Abstract

We present a wind-predictive controller for astronomical adaptive optics (AO) systems that is able to predict the motion of a single windblown layer in the presence of other, more slowly varying phase aberrations. This controller relies on fast, gradient-based optical flow estimation to identify the velocity of the translating layer and a recursive mean estimator to account for turbulence that varies on a time scale much slower than the operating speed of the AO loop. We derive the Cramer–Rao lower bound for the wind estimation problem and show that the proposed estimator is very close to achieving theoretical minimum-variance performance. We also present simulations using on-sky data that show significant Strehl increases from using this controller in realistic atmospheric conditions.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2010

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. Morzinski, D. T. Gavel, B. J. Grigsby, L. C. Johnson, D. Dillon, M. R. Reinig, B. A. Macintosh, and C. E. Max, “Performance of MEMS-based visible-light adaptive optics at Lick Observatory: closed- and open-loop control,” Proc. SPIE 7736, 77361O(2010).
[CrossRef]

L. C. Johnson, D. T. Gavel, and D. M. Wiberg, “Online wind estimation and prediction for a two-layer frozen-flow atmosphere,” Proc. SPIE 7736, 77362R (2010).
[CrossRef]

2009

2008

L. A. 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]

B. A. Macintosh, J. R. Graham, D. W. Palmer, R. Doyon, J. Dunn, D. T. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, A. Sivaramakrishnan, J. K. Wallace, B. Bauman, D. A. Erickson, C. Marois, L. A. Poyneer, and R. Soummer, “The Gemini Planet Imager: from science to design to construction,” Proc. SPIE 7015, 701518 (2008).
[CrossRef]

J.-L. Beuzit, M. Feldt, K. Dohlen, D. Mouillet, P. Puget, and F. Wildi, “SPHERE: a planet finder instrument for the VLT,” Proc. SPIE 7014, 701418 (2008).
[CrossRef]

2007

2006

D. M. Wiberg, L. C. Johnson, and D. T. Gavel, “Adaptive optics control of wind blown turbulence via translation and prediction,” Proc. SPIE 6272, 62722X (2006).
[CrossRef]

2004

L. A. Poyneer and B. A. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. Am. A 21, 810–819 (2004).
[CrossRef]

D. Robinson and P. Milanfar, “Fundamental performance limits in image registration,” IEEE Trans. Image Process. 13, 1185–1199 (2004).
[CrossRef] [PubMed]

2003

D. T. Gavel, “Suppressing anomalous localized waffle behavior in least squares wavefront reconstructors,” Proc. SPIE 4839, 972–980 (2003).
[CrossRef]

D. T. Gavel and D. M. Wiberg, “Toward Strehl-optimizing adaptive optics controllers,” Proc. SPIE 4839, 890–901 (2003).
[CrossRef]

2002

C. Li and X. Wang, “On convergence of the Gauss–Newton method for convex composite optimization,” Math. Program. 91, 349–356 (2002).
[CrossRef]

2000

1990

1984

P. Lindström and P.-A. Wedin, “A new linesearch algorithm for nonlinear least squares problems,” Math. Program. 29, 268–296 (1984).
[CrossRef]

1982

1977

Anandan, P.

J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical model-based motion estimation,” in ECCV ’92: Proceedings of the Second European Conference on Computer Vision (Springer-Verlag, 1992), pp. 237–252.

Bauman, B.

B. A. Macintosh, J. R. Graham, D. W. Palmer, R. Doyon, J. Dunn, D. T. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, A. Sivaramakrishnan, J. K. Wallace, B. Bauman, D. A. Erickson, C. Marois, L. A. Poyneer, and R. Soummer, “The Gemini Planet Imager: from science to design to construction,” Proc. SPIE 7015, 701518 (2008).
[CrossRef]

Bergen, J. R.

J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical model-based motion estimation,” in ECCV ’92: Proceedings of the Second European Conference on Computer Vision (Springer-Verlag, 1992), pp. 237–252.

Beuzit, J.-L.

