Abstract

Here we depart from the inhomogeneous solution of a lidar equation using the backward inversion algorithm that is nowadays generally referred to as the Klett method. In particular, we develop an error sensitivity study that relates errors in the user-input parameters boundary extinction and exponential term in the extinction-to-backscatter relationship to errors in the inverted extinction profile. The validity of the analysis presented is limited only by the validity of application of the inversion algorithm itself, its numerical performance having been tested for optical depths in the 0.01–10 range. Toward this end, we focus on an introductory background about how uncertainties in these two parameters can apply to a family of inverted extinction profiles rather than a single profile and on its range-dependent behavior as a function of the optical thickness of the lidar inversion range. Next, we performed a mathematical study to derive the error span of the inverted extinction profile that is due to uncertainties in the above-mentioned user calibration parameters. This takes the form of upper and lower range-dependent error bounds. Finally, appropriate inversion plots are presented as application examples of this study to a parameterized set of atmospheric scenes inverted from both synthesized elastic-backscatter lidar signals and a live signal.

© 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. Appl. Meteorol. 11, 58–67 (1954).
    [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. W. Viezee, E. E. Uthe, R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploration study,” J. Appl. Meteorol. 8, 274–283 (1969).
    [CrossRef]
  4. P. A. Davis, “Analysis of lidar signatures of cirrus clouds,” Appl. Opt. 8, 2099–2102 (1969).
    [CrossRef] [PubMed]
  5. F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
    [CrossRef]
  6. R. T. H. Collis, 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, New York, 1976), Chap. 4, pp. 71–102.
    [CrossRef]
  7. R. H. Kohl, “Discussion of the interpretation problem encountered in single-wavelength lidar transmissometers,” J. Appl. Meteorol. 17, 1034–1038 (1978).
    [CrossRef]
  8. J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1985).
    [CrossRef]
  9. S. R. Pal, W. Steinbrecht, A. I. Carswell, “Automated method for lidar determination of cloud-base height and vertical extent,” Appl. Opt. 31, 1488–1494 (1992).
    [CrossRef] [PubMed]
  10. G. J. Kunz, G. de Leeuw, “Inversion of lidar signals with the slope method,” Appl. Opt. 32, 3249–3256 (1993).
    [CrossRef] [PubMed]
  11. F. Rocadenbosch, A. Comerón, D. Pineda, “Assessment of lidar inversion errors for homogeneous atmospheres,” Appl. Opt. 37, 2199–2206 (1998).
    [CrossRef]
  12. G. J. Kunz, “Vertical atmospheric profiles measured with lidar,” Appl. Opt. 22, 1955–1957 (1983).
    [CrossRef] [PubMed]
  13. J. A. Ferguson, D. H. Stephans, “Algorithm for inverting lidar returns,” Appl. Opt. 22, 3673–3675 (1983).
    [CrossRef] [PubMed]
  14. F. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652–653 (1984).
    [CrossRef] [PubMed]
  15. Y. Sasano, “Observational study on atmospheric mixed layer and transition layer structures using Mie lidar,” J. Meteorol. Soc. Jpn. 63, 419–435 (1985).
  16. H. G. Hughes, J. A. Ferguson, D. H. Stephans, “Sensitivity of a lidar inversion algorithm to parameters relating atmospheric backscatter and extinction,” Appl. Opt. 24, 1609–1613 (1985).
    [CrossRef] [PubMed]
  17. J. D. Klett, “Extinction boundary value algorithm for lidar inversion,” Appl. Opt. 25, 2462–2464 (1986).
    [CrossRef] [PubMed]
  18. L. R. Bissonnette, “Sensitivity analysis of lidar inversion algorithms,” Appl. Opt. 25, 2122–2125 (1986).
    [CrossRef] [PubMed]
  19. 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]
  20. J. D. Klett, “Lidar inversion with variable backscatter extinction ratios,” Appl. Opt. 24, 1638–1643 (1985).
    [CrossRef] [PubMed]
  21. M. Kaestner, “Lidar inversion with variable backscatter/extinction ratios: comment,” Appl. Opt. 25, 833–835 (1986).
    [CrossRef] [PubMed]
  22. Y. Sasano, E. V. Browell, S. Ismail, “Error caused by using a constant extinction/backscattering ratio in the lidar solution,” Appl. Opt. 24, 3929–3932 (1985).
    [CrossRef] [PubMed]
  23. R. G. Pinnick, J. M. Rosen, D. J. Hofman, “Stratospheric aerosol measurements. III: Optical model calculations,” J. Atmos. Sci. 33, 304–314 (1976).
    [CrossRef]
  24. P. B. Russell, T. J. Swissler, M. P. McCormick, W. P. Chu, J. M. Livingston, T. J. Pepin, “Satellite and correlative measurements of the stratospheric aerosol. I: An optical model for data conversion,” J. Atmos. Sci. 38, 1279–1294 (1981).
    [CrossRef]
  25. V. E. Zuev, “Laser beams in the atmosphere,” translated from the Russian by J. S. Wood (Consultants Bureau, New York, 1982).
  26. Y. Sasano, H. Nakane, “Quantitative analysis of RHI lidar data by an iterative adjustment of the boundary condition term in the lidar solution,” Appl. Opt. 26, 615–616 (1987).
    [CrossRef] [PubMed]
  27. R. W. Fenn, “Correlation between atmospheric backscattering and meteorological visual range,” Appl. Opt. 5, 293–295 (1966).
    [CrossRef] [PubMed]
  28. S. Twomey, H. B. Howell, “Relative merit of white and monochromatic light for the determination of visibility by backscattering measurements,” Appl. Opt. 4, 501–506 (1965).
    [CrossRef]
  29. G. L. Kunz, “Probing of the atmosphere with lidar,” AGARD Conf. Proc. 23, 1–11 (1992).
  30. J. D. Klett, “Lidar calibration and extinction coefficients,” Appl. Opt. 22, 514–515 (1983).
    [CrossRef] [PubMed]
  31. R. T. Brown, “A new lidar for meteorological application,” J. Appl. Meteorol. 12, 698–708 (1973).
    [CrossRef]
  32. L. R. Bissonnette, D. L. Hutt, “Multiple-scattering aerosol lidar inversion method,” Can. J. Phys. 71, 39–46 (1993).
    [CrossRef]
  33. 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]

