Reconstruction of piecewise smooth wave speeds using multiple scattering

TL;DR

This paper proves that piecewise smooth wave speeds can be uniquely reconstructed from external scattering data and provides explicit formulas for the reconstruction, advancing wave imaging techniques.

Contribution

It introduces a scattering control method to uniquely determine and reconstruct piecewise smooth wave speeds from limited external data.

Findings

Unique determination of wave speeds from scattering data

Explicit reconstruction formulas provided

Method for locating interfaces in broken geodesic coordinates

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.