The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain

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February 2016, 10(1): 131-163. doi: 10.3934/ipi.2016.10.131

The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain

Masaru Ikehata ^1,

1.	Laboratory of Mathematics, Institute of Engineering, Hiroshima University, Higashi Hiroshima 739-8527

Received October 2014 Revised July 2015 Published February 2016

Abstract
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Under a Creative Commons license

In this paper, a time domain enclosure method for an inverse obstacle scattering problem of electromagnetic wave is introduced. The wave as a solution of Maxwell's equations is generated by an applied volumetric current having an orientation and supported outside an unknown obstacle and observed on the same support over a finite time interval. It is assumed that the obstacle is a perfect conductor. Two types of analytical formulae which employ a single observed wave and explicitly contain information about the geometry of the obstacle are given. In particular, an effect of the orientation of the current is catched in one of two formulae. Two corollaries concerning with the detection of the points on the surface of the obstacle nearest to the centre of the current support and curvatures at the points are also given.

Keywords: Enclosure method, inverse obstacle scattering, obstacle, electromagnetic wave, reflection, mean value theorem., Maxwell`s equations.

Mathematics Subject Classification: Primary: 35R30, 35L50, 35Q61; Secondary: 78A46, 78M3.

Citation: Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems and Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131

References:

[1]	H. Ammari, G. Bao and J. L. Fleming, An inverse source problem for Maxwell's equations in magnetoencephalography, SIAM J. Appl. Math., 62 (2002), 1369-1382. doi: 10.1137/S0036139900373927.
[2]	H. Ammari, C. Latiri-Grouz and J.-C. Nédélec, The Leontovich boundary value problem for the time-harmonic Maxwell equations, Asymptotic Analysis, 18 (1998), 33-47.
[3]	C. Athanasiadis, P. A. Martin and I. G. Stratis, On the scattering of point-generated electromagnetic waves by a perfectly conducting sphere, and related near-field inverse problems, Short Communication, ZAMM$\cdot$Z. Angew. Math. Mech. 83 (2003), 129-136. doi: 10.1002/zamm.200310012.
[4]	C. A. Balanis, Antenna Theory, Analysis and Design, $3^{rd}$ edition, Wiley-Interscience, Hoboken, New Jersey, 2005.
[5]	N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, $2^{nd}$ edition, Dover Publications, New York, 1986.
[6]	R. J. Burkholder, I. J. Gupta and J. T. Johnson, Comparison of monostatic and bistatic radar images, IEEE Antennas and Propagation Magazine, 45 (2003), 41-50.
[7]	M. Cheney and B. Borden, Fundamentals of Radar Imaging, CBMS-NSF, Regional Conference Series in Applied Mathematics, 79, SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898719291.
[8]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, $3^{rd}$ edition, New York, Springer, 2013. doi: 10.1007/978-1-4614-4942-3.
[9]	R. Courant and D. Hilbert, Methoden der Mathematischen Physik, Vol. 2, Berlin, Springer, 1937.
[10]	R. Dautray and J.-L. Lions., Mathematical Analysis and Numerical Methods for Sciences and Technology, Spectral Theory and Applications, Vol. 3, Springer-Verlag, Berlin, 1990.
[11]	R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Sciences and Technology, Vol. 5, Evolution Problems I, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1.
[12]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 ed., Springer-Verlag, Berlin, Heidelberg, New York, 2001.
[13]	M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, 15 (1999), 1231-1241. doi: 10.1088/0266-5611/15/5/308.
[14]	M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval, Inverse Problems, 26 (2010), 055010, 20pp. doi: 10.1088/0266-5611/26/5/055010.
[15]	M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval: II. Obstacles with a dissipative boundary or finite refractive index and back-scattering data, Inverse Problems, 28 (2012), 045010, 29pp. doi: 10.1088/0266-5611/28/4/045010.
[16]	M. Ikehata, An inverse acoustic scattering problem inside a cavity with dynamical back-scattering data, Inverse Problems, 28 (2012), 095016, 24pp. doi: 10.1088/0266-5611/28/9/095016.
[17]	M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval: III. Sound-soft obstacle and bistatic data, Inverse Problems, 29 (2013), 085013, 35pp. doi: 10.1088/0266-5611/29/8/085013.
[18]	M. Ikehata, Extracting the geometry of an obstacle and a zeroth-order coefficient of a boundary condition via the enclosure method using a single reflected wave over a finite time interval, Inverse Problems, 30 (2014), 045011, 24pp. doi: 10.1088/0266-5611/30/4/045011.
[19]	M. Ikehata and H. Itou, On reconstruction of a cavity in a linearized viscoelastic body from infinitely many transient boundary data, Inverse Problems, 28 (2012), 125003, 19pp. doi: 10.1088/0266-5611/28/12/125003.
[20]	M. Ikehata and M. Kawashita, On the reconstruction of inclusions in a heat conductive body from dynamical boundary data over a finite time interval, Inverse Problems, 26 (2010), 095004, 15pp. doi: 10.1088/0266-5611/26/9/095004.
[21]	V. Isakov, Inverse obstacle problems, Topical review, Inverse Problems, 25 (2009), 123002, 18pp. doi: 10.1088/0266-5611/25/12/123002.
[22]	P. D. Lax and R. S. Phillips, The scattering of sound waves by an obstacle, Comm. Pure and Appl. Math., 30 (1977), 195-233. doi: 10.1002/cpa.3160300204.
[23]	H. Liu, M. Yamamoto and J. Zou, Reflection principle for the Maxwell equations and its application to inverse electromagnetic scattering, Inverse Problems, 23 (2007), 2357-2366. doi: 10.1088/0266-5611/23/6/005.
[24]	A. Majda and M. Taylor, Inverse scattering problems for transparent obstacles, electromagnetic waves, and hyperbolic systems, Comm. in Partial Differential Equations, 2 (1977), 395-438. doi: 10.1080/03605307708820035.
[25]	J.-C. Nédélec, Acoustic and Electromagnetic Equations, Integral Representations for Harmonic Problems, Springer, New York, 2001. doi: 10.1007/978-1-4757-4393-7.
[26]	B. O'Neill, Elementary Differential Geometry, Revised, $2^{nd}$ edition, Academic Press, 2006.

