Abstract

We present an analytical formulation to compute the total-backscatter range-dependent error bars from the well-known Klett’s elastic-lidar inversion algorithm. A combined error-propagation and statistical formulation approach is used to assess inversion errors in response to the following error sources: observation noise (i.e., signal-to-noise ratio) in the reception channel, the user’s uncertainty in the backscatter calibration, and in the (range-dependent) total extinction-to-backscatter ratio provided. The method is validated using a Monte Carlo procedure, where the error bars are computed by inversion of a large population of noisy generated lidar signals, for total optical depths τ5 and typical user uncertainties, all of which yield a practical tool to compute the sought-after error bars.

© 2010 Optical Society of America

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  1. J. A. Reagan, X. Wang, and M. T. Osborn, “Spaceborne lidar calibration from cirrus and molecular backscatter returns,” IEEE Trans. Geosci. Remote Sensing 40, 2285–2290 (2002).
    [CrossRef]
  2. D. Winker, J. Pelon, and M. McCormick, “Initial results from CALIPSO,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C.Nagasawa and N.Sugimoto, eds. (IOP, 2006), pp. 991–994.
  3. H. Nett and M. Endemann, “Atmospheric Dynamics Mission: AEOLUS,” Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2004), Vol. 2, pp. 1190–1195.
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  6. W. Hitschfeld and J. Bordan, “Errors inherent in the radar measurement of rainfall at attenuating wavelengths,” J. Appl. Meteorol. 1158–67 (1954).
  7. E. W. Barrett and O. Ben-Dov, “Application of the lidar to air pollution measurements,” J. Appl. Meteorol. 6500–515 (1967).
    [CrossRef]
  8. W. Viezee, E. E. Uthe, and R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploration study,” J. Appl. Meteorol. 8, 274–283 (1969).
    [CrossRef]
  9. P. A. Davis, “The analysis of lidar signatures of cirrus clouds,” Appl. Opt. 8, 2099–2102 (1969).
    [CrossRef] [PubMed]
  10. F. G. Fernald, B. M. Herman, and J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11482–489 (1972).
    [CrossRef]
  11. R. T. H. Collis and P. B. Russell, “Lidar measurement of particles and gases by elastic backscattering and differential absorption,” in Laser Monitoring of the Atmosphere, E.D.Hinkley, ed. (Springer-Verlag, 1976), pp. 71–102.
  12. R. H. Kohl, “Discussion of the interpretation problem encountered in single-wavelength lidar transmissometers,” J. Appl. Meteorol. 17, 1034–1038 (1978).
    [CrossRef]
  13. J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1981).
    [CrossRef] [PubMed]
  14. F. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652–653 (1984).
    [CrossRef] [PubMed]
  15. Y. Sasano, E. V. Browell, and S. Ismail, “Error caused by using a constant extinction/backscattering ratio in the lidar solution,” Appl. Opt. 24, 3929–3932 (1985).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  20. A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, and W. Michaelis, “Independent measurement of extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar,” Appl. Opt. 31, 7113–7131(1992).
    [CrossRef] [PubMed]
