Inna Eduardovna Stepanova

The paper proposes a neural network approach to the joint three-dimensional inversion of gravitational and magnetic data aimed at the recovery of a single ore body. The method is based on a convolutional U-Net architecture trained on a large synthetic dataset generated by stochastic combinations of rectangular prisms simulating typical geological structures. Various loss functions are considered and compared: mean squared error (MSE), mean absolute error (MAE), Huber loss, and a loss with a forward-problem physics residual. It is shown that, in the conducted synthetic experiments, this loss provides the lowest error among the considered variants and improves the physical consistency of the recovered density and magnetization models, while data augmentation by rotations improves the reconstruction quality. The results of neural network recovery are also compared with the classical solution of the inverse problem within the framework of discrete potential theory. In the considered test example, the neural network approach achieved an error of 4.23% for the loss with a physics residual with a processing time of 0.2 s on the fixed 21³ grid. В работе предложен нейросетевой подход к совместной трехмерной инверсии гравитационных и магнитных данных, ориентированный на восстановление единого рудного тела. Метод основан на сверточной архитектуре U-Net, обучаемой на большом синтетическом датасете, сгенерированном с помощью стохастической комбинации прямоугольных призм, имитирующих типичные геологические структуры. Рассмотрены и сравнены различные функции потерь: среднеквадратичная ошибка (MSE), средняя абсолютная ошибка (MAE), функция Хьюбера и функция потерь с физической невязкой прямой задачи. Показано, что в проведенных синтетических экспериментах применение этой функции потерь обеспечивает наименьшую ошибку среди рассмотренных вариантов и повышает физическую согласованность восстановленныхnмоделей плотности и намагниченности, а аугментация данных поворотами улучшает качество реконструкции. Также сравнены результаты восстановления с помощью нейронных сетей с классическим решением обратной задачи в рамках теории дискретных потенциалов. В рассмотренном тестовом примере нейросетевой подход позволил добиться ошибки 4.23% для функции потерь с физической невязкой при времени обработки 0.2 с на фиксированной сетке 21³.

Various versions (local and regional, as well as stationary and non-stationary) of the method of linear integral representations in solving inverse problems of geophysics are considered. Particular attention is paid to the issues of unique solvability of variational problems that arise in the interpretation of geophysical data. This paper emphasizes the application of the approximation approach in constructing analytical models of any physical fields, and highlights a class of problems that can be considered conditionally solvable in closed form. This article proves theorems on the linear independence of function systems, a finite linear combination of which represents the solution to the inverse problem. Theorems of this kind are of fundamental importance both for investigating the well-posedness of the inverse problem formulation and for practical application, as choosing an observation network with prescribed properties will reduce the costs of conducting geological and other surveys.

The paper considers initial-boundary value problems for a system of MHD equations in the boundary layer of a planet’s magnetosphere within the framework of the discrete field theory. An approach is used based on representing the solution to the original problem for a system of nonlinear vector partial differential equations as a sum of vector functions satisfying certain predefined conditions, thereby obtaining, in a sense, “self-similar” solutions. A finite-difference Dirichlet boundary value problem for the magnetic field is formulated, and a matrix sweep method is used to find its solution. For the velocity field of a charged fluid (plasma), two interconnected problems are formulated, and the “final” velocity field is obtained only if certain conditions concerning the asymptotic properties of the sought-for vector functions are satisfied.

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).

Two approaches to solving inverse problems of potential fields using the measurements of the field components and their gradients are considered. Both approaches find the sources of the fields on some two-dimensional varieties. This reduces inverse problems to a two-dimensional formulation, ensuring uniqueness of the solution in a number of cases and accelerating computational procedures. By solving such problems for manifolds located at varying depths, valuable input for geophysical interpretation can be obtained. The proposed methods are suitable for both separate and joint inversion of gravity and magnetic fields. Both approaches guarantee the uniqueness of the obtained solutions and their stability with respect to data perturbations. The first approach to the inverse problems is based on the variational scheme of S-approximation and is implemented using the method of Lagrange multipliers. It allows for localization of sources on several manifolds of different nature. Using field gradient data, this approach can recover not only the sources but also the complex spectra of the fields themselves on one or multiple manifolds. The method is applicable to the processing of local geophysical data as well as for regional and global-scale analyses. The second approach is based on replacing the original inverse problem matrix (which may be ill-posed or even unique) with a nearby matrix, close in norm, that possesses a better condition number. This significantly improves the stability of the resulting approximate solutions, often by several factors. An algorithm, termed the MPMI algorithm (Minimum pseudoinverse matrix with improved condition number), is proposed for the numerical implementation of this approach. The methods for solving inverse problems of potential fields presented in this paper are applied to the processing of real, heterogeneous geophysical data.

A new version of the linear integral representation method is developed for solving inverse problems in geophysics. This approach is applied to the interpretation of anomalous time-dependent field data. The reconstruction of field elements is reduced to solving a system of linear algebraic equations (SLAE) with an approximately given right-hand side. Since the matrix elements of this system are derived analytically, the modeling process is significantly simplified. The article also analyzes how the approximation quality of a non-stationary field element depends on the observation network geometry, enabling its optimization for more accurate detection of geological properties. The proposed method for solving inverse problems for hyperbolic partial differential equations with constant coefficients can also be applied to data described by systems of nonlinear PDEs, provided the target field is represented as a composition of components differing in magnitude. Finally, the results of non-stationary gravity field modeling are presented.

The eigenvalue problem for the finite-differenced analogues of the Laplace operator in spherical coordinates is considered. Finding eigenvalues and eigenfunctions for the finite-differenced boundary settings is a useful tool when evaluating the conditions for the implementation of the so-called matrix sweep method. This method makes it possible to determine potentials in two cases: (a) when the discrete fundamental solution is known, and (b) if an additional a priori information on the boundary values of potentials is given.

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.

The paper considers the problem of uniqueness of the solution to the inverse problem of determining the viscosity and magnetic viscosity coefficients from a system of partial differential equations describing MHD phenomena. Uniqueness theorems are given in the case of known magnetic field and velocity field of a charged fluid in the zero approximation, as well as in the case of a known total magnetic field.