Abstract

Backscatter and depolarization lidar measurements from clouds and precipitation are reported as functions of the elevation angle of the pointing lidar direction. We recorded the data by scanning the lidar beam (Nd:YAG) at a constant angular speed of ∼3.5°/s while operating at a repetition rate of 10 Hz. We show that in rain there is an evident and at times spectacular dependence on the elevation angle. That dependence appears to be sensitive to raindrop size. We have developed a three-dimensional polarization-dependent ray-tracing algorithm to calculate the backscatter and the depolarization ratio by large nonspherical droplets. We have applied it to raindrop shapes derived from existing static and dynamic (oscillating) models. We show that many of the observed complex backscatter and depolarization features can be interpreted to a good extent by geometrical optics. These results suggest that there is a definite need for more extensive calculations of the scattering phase matrix elements for large deformed raindrops as functions of the direction of illumination. Obvious applications are retrieval of information on the liquid–solid phase of precipitation and on the size and the vibration state of raindrops.

© 2001 Optical Society of America

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References

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  1. R. M. Schotland, K. Sassen, R. J. Stone, “Observations by lidar of linear depolarization ratios by hydrometeors,” J. Appl. Meteorol. 10, 1011–1017 (1971).
    [CrossRef]
  2. K. Sassen, “Depolarization of laser light backscattered by artificial clouds,” J. Appl. Meteorol. 13, 923–933 (1974).
    [CrossRef]
  3. S. R. Pal, A. I. Carswell, “The polarization characteristics of lidar scattering from snow and ice crystals in the atmosphere,” J. Appl. Meteorol. 16, 70–80 (1977).
    [CrossRef]
  4. C. M. R. Platt, “Lidar observation of a mixed-phase altostratus cloud,” J. Appl. Meteorol. 16, 339–345 (1977).
    [CrossRef]
  5. K. Sassen, “The polarization lidar technique: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866 (1991).
    [CrossRef]
  6. H. R. Pruppacher, R. L. Pitter, “A semiempirical determination of the shape of cloud and raindrops,” J. Atmos. Sci. 28, 86–94 (1971).
    [CrossRef]
  7. K. V. Beard, H. T. Ochs, R. J. Kubesh, “Natural oscillation of small raindrops,” Nature (London) 432, 408–410 (1989).
    [CrossRef]
  8. V. V. Sterlyadkin, “Light scattering by rain droplets,” Atmos. Oceanic Opt. 13, 497–501 (2000).
  9. L. R. Bissonnette, G. Roy, F. Fabry, “Range-height scans of lidar depolarization for characterizing the phase of clouds and precipitation,” J. Atmos. Ocean. Technol. 18, 1429–1446 (2001).
    [CrossRef]
  10. K. Sassen, T. Chen, “The lidar dark band: an oddity of the radar bright band,” Geophys. Res. Lett. 22, 3505–3508 (1995).
    [CrossRef]
  11. G. Roy, L. R. Bissonnette, F. Fabry, “Strong dependence of rain-induced lidar depolarization on illumination angle,” in Advances in Laser Radar Remote Sensing, Selected papers presented at the 20th International Laser Radar Conference, Vichy, France, July 2000, A. Dabas, C. Loth, J. Pelon, eds. (Ecole Polytechnique, Paris, 2000).
  12. J. D. Klett, R. A. Sutherland, “Approximate methods for modeling the scattering properties of nonspherical particles: evaluation of the Wentzel–Kramer–Brillouin method,” Appl. Opt. 31, 373–386 (1992).
    [CrossRef] [PubMed]
  13. K. V. Beard, C. Chuang, “A new model for the equilibrium shape of raindrops,” J. Atmos. Sci. 44, 1509–1524 (1987).
    [CrossRef]
  14. K. V. Beard, J. Q. Feng, C. Chuang, “A simple perturbation model for the electrostatic shape of falling drops,” J. Atmos. Sci. 46, 2404–2418 (1989).
    [CrossRef]
  15. K. V. Beard, R. J. Kubesh, H. T. Ochs, “Laboratory measurement of small raindrop distortion. Part 1: Axis ratios and fall behavior,” J. Atmos. Sci. 48, 698–710 (1991).
    [CrossRef]
  16. K. V. Beard, R. J. Kubesh, “Laboratory measurement of small raindrop distortion. Part 2: Oscillation frequencies and modes,” J. Atmos. Sci. 48, 2245–2264 (1991).
    [CrossRef]
  17. K. V. Beard, D. B. Johnson, A. R. Jameson, “Collisional forcing of raindrop oscillations,” J. Atmos. Sci. 40, 455–462 (1983).
    [CrossRef]
  18. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1976), pp. 40–42.
  19. T. Nousianinen, “Scattering of light by raindrops with single-mode oscillation,” J. Atmos. Sci. 57, 789–802 (2000).
    [CrossRef]
  20. M. I. Mishchenko, “Calculation of the amplitude matrix for a nonspherical particle in a fixed orientation,” Appl. Opt. 39, 1026–1031 (2000).
    [CrossRef]
  21. J. W. F. Goddard, S. M. Cherry, “The ability of dual-polarization radar (copolar linear) to predict rainfall rate and microwave attenuation,” Radio Sci. 19, 201–208 (1984).
    [CrossRef]

