American Institute of Mathematical Sciences

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

Sampling type methods for an inverse waveguide problem

Peter Monk 1, and  Virginia Selgas 2,

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.
Keywords: inverse scattering, waveguide., Linear sampling method, reciprocity gap method.
Mathematics Subject Classification: 65N21, 74F1.
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.  Google Scholar

[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.  Google Scholar

[3]

L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: A modal formulation,, Inv. Prob., 24 (2008).   Google Scholar

[4]

F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory. An Introduction,", Interaction of Mechanics and Mathematics, (2006).   Google Scholar

[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.  Google Scholar

[6]

F. Cakoni and H. Haddar, Interior transmission problem for anisotropic media,, in, (2003), 613.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem,, Inverse Problems and Imaging, 1 (2007), 13.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[13]

B. Pincon and K. Ramdani, Selective focusing on small scatterers in acoustic waveguides using time reversal mirrors,, Inv. Prob., 23 (2007), 1.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[16]

P. Roux, B. Roman and M. Fink, Time-reversal in an ultrasonic waveguide,, Appl. Phys. Lett., 70 (1997), 1811.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: A modal formulation,, Inv. Prob., 24 (2008).   Google Scholar

[4]

F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory. An Introduction,", Interaction of Mechanics and Mathematics, (2006).   Google Scholar

[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.  Google Scholar

[6]

F. Cakoni and H. Haddar, Interior transmission problem for anisotropic media,, in, (2003), 613.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem,, Inverse Problems and Imaging, 1 (2007), 13.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[13]

B. Pincon and K. Ramdani, Selective focusing on small scatterers in acoustic waveguides using time reversal mirrors,, Inv. Prob., 23 (2007), 1.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[16]

P. Roux, B. Roman and M. Fink, Time-reversal in an ultrasonic waveguide,, Appl. Phys. Lett., 70 (1997), 1811.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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