Abstract

The problem of calculating the laser radar cross section of a randomly rough target is studied. It is shown that subject to a minimum set of assumptions, all with reasonable physical interpretation, the statistics are Rician for any degree of roughness. A rigorous theory is developed to allow the two parameters of the Rician distribution to be calculated for any target shape, using only a small set of laboratory measurements of laser scattering from two simply shaped samples of the target surface material. It is noted that it is never necessary to become involved in measurements of the statistics of the target’s microscopic roughness or in the problem of electromagnetic boundary matching conditions in order to calculate the laser radar cross section of a large complex shaped target.

© 1976 Optical Society of America

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References

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  1. Laser Speckle and Related Phenamena, edited by J. C. Dainty (Springer-Verlag, New York, 1975).
  2. J. W. Goodman, “Some Effects of Target-Induced Scintillation on Optical Radar Performance,” Proc. IEEE 53, 1688 (1965).
    [CrossRef]
  3. G. Gould, S. F. Jacobs, J. T. LaTourrette, M. Newstein, and P. Rabinowitz, “Coherent Detection of Light Scattered From a Diffusely Reflecting Surface,” Appl. Opt. 3, 648 (1964).
    [CrossRef]
  4. J. W. Goodman, in Ref. 1; Sec. 2.2.
  5. D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).
  6. S. O. Rice, in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954).
  7. C. W. Helstrom, Statistical Theory of Signal Detection2nd ed. (Pergamon, Oxford, 1968).
  8. A. A. Vuylsteke (private communications).
  9. J. C. Leader, “Bidirectional Scattering of Electromagnetic Waves From Rough Surfaces,” J. Appl. Phys. 42, 4808 (1971).
    [CrossRef]
  10. It should be noted that while the Rayleigh distribution is most easily developed from a set of perturbation effects, each following a Gaussian distribution, it can also be developed using the central-limit theorem with a large number of perturbations having any distribution with reasonably constrained variance. In the same sense, though we do not attempt it in this paper, we expect that the same results we develop here using the assumption that the surface roughness has a Gaussian distribution, could be developed using the central-limit theorem and a much less restrictive assumption about the target’s surface roughness distribution. In this sense, we believe that in practice it is not necessary that we be overly concerned that actual surface roughness of a target be Gaussianly distributed in order for us to apply the theoretical results we shall develop here to evaluation of the statistics of that target.
  11. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (U. S. GPO, Washington, D. C., 1964), Sec. 9.6.
  12. Ref. 4, Sec. 2.4.2.
  13. H. Cramér, Mathematical Methods of Statistics (Princeton U. P., Princeton, N. J., 1966). Sec. 15.4.
  14. Reference 7, Appendix F.
  15. Reference 10, Sec. 22.2.
  16. We have treated the phase variations of the wave front as a real quantity. To take account of the effects of shadowing and multiple scattering on very rough surfaces, which could cause variations in the local amplitude of the scatter wave just above the surface of the scatterer, we should allow complex phase variations. However, a careful study of the analysis in this paper will show that the same results could have been developed with this phase being a complex rather than a real Gaussian random variable.
  17. Doing this does, of course, prevent us from using any exact information we may have about the statistics of the target’s surface height variation to generate an exact statement about the statistics of the scattered laser radar signal and from that about the statistics of the target’s laser radar cross section. But as we have noted previously, we do not intend to develop any such mechanical-to-optical relationship, so nothing important is lost to us by the use of this approximate relationship.
  18. It should, of course, be noted that our analysis applies to the case of vacuum propagation. The possible effects of atmospheric turbulence have not been included in this analysis.
  19. If we wish to relax this condition to consider a closer point, it is only necessary to redefine the effective target shape function H(r), to accommodate the associated r2-dependent variation of the path length.
  20. We believe that the change in our results will be accommodated by considering the effective target to be the result of superimposing an Airy diffraction pattern (power density, not amplitude) on the true target.
  21. This is of measure zero because Δr is so much smaller than the target size.
  22. We have not attempted to show this rigorously, but it appears to be the case that we actually shall recover the negligible contribution from this neglected region when we later equate the outregion area to the total target area.
  23. Strictly speaking, we should have taken explicit note of the cases where the rj values lie in each other’s neighborhoods. However, not only is this essentially a set of measure zero, but it can be shown that the resultant correction terms have no particular influence on the value of the integral.
  24. The distribution of the cross section is more properly called, in Goodman’s terminology, (Ref. 4), a modified Rician.
  25. It should be recognized that this assumption of the smoothly-varying dependence on angle of incidence, and not the assumption of a Gaussian distribution of random phase shift (which could be been avoided by use of the central-limit theorem) is the most important and interesting assumption involved in the analysis in this paper.

