Abstract

The spatiotemporal cross-correlation function of single-star scintillation is estimated. From this estimation and using a priori knowledge of the theoretical shape of the correlation peaks, a method for simultaneously measuring horizontal velocity, altitude, and integrated CN2 value for each atmospheric turbulent layer between 2 and 20 km is described. Taylor’s hypothesis is tested for one particular layer and the lifetime of certain turbulent eddies is estimated. Results are in good agreement with two other methods.

© 1987 Optical Society of America

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References

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  1. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).
  2. J. Vernin, M. Azouit, “Traitement d’image adapté au speckle atmosphérique. I: Formation du speckle en atmosphére turbulente. Propriétés statistiques,” J. Opt. Paris 14, (1983).
  3. A. Rocca, F. Roddier, J. Vernin, “Detection of Atmospheric Turbulent Layers by Spatiotemporal and Spatioangular Correlation Measurements of Stellar-Light Scintillation,” J. Opt. Soc. Am. 64, 1000 (1974).
    [CrossRef]
  4. G. R. Ochs, T-i. 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]
  5. F. Roddier, “The Effects of Atmospheric Turbulence in Optical Astronomy,” Prog. Opt. 19, 281 (1981).
    [CrossRef]
  6. C. Coulman, “Fundamental and Applied Aspects of Astronomical Seeing,” Ann. Rev. Astron. Astrophys. 23, 19 (1985).
    [CrossRef]
  7. J. Vernin, J. Pelon, “Scidar/Lidar Description of a Gravity Wave and Associated Turbulence: Preliminary Results,” Appl. Opt. 25, 2874 (1986).
    [CrossRef] [PubMed]
  8. D. C. Fritts et al., “Research Status and Recommendations from the Alaska Workshop on Gravity Waves and Turbulence in the Middle Atmosphere,” Bull. Am. Meteorol. Soc. 65, 149 (1984).
  9. G. R. Ochs, S. F. Clifford, “Methods for Obtaining Daytime Vertical Profiles of CN2and Wind,” NOAA Tech. ERL WPL-91 (U.S. Department of Commerce, Washington, DC, 1981).
  10. H. Tennekes, J. L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, MA, 1972).
  11. A. N. Kolmogorov, “Local Structure of Turbulence in Incompressible Fluids with Very High Reynolds Number,” Dan SSSR 30, 229 (1941).
  12. J. Vernin, M. Azouit, “Traitement d’image adapté au speckle atmosphérique. II: Analyse multidimensionnelle appliquée au diagnostic à distance de la turbulence,” J. Opt. Paris 14, 131 (1983).
    [CrossRef]
  13. M. Azouit, “Corrélateur d’images video en temps réel,” J. Opt. Paris 13, 345 (1982).
    [CrossRef]
  14. L. T. Little, R. D. Ekers, “A Method for Analysing Drifting Random Patterns in Astronomy and Geophysics,” Astron. Astrophys. 10, 306 (1971).

1986

1985

C. Coulman, “Fundamental and Applied Aspects of Astronomical Seeing,” Ann. Rev. Astron. Astrophys. 23, 19 (1985).
[CrossRef]

1984

D. C. Fritts et al., “Research Status and Recommendations from the Alaska Workshop on Gravity Waves and Turbulence in the Middle Atmosphere,” Bull. Am. Meteorol. Soc. 65, 149 (1984).

1983

J. Vernin, M. Azouit, “Traitement d’image adapté au speckle atmosphérique. I: Formation du speckle en atmosphére turbulente. Propriétés statistiques,” J. Opt. Paris 14, (1983).

J. Vernin, M. Azouit, “Traitement d’image adapté au speckle atmosphérique. II: Analyse multidimensionnelle appliquée au diagnostic à distance de la turbulence,” J. Opt. Paris 14, 131 (1983).
[CrossRef]

1982

M. Azouit, “Corrélateur d’images video en temps réel,” J. Opt. Paris 13, 345 (1982).
[CrossRef]

1981

F. Roddier, “The Effects of Atmospheric Turbulence in Optical Astronomy,” Prog. Opt. 19, 281 (1981).
[CrossRef]

1976

1974

1971

L. T. Little, R. D. Ekers, “A Method for Analysing Drifting Random Patterns in Astronomy and Geophysics,” Astron. Astrophys. 10, 306 (1971).

