Abstract

Methods for simulation of two-dimensional signals with arbitrary power spectral densities and signal amplitude probability density functions are disclosed. The method relies on initially transforming a white noise sample set of random Gaussian distributed numbers into a corresponding set with the desired spectral distribution, after which this colored Gaussian probability distribution is transformed via an inverse transform into the desired probability distribution. In most cases the method provides satisfactory results and can thus be considered an engineering approach. Several illustrative examples with relevance for optics are given.

© 2012 Optical Society of America

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References

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  1. M. Borengasser, W. S. Hungate, S. William, and R. Watkins, Hyperspectral Remote Sensing: Principles and Applications (CRC Press, 2008).
  2. T. Vo-Dinh, Advanced Biomedical and Clinical Diagnostic Systems III (SPIE, 2005).
  3. R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the diffusing-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
    [CrossRef]
  4. S. Rehn, A. Planat-Chretien, M. Berger, J.-M. Dinten, C. Deumie, and A. da Silva, “Comparison of polarized light penetration depth in scattering media,” Proc. SPIE  8088, 80881I (2011).
  5. U. Schnell, J. Piot, and R. Dändliker, “Detection of movement with laser speckle patterns: statistical properties,” J. Opt. Soc. Am. A 15, 207–216 (1998).
    [CrossRef]
  6. Y. Aizu and T. Asakura, Spatial Filtering Velocimetry, Fundamentals and Applications, Vol. 116 of Springer Series in Optical Science (Springer-Verlag, 2006).
  7. V. N. Nosov and S. Yu. Pashin, “Influence of large-scale waves on the accuracy of sea surface parameters measurement by optical methods,” Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 24, 53–58 (1988).
  8. W. Wang, N. Ishii, S. G. Hanson, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
    [CrossRef]
  9. O. V. Angelsky, D. N. Burkovets, P. P. Maksimyak, and S. G. Hanson, “Applicability of the singular-optics concept for diagnostics of random and fractal rough surfaces,” Appl. Opt. 42, 4529–4540 (2003).
    [CrossRef]
  10. R. Popescu, G. Deodatis, and J. H. Prevost, “Simulation of homogeneous non-Gaussian stochastic vector fields,” Prob. Eng. Mech. 13, 1–13 (1988).
  11. F. Yamazaki and M. Shinozuka, “Digital generation of non-Gaussian stochastic fields,” J. Eng. Mech. 114, 1183–1197 (1988).
    [CrossRef]
  12. S. S. Filho, J. C. Yacoub, and M. Candido Daoud, “Coloring non-Gaussian sequences,” IEEE Trans. Signal Process. 56, 5817–5822 (2008).
    [CrossRef]
  13. G. Q. Cai and Y. K. Lin, “Generation of non-Gaussian stationary stochastic processes,” Phys. Rev. E 54, 299–303 (1996).
    [CrossRef]
  14. H. T. Yura and S. G. Hanson, “Digital simulation of an arbitrary stochastic process by spectral representation,” J. Opt. Soc. Am. A 28, 675–685 (2011).
    [CrossRef]
  15. J. M. Nichols, C. C. Olson, J. V. Michalowitz, and F. Bucholtz, “A simple algorithm for generating spectrally colored non-Gaussian signals,” Prob. Eng. Mech. 25, 315–322(2010).
    [CrossRef]
  16. P. Beckmann, Probability in Communication Engineering, (Harcourt, Brace & World, 1967), Section 6.6.
  17. F. Yamazaki and M. Shinozuka, “Digital generation of non-Gaussian stochastic fields,” J. Eng. Mech. Div., Am. Soc. Civ. Eng. 114, 1183–1197 (1988).
    [CrossRef]
  18. M. Shinozuka and G. Deodatis, “Simulation of stochastic processes by spectral representation,” Appl. Mech. Rev. 44, 191–204 (1991).
    [CrossRef]
  19. We note under certain non-isotropic conditions [e.g., stratospheric turbulence; see, for example, C. Robert, J. M. Conan, V. Michau, J. B. Renard, C. Robert, and F. Dalaudier, “Retrieving parameters of the anisoptropic index fluctuations spectrum in the stratosphere from balloon-borne observations of scintillation,” J. Opt. Soc. Am. A25, 379–393 (2008)] that the power spectra is a function of Kx2+η2Ky2, where the anisotropic parameter η≥0.
  20. Wolfram, Mathematica, Version 7 (Cambridge University, 2008).
  21. J. Mendez and S. Castanedo, “A probability distribution for depth-limited extreme wave heights in a sea state,” Coastal Eng. 54, 878–882 (2007).
    [CrossRef]
  22. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).
  23. R. Barakat, “Level-crossing statistics of aperture averaged-integrated isotropic speckle,” J. Opt. Soc. Am. A 5, 1244–1247 (1988).
    [CrossRef]
  24. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001), Chap. 2.
  25. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, 1971).
  26. The CDF of the Rice–Nakagami distribution can be expressed in terms of a Marcum Q function, which are tabulated but not supported, to the best of our knowledge, by any commercial commuter programs such as Mathematica and Matlab.
  27. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006), Sec. 3.2.2.
  28. For this case we obtain that the non linear least squares fit parameters are given by n=−0.441, a=1.24, a1=0.460, a3=−0.516, a5=0.634, and b=−0.216.

