Abstract

A method to obtain an estimate of Fried’s seeing parameter r0 from time series of an arbitrarily shaped, resolved structure that exhibits degradation resulting from atmospheric turbulence is presented. The basic idea is to evaluate the ratio of the observed squared modulus of the average Fourier transform and the observed average power spectrum. The theory of the method is developed, and the influence of noise on the ratio is discussed. The method has been applied to five consecutive time series of observations of solar granulation under different seeing conditions. The power spectra, which are reconstructed with appropriate theoretical modulation transfer functions, converge.

© 1984 Optical Society of America

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References

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  1. A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  2. D. Y. Gezari, A. Labeyrie, R. V. Stachnik, “Speckle interferometry: diffraction-limited measurements of nine stars with the 200 inch telescope,” Astrophys. J. 173, L1–L5 (1972).
    [CrossRef]
  3. K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
    [CrossRef]
  4. J. R. Fienup, G. B. Feldkamp, “Astronomical imaging by processing stellar speckle interferometry data,” Proc. Soc. Photo-Opt. Instrum. Eng. 243, 95–100 (1980).
  5. D. P. Karo, A. M. Schneidermann, “Speckle interferometry lens-atmosphere MTF measurements,”J. Opt. Soc. Am. 66, 1252–1256 (1976).
    [CrossRef]
  6. D. P. Karo, A. M. Schneidermann, “Transfer functions, correlation scales, and phase retrieval in speckle interferometry,”J. Opt. Soc. Am. 67, 1583–1587 (1977).
    [CrossRef]
  7. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (American Elsevier, New York, 1981), Vol. XIX.
    [CrossRef]
  8. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,”J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  9. D. Korff, “Analysis of a method for obtaining near-diffraction-limited information in the presence of atmospheric turbulence,”J. Opt. Soc. Am. 63, 971–980 (1973).
    [CrossRef]
  10. C. Aime, G. Ricort, C. Roddier, G. Lago, “Changes in the atmospheric-lens modulation transfer function used for calibration in solar speckle interferometry,”J. Opt. Soc. Am. 68, 1063–1066 (1978).
    [CrossRef]
  11. J. P. Mehltretter, “Balloon-borne imagery of the solar granulation: II. The lifetime of solar granulation,” Astron. Astrophys. 62, 311–316 (1978).
  12. R. V. Stachnik, P. Nisenson, R. W. Noyes, “Speckle image reconstruction of solar features,” Astrophys. J. 271, L37–L40 (1983).
    [CrossRef]
  13. R. W. Noyes, R. V. Stachnik, P. Nisenson, “Speckle image reconstruction of solar features,” (U.S. Air Force Systems Command, Hanscomb Air Force Base, Mass., May1981).
  14. J. C. Dainty, “The transfer function, signal-to-noise ratio, and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astron. Soc. 169, 631–641 (1974).
  15. F. Roddier, J. M. Gilli, G. Lund, “On the origin of speckle boiling and its effects in stellar interferometry,”J. Opt. (Paris) 13, 263–271 (1982).
    [CrossRef]
  16. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, Israel, 1971).
  17. R. Barakat, P. Nisenson, “Influence of the wave-front correlation function and deterministic wave-front aberrations on the speckle image-reconstruction problem in the high-light-level regime,”J. Opt. Soc. Am. 71, 1390–1402 (1981).
  18. S. Chandrasekhar, “A statistical basis for the theory of stellar scintillation,” Mon. Not. R. Astron. Soc. 112, 475–483 (1952).
  19. C. Roddier, “Measurements of the atmospheric attenuation of the spectral components of astronomical images,”J. Opt. Soc. Am. 66, 478–482 (1976).
    [CrossRef]
  20. J. Breckinridge, “Measurement of the amplitude of phase excursions in the earth’s atmosphere,”J. Opt. Soc. Am. 66, 143–144 (1976).
    [CrossRef]
  21. J. Borgnino, F. Martin, “Correlation between angle-of-arrival fluctuations on the entrance pupil of a solar telescope,”J. Opt. Soc. Am. 67, 1065–1072 (1977).
    [CrossRef]
  22. J. Borgnino, J. Vernin, “Experimental verification of the inertial model of atmospheric turbulence from solar limb motion,”J. Opt. Soc. Am. 68, 1056–1062 (1978).
    [CrossRef]
  23. J. Borgnino, J. Vernin, C. Aime, G. Ricort, “A study of the degradation of daytime astronomical images due to turbulence in the lower atmosphere by measurement of the standard deviation of the angle of arrival,” Sol. Phys. 64, 403–415 (1979).
    [CrossRef]
  24. P. Burlamacci, A. Consortini, “Study of atmospheric turbulence by means of a laser beam,” Opt. Acta 14, 17–26 (1967).
    [CrossRef]
  25. P. Burlamacci, A. Consortini, L. Ronchi, G. Toraldo di Francia, “Phase correlation measurements of a laser beam propagating through a turbulent medium,” Acta Frequenza 38, 149–150 (1969).
  26. P. Burlamacci, A. Consortini, L. Ronchi, G. Toraldo di Francia, in Phase and Frequency Instabilities in Electromagnetic Wage Propagation, NATO Advisory Group for Aerospace Research and Development (AGARD) Conf. Proc.33, 665–672 (1970).
  27. S. F. Clifford, G. M. B. Bouricius, G. R. Ochs, M. H. Ackley, “Phase variations in atmospheric optical propagation,”J. Opt. Soc. Am. 61, 1279–1284 (1971).
    [CrossRef]
  28. A. T. Young, “Seeing: its cause and cure,” Astrophys. J. 189, 587–604 (1974).
    [CrossRef]
  29. E. C. Crittenden, A. W. Cooper, E. A. Milne, G. W. Rodeback, S. H. Kalmbach, R. L. Armstead, “Effects of turbulence on imaging through the atmosphere,” Proc. Soc. Photo-Opt. Instrum. Eng. 142, 130–134 (1978).
  30. O. von der Lühe, “A study of a correlation tracking method to improve imaging quality of ground-based solar telescopes,” Astron. Astrophys. 119, 85–94 (1983).
  31. D. L. Fried, “Probability of getting a lucky short-exposure image through turbulence,”J. Opt. Soc. Am. 68, 1651–1658 (1978).
    [CrossRef]

