Abstract

Theoretical relations were obtained to calculate the correlation of stellar scintillation, when a star is measured simultaneously in two different photometric bands (channels). Differential color scintillation and the respective indices are proposed for practical evaluation of the correlation. Asymptotic dependence for the case of an infinitely small receiving aperture is derived and analyzed. The effect of atmospheric dispersion, which transforms the color differences into spatial differences, is examined. Additionally, the effect of the deviation of real optical turbulence from Kolmogorov turbulence is estimated. The obtained relations were compared with simultaneous measurements in three photometric bands and different receiving apertures in the range of 1 to 15cm. It is shown that the measurements and calculated estimations of the color scintillation agree well.

© 2011 Optical Society of America

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References

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  1. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
    [CrossRef]
  2. D. Dravins, L. Lindegren, E. Mezey, and A. T. Young, “Atmospheric intensity scintillation of stars. ii. dependence on optical wavelength,” Publ. Astron. Soc. Pac. 109, 725–737 (1997).
    [CrossRef]
  3. A. Rocca, F. Roddier, and J. Vernin, “Detection of atmospheric turbulent layers by spatiotemporal and spatioangular correlation measurements of stellar-light scintillation,” J. Opt. Soc. Am. 64, 1000–1004 (1974).
    [CrossRef]
  4. V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
    [CrossRef]
  5. V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
    [CrossRef]
  6. A. A. Tokovinin, “Polychromatic scintillation,” J. Opt. Soc. Am. A 20, 686–689 (2003).
    [CrossRef]
  7. W. B. Hubbard and H. J. Reitsema, “Scintillation at two optical frequencies,” Appl. Opt. 20, 3227–3232 (1981).
    [CrossRef] [PubMed]
  8. J. Caccia, J. Vernin, and M. Azouit, “Structure function C(n)-squared profiling by two-color stellar scintillation with atmospheric dispersion,” Appl. Opt. 27, 2229–2235 (1988).
    [CrossRef] [PubMed]
  9. C. Robert, J. Conan, V. Michau, J. Renard, C. Robert, and F. Dalaudier, “Retrieving parameters of the anisotropic refractive index fluctuations spectrum in the stratosphere from balloon-borne observations of stellar scintillation,” J. Opt. Soc. Am. A 25, 379–393 (2008).
    [CrossRef]
  10. V. G. Kornilov and A. A. Tokovinin, “Measurement of the turbulence in the free atmosphere above Mt. Maidanak,” Astron. Rep. 45, 395–408 (2001).
    [CrossRef]
  11. V. G. Kornilov, “The statistics of the photometric accuracy based on MASS data and the evaluation of high-altitude wind,” Astron. Lett. 37, 40–48 (2011).
    [CrossRef]
  12. A. Berdja, “On the DIMM interpretation of non-Kolmogorov turbulence in the atmospheric surface layer,” Mon. Not. R. Astron. Soc. 409, 722–726 (2010).
    [CrossRef]
  13. A. Tokovinin, “Measurement of seeing and the atmospheric time constant by differential scintillations,” Appl. Opt. 41, 957–964 (2002).
    [CrossRef] [PubMed]
  14. A. Tokovinin, V. Kornilov, N. Shatsky, and O. Voziakova, “Restoration of turbulence profile from scintillation indices,” Mon. Not. R. Astron. Soc. 343, 891–899 (2003).
    [CrossRef]
  15. S. A. Ehgamberdiev, A. K. Baijumanov, S. P. Ilyasov, M. Sarazin, Y. A. Tillayev, A. A. Tokovinin, and A. Ziad, “The astroclimate of Maidanak Observatory in Uzbekistan,” Astron. Astrophys. 145, 293–304 (2000).
    [CrossRef]
  16. V. G. Kornilov, “Four-channel stellar photometer with dichroic beam-splitters,” Baltic Astron. 7, 513–524 (1998).
  17. V. G. Kornilov and M. V. Kornilov, “The revision of the turbulence profiles restoration from MASS scintillation indices,” Exp. Astron. 29, 155–178 (2011).
    [CrossRef]

