American Institute of Mathematical Sciences

Advanced
February  2017, 11(1): 87-97. doi: 10.3934/ipi.2017005

On the measurement operator for scattering in layered media

Peter C. Gibson

Dept. of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J1P3, Canada

Received  February 2016 Revised  September 2016 Published  January 2017

We describe new mathematical structures associated with the scattering of plane waves in piecewise constant layered media, a basic model for acoustic imaging of laminated structures and in geophysics. Using explicit formulas for the reflection Green's function it is shown that the measurement operator satisfies a system of quasilinear PDE with smooth coefficients, and that the sum of the amplitude data has a simple expression in terms of inverse hyperbolic tangent of the reflection coefficients. In addition we derive a simple geometric description of the measured data, which, in the generic case, yields a natural factorization of the inverse problem.

Keywords: Layered media, discontinuous coefficients, 1D scattering, measurement operator, inverse problems.
Mathematics Subject Classification: Primary: 35R05, 74J20; Secondary: 86A22.
Citation: Peter C. Gibson. On the measurement operator for scattering in layered media. Inverse Problems & Imaging, 2017, 11 (1) : 87-97. doi: 10.3934/ipi.2017005
References:
[1]

N. Bleistein, J. K. Cohen and J. W. Stockwell Jr. , Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion vol. 13 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York, 2001, Geophysics and Planetary Sciences.  Google Scholar

[2]

L. M. Brekhovskikh and O. A. Godin, Acoustics of Layered Media I vol. 5 of Springer Series on Wave Phenomena, Springer, Heidelberg, 1990. Google Scholar

[3]

H. Bremmer, The W.K.B. approximation as the first term of a geometric-optical series, Comm. Pure Appl. Math., 4 (1951), 105-115.  doi: 10.1002/cpa.3160040111.  Google Scholar

[4]

K. P. Bube and R. Burridge, The one-dimensional inverse problem of reflection seismology, SIAM Rev. , 25 (1983), 497–559, URL http://dx.doi.org/10.1137/1025122. doi: 10.1137/1025122.  Google Scholar

[5]

J. F. Clouet and J. P. Fouque, A time-reversal method for an acoustical pulse propagating in randomly layered media, Wave Motion, 25 (1997), 361–368, URL http://dx.doi.org/10.1016/S0165-2125(97)00002-4. doi: 10.1016/S0165-2125(97)00002-4.  Google Scholar

[6]

J. -P. Fouque, J. Garnier, G. Papanicolaou and K. Solna, Wave Propagation and Time Reversal in Randomly Layered Media vol. 56 of Stochastic Modelling and Applied Probability, Springer, New York, 2007. doi: 10.1007/978-0-387-49808-9_4.  Google Scholar

[7]

P. C. Gibson, The combinatorics of scattering in layered media, SIAM J. Appl. Math. , 74 (2014), 919–938, URL http://dx.doi.org/10.1137/130923075. doi: 10.1137/130923075.  Google Scholar

[8]

P. C. Gibson, A multivariate interpolation problem arising from the scattering of waves in layered media, Dolomites Res. Notes Approx. DRNA, 7 (2014), 7-15.   Google Scholar

[9]

P. C. Gibson, Fourier expansion of disk automorphisms via scattering in layered media J. Fourier Anal. Appl. (2016), URL http://dx.doi.org/10.1007/s00041-016-9514-6. Google Scholar

[10]

K. A. Innanen, Born series forward modelling of seismic primary and multiple reflections: An inverse scattering shortcut, Geophysical Journal International, 177 (2009), 1197–1204, URL http://dx.doi.org/10.1111/j.1365-246X.2009.04131.x. doi: 10.1111/j.1365-246X.2009.04131.x.  Google Scholar

[11]

G. C. Papanicolaou, Wave propagation in a one-dimensional random medium, SIAM J. Appl. Math., 21 (1971), 13-18.  doi: 10.1137/0121002.  Google Scholar

[12]

Rakesh, An inverse problem for a layered medium with a point source, Inverse Problems, 19 (2003), 497–506, URL http://dx.doi.org/10.1088/0266-5611/19/3/301. doi: 10.1088/0266-5611/19/3/301.  Google Scholar

[13]

