Abstract

The inversion of lidar returns from homogeneous atmospheres has been done customarily through the well-known slope method. The logarithmic operation over the range-corrected and system-normalized received signal used in this method introduces a bias in the statistics of the noise-affected processed signal that can severely distort the estimates of the atmospheric attenuation and backscatter coefficients under measurement. It is shown that a fitting of the theoretically expected exponential signal to the range-corrected received one, using as the initial guess the results provided by the slope method and a least-squares iterative procedure, can yield enhanced accuracy under low signal-to-noise ratios and especially in moderate-to-high extinction conditions.

© 1998 Optical Society of America

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References

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  1. R. T. H. Collis, P. B. Russell, “Lidar measurement of particles and gases by elastic backscattering and differential absorption,” in Laser Monitoring of the Atmosphere, E. D. Hinkley, ed. (Springer-Verlag, New York, 1976), Chap. 4, pp. 71–102.
    [CrossRef]
  2. J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1985).
    [CrossRef]
  3. G. J. Kunz, G. de Leeuw, “Inversion of lidar signals with the slope method,” Appl. Opt. 32, 3249–3256 (1993).
    [CrossRef] [PubMed]
  4. M. E. Tiuri, “Radio astronomy receivers,” IEEE Trans. Antennas Propag. AP-12, 930–938 (1964).
    [CrossRef]
  5. A. B. Carlson, “Signal transmission and filtering,” in Communication Systems, 3rd ed. (McGraw-Hill, Singapore, 1986), Chap. 3, pp. 177–178.
  6. R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Krieger, Malabar, Fla., 1992), Chap. 4, pp. 138–145.
  7. H. Koschmieder, “Theorie der horizontalen Sichtweite,” Beitr. Phys. Freien Atmos. 12, 33–53 (1925).
  8. P. W. Kruse, L. D. McGlauchlin, R. B. McQuiston, Elements of Infrared Technology: Generation, Transmission and Detection (Wiley, New York, 1962).
  9. R. T. H. Collis, “Lidar: a new atmospheric probe,” Q. J. R. Meteorol. Soc. 92, 220–230 (1966).
    [CrossRef]
  10. R. J. Barlow, Statistics (Wiley, New York, 1989), Chap. 6.
  11. J. J. More, “The Levenberg–Marquardt algorithm: implementation and theory,” in Numerical Analysis, Lecture Notes in Mathematics630, G. A. Watson, ed. (Springer-Verlag, New York, 1977), pp. 105–116.
  12. W. B. Jones, Introduction to Optical Fiber Communication Systems, (Holt, Rinehart & Winston, New York, 1988), Chaps. 7, 8.
  13. R. J. McIntyre, “Multiplication noise in uniform avalanche diodes,” IEEE Trans. Electron Devices ED-13, 164–168 (1966).
    [CrossRef]

1993 (1)

1985 (1)

1966 (2)

R. T. H. Collis, “Lidar: a new atmospheric probe,” Q. J. R. Meteorol. Soc. 92, 220–230 (1966).
[CrossRef]

R. J. McIntyre, “Multiplication noise in uniform avalanche diodes,” IEEE Trans. Electron Devices ED-13, 164–168 (1966).
[CrossRef]

1964 (1)

M. E. Tiuri, “Radio astronomy receivers,” IEEE Trans. Antennas Propag. AP-12, 930–938 (1964).
[CrossRef]

1925 (1)

H. Koschmieder, “Theorie der horizontalen Sichtweite,” Beitr. Phys. Freien Atmos. 12, 33–53 (1925).

Barlow, R. J.

R. J. Barlow, Statistics (Wiley, New York, 1989), Chap. 6.

Carlson, A. B.

A. B. Carlson, “Signal transmission and filtering,” in Communication Systems, 3rd ed. (McGraw-Hill, Singapore, 1986), Chap. 3, pp. 177–178.

Collis, R. T. H.

