Abstract

Similar in principle to recent implementations of a lidar system at 355nm [Opt. Lett. 25, 1231 (2000), Appl. Opt. 44, 6023 (2005)], an incoherent-detection Mie Doppler wind lidar at 1064 nm was developed and deployed in 2005 [Opt. Rev. 12, 409 (2005)] for wind measurements in the low troposphere, taking advantage of aerosol scattering for signal enhancement. We present a number of improvements made to the original 1064nm system to increase its robustness for long-period operation. These include a multimode fiber for receiving the reference signal, a mode scrambler to allow uniform illumination over the Fabry–Perot interferometer, and a fast scannable Fabry–Perot interferometer for calibration and for the determination of outgoing laser frequency during the wind observation. With these improvements in stability, the standard deviation of peak transmission and FWHM of the Fabry–Perot interferometer was determined to be 0.49% and 0.36%, respectively. The lidar wind measurements were validated within a dynamic range of ±40m/s. Comparison experiments with both wind profiler radar and Vaisala wiresonde show good agreement with expected observation error. An example of 24 h continuous observations of wind field and aerosol backscatter coefficients in the boundary layer with 1 min and 30  m temporal and spatial resolution and 3 m∕s tolerated wind velocity error is presented and fully demonstrates the stability and robustness of this lidar.

