Abstract

The effects of turbulence induced pathlength fluctuations on the accuracy of single color and two color laser ranging systems are examined. Correlation and structure functions for the path deviations are derived using several proposed models for the variation of Cn2 with altitude. For single color systems, random pathlength fluctuations can limit the accuracy of a range measurement to a few centimeters when the turbulence is strong (Cn2 ~ 10−13 m−2/3), and the effective propagation path is long (>10 km). Two color systems can partially correct for the random path fluctuations so that in most cases their accuracy is limited to a few millimeters. However, at low elevation angles for satellite ranging (<20°) and over long horizontal paths, two color systems can also have errors approaching a few centimeters.

© 1976 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. E. Golden et al., Appl. Opt. 12, 1447 (1973).
  2. R. Kolenkiewicz et al., Inter-site Distance Measurement from Ranging to Artificial Satellites, presented at the 1975 Fall Meeting of USNC/URSI, University of Colorado, Boulder.
  3. J. E. Faller, Expected Crustal Movement Measurements using Lunar Laser Ranging, presented at the 1975 Fall Meeting of USNC/URSI, University of Colorado, Boulder.
  4. J. Marini, Radio Sci. 7, 223 (1972).
  5. J. Saastamoinen, Bulletin Geodesique 105–107 (1972), pp. 279–298, 383–397, 13–34.
  6. J. Marini, C. Murray, Correction of Laser Range Tracking Data for Atmospheric Refraction at Elevations above 10 Degrees, NASA Tech. Rep. X-591-73-351 (1973).
  7. V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, 1971).
  8. J. Bufton, A Radiosonde Thermal Sensor Technique for Measurement of Atmospheric Turbulence, NASA Tech. Rep. G-7443 (1973).
  9. J. Bufton, Appl. Opt. 12, 1785 (1973).
  10. E. Brookner, Appl. Opt. 10, 1960 (1971).
  11. P. Titterton, J. Opt. Soc. Am. 63, 439 (1973).
  12. D. P. Greenwoodand, D. O. Tarazano, A Proposed Form for the Atmospheric Microtemperature Spatial Spectrum in the Input Range, RADC Tech. Rep. RADG-TR-74-19 (1974).
  13. P. L. Bender, J. L. Owens, J. Geophys. Res. 70, 2461 (1965).
  14. K. E. Earnshaw, E. N. Hernandez, Appl. Opt. 11, 749 (1972).
  15. E. N. Hernandez, K. B. Earnshaw, J. Geophys. Res. 77, 6994 (1972).
  16. G. D. Thayer, Atmospheric Effects on Multiple-Frequency Range Measurements, ESSA Tech. Rep. IER 56-ITSA53 (1967).
  17. A. T. Young, Appl. Opt. 8, 869 (1969).
  18. R. E. Hufnagel, Restoration of Atmospherically Degraded Images, Woods Hole Summer Study (1966), Vol. 2, Appendix 3, p. 15.

1973 (3)

1972 (4)

J. Marini, Radio Sci. 7, 223 (1972).

J. Saastamoinen, Bulletin Geodesique 105–107 (1972), pp. 279–298, 383–397, 13–34.

K. E. Earnshaw, E. N. Hernandez, Appl. Opt. 11, 749 (1972).

E. N. Hernandez, K. B. Earnshaw, J. Geophys. Res. 77, 6994 (1972).

1971 (1)

1969 (1)

1966 (1)

R. E. Hufnagel, Restoration of Atmospherically Degraded Images, Woods Hole Summer Study (1966), Vol. 2, Appendix 3, p. 15.

1965 (1)

P. L. Bender, J. L. Owens, J. Geophys. Res. 70, 2461 (1965).

Bender, P. L.

P. L. Bender, J. L. Owens, J. Geophys. Res. 70, 2461 (1965).

Brookner, E.

Bufton, J.

J. Bufton, Appl. Opt. 12, 1785 (1973).

J. Bufton, A Radiosonde Thermal Sensor Technique for Measurement of Atmospheric Turbulence, NASA Tech. Rep. G-7443 (1973).

