Abstract

Scintillation effects are not negligible in the stratosphere. We present a model based on a 3D model of anisotropic and isotropic refractive index fluctuations spectra that predicts scintillation rates within the so-called small perturbation approximation. Atmospheric observations of stellar scintillation made from the AMON-RA (AMON, Absorption par les Minoritaires Ozone et NOx; RA, rapid) balloon-borne spectrometer allows us to remotely probe wave-turbulence characteristics in the stratosphere. Data reduction from these observations brings out values of the inner scale of the anisotropic spectrum. We find metric values of the inner scale that are compatible with space-based measurements. We find a major contribution of the anisotropic spectrum relative to the isotropic contribution. When the sight line plunges into the atmosphere, strong scintillation occurs as well as coupled chromatic refraction effects.

© 2008 Optical Society of America

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References

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  1. F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, "Direct evidence of 'sheets'," J. Atmos. Sci. 51, 237-248 (1994).
    [CrossRef]
  2. D. C. Fritts and M. J. Alexander, "Gravity wave dynamics and effects in the middle atmosphere," Rev. Geophys. 41, 3-1-3-63 (2003).
    [CrossRef]
  3. A. S. Gurvich and I. P. Chunchuzov, "Parameters of the fine density structure in the stratosphere obtained from spacecraft observations of stellar scintillations," J. Geophys. Res. 108(D5), 4166 ACL 6-1-ACL 6-4 (2003).
    [CrossRef]
  4. G. M. Grechko, A. S. Gurvich, V. Kan, A. I. Pakhomov, Ya. P. Podvyazni, and S. A. Savchenko, "Observations of atmospheric turbulence at altitudes of 20-70 km," Dokl. Akad. Nauk 357, 1382-1385 (1997).
  5. A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, and M. K. Manarov, "Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars," Atmos. Oceanic Phys. 26, 1-8 (1990).
  6. A. S. Gurvich and V. Kan, "Radio wave fluctuations in satellite-atmosphere-satellite links: estimates from stellar scintillation observations and their comparison with experimental data," Atmos. Oceanic Phys. 33, 284-292 (1997).
  7. A. S. Gurvich and S. V. Sokolovskii, "Two-wavelength observations of stellar scintillations for autonomous satellite navigation," Navigation 38, 359-366 (1992).
  8. A. S. Gurvich, F. Dalaudier, and V. F. Sofieva, "Study of stratospheric air density irregularities based on two wavelength observation of stellar scintillation by global ozone monitoring by occultation of stars (GOMOS) on Envisat," J. Geophys. Res. 110, D11110 1-9 (2005).
    [CrossRef]
  9. J.-B. Renard, F. Dalaudier, A. Hauchecorne, C. Robert, T. Lemaire, M. Pirre, and J.-L. Bertaux, "Measurement of stratospheric chromatic scintillation with the AMON-RA balloon-borne spectrometer," Appl. Opt. 40, 4254-4260 (2001).
    [CrossRef]
  10. C. Robert, J.-M. Conan, and V. Michau, "Scintillation from turbulence in the stratosphere for middle-infrared observations," Vision in InfraRed Astronomy, Vol. 6 of Instrumentation Measurement, Meteorology (Lavoisier, 2006), pp. 147-150
  11. D. Huguenin, "Design and performance of stratospheric balloon-borne platforms for infrared astrophysical observations," Infrared Phys. Technol. 35, 195-202 (1994).
    [CrossRef]
  12. J.-B. Renard, M. Pirre, C. Robert, D. Huguenin, G. Moreau, and M. Russell, "Nocturnal distribution of stratospheric O3, NO2, and NO3 from balloon measurements," Rev. Geophys. 101, 28793-28804 (1996).
    [CrossRef]
  13. G. Berthet, J.-B. Renard, C. Brogniez, C. Robert, M. Chartier, and M. Pirre, "Optical and physical properties of stratospheric aerosols from balloon measurements in the visible and near-infrared domains I. Analysis of aerosol extinction spectra from the AMON and SALOMON balloon-borne spectrometers," Appl. Opt. 41, 7522-7537 (2002).
    [CrossRef]
  14. J. Alfred, M. Fromm, R. Bevilacqua, G. Nedoluha, A. Strawa, L. Poole, and J. Wickert, "Observation and analysis of polar stratospheric clouds detected by POAM III and SAGE III during the SOLVE II/VINTERSOL campaign in the 2002/2003 Northern Hemisphere winter," Atmos. Chem. Phys. Discuss 6, 11391-11426 (2006).
    [CrossRef]
  15. J.-B. Renard, G. Berthet, C. Brogniez, V. Catoire, D. Fussen, F. Goutail, H. Oelhaf, J.-P. Pommereau, H. Roscoe, G. Wetzel, M. Chartier, C. Robert, J.-Y. Balois, C. Verwaerde, F. Auriol, P. François, B. Gaubicher, and P. Wursteisen, "Validation of GOMOS-Envisat vertical profiles of O3, NO2, NO3, and aerosol extinction using balloon-borne instruments," Rev. Geophys. (to be published).
  16. J. W. Goodman, Statistical Optics (Wiley, 1985), Chaps. 7 and 8.
  17. S. F. Clifford, G. R. Ochs, and R. S. Lawrence, "Saturation of optical scintillation by strong turbulence," J. Opt. Soc. Am. A 64, 148-154 (1974).
    [CrossRef]
  18. R. J. Hill and S. F. Clifford, "Theory of saturation of optical scintillation by strong turbulence for arbitrary refractive-index spectra," J. Opt. Soc. Am. A 71, 675-686 (1981).
    [CrossRef]
  19. F. Clifford, "The classical theory of wave propagation in a Turbulent Medium," in Laser Beam Propagation in the Atmosphere, J.W.Strohbehn, ed., Volume 25 of Topics in Applied Physics (Springer-Verlag, 1978) Chap. 2, pp. 9-43.
  20. A. S. Gurvich, V. V. Vorob'ev, and O. V. Fedorova, "Determination of parameters of the spectrum of internal waves in the stratosphere from space-based observations of strong stellar scintillation," Izv. Atmos. Oceanic Phys. 42, 512-523 (2006).
  21. A. S. Gurvich, "Parameters of turbulence and internal waves and the dissipation of kinetic energy in the stratosphere based on space observations," Dokl. Akad. Nauk 385, 1-7 (2002).
  22. V. F. Sofieva, A. S. Gurvich, F. Dalaudier, and V. Kan, "Reconstruction of internal gravity wave and turbulence parameters in the stratosphere using GOMOS scintillation measurements," J. Geophys. Res. 112, D12113, 1-14 (2007).
    [CrossRef]
  23. A. Abahamid, A. Jabiri, J. Vernin, Z. Benkhaldoun, M. Azouit, and A. Agabi, "Optical turbulence modeling in the boundary layer and free atmosphere using instrumented meteorological balloons," Astron. Astrophys. 416, 1193-1200 (2004).
    [CrossRef]
  24. N. M. Gavrilov, H. Luce, M. Crochet, F. Dalaudier, and S. Fukao, "Turbulence parameter estimations from high-resolution balloon temperature measurements of the MUTSI-2000 campaign," Ann. Geophys. (Germany) 23, 2401-2413 (2005).
  25. F. Dalaudier, V. Kan, and A. S. Gurvich, "Chromatic refraction with global ozone monitoring by occultation of stars. I. Description and scintillation correction," Appl. Opt. 40, 866-889 (2001).
    [CrossRef]

