Аппроксимационный оператор обратной задачи гравиметрии для горизонтального слоя
Ключевые слова:
обратная задача гравиметрии, аппроксимационный оператор, итерационное решение
Аннотация
Рассматривается метод решения трѐхмерной обратной задачи гравиметрии основанный на использовании приближенного оператора обратной задачи для горизонтального слоя. Построение приближенного оператора выполнено на основе аппроксимации точного аналитического обратного оператора для бесконечного горизонтального слоя конечной суммой простых дискретных линейных операторов. Выбор структуры приближенного обратного оператора даѐтся в спектральной форме, исходя из физической сущности задачи и учитывая естественные ограничения по верхней и нижней частоте для спектрального представления дискретно заданного поля на конечном интервале его определения, в соответствии с теоремой Котельникова. Вычисление параметров приближенного обратного оператора осуществляется на основе минимизации его отклонения от аналитического значения обратного оператора для горизонтального слоя, на конечном числе точек спектра этих функций, что обеспечивает требуемую точность практического решения обратной задачи гравиметрии. Приводятся явные аппроксимационные выражения для расчѐта параметров обратного оператора, зависящие от соотношения дискретного шага задания гравитационного поля и мощности горизонтального слоя, в котором осуществляется поиск решения обратной задачи. Общий алгоритм приближенного решения трѐхмерной обратной задачи гравиметрии на основе предложенного обратного оператора реализован в виде итерационного решении, учитывающего сведения о геометрии изучаемой среды и начального приближения плотности в модели. Существенным моментов в решении обратной задачи является априорная оценка допустимых вариаций аномальной плотности искомого решения и трѐхмерная весовая функция, определяющая меру достоверности начального приближения для изучаемой среды. Даѐтся краткое описание технологии практического применения предлагаемого подхода решении обратной задачи гравиметрии при изучении плотностного строения щитов и фундамента платформ.
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Литература
1. Tikhonov A.N., Arsenin V.Ya. Method reshenia necorrectnych zadach [Methods of solving incorrect problems]. Moscow, Nauka publ., 1979, 288 p. (In Russ.)
2. Tarantola A., Valette B. Generalized nonlinear inverse problems solved using the least square criterion. Reviews of Geophysics and Space Physics. 1982, vol. 20, no 2, pp. 219–232. DOI
3. Starostenko V.I. Ustoychivye cislennie method vie zadachakh gravimetriy [Stable numerical methods in gravity problems]. Kyiv, Naukova Dumka publ., 1978, 228 p. (In Russ.)
4. Aleksidze M.A. Priblijennye method reshenia pryamykh yi obratnyh zadach gravimetriy [Approximate methods of solving direct and inverse gravity problems]. Moscow, Nauka publ., 1987, 336 p. (In Russ.)
5. Starostenko V.I., Isaev V.I., Pyatakov Yu.V. Reshenie obratnoy zadachi gravimetriy dla kontaktov osadochnykh porod [Solution of the inverse problem of gravimetry for contacts of sedimentary rocks]. Geofizicheskij zhurnal ‒ Geophysical Journal, 1993, vol. 15, no. 1, pp. 62‒71. (In Russ.)
6. Akimova E.N., Vasin V.V., Perestoronina G.Ya., Timerkhanova L.Yu., Martyshko P.S., Koksharov D.E. Au regularion metodach reshenia obratnyh zadach gravimetry na mnogoprocessornome vychyslytelnom complexe [On regular methods of solving inverse gravimetry problems on a multiprocessor computational complex]. Vychislitel'nye metody i programmirovanie ‒ Computational methods and programming, 2007, vol. 8, no. 1, pp. 103‒112. (In Russ.)
