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August  2016, 10(3): 641-658. doi: 10.3934/ipi.2016015

## The inverse problem for electroseismic conversion: Stable recovery of the conductivity and the electrokinetic mobility parameter

 1 8817 234th St. SW, Edmonds, WA 98026, United States 2 Computational and Applied Mathematics, Rice University, Houston, TX 77005, United States

Received  May 2015 Revised  May 2016 Published  August 2016

Pride (1994, Phys. Rev. B 50 15678-96) derived the governing model of electroseismic conversion, in which Maxwell's equations are coupled with Biot's equations through an electrokinetic mobility parameter. The inverse problem of electroseismic conversion was first studied by Chen and Yang (2013, Inverse Problem 29 115006). By following the construction of Complex Geometrical Optics (CGO) solutions to a matrix Schrödinger equation introduced by Ola and Somersalo (1996, SIAM J. Appl. Math. 56 No. 4 1129-1145), we analyze the recovering of conductivity, permittivity and the electrokinetic mobility parameter in Maxwell's equations with internal measurements, while allowing the magnetic permeability $\mu$ to be a variable function. We show that knowledge of two internal data sets associated with well-chosen boundary electrical sources uniquely determines these parameters. Moreover, a Lipschitz-type stability is obtained based on the same set.
Citation: Jie Chen, Maarten de Hoop. The inverse problem for electroseismic conversion: Stable recovery of the conductivity and the electrokinetic mobility parameter. Inverse Problems and Imaging, 2016, 10 (3) : 641-658. doi: 10.3934/ipi.2016015
##### References:
 [1] G. Bal and G. Uhlmann, Inverse diffusion theory of photo-acoustics, Inverse Problems, 26 (2010), 085010, 20pp. doi: 10.1088/0266-5611/26/8/085010. [2] G. Bal and T. Zhou, Hybrid inverse problems for a system of Maxwell's equations, Inverse Problem, 30 (2014), 055013, 17pp. doi: 10.1088/0266-5611/30/5/055013. [3] M. A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid. I-Low-frequency range, Journal of the Acoustical Society of America, 28 (1956), 168-178. doi: 10.1121/1.1908239. [4] M. A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid. II-High-frequency range, Journal of the Acoustical Society of America, 28 (1956), 179-191. doi: 10.1121/1.1908241. [5] P. Caro, P. Ola and M. Salo, Inverse boundary value problem for Maxwell equations with local data, Comm. in PDE, 34 (2009), 1425-1464. doi: 10.1080/03605300903296272. [6] J. Chen and Y. Yang, Quantitative photo-acoustic tomography with partial data, Inverse Problem, 28 (2012), 115014, 15pp. doi: 10.1088/0266-5611/28/11/115014. [7] J. Chen and Y. Yang, Inverse problem of electroseismic conversion, Inverse Problem, 29 (2013), 115006, 15pp. [8] D. Colton and L. Paivarinta, The uniqueness of a solution to an inverse scattering problem for electromagnetic waves, Arch. Rational Mech. Anal., 119 (1992), 59-70. doi: 10.1007/BF00376010. [9] C. Guo and G. Bal, Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields, Inverse Problems and Imaging, 8 (2014), 1033-1051. doi: 10.3934/ipi.2014.8.1033. [10] M. W. Haartsen, Coupled Electromagnetic and Acoustic Wavefield Modeling in Poro-Elastic Media and Its Application in Geophysical Exploration, Ph.D. Thesis, Massachusetts Institute of Technology, 1995. [11] C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic maxwell equations, Duke Math. J., 157 (2011), 369-419. doi: 10.1215/00127094-1272903. [12] P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized sommerfeld potentials, SIAM J. Appl. Math., 56 (1996), 1129-1145. doi: 10.1137/S0036139995283948. [13] P. Ola, L. Päivärinta and E Somersalo, An inverse boundary value problem in electrodynamics, Duke Mathematical Journal, 70 (1993), 617-653. doi: 10.1215/S0012-7094-93-07014-7. [14] T. Plona, Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies, Appl. Phys. Lett., 36 (1980), 259-261. doi: 10.1063/1.91445. [15] S. R. Pride, Governing equations for the coupled electro-magnetics and acoustics of porous media, Phys. Rev., 50 (1994), 15678-15696. [16] S. R. Pride and M. W. Haartsen, Electroseismic wave properties, Journal of the Acoustical Society of America, 100 (1996), 1301-1315. doi: 10.1121/1.416018. [17] M. D. Schakel, Coupled Seismic and Electromagnetic Wave Propagation, Ph.D. thesis, de Technische Universiteit Delft, 2011. [18] A. Thompson and G. Gist, Geophysical applications of electro-kinetic conversion, The Leading Edge, 12 (1993), 1169-1173. [19] A. Thompson and S. Hornbostel et. al., Field tests of electroseismic hydrocarbon detection, Geophysics, 72 (2007), N1-N9. [20] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291. [21] B. White, Asymptotic theory of electro-seismic prospecting, SIAM J. Appl. Math., 65 (2005), 1443-1462. doi: 10.1137/040604108. [22] K. L. Williams, An effective density fluid model for acoustic propagation in sediments derived from Biot theory, J. Acoust. Soc. Am., 110 (2001), 2276-2281. doi: 10.1121/1.1412449. [23] Z. Zhu, M. W. Haartsen and M. N. Toksöz, Experimental studies of electro-kinetic conversions in fluid-saturated bore-hole models, Geophysics, 64 (1999), 1349-1356. [24] Z. Zhu and M. N. Toksöz, Cross hole seismoelectric measurements in bore-hole models with fractures, Geophysics, 68 (2003), 1519-1524. [25] Z. Zhu and M. N. Toksöz, Seismoelectric and seismomagnetic measurements in fractured bore-hole models, Geophysics, 70 (2005), F45-F51.

