Abstract

A new concept of spectrum analyzer is proposed for short-range lidar measurements in airborne applications. It implements a combination of two fringe-imaging Michelson interferometers to analyze the Rayleigh–Mie spectrum backscattered by molecules and particles at 355nm. The objective is to perform simultaneous measurements of four variables: the air speed, the air temperature and density, and the particle scattering ratio. The Cramer–Rao bounds are calculated to evaluate the best expectable measurement accuracies. The performance optimization shows that a Michelson interferometer with a path difference of 3cm is optimal for air speed measurements in clear air. To optimize density, temperature, and scattering ratio measurements, the second interferometer should be set to a path difference of 10cm at least; 20cm would be better to be less sensitive to the actual Rayleigh–Brillouin line shape.

© 2009 Optical Society of America

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2007 (1)

N. Cézard, A. Dolfi-Bouteyre, J.-P. Huignard, and P. H. Flamant, “Development of a fringe-imaging Michelson interferometer for wind speed measurements using a short-range 355 nm Rayleigh-Mie lidar,” Proc. SPIE 6750, 675008(2007).
[CrossRef]

2004 (2)

D. Hua, M. Uchida, and T. Kobayashi, “Ultraviolet high-spectral-resolution Rayleigh-Mie lidar with a dual-pass Fabry-Perot etalon for measuring atmospheric temperature profiles of the troposphere,” Opt. Lett. 29, 1063-1065 (2004).
[CrossRef] [PubMed]

C. B. Watkins, C. J. Richey, P. Tchoryk, Jr., G. A. Ritter, M. Dehring, P. B. Hays, C. A. Nardell, and R. Urzi, “Molecular optical air data system (MOADS) prototype II,” Proc. SPIE 5412, 10-20 (2004).
[CrossRef]

2003 (2)

2002 (3)

2001 (2)

P. Tchoryk, Jr., C. B. Watkins, S. K. Lindemann, P. B. Hays, and C. A. Nardell, “Molecular optical air data system (MOADS),” Proc. SPIE 4377, 194-204 (2001).
[CrossRef]

R. Miles, W. Rempert, and J. Forkey, “Laser Rayleigh scattering,” Meas. Sci. Technol. 12, R33-R51 (2001).
[CrossRef]

2000 (1)

1997 (1)

1996 (1)

1995 (1)

1992 (1)

1991 (1)

1990 (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]

1983 (1)

1974 (1)

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

1966 (1)

Abreu, V. J.

Ames, L. L.

Augere, B.

J. P. Cariou, B. Augere, D. Goular, J. P. Schlotterbeck, and X. Lacondemine, “All-fiber 1.5 μm CW coherent laser anemometer DALHEC--helicopter flight test analysis,” in Proceedings of 13th Coherent Laser Radar Conference (National Institute of Information and Communications Technology, 2005), pp. 157-160.

Bagley, H.

D. Soreide, R. K. Bogue, L. J. Ehernberger, and H. Bagley, “Coherent lidar turbulence measurement for gust load alleviation,” NASA Technical Memorandum104318 (NASA, 1996).

Barnes, J. E.

Bogue, R. K.

D. Soreide, R. K. Bogue, L. J. Ehernberger, and H. Bagley, “Coherent lidar turbulence measurement for gust load alleviation,” NASA Technical Memorandum104318 (NASA, 1996).

Boley, C. D.

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

Bowles, R. L.

Brockman, P.

Brown, C. M.

Bruneau, D.

Calloway, R. S.

Cardon, J. G.

Cariou, J. P.

J. P. Cariou, B. Augere, D. Goular, J. P. Schlotterbeck, and X. Lacondemine, “All-fiber 1.5 μm CW coherent laser anemometer DALHEC--helicopter flight test analysis,” in Proceedings of 13th Coherent Laser Radar Conference (National Institute of Information and Communications Technology, 2005), pp. 157-160.

Cézard, N.

