Reconstruction of piecewise smooth wave speeds using multiple scattering

Abstract

Let $c$ be a piecewise smooth wave speed on ${\mathbb{R}}^n$, unknown inside a domain $\Omega$. We are given the solution operator for the scalar wave equation $({\partial }_t^2-c^2\Delta )u=0$, but only outside $\Omega$ and only for initial data supported outside $\Omega$. Using our recently developed scattering control method, we prove that piecewise smooth wave speeds are uniquely determined by this map, and provide a reconstruction formula. In other words, the wave imaging problem is solvable in the piecewise smooth setting under mild conditions. We also illustrate a separate method, likewise constructive, for recovering the locations of interfaces in broken geodesic normal coordinates using scattering control.

18 pages, 3 figures