Partial Global Recovery in the Elastic Travel Time Tomography Problem for Transversely Isotropic Media
[Récupération globale partielle dans le problème de la tomographie à temps de parcours élastique pour les milieux isotropes transverses]
Zou, Yuzhou1
1 Department of Mathematics Stanford University Stanford, CA 94305-2125 (U.S.A.)
Annales de l'Institut Fourier, Online first, 63 p.

Nous considérons le problème de la récupération des paramètres matériaux dans un milieu transversalement isotrope à partir des temps de parcours des ondes qP et qSV, étant donné l’axe d’isotropie et les paramètres matériaux associés à la vitesse d’onde qSH. Les opérateurs obtenus à partir de l’argument de pseudo-linéarisation sont de type parabolique, et nous discutons donc des opérateurs d’inversion dont les symboles sont de type parabolique. Nous présentons des estimations de stabilité pour récupérer soit un paramètre à partir d’une vitesse d’onde, soit deux paramètres à partir de deux vitesses d’onde avec les paramètres restants connus ou avec une relation fonctionnelle connue, et ces estimations montrent l’injectivité parmi les paramètres qui diffèrent sur des ensembles de petite largeur.

We consider the problem of recovering material parameters in a transversely isotropic medium from the qP and qSV waves’ travel times, given the axis of isotropy and the material parameters associated to the qSH wave speed. The operators obtained from the pseudolinearization argument are of parabolic type, and so we discuss inverting operators whose symbols are of parabolic type. We present stability estimates for recovering either one parameter from one wave speed or two parameters from two wave speeds with the remaining parameters either known or with a known functional relationship, and these estimates provide injectivity among parameters that differ on sets of small width.

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DOI : 10.5802/aif.3617
Classification : 74J25, 86A22, 35R30
Keywords: Travel time tomography, elastic waves, microlocal analysis.
Mot clés : Tomographie à temps de parcours, ondes élastiques, analyse microlocale.
Zou, Yuzhou 1

1 Department of Mathematics Stanford University Stanford, CA 94305-2125 (U.S.A.) 
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Zou, Yuzhou. Partial Global Recovery in the Elastic Travel Time Tomography Problem for Transversely Isotropic Media. Annales de l'Institut Fourier, Online first, 63 p.

[1] Arnolʼd, Vladimir I. Mathematical methods of classical mechanics, Graduate Texts in Mathematics, 60, Springer, 1989, xvi+508 pages | DOI | MR

[2] Caday, Peter; de Hoop, Maarten V.; Katsnelson, Vitaly; Uhlmann, Gunther Recovery of discontinuous Lamé parameters from exterior Cauchy data, Commun. Partial Differ. Equations, Volume 46 (2021) no. 4, pp. 680-715 | DOI | MR | Zbl

[3] Chapman, Chris Fundamentals of seismic wave propagation, Cambridge University Press, 2004 | DOI

[4] de Hoop, Maarten V.; Uhlmann, Gunther; Vasy, András Recovery of material parameters in transversely isotropic media, Arch. Ration. Mech. Anal., Volume 235 (2020) no. 1, pp. 141-165 | DOI | MR | Zbl

[5] Hörmander, Lars The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Grundlehren der Mathematischen Wissenschaften, 256, Springer, 1990, xii+440 pages | DOI | MR | Zbl

[6] Mazzucato, Anna L.; Rachele, Lizabeth V. On uniqueness in the inverse problem for transversely isotropic elastic media with a disjoint wave mode, Wave Motion, Volume 44 (2007) no. 7-8, pp. 605-625 | DOI | MR | Zbl

[7] Melrose, Richard B. Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Spectral and scattering theory, CRC Press, 1994 | Zbl

[8] Michel, René Sur la rigidité imposée par la longueur des géodésiques, Invent. Math., Volume 65 (1981) no. 1, pp. 71-83 | DOI | MR | Zbl

[9] Boutet de Monvel, Louis Hypoelliptic operators with double characteristics and related pseudo-differential operators, Commun. Pure Appl. Math., Volume 27 (1974), pp. 585-639 | DOI | MR | Zbl

[10] Schoenberg, Michael; de Hoop, Maarten V. Approximate dispersion relations for qP-qSV waves in transversely isotropic media, Geophysics, Volume 65 (2000), pp. 919-933 | DOI

[11] Stefanov, Plamen; Uhlmann, Gunther Rigidity for metrics with the same lengths of geodesics, Math. Res. Lett., Volume 5 (1998) no. 1-2, pp. 83-96 | DOI | MR | Zbl

[12] Stefanov, Plamen; Uhlmann, Gunther; Vasy, András Boundary rigidity with partial data, J. Am. Math. Soc., Volume 29 (2016) no. 2, pp. 299-332 | DOI | MR | Zbl

[13] Stefanov, Plamen; Uhlmann, Gunther; Vasy, András Local recovery of the compressional and shear speeds from the hyperbolic DN map, Inverse Probl., Volume 34 (2018) no. 1, 014003, 13 pages | DOI | MR | Zbl

[14] Stefanov, Plamen; Uhlmann, Gunther; Vasy, András Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge, Ann. Math., Volume 194 (2021) no. 1, pp. 1-95 | DOI | MR | Zbl

[15] Taylor, Michael E. Pseudodifferential operators, Princeton Mathematical Series, 34, Princeton University Press, 1981, xi+452 pages | DOI | MR | Zbl

[16] Tsvankin, Ilya Seismic signatures and analysis of reflection data in anisotropic media, Elsevier Science Publishers, 2001

[17] Uhlmann, Gunther; Vasy, András The inverse problem for the local geodesic ray transform, Invent. Math., Volume 205 (2016) no. 1, pp. 83-120 | DOI | MR | Zbl

[18] Vasy, András A Semiclassical Approach to Geometric X-Ray Transforms in the Presence of Convexity (2020) (https://arxiv.org/abs/2012.14307)

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