Abstract

Laser backscatter data for the ocean surface near nadir have been acquired from an airborne lidar platform. The unique capability of this lidar instrument to scan both transmitted laser beam and receiver field of view up to 15° off nadir have made these data sets possible. Backscatter data were collected on eight separate missions using laser wavelengths at 337 and 532 nm and 9.5 μm. Statistics of the mean, standard deviation, and probability density function of backscatter were computed and analyzed in terms of prior analytical work that relates backscatter to wind speed and mean-square wave-slope statistics. We found the full width at half-maximum of the Gaussian-shaped mean backscatter pattern to range from 11 to 24° and the normalized standard deviation at a nadir-viewing angle to range from 0.1 to 0.6. We calibrated mean backscatter at nadir for the ocean surface in terms of an effective Lambertian reflectance by comparison of beach sand and ocean backscatter. Results were 16 and 24% reflectance on two missions where calibration was possible. Our data are compared with prior laser backscatter measurements and the general literature on optical scattering from the ocean surface.

© 1983 Optical Society of America

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References

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  1. K. I. Petri, IEEE Trans. Geosci. Electron. GE-15, 87 (1977).
    [CrossRef]
  2. A. V. Jellalian, Proc. IEEE 56, 828 (1968).
    [CrossRef]
  3. P. J. W. Swennen, J. Opt. Soc. Am. 58, 47 (1968).
    [CrossRef]
  4. C. Cox, W. Munk, J. Opt. Soc. Am. 44, 838 (1954).
    [CrossRef]
  5. J. Wu, Phys. Fluids 15, 741 (1972).
    [CrossRef]
  6. M. S. Longuet-Higgins, Proc. Cambridge Philos. Soc. 55, 91 (1959).
    [CrossRef]
  7. M. S. Longuet-Higgins, J. Opt. Soc. Am. 50, 838 (1960).
    [CrossRef]
  8. M. S. Longuet-Higgins, J. Opt. Soc. Am. 50, 845 (1960).
    [CrossRef]
  9. M. S. Longuet-Higgins, J. Opt. Soc. Am. 50, 851 (1960).
    [CrossRef]
  10. R. D. Kodis, IEEE Trans. Antennas Propag. AP-14, 77 (1966).
    [CrossRef]
  11. D. E. Barrick, IEEE Trans. Antennas Propag. AP-16, 449 (1968).
    [CrossRef]
  12. D. E. Barrick, Proc. IEEE 56, 1728 (1968).
    [CrossRef]
  13. B. M. Tsai, C. S. Gardner, Appl. Opt. 21, 3932 (1982).
    [CrossRef] [PubMed]
  14. C. S. Gardner, B. M. Tsai, K. E. Im, Appl. Opt. 22, 2571 (1983).
    [CrossRef] [PubMed]
  15. K. S. Krishnan, N. A. Peppers, “Scattering of Laser Radiation from the Ocean Surface,” Stanford Research Institute, Menlo Park, Calif., final report on ONR contract N00014-73-C-0445, Oct.1973.
  16. G. N. Plass, G. W. Kattawar, J. A. Guinn, Appl. Opt. 15, 3161 (1976).
    [CrossRef] [PubMed]
  17. J. Wu, J. Geophys. Res. 83, 1359 (1977).
    [CrossRef]
  18. H. C. Schau, Appl. Opt. 17, 15 (1978).
    [CrossRef] [PubMed]
  19. D. M. Rayner, M. Lee, A. G. Szabo, Appl. Opt. 17, 2730 (1978).
    [CrossRef] [PubMed]
