<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Publishing DTD v1.3 20210610//EN" "JATS-journalpublishing1-3.dtd">
<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="en"><front><journal-meta><journal-id journal-id-type="publisher-id">najo</journal-id><journal-title-group><journal-title xml:lang="en">Nanosystems: Physics, Chemistry, Mathematics</journal-title><trans-title-group xml:lang="ru"><trans-title>Наносистемы: физика, химия, математика</trans-title></trans-title-group></journal-title-group><issn pub-type="ppub">2220-8054</issn><issn pub-type="epub">2305-7971</issn><publisher><publisher-name>Университет ИТМО</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="doi">10.17586/2220-8054-2015-6-4-524-536</article-id><article-id custom-type="elpub" pub-id-type="custom">najo-1036</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>Boundary effect on multiple scattering of elastic waves in a half-space</article-title><trans-title-group xml:lang="ru"><trans-title>Boundary effect on multiple scattering of elastic waves in a half-space</trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Val’kov</surname><given-names>A. Yu.</given-names></name><name name-style="western" xml:lang="en"><surname>Val’kov</surname><given-names>A. Yu.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Department of Physics; Department of Mathematics</p><p>St. Petersburg</p></bio><bio xml:lang="en"><p>Department of Physics; Department of Mathematics</p><p>St. Petersburg</p></bio><email xlink:type="simple">alexvalk@mail.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Kuzmin</surname><given-names>V. L.</given-names></name><name name-style="western" xml:lang="en"><surname>Kuzmin</surname><given-names>V. L.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Department of Physics; Department of Mathematics</p><p>St. Petersburg</p></bio><bio xml:lang="en"><p>Department of Physics; Department of Mathematics</p><p>St. Petersburg</p></bio><email xlink:type="simple">kuzmin_vl@mail.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Romanov</surname><given-names>V. P.</given-names></name><name name-style="western" xml:lang="en"><surname>Romanov</surname><given-names>V. P.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Department of Physics</p><p>St. Petersburg</p></bio><bio xml:lang="en"><p>Department of Physics</p><p>St. Petersburg</p></bio><xref ref-type="aff" rid="aff-2"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Nikitina</surname><given-names>M. A.</given-names></name><name name-style="western" xml:lang="en"><surname>Nikitina</surname><given-names>M. A.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Department of Physics</p><p>St. Petersburg</p></bio><bio xml:lang="en"><p>Department of Physics</p><p>St. Petersburg</p></bio><xref ref-type="aff" rid="aff-2"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Meglinskii</surname><given-names>V.</given-names></name><name name-style="western" xml:lang="en"><surname>Meglinskii</surname><given-names>V.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Department of Physics; Faculty of Physics</p><p>Dunedin</p><p>Saratov</p></bio><bio xml:lang="en"><p>Department of Physics; Faculty of Physics</p><p>Dunedin</p><p>Saratov</p></bio><xref ref-type="aff" rid="aff-3"/></contrib></contrib-group><aff xml:lang="en" id="aff-1"><institution>St. Petersburg State University; St. Petersburg State University of Commerce and Economics</institution><country>Russian Federation</country></aff><aff xml:lang="en" id="aff-2"><institution>St. Petersburg State University</institution><country>Russian Federation</country></aff><aff xml:lang="en" id="aff-3"><institution>University of Otago; Saratov State University</institution><country>New Zealand</country></aff><pub-date pub-type="collection"><year>2015</year></pub-date><pub-date pub-type="epub"><day>16</day><month>08</month><year>2025</year></pub-date><volume>6</volume><issue>4</issue><fpage>524</fpage><lpage>536</lpage><permissions><copyright-statement>Copyright &amp;#x00A9; Val’kov A.Y., Kuzmin V.L., Romanov V.P., Nikitina M.A., Meglinskii V., 2025</copyright-statement><copyright-year>2025</copyright-year><copyright-holder xml:lang="ru">Val’kov A.Y., Kuzmin V.L., Romanov V.P., Nikitina M.A., Meglinskii V.