Abstract

The performance of a coherent Doppler lidar is determined by the statistics of the coherent Doppler signal. The derivation and calculation of the covariance of the Doppler lidar signal for random atmospheric wind fields and wind shear are presented. The signal parameters are defined for a general coherent Doppler lidar system in terms of the atmospheric parameters. There are two distinct physical regimes: one in which the transmitted pulse determines the signal statistics and the other in which the wind field and the atmospheric parameters dominate the signal statistics. When the wind fields dominate the signal statistics, Doppler lidar data are nonstationary and the signal correlation time is proportional to the operating wavelength of the lidar. The signal covariance is derived for signal-shot and multiple-shot conditions. For a single shot, the parameters of the signal covariance depend on the random, instantaneous atmospheric parameters. For multiple shots, various levels of ensemble averaging over the temporal scales of the atmospheric processes are required. The wind turbulence is described by a Kolmogorov spectrum with an outer scale of turbulence. The effects of the wind turbulence are demonstrated with calculations for a horizontal propagation path in the atmospheric surface layer.

© 1994 Optical Society of America

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References

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  1. J. W. Bilbro, C. DiMarzio, D. Fitzjarrald, S. Johnson, W. Jones, “Airborne Doppler lidar measurements,” Appl. Opt. 25, 2952–2960 (1986).
    [CrossRef]
  2. J. C. Petheram, G. Frohbeiter, A. Rosenberg, “Carbon dioxide Doppler lidar wind sensor on a space station polar platform,” Appl. Opt. 28, 834–839 (1989).
    [CrossRef] [PubMed]
  3. M. J. Post, R. E. Cupp, “Optimizing a pulse Doppler lidar,” Appl. Opt. 29, 4145–4158 (1990).
    [CrossRef] [PubMed]
  4. G. N. Pearson, B. J. Rye, “Frequency fidelity of a compact CO2 Doppler lider transmitter,” Appl. Opt. 31, 6475–6484 (1992).
    [CrossRef] [PubMed]
  5. M. J. Kavaya, S. W. Henderson, J. R. Magee, C. P. Hale, R. M. Huffaker, “Remote wind profiling with a solid-state Nd:YAG coherent lidar system,” Opt. Lett. 14, 776–778 (1989).
    [CrossRef] [PubMed]
  6. S. W. Henderson, C. P. Hale, J. R. Magee, M. J. Kavaya, A. V. Huffaker, “Eye-safe coherent laser radar system at 2.1 μm using Tm, Ho:YAG lasers,” Opt. Lett. 16, 773–775 (1991).
    [CrossRef] [PubMed]
  7. S. W. Henderson, P. J. M. Suni, C. P. Hale, S. M. Hannon, J. R. Magee, D. L. Bruns, E. H. Yuen, “Coherent laser radar at 2 μm using solid-state lasers,” IEEE Trans. Geosci. Remote Sensing 31, 4–15 (1993).
    [CrossRef]
  8. B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterdoyne lidar. I. Spectral accumulation and the Cramer–Rao lower bound,” IEEE Trans. Geosci. Remote Sensing 31, 16–27 (1993).
    [CrossRef]
  9. R. G. Frehlich, M. J. Yadlowsky, “Performance of mean frequency estimators for Doppler radar and lidar,” J. Atmos. Ocean. Technol. (to be published).
  10. R. M. Hardesty, “Performance of a discrete spectral peak frequency estimator for Doppler wind velocity measurements,” IEEE Trans. Geosci. Remote Sensing GFE-24, 777–783 (1986).
    [CrossRef]
  11. P. R. Mahapatra, D. S. Zrnic, “Practical algorithms for mean velocity estimation in pulse Doppler weather radars using a small number of samples,” IEEE Trans. Geosci. Electron. GE-21, 491–501 (1983).
  12. P. T. May, R. G. Strauch, “An examination of Wind Profiler signal processing algorithms,” J. Atmos. Oceanic Technol. 6, 731–735 (1989).
    [CrossRef]
  13. P. T. May, T. Sato, M. Yamamoto, S. Kato, T. Tsuda, S. Fukao, “Errors in the determination of wind speed by Doppler radar,” J. Atmos. Oceanic Technol. 6, 235–242 (1989).
    [CrossRef]
  14. B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. II. Correlogram accumulation,” IEEE Trans Geosci. Remote Sensing 31, 28–35 (1993).
    [CrossRef]
  15. D. S. Zrnic, “Estimation of spectral moments of weather echoes,” IEEE Trans. Geosci. Electron. GE-17, 113–128 (1979).
    [CrossRef]
  16. R. J. Doviak, D. S. Zrnic, Doppler Radar and Weather Observations (Academic, San Diego, Calif., 1984).
  17. H. L. van Trees, Detection, Estimate, and Modulation Theory, Part I (Wiley, New York, 1968).
  18. C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon, New York, 1968).
  19. B. Gold, A. V. Oppenheim, C. M. Rader, “Theory and implementation of the discrete Hilbert transformation,” in Symposium on Computer Processing in Communications, (Polytechnic, Brooklyn, N. Y., 1970), Vol. 19, pp. 235–250.
  20. V. Cizek, “Discrete Hilbert transform,” IEEE Trans. Audio Electroacoust. AU-18, 340–343 (1970).
    [CrossRef]
  21. R. G. Frehlich, “Cramer–Rao bound for Gaussian random processes and applications to radar processing of atmospheric signals,” IEEE Trans. Geosci. Remote Sensing 31, 1123–1131 (1993).
    [CrossRef]
  22. G. M. Ancellet, R. T. Menzies, “Atmospheric correlation-time measurements and effects on coherent Doppler lidar,” J. Opt. Soc. Am. A 4, 367–373 (1987).
    [CrossRef]
  23. G. M. Ancellet, R. T. Menzies, W. B. Grant, “Atmospheric velocity spectral width measurements using the statistical distribution of pulsed CO2 lidar return signal intensities,” J. Atmos. Oceanic Technol. 6, 50–58 (1989).
    [CrossRef]
  24. R. G. Frehlich, M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
    [CrossRef] [PubMed]
  25. J. H. Churnside, H. T. Yura, “Speckle statistics of atmospherically backscattered laser light,” Appl. Opt. 22, 2559–2565 (1983).
    [CrossRef] [PubMed]
  26. B. J. Rye, “Spectral correlation of atmospheric lidar returns with range-dependent backscatter,” J. Opt. Soc. Am. A 7, 2199–2207 (1990).
    [CrossRef]
  27. A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (MIT, Cambridge, Mass., 1975), Vol. 2, pp. 102, 461.
  28. J. C. Kaimal, J. C. Wyngaard, Y. Izumi, O. R. Cote, “Spectral characteristics of surface-layer turbulence,” Q. J. R. Meteorol. Soc. 98, 563–589 (1972).
    [CrossRef]

