Some spherical harmonic expressions (SHEs) of gravitational and geomagnetic field elements will become infinite when computation point approaches polar regions, as the sine function of the geocentric co-latitude contained in the denominator tends to be zero. Currently, this singularity problem has been solved for gravitational field case, however, it remains unsolved for geomagnetic vectors (GVs) and geomagnetic gradient tensors (GGTs). The reason is that the latter use Schmidt semi-normalized associated Legendre function (SNALF), which is different from fully-normalized associated Legendre function (FNALF) used in the former. To overcome this singularity problem, we derive new non-singular expressions of the first- and second-order derivatives of Schmidt SNALF, and the corresponding two kinds of spherical harmonic polynomials. When the novel expressions are applied to the traditional formulae of GVs and GGTs, more practical expressions of GVs and GGTs with non-singularity are formulated by refining the cases that the order m equals 0, 1, 2 and other values. Furthermore, to provide flexible calculation strategies for Schmidt SNALF, we derive four kinds of recursive formulae, including the standard forward row recursion (SFRR), the standard forward column recursion (SFCR), the cross degree and order recursion (CDOR), and the Belikov recursion (BR). Besides, we demonstrate the effectiveness of the new derived non-singular expressions of GVs and GGTs and analyze the computation speed and stability of the four recursive formulae of Schmidt SNALF by extensive numerical experiments. Results achieve significant improvements in solving the singularity problem of the SHEs of GVs and GGTs.

In aeromagnetic measurement, accurate compensation for the interference magnetic field generated by the aircraft platform due to the aircraft maneuvering in the Earth's magnetic field is an important prerequisite for the accurate indentification of the magnetic anomaly signal of the detection target. The flaws of the traditional calibration flight scheme in aeromagnetic interference compensation are firstly analyzed, and then an improved scheme is proposed, featuring smaller calibration flight areas, more simplified maneuvers, shorter flight route and less influence from the magnetometer's direction. The improved scheme can enhance the robustness of the aeromagnetic interference compensation matrix, and improve the compensation efficiency, and thus both the solved magnetic compensation coefficients and compensation effects are significantly improved. The experimental results of aeromagnetic interference compensation in an area of Inner Mongolia of China show that the average accuracy of the repeated survey lines, which are compensated by the improved scheme, can be increased by 5 nT and 3 nT, respectively, compared with the uncompensated results and the results compensated by the traditional scheme. Therefore, the effectiveness and superiority of the improved calibration flight scheme is fully proved.