Abstract

Multi-object adaptive optics (MOAO) systems are still in their infancy: their complex optical designs for tomographic, wide-field wavefront sensing, coupled with open-loop (OL) correction, make their calibration a challenge. The correction of a discrete number of specific directions in the field allows for streamlined application of a general class of spatio-angular algorithms, initially proposed in Whiteley et al. [J. Opt. Soc. Am. A 15, 2097 (1998)], which is compatible with partial on-line calibration. The recent Learn & Apply algorithm from Vidal et al. [J. Opt. Soc. Am. A 27, A253 (2010)] can then be reinterpreted in a broader framework of tomographic algorithms and is shown to be a special case that exploits the particulars of OL and aperture-plane phase conjugation. An extension to embed a temporal prediction step to tackle sky-coverage limitations is discussed. The trade-off between lengthening the camera integration period, therefore increasing system lag error, and the resulting improvement in SNR can be shifted to higher guide-star magnitudes by introducing temporal prediction. The derivation of the optimal predictor and a comparison to suboptimal autoregressive models is provided using temporal structure functions. It is shown using end-to-end simulations of Raven, the MOAO science, and technology demonstrator for the 8 m Subaru telescope that prediction allows by itself the use of 1-magnitude-fainter guide stars.

© 2013 Optical Society of America

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2013 (1)

2012 (3)

D. R. Andersen, K. J. Jackson, C. Blain, C. Bradley, C. Correia, M. Ito, O. Lardière, and J.-P. Véran, “Performance modeling for the raven multi-object adaptive optics demonstrator,” Publ. Astron. Soc. Pac. 124, 469–484 (2012).
[CrossRef]

A. Cortés, B. Neichel, A. Guesalaga, J. Osborn, F. Rigaut, and D. Guzman, “Atmospheric turbulence profiling using multiple laser star wavefront sensors,” Mon. Not. R. Astron. Soc. 427, 2089–2099 (2012).
[CrossRef]

G. Sivo, H.-F. Raynaud, J.-M. Conan, C. Kulcsr, E. Gendron, F. Vidal, and A. Basden, “First laboratory validation of LQG control with the CANARY MOAO pathfinder,” Proc. SPIE 8447, 84472Y (2012).
[CrossRef]

2010 (3)

2009 (1)

2008 (2)

2007 (1)

2006 (3)

2005 (1)

2004 (1)

2003 (1)

A. C. Robin, C. Reylé, S. Derrière, and S. Picaud, “A synthetic view on structure and evolution of the Milky Way,” Astron. Astrophys. 409, 523–540 (2003).
[CrossRef]

2002 (2)

R. Ragazzoni, E. Diolaiti, J. Farinato, E. Fedrigo, E. Marchetti, M. Tordi, and D. Kirkman, “Multiple field of view layer-oriented adaptive optics,” Astron. Astrophys. 396, 731–744 (2002).
[CrossRef]

B. L. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. A 19, 1803–1816 (2002).
[CrossRef]

2001 (2)

T. Fusco, J.-M. Conan, G. Rousset, L. M. Mugnier, and V. Michau, “Optimal wave-front reconstruction strategies for multiconjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2527–2538 (2001).
[CrossRef]

A. Tokovinin, M. Le Louarn, E. Viard, N. Hubin, and R. Conan, “Optimized modal tomography in adaptive optics,” Astron. Astrophys. 378, 710–721 (2001).
[CrossRef]

2000 (1)

F. J. Rigaut, B. L. Ellerbroek, and R. Flicker, “Principles, limitations, and performance of multiconjugate adaptive optics,” Proc. SPIE 4007, 1022–1031 (2000).
[CrossRef]

1999 (1)

R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999).

1998 (2)

1995 (2)

1992 (1)

F. Rigaut and E. Gendron, “Laser guide star in adaptive optics—the tilt determination problem,” Astron. Astrophys. 261, 677–684 (1992).

1991 (1)

1989 (1)

F. Chassat, “Calcul du domaine d’isoplanétisme d’un système d’optique adaptative fonctionnant à travers la turbulence atmosphérique,” J. Opt. 20, 13–23 (1989).
[CrossRef]

1979 (2)

1976 (1)

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
[CrossRef]

Andersen, D.

