Abstract

The complex amplitude of scattered radiation at a point is modeled as a superposition of random amplitude, random phases, and random number of multipaths (or correlation areas) [see Eq. (2.1)]. When the phases have uniform probability density (strong-scatterer regime) and the multipaths are governed by a negative binomial distribution, then Jakeman and Pusey [ Phys. Rev. Lett. 40, 546 ( 1978); IEEE Trans. Antennas Propag. AP-24, 806 ( 1976)] and associates have shown that the probability density of the intensity of the scattered radiation becomes K distributed as the average number of multipaths tends to infinity. This density function has the property that its normalized second moment is always greater than two. However, field measurements by Parry [ Opt. Acta 28, 715 ( 1981)] and Phillips and Andrews [ J. Opt. Soc. Am. 71, 1440 ( 1981); J. Opt. Soc. Am. 72, 864 ( 1982)] show that the normalized second moment can lie below two but must be greater than or equal to unity. The present paper is devoted to a generalization in which the phases are nonuniformly distributed (weak-scatterer regime) but the multipaths are still governed by the negative binomial distribution. In the limiting case, in which the average number of multipaths tends to infinity, the probability density of the scattered intensity is shown to be a generalization of the K-density function. This density function has the property that its second moment is greater than or equal to unity. Section 5 is devoted to the fit between this model and the Phillips–Andrews experimental data. Finally, the moments of the scattered intensity are evaluated when the average number of multipaths is finite.

© 1986 Optical Society of America

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References

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  1. V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translation, Jerusalem, 1971).
  2. A. M. Prokhorov, F. V. Bunkin, K. S. Gochlashvily, V. I. Shisov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
    [CrossRef]
  3. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
    [CrossRef]
  4. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 68, 1424–1443 (1980).
    [CrossRef]
  5. J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere (Springer-Verlag, Berlin, 1978).
    [CrossRef]
  6. V. E. Zuev, Laser Beams in the Atmosphere (Consultants Bureau, New York, 1982).
    [CrossRef]
  7. V. I. Tatarski, V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” Prog. Opt. 18, 204–256 (1980).
    [CrossRef]
  8. E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,”IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
    [CrossRef]
  9. E. Jakeman, P.N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
    [CrossRef]
  10. G. Parry, P. N. Pusey, “K distributions in atmospheric propagation of laser light,”J. Opt. Soc. Am. 69, 796–798 (1979).
    [CrossRef]
  11. E. Jakeman, “On the statistics of K distributed noise,” J. Phys. A 13, 31–48 (1980).
    [CrossRef]
  12. R. L. Phillips, L. C. Andrews, “Measured statistics for laser light scattering in atmospheric turbulence,”J. Opt. Soc. Am. 71, 1440–1445 (1981).
    [CrossRef]
  13. R. L. Phillips, L. C. Andrews, “Universal fluctuations model for irradiance fluctuations in a turbulent medium,”J. Opt. Soc. Am. 72, 864–870 (1982).
    [CrossRef]
  14. G. Parry, “Measurement of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
    [CrossRef]
  15. R. Dashen, “Path integrals for waves in random media,”J. Math. Phys. 20, 894–920 (1979).
    [CrossRef]
  16. S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission through a Fluctuating Ocean (Cambridge U. Press, Cambridge, 1979).
  17. S. M. Flatté, D. R. Bernstein, R. Dashen, “Intensity moments by paths integral techniques for wave propagation through random media, with application to sound in the ocean,” Phys. Fluids 26, 1701–1713 (1983).
    [CrossRef]
  18. S. M. Flatté, “Wave propagation through random media: contributions from ocean acoustics,” Proc. IEEE 71, 1267–1294 (1983).
    [CrossRef]
  19. R. Dashen, “Distribution of intensity in a multiple scattering medium,” Opt. Lett. 10, 110–112 (1984).
    [CrossRef]
  20. R. von Mises, “Über die ‘Ganzzähligkeit’ der Atomgewichte und verwandte Fragen,” Phys. Z. 19, 490–500 (1918). See also A. J. Viterbi, Principles of Coherent Communciation (McGraw-Hill, New York, 1966), Chap. 4.
  21. R. Barakat, “Direct derivation of intensity and phase statistics of speckle produced by a weak scatterer from the random sinusoid model,”J. Opt. Soc. Am. 71, 86–90 (1981).
    [CrossRef]
  22. R. Barakat, “Onefold photoelectron counting statistics for non-Gaussian light: scattering from an arbitrary number of weak scatterers,”J. Opt. Soc. Am. 73, 1138–1142 (1983); see App. A.
    [CrossRef]
  23. R. Barakat, “Speckle intensity due to strong and weak scatterers in the presence of amplitude correlated multipaths,” Opt. Acta 29, 947–960 (1982).
    [CrossRef]
  24. H. M. Pedersen, “Object roughness dependence of partially developed speckle patterns in coherent light,” Opt. Commun. 16, 63–67 (1976).
    [CrossRef]
  25. B. Gnedenko, A. Kolmogorov, Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, Reading, Mass., 1964).
  26. R. Barakat, “Isotropic random flights: random numbers of flights,”J. Phys. A 15, 3073–3082 (1982). See Sec. 5.
    [CrossRef]
  27. Y. Luke, Integrals of Bessel Functions (McGraw-Hill, New York, 1962), p. 330.
  28. N. George, A Jain, R. D. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
    [CrossRef]
  29. S. F. Clifford, R. J. Hill, “Relation between irradiance and log-amplitude variance for optical scintillation described by the K distribution,”J. Opt. Soc. Am. 71, 112–114 (1981).
    [CrossRef]
  30. I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry (Oliver and Boyd, London, 1956), Chap. 2.

