Abstract

The purpose of this paper is to review some of the inverse methods in electromagnetics for the reconstruction of one-dimensional complex refractive-index profiles, using transient or spectral data. Two different categories of inversion schemes, viz., the differential-inverse and integral-inverse algorithms, are discussed.

© 1987 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. J. Devaney, ed., Inverse Problems in Propagation and Scattering, feature section of J. Opt. Soc. Am. A 2, 1902–2061 (1985).
  2. A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
    [CrossRef]
  3. A. E. Yagle, “Layer stripping solutions of inverse seismic problems,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1985).
  4. A. E. Yagle, B. C. Levy, “Application of the Schur algorithm to the inverse problem for a layered acoustic medium,”J. Acoust. Soc. Am. 76, 301–308 (1984).
    [CrossRef]
  5. A. E. Yagle, B. C. Levy, “A fast algorithm solution of the inverse problem for a layered acoustic medium probed by spherical harmonic waves,”J. Acoust. Soc. Am. 78, 729–737 (1985).
    [CrossRef]
  6. A. E. Yagle, B. C. Levy, “The Schur algorithm and its applications,” Acta Appl. Math. 3, 255–284 (1985).
    [CrossRef]
  7. B. C. Levy, “Layer by layer reconstruction methods for the earth resistivity from direct current measurements,”IEEE Trans. Geosci. Remote Sensing GE-23, 841–850 (1985).
    [CrossRef]
  8. T. M. Habashy, R. Mittra, “On some inverse methods in electromagnetics,”J. Electromag. Waves Appl. (to be published).
  9. K. B. Bube, R. Burridge, “The one-dimensional inverse problem of reflection seismology,” SIAM Rev. 25, 497–559 (1983).
    [CrossRef]
  10. F. Santosa, H. Schwetlick, “The inversion of acoustical impedance profile by methods of characteristics,” Wave Motion 4, 99–110 (1982).
    [CrossRef]
  11. A. Sezginer, “Forward and inverse problems in transient electromagnetic fields,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1985).
  12. A. Sezginer, “Profile inversion by method of characteristics in a lossy, cylindrical medium,” presented at the National Radio Science Meeting, Boston, Mass., June 25–29, 1984.
  13. T. M. Habashy, A. Sezginer, W. C. Chew, “Simultaneous inversion of radially varying conductivity and permittivity profiles,” presented at the International Geoscience and Remote Sensing Symposium, Amherst, Mass., October 7–9, 1985.
  14. E. A. Robinson, “Dynamic predictive deconvolution,” Geophys. Prospect. 23, 779–797 (1975).
    [CrossRef]
  15. A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,”J. Opt. (Paris) 9, 83–90 (1978).
    [CrossRef]
  16. W. C. Chew, S. L. Chuang, “Profile inversion of a planar medium with a line source or a point source,” presented at the International Geoscience and Remote Sensing Symposium, Strasbourg, France, August 27–30, 1984.
  17. S. Coen, K. K. Mei, D. J. Angelakos, “Inverse scattering technique applied to remote sensing of layered media,”IEEE Trans. Antennas Propag. AP-29, 298–306 (1981).
    [CrossRef]
  18. A. G. Tijhuis, C. Van Der Worm, “Iterative approach to the frequency-domain solution of the inverse-scattering problem for an inhomogeneous lossless dielectric slab,”IEEE Trans. Antennas Propag. AP-32, 711–716 (1984).
    [CrossRef]
  19. D. Lesselier, “Optimization techniques and inverse problems: reconstruction of conductivity profiles in the time domain,”IEEE Trans. Antennas Propag. AP-30, 59–65 (1982).
    [CrossRef]
  20. A. G. Tijhuis, “Iterative determination of permittivity and conductivity profiles of a dielectric slab in the time domain,”IEEE Trans. Antennas Propag. AP-29, 239–245 (1981).
    [CrossRef]
  21. T. M. Habashy, W. C. Chew, E. Y. Chow, “Simultaneous reconstruction of permittivity and conductivity profiles in a radially inhomogeneous slab,” Radio Sci. 21, 635–645 (1986).
    [CrossRef]
  22. W. Tabbara, “Reconstruction of permittivity profiles from a spectral analysis of the reflection coefficient,”IEEE Trans. Antennas Propag. AP-27, 241–248 (1979).
    [CrossRef]
  23. J. A. Kong, Theory of Electromagnetic Waves (Wiley, New York, 1975).
  24. N. Bleistein, J. K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,”J. Math. Phys. 18, 194–201 (1977).
    [CrossRef]
  25. A. J. Devaney, G. C. Sherman, “Nonuniqueness in inverse source and scattering problems,”IEEE Trans. Antennas Propag. AP-30, 1034–1037 (1982).
    [CrossRef]
  26. A. J. Devaney, E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
    [CrossRef]
  27. C. T. H. Baker, The Numerical Treatment of Integral Equations (Clarendon, Oxford, 1977).
  28. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).
  29. P. C. Sabatier, ed., Applied Inverse Problems (Springer-Verlag, Berlin, 1978).
    [CrossRef]
  30. I. M. Gelfand, B. M. Levitan, “On the determination of a differential equation from its spectral measure function,” Am. Math. Soc. 1, 253–304 (1955).
  31. L. D. Faddeyev, B. Seckler, “The inverse problem in the quantum theory of scattering,”J. Math. Phys. 4, 72–103 (1963).
    [CrossRef]
  32. Z. S. Agranovich, V. A. Marchenko, The Inverse Problem of Scattering Theory (Gordon and Breach, New York, 1963).
  33. I. Kay, H. E. Moses, Inverse Scattering Papers: 1955–1963, Vol. 12 of Lie Groups: History Frontiers and Applications (Mathematical Science, Mass.1982).
  34. G. L. Lamb, Elements of Soliton Theory (Wiley-Interscience, New York, 1980).
  35. K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer-Verlag, New York, 1977).
    [CrossRef]
  36. S. Coen, “On the elastic profiles of a layered medium from reflection data. Part I. Plane-wave sources,”J. Acoust. Soc. Am. 70, 172–175 (1981).
    [CrossRef]
  37. S. Coen, “Density and compressibility profiles of a layered acoustic medium from precritical incidence data,” Geophysics 46, 1244–1246 (1981).
    [CrossRef]
  38. S. Coen, “Inverse scattering of the permittivity and permeability profiles of a plane stratified medium,”J. Math. Phys. 22, 1127–1129 (1981).
    [CrossRef]
  39. R. J. Krueger, “An inverse problem for an absorbing medium with multiple discontinuities,”Q. Appl. Math. 34, 129–147 (1976).
  40. R. J. Krueger, “An inverse problem for a dissipative hyperbolic equation with discontinuous coefficients,”Q. Appl. Math. 36, 235–253 (1978).
  41. R. J. Krueger, “Numerical aspects of a dissipative inverse problem,”IEEE Trans. Antennas Propag. AP-29, 253–261 (1981).
    [CrossRef]
  42. M. Jaulent, “Inverse scattering problems in absorbing media,”J. Math. Phys. 17, 1351–1360 (1976).
    [CrossRef]
  43. V. H. Weston, “On the inverse problem for a hyperbolic dispersive partial differential equation,”J. Math. Phys. 13, 1952–1956 (1972).
    [CrossRef]
  44. V. H. Weston, “On inverse scattering,”J. Math. Phys. 15, 209–213 (1974).
    [CrossRef]
  45. S. Coen, “The inverse problem of the direct current conductivity profile of a layered earth,” Geophysics 46, 1702–1713 (1981).
    [CrossRef]
  46. D. H. Schaubert, “Spectral domain methods for remote probing of stratified media,” Ph.D. dissertation (University of Illinois, Urbana, Ill., 1974).
  47. D. H. Schaubert, R. Mittra, “A spectral domain method for remotely probing stratified media,”IEEE Trans. Antennas Propag. AP-25, 261–265 (1977).
    [CrossRef]
  48. R. Mittra, T. M. Habashy, “Profile inversion of radially inhomogeneous media with a single frequency measurement,” presented at the National Radio Science Meeting, Boston, Mass., June 25–29, 1984.
  49. T. M. Habashy, R. Mittra, “Time domain profile inversion of a cylindrically stratified medium,” presented at the National Radio Science Meeting, Boston, Mass., June 25–29, 1984.
  50. S. Coen, “Velocity and density profiles of a layered acoustic medium from common source-point data,” Geophysics 47, 898–905 (1982).
    [CrossRef]
  51. R. Burridge, “The Gelfand–Levitan, the Marchenko, and the Gopinath–Sondhi integral equations of inverse scattering theory, regarded in the context of inverse impulse-response problems,” Wave Motion 2, 305–323 (1980).
    [CrossRef]
  52. M. M. Sondhi, B. Gopinath, “Determination of vocal-tract shape from impulse response at the lips,”J. Acoust. Soc. Am. 49, 1867–1873 (1970).
    [CrossRef]
  53. B. Gopinath, M. M. Sondhi, “Inversion of the telegraph equation and the synthesis of nonuniform lines,” Proc. IEEE 59, 383–392 (1971).
    [CrossRef]
  54. R. G. Newton, “Inversion of reflection data for layered media: a review of exact methods,” Geophys. J. R. Astron. Soc. 65, 191–215 (1981).
    [CrossRef]
  55. G. N. Balanis, “The plasma inverse problem,”J. Math. Phys. 13, 1001–1005 (1972).
    [CrossRef]
  56. S. Coen, “Inverse scattering of a layered and dispersionless dielectric half-space, part I: reflection data from plane waves at normal incidence,”IEEE Trans. Antennas Propag. AP-29, 726–732 (1981).
    [CrossRef]