1983 (2)

R. V. Stachnik, P. Nisenson, R. W. Noyes, “Speckle image reconstruction of solar features,” Astrophys. J. 271, L37–L40 (1983).
[CrossRef]

O. von der Lühe, “A study of a correlation tracking method to improve imaging quality of ground-based solar telescopes,” Astron. Astrophys. 119, 85–94 (1983).

1982 (1)

F. Roddier, J. M. Gilli, G. Lund, “On the origin of speckle boiling and its effects in stellar interferometry,”J. Opt. (Paris) 13, 263–271 (1982).
[CrossRef]

1981 (1)

1980 (1)

J. R. Fienup, G. B. Feldkamp, “Astronomical imaging by processing stellar speckle interferometry data,” Proc. Soc. Photo-Opt. Instrum. Eng. 243, 95–100 (1980).

1979 (1)

J. Borgnino, J. Vernin, C. Aime, G. Ricort, “A study of the degradation of daytime astronomical images due to turbulence in the lower atmosphere by measurement of the standard deviation of the angle of arrival,” Sol. Phys. 64, 403–415 (1979).
[CrossRef]

1978 (5)

E. C. Crittenden, A. W. Cooper, E. A. Milne, G. W. Rodeback, S. H. Kalmbach, R. L. Armstead, “Effects of turbulence on imaging through the atmosphere,” Proc. Soc. Photo-Opt. Instrum. Eng. 142, 130–134 (1978).

J. P. Mehltretter, “Balloon-borne imagery of the solar granulation: II. The lifetime of solar granulation,” Astron. Astrophys. 62, 311–316 (1978).

J. Borgnino, J. Vernin, “Experimental verification of the inertial model of atmospheric turbulence from solar limb motion,”J. Opt. Soc. Am. 68, 1056–1062 (1978).
[CrossRef]

C. Aime, G. Ricort, C. Roddier, G. Lago, “Changes in the atmospheric-lens modulation transfer function used for calibration in solar speckle interferometry,”J. Opt. Soc. Am. 68, 1063–1066 (1978).
[CrossRef]

D. L. Fried, “Probability of getting a lucky short-exposure image through turbulence,”J. Opt. Soc. Am. 68, 1651–1658 (1978).
[CrossRef]

1977 (2)

1976 (3)

1974 (3)

J. C. Dainty, “The transfer function, signal-to-noise ratio, and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astron. Soc. 169, 631–641 (1974).