7. Akimova E.N., Martyshko P.S., Misilov V.E. Algorithms reshenia structure zadachi gravimetry vie mnogosloynoy srede [Algorithms for solving the structural problem of gravimetry in a multilayer environment]. Doklady RAN ‒ Doklady RAN. 2013, vol. 453, no. 6, pp. 676‒679. DOI
8. Martyshko P., Ladovskii I., Byzov D. Parallel algorithms for solving inverse gravimetry problems: application for earth’s crust density models creation. Mathematics, 2021, vol. 9, no. 2966. DOI
9. Novoselitsky V.M. K theory opredelenia ismenenia plotnosti vie gorizontalis plaste pau anomaliamus sila tyazhesti [To the theory of determining the change in density in the horizontal layer by anomalies of gravity]. Fizika Zemli ‒ Physics of the Earth, 1965, no. 5, pp. 25–32. (In Russ.)
10. Cribb J. Application of the generalized linear inverse to the inversion of static potential data. Geophysics, 1976, vol. 41, no. 6, pp. 1365–1369. DOI
11. Savelova T.I. Ob optimalnoy regularization uravnenius tipa svertki so sluchaynymi pomechamy vie yadre yi pravoy chasti [On the optimal regularization of convolution-type equations with random interference in the kernel and the right part]. Zhurnal vychislitel'noj matematiki i matematicheskoj fiziki ‒ Journal of Computational Mathematics and Mathematical Physics, 1978, vol. 18, no. 2, pp. 275‒283. DOI
12. Kobrunov A.I., Varfolomeev V.A. Ob odnom metode ε-ecvivalentnogo pereraspredelenia yi ego ispolzovania dla interpretation gravitational polya [About one method of ε-equivalent redistribution and its use for the interpretation of the gravitational field]. Fizika Zemli ‒ Physics of the Earth, 1981, no. 10, pp. 25–44. (In Russ.)
13. Raevskiy A.B. Primenenie lineynykh transformation pri gravitational modeliridae verkhney chasti zemnoy corri na kristallicheskikh szczytakh (na primere zapadnogo rayona kolskogo poluostrova) Diss. kand. fiz.-mat. nauk [Application of linear transformations in gravitational modeling of the upper part of the earth's crust on crystalline shields (on the example of the western region of the Kola Peninsula). Diss. for academic step. cand. phys.-math. Sciences.]. Apatity, 1984, 102 p. (In Russ.)
14. Pedersen L.B. Relation between potential fields and some equivalent sources. Geophysics, 1991, vol. 56, no. 7, p. 961–971. DOI
15. Kobrunov A.I. Mathematical osnovy theory interpretation geophysical dannykh [Mathematical foundations of the theory of interpretation of geophysical data]. Moscow, TsentrLitNefteGaz publ., 2008, 288 p. (In Russ.)
16. Glaznev V.N., Raevsky A.B., Balagansky V.V., Manninen T. Trochmernaya model verkhney corri rayona kittilasodankyla, Finnish Laplandia (sever Baltiyskogo shchita) [Three-dimensional model of the upper crust of the Kittila-Sodankylä region, Finnish Lapland (north of the Baltic shield)]. Cb. materialov, posvjashhjonnyj 40-letnemu jubileju kafedry geofiziki VGU [Collection of materials dedicated to the 40th anniversary of the Department of Geophysics of VSU]. Voronezh: VSU publ., 2002, pp. 11‒20. (In Russ.)
17. Arzamastsev A.A., Glaznev V.N., Raevsky A.B., Arzamastseva L.V. Morphology and internal structure of the Kola alkaline intrusions, NE Fennoscandian Shield: 3D density modelling and geological implication. Journal of Asian Earth Sciences, 2000, vol. 18, no. 2, pp. 213‒228. DOI
18. Glaznev V.N., Zhirova A.M., Raevsky A.B. Novye dannye au glubinnom stroenia Khibinsky yi Lovozersky massivov, Kolsky poluostrov [New data on the deep structure of the Khibiny and Lovozero massifs, Kola Peninsula]. Doklady RAN ‒ Doklady RAN, 2008, vol. 422, no. 3, pp. 391‒393. DOI
19. Glaznev V.N., Mints M.V., Muravina O.M., Raevsky A.B., Osipenko L.G. Complex geological–geophysical 3D model of the crust in the southeastern Fennoscandian Shield: Nature of density layering of the crust and the crust–mantle boundary. Geodynamics & Tectonophysics, 2015, vol. 6, no. 2, pp. 133–170. DOI
20. Glaznev V.N., Mints M.V., Muravina O.M. Plotnostnoye modeliridae zemnoy corri tsentralnoy chasti vostochno-yevropeyskoy platform [Density modeling of the earth's crust of the central part of the East European Platform]. Vestnik Kamchatskoj regional'noj organizacii Uchebno-nauchnyj centr. Serija: Nauki o Zemle ‒ Vestnik of the Kamchatka Regional Organization Educational and Scientific Center. Series: Earth Science, 2016, vol. 29, no. 1, pp. 53‒63. (In Russ.)