1993 (4)

S. W. Henderson, P. J. M. Suni, C. P. Hale, S. M. Hannon, J. R. Magee, D. L. Bruns, E. H. Yuen, “Coherent laser radar at 2 μm using solid-state lasers,” IEEE Trans. Geosci. Remote Sensing 31, 4–15 (1993).
[CrossRef]

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterdoyne lidar. I. Spectral accumulation and the Cramer–Rao lower bound,” IEEE Trans. Geosci. Remote Sensing 31, 16–27 (1993).
[CrossRef]

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. II. Correlogram accumulation,” IEEE Trans Geosci. Remote Sensing 31, 28–35 (1993).
[CrossRef]

R. G. Frehlich, “Cramer–Rao bound for Gaussian random processes and applications to radar processing of atmospheric signals,” IEEE Trans. Geosci. Remote Sensing 31, 1123–1131 (1993).
[CrossRef]

1992 (1)

1991 (2)

1990 (2)

1989 (5)

J. C. Petheram, G. Frohbeiter, A. Rosenberg, “Carbon dioxide Doppler lidar wind sensor on a space station polar platform,” Appl. Opt. 28, 834–839 (1989).
[CrossRef] [PubMed]

M. J. Kavaya, S. W. Henderson, J. R. Magee, C. P. Hale, R. M. Huffaker, “Remote wind profiling with a solid-state Nd:YAG coherent lidar system,” Opt. Lett. 14, 776–778 (1989).
[CrossRef] [PubMed]

G. M. Ancellet, R. T. Menzies, W. B. Grant, “Atmospheric velocity spectral width measurements using the statistical distribution of pulsed CO2 lidar return signal intensities,” J. Atmos. Oceanic Technol. 6, 50–58 (1989).
[CrossRef]

P. T. May, R. G. Strauch, “An examination of Wind Profiler signal processing algorithms,” J. Atmos. Oceanic Technol. 6, 731–735 (1989).
[CrossRef]

P. T. May, T. Sato, M. Yamamoto, S. Kato, T. Tsuda, S. Fukao, “Errors in the determination of wind speed by Doppler radar,” J. Atmos. Oceanic Technol. 6, 235–242 (1989).
[CrossRef]

1987 (1)

1986 (2)

R. M. Hardesty, “Performance of a discrete spectral peak frequency estimator for Doppler wind velocity measurements,” IEEE Trans. Geosci. Remote Sensing GFE-24, 777–783 (1986).
[CrossRef]

J. W. Bilbro, C. DiMarzio, D. Fitzjarrald, S. Johnson, W. Jones, “Airborne Doppler lidar measurements,” Appl. Opt. 25, 2952–2960 (1986).
[CrossRef]

1983 (2)

J. H. Churnside, H. T. Yura, “Speckle statistics of atmospherically backscattered laser light,” Appl. Opt. 22, 2559–2565 (1983).
[CrossRef] [PubMed]

P. R. Mahapatra, D. S. Zrnic, “Practical algorithms for mean velocity estimation in pulse Doppler weather radars using a small number of samples,” IEEE Trans. Geosci. Electron. GE-21, 491–501 (1983).