2001

L. R. Bissonnette, G. Roy, F. Fabry, “Range-height scans of lidar depolarization for characterizing the phase of clouds and precipitation,” J. Atmos. Ocean. Technol. 18, 1429–1446 (2001).
[CrossRef]

2000

T. Nousianinen, “Scattering of light by raindrops with single-mode oscillation,” J. Atmos. Sci. 57, 789–802 (2000).
[CrossRef]

V. V. Sterlyadkin, “Light scattering by rain droplets,” Atmos. Oceanic Opt. 13, 497–501 (2000).

M. I. Mishchenko, “Calculation of the amplitude matrix for a nonspherical particle in a fixed orientation,” Appl. Opt. 39, 1026–1031 (2000).
[CrossRef]

1995

K. Sassen, T. Chen, “The lidar dark band: an oddity of the radar bright band,” Geophys. Res. Lett. 22, 3505–3508 (1995).
[CrossRef]

1992

1991

K. V. Beard, R. J. Kubesh, H. T. Ochs, “Laboratory measurement of small raindrop distortion. Part 1: Axis ratios and fall behavior,” J. Atmos. Sci. 48, 698–710 (1991).
[CrossRef]

K. V. Beard, R. J. Kubesh, “Laboratory measurement of small raindrop distortion. Part 2: Oscillation frequencies and modes,” J. Atmos. Sci. 48, 2245–2264 (1991).
[CrossRef]

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

1989

K. V. Beard, H. T. Ochs, R. J. Kubesh, “Natural oscillation of small raindrops,” Nature (London) 432, 408–410 (1989).
[CrossRef]

K. V. Beard, J. Q. Feng, C. Chuang, “A simple perturbation model for the electrostatic shape of falling drops,” J. Atmos. Sci. 46, 2404–2418 (1989).
[CrossRef]

1987

K. V. Beard, C. Chuang, “A new model for the equilibrium shape of raindrops,” J. Atmos. Sci. 44, 1509–1524 (1987).
[CrossRef]

1984

J. W. F. Goddard, S. M. Cherry, “The ability of dual-polarization radar (copolar linear) to predict rainfall rate and microwave attenuation,” Radio Sci. 19, 201–208 (1984).
[CrossRef]

1983

K. V. Beard, D. B. Johnson, A. R. Jameson, “Collisional forcing of raindrop oscillations,” J. Atmos. Sci. 40, 455–462 (1983).
[CrossRef]

1977

S. R. Pal, A. I. Carswell, “The polarization characteristics of lidar scattering from snow and ice crystals in the atmosphere,” J. Appl. Meteorol. 16, 70–80 (1977).
[CrossRef]

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

1974

K. Sassen, “Depolarization of laser light backscattered by artificial clouds,” J. Appl. Meteorol. 13, 923–933 (1974).
[CrossRef]

1971

R. M. Schotland, K. Sassen, R. J. Stone, “Observations by lidar of linear depolarization ratios by hydrometeors,” J. Appl. Meteorol. 10, 1011–1017 (1971).
[CrossRef]

H. R. Pruppacher, R. L. Pitter, “A semiempirical determination of the shape of cloud and raindrops,” J. Atmos. Sci. 28, 86–94 (1971).
[CrossRef]

Beard, K. V.

