Abstract

The low-contrast or (first-order) Born approximation is applied to the time-domain scattering of a plane scalar wave by an object of bounded extent present in a homogeneous embedding. Closed-form analytic expressions are obtained for the spherical-wave far-field scattering amplitude related to homogeneous objects of the following shapes: an ellipsoid, an elliptic cylinder of finite height, and a tetrahedron. Dispersion is included. Apart from their intrinsic interest, the results may be useful as test cases for time-domain inverse-scattering algorithms.

© 1985 Optical Society of America

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References

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  1. D. Miller, M. Oristaglio, G. Beylkin, “A new formalism and an old heuristic for seismic migration,” in Proceedings of the 54th Annual Meeting of the SEG (Society of Exploration Geo-physicists, Tulsa, Okla., 1984), pp. 704.
  2. G. Beylkin, “Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform,” J. Math. Phys. 26, 99–108 (1985).
    [CrossRef]
  3. Y. Das, W.-M. Boerner, “On radar target shape estimations using algorithms for reconstruction from projections,” IEEE Trans. Antennas Propag. AP-26, 274–279 (1978).
    [CrossRef]
  4. W.-M. Boerner, C. M. Ho, B. Y. Foo, “Use of Radon’s projection theory in electromagnetic inverse scattering,” IEEE Trans. Antennas Propag. AP-29, 336–341 (1981).
    [CrossRef]
  5. W.-M. Boerner, C. M. Ho, “Analysis of physical optics far-field inverse scattering for the limited data case using Radon theory and polarization information,” Wave Motion 3, 311–333 (1981).
    [CrossRef]
  6. J. H. Rose, J. L. Opsal, “The inverse Born approximation: exact determination of shape of convex voids,” in Review of Progress in Quantitative Nondestructive Evaluation, D. P. Thompson, D. E. Cimenti, eds. (Plenum, New York, 1983), pp. 949–959.
    [CrossRef]
  7. J. H. Rose, M. Cheney, B. DeFacio, “The connection between time- and frequency-domain three-dimensional inverse scattering methods,” J. Math. Phys. 25, 2995–3000 (1984).
    [CrossRef]
  8. N. N. Bojarski, “Three dimensional electromagnetic short pulse inverse scattering,” Special Projects Lab. Rep. No. SURC SPL 67-3 NTIS AD/845126 (Syracuse University Research Corporation, Syracuse, N.Y., February1967).
  9. N. N. Bojarski, “Electromagnetic inverse scattering theory,” Special Projects Lab. Rep. No. SURG SPL R68-70 NTIS AD/509134 (Syracuse University Research Corporation, Syracuse, N.Y., December1968).
  10. N. N. Bojarski, “Electromagnetic inverse scattering,” Naval Air Systems Command Contract N00019-72-0-0462 Rep. (Naval Air Systems Command, Washington, D.C., June1972).

1985 (1)

G. Beylkin, “Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform,” J. Math. Phys. 26, 99–108 (1985).
[CrossRef]

1984 (1)

J. H. Rose, M. Cheney, B. DeFacio, “The connection between time- and frequency-domain three-dimensional inverse scattering methods,” J. Math. Phys. 25, 2995–3000 (1984).
[CrossRef]

1981 (2)

W.-M. Boerner, C. M. Ho, B. Y. Foo, “Use of Radon’s projection theory in electromagnetic inverse scattering,” IEEE Trans. Antennas Propag. AP-29, 336–341 (1981).
[CrossRef]

W.-M. Boerner, C. M. Ho, “Analysis of physical optics far-field inverse scattering for the limited data case using Radon theory and polarization information,” Wave Motion 3, 311–333 (1981).
[CrossRef]

1978 (1)

Y. Das, W.-M. Boerner, “On radar target shape estimations using algorithms for reconstruction from projections,” IEEE Trans. Antennas Propag. AP-26, 274–279 (1978).
[CrossRef]

Beylkin, G.

G. Beylkin, “Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform,” J. Math. Phys. 26, 99–108 (1985).
[CrossRef]

D. Miller, M. Oristaglio, G. Beylkin, “A new formalism and an old heuristic for seismic migration,” in Proceedings of the 54th Annual Meeting of the SEG (Society of Exploration Geo-physicists, Tulsa, Okla., 1984), pp. 704.

