Abstract

The measurement accuracies of three-frequency resonance fluorescence Doppler lidars are limited by photon noise and uncertainties in the laser frequency and line width. We analyze the performance of Na, Fe, and He lidars using a new technique, which incorporates precise information about the absorption spectrum of the species and the pulse spectrum of the lasers. We derive the measurement errors associated with photon noise, laser frequency errors, and laser line width errors. Optimizing the lidar design, based upon the measurement requirements, can improve system performance by reducing the required integration times, enabling measurements to be made in less time or at higher altitudes where the densities and signal levels are smaller. The optimum frequency shift for observing heat and constituent transport velocities is 689 MHz (580 MHz) at night (day) for Na lidars and 774 MHz (597 MHz) for Fe lidars. The optimum frequency shift for observing winds, temperature, and He densities is 3.66 GHz (3.16 GHz) at night (day) for He lidars.

© 2014 Optical Society of America

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  1. C. S. Gardner and W. Yang, “Measurements of the dynamical cooling rate associated with the vertical transport of heat by dissipating gravity waves in the mesopause region at the Starfire Optical Range, New Mexico,” J. Geophys. Res. 103, 16909–16926 (1998).
    [CrossRef]
  2. J. Yue, C.-Y. She, and H.-L. Liu, “Large wind shears and stabilities in the mesopause region observed by Na wind-temperature lidar at mid-latitude,” J. Geophys. Res. 115, A10307 (2010).
    [CrossRef]
  3. F. J. Lübken, J. Höffner, T. P. Viehl, B. Kaifler, and R. J. Morris, “First measurements of thermal tides in the summer mesopause region at Antarctic latitudes,” Geophys. Res. Lett. 38, L24806 (2011).
    [CrossRef]
  4. R. E. Bills, C. S. Gardner, and S. F. Franke, “Na Doppler/temperature lidar: initial mesopause region observations and comparison with the Urbana MF radar,” J. Geophys. Res. 96, 22701–22707 (1991).
    [CrossRef]
  5. C. Y. She and J. R. Yu, “Simultaneous three-frequency Na lidar measurements of radial wind and temperature in the mesopause region,” Geophys. Res. Lett. 21, 1771–1774 (1994).
    [CrossRef]
  6. J. Lautenbach and J. Höffner, “Scanning iron temperature lidar for mesospheric temperature observation,” Appl. Opt. 43, 4559–4563 (2004).
    [CrossRef]
  7. T. Pfrommer and P. Hickson, “High-resolution lidar observations of mesospheric sodium and implications for adaptive optics,” J. Opt. Soc. Am. A 27, A97–A105 (2010).
    [CrossRef]
  8. C. S. Gardner and A. Z. Liu, “Measuring eddy heat and constituent fluxes with high-resolution Na and Fe Doppler lidars,” J. Geophys. Res. (in press).
  9. J. Höffner and J. S. Friedman, “The mesospheric metal layer topside: examples of simultaneous metal observations,” J. Atmos. Sol. Terr. Phys. 67, 1226–1237 (2005).
    [CrossRef]
  10. X. Chu, Z. Yu, C. S. Gardner, C. Chen, and W. Fong, “Lidar observations of neutral Fe layers and fast gravity waves in the thermosphere (110–155  km) at McMurdo (77.8°S, 166.7°E), Antarctica,” Geophys. Res. Lett. 38, L23807 (2011).
  11. A. J. Gerrard, T. J. Kane, D. D. Meisel, J. P. Thayer, and R. B. Kerr, “Investigation of a resonance lidar for measurement of thermospheric metastable helium,” J. Atmos. Sol. Terr. Phys. 59, 2023–2035 (1997).
    [CrossRef]
  12. C. G. Carlson, P. D. Dragic, R. K. Price, J. J. Coleman, and G. R. Swenson, “A narrow linewidth Yb fiber-amplified-based upper atmospheric Doppler temperature lidar,” IEEE J. Sel. Top. Quantum Electron. 15, 451–461 (2009).
    [CrossRef]
  13. B. M. Welsh and C. S. Gardner, “Nonlinear resonant absorption effects on the design of resonance fluorescence lidars and laser guide stars,” Appl. Opt. 28, 4141–4152 (1989).
    [CrossRef]
  14. P. von der Gathen, “Saturation effects in Na lidar temperature measurements,” J. Geophys. Res. 96, 3679–3690 (1991).
    [CrossRef]
  15. R. E. Bills, C. S. Gardner, and C. Y. She, “Narrowband lidar technique for Na temperature and Doppler wind observations of the upper atmosphere,” Opt. Eng. 30, 13–21 (1991).
    [CrossRef]
  16. C. Y. She, J. R. Yu, H. Latifi, and R. E. Bills, “High-spectral resolution fluorescence light detection and ranging for mesospheric sodium temperature measurements,” Appl. Opt. 31, 2095–2106 (1992).
    [CrossRef]
  17. G. C. Papen, W. Pfenninger, and D. Simonich, “Sensitivity analysis of Na narrowband wind-temperature lidar systems,” Appl. Opt. 34, 480–498 (1995).
    [CrossRef]
  18. C. S. Gardner, “Performance capabilities of middle-atmosphere temperature lidars: comparison of Na, Fe, K, Ca, Ca+, and Rayleigh systems,” Appl. Opt. 43, 4941–4956 (2004).
    [CrossRef]
  19. X. Chu and G. C. Papen, “Resonance fluorescence lidar for measurements of the middle and upper atmosphere,” in Laser Remote Sensing, T. Fujii and T. Fukuchi, eds. (CRC Press, 2005), pp. 179–432.
  20. L. Su, R. L. Collins, D. A. Krueger, and C. Y. She, “Statistical analysis of sodium Doppler wind-temperature lidar measurements of vertical heat flux,” J. Atmos. Ocean. Technol. 25, 401–415 (2008).
    [CrossRef]
  21. A. Corney, Atomic and Laser Spectroscopy (Oxford, 1977).
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    [CrossRef]
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    [CrossRef]
  26. M. S. Fee, K. Danzmann, and S. Chu, “Optical heterodyne measurement of pulsed lasers: toward high precision pulsed spectroscopy,” Phys. Rev. A 45, 4911–4924 (1992).
    [CrossRef]
  27. R. T. White, Y. He, B. J. Orr, M. Kono, and K. G. H. Baldwin, “Control of frequency chirp in nanosecond-pulsed laser spectroscopy. 1. Optical heterodyne chirp analysis techniques,” J. Opt. Soc. Am. B 21, 1577–1585 (2004).
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  28. X. Chu and W. Huang, “Fe Doppler-free spectroscopy and optical heterodyne detection for accurate frequency control of Fe-resonance Doppler lidar,” in Proceedings of 25th International Laser Radar Conference, St. Petersburg, Russia, July5–9 (Curran Associates, 2010), p. 1374.
  29. C. S. Gardner and A. Z. Liu, “Wave-induced transport of atmospheric constituents and its effect on the mesospheric Na layer,” J. Geophys. Res. 115, D20302 (2010).
    [CrossRef]
  30. K. H. Fricke and U. von Zahn, “Mesopause temperature derived from probing the hyperfine structure of the D2 resonance line of sodium by lidar,” J. Atmos. Terr. Phys. 47, 499–512 (1985).
    [CrossRef]
  31. L. S. Waldrop, R. B. Kerr, S. A. Gonzalez, M. P. Sulzer, J. Noto, and F. Kamalabadi, “Generation of metastable helium and the 1083  nm emission in the upper thermosphere,” J. Geophys. Res. 110, A08304 (2005).
    [CrossRef]
  32. C. S. Gardner and A. Z. Liu, “Seasonal variations of the vertical fluxes of heat and horizontal momentum in the mesopause region at Starfire Optical Range, New Mexico,” J. Geophys. Res. 112, D09113 (2007).
    [CrossRef]

