October  2018, 12(5): 1173-1198. doi: 10.3934/ipi.2018049

On finding a buried obstacle in a layered medium via the time domain enclosure method

1. 

Laboratory of Mathematics, Graduate School of Engineering, Hiroshima University, Higashihiroshima 739-8527, Japan

2. 

Department of Mathematics, Graduate School of Sciences, Hiroshima University, Higashihiroshima 739-8526, Japan

Received  June 2017 Revised  February 2018 Published  July 2018

An inverse obstacle problem for the wave equation in a two layered medium is considered. It is assumed that the unknown obstacle is penetrable and embedded in the lower half-space. The wave as a solution of the wave equation is generated by an initial data whose support is in the upper half-space and observed at the same place as the support over a finite time interval. From the observed wave an indicator function in the time domain enclosure method is constructed. It is shown that, one can find some information about the geometry of the obstacle together with the qualitative property in the asymptotic behavior of the indicator function.

Citation: Masaru Ikehata, Mishio Kawashita. On finding a buried obstacle in a layered medium via the time domain enclosure method. Inverse Problems & Imaging, 2018, 12 (5) : 1173-1198. doi: 10.3934/ipi.2018049
References:
[1]

J. D. Achenbach, Wave Propagation in Elastic Solids, North-Holland, 1976.  Google Scholar

[2]

E. J. Baranoski, Through-wall imaging: Historical perspective and future directions, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, (2008). doi: 10.1109/ICASSP.2008.4518824.  Google Scholar

[3]

L. M. Brekhovskikh, Waves in Layered Media, Translated from the Russian by D. Lieberman, Academic Press, New York, 1960.  Google Scholar

[4]

D. J. DanielsD. J. Gunton and H. F. Scott, Introduction to subsurface radar, IEE Proceedings F Communications, Radar and Signal Processing, 135 (1988), 278-320.  doi: 10.1049/ip-f-1.1988.0038.  Google Scholar

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R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Sciences and Technology, Evolution problems Ⅰ, Vol. 5, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1.  Google Scholar

[6]

F. Delbary, K. Erhard, R. Kress, R. Potthast and J. Schulz, Inverse electromagnetic scattering in a two-layered medium with an application to mine detection, Invesre Problems, 24 (2008), 015002, 18pp. doi: 10.1088/0266-5611/24/1/015002.  Google Scholar

[7]

L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-96750-4.  Google Scholar

[8]

M. Ikehata, Size estimation of inclusion, J. Inv. Ill-Posed Problems, 6 (1998), 127-140.  doi: 10.1515/jiip.1998.6.2.127.  Google Scholar

[9]

M. Ikehata, Extracting discontinuity in a heat conductive body. One-space dimensional case, Applicable Analysis, 86 (2007), 963-1005.  doi: 10.1080/00036810701460834.  Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

M. Ikehata, On finding an obstacle embedded in the rough background medium via the enclosure method in the time domain, Inverse Problems, 31 (2015), 085011(21pp). doi: 10.1088/0266-5611/31/8/085011.  Google Scholar

[13]

M. Ikehata, New development of the enclosure method for inverse obstacle scattering, Chapter 6 in Inverse Problems and Computational Mechanics (eds. L. Marin, L. Munteanu, V. Chiroiu), Editura Academiei, Bucharest, Romania, 2 (2016), 123-147. Google Scholar

[14]

M. Ikehata and M. Kawashita, The enclosure method for the heat equation, Inverse Problems, 25 (2009), 075005, 10pp. doi: 10.1088/0266-5611/25/7/075005.  Google Scholar

[15]

M. Ikehata and M. Kawashita, An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method, Inverse Problems and Imaging, 8 (2014), 1073-1116.  doi: 10.3934/ipi.2014.8.1073.  Google Scholar

[16]

J. B. KellerR. M. Lewis and B. D. Seckler, Asymptotic solution of some diffraction problems, Commun. Pure Appl. Math., 9 (1956), 207-265.  doi: 10.1002/cpa.3160090205.  Google Scholar