A. T. Young, “Seeing: its cause and cure,” Astrophys. J. 189, 587–604 (1974).
[CrossRef]

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

1973 (1)

1972 (1)

D. Y. Gezari, A. Labeyrie, R. V. Stachnik, “Speckle interferometry: diffraction-limited measurements of nine stars with the 200 inch telescope,” Astrophys. J. 173, L1–L5 (1972).
[CrossRef]

1971 (1)

1970 (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

1969 (1)

P. Burlamacci, A. Consortini, L. Ronchi, G. Toraldo di Francia, “Phase correlation measurements of a laser beam propagating through a turbulent medium,” Acta Frequenza 38, 149–150 (1969).

1967 (1)

P. Burlamacci, A. Consortini, “Study of atmospheric turbulence by means of a laser beam,” Opt. Acta 14, 17–26 (1967).
[CrossRef]

1966 (1)

1952 (1)

S. Chandrasekhar, “A statistical basis for the theory of stellar scintillation,” Mon. Not. R. Astron. Soc. 112, 475–483 (1952).

Ackley, M. H.

Aime, C.

J. Borgnino, J. Vernin, C. Aime, G. Ricort, “A study of the degradation of daytime astronomical images due to turbulence in the lower atmosphere by measurement of the standard deviation of the angle of arrival,” Sol. Phys. 64, 403–415 (1979).
[CrossRef]

C. Aime, G. Ricort, C. Roddier, G. Lago, “Changes in the atmospheric-lens modulation transfer function used for calibration in solar speckle interferometry,”J. Opt. Soc. Am. 68, 1063–1066 (1978).
[CrossRef]

Armstead, R. L.

E. C. Crittenden, A. W. Cooper, E. A. Milne, G. W. Rodeback, S. H. Kalmbach, R. L. Armstead, “Effects of turbulence on imaging through the atmosphere,” Proc. Soc. Photo-Opt. Instrum. Eng. 142, 130–134 (1978).

Barakat, R.

Borgnino, J.

J. Borgnino, J. Vernin, C. Aime, G. Ricort, “A study of the degradation of daytime astronomical images due to turbulence in the lower atmosphere by measurement of the standard deviation of the angle of arrival,” Sol. Phys. 64, 403–415 (1979).
[CrossRef]

J. Borgnino, J. Vernin, “Experimental verification of the inertial model of atmospheric turbulence from solar limb motion,”J. Opt. Soc. Am. 68, 1056–1062 (1978).
[CrossRef]

J. Borgnino, F. Martin, “Correlation between angle-of-arrival fluctuations on the entrance pupil of a solar telescope,”J. Opt. Soc. Am. 67, 1065–1072 (1977).
[CrossRef]

Bouricius, G. M. B.

Breckinridge, J.

Burlamacci, P.

P. Burlamacci, A. Consortini, L. Ronchi, G. Toraldo di Francia, “Phase correlation measurements of a laser beam propagating through a turbulent medium,” Acta Frequenza 38, 149–150 (1969).

P. Burlamacci, A. Consortini, “Study of atmospheric turbulence by means of a laser beam,” Opt. Acta 14, 17–26 (1967).
[CrossRef]

P. Burlamacci, A. Consortini, L. Ronchi, G. Toraldo di Francia, in Phase and Frequency Instabilities in Electromagnetic Wage Propagation, NATO Advisory Group for Aerospace Research and Development (AGARD) Conf. Proc.33, 665–672 (1970).

Chandrasekhar, S.

S. Chandrasekhar, “A statistical basis for the theory of stellar scintillation,” Mon. Not. R. Astron. Soc. 112, 475–483 (1952).

Clifford, S. F.

Consortini, A.

P. Burlamacci, A. Consortini, L. Ronchi, G. Toraldo di Francia, “Phase correlation measurements of a laser beam propagating through a turbulent medium,” Acta Frequenza 38, 149–150 (1969).