2011 (2)

V. G. Kornilov, “The statistics of the photometric accuracy based on MASS data and the evaluation of high-altitude wind,” Astron. Lett. 37, 40–48 (2011).
[CrossRef]

V. G. Kornilov and M. V. Kornilov, “The revision of the turbulence profiles restoration from MASS scintillation indices,” Exp. Astron. 29, 155–178 (2011).
[CrossRef]

2010 (1)

A. Berdja, “On the DIMM interpretation of non-Kolmogorov turbulence in the atmospheric surface layer,” Mon. Not. R. Astron. Soc. 409, 722–726 (2010).
[CrossRef]

2008 (1)

2007 (1)

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

2003 (3)

A. A. Tokovinin, “Polychromatic scintillation,” J. Opt. Soc. Am. A 20, 686–689 (2003).
[CrossRef]

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

A. Tokovinin, V. Kornilov, N. Shatsky, and O. Voziakova, “Restoration of turbulence profile from scintillation indices,” Mon. Not. R. Astron. Soc. 343, 891–899 (2003).
[CrossRef]

2002 (1)

2001 (1)

V. G. Kornilov and A. A. Tokovinin, “Measurement of the turbulence in the free atmosphere above Mt. Maidanak,” Astron. Rep. 45, 395–408 (2001).
[CrossRef]

2000 (1)

S. A. Ehgamberdiev, A. K. Baijumanov, S. P. Ilyasov, M. Sarazin, Y. A. Tillayev, A. A. Tokovinin, and A. Ziad, “The astroclimate of Maidanak Observatory in Uzbekistan,” Astron. Astrophys. 145, 293–304 (2000).
[CrossRef]

1998 (1)

V. G. Kornilov, “Four-channel stellar photometer with dichroic beam-splitters,” Baltic Astron. 7, 513–524 (1998).

1997 (1)

D. Dravins, L. Lindegren, E. Mezey, and A. T. Young, “Atmospheric intensity scintillation of stars. ii. dependence on optical wavelength,” Publ. Astron. Soc. Pac. 109, 725–737 (1997).
[CrossRef]

1988 (1)

1981 (2)

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[CrossRef]

W. B. Hubbard and H. J. Reitsema, “Scintillation at two optical frequencies,” Appl. Opt. 20, 3227–3232 (1981).
[CrossRef] [PubMed]

1974 (1)

Azouit, M.

Baijumanov, A. K.

S. A. Ehgamberdiev, A. K. Baijumanov, S. P. Ilyasov, M. Sarazin, Y. A. Tillayev, A. A. Tokovinin, and A. Ziad, “The astroclimate of Maidanak Observatory in Uzbekistan,” Astron. Astrophys. 145, 293–304 (2000).
[CrossRef]

Berdja, A.

A. Berdja, “On the DIMM interpretation of non-Kolmogorov turbulence in the atmospheric surface layer,” Mon. Not. R. Astron. Soc. 409, 722–726 (2010).
[CrossRef]

Caccia, J.

Conan, J.

Dalaudier, F.

Dravins, D.

D. Dravins, L. Lindegren, E. Mezey, and A. T. Young, “Atmospheric intensity scintillation of stars. ii. dependence on optical wavelength,” Publ. Astron. Soc. Pac. 109, 725–737 (1997).
[CrossRef]

Ehgamberdiev, S. A.

S. A. Ehgamberdiev, A. K. Baijumanov, S. P. Ilyasov, M. Sarazin, Y. A. Tillayev, A. A. Tokovinin, and A. Ziad, “The astroclimate of Maidanak Observatory in Uzbekistan,” Astron. Astrophys. 145, 293–304 (2000).
[CrossRef]

Hubbard, W. B.

Ilyasov, S. P.

S. A. Ehgamberdiev, A. K. Baijumanov, S. P. Ilyasov, M. Sarazin, Y. A. Tillayev, A. A. Tokovinin, and A. Ziad, “The astroclimate of Maidanak Observatory in Uzbekistan,” Astron. Astrophys. 145, 293–304 (2000).
[CrossRef]

Kornilov, M. V.

