{"id":4771,"date":"2016-01-18T14:22:41","date_gmt":"2016-01-18T12:22:41","guid":{"rendered":"http:\/\/www.geology.com.ua\/?page_id=4771"},"modified":"2017-10-26T15:33:51","modified_gmt":"2017-10-26T13:33:51","slug":"geoinformatika-2015-456-35-42","status":"publish","type":"page","link":"http:\/\/www.geology.com.ua\/en\/geoinformatika-2015-456-35-42\/","title":{"rendered":"(\u0423\u043a\u0440\u0430\u0457\u043d\u0441\u044c\u043a\u0430) Geoinformatika 2015; 4(56) : 35-42"},"content":{"rendered":"

Geoinformatika 2015; 4(56) : 35-42 (in Ukrainian)<\/span><\/em><\/p>\n

MODELING OF A WAVE FIELD PERTURBED BY LOCAL SOURCES IN VERTICALLY INHOMOGENEOUS HALF-SPACE AND CALCULATION OF SYNTHETIC SEISMOGRAMS<\/span><\/b><\/b><\/h4>\n
R.M. Pak <\/span><\/em><\/h5>\n

Hetman Petro Sahaidachny Army Academy, 32 Heroes of Maidan Str., Lviv 79012, Ukraine, e-mail: rpak@email.ua<\/span><\/em><\/p>\n

Purpose.<\/span><\/strong> The aim of the article is to develop methods calculate seismic waves perturbed by local sources in a vertically inhomogeneous medium. For this purpose the following tasks are set: to implement mathematical modeling process of disturbance and propagation of seismic field in a horizontal layered isotropic elastic medium; to construct an algorithm and a program for numerical calculation of synthetic seismograms; to conduct computational experiments for verification.
\n<\/span>Design\/methodology\/approach.<\/span><\/strong> The suggested methodology is based on the usage of Bessel-Mellin integrals, Thomson-Haskell matrix method and its modifications.
\n<\/span>Findings.<\/span><\/strong> We developed analytical approach to modeling of waves in vertically inhomogeneous isotropic elastic environments. It permits to analyze the influence of changing environmental parameters and sources in the form of concentrated arbitrarily directed forces on synthetic seismograms. We created an algorithm and a program to calculate synthetic seismograms at a free surface medium.
\n<\/span>Practical value\/implications.<\/span><\/strong> The methodology presented here enables wave processes occurring in layered medium to be conducted both analytically and numerically. The calculation of synthetic seismograms and allocation of various types of waves in them enable a wave pattern recorded during seismic observations to be analyzed and accurately interpreted.<\/span><\/p>\n

Keywords:<\/span><\/strong> mathematical modelling, seismic wave-field, matrix method, vertically heterogeneous medium, tensor of seismic moment, synthetic seismogram.<\/span><\/p>\n

\u00a0The full text of papers<\/span><\/a>\u00a0<\/strong><\/em><\/span><\/p>\n

References:<\/span><\/strong><\/p>\n

    \n
  1. Aki K., Richards P. Kolichestvennaya seysmologiya<\/i> [Quantitative Seismology]. Moscow, Mir<\/i>, 1983, 519 p.<\/span><\/li>\n
  2. Molotkov L.A. Matrichnyy metod v teorii rasprostraneniya voln v sloistykh, uprugikh i zhidkikh sredakh<\/i> [Matrix method in the theory of wave propagation in layered, elastic and liquid media]. Leningrad, Nauka<\/i>, 1984, pp. 201.<\/span><\/li>\n
  3. Molotkov L.A. O rasprostranenii uprugikh voln v sredakh, soderzhashchikh tonkie ploskoparallel\u2019nye sloi<\/i> [Propagation of elastic waves in media containing thin plane-parallel layers]. V knige \u201cVoprosy dinamicheskoy teorii rasprostraneniya seysmicheskikh voln\u201d<\/i>. [In the book \u201cQuestions dynamic theory of seismic wave propagation\u201d]. Leningrad, Publishing house of Leningrad University<\/i>, 1961, no. 5, pp. 240-280.<\/span><\/li>\n
  4. Pak R.M. Metodyka pidvyshchennia chyslovoi stiikosti dlia rozrakhunku khvylovoho polia na osnovi matrychnoho metodu Tomsona\u2013Khaskela<\/i> [Method of increasing the numerical stability of the wave field calculation based on the matrix method of Thomson-Haskell]. Geoinformatika<\/i>, 2014, no. 4, pp. 1-6.<\/span><\/li>\n
