Abstract

Joint estimation of extinction and backscatter simulated profiles from elastic-backscatter lidar return signals is tackled by means of an extended Kalman filter (EKF). First, we introduced the issue from a theoretical point of view by using both an EKF formulation and an appropriate atmospheric stochastic model; second, it is tested through extensive simulation and under simplified conditions; and, finally, a first real application is discussed. An atmospheric model including both temporal and spatial correlation features is introduced to describe approximate fluctuation statistics in the sought-after atmospheric optical parameters and hence to include a priori information in the algorithm. Provided that reasonable models are given for the filter, inversion errors are shown to depend strongly on the atmospheric condition (i.e., the visibility) and the signal-to-noise ratio along the exploration path in spite of modeling errors in the assumed statistical properties of the atmospheric optical parameters. This is of advantage in the performance of the Kalman filter because they are often the point of most concern in identification problems. In light of the adaptive behavior of the filter and the inversion results, the EKF approach promises a successful alternative to present-day nonmemory algorithms based on exponential-curve fitting or differential equation formulations such as Klett’s method.

© 1998 Optical Society of America

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References

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  1. R. T. H. Collis, P. B. Russell, “Laser 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. 91–102.
  2. R. M. Measures, “Laser-remote-sensor equations,” in Laser Remote Sensing: Fundamentals and Applications (Krieger, Malabar, Fla., 1992), Chap. 7, pp. 237–280.
  3. D. K. Killinger, N. Menyuk, “Laser sensing of the atmosphere,” Science 235, 37–45 (1987).
    [CrossRef] [PubMed]
  4. 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).
    [CrossRef]
  5. G. J. Kunz, G. de Leeuw, “Inversion of lidar signals with the slope method,” Appl. Opt. 32, 3249–3256 (1993).
    [CrossRef] [PubMed]
  6. J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1981).
    [CrossRef] [PubMed]
  7. R. J. Barlow, Statistics (Wiley, New York, 1989).
  8. J. J. More, “The Levenberg–Marquardt algorithm: implementation and theory,” in Numerical Analysis, Lecture Notes in Mathematics 630, G. A. Watson, ed. (Springer-Verlag, New York, 1977), pp. 105–116.
  9. J. D. Klett, “Lidar calibration and extinction coefficients,” Appl. Opt. 22, 514–515 (1983).
    [CrossRef] [PubMed]
  10. J. D. Klett, “Lidar inversion with variable backscatter/extinction ratios,” Appl. Opt. 24, 1638–1643 (1985).
    [CrossRef] [PubMed]
  11. G. J. Kunz, “Probing of the atmosphere with lidar,” in Proceedings of Remote Sensing of the Propagation Environment (AGARD-CP-502), 23, 1–11 (1992).
  12. R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. 82, 35–46 (1960).
    [CrossRef]
  13. H. W. Sorenson, Kalman Filtering Techniques. Advances in Control Systems. Theory and Applications (IEEE, New York, 1985), Vol. 3.
  14. R. G. Brown, P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering (Wiley, New York, 1992).
  15. 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]
  16. 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]
  17. R. J. McIntyre, “Multiplication noise in uniform avalanche photodiodes,” IEEE Trans. Electron Devices ED-13, 164–168 (1966).
    [CrossRef]
  18. P. S. Maybeck, Stochastic Models, Estimation and Control (Academic, New York, 1977), Vol. 1.
  19. W. B. Jones, Introduction to Optical Fiber Communication Systems (Holt, Rinehart & Winston, New York, 1988), Chap. 7, 8.
  20. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1991).
  21. D. G. Lainiotis, “Partitioned estimation algorithms. I: Nonlinear estimation,” J. Inf. Sci. 7, 203–255 (1974).
    [CrossRef]
  22. D. G. Lainiotis, “Partitioning: a unifying framework for adaptive systems. I: Estimation,” Proc. IEEE 64, 1126–1143 (1976).
    [CrossRef]
  23. H. Koshmieder, “Theorie der Horizontalen Sichtweite,” Beitr. Phys. Freien Atmos. 12, 33–53 (1924).
  24. P. W. Kruse, L. D. McGlauchlin, R. B. McQuiston, Elements of Infrared Technology: Generation, Transmission and Detection (Wiley, New York, 1962).

