Abstract

The average signal spectrum (periodogram) for coherent Doppler lidar is calculated for a turbulent wind field. Simple approximations are compared with the exact calculation. The effects of random errors in the zero velocity reference, the effects of averaging spectral estimates by use of multiple lidar pulses, and the effects of the range dependence of the lidar signal power over the range gate are included. For high spatial resolution measurements the lidar signal power is concentrated around one spectral estimate (spectral bin), and correct interpretation of the contribution from turbulence is difficult because of the effects of spectral leakage. For range gates that are larger than the lidar pulse volume, the signal power is contained in many spectral bins and the effects of turbulence can be determined accurately for constant signal power over the range gate and for the far-field range dependence of the signal power.

© 1999 Optical Society of America

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References

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  1. M. J. Post, R. E. Cupp, “Optimizing a pulsed Doppler lidar,” Appl. Opt. 29, 4145–4158 (1990).
    [CrossRef] [PubMed]
  2. 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]
  3. 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]
  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 Sens. 31, 4–15 (1993).
    [CrossRef]
  5. R. G. Frehlich, S. Hannon, S. Henderson, “Performance of a 2-µm coherent Doppler lidar for wind measurements,” J. Atmos. Oceanic Technol. 11, 1517–1528 (1994).
    [CrossRef]
  6. R. Targ, B. C. Steakley, J. G. Hawley, L. L. Ames, D. Swanson, R. Stone, R. G. Otto, V. Zarifis, P. Brockman, R. S. Calloway, P. A. Robinson, S. R. Harrell, “Coherent lidar airborne wind sensor II: flight-test results at 2 and 10 µm,” Appl. Opt. 35, 7117–7127 (1996).
    [CrossRef] [PubMed]
  7. R. G. Frehlich, S. Hannon, S. Henderson, “Coherent Doppler lidar measurements of winds in the weak signal regime,” Appl. Opt. 36, 3491–3499 (1997).
    [CrossRef] [PubMed]
  8. R. G. Frehlich, S. Hannon, S. Henderson, “Coherent Doppler lidar measurements of wind field statistics,” Boundary-Layer Meteorol. 86, 233–256 (1998).
    [CrossRef]
  9. B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I. Spectral accumulation and the Cramer-Rao lower bound,” IEEE Trans. Geosci. Remote Sens. 31, 16–27 (1993).
    [CrossRef]
  10. B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. II. Correlogram accumulation,” IEEE Trans. Geosci. Remote Sens. 31, 28–35 (1993).
    [CrossRef]
  11. R. G. Frehlich, M. J. Yadlowsky, “Performance of mean frequency estimators for Doppler radar and lidar,” J. Atmos. Oceanic Technol. 11, 1217–1230 (1994).
    [CrossRef]
  12. R. G. Frehlich, “Simulation of coherent Doppler lidar performance in the weak signal regime,” J. Atmos. Oceanic Technol. 13, 646–658 (1996).
    [CrossRef]
  13. R. G. Frehlich, “Effects of wind turbulence on coherent Doppler lidar performance,” J. Atmos. Oceanic Technol. 14, 54–75 (1997).
    [CrossRef]
  14. B. J. Rye, R. M. Hardesty, “Detection techniques for validating Doppler estimates in heterodyne lidar,” Appl. Opt. 36, 1940–1951 (1997).
    [CrossRef] [PubMed]
  15. B. J. Rye, “Spectral correlation of atmospheric lidar returns with range-dependent backscatter,” J. Opt. Soc. Am. A. 7, 2199–2207 (1990).
    [CrossRef]
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    [CrossRef]
  19. S. A. Cohn, “Radar measurements of turbulent eddy dissipation rate in the troposphere: a comparison of techniques,” J. Atmos. Oceanic Technol. 12, 85–95 (1995).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  26. B. T. Lottman, R. G. Frehlich, “Evaluation of coherent Doppler lidar velocity estimators in nonstationary regimes,” Appl. Opt. 36, 7906–7918 (1997).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  30. H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Part 1.
  31. C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon, Oxford, England, 1968).