1984 (1)

1983 (3)

S. M. Flatté, D. R. Bernstein, R. Dashen, “Intensity moments by paths integral techniques for wave propagation through random media, with application to sound in the ocean,” Phys. Fluids 26, 1701–1713 (1983).
[CrossRef]

S. M. Flatté, “Wave propagation through random media: contributions from ocean acoustics,” Proc. IEEE 71, 1267–1294 (1983).
[CrossRef]

R. Barakat, “Onefold photoelectron counting statistics for non-Gaussian light: scattering from an arbitrary number of weak scatterers,”J. Opt. Soc. Am. 73, 1138–1142 (1983); see App. A.
[CrossRef]

1982 (3)

R. Barakat, “Isotropic random flights: random numbers of flights,”J. Phys. A 15, 3073–3082 (1982). See Sec. 5.
[CrossRef]

R. L. Phillips, L. C. Andrews, “Universal fluctuations model for irradiance fluctuations in a turbulent medium,”J. Opt. Soc. Am. 72, 864–870 (1982).
[CrossRef]

R. Barakat, “Speckle intensity due to strong and weak scatterers in the presence of amplitude correlated multipaths,” Opt. Acta 29, 947–960 (1982).
[CrossRef]

1981 (4)

1980 (3)

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

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

V. I. Tatarski, V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” Prog. Opt. 18, 204–256 (1980).
[CrossRef]

1979 (2)

1978 (1)

E. Jakeman, P.N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

1976 (2)

H. M. Pedersen, “Object roughness dependence of partially developed speckle patterns in coherent light,” Opt. Commun. 16, 63–67 (1976).
[CrossRef]

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,”IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[CrossRef]

1975 (3)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochlashvily, V. I. Shisov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

N. George, A Jain, R. D. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[CrossRef]

1918 (1)

R. von Mises, “Über die ‘Ganzzähligkeit’ der Atomgewichte und verwandte Fragen,” Phys. Z. 19, 490–500 (1918). See also A. J. Viterbi, Principles of Coherent Communciation (McGraw-Hill, New York, 1966), Chap. 4.

Andrews, L. C.

Barakat, R.

Bernstein, D. R.

S. M. Flatté, D. R. Bernstein, R. Dashen, “Intensity moments by paths integral techniques for wave propagation through random media, with application to sound in the ocean,” Phys. Fluids 26, 1701–1713 (1983).
[CrossRef]

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochlashvily, V. I. Shisov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Clifford, S. F.

Dashen, R.