F. Santosa and W. W. Symes, Reconstruction of blocky impedance profiles from normalincidence reflection seismograms which are band-limited and miscalibrated, Wave Motion, 10 (1988), 209–230, URL http://dx.doi.org/10.1016/0165-2125(88)90019-4. doi: 10.1016/0165-2125(88)90019-4.  Google Scholar

[14]

J. Sylvester and D. P. Winebrenner, Linear and nonlinear inverse scattering, SIAM J. Appl. Math. , 59 (1998), 669–699, URL http://dx.doi.org/10.1137/S0036139997319773.  Google Scholar

[15]

W. W. Symes, Impedance profile inversion via the first transport equation, J. Math. Anal. Appl. , 94 (1983), 435–453, URL http://dx.doi.org/10.1016/0022-247X(83)90072-0. doi: 10.1016/0022-247X(83)90072-0.  Google Scholar

[16]

R. Weder, Spectral and Scattering Theory for Wave Propagation in Perturbed Stratified Media vol. 87 of Applied Mathematical Sciences, Springer-Verlag, New York, 1991, URL http://dx.doi.org/10.1007/978-1-4612-4430-1.  Google Scholar

show all references

References:
[1]

N. Bleistein, J. K. Cohen and J. W. Stockwell Jr. , Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion vol. 13 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York, 2001, Geophysics and Planetary Sciences.  Google Scholar

[2]

L. M. Brekhovskikh and O. A. Godin, Acoustics of Layered Media I vol. 5 of Springer Series on Wave Phenomena, Springer, Heidelberg, 1990. Google Scholar

[3]

H. Bremmer, The W.K.B. approximation as the first term of a geometric-optical series, Comm. Pure Appl. Math., 4 (1951), 105-115.  doi: 10.1002/cpa.3160040111.  Google Scholar

[4]

K. P. Bube and R. Burridge, The one-dimensional inverse problem of reflection seismology, SIAM Rev. , 25 (1983), 497–559, URL http://dx.doi.org/10.1137/1025122. doi: 10.1137/1025122.  Google Scholar

[5]

J. F. Clouet and J. P. Fouque, A time-reversal method for an acoustical pulse propagating in randomly layered media, Wave Motion, 25 (1997), 361–368, URL http://dx.doi.org/10.1016/S0165-2125(97)00002-4. doi: 10.1016/S0165-2125(97)00002-4.  Google Scholar

[6]

J. -P. Fouque, J. Garnier, G. Papanicolaou and K. Solna, Wave Propagation and Time Reversal in Randomly Layered Media vol. 56 of Stochastic Modelling and Applied Probability, Springer, New York, 2007. doi: 10.1007/978-0-387-49808-9_4.  Google Scholar

[7]

P. C. Gibson, The combinatorics of scattering in layered media, SIAM J. Appl. Math. , 74 (2014), 919–938, URL http://dx.doi.org/10.1137/130923075. doi: 10.1137/130923075.  Google Scholar

[8]

P. C. Gibson, A multivariate interpolation problem arising from the scattering of waves in layered media, Dolomites Res. Notes Approx. DRNA, 7 (2014), 7-15.   Google Scholar

[9]

P. C. Gibson, Fourier expansion of disk automorphisms via scattering in layered media J. Fourier Anal. Appl. (2016), URL http://dx.doi.org/10.1007/s00041-016-9514-6. Google Scholar

[10]

K. A. Innanen, Born series forward modelling of seismic primary and multiple reflections: An inverse scattering shortcut, Geophysical Journal International, 177 (2009), 1197–1204, URL http://dx.doi.org/10.1111/j.1365-246X.2009.04131.x. doi: 10.1111/j.1365-246X.2009.04131.x.  Google Scholar

[11]

G. C. Papanicolaou, Wave propagation in a one-dimensional random medium, SIAM J. Appl. Math., 21 (1971), 13-18.  doi: 10.1137/0121002.  Google Scholar

[12]

Rakesh, An inverse problem for a layered medium with a point source, Inverse Problems, 19 (2003), 497–506, URL http://dx.doi.org/10.1088/0266-5611/19/3/301. doi: 10.1088/0266-5611/19/3/301.  Google Scholar

[13]

F. Santosa and W. W. Symes, Reconstruction of blocky impedance profiles from normalincidence reflection seismograms which are band-limited and miscalibrated, Wave Motion, 10 (1988), 209–230, URL http://dx.doi.org/10.1016/0165-2125(88)90019-4. doi: 10.1016/0165-2125(88)90019-4.  Google Scholar