2011 (2)

F. J. Lübken, J. Höffner, T. P. Viehl, B. Kaifler, and R. J. Morris, “First measurements of thermal tides in the summer mesopause region at Antarctic latitudes,” Geophys. Res. Lett. 38, L24806 (2011).
[CrossRef]

X. Chu, Z. Yu, C. S. Gardner, C. Chen, and W. Fong, “Lidar observations of neutral Fe layers and fast gravity waves in the thermosphere (110–155  km) at McMurdo (77.8°S, 166.7°E), Antarctica,” Geophys. Res. Lett. 38, L23807 (2011).

2010 (3)

J. Yue, C.-Y. She, and H.-L. Liu, “Large wind shears and stabilities in the mesopause region observed by Na wind-temperature lidar at mid-latitude,” J. Geophys. Res. 115, A10307 (2010).
[CrossRef]

T. Pfrommer and P. Hickson, “High-resolution lidar observations of mesospheric sodium and implications for adaptive optics,” J. Opt. Soc. Am. A 27, A97–A105 (2010).
[CrossRef]

C. S. Gardner and A. Z. Liu, “Wave-induced transport of atmospheric constituents and its effect on the mesospheric Na layer,” J. Geophys. Res. 115, D20302 (2010).
[CrossRef]

2009 (2)

T. Yuan, J. Yue, C. Y. She, J. P. Sherman, M. A. White, S. D. Harrell, P. E. Acott, and D. A. Kreuger, “Wind-bias correction method for narrowband sodium Doppler lidars using iodine absorption spectroscopy,” Appl. Opt. 48, 3988–3993 (2009).
[CrossRef]

C. G. Carlson, P. D. Dragic, R. K. Price, J. J. Coleman, and G. R. Swenson, “A narrow linewidth Yb fiber-amplified-based upper atmospheric Doppler temperature lidar,” IEEE J. Sel. Top. Quantum Electron. 15, 451–461 (2009).
[CrossRef]

2008 (1)

L. Su, R. L. Collins, D. A. Krueger, and C. Y. She, “Statistical analysis of sodium Doppler wind-temperature lidar measurements of vertical heat flux,” J. Atmos. Ocean. Technol. 25, 401–415 (2008).
[CrossRef]

2007 (1)

C. S. Gardner and A. Z. Liu, “Seasonal variations of the vertical fluxes of heat and horizontal momentum in the mesopause region at Starfire Optical Range, New Mexico,” J. Geophys. Res. 112, D09113 (2007).
[CrossRef]

2005 (2)

L. S. Waldrop, R. B. Kerr, S. A. Gonzalez, M. P. Sulzer, J. Noto, and F. Kamalabadi, “Generation of metastable helium and the 1083  nm emission in the upper thermosphere,” J. Geophys. Res. 110, A08304 (2005).
[CrossRef]

J. Höffner and J. S. Friedman, “The mesospheric metal layer topside: examples of simultaneous metal observations,” J. Atmos. Sol. Terr. Phys. 67, 1226–1237 (2005).
[CrossRef]

2004 (3)

1998 (1)

C. S. Gardner and W. Yang, “Measurements of the dynamical cooling rate associated with the vertical transport of heat by dissipating gravity waves in the mesopause region at the Starfire Optical Range, New Mexico,” J. Geophys. Res. 103, 16909–16926 (1998).
[CrossRef]

1997 (1)

A. J. Gerrard, T. J. Kane, D. D. Meisel, J. P. Thayer, and R. B. Kerr, “Investigation of a resonance lidar for measurement of thermospheric metastable helium,” J. Atmos. Sol. Terr. Phys. 59, 2023–2035 (1997).
[CrossRef]

1995 (2)

1994 (3)

1992 (2)

C. Y. She, J. R. Yu, H. Latifi, and R. E. Bills, “High-spectral resolution fluorescence light detection and ranging for mesospheric sodium temperature measurements,” Appl. Opt. 31, 2095–2106 (1992).
[CrossRef]

M. S. Fee, K. Danzmann, and S. Chu, “Optical heterodyne measurement of pulsed lasers: toward high precision pulsed spectroscopy,” Phys. Rev. A 45, 4911–4924 (1992).
[CrossRef]

1991 (3)

P. von der Gathen, “Saturation effects in Na lidar temperature measurements,” J. Geophys. Res. 96, 3679–3690 (1991).
[CrossRef]

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

R. E. Bills, C. S. Gardner, and S. F. Franke, “Na Doppler/temperature lidar: initial mesopause region observations and comparison with the Urbana MF radar,” J. Geophys. Res. 96, 22701–22707 (1991).
[CrossRef]

1989 (1)

1985 (1)

K. H. Fricke and U. von Zahn, “Mesopause temperature derived from probing the hyperfine structure of the D2 resonance line of sodium by lidar,” J. Atmos. Terr. Phys. 47, 499–512 (1985).
[CrossRef]

Acott, P. E.

Baldwin, K. G. H.

Bills, R. E.

C. Y. She, J. R. Yu, H. Latifi, and R. E. Bills, “High-spectral resolution fluorescence light detection and ranging for mesospheric sodium temperature measurements,” Appl. Opt. 31, 2095–2106 (1992).
[CrossRef]

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

R. E. Bills, C. S. Gardner, and S. F. Franke, “Na Doppler/temperature lidar: initial mesopause region observations and comparison with the Urbana MF radar,” J. Geophys. Res. 96, 22701–22707 (1991).
[CrossRef]

Carlson, C. G.

