Abstract

We discuss the role of uncertainties in the knowledge of the background in inverse scattering for a buried object under the distorted Born approximation. In particular, we focus on the role played by inaccuracy in the knowledge of the dielectric permittivity of the host medium, with reference to both a lossless half-space and a lossless three-layered medium. This investigation allows us to show how reconstruction of an inhomogeneity in a three-layered medium is more critical than in the case of a half-space (two-layered) geometry.

© 2004 Optical Society of America

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References

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  1. E. Nyfors, “Industrial microwave sensors,” Subsurf. Sens. Technol. Appl. 1, 23–43 (2000).
    [CrossRef]
  2. J. Daniels, Surface Penetrating Radar (Institution of Electrical Engineers, London, 1996).
  3. D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).
  4. A. N. Tichonov, V. Y. Arsenine, Solution of Ill-Posed Problems (Winston, Washington, D.C., 1977).
  5. R. Persico, F. Soldovieri, R. Pierri, “On the convergence properties of a quadratic approach to the inverse scattering problem,” J. Opt. Soc. Am. A 19, 2424–2428 (2002).
    [CrossRef]
  6. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J., 1995).
  7. M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
    [CrossRef]
  8. B. Chen, J. Stamnes, “Validity of diffraction tomography based on first Born and first Rytov approximations,” Appl. Opt. 37, 2996–3006 (1998).
    [CrossRef]
  9. G. Leone, R. Persico, R. Pierri, “Inverse scattering under the distorted Born approximation for cylindrical geometries,” J. Opt. Soc. Am. A 16, 1779–1787 (1999); Errata, 16, 2310 (1999).
    [CrossRef]
  10. I. Akudman, M. Idemen, “On the use of Gaussian beams in one-dimensional profile inversion connected with lossy dielectric slabs,” Inverse Probl. 11, 315–328 (1995).
    [CrossRef]
  11. T. B. Hansen, M. Johansen, “Inversion scheme for ground penetrating radar that takes into account the planar air-soil interface,” IEEE Trans. Geosci. Remote Sens. 38, 496–506 (2000).
    [CrossRef]
  12. R. Persico, F. Soldovieri, “One-dimensional inverse scattering with a Born model in a three-layered medium,” J. Opt. Soc. Am. A 21, 35–45 (2004).
    [CrossRef]
  13. F. Soldovieri, R. Persico, “Reconstruction of an embedded slab with Born approximation from multifrequency data,” IEEE Trans. Antennas Propag. 13, 2348–2356 (2004).
    [CrossRef]
  14. Similar considerations can be applied to objects embedded in a very (optically) dense layer within a stratified medium.
  15. S. E. Hamran, D. T. Gjessing, J. Hielmstad, E. Aarholt, “Ground penetrating radar synthetic pulse radar: dynamic range and modes of operation,” J. Appl. Geophys. 33, 7–14 (1995).
    [CrossRef]
  16. P. D. Walker, M. R. Bell, “Subsurface permittivity estimation from ground penetrating radar measurements,” in Proceedings of the IEEE International Radar Conference: Radar 2000 (IEEE Press, Piscataway, N.J., 2000), pp. 341–346.
  17. Rigorously, some frequency weighting of the second member of Eq. (10) should be taken into account too, but we neglect it here.
  18. M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, London, 1998).
  19. We avoid showing explicitly this test for sake of brevity.
  20. D. Pozar, Microwave Engineering (Wiley, New York, 1997).

2004 (2)

R. Persico, F. Soldovieri, “One-dimensional inverse scattering with a Born model in a three-layered medium,” J. Opt. Soc. Am. A 21, 35–45 (2004).
[CrossRef]

F. Soldovieri, R. Persico, “Reconstruction of an embedded slab with Born approximation from multifrequency data,” IEEE Trans. Antennas Propag. 13, 2348–2356 (2004).
[CrossRef]

2002 (1)

2000 (2)

E. Nyfors, “Industrial microwave sensors,” Subsurf. Sens. Technol. Appl. 1, 23–43 (2000).
[CrossRef]

T. B. Hansen, M. Johansen, “Inversion scheme for ground penetrating radar that takes into account the planar air-soil interface,” IEEE Trans. Geosci. Remote Sens. 38, 496–506 (2000).
[CrossRef]

1999 (1)

1998 (1)

1995 (2)

I. Akudman, M. Idemen, “On the use of Gaussian beams in one-dimensional profile inversion connected with lossy dielectric slabs,” Inverse Probl. 11, 315–328 (1995).
[CrossRef]

S. E. Hamran, D. T. Gjessing, J. Hielmstad, E. Aarholt, “Ground penetrating radar synthetic pulse radar: dynamic range and modes of operation,” J. Appl. Geophys. 33, 7–14 (1995).
[CrossRef]

1984 (1)

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Aarholt, E.