1941

A. N. Kolmogorov, “Local Structure of Turbulence in Incompressible Fluids with Very High Reynolds Number,” Dan SSSR 30, 229 (1941).

Azouit, M.

J. Vernin, M. Azouit, “Traitement d’image adapté au speckle atmosphérique. I: Formation du speckle en atmosphére turbulente. Propriétés statistiques,” J. Opt. Paris 14, (1983).

J. Vernin, M. Azouit, “Traitement d’image adapté au speckle atmosphérique. II: Analyse multidimensionnelle appliquée au diagnostic à distance de la turbulence,” J. Opt. Paris 14, 131 (1983).
[CrossRef]

M. Azouit, “Corrélateur d’images video en temps réel,” J. Opt. Paris 13, 345 (1982).
[CrossRef]

Clifford, S. F.

G. R. Ochs, T-i. 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]

G. R. Ochs, S. F. Clifford, “Methods for Obtaining Daytime Vertical Profiles of CN2and Wind,” NOAA Tech. ERL WPL-91 (U.S. Department of Commerce, Washington, DC, 1981).

Coulman, C.

C. Coulman, “Fundamental and Applied Aspects of Astronomical Seeing,” Ann. Rev. Astron. Astrophys. 23, 19 (1985).
[CrossRef]

Ekers, R. D.

L. T. Little, R. D. Ekers, “A Method for Analysing Drifting Random Patterns in Astronomy and Geophysics,” Astron. Astrophys. 10, 306 (1971).

Fritts, D. C.

D. C. Fritts et al., “Research Status and Recommendations from the Alaska Workshop on Gravity Waves and Turbulence in the Middle Atmosphere,” Bull. Am. Meteorol. Soc. 65, 149 (1984).

Kolmogorov, A. N.

A. N. Kolmogorov, “Local Structure of Turbulence in Incompressible Fluids with Very High Reynolds Number,” Dan SSSR 30, 229 (1941).

Lawrence, R. S.

Little, L. T.

L. T. Little, R. D. Ekers, “A Method for Analysing Drifting Random Patterns in Astronomy and Geophysics,” Astron. Astrophys. 10, 306 (1971).

Lumley, J. L.

H. Tennekes, J. L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, MA, 1972).

Ochs, G. R.

G. R. Ochs, T-i. 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]

G. R. Ochs, S. F. Clifford, “Methods for Obtaining Daytime Vertical Profiles of CN2and Wind,” NOAA Tech. ERL WPL-91 (U.S. Department of Commerce, Washington, DC, 1981).

Pelon, J.

Rocca, A.

Roddier, F.

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).

Tennekes, H.

H. Tennekes, J. L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, MA, 1972).

Vernin, J.

J. Vernin, J. Pelon, “Scidar/Lidar Description of a Gravity Wave and Associated Turbulence: Preliminary Results,” Appl. Opt. 25, 2874 (1986).
[CrossRef] [PubMed]

J. Vernin, M. Azouit, “Traitement d’image adapté au speckle atmosphérique. II: Analyse multidimensionnelle appliquée au diagnostic à distance de la turbulence,” J. Opt. Paris 14, 131 (1983).
[CrossRef]

J. Vernin, M. Azouit, “Traitement d’image adapté au speckle atmosphérique. I: Formation du speckle en atmosphére turbulente. Propriétés statistiques,” J. Opt. Paris 14, (1983).

A. Rocca, F. Roddier, J. Vernin, “Detection of Atmospheric Turbulent Layers by Spatiotemporal and Spatioangular Correlation Measurements of Stellar-Light Scintillation,” J. Opt. Soc. Am. 64, 1000 (1974).
[CrossRef]

Wang, T-i.

Ann. Rev. Astron. Astrophys.

C. Coulman, “Fundamental and Applied Aspects of Astronomical Seeing,” Ann. Rev. Astron. Astrophys. 23, 19 (1985).
[CrossRef]

Appl. Opt.

Astron. Astrophys.

L. T. Little, R. D. Ekers, “A Method for Analysing Drifting Random Patterns in Astronomy and Geophysics,” Astron. Astrophys. 10, 306 (1971).

Bull. Am. Meteorol. Soc.

D. C. Fritts et al., “Research Status and Recommendations from the Alaska Workshop on Gravity Waves and Turbulence in the Middle Atmosphere,” Bull. Am. Meteorol. Soc. 65, 149 (1984).

