Abstract

A multiple-field-of-view (MFOV) lidar measurement and solution technique has been developed to exploit the retrievable particle extinction and size information contained in the multiple-scattering contributions to aerosol lidar returns. We describe the proposed solution algorithm. The primary retrieved parameters are the extinction coefficient at the lidar wavelength and the effective particle diameter from which secondary products such as the extinction at other wavelengths and the liquid-water content (LWC) of liquid-phase clouds can be derived. The solutions are compared with true values in a series of Monte Carlo simulations and with in-cloud measurements. Good agreement is obtained for the simulations. For the field experiment, the retrieved effective droplet diameter and LWC for the available seven cases studied are on average 15% and 35% (worst case) smaller than the measured data, respectively. In the latter case, the analysis shows that the differences cannot be attributed solely to lidar inversion errors. Despite the limited penetration depth (150–300 m) of the lidar pulses, the results of the studied cases indicate that the retrieved lidar solutions remain statistically representative of measurements performed over the full cloud extent. Long-term MFOV lidar monitoring could thus become a practical and economical option for cloud statistical studies but more experimentation on more varied cloud conditions, especially for LWC, is still needed.

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2001 (5)

G. A. Isaac, S. G. Cober, J. W. Strapp, A. V. Korolev, A. Tremblay, D. L. Marcotte, “Recent Canadian research on aircraft in-flight icing,” Can. Aeronaut. Space J. 47(3), 213–221 (2001).

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

G. Roy, L. R. Bissonnette, “Strong dependence of rain-induced lidar depolarization on the illumination angle: experimental evidence and geometrical-optics interpretation,” Appl. Opt. 40, 4770–4789 (2001).
[CrossRef]

S. G. Cober, G. A. Isaac, A. V. Korolev, J. W. Strapp, “Assessing cloud phase conditions,” J. Appl. Meteorol. 40, 1967–1983 (2001).
[CrossRef]

S. G. Cober, G. A. Isaac, A. V. Korolev, “Assessing the Rosemount icing detector with in-situ measurements,” J. Atmos. Oceanic Technol. 18, 515–528 (2001).
[CrossRef]

1999 (2)

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

K. Sassen, G. G. Mace, Z. Wang, M. R. Poellet, S. M. Sekelsky, R. E. McIntosh, “Continental stratus clouds: a case study of coordinated remote sensing and aircraft measurements,” J. Atmos. Sci. 56, 2345–2358 (1999).
[CrossRef]

1998 (1)

1997 (1)

1996 (1)

1995 (5)

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

C. Flesia, P. Schwendimann, eds., Topical feature on Multiple-Scattering Lidar Experiments, Appl. Phys. B 60, 315–362 (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]

S. Elouragini, “Useful algorithms to derive the optical properties of clouds from a backscatter lidar return,” J. Mod. Opt. 42, 1439–1446 (1995).
[CrossRef]

S. A. Young, “Analysis of lidar backscatter profiles in optically thin clouds,” Appl. Opt. 34, 7019–7031 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (3)

1991 (1)

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

1990 (2)

1987 (2)

1986 (1)

1984 (1)

1983 (2)

1981 (2)

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

C. M. R. Platt, “Remote sensing of high clouds III: Monte Carlo calculations of multiple-scattered lidar returns,” J. Atmos. Sci. 38, 156–167 (1981).
[CrossRef]

1980 (1)

J. D. Spinhirne, J. A. Reagan, B. M. Herman, “Vertical distribution of aerosol extinction cross section and inference of aerosol imaginary index in the troposphere by lidar technique,” J. Appl. Meteorol. 19, 426–438 (1980).
[CrossRef]

1977 (1)

Allen, R. J.

Ansmann, A.

Bastille, C.

Bissonnette, L. R.