21. Mints M.V., Glaznev V.N., Muravina O.M. Glubinnoye stroenie corri yugo-vostoka Voronezhsky cristallicheskogo massiva pau geophysical dannym: geodynamical evolution vie paleoproterozida yi sovremennoye sostoyaniye corri [Deep structure of the crust of the south-east of the Voronezh crystalline massif according to geophysical data: geodynamic evolution in paleoproterozoic and modern state of the crust]. Vestnik Voronezhskogo gosudarstvennogo universiteta. Serija: Geologija ‒ Vestnik Voronezhskogo gosudarstvennogo universiteta. Series: Geology, 2017, no. 4, pp. 5‒23. (In Russ.)
22. Glaznev V.N., Zhavoronkin V.I., Muravina O.M., Antonova I.Yu., Voronova T.A., Chereshinsky A.V., Kholin P.V. Stroenie verkhney corri Yeletsky uchastka Losevsky terrana (Voronezhsky kristallichesky massive) pau dannym plotnostnogo modeliridae [The structure of the upper crust of the Yeletsky section of the Losevsky territory (Voronezh crystalline array) according to density modeling]. Vestnik Voronezhskogo gosudarstvennogo universiteta. Serija: Geologija ‒ Vestnik Voronezhskogo gosudarstvennogo universiteta. Series: Geology, 2019, no. 3, pp. 74‒83. (In Russ.)
23. Voronova T.A., Glaznev V.N., Muravina O.M., Antonova I.Y. The Density Model of the Crystalline Crust the Southwestern Part of the Lipetsk Region. Springer Proceedings in Earth and Environmental Sciences: Practical and Theoretical Aspects of Geological Interpretation of Gravitational, Magnetic and Electric Fields. Eds. D.Nurgaliev, N.Khairullina. Springer Nature Switzerland AG, 2019, pp. 69‒76. DOI
24. Mints M.V., Glaznev V.N., Muravina O.M., Sokolova E.Yu. 3D model of Svecofennian Accretionary Orogen and Karelia Craton based on geology, reflection seismics, magnetotellurics and density modelling: Geodynamic speculations. Geoscience Frontiers, 2020, vol. 11, no. 3, pp. 999‒1023. DOI
25. Voronova T.A., Muravina O.M., Glaznev V.N., Berezneva S.I. Trochmernaya plotnostnaya model verkhney corri vie oblasts sochleneniya Losevsky yi Donskogo terranov (Voronezhsky kristallichesky massive) [Three-dimensional density model of the upper crust in the region of articulation of Losevsky and Don terranes (Voronezh crystalline massif)]. Vestnik Kamchatskoj regional'noj organizacii Uchebno-nauchnyj centr. Serija: Nauki o Zemle ‒ Vestnik of the Kamchatka Regional Organization Educational and Scientific Center. Series: Earth Sciences, 2021, no. 1 (49), pp. 24–35. DOI
26. Blakely R.J. Potential theory in gravity and magnetic applications. Cambridge University Press, 1995, 461 p. DOI
27. Vogel C.R. Computational methods for inverse problems. Frontiers in Applied Mathematics SIAM, 2002, 183 p. DOI
28. Fullagar P.K., Pears G.A., McMonnies B. Constrained inversion of geologic surfaces - Pushing the boundaries. The Leading Edge. 2008, vol. 27, no. 1, pp. 98‒105. DOI
29. Glaznev V.N. Complexnieu geophysical modelli lithosphere Fennoscandia [Complex geophysical models of the lithosphere of Fennoscandia]. Apatity, KaM. publ., 2003, 252 p. (In Russ.)
