Abstract

The modified Mueller matrix elements for electromagnetic scattering from penetrable objects buried under two-dimensional random rough surfaces are investigated. This matrix relates the incident to the scattered waves, and it contains different combinations of the fully polarimetric scattering matrix elements. The statistical average of each Mueller matrix element is computed on the basis of the Monte Carlo simulations by exploiting the speed of the three-dimensional steepest-descent fast multipole method. The numerical results clearly show that relying only on the co-polarized or the cross-polarized intensities or both (i.e., vv, hh, vh, and hv) is not sufficient for sensing the buried objects. However, examining all 16 Mueller matrix elements significantly increases the possibility of detecting these objects. This technique can be used in remote sensing of scatterers buried beneath the rough ground.

© 2003 Optical Society of America

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References

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  1. M. El-Shenawee, C. Rappaport, E. Miller, M. Silevitch, “Three-dimensional subsurface analysis of electromagnetic scattering from penetrable/PEC objects buried under rough surfaces: use of the steepest descent fast multipole method (SDFMM),” IEEE Trans. Geosci. Remote Sens. 39, 1174–1182 (2001).
    [CrossRef]
  2. M. El-Shenawee, C. Rappaport, M. Silevitch, “Monte Carlo simulations of electromagnetic wave scattering from a random rough surface with three-dimensional penetrable buried object: mine detection application using the steepest-descent fast multipole method,” J. Opt. Soc. Am. A 18, 3077–3084 (2001).
    [CrossRef]
  3. M. El-Shenawee, “Scattering from multiple objects buried under two-dimensional randomly rough surface using the steepest descent fast multipole method,” IEEE Trans. Antennas Propag. (to be published).
  4. G. Zhang, L. Tsang, K. Pak, “Angular correlation function and scattering coefficient of electromagnetic waves scattered by a buried object under a two-dimensional rough surface,” J. Opt. Soc. Am. A 15, 2995–3002 (1998).
    [CrossRef]
  5. Y. Zhang, E. Bahar, “Mueller matrix elements that characterize scattering from coated random rough surfaces,” IEEE Trans. Antennas Propag. 47, 949–955 (1999).
    [CrossRef]
  6. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  7. F. T. Ulaby, C. Elachi, Radar Polarimetry for Geoscience Applications (Artech House, Norwood, Mass., 1990).
  8. G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1852). Reprinted in Mathematical and Physical Papers (Cambridge U. Press, London, 1901), Vol. 3, pp. 233–250.
  9. L. Medgyesi-Mitschang, J. Putnam, M. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Soc. Am. A 11, 1383–1398 (1994).
    [CrossRef]
  10. S. M. Rao, D. R. Wilton, A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. AP-30, 409–418 (1982).
    [CrossRef]
  11. V. Jandhyala, “Fast multilevel algorithms for the efficient electromagnetic analysis of quasi-planar structures,” Ph.D. dissertation (University of Illinois at Urbana-Champaign, Urbana-Champaign, Ill.1998.
  12. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).
  13. P. Tran, A. A. Maradudin, “Scattering of a scalar beam from a two-dimensional randomly rough hard wall: enhanced backscatter,” Phys. Rev. B 45, 3936–3939 (1992).
    [CrossRef]
  14. R. L. Wagner, J. Song, W. C. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
    [CrossRef]

2001 (2)

M. El-Shenawee, C. Rappaport, E. Miller, M. Silevitch, “Three-dimensional subsurface analysis of electromagnetic scattering from penetrable/PEC objects buried under rough surfaces: use of the steepest descent fast multipole method (SDFMM),” IEEE Trans. Geosci. Remote Sens. 39, 1174–1182 (2001).
[CrossRef]

M. El-Shenawee, C. Rappaport, M. Silevitch, “Monte Carlo simulations of electromagnetic wave scattering from a random rough surface with three-dimensional penetrable buried object: mine detection application using the steepest-descent fast multipole method,” J. Opt. Soc. Am. A 18, 3077–3084 (2001).
[CrossRef]

1999 (1)

Y. Zhang, E. Bahar, “Mueller matrix elements that characterize scattering from coated random rough surfaces,” IEEE Trans. Antennas Propag. 47, 949–955 (1999).
[CrossRef]

1998 (1)

1997 (1)

R. L. Wagner, J. Song, W. C. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
[CrossRef]

1994 (1)

1992 (1)

P. Tran, A. A. Maradudin, “Scattering of a scalar beam from a two-dimensional randomly rough hard wall: enhanced backscatter,” Phys. Rev. B 45, 3936–3939 (1992).
[CrossRef]

1982 (1)

S. M. Rao, D. R. Wilton, A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. AP-30, 409–418 (1982).
[CrossRef]

1852 (1)

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1852). Reprinted in Mathematical and Physical Papers (Cambridge U. Press, London, 1901), Vol. 3, pp. 233–250.

