Abstract

We address the problem of highly accurate phase estimation at low light levels, as required by the Space Interferometry Mission (SIM). The most stringent SIM requirement in this regard is that the average phase error over a 30-s integration time correspond to a path-length error of approximately 30 pm. Most conventional phase-estimation algorithms exhibit significant enough bias at the signal levels at which the SIM will be operating so that some correction is necessary. Several algorithms are analyzed, and methods of compensating for their bias are developed. Another source of error in phase estimation occurs because the phase is not constant over the integration period. Errors that are due to spacecraft motion, the motion of compensating optical elements, and modulation errors are analyzed and simulated. A Kalman smoothing approach for compensating for these errors is introduced.

© 2002 Optical Society of America

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References

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  1. R. Danner, S. Unwin, The Space Interferometry Mission, Taking the Measure of the Universe (Jet Propulsion Laboratory, Pasadena, Calif., 1999) Pub. 400-811.
  2. These data are available at http://sim.jpl.nasa.gov .
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2001 (1)

1999 (1)

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

1998 (1)

1995 (3)

1994 (2)

G. S. Han, S. W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33, 7321–7325 (1994).
[CrossRef] [PubMed]

A. Quirrenbach, D. Mozurkewich, D. F. Buscher, C. A. Hummel, J. T. Armstrong, “Phase-referenced visibility averaging in optical long-baseline interferometry,” Astron. Astrophys. 286, 1019–1027 (1994).

1991 (1)

1988 (1)

1984 (1)

J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

1975 (1)

Armstrong, J. T.

A. Quirrenbach, D. Mozurkewich, D. F. Buscher, C. A. Hummel, J. T. Armstrong, “Phase-referenced visibility averaging in optical long-baseline interferometry,” Astron. Astrophys. 286, 1019–1027 (1994).

Azevedo, L. S.

A. C. Kuhnert, S. B. Shaklan, J. Shen, A. E. Carlson, L. S. Azevedo, “First tests of the interferometer in the micro-arcsecond metrology testbed (MAM),” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 815–827 (2000).
[CrossRef]

Basinger, S.

M. Milman, S. Basinger, “A comparison of dispersed fringe detection techniques for measuring the group delay in an astronomical interferometer I: monochromatic light,” JPL SIM Eng. Memo (June2000).

Boden, A. F.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

Burrow, R.

Buscher, D. F.

A. Quirrenbach, D. Mozurkewich, D. F. Buscher, C. A. Hummel, J. T. Armstrong, “Phase-referenced visibility averaging in optical long-baseline interferometry,” Astron. Astrophys. 286, 1019–1027 (1994).

Carlson, A. E.

A. C. Kuhnert, S. B. Shaklan, J. Shen, A. E. Carlson, L. S. Azevedo, “First tests of the interferometer in the micro-arcsecond metrology testbed (MAM),” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 815–827 (2000).
[CrossRef]

Coggrave, C. R.

Colavita, M.

P. R. Lawson, M. Colavita, P. J. Dumont, B. F. Lane, “Least-squares estimation and group delay in astrometric interferometers,” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 397–406 (2000).
[CrossRef]

Colavita, M. M.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics XXVI, E. Wolf, ed. (Elsever Science, Amsterdam, 1988), pp. 350–392.

Danner, R.

R. Danner, S. Unwin, The Space Interferometry Mission, Taking the Measure of the Universe (Jet Propulsion Laboratory, Pasadena, Calif., 1999) Pub. 400-811.

de Groot, P. J.

Dumont, P. J.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

P. R. Lawson, M. Colavita, P. J. Dumont, B. F. Lane, “Least-squares estimation and group delay in astrometric interferometers,” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 397–406 (2000).
[CrossRef]

Elsner, K.-E.

Frankena, H. J.

Goodwin, G. C.

G. C. Goodwin, K. S. Sin, Adaptive Filtering and Control (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Grievenkamp, J. E.

