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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. Devaney, Geophysical diffraction tomography, IEEE Transactions on Geoscience and Remote Sensing, 22 (1984), 3-13. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

R. Felea, Displacement of artefacts in inverse scattering, Inverse Problems, 23 (2007), 1519-1531. doi: 10.1088/0266-5611/23/4/009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

V. Guillemin, Some Remarks on Integral Geometry, Technical Report, MIT, 1975. Google Scholar

[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.  Google Scholar

[19]

V. Guillemin and S. Sternberg, Geometric Asymptotics, Mathematical Surveys, No. 14, American Mathematical Society, Providence, R.I., 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

C. J. Nolan, Scattering in the presence of fold caustics, SIAM J. Appl. Math., 61 (2000), 659-672. doi: 10.1137/S0036139999356107.  Google Scholar

[27]

C. J. Nolan and M. Cheney, Synthetic aperture inversion, Inverse Problems, 18 (2002), 221-235. doi: 10.1088/0266-5611/18/1/315.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

Rakesh, A linearised inverse problem for the wave equation, Comm. Partial Differential Equations, 13 (1988), 573-601. doi: 10.1080/03605308808820553.  Google Scholar

[32]

R. E. Sheriff, Encyclopedic Dictionary of Applied Geophysics, Society Of Exploration Geophysicists, 2002. doi: 10.1190/1.9781560802969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[39]

F. Trèves, Introduction to Pseudodifferential and Fourier Integral Operators, Fourier Integral Operators, Vol. 2, Plenum Press, New York-London, 1980.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. Devaney, Geophysical diffraction tomography, IEEE Transactions on Geoscience and Remote Sensing, 22 (1984), 3-13. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

R. Felea, Displacement of artefacts in inverse scattering, Inverse Problems, 23 (2007), 1519-1531. doi: 10.1088/0266-5611/23/4/009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

V. Guillemin, Some Remarks on Integral Geometry, Technical Report, MIT, 1975. Google Scholar

[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.  Google Scholar

[19]

V. Guillemin and S. Sternberg, Geometric Asymptotics, Mathematical Surveys, No. 14, American Mathematical Society, Providence, R.I., 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

C. J. Nolan, Scattering in the presence of fold caustics, SIAM J. Appl. Math., 61 (2000), 659-672. doi: 10.1137/S0036139999356107.  Google Scholar

[27]

C. J. Nolan and M. Cheney, Synthetic aperture inversion, Inverse Problems, 18 (2002), 221-235. doi: 10.1088/0266-5611/18/1/315.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

Rakesh, A linearised inverse problem for the wave equation, Comm. Partial Differential Equations, 13 (1988), 573-601. doi: 10.1080/03605308808820553.  Google Scholar

[32]

R. E. Sheriff, Encyclopedic Dictionary of Applied Geophysics, Society Of Exploration Geophysicists, 2002. doi: 10.1190/1.9781560802969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[39]

F. Trèves, Introduction to Pseudodifferential and Fourier Integral Operators, Fourier Integral Operators, Vol. 2, Plenum Press, New York-London, 1980.  Google Scholar

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