Abstract

The slope method has customarily been used and is still used for inversion of atmospheric optical parameters, extinction, and backscatter in homogeneous atmospheres from lidar returns. Our aim is to study the underlying statistics of the old slope method and ultimately to compare its inversion performance with that of the present-day nonlinear least-squares solution (the so-called exponential-curve fitting). The contents are twofold: First, an analytical study is conducted to characterize the bias and the mean-square-estimation error of the regression operator, which permits estimation of the optical parameters from the logarithm of the range-compensated lidar return. Second, universal plots for most short- and far-range tropospheric backscatter lidars are presented as a rule of thumb for obtaining the optimum regression interval length that yields unbiased estimates. As a result, the simple graphic basis of the slope method is still maintained, and its inversion performance improves up to that of the present-day computer-oriented exponential-curve fitting, which ends the controversy between these two algorithms.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. T. H. Collis, “Lidar: a new atmospheric probe,” Q. J. R. Meteorol. Soc. 92, 220–230 (1966).
    [CrossRef]
  2. 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]
  3. J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1981).
    [CrossRef] [PubMed]
  4. G. J. Kunz, “Probing of the atmosphere with lidar,” in Proceedings of the Remote Sensing of the Propagation Environment Conference, AGARD-CP-502, (AGARD, Neuilly sur Seine, France, 1992), Vol. 23, pp. 1–11.
  5. J. D. Klett, “Lidar calibration and extinction coefficients,” Appl. Opt. 22, 514–515 (1983).
    [CrossRef] [PubMed]
  6. R. T. Brown, “A new lidar for meteorological application,” J. Appl. Meteorol. 12, 698–708 (1973).
    [CrossRef]
  7. G. J. Kunz, G. de Leeuw, “Inversion of lidar signals with the slope method,” Appl. Opt. 32, 3249–3256 (1993).
    [CrossRef] [PubMed]
  8. F. Rocadenbosch, A. Comerón, D. Pineda, “Assessment of lidar inversion errors for homogeneous atmospheres,” Appl. Opt. 37, 2199–2206 (1998).
    [CrossRef]
  9. J. J. More, “The Levenberg–Marquardt algorithm: implementation and theory,” in Numerical Analysis, Vol. 630 of Springer-Verlag Lecture Notes in Mathematics Series, G. A. Watson, ed. (Springer-Verlag, New York, 1977), pp. 105–116.
  10. R. Velotta, B. Bartoli, R. Capobianco, L. Fiorani, N. Spinelli, “Analysis of the receiver response in lidar measurements,” Appl. Opt. 37, 6999–7007 (1998).
    [CrossRef]
  11. G. J. Kunz, “Effects of detector bandwidth reduction on lidar signal processing,” (Physics and Electronics Laboratory, TNO-FEL, The Hague, 1977).
  12. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1991), pp. 345–354.
  13. A. B. Carlson, “Signal transmission and filtering,” in Communication Systems, 3rd ed. (McGraw-Hill, Singapore, 1986), Chap. 3, pp. 177–178.
  14. H. Koschmieder, “Theorie der Horizontalen Sichtweite,” Beitr. Phys. Freien Atmos. 12, 33–53 (1924).
  15. P. W. Kruse, L. D. McGlauchlin, R. B. McQuiston, Elements of Infrared Technology: Generation, Transmission and Detection (Wiley, New York, 1962).
  16. R. J. Barlow, “Least squares,” in Statistics: A Guide To The Use Of Statistical Methods In The Physical Sciences, F. Mandl, R. J. Ellison, D. J. Sandiford, eds. (Wiley, New York, 1989), Chap. 6.
  17. W. B. Jones, Introduction to Optical Fiber Communication Systems (Holt, Rinehart & Winston, New York, 1988), Chaps. 7 and 8.
  18. R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Krieger, Malabar, Fla., 1992), Chap. 4, pp. 138–145.
  19. R. J. McIntyre, “Multiplication noise in uniform avalanche photodiodes,” IEEE Trans. Electron Devices ED-13, 164–168 (1966).
    [CrossRef]

1998 (2)

1993 (1)

1983 (1)

1981 (1)

1973 (1)

R. T. Brown, “A new lidar for meteorological application,” J. Appl. Meteorol. 12, 698–708 (1973).
[CrossRef]

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 photodiodes,” IEEE Trans. Electron Devices ED-13, 164–168 (1966).
[CrossRef]

1924 (1)

H. Koschmieder, “Theorie der Horizontalen Sichtweite,” Beitr. Phys. Freien Atmos. 12, 33–53 (1924).

Barlow, R. J.

