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<article article-type="research-article" dtd-version="1.3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xml:lang="ru"><front><journal-meta><journal-id journal-id-type="publisher-id">gyroscopy</journal-id><journal-title-group><journal-title xml:lang="ru">Гироскопия и навигация</journal-title><trans-title-group xml:lang="en"><trans-title>Giroskopiya i Navigatsiya / Gyroscopy and Navigation</trans-title></trans-title-group></journal-title-group><issn pub-type="ppub">0869-7035</issn><issn pub-type="epub">2075-0927</issn><publisher><publisher-name>AO «Концерн «ЦНИИ «Электроприбор»</publisher-name></publisher></journal-meta><article-meta><article-id custom-type="edn" pub-id-type="custom">VZKGED</article-id><article-id custom-type="elpub" pub-id-type="custom">gyroscopy-147</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="ru"><subject>Статьи</subject></subj-group></article-categories><title-group><article-title>Алгоритм трехмерной радиотомографии ионосферы по данным сигналов глобальной навигационной спутниковой системы</article-title><trans-title-group xml:lang="en"><trans-title>Algorithm for Three-Dimensional Ionospheric Radio Tomography Based on GNSS Data</trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0009-0007-6246-992X</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Артемьев</surname><given-names>В. М.</given-names></name><name name-style="western" xml:lang="en"><surname>Artemiev</surname><given-names>V. M.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Артемьев Валентин Михайлович. Член-корреспондент Национальной академии наук Беларуси, доктор технических наук, профессор, главный научный сотрудник</p><p>Минск</p></bio><bio xml:lang="en"><p>Minsk</p></bio><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-4624-9261</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Наумов</surname><given-names>А. О.</given-names></name><name name-style="western" xml:lang="en"><surname>Naumov</surname><given-names>A. O.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Наумов Александр Олегович. Кандидат физико-математических наук, заведующий лабораторией</p><p>Минск</p></bio><bio xml:lang="en"><p>Minsk</p></bio><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0003-3404-3917</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Хмарский</surname><given-names>П. А.</given-names></name><name name-style="western" xml:lang="en"><surname>Khmarski</surname><given-names>P. A.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Хмарский Петр Александрович. Кандидат технических наук, доцент, ведущий научный сотрудник, докторант</p><p>Минск</p></bio><bio xml:lang="en"><p>Minsk</p></bio><xref ref-type="aff" rid="aff-2"/></contrib></contrib-group><aff-alternatives id="aff-1"><aff xml:lang="ru"><institution>Институт прикладной физики Национальной академии наук Беларуси</institution><country>Беларусь</country></aff><aff xml:lang="en"><institution>Institute of Applied Physics, National Academy of Sciences of Belarus</institution><country>Belarus</country></aff></aff-alternatives><aff-alternatives id="aff-2"><aff xml:lang="ru"><institution>Институт прикладной физики Национальной академии наук Беларуси. Действительный член международной общественной организации «Академия навигации и управления  движением»</institution><country>Беларусь</country></aff><aff xml:lang="en"><institution>Institute of Applied Physics, National Academy of Sciences of Belarus</institution><country>Belarus</country></aff></aff-alternatives><pub-date pub-type="collection"><year>2025</year></pub-date><pub-date pub-type="epub"><day>30</day><month>07</month><year>2025</year></pub-date><volume>33</volume><issue>2</issue><fpage>103</fpage><lpage>121</lpage><permissions><copyright-statement>Copyright &amp;#x00A9; Артемьев В.