Abstract

In February 2005 a joint atmospheric propagation experiment was conducted between the Australian Defence Science and Technology Organisation and the University of Central Florida. A Gaussian beam was propagated along a horizontal 1500  m path near the ground. Scintillation was measured simultaneously at three receivers of diameters 1, 5, and 13  mm. Scintillation theory combined with a numerical scheme was used to infer the structure constant Cn2, the inner scale l0, and the outer scale L0 from the optical measurements. At the same time, Cn2 measurements were taken by a commercial scintillometer, set up parallel to the optical path. The Cn2 values from the inferred scheme and the commercial scintillometer predict the same behavior, but the inferred scheme consistently gives slightly smaller Cn2 values.

© 2006 Optical Society of America

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References

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  1. L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave model," Waves Random Media 11, 271-291 (2001).
    [CrossRef]
  2. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press,2001).
  3. M. E. Gracheva and A. S. Gurvich, "Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth," Izv. Vyssh. Uchebn. Zaved. Radiofiz. 8, 717-724 (1965).
  4. G. Parry, "Measurements of atmospheric turbulence-induced intensity fluctuations in a laser beam," Opt. Acta 28, 715-728 (1981).
  5. R. L. Phillips and L. C. Andrews, "Measured statistics of laser-light scattering in atmospheric turbulence," J. Opt. Soc. Am. 71, 1440-1445 (1981).
  6. A. Consortini, R. Cochetti, J. H. Churnside, and R. J. Hill, "Inner-scale effect on irradiance variance measured for weak-to-strong atmospheric scintillation," J. Opt. Soc. Am. A 10, 2354-2362 (1993).
  7. T.-I. Wang, G. R. Ochs, and S. F. Clifford, "A saturation-resistant optical scintillometer to measure Cn2," J. Opt. Soc. Am. 68, 334-338 (1978).
  8. R. G. Frehlich and G. R. Ochs, "Effects of saturation on the optical scintillometer," Appl. Opt. 29, 548-553 (1990).
  9. P. M. Livingston, "Proposed method of inner scale measurement in a turbulent atmosphere," Appl. Opt. 11, 684-687 (1972).
  10. R. J. Hill and G. R. Ochs, "Fine calibration of large-aperture optical scintillometers and an optical estimate of inner scale of turbulence," Appl. Opt. 17, 3608-3612 (1978).
  11. G. R. Ochs and R. J. Hill, "Optical-scintillation method of measuring turbulence inner scale," Appl. Opt. 24, 2430-2432 (1985).
  12. Scintec, "Boundary layer scintillometers," http://www.scintec.com/Site.1/turb.htm.
  13. M. L. Wesley and Z. I. Derzko, "Atmospheric turbulence parameters from visual resolution," Appl. Opt. 14, 847-853 (1975).
  14. M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. Moore, and M. R. Suite, "Multiple methods for measuring atmospheric turbulence," Proc. SPIE 4821, 300-309 (2002).
    [CrossRef]
  15. R. S. Lawrence, G. R. Ochs, and S. F. Clifford, "Measurements of atmospheric turbulence relevant to optical propagation," J. Opt. Soc. Am. 60, 826-830 (1970).
  16. R. J. Hill, "Inner scale effect on the irradiance of light propagating in atmospheric turbulence," in Proc. SPIE 410, 67-72 (1983).
  17. D. A. Gray and A. T. Waterman, Jr., "Measurement of fine-structure atmospheric structure using an optical propagation technique," J. Geophys. Res. 75, 1077-1082 (1970).
  18. R. G. Frehlich, "Estimation of the parameters of the atmospheric turbulence spectrum using measurements of the spatial intensity covariance," Appl. Opt. 27, 2194-2198 (1988).
  19. E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, and Z. Azar, "Optical measurement of the inner scale of turbulence," J. Phys. D 21, S41-S44 (1988).
    [CrossRef]
  20. R. J. Hill, "Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes," Waves Random Media 2, 179-201 (1992).
    [CrossRef]
  21. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, "Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum," Waves Random Media 10, 53-70 (2000).
    [CrossRef]
  22. R. J. Hill, "Models of the scalar spectrum for turbulent advection," J. Fluid Mech. 88, 541-562 (1978).
    [CrossRef]
  23. L. C. Andrews, "An analytical model for the refractive index power spectrum and its application to optical scintillation in the atmosphere," J. Mod. Opt. 39, 1849-1853 (1992).
  24. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. H. Vetterling, "Downhill simplex method in multidimensions," in Numerical Recipes in C: The Art of Scientific Computing (1988-1992), Chap. 10.4, pp. 408-412, http://www.library.cornell.edu/nr/bookcpdf/c10-4.pdf.
  25. Z. Dong, "Downhill simplex method in multidimensions,"http://freehost26.websamba.com/zhanshan2002/matlab/simplex.html.
  26. Ref. 2, Chap. 1.3.1, pp. 14-15.
  27. Ref. 2, Chap. 6.5.3, p. 195.
  28. Ref. 2, Chap. 6.5.2, p. 193.
  29. Ref. 2, Chap 5.5.1, p. 162.