© 2007 Optical Society of America

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References

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  1. World Meteorological Organization, "Preliminary statement of guidance regarding how well satellite capabilities meet WMO user requirements in several application areas," WMO Satellite Reports SAT-21, WMO/TD 913 (1998).
  2. W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. R Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, and J. McElroy, "Lidar-measured winds from space: a key component for weather and climate prediction," Bull. Am. Meteorol. Soc. 76, 869-888 (1995).
    [CrossRef]
  3. F. F. Hall, R. M. Huffaker, R. M. Hardesty, M. Jackson, T. R. Lawrence, M. Post, R. A. Richter, and B. F. Weber, "Wind measurement accuracy of the NOAA pulsed infrared Doppler," Appl. Opt. 23, 2503-2506 (1987).
    [CrossRef]
  4. R. M. Huffaker and R. M. Hardesty, "Remote sensing of atmospheric wind velocities using solid-state and CO2 coherent laser systems," Proc. IEEE 84, 181-204 (1996).
    [CrossRef]
  5. M. L. Chanin, A. Garnier, A. Hauchecorne, and J. Porteneuve, "A Doppler lidar for measuring winds in the middle atmosphere," Geophys. Res. Lett. 16, 1273-1276 (1989).
    [CrossRef]
  6. A. Garnier and M. L. Chanin, "Description of a Doppler Rayleigh lidar for measuring winds in the middle atmosphere," Appl. Phys. B 55, 35-40 (1992).
    [CrossRef]
  7. C. L. Korb, B. M. Gentry, S. X. Li, and C. Flesia, "Theory of the double-edge technique for Doppler lidar wind measurement," Appl. Opt. 37, 3097-3104 (1998).
    [CrossRef]
  8. C. Flesia and C. L. Korb, "Theory of the double-edge molecular technique for Doppler lidar wind measurement," Appl. Opt. 38, 432-440 (1999).
    [CrossRef]
  9. J. Wu, J. Wang, and P. B. Hays, "Performance of a circle-to-line optical system for a Fabry-Perot interferometer: a laboratory study," Appl. Opt. 33, 7823-7828 (1994).
    [CrossRef] [PubMed]
  10. T. D. Irgang, P. B. Hays, and W. R. Skinner, "Two-channel direct-detection Doppler lidar employing a charge-coupled device as a detector," Appl. Opt. 41, 1145-1155 (2002).
    [CrossRef] [PubMed]
  11. C. Nagasawa, Y. Shibata, M. Abo, T. Nagai, and O. Uchino, "Incoherent Doppler lidar using wavelengths for wind measurement," in Lidar Remote Sensing for Industry and Environment Monitoring, U. Singh, T. Itabe, and N. Sugimoto, eds., Proc. SPIE 4153, 338-349 (2001).
  12. Z.-S. Liu, D. Wu, J.-T. Liu, K.-L. Zhang, W.-B. Chen, X.-Q. Song, J. W. Hair, and C. Y. She, "Low-altitude atmospheric wind measurement from the combined Mie and Rayleigh backscattering by Doppler lidar with an iodine filter," Appl. Opt. 41, 7079-7086 (2002).
    [CrossRef] [PubMed]
  13. J. A. McKay, "Assessment of a multibeam Fizeau wedge interferometer for Doppler wind lidar," Appl. Opt. 41, 1760-1767 (2002).
    [CrossRef] [PubMed]
  14. D. Bruneau, A. Garnier, A. Hertzog, and J. Porteneuve, "Wind-velocity lidar measurements by use of a Mach-Zehnder interferometer, comparison with a Fabry-Perot interferometer," Appl. Opt. 43, 173-182 (2004).
    [CrossRef] [PubMed]
  15. J. A. McKay, "Modeling of direct-detection Doppler wind lidar. I. The edge technique," Appl. Opt. 37, 6480-6486 (1998).
    [CrossRef]
  16. J. A. McKay, "Modeling of direct-detection Doppler wind lidar. II. The fringe imaging technique," Appl. Opt. 37, 6487-6493 (1998).
    [CrossRef]
  17. M. J. McGill and J. D. Spinhirne, "Comparison of two direct-detection Doppler lidar techniques," Opt. Eng. 37, 2675-2686 (1998).
    [CrossRef]
  18. Y. Durand, A. Culoma, R. Meynart, D. Morançais, and F. Fabre, "Predevelopment of a direct-detection Doppler wind lidar for ADM/AEOLUS mission," in Sensors, Systems, and Next-Generation Satellites VII, R. Meynart, S. P. Neeck, H. Shimoda, J. B. Lurie, and M. L. Aten, eds., Proc. SPIE 5234, 354-363 (2004).
  19. C. Souprayen, A. Garnier, A. Hertzog, A. Hauchecorne, and J. Porteneuve, "Rayleigh-Mie Doppler wind lidar for atmospheric measurements. I. Instrumental setup, validation, and first climatological results," Appl. Opt. 38, 2410-2421 (1999).
    [CrossRef]
  20. C. Souprayen, A. Garnier, and A. Hertzog, "Rayleigh-Mie Doppler wind lidar for atmospheric measurements. II. Mie scattering effect, theory, and calibration," Appl. Opt. 38, 2422-2431 (1999).
    [CrossRef]
  21. C. Flesia, C. L. Korb, and C. Hirt, "Double-edge molecular measurement of lidar wind profiles at 355 nm," Opt. Lett. 25, 1466-1468 (2000).
    [CrossRef]
  22. B. M. Gentry, H. Chen, and S. X. Li, "Wind measurements with 355 nm molecular Doppler lidar," Opt. Lett. 25, 1231-1233 (2000).
    [CrossRef]
  23. V. J. Abreu, J. E. Barnes, and P. B. Hays, "Observations of winds with an incoherent lidar detector," Appl. Opt. 31, 4509-4514 (1992).
    [CrossRef] [PubMed]
  24. K. F. Fischer, V. J. Abreu, W. R. Skinner, J. E. Barnes, M. J. McGill, and 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]
  25. C. L. Korb, B. M. Gentry, and S. X. Li, "Edge technique Doppler lidar wind measurements with high vertical resolution," Appl. Opt. 36, 5976-5983 (1997).
    [CrossRef] [PubMed]
  26. M. J. McGill, W. R. Skinner, and T. D. Irgang, "Validation of wind profiles measured with incoherent Doppler lidar," Appl. Opt. 36, 1928-1938 (1997).
    [CrossRef] [PubMed]
  27. M. Imaki and T. Kobayashi, "Ultraviolet high-spectral-resolution Doppler lidar for measuring wind field and aerosol optical properties," Appl. Opt. 44, 6023-6030 (2005).
    [CrossRef] [PubMed]
  28. G. Tenti, C. D. Boley, and R. D. Desai, "On the kinetic model description of Rayleigh-Brillouin scattering from molecular gases," Can. J. Phys. 52, 285-290 (1974).
  29. J. D. Klett, "Stable analytical inversion solution for processing lidar returns," Appl. Opt. 20, 211-220 (1981).
    [CrossRef] [PubMed]
  30. J. D. Klett, "Lidar inversion with variable backscatter extinction ratios," Appl. Opt. 24, 1638-1643 (1985).
    [CrossRef] [PubMed]
  31. F. G. Fernald, "Analysis of atmospheric lidar observations: some comments," Appl. Opt. 23, 652-653 (1984).
    [CrossRef] [PubMed]
  32. D. Sun, Z., Zhong, J. Zhou, H. Hu, and T. Kobayashi, "Accuracy analysis of the Fabry-Perot based Doppler wind lidar," Opt. Rev. 12, 409-414 (2005).
    [CrossRef]
  33. C. J. Grand and E. Eloranta, "Fiber-optic scrambler reduces the bandpass range dependence of Fabry-Perot etalons used for spectral analysis of lidar backscatter," Appl. Opt. 30, 2668-2670 (1991).
    [CrossRef]
  34. T. Ida, M. Ando, and H. Toraya, "Extended pseudo-Voigt function for approximating the Voigt profile," J. Appl. Crystallogr. 33, 1311-1316 (2000).
    [CrossRef]
  35. H. Xia, D. Sun, Z. Zhong, B. Wang, J. Dong, F. Shen, M. Chen, and X. Zhou, "A design of verifying attachment for calibration of wind lidar," Chin. J. Lasers 33, 1412-1416 (2006) (in Chinese).
  36. B. J. Rye, "Comparative precision of distributed-backscatter Doppler lidars," Appl. Opt. 34, 8341-8344 (1995).
    [CrossRef] [PubMed]