1986

T. M. Habashy, W. C. Chew, E. Y. Chow, “Simultaneous reconstruction of permittivity and conductivity profiles in a radially inhomogeneous slab,” Radio Sci. 21, 635–645 (1986).
[CrossRef]

1985

A. E. Yagle, B. C. Levy, “A fast algorithm solution of the inverse problem for a layered acoustic medium probed by spherical harmonic waves,”J. Acoust. Soc. Am. 78, 729–737 (1985).
[CrossRef]

A. E. Yagle, B. C. Levy, “The Schur algorithm and its applications,” Acta Appl. Math. 3, 255–284 (1985).
[CrossRef]

B. C. Levy, “Layer by layer reconstruction methods for the earth resistivity from direct current measurements,”IEEE Trans. Geosci. Remote Sensing GE-23, 841–850 (1985).
[CrossRef]

A. J. Devaney, ed., Inverse Problems in Propagation and Scattering, feature section of J. Opt. Soc. Am. A 2, 1902–2061 (1985).

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

1984

A. E. Yagle, B. C. Levy, “Application of the Schur algorithm to the inverse problem for a layered acoustic medium,”J. Acoust. Soc. Am. 76, 301–308 (1984).
[CrossRef]

A. G. Tijhuis, C. Van Der Worm, “Iterative approach to the frequency-domain solution of the inverse-scattering problem for an inhomogeneous lossless dielectric slab,”IEEE Trans. Antennas Propag. AP-32, 711–716 (1984).
[CrossRef]

1983

K. B. Bube, R. Burridge, “The one-dimensional inverse problem of reflection seismology,” SIAM Rev. 25, 497–559 (1983).
[CrossRef]

1982

F. Santosa, H. Schwetlick, “The inversion of acoustical impedance profile by methods of characteristics,” Wave Motion 4, 99–110 (1982).
[CrossRef]

D. Lesselier, “Optimization techniques and inverse problems: reconstruction of conductivity profiles in the time domain,”IEEE Trans. Antennas Propag. AP-30, 59–65 (1982).
[CrossRef]

A. J. Devaney, G. C. Sherman, “Nonuniqueness in inverse source and scattering problems,”IEEE Trans. Antennas Propag. AP-30, 1034–1037 (1982).
[CrossRef]

S. Coen, “Velocity and density profiles of a layered acoustic medium from common source-point data,” Geophysics 47, 898–905 (1982).
[CrossRef]

1981

R. G. Newton, “Inversion of reflection data for layered media: a review of exact methods,” Geophys. J. R. Astron. Soc. 65, 191–215 (1981).
[CrossRef]

S. Coen, “Inverse scattering of a layered and dispersionless dielectric half-space, part I: reflection data from plane waves at normal incidence,”IEEE Trans. Antennas Propag. AP-29, 726–732 (1981).
[CrossRef]

