Abstract

We describe an analysis procedure for estimating the thermospheric winds and temperatures from the multi-order two-dimensional (2D) interferograms produced by an imaging Fabry–Perot interferometer (FPI) as imaged by a CCD detector. We also present a forward model describing the 2D interferograms. To investigate the robustness and accuracy of the analysis, we perform several Monte Carlo simulations using this forward model for an FPI that has recently been developed and deployed to northeastern Brazil. The first simulation shows that a slight cross-contamination at high temperatures exists between neighboring orders in the interferogram, introducing a bias in the estimated temperatures and increasing errors in both the estimated temperatures and winds when each order is analyzed in full. The second simulation investigates how using less than an entire order in the analysis reduces the cross contamination observed in the first set of simulations, improving the accuracy of the estimated temperatures. The last simulation investigates the effect of the signal-to-noise ratio on the errors in the estimated parameters. It is shown that, for the specific FPI simulated in this study, a signal-to-noise ratio of 1.5 is required to obtain thermospheric wind errors of 5m/s and temperature errors of 20K.

© 2011 Optical Society of America

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References

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  1. E. Kudeki, A. Akgiray, M. Milla, J. Chau, D. Hysell, “Equatorial spread-F initiation: post-sunsent vortex, thermospheric winds, gravity waves,” J. Atmos. Sol. Terr. Phys. 69 (2007).
    [CrossRef]
  2. D. Rees, “Observations and modelling of ionospheric and thermospheric disturbances during major geomagnetic storms: A review,” J. Atmos. Sol. Terr. Phys. 57, 1143–1457 (1995).
  3. H. Rishbeth, “The F-region dynamo,” J. Atmos. Terr. Phys. 43, 387–392 (1981).
    [CrossRef]
  4. M. Biondi, W. Feibelman, “Twilight and nightglow spectral line shapes of oxygen λ6300 and λ5577 radiation,” Planet. Space Sci. 16, 431–443 (1968).
    [CrossRef]
  5. G. Hernandez, “Measurement of thermospheric temperatures and winds by remote Fabry-Perot spectrometry,” Opt. Eng. 19, 518–532 (1980).
  6. M. M. Coakley, F. L. Roesler, R. J. Reynolds, S. Nossal, “Fabry-Perot CCD annular-summing spectroscopy: study and implementation for aeronomy applications,” Appl. Opt. 35, 6479–6493 (1996).
    [CrossRef] [PubMed]
  7. M. Conde, “Deriving wavelength spectra from fringe images from a fixed-gap single-etalon Fabry-Perot spectrometer,” Appl. Opt. 41, 2672–2678 (2002).
    [CrossRef] [PubMed]
  8. G. Hernandez, “Analytical description of a Fabry-Perot photoelectric spectrometer,” Appl. Opt. 5, 1745–1748 (1966).
    [CrossRef] [PubMed]
  9. P. Hays, R. Roble, “A technique for recovering Doppler line profiles from Fabry-Perot interfermoter fringes of very low intensity,” Appl. Opt. 10, 193–200 (1971).
    [CrossRef] [PubMed]
  10. J. Vaughan, The Fabry-Perot Interferometer: History, Theory, Practice, and Applications (Taylor & Francis, 1989).
  11. T. Killeen, P. Hays, “Doppler line profile analysis for a multichannel Fabry-Perot interferometer,” Appl. Opt. 23, 612–620 (1984).
    [CrossRef] [PubMed]
  12. K. Shiokawa, T. Kadota, M. Ejiri, Y. Otsuka, Y. Katoh, M. Satoh, T. Ogawa, “Three-channel imaging Fabry-Perot interferometer for measurement of mid-latitude airglow,” Appl. Opt. 40, 4286–4296 (2001).
    [CrossRef]
  13. J. Meriwether, M. Faivre, C. Fesen, P. Sherwood, O. Veliz, “New results on equatorial thermospheric winds and the midnight temperature maximum,” Ann. Geophys. 26, 447–466 (2008).
    [CrossRef]
  14. K. Krebs, A. Sauer, “Über die Intensitätsverteilung von Spektrallinien im Pérot-Fabry-Interferometer,” Ann. Phys. 448, 359–368 (1953).
    [CrossRef]
  15. J. J. Makela, J. W. Meriwether, J. P. Lima, E. S. Miller, S. J. Armstrong, “The remote equatorial nighttime observatory of ionospheric regions project and the international heliospherical year,” Earth Moon Planet 104, 211–226 (2009).
    [CrossRef]
  16. S. L. Armstrong, “Fabry-Perot data analysis and simulation for the RENOIR observatories,” Master’s thesis (University of Illinois at Urbana-Champaign, 2008).
  17. K. Levenberg, “A method for the solution of certain nonlinear problems in least squares,” Quart. Appl. Math. 2, 164–168 (1944).
  18. D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. SIAM Control 11, 431–441 (1963).
  19. M. A. Biondi, D. P. Sipler, “Horizontal and vertical winds and temperatures in the equatorial thermosphere: Measurements from Natal, Brazil during August—September 1982,” Planet. Space Sci. 33, 817–823 (1985).
    [CrossRef]
  20. R. Raghavarao, W. Hoefy, N. Spencer, L. Wharton, “Neutral temperature anomaly in the equatorial thermosphere-A source of vertical winds,” Geophys. Res. Lett. 20, 1023–1026 (1993).
    [CrossRef]
  21. D. P. Sipler, M. A. Biondi, M. E. Zipf, “Vertical winds in the midlatitude thermosphere from Fabry-Perot interferometer measurements,” J. Atmos. Terr. Phys. 57, 621–629 (1995).
    [CrossRef]
  22. G. Hernandez, Fabry-Perot Interferometers (Cambridge University, 1988).