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

G. Roy, L. R. Bissonnette, “Strong dependence of rain-induced lidar depolarization on the illumination angle: experimental evidence and geometrical-optics interpretation,” Appl. Opt. 40, 4770–4789 (2001).
[CrossRef]

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

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]

L. R. Bissonnette, G. Roy, L. Poutier, S. G. Cober, G. A. Isaac, “Lidar remote sensing of cloud liquid water content and effective droplet diameter: retrieval method and comparison with Monte Carlo simulations and in situ measurements,” TR 2002-20 (Defence Research and Development Canada Establishment Valcartier, 2459 Pie XI Blvd. North, Val-Bélair, Québec G3J 1X5, Canada), to be published.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A.

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.

S. G. Cober, G. A. Isaac, A. V. Korolev, “Assessing the Rosemount icing detector with in-situ measurements,” J. Atmos. Oceanic Technol. 18, 515–528 (2001).
[CrossRef]

G. A. Isaac, S. G. Cober, J. W. Strapp, A. V. Korolev, A. Tremblay, D. L. Marcotte, “Recent Canadian research on aircraft in-flight icing,” Can. Aeronaut. Space J. 47(3), 213–221 (2001).

S. G. Cober, G. A. Isaac, A. V. Korolev, J. W. Strapp, “Assessing cloud phase conditions,” J. Appl. Meteorol. 40, 1967–1983 (2001).
[CrossRef]

L. R. Bissonnette, G. Roy, L. Poutier, S. G. Cober, G. A. Isaac, “Lidar remote sensing of cloud liquid water content and effective droplet diameter: retrieval method and comparison with Monte Carlo simulations and in situ measurements,” TR 2002-20 (Defence Research and Development Canada Establishment Valcartier, 2459 Pie XI Blvd. North, Val-Bélair, Québec G3J 1X5, Canada), to be published.

G. A. Isaac, S. G. Cober, J. W. Strapp, D. Hudak, T. P. Ratvasky, D. L. Marcotte, F. Fabry, “Preliminary results from the Alliance Icing Research Study (AIRS),” paper AIAA-2001-0393, presented at the 39th Aerospace Science Meeting and Exhibit, Reno Nevada, 8–11 January 2001, (American Institute of Aeronautics and Astronautics, Reston, Va., 2001).

de Leeuw, G.

Durand, L.

Eloranta, E. W.

Elouragini, S.

S. Elouragini, “Useful algorithms to derive the optical properties of clouds from a backscatter lidar return,” J. Mod. Opt. 42, 1439–1446 (1995).
[CrossRef]

Fabry, F.

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

G. A. Isaac, S. G. Cober, J. W. Strapp, D. Hudak, T. P. Ratvasky, D. L. Marcotte, F. Fabry, “Preliminary results from the Alliance Icing Research Study (AIRS),” paper AIAA-2001-0393, presented at the 39th Aerospace Science Meeting and Exhibit, Reno Nevada, 8–11 January 2001, (American Institute of Aeronautics and Astronautics, Reston, Va., 2001).

Fernald, F. G.

Gutkowicz-Krusin, D.

Herman, B. M.

J. D. Spinhirne, J. A. Reagan, B. M. Herman, “Vertical distribution of aerosol extinction cross section and inference of aerosol imaginary index in the troposphere by lidar technique,” J. Appl. Meteorol. 19, 426–438 (1980).
[CrossRef]

Hudak, D.

G. A. Isaac, S. G. Cober, J. W. Strapp, D. Hudak, T. P. Ratvasky, D. L. Marcotte, F. Fabry, “Preliminary results from the Alliance Icing Research Study (AIRS),” paper AIAA-2001-0393, presented at the 39th Aerospace Science Meeting and Exhibit, Reno Nevada, 8–11 January 2001, (American Institute of Aeronautics and Astronautics, Reston, Va., 2001).

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A.

Hutt, D. L.

Isaac, G. A.

S. G. Cober, G. A. Isaac, A. V. Korolev, J. W. Strapp, “Assessing cloud phase conditions,” J. Appl. Meteorol. 40, 1967–1983 (2001).
[CrossRef]

G. A. Isaac, S. G. Cober, J. W. Strapp, A. V. Korolev, A. Tremblay, D. L. Marcotte, “Recent Canadian research on aircraft in-flight icing,” Can. Aeronaut. Space J. 47(3), 213–221 (2001).