R. J. Krueger, “Numerical aspects of a dissipative inverse problem,”IEEE Trans. Antennas Propag. AP-29, 253–261 (1981).
[CrossRef]

S. Coen, “The inverse problem of the direct current conductivity profile of a layered earth,” Geophysics 46, 1702–1713 (1981).
[CrossRef]

A. G. Tijhuis, “Iterative determination of permittivity and conductivity profiles of a dielectric slab in the time domain,”IEEE Trans. Antennas Propag. AP-29, 239–245 (1981).
[CrossRef]

S. Coen, “On the elastic profiles of a layered medium from reflection data. Part I. Plane-wave sources,”J. Acoust. Soc. Am. 70, 172–175 (1981).
[CrossRef]

S. Coen, “Density and compressibility profiles of a layered acoustic medium from precritical incidence data,” Geophysics 46, 1244–1246 (1981).
[CrossRef]

S. Coen, “Inverse scattering of the permittivity and permeability profiles of a plane stratified medium,”J. Math. Phys. 22, 1127–1129 (1981).
[CrossRef]

S. Coen, K. K. Mei, D. J. Angelakos, “Inverse scattering technique applied to remote sensing of layered media,”IEEE Trans. Antennas Propag. AP-29, 298–306 (1981).
[CrossRef]

1980

R. Burridge, “The Gelfand–Levitan, the Marchenko, and the Gopinath–Sondhi integral equations of inverse scattering theory, regarded in the context of inverse impulse-response problems,” Wave Motion 2, 305–323 (1980).
[CrossRef]

1979

W. Tabbara, “Reconstruction of permittivity profiles from a spectral analysis of the reflection coefficient,”IEEE Trans. Antennas Propag. AP-27, 241–248 (1979).
[CrossRef]

1978

A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,”J. Opt. (Paris) 9, 83–90 (1978).
[CrossRef]

R. J. Krueger, “An inverse problem for a dissipative hyperbolic equation with discontinuous coefficients,”Q. Appl. Math. 36, 235–253 (1978).

1977

D. H. Schaubert, R. Mittra, “A spectral domain method for remotely probing stratified media,”IEEE Trans. Antennas Propag. AP-25, 261–265 (1977).
[CrossRef]

N. Bleistein, J. K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,”J. Math. Phys. 18, 194–201 (1977).
[CrossRef]

1976

R. J. Krueger, “An inverse problem for an absorbing medium with multiple discontinuities,”Q. Appl. Math. 34, 129–147 (1976).

M. Jaulent, “Inverse scattering problems in absorbing media,”J. Math. Phys. 17, 1351–1360 (1976).
[CrossRef]

1975

E. A. Robinson, “Dynamic predictive deconvolution,” Geophys. Prospect. 23, 779–797 (1975).
[CrossRef]

1974

V. H. Weston, “On inverse scattering,”J. Math. Phys. 15, 209–213 (1974).
[CrossRef]

1973

A. J. Devaney, E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
[CrossRef]

1972

V. H. Weston, “On the inverse problem for a hyperbolic dispersive partial differential equation,”J. Math. Phys. 13, 1952–1956 (1972).
[CrossRef]

G. N. Balanis, “The plasma inverse problem,”J. Math. Phys. 13, 1001–1005 (1972).
[CrossRef]

1971

B. Gopinath, M. M. Sondhi, “Inversion of the telegraph equation and the synthesis of nonuniform lines,” Proc. IEEE 59, 383–392 (1971).
[CrossRef]

1970

M. M. Sondhi, B. Gopinath, “Determination of vocal-tract shape from impulse response at the lips,”J. Acoust. Soc. Am. 49, 1867–1873 (1970).
[CrossRef]

1963

L. D. Faddeyev, B. Seckler, “The inverse problem in the quantum theory of scattering,”J. Math. Phys. 4, 72–103 (1963).
[CrossRef]

1955

I. M. Gelfand, B. M. Levitan, “On the determination of a differential equation from its spectral measure function,” Am. Math. Soc. 1, 253–304 (1955).

Agranovich, Z. S.

Z. S. Agranovich, V. A. Marchenko, The Inverse Problem of Scattering Theory (Gordon and Breach, New York, 1963).

Angelakos, D. J.

S. Coen, K. K. Mei, D. J. Angelakos, “Inverse scattering technique applied to remote sensing of layered media,”IEEE Trans. Antennas Propag. AP-29, 298–306 (1981).
[CrossRef]

Baker, C. T. H.

C. T. H. Baker, The Numerical Treatment of Integral Equations (Clarendon, Oxford, 1977).

Balanis, G. N.

G. N. Balanis, “The plasma inverse problem,”J. Math. Phys. 13, 1001–1005 (1972).
[CrossRef]

Bleistein, N.

N. Bleistein, J. K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,”J. Math. Phys. 18, 194–201 (1977).
[CrossRef]

Bruckstein, A. M.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

Bube, K. B.

K. B. Bube, R. Burridge, “The one-dimensional inverse problem of reflection seismology,” SIAM Rev. 25, 497–559 (1983).
[CrossRef]

Burridge, R.

K. B. Bube, R. Burridge, “The one-dimensional inverse problem of reflection seismology,” SIAM Rev. 25, 497–559 (1983).
[CrossRef]

R. Burridge, “The Gelfand–Levitan, the Marchenko, and the Gopinath–Sondhi integral equations of inverse scattering theory, regarded in the context of inverse impulse-response problems,” Wave Motion 2, 305–323 (1980).
[CrossRef]

Cadilhac, M.

A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,”J. Opt. (Paris) 9, 83–90 (1978).
[CrossRef]

Chadan, K.

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer-Verlag, New York, 1977).
[CrossRef]

Chew, W. C.

T. M. Habashy, W. C. Chew, E. Y. Chow, “Simultaneous reconstruction of permittivity and conductivity profiles in a radially inhomogeneous slab,” Radio Sci. 21, 635–645 (1986).
[CrossRef]

W. C. Chew, S. L. Chuang, “Profile inversion of a planar medium with a line source or a point source,” presented at the International Geoscience and Remote Sensing Symposium, Strasbourg, France, August 27–30, 1984.

T. M. Habashy, A. Sezginer, W. C. Chew, “Simultaneous inversion of radially varying conductivity and permittivity profiles,” presented at the International Geoscience and Remote Sensing Symposium, Amherst, Mass., October 7–9, 1985.

