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Gravity inverse problem solution with variable rate of gradient descent
Lomonosov Moscow State University
Journal: Geophysical research
Tome: 23
Number: 1
Year: 2022
Pages: 5-19
UDK: 550.831
DOI: 10.21455/gr2022.1-1
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Chepigo L.S., Lygin I.V., Bulychev A.A. Gravity inverse problem solution with variable rate of gradient descent // . 2022. Т. 23. № 1. С. 5-19. DOI: 10.21455/gr2022.1-1
@article{ChepigoGravity2022,
author = "Chepigo, L. S. and Lygin, I. V. and Bulychev, A. A.",
title = "Gravity inverse problem solution with variable rate of gradient descent",
journal = "Geophysical research",
year = 2022,
volume = "23",
number = "1",
pages = "5-19",
doi = "10.21455/gr2022.1-1",
language = "English"
}
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Keywords: gravity exploration, inverse problem, gradient descent method, density modeling
Аnnotation: The article describes an approach to the automated solution of a linear inverse problem of gravity, which implements the construction of lateral and vertical gradient density models with the possibility to select the preferred depth of sources. The inverse problem is solved using the gradient descent method with variable rate. It is shown that if the gradient descent rate is increasing with depth, then deep cells are “included” in the process of selecting the density model. In particular, the rate of gradient descent can increase with depth as a power function. In general, the gradient descent rate depends on both depth and horizontal coordinates, and it can be specified functionally or explicitly. In the presence of a prior information, the gradient descent rate can be expressed in a more complex way, depending on the depth and on the horizontal coordinates. In this case, the maximum values of the gradient descent rate should be assigned to the cells in which, according to a prior data, density inhomogeneities are located or expected. These can be depth-velocity models, widely used in seismic exploration, so it opens the way for integration.
The article shows the application of the approach on a test model consisting of two infinite horizontal rods located at different depths. The performance of the algorithm with different values of the power function is assessed by comparing the selected depths of the centers of mass with the true depths. It is shown that the optimal results of solving the inverse problem for the selected type of model are achieved using the gradient descent rate proportional to the square of the depth.
The developed algorithm forms the basis of the author's software package GravInv [Chepigo, 2019].
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Bulychev А.А., Lygin I.V., Sokolova T.B., Kuznetsov K.M., Pryamaya zadacha gravirazvedki i magneto-razvedki (konspekt lektsii) (Forward gravimetry and magnetometry problem (lecture notes)), Moscow: “Universitetskaya kniga”, 2019, 176 p. [In Russian]. doi: 10.31453/kdu.ru.91304.0040
Chepigo L.S., Svidetel'stvo o gosudarstvennoi registratsii programmy dlya EVM № 2019662512 GravInv2D, vydano 25.09.2019. [In Russian].
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Cormen T.H., Leiserson C.E., Rivest R.L., Stein C., Introduction to Algorithms, 3rd Edition, Cambridge, Massa-chusetts: MIT Press, 2009, 1292 p. ISBN 0-262-03384-4.
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Kobrunov A.I., Varfolomeev V.A., On one method of ε-equivalent redistributions and its use in the interpretation of gravitational fields, Izvestiya AN SSSR. Fizika Zemli (Izvestia of the Academy of Sciences of the USSR. Physics of the Earth), 1981, no. 10, pp. 25-44. [In Russian].
Kuznetsov K.M., Bulychev A.A., Analysis of areal potential fields based on Poisson wavelets, Geofizika (Geo-physics), 2017, no. 6, pp. 25-32. [In Russian].
Li Y., Oldenburg W., 3-D inversion of magnetic data, Geophysics, 1996, vol. 61, no. 2, pp. 394-408.
Li Y., Oldenburg W., 3-D inversion of gravity data, Geophysics, 1998, vol. 63, no. 1, pp. 109-119.
Martyshko P.S., Akimova E.N., Misilov V.E., Solving the Structural Inverse Gravity Problem by the Modified Gradient Methods, Izvestiya. Physics of the Solid Earth, 2016, vol. 52, no. 5, pp. 704-708.
