Abstract

The lognormal (LN) and gamma–gamma (GG) distributions are compared to simulated and experimental data of the irradiance fluctuations of a Gaussian beam wave propagating through the atmosphere along a horizontal path, near the ground, in the moderate-to-strong turbulence regime. Irradiance data were collected simultaneously at three receiving apertures of different sizes. Atmospheric parameters were inferred from the measurements and scintillation theory and were used to develop the parameters for the theoretical probability density functions. Numerical simulations were produced with the same Cn2 value as the experimental data. Aperture-averaging effects were investigated by comparing the irradiance distributions for the three apertures at two different values of the structure parameter Cn2, and, hence, different values of the coherence radius ρ0. For the moderate-to-strong fluctuation regime, the GG distribution provides a good fit to the irradiance fluctuations collected by finite-sized apertures that are significantly smaller than ρ0. For apertures larger than or equal to ρ0, the irradiance fluctuations appear to be LN distributed.

© 2007 Optical Society of America

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  1. J. W. Strohbehn, T.-I. Wang, and J. P. Speck, "On the probability density distribution of line-of-sight fluctuations of optical signals," Radio Sci. 10, 59-70 (1975).
    [CrossRef]
  2. J. W. Strohbehn, "Modern theories in the propagation of optical waves in a turbulent medium," in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, 1978), pp. 97-104.
  3. L. C. Andrews and R. L. Phillips, "I-K-distribution as a universal propagation model of laser beams in atmospheric turbulence," J. Opt. Soc. Am. A 2, 160-163 (1985).
    [CrossRef]
  4. D. J. Lewinski, "Nonstationary probabilistic target and clutter scattering models," IEEE Trans. Antennas Propag. AP-31, 490-498 (1983).
    [CrossRef]
  5. E. Jakeman and P. N. Pusey, "The significance of K-distributions in scattering experiments," Phys. Rev. Lett. 40, 546-550 (1978).
    [CrossRef]
  6. G. Parry and P. N. Pusey, "K distributions in atmospheric propagation of laser light," J. Opt. Soc. Am. 69, 796-798 (1979).
    [CrossRef]
  7. E. Jakeman, "On the statistics of K-distributed noise," J. Phys. A 13, 31-48 (1980).
    [CrossRef]
  8. M. C. Teich and P. Diament, "Multiply stochastic representations for K distributions and their Poisson transforms," J. Opt. Soc. Am. A 6, 80-91 (1989).
    [CrossRef]
  9. J. H. Churnside and R. J. Hill, "Probability density of irradiance scintillations for strong path-integrated refractive turbulence," J. Opt. Soc. Am. A 4, 727-733 (1987).
    [CrossRef]
  10. L. C. Andrews and R. L. Phillips, "Mathematical genesis of the I-K-distribution for random optical fields," J. Opt. Soc. Am. A 3, 1912-1919 (1986).
    [CrossRef]
  11. J. H. Churnside and S. F. Clifford, "Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere," J. Opt. Soc. Am. A 4, 1923-1930 (1987).
    [CrossRef]
  12. R. J. Hill and R. G. Frehlich, "Probability distribution of irradiance for the onset of strong scintillation," J. Opt. Soc. Am. A 14, 1530-1540 (1997).
    [CrossRef]
  13. M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mathematical model for the irradiance PDF of a laser beam propagating through turbulent media," Opt. Eng. 40, 1554-1562 (2001).
    [CrossRef]
  14. J. H. Churnside and R. G. Frehlich, "Experimental evaluation of log-normally modulated Rician and IK models of optical scintillation in the atmosphere," J. Opt. Soc. Am. A 6, 1760-1766 (1989).
    [CrossRef]
  15. Z. Azar, H. M. Loebenstein, G. Appelbaum, E. Azoulay, U. Halavee, M. Tamir, and M. Tur, "Aperture averaging of the two-wavelength intensity covariance function in atmospheric turbulence," Appl. Opt. 24, 2401-2407 (1985).
    [CrossRef] [PubMed]
  16. A. D. Wheelon, "Skewed distribution of irradiance predicted by second-order Rytov approximation," J. Opt. Soc. Am. A 18, 2789-2798 (2001).
    [CrossRef]
  17. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering Press, 2001).
  18. S. M. Flatté, C. Bracher, and G.-Yu Wang, "Probability-density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulations," J. Opt. Soc. Am. A 11, 2080-2092 (1994).
    [CrossRef]
  19. G. J. Baker, "Gaussian beam weak scintillation: low-order turbulence effects and applicability of the Rytov method," J. Opt. Soc. Am. A 23, 395-417 (2006).
    [CrossRef]
  20. F. Dios, J. A. Rubio, A. Rodríguez, and A. Comerón, "Scintillation and beam-wander analysis in an optical ground station-satellite uplink," Appl. Opt. 43, 3866-3873 (2004).
    [CrossRef] [PubMed]
  21. J. H. Churnside, "Aperture averaging of optical scintillation in the turbulent atmosphere," Appl. Opt. 30, 1982-1994 (1991).
    [CrossRef] [PubMed]
  22. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, "Aperture averaging of optical scintillations: power fluctuations and temporal spectrum," Waves Random Media 10, 53-70 (2000).
    [CrossRef]
  23. V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave Propagation (National Technical Information Office, 1971), p. 292.
  24. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, "Theory of optical scintillation," J. Opt. Soc. Am. A 16, 1417-1429 (1999).
    [CrossRef]
  25. 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]
  26. F. Strömqvist Vetelino, B. Clare, K. Corbett, C. Young, K. Grant, and L. Andrews, "Characterizing the propagation path in moderate to strong optical turbulence," Appl. Opt. 45, 3534-3543 (2006).
    [CrossRef] [PubMed]
  27. "Downhill simplex method in multidimensions," Chap. 10.4 in Numerical Recipes in C: The Art of Scientific Computing (1988-1992), http://www.library.cornell.edu/nr/bookcpdf/c10-4.pdf.
  28. R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
    [CrossRef]
  29. J. Recolons and F. Dios, "Accurate calculation of phase screens for the modeling of laser beam propagation through atmospheric turbulence," in Proc. SPIE 5891, 51-62 (2005).