K. V. Beard, R. J. Kubesh, H. T. Ochs, “Laboratory measurement of small raindrop distortion. Part 1: Axis ratios and fall behavior,” J. Atmos. Sci. 48, 698–710 (1991).
[CrossRef]

K. V. Beard, R. J. Kubesh, “Laboratory measurement of small raindrop distortion. Part 2: Oscillation frequencies and modes,” J. Atmos. Sci. 48, 2245–2264 (1991).
[CrossRef]

K. V. Beard, J. Q. Feng, C. Chuang, “A simple perturbation model for the electrostatic shape of falling drops,” J. Atmos. Sci. 46, 2404–2418 (1989).
[CrossRef]

K. V. Beard, H. T. Ochs, R. J. Kubesh, “Natural oscillation of small raindrops,” Nature (London) 432, 408–410 (1989).
[CrossRef]

K. V. Beard, C. Chuang, “A new model for the equilibrium shape of raindrops,” J. Atmos. Sci. 44, 1509–1524 (1987).
[CrossRef]

K. V. Beard, D. B. Johnson, A. R. Jameson, “Collisional forcing of raindrop oscillations,” J. Atmos. Sci. 40, 455–462 (1983).
[CrossRef]

Bissonnette, L. R.

L. R. Bissonnette, G. Roy, F. Fabry, “Range-height scans of lidar depolarization for characterizing the phase of clouds and precipitation,” J. Atmos. Ocean. Technol. 18, 1429–1446 (2001).
[CrossRef]

G. Roy, L. R. Bissonnette, F. Fabry, “Strong dependence of rain-induced lidar depolarization on illumination angle,” in Advances in Laser Radar Remote Sensing, Selected papers presented at the 20th International Laser Radar Conference, Vichy, France, July 2000, A. Dabas, C. Loth, J. Pelon, eds. (Ecole Polytechnique, Paris, 2000).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1976), pp. 40–42.

Carswell, A. I.

S. R. Pal, A. I. Carswell, “The polarization characteristics of lidar scattering from snow and ice crystals in the atmosphere,” J. Appl. Meteorol. 16, 70–80 (1977).
[CrossRef]

Chen, T.

K. Sassen, T. Chen, “The lidar dark band: an oddity of the radar bright band,” Geophys. Res. Lett. 22, 3505–3508 (1995).
[CrossRef]

Cherry, S. M.

J. W. F. Goddard, S. M. Cherry, “The ability of dual-polarization radar (copolar linear) to predict rainfall rate and microwave attenuation,” Radio Sci. 19, 201–208 (1984).
[CrossRef]

Chuang, C.

K. V. Beard, J. Q. Feng, C. Chuang, “A simple perturbation model for the electrostatic shape of falling drops,” J. Atmos. Sci. 46, 2404–2418 (1989).
[CrossRef]

K. V. Beard, C. Chuang, “A new model for the equilibrium shape of raindrops,” J. Atmos. Sci. 44, 1509–1524 (1987).
[CrossRef]

Fabry, F.

L. R. Bissonnette, G. Roy, F. Fabry, “Range-height scans of lidar depolarization for characterizing the phase of clouds and precipitation,” J. Atmos. Ocean. Technol. 18, 1429–1446 (2001).
[CrossRef]

G. Roy, L. R. Bissonnette, F. Fabry, “Strong dependence of rain-induced lidar depolarization on illumination angle,” in Advances in Laser Radar Remote Sensing, Selected papers presented at the 20th International Laser Radar Conference, Vichy, France, July 2000, A. Dabas, C. Loth, J. Pelon, eds. (Ecole Polytechnique, Paris, 2000).

Feng, J. Q.

K. V. Beard, J. Q. Feng, C. Chuang, “A simple perturbation model for the electrostatic shape of falling drops,” J. Atmos. Sci. 46, 2404–2418 (1989).
[CrossRef]

Goddard, J. W. F.

J. W. F. Goddard, S. M. Cherry, “The ability of dual-polarization radar (copolar linear) to predict rainfall rate and microwave attenuation,” Radio Sci. 19, 201–208 (1984).
[CrossRef]

Jameson, A. R.

