Abstract

Motivated by the increasing importance of adaptive optics (AO) systems for improving the real resolution of large ground telescopes, and by the need of testing the AO system performance in realistic working conditions, in this paper we address the problem of simulating the turbulence effect on ground telescope observations at high resolution. The procedure presented here generalizes the multiscale stochastic approach introduced in our earlier paper [Appl. Opt. 50, 4124 (2011)], with respect to the previous solution, a relevant computational time reduction is obtained by exploiting a local spatial principal component analysis (PCA) representation of the turbulence. Furthermore, the turbulence at low resolution is modeled as a moving average (MA) process, while previously [Appl. Opt. 50, 4124 (2011)] the wind velocity was restricted to be directed along one of the two spatial axes, the use of such MA model allows the turbulence to evolve indifferently in all the directions. In our simulations, the proposed procedure reproduces the theoretical statistical characteristics of the turbulent phase with good accuracy.

© 2013 Optical Society of America

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  1. C. Correia, H.-F. Raynaud, C. Kulcsár, and J.-M. Conan, “On the optimal reconstruction and control of adaptive optical systems with mirror dynamics,” J. Opt. Soc. Am. A 27, 333–349 (2010).
    [CrossRef]
  2. C. Dessenne, P.-Y. Madec, and G. Rousset, “Sky implementation of modal predictive control in adaptive optics,” Opt. Lett. 24, 339–341 (1999).
    [CrossRef]
  3. E. Fedrigo, R. Muradore, and D. Zilio, “High performance adaptive optics system with fine tip/tilt control,” Control Eng. Pract. 17, 122–135 (2009).
    [CrossRef]
  4. E. Gendron and P. Léna, “Astronomical adaptive optics,” Astron. Astrophys. 291, 337–347 (1994).
  5. B. Le Roux, J. M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for classical and multiconjugate adaptive optics,” J. Opt. Soc. Am. A 21, 1261–1276 (2004).
    [CrossRef]
  6. K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven h2-optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol. 16, 381–395 (2008).
    [CrossRef]
  7. L. C. Johnson, D. T. Gavel, and D. M. Wiberg, “Bulk wind estimation and prediction for adaptive optics control systems,” J. Opt. Soc. Am. A 28, 1566–1577 (2011).
    [CrossRef]
  8. C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance prediction and control for adaptive optics,” Automatica 48, 1939–1954 (2012).
    [CrossRef]
  9. J. Tesch and S. Gibson, “Optimal and adaptive control of aero-optical wavefronts for adaptive optics,” J. Opt. Soc. Am. A 29, 1625–1638 (2012).
    [CrossRef]
  10. D. M. Wiberg, C. E. Max, and D. T. Gavel, “Geometric view of adaptive optics control,” J. Opt. Soc. Am. A 22, 870–880 (2005).
    [CrossRef]
  11. R. Fraanje, J. Rice, M. Verhaegen, and N. Doelman, “Fast reconstruction and prediction of frozen flow turbulence based on structured Kalman filtering,” J. Opt. Soc. Am. A 27, A235–A245 (2010).
    [CrossRef]
  12. L. Gilles and B. L. Ellerbroek, “Real-time turbulence profiling with a pair of laser guide star Shack–Hartmann wavefront sensors for wide-field adaptive optics systems on large to extremely large telescopes,” J. Opt. Soc. Am. A 27, A76–A83 (2010).
    [CrossRef]
  13. P. Massioni, C. Kulcsár, H.-F. Raynaud, and J.-M. Conan, “Fast computation of an optimal controller for large-scale adaptive optics,” J. Opt. Soc. Am. A 28, 2298–2309 (2011).
    [CrossRef]
  14. L. Poyneer and J.-P. Véran, “Predictive wavefront control for adaptive optics with arbitrary control loop delays,” J. Opt. Soc. Am. A 25, 1486–1496 (2008).
    [CrossRef]
  15. R. Conan, “Modelisation des effets de l’echelle externe de coherence spatiale du front d’onde pour l’observation a haute resolution angulaire en astronomie,” Ph.D. thesis (Université Nice Sophia Antipolis, 2000).
  16. F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).