2007 (1)

V. F. Sofieva, A. S. Gurvich, F. Dalaudier, and V. Kan, "Reconstruction of internal gravity wave and turbulence parameters in the stratosphere using GOMOS scintillation measurements," J. Geophys. Res. 112, D12113, 1-14 (2007).
[CrossRef]

2006 (2)

J. Alfred, M. Fromm, R. Bevilacqua, G. Nedoluha, A. Strawa, L. Poole, and J. Wickert, "Observation and analysis of polar stratospheric clouds detected by POAM III and SAGE III during the SOLVE II/VINTERSOL campaign in the 2002/2003 Northern Hemisphere winter," Atmos. Chem. Phys. Discuss 6, 11391-11426 (2006).
[CrossRef]

A. S. Gurvich, V. V. Vorob'ev, and O. V. Fedorova, "Determination of parameters of the spectrum of internal waves in the stratosphere from space-based observations of strong stellar scintillation," Izv. Atmos. Oceanic Phys. 42, 512-523 (2006).

2005 (2)

N. M. Gavrilov, H. Luce, M. Crochet, F. Dalaudier, and S. Fukao, "Turbulence parameter estimations from high-resolution balloon temperature measurements of the MUTSI-2000 campaign," Ann. Geophys. (Germany) 23, 2401-2413 (2005).