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)

1989 (1)

1987 (1)

D. K. Killinger, N. Menyuk, “Laser sensing of the atmosphere,” Science 235, 37–45 (1987).
[CrossRef] [PubMed]

1985 (1)

1983 (1)

1981 (1)

1976 (1)

D. G. Lainiotis, “Partitioning: a unifying framework for adaptive systems. I: Estimation,” Proc. IEEE 64, 1126–1143 (1976).
[CrossRef]

1974 (1)

D. G. Lainiotis, “Partitioned estimation algorithms. I: Nonlinear estimation,” J. Inf. Sci. 7, 203–255 (1974).
[CrossRef]

1966 (1)

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

1960 (1)

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. 82, 35–46 (1960).
[CrossRef]

1924 (1)

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

Barlow, R. J.

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

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).
[CrossRef]

Collis, R. T. H.

R. T. H. Collis, P. B. Russell, “Laser 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. 91–102.

de Leeuw, G.

Hardesty, R. M.

Hwang, P. Y. C.

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

Jones, W. B.

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

Kalman, R. E.

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. 82, 35–46 (1960).
[CrossRef]

Killinger, D. K.

D. K. Killinger, N. Menyuk, “Laser sensing of the atmosphere,” Science 235, 37–45 (1987).
[CrossRef] [PubMed]

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 Proceedings of Remote Sensing of the Propagation Environment (AGARD-CP-502), 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]

D. G. Lainiotis, “Partitioning: a unifying framework for adaptive systems. I: Estimation,” Proc. IEEE 64, 1126–1143 (1976).
[CrossRef]

D. G. Lainiotis, “Partitioned estimation algorithms. I: Nonlinear estimation,” J. Inf. Sci. 7, 203–255 (1974).
[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-sensor equations,” in Laser Remote Sensing: Fundamentals and Applications (Krieger, Malabar, Fla., 1992), Chap. 7, pp. 237–280.

Menyuk, N.

D. K. Killinger, N. Menyuk, “Laser sensing of the atmosphere,” Science 235, 37–45 (1987).
[CrossRef] [PubMed]

More, J. J.

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

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 Processes (McGraw-Hill, New York, 1991).

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]

Russell, P. B.

R. T. H. Collis, P. B. Russell, “Laser 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. 91–102.

Rye, B. J.

Sorenson, H. W.

H. W. Sorenson, Kalman Filtering Techniques. Advances in Control Systems. Theory and Applications (IEEE, New York, 1985), Vol. 3.

Appl. Opt. (5)

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. 82, 35–46 (1960).
[CrossRef]

J. Inf. Sci. (1)

D. G. Lainiotis, “Partitioned estimation algorithms. I: Nonlinear estimation,” J. Inf. Sci. 7, 203–255 (1974).
[CrossRef]

Proc. IEEE (1)

D. G. Lainiotis, “Partitioning: a unifying framework for adaptive systems. I: Estimation,” Proc. IEEE 64, 1126–1143 (1976).
[CrossRef]

Science (1)

D. K. Killinger, N. Menyuk, “Laser sensing of the atmosphere,” Science 235, 37–45 (1987).
[CrossRef] [PubMed]

Other (12)

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).
[CrossRef]

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

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

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

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

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

R. T. H. Collis, P. B. Russell, “Laser 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. 91–102.

R. M. Measures, “Laser-remote-sensor equations,” in Laser Remote Sensing: Fundamentals and Applications (Krieger, Malabar, Fla., 1992), Chap. 7, pp. 237–280.

H. W. Sorenson, Kalman Filtering Techniques. Advances in Control Systems. Theory and Applications (IEEE, New York, 1985), Vol. 3.

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

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

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

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Figures (13)

Fig. 1
Fig. 1

Time–space EKF correlation model.

Fig. 2
Fig. 2

Spatial correlation graph of the state-vector components.

