Abstract

We introduce a new, general formalism to model the turbulent wave-front phase by using fractional Brownian motion processes. Moreover, it extends results to non-Kolmogorov turbulence. In particular, generalized expressions for the Strehl ratio and the angle-of-arrival variance are obtained. These are dependent on the dynamic state of the turbulence.

© 2004 Optical Society of America

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  1. E. E. Silbaugh, B. M. Welsh, M. C. Roggemann, “Characterization of atmospheric turbulence phase statistics using wave-front slope measurements,” J. Opt. Soc. Am. A 13, 2453–2460 (1996).
    [CrossRef]
  2. D. S. Acton, R. J. Sharbaugh, J. R. Roehrig, D. Tiszauer, “Wave-front tilt power spectral density from the image motion of solar pores,” Appl. Opt. 31, 4280–4284 (1992).
    [CrossRef] [PubMed]
  3. D. L. Fried, “Differential angle of arrival: theory, evaluation, and measurement feasibility,” Radio Sci. 10, 71–76 (1975).
    [CrossRef]
  4. M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).
  5. D. Dayton, Bob Pierson, B. Spielbusch, J. Gonglewski, “Atmospheric structure function measurements with a Shack–Hartmann wave-front sensor,” Opt. Lett. 17, 1737–1739 (1992).
    [CrossRef] [PubMed]
  6. T. W. Nicholls, G. D. Boreman, J. C. Dainty, “Use of a Shack–Hartmann wave-front sensor to measure deviations from a Kolmogorov phase spectrum,” Opt. Lett. 20, 2460–2462 (1995).
    [CrossRef]
  7. The structure function is a basic characteristic of random fields with stationary increments. It is appropriate for analyzing the temporal and spatial structure of locally homogeneous and isotropic random fields; see V. I. Tatarskı̆, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), pp. 9–10, 19–20.
  8. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, Vol. XIX, E. Wolf, ed. (North-Holland, Amsterdam, 1981), pp. 281–376.
  9. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  10. R. G. Buser, “Interferometric determination of the distance dependence of the phase structure function for near-ground horizontal propagation at 6328 Å,” J. Opt. Soc. Am. 61, 488–491 (1971).
    [CrossRef]
  11. M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
    [CrossRef]
  12. B. E. Stribling, “Laser beam propagation in non-Kolmogorov atmospheric turbulence,” M.S. thesis (Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, 1994).
  13. C. Schwartz, G. Baum, E. N. Ribak, “Turbulence-degraded wave fronts as fractal surfaces,” J. Opt. Soc. Am. A 11, 444–451 (1994).
    [CrossRef]
  14. M. B. Jorgenson, G. J. M. Aitken, “Prediction of atmospherically induced wave-front degradations,” Opt. Lett. 17, 466–468 (1992).
    [CrossRef] [PubMed]
  15. M. Lloyd-Hart, P. C. McGuire, “Spatio-temporal prediction for adaptive optics wavefront reconstructors,” in Proceedings of the European Southern Observatory Conference on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1995), Vol. 54, pp. 95–101.
  16. D. R. McGaughey, G. J. M. Aitken, “Temporal analysis of stellar wave-front-tilt data,” J. Opt. Soc. Am. A 14, 1967–1974 (1997).
    [CrossRef]
  17. R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
    [CrossRef]
  18. L. Lakhal, A. Irbah, M. Bouzaria, J. Borgnino, F. Laclare, C. Delmas, “Error due to atmospheric turbulence effects on solar diameter measurements performed with an astrolabe,” Astron. Astrophys. Suppl. Ser. 138, 155–162 (1999).
    [CrossRef]
  19. This algorithm has a major drawback: It produces nonstationary increments, as was noted by McGaughey, Aitken, “Statistical analysis of successive random addition for generating fractional Brownian motion,” Physica A 277, 25–34 (2000).
    [CrossRef]
  20. H. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals: New Frontiers of Science (Springer-Verlag, New York, 1992).
  21. D. G. Pérez, L. Zunino, M. Garavaglia, “A fractional Brownian motion model for the turbulent refractive index in lightwave propagation,” revised for Opt. Commun., http://arxiv.org/pdf/physics/0307052 .
  22. Y. Chen, M. Ding, J. A. Scott Kelso, “Long memory processes (1/fα-type) in human coordination,” Phys. Rev. Lett. 79, 4501–4504 (1997).