N. Cézard, A. Dolfi-Bouteyre, J.-P. Huignard, and P. H. Flamant, “Development of a fringe-imaging Michelson interferometer for wind speed measurements using a short-range 355 nm Rayleigh-Mie lidar,” Proc. SPIE 6750, 675008(2007).
[CrossRef]

N. Cézard, A. Dolfi-Bouteyre, J.-P. Huignard, and P. H. Flamant, “Potential of the Michelson interferometer fringe-imaging technique for Rayleigh-Mie spectral analysis,” in Proceedings of 24th International Laser Radar Conference (2008), pp. 231-234.

Chanin, M. L.

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]

Chen, H.

Chen, W.-B.

Conway, R. R.

Dehring, M.

C. B. Watkins, C. J. Richey, P. Tchoryk, Jr., G. A. Ritter, M. Dehring, P. B. Hays, C. A. Nardell, and R. Urzi, “Molecular optical air data system (MOADS) prototype II,” Proc. SPIE 5412, 10-20 (2004).
[CrossRef]

Desai, R. C.

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

Diehl, H.

N. P. Schmitt, W. Rehm, T. Pistner, H. Diehl, P. Navé, G. Jenaro-Rabadan, P. Mirand, and M. Reymond, “Flight test of the AWIATOR airborne LIDAR turbulence sensor--first results,” in Proceedings of 14th Coherent Laser Radar Conference (2007).

N. Schmitt, W. Rehm, T. Pistner, P. Zeller, H. Diehl, and P. Navé, “Airborne direct detection UV lidar,” in Proceedings of 23rd International Laser Radar Conference (2006), pp. 167-170.

Dolfi-Bouteyre, A.

N. Cézard, A. Dolfi-Bouteyre, J.-P. Huignard, and P. H. Flamant, “Development of a fringe-imaging Michelson interferometer for wind speed measurements using a short-range 355 nm Rayleigh-Mie lidar,” Proc. SPIE 6750, 675008(2007).
[CrossRef]

N. Cézard, A. Dolfi-Bouteyre, J.-P. Huignard, and P. H. Flamant, “Potential of the Michelson interferometer fringe-imaging technique for Rayleigh-Mie spectral analysis,” in Proceedings of 24th International Laser Radar Conference (2008), pp. 231-234.

Ehernberger, L. J.

D. Soreide, R. K. Bogue, L. J. Ehernberger, and H. Bagley, “Coherent lidar turbulence measurement for gust load alleviation,” NASA Technical Memorandum104318 (NASA, 1996).

Englert, C. R.

Flamant, P. H.

N. Cézard, A. Dolfi-Bouteyre, J.-P. Huignard, and P. H. Flamant, “Development of a fringe-imaging Michelson interferometer for wind speed measurements using a short-range 355 nm Rayleigh-Mie lidar,” Proc. SPIE 6750, 675008(2007).
[CrossRef]

N. Cézard, A. Dolfi-Bouteyre, J.-P. Huignard, and P. H. Flamant, “Potential of the Michelson interferometer fringe-imaging technique for Rayleigh-Mie spectral analysis,” in Proceedings of 24th International Laser Radar Conference (2008), pp. 231-234.

Forkey, J.

R. Miles, W. Rempert, and J. Forkey, “Laser Rayleigh scattering,” Meas. Sci. Technol. 12, R33-R51 (2001).
[CrossRef]

Forney, P.

Garnier, A.

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]

Gentry, B. M.

Goular, D.

J. P. Cariou, B. Augere, D. Goular, J. P. Schlotterbeck, and X. Lacondemine, “All-fiber 1.5 μm CW coherent laser anemometer DALHEC--helicopter flight test analysis,” in Proceedings of 13th Coherent Laser Radar Conference (National Institute of Information and Communications Technology, 2005), pp. 157-160.

Grund, C. J.