  17. B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE 74, 225–233 (1976).
    [CrossRef]
  18. F. Assemat, R. W. Wilson, and E. Gendron, “Method for simulating infinitely long and non stationary phase screens with optimized memory storage,” Opt. Express 14, 988–999 (2006).
    [CrossRef]
  19. A. Beghi, A. Cenedese, and A. Masiero, “Stochastic realization approach to the efficient simulation of phase screens,” J. Opt. Soc. Am. A 25, 515–525 (2008).
    [CrossRef]
  20. D. L. Fried and T. Clark, “Extruding Kolmogorov-type phase screen ribbons,” J. Opt. Soc. Am. A 25, 463–468 (2008).
    [CrossRef]
  21. A. Beghi, A. Cenedese, and A. Masiero, “Multiscale stochastic approach for phase screens synthesis,” Appl. Opt. 50, 4124–4133 (2011).
    [CrossRef]
  22. A. Benveniste, R. Nikoukhah, and A. S. Willsky, “Multiscale system theory,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 41, 2–15 (1994).
    [CrossRef]
  23. K. Daoudi, A. B. Frakt, and A. S. Willsky, “Multiscale autoregressive models and wavelets,” IEEE Trans. Inf. Theory 45, 828–845 (1999).
    [CrossRef]
  24. A. B. Frakt and A. S. Willsky, “Computationally efficient stochastic realization for internal multiscale autoregressive models,” Multidimens. Syst. Signal Process. 12, 109–142 (2001).
    [CrossRef]
  25. W. W. Irving and A. S. Willsky, “A canonical correlations approach to multiscale stochastic realization,” IEEE Trans. Autom. Control 46, 1514–1528 (2001).
    [CrossRef]
  26. M. R. Luettgen, W. C. Karl, A. S. Willsky, and R. R. Tenney, “Multiscale representations of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3396 (1993).
    [CrossRef]
  27. R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
    [CrossRef]
  28. E. Thiebaut and M. Tallon, “Fast minimum variance wavefront reconstruction for extremely large telescope,” J. Opt. Soc. Am. A 27, 1046–1059 (2010).
    [CrossRef]
  29. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
    [CrossRef]
  30. G. Golub and C. Van Loan, Matrix Computations (Johns Hopkins University, 1989).
  31. J. E. Jackson, A User’s Guide to Principal Components (Wiley, 1991).
  32. Since xM(u,v)=ϕ(u,v), ∀  (u,v), and for each l the coefficients representing the turbulence at scale l−1 can be linearly obtained from those at scale l, then the second order statistics of xl can be obtained by means of linear combinations of values from Eq. (2).
  33. S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1999).
  34. The coefficients of the best linear predictor can be computed from the theoretical covariances (2).
  35. Only the most computationally expensive operations are considered in the reported list.
  36. A. N. Shiryaev, Probability (Springer, 1995).
  37. Discarding operations with computational cost typically much lower than filtering the (multiscale) turbulent phase with filters of size dl×dl.
  38. A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall1975).
  39. M. Lyubenova and M. Kissler-Patig, An Expanded View of the Universe: Science with the European Extremely Large Telescope (European Southern Observatory, 2009).
  40. In the complete model the neighborhood size, δl, for Eq. (11) is set to 5 for l=1,…,M, whereas δM=3 in the reduced complexity model.
  41. Let σl be computed for two positions (u,v) and (u′,v′). Then its dependence on (u−u′,v−v′) is reduced only to a dependence on |(u−u′,v−v′)| because the turbulence is homogeneous and isotropic (at each scale).
  42. M. Le Ravalec, B. Noetinger, and L. Y. Hu, “The FFT moving average (FFT-MA) generator: an efficient numerical method for generating and conditioning Gaussian simulations,” Math. Geol. 32, 701–723 (2000).
    [CrossRef]

2012

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance prediction and control for adaptive optics,” Automatica 48, 1939–1954 (2012).
[CrossRef]

J. Tesch and S. Gibson, “Optimal and adaptive control of aero-optical wavefronts for adaptive optics,” J. Opt. Soc. Am. A 29, 1625–1638 (2012).
[CrossRef]

