Inverse scattering in an asymptotically flat multilayer domain

We consider a scattering problem for a wave equation ${\mathrm{\partial }}_t^{2}u=\frac{1}{\sqrt{g}}{\mathrm{\partial }}_i(\sqrt{g}{g}^{ij}{\mathrm{\partial }}_j)u$ in a multilayer domain $\mathrm{\Omega }\subset {\mathbf{R}}_x^{n+1}={\mathbf{R}}_y^n\times {\mathbf{R}}_{x^{n+1}}^1$ of the form $\mathrm{\Omega }=\mathcal{K}\cup {\mathrm{\Omega }}_1\cup \cdots \cup {\mathrm{\Omega }}_N$, where $\mathcal{K}$ is a bounded open set and ${\mathrm{\Omega }}_k$ is asymptotically equal to a slab domain ${\mathbf{R}}^n\times (c_k,c_k+d_k)$ as $\mathrm{\mid }y\mathrm{\mid }\to \mathrm{\infty }$. Assuming that ${\mathrm{\partial }}_x^{\alpha }(g_{ij}(x)-{\delta }_{ij})=O(\mathrm{\mid }x{\mathrm{\mid }}^{-\mathrm{\mid }\alpha \mathrm{\mid }-{\delta }_0}),{\delta }_0>1,\mathrm{\forall }\alpha$, we show that $\mathrm{\Omega }$ and $g^{ij}$ are determined by one diagonal component $S_{11}(\lambda )$, for all energies, of the S-matrix associated with the slab ${\mathrm{\Omega }}_1$, provided ${\mathrm{\Omega }}_1$ is flat: ${\mathrm{\Omega }}_1\cap \{\mathrm{\mid }y\mathrm{\mid }>R\}=\{\mathrm{\mid }y\mathrm{\mid }>R\}\times (c_1,c_1+d_1)$ for some constants $c_1,d_1,R>0$, and the metric is Euclidean on ${\mathrm{\Omega }}_1\cap \{\mathrm{\mid }y\mathrm{\mid }>R\}$.