Abstract

The composite scattering from the conducting targets above and below the dielectric rough surface using the extended Propagation-Inside-Layer Expansion (EPILE) combined with the Forward-Backward method (FBM) is studied. The established integral equations are approved by comparing with the related theory. The accuracy and efficiency of the EPILE + FBM are compared with the method of moments (MOM). The influences of target size, target height/depth, target position, and the rms height, the correlation length, as well as the incident angle on the bistatic scattering coefficient (BSC) for different polarizations are also investigated. The presented algorithm is of generality for the target-rough surface composite scattering problems.

© 2011 OSA

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  2. D. Holliday, “Resolution of a controversy surrounding the Kirchhoff approach and the small perturbation method in rough surface scattering theory,” IEEE Trans. Antenn. Propag. 35(1), 120–122 (1987).
    [CrossRef]
  3. L. X. Guo and Z. S. Wu, “Application of the extended boundary condition method to electromagnetic scattering from rough dielectric fractal sea surface,” J. Electromagn. Waves Appl. 18(9), 1219–1234 (2004).
    [CrossRef]
  4. S. L. Durden and J. F. Vesecky, “A numerical study of the separation wavenumber in the two-scale scattering approximation,” IEEE Trans. Geosci. Rem. Sens. 28(2), 271–272 (1990).
    [CrossRef]
  5. D. Winebrenner and A. Ishimaru, “Investigation of a surface field phase perturbation technique for scattering from rough surfaces,” Radio Sci. 20(2), 161–170 (1985).
    [CrossRef]
  6. A. Voronovich, “Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half- spaces,” Waves Random Media 4(3), 337–367 (1994).
    [CrossRef]
  7. R. R. Lentz, “A numerical study of electromagnetic scattering from ocean-like surfaces,” Radio Sci. 9(12), 1139–1146 (1974).
    [CrossRef]
  8. R. M. Axline and A. K. Fung, “Numerical computations of scattering from a perfectly conducting random surface,” IEEE Trans. Antenn. Propag. 26(3), 482–488 (1978).
    [CrossRef]
  9. C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, “Electromagnetic scattering of waves by random rough surface: A finite- difference time-domain approach,” Microw. Opt. Technol. Lett. 4(9), 355–359 (1991).
    [CrossRef]
  10. S. H. Lou, L. Tsang, and C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1(4), 287–307 (1991).
    [CrossRef]
  11. D. A. Kapp and G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antenn. Propag. 44(5), 711–721 (1996).
    [CrossRef]
  12. V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 36(3), 738–748 (1998).
    [CrossRef]
  13. L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993).
    [CrossRef]
  14. A. Iodice, “Forward–backward method for scattering from dielectric rough surfaces,” IEEE Trans. Antenn. Propag. 50(7), 901–911 (2002).
    [CrossRef]
  15. X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003).
    [CrossRef]
  16. Y. Q. Jin and G. Li, “Detection of a scatter target over a randomly rough surface by using the angular correlation function in a finite-element approach,” Waves Random Media 10(2), 273–280 (2000).
    [CrossRef]
  17. J. Li, L. X. Guo, and H. Zeng, “FDTD investigation on the electromagnetic scattering from a target above a randomly rough sea surface,” Waves Random Media 10, 273–280 (2008).
  18. Y. Zhang, Y. E. Yang, H. Braunisch, and J. A. Kong, “Electromagnetic wave interaction of conducting object with rough surface by hybrid SPM/MOM technique,” Prog. Electromagn. Res. 22, 315–335 (1999).
    [CrossRef]
  19. H. Ye and Y. Q. Jin, “A hybrid KA-MOM algorithm for computation of scattering from a 3-D PEC target above a dielectric rough surface,” Radio Sci. 43(3), RS3005 (2008).
    [CrossRef]
  20. S. Y. He and G. Q. Zhu, “A hybrid MM-PO method combining UV technique for scattering from two-dimensional target above a rough surface,” Microw. Opt. Technol. Lett. 49(12), 2957–2960 (2007).
    [CrossRef]
  21. T. Chiu and K. Sarabandi, “Electromagnetic scattering interaction between a dielectric cylinder and a slightly rough surface,” IEEE Trans. Antenn. Propag. 47(5), 902–913 (1999).
    [CrossRef]
  22. G. Lixin and K. Cheyoung, “Light scattering models for a spherical particle above a slightly dielectric rough surface,” Microw. Opt. Technol. Lett. 33(2), 142–146 (2002).
    [CrossRef]
  23. M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998).
  24. Z. Li and Y. Q. Jin, “Bistatic scattering from a fractal dynamic rough sea surface with a ship presence at low grazing-angle incidence using the FBM/SAA,” Microw. Opt. Technol. Lett. 31(2), 146–151 (2001).
    [CrossRef]
  25. N. Déchamps, N. de Beaucoudrey, C. Bourlier, and S. Toutain, “Fast numerical method for electromagnetic scattering by rough layered interfaces: propagation-inside-layer expansion method,” J. Opt. Soc. Am. A 23(2), 359–369 (2006).
    [CrossRef]
  26. G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: Extended PILE method combined with FB-SA,” Waves Random Complex Media 18(3), 495–519 (2008).
    [CrossRef]
  27. C. Bourlier, G. Kubické, and N. Déchamps, “Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA,” J. Opt. Soc. Am. A 25(4), 891–902 (2008).
    [CrossRef]
  28. L. Tsang, and J. A. Kong, Scattering of Electromagnetic Waves- Numerical Simulations, (Wiley, 2000).
  29. E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 (1988).
    [CrossRef]
  30. M. R. Hestenes and E. Stiefel, “Method of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. 49, 409–436 (1952).
  31. G. L. G. Sleijpeny and D. R. Fokkema, “Bicgstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal. 1, 11–32 (1993).