1971 (1)

J. C. Leader, “Bidirectional Scattering of Electromagnetic Waves From Rough Surfaces,” J. Appl. Phys. 42, 4808 (1971).
[CrossRef]

1965 (1)

J. W. Goodman, “Some Effects of Target-Induced Scintillation on Optical Radar Performance,” Proc. IEEE 53, 1688 (1965).
[CrossRef]

1964 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (U. S. GPO, Washington, D. C., 1964), Sec. 9.6.

Cramér, H.

H. Cramér, Mathematical Methods of Statistics (Princeton U. P., Princeton, N. J., 1966). Sec. 15.4.

Goodman, J. W.

J. W. Goodman, “Some Effects of Target-Induced Scintillation on Optical Radar Performance,” Proc. IEEE 53, 1688 (1965).
[CrossRef]

J. W. Goodman, in Ref. 1; Sec. 2.2.

Gould, G.

Helstrom, C. W.

C. W. Helstrom, Statistical Theory of Signal Detection2nd ed. (Pergamon, Oxford, 1968).

Jacobs, S. F.

LaTourrette, J. T.

Leader, J. C.

J. C. Leader, “Bidirectional Scattering of Electromagnetic Waves From Rough Surfaces,” J. Appl. Phys. 42, 4808 (1971).
[CrossRef]

Middleton, D.

D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

Newstein, M.

Rabinowitz, P.

Rice, S. O.

S. O. Rice, in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (U. S. GPO, Washington, D. C., 1964), Sec. 9.6.

Vuylsteke, A. A.

A. A. Vuylsteke (private communications).

Appl. Opt. (1)

J. Appl. Phys. (1)

J. C. Leader, “Bidirectional Scattering of Electromagnetic Waves From Rough Surfaces,” J. Appl. Phys. 42, 4808 (1971).
[CrossRef]

Proc. IEEE (1)

J. W. Goodman, “Some Effects of Target-Induced Scintillation on Optical Radar Performance,” Proc. IEEE 53, 1688 (1965).
[CrossRef]

Other (22)

J. W. Goodman, in Ref. 1; Sec. 2.2.

D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

S. O. Rice, in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954).

C. W. Helstrom, Statistical Theory of Signal Detection2nd ed. (Pergamon, Oxford, 1968).

A. A. Vuylsteke (private communications).

It should be noted that while the Rayleigh distribution is most easily developed from a set of perturbation effects, each following a Gaussian distribution, it can also be developed using the central-limit theorem with a large number of perturbations having any distribution with reasonably constrained variance. In the same sense, though we do not attempt it in this paper, we expect that the same results we develop here using the assumption that the surface roughness has a Gaussian distribution, could be developed using the central-limit theorem and a much less restrictive assumption about the target’s surface roughness distribution. In this sense, we believe that in practice it is not necessary that we be overly concerned that actual surface roughness of a target be Gaussianly distributed in order for us to apply the theoretical results we shall develop here to evaluation of the statistics of that target.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (U. S. GPO, Washington, D. C., 1964), Sec. 9.6.

Ref. 4, Sec. 2.4.2.

H. Cramér, Mathematical Methods of Statistics (Princeton U. P., Princeton, N. J., 1966). Sec. 15.4.

Reference 7, Appendix F.

Reference 10, Sec. 22.2.

We have treated the phase variations of the wave front as a real quantity. To take account of the effects of shadowing and multiple scattering on very rough surfaces, which could cause variations in the local amplitude of the scatter wave just above the surface of the scatterer, we should allow complex phase variations. However, a careful study of the analysis in this paper will show that the same results could have been developed with this phase being a complex rather than a real Gaussian random variable.

Doing this does, of course, prevent us from using any exact information we may have about the statistics of the target’s surface height variation to generate an exact statement about the statistics of the scattered laser radar signal and from that about the statistics of the target’s laser radar cross section. But as we have noted previously, we do not intend to develop any such mechanical-to-optical relationship, so nothing important is lost to us by the use of this approximate relationship.

It should, of course, be noted that our analysis applies to the case of vacuum propagation. The possible effects of atmospheric turbulence have not been included in this analysis.

