Abstract

Single-scatter lidar signals carry information on the spatial atmospheric backscatter coefficient, attenuated by the path-integrated extinction. Assuming that the relationship between the backscatter and the extinction is known, the inverted extinction profile and the path-integrated extinction are uniquely related to the input boundary value. The integrated extinction over a certain range interval is a measure of the optical transmission along that path. In reverse, for a given transmission over the path of interest, the input boundary value is uniquely defined. An analytical expression is derived that describes the input boundary condition for the inversion of the single-scatter lidar equation in terms of the transmission losses over the path of interest. The proposed method is useful in situations in which independent transmission measurements are carried out or in situations in which targets such as multiple cloud layers or beam stops are available in the lidar path. Equations for both the forward and the backward integration method are presented. Compared with the widely accepted inversion schemes that are based on single-point reference extinction values, the proposed method is less sensitive to noise.

© 1996 Optical Society of America

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  1. 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]
  2. E. W. Barret, O. Ben-Dov, “Application of the lidar to air pollution measurements,” J. Appl. Meteorol. 6, 500–515 (1967).
    [CrossRef]
  3. P. A. Davis, “The analysis of lidar signatures of cirrus clouds,” Appl. Opt. 8, 2099–2102 (1969).
    [CrossRef] [PubMed]
  4. P. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
    [CrossRef]
  5. R. H. Kohl, “Discussion of the interpretation problem encountered in single-wavelength lidar transmissometers,” J. Appl. Meteorol. 17, 1034–1038 (1978).
    [CrossRef]
  6. J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1981).
    [CrossRef] [PubMed]
  7. P. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652–654 (1984).
    [CrossRef] [PubMed]
  8. W. P. Hooper, H. E. Gerber, “Down looking lidar inversion constrained by ocean reflection and forward scatter of laser light,” Appl. Opt. 25, 689–697 (1986).
    [CrossRef] [PubMed]
  9. R. Gonzalez, “Recursive technique for inverting the lidar equation,” Appl. Opt. 27, 2741–2745 (1988).
    [CrossRef] [PubMed]
  10. W. Hitschfeld, J. Bordan, “Errors inherent in radar measurement of rainfall at attenuating wavelength,” J. Meteorol. 11, 58–67 (1954).
    [CrossRef]
  11. D. C. Knauss, “Significance of the boundary value term in the Klett inversion formula,” Appl. Opt. 21, 4194 (1982).
    [CrossRef] [PubMed]
  12. J. D. Klett, “Lidar calibration and extinction,” Appl. Opt. 22, 514–515 (1983).
    [CrossRef] [PubMed]
  13. J. A. Ferguson, D. H. Stephens, “Algorithm for inverting lidar returns,” Appl. Opt. 22, 3673–3675 (1983).
    [CrossRef] [PubMed]
  14. Y. Sasano, H. Nakane, “Significance of the extinction/backscatter ratio and the boundary value term in the solution for the two-component lidar equation,” Appl. Opt. 23, 11–13 (1984).
    [CrossRef]
  15. 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]
  16. L. R. Bissonnette, “Sensitivity analysis of lidar inversion algorithms,” Appl. Opt. 25, 2122–2125 (1986).
    [CrossRef] [PubMed]
  17. Q. Jinhuan, “Sensitivity of lidar equation solution to boundary values and determination of the values,” Adv. Atmos. Sci. 5, 229–241 (1988).
    [CrossRef]
  18. E. E. Uthe, J. M. Livingston, “Lidar extinction methods applied to observations of obscurant events,” Appl. Opt. 25, 676–684 (1986).
    [CrossRef]
  19. Yu. S. Balin, S. I. Kavkyanov, G. M. Krekov, A. I. Razenkov, “Noise-proof inversion of lidar equation,” Opt. Lett. 12, 13–15 (1987).
    [CrossRef] [PubMed]
  20. 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]
  21. G. Roy, G. Vallée, M. Jean, “Lidar-inversion technique based on total integrated backscatter calibrated curves,” Appl. Opt. 32, 6754–6763 (1993).
    [CrossRef] [PubMed]
  22. H. G. Hughes, J. A. Ferguson, D. H. Stephens, “Sensitivity of a lidar inversion algorithm to parameters relating atmospheric backscatter and extinction,” Appl. Opt. 24, 1609–1613 (1985).
    [CrossRef] [PubMed]
  23. Y. Sasano, E. W. Browell, S. Ismail, “Error caused by using a constant extinction/backscattering ratio in the lidar solution,” Appl. Opt. 24, 3929–3932 (1985).
    [CrossRef] [PubMed]
  24. J. D. Klett, “Lidar inversion with variable backscatter/extinction ratios,” Appl. Opt. 24, 1638–1643 (1985).
    [CrossRef] [PubMed]
  25. M. Kaestner, “Lidar inversion variable backscatter/extinction ratios: comment,” Appl. Opt. 25, 833–835 (1986).
    [CrossRef] [PubMed]

