Inverse wave scattering in the time domain for point scatterers

Abstract

Let Δα,Y be the bounded from above self-adjoint realization in L2(R3) of the Laplacian with n point scatterers placed at Y={y1,…,yn}⊂R3, the parameters (α1,…αn)≡α∈Rn being related to the scattering properties of the obstacles. Let ufϵα,Y and ufϵ∅ denote the solutions of the wave equations corresponding to Δα,Y and to the free Laplacian Δ respectively, with a source term given a pulse fϵ supported in ϵ-neighborhoods of the points in XN={x1,…,xN}, XN∩Y=∅. We show that, for any fixed λ>sup⁡σ(Δα,Y)≥0, there exists N∘≥1 such that the locations of the points in Y can be determined by the knowledge of a finite-dimensional scattering data operator FλN:RN→RN, N≥N∘. Such an operator is defined in terms of the limit as ϵ↘0 of the Laplace transform of ufϵα,Y(t,xk)−ufϵ∅(t,xk), k=1,…,N. We exploit the factorized form of the resolvent difference (−Δα,Y+λ)−1−(−Δ+λ)−1 and a variation on the finite-dimensional factorization in the MUSIC algorithm. Multiple scattering effects are not neglected; our model can be interpreted as the time-domain version of a frequency-domain scattering from an array of Foldy's point-like obstacles.