K. V. Beard, D. B. Johnson, A. R. Jameson, “Collisional forcing of raindrop oscillations,” J. Atmos. Sci. 40, 455–462 (1983).
[CrossRef]

Johnson, D. B.

K. V. Beard, D. B. Johnson, A. R. Jameson, “Collisional forcing of raindrop oscillations,” J. Atmos. Sci. 40, 455–462 (1983).
[CrossRef]

Klett, J. D.

Kubesh, R. J.

K. V. Beard, R. J. Kubesh, “Laboratory measurement of small raindrop distortion. Part 2: Oscillation frequencies and modes,” J. Atmos. Sci. 48, 2245–2264 (1991).
[CrossRef]

K. V. Beard, R. J. Kubesh, H. T. Ochs, “Laboratory measurement of small raindrop distortion. Part 1: Axis ratios and fall behavior,” J. Atmos. Sci. 48, 698–710 (1991).
[CrossRef]

K. V. Beard, H. T. Ochs, R. J. Kubesh, “Natural oscillation of small raindrops,” Nature (London) 432, 408–410 (1989).
[CrossRef]

Mishchenko, M. I.

Nousianinen, T.

T. Nousianinen, “Scattering of light by raindrops with single-mode oscillation,” J. Atmos. Sci. 57, 789–802 (2000).
[CrossRef]

Ochs, H. T.

K. V. Beard, R. J. Kubesh, H. T. Ochs, “Laboratory measurement of small raindrop distortion. Part 1: Axis ratios and fall behavior,” J. Atmos. Sci. 48, 698–710 (1991).
[CrossRef]

K. V. Beard, H. T. Ochs, R. J. Kubesh, “Natural oscillation of small raindrops,” Nature (London) 432, 408–410 (1989).
[CrossRef]

Pal, S. R.

S. R. Pal, A. I. Carswell, “The polarization characteristics of lidar scattering from snow and ice crystals in the atmosphere,” J. Appl. Meteorol. 16, 70–80 (1977).
[CrossRef]

Pitter, R. L.

H. R. Pruppacher, R. L. Pitter, “A semiempirical determination of the shape of cloud and raindrops,” J. Atmos. Sci. 28, 86–94 (1971).
[CrossRef]

Platt, C. M. R.

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

Pruppacher, H. R.

H. R. Pruppacher, R. L. Pitter, “A semiempirical determination of the shape of cloud and raindrops,” J. Atmos. Sci. 28, 86–94 (1971).
[CrossRef]

Roy, G.

L. R. Bissonnette, G. Roy, F. Fabry, “Range-height scans of lidar depolarization for characterizing the phase of clouds and precipitation,” J. Atmos. Ocean. Technol. 18, 1429–1446 (2001).
[CrossRef]

G. Roy, L. R. Bissonnette, F. Fabry, “Strong dependence of rain-induced lidar depolarization on illumination angle,” in Advances in Laser Radar Remote Sensing, Selected papers presented at the 20th International Laser Radar Conference, Vichy, France, July 2000, A. Dabas, C. Loth, J. Pelon, eds. (Ecole Polytechnique, Paris, 2000).

Sassen, K.

K. Sassen, T. Chen, “The lidar dark band: an oddity of the radar bright band,” Geophys. Res. Lett. 22, 3505–3508 (1995).
[CrossRef]

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

K. Sassen, “Depolarization of laser light backscattered by artificial clouds,” J. Appl. Meteorol. 13, 923–933 (1974).
[CrossRef]

R. M. Schotland, K. Sassen, R. J. Stone, “Observations by lidar of linear depolarization ratios by hydrometeors,” J. Appl. Meteorol. 10, 1011–1017 (1971).
[CrossRef]

Schotland, R. M.

R. M. Schotland, K. Sassen, R. J. Stone, “Observations by lidar of linear depolarization ratios by hydrometeors,” J. Appl. Meteorol. 10, 1011–1017 (1971).
[CrossRef]

Sterlyadkin, V. V.

V. V. Sterlyadkin, “Light scattering by rain droplets,” Atmos. Oceanic Opt. 13, 497–501 (2000).

Stone, R. J.

R. M. Schotland, K. Sassen, R. J. Stone, “Observations by lidar of linear depolarization ratios by hydrometeors,” J. Appl. Meteorol. 10, 1011–1017 (1971).
[CrossRef]

Sutherland, R. A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1976), pp. 40–42.