1998

R. G. Frehlich, S. Hannon, S. Henderson, “Coherent Doppler lidar measurements of wind field statistics,” Boundary-Layer Meteorol. 86, 233–256 (1998).
[CrossRef]

1997

1996

1995

S. A. Cohn, “Radar measurements of turbulent eddy dissipation rate in the troposphere: a comparison of techniques,” J. Atmos. Oceanic Technol. 12, 85–95 (1995).
[CrossRef]

L. B. Cornman, C. S. Morse, G. Cunning, “Real-time estimation of atmospheric turbulence severity from in-situ aircraft measurements,” J. Aircr. 32, 171–177 (1995).
[CrossRef]

1994

R. G. Frehlich, M. J. Yadlowsky, “Performance of mean frequency estimators for Doppler radar and lidar,” J. Atmos. Oceanic Technol. 11, 1217–1230 (1994).
[CrossRef]

R. G. Frehlich, S. Hannon, S. Henderson, “Performance of a 2-µm coherent Doppler lidar for wind measurements,” J. Atmos. Oceanic Technol. 11, 1517–1528 (1994).
[CrossRef]

R. G. Frehlich, “Coherent Doppler lidar signal covariance including wind shear and wind turbulence,” Appl. Opt. 33, 6472–6481 (1994).
[CrossRef] [PubMed]

1993

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I. Spectral accumulation and the Cramer-Rao lower bound,” IEEE Trans. Geosci. Remote Sens. 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 Sens. 31, 28–35 (1993).
[CrossRef]

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 Sens. 31, 4–15 (1993).
[CrossRef]

1991

1990

M. J. Post, R. E. Cupp, “Optimizing a pulsed Doppler lidar,” Appl. Opt. 29, 4145–4158 (1990).
[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]

1989

1983

1974

P. H. Hilderbrand, R. S. Sekhon, “Objective determination of the noise level in Doppler spectra,” J. Appl. Meteorol. 13, 808–811 (1974).
[CrossRef]

1970

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

1964

R. R. Rogers, B. R. Tripp, “Some radar measurements of turbulence in snow,” J. Appl. Meteorol. 3, 603–610 (1964).
[CrossRef]

Ames, L. L.

Bagley, H. R.

S. M. Hannon, H. R. Bagley, R. K. Bogue, “Airborne Doppler lidar turbulence detection: ACLAIM flight test results,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 234–241 (1999).
[CrossRef]

Banakh, V. A.

V. A. Banakh, I. N. Smalikho, “Estimation of the turbulence energy dissipation rate from pulsed Doppler lidar data,” Atmos. Oceanic Opt. 10, 957–965 (1997).

Bogue, R. K.

S. M. Hannon, H. R. Bagley, R. K. Bogue, “Airborne Doppler lidar turbulence detection: ACLAIM flight test results,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 234–241 (1999).
[CrossRef]

Brockman, P.

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 Sens. 31, 4–15 (1993).
[CrossRef]

Calloway, R. S.

Churnside, J. H.

Cizek, V.

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

Cohn, S. A.

S. A. Cohn, “Radar measurements of turbulent eddy dissipation rate in the troposphere: a comparison of techniques,” J. Atmos. Oceanic Technol. 12, 85–95 (1995).
[CrossRef]

Cornman, L. B.

L. B. Cornman, C. S. Morse, G. Cunning, “Real-time estimation of atmospheric turbulence severity from in-situ aircraft measurements,” J. Aircr. 32, 171–177 (1995).
[CrossRef]

Cunning, G.

L. B. Cornman, C. S. Morse, G. Cunning, “Real-time estimation of atmospheric turbulence severity from in-situ aircraft measurements,” J. Aircr. 32, 171–177 (1995).
[CrossRef]

Cupp, R. E.

Doviak, R. J.

