Abstract

An efficient method is described for the approximate calculation of the intensity of multiply scattered lidar returns. It divides the outgoing photons into three populations, representing those that have experienced zero, one, and more than one forward-scattering event. Each population is parameterized at each range gate by its total energy, its spatial variance, the variance of photon direction, and the covariance of photon direction and position. The result is that for an N-point profile the calculation is O(N2) efficient and implicitly includes up to N-order scattering, making it ideal for use in iterative retrieval algorithms for which speed is crucial. In contrast, models that explicitly consider each scattering order separately are at best O(Nm/m!) efficient for m-order scattering and often cannot be performed to more than the third or fourth order in retrieval algorithms. For typical cloud profiles and a wide range of lidar fields of view, the new algorithm is as accurate as an explicit calculation truncated at the fifth or sixth order but faster by several orders of magnitude.

© 2006 Optical Society of America

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    [CrossRef]
  2. M. Chiriaco, H. Chepfer, V. Noel, A. Delaval, M. Haeffelin, P. Dubuisson, and P. Yang, "Improving retrievals of cirrus cloud particle size coupling lidar and three-channel radiometric techniques," Mon. Weath. Rev. 132, 1684-1700 (2004).
    [CrossRef]
  3. D. P. Donovan, A. C. A. P. van Lammeren, H. W. J. Russchenberg, A. Apituley, R. J. Hogan, P. N. Francis, J. Testud, J. Pelon, M. Quante, and J. W. F. Goddard, "Cloud effective particle size and water content profile retrievals using combined lidar and radar observations--2. Comparison with IR radiometer and in situ measurements of ice clouds," J. Geophys. Res. 106(D21), 27449-27464 (2001).
    [CrossRef]
  4. C. Tinel, J. Testud, R. J. Hogan, A. Protat, J. Delanoe, and D. Bouniol, "The retrieval of ice cloud properties from cloud radar and lidar synergy," J. Appl. Meteorol. 44, 860-875 (2005).
    [CrossRef]
  5. R. J. Hogan, D. P. Donovan, C. Tinel, M. A. Brooks, A. J. Illingworth, and J. P. V. Poiares Baptista, "Independent evaluation of the ability of spaceborne radar and lidar to retrieve the microphysical and radiative properties of ice clouds," J. Atmos. Ocean. Technol. 23, 211-227 (2006).
    [CrossRef]
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    [CrossRef]
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  8. G. L. Stephens, D. G. Vane, R. J. Boain, G. G. Mace, K. Sassen, Z. Wang, A. J. Illingworth, E. J. O'Connor, W. B. Rossow, S. L. Durden, S. D. Miller, R. T. Austin, A. Benedetti, C. Mitrescu, and the CloudSat Science Team, "The CloudSat mission and the A-Train," Bull. Am. Meteorol. Soc. 83, 1771-1790 (2002).
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2006

R. J. Hogan, D. P. Donovan, C. Tinel, M. A. Brooks, A. J. Illingworth, and J. P. V. Poiares Baptista, "Independent evaluation of the ability of spaceborne radar and lidar to retrieve the microphysical and radiative properties of ice clouds," J. Atmos. Ocean. Technol. 23, 211-227 (2006).
[CrossRef]

2005

C. Tinel, J. Testud, R. J. Hogan, A. Protat, J. Delanoe, and D. Bouniol, "The retrieval of ice cloud properties from cloud radar and lidar synergy," J. Appl. Meteorol. 44, 860-875 (2005).
[CrossRef]

2004

M. Chiriaco, H. Chepfer, V. Noel, A. Delaval, M. Haeffelin, P. Dubuisson, and P. Yang, "Improving retrievals of cirrus cloud particle size coupling lidar and three-channel radiometric techniques," Mon. Weath. Rev. 132, 1684-1700 (2004).
[CrossRef]

2002

G. L. Stephens, D. G. Vane, R. J. Boain, G. G. Mace, K. Sassen, Z. Wang, A. J. Illingworth, E. J. O'Connor, W. B. Rossow, S. L. Durden, S. D. Miller, R. T. Austin, A. Benedetti, C. Mitrescu, and the CloudSat Science Team, "The CloudSat mission and the A-Train," Bull. Am. Meteorol. Soc. 83, 1771-1790 (2002).
[CrossRef]