</copyright-holder><copyright-holder xml:lang="en">Val’kov A.Y., Kuzmin V.L., Romanov V.P., Nikitina M.A., Meglinskii V.</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://nanojournal.ifmo.ru/jour/article/view/1036">https://nanojournal.ifmo.ru/jour/article/view/1036</self-uri><abstract><p>The scattering of elastic waves is studied in the vicinity of a vacuum-medium boundary. The Green’s function for a half-space is re-derived within the mixed 2D-Fourier representation, which is convenient for studying layered media. Monte-Carlo simulations of elastic wave scattering from random inhomogeneities within a simplified scalar model are performed, accounting for a boundary-induced term in the Green’s function. The multiply scattered elastic waves’ radiation is shown to decay with distance from the source much slower in vicinity of boundary than in an infinite medium, due to the boundary condition requirements.</p></abstract><kwd-group xml:lang="en"><kwd>Elastic waves</kwd><kwd>Boundary effect</kwd><kwd>Half-space</kwd><kwd>Multiple scattering</kwd><kwd>Monte-Carlo simulation</kwd></kwd-group><funding-group><funding-statement xml:lang="ru">This work was supported partially by the Russian Foundation for Basic Research (Grant No. 13-02-01255).</funding-statement><funding-statement xml:lang="en">This work was supported partially by the Russian Foundation for Basic Research (Grant No. 13-02-01255).</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">Ishimaru A. Wave Propagation and Scattering in Random Media. IEEE Press Series on Electromagnetic Wave Theory. Wiley, New York, 1999.</mixed-citation><mixed-citation xml:lang="en">Ishimaru A. Wave Propagation and Scattering in Random Media. IEEE Press Series on Electromagnetic Wave Theory. Wiley, New York, 1999.</mixed-citation></citation-alternatives></ref><ref id="cit2"><label>2</label><citation-alternatives><mixed-citation xml:lang="ru">Sir W. Thomson. Note on the integration of the equations of equilibrium of an elastic solid. Cambridge and Dublin Mathematical Journal, 1848, 3, P. 87–89.</mixed-citation><mixed-citation xml:lang="en">Sir W. Thomson. Note on the integration of the equations of equilibrium of an elastic solid. Cambridge and Dublin Mathematical Journal, 1848, 3, P. 87–89.</mixed-citation></citation-alternatives></ref><ref id="cit3"><label>3</label><citation-alternatives><mixed-citation xml:lang="ru">Sir G. G. Stokes. On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Trans. Cambridge Phil. Soc., 1849, 8, P. 287–319.</mixed-citation><mixed-citation xml:lang="en">Sir G. G. Stokes. On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Trans. Cambridge Phil. Soc., 1849, 8, P. 287–319.</mixed-citation></citation-alternatives></ref><ref id="cit4"><label>4</label><citation-alternatives><mixed-citation xml:lang="ru">Aki K. and Richards P. G.. Quantitative Seismology: Theory and Methods. University Science Books, Sausalito, CA, 2nd edition, 2002.</mixed-citation><mixed-citation xml:lang="en">Aki K. and Richards P. G.. Quantitative Seismology: Theory and Methods. University Science Books, Sausalito, CA, 2nd edition, 2002.</mixed-citation></citation-alternatives></ref><ref id="cit5"><label>5</label><citation-alternatives><mixed-citation xml:lang="ru">Felsen L. B. and Marcuvitz N. Radiation and Scattering of Waves. IEEE Press Series on Electromagnetic Wave Theory. John Wiley&amp; Sons, New York, 2003.</mixed-citation><mixed-citation xml:lang="en">Felsen L. B. and Marcuvitz N. Radiation and Scattering of Waves. IEEE Press Series on Electromagnetic Wave Theory. John Wiley&amp; Sons, New York, 2003.</mixed-citation></citation-alternatives></ref><ref id="cit6"><label>6</label><citation-alternatives><mixed-citation xml:lang="ru">Kachanov M. L., Shafiro B., and Tsukrov I. Handbook of Elasticity Solutions. Kluwer Academic Publisher, Dordrecht, 2003.</mixed-citation><mixed-citation xml:lang="en">Kachanov M. L., Shafiro B., and Tsukrov I. Handbook of Elasticity Solutions. Kluwer Academic Publisher, Dordrecht, 2003.