Abstract

The ratio of temporally adjacent lidar pulse returns is commonly used in differential absorption lidar (DIAL) to reduce correlated noise. These pulses typically are generated at different wavelengths with the assumption that the dominant noise is common to both. This is not the case when the mean number of laser speckle integrated per pulse by the lidar receiver is small (namely, less than 10 speckles at each wavelength). In this case a large increase in the standard deviation of the ratio data results. We demonstrate this effect both theoretically and experimentally. The theoretical value for the expected standard deviation of the pulse–pair ratio data compares well with the measured values that used a dual CO2 laser-based lidar with a hard target. Pulse averaging statistics of the pulse–pair data obey the expected σ1/N reduction in the standard deviation, σN, for N-pulse averages. We consider the ratio before average, average before ratio, and log of the ratio before average methods for noise reduction in the lidar equation. The implications of our results are discussed in the context of dual-laser versus single-laser lidar configurations.

© 1997 Optical Society of America

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References

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  1. R. M. Measures, Laser Remote Sensing, Fundamentals and Applications (Krieger, Malabar, Fla., 1992).
  2. W. B. Grant, “He–Ne and cw CO2 laser long-path systems for gas detection,” Appl. Opt. 25, 709–719 (1986).
    [CrossRef] [PubMed]
  3. D. K. Killinger, N. Menyuk, “Effect of turbulence-induced correlation on laser remote sensing errors,” Appl. Phys. Lett. 38, 968–970 (1981).
    [CrossRef]
  4. R. M. Schotland, “Errors in the lidar measurement of atmospheric gases by differential absorption,” J. Appl. Meteorol. 13, 71–77 (1974).
    [CrossRef]
  5. M. J. T. Milton, P. T. Woods, “Pulse averaging methods for a laser remote monitoring system using atmospheric backscatter,” Appl. Opt. 26, 2598–2603 (1987).
    [CrossRef] [PubMed]
  6. Y. Sasano, E. V. Browell, S. Ismail, “Error caused by using a constant extinction/backscattering ratio in the lidar solution,” Appl. Opt. 24, 3929–3932 (1985).
    [CrossRef] [PubMed]
  7. H. Ahlberg, S. Lundqvist, M. S. Shumate, U. Persson, “Analysis of errors caused by optical interference effects in wavelength-diverse CO2 laser long-path systems,” Appl. Opt. 24, 3917–3923 (1985).
    [CrossRef]
  8. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), Chap. 2.
  9. G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), Chap. 3.
  10. L. G. Shirley, E. E. Arieal, G. R. Hallerman, H. C. Payson, J. R. Vivilecchia, “Advanced techniques for target discrimination using laser speckle,” Mass. Inst. Technol. Lincoln Lab. J. 5, 367–440 (1992).
  11. T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1995), Vol. 34, pp. 183–248.
    [CrossRef]
  12. E. P. MacKerrow, M. J. Schmitt, “Measurement of integrated speckle statistics for CO2 lidar returns from a moving, non-uniform, hard-target,” Appl. Opt. 36, 6921–6937 (1997).
    [CrossRef]
  13. J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
    [CrossRef]
  14. J. A. Fox, C. R. Gautier, J. L. Ahl, “Practical consideration for the design of CO2 lidar systems,” Appl. Opt. 27, 847–855 (1988).
    [CrossRef] [PubMed]
  15. W. B. Grant, A. M. Brothers, J. R. Bogan, “Differential absorption lidar signal averaging,” Appl. Opt. 27, 1934–1938 (1988).
    [CrossRef] [PubMed]
  16. P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).
  17. W. R. Leo, Techniques for Nuclear and Particle Physics Experiments (Springer-Verlag, Berlin, 1987), Chap. 4.
    [CrossRef]
  18. F. James, M. Roos, “Errors on ratios of small number of events,” Nucl. Phys. B 172, 475–480 (1980).
    [CrossRef]
  19. R. E. Warren, “Effect of pulse-pair correlation on differential absorption lidar,” Appl. Opt. 24, 3472–3475 (1985).
    [CrossRef] [PubMed]
  20. S. L. Meyer, Data Analysis for Scientists and Engineers (Wiley, New York, 1975).
  21. This integral was evaluated using the software Mathematica Version 3.0 (Wolfram Research, Inc., Champaign, Ill., 1996).
  22. A Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), Chap. 5.
  23. J. S. Simonoff, Smoothing Methods in Statistics (Springer-Verlag, New York, 1996), Chap. 2.
    [CrossRef]
  24. G. E. P. Box, G. M. Jenkins, G. C. Reinsel, Time Series Analysis: Forecasting and Control (Prentice-Hall, Englewood Cliffs, N.J., 1994), pp. 30–31.
  25. G. M. Jenkins, D. G. Watts, Spectral analysis and its applications (Holden-Day, San Francisco, Calif., 1968), pp. 155–289.
  26. N. Menyuk, D. K. Killinger, C. R. Menyuk, “Limitations of signal averaging due to temporal correlation in laser remote-sensing measurements,” Appl. Opt. 21, 3377–3383 (1982).
    [CrossRef] [PubMed]
  27. G. D. Boreman, Y. Sun, A. B. James, “Generation of laser speckle with an integrating sphere,” Opt. Eng. 29, 339–342 (1990).
    [CrossRef]
  28. G. E. Busch, “Speckle error in transmitter energy measurement,” (Los Alamos National Laboratory, Chemical Science and Technology Division, Los Alamos, N. Mex., 1996), pp. 1–6.
  29. S. J. Czuchlewski, “Speckle effects with integrating spheres,” (Los Alamos National Laboratory, Chemical Science and Technology Division, Los Alamos, N. Mex., 1996), pp. 1–6.
  30. American National Standard for Information Systems, IEEE Standard for Binary Floating-Point Numbers, (IEEE, New York, 1985).
  31. N. Menyuk, D. K. Killinger, “Assessment of relative error sources in IR DIAL measurement accuracy,” Appl. Opt. 22, 2690–2698 (1983).
    [CrossRef] [PubMed]
  32. N. Menyuk, D. K. Killinger, C. R. Menyuk, “Error reduction in laser remote sensing: combined effects of cross correlation and signal averaging,” Appl. Opt. 24, 118–131 (1985).
    [CrossRef] [PubMed]