[14]

J. Sylvester and D. P. Winebrenner, Linear and nonlinear inverse scattering, SIAM J. Appl. Math. , 59 (1998), 669–699, URL http://dx.doi.org/10.1137/S0036139997319773.  Google Scholar

[15]

W. W. Symes, Impedance profile inversion via the first transport equation, J. Math. Anal. Appl. , 94 (1983), 435–453, URL http://dx.doi.org/10.1016/0022-247X(83)90072-0. doi: 10.1016/0022-247X(83)90072-0.  Google Scholar

[16]

R. Weder, Spectral and Scattering Theory for Wave Propagation in Perturbed Stratified Media vol. 87 of Applied Mathematical Sciences, Springer-Verlag, New York, 1991, URL http://dx.doi.org/10.1007/978-1-4612-4430-1.  Google Scholar

[1]

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
[2]

Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control & Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017
[3]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems for evolution equations with time dependent operator-coefficients. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 737-744. doi: 10.3934/dcdss.2016025
[4]

Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Local study of a renormalization operator for 1D maps under quasiperiodic forcing. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1171-1188. doi: 10.3934/dcdss.2016047
[5]

Quansen Jiu, Zhouping Xin. The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients. Kinetic & Related Models, 2008, 1 (2) : 313-330. doi: 10.3934/krm.2008.1.313
[6]

Franck Boyer, Guillaume Olive. Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients. Mathematical Control & Related Fields, 2014, 4 (3) : 263-287. doi: 10.3934/mcrf.2014.4.263
[7]

Wuming Li, Xiaojun Liu, Quansen Jiu. The decay estimates of solutions for 1D compressible flows with density-dependent viscosity coefficients. Communications on Pure & Applied Analysis, 2013, 12 (2) : 647-661. doi: 10.3934/cpaa.2013.12.647
[8]

Elena Rossi. Well-posedness of general 1D initial boundary value problems for scalar balance laws. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3577-3608. doi: 10.3934/dcds.2019147
[9]

Yu. Dabaghian, R. V. Jensen, R. Blümel. Integrability in 1D quantum chaos. Conference Publications, 2003, 2003 (Special) : 206-212. doi: 10.3934/proc.2003.2003.206
[10]

Christodoulos E. Athanasiadis, Vassilios Sevroglou, Konstantinos I. Skourogiannis. The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media. Inverse Problems & Imaging, 2015, 9 (4) : 951-970. doi: 10.3934/ipi.2015.9.951
[11]

Teemu Tyni, Valery Serov. Scattering problems for perturbations of the multidimensional biharmonic operator. Inverse Problems & Imaging, 2018, 12 (1) : 205-227. doi: 10.3934/ipi.2018008
[12]

Eliane Bécache, Laurent Bourgeois, Lucas Franceschini, Jérémi Dardé. Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case. Inverse Problems & Imaging, 2015, 9 (4) : 971-1002. doi: 10.3934/ipi.2015.9.971
[13]

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
[14]

Johannes Elschner, Guanghui Hu. Uniqueness in inverse transmission scattering problems for multilayered obstacles. Inverse Problems & Imaging, 2011, 5 (4) : 793-813. doi: 10.3934/ipi.2011.5.793
[15]

Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems & Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577
[16]

Gabriel Katz. Causal holography in application to the inverse scattering problems. Inverse Problems & Imaging, 2019, 13 (3) : 597-633. doi: 10.3934/ipi.2019028
[17]

François Delarue, Franco Flandoli. The transition point in the zero noise limit for a 1D Peano example. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4071-4083. doi: 10.3934/dcds.2014.34.4071
[18]

Harish S. Bhat, Razvan C. Fetecau. Lagrangian averaging for the 1D compressible Euler equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 979-1000. doi: 10.3934/dcdsb.2006.6.979
[19]

Yohan Penel. An explicit stable numerical scheme for the $1D$ transport equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 641-656. doi: 10.3934/dcdss.2012.5.641
[20]

Bernard Ducomet. Asymptotics for 1D flows with time-dependent external fields. Conference Publications, 2007, 2007 (Special) : 323-333. doi: 10.3934/proc.2007.2007.323

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (29)
  • HTML views (92)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences