Generic uniqueness and stability for the mixed ray transform

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 1+1 and 2 + 2 2+2 tensors fields. We show how the anisotropic perturbations of averaged isotopic travel-times of q S qS -polarized elastic waves provide partial information about the mixed ray transform of 2 + 2 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.

The Geodesic Ray Transform on Spherically Symmetric Reversible Finsler Manifolds

“…This in turn implies that the stiffness tensors c s i jkl = f s c i jkl all agree to first order in s, i.e., We note that it was shown in [18] that the linearization of the elastic travel time tomography problem for a family of isotropic stiffness tensors leads to the geodesic ray transform of scalar fields on Riemannian manifolds (and more generally to an integral geometry problem of 4-tensor fields). Our conformal linearization allows general anisotropies for c s i jkl (the background stiffness tensor c i jkl can be anisotropic) and therefore the geometry is Finslerian; in [18] the authors mainly study perturbations around isotropic elasticity (weakly anisotropic medium). Other difference is that our linearization applies to qP-waves and the linearization in [18] to S-waves and qSwaves.…”

“…(A5) The curves are symmetric with respect to the lowest point and they consist of two parts where ṙ > 0 and ṙ < 0. (A6) The curves satisfy the weak reversibility condition (18).…”

The geodesic ray transform on spherically symmetric reversible Finsler manifolds

“…We note that it was shown in [17] that the linearization of elastic travel time tomography problem for a family of isotropic stiffness tensors leads to the geodesic ray transform of scalar fields on Riemannian manifolds (and more generally to an integral geometry problem of 4-tensor fields). Our conformal linearization allows general anisotropies for c s ijkl (the background stiffness tensor c ijkl can be anisotropic) and therefore the geometry is Finslerian; in [17] the authors mainly study perturbations around isotropic elasticity (weakly anisotropic medium). Other difference is that our linearization applies to qP-waves and the linearization in [17] to S-waves and qS-waves.…”

“…Our conformal linearization allows general anisotropies for c s ijkl (the background stiffness tensor c ijkl can be anisotropic) and therefore the geometry is Finslerian; in [17] the authors mainly study perturbations around isotropic elasticity (weakly anisotropic medium). Other difference is that our linearization applies to qP-waves and the linearization in [17] to S-waves and qS-waves. For more linearization results in elastic travel time tomography see [17,18,46].…”

“…Other difference is that our linearization applies to qP-waves and the linearization in [17] to S-waves and qS-waves. For more linearization results in elastic travel time tomography see [17,18,46].…”