    [CrossRef]
  23. G. Wornell, Signal Processing with Fractals: A Wavelet-Based Approach, Vol. 35 of Graduate Texts in Mathematics (Springer-Verlag, New York, 1984).
  24. E. E. Peters, Fractal Market Analysis: Applying Chaos Theory in Investment, Wiley Finance Editions (Wiley, New York, 1994).
  25. In this paper instead of the usual frequency f, ν=2πfis used for self-consistency.
  26. P. Flandrin, “Wavelet tools for scaling processes,” lecture, http://perso.ens-Lyon.fr/patrick.flandrin/Cargese02.pdf .
  27. R. J. Elliott, J. van der Hoek, “A general fractional white noise theory and applications to finance,” Math. Finance 13, 301–330 (2003).
    [CrossRef]
  28. D. G. Pérez, “Propagación de Luz en Medios Turbulentos,” Ph.D.thesis (Universidad Nacional de La Plata, Argentina, 2003), http://arxiv.org/pdf/physics/0307144 .
  29. G. Samorodnitsky, M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance (Chapman & Hall, London, 1994).
  30. B. B. Mandelbrot, J. W. Van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 4, 422–437 (1968).
    [CrossRef]
  31. K. J. Falconer, Fractal Geometry: Mathematical Theory and Applications (Wiley, New York, 1990).
  32. K. Itô, Kiyosi Itô: Selected Papers, S. Varadhan, D. W. Strook, eds. (Springer-Verlag, New York, 1986).
  33. B. Øksendal, Stochastic Differential Equations (Springer, New York, 1998).
  34. D. Nualart, The Malliavin Calculus and Related Topics (Springer-Verlag, New York, 1995).
  35. T. Hida, Selected Papers of Takeyuki Hida, L. Accardi, H. H. Kuo, N. Obata, K. Saitô, S. Si, L. Streit, eds. (World Scientific, River Edge, N.J., 2000).
  36. H. Holden, B. Øksendal, J. Ubøe, T. Zhang, Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach: Probability and Its Applications (Birkhäuser, Boston, Mass., 1996).
  37. M. Zähle, “Integration with respect to fractal functions andstochastic calculus I,” Probab. Theory Relat. Fields 97, 333–374 (1993).
  38. M. Zähle, “Integration with respect to fractal functions and stochastic calculus II,” Math. Nachr. 225, 145–183 (2001).
    [CrossRef]
  39. L. Decreusefond, A. S. Üstünel, “Fractional Brownian motion: theory and applications,” in ESAIM: Proceedings on Fractional Differential Systems: Models, Methods and Applications, Vol. 5, pp. 75–86, http://citeseer.ist.psu.edu/decreusefond98fractional.html .
  40. H. Föllmer, P. Protter, A. N. Shiryaev, “Quadratic covariation and an extension of Itô formula,” J. Bernoulli Soc. 1, 175–169 (1995).
  41. T. E. Duncan, Y. Hu, B. Pasik-Duncan, “Stochastic calculus for fractional Brownian motion: I. theory,” SIAM J. Control Optim. 38, 582–612 (2000).
    [CrossRef]
  42. If tilt aberration is present, the axis by this definition would be normal to the plane of that tilt.
  43. D. L. Fried, “Atmospheric turbulence optical effects: understanding the adaptive-optics implications,” in Proceedings of the NATO Advanced Study Institute on Adaptive Optics for Astronomy, D. M. Alloin, J. M. Mariotti, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1993), pp. 25–27.
  44. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975).
  45. V. I. Tatarskı̆, Wave Propagation in a Turbulent Atmosphere (Nauka, Moscow, 1967) (in Russian). English translation, Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Virginia, 1971).
  46. P. N. Brandt, H. A. Mauter, R. Smartt, “Day-time seeing statistics at Sacramento Peak Observatory,” Astron. Astrophys. 188, 163–168 (1987).
  47. D. S. Acton, “Simultaneous daytime measurements of the atmospheric coherence diameter r0 with three different methods,” Appl. Opt. 34, 4526–4529 (1995).
    [CrossRef] [PubMed]
  48. S. S. Olivier, C. E. Max, D. T. Gavel, J. M. Brase, “Tip-tilt compensation: resolution limits for ground-based telescopes using laser guide star adaptive optics,” Astrophys. J. 407, 428–439 (1993).
    [CrossRef]
  49. In this paper the Fourier transform is defined as ϕˆ(ν)=1/2π∫Rdsϕ(s)exp(-iνs).
  50. J.-M. Conan, G. Rousset, P.-Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
    [CrossRef]
  51. It is equal to 1 if tfalls in the interval (a, b),zero otherwise, and -1 when b<a.

