by Joonas Ilmavirta, Keijo Mönkkönen
Abstract:
We show that the normal operator of the X-ray transform in $\mathbb R^d$, $d\geq2$, has a unique continuation property in the class of compactly supported distributions. This immediately implies uniqueness for the X-ray tomography problem with partial data and generalizes some earlier results to higher dimensions. Our proof also gives a unique continuation property for certain Riesz potentials in the space of rapidly decreasing distributions. We present applications to local and global seismology. These include linearized travel time tomography with half-local data and global tomography based on shear wave splitting in a weakly anisotropic elastic medium.
Reference:
Unique continuation of the normal operator of the X-ray transform and applications in geophysics (Joonas Ilmavirta, Keijo Mönkkönen), Inverse Problems, IOP Publishing, volume 36, number 4, pp. 045014, 2020.
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We show that the normal operator of the X-ray transform in $\mathbb R^d$, $d\geq2$, has a unique continuation property in the class of compactly supported distributions. This immediately implies uniqueness for the X-ray tomography problem with partial data and generalizes some earlier results to higher dimensions. Our proof also gives a unique continuation property for certain Riesz potentials in the space of rapidly decreasing distributions. We present applications to local and global seismology. These include linearized travel time tomography with half-local data and global tomography based on shear wave splitting in a weakly anisotropic elastic medium.
[arXiv]
Bibtex Entry:
@article{xrt-no-uc,
author = {Joonas Ilmavirta and Keijo M{\"o}nkk{\"o}nen},
title = {{Unique continuation of the normal operator of the X-ray transform and applications in geophysics}},
year = 2020,
month = mar,
publisher = {{IOP} Publishing},
journal = {Inverse Problems},
volume = {36},
number = {4},
pages = {045014},
arxiv = {1909.05585},
url={http://users.jyu.fi/~jojapeil/pub/xrt-no-uc.pdf},
doi = {10.1088/1361-6420/ab6e75},
gsid = {11944799965819298982},
abstract = {We show that the normal operator of the X-ray transform in $\mathbb R^d$, $d\geq2$, has a unique continuation property in the class of compactly supported distributions. This immediately implies uniqueness for the X-ray tomography problem with partial data and generalizes some earlier results to higher dimensions. Our proof also gives a unique continuation property for certain Riesz potentials in the space of rapidly decreasing distributions. We present applications to local and global seismology. These include linearized travel time tomography with half-local data and global tomography based on shear wave splitting in a weakly anisotropic elastic medium.}
}