J.-L. Beuzit, M. Feldt, K. Dohlen, D. Mouillet, P. Puget, and F. Wildi, “SPHERE: a planet finder instrument for the VLT,” Proc. SPIE 7014, 701418 (2008).
[CrossRef]

Björck, Å.

Å. Björck, Numerical Methods for Least Squares Problems (SIAM, 1996).
[CrossRef]

Conan, J.

Dillon, D.

K. Morzinski, D. T. Gavel, B. J. Grigsby, L. C. Johnson, D. Dillon, M. R. Reinig, B. A. Macintosh, and C. E. Max, “Performance of MEMS-based visible-light adaptive optics at Lick Observatory: closed- and open-loop control,” Proc. SPIE 7736, 77361O(2010).
[CrossRef]

Doelman, N.

Dohlen, K.

J.-L. Beuzit, M. Feldt, K. Dohlen, D. Mouillet, P. Puget, and F. Wildi, “SPHERE: a planet finder instrument for the VLT,” Proc. SPIE 7014, 701418 (2008).
[CrossRef]

Doyon, R.

B. A. Macintosh, J. R. Graham, D. W. Palmer, R. Doyon, J. Dunn, D. T. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, A. Sivaramakrishnan, J. K. Wallace, B. Bauman, D. A. Erickson, C. Marois, L. A. Poyneer, and R. Soummer, “The Gemini Planet Imager: from science to design to construction,” Proc. SPIE 7015, 701518 (2008).
[CrossRef]

Dunn, J.

B. A. Macintosh, J. R. Graham, D. W. Palmer, R. Doyon, J. Dunn, D. T. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, A. Sivaramakrishnan, J. K. Wallace, B. Bauman, D. A. Erickson, C. Marois, L. A. Poyneer, and R. Soummer, “The Gemini Planet Imager: from science to design to construction,” Proc. SPIE 7015, 701518 (2008).
[CrossRef]

Erickson, D. A.

B. A. Macintosh, J. R. Graham, D. W. Palmer, R. Doyon, J. Dunn, D. T. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, A. Sivaramakrishnan, J. K. Wallace, B. Bauman, D. A. Erickson, C. Marois, L. A. Poyneer, and R. Soummer, “The Gemini Planet Imager: from science to design to construction,” Proc. SPIE 7015, 701518 (2008).
[CrossRef]

Feldt, M.

J.-L. Beuzit, M. Feldt, K. Dohlen, D. Mouillet, P. Puget, and F. Wildi, “SPHERE: a planet finder instrument for the VLT,” Proc. SPIE 7014, 701418 (2008).
[CrossRef]

Fraanje, R.

Fried, D. L.

Gavel, D.

D. Wiberg, C. Max, and D. Gavel, “A spatial non-dynamic LQG controller: part 1, application to adaptive optics,” in 43rd IEEE Conference on Decision and Control, (IEEE, 2004), Vol.  3, pp. 3326–3332.

D. Wiberg, C. Max, and D. Gavel, “A spatial non-dynamic LQG controller: part 2, theory,” in 43rd IEEE Conference on Decision and Control, Vol.  3 (IEEE, 2004), pp. 3333–3338.

Gavel, D. T.

K. Morzinski, D. T. Gavel, B. J. Grigsby, L. C. Johnson, D. Dillon, M. R. Reinig, B. A. Macintosh, and C. E. Max, “Performance of MEMS-based visible-light adaptive optics at Lick Observatory: closed- and open-loop control,” Proc. SPIE 7736, 77361O(2010).
[CrossRef]

L. C. Johnson, D. T. Gavel, and D. M. Wiberg, “Online wind estimation and prediction for a two-layer frozen-flow atmosphere,” Proc. SPIE 7736, 77362R (2010).
[CrossRef]

B. A. Macintosh, J. R. Graham, D. W. Palmer, R. Doyon, J. Dunn, D. T. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, A. Sivaramakrishnan, J. K. Wallace, B. Bauman, D. A. Erickson, C. Marois, L. A. Poyneer, and R. Soummer, “The Gemini Planet Imager: from science to design to construction,” Proc. SPIE 7015, 701518 (2008).
[CrossRef]

D. M. Wiberg, L. C. Johnson, and D. T. Gavel, “Adaptive optics control of wind blown turbulence via translation and prediction,” Proc. SPIE 6272, 62722X (2006).
[CrossRef]

D. T. Gavel and D. M. Wiberg, “Toward Strehl-optimizing adaptive optics controllers,” Proc. SPIE 4839, 890–901 (2003).
[CrossRef]

D. T. Gavel, “Suppressing anomalous localized waffle behavior in least squares wavefront reconstructors,” Proc. SPIE 4839, 972–980 (2003).
[CrossRef]

Graham, J. R.