М., Наумов А.О., Хмарский П.А., 2025</copyright-statement><copyright-year>2025</copyright-year><copyright-holder xml:lang="ru">Артемьев В.М., Наумов А.О., Хмарский П.А.</copyright-holder><copyright-holder xml:lang="en">Artemiev V.M., Naumov A.O., Khmarski P.A.</copyright-holder><license xml:lang="ru" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>Данная работа распространяется под лицензией Creative Commons Attribution 4.0.</license-p></license><license xml:lang="en" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>This work is licensed under a Creative Commons Attribution 4.0 License.</license-p></license></permissions><self-uri xlink:href="https://www.gyroscopy.ru/jour/article/view/147">https://www.gyroscopy.ru/jour/article/view/147</self-uri><abstract><p>В статье описываются особенности построения алгоритма трехмерной томографии концентрации электронов в ионосфере по радиосигналам глобальной навигационной спутниковой системы. На модельных данных исследуются положенные в его основу алгебраические методы реконструкции. Обсуждаются способы учета априорных сведений и подходы к оптимизации параметров этих алгебраических методов. Приводятся экспериментальные результаты реконструкции ионосферы по данным спутниковой системы точного позиционирования Республики Беларусь.</p></abstract><trans-abstract xml:lang="en"><p>The paper describes the design of an algorithm for three-dimensional radio tomography of electron concentration in the ionosphere based on the GNSS data. Simulation data are used to describe the algebraic reconstruction techniques underlying the algorithm design; also considered are the methods taking into account a priori information as well as the approaches to optimization of parameters of these algebraic methods. The results of the experimental ionospheric reconstruction obtained with the use of the data obtained from the satellite system of precise positioning of the Republic of Belarus are presented.</p></trans-abstract><kwd-group xml:lang="ru"><kwd>ионосфера</kwd><kwd>глобальные навигационные спутниковые системы</kwd><kwd>радиотомография</kwd><kwd>алгебраическая реконструкция</kwd><kwd>полное электронное содержание</kwd><kwd>трехмерная реконструкция</kwd><kwd>SART-алгоритм</kwd></kwd-group><kwd-group xml:lang="en"><kwd>ionosphere</kwd><kwd>global navigation satellite systems</kwd><kwd>radio tomography</kwd><kwd>algebraic reconstruction</kwd><kwd>total electron content</kwd><kwd>three-dimensional reconstruction</kwd><kwd>SART algorithm</kwd></kwd-group><funding-group><funding-statement xml:lang="ru">Математическое моделирование для исследования алгоритма реконструкции выполнено при поддержке программы Союзного государства «Комплекс-СГ» (2023–2026 гг., мероприятие 3.3). Экспериментальные исследования по данным ССТП РБ выполнены в рамках проекта подпрограммы 6 «Исследование и использование космического пространства в мирных целях» Государственной программы «Наукоемкие технологии и техника» на 2021–2025 гг.</funding-statement></funding-group></article-meta></front><back><ref-list><title>References</title><ref id="cit1"><label>1</label><citation-alternatives><mixed-citation xml:lang="ru">Куницын В.Е., Терещенко Е.Д., Андреева Е.С. Радиотомография ионосферы. М.: Физматлит, 2007. 693 с.</mixed-citation><mixed-citation xml:lang="en">Куницын В.Е., Терещенко Е.Д., Андреева Е.С. Радиотомография ионосферы. М.: Физматлит, 2007. 693 с.</mixed-citation></citation-alternatives></ref><ref id="cit2"><label>2</label><citation-alternatives><mixed-citation xml:lang="ru">Afraimovich, E.L., Astafyeva, E.I., Demyanov, V.V., Edemskiy, Il.K., Gavrilyuk, N.S., Ishin, A.B., Kosogorov, E.A., Leonovich, L.A., Lesyuta, O.S., Palamartchouk, K.S., Perevalova, N.P., Polyakova, A.S., Smolkov, G.Y., Voeykov, S.V., Yasyukevich, Y.V., Zhivetiev, I.V., A review of GPS/GLONASS studies of the ionospheric response to natural and anthropogenic processes and phenomena, Space Weather Space Clim., 2013, vol. 3, doi: 10.1051/swsc/2013049.