2009

J. J. Makela, J. W. Meriwether, J. P. Lima, E. S. Miller, S. J. Armstrong, “The remote equatorial nighttime observatory of ionospheric regions project and the international heliospherical year,” Earth Moon Planet 104, 211–226 (2009).
[CrossRef]

2008

J. Meriwether, M. Faivre, C. Fesen, P. Sherwood, O. Veliz, “New results on equatorial thermospheric winds and the midnight temperature maximum,” Ann. Geophys. 26, 447–466 (2008).
[CrossRef]

2007

E. Kudeki, A. Akgiray, M. Milla, J. Chau, D. Hysell, “Equatorial spread-F initiation: post-sunsent vortex, thermospheric winds, gravity waves,” J. Atmos. Sol. Terr. Phys. 69 (2007).
[CrossRef]

2002

2001

1996

1995

D. Rees, “Observations and modelling of ionospheric and thermospheric disturbances during major geomagnetic storms: A review,” J. Atmos. Sol. Terr. Phys. 57, 1143–1457 (1995).

D. P. Sipler, M. A. Biondi, M. E. Zipf, “Vertical winds in the midlatitude thermosphere from Fabry-Perot interferometer measurements,” J. Atmos. Terr. Phys. 57, 621–629 (1995).
[CrossRef]

1993

R. Raghavarao, W. Hoefy, N. Spencer, L. Wharton, “Neutral temperature anomaly in the equatorial thermosphere-A source of vertical winds,” Geophys. Res. Lett. 20, 1023–1026 (1993).
[CrossRef]

1985

M. A. Biondi, D. P. Sipler, “Horizontal and vertical winds and temperatures in the equatorial thermosphere: Measurements from Natal, Brazil during August—September 1982,” Planet. Space Sci. 33, 817–823 (1985).
[CrossRef]

1984

1981

H. Rishbeth, “The F-region dynamo,” J. Atmos. Terr. Phys. 43, 387–392 (1981).
[CrossRef]

1980

G. Hernandez, “Measurement of thermospheric temperatures and winds by remote Fabry-Perot spectrometry,” Opt. Eng. 19, 518–532 (1980).