2008

J. Li, L. X. Guo, and H. Zeng, “FDTD investigation on the electromagnetic scattering from a target above a randomly rough sea surface,” Waves Random Media 10, 273–280 (2008).

H. Ye and Y. Q. Jin, “A hybrid KA-MOM algorithm for computation of scattering from a 3-D PEC target above a dielectric rough surface,” Radio Sci. 43(3), RS3005 (2008).
[CrossRef]

G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: Extended PILE method combined with FB-SA,” Waves Random Complex Media 18(3), 495–519 (2008).
[CrossRef]

C. Bourlier, G. Kubické, and N. Déchamps, “Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA,” J. Opt. Soc. Am. A 25(4), 891–902 (2008).
[CrossRef]

2007

S. Y. He and G. Q. Zhu, “A hybrid MM-PO method combining UV technique for scattering from two-dimensional target above a rough surface,” Microw. Opt. Technol. Lett. 49(12), 2957–2960 (2007).
[CrossRef]

2006

2004

L. X. Guo and Z. S. Wu, “Application of the extended boundary condition method to electromagnetic scattering from rough dielectric fractal sea surface,” J. Electromagn. Waves Appl. 18(9), 1219–1234 (2004).
[CrossRef]

2003

X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003).
[CrossRef]

2002

A. Iodice, “Forward–backward method for scattering from dielectric rough surfaces,” IEEE Trans. Antenn. Propag. 50(7), 901–911 (2002).
[CrossRef]

G. Lixin and K. Cheyoung, “Light scattering models for a spherical particle above a slightly dielectric rough surface,” Microw. Opt. Technol. Lett. 33(2), 142–146 (2002).
[CrossRef]

2001

Z. Li and Y. Q. Jin, “Bistatic scattering from a fractal dynamic rough sea surface with a ship presence at low grazing-angle incidence using the FBM/SAA,” Microw. Opt. Technol. Lett. 31(2), 146–151 (2001).
[CrossRef]

2000

Y. Q. Jin and G. Li, “Detection of a scatter target over a randomly rough surface by using the angular correlation function in a finite-element approach,” Waves Random Media 10(2), 273–280 (2000).
[CrossRef]

1999

Y. Zhang, Y. E. Yang, H. Braunisch, and J. A. Kong, “Electromagnetic wave interaction of conducting object with rough surface by hybrid SPM/MOM technique,” Prog. Electromagn. Res. 22, 315–335 (1999).
[CrossRef]

T. Chiu and K. Sarabandi, “Electromagnetic scattering interaction between a dielectric cylinder and a slightly rough surface,” IEEE Trans. Antenn. Propag. 47(5), 902–913 (1999).
[CrossRef]

1998

M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998).

