Abstract

Potter [Appl. Opt. 26, 1250 (1987)] has presented a method to determine profiles of the atmospheric aerosol extinction coefficients by use of a two-wavelength lidar with the assumptions of a constant value for the extinction-to-backscatter ratio for each wavelength and a constant value for the ratio between the two extinction coefficients at the two wavelengths. Triggered by this idea, Ackermann [Appl. Opt. 36, 5134 (1997)] expanded this method to consider lidar returns that are a composition of scattering by atmospheric aerosols and molecules, assuming that the molecular scattering is known. In both papers the method is based on the well-known solutions of Bernoulli’s differential equation in an iterative scheme with an unknown boundary transmission condition. This boundary condition is less sensitive to noise than boundary extinction conditions. My main purpose is to critically consider the principle behind Potter’s method, because it seems that there are several reasons why the number of solutions is not limited to one, as suggested by his original work.

© 1999 Optical Society of America

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References

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  1. W. Hitschfeld, J. Bordan, “Errors inherent in the radar measurement of rainfall at attenuating wavelengths,” J. Meteorol. 11, 58–67 (1954).
    [CrossRef]
  2. P. A. Davis, “The analysis of lidar signatures of cirrus clouds,” Appl. Opt. 8, 2099–2102 (1969).
    [CrossRef] [PubMed]
  3. W. Viezee, E. E. Uthe, R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploratory study,” J. Appl. Meteorol. 8, 274–283 (1969).
    [CrossRef]
  4. R. H. Kohl, “Discussion of the interpretation problem encountered in single-wavelength lidar transmissometers,” J. Appl. Meteorol. 17, 1034–1038 (1978).
    [CrossRef]
  5. J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1981).
    [CrossRef] [PubMed]
  6. F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
    [CrossRef]
  7. V. A. Kovalev, “Lidar measurement of the vertical aerosol extinction profiles with range-dependent backscatter-to-extinction ratios,” Appl. Opt. 32, 6053–6065 (1993).
    [CrossRef] [PubMed]
  8. E. E. Uthe, J. M. Livingston, “Lidar extinction methods applied to observations of obscurant events,” Appl. Opt. 25, 678–684 (1986).
    [CrossRef] [PubMed]
  9. Yu. Balin, S. I. Kavkyanov, G. M. Krekov, I. A. Razenkov, “Noise-proof inversion of lidar equation,” Opt. Lett. 12, 13–15 (1987).
    [CrossRef] [PubMed]
  10. J. A. Weinman, “Derivation of atmospheric extinction profiles and wind speed over the ocean from a satellite-borne lidar,” Appl. Opt. 27, 3994–4001 (1988).
    [CrossRef] [PubMed]
  11. G. Roy, G. Vallée, M. Jean, “Lidar-inversion technique based on total integrated backscatter calibrated curves,” Appl. Opt. 32, 6754–6763 (1993).
    [CrossRef] [PubMed]
  12. G. J. Kunz, “Transmission as input boundary value for an analytical solution of the single scatter lidar equation,” Appl. Opt. 35, 3255–3260 (1996).
    [CrossRef] [PubMed]
  13. J. D. Klett, “Extinction boundary value algorithms for lidar inversion,” Appl. Opt. 25, 2462–2464 (1986).
    [CrossRef] [PubMed]
  14. J. A. Ferguson, D. H. Stephens, “Algorithm for inverting lidar returns,” Appl. Opt. 22, 3673–3675 (1983).
    [CrossRef] [PubMed]
  15. J. F. Potter, “Two-frequency lidar inversion technique,” Appl. Opt. 26, 1250–1256 (1987).
    [CrossRef] [PubMed]
  16. J. Ackermann, “Two-wavelength lidar inversion algorithm for a two-component atmosphere,” Appl. Opt. 36, 5134–5143 (1997).
    [CrossRef] [PubMed]
  17. Y. Sasano, E. Browell, “Light scattering characteristics of various aerosol types derived from multiple wavelength lidar observation,” Appl. Opt. 28, 1670–1679 (1989).
    [CrossRef] [PubMed]
  18. P. Qing, H. Nakane, Y. Sasano, S. Kitamura, “Numerical simulation of the retrieval of aerosol size distribution from multiwavelength laser radar measurements,” Appl. Opt. 28, 5259–5265 (1989).
    [CrossRef] [PubMed]
  19. A. Papayannis, G. Ancellet, J. Pelon, G. Mégie, “Multiwavelength lidar for ozone measurements in the troposphere and the lower stratosphere,” Appl. Opt. 29, 467–476 (1990).
    [CrossRef] [PubMed]
  20. A. Papayannis, H. Kambezidis, D. Asimakopoulos, “Development of a mobile three-dimensional scanning lidar system for aerosol monitoring in rural areas in Greece,” Int. J. Remote Sens. 15, 361–368 (1994).
    [CrossRef]

1997 (1)

1996 (1)

1994 (1)

A. Papayannis, H. Kambezidis, D. Asimakopoulos, “Development of a mobile three-dimensional scanning lidar system for aerosol monitoring in rural areas in Greece,” Int. J. Remote Sens. 15, 361–368 (1994).
[CrossRef]