2001

D. P. Donovan, A. C. A. P. van Lammeren, H. W. J. Russchenberg, A. Apituley, R. J. Hogan, P. N. Francis, J. Testud, J. Pelon, M. Quante, and J. W. F. Goddard, "Cloud effective particle size and water content profile retrievals using combined lidar and radar observations--2. Comparison with IR radiometer and in situ measurements of ice clouds," J. Geophys. Res. 106(D21), 27449-27464 (2001).
[CrossRef]

1998

1997

1996

1981

C. M. R. Platt and A. C. Dilley, "Remote sounding of high clouds--4. Observed temperature variations in cirrus optical properties," J. Atmos. Sci. 38, 1069-1082 (1981).
[CrossRef]

Apituley, A.

D. P. Donovan, A. C. A. P. van Lammeren, H. W. J. Russchenberg, A. Apituley, R. J. Hogan, P. N. Francis, J. Testud, J. Pelon, M. Quante, and J. W. F. Goddard, "Cloud effective particle size and water content profile retrievals using combined lidar and radar observations--2. Comparison with IR radiometer and in situ measurements of ice clouds," J. Geophys. Res. 106(D21), 27449-27464 (2001).
[CrossRef]

Austin, R. T.

G. L. Stephens, D. G. Vane, R. J. Boain, G. G. Mace, K. Sassen, Z. Wang, A. J. Illingworth, E. J. O'Connor, W. B. Rossow, S. L. Durden, S. D. Miller, R. T. Austin, A. Benedetti, C. Mitrescu, and the CloudSat Science Team, "The CloudSat mission and the A-Train," Bull. Am. Meteorol. Soc. 83, 1771-1790 (2002).
[CrossRef]

Baptista, J. P. V. Poiares

R. J. Hogan, D. P. Donovan, C. Tinel, M. A. Brooks, A. J. Illingworth, and J. P. V. Poiares Baptista, "Independent evaluation of the ability of spaceborne radar and lidar to retrieve the microphysical and radiative properties of ice clouds," J. Atmos. Ocean. Technol. 23, 211-227 (2006).
[CrossRef]

Benedetti, A.

G. L. Stephens, D. G. Vane, R. J. Boain, G. G. Mace, K. Sassen, Z. Wang, A. J. Illingworth, E. J. O'Connor, W. B. Rossow, S. L. Durden, S. D. Miller, R. T. Austin, A. Benedetti, C. Mitrescu, and the CloudSat Science Team, "The CloudSat mission and the A-Train," Bull. Am. Meteorol. Soc. 83, 1771-1790 (2002).
[CrossRef]

Bissonnette, L. R.

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

L. R. Bissonnette, "Lidar and multiple scattering," in Lidar Range-Resolved Optical Remote Sensing of the Atmosphere, C.Weitkamp, ed. (Springer, 2005), pp. 43-103.

Boain, R. J.

G. L. Stephens, D. G. Vane, R. J. Boain, G. G. Mace, K. Sassen, Z. Wang, A. J. Illingworth, E. J. O'Connor, W. B. Rossow, S. L. Durden, S. D. Miller, R. T. Austin, A. Benedetti, C. Mitrescu, and the CloudSat Science Team, "The CloudSat mission and the A-Train," Bull. Am. Meteorol. Soc. 83, 1771-1790 (2002).
[CrossRef]

Bouniol, D.

C. Tinel, J. Testud, R. J. Hogan, A. Protat, J. Delanoe, and D. Bouniol, "The retrieval of ice cloud properties from cloud radar and lidar synergy," J. Appl. Meteorol. 44, 860-875 (2005).
[CrossRef]

Brooks, M. A.

R. J. Hogan, D. P. Donovan, C. Tinel, M. A. Brooks, A. J. Illingworth, and J. P. V. Poiares Baptista, "Independent evaluation of the ability of spaceborne radar and lidar to retrieve the microphysical and radiative properties of ice clouds," J. Atmos. Ocean. Technol. 23, 211-227 (2006).
[CrossRef]

Chepfer, H.