If we wish to relax this condition to consider a closer point, it is only necessary to redefine the effective target shape function H(r), to accommodate the associated r2-dependent variation of the path length.

We believe that the change in our results will be accommodated by considering the effective target to be the result of superimposing an Airy diffraction pattern (power density, not amplitude) on the true target.

This is of measure zero because Δr is so much smaller than the target size.

We have not attempted to show this rigorously, but it appears to be the case that we actually shall recover the negligible contribution from this neglected region when we later equate the outregion area to the total target area.

Strictly speaking, we should have taken explicit note of the cases where the rj values lie in each other’s neighborhoods. However, not only is this essentially a set of measure zero, but it can be shown that the resultant correction terms have no particular influence on the value of the integral.

The distribution of the cross section is more properly called, in Goodman’s terminology, (Ref. 4), a modified Rician.

It should be recognized that this assumption of the smoothly-varying dependence on angle of incidence, and not the assumption of a Gaussian distribution of random phase shift (which could be been avoided by use of the central-limit theorem) is the most important and interesting assumption involved in the analysis in this paper.

Laser Speckle and Related Phenamena, edited by J. C. Dainty (Springer-Verlag, New York, 1975).

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Figures (1)

FIG. 1
FIG. 1

Representation of laser radiation incident on a randomly rough target surface. The height of the target surface z ˜(r) is measured relative to some arbitrarily chosen reference surface which is normal to the incident laser radiation. This height is the sum of a macroscopic nonrandom contribution H ˜(r), which defines the target shape, and a microscopic random component h ˜(r), which constitutes the target roughness. The local surface normal is defined with respect to the macroscopic target shape and does not depend on the microscopic target details.

Equations (69)

