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.     

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.     

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.     

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.     

Uniqueness of the solution of the inverse problem of magnetotelluric sounding for two-dimensional media

Uniqueness of the solution of the inverse problem of magnetotelluric sounding in two-dimensional media is proved under certain assumptions on conductivity.Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 231–242, 1986.     

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.     

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.     

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.     

Synthesized field method in inverse problems of magnetotelluric sounding

A new method is proposed and investigated for solving the inverse three-dimensional problem of magnetotelluric sounding. The method uses the tangential electric field on the Earth's surface, which is synthesized from the known impedance.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 115–121.     

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.     

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.     

A reconstruction method for a two-dimensional inverse eigenvalue problem

In this paper we describe a method for constructing approximate solutions of a two-dimensional inverse eigenvalue problem. Here we consider the problem of recovering a functionq(x, y) from the eigenvalues of — +q(x, y) on a rectangle with Dirichlet boundary conditions. The potentialq(x, y) is assumed to be symmetric with respect to the midlines of the rectangle. Our method is a generalization of an algorithm Hald presented for the construction of symmetric potentials in the one-dimensional inverse Sturm-Liouville problem. Using a projection method, the inverse spectral problem is reduced to an inverse eigenvalue problem for a matrix. We show that if the given eigenvalues are small perturbations of simple eigenvalues ofq=0, then the matrix problem has a solution. This solution is used to construct a functionq which has the same lowest eigenvalues as the unknownq, and several numerical examples are given to illustrate the methods.     

An inverse problem of atmospheric radar sounding

The paper considers the problem of determining the velocities of targets in the atmosphere given a set of Doppler spectra produced by continuous Doppler systems. The problem of determining the velocities of particles moving in a closed trajectory (a circle) is examined. A uniqueness theorem is proved for the reconstruction of velocities from two spectra. An algorithm is proposed for determining the sought function, and a numerical experiment is conducted to establish its efficiency.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 162–167.     

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.     

A two-dimensional inverse problem for the viscoelasticity equation

For the integrodifferential equation that corresponds to the two-dimensional viscoelasticity problem, we study the problem of determining the density, the elasticity coefficient, and the spaceintegral term in the equation. We assume that the sought functions differ from the given constants only inside the unit disk D = {x ∈ ?² | |x| < 1}. As information for solving this inverse problem, we consider the one-parameter family of solutions to the integrodifferential equation corresponding to impulse sources localized on straight lines and, on the boundary of D, there are defined the traces of the solutions for some finite time interval. It is shown that the use of a comparatively small part of the given information about the kinematics and the elements of dynamics of the propagating waves makes it possible to reduce the problem under consideration to three consecutively and uniquely solvable inverse problems that together give a solution to the initial inverse problem.