Mathematical Modeling in Geophysics: Concepts and Practices

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: 31 March 2025 | Viewed by 6220

Special Issue Editor


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Guest Editor
Institute of Geophysics, Polish Academy of Sciences, 01-452 Warszawa, Poland
Interests: mathematical modeling of earthquakes and other geophysical phenomena; complex systems in geosciences; deterministic and stochastic cellular automata; discrete integrable systems

Special Issue Information

Dear Colleagues,

Modern geophysical research is increasingly dependent on mathematical modeling. Measuring devices located in research stations scattered all over the Earth, on satellites, and also in laboratories provide an incredibly large amount of data on geophysical processes, crucial for discovering the history of the Earth and predicting natural hazards in our future. These data require specialized methods of interpretation as well as a conceptual understanding. In both of these fields, geophysical research is critically based on modern mathematical methods and concepts.

Mathematical modeling in geophysics uses tools from a wide variety of mathematical findings from statistics, theory of dynamical systems, differential equations, graph theory, topology, and others to attack problems of time-series analysis, nonlinear problems in fluid dynamics, and wave propagation, to mention just a few examples. Moreover, some geophysical phenomena are of great complexity, and their comprehension is based on conceptual models created, sometimes even from first principles, in the spirit of complex systems theory. Creativity in the construction of models to describe geophysical phenomena often encounters new problems of a mathematical nature that can be a challenge and an inspiration for mathematicians.

This Special Issue is intended to bring together original research and reviews that represent the state of the art in various geophysical modeling topics, presented in a manner suitable for a mathematically oriented audience, with an emphasis on the central role of mathematical structures. Contributions regarding both theoretical and practical models of any geophysical phenomena are welcome.

Prof. Dr. Mariusz Białecki
Guest Editor

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Keywords

  • geophysics
  • mathematical modeling
  • time-series analysis
  • complex systems
  • nonlinear models
  • wave propagation
  • statistical models
  • fluid mechanics
  • solid mechanics

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Published Papers (5 papers)

