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Mathematics in Geophysical Research
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Mathematics in Geophysical Research

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E2: Control Theory and Mechanics".

Deadline for manuscript submissions: closed (30 November 2023) | Viewed by 10617

Special Issue Editors

Prof. Dr. Valentina Yakovleva

E-Mail Website
Guest Editor
School of Nuclear Science and Engineering, Tomsk Polytechnic University, 634050 Tomsk, Russia
Interests: radon; thoron; radionuclides; dosimetry; radiation safety; nuclear physics; geophysics; gamma; radiation; Geant4; physical simulation; mathematical modeling
Dr. Roman Parovik

E-Mail Website
Guest Editor
Physical Processes Modeling Laboratory, Institute of Cosmophysical Research and Radio Wave Propagation, Far Eastern Branch of the Russian Academy of Sciences, Kamchatskiy Kray, 684034 Paratunka, Russia
Interests: fractional calculus; fractional oscillators; fractional dynamics; numerical methods; mathematical modeling
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

We invite you to contribute original research articles on the theoretical or practical application of mathematical methods and tools in solving complex problems in geophysics to this Special Issue, "Mathematics in Geophysical Research".

The development of modern mathematical research tools has led to their application in the study of complex natural phenomena and processes. This Special Issue will particularly focus on the use of mathematical tools (methods for mathematical modeling (simulation), computational mathematics, processing experimental data, etc.) in the study of dynamic processes in various geospheres. Articles using computer simulation to visualize the results of geophysical process research are also welcome.

Topics that may be featured in the Special Issue include (but are not limited to):

  1. Mathematical modeling in geospheres;
  2. Mathematical methods for processing geophysical data;
  3. Simulation and computer modeling in geophysical research;
  4. Inverse problems in geophysics;
  5. Applied nuclear physics in geophysics.

Prof. Dr. Valentina Yakovleva
Dr. Roman Parovik
Guest Editors

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

  • engineering mathematics
  • mathematical and computer modeling
  • mathematical tools
  • computational geophysics
  • geospheres
  • geophysical research
  • geophysical data

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Published Papers (7 papers)

