Abstract

An integral transform technique, based on Tikhonov’s regularization method, is presented which is useful for the remote sensing of structure constant profiles from noisy log amplitude scintillation data. The inversion technique is shown to consist of two parts, one of which is to determine the average value of the structure constant profile and the other of which is to determine the inhomogeneous portion of the profile. A simple and effective function for regularization which is the core of the inversion technique is given. Numerical examples of the technique are also given.

© 1984 Optical Society of America

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Corrections

John M. Jarem, "Remote sensing of structure constant profiles using Tikhonov’s regularized Fourier integral method: erratum," Appl. Opt. 23, 3740_1-3740 (1984)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-23-21-3740_1

References

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  1. L. C. Shen, “Remote Probing of Atmospheric and Wind Velocity by Millimeter Waves,” IEEE Trans. Antennas Propag. AP-18, 493 (1970).
    [Crossref]
  2. J. M. Heneghan, A. Ishimaru, “Remote Determination of the Profiles of the Atmospheric Structure Constant and Wind Velocity Along a Line-of-Sight Path by a Statistical Inversion Procedure,” IEEE Trans. Antennas Propag. AP-22, 457 (1974).
    [Crossref]
  3. R. Barakat, T. E. Buder, “Remote Sensing of Crosswind Profiles Using the Correlation Slope Method,” J. Opt. Soc. Am. 69, 1604 (1979).
    [Crossref]
  4. M. Azouit, J. Vernin, R. Barletti, G. Ceppatelli, A. Righini, N. Speroni, “Remote Sensing of Atmospheric Turbulence by Means of a Fast Optical Method: A Comparison with Simultaneous In Situ Measurements,” J. Appl. Meteorol. 19, 834 (1980).
    [Crossref]
  5. K. Leuenberger, R. W. Lee, A. T. Waterman, “Remote Atmospheric Probing on a Line-of-Sight Path Using Spatial Filter Concepts 1. Theory,” Radio Sci. 14, 781 (1979).
    [Crossref]
  6. R. S. Lawrence, G. R. Ochs, S. F. Clifford, “Use of Scintillations to Measure Average Wind Across a Light Beam,” Appl. Opt. 11, 239, (1972).
    [Crossref] [PubMed]
  7. T. Wang, S. F. Clifford, G. R. Ochs, “Wind and Refractive-Turbulence Sensing Using Crossed Laser Beams,” Appl. Opt. 13, 2602 (1974).
    [Crossref] [PubMed]
  8. S. F. Clifford, G. R. Ochs, T. Wang, “Optical Wind Sensing by Observing the Scintillations of a Random Scene,” Appl. Opt. 14, 2844 (1975).
    [Crossref] [PubMed]
  9. G. R. Ochs, T. Wang, R. S. Lawrence, S. F. Clifford, “Refractive-Turbulence Profiles Measured by One-Dimensional Spatial Filtering of Scintillations,” Appl. Opt. 15, 2504 (1976).
    [Crossref] [PubMed]
  10. D. L. Walters, “Passive Remote Crosswind Sensor,” Appl. Opt. 16, 2625 (1977).
    [Crossref] [PubMed]
  11. T. Wang, G. R. Ochs, S. F. Clifford, “Saturation-Resistant Optical Scintillometer to Measure Cn2,” J. Opt. Soc. Am. 68, 334 (1978).
    [Crossref]
  12. G. R. Ochs, T. Wang, “Finite Aperture Optical Scintillometer for Profiling Wind Cn2,” Appl. Opt. 17, 3774 (1978).
    [Crossref] [PubMed]
  13. M-K. Tsay, T. Wang, R. S. Lawrence, R. G. Ochs, R. B. Fritz, “Wind Velocity and Convergence Measurements at the Boulder Atmospheric Observatory Using Path-Averaged Optical Wind Sensors,” J Appl Meteorol. 19, 826 (1980).
    [Crossref]
  14. T. Wang, G. R. Ochs, R. S. Lawrence, “Wind Measurements by the Temporal Cross-Correlation of the Optical Scintillations,” Appl. Opt. 20, 4073 (1981).
    [Crossref] [PubMed]
  15. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic, New York, 1978).
  16. D. L. Fried, “Remote Probing of the Optical Strength of Atmospheric Turbulence and of Wind Velocity,” Proc. IEEE 57, 415 (1969).
    [Crossref]
  17. A. Peskoff, “Remote Sensing of Clear-Air Turbulence Profiles,” J. Opt. Soc. Am. 58, 1032 (1968).
    [Crossref]
  18. A. Peskoff, “Theory for Remote Sensing of Wind-Velocity Profiles,” Proc. IEEE 59, 324 (1971).
    [Crossref]
  19. J. Jarem, “Remote Determination of the Structure Constant Profile from Amplitude Scintillation Data Using Tikhonov’s Regularized Inverse Method,” IEEE Trans. Antennas Propag. AP-31, 145 (1983).
    [Crossref]
  20. A. N. Tikhonov, V. Y. Arsenin, Solutions of I11-Posed Problems, (Wiley, New York, 1977).

