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doi: 10.3934/ipi.2022026
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Microlocal analysis of borehole seismic data

1. 

School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY, 14623, USA

2. 

Department of Mathematics and Statistics, CONFIRM, HRI, University of Limerick, Limerick, V94 T9PX, Ireland

3. 

Department of Mathematics, University of Rochester, Rochester, NY, 14627, USA

* Corresponding author: Allan Greenleaf, allan@math.rochester.edu

Dedicated to the memory of Victor Isakov, for his contributions to the field and for his friendship

Received  October 2021 Revised  March 2022 Early access May 2022

Borehole seismic data is obtained by receivers located in a well, with sources located on the surface or another well. Using microlocal analysis, we study possible approximate reconstruction, via linearized, filtered backprojection, of an isotropic sound speed in the subsurface for three types of data sets. The sources may form a dense array on the surface, or be located along a line on the surface (walkaway geometry) or in another borehole (crosswell). We show that for the dense array, reconstruction is feasible, with no artifacts in the absence of caustics in the background ray geometry, and mild artifacts in the presence of fold caustics in a sense that we define. In contrast, the walkaway and crosswell data sets both give rise to strong, nonremovable artifacts.

Citation: Raluca Felea, Romina Gaburro, Allan Greenleaf, Clifford Nolan. Microlocal analysis of borehole seismic data. Inverse Problems and Imaging, doi: 10.3934/ipi.2022026
References:
[1]

J. Ajo-FranklinJ. PetersonJ. Doetsch and T. Daleya, High-resolution characterization of a CO2 plume using crosswell seismic tomography: Cranfield, MS, USA, Int. J. Greenhouse Gas Control, 18 (2013), 497-509.  doi: 10.1016/j.ijggc.2012.12.018.

[2]

G. AmbartsoumianR. FeleaV. KrishnanC. Nolan and E. T. Quinto, A class of singular Fourier integral operators in synthetic aperture radar imaging, II: Transmitter and receiver with different speeds, SIAM J. Math. Analysis, 50 (2018), 591-621.  doi: 10.1137/17M1125741.

[3]

A. Balch and M. Lee, Vertical Seismic Profiling: Technique, Applications, and Case Histories, International Human Resources Development Corporation (Boston), 1984.

[4]

G. Beylkin, Imaging of discontinuities in the inverse problem by inversion of a generalized Radon transform, J. Math. Phys., 26 (1985), 99-108.  doi: 10.1063/1.526755.

[5]

T. DaleyE. Majer and J. Peterson, Crosswell seismic imaging in a contaminated basalt aquifer, Geophysics, 69 (2004), 16-24.  doi: 10.1190/1.1649371.

[6]

T. DaleyL. MyerJ. PetersonE. Majer and G. Hoversten, Time-lapse crosswell seismic and VSP monitoring of injected CO 2 in a brine aquifer, Environ. Geology, 54 (2008), 1657-1665. 

[7]

J. J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Inv. Math., 29 (1975), 39-79.  doi: 10.1007/BF01405172.

[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]

_, Displacement of artefacts in inverse scattering, Inverse Problems, 23 (2007), 1519. 

[10]

R. FeleaR. GaburroA. Greenleaf and C. Nolan, Microlocal analysis of Doppler synthetic aperture radar, Inverse Prob. Imaging, 13 (2019), 1283-1307.  doi: 10.3934/ipi.2019056.

[11]

R. Felea and A. Greenleaf, An FIO calculus for marine seismic imaging: Folds and cross-caps, Comm. Partial Differential Equations, 33 (2008), 45-77.  doi: 10.1080/03605300701318716.

[12]

R. Felea and A. Greenleaf, Fourier integral operators with open umbrellas and seismic inversion for cusp caustics, Math. Research Lett., 17 (2010), 867-886. 

[13]

R. FeleaA. 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.

[14]

R. FeleaV. P. KrishnanC. Nolan and E. T. Quinto, Common midpoint versus common offset acquisition geometry in seismic imaging, Inverse Probl. Imaging, 10 (2016), 87-102.  doi: 10.3934/ipi.2016.10.87.

[15]

R. Felea and C. Nolan, Monostatic SAR with fold/cusp singularities, J. Fourier Anal. Appl., 21 (2015), 799-821.  doi: 10.1007/s00041-015-9387-0.

[16]

R. Felea and E. T. Quinto, The microlocal properties of the local 3-D SPECT operator, SIAM J. Math. Anal., 43 (2011), 1145-1157.  doi: 10.1137/100807703.

