{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T16:00:51Z","timestamp":1772726451733,"version":"3.50.1"},"reference-count":36,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,18]],"date-time":"2024-06-18T00:00:00Z","timestamp":1718668800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Committee of Science of the Ministry of Higher Education and Science of the Republic of Kazakhstan","award":["AP14871252"],"award-info":[{"award-number":["AP14871252"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>The article discusses the application of quaternion Fourier transforms and quaternion algebra to transform Maxwell\u2019s equations. This makes it possible to present the problem of magnetotelluric sensing (MTS) in a more convenient form for research. Studies of the inverse MTS problem for multi-layer regions are presented using the differential evolution method, which demonstrates high convergence. For single-layer regions, a new method for solving inverse problems based on minimizing the quadratic functional using conjugate optimization methods is considered. Numerical results obtained using special Python libraries are presented, with analysis and conclusions.<\/jats:p>","DOI":"10.3390\/computation12060127","type":"journal-article","created":{"date-parts":[[2024,6,19]],"date-time":"2024-06-19T04:21:28Z","timestamp":1718770888000},"page":"127","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Solution of the Optimization Problem of Magnetotelluric Sounding in Quaternions by the Differential Evolution Method"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0097-1873","authenticated-orcid":false,"given":"Syrym E.","family":"Kasenov","sequence":"first","affiliation":[{"name":"Faculty of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan"}]},{"given":"Zhanar E.","family":"Demeubayeva","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Daulet Serikbayev East Kazakhstan Technical University, Ust-Kamenogorsk 070000, Kazakhstan"}]},{"given":"Nurlan M.","family":"Temirbekov","sequence":"additional","affiliation":[{"name":"Faculty of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan"},{"name":"National Engineering Academy of the Republic of Kazakhstan, Almaty 050010, Kazakhstan"}]},{"given":"Laura N.","family":"Temirbekova","sequence":"additional","affiliation":[{"name":"National Engineering Academy of the Republic of Kazakhstan, Almaty 050010, Kazakhstan"},{"name":"Faculty of Mathematics, Physics and Informatics, Abai Kazakh National Pedagogical University, Almaty 050010, Kazakhstan"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,18]]},"reference":[{"key":"ref_1","unstructured":"Berdichevsky, M.N., and Dmitriev, V.I. (1992). Magnetotellurgical Sounding of Horizontally Homogeneous Media, Nedra."},{"key":"ref_2","unstructured":"Kashafutdinov, O.V. (2005). Substantiation of Direct and Inverse Problems of Magnetotelluric Sounding (MTS) and Experiments to Solve Them, INM SO RAN."},{"key":"ref_3","first-page":"475","article-title":"Numerical solution of the problem of restoring electrophysical parameters based on the results of sounding and alternating current","volume":"58","author":"Krasnov","year":"2018","journal-title":"Math. Model. Numer. Methods"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Urynbassarova, D., and Urynbassarova, A. (2023). Hybrid Transforms, IntechOpen.","DOI":"10.5772\/intechopen.108186"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Urynbassarova, D., and Teali, A.A. (2023). Convolution, Correlation, and Uncertainty Principles for the Quaternion Offset Linear Canonical Transform. Mathematics, 11.","DOI":"10.3390\/math11092201"},{"key":"ref_6","first-page":"91","article-title":"Biquaternion generalizations of Maxwell\u2019s and Dirac\u2019s equations and properties of their solutions","volume":"24","author":"Alekseeva","year":"2023","journal-title":"J. Probl. Evol. Open Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1134\/S0012266122040048","article-title":"Generalized Solutions of Stationary Boundary Value Problems for Biwave Equations","volume":"58","author":"Alexeyeva","year":"2022","journal-title":"Differ. Equ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1460003","DOI":"10.1142\/S0219691314600030","article-title":"Continuous quaternion fourier and wavelet transformations","volume":"12","author":"Bahri","year":"2014","journal-title":"Int. J. Wavelets Multiresolution Inf. Process."},{"key":"ref_9","first-page":"10","article-title":"Two-dimensional quaternion wavelet transform","volume":"218","author":"Bahri","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2398","DOI":"10.1016\/j.camwa.2008.05.032","article-title":"An uncertain principle for quaternion Fourier transform","volume":"56","author":"Bahri","year":"2008","journal-title":"Comput. Math. Appl."},{"key":"ref_11","first-page":"295","article-title":"On determining electrical characteristics of the deep layers of the Earth\u2019s crust","volume":"73","author":"Tikhonov","year":"1950","journal-title":"Dokl. Akad. Nauk. SSSR"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1190\/1.1437915","article-title":"Basic theory of the magneto-telluric method of geophysical prospecting","volume":"18","author":"Cagniard","year":"1953","journal-title":"Geophysics"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"48","DOI":"10.33048\/sibjim.2019.22.305","article-title":"Numerical method for determining the dielectric constant modulo the vector of the electric intensity of the electromagnetic field","volume":"22","author":"Dedok","year":"2019","journal-title":"Sib. J. Ind. Math"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1088\/0266-5611\/7\/6\/009","article-title":"An identification problem related to the integrodifferent Maxwell\u2019s equation","volume":"7","author":"Kabanikhin","year":"1991","journal-title":"Inverse Probl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"020119","DOI":"10.1063\/1.4959733","article-title":"A conditional stability estimate of continuation problem for the Helmholtz equation","volume":"1759","author":"Kasenov","year":"2016","journal-title":"AIP Conf. Proc."