1983 (1)

J. Jarem, “Remote Determination of the Structure Constant Profile from Amplitude Scintillation Data Using Tikhonov’s Regularized Inverse Method,” IEEE Trans. Antennas Propag. AP-31, 145 (1983).
[Crossref]

1981 (1)

1980 (2)

M. Azouit, J. Vernin, R. Barletti, G. Ceppatelli, A. Righini, N. Speroni, “Remote Sensing of Atmospheric Turbulence by Means of a Fast Optical Method: A Comparison with Simultaneous In Situ Measurements,” J. Appl. Meteorol. 19, 834 (1980).
[Crossref]

M-K. Tsay, T. Wang, R. S. Lawrence, R. G. Ochs, R. B. Fritz, “Wind Velocity and Convergence Measurements at the Boulder Atmospheric Observatory Using Path-Averaged Optical Wind Sensors,” J Appl Meteorol. 19, 826 (1980).
[Crossref]

1979 (2)

K. Leuenberger, R. W. Lee, A. T. Waterman, “Remote Atmospheric Probing on a Line-of-Sight Path Using Spatial Filter Concepts 1. Theory,” Radio Sci. 14, 781 (1979).
[Crossref]

R. Barakat, T. E. Buder, “Remote Sensing of Crosswind Profiles Using the Correlation Slope Method,” J. Opt. Soc. Am. 69, 1604 (1979).
[Crossref]

1978 (2)

1977 (1)

1976 (1)

1975 (1)

1974 (2)

T. Wang, S. F. Clifford, G. R. Ochs, “Wind and Refractive-Turbulence Sensing Using Crossed Laser Beams,” Appl. Opt. 13, 2602 (1974).
[Crossref] [PubMed]

J. M. Heneghan, A. Ishimaru, “Remote Determination of the Profiles of the Atmospheric Structure Constant and Wind Velocity Along a Line-of-Sight Path by a Statistical Inversion Procedure,” IEEE Trans. Antennas Propag. AP-22, 457 (1974).
[Crossref]

1972 (1)

1971 (1)

A. Peskoff, “Theory for Remote Sensing of Wind-Velocity Profiles,” Proc. IEEE 59, 324 (1971).
[Crossref]

1970 (1)

L. C. Shen, “Remote Probing of Atmospheric and Wind Velocity by Millimeter Waves,” IEEE Trans. Antennas Propag. AP-18, 493 (1970).
[Crossref]

1969 (1)

D. L. Fried, “Remote Probing of the Optical Strength of Atmospheric Turbulence and of Wind Velocity,” Proc. IEEE 57, 415 (1969).
[Crossref]

1968 (1)

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solutions of I11-Posed Problems, (Wiley, New York, 1977).

Azouit, M.

M. Azouit, J. Vernin, R. Barletti, G. Ceppatelli, A. Righini, N. Speroni, “Remote Sensing of Atmospheric Turbulence by Means of a Fast Optical Method: A Comparison with Simultaneous In Situ Measurements,” J. Appl. Meteorol. 19, 834 (1980).
[Crossref]

Barakat, R.

Barletti, R.

M. Azouit, J. Vernin, R. Barletti, G. Ceppatelli, A. Righini, N. Speroni, “Remote Sensing of Atmospheric Turbulence by Means of a Fast Optical Method: A Comparison with Simultaneous In Situ Measurements,” J. Appl. Meteorol. 19, 834 (1980).
[Crossref]

Buder, T. E.