1998 (1)

1993 (2)

G. J. Kunz, G. de Leeuw, “Inversion of lidar signals with the slope method,” Appl. Opt. 32, 3249–3256 (1993).
[CrossRef] [PubMed]

L. R. Bissonnette, D. L. Hutt, “Multiple-scattering aerosol lidar inversion method,” Can. J. Phys. 71, 39–46 (1993).
[CrossRef]

1992 (2)

1988 (1)

1987 (1)

1986 (3)

1985 (5)

1984 (2)

1983 (3)

1981 (1)

P. B. Russell, T. J. Swissler, M. P. McCormick, W. P. Chu, J. M. Livingston, T. J. Pepin, “Satellite and correlative measurements of the stratospheric aerosol. I: An optical model for data conversion,” J. Atmos. Sci. 38, 1279–1294 (1981).
[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]

1976 (1)

R. G. Pinnick, J. M. Rosen, D. J. Hofman, “Stratospheric aerosol measurements. III: Optical model calculations,” J. Atmos. Sci. 33, 304–314 (1976).
[CrossRef]

1973 (1)

R. T. Brown, “A new lidar for meteorological application,” J. Appl. Meteorol. 12, 698–708 (1973).
[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)

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

P. A. Davis, “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]

1966 (1)

1965 (1)

1954 (1)

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

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.

L. R. Bissonnette, D. L. Hutt, “Multiple-scattering aerosol lidar inversion method,” Can. J. Phys. 71, 39–46 (1993).
[CrossRef]

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

Bordan, J.