C. J. Grund, M. Stephens, and C. Weimer, “Simultaneous profiling of aerosol optical properties, gas chemistry, and winds with optical autocovariance lidar,” in Proceedings of 24th International Laser Radar Conference (2008).

Hair, J. W.

Harlander, J. M.

Hauchecorne, A.

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]

Hawley, J. G.

Hays, P. B.

C. B. Watkins, C. J. Richey, P. Tchoryk, Jr., G. A. Ritter, M. Dehring, P. B. Hays, C. A. Nardell, and R. Urzi, “Molecular optical air data system (MOADS) prototype II,” Proc. SPIE 5412, 10-20 (2004).
[CrossRef]

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]

P. Tchoryk, Jr., C. B. Watkins, S. K. Lindemann, P. B. Hays, and C. A. Nardell, “Molecular optical air data system (MOADS),” Proc. SPIE 4377, 194-204 (2001).
[CrossRef]

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

Hilliard, R. L.

Hua, D.

Huffaker, R. M.

Huignard, J.-P.

N. Cézard, A. Dolfi-Bouteyre, J.-P. Huignard, and P. H. Flamant, “Development of a fringe-imaging Michelson interferometer for wind speed measurements using a short-range 355 nm Rayleigh-Mie lidar,” Proc. SPIE 6750, 675008(2007).
[CrossRef]

N. Cézard, A. Dolfi-Bouteyre, J.-P. Huignard, and P. H. Flamant, “Potential of the Michelson interferometer fringe-imaging technique for Rayleigh-Mie spectral analysis,” in Proceedings of 24th International Laser Radar Conference (2008), pp. 231-234.

Irgang, T. D.

Jenaro-Rabadan, G.

N. P. Schmitt, W. Rehm, T. Pistner, H. Diehl, P. Navé, G. Jenaro-Rabadan, P. Mirand, and M. Reymond, “Flight test of the AWIATOR airborne LIDAR turbulence sensor--first results,” in Proceedings of 14th Coherent Laser Radar Conference (2007).

Kattawar, G. W.

Kavaya, M. J.

Klein, S. H.

Kobayashi, T.

Lacondemine, X.

J. P. Cariou, B. Augere, D. Goular, J. P. Schlotterbeck, and X. Lacondemine, “All-fiber 1.5 μm CW coherent laser anemometer DALHEC--helicopter flight test analysis,” in Proceedings of 13th Coherent Laser Radar Conference (National Institute of Information and Communications Technology, 2005), pp. 157-160.

Li, S. X.

Lindemann, S. K.

P. Tchoryk, Jr., C. B. Watkins, S. K. Lindemann, P. B. Hays, and C. A. Nardell, “Molecular optical air data system (MOADS),” Proc. SPIE 4377, 194-204 (2001).
[CrossRef]

Liu, J.-T.

Liu, Z-S.

McDermid, I. S.

McGill, M. J.

Miles, R.

R. Miles, W. Rempert, and J. Forkey, “Laser Rayleigh scattering,” Meas. Sci. Technol. 12, R33-R51 (2001).
[CrossRef]

Mirand, P.

N. P. Schmitt, W. Rehm, T. Pistner, H. Diehl, P. Navé, G. Jenaro-Rabadan, P. Mirand, and M. Reymond, “Flight test of the AWIATOR airborne LIDAR turbulence sensor--first results,” in Proceedings of 14th Coherent Laser Radar Conference (2007).

Nardell, C. A.

C. B. Watkins, C. J. Richey, P. Tchoryk, Jr., G. A. Ritter, M. Dehring, P. B. Hays, C. A. Nardell, and R. Urzi, “Molecular optical air data system (MOADS) prototype II,” Proc. SPIE 5412, 10-20 (2004).
[CrossRef]

P. Tchoryk, Jr., C. B. Watkins, S. K. Lindemann, P. B. Hays, and C. A. Nardell, “Molecular optical air data system (MOADS),” Proc. SPIE 4377, 194-204 (2001).
[CrossRef]

Navé, P.