2011

2010

2009

E. Fedrigo, R. Muradore, and D. Zilio, “High performance adaptive optics system with fine tip/tilt control,” Control Eng. Pract. 17, 122–135 (2009).
[CrossRef]

2008

2006

2005

2004

2001

A. B. Frakt and A. S. Willsky, “Computationally efficient stochastic realization for internal multiscale autoregressive models,” Multidimens. Syst. Signal Process. 12, 109–142 (2001).
[CrossRef]

W. W. Irving and A. S. Willsky, “A canonical correlations approach to multiscale stochastic realization,” IEEE Trans. Autom. Control 46, 1514–1528 (2001).
[CrossRef]

2000

M. Le Ravalec, B. Noetinger, and L. Y. Hu, “The FFT moving average (FFT-MA) generator: an efficient numerical method for generating and conditioning Gaussian simulations,” Math. Geol. 32, 701–723 (2000).
[CrossRef]

1999

K. Daoudi, A. B. Frakt, and A. S. Willsky, “Multiscale autoregressive models and wavelets,” IEEE Trans. Inf. Theory 45, 828–845 (1999).
[CrossRef]

C. Dessenne, P.-Y. Madec, and G. Rousset, “Sky implementation of modal predictive control in adaptive optics,” Opt. Lett. 24, 339–341 (1999).
[CrossRef]

1994

E. Gendron and P. Léna, “Astronomical adaptive optics,” Astron. Astrophys. 291, 337–347 (1994).

A. Benveniste, R. Nikoukhah, and A. S. Willsky, “Multiscale system theory,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 41, 2–15 (1994).
[CrossRef]

1993

M. R. Luettgen, W. C. Karl, A. S. Willsky, and R. R. Tenney, “Multiscale representations of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3396 (1993).
[CrossRef]

1992

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

1981

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[CrossRef]

1976

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE 74, 225–233 (1976).
[CrossRef]

Assemat, F.

Beghi, A.

Benveniste, A.

A. Benveniste, R. Nikoukhah, and A. S. Willsky, “Multiscale system theory,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 41, 2–15 (1994).
[CrossRef]

Cenedese, A.

Clark, T.

Conan, J. M.

Conan, J.-M.

Conan, R.

R. Conan, “Modelisation des effets de l’echelle externe de coherence spatiale du front d’onde pour l’observation a haute resolution angulaire en astronomie,” Ph.D. thesis (Université Nice Sophia Antipolis, 2000).

Correia, C.

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]

Daoudi, K.

K. Daoudi, A. B. Frakt, and A. S. Willsky, “Multiscale autoregressive models and wavelets,” IEEE Trans. Inf. Theory 45, 828–845 (1999).
[CrossRef]

Dessenne, C.

Doelman, N.

R. Fraanje, J. Rice, M. Verhaegen, and N. Doelman, “Fast reconstruction and prediction of frozen flow turbulence based on structured Kalman filtering,” J. Opt. Soc. Am. A 27, A235–A245 (2010).
[CrossRef]

K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven h2-optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol. 16, 381–395 (2008).
[CrossRef]

Ellerbroek, B. L.

Fedrigo, E.

E. Fedrigo, R. Muradore, and D. Zilio, “High performance adaptive optics system with fine tip/tilt control,” Control Eng. Pract. 17, 122–135 (2009).
[CrossRef]

Fraanje, R.

Frakt, A. B.

A. B. Frakt and A. S. Willsky, “Computationally efficient stochastic realization for internal multiscale autoregressive models,” Multidimens. Syst. Signal Process. 12, 109–142 (2001).
[CrossRef]

K. Daoudi, A. B. Frakt, and A. S. Willsky, “Multiscale autoregressive models and wavelets,” IEEE Trans. Inf. Theory 45, 828–845 (1999).
[CrossRef]

Fried, D. L.

Fusco, T.

Gavel, D. T.

Gendron, E.

Gibson, S.

Gilles, L.

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]

Golub, G.

G. Golub and C. Van Loan, Matrix Computations (Johns Hopkins University, 1989).

Hinnen, K.