  5. Pak R.M. Modeliuvannia khvylovoho polia, zbudzhenoho hlybynnym dzherelom u vertykalno-neodnoridnomu seredovyshchi <\/i>[Modeling of wave-fields excited by deep source in a vertically-heterogeneous medium]. Geophysical Journal<\/i>, 2005, vol. 27, no. 5, pp. 887-894.<\/span><\/li>\n
  6. Pak R.M. Khvylove pole v odnoridnomu seredovyshchi dlia dzherela u vyhliadi odynarnoi syly abo podviinoi pary syl<\/i> [Wave field calculation for rupture with displacement along inner surface]. Geoinformatika<\/i>, 2004, no. 1, pp. 36-44.<\/span><\/li>\n
  7. Roganov Yu.V., Pak R.M. Predstavlenie potentsiala ot tochechnykh istochnikov dlya odnorodnoy izotropnoy sredy v vide integralov Besselya\u2013Mellina<\/i> [Representation of potentials of point sources for the homogeneous isotropic medium as Bessel-Mellin integrals]. Geophysical Journal<\/i>, 2013, vol. 35, no. 2, pp. 163-167.<\/span><\/li>\n
  8. Abo-Zena A. Dispersion function computations for unlimited frequency values. Geophysical Journal of the Royal Astronomical Society<\/i>, 1979, vol. 58, pp. 91-105.<\/span><\/li>\n
  9. Bouchon M.A. Review of the discrete wavenumber method. Pure and Applied Geophysics<\/i>, 2003, vol. 160, pp. 445-465.<\/span><\/li>\n
  10. Chapman C.H. Yet another elastic plane-wave, layer-matrix algorithm. Geophysical Journal International<\/i>, 2003, vol. 154, pp.<\/span>212-223.<\/span><\/li>\n
  11. Dunkin I.W. Computation of modal solution in layered elastic media at high frequencies. Bulletin of the Seismological Society of America<\/i>, 1965, vol. 55, pp. 335-358.<\/span><\/li>\n
  12. Kennett B.L.N. Seismic Wave Propagation in Stratified Media. Cambridge University Press<\/i>, 1983, 342 p.<\/span><\/li>\n
  13. Knopoff L.A. Matrix method for elastic wave problems. Bulletin of the Seismological Society of America<\/i>, 1964, vol. 54, pp.<\/span>431-438.<\/span><\/li>\n
  14. M<\/span>\u044c<\/span>ller G. The reflectivity method: a tutorial. Journal Geophysical<\/i>, 1985, vol. 58, pp. 153-174.<\/span><\/li>\n
  15. Thrower E.N. The computation of the dispersion of elastic waves in layered media. Journal of Sound and Vibration<\/i>, 1965, vol.<\/span>2, pp. 210-226.<\/span><\/li>\n
  16. Wang R. A simple orthonormalization method for stable and efficient computation of Green\u2019s functions. Bulletin of the Seismological Society of America<\/i>, 1999, vol. 89, pp. 733-741.<\/span><\/li>\n
  17. Watson T.N. A note on fast computation of Rayleigh wave dispersion in the multilayered halfspace. Bulletin of the Seismological Society of America<\/i>, 1970, vol. 60, pp. 161-166.<\/span><\/li>\n<\/ol>\n

    <\/p>","protected":false},"excerpt":{"rendered":"

    Geoinformatika 2015; 4(56) : 35-42 (in Ukrainian) MODELING OF A WAVE FIELD PERTURBED BY LOCAL SOURCES IN VERTICALLY INHOMOGENEOUS HALF-SPACE AND CALCULATION OF SYNTHETIC SEISMOGRAMS R.M. Pak Hetman Petro Sahaidachny Army Academy, 32 Heroes of Maidan Str., Lviv 79012, Ukraine, e-mail: rpak@email.ua Purpose. The aim of the article is to develop methods calculate seismic waves perturbed by local sources in a vertically inhomogeneous medium. For this purpose the following tasks are set: to implement mathematical modeling process of disturbance and propagation of seismic field in a horizontal layered isotropic elastic medium; to construct an algorithm and a program for numerical calculation of synthetic seismograms; to conduct computational experiments for verification. Design\/methodology\/approach. The suggested methodology is based on the usage of Bessel-Mellin integrals, Thomson-Haskell