Reconstruction of piecewise smooth wave speeds using multiple scattering

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.