Abstract

A new solution is presented for the reconstruction of profiles of aerosol volume extinction coefficients from the noisy backscattered returns of a monostatic single-wavelength lidar system. This inverse problem is solved by utilizing an information-theoretic method based on the principle of minimum cross-entropy (MCE), which represents an objective and rational approach for the effective incorporation, into the inversion procedure, of both prior information in the form of an initial estimate of the extinction coefficient and additional information in the form of the observed lidar data. A simple and robust numerical procedure, based on the ellipsoid algorithm, is developed to compute the MCE reconstruction of the extinction function. A number of numerical examples, based on noisy synthetic lidar data, are employed to demonstrate and evaluate the utility and efficacy of the inversion method.

© 1989 Optical Society of America

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References

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  1. B. T. N. Evans, “Field Evaluations of a Canadian Laser Cloud Mapper and Candidate IR Screening Aerosols,” presented at Smoke/Obscurants Symposium VI, Unclassified Section, Harry Diamond Laboratories, Adelphi, MD. (Apr. 1982).
  2. W. Hitchfeld, J. Bordan, “Errors Inherent in the Radar Measurement of Rainfall at Attenuating Wavelengths,” J. Meteorol. 11, 58 (1954).
    [CrossRef]
  3. J. D. Klett, “Stable Analytical Inversion Solution for Processing Lidar Returns,” Appl. Opt. 20, 211 (1981).
    [CrossRef] [PubMed]
  4. B. T. N. Evans, “On the Inversion of the Lidar Equation (U),” DREV R-4343/84 (Nov.1984).
  5. R. H. Kohl, “Discussion of the Interpretation Problem Encountered in Single-Wavelength Lidar Transmissometers,” J. Appl. Meteorol. 17, 1034 (1978).
    [CrossRef]
  6. G. DeLeeuw, “Mie Scattering on Particle Size Distributions: Influence of Size Limits and Complex Refractive Index on the Calculated Extinction and Backscatter Coefficients,” Physics Laboratory TNO, PHL 1982-50, The Netherlands (1982).
  7. S. Kullback, Information Theory and Statistics (Wiley, New York, 1959).
  8. M. S. Pinsker, Information and Information Stability of Random Variables and Processes (Holden-Day, San Francisco, 1964).
  9. I. J. Good, Probability and the Weighting of Evidence (Griffen, London, 1950).
  10. I. Csiszar, “I-Divergence Geometry of Probability Distributions and Minimization Problems,” Ann. Prob. 3, 146 (1975).
    [CrossRef]
  11. J. E. Shore, R. W. Johnson, “Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross-Entropy,” IEEE Trans. Inf. Theory IT-26, 26 (1980).
    [CrossRef]
  12. J. E. Shore, R. W. Johnson, “Properties of Cross-Entropy Minimization,” IEEE Trans. Inf. Theory IT-27, 472 (1981).
    [CrossRef]
  13. W. M. Elsasser, “On Quantum Measurements and the Role of the Uncertainty Relations in Statistical Mechanics,” Phys. Rev. 52, 978 (1937).
    [CrossRef]
  14. E. T. Jaynes, “Information Theory and Statistical Mechanics I,” Phys. Rev. 108, 171 (1957).
    [CrossRef]
  15. E. T. Jaynes, “Prior Probabilities,” IEEE Trans. Syst. Sci. Cybern. SSC-4, 227 (1968).
    [CrossRef]
  16. L. G. Khachian, “A Polynomial Algorithm in Linear Programming,” Sov. Math. 20, 191 (1979).
  17. N. Z. Shor, “Cut-Off Method with Space Extension in Convex Programming Problems,” Cybernetics 12, 94 (1977).
  18. N. Z. Shor, V. I. Gershovich, “Family of Algorithms for Solving Convex Programming Problems,” Cybernetics 15, 502 (1980).
    [CrossRef]
  19. R. G. Bland, D. Goldfarb, M. J. Todd, “The Ellipsoid Method: A Survey,” Oper. Res. 29, 1039 (1981).
    [CrossRef]
  20. E. P. Shettle, R. W. Fenn, “Models for the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties (U),” AFGL-TR-79-0214, Air Force Geophysics Laboratory, Hanscom AFB, MA (Sept.1979).
  21. L. R. Bissonnette, “Multiscattering Lidar Method for Determining Optical Parameters of Aerosols (U),” DREV R-4430/86 (Oct.1986).

