Abstract

Atmospheric line-of-sight (LOS) wind measurement by means of incoherent Cabannes–Mie lidar with three frequency analyzers with nearly the same maximum transmission of 80% that could be fielded at different wavelengths is analytically considered. These frequency analyzers are (a) a double-edge Fabry–Perot interferometer (FPI) at 1064  nm (IR-FPI), (b) a double-edge Fabry–Perot interferometer at 355   nm (UV-FPI), and (c) an iodine vapor filter (IVF) at 532   nm with two different methods, using either one absorption edge, single edge (se-IVF), or both absorption edges, double edge (de-IVF). The effect of the backscattered aerosol mixing ratio, Rb, defined as the ratio of the aerosol volume backscatter coefficient to molecular volume backscatter coefficient, on LOS wind uncertainty is discussed. Assuming a known aerosol mixing ratio, Rb, and 100,000 photons owing to Cabannes scattering to the receiver, in shot-noise-limited detection without sky background, the LOS wind uncertainty of the UV-FPI in the aerosol-free air (Rb=0), is lower by 16% than that of de-IVF, which has the lowest uncertainty for Rb between 0.02 and 0.08; for Rb>0.08, the IR-FPI yielded the lowest wind uncertainty. The wind uncertainty for se-IVF is always higher than that of de-IVF, but by less than a factor of 2 under all aerosol conditions, if the split between the reference and measurement channels is optimized. The design flexibility, which allows the desensitization of either aerosol or molecular scattering, exists only with the FPI system, leading to the common practice of using IR-FPI for the planetary boundary layer and using UV-FPI for higher altitudes. Without this design flexibility, there is little choice but to use a single wavelength IVF system at 532  nm for all atmospheric altitudes.

© 2007 Optical Society of America

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  3. G. J. Koch, M. Petros, B. W. Barnes, J. Y. Beyon, F. Amzajerdian, J. Yu, M. J. Kavaya, and U. N. Singh, "Validar: A testbed for advanced 2-micron Doppler lidar," Proc SPIE 5412, 87-98 (2004).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  32. M. J. McGill, W. D. Hart, J. A. McKay, and J. D. Spinhirne, "Modeling the performance of direct-detection Doppler lidar systems including cloud and solar background variability," Appl. Opt. 38, 6388-6397 (1999).
    [CrossRef]
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2007 (1)

2005 (1)

2004 (1)

G. J. Koch, M. Petros, B. W. Barnes, J. Y. Beyon, F. Amzajerdian, J. Yu, M. J. Kavaya, and U. N. Singh, "Validar: A testbed for advanced 2-micron Doppler lidar," Proc SPIE 5412, 87-98 (2004).
[CrossRef]

2002 (2)

2001 (2)

J. W. Hair, L. M. Caldwell, D. A. Krueger, and C.-Y. She, "High-spectral-resolution lidar with iodine-vapor filters: measurement of atmospheric-state and aerosol profiles," Appl. Opt. 40, 5280-5294 (2001).
[CrossRef]

C. Nagasawa, Y. Shibata, M. Abo, T. Nagai, and O. Uchino, "Incoherent Doppler lidar using wavelengths for wind measurement," Proc. SPIE 4153, 338-349 (2001).
[CrossRef]

2000 (2)

1999 (3)

1998 (1)

1997 (4)

1996 (2)

M. A. White, D. Golias, D. A. Krueger, and C. Y. She, "A frequency-agile lidar for simultaneous measurement of temperature and radial wind in the mesopause region without sodium density contamination," Proc. SPIE 2833, 136-142 (1996).
[CrossRef]

R. Targ, B. C. Steakley, J. G. Hawley, L. L. Ames, P. Forney, D. Swanson, R. Stone, R. G. Otto, V. Zarifis, P. Brockman, R. S. Calloway, S. H. Klein, and P. A. Robinson, "Coherent lidar airborne wind sensor II: flight-test results at 2 and 10 μm," Appl. Opt. 35, 7117-7128 (1996).
[CrossRef] [PubMed]

1995 (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]

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

1994 (1)

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]

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]

C. L. Korb, B. Gentry, and C. Weng, "The edge technique: theory and application to the lidar measurement of atmospheric winds," Appl. Opt. 31, 4202-4213 (1992).
[CrossRef] [PubMed]

1989 (1)

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

1986 (1)

M. J. Post and W. D. Neff, "Doppler lidar measurements of winds in a narrow mountain valley," Bull. Am. Meteorol. Soc. 67, 274-281 (1986).
[CrossRef]

1983 (1)

1982 (1)

A. T. Young, "Rayleigh scattering," Phys. Today 35, 42-48 (1982).
[CrossRef]

Abo, M.

