Abstract

The University of Michigan has developed an incoherent-detection Doppler lidar system that continuously measures vertical profiles of horizontal winds and aerosol backscatter. An overview of the instrument is given, followed by a description of improvements that have been made to control the system stability. Most notably, an active feedback system has been implemented to improve the laser frequency stability. A detailed forward model of the instrument is developed that includes many subtle effects, such as detector nonlinearity. A nonlinear least-squares inversion method is then described that permits the recovery of Doppler shift and aerosol backscatter without requiring assumptions about the molecular component of the signal. Examples of wind and aerosol backscatter profiles are shown to illustrate the capabilities of the fitting method.

© 1997 Optical Society of America

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References

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  1. G. Benedetti-Michelangeli, F. Congeduti, G. Fiocco, “Measurement of aerosol motion and wind velocity in the lower troposphere by Doppler optical radar,” J. Atmos. Sci. 29, 906–910 (1972).
    [CrossRef]
  2. V. J. Abreu, J. E. Barnes, P. B. Hays, “Observations of winds with an incoherent lidar detector,” Appl. Opt. 31, 4509–4514 (1992).
    [CrossRef] [PubMed]
  3. M. L. Chanin, A. Garnier, A. Hauchecorne, J. Porteneuve, “A Doppler lidar for measuring winds in the middle atmosphere,” Geophys. Res. Lett. 16, 1273–1276 (1989).
    [CrossRef]
  4. C. A. Tepley, S. I. Sargoytchev, C. O. Hines, “Initial Doppler Rayleigh lidar results from Arecibo,” Geophys. Res. Lett. 18, 167–170 (1991).
    [CrossRef]
  5. R. E. Bills, C. S. Gardner, C.-Y. She, “Narrowband lidar technique for sodium temperature and Doppler wind observations of the upper atmosphere,” Opt. Eng. 30, 13–21 (1991).
    [CrossRef]
  6. F. J. Marsik, K. W. Fischer, T. D. McDonald, P. J. Samson, “Comparison of methods for estimating mixing height used during the 1992 Atlanta Field Intensive,” J. Appl. Meteorol. 34, 1802–1814 (1995).
    [CrossRef]
  7. J. D. Klett, “Lidar inversion with variable backscatter/extinction ratios,” Appl. Opt. 24, 1638–1643 (1985).
    [CrossRef] [PubMed]
  8. J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1981).
    [CrossRef] [PubMed]
  9. K. W. Fischer, V. J. Abreu, W. R. Skinner, J. E. Barnes, M. J. McGill, T. D. Irgang, “Visible wavelength Doppler lidar for measurement of wind and aerosol profiles during day and night,” Opt. Eng. 34, 499–511 (1995).
    [CrossRef]
  10. P. B. Hays, V. J. Abreu, M. E. Dobbs, D. A. Gell, H. J. Grassl, W. R. Skinner, “The High-Resolution Doppler Imager on the Upper Atmosphere Research Satellite,” J. Geophys. Res. 98, 10713–10723 (1993).
    [CrossRef]
  11. P. B. Hays, J. Wang, “Image plane detector for Fabry–Perot interferometers: physical model and improvement with anticoincidence detection,” Appl. Opt. 30, 3100–3107 (1991).
    [CrossRef] [PubMed]
  12. T. L. Killeen, B. C. Kennedy, P. B. Hays, D. A. Symarrow, D. H. Ceckowski, “Image plane detector for the Dynamics Explorer Fabry–Perot interferometer,” Appl. Opt. 22, 3503–3513 (1983).
    [CrossRef] [PubMed]
  13. J. M. Vaughan, The Fabry-Perot Interferometer (Hilger, London, 1989).
  14. G. Hernandez, Fabry-Perot Interferometers (Cambridge U. Press, Cambridge, England, 1986).
  15. K-N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).
  16. R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Wiley-Interscience, New York, 1984).
  17. Y. Benayahu, A. Ben-David, S. Fastig, A. Cohen, “Cloud-droplet-size distribution from lidar multiple-scattering measurements,” Appl. Opt. 34, 1569–1578 (1995).
    [CrossRef] [PubMed]
  18. G. Fiocco, J. B. DeWolf, “Frequency spectrum of laser echoes from atmospheric constituents and determination of the aerosol content of the air,” J. Atmos. Sci. 25, 488–496 (1968).
    [CrossRef]
  19. J. M. Hammersley, D. C. Handscomb, Monte Carlo Methods (Wiley, New York, 1964).
    [CrossRef]
  20. M. J. McGill, W. R. Skinner, T. D. Irgang, “Validation of wind profiles measured using incoherent Doppler lidar,” Appl. Opt. (to be published), ms. #11493.