Appl. Opt.

Atmos. Oceanic Opt.

V. V. Sterlyadkin, “Light scattering by rain droplets,” Atmos. Oceanic Opt. 13, 497–501 (2000).

Bull. Am. Meteorol. Soc.

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

Geophys. Res. Lett.

K. Sassen, T. Chen, “The lidar dark band: an oddity of the radar bright band,” Geophys. Res. Lett. 22, 3505–3508 (1995).
[CrossRef]

J. Appl. Meteorol.

R. M. Schotland, K. Sassen, R. J. Stone, “Observations by lidar of linear depolarization ratios by hydrometeors,” J. Appl. Meteorol. 10, 1011–1017 (1971).
[CrossRef]

J. Appl. Meteorol.

K. Sassen, “Depolarization of laser light backscattered by artificial clouds,” J. Appl. Meteorol. 13, 923–933 (1974).
[CrossRef]

S. R. Pal, A. I. Carswell, “The polarization characteristics of lidar scattering from snow and ice crystals in the atmosphere,” J. Appl. Meteorol. 16, 70–80 (1977).
[CrossRef]

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

J. Atmos. Sci.

K. V. Beard, J. Q. Feng, C. Chuang, “A simple perturbation model for the electrostatic shape of falling drops,” J. Atmos. Sci. 46, 2404–2418 (1989).
[CrossRef]

J. Atmos. Ocean. Technol.

L. R. Bissonnette, G. Roy, F. Fabry, “Range-height scans of lidar depolarization for characterizing the phase of clouds and precipitation,” J. Atmos. Ocean. Technol. 18, 1429–1446 (2001).
[CrossRef]

J. Atmos. Sci.

H. R. Pruppacher, R. L. Pitter, “A semiempirical determination of the shape of cloud and raindrops,” J. Atmos. Sci. 28, 86–94 (1971).
[CrossRef]

K. V. Beard, R. J. Kubesh, H. T. Ochs, “Laboratory measurement of small raindrop distortion. Part 1: Axis ratios and fall behavior,” J. Atmos. Sci. 48, 698–710 (1991).
[CrossRef]

K. V. Beard, R. J. Kubesh, “Laboratory measurement of small raindrop distortion. Part 2: Oscillation frequencies and modes,” J. Atmos. Sci. 48, 2245–2264 (1991).
[CrossRef]

K. V. Beard, D. B. Johnson, A. R. Jameson, “Collisional forcing of raindrop oscillations,” J. Atmos. Sci. 40, 455–462 (1983).
[CrossRef]

T. Nousianinen, “Scattering of light by raindrops with single-mode oscillation,” J. Atmos. Sci. 57, 789–802 (2000).
[CrossRef]

K. V. Beard, C. Chuang, “A new model for the equilibrium shape of raindrops,” J. Atmos. Sci. 44, 1509–1524 (1987).
[CrossRef]

Nature (London)

K. V. Beard, H. T. Ochs, R. J. Kubesh, “Natural oscillation of small raindrops,” Nature (London) 432, 408–410 (1989).
[CrossRef]

Radio Sci.

J. W. F. Goddard, S. M. Cherry, “The ability of dual-polarization radar (copolar linear) to predict rainfall rate and microwave attenuation,” Radio Sci. 19, 201–208 (1984).
[CrossRef]

Other

G. Roy, L. R. Bissonnette, F. Fabry, “Strong dependence of rain-induced lidar depolarization on illumination angle,” in Advances in Laser Radar Remote Sensing, Selected papers presented at the 20th International Laser Radar Conference, Vichy, France, July 2000, A. Dabas, C. Loth, J. Pelon, eds. (Ecole Polytechnique, Paris, 2000).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1976), pp. 40–42.

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Figures (24)

Fig. 1
Fig. 1

Scanning lidar head mounted at the back of the Defence Research Establishment Valcartier (DREV) multiple field-of-view mobile lidar laboratory.

Fig. 2
Fig. 2

Schematic diagram of the lidar: SM, scanning mirror; CM, transmitter coupling mirror; TM, parabolic telescope mirror; CO, collimating optics; PC, polarizing cube; E I , electric-field component of linearly polarized transmitted beam; s I , propagation vector of transmitted beam; E b and E b, parallel and perpendicular components of the electric field of backscattered radiation, s b , propagation vector of backscattered radiation.