M. Chiriaco, H. Chepfer, V. Noel, A. Delaval, M. Haeffelin, P. Dubuisson, and P. Yang, "Improving retrievals of cirrus cloud particle size coupling lidar and three-channel radiometric techniques," Mon. Weath. Rev. 132, 1684-1700 (2004).
[CrossRef]

Chiriaco, M.

M. Chiriaco, H. Chepfer, V. Noel, A. Delaval, M. Haeffelin, P. Dubuisson, and P. Yang, "Improving retrievals of cirrus cloud particle size coupling lidar and three-channel radiometric techniques," Mon. Weath. Rev. 132, 1684-1700 (2004).
[CrossRef]

Delanoe, J.

C. Tinel, J. Testud, R. J. Hogan, A. Protat, J. Delanoe, and D. Bouniol, "The retrieval of ice cloud properties from cloud radar and lidar synergy," J. Appl. Meteorol. 44, 860-875 (2005).
[CrossRef]

Delaval, A.

M. Chiriaco, H. Chepfer, V. Noel, A. Delaval, M. Haeffelin, P. Dubuisson, and P. Yang, "Improving retrievals of cirrus cloud particle size coupling lidar and three-channel radiometric techniques," Mon. Weath. Rev. 132, 1684-1700 (2004).
[CrossRef]

Dilley, A. C.

C. M. R. Platt and A. C. Dilley, "Remote sounding of high clouds--4. Observed temperature variations in cirrus optical properties," J. Atmos. Sci. 38, 1069-1082 (1981).
[CrossRef]

Donovan, D. P.

R. J. Hogan, D. P. Donovan, C. Tinel, M. A. Brooks, A. J. Illingworth, and J. P. V. Poiares Baptista, "Independent evaluation of the ability of spaceborne radar and lidar to retrieve the microphysical and radiative properties of ice clouds," J. Atmos. Ocean. Technol. 23, 211-227 (2006).
[CrossRef]

D. P. Donovan, A. C. A. P. van Lammeren, H. W. J. Russchenberg, A. Apituley, R. J. Hogan, P. N. Francis, J. Testud, J. Pelon, M. Quante, and J. W. F. Goddard, "Cloud effective particle size and water content profile retrievals using combined lidar and radar observations--2. Comparison with IR radiometer and in situ measurements of ice clouds," J. Geophys. Res. 106(D21), 27449-27464 (2001).
[CrossRef]

Dubuisson, P.

M. Chiriaco, H. Chepfer, V. Noel, A. Delaval, M. Haeffelin, P. Dubuisson, and P. Yang, "Improving retrievals of cirrus cloud particle size coupling lidar and three-channel radiometric techniques," Mon. Weath. Rev. 132, 1684-1700 (2004).
[CrossRef]

Durden, S. L.

G. L. Stephens, D. G. Vane, R. J. Boain, G. G. Mace, K. Sassen, Z. Wang, A. J. Illingworth, E. J. O'Connor, W. B. Rossow, S. L. Durden, S. D. Miller, R. T. Austin, A. Benedetti, C. Mitrescu, and the CloudSat Science Team, "The CloudSat mission and the A-Train," Bull. Am. Meteorol. Soc. 83, 1771-1790 (2002).
[CrossRef]

Eloranta, E. W.

Francis, P. N.

D. P. Donovan, A. C. A. P. van Lammeren, H. W. J. Russchenberg, A. Apituley, R. J. Hogan, P. N. Francis, J. Testud, J. Pelon, M. Quante, and J. W. F. Goddard, "Cloud effective particle size and water content profile retrievals using combined lidar and radar observations--2. Comparison with IR radiometer and in situ measurements of ice clouds," J. Geophys. Res. 106(D21), 27449-27464 (2001).
[CrossRef]

Goddard, J. W. F.

D. P. Donovan, A. C. A. P. van Lammeren, H. W. J. Russchenberg, A. Apituley, R. J. Hogan, P. N. Francis, J. Testud, J. Pelon, M. Quante, and J. W. F. Goddard, "Cloud effective particle size and water content profile retrievals using combined lidar and radar observations--2. Comparison with IR radiometer and in situ measurements of ice clouds," J. Geophys. Res. 106(D21), 27449-27464 (2001).
[CrossRef]

Haeffelin, M.