R. T. H. Collis, “Lidar: a new atmospheric probe,” Q. J. R. Meteorol. Soc. 92, 220–230 (1966).
[CrossRef]

R. T. H. Collis, P. B. Russell, “Lidar measurement of particles and gases by elastic backscattering and differential absorption,” in Laser Monitoring of the Atmosphere, E. D. Hinkley, ed. (Springer-Verlag, New York, 1976), Chap. 4, pp. 71–102.
[CrossRef]

de Leeuw, G.

Jones, W. B.

W. B. Jones, Introduction to Optical Fiber Communication Systems, (Holt, Rinehart & Winston, New York, 1988), Chaps. 7, 8.

Klett, J. D.

Koschmieder, H.

H. Koschmieder, “Theorie der horizontalen Sichtweite,” Beitr. Phys. Freien Atmos. 12, 33–53 (1925).

Kruse, P. W.

P. W. Kruse, L. D. McGlauchlin, R. B. McQuiston, Elements of Infrared Technology: Generation, Transmission and Detection (Wiley, New York, 1962).

Kunz, G. J.

McGlauchlin, L. D.

P. W. Kruse, L. D. McGlauchlin, R. B. McQuiston, Elements of Infrared Technology: Generation, Transmission and Detection (Wiley, New York, 1962).

McIntyre, R. J.

R. J. McIntyre, “Multiplication noise in uniform avalanche diodes,” IEEE Trans. Electron Devices ED-13, 164–168 (1966).
[CrossRef]

McQuiston, R. B.

P. W. Kruse, L. D. McGlauchlin, R. B. McQuiston, Elements of Infrared Technology: Generation, Transmission and Detection (Wiley, New York, 1962).

Measures, R. M.

R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Krieger, Malabar, Fla., 1992), Chap. 4, pp. 138–145.

More, J. J.

J. J. More, “The Levenberg–Marquardt algorithm: implementation and theory,” in Numerical Analysis, Lecture Notes in Mathematics630, G. A. Watson, ed. (Springer-Verlag, New York, 1977), pp. 105–116.

Russell, P. B.

R. T. H. Collis, P. B. Russell, “Lidar measurement of particles and gases by elastic backscattering and differential absorption,” in Laser Monitoring of the Atmosphere, E. D. Hinkley, ed. (Springer-Verlag, New York, 1976), Chap. 4, pp. 71–102.
[CrossRef]

Tiuri, M. E.

M. E. Tiuri, “Radio astronomy receivers,” IEEE Trans. Antennas Propag. AP-12, 930–938 (1964).
[CrossRef]

Appl. Opt. (2)

Beitr. Phys. Freien Atmos. (1)

H. Koschmieder, “Theorie der horizontalen Sichtweite,” Beitr. Phys. Freien Atmos. 12, 33–53 (1925).

IEEE Trans. Antennas Propag. (1)

M. E. Tiuri, “Radio astronomy receivers,” IEEE Trans. Antennas Propag. AP-12, 930–938 (1964).
[CrossRef]

IEEE Trans. Electron Devices (1)

R. J. McIntyre, “Multiplication noise in uniform avalanche diodes,” IEEE Trans. Electron Devices ED-13, 164–168 (1966).
[CrossRef]

Q. J. R. Meteorol. Soc. (1)

R. T. H. Collis, “Lidar: a new atmospheric probe,” Q. J. R. Meteorol. Soc. 92, 220–230 (1966).
[CrossRef]

Other (7)

R. J. Barlow, Statistics (Wiley, New York, 1989), Chap. 6.

J. J. More, “The Levenberg–Marquardt algorithm: implementation and theory,” in Numerical Analysis, Lecture Notes in Mathematics630, G. A. Watson, ed. (Springer-Verlag, New York, 1977), pp. 105–116.

W. B. Jones, Introduction to Optical Fiber Communication Systems, (Holt, Rinehart & Winston, New York, 1988), Chaps. 7, 8.

