Abstract

When studying light propagation through the atmosphere, it is usual to rely on widely used spectra such as the modified von Kármán or Andrews-Hill. These are relatively tractable models for the fluctuations of the refractive index, and are primarily used because of their mathematical convenience. They correctly describe the fluctuations behaviour at the inertial range yet lack any physical basis outside this range. In recent years, deviations from the Obukhov-Kolmogorov theory (e. g. interminttency, partially developed turbulence, etc.) have been built upon these models through the introduction of arbitrary spectral power laws. Here we introduce a quasi-wavelet model for the refractive index fluctuations which is based on a phenomenological representation of the Richardson cascade. Under this model, the atmospheric refractive index has a correct spectral representation for the inertial range, behaves as expected outside it, and even accounts for non-Kolmogorov behaviour; moreover, it has non-Gaussian statistics. Finally, we are able to produce second order moments under the Rytov approximation for the complex phase; we estimate the angle-of-arrival as an example of application.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  7. A. Skvortsov, M. Jamriska, and T. C. DuBois, “Scaling laws of passive tracer dispersion in the turbulent surface layer,” Phys. Rev. E 82, 056304 (2010).
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    [Crossref]
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    [Crossref]
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    [Crossref]
  28. D. K. Wilson, V. E. Ostashev, and G. H. Goedecke, “Sound-wave coherence in atmospheric turbulence with intrinsic and global intermittency,” J. Acoust. Soc. Am. 124, 743–757 (2008).
    [Crossref] [PubMed]
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    [Crossref]
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2013 (1)

G. Funes, E. Figueroa, D. Gulich, L. Zunino, and D. G. Pérez, “Characterizing inertial and convective optical turbulence by detrended fluctuation analysis,” Proc. SPIE 8890, 889016 (2013).
[Crossref]

2012 (2)

2011 (1)

2010 (1)

A. Skvortsov, M. Jamriska, and T. C. DuBois, “Scaling laws of passive tracer dispersion in the turbulent surface layer,” Phys. Rev. E 82, 056304 (2010).
[Crossref]

2009 (1)

D. K. Wilson, V. E. Ostashev, and G. H. Goedecke, “Quasi-wavelet formulations of turbulence and other random fields with correlated properties,” Probabilist. Eng. Mech. 24, 343–357 (2009).
[Crossref]

2008 (4)

D. K. Wilson, V. E. Ostashev, and G. H. Goedecke, “Sound-wave coherence in atmospheric turbulence with intrinsic and global intermittency,” J. Acoust. Soc. Am. 124, 743–757 (2008).
[Crossref] [PubMed]

S. SankarRay, D. Mitra, and R. Pandit, “The universality of dynamic multiscaling in homogeneous, isotropic NavierStokes and passive-scalar turbulence,” New J. Phys. 10, 033003 (2008).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 26003 (2008).
[Crossref]

Y. Jiang, J. Ma, L. Tan, S. Yu, and W. Du, “Measurement of optical intensity fluctuation over an 118 km turbulent path,” Opt. Express 16, 6963 (2008).
[Crossref] [PubMed]

2006 (2)

G. H. Goedecke, D. K. Wilson, and V. E. Ostashev, “Quasi-wavelet models of turbulent temperature fluctuations,” Bound.-Lay. Meteorol. 120, 1–23 (2006).
[Crossref]

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlin. Process. Geophys. 13, 297–301 (2006).
[Crossref]

2005 (1)

A. Celani, M. Cencini, M. Vergassola, E. Villeramux, and D. Vincenzi, “Shear effects on passive scalar spectra,” J. Fluid Mech. 523, 99–108 (2005).
[Crossref]

2004 (1)

G. H. Goedecke, V. E. Ostashev, D. K. Wilson, and H. J. Auvermann, “Quasi-wavelet model of von Kármán spectrum of turbulent velocity fluctuations,” Bound.-Lay. Meteorol. 112, 33–56 (2004).
[Crossref]

2002 (1)