Fig. 3
Fig. 3

(Set 1) initial state of the simulation: (a) synthesized backscatter profile, (b) range-corrected return power, (c) return power as received by the lidar, (d) associated SNR.

Fig. 4
Fig. 4

(Set 1) time–space evolution of the extinction and backscatter profiles: (a) synthesized atmospheric optical parameters (extinction and backscatter), (b) EKF inverted optical parameters.

Fig. 5
Fig. 5

(Set 1) contour plots of Fig. 4 showing very good correlation between the time–space evolution of the atmospheric optical parameters and the inverted ones: (a) synthesized atmospheric optical parameters, (b) EKF inverted optical parameters.

Fig. 6
Fig. 6

(Set 1) temporal evolution of the backscatter profiles in four representative observation cells along the lidar beam path: (horizontal lines) starting backscatter values for the atmospheric simulator, (solid curves) atmospheric backscatter evolution, (circles) EKF estimates.

Fig. 7
Fig. 7

(Set 1) Inversion results after 320 iterations: (a) final atmospheric extinction (solid curve), final EKF extinction estimates (circles); (b) final atmospheric backscatter (solid curve), final EKF backscatter estimates (circles); (c) extinction relative inversion error; (d) backscatter relative inversion error.

Fig. 8
Fig. 8

(Set 2) Contour plots comparing (a) the synthesized opto-atmospheric parameters and (b) the EKF estimates.

Fig. 9
Fig. 9

(Set 2) Temporal evolution of the backscatter profiles in four representative observation cells along the lidar beam path: (horizontal lines) starting backscatter values for the atmospheric simulator, (solid curves) atmospheric backscatter evolution, (circles) EKF estimates.

Fig. 10
Fig. 10

(Set 2) Inversion results after 320 iterations: (a) final atmospheric extinction (solid curve), final EKF extinction estimates (circles); (b) final atmospheric backscatter (solid curve), final EKF backscatter estimates (circles); (c) extinction relative inversion error; (d) backscatter relative inversion error.

Fig. 11
Fig. 11

Example of a real application: (a) range-corrected received power as observables to the filter [z k = R 2 P(R)]; the plot consists of 50 signal packets (15 pulses per packet) in the range 0.78–2.65 km; (b) associated SNR at the receiver output; (c) contour plot of (a) after distance is translated into samples (each sample equals 7.5 m); by columns, we read the measurement vector z k at successive times (t k = 1, … , 50); (d) same as (c) but showing EKF range-corrected estimated power during the first 50 iterations.

Fig. 12
Fig. 12

Identification of modeling errors and convergence times. (a) Comparison between the range-corrected received power (solid curve) and the filter’s estimate (circles): The range interval approximately between 0.78 and 1.7 km is largely affected by ovf modeling errors as discussed in the text, whereas the end interval between 1.7 and 2.65 km shows much better agreement between both the EKF estimates and the received signal. (b) Time evolution of P k , P k - , Q k , and R k traces: Steady convergence is indicated by a constant value in the trace of P k between the traces of R k (measurement noise) and Q k (state-vector noise).

Fig. 13
Fig. 13

Inversion results: (a) three-dimensional plot showing the time evolution of the extinction estimates of the filter during the last 50 iterations in response to Figs. 11(a) and 11(b); the reliable inversion interval is virtually free from ovf modeling errors as discussed in the text; (b) same as (a) but showing backscatter estimates; (c) contour plot of (a) in units of km-1; the reliable inversion interval (horizontal dashed line) extents along the vertical of the instrument roughly from 1.7 km up; (d) same as (c) but for the backscatter estimates in units of km-1 sr-1.