[17]

M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Graduate Texts in Mathematics, 14, Springer-Verlag, New York-Heidelberg, 1973.

[18]

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.

[19]

V. Guillemin, On some results of Gelfand in integral geometry, Proc. Symp. Pure Math, 43 (1985), 149-155.  doi: 10.1090/pspum/043/812288.

[20]

V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Math. J., 48 (1981), 251-267. 

[21]

S. Hansen, Solution of a hyperbolic inverse problem by linearization, Comm. Partial Differential Equations, 16 (1991), 291-309.  doi: 10.1080/03605309108820760.

[22]

L. Hörmander, Fourier integral operators, I, Acta Math., 127 (1971), 79-183.  doi: 10.1007/BF02392052.

[23]

A. Kirsch and A. Rieder, On the linearization of operators related to the full waveform inversion in seismology, Math. Meth. Appl. Sci., 37 (2014), 2995-3007.  doi: 10.1002/mma.3037.

[24]

A. ten KroodeD. Smit and A. Verdel, A microlocal analysis of migration, Wave Motion, 28 (1998), 149-172.  doi: 10.1016/S0165-2125(98)00004-3.

[25]

R. Melrose and M. Taylor, Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle, Adv. in Math., 55 (1985), 242-315.  doi: 10.1016/0001-8708(85)90093-3.

[26]

R. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32 (1979), 483-519.  doi: 10.1002/cpa.3160320403.

[27]

B. Morin, Formes canoniques des singularités d'une application differentiable, I, C. R. Acad. Sc. Paris, 260 (1965), 6503-6506. 

[28]

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

[29]

C. Nolan and M. Cheney, Microlocal analysis of synthetic aperture radar, J. Fourier Anal. Appl., 10 (2004), 133-148.  doi: 10.1007/s00041-004-8008-0.

[30]

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.

[31]

O. Podgornova, Lecture at ICERM, 2015, https://icerm.brown.edu/video_archive/?play=720.

[32]

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

[33]

C. Schmelzbach, et al., Advanced seismic processing/imaging techniques and their potential for geothermal exploration, Interpretation, 4 (2016). doi: 10.1190/INT-2016-0017.1.

[34]

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]

W. W. Symes, The seismic reflection inverse problem, Inverse Problems, 25 (2009), 123008.  doi: 10.1088/0266-5611/25/12/123008.

[36]

A. Weinstein, On Maslov's quantization condition, In Fourier Integral Operators and Partial Differential Equations, Lecture Notes in Math., Springer-Verlag, New York, 459 (1975), 341–372.

[37]

H. Whitney, The general type of singularity of a set of $2n-1$ smooth functions of $n$ variables, Duke Math. J., 10 (1943), 161-172. 

show all references

References:
[1]

J. Ajo-FranklinJ. PetersonJ. Doetsch and T. Daleya, High-resolution characterization of a CO2 plume using crosswell seismic tomography: Cranfield, MS, USA, Int. J. Greenhouse Gas Control, 18 (2013), 497-509.  doi: 10.1016/j.ijggc.2012.12.018.

[2]

G. AmbartsoumianR. FeleaV. KrishnanC. Nolan and E. T. Quinto, A class of singular Fourier integral operators in synthetic aperture radar imaging, II: Transmitter and receiver with different speeds, SIAM J. Math. Analysis, 50 (2018), 591-621.  doi: 10.1137/17M1125741.

[3]

A. Balch and M. Lee, Vertical Seismic Profiling: Technique, Applications, and Case Histories, International Human Resources Development Corporation (Boston), 1984.

[4]

G. Beylkin, Imaging of discontinuities in the inverse problem by inversion of a generalized Radon transform, J. Math. Phys., 26 (1985), 99-108.  doi: 10.1063/1.526755.

[5]

T. DaleyE. Majer and J. Peterson, Crosswell seismic imaging in a contaminated basalt aquifer, Geophysics, 69 (2004), 16-24.  doi: 10.1190/1.1649371.

[6]

T. DaleyL. MyerJ. PetersonE. Majer and G. Hoversten, Time-lapse crosswell seismic and VSP monitoring of injected CO 2 in a brine aquifer, Environ. Geology, 54 (2008), 1657-1665. 

[7]

J. J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Inv. Math., 29 (1975), 39-79.  doi: 10.1007/BF01405172.

[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]

_, Displacement of artefacts in inverse scattering, Inverse Problems, 23 (2007), 1519. 