2002 (1)

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. Moore, and M. R. Suite, "Multiple methods for measuring atmospheric turbulence," Proc. SPIE 4821, 300-309 (2002).
[CrossRef]

2001 (2)

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave model," Waves Random Media 11, 271-291 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press,2001).

2000 (1)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, "Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum," Waves Random Media 10, 53-70 (2000).
[CrossRef]

1993 (1)

1992 (2)

L. C. Andrews, "An analytical model for the refractive index power spectrum and its application to optical scintillation in the atmosphere," J. Mod. Opt. 39, 1849-1853 (1992).

R. J. Hill, "Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes," Waves Random Media 2, 179-201 (1992).
[CrossRef]

1990 (1)

1988 (2)

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, and Z. Azar, "Optical measurement of the inner scale of turbulence," J. Phys. D 21, S41-S44 (1988).
[CrossRef]

R. G. Frehlich, "Estimation of the parameters of the atmospheric turbulence spectrum using measurements of the spatial intensity covariance," Appl. Opt. 27, 2194-2198 (1988).

1985 (1)

1983 (1)

R. J. Hill, "Inner scale effect on the irradiance of light propagating in atmospheric turbulence," in Proc. SPIE 410, 67-72 (1983).

1981 (2)

G. Parry, "Measurements of atmospheric turbulence-induced intensity fluctuations in a laser beam," Opt. Acta 28, 715-728 (1981).

R. L. Phillips and L. C. Andrews, "Measured statistics of laser-light scattering in atmospheric turbulence," J. Opt. Soc. Am. 71, 1440-1445 (1981).

1978 (3)

1975 (1)

1972 (1)

1970 (2)

D. A. Gray and A. T. Waterman, Jr., "Measurement of fine-structure atmospheric structure using an optical propagation technique," J. Geophys. Res. 75, 1077-1082 (1970).

R. S. Lawrence, G. R. Ochs, and S. F. Clifford, "Measurements of atmospheric turbulence relevant to optical propagation," J. Opt. Soc. Am. 60, 826-830 (1970).

1965 (1)

M. E. Gracheva and A. S. Gurvich, "Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth," Izv. Vyssh. Uchebn. Zaved. Radiofiz. 8, 717-724 (1965).

Al-Habash, M. A.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave model," Waves Random Media 11, 271-291 (2001).
[CrossRef]

Andrews, L. C.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave model," Waves Random Media 11, 271-291 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press,2001).

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, "Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum," Waves Random Media 10, 53-70 (2000).
[CrossRef]

L. C. Andrews, "An analytical model for the refractive index power spectrum and its application to optical scintillation in the atmosphere," J. Mod. Opt. 39, 1849-1853 (1992).

R. L. Phillips and L. C. Andrews, "Measured statistics of laser-light scattering in atmospheric turbulence," J. Opt. Soc. Am. 71, 1440-1445 (1981).

Azar, Z.

