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[{"key": "dc.contributor.advisor", "value": "Ilmavirta, Joonas", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "M\u00f6nkk\u00f6nen, Keijo", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2017-05-04T15:43:20Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2017-05-04T15:43:20Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2017", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1700343", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/53788", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Tutkielman p\u00e4\u00e4aiheena on maanj\u00e4ristysaaltoihin ja Maan sis\u00e4rakenteen tutkimiseen liittyv\u00e4 k\u00e4\u00e4nteinen kinemaattinen ongelma. Maapalloa mallinnetaan kolmiulotteisella kompaktilla reunallisella monistolla \\(\\bar{B}^3(0, R)\\), jonka s\u00e4de normitetaan ykk\u00f6seksi \\(R=1\\). Aaltorintamat kulkevat pitkin geodeeseja, jotka sijaitsevat kokonaan avoimessa pallossa \\(B^3(0, 1)\\) lukuun ottamatta p\u00e4\u00e4tepisteit\u00e4, jotka ovat reunalla \\(S^2(0, 1)\\). Symmetrioiden nojalla tarkastelu voidaan siirt\u00e4\u00e4 tasoon \\(\\mathbb{R}^2\\), jossa riitt\u00e4\u00e4 tutkia kiekon \\(\\bar{B}^2(0, 1)\\) geodeeseja. \u00c4\u00e4nennopeus \\(v=v(r)\\) oletetaan isotrooppiseksi ja aidosti positiiviseksi \\(C^{1,1}([0, 1])\\)-funktioksi, jolle \\(v^{\\prime}(0)=0\\). Lis\u00e4ksi silt\u00e4 vaaditaan Herglotz-ehto \\(\\frac{\\text{d}}{\\text{d}r}\\Big(\\frac{r}{v(r)}\\Big)>0\\) kaikilla \\(r\\in [0, 1]\\). N\u00e4iden oletusten vallitessa, Abel-integraalin k\u00e4\u00e4ntyvyytt\u00e4 apuna k\u00e4ytt\u00e4en, todistetaan tutkielman p\u00e4\u00e4tulos: jos aaltojen matka-ajat reunapisteiden v\u00e4lill\u00e4 tunnetaan kaikille saapumiskulmille ja reunanopeus \\(v(1)\\) tiedet\u00e4\u00e4n, \u00e4\u00e4nennopeus \\(v(r)\\) m\u00e4\u00e4r\u00e4ytyy datasta yksik\u00e4sitteisesti. Maapallon sis\u00e4-rakenteesta saadaan siten tietoa pelk\u00e4st\u00e4\u00e4n reunamittauksia tekem\u00e4ll\u00e4.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "The main subject of the thesis is the inverse kinematic problem related to seismic waves and the study of the inner structure of the Earth. The Earth is modelled by three-dimensional compact manifold with boundary \\(\\bar{B}^3(0, R)\\) whose radius is normed to one \\(R=1\\). The wave fronts travel along geodesics which completely lie in the open ball \\(B^3(0, 1)\\) except the endpoints which are on the boundary \\(S^2(0, 1)\\). By symmetry arguments the treatment can be transferred to the plane \\(\\mathbb{R}^2\\), where it is enough to study the geodesics of the disk \\(\\bar{B}^2(0, 1)\\). The speed of sound \\(v=v(r)\\) is assumed to be isotropic and strictly positive \\(C^{1,1}([0, 1])\\)-function for which \\(v^{\\prime}(0)=0\\). In addition it is required to satisfy the Herglotz-condition \\(\\frac{\\text{d}}{\\text{d}r}\\Big(\\frac{r}{v(r)}\\Big)>0\\) for all \\(r\\in [0, 1]\\). Under these assumptions, with the help of the invertibility of Abel-integral, we prove the main result of the thesis: if the travel-times between boundary points are known for all arrival angles and the boundary speed \\(v(1)\\) is known, the speed of sound \\(v(r)\\) is determined uniquely from the data. One can thus get information about the inner structure of the Earth by only doing boundary measurements.", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Keijo M\u00f6nkk\u00f6nen (kematamo) on 2017-05-04 15:43:19.921066. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija.group@korppi.jyu.fi) on 2017-05-04T15:43:20Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201705042197.pdf: 247621 bytes, checksum: f888203190eedfd2046c2ceb144e7872 (MD5)\nlicense.html: 4824 bytes, checksum: a1dc92842817301f108ab502ebef7fbb (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2017-05-04T15:43:20Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201705042197.pdf: 247621 bytes, checksum: f888203190eedfd2046c2ceb144e7872 (MD5)\nlicense.html: 4824 bytes, checksum: a1dc92842817301f108ab502ebef7fbb (MD5)\n Previous issue date: 2017", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (31 sivua)", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "riemannin geometria", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Kinemaattinen inversio-ongelma pallosymmetrisell\u00e4 monistolla", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201705042197", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.date.updated", "value": "2017-05-04T15:43:20Z", "language": null, "element": "date", "qualifier": "updated", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": null, "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": "fi", "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "inversio-ongelmat", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "seismologia", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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