V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 36(3), 738–748 (1998).
[CrossRef]

1996

D. A. Kapp and G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antenn. Propag. 44(5), 711–721 (1996).
[CrossRef]

1994

A. Voronovich, “Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half- spaces,” Waves Random Media 4(3), 337–367 (1994).
[CrossRef]

1993

L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993).
[CrossRef]

G. L. G. Sleijpeny and D. R. Fokkema, “Bicgstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal. 1, 11–32 (1993).

1991

C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, “Electromagnetic scattering of waves by random rough surface: A finite- difference time-domain approach,” Microw. Opt. Technol. Lett. 4(9), 355–359 (1991).
[CrossRef]

S. H. Lou, L. Tsang, and C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1(4), 287–307 (1991).
[CrossRef]

1990

S. L. Durden and J. F. Vesecky, “A numerical study of the separation wavenumber in the two-scale scattering approximation,” IEEE Trans. Geosci. Rem. Sens. 28(2), 271–272 (1990).
[CrossRef]

1988

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 (1988).
[CrossRef]

1987

D. Holliday, “Resolution of a controversy surrounding the Kirchhoff approach and the small perturbation method in rough surface scattering theory,” IEEE Trans. Antenn. Propag. 35(1), 120–122 (1987).
[CrossRef]

1985

D. Winebrenner and A. Ishimaru, “Investigation of a surface field phase perturbation technique for scattering from rough surfaces,” Radio Sci. 20(2), 161–170 (1985).
[CrossRef]

1978

R. M. Axline and A. K. Fung, “Numerical computations of scattering from a perfectly conducting random surface,” IEEE Trans. Antenn. Propag. 26(3), 482–488 (1978).
[CrossRef]

1974

R. R. Lentz, “A numerical study of electromagnetic scattering from ocean-like surfaces,” Radio Sci. 9(12), 1139–1146 (1974).
[CrossRef]

1952

M. R. Hestenes and E. Stiefel, “Method of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. 49, 409–436 (1952).

Axline, R. M.

R. M. Axline and A. K. Fung, “Numerical computations of scattering from a perfectly conducting random surface,” IEEE Trans. Antenn. Propag. 26(3), 482–488 (1978).
[CrossRef]

Balasubramaniam, S.

V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 36(3), 738–748 (1998).
[CrossRef]

Bourlier, C.

Braunisch, H.

Y. Zhang, Y. E. Yang, H. Braunisch, and J. A. Kong, “Electromagnetic wave interaction of conducting object with rough surface by hybrid SPM/MOM technique,” Prog. Electromagn. Res. 22, 315–335 (1999).
[CrossRef]

Brown, G. S.

D. A. Kapp and G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antenn. Propag. 44(5), 711–721 (1996).
[CrossRef]

Burkholder, R. J.

M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998).

Chan, C. H.

L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993).
[CrossRef]

C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, “Electromagnetic scattering of waves by random rough surface: A finite- difference time-domain approach,” Microw. Opt. Technol. Lett. 4(9), 355–359 (1991).
[CrossRef]

S. H. Lou, L. Tsang, and C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1(4), 287–307 (1991).
[CrossRef]

Chew, W. C.

V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 36(3), 738–748 (1998).
[CrossRef]

Cheyoung, K.

G. Lixin and K. Cheyoung, “Light scattering models for a spherical particle above a slightly dielectric rough surface,” Microw. Opt. Technol. Lett. 33(2), 142–146 (2002).
[CrossRef]

Chiu, T.

T. Chiu and K. Sarabandi, “Electromagnetic scattering interaction between a dielectric cylinder and a slightly rough surface,” IEEE Trans. Antenn. Propag. 47(5), 902–913 (1999).
[CrossRef]

de Beaucoudrey, N.

Déchamps, N.

Durden, S. L.

S. L. Durden and J. F. Vesecky, “A numerical study of the separation wavenumber in the two-scale scattering approximation,” IEEE Trans. Geosci. Rem. Sens. 28(2), 271–272 (1990).
[CrossRef]

Fokkema, D. R.

G. L. G. Sleijpeny and D. R. Fokkema, “Bicgstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal. 1, 11–32 (1993).

Fung, A. K.

R. M. Axline and A. K. Fung, “Numerical computations of scattering from a perfectly conducting random surface,” IEEE Trans. Antenn. Propag. 26(3), 482–488 (1978).
[CrossRef]

Gan, Y. B.

X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003).
[CrossRef]

Guo, L. X.

J. Li, L. X. Guo, and H. Zeng, “FDTD investigation on the electromagnetic scattering from a target above a randomly rough sea surface,” Waves Random Media 10, 273–280 (2008).