C. G. Carlson, P. D. Dragic, R. K. Price, J. J. Coleman, and G. R. Swenson, “A narrow linewidth Yb fiber-amplified-based upper atmospheric Doppler temperature lidar,” IEEE J. Sel. Top. Quantum Electron. 15, 451–461 (2009).
[CrossRef]

Chen, C.

X. Chu, Z. Yu, C. S. Gardner, C. Chen, and W. Fong, “Lidar observations of neutral Fe layers and fast gravity waves in the thermosphere (110–155  km) at McMurdo (77.8°S, 166.7°E), Antarctica,” Geophys. Res. Lett. 38, L23807 (2011).

Chu, S.

M. S. Fee, K. Danzmann, and S. Chu, “Optical heterodyne measurement of pulsed lasers: toward high precision pulsed spectroscopy,” Phys. Rev. A 45, 4911–4924 (1992).
[CrossRef]

Chu, X.

X. Chu, Z. Yu, C. S. Gardner, C. Chen, and W. Fong, “Lidar observations of neutral Fe layers and fast gravity waves in the thermosphere (110–155  km) at McMurdo (77.8°S, 166.7°E), Antarctica,” Geophys. Res. Lett. 38, L23807 (2011).

X. Chu and G. C. Papen, “Resonance fluorescence lidar for measurements of the middle and upper atmosphere,” in Laser Remote Sensing, T. Fujii and T. Fukuchi, eds. (CRC Press, 2005), pp. 179–432.

X. Chu and W. Huang, “Fe Doppler-free spectroscopy and optical heterodyne detection for accurate frequency control of Fe-resonance Doppler lidar,” in Proceedings of 25th International Laser Radar Conference, St. Petersburg, Russia, July5–9 (Curran Associates, 2010), p. 1374.

Coleman, J. J.

C. G. Carlson, P. D. Dragic, R. K. Price, J. J. Coleman, and G. R. Swenson, “A narrow linewidth Yb fiber-amplified-based upper atmospheric Doppler temperature lidar,” IEEE J. Sel. Top. Quantum Electron. 15, 451–461 (2009).
[CrossRef]

Collins, R. L.

L. Su, R. L. Collins, D. A. Krueger, and C. Y. She, “Statistical analysis of sodium Doppler wind-temperature lidar measurements of vertical heat flux,” J. Atmos. Ocean. Technol. 25, 401–415 (2008).
[CrossRef]

Corney, A.

A. Corney, Atomic and Laser Spectroscopy (Oxford, 1977).

Danzmann, K.

M. S. Fee, K. Danzmann, and S. Chu, “Optical heterodyne measurement of pulsed lasers: toward high precision pulsed spectroscopy,” Phys. Rev. A 45, 4911–4924 (1992).
[CrossRef]

Dragic, P. D.

C. G. Carlson, P. D. Dragic, R. K. Price, J. J. Coleman, and G. R. Swenson, “A narrow linewidth Yb fiber-amplified-based upper atmospheric Doppler temperature lidar,” IEEE J. Sel. Top. Quantum Electron. 15, 451–461 (2009).
[CrossRef]

Eyler, E. E.

Fee, M. S.

M. S. Fee, K. Danzmann, and S. Chu, “Optical heterodyne measurement of pulsed lasers: toward high precision pulsed spectroscopy,” Phys. Rev. A 45, 4911–4924 (1992).
[CrossRef]

Fong, W.

X. Chu, Z. Yu, C. S. Gardner, C. Chen, and W. Fong, “Lidar observations of neutral Fe layers and fast gravity waves in the thermosphere (110–155  km) at McMurdo (77.8°S, 166.7°E), Antarctica,” Geophys. Res. Lett. 38, L23807 (2011).

Franke, S. F.

R. E. Bills, C. S. Gardner, and S. F. Franke, “Na Doppler/temperature lidar: initial mesopause region observations and comparison with the Urbana MF radar,” J. Geophys. Res. 96, 22701–22707 (1991).
[CrossRef]

Fricke, K. H.

K. H. Fricke and U. von Zahn, “Mesopause temperature derived from probing the hyperfine structure of the D2 resonance line of sodium by lidar,” J. Atmos. Terr. Phys. 47, 499–512 (1985).
[CrossRef]

Friedman, J. S.

J. Höffner and J. S. Friedman, “The mesospheric metal layer topside: examples of simultaneous metal observations,” J. Atmos. Sol. Terr. Phys. 67, 1226–1237 (2005).
[CrossRef]

Gangopadhyay, S.

Gardner, C. S.

X. Chu, Z. Yu, C. S. Gardner, C. Chen, and W. Fong, “Lidar observations of neutral Fe layers and fast gravity waves in the thermosphere (110–155  km) at McMurdo (77.8°S, 166.7°E), Antarctica,” Geophys. Res. Lett. 38, L23807 (2011).

C. S. Gardner and A. Z. Liu, “Wave-induced transport of atmospheric constituents and its effect on the mesospheric Na layer,” J. Geophys. Res. 115, D20302 (2010).
[CrossRef]

C. S. Gardner and A. Z. Liu, “Seasonal variations of the vertical fluxes of heat and horizontal momentum in the mesopause region at Starfire Optical Range, New Mexico,” J. Geophys. Res. 112, D09113 (2007).
[CrossRef]

C. S. Gardner, “Performance capabilities of middle-atmosphere temperature lidars: comparison of Na, Fe, K, Ca, Ca+, and Rayleigh systems,” Appl. Opt. 43, 4941–4956 (2004).
[CrossRef]

C. S. Gardner and W. Yang, “Measurements of the dynamical cooling rate associated with the vertical transport of heat by dissipating gravity waves in the mesopause region at the Starfire Optical Range, New Mexico,” J. Geophys. Res. 103, 16909–16926 (1998).
[CrossRef]

R. E. Bills, C. S. Gardner, and S. F. Franke, “Na Doppler/temperature lidar: initial mesopause region observations and comparison with the Urbana MF radar,” J. Geophys. Res. 96, 22701–22707 (1991).
[CrossRef]

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

B. M. Welsh and C. S. Gardner, “Nonlinear resonant absorption effects on the design of resonance fluorescence lidars and laser guide stars,” Appl. Opt. 28, 4141–4152 (1989).
[CrossRef]

C. S. Gardner and A. Z. Liu, “Measuring eddy heat and constituent fluxes with high-resolution Na and Fe Doppler lidars,” J. Geophys. Res. (in press).

Gerrard, A. J.