  21. M. Sicard, P. Chazette, J. Pelon, J. Gwang-Won, and Soon-Chang Yoon, “Variational method for the retrieval of the optical thickness and the backscatter coefficient from multiangle lidar profiles,” Appl. Opt. 41, 493–502 (2002).
    [CrossRef] [PubMed]
  22. G. J. Kunz, “Transmission as an input boundary value for an analytical solution of a single-scatter lidar equation,” Appl. Opt. 35, 3255–3260 (1996).
    [CrossRef] [PubMed]
  23. 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]
  24. V. A. Kovalev, “Stable near-end solution of the lidar equation for clear atmospheres,” Appl. Opt. 42, 585–591 (2003).
    [CrossRef] [PubMed]
  25. L. R. Bissonnette, “Sensitivity analysis of lidar inversion algorithms,” Appl. Opt. 25, 2122–2125 (1986).
    [CrossRef] [PubMed]
  26. F. Rocadenbosch and A. Comerón, “Error analysis for the lidar backward inversion algorithm,” Appl. Opt. 38, 4461–4474(1999).
    [CrossRef]
  27. J. Qiu, “Sensitivity of lidar equation solution to boundary values and determination of the values,” Adv. Atmos. Sci. 5, 229–241 (1988).
    [CrossRef]
  28. M. Matsumoto and N. Takeuchi, “Effects of misestimated far-end boundary values on two common lidar inversion solutions,” Appl. Opt. 33, 6451–6456 (1994).
    [CrossRef] [PubMed]
  29. F. Rocadenbosch, A. Comerón, and D. Pineda, “Assessment of lidar inversion errors for homogeneous atmospheres,” Appl. Opt. 37, 2199–2206 (1998).
    [CrossRef]
  30. A. Comerón, F. Rocadenbosch, M. A. López, A. Rodríguez, C. Muñoz, D. García-Vizcaíno, and M. Sicard, “Effects of noise on lidar data inversion with the backward algorithm,” Appl. Opt. 43, 2572–2577 (2004).
    [CrossRef] [PubMed]
  31. M. Sicard, A. Comerón, F. Rocadenbosch, A. Rodríguez, and C. Muñoz, “Quasi-analytical determination of noise-induced error limits in lidar retrieval of aerosol backscatter coefficient by the elastic, two-component algorithm,” Appl. Opt. 48, 176–182 (2009).
    [CrossRef] [PubMed]
  32. D. C. Knauss, “Significance of the boundary value term in the Klett lidar inversion formula,” Appl. Opt. 21, 4194–4194 (1982).
    [CrossRef] [PubMed]
  33. H. G. Hughes, J. A. Ferguson, and D. H. Stephans, “Sensitivity of a lidar inversion algorithm to parameters relating atmospheric backscatter and extinction,” Appl. Opt. 24, 1609–1613(1985).
    [CrossRef] [PubMed]
  34. M. Keastner, “Lidar inversion with variable backscatter/extinction: comment,” Appl. Opt. 25, 833–835 (1986).
    [CrossRef]
  35. Y. S. Balin, S. I. Kavkyanov, G. M. Krekov, and I. A. Razenkov, “Noise-proof inversion of lidar equation,” Opt. Lett. 12, 13–15(1987).
    [CrossRef] [PubMed]
  36. J. D. Spinhirne, J. A. Reagan, and B. M. Herman, “Vertical distribution of aerosol extinction cross section and interference of aerosol imaginary index in the troposphere by lidar technique,” J. Appl. Meteorol. 19, 426–438 (1980).
    [CrossRef]
  37. R. J. Barlow, “Theoretical distributions,” in Statistics. A Guide to the Use of Statistical Methods in Physical Sciences (Wiley, 1999), pp. 28–33.
  38. M. N. Md. Reba, F. Rocadenbosch, and M. Sicard, “A straightforward signal-to-noise ratio estimator for elastic/Raman lidar signals,” Proc. SPIE 6362, 636223 (2006).
    [CrossRef]
  39. B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean. Technol. 16, 1854–1861 (1999).
    [CrossRef]