  20. N. A. Peppers, J. S. Ostrem, Appl. Opt. 17, 3450 (1978).
    [CrossRef] [PubMed]
  21. J. A. Guinn, G. N. Plass, G. W. Kattawar, Appl. Opt. 18, 842 (1979).
    [CrossRef] [PubMed]
  22. L. C. Bobb, G. Ferguson, M. Rankin, Appl. Opt. 18, 1167 (1979).
    [CrossRef] [PubMed]
  23. R. D. Chopsman, G. B. Irani, Appl. Opt. 20, 3645 (1981).
    [CrossRef]
  24. L. B. Stotts, S. Karp, Appl. Opt. 21, 978 (1982).
    [CrossRef] [PubMed]
  25. F. E. Hoge, R. N. Swift, E. B. Frederick, Appl. Opt. 19, 871 (1980).
    [CrossRef] [PubMed]
  26. F. E. Hoge, R. N. Swift, Appl. Opt. 19, 3296 (1980).
  27. F. E. Hoge, R. N. Swift, Appl. Opt. 20, 3197 (1981).
    [CrossRef] [PubMed]
  28. G. M. Hale, M. E. Querry, Appl. Opt. 12, 555 (1973).
    [CrossRef] [PubMed]
  29. A. Morel, “Optical Properties of Pure Water and Pure Sea Water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielson, Eds. (Academic, New York, 1974), Chap. 1.
  30. M. R. Querry, W. E. Holland, R. C. Waring, L. M. Earls, M. D. Querry, J. Geophys. Res. 82, 1425 (1977).
    [CrossRef]
  31. H. R. Gordon, Appl. Opt. 16, 2627 (1977).
    [CrossRef] [PubMed]
  32. C. H. Whitlock et al., Appl. Opt. 20, 517 (1981).
    [CrossRef] [PubMed]
  33. A. Morel, L. Prieur, Limnol. Oceanogr. 22, 709 (1977).
    [CrossRef]
  34. J. E. Tyler, R. C. Smith, W. H. Wilson, J. Opt. Soc. Am. 62, 83 (1972).
    [CrossRef]
  35. G. S. Brown, IEEE Tranś. Antennas Propag. AP-26, 472 (1978).
  36. R. W. Newton, IEEE Trans. Geosci. Electron. GE-19, 2 (1972).
    [CrossRef]
  37. F. C. Jackson, Radio Sci. 16, 1385 (1981).
    [CrossRef]
  38. J. Wu, J. Opt. Soc. Am. 61, 852 (1971).
    [CrossRef]
  39. W. J. Pierson, R. A. Stacy, “The Elevation Slope and Curvature Spectra of a Wind Roughened Sea Surface,” NASA Contract. Rep. CR-2247 (NASA, Washington, D.C., Dec.1973).
  40. K. S. Krishnan, Palo Alto, Calif.; private communication, Jan.1982.
  41. F. C. Jackson, “Calculation of Standard Deviation of Finite-Sample Mean Square Slope of a Short-Crested Gaussian Sea,” unpublished internal report, General Electric Corp., Valley Forge, Pa., 24Sept.1974.
  42. J. L. Bufton, T. Itabe, D. A. Grolemund, Opt. Lett. 7, 584 (1982).
    [CrossRef] [PubMed]
  43. “Final Report of the Advanced Application Flight Experiment Breadboard Pulse Compression Radar Altimeter Program,” NASA Contract. Rep. CR-141411 (NTIS, Springfield, Va., Aug.1976).
  44. J. L. MacArthur, “Design of the SEASAT-A Radar Altimeter,” in Proceedings, Ocean '76 Conference, Washington, D.C., 13–15 Sept. 1976.
  45. J. E. Kenney, E. A. Juliana, E. J. Walsh, IEEE Trans. Microwave Theory Tech. MTT-27, 1080 (1979).
    [CrossRef]
  46. R. E. Hufnagel, “Propagation Through Atmospheric Turbulence,” in The Infrared Handbook, W. L. Wolfe, G. J. Zissis, Eds. (U.S. GPO, Washington, D.C., 1978), Chap. 6.
  47. G. H. Suits, “Natural Sources,” in The Infrared Handbook, W. L. Wolfe, G. J. Zissis, Eds. (U.S. GPO, Washington, D.C., 1978), Chap. 3.