2011 (2)

S. Rehn, A. Planat-Chretien, M. Berger, J.-M. Dinten, C. Deumie, and A. da Silva, “Comparison of polarized light penetration depth in scattering media,” Proc. SPIE  8088, 80881I (2011).

H. T. Yura and S. G. Hanson, “Digital simulation of an arbitrary stochastic process by spectral representation,” J. Opt. Soc. Am. A 28, 675–685 (2011).
[CrossRef]

2010 (1)

J. M. Nichols, C. C. Olson, J. V. Michalowitz, and F. Bucholtz, “A simple algorithm for generating spectrally colored non-Gaussian signals,” Prob. Eng. Mech. 25, 315–322(2010).
[CrossRef]

2008 (1)

S. S. Filho, J. C. Yacoub, and M. Candido Daoud, “Coloring non-Gaussian sequences,” IEEE Trans. Signal Process. 56, 5817–5822 (2008).
[CrossRef]

2007 (1)

J. Mendez and S. Castanedo, “A probability distribution for depth-limited extreme wave heights in a sea state,” Coastal Eng. 54, 878–882 (2007).
[CrossRef]

2005 (1)

W. Wang, N. Ishii, S. G. Hanson, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[CrossRef]

2004 (1)

R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the diffusing-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
[CrossRef]

2003 (1)

1998 (1)

1996 (1)

G. Q. Cai and Y. K. Lin, “Generation of non-Gaussian stationary stochastic processes,” Phys. Rev. E 54, 299–303 (1996).
[CrossRef]

1991 (1)

M. Shinozuka and G. Deodatis, “Simulation of stochastic processes by spectral representation,” Appl. Mech. Rev. 44, 191–204 (1991).
[CrossRef]

1988 (5)

F. Yamazaki and M. Shinozuka, “Digital generation of non-Gaussian stochastic fields,” J. Eng. Mech. Div., Am. Soc. Civ. Eng. 114, 1183–1197 (1988).
[CrossRef]

R. Barakat, “Level-crossing statistics of aperture averaged-integrated isotropic speckle,” J. Opt. Soc. Am. A 5, 1244–1247 (1988).
[CrossRef]

V. N. Nosov and S. Yu. Pashin, “Influence of large-scale waves on the accuracy of sea surface parameters measurement by optical methods,” Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 24, 53–58 (1988).

R. Popescu, G. Deodatis, and J. H. Prevost, “Simulation of homogeneous non-Gaussian stochastic vector fields,” Prob. Eng. Mech. 13, 1–13 (1988).

F. Yamazaki and M. Shinozuka, “Digital generation of non-Gaussian stochastic fields,” J. Eng. Mech. 114, 1183–1197 (1988).
[CrossRef]

Aizu, Y.

Y. Aizu and T. Asakura, Spatial Filtering Velocimetry, Fundamentals and Applications, Vol. 116 of Springer Series in Optical Science (Springer-Verlag, 2006).

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001), Chap. 2.

Angelsky, O. V.

Asakura, T.

Y. Aizu and T. Asakura, Spatial Filtering Velocimetry, Fundamentals and Applications, Vol. 116 of Springer Series in Optical Science (Springer-Verlag, 2006).

Barakat, R.

Beckmann, P.

P. Beckmann, Probability in Communication Engineering, (Harcourt, Brace & World, 1967), Section 6.6.

Berger, M.

S. Rehn, A. Planat-Chretien, M. Berger, J.-M. Dinten, C. Deumie, and A. da Silva, “Comparison of polarized light penetration depth in scattering media,” Proc. SPIE  8088, 80881I (2011).

