Abstract

The Wigner distribution function is investigated as a signal processing tool to detect subsurface targets closely located beneath a randomly rough surface. Information provided by a bistatic arrangement of sources and detectors can be used to discriminate target and surface response based on their scattering behavior. It is shown that the bilinearity of the Wigner distribution function can be exploited for nonlinear amplification of the target response. This is achieved by averaging the Wigner distribution of the detected signal for different source locations. Target detection is further improved by numerically backpropagating the detected signal to the surface. A statistical evaluation based on simulated data sets is used to evaluate the performance of the detection method.

© 2009 Optical Society of America

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References

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  1. G. Matz and F. Hlawatsch, “Wigner distributions (nearly) everywhere: time-frequency analysis of signals, systems, random processes, signal spaces, and frames,” Signal Process. 83, 1355-1378 (2003).
    [CrossRef]
  2. P. Flandrin, Time-Frequency/Time-Scale Analysis (Academic, 1999).
  3. A. Torre, Linear Ray and Wave Optics (Elesevier, 2005).
  4. H. Brunzell, “Detection of shallowly buried objects using impulse radar,” IEEE Trans. Geosci. Remote Sens. 37, 875-886 (1999).
    [CrossRef]
  5. J. D. Kekis, M. Testorf, M. A. Fiddy, and R. H. Giles, “Detecting 1/10th scaled structures in dielectric media using monostatic X-band radar scattering measurements,” Proc. SPIE 4123, 13-24 (2000).
    [CrossRef]
  6. T. Dogaru and L. Carin, “Time-domain sensing of targets buried under a Gaussian, exponential, or fractal rough interface,” IEEE Trans. Geosci. Remote Sens. 39, 1807-1819 (2001).
    [CrossRef]
  7. D. Potin, B. Duflos, and P. Vanheeghe, “Landmines ground-penetrating radar signal enhancement by digital filtering,” IEEE Trans. Geosci. Remote Sens. 44, 2393-2406 (2006).
    [CrossRef]
  8. P. D. Gader, M. Mystkowski, and Y. Zhao, “Landmine detection with ground penetrating radar using hidden Markov models,” IEEE Trans. Geosci. Remote Sens. 39, 1231-1244 (2001).
    [CrossRef]
  9. O. Cmielewski, M. Saillard, K. Belkebir, and H. Tortel, “On the characterization of buried targets under a rough surface using the Wigner-Ville transform,” IEEE Geosci. Remote Sens. Lett. 3, 442-446 (2006).
    [CrossRef]
  10. O. Cmielewski, M. Saillard, and H. Tortel, “Detection of buried objects beneath a rough surface,” Waves Random Complex Media 16, 417-431 (2006).
    [CrossRef]
  11. M. Nieto-Vesperinas and J. A. Sánchez-Gil, “Intensity angular correlations of light multiply scattered from random rough surfaces,” J. Opt. Soc. Am. A 10, 150-157 (1993).
    [CrossRef]
  12. T.-K. Chan, Y. Kuga, and A. Ishimary, “Subsurface detection of a buried object using angular correlation function measurements,” Waves Random Complex Media 7, 457-465 (1997).
  13. G. Zhang and L. Tsang, “Application of angular correlation function of clutter scattering and correlation imaging in target detection,” IEEE Trans. Geosci. Remote Sens. 36, 1485-1493 (1998).
    [CrossRef]
  14. K. O'Neill, “Broadband bistatic coherent and incoherent detection of buried objects beneath randomly rough surfaces,” IEEE Trans. Geosci. Remote Sens. 38, 891-898 (2000).
    [CrossRef]
  15. M. Testorf and M. Saillard, “The Wigner distribution function applied to the detection of subsurface objects,” Proc. SPIE 6316, 63160L (2006).
    [CrossRef]
  16. L. Onural and M. T. Özgen, “Extraction of three-dimensional object location information directly from in-line holograms using Wigner analysis,” J. Opt. Soc. Am. A 9, 252-260 (1992).
    [CrossRef]
  17. M. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution--Theory and Applications in Signal Processing, W.Mecklenbräuker and F.Hlawatsch, eds. (Elsevier, 1997), pp. 375-426.
  18. J. A. DeSanto, Scalar Wave Theory (Springer, 1992).
    [CrossRef]
  19. A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice Hall, 1991).
  20. K.-H. Brenner and A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310-314 (1982).
    [CrossRef]
  21. M. Saillard and G. Toso, “Electromagnetic scattering from bounded or infinite subsurface bodies,” Radio Sci. 32, 1347-1360 (1997).
    [CrossRef]
  22. M. W. J. Davidson, T. L. Toan, F. Mattia, G. Satalino, T. Manninen, and M. Borgeaud, “On the characterisation of agricultural soil roughness for remote sensing studies,” IEEE Trans. Geosci. Remote Sens. 38, 630-640 (2000).
    [CrossRef]
  23. L. Scharf, Statistical Signal Processing (Prentice Hall, 1990).

