Linearization method in two-dimensional inverse problem of magnetotelluric sounding

An algorithm is described for the two-dimensional inverse problem of magnetotelluric sounding with E-polarization, based on minimization of the Tikhonov smoothing functional by the Newton—Kantorovich method (linearization method). The method of integral equations is used in the algorithm to solve the direct problem of magnetotelluric sounding. An iterative method is constructed for the solution of the integral equation in the direct problem and a rate of convergence bound is derived for this method.Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 199–215, 1986.     

Quasi-one-dimensional method for the two-dimensional magnetotelluric sounding inverse problem

The two-dimensional magnetotelluric sounding inverse problem is solved by a quasi-one-dimensional method. A linearization method is proposed for one-dimensional inverse problems. The appearance of false conducting structures in the solution of the inverse problem is considered and some counter-measures are suggested. The quasi-one-dimensional method is applicable to many-dimensional magnetotelluric sounding inverse problems if the electrical conductivity is a priori known to vary slowly along the Earth’s surface. __________ Translated from Prikladnaya Matematika i Informatika, No. 22, pp. 5–11, 2005.     

A solution method for the inverse problem of magnetotelluric sounding

We consider the inverse problem of magnetotelluric sounding for integral conductivity. The problem is proved to be stable to changes in input data.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 165–174, 1985.     

Quasi-one-dimensional method for the two-dimensional inverse problem of magnetotelluric sounding

The article presents a quasi-one-dimensional method for solving the inverse problem of electromagnetic sounding. The quasi-one-dimensional method is an iteration process that in each iteration solves a parametric one-dimensional inverse problem and a two-dimensional direct problem. The solution results of these problems are applied to update the input values for the parametric one-dimensional inverse problem in the next iteration. The method has been implemented for a two-dimensional inverse problem of magnetotelluric sounding in a quasi-layered medium.     

An approach to the solution of two-dimensional inverse problems of magnetotelluric sounding

Translated from Metody Matematicheskogo Modelirovaniya i Vychislitel'noi Diagnostika, pp. 45–50, Izd. Moskovskogo Universiteta, Moscow, 1990.     

Integral characteristics in the inverse problem of magnetotelluric sounding

This article examines some aspects of the one-dimensional inverse problem of magnetotelluric sounding. A uniqueness theorem is proved in the presence ofS-surfaces. A numerical algorithm based on transformation formulas is proposed. This research was partially supported by Russian Foundation for Basic Research (grant No. 96-01-00410) and by the State Scientific-Technical Program “Future Information Technologies” (grant No. 0201.06.010). Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 53–66, 1998.     

Solution of the inverse magnetotelluric sounding problem by means of a method using the synthesized magnetic field

A method of solution of the inverse magnetotelluric sounding problem is considered. The method uses the known frequency-dependent magnetic field on the Earth’s surface. The magnetic field on the Earth’s surface is found with the help of a special technique from the known impedance on the Earth’s surface. The method is applied to solve typical inverse problems and provides for efficient determination of final solutions.     

Unique solvability of the inverse two-dimensional problem of magnetotelluric sounding forH-polarization

Translated from Matematicheskie Modeli i Optimizatsiya Vychislitel'nykh Algoritmov, pp. 86–94, 1994.     

The inverse problem of magnetotelluric sounding of the Earth's interior in the presence of fractures

The paper considers the inverse problem of magnetotelluric sounding for the case of H-polarization in the presence of tectonic fractures in the Earth's interior.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach Matematicheskoi Fiziki, pp. 5–12, 1993.     

Mathematical models of marine magnetotelluric sounding

A mathematical model of magnetotelluric sounding (MTS) is constructed for a layered medium with a nonhomogeneous inclusion. The model is intended for marine studies. The problem is solved by the integral electric current method. A modified integral current method is proposed capable of generating a solution with a small number of partitions of the nohomogeneity region. The efficiency of the proposed method is demonstrated for H-polarized field. The behavior of apparent resistivity and magnetic parameter curves is investigated for marine MTS. Translated from Prikladnaya Matematika i Informatika, No. 29, pp. 19–28, 2008.     

Inverse problem of magnetotelluric sounding in a medium with faults

This article considers a method for solving the direct and the inverse problems in the presence ofH-polarization for the fundamental three-layer model of deep magnetotelluric sounding with vertical tectonic faults and horizontal high-conductivityS-layers in a poorly conducting crystalline matrix. The conductivities may vanish on parts of the vertical cracks or the horizontalS-layers, i.e., in this modelS-layers and vertical faults may degenerate into an insulator. This research was partially supported by the Russian Foundation for Basic Research (grant No. 96-95-64340) and by the State Scientific-Technical Program “Future Information Technologies” (grant No. 0201.06.010). Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 53–66, 1998.     

Three-dimensional models of marine magnetotelluric sounding of nonhomogeneous media

We consider the direct problem of three-dimensional modeling of marine magnetotelluric sounding of nonhomogeneous media in the presence of a vertical magnetic field component in the primary field. The problem is reduced to a system of volume integral equations. __________ Translated from Prikladnaya Matematika i Informatika, No. 27, pp. 46–53, 2007.     

Depth sounding: an illustration of some of the pitfalls of inverse scattering problems

The principal objective of this work is to show how various “connections” between the estimator and the predictor affect the solution of an inverse scattering problem as it is formulated in the frequency domain. We show that when there is little or no connection, it is impossible to obtain a solution. The other extreme, i.e., identity of the estimator and predictor (inverse crime [1]), enables solutions to be obtained, whatever the particular choices of the estimator or predictor, but these solutions are not trivial, as is written in [1], in that they are not unique. Moreover, we show that by a suitable change of external variables (e.g., frequency), one can lift the degeneracy and thereby spot the correct solution, which is unique. In this respect, the inverse crime turns out to be useful in that it enables one to devise methods for resolving the nonuniqueness issue of inverse problems. More generally, we show that successful inversion, in both the frequency and time domains, can be accomplished only when the discrepancy between the estimator and the predictor is small.     

Spectral method for the inverse vertical electrical sounding problem in two-dimensional quasi-layered media

The spectral method is considered for solving the direct and inverse problems of vertical electrical sounding (VES) in two-dimensional quasi-layered media. The behavior of the anomalous potential spectrum is analyzed. An example is presented applying the spectral method to solve the inverse VES problem. Translated from Prikladnaya Matematika i Informatika, No. 2, pp. 34–43, 1999.