R. J. Barlow, “Least squares,” in Statistics: A Guide To The Use Of Statistical Methods In The Physical Sciences, F. Mandl, R. J. Ellison, D. J. Sandiford, eds. (Wiley, New York, 1989), Chap. 6.

Bartoli, B.

Brown, R. T.

R. T. Brown, “A new lidar for meteorological application,” J. Appl. Meteorol. 12, 698–708 (1973).
[CrossRef]

Capobianco, R.

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]

Comerón, A.

de Leeuw, G.

Fiorani, L.

Jones, W. B.

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

Klett, J. D.

Koschmieder, H.

H. Koschmieder, “Theorie der Horizontalen Sichtweite,” Beitr. Phys. Freien Atmos. 12, 33–53 (1924).

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.

G. J. Kunz, G. de Leeuw, “Inversion of lidar signals with the slope method,” Appl. Opt. 32, 3249–3256 (1993).
[CrossRef] [PubMed]

G. J. Kunz, “Probing of the atmosphere with lidar,” in Proceedings of the Remote Sensing of the Propagation Environment Conference, AGARD-CP-502, (AGARD, Neuilly sur Seine, France, 1992), Vol. 23, pp. 1–11.

G. J. Kunz, “Effects of detector bandwidth reduction on lidar signal processing,” (Physics and Electronics Laboratory, TNO-FEL, The Hague, 1977).

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 photodiodes,” 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, Vol. 630 of Springer-Verlag Lecture Notes in Mathematics Series, G. A. Watson, ed. (Springer-Verlag, New York, 1977), pp. 105–116.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1991), pp. 345–354.

Pineda, D.

Rocadenbosch, F.

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]

Spinelli, N.

Velotta, R.

Appl. Opt. (5)

Beitr. Phys. Freien Atmos. (1)

H. Koschmieder, “Theorie der Horizontalen Sichtweite,” Beitr. Phys. Freien Atmos. 12, 33–53 (1924).

IEEE Trans. Electron Devices (1)

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

J. Appl. Meteorol. (1)

R. T. Brown, “A new lidar for meteorological application,” J. Appl. Meteorol. 12, 698–708 (1973).
[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 (10)

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]

G. J. Kunz, “Probing of the atmosphere with lidar,” in Proceedings of the Remote Sensing of the Propagation Environment Conference, AGARD-CP-502, (AGARD, Neuilly sur Seine, France, 1992), Vol. 23, pp. 1–11.

J. J. More, “The Levenberg–Marquardt algorithm: implementation and theory,” in Numerical Analysis, Vol. 630 of Springer-Verlag Lecture Notes in Mathematics Series, G. A. Watson, ed. (Springer-Verlag, New York, 1977), pp. 105–116.

G. J. Kunz, “Effects of detector bandwidth reduction on lidar signal processing,” (Physics and Electronics Laboratory, TNO-FEL, The Hague, 1977).

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1991), pp. 345–354.

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

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

R. J. Barlow, “Least squares,” in Statistics: A Guide To The Use Of Statistical Methods In The Physical Sciences, F. Mandl, R. J. Ellison, D. J. Sandiford, eds. (Wiley, New York, 1989), Chap. 6.

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

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Illustration of the bias effect for the slope method: solid curve, noisy realization of the log- and range-corrected signal S(R) in Eq. (2); dotted line, unbiased ideal line built from the true atmospheric parameters in the simulator (α = 1 km-1, β = 3 × 10-2 km-1 sr-1); solid line, regressed line from the noisy S(R) in application of the slope method. The line becomes tilted downward as a result of the estimator bias; dashed–dotted curves, 80% noise confidence limits for N i ′ [Eq. (8)].

Fig. 2
Fig. 2

(a) Equation (6) functional relationship between N i ′ and N i random variables. Note that the symmetric interval [γ l,i , γ u,i ] for the Gaussian variable N i translates into an asymmetric one [γ l,i ′, γ u,i ′] for variable N i ′. As a result E[ N i ′] ≠ 0 and the estimator becomes biased. (b) PDF of the threshold-limited Gaussian random variable N i .

Fig. 3
Fig. 3

Equation (22) plot of the relative inversion error [as defined in Eq. (27)] versus N. The study parameters are α = 1 km-1, SNR(R min = 100, R min = 0.2 km, ΔR = 7.5 m (sampling rate 20 × 106 samples/s), ρ = 2 × 10-5 Hz-1/2.