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

Browell, E. V.

Brown, R. T.

R. T. Brown, “A new lidar for meteorological application,” J. Appl. Meteorol. 12, 698–708 (1973).
[CrossRef]

Carswell, A. I.

Chu, W. P.

P. B. Russell, T. J. Swissler, M. P. McCormick, W. P. Chu, J. M. Livingston, T. J. Pepin, “Satellite and correlative measurements of the stratospheric aerosol. I: An optical model for data conversion,” J. Atmos. Sci. 38, 1279–1294 (1981).
[CrossRef]

Collis, R. T. H.

W. Viezee, E. E. Uthe, 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, 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, New York, 1976), Chap. 4, pp. 71–102.
[CrossRef]

Comerón, A.

Davis, P. A.

de Leeuw, G.

Fenn, R. W.

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, 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. Appl. Meteorol. 11, 58–67 (1954).
[CrossRef]

Hofman, D. J.

R. G. Pinnick, J. M. Rosen, D. J. Hofman, “Stratospheric aerosol measurements. III: Optical model calculations,” J. Atmos. Sci. 33, 304–314 (1976).
[CrossRef]

Howell, H. B.

Hughes, H. G.

Hutt, D. L.

L. R. Bissonnette, D. L. Hutt, “Multiple-scattering aerosol lidar inversion method,” Can. J. Phys. 71, 39–46 (1993).
[CrossRef]

Ismail, S.

Kaestner, M.

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]

Kunz, G. J.

Kunz, G. L.

G. L. Kunz, “Probing of the atmosphere with lidar,” AGARD Conf. Proc. 23, 1–11 (1992).

Livingston, J. M.

P. B. Russell, T. J. Swissler, M. P. McCormick, W. P. Chu, J. M. Livingston, T. J. Pepin, “Satellite and correlative measurements of the stratospheric aerosol. I: An optical model for data conversion,” J. Atmos. Sci. 38, 1279–1294 (1981).
[CrossRef]

McCormick, M. P.

P. B. Russell, T. J. Swissler, M. P. McCormick, W. P. Chu, J. M. Livingston, T. J. Pepin, “Satellite and correlative measurements of the stratospheric aerosol. I: An optical model for data conversion,” J. Atmos. Sci. 38, 1279–1294 (1981).
[CrossRef]

Nakane, H.

Pal, S. R.

Pepin, T. J.

P. B. Russell, T. J. Swissler, M. P. McCormick, W. P. Chu, J. M. Livingston, T. J. Pepin, “Satellite and correlative measurements of the stratospheric aerosol. I: An optical model for data conversion,” J. Atmos. Sci. 38, 1279–1294 (1981).
[CrossRef]

Pineda, D.

Pinnick, R. G.

R. G. Pinnick, J. M. Rosen, D. J. Hofman, “Stratospheric aerosol measurements. III: Optical model calculations,” J. Atmos. Sci. 33, 304–314 (1976).
[CrossRef]

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]

Rocadenbosch, F.

Rosen, J. M.

R. G. Pinnick, J. M. Rosen, D. J. Hofman, “Stratospheric aerosol measurements. III: Optical model calculations,” J. Atmos. Sci. 33, 304–314 (1976).
[CrossRef]

Russell, P. B.

P. B. Russell, T. J. Swissler, M. P. McCormick, W. P. Chu, J. M. Livingston, T. J. Pepin, “Satellite and correlative measurements of the stratospheric aerosol. I: An optical model for data conversion,” J. Atmos. Sci. 38, 1279–1294 (1981).
[CrossRef]

R. T. H. Collis, 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, New York, 1976), Chap. 4, pp. 71–102.
[CrossRef]

Sasano, Y.

Steinbrecht, W.

Stephans, D. H.

Swissler, T. J.