M. Chiriaco, H. Chepfer, V. Noel, A. Delaval, M. Haeffelin, P. Dubuisson, and P. Yang, "Improving retrievals of cirrus cloud particle size coupling lidar and three-channel radiometric techniques," Mon. Weath. Rev. 132, 1684-1700 (2004).
[CrossRef]

Hogan, R. J.

R. J. Hogan, D. P. Donovan, C. Tinel, M. A. Brooks, A. J. Illingworth, and J. P. V. Poiares Baptista, "Independent evaluation of the ability of spaceborne radar and lidar to retrieve the microphysical and radiative properties of ice clouds," J. Atmos. Ocean. Technol. 23, 211-227 (2006).
[CrossRef]

C. Tinel, J. Testud, R. J. Hogan, A. Protat, J. Delanoe, and D. Bouniol, "The retrieval of ice cloud properties from cloud radar and lidar synergy," J. Appl. Meteorol. 44, 860-875 (2005).
[CrossRef]

D. P. Donovan, A. C. A. P. van Lammeren, H. W. J. Russchenberg, A. Apituley, R. J. Hogan, P. N. Francis, J. Testud, J. Pelon, M. Quante, and J. W. F. Goddard, "Cloud effective particle size and water content profile retrievals using combined lidar and radar observations--2. Comparison with IR radiometer and in situ measurements of ice clouds," J. Geophys. Res. 106(D21), 27449-27464 (2001).
[CrossRef]

Illingworth, A. J.

R. J. Hogan, D. P. Donovan, C. Tinel, M. A. Brooks, A. J. Illingworth, and J. P. V. Poiares Baptista, "Independent evaluation of the ability of spaceborne radar and lidar to retrieve the microphysical and radiative properties of ice clouds," J. Atmos. Ocean. Technol. 23, 211-227 (2006).
[CrossRef]

G. L. Stephens, D. G. Vane, R. J. Boain, G. G. Mace, K. Sassen, Z. Wang, A. J. Illingworth, E. J. O'Connor, W. B. Rossow, S. L. Durden, S. D. Miller, R. T. Austin, A. Benedetti, C. Mitrescu, and the CloudSat Science Team, "The CloudSat mission and the A-Train," Bull. Am. Meteorol. Soc. 83, 1771-1790 (2002).
[CrossRef]

Katsev, I. L.

Mace, G. G.

G. L. Stephens, D. G. Vane, R. J. Boain, G. G. Mace, K. Sassen, Z. Wang, A. J. Illingworth, E. J. O'Connor, W. B. Rossow, S. L. Durden, S. D. Miller, R. T. Austin, A. Benedetti, C. Mitrescu, and the CloudSat Science Team, "The CloudSat mission and the A-Train," Bull. Am. Meteorol. Soc. 83, 1771-1790 (2002).
[CrossRef]

McCormick, M. P.

D. M. Winker, J. Pelon, and M. P. McCormick, "The CALIPSO mission: spaceborne lidar for observation of aerosols and clouds," in Proc. SPIE 4893,1-11 (2003).

Miller, S. D.

G. L. Stephens, D. G. Vane, R. J. Boain, G. G. Mace, K. Sassen, Z. Wang, A. J. Illingworth, E. J. O'Connor, W. B. Rossow, S. L. Durden, S. D. Miller, R. T. Austin, A. Benedetti, C. Mitrescu, and the CloudSat Science Team, "The CloudSat mission and the A-Train," Bull. Am. Meteorol. Soc. 83, 1771-1790 (2002).
[CrossRef]

Mitrescu, C.

G. L. Stephens, D. G. Vane, R. J. Boain, G. G. Mace, K. Sassen, Z. Wang, A. J. Illingworth, E. J. O'Connor, W. B. Rossow, S. L. Durden, S. D. Miller, R. T. Austin, A. Benedetti, C. Mitrescu, and the CloudSat Science Team, "The CloudSat mission and the A-Train," Bull. Am. Meteorol. Soc. 83, 1771-1790 (2002).
[CrossRef]

Noel, V.

M. Chiriaco, H. Chepfer, V. Noel, A. Delaval, M. Haeffelin, P. Dubuisson, and P. Yang, "Improving retrievals of cirrus cloud particle size coupling lidar and three-channel radiometric techniques," Mon. Weath. Rev. 132, 1684-1700 (2004).
[CrossRef]

O'Connor, E. J.

