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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

 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.
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 $\mathbb{R}^2$ and $\mathbb{R}^3$, 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 $\mathbb{R}^2$ and $\mathbb{R}^3$, 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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