1979 (1)

D. S. Zrnic, “Estimation of spectral moments of weather echoes,” IEEE Trans. Geosci. Electron. GE-17, 113–128 (1979).
[CrossRef]

1972 (1)

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, O. R. Cote, “Spectral characteristics of surface-layer turbulence,” Q. J. R. Meteorol. Soc. 98, 563–589 (1972).
[CrossRef]

1970 (1)

V. Cizek, “Discrete Hilbert transform,” IEEE Trans. Audio Electroacoust. AU-18, 340–343 (1970).
[CrossRef]

Ancellet, G. M.

G. M. Ancellet, R. T. Menzies, W. B. Grant, “Atmospheric velocity spectral width measurements using the statistical distribution of pulsed CO2 lidar return signal intensities,” J. Atmos. Oceanic Technol. 6, 50–58 (1989).
[CrossRef]

G. M. Ancellet, R. T. Menzies, “Atmospheric correlation-time measurements and effects on coherent Doppler lidar,” J. Opt. Soc. Am. A 4, 367–373 (1987).
[CrossRef]

Bilbro, J. W.

J. W. Bilbro, C. DiMarzio, D. Fitzjarrald, S. Johnson, W. Jones, “Airborne Doppler lidar measurements,” Appl. Opt. 25, 2952–2960 (1986).
[CrossRef]

Bruns, D. L.

S. W. Henderson, P. J. M. Suni, C. P. Hale, S. M. Hannon, J. R. Magee, D. L. Bruns, E. H. Yuen, “Coherent laser radar at 2 μm using solid-state lasers,” IEEE Trans. Geosci. Remote Sensing 31, 4–15 (1993).
[CrossRef]

Churnside, J. H.

Cizek, V.

V. Cizek, “Discrete Hilbert transform,” IEEE Trans. Audio Electroacoust. AU-18, 340–343 (1970).
[CrossRef]

Cote, O. R.

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, O. R. Cote, “Spectral characteristics of surface-layer turbulence,” Q. J. R. Meteorol. Soc. 98, 563–589 (1972).
[CrossRef]

Cupp, R. E.

DiMarzio, C.

J. W. Bilbro, C. DiMarzio, D. Fitzjarrald, S. Johnson, W. Jones, “Airborne Doppler lidar measurements,” Appl. Opt. 25, 2952–2960 (1986).
[CrossRef]

Doviak, R. J.

R. J. Doviak, D. S. Zrnic, Doppler Radar and Weather Observations (Academic, San Diego, Calif., 1984).

Fitzjarrald, D.

J. W. Bilbro, C. DiMarzio, D. Fitzjarrald, S. Johnson, W. Jones, “Airborne Doppler lidar measurements,” Appl. Opt. 25, 2952–2960 (1986).
[CrossRef]

Frehlich, R. G.

R. G. Frehlich, “Cramer–Rao bound for Gaussian random processes and applications to radar processing of atmospheric signals,” IEEE Trans. Geosci. Remote Sensing 31, 1123–1131 (1993).
[CrossRef]

R. G. Frehlich, M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
[CrossRef] [PubMed]

R. G. Frehlich, M. J. Yadlowsky, “Performance of mean frequency estimators for Doppler radar and lidar,” J. Atmos. Ocean. Technol. (to be published).

Frohbeiter, G.

Fukao, S.

P. T. May, T. Sato, M. Yamamoto, S. Kato, T. Tsuda, S. Fukao, “Errors in the determination of wind speed by Doppler radar,” J. Atmos. Oceanic Technol. 6, 235–242 (1989).
[CrossRef]

Gold, B.

B. Gold, A. V. Oppenheim, C. M. Rader, “Theory and implementation of the discrete Hilbert transformation,” in Symposium on Computer Processing in Communications, (Polytechnic, Brooklyn, N. Y., 1970), Vol. 19, pp. 235–250.

Grant, W. B.

G. M. Ancellet, R. T. Menzies, W. B. Grant, “Atmospheric velocity spectral width measurements using the statistical distribution of pulsed CO2 lidar return signal intensities,” J. Atmos. Oceanic Technol. 6, 50–58 (1989).
[CrossRef]

Hale, C. P.

Hannon, S. M.

S. W. Henderson, P. J. M. Suni, C. P. Hale, S. M. Hannon, J. R. Magee, D. L. Bruns, E. H. Yuen, “Coherent laser radar at 2 μm using solid-state lasers,” IEEE Trans. Geosci. Remote Sensing 31, 4–15 (1993).
[CrossRef]

Hardesty, R. M.

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterdoyne lidar. I. Spectral accumulation and the Cramer–Rao lower bound,” IEEE Trans. Geosci. Remote Sensing 31, 16–27 (1993).
[CrossRef]

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. II. Correlogram accumulation,” IEEE Trans Geosci. Remote Sensing 31, 28–35 (1993).
[CrossRef]

R. M. Hardesty, “Performance of a discrete spectral peak frequency estimator for Doppler wind velocity measurements,” IEEE Trans. Geosci. Remote Sensing GFE-24, 777–783 (1986).
[CrossRef]

Helstrom, C. W.

