Abstract

The conditions for optimizing the precision of heterodyne atmospheric lidar measurements using extended (deep) targets are investigated. The minimum standard deviation of each unknown (return power, Doppler shift, and signal bandwidth) is approximately twice the optical limit at best and is only weakly dependent on knowledge of the other parameters at optimal power levels. Somewhat stronger signal power levels are needed for bandwidth estimation. Results are displayed as a function of a time–bandwidth product to clarify the trade-off between estimate precision and range weighting. Realization under ideal conditions is confirmed by use of simulations.

© 2000 Optical Society of America

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References

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  1. B. J. Rye, R. M. Hardesty, “Estimate optimization parameters for incoherent backscatter lidar,” Appl. Opt. 36, 9425–9436 (1997); errata, 37, 4016 (1998).
  2. B. J. Rye, R. M. Hardesty, “Deteciton techniques for validating Doppler estimates in heterodyne lidar,” Appl. Opt. 36, 1940–1951 (1997).
    [CrossRef] [PubMed]
  3. B. J. Rye, “Estimation of return signal spectral width in incoherent backscatter heterodyne lidar,” in Proceedings of Tenth Biennial coherent laser Radar Conference (University Space Research Association, 4950 Corporate Drive, Suite 100, Huntsville, Ala. 35808, 1999), pp. 195–197.
  4. L. Lading, A. S. Jensen, “Estimating the spectral width of a narrowband optical signal,” Appl. Opt. 19, 2750–2756 (1980).
  5. R. G. Seasholtz, “High-speed anemometry based on spectrally resolved Rayleigh scattering,” in Fourth International Conference on Laser Anemometry, NASA Tech. Memo. 104522 (NASA Lewis Research Center, Cleveland, Ohio 44135, 1991).
  6. R. G. Seasholtz, “Gas temperature and density measurements based on spectrally resolved Rayleigh-Brillouin scattering,” presented at the Proceedings of the Measurement Technology Conference (NASA Langley Research Center, Hampton, Virginia, 1992).
  7. B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in Doppler lidar. II: Incoherent correlogram accumulation,” IEEE Trans. Geosci. Remote Sens. 31, 28–35 (1993).
    [CrossRef]
  8. J.-M. Gagne, J.-P. Saint-Dizier, M. Picard, “Methode d’echantillonnage des fonctions deterministes en spectroscopie: application à un spectromètre multicanal par comptage photonique,” Appl. Opt. 13, 581–588 (1974).
    [CrossRef]
  9. H. Cramer, Mathematical Methods of Statistics (Princeton University, Princeton, N.J., 1946).
  10. H. Z. Cummins, R. L. Swinney, “Light beating spectroscopy,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1970), vol. 8, pp. 133–200.
  11. M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1962), Chap. 9 (Chap. 10 in 2nd ed., 1980).
  12. B. J. Rye, “Molecular backscatter heterodyne lidar: a computational evaluation,” Appl. Opt. 37, 6321–6328 (1998); see qualitative discussion in Appendix.
  13. M. J. Levin, “Power spectrum parameter estimation,” IEEE Trans. Inf. Theory IT-11, 100–107 (1965).
    [CrossRef]
  14. D. S. Zrnic, “Spectral statistics for complex colored discrete-time sequences,” IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 596–599 (1980).
    [CrossRef]
  15. A. Arcese, E. W. Trombini, “Variances of spectral parameters with a Gaussian shape,” IEEE Trans. Inf. Theory IT-17, 200–201 (1971).
    [CrossRef]
  16. D. S. Zrnic, “Estimation of spectral moments for weather echoes,” IEEE Trans. Geosci. Electron. GE-17, 113–128 (1979).
    [CrossRef]
  17. O. Brovko, “The structure of a maximum likelihood center frequency estimate,” (Hughes Aircraft Co., 1977).
  18. B. J. Rye, “The spectral correlation of atmospheric lidar returns with range-dependent backscatter,” J. Opt. Soc. Am. A 7, 2199–2207 (1990).
  19. E. S. Chornoboy, “Optimal mean velocity estimation for Doppler weather radars,” IEEE Trans. Geosci. Remote Sens. 31, 575–586 (1993).
    [CrossRef]
  20. R. G. Frehlich, “Cramer-Rao bound for Gaussian random processes and applications to radar processing of atmospheric signals,” IEEE Trans. Geosci. Remote Sens 31, 1123–1131 (1993).
    [CrossRef]
  21. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press. Cambridge, England, 1992).
  22. K. S. Miller, M. M. Rochwarger, “A covariance approach to spectral moment estimation,” IEEE Trans. Inf. Theory IT-18, 588–596 (1972).
  23. R. Frehlich, L. Cornman, “Coherent Doppler lidar signal spectrum with wind turbulence,” Appl. Opt. 38, 7456–7466 (1999).
    [CrossRef]
  24. B. Porat, B. Friedlander, “Computation of the exact information matrix of Gaussian time series with stationary random components,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 118–130 (1986); see Appendix A for derivation of expression for information matrix of Gaussian time series.
  25. R. Frehlich, “Performance of maximum likelihood estimators of mean power and Doppler velocity with a priori knowledge of spectral width,” J. Atmos. Oceanic Technol. 16, 1702–1709 (1999).
    [CrossRef]

