Abstract

Deconvolution techniques are developed for improving lidar resolution when the sampling intervals are shorter than the sensing laser pulse. Such approaches permit the maximum-resolved lidar return in the case of arbitrary-shaped long laser pulses such as those used in CO2 lidars. The general algorithms are based on the Fourier-deconvolution technique as well as on the solution of the first kind of Volterra integral equation. In the case of rectangular pulses a simple and convenient recurrence algorithm is proposed and is analyzed in detail. The effect of stationary additive noise on algorithm performance is investigated. The theoretical analysis is supported by computer simulations demonstrating the increased resolution of the retrieved lidar profiles.

© 1993 Optical Society of America

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References

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  1. R. M. Measures, Laser Remote Sensing Fundamentals and Applications (Wiley, New York, 1984).
  2. P. W. Baker, “Atmospheric water vapor differential absorption measurements on vertical paths with a CO2lidar,” Appl. Opt. 22, 2257–2264 (1983).
    [CrossRef] [PubMed]
  3. M. J. Kavaya, R. T. Menzies, “Lidar aerosol backscattering measurements: systematic, modeling, and calibration error considerations,” Appl. Opt. 24, 3444–3453 (1985).
    [CrossRef] [PubMed]
  4. Y. Zhao, T. K. Lea, R. M. Schotland, “Correction function in the lidar equation and some techniques for incoherent CO2lidar data reduction,” Appl. Opt. 27, 2730–2740 (1988).
    [CrossRef] [PubMed]
  5. Y. Zhao, R. M. Hardesty, “Technique for correcting effects of long CO2laser pulses in aerosol-backscattered coherent lidar returns,” Appl. Opt. 27, 2719–2729 (1988).
    [CrossRef] [PubMed]
  6. K. J. Andrews, P. E. Dyer, D. J. James, “A rate equation model for the design of TEA CO2oscillators,” J. Phys. E 8, 493–497 (1975).
    [CrossRef]
  7. J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1981).
    [CrossRef] [PubMed]
  8. R. Gonzalez, “Recursive technique for inverting the lidar equation,” Appl. Opt. 27, 2741–2745 (1988).
    [CrossRef] [PubMed]
  9. V. Volterra, Theory of Functionals and of Integral and of Integro-Differential Equations (Dover, New York, 1958).
  10. I. V. Samokhvalov, “Double-scattering approximation of lidar equation for inhomogeneous atmosphere,” Opt. Lett. 4, 12–14 (1979).
    [CrossRef] [PubMed]
  11. M. R. Harris, D. V. Willetts, “Performance characteristics of a TE CO2laser with a long excitation pulse,” in Coherent Laser Radar: Technology and Applications, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 5–7.
  12. C. P. Hale, S. W. Henderson, J. R. Magee, S. R. Vetorino, “Compact high-energy Nd:YAG coherent laser radar transceiver,” in Coherent Laser Radar: Technology and Applications, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 133–135.
  13. F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956).

1988 (3)

1985 (1)

1983 (1)

1981 (1)

1979 (1)

1975 (1)

K. J. Andrews, P. E. Dyer, D. J. James, “A rate equation model for the design of TEA CO2oscillators,” J. Phys. E 8, 493–497 (1975).
[CrossRef]

Andrews, K. J.

K. J. Andrews, P. E. Dyer, D. J. James, “A rate equation model for the design of TEA CO2oscillators,” J. Phys. E 8, 493–497 (1975).
[CrossRef]

Baker, P. W.

Dyer, P. E.

K. J. Andrews, P. E. Dyer, D. J. James, “A rate equation model for the design of TEA CO2oscillators,” J. Phys. E 8, 493–497 (1975).
[CrossRef]

Gonzalez, R.

Hale, C. P.

C. P. Hale, S. W. Henderson, J. R. Magee, S. R. Vetorino, “Compact high-energy Nd:YAG coherent laser radar transceiver,” in Coherent Laser Radar: Technology and Applications, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 133–135.

Hardesty, R. M.

Harris, M. R.

