Численный алгоритм решения задач электрического зондирования среды с рельефом земной поверхности
DOI:
https://doi.org/10.26577/JMMCS-2018-4-561Ключевые слова:
метод интегральных уравнений, численный алгоритм, прямая задача электроразведки, рельеф земной поверхностиАннотация
На сегодняшний день проблема влияния рельефа земной поверхности на распределение электрического поля является актуальным вопросом в интерпретации электромагнитных полей, изучаемых в электроразведке постоянным током. В работе рассматривается тестирование алгоритма численного решения задачи электрического зондирования среды с рельефом земной поверхности путем моделирования методом интегральных уравнений. Идея метода интегральных уравнений заключается в представлении электрического поля, как суммы первичного поля и поля вторичных зарядов. Контактные границы и поверхность геоэлектрического разреза выступают как вторичные возбудители электрического поля. Задача расчета полей сводится к системе интегральных уравнений на плотности вторичных источников, индуцируемых на поверхностях контакта проводящих сред и на рельефной поверхности среды. Математическое описание этого явления приводит к уравнениям Фредгольма II рода с полярным ядром. Алгоритм расчета протестирован путем сравнения результатов с решениями, приведенными в работах в открытом доступе по учету влияния рельефа, задавая такие же параметры среды. Выполнены сравнения полученными разными подходами конечно-элементных и конечно-разностных методов для учета влияния рельефа в 2D средах. Также выполнены сравнения полученными данными физического моделирования. Можно сказать, что выполненные нами результаты расчетов кривых кажущихся сопротивлений находятся в хорошем согласии с имеющимися исследованиями в этой области.
Библиографические ссылки
[1] Shevnin V.A. "Elektrorazvedka. Posobie po ehlektrorazvedochnoj praktike dlya studentov geofizicheskikh spetsial’nostej [Electrical prospecting. Manual on electrical exploration practice for students of geophysical specialties] Moscow, (2013), 124.
[2] Loke M.H., Barker R.D. "Practical techniques for 3D resistivity surveys and data inversion."Geophysical Prospecting 44 (1996): 499-523.
[3] Gunther Т., Rucker С. Boundless Electrical Resistivity Tomography. BERT 2 - the user tutorial( 2013).
[4] Baranchuk K.I., Mirgalikyzy T., Modin I.N., Mukanova B.G. "Fizicheskoe modelirovanie ehlektricheskoj tomografii na poverkhnosti so slozhnym rel’efom [Physical modeling of electrical tomography on the surface with a complex relief]."Engineering Surveys 11 (2017): 56-65.
[5] Veshev A.V. "Elektroprofilirovanie na postoyannom i peremennom toke [Electrical profiling on direct and alternating current]."2 (1980).
[6] Bobachev A.A. "Reshenie pryamykh i obratnykh zadach elektrorazvedki metodom soprotivlenij dlya slozhno-postroennykh sred [The solution of direct and inverse problems of electrical prospecting by the method of resistance for complex-built environments]"( Thesis for the degree of physical and mathematical sciences, MSU, 2003).
[7] Mirgalikyzy T., Mukanova B., Modin I. "Method of Integral Equations for the Problem of Electrical Tomography in a Medium with Ground Surface Relief."Journal of Applied Mathematic(2015).
[8] Balgaisha Mukanova, Tolkyn Mirgalikyzy and Dilyara Rakisheva "Modelling the Influence of Ground Surface Relief on Electric Sounding Curves Using the Integral Equations Method."Mathematical Problems in Engineering (2017).
[9] Zaporozhets V.M. "Vliyanie rel’efa na rezul’taty zamerov soprotivleniya (po rabotam S. G. Komarova i L.P. Gorbenko) [The influence of the relief on the results of resistance measurements]."Elkgr 4 (12)(1938).
[10] Chanturishvili L.S. "O kolichestvennom uchete vliyaniya rel’efa dlya nekotorykh sluchaev razvedki postoyannym tokom [On the quantitative account of the influence of the relief for some cases of direct current intelligence]."Proceedings of the Institute of Geophysics 14(1955): 199-209.
