Common midpoint versus common offset acquisition geometry in seismic imaging

Previous Article
Factorization method in inverse interaction problems with bi-periodic interfaces between acoustic and elastic waves
IPI Home
This Issue
Next Article
The topological gradient method for semi-linear problems and application to edge detection and noise removal

February 2016, 10(1): 87-102. doi: 10.3934/ipi.2016.10.87

Common midpoint versus common offset acquisition geometry in seismic imaging

Raluca Felea ^1, , Venkateswaran P. Krishnan ^2, , Clifford J. Nolan ^3, and Eric Todd Quinto ^4,

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

Received December 2014 Revised June 2015 Published February 2016

Abstract
Related Papers

We compare and contrast the qualitative nature of backprojected images obtained in seismic imaging when common offset data are used versus when common midpoint data are used. Our results show that the image obtained using common midpoint data contains artifacts which are not present with common offset data. Although there are situations where one would still want to use common midpoint data, this result points out a shortcoming that should be kept in mind when interpreting the images.

Keywords: Fourier integral operators, fold and blowdown singularities, geophysics., microlocal analysis, Seismic imaging, paired Lagrangians.

Mathematics Subject Classification: 58J40, 57R45, 86A15, 86A22, 35S3.

Citation: Raluca Felea, Venkateswaran P. Krishnan, Clifford J. Nolan, Eric Todd Quinto. Common midpoint versus common offset acquisition geometry in seismic imaging. Inverse Problems and Imaging, 2016, 10 (1) : 87-102. doi: 10.3934/ipi.2016.10.87

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.

[1]	Raluca Felea, Romina Gaburro, Allan Greenleaf, Clifford Nolan. Microlocal analysis of borehole seismic data. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022026
[2]	Elena Cordero, Fabio Nicola, Luigi Rodino. Time-frequency analysis of fourier integral operators. Communications on Pure and Applied Analysis, 2010, 9 (1) : 1-21. doi: 10.3934/cpaa.2010.9.1
[3]	Daniela Calvetti, Erkki Somersalo. Microlocal sequential regularization in imaging. Inverse Problems and Imaging, 2007, 1 (1) : 1-11. doi: 10.3934/ipi.2007.1.1
[4]	Kanghui Guo and Demetrio Labate. Sparse shearlet representation of Fourier integral operators. Electronic Research Announcements, 2007, 14: 7-19. doi: 10.3934/era.2007.14.7
[5]	James W. Webber, Sean Holman. Microlocal analysis of a spindle transform. Inverse Problems and Imaging, 2019, 13 (2) : 231-261. doi: 10.3934/ipi.2019013
[6]	Venkateswaran P. Krishnan, Eric Todd Quinto. Microlocal aspects of common offset synthetic aperture radar imaging. Inverse Problems and Imaging, 2011, 5 (3) : 659-674. doi: 10.3934/ipi.2011.5.659
[7]	Radjesvarane Alexandre, Lingbing He. Integral estimates for a linear singular operator linked with Boltzmann operators part II: High singularities $1\le\nu<2$. Kinetic and Related Models, 2008, 1 (4) : 491-513. doi: 10.3934/krm.2008.1.491
[8]	Raluca Felea, Romina Gaburro, Allan Greenleaf, Clifford Nolan. Microlocal analysis of Doppler synthetic aperture radar. Inverse Problems and Imaging, 2019, 13 (6) : 1283-1307. doi: 10.3934/ipi.2019056
[9]	G. Infante. Positive solutions of some nonlinear BVPs involving singularities and integral BCs. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 99-106. doi: 10.3934/dcdss.2008.1.99
[10]	Ennio Fedrizzi. High frequency analysis of imaging with noise blending. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 979-998. doi: 10.3934/dcdsb.2014.19.979
[11]	Peter W. Bates, Ji Li, Mingji Zhang. Singular fold with real noise. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2091-2107. doi: 10.3934/dcdsb.2016038
[12]	Dorota Bors, Andrzej Skowron, Stanisław Walczak. Systems described by Volterra type integral operators. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2401-2416. doi: 10.3934/dcdsb.2014.19.2401
[13]	Patricio Felmer, Alexander Quaas. Fundamental solutions for a class of Isaacs integral operators. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 493-508. doi: 10.3934/dcds.2011.30.493
[14]	Hermann Brunner. On Volterra integral operators with highly oscillatory kernels. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 915-929. doi: 10.3934/dcds.2014.34.915
[15]	Ahmad Al-Salman. Marcinkiewicz integral operators along twisted surfaces. Communications on Pure and Applied Analysis, 2022, 21 (1) : 159-181. doi: 10.3934/cpaa.2021173
[16]	Gary Froyland, Cecilia González-Tokman, Anthony Quas. Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools. Journal of Computational Dynamics, 2014, 1 (2) : 249-278. doi: 10.3934/jcd.2014.1.249
[17]	Jorge J. Betancor, Alejandro J. Castro, Marta De León-Contreras. Variation operators for semigroups associated with Fourier-Bessel expansions. Communications on Pure and Applied Analysis, 2022, 21 (1) : 239-273. doi: 10.3934/cpaa.2021176
[18]	Shanshan Wang, Yanxia Chen, Taohui Xiao, Lei Zhang, Xin Liu, Hairong Zheng. LANTERN: Learn analysis transform network for dynamic magnetic resonance imaging. Inverse Problems and Imaging, 2021, 15 (6) : 1363-1379. doi: 10.3934/ipi.2020051
[19]	Joshua L. Mike, Vasileios Maroulas. Combinatorial Hodge theory for equitable kidney paired donation. Foundations of Data Science, 2019, 1 (1) : 87-101. doi: 10.3934/fods.2019004
[20]	Congming Li, Jisun Lim. The singularity analysis of solutions to some integral equations. Communications on Pure and Applied Analysis, 2007, 6 (2) : 453-464. doi: 10.3934/cpaa.2007.6.453

2020 Impact Factor: 1.639

Tools

Metrics

Other articles
by authors

on AIMS
Raluca Felea Venkateswaran P. Krishnan Clifford J. Nolan Eric Todd Quinto
on Google Scholar
Raluca Felea Venkateswaran P. Krishnan Clifford J. Nolan Eric Todd Quinto

[Back to Top]