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Prob ( a < α < a + d a ) = ( a κ X R ) exp ( - 1 2 ( a κ ) 2 + X S X R ) I 0 ( 2 1 / 2 a κ X S 1 / 2 X R ) κ d a ,
X = 1 2 ( a κ ) 2 .
Prob ( X < χ < X + d X ) = P Rice ( X ) d X ,
P Rice ( X ) = X R - 1 exp ( - X + X S X R ) I 0 ( 2 ( X X S ) 1 / 2 X R ) .
( α Rice ) 2 N = ( X R / κ 2 ) N 2 N N ! L N ( - X S / X R ) .
( X Rice ) N = ( X R ) N N ! L N ( - X S / X R ) .
L N ( z ) = J = 0 N ( - z ) J N ! ( J ! ) 2 ( N - J ) ! ,
( X Rice ) N = ( X R ) N J = 0 N ( X S X R ) J ( N ! J ! ) 2 1 ( N - J ) ! .
H ˜ ( r ) = z ˜ ( r ) ,
θ ( r ) = H ˜ ( r ) .
h ˜ ( r ) = z ˜ ( r ) - H ˜ ( r ) .
u ( r ) = exp { - 2 i k f [ θ ( r ) ] z ( r ) } ,
z ( r ) = z ˜ ( r ) = H ˜ ( r ) .
z ( r ) = H ( r ) ,
h ( r ) = z ( r ) - H ( r ) ,
h ( r ) = 0.
σ h 2 = [ h ( r ) ] 2 ,
D h ( ρ ) = [ h ( r ) - h ( r ) ] 2 ,
ρ = r - r .
D h ( ρ ) 2 σ h 2             for ρ > Δ r
h ( r ) h ( r ) 0             for ρ > Δ r .
U = target d r u ( r ) ,
U = target d r exp { - 2 i k f [ θ ( r ) ] [ H ( r ) + h ( r ) ] } ;
I = 1 2 U * U = 1 2 ( target ) 2 d r d r × exp ( - 2 i k { f [ θ ( r ) ] H ( r ) - f [ θ ( r ) ] H ( r ) } ) × exp ( - 2 i k { f [ θ ( r ) ] h ( r ) - f [ θ ( r ) ] h ( r ) } ) .
X = κ 2 I ,
X = κ 2 2 ( target ) 2 d r d r × exp ( - 2 i k { f [ θ ( r ) ] H ( r ) - f [ θ ( r ) ] H ( r ) } ) × exp ( - 2 i k { f [ θ ( r ) ] h ( r ) - f [ θ ( r ) ] h ( r ) } ) .
X N = ( κ 2 2 ) N ( target ) 2 N d r 1 d r 1 d r 2 d r 2 d r N d r N × exp ( - 2 i k j = 1 N { f [ θ ( r j ) ] H ( r j ) - f [ θ ( r j ) ] H ( r j ) } ) × exp ( - 2 i k j = 1 N { f [ θ ( r j ) ] h ( r j ) - f [ θ ( r j ) ] h ( r j ) } ) .
h ( r ) 0             ( polished target ) .
( X polished ) N = ( κ 2 2 ) N × ( target ) 2 N d r 1 d r 1 d r 2 d r 2 d r N d r N × exp ( - 2 i k j = 1 N { f [ θ ( r j ) ] H ( r j ) - f [ θ ( r j ) ] H ( r j ) } ) .
( X polished ) 2 = [ ( κ 2 2 ) ( target ) 2 d r d r × exp ( - 2 i k { f [ θ ( r ) ] H ( r ) - f [ θ ( r ) ] H ( r ) } ) ] N .
X spec = κ 2 2 ( target ) 2 d r d r × exp ( - 2 i k { f [ θ ( r ) ] - f [ θ ( r ) ] H ( r ) } ) ,
( X polished ) N = ( X spec ) N .
X spec = κ 2 2 | target d r exp { - 2 i k f [ θ ( r ) ] H ( r ) } | 2 .
f 0 = f [ θ ( r ) ]             for θ ( r ) 0 ,
X spec = κ 2 2 | target d r exp [ - 2 i k f 0 H ( r ) ] | 2 ,
X spec = κ 2 2 ( target ) 2 d r d r exp { - 2 i k f 0 [ H ( r ) - H ( r ) ] } .
exp ( α χ ) = exp ( 1 2 α 2 χ 2 ) .
X N = ( κ 2 2 ) N ( target ) 2 N d r 1 d r 1 d r 2 d r 2 d r N d r N × exp ( - 2 i k j = 1 N { f [ θ ( r j ) ] H ( r j ) - f [ θ ( r j ) ] H ( r j ) } ) exp [ - 2 k 2 ( j = 1 N { f [ θ ( r j ) ] h ( r j ) - f [ θ ( r j ) ] h ( r j ) } ) 2 ] .
X N = ( κ 2 2 ) N ( target ) N d r 1 d r 2 d r N { ( target ) N d r 1 d r 2 d r N × exp ( - 2 i k j = 1 N { f [ θ ( r j ) ] H ( r j ) - f [ θ ( r j ) ] H ( r j ) } ) exp [ - 2 k 2 ( j = 1 N { f [ θ ( r j ) ] h ( r j ) - f [ θ ( r j ) ] h ( r j ) } ) 2 ] } ,
X N = J = 0 N X J N ,
X J N = ( κ 2 2 ) N ( target ) N d r 1 d r 2 d r N target ( N , J ) d r 1 d r 2 d r N × exp ( - 2 i k j = 1 N { f [ θ ( r j ) ] H ( r j ) - f [ θ ( r j ) ] H ( r j ) } ) exp [ - 2 k 2 ( j = 1 N { f [ θ ( r j ) ] h ( r j ) - f [ θ ( r j ) ] h ( r j ) } ) 2 ] .
