Abstract

It has been demonstrated by several authors that the optical turbulence parameters associated with a given adaptive optics (AO) run—the seeing angle and outer scale—can be determined from a statistical analysis of the commands of the system’s deformable mirror (DM). The higher the accuracy on these parameters, the more we can make use of them, allowing for instance a better estimation of the seeing statistics at the telescope location or a more accurate assessment of the performance of the AO system. In the context of a point spread function reconstruction project (PSF-R) for the W. M. Keck observatory AO system, we decided to identify, in the most exhaustive way, all the sources of systematic and random errors affecting the determination of the seeing angle and outer scale from the DM telemetry, and find ways to compensate/mitigate these errors to keep them under 10%. The seeing estimated using our improved DM-seeing method was compared with more than 70 nearly simultaneous seeing measurements from open-loop PSFs on the same optical axis, and with independent seeing-monitor measurements acquired at the same time but far from the telescope (DIMM/MASS): the correlation with the open-loop PSF is very good (the error is about 10%), validating the DM-seeing method for accurate seeing determination, while it is weak and sometimes completely uncorrelated with the DIMM/MASS seeing monitor data. We concluded that DM-based seeing can be very accurate if all the error terms are considered in the DM data processing, but that seeing taken from non-collocated seeing monitors is of no use even when moderate accuracy is required.

© 2018 Optical Society of America

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References

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    [Crossref]
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2017 (1)

Y. H. Ono, C. M. Correia, D. R. Andersen, O. Lardière, S. Oya, M. Akiyama, K. Jackson, and C. Bradley, “Statistics of turbulence parameters at Mauna Kea using the multiple wavefront sensor data of RAVEN,” Mon. Not. R. Astron. Soc. 465, 4931–4941 (2017).
[Crossref]

2012 (1)

J. Kolb, N. Muller, E. Aller-Carpentier, P. Andrade, and J. Girard, “What can be retrieved from adaptive optics real-time data?” Proc. SPIE 8447, 84475U (2012).
[Crossref]

2010 (3)

J. R. Lu, A. M. Ghez, S. Yelda, T. Do, W. Clarkson, N. McCrady, and M. Morris, “Recent results and perspectives for precision astrometry and photometry with adaptive optics,” Proc. SPIE 7736, 77361I (2010).
[Crossref]

L. Jolissaint, “Synthetic modeling of astronomical closed loop adaptive optics,” J. Eur. Opt. Soc. Rapid Publ. 5, 10055 (2010).
[Crossref]

T.-J. Brennan and D. C. Mann, “Estimation of optical turbulence characteristics from Shack Hartmann wavefront sensor measurements,” Proc. SPIE 7816, 781602 (2010).
[Crossref]

2007 (1)

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[Crossref]

2006 (1)

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[Crossref]

2004 (2)

L. Jolissaint, J.-P. Veran, and J. Marino, “OPERA, an automatic PSF reconstruction software for Shack-Hartmann AO systems: application to Altair,” Proc. SPIE 5490, 151–163 (2004).
[Crossref]

T. Fusco, G. Rousset, D. Rabaud, E. Gendron, D. Mouillet, F. Lacombe, G. Zins, P.-Y. Madec, A.-M. Lagrange, J. Charton, D. Rouan, N. Hubin, and N. Ageorges, “NAOS on-line characterization of turbulence parameters and adaptive optics performance,” J. Opt. A 6, 585–596 (2004).
[Crossref]

2003 (1)

2002 (1)

R. Conan, R. Avila, L. J. Sánchez, A. Ziad, F. Martin, J. Borgnino, O. Harris, S. I. González, R. Michel, and D. Hiriart, “Wavefront outer scale and seeing measurements at San Pedro Mártir observatory,” Astron. Astrophys. 396, 723–730 (2002).
[Crossref]

1997 (2)

1995 (1)

1991 (2)

D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J. Opt. Soc. Am. A 8, 1568–1573 (1991).
[Crossref]

F. Rigaut, G. Rousset, P. Kern, J.-C. Fontanella, J.-P. Gaffard, F. Merkle, and P. Léna, “Adaptive optics on a 3.6-m telescope—results and performance,” Astron. Astrophys. 250, 280–290 (1991).

1990 (1)

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

1985 (1)

C. E. Coulman, “Fundamental and applied aspects of astronomical seeing,” Appl. Opt. 23, 19–57 (1985).
[Crossref]

1976 (1)

1966 (1)

Ageorges, N.

T. Fusco, G. Rousset, D. Rabaud, E. Gendron, D. Mouillet, F. Lacombe, G. Zins, P.-Y. Madec, A.-M. Lagrange, J. Charton, D. Rouan, N. Hubin, and N. Ageorges, “NAOS on-line characterization of turbulence parameters and adaptive optics performance,” J. Opt. A 6, 585–596 (2004).
[Crossref]

Akiyama, M.

Y. H. Ono, C. M. Correia, D. R. Andersen, O. Lardière, S. Oya, M. Akiyama, K. Jackson, and C. Bradley, “Statistics of turbulence parameters at Mauna Kea using the multiple wavefront sensor data of RAVEN,” Mon. Not. R. Astron. Soc. 465, 4931–4941 (2017).
[Crossref]

Aller-Carpentier, E.