A. J. Gerrard, T. J. Kane, D. D. Meisel, J. P. Thayer, and R. B. Kerr, “Investigation of a resonance lidar for measurement of thermospheric metastable helium,” J. Atmos. Sol. Terr. Phys. 59, 2023–2035 (1997).
[CrossRef]

Gonzalez, S. A.

L. S. Waldrop, R. B. Kerr, S. A. Gonzalez, M. P. Sulzer, J. Noto, and F. Kamalabadi, “Generation of metastable helium and the 1083  nm emission in the upper thermosphere,” J. Geophys. Res. 110, A08304 (2005).
[CrossRef]

Harrell, S. D.

He, Y.

Hickson, P.

Höffner, J.

F. J. Lübken, J. Höffner, T. P. Viehl, B. Kaifler, and R. J. Morris, “First measurements of thermal tides in the summer mesopause region at Antarctic latitudes,” Geophys. Res. Lett. 38, L24806 (2011).
[CrossRef]

J. Höffner and J. S. Friedman, “The mesospheric metal layer topside: examples of simultaneous metal observations,” J. Atmos. Sol. Terr. Phys. 67, 1226–1237 (2005).
[CrossRef]

J. Lautenbach and J. Höffner, “Scanning iron temperature lidar for mesospheric temperature observation,” Appl. Opt. 43, 4559–4563 (2004).
[CrossRef]

Huang, W.

X. Chu and W. Huang, “Fe Doppler-free spectroscopy and optical heterodyne detection for accurate frequency control of Fe-resonance Doppler lidar,” in Proceedings of 25th International Laser Radar Conference, St. Petersburg, Russia, July5–9 (Curran Associates, 2010), p. 1374.

Kaifler, B.

F. J. Lübken, J. Höffner, T. P. Viehl, B. Kaifler, and R. J. Morris, “First measurements of thermal tides in the summer mesopause region at Antarctic latitudes,” Geophys. Res. Lett. 38, L24806 (2011).
[CrossRef]

Kamalabadi, F.

L. S. Waldrop, R. B. Kerr, S. A. Gonzalez, M. P. Sulzer, J. Noto, and F. Kamalabadi, “Generation of metastable helium and the 1083  nm emission in the upper thermosphere,” J. Geophys. Res. 110, A08304 (2005).
[CrossRef]

Kane, T. J.

A. J. Gerrard, T. J. Kane, D. D. Meisel, J. P. Thayer, and R. B. Kerr, “Investigation of a resonance lidar for measurement of thermospheric metastable helium,” J. Atmos. Sol. Terr. Phys. 59, 2023–2035 (1997).
[CrossRef]

Kerr, R. B.

L. S. Waldrop, R. B. Kerr, S. A. Gonzalez, M. P. Sulzer, J. Noto, and F. Kamalabadi, “Generation of metastable helium and the 1083  nm emission in the upper thermosphere,” J. Geophys. Res. 110, A08304 (2005).
[CrossRef]

A. J. Gerrard, T. J. Kane, D. D. Meisel, J. P. Thayer, and R. B. Kerr, “Investigation of a resonance lidar for measurement of thermospheric metastable helium,” J. Atmos. Sol. Terr. Phys. 59, 2023–2035 (1997).
[CrossRef]

Kono, M.

Kreuger, D. A.

Krueger, D. A.

L. Su, R. L. Collins, D. A. Krueger, and C. Y. She, “Statistical analysis of sodium Doppler wind-temperature lidar measurements of vertical heat flux,” J. Atmos. Ocean. Technol. 25, 401–415 (2008).
[CrossRef]

Latifi, H.

Lautenbach, J.

Liu, A. Z.

C. S. Gardner and A. Z. Liu, “Wave-induced transport of atmospheric constituents and its effect on the mesospheric Na layer,” J. Geophys. Res. 115, D20302 (2010).
[CrossRef]

C. S. Gardner and A. Z. Liu, “Seasonal variations of the vertical fluxes of heat and horizontal momentum in the mesopause region at Starfire Optical Range, New Mexico,” J. Geophys. Res. 112, D09113 (2007).
[CrossRef]

C. S. Gardner and A. Z. Liu, “Measuring eddy heat and constituent fluxes with high-resolution Na and Fe Doppler lidars,” J. Geophys. Res. (in press).

Liu, H.-L.

J. Yue, C.-Y. She, and H.-L. Liu, “Large wind shears and stabilities in the mesopause region observed by Na wind-temperature lidar at mid-latitude,” J. Geophys. Res. 115, A10307 (2010).
[CrossRef]

Lübken, F. J.

F. J. Lübken, J. Höffner, T. P. Viehl, B. Kaifler, and R. J. Morris, “First measurements of thermal tides in the summer mesopause region at Antarctic latitudes,” Geophys. Res. Lett. 38, L24806 (2011).
[CrossRef]

Meisel, D. D.

A. J. Gerrard, T. J. Kane, D. D. Meisel, J. P. Thayer, and R. B. Kerr, “Investigation of a resonance lidar for measurement of thermospheric metastable helium,” J. Atmos. Sol. Terr. Phys. 59, 2023–2035 (1997).
[CrossRef]

Melikechi, N.

Morris, R. J.

F. J. Lübken, J. Höffner, T. P. Viehl, B. Kaifler, and R. J. Morris, “First measurements of thermal tides in the summer mesopause region at Antarctic latitudes,” Geophys. Res. Lett. 38, L24806 (2011).
[CrossRef]

Noto, J.

L. S. Waldrop, R. B. Kerr, S. A. Gonzalez, M. P. Sulzer, J. Noto, and F. Kamalabadi, “Generation of metastable helium and the 1083  nm emission in the upper thermosphere,” J. Geophys. Res. 110, A08304 (2005).
[CrossRef]

Orr, B. J.

Papen, G. C.

G. C. Papen, W. Pfenninger, and D. Simonich, “Sensitivity analysis of Na narrowband wind-temperature lidar systems,” Appl. Opt. 34, 480–498 (1995).
[CrossRef]

X. Chu and G. C. Papen, “Resonance fluorescence lidar for measurements of the middle and upper atmosphere,” in Laser Remote Sensing, T. Fujii and T. Fukuchi, eds. (CRC Press, 2005), pp. 179–432.

Pfenninger, W.

Pfrommer, T.

Price, R. K.

C. G. Carlson, P. D. Dragic, R. K. Price, J. J. Coleman, and G. R. Swenson, “A narrow linewidth Yb fiber-amplified-based upper atmospheric Doppler temperature lidar,” IEEE J. Sel. Top. Quantum Electron. 15, 451–461 (2009).
[CrossRef]

She, C. Y.

