Abstract

In the weak signal regime coherent Doppler lidar velocity estimates are characterized by a localized distribution around the true mean velocity and a uniform distribution of random outliers over the velocity search space. The performance of velocity estimators is defined by the standard deviation of the good estimates around the true mean velocity and the fraction of random outliers. The quality of velocity estimates is improved with pulse accumulation. The performance of velocity estimates from two different coherent Doppler lidars in the weak signal regime is compared with the predictions of computer simulations for pulse accumulation from 1 to 100 pulses.

© 1997 Optical Society of America

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References

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  1. 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]
  2. 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]
  3. 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]
  4. R. G. Frehlich, S. Hannon, S. Henderson, “Performance of a 2-µm coherent Doppler lidar for wind measurements,” J. Atmos. Ocean. Technol. 11, 1517–1528 (1994).
    [CrossRef]
  5. 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 Sensing 31, 16–27 (1993).
    [CrossRef]
  6. 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]
  7. R. G. Frehlich, M. J. Yadlowsky, “Performance of mean frequency estimators for Doppler radar and lidar,” J. Atmos. Ocean. Technol. 11, 1217–1230 (1994).
    [CrossRef]
  8. R. G. Frehlich, “Simulation of coherent Doppler lidar performance in the weak signal regime,” J. Atmos. Ocean. Technol. 13, 646–658 (1996).
    [CrossRef]
  9. R. G. Frehlich, “Effects of wind turbulence on coherent Doppler lidar performance,” J. Atmos. Ocean. Technol. 14, 54–75 (1997).
    [CrossRef]
  10. B. J. Rye, R. M. Hardesty, “Detection techniques for validating Doppler estimates in heterodyne lidar,” Appl. Opt. 36, 1940–1951 (1997).
    [CrossRef] [PubMed]
  11. B. J. Rye, “Spectral correlation of atmospheric lidar returns with range-dependent backscatter,” J. Opt. Soc. Am. A 7, 2199–2207 (1990).
    [CrossRef]
  12. 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 µm and 10 µm,” Appl. Opt. 35, 7117–7127 (1996).
    [CrossRef] [PubMed]
  13. H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968).
  14. C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon, Oxford, England, 1968).
  15. 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]
  16. B. Gold, A. V. Oppenheim, C. M. Rader, “Theory and implementation of the discrete Hilbert transformation,” in Symposium on Computer Processing in Communications (Polytechnic Press, Brooklyn, N.Y., 1970), Vol. 19, pp. 235–250.
  17. V. Cizek, “Discrete Hilbert transform,” IEEE Trans. Audio Electroacoust. AU-18, 340–343 (1970).
    [CrossRef]
  18. P. H. Hilderbrand, R. S. Sekhon, “Objective determination of the noise level in Doppler spectra,” J. Appl. Meteorl. 13, 808–811 (1974).
    [CrossRef]
  19. J. H. Churnside, H. T. Yura, “Speckle statistics of atmospherically backscattered laser light,” Appl. Opt. 22, 2559–2565 (1983).
    [CrossRef] [PubMed]
  20. R. G. Frehlich, “Coherent Doppler lidar signal covariance including wind shear and wind turbulence,” Appl. Opt. 33, 6472–6481 (1994).
    [CrossRef] [PubMed]

1997 (2)

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

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

1996 (2)

1994 (3)

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, M. J. Yadlowsky, “Performance of mean frequency estimators for Doppler radar and lidar,” J. Atmos. Ocean. 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. Ocean. Technol. 11, 1517–1528 (1994).
[CrossRef]

1993 (4)

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 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]

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]

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]

1991 (1)

1990 (1)

1989 (1)

1983 (1)

1974 (1)

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

1970 (1)

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

Ames, L. L.

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 Sensing 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]

Frehlich, R. G.

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

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

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

R. G. Frehlich, M. J. Yadlowsky, “Performance of mean frequency estimators for Doppler radar and lidar,” J. Atmos. Ocean. 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, “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]

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 Press, Brooklyn, N.Y., 1970), Vol. 19, pp. 235–250.

Hale, C. P.

Hannon, S.

R. G. Frehlich, S. Hannon, S. Henderson, “Performance of a 2-µm coherent Doppler lidar for wind measurements,” J. Atmos. Ocean. 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 Sensing 31, 4–15 (1993).
[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 Sensing 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 Sensing 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, “Performance of a 2-µm coherent Doppler lidar for wind measurements,” J. Atmos. Ocean. 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. Meteorl. 13, 808–811 (1974).
[CrossRef]

Huffaker, A. V.

Huffaker, R. M.

Kavaya, M. J.

Magee, J. R.

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 Press, Brooklyn, N.Y., 1970), Vol. 19, pp. 235–250.

Otto, R. G.

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 Press, Brooklyn, N.Y., 1970), Vol. 19, pp. 235–250.

Robinson, P. A.

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. 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]

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. Meteorl. 13, 808–811 (1974).
[CrossRef]

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 Sensing 31, 4–15 (1993).
[CrossRef]

Swanson, D.

Targ, R.

Van Trees, H. L.

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

Yadlowsky, M. J.

R. G. Frehlich, M. J. Yadlowsky, “Performance of mean frequency estimators for Doppler radar and lidar,” J. Atmos. Ocean. 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 Sensing 31, 4–15 (1993).
[CrossRef]

Yura, H. T.

Zarifis, V.