J. Kolb, N. Muller, E. Aller-Carpentier, P. Andrade, and J. Girard, “What can be retrieved from adaptive optics real-time data?” Proc. SPIE 8447, 84475U (2012).
[Crossref]

Andersen, D. R.

Y. H. Ono, C. M. Correia, D. R. Andersen, O. Lardière, S. Oya, M. Akiyama, K. Jackson, and C. Bradley, “Statistics of turbulence parameters at Mauna Kea using the multiple wavefront sensor data of RAVEN,” Mon. Not. R. Astron. Soc. 465, 4931–4941 (2017).
[Crossref]

Andrade, P.

J. Kolb, N. Muller, E. Aller-Carpentier, P. Andrade, and J. Girard, “What can be retrieved from adaptive optics real-time data?” Proc. SPIE 8447, 84475U (2012).
[Crossref]

Avila, R.

R. Conan, R. Avila, L. J. Sánchez, A. Ziad, F. Martin, J. Borgnino, O. Harris, S. I. González, R. Michel, and D. Hiriart, “Wavefront outer scale and seeing measurements at San Pedro Mártir observatory,” Astron. Astrophys. 396, 723–730 (2002).
[Crossref]

Ben Abdallah, K.

A. Ziad, J. Maire, J. Borgnino, W. Dali Ali, A. Berdja, K. Ben Abdallah, F. Martin, and M. Sarazin, “MOSP: monitor of outer scale profile,” in Adaptative Optics for Extremely Large Telescopes (AO4ELT-I), 2010.

Berdja, A.

A. Ziad, J. Maire, J. Borgnino, W. Dali Ali, A. Berdja, K. Ben Abdallah, F. Martin, and M. Sarazin, “MOSP: monitor of outer scale profile,” in Adaptative Optics for Extremely Large Telescopes (AO4ELT-I), 2010.

Borgnino, J.

R. Conan, R. Avila, L. J. Sánchez, A. Ziad, F. Martin, J. Borgnino, O. Harris, S. I. González, R. Michel, and D. Hiriart, “Wavefront outer scale and seeing measurements at San Pedro Mártir observatory,” Astron. Astrophys. 396, 723–730 (2002).
[Crossref]

A. Ziad, J. Maire, J. Borgnino, W. Dali Ali, A. Berdja, K. Ben Abdallah, F. Martin, and M. Sarazin, “MOSP: monitor of outer scale profile,” in Adaptative Optics for Extremely Large Telescopes (AO4ELT-I), 2010.

Bouchez, A. H.

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[Crossref]

Bradley, C.

Y. H. Ono, C. M. Correia, D. R. Andersen, O. Lardière, S. Oya, M. Akiyama, K. Jackson, and C. Bradley, “Statistics of turbulence parameters at Mauna Kea using the multiple wavefront sensor data of RAVEN,” Mon. Not. R. Astron. Soc. 465, 4931–4941 (2017).
[Crossref]

Brennan, T.-J.

T.-J. Brennan and D. C. Mann, “Estimation of optical turbulence characteristics from Shack Hartmann wavefront sensor measurements,” Proc. SPIE 7816, 781602 (2010).
[Crossref]

Campbell, R. D.

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[Crossref]

Chanan, G. A.

Charton, J.

T. Fusco, G. Rousset, D. Rabaud, E. Gendron, D. Mouillet, F. Lacombe, G. Zins, P.-Y. Madec, A.-M. Lagrange, J. Charton, D. Rouan, N. Hubin, and N. Ageorges, “NAOS on-line characterization of turbulence parameters and adaptive optics performance,” J. Opt. A 6, 585–596 (2004).
[Crossref]

Chin, J. C. Y.

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[Crossref]

Clarkson, W.

J. R. Lu, A. M. Ghez, S. Yelda, T. Do, W. Clarkson, N. McCrady, and M. Morris, “Recent results and perspectives for precision astrometry and photometry with adaptive optics,” Proc. SPIE 7736, 77361I (2010).
[Crossref]

Conan, R.

R. Conan, R. Avila, L. J. Sánchez, A. Ziad, F. Martin, J. Borgnino, O. Harris, S. I. González, R. Michel, and D. Hiriart, “Wavefront outer scale and seeing measurements at San Pedro Mártir observatory,” Astron. Astrophys. 396, 723–730 (2002).
[Crossref]

Correia, C. M.

Y. H. Ono, C. M. Correia, D. R. Andersen, O. Lardière, S. Oya, M. Akiyama, K. Jackson, and C. Bradley, “Statistics of turbulence parameters at Mauna Kea using the multiple wavefront sensor data of RAVEN,” Mon. Not. R. Astron. Soc. 465, 4931–4941 (2017).
[Crossref]

Coulman, C. E.

C. E. Coulman, “Fundamental and applied aspects of astronomical seeing,” Appl. Opt. 23, 19–57 (1985).
[Crossref]

Cuevas, S.

Dali Ali, W.

A. Ziad, J. Maire, J. Borgnino, W. Dali Ali, A. Berdja, K. Ben Abdallah, F. Martin, and M. Sarazin, “MOSP: monitor of outer scale profile,” in Adaptative Optics for Extremely Large Telescopes (AO4ELT-I), 2010.

Do, T.

J. R. Lu, A. M. Ghez, S. Yelda, T. Do, W. Clarkson, N. McCrady, and M. Morris, “Recent results and perspectives for precision astrometry and photometry with adaptive optics,” Proc. SPIE 7736, 77361I (2010).
[Crossref]

Fontanella, J.-C.