T. Yuan, J. Yue, C. Y. She, J. P. Sherman, M. A. White, S. D. Harrell, P. E. Acott, and D. A. Kreuger, “Wind-bias correction method for narrowband sodium Doppler lidars using iodine absorption spectroscopy,” Appl. Opt. 48, 3988–3993 (2009).
[CrossRef]

L. Su, R. L. Collins, D. A. Krueger, and C. Y. She, “Statistical analysis of sodium Doppler wind-temperature lidar measurements of vertical heat flux,” J. Atmos. Ocean. Technol. 25, 401–415 (2008).
[CrossRef]

C. Y. She and J. R. Yu, “Doppler-free saturation fluorescence spectroscopy of Na atoms for atmospheric applications,” Appl. Opt. 34, 1063–1075 (1995).
[CrossRef]

C. Y. She and J. R. Yu, “Simultaneous three-frequency Na lidar measurements of radial wind and temperature in the mesopause region,” Geophys. Res. Lett. 21, 1771–1774 (1994).
[CrossRef]

C. Y. She, J. R. Yu, H. Latifi, and R. E. Bills, “High-spectral resolution fluorescence light detection and ranging for mesospheric sodium temperature measurements,” Appl. Opt. 31, 2095–2106 (1992).
[CrossRef]

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

She, C.-Y.

J. Yue, C.-Y. She, and H.-L. Liu, “Large wind shears and stabilities in the mesopause region observed by Na wind-temperature lidar at mid-latitude,” J. Geophys. Res. 115, A10307 (2010).
[CrossRef]

Sherman, J. P.

Simonich, D.

Su, L.

L. Su, R. L. Collins, D. A. Krueger, and C. Y. She, “Statistical analysis of sodium Doppler wind-temperature lidar measurements of vertical heat flux,” J. Atmos. Ocean. Technol. 25, 401–415 (2008).
[CrossRef]

Sulzer, M. P.

L. S. Waldrop, R. B. Kerr, S. A. Gonzalez, M. P. Sulzer, J. Noto, and F. Kamalabadi, “Generation of metastable helium and the 1083  nm emission in the upper thermosphere,” J. Geophys. Res. 110, A08304 (2005).
[CrossRef]

Swenson, G. R.

C. G. Carlson, P. D. Dragic, R. K. Price, J. J. Coleman, and G. R. Swenson, “A narrow linewidth Yb fiber-amplified-based upper atmospheric Doppler temperature lidar,” IEEE J. Sel. Top. Quantum Electron. 15, 451–461 (2009).
[CrossRef]

Thayer, J. P.

A. J. Gerrard, T. J. Kane, D. D. Meisel, J. P. Thayer, and R. B. Kerr, “Investigation of a resonance lidar for measurement of thermospheric metastable helium,” J. Atmos. Sol. Terr. Phys. 59, 2023–2035 (1997).
[CrossRef]

Viehl, T. P.

F. J. Lübken, J. Höffner, T. P. Viehl, B. Kaifler, and R. J. Morris, “First measurements of thermal tides in the summer mesopause region at Antarctic latitudes,” Geophys. Res. Lett. 38, L24806 (2011).
[CrossRef]

von der Gathen, P.

P. von der Gathen, “Saturation effects in Na lidar temperature measurements,” J. Geophys. Res. 96, 3679–3690 (1991).
[CrossRef]

von Zahn, U.

K. H. Fricke and U. von Zahn, “Mesopause temperature derived from probing the hyperfine structure of the D2 resonance line of sodium by lidar,” J. Atmos. Terr. Phys. 47, 499–512 (1985).
[CrossRef]

Waldrop, L. S.

L. S. Waldrop, R. B. Kerr, S. A. Gonzalez, M. P. Sulzer, J. Noto, and F. Kamalabadi, “Generation of metastable helium and the 1083  nm emission in the upper thermosphere,” J. Geophys. Res. 110, A08304 (2005).
[CrossRef]

Welsh, B. M.

White, M. A.

White, R. T.

Yang, W.

C. S. Gardner and W. Yang, “Measurements of the dynamical cooling rate associated with the vertical transport of heat by dissipating gravity waves in the mesopause region at the Starfire Optical Range, New Mexico,” J. Geophys. Res. 103, 16909–16926 (1998).
[CrossRef]

Yu, J. R.

Yu, Z.

X. Chu, Z. Yu, C. S. Gardner, C. Chen, and W. Fong, “Lidar observations of neutral Fe layers and fast gravity waves in the thermosphere (110–155  km) at McMurdo (77.8°S, 166.7°E), Antarctica,” Geophys. Res. Lett. 38, L23807 (2011).

Yuan, T.

Yue, J.

J. Yue, C.-Y. She, and H.-L. Liu, “Large wind shears and stabilities in the mesopause region observed by Na wind-temperature lidar at mid-latitude,” J. Geophys. Res. 115, A10307 (2010).
[CrossRef]

T. Yuan, J. Yue, C. Y. She, J. P. Sherman, M. A. White, S. D. Harrell, P. E. Acott, and D. A. Kreuger, “Wind-bias correction method for narrowband sodium Doppler lidars using iodine absorption spectroscopy,” Appl. Opt. 48, 3988–3993 (2009).
[CrossRef]

Appl. Opt. (7)

Geophys. Res. Lett. (3)

X. Chu, Z. Yu, C. S. Gardner, C. Chen, and W. Fong, “Lidar observations of neutral Fe layers and fast gravity waves in the thermosphere (110–155  km) at McMurdo (77.8°S, 166.7°E), Antarctica,” Geophys. Res. Lett. 38, L23807 (2011).

C. Y. She and J. R. Yu, “Simultaneous three-frequency Na lidar measurements of radial wind and temperature in the mesopause region,” Geophys. Res. Lett. 21, 1771–1774 (1994).
[CrossRef]

F. J. Lübken, J. Höffner, T. P. Viehl, B. Kaifler, and R. J. Morris, “First measurements of thermal tides in the summer mesopause region at Antarctic latitudes,” Geophys. Res. Lett. 38, L24806 (2011).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

C. G. Carlson, P. D. Dragic, R. K. Price, J. J. Coleman, and G. R. Swenson, “A narrow linewidth Yb fiber-amplified-based upper atmospheric Doppler temperature lidar,” IEEE J. Sel. Top. Quantum Electron. 15, 451–461 (2009).
[CrossRef]

J. Atmos. Ocean. Technol. (1)

L. Su, R. L. Collins, D. A. Krueger, and C. Y. She, “Statistical analysis of sodium Doppler wind-temperature lidar measurements of vertical heat flux,” J. Atmos. Ocean. Technol. 25, 401–415 (2008).
[CrossRef]

