Abstract

A simple analytical method is presented that shows some potential for application to the problem of extracting attenuation and backscatter coefficients in an inhomogeneous atmosphere from the return signal of a monostatic single-wavelength lidar system. The method assumes the validity of the single-scattering lidar equation and a power law relationship between backscatter and attenuation. For optical depths greater than unity the inversion method can be applied in principle using only information contained in the signal itself. In contrast to a well-known related analytical inversion solution, the new solution form is shown to be stable with respect to perturbations in the signal, the postulated relationship between backscatter and attenuation, and the assumed or estimated boundary value of attenuation.

© 1981 Optical Society of America

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References

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  1. R. T. H. Collis, P. B. Russell, in Laser Monitoring of the Atmosphere, E. D. Hinkley, Ed. (Springer, New York, 1976), p. 117.
  2. R. T. H. Collis, Q. J. R. Meteorol. Soc. 92, 220 (1966).
    [CrossRef]
  3. W. Viezee, E. E. Uthe, R. T. H. Collis, J. Appl. Meteorol. 8, 274 (1969).
    [CrossRef]
  4. H. R. Pruppacher, J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, Holland, 1978), pp. 20–21.
  5. T. Okita, J. Meteorol. Soc. Jpn. 40, 39 (1962).
  6. R. T. Brown, J. Appl. Meteorol. 12, 698 (1973).
    [CrossRef]
  7. W. Viezee, J. Oblanas, R. T. H. Collis, AFCRL-TR-73-0708, Air Force Cambridge Research Laboratories, Bedford, Mass. (1973), NTIS 776 054.
  8. R. H. Kohl, J. Appl. Meteorol. 17, 1034 (1978).
    [CrossRef]
  9. R. H. Kohl, J. Appl. Meteorol. 18, 712 (1979).
    [CrossRef]
  10. R. T. Brown, J. Appl. Meteorol. 18, 711 (1979).
    [CrossRef]
  11. J. A. Curcio, G. L. Knestrick, J. Opt. Soc. Am. 48, 686 (1958).
    [CrossRef]
  12. R. W. Fenn, Appl. Opt. 5, 293 (1966).
    [CrossRef] [PubMed]
  13. O. D. Barteneva, Bull. Acad. Sci. USSR No. 12, 1, 852 (1960).
  14. S. Twomey, H. B. Howell, Appl. Opt. 4, 501 (1965).
    [CrossRef]
  15. R. G. Pinnick, S. G. Jennings, P. Chýlek, C. V. Hamm, to be submitted to J. Atmos Sci.
  16. E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956).
  17. W. Hitschfeld, J. Bordan, J. Meteorol. 11, 58 (1954).
    [CrossRef]
  18. E. W. Barnett, O. Ben-Dov, J. Appl. Meteorol. 6, 500 (1967).
    [CrossRef]
  19. P. A. Davis, Appl. Opt. 10, 2099 (1969).
    [CrossRef]
  20. F. G. Fernald, B. M. Herman, J. A. Reagan, J. Appl. Meteorol. 11, 482 (1972).
    [CrossRef]
  21. H. Herrmann, Alta Freq. 9, 732 (1974).
  22. K. N. Liou, R. M. Schotland, J. Atmos. Sci. 28, 772 (1971).
    [CrossRef]
  23. E. W. Eloranta, Ph.D. Thesis, Department of Meteorology, U. Wisconsin (1972).
  24. B. M. Golubitskiy, T. M. Zhadko, M. V. Tantashev, Izv. Atmos. Oceanic Phys. 8, 1226 (1972).
  25. W. G. M. Blättner, C. M. Lampley, Radiation Research Associates Report RRA-47706, Fort Worth, Tex. (1977).
  26. I. V. Samokhvalov, Opt. Lett. 4, 12 (1979).
    [CrossRef] [PubMed]
  27. R. S. Bonner, W. J. Lentz, Atmospheric Sciences Laboratory Report ASL-TR-0042, White Sands Missile Range, N.Mex. (1979).

1979 (3)

R. H. Kohl, J. Appl. Meteorol. 18, 712 (1979).
[CrossRef]

R. T. Brown, J. Appl. Meteorol. 18, 711 (1979).
[CrossRef]

I. V. Samokhvalov, Opt. Lett. 4, 12 (1979).
[CrossRef] [PubMed]

1978 (1)

R. H. Kohl, J. Appl. Meteorol. 17, 1034 (1978).
[CrossRef]

1974 (1)

H. Herrmann, Alta Freq. 9, 732 (1974).

1973 (1)

R. T. Brown, J. Appl. Meteorol. 12, 698 (1973).
[CrossRef]

1972 (2)