matrix method and its modifications. Findings. We developed analytical approach to modeling of waves in vertically inhomogeneous isotropic elastic environments. It permits to analyze the influence of changing environmental parameters and sources in the form of concentrated arbitrarily directed forces on synthetic seismograms. We created an algorithm and a program to calculate synthetic seismograms at a free surface medium. Practical value\/implications. The methodology presented here enables wave processes occurring in layered medium to be conducted both analytically and numerically. The calculation of synthetic seismograms and allocation of various types of waves in them enable a wave pattern recorded during seismic observations to be analyzed and accurately interpreted. Keywords: mathematical modelling, seismic wave-field, matrix method, vertically heterogeneous medium, tensor of seismic moment, synthetic seismogram. \u00a0The full text of papers\u00a0 References: Aki K., Richards P. Kolichestvennaya seysmologiya [Quantitative Seismology]. Moscow, Mir, 1983, 519 p. Molotkov L.A. Matrichnyy metod v teorii rasprostraneniya voln v sloistykh, uprugikh i zhidkikh sredakh [Matrix method in the theory of wave propagation in layered, elastic and liquid media]. Leningrad, Nauka, 1984, pp. 201. Molotkov L.A. O rasprostranenii uprugikh voln v sredakh, soderzhashchikh tonkie ploskoparallel\u2019nye sloi [Propagation of elastic waves in media containing thin plane-parallel layers]. V knige \u201cVoprosy dinamicheskoy teorii rasprostraneniya seysmicheskikh voln\u201d. [In the book \u201cQuestions dynamic theory of seismic wave propagation\u201d]. Leningrad, Publishing house of Leningrad University, 1961, no. 5, pp. 240-280. Pak R.M. Metodyka pidvyshchennia chyslovoi stiikosti dlia rozrakhunku khvylovoho polia na osnovi matrychnoho metodu Tomsona\u2013Khaskela [Method of increasing the numerical stability of the wave field calculation based on the matrix method of Thomson-Haskell]. Geoinformatika, 2014, no. 4, pp. 1-6. Pak R.M. Modeliuvannia khvylovoho polia, zbudzhenoho hlybynnym dzherelom u vertykalno-neodnoridnomu seredovyshchi [Modeling of wave-fields excited by deep source in a vertically-heterogeneous medium]. Geophysical Journal, 2005, vol. 27, no. 5, pp. 887-894. Pak R.M. Khvylove pole v odnoridnomu seredovyshchi dlia dzherela u vyhliadi odynarnoi syly abo podviinoi pary syl [Wave field calculation for rupture with displacement along inner surface]. Geoinformatika, 2004, no. 1, pp. 36-44. Roganov Yu.V., Pak R.M. Predstavlenie potentsiala ot tochechnykh istochnikov dlya odnorodnoy izotropnoy sredy v vide integralov Besselya\u2013Mellina [Representation of potentials of point sources for the homogeneous isotropic medium as Bessel-Mellin integrals]. Geophysical Journal, 2013, vol. 35, no. 2, pp. 163-167. Abo-Zena A. Dispersion function computations for unlimited frequency values. Geophysical Journal of the Royal Astronomical Society, 1979, vol. 58, pp. 91-105. Bouchon M.A. Review of the discrete wavenumber method. Pure and Applied Geophysics, 2003, vol. 160, pp. 445-465. Chapman C.H. Yet another elastic plane-wave, layer-matrix algorithm. Geophysical Journal International, 2003, vol. 154, pp.212-223. Dunkin I.W. Computation of modal solution in layered elastic media at high frequencies. Bulletin of the Seismological Society of America, 1965, vol. 55, pp. 335-358. Kennett B.L.N. Seismic Wave Propagation in Stratified Media. Cambridge University Press, 1983, 342 p. Knopoff L.A. Matrix method for elastic wave problems. Bulletin of the Seismological Society of America, 1964, vol. 54, pp.431-438. M\u044cller G. The reflectivity method: a tutorial. Journal Geophysical, 1985, vol. 58, pp. 153-174. Thrower E.N. The computation of the dispersion of elastic waves in layered media. Journal of Sound and Vibration, 1965, vol.2, pp. 210-226. Wang R. A simple orthonormalization method for stable and efficient computation of Green\u2019s functions. Bulletin of the Seismological Society of America, 1999, vol. 89, pp. 733-741. Watson T.N. A note on fast computation of Rayleigh wave dispersion in the multilayered halfspace. Bulletin of the Seismological Society of America, 1970, vol. 60, pp. 161-166.