1995

F. J. Marsik, K. W. Fischer, T. D. McDonald, P. J. Samson, “Comparison of methods for estimating mixing height used during the 1992 Atlanta Field Intensive,” J. Appl. Meteorol. 34, 1802–1814 (1995).
[CrossRef]

K. W. Fischer, V. J. Abreu, W. R. Skinner, J. E. Barnes, M. J. McGill, T. D. Irgang, “Visible wavelength Doppler lidar for measurement of wind and aerosol profiles during day and night,” Opt. Eng. 34, 499–511 (1995).
[CrossRef]

Y. Benayahu, A. Ben-David, S. Fastig, A. Cohen, “Cloud-droplet-size distribution from lidar multiple-scattering measurements,” Appl. Opt. 34, 1569–1578 (1995).
[CrossRef] [PubMed]

1993

P. B. Hays, V. J. Abreu, M. E. Dobbs, D. A. Gell, H. J. Grassl, W. R. Skinner, “The High-Resolution Doppler Imager on the Upper Atmosphere Research Satellite,” J. Geophys. Res. 98, 10713–10723 (1993).
[CrossRef]

1992

1991

P. B. Hays, J. Wang, “Image plane detector for Fabry–Perot interferometers: physical model and improvement with anticoincidence detection,” Appl. Opt. 30, 3100–3107 (1991).
[CrossRef] [PubMed]

C. A. Tepley, S. I. Sargoytchev, C. O. Hines, “Initial Doppler Rayleigh lidar results from Arecibo,” Geophys. Res. Lett. 18, 167–170 (1991).
[CrossRef]

R. E. Bills, C. S. Gardner, C.-Y. She, “Narrowband lidar technique for sodium temperature and Doppler wind observations of the upper atmosphere,” Opt. Eng. 30, 13–21 (1991).
[CrossRef]

1989

M. L. Chanin, A. Garnier, A. Hauchecorne, J. Porteneuve, “A Doppler lidar for measuring winds in the middle atmosphere,” Geophys. Res. Lett. 16, 1273–1276 (1989).
[CrossRef]

1985

1983

1981

1972

G. Benedetti-Michelangeli, F. Congeduti, G. Fiocco, “Measurement of aerosol motion and wind velocity in the lower troposphere by Doppler optical radar,” J. Atmos. Sci. 29, 906–910 (1972).
[CrossRef]

1968

G. Fiocco, J. B. DeWolf, “Frequency spectrum of laser echoes from atmospheric constituents and determination of the aerosol content of the air,” J. Atmos. Sci. 25, 488–496 (1968).
[CrossRef]

Abreu, V. J.

K. W. Fischer, V. J. Abreu, W. R. Skinner, J. E. Barnes, M. J. McGill, T. D. Irgang, “Visible wavelength Doppler lidar for measurement of wind and aerosol profiles during day and night,” Opt. Eng. 34, 499–511 (1995).
[CrossRef]

P. B. Hays, V. J. Abreu, M. E. Dobbs, D. A. Gell, H. J. Grassl, W. R. Skinner, “The High-Resolution Doppler Imager on the Upper Atmosphere Research Satellite,” J. Geophys. Res. 98, 10713–10723 (1993).
[CrossRef]

V. J. Abreu, J. E. Barnes, P. B. Hays, “Observations of winds with an incoherent lidar detector,” Appl. Opt. 31, 4509–4514 (1992).
[CrossRef] [PubMed]

Barnes, J. E.