P. Burlamacci, A. Consortini, “Study of atmospheric turbulence by means of a laser beam,” Opt. Acta 14, 17–26 (1967).
[CrossRef]

P. Burlamacci, A. Consortini, L. Ronchi, G. Toraldo di Francia, in Phase and Frequency Instabilities in Electromagnetic Wage Propagation, NATO Advisory Group for Aerospace Research and Development (AGARD) Conf. Proc.33, 665–672 (1970).

Cooper, A. W.

E. C. Crittenden, A. W. Cooper, E. A. Milne, G. W. Rodeback, S. H. Kalmbach, R. L. Armstead, “Effects of turbulence on imaging through the atmosphere,” Proc. Soc. Photo-Opt. Instrum. Eng. 142, 130–134 (1978).

Crittenden, E. C.

E. C. Crittenden, A. W. Cooper, E. A. Milne, G. W. Rodeback, S. H. Kalmbach, R. L. Armstead, “Effects of turbulence on imaging through the atmosphere,” Proc. Soc. Photo-Opt. Instrum. Eng. 142, 130–134 (1978).

Dainty, J. C.

J. C. Dainty, “The transfer function, signal-to-noise ratio, and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astron. Soc. 169, 631–641 (1974).

Feldkamp, G. B.

J. R. Fienup, G. B. Feldkamp, “Astronomical imaging by processing stellar speckle interferometry data,” Proc. Soc. Photo-Opt. Instrum. Eng. 243, 95–100 (1980).

Fienup, J. R.

J. R. Fienup, G. B. Feldkamp, “Astronomical imaging by processing stellar speckle interferometry data,” Proc. Soc. Photo-Opt. Instrum. Eng. 243, 95–100 (1980).

Fried, D. L.

Gezari, D. Y.

D. Y. Gezari, A. Labeyrie, R. V. Stachnik, “Speckle interferometry: diffraction-limited measurements of nine stars with the 200 inch telescope,” Astrophys. J. 173, L1–L5 (1972).
[CrossRef]

Gilli, J. M.

F. Roddier, J. M. Gilli, G. Lund, “On the origin of speckle boiling and its effects in stellar interferometry,”J. Opt. (Paris) 13, 263–271 (1982).
[CrossRef]

Kalmbach, S. H.

E. C. Crittenden, A. W. Cooper, E. A. Milne, G. W. Rodeback, S. H. Kalmbach, R. L. Armstead, “Effects of turbulence on imaging through the atmosphere,” Proc. Soc. Photo-Opt. Instrum. Eng. 142, 130–134 (1978).

Karo, D. P.

Knox, K. T.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

Korff, D.

Labeyrie, A.

D. Y. Gezari, A. Labeyrie, R. V. Stachnik, “Speckle interferometry: diffraction-limited measurements of nine stars with the 200 inch telescope,” Astrophys. J. 173, L1–L5 (1972).
[CrossRef]

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Lago, G.

Lund, G.

F. Roddier, J. M. Gilli, G. Lund, “On the origin of speckle boiling and its effects in stellar interferometry,”J. Opt. (Paris) 13, 263–271 (1982).
[CrossRef]

Martin, F.

Mehltretter, J. P.

J. P. Mehltretter, “Balloon-borne imagery of the solar granulation: II. The lifetime of solar granulation,” Astron. Astrophys. 62, 311–316 (1978).

Milne, E. A.

E. C. Crittenden, A. W. Cooper, E. A. Milne, G. W. Rodeback, S. H. Kalmbach, R. L. Armstead, “Effects of turbulence on imaging through the atmosphere,” Proc. Soc. Photo-Opt. Instrum. Eng. 142, 130–134 (1978).

Nisenson, P.

R. V. Stachnik, P. Nisenson, R. W. Noyes, “Speckle image reconstruction of solar features,” Astrophys. J. 271, L37–L40 (1983).
[CrossRef]

R. Barakat, P. Nisenson, “Influence of the wave-front correlation function and deterministic wave-front aberrations on the speckle image-reconstruction problem in the high-light-level regime,”J. Opt. Soc. Am. 71, 1390–1402 (1981).

R. W. Noyes, R. V. Stachnik, P. Nisenson, “Speckle image reconstruction of solar features,” (U.S. Air Force Systems Command, Hanscomb Air Force Base, Mass., May1981).

Noyes, R. W.