1993 (2)

1990 (1)

1989 (2)

1988 (1)

1987 (2)

1986 (2)

1983 (1)

1981 (1)

1978 (1)

R. H. Kohl, “Discussion of the interpretation problem encountered in single-wavelength lidar transmissometers,” J. Appl. Meteorol. 17, 1034–1038 (1978).
[CrossRef]

1972 (1)

F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
[CrossRef]

1969 (2)

P. A. Davis, “The analysis of lidar signatures of cirrus clouds,” Appl. Opt. 8, 2099–2102 (1969).
[CrossRef] [PubMed]

W. Viezee, E. E. Uthe, R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploratory study,” J. Appl. Meteorol. 8, 274–283 (1969).
[CrossRef]

1954 (1)

W. Hitschfeld, J. Bordan, “Errors inherent in the radar measurement of rainfall at attenuating wavelengths,” J. Meteorol. 11, 58–67 (1954).
[CrossRef]

Ackermann, J.

Ancellet, G.

Asimakopoulos, D.

A. Papayannis, H. Kambezidis, D. Asimakopoulos, “Development of a mobile three-dimensional scanning lidar system for aerosol monitoring in rural areas in Greece,” Int. J. Remote Sens. 15, 361–368 (1994).
[CrossRef]

Balin, Yu.

Bordan, J.

W. Hitschfeld, J. Bordan, “Errors inherent in the radar measurement of rainfall at attenuating wavelengths,” J. Meteorol. 11, 58–67 (1954).
[CrossRef]

Browell, E.

Collis, R. T. H.

W. Viezee, E. E. Uthe, R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploratory study,” J. Appl. Meteorol. 8, 274–283 (1969).
[CrossRef]

Davis, P. A.

Ferguson, J. A.

Fernald, F. G.

F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
[CrossRef]

Herman, B. M.

F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
[CrossRef]

Hitschfeld, W.

W. Hitschfeld, J. Bordan, “Errors inherent in the radar measurement of rainfall at attenuating wavelengths,” J. Meteorol. 11, 58–67 (1954).
[CrossRef]

Jean, M.

Kambezidis, H.

A. Papayannis, H. Kambezidis, D. Asimakopoulos, “Development of a mobile three-dimensional scanning lidar system for aerosol monitoring in rural areas in Greece,” Int. J. Remote Sens. 15, 361–368 (1994).
[CrossRef]

Kavkyanov, S. I.

Kitamura, S.

Klett, J. D.

Kohl, R. H.

R. H. Kohl, “Discussion of the interpretation problem encountered in single-wavelength lidar transmissometers,” J. Appl. Meteorol. 17, 1034–1038 (1978).
[CrossRef]

Kovalev, V. A.

Krekov, G. M.

Kunz, G. J.

Livingston, J. M.

Mégie, G.

Nakane, H.

Papayannis, A.

A. Papayannis, H. Kambezidis, D. Asimakopoulos, “Development of a mobile three-dimensional scanning lidar system for aerosol monitoring in rural areas in Greece,” Int. J. Remote Sens. 15, 361–368 (1994).
[CrossRef]

A. Papayannis, G. Ancellet, J. Pelon, G. Mégie, “Multiwavelength lidar for ozone measurements in the troposphere and the lower stratosphere,” Appl. Opt. 29, 467–476 (1990).
[CrossRef] [PubMed]

Pelon, J.

Potter, J. F.

Qing, P.

Razenkov, I. A.

Reagan, J. A.

F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
[CrossRef]

Roy, G.

Sasano, Y.

Stephens, D. H.

Uthe, E. E.

E. E. Uthe, J. M. Livingston, “Lidar extinction methods applied to observations of obscurant events,” Appl. Opt. 25, 678–684 (1986).
[CrossRef] [PubMed]

W. Viezee, E. E. Uthe, R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploratory study,” J. Appl. Meteorol. 8, 274–283 (1969).
[CrossRef]

Vallée, G.

Viezee, W.

W. Viezee, E. E. Uthe, R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploratory study,” J. Appl. Meteorol. 8, 274–283 (1969).
[CrossRef]

Weinman, J. A.

Appl. Opt. (14)

J. A. Weinman, “Derivation of atmospheric extinction profiles and wind speed over the ocean from a satellite-borne lidar,” Appl. Opt. 27, 3994–4001 (1988).
[CrossRef] [PubMed]

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

G. J. Kunz, “Transmission as input boundary value for an analytical solution of the single scatter lidar equation,” Appl. Opt. 35, 3255–3260 (1996).
[CrossRef] [PubMed]

J. D. Klett, “Extinction boundary value algorithms for lidar inversion,” Appl. Opt. 25, 2462–2464 (1986).
[CrossRef] [PubMed]

J. A. Ferguson, D. H. Stephens, “Algorithm for inverting lidar returns,” Appl. Opt. 22, 3673–3675 (1983).
[CrossRef] [PubMed]

