Abstract

A technique for recovering doppler line profiles from Fabry-Perot interferometer fringes of very low intensity is described. The technique is based on a fourier decomposition of the data and a subsequent nonlinear least squares fit of the low order fourier coefficients to the fourier decomposition of an ideal instrument function. The ideal instrument function is expressed by the convolution of various instrument broadening functions and includes a parametric representation of the actual instrument. The method for recovering doppler temperature, emission line intensity, and mass motion of the emitting molecules is described. A theoretical analysis of errors for doppler temperature and emission line intensity is made for a statistical noise distribution superimposed upon a fringe profile of very low intensity. These errors are related to the emission line intensity, number of data points per fringe, background continuum level, and instrument parameters. As a specific example, the errors in retrieving the doppler temperature from the 6300-Å atomic oxygen emission line OI(1D3P) in the nightglow are determined for the 15-cm Fabry-Perot interferometer at the University of Michigan Airglow Observatory.

© 1971 Optical Society of America

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References

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    [CrossRef]
  2. D. Q. Wark, J. M. Stone, Nature 175, 254 (1955).
    [CrossRef]
  3. J. Cabannes, D. Dufay, in The Airglow and the Aurorae (Pergamon Press, New York, 1956).
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  6. R. V. Karandikar, in The Airglow and the Aurorae (Pergamon Press, New York, 1956), p. 374.
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    [CrossRef]
  8. J. A. Nilson, G. G. Shepherd, Planet. Space Sci. 5, 299 (1961).
    [CrossRef]
  9. E. C. Turgeon, G. G. Shepherd, Planet. Space Sci. 9, 925 (1962).
    [CrossRef]
  10. G. J. Hernandez, J. P. Turtle, Planet. Space Sci. 13, 901 (1965).
    [CrossRef]
  11. G. G. Shepherd, C. W. Lake, J. R. Miiler, L. L. Cogger, Appl. Opt. 4, 267 (1965).
    [CrossRef]
  12. G. J. Hernandez, J. P. Turtle, Aurora and Airglow (ReinholdNew York, 1967).
  13. E. B. Armstrong, Planet. Space Sci. 16, 211 (1968).
    [CrossRef]
  14. H. H. Zwick, G. G. Shepherd, Can. J. Phys. 41, 343 (1963).
    [CrossRef]
  15. L. Vegard, Phil. Mag. 24, 588 (1937).
  16. T. M. Mulyarchick, Izv. Akad. Nauk. S.S.R. 3, 449 (1960).
  17. D. Q. Wark, Astrophys. J. 131, 491 (1960).
    [CrossRef]
  18. A. H. Jarret, M. J. Hoey, L. Paffrath, Planet. Space Sci. 12, 591 (1964).
    [CrossRef]
  19. A. H. Jarrett, M. J. Hoey, J. Atmos. Terrest. Phys. 28, 175 (1966).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. R. G. Roble, A Theoretical and Experimental Study of the Stable Midlatitude Red Arc (SAR-arc) Ph.D. thesis, University of Michigan (1969).
  23. E. B. Armstrong, Planet. Space Sci. 17, 957 (1969).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  26. R. G. Roble, P. B. Hays, A. F. Nagy, Planet. Space Sci. 16, 1109 (1968).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  32. K. Krebs, A. Sauer, Ann. Phys. 13, 359 (1953).
    [CrossRef]
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1970