R. V. Stachnik, P. Nisenson, R. W. Noyes, “Speckle image reconstruction of solar features,” Astrophys. J. 271, L37–L40 (1983).
[CrossRef]

R. W. Noyes, R. V. Stachnik, P. Nisenson, “Speckle image reconstruction of solar features,” (U.S. Air Force Systems Command, Hanscomb Air Force Base, Mass., May1981).

Ochs, G. R.

Ricort, G.

J. Borgnino, J. Vernin, C. Aime, G. Ricort, “A study of the degradation of daytime astronomical images due to turbulence in the lower atmosphere by measurement of the standard deviation of the angle of arrival,” Sol. Phys. 64, 403–415 (1979).
[CrossRef]

C. Aime, G. Ricort, C. Roddier, G. Lago, “Changes in the atmospheric-lens modulation transfer function used for calibration in solar speckle interferometry,”J. Opt. Soc. Am. 68, 1063–1066 (1978).
[CrossRef]

Roddier, C.

Roddier, F.

F. Roddier, J. M. Gilli, G. Lund, “On the origin of speckle boiling and its effects in stellar interferometry,”J. Opt. (Paris) 13, 263–271 (1982).
[CrossRef]

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (American Elsevier, New York, 1981), Vol. XIX.
[CrossRef]

Rodeback, G. W.

E. C. Crittenden, A. W. Cooper, E. A. Milne, G. W. Rodeback, S. H. Kalmbach, R. L. Armstead, “Effects of turbulence on imaging through the atmosphere,” Proc. Soc. Photo-Opt. Instrum. Eng. 142, 130–134 (1978).

Ronchi, L.

P. Burlamacci, A. Consortini, L. Ronchi, G. Toraldo di Francia, “Phase correlation measurements of a laser beam propagating through a turbulent medium,” Acta Frequenza 38, 149–150 (1969).

P. Burlamacci, A. Consortini, L. Ronchi, G. Toraldo di Francia, in Phase and Frequency Instabilities in Electromagnetic Wage Propagation, NATO Advisory Group for Aerospace Research and Development (AGARD) Conf. Proc.33, 665–672 (1970).

Schneidermann, A. M.

Stachnik, R. V.

R. V. Stachnik, P. Nisenson, R. W. Noyes, “Speckle image reconstruction of solar features,” Astrophys. J. 271, L37–L40 (1983).
[CrossRef]

D. Y. Gezari, A. Labeyrie, R. V. Stachnik, “Speckle interferometry: diffraction-limited measurements of nine stars with the 200 inch telescope,” Astrophys. J. 173, L1–L5 (1972).
[CrossRef]

R. W. Noyes, R. V. Stachnik, P. Nisenson, “Speckle image reconstruction of solar features,” (U.S. Air Force Systems Command, Hanscomb Air Force Base, Mass., May1981).

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, Israel, 1971).

Thompson, B. J.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

Toraldo di Francia, G.

P. Burlamacci, A. Consortini, L. Ronchi, G. Toraldo di Francia, “Phase correlation measurements of a laser beam propagating through a turbulent medium,” Acta Frequenza 38, 149–150 (1969).

P. Burlamacci, A. Consortini, L. Ronchi, G. Toraldo di Francia, in Phase and Frequency Instabilities in Electromagnetic Wage Propagation, NATO Advisory Group for Aerospace Research and Development (AGARD) Conf. Proc.33, 665–672 (1970).

Vernin, J.

J. Borgnino, J. Vernin, C. Aime, G. Ricort, “A study of the degradation of daytime astronomical images due to turbulence in the lower atmosphere by measurement of the standard deviation of the angle of arrival,” Sol. Phys. 64, 403–415 (1979).
[CrossRef]

J. Borgnino, J. Vernin, “Experimental verification of the inertial model of atmospheric turbulence from solar limb motion,”J. Opt. Soc. Am. 68, 1056–1062 (1978).
[CrossRef]

von der Lühe, O.

O. von der Lühe, “A study of a correlation tracking method to improve imaging quality of ground-based solar telescopes,” Astron. Astrophys. 119, 85–94 (1983).

Young, A. T.

A. T. Young, “Seeing: its cause and cure,” Astrophys. J. 189, 587–604 (1974).
[CrossRef]

Acta Frequenza (1)

P. Burlamacci, A. Consortini, L. Ronchi, G. Toraldo di Francia, “Phase correlation measurements of a laser beam propagating through a turbulent medium,” Acta Frequenza 38, 149–150 (1969).

