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The topological gradient method for semi-linear problems and application to edge detection and noise removal
Common midpoint versus common offset acquisition geometry in seismic imaging
1. | School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY 14623, United States |
2. | TIFR Centre for Applicable Mathematics, Post Bag No. 6503, GKVK Post Office, Sharada Nagar, Chikkabommasandra, Bangalore, Karnataka 560065, India |
3. | Department of Mathematics and Statistics, University of Limerick, Castletroy, Co. Limerick, Ireland |
4. | Department of Mathematics, Tufts University, Medford, MA 02155, United States |
References:
[1] |
G. Ambartsoumian, R. Felea, V. P. Krishnan, C. Nolan and E. T. Quinto, A class of singular Fourier integral operators in synthetic aperture radar imaging, J. Funct. Anal., 264 (2013), 246-269.
doi: 10.1016/j.jfa.2012.10.008. |
[2] |
G. Beylkin, Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform, J. Math. Phys., 26 (1985), 99-108.
doi: 10.1063/1.526755. |
[3] |
N. Bleistein, J. Cohen and J. J.W. Stockhwell, Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion, Interdisciplinary Applied Mathematics, 13, Springer, New York, 2001.
doi: 10.1007/978-1-4613-0001-4. |
[4] |
M. V. de Hoop, H. Smith, G. Uhlmann and R. D. van der Hilst, Seismic imaging with the generalized Radon transform: A curvelet transform perspective, Inverse Problems, 25 (2009), 025005, 21pp.
doi: 10.1088/0266-5611/25/2/025005. |
[5] |
M. V. de Hoop, Microlocal analysis of seismic inverse scattering, in Inside Out: Inverse Problems and Applications, Math. Sci. Res. Inst. Publ., 47, Cambridge Univ. Press, Cambridge, 2003, 219-296. |
[6] |
A. Devaney, Geophysical diffraction tomography, IEEE Transactions on Geoscience and Remote Sensing, 22 (1984), 3-13. |
[7] |
R. Felea and A. Greenleaf, An FIO calculus for marine seismic imaging: folds and crosscaps, Communications in Partial Differential Equations, 33 (2008), 45-77.
doi: 10.1080/03605300701318716. |
[8] |
R. Felea, Composition of Fourier integral operators with fold and blowdown singularities, Comm. Partial Differential Equations, 30 (2005), 1717-1740.
doi: 10.1080/03605300500299968. |
[9] |
R. Felea, Displacement of artefacts in inverse scattering, Inverse Problems, 23 (2007), 1519-1531.
doi: 10.1088/0266-5611/23/4/009. |
[10] |
R. Felea and A. Greenleaf, Fourier integral operators with open umbrellas and seismic inversion for cusp caustics, Math. Res. Lett., 17 (2010), 867-886.
doi: 10.4310/MRL.2010.v17.n5.a6. |
[11] |
R. Felea, A. Greenleaf and M. Pramanik, An FIO calculus for marine seismic imaging, II: Sobolev estimates, Math. Ann., 352 (2012), 293-337.
doi: 10.1007/s00208-011-0644-5. |
[12] |
J. Frikel and E. T. Quinto, Characterization and reduction of artifacts in limited angle tomography, Inverse Problems, 29 (2013), 125007, 21pp.
doi: 10.1088/0266-5611/29/12/125007. |
[13] |
A. Greenleaf and G. Uhlmann, Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms, Ann. Inst. Fourier (Grenoble), 40 (1990), 443-466.
doi: 10.5802/aif.1220. |
[14] |
A. Greenleaf and G. Uhlmann, Non-local inversion formulas for the X-ray transform, Duke Math. J., 58 (1989), 205-240.
doi: 10.1215/S0012-7094-89-05811-0. |
[15] |
A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, J. Funct. Anal., 89 (1990), 202-232.
doi: 10.1016/0022-1236(90)90011-9. |
[16] |
A. Greenleaf and G. Uhlmann, Microlocal techniques in integral geometry, in Integral Geometry and Tomography (Arcata, CA, 1989), Contemp. Math., Amer. Math. Soc., 113, Providence, RI, 1990, 121-135.