J. Atmos. Sol. Terr. Phys. (2)

J. Höffner and J. S. Friedman, “The mesospheric metal layer topside: examples of simultaneous metal observations,” J. Atmos. Sol. Terr. Phys. 67, 1226–1237 (2005).
[CrossRef]

A. J. Gerrard, T. J. Kane, D. D. Meisel, J. P. Thayer, and R. B. Kerr, “Investigation of a resonance lidar for measurement of thermospheric metastable helium,” J. Atmos. Sol. Terr. Phys. 59, 2023–2035 (1997).
[CrossRef]

J. Atmos. Terr. Phys. (1)

K. H. Fricke and U. von Zahn, “Mesopause temperature derived from probing the hyperfine structure of the D2 resonance line of sodium by lidar,” J. Atmos. Terr. Phys. 47, 499–512 (1985).
[CrossRef]

J. Geophys. Res. (7)

L. S. Waldrop, R. B. Kerr, S. A. Gonzalez, M. P. Sulzer, J. Noto, and F. Kamalabadi, “Generation of metastable helium and the 1083  nm emission in the upper thermosphere,” J. Geophys. Res. 110, A08304 (2005).
[CrossRef]

C. S. Gardner and A. Z. Liu, “Seasonal variations of the vertical fluxes of heat and horizontal momentum in the mesopause region at Starfire Optical Range, New Mexico,” J. Geophys. Res. 112, D09113 (2007).
[CrossRef]

C. S. Gardner and A. Z. Liu, “Wave-induced transport of atmospheric constituents and its effect on the mesospheric Na layer,” J. Geophys. Res. 115, D20302 (2010).
[CrossRef]

P. von der Gathen, “Saturation effects in Na lidar temperature measurements,” J. Geophys. Res. 96, 3679–3690 (1991).
[CrossRef]

R. E. Bills, C. S. Gardner, and S. F. Franke, “Na Doppler/temperature lidar: initial mesopause region observations and comparison with the Urbana MF radar,” J. Geophys. Res. 96, 22701–22707 (1991).
[CrossRef]

C. S. Gardner and W. Yang, “Measurements of the dynamical cooling rate associated with the vertical transport of heat by dissipating gravity waves in the mesopause region at the Starfire Optical Range, New Mexico,” J. Geophys. Res. 103, 16909–16926 (1998).
[CrossRef]

J. Yue, C.-Y. She, and H.-L. Liu, “Large wind shears and stabilities in the mesopause region observed by Na wind-temperature lidar at mid-latitude,” J. Geophys. Res. 115, A10307 (2010).
[CrossRef]

J. Opt. Soc. Am. A (1)

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

Opt. Eng. (1)

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

Phys. Rev. A (1)

M. S. Fee, K. Danzmann, and S. Chu, “Optical heterodyne measurement of pulsed lasers: toward high precision pulsed spectroscopy,” Phys. Rev. A 45, 4911–4924 (1992).
[CrossRef]

Other (4)

X. Chu and W. Huang, “Fe Doppler-free spectroscopy and optical heterodyne detection for accurate frequency control of Fe-resonance Doppler lidar,” in Proceedings of 25th International Laser Radar Conference, St. Petersburg, Russia, July5–9 (Curran Associates, 2010), p. 1374.

A. Corney, Atomic and Laser Spectroscopy (Oxford, 1977).

X. Chu and G. C. Papen, “Resonance fluorescence lidar for measurements of the middle and upper atmosphere,” in Laser Remote Sensing, T. Fujii and T. Fukuchi, eds. (CRC Press, 2005), pp. 179–432.

C. S. Gardner and A. Z. Liu, “Measuring eddy heat and constituent fluxes with high-resolution Na and Fe Doppler lidars,” J. Geophys. Res. (in press).

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Figures (3)

Fig. 1.
Fig. 1.

Na D2 absorption cross-section near 589 nm plotted for a temperature of 185 K. The long vertical lines denote the six hyperfine lines, which form the D2a and D2b groups (see Table 4). The laser frequencies used for nighttime (daytime) observations are denoted by closed circles (pluses) (see Table 7).

Fig. 2.
Fig. 2.

Fe absorption cross-section near 372 nm plotted for a temperature of 185 K. The long vertical lines denote the Fe54, Fe56, Fe57, and Fe58 isotope lines (see Table 5). The laser frequencies used for nighttime (daytime) observations are denoted by closed circles (pluses) (see Table 7).

Fig. 3.
Fig. 3.

He4(2S3) absorption cross-section near 1083 nm plotted for a temperature of 850 K. The long vertical lines denote the 1083.034 and 1083.025 nm transition lines (see Table 6). The laser frequencies used for nighttime (daytime) observations are denoted by closed circles (pluses) (see Table 7).

Tables (9)

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Table 1. Lidar Photon Count Sensitivities for a Gaussian Absorption Line Shape f0=fM, w=0, α=fδ2/(σM2+σLaser2)

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Table 2. Optimum Values for the Frequency Shift and Dwell Time for a Gaussian Absorption Line Shape f0=fM, w=0, α=[fδ2/(σM2+σLaser2)]

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Table 3. Photon Noise Errors for the Lidar Configuration that Optimizes Measurements of Heat and Constituent Transport: Gaussian Absorption Line Shape αOpt=[fδ2/(σM2+σLaser2)]={3Nighttime1.7323Daytime χOpt={25.5%Nighttime19.7%Daytime

Tables Icon

Table 4. Na23 589 nm D2 Transition Line Parameters

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Table 5. Fe 372 nm Transition Line Parameters

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Table 6. He4(2S3) 1083 nm Transition Line Parameters

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Table 7. Operating Parameters for Three-Frequency Na, Fe, and He Lidars

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Table 8. Photon Noise Errors and Flux Biases for the Optimum Lidar Configurations.