R. Dashen, “Distribution of intensity in a multiple scattering medium,” Opt. Lett. 10, 110–112 (1984).
[CrossRef]

S. M. Flatté, D. R. Bernstein, R. Dashen, “Intensity moments by paths integral techniques for wave propagation through random media, with application to sound in the ocean,” Phys. Fluids 26, 1701–1713 (1983).
[CrossRef]

R. Dashen, “Path integrals for waves in random media,”J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission through a Fluctuating Ocean (Cambridge U. Press, Cambridge, 1979).

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Flatté, S. M.

S. M. Flatté, D. R. Bernstein, R. Dashen, “Intensity moments by paths integral techniques for wave propagation through random media, with application to sound in the ocean,” Phys. Fluids 26, 1701–1713 (1983).
[CrossRef]

S. M. Flatté, “Wave propagation through random media: contributions from ocean acoustics,” Proc. IEEE 71, 1267–1294 (1983).
[CrossRef]

S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission through a Fluctuating Ocean (Cambridge U. Press, Cambridge, 1979).

George, N.

N. George, A Jain, R. D. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[CrossRef]

Gnedenko, B.

B. Gnedenko, A. Kolmogorov, Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, Reading, Mass., 1964).

Gochlashvily, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochlashvily, V. I. Shisov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Hill, R. J.

Jain, A

N. George, A Jain, R. D. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[CrossRef]

Jakeman, E.

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

E. Jakeman, P.N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,”IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[CrossRef]

Kolmogorov, A.

B. Gnedenko, A. Kolmogorov, Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, Reading, Mass., 1964).

Luke, Y.

Y. Luke, Integrals of Bessel Functions (McGraw-Hill, New York, 1962), p. 330.

Melville, R. D.

N. George, A Jain, R. D. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[CrossRef]

Munk, W. H.

S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission through a Fluctuating Ocean (Cambridge U. Press, Cambridge, 1979).

Parry, G.

G. Parry, “Measurement of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
[CrossRef]

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

Pedersen, H. M.

H. M. Pedersen, “Object roughness dependence of partially developed speckle patterns in coherent light,” Opt. Commun. 16, 63–67 (1976).
[CrossRef]

Phillips, R. L.

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochlashvily, V. I. Shisov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Pusey, P. N.

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

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,”IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[CrossRef]

Pusey, P.N.

E. Jakeman, P.N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

Shisov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochlashvily, V. I. Shisov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Sneddon, I. N.

I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry (Oliver and Boyd, London, 1956), Chap. 2.

Tatarski, V. I.

V. I. Tatarski, V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” Prog. Opt. 18, 204–256 (1980).
[CrossRef]

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translation, Jerusalem, 1971).

von Mises, R.

R. von Mises, “Über die ‘Ganzzähligkeit’ der Atomgewichte und verwandte Fragen,” Phys. Z. 19, 490–500 (1918). See also A. J. Viterbi, Principles of Coherent Communciation (McGraw-Hill, New York, 1966), Chap. 4.

Watson, K. M.

S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission through a Fluctuating Ocean (Cambridge U. Press, Cambridge, 1979).

Zachariasen, F.

S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission through a Fluctuating Ocean (Cambridge U. Press, Cambridge, 1979).

Zavorotnyi, V. U.

V. I. Tatarski, V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” Prog. Opt. 18, 204–256 (1980).
[CrossRef]

Zuev, V. E.

V. E. Zuev, Laser Beams in the Atmosphere (Consultants Bureau, New York, 1982).
[CrossRef]

Appl. Phys. (1)

N. George, A Jain, R. D. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,”IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[CrossRef]

J. Math. Phys. (1)

R. Dashen, “Path integrals for waves in random media,”J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

J. Opt. Soc. Am. (6)

J. Phys. A (2)

R. Barakat, “Isotropic random flights: random numbers of flights,”J. Phys. A 15, 3073–3082 (1982). See Sec. 5.
[CrossRef]

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

Opt. Acta (2)

R. Barakat, “Speckle intensity due to strong and weak scatterers in the presence of amplitude correlated multipaths,” Opt. Acta 29, 947–960 (1982).
[CrossRef]

G. Parry, “Measurement of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
[CrossRef]

Opt. Commun. (1)

H. M. Pedersen, “Object roughness dependence of partially developed speckle patterns in coherent light,” Opt. Commun. 16, 63–67 (1976).
[CrossRef]

Opt. Lett. (1)

