## Generic uniqueness and stability for the mixed ray transform

HTML articles powered by AMS MathViewer

- by Maarten V. de Hoop, Teemu Saksala, Gunther Uhlmann and Jian Zhai PDF
- Trans. Amer. Math. Soc.
**374**(2021), 6085-6144 Request permission

## Abstract:

We consider the mixed ray transform of tensor fields on a three-dimensional compact simple Riemannian manifold with boundary. We prove the injectivity of the transform, up to natural obstructions, and establish stability estimates for the normal operator on generic three dimensional simple manifold in the case of $1+1$ and $2+2$ tensors fields.

We show how the anisotropic perturbations of averaged isotopic travel-times of $qS$-polarized elastic waves provide partial information about the mixed ray transform of $2+2$ tensors fields. If in addition we include the measurement of the shear wave amplitude, the complete mixed ray transform can be recovered. We also show how one can obtain the mixed ray transform from an anisotropic perturbation of the \text{Dirichlet-to-Neumann} map of an isotropic elastic wave equation on a smooth and bounded domain in three dimensional Euclidean space.

## References

- Robert A. Adams and John J. F. Fournier,
*Sobolev spaces*, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR**2424078** - Yu. E. Anikonov and V. G. Romanov,
*On uniqueness of determination of a form of first degree by its integrals along geodesics*, J. Inverse Ill-Posed Probl.**5**(1997), no. 6, 487–490 (1998). MR**1623603**, DOI 10.1515/jiip.1997.5.6.487 - Vassily M. Babich and Aleksei P. Kiselev,
*Elastic waves*, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2018. High frequency theory; Translated from the 2014 Russian edition by Irina A. So. MR**3822378**, DOI 10.1201/b21845 - M. I. Belishev,
*An approach to multidimensional inverse problems for the wave equation*, Dokl. Akad. Nauk SSSR**297**(1987), no. 3, 524–527 (Russian); English transl., Soviet Math. Dokl.**36**(1988), no. 3, 481–484. MR**924687** - Michael I. Belishev and Yaroslav V. Kurylev,
*To the reconstruction of a Riemannian manifold via its spectral data (BC-method)*, Comm. Partial Differential Equations**17**(1992), no. 5-6, 767–804. MR**1177292**, DOI 10.1080/03605309208820863 - Sombuddha Bhattacharyya,
*Local uniqueness of the density from partial boundary data for isotropic elastodynamics*, Inverse Problems**34**(2018), no. 12, 125001, 10. MR**3861940**, DOI 10.1088/1361-6420/aade76 - Dmitri Burago and Sergei Ivanov,
*Boundary rigidity and filling volume minimality of metrics close to a flat one*, Ann. of Math. (2)**171**(2010), no. 2, 1183–1211. MR**2630062**, DOI 10.4007/annals.2010.171.1183 - V. Červený,
*Seismic ray theory*, Cambridge University Press, Cambridge, 2001. MR**1838111**, DOI 10.1017/CBO9780511529399 - V. Červeny and J. Jech,
*Linearized solutions of kinematic problems of seismic body waves in inhomogeneous slightly anisotropic media*, Journal of Geophysics, 51(1):96–104, 1982. - C. H. Chapman and R. G. Pratt,
*Traveltime tomography in anisotropic media – I. Theory*, Geophysical Journal International, 109(1):1–19, 1992. - Myung Seob Son and Yeon June Kang,
*Shear wave propagation in a layered poroelastic structure*, Wave Motion**49**(2012), no. 4, 490–500. MR**2911501**, DOI 10.1016/j.wavemoti.2012.02.001 - Christopher B. Croke,
*Rigidity and the distance between boundary points*, J. Differential Geom.**33**(1991), no. 2, 445–464. MR**1094465** - N. S. Dairbekov and V. A. Sharafutdinov,
*Conformal Killing symmetric tensor fields on Riemannian manifolds*, Mat. Tr.**13**(2010), no. 1, 85–145 (Russian, with Russian summary); English transl., Siberian Adv. Math.**21**(2011), no. 1, 1–41. MR**2682769**, DOI 10.3103/s1055134411010019 - Nurlan S. Dairbekov,
*Integral geometry problem for nontrapping manifolds*, Inverse Problems**22**(2006), no. 2, 431–445. MR**2216407**, DOI 10.1088/0266-5611/22/2/003 - M. V. De Hoop, J. Ilmavirta, M. Lassas, and T. Saksala,
*Inverse problem for compact Finsler manifolds with the boundary distance map*, arXiv:1901.03902, 2019. - Maarten V. de Hoop, Gen Nakamura, and Jian Zhai,
*Unique recovery of piecewise analytic density and stiffness tensor from the elastic-wave Dirichlet-to-Neumann map*, SIAM J. Appl. Math.**79**(2019), no. 6, 2359–2384. MR**4039538**, DOI 10.1137/18M1232802 - Maarten V. de Hoop, Teemu Saksala, and Jian Zhai,
*Mixed ray transform on simple 2-dimensional Riemannian manifolds*, Proc. Amer. Math. Soc.**147**(2019), no. 11, 4901–4913. MR**4011522**, DOI 10.1090/proc/14601 - Maarten V. de Hoop, Gunther Uhlmann, and Jian Zhai,
*Inverting the local geodesic ray transform of higher rank tensors*, Inverse Problems**35**(2019), no. 11, 115009, 27. MR**4027420**, DOI 10.1088/1361-6420/ab1ace - J. J. Duistermaat and L. Hörmander,
*Fourier integral operators. II*, Acta Math.**128**(1972), no. 3-4, 183–269. MR**388464**, DOI 10.1007/BF02392165 - Lawrence C. Evans,
*Partial differential equations*, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR**2597943**, DOI 10.1090/gsm/019 - A. Feizmohammadi, J. Ilmavirta, Y. Kian, and L. Oksanen,
*Recovery of time dependent coefficients from boundary data for hyperbolic equations*, arXiv:1901.04211, 2019. - A. Feizmohammadi, J. Ilmavirta, and L. Oksanen,
*The light ray transform in stationary and static Lorentzian geometries*, J. Geom. Anal.**31**(2021) 3656-3682, DOI: 10.1007/s12220-020-00409-y. - Allan Greenleaf and Gunther Uhlmann,
*Recovering singularities of a potential from singularities of scattering data*, Comm. Math. Phys.**157**(1993), no. 3, 549–572. MR**1243710** - Sönke Hansen and Gunther Uhlmann,