C. Nagasawa, Y. Shibata, M. Abo, T. Nagai, and O. Uchino, "Incoherent Doppler lidar using wavelengths for wind measurement," Proc. SPIE 4153, 338-349 (2001).
[CrossRef]

Y. Shibata, C. Nagasawa, and M. Abo, "Development of two-wavelength Doppler lidar for wind measurement using an iodine edge filter," in Advances in Laser Remote Sensing--Selected Papers Presented at the 20th International Laser Radar Conference, A. Dabas, C. Loth, and J. Pelon, eds. (Ecole Polytechnique, 2000), pp. 85-88.

Abreu, V. J.

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]

Ames, L. L.

Amzajerdian, F.

G. J. Koch, M. Petros, B. W. Barnes, J. Y. Beyon, F. Amzajerdian, J. Yu, M. J. Kavaya, and U. N. Singh, "Validar: A testbed for advanced 2-micron Doppler lidar," Proc SPIE 5412, 87-98 (2004).
[CrossRef]

Barnes, B. W.

G. J. Koch, M. Petros, B. W. Barnes, J. Y. Beyon, F. Amzajerdian, J. Yu, M. J. Kavaya, and U. N. Singh, "Validar: A testbed for advanced 2-micron Doppler lidar," Proc SPIE 5412, 87-98 (2004).
[CrossRef]

Barnes, J. E.

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]

Beyon, J. Y.

G. J. Koch, M. Petros, B. W. Barnes, J. Y. Beyon, F. Amzajerdian, J. Yu, M. J. Kavaya, and U. N. Singh, "Validar: A testbed for advanced 2-micron Doppler lidar," Proc SPIE 5412, 87-98 (2004).
[CrossRef]

Brockman, P.

Caldwell, L. M.

Calloway, R. S.

Castleberg, P. A.

Chanin, M. L.

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]

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

Chen, H.

Chen, W. B.

Z. S. Liu, W. B. Chen, T. L. Zhang, J. W. Hair, and C. Y. She, "An incoherent Doppler lidar for ground-based atmospheric wind profiling," Appl. Phys. B 64, 561-566 (1997).
[CrossRef]

Chen, W.-B.

Fischer, K. F.

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]

Flesia, C.

Forkey, J. N.

Forney, P.

Friedman, J. S.

Galbraith, A. E.

J. A. Reagan, A. E. Galbraith, and J. D. Spinhirne, "Micro pulse lidar daytime performance: simulations and observations," in Geoscience and Remote Sensing Symposium, Remote Sensing for a Sustainable Future (IEEE, 1996), pp. 683-685.

Gariner, A.

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

Garnier, A.

Gentry, B.

Gentry, B. M.

Golias, D.

M. A. White, D. Golias, D. A. Krueger, and C. Y. She, "A frequency-agile lidar for simultaneous measurement of temperature and radial wind in the mesopause region without sodium density contamination," Proc. SPIE 2833, 136-142 (1996).
[CrossRef]

Guo, J.-J.

Hair, J. W.

Hart, W. D.

Hauchecorne, A.

C. Souprayen, A. Garnier, A. Hertzog, A. Hauchecorne, and J. Portenuve, "Rayleigh-Mie Doppler wind lidar for atmospheric measurements. 2. Mie scattering effect, theory, and calibration," Appl. Opt. 38, 2422-2431 (1999).
[CrossRef]

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

Hawley, J. G.

Hecht, E.

E. Hecht, Optics, 3rd ed. (Addison-Wesley, 1998), pp. 413-417.

Hertzog, A.

Imaki, M.

Irgang, T. D.

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]

Kavaya, M. J.

G. J. Koch, M. Petros, B. W. Barnes, J. Y. Beyon, F. Amzajerdian, J. Yu, M. J. Kavaya, and U. N. Singh, "Validar: A testbed for advanced 2-micron Doppler lidar," Proc SPIE 5412, 87-98 (2004).
[CrossRef]

Klein, S. H.

Kobayashi, T.

Koch, G. J.