1997 (1)

1992 (1)

L. G. Shirley, E. E. Arieal, G. R. Hallerman, H. C. Payson, J. R. Vivilecchia, “Advanced techniques for target discrimination using laser speckle,” Mass. Inst. Technol. Lincoln Lab. J. 5, 367–440 (1992).

1990 (1)

G. D. Boreman, Y. Sun, A. B. James, “Generation of laser speckle with an integrating sphere,” Opt. Eng. 29, 339–342 (1990).
[CrossRef]

1988 (2)

1987 (1)

1986 (1)

1985 (4)

1983 (1)

1982 (1)

1981 (1)

D. K. Killinger, N. Menyuk, “Effect of turbulence-induced correlation on laser remote sensing errors,” Appl. Phys. Lett. 38, 968–970 (1981).
[CrossRef]

1980 (1)

F. James, M. Roos, “Errors on ratios of small number of events,” Nucl. Phys. B 172, 475–480 (1980).
[CrossRef]

1974 (1)

R. M. Schotland, “Errors in the lidar measurement of atmospheric gases by differential absorption,” J. Appl. Meteorol. 13, 71–77 (1974).
[CrossRef]

1965 (1)

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

Ahl, J. L.

Ahlberg, H.

Arieal, E. E.

L. G. Shirley, E. E. Arieal, G. R. Hallerman, H. C. Payson, J. R. Vivilecchia, “Advanced techniques for target discrimination using laser speckle,” Mass. Inst. Technol. Lincoln Lab. J. 5, 367–440 (1992).

Asakura, T.

T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1995), Vol. 34, pp. 183–248.
[CrossRef]

Bevington, P. R.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).

Bogan, J. R.

Boreman, G. D.

G. D. Boreman, Y. Sun, A. B. James, “Generation of laser speckle with an integrating sphere,” Opt. Eng. 29, 339–342 (1990).
[CrossRef]

Box, G. E. P.

G. E. P. Box, G. M. Jenkins, G. C. Reinsel, Time Series Analysis: Forecasting and Control (Prentice-Hall, Englewood Cliffs, N.J., 1994), pp. 30–31.

Brothers, A. M.

Browell, E. V.

Busch, G. E.