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Research

19 pages, 6766 KiB  
Article
PDE-Based Two-Dimensional Radiomagnetotelluric forward Modelling Using Vertex-Centered Finite-Volume Scheme
by Wei Xie, Wendi Zhu, Xiaozhong Tong and Huiying Ma
Mathematics 2024, 12(13), 2096; https://doi.org/10.3390/math12132096 - 3 Jul 2024
Viewed by 500
Abstract
An efficient finite-volume algorithm, based on the vertex-centered technique, is proposed for solving two-dimensional radiomagnetotelluric forward modeling. Firstly, we derive the discrete expressions of the radiomagnetotelluric Helmholtz-type equation and the corresponding mixed boundary conditions using the vertex-centered finite-volume technique. Then, the corresponding approximate [...] Read more.
An efficient finite-volume algorithm, based on the vertex-centered technique, is proposed for solving two-dimensional radiomagnetotelluric forward modeling. Firstly, we derive the discrete expressions of the radiomagnetotelluric Helmholtz-type equation and the corresponding mixed boundary conditions using the vertex-centered finite-volume technique. Then, the corresponding approximate solutions of the radiomagnetotelluric forward problem can be calculated by applying the finite-volume scheme to treat the boundary conditions. Secondly, we apply the finite-volume algorithm to solve two-dimensional Helmholtz equations and the resistivity half-space model. Numerical experiments demonstrate the high accuracy of the proposed approach. Finally, we summarize the radiomagnetotelluric responses through a numerical simulation of a two-dimensional model, which enables qualitative interpretation of field data. Furthermore, our numerical method can be extended and implemented for three-dimensional radiomagnetotelluric forward modeling to achieve more accurate computation. Full article
(This article belongs to the Special Issue Mathematical Modeling in Geophysics: Concepts and Practices)
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0 pages, 20145 KiB  
Article
A Legendre Spectral-Element Method to Incorporate Topography for 2.5D Direct-Current-Resistivity Forward Modeling
by Wei Xie, Wendi Zhu, Xiaozhong Tong and Huiying Ma
Mathematics 2024, 12(12), 1864; https://doi.org/10.3390/math12121864 - 14 Jun 2024
Viewed by 516
Abstract
An effective and accurate solver for the direct-current-resistivity forward-modeling problem has become a cutting-edge research topic. However, computational limitations arise due to the substantial amount of data involved, hindering the widespread use of three-dimensional forward modeling, which is otherwise considered the most effective [...] Read more.
An effective and accurate solver for the direct-current-resistivity forward-modeling problem has become a cutting-edge research topic. However, computational limitations arise due to the substantial amount of data involved, hindering the widespread use of three-dimensional forward modeling, which is otherwise considered the most effective approach for identifying geo-electrical anomalies. An efficient compromise, or potentially an alternative, is found in two-and-a-half-dimensional (2.5D) modeling, which employs a three-dimensional current source within a two-dimensional subsurface medium. Consequently, a Legendre spectral-element algorithm is developed specifically for 2.5D direct-current-resistivity forward modeling, taking into account the presence of topography. This numerical algorithm can combine the complex geometric flexibility of the finite-element method with the high precision of the spectral method. To solve the wavenumber-domain electrical potential variational problem, which is converted into the two-dimensional Helmholtz equation with mixed boundary conditions, the Gauss–Lobatto–Legendre (GLL) quadrature is employed in all discrete quadrilateral spectral elements, ensuring identical Legendre polynomial interpolation and quadrature points. The Legendre spectral-element method is applied to solve a two-dimensional Helmholtz equation and a resistivity half-space model. Numerical experiments demonstrate that the proposed approach yields highly accurate numerical results, even with a coarse mesh. Additionally, the Legendre spectral-element algorithm is employed to simulate the apparent resistivity distortions caused by surface topographical variations in the direct-current resistivity Wenner-alpha array. These numerical results affirm the substantial impact of topographical variations on the apparent resistivity data obtained in the field. Consequently, when interpreting field data, it is crucial to consider topographic effects to the extent they can be simulated. Moreover, our numerical method can be extended and implemented for a more accurate computation of three-dimensional direct-current-resistivity forward modeling. Full article
(This article belongs to the Special Issue Mathematical Modeling in Geophysics: Concepts and Practices)
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12 pages, 284 KiB  
Article
Stochastic Process Leading to Catalan Number Recurrence
by Mariusz Białecki
Mathematics 2023, 11(24), 4953; https://doi.org/10.3390/math11244953 - 14 Dec 2023
Viewed by 1541
Abstract
Motivated by a simple model of earthquake statistics, a finite random discrete dynamical system is defined in order to obtain Catalan number recurrence by describing the stationary state of the system in the limit of its infinite size. Equations describing dynamics of the [...] Read more.
Motivated by a simple model of earthquake statistics, a finite random discrete dynamical system is defined in order to obtain Catalan number recurrence by describing the stationary state of the system in the limit of its infinite size. Equations describing dynamics of the system, represented by partitions of a subset of {1,2,,N}, are derived using basic combinatorics. The existence and uniqueness of a stationary state are shown using Markov chains terminology. A well-defined mean-field type approximation is used to obtain block size distribution and the consistency of the approach is verified. It is shown that this recurrence asymptotically takes the form of Catalan number recurrence for particular dynamics parameters of the system. Full article
(This article belongs to the Special Issue Mathematical Modeling in Geophysics: Concepts and Practices)
16 pages, 2878 KiB  
Article
Optimal Transport and Seismic Rays
by Fabrizio Magrini and Malcolm Sambridge
Mathematics 2023, 11(22), 4686; https://doi.org/10.3390/math11224686 - 17 Nov 2023
Viewed by 1214
Abstract
We present a theoretical framework that links Fermat’s principle of least time to optimal transport theory via a cost function that enforces local transport. The proposed cost function captures the physical constraints inherent in wave propagation; when paired with specific mass distributions, it [...] Read more.
We present a theoretical framework that links Fermat’s principle of least time to optimal transport theory via a cost function that enforces local transport. The proposed cost function captures the physical constraints inherent in wave propagation; when paired with specific mass distributions, it yields shortest paths in the considered media through the optimal transport plans. In the discrete setting, our formulation results in physically significant optimal couplings, whose off-diagonal entries identify shortest paths in both directed and undirected graphs. For undirected graphs with positive edge weights, commonly used to parameterize seismic media, our method provides solutions to the Eikonal equation consistent with those from the Dijkstra algorithm. For directed negative-weight graphs, corresponding to transportation cost matrices with negative entries, our approach aligns with the Bellman–Ford algorithm but offers considerable computational advantages. We also highlight potential research directions. These include the use of sparse cost matrices to reduce the number of unknowns and constraints in the considered transportation problem, and solving specific classes of optimal transport problems through the Dijkstra algorithm to enhance computational efficiency. Full article
(This article belongs to the Special Issue Mathematical Modeling in Geophysics: Concepts and Practices)
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12 pages, 294 KiB  
Article
Seismological Problem, Seismic Waves and the Seismic Mainshock
by Bogdan Felix Apostol
Mathematics 2023, 11(17), 3777; https://doi.org/10.3390/math11173777 - 2 Sep 2023
Viewed by 988
Abstract
The elastic wave equation with seismic tensorial force is solved in a homogeneous and isotropic medium (the Earth). Spherical-shell waves are obtained, which are associated to the primary P and S seismic waves. It is shown that these waves produce secondary waves with [...] Read more.
The elastic wave equation with seismic tensorial force is solved in a homogeneous and isotropic medium (the Earth). Spherical-shell waves are obtained, which are associated to the primary P and S seismic waves. It is shown that these waves produce secondary waves with sources on the plane surface of a half-space, which have the form of abrupt walls with a long tail, propagating in the interior and on the surface of the half-space. These secondary waves are associated to the seismic mainshock. The results, previously reported, are re-derived using Fourier transformations and specific regularization procedures. The relevance of this seismic motion for the ground motion, the seismographs’ recordings and the effect of the inhomogeneities in the medium are discussed. Full article
(This article belongs to the Special Issue Mathematical Modeling in Geophysics: Concepts and Practices)
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