R. G. Roble, P. B. Hays, A. F. Nagy, Planet. Space Sci. 18, 431 (1970).
[CrossRef]

1969

E. B. Armstrong, Planet. Space Sci. 17, 957 (1969).
[CrossRef]

P. B. Hays, A. F. Nagy, R. G. Roble, J. Geophys. Res. 74, 4162 (1969).
[CrossRef]

1968

E. B. Armstrong, Planet. Space Sci. 16, 211 (1968).
[CrossRef]

R. G. Roble, P. B. Hays, A. F. Nagy, Planet. Space Sci. 16, 1109 (1968).
[CrossRef]

M. A. Biondi, W. A. Feibelman, Planet Space Sci. 16, 431 (1968).
[CrossRef]

1967

1966

1965

1964

A. H. Jarret, M. J. Hoey, L. Paffrath, Planet. Space Sci. 12, 591 (1964).
[CrossRef]

1963

H. H. Zwick, G. G. Shepherd, Can. J. Phys. 41, 343 (1963).
[CrossRef]

F. Bayer-Helms, Z. Angew. Phys. 15, 330 (1963).

1962

E. C. Turgeon, G. G. Shepherd, Planet. Space Sci. 9, 925 (1962).
[CrossRef]

1961

J. A. Nilson, G. G. Shepherd, Planet. Space Sci. 5, 299 (1961).
[CrossRef]

1960

T. M. Mulyarchick, Izv. Akad. Nauk. S.S.R. 3, 449 (1960).

D. Q. Wark, Astrophys. J. 131, 491 (1960).
[CrossRef]

1958

E. B. Armstrong, J. Phys. Rad. 19, 358 (1958).
[CrossRef]

1955

D. Q. Wark, J. M. Stone, Nature 175, 254 (1955).
[CrossRef]

1953

R. Chabbal, J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue, Paris, 24, 138 (1953).

K. Krebs, A. Sauer, Ann. Phys. 13, 359 (1953).
[CrossRef]

1937

L. Vegard, Phil. Mag. 24, 588 (1937).

1923

H. D. Babcock, Astrophys J. 57, 254 (1923).
[CrossRef]

Andrew, K. L.

Armstrong, E. B.

E. B. Armstrong, Planet. Space Sci. 17, 957 (1969).
[CrossRef]

E. B. Armstrong, Planet. Space Sci. 16, 211 (1968).
[CrossRef]

E. B. Armstrong, J. Phys. Rad. 19, 358 (1958).
[CrossRef]

E. B. Armstrong, in The Airglow and the Aurorae (Pergamon Press, New York, 1956), p. 366.

Babcock, H. D.

H. D. Babcock, Astrophys J. 57, 254 (1923).
[CrossRef]

Ballik, E. A.

Bayer-Helms, F.

F. Bayer-Helms, Z. Angew. Phys. 15, 330 (1963).

Biondi, M. A.

M. A. Biondi, W. A. Feibelman, Planet Space Sci. 16, 431 (1968).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965).

Cabannes, J.

J. Cabannes, D. Dufay, in The Airglow and the Aurorae (Pergamon Press, New York, 1956).

Chabbal, R.

R. Chabbal, J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue, Paris, 24, 138 (1953).

Chamberlain, J. W.

J. W. Chamberlain, Physics of the Aurora and Airglow (Academic Press, New York, 1961).

Cogger, L. L.

Dufay, D.

J. Cabannes, D. Dufay, in The Airglow and the Aurorae (Pergamon Press, New York, 1956).

Feibelman, W. A.

M. A. Biondi, W. A. Feibelman, Planet Space Sci. 16, 431 (1968).
[CrossRef]

Hays, P. B.

R. G. Roble, P. B. Hays, A. F. Nagy, Planet. Space Sci. 18, 431 (1970).
[CrossRef]

P. B. Hays, A. F. Nagy, R. G. Roble, J. Geophys. Res. 74, 4162 (1969).
[CrossRef]

R. G. Roble, P. B. Hays, A. F. Nagy, Planet. Space Sci. 16, 1109 (1968).
[CrossRef]

Hernandez, G.

Hernandez, G. J.

G. J. Hernandez, J. P. Turtle, Planet. Space Sci. 13, 901 (1965).
[CrossRef]

G. J. Hernandez, J. P. Turtle, Aurora and Airglow (ReinholdNew York, 1967).

Hoey, M. J.