G. E. Busch, “Speckle error in transmitter energy measurement,” (Los Alamos National Laboratory, Chemical Science and Technology Division, Los Alamos, N. Mex., 1996), pp. 1–6.

Czuchlewski, S. J.

S. J. Czuchlewski, “Speckle effects with integrating spheres,” (Los Alamos National Laboratory, Chemical Science and Technology Division, Los Alamos, N. Mex., 1996), pp. 1–6.

Fox, J. A.

Gautier, C. R.

Goodman, J. W.

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), Chap. 2.

Grant, W. B.

Hallerman, G. R.

L. G. Shirley, E. E. Arieal, G. R. Hallerman, H. C. Payson, J. R. Vivilecchia, “Advanced techniques for target discrimination using laser speckle,” Mass. Inst. Technol. Lincoln Lab. J. 5, 367–440 (1992).

Ismail, S.

James, A. B.

G. D. Boreman, Y. Sun, A. B. James, “Generation of laser speckle with an integrating sphere,” Opt. Eng. 29, 339–342 (1990).
[CrossRef]

James, F.

F. James, M. Roos, “Errors on ratios of small number of events,” Nucl. Phys. B 172, 475–480 (1980).
[CrossRef]

Jenkins, G. M.

G. M. Jenkins, D. G. Watts, Spectral analysis and its applications (Holden-Day, San Francisco, Calif., 1968), pp. 155–289.

G. E. P. Box, G. M. Jenkins, G. C. Reinsel, Time Series Analysis: Forecasting and Control (Prentice-Hall, Englewood Cliffs, N.J., 1994), pp. 30–31.

Killinger, D. K.

Leo, W. R.

W. R. Leo, Techniques for Nuclear and Particle Physics Experiments (Springer-Verlag, Berlin, 1987), Chap. 4.
[CrossRef]

Lundqvist, S.

MacKerrow, E. P.

Measures, R. M.

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

Menyuk, C. R.

Menyuk, N.

Meyer, S. L.

S. L. Meyer, Data Analysis for Scientists and Engineers (Wiley, New York, 1975).

Milton, M. J. T.

Okamoto, T.

T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1995), Vol. 34, pp. 183–248.
[CrossRef]

Papoulis, A

A Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), Chap. 5.

Parry, G.

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), Chap. 3.

Payson, H. C.

L. G. Shirley, E. E. Arieal, G. R. Hallerman, H. C. Payson, J. R. Vivilecchia, “Advanced techniques for target discrimination using laser speckle,” Mass. Inst. Technol. Lincoln Lab. J. 5, 367–440 (1992).

Persson, U.

Reinsel, G. C.

G. E. P. Box, G. M. Jenkins, G. C. Reinsel, Time Series Analysis: Forecasting and Control (Prentice-Hall, Englewood Cliffs, N.J., 1994), pp. 30–31.

Roos, M.

F. James, M. Roos, “Errors on ratios of small number of events,” Nucl. Phys. B 172, 475–480 (1980).
[CrossRef]

Sasano, Y.

Schmitt, M. J.

Schotland, R. M.

R. M. Schotland, “Errors in the lidar measurement of atmospheric gases by differential absorption,” J. Appl. Meteorol. 13, 71–77 (1974).
[CrossRef]

Shirley, L. G.

L. G. Shirley, E. E. Arieal, G. R. Hallerman, H. C. Payson, J. R. Vivilecchia, “Advanced techniques for target discrimination using laser speckle,” Mass. Inst. Technol. Lincoln Lab. J. 5, 367–440 (1992).

Shumate, M. S.

Simonoff, J. S.

J. S. Simonoff, Smoothing Methods in Statistics (Springer-Verlag, New York, 1996), Chap. 2.
[CrossRef]

Sun, Y.

G. D. Boreman, Y. Sun, A. B. James, “Generation of laser speckle with an integrating sphere,” Opt. Eng. 29, 339–342 (1990).
[CrossRef]

Vivilecchia, J. R.

L. G. Shirley, E. E. Arieal, G. R. Hallerman, H. C. Payson, J. R. Vivilecchia, “Advanced techniques for target discrimination using laser speckle,” Mass. Inst. Technol. Lincoln Lab. J. 5, 367–440 (1992).

Warren, R. E.

Watts, D. G.

G. M. Jenkins, D. G. Watts, Spectral analysis and its applications (Holden-Day, San Francisco, Calif., 1968), pp. 155–289.