A. S. Gurvich, F. Dalaudier, and V. F. Sofieva, "Study of stratospheric air density irregularities based on two wavelength observation of stellar scintillation by global ozone monitoring by occultation of stars (GOMOS) on Envisat," J. Geophys. Res. 110, D11110 1-9 (2005).
[CrossRef]

2004 (1)

A. Abahamid, A. Jabiri, J. Vernin, Z. Benkhaldoun, M. Azouit, and A. Agabi, "Optical turbulence modeling in the boundary layer and free atmosphere using instrumented meteorological balloons," Astron. Astrophys. 416, 1193-1200 (2004).
[CrossRef]

2003 (2)

D. C. Fritts and M. J. Alexander, "Gravity wave dynamics and effects in the middle atmosphere," Rev. Geophys. 41, 3-1-3-63 (2003).
[CrossRef]

A. S. Gurvich and I. P. Chunchuzov, "Parameters of the fine density structure in the stratosphere obtained from spacecraft observations of stellar scintillations," J. Geophys. Res. 108(D5), 4166 ACL 6-1-ACL 6-4 (2003).
[CrossRef]

2002 (2)

2001 (2)

1997 (2)

G. M. Grechko, A. S. Gurvich, V. Kan, A. I. Pakhomov, Ya. P. Podvyazni, and S. A. Savchenko, "Observations of atmospheric turbulence at altitudes of 20-70 km," Dokl. Akad. Nauk 357, 1382-1385 (1997).

A. S. Gurvich and V. Kan, "Radio wave fluctuations in satellite-atmosphere-satellite links: estimates from stellar scintillation observations and their comparison with experimental data," Atmos. Oceanic Phys. 33, 284-292 (1997).

1996 (1)

J.-B. Renard, M. Pirre, C. Robert, D. Huguenin, G. Moreau, and M. Russell, "Nocturnal distribution of stratospheric O3, NO2, and NO3 from balloon measurements," Rev. Geophys. 101, 28793-28804 (1996).
[CrossRef]

1994 (2)

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, "Direct evidence of 'sheets'," J. Atmos. Sci. 51, 237-248 (1994).
[CrossRef]

D. Huguenin, "Design and performance of stratospheric balloon-borne platforms for infrared astrophysical observations," Infrared Phys. Technol. 35, 195-202 (1994).
[CrossRef]

1992 (1)

A. S. Gurvich and S. V. Sokolovskii, "Two-wavelength observations of stellar scintillations for autonomous satellite navigation," Navigation 38, 359-366 (1992).

1990 (1)

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, and M. K. Manarov, "Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars," Atmos. Oceanic Phys. 26, 1-8 (1990).

1981 (1)

R. J. Hill and S. F. Clifford, "Theory of saturation of optical scintillation by strong turbulence for arbitrary refractive-index spectra," J. Opt. Soc. Am. A 71, 675-686 (1981).
[CrossRef]

1974 (1)

S. F. Clifford, G. R. Ochs, and R. S. Lawrence, "Saturation of optical scintillation by strong turbulence," J. Opt. Soc. Am. A 64, 148-154 (1974).
[CrossRef]

Ann. Geophys. (Germany) (1)

N. M. Gavrilov, H. Luce, M. Crochet, F. Dalaudier, and S. Fukao, "Turbulence parameter estimations from high-resolution balloon temperature measurements of the MUTSI-2000 campaign," Ann. Geophys. (Germany) 23, 2401-2413 (2005).

Appl. Opt. (3)

Astron. Astrophys. (1)

A. Abahamid, A. Jabiri, J. Vernin, Z. Benkhaldoun, M. Azouit, and A. Agabi, "Optical turbulence modeling in the boundary layer and free atmosphere using instrumented meteorological balloons," Astron. Astrophys. 416, 1193-1200 (2004).
[CrossRef]

Atmos. Chem. Phys. Discuss (1)

J. Alfred, M. Fromm, R. Bevilacqua, G. Nedoluha, A. Strawa, L. Poole, and J. Wickert, "Observation and analysis of polar stratospheric clouds detected by POAM III and SAGE III during the SOLVE II/VINTERSOL campaign in the 2002/2003 Northern Hemisphere winter," Atmos. Chem. Phys. Discuss 6, 11391-11426 (2006).
[CrossRef]

Atmos. Oceanic Phys. (2)

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, and M. K. Manarov, "Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars," Atmos. Oceanic Phys. 26, 1-8 (1990).