Earnshaw, K. B.

E. N. Hernandez, K. B. Earnshaw, J. Geophys. Res. 77, 6994 (1972).

Earnshaw, K. E.

Faller, J. E.

J. E. Faller, Expected Crustal Movement Measurements using Lunar Laser Ranging, presented at the 1975 Fall Meeting of USNC/URSI, University of Colorado, Boulder.

Golden, K. E.

Greenwoodand, D. P.

D. P. Greenwoodand, D. O. Tarazano, A Proposed Form for the Atmospheric Microtemperature Spatial Spectrum in the Input Range, RADC Tech. Rep. RADG-TR-74-19 (1974).

Hernandez, E. N.

K. E. Earnshaw, E. N. Hernandez, Appl. Opt. 11, 749 (1972).

E. N. Hernandez, K. B. Earnshaw, J. Geophys. Res. 77, 6994 (1972).

Hufnagel, R. E.

R. E. Hufnagel, Restoration of Atmospherically Degraded Images, Woods Hole Summer Study (1966), Vol. 2, Appendix 3, p. 15.

Kolenkiewicz, R.

R. Kolenkiewicz et al., Inter-site Distance Measurement from Ranging to Artificial Satellites, presented at the 1975 Fall Meeting of USNC/URSI, University of Colorado, Boulder.

Marini, J.

J. Marini, Radio Sci. 7, 223 (1972).

J. Marini, C. Murray, Correction of Laser Range Tracking Data for Atmospheric Refraction at Elevations above 10 Degrees, NASA Tech. Rep. X-591-73-351 (1973).

Murray, C.

J. Marini, C. Murray, Correction of Laser Range Tracking Data for Atmospheric Refraction at Elevations above 10 Degrees, NASA Tech. Rep. X-591-73-351 (1973).

Owens, J. L.

P. L. Bender, J. L. Owens, J. Geophys. Res. 70, 2461 (1965).

Saastamoinen, J.

J. Saastamoinen, Bulletin Geodesique 105–107 (1972), pp. 279–298, 383–397, 13–34.

Tarazano, D. O.

D. P. Greenwoodand, D. O. Tarazano, A Proposed Form for the Atmospheric Microtemperature Spatial Spectrum in the Input Range, RADC Tech. Rep. RADG-TR-74-19 (1974).

Tatarski, V. I.

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, 1971).

Thayer, G. D.

G. D. Thayer, Atmospheric Effects on Multiple-Frequency Range Measurements, ESSA Tech. Rep. IER 56-ITSA53 (1967).

Titterton, P.

Young, A. T.

Appl. Opt. (5)

Bulletin Geodesique 105–107 (1)

J. Saastamoinen, Bulletin Geodesique 105–107 (1972), pp. 279–298, 383–397, 13–34.

J. Geophys. Res. (2)

E. N. Hernandez, K. B. Earnshaw, J. Geophys. Res. 77, 6994 (1972).

P. L. Bender, J. L. Owens, J. Geophys. Res. 70, 2461 (1965).

J. Opt. Soc. Am. (1)

Radio Sci. (1)

J. Marini, Radio Sci. 7, 223 (1972).

Restoration of Atmospherically Degraded Images (1)

R. E. Hufnagel, Restoration of Atmospherically Degraded Images, Woods Hole Summer Study (1966), Vol. 2, Appendix 3, p. 15.

Other (7)

G. D. Thayer, Atmospheric Effects on Multiple-Frequency Range Measurements, ESSA Tech. Rep. IER 56-ITSA53 (1967).

R. Kolenkiewicz et al., Inter-site Distance Measurement from Ranging to Artificial Satellites, presented at the 1975 Fall Meeting of USNC/URSI, University of Colorado, Boulder.

J. E. Faller, Expected Crustal Movement Measurements using Lunar Laser Ranging, presented at the 1975 Fall Meeting of USNC/URSI, University of Colorado, Boulder.