1983 (1)

1982 (3)

1981 (4)

1980 (2)

1979 (3)

1978 (4)

1977 (5)

H. R. Gordon, Appl. Opt. 16, 2627 (1977).
[CrossRef] [PubMed]

M. R. Querry, W. E. Holland, R. C. Waring, L. M. Earls, M. D. Querry, J. Geophys. Res. 82, 1425 (1977).
[CrossRef]

A. Morel, L. Prieur, Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

K. I. Petri, IEEE Trans. Geosci. Electron. GE-15, 87 (1977).
[CrossRef]

J. Wu, J. Geophys. Res. 83, 1359 (1977).
[CrossRef]

1976 (1)

1973 (1)

1972 (3)

R. W. Newton, IEEE Trans. Geosci. Electron. GE-19, 2 (1972).
[CrossRef]

J. Wu, Phys. Fluids 15, 741 (1972).
[CrossRef]

J. E. Tyler, R. C. Smith, W. H. Wilson, J. Opt. Soc. Am. 62, 83 (1972).
[CrossRef]

1971 (1)

1968 (4)

P. J. W. Swennen, J. Opt. Soc. Am. 58, 47 (1968).
[CrossRef]

D. E. Barrick, IEEE Trans. Antennas Propag. AP-16, 449 (1968).
[CrossRef]

D. E. Barrick, Proc. IEEE 56, 1728 (1968).
[CrossRef]

A. V. Jellalian, Proc. IEEE 56, 828 (1968).
[CrossRef]

1966 (1)

R. D. Kodis, IEEE Trans. Antennas Propag. AP-14, 77 (1966).
[CrossRef]

1960 (3)

1959 (1)

M. S. Longuet-Higgins, Proc. Cambridge Philos. Soc. 55, 91 (1959).
[CrossRef]

1954 (1)

Barrick, D. E.

D. E. Barrick, IEEE Trans. Antennas Propag. AP-16, 449 (1968).
[CrossRef]

D. E. Barrick, Proc. IEEE 56, 1728 (1968).
[CrossRef]

Bobb, L. C.

Brown, G. S.

G. S. Brown, IEEE Tranś. Antennas Propag. AP-26, 472 (1978).

Bufton, J. L.

Chopsman, R. D.

Cox, C.

Earls, L. M.

M. R. Querry, W. E. Holland, R. C. Waring, L. M. Earls, M. D. Querry, J. Geophys. Res. 82, 1425 (1977).
[CrossRef]

Ferguson, G.

Frederick, E. B.

Gardner, C. S.

Gordon, H. R.

Grolemund, D. A.

Guinn, J. A.

Hale, G. M.

Hoge, F. E.

Holland, W. E.

M. R. Querry, W. E. Holland, R. C. Waring, L. M. Earls, M. D. Querry, J. Geophys. Res. 82, 1425 (1977).
[CrossRef]

Hufnagel, R. E.

R. E. Hufnagel, “Propagation Through Atmospheric Turbulence,” in The Infrared Handbook, W. L. Wolfe, G. J. Zissis, Eds. (U.S. GPO, Washington, D.C., 1978), Chap. 6.

Im, K. E.

Irani, G. B.

Itabe, T.

Jackson, F. C.

F. C. Jackson, Radio Sci. 16, 1385 (1981).
[CrossRef]

F. C. Jackson, “Calculation of Standard Deviation of Finite-Sample Mean Square Slope of a Short-Crested Gaussian Sea,” unpublished internal report, General Electric Corp., Valley Forge, Pa., 24Sept.1974.

Jellalian, A. V.

A. V. Jellalian, Proc. IEEE 56, 828 (1968).
[CrossRef]

Juliana, E. A.

J. E. Kenney, E. A. Juliana, E. J. Walsh, IEEE Trans. Microwave Theory Tech. MTT-27, 1080 (1979).
[CrossRef]

Karp, S.

Kattawar, G. W.

Kenney, J. E.

J. E. Kenney, E. A. Juliana, E. J. Walsh, IEEE Trans. Microwave Theory Tech. MTT-27, 1080 (1979).
[CrossRef]

Kodis, R. D.

R. D. Kodis, IEEE Trans. Antennas Propag. AP-14, 77 (1966).
[CrossRef]

Krishnan, K. S.

K. S. Krishnan, N. A. Peppers, “Scattering of Laser Radiation from the Ocean Surface,” Stanford Research Institute, Menlo Park, Calif., final report on ONR contract N00014-73-C-0445, Oct.1973.

K. S. Krishnan, Palo Alto, Calif.; private communication, Jan.1982.

Lee, M.

Longuet-Higgins, M. S.

MacArthur, J. L.

J. L. MacArthur, “Design of the SEASAT-A Radar Altimeter,” in Proceedings, Ocean '76 Conference, Washington, D.C., 13–15 Sept. 1976.

Morel, A.

A. Morel, L. Prieur, Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

A. Morel, “Optical Properties of Pure Water and Pure Sea Water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielson, Eds. (Academic, New York, 1974), Chap. 1.

Munk, W.

Newton, R. W.

R. W. Newton, IEEE Trans. Geosci. Electron. GE-19, 2 (1972).
[CrossRef]

Ostrem, J. S.

Peppers, N. A.

N. A. Peppers, J. S. Ostrem, Appl. Opt. 17, 3450 (1978).
[CrossRef] [PubMed]

K. S. Krishnan, N. A. Peppers, “Scattering of Laser Radiation from the Ocean Surface,” Stanford Research Institute, Menlo Park, Calif., final report on ONR contract N00014-73-C-0445, Oct.1973.

Petri, K. I.

K. I. Petri, IEEE Trans. Geosci. Electron. GE-15, 87 (1977).
[CrossRef]

Pierson, W. J.

W. J. Pierson, R. A. Stacy, “The Elevation Slope and Curvature Spectra of a Wind Roughened Sea Surface,” NASA Contract. Rep. CR-2247 (NASA, Washington, D.C., Dec.1973).

Plass, G. N.

Prieur, L.

A. Morel, L. Prieur, Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

Querry, M. D.

M. R. Querry, W. E. Holland, R. C. Waring, L. M. Earls, M. D. Querry, J. Geophys. Res. 82, 1425 (1977).
[CrossRef]

Querry, M. E.

Querry, M. R.

M. R. Querry, W. E. Holland, R. C. Waring, L. M. Earls, M. D. Querry, J. Geophys. Res. 82, 1425 (1977).
[CrossRef]

Rankin, M.

Rayner, D. M.

Schau, H. C.

Smith, R. C.

Stacy, R. A.

W. J. Pierson, R. A. Stacy, “The Elevation Slope and Curvature Spectra of a Wind Roughened Sea Surface,” NASA Contract. Rep. CR-2247 (NASA, Washington, D.C., Dec.1973).

Stotts, L. B.

Suits, G. H.

G. H. Suits, “Natural Sources,” in The Infrared Handbook, W. L. Wolfe, G. J. Zissis, Eds. (U.S. GPO, Washington, D.C., 1978), Chap. 3.

Swennen, P. J. W.

Swift, R. N.

Szabo, A. G.

Tsai, B. M.

Tyler, J. E.

Walsh, E. J.

J. E. Kenney, E. A. Juliana, E. J. Walsh, IEEE Trans. Microwave Theory Tech. MTT-27, 1080 (1979).
[CrossRef]

Waring, R. C.

M. R. Querry, W. E. Holland, R. C. Waring, L. M. Earls, M. D. Querry, J. Geophys. Res. 82, 1425 (1977).
[CrossRef]

Whitlock, C. H.

Wilson, W. H.

Wu, J.

J. Wu, J. Geophys. Res. 83, 1359 (1977).
[CrossRef]

J. Wu, Phys. Fluids 15, 741 (1972).
[CrossRef]

J. Wu, J. Opt. Soc. Am. 61, 852 (1971).
[CrossRef]

Appl. Opt. (16)

IEEE Trans. Antennas Propag. (3)

R. D. Kodis, IEEE Trans. Antennas Propag. AP-14, 77 (1966).
[CrossRef]

D. E. Barrick, IEEE Trans. Antennas Propag. AP-16, 449 (1968).
[CrossRef]

G. S. Brown, IEEE Tranś. Antennas Propag. AP-26, 472 (1978).

IEEE Trans. Geosci. Electron. (2)

R. W. Newton, IEEE Trans. Geosci. Electron. GE-19, 2 (1972).
[CrossRef]

K. I. Petri, IEEE Trans. Geosci. Electron. GE-15, 87 (1977).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

J. E. Kenney, E. A. Juliana, E. J. Walsh, IEEE Trans. Microwave Theory Tech. MTT-27, 1080 (1979).
[CrossRef]

J. Geophys. Res. (2)

M. R. Querry, W. E. Holland, R. C. Waring, L. M. Earls, M. D. Querry, J. Geophys. Res. 82, 1425 (1977).
[CrossRef]

J. Wu, J. Geophys. Res. 83, 1359 (1977).
[CrossRef]

J. Opt. Soc. Am. (7)

Limnol. Oceanogr. (1)