Chow, E. Y.

T. M. Habashy, W. C. Chew, E. Y. Chow, “Simultaneous reconstruction of permittivity and conductivity profiles in a radially inhomogeneous slab,” Radio Sci. 21, 635–645 (1986).
[CrossRef]

Chuang, S. L.

W. C. Chew, S. L. Chuang, “Profile inversion of a planar medium with a line source or a point source,” presented at the International Geoscience and Remote Sensing Symposium, Strasbourg, France, August 27–30, 1984.

Coen, S.

S. Coen, “Velocity and density profiles of a layered acoustic medium from common source-point data,” Geophysics 47, 898–905 (1982).
[CrossRef]

S. Coen, “Inverse scattering of a layered and dispersionless dielectric half-space, part I: reflection data from plane waves at normal incidence,”IEEE Trans. Antennas Propag. AP-29, 726–732 (1981).
[CrossRef]

S. Coen, K. K. Mei, D. J. Angelakos, “Inverse scattering technique applied to remote sensing of layered media,”IEEE Trans. Antennas Propag. AP-29, 298–306 (1981).
[CrossRef]

S. Coen, “On the elastic profiles of a layered medium from reflection data. Part I. Plane-wave sources,”J. Acoust. Soc. Am. 70, 172–175 (1981).
[CrossRef]

S. Coen, “Density and compressibility profiles of a layered acoustic medium from precritical incidence data,” Geophysics 46, 1244–1246 (1981).
[CrossRef]

S. Coen, “Inverse scattering of the permittivity and permeability profiles of a plane stratified medium,”J. Math. Phys. 22, 1127–1129 (1981).
[CrossRef]

S. Coen, “The inverse problem of the direct current conductivity profile of a layered earth,” Geophysics 46, 1702–1713 (1981).
[CrossRef]

Cohen, J. K.

N. Bleistein, J. K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,”J. Math. Phys. 18, 194–201 (1977).
[CrossRef]

Devaney, A. J.

A. J. Devaney, G. C. Sherman, “Nonuniqueness in inverse source and scattering problems,”IEEE Trans. Antennas Propag. AP-30, 1034–1037 (1982).
[CrossRef]

A. J. Devaney, E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
[CrossRef]

Faddeyev, L. D.

L. D. Faddeyev, B. Seckler, “The inverse problem in the quantum theory of scattering,”J. Math. Phys. 4, 72–103 (1963).
[CrossRef]

Gelfand, I. M.

I. M. Gelfand, B. M. Levitan, “On the determination of a differential equation from its spectral measure function,” Am. Math. Soc. 1, 253–304 (1955).

Gopinath, B.

B. Gopinath, M. M. Sondhi, “Inversion of the telegraph equation and the synthesis of nonuniform lines,” Proc. IEEE 59, 383–392 (1971).
[CrossRef]

M. M. Sondhi, B. Gopinath, “Determination of vocal-tract shape from impulse response at the lips,”J. Acoust. Soc. Am. 49, 1867–1873 (1970).
[CrossRef]

Habashy, T. M.

T. M. Habashy, W. C. Chew, E. Y. Chow, “Simultaneous reconstruction of permittivity and conductivity profiles in a radially inhomogeneous slab,” Radio Sci. 21, 635–645 (1986).
[CrossRef]

T. M. Habashy, A. Sezginer, W. C. Chew, “Simultaneous inversion of radially varying conductivity and permittivity profiles,” presented at the International Geoscience and Remote Sensing Symposium, Amherst, Mass., October 7–9, 1985.

T. M. Habashy, R. Mittra, “On some inverse methods in electromagnetics,”J. Electromag. Waves Appl. (to be published).

R. Mittra, T. M. Habashy, “Profile inversion of radially inhomogeneous media with a single frequency measurement,” presented at the National Radio Science Meeting, Boston, Mass., June 25–29, 1984.

T. M. Habashy, R. Mittra, “Time domain profile inversion of a cylindrically stratified medium,” presented at the National Radio Science Meeting, Boston, Mass., June 25–29, 1984.

Jaulent, M.

M. Jaulent, “Inverse scattering problems in absorbing media,”J. Math. Phys. 17, 1351–1360 (1976).
[CrossRef]

Kailath, T.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

Kay, I.

I. Kay, H. E. Moses, Inverse Scattering Papers: 1955–1963, Vol. 12 of Lie Groups: History Frontiers and Applications (Mathematical Science, Mass.1982).

Kong, J. A.

J. A. Kong, Theory of Electromagnetic Waves (Wiley, New York, 1975).

Krueger, R. J.

R. J. Krueger, “Numerical aspects of a dissipative inverse problem,”IEEE Trans. Antennas Propag. AP-29, 253–261 (1981).
[CrossRef]

R. J. Krueger, “An inverse problem for a dissipative hyperbolic equation with discontinuous coefficients,”Q. Appl. Math. 36, 235–253 (1978).

R. J. Krueger, “An inverse problem for an absorbing medium with multiple discontinuities,”Q. Appl. Math. 34, 129–147 (1976).

Lamb, G. L.

G. L. Lamb, Elements of Soliton Theory (Wiley-Interscience, New York, 1980).

Lesselier, D.

D. Lesselier, “Optimization techniques and inverse problems: reconstruction of conductivity profiles in the time domain,”IEEE Trans. Antennas Propag. AP-30, 59–65 (1982).
[CrossRef]

Levitan, B. M.

I. M. Gelfand, B. M. Levitan, “On the determination of a differential equation from its spectral measure function,” Am. Math. Soc. 1, 253–304 (1955).

Levy, B. C.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

A. E. Yagle, B. C. Levy, “A fast algorithm solution of the inverse problem for a layered acoustic medium probed by spherical harmonic waves,”J. Acoust. Soc. Am. 78, 729–737 (1985).
[CrossRef]

A. E. Yagle, B. C. Levy, “The Schur algorithm and its applications,” Acta Appl. Math. 3, 255–284 (1985).
[CrossRef]

B. C. Levy, “Layer by layer reconstruction methods for the earth resistivity from direct current measurements,”IEEE Trans. Geosci. Remote Sensing GE-23, 841–850 (1985).
[CrossRef]

A. E. Yagle, B. C. Levy, “Application of the Schur algorithm to the inverse problem for a layered acoustic medium,”J. Acoust. Soc. Am. 76, 301–308 (1984).
[CrossRef]

Marchenko, V. A.