B. A. Macintosh, J. R. Graham, D. W. Palmer, R. Doyon, J. Dunn, D. T. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, A. Sivaramakrishnan, J. K. Wallace, B. Bauman, D. A. Erickson, C. Marois, L. A. Poyneer, and R. Soummer, “The Gemini Planet Imager: from science to design to construction,” Proc. SPIE 7015, 701518 (2008).
[CrossRef]

Greenwood, D. P.

Grigsby, B. J.

K. Morzinski, D. T. Gavel, B. J. Grigsby, L. C. Johnson, D. Dillon, M. R. Reinig, B. A. Macintosh, and C. E. Max, “Performance of MEMS-based visible-light adaptive optics at Lick Observatory: closed- and open-loop control,” Proc. SPIE 7736, 77361O(2010).
[CrossRef]

Hanna, K. J.

J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical model-based motion estimation,” in ECCV ’92: Proceedings of the Second European Conference on Computer Vision (Springer-Verlag, 1992), pp. 237–252.

Hingorani, R.

J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical model-based motion estimation,” in ECCV ’92: Proceedings of the Second European Conference on Computer Vision (Springer-Verlag, 1992), pp. 237–252.

Hinnen, K.

Johnson, L. C.

K. Morzinski, D. T. Gavel, B. J. Grigsby, L. C. Johnson, D. Dillon, M. R. Reinig, B. A. Macintosh, and C. E. Max, “Performance of MEMS-based visible-light adaptive optics at Lick Observatory: closed- and open-loop control,” Proc. SPIE 7736, 77361O(2010).
[CrossRef]

L. C. Johnson, D. T. Gavel, and D. M. Wiberg, “Online wind estimation and prediction for a two-layer frozen-flow atmosphere,” Proc. SPIE 7736, 77362R (2010).
[CrossRef]

D. M. Wiberg, L. C. Johnson, and D. T. Gavel, “Adaptive optics control of wind blown turbulence via translation and prediction,” Proc. SPIE 6272, 62722X (2006).
[CrossRef]

L. C. Johnson, “Wind estimation and prediction for control of adaptive optics systems,” Ph.D. thesis (University of California, Santa Cruz, 2010).

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).

Kulcsár, C.

Larkin, J.

B. A. Macintosh, J. R. Graham, D. W. Palmer, R. Doyon, J. Dunn, D. T. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, A. Sivaramakrishnan, J. K. Wallace, B. Bauman, D. A. Erickson, C. Marois, L. A. Poyneer, and R. Soummer, “The Gemini Planet Imager: from science to design to construction,” Proc. SPIE 7015, 701518 (2008).
[CrossRef]

Li, C.

C. Li and X. Wang, “On convergence of the Gauss–Newton method for convex composite optimization,” Math. Program. 91, 349–356 (2002).
[CrossRef]

Lindström, P.

P. Lindström and P.-A. Wedin, “A new linesearch algorithm for nonlinear least squares problems,” Math. Program. 29, 268–296 (1984).
[CrossRef]

Looze, D. P.

Macintosh, B. A.

K. Morzinski, D. T. Gavel, B. J. Grigsby, L. C. Johnson, D. Dillon, M. R. Reinig, B. A. Macintosh, and C. E. Max, “Performance of MEMS-based visible-light adaptive optics at Lick Observatory: closed- and open-loop control,” Proc. SPIE 7736, 77361O(2010).
[CrossRef]

B. A. Macintosh, J. R. Graham, D. W. Palmer, R. Doyon, J. Dunn, D. T. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, A. Sivaramakrishnan, J. K. Wallace, B. Bauman, D. A. Erickson, C. Marois, L. A. Poyneer, and R. Soummer, “The Gemini Planet Imager: from science to design to construction,” Proc. SPIE 7015, 701518 (2008).
[CrossRef]

L. A. Poyneer and B. A. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. Am. A 21, 810–819 (2004).
[CrossRef]

Marois, C.