F. G. Fernald, B. M. Herman, J. A. Reagan, J. Appl. Meteorol. 11, 482 (1972).
[CrossRef]

B. M. Golubitskiy, T. M. Zhadko, M. V. Tantashev, Izv. Atmos. Oceanic Phys. 8, 1226 (1972).

1971 (1)

K. N. Liou, R. M. Schotland, J. Atmos. Sci. 28, 772 (1971).
[CrossRef]

1969 (2)

P. A. Davis, Appl. Opt. 10, 2099 (1969).
[CrossRef]

W. Viezee, E. E. Uthe, R. T. H. Collis, J. Appl. Meteorol. 8, 274 (1969).
[CrossRef]

1967 (1)

E. W. Barnett, O. Ben-Dov, J. Appl. Meteorol. 6, 500 (1967).
[CrossRef]

1966 (2)

R. T. H. Collis, Q. J. R. Meteorol. Soc. 92, 220 (1966).
[CrossRef]

R. W. Fenn, Appl. Opt. 5, 293 (1966).
[CrossRef] [PubMed]

1965 (1)

1962 (1)

T. Okita, J. Meteorol. Soc. Jpn. 40, 39 (1962).

1960 (1)

O. D. Barteneva, Bull. Acad. Sci. USSR No. 12, 1, 852 (1960).

1958 (1)

1954 (1)

W. Hitschfeld, J. Bordan, J. Meteorol. 11, 58 (1954).
[CrossRef]

Barnett, E. W.

E. W. Barnett, O. Ben-Dov, J. Appl. Meteorol. 6, 500 (1967).
[CrossRef]

Barteneva, O. D.

O. D. Barteneva, Bull. Acad. Sci. USSR No. 12, 1, 852 (1960).

Ben-Dov, O.

E. W. Barnett, O. Ben-Dov, J. Appl. Meteorol. 6, 500 (1967).
[CrossRef]

Blättner, W. G. M.

W. G. M. Blättner, C. M. Lampley, Radiation Research Associates Report RRA-47706, Fort Worth, Tex. (1977).

Bonner, R. S.

R. S. Bonner, W. J. Lentz, Atmospheric Sciences Laboratory Report ASL-TR-0042, White Sands Missile Range, N.Mex. (1979).

Bordan, J.

W. Hitschfeld, J. Bordan, J. Meteorol. 11, 58 (1954).
[CrossRef]

Brown, R. T.

R. T. Brown, J. Appl. Meteorol. 18, 711 (1979).
[CrossRef]

R. T. Brown, J. Appl. Meteorol. 12, 698 (1973).
[CrossRef]

Chýlek, P.

R. G. Pinnick, S. G. Jennings, P. Chýlek, C. V. Hamm, to be submitted to J. Atmos Sci.

Collis, R. T. H.

W. Viezee, E. E. Uthe, R. T. H. Collis, J. Appl. Meteorol. 8, 274 (1969).
[CrossRef]

R. T. H. Collis, Q. J. R. Meteorol. Soc. 92, 220 (1966).
[CrossRef]

R. T. H. Collis, P. B. Russell, in Laser Monitoring of the Atmosphere, E. D. Hinkley, Ed. (Springer, New York, 1976), p. 117.

W. Viezee, J. Oblanas, R. T. H. Collis, AFCRL-TR-73-0708, Air Force Cambridge Research Laboratories, Bedford, Mass. (1973), NTIS 776 054.

Curcio, J. A.

Davis, P. A.

P. A. Davis, Appl. Opt. 10, 2099 (1969).
[CrossRef]

Eloranta, E. W.

E. W. Eloranta, Ph.D. Thesis, Department of Meteorology, U. Wisconsin (1972).

Fenn, R. W.

Fernald, F. G.

F. G. Fernald, B. M. Herman, J. A. Reagan, J. Appl. Meteorol. 11, 482 (1972).
[CrossRef]

Golubitskiy, B. M.

B. M. Golubitskiy, T. M. Zhadko, M. V. Tantashev, Izv. Atmos. Oceanic Phys. 8, 1226 (1972).

Hamm, C. V.

R. G. Pinnick, S. G. Jennings, P. Chýlek, C. V. Hamm, to be submitted to J. Atmos Sci.

Herman, B. M.

F. G. Fernald, B. M. Herman, J. A. Reagan, J. Appl. Meteorol. 11, 482 (1972).
[CrossRef]

Herrmann, H.

H. Herrmann, Alta Freq. 9, 732 (1974).

Hitschfeld, W.

W. Hitschfeld, J. Bordan, J. Meteorol. 11, 58 (1954).
[CrossRef]

Howell, H. B.

Ince, E. L.

E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956).

Jennings, S. G.

R. G. Pinnick, S. G. Jennings, P. Chýlek, C. V. Hamm, to be submitted to J. Atmos Sci.

Klett, J. D.

