  Given a compact $k$-dimensional submanifold $K \subset \mathbf R^n$,
incapsulated in a compact domain $M \subset \mathbf R^n$, we consider the
problem of determining the inner geometry of the obstacle $K$ from the
scattering data, produced by the reflections of geodesic trajectories from the
boundary of a tubular $\epsilon$-neighborhood $\mathsf T(K, \epsilon)$ of $K$
in $M$. The geodesics emanate from $\partial M$ and terminate there, after a
number of reflections from the boundary $\partial \mathsf T(K, \epsilon)$. We
use $\lceil \dim(K)/2\rceil$ many tubes $\{\mathsf T(K, \epsilon_j)\}_j$ for
detecting certain global intrinsic geometry invariants of $K$, thus the words
"layered scattering" in the title. These invariants were studied by Hermann
Weyl in his theory of tubes.