Phys. Fluids (1)

S. M. Flatté, D. R. Bernstein, R. Dashen, “Intensity moments by paths integral techniques for wave propagation through random media, with application to sound in the ocean,” Phys. Fluids 26, 1701–1713 (1983).
[CrossRef]

Phys. Rev. Lett. (1)

E. Jakeman, P.N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

Phys. Z. (1)

R. von Mises, “Über die ‘Ganzzähligkeit’ der Atomgewichte und verwandte Fragen,” Phys. Z. 19, 490–500 (1918). See also A. J. Viterbi, Principles of Coherent Communciation (McGraw-Hill, New York, 1966), Chap. 4.

Proc. IEEE (4)

S. M. Flatté, “Wave propagation through random media: contributions from ocean acoustics,” Proc. IEEE 71, 1267–1294 (1983).
[CrossRef]

A. M. Prokhorov, F. V. Bunkin, K. S. Gochlashvily, V. I. Shisov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

Prog. Opt. (1)

V. I. Tatarski, V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” Prog. Opt. 18, 204–256 (1980).
[CrossRef]

Other (7)

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translation, Jerusalem, 1971).

J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere (Springer-Verlag, Berlin, 1978).
[CrossRef]

V. E. Zuev, Laser Beams in the Atmosphere (Consultants Bureau, New York, 1982).
[CrossRef]

S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission through a Fluctuating Ocean (Cambridge U. Press, Cambridge, 1979).

B. Gnedenko, A. Kolmogorov, Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, Reading, Mass., 1964).

Y. Luke, Integrals of Bessel Functions (McGraw-Hill, New York, 1962), p. 330.

I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry (Oliver and Boyd, London, 1956), Chap. 2.

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Figures (8)

Fig. 1
Fig. 1

Normalized variance as a function of v for α = 1, 2, 3, 5, 10, ∞.

Fig. 2
Fig. 2

h3|NB〉w/〈h|NB〉w3 as a function of v for α = 1, 2, 3, 5, 10, 20, ∞.

Fig. 3
Fig. 3

I3〉 versus 〈I2〉 for generalized K-PDF.

Fig. 4
Fig. 4

fw(I) for α = 20: —, v = 0; ⋯, v = 2; — · — v = 5.

Fig. 5
Fig. 5

fw(I) for α = ∞: —, v = 0; ⋯, v = 2; — · —, v = 5.

Fig. 6
Fig. 6

fw(I) for α = 1: – – –, v = 0; —, v = 2.

Fig. 7
Fig. 7

Normalized third moment versus normalized second moment. Solid line corresponds to Fig. 3 for generalized K-PDF, whereas the solid circles correspond to the experimental data of Phillips and Andrews’ Fig. 3.12

Fig. 8
Fig. 8

Normalized third moment versus normalized second moment. Solid line corresponds to Fig. 3 for generalized K-PDF, whereas the filled circles correspond to the experimental data of Phillips and Andrews’ Fig. 4.12

Equations (86)