1981 (3)

J. D. Klett, “Stable Analytical Inversion Solution for Processing Lidar Returns,” Appl. Opt. 20, 211 (1981).
[CrossRef] [PubMed]

J. E. Shore, R. W. Johnson, “Properties of Cross-Entropy Minimization,” IEEE Trans. Inf. Theory IT-27, 472 (1981).
[CrossRef]

R. G. Bland, D. Goldfarb, M. J. Todd, “The Ellipsoid Method: A Survey,” Oper. Res. 29, 1039 (1981).
[CrossRef]

1980 (2)

N. Z. Shor, V. I. Gershovich, “Family of Algorithms for Solving Convex Programming Problems,” Cybernetics 15, 502 (1980).
[CrossRef]

J. E. Shore, R. W. Johnson, “Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross-Entropy,” IEEE Trans. Inf. Theory IT-26, 26 (1980).
[CrossRef]

1979 (1)

L. G. Khachian, “A Polynomial Algorithm in Linear Programming,” Sov. Math. 20, 191 (1979).

1978 (1)

R. H. Kohl, “Discussion of the Interpretation Problem Encountered in Single-Wavelength Lidar Transmissometers,” J. Appl. Meteorol. 17, 1034 (1978).
[CrossRef]

1977 (1)

N. Z. Shor, “Cut-Off Method with Space Extension in Convex Programming Problems,” Cybernetics 12, 94 (1977).

1975 (1)

I. Csiszar, “I-Divergence Geometry of Probability Distributions and Minimization Problems,” Ann. Prob. 3, 146 (1975).
[CrossRef]

1968 (1)

E. T. Jaynes, “Prior Probabilities,” IEEE Trans. Syst. Sci. Cybern. SSC-4, 227 (1968).
[CrossRef]

1957 (1)

E. T. Jaynes, “Information Theory and Statistical Mechanics I,” Phys. Rev. 108, 171 (1957).
[CrossRef]

1954 (1)

W. Hitchfeld, J. Bordan, “Errors Inherent in the Radar Measurement of Rainfall at Attenuating Wavelengths,” J. Meteorol. 11, 58 (1954).
[CrossRef]

1937 (1)

W. M. Elsasser, “On Quantum Measurements and the Role of the Uncertainty Relations in Statistical Mechanics,” Phys. Rev. 52, 978 (1937).
[CrossRef]

Bissonnette, L. R.

L. R. Bissonnette, “Multiscattering Lidar Method for Determining Optical Parameters of Aerosols (U),” DREV R-4430/86 (Oct.1986).

Bland, R. G.

R. G. Bland, D. Goldfarb, M. J. Todd, “The Ellipsoid Method: A Survey,” Oper. Res. 29, 1039 (1981).
[CrossRef]

Bordan, J.

W. Hitchfeld, J. Bordan, “Errors Inherent in the Radar Measurement of Rainfall at Attenuating Wavelengths,” J. Meteorol. 11, 58 (1954).
[CrossRef]

Csiszar, I.

I. Csiszar, “I-Divergence Geometry of Probability Distributions and Minimization Problems,” Ann. Prob. 3, 146 (1975).
[CrossRef]

DeLeeuw, G.

G. DeLeeuw, “Mie Scattering on Particle Size Distributions: Influence of Size Limits and Complex Refractive Index on the Calculated Extinction and Backscatter Coefficients,” Physics Laboratory TNO, PHL 1982-50, The Netherlands (1982).

Elsasser, W. M.

W. M. Elsasser, “On Quantum Measurements and the Role of the Uncertainty Relations in Statistical Mechanics,” Phys. Rev. 52, 978 (1937).
[CrossRef]

Evans, B. T. N.

B. T. N. Evans, “On the Inversion of the Lidar Equation (U),” DREV R-4343/84 (Nov.1984).

B. T. N. Evans, “Field Evaluations of a Canadian Laser Cloud Mapper and Candidate IR Screening Aerosols,” presented at Smoke/Obscurants Symposium VI, Unclassified Section, Harry Diamond Laboratories, Adelphi, MD. (Apr. 1982).

Fenn, R. W.

E. P. Shettle, R. W. Fenn, “Models for the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties (U),” AFGL-TR-79-0214, Air Force Geophysics Laboratory, Hanscom AFB, MA (Sept.1979).

Gershovich, V. I.

N. Z. Shor, V. I. Gershovich, “Family of Algorithms for Solving Convex Programming Problems,” Cybernetics 15, 502 (1980).
[CrossRef]

Goldfarb, D.

R. G. Bland, D. Goldfarb, M. J. Todd, “The Ellipsoid Method: A Survey,” Oper. Res. 29, 1039 (1981).
[CrossRef]

Good, I. J.

I. J. Good, Probability and the Weighting of Evidence (Griffen, London, 1950).

Hitchfeld, W.

W. Hitchfeld, J. Bordan, “Errors Inherent in the Radar Measurement of Rainfall at Attenuating Wavelengths,” J. Meteorol. 11, 58 (1954).
[CrossRef]

Jaynes, E. T.

E. T. Jaynes, “Prior Probabilities,” IEEE Trans. Syst. Sci. Cybern. SSC-4, 227 (1968).
[CrossRef]

E. T. Jaynes, “Information Theory and Statistical Mechanics I,” Phys. Rev. 108, 171 (1957).
[CrossRef]

Johnson, R. W.

J. E. Shore, R. W. Johnson, “Properties of Cross-Entropy Minimization,” IEEE Trans. Inf. Theory IT-27, 472 (1981).
[CrossRef]

J. E. Shore, R. W. Johnson, “Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross-Entropy,” IEEE Trans. Inf. Theory IT-26, 26 (1980).
[CrossRef]

Khachian, L. G.

L. G. Khachian, “A Polynomial Algorithm in Linear Programming,” Sov. Math. 20, 191 (1979).

Klett, J. D.

Kohl, R. H.

R. H. Kohl, “Discussion of the Interpretation Problem Encountered in Single-Wavelength Lidar Transmissometers,” J. Appl. Meteorol. 17, 1034 (1978).
[CrossRef]

Kullback, S.

S. Kullback, Information Theory and Statistics (Wiley, New York, 1959).

Pinsker, M. S.

M. S. Pinsker, Information and Information Stability of Random Variables and Processes (Holden-Day, San Francisco, 1964).

Shettle, E. P.

E. P. Shettle, R. W. Fenn, “Models for the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties (U),” AFGL-TR-79-0214, Air Force Geophysics Laboratory, Hanscom AFB, MA (Sept.1979).

Shor, N. Z.

N. Z. Shor, V. I. Gershovich, “Family of Algorithms for Solving Convex Programming Problems,” Cybernetics 15, 502 (1980).
[CrossRef]

N. Z. Shor, “Cut-Off Method with Space Extension in Convex Programming Problems,” Cybernetics 12, 94 (1977).

Shore, J. E.

J. E. Shore, R. W. Johnson, “Properties of Cross-Entropy Minimization,” IEEE Trans. Inf. Theory IT-27, 472 (1981).
[CrossRef]

J. E. Shore, R. W. Johnson, “Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross-Entropy,” IEEE Trans. Inf. Theory IT-26, 26 (1980).
[CrossRef]

Todd, M. J.

R. G. Bland, D. Goldfarb, M. J. Todd, “The Ellipsoid Method: A Survey,” Oper. Res. 29, 1039 (1981).
[CrossRef]

Ann. Prob. (1)

I. Csiszar, “I-Divergence Geometry of Probability Distributions and Minimization Problems,” Ann. Prob. 3, 146 (1975).
[CrossRef]

Appl. Opt. (1)

Cybernetics (2)

N. Z. Shor, “Cut-Off Method with Space Extension in Convex Programming Problems,” Cybernetics 12, 94 (1977).

N. Z. Shor, V. I. Gershovich, “Family of Algorithms for Solving Convex Programming Problems,” Cybernetics 15, 502 (1980).
[CrossRef]

IEEE Trans. Inf. Theory (2)

J. E. Shore, R. W. Johnson, “Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross-Entropy,” IEEE Trans. Inf. Theory IT-26, 26 (1980).
[CrossRef]

J. E. Shore, R. W. Johnson, “Properties of Cross-Entropy Minimization,” IEEE Trans. Inf. Theory IT-27, 472 (1981).
[CrossRef]

IEEE Trans. Syst. Sci. Cybern. (1)

E. T. Jaynes, “Prior Probabilities,” IEEE Trans. Syst. Sci. Cybern. SSC-4, 227 (1968).
[CrossRef]

J. Appl. Meteorol. (1)

R. H. Kohl, “Discussion of the Interpretation Problem Encountered in Single-Wavelength Lidar Transmissometers,” J. Appl. Meteorol. 17, 1034 (1978).
[CrossRef]

J. Meteorol. (1)

W. Hitchfeld, J. Bordan, “Errors Inherent in the Radar Measurement of Rainfall at Attenuating Wavelengths,” J. Meteorol. 11, 58 (1954).
[CrossRef]

Oper. Res. (1)

R. G. Bland, D. Goldfarb, M. J. Todd, “The Ellipsoid Method: A Survey,” Oper. Res. 29, 1039 (1981).
[CrossRef]

Phys. Rev. (2)

W. M. Elsasser, “On Quantum Measurements and the Role of the Uncertainty Relations in Statistical Mechanics,” Phys. Rev. 52, 978 (1937).
[CrossRef]

E. T. Jaynes, “Information Theory and Statistical Mechanics I,” Phys. Rev. 108, 171 (1957).
[CrossRef]

Sov. Math. (1)

L. G. Khachian, “A Polynomial Algorithm in Linear Programming,” Sov. Math. 20, 191 (1979).

Other (8)

E. P. Shettle, R. W. Fenn, “Models for the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties (U),” AFGL-TR-79-0214, Air Force Geophysics Laboratory, Hanscom AFB, MA (Sept.1979).

L. R. Bissonnette, “Multiscattering Lidar Method for Determining Optical Parameters of Aerosols (U),” DREV R-4430/86 (Oct.1986).

B. T. N. Evans, “Field Evaluations of a Canadian Laser Cloud Mapper and Candidate IR Screening Aerosols,” presented at Smoke/Obscurants Symposium VI, Unclassified Section, Harry Diamond Laboratories, Adelphi, MD. (Apr. 1982).

B. T. N. Evans, “On the Inversion of the Lidar Equation (U),” DREV R-4343/84 (Nov.1984).

G. DeLeeuw, “Mie Scattering on Particle Size Distributions: Influence of Size Limits and Complex Refractive Index on the Calculated Extinction and Backscatter Coefficients,” Physics Laboratory TNO, PHL 1982-50, The Netherlands (1982).

S. Kullback, Information Theory and Statistics (Wiley, New York, 1959).

M. S. Pinsker, Information and Information Stability of Random Variables and Processes (Holden-Day, San Francisco, 1964).

I. J. Good, Probability and the Weighting of Evidence (Griffen, London, 1950).

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Figures (8)

Fig. 1
Fig. 1

Schematic illustration of the geometric interpretation of the principle of minimum cross-entropy. Here, the final estimate q is the information or I-projection of the initial estimate p onto the admissible set Π. Consequently, q is as close to p as possible in the information measure sense while at the same time satisfying the new information encoded in the data.

Fig. 2
Fig. 2

Flow chart of the ellipsoid algorithm used to solve the convex optimization problem required to obtain the minimum cross-entropy recovery of the extinction function.

Fig. 3
Fig. 3

Comparison of the aerosol extinction functions recovered by the MCE and Klett inversion procedures for the case of noise-free lidar data. The true extinction function (solid line) is representative of a generic maritime aerosol cloud at 99% relative humidity. The MCE solution and the Klett solution, with a correctly specified far-end boundary condition, returned extinction profiles that coincided with the true extinction coefficient. The dashed line shows the Klett solution with an incorrectly specified far-end boundary value on the aerosol cloud.

Fig. 4
Fig. 4

Comparison of the aerosol extinction functions recovered by the MCE and Klett inversion procedures for the case of lidar data contaminated with 1.5% rms Gaussian noise. The Klett solution was obtained with the correctly specified far-end boundary value for the aerosol cloud. Note that the MCE solution is much smoother than the Klett solution.

