Abstract

The statistical characteristics of the received signal for short pulse laser altimeters are investigated. Expressions are derived for the mean and temporal covariances of the received pulse for a direct detection receiver. The effects of laser speckle, shot noise, and surface profile of the ground target are considered. The results are used to compute the means and variances of the total received energy, propagation delay, and rms width of the received pulse. These parameters are shown to be directly related to the statistics of the surface profile.

© 1982 Optical Society of America

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References

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  1. J. J. Degnan, “Airborne Laser Ranging System for Monitoring Regional Crustal Deformation,” in Proceedings, 1981 International Geoscience and Remote Sensing Symposium, 8–10 June 1981, Washington, D.C. (IEEE, New York, 1981).
  2. R. C. Brooks et al., Nature London 274, 539 (1978).
    [CrossRef]
  3. C. S. Gardner, Appl. Opt. 18, 3184 (1979).
    [CrossRef] [PubMed]
  4. Special Issue on Radio Oceanography, IEEE Trans. Antennas Propag. AP-25, 1 (1977).
  5. K. S. Krishman, N. A. Peppers, “Scattering of Laser Radiation from the Ocean Surface,” Midpoint Tech. Rept. SRI Proj. ISE 2618, Stanford Research Institute (Oct.1973).
  6. H. H. Kim, Appl. Opt. 16, 46 (1977).
    [CrossRef] [PubMed]
  7. J. B. Abshire, Appl. Opt. 19, 3436 (1980).
    [CrossRef] [PubMed]
  8. J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, New York, 1975).
    [CrossRef]
  9. A. Papoulis, IEEE Trans. Commun. Technol. COM-22, 162 (1974).
    [CrossRef]
  10. C. S. Gardner, “Speckle Noise in Satellite Based Lidar Systems,” Radio Research Lab. Tech. Rept. 488, U. Illinois, Urbana (Dec.1977).
  11. C. S. Gardner, “Analysis of Target Signatures for Laser Altimeters”, Radio Research Labs. Tech. Rept. 510, U. Illinois, Urbana (Apr.1981).

1980 (1)

1979 (1)

1978 (1)

R. C. Brooks et al., Nature London 274, 539 (1978).
[CrossRef]

1977 (2)

Special Issue on Radio Oceanography, IEEE Trans. Antennas Propag. AP-25, 1 (1977).

H. H. Kim, Appl. Opt. 16, 46 (1977).
[CrossRef] [PubMed]

1974 (1)

A. Papoulis, IEEE Trans. Commun. Technol. COM-22, 162 (1974).
[CrossRef]

Abshire, J. B.

Brooks, R. C.

R. C. Brooks et al., Nature London 274, 539 (1978).
[CrossRef]

Degnan, J. J.

J. J. Degnan, “Airborne Laser Ranging System for Monitoring Regional Crustal Deformation,” in Proceedings, 1981 International Geoscience and Remote Sensing Symposium, 8–10 June 1981, Washington, D.C. (IEEE, New York, 1981).

Gardner, C. S.

C. S. Gardner, Appl. Opt. 18, 3184 (1979).
[CrossRef] [PubMed]

C. S. Gardner, “Speckle Noise in Satellite Based Lidar Systems,” Radio Research Lab. Tech. Rept. 488, U. Illinois, Urbana (Dec.1977).

C. S. Gardner, “Analysis of Target Signatures for Laser Altimeters”, Radio Research Labs. Tech. Rept. 510, U. Illinois, Urbana (Apr.1981).

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).
[CrossRef]

Kim, H. H.

Krishman, K. S.

K. S. Krishman, N. A. Peppers, “Scattering of Laser Radiation from the Ocean Surface,” Midpoint Tech. Rept. SRI Proj. ISE 2618, Stanford Research Institute (Oct.1973).

Papoulis, A.

A. Papoulis, IEEE Trans. Commun. Technol. COM-22, 162 (1974).
[CrossRef]

Peppers, N. A.

K. S. Krishman, N. A. Peppers, “Scattering of Laser Radiation from the Ocean Surface,” Midpoint Tech. Rept. SRI Proj. ISE 2618, Stanford Research Institute (Oct.1973).

Appl. Opt. (3)

IEEE Trans. Antennas Propag. (1)

Special Issue on Radio Oceanography, IEEE Trans. Antennas Propag. AP-25, 1 (1977).

IEEE Trans. Commun. Technol. (1)

A. Papoulis, IEEE Trans. Commun. Technol. COM-22, 162 (1974).
[CrossRef]

Nature London (1)

R. C. Brooks et al., Nature London 274, 539 (1978).
[CrossRef]

Other (5)

J. J. Degnan, “Airborne Laser Ranging System for Monitoring Regional Crustal Deformation,” in Proceedings, 1981 International Geoscience and Remote Sensing Symposium, 8–10 June 1981, Washington, D.C. (IEEE, New York, 1981).

