The S-approximation method proposed by Stepanova [Inv. Probl. Sci. Eng. 16 (5) (2008), pp. 547–566] is applied for solving interpretation problems when the sphericity of the Earth is taken into account. A new method of finding stable approximate solutions of linear algebraic equation systems is described.

Enhancing the reliability of inversion results has always been a prominent issue in the field of geophysics. In recent years, data-driven inversion methods leveraging deep neural networks (DNNs) have gained prominence for their ability to address non-uniqueness issues and reduce computational costs compared to traditional physically model-driven methods. In this study, we propose a GMNet machine learning method, i.e., a CNN-based inversion method for gravity and magnetic field data. This method relies more on data-driven training, and in the prediction phase after the model is trained, it does not heavily depend on a priori assumptions, unlike traditional methods. By forward modeling gravity and magnetic fields, we obtain a substantial dataset to train the CNN model, enabling the direct mapping from field data to subsurface property distribution. Applying this method to synthetic data and one-field data yields promising inversion results.

A model distribution of anomalous masses in the southwestern part of Elysium Planitia under the surface of the site of the InSight mission seismometer has been obtained. Numerical simulations are performed with the S-approximation method by expansions of the data on the Martian topography and gravitational field (the MRO120D model) in spherical harmonics to the 90th degree and order. Inhomogeneity of the density structure in the form of zones, containing blocks of higher and lower density in the subsurface layers of the study area, has been revealed.

Ways to solve the inverse linear problems of gravimetry and constructing linear transformations of the anomalous physical fields based on the modified method of S-approximations of the initial field element are discussed. The results of the mathematical experiment are presented.

A modified method of S-approximations of the initial field element is described for the regional version of the problem when the sphericity of the Earth should be taken into account. The application of the method is illustrated by the results of the mathematical experiment.

This review describes the specifics of the application of the approximation approach in solving the linear and nonlinear inverse problems of geophysics, geodesy, and geomorphology. Within the...