A. H. Jarrett, M. J. Hoey, J. Atmos. Terrest. Phys. 28, 175 (1966).
[CrossRef]

A. H. Jarret, M. J. Hoey, L. Paffrath, Planet. Space Sci. 12, 591 (1964).
[CrossRef]

Huten, D. M.

Jarret, A. H.

A. H. Jarret, M. J. Hoey, L. Paffrath, Planet. Space Sci. 12, 591 (1964).
[CrossRef]

Jarrett, A. H.

A. H. Jarrett, M. J. Hoey, J. Atmos. Terrest. Phys. 28, 175 (1966).
[CrossRef]

Karandikar, R. V.

R. V. Karandikar, in The Airglow and the Aurorae (Pergamon Press, New York, 1956), p. 374.

Krebs, K.

K. Krebs, A. Sauer, Ann. Phys. 13, 359 (1953).
[CrossRef]

Lake, C. W.

Larson, H. P.

Miiler, J. R.

Mulyarchick, T. M.

T. M. Mulyarchick, Izv. Akad. Nauk. S.S.R. 3, 449 (1960).

Nagy, A. F.

R. G. Roble, P. B. Hays, A. F. Nagy, Planet. Space Sci. 18, 431 (1970).
[CrossRef]

P. B. Hays, A. F. Nagy, R. G. Roble, J. Geophys. Res. 74, 4162 (1969).
[CrossRef]

R. G. Roble, P. B. Hays, A. F. Nagy, Planet. Space Sci. 16, 1109 (1968).
[CrossRef]

Nilson, J. A.

J. A. Nilson, G. G. Shepherd, Planet. Space Sci. 5, 299 (1961).
[CrossRef]

Paffrath, L.

A. H. Jarret, M. J. Hoey, L. Paffrath, Planet. Space Sci. 12, 591 (1964).
[CrossRef]

Phillips, J. G.

J. G. Phillips, in The Airglow and the Aurorae (Pergamon Press, New York, 1956), p. 67.

Roble, R. G.

R. G. Roble, P. B. Hays, A. F. Nagy, Planet. Space Sci. 18, 431 (1970).
[CrossRef]

P. B. Hays, A. F. Nagy, R. G. Roble, J. Geophys. Res. 74, 4162 (1969).
[CrossRef]

R. G. Roble, P. B. Hays, A. F. Nagy, Planet. Space Sci. 16, 1109 (1968).
[CrossRef]

R. G. Roble, A Theoretical and Experimental Study of the Stable Midlatitude Red Arc (SAR-arc) Ph.D. thesis, University of Michigan (1969).

Rundle, H. N.

Sauer, A.

K. Krebs, A. Sauer, Ann. Phys. 13, 359 (1953).
[CrossRef]

Shepherd, G. G.

D. M. Huten, H. N. Rundle, G. G. Shepherd, A. Vallance-Jones, Appl. Opt. 6, 1609 (1967).
[CrossRef]

G. G. Shepherd, J. Phys. 28, 301 (1967).
[CrossRef]

G. G. Shepherd, C. W. Lake, J. R. Miiler, L. L. Cogger, Appl. Opt. 4, 267 (1965).
[CrossRef]

H. H. Zwick, G. G. Shepherd, Can. J. Phys. 41, 343 (1963).
[CrossRef]

E. C. Turgeon, G. G. Shepherd, Planet. Space Sci. 9, 925 (1962).
[CrossRef]

J. A. Nilson, G. G. Shepherd, Planet. Space Sci. 5, 299 (1961).
[CrossRef]

Stone, J. M.

D. Q. Wark, J. M. Stone, Nature 175, 254 (1955).
[CrossRef]

Turgeon, E. C.

E. C. Turgeon, G. G. Shepherd, Planet. Space Sci. 9, 925 (1962).
[CrossRef]

Turtle, J. P.

G. J. Hernandez, J. P. Turtle, Planet. Space Sci. 13, 901 (1965).
[CrossRef]

G. J. Hernandez, J. P. Turtle, Aurora and Airglow (ReinholdNew York, 1967).

Vallance-Jones, A.