J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

Grzana, J.

Gubler, J.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

Gursel, Y.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

Han, G. S.

Hines, B. E.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

Hummel, C. A.

A. Quirrenbach, D. Mozurkewich, D. F. Buscher, C. A. Hummel, J. T. Armstrong, “Phase-referenced visibility averaging in optical long-baseline interferometry,” Astron. Astrophys. 286, 1019–1027 (1994).

Huntley, J. M.

Kaufmann, G. H.

Kim, S. W.

Kinnstaetter, K.

Koresko, C. D.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

Kuhnert, A. C.

A. C. Kuhnert, S. B. Shaklan, J. Shen, A. E. Carlson, L. S. Azevedo, “First tests of the interferometer in the micro-arcsecond metrology testbed (MAM),” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 815–827 (2000).
[CrossRef]

Kulkarni, S. R.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

Lane, B. F.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

P. R. Lawson, M. Colavita, P. J. Dumont, B. F. Lane, “Least-squares estimation and group delay in astrometric interferometers,” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 397–406 (2000).
[CrossRef]

Lawson, P. R.

P. R. Lawson, M. Colavita, P. J. Dumont, B. F. Lane, “Least-squares estimation and group delay in astrometric interferometers,” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 397–406 (2000).
[CrossRef]

Linfield, R.

R. Linfield, Jet Propulsion Laboratory Pasadena, Calif. 91109 (personal communication, January2000).

Lohmann, A. W.

Malbet, F.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

Merkel, K.

Milman, M.

M. Milman, S. Basinger, “A comparison of dispersed fringe detection techniques for measuring the group delay in an astronomical interferometer I: monochromatic light,” JPL SIM Eng. Memo (June2000).

Mobley, D. W.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

Mozurkewich, D.

A. Quirrenbach, D. Mozurkewich, D. F. Buscher, C. A. Hummel, J. T. Armstrong, “Phase-referenced visibility averaging in optical long-baseline interferometry,” Astron. Astrophys. 286, 1019–1027 (1994).

Palmer, D. L.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

Pan, X. P.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

Quirrenbach, A.

A. Quirrenbach, D. Mozurkewich, D. F. Buscher, C. A. Hummel, J. T. Armstrong, “Phase-referenced visibility averaging in optical long-baseline interferometry,” Astron. Astrophys. 286, 1019–1027 (1994).

Ruiz, P. D.

Schwider, J.

Shaklan, S. B.

A. C. Kuhnert, S. B. Shaklan, J. Shen, A. E. Carlson, L. S. Azevedo, “First tests of the interferometer in the micro-arcsecond metrology testbed (MAM),” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 815–827 (2000).
[CrossRef]

Shao, M.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

Shen, J.

A. C. Kuhnert, S. B. Shaklan, J. Shen, A. E. Carlson, L. S. Azevedo, “First tests of the interferometer in the micro-arcsecond metrology testbed (MAM),” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 815–827 (2000).
[CrossRef]

Shen, Y.

Sin, K. S.

G. C. Goodwin, K. S. Sin, Adaptive Filtering and Control (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Smorenburg, C.

Spolaczyk, R.

Streibl, N.

Surrel, Y.

B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

Unwin, S.

R. Danner, S. Unwin, The Space Interferometry Mission, Taking the Measure of the Universe (Jet Propulsion Laboratory, Pasadena, Calif., 1999) Pub. 400-811.

van Belle, G. T.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

Van Wingerden, J.

Wallace, J. K.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

Wyant, J.

Yu, J. W.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

Zhao, B.

B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

Appl. Opt. (7)

Astron. Astrophys. (1)

A. Quirrenbach, D. Mozurkewich, D. F. Buscher, C. A. Hummel, J. T. Armstrong, “Phase-referenced visibility averaging in optical long-baseline interferometry,” Astron. Astrophys. 286, 1019–1027 (1994).

