Abstract

The Bernoulli solution of the lidar equation with the assumption of a constant extinction/backscattering ratio can lead to errors in the derived aerosol extinction and backscattering profiles. This paper presents a general theoretical analysis of the errors that result from differences between the assumed and actual extinction/backscattering ratio profiles. Examples of the influence of the constant extinction/backscattering ratio assumption on the lidar derived aerosol extinction profile are presented for various laser wavelengths.

© 1985 Optical Society of America

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References

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  1. 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. Hjnkley, Ed. (Springer-Verlag, New York, 1976), p. 117.
  2. J. D. Klett, “Lidar Inversion with Variable/Extinction Ratios,” Appl. Opt. 24, 1638 (1985).
    [CrossRef] [PubMed]
  3. J. D. Klett, “Stable Analytical Inversion Solution for Processing Lidar Returns,” Appl. Opt. 20, 211 (1981).
    [CrossRef] [PubMed]
  4. F. G. Fernald, “Analysis of Atmospheric Lidar Observations: Some Comments,” Appl. Opt. 23, 652 (1984).
    [CrossRef] [PubMed]
  5. C. Braun, “General Formula for the Errors in Aerosol Properties Determined from Lidar Measurements at a Single Wavelength,” Appl. Opt. 24, 925 (1985).
    [CrossRef] [PubMed]
  6. H. Salemink, P. Schotanus, J. B. Bergwerff, “Quantitative Lidar at 532 nm for Vertical Extinction Profiles and the Effect of Relative Humidity,” Appl. Phys. B34, 187 (1984).

1985 (2)

1984 (2)

H. Salemink, P. Schotanus, J. B. Bergwerff, “Quantitative Lidar at 532 nm for Vertical Extinction Profiles and the Effect of Relative Humidity,” Appl. Phys. B34, 187 (1984).

F. G. Fernald, “Analysis of Atmospheric Lidar Observations: Some Comments,” Appl. Opt. 23, 652 (1984).
[CrossRef] [PubMed]

1981 (1)

Bergwerff, J. B.

H. Salemink, P. Schotanus, J. B. Bergwerff, “Quantitative Lidar at 532 nm for Vertical Extinction Profiles and the Effect of Relative Humidity,” Appl. Phys. B34, 187 (1984).

Braun, C.

Collis, R. T. H.

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. Hjnkley, Ed. (Springer-Verlag, New York, 1976), p. 117.

Fernald, F. G.

Klett, J. D.

Russell, P. B.

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. Hjnkley, Ed. (Springer-Verlag, New York, 1976), p. 117.

Salemink, H.

H. Salemink, P. Schotanus, J. B. Bergwerff, “Quantitative Lidar at 532 nm for Vertical Extinction Profiles and the Effect of Relative Humidity,” Appl. Phys. B34, 187 (1984).

Schotanus, P.

H. Salemink, P. Schotanus, J. B. Bergwerff, “Quantitative Lidar at 532 nm for Vertical Extinction Profiles and the Effect of Relative Humidity,” Appl. Phys. B34, 187 (1984).

Appl. Opt. (4)

Appl. Phys. (1)

H. Salemink, P. Schotanus, J. B. Bergwerff, “Quantitative Lidar at 532 nm for Vertical Extinction Profiles and the Effect of Relative Humidity,” Appl. Phys. B34, 187 (1984).

Other (1)

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. Hjnkley, Ed. (Springer-Verlag, New York, 1976), p. 117.

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Figures (1)

Fig. 1
Fig. 1

Reconstructed profiles of aerosol extinction coefficient. Boundary conditions were correctly given at 1500 m, and the backward integration was applied.

Equations (20)