2006 (4)

D. Potin, B. Duflos, and P. Vanheeghe, “Landmines ground-penetrating radar signal enhancement by digital filtering,” IEEE Trans. Geosci. Remote Sens. 44, 2393-2406 (2006).
[CrossRef]

O. Cmielewski, M. Saillard, K. Belkebir, and H. Tortel, “On the characterization of buried targets under a rough surface using the Wigner-Ville transform,” IEEE Geosci. Remote Sens. Lett. 3, 442-446 (2006).
[CrossRef]

O. Cmielewski, M. Saillard, and H. Tortel, “Detection of buried objects beneath a rough surface,” Waves Random Complex Media 16, 417-431 (2006).
[CrossRef]

M. Testorf and M. Saillard, “The Wigner distribution function applied to the detection of subsurface objects,” Proc. SPIE 6316, 63160L (2006).
[CrossRef]

2003 (1)

G. Matz and F. Hlawatsch, “Wigner distributions (nearly) everywhere: time-frequency analysis of signals, systems, random processes, signal spaces, and frames,” Signal Process. 83, 1355-1378 (2003).
[CrossRef]

2001 (2)

T. Dogaru and L. Carin, “Time-domain sensing of targets buried under a Gaussian, exponential, or fractal rough interface,” IEEE Trans. Geosci. Remote Sens. 39, 1807-1819 (2001).
[CrossRef]

P. D. Gader, M. Mystkowski, and Y. Zhao, “Landmine detection with ground penetrating radar using hidden Markov models,” IEEE Trans. Geosci. Remote Sens. 39, 1231-1244 (2001).
[CrossRef]

2000 (3)

J. D. Kekis, M. Testorf, M. A. Fiddy, and R. H. Giles, “Detecting 1/10th scaled structures in dielectric media using monostatic X-band radar scattering measurements,” Proc. SPIE 4123, 13-24 (2000).
[CrossRef]

K. O'Neill, “Broadband bistatic coherent and incoherent detection of buried objects beneath randomly rough surfaces,” IEEE Trans. Geosci. Remote Sens. 38, 891-898 (2000).
[CrossRef]

M. W. J. Davidson, T. L. Toan, F. Mattia, G. Satalino, T. Manninen, and M. Borgeaud, “On the characterisation of agricultural soil roughness for remote sensing studies,” IEEE Trans. Geosci. Remote Sens. 38, 630-640 (2000).
[CrossRef]

1999 (1)

H. Brunzell, “Detection of shallowly buried objects using impulse radar,” IEEE Trans. Geosci. Remote Sens. 37, 875-886 (1999).
[CrossRef]

1998 (1)

G. Zhang and L. Tsang, “Application of angular correlation function of clutter scattering and correlation imaging in target detection,” IEEE Trans. Geosci. Remote Sens. 36, 1485-1493 (1998).
[CrossRef]

1997 (2)

M. Saillard and G. Toso, “Electromagnetic scattering from bounded or infinite subsurface bodies,” Radio Sci. 32, 1347-1360 (1997).
[CrossRef]

T.-K. Chan, Y. Kuga, and A. Ishimary, “Subsurface detection of a buried object using angular correlation function measurements,” Waves Random Complex Media 7, 457-465 (1997).