K. Jackson, C. Correia, O. Lardière, D. Andersen, and C. Bradley, “Tomography for Raven, a multi-object adaptive optics science and technology demonstrator,” in Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS 2012), Maui, Hawaii, September11–14, 2012 (Curran Associates, 2012).

Andersen, D. R.

D. R. Andersen, K. J. Jackson, C. Blain, C. Bradley, C. Correia, M. Ito, O. Lardière, and J.-P. Véran, “Performance modeling for the raven multi-object adaptive optics demonstrator,” Publ. Astron. Soc. Pac. 124, 469–484 (2012).
[CrossRef]

Anderson, B. D. O.

B. D. O. Anderson and J. B. Moore, Optimal Filtering (Dover, 1995).

Assémat, F.

Basden, A.

G. Sivo, H.-F. Raynaud, J.-M. Conan, C. Kulcsr, E. Gendron, F. Vidal, and A. Basden, “First laboratory validation of LQG control with the CANARY MOAO pathfinder,” Proc. SPIE 8447, 84472Y (2012).
[CrossRef]

Beghi, A.

Blain, C.

D. R. Andersen, K. J. Jackson, C. Blain, C. Bradley, C. Correia, M. Ito, O. Lardière, and J.-P. Véran, “Performance modeling for the raven multi-object adaptive optics demonstrator,” Publ. Astron. Soc. Pac. 124, 469–484 (2012).
[CrossRef]

Bradley, C.

D. R. Andersen, K. J. Jackson, C. Blain, C. Bradley, C. Correia, M. Ito, O. Lardière, and J.-P. Véran, “Performance modeling for the raven multi-object adaptive optics demonstrator,” Publ. Astron. Soc. Pac. 124, 469–484 (2012).
[CrossRef]

K. Jackson, C. Correia, O. Lardière, D. Andersen, and C. Bradley, “Tomography for Raven, a multi-object adaptive optics science and technology demonstrator,” in Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS 2012), Maui, Hawaii, September11–14, 2012 (Curran Associates, 2012).

Cenedese, A.

Chassat, F.

F. Chassat, “Calcul du domaine d’isoplanétisme d’un système d’optique adaptative fonctionnant à travers la turbulence atmosphérique,” J. Opt. 20, 13–23 (1989).
[CrossRef]

Clare, R. M.

Clark, T.

Conan, J.-M.

Conan, R.

A. Tokovinin, M. Le Louarn, E. Viard, N. Hubin, and R. Conan, “Optimized modal tomography in adaptive optics,” Astron. Astrophys. 378, 710–721 (2001).
[CrossRef]

Correia, C.

C. Correia, J.-P. Véran, G. Herriot, B. L. Ellerbroek, L. Wang, and L. Gilles, “Increased sky coverage with optimal correction of tilt and tilt-anisoplanatism modes in laser-guide-star multiconjugate adaptive optics,” J. Opt. Soc. Am. A 30, 604–615 (2013).
[CrossRef]

D. R. Andersen, K. J. Jackson, C. Blain, C. Bradley, C. Correia, M. Ito, O. Lardière, and J.-P. Véran, “Performance modeling for the raven multi-object adaptive optics demonstrator,” Publ. Astron. Soc. Pac. 124, 469–484 (2012).
[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]

K. Jackson, C. Correia, O. Lardière, D. Andersen, and C. Bradley, “Tomography for Raven, a multi-object adaptive optics science and technology demonstrator,” in Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS 2012), Maui, Hawaii, September11–14, 2012 (Curran Associates, 2012).

Cortés, A.

A. Cortés, B. Neichel, A. Guesalaga, J. Osborn, F. Rigaut, and D. Guzman, “Atmospheric turbulence profiling using multiple laser star wavefront sensors,” Mon. Not. R. Astron. Soc. 427, 2089–2099 (2012).
[CrossRef]

de Lesegno, P. V.

Derrière, S.

A. C. Robin, C. Reylé, S. Derrière, and S. Picaud, “A synthetic view on structure and evolution of the Milky Way,” Astron. Astrophys. 409, 523–540 (2003).
[CrossRef]

Diolaiti, E.

R. Ragazzoni, E. Diolaiti, J. Farinato, E. Fedrigo, E. Marchetti, M. Tordi, and D. Kirkman, “Multiple field of view layer-oriented adaptive optics,” Astron. Astrophys. 396, 731–744 (2002).
[CrossRef]

Ellerbroek, B. L.