2009 (1)

2006 (1)

M. N. Md. Reba, F. Rocadenbosch, and M. Sicard, “A straightforward signal-to-noise ratio estimator for elastic/Raman lidar signals,” Proc. SPIE 6362, 636223 (2006).
[CrossRef]

2004 (1)

2003 (1)

2002 (2)

M. Sicard, P. Chazette, J. Pelon, J. Gwang-Won, and Soon-Chang Yoon, “Variational method for the retrieval of the optical thickness and the backscatter coefficient from multiangle lidar profiles,” Appl. Opt. 41, 493–502 (2002).
[CrossRef] [PubMed]

J. A. Reagan, X. Wang, and M. T. Osborn, “Spaceborne lidar calibration from cirrus and molecular backscatter returns,” IEEE Trans. Geosci. Remote Sensing 40, 2285–2290 (2002).
[CrossRef]

1999 (2)

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean. Technol. 16, 1854–1861 (1999).
[CrossRef]

F. Rocadenbosch and A. Comerón, “Error analysis for the lidar backward inversion algorithm,” Appl. Opt. 38, 4461–4474(1999).
[CrossRef]

1998 (1)

1996 (1)

1994 (1)

1993 (1)

1992 (1)

1988 (1)

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

1987 (1)

1986 (3)

1985 (3)

1984 (1)

1983 (1)

1982 (1)

1981 (1)

1980 (1)

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

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, and J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11482–489 (1972).
[CrossRef]

1969 (2)

W. Viezee, E. E. Uthe, and R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploration 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. Barrett and O. Ben-Dov, “Application of the lidar to air pollution measurements,” J. Appl. Meteorol. 6500–515 (1967).
[CrossRef]

1954 (1)

W. Hitschfeld and J. Bordan, “Errors inherent in the radar measurement of rainfall at attenuating wavelengths,” J. Appl. Meteorol. 1158–67 (1954).

Ansmann, A.

Balin, Y. S.

Barlow, R. J.

R. J. Barlow, “Theoretical distributions,” in Statistics. A Guide to the Use of Statistical Methods in Physical Sciences (Wiley, 1999), pp. 28–33.

Barrett, E. W.

E. W. Barrett and O. Ben-Dov, “Application of the lidar to air pollution measurements,” J. Appl. Meteorol. 6500–515 (1967).
[CrossRef]

Ben-Dov, O.

E. W. Barrett and O. Ben-Dov, “Application of the lidar to air pollution measurements,” J. Appl. Meteorol. 6500–515 (1967).
[CrossRef]

Bissonnette, L. R.

Böckmann, C.

C. Böckmann, D. Müller, L. Osterloh, P. Pornsawad, and A. Papayannis, “From EARLINET-ASOS Raman-Lidar signals to microphysical aerosol properties via advances regularizing software,” in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2008), pp. (II-422)–(II-425).

Bodhaine, B. A.

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean. Technol. 16, 1854–1861 (1999).
[CrossRef]

Bordan, J.

W. Hitschfeld and J. Bordan, “Errors inherent in the radar measurement of rainfall at attenuating wavelengths,” J. Appl. Meteorol. 1158–67 (1954).

Browell, E. V.

Chazette, P.

Collis, R. T. H.

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

R. T. H. Collis and P. B. Russell, “Lidar measurement of particles and gases by elastic backscattering and differential absorption,” in Laser Monitoring of the Atmosphere, E.D.Hinkley, ed. (Springer-Verlag, 1976), pp. 71–102.

Comerón, A.

Davis, P. A.

Dutton, E. G.

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean. Technol. 16, 1854–1861 (1999).
[CrossRef]

Endemann, M.

H. Nett and M. Endemann, “Atmospheric Dynamics Mission: AEOLUS,” Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2004), Vol. 2, pp. 1190–1195.
[CrossRef]

Ferguson, J. A.

Fernald, F. G.

F. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652–653 (1984).
[CrossRef] [PubMed]

F. G. Fernald, B. M. Herman, and J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11482–489 (1972).
[CrossRef]

García-Vizcaíno, D.

Gwang-Won, J.

Herman, B. M.

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

F. G. Fernald, B. M. Herman, and J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11482–489 (1972).
[CrossRef]

Hitschfeld, W.

W. Hitschfeld and J. Bordan, “Errors inherent in the radar measurement of rainfall at attenuating wavelengths,” J. Appl. Meteorol. 1158–67 (1954).

Hughes, H. G.

Ismail, S.

Kavkyanov, S. I.

Keastner, M.

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.

Kunz, G. J.

López, M. A.

Matsumoto, M.

McCormick, M.

D. Winker, J. Pelon, and M. McCormick, “Initial results from CALIPSO,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C.Nagasawa and N.Sugimoto, eds. (IOP, 2006), pp. 991–994.

