Pade Approximant ELF VED Fields in the Earth‐Ionosphere Cavity

Abstract

Electromagnetic (EM) fields in the extremely low‐frequency (ELF) domain that can be excited by the vertical electric dipole (VED) and propagated in the Earth‐ionosphere spherical cavity have received tremendous attention owing to their exploitation in many fields such as communication, earthquake prediction, and geophysical investigations. Some researchers use the planar model and curvature correction terms to simulate EM fields in a spherical cavity, while others directly solve the differential functions and estimate the excited EM fields represented by spherical Bessel and associated Legendre functions. Regarding the divergence problem of a series when simulating EM fields, several numerical methods are available such as Watson transformation method, W. K. B. J method, and speeding algorithm. In this study, Pade approximant is utilized to estimate EM fields propagating in the Earth‐ionosphere spherical cavity, where new asymptotic expansions of spherical Bessel functions are introduced for estimating their products for high orders. These EM fields are then interpreted as a sum of the dominant incident fields and the increase of multiple scattering fields. The convergence of series expressions of EM fields estimated by Pade approximant is theoretically guaranteed, with its computational complexity lower than that of the speeding algorithm. The EM fields are further verified by numerical tests. The effects of the height and conductivity of the ionosphere, the conductivity of air, the VED height, and the frequencies of the resonance phenomenon are also highlighted. The proposed method is helpful to capture the true nature of the field distribution within a spherical cavity.

SLF/ELF Fields From VED Solved by Normalized Spherical Harmonic Functions

“…The dark blue line is from the method of this study, the light blue line is from the method from Wang et al. (2008), and the red line is from the Pade approximation method (Liang et al., 2021).…”

“…Also, the W.K.B.J method is used to approximate the EM fields excited by a VED (W. Pan & Li, 2014). Another method is developed to solve the non‐convergence problem for series summation expression of EM fields excited by a VED by using the Pade approximant method (Liang et al., 2021).…”

“…𝐴𝐴 𝐴𝐴 varies from −90° (the North Pole) to 0° (equator) and then increases to 90° (the South Pole). The dark blue line is from the method of this study, the light blue line is from the method fromWang et al (2008), and the red line is from the Pade approximation method(Liang et al, 2021).…”

“…2 of 16 method is developed to solve the non-convergence problem for series summation expression of EM fields excited by a VED by using the Pade approximant method (Liang et al, 2021).…”