[17]

J. Li, P. Li, H. Liu and X. Liu, Recovering multiscale buried anomalies in a two-layered medium, Inverse Problems, 31 (2015), 105006, 28pp. doi: 10.1088/0266-5611/31/10/105006.  Google Scholar

[18]

X. Liu and B. Zhang, A uniqueness result for the inverse electromagnetic scattering problem in a two-layered medium, Inverse Problems, 26 (2010), 105007, 11pp. doi: 10.1088/0266-5611/26/10/105007.  Google Scholar

show all references

References:
[1]

J. D. Achenbach, Wave Propagation in Elastic Solids, North-Holland, 1976.  Google Scholar

[2]

E. J. Baranoski, Through-wall imaging: Historical perspective and future directions, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, (2008). doi: 10.1109/ICASSP.2008.4518824.  Google Scholar

[3]

L. M. Brekhovskikh, Waves in Layered Media, Translated from the Russian by D. Lieberman, Academic Press, New York, 1960.  Google Scholar

[4]

D. J. DanielsD. J. Gunton and H. F. Scott, Introduction to subsurface radar, IEE Proceedings F Communications, Radar and Signal Processing, 135 (1988), 278-320.  doi: 10.1049/ip-f-1.1988.0038.  Google Scholar

[5]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Sciences and Technology, Evolution problems Ⅰ, Vol. 5, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1.  Google Scholar

[6]

F. Delbary, K. Erhard, R. Kress, R. Potthast and J. Schulz, Inverse electromagnetic scattering in a two-layered medium with an application to mine detection, Invesre Problems, 24 (2008), 015002, 18pp. doi: 10.1088/0266-5611/24/1/015002.  Google Scholar

[7]

L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-96750-4.  Google Scholar

[8]

M. Ikehata, Size estimation of inclusion, J. Inv. Ill-Posed Problems, 6 (1998), 127-140.  doi: 10.1515/jiip.1998.6.2.127.  Google Scholar

[9]

M. Ikehata, Extracting discontinuity in a heat conductive body. One-space dimensional case, Applicable Analysis, 86 (2007), 963-1005.  doi: 10.1080/00036810701460834.  Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

M. Ikehata, On finding an obstacle embedded in the rough background medium via the enclosure method in the time domain, Inverse Problems, 31 (2015), 085011(21pp). doi: 10.1088/0266-5611/31/8/085011.  Google Scholar

[13]

M. Ikehata, New development of the enclosure method for inverse obstacle scattering, Chapter 6 in Inverse Problems and Computational Mechanics (eds. L. Marin, L. Munteanu, V. Chiroiu), Editura Academiei, Bucharest, Romania, 2 (2016), 123-147. Google Scholar

[14]

M. Ikehata and M. Kawashita, The enclosure method for the heat equation, Inverse Problems, 25 (2009), 075005, 10pp. doi: 10.1088/0266-5611/25/7/075005.  Google Scholar

[15]

M. Ikehata and M. Kawashita, An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method, Inverse Problems and Imaging, 8 (2014), 1073-1116.  doi: 10.3934/ipi.2014.8.1073.  Google Scholar

[16]

J. B. KellerR. M. Lewis and B. D. Seckler, Asymptotic solution of some diffraction problems, Commun. Pure Appl. Math., 9 (1956), 207-265.  doi: 10.1002/cpa.3160090205.  Google Scholar

[17]

J. Li, P. Li, H. Liu and X. Liu, Recovering multiscale buried anomalies in a two-layered medium, Inverse Problems, 31 (2015), 105006, 28pp. doi: 10.1088/0266-5611/31/10/105006.  Google Scholar

[18]

X. Liu and B. Zhang, A uniqueness result for the inverse electromagnetic scattering problem in a two-layered medium, Inverse Problems, 26 (2010), 105007, 11pp. doi: 10.1088/0266-5611/26/10/105007.  Google Scholar

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