Abstract

High spatial resolution vertical profiles of refractive turbulence Cn2 can be obtained using a translating airborne light source. From spatially filtered observations of the optical scintillation pattern on the ground, caused by atmospheric density fluctuations, it is possible to infer both vertical profiles of Cn2 and the shape of the refractive turbulence spectrum. Profiles of key parameters such as the turbulence microscale are thereby accessible from ground-based measurements. The required signal-to-noise ratio and resultant spatial and spectral resolutions are determined.

© 1987 Optical Society of America

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References

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  1. S. F. Clifford, “The Classical Theory of Wave Propagation in a Turbulent Medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, Ed. (Springer-Verlag, New York, 1978).
    [CrossRef]
  2. N. Ben-Yosef, E. Tirosh, A. Weitz, E. Pinsky, “Refractive-Index Structure Constant Dependence on Height,” J. Opt. Soc. Am. 69, 1616 (1979).
    [CrossRef]
  3. R. E. Good, J. H. Brown, A. F. Questada, “Measurement of High Altitude Resolution Cn2 Profiles and Their Importance on Coherence Lengths,” Proc. Soc. Photo-Opt. Instrum. Eng. 365, 105 (1982).
  4. P. H. Pasricha, M. Mohan, D. K. Tiwari, B. M. Reddy, “Height Distribution of Refractivity Structure Constant from Radiosonde Observations,” Indian J. Radio Space Phys. 12, 12 (1983).
  5. E. A. Murphy, E. M. Dewan, S. M. Sheldon, “Daytime Comparisons of Cn2 Models to Measurements in a Desert Location,” Proc. Soc. Photo-Opt. Instrum. Eng. 547, 60 (1985).
  6. R. W. Lee, J. C. Harp, “Weak Scattering in Random Media, with Applications to Remote Probing,” Proc. IEEE 57, 375 (1969).
    [CrossRef]
  7. S. E. Clifford, G. R. Ochs, T. I. Wang, “Optical Wind Sensing by Observing the Scintillations of a Random Scene,” Appl. Opt. 14, 2844 (1975).
    [CrossRef] [PubMed]
  8. G. R. Ochs, T. I. Wang, R. S. Lawrence, S. F. Clifford, “Refractive-Turbulence Profiles Measured by One-Dimensional Spatial Filtering of Scintillations,” Appl. Opt. 15, 2504 (1976).
    [CrossRef] [PubMed]
  9. J. W. Strohbehn, “Modern Theories in the Propagation of Optical Waves in a Turbulent Medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, Ed. (Springer-Verlag, New York, 1978).
    [CrossRef]
  10. R. A. Kazaryan, V. M. Dzhulakyan, “Experimental Investigation of Longitudinal Correlation of Laser Signals in a Turbulent Atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 23, 718 (1980).
  11. S. I. Belusov, V. M. Dzhulakyan, V. U. Zavbrotnyi, “Longitudinal Correlation of the Fluctuations of Laser Radiation in a Model Turbulent Medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 24, 1345 (1981).

1985

E. A. Murphy, E. M. Dewan, S. M. Sheldon, “Daytime Comparisons of Cn2 Models to Measurements in a Desert Location,” Proc. Soc. Photo-Opt. Instrum. Eng. 547, 60 (1985).

1983

P. H. Pasricha, M. Mohan, D. K. Tiwari, B. M. Reddy, “Height Distribution of Refractivity Structure Constant from Radiosonde Observations,” Indian J. Radio Space Phys. 12, 12 (1983).

1982

R. E. Good, J. H. Brown, A. F. Questada, “Measurement of High Altitude Resolution Cn2 Profiles and Their Importance on Coherence Lengths,” Proc. Soc. Photo-Opt. Instrum. Eng. 365, 105 (1982).

1981

S. I. Belusov, V. M. Dzhulakyan, V. U. Zavbrotnyi, “Longitudinal Correlation of the Fluctuations of Laser Radiation in a Model Turbulent Medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 24, 1345 (1981).

1980

R. A. Kazaryan, V. M. Dzhulakyan, “Experimental Investigation of Longitudinal Correlation of Laser Signals in a Turbulent Atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 23, 718 (1980).