30. Martyshko P.S., Prutkin I.L. Technology razdeleniya istochnikov gravitational polya pau glubine [Technology of separation of gravity field sources by depth]. Geofizicheskij zhurnal ‒ Geophysical Journal. 2003, vol. 25, no. 3. pp. 159-169. (In Russ.)
31. Voronova T.A., Glaznev V.N., Muravina O.M. Testirovania approximation algorithms reshenia obratnoy zadachi gravimetry [Testing of the approximation algorithm for solving the inverse gravimetry problem]. Glubinnoe stroenie, geodinamika, teplovoe pole Zemli, interpretacija geofizicheskih polej. Desjatye nauchnye chtenija pamjati Ju.P. Bulashevicha: [Deep structure, geodynamics, thermal field of the Earth, interpretation of geophysical fields. Tenth scientific readings in memory of Yu.P. Bulashevich]. Ekaterinburg, IGF UrO RAN publ., 2019, pp. 77‒80. (In Russ.)
32. Voronova T.A., Glaznev V.N., Muravina O.M. Resultate chislennogo modeliridae s tselyu optimization rabota algorithms inversion gravitational polya [Results of numerical modeling in order to optimize the work of the algorithm for inversion of the gravitational field]. Voprosy teorii i praktiki geologicheskoj interpretacii geofizicheskih polej: materials of the 47th session of the International Scientific Seminar of D.G. Uspensky‒V.N. Strakhov [Issues of theory and practice of geological interpretation of geophysical fields]. Voronezh, Nauchnaya kniga publ., 2020, pp. 76‒79. (In Russ.)
33. Mints M.V., Glaznev V.N., Raevsky A.B. Trochmernaya
model geological stroenia verkhney corri rayona kolskoy superchubockeu skvazhiny yi sopredelnykh territorium kolskogo poluostrova [Three-dimensional model of the geological structure of the upper crust of the Kola ultra-deep well area and adjacent territories of the Kola Peninsula]. Geotektonika ‒ Geotectonics. 1994, no. 6, pp. 3‒22. (In Russ.)
34. Glaznev V.N., Mints M.V., Yakuba I.A. Trochmernaya plotnostnaya model zemnoy corri territories Respubliki Niger [Three-dimensional density model of the earth's crust of the territory of the Republic of Niger]. Vestnik Kamchatskoj regional'noj organizacii Uchebno-nauchnyj centr. Serija: Nauki o Zemle ‒ Vestnik of the Kamchatka Regional Organization Educational and Scientific Center. Series: Earth Sciences, 2021, no. 4 (52), pp. 6‒21. DOI
35. Muravina O.M., Glaznev V.N., Voronova T.A., Terent'ev R.A. Trochmernaya plotnostnaya model verkhney corri vie oblasts sochleneniya Losevsky yi Vorontsovsky terranov (Voronezhsky kristallichesky massive) [Three-dimensional density model of the upper crust in the region of articulation of Losevsky and Vorontsovsky terranes (Voronezh crystalline massif)]. Vestnik Kamchatskoj regional'noj organizacii Uchebnonauchnyj centr. Serija: Nauki o Zemle ‒ Vestnik of the Kamchatka Regional Organization Educational and Scientific Center. Series: Earth Sciences, 2021, vol. 55, no 3, pp. 45‒57. DOI
2. Tarantola A., Valette B. Generalized nonlinear inverse problems solved using the least square criterion. Reviews of Geophysics and Space Physics. 1982, vol. 20, no 2, pp. 219–232. DOI
3. Starostenko V.I. Ustoychivye cislennie method vie zadachakh gravimetriy [Stable numerical methods in gravity problems]. Kyiv, Naukova Dumka publ., 1978, 228 p. (In Russ.)
4. Aleksidze M.A. Priblijennye method reshenia pryamykh yi obratnyh zadach gravimetriy [Approximate methods of solving direct and inverse gravity problems]. Moscow, Nauka publ., 1987, 336 p. (In Russ.)