P. B. Russell, T. J. Swissler, M. P. McCormick, W. P. Chu, J. M. Livingston, T. J. Pepin, “Satellite and correlative measurements of the stratospheric aerosol. I: An optical model for data conversion,” J. Atmos. Sci. 38, 1279–1294 (1981).
[CrossRef]

Twomey, S.

Uthe, E. E.

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

Weinman, J. A.

Zuev, V. E.

V. E. Zuev, “Laser beams in the atmosphere,” translated from the Russian by J. S. Wood (Consultants Bureau, New York, 1982).

AGARD Conf. Proc. (1)

G. L. Kunz, “Probing of the atmosphere with lidar,” AGARD Conf. Proc. 23, 1–11 (1992).

Appl. Opt. (20)

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

Y. Sasano, H. Nakane, “Quantitative analysis of RHI lidar data by an iterative adjustment of the boundary condition term in the lidar solution,” Appl. Opt. 26, 615–616 (1987).
[CrossRef] [PubMed]

R. W. Fenn, “Correlation between atmospheric backscattering and meteorological visual range,” Appl. Opt. 5, 293–295 (1966).
[CrossRef] [PubMed]

S. Twomey, H. B. Howell, “Relative merit of white and monochromatic light for the determination of visibility by backscattering measurements,” Appl. Opt. 4, 501–506 (1965).
[CrossRef]

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

J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1985).
[CrossRef]

S. R. Pal, W. Steinbrecht, A. I. Carswell, “Automated method for lidar determination of cloud-base height and vertical extent,” Appl. Opt. 31, 1488–1494 (1992).
[CrossRef] [PubMed]

G. J. Kunz, G. de Leeuw, “Inversion of lidar signals with the slope method,” Appl. Opt. 32, 3249–3256 (1993).
[CrossRef] [PubMed]

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

G. J. Kunz, “Vertical atmospheric profiles measured with lidar,” Appl. Opt. 22, 1955–1957 (1983).
[CrossRef] [PubMed]

J. A. Ferguson, D. H. Stephans, “Algorithm for inverting lidar returns,” Appl. Opt. 22, 3673–3675 (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, 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, “Extinction boundary value algorithm for lidar inversion,” Appl. Opt. 25, 2462–2464 (1986).
[CrossRef] [PubMed]

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

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

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

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

Can. J. Phys. (1)

L. R. Bissonnette, D. L. Hutt, “Multiple-scattering aerosol lidar inversion method,” Can. J. Phys. 71, 39–46 (1993).
[CrossRef]

J. Appl. Meteorol. (6)

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

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

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

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

R. T. Brown, “A new lidar for meteorological application,” J. Appl. Meteorol. 12, 698–708 (1973).
[CrossRef]

J. Atmos. Sci. (2)

R. G. Pinnick, J. M. Rosen, D. J. Hofman, “Stratospheric aerosol measurements. III: Optical model calculations,” J. Atmos. Sci. 33, 304–314 (1976).
[CrossRef]

P. B. Russell, T. J. Swissler, M. P. McCormick, W. P. Chu, J. M. Livingston, T. J. Pepin, “Satellite and correlative measurements of the stratospheric aerosol. I: An optical model for data conversion,” J. Atmos. Sci. 38, 1279–1294 (1981).
[CrossRef]

J. Meteorol. Soc. Jpn. (1)

Y. Sasano, “Observational study on atmospheric mixed layer and transition layer structures using Mie lidar,” J. Meteorol. Soc. Jpn. 63, 419–435 (1985).

Other (2)

V. E. Zuev, “Laser beams in the atmosphere,” translated from the Russian by J. S. Wood (Consultants Bureau, New York, 1982).

R. T. H. Collis, 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, New York, 1976), Chap. 4, pp. 71–102.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Trapezium profile. Solutions of the synthesized lidar signals according to Eq. (10) with a common shape of the extinction profile (true ext.) by use of different values for k c (0.67 ≤ k c ≤ 1.34) and the boundary condition α c = 2α m . The × indicates the user-selected boundary extinction α c at R m . Simulations are parameterized for optical thicknesses of (a) τ = 7.4 and (b) τ = 0.74.

