Abstract

A detailed study using both analytical and numerical calculations of direct and heterodyne differential absorption lidar (DIAL) techniques is conducted to complement previous studies. The DIAL measurement errors depend on key experimental parameters, some of which can be adjusted to minimize the statistical error. Accordingly, the pertinent criteria on optical thickness, the number of photons emitted at the on and off wavelengths, are discussed to reduce the relative error on the total column content or range-resolved measurements that rely on either hard target or atmospheric backscatter returns. In direct detection, the optimal optical thickness decreases from 1.3 to 0.8 when the background increases while the on-line-to-off-line optimal energy ratio decreases from 3.6 to 2.7. In heterodyne detection, the minimum error is obtained for an optical thickness of 1.2 and an energy ratio of 4.3.

© 2006 Optical Society of America

Full Article  |  PDF Article

Corrections

Didier Bruneau, Fabien Gibert, Pierre H. Flamant, and Jacques Pelon, "Complementary study of differential absorption lidar optimization in direct and heterodyne detections: erratum," Appl. Opt. 46, 428-428 (2007)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-46-3-428

References

  • View by:
  • |
  • |
  • |

  1. E. E. Remsberg and L. L. Gordley, "Analysis of differential absorption lidar from the Space Shuttle," Appl. Opt. 17, 624-630 (1978).
    [CrossRef] [PubMed]
  2. G. Mégie and R. T. Menzies, "Complementarity of UV and IR differential absorption lidar for global measurements of atmospheric species," Appl. Opt. 19, 1173-1183 (1980).
    [CrossRef] [PubMed]
  3. R. M. Schotland, "Errors in the lidar measurements of atmospheric gases by differential absorption," J. Appl. Meteorol. 13, 71-77 (1974).
    [CrossRef]
  4. B. Saleh, Photoelectron Statistics (Springer, 1978), pp. 195-196.
  5. B. J. Rye and R. M. Hardesty, "Estimate optimization parameters for incoherent backscatter heterodyne lidar," Appl. Opt. 36, 9425-9436 (1997).
    [CrossRef]

1997 (1)

1980 (1)

1978 (1)

1974 (1)

R. M. Schotland, "Errors in the lidar measurements of atmospheric gases by differential absorption," J. Appl. Meteorol. 13, 71-77 (1974).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Relative optical thickness error in direct detection as a function of optical thickness τ and on-line-to-off-line emitted energy ratio α. Left plot, for SNL conditions. Right plot, for NB on NT = NB off NT = 0.2 with optimized NT Md. The error is divided by the DELB.

Fig. 2
Fig. 2

Optimum optical thickness τ for direct detection integrated measurements as a function of NB on Non and NB off Noff .

Fig. 3
Fig. 3

Optimum on-line-to-off-line emitted energy ratio α for direct detection integrated measurements as a function of NB on Non and NB off Noff .

Fig. 4
Fig. 4

Optimum number of off-line DDPs per DOF, Noff Md , for direct detection integrated measurements as a function of NB on Non and NB off Noff .

Fig. 5
Fig. 5

Optical thickness relative error for direct detection integrated measurements in optimal conditions as a function of NB on Non and NB off Noff , divided by the DELB.

Fig. 6
Fig. 6

Optimum CNR for integrated heterodyne detection measurements as a function of optical thickness τ and on-line-to-off-line emitted energy ratio α. Left, CNRon; right, CNRoff.

Fig. 7
Fig. 7

Relative optical thickness error for heterodyne detection integrated measurements as a function of optical thickness τ and on-line-to-off-line emitted energy ratio α with optimized CNR.

Fig. 8
Fig. 8

Optimum optical thickness at the most distant range τ2 and on-line-to-off-line emitted energy ratio α as a function of the relative LOT δττ 2 for range-resolved direct detection measurements with different BSRs. NB off NT = NB on NT = 0 (solid curve), NB off NT = NB on NT = 0.01 (dashed curve), NB off NT = NB on NT = 0.2 (dotted–dashed curve), and for a signal ratio of k = 1.