doi: 10.1090/conm/113/1108649. |
[17] |
V. Guillemin, Some Remarks on Integral Geometry, Technical Report, MIT, 1975. |
[18] |
V. Guillemin, Cosmology in $(2 + 1)$-dimensions, Cyclic Models, and Deformations of $M_{2,1}$, Annals of Mathematics Studies, 121, Princeton University Press, Princeton, NJ, 1989. |
[19] |
V. Guillemin and S. Sternberg, Geometric Asymptotics, Mathematical Surveys, No. 14, American Mathematical Society, Providence, R.I., 1977. |
[20] |
V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Math. J., 48 (1981), 251-267.
doi: 10.1215/S0012-7094-81-04814-6. |
[21] |
A. I. Katsevich, Local Tomography for the Limited-Angle Problem, Journal of mathematical analysis and applications, 213 (1997), 160-182.
doi: 10.1006/jmaa.1997.5412. |
[22] |
V. P. Krishnan and E. T. Quinto, Microlocal aspects of bistatic synthetic aperture radar imaging, Inverse Problems and Imaging, 5 (2011), 659-674.
doi: 10.3934/ipi.2011.5.659. |
[23] |
A. Malcolm, B. Ursin and M. de Hoop, Seismic imaging and illumination with internal multiples, Geophysical Journal International, 176 (2009), 847-864.
doi: 10.1111/j.1365-246X.2008.03992.x. |
[24] |
R. B. Melrose and G. A. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32 (1979), 483-519.
doi: 10.1002/cpa.3160320403. |
[25] |
C. J. Nolan and W. W. Symes, Global solution of a linearized inverse problem for the wave equation, Comm. Partial Differential Equations, 22 (1997), 919-952.
doi: 10.1080/03605309708821289. |
[26] |
C. J. Nolan, Scattering in the presence of fold caustics, SIAM J. Appl. Math., 61 (2000), 659-672.
doi: 10.1137/S0036139999356107. |
[27] |
C. J. Nolan and M. Cheney, Synthetic aperture inversion, Inverse Problems, 18 (2002), 221-235.
doi: 10.1088/0266-5611/18/1/315. |
[28] |
C. J. Nolan and M. Cheney, Microlocal analysis of synthetic aperture radar imaging, J. Fourier Anal. Appl., 10 (2004), 133-148.
doi: 10.1007/s00041-004-8008-0. |
[29] |
E. T. Quinto, Singularities of the X-ray transform and limited data tomography in $\mathbbR^2$ and $\mathbbR^3$, SIAM J. Math. Anal., 24 (1993), 1215-1225.
doi: 10.1137/0524069. |
[30] |
E. T. Quinto, A. Rieder and T. Schuster, Local inversion of the sonar transform regularized by the approximate inverse, Inverse Problems, 27 (2011), 035006, 18pp.
doi: 10.1088/0266-5611/27/3/035006. |
[31] |
Rakesh, A linearised inverse problem for the wave equation, Comm. Partial Differential Equations, 13 (1988), 573-601.
doi: 10.1080/03605308808820553. |
[32] |
R. E. Sheriff, Encyclopedic Dictionary of Applied Geophysics, Society Of Exploration Geophysicists, 2002.
doi: 10.1190/1.9781560802969. |
[33] |
P. Stefanov and G. Uhlmann, Is a curved flight path in {SAR} better than a straight one?, SIAM J. Appl. Math., 73 (2013), 1596-1612.
doi: 10.1137/120882639. |
[34] |
C. C. Stolk, Microlocal analysis of a seismic linearized inverse problem, Wave Motion, 32 (2000), 267-290.
doi: 10.1016/S0165-2125(00)00043-3. |
[35] |
C. C. Stolk and M. V. de Hoop, Microlocal analysis of seismic inverse scattering in anisotropic elastic media, Comm. Pure Appl. Math., 55 (2002), 261-301.
doi: 10.1002/cpa.10019. |
[36] |
C. C. Stolk and M. V. de Hoop, Seismic inverse scattering in the downward continuation approach, Wave Motion, 43 (2006), 579-598.