Ceppatelli, G.

M. Azouit, J. Vernin, R. Barletti, G. Ceppatelli, A. Righini, N. Speroni, “Remote Sensing of Atmospheric Turbulence by Means of a Fast Optical Method: A Comparison with Simultaneous In Situ Measurements,” J. Appl. Meteorol. 19, 834 (1980).
[Crossref]

Clifford, S. F.

Fried, D. L.

D. L. Fried, “Remote Probing of the Optical Strength of Atmospheric Turbulence and of Wind Velocity,” Proc. IEEE 57, 415 (1969).
[Crossref]

Fritz, R. B.

M-K. Tsay, T. Wang, R. S. Lawrence, R. G. Ochs, R. B. Fritz, “Wind Velocity and Convergence Measurements at the Boulder Atmospheric Observatory Using Path-Averaged Optical Wind Sensors,” J Appl Meteorol. 19, 826 (1980).
[Crossref]

Heneghan, J. M.

J. M. Heneghan, A. Ishimaru, “Remote Determination of the Profiles of the Atmospheric Structure Constant and Wind Velocity Along a Line-of-Sight Path by a Statistical Inversion Procedure,” IEEE Trans. Antennas Propag. AP-22, 457 (1974).
[Crossref]

Ishimaru, A.

J. M. Heneghan, A. Ishimaru, “Remote Determination of the Profiles of the Atmospheric Structure Constant and Wind Velocity Along a Line-of-Sight Path by a Statistical Inversion Procedure,” IEEE Trans. Antennas Propag. AP-22, 457 (1974).
[Crossref]

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic, New York, 1978).

Jarem, J.

J. Jarem, “Remote Determination of the Structure Constant Profile from Amplitude Scintillation Data Using Tikhonov’s Regularized Inverse Method,” IEEE Trans. Antennas Propag. AP-31, 145 (1983).
[Crossref]

Lawrence, R. S.

Lee, R. W.

K. Leuenberger, R. W. Lee, A. T. Waterman, “Remote Atmospheric Probing on a Line-of-Sight Path Using Spatial Filter Concepts 1. Theory,” Radio Sci. 14, 781 (1979).
[Crossref]

Leuenberger, K.

K. Leuenberger, R. W. Lee, A. T. Waterman, “Remote Atmospheric Probing on a Line-of-Sight Path Using Spatial Filter Concepts 1. Theory,” Radio Sci. 14, 781 (1979).
[Crossref]

Ochs, G. R.

Ochs, R. G.

M-K. Tsay, T. Wang, R. S. Lawrence, R. G. Ochs, R. B. Fritz, “Wind Velocity and Convergence Measurements at the Boulder Atmospheric Observatory Using Path-Averaged Optical Wind Sensors,” J Appl Meteorol. 19, 826 (1980).
[Crossref]

Peskoff, A.

A. Peskoff, “Theory for Remote Sensing of Wind-Velocity Profiles,” Proc. IEEE 59, 324 (1971).
[Crossref]

A. Peskoff, “Remote Sensing of Clear-Air Turbulence Profiles,” J. Opt. Soc. Am. 58, 1032 (1968).
[Crossref]

Righini, A.

M. Azouit, J. Vernin, R. Barletti, G. Ceppatelli, A. Righini, N. Speroni, “Remote Sensing of Atmospheric Turbulence by Means of a Fast Optical Method: A Comparison with Simultaneous In Situ Measurements,” J. Appl. Meteorol. 19, 834 (1980).
[Crossref]

Shen, L. C.

L. C. Shen, “Remote Probing of Atmospheric and Wind Velocity by Millimeter Waves,” IEEE Trans. Antennas Propag. AP-18, 493 (1970).
[Crossref]

Speroni, N.

M. Azouit, J. Vernin, R. Barletti, G. Ceppatelli, A. Righini, N. Speroni, “Remote Sensing of Atmospheric Turbulence by Means of a Fast Optical Method: A Comparison with Simultaneous In Situ Measurements,” J. Appl. Meteorol. 19, 834 (1980).
[Crossref]

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solutions of I11-Posed Problems, (Wiley, New York, 1977).

Tsay, M-K.