C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon, New York, 1968).

Henderson, S. W.

Huffaker, A. V.

Huffaker, R. M.

Izumi, Y.

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, O. R. Cote, “Spectral characteristics of surface-layer turbulence,” Q. J. R. Meteorol. Soc. 98, 563–589 (1972).
[CrossRef]

Johnson, S.

J. W. Bilbro, C. DiMarzio, D. Fitzjarrald, S. Johnson, W. Jones, “Airborne Doppler lidar measurements,” Appl. Opt. 25, 2952–2960 (1986).
[CrossRef]

Jones, W.

J. W. Bilbro, C. DiMarzio, D. Fitzjarrald, S. Johnson, W. Jones, “Airborne Doppler lidar measurements,” Appl. Opt. 25, 2952–2960 (1986).
[CrossRef]

Kaimal, J. C.

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, O. R. Cote, “Spectral characteristics of surface-layer turbulence,” Q. J. R. Meteorol. Soc. 98, 563–589 (1972).
[CrossRef]

Kato, S.

P. T. May, T. Sato, M. Yamamoto, S. Kato, T. Tsuda, S. Fukao, “Errors in the determination of wind speed by Doppler radar,” J. Atmos. Oceanic Technol. 6, 235–242 (1989).
[CrossRef]

Kavaya, M. J.

Magee, J. R.

Mahapatra, P. R.

P. R. Mahapatra, D. S. Zrnic, “Practical algorithms for mean velocity estimation in pulse Doppler weather radars using a small number of samples,” IEEE Trans. Geosci. Electron. GE-21, 491–501 (1983).

May, P. T.

P. T. May, R. G. Strauch, “An examination of Wind Profiler signal processing algorithms,” J. Atmos. Oceanic Technol. 6, 731–735 (1989).
[CrossRef]

P. T. May, T. Sato, M. Yamamoto, S. Kato, T. Tsuda, S. Fukao, “Errors in the determination of wind speed by Doppler radar,” J. Atmos. Oceanic Technol. 6, 235–242 (1989).
[CrossRef]

Menzies, R. T.

G. M. Ancellet, R. T. Menzies, W. B. Grant, “Atmospheric velocity spectral width measurements using the statistical distribution of pulsed CO2 lidar return signal intensities,” J. Atmos. Oceanic Technol. 6, 50–58 (1989).
[CrossRef]

G. M. Ancellet, R. T. Menzies, “Atmospheric correlation-time measurements and effects on coherent Doppler lidar,” J. Opt. Soc. Am. A 4, 367–373 (1987).
[CrossRef]

Monin, A. S.

A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (MIT, Cambridge, Mass., 1975), Vol. 2, pp. 102, 461.

Oppenheim, A. V.

B. Gold, A. V. Oppenheim, C. M. Rader, “Theory and implementation of the discrete Hilbert transformation,” in Symposium on Computer Processing in Communications, (Polytechnic, Brooklyn, N. Y., 1970), Vol. 19, pp. 235–250.

Pearson, G. N.

Petheram, J. C.

Post, M. J.

Rader, C. M.

B. Gold, A. V. Oppenheim, C. M. Rader, “Theory and implementation of the discrete Hilbert transformation,” in Symposium on Computer Processing in Communications, (Polytechnic, Brooklyn, N. Y., 1970), Vol. 19, pp. 235–250.

Rosenberg, A.

Rye, B. J.

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. II. Correlogram accumulation,” IEEE Trans Geosci. Remote Sensing 31, 28–35 (1993).
[CrossRef]

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterdoyne lidar. I. Spectral accumulation and the Cramer–Rao lower bound,” IEEE Trans. Geosci. Remote Sensing 31, 16–27 (1993).
[CrossRef]

G. N. Pearson, B. J. Rye, “Frequency fidelity of a compact CO2 Doppler lider transmitter,” Appl. Opt. 31, 6475–6484 (1992).
[CrossRef] [PubMed]

B. J. Rye, “Spectral correlation of atmospheric lidar returns with range-dependent backscatter,” J. Opt. Soc. Am. A 7, 2199–2207 (1990).
[CrossRef]

Sato, T.

P. T. May, T. Sato, M. Yamamoto, S. Kato, T. Tsuda, S. Fukao, “Errors in the determination of wind speed by Doppler radar,” J. Atmos. Oceanic Technol. 6, 235–242 (1989).
[CrossRef]

Strauch, R. G.

P. T. May, R. G. Strauch, “An examination of Wind Profiler signal processing algorithms,” J. Atmos. Oceanic Technol. 6, 731–735 (1989).
[CrossRef]

Suni, P. J. M.

S. W. Henderson, P. J. M. Suni, C. P. Hale, S. M. Hannon, J. R. Magee, D. L. Bruns, E. H. Yuen, “Coherent laser radar at 2 μm using solid-state lasers,” IEEE Trans. Geosci. Remote Sensing 31, 4–15 (1993).
[CrossRef]

Tsuda, T.