Borengasser, M.

M. Borengasser, W. S. Hungate, S. William, and R. Watkins, Hyperspectral Remote Sensing: Principles and Applications (CRC Press, 2008).

Bucholtz, F.

J. M. Nichols, C. C. Olson, J. V. Michalowitz, and F. Bucholtz, “A simple algorithm for generating spectrally colored non-Gaussian signals,” Prob. Eng. Mech. 25, 315–322(2010).
[CrossRef]

Burkovets, D. N.

Cai, G. Q.

G. Q. Cai and Y. K. Lin, “Generation of non-Gaussian stationary stochastic processes,” Phys. Rev. E 54, 299–303 (1996).
[CrossRef]

Carminati, R.

R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the diffusing-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
[CrossRef]

Castanedo, S.

J. Mendez and S. Castanedo, “A probability distribution for depth-limited extreme wave heights in a sea state,” Coastal Eng. 54, 878–882 (2007).
[CrossRef]

Conan, J. M.

We note under certain non-isotropic conditions [e.g., stratospheric turbulence; see, for example, C. Robert, J. M. Conan, V. Michau, J. B. Renard, C. Robert, and F. Dalaudier, “Retrieving parameters of the anisoptropic index fluctuations spectrum in the stratosphere from balloon-borne observations of scintillation,” J. Opt. Soc. Am. A25, 379–393 (2008)] that the power spectra is a function of Kx2+η2Ky2, where the anisotropic parameter η≥0.

da Silva, A.

S. Rehn, A. Planat-Chretien, M. Berger, J.-M. Dinten, C. Deumie, and A. da Silva, “Comparison of polarized light penetration depth in scattering media,” Proc. SPIE  8088, 80881I (2011).

Dalaudier, F.

We note under certain non-isotropic conditions [e.g., stratospheric turbulence; see, for example, C. Robert, J. M. Conan, V. Michau, J. B. Renard, C. Robert, and F. Dalaudier, “Retrieving parameters of the anisoptropic index fluctuations spectrum in the stratosphere from balloon-borne observations of scintillation,” J. Opt. Soc. Am. A25, 379–393 (2008)] that the power spectra is a function of Kx2+η2Ky2, where the anisotropic parameter η≥0.

Dändliker, R.

Daoud, M. Candido

S. S. Filho, J. C. Yacoub, and M. Candido Daoud, “Coloring non-Gaussian sequences,” IEEE Trans. Signal Process. 56, 5817–5822 (2008).
[CrossRef]

Deodatis, G.

M. Shinozuka and G. Deodatis, “Simulation of stochastic processes by spectral representation,” Appl. Mech. Rev. 44, 191–204 (1991).
[CrossRef]

R. Popescu, G. Deodatis, and J. H. Prevost, “Simulation of homogeneous non-Gaussian stochastic vector fields,” Prob. Eng. Mech. 13, 1–13 (1988).

Deumie, C.

S. Rehn, A. Planat-Chretien, M. Berger, J.-M. Dinten, C. Deumie, and A. da Silva, “Comparison of polarized light penetration depth in scattering media,” Proc. SPIE  8088, 80881I (2011).

Dinten, J.-M.

S. Rehn, A. Planat-Chretien, M. Berger, J.-M. Dinten, C. Deumie, and A. da Silva, “Comparison of polarized light penetration depth in scattering media,” Proc. SPIE  8088, 80881I (2011).

Elaloufi, R.

R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the diffusing-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
[CrossRef]

Filho, S. S.

S. S. Filho, J. C. Yacoub, and M. Candido Daoud, “Coloring non-Gaussian sequences,” IEEE Trans. Signal Process. 56, 5817–5822 (2008).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006), Sec. 3.2.2.

Greffet, J.-J.

R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the diffusing-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
[CrossRef]

Hanson, S. G.

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001), Chap. 2.

Hungate, W. S.

M. Borengasser, W. S. Hungate, S. William, and R. Watkins, Hyperspectral Remote Sensing: Principles and Applications (CRC Press, 2008).

Ishii, N.

W. Wang, N. Ishii, S. G. Hanson, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[CrossRef]

Lin, Y. K.

G. Q. Cai and Y. K. Lin, “Generation of non-Gaussian stationary stochastic processes,” Phys. Rev. E 54, 299–303 (1996).
[CrossRef]

Maksimyak, P. P.

Mendez, J.