Z. S. Agranovich, V. A. Marchenko, The Inverse Problem of Scattering Theory (Gordon and Breach, New York, 1963).

Maystre, D.

A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,”J. Opt. (Paris) 9, 83–90 (1978).
[CrossRef]

Mei, K. K.

S. Coen, K. K. Mei, D. J. Angelakos, “Inverse scattering technique applied to remote sensing of layered media,”IEEE Trans. Antennas Propag. AP-29, 298–306 (1981).
[CrossRef]

Mittra, R.

D. H. Schaubert, R. Mittra, “A spectral domain method for remotely probing stratified media,”IEEE Trans. Antennas Propag. AP-25, 261–265 (1977).
[CrossRef]

R. Mittra, T. M. Habashy, “Profile inversion of radially inhomogeneous media with a single frequency measurement,” presented at the National Radio Science Meeting, Boston, Mass., June 25–29, 1984.

T. M. Habashy, R. Mittra, “Time domain profile inversion of a cylindrically stratified medium,” presented at the National Radio Science Meeting, Boston, Mass., June 25–29, 1984.

T. M. Habashy, R. Mittra, “On some inverse methods in electromagnetics,”J. Electromag. Waves Appl. (to be published).

Moses, H. E.

I. Kay, H. E. Moses, Inverse Scattering Papers: 1955–1963, Vol. 12 of Lie Groups: History Frontiers and Applications (Mathematical Science, Mass.1982).

Newton, R. G.

R. G. Newton, “Inversion of reflection data for layered media: a review of exact methods,” Geophys. J. R. Astron. Soc. 65, 191–215 (1981).
[CrossRef]

Robinson, E. A.

E. A. Robinson, “Dynamic predictive deconvolution,” Geophys. Prospect. 23, 779–797 (1975).
[CrossRef]

Roger, A.

A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,”J. Opt. (Paris) 9, 83–90 (1978).
[CrossRef]

Sabatier, P. C.

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer-Verlag, New York, 1977).
[CrossRef]

Santosa, F.

F. Santosa, H. Schwetlick, “The inversion of acoustical impedance profile by methods of characteristics,” Wave Motion 4, 99–110 (1982).
[CrossRef]

Schaubert, D. H.

D. H. Schaubert, R. Mittra, “A spectral domain method for remotely probing stratified media,”IEEE Trans. Antennas Propag. AP-25, 261–265 (1977).
[CrossRef]

D. H. Schaubert, “Spectral domain methods for remote probing of stratified media,” Ph.D. dissertation (University of Illinois, Urbana, Ill., 1974).

Schwetlick, H.

F. Santosa, H. Schwetlick, “The inversion of acoustical impedance profile by methods of characteristics,” Wave Motion 4, 99–110 (1982).
[CrossRef]

Seckler, B.

L. D. Faddeyev, B. Seckler, “The inverse problem in the quantum theory of scattering,”J. Math. Phys. 4, 72–103 (1963).
[CrossRef]

Sezginer, A.

A. Sezginer, “Forward and inverse problems in transient electromagnetic fields,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1985).

A. Sezginer, “Profile inversion by method of characteristics in a lossy, cylindrical medium,” presented at the National Radio Science Meeting, Boston, Mass., June 25–29, 1984.

T. M. Habashy, A. Sezginer, W. C. Chew, “Simultaneous inversion of radially varying conductivity and permittivity profiles,” presented at the International Geoscience and Remote Sensing Symposium, Amherst, Mass., October 7–9, 1985.

Sherman, G. C.

A. J. Devaney, G. C. Sherman, “Nonuniqueness in inverse source and scattering problems,”IEEE Trans. Antennas Propag. AP-30, 1034–1037 (1982).
[CrossRef]

Sondhi, M. M.

B. Gopinath, M. M. Sondhi, “Inversion of the telegraph equation and the synthesis of nonuniform lines,” Proc. IEEE 59, 383–392 (1971).
[CrossRef]

M. M. Sondhi, B. Gopinath, “Determination of vocal-tract shape from impulse response at the lips,”J. Acoust. Soc. Am. 49, 1867–1873 (1970).
[CrossRef]

Tabbara, W.

W. Tabbara, “Reconstruction of permittivity profiles from a spectral analysis of the reflection coefficient,”IEEE Trans. Antennas Propag. AP-27, 241–248 (1979).
[CrossRef]

Tijhuis, A. G.

A. G. Tijhuis, C. Van Der Worm, “Iterative approach to the frequency-domain solution of the inverse-scattering problem for an inhomogeneous lossless dielectric slab,”IEEE Trans. Antennas Propag. AP-32, 711–716 (1984).
[CrossRef]

A. G. Tijhuis, “Iterative determination of permittivity and conductivity profiles of a dielectric slab in the time domain,”IEEE Trans. Antennas Propag. AP-29, 239–245 (1981).
[CrossRef]

Twomey, S.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

Van Der Worm, C.

A. G. Tijhuis, C. Van Der Worm, “Iterative approach to the frequency-domain solution of the inverse-scattering problem for an inhomogeneous lossless dielectric slab,”IEEE Trans. Antennas Propag. AP-32, 711–716 (1984).
[CrossRef]

Weston, V. H.

V. H. Weston, “On inverse scattering,”J. Math. Phys. 15, 209–213 (1974).
[CrossRef]

V. H. Weston, “On the inverse problem for a hyperbolic dispersive partial differential equation,”J. Math. Phys. 13, 1952–1956 (1972).
[CrossRef]

Wolf, E.

A. J. Devaney, E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
[CrossRef]

Yagle, A. E.

A. E. Yagle, B. C. Levy, “A fast algorithm solution of the inverse problem for a layered acoustic medium probed by spherical harmonic waves,”J. Acoust. Soc. Am. 78, 729–737 (1985).
[CrossRef]

A. E. Yagle, B. C. Levy, “The Schur algorithm and its applications,” Acta Appl. Math. 3, 255–284 (1985).
[CrossRef]

A. E. Yagle, B. C. Levy, “Application of the Schur algorithm to the inverse problem for a layered acoustic medium,”J. Acoust. Soc. Am. 76, 301–308 (1984).
[CrossRef]

A. E. Yagle, “Layer stripping solutions of inverse seismic problems,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1985).