G. J. Koch, M. Petros, B. W. Barnes, J. Y. Beyon, F. Amzajerdian, J. Yu, M. J. Kavaya, and U. N. Singh, "Validar: A testbed for advanced 2-micron Doppler lidar," Proc SPIE 5412, 87-98 (2004).
[CrossRef]

Korb, C. L.

Krueger, D. A.

J. W. Hair, L. M. Caldwell, D. A. Krueger, and C.-Y. She, "High-spectral-resolution lidar with iodine-vapor filters: measurement of atmospheric-state and aerosol profiles," Appl. Opt. 40, 5280-5294 (2001).
[CrossRef]

M. A. White, D. Golias, D. A. Krueger, and C. Y. She, "A frequency-agile lidar for simultaneous measurement of temperature and radial wind in the mesopause region without sodium density contamination," Proc. SPIE 2833, 136-142 (1996).
[CrossRef]

Lee, S. A.

Lempert, W. R.

Li, S. X.

Li, X.

Li, Z.-G.

B.-Y. Liu, Z.-S. Liu, Z.-G., Li, Z.-A. Yan, R.-B. Wang, and Z.-B. Sun, "Wind measurements with incoherent Doppler lidar based on iodine filters at night and day," in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (2006), pp. 55-58.

Liu, B.-Y.

B.-Y. Liu, Z.-S. Liu, Z.-G., Li, Z.-A. Yan, R.-B. Wang, and Z.-B. Sun, "Wind measurements with incoherent Doppler lidar based on iodine filters at night and day," in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (2006), pp. 55-58.

Liu, J.-T.

Liu, Z. S.

Z. S. Liu, W. B. Chen, T. L. Zhang, J. W. Hair, and C. Y. She, "An incoherent Doppler lidar for ground-based atmospheric wind profiling," Appl. Phys. B 64, 561-566 (1997).
[CrossRef]

Liu, Z.-S.

McGill, M. J.

M. J. McGill, W. D. Hart, J. A. McKay, and J. D. Spinhirne, "Modeling the performance of direct-detection Doppler lidar systems including cloud and solar background variability," Appl. Opt. 38, 6388-6397 (1999).
[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]

McKay, J.

J. McKay, "Comparing the intrinsic, photon shot noise limited sensitivity of coherent and direct detection Doppler wind lidar," presented at the Tenth Biennial Coherent Laser Radar Technology and Applications Conference, Mount Hood, Oregon, USA, June 28-July 2 1999, pp. 106-109.

Mckay, J. A.

Miles, R. B.

Nagai, T.

C. Nagasawa, Y. Shibata, M. Abo, T. Nagai, and O. Uchino, "Incoherent Doppler lidar using wavelengths for wind measurement," Proc. SPIE 4153, 338-349 (2001).
[CrossRef]

Nagasawa, C.

C. Nagasawa, Y. Shibata, M. Abo, T. Nagai, and O. Uchino, "Incoherent Doppler lidar using wavelengths for wind measurement," Proc. SPIE 4153, 338-349 (2001).
[CrossRef]

Y. Shibata, C. Nagasawa, and M. Abo, "Development of two-wavelength Doppler lidar for wind measurement using an iodine edge filter," in Advances in Laser Remote Sensing--Selected Papers Presented at the 20th International Laser Radar Conference, A. Dabas, C. Loth, and J. Pelon, eds. (Ecole Polytechnique, 2000), pp. 85-88.

Neff, W. D.

M. J. Post and W. D. Neff, "Doppler lidar measurements of winds in a narrow mountain valley," Bull. Am. Meteorol. Soc. 67, 274-281 (1986).
[CrossRef]

Otto, R. G.

Petros, M.

G. J. Koch, M. Petros, B. W. Barnes, J. Y. Beyon, F. Amzajerdian, J. Yu, M. J. Kavaya, and U. N. Singh, "Validar: A testbed for advanced 2-micron Doppler lidar," Proc SPIE 5412, 87-98 (2004).
[CrossRef]

Porteneuve, J.

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

Portenuve, J.

Post, M. J.

M. J. Post and W. D. Neff, "Doppler lidar measurements of winds in a narrow mountain valley," Bull. Am. Meteorol. Soc. 67, 274-281 (1986).
[CrossRef]

Reagan, J. A.