K. S. Krishman, N. A. Peppers, “Scattering of Laser Radiation from the Ocean Surface,” Midpoint Tech. Rept. SRI Proj. ISE 2618, Stanford Research Institute (Oct.1973).

C. S. Gardner, “Speckle Noise in Satellite Based Lidar Systems,” Radio Research Lab. Tech. Rept. 488, U. Illinois, Urbana (Dec.1977).

C. S. Gardner, “Analysis of Target Signatures for Laser Altimeters”, Radio Research Labs. Tech. Rept. 510, U. Illinois, Urbana (Apr.1981).

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

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Figures (2)

Fig. 1
Fig. 1

Geometry of the laser altimeter and ground target for normal incidence.

Fig. 2
Fig. 2

Geometry of the laser altimeter and ground target for non-normal incidence.

Equations (65)

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u ( r , z , t ) = a ( r , z , t ) exp ( i ω 0 t ) ,
p ( t ) = d 2 r w ( r ) a ( r , z , t ) 2
E [ S ( t ) ξ ] = η h f 0 E [ p ( t ) ξ ] * h ( t ) ,
C S ξ ( t 1 , t 2 ) = η h f 0 - d τ E [ p ( τ ) ξ ] h ( t 1 - τ ) h ( t 2 - τ ) + ( η h f 0 ) 2 - d τ 1 - d τ 2 C p ξ ( 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 ) ,
C p ξ ( t 1 , t 2 ) = d 2 r 1 d 2 r 2 J a ( r 1 , t 1 ; r 2 , t 2 ) 2 × w ( r 1 ) w ( r 2 ) ,
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 1 ; r 2 , t 2 ) = T a 2 β r z - 2 d 2 ρ a i ( ρ , z ) 2 × 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 ,
E [ p ( t ) ξ ] = A R T a 2 β r z - 2 d 2 ρ a i ( ρ , z ) 2 f ( t - ψ ) 2 .
E [ p F ( t ) ] = A R T a 2 β r z - 2 d 2 ρ a i ( ρ , z ) 2 f ( t - 2 z c - ρ 2 c z ) | 2 .
E [ p ( t ) ] = c 2 E [ p F ( t ) ] * P ξ ( c t 2 ) ,
E [ S ( t ) ] = c 2 E [ S F ( t ) ] * P ξ ( c t 2 ) ,
E [ S F ( t ) ] = η h f 0 E [ p F ( t ) ] * h ( t ) .
C p ξ ( t 1 , t 2 ) = d 2 r J a ( r / 2 , t 1 ; - r / 2 , t 2 ) 2 R w ( r ) ,
R w ( r ) = d 2 ρ w ( ρ ) w ( ρ + r ) .
C p ξ ( t 1 , t 2 ) R w ( 0 ) d 2 r J a ( r / 2 , t 1 ; - r / 2 , t 2 ) 2 .
C p ξ ( t 1 , t 2 ) λ 0 2 A R T a 4 β r 2 z - 2 d 2 ρ a i ( ρ , z ) 4 × f ( t 1 - ψ ) 2 f ( t 2 - ψ ) 2 .
m k = - d t t k S ( t ) .
E ( m 0 ξ ) = m 0 = N G ,
N = η h f 0 Q T a 2 β r A R z - 2 ,
G = - d t h ( t ) ,
Q = d 2 ρ a T ( ρ , z ) 2 - d t f ( t ) 2 .
E [ S ( t ) ξ ] = N d 2 ρ b 2 ( ρ , z ) g ( t - ψ ) ,
C s ξ ( t 1 , t 2 ) = N d 2 ρ b 2 ( ρ , z ) - d τ f ( τ - ψ ) 2 h ( t 1 - τ ) + N K 2 S - 1 d 2 ρ b 4 ( ρ , z ) g ( t 1 - ψ ) g ( t 2 - ψ ) ,
g ( t ) = f ( t ) 2 * h ( t ) ,
b n ( ρ , z ) = a i ( ρ , z ) n / d 2 ρ a i ( ρ , z ) n , n = 2 , 4 ,
K S = A R ( λ 0 z ) - 2 [ d 2 ρ a i ( ρ , z ) 2 ] 2 × [ d 2 ρ a i ( ρ , z ) 4 ] - 1 .
Var ( m 0 ξ ) = N G 2 + N G 2 2 K s - 1 .