1971

1968

M. Biondi, W. Feibelman, “Twilight and nightglow spectral line shapes of oxygen λ6300 and λ5577 radiation,” Planet. Space Sci. 16, 431–443 (1968).
[CrossRef]

1966

1963

D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. SIAM Control 11, 431–441 (1963).

1953

K. Krebs, A. Sauer, “Über die Intensitätsverteilung von Spektrallinien im Pérot-Fabry-Interferometer,” Ann. Phys. 448, 359–368 (1953).
[CrossRef]

1944

K. Levenberg, “A method for the solution of certain nonlinear problems in least squares,” Quart. Appl. Math. 2, 164–168 (1944).

Akgiray, A.

E. Kudeki, A. Akgiray, M. Milla, J. Chau, D. Hysell, “Equatorial spread-F initiation: post-sunsent vortex, thermospheric winds, gravity waves,” J. Atmos. Sol. Terr. Phys. 69 (2007).
[CrossRef]

Armstrong, S. J.

J. J. Makela, J. W. Meriwether, J. P. Lima, E. S. Miller, S. J. Armstrong, “The remote equatorial nighttime observatory of ionospheric regions project and the international heliospherical year,” Earth Moon Planet 104, 211–226 (2009).
[CrossRef]

Armstrong, S. L.

S. L. Armstrong, “Fabry-Perot data analysis and simulation for the RENOIR observatories,” Master’s thesis (University of Illinois at Urbana-Champaign, 2008).

Biondi, M.

M. Biondi, W. Feibelman, “Twilight and nightglow spectral line shapes of oxygen λ6300 and λ5577 radiation,” Planet. Space Sci. 16, 431–443 (1968).
[CrossRef]

Biondi, M. A.

D. P. Sipler, M. A. Biondi, M. E. Zipf, “Vertical winds in the midlatitude thermosphere from Fabry-Perot interferometer measurements,” J. Atmos. Terr. Phys. 57, 621–629 (1995).
[CrossRef]

M. A. Biondi, D. P. Sipler, “Horizontal and vertical winds and temperatures in the equatorial thermosphere: Measurements from Natal, Brazil during August—September 1982,” Planet. Space Sci. 33, 817–823 (1985).
[CrossRef]

Chau, J.

E. Kudeki, A. Akgiray, M. Milla, J. Chau, D. Hysell, “Equatorial spread-F initiation: post-sunsent vortex, thermospheric winds, gravity waves,” J. Atmos. Sol. Terr. Phys. 69 (2007).
[CrossRef]

Coakley, M. M.

Conde, M.

Ejiri, M.

Faivre, M.

J. Meriwether, M. Faivre, C. Fesen, P. Sherwood, O. Veliz, “New results on equatorial thermospheric winds and the midnight temperature maximum,” Ann. Geophys. 26, 447–466 (2008).
[CrossRef]

Feibelman, W.

M. Biondi, W. Feibelman, “Twilight and nightglow spectral line shapes of oxygen λ6300 and λ5577 radiation,” Planet. Space Sci. 16, 431–443 (1968).
[CrossRef]

Fesen, C.

J. Meriwether, M. Faivre, C. Fesen, P. Sherwood, O. Veliz, “New results on equatorial thermospheric winds and the midnight temperature maximum,” Ann. Geophys. 26, 447–466 (2008).
[CrossRef]

Hays, P.

Hernandez, G.

G. Hernandez, “Measurement of thermospheric temperatures and winds by remote Fabry-Perot spectrometry,” Opt. Eng. 19, 518–532 (1980).

G. Hernandez, “Analytical description of a Fabry-Perot photoelectric spectrometer,” Appl. Opt. 5, 1745–1748 (1966).
[CrossRef] [PubMed]

G. Hernandez, Fabry-Perot Interferometers (Cambridge University, 1988).

Hoefy, W.

R. Raghavarao, W. Hoefy, N. Spencer, L. Wharton, “Neutral temperature anomaly in the equatorial thermosphere-A source of vertical winds,” Geophys. Res. Lett. 20, 1023–1026 (1993).
[CrossRef]

Hysell, D.

