Abstract

The reflection of short laser pulses from the ocean surface is analyzed based on the specular point theory of scattering. The expressions for the averaged received signal, shot noise, and speckle-induced noise are derived for a direct detection receiver system. It is found that the reflected laser pulses have an average shape which is closely related to the probability density function associated with the ocean surface profile. The result is used to estimate the mean sea level and significant wave height from temporal moments of the reflected laser pulse.

© 1982 Optical Society of America

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References

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  1. C. S. Gardner, Appl. Opt. 21, 448 (1982).
    [CrossRef] [PubMed]
  2. J. L. Bufton, T. Itabe, D. Grolemund, “Airborne Remote Sensing Measurements with a Pulsed CO2 DIAL System,” at Army Research Office Workshop on Optical and Laser Remote Sensing, Monterey, Calif.9–11 Feb. 1982.
  3. W. F. Townsend, IEEE J. Oceanic Eng. OE-5, 80 (1980).
    [CrossRef]
  4. L. S. Fedor, T. W. Godbey, J. F. R. Gower, R. Guptill, G. S. Hayne, C. L. Rufenach, E. J. Walsh, J. Geophys. Res. 84, 3991 (1979).
    [CrossRef]
  5. F. C. Jackson, Radio Sci. 16, 1385 (1981).
    [CrossRef]
  6. F. C. Jackson, W. T. Walton, P. L. Baker, “Directional Spectra from Airborne and Spaceborne Radars,” presented at ASCE and ECOR Symposium on Directional Wave Spectral Applications, Berkeley, Calif.14–17 Sept. 1981.
  7. R. D. Kodis, IEEE Trans. Antennas Propag. AP-14, 77 (1966).
    [CrossRef]
  8. M. I. Skolnik, Introduction to Radar System (McGraw-Hill, New York, 1980), p. 34.
  9. B. Tsai, C. S. Gardner, “Remote Sensing of Sea State by Laser Altimeters,” RRL Publication 514, Radio Research Laboratory, Department of Electrical Engineering, U. Illinois, Urbana-Champaign.
  10. D. E. Barrick, IEEE Trans. Antennas Propag. AP-16, 449 (1968).
    [CrossRef]
  11. A. Papoulis, IEEE Trans. Commun. COM-22, (1974).
  12. J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, New York, 1975), pp. 9–65.
    [CrossRef]
  13. C. S. Gardner, Appl. Opt. 16, 2427 (1977).
    [CrossRef] [PubMed]
  14. O. M. Phillips, The Dynamics of the Upper Ocean (Cambridge U. P., London, 1977), Chap. 4.
  15. D. E. Barrick, “Remote Sensing of Sea State by Radar,” in Remote Sensing of the Troposphere, V. E. Derr, Ed. (U.S. Washington, D.C.,), Chap. 12.
  16. M. S. Longuet-Higgins, J. Fluid Mech. 17, 459 (1963).
    [CrossRef]
  17. C. Cox, W. Munk, J. Opt. Soc. Am. 44, 838 (1954).
    [CrossRef]
  18. B. S. Yaplee, A. Shapiro, D. L. Hammond, B. D. Au, E. A. Uliana, IEEE Trans. Geosci. Electron. GE-9, 170 (1971).
    [CrossRef]
  19. F. C. Jackson, J. Geophys. Res. 84, 4939 (1979).
    [CrossRef]
  20. D. L. Hammond, R. A. Mennella, E. J. Walsh, IEEE Trans. Antennas Propag. AP-25, (1977).
  21. R. C. Smith, K. S. Baker, Appl. Opt. 20, 177 (1981).
    [CrossRef] [PubMed]

1982 (1)

1981 (2)

1980 (1)

W. F. Townsend, IEEE J. Oceanic Eng. OE-5, 80 (1980).
[CrossRef]

1979 (2)

L. S. Fedor, T. W. Godbey, J. F. R. Gower, R. Guptill, G. S. Hayne, C. L. Rufenach, E. J. Walsh, J. Geophys. Res. 84, 3991 (1979).
[CrossRef]

