# American Institute of Mathematical Sciences

August  2019, 13(4): 745-754. doi: 10.3934/ipi.2019034

## Magnetic parameters inversion method with full tensor gradient data

 1 Key Laboratory of Petroleum Resources Research, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China 2 University of the Chinese Academy of Sciences, Beijing 100049, China 3 Institutions of Earth Science, Chinese Academy of Sciences, Beijing 100029, China 4 Department of Mathematics, Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia

* Corresponding author: Yanfei Wang

Received  April 2018 Revised  January 2019 Published  May 2019

Retrieval of magnetization parameters using magnetic tensor gradient measurements receives attention in recent years. Determination of subsurface properties from the observed potential field measurements is referred to as inversion. Little regularizing inversion results using full tensor magnetic gradient modeling so far has been reported in the literature. Traditional magnetic inversion is based on the total magnetic intensity (TMI) data and solving the corresponding mathematical physical model. In recent years, with the development of the advanced technology, acquisition of the full tensor gradient magnetic data becomes available. In this paper, we study invert the magnetic parameters using the full tensor magnetic gradient data. A sparse Tikhonov regularization model is established. In solving the minimization model, the conjugate gradient method is addressed. Numerical and field data experiments are performed to show feasibility of our algorithm.

Citation: Yanfei Wang, Dmitry Lukyanenko, Anatoly Yagola. Magnetic parameters inversion method with full tensor gradient data. Inverse Problems & Imaging, 2019, 13 (4) : 745-754. doi: 10.3934/ipi.2019034
##### References:
 [1] O. M. Alifanov, E. A. Artuhin and S. V. Rumyantsev, Extreme Methods for the Solution of Ill-Posed Problems, Moscow: Nauka, 1988. [2] A. V. Goncharskii, A. S. Leonov and A. G. Yagola, A generalized discrepancy principle, USSR Computational Mathematics and Mathematical Physics, 13 (1973), 25-37. [3] P. Heath, G. Heinson and S. Greenhalgh, Some comments on potential field tensor data, Exploration Geophysics, 34 (2003), 57-62. doi: 10.1071/EG03057. [4] S. X. Ji, Y. F. Wang and A. Q. Zou, Regularizing inversion of susceptibility with projection onto convex set using full tensor magnetic gradient data, Inverse Problems in Science and Engineering, 25 (2017), 202-217. doi: 10.1080/17415977.2016.1160390. [5] Y. G. Li and D. W. Oldenburg, 3-D inversion of magnetic data, Geophysics, 61 (1996), 394-408. doi: 10.1190/1.1822498. [6] D. V. Lukyanenko, A. G. Yagola and N. A. Evdokimova, Application of inversion methods in solving ill-posed problems for magnetic parameter identification of steel hull vessel, Journal of Inverse and Ill-Posed Problems, 18 (2011), 1013-1029. doi: 10.1515/JIIP.2011.018. [7] D. V. Lukyanenko and A. G. Yagola, Some methods for solving of 3d inverse problem of magnetometry, Eurasian Journal of Mathematical and Computer Applications, 4 (2016), 4-14. [8] A. Pignatelli, I. Nicolosi and M. Chiappini, An alternative 3D inversion method for magnetic anomalies with depth resolution, Annals of Geophysics, 49 (2006), 1021-1027. [9] O. Portniaguine and M. S. Zhdanov, Focusing geophysical inversion images, Geophysics, 64 (1999), 874-887. doi: 10.1190/1.1444596. [10] O. Portniaguine and M. S. Zhdanov, 3-D magnetic inversion with data compression and image focusing, Geophysics, 67 (2002), 1532-1541. doi: 10.1190/1.1816073. [11] V. Sadovnichy, A. Tikhonravov, Vl. Voevodin and V. Opanasenko, "Lomonosov": Supercomputing at Moscow State University. In Contemporary High Performance Computing: From Petascale toward Exascale, Chapman & Hall/CRC Computational Science, Boca Raton, USA, CRC Press, (2013), 283–307. [12] M. Schiffler, M. Queitsch, R. Stolz, A. Chwala, W. Krech, H.-G. Meyer and N. Kukowski, Calibration of SQUID vector magnetometers in full tensor gradiometry systems, Geophysical Journal International, 198 (2014), 954-964. doi: 10.1093/gji/ggu173. [13] P. W. Schmidt and D. A. Clark, Advantages of measuring the magnetic gradient tensor, Advantages of measuring the magnetic gradient tensor, (2000), 26–30. [14] P. W. Schmidt, D. A. Clark, K. E. Leslie, M. Bick and D. L. Tilbrook, GETMAG-a SQUID magnetic tensor gradiometer for mineral and oil exploration, Exploration Geophysics, 35 (2004), 297-305. doi: 10.1071/EG04297. [15] A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, Philadelphia, 2005. doi: 10.1137/1.9780898717921. [16] A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer, Dordrecht, 1995. doi: 10.1007/978-94-015-8480-7. [17] A. G. Yagola, Y. Wang, I. E. Stepanova and V. N. Titarenko, Inverse Problems and Recommended Solutions. Applications to Geophysics, Moscow: BINOM, 2014. [18] X. Wang and R. O. Hansen, Inversion for magnetic anomalies of arbitrary three-dimensional bodies, Geophysics, 55 (1990), 1321-1326. doi: 10.1190/1.1892384.