Astrophys. J. (1)

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar testbed interferometer,” Astrophys. J. 510, 505–520 (1999).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Eng. (2)

B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

Other (8)

A. C. Kuhnert, S. B. Shaklan, J. Shen, A. E. Carlson, L. S. Azevedo, “First tests of the interferometer in the micro-arcsecond metrology testbed (MAM),” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 815–827 (2000).
[CrossRef]

M. Milman, S. Basinger, “A comparison of dispersed fringe detection techniques for measuring the group delay in an astronomical interferometer I: monochromatic light,” JPL SIM Eng. Memo (June2000).

G. C. Goodwin, K. S. Sin, Adaptive Filtering and Control (Prentice-Hall, Englewood Cliffs, N.J., 1984).

P. R. Lawson, M. Colavita, P. J. Dumont, B. F. Lane, “Least-squares estimation and group delay in astrometric interferometers,” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 397–406 (2000).
[CrossRef]

R. Linfield, Jet Propulsion Laboratory Pasadena, Calif. 91109 (personal communication, January2000).

R. Danner, S. Unwin, The Space Interferometry Mission, Taking the Measure of the Universe (Jet Propulsion Laboratory, Pasadena, Calif., 1999) Pub. 400-811.

These data are available at http://sim.jpl.nasa.gov .

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics XXVI, E. Wolf, ed. (Elsever Science, Amsterdam, 1988), pp. 350–392.

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Figures (8)

Fig. 1
Fig. 1

Typical modulated intensity pattern.

Fig. 2
Fig. 2

Comparison of four-bucket algorithms: with and without bias correction.

Fig. 3
Fig. 3

(a) Comparison of rms shot-noise error for four algorithms (matched stroke and wavelength). (b) Comparison of rms shot-noise error for four algorithms (unmatched stroke and wavelength).

Fig. 4
Fig. 4

Realization of random phase over a 1-s period.

Fig. 5
Fig. 5

Error in estimation of phase that is due to a constant nonzero phase velocity.

Fig. 6
Fig. 6

Realization of 8-s delay profile for Monte Carlo simulations of the Kalman smoothing algorithm.

Fig. 7
Fig. 7

Comparison of mean values of errors between the least-squares method and the Kalman smoothing method at measurement time steps.

Fig. 8
Fig. 8

Comparison of rms errors between the least-squares method and the Kalman smoothing method at measurement time steps.

Tables (2)

Tables Icon

Table 1 Simulation Results for Four Different Bias-Correcting Algorithmsa

Tables Icon

Table 2 Simulation Results for Bias-Correcting Algorithms with Time-Varying Phasea

Equations (117)