E. Kudeki, A. Akgiray, M. Milla, J. Chau, D. Hysell, “Equatorial spread-F initiation: post-sunsent vortex, thermospheric winds, gravity waves,” J. Atmos. Sol. Terr. Phys. 69 (2007).
[CrossRef]

Kadota, T.

Katoh, Y.

Killeen, T.

Krebs, K.

K. Krebs, A. Sauer, “Über die Intensitätsverteilung von Spektrallinien im Pérot-Fabry-Interferometer,” Ann. Phys. 448, 359–368 (1953).
[CrossRef]

Kudeki, E.

E. Kudeki, A. Akgiray, M. Milla, J. Chau, D. Hysell, “Equatorial spread-F initiation: post-sunsent vortex, thermospheric winds, gravity waves,” J. Atmos. Sol. Terr. Phys. 69 (2007).
[CrossRef]

Levenberg, K.

K. Levenberg, “A method for the solution of certain nonlinear problems in least squares,” Quart. Appl. Math. 2, 164–168 (1944).

Lima, J. P.

J. J. Makela, J. W. Meriwether, J. P. Lima, E. S. Miller, S. J. Armstrong, “The remote equatorial nighttime observatory of ionospheric regions project and the international heliospherical year,” Earth Moon Planet 104, 211–226 (2009).
[CrossRef]

Makela, J. J.

J. J. Makela, J. W. Meriwether, J. P. Lima, E. S. Miller, S. J. Armstrong, “The remote equatorial nighttime observatory of ionospheric regions project and the international heliospherical year,” Earth Moon Planet 104, 211–226 (2009).
[CrossRef]

Marquardt, D.

D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. SIAM Control 11, 431–441 (1963).

Meriwether, J.

J. Meriwether, M. Faivre, C. Fesen, P. Sherwood, O. Veliz, “New results on equatorial thermospheric winds and the midnight temperature maximum,” Ann. Geophys. 26, 447–466 (2008).
[CrossRef]

Meriwether, J. W.

J. J. Makela, J. W. Meriwether, J. P. Lima, E. S. Miller, S. J. Armstrong, “The remote equatorial nighttime observatory of ionospheric regions project and the international heliospherical year,” Earth Moon Planet 104, 211–226 (2009).
[CrossRef]

Milla, M.

E. Kudeki, A. Akgiray, M. Milla, J. Chau, D. Hysell, “Equatorial spread-F initiation: post-sunsent vortex, thermospheric winds, gravity waves,” J. Atmos. Sol. Terr. Phys. 69 (2007).
[CrossRef]

Miller, E. S.

J. J. Makela, J. W. Meriwether, J. P. Lima, E. S. Miller, S. J. Armstrong, “The remote equatorial nighttime observatory of ionospheric regions project and the international heliospherical year,” Earth Moon Planet 104, 211–226 (2009).
[CrossRef]

Nossal, S.

Ogawa, T.

Otsuka, Y.

Raghavarao, R.

R. Raghavarao, W. Hoefy, N. Spencer, L. Wharton, “Neutral temperature anomaly in the equatorial thermosphere-A source of vertical winds,” Geophys. Res. Lett. 20, 1023–1026 (1993).
[CrossRef]

Rees, D.

D. Rees, “Observations and modelling of ionospheric and thermospheric disturbances during major geomagnetic storms: A review,” J. Atmos. Sol. Terr. Phys. 57, 1143–1457 (1995).

Reynolds, R. J.

Rishbeth, H.

H. Rishbeth, “The F-region dynamo,” J. Atmos. Terr. Phys. 43, 387–392 (1981).
[CrossRef]

Roble, R.

Roesler, F. L.

Satoh, M.

Sauer, A.

K. Krebs, A. Sauer, “Über die Intensitätsverteilung von Spektrallinien im Pérot-Fabry-Interferometer,” Ann. Phys. 448, 359–368 (1953).
[CrossRef]

Sherwood, P.