G. L. Stephens, D. G. Vane, R. J. Boain, G. G. Mace, K. Sassen, Z. Wang, A. J. Illingworth, E. J. O'Connor, W. B. Rossow, S. L. Durden, S. D. Miller, R. T. Austin, A. Benedetti, C. Mitrescu, and the CloudSat Science Team, "The CloudSat mission and the A-Train," Bull. Am. Meteorol. Soc. 83, 1771-1790 (2002).
[CrossRef]

Pelon, J.

D. P. Donovan, A. C. A. P. van Lammeren, H. W. J. Russchenberg, A. Apituley, R. J. Hogan, P. N. Francis, J. Testud, J. Pelon, M. Quante, and J. W. F. Goddard, "Cloud effective particle size and water content profile retrievals using combined lidar and radar observations--2. Comparison with IR radiometer and in situ measurements of ice clouds," J. Geophys. Res. 106(D21), 27449-27464 (2001).
[CrossRef]

D. M. Winker, J. Pelon, and M. P. McCormick, "The CALIPSO mission: spaceborne lidar for observation of aerosols and clouds," in Proc. SPIE 4893,1-11 (2003).

Platt, C. M. R.

C. M. R. Platt and A. C. Dilley, "Remote sounding of high clouds--4. Observed temperature variations in cirrus optical properties," J. Atmos. Sci. 38, 1069-1082 (1981).
[CrossRef]

Polonsky, I. N.

Prikhach, A. S.

Protat, A.

C. Tinel, J. Testud, R. J. Hogan, A. Protat, J. Delanoe, and D. Bouniol, "The retrieval of ice cloud properties from cloud radar and lidar synergy," J. Appl. Meteorol. 44, 860-875 (2005).
[CrossRef]

Quante, M.

D. P. Donovan, A. C. A. P. van Lammeren, H. W. J. Russchenberg, A. Apituley, R. J. Hogan, P. N. Francis, J. Testud, J. Pelon, M. Quante, and J. W. F. Goddard, "Cloud effective particle size and water content profile retrievals using combined lidar and radar observations--2. Comparison with IR radiometer and in situ measurements of ice clouds," J. Geophys. Res. 106(D21), 27449-27464 (2001).
[CrossRef]

Rodgers, C. D.

C. D. Rodgers, Inverse Methods for Atmospheric Sounding: Theory and Practice (World Scientific, 2000).
[CrossRef]

Rossow, W. B.

G. L. Stephens, D. G. Vane, R. J. Boain, G. G. Mace, K. Sassen, Z. Wang, A. J. Illingworth, E. J. O'Connor, W. B. Rossow, S. L. Durden, S. D. Miller, R. T. Austin, A. Benedetti, C. Mitrescu, and the CloudSat Science Team, "The CloudSat mission and the A-Train," Bull. Am. Meteorol. Soc. 83, 1771-1790 (2002).
[CrossRef]

Russchenberg, H. W. J.

D. P. Donovan, A. C. A. P. van Lammeren, H. W. J. Russchenberg, A. Apituley, R. J. Hogan, P. N. Francis, J. Testud, J. Pelon, M. Quante, and J. W. F. Goddard, "Cloud effective particle size and water content profile retrievals using combined lidar and radar observations--2. Comparison with IR radiometer and in situ measurements of ice clouds," J. Geophys. Res. 106(D21), 27449-27464 (2001).
[CrossRef]

Sassen, K.

G. L. Stephens, D. G. Vane, R. J. Boain, G. G. Mace, K. Sassen, Z. Wang, A. J. Illingworth, E. J. O'Connor, W. B. Rossow, S. L. Durden, S. D. Miller, R. T. Austin, A. Benedetti, C. Mitrescu, and the CloudSat Science Team, "The CloudSat mission and the A-Train," Bull. Am. Meteorol. Soc. 83, 1771-1790 (2002).
[CrossRef]

Stephens, G. L.

G. L. Stephens, D. G. Vane, R. J. Boain, G. G. Mace, K. Sassen, Z. Wang, A. J. Illingworth, E. J. O'Connor, W. B. Rossow, S. L. Durden, S. D. Miller, R. T. Austin, A. Benedetti, C. Mitrescu, and the CloudSat Science Team, "The CloudSat mission and the A-Train," Bull. Am. Meteorol. Soc. 83, 1771-1790 (2002).
[CrossRef]

Testud, J.