1979

1976

1975

1969

R. W. Lee, J. C. Harp, “Weak Scattering in Random Media, with Applications to Remote Probing,” Proc. IEEE 57, 375 (1969).
[CrossRef]

Belusov, S. I.

S. I. Belusov, V. M. Dzhulakyan, V. U. Zavbrotnyi, “Longitudinal Correlation of the Fluctuations of Laser Radiation in a Model Turbulent Medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 24, 1345 (1981).

Ben-Yosef, N.

Brown, J. H.

R. E. Good, J. H. Brown, A. F. Questada, “Measurement of High Altitude Resolution Cn2 Profiles and Their Importance on Coherence Lengths,” Proc. Soc. Photo-Opt. Instrum. Eng. 365, 105 (1982).

Clifford, S. E.

Clifford, S. F.

G. R. Ochs, T. I. Wang, R. S. Lawrence, S. F. Clifford, “Refractive-Turbulence Profiles Measured by One-Dimensional Spatial Filtering of Scintillations,” Appl. Opt. 15, 2504 (1976).
[CrossRef] [PubMed]

S. F. Clifford, “The Classical Theory of Wave Propagation in a Turbulent Medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, Ed. (Springer-Verlag, New York, 1978).
[CrossRef]

Dewan, E. M.

E. A. Murphy, E. M. Dewan, S. M. Sheldon, “Daytime Comparisons of Cn2 Models to Measurements in a Desert Location,” Proc. Soc. Photo-Opt. Instrum. Eng. 547, 60 (1985).

Dzhulakyan, V. M.

S. I. Belusov, V. M. Dzhulakyan, V. U. Zavbrotnyi, “Longitudinal Correlation of the Fluctuations of Laser Radiation in a Model Turbulent Medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 24, 1345 (1981).

R. A. Kazaryan, V. M. Dzhulakyan, “Experimental Investigation of Longitudinal Correlation of Laser Signals in a Turbulent Atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 23, 718 (1980).

Good, R. E.

R. E. Good, J. H. Brown, A. F. Questada, “Measurement of High Altitude Resolution Cn2 Profiles and Their Importance on Coherence Lengths,” Proc. Soc. Photo-Opt. Instrum. Eng. 365, 105 (1982).

Harp, J. C.

R. W. Lee, J. C. Harp, “Weak Scattering in Random Media, with Applications to Remote Probing,” Proc. IEEE 57, 375 (1969).
[CrossRef]

Kazaryan, R. A.

R. A. Kazaryan, V. M. Dzhulakyan, “Experimental Investigation of Longitudinal Correlation of Laser Signals in a Turbulent Atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 23, 718 (1980).

Lawrence, R. S.

Lee, R. W.

R. W. Lee, J. C. Harp, “Weak Scattering in Random Media, with Applications to Remote Probing,” Proc. IEEE 57, 375 (1969).
[CrossRef]

Mohan, M.

P. H. Pasricha, M. Mohan, D. K. Tiwari, B. M. Reddy, “Height Distribution of Refractivity Structure Constant from Radiosonde Observations,” Indian J. Radio Space Phys. 12, 12 (1983).

Murphy, E. A.

E. A. Murphy, E. M. Dewan, S. M. Sheldon, “Daytime Comparisons of Cn2 Models to Measurements in a Desert Location,” Proc. Soc. Photo-Opt. Instrum. Eng. 547, 60 (1985).

Ochs, G. R.

Pasricha, P. H.

P. H. Pasricha, M. Mohan, D. K. Tiwari, B. M. Reddy, “Height Distribution of Refractivity Structure Constant from Radiosonde Observations,” Indian J. Radio Space Phys. 12, 12 (1983).

Pinsky, E.

Questada, A. F.

R. E. Good, J. H. Brown, A. F. Questada, “Measurement of High Altitude Resolution Cn2 Profiles and Their Importance on Coherence Lengths,” Proc. Soc. Photo-Opt. Instrum. Eng. 365, 105 (1982).

Reddy, B. M.

P. H. Pasricha, M. Mohan, D. K. Tiwari, B. M. Reddy, “Height Distribution of Refractivity Structure Constant from Radiosonde Observations,” Indian J. Radio Space Phys. 12, 12 (1983).