Dan SSSR

A. N. Kolmogorov, “Local Structure of Turbulence in Incompressible Fluids with Very High Reynolds Number,” Dan SSSR 30, 229 (1941).

J. Opt. Paris

J. Vernin, M. Azouit, “Traitement d’image adapté au speckle atmosphérique. II: Analyse multidimensionnelle appliquée au diagnostic à distance de la turbulence,” J. Opt. Paris 14, 131 (1983).
[CrossRef]

M. Azouit, “Corrélateur d’images video en temps réel,” J. Opt. Paris 13, 345 (1982).
[CrossRef]

J. Vernin, M. Azouit, “Traitement d’image adapté au speckle atmosphérique. I: Formation du speckle en atmosphére turbulente. Propriétés statistiques,” J. Opt. Paris 14, (1983).

J. Opt. Soc. Am.

Prog. Opt.

F. Roddier, “The Effects of Atmospheric Turbulence in Optical Astronomy,” Prog. Opt. 19, 281 (1981).
[CrossRef]

Other

G. R. Ochs, S. F. Clifford, “Methods for Obtaining Daytime Vertical Profiles of CN2and Wind,” NOAA Tech. ERL WPL-91 (U.S. Department of Commerce, Washington, DC, 1981).

H. Tennekes, J. L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, MA, 1972).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).

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Figures (7)

Fig. 1
Fig. 1

Radial cut of the theoretical autocorrelation function of the atmospheric speckle pattern to an arbitrary scale, assuming a thin turbulent layer at the altitudes indicated on each curve: 2.5, 5, 7.5, and 10 km; C N 2 Δ h i is assumed to be the same in each case (curves from Ref. 5).

Fig. 2
Fig. 2

Schematic representation of the experimental apparatus. The atmospheric speckle patterns at the telescope entrance pupil are imaged onto both a TV camera and a photomultiplier pinhole.

Fig. 3
Fig. 3

Two-dimensional representation of an experimental spatiotemporal cross-correlation function obtained with the star Capella. The integration time is 5 min, 30 s, the exposure time of the atmospheric speckles is 0.5 ms and the time lag is 6 ms. Black pixels refer to the lowest correlation values whereas the white pixels refer to the highest. The orientation and the scale are indicated in the figure. For example, the upper right correlation peak is 0.36 m apart from the origin, corresponding to a 60-m/s wind speed with a northeasterly orientation.

Fig. 4
Fig. 4

Experimental values of Ci(uiτ,viτ,τ) for a delay ranging from 9 ms to 22 ms. In this case ui and vi are, respectively, 9 and 10 m/s. The shaded part shows the standard deviation σN of the noise. The two straight dotted lines represent the two linear regressions obtained from our experimental data. These straight lines reach the horizontal axis at ~40 and 80 ms, respectively.

Fig. 5
Fig. 5

Comparison between our experimental hodograph (full line) and that of the Meteorological Station of Nîmes (dotted line). The numbers written along the curves refer to the altitudes expressed in kilometers. The two layers found at 12 km are inside the vertical resolution (1 km).

Fig. 6
Fig. 6

Experimental vertical profile of C N 2 Δ h i. The shaded parts represent the detectivity threshold of C N 2 Δ h i for each altitude detected. Our method found two correlation peaks corresponding to 12 km; the lowest value of C N 2 Δ h i is indicated by the vertical dotted line.

Fig. 7
Fig. 7

Function μ(uiτ,viτ,h) is plotted, in the case of theoretical peaks, for four different layers located at 4, 6, 8, and 10 km; ui,vi, and τ are arbitrary constants and the curves are given to an arbitrary scale. These curves indicate the sensitivity of our method for the detection of turbulent layer altitudes.