P. T. May, T. Sato, M. Yamamoto, S. Kato, T. Tsuda, S. Fukao, “Errors in the determination of wind speed by Doppler radar,” J. Atmos. Oceanic Technol. 6, 235–242 (1989).
[CrossRef]

van Trees, H. L.

H. L. van Trees, Detection, Estimate, and Modulation Theory, Part I (Wiley, New York, 1968).

Wyngaard, J. C.

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, O. R. Cote, “Spectral characteristics of surface-layer turbulence,” Q. J. R. Meteorol. Soc. 98, 563–589 (1972).
[CrossRef]

Yadlowsky, M. J.

R. G. Frehlich, M. J. Yadlowsky, “Performance of mean frequency estimators for Doppler radar and lidar,” J. Atmos. Ocean. Technol. (to be published).

Yaglom, A. M.

A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (MIT, Cambridge, Mass., 1975), Vol. 2, pp. 102, 461.

Yamamoto, M.

P. T. May, T. Sato, M. Yamamoto, S. Kato, T. Tsuda, S. Fukao, “Errors in the determination of wind speed by Doppler radar,” J. Atmos. Oceanic Technol. 6, 235–242 (1989).
[CrossRef]

Yuen, E. H.

S. W. Henderson, P. J. M. Suni, C. P. Hale, S. M. Hannon, J. R. Magee, D. L. Bruns, E. H. Yuen, “Coherent laser radar at 2 μm using solid-state lasers,” IEEE Trans. Geosci. Remote Sensing 31, 4–15 (1993).
[CrossRef]

Yura, H. T.

Zrnic, D. S.

P. R. Mahapatra, D. S. Zrnic, “Practical algorithms for mean velocity estimation in pulse Doppler weather radars using a small number of samples,” IEEE Trans. Geosci. Electron. GE-21, 491–501 (1983).

D. S. Zrnic, “Estimation of spectral moments of weather echoes,” IEEE Trans. Geosci. Electron. GE-17, 113–128 (1979).
[CrossRef]

R. J. Doviak, D. S. Zrnic, Doppler Radar and Weather Observations (Academic, San Diego, Calif., 1984).

Appl. Opt. (6)

IEEE Trans Geosci. Remote Sensing (1)

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. II. Correlogram accumulation,” IEEE Trans Geosci. Remote Sensing 31, 28–35 (1993).
[CrossRef]

IEEE Trans. Audio Electroacoust. (1)

V. Cizek, “Discrete Hilbert transform,” IEEE Trans. Audio Electroacoust. AU-18, 340–343 (1970).
[CrossRef]

IEEE Trans. Geosci. Electron. (2)

D. S. Zrnic, “Estimation of spectral moments of weather echoes,” IEEE Trans. Geosci. Electron. GE-17, 113–128 (1979).
[CrossRef]

P. R. Mahapatra, D. S. Zrnic, “Practical algorithms for mean velocity estimation in pulse Doppler weather radars using a small number of samples,” IEEE Trans. Geosci. Electron. GE-21, 491–501 (1983).

IEEE Trans. Geosci. Remote Sensing (4)

S. W. Henderson, P. J. M. Suni, C. P. Hale, S. M. Hannon, J. R. Magee, D. L. Bruns, E. H. Yuen, “Coherent laser radar at 2 μm using solid-state lasers,” IEEE Trans. Geosci. Remote Sensing 31, 4–15 (1993).
[CrossRef]

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterdoyne lidar. I. Spectral accumulation and the Cramer–Rao lower bound,” IEEE Trans. Geosci. Remote Sensing 31, 16–27 (1993).
[CrossRef]

R. G. Frehlich, “Cramer–Rao bound for Gaussian random processes and applications to radar processing of atmospheric signals,” IEEE Trans. Geosci. Remote Sensing 31, 1123–1131 (1993).
[CrossRef]

R. M. Hardesty, “Performance of a discrete spectral peak frequency estimator for Doppler wind velocity measurements,” IEEE Trans. Geosci. Remote Sensing GFE-24, 777–783 (1986).
[CrossRef]

J. Atmos. Oceanic Technol. (3)

G. M. Ancellet, R. T. Menzies, W. B. Grant, “Atmospheric velocity spectral width measurements using the statistical distribution of pulsed CO2 lidar return signal intensities,” J. Atmos. Oceanic Technol. 6, 50–58 (1989).
[CrossRef]

P. T. May, R. G. Strauch, “An examination of Wind Profiler signal processing algorithms,” J. Atmos. Oceanic Technol. 6, 731–735 (1989).
[CrossRef]

P. T. May, T. Sato, M. Yamamoto, S. Kato, T. Tsuda, S. Fukao, “Errors in the determination of wind speed by Doppler radar,” J. Atmos. Oceanic Technol. 6, 235–242 (1989).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Lett. (2)

Q. J. R. Meteorol. Soc. (1)

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, O. R. Cote, “Spectral characteristics of surface-layer turbulence,” Q. J. R. Meteorol. Soc. 98, 563–589 (1972).
[CrossRef]

Other (6)

A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (MIT, Cambridge, Mass., 1975), Vol. 2, pp. 102, 461.