Sheldon, S. M.

E. A. Murphy, E. M. Dewan, S. M. Sheldon, “Daytime Comparisons of Cn2 Models to Measurements in a Desert Location,” Proc. Soc. Photo-Opt. Instrum. Eng. 547, 60 (1985).

Strohbehn, J. W.

J. W. Strohbehn, “Modern Theories in the Propagation of Optical Waves in a Turbulent Medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, Ed. (Springer-Verlag, New York, 1978).
[CrossRef]

Tirosh, E.

Tiwari, D. K.

P. H. Pasricha, M. Mohan, D. K. Tiwari, B. M. Reddy, “Height Distribution of Refractivity Structure Constant from Radiosonde Observations,” Indian J. Radio Space Phys. 12, 12 (1983).

Wang, T. I.

Weitz, A.

Zavbrotnyi, V. U.

S. I. Belusov, V. M. Dzhulakyan, V. U. Zavbrotnyi, “Longitudinal Correlation of the Fluctuations of Laser Radiation in a Model Turbulent Medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 24, 1345 (1981).

Appl. Opt.

Indian J. Radio Space Phys.

P. H. Pasricha, M. Mohan, D. K. Tiwari, B. M. Reddy, “Height Distribution of Refractivity Structure Constant from Radiosonde Observations,” Indian J. Radio Space Phys. 12, 12 (1983).

Izv. Vyssh. Uchebn. Zaved. Radiofiz.

R. A. Kazaryan, V. M. Dzhulakyan, “Experimental Investigation of Longitudinal Correlation of Laser Signals in a Turbulent Atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 23, 718 (1980).

S. I. Belusov, V. M. Dzhulakyan, V. U. Zavbrotnyi, “Longitudinal Correlation of the Fluctuations of Laser Radiation in a Model Turbulent Medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 24, 1345 (1981).

J. Opt. Soc. Am.

Proc. IEEE

R. W. Lee, J. C. Harp, “Weak Scattering in Random Media, with Applications to Remote Probing,” Proc. IEEE 57, 375 (1969).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

R. E. Good, J. H. Brown, A. F. Questada, “Measurement of High Altitude Resolution Cn2 Profiles and Their Importance on Coherence Lengths,” Proc. Soc. Photo-Opt. Instrum. Eng. 365, 105 (1982).

E. A. Murphy, E. M. Dewan, S. M. Sheldon, “Daytime Comparisons of Cn2 Models to Measurements in a Desert Location,” Proc. Soc. Photo-Opt. Instrum. Eng. 547, 60 (1985).

Other

S. F. Clifford, “The Classical Theory of Wave Propagation in a Turbulent Medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, Ed. (Springer-Verlag, New York, 1978).
[CrossRef]

J. W. Strohbehn, “Modern Theories in the Propagation of Optical Waves in a Turbulent Medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, Ed. (Springer-Verlag, New York, 1978).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Synthetic aperture spatial filtering geometry.

Fig. 2
Fig. 2

Effect of a single component of turbulence on the received irradiance.

Fig. 3
Fig. 3

Schematic diagram of single-aperture zero-sum receiver.

Fig. 4
Fig. 4

Single-aperture zero-sum receiver filter function (dashed curve) and approximation used for calculations (solid curve).

Fig. 5
Fig. 5

Normalized path weighting functions W(u)/W(u0) of path position u for u0 = 0.1, 0.2, 0.3, … 0.9. In this example, K R r = 5, and K R 2 h / k = 1 .

Fig. 6
Fig. 6

Single-aperture nonzero-sum receiver filter function (dashed curve) and approximation used for calculations (solid curve).

Fig. 7
Fig. 7

Eleven-aperture zero-sum receiver filter (dashed curve) and approximation used for calculations (solid curve).

Fig. 8
Fig. 8

Five-aperture nonzero-sum receiver filter function (dashed curve) and approximation used for calculations (solid curve).

Fig. 9
Fig. 9

Normalized path weighting functions W(u)/W(u0) of path position u for u0 = 0.2, 0.4, 0.6, and 0.8. In this example, eight detectors were used with a separation of 5 times their diameter, and K R 2 h / k = 1 .

