Abstract
Similar to the gravitational curvatures in the gravity field, the magnetic curvatures (i.e., third-order derivatives of the magnetic potential) have been recently proposed in the magnetic field. The components of the magnetic curvatures are all involved in the expressions for the first-order derivatives of invariants of the magnetic gradient tensor (MGT), whose physical meaning is the change rate of invariants of the MGT and their theoretical models have not yet been fully developed. In this contribution, the general expressions for the First-order derivatives of Principal and Main Invariants of the Magnetic Gradient Tensor (i.e., FPIMGT and FMIMGT) with combined components of the MGT and magnetic curvatures are presented. Specifically, the expressions for the FPIMGT and FMIMGT of a uniformly magnetized tesseroid and spherical shell are derived as examples for basic mass bodies in the spherical coordinate system in the spatial domain for the magnetic field modeling. The near zone and polar singularity problems are numerically investigated for these newly derived expressions. Numerical experiments show that the near zone problem has been found for the FPIMGT and principal invariants of the magnetic gradient tensor (PIMGT), whereas the polar singularity problem does not occur for the FPIMGT and PIMGT when using the Cartesian integral kernels for different heights and grid resolutions. This study shows that the calculation strategy by substituting the calculated values of the MGT and magnetic curvatures into the general formulae of the PIMGT and FPIMGT can provide proper numerical precision measured by relative approximation errors dependent on the computation point’s height and latitude for the FPIMGT and PIMGT. For instance, using the grid size of \(1^{\circ }\times 1^{\circ }\) at a satellite height of 260 km, relative approximation errors in \(Log_{10}\) scale have been reduced at a level lower than − 1 for the evaluation of the FPIMGT and PIMGT. The first-order derivatives of principal and main invariants of the MGT will be applied in related magnetic field studies (e.g., magnetic detection, inversion, location, position, characterization, navigation, and exploration) to present additional information (i.e., more detailed geophysical features in terms of change rates) compared to the principal and main invariants of the magnetic gradient tensor.
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We are very grateful to Prof. Michael J. Rycroft, Editor, and two anonymous reviewers for their valuable comments and suggestions, which helped to improve the manuscript. This study is supported by China Postdoctoral Science Foundation (Grant No. 2021M691402) and National Natural Science Foundation of China (Grant Nos. 42174096 and 41974094).
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Deng, XL., Shen, WB., Yang, M. et al. First-Order Derivatives of Principal and Main Invariants of Magnetic Gradient Tensor of a Uniformly Magnetized Tesseroid and Spherical Shell. Surv Geophys 43, 1233–1262 (2022). https://doi.org/10.1007/s10712-022-09697-1
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DOI: https://doi.org/10.1007/s10712-022-09697-1