Bahar, E.

Y. Zhang, E. Bahar, “Mueller matrix elements that characterize scattering from coated random rough surfaces,” IEEE Trans. Antennas Propag. 47, 949–955 (1999).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).

Chew, W. C.

R. L. Wagner, J. Song, W. C. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
[CrossRef]

Elachi, C.

F. T. Ulaby, C. Elachi, Radar Polarimetry for Geoscience Applications (Artech House, Norwood, Mass., 1990).

El-Shenawee, M.

M. El-Shenawee, C. Rappaport, M. Silevitch, “Monte Carlo simulations of electromagnetic wave scattering from a random rough surface with three-dimensional penetrable buried object: mine detection application using the steepest-descent fast multipole method,” J. Opt. Soc. Am. A 18, 3077–3084 (2001).
[CrossRef]

M. El-Shenawee, C. Rappaport, E. Miller, M. Silevitch, “Three-dimensional subsurface analysis of electromagnetic scattering from penetrable/PEC objects buried under rough surfaces: use of the steepest descent fast multipole method (SDFMM),” IEEE Trans. Geosci. Remote Sens. 39, 1174–1182 (2001).
[CrossRef]

M. El-Shenawee, “Scattering from multiple objects buried under two-dimensional randomly rough surface using the steepest descent fast multipole method,” IEEE Trans. Antennas Propag. (to be published).

Gedera, M.

Glisson, A. W.

S. M. Rao, D. R. Wilton, A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. AP-30, 409–418 (1982).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Jandhyala, V.

V. Jandhyala, “Fast multilevel algorithms for the efficient electromagnetic analysis of quasi-planar structures,” Ph.D. dissertation (University of Illinois at Urbana-Champaign, Urbana-Champaign, Ill.1998.

Maradudin, A. A.

P. Tran, A. A. Maradudin, “Scattering of a scalar beam from a two-dimensional randomly rough hard wall: enhanced backscatter,” Phys. Rev. B 45, 3936–3939 (1992).
[CrossRef]

Medgyesi-Mitschang, L.

Miller, E.

M. El-Shenawee, C. Rappaport, E. Miller, M. Silevitch, “Three-dimensional subsurface analysis of electromagnetic scattering from penetrable/PEC objects buried under rough surfaces: use of the steepest descent fast multipole method (SDFMM),” IEEE Trans. Geosci. Remote Sens. 39, 1174–1182 (2001).
[CrossRef]

Pak, K.

Putnam, J.

Rao, S. M.

S. M. Rao, D. R. Wilton, A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. AP-30, 409–418 (1982).
[CrossRef]

Rappaport, C.

M. El-Shenawee, C. Rappaport, M. Silevitch, “Monte Carlo simulations of electromagnetic wave scattering from a random rough surface with three-dimensional penetrable buried object: mine detection application using the steepest-descent fast multipole method,” J. Opt. Soc. Am. A 18, 3077–3084 (2001).
[CrossRef]

M. El-Shenawee, C. Rappaport, E. Miller, M. Silevitch, “Three-dimensional subsurface analysis of electromagnetic scattering from penetrable/PEC objects buried under rough surfaces: use of the steepest descent fast multipole method (SDFMM),” IEEE Trans. Geosci. Remote Sens. 39, 1174–1182 (2001).
[CrossRef]

Silevitch, M.

M. El-Shenawee, C. Rappaport, E. Miller, M. Silevitch, “Three-dimensional subsurface analysis of electromagnetic scattering from penetrable/PEC objects buried under rough surfaces: use of the steepest descent fast multipole method (SDFMM),” IEEE Trans. Geosci. Remote Sens. 39, 1174–1182 (2001).
[CrossRef]

M. El-Shenawee, C. Rappaport, M. Silevitch, “Monte Carlo simulations of electromagnetic wave scattering from a random rough surface with three-dimensional penetrable buried object: mine detection application using the steepest-descent fast multipole method,” J. Opt. Soc. Am. A 18, 3077–3084 (2001).
[CrossRef]

Song, J.

R. L. Wagner, J. Song, W. C. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
[CrossRef]

Stokes, G. G.

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1852). Reprinted in Mathematical and Physical Papers (Cambridge U. Press, London, 1901), Vol. 3, pp. 233–250.

