Convergence Acceleration of Infinite Series Involving the Product of Riccati–Bessel Function and Its Application for the Electromagnetic Field: Using the Continued Fraction Expansion Method

Authors

  • Zheng Fanghua 1CAS Engineering Laboratory for Deep Resources Equipment and Technology, Institute of Geology and Geophysic, Chinese Academy of Sciences, Beijing 100029, China 2Innovation Academy for Earth Science, Chinese Academy of Sciences, Beijing 100029, China 3University of Chinese Academy of Sciences, Beijing 100049, China
  • Di Qingyun 1CAS Engineering Laboratory for Deep Resources Equipment and Technology, Institute of Geology and Geophysic, Chinese Academy of Sciences, Beijing 100029, China 2Innovation Academy for Earth Science, Chinese Academy of Sciences, Beijing 100029, China 3University of Chinese Academy of Sciences, Beijing 100049, China
  • Yun Zhe 1CAS Engineering Laboratory for Deep Resources Equipment and Technology, Institute of Geology and Geophysic, Chinese Academy of Sciences, Beijing 100029, China 2Innovation Academy for Earth Science, Chinese Academy of Sciences, Beijing 100029, China 3University of Chinese Academy of Sciences, Beijing 100049, China
  • Gao Ya 1CAS Engineering Laboratory for Deep Resources Equipment and Technology, Institute of Geology and Geophysic, Chinese Academy of Sciences, Beijing 100029, China 2Innovation Academy for Earth Science, Chinese Academy of Sciences, Beijing 100029, China 3University of Chinese Academy of Sciences, Beijing 100049, China

DOI:

https://doi.org/10.13052/2021.ACES.J.361202

Keywords:

infinite series; Riccati-Bessel function; Mie scattering; electromagnetic prospecting

Abstract

A summation technique has been developed based on the continuous fractional expansion to accelerate the convergence of infinite series involving the product of Riccati–Bessel functions, which are common to electromagnetic applications. The series is transformed into a new and faster convergent sequence with a continued fraction form, and then the continued fraction approximation is used to accelerate the calculation. The well-known addition theorem formula for spherical wave function is used to verify the correctness of the algorithm. Then, some fundamental aspects of the practical application of continuous fractional expansion for Mie scattering theory and electromagnetic exploration are considered. The results of different models show that this new technique can be applied reliably, especially in the electromagnetic field excited by the vertical electric dipole (VED) source in the “earth-ionospheric” cavity. The comparison among the new technology, the Watson-transform, and the spherical harmonic series summation algorithm shows that this new technology only needs less than 120 series items which is already enough to obtain a small relative error, which greatly improves the convergence speed, and provides a new way to solve the problem.

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Author Biographies

Zheng Fanghua, 1CAS Engineering Laboratory for Deep Resources Equipment and Technology, Institute of Geology and Geophysic, Chinese Academy of Sciences, Beijing 100029, China 2Innovation Academy for Earth Science, Chinese Academy of Sciences, Beijing 100029, China 3University of Chinese Academy of Sciences, Beijing 100049, China

Fang-hua Zheng was born in Luzhou, Sichuan, China, in 1995. He received a bachelor’s degree in exploration technology and engineering from the Southwest Petroleum University in Chengdu, China, in 2018. He is currently pursuing the Ph.D. degree in geophysics at University of Chinese Academy of Sciences, Beijing, China. His current research interests include the propagation of SLF/ELF electromagnetic waves in the “earth-ionosphere” cavity and its applications in geophysical prospecting.

Di Qingyun, 1CAS Engineering Laboratory for Deep Resources Equipment and Technology, Institute of Geology and Geophysic, Chinese Academy of Sciences, Beijing 100029, China 2Innovation Academy for Earth Science, Chinese Academy of Sciences, Beijing 100029, China 3University of Chinese Academy of Sciences, Beijing 100049, China

Qing-yun Di received the bachelor’s and master’s degrees from the Changchun College of Geology (now Jilin University), Jilin, China, in 1987 and 1990, respectively, and the Ph.D. degree from the Institute of Geology and Geophysics (IGG), Chinese Academy of Sciences (CAS), Beijing, China, in 1998. Dr. Di is a research fellow of geophysics at the CAS Engineering Laboratory for Deep Resources Equipment and Technology, IGGCAS. She has received a number of awards from the Chinese Government and CAS, including the National Science and Technology Progress Second Prize and the Outstanding Science and Technology Achievement Prize of CAS. She is researching the propagation characteristics of controlled-source EM waves, putting into consideration the ionosphere, atmosphere, and the earth. Her research activities are mainly devoted to research and development of EM method instruments, forward modeling, and inversion of controlled source audio-frequency magnetotellurics (CSAMT), electrical resistivity tomography (ERT), and ground-penetrating radar (GPR) methods.

Yun Zhe, 1CAS Engineering Laboratory for Deep Resources Equipment and Technology, Institute of Geology and Geophysic, Chinese Academy of Sciences, Beijing 100029, China 2Innovation Academy for Earth Science, Chinese Academy of Sciences, Beijing 100029, China 3University of Chinese Academy of Sciences, Beijing 100049, China

Zhe Yun received the B.S. degree in Geo-exploration Technology and Engineering from the Ocean University of China, Qingdao, China, in 2020. He is currently pursuing the M.S. degree in geophysics at the Institute of Geology and Geophysics (subordinate to the Innovation Academy for Earth Science), and the College of Earth and Planetary Sciences, University of the Chinese Academy of Sciences. His research interests include the controlled source electromagnetic data forward modeling and inversion.

Gao Ya, 1CAS Engineering Laboratory for Deep Resources Equipment and Technology, Institute of Geology and Geophysic, Chinese Academy of Sciences, Beijing 100029, China 2Innovation Academy for Earth Science, Chinese Academy of Sciences, Beijing 100029, China 3University of Chinese Academy of Sciences, Beijing 100049, China

Ya Gao received the B.S. degree in Geophysics from the Jilin University, Jilin, China, in 2017. She is currently pursuing the Ph.D. degree at the Key Laboratory of Shale Gas and Engineering, Institute of Geology and Geophysics (subordinate to the Innovation Academy for Earth Science), and the College of Earth and Planetary Sciences, University of the Chinese Academy of Sciences. Her research interests cover the controlled source electromagnetic data forward modeling and inversion.

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Published

2022-03-10

How to Cite

[1]
Z. . Fanghua, D. . Qingyun, Y. . Zhe, and G. . Ya, “Convergence Acceleration of Infinite Series Involving the Product of Riccati–Bessel Function and Its Application for the Electromagnetic Field: Using the Continued Fraction Expansion Method”, ACES Journal, vol. 36, no. 12, pp. 1518–1525, Mar. 2022.

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2021: Vol 36 Iss 12

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