S. E. Hamran, D. T. Gjessing, J. Hielmstad, E. Aarholt, “Ground penetrating radar synthetic pulse radar: dynamic range and modes of operation,” J. Appl. Geophys. 33, 7–14 (1995).
[CrossRef]

Akudman, I.

I. Akudman, M. Idemen, “On the use of Gaussian beams in one-dimensional profile inversion connected with lossy dielectric slabs,” Inverse Probl. 11, 315–328 (1995).
[CrossRef]

Arsenine, V. Y.

A. N. Tichonov, V. Y. Arsenine, Solution of Ill-Posed Problems (Winston, Washington, D.C., 1977).

Bell, M. R.

P. D. Walker, M. R. Bell, “Subsurface permittivity estimation from ground penetrating radar measurements,” in Proceedings of the IEEE International Radar Conference: Radar 2000 (IEEE Press, Piscataway, N.J., 2000), pp. 341–346.

Bertero, M.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, London, 1998).

Boccacci, P.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, London, 1998).

Chen, B.

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J., 1995).

Colton, D.

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).

Daniels, J.

J. Daniels, Surface Penetrating Radar (Institution of Electrical Engineers, London, 1996).

Gjessing, D. T.

S. E. Hamran, D. T. Gjessing, J. Hielmstad, E. Aarholt, “Ground penetrating radar synthetic pulse radar: dynamic range and modes of operation,” J. Appl. Geophys. 33, 7–14 (1995).
[CrossRef]

Hamran, S. E.

S. E. Hamran, D. T. Gjessing, J. Hielmstad, E. Aarholt, “Ground penetrating radar synthetic pulse radar: dynamic range and modes of operation,” J. Appl. Geophys. 33, 7–14 (1995).
[CrossRef]

Hansen, T. B.

T. B. Hansen, M. Johansen, “Inversion scheme for ground penetrating radar that takes into account the planar air-soil interface,” IEEE Trans. Geosci. Remote Sens. 38, 496–506 (2000).
[CrossRef]

Hielmstad, J.

S. E. Hamran, D. T. Gjessing, J. Hielmstad, E. Aarholt, “Ground penetrating radar synthetic pulse radar: dynamic range and modes of operation,” J. Appl. Geophys. 33, 7–14 (1995).
[CrossRef]

Idemen, M.

I. Akudman, M. Idemen, “On the use of Gaussian beams in one-dimensional profile inversion connected with lossy dielectric slabs,” Inverse Probl. 11, 315–328 (1995).
[CrossRef]

Johansen, M.

T. B. Hansen, M. Johansen, “Inversion scheme for ground penetrating radar that takes into account the planar air-soil interface,” IEEE Trans. Geosci. Remote Sens. 38, 496–506 (2000).
[CrossRef]

Kak, A. C.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Kress, R.

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).

Larsen, L. E.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Leone, G.

Nyfors, E.

E. Nyfors, “Industrial microwave sensors,” Subsurf. Sens. Technol. Appl. 1, 23–43 (2000).
[CrossRef]

Persico, R.

Pierri, R.

Pozar, D.

D. Pozar, Microwave Engineering (Wiley, New York, 1997).

Slaney, M.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Soldovieri, F.

Stamnes, J.

Tichonov, A. N.

A. N. Tichonov, V. Y. Arsenine, Solution of Ill-Posed Problems (Winston, Washington, D.C., 1977).

Walker, P. D.

P. D. Walker, M. R. Bell, “Subsurface permittivity estimation from ground penetrating radar measurements,” in Proceedings of the IEEE International Radar Conference: Radar 2000 (IEEE Press, Piscataway, N.J., 2000), pp. 341–346.