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

E = n = 1 N a n exp ( i θ n ) .
f ( θ ) = 1 2 π , - π < θ < π = 0 , elsewhere .
f ( θ ) = [ 2 π I 0 ( v ) ] - 1 exp ( v cos θ ) ,             - π < θ < π = 0 ,             elsewhere .
f ( θ ) ~ ( 2 π v - 1 ) - 1 / 2 exp ( - θ 2 / 2 v - 1 ) ,
f w ( h N ) = I 0 - N ( v ) I 0 ( v h 1 / 2 / a ) 1 2 0 J 0 N ( a ω ) J 0 ( h 1 / 2 ω ) ω d ω
f w ( h 0 ) = I 0 ( v h 1 / 2 / a ) δ ( h ) ,
f w ( h 1 ) = I 0 - 1 ( v ) I 0 ( v h 1 / 2 / a ) δ ( h - a 2 ) .
f w ( h ) = N = 0 f ( h N ) P ( N ) .
P ( N ) = N N e - N N ! ,             N = 0 , 1 , ,
P ( N ) = ( N + α - 1 N ) ( N / α ) N ( 1 + N / α ) N + α ,             N = 0 , 1 , ,
a )     Poisson : var ( N ) = N , b )     NB : var ( N ) = N ( 1 + N α ) .
f ( h ) = 1 2 I 0 ( v h 1 / 2 / a ) 0 d ω ω J 0 ( h 1 / 2 ω ) × N = 0 ( N + α - 1 N ) ( N J 0 ( a ω ) α I 0 ( v ) ) N ( 1 + N α ) - N - α = 1 2 I 0 ( v h 1 / 2 / a ) 0 Φ ( ω ) J 0 ( h 1 / 2 ω ) ω d ω ,
Φ ( ω ) { 1 + N α [ 1 - J 0 ( a ω ) I 0 ( v ) ] } - α
Φ ( ω ) = exp { - N [ 1 - J 0 ( a ω ) I 0 ( v ) ] } .
h k NB = N = 0 h k N P ( N ) ,
N ( N - 1 ) ( N - k + 1 ) = N k l = 1 k - 1 ( 1 + l α ) N k g k - 1 ( α ) ,             k 2 ,
h N = N a 2 + N ( N - 1 ) a 2 ϕ ( 1 ) 2 ,
h 2 N = N a 4 + N ( N - 1 ) a 4 A 2 + N ( N - 1 ) ( N - 2 ) a 4 B 2 + N ( N - 1 ) ( N - 2 ) ( N - 3 ) a 4 ϕ ( 1 ) 4 ,
h 3 N = N a 6 + N ( N - 1 ) a 6 A 3 + N ( N - 1 ) ( N - 2 ) a 6 B 3 + N ( N - 1 ) ( N - 2 ) ( N - 3 ) a 6 C 3 + N ( N - 1 ) ( N - 2 ) ( N - 3 ) ( N - 4 ) a 6 D 3 + N ( N - 1 ) ( N - 2 ) ( N - 3 ) ( N - 4 ) × ( N - 5 ) a 6 ϕ ( 1 ) 6 ,
A 2 2 + 4 ϕ ( 1 ) 2 + ϕ ( 2 ) 2 ,
B 2 4 ϕ ( 1 ) 2 + 2 Re ϕ 2 ( 1 ) ϕ * ( 2 ) ,
A 3 9 + 6 ϕ ( 2 ) 2 + 6 ϕ ( 1 ) 2 ,
B 3 6 + 9 ϕ ( 2 ) 2 + 45 ϕ ( 1 ) 2 + 12 ϕ ( 2 ) ϕ 2 ( - 1 ) + 12 ϕ ( - 2 ) ϕ 2 ( 1 ) + 3 ϕ ( - 3 ) ϕ ( 2 ) ϕ ( 1 ) + 3 ϕ ( 3 ) ϕ ( - 2 ) ϕ ( - 1 ) ,
C 3 9 ϕ ( 2 ) 2 ϕ ( 1 ) 2 + 18 ϕ ( 1 ) 4 + 18 ϕ ( 1 ) 2 + 9 ϕ ( 2 ) ϕ 2 ( - 1 ) + 9 ϕ ( - 2 ) ϕ 2 ( 1 ) + ϕ ( 3 ) ϕ 3 ( - 1 ) + ϕ ( - 3 ) ϕ 3 ( 1 ) ,
D 3 9 ϕ ( 1 ) 4 + 3 ϕ ( 2 ) ϕ ( 1 ) ϕ 3 ( - 1 ) + 3 ϕ ( - 2 ) ϕ ( - 1 ) ϕ 3 ( 1 ) .