R. J. Doviak, D. S. Zrnic, Doppler Radar and Weather Observations (Academic, San Diego, Calif., 1993).

Frehlich, R. G.

R. G. Frehlich, S. Hannon, S. Henderson, “Coherent Doppler lidar measurements of wind field statistics,” Boundary-Layer Meteorol. 86, 233–256 (1998).
[CrossRef]

R. G. Frehlich, S. Hannon, S. Henderson, “Coherent Doppler lidar measurements of winds in the weak signal regime,” Appl. Opt. 36, 3491–3499 (1997).
[CrossRef] [PubMed]

R. G. Frehlich, “Effects of wind turbulence on coherent Doppler lidar performance,” J. Atmos. Oceanic Technol. 14, 54–75 (1997).
[CrossRef]

B. T. Lottman, R. G. Frehlich, “Evaluation of coherent Doppler lidar velocity estimators in nonstationary regimes,” Appl. Opt. 36, 7906–7918 (1997).
[CrossRef]

R. G. Frehlich, “Simulation of coherent Doppler lidar performance in the weak signal regime,” J. Atmos. Oceanic Technol. 13, 646–658 (1996).
[CrossRef]

R. G. Frehlich, M. J. Yadlowsky, “Performance of mean frequency estimators for Doppler radar and lidar,” J. Atmos. Oceanic Technol. 11, 1217–1230 (1994).
[CrossRef]

R. G. Frehlich, “Coherent Doppler lidar signal covariance including wind shear and wind turbulence,” Appl. Opt. 33, 6472–6481 (1994).
[CrossRef] [PubMed]

R. G. Frehlich, S. Hannon, S. Henderson, “Performance of a 2-µm coherent Doppler lidar for wind measurements,” J. Atmos. Oceanic Technol. 11, 1517–1528 (1994).
[CrossRef]

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

Gold, B.

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

Hale, C. P.

Hannon, S.

R. G. Frehlich, S. Hannon, S. Henderson, “Coherent Doppler lidar measurements of wind field statistics,” Boundary-Layer Meteorol. 86, 233–256 (1998).
[CrossRef]

R. G. Frehlich, S. Hannon, S. Henderson, “Coherent Doppler lidar measurements of winds in the weak signal regime,” Appl. Opt. 36, 3491–3499 (1997).
[CrossRef] [PubMed]

R. G. Frehlich, S. Hannon, S. Henderson, “Performance of a 2-µm coherent Doppler lidar for wind measurements,” J. Atmos. Oceanic Technol. 11, 1517–1528 (1994).
[CrossRef]

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 Sens. 31, 4–15 (1993).
[CrossRef]

S. M. Hannon, H. R. Bagley, R. K. Bogue, “Airborne Doppler lidar turbulence detection: ACLAIM flight test results,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 234–241 (1999).
[CrossRef]

Hardesty, R. M.

B. J. Rye, R. M. Hardesty, “Detection techniques for validating Doppler estimates in heterodyne lidar,” Appl. Opt. 36, 1940–1951 (1997).
[CrossRef] [PubMed]

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

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

Harrell, S. R.

Hawley, J. G.

Helstrom, C. W.

C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon, Oxford, England, 1968).

Henderson, S.

R. G. Frehlich, S. Hannon, S. Henderson, “Coherent Doppler lidar measurements of wind field statistics,” Boundary-Layer Meteorol. 86, 233–256 (1998).
[CrossRef]

R. G. Frehlich, S. Hannon, S. Henderson, “Coherent Doppler lidar measurements of winds in the weak signal regime,” Appl. Opt. 36, 3491–3499 (1997).
[CrossRef] [PubMed]

R. G. Frehlich, S. Hannon, S. Henderson, “Performance of a 2-µm coherent Doppler lidar for wind measurements,” J. Atmos. Oceanic Technol. 11, 1517–1528 (1994).
[CrossRef]

Henderson, S. W.

Hilderbrand, P. H.