N. Schmitt, W. Rehm, T. Pistner, P. Zeller, H. Diehl, and P. Navé, “Airborne direct detection UV lidar,” in Proceedings of 23rd International Laser Radar Conference (2006), pp. 167-170.

N. P. Schmitt, W. Rehm, T. Pistner, H. Diehl, P. Navé, G. Jenaro-Rabadan, P. Mirand, and M. Reymond, “Flight test of the AWIATOR airborne LIDAR turbulence sensor--first results,” in Proceedings of 14th Coherent Laser Radar Conference (2007).

Otto, R. G.

Pelon, J.

Pistner, T.

N. Schmitt, W. Rehm, T. Pistner, P. Zeller, H. Diehl, and P. Navé, “Airborne direct detection UV lidar,” in Proceedings of 23rd International Laser Radar Conference (2006), pp. 167-170.

N. P. Schmitt, W. Rehm, T. Pistner, H. Diehl, P. Navé, G. Jenaro-Rabadan, P. Mirand, and M. Reymond, “Flight test of the AWIATOR airborne LIDAR turbulence sensor--first results,” in Proceedings of 14th Coherent Laser Radar Conference (2007).

Porteneuve, J.

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]

Rahm, S.

Rees, D.

Rehm, W.

N. P. Schmitt, W. Rehm, T. Pistner, H. Diehl, P. Navé, G. Jenaro-Rabadan, P. Mirand, and M. Reymond, “Flight test of the AWIATOR airborne LIDAR turbulence sensor--first results,” in Proceedings of 14th Coherent Laser Radar Conference (2007).

N. Schmitt, W. Rehm, T. Pistner, P. Zeller, H. Diehl, and P. Navé, “Airborne direct detection UV lidar,” in Proceedings of 23rd International Laser Radar Conference (2006), pp. 167-170.

Rempert, W.

R. Miles, W. Rempert, and J. Forkey, “Laser Rayleigh scattering,” Meas. Sci. Technol. 12, R33-R51 (2001).
[CrossRef]

Reymond, M.

N. P. Schmitt, W. Rehm, T. Pistner, H. Diehl, P. Navé, G. Jenaro-Rabadan, P. Mirand, and M. Reymond, “Flight test of the AWIATOR airborne LIDAR turbulence sensor--first results,” in Proceedings of 14th Coherent Laser Radar Conference (2007).

Richey, C. J.

C. B. Watkins, C. J. Richey, P. Tchoryk, Jr., G. A. Ritter, M. Dehring, P. B. Hays, C. A. Nardell, and R. Urzi, “Molecular optical air data system (MOADS) prototype II,” Proc. SPIE 5412, 10-20 (2004).
[CrossRef]

Ritter, G. A.

C. B. Watkins, C. J. Richey, P. Tchoryk, Jr., G. A. Ritter, M. Dehring, P. B. Hays, C. A. Nardell, and R. Urzi, “Molecular optical air data system (MOADS) prototype II,” Proc. SPIE 5412, 10-20 (2004).
[CrossRef]

Robinson, P. A.

Roesler, F. L.

Schlotterbeck, J. P.

J. P. Cariou, B. Augere, D. Goular, J. P. Schlotterbeck, and X. Lacondemine, “All-fiber 1.5 μm CW coherent laser anemometer DALHEC--helicopter flight test analysis,” in Proceedings of 13th Coherent Laser Radar Conference (National Institute of Information and Communications Technology, 2005), pp. 157-160.

Schmitt, N.

N. Schmitt, W. Rehm, T. Pistner, P. Zeller, H. Diehl, and P. Navé, “Airborne direct detection UV lidar,” in Proceedings of 23rd International Laser Radar Conference (2006), pp. 167-170.

Schmitt, N. P.