B. A. Macintosh, J. R. Graham, D. W. Palmer, R. Doyon, J. Dunn, D. T. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, A. Sivaramakrishnan, J. K. Wallace, B. Bauman, D. A. Erickson, C. Marois, L. A. Poyneer, and R. Soummer, “The Gemini Planet Imager: from science to design to construction,” Proc. SPIE 7015, 701518 (2008).
[CrossRef]

Max, C.

D. Wiberg, C. Max, and D. Gavel, “A spatial non-dynamic LQG controller: part 1, application to adaptive optics,” in 43rd IEEE Conference on Decision and Control, (IEEE, 2004), Vol.  3, pp. 3326–3332.

D. Wiberg, C. Max, and D. Gavel, “A spatial non-dynamic LQG controller: part 2, theory,” in 43rd IEEE Conference on Decision and Control, Vol.  3 (IEEE, 2004), pp. 3333–3338.

Max, C. E.

K. Morzinski, D. T. Gavel, B. J. Grigsby, L. C. Johnson, D. Dillon, M. R. Reinig, B. A. Macintosh, and C. E. Max, “Performance of MEMS-based visible-light adaptive optics at Lick Observatory: closed- and open-loop control,” Proc. SPIE 7736, 77361O(2010).
[CrossRef]

Milanfar, P.

D. Robinson and P. Milanfar, “Fundamental performance limits in image registration,” IEEE Trans. Image Process. 13, 1185–1199 (2004).
[CrossRef] [PubMed]

Morzinski, K.

K. Morzinski, D. T. Gavel, B. J. Grigsby, L. C. Johnson, D. Dillon, M. R. Reinig, B. A. Macintosh, and C. E. Max, “Performance of MEMS-based visible-light adaptive optics at Lick Observatory: closed- and open-loop control,” Proc. SPIE 7736, 77361O(2010).
[CrossRef]

Mouillet, D.

J.-L. Beuzit, M. Feldt, K. Dohlen, D. Mouillet, P. Puget, and F. Wildi, “SPHERE: a planet finder instrument for the VLT,” Proc. SPIE 7014, 701418 (2008).
[CrossRef]

Oppenheimer, B.

B. A. Macintosh, J. R. Graham, D. W. Palmer, R. Doyon, J. Dunn, D. T. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, A. Sivaramakrishnan, J. K. Wallace, B. Bauman, D. A. Erickson, C. Marois, L. A. Poyneer, and R. Soummer, “The Gemini Planet Imager: from science to design to construction,” Proc. SPIE 7015, 701518 (2008).
[CrossRef]

Palmer, D. W.

B. A. Macintosh, J. R. Graham, D. W. Palmer, R. Doyon, J. Dunn, D. T. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, A. Sivaramakrishnan, J. K. Wallace, B. Bauman, D. A. Erickson, C. Marois, L. A. Poyneer, and R. Soummer, “The Gemini Planet Imager: from science to design to construction,” Proc. SPIE 7015, 701518 (2008).
[CrossRef]

Petit, C.

Poyneer, L. A.

Puget, P.

J.-L. Beuzit, M. Feldt, K. Dohlen, D. Mouillet, P. Puget, and F. Wildi, “SPHERE: a planet finder instrument for the VLT,” Proc. SPIE 7014, 701418 (2008).
[CrossRef]

Raynaud, H.

Reinig, M. R.

K. Morzinski, D. T. Gavel, B. J. Grigsby, L. C. Johnson, D. Dillon, M. R. Reinig, B. A. Macintosh, and C. E. Max, “Performance of MEMS-based visible-light adaptive optics at Lick Observatory: closed- and open-loop control,” Proc. SPIE 7736, 77361O(2010).
[CrossRef]

Rice, J.