C. Tinel, J. Testud, R. J. Hogan, A. Protat, J. Delanoe, and D. Bouniol, "The retrieval of ice cloud properties from cloud radar and lidar synergy," J. Appl. Meteorol. 44, 860-875 (2005).
[CrossRef]

D. P. Donovan, A. C. A. P. van Lammeren, H. W. J. Russchenberg, A. Apituley, R. J. Hogan, P. N. Francis, J. Testud, J. Pelon, M. Quante, and J. W. F. Goddard, "Cloud effective particle size and water content profile retrievals using combined lidar and radar observations--2. Comparison with IR radiometer and in situ measurements of ice clouds," J. Geophys. Res. 106(D21), 27449-27464 (2001).
[CrossRef]

Tinel, C.

R. J. Hogan, D. P. Donovan, C. Tinel, M. A. Brooks, A. J. Illingworth, and J. P. V. Poiares Baptista, "Independent evaluation of the ability of spaceborne radar and lidar to retrieve the microphysical and radiative properties of ice clouds," J. Atmos. Ocean. Technol. 23, 211-227 (2006).
[CrossRef]

C. Tinel, J. Testud, R. J. Hogan, A. Protat, J. Delanoe, and D. Bouniol, "The retrieval of ice cloud properties from cloud radar and lidar synergy," J. Appl. Meteorol. 44, 860-875 (2005).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

van Lammeren, A. C. A. P.

D. P. Donovan, A. C. A. P. van Lammeren, H. W. J. Russchenberg, A. Apituley, R. J. Hogan, P. N. Francis, J. Testud, J. Pelon, M. Quante, and J. W. F. Goddard, "Cloud effective particle size and water content profile retrievals using combined lidar and radar observations--2. Comparison with IR radiometer and in situ measurements of ice clouds," J. Geophys. Res. 106(D21), 27449-27464 (2001).
[CrossRef]

Vane, D. G.

G. L. Stephens, D. G. Vane, R. J. Boain, G. G. Mace, K. Sassen, Z. Wang, A. J. Illingworth, E. J. O'Connor, W. B. Rossow, S. L. Durden, S. D. Miller, R. T. Austin, A. Benedetti, C. Mitrescu, and the CloudSat Science Team, "The CloudSat mission and the A-Train," Bull. Am. Meteorol. Soc. 83, 1771-1790 (2002).
[CrossRef]

Wang, Z.

G. L. Stephens, D. G. Vane, R. J. Boain, G. G. Mace, K. Sassen, Z. Wang, A. J. Illingworth, E. J. O'Connor, W. B. Rossow, S. L. Durden, S. D. Miller, R. T. Austin, A. Benedetti, C. Mitrescu, and the CloudSat Science Team, "The CloudSat mission and the A-Train," Bull. Am. Meteorol. Soc. 83, 1771-1790 (2002).
[CrossRef]

Winker, D. M.

D. M. Winker, J. Pelon, and M. P. McCormick, "The CALIPSO mission: spaceborne lidar for observation of aerosols and clouds," in Proc. SPIE 4893,1-11 (2003).

Yang, P.

M. Chiriaco, H. Chepfer, V. Noel, A. Delaval, M. Haeffelin, P. Dubuisson, and P. Yang, "Improving retrievals of cirrus cloud particle size coupling lidar and three-channel radiometric techniques," Mon. Weath. Rev. 132, 1684-1700 (2004).
[CrossRef]

Zege, E. P.

Appl. Opt.

Bull. Am. Meteorol. Soc.