P. F. Lazorenko, “Differential image motion at non-Kolmogorov distortions of the turbulent wave-front,” Astron. Astrophys. 382, 1125–1137 (2002).
[Crossref]

2000 (1)

E. S. C. Ching, “Intermittency of temperature field in turbulent convection,” Phys. Fluids E 61, R33–R36 (2000).

1998 (1)

U. Frisch, A. Mazzino, and M. Vergassola, “Intermittency in passive scalar advection,” Phys. Rev. Lett. 80, 5532–5535 (1998).
[Crossref]

1996 (1)

K. R. Sreenivasan, “The passive scalar spectrum and the Obukhov-Corrsin constant,” Phys. Fluids 8, 189 (1996).
[Crossref]

1995 (1)

1992 (3)

1984 (1)

E. Vilar and J. Haddon, “Measurement and modeling of scintillation intensity to estimate turbulence parameters in an Earth-space path,” IEEE Trans. Antennas Propag. 32, 340–346 (1984).
[Crossref]

1978 (1)

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541 (1978).
[Crossref]

1971 (1)

1968 (1)

J. Strohbehn, “Line-of-sight wave propagation through the turbulent atmosphere,” Proc. IEEE 56, 1301–1318 (1968).
[Crossref]

1951 (1)

S. Corrsin, “On the spectrum of isotropic temperature fluctuations in an isotropic turbulence,” J. Appl. Phys. 22, 469 (1951).
[Crossref]

1949 (1)

A. M. Obukhov, “Temperature field structure in a turbulent flow,” Izv. Akad. Nauk. SSSR Ser. Geogr. Geophys. 13, 58–69 (1949).

Acton, D. S.

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 26003 (2008).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Auvermann, H. J.

G. H. Goedecke, V. E. Ostashev, D. K. Wilson, and H. J. Auvermann, “Quasi-wavelet model of von Kármán spectrum of turbulent velocity fluctuations,” Bound.-Lay. Meteorol. 112, 33–56 (2004).
[Crossref]

Bai, X.

Bester, M.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, and C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[Crossref]

Boreman, G. D.

Branover, H.

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlin. Process. Geophys. 13, 297–301 (2006).
[Crossref]

Buser, R. G.

Cao, X.

Celani, A.

A. Celani, M. Cencini, M. Vergassola, E. Villeramux, and D. Vincenzi, “Shear effects on passive scalar spectra,” J. Fluid Mech. 523, 99–108 (2005).
[Crossref]

Cencini, M.

A. Celani, M. Cencini, M. Vergassola, E. Villeramux, and D. Vincenzi, “Shear effects on passive scalar spectra,” J. Fluid Mech. 523, 99–108 (2005).
[Crossref]

Ching, E. S. C.

E. S. C. Ching, “Intermittency of temperature field in turbulent convection,” Phys. Fluids E 61, R33–R36 (2000).

Corrsin, S.

S. Corrsin, “On the spectrum of isotropic temperature fluctuations in an isotropic turbulence,” J. Appl. Phys. 22, 469 (1951).
[Crossref]

Cui, L.

Dainty, J. C.

Danchi, W. C.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, and C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[Crossref]

Dayton, D.

Degiacomi, C. G.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, and C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[Crossref]

Du, W.

DuBois, T. C.

A. Skvortsov, M. Jamriska, and T. C. DuBois, “Scaling laws of passive tracer dispersion in the turbulent surface layer,” Phys. Rev. E 82, 056304 (2010).
[Crossref]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 26003 (2008).
[Crossref]

Figueroa, E.

G. Funes, E. Figueroa, D. Gulich, L. Zunino, and D. G. Pérez, “Characterizing inertial and convective optical turbulence by detrended fluctuation analysis,” Proc. SPIE 8890, 889016 (2013).
[Crossref]

Frisch, U.

U. Frisch, A. Mazzino, and M. Vergassola, “Intermittency in passive scalar advection,” Phys. Rev. Lett. 80, 5532–5535 (1998).
[Crossref]

U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, 1995).

Funes, G.