Vegard, L.

L. Vegard, Phil. Mag. 24, 588 (1937).

Wark, D. Q.

D. Q. Wark, Astrophys. J. 131, 491 (1960).
[CrossRef]

D. Q. Wark, J. M. Stone, Nature 175, 254 (1955).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965).

Zwick, H. H.

H. H. Zwick, G. G. Shepherd, Can. J. Phys. 41, 343 (1963).
[CrossRef]

Ann. Phys.

K. Krebs, A. Sauer, Ann. Phys. 13, 359 (1953).
[CrossRef]

Appl. Opt.

Astrophys J.

H. D. Babcock, Astrophys J. 57, 254 (1923).
[CrossRef]

Astrophys. J.

D. Q. Wark, Astrophys. J. 131, 491 (1960).
[CrossRef]

Can. J. Phys.

H. H. Zwick, G. G. Shepherd, Can. J. Phys. 41, 343 (1963).
[CrossRef]

Izv. Akad. Nauk. S.S.R.

T. M. Mulyarchick, Izv. Akad. Nauk. S.S.R. 3, 449 (1960).

J. Atmos. Terrest. Phys.

A. H. Jarrett, M. J. Hoey, J. Atmos. Terrest. Phys. 28, 175 (1966).
[CrossRef]

J. Geophys. Res.

P. B. Hays, A. F. Nagy, R. G. Roble, J. Geophys. Res. 74, 4162 (1969).
[CrossRef]

J. Phys.

G. G. Shepherd, J. Phys. 28, 301 (1967).
[CrossRef]

J. Phys. Rad.

E. B. Armstrong, J. Phys. Rad. 19, 358 (1958).
[CrossRef]

J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue, Paris

R. Chabbal, J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue, Paris, 24, 138 (1953).

Nature

D. Q. Wark, J. M. Stone, Nature 175, 254 (1955).
[CrossRef]

Phil. Mag.

L. Vegard, Phil. Mag. 24, 588 (1937).

Planet Space Sci.

M. A. Biondi, W. A. Feibelman, Planet Space Sci. 16, 431 (1968).
[CrossRef]

Planet. Space Sci.

A. H. Jarret, M. J. Hoey, L. Paffrath, Planet. Space Sci. 12, 591 (1964).
[CrossRef]

E. B. Armstrong, Planet. Space Sci. 16, 211 (1968).
[CrossRef]

J. A. Nilson, G. G. Shepherd, Planet. Space Sci. 5, 299 (1961).
[CrossRef]

E. C. Turgeon, G. G. Shepherd, Planet. Space Sci. 9, 925 (1962).
[CrossRef]

G. J. Hernandez, J. P. Turtle, Planet. Space Sci. 13, 901 (1965).
[CrossRef]

E. B. Armstrong, Planet. Space Sci. 17, 957 (1969).
[CrossRef]

R. G. Roble, P. B. Hays, A. F. Nagy, Planet. Space Sci. 18, 431 (1970).
[CrossRef]

R. G. Roble, P. B. Hays, A. F. Nagy, Planet. Space Sci. 16, 1109 (1968).
[CrossRef]

Z. Angew. Phys.

F. Bayer-Helms, Z. Angew. Phys. 15, 330 (1963).

Other

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965).

J. W. Chamberlain, Physics of the Aurora and Airglow (Academic Press, New York, 1961).

G. J. Hernandez, J. P. Turtle, Aurora and Airglow (ReinholdNew York, 1967).

J. Cabannes, D. Dufay, in The Airglow and the Aurorae (Pergamon Press, New York, 1956).

J. G. Phillips, in The Airglow and the Aurorae (Pergamon Press, New York, 1956), p. 67.

E. B. Armstrong, in The Airglow and the Aurorae (Pergamon Press, New York, 1956), p. 366.

R. V. Karandikar, in The Airglow and the Aurorae (Pergamon Press, New York, 1956), p. 374.

R. G. Roble, A Theoretical and Experimental Study of the Stable Midlatitude Red Arc (SAR-arc) Ph.D. thesis, University of Michigan (1969).