1993 (1)

1992 (1)

1982 (1)

K.-H. Brenner and A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310-314 (1982).
[CrossRef]

Bastiaans, M.

M. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution--Theory and Applications in Signal Processing, W.Mecklenbräuker and F.Hlawatsch, eds. (Elsevier, 1997), pp. 375-426.

Belkebir, K.

O. Cmielewski, M. Saillard, K. Belkebir, and H. Tortel, “On the characterization of buried targets under a rough surface using the Wigner-Ville transform,” IEEE Geosci. Remote Sens. Lett. 3, 442-446 (2006).
[CrossRef]

Borgeaud, M.

M. W. J. Davidson, T. L. Toan, F. Mattia, G. Satalino, T. Manninen, and M. Borgeaud, “On the characterisation of agricultural soil roughness for remote sensing studies,” IEEE Trans. Geosci. Remote Sens. 38, 630-640 (2000).
[CrossRef]

Brenner, K.-H.

K.-H. Brenner and A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310-314 (1982).
[CrossRef]

Brunzell, H.

H. Brunzell, “Detection of shallowly buried objects using impulse radar,” IEEE Trans. Geosci. Remote Sens. 37, 875-886 (1999).
[CrossRef]

Carin, L.

T. Dogaru and L. Carin, “Time-domain sensing of targets buried under a Gaussian, exponential, or fractal rough interface,” IEEE Trans. Geosci. Remote Sens. 39, 1807-1819 (2001).
[CrossRef]

Chan, T.-K.

T.-K. Chan, Y. Kuga, and A. Ishimary, “Subsurface detection of a buried object using angular correlation function measurements,” Waves Random Complex Media 7, 457-465 (1997).

Cmielewski, O.

O. Cmielewski, M. Saillard, and H. Tortel, “Detection of buried objects beneath a rough surface,” Waves Random Complex Media 16, 417-431 (2006).
[CrossRef]

O. Cmielewski, M. Saillard, K. Belkebir, and H. Tortel, “On the characterization of buried targets under a rough surface using the Wigner-Ville transform,” IEEE Geosci. Remote Sens. Lett. 3, 442-446 (2006).
[CrossRef]

Davidson, M. W. J.

M. W. J. Davidson, T. L. Toan, F. Mattia, G. Satalino, T. Manninen, and M. Borgeaud, “On the characterisation of agricultural soil roughness for remote sensing studies,” IEEE Trans. Geosci. Remote Sens. 38, 630-640 (2000).
[CrossRef]

DeSanto, J. A.

J. A. DeSanto, Scalar Wave Theory (Springer, 1992).
[CrossRef]

Dogaru, T.

T. Dogaru and L. Carin, “Time-domain sensing of targets buried under a Gaussian, exponential, or fractal rough interface,” IEEE Trans. Geosci. Remote Sens. 39, 1807-1819 (2001).
[CrossRef]

Duflos, B.

D. Potin, B. Duflos, and P. Vanheeghe, “Landmines ground-penetrating radar signal enhancement by digital filtering,” IEEE Trans. Geosci. Remote Sens. 44, 2393-2406 (2006).
[CrossRef]

Fiddy, M. A.

J. D. Kekis, M. Testorf, M. A. Fiddy, and R. H. Giles, “Detecting 1/10th scaled structures in dielectric media using monostatic X-band radar scattering measurements,” Proc. SPIE 4123, 13-24 (2000).
[CrossRef]

Flandrin, P.

P. Flandrin, Time-Frequency/Time-Scale Analysis (Academic, 1999).

Gader, P. D.

P. D. Gader, M. Mystkowski, and Y. Zhao, “Landmine detection with ground penetrating radar using hidden Markov models,” IEEE Trans. Geosci. Remote Sens. 39, 1231-1244 (2001).
[CrossRef]

Giles, R. H.