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X ( R ) P ( R ) R 2 = C [ β 1 ( R ) + β 2 ( R ) ] exp { 2 0 R [ α 1 ( r ) + α 2 ( r ) ] d r } ,
S 1 ( R ) α ( R ) / β 1 ( R ) ,
S 2 α 2 ( R ) / β 2 ( R ) .
X ( R ) = C S 1 ( R ) [ α 1 ( R ) + S 1 ( R ) S 2 α 2 ( R ) ] × exp { 2 0 R [ α 1 ( r ) + α 2 ( r ) ] d r } .
y ( R ) α 1 ( R ) + S 1 ( R ) S 2 α 2 ( R ) ,
S 1 ( R ) X ( R ) exp { 2 0 R [ S 1 ( r ) S 2 1 ] α 2 ( r ) d r } = C y ( R ) exp [ 2 0 R y ( r ) d r ] .
d ln ( [ S 1 ( R ) X ( R ) ] exp { 2 0 R [ S 1 ( r ) S 2 1 ] α 2 ( r ) d r } ) d R = 1 y ( R ) d y ( R ) d R 2 y ( R ) .
α 1 ( R ) + S 1 ( R ) S 2 α 2 ( R ) = S 1 ( R ) X ( R ) exp { 2 0 R [ S 1 ( r ) S 2 1 ] α 2 ( r ) d r } S 1 ( R o ) X ( R o ) α 1 ( R o ) + S 1 ( R o ) S 2 α 2 ( R o ) 2 R o R S 1 ( r ) X ( r ) exp { 2 R o r [ S 1 ( r ) S 2 1 ] α 2 ( r ) d r } d r ,
y ( R o ) = α 1 ( R o ) + S 1 ( R o ) S 2 α 2 ( R o ) .
β 1 ( R ) + β 2 ( R ) = X ( R ) exp { 2 R o R [ S 1 ( r ) S 2 ] β 2 ( r ) d r } S ( R o ) β 1 ( R o ) + β 2 ( R o ) 2 R o R S 1 ( r ) X ( r ) exp { 2 R o r [ S 1 ( r ) S 2 ] β 2 ( r ) d r } d r .
A ( I , I 1 ) = { [ S 1 ( I 1 ) S 2 ] β 2 ( I 1 ) + [ S 1 ( I ) S 2 ] β 2 ( I ) } Δ R .
α 1 ( I 1 ) + S 1 ( I 1 ) S 2 α 2 ( I 1 ) = S 1 ( I 1 ) X ( I 1 ) exp [ A ( I , I 1 ) ] S 1 ( I ) X ( I ) α 1 ( I ) + S 1 ( I ) S 2 α 2 ( I ) + { S 1 ( I ) X ( I ) + S 1 ( I 1 ) X ( I 1 ) exp [ A ( I , I 1 ) ] } Δ R ,
β 1 ( I 1 ) + β 2 ( I 1 ) = X ( I 1 ) exp [ A ( I , I 1 ) ] X ( I ) β 1 ( I ) + β 2 ( I ) + { S 1 ( I ) X ( I ) + S 1 ( I 1 ) X ( I 1 ) exp [ A ( I , I 1 ) ] } Δ R .
S ˆ 1 ( R ) X ˆ ( R ) exp { 2 0 R [ S ˆ 1 ( r ) S 2 1 ] α ˆ 2 ( r ) d r } = C y ˆ ( R ) exp [ 2 0 R y ˆ ( r ) d r ] .
σ ( R ) ξ ( R ) exp ( 2 0 R { [ σ ( r ) δ ( r ) 1 ] S ˆ 1 ( r ) S 2 α ˆ 2 ( r ) [ δ ( r ) 1 ] α ˆ 2 ( r ) } d r ) = η ( R ) exp { 2 0 R [ η ( r ) 1 ] y ˆ ( r ) d r } .
1 η d η d R 2 y ˆ η = 2 y ˆ + d ln ( σ ξ ) d r 2 ( σ δ 1 ) S ˆ 1 α ˆ 2 S 2 + 2 ( δ 1 ) α ˆ 2 ,
η ( R ) = [ exp [ 2 Ro R { y ˆ ( r ) + ( σ ( r ) δ ( r ) 1 ) S ˆ 1 ( r ) S 2 α ˆ 2 ( r ) ( δ ( r ) 1 ) α ˆ 2 ( r ) } dr ] σ ( R ) ξ ( R ) / ( σ ( Ro ) ξ ( Ro ) ) ] [ 1 / η ( Ro ) 2 Ro R y ˆ ( r ) exp [ 2 Ro R { y ˆ ( r ) + ( σ ( r ) δ ( r ) 1 ) S ˆ 1 ( r ˙ ) S 2 α ˆ 2 ( r ) ( δ ( r ) 1 ) α ˆ 2 ( r ) } d r ] σ ( r ) ξ ( r ) / ( σ ( Ro ) ξ ( Ro ) ) dr ]
η ( R ) = η ( R o ) σ ( R ) σ ( R o )
α 1 ( R ) + S 1 ( R ) S 2 α 2 ( R ) α ˆ 1 ( R ) + S ˆ 1 ( R ) S 2 α ˆ 2 ( R ) = α 1 ( R o ) + S 1 ( R o ) S 2 α 2 ( R o ) α ˆ 1 ( R o ) + S ˆ 1 ( R o ) S 2 α ˆ 2 ( R o ) [ S 1 ( R ) S ˆ 1 ( R ) / S 1 ( R o ) S ˆ 1 ( R o ) ]
α 1 ( R ) α ˆ 1 ( R ) = α 1 ( R o ) α ˆ 1 ( R o ) [ S 1 ( R ) S ˆ 1 ( R ) / S 1 ( R o ) S ˆ 1 ( R o ) ] .

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