P. H. Hilderbrand, R. S. Sekhon, “Objective determination of the noise level in Doppler spectra,” J. Appl. Meteorol. 13, 808–811 (1974).
[CrossRef]

Huffaker, A. V.

Huffaker, R. M.

Kavaya, M. J.

Lottman, B. T.

Magee, J. R.

Morse, C. S.

L. B. Cornman, C. S. Morse, G. Cunning, “Real-time estimation of atmospheric turbulence severity from in-situ aircraft measurements,” J. Aircr. 32, 171–177 (1995).
[CrossRef]

Oppenheim, A. V.

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

Otto, R. G.

Post, M. J.

Rader, C. M.

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

Robinson, P. A.

Rogers, R. R.

R. R. Rogers, B. R. Tripp, “Some radar measurements of turbulence in snow,” J. Appl. Meteorol. 3, 603–610 (1964).
[CrossRef]

Rye, B. J.

B. J. Rye, R. M. Hardesty, “Detection techniques for validating Doppler estimates in heterodyne lidar,” Appl. Opt. 36, 1940–1951 (1997).
[CrossRef] [PubMed]

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

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

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

Sekhon, R. S.

P. H. Hilderbrand, R. S. Sekhon, “Objective determination of the noise level in Doppler spectra,” J. Appl. Meteorol. 13, 808–811 (1974).
[CrossRef]

Smalikho, I. N.

V. A. Banakh, I. N. Smalikho, “Estimation of the turbulence energy dissipation rate from pulsed Doppler lidar data,” Atmos. Oceanic Opt. 10, 957–965 (1997).

Steakley, B. C.

Stone, R.

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 Sens. 31, 4–15 (1993).
[CrossRef]

Swanson, D.

Targ, R.

Tripp, B. R.

R. R. Rogers, B. R. Tripp, “Some radar measurements of turbulence in snow,” J. Appl. Meteorol. 3, 603–610 (1964).
[CrossRef]

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Radar-Sonar Signal Processing and Gaussian Signals in Noise (Wiley, New York, 1992).

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Part 1.

Yadlowsky, M. J.

R. G. Frehlich, M. J. Yadlowsky, “Performance of mean frequency estimators for Doppler radar and lidar,” J. Atmos. Oceanic Technol. 11, 1217–1230 (1994).
[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 Sens. 31, 4–15 (1993).
[CrossRef]

Yura, H. T.

Zarifis, V.

Zrnic, D. S.

R. J. Doviak, D. S. Zrnic, Doppler Radar and Weather Observations (Academic, San Diego, Calif., 1993).

Appl. Opt.

Atmos. Oceanic Opt.

V. A. Banakh, I. N. Smalikho, “Estimation of the turbulence energy dissipation rate from pulsed Doppler lidar data,” Atmos. Oceanic Opt. 10, 957–965 (1997).

Boundary-Layer Meteorol.

R. G. Frehlich, S. Hannon, S. Henderson, “Coherent Doppler lidar measurements of wind field statistics,” Boundary-Layer Meteorol. 86, 233–256 (1998).
[CrossRef]

IEEE Trans. Audio Electroacoust.

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

IEEE Trans. Geosci. Remote Sens.

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I. Spectral accumulation and the Cramer-Rao lower bound,” IEEE Trans. Geosci. Remote Sens. 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 Sens. 31, 28–35 (1993).
[CrossRef]

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 Sens. 31, 4–15 (1993).
[CrossRef]

J. Aircr.

L. B. Cornman, C. S. Morse, G. Cunning, “Real-time estimation of atmospheric turbulence severity from in-situ aircraft measurements,” J. Aircr. 32, 171–177 (1995).
[CrossRef]

J. Appl. Meteorol.

R. R. Rogers, B. R. Tripp, “Some radar measurements of turbulence in snow,” J. Appl. Meteorol. 3, 603–610 (1964).
[CrossRef]

P. H. Hilderbrand, R. S. Sekhon, “Objective determination of the noise level in Doppler spectra,” J. Appl. Meteorol. 13, 808–811 (1974).
[CrossRef]