Fig. 2
Fig. 2

Trapezium profile. Same as Fig. 1 but k c and α c span the user-error margin α m /2 ≤ α c ≤ 2α m and 0.67 ≤ k c ≤ 1.34 [expression (9)].

Fig. 3
Fig. 3

Flow diagram indicating inputs and outputs of the error bound computation algorithms for the inverted extinction given a priori uncertainty information.

Fig. 4
Fig. 4

Trapezium profile. Comparison between the absolute bounds listed in Section 3 and the closest bounds in Section 4 relative to Fig. 2. Subplots (a) and (b), details for τ = 7.4 and τ = 0.74, respectively. Subplots (c) and (d), reproduction of the closest bounds depicted in Fig. 2 (same scale) with superimposed absolute bounds of subplots (a) and (b) for comparison. Performance of the absolute bounds improves when the optical depths decrease.

Fig. 5
Fig. 5

Absolute and relative extrema of α̂(R 0, γ, a) in the feasible domain (γ, a). Curves within circles correspond to αfit(R 0, γ, a min) and αfit(R 0, γ, a max) [Eq. (31)], for which function α̂(R 0, γ, a) always attains its absolute maximum and minimum. In this example, they correspond to corner points (γmax, a min) and (γmin, a max), respectively within solid boxes). Corner points (γmin, a min) and (γmax, a max) and local maximum (γ c , a max) (dotted box) are failed candidates of the absolute extrema. (The plot has been computed from the real live scene shown in Fig. 10, where R 0 = 497.5m).

Fig. 6
Fig. 6

Parametric study versus optical depth similar to that in Figs. 2 and 4, with Eq. (37) as the common shape of the extinction profile: (a) τ = 4.9, (b) τ = 0.49, (c) τ = 0.049 (a1)–(c1) show families of inverted solutions with superimposed closest bounds: ○, upper bounds; ×, lower bounds. (a2)–(c2) show related error bounds: closest bounds as indicated; absolute bounds represented by a dashed curve). This is case study (3) in Table 1.

Fig. 7
Fig. 7

Parametric study of the closeness error for the absolute bounds illustrated in Figs. 6(a2)6(c2). Here, (a1)–(c1) show ∊up(R) [Eq. (33)] and (a2)–(c2) show ∊low(R) [Eq. (34)]. Note that ∊(R) is computed as a per-unit variation.

Fig. 8
Fig. 8

Comparative plots showing closeness errors of the absolute and closest bounds (Sections 3 and 4, respectively) for optical depths in the range of τ = 0.01–10 and by use of two different extinction scenes as inputs: (a), (b) trapezium profile; (c), (d) Eq. (37) profile: × closeness error ‖∊up‖ [Eqs. (33) and (36)]; ○, ‖∊low‖ [Eq. (34) and (36)]; labels 1, 2, and 3 are explained in Table 1. In all cases, performance of the closest bounds is much better than that of the absolute bounds with ‖∊up‖ typically well below 10-3 (0.1%) and ‖∊low‖ virtually negligible. For the absolute bounds, the results are also much more dependent on optical thickness.

Fig. 9
Fig. 9

Error margin on the retrieved optical thickness for the two simulation sets studied: (a) trapezium profile input, (b) Eq. (37) profile input; × ∊τ,max [Eq. (39)]; ○, ∊τ,min [Eq. (40)]; labels 1, 2 and 3 are explained in Table 1.

Fig. 10
Fig. 10

Normalized R 2-corrected function x(R) corresponding to a real inversion example (445–895 m). The profile shows two storm cloud layers located between 450 and 550 m and between 785 and 835 m.

Fig. 11
Fig. 11

Error bounds for the live scene of Fig. 10: (a) comparison between the absolute bounds of Section 3, Eq. (24) (dashed curves) and the closest bounds of Section 4, Eq. (32) (curves with circles). (b) Zoom-in detail of the range interval between the two cloud layers (550 and 800 m).