F. Rigaut, G. Rousset, P. Kern, J.-C. Fontanella, J.-P. Gaffard, F. Merkle, and P. Léna, “Adaptive optics on a 3.6-m telescope—results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Fried, D. L.

Fusco, T.

T. Fusco, G. Rousset, D. Rabaud, E. Gendron, D. Mouillet, F. Lacombe, G. Zins, P.-Y. Madec, A.-M. Lagrange, J. Charton, D. Rouan, N. Hubin, and N. Ageorges, “NAOS on-line characterization of turbulence parameters and adaptive optics performance,” J. Opt. A 6, 585–596 (2004).
[Crossref]

Gaffard, J.-P.

F. Rigaut, G. Rousset, P. Kern, J.-C. Fontanella, J.-P. Gaffard, F. Merkle, and P. Léna, “Adaptive optics on a 3.6-m telescope—results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Gendron, E.

T. Fusco, G. Rousset, D. Rabaud, E. Gendron, D. Mouillet, F. Lacombe, G. Zins, P.-Y. Madec, A.-M. Lagrange, J. Charton, D. Rouan, N. Hubin, and N. Ageorges, “NAOS on-line characterization of turbulence parameters and adaptive optics performance,” J. Opt. A 6, 585–596 (2004).
[Crossref]

Ghez, A. M.

J. R. Lu, A. M. Ghez, S. Yelda, T. Do, W. Clarkson, N. McCrady, and M. Morris, “Recent results and perspectives for precision astrometry and photometry with adaptive optics,” Proc. SPIE 7736, 77361I (2010).
[Crossref]

Girard, J.

J. Kolb, N. Muller, E. Aller-Carpentier, P. Andrade, and J. Girard, “What can be retrieved from adaptive optics real-time data?” Proc. SPIE 8447, 84475U (2012).
[Crossref]

González, S. I.

R. Conan, R. Avila, L. J. Sánchez, A. Ziad, F. Martin, J. Borgnino, O. Harris, S. I. González, R. Michel, and D. Hiriart, “Wavefront outer scale and seeing measurements at San Pedro Mártir observatory,” Astron. Astrophys. 396, 723–730 (2002).
[Crossref]

Harris, O.

R. Conan, R. Avila, L. J. Sánchez, A. Ziad, F. Martin, J. Borgnino, O. Harris, S. I. González, R. Michel, and D. Hiriart, “Wavefront outer scale and seeing measurements at San Pedro Mártir observatory,” Astron. Astrophys. 396, 723–730 (2002).
[Crossref]

Hartman, S. K.

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[Crossref]

Hiriart, D.

R. Conan, R. Avila, L. J. Sánchez, A. Ziad, F. Martin, J. Borgnino, O. Harris, S. I. González, R. Michel, and D. Hiriart, “Wavefront outer scale and seeing measurements at San Pedro Mártir observatory,” Astron. Astrophys. 396, 723–730 (2002).
[Crossref]

Hubin, N.

T. Fusco, G. Rousset, D. Rabaud, E. Gendron, D. Mouillet, F. Lacombe, G. Zins, P.-Y. Madec, A.-M. Lagrange, J. Charton, D. Rouan, N. Hubin, and N. Ageorges, “NAOS on-line characterization of turbulence parameters and adaptive optics performance,” J. Opt. A 6, 585–596 (2004).
[Crossref]

Jackson, K.

Y. H. Ono, C. M. Correia, D. R. Andersen, O. Lardière, S. Oya, M. Akiyama, K. Jackson, and C. Bradley, “Statistics of turbulence parameters at Mauna Kea using the multiple wavefront sensor data of RAVEN,” Mon. Not. R. Astron. Soc. 465, 4931–4941 (2017).
[Crossref]

Johansson, E. M.

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[Crossref]

Jolissaint, L.

L. Jolissaint, “Synthetic modeling of astronomical closed loop adaptive optics,” J. Eur. Opt. Soc. Rapid Publ. 5, 10055 (2010).
[Crossref]

L. Jolissaint, J.-P. Veran, and J. Marino, “OPERA, an automatic PSF reconstruction software for Shack-Hartmann AO systems: application to Altair,” Proc. SPIE 5490, 151–163 (2004).
[Crossref]

Kern, P.

F. Rigaut, G. Rousset, P. Kern, J.-C. Fontanella, J.-P. Gaffard, F. Merkle, and P. Léna, “Adaptive optics on a 3.6-m telescope—results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Kolb, J.

J. Kolb, N. Muller, E. Aller-Carpentier, P. Andrade, and J. Girard, “What can be retrieved from adaptive optics real-time data?” Proc. SPIE 8447, 84475U (2012).
[Crossref]

Kornilov, V.

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[Crossref]

Lacombe, F.

T. Fusco, G. Rousset, D. Rabaud, E. Gendron, D. Mouillet, F. Lacombe, G. Zins, P.-Y. Madec, A.-M. Lagrange, J. Charton, D. Rouan, N. Hubin, and N. Ageorges, “NAOS on-line characterization of turbulence parameters and adaptive optics performance,” J. Opt. A 6, 585–596 (2004).
[Crossref]

Lafon, R. E.

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[Crossref]

Lagrange, A.-M.