J. Atmos. Oceanic Technol.

S. A. Cohn, “Radar measurements of turbulent eddy dissipation rate in the troposphere: a comparison of techniques,” J. Atmos. Oceanic Technol. 12, 85–95 (1995).
[CrossRef]

R. G. Frehlich, S. Hannon, S. Henderson, “Performance of a 2-µm coherent Doppler lidar for wind measurements,” J. Atmos. Oceanic Technol. 11, 1517–1528 (1994).
[CrossRef]

R. G. Frehlich, M. J. Yadlowsky, “Performance of mean frequency estimators for Doppler radar and lidar,” J. Atmos. Oceanic Technol. 11, 1217–1230 (1994).
[CrossRef]

R. G. Frehlich, “Simulation of coherent Doppler lidar performance in the weak signal regime,” J. Atmos. Oceanic Technol. 13, 646–658 (1996).
[CrossRef]

R. G. Frehlich, “Effects of wind turbulence on coherent Doppler lidar performance,” J. Atmos. Oceanic Technol. 14, 54–75 (1997).
[CrossRef]

J. Opt. Soc. Am. A.

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

Opt. Lett.

Other

R. J. Doviak, D. S. Zrnic, Doppler Radar and Weather Observations (Academic, San Diego, Calif., 1993).

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Radar-Sonar Signal Processing and Gaussian Signals in Noise (Wiley, New York, 1992).

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

S. M. Hannon, H. R. Bagley, R. K. Bogue, “Airborne Doppler lidar turbulence detection: ACLAIM flight test results,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 234–241 (1999).
[CrossRef]

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Part 1.

C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon, Oxford, England, 1968).

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Figures (10)

Fig. 1
Fig. 1

Wind turbulence function G(z) Eq. (57) for various parameters Δp/ L 0. The Kolmogorov model Eq. (56) is indicated by Δp/ L 0 = 0.

Fig. 2
Fig. 2

Average periodogram (circle) Eq. (55) for a Gaussian pulse with no chirp, Kolmogorov wind turbulence Eq. (56) (L 0 = ∞), frequency bin offset δ = 0.0, and with the parameters of Table 1. The case of no turbulence (triangle), the Gaussian covariance approximation (plus) Eq. (58), and the Gaussian spectrum approximation (square) Eq. (61) are also shown.

Fig. 3
Fig. 3

Average periodogram (circle) Eq. (55) for a Gaussian pulse with no chirp, Kolmogorov wind turbulence Eq. (56) (L 0 = ∞), frequency bin offset δ = 0.5, and with the parameters of Table 1. The case of no turbulence (triangle), the Gaussian covariance approximation (plus) Eq. (58), and the Gaussian spectrum approximation (square) Eq. (61) are also shown.

Fig. 4
Fig. 4

Average periodogram (circle) Eq. (55) for a Gaussian pulse with chirp, Kolmogorov wind turbulence Eq. (56) (L 0 = ∞), frequency bin offset δ = 0.0, and with the parameters of Table 1. The case of no turbulence (triangle), the Gaussian covariance approximation (plus) Eq. (58), and the Gaussian spectrum approximation (square) Eq. (61) are also shown.

Fig. 5
Fig. 5

Average periodogram (circle) Eq. (55) for a Gaussian pulse with chirp, Kolmogorov wind turbulence Eq. (56) (L 0 = ∞), frequency bin offset δ = 0.5, and with the parameters of Table 1. The case of no turbulence (triangle), the Gaussian covariance approximation (plus) Eq. (58), and the Gaussian spectrum approximation (square) Eq. (61) are also shown.

Fig. 6
Fig. 6

Average periodogram (circle) Eq. (55) for a Gaussian pulse with no chirp, Kolmogorov wind turbulence Eq. (56) (L 0 = ∞), frequency bin offset δ = 0.0, and a large range gate with the parameters of Table 1. The case of no turbulence (triangle), the Gaussian covariance approximation (plus) Eq. (58), and the Gaussian spectrum approximation (square) Eq. (61) are also shown.