V. G. Kornilov and M. V. Kornilov, “The revision of the turbulence profiles restoration from MASS scintillation indices,” Exp. Astron. 29, 155–178 (2011).
[CrossRef]

Kornilov, V.

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

A. Tokovinin, V. Kornilov, N. Shatsky, and O. Voziakova, “Restoration of turbulence profile from scintillation indices,” Mon. Not. R. Astron. Soc. 343, 891–899 (2003).
[CrossRef]

Kornilov, V. G.

V. G. Kornilov and M. V. Kornilov, “The revision of the turbulence profiles restoration from MASS scintillation indices,” Exp. Astron. 29, 155–178 (2011).
[CrossRef]

V. G. Kornilov, “The statistics of the photometric accuracy based on MASS data and the evaluation of high-altitude wind,” Astron. Lett. 37, 40–48 (2011).
[CrossRef]

V. G. Kornilov and A. A. Tokovinin, “Measurement of the turbulence in the free atmosphere above Mt. Maidanak,” Astron. Rep. 45, 395–408 (2001).
[CrossRef]

V. G. Kornilov, “Four-channel stellar photometer with dichroic beam-splitters,” Baltic Astron. 7, 513–524 (1998).

Lindegren, L.

D. Dravins, L. Lindegren, E. Mezey, and A. T. Young, “Atmospheric intensity scintillation of stars. ii. dependence on optical wavelength,” Publ. Astron. Soc. Pac. 109, 725–737 (1997).
[CrossRef]

Mezey, E.

D. Dravins, L. Lindegren, E. Mezey, and A. T. Young, “Atmospheric intensity scintillation of stars. ii. dependence on optical wavelength,” Publ. Astron. Soc. Pac. 109, 725–737 (1997).
[CrossRef]

Michau, V.

Potanin, S.

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

Potanin, S. F.

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

Reitsema, H. J.

Renard, J.

Robert, C.

Rocca, A.

Roddier, F.

Safonov, B.

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

Sarazin, M.

S. A. Ehgamberdiev, A. K. Baijumanov, S. P. Ilyasov, M. Sarazin, Y. A. Tillayev, A. A. Tokovinin, and A. Ziad, “The astroclimate of Maidanak Observatory in Uzbekistan,” Astron. Astrophys. 145, 293–304 (2000).
[CrossRef]

Sarazin, M. S.

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

Shatsky, N.

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

A. Tokovinin, V. Kornilov, N. Shatsky, and O. Voziakova, “Restoration of turbulence profile from scintillation indices,” Mon. Not. R. Astron. Soc. 343, 891–899 (2003).
[CrossRef]

Tillayev, Y. A.

S. A. Ehgamberdiev, A. K. Baijumanov, S. P. Ilyasov, M. Sarazin, Y. A. Tillayev, A. A. Tokovinin, and A. Ziad, “The astroclimate of Maidanak Observatory in Uzbekistan,” Astron. Astrophys. 145, 293–304 (2000).
[CrossRef]

Tokovinin, A.

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

A. Tokovinin, V. Kornilov, N. Shatsky, and O. Voziakova, “Restoration of turbulence profile from scintillation indices,” Mon. Not. R. Astron. Soc. 343, 891–899 (2003).
[CrossRef]

A. Tokovinin, “Measurement of seeing and the atmospheric time constant by differential scintillations,” Appl. Opt. 41, 957–964 (2002).
[CrossRef] [PubMed]

Tokovinin, A. A.

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

A. A. Tokovinin, “Polychromatic scintillation,” J. Opt. Soc. Am. A 20, 686–689 (2003).
[CrossRef]

V. G. Kornilov and A. A. Tokovinin, “Measurement of the turbulence in the free atmosphere above Mt. Maidanak,” Astron. Rep. 45, 395–408 (2001).
[CrossRef]

S. A. Ehgamberdiev, A. K. Baijumanov, S. P. Ilyasov, M. Sarazin, Y. A. Tillayev, A. A. Tokovinin, and A. Ziad, “The astroclimate of Maidanak Observatory in Uzbekistan,” Astron. Astrophys. 145, 293–304 (2000).
[CrossRef]

Vernin, J.