K. W. Fischer, V. J. Abreu, W. R. Skinner, J. E. Barnes, M. J. McGill, T. D. Irgang, “Visible wavelength Doppler lidar for measurement of wind and aerosol profiles during day and night,” Opt. Eng. 34, 499–511 (1995).
[CrossRef]

V. J. Abreu, J. E. Barnes, P. B. Hays, “Observations of winds with an incoherent lidar detector,” Appl. Opt. 31, 4509–4514 (1992).
[CrossRef] [PubMed]

Benayahu, Y.

Ben-David, A.

Benedetti-Michelangeli, G.

G. Benedetti-Michelangeli, F. Congeduti, G. Fiocco, “Measurement of aerosol motion and wind velocity in the lower troposphere by Doppler optical radar,” J. Atmos. Sci. 29, 906–910 (1972).
[CrossRef]

Bills, R. E.

R. E. Bills, C. S. Gardner, C.-Y. She, “Narrowband lidar technique for sodium temperature and Doppler wind observations of the upper atmosphere,” Opt. Eng. 30, 13–21 (1991).
[CrossRef]

Ceckowski, D. H.

Chanin, M. L.

M. L. Chanin, A. Garnier, A. Hauchecorne, J. Porteneuve, “A Doppler lidar for measuring winds in the middle atmosphere,” Geophys. Res. Lett. 16, 1273–1276 (1989).
[CrossRef]

Cohen, A.

Congeduti, F.

G. Benedetti-Michelangeli, F. Congeduti, G. Fiocco, “Measurement of aerosol motion and wind velocity in the lower troposphere by Doppler optical radar,” J. Atmos. Sci. 29, 906–910 (1972).
[CrossRef]

DeWolf, J. B.

G. Fiocco, J. B. DeWolf, “Frequency spectrum of laser echoes from atmospheric constituents and determination of the aerosol content of the air,” J. Atmos. Sci. 25, 488–496 (1968).
[CrossRef]

Dobbs, M. E.

P. B. Hays, V. J. Abreu, M. E. Dobbs, D. A. Gell, H. J. Grassl, W. R. Skinner, “The High-Resolution Doppler Imager on the Upper Atmosphere Research Satellite,” J. Geophys. Res. 98, 10713–10723 (1993).
[CrossRef]

Fastig, S.

Fiocco, G.

G. Benedetti-Michelangeli, F. Congeduti, G. Fiocco, “Measurement of aerosol motion and wind velocity in the lower troposphere by Doppler optical radar,” J. Atmos. Sci. 29, 906–910 (1972).
[CrossRef]

G. Fiocco, J. B. DeWolf, “Frequency spectrum of laser echoes from atmospheric constituents and determination of the aerosol content of the air,” J. Atmos. Sci. 25, 488–496 (1968).
[CrossRef]

Fischer, K. W.

F. J. Marsik, K. W. Fischer, T. D. McDonald, P. J. Samson, “Comparison of methods for estimating mixing height used during the 1992 Atlanta Field Intensive,” J. Appl. Meteorol. 34, 1802–1814 (1995).
[CrossRef]

K. W. Fischer, V. J. Abreu, W. R. Skinner, J. E. Barnes, M. J. McGill, T. D. Irgang, “Visible wavelength Doppler lidar for measurement of wind and aerosol profiles during day and night,” Opt. Eng. 34, 499–511 (1995).
[CrossRef]

Gardner, C. S.

R. E. Bills, C. S. Gardner, C.-Y. She, “Narrowband lidar technique for sodium temperature and Doppler wind observations of the upper atmosphere,” Opt. Eng. 30, 13–21 (1991).
[CrossRef]

Garnier, A.

M. L. Chanin, A. Garnier, A. Hauchecorne, J. Porteneuve, “A Doppler lidar for measuring winds in the middle atmosphere,” Geophys. Res. Lett. 16, 1273–1276 (1989).
[CrossRef]

Gell, D. A.

