Ellipsoidal corrections to gravitational anomalies in determining the disturbing gravitational potential of the Earth
Heading:
| 1Sohor, AR, 1Brydun, AM 1Lviv Polytechnic National University, Lviv, Ukraine |
| Space Sci. & Technol. 2025, 31 ;(6):106-111 |
| https://doi.org/10.15407/knit2025.06.106 |
| Publication Language: English |
Abstract: Elements of the anomalous gravity field, such as the gravity anomaly or the geoid height, are relatively small. The formulas relating these quantities neglect the compression of the reference ellipsoid, using a spherical approximation for calculations. The spherical approximation serves as the basis for nearly all formulas in physical geodesy. The spherical approximation gives an error that is neglected in most practical applications. Thus, D. Lelgemann showed that when applying the Stokes formula, this error for geoid heights averaged over the entire Earth gives a value of 0.2 m. This was an order of magnitude less than the accuracy of modern gravimetric satellite data at that time. To obtain higher accuracies, ellipsoidal corrections should be taken into account. This can be done if an arbitrary element of the anomalous gravitational field (disturbing potential, geoid height, gravitational anomaly) is expanded into a series by a small parameter that characterizes the deviation of the reference ellipsoid from the sphere. In this work, the numerical values of ellipsoidal corrections to the spherical approximation in the case of determining the components of the perturbing gravitational potential of the Earth are obtained.
The expediency of including ellipsoidal corrections in calculating gravity anomalies has been proven by comparing the obtained results with the accuracy provided by modern gravimetric satellite data. The results of the performed calculations showed that the gravity anomaly practically does not depend on the displacement of the coordinate system of the reference ellipsoid relative to the center of mass of the Earth; therefore, when determining the component of the anomalous field of the Earth’s gravity, the ellipsoid correction of the first-degree can be disregarded. However, the gravity anomaly strongly depends on the С20 coefficient of the second-order zonal harmonic of its decomposition into a trigonometric series of spherical functions. Therefore, the ellipsoidal correction of the second degree must be taken into account when finding the component of the anomalous gravitational field of the Earth, since the error of such calculations can have a value corresponding to modern high-precision gravimetric satellite data. |
| Keywords: anomalous gravitational field, ellipsoidal correction, gravimetric satellite data, gravity anomaly, spherical functions |
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3. Fys M., Brydun A., Sohor A., Lozynskyi V. (2023). Presentation of the gravity field of celestial bodies using the potentials of
flat ellipsoidal discs. Space Science and Technology, 29(2), 78—85.
4. Fys M., Yurkiv M., Brydun A., Sohor A. (2024). Representation of three-dimensional mass distribution of the Earth’s interior
by biorthogonal series and its use for studying internal structure of the planet. Geodesy and Geodynamics, 15(3), 264—275.
https://doi.org/10.1016/j.geog.2023.11.003
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Space Science and Technology, 28(4), 71—77.
6. Heiskanen W., Moritz H. (1967). Physical Geodesy. San Francisco, California: W. H. Freeman and Company.
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8. Lelgemann D. (1973). Spherical Approximation and the Combination of Gravimetric and Satellite Data. Boll. Geold. Sci.
Affini, 32, 241—250.
9. Lemoine J., Bourgogne S., Biancal R., Reinquin F., Bruinsma S. (2019). EIGEN-GRGS.RL04.MEAN-FIELD. Model of
the Earth’s Gravitational Field with Time Variable Part CNES/GRGS RL04. URL: https://grace.obs-mip.fr/variable-modelsgrace-
lageos/mean-fields/release-04 (Last accessed: 12.08. 2019).
10. Meshcheryakov G. A., Tserklevich A. L. (1987). Gravitational field, figure and internal structure of Mars. Kiev: Naukova
Dumka, 240 p. [in Russian].
11. Molodenskiy M. S., Eremeyev V. F., Yurkina M. I. (1960). Methods for studying the external gravitational field and the
figure of the Earth. Trudy TSNIIGAiK, 131, 250—251 [in Russian].
12. Moritz H. (1983). Advanced physical geodesy. M.: Nedra, 392 p. [in Russian].
13. Sohor A., Brydun A., Fys M., Dzhuman B. (2022). Calculation of corrections to the spherical approximation of the
components of the Earth’s anomalous gravity field. GeoTerrace–2022 (October 3—5, Lviv, Ukraine). https://doi.
org/10.3997/2214-4609.2022590061
14. Sohor A., Marchenko D., Kryva Kh. (2024). Calculation of the regional ellipsoid for Ukraine and its efficiency. Space
Science and Technology, 30(5), 87-95. https://doi.org/10.15407/knit2024.05.087
15. Tikhonov A. N., Samarskiy A. A. (1966). Equations of mathematical physics. M.: Nauka, 724 p. [in Russian].
16. Zagrebin D.V. (1952). Theory of regularized geoid. Trudy ITA, 1, 52—61. [in Russian].
17. Zingerle P., Brockmann J., Pail R., Gruber Т., Willberg M. (2019). GO CONS GCF-2 TIM R6e. Polar Model of Extended
Gravitational Field TIM_R6. doi: 10.5880/ICGEM.2019.005