K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven h2-optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol. 16, 381–395 (2008).
[CrossRef]

Hu, L. Y.

M. Le Ravalec, B. Noetinger, and L. Y. Hu, “The FFT moving average (FFT-MA) generator: an efficient numerical method for generating and conditioning Gaussian simulations,” Math. Geol. 32, 701–723 (2000).
[CrossRef]

Irving, W. W.

W. W. Irving and A. S. Willsky, “A canonical correlations approach to multiscale stochastic realization,” IEEE Trans. Autom. Control 46, 1514–1528 (2001).
[CrossRef]

Jackson, J. E.

J. E. Jackson, A User’s Guide to Principal Components (Wiley, 1991).

Johnson, L. C.

Karl, W. C.

M. R. Luettgen, W. C. Karl, A. S. Willsky, and R. R. Tenney, “Multiscale representations of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3396 (1993).
[CrossRef]

Kissler-Patig, M.

M. Lyubenova and M. Kissler-Patig, An Expanded View of the Universe: Science with the European Extremely Large Telescope (European Southern Observatory, 2009).

Kulcsar, C.

Kulcsár, C.

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]

Le Ravalec, M.

M. Le Ravalec, B. Noetinger, and L. Y. Hu, “The FFT moving average (FFT-MA) generator: an efficient numerical method for generating and conditioning Gaussian simulations,” Math. Geol. 32, 701–723 (2000).
[CrossRef]

Le Roux, B.

Léna, P.

E. Gendron and P. Léna, “Astronomical adaptive optics,” Astron. Astrophys. 291, 337–347 (1994).

Luettgen, M. R.

M. R. Luettgen, W. C. Karl, A. S. Willsky, and R. R. Tenney, “Multiscale representations of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3396 (1993).
[CrossRef]

Lyubenova, M.

M. Lyubenova and M. Kissler-Patig, An Expanded View of the Universe: Science with the European Extremely Large Telescope (European Southern Observatory, 2009).

Madec, P.-Y.

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1999).

Masiero, A.

Massioni, P.

Max, C. E.

McGlamery, B. L.

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE 74, 225–233 (1976).
[CrossRef]

Mugnier, L. M.

Muradore, R.

E. Fedrigo, R. Muradore, and D. Zilio, “High performance adaptive optics system with fine tip/tilt control,” Control Eng. Pract. 17, 122–135 (2009).
[CrossRef]

Nikoukhah, R.

A. Benveniste, R. Nikoukhah, and A. S. Willsky, “Multiscale system theory,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 41, 2–15 (1994).
[CrossRef]

Noetinger, B.

M. Le Ravalec, B. Noetinger, and L. Y. Hu, “The FFT moving average (FFT-MA) generator: an efficient numerical method for generating and conditioning Gaussian simulations,” Math. Geol. 32, 701–723 (2000).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall1975).

Petit, C.

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance prediction and control for adaptive optics,” Automatica 48, 1939–1954 (2012).
[CrossRef]

Poyneer, L.

Raynaud, H.-F.

Rice, J.

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[CrossRef]

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).

Rousset, G.

Schafer, R. W.

A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall1975).

Shiryaev, A. N.

A. N. Shiryaev, Probability (Springer, 1995).

Tallon, M.

Tenney, R. R.

M. R. Luettgen, W. C. Karl, A. S. Willsky, and R. R. Tenney, “Multiscale representations of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3396 (1993).
[CrossRef]

Tesch, J.

Thiebaut, E.

Van Loan, C.

G. Golub and C. Van Loan, Matrix Computations (Johns Hopkins University, 1989).

Véran, J.-P.

Verhaegen, M.

R. Fraanje, J. Rice, M. Verhaegen, and N. Doelman, “Fast reconstruction and prediction of frozen flow turbulence based on structured Kalman filtering,” J. Opt. Soc. Am. A 27, A235–A245 (2010).
[CrossRef]

K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven h2-optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol. 16, 381–395 (2008).
[CrossRef]

Wiberg, D. M.

Willsky, A. S.