R. G. Frehlich, M. J. Yadlowsky, “Performance of mean frequency estimators for Doppler radar and lidar,” J. Atmos. Ocean. Technol. (to be published).

R. J. Doviak, D. S. Zrnic, Doppler Radar and Weather Observations (Academic, San Diego, Calif., 1984).

H. L. van Trees, Detection, Estimate, and Modulation Theory, Part I (Wiley, New York, 1968).

C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon, New York, 1968).

B. Gold, A. V. Oppenheim, C. M. Rader, “Theory and implementation of the discrete Hilbert transformation,” in Symposium on Computer Processing in Communications, (Polytechnic, Brooklyn, N. Y., 1970), Vol. 19, pp. 235–250.

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Figures (6)

Fig. 1
Fig. 1

Effects of wind shear on the normalized signal covariance [Eqs. (9), (30), (45), γ = f m = ϕ = 0] as a function of τ/τ A with constant SNR over the observation interval and —, μ = t 0, with no wind shear; - - -, μ − t 0 = σ and τ A = τws; – – –, μ − t 0 = 2σ and τ A = τWS - · - ·, μ − t 0 = σ and τ A = 2τws;and · · - · ·, μ − t 0 = 2σ and τ A = 2τws.

Fig. 2
Fig. 2

Effects of wind shear and gradients in SNR over the observation interval on the normalized signal covariance [Eqs. (9),(30),(45), f m = ϕ = 0] as a function of τ/τ A compared with constant SNR over the observation interval and —, no wind shear; - - -, μ − t 0 = σ, τ A = 2τws, and γ = 0; · · - · · -, μ − t 0 = 2σ,τ A = 2τws, and γ = 0.4.

Fig. 3
Fig. 3

Effects of the finite transverse dimensions of a lidar beam and wind turbulence on the normalized signal covariance [Eqs. (9), (30), (55), f m = ϕ = 0] as a function of τ/τ A with constant SNR over the observation interval and (a) μ = t 0 with—, no wind turbulence; - - -, τ E = τ A /2 and σ B r p ; – – –, τ E = τ A /2 and σ B = r p /2;– · – ·, τ E = τ A /2 and σ B = r p ;– – · – – ·, τE = τA/2 and σB = 2r p . (b) Same as (a) but with μ. − t 0 = σ.

Fig. 4
Fig. 4

Effects of wind turbulence on the normalized signal covariance [Eqs. (9), (30), (59), f m = ϕ = 0] as a function of τ/τ A with a constant SNR over the observation interval and a narrow transmitted beam compared with the range resolution (σ B r P ) for (a) μ = t 0 with —, no wind turbulence; - - -, τ E = τ A /2;– – –, τ E = τ A /4. (b) Same as (a) but with μ − t 0 = σ.

Fig. 5
Fig. 5

Normalized structure function of velocity fluctuations in the direction of the mean velocity Λ(x) versus x for neutral stability in the atmospheric surface layer.

Fig. 6
Fig. 6

Normalized signal covariance [Eqs. (9), (30), (61), (62), (64), f m = ϕ = 0] for a 10- and a 2-μm lidar as a function of τ for μ = t 0 with a horizontal propagation path at a height Z H = 2 m in the atmospheric surface layer under neutral stability and a standard deviation of the radial component of velocity fluctuations σ r = 0.5 m/s. The transmitted pulse has a Gaussian temporal profile with l/e intensity radius σ of —; 0.1 μs - - -; 0.2 μs – – –; 0.5 μs – · – ·,1.0 μs; · · – · · –, 2.0 μs; – – · – – ·, 5.0 μs.

Equations (70)

