Abstract
We propose and investigate the inverse problem of identifying magnetized anomalies beneath the Earth using geomagnetic monitoring. Suppose a collection of magnetized anomalies are presented in the shell of the Earth. The presence of the anomalies interrupts the magnetic field of the Earth, as monitored from above the Earth. Using the difference of the magnetic fields before and after the presence of the magnetized anomalies, we show that one can uniquely recover the locations as well as their material parameters of the anomalies. Our study provides a rigorous mathematical theory for the geomagnetic detection technology that has been used in practice.
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Ammari H., Ciraolo G., Kang H., Lee H., Milton G.: Anomalous localized resonance using a folded geometry in three dimensions. Proc. R. Soc. A 469, 20130048 (2013)
Ammari H., Deng Y., Kang H., Lee H.: Reconstruction of Inhomogeneous conductivities via the concept of generalized polarization tensors. Ann. Inst. Henri Poincaré 31, 877–897 (2014)
Ammari H., Deng Y., Millien P.: Surface plasmon resonance of nanoparticles and applications in imaging. Arch. Ration. Mech. Anal. 220, 109–153 (2016)
Ammari H., Kang H.: Polarization and Moment Tensors With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences. Springer, Berlin (2007)
Backus G., Parker R., Constable C.: Foundations of Geomagnetism. Cambridge University Press, Cambridge (1996)
Colton D., Kress R.: Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edn. Springer, Berlin (1998)
Deng, Y., Liu, H., Liu, X.: Recovery of an Embedded Obstacle and the Surrounding Medium for Maxwell’s System. arXiv:1801.02008
Deng, Y., Liu, H., Uhlmann, G.: On an Inverse Boundary Problem Arising in Brain Imaging. arXiv:1702.00154
Fang X., Deng Y., Li J.: Plasmon resonance and heat generation in nanostructures. Math. Method Appl. Sci. 38, 4663–4672 (2015)
Feynman R., Leighton R., Sands M.: The Feynman Lectures on Physics. The New Millennium Edition, New York (2010)
Leis R.: Zur Theorie elektromagnetischer Schwingungen in anisotropen inhomogenene Medien. Math. Z. 106, 213–224 (1968)
Leis R.: Initial Boundary Value Problems in Mathematical Physics. Wiley, Chichester (1986)
Liu, H., Rondi, L., Xiao, J.: Mosco convergence for H(curl) spaces, higher integrability for Maxwell’s equations, and stability in direct and inverse EM scattering problems. J. Eur. Math. Soc. (in press) (2017)
Nédélec J.C.: Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems. Springer, New York (2001)
Torres R.H.: Maxwell’s equations and dielectric obstacles with Lipschitz boundaries. J. Lond. Math. Soc. 57(2), 157–169 (1998)
Weiss, N.: Dynamos in planets, stars and galaxies. Astron. Geophys. 43, 3.09–3.15 (2002)
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Communicated by P.-L. Lions
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Deng, Y., Li, J. & Liu, H. On Identifying Magnetized Anomalies Using Geomagnetic Monitoring. Arch Rational Mech Anal 231, 153–187 (2019). https://doi.org/10.1007/s00205-018-1276-7
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DOI: https://doi.org/10.1007/s00205-018-1276-7