</mixed-citation></citation-alternatives></ref><ref id="cit7"><label>7</label><citation-alternatives><mixed-citation xml:lang="ru">Aki K. Analysis of the seismic coda of local earthquakes as scattered waves. Journal of Geophysical Research, 1969, 74(2), P. 615–631.</mixed-citation><mixed-citation xml:lang="en">Aki K. Analysis of the seismic coda of local earthquakes as scattered waves. Journal of Geophysical Research, 1969, 74(2), P. 615–631.</mixed-citation></citation-alternatives></ref><ref id="cit8"><label>8</label><citation-alternatives><mixed-citation xml:lang="ru">Herraiz M. and Espinosa A. F. Coda waves: A review. Pure and Applied Geophysics, 1987, 125(4), P. 499–577.</mixed-citation><mixed-citation xml:lang="en">Herraiz M. and Espinosa A. F. Coda waves: A review. Pure and Applied Geophysics, 1987, 125(4), P. 499–577.</mixed-citation></citation-alternatives></ref><ref id="cit9"><label>9</label><citation-alternatives><mixed-citation xml:lang="ru">Snieder R. Coda wave interferometry and the equilibration of energy in elastic media. Phys. Rev. E, 2002, 66, P. 046615.</mixed-citation><mixed-citation xml:lang="en">Snieder R. Coda wave interferometry and the equilibration of energy in elastic media. Phys. Rev. E, 2002, 66, P. 046615.</mixed-citation></citation-alternatives></ref><ref id="cit10"><label>10</label><citation-alternatives><mixed-citation xml:lang="ru">Snieder R. Extracting the green’s function from the correlation of coda waves: A derivation based on stationary phase. Phys. Rev. E, 2004, 69, P. 046610.</mixed-citation><mixed-citation xml:lang="en">Snieder R. Extracting the green’s function from the correlation of coda waves: A derivation based on stationary phase. Phys. Rev. E, 2004, 69, P. 046610.</mixed-citation></citation-alternatives></ref><ref id="cit11"><label>11</label><citation-alternatives><mixed-citation xml:lang="ru">Anache-M´enier D., van Tiggelen B. A., and Margerin L. Phase statistics of seismic coda waves. Phys. Rev. Lett., 2009, 102, P. 248501.</mixed-citation><mixed-citation xml:lang="en">Anache-M´enier D., van Tiggelen B. A., and Margerin L. Phase statistics of seismic coda waves. Phys. Rev. Lett., 2009, 102, P. 248501.</mixed-citation></citation-alternatives></ref><ref id="cit12"><label>12</label><citation-alternatives><mixed-citation xml:lang="ru">van Tiggelen B. A. and Skipetrov S. E., editors. Wave Scattering in Complex Media: From Theory to Applications, volume 107 of NATO Science Series II: Mathematics, Physics and Chemistry, Kluwer Academic Publisher, London, 2003.</mixed-citation><mixed-citation xml:lang="en">van Tiggelen B. A. and Skipetrov S. E., editors. Wave Scattering in Complex Media: From Theory to Applications, volume 107 of NATO Science Series II: Mathematics, Physics and Chemistry, Kluwer Academic Publisher, London, 2003.</mixed-citation></citation-alternatives></ref><ref id="cit13"><label>13</label><citation-alternatives><mixed-citation xml:lang="ru">van Rossum M. C. W. and Nieuwenhuizen Th. M. Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion. Rev. Mod. Phys., 1999, 71, P. 313–371.</mixed-citation><mixed-citation xml:lang="en">van Rossum M. C. W. and Nieuwenhuizen Th. M. Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion. Rev. Mod. Phys., 1999, 71, P. 313–371.</mixed-citation></citation-alternatives></ref><ref id="cit14"><label>14</label><citation-alternatives><mixed-citation xml:lang="ru">Abubakirov I. R. and Gusev A.A. Estimation of scattering properties of lithosphere of kamchatka based on monte-carlo simulation of record envelope of a near earthquake. Physics of the Earth and Planetary Interiors, 1990, 64(1), P. 52–67.</mixed-citation><mixed-citation xml:lang="en">Abubakirov I. R. and Gusev A.A. Estimation of scattering properties of lithosphere of kamchatka based on monte-carlo simulation of record envelope of a near earthquake. Physics of the Earth and Planetary Interiors, 1990, 64(1), P. 52–67.