M. R. Harris, D. V. Willetts, “Performance characteristics of a TE CO2laser with a long excitation pulse,” in Coherent Laser Radar: Technology and Applications, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 5–7.

Henderson, S. W.

C. P. Hale, S. W. Henderson, J. R. Magee, S. R. Vetorino, “Compact high-energy Nd:YAG coherent laser radar transceiver,” in Coherent Laser Radar: Technology and Applications, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 133–135.

Hildebrand, F. B.

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956).

James, D. J.

K. J. Andrews, P. E. Dyer, D. J. James, “A rate equation model for the design of TEA CO2oscillators,” J. Phys. E 8, 493–497 (1975).
[CrossRef]

Kavaya, M. J.

Klett, J. D.

Lea, T. K.

Magee, J. R.

C. P. Hale, S. W. Henderson, J. R. Magee, S. R. Vetorino, “Compact high-energy Nd:YAG coherent laser radar transceiver,” in Coherent Laser Radar: Technology and Applications, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 133–135.

Measures, R. M.

R. M. Measures, Laser Remote Sensing Fundamentals and Applications (Wiley, New York, 1984).

Menzies, R. T.

Samokhvalov, I. V.

Schotland, R. M.

Vetorino, S. R.

C. P. Hale, S. W. Henderson, J. R. Magee, S. R. Vetorino, “Compact high-energy Nd:YAG coherent laser radar transceiver,” in Coherent Laser Radar: Technology and Applications, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 133–135.

Volterra, V.

V. Volterra, Theory of Functionals and of Integral and of Integro-Differential Equations (Dover, New York, 1958).

Willetts, D. V.

M. R. Harris, D. V. Willetts, “Performance characteristics of a TE CO2laser with a long excitation pulse,” in Coherent Laser Radar: Technology and Applications, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 5–7.

Zhao, Y.

Appl. Opt. (6)

J. Phys. E (1)

K. J. Andrews, P. E. Dyer, D. J. James, “A rate equation model for the design of TEA CO2oscillators,” J. Phys. E 8, 493–497 (1975).
[CrossRef]

Opt. Lett. (1)

Other (5)

R. M. Measures, Laser Remote Sensing Fundamentals and Applications (Wiley, New York, 1984).

V. Volterra, Theory of Functionals and of Integral and of Integro-Differential Equations (Dover, New York, 1958).

M. R. Harris, D. V. Willetts, “Performance characteristics of a TE CO2laser with a long excitation pulse,” in Coherent Laser Radar: Technology and Applications, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 5–7.

C. P. Hale, S. W. Henderson, J. R. Magee, S. R. Vetorino, “Compact high-energy Nd:YAG coherent laser radar transceiver,” in Coherent Laser Radar: Technology and Applications, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 133–135.

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956).

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Figures (8)

Fig. 1
Fig. 1

Graph of the testing profile Φ given by Eq. (31), as a function of sample number; the relative width of the rectangular pulse with duration τ = 2T is given.

Fig. 2
Fig. 2

Original lidar returns F corresponding to the given testing profile Φ [Eq. (31)] for the rectangular pulse with duration τ = 2T (dashed curve) and for the pulse shape f(t) given by Eq. (33) (solid curve).

Fig. 3
Fig. 3

Graph of the pulse shape f(t) given by Eq. (33).

Fig. 4
Fig. 4

Theoretically predicted (solid curve) and calculated (separate points) systematic errors versus sample number for the cases of (a) Fourier deconvolution, (b) Volterra deconvolution, and (c) rectangular pulse. (d) Curves 1, 2, and 3 show the calculated relative errors δ/Φ for curves (a), (b), and (c), respectively. The computing step Δt = 4Δt0. a.u. is arbitrary units.

Fig. 5
Fig. 5

Profile Φc versus sample number, restored by Fourier deconvolution in the presence of white noise with SNR0in = 10, for (a) Δt = Δt0, (b) Δt = 4Δt0. The profile Φ is given for comparison.

Fig. 6
Fig. 6

Φc versus sample number in the case of a rectangular pulse with duration τ = 2T in the presence of white noise with SNR0in = 10, for (a) Δt = Δt0 and (b) Δt = 4Δt0. The profile Δ is also given.

