Abstract
The work is devoted to the approximate methods for solution direct and inverse problems of gravity exploration on bodies with a fractal structure. It is known that in order to construct mathematical models adequate to the geological reality, it is necessary to take into account the orderliness inherent in geological environments. One of the manifestations of orderliness is self-similarity, which remains during the transition from the microlevel to the macrolevel. Scaling of geological media can be traced in petrophysical data and in anomalous fields. It should be noted that in real structures there is no infinite self-similarity and scaling must be considered in a certain range. The work investigates analytical and numerical methods for solving inverse contact problems of the logarithmic and Newtonian potential in the generalized setting. In the case of a Newtonian potential, the problem is formulated as follows. It is required, having three independent functionals of the gravity field above the Earth's surface and additional information on the self-similarity of the disturbing body, to determine the depth, the density and the surface of the perturbing body.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
E.A. Mudretsova (ed.), Gravity Prospecting, M. Sci. (1981) [in Russian]
I.E. Stepanova, On one stable method for recovering ellipsoids. Phys. Earth 1, 101–106 (2001) [in Russian].
V.S. Sizikov, I.N. Golov, Modeling of deposits by spheroids. Phys. Earth 3, 83–96 (2009) [in Russian]
A.J. Katz, A.H. Thompson, Fractal sandstone pores: implications for conductivity and pore formation. Phys. Rev. Lett. 54, 1325–1328 (1985)
C.E. Krohn, Fractal measurement of sandstones, shales, and carbonates. J. Geophys. Res. 93, 3297–3305 (1988)
J.P. Hansen, A.T. Skjeltorp, Fractal pore space and rock permeability implications. Phys. Rev. B38, 2635–2638 (1988)
D. Turcotte, Fractals and Chaos in Geology and Geophysics (Cambridge University Press, Cambridge, 1992)
E.V. Utemov, Flicker-noise Structure of Energy Spectra of Natural Magnetic Anomalies. Monitoring of the Geological Environment: Active Endogenous and Exogenous Processes (Kazan University Publishing House, Kazan, 2000). ([in Russian])
A.A. Potapov, Yu.V. Gulyaev, S.A. Nikitov, A.A. Pakhomov, V.A. German, The Modern Methods of Image Processing. ed. by A.A. Potapov (FIZMATLIT, 2008) [in Russian]
A.H. Thompson, A.J. Katz, C.E. Krohn, The microgeometry and transport properties of sedimentary rock. Adv. Phys. 625–694 (1987)
D.L. Brakenseik, W.J. Rawles, S.D. Logsdon, W.M. Edwards, Fractal description of macroporosity. Soil Sci. Soc. Am. J. 56, 1721–1723 (1992)
S.N. Kulkov, T. Yan, S.P. Buyakova, Fractal dimension of surfaces of porous ceramic materials. Tech. Phys. Lett. 32(2), 51–55 (2006) [in Russian]
P.S. Babayants, Yu.I. Blokh, A.A. Trusov, Fundamentals of modeling potential fields of fractal geological objects. Questions of Theory and Practice of Geological Interpretation of Gravitational, Magnetic and Electric Fields: Proceedings of the 32nd session of the international seminar named after D.G. Uspensky (Mining Institute of the Ural Branch of the Russian Academy of Sciences, Perm, 2005), pp. 12–14 [in Russian]
I.V. Boikov, V.A. Ryazantsev, On Simultaneous restoration of density and surface equation in an inverse gravimetry problem for a contact surface. Numer. Anal. Appl. 13(3), 241–257 (2020)
Yu.L. Daletskii, M.G. Krein, Stability of solutions of differential equations in Banach Space, Nauka, 1970 [in Russian], English transl. Transl. Math. Monographs. V. 43. Amer. Math. Soc. Providence. RI. (1974)
K. Dekker, J. Verver, Stability of Runge-Kutta methods for stiff nonlinear differential equations, Mir. (1988) [in Russian]
I.V. Boikov, On a continuous method for solving nonlinear operator equations. Differ. Equ. 48(9), 1308–1314 (2012)
I.V. Boykov, V.A. Roudnev, A.I. Boykova, O.A. Baulina, New iterative method for solving linear and nonlinear hypersingular integral equations. Appl. Numer. Math. 127, 280–305 (2018)
S.M. Lozinskii, Note on a paper by V.S. Godlevskii, USSR computational mathematics and mathematical physics, 13(2), 232–234 (1973)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Boykov, I.V., Potapov, A.A., Rassadin, A.E., Ryazantsev, V.A. (2022). Approximate Solution of Inverse Problems of Gravity Exploration on Fractals. In: Skiadas, C.H., Dimotikalis, Y. (eds) 14th Chaotic Modeling and Simulation International Conference. CHAOS 2021. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-030-96964-6_8
Download citation
DOI: https://doi.org/10.1007/978-3-030-96964-6_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-96963-9
Online ISBN: 978-3-030-96964-6
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)