T. Fusco, G. Rousset, D. Rabaud, E. Gendron, D. Mouillet, F. Lacombe, G. Zins, P.-Y. Madec, A.-M. Lagrange, J. Charton, D. Rouan, N. Hubin, and N. Ageorges, “NAOS on-line characterization of turbulence parameters and adaptive optics performance,” J. Opt. A 6, 585–596 (2004).
[Crossref]

Lardière, O.

Y. H. Ono, C. M. Correia, D. R. Andersen, O. Lardière, S. Oya, M. Akiyama, K. Jackson, and C. Bradley, “Statistics of turbulence parameters at Mauna Kea using the multiple wavefront sensor data of RAVEN,” Mon. Not. R. Astron. Soc. 465, 4931–4941 (2017).
[Crossref]

Le Mignant, D.

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[Crossref]

M. Schöck, D. Le Mignant, G. A. Chanan, P. L. Wizinowich, and M. A. van Dam, “Atmospheric turbulence characterization with the Keck adaptive optics systems. I. Open-loop data,” Appl. Opt. 42, 3705–3720 (2003).
[Crossref]

Léna, P.

F. Rigaut, G. Rousset, P. Kern, J.-C. Fontanella, J.-P. Gaffard, F. Merkle, and P. Léna, “Adaptive optics on a 3.6-m telescope—results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Lu, J. R.

J. R. Lu, A. M. Ghez, S. Yelda, T. Do, W. Clarkson, N. McCrady, and M. Morris, “Recent results and perspectives for precision astrometry and photometry with adaptive optics,” Proc. SPIE 7736, 77361I (2010).
[Crossref]

Madec, P.-Y.

T. Fusco, G. Rousset, D. Rabaud, E. Gendron, D. Mouillet, F. Lacombe, G. Zins, P.-Y. Madec, A.-M. Lagrange, J. Charton, D. Rouan, N. Hubin, and N. Ageorges, “NAOS on-line characterization of turbulence parameters and adaptive optics performance,” J. Opt. A 6, 585–596 (2004).
[Crossref]

Maire, J.

A. Ziad, J. Maire, J. Borgnino, W. Dali Ali, A. Berdja, K. Ben Abdallah, F. Martin, and M. Sarazin, “MOSP: monitor of outer scale profile,” in Adaptative Optics for Extremely Large Telescopes (AO4ELT-I), 2010.

Maître, H.

Mann, D. C.

T.-J. Brennan and D. C. Mann, “Estimation of optical turbulence characteristics from Shack Hartmann wavefront sensor measurements,” Proc. SPIE 7816, 781602 (2010).
[Crossref]

Marino, J.

L. Jolissaint, J.-P. Veran, and J. Marino, “OPERA, an automatic PSF reconstruction software for Shack-Hartmann AO systems: application to Altair,” Proc. SPIE 5490, 151–163 (2004).
[Crossref]

Martin, F.

R. Conan, R. Avila, L. J. Sánchez, A. Ziad, F. Martin, J. Borgnino, O. Harris, S. I. González, R. Michel, and D. Hiriart, “Wavefront outer scale and seeing measurements at San Pedro Mártir observatory,” Astron. Astrophys. 396, 723–730 (2002).
[Crossref]

A. Ziad, J. Maire, J. Borgnino, W. Dali Ali, A. Berdja, K. Ben Abdallah, F. Martin, and M. Sarazin, “MOSP: monitor of outer scale profile,” in Adaptative Optics for Extremely Large Telescopes (AO4ELT-I), 2010.

McCrady, N.

J. R. Lu, A. M. Ghez, S. Yelda, T. Do, W. Clarkson, N. McCrady, and M. Morris, “Recent results and perspectives for precision astrometry and photometry with adaptive optics,” Proc. SPIE 7736, 77361I (2010).
[Crossref]

Merkle, F.

F. Rigaut, G. Rousset, P. Kern, J.-C. Fontanella, J.-P. Gaffard, F. Merkle, and P. Léna, “Adaptive optics on a 3.6-m telescope—results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Michel, R.

R. Conan, R. Avila, L. J. Sánchez, A. Ziad, F. Martin, J. Borgnino, O. Harris, S. I. González, R. Michel, and D. Hiriart, “Wavefront outer scale and seeing measurements at San Pedro Mártir observatory,” Astron. Astrophys. 396, 723–730 (2002).
[Crossref]

Morris, M.

J. R. Lu, A. M. Ghez, S. Yelda, T. Do, W. Clarkson, N. McCrady, and M. Morris, “Recent results and perspectives for precision astrometry and photometry with adaptive optics,” Proc. SPIE 7736, 77361I (2010).
[Crossref]

Mouillet, D.

T. Fusco, G. Rousset, D. Rabaud, E. Gendron, D. Mouillet, F. Lacombe, G. Zins, P.-Y. Madec, A.-M. Lagrange, J. Charton, D. Rouan, N. Hubin, and N. Ageorges, “NAOS on-line characterization of turbulence parameters and adaptive optics performance,” J. Opt. A 6, 585–596 (2004).
[Crossref]

Muller, N.

J. Kolb, N. Muller, E. Aller-Carpentier, P. Andrade, and J. Girard, “What can be retrieved from adaptive optics real-time data?” Proc. SPIE 8447, 84475U (2012).
[Crossref]

Noll, R. J.