Voziakova, O.

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

A. Tokovinin, V. Kornilov, N. Shatsky, and O. Voziakova, “Restoration of turbulence profile from scintillation indices,” Mon. Not. R. Astron. Soc. 343, 891–899 (2003).
[CrossRef]

Vozyakova, O.

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

Young, A. T.

D. Dravins, L. Lindegren, E. Mezey, and A. T. Young, “Atmospheric intensity scintillation of stars. ii. dependence on optical wavelength,” Publ. Astron. Soc. Pac. 109, 725–737 (1997).
[CrossRef]

Zaitsev, A.

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

Ziad, A.

S. A. Ehgamberdiev, A. K. Baijumanov, S. P. Ilyasov, M. Sarazin, Y. A. Tillayev, A. A. Tokovinin, and A. Ziad, “The astroclimate of Maidanak Observatory in Uzbekistan,” Astron. Astrophys. 145, 293–304 (2000).
[CrossRef]

Appl. Opt. (3)

Astron. Astrophys. (1)

S. A. Ehgamberdiev, A. K. Baijumanov, S. P. Ilyasov, M. Sarazin, Y. A. Tillayev, A. A. Tokovinin, and A. Ziad, “The astroclimate of Maidanak Observatory in Uzbekistan,” Astron. Astrophys. 145, 293–304 (2000).
[CrossRef]

Astron. Lett. (1)

V. G. Kornilov, “The statistics of the photometric accuracy based on MASS data and the evaluation of high-altitude wind,” Astron. Lett. 37, 40–48 (2011).
[CrossRef]

Astron. Rep. (1)

V. G. Kornilov and A. A. Tokovinin, “Measurement of the turbulence in the free atmosphere above Mt. Maidanak,” Astron. Rep. 45, 395–408 (2001).
[CrossRef]

Baltic Astron. (1)

V. G. Kornilov, “Four-channel stellar photometer with dichroic beam-splitters,” Baltic Astron. 7, 513–524 (1998).

Exp. Astron. (1)

V. G. Kornilov and M. V. Kornilov, “The revision of the turbulence profiles restoration from MASS scintillation indices,” Exp. Astron. 29, 155–178 (2011).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Mon. Not. R. Astron. Soc. (3)

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

A. Berdja, “On the DIMM interpretation of non-Kolmogorov turbulence in the atmospheric surface layer,” Mon. Not. R. Astron. Soc. 409, 722–726 (2010).
[CrossRef]

A. Tokovinin, V. Kornilov, N. Shatsky, and O. Voziakova, “Restoration of turbulence profile from scintillation indices,” Mon. Not. R. Astron. Soc. 343, 891–899 (2003).
[CrossRef]

Proc. SPIE (1)

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

Prog. Opt. (1)

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

D. Dravins, L. Lindegren, E. Mezey, and A. T. Young, “Atmospheric intensity scintillation of stars. ii. dependence on optical wavelength,” Publ. Astron. Soc. Pac. 109, 725–737 (1997).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Correlation coefficient ρ of the scintillations in two monochromatic channels for the case of an infinitely small aperture as a function of the wavelength ratio for zero (solid curve) and long (dashed curve) exposures. The dotted curve denotes the case of non-Kolmogorov turbulence with a 2D spectrum f 9 / 3 .

Fig. 2
Fig. 2

WFs for 1.35 (1), 2.66 (2), 9.82 (3), and 15.1 cm (4) apertures for the B photometric band (thin solid curve). The color WFs Q B , R ( z ) are shown with thick curves for the same set of apertures. The dashed curve denotes the monochromatic asymptote D 0 for wavelength λ = λ B . The dashed–dotted curve is the asymptote for the color scintillation with λ 1 = λ B and λ 2 = λ R .