<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-4771","page","type-page","status-publish","hentry"],"yoast_head":"\n(\u0423\u043a\u0440\u0430\u0457\u043d\u0441\u044c\u043a\u0430) Geoinformatika 2015; 4(56) : 35-42 - \u0421\u0430\u0439\u0442 \u0436\u0443\u0440\u043d\u0430\u043b\u0443 \u00ab\u0413\u0435\u043e\u0456\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\u00bb<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/www.geology.com.ua\/en\/geoinformatika-2015-456-35-42\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"(\u0423\u043a\u0440\u0430\u0457\u043d\u0441\u044c\u043a\u0430) Geoinformatika 2015; 4(56) : 35-42 - \u0421\u0430\u0439\u0442 \u0436\u0443\u0440\u043d\u0430\u043b\u0443 \u00ab\u0413\u0435\u043e\u0456\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\u00bb\" \/>\n<meta property=\"og:description\" content=\"Geoinformatika 2015; 4(56) : 35-42 (in Ukrainian) MODELING OF A WAVE FIELD PERTURBED BY LOCAL SOURCES IN VERTICALLY INHOMOGENEOUS HALF-SPACE AND CALCULATION OF SYNTHETIC SEISMOGRAMS R.M. Pak Hetman Petro Sahaidachny Army Academy, 32 Heroes of Maidan Str., Lviv 79012, Ukraine, e-mail: rpak@email.ua Purpose. The aim of the article is to develop methods calculate seismic waves perturbed by local sources in a vertically inhomogeneous medium. For this purpose the following tasks are set: to implement mathematical modeling process of disturbance and propagation of seismic field in a horizontal layered isotropic elastic medium; to construct an algorithm and a program for numerical calculation of synthetic seismograms; to conduct computational experiments for verification. Design\/methodology\/approach. The suggested methodology is based on the usage of Bessel-Mellin integrals, Thomson-Haskell matrix method and its modifications. Findings. We developed analytical approach to modeling of waves in vertically inhomogeneous isotropic elastic environments. It permits to analyze the influence of changing environmental parameters and sources in the form of concentrated arbitrarily directed forces on synthetic seismograms. We created an algorithm and a program to calculate synthetic seismograms at a free surface medium. Practical value\/implications. The methodology presented here enables wave processes occurring in layered medium to be conducted both analytically and numerically. The calculation of synthetic seismograms and allocation of various types of waves in them enable a wave pattern recorded during seismic observations to be analyzed and accurately interpreted. Keywords: mathematical modelling, seismic wave-field, matrix method, vertically heterogeneous medium, tensor of seismic moment, synthetic seismogram. \u00a0The full text of papers\u00a0 References: Aki K., Richards P. Kolichestvennaya seysmologiya [Quantitative Seismology]. Moscow, Mir, 1983, 519 p. Molotkov L.A. Matrichnyy metod v teorii rasprostraneniya voln v sloistykh, uprugikh i zhidkikh sredakh [Matrix method in the theory of wave propagation in layered, elastic and liquid media]. Leningrad, Nauka, 1984, pp. 201. Molotkov L.A. O rasprostranenii uprugikh voln v sredakh, soderzhashchikh tonkie ploskoparallel\u2019nye sloi [Propagation of elastic waves