Farinato, J.

R. Ragazzoni, E. Diolaiti, J. Farinato, E. Fedrigo, E. Marchetti, M. Tordi, and D. Kirkman, “Multiple field of view layer-oriented adaptive optics,” Astron. Astrophys. 396, 731–744 (2002).
[CrossRef]

Fedrigo, E.

R. Ragazzoni, E. Diolaiti, J. Farinato, E. Fedrigo, E. Marchetti, M. Tordi, and D. Kirkman, “Multiple field of view layer-oriented adaptive optics,” Astron. Astrophys. 396, 731–744 (2002).
[CrossRef]

Flicker, R.

F. J. Rigaut, B. L. Ellerbroek, and R. Flicker, “Principles, limitations, and performance of multiconjugate adaptive optics,” Proc. SPIE 4007, 1022–1031 (2000).
[CrossRef]

Fried, D. L.

Fusco, T.

Gendron, E.

G. Sivo, H.-F. Raynaud, J.-M. Conan, C. Kulcsr, E. Gendron, F. Vidal, and A. Basden, “First laboratory validation of LQG control with the CANARY MOAO pathfinder,” Proc. SPIE 8447, 84472Y (2012).
[CrossRef]

F. Vidal, E. Gendron, and G. Rousset, “Tomography approach for multi-object adaptive optics,” J. Opt. Soc. Am. A 27, A253–A264 (2010).
[CrossRef]

F. Assémat, R. 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]

F. Rigaut and E. Gendron, “Laser guide star in adaptive optics—the tilt determination problem,” Astron. Astrophys. 261, 677–684 (1992).

Gilles, L.

Guesalaga, A.

A. Cortés, B. Neichel, A. Guesalaga, J. Osborn, F. Rigaut, and D. Guzman, “Atmospheric turbulence profiling using multiple laser star wavefront sensors,” Mon. Not. R. Astron. Soc. 427, 2089–2099 (2012).
[CrossRef]

Guzman, D.

A. Cortés, B. Neichel, A. Guesalaga, J. Osborn, F. Rigaut, and D. Guzman, “Atmospheric turbulence profiling using multiple laser star wavefront sensors,” Mon. Not. R. Astron. Soc. 427, 2089–2099 (2012).
[CrossRef]

Herriot, G.

Hubin, N.

A. Tokovinin, M. Le Louarn, E. Viard, N. Hubin, and R. Conan, “Optimized modal tomography in adaptive optics,” Astron. Astrophys. 378, 710–721 (2001).
[CrossRef]

Ito, M.

D. R. Andersen, K. J. Jackson, C. Blain, C. Bradley, C. Correia, M. Ito, O. Lardière, and J.-P. Véran, “Performance modeling for the raven multi-object adaptive optics demonstrator,” Publ. Astron. Soc. Pac. 124, 469–484 (2012).
[CrossRef]

Jackson, K.

K. Jackson, C. Correia, O. Lardière, D. Andersen, and C. Bradley, “Tomography for Raven, a multi-object adaptive optics science and technology demonstrator,” in Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS 2012), Maui, Hawaii, September11–14, 2012 (Curran Associates, 2012).

Jackson, K. J.

D. R. Andersen, K. J. Jackson, C. Blain, C. Bradley, C. Correia, M. Ito, O. Lardière, and J.-P. Véran, “Performance modeling for the raven multi-object adaptive optics demonstrator,” Publ. Astron. Soc. Pac. 124, 469–484 (2012).
[CrossRef]

Kirkman, D.

R. Ragazzoni, E. Diolaiti, J. Farinato, E. Fedrigo, E. Marchetti, M. Tordi, and D. Kirkman, “Multiple field of view layer-oriented adaptive optics,” Astron. Astrophys. 396, 731–744 (2002).
[CrossRef]

Kulcsár, C.

Kulcsr, C.

G. Sivo, H.-F. Raynaud, J.-M. Conan, C. Kulcsr, E. Gendron, F. Vidal, and A. Basden, “First laboratory validation of LQG control with the CANARY MOAO pathfinder,” Proc. SPIE 8447, 84472Y (2012).
[CrossRef]

Lardière, O.