Equations (59)

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d χ ( ρ 1 , ρ 2 , z , t ) = k d z d ν ( K , z ) sin [ K 2 z ( h - z ) 2 k h ] × exp { i [ K · ρ 1 ( 1 - z / h ) + K · ( ρ 2 + v S t ) z / h + K · v l t ] } ,
χ f ( ρ 1 , ρ 2 , t ) = d 2 ρ 1 f R ( ρ 1 - ρ 1 ) 0 h d χ ( ρ 1 , ρ 2 , z ) ,
C χ f ( τ ) [ χ f ( t ) - χ f ( t ) ] [ χ f * ( t + τ ) - χ f * ( t + τ ) ] ,
S χ f ( ω ) = 1 2 π - d τ exp ( - i ω τ ) C χ f ( τ ) .
C χ f ( τ ) = k 2 d 2 ρ 1 f R ( ρ 1 - ρ 1 ) d 2 ρ 1 f R ( ρ 1 - ρ 1 ) 0 h d z 1 0 h d z 2 × d ν ( K , z 1 ) d ν * ( K , z 2 ) sin [ K 2 z 1 ( h - z 1 ) 2 k h ] × sin [ K 2 z 2 ( h - z 2 ) 2 k h ] × exp { i [ K · ρ 1 ( 1 - z 1 / h ) - K · ρ 1 ( 1 - z 2 / h ) ] } × exp { i [ K · ( ρ 2 + v S t z 1 / h ) - K · ( ρ 2 + v S ( t + τ ) z 2 / h ) + i K · v l t - i K · v l ( t + τ ) ] } .
d ν ( K , z 1 ) d ν * ( K , z 2 ) = 2 π δ ( z 1 - z 2 ) Φ n ( K , 0 ) δ ( K - K ) d 2 K d 2 K ,
C χ f ( τ ) = 2 π k 2 h 0 1 d u d 2 K Φ n ( K , u ) sin 2 [ K 2 h u ( 1 - u ) 2 k ] × F R [ K ( 1 - u ) ] 2 exp [ i K · ( v S τ u - v l τ ) ] ,
F R ( Λ ) = d 2 ρ exp ( i Λ · ρ ) f R ( ρ ) .
S χ f ( ω ) = π k 2 h 0 1 d u d 2 K sin 2 [ K 2 h u ( 1 - u ) 2 k ] Φ n ( K , u ) × F R [ K ( 1 - u ) ] 2 δ ( ω - K · v S u + K · v l ) ,
S χ f ( ω ) = 1 2 π - d τ exp ( - i ω τ ) C χ f ( τ ) .
F R 2 1 2 δ [ K y ( 1 - u ) ] { δ [ K x ( 1 - u ) - K R ] + δ [ K x ( 1 - u ) + K R ] } ,
S χ f ( ω ) π k 2 h 0 1 d u ( 1 - u ) - d K x × sin 2 [ K x 2 h u ( 1 - u ) 2 k ] { δ [ K x ( 1 - u ) - K R ] + δ [ K x ( 1 - u ) + K R ] } Φ n ( K x , 0 , u ) δ [ ω - K x v S u + K x v l ] .
S χ f ( ω ) 2 π k 2 h v S - v l 0 1 d u ( 1 - u ) δ ( u - u 0 ) - d K x K x Φ n ( K x , 0 , u ) × sin 2 [ K x 2 h u ( 1 - u ) 2 k ] δ ( K x - K 0 ) ,
K 0 K R + ω / v S , u 0 = ( 1 + K R v S / ω ) - 1 .
S χ f ( ω ) 2 π k 2 h K R v S Φ n ( K 0 , 0 , u 0 ) sin 2 [ K 0 2 h u 0 ( 1 - u 0 ) 2 k ] .
K R = K ( 1 - z / h ) , ω = K v , v ( z ) = v S z / h .
K 0 = K R + ω / v S ,             u 0 = ( 1 + K R v S / ω ) - 1 ,
f R ( x , y ) = ( π r 2 ) - 1 exp ( - x 2 + y 2 2 r 2 ) cos ( K R x ) .
