Abstract
The Trefftz method is applied to the numerical solution of the three-dimensional (3D) forward problem for an electromagnetic field harmonically oscillating with time in a 3D environment, which ensures the possibility to solve a 3D inverse problem. Here, the known idea of simultaneous joint interpretation of the tangential components of electric and magnetic fields measured on the Earth's surface, which was suggested by A.N. Tikhonov for the development of the electromagnetic sounding method, is employed in the context of the numerical-analytical representation of the forward problem solution in accordance with the Trefftz method. Application of this method allows enables one to generalize the basic plane-stratified vertically 1D model of the medium by allowing for lateral variations in electric conductivity.
Similar content being viewed by others
References
Bazaraa, M.S. and Shetty, C.M., Nonlinear Programming: Theory and Algorithms, New York: Wiley, 1979; Moscow: Mir, 1982.
Berdichevskii, M.N. and Dmitriev, V.I., Magnitotelluricheskoe zondirovanie gorizontal'no odnorodnykh sred (Magnetotelluric Sounding of Laterally Uniform Media), Moscow: Nedra, 1991.
Berdichevsky, M.N., Marginal Notes on Magnetotellurics, Surv. Geophys., 1999, vol. 20, pp. 341–375.
Berdichevsky, M.N. and Dmitriev, V.I., Modeli i metody magnitotelluriki (Models and Methods of Magnetotellurics), Moscow: Nauchnyi Mir, 2009; Berlin: Springer-Verlag, 2008.
Dmitriev, V.I., Inverse Problems of Electromagnetic Sounding, Izv. Fiz. Zemli, 1977, no. 1, pp. 19–23.
Hahn, W.C., A New Method for Calculation of Cavity Resonators, J. Appl. Phys., 1941, vol. 12, no. 1, pp. 62–68.
Ikramov, Kh.D., Nesimmetrichnaya problema sobstvennykh znachenii (Non-Symmetric Eigenvalue Problem), Moscow: Nauka, 1991.
Il'in, V.P., On the Iterative Kaczmarz Method and Its Generalizations, Sib. Zh. Industr. Mat., 2006, vol. 9, no. 3, pp. 39–49.
Il'in, V.P., Biconjugate Direction Methods in Krylov Subspaces, Sib. Zh. Industr. Mat., 2008, vol. 11, no. 4, pp. 47–60.
Kaczmarz, S., Angenaherte Auflosung von Systemen linearer Gleichunger, Bulletin International de l'Academie Polonaise des Sciences, 1937, Lett. A35, pp. 355–357.
Konovalov, A.N., Vvedenie v vychislitel'nye metoy lineinoi algebry (Introduction to Computational Methods of Linear Algebra), Novosibirsk: VO Nauka, 1993.
Mitsuhata, Y. and Uchida, T., 3D Magnetotelluric Modeling using the T-Ω Finite-Element Method, Geophysics, 2004, vol. 69, no. 1, pp. 108–119.
Nikol'skii, V.V. and Lavrova, T.A., The Method of Minimum Autonomous Blocks and its Application to Waveguide Diffraction Problems, Radio Eng. Electron. Phys., 1978, vol. 23, no. 2, pp. 1–10.
Nikol'skii, V.V. and Nikol'skaya, T.I., Dekompozitcionnyi podkhod v zadachakh elektrodinamiki (Decomposition Approach in the Problems of Electrodynamics), Moscow: Nauka, 1983.
Ortega, J., Vvedenie v parallel'nye i vektornye metody resheniya lineinykh sistem (Introduction to Parallel and Vector Solution of Linear Systems), New York: Plenum, 1988; Moscow: Mir, 1991.
Sobolev, V.A., Solving Incompatible Systems Sorosovskii Obrazovatel'nyi Zhurnal, 2000, vol. 6, no. 4, pp. 116–119.
Strohmer, T. and Vershynin, R., A Randomized Kaczmarz Algorithm with Exponential Convergence, Journal of Fourier Analysis and Applications, 2009, vol. 15, no. 1, pp. 262–278.
Tanabe, K., Projection Method for Solving a Singular System of Linear Equations and its Applications, Numer. Math., 1971, vol. 17, pp. 203–214.
Tikhonov, A.N., On the Determination of Electric Characteristics of the Deep Crustal Layers, Dokl. Akad. Nauk SSSR, 1950, vol. 73, pp. 295–297.
Tikhonov, A.N. and Arsenin, V.Ya., Metody resheniya nekorrektnykh zadach (Methods of Solution of Ill-posed Problems), Moscow: Nauka, 1979.
Trefftz, E., Ein Gegenstuck Zum Ritzchen Verfahren, Intern. Kongress Technische Mechanik. Verhandl. d. 2, Zurich, 1926, pp. 131–137.
Vaganov, R.B. and Katsenelenbaum, B.Z., Osnovy teorii difraktsii (Principles of the Diffraction Theory), Moscow: Nauka, 1982.
Yegorov, I.V., Solution of the Three-Dimensional Problems of Geoelectrics based on the Trefftz Method, in II Vserossiiskaya Shkola-Seminar Po Elektromagnitnym Zondirovaniyam Zemli. Tezisy dokl. (Abstracts of Papers. II Russian School-Seminar on Electromagnetic Sounding of the Earth), Moscow: MAKS Press, 2005, pp. 62–63.
Yegorov, I.V., 3-D Numerical Modeling of an Electromagnetic Field in Geoelectrics Using the Trefftz Method, Fiz. Zemli, 2009a no. 9, pp. 86–96 [Izv. Phys. Earth (Engl. Transl.), 2009a vol. 45, no. 9, pp. 812–821].
Yegorov, I.V., Trefftz Method for the Solution of Three-Dimensional Forward and Inverse Problems of Geoelectrics, Elektromagnitnye issledovaniya Zemli. Tezisy dokl. IV Vserossiiskaya shkola-seminar po elektromagnitnym zondirovaniyam Zemli (Electromagnetic Studies of the Earth. Abstracts of Papers. IV Russian School-Seminar on Electromagnetic Sounding of the Earth), Moscow: RAN, IFZ, TsGEMI, 2009b p. 95.
Author information
Authors and Affiliations
Additional information
Original Russian Text © I.V. Yegorov, 2011, published in Fizika Zemli, 2011, No. 2, pp. 15–26.
Rights and permissions
About this article
Cite this article
Yegorov, I.V. Trefftz method for the solution of three-dimensional forward and inverse problems of geoelectrics. Izv., Phys. Solid Earth 47, 90–100 (2011). https://doi.org/10.1134/S1069351311010034
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1069351311010034