L. X. Guo and Z. S. Wu, “Application of the extended boundary condition method to electromagnetic scattering from rough dielectric fractal sea surface,” J. Electromagn. Waves Appl. 18(9), 1219–1234 (2004).
[CrossRef]

He, S. Y.

S. Y. He and G. Q. Zhu, “A hybrid MM-PO method combining UV technique for scattering from two-dimensional target above a rough surface,” Microw. Opt. Technol. Lett. 49(12), 2957–2960 (2007).
[CrossRef]

Hestenes, M. R.

M. R. Hestenes and E. Stiefel, “Method of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. 49, 409–436 (1952).

Holliday, D.

D. Holliday, “Resolution of a controversy surrounding the Kirchhoff approach and the small perturbation method in rough surface scattering theory,” IEEE Trans. Antenn. Propag. 35(1), 120–122 (1987).
[CrossRef]

Iodice, A.

A. Iodice, “Forward–backward method for scattering from dielectric rough surfaces,” IEEE Trans. Antenn. Propag. 50(7), 901–911 (2002).
[CrossRef]

Ishimaru, A.

L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993).
[CrossRef]

D. Winebrenner and A. Ishimaru, “Investigation of a surface field phase perturbation technique for scattering from rough surfaces,” Radio Sci. 20(2), 161–170 (1985).
[CrossRef]

Jandhyala, V.

V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 36(3), 738–748 (1998).
[CrossRef]

Jin, Y. Q.

H. Ye and Y. Q. Jin, “A hybrid KA-MOM algorithm for computation of scattering from a 3-D PEC target above a dielectric rough surface,” Radio Sci. 43(3), RS3005 (2008).
[CrossRef]

Z. Li and Y. Q. Jin, “Bistatic scattering from a fractal dynamic rough sea surface with a ship presence at low grazing-angle incidence using the FBM/SAA,” Microw. Opt. Technol. Lett. 31(2), 146–151 (2001).
[CrossRef]

Y. Q. Jin and G. Li, “Detection of a scatter target over a randomly rough surface by using the angular correlation function in a finite-element approach,” Waves Random Media 10(2), 273–280 (2000).
[CrossRef]

Kapp, D. A.

D. A. Kapp and G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antenn. Propag. 44(5), 711–721 (1996).
[CrossRef]

Kong, J. A.

Y. Zhang, Y. E. Yang, H. Braunisch, and J. A. Kong, “Electromagnetic wave interaction of conducting object with rough surface by hybrid SPM/MOM technique,” Prog. Electromagn. Res. 22, 315–335 (1999).
[CrossRef]

C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, “Electromagnetic scattering of waves by random rough surface: A finite- difference time-domain approach,” Microw. Opt. Technol. Lett. 4(9), 355–359 (1991).
[CrossRef]

Kubické, G.

G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: Extended PILE method combined with FB-SA,” Waves Random Complex Media 18(3), 495–519 (2008).
[CrossRef]

C. Bourlier, G. Kubické, and N. Déchamps, “Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA,” J. Opt. Soc. Am. A 25(4), 891–902 (2008).
[CrossRef]

Landesa, L.

M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998).

Lentz, R. R.

R. R. Lentz, “A numerical study of electromagnetic scattering from ocean-like surfaces,” Radio Sci. 9(12), 1139–1146 (1974).
[CrossRef]

Li, G.

Y. Q. Jin and G. Li, “Detection of a scatter target over a randomly rough surface by using the angular correlation function in a finite-element approach,” Waves Random Media 10(2), 273–280 (2000).
[CrossRef]

Li, J.

J. Li, L. X. Guo, and H. Zeng, “FDTD investigation on the electromagnetic scattering from a target above a randomly rough sea surface,” Waves Random Media 10, 273–280 (2008).

Li, L. W.

X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003).
[CrossRef]

Li, Z.

Z. Li and Y. Q. Jin, “Bistatic scattering from a fractal dynamic rough sea surface with a ship presence at low grazing-angle incidence using the FBM/SAA,” Microw. Opt. Technol. Lett. 31(2), 146–151 (2001).
[CrossRef]

Lixin, G.

G. Lixin and K. Cheyoung, “Light scattering models for a spherical particle above a slightly dielectric rough surface,” Microw. Opt. Technol. Lett. 33(2), 142–146 (2002).
[CrossRef]

Lou, S. H.