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E L ( u , 0 , t ) = U L A L ( t ) e L ( u , 0 ) exp ( 2 π i f L t ) ,
A L ( t ) = J L 1 / 2 ( t ) exp [ 2 π i 0 t f C ( t ) d t ]
- A L ( t ) 2 d t = 1.
E LO ( v , 0 ) = P LO 1 / 2 e LO ( v , 0 ) exp ( 2 π i f LO t ) ,
- e L , LO ( u , 0 ) d u = 1 ,
X ( t ) = S ( t ) + N ( t ) = Re [ x ( t ) ] ,
B ξ ( t 1 , t 2 ) = ξ ( t 1 ) ξ ( t 2 ) ,
x ( t 1 ) x ( t 2 ) = 0.
B X ( t 1 , t 2 ) = ½ Re [ B x ( t 1 , t 2 ) ] = Re [ R x ( t 1 , t 2 ) ] ,
B X ( t 1 , t 2 ) = B S ( t 1 , t 2 ) + B N ( t 1 , t 2 ) .
B S ( t , t ) = ½ B s ( t , t ) = R s ( t , t ) = SNR ( t ) ,
R s ( t 1 , t 2 ) = η Q U L h ν B 0 - A L ( t 1 - 2 z / c ) A L * ( t 2 - 2 z / c ) × K 2 ( p , z ) β ( p , z ) Q ( p , z ) 2 × exp { 2 π i ( t 1 - t 2 ) [ Δ f - 2 v r ( p , z ) / λ ] } d p d z ,
Q ( p , z ) = λ - - W T ( u ) W R ( v ) e L ( u , 0 ) × e LO * ( v , 0 ) G ( p ; u , z ) G ( p ; v , z ) d u d v ,
R x ( t 1 , t 2 ) = R x * ( t 2 , t 1 ) .
R s ( t 1 , t 2 ) = η Q U L h ν B 0 - A L ( t 1 - 2 z / c ) A L * ( t 2 - 2 z / c ) O ( p , z ) × exp { 2 π i ( t 1 - t 2 ) [ Δ f - 2 v r ( p , z ) / λ ] } d p d z ,
O ( p , z ) = K 2 ( p , z ) β ( p , z ) c ( p , z ) ,
c ( p , z ) = Q ( p , z ) 2 .
G ( p ; u , z ) = G f ( p ; u , z ) = i k 2 π z exp [ - i k 2 z ( p - u ) 2 ] ,
R s ( t 1 , t 2 ) = η Q U L h ν B 0 - A L ( t 1 - 2 z / c ) A L * ( t 2 - 2 z / c ) O ( p , z ) × exp [ 2 π i ( t 1 - t 2 ) ( Δ f - 2 v ¯ z / λ ) - 8 π 2 σ z 2 / λ 2 ] d p d z .
R s ( t 1 , t 2 ) = η Q U L h ν B 0 H ( z ) A L ( t 1 - 2 z / c ) A L * ( t 2 - 2 z / c ) × exp { 2 π i ( t 1 - t 2 ) [ Δ f - 2 v r ( 0 , z ) / λ ] } d z ,
H ( z ) = K 2 ( 0 , z ) β ( 0 , z ) C ( z ) ,
C ( z ) = - c ( p , z ) d p
z 0 = c t 0 / 2 ,
R s ( t 1 , t 2 ) = R s ( τ ) = SNR ( z 0 ) exp ( 2 π i τ f m ) × - A L ( t ) A L * ( t - τ ) d t ,
SNR ( z 0 ) = η Q c U L 2 h ν B H ( z 0 )
f m = Δ f - 2 v r ( 0 , z 0 ) / λ ,
Φ X ( f ) = - B X ( τ ) exp ( - 2 π i τ f ) d τ
Φ X ( f ) = SNR ( z 0 ) Ω ( f - f m ) ,
Ω ( f ) = | - A L ( t ) exp ( - 2 π i f t ) d t | 2
A L ( t ) = 1 π 1 / 4 σ 1 / 2 exp ( - t 2 2 σ 2 + π i ϕ t 2 ) ,
R s ( τ ) = SNR ( z 0 ) exp ( 2 π i f m - τ 2 / τ P 2 ) ,
1 τ P 2 = 1 τ A 2 + 1 τ C 2 ,
τ A = 2 σ ,
τ C = 1 π ϕ σ .
v r ( p , z ) = v r ( p , z 0 ) + g ( p , z 0 ) ( z - z 0 ) ,
g ( p , z ) = v r ( p , z ) z .
O ( p , z ) = O ( p , z 0 ) [ 1 + b ( p , z 0 ) ( z - z 0 ) ] ,
b ( p , z ) = 1 O ( p , z ) O ( p , z ) z .
R s ( μ , τ ) = η Q c U L 2 h ν B - O ( p , z 0 ) [ 1 + r P b ( p , z 0 ) ( μ - t 0 σ - i τ τ 1 ) ] × exp [ 2 π i Δ f τ - 4 π i λ v r ( p , z 0 ) τ - 2 i ( μ - t 0 ) σ τ τ WS - τ 2 τ T 2 ] d p ,
μ = ( t 1 + t 2 ) / 2 ,
τ WS = λ 2 π r P g ( p , z 0 )
r P = c σ / 2
1 τ T 2 = 1 τ 1 2 + 1 τ A 2 ,
1 τ 1 = 1 τ C + 1 τ WS .
