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.

The Tonga volcano erupted with a mass of energy released into the plate and atmosphere, causing the variations of the relevant geophysical parameters. Thus, we creatively apply the observations from space-borne and ground-based sensors to reveal the characterizations of up to four items (tropospheric response, sea surface temperature, plate movement, and ionospheric reaction). The results of precipitable water vapor from space-borne and ground-based data show that tropospheric response has a trend of rising first, after volcano event dropping sharply, then recovering to a normal level. The examination of ground-based GNSS station coordinates illustrates that plate movements up to 20 dm are caused by volcano eruption. Eventually, for the ionospheric reaction a similar characterization is found as the tropospheric response through total electron content variation. These are the first comprehensive analyses showing the impacts of the Tonga volcano on multiple types of parameters from both space-borne and ground-based data.