A. B. Carlson, “Signal transmission and filtering,” in Communication Systems, 3rd ed. (McGraw-Hill, Singapore, 1986), Chap. 3, pp. 177–178.

R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Krieger, Malabar, Fla., 1992), Chap. 4, pp. 138–145.

P. W. Kruse, L. D. McGlauchlin, R. B. McQuiston, Elements of Infrared Technology: Generation, Transmission and Detection (Wiley, New York, 1962).

R. T. H. Collis, P. B. Russell, “Lidar measurement of particles and gases by elastic backscattering and differential absorption,” in Laser Monitoring of the Atmosphere, E. D. Hinkley, ed. (Springer-Verlag, New York, 1976), Chap. 4, pp. 71–102.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Maximum range of the lidar, expressed in the required SNR at R min for different extinction coefficients.

Fig. 2
Fig. 2

System constants versus SNR(R min) for different atmospheric extinctions (α = 10, 1, 0.1, and 0.01 km-1). Solid horizontal lines indicate a typical system constant range.

Fig. 3
Fig. 3

Block diagram for the error evaluation procedure.

Fig. 4
Fig. 4

(a) Ideal range-corrected function S(R), (b) simulated noise-corrupted S(R), (c) S(R) as inverted by the slope method [simulation parameters: α = 1 km-1, β = 3 × 10-2 km-1 sr-1, SNR(R min) = 50].

Fig. 5
Fig. 5

Comparison between extinction and backscatter inversion errors by use of the slope method and exponential-curve fitting. The final results are shown as solid curves, and the results obtained by filtering only the noise component are shown as dashed curves. Vertical solid lines indicate the SNR(R min) margin for typical system constants (simulation parameters: α = 10 km-1, floor = -23).

Fig. 6
Fig. 6

Same as Fig. 5, but for α = 1 km-1.

Fig. 7
Fig. 7

Same as Fig. 5, but for α = 0.1 km-1.

Fig. 8
Fig. 8

Same as Fig. 5, but for α = 0.01 km-1.

Fig. 9
Fig. 9

Same as Fig. 5, but for floor = -30.

Tables (2)

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Table 1 Default Values Used in the Simulations

Tables Icon

Table 2 Assumed Optical Parameters for Different Visibilities

Equations (18)

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P R = K R 2   β R exp - 2   0 R   α r d r ,
K = 10 - 9 Ec / 2 A r     W   km 3 ,
d S R d R = 1 β R d β R d R - 2 α R ,
S R ¯ ln R 2 P R .
1 β R d β R d R   2 α ,
SNR R = R v LP R σ eq R B N = R v LP R σ sh , s 2 R B N + σ sh , d 2 B N + σ th 2 B N 1 / 2 V V ,
B N = 0   S nn | H f | 2 d f S nn | H f | max 2 = 0   | H f | 2 d f | H f | max 2 ,
B N = π B 2 n   sin π / 2 n .
R min = R ovf + Δ R min .
R max ¯ min R     for   which   SNR R = 1 5 km .
e r , α = 1 M i = 1 M e r , α i 2 1 / 2 × 100 % ,
e r , α i = α e i - α α ,
S R = ln ( R 2 P R + n R ) = ln K β - 2 α R + ln 1 + n R P R ,
S R - mR - c 2 = i = 1 N S R i - mR i - c 2 ,
R 2 P R - b   exp - aR 2 = i = 1 N R i 2 P R i - b   exp - aR i 2 ,
P R = K R 2   β   exp - 2 α R + n R K R 2   β 1 - 2 α R + n R .
σ sh , s 2 R = 2 qG T 2 FM 2 R io P R + P back L , σ sh , d 2 = 2 qG T 2 I ds + FM 2 I db , σ th 2 = σ th , i 2 G T 2 ,
S N = R v LP R [ 2 qG T 2 ( I ds + FM 2 I db + R io L P R + P back ) B N + σ th 2 B N ] 1 / 2   V V ,

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