</mixed-citation><mixed-citation xml:lang="en">Afraimovich, E.L., Astafyeva, E.I., Demyanov, V.V., Edemskiy, Il.K., Gavrilyuk, N.S., Ishin, A.B., Kosogorov, E.A., Leonovich, L.A., Lesyuta, O.S., Palamartchouk, K.S., Perevalova, N.P., Polyakova, A.S., Smolkov, G.Y., Voeykov, S.V., Yasyukevich, Y.V., Zhivetiev, I.V., A review of GPS/GLONASS studies of the ionospheric response to natural and anthropogenic processes and phenomena, Space Weather Space Clim., 2013, vol. 3, doi: 10.1051/swsc/2013049.</mixed-citation></citation-alternatives></ref><ref id="cit3"><label>3</label><citation-alternatives><mixed-citation xml:lang="ru">Hofmann-Wellenhof, B., Lichtenegger, H., Wasle, E., GNSS – Global Navigation Satellite Systems. GPS, GLONASS, Galileo, and More, Springer, 2008, 516 p.</mixed-citation><mixed-citation xml:lang="en">Hofmann-Wellenhof, B., Lichtenegger, H., Wasle, E., GNSS – Global Navigation Satellite Systems. GPS, GLONASS, Galileo, and More, Springer, 2008, 516 p.</mixed-citation></citation-alternatives></ref><ref id="cit4"><label>4</label><citation-alternatives><mixed-citation xml:lang="ru">Solonar, A.S., Khmarski, P.A., Tsuprik, S.V., Tracking estimator of the ground target coordinates and motion parameters using onboard optical location system data, Gyroscopy and Navigation, 2023, vol. 14, no. 3, pp. 244–258, doi: 10.1134/S2075108723030082.</mixed-citation><mixed-citation xml:lang="en">Solonar, A.S., Khmarski, P.A., Tsuprik, S.V., Tracking estimator of the ground target coordinates and motion parameters using onboard optical location system data, Gyroscopy and Navigation, 2023, vol. 14, no. 3, pp. 244–258, doi: 10.1134/S2075108723030082.</mixed-citation></citation-alternatives></ref><ref id="cit5"><label>5</label><citation-alternatives><mixed-citation xml:lang="ru">Solonar, A.S., Khmarski, P.A., Naumov, A.O., Juraev, D.A., Muxammedov, B.M., The use of numerical Monte Carlo integration to verify the physical feasibility of a trajectory based on surveillance radar data, Stochastic Modelling and Computational Sciences, 2023, vol. 3, no. 1, pp. 59–73, doi: 10.61485/smcs.27523829/v3n1p5.</mixed-citation><mixed-citation xml:lang="en">Solonar, A.S., Khmarski, P.A., Naumov, A.O., Juraev, D.A., Muxammedov, B.M., The use of numerical Monte Carlo integration to verify the physical feasibility of a trajectory based on surveillance radar data, Stochastic Modelling and Computational Sciences, 2023, vol. 3, no. 1, pp. 59–73, doi: 10.61485/smcs.27523829/v3n1p5.</mixed-citation></citation-alternatives></ref><ref id="cit6"><label>6</label><citation-alternatives><mixed-citation xml:lang="ru">Solonar, A.S., Khmarski, P.A., Main problems of trajectory processing and approaches to their solution within the framework of multitarget tracking, J. Phys. Conf. Ser., 2021, vol. 1864, doi: 10.1088/1742-6596/1864/1/012004.</mixed-citation><mixed-citation xml:lang="en">Solonar, A.S., Khmarski, P.A., Main problems of trajectory processing and approaches to their solution within the framework of multitarget tracking, J. Phys. Conf. Ser., 2021, vol. 1864, doi: 10.1088/1742-6596/1864/1/012004.</mixed-citation></citation-alternatives></ref><ref id="cit7"><label>7</label><citation-alternatives><mixed-citation xml:lang="ru">Yasyukevich, Y.V., Zhang, B., Devanaboyina, V.R., Advances in GNSS positioning and GNSS remote sensing, Sensors, 2024, vol. 24, no. 4, pp. 1200, doi: 10.3390/s24041200.