2003 (1)

R. J. Elliott, J. van der Hoek, “A general fractional white noise theory and applications to finance,” Math. Finance 13, 301–330 (2003).
[CrossRef]

2001 (1)

M. Zähle, “Integration with respect to fractal functions and stochastic calculus II,” Math. Nachr. 225, 145–183 (2001).
[CrossRef]

2000 (2)

T. E. Duncan, Y. Hu, B. Pasik-Duncan, “Stochastic calculus for fractional Brownian motion: I. theory,” SIAM J. Control Optim. 38, 582–612 (2000).
[CrossRef]

This algorithm has a major drawback: It produces nonstationary increments, as was noted by McGaughey, Aitken, “Statistical analysis of successive random addition for generating fractional Brownian motion,” Physica A 277, 25–34 (2000).
[CrossRef]

1999 (1)

L. Lakhal, A. Irbah, M. Bouzaria, J. Borgnino, F. Laclare, C. Delmas, “Error due to atmospheric turbulence effects on solar diameter measurements performed with an astrolabe,” Astron. Astrophys. Suppl. Ser. 138, 155–162 (1999).
[CrossRef]

1997 (2)

Y. Chen, M. Ding, J. A. Scott Kelso, “Long memory processes (1/fα-type) in human coordination,” Phys. Rev. Lett. 79, 4501–4504 (1997).
[CrossRef]

D. R. McGaughey, G. J. M. Aitken, “Temporal analysis of stellar wave-front-tilt data,” J. Opt. Soc. Am. A 14, 1967–1974 (1997).
[CrossRef]

1996 (1)

1995 (4)

1994 (1)

1993 (2)

M. Zähle, “Integration with respect to fractal functions andstochastic calculus I,” Probab. Theory Relat. Fields 97, 333–374 (1993).

S. S. Olivier, C. E. Max, D. T. Gavel, J. M. Brase, “Tip-tilt compensation: resolution limits for ground-based telescopes using laser guide star adaptive optics,” Astrophys. J. 407, 428–439 (1993).
[CrossRef]

1992 (5)

1990 (1)

M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).

1987 (1)

P. N. Brandt, H. A. Mauter, R. Smartt, “Day-time seeing statistics at Sacramento Peak Observatory,” Astron. Astrophys. 188, 163–168 (1987).

1975 (1)

D. L. Fried, “Differential angle of arrival: theory, evaluation, and measurement feasibility,” Radio Sci. 10, 71–76 (1975).
[CrossRef]

1971 (1)

1968 (1)

B. B. Mandelbrot, J. W. Van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 4, 422–437 (1968).
[CrossRef]

1966 (1)

Acton, D. S.

Aitken,

This algorithm has a major drawback: It produces nonstationary increments, as was noted by McGaughey, Aitken, “Statistical analysis of successive random addition for generating fractional Brownian motion,” Physica A 277, 25–34 (2000).
[CrossRef]

Aitken, G. J. M.

Baum, G.

Bester, M.

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

Boreman, G. D.

Borgnino, J.

L. Lakhal, A. Irbah, M. Bouzaria, J. Borgnino, F. Laclare, C. Delmas, “Error due to atmospheric turbulence effects on solar diameter measurements performed with an astrolabe,” Astron. Astrophys. Suppl. Ser. 138, 155–162 (1999).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975).

Bouzaria, M.

L. Lakhal, A. Irbah, M. Bouzaria, J. Borgnino, F. Laclare, C. Delmas, “Error due to atmospheric turbulence effects on solar diameter measurements performed with an astrolabe,” Astron. Astrophys. Suppl. Ser. 138, 155–162 (1999).
[CrossRef]

Brandt, P. N.

P. N. Brandt, H. A. Mauter, R. Smartt, “Day-time seeing statistics at Sacramento Peak Observatory,” Astron. Astrophys. 188, 163–168 (1987).

Brase, J. M.

S. S. Olivier, C. E. Max, D. T. Gavel, J. M. Brase, “Tip-tilt compensation: resolution limits for ground-based telescopes using laser guide star adaptive optics,” Astrophys. J. 407, 428–439 (1993).
[CrossRef]

Buser, R. G.