W. W. Irving and A. S. Willsky, “A canonical correlations approach to multiscale stochastic realization,” IEEE Trans. Autom. Control 46, 1514–1528 (2001).
[CrossRef]

A. B. Frakt and A. S. Willsky, “Computationally efficient stochastic realization for internal multiscale autoregressive models,” Multidimens. Syst. Signal Process. 12, 109–142 (2001).
[CrossRef]

K. Daoudi, A. B. Frakt, and A. S. Willsky, “Multiscale autoregressive models and wavelets,” IEEE Trans. Inf. Theory 45, 828–845 (1999).
[CrossRef]

A. Benveniste, R. Nikoukhah, and A. S. Willsky, “Multiscale system theory,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 41, 2–15 (1994).
[CrossRef]

M. R. Luettgen, W. C. Karl, A. S. Willsky, and R. R. Tenney, “Multiscale representations of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3396 (1993).
[CrossRef]

Wilson, R. W.

Zilio, D.

E. Fedrigo, R. Muradore, and D. Zilio, “High performance adaptive optics system with fine tip/tilt control,” Control Eng. Pract. 17, 122–135 (2009).
[CrossRef]

Appl. Opt.

Astron. Astrophys.

E. Gendron and P. Léna, “Astronomical adaptive optics,” Astron. Astrophys. 291, 337–347 (1994).

Automatica

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance prediction and control for adaptive optics,” Automatica 48, 1939–1954 (2012).
[CrossRef]

Control Eng. Pract.

E. Fedrigo, R. Muradore, and D. Zilio, “High performance adaptive optics system with fine tip/tilt control,” Control Eng. Pract. 17, 122–135 (2009).
[CrossRef]

IEEE Trans. Autom. Control

W. W. Irving and A. S. Willsky, “A canonical correlations approach to multiscale stochastic realization,” IEEE Trans. Autom. Control 46, 1514–1528 (2001).
[CrossRef]

IEEE Trans. Circuits Syst. I Fundam. Theory Appl.

A. Benveniste, R. Nikoukhah, and A. S. Willsky, “Multiscale system theory,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 41, 2–15 (1994).
[CrossRef]

IEEE Trans. Control Syst. Technol.

K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven h2-optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol. 16, 381–395 (2008).
[CrossRef]

IEEE Trans. Inf. Theory

K. Daoudi, A. B. Frakt, and A. S. Willsky, “Multiscale autoregressive models and wavelets,” IEEE Trans. Inf. Theory 45, 828–845 (1999).
[CrossRef]

IEEE Trans. Signal Process.

M. R. Luettgen, W. C. Karl, A. S. Willsky, and R. R. Tenney, “Multiscale representations of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3396 (1993).
[CrossRef]

J. Opt. Soc. Am. A

A. Beghi, A. Cenedese, and A. Masiero, “Stochastic realization approach to the efficient simulation of phase screens,” J. Opt. Soc. Am. A 25, 515–525 (2008).
[CrossRef]

D. L. Fried and T. Clark, “Extruding Kolmogorov-type phase screen ribbons,” J. Opt. Soc. Am. A 25, 463–468 (2008).
[CrossRef]

E. Thiebaut and M. Tallon, “Fast minimum variance wavefront reconstruction for extremely large telescope,” J. Opt. Soc. Am. A 27, 1046–1059 (2010).
[CrossRef]

L. C. Johnson, D. T. Gavel, and D. M. Wiberg, “Bulk wind estimation and prediction for adaptive optics control systems,” J. Opt. Soc. Am. A 28, 1566–1577 (2011).
[CrossRef]

B. Le Roux, J. M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for classical and multiconjugate adaptive optics,” J. Opt. Soc. Am. A 21, 1261–1276 (2004).
[CrossRef]

C. Correia, H.-F. Raynaud, C. Kulcsár, and J.-M. Conan, “On the optimal reconstruction and control of adaptive optical systems with mirror dynamics,” J. Opt. Soc. Am. A 27, 333–349 (2010).
[CrossRef]

J. Tesch and S. Gibson, “Optimal and adaptive control of aero-optical wavefronts for adaptive optics,” J. Opt. Soc. Am. A 29, 1625–1638 (2012).
[CrossRef]