2006 (1)

H. Xia, D. Sun, Z. Zhong, B. Wang, J. Dong, F. Shen, M. Chen, and X. Zhou, "A design of verifying attachment for calibration of wind lidar," Chin. J. Lasers 33, 1412-1416 (2006) (in Chinese).

2005 (2)

D. Sun, Z., Zhong, J. Zhou, H. Hu, and T. Kobayashi, "Accuracy analysis of the Fabry-Perot based Doppler wind lidar," Opt. Rev. 12, 409-414 (2005).
[CrossRef]

M. Imaki and T. Kobayashi, "Ultraviolet high-spectral-resolution Doppler lidar for measuring wind field and aerosol optical properties," Appl. Opt. 44, 6023-6030 (2005).
[CrossRef] [PubMed]

2004 (1)

2002 (3)

2000 (3)

1999 (3)

1998 (4)

1997 (2)

1996 (1)

R. M. Huffaker and R. M. Hardesty, "Remote sensing of atmospheric wind velocities using solid-state and CO2 coherent laser systems," Proc. IEEE 84, 181-204 (1996).
[CrossRef]

1995 (3)

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. R Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, and J. McElroy, "Lidar-measured winds from space: a key component for weather and climate prediction," Bull. Am. Meteorol. Soc. 76, 869-888 (1995).
[CrossRef]

K. F. Fischer, V. J. Abreu, W. R. Skinner, J. E. Barnes, M. J. McGill, and 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]

B. J. Rye, "Comparative precision of distributed-backscatter Doppler lidars," Appl. Opt. 34, 8341-8344 (1995).
[CrossRef] [PubMed]

1994 (1)

1992 (2)

A. Garnier and M. L. Chanin, "Description of a Doppler Rayleigh lidar for measuring winds in the middle atmosphere," Appl. Phys. B 55, 35-40 (1992).
[CrossRef]

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

1991 (1)

1989 (1)

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

1987 (1)

1985 (1)

1984 (1)

1981 (1)

1974 (1)

G. Tenti, C. D. Boley, and R. D. Desai, "On the kinetic model description of Rayleigh-Brillouin scattering from molecular gases," Can. J. Phys. 52, 285-290 (1974).

