Abstract

A first inversion of the backscatter profile and extinction-to-backscatter ratio from pulsed elastic-backscatter lidar returns is treated by means of an extended Kalman filter (EKF). The EKF approach enables one to overcome the intrinsic limitations of standard straightforward nonmemory procedures such as the slope method, exponential curve fitting, and the backward inversion algorithm. Whereas those procedures are inherently not adaptable because independent inversions are performed for each return signal and neither the statistics of the signals nor a priori uncertainties (e.g., boundary calibrations) are taken into account, in the case of the Kalman filter the filter updates itself because it is weighted by the imbalance between the a priori estimates of the optical parameters (i.e., past inversions) and the new estimates based on a minimum-variance criterion, as long as there are different lidar returns. Calibration errors and initialization uncertainties can be assimilated also. The study begins with the formulation of the inversion problem and an appropriate atmospheric stochastic model. Based on extensive simulation and realistic conditions, it is shown that the EKF approach enables one to retrieve the optical parameters as time-range-dependent functions and hence to track the atmospheric evolution; the performance of this approach is limited only by the quality and availability of the a priori information and the accuracy of the atmospheric model used. The study ends with an encouraging practical inversion of a live scene measured at the Nd:YAG elastic-backscatter lidar station at our premises at the Polytechnic University of Catalonia, Barcelona.

© 1999 Optical Society of America

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References

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  1. A. I. Carswell, “Lidar remote sensing of atmospheric aerosols,” in Propagation Engineering: Third in a Series, L. R. Bissonnette, W. B. Miller, eds., Proc. SPIE1312, 206–220 (1990).
  2. G. J. Kunz, G. de Leeuw, “Inversion of lidar signals with the slope method,” Appl. Opt. 32, 3249–3256 (1993).
    [CrossRef] [PubMed]
  3. F. Rocadenbosch, A. Comerón, D. Pineda, “Assessment of lidar inversion errors for homogeneous atmospheres,” Appl. Opt. 37, 2199–2206 (1998).
    [CrossRef]
  4. J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1985).
    [CrossRef]
  5. R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Krieger, Malabar, Fla., 1992), Chap. 4, pp. 138–145.
  6. J. D. Klett, “Lidar inversion with variable backscatter/extinction ratios,” Appl. Opt. 24, 1638–1643 (1985).
    [CrossRef] [PubMed]
  7. J. D. Klett, “Lidar calibration and extinction coefficients,” Appl. Opt. 20, 514–515 (1983).
    [CrossRef]
  8. F. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652–653 (1984).
    [CrossRef] [PubMed]
  9. Y. Sasano, H. Nakane, “Significance of the extinction/backscatter ratio and the boundary value term in the solution for the two-component lidar equation,” Appl. Opt. 23, 11–13 (1984).
    [CrossRef]
  10. G. J. Kunz, “Probing of the atmosphere with lidar,” in AGARD Conf. Proc. 23, 1–11 (1992).
  11. R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng.35–46 (1960).
    [CrossRef]
  12. H. W. Sorenson, “Kalman filtering techniques,” in Theory and Applications, Vol. 3 of Advances in Control Systems, IEEE Press Selected Reprint Series, H. W. Sorenson, ed. (IEEE Press, New York, 1985).
  13. R. G. Brown, P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering (Wiley, New York, 1992).
  14. B. J. Rye, R. M. Hardesty, “Nonlinear Kalman filtering techniques for incoherent backscatter lidar: return power and log power estimation,” Appl. Opt. 28, 3908–3917 (1989).
    [CrossRef] [PubMed]
  15. D. G. Lainiotis, P. Papaparaskeva, G. Kothapalli, K. Plataniotis, “Adaptive filter applications to lidar: return power and log power estimation,” IEEE Trans. Geosci. Remote Sens. 34, 886–891 (1996).
    [CrossRef]
  16. R. J. McIntyre, “Multiplication noise in uniform avalanche photodiodes,” IEEE Trans. Electron Devices ED-13, 164–168 (1966).
    [CrossRef]
  17. P. S. Maybeck, Stochastic Models, Estimation and Control (Academic, New York, 1977), Vol. 1.
  18. F. G. Smith, Atmospheric Propagation of Radiation, Vol. 2 of The Infrared and Electro-Optical System Handbook (SPIE, Bellingham, Wash., 1993).
  19. R. W. Fenn, “Correlation between atmospheric backscattering and meteorological visual range,” Appl. Opt. 5, 293–295 (1966).
    [CrossRef] [PubMed]
  20. S. Twomey, H. B. Howell, “The Relative merits of white and monochromatic light for the determination of visibility by backscattering measurements,” Appl. Opt. 4, 501–506 (1965).
    [CrossRef]
  21. F. Spitzer, Principles of Random Walk (Van Nostrand, Princeton, N.J., 1964).
    [CrossRef]
  22. A. Papoulis, Probability, Random Variables and Stochastic ProcessesMcGraw–Hill, New York, 1991), pp. 345–354.
  23. J. H. Wilkinson, The Algebraic Eigenvalue Problem, Monographs on Numerical Analysis (Clarendon, Oxford, 1965).
  24. H. Koshmieder, “Theorie der horizontalen Sichtweite,” Beitr. Phys. Freien Atmos. 12, 33–53 (1924).
  25. P. W. Kruse, L. D. McGlauchlin, R. B. McQuiston, Elements of Infrared Technology: Generation, Transmission and Detection (Wiley, New York, 1962).