D. P. Greenwoodand, D. O. Tarazano, A Proposed Form for the Atmospheric Microtemperature Spatial Spectrum in the Input Range, RADC Tech. Rep. RADG-TR-74-19 (1974).

J. Marini, C. Murray, Correction of Laser Range Tracking Data for Atmospheric Refraction at Elevations above 10 Degrees, NASA Tech. Rep. X-591-73-351 (1973).

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, 1971).

J. Bufton, A Radiosonde Thermal Sensor Technique for Measurement of Atmospheric Turbulence, NASA Tech. Rep. G-7443 (1973).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Geometry of the laser ranging site and targets.

Fig. 2
Fig. 2

Root mean square path deviation given by Eq. (16) for the von Karman spectrum. Cn2 is the refractive index structure constant, L0 is the outer scale of turbulence, and Le is the effective pathlength in turbulence defined by Eq. (15).

Fig. 3
Fig. 3

Root mean square path deviation given by Eq. (16) for the von Karman spectrum. Cn2 is the refractive index structure constant, L0 is the outer scale of turbulence and Le is the effective pathlength in turbulence defined by Eq. (15).

Fig. 4
Fig. 4

Angular path structure functions: (a) von Karman spectrum, Eq. (25); (b) Greenwood and Tarazano spectrum.

Fig. 5
Fig. 5

Angular path correlation function calculated using Brookner’s10 improved model for Cn2 (h) given in Eq. (27): (a) sunny day; (b) clear night; (c) dawn–dusk minimums.

Fig. 6
Fig. 6

Angular path structure function calculated using Brookner’s10 improved model for Cn2(h) given in Eq. (27): (a) sunnny day; (b) clear night; (c) dawn–dusk minimums.

Fig. 7
Fig. 7

Ratio of the rms path deviation for a two color system to the rms deviation for a single color system [Eq. (29)]. ψ is the satellite elevation angle. The results were calculated for λ1 = 2λ2 and apply to the case where 0.4 μm ≤ λ1 ≤ 1 μm.

Fig. 8
Fig. 8

Ratio of the rms path deviation for a two color system to the rms deviation for a single color system [Eq. (31)]. L is the actual range. he results were calculated for λ1 = 0.6328 μm and λ2 = 0.3164 μm.

Equations (35)