Robinson, D.

D. Robinson and P. Milanfar, “Fundamental performance limits in image registration,” IEEE Trans. Image Process. 13, 1185–1199 (2004).
[CrossRef] [PubMed]

Saddlemyer, L.

B. A. Macintosh, J. R. Graham, D. W. Palmer, R. Doyon, J. Dunn, D. T. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, A. Sivaramakrishnan, J. K. Wallace, B. Bauman, D. A. Erickson, C. Marois, L. A. Poyneer, and R. Soummer, “The Gemini Planet Imager: from science to design to construction,” Proc. SPIE 7015, 701518 (2008).
[CrossRef]

Schöck, M.

Sivaramakrishnan, A.

B. A. Macintosh, J. R. Graham, D. W. Palmer, R. Doyon, J. Dunn, D. T. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, A. Sivaramakrishnan, J. K. Wallace, B. Bauman, D. A. Erickson, C. Marois, L. A. Poyneer, and R. Soummer, “The Gemini Planet Imager: from science to design to construction,” Proc. SPIE 7015, 701518 (2008).
[CrossRef]

Soummer, R.

B. A. Macintosh, J. R. Graham, D. W. Palmer, R. Doyon, J. Dunn, D. T. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, A. Sivaramakrishnan, J. K. Wallace, B. Bauman, D. A. Erickson, C. Marois, L. A. Poyneer, and R. Soummer, “The Gemini Planet Imager: from science to design to construction,” Proc. SPIE 7015, 701518 (2008).
[CrossRef]

Spillar, E. J.

Tyler, G. A.

van Dam, M.

Véran, J.-P.

Verhaegen, M.

Wallace, J. K.

B. A. Macintosh, J. R. Graham, D. W. Palmer, R. Doyon, J. Dunn, D. T. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, A. Sivaramakrishnan, J. K. Wallace, B. Bauman, D. A. Erickson, C. Marois, L. A. Poyneer, and R. Soummer, “The Gemini Planet Imager: from science to design to construction,” Proc. SPIE 7015, 701518 (2008).
[CrossRef]

Wang, X.

C. Li and X. Wang, “On convergence of the Gauss–Newton method for convex composite optimization,” Math. Program. 91, 349–356 (2002).
[CrossRef]

Wedin, P.-A.

P. Lindström and P.-A. Wedin, “A new linesearch algorithm for nonlinear least squares problems,” Math. Program. 29, 268–296 (1984).
[CrossRef]

Wiberg, D.

D. Wiberg, C. Max, and D. Gavel, “A spatial non-dynamic LQG controller: part 1, application to adaptive optics,” in 43rd IEEE Conference on Decision and Control, (IEEE, 2004), Vol.  3, pp. 3326–3332.

D. Wiberg, C. Max, and D. Gavel, “A spatial non-dynamic LQG controller: part 2, theory,” in 43rd IEEE Conference on Decision and Control, Vol.  3 (IEEE, 2004), pp. 3333–3338.

Wiberg, D. M.

L. C. Johnson, D. T. Gavel, and D. M. Wiberg, “Online wind estimation and prediction for a two-layer frozen-flow atmosphere,” Proc. SPIE 7736, 77362R (2010).
[CrossRef]

D. M. Wiberg, L. C. Johnson, and D. T. Gavel, “Adaptive optics control of wind blown turbulence via translation and prediction,” Proc. SPIE 6272, 62722X (2006).
[CrossRef]

D. T. Gavel and D. M. Wiberg, “Toward Strehl-optimizing adaptive optics controllers,” Proc. SPIE 4839, 890–901 (2003).
[CrossRef]

Wildi, F.

J.-L. Beuzit, M. Feldt, K. Dohlen, D. Mouillet, P. Puget, and F. Wildi, “SPHERE: a planet finder instrument for the VLT,” Proc. SPIE 7014, 701418 (2008).
[CrossRef]

IEEE Trans. Image Process.

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Figures (10)

Fig. 1
Fig. 1

Predictive time-update corrects for time delay in the control loop. The parameter estimator can use either the estimated wavefront phase (solid line) or slopes (dotted line) as input.