Chen, Y.

Y. Chen, M. Ding, J. A. Scott Kelso, “Long memory processes (1/fα-type) in human coordination,” Phys. Rev. Lett. 79, 4501–4504 (1997).
[CrossRef]

Conan, J.-M.

Dainty, J. C.

Danchi, W. C.

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

Delmas, C.

L. Lakhal, A. Irbah, M. Bouzaria, J. Borgnino, F. Laclare, C. Delmas, “Error due to atmospheric turbulence effects on solar diameter measurements performed with an astrolabe,” Astron. Astrophys. Suppl. Ser. 138, 155–162 (1999).
[CrossRef]

Ding, M.

Y. Chen, M. Ding, J. A. Scott Kelso, “Long memory processes (1/fα-type) in human coordination,” Phys. Rev. Lett. 79, 4501–4504 (1997).
[CrossRef]

Duncan, T. E.

T. E. Duncan, Y. Hu, B. Pasik-Duncan, “Stochastic calculus for fractional Brownian motion: I. theory,” SIAM J. Control Optim. 38, 582–612 (2000).
[CrossRef]

Elliott, R. J.

R. J. Elliott, J. van der Hoek, “A general fractional white noise theory and applications to finance,” Math. Finance 13, 301–330 (2003).
[CrossRef]

Falconer, K. J.

K. J. Falconer, Fractal Geometry: Mathematical Theory and Applications (Wiley, New York, 1990).

Föllmer, H.

H. Föllmer, P. Protter, A. N. Shiryaev, “Quadratic covariation and an extension of Itô formula,” J. Bernoulli Soc. 1, 175–169 (1995).

Fried, D. L.

D. L. Fried, “Differential angle of arrival: theory, evaluation, and measurement feasibility,” Radio Sci. 10, 71–76 (1975).
[CrossRef]

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
[CrossRef]

D. L. Fried, “Atmospheric turbulence optical effects: understanding the adaptive-optics implications,” in Proceedings of the NATO Advanced Study Institute on Adaptive Optics for Astronomy, D. M. Alloin, J. M. Mariotti, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1993), pp. 25–27.

Gavel, D. T.

S. S. Olivier, C. E. Max, D. T. Gavel, J. M. Brase, “Tip-tilt compensation: resolution limits for ground-based telescopes using laser guide star adaptive optics,” Astrophys. J. 407, 428–439 (1993).
[CrossRef]

Glindemann, A.

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

Gonglewski, J.

Greenhill, L. J.

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

Hida, T.

T. Hida, Selected Papers of Takeyuki Hida, L. Accardi, H. H. Kuo, N. Obata, K. Saitô, S. Si, L. Streit, eds. (World Scientific, River Edge, N.J., 2000).

Holden, H.

H. Holden, B. Øksendal, J. Ubøe, T. Zhang, Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach: Probability and Its Applications (Birkhäuser, Boston, Mass., 1996).

Hu, Y.

T. E. Duncan, Y. Hu, B. Pasik-Duncan, “Stochastic calculus for fractional Brownian motion: I. theory,” SIAM J. Control Optim. 38, 582–612 (2000).
[CrossRef]

Irbah, A.

L. Lakhal, A. Irbah, M. Bouzaria, J. Borgnino, F. Laclare, C. Delmas, “Error due to atmospheric turbulence effects on solar diameter measurements performed with an astrolabe,” Astron. Astrophys. Suppl. Ser. 138, 155–162 (1999).
[CrossRef]

Itô, K.

K. Itô, Kiyosi Itô: Selected Papers, S. Varadhan, D. W. Strook, eds. (Springer-Verlag, New York, 1986).

Jorgenson, M. B.

Jürgens, H.

H. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals: New Frontiers of Science (Springer-Verlag, New York, 1992).

Laclare, F.

L. Lakhal, A. Irbah, M. Bouzaria, J. Borgnino, F. Laclare, C. Delmas, “Error due to atmospheric turbulence effects on solar diameter measurements performed with an astrolabe,” Astron. Astrophys. Suppl. Ser. 138, 155–162 (1999).
[CrossRef]

Lakhal, L.

L. Lakhal, A. Irbah, M. Bouzaria, J. Borgnino, F. Laclare, C. Delmas, “Error due to atmospheric turbulence effects on solar diameter measurements performed with an astrolabe,” Astron. Astrophys. Suppl. Ser. 138, 155–162 (1999).
[CrossRef]

Lane, R. G.