J. D. Kekis, M. Testorf, M. A. Fiddy, and R. H. Giles, “Detecting 1/10th scaled structures in dielectric media using monostatic X-band radar scattering measurements,” Proc. SPIE 4123, 13-24 (2000).
[CrossRef]

Hlawatsch, F.

G. Matz and F. Hlawatsch, “Wigner distributions (nearly) everywhere: time-frequency analysis of signals, systems, random processes, signal spaces, and frames,” Signal Process. 83, 1355-1378 (2003).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice Hall, 1991).

Ishimary, A.

T.-K. Chan, Y. Kuga, and A. Ishimary, “Subsurface detection of a buried object using angular correlation function measurements,” Waves Random Complex Media 7, 457-465 (1997).

Kekis, J. D.

J. D. Kekis, M. Testorf, M. A. Fiddy, and R. H. Giles, “Detecting 1/10th scaled structures in dielectric media using monostatic X-band radar scattering measurements,” Proc. SPIE 4123, 13-24 (2000).
[CrossRef]

Kuga, Y.

T.-K. Chan, Y. Kuga, and A. Ishimary, “Subsurface detection of a buried object using angular correlation function measurements,” Waves Random Complex Media 7, 457-465 (1997).

Lohmann, A. W.

K.-H. Brenner and A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310-314 (1982).
[CrossRef]

Manninen, T.

M. W. J. Davidson, T. L. Toan, F. Mattia, G. Satalino, T. Manninen, and M. Borgeaud, “On the characterisation of agricultural soil roughness for remote sensing studies,” IEEE Trans. Geosci. Remote Sens. 38, 630-640 (2000).
[CrossRef]

Mattia, F.

M. W. J. Davidson, T. L. Toan, F. Mattia, G. Satalino, T. Manninen, and M. Borgeaud, “On the characterisation of agricultural soil roughness for remote sensing studies,” IEEE Trans. Geosci. Remote Sens. 38, 630-640 (2000).
[CrossRef]

Matz, G.

G. Matz and F. Hlawatsch, “Wigner distributions (nearly) everywhere: time-frequency analysis of signals, systems, random processes, signal spaces, and frames,” Signal Process. 83, 1355-1378 (2003).
[CrossRef]

Mystkowski, M.

P. D. Gader, M. Mystkowski, and Y. Zhao, “Landmine detection with ground penetrating radar using hidden Markov models,” IEEE Trans. Geosci. Remote Sens. 39, 1231-1244 (2001).
[CrossRef]

Nieto-Vesperinas, M.

O'Neill, K.

K. O'Neill, “Broadband bistatic coherent and incoherent detection of buried objects beneath randomly rough surfaces,” IEEE Trans. Geosci. Remote Sens. 38, 891-898 (2000).
[CrossRef]

Onural, L.

Özgen, M. T.

Potin, D.

D. Potin, B. Duflos, and P. Vanheeghe, “Landmines ground-penetrating radar signal enhancement by digital filtering,” IEEE Trans. Geosci. Remote Sens. 44, 2393-2406 (2006).
[CrossRef]

Saillard, M.

O. Cmielewski, M. Saillard, K. Belkebir, and H. Tortel, “On the characterization of buried targets under a rough surface using the Wigner-Ville transform,” IEEE Geosci. Remote Sens. Lett. 3, 442-446 (2006).
[CrossRef]

O. Cmielewski, M. Saillard, and H. Tortel, “Detection of buried objects beneath a rough surface,” Waves Random Complex Media 16, 417-431 (2006).
[CrossRef]

M. Testorf and M. Saillard, “The Wigner distribution function applied to the detection of subsurface objects,” Proc. SPIE 6316, 63160L (2006).
[CrossRef]

M. Saillard and G. Toso, “Electromagnetic scattering from bounded or infinite subsurface bodies,” Radio Sci. 32, 1347-1360 (1997).
[CrossRef]

Sánchez-Gil, J. A.