in media containing thin plane-parallel layers]. V knige \u201cVoprosy dinamicheskoy teorii rasprostraneniya seysmicheskikh voln\u201d. [In the book \u201cQuestions dynamic theory of seismic wave propagation\u201d]. Leningrad, Publishing house of Leningrad University, 1961, no. 5, pp. 240-280. Pak R.M. Metodyka pidvyshchennia chyslovoi stiikosti dlia rozrakhunku khvylovoho polia na osnovi matrychnoho metodu Tomsona\u2013Khaskela [Method of increasing the numerical stability of the wave field calculation based on the matrix method of Thomson-Haskell]. Geoinformatika, 2014, no. 4, pp. 1-6. Pak R.M. Modeliuvannia khvylovoho polia, zbudzhenoho hlybynnym dzherelom u vertykalno-neodnoridnomu seredovyshchi [Modeling of wave-fields excited by deep source in a vertically-heterogeneous medium]. Geophysical Journal, 2005, vol. 27, no. 5, pp. 887-894. Pak R.M. Khvylove pole v odnoridnomu seredovyshchi dlia dzherela u vyhliadi odynarnoi syly abo podviinoi pary syl [Wave field calculation for rupture with displacement along inner surface]. Geoinformatika, 2004, no. 1, pp. 36-44. Roganov Yu.V., Pak R.M. Predstavlenie potentsiala ot tochechnykh istochnikov dlya odnorodnoy izotropnoy sredy v vide integralov Besselya\u2013Mellina [Representation of potentials of point sources for the homogeneous isotropic medium as Bessel-Mellin integrals]. Geophysical Journal, 2013, vol. 35, no. 2, pp. 163-167. Abo-Zena A. Dispersion function computations for unlimited frequency values. Geophysical Journal of the Royal Astronomical Society, 1979, vol. 58, pp. 91-105. Bouchon M.A. Review of the discrete wavenumber method. Pure and Applied Geophysics, 2003, vol. 160, pp. 445-465. Chapman C.H. Yet another elastic plane-wave, layer-matrix algorithm. Geophysical Journal International, 2003, vol. 154, pp.212-223. Dunkin I.W. Computation of modal solution in layered elastic media at high frequencies. Bulletin of the Seismological Society of America, 1965, vol. 55, pp. 335-358. Kennett B.L.N. Seismic Wave Propagation in Stratified Media. Cambridge University Press, 1983, 342 p. Knopoff L.A. Matrix method for elastic wave problems. Bulletin of the Seismological Society of America, 1964, vol. 54, pp.431-438. M\u044cller G. The reflectivity method: a tutorial. Journal Geophysical, 1985, vol. 58, pp. 153-174. Thrower E.N. The computation of the dispersion of elastic waves in layered media. Journal of Sound and Vibration, 1965, vol.2, pp. 210-226. Wang R. A simple orthonormalization method for stable and efficient computation of Green\u2019s functions. Bulletin of the Seismological Society of America, 1999, vol. 89, pp. 733-741. Watson T.N. A note on fast computation of Rayleigh wave dispersion in the multilayered halfspace. Bulletin of the Seismological Society of America, 1970, vol. 60, pp. 161-166.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/www.geology.com.ua\/en\/geoinformatika-2015-456-35-42\/\" \/>\n<meta property=\"og:site_name\" content=\"\u0421\u0430\u0439\u0442 \u0436\u0443\u0440\u043d\u0430\u043b\u0443 \u00ab\u0413\u0435\u043e\u0456\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\u00bb\" \/>\n<meta property=\"article:modified_time\" content=\"2017-10-26T13:33:51+00:00\" \/>\n<meta name=\"twitter:label1\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data1\" content=\"4 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/www.geology.com.ua\\\/en\\\/geoinformatika-2015-456-35-42\\\/\",\"url\":\"http:\\\/\\\/www.geology.com.ua\\\/en\\\/geoinformatika-2015-456-35-42\\\/\",\"name\":\"(\u0423\u043a\u0440\u0430\u0457\u043d\u0441\u044c\u043a\u0430) Geoinformatika 2015; 4(56) : 35-42 - \u0421\u0430\u0439\u0442 \u0436\u0443\u0440\u043d\u0430\u043b\u0443 \u00ab\u0413\u0435\u043e\u0456\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\u00bb\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/www.geology.com.ua\\\/en\\\/#website\"},\"datePublished\":\"2016-01-18T12:22:41+00:00\",\"dateModified\":\"2017-10-26T13:33:51+00:00\",\"breadcrumb\":{\"@id\":\"http:\\\/\\\/www.geology.com.ua\\\/en\\\/geoinformatika-2015-456-35-42\\\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[[\"http:\\\/\\\/www.geology.com.ua\\\/en\\\/geoinformatika-2015-456-35-42\\\/\"]]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"http:\\\/\\\/www.geology.com.ua\\\/en\\\/geoinformatika-2015-456-35-42\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"http:\\\/\\\/www.geology.com.ua\\\/en\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"(\u0423\u043a\u0440\u0430\u0457\u043d\u0441\u044c\u043a\u0430) Geoinformatika 2015; 4(56) : 35-42\"}]},{\"@type\":\"WebSite\",\"@id\":\"http:\\\/\\\/www.geology.com.ua\\\/en\\\/#website\",\"url\":\"http:\\\/\\\/www.geology.com.ua\\\/en\\\/\",\"name\":\"\u0421\u0430\u0439\u0442 \u0436\u0443\u0440\u043d\u0430\u043b\u0443 \u00ab\u0413\u0435\u043e\u0456\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\u00bb\",\"description\":\"\u0426\u0435\u043d\u0442\u0440 \u043c\u0435\u043d\u0435\u0434\u0436\u043c\u0435\u043d\u0442\u0443 \u0442\u0430 \u043c\u0430\u0440\u043a\u0435\u0442\u0438\u043d\u0433\u0443 \u0432 \u0433\u0430\u043b\u0443\u0437\u0456 \u043d\u0430\u0443\u043a \u043f\u0440\u043e \u0417\u0435\u043c\u043b\u044e\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"http:\\\/\\\/www.geology.com.ua\\\/en\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"(\u0423\u043a\u0440\u0430\u0457\u043d\u0441\u044c\u043a\u0430) Geoinformatika 2015; 4(56) : 35-42 - \u0421\u0430\u0439\u0442 \u0436\u0443\u0440\u043d\u0430\u043b\u0443 \u00ab\u0413\u0435\u043e\u0456\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\u00bb","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"http:\/\/www.geology.com.ua\/en\/geoinformatika-2015-456-35-42\/","og_locale":"en_US","og_type":"article","og_title":"(\u0423\u043a\u0440\u0430\u0457\u043d\u0441\u044c\u043a\u0430) Geoinformatika 2015; 4(56) : 35-42 - \u0421\u0430\u0439\u0442 \u0436\u0443\u0440\u043d\u0430\u043b\u0443 \u00ab\u0413\u0435\u043e\u0456\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\u00bb","og_description":"Geoinformatika 2015; 4(56) : 35-42 (in Ukrainian) MODELING OF A WAVE FIELD PERTURBED BY LOCAL SOURCES IN VERTICALLY INHOMOGENEOUS HALF-SPACE AND CALCULATION OF SYNTHETIC SEISMOGRAMS R.M. Pak Hetman Petro Sahaidachny Army Academy, 32 Heroes of Maidan Str., Lviv 79012, Ukraine, e-mail: rpak@email.ua Purpose. The aim of the article is to develop methods calculate seismic waves perturbed by local sources in a vertically inhomogeneous medium. For this purpose the following tasks are set: to implement mathematical modeling process of disturbance and propagation of seismic field in a horizontal layered isotropic elastic medium; to construct an algorithm and a program for numerical calculation of synthetic seismograms; to conduct computational experiments for verification. Design\/methodology\/approach. The suggested methodology is based on the usage of Bessel-Mellin integrals, Thomson-Haskell matrix method and its modifications. Findings. We developed analytical approach to modeling of waves in vertically inhomogeneous isotropic elastic environments. It permits to analyze the influence of changing environmental parameters and sources in the form of concentrated arbitrarily directed forces on synthetic seismograms. We created an algorithm and a program to calculate synthetic seismograms at a free surface medium. Practical value\/implications. The methodology presented here enables