F. C. Jackson, J. Geophys. Res. 84, 4939 (1979).
[CrossRef]

1977 (2)

D. L. Hammond, R. A. Mennella, E. J. Walsh, IEEE Trans. Antennas Propag. AP-25, (1977).

C. S. Gardner, Appl. Opt. 16, 2427 (1977).
[CrossRef] [PubMed]

1974 (1)

A. Papoulis, IEEE Trans. Commun. COM-22, (1974).

1971 (1)

B. S. Yaplee, A. Shapiro, D. L. Hammond, B. D. Au, E. A. Uliana, IEEE Trans. Geosci. Electron. GE-9, 170 (1971).
[CrossRef]

1968 (1)

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

1966 (1)

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

1963 (1)

M. S. Longuet-Higgins, J. Fluid Mech. 17, 459 (1963).
[CrossRef]

1954 (1)

Au, B. D.

B. S. Yaplee, A. Shapiro, D. L. Hammond, B. D. Au, E. A. Uliana, IEEE Trans. Geosci. Electron. GE-9, 170 (1971).
[CrossRef]

Baker, K. S.

Baker, P. L.

F. C. Jackson, W. T. Walton, P. L. Baker, “Directional Spectra from Airborne and Spaceborne Radars,” presented at ASCE and ECOR Symposium on Directional Wave Spectral Applications, Berkeley, Calif.14–17 Sept. 1981.

Barrick, D. E.

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

D. E. Barrick, “Remote Sensing of Sea State by Radar,” in Remote Sensing of the Troposphere, V. E. Derr, Ed. (U.S. Washington, D.C.,), Chap. 12.

Bufton, J. L.

J. L. Bufton, T. Itabe, D. Grolemund, “Airborne Remote Sensing Measurements with a Pulsed CO2 DIAL System,” at Army Research Office Workshop on Optical and Laser Remote Sensing, Monterey, Calif.9–11 Feb. 1982.

Cox, C.

Fedor, L. S.

L. S. Fedor, T. W. Godbey, J. F. R. Gower, R. Guptill, G. S. Hayne, C. L. Rufenach, E. J. Walsh, J. Geophys. Res. 84, 3991 (1979).
[CrossRef]

Gardner, C. S.

C. S. Gardner, Appl. Opt. 21, 448 (1982).
[CrossRef] [PubMed]

C. S. Gardner, Appl. Opt. 16, 2427 (1977).
[CrossRef] [PubMed]

B. Tsai, C. S. Gardner, “Remote Sensing of Sea State by Laser Altimeters,” RRL Publication 514, Radio Research Laboratory, Department of Electrical Engineering, U. Illinois, Urbana-Champaign.

Godbey, T. W.

L. S. Fedor, T. W. Godbey, J. F. R. Gower, R. Guptill, G. S. Hayne, C. L. Rufenach, E. J. Walsh, J. Geophys. Res. 84, 3991 (1979).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, New York, 1975), pp. 9–65.
[CrossRef]

Gower, J. F. R.

L. S. Fedor, T. W. Godbey, J. F. R. Gower, R. Guptill, G. S. Hayne, C. L. Rufenach, E. J. Walsh, J. Geophys. Res. 84, 3991 (1979).
[CrossRef]

Grolemund, D.

J. L. Bufton, T. Itabe, D. Grolemund, “Airborne Remote Sensing Measurements with a Pulsed CO2 DIAL System,” at Army Research Office Workshop on Optical and Laser Remote Sensing, Monterey, Calif.9–11 Feb. 1982.

Guptill, R.

L. S. Fedor, T. W. Godbey, J. F. R. Gower, R. Guptill, G. S. Hayne, C. L. Rufenach, E. J. Walsh, J. Geophys. Res. 84, 3991 (1979).
[CrossRef]

Hammond, D. L.

D. L. Hammond, R. A. Mennella, E. J. Walsh, IEEE Trans. Antennas Propag. AP-25, (1977).

B. S. Yaplee, A. Shapiro, D. L. Hammond, B. D. Au, E. A. Uliana, IEEE Trans. Geosci. Electron. GE-9, 170 (1971).
[CrossRef]

Hayne, G. S.