Tran, P.

P. Tran, A. A. Maradudin, “Scattering of a scalar beam from a two-dimensional randomly rough hard wall: enhanced backscatter,” Phys. Rev. B 45, 3936–3939 (1992).
[CrossRef]

Tsang, L.

Ulaby, F. T.

F. T. Ulaby, C. Elachi, Radar Polarimetry for Geoscience Applications (Artech House, Norwood, Mass., 1990).

Wagner, R. L.

R. L. Wagner, J. Song, W. C. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
[CrossRef]

Wilton, D. R.

S. M. Rao, D. R. Wilton, A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. AP-30, 409–418 (1982).
[CrossRef]

Zhang, G.

Zhang, Y.

Y. Zhang, E. Bahar, “Mueller matrix elements that characterize scattering from coated random rough surfaces,” IEEE Trans. Antennas Propag. 47, 949–955 (1999).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

Y. Zhang, E. Bahar, “Mueller matrix elements that characterize scattering from coated random rough surfaces,” IEEE Trans. Antennas Propag. 47, 949–955 (1999).
[CrossRef]

S. M. Rao, D. R. Wilton, A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. AP-30, 409–418 (1982).
[CrossRef]

R. L. Wagner, J. Song, W. C. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

M. El-Shenawee, C. Rappaport, E. Miller, M. Silevitch, “Three-dimensional subsurface analysis of electromagnetic scattering from penetrable/PEC objects buried under rough surfaces: use of the steepest descent fast multipole method (SDFMM),” IEEE Trans. Geosci. Remote Sens. 39, 1174–1182 (2001).
[CrossRef]

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

Phys. Rev. B (1)

P. Tran, A. A. Maradudin, “Scattering of a scalar beam from a two-dimensional randomly rough hard wall: enhanced backscatter,” Phys. Rev. B 45, 3936–3939 (1992).
[CrossRef]

Trans. Cambridge Philos. Soc. (1)

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1852). Reprinted in Mathematical and Physical Papers (Cambridge U. Press, London, 1901), Vol. 3, pp. 233–250.

Other (5)

V. Jandhyala, “Fast multilevel algorithms for the efficient electromagnetic analysis of quasi-planar structures,” Ph.D. dissertation (University of Illinois at Urbana-Champaign, Urbana-Champaign, Ill.1998.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).

M. El-Shenawee, “Scattering from multiple objects buried under two-dimensional randomly rough surface using the steepest descent fast multipole method,” IEEE Trans. Antennas Propag. (to be published).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

F. T. Ulaby, C. Elachi, Radar Polarimetry for Geoscience Applications (Artech House, Norwood, Mass., 1990).

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Figures (9)

Fig. 1
Fig. 1

(a) Cross section of a general two-dimensional rough ground with two buried objects. (b) Top view of the geometry. (c) Three-dimensional geometry for the buried spheroid and the 30°-tilted horizontal cylinder. (d) Three-dimensional geometry for the buried 90°-tilted horizontal cylinder. (e) Three-dimensional geometry for the buried 20°-tilted disk. All figures in (c)–(e) show exact locations of the objects.

Fig. 2
Fig. 2

Normalized bistatic modified Mueller matrix elements (total intensity): m11, m12,  , m44. Solid curve, for the rough ground only; cross symbol, for the rough ground with the two buried objects [the spheroid and the horizontal 30°-tilted cylinder, Fig. 1(c)] for incident angles θi=0° and ϕi=0°. Monte Carlo for 100 rough surface realizations.

Fig. 3
Fig. 3

Normalized bistatic modified Mueller matrix elements (total intensity): m11, m12,  , m44. Solid curve, for the rough ground only; cross symbol, for the rough ground with only the second object [horizontal 30°-tilted cylinder, Fig. 1(c)] for the incident angles θi=0° and ϕi=0°. Monte Carlo for 100 rough surface realizations.

Fig. 4
Fig. 4

Normalized bistatic modified Mueller matrix elements (total intensity): m11, m12,  , m44. Solid curve, for the rough ground only; the cross symbol, for the rough ground with only the first object [the spheroid, Fig. 1(c)] for the incident angles θi=0° and ϕi=0°. Monte Carlo for 100 rough surface realizations.

Fig. 5
Fig. 5

Normalized bistatic modified Mueller matrix elements (total intensity): m11, m12,  , m44. Solid curve, for the rough ground only; cross symbol, for the rough ground with only the second object [horizontal 90°-tilted cylinder, Fig. 1(d)] for the incident angles θi=0° and ϕi=0°. Monte Carlo for 100 rough surface realizations.

