Inverse wave scattering in the time domain for point scatterers

Abstract

Let ${\Delta }_{\alpha ,Y}$ be the bounded from above self-adjoint realization in $L^2({\mathbb{R}}^3)$ of the Laplacian with $n$ point scatterers placed at $Y=\{y_1,\dots ,y_n\}\subset {\mathbb{R}}^3$, the parameters $({\alpha }_1,\dots {\alpha }_n)\equiv \alpha \in {\mathbb{R}}^n$ being related to the scattering properties of the obstacles. Let $u_{f_{\epsilon }}^{\alpha ,Y}$ and $u_{f_{\epsilon }}^{\varnothing }$ denote the solutions of the wave equations corresponding to ${\Delta }_{\alpha ,Y}$ and to the free Laplacian $\Delta$ respectively, with a source term given by the pulse $f_{\epsilon }(x)={\sum }_{k=1}^N{f}_k{\phi }_{\epsilon }(x-x_k)$ supported in $\epsilon$-neighborhoods of the points in $X_N=\{x_1,\dots ,x_N\}$, $X_N\cap Y=\varnothing$. We show that, for any fixed $\lambda >\sup \sigma ({\Delta }_{\alpha ,Y})$, there exits $N_{\circ }\ge 1$ such that the locations of the points in $Y$ can be determined by the knowledge of the finite-dimensional scattering data operator $F_{\lambda }^N:{\mathbb{R}}^N\to {\mathbb{R}}^N$, $N\ge N_{\circ }$, $$(F_{\lambda }^{N}f{)}_k:=\underset{\epsilon \searrow 0}{\lim }{\int }_0^{\infty }{e}^{-\sqrt{\lambda }t}(u_{f_{\epsilon }}^{\alpha ,Y}(t,x_k)-u_{f_{\epsilon }}^{\varnothing }(t,x_k))dt.$$ We exploit the factorized form of the resolvent difference $(-{\Delta }_{\alpha ,Y}+\lambda {)}^{-1}-(-\Delta +\lambda {)}^{-1}$ and a variation on the finite-dimensional factorization in the MUSIC algorithm; multiple scattering effects are not neglected.

Final version

References in corpus (4)

• Limiting Absorption Principle, Generalized Eigenfunctions and Scattering Matrix for Laplace Operators with Boundary conditions on Hypersurfaces
• Asymptotic Completeness and S-Matrix for Singular Perturbations
• Inverse Scattering for the Laplace operator with boundary conditions on Lipschitz surfaces
• Inverse wave scattering in the Laplace domain: a factorization method approach