G. Funes, E. Figueroa, D. Gulich, L. Zunino, and D. G. Pérez, “Characterizing inertial and convective optical turbulence by detrended fluctuation analysis,” Proc. SPIE 8890, 889016 (2013).
[Crossref]

D. G. Pérez and G. Funes, “Beam wandering statistics of twin thin laser beam propagation under generalized atmospheric conditions,” Opt. Express 20, 27766 (2012).
[Crossref] [PubMed]

Goedecke, G. H.

D. K. Wilson, V. E. Ostashev, and G. H. Goedecke, “Quasi-wavelet formulations of turbulence and other random fields with correlated properties,” Probabilist. Eng. Mech. 24, 343–357 (2009).
[Crossref]

D. K. Wilson, V. E. Ostashev, and G. H. Goedecke, “Sound-wave coherence in atmospheric turbulence with intrinsic and global intermittency,” J. Acoust. Soc. Am. 124, 743–757 (2008).
[Crossref] [PubMed]

G. H. Goedecke, D. K. Wilson, and V. E. Ostashev, “Quasi-wavelet models of turbulent temperature fluctuations,” Bound.-Lay. Meteorol. 120, 1–23 (2006).
[Crossref]

G. H. Goedecke, V. E. Ostashev, D. K. Wilson, and H. J. Auvermann, “Quasi-wavelet model of von Kármán spectrum of turbulent velocity fluctuations,” Bound.-Lay. Meteorol. 112, 33–56 (2004).
[Crossref]

Golbraikh, E.

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlin. Process. Geophys. 13, 297–301 (2006).
[Crossref]

Gonglewski, J.

Greenhill, L. J.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, and C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[Crossref]

Gulich, D.

G. Funes, E. Figueroa, D. Gulich, L. Zunino, and D. G. Pérez, “Characterizing inertial and convective optical turbulence by detrended fluctuation analysis,” Proc. SPIE 8890, 889016 (2013).
[Crossref]

Haddon, J.

E. Vilar and J. Haddon, “Measurement and modeling of scintillation intensity to estimate turbulence parameters in an Earth-space path,” IEEE Trans. Antennas Propag. 32, 340–346 (1984).
[Crossref]

Hill, R. J.

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541 (1978).
[Crossref]

Jamriska, M.

A. Skvortsov, M. Jamriska, and T. C. DuBois, “Scaling laws of passive tracer dispersion in the turbulent surface layer,” Phys. Rev. E 82, 056304 (2010).
[Crossref]

Jiang, Y.

Kopeika, N. S.

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlin. Process. Geophys. 13, 297–301 (2006).
[Crossref]

Lazorenko, P. F.

P. F. Lazorenko, “Differential image motion at non-Kolmogorov distortions of the turbulent wave-front,” Astron. Astrophys. 382, 1125–1137 (2002).
[Crossref]

Ma, J.

Mazzino, A.

U. Frisch, A. Mazzino, and M. Vergassola, “Intermittency in passive scalar advection,” Phys. Rev. Lett. 80, 5532–5535 (1998).
[Crossref]

Mitra, D.

S. SankarRay, D. Mitra, and R. Pandit, “The universality of dynamic multiscaling in homogeneous, isotropic NavierStokes and passive-scalar turbulence,” New J. Phys. 10, 033003 (2008).
[Crossref]

Nicholls, T. W.

Obukhov, A. M.

A. M. Obukhov, “Temperature field structure in a turbulent flow,” Izv. Akad. Nauk. SSSR Ser. Geogr. Geophys. 13, 58–69 (1949).

Ostashev, V. E.