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

Lloyd-Hart, M.

M. Lloyd-Hart, P. C. McGuire, “Spatio-temporal prediction for adaptive optics wavefront reconstructors,” in Proceedings of the European Southern Observatory Conference on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1995), Vol. 54, pp. 95–101.

Madec, P.-Y.

Mandelbrot, B. B.

B. B. Mandelbrot, J. W. Van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 4, 422–437 (1968).
[CrossRef]

Mauter, H. A.

P. N. Brandt, H. A. Mauter, R. Smartt, “Day-time seeing statistics at Sacramento Peak Observatory,” Astron. Astrophys. 188, 163–168 (1987).

Max, C. E.

S. S. Olivier, C. E. Max, D. T. Gavel, J. M. Brase, “Tip-tilt compensation: resolution limits for ground-based telescopes using laser guide star adaptive optics,” Astrophys. J. 407, 428–439 (1993).
[CrossRef]

McGaughey,

This algorithm has a major drawback: It produces nonstationary increments, as was noted by McGaughey, Aitken, “Statistical analysis of successive random addition for generating fractional Brownian motion,” Physica A 277, 25–34 (2000).
[CrossRef]

McGaughey, D. R.

McGuire, P. C.

M. Lloyd-Hart, P. C. McGuire, “Spatio-temporal prediction for adaptive optics wavefront reconstructors,” in Proceedings of the European Southern Observatory Conference on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1995), Vol. 54, pp. 95–101.

Nicholls, T. W.

Nualart, D.

D. Nualart, The Malliavin Calculus and Related Topics (Springer-Verlag, New York, 1995).

Øksendal, B.

B. Øksendal, Stochastic Differential Equations (Springer, New York, 1998).

H. Holden, B. Øksendal, J. Ubøe, T. Zhang, Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach: Probability and Its Applications (Birkhäuser, Boston, Mass., 1996).

Olivier, S. S.

S. S. Olivier, C. E. Max, D. T. Gavel, J. M. Brase, “Tip-tilt compensation: resolution limits for ground-based telescopes using laser guide star adaptive optics,” Astrophys. J. 407, 428–439 (1993).
[CrossRef]

Pasik-Duncan, B.

T. E. Duncan, Y. Hu, B. Pasik-Duncan, “Stochastic calculus for fractional Brownian motion: I. theory,” SIAM J. Control Optim. 38, 582–612 (2000).
[CrossRef]

Peitgen, H.

H. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals: New Frontiers of Science (Springer-Verlag, New York, 1992).

Pérez, D. G.

D. G. Pérez, “Propagación de Luz en Medios Turbulentos,” Ph.D.thesis (Universidad Nacional de La Plata, Argentina, 2003), http://arxiv.org/pdf/physics/0307144 .

Peters, E. E.

E. E. Peters, Fractal Market Analysis: Applying Chaos Theory in Investment, Wiley Finance Editions (Wiley, New York, 1994).

Pierson, Bob

Protter, P.

H. Föllmer, P. Protter, A. N. Shiryaev, “Quadratic covariation and an extension of Itô formula,” J. Bernoulli Soc. 1, 175–169 (1995).

Ribak, E. N.

Roddier, F.

M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, Vol. XIX, E. Wolf, ed. (North-Holland, Amsterdam, 1981), pp. 281–376.

Roehrig, J. R.

Roggemann, M. C.

Rousset, G.

Samorodnitsky, G.

G. Samorodnitsky, M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance (Chapman & Hall, London, 1994).

Sarazin, M.

M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).

Saupe, D.

H. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals: New Frontiers of Science (Springer-Verlag, New York, 1992).

Schwartz, C.

Scott Kelso, J. A.

Y. Chen, M. Ding, J. A. Scott Kelso, “Long memory processes (1/fα-type) in human coordination,” Phys. Rev. Lett. 79, 4501–4504 (1997).
[CrossRef]

Sharbaugh, R. J.

Shiryaev, A. N.

H. Föllmer, P. Protter, A. N. Shiryaev, “Quadratic covariation and an extension of Itô formula,” J. Bernoulli Soc. 1, 175–169 (1995).

Silbaugh, E. E.

Smartt, R.

P. N. Brandt, H. A. Mauter, R. Smartt, “Day-time seeing statistics at Sacramento Peak Observatory,” Astron. Astrophys. 188, 163–168 (1987).