</mixed-citation></citation-alternatives></ref><ref id="cit15"><label>15</label><citation-alternatives><mixed-citation xml:lang="ru">Sato H. Multiple isotropic scattering model including p-s conversions for the seismogram envelope formation. Geophysical Journal International, 1994, 117(2), P. 487–494.</mixed-citation><mixed-citation xml:lang="en">Sato H. Multiple isotropic scattering model including p-s conversions for the seismogram envelope formation. Geophysical Journal International, 1994, 117(2), P. 487–494.</mixed-citation></citation-alternatives></ref><ref id="cit16"><label>16</label><citation-alternatives><mixed-citation xml:lang="ru">Shapiro S. A. and Treitel S. Multiple scattering of seismic waves in multilayered structures. Physics of the Earth and Planetary Interiors, 1997, 104(1–3), P. 147–159.</mixed-citation><mixed-citation xml:lang="en">Shapiro S. A. and Treitel S. Multiple scattering of seismic waves in multilayered structures. Physics of the Earth and Planetary Interiors, 1997, 104(1–3), P. 147–159.</mixed-citation></citation-alternatives></ref><ref id="cit17"><label>17</label><citation-alternatives><mixed-citation xml:lang="ru">Tr´egour`es N. P., Hennino R., Lacombe C., Shapiro N. M., Margerin L., Campillo M., and van Tiggelen B. A. Multiple scattering of seismic waves. Ultrasonics, 2002, 40(1–8), P. 269–274.</mixed-citation><mixed-citation xml:lang="en">Tr´egour`es N. P., Hennino R., Lacombe C., Shapiro N. M., Margerin L., Campillo M., and van Tiggelen B. A. Multiple scattering of seismic waves. Ultrasonics, 2002, 40(1–8), P. 269–274.</mixed-citation></citation-alternatives></ref><ref id="cit18"><label>18</label><citation-alternatives><mixed-citation xml:lang="ru">Malcolm A. E., Scales J. A., and van Tiggelen B. A. Extracting the green function from diffuse, equipartitioned waves. Phys. Rev. E, 2004, 70, P. 015601.</mixed-citation><mixed-citation xml:lang="en">Malcolm A. E., Scales J. A., and van Tiggelen B. A. Extracting the green function from diffuse, equipartitioned waves. Phys. Rev. E, 2004, 70, P. 015601.</mixed-citation></citation-alternatives></ref><ref id="cit19"><label>19</label><citation-alternatives><mixed-citation xml:lang="ru">Trujilio L., Peniche F., and Jia X. Waves in Fluids and Solids, chapter 5. Multiple Scattering of Elastic Waves in Granular Media: Theory and Experiments, pages 127–152. Intech, Chanhai, China, 2011.</mixed-citation><mixed-citation xml:lang="en">Trujilio L., Peniche F., and Jia X. Waves in Fluids and Solids, chapter 5. Multiple Scattering of Elastic Waves in Granular Media: Theory and Experiments, pages 127–152. Intech, Chanhai, China, 2011.</mixed-citation></citation-alternatives></ref><ref id="cit20"><label>20</label><citation-alternatives><mixed-citation xml:lang="ru">Margerin L. Diffusion approximation with polarization and resonance effects for the modelling of seismic waves in strongly scattering small-scale media. Geophysical Journal International, 2013, 192(1), P. 326– 345.</mixed-citation><mixed-citation xml:lang="en">Margerin L. Diffusion approximation with polarization and resonance effects for the modelling of seismic waves in strongly scattering small-scale media. Geophysical Journal International, 2013, 192(1), P. 326– 345.</mixed-citation></citation-alternatives></ref><ref id="cit21"><label>21</label><citation-alternatives><mixed-citation xml:lang="ru">Hoshiba M. Simulation of multiple-scattered coda wave excitation based on the energy conservation law. Physics of the Earth and Planetary Interiors, 1991, 67(12), P. 123–136.</mixed-citation><mixed-citation xml:lang="en">Hoshiba M. Simulation of multiple-scattered coda wave excitation based on the energy conservation law. Physics of the Earth and Planetary Interiors, 1991, 67(12), P. 123–136.</mixed-citation></citation-alternatives></ref><ref id="cit22"><label>22</label><citation-alternatives><mixed-citation xml:lang="ru">Gusev A. A. and Abubakirov I. R. Vertical profile of effective turbidity reconstructed from broadening of incoherent body-wave pulses — I. General approach and the inversion procedure. Geophysical Journal International, 1999, 136(2), P. 295–308.