A. S. Gurvich and V. Kan, "Radio wave fluctuations in satellite-atmosphere-satellite links: estimates from stellar scintillation observations and their comparison with experimental data," Atmos. Oceanic Phys. 33, 284-292 (1997).

Dokl. Akad. Nauk (2)

G. M. Grechko, A. S. Gurvich, V. Kan, A. I. Pakhomov, Ya. P. Podvyazni, and S. A. Savchenko, "Observations of atmospheric turbulence at altitudes of 20-70 km," Dokl. Akad. Nauk 357, 1382-1385 (1997).

A. S. Gurvich, "Parameters of turbulence and internal waves and the dissipation of kinetic energy in the stratosphere based on space observations," Dokl. Akad. Nauk 385, 1-7 (2002).

Infrared Phys. Technol. (1)

D. Huguenin, "Design and performance of stratospheric balloon-borne platforms for infrared astrophysical observations," Infrared Phys. Technol. 35, 195-202 (1994).
[CrossRef]

Izv. Atmos. Oceanic Phys. (1)

A. S. Gurvich, V. V. Vorob'ev, and O. V. Fedorova, "Determination of parameters of the spectrum of internal waves in the stratosphere from space-based observations of strong stellar scintillation," Izv. Atmos. Oceanic Phys. 42, 512-523 (2006).

J. Atmos. Sci. (1)

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, "Direct evidence of 'sheets'," J. Atmos. Sci. 51, 237-248 (1994).
[CrossRef]

J. Geophys. Res. (3)

A. S. Gurvich and I. P. Chunchuzov, "Parameters of the fine density structure in the stratosphere obtained from spacecraft observations of stellar scintillations," J. Geophys. Res. 108(D5), 4166 ACL 6-1-ACL 6-4 (2003).
[CrossRef]

A. S. Gurvich, F. Dalaudier, and V. F. Sofieva, "Study of stratospheric air density irregularities based on two wavelength observation of stellar scintillation by global ozone monitoring by occultation of stars (GOMOS) on Envisat," J. Geophys. Res. 110, D11110 1-9 (2005).
[CrossRef]

V. F. Sofieva, A. S. Gurvich, F. Dalaudier, and V. Kan, "Reconstruction of internal gravity wave and turbulence parameters in the stratosphere using GOMOS scintillation measurements," J. Geophys. Res. 112, D12113, 1-14 (2007).
[CrossRef]

J. Opt. Soc. Am. A (2)

S. F. Clifford, G. R. Ochs, and R. S. Lawrence, "Saturation of optical scintillation by strong turbulence," J. Opt. Soc. Am. A 64, 148-154 (1974).
[CrossRef]

R. J. Hill and S. F. Clifford, "Theory of saturation of optical scintillation by strong turbulence for arbitrary refractive-index spectra," J. Opt. Soc. Am. A 71, 675-686 (1981).
[CrossRef]

Navigation (1)

A. S. Gurvich and S. V. Sokolovskii, "Two-wavelength observations of stellar scintillations for autonomous satellite navigation," Navigation 38, 359-366 (1992).

Rev. Geophys. (3)

J.-B. Renard, G. Berthet, C. Brogniez, V. Catoire, D. Fussen, F. Goutail, H. Oelhaf, J.-P. Pommereau, H. Roscoe, G. Wetzel, M. Chartier, C. Robert, J.-Y. Balois, C. Verwaerde, F. Auriol, P. François, B. Gaubicher, and P. Wursteisen, "Validation of GOMOS-Envisat vertical profiles of O3, NO2, NO3, and aerosol extinction using balloon-borne instruments," Rev. Geophys. (to be published).

D. C. Fritts and M. J. Alexander, "Gravity wave dynamics and effects in the middle atmosphere," Rev. Geophys. 41, 3-1-3-63 (2003).
[CrossRef]

J.-B. Renard, M. Pirre, C. Robert, D. Huguenin, G. Moreau, and M. Russell, "Nocturnal distribution of stratospheric O3, NO2, and NO3 from balloon measurements," Rev. Geophys. 101, 28793-28804 (1996).
[CrossRef]

Other (3)

J. W. Goodman, Statistical Optics (Wiley, 1985), Chaps. 7 and 8.

F. Clifford, "The classical theory of wave propagation in a Turbulent Medium," in Laser Beam Propagation in the Atmosphere, J.W.Strohbehn, ed., Volume 25 of Topics in Applied Physics (Springer-Verlag, 1978) Chap. 2, pp. 9-43.