J. A. Reagan, A. E. Galbraith, and J. D. Spinhirne, "Micro pulse lidar daytime performance: simulations and observations," in Geoscience and Remote Sensing Symposium, Remote Sensing for a Sustainable Future (IEEE, 1996), pp. 683-685.

Robinson, P. A.

Roe, H.

Rye, B. J.

She, C. Y.

Z. S. Liu, W. B. Chen, T. L. Zhang, J. W. Hair, and C. Y. She, "An incoherent Doppler lidar for ground-based atmospheric wind profiling," Appl. Phys. B 64, 561-566 (1997).
[CrossRef]

M. A. White, D. Golias, D. A. Krueger, and C. Y. She, "A frequency-agile lidar for simultaneous measurement of temperature and radial wind in the mesopause region without sodium density contamination," Proc. SPIE 2833, 136-142 (1996).
[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]

H. Shimizu, S. A. Lee, and C. Y. She, "A high resolution lidar system with atomic blocking filters for measuring atmospheric parameters," Appl. Opt. 22, 1373-1381 (1983).
[CrossRef] [PubMed]

She, C.-Y.

Shibata, Y.

C. Nagasawa, Y. Shibata, M. Abo, T. Nagai, and O. Uchino, "Incoherent Doppler lidar using wavelengths for wind measurement," Proc. SPIE 4153, 338-349 (2001).
[CrossRef]

Y. Shibata, C. Nagasawa, and M. Abo, "Development of two-wavelength Doppler lidar for wind measurement using an iodine edge filter," in Advances in Laser Remote Sensing--Selected Papers Presented at the 20th International Laser Radar Conference, A. Dabas, C. Loth, and J. Pelon, eds. (Ecole Polytechnique, 2000), pp. 85-88.

Shimizu, H.

Singh, U. N.

G. J. Koch, M. Petros, B. W. Barnes, J. Y. Beyon, F. Amzajerdian, J. Yu, M. J. Kavaya, and U. N. Singh, "Validar: A testbed for advanced 2-micron Doppler lidar," Proc SPIE 5412, 87-98 (2004).
[CrossRef]

Skinner, W. R.

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]

Song, X.-Q.

Souprayen, C.

Spinhirne, J. D.

M. J. McGill, W. D. Hart, J. A. McKay, and J. D. Spinhirne, "Modeling the performance of direct-detection Doppler lidar systems including cloud and solar background variability," Appl. Opt. 38, 6388-6397 (1999).
[CrossRef]

J. A. Reagan, A. E. Galbraith, and J. D. Spinhirne, "Micro pulse lidar daytime performance: simulations and observations," in Geoscience and Remote Sensing Symposium, Remote Sensing for a Sustainable Future (IEEE, 1996), pp. 683-685.

Steakley, B. C.

Stone, R.

Sun, Z.-B.

B.-Y. Liu, Z.-S. Liu, Z.-G., Li, Z.-A. Yan, R.-B. Wang, and Z.-B. Sun, "Wind measurements with incoherent Doppler lidar based on iodine filters at night and day," in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (2006), pp. 55-58.

Swanson, D.

Targ, R.

Tepley, C. A.

Uchino, O.

C. Nagasawa, Y. Shibata, M. Abo, T. Nagai, and O. Uchino, "Incoherent Doppler lidar using wavelengths for wind measurement," Proc. SPIE 4153, 338-349 (2001).
[CrossRef]

Wang, R.-B.

B.-Y. Liu, Z.-S. Liu, Z.-G., Li, Z.-A. Yan, R.-B. Wang, and Z.-B. Sun, "Wind measurements with incoherent Doppler lidar based on iodine filters at night and day," in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (2006), pp. 55-58.

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Figures (5)

Fig. 1
Fig. 1

(Color online) Transmission functions of Fabry–Perot interferometer (FPI) or iodine vapor filter (IVF) together with aerosol and Cabannes scattering spectrum for (a) IR-FPI at 1064   nm , (b) UV-FPI at 355   nm , (c) se-IVF at 532   nm , and (d) de-IVF at 532   nm .

Fig. 2
Fig. 2

(Color online) (a) Simulated transmission function of the 1109 iodine absorption line with a 10 cm long vapor cell. The cell body and finger temperatures were chosen to yield FWHM of 1 .92   GHz . (b) Signal-to-noise ratio as a function of fraction of light into the measurement channel 1 for the four analysis methods, calculated with 100,000 photons from Cabannes scattering into the receiver of UV-FPI and se-IVF, 90,000 photons into de-IVF, and 100,000 photons due to aerosol scattering (or R b = 1 ) for IR-FPI at 1064   nm . The decimal numbers in the legend indicate the transmission factor of Cabannes scattering (aerosol for IR-FPI) at zero wind.