Boerner, W.-M.

W.-M. Boerner, C. M. Ho, B. Y. Foo, “Use of Radon’s projection theory in electromagnetic inverse scattering,” IEEE Trans. Antennas Propag. AP-29, 336–341 (1981).
[CrossRef]

W.-M. Boerner, C. M. Ho, “Analysis of physical optics far-field inverse scattering for the limited data case using Radon theory and polarization information,” Wave Motion 3, 311–333 (1981).
[CrossRef]

Y. Das, W.-M. Boerner, “On radar target shape estimations using algorithms for reconstruction from projections,” IEEE Trans. Antennas Propag. AP-26, 274–279 (1978).
[CrossRef]

Bojarski, N. N.

N. N. Bojarski, “Three dimensional electromagnetic short pulse inverse scattering,” Special Projects Lab. Rep. No. SURC SPL 67-3 NTIS AD/845126 (Syracuse University Research Corporation, Syracuse, N.Y., February1967).

N. N. Bojarski, “Electromagnetic inverse scattering theory,” Special Projects Lab. Rep. No. SURG SPL R68-70 NTIS AD/509134 (Syracuse University Research Corporation, Syracuse, N.Y., December1968).

N. N. Bojarski, “Electromagnetic inverse scattering,” Naval Air Systems Command Contract N00019-72-0-0462 Rep. (Naval Air Systems Command, Washington, D.C., June1972).

Cheney, M.

J. H. Rose, M. Cheney, B. DeFacio, “The connection between time- and frequency-domain three-dimensional inverse scattering methods,” J. Math. Phys. 25, 2995–3000 (1984).
[CrossRef]

Das, Y.

Y. Das, W.-M. Boerner, “On radar target shape estimations using algorithms for reconstruction from projections,” IEEE Trans. Antennas Propag. AP-26, 274–279 (1978).
[CrossRef]

DeFacio, B.

J. H. Rose, M. Cheney, B. DeFacio, “The connection between time- and frequency-domain three-dimensional inverse scattering methods,” J. Math. Phys. 25, 2995–3000 (1984).
[CrossRef]

Foo, B. Y.

W.-M. Boerner, C. M. Ho, B. Y. Foo, “Use of Radon’s projection theory in electromagnetic inverse scattering,” IEEE Trans. Antennas Propag. AP-29, 336–341 (1981).
[CrossRef]

Ho, C. M.

W.-M. Boerner, C. M. Ho, “Analysis of physical optics far-field inverse scattering for the limited data case using Radon theory and polarization information,” Wave Motion 3, 311–333 (1981).
[CrossRef]

W.-M. Boerner, C. M. Ho, B. Y. Foo, “Use of Radon’s projection theory in electromagnetic inverse scattering,” IEEE Trans. Antennas Propag. AP-29, 336–341 (1981).
[CrossRef]

Miller, D.

D. Miller, M. Oristaglio, G. Beylkin, “A new formalism and an old heuristic for seismic migration,” in Proceedings of the 54th Annual Meeting of the SEG (Society of Exploration Geo-physicists, Tulsa, Okla., 1984), pp. 704.

Opsal, J. L.

J. H. Rose, J. L. Opsal, “The inverse Born approximation: exact determination of shape of convex voids,” in Review of Progress in Quantitative Nondestructive Evaluation, D. P. Thompson, D. E. Cimenti, eds. (Plenum, New York, 1983), pp. 949–959.
[CrossRef]

Oristaglio, M.

D. Miller, M. Oristaglio, G. Beylkin, “A new formalism and an old heuristic for seismic migration,” in Proceedings of the 54th Annual Meeting of the SEG (Society of Exploration Geo-physicists, Tulsa, Okla., 1984), pp. 704.

Rose, J. H.