D. M. Wiberg, C. E. Max, and D. T. Gavel, “Geometric view of adaptive optics control,” J. Opt. Soc. Am. A 22, 870–880 (2005).
[CrossRef]

R. Fraanje, J. Rice, M. Verhaegen, and N. Doelman, “Fast reconstruction and prediction of frozen flow turbulence based on structured Kalman filtering,” J. Opt. Soc. Am. A 27, A235–A245 (2010).
[CrossRef]

L. Gilles and B. L. Ellerbroek, “Real-time turbulence profiling with a pair of laser guide star Shack–Hartmann wavefront sensors for wide-field adaptive optics systems on large to extremely large telescopes,” J. Opt. Soc. Am. A 27, A76–A83 (2010).
[CrossRef]

P. Massioni, C. Kulcsár, H.-F. Raynaud, and J.-M. Conan, “Fast computation of an optimal controller for large-scale adaptive optics,” J. Opt. Soc. Am. A 28, 2298–2309 (2011).
[CrossRef]

L. Poyneer and J.-P. Véran, “Predictive wavefront control for adaptive optics with arbitrary control loop delays,” J. Opt. Soc. Am. A 25, 1486–1496 (2008).
[CrossRef]

Math. Geol.

M. Le Ravalec, B. Noetinger, and L. Y. Hu, “The FFT moving average (FFT-MA) generator: an efficient numerical method for generating and conditioning Gaussian simulations,” Math. Geol. 32, 701–723 (2000).
[CrossRef]

Multidimens. Syst. Signal Process.

A. B. Frakt and A. S. Willsky, “Computationally efficient stochastic realization for internal multiscale autoregressive models,” Multidimens. Syst. Signal Process. 12, 109–142 (2001).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE 74, 225–233 (1976).
[CrossRef]

Prog. Opt.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[CrossRef]

Waves Random Media

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

Other

G. Golub and C. Van Loan, Matrix Computations (Johns Hopkins University, 1989).

J. E. Jackson, A User’s Guide to Principal Components (Wiley, 1991).

Since xM(u,v)=ϕ(u,v), ∀  (u,v), and for each l the coefficients representing the turbulence at scale l−1 can be linearly obtained from those at scale l, then the second order statistics of xl can be obtained by means of linear combinations of values from Eq. (2).

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1999).

The coefficients of the best linear predictor can be computed from the theoretical covariances (2).

Only the most computationally expensive operations are considered in the reported list.

A. N. Shiryaev, Probability (Springer, 1995).

Discarding operations with computational cost typically much lower than filtering the (multiscale) turbulent phase with filters of size dl×dl.

A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall1975).

M. Lyubenova and M. Kissler-Patig, An Expanded View of the Universe: Science with the European Extremely Large Telescope (European Southern Observatory, 2009).

In the complete model the neighborhood size, δl, for Eq. (11) is set to 5 for l=1,…,M, whereas δM=3 in the reduced complexity model.

Let σl be computed for two positions (u,v) and (u′,v′). Then its dependence on (u−u′,v−v′) is reduced only to a dependence on |(u−u′,v−v′)| because the turbulence is homogeneous and isotropic (at each scale).

R. Conan, “Modelisation des effets de l’echelle externe de coherence spatiale du front d’onde pour l’observation a haute resolution angulaire en astronomie,” Ph.D. thesis (Université Nice Sophia Antipolis, 2000).

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).

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Figures (12)

Fig. 1.
Fig. 1.

(a) Coordinates on the telescope image domain. (b) Two points, (u,v) and (u,v), separated by a distance r on the telescope aperture plane. (c) An example of phase screen.

Fig. 2.
Fig. 2.

Simulated turbulence samples. The temporal evolution is obtained by generating new rows and columns of the phase screen matrix and shifting the window corresponding to the telescope aperture of v/fs pixels.

Fig. 3.
Fig. 3.

Normalized 1D spatial correlations of x0 with (a) M=0, (b) M=3, and (c) M=7.

Fig. 4.
Fig. 4.