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

L = C d r n 0 ( r ) + C d r n 1 ( r ) = L + Δ L ,
Δ L = C d r n 1 ( r ) = 0.
B Δ L = C 1 d r 1 C 2 d r 2 n 1 ( r 1 ) n 1 ( r 2 ) .
n 1 ( r 1 ) n 1 ( r 2 ) = B n ν ( r 1 + r 1 2 ) B n 0 ( r 1 - r 2 ) .
B Δ L ( d ) = 8 0 L d ξ B n ν ( ξ ) 0 d ρ B n 0 [ ( ρ 2 + d 2 ) 1 / 2 ] .
B n 0 ( r ) = 4 π 0 d K K 2 Φ n 0 ( K ) sin ( K r ) K r .
Φ n 0 ( K ) = α exp ( - K 2 / K m 2 ) ( 1 + K 2 L 0 2 ) 11 / 6 , K m = 5.92 / l 0 , α = L 0 3 π 3 / 2 Γ ( 11 / 6 ) Γ ( 1 / 3 ) ,
B Δ L ( d ) = 16 π 2 0 L d ξ B n ν ( ξ ) 0 d K K Φ n 0 ( K ) J 0 ( K d ) .
D Δ L ( d ) = 2 [ B Δ L ( 0 ) - B Δ L ( d ) ] = 32 π 2 × 0 L d ξ B n ν ( ξ ) 0 d K K Φ n 0 ( K ) [ 1 - J 0 ( K d ) ] .
B Δ L ( ϕ ) = 16 π 2 cos 2 ( ϕ / 2 ) × 0 d K K ϕ n 0 ( K ) 0 d ξ B n ν ( ξ ) J 0 [ 2 K ξ tan ( ϕ / 2 ) ] ;
D Δ L ( ϕ ) = 32 π 2 0 d K K Φ n 0 ( K ) 0 d ξ B n ν ( ξ ) × { 1 - J 0 [ 2 K ξ tan ( ϕ / 2 ) ] cos 2 ( ϕ / 2 ) } .
D Δ L = [ 4 / ( k 2 ) ] D w ,
Δ L 2 = 16 π 2 0 L d ξ B n ν ( ξ ) 0 d K K Φ n 0 ( K ) .
B n ν ( r ) = 0.033 C n 2 ( r ) π 3 / 2 Γ ( 1 / 3 ) Γ ( 11 / 6 ) L 0 2 / 3 = 0.523 C n 2 ( r ) L 0 2 / 3 .
L e = 1 C n 2 ( 0 ) 0 L d ξ C n 2 ( ξ ) .
Δ L 2 = 0.033 C n 2 ( 0 ) 48 π 2 5 L 0 5 / 3 L e = 3.127 C n 2 ( 0 ) L 0 5 / 3 L e .
C n 2 ( h ) = C n 2 ( 0 ) exp ( - h / h s ) ,
L e = h s csc ψ ,
Φ n 0 ( K ) ( K 2 L 0 2 + K L 0 ) - 11 / 6 .
Δ L 2 G - T = 2 π 1 / 2 Γ ( 1 / 3 ) Γ ( 5 / 6 ) [ 0.033 C n 2 ( 0 ) 48 π 2 5 L 0 5 / 3 L e ] = 26.31 C n 2 ( 0 ) L 0 5 / 3 L e .
[ 2 π 1 / 2 Γ ( 1 / 3 ) Γ ( 5 / 6 ) ] 1 / 2 2.9.
L L 1 - n ( λ 1 ) - 1 Δ n 21 Δ L 21
L L 2 + n ( λ 2 ) - 1 Δ n 12 Δ L 12 ,
[ Δ L ( λ 1 ) - Δ L ( λ 2 ) ] 2 = 2 ( 1 - ρ 12 ) Δ L 2 .
ϕ = Δ n 12 cos ψ .
L error = [ ( n - 1 ) / Δ n ] [ D Δ L ( ϕ = Δ n cot ψ ) ] 1 / 2 .
D Δ L ( ϕ ) D Δ L ( π ) = 1 - 1 cos 2 ( ϕ / 2 ) 5 8 F 2 1 [ ½ , 1 , 7 / 3 , 1 - ρ s 2 L 0 2 tan 2 ( ϕ / 2 ) ] , ρ s = 2 h s cos ( ψ + ϕ / 2 ) ,
D Δ L ( ϕ ϕ s ) D Δ L ( π ) 2.804 ( ϕ / ϕ s ) 5 / 3 ϕ s = L 0 h s sin ψ .
C n 2 ( h ) = C n 0 2 h - b exp ( - h / h s ) + C n T + δ ( h - h T ) .
L error ( Δ n ) - 1 / 6 ( n - 1 ) β 1 / 2 ( cot ψ ϕ s ) 5 / 6 ,
L error Δ L rms = 2 1 / 2 ( n - 1 ) Δ n [ D Δ L ( ϕ = Δ n cot ψ ) D Δ L ( π ) ] 1 / 2 .
Δ h L 2 12 [ ( d n d h ) λ 1 - ( d n d h ) λ 2 ] ,
L error Δ L rms 2 1 / 2 ( n - 1 ) Δ n [ D Δ L ( d = Δ h ) D Δ L ( ) ] .
D Δ L ( d ) D Δ L ( ) = 1 - 2 1 / 6 Γ ( 5 / 6 ) ( d L 0 ) 5 / 6 K 5 / 6 ( d L 0 ) 1.864 ( d L 0 ) 5 / 3 , d L 0 ,
D Δ L ( d ) D Δ L ( ) = 1 - 1 π 0 π d θ U ( 1 / 6 , - 2 / 3 , - i d L 0 cos θ ) U ( 1 / 6 , - 2 / 3 , 0 ) , 0.2093 ( d L 0 ) 5 / 3 , d L 0 .

Metrics