Abstract

The lidar data-inversion algorithm widely known as the Klett method (and its more elaborate variants) has long been used to invert elastic-lidar data obtained from atmospheric sounding systems. The Klett backward algorithm has also been shown to be robust in the face of uncertainties concerning the boundary condition. Nevertheless electrical noise at the photoreceiver output unavoidably has an impact on the data-inversion process, and describing in an explicit way how it affects retrieval of the atmospheric optical coefficients can contribute to improvement in inversion quality. We examine formally the way noise disturbs backscatter-coefficient retrievals done with the Klett backward algorithm, derive a mathematical expression for the retrieved backscatter coefficient in the presence of noise affecting the signal, and assess the noise impact and suggest ways to limit it.

© 2004 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, 211–220 (1981).
    [CrossRef] [PubMed]
  2. F. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652–653 (1984).
    [CrossRef] [PubMed]
  3. L. R. Bissonnette, “Sensitivity analysis of lidar inversion algorithms,” Appl. Opt. 25, 2122–2125 (1986).
    [CrossRef] [PubMed]
  4. F. Rocadenbosch, A. Comerón, “Error analysis for the lidar backward inversion algorithm,” Appl. Opt. 38, 4461–4474 (1999).
    [CrossRef]
  5. J. D. Klett, “Lidar inversion with variable backscatter/extinction ratios,” Appl. Opt. 24, 1638–1643 (1985).
    [CrossRef] [PubMed]
  6. 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]
  7. Q. Jinhuan, “Sensitivity of lidar equation solution to boundary values and determination of the values,” Adv. Atmos. Sci. 5, 229–241 (1988).
    [CrossRef]
  8. M. Matsumoto, N. Takeuchi, “Effects of misestimated far-end boundary values on two common lidar inversion solutions,” Appl. Opt. 33, 6451–6456 (1994).
    [CrossRef] [PubMed]

1999

1994

1988

Q. Jinhuan, “Sensitivity of lidar equation solution to boundary values and determination of the values,” Adv. Atmos. Sci. 5, 229–241 (1988).
[CrossRef]

1986

1985

1984

1981

Bissonnette, L. R.

Comerón, A.

Fernald, F. G.

Jinhuan, Q.

Q. Jinhuan, “Sensitivity of lidar equation solution to boundary values and determination of the values,” Adv. Atmos. Sci. 5, 229–241 (1988).
[CrossRef]

Klett, J. D.

Matsumoto, M.

Nakane, H.

Rocadenbosch, F.

Sasano, Y.

Takeuchi, N.

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Figures (5)

Fig. 1
Fig. 1

Lidar range-corrected signal between 600 and 6180 m for 4000 pulses integrated at 1064 nm.

Fig. 2
Fig. 2

Estimate of the signal-to-noise ratio for the example lidar signal being studied. The discontinuous line takes into account only photoreceiver noise and background-radiation shot noise. The continuous curve also shows the effect of the signal-induced shot noise.

Fig. 3
Fig. 3

Estimate of the ratio [σ n max(R)]/σ nm for the lidar-signal example being studied.

Fig. 4
Fig. 4

Backscatter coefficient inversions starting from R m = 6000 to R m = 6120 m. (a) (R m ) is taken directly as the value contained in the resolution cell corresponding to R m . (b) (R m ) is taken as the average of R m and the R m ± 7.5 m cells (N = 3 cells). (c) (R m ) is taken as the average of the cells between R m - 30 and R m + 30 m (N = 9 cells). (d) (R m ) is taken as the average of the cells between R m - 60 and R m + 60 m (N = 17 cells).

Fig. 5
Fig. 5

Estimated 68% confidence interval, dotted curves, for the inverted backscatter coefficient: (a) N = 1, (b) N = 17.

Equations (20)

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βR=βmR2PRRm2PRm+2βmRRmz2PzCzdz,
PˆR=PR+nR,
1βˆR=1R2PˆR1βm Rm2PˆRm+2 RRmz2PˆzCzdz.
1βˆR=1βR+1βmRm2nRmR2PR+2R2PRRRmz2nzCzdzPRPR+nR.
mR=1βmRm2nRmR2PR+2R2PRRRmz2nzCzdz.
βˆR=1+nRPRβR1+mRβR.
1+nRPR
βRmR=ζmR+ζiR
ζmR=βRβmRm2nRmR2PR
ζiR=2βRRmRm2R/Rm1 ξ2nnξCnξdξR2PR,
ζm1/2=βRβmRm2σnmR2PR,
ζi2R=4β2RRm6R4P2RR/Rm1R/Rm1×ξ12ξ22Cnξ1Cnξ2 rξnnξ1, ξ2dξ1dξ2,
rznnz1, z2=σn2z1ΔRδz1-z2,
ζi2R1/2<2βRRmCmin15ΔRRm×1-RRm51/2Rm2σn maxRR2PR,
R2PR=AβRexp-2 0R αxdx,
ζm2R1/2=σnmPRmexp-2 RRm αxdx.
ζi2R1/2ζm2R1/2 <2RmβmNCminσn maxRσnm×15ΔRRm1-RRm51/2,
2RmβmCmin15ΔRRm
2RmβmCmin15ΔRRm1/2=0.24.
ζi2R1/2ζm2R1/2.

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