N. P. Schmitt, W. Rehm, T. Pistner, H. Diehl, P. Navé, G. Jenaro-Rabadan, P. Mirand, and M. Reymond, “Flight test of the AWIATOR airborne LIDAR turbulence sensor--first results,” in Proceedings of 14th Coherent Laser Radar Conference (2007).

She, C.-Y.

Shepherd, G. G.

Skinner, W. R.

Song, X.-Q.

Soreide, D.

D. Soreide, R. K. Bogue, L. J. Ehernberger, and H. Bagley, “Coherent lidar turbulence measurement for gust load alleviation,” NASA Technical Memorandum104318 (NASA, 1996).

Steakley, B. C.

Stephens, M.

C. J. Grund, M. Stephens, and C. Weimer, “Simultaneous profiling of aerosol optical properties, gas chemistry, and winds with optical autocovariance lidar,” in Proceedings of 24th International Laser Radar Conference (2008).

Stone, R.

Swanson, D.

Targ, R.

Tchoryk, P.

C. B. Watkins, C. J. Richey, P. Tchoryk, Jr., G. A. Ritter, M. Dehring, P. B. Hays, C. A. Nardell, and R. Urzi, “Molecular optical air data system (MOADS) prototype II,” Proc. SPIE 5412, 10-20 (2004).
[CrossRef]

P. Tchoryk, Jr., C. B. Watkins, S. K. Lindemann, P. B. Hays, and C. A. Nardell, “Molecular optical air data system (MOADS),” Proc. SPIE 4377, 194-204 (2001).
[CrossRef]

Tenti, G.

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

Uchida, M.

Urzi, R.

C. B. Watkins, C. J. Richey, P. Tchoryk, Jr., G. A. Ritter, M. Dehring, P. B. Hays, C. A. Nardell, and R. Urzi, “Molecular optical air data system (MOADS) prototype II,” Proc. SPIE 5412, 10-20 (2004).
[CrossRef]

Van Trees, H. L.

H. L. Van Trees, in Detection, Estimation and Modulation Theory. Part I: Detection, Estimation, and Linear Modulation Theory (Wiley, 1968), pp. 52-86.

Watkins, C. B.

C. B. Watkins, C. J. Richey, P. Tchoryk, Jr., G. A. Ritter, M. Dehring, P. B. Hays, C. A. Nardell, and R. Urzi, “Molecular optical air data system (MOADS) prototype II,” Proc. SPIE 5412, 10-20 (2004).
[CrossRef]

P. Tchoryk, Jr., C. B. Watkins, S. K. Lindemann, P. B. Hays, and C. A. Nardell, “Molecular optical air data system (MOADS),” Proc. SPIE 4377, 194-204 (2001).
[CrossRef]

Weimer, C.

C. J. Grund, M. Stephens, and C. Weimer, “Simultaneous profiling of aerosol optical properties, gas chemistry, and winds with optical autocovariance lidar,” in Proceedings of 24th International Laser Radar Conference (2008).

Wimperis, J.

Wu, D.

Young, A. T.

Zarifis, V.

Zeller, P.

N. Schmitt, W. Rehm, T. Pistner, P. Zeller, H. Diehl, and P. Navé, “Airborne direct detection UV lidar,” in Proceedings of 23rd International Laser Radar Conference (2006), pp. 167-170.

Zhang, K.-L.

Appl. Opt. (11)

R. Targ, M. J. Kavaya, R. M. Huffaker, and R. L. Bowles, “Coherent lidar airborne windshear sensor: performance evaluation,” Appl. Opt. 30, 2013-2026 (1991).
[CrossRef] [PubMed]

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Figures (11)

Fig. 1
Fig. 1

Schematic view of a Rayleigh–Mie lidar implementing a dual-FIMI as receiver. L 1 , collimation lens; PC, polarizing cube; BS, beam splitter; FWP, field-widening plate; M − M , pair of mirrors; L 2 , 3 , relay optics.