</mixed-citation><mixed-citation xml:lang="en">Yasyukevich, Y.V., Zhang, B., Devanaboyina, V.R., Advances in GNSS positioning and GNSS remote sensing, Sensors, 2024, vol. 24, no. 4, pp. 1200, doi: 10.3390/s24041200.</mixed-citation></citation-alternatives></ref><ref id="cit8"><label>8</label><citation-alternatives><mixed-citation xml:lang="ru">Astafyeva, E., Yasyukevich, Y., Maksikov, A., Zhivetiev, I., Geomagnetic storms, super-storms, and their impacts on GPS-based navigation systems, Space Weather, 2014, vol. 12, pp. 508–525, doi: 10.1002/2014SW001072.</mixed-citation><mixed-citation xml:lang="en">Astafyeva, E., Yasyukevich, Y., Maksikov, A., Zhivetiev, I., Geomagnetic storms, super-storms, and their impacts on GPS-based navigation systems, Space Weather, 2014, vol. 12, pp. 508–525, doi: 10.1002/2014SW001072.</mixed-citation></citation-alternatives></ref><ref id="cit9"><label>9</label><citation-alternatives><mixed-citation xml:lang="ru">Yasyukevich, Yu.V., Mylnikova, A.A., Polyakova, A.S., Estimating the total electron content absolute value from the GPS/GLONASS data, Results in Physics, 2015, vol. 5, pp. 32–33, doi: 10.1016/j.rinp.2014.12.006.</mixed-citation><mixed-citation xml:lang="en">Yasyukevich, Yu.V., Mylnikova, A.A., Polyakova, A.S., Estimating the total electron content absolute value from the GPS/GLONASS data, Results in Physics, 2015, vol. 5, pp. 32–33, doi: 10.1016/j.rinp.2014.12.006.</mixed-citation></citation-alternatives></ref><ref id="cit10"><label>10</label><citation-alternatives><mixed-citation xml:lang="ru">Nesterov, I.A., Kunitsyn, V.E., GNSS radio tomography of the ionosphere: The problem with essentially incomplete data, Advances in Space Research, 2011, vol. 47, issue 10, pp. 1789–1803, doi: 10.1016/j.asr.2010.11.034.</mixed-citation><mixed-citation xml:lang="en">Nesterov, I.A., Kunitsyn, V.E., GNSS radio tomography of the ionosphere: The problem with essentially incomplete data, Advances in Space Research, 2011, vol. 47, issue 10, pp. 1789–1803, doi: 10.1016/j.asr.2010.11.034.</mixed-citation></citation-alternatives></ref><ref id="cit11"><label>11</label><citation-alternatives><mixed-citation xml:lang="ru">Kunitsyn, V.E., Nesterov, I.A., Padokhin, A.M., et al., Ionospheric radio tomography based on the GPS/GLONASS navigation systems, Journal of Communications Technology and Electronics, 2011, vol. 56, pp. 1269–1281, doi: 10.1134/S1064226911100147.</mixed-citation><mixed-citation xml:lang="en">Kunitsyn, V.E., Nesterov, I.A., Padokhin, A.M., et al., Ionospheric radio tomography based on the GPS/GLONASS navigation systems, Journal of Communications Technology and Electronics, 2011, vol. 56, pp. 1269–1281, doi: 10.1134/S1064226911100147.</mixed-citation></citation-alternatives></ref><ref id="cit12"><label>12</label><citation-alternatives><mixed-citation xml:lang="ru">Kunitsyn, V.E., Andreeva, E., Nesterov, I.A., Padokhin, A.M., Ionospheric sounding and tomography by GNSS, Geodetic Sciences – Observations, Modeling and Applications, 2013, doi: 10.5772/54589.</mixed-citation><mixed-citation xml:lang="en">Kunitsyn, V.E., Andreeva, E., Nesterov, I.A., Padokhin, A.M., Ionospheric sounding and tomography by GNSS, Geodetic Sciences – Observations, Modeling and Applications, 2013, doi: 10.5772/54589.</mixed-citation></citation-alternatives></ref><ref id="cit13"><label>13</label><citation-alternatives><mixed-citation xml:lang="ru">Naumov, A.O., Khmarskiy, P.A., Byshnev, N.I., Piatrouski, M.A., Determination of total electron content in the ionosphere over the territory of the Republic of Belarus based on global navigation satellite systems data, Vestsi Natsyyanal’nai akademii navuk Belarusi. Seryya fizika-technichnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physical-technical series, 2024, vol. 69, no. 1, pp. 53–64, doi: 10.29235/1561-8358-2024-69-1-53-64.