Fig. 7
Fig. 7

Graph of the testing profile Φ1 as a function of sample number.

Fig. 8
Fig. 8

Profile Φc1 versus sample number, restored by Fourier deconvolution in the presence of white noise for SNR0in = 10 and Δt = 4Δt0: (a) without filtering, (b) with filtering, (c) for SNR0in = 2 and (d) for SNR0in = 1 with preliminary filtering and the step Δt = 8Δt0. The profile Φ1 is also represented.

Tables (1)

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Table 1 Model Parameters of Short-Pulse Profiles and Pulse Shape

Equations (62)

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F ( t ) = c ( t - τ ) / 2 c t / 2 f ( t - 2 z / c ) Φ ( z ) d z ,
F N ( t ) = K Φ ( z c t / 2 ) , K = ( c / 2 ) 0 τ f ( t ) d t ,
F ( t ) = ( c / 2 ) 0 μ f ( θ ) Φ [ c ( t - θ ) / 2 ] d θ ,
F ˜ ( ω ) = f ˜ ( ω ) Φ ˜ ( k ) ,
F ˜ ( ω ) = - F ( t ) exp ( j ω t ) d t ,
f ˜ ( ω ) = - f ( t ) exp ( j ω t ) d t = 0 τ f ( t ) exp ( j ω t ) d t ,
Φ ˜ ( k ) = - Φ ( z ) exp ( j k z ) d z = z 0 Φ ( z ) exp ( j k z ) d z
Φ ( z ) = ( 2 π ) - 1 - Φ ˜ ( k ) exp ( - j k z ) d k = ( π c ) - 1 - [ F ˜ ( ω ) / f ˜ ( ω ) ] exp ( - j 2 ω z / c ) d ω Φ ( t ) = ( π c ) - 1 - [ F ˜ ( ω ) / f ˜ ( ω ) ] exp ( - j ω t ) d ω .
δ ( z ) = Φ c ( z ) - Φ ( z ) κ Φ II ( z ) ( Δ z ) 2 ,
ɛ ( t ) = ( π c ) - 1 - [ n ˜ ( ω ) / f ˜ ( ω ) ] exp ( - j ω t ) d ω ,
n ˜ ( ω ) = - t l t l n ( t ) exp ( j ω t ) d t .
D ɛ = ( π c ) - 2 2 π - [ I n ( ω ) / I f ( ω ) ] d ω ,
D ɛ = ( π c ) - 2 2 π - π / Δ t π / Δ t [ I n ( ω ) / I f ( ω ) ] d ω .
Δ u ( z ) = Φ r ( z ) - Φ ( z ) = - Φ ˜ ( ω ) Δ ˜ f ( ω ) [ f ˜ ( ω ) + Δ ˜ ( ω ) ] - 1 exp ( - j ω t ) d ω ,
F ( t ) = z 0 c t / 2 f ( t - 2 z / c ) Φ ( z ) d z .
Φ ( c t / 2 ) = F ( t ) = t 0 t K ( t - t ) Φ ( c t / 2 ) d t ,
Φ ( z = c t / 2 ) = F ( t ) + 0 t - t 0 R ( ζ ) F ( t - ζ ) d ζ ,
δ ( z = c t / 2 ) = - ( 2 / 30 ) ( Δ t ) 4 [ Φ IV ( t ) - F I ( t 0 ) R II ( t - t 0 ) - F II ( t 0 ) R I ( t - t 0 ) - F III ( t 0 ) R ( t - t 0 ) ] .
ɛ ( t ) = l ( t ) + 0 t - t 0 R ( ζ ) l ( t - ζ ) d ζ ,
D ɛ = ɛ ( t ) 2 = D l [ 1 + 2 0 t - t 0 R ( ζ ) K l ( ζ ) d ζ + t - t 0 0 R ( ζ ) R ( ζ ) K l ( ζ - ζ ) d ζ d ζ ] ,
