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Mathematics
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Mathematical Modeling in Geophysics: Concepts and Practices
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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 August 2024 | Viewed by 4354

Special Issue Editor

Prof. Dr. Mariusz Białecki

E-Mail Website
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

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

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

Published Papers (3 papers)

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Research

12 pages, 284 KiB  
Article
Stochastic Process Leading to Catalan Number Recurrence
by Mariusz Białecki
Mathematics 2023, 11(24), 4953; https://0-doi-org.brum.beds.ac.uk/10.3390/math11244953 - 14 Dec 2023
Viewed by 1285
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://0-doi-org.brum.beds.ac.uk/10.3390/math11224686 - 17 Nov 2023
Viewed by 1046
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://0-doi-org.brum.beds.ac.uk/10.3390/math11173777 - 2 Sep 2023
Viewed by 836
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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