*Propagation of polarization in elastodynamics with residual stress and travel times*, Math. Ann.**326**(2003), no. 3, 563–587. MR**1992278**, DOI 10.1007/s00208-003-0437-6 - Sean Holman,
*Generic uniqueness in polarization tomography*, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–University of Washington. MR**2718047** - Lars Hörmander,
*The analysis of linear partial differential operators. III*, Classics in Mathematics, Springer, Berlin, 2007. Pseudo-differential operators; Reprint of the 1994 edition. MR**2304165**, DOI 10.1007/978-3-540-49938-1 - Erwin Kreyszig,
*Introductory functional analysis with applications*, John Wiley & Sons, New York-London-Sydney, 1978. MR**0467220** - Venkateswaran P. Krishnan, Rohit K. Mishra, and François Monard,
*On solenoidal-injective and injective ray transforms of tensor fields on surfaces*, J. Inverse Ill-Posed Probl.**27**(2019), no. 4, 527–538. MR**3987895**, DOI 10.1515/jiip-2018-0067 - Matti Lassas, Vladimir Sharafutdinov, and Gunther Uhlmann,
*Semiglobal boundary rigidity for Riemannian metrics*, Math. Ann.**325**(2003), no. 4, 767–793. MR**1974568**, DOI 10.1007/s00208-002-0407-4 - René Michel,
*Sur la rigidité imposée par la longueur des géodésiques*, Invent. Math.**65**(1981/82), no. 1, 71–83 (French). MR**636880**, DOI 10.1007/BF01389295 - R. G. Mukhometov,
*On the problem of integral geometry (Russian)*, Math. problems of geophysics. Akad. Nauk SSSR, Sibirsk., Otdel., Vychisl., Tsentr, Novosibirsk, 6, 1975. - R. G. Muhometov,
*On a problem of reconstructing Riemannian metrics*, Sibirsk. Mat. Zh.**22**(1981), no. 3, 119–135, 237 (Russian). MR**621466** - L. Oksanen, M. Salo, P. Stefanov, and G. Uhlmann,
*Inverse problems for real principal type operators*, arXiv:2001.07599, 2020. - Jean-Pierre Otal,
*Le spectre marqué des longueurs des surfaces à courbure négative*, Ann. of Math. (2)**131**(1990), no. 1, 151–162 (French). MR**1038361**, DOI 10.2307/1971511 - Gabriel P. Paternain, Mikko Salo, and Gunther Uhlmann,
*Tensor tomography on surfaces*, Invent. Math.**193**(2013), no. 1, 229–247. MR**3069117**, DOI 10.1007/s00222-012-0432-1 - L. Pestov,
*Well-posedness questions of the ray tomography problems*(in Russian). Siberian Science Press, Novosibirsk, 2003. - L. N. Pestov and V. A. Sharafutdinov,
*Integral geometry of tensor fields on a manifold of negative curvature*, Sibirsk. Mat. Zh.**29**(1988), no. 3, 114–130, 221 (Russian); English transl., Siberian Math. J.**29**(1988), no. 3, 427–441 (1989). MR**953028**, DOI 10.1007/BF00969652 - Leonid Pestov and Gunther Uhlmann,
*Two dimensional compact simple Riemannian manifolds are boundary distance rigid*, Ann. of Math. (2)**161**(2005), no. 2, 1093–1110. MR**2153407**, DOI 10.4007/annals.2005.161.1093 - Lizabeth V. Rachele,
*An inverse problem in elastodynamics: uniqueness of the wave speeds in the interior*, J. Differential Equations**162**(2000), no. 2, 300–325. MR**1751708**, DOI 10.1006/jdeq.1999.3657 - Lizabeth V. Rachele,
*Uniqueness of the density in an inverse problem for isotropic elastodynamics*, Trans. Amer. Math. Soc.**355**(2003), no. 12, 4781–4806. MR**1997584**, DOI 10.1090/S0002-9947-03-03268-9 - Vladimir Sharafutdinov,
*Ray transform and some rigidity problems for Riemannian metrics*, Geometric methods in inverse problems and PDE control, IMA Vol. Math. Appl., vol. 137, Springer, New York, 2004, pp. 215–238. MR**2169905**, DOI 10.1007/978-1-4684-9375-7_{7} - V. A. Sharafutdinov,
*Integral geometry of tensor fields*, Inverse and Ill-posed Problems Series, VSP, Utrecht, 1994. MR**1374572**, DOI 10.1515/9783110900095 - Plamen Stefanov and Gunther Uhlmann,
*Stability estimates for the X-ray transform of tensor fields and boundary rigidity*, Duke Math. J.**123**(2004), no. 3, 445–467. MR**2068966**, DOI 10.1215/S0012-7094-04-12332-2 - Plamen Stefanov and Gunther Uhlmann,
*Boundary rigidity and stability for generic simple metrics*, J. Amer. Math. Soc.**18**(2005), no. 4, 975–1003. MR**2163868**, DOI 10.1090/S0894-0347-05-00494-7 - Plamen Stefanov, Gunther Uhlmann, and Andras Vasy,
*Boundary rigidity with partial data*, J. Amer. Math. Soc.**29**(2016), no. 2, 299–332. MR**3454376**, DOI 10.1090/jams/846 - Plamen Stefanov, Gunther Uhlmann, and Andras Vasy,
*Local recovery of the compressional and shear speeds from the hyperbolic DN map*, Inverse Problems**34**(2018), no. 1, 014003, 13. MR**3742360**, DOI 10.1088/1361-6420/aa9833 - Plamen Stefanov, Gunther Uhlmann, and András Vasy,
*Inverting the local geodesic X-ray transform on tensors*, J. Anal. Math.**136**(2018), no. 1, 151–208. MR**3892472**, DOI 10.1007/s11854-018-0058-3 - François Trèves,
*Introduction to pseudodifferential and Fourier integral operators. Vol. 1*, University Series in Mathematics, Plenum Press, New York-London, 1980. Pseudodifferential operators. MR**597144** - Gunther Uhlmann and András Vasy,
*The inverse problem for the local geodesic ray transform*, Invent. Math.**205**(2016), no. 1, 83–120. MR**3514959**, DOI 10.1007/s00222-015-0631-7 - G. Uhlmann and J. Zhai,
*On an inverse boundary value problem for a nonlinear elastic wave equation*, arXiv:1912.11756, 2019. - J. T. Wloka, B. Rowley, and B. Lawruk,
*Boundary value problems for elliptic systems*, Cambridge University Press, Cambridge, 1995. MR**1343490**, DOI 10.1017/CBO9780511662850 - Yang Yang and Jian Zhai,
*Unique determination of a transversely isotropic perturbation in a linearized inverse boundary value problem for elasticity*, Inverse Probl. Imaging**13**(2019), no. 6, 1309–1325. MR**4027034**, DOI 10.3934/ipi.2019057