2006 (2)

2005 (1)

J. Recolons and F. Dios, "Accurate calculation of phase screens for the modeling of laser beam propagation through atmospheric turbulence," in Proc. SPIE 5891, 51-62 (2005).

2004 (1)

2001 (3)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mathematical model for the irradiance PDF of a laser beam propagating through turbulent media," Opt. Eng. 40, 1554-1562 (2001).
[CrossRef]

A. D. Wheelon, "Skewed distribution of irradiance predicted by second-order Rytov approximation," J. Opt. Soc. Am. A 18, 2789-2798 (2001).
[CrossRef]

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]

2000 (1)

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

1999 (1)

1997 (1)

1994 (1)

1992 (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
[CrossRef]

1991 (1)

1989 (2)

1987 (2)

1986 (1)

1985 (2)

1983 (1)

D. J. Lewinski, "Nonstationary probabilistic target and clutter scattering models," IEEE Trans. Antennas Propag. AP-31, 490-498 (1983).
[CrossRef]

1980 (1)

E. Jakeman, "On the statistics of K-distributed noise," J. Phys. A 13, 31-48 (1980).
[CrossRef]

1979 (1)

1978 (1)

E. Jakeman and P. N. Pusey, "The significance of K-distributions in scattering experiments," Phys. Rev. Lett. 40, 546-550 (1978).
[CrossRef]

1975 (1)

J. W. Strohbehn, T.-I. Wang, and J. P. Speck, "On the probability density distribution of line-of-sight fluctuations of optical signals," Radio Sci. 10, 59-70 (1975).
[CrossRef]

Al-Habash, M. A.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mathematical model for the irradiance PDF of a laser beam propagating through turbulent media," Opt. Eng. 40, 1554-1562 (2001).
[CrossRef]

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, C. Y. Hopen, and M. A. Al-Habash, "Theory of optical scintillation," J. Opt. Soc. Am. A 16, 1417-1429 (1999).
[CrossRef]

Andrews, L.