A. Morel, L. Prieur, Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

Opt. Lett. (1)

Phys. Fluids (1)

J. Wu, Phys. Fluids 15, 741 (1972).
[CrossRef]

Proc. Cambridge Philos. Soc. (1)

M. S. Longuet-Higgins, Proc. Cambridge Philos. Soc. 55, 91 (1959).
[CrossRef]

Proc. IEEE (2)

D. E. Barrick, Proc. IEEE 56, 1728 (1968).
[CrossRef]

A. V. Jellalian, Proc. IEEE 56, 828 (1968).
[CrossRef]

Radio Sci. (1)

F. C. Jackson, Radio Sci. 16, 1385 (1981).
[CrossRef]

Other (9)

W. J. Pierson, R. A. Stacy, “The Elevation Slope and Curvature Spectra of a Wind Roughened Sea Surface,” NASA Contract. Rep. CR-2247 (NASA, Washington, D.C., Dec.1973).

K. S. Krishnan, Palo Alto, Calif.; private communication, Jan.1982.

F. C. Jackson, “Calculation of Standard Deviation of Finite-Sample Mean Square Slope of a Short-Crested Gaussian Sea,” unpublished internal report, General Electric Corp., Valley Forge, Pa., 24Sept.1974.

“Final Report of the Advanced Application Flight Experiment Breadboard Pulse Compression Radar Altimeter Program,” NASA Contract. Rep. CR-141411 (NTIS, Springfield, Va., Aug.1976).

J. L. MacArthur, “Design of the SEASAT-A Radar Altimeter,” in Proceedings, Ocean '76 Conference, Washington, D.C., 13–15 Sept. 1976.

K. S. Krishnan, N. A. Peppers, “Scattering of Laser Radiation from the Ocean Surface,” Stanford Research Institute, Menlo Park, Calif., final report on ONR contract N00014-73-C-0445, Oct.1973.

A. Morel, “Optical Properties of Pure Water and Pure Sea Water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielson, Eds. (Academic, New York, 1974), Chap. 1.

R. E. Hufnagel, “Propagation Through Atmospheric Turbulence,” in The Infrared Handbook, W. L. Wolfe, G. J. Zissis, Eds. (U.S. GPO, Washington, D.C., 1978), Chap. 6.

G. H. Suits, “Natural Sources,” in The Infrared Handbook, W. L. Wolfe, G. J. Zissis, Eds. (U.S. GPO, Washington, D.C., 1978), Chap. 3.

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Figures (11)

Fig. 1
Fig. 1

Lidar backscatter measurement geometry.

Fig. 2
Fig. 2

Theoretical mean backscatter factor ρ/Ω of Eq. (12) vs offnadir angle θ.

Fig. 3
Fig. 3

Estimates of normalized standard deviation of backscatter from Eq. (23) (- - -) and from Eq. (26) and a tanθ dependence (—) for various values of wind speed and a footprint area AF = 1 m2.

Fig. 4
Fig. 4

Optical schematic of Airborne Oceanographic Lidar system.

Fig. 5
Fig. 5

Laser backscatter receiver electronics and waveform digitizer in Airborne Oceanographic Lidar bathymetry mode.

Fig. 6
Fig. 6

Normalized mean backscatter 〈ER〉/〈ER vs off-nadir angle θ for missions (see symbol key in Table IV) in the Wallops Flight Facility vicinity at 532-nm wavelength and theoretical result (- - -) of Eq. (14) for wind speed U10 = 2 m/sec.

Fig. 7
Fig. 7

Normalized mean backscatter 〈ER〉/〈ER0° vs off-nadir angle θ for open ocean missions (see symbol key in Table IV) at 337-nm wavelength and theoretical result (- - -) of Eq. (14) for wind speed U10 = 10 m/sec.

Fig. 8
Fig. 8

Calibrated ocean backscatter in terms of effective Lambertian reflectance ρeff for 532-nm wavelength laser backscatter data (see symbol key in Table IV) in the Wallops Flight Facility vicinity vs off-nadir angle θ and theoretical predictions (- - -) for this quantity based on Eq. (14), Fresnel reflectance of 0.02, and wind speeds U10 of 2 and 5 m/sec.