Fig. 3
Fig. 3

Measured depolarization ratio as a function of the zenith angle obtained in light fog and in moderate snow precipitation.

Fig. 4
Fig. 4

(a) Range-height color-coded plots of, upper panel, range-corrected and, lower panel, the linear depolarization ratio measured at 1.06 µm and a receiver FOV of 1 mrad. Conditions at ground level were continuous rain, light winds, and a temperature of 4 °C. The scales are 500 m/div and 15°/div. (b) Range-averaged lidar return and depolarization ratio as functions of zenith angle. Averaging was performed between 300 and 450 m.

Fig. 5
Fig. 5

(a) Range-height color-coded plots of, upper panel, range-corrected and, lower panel, the linear depolarization ratio measured at 1.06 µm and a receiver FOV of 1 mrad. Conditions at ground level were continuous rain, light winds, and a temperature of 0 °C. The scales are 500 m/div and 15°/div. (b) Range-averaged lidar return and depolarization ratio as functions of zenith angle. Averaging was performed between 300 and 450 m.

Fig. 6
Fig. 6

(a) Range-height color-coded plots of, upper panel, range-corrected and, lower panel, the linear depolarization ratio measured at 1.06 µm and a receiver FOV of 1 mrad. Conditions at ground level were continuous rain, light winds, and a temperature of 4 °C. The scales are 500 m/div and 15°/div. (b) Range-averaged lidar return and depolarization ratio as functions of the zenith angle. Averaging was performed between 200 and 350 m.

Fig. 7
Fig. 7

(a) Range-height color-coded plots of, upper panel, range-corrected and, lower panel, the linear depolarization ratio measured at 1.06 µm and the receiver FOV of 1 mrad. Conditions at ground level were continuous rain, light winds, and a temperature of 4 °C. The scales are 150 m/div and 15°/div. (b) Range-averaged lidar return and depolarization ratio as functions of the zenith angle. The average is performed between 300 and 450 m.

Fig. 8
Fig. 8

(a) Range-height color-coded plots of, upper panel, range-corrected and, lower panel, the linear depolarization ratio measured at 1.06 µm and the receiver FOV of 1 mrad. Conditions at ground level were continuous rain, light winds, and a temperature of 4 °C. The scales are 500 m/div and 15°/div. (b) Range-averaged lidar return and the depolarization ratio as functions of zenith angle. Averaging was performed between 200 and 300 m.

Fig. 9
Fig. 9

(a) Range-height color-coded plots of, upper panel, range-corrected and, lower panel, the linear depolarization ratio measured at 1.06 µm and the receiver FOV of 1 mrad. Conditions at ground level were continuous rain, high winds, and a temperature of 4 °C. The scales are 500 m/div and 15°/div. (b) Range-averaged lidar return and the depolarization ratio as functions of zenith angle. Averaging was performed between 300 and 400 m.

Fig. 10
Fig. 10

Ray geometry for determining the point of intersection with the droplet surface, vector b, of the transmitted or refracted ray αŝ T , where α is the length of the vector joining the point of entry a to b. Vectors n 1 and n 3 are in the scattering plane, and n 2 is perpendicular to it; n 1 is normal to the raindrop surface at the entry point.

Fig. 11
Fig. 11

Definitions of the variables used to solve for the raindrop surface point a N where the normal is parallel to the direction of illumination. This point is used as reference to the definition of all entry points: a, arbitrary point of entry; d, distance normal to the direction of illumination between an arbitrary ray and the one ray entering the raindrop at normal incidence; R, ●●●.

Fig. 12
Fig. 12

Definitions of the electric-field components parallel and perpendicular to the scattering plane for the reflected and the transmitted (refracted) fields: open circles, vectors defined in the paper.

Fig. 13
Fig. 13

Definition of the backscatter-collection angle.

Fig. 14
Fig. 14

Calculated phase function and depolarization ratio at 180° as functions of the zenith angle for a sphere and a deformed free-falling droplet of 0.342-mm diameter.

