Modeling of regional magnetic field applying spherical functions: theoretical aspect

Authors

  • Yu. P. Sumaruk Subbotin Institute of Geophysics, National Academy of Sciences of Ukraine, Ukraine
  • L. M. Yankiv-Vitkovska Department of Higher Geodesy and Astronomy, Lviv Polytechnic National University, Ukraine
  • B. B. Dzuman Department of Higher Geodesy and Astronomy, Lviv Polytechnic National University, Ukraine

DOI:

https://doi.org/10.24028/gzh.0203-3100.v41i1.2019.158872

Keywords:

regional magnetic field, spherical functions, modeling

Abstract

Observation of geomagnetic field, measurement of its components values and establishment on their base models of geomagnetic field as well as geomagnetic mapping are the main trend of geomagnetic studies. Analytical model of geomagnetic field allows calculating the value of any component of geomagnetic field at any point of near-earth space and on the Earth. A new method has been proposed for construction of geomagnetic potential regional model. In accordance with Gauss’ fundamental studies, classical concept of geomagnetic potential became its record as endless series of Legeandre functions. A program of writing the series of geomagnetic potentials according to their spherical or ellipsoidal functions is used for most cases for modeling of global (normal) geomagnetic field with the length of a series equal to 9—13 harmonics. However in case when the sphere is not complete and only its part (a segment of the sphere or its trapezium) is taken into account the spherical Legeandre functions lose their orthogonality. In this connection for elaboration of regional field models different modifications of spherical harmonic analysis with application of spherical Legeandre functions even grade and real order are used. Such functions form an orthogonal by weight system of functions on arbitrary spherical trapezium but do not have recurrent correlation therefore we are to use for their calculation an arrangement as hypergeometric series. The area of determination of such functions is spherical segment. Working formulae have been obtained for constructing the model mentioned above. As front-end data for constructing a model of regional magnetic field the values of its components obtained at geomagnetic observatories have been used. It has also been suggested to make calculations of regional model of geomagnetic potential by a method proposed within the limits of a procedure deletion-computing-restoration. Initially a systematic product of components is found using a global model of geomagnetic field. Then abnormal values of components are calculated. A model of regional abnormal geomagnetic field is computed using basic functions. A parameter of Tikhonov’s regularizing is introduced to stabilize the solution.

References

Orlyuk, M. I. (2000). Spatial and spatial-temporal magnetic models of different-rank structures of the lithosphere of continental type. Geofizicheskiy zhurnal, 22(6), 148—165 (in Russian).

Orlyuk, M. I., Marchenko, A. V., & Romenets, A. A. (2017). Spatial-temporeral changes in the geomagnetic field and seismisity. Geofizicheskiy zhurnal, 39(6), 84—105. https://doi.org/10.24028/gzh.0203-3100.v39i6.2017.116371 (in Russian).

Smirnov, V. (1953). Course of higher mathematics. Vol. III. Pt. 2. Moscow: Nauka, 676 p. (in Russian).

Beggan, C. D., Saarimäki, J., Whaler, K. A. & Simons, F. J. (2013). Spectral and spatial decomposition of lithospheric magnetic field models using spherical Slepian functions. Geophysical Journal International, 193(1), 136—148. https://doi.org/10.1093/gji/ggs122.

De Santis, A. (1992). Conventional spherical harmonic analysis for regional modeling of the geomagnetic field.. Geophysical Research Letters, 19(10), 1065—1067. https://doi.org/10.1029/92GL01068.

De Santis, A. (1991). Translated origin spherical cap harmonic analysis. Geophysical Journal International, 106(1), 253—263. https://doi.org/10.1111/j.1365-246X.1991.tb04615.x.

De Santis, A. & Torta, J. (1997). Spherical cap harmonic analysis: a comment on its proper use for local gravity field representation. Journal of Geodesy, 71(9), 526—532. https://doi.org/10.1007/s001900050120.

Düzgit, Z., & Malin, S. R. C. (2000). Assessment of regional geomagnetic field modeling methods using a standard data set: spherical cap harmonic analysis. Geophysical Journal International, 141(3), 829—831. https://doi.org/10.1046/j.1365-246x.2000.00099.x.

