# American Institute of Mathematical Sciences

February  2012, 6(1): 77-94. doi: 10.3934/ipi.2012.6.77

## Inverse obstacle scattering with limited-aperture data

 1 Department of Mathematics, Graduate School of Engineering, Gunma University, Japan 2 Department of Mathematics and Statistics, University of Helsinki, Finland

Received  March 2011 Revised  October 2011 Published  February 2012

Inverse obstacle scattering aims to extract information about distant and unknown targets using wave propagation. This study concentrates on a two-dimensional setting using time-harmonic acoustic plane waves as incident fields and taking the obstacles to be sound-hard with smooth or polygonal boundary. Measurement data is simulated by sending one incident wave towards the area of interest and computing the far field pattern (1) on the whole circle of observation directions, (2) only in directions close to backscattering, and (3) only in directions close to forward-scattering. A variant of the enclosure method is introduced, based on applying the far field operator to an explicitly constructed density, yielding information about the convex hull of the obstacle. The numerical evidence presented suggests that the convex hull of obstacles can be approximately recovered from noisy limited-aperture far field data.
Citation: Masaru Ikehata, Esa Niemi, Samuli Siltanen. Inverse obstacle scattering with limited-aperture data. Inverse Problems and Imaging, 2012, 6 (1) : 77-94. doi: 10.3934/ipi.2012.6.77
##### References:
 [1] M. Bruhl and M. Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography, Inverse Problems, 16 (2000), 1029-1042. doi: 10.1088/0266-5611/16/4/310. [2] F. Cakoni, D. Colton and P. Monk, "The Linear Sampling Method in Inverse Electromagnetic Scattering," CBMS-NSF Regional Conference Series in Applied Mathematics, 80, SIAM, Philadelphia, PA, 2011. [3] D. Colton, J. Coyle and P. Monk, Recent developments in inverse acoustic scattering theory, SIAM Review, 42 (2000), 369-414. doi: 10.1137/S0036144500367337. [4] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003. [5] D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. [6] D. Colton and R. Kress, Eigenvalues of the far field operator for the Helmholtz equation in an absorbing medium, SIAM J. Appl. Math., 55 (1995), 1724-1735. doi: 10.1137/S0036139993256114. [7] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," 2nd edition, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998. [8] D. Colton and R. Kress, Using fundamental solutions in inverse scattering, Inverse Problems, 22 (2006), R49-R66. doi: 10.1088/0266-5611/22/3/R01. [9] D. Colton and P. Monk, A linear sampling method for the detection of leukemia using microwaves, SIAM J. Appl. Math., 58 (1998), 926-941. doi: 10.1137/S0036139996308005. [10] F. Hettlich, On the uniqueness of the inverse conductive scattering problem for the Helmholtz equation, Inverse Problems, 10 (1994), 129-144. doi: 10.1088/0266-5611/10/1/010. [11] M. Ikehata, Reconstruction of an obstacle from the scattering amplitude at a fixed frequency, Inverse Problems, 14 (1998), 949-954. doi: 10.1088/0266-5611/14/4/012. [12] M. Ikehata, Reconstruction of a source domain from the Cauchy data, Inverse Problems, 15 (1999), 637-645. doi: 10.1088/0266-5611/15/2/019. [13] M. Ikehata, Reconstruction of obstacle from boundary measurements, Wave Motion, 30 (1999), 205-223. doi: 10.1016/S0165-2125(99)00006-2. [14] M. Ikehata, How to draw a picture of an unknown inclusion from boundary measurements. Two mathematical inversion algorithms, J. Inv. Ill-Posed Problems, 7 (1999), 255-271. doi: 