J. H. Rose, M. Cheney, B. DeFacio, “The connection between time- and frequency-domain three-dimensional inverse scattering methods,” J. Math. Phys. 25, 2995–3000 (1984).
[CrossRef]

J. H. Rose, J. L. Opsal, “The inverse Born approximation: exact determination of shape of convex voids,” in Review of Progress in Quantitative Nondestructive Evaluation, D. P. Thompson, D. E. Cimenti, eds. (Plenum, New York, 1983), pp. 949–959.
[CrossRef]

IEEE Trans. Antennas Propag. (2)

Y. Das, W.-M. Boerner, “On radar target shape estimations using algorithms for reconstruction from projections,” IEEE Trans. Antennas Propag. AP-26, 274–279 (1978).
[CrossRef]

W.-M. Boerner, C. M. Ho, B. Y. Foo, “Use of Radon’s projection theory in electromagnetic inverse scattering,” IEEE Trans. Antennas Propag. AP-29, 336–341 (1981).
[CrossRef]

J. Math. Phys. (2)

J. H. Rose, M. Cheney, B. DeFacio, “The connection between time- and frequency-domain three-dimensional inverse scattering methods,” J. Math. Phys. 25, 2995–3000 (1984).
[CrossRef]

G. Beylkin, “Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform,” J. Math. Phys. 26, 99–108 (1985).
[CrossRef]

Wave Motion (1)

W.-M. Boerner, C. M. Ho, “Analysis of physical optics far-field inverse scattering for the limited data case using Radon theory and polarization information,” Wave Motion 3, 311–333 (1981).
[CrossRef]

Other (5)

J. H. Rose, J. L. Opsal, “The inverse Born approximation: exact determination of shape of convex voids,” in Review of Progress in Quantitative Nondestructive Evaluation, D. P. Thompson, D. E. Cimenti, eds. (Plenum, New York, 1983), pp. 949–959.
[CrossRef]

N. N. Bojarski, “Three dimensional electromagnetic short pulse inverse scattering,” Special Projects Lab. Rep. No. SURC SPL 67-3 NTIS AD/845126 (Syracuse University Research Corporation, Syracuse, N.Y., February1967).

N. N. Bojarski, “Electromagnetic inverse scattering theory,” Special Projects Lab. Rep. No. SURG SPL R68-70 NTIS AD/509134 (Syracuse University Research Corporation, Syracuse, N.Y., December1968).

N. N. Bojarski, “Electromagnetic inverse scattering,” Naval Air Systems Command Contract N00019-72-0-0462 Rep. (Naval Air Systems Command, Washington, D.C., June1972).

D. Miller, M. Oristaglio, G. Beylkin, “A new formalism and an old heuristic for seismic migration,” in Proceedings of the 54th Annual Meeting of the SEG (Society of Exploration Geo-physicists, Tulsa, Okla., 1984), pp. 704.

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Equations (46)