</mixed-citation><mixed-citation xml:lang="en">Naumov, A.O., Khmarskiy, P.A., Byshnev, N.I., Piatrouski, M.A., Determination of total electron content in the ionosphere over the territory of the Republic of Belarus based on global navigation satellite systems data, Vestsi Natsyyanal’nai akademii navuk Belarusi. Seryya fizika-technichnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physical-technical series, 2024, vol. 69, no. 1, pp. 53–64, doi: 10.29235/1561-8358-2024-69-1-53-64.</mixed-citation></citation-alternatives></ref><ref id="cit14"><label>14</label><citation-alternatives><mixed-citation xml:lang="ru">Stankov, S.M., Stegen, K., Muhtarov, P., Warnant, R., Local ionospheric electron density profile reconstruction in real time from simultaneous ground-based GNSS and ionosonde measurements, Advances in Space Research, 2011, vol. 47, issue 7, pp. 1172–1180, doi: 10.1016/j.asr.2010.11.039.</mixed-citation><mixed-citation xml:lang="en">Stankov, S.M., Stegen, K., Muhtarov, P., Warnant, R., Local ionospheric electron density profile reconstruction in real time from simultaneous ground-based GNSS and ionosonde measurements, Advances in Space Research, 2011, vol. 47, issue 7, pp. 1172–1180, doi: 10.1016/j.asr.2010.11.039.</mixed-citation></citation-alternatives></ref><ref id="cit15"><label>15</label><citation-alternatives><mixed-citation xml:lang="ru">Naumov, A.O., Khmarski, P.A., Aronov, G.A., Kotov, D.S., Results of studies on processes occurring in the ionosphere and Earth’s magnetic field over the territory of the Republic of Belarus for the year 2023, Nonlinear Phenomena in Complex Systems, 2024, vol. 27, no. 3, pp. 225–233, doi: 10.5281/zenodo.13960570.</mixed-citation><mixed-citation xml:lang="en">Naumov, A.O., Khmarski, P.A., Aronov, G.A., Kotov, D.S., Results of studies on processes occurring in the ionosphere and Earth’s magnetic field over the territory of the Republic of Belarus for the year 2023, Nonlinear Phenomena in Complex Systems, 2024, vol. 27, no. 3, pp. 225–233, doi: 10.5281/zenodo.13960570.</mixed-citation></citation-alternatives></ref><ref id="cit16"><label>16</label><citation-alternatives><mixed-citation xml:lang="ru">Chen, C., Pavlov, I., Padokhin, A., Yasyukevich, Y., Demyanov, V., Danilchuk, E., Vesnin, A., Galileo and BeiDou AltBOC Signals and Their Perspectives for Ionospheric TEC Studies, Sensors, 2024, vol. 24, no. 6472, doi: 10.3390/s24196472.</mixed-citation><mixed-citation xml:lang="en">Chen, C., Pavlov, I., Padokhin, A., Yasyukevich, Y., Demyanov, V., Danilchuk, E., Vesnin, A., Galileo and BeiDou AltBOC Signals and Their Perspectives for Ionospheric TEC Studies, Sensors, 2024, vol. 24, no. 6472, doi: 10.3390/s24196472.</mixed-citation></citation-alternatives></ref><ref id="cit17"><label>17</label><citation-alternatives><mixed-citation xml:lang="ru">Artemiev, V.M., Naumov, A.O., Stepanov, V.L., Murashko, N.I., Method and results of real time modeling of ionosphere radiotomography on the basis of the Kalman filter theory, Journal of Automation and Information Sciences, 2008, vol. 40, no. 2, pp. 52–62.</mixed-citation><mixed-citation xml:lang="en">Artemiev, V.M., Naumov, A.O., Stepanov, V.L., Murashko, N.I., Method and results of real time modeling of ionosphere radiotomography on the basis of the Kalman filter theory, Journal of Automation and Information Sciences, 2008, vol. 40, no. 2, pp. 52–62.</mixed-citation></citation-alternatives></ref><ref id="cit18"><label>18</label><citation-alternatives><mixed-citation xml:lang="ru">Herman, G.T., Fundamentals of Computerized Tomography, Image Reconstruction from Projections, Springer, New York, 2009, 297 p.</mixed-citation><mixed-citation xml:lang="en">Herman, G.T., Fundamentals of Computerized Tomography, Image Reconstruction from Projections, Springer, New York, 2009, 297 p.