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Research

22 pages, 5683 KiB  
Article
Fictitious Point Technique Based on Finite-Difference Method for 2.5D Direct-Current Resistivity Forward Problem
by Xiaozhong Tong and Ya Sun
Mathematics 2024, 12(2), 269; https://doi.org/10.3390/math12020269 - 14 Jan 2024
Cited by 3 | Viewed by 1408
Abstract
With the widespread application of the direct-current resistivity method, searching for accurate and fast-forward algorithms has become the focus of research for geophysicists and engineers. Three-dimensional forward modeling can be the best way to identify geo-electrical anomalies but are hampered by computational limitations [...] Read more.
With the widespread application of the direct-current resistivity method, searching for accurate and fast-forward algorithms has become the focus of research for geophysicists and engineers. Three-dimensional forward modeling can be the best way to identify geo-electrical anomalies but are hampered by computational limitations because of the large amount of data. A practical compromise, or even alternative, is represented by 2.5D modeling characterized using a 3D source in a 2D medium. Thus, we develop a 2.5D direct-current resistivity forward modeling algorithm. The algorithm incorporates the finite-difference approximation and fictitious point technique that can improve the efficiency and accuracy of numerical simulation. Firstly, from the boundary value problem of the electric potential generated by the point source, the discrete expressions of the governing equation are derived from the finite-difference approach. The numerical solutions of the discrete electric potential are calculated after the approximate treatment of the boundary conditions with a finite-difference method based on a fictitious point scheme. Secondly, through the simulation of a homogeneous half-space model and a one-dimensional model, and compared with the analytical results, the correctness and stability of the finite-difference forward algorithm are verified. Lastly, through the numerical simulation for a two-dimensional model, 2.5D direct-current sounding responses are summarized, which can provide a qualitative interpretation of field data. Full article
(This article belongs to the Special Issue Mathematics in Geophysical Research)
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22 pages, 7077 KiB  
Article
Second-Order Approximate Reflection Coefficient of Thin Interbeds with Vertical Fractures
by Shiwei Cui, Ya Sun and Pu Wang
Mathematics 2024, 12(2), 232; https://doi.org/10.3390/math12020232 - 11 Jan 2024
Viewed by 1016
Abstract
The horizontal fractures in the strata will close in the compaction effect of overlying strata, while the vertical cracks are widely developed, which can be equivalent to HTI (transverse isotropy with a horizontal axis of symmetry) medium. When an S-wave propagates into HTI [...] Read more.
The horizontal fractures in the strata will close in the compaction effect of overlying strata, while the vertical cracks are widely developed, which can be equivalent to HTI (transverse isotropy with a horizontal axis of symmetry) medium. When an S-wave propagates into HTI media, the shear wave will divide into two types of waves: a fast S-wave and slow S-wave. When the strata of HTI are thin and overlapping, called the thin interbeds model, the wave field exhibits complex primary reflections, converted waves, and multiples. We introduce a new second-order approximation of the total reflection coefficient, with the incidence angle lower than the critical angle in thin-interbed HTI media using a recursive algorithm. We verify the effectiveness of the second-order approximation by analyzing the energy of multiples. Comparing the second-order approximate solution that degenerates the HTI medium into isotropic and Kennett’s exact solution, we find that our solution has an accuracy of over 99.9% in any azimuth, with the incidence angle lower than the critical angle under P-wave incidence. However, our solution of the SP wave field is suitable for incidence azimuth angles between 0–75° and 120–180°, with the lowest accuracy occurring at an incidence angle of 25° and a relative error of 6.4%. The approximate solution in the SS wave field has the same applicable range as the SP wave, with the maximum error of 6.3% occurring at the incident angle of 1°. This new second-order approximate formula for the total reflection coefficient of thin interbeds composed of HTI helps us to understand the reflection characteristics of complex thin interbeds. It also lays a theoretical foundation for the development of AVO (Amplitude Versus Offset) analysis and inversion techniques for lithological and stratigraphic oil and gas reservoirs. Full article
(This article belongs to the Special Issue Mathematics in Geophysical Research)
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14 pages, 17995 KiB  
Article
A Hybrid Chebyshev Pseudo-Spectral Finite-Difference Time-Domain Method for Numerical Simulation of 2D Acoustic Wave Propagation
by Xiaozhong Tong and Ya Sun
Mathematics 2024, 12(1), 117; https://doi.org/10.3390/math12010117 - 29 Dec 2023
Cited by 4 | Viewed by 1381
Abstract
In this study, a hybrid Chebyshev pseudo-spectral finite-difference time-domain (CPS-FDTD) algorithm is proposed for simulating 2D acoustic wave propagation in heterogeneous media, which is different from the other traditional numerical schemes such as finite element and finite difference. This proposed hybrid method integrates [...] Read more.
In this study, a hybrid Chebyshev pseudo-spectral finite-difference time-domain (CPS-FDTD) algorithm is proposed for simulating 2D acoustic wave propagation in heterogeneous media, which is different from the other traditional numerical schemes such as finite element and finite difference. This proposed hybrid method integrates the efficiency of the FDTD approach in the time domain and the high accuracy of the CPS technique in the spatial domain. We present the calculation formulas of this novel approach and conduct simulation experiments to test it. The biconjugate gradient is solved by combining the large symmetric sparse systems stabilized algorithm with an incomplete LU factorization. Three numerical experiments are further presented to illustrate the accuracy, efficiency, and flexibility of the hybrid CPS-FDTD algorithm. Full article
(This article belongs to the Special Issue Mathematics in Geophysical Research)
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34 pages, 983 KiB  
Article
Computational Technology for the Basis and Coefficients of Geodynamo Spectral Models in the Maple System
by Gleb Vodinchar and Liubov Feshchenko
Mathematics 2023, 11(13), 3000; https://doi.org/10.3390/math11133000 - 5 Jul 2023
Cited by 4 | Viewed by 1025
Abstract
Spectral models are often used in the study of geodynamo problems. Physical fields in these models are presented as stationary basic modes combinations with time-dependent amplitudes. To construct a model it is necessary to calculate the modes parameters, and to calculate the model [...] Read more.