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

dt=bt·s+νt,
d=s·b+ν,
d=s·b*+ν+s·δb,
I=I01+V coskx+ϕ,
Ii=I01+V coskxi+ϕ
FiI0, V, ϕ=I01+V coskxi+ϕ,
mini,V,ϕ  |FiI0, V, ϕ-Ii|2.
Ii=I01+Vcoskxicosϕ-sinkxisinϕ,
I1IN=1coskx1-sinkx11coskxN-sinkxNI0I0V cosϕI0V sinϕ.
ϕˆ=arctanxˆ3/xˆ2.
ϕˆ=arctanB-DA-C-π4.
ϕˆ=arctanX+YY-X.
ϕˆ=arctanγ X+YY-X,
γ=2 sins/4-sins/21-coss/2
ϕˆ=arctanΣh1jIjΣh2jIj,
ϕ=arctanΣEh1jIjΣEh2jIj.
Eϕˆ=   arctanΣh1jxjΣh2jxjp1x1  pNxN×dx1  dxN.
Eϕˆ=ϕ+12     2xi2arctanΣh1jxjΣh2jxj×xj-x¯j2p1x1  pNxNdx1  dxN=ϕ+12 2xi2arctanΣh1jxjΣh2jxjσxi2,
b=2σread2+Ā+C¯×2γ1-γ2Y¯3-4γX¯Y¯2Y¯-1+γ2X¯Y¯-X¯2+γ2X¯+Y¯22Y¯-X¯+2σread2+B¯+D¯×2γ1-γ2X¯3-4γX¯Y¯2X¯-1+γ2Y¯Y¯-X¯2+γ2X¯+Y¯22Y¯-X¯,
bϕˆ, N=2σread2+A+C×2γ1-γ2Y3-4γXY2Y-1+γ2XY-X2+γ2X+Y22Y-X+2σread2+B+D×2γ1-γ2X3-4γXY2X-1+γ2YY-X2+γ2X+Y22Y-X.
eϕ, N=bϕ, N¯-bϕˆ, N.
ϕx=arctanx3/x2.
Eϕ|I1,, IN= ϕx1, x2, x3px1, x2, x3|I× dx1dx2dx3,
ϕˆoptarctanx¯3x¯2+12i,j=2,32ϕxixj  xi-x¯i×xj-x¯jpx1, x2, x3|Idx1dx2dx3,
ϕˆoptarctanxˆ3xˆ2+p22-p33xˆ2xˆ3xˆ22+xˆ322-p23xˆ22-xˆ32xˆ22+xˆ322.
ϕ¯=1Ni=1N ϕi, sinϕ¯=1Ni=1Nsinϕi,cosϕ¯=1Ni=1Ncosϕi.
ϕ¯=arctansinϕ¯cosϕ¯+16Ncos2ϕ¯i=1N δϕi3+O|δϕi|4,
EXi=κ sinϕi, EYi=κ cosϕi,
ϕ¯=arctanE Σi=1N XiE Σi=1N Yi+cos2ϕ¯6N  δϕi3.
ϕ-ϕˆ=arctanx3/x2-arctanxˆ3/xˆ2,
ϕ-ϕˆ=-sinϕx2-xˆ2+cosϕx3-xˆ3I0V.
E|ϕ-ϕˆ|2=p22 cos2ϕ+p33 sin2ϕ-p23 sin2ϕI02V2,
E|ϕˆ-ϕ|2=  arctanγ x+yy-x-ϕ2×pXxpYydxdy,
ϕ=arctanγ x¯+y¯y¯-x¯,
E|ϕˆ-ϕ|2=fx2σX2+fy2σY2.
E|ϕˆ-ϕ|2=π24NV2+π2σread2N2V2,
1000
Ii=ui-Δ/2ui+Δ/2 I01+V cosu+ϕudu,
Ii=I0Δ+2V sinΔ/2cosui+ϕ.
I1IN=Δcosu1-sinu1ΔcosuN-sinuNI0I0V˜ cosϕI0V˜ sinϕ,
I=Ax.
 ui-Δ/2ui+Δ/2 δϕudu=0.
Ii=ui-Δ/2ui+Δ/2 I01+V cosu+ϕ¯+δϕudu=I0Δ+2V sinΔ/2cosui+ϕ¯-I0V cosϕ¯ui-Δ/2ui+Δ/2sinuδϕudu-I0V sinϕ¯ui-Δ/2ui+Δ/2cosuδϕudu.