L. S. Fedor, T. W. Godbey, J. F. R. Gower, R. Guptill, G. S. Hayne, C. L. Rufenach, E. J. Walsh, J. Geophys. Res. 84, 3991 (1979).
[CrossRef]

Itabe, T.

J. L. Bufton, T. Itabe, D. Grolemund, “Airborne Remote Sensing Measurements with a Pulsed CO2 DIAL System,” at Army Research Office Workshop on Optical and Laser Remote Sensing, Monterey, Calif.9–11 Feb. 1982.

Jackson, F. C.

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

F. C. Jackson, J. Geophys. Res. 84, 4939 (1979).
[CrossRef]

F. C. Jackson, W. T. Walton, P. L. Baker, “Directional Spectra from Airborne and Spaceborne Radars,” presented at ASCE and ECOR Symposium on Directional Wave Spectral Applications, Berkeley, Calif.14–17 Sept. 1981.

Kodis, R. D.

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

Longuet-Higgins, M. S.

M. S. Longuet-Higgins, J. Fluid Mech. 17, 459 (1963).
[CrossRef]

Mennella, R. A.

D. L. Hammond, R. A. Mennella, E. J. Walsh, IEEE Trans. Antennas Propag. AP-25, (1977).

Munk, W.

Papoulis, A.

A. Papoulis, IEEE Trans. Commun. COM-22, (1974).

Phillips, O. M.

O. M. Phillips, The Dynamics of the Upper Ocean (Cambridge U. P., London, 1977), Chap. 4.

Rufenach, C. L.

L. S. Fedor, T. W. Godbey, J. F. R. Gower, R. Guptill, G. S. Hayne, C. L. Rufenach, E. J. Walsh, J. Geophys. Res. 84, 3991 (1979).
[CrossRef]

Shapiro, A.

B. S. Yaplee, A. Shapiro, D. L. Hammond, B. D. Au, E. A. Uliana, IEEE Trans. Geosci. Electron. GE-9, 170 (1971).
[CrossRef]

Skolnik, M. I.

M. I. Skolnik, Introduction to Radar System (McGraw-Hill, New York, 1980), p. 34.

Smith, R. C.

Townsend, W. F.

W. F. Townsend, IEEE J. Oceanic Eng. OE-5, 80 (1980).
[CrossRef]

Tsai, B.

B. Tsai, C. S. Gardner, “Remote Sensing of Sea State by Laser Altimeters,” RRL Publication 514, Radio Research Laboratory, Department of Electrical Engineering, U. Illinois, Urbana-Champaign.

Uliana, E. A.

B. S. Yaplee, A. Shapiro, D. L. Hammond, B. D. Au, E. A. Uliana, IEEE Trans. Geosci. Electron. GE-9, 170 (1971).
[CrossRef]

Walsh, E. J.

L. S. Fedor, T. W. Godbey, J. F. R. Gower, R. Guptill, G. S. Hayne, C. L. Rufenach, E. J. Walsh, J. Geophys. Res. 84, 3991 (1979).
[CrossRef]

D. L. Hammond, R. A. Mennella, E. J. Walsh, IEEE Trans. Antennas Propag. AP-25, (1977).

Walton, W. T.

F. C. Jackson, W. T. Walton, P. L. Baker, “Directional Spectra from Airborne and Spaceborne Radars,” presented at ASCE and ECOR Symposium on Directional Wave Spectral Applications, Berkeley, Calif.14–17 Sept. 1981.

Yaplee, B. S.

B. S. Yaplee, A. Shapiro, D. L. Hammond, B. D. Au, E. A. Uliana, IEEE Trans. Geosci. Electron. GE-9, 170 (1971).
[CrossRef]

Appl. Opt. (3)

IEEE J. Oceanic Eng. (1)

W. F. Townsend, IEEE J. Oceanic Eng. OE-5, 80 (1980).
[CrossRef]

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]

D. L. Hammond, R. A. Mennella, E. J. Walsh, IEEE Trans. Antennas Propag. AP-25, (1977).

IEEE Trans. Commun. (1)