1998 (1)

1996 (1)

D. G. Lainiotis, P. Papaparaskeva, G. Kothapalli, K. Plataniotis, “Adaptive filter applications to lidar: return power and log power estimation,” IEEE Trans. Geosci. Remote Sens. 34, 886–891 (1996).
[CrossRef]

1993 (1)

1992 (1)

G. J. Kunz, “Probing of the atmosphere with lidar,” in AGARD Conf. Proc. 23, 1–11 (1992).

1989 (1)

1985 (2)

1984 (2)

1983 (1)

1966 (2)

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

R. W. Fenn, “Correlation between atmospheric backscattering and meteorological visual range,” Appl. Opt. 5, 293–295 (1966).
[CrossRef] [PubMed]

1965 (1)

1960 (1)

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng.35–46 (1960).
[CrossRef]

1924 (1)

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

Brown, R. G.

R. G. Brown, P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering (Wiley, New York, 1992).

Carswell, A. I.

A. I. Carswell, “Lidar remote sensing of atmospheric aerosols,” in Propagation Engineering: Third in a Series, L. R. Bissonnette, W. B. Miller, eds., Proc. SPIE1312, 206–220 (1990).

Comerón, A.

de Leeuw, G.

Fenn, R. W.

Fernald, F. G.

Hardesty, R. M.

Howell, H. B.

Hwang, P. Y. C.

R. G. Brown, P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering (Wiley, New York, 1992).

Kalman, R. E.

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng.35–46 (1960).
[CrossRef]

Klett, J. D.

Koshmieder, H.

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

Kothapalli, G.

D. G. Lainiotis, P. Papaparaskeva, G. Kothapalli, K. Plataniotis, “Adaptive filter applications to lidar: return power and log power estimation,” IEEE Trans. Geosci. Remote Sens. 34, 886–891 (1996).
[CrossRef]

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 AGARD Conf. Proc. 23, 1–11 (1992).

Lainiotis, D. G.

D. G. Lainiotis, P. Papaparaskeva, G. Kothapalli, K. Plataniotis, “Adaptive filter applications to lidar: return power and log power estimation,” IEEE Trans. Geosci. Remote Sens. 34, 886–891 (1996).
[CrossRef]

Maybeck, P. S.

P. S. Maybeck, Stochastic Models, Estimation and Control (Academic, New York, 1977), Vol. 1.

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.

Nakane, H.

Papaparaskeva, P.

D. G. Lainiotis, P. Papaparaskeva, G. Kothapalli, K. Plataniotis, “Adaptive filter applications to lidar: return power and log power estimation,” IEEE Trans. Geosci. Remote Sens. 34, 886–891 (1996).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic ProcessesMcGraw–Hill, New York, 1991), pp. 345–354.