1999

R. Frehlich, L. Cornman, “Coherent Doppler lidar signal spectrum with wind turbulence,” Appl. Opt. 38, 7456–7466 (1999).
[CrossRef]

R. Frehlich, “Performance of maximum likelihood estimators of mean power and Doppler velocity with a priori knowledge of spectral width,” J. Atmos. Oceanic Technol. 16, 1702–1709 (1999).
[CrossRef]

1998

1997

1993

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

E. S. Chornoboy, “Optimal mean velocity estimation for Doppler weather radars,” IEEE Trans. Geosci. Remote Sens. 31, 575–586 (1993).
[CrossRef]

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

1990

1986

B. Porat, B. Friedlander, “Computation of the exact information matrix of Gaussian time series with stationary random components,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 118–130 (1986); see Appendix A for derivation of expression for information matrix of Gaussian time series.

1980

L. Lading, A. S. Jensen, “Estimating the spectral width of a narrowband optical signal,” Appl. Opt. 19, 2750–2756 (1980).

D. S. Zrnic, “Spectral statistics for complex colored discrete-time sequences,” IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 596–599 (1980).
[CrossRef]

1979

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

1974

1972

K. S. Miller, M. M. Rochwarger, “A covariance approach to spectral moment estimation,” IEEE Trans. Inf. Theory IT-18, 588–596 (1972).

1971

A. Arcese, E. W. Trombini, “Variances of spectral parameters with a Gaussian shape,” IEEE Trans. Inf. Theory IT-17, 200–201 (1971).
[CrossRef]

1965

M. J. Levin, “Power spectrum parameter estimation,” IEEE Trans. Inf. Theory IT-11, 100–107 (1965).
[CrossRef]

Arcese, A.

A. Arcese, E. W. Trombini, “Variances of spectral parameters with a Gaussian shape,” IEEE Trans. Inf. Theory IT-17, 200–201 (1971).
[CrossRef]

Brovko, O.

O. Brovko, “The structure of a maximum likelihood center frequency estimate,” (Hughes Aircraft Co., 1977).

Chornoboy, E. S.

E. S. Chornoboy, “Optimal mean velocity estimation for Doppler weather radars,” IEEE Trans. Geosci. Remote Sens. 31, 575–586 (1993).
[CrossRef]

Cornman, L.

Cramer, H.

H. Cramer, Mathematical Methods of Statistics (Princeton University, Princeton, N.J., 1946).

Cummins, H. Z.

H. Z. Cummins, R. L. Swinney, “Light beating spectroscopy,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1970), vol. 8, pp. 133–200.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press. Cambridge, England, 1992).

Frehlich, R.

R. Frehlich, L. Cornman, “Coherent Doppler lidar signal spectrum with wind turbulence,” Appl. Opt. 38, 7456–7466 (1999).
[CrossRef]

R. Frehlich, “Performance of maximum likelihood estimators of mean power and Doppler velocity with a priori knowledge of spectral width,” J. Atmos. Oceanic Technol. 16, 1702–1709 (1999).
[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 Sens 31, 1123–1131 (1993).
[CrossRef]

Friedlander, B.

