# American Institute of Mathematical Sciences

November  2012, 6(4): 709-747. doi: 10.3934/ipi.2012.6.709

## Sampling type methods for an inverse waveguide problem

 1 Department of Mathematics, University of Delaware, Newark, DE 19716 2 Departamento de Matemáticas, Escuela Politécnica de Ingeniería de Gijón, Universidad de Oviedo, 33203 Gijón, Spain

Received  November 2011 Revised  August 2012 Published  November 2012

We consider the problem of locating a penetrable obstacle in an acoustic waveguide from measurements of pressure waves due to point sources inside the waveguide. More precisely, we assume that we are given the scattered field and its normal derivative for any source point and receiver placed on a pair of surfaces known as the source and the measurement surfaces, respectively. A novel feature of this work is that the obstacle is allowed to touch the boundary of the pipe.
We first analyze the associated interior transmission problem. Then, we adapt and analyze the Reciprocity Gap Method (RGM) and the Linear Sampling Method (LSM) to deal with the inverse problem. We also study the relationship between these two methods and provide numerical results.
Citation: Peter Monk, Virginia Selgas. Sampling type methods for an inverse waveguide problem. Inverse Problems & Imaging, 2012, 6 (4) : 709-747. doi: 10.3934/ipi.2012.6.709
##### References:
 [1] T. Arens, D. Gintides and A. Lechleiter, Direct and inverse medium scattering in a 3D homogeneous planar waveguide,, SIAM J. Appl. Math., 71 (2011), 753. doi: 10.1137/100806333. [2] T. Arens, D. Gintides and A. Lechleiter, Variational formulations for scattering in a three-dimensional acoustic waveguide,, Math. Meth. Appl. Sci., 31 (2007), 821. doi: 10.1002/mma.947. [3] L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: A modal formulation,, Inv. Prob., 24 (2008). [4] F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory. An Introduction,", Interaction of Mechanics and Mathematics, (2006). [5] F. Cakoni, M. Fares and H. Haddar, Analysis of two linear sampling methods applied to electromagnetic imaging of buried objects,, Inv. Prob., 22 (2006), 845. doi: 10.1088/0266-5611/22/3/007. [6] F. Cakoni and H. Haddar, Interior transmission problem for anisotropic media,, in, (2003), 613. [7] D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory,, Inv. Prob., 21 (2005), 383. doi: 10.1088/0266-5611/21/1/023. [8] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region,, Inv. Prob., 12 (1996), 383. doi: 10.1088/0266-5611/12/4/003. [9] D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem,, Inverse Problems and Imaging, 1 (2007), 13. [10] D. Colton, M. Piana and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problems,, Inv. Prob., 13 (1997), 1477. doi: 10.1088/0266-5611/13/6/005. [11] K. Horoshenkov, R. Ashley and J. Blanksby, Determination of sewer roughness and sediment properties using acoustic techniques,, Water Science and Technology, 47 (2003), 87. [12] P. Monk and V. Selgas, Near field sampling type methods for the inverse fluid-solid interaction problem,, Inverse Problems and Imaging, 5 (2011), 465. [13] B. Pincon and K. Ramdani, Selective focusing on small scatterers in acoustic waveguides using time reversal mirrors,, Inv. Prob., 23 (2007), 1. [14] F. Podd, M. Ali, K. Horoshenkov, A. Wood, S. Tait, J. Boot, R. Long and A. Saul, Rapid sonic characterisation of sewer change and obstructions,, Water Science and Technology, 56 (2007), 131. [15] P. Roux and M. Fink, Time reversal in a waveguide: Study of the temporal and spatial focusing,, J. Acoust. Soc. Am., 107 (2000), 2418. doi: 10.1121/1.428628. [16] P. Roux, B. Roman and M. Fink, Time-reversal in an ultrasonic waveguide,, Appl. Phys. Lett., 70 (1997), 1811. [17] A. Tolstoy, K. Horoshenkov and M. Bin Ali, Detecting pipe changes via acoustic matched field processing,, Applied Acoustics, 70 (2009), 695. doi: 10.1016/j.apacoust.2008.08.007. [18] Y. Xu, C. Matawa and W. Lin, Generalized dual space indicator method for underwater imaging,, Inv. Prob., 16 (2000), 1761. doi: 10.1088/0266-5611/16/6/311.

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##### References:
 [1] T. Arens, D. Gintides and A. Lechleiter, Direct and inverse medium scattering in a 3D homogeneous planar waveguide,, SIAM J. Appl. Math., 71 (2011), 753. doi: 10.1137/100806333. [2] T. Arens, D. Gintides and A. Lechleiter, Variational formulations for scattering in a three-dimensional acoustic waveguide,, Math. Meth. Appl. Sci., 31 (2007), 821. doi: 10.1002/mma.947. [3] L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: A modal formulation,, Inv. Prob., 24 (2008). [4] F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory. An Introduction,", Interaction of Mechanics and Mathematics, (2006). [5] F. Cakoni, M. Fares and H. Haddar, Analysis of two linear sampling methods applied to electromagnetic imaging of buried objects,, Inv. Prob., 22 (2006), 845. doi: 10.1088/0266-5611/22/3/007. [6] F. Cakoni and H. Haddar, Interior transmission problem for anisotropic media,, in, (2003), 613. [7] D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory,, Inv. Prob., 21 (2005), 383. doi: 10.1088/0266-5611/21/1/023. [8] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region,, Inv. Prob., 12 (1996), 383. doi: 10.1088/0266-5611/12/4/003. [9] D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem,, Inverse Problems and Imaging, 1 (2007), 13. [10] D. Colton, M. Piana and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problems,, Inv. Prob., 13 (1997), 1477. doi: 10.1088/0266-5611/13/6/005. [11] K. Horoshenkov, R. Ashley and J. Blanksby, Determination of sewer roughness and sediment properties using acoustic techniques,, Water Science and Technology, 47 (2003), 87. [12] P. Monk and V. Selgas, Near field sampling type methods for the inverse fluid-solid interaction problem,, Inverse Problems and Imaging, 5 (2011), 465. [13] B. Pincon and K. Ramdani, Selective focusing on small scatterers in acoustic waveguides using time reversal mirrors,, Inv. Prob., 23 (2007), 1. [14] F. Podd, M. Ali, K. Horoshenkov, A. Wood, S. Tait, J. Boot, R. Long and A. Saul, Rapid sonic characterisation of sewer change and obstructions,, Water Science and Technology, 56 (2007), 131. [15] P. Roux and M. Fink, Time reversal in a waveguide: Study of the temporal and spatial focusing,, J. Acoust. Soc. Am., 107 (2000), 2418. doi: 10.1121/1.428628. [16] P. Roux, B. Roman and M. Fink, Time-reversal in an ultrasonic waveguide,, Appl. Phys. Lett., 70 (1997), 1811. [17] A. Tolstoy, K. Horoshenkov and M. Bin Ali, Detecting pipe changes via acoustic matched field processing,, Applied Acoustics, 70 (2009), 695. doi: 10.1016/j.apacoust.2008.08.007. [18] Y. Xu, C. Matawa and W. Lin, Generalized dual space indicator method for underwater imaging,, Inv. Prob., 16 (2000), 1761. doi: 10.1088/0266-5611/16/6/311.
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