S. H. Lou, L. Tsang, and C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1(4), 287–307 (1991).
[CrossRef]

C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, “Electromagnetic scattering of waves by random rough surface: A finite- difference time-domain approach,” Microw. Opt. Technol. Lett. 4(9), 355–359 (1991).
[CrossRef]

Michielssen, E.

V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 36(3), 738–748 (1998).
[CrossRef]

Obelleiro, F.

M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998).

Phu, P.

L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993).
[CrossRef]

Pino, M. R.

M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998).

Rodriguez, J. L.

M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998).

Saillard, J.

G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: Extended PILE method combined with FB-SA,” Waves Random Complex Media 18(3), 495–519 (2008).
[CrossRef]

Sangani, H.

L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993).
[CrossRef]

Sarabandi, K.

T. Chiu and K. Sarabandi, “Electromagnetic scattering interaction between a dielectric cylinder and a slightly rough surface,” IEEE Trans. Antenn. Propag. 47(5), 902–913 (1999).
[CrossRef]

Sleijpeny, G. L. G.

G. L. G. Sleijpeny and D. R. Fokkema, “Bicgstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal. 1, 11–32 (1993).

Stiefel, E.

M. R. Hestenes and E. Stiefel, “Method of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. 49, 409–436 (1952).

Thorsos, E. I.

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 (1988).
[CrossRef]

Toutain, S.

Tsang, L.

L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993).
[CrossRef]

C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, “Electromagnetic scattering of waves by random rough surface: A finite- difference time-domain approach,” Microw. Opt. Technol. Lett. 4(9), 355–359 (1991).
[CrossRef]

S. H. Lou, L. Tsang, and C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1(4), 287–307 (1991).
[CrossRef]

Vesecky, J. F.

S. L. Durden and J. F. Vesecky, “A numerical study of the separation wavenumber in the two-scale scattering approximation,” IEEE Trans. Geosci. Rem. Sens. 28(2), 271–272 (1990).
[CrossRef]

Voronovich, A.

A. Voronovich, “Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half- spaces,” Waves Random Media 4(3), 337–367 (1994).
[CrossRef]

Wang, C. F.

X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003).
[CrossRef]

Wang, X.

X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003).
[CrossRef]

Winebrenner, D.

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Figures (14)

Fig. 1
Fig. 1

Geometric model of targets located both above and below the dielectric rough surface.

Fig. 2
Fig. 2

Physical interpretation of the EPILE for targets both above and below the dielectric rough surface.

Fig. 4
Fig. 4

BSC versus the scattering angle (HH polarization).

Fig. 5
Fig. 5

BSC versus the scattering angle (VV polarization).

Fig. 3
Fig. 3

Comparison of the angular distribution of BSC by our scheme and X. Wang’s method.

Fig. 6
Fig. 6

BSC versus the scattering angle (VV polarization).

Fig. 7
Fig. 7

BSC versus the scattering angle (different Xu).

Fig. 8
Fig. 8

BSC versus the scattering angle (different Hu).

Fig. 9
Fig. 9

BSC versus the scattering angle (different Rd).

Fig. 11
Fig. 11

BSC versus the scattering angle (different Dd).

Fig. 10
Fig. 10

BSC versus the scattering angle (different Xd).

Fig. 12
Fig. 12

BSC versus the scattering angle (different δ).

Fig. 13
Fig. 13

BSC versus the scattering angle (different l).

Fig. 14
Fig. 14

BSC versus the scattering angle (different θi ) .

Tables (2)

Tables Icon

Table 1 Comparison of Different Order EPILE Combined with Different Iteration Number of FBM in Relative Residual Error and Computational Time for One Dielectric Rough Surface Realization (HH Polarization)

Tables Icon

Table 2 Comparison of Different Order EPILE Combined with Different Iteration Number of FBM in Relative Residual Error and Computational Time for One Dielectric Rough Surface Realization (VV Polarization)

Equations (49)