S. G. Cober, G. A. Isaac, A. V. Korolev, “Assessing the Rosemount icing detector with in-situ measurements,” J. Atmos. Oceanic Technol. 18, 515–528 (2001).
[CrossRef]

L. R. Bissonnette, G. Roy, L. Poutier, S. G. Cober, G. A. Isaac, “Lidar remote sensing of cloud liquid water content and effective droplet diameter: retrieval method and comparison with Monte Carlo simulations and in situ measurements,” TR 2002-20 (Defence Research and Development Canada Establishment Valcartier, 2459 Pie XI Blvd. North, Val-Bélair, Québec G3J 1X5, Canada), to be published.

G. A. Isaac, S. G. Cober, J. W. Strapp, D. Hudak, T. P. Ratvasky, D. L. Marcotte, F. Fabry, “Preliminary results from the Alliance Icing Research Study (AIRS),” paper AIAA-2001-0393, presented at the 39th Aerospace Science Meeting and Exhibit, Reno Nevada, 8–11 January 2001, (American Institute of Aeronautics and Astronautics, Reston, Va., 2001).

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]

Jean, M.

Katsev, I. L.

Klett, J. D.

Korolev, A. V.

S. G. Cober, G. A. Isaac, A. V. Korolev, J. W. Strapp, “Assessing cloud phase conditions,” J. Appl. Meteorol. 40, 1967–1983 (2001).
[CrossRef]

G. A. Isaac, S. G. Cober, J. W. Strapp, A. V. Korolev, A. Tremblay, D. L. Marcotte, “Recent Canadian research on aircraft in-flight icing,” Can. Aeronaut. Space J. 47(3), 213–221 (2001).

S. G. Cober, G. A. Isaac, A. V. Korolev, “Assessing the Rosemount icing detector with in-situ measurements,” J. Atmos. Oceanic Technol. 18, 515–528 (2001).
[CrossRef]

Kunz, G. J.

Mace, G. G.

K. Sassen, G. G. Mace, Z. Wang, M. R. Poellet, S. M. Sekelsky, R. E. McIntosh, “Continental stratus clouds: a case study of coordinated remote sensing and aircraft measurements,” J. Atmos. Sci. 56, 2345–2358 (1999).
[CrossRef]

Marcotte, D. L.

G. A. Isaac, S. G. Cober, J. W. Strapp, A. V. Korolev, A. Tremblay, D. L. Marcotte, “Recent Canadian research on aircraft in-flight icing,” Can. Aeronaut. Space J. 47(3), 213–221 (2001).

G. A. Isaac, S. G. Cober, J. W. Strapp, D. Hudak, T. P. Ratvasky, D. L. Marcotte, F. Fabry, “Preliminary results from the Alliance Icing Research Study (AIRS),” paper AIAA-2001-0393, presented at the 39th Aerospace Science Meeting and Exhibit, Reno Nevada, 8–11 January 2001, (American Institute of Aeronautics and Astronautics, Reston, Va., 2001).

McIntosh, R. E.

K. Sassen, G. G. Mace, Z. Wang, M. R. Poellet, S. M. Sekelsky, R. E. McIntosh, “Continental stratus clouds: a case study of coordinated remote sensing and aircraft measurements,” J. Atmos. Sci. 56, 2345–2358 (1999).
[CrossRef]

Nakane, H.

Platt, C. M. R.

Poellet, M. R.

K. Sassen, G. G. Mace, Z. Wang, M. R. Poellet, S. M. Sekelsky, R. E. McIntosh, “Continental stratus clouds: a case study of coordinated remote sensing and aircraft measurements,” J. Atmos. Sci. 56, 2345–2358 (1999).
[CrossRef]

Polonsky, I. N.

Poutier, L.

L. Poutier, “Evaluation de la technique de sondage par lidar à champs de vue multiples,” Technical Report No. RTS 2/05101 DOTA (ONERA, Office National d’Etudes et Recherches Aéronautiques, Centre de Toulouse, 2 ave Edouard Belin, 31055 Toulouse, France, 2001).