ϕ ( n ) e i n θ = - π π e i n θ f ( θ ) d θ .
ϕ ( n ) = I n ( v ) / I 0 ( v ) ,
ϕ ( 0 ) = 1 ,             ϕ ( n ) = 0             for             n = ± 1 , ± 2 , .
h NB w = N a 2 + N 2 a 2 ϕ ( 1 ) 2 g 1 ( α ) ,
h 2 NB w = N a 4 + N 2 a 4 A 2 g 1 ( α ) + N 3 a 4 B 2 g 2 ( α ) + N 4 a 4 ϕ ( 1 ) 4 g 3 ( α ) ,
h 3 NB w = N a 6 + N 2 a 6 A 3 g 1 ( α ) + N 3 a 6 B 3 g 2 ( α ) + N 4 a 6 C 3 g 3 ( α ) + N 5 a 6 D 3 g 4 ( α ) + N 6 a 6 ϕ ( 1 ) 6 g 5 ( α ) ,
h NB s = N a 2 ,
h 2 NB s = N a 4 + 2 N 2 a 4 g 1 ( α ) ,
h 3 NB s = N a 6 + 9 N 2 a 4 g 1 ( α ) + 6 N 3 a 6 g 2 ( α ) .
var ( h NB ) w = N a 4 + N 2 a 4 ( A 2 g 1 - 1 ) + N 3 a 4 ( B 2 g 2 - 2 ϕ ( 1 ) 2 g 1 ) + N 4 a 4 ϕ ( 1 ) 4 ( g 3 - g 1 2 ) ,
var ( h NB ) s = N a 4 + N 2 a 4 ( 1 + 2 α ) .
var ( h P ) w = N a 4 + N 2 a 4 ( A 2 - 1 ) + N 3 a 4 ( B 2 - 2 ϕ ( 1 ) 2 ) ,
var ( h P ) s = N a 4 + N 2 a 4 .
v N 1 / 2 v , η N 1 / 2 a
ϕ ( n ) 1 n ! ( v 2 ) n ,
h NB w ~ [ 1 + 1 4 g 1 ( α ) v 2 ] η 2 ,
h 2 NB w ~ [ 2 g 1 ( α ) + g 2 ( α ) v 2 + 1 16 g 3 ( α ) v 4 ] η 4 ,
h 3 NB w ~ [ 6 g 2 ( α ) + 9 2 g 3 ( α ) v 2 + 9 16 g 4 ( α ) v 4 + 1 64 g 5 ( α ) v 6 ] η 6 .
h P w ~ [ 1 + 1 4 v 2 ] η 2 ,
h 2 P w ~ [ 2 + v 2 + 1 16 v 4 ] η 4 ,
h 3 P w ~ [ 6 + 9 2 v 2 + 9 16 v 4 + 1 64 v 6 ] η 6 .
h NB s ~ η 2 , h P s ~ η 2 , h 2 NB s ~ 2 g 1 ( α ) η 4 , h 2 P s ~ 2 η 4 , h 3 NB s ~ 6 g 2 ( α ) η 6 , h 3 P s ~ 6 η 6 .
var ( h NB ) w ~ [ ( 2 g 1 - 1 ) + ( g 2 - 1 2 g 1 ) v 2 + 1 16 ( g 3 - g 1 2 ) v 4 ] η 4 ,
var ( h NB ) s ~ ( 2 g 1 - 1 ) η 4 ,
var ( h P ) w ~ [ 1 + 1 2 v 2 ] η 4 ,
var ( h P ) w ~ η 4 ,
c var ( h ) h 2 ,
c ( NB ) s = 1 N + ( 1 + 2 α ) .
c ( NB ) w = N + N 2 ( A 2 g 1 - 1 ) + N 3 [ B 2 g 2 - 2 ϕ ( 1 ) 2 g 1 ] + N 4 [ ϕ ( 1 ) 4 ( g 3 - g 1 2 ) ] N 2 + N 3 [ 2 ϕ ( 1 ) 2 g 1 ] + N 4 ( ϕ ( 1 ) 4 g 1 2 )
c ( P ) w = 1 + N ( A 2 - 1 ) + N 2 [ B 2 - 2 ϕ ( 1 ) 2 ] N + N 2 [ 2 ϕ ( 1 ) 2 ] + N 3 [ ϕ ( 1 ) 4 ] ,
c ( NB ) w ~ ( 2 g 1 - 1 ) + ( g 2 - 1 2 g 1 ) v 2 + 1 16 ( g 3 - g 1 2 ) v 4 ( 1 + 1 4 g 1 v 2 ) 2 ,
c ( P ) w ~ 1 + 1 2 v 2 ( 1 + 1 4 v 2 ) 2 ,
c ( NB ) s ~ 2 g 1 - 1 ,
c ( P ) s ~ 1.
c ( NB ) w ~ g 3 - g 1 2 g 1 2 ,