D. K. Wilson, V. E. Ostashev, and G. H. Goedecke, “Quasi-wavelet formulations of turbulence and other random fields with correlated properties,” Probabilist. Eng. Mech. 24, 343–357 (2009).
[Crossref]

D. K. Wilson, V. E. Ostashev, and G. H. Goedecke, “Sound-wave coherence in atmospheric turbulence with intrinsic and global intermittency,” J. Acoust. Soc. Am. 124, 743–757 (2008).
[Crossref] [PubMed]

G. H. Goedecke, D. K. Wilson, and V. E. Ostashev, “Quasi-wavelet models of turbulent temperature fluctuations,” Bound.-Lay. Meteorol. 120, 1–23 (2006).
[Crossref]

G. H. Goedecke, V. E. Ostashev, D. K. Wilson, and H. J. Auvermann, “Quasi-wavelet model of von Kármán spectrum of turbulent velocity fluctuations,” Bound.-Lay. Meteorol. 112, 33–56 (2004).
[Crossref]

Pandit, R.

S. SankarRay, D. Mitra, and R. Pandit, “The universality of dynamic multiscaling in homogeneous, isotropic NavierStokes and passive-scalar turbulence,” New J. Phys. 10, 033003 (2008).
[Crossref]

Pérez, D. G.

G. Funes, E. Figueroa, D. Gulich, L. Zunino, and D. G. Pérez, “Characterizing inertial and convective optical turbulence by detrended fluctuation analysis,” Proc. SPIE 8890, 889016 (2013).
[Crossref]

D. G. Pérez and G. Funes, “Beam wandering statistics of twin thin laser beam propagation under generalized atmospheric conditions,” Opt. Express 20, 27766 (2012).
[Crossref] [PubMed]

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 26003 (2008).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Pierson, B.

Roehrig, J. R.

SankarRay, S.

S. SankarRay, D. Mitra, and R. Pandit, “The universality of dynamic multiscaling in homogeneous, isotropic NavierStokes and passive-scalar turbulence,” New J. Phys. 10, 033003 (2008).
[Crossref]

Sharbaugh, R. J.

Skvortsov, A.

A. Skvortsov, M. Jamriska, and T. C. DuBois, “Scaling laws of passive tracer dispersion in the turbulent surface layer,” Phys. Rev. E 82, 056304 (2010).
[Crossref]

Spielbusch, B.

Sreenivasan, K. R.

K. R. Sreenivasan, “The passive scalar spectrum and the Obukhov-Corrsin constant,” Phys. Fluids 8, 189 (1996).
[Crossref]

Strohbehn, J.

J. Strohbehn, “Line-of-sight wave propagation through the turbulent atmosphere,” Proc. IEEE 56, 1301–1318 (1968).
[Crossref]

Tan, L.

Tatarski, V. I.

V. I. Tatarskĭ, Wave Propagation in a Turbulent Atmosphere (Nauka Press, 1967).

Tatarskii, V.I.

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

Tiszauer, D.

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 26003 (2008).
[Crossref]

Townes, C. H.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, and C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[Crossref]

Vergassola, M.

A. Celani, M. Cencini, M. Vergassola, E. Villeramux, and D. Vincenzi, “Shear effects on passive scalar spectra,” J. Fluid Mech. 523, 99–108 (2005).
[Crossref]

U. Frisch, A. Mazzino, and M. Vergassola, “Intermittency in passive scalar advection,” Phys. Rev. Lett. 80, 5532–5535 (1998).
[Crossref]

Vilar, E.

E. Vilar and J. Haddon, “Measurement and modeling of scintillation intensity to estimate turbulence parameters in an Earth-space path,” IEEE Trans. Antennas Propag. 32, 340–346 (1984).
[Crossref]

Villeramux, E.

A. Celani, M. Cencini, M. Vergassola, E. Villeramux, and D. Vincenzi, “Shear effects on passive scalar spectra,” J. Fluid Mech. 523, 99–108 (2005).
[Crossref]

Vincenzi, D.

A. Celani, M. Cencini, M. Vergassola, E. Villeramux, and D. Vincenzi, “Shear effects on passive scalar spectra,” J. Fluid Mech. 523, 99–108 (2005).
[Crossref]

Wilson, D. K.