J. Meriwether, M. Faivre, C. Fesen, P. Sherwood, O. Veliz, “New results on equatorial thermospheric winds and the midnight temperature maximum,” Ann. Geophys. 26, 447–466 (2008).
[CrossRef]

Shiokawa, K.

Sipler, D. P.

D. P. Sipler, M. A. Biondi, M. E. Zipf, “Vertical winds in the midlatitude thermosphere from Fabry-Perot interferometer measurements,” J. Atmos. Terr. Phys. 57, 621–629 (1995).
[CrossRef]

M. A. Biondi, D. P. Sipler, “Horizontal and vertical winds and temperatures in the equatorial thermosphere: Measurements from Natal, Brazil during August—September 1982,” Planet. Space Sci. 33, 817–823 (1985).
[CrossRef]

Spencer, N.

R. Raghavarao, W. Hoefy, N. Spencer, L. Wharton, “Neutral temperature anomaly in the equatorial thermosphere-A source of vertical winds,” Geophys. Res. Lett. 20, 1023–1026 (1993).
[CrossRef]

Vaughan, J.

J. Vaughan, The Fabry-Perot Interferometer: History, Theory, Practice, and Applications (Taylor & Francis, 1989).

Veliz, O.

J. Meriwether, M. Faivre, C. Fesen, P. Sherwood, O. Veliz, “New results on equatorial thermospheric winds and the midnight temperature maximum,” Ann. Geophys. 26, 447–466 (2008).
[CrossRef]

Wharton, L.

R. Raghavarao, W. Hoefy, N. Spencer, L. Wharton, “Neutral temperature anomaly in the equatorial thermosphere-A source of vertical winds,” Geophys. Res. Lett. 20, 1023–1026 (1993).
[CrossRef]

Zipf, M. E.

D. P. Sipler, M. A. Biondi, M. E. Zipf, “Vertical winds in the midlatitude thermosphere from Fabry-Perot interferometer measurements,” J. Atmos. Terr. Phys. 57, 621–629 (1995).
[CrossRef]

Ann. Geophys.

J. Meriwether, M. Faivre, C. Fesen, P. Sherwood, O. Veliz, “New results on equatorial thermospheric winds and the midnight temperature maximum,” Ann. Geophys. 26, 447–466 (2008).
[CrossRef]

Ann. Phys.

K. Krebs, A. Sauer, “Über die Intensitätsverteilung von Spektrallinien im Pérot-Fabry-Interferometer,” Ann. Phys. 448, 359–368 (1953).
[CrossRef]

Appl. Opt.

Earth Moon Planet

J. J. Makela, J. W. Meriwether, J. P. Lima, E. S. Miller, S. J. Armstrong, “The remote equatorial nighttime observatory of ionospheric regions project and the international heliospherical year,” Earth Moon Planet 104, 211–226 (2009).
[CrossRef]

Geophys. Res. Lett.

R. Raghavarao, W. Hoefy, N. Spencer, L. Wharton, “Neutral temperature anomaly in the equatorial thermosphere-A source of vertical winds,” Geophys. Res. Lett. 20, 1023–1026 (1993).
[CrossRef]

J. Atmos. Sol. Terr. Phys.

E. Kudeki, A. Akgiray, M. Milla, J. Chau, D. Hysell, “Equatorial spread-F initiation: post-sunsent vortex, thermospheric winds, gravity waves,” J. Atmos. Sol. Terr. Phys. 69 (2007).
[CrossRef]

D. Rees, “Observations and modelling of ionospheric and thermospheric disturbances during major geomagnetic storms: A review,” J. Atmos. Sol. Terr. Phys. 57, 1143–1457 (1995).

J. Atmos. Terr. Phys.

H. Rishbeth, “The F-region dynamo,” J. Atmos. Terr. Phys. 43, 387–392 (1981).
[CrossRef]

D. P. Sipler, M. A. Biondi, M. E. Zipf, “Vertical winds in the midlatitude thermosphere from Fabry-Perot interferometer measurements,” J. Atmos. Terr. Phys. 57, 621–629 (1995).
[CrossRef]

J. SIAM Control

D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. SIAM Control 11, 431–441 (1963).