</mixed-citation><mixed-citation xml:lang="en">Gusev A. A. and Abubakirov I. R. Vertical profile of effective turbidity reconstructed from broadening of incoherent body-wave pulses — I. General approach and the inversion procedure. Geophysical Journal International, 1999, 136(2), P. 295–308.</mixed-citation></citation-alternatives></ref><ref id="cit23"><label>23</label><citation-alternatives><mixed-citation xml:lang="ru">Ryzhik L., Papanicolaou G., and Keller J. B.. Transport equations for elastic and other waves in random media. Wave Motion, 1996, 24(4), P. 327–370.</mixed-citation><mixed-citation xml:lang="en">Ryzhik L., Papanicolaou G., and Keller J. B.. Transport equations for elastic and other waves in random media. Wave Motion, 1996, 24(4), P. 327–370.</mixed-citation></citation-alternatives></ref><ref id="cit24"><label>24</label><citation-alternatives><mixed-citation xml:lang="ru">Margerin L., Campillo M., and van Tiggelen B. A. Monte carlo simulation of multiple scattering of elastic waves. Journal of Geophysical Research, 2000, 105(B4), P. 7873–7892.</mixed-citation><mixed-citation xml:lang="en">Margerin L., Campillo M., and van Tiggelen B. A. Monte carlo simulation of multiple scattering of elastic waves. Journal of Geophysical Research, 2000, 105(B4), P. 7873–7892.</mixed-citation></citation-alternatives></ref><ref id="cit25"><label>25</label><citation-alternatives><mixed-citation xml:lang="ru">Meglinski I. V., Kuzmin V. L., Churmakov D. Y., and Greenhalgh D. A. Monte carlo simulation of coherent effects in multiple scattering. Proceeding of the Royal Society A, 2005, 461(2053), P. 43–53.</mixed-citation><mixed-citation xml:lang="en">Meglinski I. V., Kuzmin V. L., Churmakov D. Y., and Greenhalgh D. A. Monte carlo simulation of coherent effects in multiple scattering. Proceeding of the Royal Society A, 2005, 461(2053), P. 43–53.</mixed-citation></citation-alternatives></ref><ref id="cit26"><label>26</label><citation-alternatives><mixed-citation xml:lang="ru">Landau L. D. and Lifshitz E. M. Theory of Elasticity. Butterworth-Heinemann, Oxford, 3rd edition, 2007.</mixed-citation><mixed-citation xml:lang="en">Landau L. D. and Lifshitz E. M. Theory of Elasticity. Butterworth-Heinemann, Oxford, 3rd edition, 2007.</mixed-citation></citation-alternatives></ref><ref id="cit27"><label>27</label><citation-alternatives><mixed-citation xml:lang="ru">Jackson J. D. Classical Electrodynamics. Wiley, New York, 3rd. edition, 1998.</mixed-citation><mixed-citation xml:lang="en">Jackson J. D. Classical Electrodynamics. Wiley, New York, 3rd. edition, 1998.</mixed-citation></citation-alternatives></ref><ref id="cit28"><label>28</label><citation-alternatives><mixed-citation xml:lang="ru">Johnson L. R.. Green’s function for Lamb’s problem. Geophysical Journal of the Royal Astronomical Society, 1974, 37(1), P. 99–131.</mixed-citation><mixed-citation xml:lang="en">Johnson L. R.. Green’s function for Lamb’s problem. Geophysical Journal of the Royal Astronomical Society, 1974, 37(1), P. 99–131.</mixed-citation></citation-alternatives></ref><ref id="cit29"><label>29</label><citation-alternatives><mixed-citation xml:lang="ru">Lax M. and Nelson D. F.. Adventures in Green’s land: Light scattering in. anisotropic media. In L. Mandel and Emil E. Wolf, editors, Proceedings of the Third Rochester Conference on Coherence and Quantum Optics, Plenum Press, New York, 1973, P. 415–445.</mixed-citation><mixed-citation xml:lang="en">Lax M. and Nelson D. F.. Adventures in Green’s land: Light scattering in. anisotropic media. In L. Mandel and Emil E. Wolf, editors, Proceedings of the Third Rochester Conference on Coherence and Quantum Optics, Plenum Press, New York, 1973, P. 415–445.</mixed-citation></citation-alternatives></ref><ref id="cit30"><label>30</label><citation-alternatives><mixed-citation xml:lang="ru">Lax M. and Nelson D. F. Light scattering in crystals with surface corrections. In B. Bendow, V. M. Agranovich, and J. L. Birman, editors, Theory of Light Scattering in Condensed Media, Proc. of the First Joint USA-USSR Symp., Plenum Press, New York, 1976, P. 371–90.