</mixed-citation></citation-alternatives></ref><ref id="cit19"><label>19</label><citation-alternatives><mixed-citation xml:lang="ru">Zolotarev, S.A., Ahmed Talat Taufik Taruat, Bilenko, E.G., Taking into account a priori information in the iterative reconstruction of images of foundry products, Vestsi Natsyyanal’nai akademii navuk Belarusi. Seryya fizika-technichnykhnavuk = Proceedings of the National Academy of Sciences of Belarus. Physical-technical series, 2023, vol. 68, no. 3, pp. 242–251, doi: 10.29 235/1561-8358-2023-68-3-242-251.</mixed-citation><mixed-citation xml:lang="en">Zolotarev, S.A., Ahmed Talat Taufik Taruat, Bilenko, E.G., Taking into account a priori information in the iterative reconstruction of images of foundry products, Vestsi Natsyyanal’nai akademii navuk Belarusi. Seryya fizika-technichnykhnavuk = Proceedings of the National Academy of Sciences of Belarus. Physical-technical series, 2023, vol. 68, no. 3, pp. 242–251, doi: 10.29 235/1561-8358-2023-68-3-242-251.</mixed-citation></citation-alternatives></ref><ref id="cit20"><label>20</label><citation-alternatives><mixed-citation xml:lang="ru">Zolotarev, S.A., Vengrinovich, V.L., Smagin, S.I., Iterative tomography of pipes during operation, Vestsi Natsyyanal’nai akademii navuk Belarusi. Seryya fizika-technichnykh navuk, 2021, vol. 66, no. 4, pp. 505–512, doi: 10.29235/1561-8358-2021-66-4-505-512.</mixed-citation><mixed-citation xml:lang="en">Zolotarev, S.A., Vengrinovich, V.L., Smagin, S.I., Iterative tomography of pipes during operation, Vestsi Natsyyanal’nai akademii navuk Belarusi. Seryya fizika-technichnykh navuk, 2021, vol. 66, no. 4, pp. 505–512, doi: 10.29235/1561-8358-2021-66-4-505-512.</mixed-citation></citation-alternatives></ref><ref id="cit21"><label>21</label><citation-alternatives><mixed-citation xml:lang="ru">Bust, G.S., Garner, T.W., Gaussiran, T.L., II Ionospheric Data Assimilation Three-Dimensional (IDA3D): A global, multisensor, electron density specification algorithm, Journal of Geophysical Research, 2004, vol. 109, A11, doi: 10.1029/2003JA010234.</mixed-citation><mixed-citation xml:lang="en">Bust, G.S., Garner, T.W., Gaussiran, T.L., II Ionospheric Data Assimilation Three-Dimensional (IDA3D): A global, multisensor, electron density specification algorithm, Journal of Geophysical Research, 2004, vol. 109, A11, doi: 10.1029/2003JA010234.</mixed-citation></citation-alternatives></ref><ref id="cit22"><label>22</label><citation-alternatives><mixed-citation xml:lang="ru">Bruno, J., Mitchell, C.N., Bolmgren, K.H.A., Witvliet, B.A., A realistic simulation framework to evaluate ionospheric tomography, Advances in Space Research, 2020, vol. 65, issue 3, pp. 891–901, doi: 10.1016/j.asr.2019.11.015.</mixed-citation><mixed-citation xml:lang="en">Bruno, J., Mitchell, C.N., Bolmgren, K.H.A., Witvliet, B.A., A realistic simulation framework to evaluate ionospheric tomography, Advances in Space Research, 2020, vol. 65, issue 3, pp. 891–901, doi: 10.1016/j.asr.2019.11.015.</mixed-citation></citation-alternatives></ref><ref id="cit23"><label>23</label><citation-alternatives><mixed-citation xml:lang="ru">Bust, G.S., Mitchell, C.N., History, current state, and future directions of ionospheric imaging, Reviews of Geophysics, 2008, vol. 46, RG1003, doi: 10.1029/2006RG000212.</mixed-citation><mixed-citation xml:lang="en">Bust, G.S., Mitchell, C.N., History, current state, and future directions of ionospheric imaging, Reviews of Geophysics, 2008, vol. 46, RG1003, doi: 10.1029/2006RG000212.</mixed-citation></citation-alternatives></ref><ref id="cit24"><label>24</label><citation-alternatives><mixed-citation xml:lang="ru">Mitchell, C.N., Cannon, P.S., Multi-Instrument Data Analysis System (MIDAS) Imaging of the Ionosphere, Advances in Space Research, 2002, pp. 147–152.