History

The Laboratory of Mathematical Geophysics was established within the Institute in 1961. Its first head, Georgy Mitrofanovich Voskoboynikov, Doctor of Physical and Mathematical Sciences, led the laboratory until 1981. From 1981 to 1990, the laboratory was headed by Alexander Veniaminovich Tsirulsky, Doctor of Physical and Mathematical Sciences. From 1990 to 1992, Faina Iosifovna Nikonova, Candidate of Physical and Mathematical Sciences, served as Acting Head of the Laboratory. Since 1992, the laboratory has been headed by Petr Sergeevich Martyshko, Corresponding Member of the Russian Academy of Sciences and Doctor of Physical and Mathematical Sciences.

The laboratory was tasked with developing the mathematical theory and methods for interpreting geophysical fields. Its first major results were obtained in the study of potential fields. To solve the two-dimensional inverse problem, G. M. Voskoboynikov proposed the method of singular points. A. V. Tsirulsky derived a representation of the complex gravitational field intensity in the form of a Cauchy-type integral over the boundary of a domain and introduced an equation for curves in complex form.

Subsequently, G. M. Voskoboynikov, together with Nikolai Ivanovich Nachapkin, developed a stable algorithm for solving the problem of locating singularities in gravity and magnetic fields. They established a relationship between the shapes of two-dimensional anomaly-producing bodies and the parameters of their nearest singular points. In collaboration with Alexei Fedorovich Shestakov, Voskoboynikov extended the method of singular points to three-dimensional geophysical fields described by the Laplace, Helmholtz, and Lamé differential equations.

A. V. Tsirulsky introduced the concept of the theoretical inverse problem, constructed families of equivalent solutions to the inverse problem for the first time, and developed the theory of two-stage methods for interpreting anomalous gravitational and magnetic fields in two dimensions.

Under his supervision, F. I. Nikonova obtained significant theoretical results in the study of the two-dimensional inverse problem of potential theory for bounded bodies and implemented interpretation algorithms in software. Similar algorithms for two-dimensional structural inverse problems were developed by Natalia Vasilievna Fedorova.

This research team developed theoretical and practical examples of interpreting profile data. Based on the results described above, the SIGMA system for interpreting gravitational and magnetic anomalies was developed. The system enables the rapid construction and analysis of alternative density and magnetic cross-sectional models. It was successfully used to investigate the structure of the Earth's crust along several latitudinal transects across the Urals and adjacent platforms and was adopted by scientific and industrial organizations.

A. V. Tsirulsky, together with Ilya Leonidovich Prutkin, derived new, simpler equations for three-dimensional inverse problems in gravity and magnetic exploration.

P. S. Martyshko and I. L. Prutkin proposed and implemented algorithms for separating potential-field sources according to depth. I. L. Prutkin developed an original method of local corrections, along with algorithms and software for determining the geometry of bounded three-dimensional bodies and interfaces from gravity or magnetic field measurements collected over an area.

The method of local corrections was subsequently substantially modified for solving both three-dimensional structural (nonlinear) inverse problems and linear inverse problems, including density determination and potential-field transformation, by P. S. Martyshko, I. V. Ladovsky, A. G. Tsidaev, and D. D. Byzov.

P. S. Martyshko derived equations for theoretical inverse problems, with explicitly defined operators, for geophysical fields satisfying the Laplace, Helmholtz, diffusion, and telegraph equations. On this basis, examples of solutions to the three-dimensional inverse problem of direct-current electrical prospecting were constructed for the first time. Under his supervision, A. L. Rublev developed software and constructed examples of solutions to the inverse problem for the Helmholtz equation. Similar equations were derived by P. S. Martyshko and M. P. Martyshko for the inverse problem of magnetometry, taking demagnetization into account.

N. V. Fedorova conducts internationally recognized research on the interpretation of magnetic anomalies, with the participation of A. L. Rublev, V. A. Pyankov, and D. Gemaidinov. The results of this research are presented below. She also participates in joint projects with archaeologists supported by the Russian Science Foundation (RSF) and the Russian Foundation for Basic Research (RFBR). I. V. Ladovsky makes a significant contribution to all areas of the laboratory's research.

Since 2006, the laboratory has been joined by graduates of the Department of Computational Methods and Equations of Mathematical Geophysics at Ural Federal University (UrFU), including A. G. Tsidaev, D. V. Gemaidinov, D. D. Byzov, and A. I. Chernoskutov. The department was headed by P. S. Martyshko, Corresponding Member of the Russian Academy of Sciences, from 2002 until the restructuring of the Faculty of Radio Engineering in 2016.

N. V. Fadeeva, a graduate of Ural State University, also participates in the laboratory's research.

The Ural School of Mathematical Geophysics has gained international recognition. In 2014, the laboratory was awarded a Russian Science Foundation grant under a program supporting established laboratories. From 2020 to 2025, it received RSF funding for research groups under the leadership of P. S. Martyshko.

Based on new grid-based algorithms, a method has been developed for interpreting gravitational (and magnetic) anomalies by identifying local heterogeneities. Using anomalies associated with sources separated into horizontal layers, a three-dimensional distribution of density (magnetization) is constructed within the volume under investigation in the form of grid functions.