Equations (34)

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W i ( f x , f y ) = 1.54 λ - 2 ( f x 2 + f y 2 ) - 11 / 6 h i - Δ h i / 2 h i + Δ h i / 2 C 2
W i ( f x , f y ) = 1.54 λ - 2 ( f x 2 + f y 2 ) - 11 / 6 C N 2 Δ h i sin 2 π λ h i ( f x 2 + f y 2 ) .
C i ( x , y , 0 ) = - + W i ( f x , f y ) exp [ - 2 i π ( x f x + y f y ) ] d f x d f y .
C i ( x , y , τ ) = C i ( x - u i τ , y - v i τ , 0 ) .
C i ( u i τ , v i τ , τ ) = C i ( 0 , 0 , 0 ) ,
Γ ( x , y , h ) = 1.54 λ - 2 - + ( f x 2 + f y 2 ) - 11 / 6 sin 2 π λ h ( f x 2 + f y 2 ) · exp [ - 2 i π ( x f x + y f y ) ] d f x d f y ,
C i ( x , y , τ ) = C N 2 Δ h i Γ ( x - u i τ , y - v i τ , h i ) .
C ( x , y , τ ) = i = 1 N C i ( x , y , τ ) ,
C ( x , y , τ ) = i = 1 N C N 2 Δ h i Γ ( x - u i τ , y - v i τ , h i ) .
C ( 0 , 0 , 0 ) = i = 1 N C i ( u i τ , v i τ , τ ) ,
C * ( x , y , τ ) = I ( x , y ) - 2 · I ( x + x 0 , y + y 0 , t ) · - + I ( x , y , t + τ ) · H ( x - x 0 , y - y 0 ) d x d y
C * ( x , y , τ ) = I ( x , y ) - 2 · - + I ( x + x 0 , y + y 0 , t ) · I ( x , y , t + τ ) H ( x - x 0 , y - y 0 ) d x d y .
C ( x + x 0 - x , y + y 0 - y , τ ) = I ( x , y ) - 2 I ( x + x 0 , y + y 0 , t ) · I ( x , y , t + τ ) .
C * ( x , y , τ ) = - + C ( x - X , y - Y , τ ) · H ( X , Y ) d X d Y ,
C * ( x , y , τ ) = C ( x , y , τ ) * H ( x , y ) ,
C * ( x , y , τ ) = C ( x , y , τ ) * H ( x , y ) + n ( x , y ) .
C * ( x , y , τ ) = i = 1 N C N 2 Δ h i Γ ( x - u i τ , y - v i τ , h i ) * H ( x , y ) + n ( x , y ) .
h > d 2 / λ .
C i ( 0 , 0 , 0 ) = 19.2 λ - 7 / 6 h i 5 / 6 C N 2 Δ h i .
C N 2 Δ h i ( min ) = 2.5 σ N / 19.2 λ - 7 / 6 h i 5 / 6 .
0.05 × 10 - 12 m 1 / 3 < 4 km 14 km C N 2 ( h ) d h < 0.50 × 10 - 12 m 1 / 3 ,
4 km 18 km C N 2 Δ h i = 0.41 × 10 - 12 m 1 / 3 .
i = 1 8 C i ( u i τ , v i τ , τ ) = 0.27 ,
C ( 0 , 0 , 0 ) = 0.29.
C * ( x , y , τ ) = i = 1 N C N 2 Δ h i Γ s ( x - u i τ , y - v i τ i h i ) + n ( x , y ) ,
Γ s ( x - u i τ , y - v i τ , h i ) = Γ ( x - u i τ , y - v i τ , h i ) * H ( x , y )
W ( x - a , y - β ) = 1 if ( x - α ) 2 + ( y - β ) 2 r 0 2 = 0 elsewhere ,
μ ( α , β , h ) = x , y W ( x - α , y - β ) [ C * ( x , y , τ ) - γ ( α , β ) Γ s ( x - α , y - β , h ) ] 2 ,
γ ( α , β ) = x , y W ( x - α , y - β ) C * ( x , y , τ ) x , y W ( x - α , y - β ) Γ s ( x - α , y - β , h )
W ( x - α , y - β ) C * ( x , y , τ ) = W ( x - α , y - β ) [ C N 2 Δ h i Γ s ( x - u i τ , y - v i τ , h i ) + n ( x , y ) ] .
γ ( α , β ) = C N 2 Δ h i · x , y W ( x - α , y - β ) Γ s ( x - u i τ , y - v i τ , h i ) x , y W ( x - α , y - β ) Γ s ( x - α , y - β , h ) .
μ ( u i τ , v i τ , h i ) = x , y W ( x - u i τ , y - v i τ ) n 2 ( x , y ) ,
u i = α / τ             and             v i = β / τ ,
γ ( u i τ , v i τ , h i ) = C N 2 Δ h i .

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