J. Mendez and S. Castanedo, “A probability distribution for depth-limited extreme wave heights in a sea state,” Coastal Eng. 54, 878–882 (2007).
[CrossRef]

Michalowitz, J. V.

J. M. Nichols, C. C. Olson, J. V. Michalowitz, and F. Bucholtz, “A simple algorithm for generating spectrally colored non-Gaussian signals,” Prob. Eng. Mech. 25, 315–322(2010).
[CrossRef]

Michau, V.

We note under certain non-isotropic conditions [e.g., stratospheric turbulence; see, for example, C. Robert, J. M. Conan, V. Michau, J. B. Renard, C. Robert, and F. Dalaudier, “Retrieving parameters of the anisoptropic index fluctuations spectrum in the stratosphere from balloon-borne observations of scintillation,” J. Opt. Soc. Am. A25, 379–393 (2008)] that the power spectra is a function of Kx2+η2Ky2, where the anisotropic parameter η≥0.

Nichols, J. M.

J. M. Nichols, C. C. Olson, J. V. Michalowitz, and F. Bucholtz, “A simple algorithm for generating spectrally colored non-Gaussian signals,” Prob. Eng. Mech. 25, 315–322(2010).
[CrossRef]

Nosov, V. N.

V. N. Nosov and S. Yu. Pashin, “Influence of large-scale waves on the accuracy of sea surface parameters measurement by optical methods,” Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 24, 53–58 (1988).

Olson, C. C.

J. M. Nichols, C. C. Olson, J. V. Michalowitz, and F. Bucholtz, “A simple algorithm for generating spectrally colored non-Gaussian signals,” Prob. Eng. Mech. 25, 315–322(2010).
[CrossRef]

Pashin, S. Yu.

V. N. Nosov and S. Yu. Pashin, “Influence of large-scale waves on the accuracy of sea surface parameters measurement by optical methods,” Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 24, 53–58 (1988).

Phillips, R. L.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001), Chap. 2.

Piot, J.

Planat-Chretien, A.

S. Rehn, A. Planat-Chretien, M. Berger, J.-M. Dinten, C. Deumie, and A. da Silva, “Comparison of polarized light penetration depth in scattering media,” Proc. SPIE  8088, 80881I (2011).

Popescu, R.

R. Popescu, G. Deodatis, and J. H. Prevost, “Simulation of homogeneous non-Gaussian stochastic vector fields,” Prob. Eng. Mech. 13, 1–13 (1988).

Prevost, J. H.

R. Popescu, G. Deodatis, and J. H. Prevost, “Simulation of homogeneous non-Gaussian stochastic vector fields,” Prob. Eng. Mech. 13, 1–13 (1988).

Rehn, S.

S. Rehn, A. Planat-Chretien, M. Berger, J.-M. Dinten, C. Deumie, and A. da Silva, “Comparison of polarized light penetration depth in scattering media,” Proc. SPIE  8088, 80881I (2011).

Renard, J. B.

We note under certain non-isotropic conditions [e.g., stratospheric turbulence; see, for example, C. Robert, J. M. Conan, V. Michau, J. B. Renard, C. Robert, and F. Dalaudier, “Retrieving parameters of the anisoptropic index fluctuations spectrum in the stratosphere from balloon-borne observations of scintillation,” J. Opt. Soc. Am. A25, 379–393 (2008)] that the power spectra is a function of Kx2+η2Ky2, where the anisotropic parameter η≥0.

Robert, C.

We note under certain non-isotropic conditions [e.g., stratospheric turbulence; see, for example, C. Robert, J. M. Conan, V. Michau, J. B. Renard, C. Robert, and F. Dalaudier, “Retrieving parameters of the anisoptropic index fluctuations spectrum in the stratosphere from balloon-borne observations of scintillation,” J. Opt. Soc. Am. A25, 379–393 (2008)] that the power spectra is a function of Kx2+η2Ky2, where the anisotropic parameter η≥0.

We note under certain non-isotropic conditions [e.g., stratospheric turbulence; see, for example, C. Robert, J. M. Conan, V. Michau, J. B. Renard, C. Robert, and F. Dalaudier, “Retrieving parameters of the anisoptropic index fluctuations spectrum in the stratosphere from balloon-borne observations of scintillation,” J. Opt. Soc. Am. A25, 379–393 (2008)] that the power spectra is a function of Kx2+η2Ky2, where the anisotropic parameter η≥0.

Schnell, U.

Shinozuka, M.