Spectral models are often used in the study of geodynamo problems. Physical fields in these models are presented as stationary basic modes combinations with time-dependent amplitudes. To construct a model it is necessary to calculate the modes parameters, and to calculate the model coefficients (the Galerkin coefficients). These coefficients are integrals of complex multiplicative combinations of modes and differential operators. The paper proposes computing technology for the calculation of parameters, the derivation of integrands and the calculation of the integrals themselves. The technology is based on computer algebra methods. The main elements for implementation of technology in the Maple system are described. The proposed computational technology makes it possible to quickly and accurately construct fairly wide classes of new geodynamo spectral models. Full article
(This article belongs to the Special Issue Mathematics in Geophysical Research)
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17 pages, 2128 KiB  
Article
Synchrosqueezing Transform Based on Frequency-Domain Gaussian-Modulated Linear Chirp Model for Seismic Time–Frequency Analysis
by Pingping Bing, Wei Liu, Haoqi Zhang, Li Zhu, Guiping Zhu, Jun Zhou and Binsheng He
Mathematics 2023, 11(13), 2904; https://doi.org/10.3390/math11132904 - 28 Jun 2023
Cited by 1 | Viewed by 1884
Abstract
The synchrosqueezing transform (SST) has attracted much attention as a post-processing technique since it was proposed. In recent years, improvements to SST have been made. However, the existing methods are mainly based on the time-domain signal model, and the weak frequency modulation assumption [...] Read more.
The synchrosqueezing transform (SST) has attracted much attention as a post-processing technique since it was proposed. In recent years, improvements to SST have been made. However, the existing methods are mainly based on the time-domain signal model, and the weak frequency modulation assumption for the components composing the signal is always taken into account. Thus, the signals characterized by a rapidly changing instantaneous frequency (IF) may fail to be adequately tackled. To address this problem, the paper presents a novel seismic time–frequency analysis method via synchrosqueezing transform where a frequency-domain Gaussian modulated linear chirp model is utilized to deduce the SST. The group delay (GD) rather than the IF estimator is implemented to compute an estimation of the ridge. Furthermore, a new synchrosqueezing operator is constructed to rearrange the energy around the ridge. A synthetic example verifies the efficiency and robustness of the proposed SST method, which generates better results than some classic time–frequency analysis (TFA) approaches, e.g., short-time Fourier transform (STFT) and STFT-based SST (FSST). A field dataset further demonstrates this method’s potential in the delineation of subsurface geological structures. Full article
(This article belongs to the Special Issue Mathematics in Geophysical Research)
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14 pages, 962 KiB  
Article
Magnetic Field Dynamical Regimes in a Large-Scale Low-Mode αΩ-Dynamo Model with Hereditary α-Quenching by Field Energy
by Olga Sheremetyeva
Mathematics 2023, 11(10), 2297; https://doi.org/10.3390/math11102297 - 15 May 2023
Cited by 2 | Viewed by 1599
Abstract
The article considers a large-scale model of an αΩ-dynamo in the low-mode approximation. The intensity of the α-effect is regulated by a process that depends on the energy of the magnetic field and has hereditarity properties (finite “memory”). The regulation [...] Read more.
The article considers a large-scale model of an αΩ-dynamo in the low-mode approximation. The intensity of the α-effect is regulated by a process that depends on the energy of the magnetic field and has hereditarity properties (finite “memory”). The regulation process is included in the MHD-system in the form of an additive correction. The action character of the process is defined by the alternating kernel with variable parameters: the damping frequency and the damping coefficient. The Reynolds number and the α-effect measure are the control parameters of the system. Information about the action of a large-scale generator is contained in the Reynolds number, and that about the action of a turbulent one is contained in the measure of the α-effect. The stability of the solution of the MHD-system is studied depending on the values of the control parameters and the parameters of the alternating kernel. Based on the results of numerical simulation of the dynamical regimes, limitations are determined for the values of the model parameters at which the regimes are reproduced against the background of small oscillations of the viscous liquid velocity field. The results of the study of the stability of solutions and numerical simulations of the dynamical regimes are represented on the phase plane of the control parameters. The paper investigates the question of changing the pattern on the phase plane depending on the values of the damping coefficient, the damping frequency, and the waiting time. A comparison is made with the results obtained earlier, when the α-effect intensity is a constant or is regulated by a process with an exponential kernel and the same values of the damping coefficient. Full article
(This article belongs to the Special Issue Mathematics in Geophysical Research)
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14 pages, 745 KiB  
Article
Studies of the Fractional Selkov Dynamical System for Describing the Self-Oscillatory Regime of Microseisms
by Roman Ivanovich Parovik
Mathematics 2022, 10(22), 4208; https://doi.org/10.3390/math10224208 - 10 Nov 2022
Cited by 4 | Viewed by 1331
Abstract
A non-linear fractional Selkov dynamic system for mathematical modeling of microseismic phenomena is proposed. This system is a generalization of the previously known Selkov system, which has self-oscillatory modes and is used in biology to describe glycolytic vibrations of the substrate and product. [...] Read more.
A non-linear fractional Selkov dynamic system for mathematical modeling of microseismic phenomena is proposed. This system is a generalization of the previously known Selkov system, which has self-oscillatory modes and is used in biology to describe glycolytic vibrations of the substrate and product. The Selkov fractional dynamical system takes into account the influence of heredity and is described using derivative fractional orders. The article investigates the Selkov fractional dynamic model using the Adams–Bashforth–Moulton numerical method, constructs oscillograms and phase trajectories, and studies the equilibrium points. Based on the spectra of the maximum Lyapunov exponents, it is shown that in the fractional dynamic model there can be relaxation and damped oscillations. Full article
(This article belongs to the Special Issue Mathematics in Geophysical Research)
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