D. K. Wilson, V. E. Ostashev, and G. H. Goedecke, “Quasi-wavelet formulations of turbulence and other random fields with correlated properties,” Probabilist. Eng. Mech. 24, 343–357 (2009).
[Crossref]

D. K. Wilson, V. E. Ostashev, and G. H. Goedecke, “Sound-wave coherence in atmospheric turbulence with intrinsic and global intermittency,” J. Acoust. Soc. Am. 124, 743–757 (2008).
[Crossref] [PubMed]

G. H. Goedecke, D. K. Wilson, and V. E. Ostashev, “Quasi-wavelet models of turbulent temperature fluctuations,” Bound.-Lay. Meteorol. 120, 1–23 (2006).
[Crossref]

G. H. Goedecke, V. E. Ostashev, D. K. Wilson, and H. J. Auvermann, “Quasi-wavelet model of von Kármán spectrum of turbulent velocity fluctuations,” Bound.-Lay. Meteorol. 112, 33–56 (2004).
[Crossref]

Xue, B.

Xue, W.

Yu, S.

Zheng, S.

Zhou, F.

Zilberman, A.

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlin. Process. Geophys. 13, 297–301 (2006).
[Crossref]

Zunino, L.

G. Funes, E. Figueroa, D. Gulich, L. Zunino, and D. G. Pérez, “Characterizing inertial and convective optical turbulence by detrended fluctuation analysis,” Proc. SPIE 8890, 889016 (2013).
[Crossref]

Appl. Opt. (1)

Astron. Astrophys. (1)

P. F. Lazorenko, “Differential image motion at non-Kolmogorov distortions of the turbulent wave-front,” Astron. Astrophys. 382, 1125–1137 (2002).
[Crossref]

Astrophys. J. (1)

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, and C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[Crossref]

Bound.-Lay. Meteorol. (2)

G. H. Goedecke, D. K. Wilson, and V. E. Ostashev, “Quasi-wavelet models of turbulent temperature fluctuations,” Bound.-Lay. Meteorol. 120, 1–23 (2006).
[Crossref]

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Figures (3)

Fig. 1
Fig. 1 Comparison between QW and von Kármán spectra for L0 = 28.1 m, 0 = 0.0054 m, and C n 2 = 1 m 2 H: (a) corresponds to the Obukhov-Kolmogorov case (α = −11/3, H = 1/3), QW spectrum depicted following (20) with β = 2πand β = 2.4π; (b) exemplifies a NOK case for H = 1/6 (α = −10/3), QW spectrum defined as in (a); (c) and (d) show the low- and high-frequency regions, respectively, for the OK case—H = 1/3; (e) and (f) show the low- and high-frequency regions, respectively, of different NOK spectrum.
Fig. 2
Fig. 2 Comparison between von Kármán, generalized atmospheric spectrum, Andrews & Phillips approximation, and quasi-wavelet wavefront structure functions: (a) corresponds to the OK case (α = −11/3 and H = 1/3), the QW spectra were calculated under the same phase variance condition with a1 = 0.546L0, and Eq. (20) with β= 2π; (b) NOK case for H = 1/6, the QW spectra were calculated under the same phase variance condition with a1 = 0.522L0, and Eq. (20) with β = 2π—in all evaluations L0 = 10 m, 0 = 1 cm, L = 10 m and C n 2 = 10 14 m 2 H.
Fig. 3
Fig. 3 Angle-of-arrival variance for different models, normalized by C n 2 R 1 / 3 (the variance estimated for the OK model), with L0 = 33 m, 0 = 0.009 m, L = 10 m, and C n 2 = 1 m 2 H: estimated for OK turbulence, H = 1/3, (a) from Tatarskĭ’s approximation, Eq. (34), and (b) from aperture averaging, Eq. (35); then, estimated for NOK turbulence, H = 1/6, (c) from Tatarskĭ’s approximation and (d) from aperture averaging.