[10]

R. FeleaR. GaburroA. Greenleaf and C. Nolan, Microlocal analysis of Doppler synthetic aperture radar, Inverse Prob. Imaging, 13 (2019), 1283-1307.  doi: 10.3934/ipi.2019056.

[11]

R. Felea and A. Greenleaf, An FIO calculus for marine seismic imaging: Folds and cross-caps, Comm. Partial Differential Equations, 33 (2008), 45-77.  doi: 10.1080/03605300701318716.

[12]

R. Felea and A. Greenleaf, Fourier integral operators with open umbrellas and seismic inversion for cusp caustics, Math. Research Lett., 17 (2010), 867-886. 

[13]

R. FeleaA. 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.

[14]

R. FeleaV. P. KrishnanC. Nolan and E. T. Quinto, Common midpoint versus common offset acquisition geometry in seismic imaging, Inverse Probl. Imaging, 10 (2016), 87-102.  doi: 10.3934/ipi.2016.10.87.

[15]

R. Felea and C. Nolan, Monostatic SAR with fold/cusp singularities, J. Fourier Anal. Appl., 21 (2015), 799-821.  doi: 10.1007/s00041-015-9387-0.

[16]

R. Felea and E. T. Quinto, The microlocal properties of the local 3-D SPECT operator, SIAM J. Math. Anal., 43 (2011), 1145-1157.  doi: 10.1137/100807703.

[17]

M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Graduate Texts in Mathematics, 14, Springer-Verlag, New York-Heidelberg, 1973.

[18]

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.

[19]

V. Guillemin, On some results of Gelfand in integral geometry, Proc. Symp. Pure Math, 43 (1985), 149-155.  doi: 10.1090/pspum/043/812288.

[20]

V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Math. J., 48 (1981), 251-267. 

[21]

S. Hansen, Solution of a hyperbolic inverse problem by linearization, Comm. Partial Differential Equations, 16 (1991), 291-309.  doi: 10.1080/03605309108820760.

[22]

L. Hörmander, Fourier integral operators, I, Acta Math., 127 (1971), 79-183.  doi: 10.1007/BF02392052.

[23]

A. Kirsch and A. Rieder, On the linearization of operators related to the full waveform inversion in seismology, Math. Meth. Appl. Sci., 37 (2014), 2995-3007.  doi: 10.1002/mma.3037.

[24]

A. ten KroodeD. Smit and A. Verdel, A microlocal analysis of migration, Wave Motion, 28 (1998), 149-172.  doi: 10.1016/S0165-2125(98)00004-3.

[25]

R. Melrose and M. Taylor, Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle, Adv. in Math., 55 (1985), 242-315.  doi: 10.1016/0001-8708(85)90093-3.

[26]

R. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32 (1979), 483-519.  doi: 10.1002/cpa.3160320403.

[27]

B. Morin, Formes canoniques des singularités d'une application differentiable, I, C. R. Acad. Sc. Paris, 260 (1965), 6503-6506. 

[28]

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

[29]

C. Nolan and M. Cheney, Microlocal analysis of synthetic aperture radar, J. Fourier Anal. Appl., 10 (2004), 133-148.  doi: 10.1007/s00041-004-8008-0.

[30]

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.

[31]

O. Podgornova, Lecture at ICERM, 2015, https://icerm.brown.edu/video_archive/?play=720.

[32]

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

[33]

C. Schmelzbach, et al., Advanced seismic processing/imaging techniques and their potential for geothermal exploration, Interpretation, 4 (2016). doi: 10.1190/INT-2016-0017.1.

[34]

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]

W. W. Symes, The seismic reflection inverse problem, Inverse Problems, 25 (2009), 123008.  doi: 10.1088/0266-5611/25/12/123008.

[36]

A. Weinstein, On Maslov's quantization condition, In Fourier Integral Operators and Partial Differential Equations, Lecture Notes in Math., Springer-Verlag, New York, 459 (1975), 341–372.

[37]

H. Whitney, The general type of singularity of a set of $2n-1$ smooth functions of $n$ variables, Duke Math. J., 10 (1943), 161-172. 

Figure 1.  Illustration of data acquisition geometry and filtering: contributions to the data from unbroken rays such as that illustrated here are filtered out by removing data associated to those rays arriving from nearby directions, as indicated by the gray cone
Figure 2.  Construction of the $ y $-coordinates, with the $ y_1 $-direction being tangent to the ray connecting $ y $ to a source $ (s, 0)\in\Sigma_S $
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