P. B. Hays, V. J. Abreu, M. E. Dobbs, D. A. Gell, H. J. Grassl, W. R. Skinner, “The High-Resolution Doppler Imager on the Upper Atmosphere Research Satellite,” J. Geophys. Res. 98, 10713–10723 (1993).
[CrossRef]

Grassl, H. J.

P. B. Hays, V. J. Abreu, M. E. Dobbs, D. A. Gell, H. J. Grassl, W. R. Skinner, “The High-Resolution Doppler Imager on the Upper Atmosphere Research Satellite,” J. Geophys. Res. 98, 10713–10723 (1993).
[CrossRef]

Hammersley, J. M.

J. M. Hammersley, D. C. Handscomb, Monte Carlo Methods (Wiley, New York, 1964).
[CrossRef]

Handscomb, D. C.

J. M. Hammersley, D. C. Handscomb, Monte Carlo Methods (Wiley, New York, 1964).
[CrossRef]

Hauchecorne, A.

M. L. Chanin, A. Garnier, A. Hauchecorne, J. Porteneuve, “A Doppler lidar for measuring winds in the middle atmosphere,” Geophys. Res. Lett. 16, 1273–1276 (1989).
[CrossRef]

Hays, P. B.

Hernandez, G.

G. Hernandez, Fabry-Perot Interferometers (Cambridge U. Press, Cambridge, England, 1986).

Hines, C. O.

C. A. Tepley, S. I. Sargoytchev, C. O. Hines, “Initial Doppler Rayleigh lidar results from Arecibo,” Geophys. Res. Lett. 18, 167–170 (1991).
[CrossRef]

Irgang, T. D.

K. W. Fischer, V. J. Abreu, W. R. Skinner, J. E. Barnes, M. J. McGill, T. D. Irgang, “Visible wavelength Doppler lidar for measurement of wind and aerosol profiles during day and night,” Opt. Eng. 34, 499–511 (1995).
[CrossRef]

M. J. McGill, W. R. Skinner, T. D. Irgang, “Validation of wind profiles measured using incoherent Doppler lidar,” Appl. Opt. (to be published), ms. #11493.

Kennedy, B. C.

Killeen, T. L.

Klett, J. D.

Liou, K-N.

K-N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).

Marsik, F. J.

F. J. Marsik, K. W. Fischer, T. D. McDonald, P. J. Samson, “Comparison of methods for estimating mixing height used during the 1992 Atlanta Field Intensive,” J. Appl. Meteorol. 34, 1802–1814 (1995).
[CrossRef]

McDonald, T. D.

F. J. Marsik, K. W. Fischer, T. D. McDonald, P. J. Samson, “Comparison of methods for estimating mixing height used during the 1992 Atlanta Field Intensive,” J. Appl. Meteorol. 34, 1802–1814 (1995).
[CrossRef]

McGill, M. J.

K. W. Fischer, V. J. Abreu, W. R. Skinner, J. E. Barnes, M. J. McGill, T. D. Irgang, “Visible wavelength Doppler lidar for measurement of wind and aerosol profiles during day and night,” Opt. Eng. 34, 499–511 (1995).
[CrossRef]

M. J. McGill, W. R. Skinner, T. D. Irgang, “Validation of wind profiles measured using incoherent Doppler lidar,” Appl. Opt. (to be published), ms. #11493.

Measures, R. M.

R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Wiley-Interscience, New York, 1984).

Porteneuve, J.

M. L. Chanin, A. Garnier, A. Hauchecorne, J. Porteneuve, “A Doppler lidar for measuring winds in the middle atmosphere,” Geophys. Res. Lett. 16, 1273–1276 (1989).
[CrossRef]

Samson, P. J.

F. J. Marsik, K. W. Fischer, T. D. McDonald, P. J. Samson, “Comparison of methods for estimating mixing height used during the 1992 Atlanta Field Intensive,” J. Appl. Meteorol. 34, 1802–1814 (1995).
[CrossRef]

Sargoytchev, S. I.

