appl. Meteorol.

D. Deirmendjian, “Far-infrared and submillimeter wave attenuation by clouds and rain,” J. Appl. Meteorol. 14, 1584–1593 (1975).
[CrossRef]

J. Atmos. Sci.

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

C. M. R. Platt, “Lidar and radiometer observations of cirrus clouds,” J. Atmos. Sci. 30, 1191–1204 (1973).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Other

G. Roy, L. R. Bissonnette, C. Bastille, “Efficient field-of-view control for multiple-field-of-view lidar receivers,” in Proceedings of Nineteenth International Laser Radar Conference, (U.S. Government Printing Office, 1998), pp. 767–770.

C. Flesia, P. Schwendimann, eds., “Topical feature on multiple-scattering lidar experiments,” Appl. Phys. B60, 315–362 (1995).

E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their physical properties,” (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1979).

G. A. Isaac, J. K. Ayers, M. Bailey, L. R. Bissonnette, B. C. Bernstein, S. G. Cober, N. Dreidger, W. F. J. Evans, F. Fabry, A. Glazer, I. Gutelpe, J. Hallett, D. Hudak, A. V. Korolev, D. L. Marcotte, P. Minnis, J. Murray, L. Nguyen, T. P. Ratvasky, A. Reehorst, J. Reid, P. Rodriguez, T. Schneider, B. E. Sheppard, J. W. Strapp, M. Wolde, “First results from the Alliance Icing Research Study II,” paper AIAA-2005-0252, presented at the 43rd American Institute for Aeronautics and Astronautics Aerospace Sciences Meeting, Reno, Nev., 11–13 January 2005.

J. L. Lumley, H. A. Panofsky, The Structure of Atmospheric Turbulence (Wiley-Interscience, 1964).

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Figures (14)

Fig. 1
Fig. 1

Schematic diagram of the MFOV receiver of the lidar used in experiments reported here.

Fig. 2
Fig. 2

Example of measured multiply scattered lidar returns plotted as functions of range for several FOVs (left) and as functions of FOV for several ranges (right).

Fig. 3
Fig. 3

Schematic representation of a photon trajectory contributing to divergence Θ of the radiation forward scattered from the lidar beam.

Fig. 4
Fig. 4

Profile of angular width θd of the phase function forward diffraction peak as a function of distance z along the lidar axis for a hypothetical vertical distribution of clouds and precipitation.

Fig. 5
Fig. 5

Histograms of the ratio of lidar-derived to true values of the extinction coefficient (left) and effective droplet diameter (right) for a series of Monte Carlo simulations of MFOV lidar returns from water clouds; μ is the average value and σ, the standard deviation of the ratio.

Fig. 6
Fig. 6

Comparison of aircraft-measured and lidar-derived profiles of LWC: circles, 30 s aircraft averages; crosses, lidar instantaneous values; thinner curves, average profiles through aircraft data; thicker curves, average profiles through lidar data. Left: 10 December 1999, 17h40–18h10 UTC; right, 16 December 1999, 18h50–21h30 UTC.

Fig. 7
Fig. 7

Color-coded time–height plots of lidar-derived solutions. Counterclockwise from upper left corner: extinction coefficient, particle effective diameter, depolarization ratio, and particle volume mixing ratio. Event of 6 November 2003.

Fig. 8
Fig. 8

Same as Fig. 7 but for 10 December 2003.

Fig. 9
Fig. 9

Time series of the cloud base height (small filled circles), the cloud-averaged extinction coefficient (filled squares), and the effective droplet diameter (open squares) for the event of 6 December 1999. Refer to text for definitions of parameters.

Fig. 10
Fig. 10

Time-correlation functions of the cloud-averaged extinction coefficient (filled squares) and effective droplet diameter (open squares). Refer to text for definition of cloud averaging. Left, 6 December 1999, 18h50–20h31 UTC; right, 12 November 2003, 12h58–1459 UTC.

Fig. 11
Fig. 11

Same as Fig. 10 but for left, 13 November 2003, 16h06–18h13 UTC; right, 11 November 2003, 21h27–22h23 UTC.

Fig. 12
Fig. 12

Histogram of the measured cross correlation between the cloud-averaged extinction coefficient and effective droplet diameter, 〈 α ¯ d e ¯ 〉, for all continuous lidar measurement periods of durations greater than 30 min. Refer to text for definition of cloud averaging.

Fig. 13
Fig. 13

Power spectra of the cloud-averaged extinction coefficient (filled squares) and effective droplet diameter (open squares) calculated from the time-correlation functions of Fig. 10. Refer to text for definition of cloud averaging. Continuous curves show Kolmororov −5/3 power law dependence. Left, 6 December 1999, 18h50–20h31 UTC; right, 12 November 2003, 12h58–1459 UTC.

