Nonsingular Calculation of Global Magnetic Gradient Tensor Based on Oblate Spheroidal Harmonics Representation

A high-resolution global magnetic gradient tensor (MGT) model is an essential tool for aided navigation. However, the high-order expansion using the traditional spherical harmonic (SH) method will lead to nonconvergence in calculating locations near the Earth’s polar regions, and poles are treated as singularities, making it challenging to meet navigation’s high-resolution requirements. This article proposes a calculation method for the nonsingular MGT model based on oblate spheroidal harmonics (OSH). First, we introduce a semi-normalization factor to the second-kind associated Legendre function to ensure equal contribution from each degree function. To avoid calculating complex numbers in the above procedures, a renormalization process is used alongside a recursive expansion of the operation, utilizing a suitable hypergeometric transformation. The first- and second-order derivative explicit equations are derived. The effectiveness of higher degree forms (up to 2000) is evaluated to verify the model’s accuracy, and we establish the conversion relation between OSH and SH under semi-normalized coefficients. For the pole singularity, the differential constant equation of
$P_{n}(x)$
is used to eliminate the sinusoidal co-latitude contained in the denominator of MGT. Finally, the high-resolution modeling of MGT is realized. We discuss the distribution of the MGT at the heights of 0.2 and 300 km and draw their spectra. The results of the numerical experiments verify the validity of the model and the accuracy of the calculations at the geographic poles, which are difficult for conventional methods.