B. Porat, B. Friedlander, “Computation of the exact information matrix of Gaussian time series with stationary random components,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 118–130 (1986); see Appendix A for derivation of expression for information matrix of Gaussian time series.

Gagne, J.-M.

Hardesty, R. M.

Jensen, A. S.

Lading, L.

Levin, M. J.

M. J. Levin, “Power spectrum parameter estimation,” IEEE Trans. Inf. Theory IT-11, 100–107 (1965).
[CrossRef]

Miller, K. S.

K. S. Miller, M. M. Rochwarger, “A covariance approach to spectral moment estimation,” IEEE Trans. Inf. Theory IT-18, 588–596 (1972).

Picard, M.

Porat, B.

B. Porat, B. Friedlander, “Computation of the exact information matrix of Gaussian time series with stationary random components,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 118–130 (1986); see Appendix A for derivation of expression for information matrix of Gaussian time series.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press. Cambridge, England, 1992).

Rochwarger, M. M.

K. S. Miller, M. M. Rochwarger, “A covariance approach to spectral moment estimation,” IEEE Trans. Inf. Theory IT-18, 588–596 (1972).

Rye, B. J.

B. J. Rye, “Molecular backscatter heterodyne lidar: a computational evaluation,” Appl. Opt. 37, 6321–6328 (1998); see qualitative discussion in Appendix.

B. J. Rye, R. M. Hardesty, “Estimate optimization parameters for incoherent backscatter lidar,” Appl. Opt. 36, 9425–9436 (1997); errata, 37, 4016 (1998).

B. J. Rye, R. M. Hardesty, “Deteciton 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 Doppler lidar. II: Incoherent correlogram accumulation,” IEEE Trans. Geosci. Remote Sens. 31, 28–35 (1993).
[CrossRef]

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

B. J. Rye, “Estimation of return signal spectral width in incoherent backscatter heterodyne lidar,” in Proceedings of Tenth Biennial coherent laser Radar Conference (University Space Research Association, 4950 Corporate Drive, Suite 100, Huntsville, Ala. 35808, 1999), pp. 195–197.

Saint-Dizier, J.-P.

Seasholtz, R. G.

R. G. Seasholtz, “High-speed anemometry based on spectrally resolved Rayleigh scattering,” in Fourth International Conference on Laser Anemometry, NASA Tech. Memo. 104522 (NASA Lewis Research Center, Cleveland, Ohio 44135, 1991).

R. G. Seasholtz, “Gas temperature and density measurements based on spectrally resolved Rayleigh-Brillouin scattering,” presented at the Proceedings of the Measurement Technology Conference (NASA Langley Research Center, Hampton, Virginia, 1992).

Skolnik, M. I.

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1962), Chap. 9 (Chap. 10 in 2nd ed., 1980).

Swinney, R. L.

H. Z. Cummins, R. L. Swinney, “Light beating spectroscopy,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1970), vol. 8, pp. 133–200.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press. Cambridge, England, 1992).

Trombini, E. W.

A. Arcese, E. W. Trombini, “Variances of spectral parameters with a Gaussian shape,” IEEE Trans. Inf. Theory IT-17, 200–201 (1971).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press. Cambridge, England, 1992).

Zrnic, D. S.

D. S. Zrnic, “Spectral statistics for complex colored discrete-time sequences,” IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 596–599 (1980).
[CrossRef]

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

Appl. Opt.

IEEE Trans. Acoust. Speech Signal Process

B. Porat, B. Friedlander, “Computation of the exact information matrix of Gaussian time series with stationary random components,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 118–130 (1986); see Appendix A for derivation of expression for information matrix of Gaussian time series.

IEEE Trans. Acoust. Speech Signal Process.

D. S. Zrnic, “Spectral statistics for complex colored discrete-time sequences,” IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 596–599 (1980).
[CrossRef]

IEEE Trans. Geosci. Electron.

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

IEEE Trans. Geosci. Remote Sens

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

IEEE Trans. Geosci. Remote Sens.