Fig. 7
Fig. 7

Average periodogram (circle) Eq. (55) for a Gaussian pulse with no chirp, Kolmogorov wind turbulence Eq. (56) (L 0 = ∞), frequency bin offset δ = 0.5, and a large range gate with the parameters of Table 1. The case of no turbulence (triangle), the Gaussian covariance approximation (plus) Eq. (58), and the Gaussian spectrum approximation (square) Eq. (61) are also shown.

Fig. 8
Fig. 8

Average periodogram (circle) Eq. (55) for a Gaussian pulse with chirp, Kolmogorov wind turbulence Eq. (56) (L 0 = ∞), frequency bin offset δ = 0.0, and a large range gate with the parameters of Table 1. The case of no turbulence (triangle), the Gaussian covariance approximation (plus) Eq. (58), and the Gaussian spectrum approximation (square) Eq. (61) are also shown.

Fig. 9
Fig. 9

Average periodogram (solid curve) Eq. (55) for a Gaussian pulse with no chirp, Kolmogorov wind turbulence Eq. (56) (L 0 = ∞), frequency bin offset δ = 0.0, a large range gate centered at a range of 2 km, the far-field lidar response H(z) ∝ z -2, and with the parameters of Table 1. The case of no turbulence (dashed curve), the Gaussian covariance approximation (dotted curve) Eq. (58), and the Gaussian spectrum approximation (dash–dot curve) Eq. (61) are also shown.

Fig. 10
Fig. 10

Average periodogram (solid curve) Eq. (55) for a Gaussian pulse with chirp, Kolmogorov wind turbulence Eq. (56) (L 0 = ∞), frequency bin offset δ = 0.0, a large range gate centered at a range of 2 km, the far-field lidar response H(z) ∝ z -2, and with the parameters of Table 1. The case of no turbulence (dashed curve), the Gaussian covariance approximation (dotted curve) Eq. (58), and the Gaussian spectrum approximation (dash–dot curve) Eq. (61) are also shown.

Tables (1)

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Table 1 Lidar Parameters

Equations (63)