</mixed-citation><mixed-citation xml:lang="en">Mitchell, C.N., Cannon, P.S., Multi-Instrument Data Analysis System (MIDAS) Imaging of the Ionosphere, Advances in Space Research, 2002, pp. 147–152.</mixed-citation></citation-alternatives></ref><ref id="cit25"><label>25</label><citation-alternatives><mixed-citation xml:lang="ru">Hobiger, T., Kondo, T., Koyama, Y., Constrained simultaneous algebraic reconstruction technique (C-SART) – a new and simple algorithm applied to ionospheric tomography, Earth Planet., 2008, vol. 60, pp. 727–735, doi: 10.1186/BF03352821.</mixed-citation><mixed-citation xml:lang="en">Hobiger, T., Kondo, T., Koyama, Y., Constrained simultaneous algebraic reconstruction technique (C-SART) – a new and simple algorithm applied to ionospheric tomography, Earth Planet., 2008, vol. 60, pp. 727–735, doi: 10.1186/BF03352821.</mixed-citation></citation-alternatives></ref><ref id="cit26"><label>26</label><citation-alternatives><mixed-citation xml:lang="ru">Wen, D., Liu, S., Tang, P., Tomographic reconstruction of ionospheric electron density based on constrained algebraic reconstruction technique, GPS Solution, 2010, vol. 14, pp. 375–380, doi:10.1007/s10291-010-0161-0.</mixed-citation><mixed-citation xml:lang="en">Wen, D., Liu, S., Tang, P., Tomographic reconstruction of ionospheric electron density based on constrained algebraic reconstruction technique, GPS Solution, 2010, vol. 14, pp. 375–380, doi:10.1007/s10291-010-0161-0.</mixed-citation></citation-alternatives></ref><ref id="cit27"><label>27</label><citation-alternatives><mixed-citation xml:lang="ru">Khmarski, P.A., Naumov, A.О., Algorithms for three-dimensional reconstruction of electron concentration fields in the ionosphere using data from the global navigation satellite system, Proceedings 31st Saint Petersburg International Conference, Saint-Petersburg, 2024, pp. 185–188.</mixed-citation><mixed-citation xml:lang="en">Khmarski, P.A., Naumov, A.О., Algorithms for three-dimensional reconstruction of electron concentration fields in the ionosphere using data from the global navigation satellite system, Proceedings 31st Saint Petersburg International Conference, Saint-Petersburg, 2024, pp. 185–188.</mixed-citation></citation-alternatives></ref><ref id="cit28"><label>28</label><citation-alternatives><mixed-citation xml:lang="ru">Lu, W., Ma, G., Wan, Q., A review of voxel-based computerized ionospheric tomography with GNSS ground receivers, Remote Sensing, 2021, vol. 13, no. 3432, doi: 10.3390/rs13173432.</mixed-citation><mixed-citation xml:lang="en">Lu, W., Ma, G., Wan, Q., A review of voxel-based computerized ionospheric tomography with GNSS ground receivers, Remote Sensing, 2021, vol. 13, no. 3432, doi: 10.3390/rs13173432.</mixed-citation></citation-alternatives></ref><ref id="cit29"><label>29</label><citation-alternatives><mixed-citation xml:lang="ru">Sutton, E., Na, H., Comparison of geometries for ionospheric tomography, Radio Sci., 1995, vol. 30, pp. 115–125.</mixed-citation><mixed-citation xml:lang="en">Sutton, E., Na, H., Comparison of geometries for ionospheric tomography, Radio Sci., 1995, vol. 30, pp. 115–125.</mixed-citation></citation-alternatives></ref><ref id="cit30"><label>30</label><citation-alternatives><mixed-citation xml:lang="ru">Semeter, J., Kamalabadi, F., A natural pixel decomposition for tomographic imaging of the ionosphere, In Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, Seattle, 1998, vol. 5, pp. 2913–2916.</mixed-citation><mixed-citation xml:lang="en">Semeter, J., Kamalabadi, F., A natural pixel decomposition for tomographic imaging of the ionosphere, In Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, Seattle, 1998, vol. 5, pp. 2913–2916.