doi: 10.1016/j.wavemoti.2006.05.003. |
[37] |
W. W. Symes, Mathematics of Reflection Seismology, Technical Report, Department of Computational and Applied Mathematics, Rice University, Houston, Texas, 1990, Technical Report TR90-02. |
[38] |
A. P. E. ten Kroode, D.-J. Smit and A. R. Verdel, A microlocal analysis of migration, Wave Motion, 28 (1998), 149-172.
doi: 10.1016/S0165-2125(98)00004-3. |
[39] |
F. Trèves, Introduction to Pseudodifferential and Fourier Integral Operators, Fourier Integral Operators, Vol. 2, Plenum Press, New York-London, 1980. |
show all references
References:
[1] |
G. Ambartsoumian, R. Felea, V. P. Krishnan, C. Nolan and E. T. Quinto, A class of singular Fourier integral operators in synthetic aperture radar imaging, J. Funct. Anal., 264 (2013), 246-269.
doi: 10.1016/j.jfa.2012.10.008. |
[2] |
G. Beylkin, Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform, J. Math. Phys., 26 (1985), 99-108.
doi: 10.1063/1.526755. |
[3] |
N. Bleistein, J. Cohen and J. J.W. Stockhwell, Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion, Interdisciplinary Applied Mathematics, 13, Springer, New York, 2001.
doi: 10.1007/978-1-4613-0001-4. |
[4] |
M. V. de Hoop, H. Smith, G. Uhlmann and R. D. van der Hilst, Seismic imaging with the generalized Radon transform: A curvelet transform perspective, Inverse Problems, 25 (2009), 025005, 21pp.
doi: 10.1088/0266-5611/25/2/025005. |
[5] |
M. V. de Hoop, Microlocal analysis of seismic inverse scattering, in Inside Out: Inverse Problems and Applications, Math. Sci. Res. Inst. Publ., 47, Cambridge Univ. Press, Cambridge, 2003, 219-296. |
[6] |
A. Devaney, Geophysical diffraction tomography, IEEE Transactions on Geoscience and Remote Sensing, 22 (1984), 3-13. |
[7] |
R. Felea and A. Greenleaf, An FIO calculus for marine seismic imaging: folds and crosscaps, Communications in Partial Differential Equations, 33 (2008), 45-77.
doi: 10.1080/03605300701318716. |
[8] |
R. Felea, Composition of Fourier integral operators with fold and blowdown singularities, Comm. Partial Differential Equations, 30 (2005), 1717-1740.
doi: 10.1080/03605300500299968. |
[9] |
R. Felea, Displacement of artefacts in inverse scattering, Inverse Problems, 23 (2007), 1519-1531.
doi: 10.1088/0266-5611/23/4/009. |
[10] |
R. Felea and A. Greenleaf, Fourier integral operators with open umbrellas and seismic inversion for cusp caustics, Math. Res. Lett., 17 (2010), 867-886.
doi: 10.4310/MRL.2010.v17.n5.a6. |
[11] |
R. Felea, A. Greenleaf and M. Pramanik, An FIO calculus for marine seismic imaging, II: Sobolev estimates, Math. Ann., 352 (2012), 293-337.
doi: 10.1007/s00208-011-0644-5. |
[12] |
J. Frikel and E. T. Quinto, Characterization and reduction of artifacts in limited angle tomography, Inverse Problems, 29 (2013), 125007, 21pp.
doi: 10.1088/0266-5611/29/12/125007. |
[13] |
A. Greenleaf and G. Uhlmann, Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms, Ann. Inst. Fourier (Grenoble), 40 (1990), 443-466.
doi: 10.5802/aif.1220. |
[14] |
A. Greenleaf and G. Uhlmann, Non-local inversion formulas for the X-ray transform, Duke Math. J., 58 (1989), 205-240.
doi: 10.1215/S0012-7094-89-05811-0. |
[15] |
A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, J. Funct. Anal., 89 (1990), 202-232.
doi: 10.1016/0022-1236(90)90011-9. |
[16] |
A. Greenleaf and G. Uhlmann, Microlocal techniques in integral geometry, in Integral Geometry and Tomography (Arcata, CA, 1989), Contemp. Math., Amer. Math. Soc., 113, Providence, RI, 1990, 121-135.
doi: 10.1090/conm/113/1108649. |
[17] |
V. Guillemin, Some Remarks on Integral Geometry, Technical Report, MIT, 1975. |
[18] |
V. Guillemin, Cosmology in $(2 + 1)$-dimensions, Cyclic Models, and Deformations of $M_{2,1}$, Annals of Mathematics Studies, 121, Princeton University Press, Princeton, NJ, 1989. |
[19] |
V. Guillemin and S. Sternberg, Geometric Asymptotics, Mathematical Surveys, No. 14, American Mathematical Society, Providence, R.I., 1977. |
[20] |
V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Math. J., 48 (1981), 251-267.
