Abstract

Properties of small sample estimators for the return signal power ratio or log ratio in direct detection incoherent backscatter lidar systems are analyzed. As for heterodyne receivers it is usually preferable to form an estimator from the logarithmic difference of the sample averages rather than their ratio. Calculated values of bias and noise figures are confirmed using simulated data based on constant signal models and compared with the estimates obtained from nonlinear Kalman filters. The latter generally provide the least bias at high noise levels at the cost of greater computational complexity.

© 1989 Optical Society of America

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References

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  1. B. J. Rye, “Power Ratio Estimation in Incoherent Backscatter Lidar: Heterodyne Receiver with Square Law Detection,” J. Climate Appl. Meteorol. 22, 1899–1913 (1983).
    [CrossRef]
  2. B. J. Rye, R. M. Hardesty, “Time Series Identification and Kalman Filtering Techniques for Doppler Lidar Velocity Estimation,” Appl. Opt. 28, 879–891 (1989).
    [CrossRef] [PubMed]
  3. R. E. Warren, “Detection and Discrimination Using Multiple-Wavelength Differential Absorption Lidar,” Appl. Opt. 24, 3541–3545 (1985).
    [CrossRef] [PubMed]
  4. B. J. Rye, R. M. Hardesty, “Nonlinear Kalman Filtering Techniques for Incoherent Backscatter Lidar: Return Power and Log Power Estimation,” Appl. Opt.28, to be published (1989).
    [CrossRef] [PubMed]
  5. J. W. van Dijk, J. F. Kusters, A. Layfield, B. J. Rye, “The Use of TEA and Multiatmosphere CO2 Lasers in Active Remote Sensing,” in Proceedings, ESA Workshop on Space Laser Applications and Technology, ESA SP-202, Les Diablerets, 225–30 (1984).
  6. A. Layfield, B. J. Rye, J. W. van Dijk, “Application of Optimal Estimation Techniques in DIAL,” in Proceedings, Third International Topical Meeting on Coherent Lidar: Technology and Applications, Malvern (1985).
  7. A. Layfield, B. J. Rye, “Software Filtering of Differential Absorption Lidar Returns,” in Proceedings, Workshop on DIAL Data Collection and Analysis, Virginia Beach (Nov.1985), to be published as a NASA report.
  8. J. W. Goodman, “Some Effects of Target-Induced Scintillation on Optical Radar Performance,” Proc. IEEE 53, 1688–1700 (1965).
    [CrossRef]
  9. M. Elbaum, P. Diamant, “Signal-to-Noise Ratio in Photo-counting Images of Rough Objects in Partially Coherent Light,” Appl. Opt. 15, 2268–2275 (1976).
    [CrossRef] [PubMed]
  10. J. H. Shapiro, “Target Detection with a Direct-Reception Optical Radar,” MIT Lincoln Laboratory Report TST-27 (1978).
  11. T. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, translated by A. Jeffrey (Academic, New York, 1965).
  12. A. P. Sage, J. L. Melsa, Estimation Theory with Applications to Communications and Control (McGraw-Hill, New York, 1971).
  13. A. Gelb, Ed., Applied Optimal Estimation (MIT Press, Cambridge, MA, 1974).

1989 (1)

1985 (1)

1983 (1)

B. J. Rye, “Power Ratio Estimation in Incoherent Backscatter Lidar: Heterodyne Receiver with Square Law Detection,” J. Climate Appl. Meteorol. 22, 1899–1913 (1983).
[CrossRef]

1976 (1)

1965 (1)

J. W. Goodman, “Some Effects of Target-Induced Scintillation on Optical Radar Performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

Diamant, P.

Elbaum, M.

Goodman, J. W.

J. W. Goodman, “Some Effects of Target-Induced Scintillation on Optical Radar Performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

Gradshteyn, T. S.

T. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, translated by A. Jeffrey (Academic, New York, 1965).

Hardesty, R. M.

B. J. Rye, R. M. Hardesty, “Time Series Identification and Kalman Filtering Techniques for Doppler Lidar Velocity Estimation,” Appl. Opt. 28, 879–891 (1989).
[CrossRef] [PubMed]

B. J. Rye, R. M. Hardesty, “Nonlinear Kalman Filtering Techniques for Incoherent Backscatter Lidar: Return Power and Log Power Estimation,” Appl. Opt.28, to be published (1989).
[CrossRef] [PubMed]

Kusters, J. F.

