# On the constraction of local gravitational field model

2013;
: pp. 29-33

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

Methods of local gravitational field presentation using nonorthogonal functions are considered. Analysis of SCHA, ASHA and TOSCHA techniques of local field modeling on spherical cap and spherical segment is accomplished according to density distribution of initial data. Approximate formula for finding eigenvalues of differential equation of associated spherical functions   is found and compared with other formulas. This approach involves using of  associated Legendre functions of integer degree and noninteger order. These functions form two sets of functions. They are mutually orthogonal over the spherical cap in each set. However, in general these functions are not orthogonal. Thus, for using both of these sets of functions it is traditionally used least squares method. However, for higher orders it is quite difficult to compute eigenvalues and norms of these functions. such case it is possible to project the initial data on the hemisphere and to use associated Legendre functions of integer degree and integer order. The properties of these functions are similar to properties of functions on the spherical cap.  Traditionally, initial data is selected in the nodes of grid, especially for fast computations. There are many kinds of uniform grids, which allow to speed up the process of computation the unknown harmonic coefficients. Among these grids it is possible to allocate the geographical grid, Gauss grid and others. Thus, grid is developed to accommodate the initial data and is defined its basic properties in the segment of sphere and hemisphere .

1. Smirnov V. Kurs vysshej matematiki. Tom ІІ [course of higher mathematics. Volume II]. Moscow: Nauka, 1954, 627 p..
2. Churchill R.V. Fourier Series and Boundary Value Prob¬lems, 2nd ed. - New York: McGraw-Hill, 1963.
3. De Santis A. Conventional spherical harmonic analysis for regional modeling of the geomagnetic feld. Geophys. Res. Lett. 1992, 19, pp. 1065-1067.
https://doi.org/10.1029/92GL01068
4. De Santis A. Translated origin spherical cap harmonic analysis. Geophys. J. Int. 1991, 106, pp. 253-263.
https://doi.org/10.1111/j.1365-246X.1991.tb04615.x
5. De Santis A., Torta J.M. Spherical cap harmonic analysis: a comment on its proper use for local gravity field representation. J. of Geodesy - 1997, 71. - P. 526-532.
https://doi.org/10.1007/s001900050120
6. Earth Gravitational Model 2008 (EGM 2008). - http://earthinfo.nga.mil/GandG/wgs84/gravitymod/egm2008/
7. Haines G.V. Computer programs for spherical cap harmonic analysis of potential and general felds. Comput. Geosci. - 1988, 14, pp. 413-447.
https://doi.org/10.1016/0098-3004(88)90027-1
8. Haines G.V. Spherical cap harmonic analysis. J. Geophys. Res. 1985, 90, pp. 2583-2591.
https://doi.org/10.1029/JB090iB03p02583
9. Hobson E.W. The Theory of Spherical and Ellipsoidal Harmonics. - New York: Cambridge Univ. Press, 1931.
10. Hofmann-Wellenhof B., Moritz H. Physical Geodesy. Wien New York: Springer Science + Busines Me¬dia, 2005, 403 p.
11. Hwang C., Chen S. Fully normalized spherical cap harmonics: application to the analysis of sea-level data from TOPEX/POSEIDON and ERS-. Geophys. J. Int. 1997, 129, pp. 450-460.
https://doi.org/10.1111/j.1365-246X.1997.tb01595.x
12. Jiancheng L., Dingbo C., Jinsheng N. Spherical cap harmonic expansion for local gravity field representation. Manuscr. Geod, 1995, 20, pp. 265-277.
13. Jong Sun Hwang, Hyun-Chul Han, Shin-Chan Han, Kyong-O Kim, Jin-Ho Kim, Moo Hee Kang, Chang Hwan Kim. Gravity and geoid model in South Korea and its vicinity by spherical cap harmonic analysis // J. of Geodynamics - 2012, 53, pp. 27-33.
https://doi.org/10.1016/j.jog.2011.08.001
14. Kelvin L., Tait P. Treatise on natural philosophy. - New York: Cambridge Univ. Press., 1896, 536 p.
15. Macdonald H.M. Zeroes of the spherical harmonic considered as a function of n. Proc. London Math. Soc., 1900, 31, pp. 264-278.
https://doi.org/10.1112/plms/s1-31.1.264
16. Pal B. On the numerical calculation of the roots of the equation and regarded as equations in n . Bull. Calcutta Math. Soc., 1919, 9, pp. 85-95.
17. Smythe W.R. Static and Dynamic Electricity, 2nd ed. - New York: McGraw-Hill, 1950, 616 p.
18. Stening R.J., Reztsova T., Ivers D., Turner J. and Winch D.E. Spherical cap harmonic analysis of magnetic variations data from mainland Australia. Earth Planets Space, 2008, 60, pp. 1177-1186.
https://doi.org/10.1186/BF03352875
19. Zhenchang An. Spherical cap harmonic analysis of the geomagnetic field and its secular variation in China for 2000. Chinese J. of Geophysics, 2003, 46, pp. 85-91.
https://doi.org/10.1002/cjg2.319