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( · ) u c 0 2 t 2 u = 0 ,
( · ) u c 0 2 t 2 [ u ( r , t ) ] + 0 χ ( r , t ) u ( r , t t ) d t ] = 0 .
χ = { 0 when < t < 0 ( ω p 2 / Ω ) exp ( Γ t ) sin ( Ω t ) when 0 < t <
ω p = ( N q / m 0 ) 1 / 2
Ω = ( ω 0 2 + ω p 2 / 3 Γ 2 ) 1 / 2 .
χ ̂ = ω p 2 ( ω 2 2 i ω Γ + ω 0 2 + ω p 2 / 3 ) 1 ,
u i = f ( t α ̂ · r / c 0 ) .
u s = u u i .
( · ) u s c 0 2 t 2 u s = q s ,
q s = c 0 2 t 2 0 χ ( r , t ) u ( r , t t ) d t .
u s ( r , t ) = r V q s ( r , t | r r | / c 0 ) 4 π | r r | d V .
u s ( r , t ) A ( β ̂ , t | r | / c 0 ) 4 π | r | , ( | r | ) ,
A ( β ̂ , t ) = r V q s ( r , t + β ̂ · r / c 0 ) d V ,
q s = B c 0 2 t 2 0 χ ( r , t ) f ( t t α ̂ · r / c 0 ) d t .
A ( β ̂ , t ) = V 0 χ ( t ) ϒ ( β ̂ α ̂ , t t ) d t ,
D n f = t n f with n = 1 , 2 ,
I f = t 0 t f ( t ) d t ,
ϒ ( s , t ) = c 0 2 V 1 r V t 2 f ( t + s · r ) d V ,
ϒ ( 0 , t ) = c 0 2 D 2 f ( t )
ϒ ( s , t ) = V 1 r V ( υ ̂ · s ) D f ( t + s · r ) d A ,
A ( β ̂ , t ) = V 0 χ ( t ) ϒ ( β ̂ , t t ) d t ,
ϒ ( β ̂ , t ) = c 0 2 V 1 r V t 2 u i d V .
ϒ ( β ̂ , t ) = V 1 r V ( υ ̂ · ) u i d A ,
V = { r ; 0 x 2 / a 2 + y 2 / b 2 + z 2 / c 2 < 1 } .
V = 4 π a b c / 3 .
ξ = x / a , η = y / b , ζ = z / c .
s · r = ( s x a ) ξ + ( s y b ) η + ( s z c ) ζ = γ ρ cos ( θ ) ,
γ = ( s x 2 a 2 + s y 2 b 2 + s z 2 c 2 ) 1 / 2 > 0 ,
d V = abc ρ 2 sin ( θ ) d ρ d θ d ϕ .
ϒ = ( 3 c 0 2 / 2 γ 3 ) { γ [ f ( t + γ ) + f ( t γ ) ] I f ( t + γ ) + I f ( t γ ) } .
V = { r ; 0 x 2 / a 2 + y 2 / b 2 < 1 , h / 2 < z < h / 2 } .
V = π abh .
ξ = x / a , η = y / b .
s x x + s y y = ( s x a ) ξ + ( s y b ) η = γ ρ cos ( θ ) ,
γ = ( s x 2 a 2 + s y 2 a 2 ) 1 / 2 > 0 ,
d x d y = a b ρ d ρ d θ .
ϒ = [ 2 / π c 0 2 s z h ] 1 1 ( 1 τ 2 ) 1 / 2 { D f [ t + γ τ + s z h / 2 ] D f [ t + γ τ + s z h / 2 ] } d τ .
ϒ = ( 2 / π c 0 2 ) 1 1 D 2 f ( t + γ τ ) ( 1 τ 2 ) 1 / 2 d τ .
ϒ = ( c 0 2 / s z h ) [ D f ( t + s z h / 2 ) D f ( t s z h / 2 ) ] .
V = { r ; r = i = 1 4 λ i r i , 0 λ i 1 , i = 1 4 λ i = 1 , } .
V = | r 1 · ( r 2 × r 3 ) + r 2 · ( r 3 × r 4 ) r 3 · ( r 4 × r 1 ) + r 4 · ( r 1 × r 2 ) | / 6 .
ϒ = 6 c 0 2 [ ( γ 12 γ 13 γ 14 ) 1 I f ( t + s · r 1 ) + ( γ 21 γ 23 γ 24 ) 1 I f ( t + s · r 2 ) + ( γ 31 γ 32 γ 33 ) 1 I f ( t + s · r 3 ) + ( γ 41 γ 42 γ 43 ) 1 I f ( t + s · r 4 ) ] ,
γ i j = s · ( r i r j ) = γ j i .
ϒ = 6 c 0 2 { ( γ 13 γ 14 ) 1 [ f ( t + s · r 1 ) ( γ 13 1 + γ 14 1 ) × I f ( t + s · r 1 ) ] + ( γ 31 2 + γ 34 ) 1 If ( t + s · r 3 ) + ( γ 41 2 γ 43 ) 1 I f ( t + s · r 4 } .
ϒ = 6 c 0 2 { γ 14 3 [ ½ γ 14 2 Df ( t + s · r 1 ) γ 14 f ( t + s · r 1 ) + If ( t + s · r 1 ) ] + γ 14 3 If ( t + s · r 4 ) } .
ϒ = 6 c 0 2 { γ 31 2 [ f ( t + s · r 3 ) + f ( t + s · r 1 ) ] 2 γ 31 3 [ If ( t + s · r 3 ) If ( t + s · r 1 ) ] } .

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