Tables (1)

Tables Icon

Table 1 Simulation Ranges and Labels for Figs. 8 and 9

Equations (50)

Equations on this page are rendered with MathJax. Learn more.

PR=AR2 βRexp-2 0R αrdr,
dSRdR=1βRdβRdR-2αR,
SR=lnR2PR.
βR=BαRk,
dSRdR=kαRdαRdR-2αR.
SRm=Sm  αRm=αm,
αR=expS-Sm/kαm-1+2kRRm expS-Sm/kdr.
mina,bR2PR-b exp-aR2=mina,bi=1NRi2PRi-b exp-aRi2,
kc,min<kc<kc,max,  αc,min<αc<αc,max.
αˆR, kc, αc=αRk/kcIR, kc,ααmk/kcαc IRm, kc, α + 2kc FR, k, kc, α,
IR, kc, α=exp-2kcR0R αrdr,
FR, k, kc, α=RRm αrk/kcIr, kc, αdr.
τRmin, Rmax=RminRmax αrdr
T=10 logexp-2τ=-8.686τ (dB
R, kc, αc, k, αm=1-αˆR, kc, αcαR, k, αm100%.
αlowRαˆR, kc, αcαupR,
xR=expSR-Sm=R2PRRm2PRm,
γ=1/kc;  a=1/αc,
αˆR, γ, a=xRγa+2γ RRm xrγdr.
xRγmin<xRγ<xRγmax for xR>1,  xRγmax<xRγ<xRγmin for xR<1.
ux1x10otherwise,
δx1x=00otherwise,
yupR=xγmaxux-1+xγminu1-x-δx-1,  ylowR=xγminux-1+xγmaxu1-x-δx-1,
ylowRamax+2γmaxRRm yuprdrαˆR, γ, ayupRamin+2γminRRm ylowrdr,
(γopt(max, αopt(max)  (γopt(min, αopt(min),
αˆR0, γopt(max, αopt(max
αˆ(R0, γopt(max, αopt(max)  αup(R0),
αˆ(R0, γopt(min, αopt(min) =αlow(R0).
αˆR, γoptmin, amax<αˆR, γ, a<αˆR, γoptmax, amin.
dαˆdγa=amaxR=R0=0;  dαˆdγa=aminR=R0=0.
αˆR, γ, a=xRγa+qR, γ,
qR, γ=2γ RRm xrγdr.
q0γ2γc0+2γγ-1c1+γγ-12c2,
cn=R0Rm xrlnnxrdr.
αfitR0, γ, a=x0a+2c0+x0a+2c0lnx0-2c0+c1a+2c0×γ-1+x0a+2c0½ ln2x0-2c1+c2a+2c0-ln x0-2c0+c1a+2c0×2c0+c1a+2c0γ-12+Oγ-13,
αupR=maxαˆR, γi, aminαlowR=minαˆR, γi, amax;  γi=γmax, γmin, γc,1R, γc,2R.
upR=αupR-maxαR, γ, amaxαR, γ, a.
lowR=αlowR-minαR, γ, aminαR, γ, a
ΛR=|αupR-αlowR|maxαR, γ, a-minαR, γ, a.
=1Ni=1N Ri21/2.
αR=α¯faR+b,
ft=gt+c sindtgt+e,  gt=g01t-t02+t1+1t-t22+t3,
τ,max=ταup-τ0τ0,
τ,min=τ0-ταlowτ0,
αˆR, kc, αc=expS-Sm/kcαc-1+2kcRRm expS-Sm/kcdr.
xR=expS-Sm=αRk exp-2 R0Rαrdrαmk exp-2 R0Rm αrdr,
FR, α=RRm αrexp-2kR0r αzdzdr.
ur=exp-2kR0r αzdz
FR, α=k2IR, k-IRm, k.
αˆR, k, αc=αR1+1c-1IRm, k, αIR, k, α; c=αcαm.

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