Moskva, Moscow, Russian Federation
Russian Federation
UDC 55
UDC 550.34
UDC 550.383
CSCSTI 37.01
CSCSTI 37.15
CSCSTI 37.25
CSCSTI 37.31
CSCSTI 38.01
CSCSTI 36.00
CSCSTI 37.00
CSCSTI 38.00
CSCSTI 39.00
CSCSTI 52.00
Russian Classification of Professions by Education 05.00.00
Russian Library and Bibliographic Classification 26
Russian Trade and Bibliographic Classification 63
BISAC SCI SCIENCE
The paper deals with the possibility of finding solutions to direct and inverse gravimetry problems by means of the apparatus of eigenfunctions of the Laplace operator. Data acquired through remote sensing of the Earth’s physical fields are of a discrete character (often in the form of bit chains). The discrete gravitational potential theory allows restoring, with a relatively high degree of reliability, grid mass distributions that are equivalent with respect to an external field complicated by random noise, as well as solving various direct and inverse gravimetry problems. Construction of eigenfunctions of a discrete analogue of the Laplace operator in three-dimensional domains with complex structure eliminates the need for multiple solutions of ill-conditioned systems of linear algebraic equations (SLAE).
discrete potential, complex geological structure, eigenfunctions of the Laplace operator
1. Baranov Yu. V. Analysis of Geophysical Data to Highlight Areas of Possible Earthquakes in the Eastern Margin of the East European Platform // Bulletin of Perm University. Geology. — 2016. — Vol. 4, no. 33. — P. 36–40. — https://doi.org/10.17072/psu.geol.33.36 — (In Russian).
2. Blinova T. S. Formation of unified database for forecasting geodynamically unstable zones of low seismic regions // Geophysical Research. — 2021. — Vol. 22, no. 2. — P. 5–30. — https://doi.org/10.21455/gr2021.2-1 — (In Russian).
3. Samarskii A. A. and Nikolaev E. S. Methods for solving grid equations. — M. : Nauka, 1978. — 592 p. — (In Russian).
4. Stepanova I. E. and Kolotov I. I. Combinatoric-Variational Approach to Solving Inverse Problems in Geophysics Within the Framework of the Discrete Potential Theory. Complex Geological Environment // Russian Journal of Earth Sciences. — 2025. — No. 3. — ES3002. — https://doi.org/10.2205/2025es001000 — (In Russian).
5. Strakhov V. N. On the equivalence in the inverse problem of gravimetry at variable mass density // Doklady AN SSSR. — 1977. — Vol. 236, no. 2. — P. 329–331. — (In Russian).
6. Strakhov V. N., Stepanova I. E. and Grichuk L. V. Theory of discrete gravitational potential and its use in gravimetry // "Issues of theory and practice of geological interpretation of gravitational, magnetic and electric fields", Proceedings of the international conference. — Voronezh : Voronezh State University, 1996. — P. 49–71. — (In Russian).
7. Arsanukaev Z. Z. Calculation of spatial elements of anomalous fields based on the theory of discrete gravitational fields // Izvestiya, Physics of the Solid Earth. — 2004. — Vol. 40, no. 11. — P. 935–955.
8. Cain J. C., Wang Z., Kluth C., et al. Derivation of A Geomagnetic Model to 𝑁 = 63 // Geophysical Journal International. — 1989. — Vol. 97, no. 3. — P. 431–441. — https://doi.org/10.1111/j.1365-246x.1989.tb00513.x
9. Dyment J. and Arkani-Hamed J. Equivalent source magnetic dipoles revisited // Geophysical Research Letters. — 1998. — Vol. 25, no. 11. — P. 2003–2006. — https://doi.org/10.1029/98gl51331
10. Emilia D. A. Equivalent Sources Used As An Analytic Base For Processing Total Magnetic Field Profiles // Geophysics. — 1973. — Vol. 38, no. 2. — P. 339–348. — https://doi.org/10.1190/1.1440344
11. Frik P. G., Sokoloff D. D. and Stepanov R. A. Wavelets for the space-time structure analysis of physical fields // PhysicsUspekhi. — 2022. — Vol. 65, no. 1. — P. 62–89. — https://doi.org/10.3367/ufne.2020.10.038859
12. Hood L. L., Oliveira J. S., Galluzzi V. D., et al. Investigating Sources of Mercury’s Crustal Magnetic Field: Further Mapping of MESSENGER Magnetometer Data // Journal of Geophysical Research: Planets. — 2018. — Vol. 123, no. 10. — P. 2647–2666. — https://doi.org/10.1029/2018je005683
13. Langlais B., Purucker M. E. and Mandea M. Crustal magnetic field of Mars // Journal of Geophysical Research: Planets. — 2004. — Vol. 109, E2. — E02008. — https://doi.org/10.1029/2003je002048
14. Martyshko P. S., Ladovskii I. V., Byzov D. D., et al. Gravity Data Inversion with Method of Local Corrections for Finite Elements Models // Geosciences. — 2018a. — Vol. 8, no. 10. — P. 373. — https://doi.org/10.3390/geosciences8100373
15. Martyshko P. S., Ladovskij I. V., Byzov D. D., et al. On Solving the Forward Problem of Gravimetry in Curvilinear and Cartesian Coordinates: Krasovskii’s Ellipsoid and Plane Modeling // Izvestiya, Physics of the Solid Earth. — 2018b. — Vol. 54, no. 4. — P. 565–573. — https://doi.org/10.1134/s1069351318040079
16. Mikhailov V. O., Timoshkina E. P., Kiseleva E. A., et al. Problems of the Joint Inversion of Temporal Gravity Variations with the Data on Land and Seafloor Displacements: a Case Study of the Tohoku-Oki Earthquake of March 11, 2011 // Izvestiya, Physics of the Solid Earth. — 2019. — Vol. 55, no. 5. — P. 746–752. — https://doi.org/10.1134/s1069351319050070
17. Stepanova I. E. and Kolotov I. I. The Combinatorial-Variational Approach to Solving Linear and Nonlinear Inverse Problems in Geophysics // Doklady Earth Sciences. — 2024. — Vol. 518, no. 1. — P. 1534–1545. — https://doi.org/10.1134/s1028334x24602499
18. Stepanova I. E., Kolotov I. I., Levashov A. N., et al. On Finding Stable Approximate Solutions of Linear Inverse Problems of Gravimetry in the Discrete Potential Theory: A Local Version // Izvestiya, Physics of the Solid Earth. — 2025. — Vol. 61, no. 1. — P. 126–141. — https://doi.org/10.1134/s1069351325700089
19. Stepanova I. E. and Strakhov V. N. Solution of gravity problems by the S-approximation method (regional version) // Izvestiya, Physics of the Solid Earth. — 2002. — Vol. 38, no. 7. — P. 535–544.
20. Thébault E., Schott J. J. and Mandea M. Revised spherical cap harmonic analysis (R-SCHA): Validation and properties // Journal of Geophysical Research: Solid Earth. — 2006. — Vol. 111, B1. — B01102. — https://doi.org/10.1029/200 5jb003836
21. Uno H., Johnson C. L., Anderson B. J., et al. Modeling Mercury’s internal magnetic field with smooth inversions // Earth and Planetary Science Letters. — 2009. — Vol. 285, no. 3/4. — P. 328–339. — https://doi.org/10.1016/j.epsl.2009.02.032