Equations (36)

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n 1 ( r ) = α = 1 N n = 1 N α n 1 α n ( r ) ,
n 1 α n ( r ) = N α n Ψ ( r R α n a α )
N α n = 0 , and N α n N β m = δ α β δ n m ν α 2 .
n 1 ( r ) n 1 ( r ) = α = 1 N N α ν α 2 Ψ ( r R α n a α ) Ψ ( r R α n a α )
Ψ ( r R α n a α ) Ψ ( r R α n a α ) = a α 6 V 3 d 3 f | Ψ ^ ( a α f ) | 2 exp [ i 2 π f ( r r ) ] ,
n 1 ( r ) n 1 ( r ) = 3 d 3 f Φ n ( f ) exp [ 2 π i f ( r r ) ] ,
Φ n ( f ) = α = 1 N ν α 2 φ α a α 3 | Ψ ^ ( a a f ) | 2 .
φ α = φ 1 ( a α / a 1 ) δ h , a N < a α < a 1 ,
ν α 2 = ν 1 2 ( a α / a 1 ) 2 h , a N < a α < a 1 ,
Φ n ( f ) = ν 1 2 φ 1 a 1 3 μ ( f a 1 ) 3 + δ h + 2 h f a N f a 1 d s s 2 + δ h + 2 h | Ψ ^ ( s n ^ ) | 2 .
Φ n ( f ) = ν 1 2 φ 1 a 1 3 2 μ γ [ [ 2 H + 3 ] 2 ; ( π f a N ) 2 ] [ γ [ 2 H + 3 ] 2 ; ( π f a 1 ) 2 ] ( π f a 1 ) 2 H + 3 ,
D n ( r r ) = 8 π 0 d f f 2 Φ n ( f ) [ 1 sin ( 2 π f r r ) ( 2 π f r r ) ] ;
D n ( r r ) = ν 1 2 φ 1 μ H a 1 2 H { a N 2 H [ H E H + 1 ( π r r 2 a N 2 ) 1 ] + a 1 2 H [ H E H + 1 ( π r r 2 a 1 2 ) 1 ] }
D n ( r r ) = ν 1 2 φ 1 π H + 1 μ sin ( π H ) Γ ( 1 + H ) r r 2 H a 1 2 H ;
D n ( r r ) = ν 1 2 φ 1 π ( a N / a 1 ) 2 H μ ( 1 H ) r r 2 a N 2 .
D n ( r r ) = { C n 2 ( H ) 0 2 H r r 2 0 2 , r r 0 , C n 2 ( H ) r r 2 H , 0 r r
0 = [ sin ( π H ) Γ ( 1 + H ) π H ( 1 H ) ] 1 2 H 2 a N , and C n 2 ( H ) = ν 1 2 φ 1 π 1 + H μ sin ( π H ) Γ ( 1 + H ) a 1 2 H
C n 2 ( H ) = 2 H + 1 sin ( π H ) Γ 2 ( H ) π κ 0 2 H
D n ( r r ) = ν 1 2 ϕ 1 μ H [ 1 ( a N a 1 ) 2 H ] .
L 0 = β ( 2 π ) 1 / 2 [ 2 Γ ( H ) ] 1 2 H [ 1 ( a N a 1 ) 2 H ] 1 2 H a 1 ,
L 0 = β 2 π 1 / 2 [ Γ ( H + 5 2 ) ] 1 2 H + 3 [ 1 ( a N a 1 ) 2 H + 3 ] 1 2 H + 3 a 1 .
n 1 ( r 1 ) n 1 ( r 2 ) n 1 ( r 3 ) n 1 ( r 4 ) = all distinct pairs i , j , k , l α , β = 1 N ν α 2 ϕ α ν β 2 ϕ β F α ( r i r j ) F β ( r k r l ) + + α = 1 N ν α 4 ϕ α G α ( r 1 r 4 , r 2 r 4 , r 3 r 4 ) α = 1 N ν α 4 ϕ α α 3 V all distinct pairs i , j , k , l F α ( r i r j ) F α ( r k r l ) ,