Woods, P. T.

Appl. Opt. (11)

N. Menyuk, D. K. Killinger, C. R. Menyuk, “Limitations of signal averaging due to temporal correlation in laser remote-sensing measurements,” Appl. Opt. 21, 3377–3383 (1982).
[CrossRef] [PubMed]

N. Menyuk, D. K. Killinger, “Assessment of relative error sources in IR DIAL measurement accuracy,” Appl. Opt. 22, 2690–2698 (1983).
[CrossRef] [PubMed]

N. Menyuk, D. K. Killinger, C. R. Menyuk, “Error reduction in laser remote sensing: combined effects of cross correlation and signal averaging,” Appl. Opt. 24, 118–131 (1985).
[CrossRef] [PubMed]

R. E. Warren, “Effect of pulse-pair correlation on differential absorption lidar,” Appl. Opt. 24, 3472–3475 (1985).
[CrossRef] [PubMed]

H. Ahlberg, S. Lundqvist, M. S. Shumate, U. Persson, “Analysis of errors caused by optical interference effects in wavelength-diverse CO2 laser long-path systems,” Appl. Opt. 24, 3917–3923 (1985).
[CrossRef]

Y. Sasano, E. V. Browell, S. Ismail, “Error caused by using a constant extinction/backscattering ratio in the lidar solution,” Appl. Opt. 24, 3929–3932 (1985).
[CrossRef] [PubMed]

W. B. Grant, “He–Ne and cw CO2 laser long-path systems for gas detection,” Appl. Opt. 25, 709–719 (1986).
[CrossRef] [PubMed]

M. J. T. Milton, P. T. Woods, “Pulse averaging methods for a laser remote monitoring system using atmospheric backscatter,” Appl. Opt. 26, 2598–2603 (1987).
[CrossRef] [PubMed]

J. A. Fox, C. R. Gautier, J. L. Ahl, “Practical consideration for the design of CO2 lidar systems,” Appl. Opt. 27, 847–855 (1988).
[CrossRef] [PubMed]

W. B. Grant, A. M. Brothers, J. R. Bogan, “Differential absorption lidar signal averaging,” Appl. Opt. 27, 1934–1938 (1988).
[CrossRef] [PubMed]

E. P. MacKerrow, M. J. Schmitt, “Measurement of integrated speckle statistics for CO2 lidar returns from a moving, non-uniform, hard-target,” Appl. Opt. 36, 6921–6937 (1997).
[CrossRef]

Appl. Phys. Lett. (1)

D. K. Killinger, N. Menyuk, “Effect of turbulence-induced correlation on laser remote sensing errors,” Appl. Phys. Lett. 38, 968–970 (1981).
[CrossRef]

J. Appl. Meteorol. (1)

R. M. Schotland, “Errors in the lidar measurement of atmospheric gases by differential absorption,” J. Appl. Meteorol. 13, 71–77 (1974).
[CrossRef]

Mass. Inst. Technol. Lincoln Lab. J. (1)

L. G. Shirley, E. E. Arieal, G. R. Hallerman, H. C. Payson, J. R. Vivilecchia, “Advanced techniques for target discrimination using laser speckle,” Mass. Inst. Technol. Lincoln Lab. J. 5, 367–440 (1992).

Nucl. Phys. B (1)

F. James, M. Roos, “Errors on ratios of small number of events,” Nucl. Phys. B 172, 475–480 (1980).
[CrossRef]

Opt. Eng. (1)

G. D. Boreman, Y. Sun, A. B. James, “Generation of laser speckle with an integrating sphere,” Opt. Eng. 29, 339–342 (1990).
[CrossRef]

Proc. IEEE (1)

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

Other (15)

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).

W. R. Leo, Techniques for Nuclear and Particle Physics Experiments (Springer-Verlag, Berlin, 1987), Chap. 4.
[CrossRef]

T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1995), Vol. 34, pp. 183–248.
[CrossRef]

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), Chap. 2.

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), Chap. 3.

G. E. Busch, “Speckle error in transmitter energy measurement,” (Los Alamos National Laboratory, Chemical Science and Technology Division, Los Alamos, N. Mex., 1996), pp. 1–6.