## Additional Information

**Maarten V. de Hoop**- Affiliation: Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005
- MR Author ID: 311568
- Email: mdehoop@rice.edu
**Teemu Saksala**- Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695
- MR Author ID: 1277799
- ORCID: 0000-0002-3785-9623
- Email: tssaksal@ncsu.edu
**Gunther Uhlmann**- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195; and Institute for Advanced Study, The Hong Kong University of Science and Technology, Kowloon, Hong Kong, People’s Republic of China
- MR Author ID: 175790
- Email: gunther@math.washington.edu
**Jian Zhai**- Affiliation: Institute for Advanced Study, The Hong Kong University of Science and Technology, Kowloon, Hong Kong, People’s Republic of China
- MR Author ID: 1206056
- ORCID: 0000-0002-2374-8922
- Email: jian.zhai@outlook.com, iasjzhai@ust.hk
- Received by editor(s): April 1, 2020
- Received by editor(s) in revised form: June 16, 2020, and August 16, 2020
- Published electronically: June 7, 2021
- Additional Notes: The first author was supported by the Simons Foundation under the MATH + X program, the National Science Foundation under grant DMS-1815143, and the corporate members of the Geo-Mathematical Imaging Group at Rice University.

The second author was supported by the Simons Foundation under the MATH + X program and the corporate members of the Geo-Mathematical Imaging Group at Rice University. Part of this work was carried out during his visit to University of Washington, and he is grateful for hospitality and support.

The third author was partially supported by NSF, a Walker Professorship at UW and a Si-Yuan Professorship at IAS, HKUST - © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 6085-6144 - MSC (2020): Primary 53C22, 53C65
- DOI: https://doi.org/10.1090/tran/8342
- MathSciNet review: 4302156