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October  2019, 13(5): 959-981. doi: 10.3934/ipi.2019043

## On finding a buried obstacle in a layered medium via the time domain enclosure method in the case of possible total reflection phenomena

 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  July 2018 Revised  March 2019 Published  July 2019

An inverse obstacle problem for the wave governed by the wave equation in a two layered medium is considered under the framework of the time domain enclosure method. The wave is generated by an initial data supported on a closed ball in the upper half-space, and observed on the same ball over a finite time interval. The unknown obstacle is penetrable and embedded in the lower half-space. It is assumed that the propagation speed of the wave in the upper half-space is greater than that of the wave in the lower half-space, which is excluded in the previous study: Ikehata and Kawashita, Inverse Problems and Imaging 12 (2018), no.5, 1173-1198. In the present case, when the reflected waves from the obstacle enter the upper half-space, the total reflection phenomena occur, which give singularities to the integral representation of the fundamental solution for the reduced transmission problem in the background medium. This fact makes the problem more complicated. However, it is shown that these waves do not have any influence on the leading profile of the indicator function of the time domain enclosure method.

Citation: Masaru Ikehata, Mishio Kawashita, Wakako Kawashita. On finding a buried obstacle in a layered medium via the time domain enclosure method in the case of possible total reflection phenomena. Inverse Problems and Imaging, 2019, 13 (5) : 959-981. doi: 10.3934/ipi.2019043
##### References:
 [1] 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. [2] D. J. Daniels, D. J. Gunton and H. F. Scott, Introduction to subsurface radar, IEE Proceedings, 135 (1988), 278-320.  doi: 10.1049/ip-f-1.1988.0038. [3] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Sciences and Technology, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1. [4] 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, Inverse Problems, 24 (2008), 015002 (18pp). doi: 10.1088/0266-5611/24/1/015002. [5] M. Ikehata, Extracting discontinuity in a heat conductive body. One-space dimensional case, Applicable Analysis, 86 (2007), 963-1005.  doi: 10.1080/00036810701460834. [6] 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. [7] 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. [8] 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. [9] M. Ikehata, New development of the enclosure method for inverse obstacle scattering, as Chapter 6, in Inverse Problems and Computational Mechanics (eds. Marin, L., Munteanu, L., Chiroiu, V.), 2, Editura Academiei, 123–147, Bucharest, Romania, 2016. [10] M. Ikehata and M. Kawashita, On finding a buried obstacle in a layered medium via the time domain enclosure method, Inverse Problems and Imaging, 12 (2018), 1173-1198.  doi: 10.3934/ipi.2018049. [11] 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. [12] 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.

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##### References:
 [1] 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. [2] D. J. Daniels, D. J. Gunton and H. F. Scott, Introduction to subsurface radar, IEE Proceedings, 135 (1988), 278-320.  doi: 10.1049/ip-f-1.1988.0038. [3] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Sciences and Technology, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1. [4] 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, Inverse Problems, 24 (2008), 015002 (18pp). doi: 10.1088/0266-5611/24/1/015002. [5] M. Ikehata, Extracting discontinuity in a heat conductive body. One-space dimensional case, Applicable Analysis, 86 (2007), 963-1005.  doi: 10.1080/00036810701460834. [6] 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. [7] 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. [8] 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. [9] M. Ikehata, New development of the enclosure method for inverse obstacle scattering, as Chapter 6, in Inverse Problems and Computational Mechanics (eds. Marin, L., Munteanu, L., Chiroiu, V.), 2, Editura Academiei, 123–147, Bucharest, Romania, 2016. [10] M. Ikehata and M. Kawashita, On finding a buried obstacle in a layered medium via the time domain enclosure method, Inverse Problems and Imaging, 12 (2018), 1173-1198.  doi: 10.3934/ipi.2018049. [11] 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. [12] 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.
Setting of the problem
Contour of the integrals
Propagation from the lower half-space
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