Md. Reba, M. N.

M. N. Md. Reba, F. Rocadenbosch, and M. Sicard, “A straightforward signal-to-noise ratio estimator for elastic/Raman lidar signals,” Proc. SPIE 6362, 636223 (2006).
[CrossRef]

Michaelis, W.

Müller, D.

C. Böckmann, D. Müller, L. Osterloh, P. Pornsawad, and A. Papayannis, “From EARLINET-ASOS Raman-Lidar signals to microphysical aerosol properties via advances regularizing software,” in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2008), pp. (II-422)–(II-425).

Muñoz, C.

Nett, H.

H. Nett and M. Endemann, “Atmospheric Dynamics Mission: AEOLUS,” Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2004), Vol. 2, pp. 1190–1195.
[CrossRef]

Osborn, M. T.

J. A. Reagan, X. Wang, and M. T. Osborn, “Spaceborne lidar calibration from cirrus and molecular backscatter returns,” IEEE Trans. Geosci. Remote Sensing 40, 2285–2290 (2002).
[CrossRef]

Osterloh, L.

C. Böckmann, D. Müller, L. Osterloh, P. Pornsawad, and A. Papayannis, “From EARLINET-ASOS Raman-Lidar signals to microphysical aerosol properties via advances regularizing software,” in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2008), pp. (II-422)–(II-425).

Papayannis, A.

C. Böckmann, D. Müller, L. Osterloh, P. Pornsawad, and A. Papayannis, “From EARLINET-ASOS Raman-Lidar signals to microphysical aerosol properties via advances regularizing software,” in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2008), pp. (II-422)–(II-425).

Pelon, J.

M. Sicard, P. Chazette, J. Pelon, J. Gwang-Won, and Soon-Chang Yoon, “Variational method for the retrieval of the optical thickness and the backscatter coefficient from multiangle lidar profiles,” Appl. Opt. 41, 493–502 (2002).
[CrossRef] [PubMed]

D. Winker, J. Pelon, and M. McCormick, “Initial results from CALIPSO,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C.Nagasawa and N.Sugimoto, eds. (IOP, 2006), pp. 991–994.

Pineda, D.

Pornsawad, P.

C. Böckmann, D. Müller, L. Osterloh, P. Pornsawad, and A. Papayannis, “From EARLINET-ASOS Raman-Lidar signals to microphysical aerosol properties via advances regularizing software,” in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2008), pp. (II-422)–(II-425).

Qiu, J.

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

Razenkov, I. A.

Reagan, J. A.

J. A. Reagan, X. Wang, and M. T. Osborn, “Spaceborne lidar calibration from cirrus and molecular backscatter returns,” IEEE Trans. Geosci. Remote Sensing 40, 2285–2290 (2002).
[CrossRef]

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

F. G. Fernald, B. M. Herman, and J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11482–489 (1972).
[CrossRef]

Riebesell, M.

Rocadenbosch, F.

Rodríguez, A.

Russell, P. B.

R. T. H. Collis and P. B. Russell, “Lidar measurement of particles and gases by elastic backscattering and differential absorption,” in Laser Monitoring of the Atmosphere, E.D.Hinkley, ed. (Springer-Verlag, 1976), pp. 71–102.

Sasano, Y.

Sicard, M.

Slusser, J. R.

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean. Technol. 16, 1854–1861 (1999).
[CrossRef]

Spinhirne, J. D.

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

Stephans, D. H.

Takeuchi, N.

Uthe, E. E.

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

Viezee, W.

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

Wandinger, U.

Wang, X.

J. A. Reagan, X. Wang, and M. T. Osborn, “Spaceborne lidar calibration from cirrus and molecular backscatter returns,” IEEE Trans. Geosci. Remote Sensing 40, 2285–2290 (2002).
[CrossRef]

Weitkamp, C.

Winker, D.