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Figures (7)

Fig. 1
Fig. 1

Calculated 6300-Å fringe profiles for the instrument given in Table I. The doppler temperature of the 6300-Å emission line is 1000 K. The solid curve represents the theoretical fringe. The scattered fringe consists of the theoretical fringe with statistical noise superimposed. The dotted curve shows the retrieved fringe profile calculated from the first five fourier coefficients of the scattered fringe profile.

Fig. 2
Fig. 2

The fourier transform coefficients plotted as a function of wavenumber. The solid curve represents the coefficients calculated for the theoretical fringe and the dashed curve gives the coefficients for the scattered fringe shown in Fig. 1.

Fig. 3
Fig. 3

The coefficients α and β, plotted as a function of 6300-Å emission line doppler temperature. These coefficients are related to the standard deviation of the temperature error in measuring the 6300-Å doppler temperature with the instrument given in Table I.

Fig. 4
Fig. 4

The coefficients α′ and β′ plotted as a function of the 6300-Å emission line doppler temperature. These coefficients are related to the standard deviation of the intensity error in measuring the 6300-Å emission line intensity with the instrument given in Table I.

Fig. 5
Fig. 5

The retrieved doppler temperature as a function of intensity variations during a scan of one order. Curve (a) represents the temperature error calculated for a constant emission line intensity and a variable background continuum. Curve (b) represents the temperature error due to a variable emission line intensity and a variable background continuum.

Fig. 6
Fig. 6

The standard deviation of the retrieved doppler temperature as a function of the doppler temperature for fringes with a peak signal of one hundred counts. The solid curve represents the results of a theoretical analysis of errors and the open circles are the results of the numerical analysis of errors.

Fig. 7
Fig. 7

The standard deviation of the retrieved doppler temperature as a function of peak signal for a 6300-Å emission line having a doppler temperature of 1000 K. The solid curve and the dashed curve represent the results of the theoretical analysis of errors for two continuum levels. The crosses and open squares are the results of the numerical analysis of errors for the same continuum levels.

Tables (1)

Tables Icon

Table I Operating Parameters of the Fabry-Perot Interferometer at the University of Michigan Airglow Observatory

Equations (27)