L. R. Bissonnette, G. Roy, L. Poutier, S. G. Cober, G. A. Isaac, “Lidar remote sensing of cloud liquid water content and effective droplet diameter: retrieval method and comparison with Monte Carlo simulations and in situ measurements,” TR 2002-20 (Defence Research and Development Canada Establishment Valcartier, 2459 Pie XI Blvd. North, Val-Bélair, Québec G3J 1X5, Canada), to be published.

Prikhach, A. S.

Ratvasky, T. P.

G. A. Isaac, S. G. Cober, J. W. Strapp, D. Hudak, T. P. Ratvasky, D. L. Marcotte, F. Fabry, “Preliminary results from the Alliance Icing Research Study (AIRS),” paper AIAA-2001-0393, presented at the 39th Aerospace Science Meeting and Exhibit, Reno Nevada, 8–11 January 2001, (American Institute of Aeronautics and Astronautics, Reston, Va., 2001).

Reagan, J. A.

J. D. Spinhirne, J. A. Reagan, B. M. Herman, “Vertical distribution of aerosol extinction cross section and inference of aerosol imaginary index in the troposphere by lidar technique,” J. Appl. Meteorol. 19, 426–438 (1980).
[CrossRef]

Riebesell, M.

Roesler, F. L.

Roy, G.

G. Roy, L. R. Bissonnette, “Strong dependence of rain-induced lidar depolarization on the illumination angle: experimental evidence and geometrical-optics interpretation,” Appl. Opt. 40, 4770–4789 (2001).
[CrossRef]

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

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

G. Roy, G. Vallée, M. Jean, “Lidar-inversion technique based on total integrated backscatter calibrated curves,” Appl. Opt. 32, 6754–6763 (1993).
[CrossRef] [PubMed]

L. R. Bissonnette, G. Roy, L. Poutier, S. G. Cober, G. A. Isaac, “Lidar remote sensing of cloud liquid water content and effective droplet diameter: retrieval method and comparison with Monte Carlo simulations and in situ measurements,” TR 2002-20 (Defence Research and Development Canada Establishment Valcartier, 2459 Pie XI Blvd. North, Val-Bélair, Québec G3J 1X5, Canada), to be published.

Sasano, Y.

Sassen, K.

K. Sassen, G. G. Mace, Z. Wang, M. R. Poellet, S. M. Sekelsky, R. E. McIntosh, “Continental stratus clouds: a case study of coordinated remote sensing and aircraft measurements,” J. Atmos. Sci. 56, 2345–2358 (1999).
[CrossRef]

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

Sekelsky, S. M.

K. Sassen, G. G. Mace, Z. Wang, M. R. Poellet, S. M. Sekelsky, R. E. McIntosh, “Continental stratus clouds: a case study of coordinated remote sensing and aircraft measurements,” J. Atmos. Sci. 56, 2345–2358 (1999).
[CrossRef]

Shipley, S. T.

Spinhirne, J. D.

J. D. Spinhirne, J. A. Reagan, B. M. Herman, “Vertical distribution of aerosol extinction cross section and inference of aerosol imaginary index in the troposphere by lidar technique,” J. Appl. Meteorol. 19, 426–438 (1980).
[CrossRef]

Strapp, J. W.

S. G. Cober, G. A. Isaac, A. V. Korolev, J. W. Strapp, “Assessing cloud phase conditions,” J. Appl. Meteorol. 40, 1967–1983 (2001).
[CrossRef]

G. A. Isaac, S. G. Cober, J. W. Strapp, A. V. Korolev, A. Tremblay, D. L. Marcotte, “Recent Canadian research on aircraft in-flight icing,” Can. Aeronaut. Space J. 47(3), 213–221 (2001).

G. A. Isaac, S. G. Cober, J. W. Strapp, D. Hudak, T. P. Ratvasky, D. L. Marcotte, F. Fabry, “Preliminary results from the Alliance Icing Research Study (AIRS),” paper AIAA-2001-0393, presented at the 39th Aerospace Science Meeting and Exhibit, Reno Nevada, 8–11 January 2001, (American Institute of Aeronautics and Astronautics, Reston, Va., 2001).