Melikhov V.R., Bulychev A.A., Sastri R.G., Solving the forward gravity problem using the fast Fourier transform, in Materialy 6-i konferentsii aspirantov i molodykh uchenykh, sektsiya “Geofizika”, MGU (Materials of the 6th conference of graduate students and young scientists, section “Geophysics”, MSU), Moscow: VINITI RAN, 1979, pp. 97-108. [In Russian].
Petrishchevsky A.M., Geological problems solved with a probabilistic-deterministic approach to the interpreta-tion of gravity anomalies, Geofizika (Geophysics), 2021, no. 2, pp. 89-99. [In Russian].
Reid A.B., Allsop J.M., Granser H., Millett A.J., Somerton I.W., Magnetic interpretation in three dimensions using Euler Deconvolution, Geophysics, 1990, vol. 55, pp. 80-91.
Sipser M., Introduction to the Theory of Computation, Course Technology Inc, 2006. ISBN 0-619-21764-2.
Starostenko V.I., Legostaeva O.V., Makarenko I.B., Savchenko A.S., Complex for automated interpretation of potential field data (GMT-Auto), Geofizicheskii zhurnal (Geophysical journal), 2015, vol. 37, no. 1, pp. 42-52. [In Russian].
Strakhov V.N., Luchitskii A.I., On solving the forward 2D gravimetry and magnetometry problems, Izvestiya AN SSSR. Fizika Zemli (Izvestia of the Academy of Sciences of the USSR. Physics of the Earth), 1980a, no. 8, pp. 65-83. [In Russian].
Strakhov V.N., Luchitskii A.I., Solution of forward problem of gravimetry and magnetometry for some classes of mass distribution, Izvestiya AN SSSR. Fizika Zemli (Izvestia of the Academy of Sciences of the USSR. Physics of the Earth), 1980b, no. 10, pp. 48-64. [In Russian].
Tikhonov A.N., Arsenin V.Y., Metody resheniya nekorrektnykh zadach (Methods for solving ill-posed problems), Moscow: Nauka, 1979, 283 p. [In Russian].
Zhou X., Analytic solution of the gravity anomaly of irregular 2D masses with density contrast varying as a 2D polynomial function, Geophysics, 2010, vol. 75, no. 2, pp. I11-I19.
Blokh Y.I., Interpretatsiya gravitatsionnykh i magnitnykh anomalii. Uchebnoe posobie (Interpretation of gravity and magnetic anomalies. Lecture notes), Moscow: MGGA, 2009, 232 p. [In Russian]. www.sigma3d.com/
pdf/books/blokh-interp.pdf
Bulychev А.А., Lygin I.V., Sokolova T.B., Kuznetsov K.M., Pryamaya zadacha gravirazvedki i magneto-razvedki (konspekt lektsii) (Forward gravimetry and magnetometry problem (lecture notes)), Moscow: “Universitetskaya kniga”, 2019, 176 p. [In Russian]. doi: 10.31453/kdu.ru.91304.0040
Chepigo L.S., Svidetel'stvo o gosudarstvennoi registratsii programmy dlya EVM № 2019662512 GravInv2D, vydano 25.09.2019. [In Russian].
Chepigo L.S., Lygin I.V., Bulychev A.A., A 2D forward gravimetry problem for a polygon with parabolic density, Moscow University Geology Bulletin, 2019, vol. 74, no. 5, pp. 516-520.
Cormen T.H., Leiserson C.E., Rivest R.L., Stein C., Introduction to Algorithms, 3rd Edition, Cambridge, Massa-chusetts: MIT Press, 2009, 1292 p. ISBN 0-262-03384-4.
D’Urso M.G., The Gravity Anomaly of a 2D polygonal body having density contrast given by polynomial func-tions, Surv. Geophys., 2015, vol. 36, no. 3, pp. 391-425.