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, and Z. Azar, "Optical measurement of the inner scale of turbulence," J. Phys. D 21, S41-S44 (1988).
[CrossRef]

Azoulay, E.

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, and Z. Azar, "Optical measurement of the inner scale of turbulence," J. Phys. D 21, S41-S44 (1988).
[CrossRef]

Burris, H. R.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. Moore, and M. R. Suite, "Multiple methods for measuring atmospheric turbulence," Proc. SPIE 4821, 300-309 (2002).
[CrossRef]

Churnside, J. H.

Clifford, S. F.

Cochetti, R.

Consortini, A.

Derzko, Z. I.

Dong, Z.

Z. Dong, "Downhill simplex method in multidimensions,"http://freehost26.websamba.com/zhanshan2002/matlab/simplex.html.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. H. Vetterling, "Downhill simplex method in multidimensions," in Numerical Recipes in C: The Art of Scientific Computing (1988-1992), Chap. 10.4, pp. 408-412, http://www.library.cornell.edu/nr/bookcpdf/c10-4.pdf.

Frehlich, R. G.

Gracheva, M. E.

M. E. Gracheva and A. S. Gurvich, "Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth," Izv. Vyssh. Uchebn. Zaved. Radiofiz. 8, 717-724 (1965).

Gray, D. A.

D. A. Gray and A. T. Waterman, Jr., "Measurement of fine-structure atmospheric structure using an optical propagation technique," J. Geophys. Res. 75, 1077-1082 (1970).

Gurvich, A. S.

M. E. Gracheva and A. S. Gurvich, "Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth," Izv. Vyssh. Uchebn. Zaved. Radiofiz. 8, 717-724 (1965).

Hill, R. J.

A. Consortini, R. Cochetti, J. H. Churnside, and R. J. Hill, "Inner-scale effect on irradiance variance measured for weak-to-strong atmospheric scintillation," J. Opt. Soc. Am. A 10, 2354-2362 (1993).

R. J. Hill, "Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes," Waves Random Media 2, 179-201 (1992).
[CrossRef]

G. R. Ochs and R. J. Hill, "Optical-scintillation method of measuring turbulence inner scale," Appl. Opt. 24, 2430-2432 (1985).

R. J. Hill, "Inner scale effect on the irradiance of light propagating in atmospheric turbulence," in Proc. SPIE 410, 67-72 (1983).

R. J. Hill, "Models of the scalar spectrum for turbulent advection," J. Fluid Mech. 88, 541-562 (1978).
[CrossRef]

R. J. Hill and G. R. Ochs, "Fine calibration of large-aperture optical scintillometers and an optical estimate of inner scale of turbulence," Appl. Opt. 17, 3608-3612 (1978).

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press,2001).

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave model," Waves Random Media 11, 271-291 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, "Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum," Waves Random Media 10, 53-70 (2000).
[CrossRef]

Jetter, A.

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, and Z. Azar, "Optical measurement of the inner scale of turbulence," J. Phys. D 21, S41-S44 (1988).
[CrossRef]

Kohnle, A.

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, and Z. Azar, "Optical measurement of the inner scale of turbulence," J. Phys. D 21, S41-S44 (1988).
[CrossRef]

Lawrence, R. S.

Livingston, P. M.

Moore, C.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. Moore, and M. R. Suite, "Multiple methods for measuring atmospheric turbulence," Proc. SPIE 4821, 300-309 (2002).
[CrossRef]

Ochs, G. R.

Parry, G.

G. Parry, "Measurements of atmospheric turbulence-induced intensity fluctuations in a laser beam," Opt. Acta 28, 715-728 (1981).

Phillips, R. L.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press,2001).

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave model," Waves Random Media 11, 271-291 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, "Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum," Waves Random Media 10, 53-70 (2000).
[CrossRef]

R. L. Phillips and L. C. Andrews, "Measured statistics of laser-light scattering in atmospheric turbulence," J. Opt. Soc. Am. 71, 1440-1445 (1981).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. H. Vetterling, "Downhill simplex method in multidimensions," in Numerical Recipes in C: The Art of Scientific Computing (1988-1992), Chap. 10.4, pp. 408-412, http://www.library.cornell.edu/nr/bookcpdf/c10-4.pdf.