Stroga, J. T.

Takashima, T.

Tracy, D. H.

Trauger, J. T.

Tremblay, A.

G. A. Isaac, S. G. Cober, J. W. Strapp, A. V. Korolev, A. Tremblay, D. L. Marcotte, “Recent Canadian research on aircraft in-flight icing,” Can. Aeronaut. Space J. 47(3), 213–221 (2001).

Tryon, P. J.

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[CrossRef]

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Other (4)

L. R. Bissonnette, G. Roy, L. Poutier, S. G. Cober, G. A. Isaac, “Lidar remote sensing of cloud liquid water content and effective droplet diameter: retrieval method and comparison with Monte Carlo simulations and in situ measurements,” TR 2002-20 (Defence Research and Development Canada Establishment Valcartier, 2459 Pie XI Blvd. North, Val-Bélair, Québec G3J 1X5, Canada), to be published.

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Figures (15)

Fig. 1
Fig. 1

Schematic of MFOV detection of multiply scattered lidar return: T, receiver telescope; z f , focal distance; OFOV, aperture defining receiver FOV; D, detector; z b , range to cloud base; z, range to backscatter event; ϕ i , forward-scattering angles; ϕ b , backscattering angle; ψ, angle of collected photon; and θ, semiangle of receiver FOV. We achieved MFOV by varying OFOV; different approaches can be used.

Fig. 2
Fig. 2

Same as Fig. 1 except that the multiple scatterings are drawn in the equivalent fictitious medium defined by Katsev et al.19

Fig. 3
Fig. 3

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

Fig. 4
Fig. 4

Comparison of true and retrieved extinction coefficients and effective droplet diameters for Monte Carlo simulated lidar returns from a vertically stratified cloud at 1000 m. Continuous curves for true values and symbols for retrievals: filled for extinction and open for droplet diameter.

Fig. 5
Fig. 5

Retrieved versus true extinction coefficients for a series of Monte Carlo simulated returns from clouds of various extinction coefficients, droplet sizes, profiles, and ranges to cloud base. Retrievals and true values are sampled at a range resolution of 3 m. The straight line represents the least-squares fit to the data points.

Fig. 6
Fig. 6

Retrieved versus true effective droplet diameters for a series of Monte Carlo simulated returns from clouds of various extinction coefficients, droplet sizes, profiles, and ranges to cloud base. Retrievals and true values are sampled at a range resolution of 3 m. The straight line represents the identity line that was drawn for reference.

Fig. 7
Fig. 7

Same as Fig. 2 except that the Monte Carlo simulations were noise corrupted as described by Eq. (27) with σ n = 0.045 and a NEP of 3.4 × 10-6 W/J m2.

Fig. 8
Fig. 8

Example of lidar solutions for LWC and effective droplet diameter d e as functions of height from the base of a stratus cloud deck: (a) instantaneous solutions at 21:00 UTC on 3 December 1999 and (b) averaged solutions over a 1-h period, i.e., 20:50–21:50 UTC on 3 December 1999. The vertical bars indicate ± one standard deviation of temporal fluctuations. The height resolution is 3 m.

Fig. 9
Fig. 9

Horizontal projection of a typical aircraft track for 20:50–21:50 UTC on 3 December 1999. The lidar position is at (0,0).

Fig. 10
Fig. 10

LWC average properties as functions of flight number. The flight numbers, dates, and overlap periods over which averages were calculated are listed in Table 1. In situ data are from the complete layer probed by aircraft: filled triangles, lidar; filled circles, King probe; filled diamonds, Nevzorov probe; filled squares, size spectrometer; (a) average value and (b) standard deviation.

Fig. 11
Fig. 11

Effective droplet diameter average properties as functions of flight number. The flight numbers, dates, and overlap periods over which averages were calculated are listed in Table 1. In situ data are from the complete layer probed by aircraft: filled triangles, lidar; filled squares, size spectrometer; (a) average value and (b) standard deviation.

Fig. 12
Fig. 12

Average and instantaneous LWC profiles for 18:50–20:30 on 6 December 1999, flight 102: continuous curve, average lidar profile; dot–dash curve, average King probe profile; pluses, instantaneous lidar solutions; open circles, instantaneous King probe measurements.