S. J. Czuchlewski, “Speckle effects with integrating spheres,” (Los Alamos National Laboratory, Chemical Science and Technology Division, Los Alamos, N. Mex., 1996), pp. 1–6.

American National Standard for Information Systems, IEEE Standard for Binary Floating-Point Numbers, (IEEE, New York, 1985).

S. L. Meyer, Data Analysis for Scientists and Engineers (Wiley, New York, 1975).

This integral was evaluated using the software Mathematica Version 3.0 (Wolfram Research, Inc., Champaign, Ill., 1996).

A Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), Chap. 5.

J. S. Simonoff, Smoothing Methods in Statistics (Springer-Verlag, New York, 1996), Chap. 2.
[CrossRef]

G. E. P. Box, G. M. Jenkins, G. C. Reinsel, Time Series Analysis: Forecasting and Control (Prentice-Hall, Englewood Cliffs, N.J., 1994), pp. 30–31.

G. M. Jenkins, D. G. Watts, Spectral analysis and its applications (Holden-Day, San Francisco, Calif., 1968), pp. 155–289.

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

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Figures (14)

Fig. 1
Fig. 1

Probability density function for single-wavelength data, Eq. (14), pulse–pair ratio data, Eq. (21), and the logarithm of the pulse–pair ratio, Eq. (31). The probability densities are for M = 1.05 integrated speckle (top) and for M = 10 integrated speckle (bottom). Note that the abscissa is the pulse–pair ratio intensity normalized to the mean of the pulse–pair ratio intensity (except for the single-wavelength data). The tails of these distributions display the singularities associated with taking the pulse–pair ratio.

Fig. 2
Fig. 2

Standard deviation for the pulse–pair ratio of lidar signals described by the gamma distribution, Eq. (14). Three cases are shown: dashed line, σ 1 / M for single-wavelength (i.e., no ratio) data; thick solid curve, the standard deviation predicted for the pulse–pair ratio, Eq. (27); thin solid curve, the standard deviation expected for the natural logarithm of the ratio.

Fig. 3
Fig. 3

Schematic of the lidar system used for this experiment: BS, beam splitter (R = 90%, T = 10%); W, wedge; OT, optical trigger; EM, energy monitor; P, periscope; BC, beam combiner (90% in main beam, 10% for diagnostics); PC, pyroelectric camera (for beam far-field profile and pointing); BE, beam expander (to 6-in.-(15-cm)-diameter); RO, relay optics; D, detector (liquid-N2-cooled mercury cadmium telluride, 1 mm × 1 mm square). The aperture used to limit the number of integrated speckle was placed directly in front of the telescope.

Fig. 4
Fig. 4

Samples of time series for the return signals from the rotating drum target with (a) a small-aperture receiver and (b) a large-aperture receiver. Speckle noise on the small-aperture signal dwarfs the reflectivity variations of the target, which are seen as periodic fluctuations in the top trace for the large-aperture data.

Fig. 5
Fig. 5

Lidar data that are dominated by speckle noise, M = 1.03, for (a) laser 0 operating at the 12C16O2(10P16) laser line and (b) laser 1 operating at 10P20. The pulse–pair ratio, each pulse in (a) divided by the temporally adjacent pulse in (b), is shown in (c). The large spikes in the data in (c) are due to the mathematical singularities associated with taking the pulse–pair ratio of lidar data dominated by speckle noise. Note that approximation (13) does not hold for these data.

Fig. 6
Fig. 6

Measured probability density functions, estimated by histograms, for the pulse–pair ratio of laser 0 pulses operating at the 12C16O2(10P16) line to laser 0 pulses at the 12C16O2(10P20) line. The aperture diameters used on the lidar receiver are given for each case along with the number of integrated speckle M, obtained from fitting Eq. (21) to the histograms.

Fig. 7
Fig. 7

Measured standard deviation versus the fitted values of M, the gamma speckle distribution parameter. The plot shows both single-wavelength and pulse–pair ratio data. The dashed curve is the theoretical relationship, Eq. (14), between the standard deviation and M for single-wavelength (single polarization) lidar data. The solid curve is the theoretical relationship, Eq. (21), between the standard deviation and M for pulse–pair ratio data. The data shown here are listed in Table 2. The standard deviation of the ratio data for M ≤ 2 was measured as infinite, since the true values of σ were outside the range of double-precision floating-point numbers.