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φ i n c ( r ) = exp ( j k i r ( 1 + [ 2 ( x + z tan θ i ) 2 / g 2 1 ] / ( k g cos θ i ) 2 ) ) exp ( ( x + z tan θ i ) 2 / g 2 ) ,
1 2 E 0 ( r ) S r [ E 0 ( r ' ) n ' G 0 ( r , r ' ) G 0 ( r , r ' ) n ' E 0 ( r ' ) ] d s ' +     S t 1 G 0 ( r , r ' ) n ' E 0 ( r ' ) d s '       = E i ( r )         r S r ,
S r [ E 0 ( r ' ) n ' G 0 ( r , r ' ) G 0 ( r , r ' ) n ' E 0 ( r ' ) ] d s '     S t 1 G 0 ( r , r ' ) n ' E 0 ( r ' ) d s '       = E i ( r )           r S t 1 ,
1 2 E 1 ( r ) + S r [ E 1 ( r ' ) n ' G 1 ( r , r ' ) G 1 ( r , r ' ) n ' E 1 ( r ' ) ρ ] d s ' S t 2 G 1 ( r , r ' ) n ' E 1 ( r ' ) d s '     = 0         r S r ,
S r [ E 1 ( r ' ) n ' G 1 ( r , r ' ) G 1 ( r , r ' ) n ' E 1 ( r ' ) ρ ] d s ' S t 2 G 1 ( r , r ' ) n ' E 1 ( r ' ) d s ' = 0           r S t 2 ,
1 2 H 0 ( r ) S r [ H 0 ( r ' ) n ' G 0 ( r , r ' ) G 0 ( r , r ' ) n ' H 0 ( r ' ) ] d s ' S t 1 H 0 ( r ' ) n ' G 0 ( r , r ' ) d s '       = H i ( r )             r S r / S t 1 ,
1 2 H 1 ( r ) + S r [ H 1 ( r ' ) n ' G 1 ( r , r ' ) G 1 ( r , r ' ) n ' H 1 ( r ' ) ρ ] d s ' + S t 2 [ H 1 ( r ' ) n ' G 1 ( r , r ' ) G 1 ( r , r ' ) n ' H 1 ( r ' ) ] d s '     = 0         r S r ,
1 2 H 1 ( r ) + S r [ H 1 ( r ' ) n ' G 1 ( r , r ' ) G 1 ( r , r ' ) n ' H 1 ( r ' ) ρ ] d s ' + S t 2 H 1 ( r ' ) n ' G 1 ( r , r ' ) d s ' = 0         r S t 2 ,
Z ¯ ( N t 1 + N t 2 + 2 N r ) × ( N t 1 + N t 2 + 2 N r ) X ( N t 1 + N t 2 + 2 N r ) = S ( N t 1 + N t 2 + 2 N r ) ,
Z ¯     ( N t 1 + N t 2 + 2 N r ) × ( N t 1 + N t 2 + 2 N r ) = [ A N t 1 × N t 1 0 N t 1 × N t 2 B N t 1 × N r C N t 1 × N r 0 N t 2 × N t 1 D N t 2 × N t 2 E N t 2 × N r ρ F N t 2 × N r G N r × N t 1 0 N r × N t 2 H N r × N r I N r × N r 0 N r × N t 1 J N r × N t 2 K N r × N r ρ L N r × N r ] .
Z ¯     ( N t 1 + N t 2 + 2 N r ) × ( N t 1 + N t 2 + 2 N r ) = [ A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r A ¯ ( 2 N r ) × ( 2 N r ) r ] ,
  A ¯     ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 = [ A N t 1 × N t 1 0 N t 2 × N t 2 0 N t 2 × N t 1 D N t 2 × N t 1 ]        A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 = [ B N t 1 × N r C N t 1 × N r E N t 2 × N r ρ F N t 2 × N r ] ,
A ¯     ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r = [ G N r × N t 1         0 N r × N t 2 0 N r × N t 1       J N r × N t 2 ]        A ¯ ( 2 N r ) × ( 2 N r ) r = [ H N r × N r I N r × N r K N r × N r ρ L N r × N r ] .
X ( N t 1 + N t 2 + 2 N r ) T = [ X t 1 , t 2 ( N t 1 + N t 2 ) T X r ( 2 N r ) T ]       S ( N t 1 + N t 2 + 2 N r ) T = [ S t 1 , t 2 ( N t 1 + N t 2 ) T S r ( 2 N r ) T ] ,
X t 1 , t 2 ( N t 1 + N t 2 ) T = [ ψ 0 ( r t 1 1 ) ψ 0 ( r t 1 N t 1 ) , ψ 1 ( r t 2 1 ) ψ 1 ( r t 2 N t 2 ) ]      X r ( 2 N r ) T = [ ψ 0 ( r r 1 ) ψ 0 ( r r N r ) , ψ 0 ( r r 1 ) n r ψ 0 ( r r N r ) n r ] ,
S t 1 , t 2 ( N t 1 + N t 2 ) T = [ ψ i ( r t 1 1 ) ψ i ( r t 1 N t 1 ) , 0 1 0 N t 2 ]      S r ( 2 N r ) T = [ ψ i ( r r 1 ) ψ i ( r r N r ) , 0 1 0 N r ] ,
A p q ( H H ) = { γ q Δ x j 4 H 0 ( 1 ) ( k 0 | r p r q | )                      p q γ q Δ x j 4 H 0 ( 1 ) [ k 0 Δ x γ q / ( 2 e ) ]       p = q ,     A p q ( V V ) = { γ q Δ x j k 0 4 ( n ^ R p q ) H 1 ( 1 ) ( k 0 | r p r q | )         p q                     1 2 z t 1 ( x p ) Δ x 4 π γ p 2                                                                                                                       p = q ,