Satalino, G.

M. W. J. Davidson, T. L. Toan, F. Mattia, G. Satalino, T. Manninen, and M. Borgeaud, “On the characterisation of agricultural soil roughness for remote sensing studies,” IEEE Trans. Geosci. Remote Sens. 38, 630-640 (2000).
[CrossRef]

Scharf, L.

L. Scharf, Statistical Signal Processing (Prentice Hall, 1990).

Testorf, M.

M. Testorf and M. Saillard, “The Wigner distribution function applied to the detection of subsurface objects,” Proc. SPIE 6316, 63160L (2006).
[CrossRef]

J. D. Kekis, M. Testorf, M. A. Fiddy, and R. H. Giles, “Detecting 1/10th scaled structures in dielectric media using monostatic X-band radar scattering measurements,” Proc. SPIE 4123, 13-24 (2000).
[CrossRef]

Toan, T. L.

M. W. J. Davidson, T. L. Toan, F. Mattia, G. Satalino, T. Manninen, and M. Borgeaud, “On the characterisation of agricultural soil roughness for remote sensing studies,” IEEE Trans. Geosci. Remote Sens. 38, 630-640 (2000).
[CrossRef]

Torre, A.

A. Torre, Linear Ray and Wave Optics (Elesevier, 2005).

Tortel, H.

O. Cmielewski, M. Saillard, and H. Tortel, “Detection of buried objects beneath a rough surface,” Waves Random Complex Media 16, 417-431 (2006).
[CrossRef]

O. Cmielewski, M. Saillard, K. Belkebir, and H. Tortel, “On the characterization of buried targets under a rough surface using the Wigner-Ville transform,” IEEE Geosci. Remote Sens. Lett. 3, 442-446 (2006).
[CrossRef]

Toso, G.

M. Saillard and G. Toso, “Electromagnetic scattering from bounded or infinite subsurface bodies,” Radio Sci. 32, 1347-1360 (1997).
[CrossRef]

Tsang, L.

G. Zhang and L. Tsang, “Application of angular correlation function of clutter scattering and correlation imaging in target detection,” IEEE Trans. Geosci. Remote Sens. 36, 1485-1493 (1998).
[CrossRef]

Vanheeghe, P.

D. Potin, B. Duflos, and P. Vanheeghe, “Landmines ground-penetrating radar signal enhancement by digital filtering,” IEEE Trans. Geosci. Remote Sens. 44, 2393-2406 (2006).
[CrossRef]

Zhang, G.

G. Zhang and L. Tsang, “Application of angular correlation function of clutter scattering and correlation imaging in target detection,” IEEE Trans. Geosci. Remote Sens. 36, 1485-1493 (1998).
[CrossRef]

Zhao, Y.

P. D. Gader, M. Mystkowski, and Y. Zhao, “Landmine detection with ground penetrating radar using hidden Markov models,” IEEE Trans. Geosci. Remote Sens. 39, 1231-1244 (2001).
[CrossRef]

IEEE Geosci. Remote Sens. Lett. (1)

O. Cmielewski, M. Saillard, K. Belkebir, and H. Tortel, “On the characterization of buried targets under a rough surface using the Wigner-Ville transform,” IEEE Geosci. Remote Sens. Lett. 3, 442-446 (2006).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (7)

T. Dogaru and L. Carin, “Time-domain sensing of targets buried under a Gaussian, exponential, or fractal rough interface,” IEEE Trans. Geosci. Remote Sens. 39, 1807-1819 (2001).
[CrossRef]

D. Potin, B. Duflos, and P. Vanheeghe, “Landmines ground-penetrating radar signal enhancement by digital filtering,” IEEE Trans. Geosci. Remote Sens. 44, 2393-2406 (2006).
[CrossRef]

P. D. Gader, M. Mystkowski, and Y. Zhao, “Landmine detection with ground penetrating radar using hidden Markov models,” IEEE Trans. Geosci. Remote Sens. 39, 1231-1244 (2001).
[CrossRef]

