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November  2020, 40(11): 6411-6440. doi: 10.3934/dcds.2020285

Identifying varying magnetic anomalies using geomagnetic monitoring

1. 

School of Mathematics and Statistics, Central South University, Changsha, Hunan, China

2. 

Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong SAR, China

3. 

Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong SAR, China

* Corresponding author: Hongyu Liu

Received  January 2020 Revised  June 2020 Published  July 2020

We are concerned with the inverse problem of identifying magnetic anomalies with varying parameters beneath the Earth using geomagnetic monitoring. Observations of the change in Earth's magnetic field–the secular variation–provide information about the anomalies as well as their variations. In this paper, we rigorously establish the unique recovery results for this magnetic anomaly detection problem. We show that one can uniquely recover the locations, the variation parameters including the growth or decaying rates as well as their material parameters of the anomalies. This paper extends the existing results in [9] by two of the authors to the more practical and challenging scenario with varying anomalies.

Citation: Youjun Deng, Hongyu Liu, Wing-Yan Tsui. Identifying varying magnetic anomalies using geomagnetic monitoring. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6411-6440. doi: 10.3934/dcds.2020285
References:
[1]

H. AmmariY. Deng and P. Millien, Surface plasmon resonance of nanoparticles and applications in imaging, Arch. Ration. Mech. Anal., 220 (2016), 109-153.  doi: 10.1007/s00205-015-0928-0.

[2]

H. Ammari and H. Kang, Polarization and Moment Tensors, Applied Mathematical Sciences, 162, Springer, New York, 2007. doi: 10.1007/978-0-387-71566-7.

[3]

H. Ammari and J.-C. Nédélec, Low-frequency electromagnetic scattering, SIAM J. Math. Anal., 31 (2000), 836-861.  doi: 10.1137/S0036141098343604.

[4]

H. AmmariM. S. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter. Ⅱ. The full Maxwell equations, J. Math. Pures Appl. (9), 80 (2001), 769-814.  doi: 10.1016/S0021-7824(01)01217-X.

[5] G. BackusR. Parker and and C. Constable, Foundations of Geomagnetism, Cambridge University Press, 1996. 
[6]

A. Coghlan, Molten Iron River Discovered Speeding Beneath Russia and Canada, 2016. Available from: https://www.newscientist.com/article/2116536-molten-iron-river-discovered-speeding-beneath-russia-and-canada.

[7]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.

[8]

G. Dassios, Low-frequency scattering, in Scattering: Scattering and Inverse Scattering in Pure and Applied Science, Academic Press, 2002, 230-244. doi: 10.1016/B978-012613760-6/50014-0.

[9]

Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring, Arch. Ration. Mech. Anal., 231 (2019), 153-187.  doi: 10.1007/s00205-018-1276-7.

[10]

Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring within a magnetohydrodynamic model, Arch. Ration. Mech. Anal., 235 (2020), 691-721.  doi: 10.1007/s00205-019-01429-x.

[11]

Y. DengH. Liu and X. Liu, Recovery of an embedded obstacle and the surrounding medium for Maxwell's system, J. Differential Equations, 267 (2019), 2192-2209.  doi: 10.1016/j.jde.2019.03.009.

[12]

Y. DengH. Liu and G. Uhlmann, On an inverse boundary problem arising in brain imaging, J. Differential Equations, 267 (2019), 2471-2502.  doi: 10.1016/j.jde.2019.03.019.

[13]

R. E. Kleinman, Low frequency electromagnetic scattering, in Electromagnetic Scattering, Academic Press, New York-London, 1978, 1-28.

[14]

R. Leis, Initial-Boundary Value Problems in Mathematical Physics, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986. doi: 10.1007/978-3-663-10649-4.

[15]

H. LiuL. Rondi and J. Xiao, Mosco convergence for $H({\rm curl})$ spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems, J. Eur. Math. Soc. (JEMS), 21 (2019), 2945-2993.  doi: 10.4171/JEMS/895.

[16]

P. W. LivermoreR. Hollerbach and C. C. Finlay, An accelerating high-latitude jet in Earth's core, Nature Geoscience, 10 (2017), 62-68.  doi: 10.1038/ngeo2859.

[17]

J.-C. Nédélec, Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems, Applied Mathematical Sciences, 144, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4393-7.