n 1 ( r 1 ) n 1 ( r 2 ) n 1 ( r 3 ) n 1 ( r 4 ) | G = all distinct pairs i , j , k , l n 1 ( r i ) n 1 ( r j ) n 1 ( r k ) n 1 ( r l ) = all distinct pairs i , j , k , l α , β = 1 N ν α 2 ϕ α ν β 2 ϕ β F α ( r i r j ) F β ( r k r l ) .
γ 2 = α = 1 N ν α 4 ϕ α α = 1 N ν α 2 ϕ α ( μ H ϕ 1 ) 1 1 δ h / 4 H ,
ψ 1 ( ρ , L ) = 0 1 d ξ 2 d 2 s n [ s , L ( 1 ξ ) ] h ( ρ , L ; s , ξ )
Φ ( ρ , L ) = 0 1 d ξ 2 d 2 s n [ s , L ( 1 ξ ) ] h ( ρ , L ; s , ξ ) ψ 1 [ s , L ( 1 ξ ) ]
h ( ρ , L ; s , ξ ) = k 2 2 π γ ξ exp [ i k L ξ ( γ ρ 2 + γ 1 s 2 2 ρ s ) ] ,
ψ 1 ( ρ , L ) = α = 1 N n = 1 N α ψ 1 α n ( ρ , L ) ,
ψ 1 α n ( ρ , L ) = i k L N α n a α 2 2 d 2 κ d f z Ψ ^ ( a α κ , a α f z ) × × 0 1 d ξ exp { 2 π i [ ( γ ρ ρ α n ) κ + ( L α n L ξ ) f z ] } exp ( i π λ γ κ 2 ξ ) .
ψ 1 ( ρ , L ) = i k L 2 α = 1 N n = 1 N α N α n 1 1 d σ a α 2 a α 2 + i λ L ( 1 σ ) × × exp ( 2 π a α 2 { a α 2 ρ α n ρ 2 a α 2 + i λ L ( 1 σ ) + [ ( 1 + σ ) 2 L L α n ] 2 } )
B ψ ( ρ ρ , L ) = ψ 1 ( ρ , L ) ψ 1 * ( ρ , L ) = k 2 L 2 4 α = 1 N ν α 2 ϕ α a α 2 1 1 1 + | δ | 1 | δ | d δ d t a α 2 + i λ L δ exp ( π L 2 a α 2 δ 2 ) exp ( π Δ 2 a α 2 + i λ L δ ) ;
B ψ ( ρ , ρ , L ) = k 2 L α = 1 N ν α 2 ϕ α a α ( 1 a α π L ) exp ( π Δ 2 a α 2 ) .
B ψ ( ρ ρ , L ) = k 2 L C n 2 ( H ) sin ( π H ) Γ ( 1 + H ) 2 π 1 + H × × [ a 1 2 H + 1 E H + 3 2 ( π Δ 2 a 1 2 ) a N 2 H + 1 E H + 3 2 ( π Δ 2 a N 2 ) ] ,
D ψ ( ρ ρ , L ) = k 2 L C n 2 ( H ) sin ( π H ) Γ ( 1 + H ) π 1 + H { a 1 2 H + 1 a N 2 H + 1 H + 1 / 2 [ a 1 2 H + 1 E H + 3 2 ( π Δ 2 a 1 2 ) a N 2 H + 1 E H + 3 2 ( π Δ 2 a N 2 ) ] } .
σ AoA 2 D ψ ( 2 R , L ) k 2 ( 2 R ) 2 = C n 2 ( H ) L sin ( π H ) Γ ( 1 + H ) 4 R 2 π 1 + H { a 1 2 H + 1 a N 2 H + 1 H + 1 / 2 [ a 1 2 H + 1 E H + 3 2 ( π 4 R 2 a 1 2 ) a N 2 H + 1 E H + 3 2 ( π 4 R 2 a N 2 ) ] }
σ AoA 2 = 1 k 2 π R 2 0 2 R ρ d ρ [ arccos ( ρ 2 R ) ρ 2 R 1 ( ρ 2 R ) 2 ] [ D ψ ( ρ , L ) + ρ 1 D ψ ( ρ , L ) ] .

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