Fig. 14
Fig. 14

Same as Fig. 13 but for 10 December 2003, 16h44–18h09 UTC.

Equations (53)

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P ( z , θ ) = P ss ( z ) M ( z , θ ) = P ss ( z ) [ 1 + F d ( z , θ ) + F g ( z , θ ) ] ,
P ss ( z ) = K O ( z ) P 0 A z 2 c t 2 β ( z ) exp [ - 2 τ ( z ) ] ,
Θ = ρ 2 / 2 ,
ρ 2 ( z ) = [ z 0 z ϑ ( z ) cos φ ( z ) d z ] 2 + [ z 0 z ϑ ( z ) sin φ ( z ) d z ] 2 ,
ρ 2 ( z ) = z 0 z d z 1 z 0 z d z 1 ϑ ( z 1 ) ϑ ( z 2 ) cos φ ( z 1 ) cos φ ( z 2 ) + z 0 z d z 1 z 0 z d z 2 ϑ ( z 1 ) ϑ ( z 2 ) sin φ ( z 1 ) sin φ ( z 2 ) .
ϑ ( z 1 ) ϑ ( z 2 ) cos φ ( z 1 ) cos φ ( z 2 ) = ϑ ( z 1 ) ϑ ( z 2 ) × cos φ ( z 1 ) cos φ ( z 2 ) ,
ϑ ( z 1 ) ϑ ( z 2 ) sin φ ( z 1 ) sin φ ( z 2 ) = ϑ ( z 1 ) ϑ ( z 2 ) × sin φ ( z 1 ) sin φ ( z 2 ) ,
cos φ ( z 1 ) cos φ ( z 2 ) cos 2 φ ( z ¯ ) f φ ( z ¯ , u ) ,
sin φ ( z 1 ) sin φ ( z 2 ) sin 2 φ ( z ¯ ) f φ ( z ¯ , u ) ,
ϑ ( z 1 ) ϑ ( z 2 ) ϑ 2 ( z ¯ ) f ϑ ( z ¯ , u ) ,
ρ 2 ( z ) = z 0 z d z ¯ ϑ 2 ( z ¯ ) - L ( z ¯ ; z 0 , z ) L ( z ¯ ; z 0 , z ) d u f φ ( z ¯ , u ) f ϑ ( z ¯ , u ) ,
L ( z ¯ ; z 0 , z ) = { 2 ( z ¯ - z 0 ) , z 0 z ¯ < ( z + z 0 ) / 2 2 ( z - z ¯ ) , ( z + z 0 ) / 2 z ¯ z .
l ( z ¯ ; z 0 , z ) = - L ( z ¯ ; z 0 , z ) L ( z ¯ ; z 0 , z ) d u f φ ( z ¯ , u ) f ϑ ( z ¯ , u ) .
Θ 2 ( z ) = 1 z 2 z 0 z d z ¯ ϑ 2 ( z ¯ ) l ( z ¯ ; z 0 , z ) .
ϑ 2 ( z ¯ ) ν ( z ¯ ) θ d 2 ( z ¯ ) ,
ν ( z ¯ ) 1 + τ ( z ¯ ) .
l ( z ¯ ; z 0 , z ) 2 L ( z ¯ ; z 0 , z ) [ 1 + α 2 ( z ¯ ) L 2 ( z ¯ ; z 0 , z ) ] 1 / 2 .
Θ 2 ( z ) = 2 z 2 z 0 z d z ¯ θ d 2 ( z ¯ ) [ 1 + τ ( z ¯ ) ] L ( z ¯ ; z 0 , z ) [ 1 + α 2 ( z ¯ ) L 2 ( z ¯ ; z 0 , z ) ] 1 / 2 .
Θ 2 ( z ) = 2 z 2 [ f ( z ) θ d 2 ( z ) - z 0 z f ( z ) d θ d 2 d z ( z ) d z ] ,
f ( z ) = z 0 z d z ¯ [ 1 + τ ( z ¯ ) ] L ( z ¯ ; z 0 , z ) [ 1 + α 2 ( z ¯ ) L 2 ( z ¯ ; z 0 , z ) ] 1 / 2 .
z 0 - z f ( z ) d θ d 2 d z ( z ) d z f ( z 0 ) [ θ d 2 ( z 1 - ) - θ d 2 ( z 0 - ) ] + f ( z 1 ) [ θ d 2 ( z 2 - ) - θ d 2 ( z 1 - ) ] + + f ( z j ) [ θ d 2 ( z ) ] - θ d 2 ( z j - ) ] , z j < z < z j + 1 ,
Θ 2 ( z ) = 2 z 2 { θ d 2 [ f ( z ) - f ( z j ) ] + θ d 2 ( z j - ) [ f ( z j ) - f ( z j - 1 ) ] + + θ d 2 ( z 1 - ) [ f ( z 1 ) - f ( z 0 ) ] + θ d 2 ( z 0 - ) f ( z 0 ) } ,             z j < z < z j + 1 .