E. S. Chornoboy, “Optimal mean velocity estimation for Doppler weather radars,” IEEE Trans. Geosci. Remote Sens. 31, 575–586 (1993).
[CrossRef]

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

IEEE Trans. Inf. Theory

A. Arcese, E. W. Trombini, “Variances of spectral parameters with a Gaussian shape,” IEEE Trans. Inf. Theory IT-17, 200–201 (1971).
[CrossRef]

M. J. Levin, “Power spectrum parameter estimation,” IEEE Trans. Inf. Theory IT-11, 100–107 (1965).
[CrossRef]

K. S. Miller, M. M. Rochwarger, “A covariance approach to spectral moment estimation,” IEEE Trans. Inf. Theory IT-18, 588–596 (1972).

J. Atmos. Oceanic Technol.

R. Frehlich, “Performance of maximum likelihood estimators of mean power and Doppler velocity with a priori knowledge of spectral width,” J. Atmos. Oceanic Technol. 16, 1702–1709 (1999).
[CrossRef]

J. Opt. Soc. Am. A

Other

O. Brovko, “The structure of a maximum likelihood center frequency estimate,” (Hughes Aircraft Co., 1977).

H. Cramer, Mathematical Methods of Statistics (Princeton University, Princeton, N.J., 1946).

H. Z. Cummins, R. L. Swinney, “Light beating spectroscopy,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1970), vol. 8, pp. 133–200.

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1962), Chap. 9 (Chap. 10 in 2nd ed., 1980).

R. G. Seasholtz, “High-speed anemometry based on spectrally resolved Rayleigh scattering,” in Fourth International Conference on Laser Anemometry, NASA Tech. Memo. 104522 (NASA Lewis Research Center, Cleveland, Ohio 44135, 1991).

R. G. Seasholtz, “Gas temperature and density measurements based on spectrally resolved Rayleigh-Brillouin scattering,” presented at the Proceedings of the Measurement Technology Conference (NASA Langley Research Center, Hampton, Virginia, 1992).

B. J. Rye, “Estimation of return signal spectral width in incoherent backscatter heterodyne lidar,” in Proceedings of Tenth Biennial coherent laser Radar Conference (University Space Research Association, 4950 Corporate Drive, Suite 100, Huntsville, Ala. 35808, 1999), pp. 195–197.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press. Cambridge, England, 1992).

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Figures (6)

Fig. 1
Fig. 1

Plots of the following normalized standard deviations calculated for heterodyne systems by use of the Brovko–Zrnic formulas (see Appendix A) versus the effective photocount N: thin curve, σBZ(δ|f 1, f 2)/δ; ●, σ BZ (f 1|δ, f 2)/f 1; □, √2σBZ(f 2|δ, f 1)/f 2; and bold curve, the ideal optical value for these ratios that, in each case, is 1/√N [Eqs. (2)].

Fig. 2
Fig. 2

Dependence of estimator noise figure F E on the tuning parameter α for estimates of frequency shift f 1, the wideband SNR δ, and bandwidth f 2, with different parameters known a priori. The minima in these curves are the optimal operating points. The CRLB’s were computed with the Brovko–Zrnic formulas (see Appendix A) with M = 10, f 2 = 0.05, and different values of N. Curves are as follows: ×, F E (δ|f 2); ⊠, F E (δ); ■, F E (f 1|δ, f 2); □, F E (f 1) (no a priori knowledge); +, F E (f 2|δ); ⊞, F E (f 2). The reduction factor of Eq. (A14) is shown as a dotted curve with open circles.

Fig. 3
Fig. 3

(a) Optimal operation (where the noise figure F E is minimized) is characterized by values of the tuning parameter α = δ/[√(2π)f 1] as a function of f 2 M for different M. Values of α are calculated with the BZ formula for the CRLB assuming that the parameters (other than the one that is estimated) are known a priori. Curve markers are as follows: (i) M = 10: solid curve, αmin(δ|f 1, f 2); ●, αmin(f 1|δ, f 2); ■, αmin(f 2|δ, f 1); (ii) M = 64: thin curve, αmin(δ|f 1, f 2); ○, αmin(f 1|δ, f 2); □, αmin(f 2|δ, f 1). Curves for αmin(δ|f 1, f 2) and αmin(f 1|δ, f 2) are identical to those shown in Ref. 1. The sloping straight dashed lines indicate values of N = 1 (leftmost line), N = 10, and N = 64 (rightmost line). That for N = 1 is the asymptotic limit for power (δ) estimates at low f 2 M, and each line is an asymptotic limit to the curve with M = N for large f 2 M.1 Horizontal straight dashed lines show the Levin values of α for optimal operation with untruncated Gaussian spectra in the order αmin(δ|f 1, f 2) (lowest), αmin(f 1|δ, f 2), and αmin(f 2|δ, f 1) (highest) (numerical values are given in Table 1). These lines would be asymptotic limits for large f 2 M if the fade count was not limited, for discretely sampled data, to a maximum value equal to M.1 (b) The CRLB’s used for δ and f 2 are calculated assuming that all other parameters are unknown a priori (for f 1 knowledge of other parameters makes no difference, see Appendix A). However, the horizontal lines for the Levin estimator are the same as those shown in (a), i.e., they are calculated with other parameters known so as to facilitate comparison.