D. R. Andersen, K. J. Jackson, C. Blain, C. Bradley, C. Correia, M. Ito, O. Lardière, and J.-P. Véran, “Performance modeling for the raven multi-object adaptive optics demonstrator,” Publ. Astron. Soc. Pac. 124, 469–484 (2012).
[CrossRef]

K. Jackson, C. Correia, O. Lardière, D. Andersen, and C. Bradley, “Tomography for Raven, a multi-object adaptive optics science and technology demonstrator,” in Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS 2012), Maui, Hawaii, September11–14, 2012 (Curran Associates, 2012).

Le Louarn, M.

A. Tokovinin, M. Le Louarn, E. Viard, N. Hubin, and R. Conan, “Optimized modal tomography in adaptive optics,” Astron. Astrophys. 378, 710–721 (2001).
[CrossRef]

Le Roux, B.

Lukin, V. P.

Madec, P.-Y.

Marchetti, E.

R. Ragazzoni, E. Diolaiti, J. Farinato, E. Fedrigo, E. Marchetti, M. Tordi, and D. Kirkman, “Multiple field of view layer-oriented adaptive optics,” Astron. Astrophys. 396, 731–744 (2002).
[CrossRef]

R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999).

Masiero, A.

Michau, V.

Moore, J. B.

B. D. O. Anderson and J. B. Moore, Optimal Filtering (Dover, 1995).

Mugnier, L. M.

Neichel, B.

A. Cortés, B. Neichel, A. Guesalaga, J. Osborn, F. Rigaut, and D. Guzman, “Atmospheric turbulence profiling using multiple laser star wavefront sensors,” Mon. Not. R. Astron. Soc. 427, 2089–2099 (2012).
[CrossRef]

Noll, R. J.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
[CrossRef]

Osborn, J.

A. Cortés, B. Neichel, A. Guesalaga, J. Osborn, F. Rigaut, and D. Guzman, “Atmospheric turbulence profiling using multiple laser star wavefront sensors,” Mon. Not. R. Astron. Soc. 427, 2089–2099 (2012).
[CrossRef]

Petit, C.

Piatrou, P.

Picaud, S.

A. C. Robin, C. Reylé, S. Derrière, and S. Picaud, “A synthetic view on structure and evolution of the Milky Way,” Astron. Astrophys. 409, 523–540 (2003).
[CrossRef]

Ragazzoni, R.

R. Ragazzoni, E. Diolaiti, J. Farinato, E. Fedrigo, E. Marchetti, M. Tordi, and D. Kirkman, “Multiple field of view layer-oriented adaptive optics,” Astron. Astrophys. 396, 731–744 (2002).
[CrossRef]

R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999).

Raynaud, H.-F.

Reylé, C.

A. C. Robin, C. Reylé, S. Derrière, and S. Picaud, “A synthetic view on structure and evolution of the Milky Way,” Astron. Astrophys. 409, 523–540 (2003).
[CrossRef]

Rigaut, F.

A. Cortés, B. Neichel, A. Guesalaga, J. Osborn, F. Rigaut, and D. Guzman, “Atmospheric turbulence profiling using multiple laser star wavefront sensors,” Mon. Not. R. Astron. Soc. 427, 2089–2099 (2012).
[CrossRef]

R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999).

F. Rigaut and E. Gendron, “Laser guide star in adaptive optics—the tilt determination problem,” Astron. Astrophys. 261, 677–684 (1992).

Rigaut, F. J.

F. J. Rigaut, B. L. Ellerbroek, and R. Flicker, “Principles, limitations, and performance of multiconjugate adaptive optics,” Proc. SPIE 4007, 1022–1031 (2000).
[CrossRef]

Robin, A. C.

A. C. Robin, C. Reylé, S. Derrière, and S. Picaud, “A synthetic view on structure and evolution of the Milky Way,” Astron. Astrophys. 409, 523–540 (2003).
[CrossRef]

Roggemann, M. C.

Rousset, G.

Sivo, G.

G. Sivo, H.-F. Raynaud, J.-M. Conan, C. Kulcsr, E. Gendron, F. Vidal, and A. Basden, “First laboratory validation of LQG control with the CANARY MOAO pathfinder,” Proc. SPIE 8447, 84472Y (2012).
[CrossRef]

Takato, N.

Tokovinin, A.

A. Tokovinin, M. Le Louarn, E. Viard, N. Hubin, and R. Conan, “Optimized modal tomography in adaptive optics,” Astron. Astrophys. 378, 710–721 (2001).
[CrossRef]

Tordi, M.

