Recovery of discontinuous Lamé parameters from exterior Cauchy data

Abstract

Consider an isotropic elastic medium $\Omega \subset {\mathbb{R}}^3$ whose Lamé parameters are piecewise smooth. In the elastic wave initial value inverse problem, we are given the solution operator for the elastic wave equation, but only outside Ω and only for initial data supported outside Ω. Using the recently introduced scattering control series in the acoustic case, we prove that piecewise smooth Lamé parameters are uniquely determined by this map under certain geometric conditions.

Keywords:

• Elastic wave equation
• inverse problems
• microlocal analysis

Acknowledgements

The authors greatly appreciate the detailed suggestions made by two anonymous referees which improved this paper.

Notes

1 The fact that the interfaces are not dense makes this possible theoretically in the sense that there will exist an open set of rays at the current layer that do not cross any interfaces after a finite time when they are close to being tangent to the layer.

2 That is, its projection to $T^{*}\partial {\Omega }_{\pm ,j}$ when it hits $\partial T^{*}{\Omega }_{\pm ,j}$ is β, but we abuse notation.

3 Here, the nonglancing, nonelliptic region refers to covectors in $T^{*}\Gamma$ that are not in the glancing set of either ${\partial }_t^2-c_P^2\Delta$ or ${\partial }_t^2-c_S^2\Delta ,$ but is in the hyperbolic set of one of the operators.

4 In fact, we can use unique continuation to obtain the same result since we are allowed to measure outside Θ for an unlimited amount of time. Nevertheless, this is overkill for what we need here, which is a microlocal equivalence.

5 The following argument is necessary to ensure that we match a P travel time associated to Q with the corresponding one associated to $\tilde{Q}$ rather than an s travel time associated to $\tilde{Q}.$

Additional information

Funding

P.C. and V.K. were supported by the Simons Foundation under the MATH + X program. M.V.dH. was partially supported by the Simons Foundation under the MATH + X program, the National Science Foundation under grant DMS-1815143, and by the members of the Geo-Mathematical Group at Rice University. G.U. was partially supported by NSF, a Walker Professorship at UW and a Si-Yuan Professorship at IAS, HKUST.