θ d 2 ( z ) = 1 2 [ f ( z ) - f ( z j ) ] { z 2 Θ 2 ( z ) - 2 [ f ( z j ) - f ( z j - 1 ) ] × θ d 2 ( z j - ) - 2 [ f ( z j - 1 ) - f ( z j - 2 ) ] θ d 2 ( z j - 1 - ) - - 2 f ( z 0 ) θ d 2 ( z 0 - 1 ) } ,             z j < z < z i .
θ d 2 ( z j - ) = 1 2 [ f ( z j ) - f ( z j - 1 ) ] { z j 2 Θ 2 ( z j ) - 2 [ f ( z j - 1 ) - f ( z j - 2 ) ] θ d 2 ( z j - 1 - ) - - 2 f ( z 0 ) θ d 2 ( z 0 - ) } .
θ d 2 ( z ) = 1 2 [ f ( z ) - f ( z j ) ] [ z 2 Θ 2 ( z ) - z j 2 Θ 2 ( z j ) ] ,             z j < z < z j + 1 .
P d ( z , θ ) = P ss ( z ) [ 1 + F d ( z , θ ) ] = P ( z , θ ) - P ss ( z ) F g ( z , θ ) .
P d ( z , θ ) = A + B Φ ( θ ɛ / θ md ɛ )
Θ = k θ md ,
θ d 2 ( z ) = k 2 2 [ f ( z ) - f ( z j ) ] [ z 2 θ md 2 ( z ) - z j 2 θ md 2 ( z j ) ] ,             z j < z < z j + 1 .
θ d 0.589 λ / d e ,
d e 2 = 0.694 λ 2 k 2 [ f ( z ) - f ( z j ) ] [ z 2 θ md 2 ( z ) - z j 2 θ md 2 ( z j ) ] ,             z j < z < z j + 1 .
1 N 0 d N d r = r a exp ( - b r ) ,
d e = 2 ( a + 3 ) / b .
a = { 1.163 + 3.715 b 0.65 b 1.5 6.0 b > 1.5 .
p ( z , θ max ) / P ( z , θ min ) 1 + ( β ¯ / β ) { exp [ τ ( z ) ] - 1 } ,
τ ( z ) ln [ 1 + β β ¯ P ( z , θ max ) - P ( z , θ min ) P ( z , θ min ) ] .
α ( z ) = S ( z ) { exp [ τ ( z i ) ] - 1 } z 0 z i S ( z ) d z + { exp [ τ ( z i ) ] - 1 } z z i S ( z ) d z ,             z z i ,
α ( z ) = 1 K O ( z ) z 2 P 0 A 2 c t κ ( z ) M ( z , θ min ) M 2 ( z , θ max ) P 2 ( z , θ max ) P ( z , θ min ) × exp [ 2 τ ( z ) ] .
S θ ( z ) = z 2 κ ( z ) P ( z , θ ) / M ( z , θ ) O ( z ) .
1 α d α d z - 2 α = 1 S θ ( z ) d S θ ( z ) d z ,
α ( z ) = S θ ( z ) / [ α f / S θ ( z f ) + z z f S θ ( z ) d z ] ,
θ md 2 ( z ) = z j 2 z 2 θ md 2 ( z j ) + 0.694 λ 2 k 2 d e 2 [ f ( z ) - f ( z j ) ] z 2 ,             z j < z < z j + 1 .
θ md 2 ( z ) = 0.694 λ 2 k 2 d e 2 f ( z ) z 2 .
θ min < 0.65 λ d e f ( z ) z < θ max .
f ( z ) = ( z - z 0 ) 2 ( 2 + τ ) τ 2 ( 1 + τ 2 - 1 ) ,
θ min < 0.46 λ d e ( z - z 0 ) z < θ max .
R g g ( t ) = 1 σ 1 σ 2 ( T - t ) × t 0 t 0 + T - t [ g ( t ) - m 1 ] [ g ( t + t ) - m 2 ] d t ,
m 1 = 1 T - t t 0 t 0 + T - t g ( t ) d t ,
m 2 = 1 T - t t 0 + t t 0 + T g ( t ) d t ,
σ 1 2 = 1 T - t t 0 t 0 + T - t [ g ( t ) - m 1 ] 2 d t ,
σ 2 2 = 1 T - t t 0 + t t 0 + T [ g ( t ) - m 2 ] 2 d t .
α ¯ d e ¯ = 1 σ 1 α ¯ σ 1 d e ¯ T t 0 t 0 + T [ α ¯ ( t ) - m 1 α ¯ ] [ d e ¯ ( t ) - m 1 d e ¯ ] d t ,
α d e N 0 d 2 d 3 / d 2 N 0 d 3 LWC .

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