10.1515/jiip.1999.7.3.255. [15] M. Ikehata, Reconstruction of the support function for inclusion from boundary measurements, J. Inv. Ill-Posed Problems, 8 (2000), 367-378. [16] M. Ikehata, Complex geometrical optics solutions and inverse crack problems, Inverse Problems, 19 (2003), 1385-1405. doi: 10.1088/0266-5611/19/6/009. [17] M. Ikehata, Inverse scattering problems and the enclosure method, Inverse Problems, 20 (2004), 533-551. doi: 10.1088/0266-5611/20/2/014. [18] M. Ikehata, The Herglotz wave function, the Vekua transform and the enclosure method, Hiroshima Math. J., 35 (2005), 485-506. [19] M. Ikehata, The probe and enclosure methods for inverse obstacle scattering problems. The past and present, New developments of functional equations in Mathematical Analysis, RIMS Kôkyûroku, 1702 (2010), 1-22. [20] M. Ikehata and T. Ohe, A numerical method for finding the convex hull of polygonal cavities using the enclosure method, Inverse Problems, 18 (2002), 111-124. doi: 10.1088/0266-5611/18/1/308. [21] M. Ikehata and S. Siltanen, Numerical method for finding the convex hull of an inclusion in conductivity from boundary measurements, Inverse Problems, 16 (2000), 1043-1052. doi: 10.1088/0266-5611/16/4/311. [22] V. Isakov, "Inverse Problems for Partial Differential Equations," 2nd edition, Applied Mathematical Sciences, 127, Springer, New York, 2006. [23] A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512. doi: 10.1088/0266-5611/14/6/009. [24] A. Kirsch, New characterizations of solutions in inverse scattering theory, Appl. Anal., 76 (2000), 319-350. doi: 10.1080/00036810008840888. [25] A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford Lecture Series in Mathematics and its Applications, 36, Oxford University Press, Oxford, 2008. [26] R. Kress, On the numerical solution of a hypersingular integral equation in scattering theory, Journal of Computational and Applied Mathematics, 61 (1995), 345-360. doi: 10.1016/0377-0427(94)00073-7. [27] S. Kusiak and J. Sylvester, The scattering support, Comm. Pure Appl. Math., 56 (2003), 1525-1548. doi: 10.1002/cpa.3038. [28] S. Kusiak and J. Sylvester, The convex scattering support in a background medium, SIAM J. Math. Anal., 36 (2005), 1142-1158. doi: 10.1137/S0036141003433577. [29] C. Liu, The Helmholtz equation on Lipshitz domains, preprint, 1995. [30] D. R. Luke and R. Potthast, The no response test-a sampling method for inverse scattering problems, SIAM J. Appl. Math., 63 (2003), 1292-1312. doi: 10.1137/S0036139902406887. [31] F. W. J. Olver, "Asymptotics and Special Functions," Computer Science and Applied Mathematics, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. [32] R. Potthast, A point source method for inverse acoustic and electromagnetic obstacle scattering problems, IMA J. Appl. Math., 61 (1998), 119-140. doi: 10.1093/imamat/61.2.119. [33] R. Potthast, Stability estimates and reconstructions in inverse scattering using singular sources, J. Comp. Appl. Math., 114 (2000), 247-274. doi: 10.1016/S0377-0427(99)00201-0. [34] R. Potthast, On the convergence of the no reponse test, SIAM J. Math. Anal., 38 (2007), 1808-1824. doi: 10.1137/S0036141004441003. [35] R. Potthast, J. Sylvester and S. Kusiak, A 'range test' for determining scatterers with unknown physical properties, Inverse Problems, 19 (2003), 533-547. doi: 10.1088/0266-5611/19/3/304. [36] I. Vekua, On the solution of the equation $\Delta u+\lambda^2 u=0$, Bull. Acad. Sci. Georgian SSR, 3 (1942), 307-314. [37] I. Vekua, Modification of an integral transformation and some of its properties, Bull. Acad. Sci. Georgian SSR, 6 (1945), 177-183.

