Recovery of Material Parameters in Transversely Isotropic Media

Abstract

In this paper we show that in anisotropic elasticity, in the particular case of transversely isotropic media, under appropriate convexity conditions, knowledge of the qSH wave travel times determines the tilt of the axis of isotropy as well as some of the elastic material parameters, and the knowledge of qP and qSV travel times conditionally determines a subset of the remaining parameters, in the sense if some of the remaining parameters are known, the rest are determined, or if the remaining parameters satisfy a suitable relation, they are all determined, under certain non-degeneracy conditions. Furthermore, we give a precise description of the additional issues, which are a subject of ongoing work, that need to be resolved for a full treatment.

Reconstruction along a geodesic from sphere data in Finsler geometry and anisotropic elasticity

“…Nevertheless, inverse problems for anisotropic elasticity have received substantial attention using different approaches in the past. Of these we mention problems of identifying inclusions [12] or cracks [11], tomography for residual stresses [16], and various problems where the stiffness tensor field itself is to be reconstructed [8,9,13,14]. A different geometric approach, using tools of metric rather than smooth geometry, was recently used to approximately reconstruct a manifold from a finite number of seismic sources at unknown times [7].…”

Partial Global Recovery in the Elastic Travel Time Tomography Problem for Transversely Isotropic Media

“…In this case there is multiplicity for the wave speeds as well: the largest eigenvalue has multiplicity 1 and is called the p wave speed, while the other two eigenvalues coincide and is called the s wave speed; these two wave speeds can be described explicitly in terms of λ and µ. We will instead study the case of transversely isotropic elasticity, and we follow the notational conventions of [3], which in turn borrows conventions from [12]. In this case, there is an axis of isotropy around which the material behaves isotropically.…”

“…We will denote this axis as a covector field ξ(x) normalized under the dual metric function on T * R 3 associated to the Euclidean metric to have norm 1. (In [3], this axis was denoted by ω; we will reserve ω for use as a spherical variable.) In addition, there are 5 independent components of the elasticity tensor, which we denote by a 11 , a 33 , a 55 , a 66 , and E 2 (with E 2 = (a 11 − a 55 )(a 33 − a 55 ) − (a 13 + a 55 ) 2 in the notation of [12]); they will be referred to as the "material parameters" for the elastic material.…”

“…Note that fully isotropic elasticity is a special case of transversely isotropic elasticity, with a 11 = a 33 = λ + 2µ, a 55 = a 66 = µ, and E 2 = 0. Transversely isotropic elastic materials appear naturally in the Earth, where rocks are formed in layers over time; within each layer there is isotropic behavior, but the composition is not isotropic across different layers (see Section 1 of [3] for more examples and details).…”

“…the qSV wave speed, or all three wave speeds if E 2 ≡ 0) that two of the five parameters can be determined from the travel time data, using techniques from boundary rigidity. In [3], the authors showed that the axis of isotropy ξ and the parameters a 55 and a 66 can be recovered from the qSH wave speed (in part due to the qSH wave speed being a quadratic form in ξ), assuming that the kernel of the axis of isotropy ξ is an integrable hyperplane distribution, i.e. ξ is a smooth multiple of some closed 1-form (locally representable as df for some layer function f ), as well as geometric conditions such as a "convex foliation" condition.…”