Opt. Eng.

G. Hernandez, “Measurement of thermospheric temperatures and winds by remote Fabry-Perot spectrometry,” Opt. Eng. 19, 518–532 (1980).

Planet. Space Sci.

M. Biondi, W. Feibelman, “Twilight and nightglow spectral line shapes of oxygen λ6300 and λ5577 radiation,” Planet. Space Sci. 16, 431–443 (1968).
[CrossRef]

M. A. Biondi, D. P. Sipler, “Horizontal and vertical winds and temperatures in the equatorial thermosphere: Measurements from Natal, Brazil during August—September 1982,” Planet. Space Sci. 33, 817–823 (1985).
[CrossRef]

Quart. Appl. Math.

K. Levenberg, “A method for the solution of certain nonlinear problems in least squares,” Quart. Appl. Math. 2, 164–168 (1944).

Other

S. L. Armstrong, “Fabry-Perot data analysis and simulation for the RENOIR observatories,” Master’s thesis (University of Illinois at Urbana-Champaign, 2008).

J. Vaughan, The Fabry-Perot Interferometer: History, Theory, Practice, and Applications (Taylor & Francis, 1989).

G. Hernandez, Fabry-Perot Interferometers (Cambridge University, 1988).

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Figures (11)

Fig. 1
Fig. 1

(Left) Example of a measured (blue circle) laser fringe and that reconstructed from Fourier coefficients (red solid curve). The result of the reconstructed fringe utilizing shifted Fourier coefficients (green dashed curve) is also shown. (Right) Example of the first 22 Fourier coefficients obtained for the laser fringe shown to the left (red crosses), as well as the Fourier coefficients obtained by shifting the laser fringe using Eqs. (9, 10, 11) (green circles).

Fig. 2
Fig. 2

(Left) Example showing the change in the first four Fourier coefficients in a laser interferogram as a function of order. (Right) Normalized and shifted cosine ( a n i * ) and sine ( b n i * ) Fourier coefficients plotted as a function of Fourier harmonic. Each line represents the coefficients obtained from analyzing an individual order in the same laser interferogram.

Fig. 3
Fig. 3

(Top) Example from an experiment in Cajazeiras, Brazil, on Sept. 22, 2009, using a 300 s integration time showing the measured sky-emission fringe (blue circles) and the reconstructed fringe (red solid curve) using the optimal U parameters from the Levenberg–Marquardt nonlinear least-squares fitting algorithm. (Bottom) Residual difference between the measured and estimated fringe.

Fig. 4
Fig. 4

Example of a measured (left) and simulated (right) laser 2D interferogram using a 30 s integration time. The mean and standard deviation for each image are given in the figure title. The images have been scaled to use the same dynamic range.

Fig. 5
Fig. 5

Example of a measured (left) and simulated (right) 2D interferogram of the 630.0 nm emission using 300 s integration time. The measured image was obtained from Cajazeiras, Brazil, on Sept. 22, 2009. The simulated spectra is for a temperature of 700 K . The mean and standard deviation for each image are given in the figure title. The images have been scaled to use the same dynamic range.

Fig. 6
Fig. 6

Results of the annular summation of the measured and simulated 2D interferograms presented in Figs. 4 (left) and 5 (right).

Fig. 7
Fig. 7

Simulation space of Doppler velocities and temperatures considered for the Monte Carlo simulations. Each dot represents the parameters for an individual simulation.

Fig. 8
Fig. 8

(blue dots) Comparison of the differences between the input to the forward model and the estimated param eters for each run of the Monte Carlo simulation. (red crosses) Estimated error from the analysis routine for each run of the Monte Carlo simulation. Results from the simulation for the Doppler velocity (top) and temperature (bottom) are shown.

Fig. 9
Fig. 9

Same as Fig. 8 except using a “fringe factor” of 60%.

Fig. 10
Fig. 10

Mean error (red squares) and absolute errors (blue circles) for 1000-iteration Monte Carlo simulations used to investigate the effect of changing the “fringe factor” on the analysis procedure. Results for both Doppler velocity (top) and temperature (bottom) are shown.