Tables (3)

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Table 1 Simulation Parameters

Tables Icon

Table 2 Lidar System Specifications

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Table 3 Inversion Parameters for Section 5

Equations (52)

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P R = A R 2   β R exp - 2   0 R   α r d r ,
S R = ln R 2 P R ,
min α , β S R - ln A β - 2 α R 2 .
F R = R 2 P R ,
min α , β F R - A β   exp - 2 α R 2 .
β R = B 0 α R γ ,
α R = exp S - S m / γ α m - 1 + 2 γ R R m exp S - S m / γ d r .
Δ R = c 2 f s .
R i = R min + i - 1 Δ R ,     i = 1 , ,   N ,
x ˆ k = H k - 1 z k ,
α i = α R min + i - 1 M Δ R ,     i = 1 , ,   N M , β i = β R min + i - 1 M Δ R ,     i = 1 , ,   N M .
x k α 1   α 2     α N / M   β 1   β 2     β N / M T ,
P 1 = A R 1 2   β 1   exp - 2 α 1 R min ,
P M = A R M 2   β 1   exp - 2 α 1 R min + M - 1 Δ R ,
P M + 1 = A R M + 1 2   β 2   exp - 2 α 1 R min + M - 1 Δ R - 2 α 2 Δ R ,
P N = A R N 2   β N / M   exp - 2 α 1 R min + M - 1 Δ R - 2   i = 2 N / M   α i M Δ R .
z k F 1 x k   F 2 x k     F N x k T .
H ij 1 = F i α j x = x ˆ k - ,     H ij 2 = F i β j x = x ˆ k - .
H 1 = - 2 R min F 1 0 0 0 - 2 R min + Δ R F 2 0 0 0 - 2 R min + M - 1 Δ R F M 0 0 0 - 2 R min + M - 1 Δ R F M + 1 - 2 Δ RF M + 1 0 0 - 2 R min + M - 1 Δ R F N - 2 M Δ RF N - 2 M Δ RF N - 2 M Δ RF N N × N / M ,
H 2 = F 1 x N / M + 1 0 0 0 F 2 x N / M + 1 0 0 0 F M x N / M + 1 0 0 0 0 F M + 1 x N / M + 2 0 0 0 0 0 F N x 2 N / M N × N / M ,
R k = E v k v k T = σ r 2 R 1 R 1 4 0 0 σ r 2 R N R N 4 .
σ r 2 R = a P R + P back + b ,
R y τ = σ m 2 exp - δ | τ | ,
y k + 1 = exp - 1 / L c y k + w k ,
L c = 1 / δ ,
Φ k = exp - 1 / L c I ,
σ w = σ m 1 - exp - 2 / L c .
σ α i = p 2.5   α i 1 - exp - 2 L c ,     i = 1 , ,   N M .
C w = C α α C α β C β α C β β .
C α α = σ α 1 2 ρ σ α 1 σ α 2 ρ n - 1 σ α 1 σ α n σ α 2 2 ρ n - 2 σ α 2 σ α n σ α n 2 ,
C α β = ρ σ α 1 σ β 1 ρ ρ σ α 1 σ β 2 ρ ρ n - 1 σ α 1 σ β n ρ σ α 2 σ β 2 ρ ρ n - 2 σ α 2 σ β n ρ σ α n σ β n ,
Q k = C w 1 - exp - 2 / L c ,
P 0 - = μ Q 0 ,     μ 1 .
x k + 1 = f x k + w k ,
Q k = E w k w k T .
z k = h k x k + v k ,
R k = E v k v k T .
f k x k f k x ˆ k + f k x x x = x ˆ k x k - x ˆ k ,
h k x k h k x ˆ k - + h k x x x = x ˆ k - x k - x ˆ k - .
F k = f k x x x = x ˆ k ,
H k = h k x x x = x ˆ k - .
x k + 1 f k x ˆ k + F k x k - x ˆ k + w k ,
z k h k x ˆ k - + H k x k - x ˆ k - + v k .
Δ x k = x k + 1 - f k x ˆ k ,
Δ z k = z k - h k x ˆ k - .
x ˆ k = x ˆ k - + K k z k - h k x ˆ k - ,
P k = I - K k H k P k - ,
x ˆ k + 1 - = f k x ˆ k ,
P k + 1 - = F k P k F k T + Q k ,
K k = P k - H k T H k P k - H k T + R k - 1 ,
P k - = E e k - e k - T = E x k - x ˆ k - x k - x ˆ k - T ,
x k + 1 = Φ k x k + w k ,

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