show all references

##### References:
 [1] G. Bal and G. Uhlmann, Inverse diffusion theory of photo-acoustics, Inverse Problems, 26 (2010), 085010, 20pp. doi: 10.1088/0266-5611/26/8/085010. [2] G. Bal and T. Zhou, Hybrid inverse problems for a system of Maxwell's equations, Inverse Problem, 30 (2014), 055013, 17pp. doi: 10.1088/0266-5611/30/5/055013. [3] M. A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid. I-Low-frequency range, Journal of the Acoustical Society of America, 28 (1956), 168-178. doi: 10.1121/1.1908239. [4] M. A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid. II-High-frequency range, Journal of the Acoustical Society of America, 28 (1956), 179-191. doi: 10.1121/1.1908241. [5] P. Caro, P. Ola and M. Salo, Inverse boundary value problem for Maxwell equations with local data, Comm. in PDE, 34 (2009), 1425-1464. doi: 10.1080/03605300903296272. [6] J. Chen and Y. Yang, Quantitative photo-acoustic tomography with partial data, Inverse Problem, 28 (2012), 115014, 15pp. doi: 10.1088/0266-5611/28/11/115014. [7] J. Chen and Y. Yang, Inverse problem of electroseismic conversion, Inverse Problem, 29 (2013), 115006, 15pp. [8] D. Colton and L. Paivarinta, The uniqueness of a solution to an inverse scattering problem for electromagnetic waves, Arch. Rational Mech. Anal., 119 (1992), 59-70. doi: 10.1007/BF00376010. [9] C. Guo and G. Bal, Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields, Inverse Problems and Imaging, 8 (2014), 1033-1051. doi: 10.3934/ipi.2014.8.1033. [10] M. W. Haartsen, Coupled Electromagnetic and Acoustic Wavefield Modeling in Poro-Elastic Media and Its Application in Geophysical Exploration, Ph.D. Thesis, Massachusetts Institute of Technology, 1995. [11] C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic maxwell equations, Duke Math. J., 157 (2011), 369-419. doi: 10.1215/00127094-1272903. [12] P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized sommerfeld potentials, SIAM J. Appl. Math., 56 (1996), 1129-1145. doi: 10.1137/S0036139995283948. [13] P. Ola, L. Päivärinta and E Somersalo, An inverse boundary value problem in electrodynamics, Duke Mathematical Journal, 70 (1993), 617-653. doi: 10.1215/S0012-7094-93-07014-7. [14] T. Plona, Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies, Appl. Phys. Lett., 36 (1980), 259-261. doi: 10.1063/1.91445. [15] S. R. Pride, Governing equations for the coupled electro-magnetics and acoustics of porous media, Phys. Rev., 50 (1994), 15678-15696. [16] S. R. Pride and M. W. Haartsen, Electroseismic wave properties, Journal of the Acoustical Society of America, 100 (1996), 1301-1315. doi: 10.1121/1.416018. [17] M. D. Schakel, Coupled Seismic and Electromagnetic Wave Propagation, Ph.D. thesis, de Technische Universiteit Delft, 2011. [18] A. Thompson and G. Gist, Geophysical applications of electro-kinetic conversion, The Leading Edge, 12 (1993), 1169-1173. [19] A. Thompson and S. Hornbostel et. al., Field tests of electroseismic hydrocarbon detection, Geophysics, 72 (2007), N1-N9. [20] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291. [21] B. White, Asymptotic theory of electro-seismic prospecting, SIAM J. Appl. Math., 65 (2005), 1443-1462. doi: 10.1137/040604108. [22] K. L. Williams, An effective density fluid model for acoustic propagation in sediments derived from Biot theory, J. Acoust. Soc. Am., 110 (2001), 2276-2281. doi: 10.1121/1.1412449. [23] Z. Zhu, M. W. Haartsen and M. N. Toksöz, Experimental studies of electro-kinetic conversions in fluid-saturated bore-hole models, Geophysics, 64 (1999), 1349-1356. [24] Z. Zhu and M. N. Toksöz, Cross hole seismoelectric measurements in bore-hole models with fractures, Geophysics, 68 (2003), 1519-1524. [25] Z. Zhu and M. N. Toksöz, Seismoelectric and seismomagnetic measurements in fractured bore-hole models, Geophysics, 70 (2005), F45-F51.
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