F R ( K x , K y ) = exp [ - 1 2 r 2 ( K x - K R ) 2 - 1 2 r 2 K y 2 ] + exp [ - 1 2 r 2 ( K x + K R ) 2 - 1 2 r 2 K y 2 ] .
S χ f ( ω ) = 4 π k 2 h v S 0 1 d u u s { 1 + cosh [ 2 ω r 2 ( 1 - u ) K R v S u ] } × exp [ - ω 2 r 2 ( 1 - u ) 2 v S 2 u 2 - r 2 K R 2 ] - d K y × exp [ - r 2 ( 1 - u ) 2 K y 2 ] Φ n ( ω v S u , K y ) × sin 2 [ 1 2 k u ( 1 - u ) h ( ω 2 v S 2 u 2 + K y 2 ) ] .
Φ n ( K x , K y ) = 0.033 C n 2 ( u ) ( K x 2 + K y 2 ) - 11 / 6 ,
S χ f ( ω ) = 0.132 π k 2 h v S 0 1 d u u C n 2 ( u ) × { 1 + cosh [ 2 ( 1 - u ) ω K R r 2 u v S ] } × exp [ - ( 1 - u ) 2 ω 2 r 2 u 2 v S 2 - K R 2 r 2 ] × - d K y exp [ - ( 1 - u ) 2 K y 2 r 2 ] [ ( ω u v S ) 2 + K y 2 ] - 11 / 6 × sin 2 { u ( 1 - u ) h 2 k [ ( ω u v S ) 2 + K y 2 ] } .
( 1 - u ) ω r u v S 1.
S χ f ( ω ) = 0.132 π 1 / 2 k 2 r - 1 h v S 8 / 3 ω - 11 / 3 exp ( - K R 2 r 2 ) × 0 1 d u u 8 / 3 1 - u { 1 + cosh [ 2 ( 1 - u ) ω K R r 2 u v S ] } × exp [ - ( 1 - u ) 2 ω 2 r 2 u 2 v S 2 ] sin 2 [ ( 1 - u ) h ω 2 2 u k v S 2 ] C n 2 ( u ) .
S χ f ( ω ) = 0.066 π 1 / 2 k 2 r - 1 h v S 8 / 3 ω - 11 / 3 × 0 1 d u u 8 / 3 1 - u sin 2 [ ( 1 - u ) h ω 2 2 u k v S 2 ] × exp { - [ 1 - ( 1 - u ) ω u K R v S ] 2 K R 2 r 2 } C n 2 ( u ) .
W ( u ) = u 8 / 3 1 - u exp { - [ 1 - ( 1 - u ) u ω K R v S ] 2 K R 2 r 2 } × sin 2 [ 1 - u 2 u ( ω 2 K R 2 v S 2 ) K R 2 h k ] ,
1 2 1 - u u ( u 0 1 - u 0 ) 2 K R 2 h k + n π ,
u 0 1 - u 0 K R 2 h k < 2 π .
W ( u ) W ( u 0 ) = exp [ - K R 2 r 2 u 0 2 ( 1 - u 0 ) 2 ( u - u 0 ) 2 ] ,
σ u = u 0 ( 1 - u 0 ) K R r ,
S χ f ( ω ) 0.066 π 1 / 2 k 2 r - 1 h v S 8 / 3 ω - 11 / 3 u 0 8 / 3 1 - u 0 × sin 2 [ ( 1 - u 0 ) h ω 2 2 u 0 k v S 2 ] C n 2 ( u 0 ) × 0 1 d u exp { - [ 1 - ( 1 - u ) ω u K R v S ] 2 K R 2 r 2 }
S χ f ( ω ) 0.066 π 2 k 2 r - 2 h K R v 0 ( K R + ω v S ) - 11 / 3 × sin 2 ( h ω K R 2 k v S ) C n 2 ( u 0 ) ,
S ^ N ( ω ) = 1 Δ ω T 0 T d t [ P S ( t ) P N + P n 1 ( t ) P N + P n 2 ( t ) P N ] ,
S N ( ω ) = 2 π r 2 S χ f ( ω ) ,
S N ( ω ) = A ( ω ) C n 2 ( u 0 ) ,
C ^ n 2 ( ω ) = S ^ N ( ω ) / A ( ω ) .