Appl. Opt. (21)

F. F. Hall, R. M. Huffaker, R. M. Hardesty, M. Jackson, T. R. Lawrence, M. Post, R. A. Richter, and B. F. Weber, "Wind measurement accuracy of the NOAA pulsed infrared Doppler," Appl. Opt. 23, 2503-2506 (1987).
[CrossRef]

C. L. Korb, B. M. Gentry, S. X. Li, and C. Flesia, "Theory of the double-edge technique for Doppler lidar wind measurement," Appl. Opt. 37, 3097-3104 (1998).
[CrossRef]

C. Flesia and C. L. Korb, "Theory of the double-edge molecular technique for Doppler lidar wind measurement," Appl. Opt. 38, 432-440 (1999).
[CrossRef]

J. Wu, J. Wang, and P. B. Hays, "Performance of a circle-to-line optical system for a Fabry-Perot interferometer: a laboratory study," Appl. Opt. 33, 7823-7828 (1994).
[CrossRef] [PubMed]

T. D. Irgang, P. B. Hays, and W. R. Skinner, "Two-channel direct-detection Doppler lidar employing a charge-coupled device as a detector," Appl. Opt. 41, 1145-1155 (2002).
[CrossRef] [PubMed]

Z.-S. Liu, D. Wu, J.-T. Liu, K.-L. Zhang, W.-B. Chen, X.-Q. Song, J. W. Hair, and C. Y. She, "Low-altitude atmospheric wind measurement from the combined Mie and Rayleigh backscattering by Doppler lidar with an iodine filter," Appl. Opt. 41, 7079-7086 (2002).
[CrossRef] [PubMed]

J. A. McKay, "Assessment of a multibeam Fizeau wedge interferometer for Doppler wind lidar," Appl. Opt. 41, 1760-1767 (2002).
[CrossRef] [PubMed]

D. Bruneau, A. Garnier, A. Hertzog, and J. Porteneuve, "Wind-velocity lidar measurements by use of a Mach-Zehnder interferometer, comparison with a Fabry-Perot interferometer," Appl. Opt. 43, 173-182 (2004).
[CrossRef] [PubMed]

J. A. McKay, "Modeling of direct-detection Doppler wind lidar. I. The edge technique," Appl. Opt. 37, 6480-6486 (1998).
[CrossRef]

J. A. McKay, "Modeling of direct-detection Doppler wind lidar. II. The fringe imaging technique," Appl. Opt. 37, 6487-6493 (1998).
[CrossRef]

C. Souprayen, A. Garnier, A. Hertzog, A. Hauchecorne, and J. Porteneuve, "Rayleigh-Mie Doppler wind lidar for atmospheric measurements. I. Instrumental setup, validation, and first climatological results," Appl. Opt. 38, 2410-2421 (1999).
[CrossRef]

C. Souprayen, A. Garnier, and A. Hertzog, "Rayleigh-Mie Doppler wind lidar for atmospheric measurements. II. Mie scattering effect, theory, and calibration," Appl. Opt. 38, 2422-2431 (1999).
[CrossRef]

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

C. L. Korb, B. M. Gentry, and S. X. Li, "Edge technique Doppler lidar wind measurements with high vertical resolution," Appl. Opt. 36, 5976-5983 (1997).
[CrossRef] [PubMed]