X J N = C ( N , J ) ( κ 2 2 ) N ( target ) N d r 1 d r 2 d r N ( Δ r ) N - J d ρ 1 d ρ 2 d ρ N - J × exp ( - 2 i k j = 1 N - J { f [ θ ( r j ) ] H ( r j ) - f [ θ ( r j + ρ j ) ] H ( r j + ρ j ) } ) exp [ - 2 k 2 ( j = 1 N - J { f [ θ ( r j ) ] h ( r j ) - f [ θ ( r j + ρ j ) ] h ( r j + ρ j ) } ) 2 ] × ( target ) J d r N - J + 1 d r N - J + 2 d r N exp ( - 2 i k j = N - J + 1 N { f [ θ ( r j ) ] H ( r j ) - f [ θ ( r j ) ] H ( r j ) } ) × exp [ - 2 k 2 ( j = N - J + 1 N { f [ θ ( r j ) ] h ( r j ) - f [ θ ( r j ) ] h ( r j ) } ) 2 ] ,
( j = 1 N - J { f [ θ ( r j ) ] h ( r j ) - f [ θ ( r j + ρ j ) ] h ( r j + ρ j ) } ) 2 = j = 1 N - J { f [ θ ( r j ) ] h ( r j ) - f [ θ ( r j + ρ j ) ] h ( r j + ρ j ) } 2
( j = N - J + 1 N { f [ θ ( r j ) ] h ( r j ) - f [ θ ( r j ) ] h ( r j ) } ) 2 = j = N - J + 1 N ( { f [ θ ( r j ) ] } 2 [ h ( r j ) ] 2 + { f [ θ ( r j ) ] } 2 [ h ( r j ) ] 2 ) .
( j = N - J + 1 N { f [ θ ( r j ) ] h ( r j ) - f [ θ ( r j ) ] h ( r j ) } ) 2 = σ h 2 j = N - J + 1 N ( { f [ θ ( r j ) ] } 2 + { f [ θ ( r j ) ] } 2 ) .
θ ( r j ) θ ( r j + ρ j )             for ρ j Δ r .
( j = 1 N - J { f [ θ ( r j ) ] h ( r j ) - f [ θ ( r j + ρ j ) ] h ( r j + ρ j ) } ) 2 = j = 1 N - J { f [ θ ( r j ) ] } 2 D h ( ρ j ) .
H ( r + ρ ) H ( r ) + ρ · θ ( r ) ,
X J N = C ( N , J ) ( κ 2 2 ) N ( target ) N d r 1 d r 2 d r N ( Δ r ) N - J d ρ 1 d ρ 2 d ρ N - J exp ( 2 i k j = 1 N - J { f [ θ ( r j ) ] ρ j · θ ( r j ) } ) × exp ( - 2 k 2 j = 1 N - J ( { f [ θ ( r j ) ] } 2 D h ( ρ j ) ) ) ( target ) J d r N - J + 1 d r N - J + 2 d r N × exp ( - 2 i k j = N - J + 1 N { f [ θ ( r j ) ] H ( r j ) - f [ θ ( r j ) ] H ( r j ) } ) exp ( - 2 k 2 σ h 2 j = N - J + 1 N ( f { [ θ ( r j ) ] } 2 + { f [ θ ( r j ) ] } 2 ) ) .
P = κ 2 2 target d r Δ r d ρ exp ( 2 i k f [ θ ( r ) ] ρ · θ ( r ) - 2 k 2 { f [ θ ( r ) ] } 2 D h ( ρ ) )
Q = κ 2 2 ( target ) 2 d r d r exp ( - 2 i k { f [ θ ( r ) ] H ( r ) - f [ θ ( r ) ] H ( r ) } ) exp [ - 2 k 2 σ h 2 ( { f [ θ ( r ) ] } 2 + { f [ θ ( r ) ] } 2 ) ] ,
X J N = C ( N , J ) P N - J Q J .
C ( N , J ) = ( N ! ( N - J ) ! J ! ) 2 ( N - J ) ! = ( N ! ) 2 ( N - J ) ! ( J ! ) 2 .
X N = J = 1 N ( N ! ) 2 ( N - J ) ! ( J ! ) 2 P N - J Q J .
P = target d r [ θ ( r ) ] ,
( θ ) = κ 2 2 Δ r d ρ exp { 2 i k f ( θ ) ρ · θ - 2 k 2 [ f ( θ ) ] 2 D h ( ρ ) } .
P FP = A FP ( θ ) ,
Q FP = κ 2 2 exp [ - 4 k 2 σ h 2 f ( θ ) ] × | target d r exp [ - 2 i k f ( θ ) θ · r ] | 2 .
Q FP = 0             for θ 0.
X FP ( θ ) = P FP .
P = target d r X FP [ θ ( r ) ] A FP .
X R = target d r X FP [ θ ( r ) ] A FP .
Q = κ 2 2 | target d r exp ( - 2 i k f [ θ ( r ) ] H ( r ) - 2 k 2 σ h 2 { f [ θ ( r ) ] } 2 ) | 2 .
Q = κ 2 2 | target d r exp [ - 2 i k f 0 H ( r ) - 2 k 2 σ h 2 f 0 2 ] | 2 = exp ( - 4 k 2 σ h 2 f 0 2 ) × ( κ 2 2 | target d r exp [ - 2 i k f 0 H ( r ) ] | 2 ) .
Q = X spec ,
= exp ( - 4 k 2 σ h 2 f 0 2 ) .
= ( X S ) sph / ( X spec ) sph ,
Q = [ ( X S ) sph / ( X spec ) sph ] X spec ,
X S = [ ( X S ) sph / ( X spec ) sph ] X spec .