Low resolution turbulence at time t1 and t2>t1. The boundaries of the neighborhood for the current phase screen are reported as the black line. In the figure on the right, the boundaries of the neighborhood at time t1 are reported as a gray line. When generating the turbulence at time t2, new values of ϵ0 are sampled in the shaded area.

Fig. 5.
Fig. 5.

Energy σl,42 of the 4th component in the multiscale PCA representation varying the scale l from 1 to M=7. Turbulence parameters r0=0.11m, L0=60m, and ps=8.2mm.

Fig. 6.
Fig. 6.

Upper bound on the realized structure function error computed by means of Eq. (26) as a function of the spatial separation [r in Equation (26)]. Turbulence parameters r0=0.11m, L0=60m, ps=8.2mm, and M=7, l¯=5.

Fig. 7.
Fig. 7.

Nonstationary turbulence simulation. Turbulence evolves along the horizontal axis of the figure and its spatial parameters switch from r0=0.1m, L0=80m to r0=0.12m, L0=50m. The plotted region correspond to the values of x0 on a 845m×1037m area.

Fig. 8.
Fig. 8.

(a) Comparison of the theoretical low resolution normalized correlations (red dotted line) with the asymptotic normalized correlations reproduced by the low resolution dynamic model in [21] (blue solid line), and those of the MA model of Section 3 for case I in Table 1. (b) Error with respect to the theoretical low resolution normalized correlations for case I in Table 1.

Fig. 9.
Fig. 9.

Comparison of the asymptotic normalized correlations reproduced by the low resolution dynamic model in [21] (solid blue line), and those of the MA model of Section 3. Error with respect to the theoretical low resolution normalized correlations in (a) case II, (b) case III, and (c) case IV of Table 1, respectively.

Fig. 10.
Fig. 10.

Comparison of mean absolute errors on normalized correlations estimated from samples generated with the low resolution models. (a) LD model as in [21] (blue solid line), and (b) MA model of Section 3 (black dashed line). The number of columns in the phase screen matrix used to estimate the correlation varies from d/(2Mps) to 2000 pix.

Fig. 11.
Fig. 11.

(a) Comparison of the theoretical structure function (red dotted line) of case I in Table 1 with the sample structure function obtained by means of the complete MAR-MA model (blue solid line), and the reduced complexity MAR-MA (black dashed line), estimated from a 40m×20,000m phase screen. (b) Error with respect to the theoretical structure function. (c) Difference between the structure functions obtained by the complete and the reduced complexity MAR-MA model.

Fig. 12.
Fig. 12.

Top row: error of the reproduced structure functions, estimated from a 40m×20,000m phase screen, with respect to the theoretical one. Comparison of the complete MAR-MA model (blue solid line), and the reduced complexity MAR-MA (black dashed line). Bottom row: difference between the structure functions obtained by the complete and the reduced complexity MAR-MA model. Results are reported for case II (left), III (middle), and IV (right) in Table 1.

Tables (1)

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Table 1. Case Studies

Equations (41)