1993 (2)

1988 (3)

1987 (1)

1986 (4)

1985 (3)

1984 (2)

1983 (2)

1982 (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)

P. 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)

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]

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

1967 (1)

E. W. Barret, O. Ben-Dov, “Application of the lidar to air pollution measurements,” J. Appl. Meteorol. 6, 500–515 (1967).
[CrossRef]

1954 (1)

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

Balin, Yu. S.

Barret, E. W.

E. W. Barret, O. Ben-Dov, “Application of the lidar to air pollution measurements,” J. Appl. Meteorol. 6, 500–515 (1967).
[CrossRef]

Ben-Dov, O.

E. W. Barret, O. Ben-Dov, “Application of the lidar to air pollution measurements,” J. Appl. Meteorol. 6, 500–515 (1967).
[CrossRef]

Bissonnette, L. R.

Bordan, J.

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

Browell, E. W.

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, P. G.

P. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652–654 (1984).
[CrossRef] [PubMed]

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

Gerber, H. E.

Gonzalez, R.

Herman, B. M.

P. 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 radar measurement of rainfall at attenuating wavelength,” J. Meteorol. 11, 58–67 (1954).
[CrossRef]

Hooper, W. P.

Hughes, H. G.

Ismail, S.

Jean, M.

Jinhuan, Q.

Q. Jinhuan, “Sensitivity of lidar equation solution to boundary values and determination of the values,” Adv. Atmos. Sci. 5, 229–241 (1988).
[CrossRef]

Kaestner, M.

Kavkyanov, S. I.

Klett, J. D.

Knauss, D. C.

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.

Livingston, J. M.

E. E. Uthe, J. M. Livingston, “Lidar extinction methods applied to observations of obscurant events,” Appl. Opt. 25, 676–684 (1986).
[CrossRef]

Nakane, H.

Razenkov, A. I.

Reagan, J. A.

P. 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, 676–684 (1986).
[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]

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.

Adv. Atmos. Sci. (1)

Q. Jinhuan, “Sensitivity of lidar equation solution to boundary values and determination of the values,” Adv. Atmos. Sci. 5, 229–241 (1988).
[CrossRef]

Appl. Opt. (18)

E. E. Uthe, J. M. Livingston, “Lidar extinction methods applied to observations of obscurant events,” Appl. Opt. 25, 676–684 (1986).
[CrossRef]

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]

D. C. Knauss, “Significance of the boundary value term in the Klett inversion formula,” Appl. Opt. 21, 4194 (1982).
[CrossRef] [PubMed]

J. D. Klett, “Lidar calibration and extinction,” Appl. Opt. 22, 514–515 (1983).
[CrossRef] [PubMed]

P. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652–654 (1984).
[CrossRef] [PubMed]

H. G. Hughes, J. A. Ferguson, D. H. Stephens, “Sensitivity of a lidar inversion algorithm to parameters relating atmospheric backscatter and extinction,” Appl. Opt. 24, 1609–1613 (1985).
[CrossRef] [PubMed]

J. D. Klett, “Lidar inversion with variable backscatter/extinction ratios,” Appl. Opt. 24, 1638–1643 (1985).
[CrossRef] [PubMed]

Y. Sasano, E. W. Browell, S. Ismail, “Error caused by using a constant extinction/backscattering ratio in the lidar solution,” Appl. Opt. 24, 3929–3932 (1985).
[CrossRef] [PubMed]

W. P. Hooper, H. E. Gerber, “Down looking lidar inversion constrained by ocean reflection and forward scatter of laser light,” Appl. Opt. 25, 689–697 (1986).
[CrossRef] [PubMed]