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Δp=Tobsc/2,
PˆunF=1Mk=0M-1 wkTSutref+kTS×exp-2πikn/M2,
Q¯unF=P¯unFP¯NnF,
RNkTS, lTS=NkTSN*lTS=δk-l,
Φˆ1=k=kmax-k1kmax+k2Q¯ukF-1,
PunF=PˆunF=1Mk=0M-1l=0M-1 wkTSw*lTSutref+kTS×u*tref+lTSexp-2πink-l/M.
PunF=PsnF+1,
PsnF=PˆsnF=1Mk=0M-1l=0M-1 wkTSw*lTSRstref+kTS, tref+lTSexp-2πink-l/M.
Rs1t1, t2=ηQULhνB- HzALt1-2z/cAL*t2-2z/c×exp2πit1-t2Δf+fe-2vrz/λdz,
Hz=K2zβzCz,
- |ALt|2dt=1,
ALt=1π1/4σP1/2exp-t22σP2+πiϕt2.
Rs1t, t=SNRt=ηQULhνB- Hz|ALt-2z/c|2dz.
rP=cσP/2
Δr=ln 2cσP.
expix=exp-x2/2.
Rs1t1, t2=ηQULhνBexp-2π2σe2t1-t22×- HzALt1-2z/cAL*t2-2z/c×exp2πit1-t2Δf-2vrz/λdz,
Rst1, t2=SNR exp2πifmt1-t2-t1-t221/4σP2+πϕσP2+2π2σe2,
fm=Δf-2vr/λ
Sf=SNR2πwexp-f-fm2/2w2,
w=18πσP1+4π2ϕ2σP4+8π2σe2σP21/2.
wv=λw/2.
Rst1, t2=SNR exp2πifmt1-t2-2π2w2t1-t22.
vlin=1ΔpR-Δp/2R+Δp/2 vrzdz,
vwgt=1ΔpR-Δp/2R+Δp/2 vpulsezdz,
vpulsez=- vrsInz-sds,
Inz=Wz- Wrdr,
Wz=|AL2z/c|2
Rs1t1, t2=ηQULhνB×exp2πit1-t2fm-2π2σe2t1-t22×- HzALt1-2z/cAL*t2-2z/c×exp4πit1-t2vrz/λdz,
fm=Δf-2vlin/λ
vrz=vrz-vlin
Rst1, t2=ηQULhνB×exp2πit1-t2fm-2π2σe2t1-t22×- HzaALt1-2z/cAL*t2-2z/c×exp4πit1-t2vrz/λvdz,
Rst1, t2=ηQULhνBexp2πit1-t2fm×- HzaALt1-2z/cAL*t2-2z/c×exp-8π2t1-t22vr2zv/λ2dz.
Rvs=vrz0vrz0+sv,
Dvs=vrz0-vrz0+s2v=2σv2-Rvs.
Dvs=2σv2Λs/L0,
Λx=1-22/3Γ1/3 x1/3K1/3x=1.0-0.5925485 x1/3K1/3x,
Dvs=Cv2/3s2/3,
σturb2=1Δpz0z0+Δp vrz-vlin2vdz=1Δp0Δp1-s/ΔpDvsds.
σturb2=9Cv40Δp2/3,
σturb2=2σv2011-sΛsΔp/L0ds.
Θ=σturbwv.
vr2zv=Gz-z0,
Gx=1Δp0Δp Dvx-sds-σturb2,
Rst1, t2=ηQULhνB×exp2πit1-t2fm-2π2σe2t1-t22×- HzaALt1-2z/cAL*t2-2z/c×exp-8π2t1-t22Gz-z0/λ2dz.
Rsμ, τ=ηQULc2hνBπ×exp2πiτfm-τ2/4σP2-2π2σe2τ2×- Hz0+xrPa×exp-8π2τ2GxrP/λ2×exp-μ-t0/σP-x2+2πiϕσPτμ-t0/σP-xdx,
μ=t1+t2/2,
τ=t1-t2,
t0=2z0/c,
x=z-z0/rP.
Rsμ, τ=SNRπ×exp2πiτfm-τ2/4σP2-2π2σe2τ2×-exp-8π2τ2GxrP/λ2×exp-μ-t0/σP-x2+2πiϕσPτμ-t0/σP-xdx.
t1=tref+kTS, t2=tref+lTS,
τ=k-lTS, μ=t0+k+l+1TS/2,
Rs,kl=SNRπ×exp2πiτfm-τ2/4σP2-2π2σe2τ2×-exp-8π2τ2GxrP/λ2×exp-χ/σP-x2+2πiϕσPτχ/σP-xdx,
PsnF=SNRMπk=0M-1l=0M-1 wkTSw*lTS×exp2πik-lkm+δ-n/M×exp-τ2/4σP2-2π2σe2τ2×-exp-8π2τ2GxrP/λ2×exp-χ/σP-x2+2πiϕσPτχ/σP-xdx.
Gz=σturb28/31+|z|/Δp5/3-8/3|z|/Δp5/3-1, z<0, Gz=σturb28/31-z/Δp5/3+ 8/3z/Δp5/3-1, 0<z<Δp, Gz=σturb28/3z/Δp5/3- 8/3z/Δp-15/3-1, z>Δp,
Gz=σturb201 Λ|z-sΔp|/L0ds011-sΛsΔp/L0ds-1.
Rs,kl=SNR exp2πiτfm-2π2weff2τ2,
weff2=w2+4σturb2/λ2
wveff2=λ2weff2/4=wv2+σturb2=wv21+Θ2.
PsnF=SNR2πTSweff×exp-F2n-km-δ2/2weff2,
PˆunF=1Nkk=1Nk PˆshiftnF, k,
σeff2=σe2+σf2+F2/12,

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