Acta Appl. Math.

A. E. Yagle, B. C. Levy, “The Schur algorithm and its applications,” Acta Appl. Math. 3, 255–284 (1985).
[CrossRef]

Am. Math. Soc.

I. M. Gelfand, B. M. Levitan, “On the determination of a differential equation from its spectral measure function,” Am. Math. Soc. 1, 253–304 (1955).

Geophys. J. R. Astron. Soc.

R. G. Newton, “Inversion of reflection data for layered media: a review of exact methods,” Geophys. J. R. Astron. Soc. 65, 191–215 (1981).
[CrossRef]

Geophys. Prospect.

E. A. Robinson, “Dynamic predictive deconvolution,” Geophys. Prospect. 23, 779–797 (1975).
[CrossRef]

Geophysics

S. Coen, “Density and compressibility profiles of a layered acoustic medium from precritical incidence data,” Geophysics 46, 1244–1246 (1981).
[CrossRef]

S. Coen, “Velocity and density profiles of a layered acoustic medium from common source-point data,” Geophysics 47, 898–905 (1982).
[CrossRef]

S. Coen, “The inverse problem of the direct current conductivity profile of a layered earth,” Geophysics 46, 1702–1713 (1981).
[CrossRef]

IEEE Trans. Antennas Propag.

D. H. Schaubert, R. Mittra, “A spectral domain method for remotely probing stratified media,”IEEE Trans. Antennas Propag. AP-25, 261–265 (1977).
[CrossRef]

R. J. Krueger, “Numerical aspects of a dissipative inverse problem,”IEEE Trans. Antennas Propag. AP-29, 253–261 (1981).
[CrossRef]

S. Coen, “Inverse scattering of a layered and dispersionless dielectric half-space, part I: reflection data from plane waves at normal incidence,”IEEE Trans. Antennas Propag. AP-29, 726–732 (1981).
[CrossRef]

W. Tabbara, “Reconstruction of permittivity profiles from a spectral analysis of the reflection coefficient,”IEEE Trans. Antennas Propag. AP-27, 241–248 (1979).
[CrossRef]

A. J. Devaney, G. C. Sherman, “Nonuniqueness in inverse source and scattering problems,”IEEE Trans. Antennas Propag. AP-30, 1034–1037 (1982).
[CrossRef]

S. Coen, K. K. Mei, D. J. Angelakos, “Inverse scattering technique applied to remote sensing of layered media,”IEEE Trans. Antennas Propag. AP-29, 298–306 (1981).
[CrossRef]

A. G. Tijhuis, C. Van Der Worm, “Iterative approach to the frequency-domain solution of the inverse-scattering problem for an inhomogeneous lossless dielectric slab,”IEEE Trans. Antennas Propag. AP-32, 711–716 (1984).
[CrossRef]

D. Lesselier, “Optimization techniques and inverse problems: reconstruction of conductivity profiles in the time domain,”IEEE Trans. Antennas Propag. AP-30, 59–65 (1982).
[CrossRef]

A. G. Tijhuis, “Iterative determination of permittivity and conductivity profiles of a dielectric slab in the time domain,”IEEE Trans. Antennas Propag. AP-29, 239–245 (1981).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing

B. C. Levy, “Layer by layer reconstruction methods for the earth resistivity from direct current measurements,”IEEE Trans. Geosci. Remote Sensing GE-23, 841–850 (1985).
[CrossRef]

Inverse Problems in Propagation and Scattering, feature section of J. Opt. Soc. Am. A

A. J. Devaney, ed., Inverse Problems in Propagation and Scattering, feature section of J. Opt. Soc. Am. A 2, 1902–2061 (1985).

J. Acoust. Soc. Am.

A. E. Yagle, B. C. Levy, “Application of the Schur algorithm to the inverse problem for a layered acoustic medium,”J. Acoust. Soc. Am. 76, 301–308 (1984).
[CrossRef]

A. E. Yagle, B. C. Levy, “A fast algorithm solution of the inverse problem for a layered acoustic medium probed by spherical harmonic waves,”J. Acoust. Soc. Am. 78, 729–737 (1985).
[CrossRef]

S. Coen, “On the elastic profiles of a layered medium from reflection data. Part I. Plane-wave sources,”J. Acoust. Soc. Am. 70, 172–175 (1981).
[CrossRef]

M. M. Sondhi, B. Gopinath, “Determination of vocal-tract shape from impulse response at the lips,”J. Acoust. Soc. Am. 49, 1867–1873 (1970).
[CrossRef]

J. Math. Phys.

G. N. Balanis, “The plasma inverse problem,”J. Math. Phys. 13, 1001–1005 (1972).
[CrossRef]

M. Jaulent, “Inverse scattering problems in absorbing media,”J. Math. Phys. 17, 1351–1360 (1976).
[CrossRef]

V. H. Weston, “On the inverse problem for a hyperbolic dispersive partial differential equation,”J. Math. Phys. 13, 1952–1956 (1972).
[CrossRef]

V. H. Weston, “On inverse scattering,”J. Math. Phys. 15, 209–213 (1974).
[CrossRef]

N. Bleistein, J. K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,”J. Math. Phys. 18, 194–201 (1977).
[CrossRef]

S. Coen, “Inverse scattering of the permittivity and permeability profiles of a plane stratified medium,”J. Math. Phys. 22, 1127–1129 (1981).
[CrossRef]

L. D. Faddeyev, B. Seckler, “The inverse problem in the quantum theory of scattering,”J. Math. Phys. 4, 72–103 (1963).
[CrossRef]

J. Opt. (Paris)

A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,”J. Opt. (Paris) 9, 83–90 (1978).
[CrossRef]

Phys. Rev. D

A. J. Devaney, E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
[CrossRef]

Proc. IEEE

B. Gopinath, M. M. Sondhi, “Inversion of the telegraph equation and the synthesis of nonuniform lines,” Proc. IEEE 59, 383–392 (1971).
[CrossRef]

Q. Appl. Math.

R. J. Krueger, “An inverse problem for an absorbing medium with multiple discontinuities,”Q. Appl. Math. 34, 129–147 (1976).