H. Brunzell, “Detection of shallowly buried objects using impulse radar,” IEEE Trans. Geosci. Remote Sens. 37, 875-886 (1999).
[CrossRef]

G. Zhang and L. Tsang, “Application of angular correlation function of clutter scattering and correlation imaging in target detection,” IEEE Trans. Geosci. Remote Sens. 36, 1485-1493 (1998).
[CrossRef]

K. O'Neill, “Broadband bistatic coherent and incoherent detection of buried objects beneath randomly rough surfaces,” IEEE Trans. Geosci. Remote Sens. 38, 891-898 (2000).
[CrossRef]

M. W. J. Davidson, T. L. Toan, F. Mattia, G. Satalino, T. Manninen, and M. Borgeaud, “On the characterisation of agricultural soil roughness for remote sensing studies,” IEEE Trans. Geosci. Remote Sens. 38, 630-640 (2000).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

K.-H. Brenner and A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310-314 (1982).
[CrossRef]

Proc. SPIE (2)

M. Testorf and M. Saillard, “The Wigner distribution function applied to the detection of subsurface objects,” Proc. SPIE 6316, 63160L (2006).
[CrossRef]

J. D. Kekis, M. Testorf, M. A. Fiddy, and R. H. Giles, “Detecting 1/10th scaled structures in dielectric media using monostatic X-band radar scattering measurements,” Proc. SPIE 4123, 13-24 (2000).
[CrossRef]

Radio Sci. (1)

M. Saillard and G. Toso, “Electromagnetic scattering from bounded or infinite subsurface bodies,” Radio Sci. 32, 1347-1360 (1997).
[CrossRef]

Signal Process. (1)

G. Matz and F. Hlawatsch, “Wigner distributions (nearly) everywhere: time-frequency analysis of signals, systems, random processes, signal spaces, and frames,” Signal Process. 83, 1355-1378 (2003).
[CrossRef]

Waves Random Complex Media (2)

O. Cmielewski, M. Saillard, and H. Tortel, “Detection of buried objects beneath a rough surface,” Waves Random Complex Media 16, 417-431 (2006).
[CrossRef]

T.-K. Chan, Y. Kuga, and A. Ishimary, “Subsurface detection of a buried object using angular correlation function measurements,” Waves Random Complex Media 7, 457-465 (1997).

Other (6)

P. Flandrin, Time-Frequency/Time-Scale Analysis (Academic, 1999).

A. Torre, Linear Ray and Wave Optics (Elesevier, 2005).

L. Scharf, Statistical Signal Processing (Prentice Hall, 1990).

M. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution--Theory and Applications in Signal Processing, W.Mecklenbräuker and F.Hlawatsch, eds. (Elsevier, 1997), pp. 375-426.

J. A. DeSanto, Scalar Wave Theory (Springer, 1992).
[CrossRef]

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice Hall, 1991).

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Figures (9)

Fig. 1
Fig. 1

Bistatic data acquisition geometry for subsurface target detection.

Fig. 2
Fig. 2

Simplified model of the working principle. The response of the point target remains independent of the source location, while the surface response experiences a linear phase change as well as a lateral shift as a function of the source location.

Fig. 3
Fig. 3

Typical shape of the target autoterm in the WDF domain. Line R corresponds to the high-frequency approximation, or group delay function; line P corresponds to the paraxial approximation of R.

Fig. 4
Fig. 4

Paraxial approximation of AWDF of the target-surface cross-terms. The vertical distance between target and surface is chosen to be (a) 1.5 λ , (b) 0.3 λ , (c) 0.03 λ .

Fig. 5
Fig. 5

Validation based on rigorous simulations (flat surface): (a) AWDF of background without target, (b) AWDF of target signature, (c) AWDF of cross-terms, (d) AWDF of background material with embedded target.