Astron. Astrophys. (3)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

J. P. Mehltretter, “Balloon-borne imagery of the solar granulation: II. The lifetime of solar granulation,” Astron. Astrophys. 62, 311–316 (1978).

O. von der Lühe, “A study of a correlation tracking method to improve imaging quality of ground-based solar telescopes,” Astron. Astrophys. 119, 85–94 (1983).

Astrophys. J. (4)

R. V. Stachnik, P. Nisenson, R. W. Noyes, “Speckle image reconstruction of solar features,” Astrophys. J. 271, L37–L40 (1983).
[CrossRef]

D. Y. Gezari, A. Labeyrie, R. V. Stachnik, “Speckle interferometry: diffraction-limited measurements of nine stars with the 200 inch telescope,” Astrophys. J. 173, L1–L5 (1972).
[CrossRef]

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

A. T. Young, “Seeing: its cause and cure,” Astrophys. J. 189, 587–604 (1974).
[CrossRef]

J. Opt. (Paris) (1)

F. Roddier, J. M. Gilli, G. Lund, “On the origin of speckle boiling and its effects in stellar interferometry,”J. Opt. (Paris) 13, 263–271 (1982).
[CrossRef]

J. Opt. Soc. Am. (12)

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,”J. Opt. Soc. Am. 56, 1372–1379 (1966).
[CrossRef]

S. F. Clifford, G. M. B. Bouricius, G. R. Ochs, M. H. Ackley, “Phase variations in atmospheric optical propagation,”J. Opt. Soc. Am. 61, 1279–1284 (1971).
[CrossRef]

D. Korff, “Analysis of a method for obtaining near-diffraction-limited information in the presence of atmospheric turbulence,”J. Opt. Soc. Am. 63, 971–980 (1973).
[CrossRef]

J. Breckinridge, “Measurement of the amplitude of phase excursions in the earth’s atmosphere,”J. Opt. Soc. Am. 66, 143–144 (1976).
[CrossRef]

C. Roddier, “Measurements of the atmospheric attenuation of the spectral components of astronomical images,”J. Opt. Soc. Am. 66, 478–482 (1976).
[CrossRef]

D. P. Karo, A. M. Schneidermann, “Speckle interferometry lens-atmosphere MTF measurements,”J. Opt. Soc. Am. 66, 1252–1256 (1976).
[CrossRef]

J. Borgnino, F. Martin, “Correlation between angle-of-arrival fluctuations on the entrance pupil of a solar telescope,”J. Opt. Soc. Am. 67, 1065–1072 (1977).
[CrossRef]

D. P. Karo, A. M. Schneidermann, “Transfer functions, correlation scales, and phase retrieval in speckle interferometry,”J. Opt. Soc. Am. 67, 1583–1587 (1977).
[CrossRef]

J. Borgnino, J. Vernin, “Experimental verification of the inertial model of atmospheric turbulence from solar limb motion,”J. Opt. Soc. Am. 68, 1056–1062 (1978).
[CrossRef]

C. Aime, G. Ricort, C. Roddier, G. Lago, “Changes in the atmospheric-lens modulation transfer function used for calibration in solar speckle interferometry,”J. Opt. Soc. Am. 68, 1063–1066 (1978).
[CrossRef]

D. L. Fried, “Probability of getting a lucky short-exposure image through turbulence,”J. Opt. Soc. Am. 68, 1651–1658 (1978).
[CrossRef]

R. Barakat, P. Nisenson, “Influence of the wave-front correlation function and deterministic wave-front aberrations on the speckle image-reconstruction problem in the high-light-level regime,”J. Opt. Soc. Am. 71, 1390–1402 (1981).

Mon. Not. R. Astron. Soc. (2)

S. Chandrasekhar, “A statistical basis for the theory of stellar scintillation,” Mon. Not. R. Astron. Soc. 112, 475–483 (1952).

J. C. Dainty, “The transfer function, signal-to-noise ratio, and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astron. Soc. 169, 631–641 (1974).

