Abstract

Exact inverse-scattering equations are derived for the time-domain plasma-wave equation. Care is taken to motivate each step of the derivation by elementary physical arguments. The inverse method in this formulation is shown to depend on (1) causality, (2) the far-field properties of the Green function, and (3) the representation theorem.

© 1985 Optical Society of America

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References

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  1. P. C. Sabatier, ed., Problèmes Inverses, Cah. Math. Montpellier, R.C.P. 264, Recontre 1982 (1983);Recontre 1983 (1984);R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed. (Springer-Verlag, New York, 1982).
  2. E. Vogelzang, D. Yevick, H. A. Ferwerda, Opt. Commun. 45, 376 (1983).
    [CrossRef]
  3. A. J. Devaney, ed., Inverse Optics, Proc. Soc. Photo-Opt. Instrum. Eng.413 (1983).
  4. K. M. Hansen, G. W. Wecksung, J. Opt. Soc. Am. 73, 1501 (1983).
    [CrossRef]
  5. D. O. Thompson, D. E. Chimenti, Review of Progress in Nondestructive Evaluation (Plenum, New York, 1982–1984), Vols. 1–3.
    [CrossRef]
  6. R. G. Newton, J. Math Phys. 23, 594 (1982), and references therein.
    [CrossRef]
  7. K. Budden, Radio Waves in the Ionosphere (Cambridge U. Press, London, 1961);A. K. Jordan, S. Ahn, Proc. IRE 126, 945 (1979);G. N. Balanis, J. Math. Phys. 13, 1001 (1972).
    [CrossRef]
  8. J. H. Rose, M. Cheney, B. DeFacio, J. Math Phys. 25, 2995 (1984).
    [CrossRef]
  9. C. S. Morawetz, Comput. Math. Appl. 7, 319 (1981);C. J. Callias, G. A. Uhlmann, MSRI Rep. 034-83 (Berkeley, Calif., to be published).
    [CrossRef]
  10. J. H. Rose, M. Cheney, B. DeFacio, “Three-dimensional inverse scattering: plasma and variable velocity wave equations,” J. Math. Phys. (to be published).
  11. B. DeFacio, J. H. Rose, in Review of Progress in Quantitative Nondestructive Evaluation, D. O. Thompson, D. E. Chimenti, eds. (Plenum, New York, 1986).

1984

J. H. Rose, M. Cheney, B. DeFacio, J. Math Phys. 25, 2995 (1984).
[CrossRef]

1983

E. Vogelzang, D. Yevick, H. A. Ferwerda, Opt. Commun. 45, 376 (1983).
[CrossRef]

K. M. Hansen, G. W. Wecksung, J. Opt. Soc. Am. 73, 1501 (1983).
[CrossRef]

1982

R. G. Newton, J. Math Phys. 23, 594 (1982), and references therein.
[CrossRef]

1981

C. S. Morawetz, Comput. Math. Appl. 7, 319 (1981);C. J. Callias, G. A. Uhlmann, MSRI Rep. 034-83 (Berkeley, Calif., to be published).
[CrossRef]

Budden, K.

K. Budden, Radio Waves in the Ionosphere (Cambridge U. Press, London, 1961);A. K. Jordan, S. Ahn, Proc. IRE 126, 945 (1979);G. N. Balanis, J. Math. Phys. 13, 1001 (1972).
[CrossRef]

Cheney, M.

J. H. Rose, M. Cheney, B. DeFacio, J. Math Phys. 25, 2995 (1984).
[CrossRef]

J. H. Rose, M. Cheney, B. DeFacio, “Three-dimensional inverse scattering: plasma and variable velocity wave equations,” J. Math. Phys. (to be published).

Chimenti, D. E.

D. O. Thompson, D. E. Chimenti, Review of Progress in Nondestructive Evaluation (Plenum, New York, 1982–1984), Vols. 1–3.
[CrossRef]

DeFacio, B.

J. H. Rose, M. Cheney, B. DeFacio, J. Math Phys. 25, 2995 (1984).
[CrossRef]

J. H. Rose, M. Cheney, B. DeFacio, “Three-dimensional inverse scattering: plasma and variable velocity wave equations,” J. Math. Phys. (to be published).

B. DeFacio, J. H. Rose, in Review of Progress in Quantitative Nondestructive Evaluation, D. O. Thompson, D. E. Chimenti, eds. (Plenum, New York, 1986).

Ferwerda, H. A.

E. Vogelzang, D. Yevick, H. A. Ferwerda, Opt. Commun. 45, 376 (1983).
[CrossRef]

Hansen, K. M.

Morawetz, C. S.

C. S. Morawetz, Comput. Math. Appl. 7, 319 (1981);C. J. Callias, G. A. Uhlmann, MSRI Rep. 034-83 (Berkeley, Calif., to be published).
[CrossRef]

Newton, R. G.

R. G. Newton, J. Math Phys. 23, 594 (1982), and references therein.
[CrossRef]

Rose, J. H.