[11] Veshev A.V. "Vliyanie rel’efa na rezul’taty rabot kombinirovannym ehlektroprofilirovaniem [Influence of relief on the results of work by combined electric profiling]."Scientific notes of Leningrad State University 278 (1959).
[12] Loke M.H., Barker R.D. "Rapid least-squares inversion of apparent resistivity pseudo sections using a quasi-Newton method."Geophysical Prospecting 44(1)(1996): 131-152.
[13] Coggon J.H. "Electromagnetic and electrical modeling by the finite element method."Geophysics 36(1) 1971: 132-155.
[14] Mufti I.R. "Finite-difference modeling for arbitrary-shaped two dimensional structures."Geophysics 41(1)(1976): 62-78.
[15] Pelton W.H., Rijo L., Swift C.M. "Inversion of two dimensional resistivity and Induced Polarization data."Geophysics 43(4) (1978): 788-803.
[16] Dey A., Morrison H.F. "Resistivity modeling for arbitrary shaped two-dimensional structures."Geophysical Prospecting 27(1) (1979): 106-136.
[17] Loke M.H., Barker R.D. "Least-squares deconvolution of apparent resistivity pseudosections."Geophysics 60(6) (1995): 1682-1690.
[18] Loke M.H. "Topographic modelling in resistivity imaging inversion."62nd EAGE Conference and Technical Exhibition, Extended Abstracts (2000).
[19] Erdogan E., Demirci I., Candasayar M.E. "Incorporating topography into 2D resistivity modeling using finite-element and finite-difference approaches."Geophysics 73(3) (2008): 135-142.
[20] Demirci I., Erdogan E., Candasayar M.E. "Two-dimensional inversion of direct current resistivity data incorporating topography by using finite difference techniques with triangle cells: Investigation of Kera fault zone in western Crete."Geophysics 77(1) (2012): 67-75.
[21] Penz S., Chauris H., Donno D., Mehl C. "Resistivity modeling with topography."Geophys. J. Int. 194(3) (2013): 1486-1497.
[22] Fox R.C., Hohmann G.W., Killpack T.J., Rijo L. "Topographic effects in resistivity and induced-polarization surveys."Geophysics 45(1) (1980): 75-93.
[23] Tsourlos P.I., Szymanski J.E., Tsokas G.N. "The effect of topography on commonly used resistivity arrays."Geophysics 64(5) (1999): 1357-1363.
[24] Plattner A.D. "Adaptive wavelet methods for geoelectric modeling and inversion"(Dissertation, 2011).
[25] Gunther Т., Rucker С., Spitzer K. "Three-dimensional modelling and inversion of dc resistivity data incorporating topography - I. Modellingn."Geophys. J. Int. 166 (2006): 495-505.
[26] Gunther Т., Rucker С. "Boundless Electrical Resistivity Tomography."BERT 2 - the user tutorial (2013).
[27] Alpin L.M. "Istochniki polya v teorii ehlektricheskoj razvedki [Sources of field in the theory of electrical intelligence]."Applied Geophysics 3 (1947): 56-200.
[28] Dieter K., Paterson N.R and Grant F.S. "KP and resistivity type awes for three-dimensional bodies."Geophysics 34 (1969): 615-632.
[29] Hohmann G.W. "Three dimensional induced polarization and electromagnetic modeling."Geophysics 40 (1975): 309-324.
[30] Eloranta E. "A method for calculation mise-a-la-masse anomalies in the case of high conductivity contrast by the integral equation technique."Geoexploration 22 (1984): 77-88.
[31] Schenkel C.J. "The Electrical Resistivity Method in Cased Boreholes"(Phodissertation, University of California, 1991).
[32] Orunkhanov M., Mukanova B. "The integral equations method in problems of electrical sounding."Advances in High Performance Computing and Computational Sciences (2006): P.15-21.
