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.

References

G. Ruello and R. Lattanzi, “Scattering from spheres: A new look into an old problem,” Electronics, vol. 10, no. 2, pp. 216, 2021.

W. Zhuang, R. Li, J. Liang, and Y. Jia, “Debye series expansion for light scattering by a charged sphere,” Applied Optics, vol. 60, no. 7, pp. 1903-1915, 2021.

C. Gao, B. Sun, and Y. Zhang, “Electromagnetic wave scattering by charged coated spheres,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 272, pp. 107757, 2021.

S. Batool, F. Frezza, F. Mangini, and X. Yu-Lin, “Scattering from multiple PEC sphere using translation addition theorems for spherical vector wave function,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 248, pp. 106905, 2020.

C. Mätzler, MATLAB Functions for Mie Scattering and Absorption, Version 2, 2002.

M. E. Nazari and W. Huang, “Asymptotic solution for the electromagnetic scattering of a vertical dipole over plasmonic and non-plasmonic half-spaces,” IET Microwaves, Antennas & Propagation, vol. 15, pp. 704-717, 2021.

B. Beiranvand, A. S. Sobolev, and A. Sheikhaleh, “A proposal for a dual-band tunable plasmonic absorber using concentric-rings resonators and mono-layer graphene,” Optik, vol. 223, pp. 165587, 2020.

M. Gingins, M. Cuevas, and R. Depine, “Surface plasmon dispersion engineering for optimizing scattering, emission, and radiation properties on a graphene spherical device,” Applied Optics, vol. 59, no. 14, pp. 4254-4262, 2020.

S. Zhao, X. Shen, C. Zhou, L. Liao, Z. Zhima, and F. Wang, “The influence of the ionospheric disturbance on the ground based VLF transmitter signal recorded by LEO satellite–Insight from full wave simulation,” Results in Physics, vol. 19, pp. 103391, 2020.

M. Hayakawa, A. P. Nickolaenko, Y. P. Galuk, and I. G. Kudintseva, “Scattering of extremely low frequency electromagnetic waves by a localized seismogenic ionospheric perturbation: observation and interpretation,” Radio Science, vol. 55, no. 12, pp. 1-26, 2020.

R. Song, K. Hattori, X. Zhang, and S. Sanaka, “Seismic-ionospheric effects prior to four earthquakes in Indonesia detected by the China seismo-electromagnetic satellite,” Journal of Atmospheric and Solar-Terrestrial Physics, vol. 205, pp. 105291, 2020.

S. Chowdhury, S. Kundu, T. Basak, S. Ghosh, M. Hayakawa, S. Chakraborty, and S. Sasmal, “Numerical simulation of lower ionospheric reflection parameters by using international reference ionosphere (IRI) model and validation with very low frequency (VLF) radio signal characteristics,” Advances in Space Research, vol. 67, no. 5, pp. 1599-1611, 2021.

C. Dong, Y. He, M. Li, C. Tu, Z. Chu, X. Liang, and N. X. Sun, “A portable very low frequency (VLF) communication system based on acoustically actuated magnetoelectric antennas,” IEEE Antennas and Wireless Propagation Letters, vol. 19, no. 3, pp. 398-402, 2020.

W. Pan and K. Li, Propagation of SLF/ELF Electromagnetic Waves. Springer Science & Business Media, 2014.

J. R. Wait, Electromagnetic Waves in Stratified Media: Revised Edition Including Supplemented Material. Elsevier, 2013.

Y. Gao, Q. Di, R. Wang, C. Fu, et al., “Strength of the electric dipole source field in multilayer spherical media,” IEEE Transactions on Geoscience and Remote Sensing, 2021.

Q. Di, C. Fu, Z. An, et al., “An application of CSAMT for detecting weak geological structures near the deeply buried long tunnel of the Shijiazhuang-Taiyuan passenger railway line in the Taihang Mountains,” Engineering Geology, vol. 268, pp. 105517, 2020.

Q. Di, G. Xue, C. Fu, and R. Wang, “An alternative tool to controlled-source audio-frequency magnetotellurics method for prospecting deeply buried ore deposits,” Science Bulletin, vol. 65, no. 8, pp. 611-615, 2020.

Q. Di, M. Wang, C. Fu, et al., Study on the Characteristics of Electromagnetic Wave Propagation in Earth-Ionosphere Mode, Science Press, 2013.

D. Shreeja, et al. “Application of Fracture Induced Electromagnetic Radiation (FEMR) technique to detect landslide-prone slip planes,” Natural Hazards, vol. 101, no. 2, 2020.

G. Mie, “Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,” Ann. Phys, vol. 330, pp. 377-445, 1908.

Y. X. Wang, R. H. Jin, J. P. Geng, et al., “Exact SLF/ELF underground HED field strengths in earth-ionosphere cavity and Schumann resonance,” IEEE Transactions on Antennas and Propagation, vol. 59, no. 8, pp. 3031-3039, 2011.

M. Tezer, “On the numerical evaluation of an oscillating infinite series,” Journal of Computational and Applied Mathematics, vol. 28, pp. 383-390, 1989.

M. Toyoda and T. Ozaki, “Fast spherical Bessel transform via fast Fourier transform and recurrence formula,” Computer Physics Communications, vol. 181, no. 2, pp. 277-282, 2010.

W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Applied Optics, vol. 15, no. 3, pp. 668-671, 1976.

D. M. O’Brien, “Spherical Bessel functions of large order,” Journal of Computational Physics, vol. 36, no. 1, pp. 128-132, 1980.

P. Hänggi, F. Rösel, and D. Trautmann, “Continued fraction expansions in scattering theory and statistical non-equilibrium mechanics,” Zeitschrift Für Naturforschung A, vol. 33, no. 4, pp. 402-417, 1978.

N. Kinayman and M. I. Aksun, “Comparative study of acceleration techniques for integrals and series in electromagnetic problems,” Radio Science, vol. 30, no. 6, pp. 1713-1722, 1995.

F. G. Mitri, “Partial-wave series expansions in spherical coordinates for the acoustic field of vortex beams generated from a finite circular aperture,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 61, no. 12, pp. 2089-2097, 2014.

W. J. Wiscombe, “Improved Mie scattering algorithms,” Applied Optics, vol. 19, no. 9, pp. 1505-1509, 1980.

J. R. Allardice and R. E. C. Le, “Convergence of Mie theory series: criteria for far-field and near-field properties,” Applied Optics, vol. 53, no. 31, pp. 7224-7229, 2014.

V. A. Fock, Electromagnetic Diffraction and Propagation Problems, Pergamon Press, 1965.

W. H. Press, W. H. Press, B. P. Flannery, et al., Numerical Recipes in Pascal: The Art of Scientific Computing, Cambridge University Press, 1989.

J. Wallis and W. Johannis, Opera Mathematica, vol. 3, 1972.

Y. Yuan, Propagation and Noise of Ultra-Low Frequency and Extremely Low Frequency Electromagnetic Waves, National Defense Industry Press, 2011.

D. E. Barrick, “Exact ULF/ELF dipole field strengths in the earth-ionosphere cavity over the Schumann resonance region: idealized boundaries,” Radio Science, vol. 34, no. 1, pp. 209-227, 1999.

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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.

Issue

2021: Vol 36 Iss 12

Section

General Submission

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