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]

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mathematical model for the irradiance PDF of a laser beam propagating through turbulent media," Opt. Eng. 40, 1554-1562 (2001).
[CrossRef]

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

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, "Theory of optical scintillation," J. Opt. Soc. Am. A 16, 1417-1429 (1999).
[CrossRef]

L. C. Andrews and R. L. Phillips, "Mathematical genesis of the I-K-distribution for random optical fields," J. Opt. Soc. Am. A 3, 1912-1919 (1986).
[CrossRef]

L. C. Andrews and R. L. Phillips, "I-K-distribution as a universal propagation model of laser beams in atmospheric turbulence," J. Opt. Soc. Am. A 2, 160-163 (1985).
[CrossRef]

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

Appelbaum, G.

Azar, Z.

Azoulay, E.

Baker, G. J.

Bracher, C.

Churnside, J. H.

Clare, B.

Clifford, S. F.

Comerón, A.

Corbett, K.

Dainty, J. C.

R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
[CrossRef]

Diament, P.

Dios, F.

J. Recolons and F. Dios, "Accurate calculation of phase screens for the modeling of laser beam propagation through atmospheric turbulence," in Proc. SPIE 5891, 51-62 (2005).

F. Dios, J. A. Rubio, A. Rodríguez, and A. Comerón, "Scintillation and beam-wander analysis in an optical ground station-satellite uplink," Appl. Opt. 43, 3866-3873 (2004).
[CrossRef] [PubMed]

Flatté, S. M.

Frehlich, R. G.

Glindemann, A.

R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
[CrossRef]

Grant, K.

Halavee, U.

Hill, R. J.

Hopen, C. Y.

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 temporal spectrum," Waves Random Media 10, 53-70 (2000).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, "Theory of optical scintillation," J. Opt. Soc. Am. A 16, 1417-1429 (1999).
[CrossRef]

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

Jakeman, E.

E. Jakeman, "On the statistics of K-distributed noise," J. Phys. A 13, 31-48 (1980).
[CrossRef]

E. Jakeman and P. N. Pusey, "The significance of K-distributions in scattering experiments," Phys. Rev. Lett. 40, 546-550 (1978).
[CrossRef]

Lane, R. G.

R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
[CrossRef]

Lewinski, D. J.

D. J. Lewinski, "Nonstationary probabilistic target and clutter scattering models," IEEE Trans. Antennas Propag. AP-31, 490-498 (1983).
[CrossRef]

Loebenstein, H. M.

Parry, G.

Phillips, R. L.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mathematical model for the irradiance PDF of a laser beam propagating through turbulent media," Opt. Eng. 40, 1554-1562 (2001).
[CrossRef]

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 temporal spectrum," Waves Random Media 10, 53-70 (2000).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, "Theory of optical scintillation," J. Opt. Soc. Am. A 16, 1417-1429 (1999).
[CrossRef]

L. C. Andrews and R. L. Phillips, "Mathematical genesis of the I-K-distribution for random optical fields," J. Opt. Soc. Am. A 3, 1912-1919 (1986).
[CrossRef]

L. C. Andrews and R. L. Phillips, "I-K-distribution as a universal propagation model of laser beams in atmospheric turbulence," J. Opt. Soc. Am. A 2, 160-163 (1985).
[CrossRef]

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

Pusey, P. N.

G. Parry and P. N. Pusey, "K distributions in atmospheric propagation of laser light," J. Opt. Soc. Am. 69, 796-798 (1979).
[CrossRef]

E. Jakeman and P. N. Pusey, "The significance of K-distributions in scattering experiments," Phys. Rev. Lett. 40, 546-550 (1978).
[CrossRef]

Recolons, J.

J. Recolons and F. Dios, "Accurate calculation of phase screens for the modeling of laser beam propagation through atmospheric turbulence," in Proc. SPIE 5891, 51-62 (2005).