J. W. van Dijk, J. F. Kusters, A. Layfield, B. J. Rye, “The Use of TEA and Multiatmosphere CO2 Lasers in Active Remote Sensing,” in Proceedings, ESA Workshop on Space Laser Applications and Technology, ESA SP-202, Les Diablerets, 225–30 (1984).

Layfield, A.

J. W. van Dijk, J. F. Kusters, A. Layfield, B. J. Rye, “The Use of TEA and Multiatmosphere CO2 Lasers in Active Remote Sensing,” in Proceedings, ESA Workshop on Space Laser Applications and Technology, ESA SP-202, Les Diablerets, 225–30 (1984).

A. Layfield, B. J. Rye, J. W. van Dijk, “Application of Optimal Estimation Techniques in DIAL,” in Proceedings, Third International Topical Meeting on Coherent Lidar: Technology and Applications, Malvern (1985).

A. Layfield, B. J. Rye, “Software Filtering of Differential Absorption Lidar Returns,” in Proceedings, Workshop on DIAL Data Collection and Analysis, Virginia Beach (Nov.1985), to be published as a NASA report.

Melsa, J. L.

A. P. Sage, J. L. Melsa, Estimation Theory with Applications to Communications and Control (McGraw-Hill, New York, 1971).

Rye, B. J.

B. J. Rye, R. M. Hardesty, “Time Series Identification and Kalman Filtering Techniques for Doppler Lidar Velocity Estimation,” Appl. Opt. 28, 879–891 (1989).
[CrossRef] [PubMed]

B. J. Rye, “Power Ratio Estimation in Incoherent Backscatter Lidar: Heterodyne Receiver with Square Law Detection,” J. Climate Appl. Meteorol. 22, 1899–1913 (1983).
[CrossRef]

J. W. van Dijk, J. F. Kusters, A. Layfield, B. J. Rye, “The Use of TEA and Multiatmosphere CO2 Lasers in Active Remote Sensing,” in Proceedings, ESA Workshop on Space Laser Applications and Technology, ESA SP-202, Les Diablerets, 225–30 (1984).

A. Layfield, B. J. Rye, “Software Filtering of Differential Absorption Lidar Returns,” in Proceedings, Workshop on DIAL Data Collection and Analysis, Virginia Beach (Nov.1985), to be published as a NASA report.

A. Layfield, B. J. Rye, J. W. van Dijk, “Application of Optimal Estimation Techniques in DIAL,” in Proceedings, Third International Topical Meeting on Coherent Lidar: Technology and Applications, Malvern (1985).

B. J. Rye, R. M. Hardesty, “Nonlinear Kalman Filtering Techniques for Incoherent Backscatter Lidar: Return Power and Log Power Estimation,” Appl. Opt.28, to be published (1989).
[CrossRef] [PubMed]

Ryzhik, I. M.

T. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, translated by A. Jeffrey (Academic, New York, 1965).

Sage, A. P.

A. P. Sage, J. L. Melsa, Estimation Theory with Applications to Communications and Control (McGraw-Hill, New York, 1971).

Shapiro, J. H.

J. H. Shapiro, “Target Detection with a Direct-Reception Optical Radar,” MIT Lincoln Laboratory Report TST-27 (1978).

van Dijk, J. W.

A. Layfield, B. J. Rye, J. W. van Dijk, “Application of Optimal Estimation Techniques in DIAL,” in Proceedings, Third International Topical Meeting on Coherent Lidar: Technology and Applications, Malvern (1985).

J. W. van Dijk, J. F. Kusters, A. Layfield, B. J. Rye, “The Use of TEA and Multiatmosphere CO2 Lasers in Active Remote Sensing,” in Proceedings, ESA Workshop on Space Laser Applications and Technology, ESA SP-202, Les Diablerets, 225–30 (1984).

Warren, R. E.

Appl. Opt. (3)

J. Climate Appl. Meteorol. (1)

B. J. Rye, “Power Ratio Estimation in Incoherent Backscatter Lidar: Heterodyne Receiver with Square Law Detection,” J. Climate Appl. Meteorol. 22, 1899–1913 (1983).
[CrossRef]

Proc. IEEE (1)

J. W. Goodman, “Some Effects of Target-Induced Scintillation on Optical Radar Performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

Other (8)

J. H. Shapiro, “Target Detection with a Direct-Reception Optical Radar,” MIT Lincoln Laboratory Report TST-27 (1978).

T. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, translated by A. Jeffrey (Academic, New York, 1965).