Fig. 4
Fig. 4

(a) Example of a regression plot used to compute the universal plots. This plot solves N opt/N given SNR(R min) for a simulated atmospheric extinction (α = 10 km-1 here). In this example, the regressed data sweep only ρ from Table 1: +, ρ = 6 × 10-6 Hz-1/2; *, ρ = 2 × 10-5 Hz-1/2; and ○, ρ = 7 × 10-5 Hz-1/2. Note that for most of the trials N opt/N is within a ±0.1 interval, which can be fitted into the flat zone of Fig. 3. (b) Binned data used to estimate the regression error quality, which is typically ∼5% for α = 10 km-1.

Fig. 5
Fig. 5

Universal plots to solve N opt for minimum MSE in the slope method given SNR(R min), an estimate of the order of magnitude of the atmospheric extinction α, and N, which is defined as that range where the received power equals the noise-equivalent power of the receiver [i.e., SNR(R N ) = 1, as explained in Subsection 3.A].

Fig. 6
Fig. 6

Comparison of the relative inversion errors [Eq. (27)] among (a) the customary nonoptimized slope method, which uses all the available samples for inversion; (b) the optimized slope method, which uses the optimum interval length N opt; (c) the NLSQ exponential fitting described in Section 1. It is shown that the performances of (b) and (c) algorithms unify in similar inversion errors, which ends an old controversy between the slope method and the exponential-curve fitting.

Tables (1)

Tables Icon

Table 1 Parameter Sets Used in the Compilation of the Universal Plots

Equations (42)

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

PR=KR2 βRexp-2 0R αrdr,
SR=lnR2PR+nR=mR+Cideal line+ln1+nRPRnoisy term,
m=-2α,  c=lnKβ.
minSR-b exp-aR2a,b=mini=1NSRi-b exp-aR2a,b,
Si=lnRi2Pi+Ni=c+mRi+Ni,
Ni=ln1+NiPi.
S0R-γlR,  S0R+γuR
pNiγl,i=10%,  pNiγu,i=10%,
pNini=pNini=12+12erfni2σi,
pNiγl,i=0.1  γl,i=ln1-1.28σRiPRi.
pγl,iNiγu,i=80% p-1.28σiNi1.28σi=80%.
γl,i=ln1-1.28SNRRi,  γu,i=ln1+1.28SNRRi.
Si=vl  nl,i=Piexpcl,i-1,
cl,i=νl-c-mRi=ln1+nl,iPi.
νl=S1-2αˆmaxRmax-R1-1,
mˆ=xy¯-xy¯x2¯-x¯2,  cˆ=y¯-mˆx¯,
bias=Emˆ-m=Emˆ-m,
εmˆ2=Emˆ-m2=Emˆ2-2mEmˆ+m2.
Emˆ=m+1NQi=1NRi-R¯ENi,
Emˆ2=Emˆ2+1N2Q2i=1NRi-R¯2ENi2-ENi2.
bias=1NQi=1NRi-R¯ENi,
εmˆ2=1N2Q2A+B,
A=i=1NRi-R¯ENi2,
B=i=1NRi-R¯2ENi2-E2Ni=i=1NRi-R¯2σNi2,
Q=1Ni=1N Ri2-R¯2.
dεmˆ2dN=0  N=Nopt.
er,α=MSEmˆ2α×100%.
ρ=2qFMσsh_d,i2+σth,i21/2Hz-1/2,
ENi=-ln1+nPifNindn.
p=-nl,i12πσn,iexp-n22σn,i2dn=12erf-nl,i2σn,i,
ENi=-nl,iln1+nPipδn-nl,idn+nl,i ln1+nPi12πσn,iexp-12nσn,i2dnmain term.
x=ln1+nPi,  n=Piex-1,  dn=Piexdx,
ENi=pcl,i+cl,i xexPi2πσn,i×exp-12Piex-1σn,i2dx.
ENi2=pcl,i2+cl,i x2exPi2πσn,i×exp-12Piex-1σn,i2dx.
nl,i  -5σn,i  p0.
ENinl,i ln1+nPi12πσn,iexp-12nσn,i2dn,
ln1+x=x-x22+x33-x44+Ox5,  |x|  1,
nmaxPi  1  5σn,iPi  1  SNRi  5,
ENiENiPi-ENi22Pi2+ENi33Pi3-ENi44Pi4.
EXn=σnn!2n/2n=even0n=odd.
ENi-121SNRi2-321SNRi4.
ENi21SNRi2+1121SNRi4+13781SNRi6.

Metrics