Reed, A. E.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. Moore, and M. R. Suite, "Multiple methods for measuring atmospheric turbulence," Proc. SPIE 4821, 300-309 (2002).
[CrossRef]

Scharpf, W. J.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. Moore, and M. R. Suite, "Multiple methods for measuring atmospheric turbulence," Proc. SPIE 4821, 300-309 (2002).
[CrossRef]

Suite, M. R.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. Moore, and M. R. Suite, "Multiple methods for measuring atmospheric turbulence," Proc. SPIE 4821, 300-309 (2002).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. H. Vetterling, "Downhill simplex method in multidimensions," in Numerical Recipes in C: The Art of Scientific Computing (1988-1992), Chap. 10.4, pp. 408-412, http://www.library.cornell.edu/nr/bookcpdf/c10-4.pdf.

Thiermann, V.

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, and Z. Azar, "Optical measurement of the inner scale of turbulence," J. Phys. D 21, S41-S44 (1988).
[CrossRef]

Vetterling, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. H. Vetterling, "Downhill simplex method in multidimensions," in Numerical Recipes in C: The Art of Scientific Computing (1988-1992), Chap. 10.4, pp. 408-412, http://www.library.cornell.edu/nr/bookcpdf/c10-4.pdf.

Vilcheck, M. J.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. Moore, and M. R. Suite, "Multiple methods for measuring atmospheric turbulence," Proc. SPIE 4821, 300-309 (2002).
[CrossRef]

Wang, T.-I.

Waterman, A. T.

D. A. Gray and A. T. Waterman, Jr., "Measurement of fine-structure atmospheric structure using an optical propagation technique," J. Geophys. Res. 75, 1077-1082 (1970).

Wesley, M. L.

Appl. Opt. (6)

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (1)

M. E. Gracheva and A. S. Gurvich, "Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth," Izv. Vyssh. Uchebn. Zaved. Radiofiz. 8, 717-724 (1965).

J. Fluid Mech. (1)

R. J. Hill, "Models of the scalar spectrum for turbulent advection," J. Fluid Mech. 88, 541-562 (1978).
[CrossRef]

J. Geophys. Res. (1)

D. A. Gray and A. T. Waterman, Jr., "Measurement of fine-structure atmospheric structure using an optical propagation technique," J. Geophys. Res. 75, 1077-1082 (1970).

J. Mod. Opt. (1)

L. C. Andrews, "An analytical model for the refractive index power spectrum and its application to optical scintillation in the atmosphere," J. Mod. Opt. 39, 1849-1853 (1992).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

J. Phys. D (1)

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, and Z. Azar, "Optical measurement of the inner scale of turbulence," J. Phys. D 21, S41-S44 (1988).
[CrossRef]

Opt. Acta (1)

G. Parry, "Measurements of atmospheric turbulence-induced intensity fluctuations in a laser beam," Opt. Acta 28, 715-728 (1981).

Proc. SPIE (2)

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. Moore, and M. R. Suite, "Multiple methods for measuring atmospheric turbulence," Proc. SPIE 4821, 300-309 (2002).
[CrossRef]

R. J. Hill, "Inner scale effect on the irradiance of light propagating in atmospheric turbulence," in Proc. SPIE 410, 67-72 (1983).

Waves Random Media (3)

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave model," Waves Random Media 11, 271-291 (2001).
[CrossRef]

R. J. Hill, "Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes," Waves Random Media 2, 179-201 (1992).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, "Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum," Waves Random Media 10, 53-70 (2000).
[CrossRef]

Other (8)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press,2001).

Scintec, "Boundary layer scintillometers," http://www.scintec.com/Site.1/turb.htm.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. H. Vetterling, "Downhill simplex method in multidimensions," in Numerical Recipes in C: The Art of Scientific Computing (1988-1992), Chap. 10.4, pp. 408-412, http://www.library.cornell.edu/nr/bookcpdf/c10-4.pdf.