Pineda, D.

Plataniotis, K.

D. G. Lainiotis, P. Papaparaskeva, G. Kothapalli, K. Plataniotis, “Adaptive filter applications to lidar: return power and log power estimation,” IEEE Trans. Geosci. Remote Sens. 34, 886–891 (1996).
[CrossRef]

Rocadenbosch, F.

Rye, B. J.

Sasano, Y.

Smith, F. G.

F. G. Smith, Atmospheric Propagation of Radiation, Vol. 2 of The Infrared and Electro-Optical System Handbook (SPIE, Bellingham, Wash., 1993).

Sorenson, H. W.

H. W. Sorenson, “Kalman filtering techniques,” in Theory and Applications, Vol. 3 of Advances in Control Systems, IEEE Press Selected Reprint Series, H. W. Sorenson, ed. (IEEE Press, New York, 1985).

Spitzer, F.

F. Spitzer, Principles of Random Walk (Van Nostrand, Princeton, N.J., 1964).
[CrossRef]

Twomey, S.

Wilkinson, J. H.

J. H. Wilkinson, The Algebraic Eigenvalue Problem, Monographs on Numerical Analysis (Clarendon, Oxford, 1965).

AGARD Conf. Proc. (1)

G. J. Kunz, “Probing of the atmosphere with lidar,” in AGARD Conf. Proc. 23, 1–11 (1992).

Appl. Opt. (10)

B. J. Rye, R. M. Hardesty, “Nonlinear Kalman filtering techniques for incoherent backscatter lidar: return power and log power estimation,” Appl. Opt. 28, 3908–3917 (1989).
[CrossRef] [PubMed]

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

F. Rocadenbosch, A. Comerón, D. Pineda, “Assessment of lidar inversion errors for homogeneous atmospheres,” Appl. Opt. 37, 2199–2206 (1998).
[CrossRef]

J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1985).
[CrossRef]

J. D. Klett, “Lidar inversion with variable backscatter/extinction ratios,” Appl. Opt. 24, 1638–1643 (1985).
[CrossRef] [PubMed]

J. D. Klett, “Lidar calibration and extinction coefficients,” Appl. Opt. 20, 514–515 (1983).
[CrossRef]

F. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652–653 (1984).
[CrossRef] [PubMed]

Y. Sasano, H. Nakane, “Significance of the extinction/backscatter ratio and the boundary value term in the solution for the two-component lidar equation,” Appl. Opt. 23, 11–13 (1984).
[CrossRef]

R. W. Fenn, “Correlation between atmospheric backscattering and meteorological visual range,” Appl. Opt. 5, 293–295 (1966).
[CrossRef] [PubMed]

S. Twomey, H. B. Howell, “The Relative merits of white and monochromatic light for the determination of visibility by backscattering measurements,” Appl. Opt. 4, 501–506 (1965).
[CrossRef]

Beitr. Phys. Freien Atmos. (1)

H. Koshmieder, “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]

IEEE Trans. Geosci. Remote Sens. (1)

D. G. Lainiotis, P. Papaparaskeva, G. Kothapalli, K. Plataniotis, “Adaptive filter applications to lidar: return power and log power estimation,” IEEE Trans. Geosci. Remote Sens. 34, 886–891 (1996).
[CrossRef]

J. Basic Eng. (1)

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng.35–46 (1960).
[CrossRef]

Other (10)

H. W. Sorenson, “Kalman filtering techniques,” in Theory and Applications, Vol. 3 of Advances in Control Systems, IEEE Press Selected Reprint Series, H. W. Sorenson, ed. (IEEE Press, New York, 1985).

R. G. Brown, P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering (Wiley, New York, 1992).

P. S. Maybeck, Stochastic Models, Estimation and Control (Academic, New York, 1977), Vol. 1.

F. G. Smith, Atmospheric Propagation of Radiation, Vol. 2 of The Infrared and Electro-Optical System Handbook (SPIE, Bellingham, Wash., 1993).

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).

