Igor Kolotov

The paper deals with the possibility of finding solutions to direct and inverse gravimetry problems by means of the apparatus of eigenfunctions of the Laplace operator. Data acquired through remote sensing of the Earth’s physical fields are of a discrete character (often in the form of bit chains). The discrete gravitational potential theory allows restoring, with a relatively high degree of reliability, grid mass distributions that are equivalent with respect to an external field complicated by random noise, as well as solving various direct and inverse gravimetry problems. Construction of eigenfunctions of a discrete analogue of the Laplace operator in three-dimensional domains with complex structure eliminates the need for multiple solutions of ill-conditioned systems of linear algebraic equations (SLAE).

The MHD (magnetic hydrodynamics) boundary problem in three-dimensional domains of certain types is considered within the framework of discrete potential theory. The discrete character of the information obtained from remote sensing of the Earth and planets of the Solar System can be taken into account when using the basic principles of this theory. This approach makes it possible to reconstruct the spatial distribution of magnetic fields and the velocity field with relatively high accuracy using the heterogeneous data in some network points. In order to restore the magnetic image of a planet with a so-called dynamo, the subsequent approximations approach is implemented. The unknown physical field is represented as a sum of terms of different magnitudes. Such an approach allows us to simplify the nonlinear partial differential equation system of magnetic hydrodynamics and extend it to discrete magnetic field and velocity vectors. The solution of the simplified MHD equation system is constructed for some classes of bounded domains in Cartesian coordinates in three-dimensional space.

Variational-combinatoric approach is used when solving linear and nonlinear inverse problems of geophysics within the framework of discrete potential. The acquired data about physical fields and their sources are of discrete character. Combinatoric methods of discrete mathematics allow us to organize some selective choice of the information on the physical field elements. Analytical approximations are built only by means of verified data which can be recognized as useful signal, not as an arbitrary or systematic noise.

This paper considers the problem of the uniqueness of the solution to the coefficient inverse problem for the system of equations of magneto-hydrodynamics, the solution to which allows more accurately describing the processes of generating the magnetic field of planets with a magneto-hydrodynamic dynamo. The conditions under which it is possible to determine three components of the magnetic induction vector and the magnetic field diffusion coefficient are determined.

This article deals with the question of the unique solvability of systems of linear algebraic equations, to the solution of which many inverse problems of geophysics are reduced as a result of discretization when applying the methods of integral equations or integral representations. Examples are given of degenerate and nondegenerate systems of different dimensions that arise in the processing of magnetometric and gravimetric data from experimental observations. Conclusions are drawn about the methods for constructing the optimal grid of experimental observation points.

A new technique for reconstructing the magnetic image of Mercury from satellite observation data is proposed. The technique is based on extracting from the general observational data only the part that is associated with magnetic field sources concentrated in Mercury’s crust. The extracted data are used to solve the inverse problem of reconstructing the near-surface magnetic image of Mercury. The results of processing real experimental data obtained via the MESSENGER interplanetary mission are presented.

One of the possible approaches to reconstructing the map of the distribution of magnetization parameters in the crust of Mars from the data of the Mars MAVEN orbiter mission is considered. Possible ways of increasing the accuracy of reconstruction of the magnetic image of Mars are discussed.