B=12 sinΔ/20ui-Δ/2ui+Δ/2sinuδϕuduui-Δ/2ui+Δ/2cosuδϕudu.
I=Ax-Bx.
xˆ=ATQ-1A-1ATQ-1I.
x=ATQ-1A-1ATQ-1I+Bx,
e=ATQ-1A-1ATQ-1Bx.
ϕx=tan-1x3/x2,
ϕx-ϕxˆ=ϕxx-xˆ=ϕxe=-x3x22+x32 e2+x2x22+x32 e3=-sinϕ¯e2I0V˜+cosϕ¯e3I0V˜.
e2=I0V˜Ncosϕ¯  δϕi sin2ui+sinϕ¯  δϕi cos2ui,
e3=I0V˜Ncosϕ¯  δϕi cos2ui-sinϕ¯  δϕi sin2ui,
ϕx-ϕxˆ=1Ncos2ϕ¯  δϕi cos2ui-sin2ϕ¯  δϕi sin2ui.
δϕi=κ cos2ui,
ϕx-ϕxˆ=1Ncos2ϕ¯C2-sin2ϕ¯S2δϕ,
C2=cos2u1, cos2u2,, cos2uN,S2=sin2u1,, sin2uN.
E|ϕx-ϕxˆ|2=σ2N2 trC2C2T+S2S2T
=σ2N.
ϕx-ϕxˆ=12N sinΔ/2i=1N-sin2ϕ¯×ui-Δ/2ui+Δ/2cosui-usin2u+sinui-ucos2uδϕudu-cos2ϕ¯ui-Δ/2ui+Δ/2cosui-ucos2u-sinui-usin2uδϕudu-ui-Δ/2ui+Δ/2cosui-uδϕudu.
ϕx-ϕxˆ12N sinΔ/2-sin2ϕ¯×02πsin2uδϕudu+cos2ϕ¯02πcos2uδϕudu.
1/8
δϕu=δϕ0+δϕ0u+12δϕ0u2.
ϕx-ϕxˆπ2N sinΔ/2cos2ϕ¯δϕ02-sin2ϕ¯δϕ0.
Zi+1=AZi+Bηi,
yi=CiZi+νi,
A=10000010Δ00010Δ0001000001, Ci=1 coskxi-sinkxi 0 0.
ϕsmooth=arctanZ3smoothZ2smooth.
ϕ˙smooth=1Z2smooth2+Z3smooth2×-Z3smoothZ4smooth+Z2smoothZ5smooth.
ϕ¯=12ϕsmoothti+ϕsmoothti+1+Δ12ϕ˙smoothti-ϕ˙smoothti+1.
dk+1=.985dk+.2ηk+.2,
d=λi2π ϕi+niλi, i=1,, r,
d=λi2π ϕi.
ϕ¯=1Ni=1N ϕi, sinϕ¯=1Ni=1Nsinϕi,cosϕ¯=1Ni=1Ncosϕi.
ϕ¯=arctansinϕ¯cosϕ¯+16Ncos2ϕ¯i=1N δϕi3+O|δϕi|4,
ϕ*=arctansinϕ¯cosϕ¯,
δs=sinϕ¯-sinϕ¯
=sinϕ¯-1N  sinϕ¯+δϕi
=sinϕ¯-1N  sinϕ¯cosδϕi+sinδϕicosϕ¯
=12Nsinϕ¯ δϕi2+cosϕ¯6N δϕi3.
δc=12Ncosϕ¯  δϕi2+sinϕ¯6N  δϕi3.
ϕ¯-ϕ*=xarctanxyx=sinϕ¯, y=cosϕ¯δs+yarctanxyx=sinϕ¯, y=cosϕ¯δy+O|δc|2, |δs|2, |δc| |δs|.
ϕ¯-ϕ*=cosϕ¯δs-sinϕ¯δc
=16Ncos2ϕ¯ δϕ3.
EXi=κ sinϕi, EYi=κ cosϕi,
ϕ¯=arctanE Σi=1N XiE Σi=1N Yi+cos2ϕ¯6N δϕi3.
XM=1Mi=1M Xi, XˆM=1Mi=1M Xˆi,
YM=1Mi=1M Yi, ŶM=1Mi=1M Ŷi.
ϕ¯*=arctanXˆMŶM.
PiXX=E|Xi-Xˆi|2, PiYY=E|Yi-Ŷi|2,iXY=EXi-XˆiYi-Ŷi.
