Mathematical model of calculating metric tensor and GNSS-observations errors taking into account relativistic effects

Ihor Bialyk, Olha Stepanchenko, Waldemar Wójcik


Study of the trajectories of the motion of satellites remains an urgent task for modern science. This is especially true for GNSS systems and for satellites intended for Earth remote sensing. The basis of their operation is to accurately determine the position of the satellite, and the parameters of signal propagation. Considering the great distances and speeds of both satellites and the Earth in calculating these parameters, it is necessary to take into account the special and general theory of relativity. In the article formulas have been derived for calculating additional corrections for relativistic effects. A mathematical model for calculating the metric tensor was created. A sequence of correction was also proposed.

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N. Ashby, B. Berdotti, “Relativistic perturbations of an Earth Satellite,” Phys. Rev. 52(7), 1984, pp. 485–495.

N. Ashby, B. Berdotti, “Relativistic effects in local inertial frames,” Phys. Rev. D34(8), 1986, pp 2246-2258.

N. Ashby, J.J. Spilker Jr., “Introduction to relativistic effects on the Global Positioning System,” Global Positioning System: Theory and Applications, 1 (18), 1996, pp 623–697.

N. Ashby, M. Weiss, Global Positioning Receivers and Relativity, Report no. 1385 NIST Technical Note, Washington, 1999, pp. 1–56.

R. S. DiEsposti, H. F. Fliegel, “GPS and Relativity: an Engineering Overview”, Aerospace Corp. Report no. ATR-97(3389)-1, 20 Dec. (1996).

M. Doligalski, S. Nikiel, J. Żołyński, “GPS system disciplined time and frequency standard,” Przegląd Elektrotechniczny 2012(8), 2012, pp. 152–155.

R. A. Hulse, J. H. Taylor, “Discovery of a pulsar in a binary system,” Astrophys. J., vol.195, 1975, pp. 51–53.

T. Hartmann, M. Leins, V.S. Schwarze, M.H. Soffel, “Relativistic effects in satellite positioning,” Manuscripta Geodaetica 18(3), 1993, pp. 306–316.

P. Wolf, G. Petit, “Satellite test of special relativity using the Global Positioning System,” Phys. Rev. A56, 1997, pp. 4405-4409.

IAU (International Astronomical Union) Resolutions adopted at the 24th General Assembly, Manchester, August 2000, B 1.5 “Extended relativistic framework for time transformations and realization of coordinate times in the solar system”.

N. Ashby, “Relativity in the Global Positioning System,” Living Rev. Relativ. 6(1) 2003,

K. Krasuski, “How stable are instrumental biases P2-C2?,” IAPGOŚ 5(3), 2015, pp. 51–52.

V. Iu. Puchkov, V. S. Sherbaevych, “Uchot reliatyvystskykh y hravytatsyonnykh effektov pry obrabotke rezultatov izmerenyi v systeme Navstar,“ Zarubezhnaia radioelektronyka 1(253), 1989, pp. 54–59.

F. T. Geyling, H. R. Westerman, Introduction to Orbital Mechanics, Addison Wesley Publishing Company, 1971.

N. Ashby, “Practical Implications of Relativity for a global coordinate time scale,” Radio Science, 14(4), 1979, pp. 649–669.

N. Ashby, “Relativity and the Global Positioning System,” Phys. Today, 55(5), 2002, pp. 41–47.

I. Byalyk, M. Byalyk, P. Chernyaha, “Influence of relativistic distortion of space-time on preciseness of GPS-measurements,” Modern achievements of geodesic science and industry, 1(7), 2002, pp. 85–88.

I. Byalyk, M. Byalyk, P. Chernyaha, “Theoretical analysis of the influence of relativistic distortion of space-time on preciseness of GPS-measurements,” Modern achievements of geodesic science and industry, 1(8), 2003, pp. 55–66.

L. D. Landau, E. M. Lyfshyts, Teoriya polia, Nauka, Moscow, 1973.

I. M. Bialyk, “Calculation and analysis of matrix whose determine the errors for relativistic effects on the GPS-observations,” Visnyk heodezii ta kartohrafii 2, 2002, pp. 10–14.

N. A. Belova, Kurs sferycheskoi astronomyy. Nedra, Moscow, 1971.

I. M. Bialyk, “Rozrakhunok ta analiz pokhybok za reliatyvistski efekty pry GPS-vymiriuvanniakh,” Visnyk Natsionalnoho universytetu vodnoho hospodarstva i pryrodokorystuvannia, 1(33), 2006, pp. 186–194.


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