C. Robert, J.-M. Conan, and V. Michau, "Scintillation from turbulence in the stratosphere for middle-infrared observations," Vision in InfraRed Astronomy, Vol. 6 of Instrumentation Measurement, Meteorology (Lavoisier, 2006), pp. 147-150

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Figures (15)

Fig. 1
Fig. 1

Geocentric axis ( X , Y , Z ) and optical axis ( x , y , z ) referentials.

Fig. 2
Fig. 2

Numerical modeling of 2D anisotropic scintillation spectrum PSD χ . Profiles of spectral energy density f × PSD χ as a function of spatial frequency along the vertical axis (upper curve) and the horizontal axis (lower curve).

Fig. 3
Fig. 3

Anisotropic spectrum (upper) and isotropic spectrum (lower). Numerical modeling of spatial PSD χ ( k x , k y ) (linear scale) as a function of k x , k y (linear scale in normalized unit; same grid for both spectra).

Fig. 4
Fig. 4

Geometry of observation used in scintillation predictions. Receiver is located at an altitude of 20 km , and propagation range is L = 500 km . Three angles of the sight line are considered, θ = 88 ° (top); θ = 90 ° (middle), i.e., horizontal; and θ = 92 ° (bottom). The optical paths (solid curve, left scale) are associated with structure constants (symbols, right scale) in this double-scale graph.

Fig. 5
Fig. 5

Influence of sight line obliquity on balance between anisotropic and isotropic scintillation. A pointlike source moving vertically is considered. Receiver altitude, 20 km ; propagation range, L = 500 km ; angle of sight line from zenith, θ = 88 ° (top); θ = 90 ° (middle), and θ = 92 ° (bottom). Structure constants of each spectrum are as in Fig. 4.

Fig. 6
Fig. 6

Influence of sight line obliquity on balance between anisotropic and isotropic scintillation. A 3 m disk source with vertical motion is considered. Receiver altitude, 20 km ; propagation range, L = 500 km ; angle of sight line from zenith θ = 88 ° (top), θ = 90 ° (middle), and θ = 92 ° (bottom). Structure constants of each spectrum are as in Fig. 4.

Fig. 7
Fig. 7

Geometry of observation of Sirius as function of time in seconds. Double-scale coordinate graph: on left, elevation in degrees is plotted with a solid line; on right, altitude of perigee in kilometers is plotted with a dotted curve. Numbers on the left scale stand for the sight line angle in respect to horizontal from the balloon and do not take refraction into account.

Fig. 8
Fig. 8

Evolution of flux (in arbitrary units) of 14 spectral channels for the Sirius star set. Evolution of the signal level comes from the shape of the star spectrum, the CCD response, and the Rayleigh and atmospheric ozone attenuation. Nine data sets (i.e., sequences for 580 s ) are delimited by short gaps for 25 s due to transfer to the onboard computer.

Fig. 9
Fig. 9

Comparison of histograms (black curves) and log-normal density functions [gray (red online) curves] for intensity at wavelength λ = 525 550 nm . Log-normal density functions are parameterized with experimental values of χ ¯ and σ χ .

Fig. 10
Fig. 10

Histogram of normalized intensity during sequence 9 with a negative exponential law superimposed.

Fig. 11
Fig. 11

Experimental scintillation variance σ I I ¯ 2 as a function of 4 σ χ 2 calculated according to the log-normal fit to data (cf. Subsection 3C). Wavelengths in the legend correspond to 14 AMON-RA spectral channels. Black solid curve represents exp ( 4 σ χ 2 ) 1 , which gives σ I I ¯ 2 if intensity I is log normal. Black dashed-dotted curve plots σ I I ¯ 2 = 4 σ χ 2 , a good approximation for σ χ 2 1 .

Fig. 12
Fig. 12

Estimation of temporal spectra of recorded intensity using the Welch estimator. Cutoff frequency on sequences 6 to 8 allows us to recover inner scale l W . In sequence 9, the temporal spectrum at a frequency near 5 Hz presents a less energetic power law in ν 0 that is probably associated with the isotropic spectrum.

Fig. 13
Fig. 13

Simulation of temporal spectra PSD ( ν ) during sequence 6. Solid curve, anisotropic spectrum; dashed curve, isotropic spectrum. Two values of outer scale L 0 are considered: left, L 0 = 5 m ; right, L 0 = 0.5 m .

