# Modeling of the Earth gravitational field using spherical functions

2017;
: 5-10

Authors:
1
Department of Higher geodesy and astronomy, Lviv polytechnic National University

Aim. There are many methods for modeling a regional gravitational field in which the Legendre spherical functions of integer degree of the real order are used. They relate, however, mainly to the region which form represents a segment of the sphere. In addition, for their use, the input data must be transformed into a sphere segment with its center at the north pole. The aim of this work is to find a system of functions that would have orthogonal properties on an arbitrary spherical trapezium, as well as researching the properties of such a system. Method. Based on the Legendre spherical functions on the spherical segment, an orthogonal system of functions to an arbitrary spherical trapezoid was developed. Such functions can not be explicitly stated, nor do they have recurring relationships. Results. This article examines the associated Legandre spherical functions on the spherical trapezium where the functions are orthogonal and provide the formulas for defining the norms of these functions. The obtained functions can be used to build regional models of the gravitational fields on the arbitrary spherical trapezium. The orthogonality of the functions ensures a sustainable solution when determining the unknown model coefficients. To model the regional gravitational field with high accuracy, it is necessary to grid the input data (define the transformants of the geopotential), and then use the partial discrete orthogonality of these functions in longitudial direction or full discrete orthogonality similar to the second Neumann’s method. This allows significant reduction of computing time without any loss of accuracy, as the functions under study do not have any recursive relations and it is required to use the decomposition into the hypergeometric series to define these functions. The scientific novelty and practical significance. In this paper we first obtained a system of functions that were orthogonally  consistent to an arbitrary spherical trapezium. It can be used to construct a regional gravitational field, a regional magnetic field, and also for high-precision interpolation or approximation tasks, for example the construction of a regional ionosphere model.

1. De Santis, A. Translated origin spherical cap harmonic analysis, Geophys. J. Int., 1991, 106, 253–263.
https://doi.org/10.1111/j.1365-246X.1991.tb04615.x
2. De Santis, A. Conventional spherical harmonic analysis for regional modeling of the geomagnetic feld, Geophys. Res. Lett., 1992, 19, 1065–1067.
https://doi.org/10.1029/92GL01068
3. De Santis, A. & Torta, J. Spherical cap harmonic analysis: a comment on its proper use for local gravity field representation, J. of Geodesy, 1997, 71, 526-532.
https://doi.org/10.1007/s001900050120
4. Haines, G. Spherical cap harmonic analysis, J. Geophys. Res., 1985, 90, 2583–2591.
https://doi.org/10.1029/JB090iB03p02583
5. Haines, G. Computer programs for spherical cap harmonic analysis of potential and general felds, Comput. Geosci., 1988, 14, 413–447.
https://doi.org/10.1016/0098-3004(88)90027-1
6. Hobson, E. The theory of spherical and ellipsoidal harmonics, New York : Cambridge Univ. Press, 1931, 476 p.
7. Hwang, C. Spectral analysis using orthonormal functions with a case study on the sea surface topography, Geophys. J. Int., 1993, 115, 148–1160.
https://doi.org/10.1111/j.1365-246X.1993.tb01517.x
8. Hwang, C. & Chen, S. Fully normalized spherical cap harmonics: application to the analysis of sea-level data from TOPEX/POSEIDON and ERS-1, Geophys. J. Int., 1997, 129, 450–460.
https://doi.org/10.1111/j.1365-246X.1997.tb01595.x
9. Macdonald, H. Zeroes of the spherical harmonic considered as a function of n, Proc. London Math. Soc., 1900, 31, 264–278.
10. Marchenko, A. & Dzhuman, B. Regional quasigeoid determination: an application to arctic gravity project, Geodynamics, 2015, 18, 7–17.
11. Pavlis, N. K., Holmes, S. A., Kenyon, S. C. & Factor, J. K. The development and evaluation of the Earth Gravitational Model 2008 (EGM2008), J. geophys. Res., 2012, 117, B04406. doi:10.1029/2011JB008916.
https://doi.org/10.1029/2011JB008916
12. Smythe, W. Static and dynamic electricity, New York : McGraw-Hill, 1950, 635 p.
13. Sneeuw, N. Global spherical harmonic analysis by least-squares and numerical quadrature methods in historical perspective, Geophys. J. Int., 1994, 118, 707–716.
https://doi.org/10.1111/j.1365-246X.1994.tb03995.x
14. Thebault, E., Mandea, M. & Schott, J. Modeling the lithospheric magnetic field over France by means of revised spherical cap harmonic analysis (R-SCHA), J. geophys. Res., 2006, 111, 111–113.
https://doi.org/10.1029/2005JB004110