Appl. Opt. (4)

IEEE Trans. Audio Electroacoust. (1)

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

IEEE Trans. Geosci. Remote Sensing (4)

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]

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 heterodyne 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]

J. Appl. Meteorl. (1)

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

J. Atmos. Ocean. Technol. (4)

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

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

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

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

J. Opt. Soc. Am. A (1)

Opt. Lett. (2)

Other (3)

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

C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon, Oxford, England, 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 Press, Brooklyn, N.Y., 1970), Vol. 19, pp. 235–250.

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Figures (12)

Fig. 1
Fig. 1

Monitor pulses for the (a) λ = 2.09-µm flash-lamp-pumped lidar and the (b) λ = 2.02-µm diode-pumped lidar. The solid curve is the best-fit model Eq. (1) and the solid circles are the data. See Table 1 for pulse parameters.

Fig. 2
Fig. 2

Noise spectrum and signal spectrum after noise correction for the λ = 2.09-µm flash-lamp-pumped lidar. The signal spectrum is for a range gate centered at R = 7 km with M = 16 complex data points per range gate and 10,000 lidar pulses. SNR is the signal-to-noise ratio.

Fig. 3
Fig. 3

Noise spectrum and signal spectrum after noise correction for the λ = 2.02-µm flash-lamp-pumped lidar. The signal spectrum is for a range gate centered at R = 8 km with M = 64 complex data points per range gate and 15,000 lidar pulses.

Fig. 4
Fig. 4

Average number of coherent photoelectrons per estimate Φ1 as a function of time for various range gates with the λ = 2.09-µm flash-lamp-pumped lidar.

Fig. 5
Fig. 5

ML velocity estimates from the λ = 2.09-µm flash-lamp-pumped lidar for a range gate centered at K = 1, 5, and 10 accumulated pulses. The low-frequency filtered velocity for K = 10 is shown as a solid curve (see Section 3).

Fig. 6
Fig. 6

ML velocity estimates for a range gate centered at K = 1, 5, 25, and 100 accumulated pulses. The low-frequency filtered velocity for K = 100 is shown as a solid curve (see Section 3).

Fig. 7
Fig. 7

Performance of ML velocity estimates with pulse accumulation order K as a function of the average number of coherent photoelectrons per estimate for a single pulse Φ1. M = 16 complex data points were used for each velocity estimate. The results for computer simulation are shown as open circles. The standard deviation of the good estimates g and the fraction b of random outliers that were extracted from the data are shown as solid circles. One sigma error bars are shown when they are larger than the symbols.

Fig. 8
Fig. 8

Performance of ML velocity estimates with pulse accumulation order K as a function of Φ1, the average number of coherent photoelectrons per estimate for a single pulse. M = 32 complex data points were used for each velocity estimate. The results for computer simulation are shown as open circles. The standard deviation of the good estimates g and the fraction b of random outliers that were extracted from the data are shown as solid circles. One sigma error bars are shown when they are larger than the symbols.

Fig. 9
Fig. 9

Average number of coherent photoelectrons per estimate Φ1 as a function of time for various range gates with the λ = 2.02-µm diode-pumped lidar.

Fig. 10
Fig. 10

ML velocity estimates from the λ = 2.02-µm diode-pumped lidar for a range gate centered at K = 1, 5, 25, and 100 accumulated pulses. The low-frequency filtered velocity for K = 100 is shown as a solid curve (see Section 3).

Fig. 11
Fig. 11

Performance of ML velocity estimates with pulse accumulation order K as a function of the average number of coherent photoelectrons per estimate for a single pulse Φ1. M = 64 complex data points were used for each velocity estimate. The results for computer simulation are shown as open circles. The standard deviation of the good estimates g and the fraction b of random outliers that were extracted from the data are shown as solid circles. One sigma error bars are shown when they are larger than the symbols.

Fig. 12
Fig. 12

Performance of ML velocity estimates with pulse accumulation order K as a function of the average number of coherent photoelectrons per estimate for a single pulse Φ1. M = 32 complex data points were used for each velocity estimate. The results for computer simulation are shown as open circles. The standard deviation of the good estimates g and the fraction b of random outliers that were extracted from the data are shown as solid circles. One sigma error bars are shown when they are larger than the symbols.

Tables (2)

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Table 1 Lidar Pulse Parameters

Tables Icon

Table 2 Velocity Estimation Parameters

Equations (18)

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xt=A exp-t2/2σ21+c1t+c2t2+c3t3×cos2πfmont+πϕt2+θ.
SˆnkF=TRLKNm=1KNNmkF2,
NmkF=l=0L-1nmlTRexp-2πikl/L
dmkTR=1Lj=0L-1XmjFN¯j exp2πikj/L,
vsearch=vmax/2=λfmax/4.
SˆnckF=SˆzkF/SˆnkF,
Φˆ1=k=kmax-k1kmax+k2SˆnkF-1=SNˆR M,
SDΦˆ12Φ1+4/KN1/2.
w=18πσ1+4π2ϕ2σ41/218πσ
Ω=w/F=wTOBS=wMTS
Ω=ln2/21/2πΔpΔr=0.1874ΔpΔr,
wkTD=vkTD-lkTD,  if lkTD-ΔV/2<vkTD<lkTD+ΔV/2=0,  otherwise.
bˆ=αˆvsearchvsearch-ΔV,
WkF2/L=σew2=1-bg2+bΔV3/12vsearch,
σˆew2=1LHFk=L/2-LHFL/2-1WkF2L,
gˆ2=σˆew2-b¯ΔV3/12vsearch/1-b¯.
VSR, t=ΦˆaR, t-ΦˆbR, tΦˆaR, t+ΦˆbR, t,
VTR, t=Φˆ1R, t+TB-Φˆ1R, tΦˆ1R, t+TB+Φˆ1R, t,

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