R. J. Krueger, “An inverse problem for a dissipative hyperbolic equation with discontinuous coefficients,”Q. Appl. Math. 36, 235–253 (1978).

Radio Sci.

T. M. Habashy, W. C. Chew, E. Y. Chow, “Simultaneous reconstruction of permittivity and conductivity profiles in a radially inhomogeneous slab,” Radio Sci. 21, 635–645 (1986).
[CrossRef]

SIAM J. Appl. Math.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

SIAM Rev.

K. B. Bube, R. Burridge, “The one-dimensional inverse problem of reflection seismology,” SIAM Rev. 25, 497–559 (1983).
[CrossRef]

Wave Motion

F. Santosa, H. Schwetlick, “The inversion of acoustical impedance profile by methods of characteristics,” Wave Motion 4, 99–110 (1982).
[CrossRef]

R. Burridge, “The Gelfand–Levitan, the Marchenko, and the Gopinath–Sondhi integral equations of inverse scattering theory, regarded in the context of inverse impulse-response problems,” Wave Motion 2, 305–323 (1980).
[CrossRef]

Other

R. Mittra, T. M. Habashy, “Profile inversion of radially inhomogeneous media with a single frequency measurement,” presented at the National Radio Science Meeting, Boston, Mass., June 25–29, 1984.

T. M. Habashy, R. Mittra, “Time domain profile inversion of a cylindrically stratified medium,” presented at the National Radio Science Meeting, Boston, Mass., June 25–29, 1984.

D. H. Schaubert, “Spectral domain methods for remote probing of stratified media,” Ph.D. dissertation (University of Illinois, Urbana, Ill., 1974).

A. Sezginer, “Forward and inverse problems in transient electromagnetic fields,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1985).

A. Sezginer, “Profile inversion by method of characteristics in a lossy, cylindrical medium,” presented at the National Radio Science Meeting, Boston, Mass., June 25–29, 1984.

T. M. Habashy, A. Sezginer, W. C. Chew, “Simultaneous inversion of radially varying conductivity and permittivity profiles,” presented at the International Geoscience and Remote Sensing Symposium, Amherst, Mass., October 7–9, 1985.

A. E. Yagle, “Layer stripping solutions of inverse seismic problems,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1985).

W. C. Chew, S. L. Chuang, “Profile inversion of a planar medium with a line source or a point source,” presented at the International Geoscience and Remote Sensing Symposium, Strasbourg, France, August 27–30, 1984.

T. M. Habashy, R. Mittra, “On some inverse methods in electromagnetics,”J. Electromag. Waves Appl. (to be published).

J. A. Kong, Theory of Electromagnetic Waves (Wiley, New York, 1975).

C. T. H. Baker, The Numerical Treatment of Integral Equations (Clarendon, Oxford, 1977).

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

P. C. Sabatier, ed., Applied Inverse Problems (Springer-Verlag, Berlin, 1978).
[CrossRef]

Z. S. Agranovich, V. A. Marchenko, The Inverse Problem of Scattering Theory (Gordon and Breach, New York, 1963).

I. Kay, H. E. Moses, Inverse Scattering Papers: 1955–1963, Vol. 12 of Lie Groups: History Frontiers and Applications (Mathematical Science, Mass.1982).

G. L. Lamb, Elements of Soliton Theory (Wiley-Interscience, New York, 1980).

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer-Verlag, New York, 1977).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

The two characteristic curves for the wave equation.

Fig. 2
Fig. 2

Geometrical configuration of the inversion problem for the method of characteristics and the distorted Born approach.

Fig. 3
Fig. 3

The results of the simultaneous inversion of the wave speed and loss profiles, obtained by using the method of characteristics.11,13 (a) The speed profile. (b) The two-parameter inversion for the loss profile.

Fig. 4
Fig. 4

The recursive patterns for the continuation of (a) the downward-propagating component D(τ, t) and (b) the upward-propagating component U(τ, t).

Fig. 5
Fig. 5

Plots of the actual and reconstructed acoustic wave speeds retrieved by using the Cholesky algorithm.3

Fig. 6
Fig. 6

Flow chart of the distorted Born inversion algorithm. FIE, field integral equation.

Fig. 7
Fig. 7

Simultaneous reconstruction of sinusoidally varying permittivity and conductivity profiles with a hump by the use of the distorted Born approach.21 The contrast is 1:40; the annulus thickness is 30 cm. (a) Dielectric constant. (b) Loss tangent.

Fig. 8
Fig. 8

Simultaneous reconstruction of sinusoidally varying permittivity and conductivity profiles with a dip. The contrast is 1:40; the annulus thickness is 30 cm. (a) Dielectric constant. (b) Loss tangent.

Fig. 9
Fig. 9

Reconstruction of a step profile. The contrast is 1:40; the annulus thickness is 30 cm. (a) Dielectric constant. (b) Loss tangent.

Equations (78)

Equations on this page are rendered with MathJax. Learn more.