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

R. Gonzalez, “Recursive technique for inverting the lidar equation,” Appl. Opt. 27, 2741–2745 (1988).
[CrossRef] [PubMed]

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]

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]

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

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

Y. Sasano, H. Nakane, “Significance of the extinction/backscatter ratio and the boundary value term in the solution for the two-component lidar equation,” Appl. Opt. 23, 11–13 (1984).
[CrossRef]

M. Kaestner, “Lidar inversion variable backscatter/extinction ratios: comment,” Appl. Opt. 25, 833–835 (1986).
[CrossRef] [PubMed]

J. Appl. Meteorol. (4)

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

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

E. W. Barret, O. Ben-Dov, “Application of the lidar to air pollution measurements,” J. Appl. Meteorol. 6, 500–515 (1967).
[CrossRef]

J. Meteorol. (1)

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

Opt. Lett. (1)

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Figures (2)

Fig. 1
Fig. 1

Extinction profiles calculated from simulated lidar signals by using the forward integration method. The input extinction coefficients in the range from 200 to 800 m were 1, 10, and 0.2 km−1. The dotted curves represent results based on the single-point input boundary extinction value at 200 m; the solid curves are based on calculations with the input boundary transmission value over the interval between 200 and 800 m. The calculations are repeated for +10% and −10% error in the input boundary condition. The different results with different noise realizations coincide except between 600 and 800 m. The correct boundary values and the SNR's in the lidar waveform are also in the figure.

Fig. 2
Fig. 2

Extinction profiles calculated from simulated lidar signals by using the backward integration method. The input extinction coefficients in the range from 200 to 800 m were 1, 10, and 0.2 km−1. The dotted curves represent results based on the single-point input boundary extinction value at 800 m. The solid curves are based on calculations with the input boundary transmission value over the interval between 200 and 800 m. The profiles obtained with this method for the different noise realizations almost coincide. The calculations are repeated for +10% and −10% error in the input boundary condition and different noise realizations. (C) indicates that the noise in the range bin at 800 m is in the deepest crest. (P) indicates that the noise in the range bin at 800 m is at the highest peak of the noise. (Z) indicates that the noise in the range bin at 800 m is approximately zero. The correct boundary values and the SNR's in the lidar waveform are also in the figure.

Equations (15)

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P ( R ) = c 2 E A R 2 β ( R ) exp [ 2 0 R σ ( r ) d r ] , R c Δ t ,
β = C σ k ,
S ( R ) = ln [ β ( R ) ] 2 0 R σ ( r ) d r .
σ ( R ) = exp [ S ( R ) / k ] exp [ S ( R 0 ) / k ] σ ( R 0 ) 2 k R 0 R exp [ S ( r ) / k ] d r .
T R 0 R e = exp [ R 0 R e σ ( r ) d r ] .
Λ = ln { exp [ S ( R 0 ) / k ] σ ( R 0 ) 2 k R 0 R exp [ S ( r ) / k ] d r } .
d Λ d R = 2 k exp [ S ( R ) / k ] exp [ S ( R 0 ) / k ] σ ( R 0 ) 2 k R 0 R exp [ S ( r ) / k ] d r .
σ ( R ) = k 2 d Λ d R .
ln [ T R 0 R e ] = k 2 R 0 R e d Λ .
ln [ T R 0 R e ] = k 2 ln { exp [ S ( R 0 ) / k ] σ ( R 0 ) 2 k R 0 R e exp [ S ( r ) / k ] d r exp [ S ( R 0 ) / k ] σ ( R 0 ) } .
σ ( R 0 ) = k 2 ( 1 T R 0 R e 2 / k ) exp [ S ( R 0 ) / k ] R 0 R e exp [ S ( r ) / k ] d r .
σ ( R ) = k 2 exp [ S ( R ) / k ] R 0 R e exp [ S ( r ) / k ] d r ( 1 T R 0 R e 2 / k ) R 0 R exp [ S ( r ) / k ] d r .
R 0 R exp [ S ( r ) / k ] d r
R 0 R m exp [ S ( r ) / k ] d r R R m exp [ S ( r ) / k ] d r ,
σ ( R ) = k 2 exp [ S ( R ) / k ] R 0 R m exp [ S ( r ) / k ] d r ( 1 T R 0 R e 2 / k 1 ) + R R m exp [ S ( r ) / k ] d r .

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