show all references

References:

[1]	H. Ammari, G. Bao and J. L. Fleming, An inverse source problem for Maxwell's equations in magnetoencephalography, SIAM J. Appl. Math., 62 (2002), 1369-1382. doi: 10.1137/S0036139900373927.
[2]	H. Ammari, C. Latiri-Grouz and J.-C. Nédélec, The Leontovich boundary value problem for the time-harmonic Maxwell equations, Asymptotic Analysis, 18 (1998), 33-47.
[3]	C. Athanasiadis, P. A. Martin and I. G. Stratis, On the scattering of point-generated electromagnetic waves by a perfectly conducting sphere, and related near-field inverse problems, Short Communication, ZAMM$\cdot$Z. Angew. Math. Mech. 83 (2003), 129-136. doi: 10.1002/zamm.200310012.
[4]	C. A. Balanis, Antenna Theory, Analysis and Design, $3^{rd}$ edition, Wiley-Interscience, Hoboken, New Jersey, 2005.
[5]	N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, $2^{nd}$ edition, Dover Publications, New York, 1986.
[6]	R. J. Burkholder, I. J. Gupta and J. T. Johnson, Comparison of monostatic and bistatic radar images, IEEE Antennas and Propagation Magazine, 45 (2003), 41-50.
[7]	M. Cheney and B. Borden, Fundamentals of Radar Imaging, CBMS-NSF, Regional Conference Series in Applied Mathematics, 79, SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898719291.
[8]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, $3^{rd}$ edition, New York, Springer, 2013. doi: 10.1007/978-1-4614-4942-3.
[9]	R. Courant and D. Hilbert, Methoden der Mathematischen Physik, Vol. 2, Berlin, Springer, 1937.
[10]	R. Dautray and J.-L. Lions., Mathematical Analysis and Numerical Methods for Sciences and Technology, Spectral Theory and Applications, Vol. 3, Springer-Verlag, Berlin, 1990.
[11]	R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Sciences and Technology, Vol. 5, Evolution Problems I, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1.
[12]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 ed., Springer-Verlag, Berlin, Heidelberg, New York, 2001.
[13]	M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, 15 (1999), 1231-1241. doi: 10.1088/0266-5611/15/5/308.
[14]	M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval, Inverse Problems, 26 (2010), 055010, 20pp. doi: 10.1088/0266-5611/26/5/055010.
[15]	M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval: II. Obstacles with a dissipative boundary or finite refractive index and back-scattering data, Inverse Problems, 28 (2012), 045010, 29pp. doi: 10.1088/0266-5611/28/4/045010.
[16]	M. Ikehata, An inverse acoustic scattering problem inside a cavity with dynamical back-scattering data, Inverse Problems, 28 (2012), 095016, 24pp. doi: 10.1088/0266-5611/28/9/095016.
[17]	M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval: III. Sound-soft obstacle and bistatic data, Inverse Problems, 29 (2013), 085013, 35pp. doi: 10.1088/0266-5611/29/8/085013.
[18]	M. Ikehata, Extracting the geometry of an obstacle and a zeroth-order coefficient of a boundary condition via the enclosure method using a single reflected wave over a finite time interval, Inverse Problems, 30 (2014), 045011, 24pp. doi: 10.1088/0266-5611/30/4/045011.
[19]	M. Ikehata and H. Itou, On reconstruction of a cavity in a linearized viscoelastic body from infinitely many transient boundary data, Inverse Problems, 28 (2012), 125003, 19pp. doi: 10.1088/0266-5611/28/12/125003.
[20]	M. Ikehata and M. Kawashita, On the reconstruction of inclusions in a heat conductive body from dynamical boundary data over a finite time interval, Inverse Problems, 26 (2010), 095004, 15pp. doi: 10.1088/0266-5611/26/9/095004.
[21]	V. Isakov, Inverse obstacle problems, Topical review, Inverse Problems, 25 (2009), 123002, 18pp. doi: 10.1088/0266-5611/25/12/123002.
[22]	P. D. Lax and R. S. Phillips, The scattering of sound waves by an obstacle, Comm. Pure and Appl. Math., 30 (1977), 195-233. doi: 10.1002/cpa.3160300204.
[23]	H. Liu, M. Yamamoto and J. Zou, Reflection principle for the Maxwell equations and its application to inverse electromagnetic scattering, Inverse Problems, 23 (2007), 2357-2366. doi: 10.1088/0266-5611/23/6/005.
[24]	A. Majda and M. Taylor, Inverse scattering problems for transparent obstacles, electromagnetic waves, and hyperbolic systems, Comm. in Partial Differential Equations, 2 (1977), 395-438. doi: 10.1080/03605307708820035.
[25]	J.-C. Nédélec, Acoustic and Electromagnetic Equations, Integral Representations for Harmonic Problems, Springer, New York, 2001. doi: 10.1007/978-1-4757-4393-7.
[26]	B. O'Neill, Elementary Differential Geometry, Revised, $2^{nd}$ edition, Academic Press, 2006.

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