B p n = γ n Δ x j k 0 4 ( n ^ R p n ) H 1 ( 1 ) ( k 0 | r p r n | ) ,     C p n = γ n Δ x j 4 H 0 ( 1 ) ( k 0 | r p r n | ) ,
D v w ( H H ) = { γ w Δ x j 4 H 0 ( 1 ) ( k 1 | r v r w | )                         v w γ w Δ x j 4 H 0 ( 1 ) [ k 1 Δ x γ w / ( 2 e ) ]       v = w ,     D v w ( V V ) = { γ w Δ x j k 1 4 ( n ^ R v w ) H 1 ( 1 ) ( k 1 | r v r w | )         v w                   1 2 + z t 2 ( x v ) Δ x 4 π γ v 2                                                                                                           v = w ,
E v n = γ n Δ x j k 1 4 ( n ^ R v n ) H 1 ( 1 ) ( k 1 | r v r n | ) ,     F v n = γ n Δ x j 4 H 0 ( 1 ) ( k 1 | r v r n | ) ,
G m q ( H H ) = γ q Δ x j 4 H 0 ( 1 ) ( k 0 | r m - r q | ) ,     G m q ( V V ) = γ q Δ x j k 0 4 ( n ^ × R m q ) H 1 ( 1 ) ( k 0 | r m - r q | ) ,
H m n = { γ n Δ x j k 0 4 ( n ^ R m n ) H 1 ( 1 ) ( k 0 | r m r n | )         m n                     1 2 z r ( x m ) Δ x 4 π γ m 2                                                                                                                         m = n ,     I m n = { γ n Δ x j 4 H 0 ( 1 ) ( k 0 | r m r n | )                              m n γ m Δ x j 4 H 0 ( 1 ) [ k 0 Δ x γ m / ( 2 e ) ]       m = n ,
J m v ( H H ) = γ v Δ x j 4 H 0 ( 1 ) ( k 1 | r m r v | ) ,     J m v ( V V ) = γ v Δ x j k 1 4 ( n ^ R m v ) H 1 ( 1 ) ( k 1 | r m r v | ) ,
K m n = { γ n Δ x j k 1 4 ( n ^ R m n ) H 1 ( 1 ) ( k 1 | r m r n | )       m n                   1 2 z r ( x m ) Δ x 4 π γ m 2                                                                                                         m = n ,      L m n = { γ n Δ x j 4 H 0 ( 1 ) ( k 1 | r m r n | )                              m n γ m Δ x j 4 H 0 ( 1 ) [ k 1 Δ x γ m / ( 2 e ) ]         m = n ,
ψ s ( r ) = e j k 0 r r ψ s N ( θ s , θ i ) .
ψ s N ( θ s , θ i ) ( TE , HH ) = j 4 2 π k 0 e j π 4 { S r exp ( j k s r ) [ j ( n ^ k s ) X r 1 ~ N r ( x ) X r N r + 1 ~ 2 N r ( x ) ]                                     1 + [ z r ( x ) ] 2 d x S t 1 X t 1 ( x ) exp ( j k s r ) 1 + [ z t 1 ( x ) ] 2 d x },
ψ s N ( θ s , θ i ) ( TM , VV ) = j 4 2 π k 0 e j π 4 { S r [ j ( n ^ k s ) X r 1 ~ N r ( x ) X r N r + 1 ~ 2 N r ( x ) ] exp ( j k s r ) 1 + [ z r ( x ) ] 2 d x                                     S t 1 j ( n ^ t 1 k s ) X t 1 ( x ) exp ( j k s r ) 1 + [ z t 1 ( x ) ] 2 d x },
σ s ( θ s , θ i ) = | ψ s N ( θ s , θ i ) | 2 g π / 2 cos θ i ( 1 ( 1 + 2 tan 2 θ i ) / ( 2 k 0 2 g 2 cos 2 θ i ) ) .
Z ¯ ( 2 N r + N t 1 + N t 2 ) × ( 2 N r + N t 1 + N t 2 ) 1 = [ T ¯ U ¯ V ¯ W ¯ ] ,
T ¯ = [ A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ] 1 ,
U ¯ = [ A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ] 1 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 ,
V ¯ = ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r [ A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ] 1 ,
W ¯ = ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r [ A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2             ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ] 1 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 .
