# Numerical algorithm for solving of electrical prospecting problems for medium with a surface relief

### Abstract

Nowadays the problem of the influence of the ground surface relief on the distribution of the electric field is a pressing issue in the interpretation of electromagnetic fields studied in the direct current electrical prospecting. The paper deals with testing the algorithm for the numerical solution of the problem of electrical sensing of a medium with the ground surface relief by means of modeling using integral equations. The idea of the method of integral equations is to represent the electric field as the sum of the primary field and the field of the secondary charges. The contact boundaries and the surface of the geoelectric section act as a secondary creators of the electric field. The problem of calculating the fields is reduced to the systems of integral equations on the density of secondary sources induced on the contact surfaces of conducting media and on the relief surface of the medium. A mathematical description of this phenomenon leads to Fredholm equations of the second kind with a polar core. The calculation algorithm was tested by comparing the results with the solutions given in the open access works on taking into account the influence of the relief, setting the same environmental parameters. Comparisons are made using the finite-element and finite-difference methods obtained by different approaches to take into account the effect of relief in 2D environments. Comparisons are also made with the physical modeling data obtained. We can say that our results of calculations curves of apparent resistivity are in good agreement with the available research in this area.### References

[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.

How to Cite

MIRGALIKYZY, Tolkyn; MUKANOVA, Balgaisha.
Numerical algorithm for solving of electrical prospecting problems for medium with a surface relief.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 100, n. 4, p. 103-116, jan. 2019. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/561>. Date accessed: 20 jan. 2021. doi: https://doi.org/10.26577/JMMCS-2018-4-561.
Section

Applied Mathematics

Keywords
method of integral equations, numerical algorithm, direct problem of electrical prospecting, surface relief