Ono, Y. H.

Y. H. Ono, C. M. Correia, D. R. Andersen, O. Lardière, S. Oya, M. Akiyama, K. Jackson, and C. Bradley, “Statistics of turbulence parameters at Mauna Kea using the multiple wavefront sensor data of RAVEN,” Mon. Not. R. Astron. Soc. 465, 4931–4941 (2017).
[Crossref]

Oya, S.

Y. H. Ono, C. M. Correia, D. R. Andersen, O. Lardière, S. Oya, M. Akiyama, K. Jackson, and C. Bradley, “Statistics of turbulence parameters at Mauna Kea using the multiple wavefront sensor data of RAVEN,” Mon. Not. R. Astron. Soc. 465, 4931–4941 (2017).
[Crossref]

Potanin, S.

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[Crossref]

Rabaud, D.

T. Fusco, G. Rousset, D. Rabaud, E. Gendron, D. Mouillet, F. Lacombe, G. Zins, P.-Y. Madec, A.-M. Lagrange, J. Charton, D. Rouan, N. Hubin, and N. Ageorges, “NAOS on-line characterization of turbulence parameters and adaptive optics performance,” J. Opt. A 6, 585–596 (2004).
[Crossref]

Rigaut, F.

J.-P. Véran, F. Rigaut, H. Maître, and D. Rouan, “Estimation of the adaptive optics long-exposure point spread function using control loop data,” J. Opt. Soc. Am. A 14, 3057–3069 (1997).
[Crossref]

F. Rigaut, G. Rousset, P. Kern, J.-C. Fontanella, J.-P. Gaffard, F. Merkle, and P. Léna, “Adaptive optics on a 3.6-m telescope—results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Roddier, F.

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

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

Rouan, D.

T. Fusco, G. Rousset, D. Rabaud, E. Gendron, D. Mouillet, F. Lacombe, G. Zins, P.-Y. Madec, A.-M. Lagrange, J. Charton, D. Rouan, N. Hubin, and N. Ageorges, “NAOS on-line characterization of turbulence parameters and adaptive optics performance,” J. Opt. A 6, 585–596 (2004).
[Crossref]

J.-P. Véran, F. Rigaut, H. Maître, and D. Rouan, “Estimation of the adaptive optics long-exposure point spread function using control loop data,” J. Opt. Soc. Am. A 14, 3057–3069 (1997).
[Crossref]

Rousset, G.

T. Fusco, G. Rousset, D. Rabaud, E. Gendron, D. Mouillet, F. Lacombe, G. Zins, P.-Y. Madec, A.-M. Lagrange, J. Charton, D. Rouan, N. Hubin, and N. Ageorges, “NAOS on-line characterization of turbulence parameters and adaptive optics performance,” J. Opt. A 6, 585–596 (2004).
[Crossref]

F. Rigaut, G. Rousset, P. Kern, J.-C. Fontanella, J.-P. Gaffard, F. Merkle, and P. Léna, “Adaptive optics on a 3.6-m telescope—results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Safonov, B.

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[Crossref]

Sánchez, L. J.

R. Conan, R. Avila, L. J. Sánchez, A. Ziad, F. Martin, J. Borgnino, O. Harris, S. I. González, R. Michel, and D. Hiriart, “Wavefront outer scale and seeing measurements at San Pedro Mártir observatory,” Astron. Astrophys. 396, 723–730 (2002).
[Crossref]

Sarazin, M.

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

A. Ziad, J. Maire, J. Borgnino, W. Dali Ali, A. Berdja, K. Ben Abdallah, F. Martin, and M. Sarazin, “MOSP: monitor of outer scale profile,” in Adaptative Optics for Extremely Large Telescopes (AO4ELT-I), 2010.

Schöck, M.

Shatsky, N.

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[Crossref]

Stomski, P. J.

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[Crossref]

Summers, D. M.

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[Crossref]

Tatarski, V.

V. Tatarski, Wave Propagation in a Turbulent Medium (Dover, 1961), translated by R. A. Silverman.

Tokovinin, A.

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[Crossref]

van Dam, M. A.

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[Crossref]

M. Schöck, D. Le Mignant, G. A. Chanan, P. L. Wizinowich, and M. A. van Dam, “Atmospheric turbulence characterization with the Keck adaptive optics systems. I. Open-loop data,” Appl. Opt. 42, 3705–3720 (2003).
[Crossref]

Veran, J.-P.

L. Jolissaint, J.-P. Veran, and J. Marino, “OPERA, an automatic PSF reconstruction software for Shack-Hartmann AO systems: application to Altair,” Proc. SPIE 5490, 151–163 (2004).
[Crossref]

Véran, J.-P.

Voitsekhovich, V. V.

Voziakova, O.

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[Crossref]

Winker, D. M.

Wizinowich, P. L.

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[Crossref]

M. Schöck, D. Le Mignant, G. A. Chanan, P. L. Wizinowich, and M. A. van Dam, “Atmospheric turbulence characterization with the Keck adaptive optics systems. I. Open-loop data,” Appl. Opt. 42, 3705–3720 (2003).
[Crossref]

Yelda, S.

J. R. Lu, A. M. Ghez, S. Yelda, T. Do, W. Clarkson, N. McCrady, and M. Morris, “Recent results and perspectives for precision astrometry and photometry with adaptive optics,” Proc. SPIE 7736, 77361I (2010).
[Crossref]

Zago, L.