A. P. Sage, J. L. Melsa, Estimation Theory with Applications to Communications and Control (McGraw-Hill, New York, 1971).

A. Gelb, Ed., Applied Optimal Estimation (MIT Press, Cambridge, MA, 1974).

B. J. Rye, R. M. Hardesty, “Nonlinear Kalman Filtering Techniques for Incoherent Backscatter Lidar: Return Power and Log Power Estimation,” Appl. Opt.28, to be published (1989).
[CrossRef] [PubMed]

J. W. van Dijk, J. F. Kusters, A. Layfield, B. J. Rye, “The Use of TEA and Multiatmosphere CO2 Lasers in Active Remote Sensing,” in Proceedings, ESA Workshop on Space Laser Applications and Technology, ESA SP-202, Les Diablerets, 225–30 (1984).

A. Layfield, B. J. Rye, J. W. van Dijk, “Application of Optimal Estimation Techniques in DIAL,” in Proceedings, Third International Topical Meeting on Coherent Lidar: Technology and Applications, Malvern (1985).

A. Layfield, B. J. Rye, “Software Filtering of Differential Absorption Lidar Returns,” in Proceedings, Workshop on DIAL Data Collection and Analysis, Virginia Beach (Nov.1985), to be published as a NASA report.

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Figures (6)

Fig. 1
Fig. 1

Block diagrams of two channel models for estimation of (i) constant ratio x1 and (ii) constant log ratio x1 from measurements y1,2 in the presence of constant reference signal x2, speckle s1,2, and additive measurement noise υ1,2.

Fig. 2
Fig. 2

Bias and noise figures of estimators as a function of SNR. (i) Bias of the z estimator: 1,1/m = 0, δ1 = 0 dB; 2, m = 10, δ1 = 0 dB; 3,1/m = 0, δ1 = 6 dB; 4, m = 10, δ1 = 6 dB. (ii) Bias of the F estimator: 1, 1/m = 0, δ1 = 0 dB, f = 0.1; 2, l/m = 0, δ1 = 0 dB, f = 0.01; 3, m = 10, δ1 = 0 dB, f = 0.1; 4, l/m = 0, δ1 = 6 dB, f = 0.1; 5, l/m = 0, δ1 = 6 dB, f = 0.01; 6, m = 10, δ1 = 6 dB, f = 0.1. (iii) Noise figures N L (—) and N L + N eff (— — —) of the z estimator: 1, 1/m = 0, δ1 = 0 dB; 2, m = 10, δ1 = 0 dB; 3, 1/m = 0, δ1 = 6 dB; 4, m = 10, δ1 = 6 dB. (iv) Noise figures N R (—) and N R + N eff (— — —) of the F estimator: l, l/m = 0, δ1 = 0 dB, f = 0.1; 2, l/m = 0, δ1 = 0 dB, f = 0.01; 3, m = 10, δ1 = 0 dB,f = 0.1. (v) Noise figures N R (—) and N R + N eff (— — —) of the F estimator: 1, 1/m = 0, δ1 = 6 dB, f = 0.1; 2, 1/m = 0, δ1 = 6 dB, f = 0.01; 3, m = 10, δ1 = 6 dB, f = 0.1.

Fig. 3
Fig. 3

Generalized curves for computing bias and noise figures of the z estimator as a function of numerator S/N δ1. (i) Bias curve: 1, 1/m = 0; 2, m = 10; 3, m = 100. (ii) Noise figure curves for Nl (—) and N L + N eff (— — —) as in (i).

Fig. 4
Fig. 4

Noise figures for N L , F (—) and N L , F + Neff (— — —) of estimators as a function of numerator S/N δ1. (i) z estimator: 1,1/m = 0, ρ = 0.9; 2,1/m = 0, ρ= 0.1;3, m = 10, ρ = 0.9;4, m = 10, ρ = 0.1. (ii)F estimator 1/m = 0: l, ρ = 0.9, f = 0.1;2, ρ = 0.1, f = 0.1;3, ρ = 0.9, f = 0.01;4, ρ = 0.1, f = 0.01. (iii)F estimator m = 10: l, ρ = 0.9, f = 0.1; 2, ρ = 0.1, f = 0.1; 3, ρ = 0.9, f = 0.01.