Spielbusch, B.

Stribling, B. E.

B. E. Stribling, “Laser beam propagation in non-Kolmogorov atmospheric turbulence,” M.S. thesis (Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, 1994).

Taqqu, M. S.

G. Samorodnitsky, M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance (Chapman & Hall, London, 1994).

Tatarski?, V. I.

The structure function is a basic characteristic of random fields with stationary increments. It is appropriate for analyzing the temporal and spatial structure of locally homogeneous and isotropic random fields; see V. I. Tatarskı̆, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), pp. 9–10, 19–20.

V. I. Tatarskı̆, Wave Propagation in a Turbulent Atmosphere (Nauka, Moscow, 1967) (in Russian). English translation, Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Virginia, 1971).

Tiszauer, D.

Townes, C. H.

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

Ubøe, J.

H. Holden, B. Øksendal, J. Ubøe, T. Zhang, Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach: Probability and Its Applications (Birkhäuser, Boston, Mass., 1996).

van der Hoek, J.

R. J. Elliott, J. van der Hoek, “A general fractional white noise theory and applications to finance,” Math. Finance 13, 301–330 (2003).
[CrossRef]

Van Ness, J. W.

B. B. Mandelbrot, J. W. Van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 4, 422–437 (1968).
[CrossRef]

Welsh, B. M.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975).

Wornell, G.

G. Wornell, Signal Processing with Fractals: A Wavelet-Based Approach, Vol. 35 of Graduate Texts in Mathematics (Springer-Verlag, New York, 1984).

Zähle, M.

M. Zähle, “Integration with respect to fractal functions and stochastic calculus II,” Math. Nachr. 225, 145–183 (2001).
[CrossRef]

M. Zähle, “Integration with respect to fractal functions andstochastic calculus I,” Probab. Theory Relat. Fields 97, 333–374 (1993).

Zhang, T.

H. Holden, B. Øksendal, J. Ubøe, T. Zhang, Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach: Probability and Its Applications (Birkhäuser, Boston, Mass., 1996).

Appl. Opt. (2)

Astron. Astrophys. (2)

P. N. Brandt, H. A. Mauter, R. Smartt, “Day-time seeing statistics at Sacramento Peak Observatory,” Astron. Astrophys. 188, 163–168 (1987).

M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).

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It is equal to 1 if tfalls in the interval (a, b),zero otherwise, and -1 when b<a.

In this paper the Fourier transform is defined as ϕˆ(ν)=1/2π∫Rdsϕ(s)exp(-iνs).

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Figures (1)

Fig. 1
Fig. 1

Decomposition of the vector ρ-ρ in terms of δθ and δr.

Equations (63)