},{"key":"ref_16","first-page":"197","article-title":"Numerical algorithm for solving the inverse problem for the Helmholtz equation","volume":"Volume 998","author":"Shishlenin","year":"2019","journal-title":"Proceedings of the 9th International Conference on CITech 2018"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1046\/j.1365-246x.2000.00007.x","article-title":"Three-dimensional magnetotelluric inversion using non-linear conjugate gradients","volume":"140","author":"Newman","year":"2000","journal-title":"Geophys. J. Int."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1190\/1.1444893","article-title":"Nonlinear conjugate gradients algorithm for 2-D magnetotelluric inversion","volume":"66","author":"Rodi","year":"2001","journal-title":"Geophysics"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Isaev, I., Obornev, E., Obornev, I., Shimelevich, M., and Dolenko, S. (2019). Neural network recognition of the type of parameterization scheme for magnetotelluric data. Advances in Neural Computation, Machine Learning, and Cognitive Research II: Selected Papers from the XX International Conference on Neuroinformatics, Springer International Publishing.","DOI":"10.1007\/978-3-030-01328-8_19"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1025","DOI":"10.1134\/S096554252006007X","article-title":"Application of Neural Networks in Nonlinear Inverse Problems of Geophysics","volume":"60","author":"Obornev","year":"2020","journal-title":"Comput. Math. Math. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Isaev, I., Obornev, I., Obornev, E., Rodionov, E., Shimelevich, M., and Dolenko, S. (2022, January 6\u20138). Neural network recovery of missing data of one geophysical method from known data of another one in solving inverse problems of exploration geophysics. Proceedings of the 6th International Workshop on Deep Learning in Computational Physics, Dubna, Russia.","DOI":"10.22323\/1.429.0018"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Deng, F., Hu, J., Wang, X., Yu, S., Zhang, B., Li, S., and Li, X. (2023). Magnetotelluric Deep Learning Forward Modeling and Its Application in Inversion. Remote Sens., 15.","DOI":"10.3390\/rs15143667"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Temirbekov, N., Kasenov, S., Berkinbayev, G., Temirbekov, A., Tamabay, D., and Temirbekova, M. (2023). Analysis of Data on Air Pollutants in the City by Machine-Intelligent Methods Considering Climatic and Geographical Features. Atmosphere, 14.","DOI":"10.3390\/atmos14050892"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1190\/1.1442303","article-title":"Occam\u2019s inversion: A practical algorithm for generating smooth models from electromagnetic sounding data","volume":"52","author":"Constable","year":"1987","journal-title":"Geophysics"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1613","DOI":"10.1190\/1.1442813","article-title":"Occam\u2019s inversion to generate smooth, two-dimensional models from magnetotelluric data","volume":"55","author":"Constable","year":"1990","journal-title":"Geophysics"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3905","DOI":"10.1029\/90JB02416","article-title":"Rapid Inversion of two and three dimensional magnetotelluric data","volume":"96","author":"Smith","year":"1991","journal-title":"J. Geophys. Res."},{"key":"ref_27","unstructured":"Tikhonov, A.N. (1999). Mathematical Geophysics, Publishing House of the Joint Institute of Earth\u2019s Physics of RAS."},{"key":"ref_28","first-page":"298","article-title":"Differentiation of functions of a quaternion variable","volume":"20","author":"Polyakova","year":"2019","journal-title":"Chebyshevsky Collect."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Ell, T.A., Le Bihan, N., and Sangwine, S.J. (2014). Quaternion Fourier Transforms for Signal and Image Processing, John Wiley & Sons. Wiley-ISTE.","DOI":"10.1002\/9781118930908"},{"key":"ref_30","first-page":"587","article-title":"On electromagnetic sounding of the deep layers of the Earth","volume":"140","author":"Tikhonov","year":"1961","journal-title":"Rep. Acad. Sci. Russ. Acad. Sci."},{"key":"ref_31","first-page":"1953","article-title":"Empowering geological data analysis with specialized software GIS modules","volume":"34","author":"Baigereyev","year":"2024","journal-title":"Indones. J. Electr. Eng. Comput. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1134\/S1990478923010118","article-title":"Simulation of COVID-19 Spread Scenarios in the Republic of Kazakhstan Based on Regularization of the Agent-Based Model","volume":"17","author":"Krivorotko","year":"2023","journal-title":"J. Appl. Ind. Math."},{"key":"ref_33","first-page":"234","article-title":"Gradient and direct method of solving Gelfand-Levitan integral equation","volume":"12","author":"Temirbekova","year":"2013","journal-title":"Appl. Comput. Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2003522","DOI":"10.1080\/23311916.2021.2003522","article-title":"Numerical simulation of inverse geochemistry problems by regularizing algorithms","volume":"9","author":"Temirbekov","year":"2022","journal-title":"Cogent Eng."},{"key":"ref_35","first-page":"137","article-title":"Module of the geoinformation system for analysis of geochemical fields based on mathematical modeling and digital prediction methods, News of the academy of sciences of the republic of Kazakhstan","volume":"5","author":"Temirbekov","year":"2021","journal-title":"Ser. Geol. Tech. Sci."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1023\/A:1008202821328","article-title":"Differential Evolution\u2014A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces","volume":"11","author":"Storn","year":"1997","journal-title":"J. Glob. 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