doi: 10.1215/S0012-7094-81-04814-6. |
[21] |
A. I. Katsevich, Local Tomography for the Limited-Angle Problem, Journal of mathematical analysis and applications, 213 (1997), 160-182.
doi: 10.1006/jmaa.1997.5412. |
[22] |
V. P. Krishnan and E. T. Quinto, Microlocal aspects of bistatic synthetic aperture radar imaging, Inverse Problems and Imaging, 5 (2011), 659-674.
doi: 10.3934/ipi.2011.5.659. |
[23] |
A. Malcolm, B. Ursin and M. de Hoop, Seismic imaging and illumination with internal multiples, Geophysical Journal International, 176 (2009), 847-864.
doi: 10.1111/j.1365-246X.2008.03992.x. |
[24] |
R. B. Melrose and G. A. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32 (1979), 483-519.
doi: 10.1002/cpa.3160320403. |
[25] |
C. J. Nolan and W. W. Symes, Global solution of a linearized inverse problem for the wave equation, Comm. Partial Differential Equations, 22 (1997), 919-952.
doi: 10.1080/03605309708821289. |
[26] |
C. J. Nolan, Scattering in the presence of fold caustics, SIAM J. Appl. Math., 61 (2000), 659-672.
doi: 10.1137/S0036139999356107. |
[27] |
C. J. Nolan and M. Cheney, Synthetic aperture inversion, Inverse Problems, 18 (2002), 221-235.
doi: 10.1088/0266-5611/18/1/315. |
[28] |
C. J. Nolan and M. Cheney, Microlocal analysis of synthetic aperture radar imaging, J. Fourier Anal. Appl., 10 (2004), 133-148.
doi: 10.1007/s00041-004-8008-0. |
[29] |
E. T. Quinto, Singularities of the X-ray transform and limited data tomography in $\mathbbR^2$ and $\mathbbR^3$, SIAM J. Math. Anal., 24 (1993), 1215-1225.
doi: 10.1137/0524069. |
[30] |
E. T. Quinto, A. Rieder and T. Schuster, Local inversion of the sonar transform regularized by the approximate inverse, Inverse Problems, 27 (2011), 035006, 18pp.
doi: 10.1088/0266-5611/27/3/035006. |
[31] |
Rakesh, A linearised inverse problem for the wave equation, Comm. Partial Differential Equations, 13 (1988), 573-601.
doi: 10.1080/03605308808820553. |
[32] |
R. E. Sheriff, Encyclopedic Dictionary of Applied Geophysics, Society Of Exploration Geophysicists, 2002.
doi: 10.1190/1.9781560802969. |
[33] |
P. Stefanov and G. Uhlmann, Is a curved flight path in {SAR} better than a straight one?, SIAM J. Appl. Math., 73 (2013), 1596-1612.
doi: 10.1137/120882639. |
[34] |
C. C. Stolk, Microlocal analysis of a seismic linearized inverse problem, Wave Motion, 32 (2000), 267-290.
doi: 10.1016/S0165-2125(00)00043-3. |
[35] |
C. C. Stolk and M. V. de Hoop, Microlocal analysis of seismic inverse scattering in anisotropic elastic media, Comm. Pure Appl. Math., 55 (2002), 261-301.
doi: 10.1002/cpa.10019. |
[36] |
C. C. Stolk and M. V. de Hoop, Seismic inverse scattering in the downward continuation approach, Wave Motion, 43 (2006), 579-598.
doi: 10.1016/j.wavemoti.2006.05.003. |
[37] |
W. W. Symes, Mathematics of Reflection Seismology, Technical Report, Department of Computational and Applied Mathematics, Rice University, Houston, Texas, 1990, Technical Report TR90-02. |
[38] |
A. P. E. ten Kroode, D.-J. Smit and A. R. Verdel, A microlocal analysis of migration, Wave Motion, 28 (1998), 149-172.
doi: 10.1016/S0165-2125(98)00004-3. |
[39] |
F. Trèves, Introduction to Pseudodifferential and Fourier Integral Operators, Fourier Integral Operators, Vol. 2, Plenum Press, New York-London, 1980. |
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