A. Papoulis, IEEE Trans. Commun. COM-22, (1974).

IEEE Trans. Geosci. Electron. (1)

B. S. Yaplee, A. Shapiro, D. L. Hammond, B. D. Au, E. A. Uliana, IEEE Trans. Geosci. Electron. GE-9, 170 (1971).
[CrossRef]

J. Fluid Mech. (1)

M. S. Longuet-Higgins, J. Fluid Mech. 17, 459 (1963).
[CrossRef]

J. Geophys. Res. (2)

F. C. Jackson, J. Geophys. Res. 84, 4939 (1979).
[CrossRef]

L. S. Fedor, T. W. Godbey, J. F. R. Gower, R. Guptill, G. S. Hayne, C. L. Rufenach, E. J. Walsh, J. Geophys. Res. 84, 3991 (1979).
[CrossRef]

J. Opt. Soc. Am. (1)

Radio Sci. (1)

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

Other (7)

F. C. Jackson, W. T. Walton, P. L. Baker, “Directional Spectra from Airborne and Spaceborne Radars,” presented at ASCE and ECOR Symposium on Directional Wave Spectral Applications, Berkeley, Calif.14–17 Sept. 1981.

M. I. Skolnik, Introduction to Radar System (McGraw-Hill, New York, 1980), p. 34.

B. Tsai, C. S. Gardner, “Remote Sensing of Sea State by Laser Altimeters,” RRL Publication 514, Radio Research Laboratory, Department of Electrical Engineering, U. Illinois, Urbana-Champaign.

J. L. Bufton, T. Itabe, D. Grolemund, “Airborne Remote Sensing Measurements with a Pulsed CO2 DIAL System,” at Army Research Office Workshop on Optical and Laser Remote Sensing, Monterey, Calif.9–11 Feb. 1982.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, New York, 1975), pp. 9–65.
[CrossRef]

O. M. Phillips, The Dynamics of the Upper Ocean (Cambridge U. P., London, 1977), Chap. 4.

D. E. Barrick, “Remote Sensing of Sea State by Radar,” in Remote Sensing of the Troposphere, V. E. Derr, Ed. (U.S. Washington, D.C.,), Chap. 12.

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Figures (7)

Fig. 1
Fig. 1

Geometry of the laser altimeter and ocean surface for normal incidence.

Fig. 2
Fig. 2

Expected received pulse shapes for several values of the laser beam divergence angle. The laser altimeter is pointed at nadir, and the SWH is 1 m.

Fig. 3
Fig. 3

Expected received pulse shape for several values of SWH. The laser altimeter is pointed at nadir, and the beam divergence is 1 mrad.

Fig. 4
Fig. 4

Expected received pulse shape for several values of SWH. The laser altimeter is pointed at nadir, and the beam divergence is 10 mrad.

Fig. 5
Fig. 5

Expected received pulse shape for several values of the skewness coefficient. The laser altimeter is pointed at nadir, and the beam divergence is 1 mrad.

Fig. 6
Fig. 6

Expected received pulse shape for several values of the skewness coefficient. The laser altimeter is pointed at nadir, and the beam divergence is 1 mrad.

Fig. 7
Fig. 7

Geometry of the laser altimeter and ocean surface for non-normal incidence.

Equations (69)