The construction of density models involves solving forward and inverse problems of gravimetry. Original, highly efficient algorithms have been developed for the fast solution of the forward gravimetric problem on large-scale grids. These algorithms have been successfully applied to implement functional and iterative schemes for solving inverse problems. Solutions are sought within practically meaningful sets of admissible solutions, with an appropriate choice of initial density models. A simple approach has been proposed for extending the mass distribution beyond the lateral boundaries of a density model and selecting the background density (reference density) distribution used to calculate gravitational field anomalies.

Based on the concept of local corrections, iterative algorithms with adaptive regularization have been developed for the stable solution of linear and nonlinear (structural) three-dimensional inverse gravimetric problems. An original technique for layer-by-layer density correction within a heterogeneous parallelepiped has also been proposed.

This technique has been implemented in a computer system for interpreting gravity data and applied to practical problems of seismic-density modeling. All stages of data processing and interpretation are integrated into a single software package. The process begins with the construction of an initial three-dimensional density model based on seismic sections and concludes with density adjustments using the authors' method for solving the gravitational modeling problem.

An additional boundary condition imposing isostatic equilibrium at depth makes it possible to delineate mantle blocks containing compensating masses and optimize the density distribution within them. The boundaries of these mantle blocks were determined from residual (mantle) gravitational anomalies and refined by analyzing variations in lithostatic pressure anomalies at a depth of 80 km.

The performance of the linear inversion algorithm was demonstrated using a particularly challenging synthetic test case. Its computational speed and efficiency were further demonstrated through practical applications involving the construction of three-dimensional geological and geophysical models of the Earth's crust and upper mantle across extensive areas of the Ural region.

Algorithms, including parallel algorithms, and software packages, including GPU-based implementations, have been developed for approximating gravitational and magnetic data using fields generated by singular sources.

To construct a seismic-density model of the subpolar Urals and adjacent regions, the results of seismic data interpretation from deep seismic sounding (DSS) profiles and segments of extended geotraverses were used. The dataset comprised eleven profiles located within the study area, bounded by geographic coordinates 60–68° N and 48–72° E.

The input data consisted of gradient velocity sections represented as grid functions and constructed from seismic travel-time fields. The advantages of using gridded velocity sections for solving inverse gravitational modeling problems have been demonstrated.

Interpretation of the magnetic data showed that the identified regional magnetic anomalies cannot be explained by variations in the topography of the upper and lower boundaries of the crystalline crust.

Modeling of seismic wave velocity and magnetization distributions in the lithosphere along the regional DSS profiles made it possible to divide the Earth's crust into two layers according to their magnetic properties:

• Upper layer: Low magnetization, below 0.3 A/m.

• Lower layer: High average magnetization, ranging from 2 to 4 A/m.

The average depth of the boundary between these layers is approximately 20 km. The topography of this boundary closely corresponds to the 6.5 km/s velocity contour in seismic sections, which marks the boundary of the so-called basaltic layer.

Numerous magnetized blocks of various sizes have been identified in the upper crystalline crust of the Urals and Western Siberia. Along the eastern boundary of the Ural Orogen and the West Siberian Plate, the magnetized sources exhibit a distinct eastward dip. These sources are apparently associated with a deep-seated fault and may be rooted in a large block within the middle crust.

As a result of the integrated interpretation of seismic, gravitational, and magnetic data, three-dimensional models of density and magnetization distributions in the upper lithosphere have been constructed, along with maps showing the block structure of the regional crust at different depths.

The results presented above are aimed at improving methods for constructing geological and geophysical models of the Earth's crust and upper mantle. Their practical significance lies in their application to the investigation of deep geological structures in regions with oil and gas exploration potential.

The research results have been presented at the following international scientific conferences and assemblies:

• General Assemblies of the International Union of Geodesy and Geophysics (IUGG): 1999, 2007, 2011, 2015.

• International Association of Geomagnetism and Aeronomy (IAGA): 2009, 2013.

• International Association for Mathematical Geosciences (IAMG): 2015.

• Conferences on Mathematical Geophysics (CMG): 2012, 2018.

• American Geophysical Union (AGU) Fall Meetings: 2010, 2021.

• European Association of Geoscientists and Engineers (EAGE): 1997, 1999, 2001, 2004, 2010.

• International Multidisciplinary Scientific GeoConference (SGEM): 2014, 2015, 2016, 2017, 2018.

• International Conference of Numerical Analysis and Applied Mathematics (ICNAAM): 2014–2025.

• Asia Oceania Geosciences Society (AOGS): 2011, 2019, 2020.

• Geoinformatics conferences: 2011–2021.

• Electromagnetic Induction (EMI) Workshops: 1996, 1998, 2000, 2006.

• International Symposia on Three-Dimensional Electromagnetics: 1995, 1999, 2007.

• Numerical Computations: Theory and Algorithms (NUMTA): 2026.

The research results are also presented annually at the Uspensky–Strakhov Seminar, Problems of the Theory and Practice of Interpretation, as well as at other scientific conferences held in Russia.