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##### References:
 [1] O. M. Alifanov, E. A. Artuhin and S. V. Rumyantsev, Extreme Methods for the Solution of Ill-Posed Problems, Moscow: Nauka, 1988. [2] A. V. Goncharskii, A. S. Leonov and A. G. Yagola, A generalized discrepancy principle, USSR Computational Mathematics and Mathematical Physics, 13 (1973), 25-37. [3] P. Heath, G. Heinson and S. Greenhalgh, Some comments on potential field tensor data, Exploration Geophysics, 34 (2003), 57-62. doi: 10.1071/EG03057. [4] S. X. Ji, Y. F. Wang and A. Q. Zou, Regularizing inversion of susceptibility with projection onto convex set using full tensor magnetic gradient data, Inverse Problems in Science and Engineering, 25 (2017), 202-217. doi: 10.1080/17415977.2016.1160390. [5] Y. G. Li and D. W. Oldenburg, 3-D inversion of magnetic data, Geophysics, 61 (1996), 394-408. doi: 10.1190/1.1822498. [6] D. V. Lukyanenko, A. G. Yagola and N. A. Evdokimova, Application of inversion methods in solving ill-posed problems for magnetic parameter identification of steel hull vessel, Journal of Inverse and Ill-Posed Problems, 18 (2011), 1013-1029. doi: 10.1515/JIIP.2011.018. [7] D. V. Lukyanenko and A. G. Yagola, Some methods for solving of 3d inverse problem of magnetometry, Eurasian Journal of Mathematical and Computer Applications, 4 (2016), 4-14. [8] A. Pignatelli, I. Nicolosi and M. Chiappini, An alternative 3D inversion method for magnetic anomalies with depth resolution, Annals of Geophysics, 49 (2006), 1021-1027. [9] O. Portniaguine and M. S. Zhdanov, Focusing geophysical inversion images, Geophysics, 64 (1999), 874-887. doi: 10.1190/1.1444596. [10] O. Portniaguine and M. S. Zhdanov, 3-D magnetic inversion with data compression and image focusing, Geophysics, 67 (2002), 1532-1541. doi: 10.1190/1.1816073. [11] V. Sadovnichy, A. Tikhonravov, Vl. Voevodin and V. Opanasenko, "Lomonosov": Supercomputing at Moscow State University. In Contemporary High Performance Computing: From Petascale toward Exascale, Chapman & Hall/CRC Computational Science, Boca Raton, USA, CRC Press, (2013), 283–307. [12] M. Schiffler, M. Queitsch, R. Stolz, A. Chwala, W. Krech, H.-G. Meyer and N. Kukowski, Calibration of SQUID vector magnetometers in full tensor gradiometry systems, Geophysical Journal International, 198 (2014), 954-964. doi: 10.1093/gji/ggu173. [13] P. W. Schmidt and D. A. Clark, Advantages of measuring the magnetic gradient tensor, Advantages of measuring the magnetic gradient tensor, (2000), 26–30. [14] P. W. Schmidt, D. A. Clark, K. E. Leslie, M. Bick and D. L. Tilbrook, GETMAG-a SQUID magnetic tensor gradiometer for mineral and oil exploration, Exploration Geophysics, 35 (2004), 297-305. doi: 10.1071/EG04297. [15] A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, Philadelphia, 2005. doi: 10.1137/1.9780898717921. [16] A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer, Dordrecht, 1995. doi: 10.1007/978-94-015-8480-7. [17] A. G. Yagola, Y. Wang, I. E. Stepanova and V. N. Titarenko, Inverse Problems and Recommended Solutions. Applications to Geophysics, Moscow: BINOM, 2014. [18] X. Wang and R. O. Hansen, Inversion for magnetic anomalies of arbitrary three-dimensional bodies, Geophysics, 55 (1990), 1321-1326. doi: 10.1190/1.1892384.
Results of testing calculations: a) model solution (the normalized value of the magnitude of the magnetic moment density ${\mathit{\boldsymbol{M}}}$), b) retrieved solution for the TMI-model, c) retrieved solution for the MGT-model, d) retrieved solution for the TMI+MGT-model. The MGT-model produces the better reconstruction for the magnitude of the small details of the model solution. The use of the combined TMI+MGT-data does not give any advantages in reconstruction quality comparing with the using of TMI-data only
Results of calculation for Real Field Example 1. The main figure shows the result of preliminary allocation of the magnetic sources using quite large area as integral domain for calculations. The mini-figure shows more accurate results of second calculations for the integral domain with specified dimensions
Results of calculation for Real Field Example 2. The main figure shows the result of preliminary allocation of the magnetic sources using quite large area as integral domain for calculations. The mini-figure shows more accurate results of second calculations for the integral domain with specified dimensions
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