Fig. 2
Fig. 2

Comparison of a Rayleigh Gaussian spectrum (black curve, T = 273 K , α = 1 ) and a Rayleigh–Mie spectrum (gray curve, T = 273 K , α = 1.1 , γ L = 150 MHz ). The spectra are plotted on a centered frequency axis (relative frequency with respect to line center).

Fig. 3
Fig. 3

Description of the dual-FIMI as two distinct multichannel spectral analyzers. The Rayleigh–Mie spectrum (black curve) is transmitted differently by the pixels of a 3 cm FIMI (solid gray curves) and by the pixels of a 10 cm FIMI (dotted gray curves). The pixel transmissions functions are calculated by using contrast factors equal to unity and arbitrary amplitudes of 0.25.

Fig. 4
Fig. 4

Description of the dual-FIMI as a two-segment coherence analyzer. The degree of temporal coherence is plotted as a function of OPD for the two spectra displayed in Fig. 2 (Rayleigh spectrum, black; Rayleigh–Mie spectrum, gray). The shaded rectangles represent the two segments sampled by the dual-FIMI ( Δ 01 = 3 cm and Δ 02 = 10 cm ). The sine wave function and segment scales are exaggerated for a better visibility.

Fig. 5
Fig. 5

Histogram of the 5000 temperature estimates provided by the MLE, compared with a Gaussian probability density function (solid curve). The histogram mean value is T = 250.2 K (true value to retrieve, 250 K ). The standard deviation is 2.1 K (equal to the CRB).

Fig. 6
Fig. 6

Optimization of a single-FIMI for speed measurements. The penalty factor for speed measurement is plotted as a function of OPD Δ 01 for different values of the experimental contrast factor V ( V = 1 , solid curve; V = 0.8 , dashed; V = 0.6 , dotted). The penalty factors are calculated assuming clear air conditions ( α = 1 ) at temperature T = 273 K .

Fig. 7
Fig. 7

Optimization of the second FIMI for temperature and density measurements. The penalty factors for measurements of temperature (solid curve) and density (dashed) are plotted as a function of OPD Δ 02 , assuming clear air conditions ( α = 1 ) at temperature T = 273 K . The first interferometer is set to Δ 01 = 3 cm .

Fig. 8
Fig. 8

Influence of scattering ratio on the penalty factors. The normalized penalty factors for measurements of speed (black curve), temperature (light gray) and density (dark gray) are plotted as a function of scattering ratio. The lower the penalty factors, the better are the performances.

Fig. 9
Fig. 9

Influence of atmospheric temperature on the penalty factors. The normalized penalty factors for measurements of speed (black curve), temperature (light gray) and density (dark gray) are plotted as a function of temperature. The lower the penalty factors, the better are the performances.

Fig. 10
Fig. 10

Comparison of a Gaussian line shape (solid curve, T = 288 K ) and a Rayleigh–Brillouin spectrum (dashed curve) computed with the Tenti S6 model ( λ = 355 nm , P r = 1 atm , T = 288 K , y = 0.4 ). The spectra are plotted on a centered frequency axis.

Fig. 11
Fig. 11

Envelop of the temporal coherence functions of the two spectra displayed in Fig. 10, as a function of OPD (Gaussian spectrum, black curve; Tenti S6 spectrum, gray curve). The shaded rectangles represent the two segments sampled by the dual-FIMI ( Δ 01 = 3 cm and Δ 02 = 10 cm ).