L. Zago, “Engineering handbook for local and dome seeing,” Proc. SPIE 2871, 726–736 (1997).
[Crossref]

Ziad, A.

R. Conan, R. Avila, L. J. Sánchez, A. Ziad, F. Martin, J. Borgnino, O. Harris, S. I. González, R. Michel, and D. Hiriart, “Wavefront outer scale and seeing measurements at San Pedro Mártir observatory,” Astron. Astrophys. 396, 723–730 (2002).
[Crossref]

A. Ziad, J. Maire, J. Borgnino, W. Dali Ali, A. Berdja, K. Ben Abdallah, F. Martin, and M. Sarazin, “MOSP: monitor of outer scale profile,” in Adaptative Optics for Extremely Large Telescopes (AO4ELT-I), 2010.

Zins, G.

T. Fusco, G. Rousset, D. Rabaud, E. Gendron, D. Mouillet, F. Lacombe, G. Zins, P.-Y. Madec, A.-M. Lagrange, J. Charton, D. Rouan, N. Hubin, and N. Ageorges, “NAOS on-line characterization of turbulence parameters and adaptive optics performance,” J. Opt. A 6, 585–596 (2004).
[Crossref]

Appl. Opt. (2)

Astron. Astrophys. (3)

R. Conan, R. Avila, L. J. Sánchez, A. Ziad, F. Martin, J. Borgnino, O. Harris, S. I. González, R. Michel, and D. Hiriart, “Wavefront outer scale and seeing measurements at San Pedro Mártir observatory,” Astron. Astrophys. 396, 723–730 (2002).
[Crossref]

F. Rigaut, G. Rousset, P. Kern, J.-C. Fontanella, J.-P. Gaffard, F. Merkle, and P. Léna, “Adaptive optics on a 3.6-m telescope—results and performance,” Astron. Astrophys. 250, 280–290 (1991).

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

J. Eur. Opt. Soc. Rapid Publ. (1)

L. Jolissaint, “Synthetic modeling of astronomical closed loop adaptive optics,” J. Eur. Opt. Soc. Rapid Publ. 5, 10055 (2010).
[Crossref]

J. Opt. A (1)

T. Fusco, G. Rousset, D. Rabaud, E. Gendron, D. Mouillet, F. Lacombe, G. Zins, P.-Y. Madec, A.-M. Lagrange, J. Charton, D. Rouan, N. Hubin, and N. Ageorges, “NAOS on-line characterization of turbulence parameters and adaptive optics performance,” J. Opt. A 6, 585–596 (2004).
[Crossref]

J. Opt. Soc. Am. (2)

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

Mon. Not. R. Astron. Soc. (2)

Y. H. Ono, C. M. Correia, D. R. Andersen, O. Lardière, S. Oya, M. Akiyama, K. Jackson, and C. Bradley, “Statistics of turbulence parameters at Mauna Kea using the multiple wavefront sensor data of RAVEN,” Mon. Not. R. Astron. Soc. 465, 4931–4941 (2017).
[Crossref]

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[Crossref]

Proc. SPIE (5)

J. R. Lu, A. M. Ghez, S. Yelda, T. Do, W. Clarkson, N. McCrady, and M. Morris, “Recent results and perspectives for precision astrometry and photometry with adaptive optics,” Proc. SPIE 7736, 77361I (2010).
[Crossref]

J. Kolb, N. Muller, E. Aller-Carpentier, P. Andrade, and J. Girard, “What can be retrieved from adaptive optics real-time data?” Proc. SPIE 8447, 84475U (2012).
[Crossref]

L. Jolissaint, J.-P. Veran, and J. Marino, “OPERA, an automatic PSF reconstruction software for Shack-Hartmann AO systems: application to Altair,” Proc. SPIE 5490, 151–163 (2004).
[Crossref]

T.-J. Brennan and D. C. Mann, “Estimation of optical turbulence characteristics from Shack Hartmann wavefront sensor measurements,” Proc. SPIE 7816, 781602 (2010).
[Crossref]

L. Zago, “Engineering handbook for local and dome seeing,” Proc. SPIE 2871, 726–736 (1997).
[Crossref]

Publ. Astron. Soc. Pac. (1)

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[Crossref]

Other (4)

R. Flicker, 2008, https://www.lao.ucolick.org/static/PsfReconstruction/psf_rev.pdf .

V. Tatarski, Wave Propagation in a Turbulent Medium (Dover, 1961), translated by R. A. Silverman.

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

A. Ziad, J. Maire, J. Borgnino, W. Dali Ali, A. Berdja, K. Ben Abdallah, F. Martin, and M. Sarazin, “MOSP: monitor of outer scale profile,” in Adaptative Optics for Extremely Large Telescopes (AO4ELT-I), 2010.

