Numerical solution of the Molodensky boundary value problem with a fixed boundary

Geodesy
UDC: 
528.2
DOI: 
10.22389/0016-7126-2025-1016-2-2-14
1 Neiman Yu.M.
2 Sugaipova L.S.
3 Nepoklonov V.B.
Year: 
2025
№: 
2
1016
Pages: 
2-14

Roskadastr, PLC

1, 
2, 
3, 
Abstract:
The functions of modern global navigation satellite systems (GNSS) allow us to consider the Earth’s surface as partially known at least. So, you can directly use the results of measurements on the real Earth surface, which significantly simplifies the entire theory of physical geodesy. The authors describe the theory of the Molodensky boundary value problem with a fixed boundary and its possible approximations briefly. It is proposed to look for a practical solution in the form of a deep neural network. Its choice represents a definite alternative to modern methods of physical geodesy and is of fundamental importance, since using neural networks with their unique flexibility and ability to learn from specific data enables significant expansion of computing capabilities and expecting the existence of a fairly reliable localized solution. In addition to theoretical considerations, the results of a numerical experiment to determine the high-frequency part of the disturbing potential in a local areа are described in detail
The study was carried out within the framework of the Federal Project "Maintenance, Development and Use of the GLONASS System" (EGISU No. 1210806000081-5) and the state assignment FSFE-2023-0005 the RF Education and Science Ministry (123021300031-4)
References: 
1.   Gofman-Vellengof B., Morits G. Fizicheskaya geodeziya. Per. s angl. pod red. Yu. M. Neimana. Moskva: Izd-vo MIIGAiK, 2007, 426 p.
2.   Kallan R. Osnovnye kontseptsii neironnykh setei. Per. s angl. A. G. Sivaka. Moskva: Vil'yamc, 2001, 287 p.
3.   Koshlyakov N. S., Gliner E. B., Smirnov M. M. Osnovnye differentsial'nye uravneniya matematicheskoi fiziki. Moskva: Fizmatgiz, 1962, 768 p.
4.   Morits G. Sovremennaya fizicheskaya geodeziya. Moskva: Nedra, 1983, 392 p.
5.   Moritz G. (2001) Theory of Molodensky and GPS. Geodezia i Kartografia, 62(6), pp. 7–17.
6.   Neiman Yu. M., Sugaipova L. S. Reshenie differentsial'nogo uravneniya Laplasa v vide glubokoi neiroseti kak edinyi algoritm priblizhennogo resheniya zadach fizicheskoi geodezii v lokal'nom raione. Izvestia vuzov. Geodesy and Aerophotosurveying, 2023, Vol. 67, no. 1, pp. 104–106.
7.   Neiman Yu. M., Sugaipova L. S., Koneshov V. N., Nepoklonov V. B. O reshenii kraevykh zadach fizicheskoi geodezii v vide glubokikh neirosetei. Geofizicheskie issledovaniya, 2024, Vol. 25, no. 2, pp. 5–19.
8.   Nikolenko S., Kadurin A., Arkhangel'skaya E. Glubokoe obuchenie: pogruzhenie v mir neironnykh setei. SPb.: Piter, 2018, 480 p.
9.   Ogorodova L. V. Normal'noe pole i opredelenie anomal'nogo potentsiala. Moskva: MIIGAiK, 2011, 106 p.
10.   Pavlenko D. Vvedenie v mashinnoe obuchenie i iskusstvennye neironnye seti. URL: https://clck.ru/3GUyJ6 (accessed: 25.01.2025).
11.   Axler S., Shin P. J. (2018) The Neumann problem on ellipsoids. Journal of Applied Mathematics and Computing, no. 57, pp. 261–278. DOI: 10.1007/s12190-017-1105-4.
12.   Backus G. E. (1968) Application of a non-linear boundary-value problem for Laplace's equation to gravity and geomagnetic intensity surveys. The Quarterly Journal of Mechanics and Applied Mathematics, Volume 21, no. 2, pp. 195–221. DOI: 10.1093/qjmam/21.2.195.
13.   Bjerhammar A., Svensson L. (1983) On the geodetic boundary value problem for a fixed boundary surface – a satellite approach. Bulletin Geodesique, no. 57, pp. 382–393.
14.   Heck B. (1989) On the non-linear geodetic boundary value problem for a fixed boundary surface. Bulletin Geodesique, no. 63, pp. 57–67.
15.   Hornik K., Stinchcombe M., White H. (1989) Multilayer feedforward networks are universal approximators. Neural Network, no. 2, pp. 359-366.
16.   Koch K. R., Pope A. J. (1972) Uniqueness and existence for the geodetic boundary value problem using the known surface of the Earth. Bulletin Geodesique, no. 106, pp. 467–476.
17.   Leshno M., Lin V. Y., Pinkus A., Schocken S. (1993) Multilayer feedforward networks with nonpolynomial activation function can approximate any function. Neural Networks, no. 6, pp. 861–867.
18.   Najafi-Alamdari M., Ardalan A. A., Emadi S.-R. (2012) Ellipsoidal Neumann geodetic boundary-value problem based on surface gravity disturbances: case study of Iran. Studia Geophysica et Geodaetica, no. 56, pp. 153–170. DOI: 10.1007/s11200-010-0098-3.
19.   Raissi M., Perdikaris P., Karniadakis G. E. (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, no. 378, pp. 686–707. DOI: 10.1016/J.JCP.2018.10.045.
20.   Sansò F., Venuti G. (2008) On the explicit determination of stability constants for the linearized geodetic boundary value problems. Journal of Geodesy, no. 82, pp. 909–916.
21.   Yu J., Jekeli C., Zhu M. (2002) The analytical solutions of the Dirichlet and Neumann boundary value problems with ellipsoidal boundary. Journal of Geodesy, no. 76, pp. 653–667.
22.   Zingerle P., Pail R., Gruber T., Oikonomidou X. (2020) The combined global gravity field model XGM2019e. Journal of Geodesy, Volume 94, no. 66, DOI: 10.1007/s00190-020-01398-0.
Citation:
Neiman Yu.M., 
Sugaipova L.S., 
Nepoklonov V.B., 
(2025) Numerical solution of the Molodensky boundary value problem with a fixed boundary. Geodesy and cartography = Geodezia i Kartografia, 86(2), pp. 2-14. (In Russian). DOI: 10.22389/0016-7126-2025-1016-2-2-14
Publication History
Received: 09.01.2025
Accepted: 24.02.2025
Published: 20.03.2025

Content

2025 February DOI:
10.22389/0016-7126-2025-1016-2