## Journal of Applied Mathematics

### Method of Integral Equations for the Problem of Electrical Tomography in a Medium with Ground Surface Relief

#### Abstract

The direct task of the subsurface exploration of a homogeneous medium with surface relief by the resistivity method is analyzed. To calculate the resistivity field for such a medium, the method of integral equations was successfully applied for the first time. The corresponding integral equation for the density of secondary current sources on the surface of the medium was established. The method of computational grid construction, adapted to the characteristics of the surface relief, was developed for the numerical solution of the integral equation. This method enables the calculation of the resistivity field of a point source on a surface that is not smooth and allows for steep ledges. Numerical examples of the calculation of resistivity fields and apparent resistivity are shown. The anomalies of apparent resistivity arising from the deviation of the surface shape from a flat medium were quantitatively established as model examples. Calculations of apparent resistivity for the direct current sounding method were carried out using modifications of the electrical tomography approach.

#### Article information

Source
J. Appl. Math., Volume 2015 (2015), Article ID 207021, 10 pages.

Dates
First available in Project Euclid: 17 August 2015

https://projecteuclid.org/euclid.jam/1439816409

Digital Object Identifier
doi:10.1155/2015/207021

#### Citation

Mirgalikyzy, Tolkyn; Mukanova, Balgaisha; Modin, Igor. Method of Integral Equations for the Problem of Electrical Tomography in a Medium with Ground Surface Relief. J. Appl. Math. 2015 (2015), Article ID 207021, 10 pages. doi:10.1155/2015/207021. https://projecteuclid.org/euclid.jam/1439816409

#### References

• J. H. Coggon, “Electromagnetic and electrical modeling by the finite element method,” Geophysics, vol. 36, no. 1, pp. 132–155, 1971.
• I. R. Mufti, “Finite-difference modeling for arbitrary-shaped two dimensional structures,” Geophysics, vol. 41, no. 1, pp. 62–78, 1976.
• W. H. Pelton, L. Rijo, and C. M. Swift Jr., “Inversion of two dimensional resistivity and Induced Polarization data,” Geophysics, vol. 43, no. 4, pp. 788–803, 1978.
• A. Dey and H. F. Morrison, “Resistivity modelling for arbitrarily shaped two-dimensional structures,” Geophysical Prospecting, vol. 27, no. 1, pp. 106–136, 1979.
• M. H. Loke and R. D. Barker, “Least-squares deconvolution of apparent resistivity pseudosections,” Geophysics, vol. 60, no. 6, pp. 1682–1690, 1995.
• M. H. Loke and R. D. Barker, “Rapid least-squares inversion of apparent resistivity pseudosections by a quasi-Newton method,” Geophysical Prospecting, vol. 44, no. 1, pp. 131–152, 1996.
• M. H. Loke, “Topographic modelling in resistivity imaging inversion,” in Proceedings of the 62nd EAGE Conference and Technical Exhibition, Glasgow, UK, May-June 2000.
• E. Erdogan, I. Demirci, and M. E. Candansayar, “Incorporating topography into 2D resistivity modeling using finite-element and finite-difference approaches,” Geophysics, vol. 73, no. 3, pp. F135–F142, 2008.
• I. Demirci, E. Erdogan, and M. E. Candansayar, “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, vol. 77, no. 1, pp. E67–E75, 2012.
• Ś. Penz, H. Chauris, D. Donno, and C. Mehl, “Resistivity modelling with topography,” Geophysical Journal International, vol. 194, no. 3, pp. 1486–1497, 2013.
• R. C. Fox, G. W. Hohman, T. J. Killpack, and L. Rijo, “Topographic effects in resistivity and induced-polarization surveys,” Geophysics, vol. 45, no. 1, pp. 75–93, 1980.
• P. Queralt, J. Pous, and A. Marcuello, “2D resistivity modeling: an approach to arrays parallel to the strike direction,” Geophysics, vol. 56, no. 7, pp. 941–950, 1991.
• P. I. Tsourlos, J. E. Szymanski, and G. N. Tsokas, “The effect of terrain topography on commonly used resistivity arrays,” Geophysics, vol. 64, no. 5, pp. 1357–1363, 1999.
• C. Rücker, T. Günther, and K. Spitzer, “Three-dimensional modelling and inversion of dc resistivity data incorporating topography–-I. Modelling,” Geophysical Journal International, vol. 166, no. 2, pp. 495–505, 2006.
• ${\CYRT}$. Gunther and ${\CYRS}$. Rucker, “Boundless electrical resistivity tomography,” in BERT 2–-The User Tutorial, Version 2.0, 2013.
• M. K. Orunkhanov, B. G. Mukanova, and B. K. Sarbasova, “Numerical modelling of electrical sounding problem,” Computational Technologies, vol. 9, part 3, pp. 259–263, 2004 (Russian), Proceedings of the International Conference on “Computational and Informational Technologies in Science, Engineering and Education”, Novosibirsk, Russia, October 2004.
• L. M. Alpin, “Field source in the theory of electrical prospecting,” Applied Geophysics, vol. 3, pp. 56–200, 1947 (Russian).
• K. Dieter, N. R. Paterson, and F. S. Grant, “KP and resistivity type awes for three-dimensional bodies,” Geophysics, vol. 34, no. 4, pp. 615–632, 1969.
• G. W. Hohmann, “Three dimensional induced polarization and electromagnetic modeling,” Geophysics, vol. 40, no. 2, pp. 309–324, 1975.
• M. Hvoždara and V. Petr, “Electric and magnetic field of a stationary current in a stratified medium with a three-dimensional conductivity inhomogeneity,” Studia Geophysica et Geodaetica, vol. 27, no. 1, pp. 59–84, 1983.
• M. Hvoždara and V. Petr, “Potential field of a stationary electric current in a stratified medium with a three-dimensional perturbing body,” Studia Geophysica et Geodaetica, vol. 26, no. 2, pp. 160–172, 1982.
• M. K. Orunkhanov, B. G. Mukanova, and B. K. Sarbasova, “Convergence of an integral equation on geoelectric sounding problem above a local patch,” Computational Technologies, vol. 9, no. 6, pp. 68–72, 2004 (Russian).
• M. Orunkhanov and B. Mukanova, “The integral equations method in problems of electrical sounding,” in Advances in High Performance Computing and Computational Sciences, vol. 93 of Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), pp. 15–21, Springer, Berlin, Germany, 2006.
• M. K. Orunkhanov, B. G. Mukanova, and B. K. Sarbasova, “Convergence of the method of integral equations for quasi three-dimensional problem of electrical sounding,” in Computational Science and High Performance Computing II, vol. 91 of Notes on Numerical Fluid Mechanics and Multidisciplinary Design, pp. 175–180, Springer, Berlin, Germany, 2006.
• S. Yilmaz and N. Coşkun, “A study of the terrain-correction technique for the inhomogeneous case of resistivity surveys,” Scientific Research and Essays, vol. 6, no. 24, pp. 5213–5223, 2011.
• M. K. Orunkhanov, B. G. Mukanova, and B. K. Sarbasova, “Numerical implementation of method of potentials for sounding above an inclined plane,” Computational Technologies, vol. 9, pp. 45–48, 2004 (Russian). \endinput