Abstract
The article examines the possibility of simultaneous reconstruction of sources equivalent in the external field and spectral characteristics of the useful signal. Examples of variational formulations are given for various versions of the method of linear integral representations, and the problem is formulated of finding the distribution density of gravitating or magnetic masses on several horizontal planes and the Fourier transform of an anomalous field element from the values of a disturbed signal known at points of a certain observation network.
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REFERENCES
V. N. Strakhov and I. E. Stepanova, “The S-approximation method and its application to gravity problems,” Izv. Phys. Solid Earth 38 (2), 91–107 (2002).
V. N. Strakhov and I. E. Stepanova, “Solution of gravity problems by the S-approximation method (regional version),” Izv. Phys. Solid Earth 38 (7), 535–544 (2002).
I. E. Stepanova, I. A. Kerimov, and A. G. Yagola, “Approximation approach in various modifications of the method of linear integral representations,” Izv. Phys. Solid Earth 55 (2), 218–231 (2019).
V. N. Strakhov, I. A. Kerimov, and I. E. Stepanova, Development of Theory and Computer Technology for Constructing Linear Analytical Approximations of Gravity and Magnetic Fields (Inst. Fiz. Zemli, Ross. Akad. Nauk, Moscow, 2009) [in Russian].
D. N. Raevskiy and I. E. Stepanova, “On the solution of inverse problems of gravimetry by the modified method of S-approximations,” Izv. Phys. Solid Earth 51 (2), 207–218 (2015).
D. N. Raevskiy and I. E. Stepanova, “The modified method of S-approximations: Regional version,” Izv. Phys. Solid Earth 51 (2), 197–206 (2015).
I. A. Kerimov, F-Approximation Method in Solving Gravimetry and Magnetometry Problems (Fizmatlit, Moscow, 2011) [in Russian].
I. E. Stepanova, “The R-approximation method and its application to the interpretation of detailed gravity and magnetic data,” Izv. Phys. Solid Earth 45 (4), 287–300 (2009).
N. S. Koshlyakov, E. B. Gliner, and N. M. Smirnov, Basic Differential Equations of Mathematical Physics (Fizmatgiz, Moscow, 1962) [in Russian].
M. A. Lavrent’ev and L. A. Lyusternik, A Course in Variational Calculus (Gosopttekhizdat, Moscow, 1950) [in Russian].
A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Nauka, Moscow, 1990; Kluwer Academic, Dordrecht, 1995).
A. G. Yagola, I. E. Stepanova, Y. Wang, and V. N. Titarenko, Inverse Problems in Geophysics and Solution Methods (Binom, Moscow, 2014) [in Russian].
N. Ya. Vilenkin, Special Functions and the Theory of Group Representations (Nauka, Moscow, 1965; Am. Math. Soc., Providence, R.I., 1968).
E. K. Leinartas and M. E. Petrochenko, “Multidimensional analogues of the Euler–Maclaurin summation formula and the Borel transform of power series,” Sib. Elektron. Mat. Izv. 19 (1), 91–100 (2022). https://doi.org/10.33048/semi.2022.19.008
V. N. Sachkov, Combinatorial Methods of Discrete Mathematics (Nauka, Moscow, 1977) [in Russian].
I. I. Kolotov, D. V. Lukyanenko, I. E. Stepanova, A. V. Shchepetilov, and A. G. Yagola, “On the uniqueness of solution to systems of linear algebraic equations to which the inverse problems of gravimetry and magnetometry are reduced: A regional variant,” Comput. Math. Math. Phys. 63 (9), 1588–1599 (2023).
I. I. Kolotov, D. V. Lukyanenko, I. E. Stepanova, and A. G. Yagola, “On the uniqueness of solutions to systems of linear algebraic equations resulting from the reduction of linear inverse problems of gravimetry and magnetometry: A local case,” Comput. Math. Math. Phys. 63 (8), 1452–1465 (2023).
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This work was supported by the Russian Science Foundation (grant no. 23-41-00002).
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Translated by E. Chernokozhin
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Stepanova, I.E., Lukyanenko, D.V., Kolotov, I.I. et al. On the Simultaneous Determination of the Distribution Density of Sources Equivalent in the External Field and the Spectrum of the Useful Signal. Comput. Math. and Math. Phys. 64, 1089–1102 (2024). https://doi.org/10.1134/S0965542524700301
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DOI: https://doi.org/10.1134/S0965542524700301