wave processes occurring in layered medium to be conducted both analytically and numerically. The calculation of synthetic seismograms and allocation of various types of waves in them enable a wave pattern recorded during seismic observations to be analyzed and accurately interpreted. Keywords: mathematical modelling, seismic wave-field, matrix method, vertically heterogeneous medium, tensor of seismic moment, synthetic seismogram. \u00a0The full text of papers\u00a0 References: Aki K., Richards P. Kolichestvennaya seysmologiya [Quantitative Seismology]. Moscow, Mir, 1983, 519 p. Molotkov L.A. Matrichnyy metod v teorii rasprostraneniya voln v sloistykh, uprugikh i zhidkikh sredakh [Matrix method in the theory of wave propagation in layered, elastic and liquid media]. Leningrad, Nauka, 1984, pp. 201. Molotkov L.A. O rasprostranenii uprugikh voln v sredakh, soderzhashchikh tonkie ploskoparallel\u2019nye sloi [Propagation of elastic waves in media containing thin plane-parallel layers]. V knige \u201cVoprosy dinamicheskoy teorii rasprostraneniya seysmicheskikh voln\u201d. [In the book \u201cQuestions dynamic theory of seismic wave propagation\u201d]. Leningrad, Publishing house of Leningrad University, 1961, no. 5, pp. 240-280. Pak R.M. Metodyka pidvyshchennia chyslovoi stiikosti dlia rozrakhunku khvylovoho polia na osnovi matrychnoho metodu Tomsona\u2013Khaskela [Method of increasing the numerical stability of the wave field calculation based on the matrix method of Thomson-Haskell]. Geoinformatika, 2014, no. 4, pp. 1-6. Pak R.M. Modeliuvannia khvylovoho polia, zbudzhenoho hlybynnym dzherelom u vertykalno-neodnoridnomu seredovyshchi [Modeling of wave-fields excited by deep source in a vertically-heterogeneous medium]. Geophysical Journal, 2005, vol. 27, no. 5, pp. 887-894. Pak R.M. Khvylove pole v odnoridnomu seredovyshchi dlia dzherela u vyhliadi odynarnoi syly abo podviinoi pary syl [Wave field calculation for rupture with displacement along inner surface]. Geoinformatika, 2004, no. 1, pp. 36-44. Roganov Yu.V., Pak R.M. Predstavlenie potentsiala ot tochechnykh istochnikov dlya odnorodnoy izotropnoy sredy v vide integralov Besselya\u2013Mellina [Representation of potentials of point sources for the homogeneous isotropic medium as Bessel-Mellin integrals]. Geophysical Journal, 2013, vol. 35, no. 2, pp. 163-167. Abo-Zena A. Dispersion function computations for unlimited frequency values. Geophysical Journal of the Royal Astronomical Society, 1979, vol. 58, pp. 91-105. Bouchon M.A. Review of the discrete wavenumber method. Pure and Applied Geophysics, 2003, vol. 160, pp. 445-465. Chapman C.H. Yet another elastic plane-wave, layer-matrix algorithm. Geophysical Journal International, 2003, vol. 154, pp.212-223. Dunkin I.W. Computation of modal solution in layered elastic media at high frequencies. Bulletin of the Seismological Society of America, 1965, vol. 55, pp. 335-358. Kennett B.L.N. Seismic Wave Propagation in Stratified Media. Cambridge University Press, 1983, 342 p. Knopoff L.A. Matrix method for elastic wave problems. Bulletin of the Seismological Society of America, 1964, vol. 54, pp.431-438. M\u044cller G. The reflectivity method: a tutorial. Journal Geophysical, 1985, vol. 58, pp. 153-174. Thrower E.N. The computation of the dispersion of elastic waves in layered media. Journal of Sound and Vibration, 1965, vol.2, pp. 210-226. Wang R. A simple orthonormalization method for stable and efficient computation of Green\u2019s functions. Bulletin of the Seismological Society of America, 1999, vol. 89, pp. 733-741. Watson T.N. A note on fast computation of Rayleigh wave dispersion in the multilayered halfspace. 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