Dzhuman, B. B. (2014). Approximation of gravity anomalies by method of ASHA on Arctic area. Geodesy, cartography and aerial photography, (80), 62—68.

Dzhuman, B. B. (2017). Modeling of the Earth’s gravitational field using spherical function. Geodesy, cartography and aerial photography, (86), 5—10. https://doi.org/10.23939/istcgcap2017.02.005.

Gao, Y., & Liu, Z. (2002). Precise Ionosphere Modeling Using Regional GPS Network Data. Journal of Global Positioning Systems, 1(1), 18—24.

Haines, G. (1988). Computer programs for spherical cap harmonic analysis of potential and general fields. Computers & Geosciences, 14(4), 413—447. https://doi.org/10.1016/0098-3004 (88)90027-1.

Haines, G. (1985). Spherical cap harmonic analysis. Journal of Geophysical Research: Solid Earth, 90(B3), 2583—2591. https://doi.org/10.1029/JB090iB03p02583.

Hwang, C. & Chen, S. (1997). Fully normalized spherical cap harmonics: application to the analysis of sea-level data from TOPEX/POSEIDON and ERS-1. Geophysical Journal International, 129(2), 450—460. doi: 10.1111/j.1365-246X.1997.tb01595.x.

Kelvin, L. & Tait, P. (1896). Treatise on natural philosophy. New York: Cambridge Univer. Press., 852 p.

Kotzé, P. B. (2001). Spherical Cap Modelling of Řrsted Magnetic Field Vectors over Southern Africa. Earth, Planets and Space, 53(5), 357—361. https://doi.org/10.1186/BF03352392.

Liu, J., Chen, R., An, J., Wang, Z. & Hyyppa, J. (2014). Spherical cap harmonic analysis of the Arctic ionospheric TEC for one solar cycle. Journal of Geophysical Research: Space Physics, 119(1), 601—619. https://doi.org/10.1002/2013JA019501.

Macdonald, H., (1900). Zeroes of the spherical harmonic considered as a function of n. Proceedings of the London Mathematical Society, 31(1), 264—278. https://doi.org/10.1112/plms/s1-31.1.264.

Stening, R. J., Reztsova, T., Ivers, D., Turner, J., & Winch, D. E. (2008). Spherical cap harmonic analysis of magnetic variations data from mainland Australia. Earth, Planets and Space, 60(12), 1177—1186. https://doi.org/10.1186/BF03352875.

Thébault, E., & Gaya-Piqué, L. (2008). Applied comparisons between SCHA and R-SCHA regional modeling techniques. Geochemistry, Geophysics, Geosystems, 9(7), Q07005, doi: 10.1029/2008GC001953.

Thébault, E., Mandea, M. & Schott, J. (2006). Modeling the lithospheric magnetic field over France by means of revised spherical cap harmonic analysis (R-SCHA). Journal of Geophysical Research: Solid Earth, 111(B5), 111—113. https://doi.org/10.1029/2005JB004110.

Thébault, E., Purucker, M., Whaler, K. A., Langlais, B., & Sabaka, T. J. (2010). The Magnetic Field of the Earth’s Lithosphere. Space Science Reviews, 155(1-4), 95—127. https://doi.org/10.1007/s11214-010-9667-6.

Yankiv-Vitkovska, L. M. & Dzhuman, B. B. (2017). Constructing of regional model of ionosphere parameters. Geodesy, cartography and aerial photography, 85, 27—35. https://doi.org/10.23939/istcgcap2017.01.027.

Younis, A., Jäger, R., & Becker, M. (2013). Transformation of global spherical harmonic models of the gravity field to a local adjusted spherical cap harmonic model. Arabian Journal of Geosciences, 6(2), 375—381. https://doi.org/10.1007/s12517-011-0352-1.

Published

2019-03-18

How to Cite

Sumaruk, Y. P., Yankiv-Vitkovska, L. M., & Dzuman, B. B. (2019). Modeling of regional magnetic field applying spherical functions: theoretical aspect. Geofizicheskiy Zhurnal, 41(1), 180–191. https://doi.org/10.24028/gzh.0203-3100.v41i1.2019.158872

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Articles