R s ( μ , τ ) = SNR ( z 0 ) [ 1 + γ ( μ - t 0 σ - i τ τ 1 ) ] × exp [ 2 π i f m τ - 2 i ( μ - t 0 ) σ τ τ WS - τ 2 τ T 2 ] ,
γ = b ( 0 , z 0 ) r P = [ O ( 0 , z 0 + r P ) - O ( 0 , z 0 ) ] / O ( 0 , z 0 )
τ A / τ WS = 2 π c σ 2 g ( 0 , z 0 ) / λ > 1 ,
R s ( t 1 , t 2 ) = η Q U L h ν B 0 - A L ( t 1 - 2 z / c ) A L * ( t 2 - 2 z / c ) O ( p , z ) × exp [ 2 π i f m τ - 4 π i λ Δ v r ( p , z ) τ ] d p d z ,
Δ v r ( p , z , z 0 ) = v r ( p , z ) - v r ( 0 , z 0 )
R s ( t 1 , t 2 ) = η Q U L h ν B 0 - A L ( t 1 - 2 z / c ) A L * ( t 2 - 2 z / c ) O ( p , z ) × exp [ 2 π i τ f m - 8 π 2 τ 2 λ 2 D r ( p , z , z 0 ) ] d p d z ,
D r ( p , z , z 0 ) = Δ v r ( p , z , z 0 ) 2
R s ( t 1 , t 2 ) = η Q U L h ν B 0 A L ( t 1 - 2 z / c ) A L * ( t 2 - 2 z / c ) H ( z ) × exp [ 2 π i τ f m - 8 π 2 τ 2 λ 2 D r ( 0 , z , z 0 ) ] d z .
R s ( t 1 , t 2 ) = SNR ( z 0 ) 0 A L ( t 1 - 2 z / c ) A L * ( t 2 - 2 z / c ) × exp [ 2 π i τ f m - 8 π 2 τ 2 λ 2 D r ( 0 , z , z 0 ) ] d z .
D r ( p , z , z 0 ) = C v 2 / 3 ( z 0 ) r 2 / 3 [ ( z - z 0 ) 2 + 4 p 2 / 3 ] / r 2 ,
R s ( μ , τ ) = η Q U L c r P 2 2 π 1 / 2 h ν B exp ( 2 π i τ f m - τ 2 / τ A 2 ) × - - O ( r P q , r P ζ + z 0 ) exp { - [ ( μ - t 0 ) / σ - ζ ] 2 + 2 π i ϕ σ [ ( μ - t 0 ) / σ - ζ ] τ - κ - 4 / 3 ( ζ 2 + 4 q 2 / 3 ) τ 2 / τ E 2 } d q d ζ ,
1 τ E 2 = 8 π 2 C v ( r P ) 2 / 3 / λ 2 = 8 π 2 D r ( 0 , z 0 + r P , z 0 ) / λ 2 .
O ( r P q , z ) = C ( z ) π σ B 2 ( z ) exp [ - r P 2 q 2 / σ B 2 ( z ) ] ,
R s ( μ , τ ) = η Q U L c r P 2 2 π 1 / 2 h ν B exp ( 2 π i τ f m - τ 2 τ A 2 ) × - C ( r P ζ + z 0 ) exp { - [ ( μ - t 0 ) / σ - ζ ] 2 + 2 π i ϕ σ [ ( μ - t 0 ) / σ - ζ ] τ - ζ 2 / 3 τ 2 / τ E 2 } d ζ .
R s ( μ , τ ) = SNR ( z 0 ) π 1 / 2 exp ( 2 π i τ f m - τ 2 τ A 2 ) × - exp { - [ ( μ - t 0 ) / σ - ζ ] 2 + 2 π i ϕ σ [ ( μ - t 0 ) / σ - ζ ] τ - ζ 2 / 3 τ 2 / τ E 2 } d ζ .
τ A / τ E = 4 π C v 1 / 2 σ ( r P ) 1 / 3 / λ > 1.
D r ( 0 , z 0 + r , z 0 ) = 2 σ r 2 Λ ( r / z H ) ,
Λ ( x ) = ( α x ) 2 / 3 [ 1 + ( α x ) ρ ] - 2 / ( 3 ρ ) ,
= ( 2 σ r 2 / C v ) 3 / 2 α / z H .
R s ( μ , τ ) = SNR ( z 0 ) π 1 / 2 exp ( 2 π i τ f m - τ 2 τ A 2 ) × - exp { - [ ( μ - t 0 ) / σ - ζ ] 2 + 2 π i ϕ σ × [ ( μ - t 0 ) / σ - ζ ] τ - Λ ( r P ζ / z H ) τ 2 / τ H 2 } d ζ ,
τ H = λ 4 π σ r .
y ( t ) = 2 G D e η Q h ν D η Q ( w ) E S ( w , L , t ) E LO * ( w , L ) × exp ( 2 π i Δ f t + i θ S ) d w ,
y ( t ) = 2 G D e η Q h ν - E S ( v , 0 , t ) E LO * ( v , 0 ) W R ( v ) × exp ( 2 π i Δ f t + i θ S ) d v ,
i N 2 ( t ) = 2 G D 2 e 2 B η Q P LO / ( h ν ) ,
s ( t ) = y ( t ) / i N 2 ( t ) 1 / 2 .
E S ( v , 0 , t ) = λ σ S 1 / 2 K ( p , z ) - E L ( u , 0 , t - 2 z / c ) × W T ( u ) G ( p ; u , z ) G ( p ; v , z ) × exp [ i θ ( p , z ) - 4 π i t v r ( p , z ) / λ ] d u ,

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