Reconstruction of piecewise smooth wave speeds using multiple scattering

Caday, Peter; de Hoop, Maarten; Katsnelson, Vitaly; Uhlmann, Gunther

doi:10.1090/tran/7632

Reconstruction of piecewise smooth wave speeds using multiple scattering
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by Peter Caday, Maarten V. de Hoop, Vitaly Katsnelson and Gunther Uhlmann PDF

Trans. Amer. Math. Soc. 372 (2019), 1213-1235 Request permission

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.

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