Fig. 13
Fig. 13

Average optical depth profile for 18:50–20:30 on 6 December 1999: solid curve, calculated from the average lidar solution for extinction coefficients; dot–dash curve, calculated from the King probe average profile of Fig. 12 according to Eqs. (29) and (30); dashed curve (top axis), relative number of instantaneous lidar solution points that contribute, at each height, to the calculated average profile.

Fig. 14
Fig. 14

Scatterplot of lidar-derived versus full-layer (a) in situ measurements of LWC and (b) effective droplet diameter. The plotted values are the data of Figs. 10(a) and 11(a). Filled circles, King probe; filled diamonds, Nevzorov probe; filled squares, size spectrometer.

Fig. 15
Fig. 15

Schematic representation of the forward-scattering events that contribute to the average angle θ md subtended at the transceiver by the diffraction-scattered photons.

Tables (2)

Tables Icon

Table 1 Validated Liquid-Phase Data Setsa

Tables Icon

Table 2 Least-Squares Fitted Proportionality Constantsa

Equations (43)

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Pz, θ=PsszMz, θ=Pssz1+Fdz, θ+Fgz, θ,
Pssz=Kz2βzexp-2τz,
pz, ϕ=pdz, ϕ+pgz, ϕ+pbz, ϕ,
pdz, ϕ=12ω1πθd2zexp-ϕ2/θd2z,
pgz, ϕ=A42ω-12ω1πθg2zexp-ϕ2/θg2z,
αz=αsz+αaz,
αsz=αdz+αgz+αbz,
τzln1+1δdPz, θmax-Pz, θminPz, θmin for zzms,
1.2Pz, θmaxPz, θmin1.5,
θd=0.585λ/de,
Pdz, θ=Pssz1+Fdz, θ=Pz, θ-PsszFgz, θ,
Pdz, θ=Pz, θ-Pz, θmaxFgz, θ/1+Fdz, θmax+Fgz, θmax.
Fdz, θmaxδdexpτz-1 for zzms,
Pdz, θ=A+BΦθa/θmda,
a=2-1/1+0.3τ
θd2z=2b2αzzτzθmd2z1+zddzln θmdz,
1+zddzln θmdz0.15z1+zz-zb-3.4if greater than 1+z2/z1+z2/zotherwise,
z1=28/αz,
z2=z1θd¯/352,
αz=1Kz2kzexp2τz×Mz, θminsMz, θmax1+sP1+sz, θmaxPsz, θmin.
M1+sz, θmaxMsz, θmin=exp2τ.
sz=2τz-ln Mz, θmaxln Mz, θmax-ln Mz, θmin.
Pz, θmaxPz, θmin=Mz, θmaxMz, θmin
Pz, θmaxPz, θminMz, θminMz, θmax1,
Sz=1KPz, θMz, θz2kz=αzexp-2τz.
αz=SzSzf/αf+2 zzfSzdz,
Pz, θ=Pmcz, θ1+σnsf1θ+NEPf2z, θ,
LWCh=13ραhdeh,
CIWkh=0hLWCkhdh,
ODkh=3ρdeCIWkh=0.22CIWkh,
θmd=bρ2z,
ρ2z=z0zξzcos ϕzdz2+z0zξzsin ϕzdz2,
ξz1ξz2ξ2z1fξz1-z2,
cos ϕz1cos ϕz2cos2 ϕz1fϕz1-z2,
sin ϕz1sin ϕz2sin2 ϕz1fϕz1-z2,
ρ2z=z0zdz1ξ2z1 z1-zz1-z0dufξufϕu.
lz1=z1-zz1-z0dufξufϕu.
θmd2z=b2z2z0zdz1ξ2z1lz1.
ξ2z=2b2zθmd2zlz1+zddzln θmdz.
ξ2zNzθd2z,
Nz=τz/1-exp-τz,
lz=1-exp-τz/αz.
θd2z=2b2αzzτzθmd2z1+zddzln θmdz.

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