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. 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

[1]

Peter Monk, Jiguang Sun. Inverse scattering using finite elements and gap reciprocity. Inverse Problems & Imaging, 2007, 1 (4) : 643-660. doi: 10.3934/ipi.2007.1.643

[2]

Fang Zeng, Xiaodong Liu, Jiguang Sun, Liwei Xu. The reciprocity gap method for a cavity in an inhomogeneous medium. Inverse Problems & Imaging, 2016, 10 (3) : 855-868. doi: 10.3934/ipi.2016024

[3]

Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems & Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757

[4]

Roland Griesmaier. Reciprocity gap music imaging for an inverse scattering problem in two-layered media. Inverse Problems & Imaging, 2009, 3 (3) : 389-403. doi: 10.3934/ipi.2009.3.389

[5]

Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems & Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577

[6]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291

[7]

Qinghua Wu, Guozheng Yan. The factorization method for a partially coated cavity in inverse scattering. Inverse Problems & Imaging, 2016, 10 (1) : 263-279. doi: 10.3934/ipi.2016.10.263

[8]

Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems & Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681

[9]

Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems & Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027

[10]

Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems & Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131

[11]

Pedro Serranho. A hybrid method for inverse scattering for Sound-soft obstacles in R3. Inverse Problems & Imaging, 2007, 1 (4) : 691-712. doi: 10.3934/ipi.2007.1.691

[12]

Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems & Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042

[13]

Jun Guo, Qinghua Wu, Guozheng Yan. The factorization method for cracks in elastic scattering. Inverse Problems & Imaging, 2018, 12 (2) : 349-371. doi: 10.3934/ipi.2018016

[14]

Jingzhi Li, Hongyu Liu, Qi Wang. Fast imaging of electromagnetic scatterers by a two-stage multilevel sampling method. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 547-561. doi: 10.3934/dcdss.2015.8.547

[15]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[16]

Keji Liu. Scattering by impenetrable scatterer in a stratified ocean waveguide. Inverse Problems & Imaging, 2019, 13 (6) : 1349-1365. doi: 10.3934/ipi.2019059

[17]

Teemu Tyni, Valery Serov. Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line. Inverse Problems & Imaging, 2019, 13 (1) : 159-175. doi: 10.3934/ipi.2019009

[18]

Mohsen Tadi. A computational method for an inverse problem in a parabolic system. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 205-218. doi: 10.3934/dcdsb.2009.12.205

[19]

Beatrice Bugert, Gunther Schmidt. Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 435-473. doi: 10.3934/dcdss.2015.8.435

[20]

Georgy L. Alfimov, Pavel P. Kizin, Dmitry A. Zezyulin. Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: Coding and method for computation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1207-1229. doi: 10.3934/dcdsb.2017059

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]