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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 and 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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

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