C. A. Tepley, S. I. Sargoytchev, C. O. Hines, “Initial Doppler Rayleigh lidar results from Arecibo,” Geophys. Res. Lett. 18, 167–170 (1991).
[CrossRef]

She, C.-Y.

R. E. Bills, C. S. Gardner, C.-Y. She, “Narrowband lidar technique for sodium temperature and Doppler wind observations of the upper atmosphere,” Opt. Eng. 30, 13–21 (1991).
[CrossRef]

Skinner, W. R.

K. W. Fischer, V. J. Abreu, W. R. Skinner, J. E. Barnes, M. J. McGill, T. D. Irgang, “Visible wavelength Doppler lidar for measurement of wind and aerosol profiles during day and night,” Opt. Eng. 34, 499–511 (1995).
[CrossRef]

P. B. Hays, V. J. Abreu, M. E. Dobbs, D. A. Gell, H. J. Grassl, W. R. Skinner, “The High-Resolution Doppler Imager on the Upper Atmosphere Research Satellite,” J. Geophys. Res. 98, 10713–10723 (1993).
[CrossRef]

M. J. McGill, W. R. Skinner, T. D. Irgang, “Validation of wind profiles measured using incoherent Doppler lidar,” Appl. Opt. (to be published), ms. #11493.

Symarrow, D. A.

Tepley, C. A.

C. A. Tepley, S. I. Sargoytchev, C. O. Hines, “Initial Doppler Rayleigh lidar results from Arecibo,” Geophys. Res. Lett. 18, 167–170 (1991).
[CrossRef]

Vaughan, J. M.

J. M. Vaughan, The Fabry-Perot Interferometer (Hilger, London, 1989).

Wang, J.

Appl. Opt.

Geophys. Res. Lett.

M. L. Chanin, A. Garnier, A. Hauchecorne, J. Porteneuve, “A Doppler lidar for measuring winds in the middle atmosphere,” Geophys. Res. Lett. 16, 1273–1276 (1989).
[CrossRef]

C. A. Tepley, S. I. Sargoytchev, C. O. Hines, “Initial Doppler Rayleigh lidar results from Arecibo,” Geophys. Res. Lett. 18, 167–170 (1991).
[CrossRef]

J. Appl. Meteorol.

F. J. Marsik, K. W. Fischer, T. D. McDonald, P. J. Samson, “Comparison of methods for estimating mixing height used during the 1992 Atlanta Field Intensive,” J. Appl. Meteorol. 34, 1802–1814 (1995).
[CrossRef]

J. Atmos. Sci.

G. Benedetti-Michelangeli, F. Congeduti, G. Fiocco, “Measurement of aerosol motion and wind velocity in the lower troposphere by Doppler optical radar,” J. Atmos. Sci. 29, 906–910 (1972).
[CrossRef]

G. Fiocco, J. B. DeWolf, “Frequency spectrum of laser echoes from atmospheric constituents and determination of the aerosol content of the air,” J. Atmos. Sci. 25, 488–496 (1968).
[CrossRef]

J. Geophys. Res.

P. B. Hays, V. J. Abreu, M. E. Dobbs, D. A. Gell, H. J. Grassl, W. R. Skinner, “The High-Resolution Doppler Imager on the Upper Atmosphere Research Satellite,” J. Geophys. Res. 98, 10713–10723 (1993).
[CrossRef]

Opt. Eng.

K. W. Fischer, V. J. Abreu, W. R. Skinner, J. E. Barnes, M. J. McGill, T. D. Irgang, “Visible wavelength Doppler lidar for measurement of wind and aerosol profiles during day and night,” Opt. Eng. 34, 499–511 (1995).
[CrossRef]

R. E. Bills, C. S. Gardner, C.-Y. She, “Narrowband lidar technique for sodium temperature and Doppler wind observations of the upper atmosphere,” Opt. Eng. 30, 13–21 (1991).
[CrossRef]

Other

J. M. Vaughan, The Fabry-Perot Interferometer (Hilger, London, 1989).

G. Hernandez, Fabry-Perot Interferometers (Cambridge U. Press, Cambridge, England, 1986).

K-N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).

R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Wiley-Interscience, New York, 1984).

J. M. Hammersley, D. C. Handscomb, Monte Carlo Methods (Wiley, New York, 1964).
[CrossRef]

M. J. McGill, W. R. Skinner, T. D. Irgang, “Validation of wind profiles measured using incoherent Doppler lidar,” Appl. Opt. (to be published), ms. #11493.

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Figures (15)

Fig. 1
Fig. 1

Lidar viewing geometry.

Fig. 2
Fig. 2

Data-collection timing diagram.

Fig. 3
Fig. 3

Lidar optical system: F’s, filters; L’s, lenses; M’s, mirrors; W’s, clear windows.

Fig. 4
Fig. 4

Measuring Doppler shifts with a 12-channel detector, as illustrated with modeled spectra. Each channel has a width of 36.66 m/s. (a) The solid black spectrum represents the unshifted reference; the cross-hatched spectrum is shifted by 10 m/s toward the observing telescope. (b) Change in signal between the two spectra.

Fig. 5
Fig. 5

Reference spectrum peak position over 4 h, showing laser frequency drift with inactive seed laser control.

Fig. 6
Fig. 6

(a) Reference peak position with the set point at channel 4.0. (b) Voltage applied to seed laser thermoelectric heater–cooler module (0.01 V ≈ 10 m/s). (c) Étalon temperatures over an 18-h period with active seed laser control.

Fig. 7
Fig. 7

(a) Transmission of a perfect Fabry–Perot interferometer with R = 0.88. The FSR is the distance between adjacent peaks, and finesse is defined as the FSR divided by the FWHM. (b) Transmission of a Fabry–Perot [from Eq. (17)] for ΔdD = 0 (solid curve) and ΔdD = 30 nm defect (dotted curve). Both have R = 0.88, and both include aperture broadening effects.

Fig. 8
Fig. 8

Three pieces of information are contained in a measured spectrum, each having a distinct functional shape [Eq. (27)]. A least-squares technique can be used to determine uniquely the three parameters [Eq. (29)].

Fig. 9
Fig. 9

Detector normalization curve. Channel 1 is normalized to 1.0.

Fig. 10
Fig. 10

HRE response function for (a) channel 1, (b) channel 8. The dashed curve is the measured instrument response function; the solid curve is the modeled curve. The best-fit model curves have defect parameters of 8 nm for channel 1 and 32 nm for channel 8.

Fig. 11
Fig. 11

Instrument defect function for the 10-cm Fabry–Perot étalon and second-order polynomial fit to the raw data points.

Fig. 12
Fig. 12

Measured (dashed) and dead-time-corrected (solid) spectra for the (a) atmospheric bin, (b) reference bin.

Fig. 13
Fig. 13

Fitted measured spectra for three different aerosol–molecular ratios: (a) 1.6, (b) 2.6, (c) 3.7. In each case, the solid curve is the measured spectrum and the dotted curve is the fitted spectrum. The lower dashed curve is the fitted molecular component of the signal.

Fig. 14
Fig. 14

Horizontal wind field during the evening of 13 September 1994. Vectors are spaced 6 min apart. As denoted by the compass, the positive x axis represents East and the positive y axis represents North. For reference, a 10 m/s scale vector is drawn in the lower left corner.

Fig. 15
Fig. 15

Profiles of aerosol–molecular ratio during the night of 11 September 1994. The top of the aerosol layer is seen to decay from 1100 m down to ∼700 m, as shown by the dashed curve.

