Abstract

Poisson statistics are traditionally used to estimate the mean and standard deviation of the mean in time–range realizations of received photon counts from stationary processes in incoherent-detection lidar systems. However, this approach must be modified if the process under study is measurably nonstationary to account for any additional (and potentially unanticipated) variability. We demonstrate that the modified approach produces a different form for the estimated standard deviation of the mean for lidar return counts, which can also be applied to binning of higher-order data products. This modified technique also serves to determine optimum time–range integrations, diagnose system stability, and constrain operational modes.

© 2001 Optical Society of America

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References

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  1. P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, Boston, 1992).
  2. J. R. Taylor, An Introduction to Error Analysis; The Study of Uncertainties in Physical Measurements (University Science, Mill Valley, Calif., 1982).
  3. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).
  4. R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Krieger, Malabar, Fla., 1992).
  5. J. P. Thayer, N. B. Nielsen, R. E. Warren, C. J. Heinselman, J. Sohn, “Rayleigh lidar systems for middle atmosphere research in the Arctic,” Opt. Eng. 36, 2045–2061 (1997).
    [CrossRef]
  6. E. Durieux, L. Fiorani, “Measurement of the lidar signal fluctuation with a shot-per-shot instrument,” App. Opt. 37, 7128–7131 (1998).
    [CrossRef]
  7. C. S. Ruf, S. E. Beus, “Retrieval of tropospheric water vapor scale height from horizontal turbulence structure,” IEEE Trans. Geosci. Remote Sens. 35, 203–211 (1997).
    [CrossRef]
  8. C. S. Gardner, D. C. Senft, T. J. Beatty, R. E. Bills, C. A. Hostetler, “Rayleigh and sodium lidar techniques for measuring middle atmosphere density, temperature, and wind perturbations and their spectra,” in World Ionosphere/Thermosphere Study Handbook, S. H. Liu, B. Edwards, eds. (University of Illinois, Urbana-Champaign, Ill., 1989).
  9. C. S. Gardner, “Sodium resonance fluorescence lidar applications in atmospheric science and astronomy,” Proc. IEEE 77, 408–418 (1989).
    [CrossRef]

1998 (1)

E. Durieux, L. Fiorani, “Measurement of the lidar signal fluctuation with a shot-per-shot instrument,” App. Opt. 37, 7128–7131 (1998).
[CrossRef]

1997 (2)

C. S. Ruf, S. E. Beus, “Retrieval of tropospheric water vapor scale height from horizontal turbulence structure,” IEEE Trans. Geosci. Remote Sens. 35, 203–211 (1997).
[CrossRef]

J. P. Thayer, N. B. Nielsen, R. E. Warren, C. J. Heinselman, J. Sohn, “Rayleigh lidar systems for middle atmosphere research in the Arctic,” Opt. Eng. 36, 2045–2061 (1997).
[CrossRef]

1989 (1)

C. S. Gardner, “Sodium resonance fluorescence lidar applications in atmospheric science and astronomy,” Proc. IEEE 77, 408–418 (1989).
[CrossRef]

Beatty, T. J.

C. S. Gardner, D. C. Senft, T. J. Beatty, R. E. Bills, C. A. Hostetler, “Rayleigh and sodium lidar techniques for measuring middle atmosphere density, temperature, and wind perturbations and their spectra,” in World Ionosphere/Thermosphere Study Handbook, S. H. Liu, B. Edwards, eds. (University of Illinois, Urbana-Champaign, Ill., 1989).

Beus, S. E.

C. S. Ruf, S. E. Beus, “Retrieval of tropospheric water vapor scale height from horizontal turbulence structure,” IEEE Trans. Geosci. Remote Sens. 35, 203–211 (1997).
[CrossRef]

Bevington, P. R.

P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, Boston, 1992).

Bills, R. E.

C. S. Gardner, D. C. Senft, T. J. Beatty, R. E. Bills, C. A. Hostetler, “Rayleigh and sodium lidar techniques for measuring middle atmosphere density, temperature, and wind perturbations and their spectra,” in World Ionosphere/Thermosphere Study Handbook, S. H. Liu, B. Edwards, eds. (University of Illinois, Urbana-Champaign, Ill., 1989).

Durieux, E.

E. Durieux, L. Fiorani, “Measurement of the lidar signal fluctuation with a shot-per-shot instrument,” App. Opt. 37, 7128–7131 (1998).
[CrossRef]

Fiorani, L.

E. Durieux, L. Fiorani, “Measurement of the lidar signal fluctuation with a shot-per-shot instrument,” App. Opt. 37, 7128–7131 (1998).
[CrossRef]

Gardner, C. S.

C. S. Gardner, “Sodium resonance fluorescence lidar applications in atmospheric science and astronomy,” Proc. IEEE 77, 408–418 (1989).
[CrossRef]

C. S. Gardner, D. C. Senft, T. J. Beatty, R. E. Bills, C. A. Hostetler, “Rayleigh and sodium lidar techniques for measuring middle atmosphere density, temperature, and wind perturbations and their spectra,” in World Ionosphere/Thermosphere Study Handbook, S. H. Liu, B. Edwards, eds. (University of Illinois, Urbana-Champaign, Ill., 1989).

Heinselman, C. J.

J. P. Thayer, N. B. Nielsen, R. E. Warren, C. J. Heinselman, J. Sohn, “Rayleigh lidar systems for middle atmosphere research in the Arctic,” Opt. Eng. 36, 2045–2061 (1997).
[CrossRef]

Hostetler, C. A.

C. S. Gardner, D. C. Senft, T. J. Beatty, R. E. Bills, C. A. Hostetler, “Rayleigh and sodium lidar techniques for measuring middle atmosphere density, temperature, and wind perturbations and their spectra,” in World Ionosphere/Thermosphere Study Handbook, S. H. Liu, B. Edwards, eds. (University of Illinois, Urbana-Champaign, Ill., 1989).