Rodríguez, A.

Rubio, J. A.

Speck, J. P.

J. W. Strohbehn, T.-I. Wang, and J. P. Speck, "On the probability density distribution of line-of-sight fluctuations of optical signals," Radio Sci. 10, 59-70 (1975).
[CrossRef]

Strohbehn, J. W.

J. W. Strohbehn, T.-I. Wang, and J. P. Speck, "On the probability density distribution of line-of-sight fluctuations of optical signals," Radio Sci. 10, 59-70 (1975).
[CrossRef]

J. W. Strohbehn, "Modern theories in the propagation of optical waves in a turbulent medium," in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, 1978), pp. 97-104.

Tamir, M.

Tatarskii, V. I.

V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave Propagation (National Technical Information Office, 1971), p. 292.

Teich, M. C.

Tur, M.

Vetelino, F. Strömqvist

Wang, G.-Yu

Wang, T.-I.

J. W. Strohbehn, T.-I. Wang, and J. P. Speck, "On the probability density distribution of line-of-sight fluctuations of optical signals," Radio Sci. 10, 59-70 (1975).
[CrossRef]

Wheelon, A. D.

Young, C.

Appl. Opt. (4)

IEEE Trans. Antennas Propag. (1)

D. J. Lewinski, "Nonstationary probabilistic target and clutter scattering models," IEEE Trans. Antennas Propag. AP-31, 490-498 (1983).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (11)

L. C. Andrews and R. L. Phillips, "I-K-distribution as a universal propagation model of laser beams in atmospheric turbulence," J. Opt. Soc. Am. A 2, 160-163 (1985).
[CrossRef]

J. H. Churnside and R. G. Frehlich, "Experimental evaluation of log-normally modulated Rician and IK models of optical scintillation in the atmosphere," J. Opt. Soc. Am. A 6, 1760-1766 (1989).
[CrossRef]

A. D. Wheelon, "Skewed distribution of irradiance predicted by second-order Rytov approximation," J. Opt. Soc. Am. A 18, 2789-2798 (2001).
[CrossRef]

S. M. Flatté, C. Bracher, and G.-Yu Wang, "Probability-density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulations," J. Opt. Soc. Am. A 11, 2080-2092 (1994).
[CrossRef]

G. J. Baker, "Gaussian beam weak scintillation: low-order turbulence effects and applicability of the Rytov method," J. Opt. Soc. Am. A 23, 395-417 (2006).
[CrossRef]

M. C. Teich and P. Diament, "Multiply stochastic representations for K distributions and their Poisson transforms," J. Opt. Soc. Am. A 6, 80-91 (1989).
[CrossRef]

J. H. Churnside and R. J. Hill, "Probability density of irradiance scintillations for strong path-integrated refractive turbulence," J. Opt. Soc. Am. A 4, 727-733 (1987).
[CrossRef]

L. C. Andrews and R. L. Phillips, "Mathematical genesis of the I-K-distribution for random optical fields," J. Opt. Soc. Am. A 3, 1912-1919 (1986).
[CrossRef]

J. H. Churnside and S. F. Clifford, "Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere," J. Opt. Soc. Am. A 4, 1923-1930 (1987).
[CrossRef]

R. J. Hill and R. G. Frehlich, "Probability distribution of irradiance for the onset of strong scintillation," J. Opt. Soc. Am. A 14, 1530-1540 (1997).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, "Theory of optical scintillation," J. Opt. Soc. Am. A 16, 1417-1429 (1999).
[CrossRef]

J. Phys. A (1)

E. Jakeman, "On the statistics of K-distributed noise," J. Phys. A 13, 31-48 (1980).
[CrossRef]

Opt. Eng. (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mathematical model for the irradiance PDF of a laser beam propagating through turbulent media," Opt. Eng. 40, 1554-1562 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

E. Jakeman and P. N. Pusey, "The significance of K-distributions in scattering experiments," Phys. Rev. Lett. 40, 546-550 (1978).
[CrossRef]

Radio Sci. (1)

J. W. Strohbehn, T.-I. Wang, and J. P. Speck, "On the probability density distribution of line-of-sight fluctuations of optical signals," Radio Sci. 10, 59-70 (1975).
[CrossRef]

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]

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

R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
[CrossRef]

Other (5)

J. Recolons and F. Dios, "Accurate calculation of phase screens for the modeling of laser beam propagation through atmospheric turbulence," in Proc. SPIE 5891, 51-62 (2005).