D. Winker, J. Pelon, and M. McCormick, “Initial results from CALIPSO,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C.Nagasawa and N.Sugimoto, eds. (IOP, 2006), pp. 991–994.

Wood, N. B.

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean. Technol. 16, 1854–1861 (1999).
[CrossRef]

Yoon, Soon-Chang

Adv. Atmos. Sci. (1)

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

Appl. Opt. (21)

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 lidar inversion formula,” Appl. Opt. 21, 4194–4194 (1982).
[CrossRef] [PubMed]

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

F. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652–653 (1984).
[CrossRef] [PubMed]

H. G. Hughes, J. A. Ferguson, and D. H. Stephans, “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. V. Browell, and S. Ismail, “Error caused by using a constant extinction/backscattering ratio in the lidar solution,” Appl. Opt. 24, 3929–3932 (1985).
[CrossRef] [PubMed]

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

A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, and W. Michaelis, “Independent measurement of extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar,” Appl. Opt. 31, 7113–7131(1992).
[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]

M. Matsumoto and N. Takeuchi, “Effects of misestimated far-end boundary values on two common lidar inversion solutions,” Appl. Opt. 33, 6451–6456 (1994).
[CrossRef] [PubMed]

F. Rocadenbosch, A. Comerón, and D. Pineda, “Assessment of lidar inversion errors for homogeneous atmospheres,” Appl. Opt. 37, 2199–2206 (1998).
[CrossRef]

F. Rocadenbosch and A. Comerón, “Error analysis for the lidar backward inversion algorithm,” Appl. Opt. 38, 4461–4474(1999).
[CrossRef]

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

M. Sicard, P. Chazette, J. Pelon, J. Gwang-Won, and Soon-Chang Yoon, “Variational method for the retrieval of the optical thickness and the backscatter coefficient from multiangle lidar profiles,” Appl. Opt. 41, 493–502 (2002).
[CrossRef] [PubMed]

V. A. Kovalev, “Stable near-end solution of the lidar equation for clear atmospheres,” Appl. Opt. 42, 585–591 (2003).
[CrossRef] [PubMed]

A. Comerón, F. Rocadenbosch, M. A. López, A. Rodríguez, C. Muñoz, D. García-Vizcaíno, and M. Sicard, “Effects of noise on lidar data inversion with the backward algorithm,” Appl. Opt. 43, 2572–2577 (2004).
[CrossRef] [PubMed]

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

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

M. Sicard, A. Comerón, F. Rocadenbosch, A. Rodríguez, and C. Muñoz, “Quasi-analytical determination of noise-induced error limits in lidar retrieval of aerosol backscatter coefficient by the elastic, two-component algorithm,” Appl. Opt. 48, 176–182 (2009).
[CrossRef] [PubMed]

IEEE Trans. Geosci. Remote Sensing (1)

J. A. Reagan, X. Wang, and M. T. Osborn, “Spaceborne lidar calibration from cirrus and molecular backscatter returns,” IEEE Trans. Geosci. Remote Sensing 40, 2285–2290 (2002).
[CrossRef]

J. Appl. Meteorol. (6)

W. Hitschfeld and J. Bordan, “Errors inherent in the radar measurement of rainfall at attenuating wavelengths,” J. Appl. Meteorol. 1158–67 (1954).

E. W. Barrett and O. Ben-Dov, “Application of the lidar to air pollution measurements,” J. Appl. Meteorol. 6500–515 (1967).
[CrossRef]

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

F. G. Fernald, B. M. Herman, and J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11482–489 (1972).
[CrossRef]

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

R. H. Kohl, “Discussion of the interpretation problem encountered in single-wavelength lidar transmissometers,” J. Appl. Meteorol. 17, 1034–1038 (1978).
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Figures (6)

Fig. 1
Fig. 1

Simulation scenario at 532 nm wavelength (R is the slant path distance, elevation angle 54 ° , sounding range 0.2 to 6 km ): (a) atmospheric backscatter components: (dashed black) noiseless total backscatter used as the simulation input, (solid gray) example of a noise-corrupted inverted backscatter representative, and (dashed gray) Rayleigh backscatter component and (b) (solid trace) range-corrected return power after 1 s integration time, (dotted trace) SNR [ SNR ( R min ) = 5 × 10 3 and SNR ( R max ) = 5 ].