Fig. 11
Fig. 11

Results of a 1000-iteration Monte Carlo simulation to study the effect of signal-to-noise ratio on the absolute error in estimated Doppler temperature (left) and velocity (right). The difference between the input to the forward model and the estimated parameters (blue dots) and the estimated error (red crosses) are shown.

Tables (4)

Tables Icon

Table 1 Variables Used in Describing the Instrument and Its Spectral Response for the Analysis Procedure, mks Units

Tables Icon

Table 2 Description of Variables Used in Analyzing the 2D Interferograms Using the Methodology of [11] and the Levenberg–Marquardt Technique, mks Units

Tables Icon

Table 3 Physical Constants and Parameters Used in the Forward Model, mks Units

Tables Icon

Table 4 System Parameters for the MiniME Imaging FPI Simulated in This Work

Equations (35)

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

r i = i A i π ,
A i = A n ann uli .
N i = A Ω i t Q i T o i × 10 6 4 π 0 T F ( λ ) Ψ ( λ , θ i ) Y ( λ ) d λ + B i ,
Ψ ( λ , θ i ) = a 0 i + n = 1 [ a n i cos ( 2 π n Δ λ 0 ( λ l λ r ) + 2 π n ϕ i ) + b n i sin ( 2 π n Δ λ 0 ( λ l λ r ) + 2 π n ϕ i ) ] ,
ϕ i = ( μ d λ l ) ( θ 1 2 θ i 2 )
θ i = tan 1 [ 1 f 0 ( r i 1 2 + r i 2 2 ) 1 / 2 ]
1 f 0 ( r i 1 2 + r i 2 2 ) 1 / 2
λ l = λ 0 ( 1 + v c )
a n i = a n i cos β n + b n i sin β n ,
b n i = a n i sin β n + b n i cos β n ,
β n = 2 π ( n 1 ) ( 1 + j l K laser ) ,
a n i * = a n i / a 0 i
b n i * = b n i / a 0 i
j = i i 0 ,
ϕ i = ( μ d λ l ) ( θ 1 2 θ i 2 )
= ( μ d λ l f 0 2 ) ( r 1 2 r i 2 )
= ( μ d λ l f 0 2 π ) ( A 1 i A i )
= ( A 1 i A i ) A FSR ,
A FSR = π f 0 2 λ l μ d .
ϕ i = ( i 1 ) K sky
= ( j + i 0 1 ) K sky .
K sky = λ l 2 λ laser 2 K laser .
N j = U 1 + U 2 [ 1 + n = 1 n coef ( a n j * cos [ 2 π n K sky ( j K sky U 3 ) ] + b n j * sin [ 2 π n K sky ( j K sky U 3 ) ] ) exp ( n 2 U 4 2 ) ] ,
U 1 = C o j t Δ λ F ¯ λ a 0 j ( 1 R 1 + R ) ,
U 2 = C o j a 0 j t T F o 0 ,
U 3 = K sky λ l λ r Δ λ 0 i 0 + 1 ,
U 4 = π c λ 0 Δ λ 0 2 k T m ,
v ^ = c ( U ^ 3 Δ λ 0 + λ r λ 0 1 ) v ref ,
T ^ = ( U ^ 4 c π Δ λ 0 λ 0 ) 2 m 2 k ,
σ v = ( c Δ λ 0 λ 0 σ U 3 ) 2 + σ v ref 2
σ T = c π Δ λ 0 λ 0 σ U 4 2 m T k .
I ( θ ) = λ I ( λ ) 1 R 2 1 + R 2 2 R cos ( δ ( θ , λ ) ) ,
δ ( θ , λ ) = 2 π λ 2 μ d cos ( θ ) ,
I ( λ ) = 1 Δ λ 2 π exp ( 1 2 · λ λ 0 1 + v Dop pler c Δ λ 2 ) ,
Δ λ = λ 0 k T m c 2 ,

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