A. I. Carswell, “Lidar remote sensing of atmospheric aerosols,” in Propagation Engineering: Third in a Series, L. R. Bissonnette, W. B. Miller, eds., Proc. SPIE1312, 206–220 (1990).

F. Spitzer, Principles of Random Walk (Van Nostrand, Princeton, N.J., 1964).
[CrossRef]

A. Papoulis, Probability, Random Variables and Stochastic ProcessesMcGraw–Hill, New York, 1991), pp. 345–354.

J. H. Wilkinson, The Algebraic Eigenvalue Problem, Monographs on Numerical Analysis (Clarendon, Oxford, 1965).

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Figures (13)

Fig. 1
Fig. 1

Time–space atmospheric correlation model for the EKF. Both backscatter and extinction-to-backscatter ratio models, β (x i,k , i = 1 … N/ M components) and C (x N/ M+1,k ), are shown. The former is a Gauss–Markov vector process; the latter, a scalar random walk. For the backscatter components the spatial correlator is implemented by linear system A; the time correlators, by an array of digital filters (IIR1–IIR N/ M ). The system is driven by an array of N/ M + 1 white-noise uncorrelated components. w i is the same array but with N/ M correlated components for the β model plus another one uncorrelated for the C model (n N/ M+1).

Fig. 2
Fig. 2

Spatial correlation.

Fig. 3
Fig. 3

First simulation set: (a) input backscatter profile, (b) synthesized range-corrected return power, (c) associated SNR.

Fig. 4
Fig. 4

First simulation set. Comparison of atmospheric and inverted optical parameters: (a) time animation of the atmospheric backscatter profile; (c) time animation of the atmospheric extinction-to-backscatter ratio; (b), (d) EKF inversion results for (a) and (c), respectively.

Fig. 5
Fig. 5

First simulation set: contour plots showing (a) time–space evolution of the atmospheric backscatter and (b) the filter’s backscatter estimates.

Fig. 6
Fig. 6

First simulation set. Relative inversion errors after 150 iterations: (a) atmospheric backscatter error indexed by number of inversion cells, (b) time evolution of the extinction-to-backscatter inversion error.

Fig. 7
Fig. 7

Second simulation set: (a) input backscatter profile; (b), (c) synthesized return and range-corrected return powers, respectively; (d) associated SNR.

Fig. 8
Fig. 8

Second simulation set. Time–space evolution of the backscatter state vector: (a) synthesized atmospheric backscatter, (b) EKF estimates.

Fig. 9
Fig. 9

Second simulation set. Temporal evolution of the backscatter in observation cells 1, 6, 13, and 20. Solid curves, atmospheric backscatter; circles, EKF estimates; solid horizontal lines, backscatter values at the beginning of the simulation.

Fig. 10
Fig. 10

Same as Fig. 6 but for the second simulation set.

Fig. 11
Fig. 11

Actual application example. (a) Range-corrected received power as observables to the filter (z k ). Fifty signal packets are shown in the range 2.28–2.65 km. (b) Comparison of the range-corrected measured power and the filter’s estimated power, showing good tracking.

Fig. 12
Fig. 12

Actual inversion. Inversion results in response to the scene of Fig. 11: (a) time–space inversion of the atmospheric backscatter, (b) contour plot of (a), (c) inversion of the extinction-to-backscatter ratio as a function of time (solid curve) and ± 1σ confidence margins.

Fig. 13
Fig. 13

Actual inversion. Convergence and tracking indicators: (a) time evolution of P k , P k -, and Q k β traces, (b) C-trace time plot. Steady convergence is indicated by a virtually constant figure from the 200th iteration onward.