Fig. 2
Fig. 2

Error source (iii), σ ε j , 3 (see Table 1 and Subsection 3B3): noise corrupting the return power in all range cells except at the calibration cell. (a) MC total backscatter inversion with superimposed 3 σ analytical error bars (vertical error bar lines). The set of 100 noise-corrupted lidar returns inverted in the MC method appears as a growing shadowed area from 2000 m onwards. (b) Comparison between the backscatter error bar amplitudes derived from both the MC and the analytical method as a function of range. Upper and lower MC backscatter error amplitudes are plotted in dark and light gray solid traces (noisy and nearly coincident), respectively. Analytical error bars are plotted in dark dashed traces for comparison.

Fig. 3
Fig. 3

Error source (iv), σ ε j , 4 (see Table 1 and Subsection 3B4): noise corrupting the return power at the calibration range [ SNR ( R cal ) = 5 , SNR ( R ) , R R cal ]. (a)–(b)–Same descriptors as in Fig. 2. In contrast to Fig. 2, the noise effects due to this error source backpropagate to all inversion cells and become more important, as shown by a considerably large spread of the inverted backscatter family.

Fig. 4
Fig. 4

Counteracting noise in practical lidar inversion: corroboration of previously published results. Same descriptors as in Fig. 2 for subsets (a)–(b) and (c)–(d). Subset (a)–(b) corresponds to the superposition of error sources (3, 4), σ ε j , 3 and σ ε j , 4 (see Subsections 3B3, 3B4), which stands for noise corrupting all the cells along the data inversion range [SNR, as in Fig. 1b]. Simulation conditions for subset (c)–(d) are identical to (a)–(b) except that the SNR at the calibration range has been enhanced to SNR ( R cal ) = SNR ( R cal ) N = 25 by spatially averaging N = 25 neighbor return-power cells.

Fig. 5
Fig. 5

Error source (i), σ ε j , 1 (see Table 1 and Subsection 3B1): total backscatter calibration at the calibration range ( R cal = R max = 6 km ). Error strength: ± 10 % Gaussian fluctuation over the nominal backscatter Rayleigh level at the calibration range. (a)–(b) Same descriptors as in Fig. 2.

Fig. 6
Fig. 6

Error source (2), σ ε j , 2 C (see Table 1 and Subsection 3B2): Range-dependent lidar ratio, S ( R ) (correlated errors). Error strength: 10 % Gaussian fluctuation over the nominal total lidar ratio. (a)–(b) Same descriptors as in Fig. 2. In (b), analytical upper and lower error amplitudes are, respectively, plotted in dashed and dotted traces. MC upper and lower error amplitudes are plotted in a solid trace.

Tables (3)

Tables Icon

Table 1 Summary Table to Compute Total-Backscatter Analytical Error Bars in Klett’s Backward Inversion Method Due to Error Sources (i–iv) in Subsection 3B a

Tables Icon

Table 2 Estimated Mean Relative Error and 1 σ Dispersion on Backscatter Analytical Error Bars for Different Optical Depths and Signal-to-Noise Ratios at Calibration Range [Error Source (ii) σ ε j , 4 , Only]

Tables Icon

Table 3 Estimated Mean Relative Error and 1 σ Dispersion on Backscatter Analytical Error Bars for Different Optical Depths and Relative Error p in Assumed Total Lidar Ratio Eq. (14) [Error Source (ii) σ ε j , 2 , Only] a

Equations (39)