X r = [ A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ] 1 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 S r ,
X t 1 , t 2 = T S t 1 , t 2 = [ A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ] 1 S t 1 , t 2 .
M c ( r ) = ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ( A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 ) 1 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2
X r ( 2 N r ) ( p ) = ( p = 0 P M c ( r ) p ) ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 ( S r A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ( A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 ) 1 S t 1 , t 2 ) ,
M c ( t 1 , t 2 ) = ( A ¯ ( 2 N t 1 ) × ( N t 1 + N t 2 ) t 1 , t 2 ) 1 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r .
X t 1 , t 2 ( N t 1 + N t 2 ) ( p ) = ( p = 0 P M c ( t 1 , t 2 ) p ) ( A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 ) 1 ( S t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 S r ) .
X r ( 2 N r ) ( p ) = p = 0 P Y ( r ) ( p ) ,     X t 1 , t 2 ( N t 1 + N t 2 ) ( p ) = p = 0 P Y ( t 1 , t 2 ) ( p ) .
Y r ( 0 ) = ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 ( S r A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ( A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 ) 1 S t 1 , t 2 ) ,     Y r ( p ) = M c ( r ) Y r ( p 1 ) ,
Y ( t 1 , t 2 ) ( 0 ) = ( A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 ) 1 ( S t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 S r ) ,     Y ( t 1 , t 2 ) ( p ) = M c ( t 1 , t 2 ) Y ( t 1 , t 2 ) ( p 1 ) .
A ¯ r ( ( 2 N r ) × ( 2 N r ) ) = A ¯ r ( ( 2 N r ) × ( 2 N r ) ) f + A ¯ r ( ( 2 N r ) × ( 2 N r ) ) d + A ¯ r ( ( 2 N r ) × ( 2 N r ) ) b ,     ξ r ( 2 N r ) = ξ r ( 2 N r ) f + ξ r ( 2 N r ) b ,
[ H N r × N r I N r × N r K N r × N r ρ L N r × N r ] [ ξ 1 r ( N r ) T ξ 2 r ( N r ) T ] = [ ζ 1 r ( N r ) T ζ 2 r ( N r ) T ] ,
H N r × N r = H N r × N r f + H N r × N r s + H N r × N r b ,     I N r × N r = I N r × N r f + I N r × N r s + I N r × N r b ,
K N r × N r = K N r × N r f + K N r × N r s + K N r × N r b ,     L N r × N r = L N r × N r f + L N r × N r s + L N r × N r b .
RRE = 90 90 | σ ( EPILE + FBM ) σ ( MOM ( CGM / LU / BCGM ) ) | 2 90 90 | σ MOM ( CGM / LU / BCGM ) | 2 .
M c ( r ) Y r ( p 1 ) = ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ( A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 ) 1 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 Y r ( p 1 ) ο ( ( N t 1 + N t 2 ) × ( 2 N r ) )     ( a ) ο ( M i t e r 12 ( N t 1 + N t 2 ) 2 ) o r ο ( 4 ( N t 1 / 3 + N t 2 / 3 ) 3 )     ( b ) ο ( ( 2 N r ) × ( N t 1 + N t 2 ) )     ( c ) ο ( ( 2 N r ) 2 )     ( d ) ,
M c ( t 1 , t 2 ) Y ( t 1 , t 2 ) ( p 1 ) = ( A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 ) 1 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r Y ( t 1 , t 2 ) ( p 1 ) ο ( ( 2 N r ) × ( N t 1 + N t 2 ) )     ( e ) ο ( ( 2 N r ) 2 )     ( f ) ο ( ( N t 1 + N t 2 ) × ( 2 N r ) )     ( g ) ο ( M i t e r 12 ( N t 1 + N t 2 ) 2 ) o r ο ( 4 ( N t 1 / 3 + N t 2 / 3 ) 3 )     ( h ) .

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