Advertisement

Archive for Rational Mechanics and Analysis

, Volume 231, Issue 1, pp 153–187 | Cite as

On Identifying Magnetized Anomalies Using Geomagnetic Monitoring

  • Youjun Deng
  • Jinhong Li
  • Hongyu LiuEmail author
Article

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ammari H., Deng Y., Millien P.: Surface plasmon resonance of nanoparticles and applications in imaging. Arch. Ration. Mech. Anal. 220, 109–153 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ammari H., Kang H.: Polarization and Moment Tensors With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences. Springer, Berlin (2007)zbMATHGoogle Scholar
  5. 5.
    Backus G., Parker R., Constable C.: Foundations of Geomagnetism. Cambridge University Press, Cambridge (1996)Google Scholar
  6. 6.
    Colton D., Kress R.: Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edn. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  7. 7.
    Deng, Y., Liu, H., Liu, X.: Recovery of an Embedded Obstacle and the Surrounding Medium for Maxwell’s System. arXiv:1801.02008
  8. 8.
    Deng, Y., Liu, H., Uhlmann, G.: On an Inverse Boundary Problem Arising in Brain Imaging. arXiv:1702.00154
  9. 9.
    Fang X., Deng Y., Li J.: Plasmon resonance and heat generation in nanostructures. Math. Method Appl. Sci. 38, 4663–4672 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Feynman R., Leighton R., Sands M.: The Feynman Lectures on Physics. The New Millennium Edition, New York (2010)zbMATHGoogle Scholar
  11. 11.
    Leis R.: Zur Theorie elektromagnetischer Schwingungen in anisotropen inhomogenene Medien. Math. Z. 106, 213–224 (1968)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Leis R.: Initial Boundary Value Problems in Mathematical Physics. Wiley, Chichester (1986)CrossRefzbMATHGoogle Scholar
  13. 13.
    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)Google Scholar
  14. 14.
    Nédélec J.C.: Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  15. 15.
    Torres R.H.: Maxwell’s equations and dielectric obstacles with Lipschitz boundaries. J. Lond. Math. Soc. 57(2), 157–169 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Weiss, N.: Dynamos in planets, stars and galaxies. Astron. Geophys. 43, 3.09–3.15 (2002)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaChina
  2. 2.School of ScienceQilu University of Technology (Shandong Academy of Sciences)JinanChina
  3. 3.Department of MathematicsHong Kong Baptist UniversityKowloonChina

Personalised recommendations