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P ( R ) = A R 2 β ( R ) exp [ 2 0 R α ( r ) d r ] ξ ( R ) ,
β ( R ) = β m [ R 2 P ( R ) ] [ R m 2 P ( R m ) ] + 2 β m R R m S ( r ) [ r 2 P ( r ) ] d r ,
{ U ( R ) = R 2 P ( R ) G ( R ) = R R m S ( r ) U ( r ) d r ,
β j ( β N , S , U ) = β N U j U N + 2 β N G j ( S , U ) ,
{ G j = i = j N w i S i U i j < N G N = 0 j = N .
β j ( β 1 , S , U ) = β 1 U j U 1 2 β 1 G j ( S , U ) .
{ | d β j | = | β j β N d β N | + k = 1 N | β j S k d S k | + k = 1 N 1 | β j P k d P k | + | β j P N d P N | ; j < N | d β j | = | d β N | ; j = N ,
ε j , 1 = β j β N d β N = ( β j β N ) 2 U N U j d β N ,
ε j , 2 = k = 1 N β j S k d S k = 2 β j 2 U j k = j N w k U k d S k ,
ε j , 3 = k = 1 N 1 β j P k d P k = β j U j d U j 2 β j 2 U j k = j N w k S k d U k ,
ε j , 4 = β j P N d P N = β j 2 β N U j d U N 2 β j 2 U j w N S N d U N β j 2 β N U j d U N ,
σ β j = { ( σ ε j , 1 2 + σ ε j , 2 2 + σ ε j , 3 2 + σ ε j , 4 2 ) 1 2 , j < N σ β N , j = N ,
σ ε j , 1 = | ( β j β N ) 2 U N U j | σ β N .
d S ( R ) = p S ( R ) d S k = p S k .
σ ε j , 2 = | p 2 β j 2 U j G j | .
σ ε j , 2 = | ± p 2 β j 2 U j G j + p 2 4 β j 3 U j 2 G j 2 | .
E [ d S i d S j ] = 0 i j .
I j G S = k = j N w k U k d S k ,
σ G S , j 2 = k = j N c k 2 σ S k 2 = k = j N ( w k U k ) 2 σ S k 2 .
σ ε j , 2 = | 2 β j 2 U j | σ G S , j .
E [ d P i d P j ] = 0 i j .
I j G U = k = j N w k S k d U k ,
σ G U , j 2 = k = j N ( w k S k ) 2 σ U k 2 .
σ ε j , 3 = [ ( β j U j ) 2 σ U j 2 + ( 2 β j 2 U j ) 2 σ G U , j 2 ] 1 2 .
SNR j = U j σ U j ,
σ ε j , 4 = ( | β j 2 β N U j | + | 2 β j 2 U j w N S N | ) σ U N | β j 2 β N U j | σ U N ,
S ( R ) = S aer ( R ) β aer ( R ) + 8 π 3 β mol ( R ) β aer ( R ) + β mol ( R ) .
σ ε j , 1 σ ε j , 4 | U N β N | σ β N σ U N = SNR N ε r β N ,
δ r β | u / l = 1 M N i = 1 M j = 1 N δ r , i β j | u / l ; δ r , i β j | u / l = σ β j A i | u / l σ β j M C i | u / l β j ,
G j S k = { 0 k < j w k U k k j ,
ε j , 2 = k = 1 N β j S k d S k = β j G j k = j N G j S k d S k .
ε j , 3 = k = 1 N 1 β j P k d P k = k = 1 N 1 ( β j U j U j P k + β j G j G j U k U k P k ) d P k .
ε j , 3 = β j U j d U j + β j G j k = j N G j U k d U k .
ε j , 4 = β j P N d P N = β j U N d U N + β j G j G j U N d U N ,
G j U N = { w N S N j < N 0 j = N .
β j ( β N , S , U ) = β N U j U N + 2 β N ( 1 + p ) G j ( S , U ) ,
| Δ β j ( p ) | = | β j ( p ) β j ( p = 0 ) | | β j p | p = 0 p + 1 2 2 β j p 2 | p = 0 p 2 | ,
β j p | p = 0 = 2 β j 2 U j G j ,
2 β j p 2 | p = 0 = 8 β j 3 U j 2 G j 2 .

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