5. Starostenko V.I., Isaev V.I., Pyatakov Yu.V. Reshenie obratnoy zadachi gravimetriy dla kontaktov osadochnykh porod [Solution of the inverse problem of gravimetry for contacts of sedimentary rocks]. Geofizicheskij zhurnal ‒ Geophysical Journal, 1993, vol. 15, no. 1, pp. 62‒71. (In Russ.)
6. Akimova E.N., Vasin V.V., Perestoronina G.Ya., Timerkhanova L.Yu., Martyshko P.S., Koksharov D.E. Au regularion metodach reshenia obratnyh zadach gravimetry na mnogoprocessornome vychyslytelnom complexe [On regular methods of solving inverse gravimetry problems on a multiprocessor computational complex]. Vychislitel'nye metody i programmirovanie ‒ Computational methods and programming, 2007, vol. 8, no. 1, pp. 103‒112. (In Russ.)
7. Akimova E.N., Martyshko P.S., Misilov V.E. Algorithms reshenia structure zadachi gravimetry vie mnogosloynoy srede [Algorithms for solving the structural problem of gravimetry in a multilayer environment]. Doklady RAN ‒ Doklady RAN. 2013, vol. 453, no. 6, pp. 676‒679. DOI
8. Martyshko P., Ladovskii I., Byzov D. Parallel algorithms for solving inverse gravimetry problems: application for earth’s crust density models creation. Mathematics, 2021, vol. 9, no. 2966. DOI
9. Novoselitsky V.M. K theory opredelenia ismenenia plotnosti vie gorizontalis plaste pau anomaliamus sila tyazhesti [To the theory of determining the change in density in the horizontal layer by anomalies of gravity]. Fizika Zemli ‒ Physics of the Earth, 1965, no. 5, pp. 25–32. (In Russ.)
10. Cribb J. Application of the generalized linear inverse to the inversion of static potential data. Geophysics, 1976, vol. 41, no. 6, pp. 1365–1369. DOI
11. Savelova T.I. Ob optimalnoy regularization uravnenius tipa svertki so sluchaynymi pomechamy vie yadre yi pravoy chasti [On the optimal regularization of convolution-type equations with random interference in the kernel and the right part]. Zhurnal vychislitel'noj matematiki i matematicheskoj fiziki ‒ Journal of Computational Mathematics and Mathematical Physics, 1978, vol. 18, no. 2, pp. 275‒283. DOI
12. Kobrunov A.I., Varfolomeev V.A. Ob odnom metode ε-ecvivalentnogo pereraspredelenia yi ego ispolzovania dla interpretation gravitational polya [About one method of ε-equivalent redistribution and its use for the interpretation of the gravitational field]. Fizika Zemli ‒ Physics of the Earth, 1981, no. 10, pp. 25–44. (In Russ.)
13. Raevskiy A.B. Primenenie lineynykh transformation pri gravitational modeliridae verkhney chasti zemnoy corri na kristallicheskikh szczytakh (na primere zapadnogo rayona kolskogo poluostrova) Diss. kand. fiz.-mat. nauk [Application of linear transformations in gravitational modeling of the upper part of the earth's crust on crystalline shields (on the example of the western region of the Kola Peninsula). Diss. for academic step. cand. phys.-math. Sciences.]. Apatity, 1984, 102 p. (In Russ.)
14. Pedersen L.B. Relation between potential fields and some equivalent sources. Geophysics, 1991, vol. 56, no. 7, p. 961–971. DOI
15. Kobrunov A.I. Mathematical osnovy theory interpretation geophysical dannykh [Mathematical foundations of the theory of interpretation of geophysical data]. Moscow, TsentrLitNefteGaz publ., 2008, 288 p. (In Russ.)