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U T ( r , t ) = f ( t ) a T ( r ) exp ( i ω 0 t ) ,
U i ( ρ , z , t ) = T a 1 / 2 f ( t - z c - ρ 2 2 c z ) × a i ( ρ , z ) exp [ i ( ω 0 t - k 0 z - k 0 2 z ρ 2 ) ] ,
a i ( ρ , z ) = 1 λ 0 z d 2 r a T ( r ) exp [ - i k 0 z ( r 2 2 - r · ρ ) ] ,
U s ( ρ , z , t ) = R ( ϕ ) ( π n r a r b ) 1 / 2 U i [ ρ , z , t + 2 ξ ( ρ ) c ] ,
U ( r , z , t ) = a ( r , z , t ) exp ( i ω 0 t ) = T a λ 0 z exp [ i ( ω 0 t - 2 k 0 z - k 0 2 z r 2 ) ] · d 2 ρ a i ( ρ , z ) R ( 0 ) ( π n r a r b ) 1 / 2 f [ t - 2 z c - ρ 2 c z + 2 ξ ( ρ ) c ] · exp { - i k 0 [ ρ 2 2 z - 2 ξ ( ρ ) - ρ · r z ] } .
J a ( r 1 , t 1 ; r 2 ; t 2 ) = a ( r 1 , z , t 1 ) a * ( r 2 , z , t 2 ) ,
J a ( r 1 , t ; r 2 , t 2 ) = T a 2 z - 2 exp [ - i k 0 2 z ( r 1 2 - r 2 2 ) ] R ( 0 ) 2 π d 2 ρ · a i ( ρ , z ) 2 ( 1 + ρ 2 z 2 ) 2 × p [ ξ x , ξ y ξ ( p ) ] f ( t 1 - ψ ) f * ( t 2 - ψ ) · exp [ i k 0 z ρ · ( r 1 - r 2 ) ] ,
ψ = 2 z c + ρ 2 c z - 2 ξ ( ρ ) c .
n ( ρ , ξ ) r a r b = ( 1 + ρ 2 z 2 ) 2 p [ ξ x , ξ y ξ ( ρ ) ] ,
P ( t ) = d 2 r W ( r ) a ( r , z , t ) 2 ,
E [ S ( t ) ] = η h f 0 E [ P ( t ) ] * h ( t ) ,
C s ( t 1 , t 2 ) = η h f 0 - d τ E [ P ( τ ) ] h ( t 1 - τ ) h ( t 2 - τ ) + ( η h f 0 ) 2 - d τ 1 - d τ 2 C p ( τ 1 τ 2 ) × h ( t 1 - τ 1 ) h ( t 2 - τ 2 ) .
E [ P ( t ) ξ ] = d 2 r J a ( r , t ; r , t ) W ( r ) .
E [ S ( t ) ] = η h f 0 A R T a 2 z - 2 R ( 0 ) 2 π d 2 ρ a i ( ρ , z ) 2 ( 1 + ρ 2 z 2 ) 2 × d ξ p ( ξ x , ξ y , ξ ) g [ t - 2 z c - ρ 2 c z + 2 ξ ( ρ ) c ] ,
C p ( t 1 , t 2 ) = d 2 r 1 d 2 r 2 W ( r 1 ) W ( r 2 ) E [ J a ( r 1 , t 1 ; r 2 ; t 2 ) 2 ] ,
C p ( t 1 , t 2 ) = λ 0 2 T a 4 z - 2 R ( 0 ) 4 π 2 A R d 2 ρ a i ( ρ , z ) 4 ( 1 + ρ 2 z 2 ) 4 × d ξ p 2 ( ξ x , ξ y ξ ) p ( ξ ) f ( t 1 - ψ ) 2 f ( t 2 - ψ ) 2 .
C s ( t 1 , t 2 ) = ( η h f 0 ) T a 2 z - 2 R ( 0 ) 2 π A R d 2 ρ a i ( ρ , z ) 2 ( 1 + ρ 2 z 2 ) 2 × d ξ p ( ξ x , ξ y , ξ ) - d τ f ( τ - ψ ) 2 h ( t 1 - τ ) h ( t 2 - τ ) + ( η h f 0 ) 2 λ 0 2 T a 4 z - 2 R ( 0 ) 4 π 2 A R d 2 ρ a i ( ρ , z ) 4 ( 1 + ρ 2 z 2 ) 4 × d ξ p 2 ( ξ x , ξ y ξ ) p ( ξ ) g ( t 1 - ψ ) g ( t 2 - ψ ) .