G. L. Stephens, D. G. Vane, R. J. Boain, G. G. Mace, K. Sassen, Z. Wang, A. J. Illingworth, E. J. O'Connor, W. B. Rossow, S. L. Durden, S. D. Miller, R. T. Austin, A. Benedetti, C. Mitrescu, and the CloudSat Science Team, "The CloudSat mission and the A-Train," Bull. Am. Meteorol. Soc. 83, 1771-1790 (2002).
[CrossRef]

J. Appl. Meteorol.

C. Tinel, J. Testud, R. J. Hogan, A. Protat, J. Delanoe, and D. Bouniol, "The retrieval of ice cloud properties from cloud radar and lidar synergy," J. Appl. Meteorol. 44, 860-875 (2005).
[CrossRef]

J. Atmos. Ocean. Technol.

R. J. Hogan, D. P. Donovan, C. Tinel, M. A. Brooks, A. J. Illingworth, and J. P. V. Poiares Baptista, "Independent evaluation of the ability of spaceborne radar and lidar to retrieve the microphysical and radiative properties of ice clouds," J. Atmos. Ocean. Technol. 23, 211-227 (2006).
[CrossRef]

J. Atmos. Sci.

C. M. R. Platt and A. C. Dilley, "Remote sounding of high clouds--4. Observed temperature variations in cirrus optical properties," J. Atmos. Sci. 38, 1069-1082 (1981).
[CrossRef]

J. Geophys. Res.

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M. Chiriaco, H. Chepfer, V. Noel, A. Delaval, M. Haeffelin, P. Dubuisson, and P. Yang, "Improving retrievals of cirrus cloud particle size coupling lidar and three-channel radiometric techniques," Mon. Weath. Rev. 132, 1684-1700 (2004).
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D. M. Winker, J. Pelon, and M. P. McCormick, "The CALIPSO mission: spaceborne lidar for observation of aerosols and clouds," in Proc. SPIE 4893,1-11 (2003).

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Figures (5)

Fig. 1
Fig. 1

Schematic showing the trajectory of a photon emitted by the laser such that when it arrives at range r i it has a lateral displacement s i and an angle ζ i relative to the lidar axis. It is then scattered by an angle θ i such that on reaching range r j it has lateral displacement s j and angle ζ j . In Cartesian coordinates the displacement is written as ( x j , y j ).

Fig. 2
Fig. 2

Schematic illustrating the representation of the outgoing photon distribution E as the sum of three populations. In this example there are three infinitesimally thin clouds A–C, each with an optical depth (in the equivalent medium) of 0.69 such that, of the incoming radiation, half is unscattered, one fourth is forward scattered and is transferred to the next population (indicated by the dotted lines), and the remaining one fourth is scattered in other directions and lost to the system (except for the very small fraction that is backscattered to the detector). The numbers in bold indicate the fractions of the initial energy (i.e., P / P 0 ) that remain at each stage. The thin solid lines depict the standard deviations of each population, with the notches in the Es and Em cones indicating where the standard deviation has been reduced by the addition of photons from one of the other populations. The three distributions in gray at the top show the representation of each outgoing population by a Gaussian with the correct mean and standard deviation.

Fig. 3
Fig. 3

Comparison of the new algorithm and the Eloranta (Ref. 9) formulation for ground-based 532 nm lidar observing an ice cloud with extinction coefficients of 2, 0.5, 0.1, and 0.5 km−1 in the height ranges 4–5, 5–6, 6–7, and 7–8 km, respectively. The black lines show the apparent backscatter estimated by the new algorithm, β ^ (thick line), together with the theoretical extremes of single scattering only (thin solid line), and all the forward-scattered photons remaining in the telescope field of view (dotted line). The gray lines show the predictions by the Eloranta model including successively higher orders of scattering.

Fig. 4
Fig. 4

As in Fig. 3, but showing the three separate contributions toward the apparent backscatter in the new algorithm: The single- and double-scattering components, β ^ 1   and   β ^ 2 , are common between the new algorithm and that of Eloranta so only one line is shown for each. The prediction of third- and higher-order scattering by the new algorithm, β ^ 3 + (black dashed line), is compared with the same prediction by Eloranta's model for scattering up to the seventh order (gray dashed line).

Fig. 5
Fig. 5

As in Fig. 3, but for the spaceborne 532   mm CALIPSO lidar at an altitude of 700   km . The ice cloud between 4 and 7 km has a constant extinction coefficient of 1 km−1 and aG of 100 μm. It overlies a 1 km thick aerosol layer with an extinction coefficient of 0.5 km−1 and aG of 0.5 μm. The crosses show the apparent backscatter when forward scattering from the aerosol layer is included in the calculation of β ^ 3 + , i.e., without the aerosol adjustment described at the end of Section 5.

