Purpose. Using known and fixed Earth potential, presented asthe biorthogonal expansion, to culculate the geoid surface, which describes the actual shape of the planet. The external gravitational field is generally described by the series of spherical functions. Since the geoid is determined with the help of such functions, a question arises converning the identity to define the shape, moreover its several points does not belong to the region of convergence. Methodology and results. We consider representation of potential by convergent series everywhere, which makes it possible to find the geoid without specifying the location of points on the surface, although the geoid heights calculation is carried out by various relations. According to the known function of the mass distribution of the Earth, represented by the second degree polynomial, internal and external potential of elliptical planet are defined and the equipotential surfaces are found. Calculated values via these formulas and their degree of coincidence was analyzed. Defined in two ways surfaces do not coincide with each other because the difference in the values of the radius-vector amouts up to ten meters. So, when applying biorthogonal expansions of higher orders in constructing equipotential surfaces based on information about the external gravitational field it is necessary to take into account characteristics of expansion. Originality. Method of determining the shape of the Earth using the biorthogonal expansions of mass distribution function is proposed. This representation is characterized by a convergence for considered series and gives the opportunity to build digital models of the geoid (volumetric or as an isolines map). Practical significance. The results of numerical experiments, described in the article, led to the conclusion about the possibility of determining the equipotential surfaces that adequately describe the physical surface of the planet not only of the second but higher orders using biorthogonal expansions only with additional investigations. Calculation of geoid heights with high accuracy opens the way to observe many regional and local geodynamic phenomena, such as the movement of tectonic plates, and high accuracy leveling using GPS technology can solve a number of geodetic problems.
1. Antonov V. A, Timoshkova Ye.I., Kholshevnikov K.V. Vvedenie v teoriu Njutonovskogo potenciala. (Glav.red. Phiz.-mat. Lit.), 1988, 272 p.
2. Balmino D. Predstavlenie potenciala Zemli s pomoschiu sovokupnosti tochechnyh mass, nahodiashchihsia vnutri Zemli. Ispolzovanie iskustviennyh suputnikov dlia geodezii. Moscow: Mir, 1975, pp. 175–183.
3. Zagrebin D. V. Vvedenie v teoreticheskuiu gravimetriu. Leningr. Otd. Izd. Nauka, 1976, 292 p.
4. Marchenko A. N. Approksimacia globalnogo, regio-nalnogo I lokalnogo gravitacionnogo polia Zemli sistemoi potencialov necentralnyh multipolei. Tr. II Orlovskoi konferencii "izuchenie Zemli kak planet metodami astronomii, geodezii I geophiziki". Kyiv: Naukova dumka, 1982, pp. 56–59.
5. Marchenko A. N. Hilbertovy prostranstva funkcii, harmonicheskih vne sphery Bjerhamera, globalnaia funkcia anomalnogo polia. Kyiv: Nauka, 1983, 22 p. (Rukopis dep. v UkrNIINTI, № 292 Ук-Д83).
6. Meshcheriakov G. A. Zadachi teorii potenciala i obobshchenaia Zemlia. Мoscow: Nauka, 1991, 216 p. (Glav.red. Phiz.-mat. Lit.).
7. Moritc G. Sovremennaia phizicheskaia geodezia. Мoscow: Nedra, 1983, 392 p.
8. Pellinen L. P. Vysshaia geodezia(teoreticheskaia geodezia . Мoscow: Nedra, 1978, 264 p.
9. Fys M. M., Fotca R. S., Sohor A. R., Volos V. O. Metod znahodjennia gustyny rozpodilu mas planet z urahuvanniam stoksovyh stalyh do chetvertogo stepenia. Geodynamics. Lviv, Lviv Polytechnic Publishing House, 2008, vol. 1(7), pp. 25–34.
10. Fys M. M., Gubar Yu. P., Pokotylo I. Ya. Netradyciinyi metod pobudovy poverhon' rivnia planety (geoida, selenoida, aeroida) za ii zovnishnim gravitaciinym polem. Zbirnyk naukovyh prats konferencii "Suchasni dosjaghnennja gheodezychnoji nauky ta vyrobnyctva v Ukraini". Lviv, 1997, pp. 39–42.
11. Fys M. M. Pro odyn class neortogonalnyh dla elipsoida harmoniinyh funkcii. Zbirnyk naukovyh prats zahidnogo geodezychnogo tovarystva "Suchasni dosjaghnennja gheodezychnoji nauky ta vyrobnyctva" [Modern achievements in geodetic science and industry]. Lviv, 2006, vol. І (11, pp. 126–130.
12. Fys M. M. O shodimosti v sredniem biortogonalnyh riadov vnutri elipsoida. Differencialnye uravnienia i ih prilogenia. Lviv: "Vyshcha shkola", 1983, Vol. 172, 2 p.
13. Cherniaha P. G., Fys M .M., Golubinka Yu. I., Yurkiv M. I. Porivniannia odnogo klasu harmonichnyh ta kuliovyh funkcii pry predstavlenni potencialu planet. Zbirnyk naukovyh prats zahidnogo geodezychnogo tovarystva "Suchasni dosjaghnennja gheodezych¬noji nauky ta vyrobnyctva" [Modern achievements in geodetic science and industry]. Lviv, 2014, vol. II (28), pp. 19–23.
14. Konopliv A., Banerdt W., Sjogren W. Venus gravity: 180 th degree and order model. Icarus, 1999, no. 139, pp. 3–18.
15. Konopliv A., Asmar S., Yuang D. Resent gravity models as a result of the Lunar Prospector mission. Icarus, 2001, no. 150, pp. 1–18.
16. Kraup T. A contribution to the mathematical foundation of physical geodesity. Danish Geodetic Institute, 1969, Copenhagen, vol. 44.
17. Lerch F. I. Model improvement using GEOS-3 (GEM 9 and 10). J. Geophys. Res. 1979, no. 138, pp. 3897–3916.
18. Marchenko A. N. Parameterization of the Earth's Gravity Field: Point and Line Singularities. Published by Lviv Astronomical and Geodetic Society. Lviv, Ukraine, 1998, 210 p.
19. Moritz G. Fundamental geodetic constant. Proceedings of the IAG XVII Gener. Assemb. IUGG/IAG, 1979, p. 24.
20. Pavlis N. K, Holmes S. A., Kenyon S. C. An Earth Gravitational Model to degree 2160: EGM2008. EGU General Assembly. Geophysical Reaseach Abstracts. 2008, no. 10, p. 2.
21. Yung D. N., Sjogren W., Konopliv A. S. Gravity field of Mars: 75 degree and order model. Geophys. Res. 2001, no. 10, pp. 23377–2340.