[18]

R. H. Torres, Maxwell's equations and dielectric obstacles with Lipschitz boundaries, J. London Math. Soc. (2), 57 (1998), 157-169.  doi: 10.1112/S0024610798005900.

[19]

The Giant Underground Iron River Between Russia and Canada is 3 Times Faster, 2016. Available from: http://www.ultimatescience.org/giant-underground-iron-river-russia-canada-3-times-faster.

[20]

Wikipedia, https://en.wikipedia.org/wiki/Magnetic_anomaly_detector.

show all references

References:
[1]

H. AmmariY. Deng and P. Millien, Surface plasmon resonance of nanoparticles and applications in imaging, Arch. Ration. Mech. Anal., 220 (2016), 109-153.  doi: 10.1007/s00205-015-0928-0.

[2]

H. Ammari and H. Kang, Polarization and Moment Tensors, Applied Mathematical Sciences, 162, Springer, New York, 2007. doi: 10.1007/978-0-387-71566-7.

[3]

H. Ammari and J.-C. Nédélec, Low-frequency electromagnetic scattering, SIAM J. Math. Anal., 31 (2000), 836-861.  doi: 10.1137/S0036141098343604.

[4]

H. AmmariM. S. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter. Ⅱ. The full Maxwell equations, J. Math. Pures Appl. (9), 80 (2001), 769-814.  doi: 10.1016/S0021-7824(01)01217-X.

[5] G. BackusR. Parker and and C. Constable, Foundations of Geomagnetism, Cambridge University Press, 1996. 
[6]

A. Coghlan, Molten Iron River Discovered Speeding Beneath Russia and Canada, 2016. Available from: https://www.newscientist.com/article/2116536-molten-iron-river-discovered-speeding-beneath-russia-and-canada.

[7]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.

[8]

G. Dassios, Low-frequency scattering, in Scattering: Scattering and Inverse Scattering in Pure and Applied Science, Academic Press, 2002, 230-244. doi: 10.1016/B978-012613760-6/50014-0.

[9]

Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring, Arch. Ration. Mech. Anal., 231 (2019), 153-187.  doi: 10.1007/s00205-018-1276-7.

[10]

Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring within a magnetohydrodynamic model, Arch. Ration. Mech. Anal., 235 (2020), 691-721.  doi: 10.1007/s00205-019-01429-x.

[11]

Y. DengH. Liu and X. Liu, Recovery of an embedded obstacle and the surrounding medium for Maxwell's system, J. Differential Equations, 267 (2019), 2192-2209.  doi: 10.1016/j.jde.2019.03.009.

[12]

Y. DengH. Liu and G. Uhlmann, On an inverse boundary problem arising in brain imaging, J. Differential Equations, 267 (2019), 2471-2502.  doi: 10.1016/j.jde.2019.03.019.

[13]

R. E. Kleinman, Low frequency electromagnetic scattering, in Electromagnetic Scattering, Academic Press, New York-London, 1978, 1-28.

[14]

R. Leis, Initial-Boundary Value Problems in Mathematical Physics, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986. doi: 10.1007/978-3-663-10649-4.

[15]

H. LiuL. Rondi and J. Xiao, Mosco convergence for $H({\rm curl})$ spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems, J. Eur. Math. Soc. (JEMS), 21 (2019), 2945-2993.  doi: 10.4171/JEMS/895.

[16]

P. W. LivermoreR. Hollerbach and C. C. Finlay, An accelerating high-latitude jet in Earth's core, Nature Geoscience, 10 (2017), 62-68.  doi: 10.1038/ngeo2859.

[17]

J.-C. Nédélec, Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems, Applied Mathematical Sciences, 144, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4393-7.

[18]

R. H. Torres, Maxwell's equations and dielectric obstacles with Lipschitz boundaries, J. London Math. Soc. (2), 57 (1998), 157-169.  doi: 10.1112/S0024610798005900.

[19]

The Giant Underground Iron River Between Russia and Canada is 3 Times Faster, 2016. Available from: http://www.ultimatescience.org/giant-underground-iron-river-russia-canada-3-times-faster.

[20]

Wikipedia, https://en.wikipedia.org/wiki/Magnetic_anomaly_detector.

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