ϕ¯-ϕ¯*=XNarctanXM-XˆM+YNarctanYM-ŶM
=1DYMXM-XˆM+XMYM-ŶM,
E|ϕ¯-ϕ¯*|21D2YM2E|XM-XˆM|2+XM2E|YM-ŶM|2+XMYMEXM-XˆMYM-ŶM.
E|XM-XˆM|2=1M2i=1M PiXX, E|YM-ŶM|2=1M2i=1M PiYY,
EXM-xˆMYM-ŶM=1M2i=1M PiXY,
E|ϕ¯-ϕ¯*|21D2YM2MP¯XX+XM2MP¯YY+2XMYMMP¯XY,
E|ϕ¯-ϕ¯*|21MI02V2cosϕ¯2P¯XX+sinϕ¯2P¯YY+sin2ϕ¯P¯XY.
cosui=cosiΔ-Δ/2
=cosiΔcosΔ/2+siniΔsinΔ/2,
sinui=siniΔcosΔ/2-cosiΔsinΔ/2,
A=ΔcosΔcosΔ/2+sinΔsinΔ/2-sinΔcosΔ/2-cosΔsinΔ/2ΔcosNΔcosΔ/2+sinNΔsinΔ/2-sinNΔcosΔ/2-cosNΔsinΔ/2.
k=1NcoskΔ=k=1NsinkΔ=0,
k=1NcoskΔsinkΔ=0,
k=1Ncos2kΔ=k=1Nsin2kΔ=N/2.
ATA=diagNΔ2, N/2, N/2.
ATBx=I0V˜×ΔΣcosϕ¯sinui+sinϕ¯cosuiδϕi12cosϕ¯Σδϕi sin2ui+12sinϕ¯Σδϕi cos2ui12cosϕ¯Σδϕi cos2ui-12sinϕ¯Σδϕi sin2ui.
csu=i=1N χui-Δ/2, ui+Δ/2ucosuisinu,
ccu=i=1N χui-Δ/2, ui+Δ/2ucosuicosu,
scu=i=1N χui-Δ/2, ui+Δ/2usinuicosu,
ssu=i=1N χui-Δ/2, ui+Δ/2usinuisinu,
e2=I0V˜N sinΔ/2cosϕ¯02π csuδϕudu+sinϕ¯02π ccuδϕudu,
e3=-I0V˜N sinΔ/2cosϕ¯02π ssuδϕudu+sinϕ¯02π scuδϕudu.
ssu=12sin2usinui-u+sin2ucosui-u;uui-Δ/2, ui+Δ/2,
ccu=-12sin2usinui-u+cos2ucosui-u;uui-Δ/2, ui+Δ/2,
csu+scu=sin2ucosui-u+cos2usinui-u;uui-Δ/2, ui+Δ/2.
ϕx-ϕxˆ=1I0V˜-sinϕ¯e2+cosϕ¯e3) =12N sinΔ/2i=1N-sin2ϕ¯ ×ui-Δ/2ui+Δ/2cosui-usin2u+sinui-ucos2uδϕudu+ui-Δ/2ui+Δ/2sin2ϕ¯sin2usinui-u-cos2ϕ¯sin2usinui-uδϕudu-ui-Δ/2ui+Δ/2cos2ϕ¯sin2ucosui-u+sin2ϕ¯cos2ucosui-uδϕudu.
ϕx-ϕxˆ=12N sinΔ/2i=1N-sin2ϕ¯×ui-Δ/2ui+Δ/2cosui-usin2u+sinui-ucos2uδϕudu-cos2ϕ¯ui-Δ/2ui+Δ/2sinui-u×sin2uδϕu+cos2ϕ¯×ui-Δ/2ui+Δ/2cosui-ucos2uδϕu-ui-Δ/2ui+Δ/2cosui-uδϕudu.
ϕx-ϕxˆ=12N sinΔ/2i=1N-sin2ϕ¯×ui-Δ/2ui+Δ/2cosui-usin2u+sinui-ucos2uδϕudu-cos2ϕ¯ui-Δ/2ui+Δ/2cosui-ucos2u-sinui-usin2uδϕudu-ui-Δ/2ui+Δ/2cosui-uδϕudu.

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