Fig. 3
Fig. 3

a) Dependence of ratio Q B ( h ) / Q V ( h ) and Q R ( h ) / Q V ( h ) on altitude. b) Dependence of ratio of color WFs Q B , R ( h ) to Q V ( h ) on altitude. Solid curves, for 1.35 cm aperture; dashed curves, for 2.66 cm ; dashed–dotted curve, for 4.59 cm ; dotted–dashed–dashed curve, for 6.91 cm ; dashed–dotted–dashed curve, for 9.82 cm ; dotted curve, for 15.1 cm aperture.

Fig. 4
Fig. 4

Ratios of the measured indices s B 2 (black higher), s R 2 (black lower curve) and the color index s B , R 2 (gray curve) to the index s V 2 for the a) 2.66 and b)  9.82 cm apertures. White circles show the median over 200 points. The straight lines indicate the limits corresponding to extremely low-altitude (dashed–dotted line) and extremely high-altitude (dashed line) turbulence. The solid curves depict the weight-average ratio.

Fig. 5
Fig. 5

Dependence of the median (in groups of 300 points) of the measured color indices s B V 2 , s B R 2 , and s V R 2 on the air mass for the 15.1 cm aperture.

Tables (1)

Tables Icon

Table 1 Medians of the Scintillation Indices in Different Apertures and Photometric Bands Measured in August–October 1999

Equations (19)

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

s 2 = C n 2 ( z ) Q ( z ) d z ,
c 1 , 2 = I 1 I 2 I 1 I 2 4 χ ¯ 1 ( λ 1 ) χ ¯ 2 ( λ 2 ) F 1 ( λ 1 ) F 2 ( λ 2 ) d λ 1 d λ 2 ,
R 1 , 2 ( z ) = 9.61 0 f 8 / 3 S 1 , 2 ( z , f ) A ( f ) d f ,
S 1 , 2 ( z , f ) = F 1 F 1 ( λ 1 ) λ 1 sin ( π λ 1 z f 2 ) d λ 1 F 2 F 2 ( λ 2 ) λ 2 sin ( π λ 2 z f 2 ) d λ 2 .
Q ( z ) = 9.61 0 f 8 / 3 S ( z , f ) A ( f ) d f ,
S ( z , f ) = [ F F ( λ ) λ sin ( π λ z f 2 ) d λ ] 2 .
c 1 , 2 = C n 2 ( z ) R 12 ( z ) d z .
ρ 1 , 2 = c 1 , 2 ( s 1 2 s 2 2 ) 1 / 2 .
R 1 , 2 ( z ) = 9.61 ( λ 1 λ 2 ) 1 0 f 8 / 3 sin ( π λ 1 z f 2 ) sin ( π λ 2 z f 2 ) A ( f ) d f .
R 1 , 2 ( z ) = 12.47 · ( λ 1 λ 2 ) 7 / 12 z 5 / 6 0 u 11 / 6 sin ( a u ) sin ( u / a ) d u ,
Y ( a ) = 0.8644 [ ( 1 a + a ) 5 / 6 | 1 a a | 5 / 6 ] .
s d 2 = s 1 2 + s 2 2 2 c 1 , 2 ,
Q d ( z ) = Q 1 ( z ) + Q 2 ( z ) 2 R 1 , 2 ( z ) .
Q d ( z ) = 9.61 0 f 8 / 3 T 1 , 2 ( z , f ) A ( f ) d f ,
T 1 , 2 ( z , f ) = F 1 F 2 ( F 1 ( λ 1 ) λ 1 sin ( π λ 1 z f 2 ) F 2 ( λ 2 ) λ 2 sin ( π λ 2 z f 2 ) ) 2 d λ 2 d λ 1 .
x = tan γ sec γ Δ n 1 , 2 e h o / H 0 ( H 0 ( H 0 + h ) e h / H 0 ) ,
s s 2 = s A , 1 2 + s A , 2 2 2 c A , A , 1 , 2 .
Δ s d 2 = s s 2 s d 2 = 2 c A , A , 1 , 2 2 c A , 1 , 2 .
Q Δ ( z ) = 9.61 0 f 8 / 3 S 1 , 2 ( z , f ) ( 2 2 J 0 ( 2 π x f ) ) A ( f ) d f .

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