16. Glaznev V.N., Raevsky A.B., Balagansky V.V., Manninen T. Trochmernaya model verkhney corri rayona kittilasodankyla, Finnish Laplandia (sever Baltiyskogo shchita) [Three-dimensional model of the upper crust of the Kittila-Sodankylä region, Finnish Lapland (north of the Baltic shield)]. Cb. materialov, posvjashhjonnyj 40-letnemu jubileju kafedry geofiziki VGU [Collection of materials dedicated to the 40th anniversary of the Department of Geophysics of VSU]. Voronezh: VSU publ., 2002, pp. 11‒20. (In Russ.)
17. Arzamastsev A.A., Glaznev V.N., Raevsky A.B., Arzamastseva L.V. Morphology and internal structure of the Kola alkaline intrusions, NE Fennoscandian Shield: 3D density modelling and geological implication. Journal of Asian Earth Sciences, 2000, vol. 18, no. 2, pp. 213‒228. DOI
18. Glaznev V.N., Zhirova A.M., Raevsky A.B. Novye dannye au glubinnom stroenia Khibinsky yi Lovozersky massivov, Kolsky poluostrov [New data on the deep structure of the Khibiny and Lovozero massifs, Kola Peninsula]. Doklady RAN ‒ Doklady RAN, 2008, vol. 422, no. 3, pp. 391‒393. DOI
19. Glaznev V.N., Mints M.V., Muravina O.M., Raevsky A.B., Osipenko L.G. Complex geological–geophysical 3D model of the crust in the southeastern Fennoscandian Shield: Nature of density layering of the crust and the crust–mantle boundary. Geodynamics & Tectonophysics, 2015, vol. 6, no. 2, pp. 133–170. DOI
20. Glaznev V.N., Mints M.V., Muravina O.M. Plotnostnoye modeliridae zemnoy corri tsentralnoy chasti vostochno-yevropeyskoy platform [Density modeling of the earth's crust of the central part of the East European Platform]. Vestnik Kamchatskoj regional'noj organizacii Uchebno-nauchnyj centr. Serija: Nauki o Zemle ‒ Vestnik of the Kamchatka Regional Organization Educational and Scientific Center. Series: Earth Science, 2016, vol. 29, no. 1, pp. 53‒63. (In Russ.)
21. Mints M.V., Glaznev V.N., Muravina O.M. Glubinnoye stroenie corri yugo-vostoka Voronezhsky cristallicheskogo massiva pau geophysical dannym: geodynamical evolution vie paleoproterozida yi sovremennoye sostoyaniye corri [Deep structure of the crust of the south-east of the Voronezh crystalline massif according to geophysical data: geodynamic evolution in paleoproterozoic and modern state of the crust]. Vestnik Voronezhskogo gosudarstvennogo universiteta. Serija: Geologija ‒ Vestnik Voronezhskogo gosudarstvennogo universiteta. Series: Geology, 2017, no. 4, pp. 5‒23. (In Russ.)
22. Glaznev V.N., Zhavoronkin V.I., Muravina O.M., Antonova I.Yu., Voronova T.A., Chereshinsky A.V., Kholin P.V. Stroenie verkhney corri Yeletsky uchastka Losevsky terrana (Voronezhsky kristallichesky massive) pau dannym plotnostnogo modeliridae [The structure of the upper crust of the Yeletsky section of the Losevsky territory (Voronezh crystalline array) according to density modeling]. Vestnik Voronezhskogo gosudarstvennogo universiteta. Serija: Geologija ‒ Vestnik Voronezhskogo gosudarstvennogo universiteta. Series: Geology, 2019, no. 3, pp. 74‒83. (In Russ.)