Fig. 8
Fig. 8

Standard deviation of the mean for data with no pulse–pair ratio taken. Data are shown for laser 0 operating at the 12C16O2(10P16) line (10.55 μm), indicated as L0(10P16), and laser 1 operating at the 12C16O2(10P20) line (10.59 μm), indicated by L1(10P20). The values of M are those obtained in Ref. 12. The solid lines represent the expected standard deviation of the mean for independent speckle averaging; i.e., σ N = σ 1 / N.

Fig. 9
Fig. 9

Pulse–pair ratio averaging statistics for single-laser data taken with the smallest sized receiver apertures. The receiver aperture diameters and the number of integrated speckles, calculated in Ref. 12, are indicated. The solid line is the σ N = σ 1 / N averaging expected for independent speckle. The ratios taken were with pulses from laser 0 operating at the 12C16O2(10P16) line (10.55 μm), ratioed to pulses from laser 0 operating at the 12C16O2(10P20) line (10.59 μm). This ratio is indicated as L0(10P16)/L0(10P20) and similarly for L1(10P20)/L1(10P16). The pulses from each laser were separated by 100 ms.

Fig. 10
Fig. 10

Pulse–pair ratio averaging statistics for dual-laser data measured with the smallest sized receiver apertures. The receiver aperture diameters and the number of integrated speckles, calculated in Ref. 12, are indicated. The ratios taken were with pulses from laser 0 operating at the 12C16O2(10P16) line (10.55 μm), ratioed to pulses from laser 1 operating at the 12C16O2(10P20) line (10.59 μm). This ratio is indicated as L0(10P16)/L1(10P20) and similarly for L0(10P20)/L1(10P16). The separation between the pulses from each laser was 30 μs. The solid line is the σ 1 / N averaging expected for independent speckle.

Fig. 11
Fig. 11

Same data as shown in Fig. 5 except that it has been averaged with a ten-pulse-segmented average. The standard deviation of the data in (a) is smaller than for the data in Fig. 5(a) by approximately a factor of 10, as expected for averaging of uncorrelated speckle. The trend in (b) limits the noise reduction gained by averaging. The data in (c) represent the ratio of the data in (a) to the data in (b). The variance predicted by approximation (13) agrees fairly well with this averaged data.

Fig. 12
Fig. 12

Averaging statistics with the average before ratio method for single-laser lidar data [the same data used in Figs. 9(a)–9(h)]. The singular values for the standard deviation are the result of division by small numbers, regardless of averaging before taking the DIAL ratio.

Fig. 13
Fig. 13

Averaging statistics with the average before ratio method for dual-laser lidar data [the same data used in Figs. 10(a)–10(h)]. The singular values for the standard deviation are the results of division by small numbers, regardless of averaging before taking the DIAL ratio.

Fig. 14
Fig. 14

Averaging statistics for the natural logarithm of the ratio of single-laser pulse pairs. The ratio was taken as L0(0)/L0(1) ≡ L0(10P16)/L0(10P20) and L1(0)/L1(1) ≡ L1(10P16)/L1(10P20).

Tables (5)

Tables Icon

Table 1 Number of Speckle Integrated by the Lidar Receiver as a Function of the Receiver Diametera

Tables Icon

Table 2 Standard Deviation for Single-Wavelength Data and Pulse–Pair Ratio Dataa

Tables Icon

Table 3 Number of Infinite Data Values Over the Total Number of Pulses Measured for Each Value of Ma

Tables Icon

Table 4 Number of Negative Values for the Pulse–Pair Ratio That Were Removed from the Data before Using the Log of the Ratio Before Average Method, Eq. (5)a

Tables Icon

Table 5 Standard Deviation for Pulse–Pair Ratio Data with All the Negative Ratio Values Removeda

Equations (39)