Fig. 6
Fig. 6

Validation based on rigorous simulations (rough surface): (a) AWDF of background without target, (b) AWDF of target signature, (c) AWDF of cross-terms, (d) AWDF of background material with embedded target.

Fig. 7
Fig. 7

AWDF of backpropagated signal: (a) AWDF of flat surface geometry with embedded target, (b) AWDF in (a) integrated over different bandwidths, (c) AWDF of rough surface with embedded target, (d) AWDF in (c) integrated over different bandwidths.

Fig. 8
Fig. 8

ROC curve of the (a) averaged correlation technique, (b) the AWDF for 104 surface realizations.

Fig. 9
Fig. 9

Position of the buried object estimated as the location of the output signal maximum for all surface realizations: (a) average correlation technique (NFCT), (b) AWDF.

Equations (22)

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

W ( x d , ν ; x s ) = u ( x d + x 2 ; x s ) u * ( x d x 2 ; x s ) exp ( i 2 π ν x ) d x .
F ( x d , ν ) = W ( x d , ν ; x s ) d x s .
u s ( x d ; x s ) = G ( s d ) ( x d x s , 2 z s a ) ,
u t ( x d ; x s ) = G ( s t ) ( x t x s , z a , z t ) G ( t d ) ( x d x t , z a , z t ) ,
G ( x x , Δ z ) = A ( ν ) exp [ i 2 π ( x x ) ν + i 2 π β ( ν ) Δ z ] d ν ,
A ( ν ) = A ( s ) ( ν ) = i 2 β 0 ( ν ) β 0 ( ν ) β g ( ν ) β 0 ( ν ) + β g ( ν ) ,
u ̃ s ( ν ; x s ) = A ( s ) ( ν ) exp [ i 2 π x s ν + i 2 π β 0 ( ν ) ( 2 z s a ) ] .
u ̃ t ( ν ; x s ) = G ( s t ) ( x t x s , z t a ) A ( t a ) ( ν ) exp [ i 2 π x t ν + i 2 π β 0 ( ν ) z t a ] ,
G ( s t ) ( x t x s , z t a ) = A ( a t ) ( ζ ) exp [ i 2 π ( x t x s ) ζ + i 2 π β 0 ( ζ ) z t a ] d ζ .
A ( a t ) ( ν ) = i 2 β 0 ( ν ) 2 β 0 ( ν ) β 0 ( ν ) + β g ( ν ) ,
A ( t a ) ( ν ) = i 2 β 0 ( ν ) 2 β g ( ν ) β 0 ( ν ) + β g ( ν ) .
W a + b ( x , ν ) = W a a + W b b + W a b + W b a ,
W a b ( x , ν ) = u ̃ a ( ν + ν 2 ) u ̃ b * ( ν ν 2 ) exp ( i 2 π x ν ) d ν
F s s ( x , ν ) = A ( s ) ( ν ) 2 .
W t t ( x , ν ) = A ( t ) 2 exp [ i ϕ ( ν + ν 2 ) i ϕ ( ν ν 2 ) + i 2 π x ν ] d ν ,
W t t ( x , ν ) = A ( t ) 2 δ ( x + 1 2 π d ϕ d ν ) = A ( t ) 2 δ [ x x t z t a ν λ 1 ( λ ν ) 2 ] .
W t t γ Ai [ γ ( x + 1 2 π d ϕ d ν ) ] ,
F t s ( x , ν ) + F s t ( x , ν ) = 2 Re { A ( t a ) A ( a t ) A ( s ) * } δ [ x x t z t a ν λ 1 ( λ ν ) 2 ] .
F t s ( x , ν ) + F s t ( x , ν ) 2 π λ ( z t z s ) cos [ 2 π λ ( z t z s ) ν 2 2 π λ ( z t z s ) ( x x t λ z s ν ) π 4 ] .
P ( x ) = d rms 2 exp ( x L c ) ,
F ( x , ν ) = x s x d u ( x + x d ) u * ( x x d ) exp ( i 4 π ν x d ) ,
λ ν < 1 1 + ( 4 h A ) 2 .

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