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Reducing spatially varying out-of-focus blur from natural image
On the measurement operator for scattering in layered media
Dept. of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J1P3, Canada |
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.
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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