M-K. Tsay, T. Wang, R. S. Lawrence, R. G. Ochs, R. B. Fritz, “Wind Velocity and Convergence Measurements at the Boulder Atmospheric Observatory Using Path-Averaged Optical Wind Sensors,” J Appl Meteorol. 19, 826 (1980).
[Crossref]

Vernin, J.

M. Azouit, J. Vernin, R. Barletti, G. Ceppatelli, A. Righini, N. Speroni, “Remote Sensing of Atmospheric Turbulence by Means of a Fast Optical Method: A Comparison with Simultaneous In Situ Measurements,” J. Appl. Meteorol. 19, 834 (1980).
[Crossref]

Walters, D. L.

Wang, T.

Waterman, A. T.

K. Leuenberger, R. W. Lee, A. T. Waterman, “Remote Atmospheric Probing on a Line-of-Sight Path Using Spatial Filter Concepts 1. Theory,” Radio Sci. 14, 781 (1979).
[Crossref]

Appl. Opt. (7)

IEEE Trans. Antennas Propag. (3)

J. Jarem, “Remote Determination of the Structure Constant Profile from Amplitude Scintillation Data Using Tikhonov’s Regularized Inverse Method,” IEEE Trans. Antennas Propag. AP-31, 145 (1983).
[Crossref]

L. C. Shen, “Remote Probing of Atmospheric and Wind Velocity by Millimeter Waves,” IEEE Trans. Antennas Propag. AP-18, 493 (1970).
[Crossref]

J. M. Heneghan, A. Ishimaru, “Remote Determination of the Profiles of the Atmospheric Structure Constant and Wind Velocity Along a Line-of-Sight Path by a Statistical Inversion Procedure,” IEEE Trans. Antennas Propag. AP-22, 457 (1974).
[Crossref]

J Appl Meteorol. (1)

M-K. Tsay, T. Wang, R. S. Lawrence, R. G. Ochs, R. B. Fritz, “Wind Velocity and Convergence Measurements at the Boulder Atmospheric Observatory Using Path-Averaged Optical Wind Sensors,” J Appl Meteorol. 19, 826 (1980).
[Crossref]

J. Appl. Meteorol. (1)

M. Azouit, J. Vernin, R. Barletti, G. Ceppatelli, A. Righini, N. Speroni, “Remote Sensing of Atmospheric Turbulence by Means of a Fast Optical Method: A Comparison with Simultaneous In Situ Measurements,” J. Appl. Meteorol. 19, 834 (1980).
[Crossref]

J. Opt. Soc. Am. (3)

Proc. IEEE (2)

A. Peskoff, “Theory for Remote Sensing of Wind-Velocity Profiles,” Proc. IEEE 59, 324 (1971).
[Crossref]

D. L. Fried, “Remote Probing of the Optical Strength of Atmospheric Turbulence and of Wind Velocity,” Proc. IEEE 57, 415 (1969).
[Crossref]

Radio Sci. (1)

K. Leuenberger, R. W. Lee, A. T. Waterman, “Remote Atmospheric Probing on a Line-of-Sight Path Using Spatial Filter Concepts 1. Theory,” Radio Sci. 14, 781 (1979).
[Crossref]

Other (2)

A. N. Tikhonov, V. Y. Arsenin, Solutions of I11-Posed Problems, (Wiley, New York, 1977).

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic, New York, 1978).

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Figures (5)

Fig. 1
Fig. 1

Four inhomogeneous profiles (marked solid lines) all with the same path averaged value of C N 2 ¯ / 2 = 1. A uniform profile (circles) with C N 2 / 2 = 1 is also shown for comparison.

Fig. 2
Fig. 2

Plane wave log amplitude scintillation curves which correspond to the sample profiles of Fig. 1. The broken curve (circles) corresponds to the scintillation curve of the uniform profile of Fig. 1.

Fig. 3
Fig. 3

Root-mean-square error for all four profiles as given by Eq. (16) as a function of log(α). Each curve corresponds to a different maximum transverse distance D.

Fig. 4
Fig. 4

Inversion of sample 3 for maximum transverse distance values of D = 1.5, 2, 2.5, 3, and 4 ( λ L ) is shown.

Fig. 5
Fig. 5

Sample 1 profile is compared with several Fourier inversions for D = 2.5 λ L. The circle is the exact Fourier inversion, the triangle is the best regularized inversion Eq. (14), and the plus is regularization with a fixed number α = 10−10 [Eq. (15)]. The cross is the result when no Fourier regularization is used.

Tables (1)

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Table I a

Equations (18)

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B χ ( ρ ) = 2 π 2 k 2 - d Z C n 2 ( Z ) 0 κ d κ Φ n ( 0 ) ( κ ) J 0 ( κ ρ ) sin 2 κ 2 Z 2 k ,
B X ( ρ ) = 1 π - d Z C N 2 ( Z ) 0 d κ Φ ( 0 ) ( κ ) J 0 ( a κ ρ ) sin 2 κ Z 2 ,
B X T ( κ ) = π a 2 0 B X ( ρ ) J 0 ( a κ ρ ) ρ d ρ = 2 Φ ( 0 ) ( κ ) - C N 2 ( Z ) sin 2 κ Z 2 d Z ,
B X T ˜ ( κ ) = [ B X T ( κ ) - Φ ( 0 ) ( κ ) C N 2 ¯ ] = - Φ ( 0 ) ( κ ) C N T 2 ( κ ) ,
C N T 2 ( κ ) = - C N 2 ( Z ) exp ( - j κ Z ) d Z , C N 2 ¯ = - C N 2 ( Z ) d Z .
C N f 2 ( Z ) = 1 2 π - f ( κ ) [ B X T ( κ ) - Φ ( 0 ) ( κ ) C N 2 ¯ ] - Φ ( 0 ) ( κ ) exp ( j κ Z ) d κ ,
f ( κ ) = Φ ( 0 ) ( κ ) 2 Φ ( 0 ) ( κ ) 2 + M ( κ ) ,
C N 2 ¯ ( 0 ) = κ min κ max B X T ( κ ) Φ ( 0 ) ( κ ) d κ / ( κ max - κ min ) ,
ɛ av = 1 2 π κ min κ max C NT f 2 ( κ ) 2 d κ = 1 2 π κ min κ max [ B X T ( κ ) - Φ ( 0 ) ( κ ) C N 2 ¯ ] 2 [ Φ ( 0 ) ( κ ) + M ( κ ) / Φ ( 0 ) ( κ ) ] 2 d κ
C N 2 ¯ ( 1 ) = κ min κ max B X T ( κ ) Φ ( 0 ) ( κ ) [ Φ ( 0 ) ( κ ) + M ( κ ) / Φ ( 0 ) ( κ ) ] 2 d κ κ min κ max Φ ( 0 ) ( κ ) 2 [ Φ ( 0 ) ( κ ) + M ( κ ) / Φ ( 0 ) ( κ ) ] 2 d κ .
ɛ = - [ C N f 2 ( Z ) - C N 2 T ( Z ) ] 2 d Z = 1 2 π - [ - Φ ( 0 ) ( κ ) B X T N ( κ ) - M ( κ ) C N T 2 T ( κ ) ] 2 [ Φ ( 0 ) ( κ ) 2 + M ( κ ) ] 2 d κ ,
B X T T ˜ ( κ ) = B X T T ( κ ) - Φ ( 0 ) ( κ ) C N 2 ¯ = - Φ ( 0 ) ( κ ) C N T 2 T ( κ ) .
2 - M ( κ ) B X T N ( κ ) B X T T ˜ ( κ ) ] [ Φ ( 0 ) ( κ ) 2 + M ( κ ) ] 2 d κ = 0
ɛ = 1 2 π - [ Φ ( 0 ) ( κ ) B X T N ( κ ) ] 2 + [ M ( κ ) C N T 2 T ( κ ) ] 2 [ Φ ( 0 ) ( κ ) 2 + M ( κ ) ] 2 d κ
M ( κ ) = [ B X T N ( κ ) C N T 2 T ( κ ) ] 2 .
C N T 2 T ( κ ) = A sin κ κ
M ( κ ) = 2 S A 2 κ 2 = α κ 2 .
ɛ rms = { 1 4 i = 1 4 - 1 1 [ C N i 2 T ( Z ) - C N f i 2 ( Z ) ] 2 d Z } 1 / 2 ,

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