Tables (5)

Tables Icon

Table 1 Lidar System Parameters

Tables Icon

Table 2 Model Input Parameters

Tables Icon

Table 3 Recovered Input Values and Percent Error Obtained with a Monte Carlo Technique

Tables Icon

Table 4 Recovered Error Estimates and Percent Error Obtained with a Monte Carlo Technique

Tables Icon

Table 5 Recovered Parameters as a Function of Peak Signal Level Obtained with a Monte Carlo Technique

Equations (36)

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

TM=1-1-R21-R1-2R cos 2πM+R2,
TM=1-1-R21-R1+R×1+2 n=1 Rn cos 2πnM,
M=2μdν cos θ,
M=M0+ΔM,
ΔM=2μd0Δν+2μν0Δd-2μd0ν0θ22.
ΔνFSR=12μd0
ΔM=ΔνΔνFSR-ν0ΔνFSRθ22+2μΔdν0.
PΔd=1πΔdDexp-Δd2ΔdD2,
TΔν, θ=-+TΔν, θ, ΔdPΔddΔd-+PΔddΔd.
TΔν, θ=1-1-R21-R1+R1+2 n=1 Rn×exp-4π2n2ΔdD2ν02×cos 2πnΔνΔνFSR-ν0ΔνFSRθ22.
An=1-1-R21-R1+R for n=0, An=21-1-R21-R1+R×Rn exp-4π2n2ΔdD2ν02  forn>0.
TΔν, θ=n=0An cos 2πnΔνΔνFSR-ν0ΔνFSRθ22.
TΔν, Δθ, θ0=θ0-Δθ/2θ0+Δθ/2TΔν, θθdθθ0-Δθ/2θ0+Δθ/2θdθ.
TΔν, Δθ, θ0=n=0An sincnν0ΔνFSRθ0Δθ×cos 2πnΔνΔνFSR-ν0ΔνFSRθ02+Δθ24.
NFSR=ΩFSRΩchannel=2πΔνFSRν012πθ0Δθ=ΔνFSRν0θ0Δθ
j=θ0,j2+Δθj24θ0Δθ
TΔν, j=n=0An cos2πnΔνΔνFSR+jNFSRsincnNFSR.
Er=ETεΔtOArAT4πr2ΔhQETOTFν×exp-20rβAr+βMrdr×PAπ, rβArGL*LA+PMπ, rβMrGL*GM,
GLΔν=1πΔνLexp-Δν2ΔνL2, GMΔν=1πΔνMexp-Δν2ΔνM2,
Nr, j=ETλεΔthcOArAT4πr2ΔhQETOTFνηjnc×exp-20rβAr+βMrdr×n=0An, j sincnNFSRexp-π2n2ΔνL2ΔνFSR2×cos 2πnj-j0rNFSRPAπ, rβAr+PMπ, rβMrexp-π2n2ΔνM2ΔνFSR2+Bj,
j0r=NFSRΔνFSRν0-νC-2UHrν0 sin ϕc,
ΔνDr=2UHrν0 sin ϕc,
Nm,i=Na,i1+Na,iτ/Δt,
Nr, j=PTλΔthcOArAT4πr2ΔhQETOTFν×ηjnCn=0An,j cos 2πnj-j0rNFSR×sincnNFSRexp-π2n2ΔνL2ΔνFSR2×αr+ωrexp-π2n2ΔνM2ΔνFSR2,
αr=PAπ, rβArexp-2 0rβAr+βMrdr,
ωr=PMπ, rβMrexp-2 0rβAr+βMrdr,
Nj, r, α, ω, j0=N0j, r, α0, ω0, j0+Nj0j0,0j0-j0,0+Nαα0α-α0+Nωω0ω-ω0.
N1-N0,1N12-N0,12=N1j0j0,0N1αα0N1ωω0N12j0j0,0N12αα0N12ωω0j0-j0,0α-α0ω-ω0.
Δxest=GTWG-1GTWΔN,
cov Δx=GTWG-1,
σR=1ω2σα2+α2ω4σω2-2αω3σαω21/2,
UH=-cΔνFSRj0r-jrefν0NFSR sin ϕ.
σU,LOSi=cΔνFSRν0NFSR sin ϕσj,ref2+σjr2i1/2.
Na,i=Nm,i1-Nm,iτ/Δt.
τ=Nm,2r-εNm,1rΔtNm,2rNm,1r1-ε,
Na,i=Nm,i1.0112-1.81×10-4Nm,i.

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