M. Shinozuka and G. Deodatis, “Simulation of stochastic processes by spectral representation,” Appl. Mech. Rev. 44, 191–204 (1991).
[CrossRef]

F. Yamazaki and M. Shinozuka, “Digital generation of non-Gaussian stochastic fields,” J. Eng. Mech. Div., Am. Soc. Civ. Eng. 114, 1183–1197 (1988).
[CrossRef]

F. Yamazaki and M. Shinozuka, “Digital generation of non-Gaussian stochastic fields,” J. Eng. Mech. 114, 1183–1197 (1988).
[CrossRef]

Takeda, M.

W. Wang, N. Ishii, S. G. Hanson, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[CrossRef]

Tatarskii, V. I.

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

Vo-Dinh, T.

T. Vo-Dinh, Advanced Biomedical and Clinical Diagnostic Systems III (SPIE, 2005).

Wang, W.

W. Wang, N. Ishii, S. G. Hanson, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[CrossRef]

Watkins, R.

M. Borengasser, W. S. Hungate, S. William, and R. Watkins, Hyperspectral Remote Sensing: Principles and Applications (CRC Press, 2008).

William, S.

M. Borengasser, W. S. Hungate, S. William, and R. Watkins, Hyperspectral Remote Sensing: Principles and Applications (CRC Press, 2008).

Yacoub, J. C.

S. S. Filho, J. C. Yacoub, and M. Candido Daoud, “Coloring non-Gaussian sequences,” IEEE Trans. Signal Process. 56, 5817–5822 (2008).
[CrossRef]

Yamazaki, F.

F. Yamazaki and M. Shinozuka, “Digital generation of non-Gaussian stochastic fields,” J. Eng. Mech. 114, 1183–1197 (1988).
[CrossRef]

F. Yamazaki and M. Shinozuka, “Digital generation of non-Gaussian stochastic fields,” J. Eng. Mech. Div., Am. Soc. Civ. Eng. 114, 1183–1197 (1988).
[CrossRef]

Yura, H. T.

Appl. Mech. Rev. (1)

M. Shinozuka and G. Deodatis, “Simulation of stochastic processes by spectral representation,” Appl. Mech. Rev. 44, 191–204 (1991).
[CrossRef]

Appl. Opt. (1)

Coastal Eng. (1)

J. Mendez and S. Castanedo, “A probability distribution for depth-limited extreme wave heights in a sea state,” Coastal Eng. 54, 878–882 (2007).
[CrossRef]

IEEE Trans. Signal Process. (1)

S. S. Filho, J. C. Yacoub, and M. Candido Daoud, “Coloring non-Gaussian sequences,” IEEE Trans. Signal Process. 56, 5817–5822 (2008).
[CrossRef]

Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana (1)

V. N. Nosov and S. Yu. Pashin, “Influence of large-scale waves on the accuracy of sea surface parameters measurement by optical methods,” Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 24, 53–58 (1988).

J. Eng. Mech. (1)

F. Yamazaki and M. Shinozuka, “Digital generation of non-Gaussian stochastic fields,” J. Eng. Mech. 114, 1183–1197 (1988).
[CrossRef]

J. Eng. Mech. Div., Am. Soc. Civ. Eng. (1)

F. Yamazaki and M. Shinozuka, “Digital generation of non-Gaussian stochastic fields,” J. Eng. Mech. Div., Am. Soc. Civ. Eng. 114, 1183–1197 (1988).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

W. Wang, N. Ishii, S. G. Hanson, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[CrossRef]

Phys. Rev. E (1)

G. Q. Cai and Y. K. Lin, “Generation of non-Gaussian stationary stochastic processes,” Phys. Rev. E 54, 299–303 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the diffusing-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
[CrossRef]

Prob. Eng. Mech. (2)

R. Popescu, G. Deodatis, and J. H. Prevost, “Simulation of homogeneous non-Gaussian stochastic vector fields,” Prob. Eng. Mech. 13, 1–13 (1988).

J. M. Nichols, C. C. Olson, J. V. Michalowitz, and F. Bucholtz, “A simple algorithm for generating spectrally colored non-Gaussian signals,” Prob. Eng. Mech. 25, 315–322(2010).
[CrossRef]

Proc. SPIE (1)

S. Rehn, A. Planat-Chretien, M. Berger, J.-M. Dinten, C. Deumie, and A. da Silva, “Comparison of polarized light penetration depth in scattering media,” Proc. SPIE  8088, 80881I (2011).