Fig. 4
Fig. 4

Minimum noise figures F E for various estimators. (a) The BZ formula is used for the CRLB assuming that all other parameters are known. The curve markers are the same as in Fig. 3, with F E replacing α. Horizontal lines show the Levin values for untruncated Gaussian spectra. The curves for (F E )min(δ|f 1, f 2) and (F E )min(f 1|,δ, f 2) are identical to those shown in Ref. 1. (b) Minimum noise figures are calculated with the BZ formulas for the CRLB with all other parameters unknown a priori. Levin lines are the same as in Fig. 4(a). (a) and (b) indicate that prior knowledge has little effect on the minimum noise figures. This contrasts with the more significant effect on α shown in the curves of Fig. 3.

Fig. 5
Fig. 5

(a)–(c) Simulated data performance of BZ and Levin estimators given no prior knowledge of other parameters are compared with the CRLB’s calculated with the BZ formula based on the same assumption and shown as a function of the effective photocount N. (a) The relative standard deviations and relative biases found by dividing σ(BZ)(δ) and σ(Lev)(δ) and B (BZ)(δ) and B (Lev)(δ) for estimates of δ by the CRLB σBZ(δ) on the standard deviation calculated with the formulas of Appendix A, are shown as a function of the effective photocount N for M = 16 and f 2 = 0.1. (b) and (c) are similar with f 1 and f 2, respectively, replacing δ. Curve markers are as follows: ■, B (BZ)BZ; ●, σ(BZ)BZ; □, B (Lev)BZ; ○, σ(Lev)BZ. These ML estimators appear to work as well as might be expected, that is, their standard deviation is approximately the same as, and any bias is much less than, the CRLB. There is little difference between the performance of the BZ and Levin estimators in these cases. The data are restricted to N > 70 because at lower signal levels our two-parameter optimization procedure became unstable. This is not unduly limiting because, at slightly lower photocounts, both the BZ and the Levin estimates obtained with spectral peak picking are expected to become biased by noise-related peaks. The CRLB’s used here then cease to be a useful guide to performance unless estimates are thresholded at a suitable level of the log likelihood ratio [Eqs. (4) and (5)]. The triangles show performance of the estimators based on a single lag in the ACF (the downward-pointing triangle showing BBZ and the upward-pointing triangle showing σ/σBZ). The estimators are the squarer for δ, the pulse pair for f 1, and the first lag f 2 estimator of Eq. (6). The last is clearly unreliable but no effort has been made to determine whether choice of a lag other than the first would improve matters. (d) Identical to (c) except that the estimators of f 2 and the CRLB’s used are based on the assumption that δ is known. For example, the filled squares indicate B (BZ)(f 2|δ)/σBZ(f 2|δ) rather than B (BZ)(f 2)/σBZ(f 2), etc. There appears to be little difference in the results, except that σ(Lev)(f 2|δ) becomes rather larger than σ(BZ)(f 2|δ) ≈ σBZ(f 2|δ) at high signal levels, which is not unexpected given the suboptimal nature of the Levin algorithm in this limit.