Fig. 5
Fig. 5

Bias of estimators as a function of numerator S/N δ1 Calculated values for F(— — —) and z (—) estimators are compared with results obtained from simulations using F (circles), z (triangles), and filter (crosses) algorithms; bars indicate standard deviation of sample estimates for the latter where these are significant: (i) 1/m = 0, ρ = 0.1; (ii) 1/m = 0, ρ = 0.9; (iii) m = 10, ρ = 0.1; (iv) m = 10, ρ = 0.9.

Fig. 6
Fig. 6

Bias and noise figures for exponentially distributed speckle noise as a function of numerator S/N δ1. Calculated values for the z estimator are presented as lines and values obtained from simulations as in Fig. 5: (i) bias, ρ = 0.1; (ii) bias, ρ = 0.9; (iii) noise figures N L (—) and N L + N eff (— — —): 1, ρ = 0.1; 2, ρ = 0.9.

Tables (1)

Tables Icon

Table I Limiting S/N in the Ratio Numerator δ1 for the Absolute Bias to be Below Bmin, Determined for Fand z Estimators

Equations (32)

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var ( I N ) = I N = N 2 .
var ( I ) speckle = I s 2 / m = S 2 / m ,
var ( I ) shot noise = S 2 / n ,
n var ( I ) = var ( I ) / S 2 = 1 / δ 2 + 1 / n + 1 / m ,
n var ( I ) = m ( 1 / n + 1 / m ) 2 = ( 1 / m ) ( 1 + 1 / δ ) 2
var ( I ) = n + n 2 / m + n a + n b + n b 2 / m ,
SNR = 1 / δ s 2 = ( 1 / δ 2 + 1 / m ) 1 , ( SNR ) sat = m , CNR = δ 2
p ( I ) = 1 ( 2 π ) 1 / 2 N exp [ ( I S ) 2 / ( 2 N 2 ) ] .
ω = I S = I / N δ .
p ( ω | δ ) = δ ( 2 π ) 1 / 2 exp [ δ ( ω 1 ) 2 / 2 ] .
p ( I ) = ( m / S ) m I m 1 exp [ m I / S ] / Γ ( m ) , I > 0 ,
p ( ω | δ ) = m m ω m 1 exp [ m ω ] / Γ ( m ) , ω > 0 .
p ( I ) = 1 [ 2 π ( N 2 + S 2 / m ) ] 1 / 2 exp [ ( I S ) 2 2 ( N 2 + S 2 / m ) ] ,
p ( ω | δ ) = δ s / ( 2 π ) 1 / 2 exp [ δ s 2 ( ω 1 ) 2 / 2 ] .
p ( I ) = ( 1 / S ) exp [ I / S ] , I > 0 ,
p ( I ) = 1 ( 2 π ) 1 / 2 S N | exp [ I / S ] exp [ ( I I ) 2 / ( 2 N 2 ) ] d I ,
p ( ω | δ ) = 1 2 exp [ ( ω 1 / δ 2 ) ] erfc [ δ ( ω 1 / δ 2 ) / 2 1 / 2 ] ,
erfc [ x ] = ( 2 / ( π ) 1 / 2 ) x exp [ t 2 ] d t
ρ = δ 1 / δ 2 .
R * | δ 1 , ρ = ρ ω 1 / ω 2 | δ 1 , ρ ,
L * | δ 1 , ρ = λ + ln ( ω 1 / ω 2 ) .
B R = ω 1 / ω 2 | δ 1 , ρ 1 ,
B L = ln ( ω 1 ) | δ 1 ln ( ω 2 ) | ( δ 1 / ρ ) .
N R * = 10 log 10 { [ n var ( R * / δ 1 , ρ ) ] / 2 } ,
N L * = 10 log 10 { [ n var ( L * / δ 1 , ρ ) ] / 2 } ,
2 = 2 / m + 1 / δ 1 2 + 1 / δ 2 2
P ( ω / δ ) = { a p ( ω / δ ) , ω > = ω min , 0 , otherwise
a = 1 / ω min p ( ω | δ ) d ω
N eff * ( dB ) = 10 log 10 ( a 1 a 2 )
y ( k ) = h [ x ( k ) ] + v ( k ) ,
ln ( ω / δ ) = ln I ln S ,
ln 2 ( ω / δ ) ln ( ω / δ ) 2 = ln 2 I ( ln S ) 2 .

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