Tables (2)

Tables Icon

Table 1 Statistical MLE Performances

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Table 2 Expectable Bias Values on Temperature and Density Measurements

Equations (29)

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R ( ν ) = K ρ * α R ˜ ( ν ν c ) ,
R ˜ ( ν ) = 1 α γ G π exp [ ( ν γ G ) 2 ] + ( 1 1 / α ) γ L π exp [ ( ν γ L ) 2 ] .
γ G ( T ) = ( γ L 2 + σ 2 T ) 1 / 2 ,
σ = 2 ν L c 2 k B M mol / N av .
T ( x , y , ν ) = 1 4 1 π r 2 [ 1 + V cos ( 2 π ν c ( Δ 0 2 ψ x ) ) ] ,
T i , j ( ν ) = 1 4 M [ 1 + V sinc ( 1 P ) cos ( 2 π ν c ( Δ 0 2 ψ x i ) ) ] .
N i , j = η i , j + T i , j ( ν ) R ( ν ) d ν ,
N i , j = A i , j ρ * α [ 1 + V sinc ( 1 P ) + R ˜ ( ν ν c ) cos ( 2 π ν Δ 0 2 ψ x i c ) d ν ] .
N i , j = A i , j ρ * α [ 1 + V sinc ( 1 P ) G ( Δ 0 ) cos ( 2 π ν c Δ 0 2 ψ x i c ) ] ,
G ( Δ 0 ) = [ FT ( R ˜ ( ν ) ) ] ( Δ 0 / c ) = exp [ ( π Δ 0 γ L c ) 2 ] ( 1 α exp [ ( π Δ 0 σ c ) 2 T ] + ( 1 1 α ) ) .
N k ¯ ( θ ) = A k ρ * α [ 1 + W ( T , α ) cos ( d ϕ d u r u r + ϕ k ) ] + B k ,
W = V sinc ( 1 / P ) G ( Δ 0 )
ϕ k = 2 π / λ L ( Δ 0 2 ψ x k )
d ϕ / d u r = 4 π Δ 0 / c λ L
P ( N k = n k ) = N k ¯ n k exp ( N k ¯ ) / n k !
Λ ( θ ) = k = 1 2 M P ( N k ( θ ) = n k .
θ i + 1 = θ i H f 1 ( θ i ) grad f ( θ i ) ,
θ true : ( u r = 100 m / s , ρ *= 0.5 , T = 250 K , α = 1 . 2 ) , θ guess : ( u r = 140 m / s , ρ *= 0.4 , T = 230 K , α = 1 . 0 ) .
ε CRB θ i = [ ( F 1 ) i i ] 1 / 2 ,
F i j = E [ d 2 ln Λ d θ i d θ j ] ,
F i j = k = 1 2 M 1 N k ¯ d N k ¯ d θ i d N k ¯ d θ j .
N ¯ ( θ , x ) = η K 4 L ρ * α [ 1 + W ( T , α ) cos ( d ϕ d u r u r + ϕ ( x ) ) ]
F i j = 0 L 1 N 1 ( x ) ¯ d N 1 ( x ) ¯ d θ i d N 1 ( x ) ¯ d θ j d x + 0 L 1 N 2 ( x ) ¯ d N 2 ( x ) ¯ d θ i d N 2 ( x ) ¯ d θ j d x ,
F tot = [ F 11 0 0 0 0 F 22 0 F 24 0 0 F 33 F 34 0 F 24 F 34 F 44 ] 1 + [ F 11 0 0 0 0 F 22 0 F 24 0 0 F 33 F 34 0 F 24 F 34 F 44 ] 2 .
ε CRB u r = c λ L 4 π Δ 01 4 K η ρ * ( 1 1 V 2 exp [ 2 ( π Δ 01 σ c ) 2 T ] ) 1 / 2 .
κ u r = ( ε CRB u r ) dual FIMI ( ε CRB u r ) ISA = 1 2 d c Δ 01 ( 1 1 V 2 exp [ 8 ( Δ 01 d c ) 2 ] ) 1 / 2 ,
y = 0.23 T + 111 T 2 P r ( atm ) λ ( nm ) .
S ( ν ) = N ISA π σ T exp [ ( ν ν c ( u r ) σ T ) 2 ] ,
ε u r = λ L 2 σ T 2 N ISA , ε ρ * = ρ * N ISA , ε T = T 2 N ISA ,

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