M. J. McGill, W. R. Skinner, and T. D. Irgang, "Validation of wind profiles measured with incoherent Doppler lidar," Appl. Opt. 36, 1928-1938 (1997).
[CrossRef] [PubMed]

M. Imaki and T. Kobayashi, "Ultraviolet high-spectral-resolution Doppler lidar for measuring wind field and aerosol optical properties," Appl. Opt. 44, 6023-6030 (2005).
[CrossRef] [PubMed]

J. D. Klett, "Stable analytical inversion solution for processing lidar returns," Appl. Opt. 20, 211-220 (1981).
[CrossRef] [PubMed]

J. D. Klett, "Lidar inversion with variable backscatter extinction ratios," Appl. Opt. 24, 1638-1643 (1985).
[CrossRef] [PubMed]

F. G. Fernald, "Analysis of atmospheric lidar observations: some comments," Appl. Opt. 23, 652-653 (1984).
[CrossRef] [PubMed]

C. J. Grand and E. Eloranta, "Fiber-optic scrambler reduces the bandpass range dependence of Fabry-Perot etalons used for spectral analysis of lidar backscatter," Appl. Opt. 30, 2668-2670 (1991).
[CrossRef]

B. J. Rye, "Comparative precision of distributed-backscatter Doppler lidars," Appl. Opt. 34, 8341-8344 (1995).
[CrossRef] [PubMed]

Appl. Phys. B (1)

A. Garnier and M. L. Chanin, "Description of a Doppler Rayleigh lidar for measuring winds in the middle atmosphere," Appl. Phys. B 55, 35-40 (1992).
[CrossRef]

Bull. Am. Meteorol. Soc. (1)

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. R Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, and J. McElroy, "Lidar-measured winds from space: a key component for weather and climate prediction," Bull. Am. Meteorol. Soc. 76, 869-888 (1995).
[CrossRef]

Can. J. Phys. (1)

G. Tenti, C. D. Boley, and R. D. Desai, "On the kinetic model description of Rayleigh-Brillouin scattering from molecular gases," Can. J. Phys. 52, 285-290 (1974).

Chin. J. Lasers (1)

H. Xia, D. Sun, Z. Zhong, B. Wang, J. Dong, F. Shen, M. Chen, and X. Zhou, "A design of verifying attachment for calibration of wind lidar," Chin. J. Lasers 33, 1412-1416 (2006) (in Chinese).

Geophys. Res. Lett. (1)

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

J. Appl. Crystallogr. (1)

T. Ida, M. Ando, and H. Toraya, "Extended pseudo-Voigt function for approximating the Voigt profile," J. Appl. Crystallogr. 33, 1311-1316 (2000).
[CrossRef]

Opt. Eng. (2)

K. F. Fischer, V. J. Abreu, W. R. Skinner, J. E. Barnes, M. J. McGill, and 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 and J. D. Spinhirne, "Comparison of two direct-detection Doppler lidar techniques," Opt. Eng. 37, 2675-2686 (1998).
[CrossRef]

Opt. Lett. (2)

Opt. Rev. (1)

D. Sun, Z., Zhong, J. Zhou, H. Hu, and T. Kobayashi, "Accuracy analysis of the Fabry-Perot based Doppler wind lidar," Opt. Rev. 12, 409-414 (2005).
[CrossRef]

Proc. IEEE (1)

R. M. Huffaker and R. M. Hardesty, "Remote sensing of atmospheric wind velocities using solid-state and CO2 coherent laser systems," Proc. IEEE 84, 181-204 (1996).
[CrossRef]

Other (3)

World Meteorological Organization, "Preliminary statement of guidance regarding how well satellite capabilities meet WMO user requirements in several application areas," WMO Satellite Reports SAT-21, WMO/TD 913 (1998).

C. Nagasawa, Y. Shibata, M. Abo, T. Nagai, and O. Uchino, "Incoherent Doppler lidar using wavelengths for wind measurement," in Lidar Remote Sensing for Industry and Environment Monitoring, U. Singh, T. Itabe, and N. Sugimoto, eds., Proc. SPIE 4153, 338-349 (2001).