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##### References:
 [1] M. Bruhl and M. Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography, Inverse Problems, 16 (2000), 1029-1042. doi: 10.1088/0266-5611/16/4/310. [2] F. Cakoni, D. Colton and P. Monk, "The Linear Sampling Method in Inverse Electromagnetic Scattering," CBMS-NSF Regional Conference Series in Applied Mathematics, 80, SIAM, Philadelphia, PA, 2011. [3] D. Colton, J. Coyle and P. Monk, Recent developments in inverse acoustic scattering theory, SIAM Review, 42 (2000), 369-414. doi: 10.1137/S0036144500367337. [4] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003. [5] D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. [6] D. Colton and R. Kress, Eigenvalues of the far field operator for the Helmholtz equation in an absorbing medium, SIAM J. Appl. Math., 55 (1995), 1724-1735. doi: 10.1137/S0036139993256114. [7] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," 2nd edition, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998. [8] D. Colton and R. Kress, Using fundamental solutions in inverse scattering, Inverse Problems, 22 (2006), R49-R66. doi: 10.1088/0266-5611/22/3/R01. [9] D. Colton and P. Monk, A linear sampling method for the detection of leukemia using microwaves, SIAM J. Appl. Math., 58 (1998), 926-941. doi: 10.1137/S0036139996308005. [10] F. Hettlich, On the uniqueness of the inverse conductive scattering problem for the Helmholtz equation, Inverse Problems, 10 (1994), 129-144. doi: 10.1088/0266-5611/10/1/010. [11] M. Ikehata, Reconstruction of an obstacle from the scattering amplitude at a fixed frequency, Inverse Problems, 14 (1998), 949-954. doi: 10.1088/0266-5611/14/4/012. [12] M. Ikehata, Reconstruction of a source domain from the Cauchy data, Inverse Problems, 15 (1999), 637-645. doi: 10.1088/0266-5611/15/2/019. [13] M. Ikehata, Reconstruction of obstacle from boundary measurements, Wave Motion, 30 (1999), 205-223. doi: 10.1016/S0165-2125(99)00006-2. [14] M. Ikehata, How to draw a picture of an unknown inclusion from boundary measurements. Two mathematical inversion algorithms, J. Inv. Ill-Posed Problems, 7 (1999), 255-271. doi: 10.1515/jiip.1999.7.3.255. [15] M. Ikehata, Reconstruction of the support function for inclusion from boundary measurements, J. Inv. Ill-Posed Problems, 8 (2000), 367-378. [16] M. Ikehata, Complex geometrical optics solutions and inverse crack problems, Inverse Problems, 19 (2003), 1385-1405. doi: 10.1088/0266-5611/19/6/009. [17] M. Ikehata, Inverse scattering problems and the enclosure method, Inverse Problems, 20 (2004), 533-551. doi: 10.1088/0266-5611/20/2/014. [18] M. Ikehata, The Herglotz wave function, the Vekua transform and the enclosure method, Hiroshima Math. J., 35 (2005), 485-506. [19] M. Ikehata, The probe and enclosure methods for inverse obstacle scattering problems. The past and present, New developments of functional equations in Mathematical Analysis, RIMS Kôkyûroku, 1702 (2010), 1-22. [20] M. Ikehata and T. Ohe, A numerical method for finding the convex hull of polygonal cavities using the enclosure method, Inverse Problems, 18 (2002), 111-124. doi: 10.1088/0266-5611/18/1/308. [21] M. Ikehata and S. Siltanen, Numerical method for finding the convex hull of an inclusion in conductivity from boundary measurements, Inverse Problems, 16 (2000), 1043-1052. doi: 10.1088/0266-5611/16/4/311. [22] V. Isakov, "Inverse Problems for Partial Differential Equations," 2nd edition, Applied Mathematical Sciences, 127, Springer, New York, 2006. [23] A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512. doi: 10.1088/0266-5611/14/6/009. [24] A. Kirsch, New characterizations of solutions in inverse scattering theory, Appl. Anal., 76 (2000), 319-350. doi: 10.1080/00036810008840888. [25] A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford Lecture Series in Mathematics and its Applications, 36, Oxford University Press, Oxford, 2008. [26] R. Kress, On the numerical solution of a hypersingular integral equation in scattering theory, Journal of Computational and Applied Mathematics, 61 (1995), 345-360. doi: 10.1016/0377-0427(94)00073-7. [27] S. Kusiak and J. Sylvester, The scattering support, Comm. Pure Appl. Math., 56 (2003), 1525-1548. doi: 10.1002/cpa.3038. [28] S. Kusiak and J. Sylvester, The convex scattering support in a background medium, SIAM J. Math. Anal., 36 (2005), 1142-1158. doi: 10.1137/S0036141003433577. [29] C. Liu, The Helmholtz equation on Lipshitz domains, preprint, 1995. [30] D. R. Luke and R. Potthast, The no response test-a sampling method for inverse scattering problems, SIAM J. Appl. Math., 63 (2003), 1292-1312. doi: 10.1137/S0036139902406887. [31] F. W. J. Olver, "Asymptotics and Special Functions," Computer Science and Applied Mathematics, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. [32] R. Potthast, A point source method for inverse acoustic and electromagnetic obstacle scattering problems, IMA J. Appl. Math., 61 (1998), 119-140. doi: 10.1093/imamat/61.2.119. [33] R. Potthast, Stability estimates and reconstructions in inverse scattering using singular sources, J. Comp. Appl. Math., 114 (2000), 247-274. doi: 10.1016/S0377-0427(99)00201-0. [34] R. Potthast, On the convergence of the no reponse test, SIAM J. Math. Anal., 38 (2007), 1808-1824. doi: 10.1137/S0036141004441003. [35] R. Potthast, J. Sylvester and S. Kusiak, A 'range test' for determining scatterers with unknown physical properties, Inverse Problems, 19 (2003), 533-547. doi: 10.1088/0266-5611/19/3/304. [36] I. Vekua, On the solution of the equation $\Delta u+\lambda^2 u=0$, Bull. Acad. Sci. Georgian SSR, 3 (1942), 307-314. [37] I. Vekua, Modification of an integral transformation and some of its properties, Bull. Acad. Sci. Georgian SSR, 6 (1945), 177-183.
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