cov l c ( θ ) ( 1.12 E ) 2 D n K n ( θ ) / ( Δ t ) 4 .
K l ( θ ) = K n IV ( θ ) / K n IV ( 0 ) ,             D l = E 2 D n K n IV ( 0 ) ,
K l c ( θ ) = K n ( θ ) ,             D l c = ( 1.12 E ) 2 D n / ( Δ t ) 4 ,
( 2 / c ) F I ( t ) + Φ [ c ( t - τ ) / 2 ] = Φ ( c t / 2 ) ,
Φ ( c t / 2 ) = 2 c i = 0 Q F I ( t - i τ ) + Φ { c [ t - ( Q + 1 ) τ ] / 2 } ,
δ ( t = 2 z / c ) = Φ c ( c t / 2 ) - Φ ( c t / 2 ) = 2 c i = 0 Q [ F c I ( t - i τ ) - F I ( t - i τ ) ] = - ( 2 / 30 c ) ( Δ t ) 4 i = 0 Q F V ( t - i τ ) ,
F V ( t ) = ( c / 2 ) 5 { Φ IV ( c t / 2 ) - Φ IV [ c ( t - τ ) / 2 ] } ,
δ ( t ) = - ( 1 / 30 ) ( c Δ t / 2 ) 4 Φ IV ( c t / 2 ) .
ɛ ( t ) = 2 c i = 0 Q n I ( t - i τ ) ,
D ɛ ( t ) = ɛ 2 ( t ) = ( 4 / c 2 ) ( Q + 1 ) D n I + ( 4 / c 2 ) i q = 0 Q cov n I [ ( q - i ) τ ] ,
D n I = [ n I ( t ) ] 2 = - D n K n II ( θ ) θ = 0 , cov n I [ ( q - i ) τ ] = n I ( t - i τ ) n I ( t - q τ ) = - D n K n II ( θ ) θ = ( q - i ) τ .
D ɛ = - ( 4 / c 2 ) ( Q + 1 ) D n K n II ( θ ) θ = 0 .
D ɛ = D ɛ c 0.9 ( 4 / c 2 ) ( Q + 1 ) D n / ( Δ t ) 2 .
Δ u ( z = c t / 2 ) = Φ r ( z ) - Φ ( z ) = - [ φ a ( t ) + φ a ( t - τ ) + φ a ( t - 2 τ ) + ] = Δ f Φ ( z = c t / 2 ) .
Δ u ( z = c t / 2 ) = Δ f ( 0 ) Φ ( c t / 2 ) - [ Δ f ( τ ) - Δ f ( 0 ) ] × i = 1 Q Φ [ c ( t - i τ ) ] + ( 2 / c ) i = 0 Q Δ I ( ξ i ) F ( t - i τ ) ,
Φ ( z = c t / 2 ) = W { 0 for t t 0 , A ( t - t 0 ) - 3 exp [ - G / ( t - t 0 ) ] + C sin 2 [ 2 π ( t - t 0 ) / T ] for t 0 < t G + t 0 , A ( t - t 0 ) - 3 exp [ - G / ( t - t 0 ) ] for G + t 0 < t t a , A ( t - t 0 ) - 3 exp [ - G / ( t - t 0 ) ] + B + b 2 - ( t - t a - b T 0 ) 2 / T 0 2 for t a < t t a + 2 b T 0 , A ( t - t 0 ) - 3 exp [ - G / ( t - t 0 ) ] + B + b 2 - ( t - t a - 3 b T 0 ) 2 / T 0 2 for t a + 2 b T 0 < t t a + 4 b T 0 , A ( t - t 0 ) - 3 exp [ - G / ( t - t 0 ) ] for t > t a + 4 b T 0 ,