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A ( x ) = 1 2 π [ ( 1 - R 2 ) 1 - 2 R cos x + R ] ,
D g ( x ) = D - 1 π - 1 2 exp ( - x 2 D - 2 ) ,
D f ( x ) = Δ σ ( 4 π d f ) - 1 π ( x ) , w h e r e π ( x ) , = 1 1 2 0 when 2 π d f ( Δ σ ) - 1 = > < | x x x | ,
f = σ d 2 ( 4 L ) - 2 ,
Y ( x ) = 7.95 × 10 4 ( A Ω R y τ a T t P e ) ( 1 - R ) ( 1 + R ) - 1 { 1 + 2 n = 1 R n exp [ - n 2 4 ( G 2 + D 2 ) ] × sinc ( 2 nd f Δ σ ) sinc ( 2 π f Δ σ ) cos ( n x ) } ,
Y ˜ c m = 1 π x 1 x 2 cos ( m x ) Y ( x ) d x ,
Y ˜ c m = 2 Δ p 0 Δ p cos [ 2 π m ( p / Δ p ) ] Y ( p ) d p ,
Y c m = 1 π m i = 1 N Y ( p i ) [ 2 sin ( 2 π m P i Δ p ) sin 2 ( m π δ p i Δ p ) + cos ( 2 π m p i Δ p ) sin ( 2 π m δ p i Δ p ) ] · 1 δ t ,
Y ˜ s m = - 1 π m i = 1 N Y ( p i ) [ 2 cos ( 2 π m p i Δ p ) sin 2 ( π m δ p i Δ p ) - sin ( 2 π m p i Δ p ) sin ( 2 π m Δ p i Δ p ) ] · 1 δ t .
Y s ( x ) = Y ˜ c o 2 + m = 1 c o Y ˜ m cos [ m ( x + α ) ] ,
Y ˜ m = ( Y ˜ c m 2 + Y ˜ s m 2 ) 1 2 and α = tan - 1 ( Y ˜ s m / Y ˜ c m ) .
Y ( x ) = I 0 [ A 0 + n = 1 A n exp ( - n 2 4 γ T n ) cos n x ] + C ,
δ Y 2 = - ( Δ x / 2 ) Δ x / 2 [ Y s ( x ) - Y ( x ) ] 2 d x .
δ Y 2 = m = 1 M Y ˜ m 2 - [ m = 1 M A m Y ˜ m exp ( - m 2 4 γ T n ) ] 2 m = 1 M A m 2 exp ( - m 2 2 γ T n )
I 0 = m = 1 M A m Y ˜ m exp ( - m 2 4 γ T n ) [ m = 1 M A m 2 exp ( - m 2 2 γ T n ) ] - 1 and C = Y ˜ c o 2 - A 0 I 0 .
δ Y 2 = A + B T + C T 2 ,
R y = I 0 ( 1 + R ) ( 2 × 10 - 6 ) A Ω τ a T t ( 1 - R ) P e .
m = 1 M m A m exp ( - m 2 4 γ T n ) { Y ˜ c m sin [ 2 π m Δ p ( p c - p i ) ] - Y ˜ s m cos [ 2 π m Δ p ( p c - p i ) ] } = 0 ,
v = C Δ σ ( p c - p c o ) ( Δ p σ ) - 1 ,
δ Y 2 * = m = 1 M [ I 0 A m exp ( - m 2 4 γ T n ) - Y ˜ m ] 2 .
Y 2 * / Δ T = 0 = Y 2 * / Δ I 0 ,
Δ T = 4 π γ I 0 m = 1 M i = 1 N Δ Y i cos ( m x i ) δ x i × [ A m exp ( - m 2 4 γ T n ) { A m 2 } - m 2 A m exp ( - m 2 4 γ T n ) { m 2 A m 2 } { m 4 A m 2 } { m 2 A m 2 } - { m 2 A m 2 } { A m 2 } ] ,
{ Q m } = m = 1 M Q m exp ( - m 2 2 γ T n ) and Δ Y ˜ m = 1 π i = 1 N Δ Y i cos ( m x i ) δ x i
Δ T ¯ = [ α N I 0 ( 1 + β α C I 0 ) ] 1 2 ,
α = 1 π 2 0 Δ x Y i ( x i ) I 0 · ψ ( x i ) d x i , β = 1 π 2 0 Δ x ψ ( x i ) d x i , ψ ( x i ) = 16 Δ x γ 2 m = 1 M cos 2 ( m x i ) × [ A m exp ( - m 2 4 γ T n ) { A m 2 } - m 2 A m exp ( - m 2 4 γ T n ) { m 2 A m 2 } { m 4 A m 2 } { m 2 A m 2 } - { m 2 A m 2 } { A m 2 } ] 2 .
Δ I ¯ = [ I 0 N ( α + β C I 0 ) ] 1 2 ,
α = 1 π 2 0 Δ x Y i ( x i ) I 0 ψ ( x i ) d x i , β = 1 π 2 0 Δ x ψ ( x i ) d x i , ψ ( x i ) = Δ x m = 1 M cos 2 ( m x i ) × [ A m exp ( - m 2 4 γ T n ) { m 2 A m 2 } - m 2 A m exp ( - m 2 4 γ T n ) { m 4 A m 2 } { A m 2 } { m 2 A m 2 } - { m 2 A m 2 } { m 2 A m 2 } ] 2 .

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