V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave Propagation (National Technical Information Office, 1971), p. 292.

"Downhill simplex method in multidimensions," Chap. 10.4 in Numerical Recipes in C: The Art of Scientific Computing (1988-1992), http://www.library.cornell.edu/nr/bookcpdf/c10-4.pdf.

J. W. Strohbehn, "Modern theories in the propagation of optical waves in a turbulent medium," in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, 1978), pp. 97-104.

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

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Figures (4)

Fig. 1
Fig. 1

PDF of the normalized log irradiance, ln(I), based on the experimental data (circles) and the simulation data (crosses) for the receiving apertures of diameter (a), (b) 1 mm, (c), (d) 5 mm, and (e), (f) 13 mm. Two cases of turbulence strength are shown: (a), (c), (e) C n 2 = 6.47 × 10 14 and (b), (d), (f) C n 2 = 4.58 × 10 13 .

Fig. 2
Fig. 2

PDF of the normalized log irradiance ln(I) for the receiving apertures of diameter (a), (b) 1 mm, (c), (d) 5 mm, and (e), (f) 13 mm. Comparisons are made between the experimental data and the GG distribution and the LN distribution for (a), (c), (e) C n 2 = 6.47 × 10 14 and (b), (d), (f) C n 2 = 4.58 × 10 13 .

Fig. 3
Fig. 3

PDF of the normalized log irradiance ln(I) for the receiving apertures of diameter (a), (b) 1 mm, (c), (d) 5 mm, and (e), (f) 13 mm. Comparisons are made between the simulation data and the GG distribution and the LN distribution for (a), (c), (e) C n 2 = 6.47 × 10 14 and (b), (d), (f) C n 2 = 4.58 × 10 13 .

Fig. 4
Fig. 4

Three receiver-aperture diameters (Ap. Diam) compared to the Fresnel zone, scattering disk and coherence radius for different values of C n 2 .

Tables (2)

Tables Icon

Table 1 Atmospheric Parameters Inferred from the Experimental Data and the PDF Parameters Used to Calculate the LN and GG Distributions when Compared to the Experimental Data and the Simulation Data

Tables Icon

Table 2 PDF of the Irradiance Fluctuations for the Three Receiver Aperture Diameters and the Two Different Cases of Turbulence Based on the Experimental Data and Simulation Data, Respectively a

Equations (12)

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

p I ( I ) = 1 I 2 π σ ln I 2 exp [ [ ln ( I ) + 1 2 σ ln I 2 ] 2 2 σ ln I 2 ] , I > 0 ,
σ ln I 2 = ln ( σ I 2 + 1 ) ,
I = x y ,
p I ( I ) = 2 Γ ( α ) Γ ( β ) I ( α β I ) ( α + β ) / 2 K α β ( 2 α β I ) , I > 0 ,
I 2 = ( 1 + 1 α ) ( 1 + 1 β ) .
I 2 = x 2 y 2 = ( 1 + σ x 2 ) ( 1 + σ y 2 ) ,
α = 1 σ x 2 = 1 exp ( σ ln x 2 ) 1 ,
β = 1 σ y 2 = 1 exp ( σ ln y 2 ) 1 .
σ I 2 = exp ( σ ln x 2 + σ ln y 2 ) 1.
p z ( z ) = p I ( I ) | d I d z | I = e z = e z p I ( e z ) ,
| A μ ν | 2 = 2 π k 2 Δ z Φ n ( κ μ ν ) ( Δ κ ) 2 ,
σ I 2 = ( 1 + 1 α ) ( 1 + 1 β ) 1 ,

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