[33] Orunkhanov M.K., Mukanova B.G., Sarbasova B.K. "CHislennaya realizatsiya metoda potentsialov v zadache
zondirovaniya nad naklonnym plastom [Numerical implementation of the method of potentials in the problem of probing over an inclined stratum]."Special Issue Proceedings of the Meeting of the Russian-Kazakhstan working group on computational and information technologies 9 (2004): 45-48.
[34] Rakisheva D. S., Mirgalikyzy T., Mukanova B. G. "Аpproksimatsiya poverkhnosti rel’efa dnevnoj poverkhnosti metodom RBF [Approximation of the surface relief of the day surface by the RBF method]."Bulletin of the National Academy of Sciences of the Republic of Kazakhstan 1(365) (2017): 210-215.
[2] Loke M.H., Barker R.D. "Practical techniques for 3D resistivity surveys and data inversion."Geophysical Prospecting 44 (1996): 499-523.
[3] Gunther Т., Rucker С. Boundless Electrical Resistivity Tomography. BERT 2 - the user tutorial( 2013).
[4] Baranchuk K.I., Mirgalikyzy T., Modin I.N., Mukanova B.G. "Fizicheskoe modelirovanie ehlektricheskoj tomografii na poverkhnosti so slozhnym rel’efom [Physical modeling of electrical tomography on the surface with a complex relief]."Engineering Surveys 11 (2017): 56-65.
[5] Veshev A.V. "Elektroprofilirovanie na postoyannom i peremennom toke [Electrical profiling on direct and alternating current]."2 (1980).
[6] Bobachev A.A. "Reshenie pryamykh i obratnykh zadach elektrorazvedki metodom soprotivlenij dlya slozhno-postroennykh sred [The solution of direct and inverse problems of electrical prospecting by the method of resistance for complex-built environments]"( Thesis for the degree of physical and mathematical sciences, MSU, 2003).
[7] Mirgalikyzy T., Mukanova B., Modin I. "Method of Integral Equations for the Problem of Electrical Tomography in a Medium with Ground Surface Relief."Journal of Applied Mathematic(2015).
[8] Balgaisha Mukanova, Tolkyn Mirgalikyzy and Dilyara Rakisheva "Modelling the Influence of Ground Surface Relief on Electric Sounding Curves Using the Integral Equations Method."Mathematical Problems in Engineering (2017).
[9] Zaporozhets V.M. "Vliyanie rel’efa na rezul’taty zamerov soprotivleniya (po rabotam S. G. Komarova i L.P. Gorbenko) [The influence of the relief on the results of resistance measurements]."Elkgr 4 (12)(1938).
[10] Chanturishvili L.S. "O kolichestvennom uchete vliyaniya rel’efa dlya nekotorykh sluchaev razvedki postoyannym tokom [On the quantitative account of the influence of the relief for some cases of direct current intelligence]."Proceedings of the Institute of Geophysics 14(1955): 199-209.
[11] Veshev A.V. "Vliyanie rel’efa na rezul’taty rabot kombinirovannym ehlektroprofilirovaniem [Influence of relief on the results of work by combined electric profiling]."Scientific notes of Leningrad State University 278 (1959).
[12] Loke M.H., Barker R.D. "Rapid least-squares inversion of apparent resistivity pseudo sections using a quasi-Newton method."Geophysical Prospecting 44(1)(1996): 131-152.
[13] Coggon J.H. "Electromagnetic and electrical modeling by the finite element method."Geophysics 36(1) 1971: 132-155.
[14] Mufti I.R. "Finite-difference modeling for arbitrary-shaped two dimensional structures."Geophysics 41(1)(1976): 62-78.
[15] Pelton W.H., Rijo L., Swift C.M. "Inversion of two dimensional resistivity and Induced Polarization data."Geophysics 43(4) (1978): 788-803.
[16] Dey A., Morrison H.F. "Resistivity modeling for arbitrary shaped two-dimensional structures."Geophysical Prospecting 27(1) (1979): 106-136.