R. Ragazzoni, E. Diolaiti, J. Farinato, E. Fedrigo, E. Marchetti, M. Tordi, and D. Kirkman, “Multiple field of view layer-oriented adaptive optics,” Astron. Astrophys. 396, 731–744 (2002).
[CrossRef]

Valley, G. C.

Véran, J.-P.

Viard, E.

A. Tokovinin, M. Le Louarn, E. Viard, N. Hubin, and R. Conan, “Optimized modal tomography in adaptive optics,” Astron. Astrophys. 378, 710–721 (2001).
[CrossRef]

Vidal, F.

G. Sivo, H.-F. Raynaud, J.-M. Conan, C. Kulcsr, E. Gendron, F. Vidal, and A. Basden, “First laboratory validation of LQG control with the CANARY MOAO pathfinder,” Proc. SPIE 8447, 84472Y (2012).
[CrossRef]

F. Vidal, E. Gendron, and G. Rousset, “Tomography approach for multi-object adaptive optics,” J. Opt. Soc. Am. A 27, A253–A264 (2010).
[CrossRef]

Wandzura, S. M.

Wang, L.

Welsh, B. M.

Whiteley, M. R.

Wilson, R.

Winker, D. M.

Yamaguchi, I.

Appl. Opt. (4)

Astron. Astrophys. (5)

R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999).

A. C. Robin, C. Reylé, S. Derrière, and S. Picaud, “A synthetic view on structure and evolution of the Milky Way,” Astron. Astrophys. 409, 523–540 (2003).
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

Temporal diagrams. Top: Int(Δ/Ts)=0 commands uk are conditioned to measurement sk1. Bottom: Ts<Δ<2Ts; uk is conditioned to sk2.

Fig. 2.
Fig. 2.

Temporal auto- and cross-correlation functions for the tip and focus modes. Although at t=0, these plots show the spatial values in [8], for t>0 correlations appear and vanish as shown. One can see, for the case of focus, that although it isn’t correlated to any other mode plotted for t=0, a strong (anti-) correlation appears with tilt as time elapses.

Fig. 3.
Fig. 3.

Face-on pattern of Aτ, single-layer, computed for nine radial orders (Nz=55) with |v|=15ms1 and τ=10ms, from which results a displacement of 0.15 m.

Fig. 4.
Fig. 4.

Theoretical temporal auto-correlation functions assuming frozen flow computed directly (black dash) or as the Fourier-transformed temporal PSDs (red dash). Comparison against the second-order continuous (blue dots) and discrete (green circles) predictive models fitting the initial 50 ms of the theoretical curves. Model fitting uses the Broyden–Fletcher–Goldfarb–Shanno method.

Fig. 5.
Fig. 5.

Comparison of temporal lag errors on an equivalent single layer atmosphere (see Table 1 for further parameters).

Fig. 6.
Fig. 6.

Functional optical block diagram of RAVEN. Dashed blocks are deployable. Raven consists of eight main subsystems: the deployable calibration unit, the OL NGS WFSs, the science pick-offs, the science relays, the closed-loop NGS truth/figure WFSs, the beam combiner, the LGS WFS, and the acquisition camera.

Fig. 7.
Fig. 7.

Comparison of residual wavefront error (in nm RMS) across the FoR with the SA (left column) and the explicit tomographic reconstructor (central column), for NGS asterisms 2, 1.5, and 1 ft apart. Differences plot on the right column.

Fig. 8.
Fig. 8.

Temporal lag error is traded by increased noise propagation through the wavefront reconstruction. Black markers: increase in limiting magnitude; blue markers: combined temporal plus noise propagation error; red circles indicate the increased limiting magnitudes for the minima of σ.

Fig. 9.
Fig. 9.

Left: Strehl ratio. Right: ensquared energy. Peak performance achieved in simulation for GS magnitudes from 14 to 17 using each of four algorithms: static, SA prediction, AR2 prediction, and AR1 prediction (black) and the integration time in ms (ordinates on the right) at which the peak value was reached (blue).