Z. Dong, "Downhill simplex method in multidimensions,"http://freehost26.websamba.com/zhanshan2002/matlab/simplex.html.

Ref. 2, Chap. 1.3.1, pp. 14-15.

Ref. 2, Chap. 6.5.3, p. 195.

Ref. 2, Chap. 6.5.2, p. 193.

Ref. 2, Chap 5.5.1, p. 162.

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Figures (8)

Fig. 1
Fig. 1

On-axis SI for a collimated beam wave as a function of variable propagation distance with Cn 2 = 5 × 10−13 m−2∕3 and outer-scale values L0 = ∞ and L0 = 1 m. Two cases of inner scale are shown, l0 = 2 mm and l0 = 8 mm.[1, 2]

Fig. 2
Fig. 2

Relative scale sizes of an optical wave as a function of propagation distance. The shaded region shows the range of scale sizes that are ineffective in producing scintillation under strong fluctuations.[1, 2]

Fig. 3
Fig. 3

(Color online) Three Ge photodiodes of diameter 13 (upper), 1 (lower left), and 5 mm (lower right). The center-to-center distance of the photodiodes is 3.175 cm.

Fig. 4
Fig. 4

(a) Comparison between the Cn 2 value obtained with the downhill simplex method (inferred Cn 2 ) and the Cn 2 value measured by the scintillometer (scintillometer Cn 2 ). (b) The ratio of the two Cn 2 values for all data runs.

Fig. 5
Fig. 5

Inner scale values inferred with the downhill Simplex method, while keeping the inferred Cn 2 fixed.

Fig. 6
Fig. 6

Outer scale values inferred with the downhill Simplex method, while keeping the inferred Cn 2 fixed.

Fig. 7
Fig. 7

SI for the receiver apertures of diameters 1, 5, and 13 mm. Data are the measured SI, theory corresponds to the SI calculated with the inferred atmospheric parameters, and scintillometer refers to the SI calculated with the scintillometer Cn 2 and its inferred l0 and L0 values.

Fig. 8
Fig. 8

Measured experimental SI with inner-scale values between 6 and 7 mm for the (a) and (b) 1 mm and (c) and (d) 13 mm receiver apertures is plotted versus σ1, calculated with the inferred (a) and (c) Cn 2 and the (b) and (d) scintillometer Cn 2 . The solid curves show the theoretical scintillation curve for l0 = 6 mm and l0 = 7 mm, respectively, both with L0 = 1.4 m.

Tables (1)

Tables Icon

Table 1 Atmospheric Parameters Used to Calculate the Theory (I) and the Scintillometer (II) Scintillation Indices Shown in Fig. 7