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

F. Soldovieri, R. Persico, “Reconstruction of an embedded slab with Born approximation from multifrequency data,” IEEE Trans. Antennas Propag. 13, 2348–2356 (2004).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

T. B. Hansen, M. Johansen, “Inversion scheme for ground penetrating radar that takes into account the planar air-soil interface,” IEEE Trans. Geosci. Remote Sens. 38, 496–506 (2000).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Inverse Probl. (1)

I. Akudman, M. Idemen, “On the use of Gaussian beams in one-dimensional profile inversion connected with lossy dielectric slabs,” Inverse Probl. 11, 315–328 (1995).
[CrossRef]

J. Appl. Geophys. (1)

S. E. Hamran, D. T. Gjessing, J. Hielmstad, E. Aarholt, “Ground penetrating radar synthetic pulse radar: dynamic range and modes of operation,” J. Appl. Geophys. 33, 7–14 (1995).
[CrossRef]

J. Opt. Soc. Am. A (3)

Subsurf. Sens. Technol. Appl. (1)

E. Nyfors, “Industrial microwave sensors,” Subsurf. Sens. Technol. Appl. 1, 23–43 (2000).
[CrossRef]

Other (10)

J. Daniels, Surface Penetrating Radar (Institution of Electrical Engineers, London, 1996).

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).

A. N. Tichonov, V. Y. Arsenine, Solution of Ill-Posed Problems (Winston, Washington, D.C., 1977).

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J., 1995).

P. D. Walker, M. R. Bell, “Subsurface permittivity estimation from ground penetrating radar measurements,” in Proceedings of the IEEE International Radar Conference: Radar 2000 (IEEE Press, Piscataway, N.J., 2000), pp. 341–346.

Rigorously, some frequency weighting of the second member of Eq. (10) should be taken into account too, but we neglect it here.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, London, 1998).

We avoid showing explicitly this test for sake of brevity.

D. Pozar, Microwave Engineering (Wiley, New York, 1997).

Similar considerations can be applied to objects embedded in a very (optically) dense layer within a stratified medium.

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Figures (9)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Actual contrast corresponding to an (a) exact and (b) erroneous estimate of the background permittivity.

Fig. 3
Fig. 3

‖ΔΓ‖ versus εr2e/εr2e. Solid curve, 1-m slab; dashed curve, a half-space with the same permittivity. (a) εr2a=9, frequency band 50–1050 MHz; (b) εr2a=5, frequency band 50–1050 MHz; (c) εr2a=5, frequency band 50–500 MHz; (d) εr2a=9, frequency band 50–500 MHz.

Fig. 4
Fig. 4

Reconstructed relative permittivity versus z (meters) in a half-space case with εr2a=9. (a) εr2e=9, noiseless data; (b) εr2e=9.5, ‖ΔΓ‖ neglected, noiseless data; (c) εr2e=9.5, ‖ΔΓ‖ accounted for, noiseless data; (d) εr2e=9.5, ‖ΔΓ‖ accounted for, noisy data with a SNR of 40 dB with respect to the total field.

Fig. 5
Fig. 5

Reconstructed relative permittivity versus z (meters) in a half-space case with εr2a=9. (a) εr2e=9, noiseless data; (b) εr2e=7, ‖ΔΓ‖ neglected, noiseless data; (c) εr2e=7, ‖ΔΓ‖ accounted for, noiseless data; (d) εr2e=7, ‖ΔΓ‖ accounted for, noisy data with a SNR of 40 dB with respect to the total field.

Fig. 6
Fig. 6

Reconstructed relative permittivity versus z (meters) in a slab case with εr2a=9. (a) εr2e=9, noiseless data; (b) εr2e=9.5, ‖ΔΓ‖ neglected, noiseless data; (c) εr2e=9.5, ‖ΔΓ‖ accounted for, noiseless data; (d) εr2e=9.5, ‖ΔΓ‖ accounted for, noisy data with a SNR of 40 dB with respect to the total field.

Fig. 7
Fig. 7

Reconstructed relative permittivity versus z (meters) in a slab case with εr2a=9. (a) εr2e=9, noiseless data; (b) εr2e=7, ‖ΔΓ‖ neglected, noiseless data; (c) εr2e=7, ‖ΔΓ‖ accounted for, noiseless data; (d) εr2e=7, ‖ΔΓ‖ accounted for, noisy data with a SNR of 40 dB with respect to the total field.

