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 $Ω$. We are given the solution operator for the scalar wave equation $(\partial_t^2-c^2Δ)u=0$, but only outside $Ω$ and only for initial data supported outside $Ω$. 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