Tables (1)

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Table 1 Elastic-Backscatter Lidar System Specifications

Equations (67)

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PR=AR2 βRexp-2 0R αrdr,
βR=B0αRγ,
ΔR=c/2fs,
Ri=Rmin+i-1ΔR,  i=1N,
xkβ1 β2βN/M CT,
βi=βRi,Ri=Rmin+i-1MΔR,i=1N/M,
zk=hkxk+νk,
Rk=EνkνkT.
Fi=R2PRi,
zkF1xk F2xkFNxkT+νk.
F1=Aβ1 exp-2Cβ1Rmin,
FM=Aβ1 exp-2Cβ1Rmin+M-1ΔR,
FM+1=Aβ2 exp-2Cβ1Rmin+M-1ΔR+β2ΔR,
FN=AβN/M exp-2Cβ1Rmin+M-1ΔR+i=2N/M βiMΔR.
Hij1=Fiβjx=x^k-,    Hi2FiCx=x^k-,
H1=-2CRmin+1x1F1000-2CRmin+ΔR+1x1F2000-2CRmin+M-1ΔR+1x1FM000-2CRmin+M-1ΔRFM+1-2CΔR+1x2FM+100-2CRmin+M-1ΔRFN-2CMΔRFN-2CMΔRFN-2CMΔR+1xN/MFNN×N/M,
H2=-2Rminx1F1-2Rmin+ΔRx1F2-2Rmin+M-1ΔRx1FM-2Rmin+M-1ΔRx1+ΔRx2FM+1-2Rmin+M-1ΔRx1+MΔRx2++MΔRxN/M-1+MΔRxN/MFN.
σrR2=aPR+Pback+b,
Rk=EνkνkT=σrR12R1400σrRN2RN4.
xk+1=Φkxk+wk, Φk=ΦβN/M×N/M00ΦC1×1,
Qk=EwkwkT.
yk+1=exp-1/Lcyk+wk,
ΦβN/M×N/M=exp-1/LcIN/M×N/M, ΦC1×1=1.
σw=σm1-exp-2/Lc1/2.
σm,i=p2.5 βi,
σwiσβi=p2.5 βi1-exp-2Lc1/2, i=1N/M.
Cw=Cβ00CC,
Cβ=σβ12ρσβ1σβ2ρn-1σβ1σβnσβ22ρn-2σβ2σβnσβn2,
Qk=Qβ00QC,  Qβ=Cβ1-exp-2/Lc.
P0-=μQ0,  μ1.
τRmin, Rmax=RminRmax αrdr=C RminRmax βrdr,
pˆi=EFitk-EFitk21/2EFitk,
p|C-C¯|nσ=erfn2.
xk+1=Φkxk+wk,
zk=Hkxk+νk,
EνkνiT=Rki=k0ik,
EwkwiT=Qki=k0ik,
EwkνiT=0 k, i.
ek-=xk-xˆk-,
Pk-=Eek-ek-T=Exk-xˆk-xk-xˆk-T.
xˆk=xˆk-+Kkzk-Hkxˆk-,
Pk=EekekT=Exk-xˆkxk-xˆkT.
Pk=I-KkHkPk-I-KkHkT+KkRkKkT.
dtrABdA=BT AB must be square,
dTrACATdA=2AC  C must be symmetric,
dTrPkdKk=-2HkPk-T+2KkHkPk-HkT+Rk,
Kk=Pk-HkTHkPk-HkT+Rk-1.
Pk=I-KkHkPk-.
xˆk+1-=Φkxˆk.
Pk+1-=Eek+1-ek+1-T=EΦkek+wkΦkek+wkT=ΦkPkΦkT+Qk.
xk+1=fkxk+wk,
zk=hkxk+νk,
fkxkfkxˆk+fkxxx=xˆkxk-xˆk,
hkxkhkxˆk-+hkxxx=xˆk-xk-xˆk-.
|xk-xˆk|  1,
|xk-xˆk-|  1,
Fk=fkxxx=xˆk,
Hk=hkxxx=xˆk-.
xk+1fkxˆk+Fkxk-xˆk+wk,
zkhkxˆk-+Hkxk-xˆk-+νk.
Δxk=xk+1-fkxˆk,
Δzk=zk-hkxˆk-.
xˆk=xˆk-+Kkzk-hkxˆk-.
Pk=I-KkHkPk-.
xˆk+1-=fkxˆk,
Pk+1-=FkPkFkT+Qk.
Kk=Pk-HkTHkPk-HkT+Rk-1.

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