Fig. 8
Fig. 8

Reconstructed relative permittivity versus z (meters) in a slab matched at the second interface. εr2a=9, εr2e=7, εr3a=εr3e=7.

Fig. 9
Fig. 9

Reconstructed relative permittivity versus z (meters) in the half-space (panel a) and slab (panel b) with εr2a=9 and εr2e=7.

Equations (31)

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Es(ω)=k22-Ge(z, ω)Einc(z, ω)χ(z)dz,
ωΩ,
χ(z)=εr(z)εr2-1,zD0elsewhere
Etot=E0+ΓE0+Es,
E˜s=Etot(εr2a)-Γ(εr2e)E0-E0.
E˜s=Etot(εr2a)-E0Γ(εr2a)-E0+E0Γ(εr2a)-E0Γ(εr2e)=Es(εr2a)-E0ΔΓ,
A(χ)=χˆ(2k0εr2),
χ^e(2k0εr2e)=εr2aεr2e(1+εr2e)2(1+εr2a)2 χ^a(2k0εr2a),
k0
χe(z)=εr2eεr2aεr2aεr2e(1+εr2e)2(1+εr2a)2 χazεr2eεr2a=f(εr2a)f(εr2e) χazεr2eεr2a,
Γ(εr2)=1-εr21+εr2.
χ^e(2k0εr2e)+2Γ23eexp(-j2k0dεr2e)χ^e(0)+Γ23e2exp(-j4k0dεr2e)χ^e(-2k0εr2e)=χ^a(2k0εr2a)+2Γ23aexp(-j2k0dεr2a)χ^a(0)+Γ23a2exp(-j4k0dεr2a)χ^a(-2k0εr2a),k0,
χ^e(0)=χ^a(0)1+Γ23a1+Γ23e2.
χe(z)+2Γ23eχ^a(0)1+Γ23a1+Γ23e2δ(z-d)+Γ23e2χe(2d-z)=εr2eεr2a χazεr2eεr2a+2Γ23aχ^a(0)×δz-dεr2aεr2e+Γ23a2εr2eεr2a χa2d-zεr2eεr2a.
χe(z)=εr2eεr2a χazεr2eεr2a+2Γ23aχ^a(0)×δz-dεr2aεr2e-2Γ23eχ^a(0)1+Γ23a1+Γ23e2δ(z-d)+Γ23a2εr2eεr2a χa2d-zεr2eεr2a,
z(0, d).
Γ=j(1-εr2)tg(dk0εr2)2εr2+j(1+εr2)tg(dk0εr2).
Einc(z, ω)=2E01+εr2exp(-jεr2k0z),
zD,ωΩ,
Ge(z, ω)=-j1+εr2exp(-jεr2k0z),
zD,ωΩ,
Es(ω)=-j2E0εr2k02(1+εr2)2 χˆ(2εr2k0),ωΩ,
-j2E0(ω)εr2k02(1+εr2)2=1ω.
Einc(z, ω)=2E0[exp(-jεr2k0z)+Γ23exp(-j2εr2k0d)exp(jεr2k0z)][1+Γ232exp(-2jεr2k0d)](1+εr2),zD,ωΩ,
Ge(z, ω)=-j[exp(-jεr2k0z)+Γ23exp(-j2εr2k0d)exp(jεr2k0z)][1+Γ232exp(-2jεr2k0d)](1+εr2),zD,ωΩ,
Es(ω)=-j2E0εr2k02(1+εr2)2[1+Γ232exp(-2jεr2k0d)]2 ×[χˆ(2εr2k0)+2Γ23exp(-j2εr2k0d)χˆ(0)+Γ232exp(-j4εr2k0d)χˆ(-2εr2k0)],
ωΩ
χˆ(2εr2k0)+2Γ23exp(-j2εr2k0d)χˆ(0)+Γ232exp(-j4εr2k0d)χˆ(-2εr2k0)
-j2E0(ω)εr2k02(1+εr2)2[1+Γ232exp(-2jεr2k0d)]2=1ω.
Es=Etot-E0-E0Γ(εr2e)=E0+E0Γ(εr2a)-E0-E0Γ(εr2e)=E0ΔΓ.
Γ1(εr2a)Γ(εr2a)+εr2a-εr2eεr2a+εr2eexp(-j2k0dεr2a).

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