Opt. Acta (1)

P. Burlamacci, A. Consortini, “Study of atmospheric turbulence by means of a laser beam,” Opt. Acta 14, 17–26 (1967).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

E. C. Crittenden, A. W. Cooper, E. A. Milne, G. W. Rodeback, S. H. Kalmbach, R. L. Armstead, “Effects of turbulence on imaging through the atmosphere,” Proc. Soc. Photo-Opt. Instrum. Eng. 142, 130–134 (1978).

J. R. Fienup, G. B. Feldkamp, “Astronomical imaging by processing stellar speckle interferometry data,” Proc. Soc. Photo-Opt. Instrum. Eng. 243, 95–100 (1980).

Sol. Phys. (1)

J. Borgnino, J. Vernin, C. Aime, G. Ricort, “A study of the degradation of daytime astronomical images due to turbulence in the lower atmosphere by measurement of the standard deviation of the angle of arrival,” Sol. Phys. 64, 403–415 (1979).
[CrossRef]

Other (4)

P. Burlamacci, A. Consortini, L. Ronchi, G. Toraldo di Francia, in Phase and Frequency Instabilities in Electromagnetic Wage Propagation, NATO Advisory Group for Aerospace Research and Development (AGARD) Conf. Proc.33, 665–672 (1970).

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (American Elsevier, New York, 1981), Vol. XIX.
[CrossRef]

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, Israel, 1971).

R. W. Noyes, R. V. Stachnik, P. Nisenson, “Speckle image reconstruction of solar features,” (U.S. Air Force Systems Command, Hanscomb Air Force Base, Mass., May1981).

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Figures (8)

Fig. 1
Fig. 1

Radial spectral ratio profiles calculated theoretically. The curves correspond to the modified Fried parameters: □, α = 0.04; josaa-1-5-510-i001, α = 0.08; △, α = 0.12; +, α = 0.16; ×, α = 0.20. (a) Long-exposure-time results. (b) Short-exposure-time results.

Fig. 2
Fig. 2

A sample frame (frame no. 14 from series 1): rms contrast is 0.097; the (square) field of view has a side length of 6.23 arc sec. The elongation is artificial.

Fig. 3
Fig. 3

A three-dimensional surface plot of the spectral ratio calculated from the first time series. The spatial units (x- and y-direction) are relative wave numbers. Frequency origin is in the center.

Fig. 4
Fig. 4

Observed radial spectral ratio profiles for the five sets of observations. □, set 1; josaa-1-5-510-i001, set 2; △, set 3; +, set 4; ×, set 5. The drop at q = 0 is a result of the low power there.

Fig. 5
Fig. 5

Observed radial profiles (solid lines) compared to the theoretical profiles (dashed lines) from Fig. 1. The dashed lines correspond to α values of (from left to right) 0.04, 0.08, 0.12, 0.16, and 0.20. (a) Long-exposure-time results. (b) Short-exposure-time results. Markers are the same as in Fig. 4.

Fig. 6
Fig. 6

The conventional seeing indicators plotted as a function of the observed modified Fried parameter α. ×, average rms image contrast; ○, rms radial image displacement.

Fig. 7
Fig. 7

Observed power spectra (dashed lines) and the same spectra corrected with a theoretical Korff MTF and the appropriate α (solid lines). Markers are the same as in Fig. 4.

Fig. 8
Fig. 8

(a). Seeing and telescope corrected rms image contrast as a function of the modified Fried parameter α. (b) Figure of equality Δ of the two-dimensional power spectra as a function of the absolute difference in α for all ten possible combinations of the five data sets. (c) Figure of equality Δ as a function of elapsed time between time series; all ten possible combinations.

Tables (3)

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Table 1 Coefficients for the Relation α = AqB

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Table 2 Seeing Parameters and Contrast Values for the Five Sets of Dataa

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Table 3 The rms Power Difference Δ ˜ i , j and Figure of Equality Δij for all Combinations of i and ja

Equations (31)