Other (12)

M. Borengasser, W. S. Hungate, S. William, and R. Watkins, Hyperspectral Remote Sensing: Principles and Applications (CRC Press, 2008).

T. Vo-Dinh, Advanced Biomedical and Clinical Diagnostic Systems III (SPIE, 2005).

Y. Aizu and T. Asakura, Spatial Filtering Velocimetry, Fundamentals and Applications, Vol. 116 of Springer Series in Optical Science (Springer-Verlag, 2006).

P. Beckmann, Probability in Communication Engineering, (Harcourt, Brace & World, 1967), Section 6.6.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001), Chap. 2.

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

The CDF of the Rice–Nakagami distribution can be expressed in terms of a Marcum Q function, which are tabulated but not supported, to the best of our knowledge, by any commercial commuter programs such as Mathematica and Matlab.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006), Sec. 3.2.2.

For this case we obtain that the non linear least squares fit parameters are given by n=−0.441, a=1.24, a1=0.460, a3=−0.516, a5=0.634, and b=−0.216.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).

We note under certain non-isotropic conditions [e.g., stratospheric turbulence; see, for example, C. Robert, J. M. Conan, V. Michau, J. B. Renard, C. Robert, and F. Dalaudier, “Retrieving parameters of the anisoptropic index fluctuations spectrum in the stratosphere from balloon-borne observations of scintillation,” J. Opt. Soc. Am. A25, 379–393 (2008)] that the power spectra is a function of Kx2+η2Ky2, where the anisotropic parameter η≥0.

Wolfram, Mathematica, Version 7 (Cambridge University, 2008).

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Figures (7)

Fig. 1.
Fig. 1.

The beta stochastic field generated for the Gaussian shaped PSD with the number of data points equal to 104 (N=100), and lC=10lS.

Fig. 2.
Fig. 2.

Ocean height field for a Pierson–Moskowitz PSD with an empirical model, Eq. (5), for the PDF, here for a resulting spatial field distribution section consisting of 104 sample points, and lC=10lS.

Fig. 3.
Fig. 3.

Example of a resulting fully developed speckle (m=1) spatial intensity field distribution section consisting of a section 200×200 sample points, and lC=10lS for the Gaussian (a) and circular aperture (b) PSD, respectively.

Fig. 4.
Fig. 4.

Integrated speckle field for the PSD resulting from a Gaussian aperture and m=7.5.

Fig. 5.
Fig. 5.

The log-normal normalized intensity field distribution for σlnI2=0.3 for lC=5lS.

Fig. 6.
Fig. 6.

Comparison between the inverse CDF (dotted points) obtained numerically and the analytic model (solid curve) obtained via the least squares fit for the Rice–Nakagami distribution.

Fig. 7.
Fig. 7.

The intensity field distribution for the PSD resulting from a circular aperture, 104 sample points, and lC=10lS for the Rice–Nakagami distribution.

Equations (17)

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

Z0=FZ1(FZ1[z1])
F(z)=BetaRegularized[z,α,β],for0z1,
F1(z)=InverseBetaRegularized[z,α,β],for0z1.
S(K)=σ2lC24πexp[(Kx2+Ky2)lC2/4],
p(z)=2ϕ2(κ)z(1κz)3exp[ϕ2(κ)(z1κz)2],for0z<1/κ,
F(z)=1exp[ϕ2(κ)(z1κz)2],for0z<1/κ,
F1(z)=ϕ(κ)log(1z)+κlog(1z)ϕ2(κ)+κ2log(1z),for0z1.
SPM(K)=4σ2KN5exp(KN4),
F(z)=1Γ(m,mz)Γ(m),forz0,
F1(z)=Q1(m,0,z),for0z1,
S(υ)={(22λRπD)2(cos1(υlC)υlC1(υlC)2),for0υlC10otherwise,
F(z)=12(1+erf[logz+σlnI2/22σlnI]),forz0,
F1(z)=exp[zσlnIσlnI2/2],for0z1,
S(K)=2.577ω8/3(18Γ(17/6)11Γ(7/3)K2Im[eiω2U(K)]),
U(K)=πsin(πb)(F11(a;b;iK2)Γ(b)Γ(ab+1)(iK2)1bF11(ab+1;2b;iK2)Γ(a)Γ(2b)).
p(z)=2zexp[(z2+C2)]I0(2zC),forz0,
F(z)=0z2μexp[(μ2+C2)]I0(2μC)dμ,for0z1,

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