J. H. Rose, M. Cheney, B. DeFacio, J. Math Phys. 25, 2995 (1984).
[CrossRef]

J. H. Rose, M. Cheney, B. DeFacio, “Three-dimensional inverse scattering: plasma and variable velocity wave equations,” J. Math. Phys. (to be published).

B. DeFacio, J. H. Rose, in Review of Progress in Quantitative Nondestructive Evaluation, D. O. Thompson, D. E. Chimenti, eds. (Plenum, New York, 1986).

Thompson, D. O.

D. O. Thompson, D. E. Chimenti, Review of Progress in Nondestructive Evaluation (Plenum, New York, 1982–1984), Vols. 1–3.
[CrossRef]

Vogelzang, E.

E. Vogelzang, D. Yevick, H. A. Ferwerda, Opt. Commun. 45, 376 (1983).
[CrossRef]

Wecksung, G. W.

Yevick, D.

E. Vogelzang, D. Yevick, H. A. Ferwerda, Opt. Commun. 45, 376 (1983).
[CrossRef]

Comput. Math. Appl.

C. S. Morawetz, Comput. Math. Appl. 7, 319 (1981);C. J. Callias, G. A. Uhlmann, MSRI Rep. 034-83 (Berkeley, Calif., to be published).
[CrossRef]

J. Math Phys.

J. H. Rose, M. Cheney, B. DeFacio, J. Math Phys. 25, 2995 (1984).
[CrossRef]

R. G. Newton, J. Math Phys. 23, 594 (1982), and references therein.
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

E. Vogelzang, D. Yevick, H. A. Ferwerda, Opt. Commun. 45, 376 (1983).
[CrossRef]

Other

A. J. Devaney, ed., Inverse Optics, Proc. Soc. Photo-Opt. Instrum. Eng.413 (1983).

D. O. Thompson, D. E. Chimenti, Review of Progress in Nondestructive Evaluation (Plenum, New York, 1982–1984), Vols. 1–3.
[CrossRef]

K. Budden, Radio Waves in the Ionosphere (Cambridge U. Press, London, 1961);A. K. Jordan, S. Ahn, Proc. IRE 126, 945 (1979);G. N. Balanis, J. Math. Phys. 13, 1001 (1972).
[CrossRef]

P. C. Sabatier, ed., Problèmes Inverses, Cah. Math. Montpellier, R.C.P. 264, Recontre 1982 (1983);Recontre 1983 (1984);R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed. (Springer-Verlag, New York, 1982).

J. H. Rose, M. Cheney, B. DeFacio, “Three-dimensional inverse scattering: plasma and variable velocity wave equations,” J. Math. Phys. (to be published).

B. DeFacio, J. H. Rose, in Review of Progress in Quantitative Nondestructive Evaluation, D. O. Thompson, D. E. Chimenti, eds. (Plenum, New York, 1986).

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Figures (1)

Fig. 1
Fig. 1

The scattering geometry. The data are measured on the surface of the sphere S. The direction of incidence is denoted by ê and the direction of scatter by ê′.

Equations (11)

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Δ u ( t , ê ê , x ) t t u ( t , ê , x ) V ( x ) u ( t , ê , x ) = 0 ,
u + ( t , ê , x ) = δ ( t ê x ) + 1 | x | R ( τ , ê , x ̂ ) + O ( 1 x 2 ) , t , | x | , τ = t x .
u + ( t , ê , x ) = S d S d t [ u + ( t , ê , x ) G n ( t t , x , x ) u + n ( t , ê , x ) G ( t t , x , x ) ] .
G 0 ( t t , x , x ) = δ ( x x ̂ x + t t ) 4 π x + O ( 1 x 2 ) , x x ,
G ( t t , x , x ) = u ( t t + x , x ̂ , x ) 4 π x + O ( 1 x 2 ) , x x .
u sc + ( t , ê , x ) = u sc ( t , ê , x ) 1 2 π S 2 d 2 e d d t R ( t ê x , ê , ê ) 1 2 π S 2 d 2 ê d τ d d t R ( t τ , ê , ê ) u sc ( τ , ê , x ) ,
II = lim x x 2 S 2 d 2 x ̂ d t { x [ 1 4 π x u s c × ( t t + x , x ̂ , x ) ] δ ( t ê x ) 1 4 π x u sc ( t t + x , x ̂ , x ) x δ ( t ê x ) } .
II = lim | x | x 4 π S 2 d 2 x ̂ ( 1 + ê x ) × d d t u sc ( t ê x + x , x ̂ , x ) .
u sc + ( t , ê , x ) = 1 2 π S 2 d 2 e d d t R ( t ê x , ê , ê ) 1 2 π S 2 d 2 e d τ d d t R ( t + τ , ê , ê ) u s c + ( τ , ê , x ) .
u ( t , ê , x ) = δ ( t ê x ) + H ( t ê x ) B ( ê , x ) + .
V ( x ) = 2 ê B ( ê , x ) .

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