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Figures (26)

Fig. 1.
Fig. 1. Keck telescope segmented pupil, with a superposition of the DM actuator and SH-WFS lenslet layout. The blue disc is 2.65 m in diameter and shows the central obscuration. The dashed circle shows the 9 m diameter of the pupil area within which the lenslets are always fully illuminated, independently of the pupil/DM angle. Note that this overlay shows only one possible orientation of the telescope pupil relative to the AO system. Courtesy W. M. Keck Observatory.
Fig. 2.
Fig. 2. RMS map of the DM wavefront for a typical AO run, over 35,912 loop sequences at 1 kHz. The spatial resolution is 7 pixels per DM pitch length. The white circles show the limits of the 2.65 m and 9 m diameter annular pupil.
Fig. 3.
Fig. 3. Stationarity of the DM wavefront as a function of the pupil radius. The dotted curves show each of the 79 cases for the night of August 1, 2013. The red curve shows the average of all cases.
Fig. 4.
Fig. 4. Effect of the projection and numerical sampling errors on the Zernike polynomial norm (here 1 ), up to j = 100 . The black lines show the difference between the norm computed numerically on the true Zernike polynomials and the theoretical value (which is 1). The different lines are for increased numerical resolution, defined as the number of pixels per actuator pitch. The higher the resolution, the smaller the error, in principle. There are 16 actuator pitches across the pupil in this example. The red lines show the same, but for the Zernike polynomial projection in the DM influence function basis. There is no central obscuration in this example.
Fig. 5.
Fig. 5. Zernike coefficient variances when including a central obscuration of 34% in the pupil. The example is voluntarily extreme to show the effect on the polynomials for which m = 0 and m = 1 . There is no DM projection error here. See also Fig. 1 of Véran et al. [2], showing the strong effect of the large central obscuration of the CFH telescope.
Fig. 6.
Fig. 6. Empirical model of the Xinetics Inc. deformable mirror influence function.
Fig. 7.
Fig. 7. Modified Zernike mode coefficient variances for the 9/2.65 m Keck annular pupil, for a D / r 0 = 1 and an infinite outer scale. The red line shows the true and unobstructed Zernike mode variances computed from Noll [13].
Fig. 8.
Fig. 8. Damping of the Zernike coefficient variances as a function of D / L 0 and radial order n . Diamonds show the exact computation of the damping factor (courtesy D. Winker), and the cont/dashed lines are our empirical model.
Fig. 9.
Fig. 9. Expected variance attenuation for the 9 m Keck telescope seeing estimation pupil considering an outer scale of 13 m, equal to the average value measured using the DM seeing monitor.
Fig. 10.
Fig. 10. Temporal PSD of the command to the actuator # 107. The red line shows the convergence to the white noise plateau. With a guide star magnitude 11 in R-band for this example, the signal noise is moderate. The t-PSD has been smoothed in this figure.
Fig. 11.
Fig. 11. Block diagram of the classical Keck AO system, based on a single Shack–Hartmann wavefront sensor.
Fig. 12.
Fig. 12. Red dots show raw DM command t-PSD for actuator # 107. Guide star magnitude 14.5 (R-band). Black dots: after compensating with the transfer function | W / H CL | 2 . The red line shows the estimated white noise t-PSD level.
Fig. 13.
Fig. 13. Unfiltered DM command white noise RMS as a function of the number of photons N γ per WFS integration time, for actuator number 107. The line shows the expected behavior in 1 / N γ .
Fig. 14.
Fig. 14. Top, Zernike coefficient variance for seeing and WFS aliasing. Bottom, relative excess of Zernike coefficient variance due to aliasing for the Keck AO WFS.
Fig. 15.
Fig. 15. NGS mode. Top figure: average of the variances of the modified Zernike coefficients calculated from the DM actuator commands, with 1 - σ bars, in red. In green, model prediction of the variances for the good modes. For each AO run, and the model, we have normalized the variances by the ratio ( D / r 0 ) 5 / 3 and removed the effect of the outer scale by dividing by the Winker’s factor, using the value of L 0 assessed from the data. In black, the same for the modes m = 0 (first sets of points above the green line) and m = 1 (top series of points). Bottom: a zoom in the region j = 70 120 .
Fig. 16.
Fig. 16. Example of model fitting to telemetry data. Case August 1st, 2013, data set 0004. Seeing was 0.44 ± 0.03 ( 1 - σ ) and the outer scale 12 m. Only the good modes were used for the model fit.
Fig. 17.
Fig. 17. LGS mode. Top figure: average of the variances of the modified Zernike coefficient variances calculated from the DM actuator commands, with 1 - σ bars, in red. In green, model prediction of the variances for the good modes. For each AO run and the model, we have normalized the variances by the ratio ( D / r 0 ) 5 / 3 and removed the effect of the outer scale by dividing by the Winker’s factor using the value of L 0 assessed from the data. In black, the same for the modes m = 0 (first sets of points above the green line) and m = 1 (top series of points). Bottom: a zoom in the σ j 2 region around 10 4    rad 2 showing an unexpected oscillation of the variances for the even azimuthal modes.
Fig. 18.
Fig. 18. Relative uncertainty of the seeing angle determination as a function of the variance of the noise for modified Zernike coefficient a 9 (considered as a proxy for the noise variance). In black, NGS cases; in red, LGS case.
Fig. 19.
Fig. 19. Histogram of the seeing measurements in NGS and LGS modes for the 841 experiments. Median seeing 0.62, mean seeing 0.70, standard deviation 0.3.
Fig. 20.
Fig. 20. Histogram of the outer-scale measurements in NGS and LGS modes for the 841 experiments. Median 12 m, mean 12.7 m, standard deviation 6 m.
Fig. 21.
Fig. 21. Closed-/open-loop seeing experiment. June 22/23, 2011 (top/bottom). Plus signs, DM-seeing; red squares, PSF seeing. The continuous lines show the range of values within which the PSF seeing is expected, if the PSF seeing is the same as the DM-based seeing. The dashed vertical line indicates an interruption in the data acquisition for about 10 min.
Fig. 22.
Fig. 22. DM seeing and PSF seeing for the two nights all together, plus five data points taken on October 13, 2011. The DM seeing values are an interpolation at the time of the PSF seeing acquisition. The dashed lines show a range of ± 10 % from the PSF seeing reference.
Fig. 23.
Fig. 23. Average seeing during the same period of time as the AO runs, for nine nights. Dates and seeing values are given in Table 3. MASS data is missing for night # 5.
Fig. 24.
Fig. 24. Comparison of the DIMM, MASS, and DM seeing for the night of June 22/23, 2011, when correlation of the night average was at its highest amongst our data sets.
Fig. 25.
Fig. 25. Comparison of the DIMM, MASS, and DM seeing for the night of September 14/15, 2013, when correlation of the night average was at its lowest.
Fig. 26.
Fig. 26. Comparison of the DIMM, MASS, OL PSF, and DM seeing. The DIMM/MASS seeing was interpolated at the time of the DM seeing acquisition.