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Table 9. Impact of Laser Calibration, Random Frequency, and Random Line Width Errors for the Optimum Lidar Configurations at Nighttime

Equations (28)

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NM(f+ΔfLaser,σLaser+ΔσLaser,w,T,ρM)=N¯M(f+ΔfLaser,σLaser+ΔσLaser,w,T,ρM)+ΔNMN¯M(f,σLaser,w,T,ρM)+N¯MfΔfLaser+N¯MσLaserΔσLaser+ΔNMN¯M(f,σLaser,w+Δw,T+ΔT,ρM+ΔρM)N¯M(f,σLaser,w,T,ρM)+N¯MwΔw+N¯MTΔT+N¯MρMΔρMNM(f)=NMN¯MN¯MfΔfLaser+N¯MσLaserΔσLaser+ΔNMN¯MwΔwN¯MTΔTN¯MρMΔρM.
[1N¯N¯w1N¯N¯T1N¯N¯ρM1N¯0N¯0w1N¯0N¯0T1N¯0N¯0ρM1N¯+N¯+w1N¯+N¯+T1N¯+N¯+ρM]·[ΔwΔTΔρM]=[1N¯N¯fΔf+1N¯N¯σΔσ+ΔNN¯1N¯0N¯0f0Δf0+1N¯0N¯0σ0Δσ0+ΔN0N¯01N¯+N¯+f+Δf++1N¯+N¯+σ+Δσ++ΔN+N¯+],
f=f0fδf0=f0f+=f0+fδ.
[ΔwΔTΔρM]=[ww0w+TT0T+ρρ0ρ+]·[1N¯N¯fΔf+1N¯N¯σΔσ+ΔNN¯1N¯0N¯0f0Δf0+1N¯0N¯0σ0Δσ0+ΔN0N¯01N¯+N¯+f+Δf++1N¯+N¯+σ+Δσ++ΔN+N¯+],[ww0w+TT0T+ρρ0ρ+]=[1N¯N¯w1N¯N¯T1N¯N¯ρM1N¯0N¯0w1N¯0N¯0T1N¯0N¯0ρM1N¯+N¯+w1N¯+N¯+T1N¯+N¯+ρM]1.
SNR(f)=N¯M2(f)N¯M(f)+N¯B=N¯M(f)1+β1β=N¯M(f)N¯B;{1nighttime1daytime,
Var(Δw)=[2w2(1χ)N¯0N¯(β+N¯0/N¯)(β+1)+w02χ+2w+2(1χ)N¯0N¯+(β+N¯0/N¯+)(β+1)]1SNR0Var(ΔT)=[2T2(1χ)N¯0N¯(β+N¯0/N¯)(β+1)+T02χ+2T+2(1χ)N¯0N¯+(β+N¯0/N¯+)(β+1)]1SNR0Var(ΔρM)=[2ρ2(1χ)N¯0N¯(β+N¯0/N¯)(β+1)+ρ02χ+2ρ+2(1χ)N¯0N¯+(β+N¯0/N¯+)(β+1)]1SNR0Cov(Δw,ΔT)=[2wT(1χ)N¯0N¯(β+N¯0/N¯)(β+1)+w0T0χ+2w+T+(1χ)N¯0N¯+(β+N¯0/N¯+)(β+1)]1SNR0Cov(Δw,ΔρM)=[2wρ(1χ)N¯0N¯(β+N¯0/N¯)(β+1)+w0ρ0χ+2w+ρ+(1χ)N¯0N¯+(β+N¯0/N¯+)(β+1)]1SNR0Cov(ΔT,ΔρM)=[2Tρ(1χ)N¯0N¯(β+N¯0/N¯)(β+1)+T0ρ0χ+2T+ρ+(1χ)N¯0N¯+(β+N¯0/N¯+)(β+1)]1SNR0,
ΔfLaser=ΔfCal+ΔfJitter+ΔfChirpΔσLaser=ΔσCal+ΔσJitter+ΔσChirp,
N¯M(f,z,t)ηTAtmos2PLaserATeleαeffM(f,z)σeffM(f,w,T,σLaser)ρM(z,t)ΔzΔt4πz2hf,
σeffM(f,w,T,σLaser)=(e2fosc/4ε0mec)2πexp[(ffMw/λM)22(σM2+σLaser2)]σM2+σLaser2,
σM2=γMTγM=kBλM2mM.
Δwrms=[eα(1+βeα/2)α(1χ)(1+β)]1/2·λMσM2+σLaser2SNR0ΔTrms=2α[eα(1+βeα/2)(1χ)(1+β)+1χ]1/2·(T+σLaser2/γM)SNR0(ΔρM)rms=1α[eα(1+βeα/2)(1χ)(1+β)+(α1)2χ]1/2ρMSNR0Cov(Δw,ΔT)=0Cov(Δw,ΔρM)=0Cov(ΔT,ΔρM)=2α2[eα(1+βeα/2)(1χ)(1+β)(α1)χ]·(T+σLaser2/γM)·ρMSNR0whereα=fδ2(σM2+σLaser2)f0=fM.
wBias=λMΔfCalTBias=2σLaserΔσCal/γM(ρM)Bias=0.
Δwrms=λMVar(ΔfJitter)+Var(ΔfChirp)2=α(Var(ΔfJitter)fδ2+Var(ΔfChirp)2fδ2)·λMσM2+σLaser2ΔTrms=2(ΔfChirp)rmsfδ·(T+σLaser2/γM)(ΔρM)rms=(ΔfChirp)rms2fδ·ρMCov(Δw,ΔT)=0Cov(Δw,ΔρM)=0Cov(ΔT,ΔρM)=Var(ΔfChirp)fδ2·(T+σLaser2/γM)·ρM.
Δwrms=α/2|α1|σLaserfδ(ΔσChirp)rmsfδ·λMσM2+σLaser2ΔTrms=2σLaserfδ{α2Var(ΔσJitter)fδ2+[1+(α1)22]Var(ΔσChirp)fδ2}1/2·(T+σLaser2/γM)(ΔρM)rms=3/2|α1|σLaserfδ(ΔσChirp)rmsfδ·ρMCov(Δw,ΔT)=0Cov(Δw,ΔρM)=0Cov(ΔT,ΔρM)=(α1)(α3)σLaser2fδ2Var(ΔσChirp)fδ2·(T+σLaser2/γM)·ρM.
w/T/ρMError Metric=(ΔwrmsλMσM)2+(ΔTrmsT)2+((ΔρM)rmsρM)2.
Vertical Flux of Heat=HF=Cov(w,T)Vertical Flux of SpeciesM=CF=Cov(w,ρM).