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

σ I 2 = I 2 I 2 1 ,
σ I 2 ( l 0 , L 0 , C n 2 ) = exp [ σ ln x 2     ( l 0 , L 0 , C n 2 ) + σ ln y 2     ( l 0 , C n 2 ) ] 1.
σ I 2 ( l 0 , L 0 , C n 2 , D ) = exp [ σ ln x 2     ( l 0 , L 0 , C n 2 , D ) + σ ln y 2     ( l 0 , C n 2 , D ) ] 1 ,
Φ n ( κ ) = 0.033 C n 2 κ 11 / 3 exp ( κ 2 / κ l 2 ) [ 1 + 1.80 ( κ κ l ) 0.25 ( κ κ l ) 7 / 6 ] ,
σ I 2 ( l 0 , L 0 , C n 2 , D j ) = σ I , exp 2 ( D j ) , j = 1 , 2 , 3 ,
f ( l 0 , L 0 , C n 2 ) = j = 1 3 [ | σ I 2 ( l 0 , L 0 , C n 2 , D j ) σ I , exp 2 ( D j ) | ] p ,
σ I , data 2 = σ I , data 2 ( V ¯ data ) 2 ( V ¯ data V ¯ BG ) 2 ,
W 0 = d 2 2 , F 0 = - W 0 θ d i v .
Θ 0 = 1 L F 0 , Λ 0 = 2 L k W 0 ,
Θ = Θ 0 Θ 0 2 + Λ 0 2 , Θ ¯ = 1 Θ , Λ = Λ 0 Θ 0 2 + Λ 0 2 .
σ I 2 ( l 0 , L 0 , C n 2 , D ) = exp [ σ ln x 2 ( l 0 , C n 2 , D ) σ ln x 2 ( l 0 , L 0 , C n 2 , D ) + σ ln y 2 ( l 0 , C n 2 , D ) ] 1 ,
Q l ( l 0 ) = 10.89 L k l 0 2 ,
Q 0 ( L 0 ) = 64 π 2 L k L 0 2 ,
σ I 2 ( C n 2 ) = 1.23 C n 2 k 7 / 6 L 11 / 6 ,
Ω G ( D ) = 16 L κD 2 ,
σ ln x 2     ( l 0 , C n 2 , D ) = 0.49 σ 1 2 ( Ω G Λ Ω G + Λ ) ×  ( 1 3 1 2 Θ ¯ + 1 5 Θ ¯ 2 ) ( η x d Q l η x d + Q l ) 7 / 6 ×  [ 1 + 1.75 ( η x d η x d + Q l ) 1 / 2 −  0.25 ( η x d η x d + Q l ) 7 / 12 ] ,
η x d ( l 0 , C n 2 , D ) = η x 1 + 0.40 η x ( 2 Θ ¯ ) / ( Ω G + Λ ) ,
1 η x ( l 0 , C n 2 ) = 0.38 1 3.21 Θ ¯ + 5.29 Θ ¯ 2 + 0.47 σ 1 2 Q l 1 / 6 [ ( 1 3 1 2 Θ ¯ + 1 5 Θ ¯ 2 ) 1.22 Θ ¯ ] ,
σ ln x 2 ( l 0 , L 0 , C n 2 , D ) = σ ln x 2 ( l 0 , C n 2 , D ) | η x d η x d 0 ,
η x d 0 ( l 0 , L 0 , C n 2 , D ) = η x d Q 0 η x d + Q 0 .
σ ln y 2 ( l 0 , C n 2 , D ) = 1.27 σ 1 2 η y 5 / 6 1 + 0.40 η y / ( Ω G + Λ ) ,
η y ( l 0 , C n 2 ) = 3 ( σ 1 2 / σ G 2 ) 6 / 5 [ 1 + 0.65 ( σ G 2 ) 6 / 5 ] ,
σ G 2 ( l 0 , C n 2 ) = 3.86 σ 1 2 { 0.40 [ ( 1 + 2 Θ ) 2 + ( 2 Λ + 3 / Q l ) 2 ] 11 / 12 [ ( 1 + 2 Θ ) 2 + 4 Λ 2 ] 1 / 2 × [ sin ( 11 6 φ 2 + φ 1 ) + 2.61 sin ( 4 3 φ 2 + φ 1 ) [ ( 1 + 2 Θ ) 2 Q l 2 + ( 3 + 2 Λ Q l ) 2 ] 1 / 4 0.52 sin ( 5 4 φ 2 + φ 1 ) [ ( 1 + 2 Θ ) 2 Q l 2 + ( 3 + 2 Λ Q l ) 2 ] 7 / 24 ] × 13.40 Λ Q l 11 / 6 [ ( 1 + 2 Θ ) 2 + 4 Λ 2 ] 11 6 [ ( 1 + 0.31 Λ Q l Q l ) 5 / 6 + 1.10 ( 1 + 0.31 Λ Q l ) 1 / 3 Q l     5 / 6 0.19 ( 1 + 0.24 Λ Q l ) 1 / 4 Q l 5 / 6 ] } ,
φ 1 = tan 1 ( 2 Λ 1 + 2 Θ ) , φ 2 ( l 0 ) = tan 1 [ ( 1 + 2 Θ ) Q l 3 + 2 Λ Q l ] ,

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