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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 plus 1"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">1+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 plus 2"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2+2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> tensors fields. We show how the anisotropic perturbations of averaged isotopic travel-times of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q upper S"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">qS</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-polarized elastic waves provide partial information about the mixed ray transform of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 plus 2"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2+2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.