Tables (1)

Tables Icon

Table 1 Execution Time of the Various Algorithms Relative to the Time for the New Algorithm to Compute a 25-Point Profile a , b

Equations (28)

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E u ( R , s ) = P 0 exp [ 2 δ ( R ) ] πρ l     2 R 2 exp ( s 2 ρ l     2 R 2 ) ,
d E s ( R , s ) = P 0 exp [ 2 δ ( R ) ] π ρ l     2 R 2 + π Θ 2 ( R r ) 2 × exp [ s 2 ρ l     2 R 2 + Θ 2 ( R r ) 2 ] α ( r ) d r ,
P ( θ ) P ( 0 ) = [ F ( θ ) ] 2 = [ 2 J 1 ( 2 π θ a / λ ) 2 π θ a / λ ] 2 ,
Θ 2 = λ 2 π 0 n ( G ) σ e ( G ) G d G / 0 n ( G ) σ e ( G ) d G ,
Θ 2 = λ 2 π G , or Θ = λ π a G ,
d Q ( R ) = A t β ( R ) d R R 2 0 2 π 0 ρ t R E ( R , s ) s d s d ϕ = 2 π A t β ( R ) d R R 2 0 ρ t R E ( R , s ) s d s ,
β ^ 1 ( R ) = β ( R ) exp [ 2 δ ( R ) ] .
β ^ 1 , i = 1 Δ R R R + Δ R β ( R ) exp [ 2 δ ( R ) ] d R
= β i exp [ 2 δ ( R ) ] 1 exp ( 2 α i Δ R ) 2 α i Δ R .
β ^ 2 ( R ) β ^ 1 ( R ) = d Q s d Q u = 0 ρ t R E s ( R , s ) s d s / 0 ρ t R E u ( R , s ) s d s ,
β ^ 2 ( R ) / β ^ 1 ( R ) = [ 1 exp ( ρ t 2 / ρ l 2 ) ] 1 0 R { 1 exp [ - ρ t 2 R 2 ρ l 2 R 2 + Θ i 2 ( R r ) 2 ] } α ( r ) d r .
P k ( R ) = 2 π 0 E k ( R , s ) s d s .
s 2 ¯ k ( R ) = 2 π P k ( R ) 0 s 2 E k ( R , s ) s d s .
P ^ u = 1 , s 2 ¯ u = r 2 ρ l 2 , ζ 2 ¯ u = ρ l 2 , s ζ ¯ u = r ρ l 2 .
Δ P ^ j = P ^ i α i Δ r ,
Δ s 2 ¯ j = s 2 ¯ i + ( ζ 2 ¯ i + Θ i 2 ) ( r j r i ) 2 + 2 s ζ ¯ i ( r j r i ) ,
Δ ζ 2 ¯ j = ζ 2 ¯ i + Θ i 2 ,
Δ s ζ ¯ j = s ζ ¯ i + ( ζ 2 ¯ i + Θ i 2 ) ( r j r i ) .
x j = x i + ( ζ x , i + θ x , i ) ( r j r i )
x j 2 ¯ = x i 2 ¯ + ( ζ x , i 2 ¯ + θ x , i 2 ¯ ) ( r j r i ) 2 + 2 x i ζ x , i ¯ ( r j r i ) ,
Δ P ^ s , j = α i Δ r ,
Δ s 2 ¯ s , j = ρ l     2 r j     2 + Θ i 2 ( r j r i ) 2 ,
Δ ζ 2 ¯ s , j = ρ l     2 + Θ i 2 ,
Δ s ζ ¯ s , j = ρ l     2 r j + Θ i 2 ( r j r i ) .
P ^ j P ^ j + Δ P ^ j .
( P ^ j s 2 ¯ j ) ( P ^ j s 2 ¯ j ) + Δ P ^ j × Δ s 2 ¯ j ,
E m ( R , s ) = P m π s 2 ¯ m exp ( s 2 s 2 ¯ m ) ,
β ^ 3 + ( R ) β ^ 1 ( R ) = d Q m d Q u = P ^ m 1 exp ( ρ t 2 R 2 / s 2 ¯ m ) 1 exp ( ρ t 2 / ρ l 2 ) .

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