Fig. 3
Fig. 3

(Color online) (a) Measurement sensitivity, and (b) signal-to-noise ratio for the four detection methods as functions of aerosol mixing ratio, 0 < R b < 10.0 .

Fig. 4
Fig. 4

(Color online) LOS wind uncertainty based on 100,000 Cabannes photons (90,000 for de-IVF) received by the receiver with ideal photodetectors, for the four methods as a function of aerosol mixing ratio, R b , (a) 0 < R b < 1.0 , and (b) 0 < R b < 10.0 .

Fig. 5
Fig. 5

(Color online) Comparison between two FPIs at 532   nm , vis-FPI_S scaled from UV-FPI and vis-FPI_A with significant aerosol scattering, and de-IVF. (a) SNR and sensitivity, (b) LOS wind uncertainty.

Tables (1)

Tables Icon

Table 1 Parameters of the Four Analysis Methods Employed in this Comparative Study

Equations (22)

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N m i = N ξ m i [ β a f a i + β c f c i ] ; i = 1 , 2 ,
N R = N ξ R [ β a + β c ] .
ν D = 2 V LOS λ ,
f c i ( ν D , T , P ) = i ( ν ν D , T , P ) F i ( ν ) d ν ; i ( ν ν D , T , P ) d ν = 1 ,
f a i ( ν D , T , P ) = G i ( ν ν D ) F i ( ν ) d ν ; G i ( ν ν D ) d ν = 1 .
1 ( ν ν D , T , P ) = 2 ( ν ν D , T , P ) = ( ν ν D , T , P ) ,
G 1 ( ν ν D ) = G 2 ( ν ν D ) = G ( ν ν D ) ,
F i ( ν ) = F FPI ( ν ν i ) ; i = 1 , 2 .
F FPI ( ν ν i , Δ ν FWHM ) = T M a x [ 1 + ( 2 F π ) 2 × sin 2 ( π ν ν i Δ ν FSR ) ] 1 ; i = 1 , 2 ;
where   Δ ν FWHM = 2 Δ ν FSR π sin 1 ( π 2 F ) .
i ( ν ν D , T , P ) ( ν ν D ν i , T , P ) ; i = 1 , 2 , G i ( ν ν D ) G ( ν ν D ν i ) ; i = 1 , 2 , F i ( ν ) = F IVF ( ν ν i ) ; i = 1 , 2 .
R W ( ν D , R b ) = N 1 ( ν D , R b ) N 2 ( ν D , R b ) ,
Δ R W R W = Δ N 1 N 1 Δ N 2 N 2 .
d R W ( ν D ) R W ( 0 ) S ν D d ν D = ( 1 N 1 ( 0 ) N 1 ( ν D ) ν D 1 N 2 ( 0 ) N 2 ( ν D ) ν D ) d ν D ,
S ν D = ( 1 N 1 ( 0 ) N 1 ( ν D ) ν D 1 N 2 ( 0 ) N 2 ( ν D ) ν D ) ,
d ν D = d R W ( ν D ) S ν D R W ( 0 ) .
S V LOS = S ν D d ν D d V LOS ,   with   d ν D d V LOS = 2 λ .
( δ R W ) 2 R W 2 = ( δ N 1 ) 2 N 1 2 + ( δ N 2 ) 2 N 2 2 SNR = [ ( δ N 1 ) 2 N 1 2 + ( δ N 2 ) 2 N 2 2 ] 1 / 2 .
SNR = [ 1 N 1 + 1 N 2 ] 1 / 2 = N 1 N 2 N 1 + N 2 .
SNR = N 1 N 2 N 1 + N 2 = ( 100, 000 f 1 ξ 1 ) ( 100, 000 f 2 ( 1 ξ 1 ) ) 100, 000 ( f 2 + ( f 1 f 2 ) ξ 1 ) ,
δ V LOS = ( λ 2 ) | δ ν D | = λ 2 ( δ R W ) 2 R W S ν D = λ / 2 S ν D ( SNR )
= 1 S V LOS ( SNR ) .

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