Fig. 6
Fig. 6

Normalized bistatic modified Mueller matrix elements (total intensity): m11, m12,  , m44. Solid curve, for the rough ground only; cross symbol, for the rough ground with only the second object [20°-tilted disk, Fig. 1(e)] for the incident angles θi=0° and ϕi=0°. Monte Carlo for 100 rough surface realizations.

Fig. 7
Fig. 7

Normalized bistatic modified Mueller matrix elements (total intensity): m11, m12,  , m44. Solid curve, for the rough ground only; cross symbol, for the rough ground with the two buried objects [the spheroid and the horizontal 30°-tilted cylinder, Fig. 1(c)] for the incident angles θi=30° and ϕi=120°. Monte Carlo for 100 rough surface realizations.

Fig. 8
Fig. 8

Normalized bistatic modified Mueller matrix (a) element m42, (b) element m14, and (c) m24 versus the scatter angle for θi=0° and ϕi=0° from the only one rough surface realization. No averaging.

Fig. 9
Fig. 9

Normalized bistatic modified Mueller matrix element for (a) m11 (vv), and (b) m41 versus scatter angle for 20 individual rough surface realizations chosen from the 100 samples. Solid curve, for the rough ground only; cross symbol, for the rough ground with the horizontal 30°-tilted buried cylinder of Fig. 1(c). The incident angles are θi=0° and ϕi=0°. No averaging.

Equations (18)

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Einc(r)|tang=[(L1+L2)J1-(K1+K2)M1-L3J2+K3M2-L4J3+K4M3]tang,rS1,
Hinc(r)|tang=(K1+K2)J1+L1η12+L2η22M1-K3J2-L3η22 M2-K4J3-L4η22 M3tang,rS1,
0=[-L2J1+K2M1+(L3+L5)J2-(K3+K5)M2+L4J3-K4M3]tang,rS2,
0=-K2J1-L2η22 M1+(K3+K5)J2+L3η22+L5η32M2+K4J3+L4η22 M3tang,rS2,
0=[-L2J1+K2M1+L3J2-K3M2+(L4+L6)J3-(K4+K6)M3]tang,rS3,
0=-K2J1+L2η22 M1+K3J2+L3η22 M2+(K4+K6)J3+L4η222+L6η42M3tang,rS3,
Jk(r)=n=1NkIn(k)jn(k)(r),Mk(r)=η1n=1NkI(n+Nk)(k)jn(k)(r),rSk,fork=1, 2, 3.
I=|Ev|2|Eh|22 Re(EvEh*)2 Im(EvEh*)/η1,
Is=1r2 MmIi,
Mm=|Svv|2|Svh|2Re(Svh*Svv)-Im(Svh*Svv)|Shv|2|Shh|2Re(ShvShh*)-Im(ShvShh*)2 Re(SvvShv*)2 Re(SvhShh*)Re(SvvShh*+SvhShv*)-Im(SvvShh*-SvhShv*)2 Im(SvvShv*)2 Im(SvhShh*)Im(SvvShh*+SvhShv*)Re(SvvShh*-SvhShv*),
EvsEhs=exp(jkr)r SvvSvhShvShhEviEhi,
L1,2X=S1iωμ1,2Φ1,2X(r)+iωϵ1,2 ·X(r)Φ1,2ds,K1,2X=S1X(r)×Φ1,2ds,
L3,5X=S2iωμ2,3Φ2,3X(r)+iωϵ2,3 ·X(r)Φ2,3ds,K3,5X=S2X(r)×Φ2,3ds,
L4,6X=S3iωμ2,4Φ2,4X(r)+iωϵ2,4 ·X(r)Φ2,4ds,K4,6X=S3X(r)×Φ2,4ds.
Z11Z12Z13Z21Z22Z23Z31Z32Z33I1I2I3=V100,
Z11=j1, (L1+L2)j1S1j1, -η1(K1+K2)j1S1j1, η1(K1+K2)j1S1j1, η12L1η12+L2η22j1S1,
Z12=j1, -L3 j2S1j1, η1K3 j2S1j1, -η1K3 j2S1j1, -η12L3η22j2S1,Z13=j1, -L4 j3S1j1, η1K4 j3S1j1, -η1K4 j3S1j1, -η12L4η22j3S1,
Z23=j2, L4 j3S2j2, -η1K4 j3S2j2, η1K4 j3S2j2, η12 L4η22 j3S2.

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