Kinemaattinen inversio-ongelma pallosymmetrisellä monistolla
Tutkielman pääaiheena on maanjäristysaaltoihin ja Maan sisärakenteen tutkimiseen liittyvä käänteinen kinemaattinen ongelma. Maapalloa mallinnetaan kolmiulotteisella kompaktilla reunallisella monistolla \(\bar{B}^3(0, R)\), jonka säde normitetaan ykköseksi \(R=1\). Aaltorintamat kulkevat pitkin geodeeseja, jotka sijaitsevat kokonaan avoimessa pallossa \(B^3(0, 1)\) lukuun ottamatta päätepisteitä, jotka ovat reunalla \(S^2(0, 1)\). Symmetrioiden nojalla tarkastelu voidaan siirtää tasoon \(\mathbb{R}^2\), jossa riittää tutkia kiekon \(\bar{B}^2(0, 1)\) geodeeseja. Äänennopeus \(v=v(r)\) oletetaan isotrooppiseksi ja aidosti positiiviseksi \(C^{1,1}([0, 1])\)-funktioksi, jolle \(v^{\prime}(0)=0\). Lisäksi siltä vaaditaan Herglotz-ehto \(\frac{\text{d}}{\text{d}r}\Big(\frac{r}{v(r)}\Big)>0\) kaikilla \(r\in [0, 1]\). Näiden oletusten vallitessa, Abel-integraalin kääntyvyyttä apuna käyttäen, todistetaan tutkielman päätulos: jos aaltojen matka-ajat reunapisteiden välillä tunnetaan kaikille saapumiskulmille ja reunanopeus \(v(1)\) tiedetään, äänennopeus \(v(r)\) määräytyy datasta yksikäsitteisesti. Maapallon sisä-rakenteesta saadaan siten tietoa pelkästään reunamittauksia tekemällä.
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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.
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