Non‐Singular Calculation of Geomagnetic Vectors and Geomagnetic Gradient Tensors

Abstract

AbstractSome spherical harmonic expressions 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). Because the latter use Schmidt semi‐normalized associated Legendre function (SNALF), which is different from fully‐normalized associated Legendre function 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 (order m equals to 0 or 1), and the corresponding two kinds of spherical harmonic polynomials. When the non‐singular 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 to 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, the standard forward column recursion, the cross degree and order recursion, and the Belikov recursion. The standard formula to check the calculation accuracy of various recursive formulae of SNALF are also given. Besides, we demonstrate the effectiveness and reliability of the new derived non‐singular expressions of GVs and GGTs and analyze the computation speed and stability of the four recursive formulae by extensive numerical experiments.