23. Voronova T.A., Glaznev V.N., Muravina O.M., Antonova I.Y. The Density Model of the Crystalline Crust the Southwestern Part of the Lipetsk Region. Springer Proceedings in Earth and Environmental Sciences: Practical and Theoretical Aspects of Geological Interpretation of Gravitational, Magnetic and Electric Fields. Eds. D.Nurgaliev, N.Khairullina. Springer Nature Switzerland AG, 2019, pp. 69‒76. DOI
24. Mints M.V., Glaznev V.N., Muravina O.M., Sokolova E.Yu. 3D model of Svecofennian Accretionary Orogen and Karelia Craton based on geology, reflection seismics, magnetotellurics and density modelling: Geodynamic speculations. Geoscience Frontiers, 2020, vol. 11, no. 3, pp. 999‒1023. DOI
25. Voronova T.A., Muravina O.M., Glaznev V.N., Berezneva S.I. Trochmernaya plotnostnaya model verkhney corri vie oblasts sochleneniya Losevsky yi Donskogo terranov (Voronezhsky kristallichesky massive) [Three-dimensional density model of the upper crust in the region of articulation of Losevsky and Don terranes (Voronezh crystalline massif)]. Vestnik Kamchatskoj regional'noj organizacii Uchebno-nauchnyj centr. Serija: Nauki o Zemle ‒ Vestnik of the Kamchatka Regional Organization Educational and Scientific Center. Series: Earth Sciences, 2021, no. 1 (49), pp. 24–35. DOI
26. Blakely R.J. Potential theory in gravity and magnetic applications. Cambridge University Press, 1995, 461 p. DOI
27. Vogel C.R. Computational methods for inverse problems. Frontiers in Applied Mathematics SIAM, 2002, 183 p. DOI
28. Fullagar P.K., Pears G.A., McMonnies B. Constrained inversion of geologic surfaces - Pushing the boundaries. The Leading Edge. 2008, vol. 27, no. 1, pp. 98‒105. DOI
29. Glaznev V.N. Complexnieu geophysical modelli lithosphere Fennoscandia [Complex geophysical models of the lithosphere of Fennoscandia]. Apatity, KaM. publ., 2003, 252 p. (In Russ.)
30. Martyshko P.S., Prutkin I.L. Technology razdeleniya istochnikov gravitational polya pau glubine [Technology of separation of gravity field sources by depth]. Geofizicheskij zhurnal ‒ Geophysical Journal. 2003, vol. 25, no. 3. pp. 159-169. (In Russ.)
31. Voronova T.A., Glaznev V.N., Muravina O.M. Testirovania approximation algorithms reshenia obratnoy zadachi gravimetry [Testing of the approximation algorithm for solving the inverse gravimetry problem]. Glubinnoe stroenie, geodinamika, teplovoe pole Zemli, interpretacija geofizicheskih polej. Desjatye nauchnye chtenija pamjati Ju.P. Bulashevicha: [Deep structure, geodynamics, thermal field of the Earth, interpretation of geophysical fields. Tenth scientific readings in memory of Yu.P. Bulashevich]. Ekaterinburg, IGF UrO RAN publ., 2019, pp. 77‒80. (In Russ.)
32. Voronova T.A., Glaznev V.N., Muravina O.M. Resultate chislennogo modeliridae s tselyu optimization rabota algorithms inversion gravitational polya [Results of numerical modeling in order to optimize the work of the algorithm for inversion of the gravitational field]. Voprosy teorii i praktiki geologicheskoj interpretacii geofizicheskih polej: materials of the 47th session of the International Scientific Seminar of D.G. Uspensky‒V.N. Strakhov [Issues of theory and practice of geological interpretation of geophysical fields]. Voronezh, Nauchnaya kniga publ., 2020, pp. 76‒79. (In Russ.)
33. Mints M.V., Glaznev V.N., Raevsky A.B. Trochmernaya
model geological stroenia verkhney corri rayona kolskoy superchubockeu skvazhiny yi sopredelnykh territorium kolskogo poluostrova [Three-dimensional model of the geological structure of the upper crust of the Kola ultra-deep well area and adjacent territories of the Kola Peninsula]. Geotektonika ‒ Geotectonics. 1994, no. 6, pp. 3‒22. (In Russ.)
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Опубликован
2023-03-23
Как цитировать
Глазнев, В. Н., Муравина, О. М., & Раевский, А. Б. (2023). Аппроксимационный оператор обратной задачи гравиметрии для горизонтального слоя. Вестник ВГУ. Серия: Геология, (1), 97-105. https://doi.org/10.17308/geology/1609-0691/2023/1/97-105
Выпуск
Раздел
Геофизика