[17] Loke M.H., Barker R.D. "Least-squares deconvolution of apparent resistivity pseudosections."Geophysics 60(6) (1995): 1682-1690.
[18] Loke M.H. "Topographic modelling in resistivity imaging inversion."62nd EAGE Conference and Technical Exhibition, Extended Abstracts (2000).
[19] Erdogan E., Demirci I., Candasayar M.E. "Incorporating topography into 2D resistivity modeling using finite-element and finite-difference approaches."Geophysics 73(3) (2008): 135-142.
[20] Demirci I., Erdogan E., Candasayar M.E. "Two-dimensional inversion of direct current resistivity data incorporating topography by using finite difference techniques with triangle cells: Investigation of Kera fault zone in western Crete."Geophysics 77(1) (2012): 67-75.
[21] Penz S., Chauris H., Donno D., Mehl C. "Resistivity modeling with topography."Geophys. J. Int. 194(3) (2013): 1486-1497.
[22] Fox R.C., Hohmann G.W., Killpack T.J., Rijo L. "Topographic effects in resistivity and induced-polarization surveys."Geophysics 45(1) (1980): 75-93.
[23] Tsourlos P.I., Szymanski J.E., Tsokas G.N. "The effect of topography on commonly used resistivity arrays."Geophysics 64(5) (1999): 1357-1363.
[24] Plattner A.D. "Adaptive wavelet methods for geoelectric modeling and inversion"(Dissertation, 2011).
[25] Gunther Т., Rucker С., Spitzer K. "Three-dimensional modelling and inversion of dc resistivity data incorporating topography - I. Modellingn."Geophys. J. Int. 166 (2006): 495-505.
[26] Gunther Т., Rucker С. "Boundless Electrical Resistivity Tomography."BERT 2 - the user tutorial (2013).
[27] Alpin L.M. "Istochniki polya v teorii ehlektricheskoj razvedki [Sources of field in the theory of electrical intelligence]."Applied Geophysics 3 (1947): 56-200.
[28] Dieter K., Paterson N.R and Grant F.S. "KP and resistivity type awes for three-dimensional bodies."Geophysics 34 (1969): 615-632.
[29] Hohmann G.W. "Three dimensional induced polarization and electromagnetic modeling."Geophysics 40 (1975): 309-324.
[30] Eloranta E. "A method for calculation mise-a-la-masse anomalies in the case of high conductivity contrast by the integral equation technique."Geoexploration 22 (1984): 77-88.
[31] Schenkel C.J. "The Electrical Resistivity Method in Cased Boreholes"(Phodissertation, University of California, 1991).
[32] Orunkhanov M., Mukanova B. "The integral equations method in problems of electrical sounding."Advances in High Performance Computing and Computational Sciences (2006): P.15-21.
[33] Orunkhanov M.K., Mukanova B.G., Sarbasova B.K. "CHislennaya realizatsiya metoda potentsialov v zadache
zondirovaniya nad naklonnym plastom [Numerical implementation of the method of potentials in the problem of probing over an inclined stratum]."Special Issue Proceedings of the Meeting of the Russian-Kazakhstan working group on computational and information technologies 9 (2004): 45-48.
[34] Rakisheva D. S., Mirgalikyzy T., Mukanova B. G. "Аpproksimatsiya poverkhnosti rel’efa dnevnoj poverkhnosti metodom RBF [Approximation of the surface relief of the day surface by the RBF method]."Bulletin of the National Academy of Sciences of the Republic of Kazakhstan 1(365) (2017): 210-215.
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Как цитировать
Mirgalikyzy, T., & Mukanova, B. (2019). Численный алгоритм решения задач электрического зондирования среды с рельефом земной поверхности. Вестник КазНУ. Серия математика, механика, информатика, 100(4), 103–116. https://doi.org/10.26577/JMMCS-2018-4-561
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Прикладная математика