Fig. 9
Fig. 9

Optimum optical thickness at the most-distant range of τ2 and on-line-to-off-line emitted energy ratio α as a function of the relative LOT δτ∕τ2 for range-resolved heterodyne detection measurements and for a signal ratio of k = 1.

Fig. 10
Fig. 10

Relative optical thickness error in direct detection as a function of optical thickness τ and on-line-to-off-line averaging ratio β. Left plot, for SNL conditions. Right plot, for NB on NT = NB off NT = 0.2 with optimized NT Md. The error is divided by the DELB.

Fig. 11
Fig. 11

Relative optical thickness error for heterodyne detection integrated measurements as a function of optical thickness τ and on-line-to-off-line averaging ratio β, with optimized CNR.

Tables (2)

Tables Icon

Table 1 Acronyms and Main Parameters

Tables Icon

Table 2 Optimal Conditions and Relative Error of Optical Thickness for Integrated Measurements for Direct Detection and Heterodyne Detection

Equations (45)

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

P i ( R ) = η i E i b i ( R ) A c 2 R 2 exp [ 2 τ i ( R ) 2 τ i     0 ( R ) ] ,
τ i ( R ) = 0 R n ( r ) σ ˜ i ( r ) d r ,
P i ( R ) = K i ( R ) E i exp [ 2 τ i ( R ) ] ,
K i ( R ) = η i b i ( R ) A c 2 R 2 exp [ 2 τ i     0 ( R ) ] .
K on ( R ) = K off ( R ) = K ( R ) .
P on ( R ) P off ( R ) = K on ( R ) K off ( R ) E on E off exp { 2 [ τ on ( 0 , R ) τ off ( 0 , R ) ] } .
τ ( R ) = 1 2 ln [ P off ( R ) E on P on ( R ) E off ] .
var ( τ ) = 1 4 [ var ( P on ) P on 2 + var ( P off ) P off 2 2 cov ( P on , P off ) P on P off ] ,
σ ( τ ) τ = 1 2 τ { [ σ ( P on ) P on ] 2 + [ σ ( P off ) P off ] 2 } 1 / 2 ,
σ ( P ) / P = ( 1 + N / M d N ) 1 / 2 ,
σ ( P i ) P i = [ 1 + ( N B i / N i ) + ( N i / M d ) N i ] 1 / 2 ,
N off = N T 1 1 + α ,
N on = N T α 1 + α exp ( 2 τ ) ,
σ ( τ ) τ = 1 2 τ N T 1 / 2 { [ 1 + ( 1 + α ) N B off N T ] ( 1 + α ) + [ 1 + ( 1 + α ) α N B on N T exp ( 2 τ ) ] × ( 1 + α α ) exp ( 2 τ ) + 2 N T M d } 1 / 2 .
[ σ ( τ ) τ ] SNL = 1 2 τ N T 1 / 2 { ( 1 + α ) [ 1 + exp ( 2 τ ) / α ] } 1 / 2 .
[ σ ( τ ) τ ] 0 = 1.8 N T - 1 / 2 = 3.86 N off - 1 / 2 = 7.32 N on - 1 / 2 .
F ( N T , M d , τ , α ) = { [ σ ( τ ) τ ] 2 [ σ ( τ ) τ ] SNL 2 } / [ σ ( τ ) τ ] SNL 2
F = 1 + α α N B off N T α 2 + N B on N T exp ( 4 τ ) + 2 N T M d ( α 1 + α ) 2 α + exp ( 2 τ ) .