Fig. 15
Fig. 15

Calculated phase function and the depolarization ratio at 180° as functions of the zenith angle for a free-falling droplet of 0.61 mm oscillating with instantaneous amplitude A′ ranging from -0.5% to 0.5% in the transverse mode of the fundamental oscillation.

Fig. 16
Fig. 16

Calculated phase function and the depolarization ratio at 180° as functions of the zenith angle for a free-falling droplet of 1 mm oscillating with instantaneous amplitude A′ ranging from -0.5% to 0.5% in the transverse mode of the fundamental oscillation.

Fig. 17
Fig. 17

Calculated phase function and the depolarization ratio at 180° as functions of the zenith angle for a free-falling droplet of 1.5 mm oscillating with instantaneous amplitude A′ ranging from -4% to 8% in the transverse mode of the fundamental oscillation.

Fig. 18
Fig. 18

Mean phase function and the depolarization ratio at 180° as functions of the zenith angle for a free-falling droplet of 0.61 mm. The means are calculated by summing over the instantaneous oscillation amplitudes A′ ranging from -0.5% to 0.5% with the equal probability of occurrence.

Fig. 19
Fig. 19

Mean phase function and the depolarization ratio at 180° as functions of the zenith angle for a free-falling droplet of 1 mm. The means are calculated by summing over the instantaneous oscillation amplitudes A′ ranging from -0.5% to 0.5% with equal probability of occurrence.

Fig. 20
Fig. 20

Mean phase function and the depolarization ratio at 180° as functions of the zenith angle for a free-falling droplet of 1.5 mm. The means are calculated by summing over the instantaneous oscillation amplitudes A′ ranging from -8% to 8% with equal probability of occurrence.

Fig. 21
Fig. 21

Differential backscattering cross section and the depolarization ratio at 180° as functions of the zenith angle for a distribution of nonoscillating free-falling droplets made up of 5, 3, and 1 droplets of 0.61, 1, and 1.5 mm, respectively.

Fig. 22
Fig. 22

Differential backscattering cross section and the depolarization ratio at 180° as functions of zenith angle for a distribution of oscillating free-falling droplets made up of 5, 3, and 1 droplets of 0.61, 1, and 1.5 mm, respectively.

Fig. 23
Fig. 23

Ray tracing inside a free-falling 1-mm droplet subjected to an additional -0.2% deformation caused by the transverse mode of the fundamental oscillation. Total internal reflections are indicated by triangles.

Fig. 24
Fig. 24

Ray tracing inside a free-falling 1.5-mm droplet subjected to an additional +16% deformation caused by the transverse mode of the fundamental oscillation.

Tables (3)

Tables Icon

Table 1 Trial Number, Meteorological Data, and Averaging Range Intervals for Figs. 4(b)9(b)

Tables Icon

Table 2 Summary Description of Lidar Measurements Reported in this Papera

Tables Icon

Table 3 Observations on Raindrop Size

Equations (61)