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

P x ( r ) = E x D x r 2 ρ x exp { - 2 0 r [ A x ( q ) + α x C ( q ) d q ] } ,
C L ( r ) = 0 r C ( q ) d q , C L = 1 2 ( α y - α x ) ( ln [ P x / E x P y / E y ] + ln [ D y D x ] + ln [ ρ y ρ x ] + 2 0 r [ A x ( q ) - A y ( q ) ] d q ) .
C L ( r ) 1 2 ( α y - α x ) ln [ S x S y ] ,
S x P x E x ,             S y P y E y .
C L 1 ( r ) = 1 2 ( α y - α x ) 1 N i = 1 N ln ( S x , i S y , i ) ,
C L 2 ( r ) = 1 2 ( α y - α x ) ln ( i = 1 N S x , i i = 1 N S y , i ) ,
C L 3 ( r ) = 1 2 ( α y - α x ) ln ( 1 N i = 1 N S x , i S y , i ) .
σ N = 1 M N , σ 1 N ,
Δ ν D ( cm - 1 ) ( 1 / 2 d ) ,
Δ ν D M c 2 d .
θ min sin - 1 ( 1 2 B Δ ν ) ,
S c = λ 2 z 2 π w t 2 = λ 2 π θ 1 / 2 2 ,
( σ z z ) 2 ( σ x x ) 2 + ( σ y y ) 2 - 2 ρ x y σ x σ y x y ,
p ( I ) = ( M I ) M I M - 1 Γ ( M ) exp ( - M I I ) ,
I z I x ( t ) I y ( t ) ,
p I x ( I x ) 1 Γ ( M x ) ( M x I x ) M x I x M x - 1 exp [ - M x I x I x ] , p I y ( I y ) 1 Γ ( M y ) ( M y I y ) M y I y M y - 1 exp [ - M y I y I y ] ,
P Z = x / y ( z ) = p x y ( x , y ) d x d y ,
P z = x / y ( z ) = - d y - y z p x y ( x , y ) d x , = - z d u - y p X Y ( u y , y ) d y ,
p z = x / y ( z ) = d d z [ - z d u 0 d y y p x y ( u y , y ) ] , = - d y y p x y ( y z , y ) .
p I z ( I z ) = 0 I y p I x ( I z I y ) p I y ( I y ) d I y .
p I z ( I z ) = { Γ ( M x + M y ) Γ ( M x ) Γ ( M y ) ( M x I x ) M x ( M y I y ) M y I z M x - 1 [ M y I y + M x I x I z ] - ( M x + M y ) , I z > 0 , 0 , I z 0 , Γ ( 2 M ) Γ 2 ( M ) ( I y I x ) M I z M - 1 [ 1 + I y I x I z ] - 2 M , M x = M x M
I z = 0 + d I z I z p I z ( I z ) , = M y ( M y - 1 ) ( I x I y ) .
I z 2 = Γ ( M x + M y ) M x M x M y M y Γ ( M y ) Γ ( M x ) ( I x I y ) M y × 0 I z M x + 1 [ M y I x I y + M x I z ] M x + M y d I z ,
I z 2 = ( M y M x ) 2 M x ( M x + 1 ) ( M y - 1 ) ( M y - 2 ) ( I x I y ) 2 .
σ z 2 = I z 2 - I z 2
σ z 2 = M y 2 ( M x + M y - 1 ) M x ( M y - 1 ) 2 ( M y - 2 ) ( I x I y ) 2 .
σ 1 σ z I z = ( [ 2 M - 1 M ( M - 2 ) ] 1 / 2 for M x = M y M , M > 2 , [ ( M y + M x - 1 ) M x ( M y - 2 ) ] 1 / 2 M y > 2.
I z i = M ( M - 1 ) I x i I y i .
L ln ( I x I y ) = ln ( I z )
ρ ( L ) = | d I z d L | ρ ( I z ) ,
ρ ( L ) = Γ ( 2 m ) Γ 2 ( m ) [ 2 cosh ( L - L 0 2 ) ] - 2 m .
L 0 ln ( I x I y ) .
N x = S x - S O x R x - R O x .
χ 2 = i = 1 N [ y i - f ( x i ; M ) ] 2 σ y i 2 .
σ ( σ N ) = σ N 2 ( n - 1 ) ,
ρ j = 1 Γ σ I x 2 i = 1 Γ - j ( I x i - I x ) ( I x ( i + j ) - I x ) ,
ρ j = 1 ( Γ - j ) σ I x 2 i = 1 Γ - j ( I x i - I x ) ( I x ( i + j ) - I x ) .
σ N 2 = σ 1 2 N [ 1 + 2 j = 1 N - 1 ( 1 - j N ) ρ j ]
σ N 2 = σ 1 2 N [ 1 + 2 j = 1 N - 1 ( 1 - j Γ ) ( 1 - j N ) ρ j ]

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