( N T M d ) opt = 1 2 ( 1 + α α ) 2 [ N B off N T α 2 + N B on N T exp ( 4 τ ) ] ,
( N on M d ) opt = 1 2 exp ( 2 τ ) [ N B off N off α + N B on N on exp ( 2 τ ) ] ,
( N off M d ) opt = 1 2 α [ N B off N off α + N B on N on exp ( 2 τ ) ] .
2 α 4 N B off / N T + α 3 ( 1 + 2 N B off / N T ) α [ exp ( 2 τ ) + 2 N B on / N T exp ( 4 τ ) ] 2 N B on / N T exp ( 4 τ ) = 0 .
σ ( P i ) P i = 1 M t ( 1 + 1 CNR ) ,
CNR i = γ H η P i h ν B h ,
CNR i = N i M t ,
σ ( P i ) P i = 1 M t ( 1 + M t N i ) .
σ ( τ ) τ = 1 2 τ M t { [ 1 + M t N T ( 1 + α ) ] 2 + [ 1 + M t N T ( 1 + α α ) exp ( 2 τ ) ] 2 } 1 / 2 .
N T M t = 1 + α α [ α 2 + exp ( 4 τ ) 2 ] 1 / 2 .
N off M t = CNR off = a α ,
N on M t = CNR on = a exp ( 2 τ ) ,
a = [ α 2 + exp ( 4 τ ) 2 ] 1 / 2 .
σ ( τ ) τ = 1 2 τ N T 1 / 2 [ ( 1 + α α ) a ] 1 / 2 { ( 1 + α a ) 2 + [ 1 + exp ( 2 τ ) a ] 2 } 1 / 2 .
σ ( δ τ ) δ τ = 1 δ τ [ σ 2 ( τ 1 ) + σ 2 ( τ 2 ) ] 1 / 2 .
σ ( δ τ ) δ τ = 1 2 δτ N T 2 1 / 2 { ( 1 + α ) ( 1 + k ) + 1 + α α × exp ( 2 τ 2 ) [ 1 + k exp ( 2 δ τ ) ] + ( 1 + α ) 2 ( 1 + k 2 ) N B off N T 2 + ( 1 + α α ) 2 × exp ( 4 τ 2 ) [ 1 + k 2 exp ( 4 δ τ ) ] N B on N T 2 + 4 N T 2 M d } 1 / 2 .
N T 2 M d = 1 4 ( 1 + α α ) 2 { N B off N T 2 α 2 ( 1 + k 2 ) + N B on N T 2 × exp ( 4 τ 2 ) [ 1 + k 2 exp ( 4 δ τ ) ] } .
σ ( δ τ ) δ τ = 1 2 δ τ M t { [ 1 + k M t N T 2 ( 1 + α ) ] 2 + [ 1 + k M t N T 2 ( 1 + α α ) exp ( 2 τ 1 ) ] 2 + [ 1 + M t N T 2 ( 1 + α ) ] 2 + [ 1 + M t N T 2 ( 1 + α α ) exp ( 2 τ 2 ) ] 2 } 1 / 2 .
N T 2 M t = a 2 1 + α α ,
a = { α 2 ( 1 + k 2 ) + exp ( 4 τ 2 ) [ 1 + k 2 exp ( 4 δ τ ) ] } 1 / 2 .
σ ( δ τ ) δ τ = 1 2 δτ N T 2 1 / 2 [ 1 + α α a ] 1 / 2 [ ( 1 + α a ) 2 + ( 1 + k α a ) 2 + [ 1 + exp ( 2 τ 2 ) a ] 2 + { 1 + k exp [ 2 ( τ 2 δ τ ) ] a } 2 ] 1 / 2 .
M on = β M off .
σ ( δ τ ) δ τ = 1 2 τ M P 1 / 2 N T 1 / 2 ( 1 + β ) [ 1 + exp ( 2 τ ) β + 2 N B on N T exp ( 4 τ ) β + 2 N B off N T + 1 + β 2 β N T M d ] 1 / 2 .
N T M d = 4 1 + β [ N B on N T exp ( 4 τ ) + N B off N T β ] .
σ ( δ τ ) δ τ = 1 2 τ M P 1 / 2 N T 1 / 2    ( 1 + β ) [ 1 + exp ( 2 τ ) β + 4 N B on N T exp ( 4 τ ) β + 4 N B off N T ] 1 / 2 .
σ ( τ ) τ = 1 2 τ M t M p 1 + β 2 { 1 β [ 1 + 2 M t N T exp ( 2 τ ) ] 2 + ( 1 + 2 M t N T ) 2 } 1 / 2 .
N T M t = 2 β 1 + β [ 1 + exp ( 4 τ ) ] .

Metrics