Measures, R. M.

R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Krieger, Malabar, Fla., 1992).

Nielsen, N. B.

J. P. Thayer, N. B. Nielsen, R. E. Warren, C. J. Heinselman, J. Sohn, “Rayleigh lidar systems for middle atmosphere research in the Arctic,” Opt. Eng. 36, 2045–2061 (1997).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

Robinson, D. K.

P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, Boston, 1992).

Ruf, C. S.

C. S. Ruf, S. E. Beus, “Retrieval of tropospheric water vapor scale height from horizontal turbulence structure,” IEEE Trans. Geosci. Remote Sens. 35, 203–211 (1997).
[CrossRef]

Senft, D. C.

C. S. Gardner, D. C. Senft, T. J. Beatty, R. E. Bills, C. A. Hostetler, “Rayleigh and sodium lidar techniques for measuring middle atmosphere density, temperature, and wind perturbations and their spectra,” in World Ionosphere/Thermosphere Study Handbook, S. H. Liu, B. Edwards, eds. (University of Illinois, Urbana-Champaign, Ill., 1989).

Sohn, J.

J. P. Thayer, N. B. Nielsen, R. E. Warren, C. J. Heinselman, J. Sohn, “Rayleigh lidar systems for middle atmosphere research in the Arctic,” Opt. Eng. 36, 2045–2061 (1997).
[CrossRef]

Taylor, J. R.

J. R. Taylor, An Introduction to Error Analysis; The Study of Uncertainties in Physical Measurements (University Science, Mill Valley, Calif., 1982).

Thayer, J. P.

J. P. Thayer, N. B. Nielsen, R. E. Warren, C. J. Heinselman, J. Sohn, “Rayleigh lidar systems for middle atmosphere research in the Arctic,” Opt. Eng. 36, 2045–2061 (1997).
[CrossRef]

Warren, R. E.

J. P. Thayer, N. B. Nielsen, R. E. Warren, C. J. Heinselman, J. Sohn, “Rayleigh lidar systems for middle atmosphere research in the Arctic,” Opt. Eng. 36, 2045–2061 (1997).
[CrossRef]

App. Opt. (1)

E. Durieux, L. Fiorani, “Measurement of the lidar signal fluctuation with a shot-per-shot instrument,” App. Opt. 37, 7128–7131 (1998).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

C. S. Ruf, S. E. Beus, “Retrieval of tropospheric water vapor scale height from horizontal turbulence structure,” IEEE Trans. Geosci. Remote Sens. 35, 203–211 (1997).
[CrossRef]

Opt. Eng. (1)

J. P. Thayer, N. B. Nielsen, R. E. Warren, C. J. Heinselman, J. Sohn, “Rayleigh lidar systems for middle atmosphere research in the Arctic,” Opt. Eng. 36, 2045–2061 (1997).
[CrossRef]

Proc. IEEE (1)

C. S. Gardner, “Sodium resonance fluorescence lidar applications in atmospheric science and astronomy,” Proc. IEEE 77, 408–418 (1989).
[CrossRef]

Other (5)

C. S. Gardner, D. C. Senft, T. J. Beatty, R. E. Bills, C. A. Hostetler, “Rayleigh and sodium lidar techniques for measuring middle atmosphere density, temperature, and wind perturbations and their spectra,” in World Ionosphere/Thermosphere Study Handbook, S. H. Liu, B. Edwards, eds. (University of Illinois, Urbana-Champaign, Ill., 1989).

P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, Boston, 1992).

J. R. Taylor, An Introduction to Error Analysis; The Study of Uncertainties in Physical Measurements (University Science, Mill Valley, Calif., 1982).

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Krieger, Malabar, Fla., 1992).

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Figures (2)

Fig. 1
Fig. 1

Top, schematic depicting a stationary child PDF, which maintains its mean and variance over multiple realizations and is therefore equal to the overall parent PDF. Bottom, schematic depicting potential nonstationary return that is due to the different characteristics of multiple child PDF’s. In this case, a new overall parent PDF is formed that includes the variance for all the children.

Fig. 2
Fig. 2

Left, three consecutive middle-atmospheric temperature profiles from a 30- to a 60-km altitude, each calculated from 2 h of nominal photon count realizations from a 6-h data set. The three profiles are plotted together to show the variation of the mean temperatures (dashed curves) over the total 6-h period. The standard deviation bars (dark, bounding curves) are obtained from traditional Poisson error analysis. Right, plot showing the 6-h nightly mean-temperature profile (dashed curve) calculated from the nominal photon count realizations (the same mean is obtained from both traditional Poisson and the modified technique). The uncertainty bars with both the traditional (dark bounding curves) and modified analysis (light, bounding curves) are also plotted. We note that the new error analysis better acknowledges the overall variability of the process under study.

Equations (12)

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μˆ=1Ni=1N xˆi,
σˆμˆ=μˆ/N,
σtotal2=σp2+σNS2,
σtotal2=xi-μ¯2,
σp2=σi2=μ¯,
σNS2=μi-μ¯2.
σˆNS2=σˆtotal2-σˆp21N-1i=1Nxˆi-μˆ2-μˆ.
σˆμˆμˆN+σˆNS21/2.
Rτ=NSt+ptNSt+τ+pt+τ=RˆNSτfor τ0σˆtotal2=RˆNSτ=0+σˆp2for τ=0.
σˆNS2=RˆNSτ=0Rˆτ=0-μˆ.
σˆNS2=σˆtotal2-σˆhodp21N-1i=1Nxˆi-μˆ2-1Ni σˆi2,
σμˆi σˆi2N2+σˆNS21/2,

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