Abstract

The lidar inversion algorithms based on near-end and far-end boundary conditions are reviewed and the recently proposed algorithm featuring the calculation of the boundary value at some middle range is expressed in analytic form. A theoretical assessment of the stability of all three algorithms to uncertainties in the boundary value and assumed backscatter/extinction model is then carried out. It is confirmed that the far-end solution is the most stable in practical applications and that the middle-range algorithm is only a variant of the unstable near-end solution.

© 1986 Optical Society of America

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References

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  1. J. D. Klett, “Stable Analytical Inversion Solution for Processing Lidar Returns,” Appl. Opt. 20, 211 (1981).
    [CrossRef] [PubMed]
  2. J. D. Klett, “Lidar Inversion with Variable Backscatter/Extinction Ratios,” Appl. Opt. 24, 1638 (1985).
    [CrossRef] [PubMed]
  3. 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 (1985).
    [CrossRef] [PubMed]
  4. J. A. Ferguson, D. H. Stephens, “Algorithm for Inverting Lidar Returns,” Appl. Opt. 22, 3673 (1983).
    [CrossRef] [PubMed]
  5. J. M. Mulders, “Algorithm for Inverting Lidar Returns: Comment,” Appl. Opt. 23, 2855 (1984).
    [CrossRef] [PubMed]
  6. G. J. Kunz, “Vertical Atmospheric Profiles Measured with Lidar,” Appl. Opt. 22, 1955 (1983).
    [CrossRef] [PubMed]

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Equations (16)

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P ( R ) = K β ( R ) R 2 exp [ - 2 0 R σ ( r ) d r ] ,
S ( R ) = l n [ R 2 R 0 2 P ( R ) P 0 ] ,
d S d R = 1 β d β d R - 2 σ .
β ( R ) = C ( R ) σ k ( R ) ,
d S d R = 1 C d C d R + k σ d σ d R - 2 σ .
σ n ( R ) = [ C ( R n ) C ( R ) ] 1 / k exp [ S ( R ) - S ( R n ) k ] σ 0 n - 1 - 2 k R n R [ C ( R n ) C ( r ) ] 1 / k exp [ S ( r ) - S ( R n ) k ] d r ,
σ f ( R ) = [ C ( R f ) C ( R ) ] 1 / k exp [ S ( R ) - S ( R f ) f ] σ 0 f - 1 + 2 k R R f [ C ( R f ) C ( r ) ] 1 / k exp [ S ( r ) - S ( R f ) k ] d r ,
σ - 1 ( R m ) = [ K C ( R m ) P 0 R 0 2 ] 1 / k exp [ - S ( R m ) - 2 I 0 k ] - 2 k R n R m [ C ( R m ) C ( r ) ] 1 / k exp [ S ( r ) - S ( R m ) k ] d r ,
I 0 = 0 R n σ d r ,
σ m ( R ) = [ C ( R m ) C ( R ) ] 1 / k exp [ S ( R ) - S ( R m ) k ] [ K C ( R m ) P 0 R 0 2 ] 1 / k exp [ - S ( R m ) - 2 I 0 k ] - 2 k R n R [ C ( R m ) C ( r ) ] 1 / k exp [ S ( r ) - S ( R m ) k ] d r .
δ σ n ( R ) σ n ( R ) = 1 k [ δ C ( R n ) C ( R n ) - δ C ( R ) C ( R ) ] + 2 k 2 σ n ( R ) R n R [ δ C ( R n ) C ( R n ) - δ C ( r ) C ( r ) ] × [ C ( R ) C ( r ) ] 1 / k exp [ S ( r ) - S ( R ) k ] d r ,
δ σ f ( R ) σ f ( R ) = 1 k [ δ C ( R f ) C ( R f ) - δ C ( R ) C ( R ) ] - 2 k 2 σ f ( R ) R R f [ δ C ( R f ) C ( R f ) - δ C ( r ) C ( r ) ] × [ C ( R ) C ( r ) ] 1 / k exp [ S ( r ) - S ( R ) k ] d r ,
δ σ m ( R ) σ m ( R ) = 1 k [ δ C ( R m ) C ( R m ) - δ C ( R ) C ( R ) ] - 2 k 2 σ m ( R ) R n R [ δ C ( R m ) C ( R m ) - δ C ( r ) C ( r ) ] × [ C ( R ) C ( r ) ] 1 / k exp [ S ( r ) - S ( R ) k ] d r - 1 k σ m ( R ) σ m ( R n ) [ C ( R ) C ( R n ) ] 1 / k exp [ S ( R n ) - S ( R ) k ] δ C ( R m ) C ( R m ) .
δ σ n ( R ) σ n ( R ) = [ C ( R ) C ( R n ) ] 1 / k exp [ S ( R n ) - S ( R ) k ] σ n ( R ) σ 0 n δ σ 0 n σ 0 n ,
δ σ f ( R ) σ f ( R ) = [ C ( R ) C ( R f ) ] 1 / k exp [ S ( R f ) - S ( R ) k ] σ f ( R ) σ 0 f δ σ 0 f σ 0 f ,
δ σ m ( R ) σ m ( R ) = [ C ( R ) C ( R n ) ] 1 / k exp [ S ( R n ) - S ( R ) k ] σ m ( R ) σ m ( R n ) × 1 k [ 2 δ I 0 - δ K K ] .

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