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Δ = P r / P r ,
e ˆ I = cos γ i ˆ - sin γ cos γ j ˆ + sin 2 γ k ˆ
s ˆ I = sin   γ j ˆ + cos   γ k ˆ ,
r θ ,   ϕ = r 0 F θ = r 0 1 + n = 0 N   C n   cos n π - θ ,
r t ,   θ ,   ϕ = r 0 F θ + r 0 A   sin 2 π ν t Y lm θ ,   ϕ ,
Y 20 θ ,   ϕ = 1 2 3   cos 2 θ - 1 .
n ˆ 1 θ ,   ϕ = H | H | ,
H θ ,   ϕ = r - r 0 F θ - r 0 A Y 20 θ .
α s ˆ T = - a + b .
f = | b |   -   r θ = | α s ˆ T + a |   -   r θ = 0 ,
θ = cos - 1 b 3 | b | .
α i + 1 = α i   -   f α i f α i ,
d f α d α = d | b | d α - d r θ d α .
| b | = i = 1 3   b i 2 1 / 2 ,
d | b | d α = i = 1 3   b i s i | b | ,
d r θ d α = d r θ d θ d θ d α ,
d r θ d θ = - r 0 n = 0 N   c n n   sin n π - θ -   3 r 0 A   cos θ sin θ ,
d θ d α = 1 1 - b 3 3 | b | 2 1 / 2 b 3 | b | 3     b i s i - s 3 | b | .
θ = cos - 1 b 3 | b | ,     ϕ = tan - 1 b 2 b 1 .
d = m ε cos γ j ˆ + sin γ k ˆ + l ε i ˆ ,
a = a N + d + α s ˆ I .
s ˆ I = - cos θ 1 n ˆ 1 + sin θ 1 n ˆ 3 ,
s ˆ R = cos θ 1 n ˆ 1 + sin θ 1 n ˆ 3 ,
s ˆ T = - cos θ 2 n ˆ 1 + sin θ 2 n ˆ 3 .
E I = sin θ - sin θ 1 n ˆ 1 - cos θ 1 n ˆ 3 ,
E I = cos θ n ˆ 2 ,
E R = A R sin θ - sin θ 1 n ˆ 1 + cos θ 1 n ˆ 3 ,
E R = A R E I ,
E T = A T sin θ - sin θ 2 n ˆ 1 - cos θ 2 n ˆ 3 ,
E T = A T E I .
E R = E R + E R ,
E T = E T + E T .
Δ Ω = π D 2 / 4 z 2 = π ϖ 2 .
π - cos - 1 s ˆ b · s ˆ I ϖ .
p π = k K   E Tk 2 N Δ Ω ,
E Tk 2 = E Tk 2 + E Tk 2 ,
E Tk 2 = E Tk · e ˆ I 2 = A Tk 2 cos 2   θ Tk ,
E Tk 2 = A Tk 2   sin 2   θ Tk ,
Δ = k K   E Tk 2 k K   E Tk 2 .
e ˆ 1 = 1 0 0 0 cos   γ sin   γ 0 - sin   γ cos   γ cos   ω 0 - sin   ω 0 1 0 sin   ω 0 cos   ω 1 0 0
e ˆ 1 = cos ω i ˆ + sin ω sin γ j ˆ + sin ω cos γ k ˆ .
e ˆ 2 = s ˆ I × e ˆ 1 | s ˆ I × e ˆ 1 | = cos γ j ˆ - sin γ k ˆ ,
e ˆ 3 = e ˆ 1 × e ˆ 2 | e ˆ 1 × e ˆ 2 | = - sin ω i ˆ + cos ω sin γ j ˆ + cos ω cos γ k ˆ .
s ˆ I = - | s ˆ I · e ˆ 1 | e ˆ 1 + 1 - s ˆ I · e ˆ 1 2 1 / 2 e ˆ 3 ,
s ˆ R = | s ˆ I · e ˆ 1 | e ˆ 1 + 1 - s ˆ I · e ˆ 1 2 1 / 2 e ˆ 3 .
s ˆ R = cos Ω i ˆ + sin Ω sin γ j ˆ + sin Ω cos γ k ˆ .
E I = sin   γ e ˆ 2 .
E I = cos γ + sin ω e ˆ 1 + cos ω e ˆ 3 .
E R = - E I ,
E R = cos γ + sin ω e ˆ 1 - cos ω e ˆ 3 .
E R = cos γ sin Ω i ˆ - cos γ sin γ cos Ω + 1 j ˆ + - cos 2 γ cos Ω + sin 2 γ k ˆ ,
E R = cos γ i ˆ - cos γ sin γ j ˆ + sin 2 γ k ˆ .
E R = sin γ i ˆ - cos 2 γ j ˆ + sin γ cos γ k ˆ .
H a n | H a n | · - s ˆ I = - 1 ,
f a N = H a N · s ˆ I - 1 = 0 .
H = 2 r θ sin θ + r θ cos θ cos φ i ˆ + r θ sin θ + r θ cos θ sin φ j ˆ + r θ cos θ - r θ sin θ k ˆ .
θ i + 1 = θ i   -   f θ i f θ i ,
f θ = s ˆ I - 1 · d H θ d θ ,
d H θ d θ = 2 r θ sin θ + r θ cos θ + r θ cos θ - r sin θ j ˆ + 2 r θ cos θ - r θ sin θ - r θ sin θ - r θ cos θ k ˆ ,
r θ = r 0 n = 1 N   c i n 2   cos n π - θ + 3 r 0 A sin 2 θ - cos 2 θ .
a N = r θ cos θ j ˆ + r θ sin θ k ˆ .

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