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Dφ(ρ-ρ)=|φ(ρ)-φ(ρ)|2,
Dφ(ρ-ρ)=Cφ2ρ-ρr05/3,
Dφ(ρ-ρ)=Cφ,β2ρ-ρr0,ββ-2,
X(t)=dcHX(c-1t);
X(t)=X(0),
Cov[X(t+τ), X(t)]=Cov[X(τ), X(0)],
BH(0)=0almostsurely,E[BH(t)]=0,
E[BH(t)BH(s)]=12(|t|2H+|s|2H-|t-s|2H),
abX(t)BH(t)dt
abX(t) ddt BH(t)dt  abX(t)dBH(t)
[ϕ, W(ω)]=RϕdB=-Rdϕdt Bdt=(-dϕ/dt, B);
B˜H(ρ)  BH(ρ)=BH(ρ)
φ  CφB˜Hρr0,
E[|φ(ρ)-φ(ρ)|2]=Cφ2|ρ-ρ|r02HCφ2ρ-ρr02H,
S=16π01udu[(cos u)-1-u(1-u2)1/2]exp-12 Dw(Du),
S=16π01udu[(cos u)-1-u(1-u2)1/2]exp-Cφ22Dr02Hu2H.
S=16π01udu[(cos u)-1-u(1-u2)1/2]exp-3.44Dr05/3u5/3.
Sexp-Cφ2r02H ρ2H.
z(ρ)=λ2π φ(ρ),
θi=-λ2πφxi,i=1, 2.
σm2=θ12+θ22=Rd2ν[Wθ1(ν)+Wθ2(ν)]=λ2Rν2d2νWφ(ν),
σm2λ2r0-5/3L0-1D-1dνν-2/3.
σm2λ2D-1/3r0-5/3.
σm2(6.88/2π2)λ2D-1/3r0-5/3,
z(ρ)=CzB˜H(ρ/r0),
θiH(ρ)=-zxi=-Czr0-HWH(ρ)ρ xi,
σm,H2(ρ)=E[θ12+θ22](ρ)=Cz2r0-2HE[(WH(ρ))2].
WH(ρ)WH(ρ)=WH(ρ)WH(ρ)+n=1MHξn(ρ)MHξn(ρ).
E[WH(ρ)WH(ρ)]=n=1MHξn(ρ)MHξn(ρ).
σm,H2(ρ)=Cz2r0-2Hn=1MHξn(ρ)MHξn(ρ).
MHξn^(ν)=cH|ν|1/2-Hξ^n(ν)=-cH|ν|1/2-Hin-1ξn(ν),
n=1MHξn(ρ)MHξn(ρ)
=cH22πR2dνdν|ν|1/2-H|ν|1/2-H×n=1(-1)n-1ξn(ν)ξn(ν)exp i(νρ+νρ)=cH22πR2dνdν|ν|1/2-H|ν|1/2-Hδ(ν+ν)exp i(νρ+νρ)
=cH22πRdν|ν|1-2Hexp iν(ρ-ρ),
σm,H2=Cz2r0-2HcH22πRdν|ν|1-2H
σm,H2=λ2(2π)2 Cφ2r0-2HcH2π02D-1dνν1-2H=Γ(2H+1)sin πH22Hπ(1-H)Cφ22π2 λ2r0-2HD2H-2.
WϕH(ρ)=(ϕρ, WH)=n=1(ϕρ, MHξn)Hn(ω)
E[(WϕH(ρ))2]=n=1(ϕρ, MHξn)2.
n=1(ϕρ, MHξn)(ϕρ, MHξn)
=cH2Rdν|ϕˆ(ν)|2|ν|1-2Hexp[-iν(ρ-ρ)]=2πcH2F-1[|ϕˆ|2||1-2H](ρ-ρ),
σm,H2=Cz22πcH2r0-2HF[|ϕˆ|2||1-2H](0).
Rϕ(s)=2πϕˆ(0)1.
ϕˆ(ν)=2π1/2J1(νD/2)(νD/2).
σm,H2=Cz2cH2r0-2HRdν|ν|1-2H2πJ12(νD/2)(νD/2)2=Γ(2H+1)Γ(H+1/2)Γ(1-H)sin πHπ3/222H-3Γ(H+1)Γ(H+2)×Cφ22π2 λ2r0-2HD2H-2.
MHϕ^(ν)=cH|ν|1/2-Hϕˆ(ν),
WH(ϕ)=MHϕ, ω=R(MHϕ)dB=RϕdBH;
E[MHϕ, ωMHϕ, ω]=(MHϕ, MHϕ).
X(ω)=αcαHα(ω),
ξn(x)=exp(-x2/2)Hn-1(x)[2n-1(n-1)!π1/2]1/2,
n=1ξn(x)ξn(y)=δ(x-y),
Rdxξn(x)ξm(x)=δn,m.
BH(t)(ω)=MH1(a,b), ω=n=1(MH1(a,b), ξn)ξn, ω.
(MH1(a,b), ξn)=(1(a,b), MHξn)=0tdsMHξn(s);
ddt BH(t)=n=1MHξn(t)ξn, ω=WH(t).
WH(ϕ)=n=1(ϕ, MHξn)ξn, ω=Rdsϕ(s)n=1MHξn(s)ξn, ω=Rϕ(s)WH(s)ds.
(XY)(ω)=α,βaαbβHα+β(ω).
(XY)(ω)=X(ω)Y(ω)-n=1anbn.
E{[B˜H(ρ)-B˜H(ρ)]2}=|ρ-ρ|2H.
E{[B˜H(ρ)-B˜H(ρ)]2}ρ-ρ2H.
ρ-ρ2H-δr2Hδ2H=δθ+δr2H-δ2Hδ2H=1+δθ2+2δθδrδ2H-1=Hδθ2+2δθδrδ2+Oδθ2+2δθδrδ22.
δθ2+2δθδrδ21
δθ2+2δθδrδ2θ2ρ2δ2+ρδθ2ρ2δ-2.
δ<δδ1/2δθ1

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