</mixed-citation><mixed-citation xml:lang="en">Lax M. and Nelson D. F. Light scattering in crystals with surface corrections. In B. Bendow, V. M. Agranovich, and J. L. Birman, editors, Theory of Light Scattering in Condensed Media, Proc. of the First Joint USA-USSR Symp., Plenum Press, New York, 1976, P. 371–90.</mixed-citation></citation-alternatives></ref><ref id="cit31"><label>31</label><citation-alternatives><mixed-citation xml:lang="ru">Val’kov A., Kuzmin V., Romanov V., Nikitina M., Kozhevnikov S., and Meglinski I. Field of a point source in a semi-infinite elastic medium. Waves in Random and Complex Media, 2012, 22(3), P. 423–434.</mixed-citation><mixed-citation xml:lang="en">Val’kov A., Kuzmin V., Romanov V., Nikitina M., Kozhevnikov S., and Meglinski I. Field of a point source in a semi-infinite elastic medium. Waves in Random and Complex Media, 2012, 22(3), P. 423–434.</mixed-citation></citation-alternatives></ref><ref id="cit32"><label>32</label><citation-alternatives><mixed-citation xml:lang="ru">Snieder R. Scattering: scattering and inverse scattering in pure and applied science, volume 1 of Scattering: Scattering and Inverse Scattering in Pure and Applied Science, chapter 1.7.1 — General Theory of Elastic Wave Scattering, Academic Press, London, 2002, P. 528–542.</mixed-citation><mixed-citation xml:lang="en">Snieder R. Scattering: scattering and inverse scattering in pure and applied science, volume 1 of Scattering: Scattering and Inverse Scattering in Pure and Applied Science, chapter 1.7.1 — General Theory of Elastic Wave Scattering, Academic Press, London, 2002, P. 528–542.</mixed-citation></citation-alternatives></ref><ref id="cit33"><label>33</label><citation-alternatives><mixed-citation xml:lang="ru">Maeda T., Sato H., and Nishimura T. Synthesis of coda wave envelopes in randomly inhomogeneous elastic media in a half-space: single scattering model including rayleigh waves. Geophysical Journal International, 2008, 172(1), P. 130–154.</mixed-citation><mixed-citation xml:lang="en">Maeda T., Sato H., and Nishimura T. Synthesis of coda wave envelopes in randomly inhomogeneous elastic media in a half-space: single scattering model including rayleigh waves. Geophysical Journal International, 2008, 172(1), P. 130–154.</mixed-citation></citation-alternatives></ref><ref id="cit34"><label>34</label><citation-alternatives><mixed-citation xml:lang="ru">Devroye L. Non-Uniform Random Variate Generation. Springer, New York, 1986.</mixed-citation><mixed-citation xml:lang="en">Devroye L. Non-Uniform Random Variate Generation. Springer, New York, 1986.</mixed-citation></citation-alternatives></ref><ref id="cit35"><label>35</label><citation-alternatives><mixed-citation xml:lang="ru">Sato H., Fehler M. C., and Maeda T. Seismic Wave Propagation and Scattering in the Heterogeneous Earth. Springer, Berlin, 2nd edition, 2012.</mixed-citation><mixed-citation xml:lang="en">Sato H., Fehler M. C., and Maeda T. Seismic Wave Propagation and Scattering in the Heterogeneous Earth. Springer, Berlin, 2nd edition, 2012.</mixed-citation></citation-alternatives></ref><ref id="cit36"><label>36</label><citation-alternatives><mixed-citation xml:lang="ru">Marston P. L. Generalized optical theorem for scatterers having inversion symmetry: Applications to acoustic backscattering. J. Acoust. Soc. Am., 2001, 109(4), P. 1291–1295.</mixed-citation><mixed-citation xml:lang="en">Marston P. L. Generalized optical theorem for scatterers having inversion symmetry: Applications to acoustic backscattering. J. Acoust. Soc. Am., 2001, 109(4), P. 1291–1295.</mixed-citation></citation-alternatives></ref><ref id="cit37"><label>37</label><citation-alternatives><mixed-citation xml:lang="ru">Margerin L. and Sato H. Generalized optical theorems for the reconstruction of green’s function of an inhomogeneous elastic medium. The Journal of the Acoustical Society of America, 2001, 130(6), P. 3674–3690.</mixed-citation><mixed-citation xml:lang="en">Margerin L. and Sato H. Generalized optical theorems for the reconstruction of green’s function of an inhomogeneous elastic medium. The Journal of the Acoustical Society of America, 2001, 130(6), P. 3674–3690.</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>