Y. Durand, A. Culoma, R. Meynart, D. Morançais, and F. Fabre, "Predevelopment of a direct-detection Doppler wind lidar for ADM/AEOLUS mission," in Sensors, Systems, and Next-Generation Satellites VII, R. Meynart, S. P. Neeck, H. Shimoda, J. B. Lurie, and M. L. Aten, eds., Proc. SPIE 5234, 354-363 (2004).

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Figures (13)

Fig. 1
Fig. 1

Spectral profiles of the FPI, laser, and atmospheric backscatter signals.

Fig. 2
Fig. 2

Schematic of the lidar system: BS, beam splitter; P, prism; MF, multimode fiber; FOBS, fiber-optic beam splitter; IF, interference filter; SPCM, single photon counting mode; FPI, Fabry–Perot interferemeter.

Fig. 3
Fig. 3

Data acquisition timing sequence.

Fig. 4
Fig. 4

Illumination patterns under the function of the mode scrambler.

Fig. 5
Fig. 5

FPI transmission curves for the pulsed laser.

Fig. 6
Fig. 6

Schematic of the verifying attachment.

Fig. 7
Fig. 7

Calibration: comparison between two results.

Fig. 8
Fig. 8

Velocity accuracy versus total incident photons at LOS speed of 5 m / s . The raw data are shown as circles, and the experimental standard deviations are shown as error bars. The theoretical standard deviations described in this paper are shown as solid curves. The statistical noise measurement limit proposed by McKay [15] are shown as dashed curves with squares.

Fig. 9
Fig. 9

Horizontal wind and direction profiles measured by Doppler wind lidar (solid curve) and wind profiler radar (circle).

Fig. 10
Fig. 10

Standard deviation of the LOS wind as a function of altitude for 50-shot averages with a sample of 1000 shots. The measured data at night are shown as circles and the daytime data are shown as filled squares.

Fig. 11
Fig. 11

LOS wind measured with DWL and wiresonde. The wind lidar measurements are shown as circles and the wiresonde data are shown as filled squares.

Fig. 12
Fig. 12

Histogram of differences between corresponding pairs of 1 min LOS projected horizontal wind velocities for the wiresonde and DWL.

Fig. 13
Fig. 13

24 h observation of wind field and aerosol backscatter coefficient on 27 April 2006: (a) time-height plot of horizontal wind velocity, (b) wind direction, (c) vertical wind velocity, (d) time–height plot of aerosol backscatter coefficients in logarithmic scale.

Tables (1)

Tables Icon

Table 1 Parameters of the Doppler Lidar System

Equations (36)

