Abstract
In this paper we propose a new method to determine the location and shape of an unbounded rough surface from measurements of scattered electromagnetic waves. The proposed method is based on the point source method of Potthast (IMA J. Appl. Math. 61, 119–140, 1998) for inverse scattering by bounded obstacles. We propose a version for inverse rough surface scattering which can reconstruct the total field when the incident field is not necessarily time harmonic. We present numerical results for the case of a perfectly conducting surface in TE polarization, in which case a homogeneous Dirichlet condition applies on the boundary. The results show great accuracy of reconstruction of the total field and of the prediction of the surface location.
Similar content being viewed by others
References
Arens, T.: Why linear sampling works! Inverse Problems 20, 163–173 (2004).
Arens, T., Kirsch A.: The factorization method in inverse scattering from periodic structures. Inverse Problems 19, 1195–1211 (2003).
Chandler-Wilde S. N.: The impedance boundary value problem for the Helmholtz equation in a half-plane. Math. Meth. Appl. Sci. 20, 813–840 (1997).
Chandler-Wilde S. N., Hothersall D. C.: Efficient calculation of the Green function for acoustic propagation above a homogeneous impedance plane. J. Sound Vib. 180, 705–724 (1995).
Chandler-Wilde, S. N., Ross C. R.: Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane. Math. Meth. Appl. Sci. 19, 959–976 (1996).
Chandler-Wilde S. N., Ross C. R., Zhang B.: Scattering by rough surfaces. In: Proc. 4th Int. Conf. Math. Numer. Aspects wave prop. (DeSanto, J., ed.), pp. 164–168. SIAM 1998.
Chandler-Wilde, S. N., Ross, C. R., Zhang, B.: Scattering by infinite one-dimensional rough surfaces. Proc. R. Soc. Lon. A 455, 3767–3787 (1999).
Cheney, M.: A mathematical tutorial on synthetic aperture radar. SIAM Rev. 43, 301–312 (2001).
Coifman, R., Goldberg, M., Hrycak, T., Israeli M., Rokhlin, V.: An improved operator expansion algorithm for direct and inverse scattering computations. Waves in Random Media 9, 441–457 (1999).
Colton D., Coyle J, Monk, P.: Recent developments in inverse acoustic scattering theory. SIAM Rev. 42, 369–414 (2000).
Colton, D., Kirsch, A.: A simple method for solving inverse scattering problems in the resonance region. Inverse Problems 12, 383–393 (1996).
DeSanto, J. A., Wombell, R. J.: The reconstruction of shallow rough-surface profiles from scattered field data. Inverse Problems 7, L7–L12 (1991).
Elschner, J., Yamamoto, M.: An inverse problem in periodic diffractive optics: Reconstruction of Lipschitz grating profiles. Appl. Anal. 81, 1307–1328 (2002).
Goodman, D., Conyers, L.: Ground penetrating radar. An introduction for archaeologists. AltaMira 1997.
Ikehata M.: On reconstruction in the inverse conductivity problem with one measurement. Inverse Problems 16, 785–793 (2000).
Kirsch, A.: Characterization of the shape of a scattering obstacle using the spectral data of the far field operator. Inverse Problems 14, 1489–1512 (1998).
Kress, R.: Linear integral equations. Berlin: Springer 1989.
Lines C. D.: Inverse scattering by unbounded rough surfaces. PhD thesis, Brunel University 2003.
Lines, C. D., Chandler-Wilde, S. N.: A point source method for inverse scattering by rough surfaces. (in preparation).
Meier A.: Numerical treatment of integral equations on the real line with application to Acoustic scattering by unbounded rough surfaces. PhD thesis, Brunel University 2001.
Meier, A., Arens, T., Chandler-Wilde, S. N., Kirsch, A.: A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces. J. Int. Eq. Appl. 12, 281–321 (2000).
Newland, D. E.: An introduction to random vibrations, spectral and wavelet analysis. Longman 1993.
Potthast, R.: A point source method for inverse acoustic and electromagnetic obstacle scattering problems. IMA J. Appl. Math. 61, 119–140 (1998).
Potthast R.: Point sources and multipoles in inverse scattering theory. CRC Press 2001.
Potthast, R., Luke, D.: The no response test – a sampling method for inverse scattering problems. SIAM J. Appl. Math. 63, 1292–1312 (2003).
Potthast, R., Sylvester, J., Kusiak, S.: A ‘‘range test’’ for determining scatterers with unknown physical properties. Inverse Problems 19, 533–547 (2003).
Sheppard C.: Imaging of random surfaces and inverse scattering in the Kirchhoff approximation. Waves in Random Media 8, 53–66 (1998).
Spivack M.: Direct solution of the inverse problem for rough scattering at grazing incidence. J. Phys. A: Math. Gen. 25, 3295–3302 (1992).
Wombell, R. J., DeSanto, J. A.: Reconstruction of rough-surface profiles with the Kirchhoff approximation. J. Opt. Soc. Am. 8, 1892–1897 (1991).
Ying, C., Noguchi, A.: Rough surface inverse scattering problem with Gaussian beam illumination. IEICE Trans. Electron. E77–C(11), 1781–1785 (1994).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lines, C., Chandler-Wilde, S. A Time Domain Point Source Method for Inverse Scattering by Rough Surfaces. Computing 75, 157–180 (2005). https://doi.org/10.1007/s00607-004-0109-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00607-004-0109-8