</mixed-citation></citation-alternatives></ref><ref id="cit31"><label>31</label><citation-alternatives><mixed-citation xml:lang="ru">Yasyukevich, Y.V., Zatolokin, D., Padokhin, A., Wang, N., Nava, B., Li, Z., Yuan, Y., Yasyukevich, A., Chen, C., Vesnin, A., Klobuchar, NeQuickG, BDGIM, GLONASS, IRI-2016, IRI-2012, IRI-Plas, NeQuick2, and GEMTEC ionospheric models: A comparison in total electron content and positioning domains, Sensors, 2023, vol. 23, no. 10, p. 4773, doi: 10.3390/s23104773</mixed-citation><mixed-citation xml:lang="en">Yasyukevich, Y.V., Zatolokin, D., Padokhin, A., Wang, N., Nava, B., Li, Z., Yuan, Y., Yasyukevich, A., Chen, C., Vesnin, A., Klobuchar, NeQuickG, BDGIM, GLONASS, IRI-2016, IRI-2012, IRI-Plas, NeQuick2, and GEMTEC ionospheric models: A comparison in total electron content and positioning domains, Sensors, 2023, vol. 23, no. 10, p. 4773, doi: 10.3390/s23104773</mixed-citation></citation-alternatives></ref><ref id="cit32"><label>32</label><citation-alternatives><mixed-citation xml:lang="ru">International Reference Ionosphere. URL: https://ccmc.gsfc.nasa.gov/modelweb/models/iri2016_vitmo.php (дата обращения: 07.06.2023).</mixed-citation><mixed-citation xml:lang="en">International Reference Ionosphere. URL: https://ccmc.gsfc.nasa.gov/modelweb/models/iri2016_vitmo.php (дата обращения: 07.06.2023).</mixed-citation></citation-alternatives></ref><ref id="cit33"><label>33</label><citation-alternatives><mixed-citation xml:lang="ru">Иванов В.Б., Затолокин Д.А., Горбачёв О.А. Сравнение моделей полного электронного содержания ионосферы для системы ГЛОНАСС // Гироскопия и навигация. 2017. No. 2 (97). С. 89–96.</mixed-citation><mixed-citation xml:lang="en">Иванов В.Б., Затолокин Д.А., Горбачёв О.А. Сравнение моделей полного электронного содержания ионосферы для системы ГЛОНАСС // Гироскопия и навигация. 2017. No. 2 (97). С. 89–96.</mixed-citation></citation-alternatives></ref><ref id="cit34"><label>34</label><citation-alternatives><mixed-citation xml:lang="ru">Dehghan, M., Mohebbi, A., High-order compact boundary value method for the solution of unsteady convection-diffusion problems, Mathematics and Computers in Simulation, 2008, vol. 79, pp. 683–699.</mixed-citation><mixed-citation xml:lang="en">Dehghan, M., Mohebbi, A., High-order compact boundary value method for the solution of unsteady convection-diffusion problems, Mathematics and Computers in Simulation, 2008, vol. 79, pp. 683–699.</mixed-citation></citation-alternatives></ref><ref id="cit35"><label>35</label><citation-alternatives><mixed-citation xml:lang="ru">Spotz, W.F., High-order compact finite difference schemes for computational mechanics, Ph. D. Thesis, University of Texas at Austin, 1995, 324 p.</mixed-citation><mixed-citation xml:lang="en">Spotz, W.F., High-order compact finite difference schemes for computational mechanics, Ph. D. Thesis, University of Texas at Austin, 1995, 324 p.</mixed-citation></citation-alternatives></ref><ref id="cit36"><label>36</label><citation-alternatives><mixed-citation xml:lang="ru">Jia R., Yu X., Xing J., Ning Y., Sun H., An improved method using adaptive smoothing for GNSS tomographic imaging of ionosphere, PLoS ONE, 2021, vol. 16, no. 5, e0250613, doi: 10.1371/journal.pone.0250613.</mixed-citation><mixed-citation xml:lang="en">Jia R., Yu X., Xing J., Ning Y., Sun H., An improved method using adaptive smoothing for GNSS tomographic imaging of ionosphere, PLoS ONE, 2021, vol. 16, no. 5, e0250613, doi: 10.1371/journal.pone.0250613.</mixed-citation></citation-alternatives></ref></ref-list><fn-group><fn fn-type="conflict"><p>The authors declare that there are no conflicts of interest present.</p></fn></fn-group></back></article>