2 x 2 E y + 2 z 2 E y - 1 c 2 2 c 2 E y - 1 c 2 T t E y = 0 ,
E y ( x , z = z m , t ) = e ( x , t ) ,
E y z ( x , z = z m , t ) = μ 0 h t ( x , t ) ,
c = [ μ 0 ( z ) ] - 1 / 2 ,
T = ( z ) / σ ( z ) .
E y ( x , z , t ) = 0             for t 0 ,
z E y ( x , z , t ) = 0             for t 0.
E y ( x , z , t ) = - d k x exp ( i k x x ) exp [ k x ( z - z m ) ] u ( k x , z , t ) .
d τ d z = 1 c ( z ) .
u τ τ - u t t + ( 2 k x c - c τ c ) u τ - 1 T u t = 0 ,
u t ( k x , τ = 0 , t ) = e ^ t ( k x , t ) ,
u τ ( k x , τ = 0 , t ) = c ( τ = 0 ) [ μ 0 h ^ t ( k x , t ) - k x e ^ ( k x , t ) ]
u t ( k x , τ , t ) = 0             for t 0 ,
u τ ( k x , τ , t ) = 0             for t 0.
C ± :             d τ d t = ± 1.
U ¯ ( τ + Δ , t ) = ½ A ¯ · U ¯ ( τ , t + Δ ) + ½ B ¯ · U ¯ ( τ , t - Δ ) ,
U ¯ ( τ , t ) = [ u τ ( τ , t ) u t ( τ , t ) ] ,
A 11 = A 21 = B 21 = - B 11 = 1 - [ 2 k x c ( τ ) - r ( τ ) ] Δ ,
A 12 = A 22 = 1 + Δ T ( τ ) ,
- B 12 = B 22 = 1 - Δ T ( τ ) ,
r ( τ ) = c τ ( τ ) c ( τ ) .
{ 1 - [ 2 k x c ( τ ) - r ( τ ) ] Δ } u τ ( τ , τ + 2 Δ ) + [ 1 + Δ T ( τ ) ] × u t ( τ , τ + 2 Δ ) = 0 ,
c ( τ + Δ ) = 2 c 2 ( τ ) R 1 k x ( 1 ) - k x ( 2 ) R 2 R 1 - R 2 Δ ,
T ( τ ) = Δ [ 2 c ( τ ) k x ( 1 ) - k x ( 2 ) R 1 - 1 - R 2 - 1 - 1 ] - 1 ,
R j = u τ ( τ , τ + 2 Δ ) u t ( τ , τ + 2 Δ ) ,
E z = μ 0 H t ,
μ 0 2 H z t = μ 0 2 E t 2 + μ 0 σ E t - 2 E x 2 .
e ( p , z , t ) = - d x E ( x , z , t - p x ) ,
h ( p , z , t ) = - d x H ( x , z , t - p x ) .
e z = μ 0 h t ,
μ 0 h z = 1 v 2 e t + μ 0 σ e ,
v v ( z , p ) = [ μ 0 ( z ) - p 2 ] - 1 / 2 ,
d τ d z = 1 v ( z , p ) .
D = 1 η e - η h ,
U = 1 η e + η h ,
D τ + D t = - 1 2 r U - 1 2 T ( U + D ) ,
U τ - U t = - 1 2 r D + 1 2 T ( U + D ) ,
r = v τ v ,
T = 1 μ 0 v 2 σ .
U ( τ , t ) = 0 = D ( τ , t ) ,             t τ .
D ( τ + Δ , t + Δ ) = [ 1 - 1 2 Δ T ( τ ) ] D ( τ , t ) - Δ 2 [ r ( τ ) + 1 T ( τ ) ] U ( τ , t ) ,
U ( τ + Δ , t - Δ ) = [ 1 + 1 2 Δ T ( τ ) ] U ( τ , t ) - Δ 2 [ r ( τ ) - 1 T ( τ ) ] D ( τ , t ) .
v ( τ + Δ ) = v ( τ ) { [ 1 + Δ T ( τ ) ] + 2 [ 1 + 1 2 Δ T ( τ ) ] U ( τ , τ + 2 Δ ) D ( τ , τ + 2 Δ ) } ,
E y = E y ( 0 ) + z i z 0 d z k 0 2 δ - d x G ( 0 ) ( x , z , x , z ) E y ( x , z ) ,
δ = c ( z ) - c ( 0 ) ( z )
z i z 0 d z Q ( z ) F ¯ ( z ) = H ¯ ,
F ¯ ( z ) = { F ( x i , z ) , i = 1 , 2 , , N } , H ¯ = { H ( x i ) , i = 1 , 2 , , N } , F ( x , z ) = k 0 2 - d x G ( 0 ) ( x , z , x , z m ) E y ( 0 ) ( x , z ) ,
H ( x ) = E y ( x , z m ) - E y ( 0 ) ( x , z m ) ,
z i z 0 d z Q NR ( z ) F ¯ ( z ) = 0.
C ( Q ) = | H ¯ - z i z 0 d z Q ( z ) F ¯ ( z ) | 2 + μ z i z 0 d z | d n Q ( z ) d z n | 2 .
( - 1 ) n μ d 2 n Q μ ( z ) d z 2 n + F ¯ + ( z ) · z i z 0 d z Q μ ( z ) F ¯ ( z ) = F ¯ + ( z ) · H ¯ ,
d n Q μ ( z ) d z n = d n + 1 Q μ ( z ) d z n + 1 = = d 2 n - 1 Q μ ( z ) d z 2 n - 1 = 0 ,             at z = z i , z 0 .
d τ d z = 1 c .
e ( τ , t ) = 1 η E ( τ , t ) ,
h ( τ , t ) = η H ( τ , t ) ,
e τ τ - e t t - V e = 0 ,
e ( τ = 0 , t ) = δ ( t ) + R ( t ) ,
e τ ( τ = 0 , t ) = - δ ( t ) + r ( t )
e ( τ , t ) = 0             for t < 0 e ( τ , t ) = 0             for t < τ ,
V = q τ τ / q ,
q = [ c ( τ ) ] - 1 / 2 .
q τ τ - V q = 0 ,
q ( τ = 0 ) = q 0 ,
q τ ( τ = 0 ) = 0.
h ( τ = 0 , t ) = - δ ( t ) + r ( t ) .
e 1 ( τ , t ) = δ ( t - τ ) + e ˜ 1 ( τ , t )
e 2 ( τ , t ) = δ ( t + τ ) + e ˜ 2 ( τ , t ) ,
e ˜ 1 ( τ , t ) + S 1 ( t - τ ) + S 2 ( t + τ ) + - τ τ d t [ S 1 ( t - t ) + S 2 ( t + t ) ] e ˜ 1 ( τ , t ) = 0 ,             t τ ,
S 1 ( t ) = ½ [ R ( t ) - r ( t ) ] ,
S 2 ( t ) = ½ [ R ( t ) + r ( t ) ] .
2 d e ˜ 1 d τ ( τ , τ + ) = V ( τ ) ,
C + :             d τ d t = + 1.
r ( t ) = 0.
V ( τ ) = 2 d d τ K ( τ , τ + ) ,
K ( τ , t ) + ½ [ R ( τ - t ) + R ( τ + t ) ] + ½ 0 τ d t [ R ( t - t ) + R ( t + t ) ] K ( τ , t ) = 0 ,             0 t τ .
r ( t ) = R ( t ) .
V ( τ ) = 2 d d τ K ( τ , τ + ) ,
K ( τ , t ) + R ( τ + t ) + - t τ d t R ( t + t ) K ( τ , t ) = 0 ,             t τ .

Metrics