Fig. 5
Fig. 5

Example of minimum cross-entropy recovered aerosol extinction functions. The dotted and dashed lines indicate MCE inversions with target misfit values of 20 and 32, respectively. The input for these inversions consisted of twenty lidar data degraded with 3% rms Gaussian noise.

Fig. 6
Fig. 6

Another example of minimum cross-entropy recovered aerosol extinction functions. The true extinction function (solid line) is a composite constructed by piecing together generic aerosol cloud characteristics from urban, rural, and maritime environments. The dotted line indicates the inverted extinction function for a target misfit value of 20 and a uniform initial estimate. The dashed line shows the inverted extinction function using a target misfit value of 20 and the initial estimate as the triangular extinction profile shown by the solid line in Fig. 5. These inversions were obtained from twenty lidar data which have been corrupted with 5% rms Gaussian noise.

Fig. 7
Fig. 7

Same example as in Fig. 6 but the dotted line shows the mean of fifty MCE inversions. The error bars indicate the standard deviations for the inversions computed from the fifty noise-corrupted lidar data sets. Each data set consisted twenty data that has been corrupted with 5% rms Gaussian noise.

Fig. 8
Fig. 8

Same example as in Fig. 6 but the dotted line shows the recovered extinction profile for a misspecified backscattering-to- extinction ratio. The target misfit value was set to 27 for the inversion.

Equations (25)

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P ( r ) = P 0 c τ 2 β ( r ) A r 2 exp { 2 0 r α e ( r ) d r } ,
S ( r ) = ln [ β ( r ) ] 2 0 r α e ( r ) d r .
y i = S ( r i ) + i , i = 1 , 2 , , M .
β ( r ) = κ ( r ) α e k ( r ) ,
= { q ( r ) : S ( r ) = ln [ κ ( r ) ] + k ln [ q ( r ) ] 2 0 r q ( r ) d r , i = 1 M [ y i S ( r i ) ] 2 / σ i 2 χ M 2 } ,
χ 2 i = 1 M [ y i S ( r i ) ] 2 / σ i 2 ,
H ( q , p ) = D q ( r ) ln [ q ( r ) / p ( r ) ] d r ,
H ( q , p ) = min q H ( q , p ) .
H ( α e , p ) = H ( α e , q ) + H ( q , p ) ,
min q ( r ) { D q ( r ) ln [ q ( r ) / p ( r ) ] d r } ,
q ( r ) ,
q ( r ) = i = 1 N q i μ ( r i 1 , r ) ( r ) ,
p ( r ) = i = 1 N p i μ ( r i 1 , r i ) ( r ) .
H ¯ ( q , p ) = i = 1 N q i ln [ q i / p i ] Δ r i ,
y ˜ i S ( r i ) = ln [ κ ( r i ) ] + k ln [ q i ] 2 j i q j Δ r j .
min q i = 1 N q i ln [ q i / p i ] Δ r i ,
j = 1 M ( y j y ˜ j ) 2 σ j 2 χ M 2 ,
E 0 = { q R N : ( q q 0 ) T R 0 ( q q 0 ) 1 } ,
d R k g g T R k g ,
q k + 1 = q k + d / ( N + 1 ) , R k + 1 = N 2 N 2 1 [ R k 2 N + 1 d d T ] ,
β ( r ) = { 0 . 0096 μ ( 0 , 0 . 35 ) ( r ) + 0 . 0396 μ ( 0 . 35 , 0 . 65 ) ( r ) + 0 . 021 μ ( 0 . 65 , 1 . 00 ) ( r ) } α e ( r ) .
p ( r ) = ( 1 . 0 + 10 . 0 r ) μ ( 0 , 0 . 5 ) ( r ) + ( 11 . 0 10 . 0 r ) μ ( 0 . 5 , 1 . 0 ) ( r ) .
B ( r ) = 1 50 i = 1 50 [ q i ( r ) α e ( r ) ] , V ( r ) = 1 49 i = 1 50 [ q i ( r ) q ¯ ( r ) ] 2 ,
q ¯ ( r ) = 1 50 i = 1 50 q i ( r )
β ( r ) = { 0 . 01 μ ( 0 , 0 . 35 ) ( r ) + 0 . 025 μ ( 0 . 35 , 1 . 0 ) ( r ) } α e ( r ) .

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