J. F. Potter, “Two-frequency lidar inversion technique,” Appl. Opt. 26, 1250–1256 (1987).
[CrossRef] [PubMed]

J. Ackermann, “Two-wavelength lidar inversion algorithm for a two-component atmosphere,” Appl. Opt. 36, 5134–5143 (1997).
[CrossRef] [PubMed]

Y. Sasano, E. Browell, “Light scattering characteristics of various aerosol types derived from multiple wavelength lidar observation,” Appl. Opt. 28, 1670–1679 (1989).
[CrossRef] [PubMed]

P. Qing, H. Nakane, Y. Sasano, S. Kitamura, “Numerical simulation of the retrieval of aerosol size distribution from multiwavelength laser radar measurements,” Appl. Opt. 28, 5259–5265 (1989).
[CrossRef] [PubMed]

A. Papayannis, G. Ancellet, J. Pelon, G. Mégie, “Multiwavelength lidar for ozone measurements in the troposphere and the lower stratosphere,” Appl. Opt. 29, 467–476 (1990).
[CrossRef] [PubMed]

P. A. Davis, “The analysis of lidar signatures of cirrus clouds,” Appl. Opt. 8, 2099–2102 (1969).
[CrossRef] [PubMed]

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

V. A. Kovalev, “Lidar measurement of the vertical aerosol extinction profiles with range-dependent backscatter-to-extinction ratios,” Appl. Opt. 32, 6053–6065 (1993).
[CrossRef] [PubMed]

E. E. Uthe, J. M. Livingston, “Lidar extinction methods applied to observations of obscurant events,” Appl. Opt. 25, 678–684 (1986).
[CrossRef] [PubMed]

Int. J. Remote Sens. (1)

A. Papayannis, H. Kambezidis, D. Asimakopoulos, “Development of a mobile three-dimensional scanning lidar system for aerosol monitoring in rural areas in Greece,” Int. J. Remote Sens. 15, 361–368 (1994).
[CrossRef]

J. Appl. Meteorol. (3)

F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
[CrossRef]

W. Viezee, E. E. Uthe, R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploratory study,” J. Appl. Meteorol. 8, 274–283 (1969).
[CrossRef]

R. H. Kohl, “Discussion of the interpretation problem encountered in single-wavelength lidar transmissometers,” J. Appl. Meteorol. 17, 1034–1038 (1978).
[CrossRef]

J. Meteorol. (1)

W. Hitschfeld, J. Bordan, “Errors inherent in the radar measurement of rainfall at attenuating wavelengths,” J. Meteorol. 11, 58–67 (1954).
[CrossRef]

Opt. Lett. (1)

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Equations (39)

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LLRTL0, R0βLRexp-2 R0R αLrdr,
LSRTS0, R0βSRexp-2 R0R αSrdr,
SLαLRβLR,
SSαSRβSR.
kαSRαLR.
αLR=LLRBL-2 R0R LLrdr,
αSR=LSRBS-2 R0R LSrdr,
TS0, R0=TLk0, R0.
LLR=TL0, R0αLRSL exp-2 R0R αLrdr,
LSR=TLk0, R0kαLRSS exp-2k R0R αLrdr.
LLRkLSR=1kSSSLkαLRk-1.
R0R αLrdr=-12 ln1-2BLR0R LLrdr,
R0R αSrdr=-12 ln1-2BSR0R LSrdr.
R0R αSrdr=k R0R αLrdr,
ln1-2BLR0R LLrdr=1k ln1-2BSR0R LSrdr.
LLRαLRBLk=LSRαSRBS.
BL=LLR0αLR0,
BS=LSR0αSR0,
BS=LSR0kαLR0.
LLRαLR0αLRLLR0k=LSRkαLR0kαLRLSR0,
LLRLLR0kLSR0LSR=αLRαLR0k-1.
BL=2 R1R2 LLrdr1-TL2,
BS=2 R2R1 LSrdr1-TS2,
BS=2 R1R2 LSrdr1-TL2k,
LLRkαLR 2 R1R2 LLrdr/1-TL2k=LSRkαLR2 R1R2 LSrdr/1-TL2k,
LLRkLSR=1k1-TL2k1-TL2k2 R1R2 LLrdrk2 R1R2 LSrdrαLRk-1,
LLRk2 R1R2 LLrdrk2 R1R2 LSrdrLSR=1k1-TL2k1-TL2kαLRk-1.
αˆLR=LLRBˆL-2 R0R LLrdr,
αˆSR=LSRBˆS-2 R0R LSrdr,
kˆ=αˆSRαˆLR.
BL-LLRαLR=BˆL-LLRαˆLR=2 R0R LLrdr,
BS-LSRαSR=BˆS-LSRαˆSR=2 R0R LSrdr,
BL-BˆL=LLR1αLR-1αˆLR,
BS-BˆS=LSR1αSR-1αˆSR.
BL-BˆL=CBS-BˆS,
LLR0αLR0-LLR0αˆLR0=CLSR0αSR0-LSR0αˆSR0.
kkˆ=C LSR0LLR0kˆαˆLR0-kαLR0αˆLR0-αLR0.
kˆ=kαLR0αˆLR0-kCLLR0LSR0αˆLR0-αLR0,
C=k LLR0LSR0   for αˆLR0αLR0

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