m k m 0 = - d t t k S ( t ) / - d t S ( t ) .
E ( T s ξ ) = T s = E ( m 1 m 0 | ξ ) = d 2 ρ b 2 ( ρ , z ) ψ .
T s = 2 z c + σ i 2 c z - 2 c ξ ¯ 2 ,
σ i 2 = d 2 ρ ρ 2 b 2 ( ρ , z ) ,
ξ ¯ 2 = d 2 ρ ξ ( ρ ) b 2 ( ρ , z ) .
σ S 2 = - d t / ( t - T s ) 2 S ( t ) / - d t S ( t ) = m 2 m 0 - ( m 1 m 0 ) 2 .
E ( σ s 2 ξ ) = σ s 2 = σ h 2 + d 2 ρ b 2 ( ρ , z ) × - d t ( t - T s ) 2 f ( t - ψ ) 2 ,
σ s 2 = σ h 2 + σ f 2 + 4 c 2 σ ξ 2 + ( c z ) - 2 d 2 ρ ( ρ 2 - σ i 2 ) 2 b 2 ( ρ , z ) - 4 c 2 z d 2 ρ ρ 2 [ ξ ( ρ ) - ξ ¯ 2 ] b 2 ( ρ , z ) .
σ ξ 2 = d 2 ρ [ ξ ( ρ ) - ξ ¯ 2 ] 2 b 2 ( ρ , z ) .
Var ( T s ξ ) = N - 1 d 2 ρ b 2 ( ρ , z ) - d t ( t - T s ) 2 f ( t - ψ ) 2 + K s - 1 d 2 ρ b 4 ( ρ , z ) ( ψ - T s ) 2 .
Var ( T s ξ ) = N - 1 ( σ s 2 - σ h 2 ) + K s - 1 d 2 ρ b 4 ( ρ , z ) ( ψ - T s ) 2 .
Var ( σ s 2 ξ ) = N - 1 d 2 ρ b 2 ( ρ , z ) - d t [ ( t - T s ) 2 - σ s 2 + σ h 2 ] 2 f ( t - ψ ) 2 + K s - 1 d 2 ρ b 4 ( ρ , z ) [ ( ψ - T s ) 2 - σ s 2 + σ h 2 + σ f 2 ] 2 .
z = z cos ϕ ,
ξ ( ρ ) = ρ x tan ϕ + ξ ( ρ ) cos ϕ ,
ρ x = ρ x cos ϕ + ξ ( ρ ) tan ϕ ,
ρ y = ρ y .
E ( T s ) = 2 z c cos ϕ + cos ϕ c z σ i 2 ,
E ( σ s 2 ) = σ h 2 + σ f 2 + 4 Var ( ξ ) c 2 cos 2 ϕ + 2 c 2 tan 2 ϕ σ i 2 , + ( cos ϕ c z ) 2 d 2 ρ ( ρ 2 - σ i 2 ) 2 b 2 ( ρ , z ) ,
σ i 2 = d 2 ρ ρ 2 b 2 ( ρ , z ) .
b 2 ( ρ , z ) = 1 2 π ( z tan θ T ) - 2 exp [ - ρ 2 2 ( z tan θ T ) - 2 ] .
E ( T s ) = 2 z c cos ϕ ( 1 + tan 2 θ T ) ,
E ( σ s 2 ) = σ h 2 + σ f 2 + 4 Var ( ξ ) c 2 cos 2 ϕ + 4 z 2 c 2 cos 2 ϕ + 4 z 2 c 2 cos 2 ϕ × ( tan 4 θ T + tan 2 θ T tan 2 ϕ ) .
Var ( T s ) = σ f 2 N + ( 1 N + 1 K s ) 4 Var ( ξ ) c 2 cos 2 ϕ + ( 1 N + 1 2 K s ) 4 z 2 c 2 cos 2 ϕ × ( tan 4 θ T + tan 2 θ T tan 2 ϕ ) ,
K s = π A R ( 2 tan θ T λ 0 ) 2.
u T ( r , t ) = f ( t ) a T ( r ) exp ( i ω 0 t ) ,
u i ( r , z , t ) = T a 1 / 2 f ( t - z c - r 2 2 c z ) a i ( r , z ) × exp [ i ( ω 0 t - k 0 z - k 0 2 z r 2 ) ] ,
a i ( r , z ) = 1 λ 0 z d 2 ρ a T ( ρ ) exp [ - i k 0 z ( ρ 2 / 2 - r · ρ ) ] .
c σ f A T z .
u s ( r , z , t ) = β r 1 / 2 u i [ r , z , t + 2 l ( r ) / c ] .
u ( r , z , t ) = a ( r , z , t ) exp ( i ω 0 t ) = T a β r 1 / 2 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 ) f [ t - 2 z c - ρ 2 c z + 2 l ( ρ ) c ] × exp { - i k 0 [ ρ 2 2 z - 2 l ( ρ ) - ρ · r z ] } .
c σ f A R / z .
l ( r ) = ξ ( r ) + ( r ) .
J a ( r 1 , t 1 ; r 2 , t 2 ) = a ( r 1 , z , t 1 ) a * ( r 2 , z , t 2 ) = γ d 2 ρ a i ( ρ , z ) 2 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 ,
γ = T a 2 β r z - 2 exp [ - i k 0 2 z ( r 1 2 - r 2 2 ) ] .
J a ( r , t ; r , t ) = T a 2 β r z - 2 d 2 ρ a i ( ρ , z ) 2 f ( t - ψ ) 2 .

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