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Dϕ(r)=2(Cϕ(0)Cϕ(r)).
Cϕ(r)=(L0r0)5/3η2(2πrL0)5/6K5/6(2πrL0),
[xl(2u,2v)xl(2u,2v+1)xl(2u+1,v)xl(2u+1,2v+1)].
[xl(2u,2v)xl(2u,2v+1)xl(2u+1,2v)xl(2u+1,2v+1)]=Ul[xl1(u,v)xl1,1(u,v)xl1,2(u,v)xl1,3(u,v)],
Cl={[Ul(:,13)0000]ifthe last componentisneglected at levell,Ulotherwise.
[xl(2u,2v)xl(2u,2v+1)xl(2u+1,2v)xl(2u+1,2v+1)]Cl[xl1(u,v)xl1,1(u,v)xl1,2(u,v)xl1,3(u,v)].
x0(u,v)=ku,kvθ0(ku,kv)ϵ0(uku,vkv),
Cϕ0(u,v)=ku,kvθ0(ku,kv)θ0(u+ku,v+kv)
=ku,kvθ0(ku,kv)θ0(uku,vkv)
=ku,kvθ¯0(ku,kv)θ0(uku,vkv).
xl1,1(u,v)=ku,kval,ku,kvxl1(uku,vkv)MAR+ku,kvθl(ku,kv)ϵl,1(uku,vkv)MA,(u,v),
xl1,2(u,v)=ku,kval,ku,kvxl1(uku,vkv)+ku,kval,ku,kv1xl1,1(uku,vkv)+ku,kvθl2(ku,kv)εl,2(uku,vkv),(u,v),
xl1,3(u,v)=ku,kval,ku,kvxl1(uku,vkv)+ku,kval,ku,kv1xl1,1(uku,vkv)+ku,kval,ku,kv2xl1,2(uku,vkv)+ku,kvθl3(ku,kv)ϵl,3(uku,vkv),(u,v).
ϕ^(u,v)=CM[x^M1(u/2,v/2)xM11(u/2,v/2)xM12(u/2,v/2)0],
ϕ(u,v)=l=0M1h=13ξl,hxl,h(u2Ml,v2Ml)+ξ0,0x0(u2Ml,v2Ml)
ϕ^(u,v)=l=0l¯1h=13ξl,hxl,h(u2Ml,v2Ml)+l=l¯M1h=12ξl,hxl,h(u2Ml,v2Ml)+ξ0,0x0(u2Ml,v2Ml),
E[xl,3xl,3]E[(xl,3)2]E[(xl,3)2]=σl+1,4σl+1,4.
E[ε(u,v)]=E[ϕ(u,v)ϕ^(u,v)]=0
E[(ε(u,v))2]=E[(ϕ(u,v)ϕ^(u,v))2=E[(l=l¯M1ξl,3xl,3(u2Ml,v2Ml))2]Δϕ,
|ξl,h|=1/2Ml,(l,h),
Δϕ=l=l¯+1M14Mlσl,42+l=l¯+1Ml=l¯+1llMσlσl22Mll,
D^ϕ(uu,vv)=E[(ϕ(u,v)ε(u,v)ϕ(u,v)+ε(u,v))2]=Dϕ(r)+E[(ε(u,v)ε(u,v))2]+Ξ(u,v,u,v),
Ξ(u,v,u,v)=2E[(ϕ(u,v)ϕ(u,v))(ε(u,v)ε(u,v))].
E[(ε(u,v)ε(u,v))2]2Δϕ2E[ε(u,v)ε(u,v)]4Δϕ.
|Ξ(u,v,u,v)|4E[(ϕ(u,v)ϕ(u,v))ε(u,v)]
4ΔϕDϕ(r).
|D^ϕ(uu,vv)Dϕ(r)|4Δϕ+4ΔϕDϕ(r).
l=5M14γ4Mll=0M19γ4Ml42%,
xl1,1(u,v)=ku,kval,ku,kvxl1(uku,vkv)+ku,kvθl(ku,kv)ϵl,1(uku,vkv),(u,v),
xl1,1(u,v)=x˜l1,1(u,v),(u,v)Ω,
z(u,v)=ku,kvθl(ku,kv)ϵl,1(uku,vkv),
z(u,v)=xl1,1(u,v)ku,kval,ku,kvxl1(uku,vkv).
[xl(2u,2v)xl(2u,2v+1)xl(2u+1,v)xl(2u+1,2v+1)].
Σl=[σl(0)σl(1)σl(1)σl(2)σl(1)σl(0)σl(2)σl(1)σl(1)σl(2)σl(0)σl(1)σl(2)σl(1)σl(1)σl(0)],
ζ0=[1/21/21/21/2],ζ1=[1/21/21/21/2],
ζ2=[1/21/21/21/2],ζ3=[1/21/21/21/2],
λl,0=σl(0)+2σl(1)+σl(2),λl,1=σl(0)σl(2),λl,2=σl(0)σl(2),λl,3=σl(0)2σl(1)+σl(2).
λl,0>λl,1=λl,2>λl,3,
Ul=[ζ0ζ1ζ2ζ3].
|ξl,h|=1/2Ml,(l,h).
e(u)=kθ(k)ϵ(uk),

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