ΔHFrms=Δwrms·ΔTrmsLτ/ΔzΔtΔCFrms=Δwrms·(ΔρM)rmsLτ/ΔzΔt,
HFBias=Cov(Δw,ΔT)CFBias=Cov(Δw,ΔρM).
Transport Velocity Error Metric=(Δwrms·ΔTrmsT)2+(Δwrms·(ΔρM)rmsρM)2.
Var(ΔfJitter)=Var(ΔfChirp)=Var(ΔfLaser)/2Var(ΔσJitter)=Var(ΔσChirp)=Var(ΔσLaser)/2.
|2N¯w2|Δwrms22|N¯w|Δwrms|2N¯T2|ΔTrms22|N¯T|ΔTrms|2N¯f2|ΔfLaser22|N¯f|ΔfLaser|2N¯σLaser2|ΔσLaser22|N¯σLaser|ΔσLaser.
ΔwrmsλMσM5α{λMσM/9NighttimeλMσM/7DaytimeΔTrmsT/7.5SNR0{250Nighttime900DaytimeΔfLaserσM/10ΔσLaserσLaser/5.
a(f)=1N¯M(f)N¯M(f)wb(f)=1N¯M(f)N¯M(f)Tc(f)=1N¯M(f)N¯M(f)ρM=1ρM
[1N¯N¯w1N¯N¯T1N¯N¯ρM1N¯0N¯0w1N¯0N¯0T1N¯0N¯0ρM1N¯+N¯+w1N¯+N¯+T1N¯+N¯+ρM]=[1N¯N¯w1N¯N¯T1ρM1N¯0N¯0w1N¯0N¯0T1ρM1N¯+N¯+w1N¯+N¯+T1ρM]=[abca0b0c0a+b+c+],
[ww0w+TT0T+ρρ0ρ+]=[(b+b0)(bb+)(b0b)(a0a+)(a+a)(aa0)(a+b0a0b+)ρM(ab+a+b)ρM(a0bab0)ρM][a(b+b0)+a0(bb+)+a+(b0b)].
wBias=(wN¯N¯f+w0N¯0N¯0f0+w+N¯+N¯+f+)ΔfCal+(wN¯N¯σ+w0N¯0N¯0σ0+w+N¯+N¯+σ+)ΔσCalTBias=(TN¯N¯f+T0N¯0N¯0f0+T+N¯+N¯+f+)ΔfCal+(TN¯N¯σ+T0N¯0N¯0σ0+T+N¯+N¯+σ+)ΔσCal(ρM)Bias=(ρN¯N¯f+ρ0N¯0N¯0f0+ρ+N¯+N¯+f+)ΔfCal+(ρN¯N¯σ+ρ0N¯0N¯0σ0+ρ+N¯+N¯+σ+)ΔσCal.
Var(Δw)=(wN¯N¯f+w0N¯0N¯0f0+w+N¯+N¯+f+)2Var(ΔfJitter)+[(wN¯N¯f)2+(w0N¯0N¯0f0)2+(w+N¯+N¯+f+)2]Var(ΔfChirp)Var(ΔT)=(TN¯N¯f+T0N¯0N¯0f0+T+N¯+N¯+f+)2Var(ΔfJitter)+[(TN¯N¯f)2+(T0N¯0N¯0f0)2+(T+N¯+N¯+f+)2]Var(ΔfChirp)Var(ΔρM)=(ρN¯N¯f+ρ0N¯0N¯0f0+ρ+N¯+N¯+f+)2Var(ΔfJitter)+[(ρN¯N¯f)2+(ρ0N¯0N¯0f0)2+(ρ+N¯+N¯+f+)2]Var(ΔfChirp)Cov(Δw,ΔT)=(wN¯N¯f+w0N¯0N¯0f0+w+N¯+N¯+f+)(TN¯N¯f+T0N¯0N¯0f0+T+N¯+N¯+f+)Var(ΔfJitter)+[(wN¯N¯f)(TN¯N¯f)+(w0N¯0N¯0f0)(T0N¯0N¯0f0)+(w+N¯+N¯+f+)(T+N¯+N¯+f+)]Var(ΔfChirp)Cov(Δw,ΔρM)=(wN¯N¯f+w0N¯0N¯0f0+w+N¯+N¯+f+)(ρN¯N¯f+ρ0N¯0N¯0f0+ρ+N¯+N¯+f+)Var(ΔfJitter)[(wN¯N¯f)(ρN¯N¯f)+(w0N¯0N¯0f0)(ρ0N¯0N¯0f0)+(w+N¯+N¯+f+)(ρ+N¯+N¯+f+)]Var(ΔfChirp)Cov(ΔT,ΔρM)=(TN¯N¯f+T0N¯0N¯0f0+T+N¯+N¯+f+)(ρN¯N¯f+ρ0N¯0N¯0f0+ρ+N¯+N¯+f+)Var(ΔfJitter)[(TN¯N¯f)(ρN¯N¯f)+(T0N¯0N¯0f0)(ρ0N¯0N¯0f0)+(T+N¯+N¯+f+)(ρ+N¯+N¯+f+)]Var(ΔfChirp).
Var(Δw)=(wN¯N¯σ+w0N¯0N¯0σ0+w+N¯+N¯+σ+)2Var(ΔσJitter)[(wN¯N¯σ)2+(w0N¯0N¯0σ0)2+(w+N¯+N¯+σ+)2]Var(ΔσChirp)Var(ΔT)=(TN¯N¯σ+T0N¯0N¯0σ0+T+N¯+N¯+σ+)2Var(ΔσJitter)[(TN¯N¯σ)2+(T0N¯0N¯0σ0)2+(T+N¯+N¯+σ+)2]Var(ΔσChirp)Var(ΔρM)=(ρN¯N¯σ+ρ0N¯0N¯0σ0+ρ+N¯+N¯+σ+)2Var(ΔσJitter)[(ρN¯N¯σ)2+(ρ0N¯0N¯0σ0)2+(ρ+N¯+N¯+σ+)2]Var(ΔσChirp)Cov(Δw,ΔT)=(wN¯N¯σ+w0N¯0N¯0σ0+w+N¯+N¯+σ+)(TN¯N¯σ+T0N¯0N¯0σ0+T+N¯+N¯+σ+)Var(ΔσJitter)+[(wN¯N¯σ)(TN¯N¯σ)+(w0N¯0N¯0σ0)(T0N¯0N¯0σ0)+(w+N¯+N¯+σ+)(T+N¯+N¯+σ+)]Var(ΔσChirp)Cov(Δw,ΔρM)=(wN¯N¯σ+w0N¯0N¯0σ0+w+N¯+N¯+σ+)(ρN¯N¯σ+ρ0N¯0N¯0σ0+ρ+N¯+N¯+σ+)Var(ΔσJitter)[(wN¯N¯σ)(ρN¯N¯σ)+(w0N¯0N¯0σ0)(ρ0N¯0N¯0σ0)+(w+N¯+N¯+σ+)(ρ+N¯+N¯+σ+)]Var(ΔσChirp)Cov(ΔT,ΔρM)=(TN¯N¯σ+T0N¯0N¯0σ0+T+N¯+N¯+σ+)(ρN¯N¯σ+ρ0N¯0N¯0σ0+ρ+N¯+N¯+σ+)Var(ΔσJitter)[(TN¯N¯σ)(ρN¯N¯σ)+(T0N¯0N¯0σ0)(ρ0N¯0N¯0σ0)+(T+N¯+N¯+σ+)(ρ+N¯+N¯+σ+)]Var(ΔσChirp).

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