Fig. 9
Fig. 9

Summary of normalized standard deviation of laser backscatter for all missions (see symbol key in Table IV) vs off-nadir angle θ with prediction (- - -) for this quantity based on Eqs. (25) and (26), a wind speed U10 = 5 m/sec, and a laser footprint area AF of 1 m2.

Fig. 10
Fig. 10

Probability density function of laser backscatter data for various missions and off-nadir angles.

Fig. 11
Fig. 11

Probability density function of laser backscatter data for two open ocean missions at 337-nm wavelength for off-nadir angles θ of 4,8, and 12°.

Tables (4)

Tables Icon

Table I Reflectance vs Wavelength for Clean Ocean Water

Tables Icon

Table II Relationship of Ocean Surface Slope Statistics to Wind Speed

Tables Icon

Table III Lidar Instrument Specifications

Tables Icon

Table IV Summary of Mission Parameters, Ocean Surface Parameters, and Backscatter Measurements

Equations (32)

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

E R = E T A R Z 2 L o L A 2 ( ρ Ω ) ,
L A 2 =
ρ = ( n 2 r n 2 + r ) 2 and ρ = ( r 1 r + 1 ) 2 ,
r = ( n 2 sin 2 θ 1 sin 2 θ ) ,
ρ = [ n ( λ ) 1 ] 2 + k 2 ( λ ) [ n ( λ ) + 1 ] 2 + k 2 ( λ ) ,
R ( 0 ) 0.33 b b a ,
S 2 = S u 2 + S c 2 , S 2 = 0.003 + 5.12 × 10 3 U 12.4 ,
S 2 = { ( ln U 10 + 1.2 ) 10 2 , U 10 7 m / sec ( 0.85 ln U 10 1.45 ) 10 1 , U 10 > 7 m / sec .
u * = 0.4 U z / ln ( Z / Z o ) , Z o = 0.684 u * + 4.28 × 10 5 u * 2 0.0443
m n m = 0 0 S ( k ) k u 2 n + 1 k c 2 m + 1 d k u d k c .
N A = 7.255 π 2 l 2 exp ( tan 2 θ S 2 ) , S 2 = 4 h 2 l 2 .
| r 1 r 2 | = 0.1378 π l 2 S 2 sec 4 θ ,
σ 0 = π ρ N A | r , r 2 | .
σ 0 4 π = ρ Ω = ρ N A | r , r 2 | 4 , ρ Ω = ρ sec 4 θ 4 π S 2 exp ( tan 2 θ S 2 ) ,
ρ Ω = ρ 4 sec 4 θ P s ( tan θ ) .
Ω = 4 / [ sec 4 θ P s ( tan θ ) ] ,
E R = E T A R h 2 L o L A 2 ρ 4 sec 2 θ P s ( tan θ ) .
FWHM ( degrees ) 2 arctan [ 0.83 S 2 1 / 2 ] .
ρ eff = ρ Ω L Ω = ρ π sec 5 θ 4 P s ( tan θ ) .
P s ( tan θ ) = exp [ tan 2 θ 2 ( cos 2 ϕ S u 2 + sin 2 ϕ S c 2 ) ] 2 π S u 2 1 / 2 S c 2 1 / 2 .
Ω = 4 π [ S 2 + tan 2 θ T ] ,
2 = ( E R E R 2 ) E R 2
2 = 2 | r 1 r 2 | N + N 2
2 = 2 cos 3 θ A F N A .
N A = ( m 40 m 04 + 3 m 22 2 ) 1 / 2 3 2 P s ( tan θ ) ,
= [ 3 cos 3 θ A F ( m 40 m 04 + 3 m 22 2 ) 1 / 2 P s ( tan θ ) ] 1 / 2 .
2 ( 1 E R E R θ Δ θ ) 2 ,
2 = [ 2 tan θ ( 1 sec 2 θ S 2 ) ] 2 | Δ θ | 2 ,
= 2 tan θ ( 1 sec 2 θ S 2 ) | Δ θ | .
o = 0.12 U 2 g A F ,
S u 2
S c 2

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