p ( ξ ) = ( 2 π σ ξ 2 ) - 1 / 2 exp ( - ξ 2 2 σ ξ 2 ) ,
SWH 4 σ ξ .
σ ξ = 0.016 W 2 ,
p ( ξ x , ξ y ) = 1 π S 2 exp ( - ξ x 2 + ξ y 2 S 2 ) ,
S 2 = ξ x 2 + ξ y 2 .
S 2 = 0.003 + 0.00512 W ,
p ( ξ , ξ x , ξ y ) = p ( ξ x , ξ y ) p ( ξ ) .
p ( ξ x , ξ ) = [ 2 π ( σ ξ 2 ξ x 2 ) 1 / 2 ] - 1 exp [ - 1 2 ( ξ 2 σ ξ 2 + ξ x 2 ξ x 2 ) ] × [ 1 + λ 3 6 ( ξ 3 σ ξ 3 - 9 ξ σ ξ + 6 ξ σ ξ ξ x 2 ξ x 2 ) ] ,
λ 3 = ξ 3 / σ ξ 3 .
p ( 0 , 0 , ξ ) = [ ( 2 π 3 ) 1 / 2 S 2 σ ξ ] - 1 exp ( - 1 2 ξ 2 σ ξ 2 ) × [ 1 + λ 3 6 ( ξ 3 σ ξ 3 - 9 ξ σ ξ ) ] .
p ( ξ ) = ( 2 π σ ξ 2 ) - 1 / 2 exp ( - ξ 2 2 σ ξ 2 ) [ 1 + λ 3 6 ( ξ 3 σ ξ 3 - 3 ξ σ ξ ) ] .
m k = - d t t k S ( t ) .
m 0 = N G ,
N = η h f 0 β r Q T a 2 A R z - 2 ,
G = - d t h ( t ) ,
Q = d 2 ρ a i ( ρ , z ) 2 - d t f ( t ) 2 ,
β r = Q - 1 R ( 0 ) 2 π d 2 ρ a i ( ρ , z ) 2 ( 1 + ρ 2 z 2 ) 2 p ( ξ x , ξ y ) × - d t f ( t ) 2 .
E [ S ( t ) ] = N d 2 ρ b 2 ( ρ , z ) d ξ p ( ξ ) g ( t - ψ ) ,
C s ( t 1 , t 2 ) = N d 2 ρ b 2 ( ρ , z ) d ξ p ( ξ ) - d τ f ( τ - ψ ) 2 × h ( t 1 - τ ) h ( t 2 - τ ) + N K s d 2 ρ b 4 ( ρ , z ) × d ξ p ( ξ ) g ( t 1 - ψ ) g ( t 2 - ψ ) ,
b n ( ρ , z ) = a i ( ρ , z ) n ( 1 + ρ 2 z 2 ) n p n / 2 ( ξ x , ξ y ) / d 2 ρ a i ( ρ , z ) n × ( 1 + ρ 2 z 2 ) n p n / 2 ( ξ x , ξ y ) ,
K s = A R ( λ 0 z ) - 2 [ d 2 ρ a i ( ρ , z ) 2 ( 1 + ρ 2 z 2 ) 2 p ( ξ x , ξ y ) ] 2 × [ d 2 ρ a i ( ρ , z ) 4 ( 1 + ρ 2 z 2 ) 4 p 2 ( ξ x , ξ y ) ] - 1 .
var ( m 0 ) = N G 2 + N 2 G 2 K s - 1 .
T s = - d t t S ( t ) / - d t S ( t ) .
T s = 2 z c + σ r 2 c z - 2 ξ c .
σ s 2 = - d t ( t - T s ) 2 S ( t ) / - d t S ( t ) .
E ( σ s 2 ) = σ h 2 + σ f 2 + 4 c 2 σ ξ 2 + ( c z ) - 2 d 2 ρ ( ρ 2 - σ r 2 ) 2 b 2 ( ρ , z ) ,
var ( T s ) = N - 1 E ( σ s 2 ) + K s - 1 c - 2 z - 2 d 2 ρ b 4 ( ρ , z ) ( ρ 2 - σ r 2 ) 2 ,