Fig. 2
Fig. 2

Real-time implementation of the Gauss–Newton wind estimator.

Fig. 3
Fig. 3

Wind estimator performance using both wavefront slope measurements and reconstructed phase estimates.

Fig. 4
Fig. 4

CRLB as a function of spatial frequency.

Fig. 5
Fig. 5

Estimator convergence rate as a function of σ s . The theoretical convergence rate is calculated from Eq. (25).

Fig. 6
Fig. 6

Number of iterations until the wind estimator converges to within the variance set by the CRLB.

Fig. 7
Fig. 7

Simulated Strehl performance of ViLLaGEs at 550 nm . Solid curves show the Strehl with no wind prediction, dashed curves show the Strehl with wind prediction and wind estimation, and dashed–dotted curves show the Strehl with predictive control and the exact wind velocity known a priori.

Fig. 8
Fig. 8

Simulated Strehl performance for an 8 m telescope with 60 subapertures across the primary.

Fig. 9
Fig. 9

Power spectra and wind estimates obtained from simulations using on-sky wavefronts from ViLLaGEs.

Fig. 10
Fig. 10

Simulation results using wavefronts measured at Palomar observatory on 1 April 2002.

Equations (29)

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ϕ b ( x , t ) = ϕ b ( x v , t 1 ) ,
ϕ c ( x , t ) = ϕ c ( x , t 1 ) .
ϕ ( x , t ) = ϕ b ( x , t ) + ϕ c ( x , t ) ,
s ( x , t ) = H ϕ ( x , t ) + n s ( x , t ) ,
ϕ ^ ( x , t ) = H s ( x , t ) ,
Cov { n ϕ ( x , t ) } = σ s 2 ( H T H + R ) 1 H T H ( H T H + R T ) 1 .
Cov { n ϕ ( x , t ) } = σ s 2 ( H T H ) 1 .
ϕ ^ + ( x , t ) = ϕ ( x , t ) s ( x , t ) , s ( x , t 1 ) , = ϕ ^ ( x , t ) + Ks ( x , t ) .
ϕ ^ ( x , t + t d ) = ϕ ( x , t + t d ) s ( x , t ) , s ( x , t 1 ) , = F ϕ ^ + ( x , t ) .
ϕ c ( t ) = τ c 1 τ c ϕ c ( t 1 ) + 1 τ c ϕ ( t ) .
J ( v ) = x Δ ϕ b ( x , v , t ) 2 .
v ( 1 ) = v ( 0 ) + dv ,
x [ x ϕ b T ( x , t ) x T ϕ b ( x , t ) ] dv = x [ x ϕ b T ( x , t ) Δ ϕ b ( x , v ( 0 ) , t ) ] .
ϕ ^ ( x , t ) = F ϕ ^ + ( x , t 1 ) .
F = M + VS V 1 M ˜ ,
M = ϕ mean ( t ) ϕ mean T ( t ) ϕ mean ( t ) 2 ,
M ˜ = I M .
l v T = 0 ,
σ v , CRLB 2 = ( l T v ) ( l v T ) 1 ,
P ( ϕ ^ ( x , t 1 ) ) = c exp { 1 2 r 1 T C 1 r 1 } ,
c = 1 ( 2 π ) N / 2 det ( C ) ,
P ( ϕ ^ ( x , t ) v ) = c exp { 1 2 r 2 T ( v ) C 1 r 2 ( v ) } .
l v T = r 2 T ( x , v , t ) C 1 ϕ ( x v , t ) v T .
σ v , CRLB 2 = σ s 2 [ ( ϕ T ( x , t ) x ) C 1 ( ϕ ( x , t ) x T ) ] 1 .
H ϕ ( x , t ) x T
R = lim k v ( k + 1 ) v * v ( k ) v * ,
R = ( x ϕ T x T ϕ ) 1 Δ ϕ T ( v * ) 2 ( xx T ) ϕ .
v estimate = v actual + v error ,
σ wind 2 = 6.88 ( v error τ d r 0 ) 5 / 3 ,

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