Tables (4)

Tables Icon

Table 1. Coefficients of the L 0 Variance Attenuation Model

Tables Icon

Table 2. Statistics of the DM- and PSF-Based Seeing Measurements for June 22 and 23, 2011

Tables Icon

Table 3. Mean Seeing Measured with the DM, PSF, DIMM, and MASS for the Nine Dates Indicateda

Tables Icon

Table 4. Average Dispersion Between DM Seeing and the Other Sensorsa

Equations (28)

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

r 0 = 1.6748 ( λ 2 π ) 6 / 5 [ 0 C N 2 ( h ) d h ] 3 / 5 ,
w 0 = 0.98 λ / r 0 [ rad ] .
z k z l = ( D / r 0 ) 5 / 3 N k , l W k , l ( D / L 0 ) ,
z · z t = M a z a · a t M a z t ,
Z j = Z ^ j + e j = i = 1 N a a ^ j , i A i + e j ,
ϕ DM = j = 1 N z z ^ j Z ^ j .
for each Z ^ j a ^ j = Γ 1 b j ,
where    { b j , i = P Z ^ j A i d 2 r Γ k , l = P A k A l d 2 r ,
I ( x , y ) = exp ( p 4 ρ ) ( 1 + p 5 ρ ) M , ρ = p 6 ε ( | x | ε + | y | ε ) , ε = ( p 0 + p 1 r 2 + p 2 r 4 + p 3 r 6 ) M , M = exp ( p 7 r 8 ) ,
p 0 7 = [ 2.24506 2.042278 × 10 1 1.921478 × 10 2 1.070762 × 10 3 7.693368 × 10 1 3.979560 × 10 1 1.802674 × 10 1 1.672743 × 10 4 ] .
W j , j ( x ) = exp { n [ A ( x ) ln n + B ( x ) ] exp C ( x ) } 1 + 13.6334 ( x / n ) 2 ,
A ( x ) = i = 0 5 α i ln i ( x ) , B ( x ) = i = 0 5 β i ln i ( x ) , C ( x ) = i = 0 5 γ i ln i ( x ) ,
m k = a , k + n k + a , k ,
a · a t = m · m t n · n t a · a t a · a t a · a t .
W ( ν ) = sinc ( ν / ν s ) exp ( i π ν / ν s ) .
R ( ν ) = exp ( 2 π i Δ t ν ) ,
C ( z ) = a 0 + a 1 z 1 + a 2 z 2 + a 3 z 3 1 + b 1 z 1 + b 2 z 2 + b 3 z 3 ,
m ˜ k = H CL a ˜ , k + H CL W R 1 w n ˜ k ,
n ˜ k = H CL W R 1 w n ˜ k .
H CL = W R 1 R 2 R 3 C 1 + W R 1 R 2 R 3 C .
m ˜ k m ˜ l = | H CL | 2 a ˜ , k a ˜ , l + | H CL | 2 W R 1 ( a ˜ , k w n ˜ l + a ˜ , l w n ˜ k ) + | H CL W R 1 | 2 w n ˜ k w n ˜ l .
m ˜ k m ˜ l = | H CL | 2 a ˜ , k a ˜ , l + | H CL W | 2 w n ˜ k w n ˜ l .
a ˜ , k a ˜ , l | W | 2 + w n ˜ k w n ˜ l .
n k n l 1 T t n k ( t ) n l ( t ) d t = 1 T ν n ˜ k ( ν ) n ˜ l ( ν ) d ν = w n ˜ k w n ˜ l ν | H CL ( ν ) W ( ν ) | 2 d ν .
alias cov. = a · a t + a · a t + a · a t .
η n = p 0 + p 1 n + p 2 n 2 + p 3 n 3 + p 4 n 4 ,
p = [ 0.998935 , 7.706682 × 10 4 , 1.823336 × 10 4 , 1.903760 × 10 5 , 2.762067 × 10 7 ] .
σ ( r 0 ) = 0.6 D ( D / r 0 ) 5 / 3 8 / 5 σ [ ( D / r 0 ) 5 / 3 ] ;

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