Fig. 14
Fig. 14

Optical paths as regards structure constants used in the scintillation model. Each box concerns one time sequence: We show sequences 1 to 8 from top to bottom and left to right. For each box, the graph line is plotted in a double-scale coordinate: left, the optical path height in killometers (solid curve;) right, the structure constants (symbols). Triangles represent structure constants of the anisotropic spectrum. Diamonds represent structure constants of the isotropic spectrum. The multilayer model of scintillation uses as many layers as symbols. The axis stands for the distance of one turbulent layer to receiver in kilometers.

Fig. 15
Fig. 15

Normalized intensity variance σ I I ¯ 2 as a function of sequence number. Top, composite model with anisotropic (Gurvich) and isotropic (von Karman) spectra of refractive index. Bottom, isotropic spectrum (von Karman) alone. Computation performed for all AMON-RA spectral channels. Grayscale (in color online) curves indicate variances measured from each channel. Black curves represent the range of chromatic variances obtained numerically with the scintillation model.

Tables (1)

Tables Icon

Table 1 Asymptotic Laws of the Anisotropic P S D χ along the Vertical and Horizontal a

Equations (21)

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Φ N 3 D ( K ) = η 2 N ( h ) 2 C W ( h ) K 5 1 + ( l W K ) 2 .
Φ N 3 D ( κ ) = 0.033 C n 2 ( ( 2 π L 0 ) 2 + κ 2 ) 11 6 exp { ( κ l 0 5.92 ) 2 }
PSD χ ( k x , k y ) = 0 L PSD χ z ( k x , k y ; z ) d z ,
PSD χ z ( k x , k y ; z ) = 2 π k 0 2 Φ N 2 D ( L z ( k x , k y ) ; z ) ( L z ) 2 sin 2 ( L z ( L z ) k 2 2 k 0 ) .
PSD ( ν , α ) = 1 v PSD 1 D ( k s , α ) ,
PSD 1 D ( k s , α ) = d k PSD χ ( k sin α + k s cos α , k cos α + k s sin α ) ,
v ( z ) = L z ( L z L v source + v wind ( z ) ) ,
PSD z ( ν , 0 ) = [ sinc ( v ( z ) t exp 2 k x ) ] 2 v ( z ) d k y PSD χ z ( k x , k y ) ,
filter Δ ( k x , k y ) = [ 2 J 1 ( k Δ 2 ) k Δ 2 ] 2 ,
filter d s z ( k x , k y ; z ) = [ 2 J 1 ( k D ( z ) 2 ) k D ( z ) 2 ] 2 ,
PSD χ ( k x , k y ) = filter Δ ( k x , k y ) 0 L filter d s z ( k x , k y ; z ) PSD χ z ( k x , k y ; z ) d z .
PSD χ ( k x , k y ) = 2 π 0.033 k 0 2 0 L C n 2 ( z ) d z ( ( 2 π L L 0 z ) 2 + k 2 ) 11 6 exp { ( k L l 0 5.91 z ) 2 } × ( L z ) 5 3 sin 2 ( L z ( L z ) k 2 2 k 0 ) .
κ X 2 = k y 2 ,
κ Y 2 = k z 2 sin 2 θ + k x 2 cos 2 θ 2 sin θ cos θ k x k z ,
κ Z 2 = k z 2 cos 2 θ + k x 2 sin 2 θ + 2 sin θ cos θ k x k z .
K 2 = ( η 2 cos 2 θ + sin 2 θ ) k x 2 + η 2 k y 2 + ( cos 2 θ + η 2 sin 2 θ ) k z 2 + ( 1 η 2 ) 2 sin θ cos θ k x k z .
K 2 = ( η 2 cos 2 θ + sin 2 θ ) k x 2 + η 2 k y 2 .
Φ N 2 D ( k x , k y ; z ) = η 2 N ( h ) 2 C W ( h ) K 5 1 + ( l W K ) 2 .
Φ N 2 D ( L z ( k x , k y ) ; z ) ( L z ) 2 = η 2 N ( h ) 2 C W ( h ) K 5 1 + ( ( L z ) l W K ) 2 ( L z ) 3 .
PSD χ z ( k x , k y ; z ) = 2 π k 0 2 η 2 N ( h ( z ) ) 2 C W ( h ( z ) ) K 5 1 + ( l W K ) 2 ( L z ) 3 sin 2 ( L z ( L z ) k 2 2 k 0 ) ,
p I ( I ) = 1 2 2 π σ χ I exp { ( l n ( I I 0 ) 2 χ ¯ ) 2 8 σ χ 2 } ,

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