Fig. 6
Fig. 6

Statistical significance of the simulation results like those shown in Fig. 5 is evaluated. (a) and (b) Absolute value of the bias of various estimates compared with the standard deviation of their mean, σsdm, calculated over the approximately 4000 independent trials that were run. [Because σ(BZ) ≈ σBZ (Fig. 5), σsdm ≈ σBZ/√(4000), and this test is rather stringent]. Only results for the BZ estimator are shown because those for Levin are similar. Curve markers are as follows: □, B (BZ)(f 1); ■, B (BZ)(f 1|f 2); □, B (BZ)(f 1|δ); +, B (BZ)(f 2); ×, B (BZ)(f 2|δ); ▽, B (BZ)(δ); △, B (BZ)L(,104|f2). It would be expected that the absolute value of almost all bias values should lie below 3σsdm, and in (a), plotted for f 1 = 0.1, M = 16 [as in Fig. (5)], this is the case, except for low signal levels where bias is expected because noise-related estimates occur. However, it is not the case if the bandwidth is narrow, f 1 = 0.02, M = 16 [Fig. 6(b)], when the bias in estimates of bandwidth itself, and in some of the estimates of δ with f 2 unknown, lie somewhat above 3σsdm. The magnitude of the maximum bias in f 2 in these cases is less than 0.0006 (approximately 3% of f 2), which occurs when N ∼ 100, and is likely to be the consequence of the optimization algorithm used or rounding errors, rather than anything more fundamental. (c) The magnitude of the fractional bias in estimates of δ, given no prior knowledge for f 2 = 0.1 and M = 16. Here the filled squares relate to the BZ estimator, the open circles to Levin’s, and the open triangles to the squarer. Except in the small signal regime, the bias is of the order of 0.1%. This appears to conflict with a recent result of Frehlich,25 who reports bias of up to 10% for the BZ estimator.

Tables (3)

Tables Icon

Table 1 Optimal Operating Parameters in the Levin Limita

Tables Icon

Table 2 Optimal Parameters and Precision for Pulse-Broadened Return from Lidar with Short Time–Bandwidth Producta

Tables Icon

Table 3 Optimal Parameters and Precision for Broadband Return from Lidara

Equations (20)

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

psf=δ2π f2exp-f-f122f22.
σopt2δ|f1, f2=δ2/N,σopt2f1|δ, f2=f22/N,σopt2f2|δ, f1=f22/2N,
σLevin2θi|θji=σopt2θi|θji×α2π-fixg2x, αdx-1,
ln Λ=i=1MXiϕN1-ϕ˜iSϕN+1-1 - lnϕ˜iϕN=i=1MXiϕ˜iSϕNϕ˜i-lnϕ˜iϕN,
ln Λ=1/PNZ+D˜I-ΓD˜*Z-lndet Q˜,
f˜2=1-|μ1|/PS1/22 π,
μk=PNδexp-2πf2k2/2expj2πf1k+1,
Rδ, f1, f2=rklδ, f1, f2=qklδ, f2expj2πf1k-1
Q=δG+I,
G=gkl=exp-2πk-lf22/2.
H={hmn}=trR-1RθmR-1Rθn
hmn=trAmAn=k=1Ml=1M aklmalkn.
akl0=ΓGkl=m=1M γkmgml,akl1=j2πm=1Mγkmm-lqml,akl2=-2π2f2δ m=1M γkmm-l2gml,
m-lqml=QL-LQ,
h11=-4π2trLΓ-ΓLQL-LQ=-4π2trl-kγlkk-lqkl=-4π2k=1Ml=1Mk-l2qklγlk
h01=j2π trΓQL-LQΓG=j2π/δtrΓL-LΓ=0,
h12=-j2π3f2δ trΓQL-LQΓGL-LGL- LGL-LG=-j32π3f2trLΓLLQ-QLLΓL.
H-1=h000h020h110h200h22-1=1h11h00h22-h022×h11h220h11h020h00h22-h0220h11h020h00h11.
σBZ2δ=σBZ2δ|f1=1/h00-h022/h22,σBZ2δ|f2=σBZ2δ|f1, f2=1/h00,σBZ2f1=σBZ2f1|δ=σBZ2f1|f2=σBZ2f1|δ, f2=1/h11,σBZ2f2=σBZ2f2|f1=1/h22-h022/h00,σBZ2f2|δ=σBZ2f2|δ, f1=1/h22.
σBZ2f2|δσBZ2f2=σBZ2δ|f2σBZ2δ=1-h022h00h22.

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