c ( P ) w ~ 8 / v 2 .
h 3 NB w / h NB w 3
2 < I 2 < , 6 < I 3 < ,             0 < α <
2 < I 2 < 4 , 6 < I 3 < 36 ,             1 α <
1 < I 2 < 6 , 1 < I 3 < 90.
Φ ( ω ) [ ( 1 + v 2 4 α ) + η 2 4 α ω 2 ] - α
t η 2 α 1 / 2 ω ,             z ( 1 + η 2 4 α ) 1 / 2 ,
0 Φ ( ω ) J 0 ( h 1 / 2 ω ) ω d ω = ( 4 α η 2 ) 0 J 0 ( 2 α 1 / 2 h 1 / 2 η t ) t d t ( z 2 + t 2 ) α .
0 J 0 ( b t ) t d t ( z 2 + t 2 ) α = ( b 2 z ) α - 1 1 Γ ( α ) K 1 - α ( b z ) .
f w ( h ) = 2 α Γ ( α ) η α + 1 ( α 1 + v 2 4 α ) ( α - 1 ) / 2 h ( α - 1 ) / 2 × I 0 ( v η h 1 / 2 ) K α - 1 { 2 η [ ( 1 + v 2 4 α ) α h ] 1 / 2 } .
f s ( h ) = 2 Γ ( α ) η α + 1 α ( α + 1 ) / 2 K α - 1 ( 2 η α 1 / 2 h 1 / 2 ) .
f w ( h ) = 1 η 2 exp ( - v 2 / 4 ) exp ( - h / η 2 ) I 0 ( v h 1 / 2 η ) ,
h l w = 4 α ( α + 1 ) / 2 Γ ( α ) η α + 1 ( 1 + q ) - ( α - 1 ) / 2 × 0 x α + 2 l I 0 ( v x η ) K α - 1 ( 2 η δ 1 / 2 x ) d x ,
h l w = η 2 l Γ ( α + l ) Γ ( 1 + l ) Γ ( α ) α l × ( 1 + q ) - α - l F 2 1 ( α + l , 1 + l , 1 , q 1 + q ) ,
F 2 1 ( a , b , c , x ) 1 + a b c x + a ( a + 1 ) b ( b + 1 ) 2 ! c ( c + 1 ) x 2 + .
F 2 1 ( a , b , c , x ) = ( 1 - x ) c - a - b F 2 1 ( c - a , c - b , c , x ) ,
h l w = η 2 l Γ ( α + l ) Γ ( 1 + l ) Γ ( α ) α l ( 1 + v 2 4 α ) l × F 2 1 [ 1 - l - α , - l , 1 , ( v 2 / 4 α ) ( 1 + v 2 4 α ) - 1 ] .
f w ( I ) = 2 α Γ ( α ) ( α 1 + v 2 4 α ) ( α - 1 ) / 2 ( h w η 2 ) ( α + 1 ) / 2 I ( α - 1 ) / 2 × I 0 [ v ( h w η 2 ) 1 / 2 I 1 / 2 ] K α - 1 × { 2 [ ( 1 + v 2 4 α ) α ( h w η 2 ) I ] 1 / 2 } .
f w ( I ) = ( 1 + v 2 4 ) exp ( - v 2 / 4 ) e - I I 0 [ v ( 1 + v 2 4 ) 1 / 2 I 1 / 2 ]
f ( I ) = e - I
f w ( I ) = 2 ( 1 + v 2 2 ) I 0 [ v ( 1 + v 2 2 ) 1 / 2 I 1 / 2 ] × K 0 [ 2 ( 1 + v 2 2 ) 1 / 2 ( 1 + v 2 4 ) 1 / 2 I 1 / 2 ] ,
f ( I ) = 2 K 0 ( 2 I 1 / 2 ) .
f w ( I ) ~ [ ( 1 + v 2 2 ) v ( 1 + v 2 4 ) ] 1 / 4 I - 1 / 2 exp ( - Q ( v ) I 1 / 2 ) ,
Q ( v ) ( 1 + v 2 2 ) 1 / 2 [ 2 ( 1 + v 2 4 ) 1 / 2 - v ] ,
f ( I ) ~ ( π 2 ) 1 / 2 I - 1 / 4 exp ( - 2 I 1 / 2 ) .
I 3 absorbing aerosols > I 3 nonabsorbing aerosols

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