H. R. Pruppacher, J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, Holland, 1978), pp. 20–21.

Knestrick, G. L.

Kohl, R. H.

R. H. Kohl, J. Appl. Meteorol. 18, 712 (1979).
[CrossRef]

R. H. Kohl, J. Appl. Meteorol. 17, 1034 (1978).
[CrossRef]

Lampley, C. M.

W. G. M. Blättner, C. M. Lampley, Radiation Research Associates Report RRA-47706, Fort Worth, Tex. (1977).

Lentz, W. J.

R. S. Bonner, W. J. Lentz, Atmospheric Sciences Laboratory Report ASL-TR-0042, White Sands Missile Range, N.Mex. (1979).

Liou, K. N.

K. N. Liou, R. M. Schotland, J. Atmos. Sci. 28, 772 (1971).
[CrossRef]

Oblanas, J.

W. Viezee, J. Oblanas, R. T. H. Collis, AFCRL-TR-73-0708, Air Force Cambridge Research Laboratories, Bedford, Mass. (1973), NTIS 776 054.

Okita, T.

T. Okita, J. Meteorol. Soc. Jpn. 40, 39 (1962).

Pinnick, R. G.

R. G. Pinnick, S. G. Jennings, P. Chýlek, C. V. Hamm, to be submitted to J. Atmos Sci.

Pruppacher, H. R.

H. R. Pruppacher, J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, Holland, 1978), pp. 20–21.

Reagan, J. A.

F. G. Fernald, B. M. Herman, J. A. Reagan, J. Appl. Meteorol. 11, 482 (1972).
[CrossRef]

Russell, P. B.

R. T. H. Collis, P. B. Russell, in Laser Monitoring of the Atmosphere, E. D. Hinkley, Ed. (Springer, New York, 1976), p. 117.

Samokhvalov, I. V.

Schotland, R. M.

K. N. Liou, R. M. Schotland, J. Atmos. Sci. 28, 772 (1971).
[CrossRef]

Tantashev, M. V.

B. M. Golubitskiy, T. M. Zhadko, M. V. Tantashev, Izv. Atmos. Oceanic Phys. 8, 1226 (1972).

Twomey, S.

Uthe, E. E.

W. Viezee, E. E. Uthe, R. T. H. Collis, J. Appl. Meteorol. 8, 274 (1969).
[CrossRef]

Viezee, W.

W. Viezee, E. E. Uthe, R. T. H. Collis, J. Appl. Meteorol. 8, 274 (1969).
[CrossRef]

W. Viezee, J. Oblanas, R. T. H. Collis, AFCRL-TR-73-0708, Air Force Cambridge Research Laboratories, Bedford, Mass. (1973), NTIS 776 054.

Zhadko, T. M.

B. M. Golubitskiy, T. M. Zhadko, M. V. Tantashev, Izv. Atmos. Oceanic Phys. 8, 1226 (1972).

Alta Freq. (1)

H. Herrmann, Alta Freq. 9, 732 (1974).

Appl. Opt. (3)

Bull. Acad. Sci. USSR No. 12 (1)

O. D. Barteneva, Bull. Acad. Sci. USSR No. 12, 1, 852 (1960).

Izv. Atmos. Oceanic Phys. (1)

B. M. Golubitskiy, T. M. Zhadko, M. V. Tantashev, Izv. Atmos. Oceanic Phys. 8, 1226 (1972).

J. Appl. Meteorol. (7)

E. W. Barnett, O. Ben-Dov, J. Appl. Meteorol. 6, 500 (1967).
[CrossRef]

F. G. Fernald, B. M. Herman, J. A. Reagan, J. Appl. Meteorol. 11, 482 (1972).
[CrossRef]

R. T. Brown, J. Appl. Meteorol. 12, 698 (1973).
[CrossRef]

W. Viezee, E. E. Uthe, R. T. H. Collis, J. Appl. Meteorol. 8, 274 (1969).
[CrossRef]

R. H. Kohl, J. Appl. Meteorol. 17, 1034 (1978).
[CrossRef]

R. H. Kohl, J. Appl. Meteorol. 18, 712 (1979).
[CrossRef]

R. T. Brown, J. Appl. Meteorol. 18, 711 (1979).
[CrossRef]

J. Atmos. Sci. (1)

K. N. Liou, R. M. Schotland, J. Atmos. Sci. 28, 772 (1971).
[CrossRef]

J. Meteorol. (1)

W. Hitschfeld, J. Bordan, J. Meteorol. 11, 58 (1954).
[CrossRef]

J. Meteorol. Soc. Jpn. (1)

T. Okita, J. Meteorol. Soc. Jpn. 40, 39 (1962).

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Q. J. R. Meteorol. Soc. (1)

R. T. H. Collis, Q. J. R. Meteorol. Soc. 92, 220 (1966).
[CrossRef]

Other (8)

R. T. H. Collis, P. B. Russell, in Laser Monitoring of the Atmosphere, E. D. Hinkley, Ed. (Springer, New York, 1976), p. 117.