D’Urso M.G., Trotta S., Gravity Anomaly of a polyhedral bodies having a polynomial density contrast, Surv. Geophys., 2017, vol. 38, no. 4, pp. 781-832.
Grebennikova I.V., Metody optimizatsii: uchebnoe posobie (Optimization methods: manual for graduate stu-dents), Yekaterinburg: UrFU, 2017, 148 p. [In Russian].
Kobrunov A.I., Matematicheskie osnovy teorii interpretatsii geofizicheskikh dannykh: ucheb. posobie (Mathe-matical foundations of the theory of interpretation of geophysical data: lecture notes), Ukhta: UGTU, 2007, 288 p. [In Russian].
Kobrunov A.I., Varfolomeev V.A., On one method of ε-equivalent redistributions and its use in the interpretation of gravitational fields, Izvestiya AN SSSR. Fizika Zemli (Izvestia of the Academy of Sciences of the USSR. Physics of the Earth), 1981, no. 10, pp. 25-44. [In Russian].
Kuznetsov K.M., Bulychev A.A., Analysis of areal potential fields based on Poisson wavelets, Geofizika (Geo-physics), 2017, no. 6, pp. 25-32. [In Russian].
Li Y., Oldenburg W., 3-D inversion of magnetic data, Geophysics, 1996, vol. 61, no. 2, pp. 394-408.
Li Y., Oldenburg W., 3-D inversion of gravity data, Geophysics, 1998, vol. 63, no. 1, pp. 109-119.
Martyshko P.S., Akimova E.N., Misilov V.E., Solving the Structural Inverse Gravity Problem by the Modified Gradient Methods, Izvestiya. Physics of the Solid Earth, 2016, vol. 52, no. 5, pp. 704-708.
Melikhov V.R., Bulychev A.A., Sastri R.G., Solving the forward gravity problem using the fast Fourier transform, in Materialy 6-i konferentsii aspirantov i molodykh uchenykh, sektsiya “Geofizika”, MGU (Materials of the 6th conference of graduate students and young scientists, section “Geophysics”, MSU), Moscow: VINITI RAN, 1979, pp. 97-108. [In Russian].
Petrishchevsky A.M., Geological problems solved with a probabilistic-deterministic approach to the interpreta-tion of gravity anomalies, Geofizika (Geophysics), 2021, no. 2, pp. 89-99. [In Russian].
Reid A.B., Allsop J.M., Granser H., Millett A.J., Somerton I.W., Magnetic interpretation in three dimensions using Euler Deconvolution, Geophysics, 1990, vol. 55, pp. 80-91.
Sipser M., Introduction to the Theory of Computation, Course Technology Inc, 2006. ISBN 0-619-21764-2.
Starostenko V.I., Legostaeva O.V., Makarenko I.B., Savchenko A.S., Complex for automated interpretation of potential field data (GMT-Auto), Geofizicheskii zhurnal (Geophysical journal), 2015, vol. 37, no. 1, pp. 42-52. [In Russian].
Strakhov V.N., Luchitskii A.I., On solving the forward 2D gravimetry and magnetometry problems, Izvestiya AN SSSR. Fizika Zemli (Izvestia of the Academy of Sciences of the USSR. Physics of the Earth), 1980a, no. 8, pp. 65-83. [In Russian].
Strakhov V.N., Luchitskii A.I., Solution of forward problem of gravimetry and magnetometry for some classes of mass distribution, Izvestiya AN SSSR. Fizika Zemli (Izvestia of the Academy of Sciences of the USSR. Physics of the Earth), 1980b, no. 10, pp. 48-64. [In Russian].
Tikhonov A.N., Arsenin V.Y., Metody resheniya nekorrektnykh zadach (Methods for solving ill-posed problems), Moscow: Nauka, 1979, 283 p. [In Russian].
Zhou X., Analytic solution of the gravity anomaly of irregular 2D masses with density contrast varying as a 2D polynomial function, Geophysics, 2010, vol. 75, no. 2, pp. I11-I19.