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I ( x ¯ ) = O ( x ¯ ) PSF ( x ¯ - x ¯ ) d x ¯ ,
F ( f ¯ ) = F 0 ( f ¯ ) S ( f ¯ ) ,
q ¯ = ( q 1 q 2 ) = 1 f c f ¯ , q = q ¯ ,             0 q 1.
F i ( q ¯ ) = F 0 ( q ¯ ) S i ( q ¯ )
ɛ ( q ¯ ) = F i ( q ¯ ) 2 F i ( q ¯ ) 2 = F 0 ( q ¯ ) 2 F 0 ( q ¯ ) 2 S i ( q ¯ ) 2 S i ( q ¯ ) 2
ɛ ( q ¯ ) = S i ( q ¯ ) 2 S i ( q ¯ ) 2 ,
E ( S i ( q ¯ ) L E ) = T 0 ( q ¯ ) exp [ - 1 2 D ( q ¯ ) ]
E ( S i ( q ¯ ) S E ) = T 0 ( q ¯ ) exp ( - 1 2 { D ( q ¯ ) - ( a ¯ i · q ¯ ) 2 ) } ) ,
T 0 ( q ¯ ) = 2 π ( arccos q - q 1 - q 2 )
E ( S i ( q ¯ ) 2 ) = 1 c W ( v ¯ - q ¯ ) W ( v ¯ ) W ( v ¯ ) × W ( v ¯ - q ¯ ) Q ( v ¯ - v ¯ , q ¯ ) d v ¯ d v ¯ ,
Q ( v ¯ - v ¯ , q ¯ ) = Q ( Δ v ¯ , q ¯ ) = exp { - [ D ( q ¯ ) + D ( Δ v ¯ ) ] + 1 2 [ D ( Δ v ¯ - q ¯ ) + D ( Δ v ¯ + q ¯ ) ] } .
D ( q ¯ ) = D ( q ) = 6.88 ( q / α ) 5 / 3 ,
D ( q ) = 2 b 2 [ 1 - exp ( - a 2 q 2 ) ] ,
E [ ɛ L E ( q ) ] = E [ S i ( q ) L E 2 ] E [ S i ( q ) 2 ] ,
E [ ɛ S E ( q ) ] = E [ ( S i ( q ) S E 2 ) ] E [ S i ( q ) 2 ] ,
- 3.44 ( q / α ) 5 / 3 ( 1 - q 1 / 3 ) ;
α = A · q B α 0.3.
F i ( q ¯ ) = F 0 ( q ¯ ) S i ( q ¯ ) + R i ( q ¯ ) ,             i = 1 , , N .
F i ( q ¯ ) 2 = F 0 ( q ¯ ) 2 S i ( q ¯ ) 2 + R i ( q ¯ ) 2 + cross terms proportional to S i ( q ) ,
F i ( q ¯ ) 2 = F 0 ( q ¯ ) 2 S i ( q ¯ ) 2 + R i ( q ¯ ) 2 + cross terms peroportional to S i ( q ¯ ) R i * ( q ¯ ) .
S i ( q ¯ ) R i * ( q ¯ ) S i ( q ¯ ) R max * ( q ¯ ) .
ɛ [ q ¯ ( q > α ) ] = R i ( q ¯ ) 2 F 0 ( q ¯ ) 2 S i ( q ¯ ) 2 + R i ( q ¯ ) 2 .
E ( R i ( q ¯ ) 2 ) = 1 N σ 2 ( q ¯ ) , E ( R i ( q ¯ ) 2 ) = σ 2 ( q ¯ ) .
E { ɛ [ q ¯ ( q > α ) ] } = σ 2 ( q ¯ ) N [ F 0 ( q ¯ ) 2 S i ( q ¯ ) 2 + σ 2 ( q ¯ ) ] .
SNR = F 0 ( q ¯ ) 2 S i ( q ¯ ) 2 σ 2 ( q ¯ ) ,
E { ɛ [ q ¯ ( q > α ) ] } = { N * [ SNR ( q ¯ ) + 1 ] } - 1 .
ɛ [ q ¯ ( q > α ) ] = N s ( q ¯ ) 2 F 0 ( q ¯ ) 2 S i ( q ¯ ) 2 + N s ( q ¯ ) 2 .
SNR ( q ¯ ) = F 0 ( q ¯ ) 2 S i ( q ¯ ) 2 N s ( q ¯ ) 2
ɛ [ q ¯ / ( q > α ) ] = { SNR ( q ¯ ) + 1 } - 1 .
Δ ˜ i , j = { [ P i ( q ¯ ) - P j ( q ¯ ) ] 2 d q ¯ } 7 / 2 ,             i , j = 1 , , 5 ,
Δ i , j = Δ ˜ i , j [ c ¯ rms , i * c ¯ rms , j ] 1 / 2 .

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