Travel time tomography for elliptically anisotropic elastic planar media

Abstract: We study the gauge of the travel time tomography problem for elliptically anisotropic planar media. Under this assumption the respective slowness surfaces consist of an inner ellipse and an outer circle. Hence, the singularities of seismic waves propagate on geodesics given by two Riemannian metrics of which the one related to shear waves is conformally Euclidean. We show that if both of these metrics are lens rigid then, up to boundary fixing diffeomorphisms, each elliptically anisotropic stiffness tensor is uniquely determined by the lens data whenever it has at least one isotropic point. If none of those points exists, then there are exactly two elliptically anisotropic stiffness tensors with the same lens data. Clearly, not all boundary fixing diffeomorphisms preserve the imposed structure of the slowness surface. We show that the set of well-behaved diffeomorphisms is never a subgroup of all boundary fixing diffeomorphisms. Our proofs are based on an interplay of algebra and geometry, and the key of the proofs is a characterization of the elliptic anisotropy via several polynomial conditions.