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

h ( ν ) = 0 θ max T 0 1 + 4 ( ν FSR π Δ ν 1 / 2 ) 2 sin 2 [ π ν cos ( θ ) ν FSR ] d θ .
f M ( ν ) = ( 4 ln 2 / π Δ ν M 2 ) 1 / 2 exp ( ν 2 4 ln 2 / Δ ν M 2 ) ,
f R ( ν ) = ( 4 ln 2 / π Δ ν R 2 ) 1 / 2 exp ( ν 2 4 ln 2 / Δ ν R 2 ) ,
Δ ν R = ( 32 k T a ln 2 / λ 2 M ) 1 / 2 ,
T M i ( ν ) = h i ( ν ν ) f M ( ν ) d ν = h i ( ν ν ) f L ( ν ) d ν ,
T R i ( ν ) = T M i ( ν ν ) f R ( ν ) d ν .
I 1 = a 1 [ I M T M 1 ( ν 0 + Δ ν d ) + I R T R 1 ( ν 0 + Δ ν d ) ] ,
I 2 = a 2 [ I M T M 2 ( ν 0 + Δ ν d ) + I R T R 2 ( ν 0 + Δ ν d ) ] ,
I E 1 = a 3 ( I M + I R ) ,
I E 2 = a 4 ( I M + I R ) ,
T M 1 ( ν 0 ) = a 3 I 1 a 1 I E 1 ,
T M 2 ( ν 0 ) = a 4 I 2 a 2 I E 2 .
I E = ( a 3 + a 4 ) ( I M + I R ) .
I R = I E / ( a 3 + a 4 ) I M .
F i ( Δ ν d , I M ) = I M T M i ( ν 0 + Δ ν d ) + I R T R i ( ν 0 + Δ ν d ) I i / a i .
[ δ ν d δ I M ] [ F 1 ( Δ ν d + h , I M ) F 1 ( Δ ν d , I M ) h F 1 I M F 2 ( Δ ν d + h , I M ) F 2 ( Δ ν d , I M ) h F 2 I M ] =  [ F 1 ( Δ ν d , I M ) F 2 ( Δ ν d , I M ) ] ,
Δ ν d ( N ) = Δ ν d ( N 1 ) + δ ν d ,
I M ( N ) = I M ( N 1 ) + δ I M ,
V = c 2 ν 0 Δ ν d .
V H = 2 2 3 ( V 1 2 + V 2 2 + V 3 2 V 1 V 2 V 2 V 3 V 1 V 3 ) 1 / 2 ,
V Z = 2 3 ( V 1 + V 2 + V 3 ) ,
θ = arctan ( 3 ( V 3 V 2 ) 2 V 1 V 2 V 3 ) π 2 sgn ( 2 3 ( 2 V 1 V 2 V 3 ) ) ,
ν s / C s = ν 0 / C 0 .
f EPV ( ν ) = ( 1 η L η I η P ) f G ( ν , w G ) + η L f L ( ν , w L ) + η I f I ( ν , w I ) + η P f P ( ν , w P ) ,
f G ( ν , γ G ) = ( 1 / π 1 / 2 γ G ) exp ( ν 2 / γ G 2 )
f L ( ν , γ L ) = ( 1 / π γ L ) ( 1 + ν 2 / γ L 2 ) 1
f I ( ν , γ I ) = ( 1 / 2 γ I ) [ 1 + ( ν / γ I ) 2 ] 3 / 2
f P ( ν , γ P ) = ( 2 / γ P ) [ exp ( ν / γ P ) + exp ( ν / γ P ) ] 2
T M i ( ν ) = B + A f EVP ( ν C ) ,
S = j = 1 200 [ T ( ν j ) ( B + A f EVP ( ν j C ) ) ] 2 / T ( ν j ) .
T M 2 ( ν 1 ) T M 1 ( ν 1 ) = Δ ν d ( d T M 2 d ν d T M 1 d ν ) ν 0 ,
δ ( Δ ν d ) = δ [ T M 2 ( ν 1 ) T M 1 ( ν 1 ) ] / ( d T M 2 d ν d T M 1 d ν ) ν 0 .
δ 2 ( Δ T M ) = j = 1 2 [ ( Δ T M I j ) 2 ( δ I j ) 2 + ( Δ T M I E j ) 2 ( δ I E j ) 2 ] ,
δ 1 ( Δ ν d ) = [ ( a 4 a 2 ) 2 ( I 2 2 I E 2 3 + I 2 I E 2 2 ) + ( a 3 a 1 ) 2 ( I 1 2 I E 1 3 + I 1 I E 1 2 ) ] 1 / 2 / ( d T M 2 d ν d T M 1 d ν ) ν 0 .
δ C = ( δ 1 2 ( Δ ν d ) + δ R 2 ) 1 / 2 .
δ 2 ( Δ ν d ) = ( 1 + 4 γ 0 2 ) 3 / 2 8 γ 0 ( 1 2 ρ T p k N 0 η ) 1 / 2 w ,

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