Tables (2)

Tables Icon

Table 1. Raven Baseline Configuration Parameters

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Table 2. Raven End-to-End Simulation Resultsa

Equations (41)

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ω(ρ,θ,t)=j=1NlWj(ρ+hjθ,t),
ω(ρ,θ,t)=i=2Nzϕi(θ,t)Zi(ρ/R),
ϕi(θ,t)=1R2Zi(ρ/R)ω(ρ,θ,t)Ω(ρ)d2ρ,
ϕ(θ,t)=Pθφ(t),
φ[φ0TφhL]T
E=argminEϕ(βi)ϕ^(βi)L2(Ω)2,
sα(t)=ΓPαφ(t)+η(t)
ϕ^β=PβEsα,
=PβφφTPαTΓT(ΓPαφφTPαTΓT+ηηT)1sα,
sα(t)=Γϕα(t)+η(t).
E{ϕβ|sα}Σ(ϕβ,sα)Σsα1sα=ϕ^β,
ϕ^βϕβϕαTΓT(ΓϕαϕαTΓT+ηηT)1sα.
ϕβϕαT=PβφφTPαT,
ϕαϕαT=PαφφTPαT.
ϕi(0)ϕj(ξ)=3.895(Dr0)530hmaxCn2(h)Iij(ξhR)dh0Cn2(h)dh
ϕ^βj,k+1=PβjP[φ^k,φ^k1,,φ^kn],
ω(ρ,t+τ)=ω(ρ+v·τ,t).
Aτ=argminAτφ(t+τ)Aτφ(t)L2(Ω)2,
Aτφk+1φkTφkφkT1.
φk+1=Aτφk+εkδ,
φk+1=f(φk,φkn+1)+εk,
φk+1AR1=AAR1φkAR1+εkAR1,
φk+1AR2=AAR2φkAR2+BAR2φk1AR2+εkAR2.
ΣεAR2=ΣφAAR2ΣφAAR2TBAR2ΣφBAR2TAAR2Σ1τAR2BAR2TBAR2Σ1τAR2AAR2T,
Σ1τAR2=φk+1AR2(φkAR2)TφkAR2(φk1AR2)T=(IBAR2)1AAR2Σφ,
φk+1AR3=AAR3φkAR3+BAR3φk1AR3+CAR3φk2AR3+εkAR3,
ΣεAR3=ΣφAAR3ΣφAAR3TBAR3ΣφBAR3TCAR3ΣφCAR3TAAR3Σ1τAR3BAR3TBAR3Σ1τAR3AAR3TBAR3Σ1τAR3CAR3TCAR3Σ1τAR3BAR3TAAR3Σ2τAR3CAR3TCAR3Σ2τAR3AAR3T,
Σ1τAR3=(BAR3+CAR3AAR3+CAR32)1(AAR3+CAR3BAR3)Σφ,
Σ2τAR3=BAR3Σφ+(AAR3+CAR3BAR3)Σ1τAR3.
Dt(τ)=|ω(ρ,t)ω(ρ,t+τ)|2,
Dt(τ)=|ω(ρ,t)ω(ρv·τ,t)|2ρ=Dρ(v·τ),
Dρ(ρ)=(Lor0)5/3×21/6Γ(11/6)π8/3[245Γ(65)]5/6[Γ(5/6)21/6(2πρL0)5/6K5/6(2πρL0)],
σlag2(τ,p)=Pθ(φkφ^k)L2(Ω)2,
σlag2(τ,p=0)=2trace{Pθ(ΣφΣ1τ)PθT}Dt(τ),
σlag2(τ,p=1)=trace{Pθ(Σφ+AΣφAT2Σ1τAT)PθT},
σlag2(τ,p=2)=trace{Pθ(Σφ+AAR2ΣφAAR2+BAR2ΣφBAR22AAR2Σ1τ+AAR2Σ1τBAR2T2BAR2Σ2τ)PT},
σlag2(τ,p=3)=trace{Pθ(Σφ+AAR3ΣφAAR3+BAR3ΣφBAR3+CΣφCAR32AAR3Σ1τ2BAR3Σ2τ2CAR3Σ3τ+AAR3Σ1τBAR3T+BAR3Σ1τCAR3T+AAR3Σ2τCAR3T)PθT},
εi(Np)=trace(ϕαϕαTPα(Np)φφTPα(Np)T)trace(ϕαϕαT)
σtotal2=σlag2+σnp2σ2+other terms,
σnp2=trace{EΣηET},
σ2=σnp2(mv,τ)+σlag2(τ,p=0)=σnp2(mv,τ)+σlag2(τ,p>0),

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