F ( t ) = W c 2 { 0 for t t 0 , H ( t ) + C ( t - t 0 ) / 2 - S ( t ) for t 0 < t t 0 + τ , L ( t ) + C τ / 2 - M ( t ) for t 0 + τ < t t 0 + G , L ( t ) + C ( t 0 + G + τ - t ) / 2 - ( C T / 8 π ) sin ( 4 π G / T ) + S ( t - τ ) for t 0 + G < t < t 0 + G + τ , L ( t ) for t 0 + G + τ t < t a , L ( t ) + N ( t - t a ) for t a < t t a + 2 b T 0 , L ( t ) + N ( t - t a - 2 b T 0 ) + N ( 2 b T 0 ) for t a + 2 b T 0 < t t a + 4 b T 0 , L ( t ) + 2 N ( 2 b T 0 ) for t a + 4 b T 0 < t < t a + τ , L ( t ) + N ( t a + τ + 2 b T 0 - t ) + N ( 2 b T 0 ) for t a + τ < t t a + τ + 4 b T 0 , L ( t ) for t a + τ + 4 b T 0 < t ,
f ( t ) = t [ ( 2 e / τ 1 ) exp ( - t 2 / τ 1 2 ) + ( x e / τ 2 ) exp ( - t / τ 2 ) ] / f p ,
5 ( D ɛ ) 1 / 2 = A l
Λ S = Λ SF = 125 SNR PF - 2 Λ c d ,
Λ S = Λ SV = 12 ( SNR PV ) - 1 / 2 Λ e ,
Λ S = Λ SR = 25 ( SNR PR ) - 1 Λ ;
F N ( t 2 z / c ) = F N 1 + F N 2 = K [ Φ ( z ) + Γ ( z ) ] ,
Φ c ( z ) = ( π c ) - 1 - π / Δ t π / Δ t [ F ˜ ( ω ) / f ˜ ( ω ) ] exp ( - j ω t ) d ω = Φ c ( t ) ;
F ˜ ( ω ) / f ˜ ( ω ) = ( c / 2 ) - Φ ( t ) exp ( j ω t ) d t .
Φ c ( t ) = π - 1 - Φ ( t ) { sin [ ( t - t ) π Δ t ] / ( t - t ) } d t .
δ ( t ) = Φ c ( t ) - Φ ( t ) = π - 1 - [ Φ ( t + x Δ t π ) - Φ ( t ) ] ( sin x / x ) d x ,
δ ( t ) = κ Φ II ( t ) ( Δ t ) 2 .
κ = 1 2 π 3 - x sin x { 1 + 1 12 Φ IV ( t ) Φ II ( t ) ( x Δ t π ) 2 + } d x
δ ( z ) = δ ( z = c t / 2 ) = κ Φ II ( z ) Δ z 2 .
f c I ( t ) = 1 12 Δ t [ f ( t - 2 Δ t ) - 8 f ( t - Δ t ) + 8 f ( t + Δ t ) - f ( t + 2 Δ t ) ] .
f c II ( t ) = 1 / ( 12 Δ t ) 2 { f ( t - 4 Δ t ) + f ( t + 4 Δ t ) - 16 [ f ( t - 3 Δ t ) + f ( t + 3 Δ t ) ] + 64 [ f ( t - 2 Δ t ) + f ( t + 2 Δ t ) ] + 16 [ f ( t - Δ t ) + f ( t + Δ t ) ] - 130 f ( t ) } .
f c I ( t ) = f I ( t ) = - 1 30 f V ( t ) ( Δ t ) 4 .
f c II ( t ) - f II ( t ) = - 2 30 f VI ( t ) ( Δ t ) 4 .
cov f I ( θ ) = - d 2 cov f ( θ ) d θ 2 = - D f K II ( θ ) ,
D f I = f 12 = cov f I ( 0 ) = - D f K f II ( θ ) θ = 0 .
cov f I c ( θ ) 0.9 cov f ( θ ) / ( Δ t ) 2 = 0.9 D f K f ( θ ) / ( Δ t ) 2 .
D c f I = cov f I c ( 0 ) 0.9 D f / ( Δ t ) 2 .
cov f II ( θ ) = D f K f IV ( θ ) ,
D f II = cov f II ( 0 ) = D f K f IV ( θ ) θ = 0 ,
cov f II c ( θ ) 1.25 D f K f ( θ ) / ( Δ t ) 4 ,
D c f II 1.25 D f / ( Δ t ) 4 .

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