C ^ n 2 ( ω ) = C n 2 + 2 A ( ω ) Ω S N R ,
[ Δ C ^ n 2 ( ω ) ] 2 = 2 Δ ω T [ C n 4 ( ω ) + 1 A 2 ( ω ) Ω 2 SNR 2 ] ,
σ u = ( 1 - u 0 ) u 0 Δ ω ω ,
Δ ω ω = 2 K R r
f R ( x , y ) = ( 2 π r 2 ) - 1 exp ( - x 2 + y 2 2 r 2 ) ( 1 + cos K R x ) ,
S χ f ( ω ) = 4 π k 2 h v S 0 1 d u u { 1 + exp ( - 1 2 r 2 K R 2 ) cosh [ ω r 2 ( 1 - u ) K R v S u ] } 2 × exp [ - ω 2 r 2 ( 1 - u ) 2 v S 2 u 2 ] - d K y exp [ - r 2 ( 1 - u ) 2 K y 2 ] × Φ n ( ω v S u , K y ) sin 2 [ 1 2 k u ( 1 - u ) h ( ω 2 v S 2 u 2 + K y 2 ) ] .
f r ( x , y ) = ( π r 2 ) - 1 n = 1 N ( - 1 ) n exp [ - ( x - x n ) 2 + y 2 2 r 2 ]
f r ( x , y ) = ( π r 2 ) - 1 n = 1 N exp [ - ( x - x n ) 2 + y 2 2 r 2 ]
F r ( K x , K y ) = n = 1 N ( - 1 ) n exp [ - 1 2 ( K x 2 + K y 2 ) r 2 - i K x x n ] ,
S χ f ( ω ) = 8 π k 2 h v S m = 1 N n = 1 N ( - 1 ) n + m 0 1 d u u - d K y Φ n ( ω v S u , K y ) × exp [ - ( 1 - u ) 2 r 2 ( ω 2 v S 2 u 2 + K y 2 ) - i ω ( 1 - u ) v S u ( x n - x m ) ] × sin 2 [ 1 2 k u ( 1 - u ) h ( ω 2 v S 2 u 2 + K y 2 ) ] .
S χ f ( ω ) = 0.528 π k 2 h v S ( v S ω ) 8 / 3 0 1 d u × u 5 / 3 C n 2 ( u ) exp [ - ( 1 - u u ) 2 ( ω r v S ) 2 ] sin 2 ( 1 - u u h ω 2 2 k v S 2 ) m = 1 N n = 1 N ( - 1 ) n + m exp [ - i 1 - u u ω v S ( x n - x m ) ] .
1 - u u h ω 2 k v S 2 π ,
W ( u ) = u 5 / 3 exp [ - ( ω r v S ) 2 ( 1 - u u ) 2 ] sin 2 ( h ω 2 2 k v S 2 1 - u u ) N ,
N = N + 2 n = 1 N ( - 1 ) n ( N - n ) cos ( n ω d v S 1 - u u ) .
N = sin 2 ( N ω d 2 v S 1 - u u ) cos 2 ( ω d 2 v S 1 - u u ) .
u 0 = ( 1 + π v S ω d ) - 1 .
N = sin 2 ( N ω d 2 v S 1 - u u ) ( π 2 - ω d 2 v S 1 - u u ) 2 ,
W ( u ) W ( u 0 ) = { sin [ N π ( u - u 0 ) 4 u 0 ( 1 - u 0 ) ] N π ( u - u 0 ) 4 u 0 ( 1 - u 0 ) } 2 .
σ u = 4 u 0 ( 1 - u 0 ) N .
S χ f = 2.11 π 2 N k 2 h ( ω d + π v S ) ( π d + ω v S ) - 8 / 3 exp [ - ( π r d ) 2 ] × sin 2 ( π ω h 2 k v S d ) C n 2 ( u 0 ) ,
C ^ n 2 ( ω ) = C n 2 + 4 N A ( ω ) Ω SNR ,
[ Δ C ^ n 2 ( ω ) ] 2 = 2 Δ ω T [ C n 2 + 4 N 2 A 2 ( ω ) Ω 2 ( SNR ) 2 ] ,
S χ f = 5.86 × 10 7 m 2 / 3 s C n 2 .

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