H. R. Pruppacher, J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, Holland, 1978), pp. 20–21.

R. G. Pinnick, S. G. Jennings, P. Chýlek, C. V. Hamm, to be submitted to J. Atmos Sci.

E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956).

W. Viezee, J. Oblanas, R. T. H. Collis, AFCRL-TR-73-0708, Air Force Cambridge Research Laboratories, Bedford, Mass. (1973), NTIS 776 054.

E. W. Eloranta, Ph.D. Thesis, Department of Meteorology, U. Wisconsin (1972).

W. G. M. Blättner, C. M. Lampley, Radiation Research Associates Report RRA-47706, Fort Worth, Tex. (1977).

R. S. Bonner, W. J. Lentz, Atmospheric Sciences Laboratory Report ASL-TR-0042, White Sands Missile Range, N.Mex. (1979).

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Figures (8)

Fig. 1
Fig. 1

Growth of perturbations in σ due to small errors in the determination of σ0.

Fig. 2
Fig. 2

Effect on inversions of errors in the boundary value estimates of σ.

Fig. 3
Fig. 3

Effect on inversions of simulated signal noise.

Fig. 4
Fig. 4

Effect on inversions of value of k.

Fig. 5
Fig. 5

Inversions for a low visibility strongly inhomogeneous atmosphere, including effects of signal noise.

Fig. 6
Fig. 6

Same as Fig. 5 but also including effects of variable k.

Fig. 7
Fig. 7

Inversions for a high visibility atmosphere, including effects of signal noise and variable k.

Fig. 8
Fig. 8

Inversions of a real lidar signal for k = 0.67 and 1.0.

Equations (24)

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P ( r ) = P 0 c τ 2 A β ( r ) r 2 exp [ - 2 0 r σ ( r ) d r ] ,
S ( r ) ln [ r 2 P ( r ) ] .
S - S 0 = ln β β 0 - 2 r 0 r σ d r
d S d r = 1 β d β d r - 2 σ ,
σ hom = - 1 2 d S d r .
β = const σ k ,
d S d r = k σ d σ d r - 2 σ .
σ - 1 = exp ( - r 1 k d S d r d r ) × [ C - 2 r exp k ( - r 1 k d S d r d r ) d r ] ,
σ = exp [ ( S - S 0 ) / k { σ 0 - 1 - 2 k r 0 r exp [ ( S - S 0 ) / k ] d r } ,
S - S 0 = k ln σ σ 0 - 2 r 0 r σ d r = k ln σ σ 0 - 2 r 0 r σ d r ,
( 1 + δ / σ ) = ( 1 + δ 0 / σ 0 ) exp ( 2 k r 0 r δ d r ) .
d ζ d r = 2 k σ 0 ζ ( ζ - 1 ) ,
( 1 + δ σ 0 ) - 1 = 1 - δ 0 σ 0 ( 1 + δ 0 σ 0 ) - 1 exp [ 2 σ 0 ( r - r 0 ) k ] .
Δ r = k 2 σ 0 ln ( 1 + σ 0 δ 0 ) .
σ ( r ) = exp [ ( S - S m ) / k ] { σ m - 1 + 2 k r r m exp [ ( S - S m ) / k ] d r } ,
r a r b σ d r = - k 2 ln { 1 - 2 σ a k r a r b exp [ ( S - S a ) / k ] d r } ,
r a r b σ d r = k 2 ln { 1 + 2 σ b k r a r b exp [ ( S - S b ) / k ] d r } ,
Ω = [ 1 - exp ( - Ω ) ] 2 I a b + [ exp ( Ω ) - 1 ] 2 I b a ,
Ω = [ 2 σ ¯ ( r b - r a ) ] / k ,
I a b = ( r b - r a ) - 1 r a r b exp [ ( S - S a ) / k ] d r ,
I b a = ( r b - r a ) - 1 r a r b exp [ ( S - S b ) / k ] d r = I a b exp [ ( S a - S b ) / k ] .
σ ¯ = - 3 ( S a + S b ) 16 × ( 1 + { 1 - 16 9 ( S a + S b ) 2 [ S a 2 + S b 2 + k ( S a + S b ) ] } 1 / 2 ) ,
σ m 1 2 ( S 0 - S m ) ( r m - r 0 ) .
σ m exp [ ( S b - S m ) / k ] - 1 2 k r b r m exp [ ( S - S m ) / k ] d r .

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