var ( σ s 2 ) = ( N - 1 + K s - 1 ) [ 48 c - 4 σ ξ 4 + E ( σ s 2 ) 2 ] - 2 E ( σ s 2 ) var ( T s ) + N - 1 d 2 ρ b 2 ( ρ , z ) c - 4 [ z - 4 ( ρ 2 - σ r 2 ) 4 + 12 z - 2 σ ξ 2 ( ρ 2 - σ r 2 ) 2 ] + K s - 1 d 2 ρ b 4 ( ρ , z ) c - 4 [ z - 4 ( ρ 2 - σ r 2 ) 4 + 12 z - 2 σ x 2 ( ρ 2 - σ r 2 ) 2 ] .
a i ( ρ , z ) 2 = Q ( 2 π z 2 tan 2 θ T ) - 1 exp - ( ρ 2 2 z 2 tan 2 θ T ) ,
f ( t ) 2 = ( 2 π σ f 2 ) - 1 / 2 exp ( - t 2 2 σ f 2 ] ,
E [ S ( t ) ] = N G c z 4 π σ r 2 exp [ σ 2 c 2 z 2 8 σ r 2 - c z 2 σ r 2 ( t - 2 z c ) ] · { 1 - erf [ 2 σ c z 4 σ r 2 - 1 2 σ ( t - 2 z c ) ] } ,
N = η Q h f 0 σ r 2 S 2 z 2 tan 2 θ T R ( 0 ) 2 A R z - 2 T a 2 ,
σ 2 = 4 c 2 σ ξ 2 + σ h 2 + σ f 2 ,
σ r 2 = 2 z 2 ( tan θ T - 2 + 2 S - 2 ) - 1 ,
T s = 2 z c + 2 z c ( tan - 2 θ T + 2 S - 2 ) - 1 ,
E ( σ s 2 ) = σ h 2 + σ f 2 + 4 c 2 σ ξ 2 + 4 z 2 c 2 ( tan - 2 θ T + 2 S - 2 ) - 2 .
var ( T s ) ( N - 1 + K s - 1 ) 4 c 2 σ ξ 2 + ( N - 1 + 1 2 K s - 1 ) × 4 z 2 ( tan - 2 θ T + 2 S - 2 ) - 2 c 2 ,
var ( σ s 2 ) ( N - 1 + K s - 1 ) 32 c 4 σ ξ 4 + ( N - 1 + 1 2 K s - 1 ) × 16 σ ξ 2 z 2 ( tan - 2 θ T + 2 S - 2 ) - 2 c 4 ,
K s = π A R ( 2 tan θ T λ 0 ) 2 .
σ z ½ ( N - 1 + K s - 1 ) 1 / 2 SWH ,
σ SWH 2 ( N - 1 + K s - 1 ) 1 / 4 SWH ,
p ( ξ x , ξ y , ξ ) p ( 0 , 0 , ξ ) .
T s = 2 z c + 2 z c tan 2 θ T + 2 c λ 3 σ ξ ,
E ( σ s 2 ) = σ h 2 + σ f 2 + 4 z 2 c 2 tan 4 θ T + 4 c 2 σ ξ 2 ( 1 - λ 3 2 ) ,
z = z cos ϕ ,
ξ ( p ) = x tan ϕ + ξ ( p ) cos ϕ ,
x = x cos ϕ + ξ ( ρ ) tan ϕ ,
y = y .
p ( ξ x , ξ y , ξ ) p ( tan ϕ , 0 , ξ ) .
P ( tan ϕ , 0 , ξ ) = [ ( 2 π 3 ) 1 / 2 S 2 σ ξ ] - 1 exp [ - 1 2 ( ξ 2 σ ξ 2 + 2 tan 2 ϕ S 2 ) ] × [ 1 + λ 3 6 ( ξ 3 σ ξ 3 - 9 ξ σ ξ + 6 ξ σ ξ 2 tan 2 ϕ S 2 ) ] .
T s = 2 z c cos ϕ + 2 z c cos ϕ tan 2 θ T + 2 c cos ϕ σ ξ λ 3 ( 1 - 2 tan 2 ϕ S 2 )
E ( σ s 2 ) = σ h 2 + σ f 2 + 4 z 2 c 2 cos 2 ϕ tan 4 θ T + 4 σ ξ 2 c 2 cos 2 ϕ [ 1 - λ 3 2 ( 1 - 2 tan 2 ϕ S 2 ) 2 ] + 4 z 2 c 2 cos 2 ϕ tan 2 ϕ T tan 2 ϕ .

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